Editor-in-chief I.B. FEDOROV, Corresponding Member, Russian Academy of Sciences Deputy editor-in-chief T.I. POPENCHENKO Executive secretary V.A. TOVSTONOG

Editorial board: A.M. ARKHAROV, D. Sc. (Eng.), Professor V.O. GLADYSHEV, D. Sc. (Phys.-Math.), Professor K.Ye. DEMIKHOV , D. Sc. (Eng.), Professor V.V. DEVYATKOV, D. Sc. (Eng.), Professor Yu.I. DIMITRIENKO, D. Sc. (Eng.), Professor S.F. KONOVALOV, D. Sc. (Eng.), Professor V.A. MATVEEV, D. Sc. (Eng.), Professor A.N. MOROZOV, D. Sc. (Phys.-Math.), Professor B.P. NAZARENKO, D. Sc. (Eng.), Professor I.P. NORENKOV, D. Sc. (Eng.), Professor M.I. OSIPOV, D. Sc. (Eng.), Professor B.A. ROZANOV, D. Sc. (Eng.), Professor S.T. SURZHIKOV, D. Sc. (Phys.-Math.), Professor A.S. YUSHCHENKO, D. Sc. (Eng.), Professor

Manuscripts have been submitted in English and papers are published without any redaction

Proof-reader M.V. SAMOKHINA Art design S.S. VODCHITS, N.G. STOLYAROVA

Address of the Editorial Ofﬁce: Vestnik MGTU, 5, Vtoraya Baumanskaya ul., Moscow, 105005, RUSSIA Tel: (095) 263-62-60, 263-67-98, 267-63-49 Fax: (095) 265-42-98 E-mail: [email protected], [email protected] c Bauman Moscow State Technical University ° VESTNIK. Journal of the Bauman Moscow State Technical University, 2005

Wide-section journal for publications in theoretical and applied sciences

Printed by the Bauman MSTU Press

Natural Sciences & Engineering

CONTENTS Scientiﬁc Schools of the Bauman Moscow State Technical University (Devoted to the 175th anniversary of the BMSTU foundation) . . . . . . . . . . . . . .

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Simulation of Processes & Systems H. Krier, S.T. Surzhikov. Computing Model of Burning of Bi-modal AP/HTPB Composite Solid Rockets Propellants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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V.T. Kalugin, A. Yu. Lutsenko, Ye.G. Stolyarova. Hysteresis in Aerodynamics of Flying Vehicles under Condition of Unsteady Movement . . . . . . . . . . . . . . . . .

31

Yu.I. Dimitrienko, M.L. Glazikov. Local Transport Phenomena in Porous Periodic Structures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

Control Systems A. Golubev, R. Johansson, A. Robertsson, S. Tkachev. Output Tracking for a Class of Nonlinear Nonminimum-phase Systems Using Observer Backstepping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

Laser & Optic-Electronic Systems V.N. Rozhdestvin, O.A. Smirnova. Combined Q-switch and Mode-lock in Laser Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

V.Ye. Karasik. Laser Range-gated Imaging System . . . . . . . . . . . . . . . . . . . . . . . .

88

Navigational & Gyroscopic Systems V.A. Matveev and M.A. Basarab. Numerical Modeling of Heat Diffusion Processes in the Solid-state Wave Gyro Resonator by the R-function Method . . . .

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B.S. Konovalov, S.F. Konovalov, A.V. Kuleshov, D.V. Mayorov, V.P. Podtchezertsev, V.V. Fateev. New Types of Vibrating Gyro for Rotating Carrier . . . . . . .

111

Information & Computation Technology V.V. Devyatkov. Multiagent Hierarchical Recognition on the Basis of Fuzzy Situational Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I. Balepin, S. Maltsev, J. Rowe, K. Levitt. Using Speciﬁcation-based Intrusion Detection for Automated Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

129 153

Physics S.M. Korotaev, A.N. Morozov, V.O. Serdyuk, Yu.V. Gorokhov, S.A. Pulinets, V.I. Nalivayko, A.V. Novysh, S.P. Gaidash, H.D. Kanonidy. Manifestation of Macroscopic Nonlocality in the Processes of Solar and Geomagnetic Activity M.A. Yakovlev. Generation of the Near-surface Electron Layer by Picosecond Laser Pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V.O. Gladyshev, T.M. Gladysheva, A.N. Morozov, V.E. Zubarev. The Results of the Experiment on Registrating Light Dragging Observed in an Interferometer with a Rotating Optical Disc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M.C. Duffy, V.O. Gladyshev, A.N. Morozov, P. Rowlands Physical Interpretations of Relativity Theory (On the materials of International Scientiﬁc Conference in Bauman Moscow State Technical University, 30th June – 3rd July, 2003) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

173

186

192

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Radio-Physics & Radiolocation B. Rozanov, S. Solomonov, A. Zrazhevsky, V. Nagnibeda, Ye. Kropotkina, S. Rozanov, N. Zharkova, T. Lebedjuk, I. Fetisov, M. Loukicheva. The Atmospheric and Solar Investigations at Millimeter Waves . . . . . . . . . . . . . . . . . . . . . . .

210

Ecology Problems N.P. Demenkov, V.A. Matveev. Fuzzy Systems of Ecological Monitoring and Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

227

Fundamental Problems of Mechanical Engineering Yu.N. Drozdov, Ye.G. Yudin. Wear Prediction Considering Mechanical Physical-chemical and Geometrical Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

238

Yu.N. Drozdov, Ye.G. Yudin. Tribological Problems of Mechanical System Creation for the Moon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

246

Production Process Procedures & Machines A.M. Dmitriev, A.L. Vorontsov. Analysis of Extrusion Process of Coreless Cylindrical Products with an External Core in the Bottom . . . . . . . . . . . . . . . . . .

252

I.N. Shiganov. Fusion Welding of Metallic Composite Materials . . . . . . . . . . . .

264

Yu. Bocharov, Yu. Gladkov. Integrated Control-Monitoring-Diagnosis System Development in Technology of Plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

272

S. Golovashchenko, V. Kondratenko, A. Vlasov. Experimental Investigations of Flanging Aluminium Panels as the First Stage of Hemming Process . . . . . . .

280

Cryogenic Engineering & Technology A.M. Arkharov, S.D. Glukhov, L.V. Grekhov, A.A. Zherdev, N.A. Ivashchenko, D.N. Kalinin, A.V. Sharaburin, A.A. Aleksandrov. Creation of Harmless for Environment Engines and Installations Using Dimethyl Ether . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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SCIENTIFIC SCHOOLS OF THE BAUMAN MOSCOW STATE TECHNICAL UNIVERSITY (Devoted to the 175th anniversary of the BMSTU foundation)

In 2005 the Bauman Moscow State Technical University (former Bauman Moscow Higher Technical School) will mark by celebration the jubilee — 175th anniversary of its foundation. The history of BMSTU as an oldest technical higher educational establishment of the country is closely connected with the history of Russia, the development of its economy, science, technology, engineering, the Russian Army. The University has sprung from the vocational school. In 1826 the widowed empress Mariya Fyodorovna “deigned to royally enjoin that large workshops of various handicrafts be established” for orphan boys of the Foundling Hospital. With this purpose, the Slobodskoy Palace in the German Suburb of Moscow (Nemetskaya Sloboda), having suffered from ﬁre in 1812, was reconstructed by the famous Moscow architect D.I. Zhilyardi. The building has acquired an appearance inherent in the late Moscow Empire-style as we can see it today. It has been decorated with an attic, representing the multi-ﬁgure sculptural group “Minerva” (by sculptor I.P. Vitali), that symbolizes scientiﬁc achievements and practical craftsman skills. On 1st July 1830, the emperor Nicholas I signed the Edict to open the Moscow Vocational School (MVS) for training “craftsmen in theoretical knowledge, serving for modernization of crafts and factory works” and approved the “Statute on the Vocational School”. (The existing BMSTU counts off its chronology since that year.) At the same time, two main trends in teaching alumni were determined: mechanical and chemical crafts. From the very beginning MVS became a center of the technical education in Russia, and the incipient home industry at once recognized a high level of training specialists in the new educational establishment. In 1868 the Regulations of the School were approved, in the ﬁrst section of which the following was written: “The Emperor’s Moscow Technical School (EMTS) is a higher special educational establishment having its main goal to educate civil engineers, mechanical engineers and technological engineers”. Early chairs in the school were those of higher mathematics, general mechanics, general and applied physics, machine building, construction art, technology of ﬁber substances, general chemistry, chemical technology. “Russian method” of training engineers. In XIX century, technical sciences and the higher education as a whole experienced the process of formation, which could not but have an immediate impact on shaping the learning process in the School. Everything depended, to a large measure, on people who worked within its precincts. The professorate and teaching staff of EMTS included such distinguished scientists as D.I. Mendeleev, N.Ye. Zhukovsky, P.L. Chebyshev, S.A. Chaplygin, A.S. Yershov, D.K. Sovetkin, A.V. Letnikov. An immense prestige of the professorate and teaching staff and a high level of training specialists put EMTS into a line of leading European polytechnic schools. An orderly system of professional teaching of future specialists that was accepted in EMTS — so-called “Russian method VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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of training engineers” — gained recognition all over the world. The method became well-known, especially after its successful demonstrations at international exhibitions: in Vienna (1873), where it was awarded the Big Golden Medal, and also in Philadelphia (1878, 1900), where the method was declared the best. With advances in engineering the “Russian method” of the nineteen seventies was progressively upgraded. By 1903 EMTS was generally recognized as the best higher educational establishment for mechanical engineering in Russia. It had become clear by that time that only scientiﬁcally formulated experiments conducted in a laboratory equipped with up-to-date apparatus could promote the technology development. Later, some world higher educational establishments were created according to that pattern. So, the President of the famous Massachusetts Institute of Technology J. Rounkle, having received from EMTS the collection of models, specially made for the American Institute, to teach engineers by the “Russian method”, wrote to the EMTS rector V.K. Della-Vos: “Russia is recognized to be completely successful in solving so important problem of the technical education. . . . After this, no other system will be applied in America”. He also published a small brochure titled “Russian Method of Professional Teaching of Engineers and Mechanics”. Development of scientiﬁc ties. Mutual enrichment of different systems of the technical education proceeded. International exhibitions, congresses, meetings attended by EMTS representatives contributed to expansion of scientiﬁc ties, enchanced the prestige of EMTS. Business-like relations with the largest foreign companies began taking shape; no less important were ties with Russian business circles, whose representatives were aware of higher educational establishments being centers of the contemporary science and technology. Scientiﬁc societies, proceedings and other mutual initiatives strengthened and enriched relations among scientists. Creating the Polytechnic Exhibition in 1872 became one of such initiatives, which gave birth to the known-to-everyone Polytechnic Museum. The Polytechnic Society, established by EMTS in 1877, only consisted of EMTS alumni and held a proper position in the line of Russian scientiﬁc and technical institutions. “Society for Contribution to Successes of Experimental Sciences and their Practical Applications” named after Kh.S. Ledentsov was created by the EMTS professorate and teaching staff and yielded inestimable fruit. It subsidized talented inventors and provided funds for equipping laboratories, including those having become early research institutes in Russia: the physical laboratory headed by P.N. Lebedev, I.P. Pavlov’s laboratory for psycho-physiological research, the aerodynamic laboratory headed by N.Ye. Zhukovsky, etc. This was the only society in Russia, performing the work similar to activity of Carnegie Fund or the German Society n.a. Caesar William. Turn for polytechnic aspects. In early XX century the world situation sharpened problems of the technical progress and as a consequence — problems of the Russian technical education. In 1915 the then EMTS rector V.I. Grinevetsky, wellknown scientist in thermal engineering, suggested the “Draft on EMTS Transformation to School of a Polytechnic Type”. He believed that “engineering education should develop in two ways. On the one hand, the specialization of training should

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increase, on the other hand — the interaction and close collaboration of various professions should be strengthened. Only a polytechnic type school can satisfy both requirements given it has a fairy ﬂexible management.” With this, V.I. Grinevetsky was sure that in a contemporary higher educational establishment the “encyclopedic learning — broad engineering scope” had to be connected with a “certain specialization”. Attention should be also paid to another thought, uttered by him, about the necessity of the economical education of an engineer. Today it is especially important to appreciate it. Such a profound, conﬁrmed with examples and statistics, view on the further development of the School was supported by the EMTS professorate and teaching staff, and then put into practice. Scientiﬁc schools in early XX century. At the XIX–XX century border the signiﬁcance of EMTS in the scientiﬁc and industrial life of the country was increasing. There were formed in EMTS fundamental scientiﬁc schools, called into being by the intensive industry growth and by enlistment in the School of top scientists, mainly, Moscow University alumni. The ﬁeld of theoretical mechanics and aeromechanics was presented by “Father of the Russian Aviation” — N.Ye. Zhukovsky with his disciples S.A. Chaplygin, B.N. Yuriev, V.P. Vetchinkin. They developed fundamentals of aerodynamics, laid a scientiﬁc base for engineering design in aircraft building. The Central Aerodynamic and Hydrodynamic Institute and Air Force Engineering Academy were established with their active participation. The ﬁeld of physics was presented by P.N. Lebedev, P.P. Lazarev, V.S. Shcheglyaev, S.I. Vavilov. The scientists highly contributed to the development of photometry, luminescence, roentgenoscopy, to the research in wireless telegraphy. Owing to work of A.S. Yershov, P.L. Chebyshev, N.Ye. Zhukovsky, N.I. Mertsalov the home science — theory of mechanisms and machines — was born. Studies of D.N. Lebedev, A.I. Sidorov, P.K. Khudyakov laid a foundation for the development of computations in strength of materials and parts of machines. M.M. Cherepashinsky, N.S. Streletsky, P.A. Velikhov were founders of methods of computation and design of building structures. Thermal engineering was formed by activities of V.I. Grinevetsky who developed methods for designing boilers and thermal processes in engines of internal combustion. He also became an initiator of creating diesel locomotives. In the ﬁeld of electrical engineering it is necessary to highlight works performed by K.I. Shenfer, K.A. Krug, B.I. Ugrimov and devoted to high voltages of electric systems, multi-phase motors, facilities for transferring energy over large distances. Later, the School’s scientists took the most active part in the development of GOELRO plan. Studies, conducted by A.Ye. Chichibabin, S.A. Fyodorov, Ya.Ya. Nikitinsky, N.A. Shilov and others in the ﬁeld of chemistry and chemical technology, technology of ﬁber, food, explosive and pharmaceutical substances, were of great importance. Transformations from EMTS to BMHTS. After 1917 the School experienced a number of transformations. Its name changed for Moscow Higher Technical School (MHTS). During the period of 1920–1930 some individual higher educational establishments and research institutes were extracted from

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it. Those were: aerohydrodynamic, automotive, chemical-technological laboratories, on which basis were established later the following institutes: the Central Aerodynamic and Hydrodynamic Institute (“TsAGI”), All-union Institute for Aviation Materials (“VIAM”), Central Institute for Aviation Engines (“TsIAM”), Scientiﬁc Institute for Automotive Engines (“NAMI”) and a number of others. The most serious transformations of the School occurred in 1930, when on its base some technical higher educational establishments were organized which later became the largest educational institutions: Moscow Aviation Institute (“MAI”), Moscow Institute for Power Engineering (“MEI”), Moscow Institute for Building Engineering (“MISI”), Academy for Chemical Protection and some others. The mechanical faculty, remained in the original MHTS building, was given a new denomination — Moscow Mechanical Machine-building Institute (MMMI). In December 1930 it was also named after N.E. Bauman (MMMI n.a. N.E. Bauman). Then, in 1943, its former denomination — MHTS — was restored. The renewed name signiﬁed not only the recognition of the Moscow Higher Technical School n.a. N.E. Bauman (BMHTS) as a descendant of the famous Russian engineering school but also its peculiar status in the system of the higher technical education of the country. As early as before the 1917 Revolution the Industrial School was established on the initiative of the EMTS teachers; after the Revolution its name became Moscow Chemical-Technological Institute (“MkhTI”) n.a. D.I. Mendeleev. In 1918 the MHTS-based Aviation Technical Secondary School was transformed into the Academy for Air Force Engineers. Based on the EMTS laboratory of ﬁlament technology, some scientiﬁc trends in Textile Institute began to progress. A number of the MHTS chairs moved into the Institute of Food Industry Technology, Institute of Chemical Engineering, Electric Engineering Institute for Communications, Building Engineering Academy. In 1921, the All-Union Thermal-Technical Institute n.a. V.I. Grinevetsky and K.V. Kirsh was established around the thermal-technical laboratory. The All-Union Electrical Engineering Institute was founded the same year on the initiative of the MHTS professors. In the nineteen thirties the Central Research Institute for Ferrous Metallurgy (“TsNIIChERMET”) n.a. I.P. Bardin was created on the basis of the MHTS laboratory of rolling. In the nineteenth forties, the physical engineering faculty moved into the just established Moscow Engineering Physical Institute (“MIFI”). The present-day Vladimir Polytechnic Institute was founded in XIX century on the initiative of EMTS teachers in town Vladimir as the Vocational Maltsev’s school. The Izhevsk Mechanical Institute emerged in 1943 (during the World War II) due to activities of BMHTS professors and teachers, evacuated to town Izhevsk. Those multiple transformations had no essential effect on the BMHTS scientiﬁc schools life. General science and technology chairs even though suffered from leaving of a number of leading scientists, but rapidly gained strength thanks to the talented young people who came to succeed their teachers and inherited all the

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best from them. New chairs became headed by MHTS alumni, keepers of its traditions, acute for the novelty, directed towards the future. Physical and mathematical sciences made further progress through activities of S.I. Vavilov, S.P. Finikov, A.S. Predvoditelev, A.P. Kotelnikov et al. Studies, performed by N.R. Briling, Ye.K. Mazing, A.S. Orlin, G.G. Kalish, V.Ye. Tsydzik, S.Ya. Gersh, G.F. Knorre, L.K. Ramzin, V.V. Uvarov, promoted the scientiﬁc schools development in the ﬁeld of power engineering. I.I. Kukolevsky and S.S. Rudnev created the school of hydromechanical engineering; Ye.A. Chudakov, M.K. Kristi — the school of wheeled and caterpillar vehicles; L.G. Kifer — the school of hoisting transportation machines. In the ﬁeld of technology of mechanical engineering and standardization the following scientists fruitfully worked: E.A. Satel, M.A. Saverin, V.M. Kovan. The professors K.K. Khrenov, N.N. Rykalin, G.A. Nikolaev founded the school of welding practice; A.M. Bochvar, I.I. Sidorin — the school of science of materials; N.N. Rubtsov and N.P. Aksyonov — the school of casting practice; A.I. Tselikov, A.I. Zimin, M.V. Storozhev — the school of non-cutting shaping of materials. Instrumental engineering schools developed with fast paces: precise instrumental engineering (F.V. Drozdov, S.O. Dobrogursky); optical engineering (S.I. Freiberg, I.A. Turygin); gyroscopy (S.S. Tikhmenyov, B.V. Bulgakov); radioelectronics (M.V. Shuleikin, A.M. Kugushev); automatic control device engineering (V.V. Solodovnikov). In 1938 three new faculties were set up: tank, artillery and ammunition engineering. Among the ﬁrst professors — the chairs organizers — there were M.K. Kristi, A.G. Gorst, K.P. Stanyukovich, A.A. Tolochkov, V.Ye. Slukhotsky. In 1948 a faculty of rocket technology was created, with which the activities in BMHTS of the rocket engineering founders — S.P. Korolyov, Yu.A. Pobedonostsev, V.I. Feodosiev, V.N. Chelomey, V.P. Barmin — were connected. BMHTS museum. It is impossible to enumerate here all noteworthy scientists names, to describe their contribution to the formation and development of the present-day Bauman Moscow State Technical University. One can become familiar in detail with the history of the university scientiﬁc schools development in the BMHTS museum, that was set up in 1967 and for years of its existence it became one of the most known museums of higher educational establishments in Russia. The main fund of museum objects consists of more than 1000 items. The museum book storage includes more than 3000 volumes: from XVIII century folios to ﬁrst home monographs and textbooks on thermal engineering, automotive engineering, cybernetics, rocket engineering, robotics. The museum closely collaborates with the Museum of History of Moscow and Polytechnic Museum and also with many museums of higher educational establishments in our country; it is engaged in many enlightenment activities. Educational-experimental center. No less unique exposition — engineering exhibition is presented in the BMSTU Branch — Educational-Experimental Center, situated at 87 km from Moscow, in the settlement Orevo, Dmitrovsky region of Moscow oblast. The Center construction was initiated in 1960 and it was dictated by the necessity to withdraw some laboratories from Moscow, because they could

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not be accomodated within the main BMHTS building. The Center has become the ﬁrst of such structural divisions not only in BMHTS, but in the whole Higher School of the country as well. Today it remains single of the kind due to the unique equipment it has and the research being conducted there. In one of its laboratories, for example, the unique exposition of samples of the rocket and space engineering was managed to collect, that has no analog not only in higher educational establishments but also in rocket and space centers of the country. However, the most valuable is the fact that these articles serve not only as museum exhibits, but also as acting laboratory test-benches, which are permanently used for educational purposes. State appreciation. The state highly appreciated the contribution of the BMHTS collective to formation and development of new trends in science, engineering and contemporary technologies, to creation of scientiﬁc and pedagogical schools, to training of highly skilled specialists: — in XIX century the status of Emperor’s was conferred on the School; — in 1933, according to the Decree of the Central Executive Committee of the USSR, the Bauman Moscow Mechanical and Machine-building Institute was awarded the Order of Labor Red Banner for great merits in socialism construction; — in 1955, according to the Edict of The Presidium of the Supreme Soviet of the USSR, the Bauman Moscow Higher Technical School was awarded the Order of Lenin in connection with the 125th anniversary of its foundation and for merits in development of science and technology and in training of highly skilled specialists; — in 1980, according to the Edict of The Presidium of the Supreme Soviet of the USSR, the Bauman Moscow Higher Technical School was awarded the Order of October Revolution in connection with the 150th anniversary of its foundation and for merits in scientiﬁc activity and in training of highly skilled specialists; By mutual decision of the Central Committee of the CPSU and Council of Ministers of the USSR No. 452 of 17th April 1987, the necessity was recognized to set up the higher educational establishment of a new type on the basis of BMHTS, entrusting it with training specialists according to the principles, which combine the advanced forms of the fundamental university’s education with the engineeringand- technological one. Since then a structure of the higher educational establishment has changed and remains the same up till now. By the Order of the USSR State Committee on People’s Education No. 617 of 27th July 1989 the Bauman Moscow Higher Technical School was transformed into the Bauman Moscow State Technical University (BMSTU). Thus, the organization of technical universities, a number of which nowadays exceeds 100, has been initiated in Russia. 8

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On 5th April 1993 the Decree of the Russian Federation Government was signed on the further BMSTU development, which provided some measures on sustaining and advancing the University’s intellectual potential. On 24th January 1995 by the Edict No. 64 of the Russian Federation President the BMSTU was involved in the State Fund of Greatly Valuable Objects of Cultural Heritage of Russian Federation Peoples. In 2000 the gratitude of the Russian Federation President was conferred on the University for the great contribution to the development of the higher education and to training of highly skilled specialists. The BMSTU has trained and graduated many thousands of high class engineers. It is practically impossible to name a ﬁeld in engineering and scientiﬁc activity where the BMSTU alumni would not do justice to themselves. It is also impossible within the framework of this paper to enumerate all outstanding BMSTU graduates. The following eminent organizers of industry and science are among them: S.A. Afanasiev, B.L. Vannikov, V.A. Malyshev; prominent scientists and designers: V.P. Barmin, A.A. Bochvar, N.A. Dollezhal, V.Ya. Klimov, S.P. Korolyov, S.A. Lebedev, V.M. Myasishchev, P.O. Sukhoy, A.N. Tupolev, Ye.A. Chudakov, A.I. Tselikov, V.G. Shukhov and many others. BMSTU today. In today’s BMSTU there are 18 thousand students of exclusively full form of learning and more than 4500 professors and teachers including 450 doctors of science (D. Sc.) and about 3000 candidates of science (Ph. D.). Eminent scientists from research institutes of the Russian Academy of Sciences and other Russian academies, from branch research institutions and international visiting scientists deliver lectures in BMSTU. About 1000 post-graduates of 86 scientiﬁc professions are taught and 26 dissertation councils, covering 62 scientiﬁc professions, function here. BMSTU has licenses for training bachelors and masters by 26 professions and certiﬁed specialists by 75 professions of a practically full set of ﬁelds of the contemporary mechanical and instrumental engineering. The BMSTU structure has no analog among technical universities of Russia and, in our opinion, it approximates most closely to that of a university of the future — the research university. It is destined to provide the integration of the fundamental theoretical training with a practical activity of future specialists that is connected with solving urgent scientiﬁc and engineering problems, with performing design and experimental work in research institutes and in BMSTU departments. The BMSTU structure also offers an opportunity to the professorate and teaching staff to accomplish the research work inside the university in a close cooperation with specialists of industrial institutions. The professional training of BMSTU students is based on the principles, combining advanced methods of fundamental university’s and engineering education, and is realized in seven research and education units (REU), which include educational faculties and the proper research institutes with design bureaus: • REU “Manufacturing Engineering” (Faculty “Manufacturing Engineering” and Research Institute “Structural Materials and Manufacturing Processes”);

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• REU “Information Technology and Control Systems” (Faculty “Information Technology and Control Systems”, “Instrumental Engineering” Faculty and Research Institute “Information Technology and Control Systems”); • REU “Radio-electronics, Laser and Medical Technology” (Faculty “Radioelectronics and Laser Technology”, Faculty “Biomedical Engineering”, Faculty “Optoelectronic Device Engineering”, “Radio Engineering” Faculty and Research Institute “Radio-electronics and Laser Technology”, Research Institute “Biomedical Engineering”); • REU “Robotics and Integrated Automation” (Faculty “Robotics and Integrated Automation” and Research Institute “Manufacturing Processes Automation”); • REU “Power Engineering” (Faculty “Power Engineering”, Faculty “Rocket and Space Engineering” and Research Institute “Power Engineering”); • REU “Special-purpose Mechanical Engineering” (Faculty “Special-purpose Mechanical Engineering”, “Aerospace Engineering” Faculty and Research Institute “Special-purpose Mechanical Engineering”); • REU “Fundamental Sciences” (Faculty “Fundamental Sciences”, “Linguistic” Faculty, Research Institute “Applied Mathematics and Mechanics”). A REU provides the theoretical education of future specialists in fundamental and professional disciplines as well as the practical training of students by their involvement in solving concrete engineering problems. Research institutes, included in REUs, are under the methodical supervision of the proper divisions of the Russian Academy of Sciences. The economical instruction of all students is performed by the Faculty “Engineering Business and Management” which also trains engineers (managers) in the ﬁeld of high technologies. The Faculty “Humanitarian and Social Sciences” provides education of students in humanitarian and social sciences. Besides, BMSTU has Faculty of Sports and Rehabilitation and Faculty of Military Training. BMSTU is one of fewer higher educational establishments in Russia with a six year term of training of certiﬁed specialists. This term allows the more profound fundamental knowledge and professional skills to be acquired by future graduates. Those students, who have successfully completed the basic (six year) course of training, may get the more thorough instruction (seventh year of training) to improve their professional skills or to prepare for the further study of the post-graduate course. Altogether in the University there are more than 100 chairs, annually graduating more than 2000 highly skilled specialists. The full form training by 16 basic professions is provided by BMSTU Kaluga Branch which has commemorated the 45th anniversary of the foundation. Students of the University have an opportunity to get additional educational services, e.g. another profession or specialization

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VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

(conﬁrmed by a diploma), on the contract basis. The BMSTU structure also includes the Institute for Improvement and Change of Professional Skills of Skilled Personnel and the Experimental Institute for Improvement of Skills of Teachers. Since 1934 BMSTU has been engaged in teaching of hard-of-hearing students and is the world’s ﬁrst educational establishment of a general type, where professional programs of the engineering education for this category of disabled people are successfully implemented and efﬁciently keep developing. It has become the largest education institution in Russia where training of highly skilled specialists is realized for hard-of-hearing citizens; nowadays many of them are prominent engineers and scientists. The unique system, developed in BMSTU, includes both the integrated continuous multi-level professional education, variative in a form, and the complex rehabilitation, accompanying the learning process. The suggested model of the innovation education, being integrated with the social policy and meeting the best world standards, may serve the basis for creating the federal system of the higher engineering education for disabled people. For many years BMSTU has been working systematically to organize a broad net of educational institutions within the framework of under-graduate activity. The net includes: more than 200 schools (secondary, lyceum and gymnasium type) of Moscow and Moscow oblast and other regions of Russia; Russian youth programs “Step into the Future” and “Cosmonautics”; multiple preparatory courses, various olympiads etc. For 12 years the Physical and Mathematical Lyceum has been successfully functioning at BMSTU. The Lyceum annually graduates about 300 future students for the University. The worked-through system is based on the all-year cycle and includes various forms of professional orientation and knowledge check of university entrants. Its main goals are as follows: creating optimal conditions to future BMSTU students for the high-quality completion of the secondary education process, preparation for entering the University and adaptation for further study there; attracting the talented youth; selecting the more prepared and professionally oriented university entrants. BMSTU takes an active part in accomplishing many scientiﬁc and technical programs of the Russian Federation Ministry for Education and Science and other ministries and organizations and BMSTU is frequently a leading executor of them. On University’s base the Teaching and Methodical Association on Polytechnic University’s Education functions, as well as 12 products-certiﬁcation centers, many other divisions, supporting close ties between the University and industrial and scientiﬁc institutions. The purposeful and fruitful work to create a strategy of developing the engineering education, a methodical support of the teaching process is carried out in BMSTU. Late in the nineteen nineties the Association of Technical Universities, headed by BMSTU, developed a concept of a contemporary technical university, the proper classiﬁer and educational standards. The BMSTU Press successfully functions in BMSTU. It publishes a periodical bulletin “Vestnik MGTU imeni N.E. Baumana” (in three series: Mechanical Engineering, Instrumental Engineering and Natural Sciences). Publishing series

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of textbooks, for example, “Mechanics in Technical University”, “Informatics in Technical University”, etc. is characteristic of BMSTU. The BMSTU technical library, having more than 3 million volumes in its scientiﬁc book-stock, is unique. The fund of the rare books storage is of exclusive value and includes volumes of XVIII–XX centuries, e.g. dissertation by D.I. Mendeleev, scientiﬁc papers by N.Ye. Zhukovsky and many others. BMSTU maintains and further develops international relations with practically all leading world technical universities. About 250 BMSTU students and postgraduates annually learn courses and work on probation in leading world universities. BMSTU scientiﬁc developments are successfully implemented by the largest companies of the USA, China, Republic of Korea, Japan, Canada, Great Britain and other countries. Russian engineers, who have graduated from BMSTU, are considered among the best in the world. Rationally combining the fundamental and special technical education and having a high level of the practical professional training, they successfully withstand a severe competition in intellectual labor markets of our country and abroad. The BMSTU development is inseparably linked with an inexhaustible “supply of strength” — above all, with an intellectual potential, owing to which for many dozens of years BMSTU has been holding its positions of a leader of our higher technical school in science, engineering and manufacturing procedures and process. A combination of an accurate engineering design with an engineer’s intuition, careful observance of traditions of graduating engineers in accordance with the “Russian method”, acute feeling for a novelty of scientiﬁc trends, complex social and economical approach to solving the most complicated technical problems, fundamental and humanitarian aspects of training specialists — all allow BMSTU to keep remaining on the edge of the world scientiﬁc and engineering progress, to head the Community of Russian Technical Universities and the whole higher engineering education in Russia. Commemorating the important jubilee — the 175th anniversary of the BMSTU foundation, we may wish the University to continue and develop the many years glorious traditions of leading scientiﬁc schools, arisen within its precincts and recognized by the world; to improve the training of highly skilled specialists — BMSTU alumni; to keep for oneself the high status of a leader of the Russian university’s engineering education. Editor-in-chief, Rector of BMSTU, Corresponding Member of the Russian Academy of Sciences I.B. Fedorov

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SIMULATION OF PROCESSES & SYSTEMS Herman Krier (Center for Simulation of Advanced Rockets University of Illinois at Urbana-Champaign Urbana, IL 61801, USA), Sergey T. Surzhikov (Institute for Problems in Mechanics of Russian Academy of Sciences — Bauman Moscow State Technical University)

COMPUTING MODEL OF BURNING OF BI-MODAL AP/HTPB COMPOSITE SOLID ROCKETS PROPELLANTS A numerical simulation model of thermo-gas-dynamic processes near to a burning surface of heterogeneous solid rocket propellants on the base of AP and HTPB is represented, including the heat transfer processes inside a solid phase of these materials. The model takes into account the gas phase chemical reactions within the framework of the concept of the global chemical reactions. Two global chemical reactions for ﬁve effective chemical components are accepted in the model.

Introduction. Composite solid propellants are a heterogeneous mix of crystalline oxidizer and polymer binder, plus other burning rate additives. A long-range goal is to eventually model the combustion of such heterogeneous material to resolve ﬂame characteristics that range from 5 to 500 µm, at rocket motor pressures ranged from 40 to 100 atm. A two-dimensional model for the unsteady burning of idealized mixtures of oxidizer (ammonium perchlorate, AP) and binder (hydroxyl-terminated polybutadiene, HTPB) was developed previously [1]. That study included the modeling by a reduced scheme of the chemistry of AP and binder combustion. Both the gas and solid phase concentration relations were solved, assuming a 2D planar geometry. The energy balances between phases and a pyrolysis rate relation allowed for the coupling and the prediction of the AP, binder, and mixture burning rates. The work reported in [1] showed how the gas phase micro-ﬂame structure supplied the local heat ﬂuxes which, in turn, determined to local regression rate, but always for a periodic “sandwich” of AP and binder. Figures 1–3 are typical results from the 2D model [1]. Since composite propellants are heterogeneous in all three dimensions, the work presented below is the next-step toward such simulations. Instead of regular 2D “sandwich” conﬁguration, we now consider the mixture of AP/Binder to be un-even, as depicted in Figure 4, a, to simulate a bi-modal oxidizer mixture. Mathematical Formulation of the Problem. The numerical simulation model includes groups of equations for describing mass, energy and species concentration transfer processes in the gas and solid phases of heterogeneous propellants.

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Fig. 1. Gas-phase temperature and velocity distributions at p = 40 atm and ϕ = 0.75 0.75. BDP-pyrolysis model, 8-component kinetics model of gas phase. Full 2D solid phase heat conduction model

Fig. 2. Distributions of convective heat ﬂux along burning surface at p = 40 atm and ϕ = 0.75 0.75. BDP-pyrolysis model, 8component kinetics model of gas phase. Full 2D solid phase heat conduction model

Fig. 3. Burning rate distribution at p = 60 atm and ϕ = 0.75 0.75. BDPpyrolysis model, 8-th-component kinetics model of gas phase. Full 2D solid phase heat conduction model

For solving these equations, a Cartesian coordinate system (Fig. 4) is used. The calculated domain (a − b − c − d) is shown here as a physical region which includes solid oxidizer (AP; a−e−y1 −g; y2 −f −d−h), solid binder (HTPB; y1 −y2 −h−g), gas phase region (e − b − c − f ), and also the symmetry planes (a − b; c − d).

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VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

Fig. 4. Schematic of the problem where two different rectangular AP oxidizers bounded the HTPB binder (aa ). Calculation domain (bb). According to Fig. 4, a pieces of the AP (aa − e − y1 − g and h − y2 − f − dd) have different depths

Gas phase. The gas phase description is based on the system of mass conservation, the Navier-Stokes equations, species conservation equations for gas mixture and for each chemical component, and the energy transfer equation for a chemically reacting mixture. The non-dimensional gas-phase relations were presented in [1], and are: ∂ρ + div(ρV) = 0, (1) ∂t ∂ρu ∂p + div(ρuV) = − + ∂t½ ∂x · µ ¶¸ µ ¶¾ 2 ∂ ∂ ∂u ∂v ∂ ∂u 1 ρ + − (µdivV) + µ + +2 µ − 2 , (2) 3 ∂x ∂y ∂y ∂x ∂x ∂x Re Fr ∂ρv ∂p + div(ρvV) = − + ∂t ∂y ¾ ½ ∂ ∂u ∂v ∂ ∂v 2 ∂ 1 (µdivV) + [µ( + )] + 2 (µ ) , (3) + − 3 ∂y ∂x ∂y ∂x ∂y ∂y Re ∂Yk + ρVgradYk = ∂t L 1 = , div(ρDk gradYk ) − $k Re Sc ρ0 u 0

ρ

ρcp

k = 1, 2, . . . , Nc

(4)

∂T + ρcp VgradT = ∂t N

=

c 1 1 X cp,k ρDk (gradYk · gradT ) + div(λgradT ) + Pr Re Re Sc k µ ¶ Nc X L + , hk $ k ρ0 cp,0 u0 T0

(5)

k

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where x, y are the longitudinal and transversal coordinates; u and v are the projections of the ﬂow speed V on the x and y axes; ρ, p, cp , µ, λ are the density, pressure, speciﬁc heat at constant pressure, viscosity, and thermal conductivity respectively; T is the temperature; cp,k is the speciﬁc heat capacity at constant pressure for the k-th component; Yk = ρk /ρ is the mass fraction of the k-th component; ρk is the density of the k-th component; Dk is the effective diffusion coefﬁcient of the k-th component; hk , $k are the enthalpy and reaction rate of the k-th species (Here: reaction rate means value of mass consumed or produced per unit volume per unit time, e.g. g/(cm3 ·sec)); Nc is the number of chemical components. The variables (x, y, t), (u, v), (ρ, p, T, cp , µ, λ) in the system of Eqs. (1)–(5) are nondimensional. To non-dimensionalize the variables, the following parameters were used: ρ0 , µ0 , L0 , u0 , where the “0” index is used from here on to denote the parameters at known ﬁxed conditions: (x, r) =

T =

T˜ , T0

µ=

(˜ x, r˜) , L0 µ ˜ , µ0

t=

λ=

t˜u0 , L0

˜ λ , λ0

(u, v) =

D=

˜ D , D0

(˜ u, v˜) , u0 cp =

ρ=

c˜p , cp,0

ρ˜ , ρ0 p=

p˜ . ρ0 u20

ρ 0 u0 L 0 is the Following criteria of the heat and mass transfer are used: Re = µ0 µ0 cp,0 µ0 Reynolds number, Pr = is the Prandtl number, Sc = is the Schmidt λ0 ρ 0 D0 uo number, Fr = √ is the Froude number, where g is the gravitational acceleragL0 tion. The following normalization parameters were used: u0 = 10 cm/sec, L0 = = 0.01 cm, ρ0 = 1.177 · 10−3 g/cm3 , µ0 = 1.983 · 10−4 g/(cm·sec), λ0 = 2.624 · λ0 µ0 · 10−4 W/(cm·K), T0 = 300 K, cp,0 = , D0 = . µ0 ρ0 The chemical kinetics model. A concept of the global chemical reactions for burning of heterogeneous solid propellants was discussed in [1, 2]. A 5-component model for two global chemical reactions is used for describing heat release processes in the gas phase k 1 a1,1 [FA ] + a1,2 [Ox] −→ b1,1 [P1 ],

(6a)

k 2 b2,1 [P2 ]. a2,1 [FB ] + a2,2 [P1 ] −→

(6b)

Correspondence between the conventional and real names of the chemical components is listed in Table 1. The global chemical reactions constants are expressed by the Arrhenius law as µ ¶ El kl (T ) = Al exp − , l = 1, 2, (7) R0 T where El is the activation energy; Al is the pre-exponent factor (see Table 2); R0 is the universal gas constant, R0 = 1.9858 cal/(mol·K). 16

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

Table 1 Chemical components taken into account in the 5-component numerical simulation model 1

2

3

4

5

Conventional names of the effective chemical components

FA

Ox

P1

FB

P2

Names of the effective chemical components

NH3

HClO4

P1

C4 H6

P2

(Yk,w )AP

0.05

0.04

0.91

0.0

0.0

(Yk,w )HTPB

0.0

0.0

0.0

0.3

0.7

Number of species

Table 2 Suggested input for the global chemical reactions constants

Number of the reaction

Al , (cm3 · sec)/mol

El , cal/mol

13

1

3.6 · 10

15236 11

1.8 · 10

2

14955

In this case the global reactions rates are expressed as, ¶µ ¶ µ ρ2 ρ1 , ω˙ 1 = k1 (T ) W1 W2 µ ¶µ ¶ ρ4 ρ3 ω˙ 2 = k2 (T ) , W4 W3

(8a)

(8b)

where Wk is the molecular weight of the species k. Here, the dimension of the ω˙ k is mol/(cm3 ·sec). The species reaction rates are calculated using, $1 = −a1,1 W1 ω˙ 1 ,

$2 = −a1,2 W2 ω˙ 1 ,

$4 = −a2,1 W4 ω˙ 2 ,

$5 = +b2,1 W5 ω˙ 2 .

$3 = +b1,1 W3 ω˙ 1 − a2,2 W3 ω˙ 2 , (9)

where a dimension of all reaction rates is g/(cm3 ·sec). Thermodynamical and Transport Properties of the Gas Phase. For calculating the thermophysical and transport properties of the gas mixture, the following relations were used: R0 p=ρ T, (10) WΣ Nc Nc X X ρk , h = h k Yk , (11) ρ= k

k

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Nc X

1 cp =

cp,k Yk ,

WΣ =

.

(12)

Nc X

Yk Wk

k k

Average characteristics for the momentum, heat and mass transfer coefﬁcient assumed that sµ ¶ T 3 273 + 117 µ(T ) = 1.983 · 10−4 , g/(cm · sec), (13) T + 117 273

λ(T ) =

µcp , Pr

Dk = D(T ) =

µ . ρSc

(14)

To approximate the thermodynamic properties the following approximations are used for the speciﬁc heat and the enthalpy [3]: cp,k = A + Bt + Ct2 + Dt3 + E/t2 , J/(mol · K); hk − hk,T =298.15K = 2

(15) 3

4

= At + Bt /2 + Ct /3 + Dt /4 − E/t + F −

0 ∆Hf,298.15 ,

kJ/mol,

0 is the enthalpy of formation at 298.15 K; where t = T /1000; ∆Hf,298.15 A, B, C, D, E, F are the approximation coefﬁcients given in [3]. It is assumed that average molecular weight of the global chemical reactions products and approximations for the enthalpy are WP1 = WP2 = 29 g/mol, hPk = APk + BPk T , k = 1, 2, where approximation coefﬁcients are presented in Table 3.

Table 3 Approximation coefﬁcients for products enthalpy of the global chemical reactions

Number of the reaction

APk , cal/mol

BPk , cal/(mol·K)

1

–59500

10

2

–41000

12

Boundary conditions. The boundary conditions for solving Eqs. (1)–(7) are formulated in relation to the velocity, temperature, and mass fractions of the chemical components. Boundary conditions at xg = 0 (surface of the heterogeneous material). Mass fractions distributions along the surface: at y < y1 , y > y2 (AP surface): Yk = Yk,w ,

18

k = 1, 2, 3;

Yk = 0,

k = 4, 5;

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

at y1 < y < y2 (binder surface): Yk = 0,

k = 1, 2, 3;

Yk = Yk,w ,

k = 4, 5.

The mass fractions Yk,w are presented in Table 1. It was assumed that the mass fractions for the chemical components considered (including effective product species P1 , P2 ) do not depend on external burning conditions or pressure. Note that because we assume the values for the interface mass fractions, we do not utilize the species mass ﬂux boundary condition which includes the diffusional velocity, related to gradYk,w . Temperature and blowing velocity distributions along the surface: at y < y1 (AP surface): Tw = Tw,1 ,

uw = uw,1 ;

(16)

Tw = Tw,3 ,

uw = uw,3 ;

(17)

uw = uw,2 .

(18)

at y > y2 (AP surface):

at y1 < y < y2 (binder surface): Tw = Tw,2 ,

Note that Tw,1 , Tw,2 , Tw,3 and uw,1 , uw,2 , uw,3 are determined after solving heat transfer equations for the heterogeneous material solid phases. Boundary conditions at the symmetry axis (plane) (y = 0).

y = 0 : v = 0,

∂T ∂Yk ∂u = = = 0, ∂y ∂y ∂y

k = 1, 2, . . . , Nc .

(19)

Boundary conditions at the symmetry plane (y = L).

y = L : v = 0,

∂T ∂Yk ∂u = = = 0, ∂y ∂y ∂y

k = 1, 2, . . . , Nc .

(20)

Boundary conditions at the upper boundary of the calculated domain (x → ∞).

x → ∞,

∂ 2v ∂Yk ∂u ∂T = = = 0, = 2 ∂x ∂x ∂x ∂x

k = 1, 2, . . . , Nc .

(21)

The downstream dimensions for the computational domain to approximate the inﬁnity conditions are varied and range from 100 to 300 µm, as was done in [1]. Initial Conditions. To obtain steady-state solutions, temperature, velocity and species concentration distributions inside the calculated domain are the same as at surface. Pressure is assumed to be constant. To obtain other unsteady solutions, known steady-state values were used to provide initial conditions, as discussed in [1].

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Heat Transfer Model for the Condensed Phases. The calculated domain for solving the solid phase heat transfer equations was shown in Fig. 4. Heat transfer equations are formulated for each of three sub-domains (see Fig. 4, b) µ ¶ α1,2 ∂T1 ∂T1 ∂T1 ∂ C 1 ρ1 ˙1 = (22) + C1 m (T1 − T2 ), λ1 −2 ∂t ∂xs ∂xs ∂xs ϕ1 L µ ¶ ∂ ∂T2 ∂T2 ∂T2 + C2 m ˙2 = λ2 + C 2 ρ2 ∂t ∂xs ∂xs ∂xs α2,3 α1,2 (T1 − T2 ) + 2 (T3 − T2 ), (23) +2 ϕ2 L ϕ2 L µ ¶ α2,3 ∂T3 ∂T3 ∂T3 ∂ λ3 −2 + C3 m (T3 − T2 ), C 3 ρ3 ˙3 = (24) ∂t ∂xs ∂xs ∂xs ϕ3 L where T1 , T2 , T3 are the temperatures in the ﬁrst, second, and third regions; C1 , C2 , C3 = C1 are the molar heat capacity of corresponding materials; ρ1 , ρ2 , ρ3 = ρ1 , λ1 , λ2 , λ3 = λ1 are the densities and heat conductivities of the materials. Note that α1,2 , α2,3 are the heat transfer coefﬁcients (between the 1st , 2nd, and 3rd materials). Also ϕ1 , ϕ2 , ϕ3 are deﬁned as the volume fraction of the 1st, 2nd, and 3rd materials in composite material (see Fig. 4). Coefﬁcients “2” in the right-hand side parts of these equations take into account 3D-character of the heat transfer processes. The temperature dependencies for the heat capacity and heat conductivity were borrowed from [6]. Thus, 0 00 C1,2 = C1,2 + C1,2 T,

(25a)

λ1,2 = λ01,2 + λ001,2 T,

(25b)

where the coefﬁcients are listed in Table 4. This table also contains values for the material densities. Table 4 Physical properties for the solid AP and HTPB

0 C1,2

C1,2 = cal/(g·K)

+

00 C1,2 T,

λ1,2 = λ01,2 + λ001,2 T , cal/(cm·sec·K) ρ1,2 , g/cm3

Solid AP

Solid HTPB

C10 = 0.15, C100 = 0.00041

C20 = 0.25, C200 = 0.00085

λ01 = 1.5 · 10−3 , λ001 = −9.2 · 10−7

λ02 = 4.4 · 10−4 , λ002 = 1.3 · 10−7

1.76

0.88

A signiﬁcant part of the coupled model are the boundary conditions on the burning surface of the solid material. Here xs = Hs and 20

x = xg = 0, y ∈ [0, y1 ] :

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

¶

µ λ1

∂T1 ∂x

= qw,1 + m ˙ 1 Q1,w ,

(T1 )w,s = (T1 )w,g ;

(26)

w,s

xs = Hs and x = xg = 0, y ∈ [y1 , y2 ] : ¶ µ ∂T2 λ2 = qw,2 + m ˙ 2 Q2,w , (T2 )w,s = (T2 )w,g ; ∂x w,s xs = Hs and x = xg = 0, y ∈ [y2 , L] : µ ¶ ∂T3 = qw,3 + m ˙ 3 Q3,w , (T3 )w,s = (T3 )w,g , λ3 ∂x w,s

(27)

(28)

where: Zy1 qw,1

1 = y1

1 qw dr = y1 0

¶ Zy1 µ ∂T λ dy, ∂x x=0 0

Zy2 qw,2

1 = y2 − y1

1 qw dr = y2 − y1 y1

y1

ZL qw,3

1 = L − y2

1 qw dr = L − y2 y2

¶ Zy2 µ ∂T λ dy, ∂x x=0

¶ ZL µ ∂T λ dy. ∂x x=0 y2

Q1,w , Q2,w , Q3,w are the latent heat terms for the surface phase transformations (J/g) (Table 5); m ˙ 1, m ˙ 2, m ˙ 3 are the mass burning rates for oxidizer and binder (g/(cm2 ·sec)). These mass burning rates are used in Arrhenius form: ¶ µ Eox ; (29) m ˙1=m ˙ oxidizer = Aox exp − R0 Tw,1 ¶ µ Ebin ; (30) ˙ binder = Abin exp − m ˙2=m R0 Tw,2 µ ¶ Eox ˙ oxidizer = Aox exp − m ˙3=m , (31) R0 Tw,3 where Aox , Abin are the pre-exponential factors; Eox , Ebin are the activation energies (see Table 5). Numerical Simulation Results. This can be used to calculate the burning rate of solid propellant based on AP/HTPB at different gas pressures. The calculations were performed for one model of the AP pyrolysis which was offered in [4] (see Eqs. (29), (31), and Table 5), and for two models of the HTPB pyrolysis. The ﬁrst model was offered in [4] (further we will call this model as BDP-pyrolysis model) and the second model was offered in [5] (CFD-pyrolysis model) (see Eq. (30), and Table 4). To demonstrate the inﬂuence of the heat transfer conditions in a solid phase of the composite material on a burning rate, factors of the heat exchange α1,2 , α2,3 VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

21

Table 5 Input values for calculation of the pyrolysis rates (BDP-pyrolysis model) A, g/(cm2 · sec)

E, cal/mol

Qw , cal/g

5.0 · 105

22000

100

15000

–50

Oxidizer

3

2.5 · 10

Binder

Constants for the pyrolysis mass burning rates (CFD-pyrolysis model) A, g/(cm2 · sec)

E, cal/mol

Qw , cal/g

22000

100

16900

–433

5

Oxidizer

5.0 · 10 2

2.9 · 10

Binder

were also varied. A base series of calculation was performed for α1,2 = 0.1 and α2,3 = 1. The input data for these calculations are presented in Tables 1–5. The following sizes of calculation domain (see Fig. 4, b) were set: Hg = 0.01 cm, L = 0.01 cm, Hs = 0.2 cm, y1 = 0.005 cm, y2 = 0.008 cm. Numerical simulation results corresponding to the BDP-pyrolysis model [4] are presented in Table 6 and in Figs. 5–7. Numerical simulations corresponding to the CFD-pyrolysis model [5] are presented in Table 7 and in Figs. 8–10. The following values are presented in those tables: r˙1 , r˙2 , r˙3 are the burning rates for AP (y < y1 , ϕ1 = 0.5), HTPB (y1 < y < y2 , ϕ2 = 0.3) and AP (y > y2 , ϕ3 = 0.2) respectively; r˙av is the average burning rate of the composite material, which is calculated by the following formula

r˙av =

ϕ1 ρ1 r˙1 + ϕ2 ρ2 r˙2 + ϕ3 ρ3 r˙3 , ϕ1 ρ1 + ϕ2 ρ2 + ϕ3 ρ3

(32)

Table 6 Calculated data for the BDP-pyrolysis model

r˙1 , cm/sec r˙2 , cm/sec r˙3 , cm/sec r˙av , cm/sec uw,1 , cm/sec uw,2 , cm/sec uw,3 , cm/sec Tw,1 , K Tw,2 , K Tw,3 , K

22

p = 40 atm 0.584 0.890 0.55 0.63 58.2 1.89 54.9 880 936 877

p = 60 atm 0.953 1.37 0.96 1.02 65.9 2.05 66.1 916 989 916

p = 80 atm 1.23 1.84 1.32 1.35 65.4 2.14 70.3 936 941 1030

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

Table 7 Calculated data for the CFD-pyrolysis model p = 40 atm, α1,2 = 0.1, α2,3 = 1.0

p = 60 atm, α1,2 = 0.1, α2,3 = 1.0

p = 80 atm, α1,2 = 0.1, α2,3 = 1.0

p = 40 atm, α1,2 = 0.01, α2,3 = 1.0

r˙1 , cm/sec

0.65

0.99

1.32

0.59

r˙2 , cm/sec

0.45

0.51

0.67

0.38

r˙3 , cm/sec

1.12

1.42

1.82

0.78

r˙av , cm/sec

0.72

1.00

1.32

0.60

uw,1 , cm/sec

65.3

68.4

70.6

59.1

uw,2 , cm/sec

1.31

1.01

1.03

1.07

uw,3 , cm/sec

117.0

101.0

99.7

79.9

Tw,1 , K

888

919

942

881

Tw,2 , K

1280

1310

1370

1250

Tw,3 , K

928

947

968

901

or with regard to ρ1 = ρ3 = 2ρ2 r˙av =

2ϕ1 r˙1 + ϕ2 r˙2 + 2ϕ3 r˙3 ; 2ϕ1 + ϕ2 ρ2 + 2ϕ3

(33)

uw,1 , uw,2 , uw,3 are the velocities of gas injection from the AP surface (y < y1 ), the HTPB surface (y1 < y < y2 ) and the AP surface at y > y2 respectively; Tw,1 , Tw,2 , Tw,3 are the temperatures of appropriate surfaces. Key interpretations of the quasi-3D simulation results (Figs. 5–10) include the following: 1. The burning rates of components (AP and HTPB) of the solid propellant, and also average burning rate will increase with a gas pressure buildup. This regularity is predicted by both HTPB pyrolysis models (compare Tables 6 and 7). 2. The burning rates obtained for two HTPB-pyrolysis models are similar. This can be explained, ﬁrst of all, by the low binder mass fraction. The largest temperature of a ﬂame (in local zones near to the burning surface), which grows with a pressure buildup, is achieved for the CFD-pyrolysis model. 3. A dependence of average burning rate upon pressure can be approximated ³ p ´0.86 by the following formula (for 40 < p < 80 atm) r˙av = 0.72 , cm/sec, (for 40 the CFD-pyrolysis model; pressure in atm), but using the BDP-pyrolysis model. 4. Comparison of the calculated burning rates with data obtained in [1] (where burning rate of a “sandwich” composite material with a characteristic size of AP layers — 150 µm and HTPB layers — 30 µm is studied) conﬁrms the experimentally established fact that bi-modal composite materials of a matrix type have greater burning rates [7].

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Fig. 5. a : Temperature and velocity distribution above the burning surface at p = 40 atm. The BDP-pyrolysis model is used. Magnitudes of the gas velocities above the surfaces are uw,1 = 58.2 cm/sec, uw,2 = 1.89 cm/sec, uw,3 = 54.9 cm/sec; the surface temperatures are Tw,1 = 880 K, Tw,2 = 936 K, Tw,3 = 877 K; b : Temperature distribution above the burning surface at p = 40 atm. The BDPpyrolysis model is used. Note that maximum x = 50 µ µm; ϕ1 = 0.5, ϕ2 = 0.3 0.3, ϕ3 = 0.2 0.2; c: Heat ﬂux near the burning surface at p = 40 atm. The BDP-pyrolysis model is used; ϕ1 = 0.5 0.5, ϕ2 = 0.3 0.3, ϕ3 = 0.2

5. Gas velocities from the burning AP surface are in the region from 50 to 100 cm/sec (for different pressures and different HTPB-pyrolysis models). The gas velocities of the burning HTPB surface are predicted to be approximately ∼ 1 . . . 2 cm/sec, relatively small compared to the AP. 6. In the calculations cases, one can observe penetration of the gases (injected from the burning surfaces) through a high-temperature zone above the HTPB surface. However, for other conditions (not included here), for lower gas velocities a by-pass of the high-temperature area was predicted.

24

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Fig. 6. a : Temperature and velocity distribution above the burning surface at p = = 60 atm. The BDP-pyrolysis model is used. Magnitudes of the gas velocities at the surfaces are uw,1 = 65.9 cm/sec, uw,2 = 2.05 cm/sec, uw,3 = 66.1 cm/sec; the surface temperatures are Tw,1 = 916 K, Tw,2 = 989 K, Tw,3 = 916 K; b : Temperature distribution above the burning surface at p = 60 atm. The BDPpyrolysis model is used; ϕ1 = 0.5 0.5, ϕ2 = 0.3 0.3, ϕ3 = 0.2 0.2; c : Heat ﬂux near the burning surface at p = 60 atm. The BDP-pyrolysis model is used; ϕ1 = 0.5 0.5, ϕ2 = 0.3 0.3, ϕ3 = 0.2

7. The temperature of the HTPB burning surface exceeds the temperature of the AP burning surface by about 200–300 K in the CFD-pyrolysis model. For the BDP-pyrolysis model the temperatures of the AP and HTPB surfaces are almost equal. 8. The density of isotherms near to the burning surfaces increases with pressure (compare Figs. 5, b–6, b–7, b). One can see that as this takes place, the hightemperature zone of the ﬂame approaches to the burning surface. This explains the increase of the heat ﬂuxes to the burning surface with increasing pressure (see sequentially Figs. 5, c–6, c–7, c). 9. In all prediction cases the heat ﬂuxes at the burning surface are maximum near to the boundaries of the AP and the HTPB. VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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Fig. 7. a : Temperature and velocity distribution above the burning surface at p = 80 atm. The BDP-pyrolysis model is used. Magnitudes of the gas velocities above the surfaces are uw,1 = 65.4 cm/sec, uw,2 = 2.14 cm/sec, u w,3 = 70.3 cm/sec; the surface temperatures are Tw,1 = 936 K, Tw,2 = 941 K, Tw,3 = 1030 K; b : Temperature distribution above the burning surface at p = 80 atm. The BDPpyrolysis model is used; ϕ1 = 0.5 0.5, ϕ2 = 0.3 0.3, ϕ3 = 0.2 0.2; c : Heat ﬂux near the burning surface at p = 80 atm. The BDP-pyrolysis model is used; ϕ1 = 0.5 0.5, ϕ2 = 0.3 0.3, ϕ3 = 0.2

10. The variation of the heat exchange factors (α1,2 , α2,3 ) between the components of the investigated material in the solid phase has shown that burning rates of the AP and HTPB decrease with decreasing of the α1,2 and α2,3 (compare the second and last columns in Table 7). Conclusions. A numerical simulation model of thermo-gas-dynamic processes near to a burning surface of heterogeneous solid propellants on the base of AP and HTPB is represented, including the heat transfer processes inside a solid phase of these materials. The model takes into account the gas phase chemical reactions within the framework of the concept of the global chemical reactions.

26

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Fig. 8. a : Temperature and velocity distribution above the burning surface at p = 40 atm. The CFD-pyrolysis model is used. Magnitudes of the gas velocities above the surfaces are uw,1 = 65.3 cm/sec, uw,2 = 1.31 cm/sec, uw,3 = 117.0 cm/sec; the surface temperatures are Tw,1 = 888 K, Tw,2 = 1280 K, Tw,3 = 928 K; b : Temperature distribution above the burning surface at p = 40 atm. The CFDpyrolysis model is used; ϕ1 = 0.5 0.5, ϕ2 = 0.3 0.3, ϕ3 = 0.2 0.2; c : Heat ﬂux near the burning surface at p = 40 atm. The CFD-pyrolysis model is used; ϕ1 = 0.5 0.5, ϕ2 = 0.3 0.3, ϕ3 = 0.2

Two global chemical reactions for ﬁve effective chemical components are accepted in the model. A joint coupling of the heat-mass-transfer processes in a gas phase and thermal processes in a solid phase allows us to predict burning rates of the separate components of the material and the mass averaged burning rate of the heterogeneous propellant, including the surface temperatures and the gas (injection) velocities of the burning process gas products.

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Fig. 9. a : Temperature and velocity distribution above the burning surface at p = 60 atm. The CFD-pyrolysis model is used. Magnitudes of the gas velocities above the surfaces are uw,1 = 68.4 cm/sec, uw,2 = 1.01 cm/sec, u w,3 = 101.0 cm/sec; the surface temperatures are Tw,1 = 919 K, Tw,2 = 1310 K, Tw,3 = 947 K; b : Temperature distribution above the burning surface at p = 60 atm. The CFDpyrolysis model is used; ϕ1 = 0.5 0.5, ϕ2 = 0.3 0.3, ϕ3 = 0.2 0.2; c : Heat ﬂux near the burning surface at p = 60 atm. The CFD-pyrolysis model is used; ϕ1 = 0.5 0.5, ϕ2 = 0.3 0.3, ϕ3 = 0.2

The gas dynamics of the process was studied within the framework of the full unsteady Navier-Stokes equations. Actual properties for gas and condensed phases were also taken into account. The systematic numerical research of the burning process of bi-modal composite material of a matrix structure for various binder pyrolysis models is fulﬁlled, although the next step is to model the “spherical” shaped oxidizer. Based on the calculations the approximate dependence of an average burning ³ p ´0.86 rate upon on pressure is offered r˙av = 0.72 , cm/sec (with p in atm). 40 28

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

Fig. 10. a : Temperature and velocity distribution above the burning surface at p = 80 atm. The CFD-pyrolysis model. Magnitudes of the gas velocities above the surfaces are uw,1 = 70.6 cm/sec, uw,2 = 1.03 cm/sec, uw,3 = 99.7 cm/sec; the surface temperatures are Tw,1 = 942 K, Tw,2 = 1370 K, T w,3 = 968 K; b : Temperature and velocity distribution above the burning surface at p = 80 atm. The CFD-pyrolysis model is used; ϕ1 = 0.5 0.5, ϕ2 = 0.3 0.3, ϕ3 = 0.2 0.2; c : Heat ﬂux near the burning surface at p = 80 atm. The CFD-pyrolysis model is used; ϕ1 = 0.5 0.5, ϕ2 = 0.3 0.3, ϕ3 = 0.2

It was shown that the burning rate of a bi-modal composite material of a matrix type exceeds the burning rate of materials with the “sandwich” structure (see [1]) for similar characteristic sizes of AP/binder. The study was partially supported by Russian Foundation for Basic Research grant No. 01-02-17532.

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REFERENCES 1. Surzhikov, S.T., Murphy, J.J., Krier, H. 2D Model for Unsteady Burning Heterogeneous AP/Binder Solid Propellants // AIAA Paper 2000-3573, 36-th AIAA/ASME/SAE/ASEE Joint Propulsion Conference 16–19 July, 2000/Huntsville, AL. 2. Ermolin, N.E., Korobeinichev, O.P., Fomin, V.M., et al. Study of Flame Structure For Mixed Solid Fuels Based on Ammonium Perchlorate and Polybutadiene Rubber // Combustion, Explosion, and Shock Waves, 1992, V. 28, No. 4, pp. 59–65. 3. Chemistry Web Book, http://webbook.nist.gov, 2000. 4. Beckstead, M.W., Derr, R.L., Price, C.F. A model of Composite Solid-Propellant Combustion Based on Multiple Flames // AIAA Journal, 1970, V. 8, No. 12, pp. 2200– 2207. 5. Cohen, N.S., Fleming, R.W., Derr, R.L. Role of Binders in Solid Propellant Combustion // AIAA Journal, 1974, No. 2, pp. 212–218. 6. Jeppson, M.B., Beckstead, M.W., Jing, Q. Model for the Premixed Combustion of a Fine AP/HTPB Composite Propellant. 35-th JANNAF Combustion Meeting, Tucson Arizona, 1998. 7. Price, E.W., Chakravarthy, S.R., Sigman, R.K., Freeman, J.M. Pressure Dependence of Burning Rate of Ammonium Perchlorate-Hydrocarbon Binder Solid Propellants // AIAA 97-3106, 33rd AIAA/ ASME/ SAE/ ASEE Joint Propulsion Conference & Exhibit, July 6–9, 1997, Seattle, WA.

S.T. Surzhikov graduated from the Bauman Moscow Higher Technical School in 1975. D.Sc. (Phys.-Math.), Head of the Computational Physical-Chemical and Radiative Gas Dynamics Laboratory of the Institute for Problems in Mechanics of Russian Academy of Sciences. Author of more than 300 publications in radiative gas dynamics and theory of heat and mass transfer.

BMSTU Press has published the book: Surzhikov S.T. Radiation of Heat in Gases and Plasma (in Russian) – M.: Izdatelstvo MGTU imeni N.E. Baumana, 2004. 544 p. Basic concepts of the radiant energy transfer theory in hot gases and low-temperature plasma are introduced. Phenomenological coefﬁcients and functions of the transfer theory are worded as well as their association with quantum properties. Main laws of the theory of transfer of radiation of heat are presented. The transfer equation is formulated and its most-used particular forms are given. Peculiarities of application of models of elementary radiation processes to construction of phenomenological models of radiation transfer are discussed. Methods of integrating the radiation transfer equation over a frequency and 3D-coordinates are presented. The book is intended for researchers and engineers practising heat exchange by radiation, physical gas dynamics and physics of low-temperature plasma as well as for students and post-graduates learning physical and technical trades in universities.

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VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

V.T. Kalugin, A. Yu. Lutsenko, Ye.G. Stolyarova (Bauman Moscow State Technical University)

HYSTERESIS IN AERODYNAMICS OF FLYING VEHICLES UNDER CONDITION OF UNSTEADY MOVEMENT On the basis of the experimental research of aerodynamic hysteresis in a wide range of the governing parameters M∞ , α under condition of unsteady movement, some regularities are found in transformation of structures of a ﬂow nearby blunt bodies. Inﬂuence of unsteady effects on moments of transformation of a ﬂow from separated to attached one is established. Unsteady aerodynamic characteristics (in view of movement background) and their derivatives for various frequencies and amplitudes of body oscillations are obtained.

During the movement of ﬂying vehicles their kinematic parameters (speed of ﬂight, angles of attack and sliding) and parameters of a ﬂow (pressure, density, temperature) are continuously changing. Not only the values of the above-mentioned parameters but also a sense of their change, i.e. decrease or increase, have a vital importance for the structure and characteristics of separated ﬂows. It means, that with the same set of parameters a vehicle may have various aerodynamic characteristics. One of the reasons of it is the aerodynamic hysteresis caused by transformation of a ﬂow structure. Because of rather small time of the transformation process, accompanied with a sharp change of pressure upon surfaces, there always occur shock loads. This should be taken into account in design of ﬂying vehicles. Aerodynamic hysteresis is most pronounced at transonic speeds of ﬂow around the ﬂying vehicles with generatrix breaks (combinations of cylindrical and conical surfaces; bodies of rotation with segmental and butt bluntness, with rod superstructures) when at least one of the following parameters has changed: a speed of an oncoming ﬂow, angle of attack or Reynolds number. Experimental complex investigations of a ﬂow nearby the segmental bluntnosed bodies were conducted in the transonic speed range. They included viewing ﬂow patterns and measuring aerodynamic forces and moments both under the condition of the model oscillations and with ﬁxed angles of attack. Experiments were conducted in the wind tunnel with the closed test portion at Mach numbers of 0.7. . . 1.3, Reynolds numbers of (1.2 . . . 4.1) · 106 , the undisturbed ﬂow pressure and stagnation temperature were equal to 12.6 Pa and 290 K, respectively. Models represented low-lengthening bodies with ﬂat side surfaces, the square cross section l × l = 64 · 10−4 m2 and the lengthening λ = l/b = 0.422, where b is a body length. Unsteady aerodynamic characteristics were determined with the method of forced oscillations. A principle of harmonic analysis was at the heart of experimental studies of aerodynamic derivatives of forces and longitudinal moment [1]. In these studies the models oscillated relative to the ﬁxed angles of attack with the VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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small amplitude θα = ±4◦ . The estimation of inﬂuence of the ﬂow separation under the unsteady ﬂow condition on total aerodynamic characteristics and acquisition of experimental data about the angular speed inﬂuence on the ﬂow pattern transformation were conducted for oscillations of the models with various amplitudes θ1α ≤ ±16◦ , when an oscillation span exceeded the angle of attack at which the ﬂow pattern transformation occurred. Structures of a ﬂow around a blunt body. An axisymmetric ﬂow around a blunt body, both at a subsonic and supersonic speed, is accompanied with formation of separated ﬂow zones behind a frontal part of the body. From all variety of ﬂow structures under condition of unsteady movement we can highlight the three basic ones, shown in Fig. 1 [2]. With change of Mach number M∞ of the free stream the ﬂow structure transforms. At small subsonic speeds near the point of a generatrix break the ﬂow separates with formation of an open zone of the separated ﬂow 1 on the body surface (see Fig. 1, ). In the process of M∞ increase within the range of subsonic speeds the separated ﬂow zone nestles up to the streamlined surface; the zone cross-section decreases and ﬂow pressure in the zone decreases too. The further speed increase causes a change in the exterior of the ﬂow structure. In front of the body and near the external border of the separated ﬂow zone the shock

Fig. 1. Typical structures of a separated ﬂow around blunt bodies

32

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

waves appear, but an open type of the separated ﬂow is kept. The pressure behind the generatrix break point levels off at a constant value. The ﬂow structure transformation occurs almost instantly at some ﬁrst critical Mach number (Mcr1 ) when instead of the advanced separated ﬂow, closed for a wake, a local separated zone 2 appears in the frontal part of the body (see Fig. 1, b). It is accompanied with forming nearby the streamlined surface a system of shock waves 3 due to the local separated ﬂow attachment and external ﬂow braking. The above-described ﬂow structure is referred to as a supersonic one. With the following reduction of Mach number down to 1, the type of the ﬂow structure remains the same, as there is no disturbing factor to cause the ﬂow structure transformation. However, at Mach numbers, smaller than 1, such a factor is available. It is known that the braking of a supersonic stream occurs through a shock wave. If a ﬂow speed becomes below the sound speed, a local supersonic ﬂow transforms to a subsonic one through a normal shock wave. The shock wave interaction with a boundary layer on the body surface results in its separation, the starting point of which comes nearer to the frontal part as the speed reduces. At the moment of merger of a circulating zone and a separated one, caused by the shock wave, the ﬂow structure is transforming, pressure is sharply growing, and separation, beginning in the generatrix break point, extends for the whole body surface. Such transformation of the ﬂow is achieved with the second critical Mach number, smaller than the ﬁrst critical one. Thus, at a various sense of the change of the free stream Mach number a ﬂow around the body has different structures, because transformation of ﬂow structures at the direct and reverse change of Mach number occurs under a different initial condition of the system and its reconstruction from this condition needs the additional energy which is taken away from the free stream. The variety of ﬂow structures within a range of small supersonic speeds also takes place with change of the body angle of attack. For zero and small values of the angle of attack there is a ﬂow structure with a forward shock wave, a forward local separated zone and a shock of the secondary compression of the ﬂow. Up to certain values of angle of attack the ﬂow structure varies insigniﬁcantly. On achievement of some critical value of angle of attack αtr1 the ﬂow structure transformation occurs almost instantly. It is explained by the following reasons. The increased pressure from a windward side propagates over the whole surface including a leeward side. For angles of attack, smaller than the ﬁrst critical angle, the propagating disturbances are not capable to cause separation from the leeward surface. If the angle of attack is more than the ﬁrst critical angle or equal to it, a merger of the local separated zone with that on the leeward surface occurs. Due to the ring-type separated ﬂow available the disturbances extend to a signiﬁcant part of the body surface which results in formation of the advanced separated ﬂow 4 (see Fig. 1, c). With the further increase of the value of angle of attack over the ﬁrst critical angle, only geometrical parameters of separated zones change. During the following reduction of angle of attack within the range from the initial value (corresponding to the separated ﬂow structure) to some second critical

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angle of attack, the type of a ﬂow structure remains constant. When the angle of attack becomes equal to the second critical angle, a reverse transformation of the separated ﬂow structure to that of a ﬂow with local separated zones occurs. The range of angles, where there is a variety of ﬂow structures, depends on design parameters of the body, Mach and Reynolds numbers in the free stream, but the reason for the hysteresis occurrence is the same, as in case of increase and reduction of the free stream speed [3]. Mathematical simulation. For construction of the mathematical model, determining the moment of transformation of a steady ﬂow structure into a separated one, we suppose, that in each section the ﬂow is ﬂat, steady, without heat exchange, the constant mass of gas circulates in the separated zone, and there are no longitudinal and transverse gradients of pressure in the zone. We believe, that the angle of transformation αtr1 corresponds to such ﬂow conditions when the angle of an inclination of a leeward surface is equal to the ﬂow separation angle βs (Fig. 2). If αtr1 is less than βs , the ﬂow structure is continuous; if αtr1 appears more than βs , then a separated zone arises on the leeward surface. Let us calculate the ﬂow separation angle βs with the use of a method of the ﬂow-dividing line (f dl). For the calculation of parameters of a ﬂow with separation we assume the direction β = βsd + α, corresponding to the critical ﬂow speed (M=1), as initial one. Here βsd is an angle of the ﬂow turn behind the forward shock wave, at which the speed is equal to the critical one; it is determined by the solution of the following system of equations [4]: 2M2∞ (1 − sin2 θsd ) 2 + (k − 1)M2∞ + = 1, 2 2kM2∞ sin θsd − (k − 1) 2 + (k − 1)M2∞ sin2 θsd · βsd = θsd − arctg

¸ 2 + (k − 1)M2∞ sin2 θsd tgθ sd , (k + 1)M2∞ sin2 θsd

Fig. 2. The scheme of a ﬂow for determination of αtr1 34

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

where θsd is an angle of shock wave inclination at Mach number M=1; k is the ratio of speciﬁc thermal capacities. The task of ﬁnding an angle, at which the ﬂow structure transforms, is solved with a method of iterations. We assume pp = 0,9p∞ as initial value of the pressure in the separated zone on the leeward surface (p∞ is the static pressure in the free stream) and determine with the help of gasdynamic functions the Mach and Crokko numbers (Mlw and Crlw ) behind the frontal surface generatrix break. Using the solution of the integral equation [5] +∞ Z

+∞ Z

ϕ2 dη = 1 − Cr2lw ϕ2 −∞

ϕ dη , 1 − Cr2lw ϕ2 ηf dl

the dimensionless speed, Crokko and Mach numbers on the ﬂow-dividing line are determined as Ã !0,5 Cr2f dl 2 ϕf dl = 0,5(1 + erf ηf dl ), Crf dl = ϕf dl Crlw , Mf dl = , k − 1 1 − Cr2f dl where η is a dimensionless coordinate. Then the angle of the stream turn is calculated: r r q k+1 k−1 2 arctg (Mlw − 1) − acrtg (M2lw − 1) − β. βb = k−1 k+1 Further, on the basis of the theory of shock waves, an inclination angle θsh for a shock wave, created by attachment of the separated ﬂow in the base area, and a value of the static pressure psh behind shock wave are estimated. Using the attachment criterion psh = p0fdl , the value plw is then reﬁned. If the calculated angle βb > 0, we have a continuous structure of ﬂow, if βb < 0 — a separated one. Equality βb = 0 corresponds to the transformation condition α = αtr1 . The unsteady ﬂow condition has an essential inﬂuence on aerodynamics of similar bodies. In the unsteady movement, e.g. the body oscillations (even with a small amplitude), there can be a transformation of areas of the aerodynamic characteristics ambiguity, which should be taken into consideration in aerodynamic calculations. The most widespread method of the unsteady aerodynamic characteristics determination is a method of “small amplitudes” with the governing parameters change according to the harmonic law. As is known, in this case the coefﬁcients of the aerodynamic normal force and aerodynamic longitudinal moment can be presented as follows: ¯ ¯ ¯ ¯˙ z ; Cy = Cyα · α + Cyα˙ · α ˙ + Cyω¯ z · ω ¯ z + Cyω˙ z · ω ¯

¯

¯˙ + mωz¯ z · ω ¯˙ z , ¯ z + mωz˙ z · ω mz = mαz · α + mαz˙ · α ¯

¯

¯

¯

where Cyα , Cyα˙ , Cyω¯ z , Cyω˙ z , mαz , mαz˙ , mωz¯ z , mzω˙ z are derivatives of corresponding ¯˙ z , where α ¯˙ = ¯˙ ω aerodynamic coefﬁcients with respect to parameters α, α, ¯z , ω VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

35

= α˙ · b/V∞ is a dimensionless derivative of an angle of attack with respect to 2 ¯˙ z = ω˙ z · b2 /V∞ is time; ω ¯ z = ωz · b/V∞ is a dimensionless pitch angular speed; ω a dimensionless derivative of the pitch angular speed with respect to time; V∞ is a speed of the free stream; b is a length of the body. Such representation of aerodynamic characteristics is based on the assumption of a possibility to use a principle of superposition of independent private movements for which the structure of ﬂow around ﬂying vehicles remains unchanged — with or without separation. In this case it appears possible to choose independently the main parameters inﬂuencing the aerodynamic characteristics of a body. Experimental results. Let us discuss the given below unsteady aerodynamic characteristics, obtained experimentally for a body of small lengthening with segmental bluntness for speciﬁc values of amplitude, frequency and Struhal number. These experimental results can form a basis for calculations of transition processes of ﬂying vehicles with any law of movement. Unsteady aerodynamic characteristics at a zero angle of attack. Figure 3 presents the experimental unsteady aerodynamic characteristics — derivatives Cyα , ´ ³ ´ ³ ¯ ¯ mαz , Cyω¯ z + Cyα˙ , mzω¯ z + mαz˙ for Mach numbers M∞ = 0.6 . . . 1.2 at a zero angle of attack and various positions of the conventional center-of-gravity x ¯cg (¯ xcg = xcg /b where xcg is a distance from the conventional center-of-gravity to ¯cg2 = 0.428; the body frontal surface; b is a length of the body): x ¯cg1 = 0.262; x ¯cg1 ) shows the following: x ¯cg3 = 0.76). The analysis of these curves (for x in the subsonic range of speeds (M∞ = 0.6 . . . 0.9) with the M∞ increase the value Cyα = f (M∞ ) decreases (Fig. 3, a), remaining positive; in the interval of Mach numbers M∞ = 0.93 . . . 1.1 the derivative changes its sign, with the minimal value Cyα = −0.5 achieved at M∞ = 1.0; the further increase of Mach numbers (M∞ > 1,0) results in growth of the derivative Cyα , and at M∞ > 1.1 the value Cyα > 0. It means, that with the body oscillations (relative to the zero angle of attack) in the range of M∞ = 0.9 . . . 1.1 the normal force direction has an opposite sense to change of an angle of attack (Cyα < 0). The experimental research of a steady ﬂow [3] has shown, that only with a ﬂow separation there can occur a negative aerodynamic normal force (α > 0), acting on a body, and a value of the number M∞ , at which the sign of Cyα changes from negative to positive, corresponds to the moment of transformation of the separated ﬂow (see Fig. 1, a) to the continuous one (see Fig. 1, b). In body oscillations with respect to the zero angle of attack, the ﬂow structure transformation occurs at M∞ = 1.15, while in conditions of a steady ﬂow it occurs at M∞ = 1.05. Thus, the pitch angular speed, which is characteristic of the oscillatory movement of the ﬂying vehicles, leads to some prolongation of existence of the separated ﬂow structure for the larger values of M∞ . The important practical task in investigations of a ﬂow around the oscillating bodies is to determine the inﬂuence of a position of the conventional center-ofgravity x ¯cg on their aerodynamic characteristics. The analysis of the obtained data shows that a character of change of the derivative Cyα = f (M∞ ) at various positions of the rotation axis is the same for all investigated values of x ¯cg . At subsonic 36

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

Fig. 3. Derivatives of aerodynamic force and moment at α0 = 0 (•, •0 — increase and reduction of M∞ , respectively)

speeds of the free stream (M∞ < 0,9) the aft displacement of the conventional center-of-gravity results in a small increase of the normal force derivative. It is evident most essentially in a range of numbers M∞ = 1.0 . . . 1.15 where the ﬂow has an unstable pattern. When a ﬂow around the body becomes attached, the derivative

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Cyα remains constant for various positions of the conventional center-of-gravity and practically becomes equal to the value Cyα obtained for a steady ﬂow. An area of negative values of Cyα and, consequently, of resulting negative normal forces de¯cg1 ) creases with the aft displacement of the conventional center-of-gravity (¯ xcg > x and almost disappears with its maximal approach to a base cut. As this force only occurs in the separated ﬂow mode, the range of Mach numbers, associated with the force, also reduces. Compared to the case of a forward position of the conventional center-of-gravity x ¯cg , the ﬂow structure transformation from ﬂow with separation to ﬂow without separation occurs at smaller numbers M∞ , practically equal to the Mach number of transformation under steady conditions (ωz = 0). It is connected with the inﬂuence of an angular speed of oscillations. With the forward position of the conventional center-of-gravity, e.g. x ¯cg1 , the aft linear speed distribution is such that promotes (at positive angular speeds) the separated zone expansion on the top surface of a body and its reduction on the bottom one. Therefore, the separated ﬂow exists in a wider range of M∞ . At the signiﬁcant aft displacement of the conventional center-of-gravity the distribution of linear speeds, caused by movement of a body, practically does not inﬂuence the ﬂow structure formation in this area, and the forebody ﬂow is rather steady. The analysis of dependence mαz = f (M∞ ) in the investigated range of ﬂow speeds has shown, that at M∞ < 1 the value mαz < 0, and at M∞ > 1 the derivative sign is determined by a position of the conventional center-of-gravity (see Fig. 3, b). For separated ﬂow modes at numbers M∞ < 1.15 the values of mαz , obtained under conditions of forced oscillations and a steady ﬂow, essentially differ from each other. It allows us to make a conclusion that the body oscillations result in the more appreciable transformation of the separated ﬂow structure at subsonic and transonic speeds of free stream and, consequently, in the change of body’s aerodynamic characteristics as compared to those obtained for a steady ﬂow. Figure 3, c presents the Mach number dependence of rotary derivatives (Cyω¯ z + ¯ + Cyα˙ ) with the zero angle of attack and various positions of the body’s axis of rotation (the conventional center-of-gravity). With the M∞ change in the subsonic range of speeds (M∞ < 0.9) a combination of derivatives of the normal force ¯ coefﬁcient (Cyω¯ z +Cyα˙ ) reduces continuously, and at transonic speeds the derivatives ¯ at ﬁrst increase and then decrease (the maximal value (Cyω¯ z + Cyα˙ ) is reached at M∞ = 1). Such a behavior of the M∞ -dependence of the derivatives combination can be explained by estimating individually each derivative in the combination. At speeds of the oncoming ﬂow corresponding to 0.6 6 M∞ 6 0.9 the structure of the separated ﬂow nearby the body (see Fig. 1, ) is transformed insigniﬁ¯ cantly. It is known that a value of the derivative Cyα˙ , basically caused by delay of the ﬂow structure transformation with respect to the angle of oscillations, remains practically constant. According to this, a change of a combination of derivatives ¯ (Cyω¯ z + Cyα˙ ) is determined completely by the law of the Cyω¯ z change. The rotary derivative Cyω¯ z depends on the local angles of attack with the body oscillations in the pitch plane. In the body rotation with angular speed ωz , at various points of the body surface the additional speed components of the ﬂow appear which are

38

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directed, in case of the inverted movement, opposite to the body circular speed. For the investigated position of a rotation axis (¯ xcg1 ) at ωz > 0 these speeds result in a negative angle of attack for the frontal part of a body and positive — for the rear part. The direction of the aerodynamic force, depending on ωz , will be determined, ﬁrstly, by the position of a rotation axis, secondly, by the size of a region of action of positive and negative normal forces, respectively, that depends for the investigated body on the speed of the free stream. So, for example, at x ¯cg = 0.262 ω ¯z (M∞ < 0.75) the value Cy > 0, i.e. normal forces in frontal and rear parts of the body have the same direction — positive. With increasing of the free stream speed the derivative Cyω¯ z decreases due to the extension of the body surface area (where the negative normal force Cyα < 0 is created) behind an axis of oscillations and to the occurrence of a negative force with the point of its application located downstream, behind the axis of oscillations. In a speed range corresponding to 0.9 < M∞ ≤ 1.0 the great increase in value ¯ of a combination of derivatives (Cyω¯ z + Cyα˙ ) is observed. However, it is necessary to note, that at M∞ = 1 the derivative Cyα is negative and reaches its minimum. ¯ Hence, the sharp increase of the combination of unsteady derivatives (Cyω¯ z + Cyα˙ ) ¯ is basically determined by the value Cyα˙ > 0. The subsequent growth of the free stream speed (M∞ > 1) results in reduction of the derivatives combination. Such stepwise change of unsteady derivatives corresponds to conditions under which an unstable separated ﬂow arises. In this case it occurs at M∞ = 1. With increase of the free stream speed (M∞ > 1.15) the ﬂow structure transformation to the ¯ ﬂow without separation is observed. At the same time the derivative Cyα˙ remains positive. In conditions of ﬂow without separation the pressure distribution on the body surface provides only the positive force creation (Cyα > 0). Hence, the sign and value of the derivative Cyω¯ z is determined by the sum of two differently directed forces in the frontal and rear parts of the body. The M∞ -dependence of derivatives of the aerodynamic longitudinal moment ¯ coefﬁcients (mωz¯ z + mαz˙ ) is presented in Fig. 3, d. The behavior of curves is similar ¯ to the specular reﬂection of the dependence Cyω¯ z + Cyα˙ = f (M∞ ). Position of the dynamic focus at various speeds of a ﬂow. In the analysis of unsteady aerodynamic characteristics it is interesting to investigate a position of dynamic focus at various speeds of a ﬂow. A coordinate of the dynamic focus is determined by the following formula: ¯

¯cg − x ¯Fω = xFω /b = x

mzω¯ z + mαz˙ . Cyω¯ z + Cyα¯˙

¯cg = 0.262 (α0 = 0) are presented Results of calculation of values x ¯Fω at x ¯ ¯˙ in a range of in Fig. 4. The point of applying the unsteady force (Cyω¯ z + Cyα˙ ) · α small subsonic speeds (M∞ = 0.6) is situated in a frontal part of the model, i.e. the coefﬁcient of the anti- damping moment is basically determined by the value Cyω¯ z of the frontal part of the model. With the subsequent growth of the free stream speed (M∞ 6 0.7) the point of this force application transfers outside the body. ¯ This means that an absolute value of the aerodynamic derivative Cyα˙ is greater than VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

39

Fig. 4. Change of the dynamic focus position at various ﬂow speeds

Cyω¯ z of a rear part of the model, and the point of application of the unsteady force ¯ ¯ ¯ ˙ is behind the axis of oscillations. When (Cyω¯ z + Cyα˙ ) = 0, the curve (Cyα˙ · α) x ¯Fω = f (M∞ ) has a discontinuity. At transonic speeds of the free stream (M∞ = 0.9 . . . 1.2) the combination ¯ ω ¯z (mz + mαz˙ ) >> 0. Anti-damping appears in the following cases: for (Cyω¯ z + ¯ + Cyα˙ ) < 0, if a coordinate of the unsteady force application point is more than that ¯ ¯cg , and for (Cyω¯ z + Cyα˙ ) > 0 if, of the conventional center-of-gravity, i.e. x ¯ Fω > x on the contrary, x ¯ Fω < x ¯cg . At the ﬂow speed, corresponding to the moment of the ﬂow structure transformation, x ¯Fω asymptotically tends to inﬁnity. If the structure of ﬂow remains separated, the dynamic focus is situated downstream behind the conventional center¯cg at a ﬂow without separation. With a supersonic ﬂow of-gravity, but x ¯ Fω < x ¯ (M∞ > 1.15) the combination (mzω¯ z + mαz˙ ) < 0. In this case the point of ap¯ ¯˙ which is positive, is plication of the aerodynamic unsteady force (Cyω¯ z + Cyα˙ ) · α, situated behind the axis of oscillations. When the point of x ¯Fω coincides with the position of the axis of oscillations (conventional center-of-gravity) a value of the longitudinal damping moment is equal to zero. ¯ The conducted analysis of the Mach number dependencies of Cyω¯ z +Cyα˙ , mωz¯ z + ¯ + mαz˙ and the dynamic focus position has revealed rather a complex structure of a ﬂow nearby a body under condition of the body oscillations at subsonic and transonic numbers M∞ . Inﬂuence of an initial angle of the model installation. For an estimation of inﬂuence of an initial angle of the model installation on its aerodynamic derivatives ¯ ¯ Cyα , mαz , Cyω¯ z + Cyα˙ , mzω¯ z + mαz˙ an experimental research has been carried out at x ¯cg = 0.262 and α0 = 5◦ , 10◦ . If α0 = 5◦ , the general character of the dependence Cyα = f (M∞ ) practically remains the same, as at a zero angle of the model installation (Fig. 5, ). In this case, the less intensive reduction of the derivative Cyα

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Fig. 5. Derivatives of aerodynamic force and moment at α0 6= 0

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with the free stream speed increase is due to the insigniﬁcant transformation of the separated ﬂow on a leeward side of the body. The derivative of the aerodynamic moment coefﬁcient mαz (Fig. 5,b) as against a case of α0 = 0 changes insigniﬁcantly in the whole range of the investigated Mach numbers. The sign and magni¯ tude of the derivatives combination Cyω¯ z + Cyα˙ depend on the distribution of normal forces over the body surface (Fig. 5, c). The change of Cyα is the supreme reason for the fact that in some range of transonic numbers M∞ the absolute magnitude ¯ of the combination Cyω¯ z + Cyα˙ decreases in comparison with that in the case of the zero angle of the body installation. The increase of the installation angle up to α0 = 10◦ leads to essential transformation of a ﬂow structure and, consequently, to the change of behavior of the ¯ curve Cyω¯ z + Cyα˙ = f (M∞ ) in the whole investigated range of Mach numbers. At such angle of installation a negative force does not arise in the body frontal part under conditions of the separated ﬂow, i.e. Cyα > 0; therefore a change of the value ¯ Cyω¯ z + Cyα˙ will be caused by redistribution of positive normal forces with respect to the rotation axis. As the ﬂow structure remains rather steady (α0 = 10◦ ) up to numbers M∞ 6 1.0, the sign and magnitude of the combination of derivatives ¯ ¯ Cyω¯ z + Cyα˙ are determined by the component Cyω¯ z . The sharp increase of Cyω¯ z + Cyα˙ ¯ at M∞ ≈ 1.0 is connected with a change of Cyα . The greatest increase of Cyω¯ z + Cyα˙ is caused by the fact that in a rear part of the body a system of oblique shock waves, closed in the separated zone, appears. The experimentally obtained dependence of a combination of derivatives of the ¯ longitudinal moment coefﬁcient (mzω¯ z + mαz˙ ) on various speeds of an free stream is presented in Fig. 5, d for α0 = 5, 10◦ . The redistribution of forces on the body surface, with its oscillatory movement, results in the essential change of the damping properties in comparison with a case of the zero angle of installation. We shall note that a frontal part of the body is practically neutral in creation of the damping ¯ moment. At an unsteady ﬂow the sign of the combination mωz¯ z + mαz˙ is determined by the action of forces in a rear part of the body. As distinction from the case of small initial angles of the body installation, with α0 = 10◦ the small damping is observed in a range of M∞ = 0.6 . . . 0.7, then for larger M∞ the anti-damping properties increase gradually. In the vicinity of the Mach number equal to 1.0 the sharp increase of the damping coefﬁcient is observed. With the following growth of Mach numbers anti-damping suddenly ¯ appears (mωz¯ z + mαz˙ ) > 0. There should be noted a peculiarity in recalculation of rotary derivatives from one position of the conventional center-of-gravity (axis of rotation) to another. Ac¯ cording to the linear theory, the value Cyα˙ does not depend on a position of the rotation axis and is to be constant for the given conditions of a ﬂow. Hence, the ¯ xcg ) is determined by the value Cyω¯ z = f (¯ xcg ) sum of derivatives Cyω¯ z + Cyα˙ = f (¯ and can be presented as follows: ´ ´ ³ ³ ¯ ¯ = Cyω¯ z + Cyα˙ + ∆¯ xcg · Cyα , Cyω¯ z + Cyα˙ x ¯cg

42

x ¯cg0

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´ ³ ¯ where Cyω¯ z + Cyα˙

´ ³ ¯ , Cyω¯ z + Cyα˙ x ¯cg0

are values of derivatives for the starting x ¯cg

¯cg , respectively; ∆¯ xcg0 = position of the rotation axis x ¯cg0 and the chosen one x ¯cg . =x ¯cg0 − x ¯ However, the unsteady aerodynamic characteristics of the body Cyω¯ z + Cyα˙ , obtained in experimental way, at various positions of the conventional center-ofgravity x ¯cg do not coincide with the calculation results, obtained with the help of the above-mentioned equation. This fact can be explained by the ﬂow structure transformation with the x ¯cg change for the examined range of speeds. The similar conclusion can be also made while analyzing the damping characteristics of the ¯ body mzω¯ z + mαz˙ = f (¯ xcg ), obtained at various positions of the rotation axis. The conducted analysis of experimental results, obtained with the usage of the “small amplitudes” method, has allowed us to make an important conclusion, that the representation of aerodynamic characteristics of bodies with a generatrix break within the framework of the harmonicity hypothesis is only possible at the certain ﬁxed position of the conventional center-of-gravity for some ﬁnite range of change of the determining parameters — an angle of attack, Mach number, amplitude and frequency of oscillations which have paramount inﬂuence on a ﬂow structure nearby such bodies. Using the “small amplitudes” method for determination of the aerodynamic coefﬁcients, required for transient process calculations, allows one to receive them only for the ﬂow structure of the same type (within the limits of oscillation amplitude) and does not enable to establish true moments of the ﬂow transformation. At oscillatory movement of a body with the amplitude exceeding angles of attack, corresponding to the ﬂow separation in steady conditions, the essential change of aerodynamic characteristics is observed. Total aerodynamic characteristics. To determine the aerodynamic coefﬁcients of bodies under consideration in conditions of oscillatory movement with various amplitudes, the experimental research has been carried out providing registration of aerodynamic forces and moments, corresponding to the instant structure of ﬂow. The angle-of-attack dependence of coefﬁcients of the body total aerodynamic normal force Cyt (α) and longitudinal moment mzt (α) in the unsteady movement with parameters M∞ = 0.8, θα = ±16◦ , f = 0.2 . . . 2.5 Hz are presented in Fig. 6. Here, for comparison, the research results Cy0 (α) and m0z (α), obtained for steady conditions, are shown by dashed lines (the same designations will be used in all subsequent ﬁgures). The structure of ﬂow corresponded to that of the separated one. The analysis of the obtained data shows, that the oscillatory movement results in change of the characteristics inherent to a steady ﬂow. In a range of angles of attack 0 ≤ α ≤ 3◦ (f = 0.2 Hz, θα = 16◦ ) the coefﬁcient of aerodynamic normal force is practically equal to zero. With an unsteady ﬂow the negative force occurs in the frontal part of a body for a little bit larger range of angles of attack, than under steady conditions, hence, the area of angles α extends, where Cyt is close to zero. An increase of an angle of attack (α > 3◦ ) results in an increase of Cyt (α, ωz , ω˙ z ), and it is necessary to note that a change of the oscillation frequency VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

43

Fig. 6. Total aerodynamic characteristics (•, •0 — increase and reduction of an angle of attack α , respectively)

in the investigated range of values (f = 0.2 . . . 2.5 Hz) do not inﬂuence essentially the behavior of the α-dependence of Cyt . The reason for this is a small increase of ¯ ¯ the unsteady component (Cyω¯ z + Cyα˙ ) · α ˙ at oscillation frequencies up to f = 2.5 Hz for the investigated kind of a ﬂow structure. The same reason can explain the fact 44

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that the values of Cyt at a direct and reverse change of angles of attack during the oscillatory movement are equal. Within the limits of the oscillation amplitude the total normal force coefﬁcient is a little bit less, than the Cyt (α) values for a steady ﬂow. However, the gradient of Cyt change remains the same, as for Cy0 (α). Some reduction of carrying properties of the body at the unsteady ﬂow is due to the less intensive growth of the positive normal force in the rear part of a body with the change of α. In a range of angles of attack 0 ≤ α ≤ 3◦ where t = 0, values of the total aerodynamic longitudinal moment coefﬁcient mzt (see Fig. 6, b) are close to the corresponding values at the steady ﬂow m0z . With increase of α (α > 3◦ ) within the limits of the oscillation amplitude the values of aerodynamic coefﬁcients mzt (α) and m0z (α) are slightly different, which is caused by transformation of the subsonic separated ﬂow structure at the unsteady movement of the body in comparison with a steady ﬂow. The analysis of the obtained results shows, that inﬂuence of a position of the conventional center-of-gravity xcg on aerodynamic characteristics is different for the steady and unsteady ﬂow. For the case of the unsteady movement under research, it is impossible to recalculate the moment characteristics based on some position of xcg to those based on another xcg . In the vicinity of a zero angle of attack at transonic speeds M∞ > 1 under steady conditions the supersonic ﬂow without separation is observed which is characterized by a local λ-shaped shock wave and the closed circulating zone within the body frontal part. At some angle of attack α = αtr1 (during a direct course — increasing α) the λ-shaped shock wave on a leeward side is breaking, the circulating zone is uniting with the separated area having developed by this moment on the body surface and the ﬂow structure is transforming from continuous to separated one. With reduction of an angle of attack (during a reverse course — reducing α) the transformation of the ﬂow structure to the continuous one occurs at the smaller value of this angle (α = αtr2 ) in comparison to that during the direct course. It is possible to consider that this pattern of the ﬂow structure change remains the same for an oscillatory movement with a large amplitude. Curves of the angle of attack dependence of the longitudinal moment coefﬁcient ¯cg1 = 0.262 are presented mzt for M∞ = 1.12, f = 0.2 . . . 1.0 Hz, θα = ±16◦ , x in Fig. 6, d, e. The ﬂow is without separation at the body oscillation frequency f = = 0.2 Hz in a range of angles of attack 0 ≤ α ≤ 8◦ . The increase of α within the limits of the oscillation amplitude results in the ﬂow transformation to a separated ﬂow. During the reduction of angle of attack (reverse course) the transformation of the separated ﬂow to a continuous one occurs at smaller α, so there is a ﬁeld of ambiguity of body’s aerodynamic characteristics. A change of the oscillation frequency within the limits from 0.2 z up to 0,5 Hz does not inﬂuence essentially the behavior of the dependence mzt = f (α). The increase of the oscillation frequency up to f = 1.0 Hz causes the essential change of aerodynamic characteristics and transformation of the ﬁeld of ambiguity of the longitudinal moment coefﬁcient. At such oscillation frequencies and angles

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45

of attack up to α ≤ 8◦ the absolute value of the aerodynamic longitudinal moment coefﬁcient grows (mzt < 0). No continuous ﬂow of the kind that was observed at smaller frequencies occurs now on the body. In the vicinity of a zero angle of attack the value of coefﬁcient mzt is less than that inherent to the steady ﬂow with a continuous structure. Such distinction in aerodynamic characteristics of the body is explained by formation of the separated ﬂow at its surface. However, if we compare the longitudinal moment coefﬁcients at the unsteady movement and at a stationary ﬂow for the separated ﬂow structure, it is visible that mzt > m0z . With reduction of angles of attack in a range of 3◦ ≤ α ≤ 9◦ the mzt values coincide with the results obtained for a steady ﬂow at the same Mach number (M∞ = 1.12). The analysis of the conducted research allows one to make a conclusion that in conditions of the body oscillations with the frequency f ≥ 1 the separated ﬂow may occur which remains for all angles of attack while in steady conditions there are two types of a ﬂow — with and without separation. In a steady ﬂow the ambiguity of aerodynamic characteristics (presence of hysteresis areas) is explained by the ﬂow transformation delay during a direct and reverse course of changing some governing parameter (Mach number M∞ , an angle of attack or sliding, etc.). In conditions of the body oscillations the ambiguity of its aerodynamic characteristics can be observed at the same ﬂow structure type, for example, at the separated one, which is explained by various changes of geometry of the separated zones and ﬂow parameters in them. With increase of the oscillation frequency (f ≥ 1 Hz) the behavior of the dependence mzt = f (α) within the limits of the investigated oscillation amplitudes does not change, but the area of the aerodynamic characteristics ambiguity extends to somewhat wider range of angles of attack. A change of a position of the conventional center-of-gravity of a body (its aft displacement, for example, x ¯cg2 = 0.428) under the continuous ﬂow and at the oscillation frequencies f ≤ 0.5 z does not result in the change of the total aerodynamic longitudinal moment coefﬁcients with respect to those obtained for a steady ﬂow (see Fig. 6, d). The transformation of the ﬂow to the separated one occurs in the vicinity of the angle of attack α ≈ 9◦ . At values α > 9◦ there is a ﬂow with separation on windward and leeward sides. During the reduction of an angle of attack, starting from the maximal value α = θα = +16◦ , values of the total longitudinal moment coefﬁcient mzt differ essentially from those obtained under steady conditions. For a considered range of frequencies f < 0.5 Hz in the vicinity of α ≈ 5.5◦ (reverse motion) a stepwise reduction of the aerodynamic longitudinal moment coefﬁcient mzt occurs. Such a behavior of the dependence is kept for all investigated oscillation frequencies (f ≤ 2.5 Hz). Analyzing changes of aerodynamic characteristics during the reduction of an angle of attack α, it is possible to make a conclusion, that the continuous structure ﬂow on the bottom and top surfaces of the body occurs at various angles of attack (in the vicinity of α = 5.5◦ and α = 0, respectively). The increase of the body oscillation frequency up to f = 1.0 Hz results in appearance of peculiarities of the aerodynamic longitudinal moment coefﬁcient

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change in the vicinity of a zero angle of attack (see Fig. 6, f). Both at transition from positive angles of attack to negative ones (α˙ < 0), and at their reverse change (α˙ > 0) the aerodynamic longitudinal moment coefﬁcient is not equal to zero and grows with the increase of the oscillations frequency. It can’t be only explained by existence of the moment due to the maximal angular speed, since the value of ¯ ¯ ˙ max is less by an order than mzt . Hence, the abovethe component (mωz¯ z + mαz˙ )α mentioned result can be only caused by the asymmetrical ﬂow structure owing to the delay of the ﬂow transformation from one ﬂow type to another. For example, at reduction of a positive angle of attack (α˙ < 0) the transformation of a ﬂow type from separated to continuous on the top body surface is delayed up to the reach of the region of negative angles of attack. Thus, at the unsteady movement there is no pronounced moment of the reconstruction of a continuous ﬂow on the top surface, characteristic of the steady ﬂow conditions. As a result, the behavior of the dependence mzt = f (α) at α = 0◦ has the above-mentioned peculiarity. The additional proof of existence of an asymmetrical ﬂow around a symmetric body was obtained during the unsteady movement research in which an artiﬁcial braking was applied in the vicinity of a zero angle of attack so that α˙ would change the sign. In this case the transformation duration appreciably reduces and the total aerodynamic longitudinal moment coefﬁcient mzt at a zero angle of attack becomes close to zero. The amplitude of oscillations θα is an important factor inﬂuencing the total aerodynamic characteristics of a body. Research was carried out at various values of θα in the range of 8 . . . 16◦ . The choice of oscillation amplitudes provided ﬂow regimes in which the ﬂow structure transformation inherent to the steady conditions occurred, which allowed the inﬂuence of the body oscillatory movement on the ﬂow structure transformation and the body aerodynamic characteristics to be estimated. In particular, at oscillatory movement with θα = 8◦ (¯ xcg1 = 0.262, f = 1.0 Hz), for which θα < αtr1 , there is no transformation of the ﬂow structure to the separated one. The longitudinal moment coefﬁcients mzt and mz at the unsteady movement and under steady conditions are approximately equal. The oscillation frequency variation in the range of f = 0.2 . . . 2.5 Hz does not inﬂuence the behavior of the dependence mzt = f (α). The analysis of the research results shows that in case of the unsteady movement with the amplitude θα = 10◦ (θα > αtr1 , x ¯cg1 = 0.262, f = 0.5 Hz) at a change of the current angle of attack α < αtr1 , i.e. with a continuous ﬂow structure, the absolute values of the longitudinal moment and normal force coefﬁcients, obtained both at the body oscillations and for steady conditions, are practically equal. The oscillatory process causes the ﬂow structure transformation delay, so a change of the longitudinal moment coefﬁcient mzt , corresponding to appearance of separated ﬂow, occurs at larger angles of attack than it was observed in steady conditions. Besides, parameters and sizes of separated zones on the top and bottom surfaces of the investigated body have changed which is conﬁrmed by essential distinction of the curves of the total normal force coefﬁcients obtained for ﬂow conditions under investigation. VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

47

At reduction of an angle of attack with the body oscillations the transformation of a ﬂow structure from separated to a continuous one occurs at smaller angles of attack. An increase of the oscillation frequency up to f = 2.5 Hz does not bring essential changes in behavior of curves Cyt (α), mzt (α). In case of θα = ±16◦ the character of the angle of attack dependence of the investigated aerodynamic characteristics is the same as at θα = ±10◦ and only has insigniﬁcant distinctions in the area of critical regimes. The comparison of experimental results obtained for various positions of the conventional center-of-gravity x ¯cg shows, that with the aft displacement of the rotation axis position the signiﬁcant transformation of areas of the aerodynamic characteristics ambiguity is observed, at the same time the difference between normal force coefﬁcients at the increase and reduction of the angle of attack is smaller. On the basis of the carried out analysis it is possible to make a conclusion that at the unsteady movement there is an essential transformation of both the ambiguity areas of aerodynamic characteristics and their values in comparison with a steady ﬂow; it is caused by changing of critical angles of attack at which a ﬂow transformation occurs both for a ﬂow with separation and without one. Thus, in the unsteady supersonic ﬂow such factors as frequency, amplitude of oscillations and position of the rotation axis (position of the conventional centerof-gravity) exert an inﬂuence on the aerodynamic characteristics of the body. The degree of this inﬂuence depends on the speed of the free stream. At M∞ values, close to the speeds corresponding to the moment of the ﬂow structure transformation from separated to continuous (with local separated zones), we have the greatest change of aerodynamic characteristics in comparison with those under steady conditions. Work is performed with ﬁnancial support of the Ministry of Education of the Russian Federation: grant 2-06.8-1134.

REFERENCES 1. Belotserkovsky, S.M., Skripach, B.K., Tabachnikov, V.G. Wing in Nonstationary Gas Stream (in Russian). – Moscow: “Nauka”, 1971. – 757 p. 2. Krasnov, N.F., Koshevoy, V.N., Kalugin, V.T. Aerodynamics of Separated Flows (in Russian). – Moscow: “Vysshaya Shkola”, 1988. – 349 p. 3. Kalugin, V.T., Lutsenko, A.Yu., Stolyarova, Ye.G. Experimental Research of Aerodynamic Hysteresis in Transonic Flow Around Segment-Blunted Bodies with Flat Sides (in Russian) // “Vestnik MGTU”, series “Mashinostroenie”. – 2002. – No 3, pp. 15– 30. 4. Krasnov, N.F. Aerodynamics of Body of Revolution. – New York, 10017, 52 Vanderbuilt Avenue: American Elsevier Publishing Company, Inc., 1970. – 867 p. 5. Korst, H.H. A theory for Base Pressure in Transonic and Supersonic Flow // Journal of Applied Mechanics. Vol. 23, 1956, pp. 593–600.

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VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

V.T. Kalugin (b. 1949) graduated from the Bauman Moscow Higher Technical School in 1972. D. Sc. (Eng.), professor of “Ballistics and Aerodynamics” department of the Bauman Moscow State Technical University. Author of over 200 publications in the ﬁeld of aero-gas dynamics of jet and detached ﬂows, design of ﬂight controls. A.Yu. Lutsenko graduated from the Bauman Moscow Higher Technical School in 1987. Ph. D. (Eng.), assoc. professor of “Ballistics and Aerodynamics” department of the Bauman Moscow State Technical University. Author of more than 40 publications in the ﬁeld of aero-gas dynamics of continuous and separated ﬂows. Ye.G. Stolyarova graduated from the Moscow Aviation Institute n. a. S. Ordzhonikidze in 1971. Ph. D. (Eng.), assoc. professor of “Ballistics and Aerodynamics” department of the Bauman Moscow State Technical University. Author of more than 40 publications in the ﬁeld of non-stationary aerodynamics of continuous and separated ﬂows.

BMSTU Press has published the book: Kalugin V.T. Aero-Gas Dynamics of Flying Vehicle Controls (In Russian). – M.: Izdatelstvo MGTU imeni N.E. Baumana, 2004. – 648 p. Results of study of various ways to control the aerodynamic characteristics of rockets, rocket stages and descending spacecrafts are presented. Methods of mathematical and physical simulation of ﬂows around ﬂight controls are given. The book material on aerodynamic, jet and gas dynamical controls is classiﬁed, which makes possible the creation of the data base for aero-gas dynamical design of controlling and braking devices for ﬂying vehicles. The book is intended for under-graduate and post-graduate students of aviation and rocket and spacecraft trades and technical universities. It may be also useful for engineers and researchers specializing in the ﬁeld of aero-gas dynamics and design of ﬂying vehicles.

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

49

Yu.I. Dimitrienko, M.L. Glazikov (Bauman Moscow State Technical University)

LOCAL TRANSPORT PHENOMENA IN POROUS PERIODIC STRUCTURES The present paper continues investigations devoted to application of the asymptotic averaging method to the transport theory in porous media. The method of solving so-called local problems of gas dynamics over a periodic cell has been developed there. Solution of the problems allows us to ﬁnd accurate local distributions of velocities and pressure inside a separate pore, and also to evaluate the gas-permeability coefﬁcient when only geometric shape of pores is known. Computed results are shown and they are compared with experimental data.

1. Introduction. At present in many works, heat and mass transport processes in porous media are investigated only as macroscopic phenomena: Darcy’s law or its modiﬁcations are postulated for relatively slow processes, and dynamical equations of gas/ﬂuid motion through a porous structure are considered for rapidly running processes. Considerable local effects occuring in an individual pore are taken into account only integrally (Darcy’s law includes them in the form of average speed of gas motion and the permeability coefﬁcient determined in experiments). For dynamical problems, speciﬁc hypotheses on local functions are usually introduced (for example, the hypothesis on uniform distribution of gas speeds in a pore etc.). These macroscopic approaches maintain comparative simplicity of the mathematical problem of transport (the set usually includes parabolic equations of ﬁltration or dynamical equations of gas dynamics), but they can not take into account the actual transport processes running in porous structures with complex internal geometry of pores. But the role of local processes (i.e. occuring in an individual pore) is of great importance: character of internal geometry of pores and processes of gas (ﬂuid) motion in a pore determine permeability coefﬁcients of the whole porous system, values of which are known to change within the limits of several orders of the magnitude. Moreover, with the help of reasonable construction of geometry of porous structures we can achieve desired permeability macrocharacteristics. This problem is especially actual for designing of special technical ﬁlters. Taking account of the above-mentioned local transport effects in porous systems can be efﬁciently realized with the help of the asymptotic averaging method for porous periodic structures which has been suggested in our preceding works [3–5]. This method is based on the idea of homogenization of periodic media [2, 7, 1, 4] and introduction of two scale coordinates. Works by Dimitrienko [3–5] formulated so-called local gasodynamics problem over a periodic cell within the frames of “slow” ﬂows, and work by Dimitrienko [5] presented the similar problem for dynamical rapidly running processes. 50

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

These local problems have some speciﬁc peculiarities: they are of integrodifferential type and include periodic boundary conditions. There is no numerical investigation of these problems in scientiﬁc works up to now (works [3, 4]) suggested only approximate solution). The purpose of the present work is to investigate the local gasodynamics problem over a periodic cell and to develop computation methods for solving the problem for some typical geometric structures of pores. It should be noted once again that solving these problems is conducted for the ﬁrst time here. 2. Main Assumptions. Let us consider a porous medium ﬁlled with linearly viscous gas (or compressible ﬂuid). The medium V is assumed to have a periodic structure (Fig. 1) so that a periodic cell (PC) can be separated. Domains Vg and Vs occupied by gas and solid phase, respectively, are oneconnected in the three-dimensional case. All the assumptions in detail have been given in works [3, 4]. The present paper considers the model two-dimensional case, where pores are longitudinal channels along the coordinate axis Ox3 , and domain Vs stops keeping one-connected. This scheme simulates to certain degree, for example, a set of long plane cracks ﬁlled with gas in rocks or a set of longitudinal cracks in composite ﬁbrous unidirectional materials which appear at the ﬁber-matrix interface in solidiﬁcation. Let l0 be the characteristic size of periodic cell Vξ , and x0 be the characteristic global size of the whole porous medium V where the following relationship is satisﬁed: κ = l0 /x0 ¿ 1 (κ is the small parameter). The parameter κ has been introduced in detail in works [3, 4]. Let us also introduce local coordinates ξ = x ¯/κ, where x ¯ = x/ξ0 are the dimensionless global Cartesian coordinates. Then all main functions Ω describing a gas ﬂow in pores can be considered to be quasiperiodic (i.e. depending on ξ and x ¯) and periodic in ξ. Differentiation of the functions is realized by the following

Fig. 1. A scheme of periodic structure and periodic cell (PC)

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51

rule: ¯i ) + (1/κ)(∂Ω/∂ξi ). ∂Ω(x, ξ)/∂xi −→ (∂Ω/∂ x Below the following notation for derivatives with respect to local and global coordinates will be also used: ∂Ω/∂ x ¯i ≡ Ω,i ,

∂Ω/∂ξi ≡ Ω/i .

Index i takes the values i = 1, 2, 3 for three-dimensional case, and i = 1, 2 for two-dimensional periodic structure. The gas ﬂow process in pores is assumed to be sufﬁciently slow so that inert terms in the momentum equation can be neglected, and we will consider only quasistatic equations of equilibrium. The solid phase is considered to be non-deformable, and gas (ﬂuid) is compressible, linearly viscous, perfect. Thermal effects in this work are not considered (i.e. the ﬂow is isothermal). 3. Statement of the Local Problem. Gas motion through porous medium V in the frames of the above-mentioned assumptions is described by the set of dimensionless equations (see [3, 4]) in the Cartesian coordinate system: ρv¯i ),i = 0, (∂ ρ¯/∂t) + (¯ (1) ¯ v¯i,jj = 0, x ¯i ∈ Vg , −p¯,i + µ v¯i |Σsg = 0, p¯ = Aρ¯. Here p¯, ρ¯, v¯ are the dimensionless pressure, density and velocity of gas, respectively; p, ρ, v are their dimensional values; p0 , ρ0 , v0 are their typical magnitudes so that: p¯ = p/p0 , ρ¯ = ρ/ρ0 , v¯ = v/v0 , and t¯ = tv0 /x0 is the dimensionless time, A = Rρ0 θ0 /p0 . Dimensionless viscous coefﬁcient µ ¯ is introduced as follows: µ ¯ = µv0 /(p0 x0 ), and µ is the true value of viscosity. The magnitude µ ¯ is connected to the Reynolds and Euler numbers: µ ¯ = 1/(Re Eu), where Re = ρ0 v0 x0 /µ and Eu = p0 /ρ0 v02 . Let us make one more assumption. Gas viscosity is considered to be sufﬁciently small so that the number µ ¯ can be written in the form ¯0 , µ ¯ = κ2 µ

(2)

where µ ¯0 has the order O(1). This assumption is practically always satisﬁed for real gases: so if p0 = 106 Pa, v0 = 10 m/s, x0 = 1 m, l0 = 10−5 m and µ = 1.5 · 10−5 Pa · s (gas CO2 ), then κ = 10−5 ,

µ ¯ = 1.5 · 10−10

and we obtain µ ¯ 0 = 1.5. 52

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

Since domain Vg has a complex porous structure (Figure 1), immediate computation of the problem (1) is practically impossible. However, due to periodicity of the structure, the solution may be represented in the form of asymptotic expansion in terms of parameter κ [3, 4]: (0)

(1)

v¯i = vi (x, ξ, t) + κvi (x, ξ, t) + O(κ2 ), ρ¯ = ρ(0) (x, t) + κρ(1) (x, ξ, t) + O(κ2 ),

(3)

p¯ = p(0) (x, t) + κp(1) (x, ξ, t) + O(κ2 ). Substitute the expansions (3) into set (1), taking account of the above-mentioned differentiation rules and assumptions (2) collect terms with the same powers of κ and equate them to zero. On collecting terms with lowest powers of κ, we obtain the desired local problem over periodic cell Vξg [3, 4]: (0) vi/i = 0, (1) (0) (0) (4) ¯0 vi/jj = p,i ξi ∈ Vξg , −p/i + µ (0) vi |Σξsg = 0. This problem statement includes the additional condition that average pulsations of pressure hp(1) i are equal to zero, where Z Z 1 (1) (1) p dVξg , ϕg = dVξg (5) hp i = |ϕg | Vξg Vξg is the integral over domain Vξg occupied by gas in PC. The necessity of this condition is explained in works (Dimitrienko, 1997, 1998). Moreover, the set (4) is (0) complemented by the condition on periodicity of functions p(1) and vi which can be written, for example, for 2-D case as follows: [[Ω]] = 0 =⇒ Ω(ξ1 , −1/2) = Ω(ξ1 , 1/2), Ω(1/2, ξ2 ) = Ω(−1/2, ξ2 ),

Ω = {v (0) , p(1) }.

(6)

Periodic cell Vξ in this case is a square with unit side, and the local coordinate system ξi is placed into its center. (0) Pressure gradient p,i in (4) depends only on global coordinates, and this is considered in the problem (4)–(6) as known value (input data of the problem). Due to linearity of local problem (4)–(6), its solution can be always written (0) formally as a linear function of input data, i.e. of p,i : k X (1)

p

(α)

=

P

(ξ)p(0) ,α ,

α=1

(0) vi

k 1 X (α) = W (ξ)p(0) ,α , µ ¯ 0 α=1 i

(7)

(α)

where functions P (α) (ξ) and Wi

(ξ) depend only on coordinate ξ.

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53

On substituting expression (7) into the local problem (4) - (6), after elimination of p(0) we obtain: (α) Wi/i = 0, (8) (α)

(α)

−P/i + ∆Wi

(α)

= hi , ξi ∈ Vξg ,

(9)

(α)

Wi

|Σξg = 0,

hP (α) i = 0,

(10) (11)

(α)

[[Wi

]] = 0,

[[P (α) ]] = 0

(12) (α)

— the collection of k local problems to determine functions P (α) (ξ), Wi (ξ). (α) (α) Here ∆Wi = Wi/jj is Laplacian operator. Unlike problem (4)–(6), the problems (8) do not include any properties of gas phase and do not depend explicitly on input data. Solution of problems (8) is determined only by internal geometric structure of pores, therefore their solution may be applied (taking account of the assumptions made in paragraph 1) for computations of transport processes for any gaseous and ﬂuid media. (α) The object hi is introduced as follows: ( 0, i 6= α or (i = α and p(0) ,α ≡ 0), (α) hi = (13) (0) 1, i = α and p,α 6= 0, that allows us to take into account the singular case of the structure where there are no through pore channels along coordinate direction xk . Figure 2 shows examples of singular and nonsingular (branching) porous structures for 2-D case. Singular structures for 2-D case can be 1-channel or 0-channel (completely closed porosity). (0) (k) For the closed porosity we have: p,k ≡ 0, hi = 0 ∀k. It follows from (8) that (k)

Wi ≡ 0, p(k) =0, i.e. there is no ﬂow. Therefore, this case will not be considered below.

Fig. 2. Examples of nonsingular (aa) and singular (1-channel (bb) and closed porosity (cc) porous structures

54

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

In general, the number of through channels in a porous structure is equal to number k in formula (7). 4. Solving Method of Local Problem (8) for 2-D Structures. For 2-D porous structures in (8)–(12) index i takes values 1 and 2. For every ﬁxed value of k, the problem (8)–(12) is a stationary problem of ﬂowing some imaginary linear-viscous incompressible ﬂuid but under speciﬁc additional conditions. In the ﬁrst place, since function p(1) in (7) physically means a pulsation of gas pressure p¯ with respect to the average value p(0) : it follows from (3) that κp(1) = p¯ − p¯(0) + O(κ2 ), and hp(0) i = p¯ due to hp(1) i = 0, so sign of p(1) as well as the sign of function P (k) may be both positive and negative. In the second place, since there is condition hP (k) i = 0 there, the problem (8)– (12) proves to be integrodifferential, that together with periodicity conditions for (k) functions Wi and P (k) considerably complicates its solving. The following theorem allows us to simplify solving the problem. Theorem. Let 2-D porous structure (1-channel or branching) have a mirror symmetry about planes Oξ1 ξ2 and Oξ2 ξ3 , then solution of the problem (8)–(12) (α) Wi , P (α) can be constructed with the help of symmetric or antisymmetric extenf (k) , Pe(k) which are deﬁned in a quarter sion (according to Figure 3) of functions W i of PC (in the ﬁrst quadrant Veξg ), and they are solutions of the following problems: (α)

Wi/i = 0, (α) f (α) = h(α) , −Pe/i + ∆W i i

(14) ξi ∈ Veξg ,

f (α) |Σ = 0, W ξsg i

(15) (16)

Fig. 3. Symmetric and antisymmetric extensions of solution for the problem (8)–(12) VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

55

ξα = 0 :

f (α) = 0, W α/α

f (α) = 0, Pe(α) = 0, W β

ξα = 1/2 : ξβ = 0 :

f (α) = 0, W α/β

fα(α) = 0, W

ξβ = 1/2 :

f (α) = 0, Pe (α) = 0, W β

(17) (18)

f (α) = 0, Pe (α) = 0, W β /β

Pe (α) = 0, α, β = 1, 2, α 6= β.

(19) (20)

Sign “+” in Figure 3 means that a function at passage from the ﬁrst quadrant into the considered one does not change its sign, and sign “−” means the change. Proof of the theorem will be given in Appendix. Local problems (14)–(20) are considerably easier for numerical solving because they do not include integral and periodic conditions. To solve the problems we can apply the method of curl and stream functions [6]. Introduce functions of curl ω (α) and stream ψ (α) for every of problems (14)–(20) as follows: f (α) − W f (α) , ω (α) = W (21) α/β β/α f (α) = ψ (α) , W α /β

f (α) = ψ (α) α 6= β, α, β = 1, 2, W β /α

(22)

then equations (14) are satisﬁed identically, and for ω (α) , ψ (α) and Pe (α) we obtain the following equation set in the standard way [6]: (α) ∆ω = 0, (23) ∆ψ (α) = ω (α) , ∆Pe (α) = 0, ξ ∈ Ve . i ξg However, the boundary conditions for these functions following from (16)–(20) differ from the well-known conditions for curl and stream functions [6]. On differentiating (20) with respect to ξα and then with respect to ξβ , we get: (α)

(α)

(α)

(α)

f f −W , ω/α = W α/βα β/αα

(α)

(α)

f f ω/β = W −W . α/ββ β/βα

Having substituted these expressions into equation (15) with account of (14) and (13), we ﬁnd the relationships (α) (α) ω/α = −Pe/β ,

(α) (α) ω/β = hαα + Pe/α ,

α 6= β.

(24)

On applying these relations for boundaries ξα = 0, ξα = 1/2 and ξβ = 1/2, we ﬁnd additional boundary conditions on the surfaces. Then with taking account of f (α) = 0 at ξα = 0, f (α) = 0 at ξα = 0 is always satisﬁed if W that the condition W β α/α we obtain systems of three equations (23): there are three boundary conditions at each boundary of domain Veξg : Z ξβ (α) (α) (α) α ξα = 0 and ξα = 1/2 : ψ/α = 0, ω = hα ξβ + Pe/α dξ, Pe (α) = 0, 0 (α)

ξβ = 0 :

ψ

(α)

= 0,

ω

= 0,

(α) Pe/β Z ξα

(α)

ξβ = 1/2 :

ψ/β = 0,

(α) Pe/β dξ,

ω (α) = −

(25)

= 0, Pe (α) = 0.

0

56

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

To ﬁnd boundary conditions for functions ψ (α) , ω (α) and Pe(α) on surface Σsg of contact with solid phase, we apply the expansion of the stream function in the vicinity of the surface (Roache, 1976). For example, let equation of the surface (or of its part) have the form ξβ = S(ξα ), where S(ξα ) is the one-valued differentiable function. Then, on passing to curvilinear coordinates ξ¯α connected to this surface as follows: ξ¯α = ξα , ξ¯β = aξβ /S(ξα ), 0 < a < 0.5, we obtain the desired boundary condition on surface Σξg : ξ¯2 = const :

ω (α) =

2[ψ (α) (ξ¯α , ξ¯β − ∆ξ¯β ) − 2ψ (α) (ξ¯α , ξ¯β )]S 2 , ∆ξ22 )a2 + (ξ¯β ∂S/∂ ξ¯β )2 )

(26)

ψ (α) = const. The third condition at the boundary for function Pe (α) we obtain by combining both conditions (24): (α) (α) ∂ Pe (α) /∂ ξ¯β = (ω/β − hαα )(∂ξ α /∂ ξ¯β ) − ω/α (∂ξ β /∂ ξ¯β ),

α 6= β.

(27)

In the similar way, we may ﬁnd conditions on the part of surface Σsg where there is no one-to-one connection ξα = S(ξβ ) (at vertical and horizontal walls of pores). 5. Numerical Solution of the Local Problem. The problem (23) for every of values α = 1, 2 with boundary conditions (25)–(27) is the set of three classical elliptic equations with boundary conditions of the standard type, when either the function itself or its derivative at the boundary is given. To solve the problem we may apply corresponding numerical methods for similar equations [6]. The present work demonstrates the iterative method combined with the method of rectifying coordinates ξ¯α transforming domain Veξg to the domain with rectangular boundaries. We have also applied the scheme by Pismen–Reckford with the help of the longitudinal-cross sweep method. Consider some results of computations. The computations have been conducted for one-channel porous medium (Figure 2, c). For this case, we have the only funcf (1) ≡ W f1 , tions’ set not equal to zero: ψ (1) ≡ ψ, ω (1) ≡ ω, Pe (1) ≡ Pe and W 1 (1) f2 . f ≡W W 2 The surface shape Σξg has been chosen cosine: ξ2 = S(ξ1 ) = (b + a + +(b−a)(cos 2πξ1 )1+n )/2, where n, a and b are the parameters: a = 0.04, b = 0.4, n ≥ 0. f2 depending on coordinates f1 and W Figures 4, a–e show functions ψ, ω, Pe , W ξ1 , ξ2 for n = 0. Typical values of these dimensionless functions are indicted in parentheses there. The stream function (−ψ) has a maximum on surface Σξg of solid phase, and the curl function ω — on symmetry surface ξ1 = 0 in the zone of the largest pore diameter. Pressure pulsation function Pe takes both negative values in a wide part of the pore and positive values in a narrow section of the pore. This picture may be explained physically by that in the narrow section the total gas pressure

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

57

f2 (e) depending on coordinates ξ1 , f1 (dd ) and W Fig. 4. Functions ψ (aa), ω (bb ), Pe (cc), W ξ2 for n = 0

58

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

exceeds the average pressure in PC, and in the wide section the total gas pressure is smaller than p(0) . f1 takes negative values, and its absolute magnitude has maximum Velocity W equal to 0.47 in the most narrow section of the pore near the solid wall. Transf2 has a maximum, being sufﬁciently high and equal to 0.1, in the verse velocity W zone of maximum bending the surface Σξg (from the side of narrow pore section). f2 in comparison with velocity These considerable values of transverse velocity W f1 demostrate the brightly expressed not one-dimensional character of the local W transport process in the porous medium. 6. Comparison with Poiseuille’s Flow. At a = b, for the considered example of one-channel structure we have a porous medium with plane straight channels with a constant cross-section. For this case, the problem (20)–(23) has the following exact solution: ψ = −0.5ξ2 (a2 − ξ 2 /3),

ω = ξ2 ,

w = −0.5(a2 − ξ22 ),

(28)

that precisely corresponds to Poiseuille’s ﬂow of linearly viscous ﬂuid. Table 1 shows testing results computed by formulas (28) and obtained by immediate calculation from equations (23) for a = b. So the numerical method is seen to maintain a quite satisfactory accuracy of computations. Table 1 Distributions of stream ψ, curl ω functions and speed W along coordinate ξ2 of the pore at a = b, where numerator is exact solution (28), and denominator is computed solution of system (23)

ξ2

0

0.04

0.08

0.12

0.16

0.2

0.24

0.28

0.32

0.36

0.4

−ψ · 102

0 0

0.32 0.3

0.63 0.6

0.93 0.88

1.21 1.15

1.47 1.39

1.69 1.59

1.87 1.76

2.01 1.89

2.1 1.96

2.13 1.97

ω

0 0

0.04 0.04

0.08 0.08

0.12 0.12

0.16 0.16

0.2 0.2

0.24 0.24

0.28 0.28

0.32 0.32

0.36 0.36

0.4 0.4

W · 103

8 7.6

7.92 7.52

7.68 7.28

7.28 6.88

6.72 6.32

6.0 5.6

5.12 4.7

4.08 3.68

2.88 2.48

1.52 1.12

0 0

7. Determination of Permeability Coefﬁcient. After the local transport paf (k) , Pe (α) , ψ (α) and ω (α) in Veξ have been determined, in accordance rameters W β with Theorem we may ﬁnd corresponding functions in the whole PC. Substitute the functions into relatioships (7) and average the obtained expressions over the PC. Then we ﬁnd: (0) (0) µ0 )p,j , (29) hvi i = −(K ji /¯ where (j)

K ji = −hWi i.

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

(30)

59

The relationship (29) is the ﬁltration law by Darcy works (see [3, 4]), and coefﬁcients K ji are gas permeability coefﬁcients of the porous medium. On using (α) Theorem, we obtain that matrix K ji is diagonal, because hWβ i = 0 at α 6= β due (α)

to antisymmetry of functions Wβ at α 6= β. For the special case when porosity is one-channel (for example, along axis (1) (2) Oξ1 ), then, as it has been shown above, Wi 6= 0 and Wi = 0, thus (1)

all K ji = 0 except K 11 = K = −hW1 i.

(31)

On passing to the integral over Veξ , we obtain Z1/2Za 4 K=− a

f (1) (ξ1 , ξ2 )dξ2 S(ξ1 )dξ1 . W 1 0

(32)

0

For the example considered in paragraph 5, magnitudes of dimensionless gas¯ for different values of parameter n characterizing the permeability coefﬁcient K pore shape and different porosities are given in Table 2. Table 2 Values of gas permeability coefﬁcient for different porosities and different shapes of pores

ϕg

0.44

0.48

0.52

0.60

0.80

K · 10 at n = 0

5.19

5.75

6.25

7.19

9.22

K · 103 at n = 2

3.53

4.15

5.12

6.08

9.22

3

The computed results allow us to make the following important conclusion. At n = 0 the pore has such shape that passage from maximum size at ξ1 = 0 to minimum at ξ1 = 1/2 is more smooth than at n = 2. This favours increasing f (1) and thus increasing gas permeability coefﬁcient K. Therefore, gas velocity W 1 we may change gas permeability of porous media with varying only pores’ shape without changing the total porosity. 8. Comparison with Experiments. Work by Dimitrienko [4] shows data on gas permeability of composite materials with different porosities ϕg . Pores’ shape may be considered appoximately as two-dimensional due to the presence of a set of parallel ﬁbers in a composite which promote the appearance of sufﬁciently large longitudinal cracks and pores. A typical size of these pores is close to monoﬁbers’ diameter and is equal to l0 ≈ 10−5 m. Experimental value of dimensionless permeability coefﬁcient K at porosity ϕg = 0.4 is K = 4·10−3 . Computed results shown in Table 2 give K = 5.19·10−3 . With taking account of rather undeterminate shape of pores in experimental data and statistic scattering, the obtained computed results should be considered as sufﬁciently accurate. 60

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

9. Conclusions. 1. The present paper continues investigations of works [3–5]. As a result, the local not one-dimensional problem of transport in porous periodic medium has been solved numerically here for the ﬁrst time in frames of the asymptotic averaging method. 2. Sufﬁciently general approach to solving the local problem has been developed, which allows us to compute a faithful (in the asymptotic sense: at κ −→ 0) local distributions of speeds and pressure inside a separate pore for 2-D case. 3. The calculation method has been developed for gas permeability coefﬁcient of porous medium when only internal pores’ geometry is known. 4. Computations conducted as the example have shown that a local gas ﬂow in pores is considerably nonuniform, and pores’ shape (at the same total porosity) has an essential effect on the value of the total gas-permeability coefﬁcient. 5. Comparison of computed results of gas permeability with experimental data for composite materials allows us to tell about sufﬁciently high accuracy of computations by the developed method. Appendix. Prove Theorem from paragraph 4. Consider the case when α = 1. Introduce new coordinates: ξ¯1 = −ξ1 , ξ¯2 = ξ2 and extend the problem (10)–(16) solution from the ﬁrst to the second quadrant according to Figure 2, a, i.e. assume (1) f (ξ1 , ξ2 ), W (1) (ξ¯1 , ξ¯2 ) = −W f (1) (ξ1 , ξ2 ), P (1) (ξ¯1 , ξ¯2 ) = that W1 (ξ¯1 , ξ¯2 ) = W 2 2 −Pe (1) (ξ1 , ξ2 ). For this substitution, differential operators in the second quadrant are connected to corresponding operators of the ﬁrst quadrant as follows: (1) f (1) , Wi//i = −W i/i

(1) (1) P//1 = Pe/1 ,

(1) (1) P//2 = −Pe/2 ,

and (1) (1) f (1) , W1//11 + W1//22 = ∆W 1

(1) (1) f (1) , W2//11 + W2//22 = −∆W 2

where //i = ∂/∂ ξ¯i . This means that in equations (4), (5) (when i = 2) at passage to the second quadrant all summands change in their signs (here h12 = 0), and in equation (5) (when i = 1) they do not change. Therefore, in the second quadrant equations (8)–(10) are satisﬁed. Boundary conditions (16)–(20) are evident to maintain the satisfaction of conditions (10), (12) at passage to the second quadrant. Moreover, due to antisymmetry of function P (1) , integral condition (11) is satisﬁed as well. In the similar way, we can prove that extension of the solution to the third and fourth quadrants realized according to Figure 3 maintains the satisfaction of all equations (8)–(12) in these quadrants. The above is also valid when α = 2 with taking account of symmetric and antisymmetric extensions of the functions by data of Figure 3, c.

REFERENCES 1. Bakhvalov, N. & Panasenko, G.: 1989, Homogenisation: Averaging Processes in Periodic Media (Mathematical Problems in Mechanics of Composite Materials), Kluwer Academic Publishers.

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2. Bensoussan, A., Lions, J.L. & Papanicolaou, G.: 1978, Asymptotic Analysis for Periodic Structures, North Holland, Amsterdam. 3. Dimitrienko, Yu.I.: 1997, Heat-mass transport and thermal stresses in porous charring materials, Transport in Porous Media, 27 (2), 143–170. 4. Dimitrienko, Yu.I.: 1998, Thermomechanics of Composites under High Temperatures, Kluwer Academic Publishers, Dordrecht. 5. Dimitrienko, Yu.I.: 1999, Dynamic transport phenomena in porous polymer materials under impulse thermal effects, Transport in Porous Media, 35, 299–326. 6. Roache, P.: 1976, Computational Fluid Dynamics, Hermosa Publishers, Albuquerque. 7. Sanchez-Palencia, E.: 1980, Non-homogeneous media and vibration theory. New– York, Springer-Verlag.

Yu. I. Dimitrienko graduated fom the Lomonosov Moscow State University in 1984. D.Sc (Phys.-Math.), professor, head of “Computational Mathematics and Mathematical Physics” department of the Bauman Moscow State Technical University, academician of the Russian Academy of Engineering Sciences. Author of over 100 publications in the ﬁeld of thermal mechanics, composite mechanics, multiphase media mechanics, mathematical modelling and numerical methods for problems of mathematical physics. M.L. Glazikov graduated from the Bauman Moscow State Technical University in 1999. Assistant of “Computational Mathematics and Mathematical Physics” department of the Bauman Moscow State Technical University. Author of a number of publications in the ﬁeld of numerical methods of the ﬁltration theory.

62

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CONTROL SYSTEMS A. Golubev∗ , R. Johansson∗∗ , A. Robertsson∗∗ , S. Tkachev∗ (∗ Bauman Moscow State Technical University), (∗∗ Lund Institute of Technology, Sweden)

OUTPUT TRACKING FOR A CLASS OF NONLINEAR NONMINIMUM-PHASE SYSTEMS USING OBSERVER BACKSTEPPING In this paper we address the problem of output tracking for a class of nonlinear nonminimum-phase systems with linear zero dynamics. Using observer backstepping a globally stabilizing control law is designed to force the tracking error to converge asymptotically to zero. The observer backstepping procedure is extended to global stabilization of a class of nonlinear dynamical systems in the output-feedback form with linear unstable zero dynamics. An output feedback controller is designed for a ﬂexible one-link robot arm as an illustrative example.

Introduction. The integrator backstepping [1, 2] is an effective control design tool for stabilization of nonlinear dynamical systems that can be transformed into the strict feedback form x˙ 1 = x2 + f1 (x1 ), x˙ 2 = x3 + f2 (x1 , x2 ), .. .

(1)

x˙ n−1 = xn + fn−1 (x1 , . . . , xn−1 ), x˙ n = u + fn (x1 , . . . , xn ), where u ∈ R is the control input, under the assumption that the full state x = = (x1 , . . . , xn )T ∈ Rn of the system is measured. In more realistic problems, where only part of the state is available for measurement, ﬁrst an observer system is designed, that is a dynamical system which provides sufﬁciently fast (for instance, exponentially) convergent estimates of the unmeasured states. Then the so called observer backstepping procedure [1, 2] can be used for stabilizing control design. It consists in the following. The integrator backstepping is applied to a new system in which the equations of the unmeasured states have been replaced by the corresponding equations of their estimates from the observer. At each step of the procedure observation errors are treated as disturbances and accounted for using the so called nonlinear “damping” [2]. Further, we consider the output tracking problem for nonlinear dynamical systems written in the output-feedback form

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63

x˙ 1 = x2 + ϕ1 (y), .. . x˙ r−1 = xr + ϕr−1 (y), x˙ r = xr+1 + β(y)u + ϕr (y), x˙ r+1 = xr+2 + b1 β(y)u + ϕr+1 (y), .. .

(2)

x˙ n−1 = xn + bm−1 β(y)u + ϕn−1 (y), x˙ n = bm β(y)u + ϕn (y), y = x1 , where the nonlinearities ϕi (·) ∈ C ∞ (R), i = 1, n depend only on the output y = x1 of the system; bj , j = 1, m are some constants; n − r = m; β(y) 6= 0 for all y ∈ R. In (2) only output of the system is supposed to be available for measurement. For the considered class of systems (2) our purpose is to design a stabilizing feedback control law that uses information only about the measured output y, guarantees global boundedness of any solution x(t) of the closed-loop system (2) and forces the tracking error y(t) − yd (t) to converge to zero as t → ∞ for all x(0), where yd (t) is a sufﬁciently smooth reference output trajectory, with its derivatives (n) y˙ d , . . . , yd being known and bounded on [0, ∞). Note that a global exponential observer for (2) is x ˆ˙ = Aˆ x + GC(ˆ x − x) + ϕ(y) + Bβ(y)u,

(3)

where x ˆ ∈ Rn ; the matrices A, B correspond to the system (2) written in vector form; ϕ(y) = (ϕ1 (y), . . . , ϕn (y))T ; C = (1, 0, . . . , 0); G = (g1 , . . . , gn )T is chosen so that A + GC is Hurwitz. Therefore, state estimation error e = x ˆ−x equation is e˙ = (A + GC)e (4) and its equilibrium e = 0 is globally exponentially stable [3]. By the analysis of the system (2) the notion of zero dynamics [4] is very useful. Note that for the inital conditions x1 (0) = 0, x2 (0) = −ϕ1 (0), . . ., xr (0) = = −ϕr−1 (0) and any xr+1 (0) = xr+1,0 , . . . , xn (0) = xn,0 the input u(t) that keeps the output y(t) indentically equal to zero for all t > 0 is uniquely deﬁned and given by xr+1 (t) + ϕr (0) . (5) u(t) = − β(0) The zero dynamics [4] of the system (2), that is dynamics of (2) with the control law (5) and the above initial conditions x1 (0), . . . , xn (0), have the following form: x˙ r+1 = xr+2 − b1 xr+1 + ϕr+1 (0) − b1 ϕr (0), .. . (6) x˙ n−1 = xn − bm−1 xr+1 + ϕn−1 (0) − bm−1 ϕr (0), x˙ n = −bm xr+1 + ϕn (0) − bm ϕr (0). 64

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If the system (2) is minimum-phase, that is the equilibrium point of the zero dynamics (6) is globally asymptotically stable, then one of the globally stabilizing feedback control laws obtained via the observer backstepping procedure that keep any solution x(t) of the closed-loop system (2) globally bounded and force the tracking error y(t) − yd (t) to converge asymptotically to zero is [2] u=

1 (r) (αr − x ˆr+1 − yd ), β(y)

(7)

with αi , zi , i = 1, r given by z 1 = x1 − y d , (i−2)

zi = x ˆi − αi−1 (y, x ˆ1 , . . . , x ˆi−1 , yd , . . . , yd α1 = −c1 z1 − d1 z1 − ϕ1 (y),

i−1 X ∂αi−1

i−2 X ∂αi−1

x ˆ˙ j + ∂x ˆj j=1

(j+1)

yd

−(

(j) j=1 ∂yd

,

i = 2, r,

c1 > 0, d1 > 0,

x1 − y) − ϕi (y) + αi = −ci zi − zi−1 − gi (ˆ

+

(i−1)

) − yd

∂αi−1 (ˆ x2 + ϕ1 (y))+ ∂y

∂αi−1 2 ) d i zi , ∂y

ci > 0, di > 0,

i = 2, r.

In case when (2) is nonminimum-phase, that is the equilibrium point of the zero dynamics (6) is unstable, the observer backstepping control (7) as it is given in [2] fails to achieve global boundedness of x(t). Hence, solutions x(t) of the system (2) with control (7) may be not well-deﬁned for all t > 0 and asymptotic convergence of the tracking error to zero may not take place. In this paper we solve the output tracking problem for nonlinear nonminimumphase systems of the form (2). According to the ideas presented in [5] the observer backstepping control design procedure is extended to global stabilization of nonlinear dynamical systems written in the output-feedback form (2) with linear unstable zero dynamics. As an illustrative example an output feedback controller is designed to stabilize angular position of a ﬂexible one-link robot arm. Output tracking in case of one-dimensional zero dynamics. Consider ﬁrst the system (2) with n − r = 1: x˙ 1 = x2 + ϕ1 (y), .. . (8) x˙ n−2 = xn−1 + ϕn−2 (y), x˙ n−1 = xn + β(y)u + ϕn−1 (y), x˙ n = b1 β(y)u + ϕn (y), y = x1 . The linear change of variables given by k1 k2 χ1 1 0 χ2 .. = 0 1 . .. . . . . χn 0 0

. . . kn x1 ... 0 x2 ... 0 .. . . .. . .. xn ... 1 0 k3 0 0 .. .

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(9)

65

with ki = (−1)i+1 b1 −i , i = 1, n transforms (8) into the form χ˙ 1 = −b1 χ1 + χ2 + ϕn (y), χ˙ 2 = χ3 + ϕ1 (y), .. . (10) χ˙ n−1 = χn + ϕn−2 (y), χ˙ n = LT χ + β(y)u + ϕn−1 (y), y = χ2 , n X

1 k1 kn−1 T ,− ,...,− ) ∈ Rn . kn kn kn i=1 An exponential observer for the system (10) is written as

where ϕn (y) =

ki ϕi (y); L = (

χˆ˙ = Aχˆ + GC(χˆ − χ) + φ(y) + Bβ(y)u,

(11)

where χˆ ∈ Rn ; φ(y) = (ϕn (y), ϕ1 (y), . . . , ϕn−1 (y))T ; the matrices A, B refer to the vector form of (10); C = (0, 1, 0, . . . , 0). In (11) the gain vector G = = (g1 , g2 , . . . , gn )T is chosen so that the matrix A + GC is Hurwitz. The state estimation error e = χˆ − χ equation is of the form (4), where the matrices A, G, C refer to the matrices in (11). Hence, its equilibrium e = 0 is globally exponentially stable [3]. A Lyapunov function for the system (4) is taken [3] as Vo (e) = eT Po e with its derivative satisfying V˙o (e)|(4) = −eT Qo e ≤ def

≤ −λmin (Qo )| e |2 = −l| e |2 . Here | · | denotes the Euclidean norm on Rn , the matrices Po = Po T > 0 and Qo = Qo T > 0 satisfy the Lyapunov equation (A + GC)T Po + Po (A + GC) = −Qo .

(12)

Note that, when solving the output tracking problem in question for the system (10), we cannot directly use state feedback control laws of the form u = u(χ) since only χ2 is available for measurement. Therefore, we will ﬁnd stabilizing control laws as u = u(χ2 , χ) ˆ = u(χ2 , χ + e), where the state estimate χˆ is generated by the observer (11). Further, consider the system χ˙ 1 = −b1 χ1 + χ2 + φ1 (y), χ˙ 2 = χ3 + φ2 (y), .. . (13) χ˙ n−1 = χn + φn−1 (y), χ˙ n = LT χ + β(y)u + φn (y), e˙ = (A + GC)e, y = χ2 composed of equations of the system (10) and the state estimation error e = χˆ − χ equation. Note that, if the function f (χ2 ) = χ2 + φ1 (χ2 ) deﬁnes a diffeomorphism, then def

for the system (13) the reference output trajectory y = yd (t) = χ2d (t) tracking 66

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problem is equivalent to that of the tracking of a globally bounded reference trajectory χ1 = χ1d (t) that satisﬁes the differential equation χ˙ 1d (t) + b1 χ1d (t) = χ2d (t) + φ1 (χ2d (t)). Suppose that the reference output trajectory χ2d (·) is smooth and belongs to L1 ∩ L∞ . The conditions on the right-hand side of of the ﬁrst equation in (13) under which for any globally bounded output trajectory χ2d (·) ∈ L1 ∩ L∞ there exists a unique solution χ1d (·) ∈ L1 ∩ L∞ ∩ C 0 , χ1d (±∞) = 0 of the differential equation χ˙ 1 + b1 χ1 = χ2d (t) + φ1 (χ2d (t)) are given in [6, 7]. Further we will assume that these conditions are satisﬁed and the smooth reference trajectory χ1d (t) that corresponds to the output reference trajectory χ2d (t) is found, for instance, via the procedure given in [6, 7]. Using the fact that e = χˆ − χ, the system (13) can be written in the following form: χ˙ 1 = −b1 χ1 + χˆ2 − e2 + φ1 (y), ˆ3 + g2 e2 + φ2 (y), χˆ˙ 2 = χ .. . (14) χˆ˙ n−1 = χˆn + gn−1 e2 + φn−1 (y), χˆ˙ n = LT χˆ + gn e2 + β(y)u + φn (y), e˙ = (A + GC)e, y = χˆ2 − e2 , ˆn )T is the state of the observer (11). where χˆ = (χˆ1 , . . . , χ The standard observer backstepping design as it is given in [1, 2] cannot be directly applied to stabilization of the system (14) since the ﬁrst state χ1 in the integrator chain of (14) is not measured but estimated only. To overcome this difﬁculty an extra “damping” term is used similar to what is done in the standard observer backstepping. To solve the output tracking problem for the system (14) we use the following extension of the standard observer backstepping procedure. Step 1. Consider the ﬁrst equation of the system (14) χ˙ 1 = −b1 χ1 + χˆ2 − e2 + φ1 (y) = −b1 χˆ1 + χˆ2 + φ1 (y) + b1 e1 − e2 , where e1 (t) = χˆ1 (t) − χ1 (t), e2 (t) = χˆ2 (t) − χ2 (t). If χˆ2 were the control input, then, using a control law χˆ2 = α1 (χˆ1 , χ1d , χ˙ 1d , y), one could compensate for the uncertainties e1 , e2 in the ﬁrst equation of the system (14) and force the tracking error χ1 (t) − χ1d (t) to converge to zero as t → ∞. To ˆ1 , χ1d , χ˙ 1d , y) consider the following design the “virtual” control law χ ˆ 2 = α1 ( χ positive deﬁnite function:

V1 (z1 , e) =

1 2 z + Vo (e) > 0, (z1 , e) 6= 0, 2 1

where z1 = χ1 − χ1d , Vo (e) is the Lyapunov function for the state estimation error e = χˆ − χ equation. For convenience sake deﬁne also the error variable z2 = = χˆ2 − α1 .

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The time derivative of V1 along the solutions of (14) is expressed as V˙ 1 |(14) = z1 z˙1 + V˙ o (e)|(14) ≤ z1 (−b1 χˆ1 + χ ˆ2 + φ1 (y)+ + b1 e1 − e2 − χ˙ 1d ) − l| e |2 ≤ z1 (z2 + α1 − b1 χˆ1 + φ1 (y)+ + b1 e1 − e2 − χ˙ 1d ) − le21 − le22 . By choosing the “stabilizing function” α1 as α1 = (b1 −c1 −d11 −d12 )χˆ1 −φ1 (y)+(c1 +d11 +d12 )χ1d + χ˙ 1d ,

c1 , d11 , d12 > 0

and using the fact that χˆ1 = e1 + χ1 = e1 + χ1d + z1 , one gets V˙ 1 |(14) ≤ z1 (−c1 z1 + z2 − d11 z1 − d12 z1 + (b1 − c1 − d11 − d12 )e1 − e2 )− − le21 − le22 = −c1 z12 + z1 z2 − (d11 z12 − (b1 − c1 − d11 − − d12 )z1 e1 + le21 ) − (d12 z12 + z1 e2 + le22 ) = −c1 z12 + z1 z2 − S1 , (15) µ √

¶2

κ1 d11 z1 − √ e1 2 d11 ¶

¶ ¶2 µ √ κ21 1 2 √ e + + l− d12 z1 + e2 + 4d11 1 2 d12 µ

with S1 = µ 1 + l− e2 ≥ 0, where κ1 = b1 − c1 − d11 − d12 and d11 , d12 are chosen 4d12 2 ½ ¾ κ21 1 such that l > max , . 4d11 4d12 Note that in the stabilizing function α1 (χˆ1 , χ1d , χ˙ 1d , y) the “damping” term −(d11 + d12 )χˆ1 is introduced to complete the squares in the right-hand side of the inequality (15) and in this way to compensate for the esimation errors e1 and e2 . Hence, if χˆ2 were the control input, then by the choice χˆ2 = α1 (χˆ1 , χ1d , χ˙ 1d , y) and appropriately taken positive constants d11 , d12 the derivative V˙ 1 along the solutions of the system (14) would be less or equal to the expression that contains a negative deﬁnite part plus indeﬁnite term z1 z2 . Step 2. Consider the subsystem composed of the ﬁrst two equations of (14). It can be expressed as ˆ 2 − α1 ) − e 2 , χ˙ 1 = −b1 χ1 + α1 + φ1 (y) + (χ ˙ χˆ2 = χˆ3 + g2 (χˆ2 − y) + φ2 (y). In this step χ ˆ3 is used as virtual control input instead of χˆ2 . By using an appropriately taken virtual control law χ ˆ 3 = α2 ( χ ˆ1 , χˆ2 , χ1d , χ˙ 1d , χ¨1d , y) one can force the errors z1 = χ1 − χ1d and z2 = χˆ2 − α1 to converge to zero as t → ∞. To design this virtual control law consider the following positive deﬁnite function: 1 V2 (z1 , z2 , e) = V1 (z1 , e) + z2 2 + Vo (e) > 0, (z1 , z2 , e) 6= 0. 2 After introducing the error variable z3 = χˆ3 − α2 , the time derivative of V2 along the solutions of (14) is computed as 68

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V˙ 2 |(14) = V˙ 1 |(14) + z2 z˙2 + V˙ o |(14) = z1 z˙1 + z2 z˙2 + 2V˙ o |(14) = = z1 (−b1 χ1 + α1 + φ1 (y) − e2 − χ˙ 1d ) + z1 (χ ˆ2 − α1 )+ + z2 ( χ ˆ˙ 2 − α˙ 1 ) + 2V˙ o |(14) ≤ −c1 z12 + z1 z2 − S1 + z2 (χˆ˙ 2 − α˙ 1 )+ + V˙ o |(14) ≤ −c1 z12 − S1 + z2 (z1 + χ ˆ3 + g2 (χˆ2 − y) + φ2 (y)− ∂α1 ˙ ∂α1 ∂α1 ∂α1 (χˆ3 + φ2 (y) − e3 ) − χ˙ 1d − χ¨1d ) − le21 − le23 . − χˆ1 − ∂y ∂ χˆ1 ∂χ1d ∂ χ˙ 1d

V˙ 2 |(14) ≤ −c1 z12 − S1 + z2 (χˆ1 − e1 − χ1d + µ ¶ ∂α1 ∂α1 + (z3 + α2 ) 1 − + g2 (χˆ2 − y) + φ2 (y) − (φ2 (y) − e3 )− ∂y ∂y ∂α1 ˙ ∂α1 ∂α1 − χ˙ 1d − χ ¨1d ) − le21 − le23 . (16) χˆ − ∂ χˆ1 1 ∂χ1d ∂ χ˙ 1d ¶ µ ∂α1 6= 0 for all y ∈ R. This assumption is equivalent to Suppose that 1 − ∂y µ ¶ ∂φ1 1+ 6= 0 for all y ∈ R. Then by the choice ∂y µ 1 α2 = (1 − ∂α1 /∂y)

− χˆ1 + χ1d − c2 z2 −

∂α1 ∂α1 ˙ ∂α1 φ2 (y) + χ˙ 1d + χˆ + ∂y ∂χ ˆ1 1 ∂χ1d µ ¶ ¶ ∂α1 ∂α1 2 + χ ¨1d − d21 z2 − d23 z2 , c2 , d21 , d23 > 0 ∂ χ˙ 1d ∂y

− g2 (χˆ2 − y) − φ2 (y) +

one gets ∂α1 )z2 z3 − S1 − S2 , V˙ 2 |(14) ≤ −c1 z12 − c2 z22 + (1 − ∂y µp

¶2 µp ¶2 µ ¶ ∂α1 1 1 √ with S2 = z2 − + d23 e3 + l− e21 + ∂y 4d 2 d23 21 ½ ¾ µ 1 1 1 2 , e ≥ 0, where d21 , d23 are taken such that l > max . + l− 4d23 3 4d21 4d23 Note that in this step the uncertainties e1 and e3 in the right-hand side of the ¶ µ ∂α1 2 z2 in inequality (16) are compensated by the damping term −d21 z2 − d23 ∂y the stabilizing function α2 (χˆ1 , χˆ2 , χ1d , χ˙ 1d , χ¨1d , y). Therefore, if χˆ3 were the control input, then, by choosing χˆ3 = α2 (χˆ1 , χˆ2 , χ1d , ¨1d , y) with the appropriately taken positive constants d11 , d12 , d21 , d23 , the χ˙ 1d , χ 1 d21 z2 + √ e1 2 d21 ¶

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69

derivative V˙ 2 along the solutions of the system (14) would be less µ or equal to ¶ the ex∂α1 z2 z3 . pression that contains a negative deﬁnite part plus indeﬁnite term 1 − ∂y Step 3. Consider the subsystem that is composed of the ﬁrst three equations of the system (14) and is written as ˆ 2 − α1 ) − e 2 , χ˙ 1 = −b1 χ1 + α1 + φ1 (y) + (χ ˆ2 − y) + φ2 (y) + (χˆ3 − α2 ), χˆ˙ 2 = α2 + g2 (χ χˆ˙ 3 = χˆ4 + g3 (χˆ2 − y) + φ3 (y). Now analogously to the previous steps of the design use the positive deﬁnite function 1 V3 (z1 , z2 , z3 , e) = V2 (z1 , z2 , e) + z3 2 + Vo (e) > 0, (z1 , z2 , z3 , e) 6= 0 2 (3)

ˆ2 , χ ˆ3 , χ1d , χ˙ 1d , χ ¨1d , χ1d , y). in order to ﬁnd a virtual control law χˆ4 = α3 (χˆ1 , χ Using the error variable z4 = χˆ4 − α3 , the time derivative of V3 along the solutions of (14) can be evaluated as follows: V˙ 3 |(14) = z1 z˙1 + z2 z˙2 + z3 z˙3 + 3V˙ o |(14) = µ = z1 (−b1 χ1 +α1 +φ1 (y)−e2 − χ˙ 1d )+z1 (χˆ2 −α1 )+z2 α2 +g2 (χˆ2 −y)+φ2 (y)− ¶ ∂α1 ∂α1 ˙ ∂α1 ∂α1 (α2 + φ2 (y) − e3 ) − − χ˙ 1d − χ¨1d + z2 (χˆ3 − α2 )− χˆ − ∂y ∂ χˆ1 1 ∂χ1d ∂ χ˙ 1d µ ¶ ∂α1 ∂α1 z2 (χˆ3 − α2 ) + z3 (χˆ˙ 3 − α˙ 2 ) + 3V˙ o |(14) ≤ −c1 z12 − c2 z22 + 1 − − z2 z3 − ∂y ∂y − S1 − S2 + z3 (χˆ˙ 3 − α˙ 2 ) + V˙ o |(14) ≤ −c1 z12 − c2 z22 − S1 − S2 + ¶ µµ ∂α2 ∂α1 + z3 ˆ2 − y) + φ3 (y) − z2 + z 4 + α 3 + g 3 ( χ (χˆ3 + φ2 (y) − e3 )− 1− ∂y ∂y ¶ ∂α2 ˙ ∂α2 ˙ ∂α2 ∂α2 ∂α2 (3) − le23 . (17) − χ˙ 1d − χ¨1d − χ χˆ − χˆ − ∂ χˆ1 1 ∂ χˆ2 2 ∂χ1d ∂ χ˙ 1d ∂ χ¨1d 1d By choosing ¶ ∂α1 z2 − g 3 ( χ ˆ2 − y)− α3 = −c3 z3 − 1 − ∂y ∂α2 ∂α2 ˙ ∂α2 ˙ (χˆ3 + φ2 (y) + − φ3 (y) + χˆ1 + χˆ + ∂y ∂χ ˆ1 ∂ χˆ2 2 ¶ µ ∂α2 ∂α2 (3) ∂α2 2 ∂α2 χ˙ 1d + χ¨1d + χ − d33 z3 , c3 , d33 > 0, + ∂χ1d ∂ χ˙ 1d ∂ χ¨1d 1d ∂y µ

one gets V˙ 3 |(14) ≤ −c1 z12 − c2 z22 − c3 z32 + z3 z4 − S1 − S2 − S3 , 70

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¶2

µp ∂α2 1 with S3 = d33 z3 − √ e3 ∂y 2 d33 1 such that l > . 4d33

¶ 1 e2 ≥ 0, where d33 is taken + l− 4d33 3 µ

In this step the only uncertainty that is contained in the right-hand side of the ¶ µ ∂α2 2 z3 inequality (17) is e3 and it is compensated by the damping term −d33 ∂y (3) in the stabilizing function α3 (χˆ1 , χ ˆ2 , χˆ3 , χ1d , χ˙ 1d , χ ¨1d , χ1d , y). Thus, if χˆ4 were the control input, then, by choosing χ ˆ4 = α3 (χˆ1 , χˆ2 , χˆ3 , χ1d , (3) χ˙ 1d , χ ¨1d , χ1d , y) with the appropriately taken positive constants d11 , d12 , d21 , d23 , d33 , the derivative V˙ 3 along the solutions of the system (14) would be less or equal to the expression that contains a negative deﬁnite part plus indeﬁnite term z3 z4 . Further, the steps 4 – (n − 1) of our control design procedure are analogous to the step 3 and in each step the corresponding virtual control law χ ˆi = (i−1) = αi−1 (χˆ1 , . . . , χˆi−1 , χ1d , χ˙ 1d , . . . , χ1d , y), i = 5, n is designed. Step n. To solve the reference trajectory χ1 = χ1d (t) tracking problem and, hence, the considered output tracking problem by exploiting the virtual control laws, the control input u can be used to force the errors z1 = χ1 − χ1d (t), z2 = χˆ2 − α1 , . . . , zn = χˆn − αn−1 to converge asymptotically to zero. Indeed, consider the following Lyapunov function candidate for the system (14): 1 Vn (z, e) = Vn−1 (z1 , . . . , zn−1 , e)+ zn 2 +Vo (e) > 0, (z, e) 6= 0, 2

Vn (0, 0) = 0.

The time derivative of Vn along the solutions of (14) is given by V˙ n |(14) = V˙ n−1 |(14) + zn (χˆ˙ n − α˙ n−1 ) + V˙ o |(14) ≤ n−1 X

n−1 X

ci zi2 −

≤− i=1

Si + zn (zn−1 + LT χˆ + gn (χˆ2 − y) + β(y)u + φn (y)− i=1

n−1 n−1 n X ∂αn−1 X ∂αn−1 (i+1) X ∂αn−1 2 ˙ (χˆ3 +φ2 (y)−e3 )− − χ )−le3 −l e2i . χ ˆ − (i) 1d ∂y ∂ χˆi i i=1 i=0 ∂χ1d i=4

Hence, by the choice of control law

u(χ, ˆ y) =

1 (−cn zn − zn−1 − LT χˆ − gn (χˆ2 − y)− β(y) n−1 X ∂αn−1 ∂αn−1 (χˆ3 + φ2 (y)) + − φn (y) + χˆ˙ + ∂y ∂ χˆi i i=1 n−2 X

∂αn−1

(i+1)

χ1d

+ (i) i=0 ∂χ1d

−(

∂αn−1 2 ) dn3 zn ), ∂y

cn , dn3 > 0 (18)

for the derivative V˙ n along the solutions of (14) holds the following: VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

71

n X

n X

ci zi2

V˙ n |(14) ≤ − i=1

µp

−

n X

e2i =

Si − l i=1

i=4

µ µp ¶ ¶2 κ1 κ21 1 2 = d11 z1 − √ e1 − l − d12 z1 + √ e2 − e − 4d11 1 2 d11 2 d12 i=1 µ µp ¶ ¶2 1 1 − l− d21 z2 + √ e1 − e22 − 4d12 2 d21 µp ¶2 µ ¶ ¶ µ 1 1 1 ∂α1 2 − z2 − √ e 3 − l − e −− l− e2 − d23 ∂y 4d21 1 4d23 3 2 d23 ¶2 X n µ n n X X p ∂αj−1 1 1 − dj3 (l − )e23 − l e2i . (19) z j − p e3 − ∂y 4dj3 2 dj3 n X

¶2

ci zi2 −

j=3

j=3

i=4

While taking the positive constants dij , j = 1, 3, i = 1, n such that ¾ ½ 2 1 1 1 1 1 κ1 def l > max , , , , ,..., = lmax , 4d11 4d12 4d21 4d23 4d33 4dn3 one gets V˙ n |(14) < 0,

(z, e) 6= 0.

Note that the condition l > lmax can be satisﬁed, for instance, in the following way. One can ﬁrst choose the coefﬁcients dij > 0, j = 1, 3; i = 1, n and then ﬁnd such matrix Qo that satisﬁes the Lyapunov equation (12) and the inequality l > lmax . In the error variables z1 , . . . , zn , e the system (14) in closed-loop form with the control law (18) is written as follows: z˙1 = −c1 z1 + z2 − d11 z1 − d12 z1 + (b1 − c1 − d11 − d12 )e1 − e2 , ∂α1 ∂α1 2 ∂α1 )z3 − d21 z2 − d23 ( ) z2 + e3 , z˙2 = −c2 z2 − z1 + (1 − ∂y ∂y ∂y ∂α1 ∂α2 2 ∂α2 )z2 + z4 − d33 ( ) z3 + e3 , z˙3 = −c3 z3 − (1 − ∂y ∂y ∂y ................................................................. ∂αn−1 2 ∂αn−1 ) zn + e3 , e˙ = (A + GC)e. z˙n = −cn zn − zn−1 − dn3 ( ∂y ∂y

(20)

Now if the condition l > lmax is satisﬁed, then global asymptotic stability of the equilibrium point z = 0, e = 0 of the system (20) follows from the inequality (19) and the main Lyapunov theorem [2]. Further, from the expressions z1 = χ1 − χ1d , z2 = χˆ2 − α1 , . . ., zn = χˆn − −αn−1 , e = e and global asymptotic stability of the equilibrium point z = 0, e = 0 of the system (20) it follows that any trajectory χ(t), e(t) of the closed-loop system (14) is globally bounded and the tracking error χ1 (t) − χ1d (t) converges asymptotically to zero for any initial conditions χ(0), e(0) of the closed-loop system (14). T T Then from the inequality |χ(t) − χd (t)| ≤ |(χT (t), eT (t))T − (χT d (t), 0 ) |, where χd (t) is the appropriate reference trajectory, it follows that for the closedloop system (10) the feedback control law u(χ + e, y) (18) also forces the tracking 72

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

error χ1 (t) − χ1d (t) to converge asymptotically to zero for any initial conditions χ(0). Finally, since the change of coordinates (9) is linear we conclude that for any initial conditions x(0) the trajectories x(t) of the closed-loop system (8) are globally bounded and the output reference trajectory tracking error y(t) − yd (t) converges asymptotically to zero. Output tracking in case of m -dimensional zero dynamics. Now consider the system (2) with n − r = m. The linear change of coordinates x1 χ1 · ¸ K1 K2 .. .. (21) . , . = Ir×r 0r×m χn xn 1 2 where K1 = [ kij ], i = 1, m, j = 1, r and K2 = [ kij ], i = 1, m, j = 1, m are appropriate matrices, with their elements being functions of bi , i = 1, m; Ir×r is the identity matrix of dimension r × r; 0r×m is the zero matrix of dimension r × m, transforms the system (2) into the form

χ˙ 1 = −b1 χ1 + χ2 + ϕr+1 (y), .. . χ˙ m−1 = −bm−1 χ1 + χm + ϕn−1 (y), χ˙ m = −bm χ1 + χm+1 + ϕn (y), χ˙ m+1 = χm+2 + ϕ1 (y), .. .

(22)

χ˙ n−1 = χn + ϕr−1 (y), χ˙ n = LT χ + β(y)u + ϕr (y), y = χm+1 . Here ϕi , i = r + 1, n are linear combinations of the nonlinearities ϕk , k = 1, n 1 , i = 1, m, and L ∈ Rn is a vector, with its components being functions of kij 2 j = 1, r and kij , i = 1, m, j = 1, m. A global exponential observer for the system (22) is written as follows: χˆ˙ = Aχˆ + GC(χˆ − χ) + φ(y) + Bβ(y)u,

(23)

where χˆ ∈ Rn ; φ(y) = (ϕr+1 (y), . . . , ϕn (y), ϕ1 (y), . . . , ϕr (y))T ; the matrices A, B refer to the vector form of (22); C = (0, . . . , 0, 1, 0, . . . , 0) has a one in the (m + 1)st entry. In (23) the gain vector G = (g1 , g2 , . . . , gn )T is chosen so that the matrix A + GC is Hurwitz. The state estimation error e = χ ˆ − χ equation is of the form (4) with the matrices A, G, C taken from (23) and its equilibrium e = 0 is globally exponentially stable, since the matrix A + GC is Hurwitz [3]. Note that in general case the backstepping procedure cannot be applied to stabilization of the system (22), since it is not in the strict feedback form (1). Further we will assume the folowing. Assumption 1. Let the following equations hold for the right-hand side of the system (2) with m > 1: σm (k)T (K1 ϕ(1:r) (y) + K2 ϕ(r+1:n) (y)) = 0,

k = 1, . . . , m − 1,

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

(24) 73

where σm (k) = (0, . . . , 0, 1, 0, . . . , 0)T ∈ Rm has a one in the kth position and ϕ(i:j) (y) = (ϕi (y), ϕi+1 (y), . . . , ϕj−1 (y), ϕj (y))T , j > i. The conditions (24) imply that in (22) the nonlinerities ϕi (y), i = r + 1, n − 1 satisfy ϕi (y) ≡ 0, i = r + 1, n − 1 and the system (22) preserves the structure similar to the strict feedback form (1), except that the nonlinearity fm (χ1 , . . . , χm , χm+1 ) = ϕn (χm+1 ) − bm χ1 depends also on the (m + 1)st variable. However, the observer backstepping procedure as it is applied to the stabilization of the system (10) cannot be directly used for stabilization of the system (22). Indeed, using the fact that e = χˆ − χ the system (22) is written as follows, the conditions (24) being satisﬁed: χ˙ 1 = −b1 χ1 + χˆ2 − e2 , χˆ˙ 2 = −b2 χˆ1 + g2 em+1 + χˆ3 , .. . χˆ˙ m−1 = −bm−1 χˆ1 + gm−1 em+1 + χˆm , χˆ˙ m = −bm χˆ1 + gm em+1 + χˆm+1 + ϕn (y), ˆm+2 + gm+1 em+1 + ϕ1 (y), χˆ˙ m+1 = χ .. . ˙ χˆn−1 = χˆn + gn−1 em+1 + ϕr−1 (y), χˆ˙ n = LT χˆ + gn em+1 + β(y)u + ϕr (y), e˙ = (A + GC)e, y = χˆm+1 − em+1 .

(25)

The presence of the terms gi em+1 = gi (χˆm+1 − χm+1 ), i = 2, m − 1 in the ﬁrst m − 1 equations of the system (25) implies that the system is not in the strict feedback form and the observer backstepping as it is considered in the previous case of one-dimensional zero dynamics is not applicable to the considered system. Now consider the following subsystem composed of the ﬁrst m equations of (22): (26) χ˙ (1:m) = Am χ(1:m) + Bm (χm+1 + ϕn (y)), where χ(1:m) = (χ1 , . . . , χm )T ; Am = (aij ) is a matrix of dimension m × m with its elements satisfying aij = 1 if j − i = 1, aij = −bi if j = 1 and aij = 0 if j − i 6= 1 and j 6= 1; Bm = (0, . . . , 0, 1)T ∈ Rm . Note that, if the function f (χm+1 ) = χm+1 + ϕn (χm+1 )) deﬁnes a diffeodef

morphism, then for the system (22) the reference output trajectory yd (t) = def

= χ(m+1)d (t) tracking problem is equivalent to that of the tracking of a globally bounded reference trajectory χ(1:m) = χ(1:m)d (t) that satisﬁes the system χ˙ (1:m)d = Am χ(1:m)d + Bm (χ(m+1)d + ϕn (χ(m+1)d )). The conditions on the right-hand side of (26) under which for any globally bounded reference trajectory χ(m+1)d (·) ∈ L1 ∩ L∞ there exists a unique solution χ(1:m)d (·) ∈ L1 ∩ L∞ ∩ C 0 , χ(1:m)d (±∞) = 0 of the system χ˙ (1:m)d = = Am χ(1:m)d + Bm (χ(m+1)d + ϕn (χ(m+1)d )) are given in [6, 7]. Further we will suppose that these conditions are satisﬁed and the smooth reference trajectory χ(1:m)d (t) that corresponds to the considered reference output trajectory χ(m+1)d (t) is found, for instance, following the procedure given in [6, 7]. 74

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

By using the fact that em+1 = χˆm+1 − χm+1 , the system (26) can be written as follows: χ˙ (1:m) = Am χ(1:m) + Bm (χˆm+1 − em+1 + ϕn (y)),

(27)

where χˆm+1 is the (m + 1)st component of the state vector estimate χˆ generated by the observer (23). Note that the linear subsystem χ˙ (1:m) = Am χ(1:m) +Bm χ ˆm+1 with χˆm+1 as its control input is completely controllable. Let us use χˆm+1 as a virtual control input to solve the reference trajectory χ1 (t) = χ1d (t), . . ., χm (t) = χmd (t) tracking problem for the system (27) by exploiting the notions of block backstepping [2] and vectorial observer backstepping [8]. Consider the following positive deﬁnite function: T Pm z(1:m) + 2V0 (e) > 0, Vm (z(1:m) , e) = z(1:m)

(z(1:m) , e) 6= 0,

T there z(1:m) = (z1 , . . . , zm )T = (χ1 − χ1d , . . . , χm − χmd )T , Pm = Pm > 0. ˆm+1 − αm , the time derivative of Vm along the Using the error variable zm+1 = χ solutions of (27) can be expressed as follows:

T V˙ m |(27) = 2z(1:m) Pm [Am χ(1:m) +

+ Bm (zm+1 + αm − em+1 + ϕn (y)) − χ˙ (1:m)d ] − 2eT Qo e. Then, having written the system (27) in the error variables (z1 , . . . , zm )T = = (χ1 − χ1d , . . . , χm − χmd )T , one can ﬁnd a control law αm that solves the considered reference trajectory tracking problem for the system χ˙ (1:m) = = Am χ(1:m) + Bm (αm + ϕn (y)). One of such control laws is given by m X

αm = − m X

−

ci (χi − χid )− i=1 i−1 X (i−j−1) (−bj χ1d

ci i=2

m X

+ χid −

(i−1) χ1d )

(m−i)

− ϕn (y) +

j=1

bi χ1d

(m)

+ χ1d =

i=1 i−1 X

m X

(i−j−1)

= −Cm (χ(1:m) − χ(1:m)d ) −

ci i=2

(−bj χ1d

(i−1)

+ χid − χ1d

)−

j=1 m X (m−i)

− ϕn (y) +

bi χ1d

(m)

+ χ1d ,

i=1

where Cm is chosen so that the matrix Ac = Am − Bm Cm is Hurwitz. Hence, T > 0 and Qm = QT there exist the matrices Pm = Pm m > 0 satisfying the Lyapunov equation [3] AT c Pm + Pm Ac = −Qm − 2Im , where Im is the identity matrix of dimension m × m. Note that here the matrix −Qm represents the effect of stabilization without taking care of the uncertainties VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

75

and the two identity matrices −2Im represent the effect of the extra damping to take care of the estimation errors e(1:m) = χˆ(1:m) − χ(1:m) and em+1 = χˆm+1 − χm+1 . This could be compared with the role of the constants c1 and d11 , d22 in the ﬁrst step of the one-dimensional case in the previous section. Substitute in αm the state estimate χˆ(1:m) instead of the true state χ(1:m) . Then the time derivative of Vm along the solutions of (27) can be written in the following way:

T T (AT V˙ m |(27) = z(1:m) c Pm + Pm Ac )z(1:m) − 2z(1:m) Pm Bm Cm e(1:m) − T T − 2z(1:m) Pm Bm em+1 + 2z(1:m) Pm Bm zm+1 − eT Qo e − eT Qo e ≤ T Qm z(1:m) − (z(1:m) + Pm Bm Cm e(1:m) )T (z(1:m) + Pm Bm Cm e(1:m) )− ≤ −z(1:m)

− eT (Qo − Λ)e − (z(1:m) + Pm Bm em+1 )T (z(1:m) + Pm Bm em+1 )− T T T − (l − Bm Pm Pm Bm )e2m+1 + 2z(1:m) Pm Bm zm+1 ,

where ¸

· Λ=

(Pm Bm Cm )T (Pm Bm Cm ) 0m×r 0r×m 0r×r

.

Here by the appropriate choice of the matrices Pm and Cm one can satisfy the T T inequalities Qo − Λ > 0 and l − Bm Pm Pm Bm > 0. Thus, if χˆm+1 were the control input, then, by choosing χˆm+1 = αm (χˆ(1:m) , (m)

χ(1:m)d , χ˙ 1d , . . . , χ1d , y) with the appropriately taken matrices Pm and Cm , the derivative V˙ m along the solutions of the system (27) would be less or equal to the expression that contains a negative deﬁnite part plus indeﬁnite term T 2z(1:m) Pm Bm zm+1 . Finally, the further steps of our stabilizing control design are analogous to the above case of one-dimensional zero dynamics and consist in the subsequent adding of the rest (n − m) equations of the system (22) to the considered subsystem (26). In each step the appropriate virtual control law is designed untill the whole system (22) is considered and the stabilizing control law is found. Remark 1. The assumption 1 can be relaxed to allow for linear terms of the form Bχm+1 , B ∈ Rm in the ﬁst m equations of the system (22) as long as the system χ˙ (1:m) = Am χ(1:m) + (Bm + B)χm+1 is stabilizable. Illustrative example. Consider the model for a ﬂexible one-link robot arm [9]. The dynamics are given by χ˙ 1 = χ2 , χ˙ 2 = −M1 sin χ1 − k1 (χ1 − χ3 ), χ˙ 3 = χ4 , χ˙ 4 = −b1 χ4 + k2 (χ1 − χ3 ) + u/J, y = χ3 ,

(28)

where χ1 is the angle of the arm, χ2 is the angular velocity of the arm, χ3 is the angle on the motor side, χ4 is the angular velocity on the motor side, the input signal u is the driving torque from the motor, the constants M, b1 , k1 , k2 , J are all positive. 76

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

It is supposed that only angular rotation χ3 on the motor side is available for measurements. For this system our purpose is to design a feedback control law that uses only values of angular coordinate χ3 and forces the tracking error χ3 (t) − χ3d to converge to zero as t → ∞ for all χ(0), where χ3d = const is a constant reference signal. First, let us design a global exponential observer for the system (28). To this end, use the new variables x 1 = χ 3 , x 2 = χ 4 , x 3 = k2 χ 1 − k 2 χ 3 − b 1 χ 4 , x4 = −k2 b1 χ1 + k2 χ2 + k2 b1 χ3 + (b21 − k2 )χ4 .

(29)

The linear change of coordinates χ = Φ(x), Φ(0) = 0 given by (29) deﬁnes a diffeomorphism R4 = {x} onto R4 = {χ}. Furthermore, in the new variables x the system (28) is written as follows: x˙ 1 = x2 , x˙ 2 = x3 + u/J, x˙ 3 = x4 − b1 u/J, x˙ 4 = a4 (x) + (b21 − k2 )u/J, y = x1 ,

(30)

where a4 (x) = −b1 k1 x2 −(k1 +k2 )x3 −b1 x4 −k2 M1 sin ((k2 x1 + b1 x2 + x3 )/k2 ). Note that both the system (28) and the dynamical part of the system (30) are in the strict feedback form (1). However, neither of these systems is in the outputfeedback form (2). The structure of (30) is similar to the output-feedback form (2), except that the nonlinearity a4 (x) also depends on unmeasured states. Hence, one cannot design observers for the system (30) directly in the form of (3). However, since the system (30) is written in the uniformly observable canonical form [10] and the right-hand side of the system is globally Lipschitz in x uniformly on u, a global exponential observer for the system (30) is readily constructed as [10] x ˆ˙ = Aˆ x + a(ˆ x) + GC(ˆ x − x) + Bu,

(31)

where A = (aij ), i = 1, 4, j = 1, 4 is a square matrix, with its elements satisfying x) = (0, 0, 0, a4 (ˆ x))T ; aij = 1, if j − i = 1 and aij = 0, if j − i 6= 1; a(ˆ B = (0, 1/J, −b1 /J, (b21 − k2 )/J)T ; C = (1, 0, 0, 0). In (31) the gain vector G = = (g1 , g2 , g3 , g4 )T has the following structure G = (θl1 , θ 2 l2 , θ 3 l3 , θ 4 l4 )T , where θ > 1 is some positive constant, L = (l1 , l2 , l3 , l4 )T is chosen so that the matrix A + LC is Hurwitz. In the variables χˆ = Φ(ˆ x) the system (31) is written as ˆ3 − χ3 ), χ ˆ˙ 1 = χˆ2 + (θ 3 l3 /k2 + θl1 + b1 θ 2 l2 /k2 )(χ ˆ1 − k1 (χˆ1 − χ ˆ3 )+ χ ˆ˙ 2 = −M1 sin χ +(θ 4 l4 /k2 + θ 2 l2 + b1 θ 3 l3 /k2 )(χˆ3 − χ3 ), ˙ ˆ3 − χ3 ), χ ˆ3 = χˆ4 + θl1 (χ ˙ ˆ1 − χˆ3 ) + θ 2 l2 (χ ˆ3 − χ3 ) + u/J. χ ˆ4 = −b1 χˆ4 + k2 (χ

(32)

Once the change of coordinates χ = Φ(x) given by (29) is linear, the system (32) is a global exponential observer for the system (28). VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

77

Further, to ﬁnd a stabilizing control law that is solution to the tracking problem in question via the introduced in the previous sections backstepping procedure, the following system is used: ˆ 2 − e2 , χ˙ 1 = χ χ ˆ˙ 2 = (θ 4 l4 /k2 + θ 2 l2 + b1 θ 3 l3 /k2 + k1 )χˆ3 − M1 sin χˆ1 − k1 χˆ1 − − (θ 4 l4 /k2 + θ 2 l2 + b1 θ 3 l3 /k2 )χ3 , ˆ4 + θl1 (χˆ3 − χ3 ), χ ˆ˙ 3 = χ ˙χ ˆ4 = −b1 χˆ4 + k2 (χˆ1 − χˆ3 ) + θ 2 l2 (χˆ3 − χ3 ) + u/J, ˜ + ϕ(χ) e˙ = Ae ˆ − ϕ(χ ˆ − e), y = χ ˆ 3 − e3 ,

(33)

where e = χˆ − χ, the matrix A˜ refers to the linear part of the system (32) with u ≡ 0 written in vector form, ϕ(χ) = (0, −M1 sin χ1 , 0, 0)T . Finally, the feedback control law that when applied to the system (28) forces the tracking error χ3 (t) − χ3d to converge to zero as t → ∞ for all χ(0) is given by µ µ ¶ ∂α2 + b1 χˆ4 − u(χ, ˆ χ3 ) = J −c4 z4 − z3 1 − ∂χ3 ∂α3 − k2 (χˆ1 − χˆ3 ) − θ 2 l2 (χˆ3 − χ3 ) + χˆ4 + ∂χ3 ¶ 3 X ∂α3 ˙ ∂α3 2 χˆ − d44 ( ) z4 , c4 , d44 > 0, (34) + ∂ χˆi i ∂χ3 i=1

where αi , zj , i = 1, 3, j = 1, 4 are calculated as follows: z1 = χ1 − χ1d , zi = χˆi − αi−1 (χ3 , χˆ1 , . . . , χˆi−1 , χ1d ), α1 = −(c1 + d11 + d12 )χˆ1 + (c1 + d11 + d12 )χ1d ,

i = 2, 4,

c1 > 0, d11 > 0, d12 > 0,

∂α1 ˜ ³ α2 = 1/(k1 + θ˜2 − ˆ1 + χ1d + θ˜2 χ3 + θ1 ) −c2 z2 − χ ∂ χˆ1 ´ ∂α1 (χ ˆ2 − θ˜1 χ3 ) − d21 z2 , + M1 sin χˆ1 + k1 χˆ1 + ∂ χˆ1

.³ α3 = 1

78

c2 > 0, d21 > 0,

µ µ ¶ ∂α2 ´ ∂α1 ˜ ˜ −c3 z3 − z2 k1 + θ2 − 1− θ1 − θl1 (χˆ3 − χ3 )+ ∂χ3 ∂ χˆ1 µ ¶ ¶ ∂α2 ˙ ∂α2 ˙ ∂α2 2 + z3 , c3 > 0, d34 > 0, χˆ + χˆ − d34 ∂χ ˆ1 1 ∂ χˆ2 2 ∂χ3

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

with θ˜1 = θ 3 l3 /k2 + θl1 + b1 θ 2 l2 /k2 , θ˜2 = θ 4 l4 /k2 + θ 2 l2 + b1 θ 3 l3 /k2 and the constants d11 , d12 , d21 , d34 , d44 being appropriately chosen. Conclusions. In this paper the output feedback control problem for a class of nonlinear nonminimum-phase systems is considered. The standard observer backstepping method is extended to stabilization of a class of nonlinear nonminimumphase systems in the output-feedback form with linear unstable zero dynamics. The proposed design achieves global stabilization and allows tracking of reference output trajectories. However, the main restriction on the class of systems to which the design is applicable is assumption 1, which is a condition for preserving strict feedback form in the state transformations used for the backstepping design. This work has been supported by Grant 02-01-00704 from the Russian Foundation for Basic Research, by Grant HU-2094.2003.1 of Support of Leading Scientiﬁc Schools and by the National Board for Industrial and Technical Development (NUTEK).

REFERENCES 1. Kanellakopoulos, I., Kokotovi´c, P., Morse, S. A toolkit for nonlinear feedback control // System and Control Letters. 1992. V. 18. P. 83–91. 2. Krsti´c, M., Kanellakopoulos, I., Kokotovi´c, P. Nonlinear and Adaptive Control Design. – New York: John Wiley and Sons, 1995. 3. Krasovskiy, N.N. Some problems of stability of motion theory (in Russian). – Moscow, 1959. 4. Isidori, A. Nonlinear Control Systems. 3rd edition. – Springer-Verlag, 1995. 5. Robertsson, A., Johansson, R. Observer backstepping for a class of nonminimumphase systems // Proceedings of IEEE Conference on Decision and Control. Phoenix, Arizona. 1999. P. 4866–4871. 6. Degang, C., Paden, B. Stable inversion of nonlinear nonminimum-phase systems // International Journal of Control. 1996. V. 64. No 1. P. 81–97. 7. Devasia, S., Degang, C., Paden, B. Nonlinear inversion-based output tracking // IEEE Transactions on Automatic Control. 1996. V. 41. No 7. P. 930–942. 8. Fossen, T.I., Grovlen, A. Nonlinear output feedback control of dynamically positioned ships using vectorial observer backstepping // IEEE Transactions on Control System and Technology. 1998. V. 6. P. 121–128. 9. Marino, R., Tomei, P. Nonlinear Control Design: Geometric, Adaptive and Robust. Prentice Hall information and system sciences series. – London: Prentice-Hall, 1995. 10. Gauthier, J.P., Hammouri, H., Othman, S. A simple observer for nonlinear systems. Applications to bioreactors // IEEE Trans. Autom. Contr. – 1992. – V. 37, No 6. – P. 875 – 880.

Alexey Evgenievich Golubev (b. 1978) graduated from Bauman Moscow State Technical University (BMSTU) in 2002. Currently, he is post graduate student at the “Mathematical modeling” Department, BMSTU. He specializes in the ﬁeld of mathematical control theory and has published 3 papers on output feedback control of nonlinear dynamical systems.

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Rolf Johansson received the Master of Science degree in Technical Physics in 1977, the Bachelor of Medicine degree in 1980, the doctorate in control theory in 1983, was appointed Docent in 1985, and received the Doctor of Medicine degree in 1986, all from Lund University, Lund, Scandinavia. He is member of SIAM and IEEE and is a fellow of the Swedish Society of Medicine. Currently, he is Professor of control theory at the Lund Institute of Technology and a coordinating director in robotics research with participants from several departments of Lund University. In his scientiﬁc work, he has been involved in research in adaptive system theory, mathematical modeling, system identiﬁcation, robotics and signal processing. Anders Robertsson (b. 1967) received the M.Sc. degree in Electrical Engineering and the Ph.D. degree in Automatic Control from Lund Institute of Technology, Lund, Sweden, in 1992 and 1999, respectively. During 2000 he worked as a Research Fellow and he is currently appointed as an Assistent Professor at the Department of Automatic Control, Lund. His research interests are in nonlinear control, real-time systems and robotics with particular attention to the output feedback control problem and force feedback control for robot manipulators. Sergey Borisovich Tkachev (b. 1961) graduated from Bauman Moscow Higher Technical School in 1984 Ph. D. (Phys.-Math.), Associate Professor at the “Mathematical modeling” Department of the Bauman Moscow State Technical University. He is author of more then 35 publications in the ﬁeld of mathematical control theory. His research interests include nonlinear control, nonlinear observation theory, geometric stabilization methods and the output feedback control problem.

BMSTU Press has published the book: Optical and Electronic Systems for Ecological Monitoring of Environment (In Russian) / V.I. Kozintsev, V.M. Orlov, M.L. Belov et al.; edited by V.N. Rozhdestvin. – M.: Izdatelstvo MGTU imeni N.E. Baumana, 2002. – 528 p. The book has two parts titled: “Laser Optical and Electronic Systems for Ecological Monitoring of Environment” and “Passive Optical and Electronic Systems for Ecological Monitoring of Environment”. Part I sets forth the physical background of laser sounding, concepts of building lidar systems of ecological monitoring and contains some examples of the systems. Part II is devoted to the physical background of passive optical monitoring, to concepts of organization of satellite systems of ecological monitoring and to building of passive optical and electronic devices for remote monitoring of environment. Examples of the satellite optical and electronic apparatus for ecological monitoring of environment are given. The book incorporates material of lectures delivered by the authors in the Bauman Moscow State Technical University. The book is intended for technical university students trained for “Optotechnology” profession and also for scientiﬁc and engineering workers specializing in instrumental engineering.

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LASER & OPTIC-ELECTRONIC SYSTEMS V.N. Rozhdestvin, O.A. Smirnova (Bauman Moscow State Technical University)

COMBINED Q-SWITCH AND MODE-LOCK IN LASER SYSTEMS Various laser system conﬁgurations are discussed which are based on the combined usage of Q-switch and mode-lock and allow generating ultrashort pulses with high peak power. Three different schemes of their realization are considered. The particular attention is given to the saturable absorber Cr+4 :YAG and its possible operation regimes. Rate equations describing the dynamics of processes inside the Cr+4 :YAG crystal are shown for each regime. Experimental results supporting theoretical expectations are presented.

Introduction. Pulse laser systems, generating short pulses (nanoseconds– picoseconds) of high peak power (kW0 s—GW0 s), have found their application in many ﬁelds of science and technology. Spectroscopy and dynamic studies in physics, nonlinear optics, chemistry, biology, optoelectronics were historically the ﬁrst areas of applications of ultrashort pulsed laser systems. Ultrafast lasers (pulse repetition rate up to 100 MHz) can be used in location systems, communication systems, systems for remote measurement of material properties, remote sensing, material processing, environmental studies, and monitoring production processes. Medical and industrial applications, metrology, astronomy of gravitational waves, optical databases, high deﬁnition laser television, optical computers — are only a few of their applications already proposed [1–3]. There is a number of known methods of obtaining the pulse generation in lasers. By means of pulse pumping the pulses of 10−6 . . . 10−3 s can be obtained. When signal short pulses are generated using Q-switch modulators, it is possible to achieve pulses with duration of about 10−9 . . . 10−8 s with peak power up to several GW. To obtain even shorter pulses (10−13 . . . 10−12 s) the mode-locking regime should be used [4, 5]. It is possible to generate short high power pulses by combining the Q-switch and mode-lock in one laser system. The mode-locking allows to obtain the very short pulse duration, while Q-switching provides the high output power [5–7]. In laser system design it is also important to consider the fact that the system should be inexpensive, simple (reduced number of components, diminished in size and weight), and have the high stability of output pulses parameters. There are some different system conﬁgurations based on the combined usage of Q-switch and mode-lock regimes. Methods of Combined Q-switching and Mode-Locking and their Realization Schemes. An injection seeding method for generating high peak power pulses at high repetition rates from Q-switched Nd:glass laser is described in [6]. VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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Fig. 1. Injection mode-locked and Q-switched laser system (two-cavity laser system)

A schematic diagram of the injection mode-locked and Q-switched laser system is shown in Fig. 1. This laser system was proposed by S. Basu and R. Byer from Stanford university in 1990. The system includes two cavities with a Q-switch and a mode-lock in separate cavities. The mode-locked laser output is injected into the Q-switched cavity through an optical isolator and a partial beam splitter. A full width of the Q-switch pulse is 64 ns. When 50 ps pulses from the mode-locked laser are injected, the output becomes a series of short pulses under a Q-switched envelope. As a result of the experiment, with 11 picosecond pulses they are able to extract a maximum of 6.3 mJ per pulse, which corresponds to approximate of 500 MW pulse peak power. Combined Q-switch and mode-lock in one cavity is the most common conﬁguration of combined Q-switch and mode-lock laser system (Fig. 2). In closed state the Q-switch is a bit transparent to the lasing radiation. This allows enough intensity through, for mode-locking process to reach a steady state. And when the Q-switch opens, an envelope of very high peak power mode-locked pulses will be radiated.

Fig. 2. Q-switched and mode-locked one cavity laser system

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Fig. 3. Laser system based on corner cube resonator with saturable absorber modulator

It should be speciﬁed, that in such a laser cavity conﬁguration, the interaction between two passive modulators is rather complicated and hardly compatible. As a rule, a combination of an active mode-lock and a passive Q-switch is used. For now, solid state lasers with the highest output characteristics are based on the conﬁguration with Q-switch and mode-lock in one cavity [4]. Combined Q-switch and mode-lock in one nonlinear element (Fig. 3) is the next laser conﬁguration [7, 8]. Several kinds of the passive nonlinear saturable absorbers that played double role of the Q-switch as well as the Q-switch and mode-lock at the same time are investigated. (The corner cube in the resonator conﬁguration was designed for practical purposes to be insensitive to the resonator misalignment.) Good example of such absorbers is Cr+4 :YAG crystal. Saturable Absorber Cr+4 :YAG. Solid-state laser engineers were eager to have at their disposal a crystal that would include in itself all main optical cavity parts (modulators, gain medium, etc.). This became possible with introducing saturable absorbers based on the same matrix as the active medium (such as YAG — yttrium aluminum garnet). This allowed to grow up crystals doped partially with saturable and partially with gain particles (though, ideally, we would like to have gain and saturable element spread over the whole cavity length). With mirrors added at the sides of such a crystal, and a small efﬁcient diode as a pump, we can have a solid-state high power pulsed laser that can be ﬁt in a volume smaller then a pencil. Let us consider the 5-level saturable absorber based on the Cr+4 :YAG crystal. The latter has at least two transitions at the same frequency, which can be used to absorb the laser radiation coming through the modulator. The schematic energy level diagram of the crystal [7, 8] is shown in Fig. 4. VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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Fig. 4. Schematic energy level diagram of Cr+4 :YAG crystal

The following designations are used: N1 , N2 , N3 are population densities; σg , σe are cross sections of the ground state and exciting state absorption respectively, with σg = 8.7·10−19 cm2 ; σe = 2.2·10−19 cm2 . At room temperature, τ1 , relaxation time from the state 3 T2 to the ground state 3 A2 , is 3.4 µs, and τ2 , relaxation time from the state 3 T1 to 3 T2 , is 50 ps. The rate equations that describe the dynamics of populations of the passive crystal Cr+4 :YAG energy levels are as follows: ³ σ ´ ∂N1 N2 g , = −N1 I+ ∂t 2π~ν τ1 ³ σ ´ ³ σ ´ ∂N2 N2 N3 g e − N2 , = N1 I− I+ ∂t 2π~ν τ1 2π~ν τ2 ³ σ ´ ∂N3 N3 e , = N2 I− ∂t 2π~ν τ2 ∂I = −I(σg N1 + σe N2 ), ∂z N0 = N1 + N2 + N3 , where I is intensity of laser radiation inside the cavity; z is the horizontal axis along the crystal length; N0 is concentration of the Cr+4 particles in the crystal. At the starting point for the small signal case we have N2 = N3 = 0, N1 = N0 . Let us consider the Q-switch regime ﬁrst [7, 9]. As photons start to go through the ﬁlter, the atoms jump to the lower excited level and stay there for the relaxation time τ1 . When all atoms are pumped up from the ground energy level, the absorption on the σg σe transition does not occur any more, and photons come through the ﬁlter without absorption. Since σg is 4 times larger then σe , then the probability of σe transition is small, but it occurs once in a while. As a result of this process, the maximum ﬁlter transparency in Q-switch regime cannot reach 100 %. The lifetime of the upper excited state τ2 is so small, that the population density 84

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N3 almost does not change, and can be approximated by zero. Hence, the rate equations can be rewritten in the following way: ³ σ ´ ∂N1 N2 g = −N1 I+ , ∂t 2π~ν τ1 ³ σ ´ N2 ∂N2 g = N1 I− . ∂t 2π~ν τ1 For the saturation case, N2 = N3 and the saturation intensity for Q-switch regime is W 2π~ν Isg = = 6.3 · 104 2 . σg τ1 cm Now, let us see what happens if we dramatically increase the intensity inside the laser cavity to turn it into the combined Q-switch and mode-lock regime [7, 8]. The atoms are quickly pumped up to the lower excited level but do not stay there. The intensity is high enough now to pump atoms even farther, to the upper excited level, where they will ﬁnally stay for the short period of time τ2 , allowing a group of photons to come through without absorption. After atoms relax back to the second level, they are met by the returning and ampliﬁed group photons that pump them up, again making the ﬁlter transparent for photons. In an actual experiment there can be as many as 100 of such loops or even more, which depends both on the combination of characteristics of the active and saturable elements and on the pumping rates. When the signal ceases, atoms slowly relax back, to the ground level. The very small lifetime τ2 , about 50 ps, is what makes the mode-locking process possible. During this process atoms spend most of the time in excited states, leaving N1 practically equal to zero. Therefore, the rate equations have the following representation: ³ σ ´ ∂N2 N3 e = −N2 I+ , ∂t 2π~ν τ2 ³ σ ´ N3 ∂N3 e = N2 I− . ∂t 2π~ν τ2 Again saturation intensity can be calculated as Ise =

2π~ν W = 7.5 · 109 2 , σe τ2 cm

giving us the result by ﬁve orders greater then for the Q-switch regime, which proves the necessity of much higher intensity for the combined Q-switch and modelock regime. The experimental results of the above-mentioned investigations [7, 8] for both the Q-switch and the combined Q-switch and mode-lock regimes are shown in Figs. 5 and 6. The horizontal time scale is 50 ns per division. It can be easily seen, that in combined regime the output becomes a train of short pulses under a Q-switched envelope [7]. VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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Fig. 5. Output for Nd:YAG laser passively Q-switched by Cr+4 :YAG crystal

Fig. 6. Output for Nd:YAG laser passively Q-switched and mode-locked by Cr+4 :YAG crystal

The performance of the Cr+4 :YAG crystal, that passively modulated the Nd:YAG laser, and the summarized experimental results are as follows: Q-switched pulsewidth (FWHM) . . . . . . . . . . . . . . 140 ns Q-switched pulsewidth (Full-width) . . . . . . . . . . . . 430 ns Mode-locked pulses number . . . . . . . . . . . . . . . . . . . 108 Mode-locked pulses separation. . . . . . . . . . . . . . . . . 4 ns Mode-locked pulsewidth . . . . . . . . . . . . . . . . . . . . . . 197 ps Q-switched pulse output power . . . . . . . . . . . . . . . . 0.07 MW Mode-locked pulses output power . . . . . . . . . . . . . . 0.47 MW

The achieved results might be not close to the limits, but they demonstrate the advanced capabilities of such systems, and suggest many possible applications with further improvements. Conclusions. In our age of simpliﬁcation, miniaturization and improved reliability, the described laser systems will surely be in great demand. Passive elements are relatively cheap and reliable, do not need external control systems, and can be made very small in size. Communication, medical application, high deﬁnition laser television, and optical data storage, are only a few examples of their possible applications. In this work there was presented the overview of combined Q-switching and mode-locking methods of generating ultrashort pulses with high peak power. Three different realization schemes of laser systems using the combined Q-switch and mode-lock regime were considered. In greater detail the saturable absorber on the basis of the Cr+4 :YAG crystal was discussed — rate equations, describing the dynamics of processes inside the Cr+4 :YAG crystal, were derived for each regime. Experimental results supporting theoretical expectations were also considered.

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REFERENCES 1. Clark, M., Sharples, S.D. & Somekh, M.G. Fast laser generated Rayleih wave scanning microscope, in Proceedings of the IEEE Ultrasonics Symposium, vol. 2, 1998, pp. 1317–1320. 2. Wayne H. Knox, Practical lasers will spawn varied ultra-fast applications, Laser Focus World, June 1996. 3. Brunner, F., Keller, U., Kuleshov, N.V., et. al., 240-fs pulses with 22-W average power from a mode-locked thin-disk Yb:KY(WO4 )2 laser, Optics Letters, vol. 27, no 13, pp. 1162–1164, 2002. 4. Malcolm, G.P.A., Ferguson, A.I. Diode-pumped solid-state lasers, Contemporary Physics, vol. 32, no 5, pp. 305–319, 1991. 5. Honninger, C., Paschotta, R., Morier-Genoud, F. & Keller, U. Q-switching stability limits of continuous-wave passive mode-locking, Journal of Optical Society of America, Vol. 16, no .1, pp. 46–56, 1999. 6. Basu, S. & Byer, R.L. Short pulse injection seeding of Q-switched Nd:glass laser oscillators theory and experiment, IEEE Journal of Quantum Electronics, vol. 26, no 1, pp. 149–157, 1990. 7. Chen, J., Yau, H.F., Liu, H.P., Chen, T.C., Cheng, C.C. & Liu, F.M. Passive Q-switch and mode-locking modulators for Nd: hosted lasers, Optics & Laser Technology, vol. 32, pp. 215–219, 2000. 8. Chen, J., Chen, D.K., Chen, R.S., Chen, Y.F. & Tsai, S.W. Passive Q-switch and modelocking modulators as well as ﬂashlamp-pump Nd: hosted lasers, in IEEE Paciﬁc Rim Conference on Lasers and Electro-Optics, CLEO, vol. 2, 2001, pp. II470–II471. 9. Shimony, Y., Burshtein, Z. & Kalisky, Y. Cr+4 :YAG as passive Q-switch and Brewster plate in a pulsed Nd:YAG laser, IEEE Journal of Quantum Electronics, vol. 31, no 10, pp. 1738–1741, 1995.

V.N. Rozhdestvin (b. 1940) graduated from the Bauman Moscow Higher Technical School in 1963. D. Sc. (Eng.), professor of “Radio Electronic Systems and Devices” department of the Bauman Moscow State Technical University, Winner of the State Prize of the Russian Federation, Honored Worker of Science and Technology of the Russian Federation, full member of the Russian Academy of Natural Sciences and Russian Academy of Military Sciences. Author of more than 200 publications in the ﬁeld of microwave and laser technology. O.A. Smirnova (b. 1947) graduated from the Bauman Moscow Higher Technical School in 1971. Ph. D. (Eng.), assoc. professor of “Radio Electronic Systems and Devices” department of the Bauman Moscow State Technical University, corresponding member of the Russian Academy of Informatization. Author of more than 80 publications in the ﬁeld of microwave and laser technology and physics of lasers.

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V.Ye. Karasik (Bauman Moscow State Technical University) LASER RANGE-GATED IMAGING SYSTEM Imaging systems of a new type — laser range-gated imaging systems are considered, in which as a ﬁnal link of decision making process an operator acts. The acquisition characteristics of such systems are introduced which take into account properties of the human visual system of an observer. A technique of their calculation based on determination of one of the most important parameters — the perceptible signal-to-noise ratio — is offered. For check of correctness of the offered computational relations another technique for an experimental estimation of the acquisition characteristics is developed and a prototype of laser imaging system is made with semiconductor laser pulse source and the gated receiving optical-electronic device containing the third-generation microchannelplate image intensiﬁer and the TV-camera on the basis of a photosensitive CCD-array. Experimental results of measuring the imaging system characteristics are discussed and compared with the calculated values.

Introduction. Laser imaging systems (LIS) are intended for formation an image of remote objects to detect and identify them under bad conditions of observation caused by low natural light and a various sort of clutters. Despite the recent intensive development of vision systems with automatic image processing, it should be admitted, that the man-operator, given a sufﬁcient resource of time available for decision making, copes with tasks of the object detection and identiﬁcation by its image better than any automatic system. Therefore, in overwhelming majority of LIS types the operator acts as a decision-maker, and the LIS characteristics should be determined in view of the observer0 s human visual system (HVS) properties. The laser imaging systems with pulse illumination are most popular now. Their operation principle is based on periodic illumination of an object by short pulses of laser radiation followed by photo-registration of the image in the receiving channel based on the staring CCD-array. For higher acquisition characteristics of LIS the microchannel-plate image intensiﬁer (MPII) is incorporated in the receiving channel, which increases the image brightness and carries out the gating of the received signal for clutters ﬁltering. At present, MPII of the third generation with gallium-arsenide (GaAs) photo-cathode has the greatest sensitivity. The application of such MPII makes possible to use as an illuminating laser not only pulse solid-state lasers, but also semiconductor pulse sources, which radiation spectrum falls within the maximal sensitivity area of GaAs photo-cathode. It allows creating the compact LIS with low weight and energy consumption. The main difference of LIS with pulse illumination from night vision devices with continuous laser illumination consists in the coordinated work of receiving and transmitting channels: MPII is open only for the time when the radiation pulse, reﬂected from the object, is expected to arrive. This results in the following:

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— effective time and spectrum gating of the signal is achieved with simultaneous suppression of the image background component caused by natural radiation; — contrast image still can be formed under the strong radiation dispersion caused by bad conditions of observation (smoke, fog, atmospheric precipitates); — high noise-immunity and range-gating of objects are ensured. At the same time, it is expected, that the overall dimensions and weight of LIS with semiconductor sources will not exceed essentially the similar parameters for night vision devices. This allows the competitiveness of similar systems to be predicted. Theoretical Analysis. The LIS performance is deﬁned by its acquisition characteristics, which are understood as its capabilities of detection, recognition and identiﬁcation of objects. Acquisition characteristics of LIS can be a priori estimated on the basis of its perceptible resolution with using the known Johnson0 s chart [1]. For this purpose, the equivalent mire (bar pattern, as a rule, four-bar pattern) is assigned to each level of the real object recognition. It is considered, that the probability of the real object recognition will be no less than the given one, if the perceptible signal-to-noise ratio (PSNR) µp∆ for an image of the equivalent p pt bar pattern is no less than the threshold value µpt ∆ , i.e. the condition µ∆ > µ∆ is satisﬁed. By threshold is meant such a value of PSNR, at which the probability of the bar pattern resolution is equal to 0.5; usually µpt ∆ is set equal to 2.5–3. If we know how PSNR depends on internal g1 , g2 , . . . gm and external q1 , q2 , . . . . . . qn parameters of the system, we are able to estimate a priori its acquisition characteristics, to carry out the light and energy calculations, to obtain such important system characteristics, as the minimum perceptible contrast and maximum distance of vision, to choose correctly the component base of LIS and to optimize its parameters. For this purpose, we only need to solve the equation µp∆ {(g) , (q)} = µpt ∆

(1)

for the unknown parameter. A technique of PSNR calculation is offered in [2, 3], where peculiarities of the image formation in LIS are deﬁned and, considering them, the mathematical model of LIS is developed in the form of a sequence of linear spatially-invariant parts and generators of additive homogeneous clutters. The functional model of the observer0 s human visual system in the form of a quasi-optimum ﬁltering receiver is proved. In the accepted model the object image on the monitor screen is described by the brightness distribution L00 (x, y) or its space-frequency spectrum ˜ 0 (νx , νy ), and image of a background or underlying surface — by average (SFS) L 0 brightness L0b . A signal component of the image is understood as a deviation of brightness value in the object image from the average background brightness, i.e. ∆L0 (x, y) = |L00 (x, y) − L0b |. Except for the object image, there is on the monitor screen a random additive clutter of the image characterized by the spectral density ˜ im (νx , νy ). Kr The HVS model presents an optimal ﬁlter receiver with noise and contains some linear parts with the appropriate transfer characteristics and a nonlinear part, VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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Fig. 1. Model of human visual system

which takes into account a nonlinear (as a rule, logarithmic) dependence of subjective (perceptible) brightness upon the real one. The inherent noise of HVS is taken into account by introduction of noise generators (photon noise, etc.) in the model. As a result, the sufﬁciently simple HVS model (Fig. 1) is synthesized that consists of only three parts: noise generator 3, optimal ﬁlter 4 with transfer func˜ f (νx , νy ) and the threshold device 5, which threshold of coming into action tion H depends on the chosen criterion of detection. Within the framework of the accepted model an expression for the perceptible signal-to-noise ratio will be as follows [2]: 1

Z+∞ Z

¯ ¯2 ¯ ˜0 ¯ ¯∆L (νx , νy )¯

µp∆ = K1 K2

2

dνx dνy ,

(2)

˜ im (νx , νy ) + Kr ˜ 0 Kr HV S (νx , νy ) −∞

˜ 0 (νx , νy ) is SFS of the image signal component on the monitor screen; where ∆L ˜ 0 ˜ = Krim (νx , νy ) is a spectral density of image noise; Kr HV S (νx , νy ) ˜ KrHV S (νx , νy ) = ¯ ¯2 is a spectral density of HVS noise, reduced to the monitor ¯ ¯˜ ¯Heye (νx , νy )¯ ˜ eye (νx , νy ) is an eye transfer function; K1 is a factor which takes account screen; H of the non-ideality of HVS, it is accepted equal to 0.6 on the basis of experimental data [4]; K2 is a factor which takes into account the ability of HVS for integrating the image over time [5]; νx , νy are spatial frequencies in two mutually perpendicular directions. To calculate the K2 value as applied to LIS with television reception at frame sequence frequency fs > 20, we can use the relation K2 = (fs τeye )1/2 , where τeye ≈ 0.1 is an “eye lag” (persistence of vision). The advantage of the offered model is that the spectral density of HVS noise, reduced to the monitor screen, has the same dimensions as that of image noise [cd2 ·m−4 ·rad2 ]. SFS of the image signal component is determined by the expression: ˜ 0 (νx , νy ) = ∆˜ ˜ LIS (νx , νy ), ∆L ρ(νx , νy )H

(3)

where ∆˜ ρ(νx , νy ) is a spectral density of the function ∆ρ(x, y), describing a devia˜ LIS (νx , νy ) is a transfer function of tion of reﬂection factor ρ(x, y) from average; H 90

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LIS. Believing clutters in imaging system to be additive, we can present a spectral density of the total image noise as ¯ ¯2 X ˜ i−e (νx , νy )¯¯ , ˜ im (νx , νy ) = ˜ i (νx , νy ) · ¯¯H (4) Kr Kr i

˜ i (νx , νy ) is a spectral density of noise created by i-th noise source; where Kr ˜ i−e (νx , νy ) is a transfer function of an optical-electronic path from the i-th noise H source to the monitor screen. In what follows we believe that the operator works under normal illumination condition, i.e. the brightness of adaptation is approximately equal to average of the background brightness and the image contrast can be adjusted. As the capability of low-contrast objects detection is limited by HVS noise, it is possible to assume, that this noise will not exert an appreciable effect on the contrast image identiﬁcation. It means that probability of detection of an object with using its image of sufﬁcient contrast is limited basically by image noise. Thus, when the condition ˜ im (νx , νy ) > Kr ˜ 0 Kr HV S (νx , νy ) is satisﬁed, the contrast may be increased only to a certain level, and the further increase becomes senseless and even harmful because the detection probability is decreased due to nonlinear transformations of the HVS signal component. According to the above-stated, the Eq. (2) can be put in the following form: 1 ¯2 ¯ 2 Z+∞ Z ¯¯∆L ˜ 0 (νx , νy )¯¯ . dν dν µp∆ = K1 K2 ˜ im (νx , νy ) x y Kr

(5)

−∞

Though the total noise components can have various distribution laws, the probability density of the perceptible brightness is close to the Gaussian function. It is caused by the HVS ability to integrate the image over space and time. As a result, the perceptible brightness is formed by summation of multitude of independent random values — brightness determined in various points of the image area and at various moments of time. According to the central limiting theorem, a probability density of the sum of a large number of such independent random values tends to the Gaussian function. Hence, the probability of the bar pattern resolution Pres , which is equal to the probability that the perceptible brightness will exceed the threshold value, is determined as ³ ´i 1h pt 1 + Φ µp∆ − µ∆ , (6) Pres = 2 r Zz µ 2¶ 2 x where Φ (z) = exp − dx is an integral of probabilities. π 2 0

The value µpt ∆ is determined by an acting rule of decision making and generally can change depending on a PSNR value, on a priori available information about the object, and on a level of training and experience of the operator. If a priori VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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probabilities of object occurrence and remunerations for the correct and erroneous decisions are not given, and the signal level is not known, then the decisionmaking rule is determined by the Neumann–Pearson criterion, and the threshold signal-to-noise ratio does not depend on a PSNR value. The dependence of object detection probability upon a PSNR value is shown in Fig. 2 on a perFig. 2. Dependence of bar pattern resolution unit basis for µpt = 2.5. ∆ probability upon PSNR Thus, knowing transfer functions of separate parts and the system as a whole, and also the noise spectral density of individual sources, we can calculate the image SFS and the image noise spectral density. However this calculation is rather labor-consuming and implies a knowledge of all characteristics of the system by the developer, which is not always possible. Therefore the calculation of PSNR is made with certain assumptions and approximations, resulting in signiﬁcant mistakes for a number of cases. In this connection the development of a technique for the experimental estimation of LIS acquisition characteristics and PSNR values represents the scientiﬁc and practical interest. Experimental Research. For experimental check of the above-described computational relations the prototype of LIS with a semiconductor source has been developed in the Bauman Moscow State Technical University (Fig. 3). The transmitting channel generates illuminating pulses with duration 100–200 nsec and repetition frequency up to 2 kHz. The image of object generated by receiving lens is ampliﬁed by photoreceiving device and then transferred by reproduction lens into the plane of sensitive elements of the CCD-camera. An output video signal of the CCD-camera is processed in the electronic unit of the videocontrol device and comes to the monitor. For preliminary pointing at an object under conditions of sufﬁcient natural light the optical sight is applied which is set co-axially with transmitting and receiving channels on a mobile platform. The latter can turn through some azimuth and elevation angle. To maintain LIS high acquisition characteristics and extend its functions the third-generation MPII is applied in the receiving channel. It operates in a pulse mode and carries out the gating of the signal coming at its input. The usage of gating allows to ﬁlter off effectively a back scattering clutter and background component. Besides, in the passive operational mode the gating by the “photocathode — microchannel plate” interval makes possible to limit an average current of the photo-cathode and to avoid its premature degradation. In the LIS transmitting channel the semiconductor laser source IDLP-100M-810E is applied, having a radiation wave length of 0.81 µm and pulse power of 70 W.

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Fig. 3. Structural and functional scheme of LIS experimental sample

The laser source luminescence body represents a lattice of elementary radiating p−n-junctions. The radiating source is located close to the lens focus, so that defocused images of elementary p−n-junctions are partially blocked from falling on object image. Thus the object appears to be lighted practically uniformly. The selection of distances between components of the lens and radiating source allows for the change of the target radiation angular divergence within a range of 0.600 × × 0.400 to 200 × 1.400 . The pumping of laser source is carried out by the power supply containing the initial generator G1, the shaper of pulses of the given duration G2 and the gate, through which a discharge of the integrating capacitor C to radiator D is provided by a ﬁxed value current. The univibrator G3 forms pulses synchronized with current pulses through radiating source (and, hence, and with pulses of radiation) and required for operation of the photodetecting device of the receiving channel. VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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The basic purpose of the LIS experimental research was to check the modulation transfer function (MTF), spectral density of a clutter in the image, PSNR, acquisition characteristics and to compare the measured values with calculated ones. During the LIS ﬁeld tests the generated image of the test-object — set of ten bar patterns Fig. 4. Image of test-object with the period from 16 to 160 mm (denoted in ﬁgures as No. 1, No. 2,. . . , No. 10) — was recorded using the standard VHS tape-recorder. Then separate pictures were put into the personal computer as bmp-ﬁles. ¯ pictures is shown in Fig. 4. ¯ One of the ¯ ¯˜ To obtain the experimental MTF ¯H (νx , νy )¯, the contrast k of the image of each bar pattern is determined as k = (Lmax − Lmin )/(Lmax + Lmin ), where Lmax and Lmin are maximal and minimal brightness of the bar pattern image respectively. The system MTF with frequency νx0 can be determined by the known Koltman formula [7] ¸ · ¯ ¯ π k(3νx0 ) k(5νx0 ) k(7νx0 ) ¯ ¯˜ − + + ... , k(νx0 ) + ¯H (νx0 , 0)¯ = 4kp 3 5 7

(7)

where kp is a contrast of the bar pattern; k(νx0 ) is a depth of modulation of the bar pattern image with spatial frequency νx0 . The calculated MTF of the LIS prototype and its elements as well as experimentally determined MTF values and conﬁdential intervals for the 90 % probability for several angular spatial frequencies are given in Fig. 5. The calculated curves are drawn according to the developed mathematical LIS model. From this ﬁgure we can notice a good agreement of the calculated MTF and experimentally determined values. Small discrepancies of the calculated and measured values within the frequency range of 4 . . . 6 mrad−1 can be explained by the fact that in this area the divergence between the real MTF of MPII and its approximation function, used in calculations, is observed. Spectral noise density is the second major LIS characteristic inﬂuencing the PSNR. It was determined by statistical processing of brightness for the image area, containing no bar patterns. This made possible to consider a clutter as an ergodic one and, according to the Wiener–Khinchin theorem, to ﬁnd its Fourier transformation of the correlation function Kr. For each of ten initial images of bar patterns ˜ was generated as arithmetic mean of arrays Kri . After Fourier transthe array Kr ˜ exp (νx , 0) of spectral density of image formation the one-dimensional function Kr noise was found on the basis of experimental data. It should be noted, that the automatic brightness adjustment system, available in the CCD-camera, makes impossible to measure absolute values of spectral densities. Therefore the measured and calculated spectral densities were normalized on the basis of the average image brightness square. The normalized measured and 94

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Fig. 5. Results of MTF measurements for LIS prototype: calculated curves (1 — MTF of atmosphere, 2 — MTF of receiving lens, 3 — MTF of MPII, 4 — MTF of reproduction lens, 5 — MTF of staring arrays, 6 — MTF of LIS prototype) with experimental estimates by bar pattern images marked with dots (with corresponding numbers) and vertical line segments which denote conﬁdential intervals of MTF estimates

calculated spectral densities of image clutters are given in Fig. 6. The good concurrence of the calculated and experimental curves is seen. Distinctions between them in the frequency region of 0 . . . 3 mrad−1 can be explained by presence of the transmitting channel clutter which was not taken into account during calculations. This clutter arises as a result of re-superposition of defocused images of p−n-junctions of the laser luminescence body, which spatial frequency is equal to 1.4 mrad−1 . The recession of the experimental curve of noise spectral density at frequencies above 9 mrad−1 can be explained by suppression of high-frequency components in the electronic path of the video-recorder (9 mrad−1 corresponds to 280 TV lines, which is a resolution limit for VHS video-recorders). The experimental value of PSNR was calculated by the approximated formula derived from Eq. (2): r (Lmax − Lmin )3.5Tp π p q µ∆ ≈ K1 , (8) 4 ˜ (νx0 , 0) 2 Kr ˜ where Tp is a period of the bar pattern image; Kr(ν x0 , 0) is a spectral density of image noise at the bar pattern frequency. Experimental values of PSNR measured in the image of each bar pattern and also the dependence of PSNR on spatial frequency of bar pattern, calculated in accordance with the above-mentioned technique, are given in Fig. 7, where a good concurrence of the calculated curve and experimental data, denoted by dots, can be seen. The deviation of some dots from the calculated curve is explained by nonuniformity of the test-object illumination and by difference between the real MTF of MPII and that of approximating function. Images of the test bar patterns reproduced by LIS on the monitor screen, allow for experimental estimates of the system acquisition characteristic — dependence VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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Fig. 6. Spectral density of image noises: 1 — calculated shot noise of MPII; 2 — calculated internal noise of CCD-camera; 3 — calculated total noise; 4 — total noise, determined experimentally

Fig. 7. Dependence of PSNR upon the bar pattern frequency

of the object detection probability upon PSNR. Ten images of bar patterns similar to those shown in Fig. 2 were demonstrated to ﬁve observers, and the cases were ﬁxed when the bar patterns were recognized. By results of the experiment (Fig. 8) the estimates of resolution probability for each bar pattern and conﬁdential intervals for the 90 % probability are found. The dependence of the bar pattern resolution probability upon PSNR, calculated according to Eq. (6), is also shown in Fig. 8. The PSNR threshold is accepted equal to 2.8 on a condition of the best conformity of the calculated curve to the experimental data. The obtained value µpt ∆ agrees well with the data of other researches. Let us note, that during the prototype experimental research the high efﬁciency was conﬁrmed when using the receiving channel gating for ﬁltration of a back scattering clutter and for the range selection of objects. The limiting range of LIS vision was estimated: for identiﬁcation of a car it was 800 m (Fig. 9), for its detection — 1200 m. Thus, the results of experimental research of LIS with semiconductor pulse radiating source and gated receiving channel on the basis of the third-generation

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Fig. 8. Dependence of probability of the bar pattern resolution upon PSNR

Fig. 9. The image of the car formed by LIS at a distance of 800 m

MPII and CCD-camera have shown, that these systems, at least, compare well with night vision devices, having active illumination, by efﬁciency, functionality and quality of formed images. The correctness of the offered simulation insight and computational relations is conﬁrmed experimentally with acceptable reliability. Acknowledgements. The author is grateful to the staff of the Bauman Moscow State Technical University, V. Popov and Ye. Mukhina, for their help in realization of experiments and preparation of this material for publication.

REFERENCES 1. Holst, Ger. “CCD Arrays, Cameras, and Displays”. JCD Publishing and SPIE Optical Engineering Press. Bellingham, WA, 1996. 2. Gusev, M.A., Karasik, V.E., Shestov, S.N. Investigation of acquisition characteristics of laser imaging systems (in Russian). Vestnik MGTU. Ser. Priborostroyenie. No 3, 1995, pp. 27–35, BMSTU Press, Moscow. 3. Karasik, V.E., Orlov V.M. Laser Imaging Systems (in Russian). Moscow. Izd-vo MGTU imeni N/E/ Baumana, 2001. P. 352. 4. Pavlov, N.I., Voronin, Yu. M. Probability of objects detection on the monitor screen in optical-electronic system, of supervision (in Russian). Optical magazine. 1994. No 7, pp. 3–11, Moscow. 5. Vaﬁadi, A.V. Analytical models of scanning thermo-vision devices (in Russian). Moscow. Optical magazine. 1997. No 1, pp. 32–36. 6. Mosyagin, G.M., Nemtinov, V.B., Lebedev, E.N. Theory of optical-electronic systems (in Russian). Moscow. Mashinostroenie, 1990. P. 432. 7. Kreopalova, G.V., Lazareva, N.L., Puryaev, D.T. Optical measurements (in Russian)/ Edited by D.T. Puryaev. Moscow. Mashinostroenie, 1987. P. 266. V.Ye. Karasik (b. 1939) graduated from the Bauman Moscow Higher Technical School in 1964. D. Sc. (Eng.), professor of “Laser & Optical-and-Electronic Systems” department of the Bauman Moscow State Technical University. Author of over 160 publications in the ﬁeld of the laser location and laser imaging.

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NAVIGATIONAL & GYROSCOPIC SYSTEMS V.A. Matveev and M.A. Basarab (Bauman Moscow State Technical University)

NUMERICAL MODELING OF HEAT DIFFUSION PROCESSES IN THE SOLID-STATE WAVE GYRO RESONATOR BY THE R-FUNCTION METHOD Stationary and non-stationary heat-transfer processes in a solid-state wave gyro are studied. The proposed numerical method is based on the theory of R-functions.

Introduction. The solid-state wave gyro (SWG) [1–3] belongs to the class of non-traditional gyro devices and is used as a sensitive element in navigational systems. One of the sources of SWG errors is a heat component of its drift [2, 4]. To identify its parameters, one must evaluate stationary and non-stationary thermal ﬁelds in the SWG resonator. The work is devoted to the use of the numericalanalytical method of R-functions (RFM) [5, 6] for solving the aforementioned problem. Mathematical Model of Heat-Transfer Processes in the SWG Resonator. The sensitive element of a real SWG is a hemisphere thin-shelled resonator made of melted quartz glass ensuring high isotropy of physical-mechanical properties and a small linear expansion coefﬁcient. Internal and external surfaces of the resonator are covered with a thin conducting ﬁlm and are not connected with each other. The resonator is ﬁxed in the device body with the help of the cylindrical leg. Due to the symmetry, we shall consider only the part of this system shown in Fig. 1, a. To identify parameters of the SWG drift rate model, the following model of an output signal was proposed [2]: ϑ˙ = −KΩ + Ω0 + Ωm sin 4(ϑ − ϕ0 ) + ΩT + W,

(1)

where ϑ = ϑ(t) is the standing wave rotation angle; K ≈ 0.3 is the SWG scaling factor; Ω is the angular velocity of SWG rotation around the symmetry axis; Ω0 is a constant component; Ωm is the amplitude, ϕ0 is an angle of orientation of the maximum quality factor axis; ΩT is a heat component of the drift rate, and W is a random component (white noise) of the drift rate. The constant parameters Ω0 , Ωm , ϕ0 , and the component ΩT are unknown and must be identiﬁed. The time constant ˙ T = −ΩT /T is taken in the form of a set of ﬁxed values T of the heat component Ω 0 T1 , T2 , . . . , TN . Its best approximation is determined on the set {Ti }N i=1 as a result of the most adequate identiﬁcation reconstructing the original realization ϑ = ϑ(t). The drift rate model (1) is not complete and may be considered as the basic one for an output signal and, after the stepwise complication, can be represented in the form providing high accuracy of the SWG characteristics determination.

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Fig. 1

SWG accuracy and its time of readiness are mostly deﬁned by heat exchange processes during heat radiation by internal sources and variations of external temperature. Inﬂuence of these factors yields the change of the SWG sensitive element temperature and, therefore, appearance of the thermal drift. To analyze the time of readiness and accuracy characteristics of the SWG, and to choose an appropriate thermal control system, one must know the temperature distribution in SWG elements. To investigate thermal processes in the SWG, it is necessary to — construct a mathematical model of heat diffusion processes in the SWG; — determine temperature gradients inside the SWG resonator; — develop the method for identiﬁcation of the heat model parameters with the use of experimental results; — choose a thermostat system. The main reason for appearance of the thermal drift is a shift of the SWG resonator center-of-mass caused by nonuniform linear deformations of its elements due to the nonzero temperature gradient. This drift induces radial and axial components of reactive forces in resonator supports during its main form of oscillations A part of the resonator energy is transferred to its support through the connecting leg and dissipates due to the damping effect. Damping in the support reduces the resonator quality factor Q and causes its dependence on the standing wave orientation. The latter effect yields the SWG drift proportional to the value −1 (Q−1 min − Qmax ), where Qmin and Qmax are the minimum and maximum values of the resonator quality factor. As in the dynamically adjusted gyros [4], one can distinguish the following main ways for decreasing the resonator center-of-mass shift caused by the temperature gradient: — decreasing the heat ﬂow from the device body to the sensitive element; VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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— increasing the resonator thermal conductivity; — decreasing the resonator linear expansion coefﬁcient. The problem of ensuring the mass center stability depends on the stability of holding up deep vacuum in the device cavity. Consider the following simpliﬁed model of heat diffusion processes in the SWG resonator (see Fig. 1, a). Due to the fact that the resonator is under vacuum, we shall not consider the convective heat exchange. We shall neglect also the radiation from the sensors. In places of the cylindrical leg ﬁxing, we assume the temperature is given (thermostatic conditions). Thus, the stationary temperature ﬁeld T (r, z) in the resonator must satisfy the following two-dimensional Laplace equation in cylindrical coordinates r, z [7, 8]: ∂ 2T 1 ∂T ∂2T + = 0; + ∂r2 r ∂r ∂z 2

(2)

the boundary conditions are T (R1 , z) = T1

at

z ∈ [0, d1 ],

∂T /∂n = 0

T (R1 , z) = T2

at

z ∈ [l − d2 , l],

on the remaining part of the boundary.

(3) (4)

In Eq. (4), n denotes the external normal to the domain boundary. Heat diffusion boundary-value problem (Eqs. (2)–(4)) can be solved by one of numerical or numerical-analytical methods [7–15]. In this work we shall consider an approach based on the use of the RFM. The R-function Method and General Solution Structures. The R-function method allows all prescribed boundary conditions to be satisﬁed exactly at all boundary points. The original idea underlying the RFM is due to Kantorovich [9]. He proposed that the homogeneous Dirichlet conditions may be satisﬁed exactly by representing the solution as the product of two functions: (1) a real-valued function that takes on zero values at the boundary points; and (2) an unknown function that allows one to satisfy (exactly or approximately) the differential equation of the problem. The idea appeared to have a limited use because it was not clear how such functions could be constructed for complex shapes and because the method did not seem to be generalized to other types of boundary value problems. Rvachev [5, 6] suggested that both of these obstacles may be overcome using R-functions — the real-valued functions that behave as continuous analogs of logical Boolean functions. With R-functions, it became possible to construct functions with prescribed values and derivatives at speciﬁed locations. Furthermore, the constructed functions possess desired differential properties and may be assembled into a solution structure that is guaranteed to contain solutions to the posed boundary value problems. A function z = f (x), x = (x1 , . . . , xN ), is called an R-function if its sign is completely determined by the signs (but not magnitudes) of its arguments. In this work, we shall consider the two-dimensional case (N = 2) when z = f (x, y).

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The most popular system of R-functions is the system > 0. If f is an R-function corresponding to the Boolean function F , then the implicit function for the resulting geometric domain is immediately given by Ω=(f (ω1 , . . . , ωn )>0). The function f (ω1 , . . . , ωn ) is negative outside Ω and the equation f (ω1 , . . . , ωn ) = 0 deﬁnes the boundary ∂Ω of the region Ω. It is known that the equation of the boundary ∂Ω = (f = 0) is called normal if the value of f (p) is equal to the Euclidean distance from the point p to the boundary ∂Ω. Similarly a function f that coincides with the normal function only on the boundary ∂Ω is called normalized and has a property that ¯ ∂f ¯¯ = −1, (7) ∂n ¯∂Ω where n is a vector of the external normal to ∂Ω. A normalized equation ω ˜ (x, y) = 0 of the boundary can be obtained from an ordinary one (ω(x, y) = 0) in the following way:

ω ˜ (x, y) =

ω(x, y) . |∂ω/∂n|∂Ω

(8)

To avoid indeterminacy due to division by zero, another normalizing formula can be used instead of Eq. (8): ω(x, y) ω ˜ (x, y) = q

. ω2

+

(9)

|∂ω/∂n|2∂Ω

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If every primitive implicit function ωi is normalized at the primitive boundaries ∂Ωi , then all of the above R-functions preserve this property, and the function f (ω1 , . . . , ωn ) is normalized at the boundary ∂Ω. Let us ﬁnd the solution to the operator equation Au = f

(10)

inside a bounded domain Ω ⊂ R2 with the given boundary conditions Lu|∂Ω = ϕ

(11)

on the boundary ∂Ω. The general solution structure (GSS) of a boundary-value problem is determined by the expression (12) u = B(Φ, ω, ωi , f, ϕ) satisfying the boundary conditions exactly under an arbitrary choice of the undetermined component Φ. Here, B is the operator depending on the geometry of the domain Ω and parts of its boundary ∂Ωi . The undetermined component Φ of the GSS in Eq. (12) is usually expressed by the series M X cn ψ n , (13) Φ= n=1

where ψn (x, y) are elements of a full system of basic functions (algebraic and trigonometric polynomials, splines, etc.), and cn are unknown coefﬁcients found with the help of one of variational or projection methods [9–12]. Some frequently encountered types of boundary conditions and GSSs of corresponding boundary-value problems are as follows. (i) The Dirichlet condition u|∂Ω = ϕ (14) is satisﬁed by the GSS u = ωΦ + ϕ.

(15)

∂u/∂n|∂Ω = ϕ

(16)

u = (1 − ωD)Φ − ωϕ,

(17)

(ii) The Neumann condition

is satisﬁed by the GSS

where µ D≡

¶ ∂ω ∂ ∂ω ∂ + ∂x ∂x ∂y ∂y

;

D|∂Ω

¯ ∂ ¯¯ = . ∂n ¯∂Ω

(18)

(iii) The GSS for the 3rd kind boundary condition (∂u/∂n + hu)|∂Ω = ϕ 102

(19)

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is u = [1 + ω(h − D)] Φ − ωϕ.

(20)

(iv) The GSS ¸ · ω1 ω2 ω1 ω2 (2) (h − D ) (ω1 Φ + ϕ1 ) − ϕ2 , u= 1+ ω1 + ω2 ω1 + ω2 where µ (2)

D

≡

¶ ∂ω2 ∂ ∂ω2 ∂ + ∂x ∂x ∂y ∂y

(2)

;

D

|∂Ω2

¯ ∂ ¯¯ = , ∂n ¯∂Ω2

(21)

(22)

corresponds to the mixed boundary conditions ( u|∂Ω1 = ϕ1 , (23) (∂u/∂n + hu)|∂Ω2 = ϕ2 . It should be noted that the functions ω(x, y) and ω2 (x, y) in GSSs for conditions of a differential type (ii)–(iv) are normalized. Besides, an arbitrary combination of different conditions on boundary parts ∂Ωk is possible. Here, the “glue” operation for boundary conditions must be used. Let, for example, u(x, y)|∂Ωk = ϕk (x, y), k = 1, . . . , K.

(24)

Then the GSS is represented as , K X

K X

ωk−1 ϕk

u = ωΦ + k=1

ωk−1 ,

(25)

k=1

where ωk are the equations of boundary parts ∂Ωk , such that ωk > 0 in (Ω ∪ ∂Ω) ∩ ∂Ωk . The latter expression is called the generalized Lagrange formula. In a general case, only a part of boundary conditions may be satisﬁed by GSSs, or the structures may obey known conditions on given manifolds inside the domain. The Rote Method for Solving Non-stationary Problems. Consider the parabolic equation of heat transfer ∆T =

1 ∂T − f λ−1 . κ ∂t

(26)

Here, t is time, κ = λ/(cρ) is the temperature conductance [m2 /s], λ is the thermal conductivity [W/(m × K)], c is the speciﬁc heat [J /(kg × K)], and ρ is the volume density [kg /m3 ]. Besides for Eq. (26), the initial temperature distribution must be given: (27) T |t=0 = ψ. The variational statement of boundary-value problems for Eq. (26) and methods of their solution are based on the combined use of the Rote method [16–18] and VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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variational principle for elliptic-type problems [11]. Here, the time digitization of nonstationary boundary-value problem (26) is carried out and then the obtained stationary problems are solved by variational methods, i.e. the parabolic-type problem is reduced to a set of elliptic-type problems. Consider the boundary-value problems for Eq. (26) in the domain Ω at t ∈ (0, T ]. Introduce the partition {∆pj }, j = 1, 2, . . . , p, of the time interval [0, T ] as p X ∆tpj = T, ∆tpj = tpj − tpj−1 . (1) j=1

Then replace the time derivative in Eq. (26) by the approximate difference relation ∂T (x, y, tpj )

T (x, y, tpj ) − T (x, y, tpj−1 ) ≈

.

(2)

∆tpj

∂t

Thus, the original problem is reduced to the set of the following problems: p

∆T (x, y, tpj ) −

p

1 T (x, y, tj ) 1 T (x, y, tj−1 ) =− − f λ−1 p κ κ ∆tj ∆tpj

(29)

which must be solved sequentially. At j = 1, initial condition (27) must be used. In [17] it is proven that a limit of the solutions sequence exists. This limit is called the solution of the non-stationary problem, coinciding with the classical solution if the latter exists. Equation (29) approximates Eq. (26) with an error of order of max ∆tpj according to the use of approximate expression (28). Other schemes of j

the Rote method with higher orders of accuracy with respect to the time variable also exist [16]. The Lumped Capacity Condition. A thin coating of a body may be considered as the lumped capacity adjoining to the body [19]. Assuming the uniform temperature distribution in a thin subsurface layer and integrating both parts of the heat transfer equation along the layer depth (averaging), we can pass from the original complicated model to the model with lumped capacity. From the mathematical point of view, the time derivative in the boundary conditions is characteristic of the latter model. Such assumption allows one to simplify essentially solutions of heat exchange problems in compound bodies with sharp contrast of geometrical and thermal-physical characteristics. However, unlike the heat transfer boundary-value problems with conventional types of boundary conditions, the well-posedness of the problem with lumped capacity conditions depends on a ratio between coefﬁcients at derivatives in boundary conditions. At deﬁnite values of this ratio, the stated problem becomes conditionally well-posed or well-posed by Tychonoff. Since a resonator surface is covered with a thin metal layer, then on the part of the boundary, corresponding to the hemisphere, the lumped capacity boundary condition must be given [19] as

k 104

∂T ∂T =C , ∂ν ∂t

(30)

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where k is the heat conduction; C = cρS is the lumped capacity; ρS = ρm hm is the surface density of the layer material [kg /m2 ]; ρm and hm are the volume density and thickness of the layer. In practice, the lumped capacity as well as mechanical factors is one of the reasons for sharp decrease of the SWG resonator quality factor. ∂T → 0 as t → ∞, and in the stationary case, the condition (30) is Obviously, ∂t reduced to the ordinary Neumann boundary condition (4). Evaluation of Stationary and Non-stationary Thermal Fields in the SWG resonator. According to the R-function method, let us write the general solution structure of the problem (Eqs. (2)–(4)) as

u = ωI P −

ω (1) T2 + ωI(2) T1 ωI ωII (II) D1 (ωI P ) + I (1) − ωI + ωII ωI + ωI(2) −

(1) (2) ωI ωII (II) ωI T2 + ωI T1 D1 , (31) ωI + ωII ωI(1) + ωI(2)

where u is the approximate temperature distribution, P is the unknown component; ωI is the equation of the boundary part with the Dirichlet conditions,

ωI(1)

ωI = ωI(1) ∧0 ωI(2) , µq ¶ 2 2 = (R1 − r) ∨0 −circle R1 + d1 , r, z , ωII(1) = (R1 − r) ∨0 (l − d2 − z),

∂ωII ∂ 1 ∂ωII ∂ (II) (R2 − x2 − y 2 ), D1 ≡ + 2R ∂r ∂r ∂z ∂r and ωII is the equation of the boundary part with the Neumann conditions (see Fig. 1, b), ωII = ωII(1) ∧0 (ωII(2) ∨0 ωII(3) ), circle(R, x, y) ≡

¢ 1¡ (l/2)2 − (z − l/2)2 , l (2) ωII = −circle(R2 − h/2, r, z) ∧0 circle(R2 + h/2, r, z), µq ¶ (3) 2 2 ωII = [(R1 − r) ∨0 (z − l + d2 )] ∨0 circle R1 + d1 , r, z . ωII(1) = r ∧0

The boundary-value problem is solved by the variational-difference method [5]. Instead of the differential operator in the left-hand side of Eq. (2), its ﬁnitedifference approximation of the second order on the uniform mesh is considered [15] · µ ¶ µ ¶¸ 1 1 1 r r + − u(r + ∆r, z) + u(r − ∆r, z) + r∆r ∆r 2 ∆r 2 (32) µ ¶ ¸ · 1 1 u(r, z + ∆z) + u(r, z − ∆z) −2 + u(r, z) = 0. + ∆z 2 ∆r 2 ∆z 2 VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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Approximation of derivatives in the expression for the differential operator is also realized by means of ﬁnite-difference relations. To increase the rate of convergence, the undetermined component P is represented by the following series with respect to trigonometric basic functions, a priori satisfying boundary conditions at deﬁnite parts of the boundary:

(II) D1

µ

¶

M X

Cq,s cos

P =

πqr R2 + h/2

³ πsz ´ .

cos

(33)

l

q+s=0

Substituting the structure Eq. (31) in Eq. (32), we obtain the system of linear algebraic equations with respect to unknown coefﬁcients Cq,s of expansion (33). Let us calculate the temperature distribution for the resonator with parameters R1 = 5 mm, R2 = 25 mm, h = 2.5 mm, l = 50 mm, d1 = d2 = 0.1l, T1 − T0 = = 0.5 K, T2 − T0 = 1 K. Digitization is performed on the mesh ri = i

R2 + h/2 (i = 0, 1, . . . , 25), 25

zj = j

l (j = 0, 1, . . . , 50); 50

parameters of ﬁnite difference analogs of derivatives are ∆r = ∆z = 0.001(R2 + + h/2). In Eq. (33), M = 2 is chosen that corresponds to 6 terms of the expansion. Figure 2 illustrates contour and surface plots of the stationary temperature ﬁeld in the SWG resonator. It should be noted that, in spite of the small number of expansion terms, the solution obtained adequately describes the thermal ﬁeld distribution and, in particular, satisﬁes given boundary conditions (3)–(4) exactly. Non-stationary overheat may be evaluated with the help of the combined use of the R-function and Rote methods. Calculate the thermal time constant of the SWG resonator under the following conditions. Let at the initial moment t0 = 0 the whole resonator, uniformly heated

Fig. 2

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to the temperature T0 , be placed into the medium with the given temperature Tc (t). On the boundary we have the non-stationary 3rd kind boundary condition λ

∂u + α(u − Tc ) = 0, ∂ν

(34)

where α is the thermal exchange coefﬁcient [W/(m2 · K)]. An approximate solution can be written in the form of the structure for the 3rd kind boundary condition u = −ωϕ0 + Φ − ω (D1 Φ + gΦ) , where g = α/λ, ϕ0 = αTc /λ, and µ ω=

[−circle(R2 − h/2, r, z) ∧0 circle(R2 + ¶ R21 − r2 (l/2)2 − (z − l/2)2 +h/2)] ∨0 . (35) ∧0 2R1 l Contour lines of the function (35) are shown in Fig. 3. The solution of the stated problem can be found by the Rote method according to Eq. (29). Here, we neglect the lumped capacity boundary condition (30). Consider another numerical technique for evaluation of non-stationary heat transfer processes in the SWG, based on averaged parameters [13]. By V and S denote the volume and surface area of the resonator, respectively. By deﬁnition, the average volume temperature of a body is Z 1 udV . u ¯= V

Fig. 3

V

Suppose the thermal parameters of the resonator are time-independent. Then, using the Green formula and boundary condition (34), we obtain the equation d¯ u + m¯ u = mTc (t), dt

(36)

αS . Taking into account the initial condition, the general integral of cρV Eq. (36) takes the form where m =

Zt −mt

u ¯ = T0 e

−mt

Tc (τ )emτ dτ .

+ me

(37)

0

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Consider the following three cases. 1. Tc = const. The relative overheat is (¯ u − Tc )/(T0 − Tc ) = e−mt ,

(38)

i.e. the difference between the ambient temperature and the temperature of the body varies according to the exponential law. 2. Tc = bt + Tc0 . The relative overheat is expressed as u ¯ − Tc = (T0 − Tc0 + b/m)e−mt − b/m.

(39)

As time elapses, the ﬁrst term in Eq. (39) becomes negligibly small in comparison with the second one and the difference between the temperatures tends to the constant value u ¯ − Tc = −b/m. 3. Tc = T¯c +A cos ωt, where T¯c is the average value of the ambient temperature, A and ω are the amplitude and frequency of oscillations. From Eq. (36) for the difference between the temperatures, we have u ¯ − Tc = [(T0 − Tc (0)) + A∗ sin β] e−mt + A∗ sin(ωt − β), √ where β = arctg(ω/m) and A∗ = Aω/ m2 + ω 2 = A sin β. For large time values, the regular regime is achieved when u ¯ − Tc = A∗ sin(ωt − β), or u ¯ − T¯c = A cos β cos(ωt − β). From the analysis of the aforementioned formulas, the important role played by the parameter m is seen in all typical cases. This parameter is called the cooling (heating) rate of a body and is determined either experimentally or numerically. The inverse value ε = 1/m is called the exponent of thermal inertia characterizing the rate of attainment of the thermal equilibrium. In fact, this exponent is the thermal time constant if the ambient temperature is constant and equals the time when the relative temperature increases (decreases) e times. In cases 2 (linearly changing ambient temperature) and 3 the thermal inertia is characterized by more complicated parameters bε and ωε, respectively. Another important parameter expressing inertial properties of a body is the time of the system stabilization t∗ corresponding to the moment when the difference between the ambient temperature and the temperature of the system becomes less than a deﬁnite value ∆ = u ¯ − Tc . For the constant ambient temperature ¯ ¯ ¯ T 0 − Tc ¯ ∗ ¯ ¯. t = ε ln ¯ ∆ ¯ 108

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

To calculate thermal parameters of the SWG resonator, we assume λ = = 1.5 W /(m × K), = 900 J /(kg × K), ρ = 2.2 kg / m3 . It is easily seen that the area of the resonator surface area is equal to

h S = 2π R12 + R1 (l − δ) + R2 h+ i + 2(R22 + h2 /4) − (R2 + h/2)g1 − (R2 − h/2)g2 , and its volume is ·q V = πR12 (l − δ) +

π 6

¡ ¢ (R2 + h/2)2 − R21 4(R2 + h/2)2 + 2R21 − ¸ q ¡ −

(R2 −

h/2)2

−

R12

¢ 2

4(R2 − h/2) +

2R12

.

Here, q g1 = R2 + h/2 −

(R2 + h/2)2 − R12 ,

q g2 = R2 − h/2 − (R2 − h/2)2 − R12 , q q δ = (R2 + h/2)2 − R12 − (R2 − h/2)2 − R21 . Suppose α = 1 W /(m2 K), Tc − T0 = 1 K, and ∆ = 0.1 K. Then, for the aforementioned geometrical parameters we get m = 3.65 · 10−4 s−1 , ε = 2.74 · 103 s, t∗ = 6.31 · 103 s ≈ 1 hour and 45 minutes. Now, we shall restrict ourselves to the case when heat exchange occurs only on the side of the resonator leg, i.e., z ∈ [0, d1 ] ∪ [l − d2 , l], r = R1 . On the other parts of the surface we assume α = 0 in Eq. (34) (thermal insulation conditions). Instead of S in the aforementioned expressions we must take S1 = 2πR1 (d1 + d2 ). In this case, the following values of thermal parameters are obtained: m1 = = 1.18 · 10−5 s−1 , ε = 8.51 · 104 s, t∗ = 1.96 · 105 s ≈ 54 hours and 24 minutes, i.e. due to decrease of the heat exchange surface, the thermal time constant of the SWG resonator is increased almost 30 times. Conclusion. Results of numerical experiments conﬁrm efﬁciency of the R-function method for solving heat-transfer problems in the SWG resonator. The use of solution structures allows one to take a small number of expansion terms to achieve the appropriate accuracy of the approximation solution satisfying the given boundary conditions exactly. Application of the R-function method gives the possibility to identify main thermal characteristics of the resonator, to take into account the thermal component in the drift model, and to improve the device accuracy. VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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REFERENCES 1. Zhuravlev, V.F. & Klimov, D.M. The Solid-State Wave Gyro (in Russian), Moscow: Nauka, 1985. 2. Matveev, V.A., Lipatnikov, V.I., & Alekhin, A.V. Design of the Solid-State Wave Gyro (in Russian), Moscow: Izd-vo MGTU imeni N.E. Baumana, 1997. 3. Basarab, M.A., Kravchenko,V.F., Matveev, V.A., & Pustovoit, V.I. Atomic Functions in the Problem of Evaluating the Rayleigh Functions and the Precession Coefﬁcient for the Resonator of a Wave Solid-State Gyroscope, Doklady Physics, 2001, vol. 46, no. 2, pp. 113–118. 4. Pelpor, D.S., Matveev, V.A., & Arsenyev, V.D. Dynamically Adjusted Gyroscopes (in Russian), Moscow: Mashinostroenie, 1988. 5. Rvachev, V.L. & Sheiko, T.I. R-functions in Boundary Value Problems in Mechanics, Appl. Mech. Rev., 1995, vol. 48, no. 4, pp. 151–188. 6. Rvachev, V.L., Sheiko, T.I., Shapiro, V., & Uicker, J.J. Implicit Function Modeling of Solidiﬁcation in Metal Casting, ASME Transaction, Journal of Mechanical Design, 1997, vol. 119, no. 4, pp. 466–473. 7. Luikov, A.V. Heat-Conductivity Theory (in Russian), Moscow: Nauka, 1986. 8. Kartashov, E.M. Analytical Methods in the Theory of Heat Transfer in Solid State Bodies (in Russian), Moscow: Vysshaya Shkola, 2001. 9. Kantorovich, L.V. & Krylov, V.I. Approximate Methods of Higher Analysis, Noordhof, Groningen, 1958. 10. Mikhlin, S.G. Variational Methods in Mathematical Physics. New-York: MacMillan, 1964. 11. Mikhlin, S.G. The Numerical Performance of Variational Methods. Groningen: Noordhoff, 1971. 12. Marchuk, G.I. & Agoshkov, V.I. Introduction to Projective-Difference Methods (in Russian). Moscow: Nauka, 1981. (in Russian). 13. Dulnev, G.N. Heat- and Mass-Exchange in Radio Electronic Equipment (in Russian). Moscow: Vysshaya Shkola, 1984. 14. Kotlyar, Ya. M., Sovershennyi, V.D., & Strizhenov, D.S. Methods and Problems of Heat- and Mass-Exchange (in Russian). Moscow: Mashinostroenie, 1987. 15. Samarskii, A.A. The Theory of Finite Difference Schemes (in Russian). Moscow: Nauka, 1989. 16. Liskovets, O.A. The Rote Method (Survey). Differential Equations. 1965, vol. 4, no. 12, pp. 1662–1678. 17. Mosolov, P.P. Variational Methods in Non-stationary Problems (Parabolic Case), (in Russian). Izvestiya AN SSSR, ser. Mat., 1970, vol. 34, no. 2, pp. 425–457. 18. Kalinichenko, V.I., Koshchii, A.F., & Ropavka, A.I. On A Posteriori Evaluations of Boundary-Value Problems for Some Elliptic-Type Equations and Their Applications for the Nonstationary Case (in Russian). Preprint of the Siberia Branch of Russ. Acad. Sci., 1982, no. 96. 19. Erzhanov, R.Zh., Matsevityi, Yu.M., Sultangazin, U.M., & Sheryshev, V.P. Lumped Capacity in Problems of Thermal Physics and Microelectronics (in Russian). Kiev: Naukova Dumka, 1992. Matveev Valerii Aleksandrovich (born 1939) is a Dr. Sc. (Eng.), Professor, the Head of the Research and Education Unit “Information Technology and Control Systems” (NUK IU) of the Bauman MSTU. Scientiﬁc interests: gyroscopic systems, navigational complexes, control systems. Basarab Mikhail Alekseevich (born 1970) is a Cand. Sci. (Eng.), post-doctoral student of the Bauman MSTU. Scientiﬁc interests: applied mathematics, numerical methods. 110

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B.S. Konovalov, S.F. Konovalov, A.V. Kuleshov, D.V. Mayorov, V.P. Podtchezertsev, V.V. Fateev (Bauman Moscow State Technical University)

NEW TYPES OF VIBRATING GYRO FOR ROTATING CARRIER The paper presents the information of design and experimental study of prototypes of three types of vibrating rate sensors developed by BMSTU. The sensors are constructed using the design concept of a vibrating gyro (VG) which is to be used on a fast-rotating object for measuring angular velocity components ωξ , ωη of its turning around axes perpendicular to the axis of fast rotation. Being installed on the rotating carrier the sensitive element of VG gets an angular moment and becomes capable to respond to initiation of angular rate ωξ , ωη , by making oscillations, which amplitudes are proportional to ωξ , ωη . A prototype of the given sensor was developed by A.I. Suchkov more than 50 years ago, and now similar gyro sensors are in large scale production, but they show low accuracy characteristics. The complex of activities on creation of more precise VG has been conducted at BMSTU. The experimental batch of the VG and test equipment for their testing in conditions of batch production have been made.

Introduction. Two of the main directions of gyroscope technology evolution are simpliﬁcation and miniaturization of sensors designed for measuring parameters of spatial moving objects with a sufﬁcient accuracy. Vibrating gyroscopes are best to satisfy these requirements as they allow one to construct on their basis simple in design, reliable and cheap sensors. In particular, vibrating gyroscopes (VG) are very suitable for using on a rotating carrier. To solve mentioned above tasks of modern gyroscopic technology for building miniature, cheap and reliable sensors of primary information, the most simple and reliable among rotary vibrating gyroscopes is a single-channel, two-dimensional measuring device. Real possibilities of similar device construction have been proposed and proved in publications of A.I. Suchkov [1] and evolved by L.I. Brozgul [2], U.B. Vlasov [3], M.N. Lyuty [4] and others. The angular rate of the carrier along the longitudinal axis (typically from 10 to 25 rpm) can be used by such gyroscopes installed on the rotating carrier to obtain the angular momentum of the sensing element. This allows denying a rotary engine in the device, which substantially decreases dimensions, mass and value of the gyroscope, simpliﬁes its construction and manufacturing method and raises its reliability. But, together with the mentioned advantages, a large instability of a carrier angular rate along the longitudinal axis (10 . . . 20 rps) and its small value create large difﬁculties for developing the device and guaranteeing a given accuracy of its parameters. Sensor without feedback. The now known VG for a rotating carrier and new developed devices of this class represent a sensing element in a one-axis suspension, whose plane is perpendicular to the longitudinal axis of a rotating carrier VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

111

Fig. 1. To derivation of the equation of VG sensing element motion: ξηζ — absolute frame of reference; X1 Y1 Z1 — frame of reference, bound to carrier; XY Z — frame of reference, bound to gyroscope sensing element

(Fig. 1). When the carrier turns about the cross axis the sensing element is producing harmonic vibration in suspension support with a frequency equal to the carrier rotary rate and an amplitude proportional to angular rate about the cross axis. We can get the equation of motion of such gyroscope in a frame of reference, bound to the carrier, using dynamic non-generalized Euler’s equations (see Fig. 1): B1 Ω˙ Y − (C1 − A1 ) ΩX ΩZ = −Dα α˙ − Kα + MYEx , where OX — sensing axis, OY — sensor axis, OZ — rotation axis, A1 , B1 , C1 — inertia moments of the sensing element relatively to its three axis, Dα α˙ — damping moment, Dα — damping coefﬁcient, Kα — elastic moment, K — elastic coefﬁcient (angular stiffness of suspension), α — sensing element angular deﬂection, MYEx — disturbing (harmful) moment relatively to sensor axis, caused by the sensing element misbalance or some other reasons. Using the following equations for Ωx , Ωy , Ωz , deﬁned through the projections of the carrier angular speed on the axes of an absolute frame of reference Ωx ≈ Ωξ sin ϕ − Ωη cos ϕ − αϕ Ωy ≈ α + Ωξ cos ϕ + Ωη sin ϕ , Ωz ≈ ϕ˙ 0 where ϕ˙ 0 — an angular rate of the carrier rotation along the longitudinal axis, we derive the equation of motion of the sensing element: α ¨ + 2ξω0 α˙ + ω02 α =

H ˙ Ωsin (ϕ˙ 0 t − δ1 ) − Ωsin (ϕ˙ 0 t + δ2 ) , B1

q Ω2ξ + Ω2η is a projection of the carrier angular rate on the plane, q perpendicular to the carrier own rotation vector; Ω˙ = Ω˙ 2ξ + Ω˙ 2η is a projection

where Ω =

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VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

of the angular acceleration to the plane, perpendicular to the carrier own rotation vector, and phase shifts δ1 and δ2 are angles, deﬁning appropriate positions of the angular rate vector and angular acceleration vector in the sensing plane: Ã ! µ ¶ Ω˙ ξ Ωη δ1 = arctg ; , δ2 = arctg Ωξ Ω˙ η s ω0 =

ξ=

(C1 − A1 ) ϕ˙ 20 + K ; B1

Dα Dα = q ¡ ¢; 2B1 ω0 2 (C 2 B1 ˙0 + K 1 − A1 ) ϕ H0 = (C1 − A1 + B1 ) ϕ˙ 0 .

The derived equation of the sensing element motion and its solution are common for all VGs of this type, used on a rotating carrier. Angular deﬂection of the gyroscope sensing element in the presence of a constant measurable angular rate is deﬁned by the following equation:

α = q¡

(C1 − A1 + B1 ) ϕ˙ 0 × ¢2 2 2 (C1 − A1 − B1 ) ϕ˙ 0 + K + (Dα ϕ˙ 0 ) µ µ ¶¶ Dα ϕ˙ 0 × Ω sin ϕ˙ 0 t − arctg . (C1 − A1 − B1 ) ϕ˙ 0 + K

Such devices with different construction shemes can operate in either beforeresonance or resonance mode, using dynamic tuning. Measuring the angular deﬂection or angular rate of the sensing element gives information about its position. A construction scheme of one of the ﬁrst gyroscopes is shown in Fig. 2. A sensing element of the device is made as a frame, with a signal winding round the

Fig. 2. Construction scheme of VG VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

113

brass skeleton. On the two sides of the frame two holders with semi-axis are glued, on which the frame is suspended in the centering screw with corundum bearing. Two other sides of the frame are placed in the gap of the magnetic system, which consists of a sensor foundation and magnet. When the frame is moving in the magnetic system gaps an output signal is induced in the sensing winding by the law of electromagnetic induction, and through the ﬂexible conduction this signal arrives at the sensor output. In case of free suspension of the sensing element, accuracy and stability of the resonance tuning do not depend on the unstable carrier angular rate along the longitudinal axis, and are deﬁned only by the relationship of the frame moments of inertia: C1 − A1 = B1 , as it follows from the equation of motion of the sensing element. But such relationship of inertia moments is not possible in practice, because it requires a zero thickness of the sensing element. That’s why a resonance tuning in the device can only be realized for C1 − A1 = (0.9 . . . 0.95)B1 . As a result, a device operates in a post-resonant zone in the immediate vicinity to resonance. In such work mode, the frame angular deﬂection and output sensor signal are: µ µ ¶¶ (C1 − A1 + B1 ) Dα α= q Ω sin ϕ˙ 0 t − arctg , (C1 − A1 − B1 ) ϕ˙ 0 (C1 − A1 − B1 )2 ϕ˙ 20 + Dα2

Ksw (C1 − A1 + B1 ) ϕ˙ 0 U = Ksw α˙ = q

× (C1 − A1 − B1 )2 ϕ˙ 20 + Dα2 µ µ × Ω cos ϕ˙ 0 t − arctg

¶¶ Dα (C1 − A1 − B1 ) ϕ˙ 0

.

r (C1 − A1 ) Dα Dα p ; H0 = ; ξ = = B1 2B1 ω0 2ϕ˙ 0 B1 (C1 − A1 ) = (C1 − A1 + B1 ) ϕ˙ 0 ; Ksw — transfer coefﬁcient of the signal winding. As is seen, the scale factor of the sensor depends on ϕ˙ 0 . In this case, an error is up to 30% in the range of the carrier angular rate from 12 to 22 rps. This error might be corrected by algorithmic compensation with the use of measuring the carrier angular rate ϕ˙ 0 . Information about ϕ˙ 0 is contained in the output signal. If there were a resonant mode the output signal would be in phase with the measured angular rate. The output signal phase error is caused by the inaccurate resonance tuning and is deﬁned by: r 1 − µ2 C1 − A 1 ∆χ = . , where µ = 2ξµ B1 where ω0 = ϕ˙ 0

The scale factor of the sensor depends on damping moment. This moment is created by Laurence force appearing in the short-circuit frame skeleton due to eddy currents. In this case the value of damping moment is proportional to electrical resistance of the frame skeleton, which is strongly effected by ambient temperature changes. Brass was used as a skeleton material, because it provides the best 114

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

Fig. 3. Construction scheme of VG with DF

correlation between electrical conductivity and temperature coefﬁcient of speciﬁc electric resistance. Errors of the scale factor can reach 30% in ambient temperature range of −40 · · · + 75 o C. Sensor with feedback. The errors have essentially reduced in a new scheme of VG for a rotating carrier with degenerative speed feedback (VG with DF). The sensing element of this gyroscope (Fig. 3) is an electrically non-conducting frame. On the frame two electrical coils are placed. One of them — signal coil — is a means for measuring the angular rate α˙ of the sensing element vibration. Other coil provides the moment effect on the frame. The coils connected with each other by an ampliﬁer. Thus a sufﬁcient value of damping in the device is provided using feedback. Current in the torque motor coil with its following integration provides readout of the output signal. An amplitude of the device output signal is proportional to that of the sensing element vibration and, consequently, to the measured angular rate. VG with DF works in a mode close to resonance. Bringing the feedback in the device sets a problem of providing stability in the system. In Fig. 4 there is a sensor block scheme. The scheme uses the following designations: M Ex — inertia moment, caused by presence of an angular rate of the carrier along the cross axis; Md — damping moment, created by feedback; s.e. — sensing element with transfer function Wf (s),

Wf (s) =

1 α(s) 1 1 = ; = · ∆M (s) B1 s2 +D0 s + B0 B0 T0 s2 + 2ξ0 T0 s + 1

D0 — air damping coefﬁcient; α — angular deﬂection of the sensing element vibration; SC — signal coil of the sensing element with transfer coefﬁcient Ksc ; VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

115

Fig. 4. Structural scheme of VG with generative feedback

esc — electromotive force directing in the signal coil; Ka — gain of the preampliﬁer; Kc — transfer coefﬁcient of the current ampliﬁer; Kmi — mutual induction coefﬁcient; emi — electromotive force, as a result of mutual induction of coils; TMC — torque moment coil of the sensing element with transfer coefﬁcient Ktmc ; Wi (s) — transfer function of the integrator:

Wi (s) =

(1 + T1 s) . + 2ξi T2 s + 1

T22 s2

During the research of stability and accuracy of the gyroscope in the transducerampliﬁer, consisting of pre-ampliﬁer and current ampliﬁer, the mutual induction of coils is taken into consideration. Examination of the system stability has shown that the system is structurally stable in all range of the carrier angular rate along the longitudinal axis. By corresponding choice of the device parameters we can provide phase stability with a margin, reaching tens degrees. In VG for a rotating carrier it is necessary to have information about the instantaneous angular rate value, measured by the device. This raises requirements to the phase characteristics of the device. To reduce inﬂuence from the carrier rotating rate on the phase of the output signal, a correcting unit is brought in. This provides the phase stability, not exceeding 1 deg. Considering a coefﬁcient n, which describes the coverage of mutual induction, the transfer function of a self-contained system is:

Φ(s) =

(1 + T1 s) KK i · 2 2 × Ktmc Kc T2 s + 2ξi T2 s + 1 s (1 + Td s) ¢ ¡ 2 , × 2 T0 nT c s3 + T0 + 2ξ0 T0 nTc s2 + (nT c + 2ξ0 T0 + K) s + 1

where Td — time constant of the correcting unit; Tc — time constant of the transducer-ampliﬁer: Tc = Kc Ka Kmi ; K = Ktmc Ksc Ka Kc /B0 .

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In compliance with !

Ã

Uout = |Φ(jω)| H0 Ω sin ϕ˙ 0 t + arg (Φ(jω)) , ω=ϕ˙ 0

ω=ϕ˙ 0

in case of ideal integration the values T1 and T2 are big, and with admission n = 0, ξ0 = 0 while measuring the constant angular rate we receive the amplitude of output signal: Uampl = (C1 − A1 + B1 ) ϕ˙ 20 Ω

KKi · Ktmc Kc

q¡ T22 ϕ˙ 0

T1 ϕ˙ 0 . ¢2 1 − T02 ϕ˙ 20 + (2ξT0 ϕ˙ 0 )2

If the device operated in a resonance mode the output signal amplitude would be detrmined only by feedback parameters, integrator characteristics and moments of inertia of the sensing element, whose stability is no more than some fractions of a percent and the phase is 90 deg: Uampl =

2B1 Ki T1 Ω. Ktmc Kc T22

Inaccuracy of the resonance tuning creates errors in both amplitude and phase of the output signal. Besides that, imperfection of the integrator creates an additional dependence on the carrier rotation rate, and the correcting unit adds amplitude errors. That’s why it is necessary to choose the time constant Td as a compromise between the accurate compensation of phase errors and amplitude small errors. In the developed sensor the amplitude errors are not exceeding 1% in the range of change of the carrier rotation rate along the longitudinal axis from 12 to 22 rps. That allows us to use VG with DF without measuring ϕ˙ 0 and without algorithmic ways of compensation. Temperature instability of the output signal parameters are mostly deﬁned by temperature instability of induction in gaps of the magnetic system. For magnetic materials (Nd15 Fe77 B8 ) used in the sensor the values of output signal errors do not exceed 3% by amplitude and 2 deg by phase in the temperature range from −40 to +75 o C. If necessary, the error can be decreased using thermomagnetic shunts. Investigation of inﬂuence of the carrier angular vibration on accuracy of angular rate measurements was performed by comparing two amplitude-manipulated values in the frame of reference bound with the carrier. These values are the device output signal and external inﬂuence on the sensing element. The investigation of measurement accuracy for a variable angular rate is difﬁcult because values of the sensing element rotation rate ϕ˙ 0 and measurable angular rate ν are close to each other: carrier angular vibration frequency reaches half the sensing element own rotation rate. Let’s consider a case of angular vibration of the carrier. Here both external inﬂuence on the sensing element and the sensor output signal are sums of two harmonic values with frequencies equal to a sum and difference of previously mentioned frequencies ϕ˙ 0 and ν. VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

117

µ α ¨ + 2ξω0 α˙ +

ω02 α

¶ H0 ν + 2B1 2

=

Ω sin ((ϕ˙ 0 + ν) t) + ¶ µ ν H0 Ω sin ((ϕ˙ 0 − ν) t) . + − 2B1 2

An output signal of the device can be presented as: Uout = Um sin (ϕ˙ 0 t + χm ) , where q Um =

U+ 2 + U−2 + 2U+ U− cos (2νt − χ− + χ+ ),

q χm

U+ 2 + U−2 − 2U+ U− cos (χ+ + χ− ) = arctg × q 2 U+ + U−2 + 2U+ U− cos (χ+ + χ− ) ¶¶ µ µ U+ sin χ+ + U− sin χ− sin νt + arctg U cos χ+ − U− cos χ− µ µ + ¶¶ × , U+ sin χ+ − U− sin χ− cos νt + arctg U+ cos χ+ + U− cos χ− KKi ((C1 − A1 + B1 ) ϕ˙ 0 + νB1 ) Ω r ³

U+ =

´2 ,

2

2Ktm Kc Ti

2

(2ξT0 (ϕ˙ 0 + ν)) + 1 − T02 (ϕ˙ 0 + ν)

KKi ((C1 − A1 + B1 ) ϕ˙ 0 − νB1 ) Ω r ³

U− =

´2 ,

2

2Ktm Kc Ti

(2ξT0 (ϕ˙ 0 − ν)) + 1 −

2

T02 (ϕ˙ 0

− ν)

Ã π χ+ = + arctg (Td (ϕ˙ 0 + ν)) − arctg (i (ϕ˙ 0 + ν)) − arctg 2

! 2ξ0 (ϕ˙ 0 + ν) , 1 −20 (ϕ˙ 0 + ν)2 !

Ã π χ− = +arctg (Td (ϕ˙ 0 − ν))−arctg (Ti (ϕ˙ 0 − ν))−arctg 2

˙ 0 − ν) 2ξ0 (ϕT . 1 − T02 (ϕ˙ 0 − ν)2

By analogy with the device output signal the input inﬂuence on the sensing element can be presented like that: M Ex = Mm sin (ϕ˙ 0 t + χinm ) , where s µ Mm = Ω

118

H02 ν 2 B12 + + 2 2

¶ H02 ν 2 B12 − 2 2

cos (2νt);

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

µ χinm = arctg

¶ νB1 tg (νt) . H0

At angular oscillations of the carrier, phase and amplitude depend on time. That’s why it was offered to estimate accuracy of measuring the harmonic angular rate by comparing phases and amplitudes of envelopes of the output signal and external inﬂuence on the sensing element, and also their carrier phases at the moment of envelope maximum, when useful signal is maximal. In the result the relative frequency dependence of the device output signal is deﬁned by the following equation: q m (µ2 − 1)2 + (2ξµ)2 max 2 + 1) ∆Um 1 (µ r = − max ´2 ∆Mm 2 ³ 2 2 2 + (2ξµ (1 + m)) µ − (1 + m) q m (µ2 − 1)2 + (2ξµ)2 2 (µ + 1) − r³ + ´ 2

µ2 − (1 − m)2 + (2ξµ (1 − m))2 q (µ2 − 1)2 + (2ξµ)2 + r³ + ´2 2 2 2 + (2ξµ (1 + m)) µ − (1 + m) q + r³

max ν Mm ;m= . 0) (ν = 0) ϕ˙ 0 Ratios of amplitudes of the output signal and input moment in relative values are shown in Fig. 5. max where ∆Um =

max Um max Um (ν =

(µ2 − 1)2 + (2ξµ)2 ; ´2 2 2 2 + (2ξµ (1 − m)) µ − (1 − m) max ; ∆Mm =

max Mm

Fig. 5. Relative frequency dependence of amplitude of output signal envelope

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This ratio is deﬁned only by coefﬁcients of detuning and vibration damping of the sensing element of the gyroscope. To reduce amplitude errors it is necessary to increase the damping coefﬁcient and improve the resonance tuning accuracy. On account of dependence of the vibration damping coefﬁcient on the rotation speed of a carrier the real error values can vary in the range between curves shown in the ﬁgure. The same conclusion can be made regarding phase differences of envelopes and carriers of the output signal and external inﬂuence on the sensing element. The difference of phases of envelopes for the output signal and external inﬂuence on the sensing element will become µ ¶ 1 2ξµ (1 + m) − χ0 = (arctg (Td ϕ˙ 0 (1+m)) −arctg (Ti ϕ˙ 0 (1 + m)) −arctg 2 µ2 − (1 + m)2 µ ¶¶ 2ξµ (1 − m) −arctg (Td ϕ˙ 0 (1 − m)) + arctg (Ti ϕ˙ 0 (1 − m)) + arctg . µ2 − (1 − m)2 The frequency dependence of the envelopes phase difference is shown in Fig. 6. To determine the angular vibration frequency inﬂuence on the output signal carrier it is necessary to take into consideration a frequency dependence of phase differences on the carrier frequency of the output signal and external inﬂuence on the sensing element of the device in moments of the envelope maximum, when useful signal is maximal. This dependence is deﬁned by the following equation: v ³ ´ u ((µ2 +1) +m)2 λ+ 2 + ((µ2 +1) −m)2 λ2− −2 (µ2 +1)2 −m2 λ+ λ− cos(χ+ +χ− ) u u ´ ³ × χc = arctg t ((µ2 +1) +m)2 λ+ 2 + ((µ2 +1) −m)2 λ2− +2 (µ2 +1)2 − m2 λ+ λ− cos(χ+ +χ− )

Ã sin

− Ã

× cos

−

Ã ¡¡ !! ¢ ¢ ¡¡ ¢ ¢ µ2 + 1 + m λ+ sin χ+ + µ2 + 1 − m λ− sin χ− ((µ2 + 1) + m) λ+ cos χ+ − ((µ2 + 1) − m) λ− cos χ− Ã ¡¡ !! ¢ ¢ ¡¡ ¢ ¢ , 2 2 µ + 1 + m λ+ sin χ+ − µ + 1 − m λ− sin χ− + arctg ((µ2 + 1) + m) λ+ cos χ+ + ((µ2 + 1) − m) λ− cos χ−

χ+ − χ− + arctg 2 χ+ − χ− 2

Fig. 6. Envelope phase difference

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Fig. 7. Carrier phase difference at the moments of maximum envelope amplitude

where 1 λ+ = v ! ; Ã uµ ¶ 2 2 u 2ξ (ϕ˙ 0 + ν) 2 (ϕ˙ 0 + ν) t + 1− ω0 ω02 1 λ− = v Ã ! . uµ 2 2 u 2ξ (ϕ˙ 0 − ν) ¶2 (ϕ˙ 0 − ν) t + 1− ω0 ω02

The frequency dependence of carriers phase difference is shown in Fig. 7. In VG with DF the errors, appearing during angular and circular oscillations of a carrier, are essentially less (more than 10 times), than those of the device shown in Fig. 2. Micromechanical sensor. Let us consider one more design of VG for a rotating carrier. The device (Fig. 8) consists of a silicon plate 1, suspended in the body on ﬂexible torsions 2. Relatively to the body, the plate has one degree of freedom, deﬁned by the relative angular deﬂection α. The α angle is measured by the differential capacitive pick-off 3, whose moving electrode is the silicon plate, and motionless electrode is put on the isolated substrate. The sensing element comprises a plate, torsions and foundation and represents a uniﬁed construction, made of silicon using anisotropic etching. The photo of the created device is shown in Fig. 9. In micromechanical sensors a ﬂexible suspension of the sensing element is used. In compliance with common equations of the VG motion for a rotating carrier the equation of motion of a micromechanical gyroscope and its solution with a constant measurable angular rate are: ¢ ¡ B1 α ¨ + D α˙ + (C1 − A1 ) ϕ˙ 20 + KT α = (C1 + B1 − A1 ) ϕ˙ 0 Ω sin (ϕ˙ 0 t) ,

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Fig. 8. Construction scheme of micromechanical sensor

Fig. 9. Micromechanical sensor

α = q¡

µ (C1 + B1 − A1 ) ϕ˙ 0 Ω sin ϕ˙ 0 t− ¢2 2 2 (C1 − A1 − B1 ) ϕ˙ 0 + KT + (D ϕ˙ 0 ) µ ¶¶ D ϕ˙ 0 −arctg , (C1 − A1 − B1 ) ϕ˙ 20 + KT

where KT — angular stiffness of suspension. Conditions of resonance tuning with angular stiffness depend on the carrier rotation rate along the longitudinal axis, which doesn’t allow one to ensure the tuning stability in all range of carrier rotation rate changes. This causes output characteristics to be dependent of the carrier rotation speed along the longitudinal axis. In such design of the sensor because of a small thickness (C1 − A1 − B1 ) ϕ˙ 20 ¿ KT . If during the choice of device parameters we will try to insure KT > DΩ, then the moment being measured will be balanced mostly by stiffness moment of torsions, which makes the output signal stable enough during temperature changes. 122

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But in this case the output signal will be dependent of the carrier rotation rate along the longitudinal axis which requires to use the algorithmic compensation. In the case when DΩ À KT and the moment being measured is balanced mostly by damping moment, the dependence of output signal on the carrier rotation rate, as in previous devices, is much less. The created micromechanical sensor also works in the mode, close to resonance. Equation for an amplitude Um of the output signal is: Um =

KE (C1 + B1 − A1 ) ϕ˙ 0 Ω q . KT2 + (D ϕ˙ 0 )2

where KE — transfer coefﬁcient of the electric part of the device. Because of presence of torsion angular stiffness in the last equation an inﬂuence of the carrier rotation rate along the longitudinal axis on the output signal amplitude is slightly larger than for the case of VG with DF. According to the results of theoretical calculations this inﬂuence is no more than 12% in the range of the carrier rotation rate from 12 to 22 rps. Test results. A special equipment to test sensors was created, representing two rotary stands with a device, placed on them, which simulates the carrier rotation along the longitudinal axis, where a sensor is attached. The simulator of own carrier rotation is controlled using a personal computer and contains the pulse sensor of the simulator rotation rate, whose output signal is compared to a frequency that is set in the computer. Digital control system provides the rotation rate to be constant in limits from 10 up to 25 rps. Technical parameters of the rotation simulator 1. The range of changing rotation rate . . . . . . . . . . . . . . . . . . . . . . . 2. Stability of speciﬁed angular rate . . . . . . . . . . . . . . . . . . . . . . . . . 3. Mass (with holder for mounting sensor to the stand) . . . . . . . . 4. Dimension (with holder) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Power source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Operation conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Technical parameters of the test stand 1. The range of changing stand table constant angular rate . . . . . 2. Maximum frequency of oscillations . . . . . . . . . . . . . . . . . . . . . . . 3. Maximum angular rate of oscillations at 7 Hz frequency . . . . 4. The form of table oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Measurement accuracy of table angular rate . . . . . . . . . . . . . . . . 6. Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Power source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. Operation conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10 . . . 25 rps 1.5% 3.7 kg 150 × 150 × 220 mm 45 V; 1,5 A laboratory ±300 deg/s 7 Hz 50 deg/s sinusoidal 1% 40 kg ∅450 × 245 mm 2 × 40 V; 10 A laboratory

The stand for dynamic tests (Fig. 10) provides a constant rate of angular motion velocity and vibratory motion with controllable frequency and amplitude. The stand refers to a single-axis indicator gyrostabilizer with a ﬁber-optic angular rate sensor served as a sensing element. The stand is controlled by a personal computer. VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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Fig. 10. Kinematic scheme of test equipment

The computer receives data on an angular rate of the stand table, an imitator rotation rate and output signal of the sensor under test. The stand has its own thermostat mounted on the simulator rotary table (temperature within the range from +20 to 80 o C). Photo of the stand for dynamic tests with rotation simulator and ﬁxed testing sensor is shown in Fig. 11. Forms of the output signal on the computers display at constant and variable measured rates are shown in Fig. 12. The accuracy of reproduction of an angular rate by the dynamic stand is not high because of a mechanical reducer used in the stand drive. To perform accuracy tests with a constant angular rate in a large temperature range, including negative temperatures, the second rotary stand with a mechanical variator, Fig. 11. Test stand allowing rotation with the constant angular speed within the range from 0.01 deg/s to 1100 deg/s was used.

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Fig. 12. Output signal of VG with DF on computer’s display: a — when measuring a constant angular rate; b — at a harmonic angular rate

Since commonly used thermal chambers of “Tabai” type are not suitable for testing at negative temperatures because of large errors due to temperature change when the rotary stand with a simulator of rotation is placed into the chamber, we have developed the special thermal chamber (with a volume of 1 litre), mounted on the simulator holder. The chamber can house only the object table of the simulator with VG installed on it for testing and it has a nipple for supply of cooled gas. The nipple axis is to be aligned with the rotation axis of the stand table. The scheme of the thermostatic control system is shown in Fig. 13.

Fig. 13. Thermostabilization system

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Photos of the stand in the process of its adjusting, and the simulator together with thermal chamber, located on the stand table, are shown in Figs. 14, 15 respectively. Parameters of the stand 1. Range of angular rate speciﬁed . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Minimum rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Stability of angular rate speciﬁed . . . . . . . . . . . . . . . . . . . . . . . . . Parameters of thermostatic control system 1. Range of supported temperature . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Absolute accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Irregularity of temperature distribution on an internal surface of the chamber . . . . . . . . . . . . . . . . . . . . . 4. Amplitude of temperature ﬂuctuation in adjusting . . . . . . . . . . 5. Operating time of one ﬁlled Dewar ﬂask . . . . . . . . . . . . . . . . . . . 6. Time of temperature changing from room temperature to limit of temperature range . . . . . . . . . . . . . . . . . .

1300 deg/s 0.1 deg/s 0.5% −60 · · · + 90 o C ≤ 1.0 o C 0.5 o C 0.05 o C 12 hours less than 15 min

All three schemes of the sensor were investigated using the indicated equipment. The sensors show normal performance at the following operating conditions 1. Linear acceleration of arbitrary orientation . . . . . . . . . . . . . . . . . 2. Linear acceleration on rotation axis . . . . . . . . . . . . . . . . . . . . . . . 3. Impact in the form of half sine wave . . . . . . . . . . . . . . . . . . . . . . 4. Random vibration of arbitrary orientation . . . . . . . . . . . . . . . . . . 5. Environmental temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12 g 70 g 150 g, 6 ms 5 g, f = 20 . . . 2000 Hz (−40 · · · + 75) o C

Technical parameters of the devices are summarized in Table 1. Table 1 Sensors parameters

Range of measurable speeds Threshold of sensitivity

VG scheme Fig. 2 Fig. 3 300 deg/s 300 deg/s 1 deg/s 0.3 deg/s 30 1 mV/(deg/s) mV/(deg/s) 1 deg/s 0.4 deg/s

Scale factor Zero signal (3σ) Temperature changing of scale factor (3σ) 30% 3% in the range from −40 . . . 75 o C Changing of scale factor (3σ) with carrier rotation rate change in the range from 12 to 22 rps (without algorithmic compensation) 30% 1% ∅30 × 15 Dimension (mm) without ∅30×15 with electronics electronics 40 g 40 g Mass

126

Fig. 8 300 deg/s 0.15 deg/s 30 mV/(deg/s) 0.2 deg/s 6%

12% ∅22 × 8 with electronics 12 g

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Fig. 14. Thermo-chamber

Fig. 15. Simulator with thermo-chamber

Conclusion. VGs for the rotating carrier have been developed, made and tested. The accuracy of VG with DF is signiﬁcantly (more than 10 times) exceeds the accuracy of the existing devices of the given class. The application of technology of etching of silicon for manufacturing a sensing element has resulted in creation of the micromechanical sensor of angular rate and allowed reducing weight and dimensions of the device while maintaining the satisfactory accuracy, which can be improved by algorithmic compensation of inﬂuence of the carrier rotation rate along the longitudinal axis.

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REFERENCES 1. Author’s certiﬁcate. No 108731 USSR. Gyroscopic Rate Sensor (in Russian) / A.I. Suchkov // Bulleten izobretenii. – 1957. – No 12. 2. Brozgul, L.I., Smirnov, E.L. Vibrating Gyroscopes (in Russian). – M.: Mashinostroenie, 1970. 3. Vlasov, Y.B., Filonov, O.M. Rotor Vibrating Gyroscopes in Navigation Systems. – Leningrad.: Sudostroenie, 1980. 4. Fateev, V.V., Podtchezertsev, V.P., Lyuty, M.N. Vibrating Rate Sensor (in Russian) // Vestnik MGTU. – 1999. – No 1. Boris S. Konovalov (b. 1972) graduated from Bauman Moscow State Technical University in 1996, a post-graduate student of BMSTU. Sergei F. Konovalov (b. 1941) graduated from Bauman Moscow Higher Technical School in 1964. D. Sc.(Eng.), Professor, a Head of “Orientation, Stabilization and Navigation Devices & Systems” Department of BMSTU. Author of more than 150 publications in the ﬁeld of development of navigation accelerometers, gyros, test equipment, inclinometering of gas and oil wells. Alexander V. Kuleshov (b. 1972) graduated from Bauman Moscow State Technical University in 1998, a lecturer of “Orientation, Stabilization and Navigation Devices & Systems” Department of BMSTU. Author of 6 publications in the ﬁeld of gyroscopic technology development. Denis V. Mayorov (b. 1972) graduated from Bauman Moscow State Technical University in 1996, a post-graduate student of “Orientation, Stabilization and Navigation Devices & Systems” Department of BMSTU. Victor P. Podtchezertsev (b. 1945) graduated from Bauman Moscow Higher Technical School in 1969, Ph. D., Assoc. Professor of “Orientation, Stabilization and Navigation Devices & Systems” Department of BMSTU. Author of more than 30 publications in the ﬁeld of gyroscopic technology development. Vladimir V. Fateev (b. 1940) graduated from Bauman Moscow Higher Technical School in 1963, Ph. D., Professor of “Orientation, Stabilization and Navigation Devices & Systems” Department of BMSTU. Author of more 60 publications in the ﬁeld of gyroscopic technology development.

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INFORMATION & COMPUTATION TECHNOLOGY V.V. Devyatkov (Bauman Moscow State Technical University)

MULTIAGENT HIERARCHICAL RECOGNITION ON THE BASIS OF FUZZY SITUATIONAL CALCULUS The effectiveness of images (signals, images, scenes) recognition system depends on ability to reveal properties of separate segments of images, relations between them in space and in time. It is known, that recognition can be considered as the syntactic analysis in some language. It is also known that languages in which fuzzy sets underlie semantics, can serve as the convenient tool for recognition of fuzzy and irregular images. The languages based on fuzzy rules are popular. This paper presents the approach to recognition of signals and images, based on the fuzzy situational calculus used by hierarchically organized community of intellectual agents. Each agent deals with the volume of knowledge represented in fuzzy situational calculus as ontology of this agent. In the paper, within the framework of the accepted concept of fuzzy hierarchical multiagent recognition, the essence of the hierarchical approach, the concrete language of fuzzy situational calculus, principles of solving problems of recognition by a collective of agents are considered. Illustrative examples are given.

1. Introduction. Various kinds of fuzzy calculi for modeling those or other ﬁelds of knowledge [4, 6–8, 9–15] are known. In the paper the concrete kind of situational calculus and its use by community of intellectual agents for multiagent hierarchical recognition (MHR) are considered. The basic attention in paper is given to principles of formation of ontology for recognition of signals, images and scenes (SIS) by communities of intellectual agents. The term “ontology” is widely applied in works on artiﬁcial intellect [1, 2, 25–27]. In the present paper under ontology the formalized description of any properties of images (signals, images, scenes) in language of fuzzy situational calculus (FSC) is understood [28]. The situation is understood as some property of images [3] deduced in FSC in process of multiagent hierarchical recognition (MHR) which they get in certain time or in the certain place. The situation is deﬁned by special parameters (variables). The property also can be independent of time and space. There is no uniﬁed methodology for ontology construction though on this theme there is a set of publications. In the paper rather universal methodology of the ontology creation and use on the basis of FSC for MHR is offered, including (1) principles of hierarchical multiagent recognition; (2) foundation of FSC; (3) consideration of formation ontology principles for a separate agent; (4) processes of recognition by a separate agent. Any calculus uses some language with its own syntax and semantics. The analysis or reasoning in calculus can be considered as syntactic recognition [15, VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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19]. Last years various syntactic methods of recognition were developed. Some authors offer to create a hierarchy of small, but stable knowledge bases (ontologies) with the description of pattern properties with which the image properties are compared [13–15] to achieve fast and reliable recognition,. The drawbacks of known approaches are the certain expressive restrictions of languages, impossibility to use modern hierarchical multiagent approaches when agents can solve problems of recognition, carrying out the communications with each other in a manner “question – answer” in the distributed environment, impossibility to query the recognition system using rather complex questions, demanding the modal and situational reasoning during the recognition. Irrespective of as far as good the ontology is there can be always an image with some unexpected features to which existing descriptions of pattern images do not correspond. To overcome this problem some mechanism of ontology adaptation can be used. The most obvious one is updating from outside, for example by a user. There are some solutions to this problem using neuron or genetic algorithms [16– 18]. In the paper this problem is not mentioned, but our point of view is that FSC should be ﬂexible enough to allow realization of correction, and the organization and interaction in MHR should be such that adaptation could be realized by community of agents using the certain criteria, known beforehand or developed during functioning. Further in the paper some data from area of fuzzy linear segmentation of signals and images is entered, principles of ontology creation for agents of the bottom level are shown. Then the basic concepts of MHR and FSC are introduced, the illustrative example of inference at the bottom level of the MHR is considered. In the conclusion, problems of the near future are formulated. 2. Fuzzy syntactic recognition. An image is representation of some physical reality (object, phenomenon), having at least one parameter which contains the information and is referred to as informative. This information can be measured for each parameter from a determined set of values. Values of informative parameters of images are used for reasoning about properties of the image, for example, for the purpose of diagnostics of heart diseases, if the image is a cardiogram, or for the purpose of management of the pumps rotation speed if images represent diagrams of the liquid pressure change in pipelines. Parameters of images can be continuous and discrete. The continuous parameter is given in all range of deﬁnition X. The discrete parameter is given only in separate points of a range of deﬁnition, for example, through the same step of digitization ∆x , since some initial value: x0 = = x, x1 = x + ∆x , x2 = x + 2∆x , x3 = x + 3∆x , . . . Steps of digitization can be various. Digital images, as a rule, have only discrete parameters. For simplicity we shall consider a case when the image has only two parameters x ∈ X and y ∈ Y . Parameters are in the certain relation R(x, y) ∈ X × ×Y. In Fig. 1 the example of the digital image (signal) having two parameters x and y is shown. The steps of digitization of parameters x and y are ∆x , ∆y . We shall put in conformity to each relation R(xi , yj ) the node of the graph signed with a symbol Ni , i = 0, 1, . . . , 12. (We shall designate nodes by a circle). The most left node we shall also sign with a symbol S = N0 . We connect each pair of the neighboring nodes i, i + 1 by an arch directed from node i to node i + 1. An arch directed from node i to node i + 1 we shall sign with a sym130

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Fig. 1. A digital signal (a step of digitization on abscissa axis is ∆x )

Fig. 2. Segmentation of a digital image

Fig. 3. Transitions graph of the automation M

bol ai in alphabet A = {a0 , a1 , a2 , a3 , . . . , am } (Fig. 2). If to write out designations of all arches from left to right we shall obtain a sequence of symbols a0 a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 . We denote the arches, having the same direction and length in a sequence a0 a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 with the same symbol, that appears ﬁrst at viewing the sequence from left to right. In the result we shall obtain a sequence a0 a1 a2 a3 a4 a3 a6 a7 a8 a9 a6 a0 and a graph depicted in Fig. 3. This sequence can be considered as a word of some language L = L(G), generated by automaton grammar G = {V, A, P, S}, V = {N1 , N2 , N3 , N4 , N5 , A = {a0 , a1 , a2 , a3 , a4 , a6 , a7 , a8 , a9 }, P = N6 , N7 , N8 , N9 , N10 , N11 }, = {S → a0 N1 , N1 → a1 N2 , N2 →a2 N3 , N3 → a3 N4 , N4 → a4 N5 , N5 → → a3 N6 , N6 → a6 N7 , N7 → a7 N8 , N8 → a8 N9 , N9 → a9 N10 , N10 → a6 N11 , N11 → a0 }. VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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To this grammar there corresponds the recognizing automaton device M = = (A, B, f, b0 , F ), where A is the input alphabet, B is the output alphabet, f is a function of transitions, b0 is an initial state, F is a set of ﬁnal states, and B = {b1 , b2 , b3 , b4 , b5 , b6 , b7 , b8 , b9 , b10 , b11 , b12 }, f : f (a0 , b0 ) = b1 , f (a1 , b1 ) = b2 , f (a2 , b2 ) = b3 , f (a3 , b3 ) = b4 , f (a4 , b4 ) = b5 , f(a3 , b5 ) = b6 , f (a6 , b6 ) = b7 , f (a7 , b7 ) = b8 , f (a8 , b8 ) = b9 , f (a9 , b9 ) = b10 , f(a6 , b10 ) = b11 , f (a0 , b11 ) = = b12 , F = b12 . Thus, to an arch in Fig. 2 there corresponds a transition of the transitions graph of the automation (see Fig. 3). Our task is ascription of an image to one of pattern classes of images. For these purposes we shall construct the fuzzy grammar GF using the crisp grammar G on the basis of the following principles. To each arch of graph transitions there corresponds two neighboring nodes bi and bj (Fig. 4). Coordinates of node bi are xi and yj , and coordinates of node bj are xj and yj . We allow, that coordinates of both nodes can change within the range of some rectangular areas, for example, the squares given in Fig. 4 with steps of digitization accordingly δx and δy . Such assumption means, that instead of one node bi with coordinates xi , yi we shall have a set of nodes bni ∈ B(bi ) with the coordinates varied within the range, outlining the node (in Fig. 4 it is a square), and

n=

+ − + (x− i − xi ) (yi − yi ) · . δx δy

Instead of one node bj with coordinates xj , yj we shall have a set of nodes bsj ∈ B(bj ) with the coordinates varied within the range of a square, outlined

Fig. 4. Membership functions of transitions graph nodes

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around node bj , and instead of one arch ai , connecting node bi with node bj we shall have a set of all arches ali ∈ A(ai ) = {(bri , bsj )|bri ∈ B(bi ), bsj ∈ B(bj ), l ≤ n2 , r ≤ n, s ≤ n}, connecting each node of the set B(bi ) with each node of the set B(bj ). Projections of sides of the squares, outlining sets B(bi ) and B(bj ), to axis X + − + − + − + will be, respectively, x− i , xi and xj , xj , and to axis Y : yi , yi and yj , yj . We shall set four triangular membership functions µi (x), µi (y), µj (x), µj (y) which + are given on universes X and Y by triplets of values 0, 1, 0 in points {x− i , xi , xi }, − + − + − + {yi , yi , yi }, {xj , xj , xj }, {yj , yj , yj }, respectively (see Fig. 4). These functions deﬁne a measure of nearness of the graph node coordinates of to “the best coordinates” to which the membership function value, equal to 1, corresponds. The membership function is expressed by the formula u− u − k u − u− − u − u− , if uk ≤ u ≤ uk , k k k k µk (x) = + u −u + + k , if uk ≤ u ≤ u+ + k, uk − uk uk − uk where u = x, y; k = i, j. Let us believe, that membership function of each arch ali ∈ A(ai ), incidental to nodes bri ∈ B(bi ) and bsj ∈ B(bj ), is deﬁned as follows µA(ai ) (ali ) = min{µi (xri ), µi (yri ), µj (xsj ), µj (ysj )}. The fuzzy automaton grammar GF = {V, AF , PF , SF } is derived from the crisp automaton grammar G = {V, A, P, S} as follows [ A(ai ). AF = i

The unique initial nonterminal symbol of the crisp grammar is replaced with a set of initial nonterminal symbols SF = B(b0 ). The set of rules PF of the fuzzy grammar GF is deﬁned by an expression PF = {(Ni → ali Ni+1 , µ(Ni → ali Ni+1 ) = µA(ai ) (ali )), ali ∈ A(ai ), i = 0, 1, . . . , m, l ≤ n2 }. The fuzzy automaton grammar GF generates the fuzzy language {L(GF ), RL(GF ) } (in automaton designations it will be the language {L(MF ), RL(MF ) }). L(GF ) = {l∗ |l∗ = al0 al1 al2 . . . alm , ali ∈ A(ai ), l ≤ n2 }, RL(GF ) = {µL(GF ) (l∗ )/l∗ |l∗ = al0 al1 al2 . . . alm , ali ∈ A(ai ), µL(GF ) (l∗ ) =

min

{µA(ai ) (ali )}.

a=al0 ,al1 ,...,alm

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Thus, recognition by pattern images in two-dimentional space can be organized as follows. For each pattern image the construction of pattern fuzzy grammar is carried out. The number of grammars will be equal to that of pattern images. If to be limited to a syntactic level of recognition on fuzzy pattern grammars then the process of recognition of an image consists of the following steps. 1. One of pattern images (the fuzzy pattern grammar appropriate to it) is chosen. 2. The image to be recognized is processed with the same steps of digitization, as the chosen pattern. 3. Syntactic analysis of the image to be recognized with the help of the fuzzy grammar of the chosen pattern image is carried out. 4. If syntactic analysis appears unsuccessful, next pattern image is chosen out of still unconsidered ones and all repeats from item 2. 5. If syntactic analysis appears successful then membership function of the image being recognized is calculated and memorized. 6. If all pattern images are not yet considered, one more is chosen out of them and all repeats from item 2. 7. If no pattern images remain unconsidered, and there has been no successful syntactic analysis the recognition of the image comes to an end with a failure (the image is not recognized). 8. If there have been some successful syntactic analyses then recognition ﬁnishes successfully, and the recognized image can be identiﬁed with a class of that pattern image, whose grammar provided analysis having returned the maximal membership function for the recognized image. It is known [3] that rules of any grammar are easy to represent by rules of adequate calculus. The initial construction of the grammar is a convenient form of a well-formalizable way of transition from graphic representation of pattern images or their elements (given in two-dimensional space and having the form of pieces, lines and curves) to describing the pattern ontology in fuzzy calculus. If it is difﬁcult to allocate pieces or curves in an image, the considered method of ontology formation is inapplicable. Fortunately, the allocation of pieces and curves for the practical majority of images can be always made and it is an important part of recognition process. In case of the automaton grammar, it is easy to organize inference on the basis of calculus with the help of the automaton. Now we shall proceed to considering the possible organization of multiagent hierarchical recognition. 3. Multiagent hierarchical recognition. In the present paper the problem of research and substantiations of a certain original architecture of multiagent hierarchical recognition is not put. There are many works devoted to multiagent architectures. It is enough to mention one of the latest work [22]. Intellectual agents are united in an agency for solving problems. In our case it is a problem of recognition of images. Each of agents in agency solves some individual problem according to its role. Together they solve the common problem of the agency. Each agent has, as a minimum, input and output ports, base of knowledge (ontology) and the machine of inference. Through an input port an agent receives messages and inquiries from other agents, and also the information from other sources which are not agents in

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Fig. 5. Interrelation between semantic and syntactic levels

the speciﬁed sense. Through an output port an agent gives out messages and inquiries to other agents, and also the information to the addressees, which are not agents. In a basis of the approach considered in the given paper the idea lies about the differentiation of the syntactic and semantic description of image properties in the language of situational modal fuzzy calculus [29] (Fig. 5). The syntactic description serves as a means of data exchange between agents. The semantic description is used by agents for calculations. In this connection there are two levels of data representation. The ﬁrst (syntactic) level represents the ontology in the formal language of situational modal fuzzy calculus. The second (semantic) level represents the ontology in the language of fuzzy relations [22–25]. It can be considered as an original formal system with the language, rules of inference and interpretation. The inference is made at a semantic level. Communication between these two levels is carried out by means of two procedures: (1) semantic translation which carries out translation of the text presented at a syntactic level into the language of fuzzy relations, that refers to a semantic level; (2) linguistic approximation which carries out translation of the conclusions derived at a semantic level, into the language referring to a syntactic level (see Fig. 5). Agents communicate at a syntactic level. A diagram of multiagent hierarchy for recognition of images is shown in Fig. 6. A number of levels of agents hierarchy can be anyone. Agents of the ﬁrst level are engaged in segmentation of images. An example of segmentation by considering principles of syntactic recognition is given further. Other agents calculate relations by results of linguistic approximation of relations executed by agents of the previous levels. 4. Fuzzy relations. Let us remind some concepts from area of fuzzy relations. An expression ai (Xi , Ri ), where Xi = {xi1 , xi2 , . . . , xiri }, is a set of term values of a parameter x; Ri is a fuzzy set appropriate to the attribute ai and given on Xi . The attribute ai unambiguously corresponds to a term. A special case of a term, which we shall name the elementary one, is an expression ai (xij , µ(xij )), where xij is a single objective variable of the universe and µ(xij ) is the membership function determined for this variable. An expression α(a1 (X1 , R1 ), . . . , aN (XN , RN )) is referred to as the scheme of relations or the fuzzy scheme of relations where α is VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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Fig. 6. Multiagent hierarchy of images recognition

a name or an attribute of the scheme of relations; ai (Xi , Ri )) is a term; Ri is the fuzzy set appropriate to the attribute ai . Generally, if we have some attributes we shall believe, that we can always index these attributes in some order by integers from 1 up to N , where N is a quantity of attributes. Then a subset of attributes from all such sets, consisting of i attributes whose indices run a set of values I (i) , we shall designate A(I (i) ). Attributes are always ordered according to the order of indices in the set I (i) , and I (i) is arranged in increasing order. Thus, A(I (i) ) = {aj |j ∈ I (i) }. For example, images can be given by a matrix of dimension m×n. Coordinates (i, j) of elements of this matrix, where i = 1, . . . , m; j = 1, . . . , n, form universe Umn on which the set of values of some membership function µ(i, j) is given. Each element of the matrix with coordinates (i, j) can be represented by the elementary term aij (xij , µ(xij )), where xij = (i, j), i = 1, . . . , m; j = 1, . . . , n, and all matrix is the scheme of relations initial (a11 (x11 , µ(x11 )), a12 (x12 , µ(x12 )), . . . . . . , amn (xmn , µ(xmn ))), named as initial. We shall name the set A(I (i) ) as a basis of the scheme of relations. Conformity between the set of attributes ai and fuzzy sets Ri can be expressed as a function dom(ai ) = Ri . A set of all various subsets of indices I (i) , where 1 ≤ i ≤ N , is (N ) (i) i i }, where CN is a quantity of combinations designated Ji = {Ij |j = 1, . . . , CN from N by i. In view of the entered designations the scheme of relations can be designated as α[A(I (m) )], R[A(I (m) )] or R[i |i ∈ I (m) ]: dom(A(I (i) )) = X dom(aj ), j∈I (i)

where X is a sign of operation of the Cartesian product. j∈I (i)

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The scheme of relations R[A(I (i) )] V is referred to as normal if R is a normal fuzzy set, i.e. (∃u)[u ∈ dom(A(I (i) )) (µR (u) = 1)]. The scheme of relations R[A(I(i))] is referred to as separable, if for any partition A = {ξj |j = 1, . . . , m} of sets I (i) such, that m [

I (i) =

ξj ,

j 6= i,

ξj ∩ ξi = ∅,

j=1

the following takes place: m

R[A(I (i) )] = X Projξj R[A(I (i) )], j=1

where Projξj R[A(I (i) )] is a projection of fuzzy set R[A(I (i) )] onto ξj . The separable scheme of relations R[A(I (i) )] has the remarkable properties: (1) it can be derived as the Cartesian product of schemes of relations R[ξj ], being projections of the scheme of relations R[A(I (i) )] appropriate to disjoint subsets ξj ∈ A of the set I (i) , forming the partition A; (2) schemes of relations R[ξj ] can be derived as a result of a projection of the scheme of relations R[A(I (i) )]. Let us add the following designations: a set of all bases of schemes of relations {A(I (i) )}: n [

[ A(I (i) );

(i)

{A(I )} = i=1 I (i) ∈J (N ) i

a set of all schemes of relations {R[A(I (i) )]}, having the basis A(I (i) ): {R[A(I (i) )]} = {R[A(I (i) )]|A(I (i) ) ∈ {A(I (i) )}}. Above schemes of relations the following operations can be carried out. The supplementation, designated ¬:

R0 [A(I (i) )] = ¬R[A(I (i) )],

where ∀u ∈ dom(A(I (i) )),

µR0 (u) = 1−µR (u).

The union, designated ∪: P [A(I (i) )] = R[A(I (i) )] ∪ Q[A(I (i) )], where ∀u ∈ dom(A(I (i) )), µP (u) = max(µR (u), µQ (u)). The intersection, designated ∩: P [A(I (i) )] = R[A(I (i) )] ∩ Q[A(I (i) )], where ∀u ∈ dom(A(I (i) )), µP (u) = min(µR (u), µQ (u)). The implication, designated ⊃ [23]: g

P [A(I (i) )] = [R[A(I (i) )] ⊃ Q[A(I (i) )]], g

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where ∀u ∈ dom(A(I (i) )), µP (u) = [µR (u) ⊃ µQ (u)], and g

( 1, a 6 b, a⊃b =

∀a, b ∈ [0, 1].

g

b, a > b, The cylindrical continuation, designated CA(K (j) ) : R0 [A(I (i) ∪ K (j) )] = CA(K (j) ) R[A(I (i) )], where ∀u ∈ dom(A(I (i) )), ∀v ∈ dom(A(K (j) )), µR0 (u, v) = µR (u). (j) The projection designated ProjA(K) : R0 [A(K (j) )] = ProjA(K (j) ) R[A(K (j) ∪ I (i) )], where ∀u ∈ dom(A(K (j) )), µR0 (u) = sup{µR (u, v)|v ∈ dom(A(I (i) ))}. The linear transformation, designated Λ(k,b) : R0 [A(I (i) )] = Λ(k,b) R[A(I (i) )], where ∀u ∈ dom(A(I (i) )), µR0 (u) = max(0, min(1, k · µR (u) + b)). A result of linear transformation operation depends on a ratio of its parameters. In particular, if k > 1 and (−k) < b ≤ (1 − k), or if 0 < k < 1 and (−k) < < b < 0 then membership function values of a result of linear transformation do not exceed membership function values of its argument in all range of deﬁnition of these functions. In this sense, linear transformation can be considered as a version of an operation of concentration [24]. On the contrary, if 0 < k < 1 and (1 − k) < < b < 1, or if k > 1 and 0 ≤ b < 1 then membership function values of an argument of the operation do not exceed membership function values of its result, that is linear transformation in these cases can be considered as a version of an operation of stretching [24]. At values of the parameters satisfying a condition k > 1 and (1 − k) < b < 0, an operation of linear transformation can be considered as a version of an operation of the contrast intensiﬁcation [24] as it increases those values of the membership function which are more than (−b)/(1 + k) and reduces those less than (−b)/(1 + k). The comparison, designated =: R[A(I (i) )] = P [A(K (j) )], and

if

I (i) = K (j)

∀u ∈ dom(A(I (i) )), µR (u) = µP (u).

The inclusion, designated ⊆: R[A(I (i) )] ⊆ P [A(K (j) )], and

138

if

I (i) = K (j)

∀u ∈ dom(A(I (i) )), µR (u) ≤ µP (u).

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

The consistency, designated CP: τ = CP(R[A(I (i) )], P [A(I (i) )]), µτ (t) = sup{µP (u)|u ∈

if

∀t ∈ [0, 1],

µ−1 R (t)},

where µ−1 R (t) = {u|x = µR (u), u ∈ U }. 5. Syntax of fuzzy situational calculus language. The alphabet of a language of fuzzy situational calculus, on which the formal ontology of agents is described, comprises the following groups of symbols: names of linguistic variables {X1 , X2 , . . . , Xn }; linguistic values n [

θ=

θ(Xi ) i=1

of these variables; modalities Mod = {m}; logic symbols {¬, &, ∨, ⊃}; the dyadic predicate “is” and also brackets “(”, “)”. According to [23] a linguistic variable is characterized by the quintuple < X, θB (X), U , G, M > in which X is a linguistic variable name, θB (X) is a base term-set, U is the universe, the symbol G designates a syntactic rule, and the symbol M designates a semantic rule of a linguistic variable. A base term-set is ﬁnite and usually contains a small number of linguistic values. It is given and directly appears in the linguistic variable deﬁnition. All other values are derived by the syntactic rule G and form a term-set which is designated by the symbol θ(X). Elements of this term-set are constructed of elements of θB (X) with the help of a ﬁnite set of linguistic modiﬁers and connectives. Fuzzy sets R given on the universe U are put in conformity to the linguistic values. Thus each fuzzy set expresses a sense of the appropriate concrete linguistic value. Conformity between linguistic values of the set θ(X) and the fuzzy sets is established by the semantic rule M . The modalities included in the language under consideration are expressed as a kind of conformity of the semantic contents of modal propositions with the real world, and force or gradation of this conformity. Symbols of a modality include a preﬁx, expressing a modality type, and a natural number, expressing a gradation value. Let us consider modalities: “impossibility”, “probability”, “possibility” and “necessity”. The preﬁxes, corresponding to them, are: {Impossible , Probable , Possible , Necessary }. A number after a preﬁx, expressing the gradation of the appropriate modality, should not exceed hundred. The symbols of modalities are read as follows: I mpossible N −”with the N % impossibility”; P robable N — ”with the N % probability”; P ossible N — ”with the N % possibility”; N ecessary N — ”with the N % necessity”, where N is a gradation value expressed in percentage. All 400 symbols of modalities form the set M od. Thus, any well-formed formula (wff) of a language in which a formal ontology is described, represents a chain of symbols, satisfying the following BNF: atom ::= Y is X VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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wﬀ ::=< atom > |¬ < wﬀ > |(< wﬀ > & < wﬀ >)|(< wﬀ > ∨ < wﬀ >)| (< wﬀ >⊃< wﬀ >)|m < wﬀ >, where Y ∈ {X1 , X2 , . . . , Xn }; X ∈ θ(Y ); m ∈ Mod . A procedure of semantic translation establishes conformity between expressions of a language and their semantic contents expressed by fuzzy relations. Thus it is considered, that each linguistic variable one-to-one corresponds to an attribute, and the universe of linguistic variables one-to-one corresponds to a domain. The atom a =”Y is X” corresponds to the scheme Rα [A(I (1) )] = α(a(d, Rα )), where a = Y , Rα = M (X), d = U . Conformity between other syntactic constructions and fuzzy schemes of relations is shown in Table 1 where N means gradation of the appropriate modality, parameter n is calculated under the formula n = N/100, and i, j are indices of linguistic values in formulas. Table 1 The formula of a language

The fuzzy relation

wﬀ i

Rwﬀ i [A(I (i) )]

wﬀ j

Rwﬀ j [A(K (j) )]

¬wﬀ i

¬Rwﬀ i [A(I (i) )]

(wﬀ i &wﬀ j )

CK Rwﬀ i [A(I (i) )] ∩ CI Rwﬀ j [A(K (j) )]

(wﬀ i ∨ wﬀ j )

CK Rwﬀ i [A(I (i) )] ∪ CI Rwﬀ j [A(K (j) )]

(wﬀ i → wﬀ j )

CK Rwﬀ i [A(I (i) )] ⊃ CI Rwﬀ j [A(K (j) )]

(j)

(i)

(j)

(i)

(j)

(i) g

Impossible N wﬀ i

¬Λ(1/(1−n),0) Rwﬀ i [A(I (i) )]

Probable N wﬀ i

¬Λ(1/n,−(1−n)/n) Rwﬀ i [A(I (i) )]

Possible N wﬀ i

Λ(1/n,0) Rwﬀ i [A(I (i) )]

Necessary N wﬀ i

Λ(1/(1−n),−n/(1−n)) Rwﬀ i [A(I (i) )]

6. Inference and linguistic approximation. Using the considered operations for fuzzy relations we shall enter the following rules for realization of inference. Successor rule. If fuzzy relation Rα [A(I (i) )] is speciﬁed and it is known, that Rα [A(I (i) )] ⊆ Rβ [A(I (i) )], then it is possible to deduce fuzzy relation Rβ [A(I (i) )]. To put it differently, if the formula α has the semantic contents Rα [A(I (i) )], that we shall write down as < α >= Rα [A(I (i) )], and the fuzzy relation Rβ [A(I (i) )] exists, such that Rα [A(I (i) )] ⊆ Rβ [A(I (i) )], then the formula β with semantic contents Rβ [A(I (i) )] is deduced from the formula α: α, < α >= Rα [A(I (i) )], Rβ [A(I (i) )], Rα [A(I (i) )] ⊆ Rβ [A(I (i) )] ` β, < β >= Rβ [A(I (i) )]. For example, let it be known, that the person is very young. This fact can be expressed by the formula “the person is very young”, where the person is a 140

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name of a linguistic variable “person”, young is its base linguistic value, very is a linguistic modiﬁer. The semantic contents of this formula is expressed by the fuzzy relation the person (very young (U, M (very young)). On the other hand, for example, it is known, that the fuzzy set M (young) includes M (very young), i.e. M (very young) ⊂ M (young). On this basis it is possible to establish, that the given person is young. The formula “the person is young” corresponds to this conclusion. Generalization rule. If the fuzzy relation Rα [A(I (i) )] is known then it is possible to deduce the fuzzy relation Rβ [A(I (i) ∪K (j) )] such, that Rβ [A(I (i) ∪ K (j) )] = = CA(K (j) ) Rα [A(I (i) )], where K (j) ∩ I (i) = ∅. In other words, if the formula α has semantic contents Rα [A(I (i) )] and some attributes which are not included in the scheme A(I (i) ) whose indices make the set K (j) , then the formula β, having semantic contents Rβ [A(I (i) ∪ K (j) )] = = CA(K (j) ) R[A( I (i) )], is deduced from the formula α: α, < α >= Rα [A(I (i) )], K (j) ∩ I (i) = ∅ ` β, < β >= CA(K)( j) R[A(I (i) )]. For example, if it is known, that “the person is young”, and nothing is known about the person’s weight, it is possible by a generalization rule to infer: “the person is young and weight is any”. Projection rule. If fuzzy relation Rα [A(I (i) )] is known then it is possible to deduce the fuzzy relation Rβ [A(K (j) )] such, that Rβ [A(K (j) )] = = ProjA(K (j) ) Rα [A(I (i) )], where K (j) ⊂ I (i) . That is, if the formula α has the semantic contents Rα [A(I (i) )] and some subset of attributes is allocated from the scheme A(I (i) ), whose indices make the K (j) , then the formula β with the value Rβ [A(K (j) )] = ProjA(K (j) ) Rα [A(I (i) )] is deduced from the formula α: α, < α >= Rα [A(I (i) )], K (j) ⊂ I (i) ` β, < β >= ProjA(K (j) ) Rα [A(I (i) )]. For example, if it is known, that “the person is young and weight is big” by a projection rule then it is possible to deduce formulas: “the person is young” and “weight is big”. Intersection rule. If two fuzzy relations Rα [A(I (i) )] and Rβ [A(I (i) )] are known then it is possible to deduce the third fuzzy relation Rγ [A(I (i) )] such, that Rγ [A(I (i) )] = Rα [A(I (i) )] ∩ Rβ [A(I (i) )]. To put it differently, if the formula α has the semantic contents Rα [A(I (i) )] and the formula β has the semantic contents Rβ [A(I (i) )], then the formula γ, which is coordinated to semantics of formulas α and β is deduced from them: α, < α >= Rα [A(I (i) )], β, < β >= Rβ [A(I (i) )] ` γ, < γ >= Rα [A(I (i) )] ∩ Rβ [A(I (i) )]. As an inference process we shall understand such a sequence of application of the above-listed rules that provides deducing the required output formula with the appropriate semantic contents. VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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An accountable set of formulas at a syntactic level and a continuum of the semantic contents expressed by fuzzy relations at a semantic level can be presented. In the process of inference, which is carried out at a semantic level, a fuzzy relation, having no expression at a syntactic level, can be derived. In this connection there appears a problem of linguistic approximation. Let the fuzzy relation Rβ [A(I (i) )], where A(I (i) ) = {X1 , X2 , . . . , Xi }, be obtained as a result of inference. Linguistic approximation of the relation Rβ [A(I (i) )] consists in ﬁnding such a formula α that its semantic contents < α >= Rα [A(I (i) )] would coincide with Rβ [A(I (i) )] or be close to it. For solving the problem we shall search among the formulas having the form (α1 &(. . . &(αi−1 &αi ) . . .)), in which every subformula αk (k ∈ 1, 2, . . . , i}) has the structure: (mk,1 Xk is Xk,1 &(. . . &(mk,q−1 Xk is Xk,q−1 &mk,q Xk is Xk,q ) . . .)), where {Xk,1 , . . . , Xk,q } = θB (Xk ) is the base term-set of the linguistic variable Xk ∈ A(I (i) ); mk,1 , . . . , mk,q are symbols of modalities. As the structure of the formula α is ﬁxed, linguistic approximation is reduced to ﬁnding all symbols of the modalities which are included in the subformula of a kind: mk,j Xk is Xk,j (Xk ∈ A(I (i) ), Xk,j ∈ θB (Xk )). For ﬁnding every concrete mk,j we shall carry out semantic translation of atom Xk is Xk,j and we shall construct projection Rβ [A(I (i) )] on the scheme {Xk }. We shall designate these fuzzy relations Rα [Xk ] and Rβ [Xk ]. We shall ﬁnd the degree of consistency τ = CP(Rα [Xk ], Rβ [Xk ]) of the semantic contents of the atom Xk is Xk,j , and the semantic contents of the inference concerning a value of the variable Xk . By deﬁnition of a degree of consistency, τ is a fuzzy set in [0, 1] and expresses an estimation of conﬁdence that the given inference can be expressed by the formula Xk is Xk,j . On the other hand, the modality is also an estimation of the conﬁdence, but only presented at a syntactic level. We shall note, that the modal semantics, expressed in a procedure of semantic translation through operations of linear transformation with the appropriate parameters, can be also deﬁned in another way. We shall consider the linguistic variable < confidence, Mod, [0, 1], G, M > whose base term-set contains all earlier considered symbols of modalities, syntactic rule G generates only elements of Mod, and semantic rule M establishes conformity between symbols of modalities and fuzzy sets in an interval [0, 1]. These fuzzy sets have trapezoid membership functions which can be constructed proceeding from intuitive representations about a nature of the modal concepts expressing feeling of conﬁdence. In particular, for modalities Impossible N , Probable N , Possible N and Necessary N the membership functions of the fuzzy sets generated by the semantic procedure have the following form: ∀x ∈ [0, 1]: µM (Impossible

N ) (x)

= 1 − max(0, min(1, x/(1 − n)));

µM (Probable

N ) (x)

= 1 − max(0, min(1, (x + n − 1)/n));

µM (Possible

N ) (x)

= max(0, min(1, x/n));

µM (Necessary

N ) (x)

= max(0, min(1, (x − n)/(1 − n))),

where n = N/100. 142

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Proceeding from this, linguistic approximation consists in a choice of such symbols of the modality mk,j , that difference of values µM (mk,j ) and appropriate values µτ would be minimal and ∀t ∈ [0, 1], µM (mk,j ) (t) > µτ (t). Formulas in the language of fuzzy situational calculus grow out of works of separate agents and are transferred to other agents for the subsequent use during recognition. 7. Example. Let us consider an example of evaluation of the membership function and linguistic approximation by the agent of the ﬁrst level according to results of segmentation after which in two-dimensional space the coordinate of a point (node) on the abscissa axis is determined. Let the arrangement of a node be characterized by two linguistic variables: < X, θB (X), U, G, M > and < Y, θB (Y ), V, G, M >, where X and Y are linguistic variables, describing the node position on the abscissa and ordinate axes, respectively; θB (X) = = θB (Y ) = {At the left , At the center , At the right } are base sets of linguistic values; U = V = [−1, 1] is the universe of values of coordinates on axes X and Y ; G — a syntactic rule, M — a semantic rule. Base sets of linguistic values of both linguistic variables are chosen conterminous for convenience of a statement. Identical dimensions universes values of coordinates from −1 up to 1 on both axes are chosen conterminous also for convenience of a statement. Let us believe, that U = V = {−1, −0.8, −0.6, −0.4, −0.2, 0, 0.2, 0.4, 0.6, 0.8, 1}. Semantic contents of linguistic values At the left, At the center, At the right are deﬁned by semantic rules: M (At the left ) = µL , M (At thecenter ) = µC , M (At the right ) = µR . Diagrams of membership functions µL , µC , µR are presented in Fig. 7. The ontology contains the following axioms expressing restrictions on an arrangement of nodes of the given segment, caused by its properties. (X is At the left ⊃ Y is At the right )

(1)

(X is At the center ⊃ Y is At the center )

(2)

(X is At the right ⊃ Y is At the left )

(3)

(Poss 50 X is At the center and Poss 50 X is At the right ).

(4)

It is required to ﬁnd the answer (linguistic approximation) to a question, which represents the following essence: “If the ontology, represented by axioms (1)–(4),

Fig. 7. Membership functions of node coordinates VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

143

is known, what is the unknown coordinate of a node?”. The axiom 4 determines a coordinate of the node on axis X. Therefore only a coordinate of the node on axis Y is unknown. In our calculus the question is put by the following expression (m1 Y is At the left and (m2 Y is At the center and m3 Y is At the right )) (5) In this expression the agent should deﬁne values of modalities m1 , m2 , m3 , describing an arrangement of nodes on axis Y . Let us carry out semantic translation of the ontology (1)–(4), and then linguistic approximation of the expression (5). We have two linguistic variables X, Y . For this set of linguistic variables three bases of schemes of relations {X}, {Y }, {X, Y } are possible. We shall illustrate semantic translation of the ﬁrst axiom of the ontology in detail: (X is At the left ⊃ ⊃ Y is At the right ). The given formula includes atoms X is At the left and Y is At the right. Results of semantic translation of these atoms are fuzzy relations RL [X] =< {X}, M (At the left ) > and RR [Y ] =< {Y }, M (At the right ) >, where RL = M (At the left ) is fuzzy set in universe U = [−1, 1], RR = M (At the right ) is fuzzy set in universe V = [−1, 1]. Fuzzy sets RL and RR (Fig. 7) have the membership functions presented in Tables 2 and 3. Table 2 x

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1

0.8

0.6

0.4

0.2

0

0

0

0

0

0

µRL (x)

Table 3 y

µRR (y)

−1

0

−0.8

0

The result of semantic translation of the formula (X is At the left ⊃ Y is At the right ) = is the fuzzy relation R1 [X, Y ] C{Y } RL [X] ⊃ C{X} RR [Y ], where R1 is a fuzzy g

−0.6

0

−0.4

0

−0.2

0

0

0

0.2

0.2

0.4

0.4

0.6

0.6

0.8

0.8

1

1

144

set in the universe U × V = [−1, 1] × [−1, 1]. For calculation of its membership function it is necessary to receive cylindrical continuation C{X} RR [Y ] of the fuzzy relation RL [X] on a basis of the scheme {Y } and cylindrical continuation C{X} RR [Y ] of the fuzzy relation RR [Y ] on a basis of the scheme {X}. Membership functions of these cylindrical continuations are given in Tables 4 and 5 respectively. They are represented as matrices in which rows are signed with elements of the universe V , and columns — with elements of the universe U . On crossing of columns x and rows y there are values of the membership function µ(x, y).

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

Table 4 x y −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1

1

0.8

0.6

0.4

0.2

0

0

0

0

0

0

−0.8

1

0.8

0.6

0.4

0.2

0

0

0

0

0

0

−0.6

1

0.8

0.6

0.4

0.2

0

0

0

0

0

0

−0.4

1

0.8

0.6

0.4

0.2

0

0

0

0

0

0

−0.2

1

0.8

0.6

0.4

0.2

0

0

0

0

0

0

0

1

0.8

0.6

0.4

0.2

0

0

0

0

0

0

0.2

1

0.8

0.6

0.4

0.2

0

0

0

0

0

0

0.4

1

0.8

0.6

0.4

0.2

0

0

0

0

0

0

0.6

1

0.8

0.6

0.4

0.2

0

0

0

0

0

0

0.8

1

0.8

0.6

0.4

0.2

0

0

0

0

0

0

1

1

0.8

0.6

0.4

0.2

0

0

0

0

0

0

Table 5 x y −1 −0.8 −0.6 −0.4 −0.2

0

0.2 0.4 0.6 0.8

1

−1

0

0

0

0

0

0

0

0

0

0

0

−0.8

0

0

0

0

0

0

0

0

0

0

0

−0.6

0

0

0

0

0

0

0

0

0

0

0

−0.4

0

0

0

0

0

0

0

0

0

0

0

−0.2

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0.2

0.2

0.2

0.2

0.2

0.2

0.2 0.2 0.2 0.2 0.2 0.2

0.4

0.4

0.4

0.4

0.4

0.4

0.4 0.4 0.4 0.4 0.4 0.4

0.6

0.6

0.6

0.6

0.6

0.6

0.6 0.6 0.6 0.6 0.6 0.6

0.8

0.8

0.8

0.8

0.8

0.8

0.8 0.8 0.8 0.8 0.8 0.8

1

1

1

1

1

1

1

1

1

1

1

1

The next step in evaluation of the membership function of the fuzzy relation R1 [X, Y ] is application of implication to the obtained cylindrical continuations. The membership function µR1 (x, y) is given in the matrix form in Table 6.

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Table 6 x y −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 −0.8 0 0 0 0 0 1 1 1 1 1 1 −0.6 0 0 0 0 0 1 1 1 1 1 1 −0.4 0 0 0 0 0 1 1 1 1 1 1 −0.2 0 0 0 0 0 1 1 1 1 1 1 0 0.2 0.2 0.2 0.2 1 1 1 1 1 1 1 0.2 0.4 0.4 0.4 1 1 1 1 1 1 1 1 0.4 0.6 0.6 1 1 1 1 1 1 1 1 1 0.6 0.8 1 1 1 1 1 1 1 1 1 1 0.8 1 1 1 1 1 1 1 1 1 1 1 1

We shall not carry out semantic translation of other axioms of the ontology in detail. We shall only show formulas for semantic translation (left column of Tables 7, 8) and fuzzy relations which are used for calculation (right column of Tables 7, 8). Table 7 The formula of a language X is At the left Y is At the right X is At the center Y is At the center X is At the right Y is At the left

The fuzzy relation RL [X] =< {X}, M (At the left) > RR [Y ] =< {Y }, M (At the right ) > RC [X] =< {X}, M (At the left) > RC [Y ] =< {Y }, M (At the center ) > RR [X] =< {X}, M (At the right) > RL [Y ] =< {Y }, M (At the left ) > Table 8

The formula of language

The fuzzy relation

(X is At the left ⊃ Y is At the right )

R1 [X, Y ] = C{Y } RL [X] ⊃ C{X} RR [Y ]

(X is At the center ⊃ Y is At the center )

R2 [X, Y ] = C{Y } RC [X] ⊃ C{X} RC [Y ]

(Xis At the right ⊃ Y is At the left)

R3 [X, Y ] = C{Y } RR [X] ⊃ C{X } RL [Y ]

g

g

g

(Possible 50 X is At the center and Possible 50 X is At the right)

R4 [X] = λ(2,0) RC [X] ∩ λ(2,0) RR [X]

Final results of semantic translation of the second and third axiom are given in Tables 9 and 10 (membership functions of fuzzy relations R2 [X, Y ] and R3 [X, Y ] in the matrix form). 146

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

Table 9 x y −1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

−1

1

0

0

0

0

0

0

0

0

0

1

−0.8

1

1

0.2

0.2

0.2

0.2

0.2

0.2

0.2

1

1

−0.6

1

1

1

0.4

0.4

0.4

0.4

0.4

1

1

1

−0.4

1

1

1

1

0.6

0.6

0.6

1

1

1

1

−0.2

1

1

1

1

1

0.8

1

1

1

1

1

0

1

1

1

1

1

1

1

1

1

1

1

0.2

1

1

1

1

1

0.8

1

1

1

1

1

0.4

1

1

1

1

0.6

0.6

0.6

1

1

1

1

0.6

1

1

1

0.4

0.4

0.4

0.4

0.4

1

1

1

0.8

1

1

0.2

0.2

0.2

0.2

0.2

0.2

0.2

1

1

1

1

0

0

0

0

0

0

0

0

0

1

Table 10 x y −1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

−1

1

1

1

1

1

1

1

1

1

1

1

−0.8

1

1

1

1

1

1

1

1

1

1

0.8

−0.6

1

1

1

1

1

1

1

1

1

0.6

0.6

−0.4

1

1

1

1

1

1

1

1

0.4

0.4

0.4

−0.2

1

1

1

1

1

1

0.2

0.2

0.2

0.2

0

1

1

1

1

1

1

0

0

0

0

0

0.2

1

1

1

1

1

1

0

0

0

0

0

0.4

1

1

1

1

1

1

0

0

0

0

0

0.6

1

1

1

1

1

1

0

0

0

0

0

0.8

1

1

1

1

1

1

0

0

0

0

0

1

1

1

1

1

1

1

0

0

0

0

0

It is especially needed to emphasize semantic translation of the last axiom of the ontology as it illustrates application of an operation of linear transformation. In Table 11 results of semantic translation of axiom Possible 50 X is At the center and Possible 50 X is At the right are given.

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Table 11

x −1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

µR C l(2,0) µRC (x)

0

0.2

0.4

0.6

0.8

1

0.8

0.6

0.4

0.2

0

0

0.4

0.8

1

1

1

1

1

0.8

0.4

0

µRR (x) l(2,0) µRR (x)

0

0

0

0

0

0

0.2

0.4

0.6

0.8

1

0

0

0

0

0

0

0.4

0.8

1

1

1

µR4 (x)

0

0

0

0

0

0

0.4

0.8

0.8

0.4

0

It should be noted, that the membership function µR4 (x) takes on a value of unity at x = 0.5, however this value is not given in Table 11, as value x = 0.5 drops out of consideration at the accepted digitization of the universe. Thus, semantic translation of axioms is completed and it is possible to pass to inference.

1) R1 [X, Y ],

an axiom,

2) R2 [X, Y ],

an axiom,

3) R3 [X, Y ],

an axiom,

4) R4 [X],

an axiom,

5) R5 [X, Y ] = C{Y } R4 [X],

from 4) by a generalization rule,

6) R6 [X, Y ] = R1 [X, Y ] ∩ R2 [X, Y ],

from 1) and 2) by an intersection rule,

7) R7 [X, Y ] = R6 [X, Y ] ∩ R3 [X, Y ],

from 6) and 3) by an intersection rule,

8) R8 [X, Y ] = R7 [X, Y ] ∩ R5 [X, Y ],

from 7) and 5) by an intersection rule,

9) R9 [Y ] = Proj{Y } R8 [X, Y ],

from 8) by a projection rule.

Each value of the membership function of the fuzzy relation R8 [X, Y ] represents minimum of the appropriate values of membership functions of fuzzy relations R1 [X, Y ], R2 [X, Y ], R3 [X, Y ] and cylindrical continuation onto a basis of the scheme {Y } of the fuzzy relation R4 [X]. The membership function of the fuzzy relation R4 [X] is given in the matrix form in Table 12. The conclusion concerning a value of the coordinate Y represents the fuzzy relation R9 [Y ], which is the projection R8 [X, Y ] onto a basis of the scheme {Y }. The membership function of this fuzzy relation is given in Table 13.

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Table 12 x y

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

−1 0 0 0 0 0 0 0 0 0 0 0

−0.8 0 0 0 0 0 0 0 0 0 0 0

−0.6 0 0 0 0 0 0 0 0 0 0 0

−0.4 0 0 0 0 0 0 0 0 0 0 0

−0.2 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

The result of linguistic approximation of the fuzzy relation R9 [Y ] is the formula (m1 Y is At the left &(m2 Y is At the center & m3 Y is At the right )) in which we should deﬁne modality symbols: m1 , m2 and m3 . We shall ﬁnd degrees of consistency τ1 = = CP(RL [Y ], R9 [Y ]), τ2 = CP(RC [Y ], R9 [Y ]), τ3 = CP(RR [Y ], R9 [Y ]). Their membership functions are calculated as follows: µτ1 (x) = sup{µR9 (y)|x = µL (y)}, µτ2 (x) = sup{µR9 (y)|x = µC (y)}, µτ3 (x) = sup{µR9 (y)|x = µR (y)}.

0.2 0 0.2 0.4 0.4 0.4 0 0 0 0 0 0

0.4 0 0.2 0.4 0.8 0.2 0 0 0 0 0 0

0.6 0 0.2 0.8 0.4 0.2 0 0 0 0 0 0

0.8 0 0.4 0.4 0.4 0.2 0 0 0 0 0 0

1 0 0 0 0 0 0 0 0 0 0 0

Table 13 y –1 –0.8 –0.6 –0.4 –0.2 0 0.2 0.4 0.6 0.8 1

µR9 (y) 0 0.4 0.8 0.8 0.4 0 0 0 0 0 0

The data on the basis of which the calculations are made is given in Table 14, and results of the calculations are given in Table 15. For determination of membership functions of a degree of consistency it is possible to take advantage of the following technique. For example we want to deﬁne membership function of a degree of consistency τ2 = CP(RC [Y ], R9 [Y ]) at x = 0.4. For this purpose it is necessary to determine at what values of an argument µC (y) = 0.4. In this case we have y ∈ {−0.6, 0.6}. Further, it is necessary to choose the greatest value from values µR9 (−0.6) = 0.8 and µR9 (0.6) = 0 (in this case 0.8). Consequently we have µτ2 (0.4) = 0.8. Diagrams of membership functions µτ1 , µτ2 and µτ3 are given in Fig. 8. Diagrams of membership functions approximating semantics of modalities Possible 50 and Impossible 95 are given in the same ﬁgure. VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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Table 15

Table 14 y x = µL (y) x = µC (y) x = µR (y) µR9 (y) –1 1 0 0 0 0.8 0.2 0 0.4 –0.8 0.6 0.4 0 0.8 –0.6 0.4 0.6 0 0.8 –0.4 0.2 0.8 0 0.4 –0.2 0 1 0 0 0 0 0.8 0.2 0 0.2 0 0.6 0.4 0 0.4 0 0.4 0.6 0 0.6 0 0.2 0.8 0 0.8 0 0 1 0 1

x µτ 1 (x) µτ 2 (x) µτ 3 (x) 0

0

0

1

0.2

0.4

0.4

0

0.4

0.8

0.8

0

0.6

0.8

0.8

0

0.8

0.4

0.4

0

1

0

0

0

Fig. 8. Approximation of membership functions

Thus, the result of linguistic approximation of the fuzzy relation R9 [Y ] is the formula (Possible 50 Y is At the left and (Possible 50 Y is At the center and Impossible 95 Y is At the right)) which can be interpreted as “Y is more to the left of the centre”.

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8. Conclusion. In the paper the multiagent hierarchical approach to recognition of images is stated on the basis of use of fuzzy situational calculus. The offered calculus is based on using relations between segments of images. Due to application of modalities the external language, in which the agents ontology is described, is expanded and translation rules for the modal statement are realized. The original solution to the problem of linguistic approximation of inference is also found. Further development of the present approach is supposed in the following directions. 1. Realization of the offered approach in one of the known tools designed for multiagents technology. 2. Approbation of the approach for recognition of images in various areas of applied sciences. 3. Development and substantiation of new formulas of the linguistic approximation adequate to inquiries of users. 4. Development of situational fuzzy calculus towards the insertion of sorts of parameters (situations, actions etc.). 5. Estimation of complexity of solving the recognition problems of various classes. 6. Development of procedures of adaptation and correction of ontologies. 7. Development of procedures of segmentation and preliminary processing of images.

REFERENCES 1. Ushold, M., Gruninger, M. Ontologies: Principles, Methods and Application Knowledge Engineering Review. Vol. 11, No 2, 1996. 2. Fadel, F.G., Fox, M.S., Gruninger, M. A Generic Enterprise Resource Ontology, Proceeding of the third IEEE Workshop on Enabling Technologies: Infrastructure for Collaborative Enterprises, April 1994, Morgantown, West Virginia (WET ICE ’ 94) 3. Russel, S.J., Norvig, P. Artiﬁcial Intelligence. A Modern Approach. NJ, Prentice-Hall, 1998. 4. Baldwin, J.F., Pilsworth, B. W. Axiomatic Approach to Implication for Approximate Reasoning with Fuzzy Logic, Fuzzy Sets and Systems, 1980. – Vol. 3. – PP.193–219. 5. Fu, K.S. Syntactic Pattern Recognition and applications, NJ, Prentice-Hall, 1982. 6. Gary, M.T. et al. A Fuzzy-Attributed Graph Approach to Handwritten Character Recognition, FUZZ- IEEE-93, pp. 570–575, 1993. 7. Guyon et al. UNIPEN Project of On-line Data Exchange and Recognition Benchmarks, 13th IEEE-ICPR, pp. 29–33, Israel, 1994. 8. Jorge, J.A. Fuzzy Relational Grammars for Interactive Gesture Recognition, 2nd International Conf. on Fuzzy Set Theory and Technology, Durham, NC, Oct. 13–16, 1993. 9. Keller, J.M. et al. Evidence Aggregation networks for fuzzy logic inference, IEEE T. on Neural Networks, vol.3, No. 5, pp. 761–769, Sept, 1992. 10. Lan, M.-S. et al. Character Recognition using Fuzzy Rules Extracted from Data, FUZZ-IEEE-94, pp.415–420, Oriando, June, 1994. 11. Lee, E.T., Zadeh, L.A. Note on Fuzzy Languages, Information Sciences-1, pp. 421– 434,1969.

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12. KIette, R., Zamperoni, P. Handbook of Image Processing Operators, Wiley, Chichester, 1996. 13. Malaviya et al. FOHDEL — A Fuzzy Handwriting Description Language, FUZZIEEE, June 1994. 14. Malaviya et al. Automaton Generation of Fuzzy Rule Base for Online Handwriting Recognition, EU FIT-94, Aachen, 1994. 15. Malaviya and Peters, L. Extracting Meaningful Handwriting Features with Fuzzy Aggregation Method, Conf. on Document Analysis and Recognition, Montreal Canada, 1995. 16. Malaviya and Klette, R. A Fuzzy Syntactic Method for On-line Handwriting Recognition, Lecture notes in Computer Science 1121, Springer, Advances in structural syntactical pattern recognition, SSPR 96, pp. 381–392, 1996. 17. Oram, M.W. and Perrett, D.I. Modeling Visual Recognition from Neurobiological Constraints, Neural Networks, vol. 7, No. 6, 7, pp. 945–972, 1994. 18. Shaw, A.C. A Formal Picture Description Scheme as a Basis for Picture Processing Systems, Information and Control–14, pp. 9–52, 1969. 19. Sloman On Designing a Visual System; J.Exp. Theor. A.l., No. pp.289-337,1989. 20. Yau, K.C. and Fu, K.S. A Syntactic Approach to Shape Recognition Using Attributed Grammars, IEEE-SMC-9, No. 6, pp. 334–345, 1979. 21. Tarasov, V.B. From Multiagent Systems to Intellectual Organizations (in Russian). Moscow: URSS, 2002, 348 p. 22. Zadeh, L.A. Outline of New Approach to the Analysis of complex Systems and Decision Processes // IEEE Trans. Syst. – 1973. – Vol. SMC-3. – P. 28–44. 23. Zadeh, L.A. A Theory of Approximate Reasoning // Machine Intelligence. – 1979. – Vol. 9. – P. 149–194. 24. Mizumoto, M., Fukami, S., Tanaka, K. Some Methods for Fuzzy Reasoning // Advances in Fuzzy Set. Theory and Applications / Ed. by M.M. Gupta, R.K. Ragade, R.R. – Amsterdam: North Holland, 1977. – P. 117–136. 25. Zemankova-Leech, M., Kandel, A. Fuzzy Relational Databases: A Key to Expert Systems. – Koln: Verlag TUV Rheinland, 1984. – 180 p. 26. Devyatkov, V.V., Gaidulov, V.A. Complex Ontology of Heat Supply for Analysis of Heat Supply Efﬁciency in Housing Design (in Russian). /Trudy instituta problem upravleniya (Proceeding of Institute for Control Problems). 1999, vol. 1. PP. 80–89. 27. Devyatkov, V.V. Ontologies and Design of Systems (in Russian). /Pribory i sistemy. Upravlenie, kontrol, diagnostika (devices and Systems). Vol. 1, 2000 28. Devyatkov, V.V. Ontologies and their Application (in Russian). /Programnye produkty i sistemy (Program Products and Systems). Vol. 3, 2000. 29. Devyatkov, V.V., Rumbesht, V.V. Fuzzy Modal Situational Calculus for Analysis of Complex Objects (in Russian). Vestnik MGTU. No. 3 (44), 2001, pp. 3–21. V.V. Devyatkov (b. 1939) graduated from the Leningrad Institute for Precise Mechanics and Optics in 1963. D. Sc. (Eng.), Professor, Head of “Information Systems and Telecommunications” department of the Bauman State Technical University. Academician of International Academy of Informatization. Author of over 80 publications in the ﬁeld of logical control, computer systems and complexes, technical cybernetics.

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Ivan Balepin∗ , Sergei Maltsev∗∗ , Jeff Rowe∗ and Karl Levitt∗ (∗ University of California, Davis; ∗∗ Bauman Moscow State Technical University)

USING SPECIFICATION-BASED INTRUSION DETECTION FOR AUTOMATED RESPONSE One of the most controversial issues in intrusion detection is automating responses to intrusions, which can provide an efﬁcient way to react to a progressing attack much quicker and with more precision than a human. However, it comes with several disadvantages that can lead to a waste of resources, and have so far prevented wide acceptance of automated response-enabled systems. We feel that a structured approach to the problem is needed, that will account for the above mentioned disadvantages. In this work, we brieﬂy describe what has been done in the area before. Then we start addressing the problem by coupling automated response with speciﬁcation-based, host-based intrusion detection. We describe the system map, and the map-based action cost model that give us the basis for deciding on response. We also show the process of suspending the attack, and designing the optimal response strategy, even in the presence of uncertainty. Finally, we discuss the implementation issues, our experience with the early automated response agent prototype, the Automated Response Broker (ARB), and suggest topics for further research.

1. Introduction. Automated response to intrusions is an exciting area of research in intrusion detection. A chance to develop a system that would resist attacks carried out or programmed by another human being can be approached in many ways, including the one in which we teach the machine to beat an attacker in the game of intrusion and response. Let us begin by formulating the objectives of our work. Objectives. With growing speed intensity of computer attacks [14] grows the need for quick and well-planned responses. Currently, some of the most intense intrusions are automated. A reliable automated response system, with the right approach, could certainly provide an efﬁcient protection, or a degree of tolerance for all kinds of attacks. However, automated response remains mostly an area of research due to the following issues: • primitive response systems that do not take the cost of intrusion and response into account apply response actions that cause more harm than the intrusion itself; • a large part of commercially available Intrusion Detection Systems (IDS) produces an extensive number of false positive alerts, causing numerous costly response actions [13]. Both cases lead to a denial of service to legitimate users of the system. VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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The objective of this work is to develop a consistent, organized, cost-based approach to automated response that would address these issues. An optimal response would stop the progressing intrusion at early stages, and clean up after it as much as feasible. The scheme that we describe in this work is geared to produce such response. We start addressing the problem by considering the host-based automated response. The key parts of our approach are the basis for response decisions — the system map, the cost model, and the process — response selection even in presence of uncertainty. Let us brieﬂy summarize the work previously done in the area. Related Work. Primitive automated response actions are implemented in some Intrusion Detection Systems (IDS) commercially available today. [13] However, these actions are rather simple and reﬂexive by their nature. Even with a limited response arsenal, many practitioners report that they disable the systems’ intrusion prevention capabilities due to a high number of false positives from IDS’s which give a false basis for response, and also denial of service caused by not very sophisticated response decision strategies. An interesting research work on Survivable Autonomic Response Architecture (SARA) [2] uses the term autonomic response by drawing an analogy with the autonomic nervous system, which automatically controls certain functions of an organism without any conscious input. The authors also propose to have two separate “loops” of response, a local autonomic response, and a global response carried out by the hosts in a system in co-operation. The primary focus of the work is a network with multiple hosts. Alphatech’s Light Autonomic Defense System (αLADS) attempts to utilize control theory when selecting a response [3]. The authors describe it as a part Autonomic Computing, which, according to them, is an emerging area of study of design and construction of self-managed computing systems with a minimum of human interference. Alphatech’s work was most likely oriented towards known systems and system tasks, and it is not applicable to general-purpose computer systems. The focus of this work was most likely on developing a full-scale solution that would have its own proﬁle-based intrusion detection components and it is intended to defend a very speciﬁc range of systems. The issue of compatibility with existing intrusion detection systems has not received much attention in published descriptions of αLADS. However, Alphatech’s work is of interest to further automated response research, since it is one of the few early organized approaches to the problem of quick automated responses. Also, another interesting work on network-oriented automated response that relies on the Control Theory is currently done at UC Davis [4]. Toth, et.al., [5] propose yet another interesting model for automating intrusion response. They suggest approaching the problem of response to network intrusions by constructing dependency trees that model conﬁguration of the network, and they suggest an outline of a cost model for estimating the effect of a response.

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Other signiﬁcant response works include a thorough consideration of some intrusion detection and response cost modeling aspects by Lee, et.al. [6], a response taxonomy by C. Carver and U. Pooch [7], and Fred Cohen’s work on deception [8], which is another very interesting perspective on countering malicious activity. The analysis of related work leads us to conclusion that the primary area of interest, and, consequently, the primary system to model, so far has been a computer network that consists of multiple hosts. The idea of responding at a level of a single host has received relatively little attention. Also, we note that despite the efforts to produce a working cost model for a set of protected resources, no welldeveloped and well-tested model currently exists, that guarantees a consistent and fair representation of protected resources, and their true value. This Work. This paper has the following remaining sections: Section 2 in which we describe the basis for constructing a response chain, Section 3, in which we discuss an implementation of our model, Section 4, which lists possible directions for future work, and, ﬁnally, we sum our work up in Section 5. The reason why we decided to separate the basis for response decisions from implementation notes on our prototype is to attempt to describe a model for host-based response in Section 2 that would not be tied to any particular operating system, and potentially could be used even for applications other than host-based response. 2. Basis for Automated Response. Several pieces of information are necessary in order to plan a sequence of response actions. For the system we are protecting, we need a clear representation of the most valuable resources and also the underlying resources that provide the basic functionality. The true value of some resources (for example, the TCP/IP network service) is heavily inﬂuenced by other resources that depend on them (network is needed by httpd, etc.), and we need a clear way to reﬂect these dependencies before we can decide how to deal with a compromised entity. We also need an organized way to store information about malicious and compromised entities, and to decide how they relate to our key resources. Part of this representation will be highly dynamic, since some entities reﬂected (processes, etc.) are dynamic; however, a large part of it, such as ﬁle structure, program conﬁguration (dependence on ﬁles, sockets, etc.), and system conﬁguration, can be determined statically. We narrow the scope of the problem by noting that transferring an entire computer system to a safe state is a challenging task, and limiting the scope of the problem to returning a set of critical system resources to a reasonably safe and working state. Resources we will model are anything of value in our system — system subjects and objects, ﬁles and running processes, sockets, ﬁle systems, etc. We arrange them in two different ways — the resource type hierarchy and the system map. Resource Type Hierarchy. We found it convenient to group resources by their type, since every such group most likely will have common response actions associated with it. Also, resource types can be arranged in a hierarchy similar to the one in Fig. 1.

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Fig. 1. An example of a response type hierarchy

On a Linux system, for example, we can subdivide resources into ﬁles, sockets, processes, etc., in advance. However, in certain situations, a more speciﬁc category would be appropriate. Consider a ﬁle that contains keys for automatic encryption/decryption of emails using the GNU Privacy Guard software (GPG, [9]). In addition to response action that applies to a more general node in the hierarchy (conﬁguration ﬁles, “restore the conﬁguration ﬁle from backup and restart the corresponding service”), we deﬁne a response action speciﬁc to this sub-category (“also revoke and re-issue the keys”). Since these more speciﬁc categories, and their corresponding response actions, depend highly on the system conﬁguration and nature of the software installed on the system, we cannot deﬁne all of them in advance. We should provide a way for the users of the response system to deﬁne new custom categories with response actions tailored to speciﬁc resources on the target systems. System Map. Although the response type hierarchy is useful for storing general response actions that might apply to a resource, it carries no information about the speciﬁc instances of resources on a system, merely their types. Therefore, we need an additional data structure to satisfy our requirements for a decision basis mentioned above. We suggest representing the necessary information as a directed graph, which we will refer to as the system map. The vertices of the graph, which we will refer to as map nodes, represent the resources in our system. Besides nodes, our map has node templates and edges. Map Nodes. The system map contains important nodes of all types — the system’s priorities. By “important nodes” we mean “all nodes with a non-zero cost”, with cost assigned according to our cost model described in the corresponding subsection. In addition to the priorities, our map also reﬂects the underlying basic resources that these priorities need for proper operation. For example, most applications need a working mounted ﬁle system with read/write access right in order to operate properly. Therefore, if we have applications A, B, and C that are our priorities, we place them on the map along with the node that represents the ﬁle system. We also note that the ﬁle system, an underlying basic resource, does not need to be explicitly speciﬁed as a priority itself, since in this simple example it

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does not have any value of its own. It costs only as much as the priority nodes that depend on it. Each node holds information about the resource it represents. Namely, we need to know the type of the resource according to our type hierarchy. We also need to know some type-speciﬁc information such as path, ﬁlename and inode for type and sub-types of “ﬁle”; PID, name and owner for type and sub-types of “process”, etc. Also, a node also has a cost value associated with it. Some static nodes might have several node templates associated with them in order to later construct dynamic dependent nodes. Finally, every node has a list of applicable response actions associated with it. Node Responses. Every node has a list of basic response actions that restore its functionality. Currently, we require that this list contain only the actions that completely restore the node to a working state. The node’s list of responses is constructed from response actions that are listed for this type of node and its parent types in the type hierarchy. Each such response action has an activation condition associated with it. Referring to the example we have used before, type “conﬁguration ﬁle” would inherit a response “restore from backup” from the parent type “ﬁle”. The activation condition would be, “the node of this type was a target of an illegal write system call” or “the node of this type was a target of an illegal unlink system call”. Another important property of a simple response action is what nodes it affects. Currently, we assume that an action either damages several resources, or does not. If the technique we rely upon for intrusion detection is oriented towards system calls, activation conditions for each response action will be also oriented towards system calls. Since the number of system calls is ﬁnite, and the number of node types is ﬁnite, it seems very feasible to pre-deﬁne response lists for every combination. We will discuss details of creating a response list in the implementation section. We also complement the node’s response list with a response “take no action”. That is an essential response alternative that has a certain cost, just like other responses, and by including it, we will ensure that any response action we take is not more expensive than the intrusion itself. Therefore, an entry in a node’s response list has three ﬁelds: — the action itself (a Linux command, etc.); — the activation criteria; — the list of nodes the action damages. Map Edges. Edges on our map represent dependencies between the resources. If an edge is directed from node A to node B, it means that A provides some service to B, B depends on A, and, most likely, A produces information that B consumes. However, it does not seem feasible to attempt to trace information ﬂow through our map, since it contains nodes that are often times not comparable (for example, ﬁle systems and sockets), and also nodes that obscure information ﬂow (if node A reads from node “ﬁle system”, node B writes to node “ﬁle system”, there is not necessarily an information ﬂow from B to A). Therefore, we do not use our map VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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for intrusion detection. For all information about the intrusion we rely on some detection technique. The true value of the map edges is that they allow us to properly carry out single response actions that involve several nodes (“restart the service that corresponds to this conﬁguration ﬁle”. What service? The one that consumes information from the ﬁle). Also, the edges allow us to collect information about the nodes that depend on a certain node, therefore allowing us to calculate the dynamic cost of the node in our system. Of course, relationships between the nodes can be speciﬁed with greater detail, such as “node A writes to node B that often”, or “node A writes to node B with probability N”. But for our purposes, it is sufﬁcient to only reﬂect the fact that one node provides services to another node, and therefore, the latter depends on the former. Also, some authors model dependency alternatives (node A depends on node B or node C) [5]. From a standpoint of resources of a single host, this is a relatively rare situation, so we will not consider it here. Constructing the Map. As we have mentioned before, the map will have a static part, which will consist of nodes that can be produced by static analysis of our priority resources when no processes are running. The static part of the map will have information about objects of the system, but not subjects. We begin operating just with the static part of the map. Therefore, as the system runs, we will need to add dynamic nodes to the map. In our design, there are ﬁve ways we can add a node to the map. Static nodes are added to the map upon upgrades/reconﬁgurations of the protected system. For the dynamic part, we propose to add new nodes for every subject or object mentioned in the incident alert from the IDS that was not previously on the map. Such nodes would be assigned cost 0, since they were not included in the list of priorities, and they get assigned the most speciﬁc type from the type hierarchy that we are able to determine automatically. Consequently, the node will have a response list that corresponds to its assigned type. Also, as we will describe in later sections, sometimes we will be able to classify a whole group of subjects as malicious, whereas only a few of them might have been explicitly mentioned in alerts. Such situation can occur, for example, when a malicious process caused an alert, and immediately produced a number of children processes that have not yet done anything illegal themselves. We will put the whole related group of subjects on the map, despite the fact that only a few were mentioned in the alert, and they will be treated just like the nodes mentioned in the alerts. We mark the nodes that were mentioned in the alerts, or the ones with their relatives mentioned in the alert, as suspicious, or “contaminated”. Finally, we will have some dynamic nodes that will represent our priority resources. Often times, a resource in general can be mapped to several nodes on our map. For example, a “web server” resource encompasses the executable ﬁle, a number of running processes, and dependent resources (conﬁguration ﬁles, sockets, etc.). At the time of static analysis, we will not have a running instance of a

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web server; however, we can get most of information about the web server process node at that time. By analyzing the static executable ﬁle, we can predict the cost of a running instance of a web server, build the response list, determine the type of the node that will correspond to this process, and also get information about the resources web server depends on. Therefore, at the time of static analysis, with every important executable we create a set of templates, that will characterize the subjects and objects that will later be produced by running the executable ﬁle. A node template is a prototype for building nodes that has all information in place except for the type-speciﬁc information (like PID or ﬁlename) that get ﬁlled upon use of the template. Properties and Beneﬁts of the Map. We are not proposing to build a complete kernel map, or a map of an entire system. Our map has only a few static and dynamic nodes that are critical to the system’s operation. They are not updated periodically; rather, we update them only when signiﬁcant events happen (alerts for dynamic nodes and system re-conﬁguration for static ones). Therefore, if our system runs for a long time without getting attacked, the map will not be updated in order to minimize the overhead. The nodes on the map can be of very different nature, so not always they can be compared directly (for example, ﬁle systems and processes). Let us illustrate some properties of the map with a small example. Suppose, we have a Linux system equipped with System Health and Intrusion Monitoring IDS (SHIM, [11]) that has been compromised, and now has an active malicious process A that was produced by a program B that is not supposed to make system calls from the exec family. Process A has its parent’s speciﬁcations imposed on it by SHIM. Suppose then that process A produces process C, and process C writes to a ﬁle. Since SHIM would promptly alert our response system about A, B and C being involved in several illegal exec system calls, the whole family would appear on the map, and would be marked as malicious. As far as the ﬁle that C has written to, if speciﬁcations for A allowed such behavior, then we would not get an alert about the ﬁle write, and, therefore, would not reﬂect that fact on the map. However, if it was not legal according to A’s speciﬁcations and the system policy, we would get an alert about a possibly contaminated ﬁle, place the corresponding node on the map, and plan our response strategy with that alert in mind. As we have shown above, our map contains all necessary information about our priorities, and resources they need to operate. The map also will reﬂect information about malicious entities, and their relation to our priorities. The map, as we have described it, gives us a solid basis for designing an intrusion response strategy. Cost Model. When deciding on response to a malicious action, most of the time, we will have several alternatives. As we mentioned, one response alternative is always present in any system — take no action. Most of the time, we will ﬁnd some more response actions whose activation criteria matches the current situation. We solve the problem of comparing these alternatives and selecting the optimal one VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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by introducing an action cost model. The cost model helps us pick the best response and also ensure that we don’t cause denial of service to ourselves by performing responses that are more harmful (i.e. more costly to us) than the intrusion itself. Our cost model is based on numerical cost values associated with every map node. Designing a cost model that allows us to quickly associate a number with a resource that precisely reﬂects the value of that resource, is a difﬁcult task. Most of the attempts to produce such a model left it up to the system administrators to determine cost values for their resources. Although it is true that only the system’s owner, familiar with its conﬁguration and primary functions, can point out the true value of the resources, it is very hard to assign the cost values in a consistent manner that would always guarantee optimal response, without exhaustive testing of the system. In our implementation, we rely on ordering the resources by their importance to help producing a cost conﬁguration that would yield optimal response. There are only a few priority nodes that have an actual cost values in our model. For example, let us consider a system with only one such priority — the web server. In the static part of the map, it is represented with the executable ﬁle of the web server. There will be a static node for the ﬁle itself, and it will have a cost of 0. The static node for the executable ﬁle will, however, have a template for web server processes to be created, and that template will have a cost value associated with it. In our model, all process nodes that get created according to that template, will share an equal fraction of the template’s cost with existing processes. For example, suppose the system’s owner has estimated that the web server has a cost x. When there is no web server running, according to our model, the executable ﬁle will have no cost value. If one instance of httpd gets started, its node will get assigned a cost value x. If y nodes of httpd get created, each will get a cost value of x/y. A static node can also get an explicit cost value assigned to its, and not to its templates; or it might not even have any templates. For example, on a certain system, some ﬁles might be indicated as a priority, even though they are not used by any subjects of that system. Cost-wise, another category of nodes on our map is the underlying service nodes. Most likely, these nodes will have a zero cost. However, any harmful action to these nodes will also affect the costly resources that rely on them, and by reﬂecting these dependencies on the map and accounting for them in our response actions, we will take into account the true value of the services. For example, let us consider a system with only two priorities — a web server and an ftp server. These resources depend heavily on the network service to operate. However, the network service does not have any value of its own; its true value is providing part of functionality of our priority resources; and without any dependents, the network service would be useless. Finally, we have all the resources that were not put on our map as a priority resource or its dependency. We assign all such resources cost 0; if they become

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malicious or get involved in an incident, they do get on the map, and a response action that affects these 0-cost nodes even in the most dramatic way will not be harmful for the whole system. Once we determine the cost values for our map nodes based on these factors, we then can associate a cost value with any action that an intruder or the response system takes. We deﬁne the cost of an intrusion action as the sum of costs of map nodes, previously in safe state, that get negatively affected by the action. We deﬁne the beneﬁt of a response action as the sum of costs of nodes, previously in the set of affected nodes, that this response action restores to a working state. Finally, we deﬁne the cost of a response action in terms of costs of the nodes that get negatively affected by the response action (“lost to the intruder”). The goal of a response system is to carry out the response sequence that yields the maximum beneﬁt at the minimum cost. We note that such an approach does not emphasize transferring the system to the ultimate safe state, or completely recovering from an intrusion, since there are situations in which these goals would be much more costly than the intrusion itself. With our approach, we are, however, guaranteed to come up with response strategy that is optimal for the current situation. Response Selection. Once we have the whole picture of the intrusion, our goal is then to “win” the resources on the “contaminated” side back. We start by listing all response alternatives at every contaminated node whose activation condition matches the intrusion. The goal of response selection is to build a response action sequence that will have one action out of a list of every contaminated node. That way, we ensure that every contaminated node is addressed. As mentioned before, an optimal response action is the one that that yields the maximum beneﬁt at the minimum cost. We then assume that a response sequence (response strategy) is optimal if it consists of response actions that are optimal for every node. Therefore, if we have the complete picture of the intrusion, we build the response chain from optimal responses at every node, and carry it out. Managing Uncertainty. Sometimes we might encounter situations where we do not know for certain what exactly the intruder has done. For example, suppose the capabilities allowed the intruder to perform a write call on a ﬁle, which is illegal according to the current system policy. The ﬁle could have been overwritten, appended to, or erased completely (overwritten with an empty string). In certain situations, response actions, and their cost, may vary depending on what has really happened. Then we turn to decision theory, which provides well-deﬁned ways to construct the response plan, for different requirements in presence of uncertainty. The possible results of a write call would be over-written data in the ﬁle, data appended to the ﬁle, or data completely erased from the ﬁle (the latter being a special case of the ﬁrst one). This allows us to list the possible system states. Every one of these states will have a potential damage value and a probability associated with it. Now, using the decision theory convention [10], we can describe VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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the situation with the following “gain matrix”:

A1 ... AN Q

Π1 a11

Π2 a12

aN 1 q1

aN 2 q2

Π3 a13 ... aN 3 q3

Π4 A14

Π5 a15

aN 4 q4

aN 5 q5

where Πi are the possible states, qi are the probabilities, and Ai ’s represent the response alternatives. aij in this matrix, again, represents the usefulness, or beneﬁt, of using the ith decision in case of a jth sub-state. This value can be estimated as: aij = (−εij )γ+1 Bj − ci .

(1)

where Bj is the potential damage of a sub-state; ci — response cost; εij — efﬁciency (beneﬁt) of response i in sub-state j, and γ is 0 if εij = 0, or 1 — otherwise. Considering the above parameters, we observe that the greater the value of ai j, the more useful the corresponding response alternative will be in the corresponding state. We deﬁne the risk of losing in a particular game situation (rij ) as the difference between the player’s gain for strategy Ai for conditions of Πj , and the player’s gain for the strategy he would have chosen, had he known the conditions of Πj . It is clear that had the player known the system state and its conditions in advance, he would have chosen the strategy that yields the maximum gain in its matrix column (mj ). According to our deﬁnition, rij = mj − aij ,

where

mj = max aij .

(2)

i

One important property of risk that follows is that it is always greater than zero. Deﬁned in this way, the concept of risk also reﬂects how favorable a given nature state is to us. Consequently, a risk matrix constructed similarly to the gain matrix, gives us a more complete picture than the gain matrix. Relying on probability signiﬁcantly simpliﬁes the decision making process, especially if we can produce relatively accurate probability estimates using the system history, general knowledge, anomaly analysis tools, etc. A promising way to eliminate the uncertainty, or at least, estimate the values of probability of a certain intrusion sequence, is monitoring the system for a long period of time, and building a proﬁle for important resources. For that, machine learning techniques can be used; also, much can be drawn from the anomaly- based and misuse-based intrusion detection techniques [1]. We will discuss our suggestions in more details in the Section 4. Let us take mathematical expectation of probability-based gain ai to be the effectiveness criterion W that we obviously would like to maximize ai = q1 ai1 + q2 ai2 + . . . + qn ain . 162

(3)

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The optimal strategy is the one that yields the maximum ai in the gain matrix. It would also yield a minimum average risk based on the risk matrix. As we mentioned above, special care must be taken to accurately estimate probability. Pure probability, as a statistics-based value, might not always be available. In that case, it can be subjectively estimated. Certain events might be more likely than others according to the system logs. There are several techniques available that help us quantify these subjective estimates. For cases in which we have no statistical information for the system states, we can assign equal probabilities to each possible state, i.e.: q1 = q2 = . . . = qn = 1/n.

(4)

This approach is called Laplace insufﬁcient reason criterion [10]. For another approach, we assume that we can order possible system states by their likeliness. In order to represent the probabilities in this case, we can use a converging arithmetic series: q1 : q2 : . . . : qn = n : (n − 1) : . . . : 1.

(5)

where: qi =

2(n − i + 1) . n(n + 1)

(6)

We can also rely on expert estimates. If we manage to completely eliminate uncertainty in some situations, the probability values for the determined system state becomes 1, probabilities of all other states become 0, the matrix turns into a single column, and decision making becomes trivial. The Optimal Decision Criteria. There are several methods for selecting the decision criteria in the decision theory [10]. In the Minimax risk criterion (Savage criterion) we select the strategy from the risk matrix that provides us with the minimal risk value under the most unfavorable conditions. The efﬁciency W is then estimated as W = min max rij . The Minimax risk approach allows us to i

j

avoid making the high-risk decisions. The Maximin criterion (Waldt criterion) favors strategies with the largest minimal gain (with W deﬁned differently, see [10]). The Hurwitz criterion is neither pessimistic nor optimistic. Risk-based criterion is analogous to Hurwitz. As we mentioned before, selection of criterion and its parameters is subjective. In any case, however, it is useful to analyze the situation using various approaches. If several criterions indicate that a certain strategy is optimal, it should certainly be selected. If several different criteria suggest different strategies, in the game theory, it is up to the system owner to select (or pre-select) the right strategy based on the fact that some criterion might be preferred over others. 3. Implementation. We have implemented several concepts mentioned in the previous section in a prototype response system, the Automated Response Broker VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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(ARB). ARB is developed for Linux, and it relies on a speciﬁcation-based intrusion detection system, namely, SHIM, for detection. Let us brieﬂy mention why we chose SHIM for that role. Intrusion Detection: SHIM. SHIM is speciﬁcation-based. It relies on the Generic Software Wrapper Toolkit (GSWTK,[12]) for all information about the system calls. SHIM does not try to recognize an attack as a whole. Instead, it relies on a set of speciﬁcations (for programs, or protocols, etc.) that reﬂect the system policy. An interesting feature of SHIM implementation is that it addresses a large part of intrusions by enforcing speciﬁcations for privileged Linux programs. For the majority of current speciﬁcations, system calls of interest are reported by the GSWTK, and then classiﬁed as legal or illegal according to the speciﬁcations, with an alert being issued for the latter. SHIM is a great vehicle for testing our automated response scheme. Such a ﬁne event granularity allows us to catch the exact system call that started the intrusion. Also, the fact that SHIM does not need the whole intrusion to recognize its signature, allows it to catch unknown intrusions, and intrusions that are still in progress. The last feature also gives us a chance to stop an intrusion in progress by responding to the ﬁrst few steps of it that have been detected. The underlying assumption about SHIM that we make is that it always promptly detects and reports all intrusions. Also, SHIM and GSWTK give us a capability to check if a system call is legal according to the system policy before it is executed. However, such mode of operation causes a large overhead for every system call, and does not seem feasible. Map Implementation. We build the map starting with a set of nodes we want to protect. It is the set of all programs that are constrained by SHIM (regardless of whether they are among our priorities; the cost will reﬂect that fact), and several nodes for resources that might not be constrained by SHIM, but the system owner wants to protect as well. The type hierarchy is constructed once upon installation of a system. It does not have a dynamic part and does not change, since it simply contains information about the types of nodes, not the nodes themselves. In ARB, the type hierarchy is constructed in the source code in C++. While it might be sufﬁcient for experiments and testing, obviously, a more convenient interface for conﬁguring the type hierarchy is needed. Currently, we experiment with XML for type hierarchy deﬁnitions. XML so far has proved to be powerful enough to express all the information necessary, and there is an abundance of tools for parsing the type hierarchy deﬁned in XML into our program. Below is an example of a response list of a conﬁguration ﬁle node. Event name and target constitute the activation condition for the action. The victim tag marks the damaged nodes.

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restore attributes; kill offender; offender delete self; kill offender; self

dependents offender restore from backup; ﬁre event(”httpd”,”restart”,true); ... Should XML fail to be descriptive enough for the task, we will design a new domain-speciﬁc language (DSL) for describing the type hierarchy. Similarly to the type hierarchy, the map itself is constructed manually as a collection of C++ data structures. We are also currently experimenting with more ﬂexible ways to deﬁne a map, such as, again, XML or a new DSL. As mentioned above, all components of a system in our prototype are determined manually. However, some of them can be pre-deﬁned for most systems; some can be determined by automated analysis upon installation or reconﬁguration. Eventually, our goal is to let the user of ARB specify just the custom types, responses to custom types, and the system’s priorities. The remainder of the map, such as the basic types, underlying service nodes, all dependencies and templates can be determined automatically. We list the requirements for automating the map construction, and some problems with it, in the future work section. In ARB, the map gets updated only when malicious activity happens, and node’s cost is calculated only when the actual response decision is made, otherwise, much computing resources can be wasted by keeping the map and the node costs up-to-date unnecessarily. Node’s response lists are constructed from the type hierarchy. The set of response actions that are implemented, or will be implemented in the prototype include: delete a ﬁle, restore a ﬁle from backup, restart a service, change permissions, kill process(es), reboot the system, block a connection, re-conﬁgure a ﬁrewall rule, unmount a ﬁle system, change the owner of the process(es), start checkpointing,

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slow down the process(es), roll back to a checkpoint, return a random result, perform a random action, operate on a fake ﬁle, tunnel the process(es) to a sandbox, operate on a fake socket. Node Costs. The most difﬁcult task of any implementation of a response system would be to perform a consistent cost assignment that would reﬂect the true value of resources. This part of map construction cannot be completed in advance, or even automated, since it needs input from the owner of the system. Currently, we approach the problem by ﬁrst manually ordering the key resources of the system, so that the resources (Ri ) are listed in the form, similar to the one below: R0 < R1 = R2 = R . . . < Ri−1 < Ri .

(7)

Then, the least important resource gets assigned priority 1, and the priority of all other (more important) resources is approximated as N times the priority of the next less important one: P riority(Rj ) = N ∗ P riority(Rj−1 ),

(8)

where N is an approximate value and is determined experimentally. Finally, for convenience, we obtain a cost value Ci for a resource Rj from priority values according to the following formula: X Ci = 100∗ P riority(Ri )/ P riority(Ri ), (9) X where P riority(Ri ) is a sum of priorities of all resources. Currently, the process of assignment is completely manual, and all information is reﬂected in the source code of ARB, in C++. However, we plan to add a script that would guide a user of ARB through the cost assignment process. As a result, the script would update the map description, in the language of choice, with the cost values. Also, the map customization interface naturally will provide ways to directly specify the node cost bypassing the script. The cost assignment method described above is certainly only an approximation of the real resource costs. We are working on improving the method to ensure consistency of the cost assignment. Damage Assessment and Response Selection. Let us say ARB received an alert about some malicious actions involving several nodes on our map. Currently, ARB reads alerts from a socket that SHIM writes to. Later, a closer form of integration with SHIM will be developed, since the current implementation scheme might be sufﬁcient for evaluation of response, but it is rather vulnerable to attacks. First, we need to stop the intrusion, if it is still in progress. We partition the map into two sets: a set of nodes that are affected (or might have been affected) by the incident, and a set of nodes that are not dependent on any in the ﬁrst set, and therefore, not affected by the incident at all. After such a partition is done, we then will know what resources are possibly under the attacker’s control. In order to stop the intrusion in progress, we propose to temporarily suspend the subjects on the map involved in the incident, and also all subjects reported in the alert as 166

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suspicious, their children and parents (excluding, of course, PID 1). We freeze the intrusion in progress by going through the hierarchy of suspicious processes and suspending them one by one (by sending them a kill-19 message). We also note that some of the nodes that we put on the map as contaminated were not necessarily explicitly speciﬁed in the alert. Upon an alert that, say, mentions only one subject, the damage assessment procedure of ARB puts the all children this subject may have on the map independently of further alerts. That is done in order to freeze the intrusion in progress before the children attempt to perform further intrusive action, since having a suspicious process as a parent already gives us a right to mark a child process as suspicious as well, without waiting for further alerts. ARB operates with a concept of an incident. Alerts are grouped to form a single incident if they report subjects from the same family as suspicious. ARB considers the damage assessment procedure as complete when it has constructed and has frozen the entire family of suspicious processes. All new alerts are treated as a new incident. The testing of ARB that we have done so far indicates that such approach allows us to clearly separate individual incidents, freeze an incident, assess the damage, and carry out response actions. Upon completion of the damage assessment procedure, we have the suspended intrusion, the frozen suspicious processes, and the complete picture of an intrusion in form of the partitioned map. Finally, the response strategy is built and carried out, as described in the previous sections. Example. Let us demonstrate how ARB carries out the entire process of response selection with an actual example. We will consider a classic vulnerability in the RedHat Linux 6.2 dump utility [15], which examines the ﬁles on a ﬁle system, and determines the ﬁles that need to be backed up. These ﬁles are copied to a disk, tape, or other storage medium. The dump utility depends on the environment variables TAPE and RSH. The goal of the dump exploit is to set the RSH environment variable to an executable ﬁle that will be executed with suid root privileges. In the particular case we will consider, the executable ﬁle will copy ﬁle /bin/bash to /tmp/rootshell, creating a root shell in the /tmp directory. This program will then execute the root shell. The complete speciﬁcations for the dump utility can be found with any distribution of SHIM source code. According to them, dump is allowed to make a few system calls; namely, open and read certain ﬁles, fork, and connect. Consequently, when this intrusion happens on a system that runs SHIM, but is not protected with ARB, the system administrator will get several alerts. Namely, there will be an alert about dump copying the shell executable to the /tmp directory. Another key alert is issued when dump executes ﬁle /tmp/rootshell. Finally, the last alert will be issued when the attacker uses the obtained root shell to issue the open and write system calls to the target. The target in our example will be the ﬁle secring.gpg, which contains the keys the GPG software uses for encryption/decryption.

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Let us ﬁrst show the few relevant parts of the map before the intrusion begins (Fig. 2).

Fig. 2. The map of a part of a computer system before an intrusion. Only a few essential nodes are shown

The map was built according to the type hierarchy in Fig. 1. According to the map, our only priority in the entire system is the gpg program that encrypts and decrypts email messages. However, we also put dump node on the map, since it has SHIM speciﬁcations. Experiments with the ARB prototype showed that it takes a variable period of time for SHIM to issue an alert, and for ARB to receive it and process it. For certain test cases with favorable conditions, that period was not too long, and ARB was actually able to freeze the entire attack right after the ﬁrst alert. For test cases under the least favorable conditions, however, ARB completed the damage assessment procedure only after the attacker already had access to the root shell. Regardless of the current conditions, our goal is to stop the intrusion and clean up after the actions that already happened. Let us re-draw the system map for the worst case that has been observed (Fig. 3). According to the new map (Fig. 3), there are four contaminated nodes as the result of the intrusion. A node for /tmp/rootshell appeared on the map because the ﬁle was involved in an illegal ﬁle copy by cp. However, the cp process itself is gone by the time we ﬁnish damage assessment, so it is not reﬂected on the map. ARB starts building the response sequence addressing node by node, in arbitrary order. The dump process node is an issuer of an illegal exec system call, so ARB chooses the most efﬁcient response — killing the process — since the value of nodes affected by the response is 0. The /tmp/rootshell node was created as a 168

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result of an illegal create system call, and it does not have any cost or dependencies. The matching response would be to remove the ﬁle. Finally, the response for the secring.gpg ﬁle is selected as follows. Several response alternatives apply to the ﬁle, including deleting it, and restoring it from the backup. Deleting the ﬁle would certainly damage it. By using the map, we detect that the gpg process depends on the ﬁle; therefore, deleting the key ﬁle would damage the ﬁle and the process, and the cost of such response would equal to the sum of affected nodes — namely, 100 points. Another alternative with an activation criterion that matches a write system call is restoring the ﬁle from backup, with a cost of 0. We select the second alternative as the least expensive one. Another matching response action is “restart the corresponding service(s)”, and it was inherited by the custom type “key conﬁguration ﬁle” from general type “conﬁguration ﬁle”. By using the map, we determine the corresponding service to be the gpg in this case, and we restart it with a restored key ﬁle. We also note that in this case we do not re-issue the keys, since SHIM has not indicated that the content of the ﬁle has been read. Therefore, the response alternative “re-issue all keys” does not apply, and we do not re-issue the keys. Experience with ARB. The ARB prototype was tested for several well-known attack scripts. Work is in progress to extend it and test it with the broadest range of other intrusions. ARB can only be run on Linux kernel 2.2.14, since the current version of GSWTK relies on that kernel version, and the current version of SHIM relies heavily on GSWTK. As we mentioned before, the map in ARB is built manually for only a subset of all resources that really should be on the map. The current version of ARB does not handle uncertainty in intrusions. It does, however, successfully freeze the set of test attacks, stop them, and respond to them. The attacks we handle include the two examples from this paper. The prototype so far has forced us to re-design our original approach to automated response greatly, and posed several new problems, which were not obvious

Fig. 3. The map of a part of a computer system after ARB has stopped an attack

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before. For example, one such problem was, as we mentioned before, the fact that we did not deﬁne when the damage assessment procedure is complete, and we can actually start deciding and carrying out response. In order to resolve the issue, the concept of an incident was introduced in the prototype. We currently continue to work on the prototype, and we expect promising results from the future work with ARB. 4. Future Work. Automating the Map Construction. First, since with SHIM all malicious actions that involve a map node can be expressed as Linux system calls, and the number of Linux system calls is relatively small, we can automate the generation of nodes’ response lists. For a new type of a node, we list all applicable system calls that this node can make as an activation criteria. Then, we either borrow the corresponding response actions from the type above in the hierarchy, or ask the system user to deﬁne a response action and the damaged nodes. Then we construct a list of applicable system calls that target this type of a node as activation criteria, and obtain the corresponding response and damaged node information in the similar manner. Therefore, we can simplify the task of constructing response lists by guiding the user through the process and producing the output in some convenient format like XML. Also, construction of the map itself and analysis of node dependencies can be done in part automatically. When constructing a map, we can rely, for example, on the program installation package (Linux RedHat Package Manager information, for instance), the program’s source code if available, documentation (man pages), etc., for dependency information about opened ﬁles, sockets, pipes, inter-process communication, etc. Designing a tool that would assist a system’s user with map construction presents an interesting implementation task. Other Directions. As we mentioned before, introducing nodes of type CPU, or memory, or user may allow us to model and respond to denial-of-service attacks. We did not consider the topic in this work, but it seems promising; especially when the intrusion detection technology will provide us with ways to clearly identify denial-of-service intrusions. Storing information about past intrusions and incorporating that knowledge in response also is promising. For example, a large number of attacks in a small period of time might cause the system to take extra response measures targeted at preventing future intrusions rather than responding to ones already in effect. Also, we might design a set of stricter speciﬁcations for the privileged programs that would reﬂect a stricter system policy in response to a large number of intrusions. Another option would be to implement a “pre-emptive” mode as a wrapper in GSWTK: all system calls would be checked in advance, and not carried out if illegal. This mode of operation would cause a large overhead for every single system call; however, it might be useful when trying to counter particularly severe types of intrusions. We would, again, have to carefully weight the cost of switching to such mode. Also, currently, our response model does not consider actions that partially restore a node, and it assumes that an action either damages resources, or does not.

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Considering actions that only partially restore resources and introducing a degree of damage also deserve consideration for further work. Another interesting research direction would be to attempt to combine our hostbased approach to response and network-oriented response mentioned in Section 1, to design a network-wide response system that possibly might be based on single host components, such as ARB, cooperating with each other to protect the entire network. Finally, in our opinion, the most exciting future work option is combining a speciﬁcation-based IDS, features of anomaly- and misuse-based IDS’s and the requires/provides model of intrusions [12] to form basis for response decisions. With SHIM being a “low-level”, system-call oriented IDS that ignores the intrusion as a whole, and focuses on individual constraint violations instead, it is able to catch violations that have never been seen before, and cannot be detected with signaturebased detection systems; whereas signature-based systems can see farther ahead than SHIM, since they have a signature of the entire intrusion. In a situation where we receive several SHIM alerts (which represent the ﬁrst few steps of an intrusion), we can use our system map to calculate the capabilities of the attacker, describing them in JIGSAW [12], and also browse the signature database for all signatures that match the current intrusion at least partially. By using some historical data from an anomaly-based system we can determine probability of each intrusion path (signature), and initiate a game with the intruder. By winning such a game, we will be able to prevent complex intrusions instead of responding to the ones that are already in full progress. 5. Conclusion. In this paper, we gave brief overview of related works, stated the problems associated with automated response, and began addressing them. First, we outlined the basis for our response system, and then described the current implementation, the ARB. The central parts of our approach are the system map, the action cost model, and the game approach to response, which also open up plenty of opportunity for further research.

REFERENCES 1. Amoroso, E: Intrusion Detection: introduction to Internet surveillance, correlation, trace back, traps, and response, Intrusion.net Books, New Jersey. (1999). 2. S. Lewandowski, D. Hook, O. O’Leary, J. Haines, L. Rosse, SARA: Survivable Autonomic Response Architecnire, DISCEX II’01, Anaheim, CA. (June 2001). 3. Alphatech: ALPHATECH Light Autonomic Defense System, http://www.alphatech.com/secondary/techpro/alads.html (last accessed April 6, 2003). 4. Tylutki, M.: Optimal Intrusion Recovery and Response Through Resource and Attack Modeling, Ph.D. Thesis, Davis, CA. (September 2003). 5. Toth, T., Kruegel, C.: Evaluating the impact of automated intrusion response mechanisms, 18th Annual Computer Security Applications Conference, Las Vegas, Nevada. (December 9–13, 2002).

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6. Lee, W., Fan, W., Miller, M., Stolfo, R., Zadok, E. Toward Cost-Sensitive Modeling for Intrusion Detection and Response, Joumal of Computer Security, Vol. 10, Numbers 1, 2 (2002). 7. Carver, C.A, Jr. and Pooch, U.W.: An Intrusion Response Taxonomy and its Role in Automatic Intrusion Response, Proceedings of the 2000 IEEE Workshop n Information Assurance and Security, United States Military Academy, West Point, NY. (6–7 June, 2000). 8. Fred Cohen & Associates, Deception for Protection, http://all.net/journal/deception/index.html (last accessed April 6, 2003). 9. Free Software Foundation, Inc., The GNU Privacy Guard, http://www.gnupg.org (last accessed April 6, 2003). 10. Raiffa, H.: Decision Analysis: Introductory Lectures n Choices under Uncertainty, Addison-Wesley, Reading, . (1968). 11. Ko, C.C.W.: Execution Monitoring of Security-Critical Programs in Distributed System: Specitication-Based Approach, Ph.D. Thesis, Davis, CA. (August 1996). 12. Templeton, S., Levitt, K.: A requires/provides model for computer attacks. In Proceedings of the New Security Paradigms Workshop, Cork, Ireland. (September 2000). 13. Security Focus, Mailing List: FOCUS-DS, http://www.secuitfocus./hiv/9/310579/200302-03/2003-02-09/1 (last accessed April 6, 2003). 14. Staniford, S., Paxson, V., Weaver, N.: How to Own the Internet in Your Spare Time, Proceedings of the 11th USENI.X Security Symposium (2002). 15. Red Hat, Inc.: Red Hat Security Advisory RHS-2000:100-02, http://rhn.rdht.com/rrat/RS-2000-100.html (lst accessed April 6, 2003).

BMSTU Press has published the book: Heat Technology (in Russian) / A.M. Arkharov, V.L. Bondarenko, B.P. Borisov et al. Under general editorship by A.M. Arkharov, V.I. Afanasiev. – 2nd edition, revised and supplemented. – M.: Izd-vo MGTU imeni N.E. Baumana. 2004. – 712 p.

The textbook (1st edition in 1986, edited by V.I. Krutov) has an encyclopedic nature and deals with bases of thermodynamics and heat exchange theory; fuel and its burning; schemes and elements for designing boilers, industrial furnaces, vapor and gas turbine facilities, refrigeration units and compressors, engines of internal and external combustion, rocket and aircraft engines, atom and plasma power units. Calculations of heater systems, ventilation and air conditioning systems are given. Besides, some important materials are included concerning space power facilities; heat exchange apparatus; hydro-machines; photon power systems; cryogenic systems for liqueﬁed gases, air separation, production of neon, krypton, xenon; thermo-electronic and thermo-magnetic low-temperature facilities and control systems as well. Well-known specialists were involved into the activity for writing new parts of the textbook. Great attention was paid to problems of ecology and environment protection. The textbook content corresponds to courses of lectures being delivered by the authors in the Bauman Moscow State Technical University and some other Russian and Foreign universities. For students of the higher educational establishments, learning the “Power Engineering” trade.

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PHYSICS S.M. Korotaev1,2 , A.N. Morozov1 , V.O. Serdyuk2 , Yu.V. Gorokhov3 , S.A. Pulinets3 , V.I. Nalivayko2 , A.V. Novysh2 , S.P. Gaidash2 , H.D. Kanonidy3 (1 Bauman Moscow State Technical University, 2 Geoelectromagnetic Research Institute, Russian Academy of Sciences, 3 Institute of Terrestrial Magnetism, Ionosphere and Radio Wave Propagation, Russian Academy of Sciences)

MANIFESTATION OF MACROSCOPIC NONLOCALITY IN THE PROCESSES OF SOLAR AND GEOMAGNETIC ACTIVITY The results of long-term experiments on study of macroscopic nonlocality of the solar and geomagnetic dissipative processes are presented. Advanced nonlocal detector reaction on these processes has been revealed. Inﬂuence of the solar activity prevails over geomagnetic one. Advancement proved to be of order of value 10–100 days. Level of advanced signal allows us to put forward a problem of employment of the macroscopic nonlocality effect for solar activity forecast.

Introduction. Macroscopic nonlocality consists in correlation of the dissipative processes without any local interaction carriers. At the micro-level quantum nonlocality was studied for a long time, but only recently theoretical reasoning on persisting of nonlocality in the macro-limit has been appeared [1, 2]. Experimentally macroscopic nonlocality was discovered in the early works on causal mechanics, although it was interpreted in another terms [3]. It was suggested that number of statistically reliable but classically impossible correlations between some astrophysical and geophysical processes might be explained by macroscopic nonlocality [3–5]. The examples are: inﬂuence of the solar activity on the meteorological processes, on the physical-chemical insulated lab processes and so on. These two examples point out two ways for study of macroscopic nonlocality. The ﬁrst way consists in study of correlation of the natural processes which, by energetic reasons, must be considered as local-independent. The second way is study of reaction of the insulated probe-process (detector) of well known nature on the external largescale source-processes. Taking into account complexity of the natural processes and embryo of macroscopic nonlocality theory, the ﬁrst way can not be realized so rigorously as it requires for veriﬁcation of a new physical hypothesis. The second way is more perspective and exactly it has been recently realized [6–13]. Bell-type VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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inequalities violation was established and nonlocal detectors reaction on the processes of solar, meteorological, geomagnetic and ionospheric activity was revealed. The most prominent property of these phenomena was (theoretically predicted) availability, side by side with retarded correlation, of advanced one [14, 15]. Moreover a level of the advanced correlation according to the theoretical ideas [6, 8, 9] and the experiment [6–13] proved to be higher than the retarded one (due to the lesser efﬁciency of absorption of the advanced direct particle ﬁeld by the intermediate medium). This work is devoted to more detailed study of effect of the solar activity and is most closely related with it the global geomagnetic activity. Statement of the Problem. The problem consists in study of dependence of the probe-process entropy production on the source-process one, which are related by the macroscopic nonlocality equation [6, 8, 9, 12]. Although any dissipative process can be taken as the probe one, its choice is dictated by relative value of effect and theoretical transparency, allowing us to relate the measured signal with the entropy production and to consciously take exhaustive steps on screening and/or control of all possible local noise factors (temperature, pressure, electromagnetic ﬁeld etc.). From several types of the detectors employed in the experiments [6–12], the most long-term observation series have been obtained with the detector based on the spontaneous variations of self-potentials of weakly polarized electrodes in the electrolyte. Such detector is an element of GEMRI setup for study of macroscopic nonlocality with which the ﬁrst purpose-directed experiment of annual duration was conducted in 1996–97. At present a new synchronous experiment with GEMRI setup (electrode and photocathode detectors) and CAP one (ion mobility detector) is under way. It turned out also that for study of macroscopic nonlocality the other, more prolonged measurements of the self-potentials of weakly polarized electrodes could be used. These measurements were conducted in 1993-97 with technical aim, without exhaustive steps on protection against local noise factors, but nevertheless allowing statistical detection of the macroscopic nonlocality effect [6, 8–13]. In this work all available electrode detectors data is used. As the main source-process we shall consider the solar activity. In the previous works [10–12], the question on the indices of the solar activity most adequate to the given problem was investigated. The detector signals proved to be most correlated with the solar radio wave ﬂux in the middle of the standard frequency range of 245 . . . 15400 MHz — about 1415 MHz, corresponding to radiation from the level of upper chromosphere — lower corona, that is just from the level of maximal dissipation of the sound energy. Therefore in this work we use radio ﬂux data at frequencies 610, 1415 and 2800 MHz. In works [6, 8–13] the detectors reaction on various ionospheric-magnetospheric processes and their indices was also studied. As a result [13] it has been found that the most correlated with detectors signal was Dst — index of geomagnetic activity, which therefore is used in the present work (standard international hourly data).

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The ﬁnal task is revealing the advanced detector reaction on the solar geomagnetic processes. It should be stressed that the electrode detector is not sensitive to the solar radio waves or the geomagnetic ﬁeld. They are only qualitative indices of the source entropy production ( quantitative estimation is also possible, e.g. it is possible to relate geomagnetic variation amplitude via medium impedance and temperature with the entropy production of the source electric current dissipation [6, 8–12]). Special investigation was taken, which shown absence of direct or indirect (for example, via the cosmic rays variations) inﬂuence of the solar-terrestrial electromagnetic ﬁelds on the electrode detector within its sensitivity [10–12]. Data Processing. Data was processed by both the standard methods (spectral and correlational analysis) and by the modern informational-statistical methods (causal analysis). Taking into account the comparative novelty of the causal analysis [16–17], we present here the essence of its formalism. For any processes X and Y via conditional H(X|Y ), H(Y |X) and marginal H(X), H (Y ) Shannon entropies the independence functions i are introduced: iY |X = H(Y |X)/H(Y ),

iX|Y = H(X|Y )/H(X),

0 ≤ i ≤ 1.

Values of i characterize one-sided independence of the processes. If e.g. iX|Y = 0 then X is a single-valued function of Y , if iX|Y = 1 then X is independent of Y . Roughly speaking, values of i behave inversely to module of the correlation coefﬁcient (more exactly such analogue is (1 − iY |X )(1 − iX|Y )). However in contrast to the correlation function, the independence ones equally ﬁt to any (nonlinear) type of dependence of X and Y , but the main thing is that they reﬂect the asymmetry, characteristic of causal-effect relationship. It allows us to introduce the causality function γ: γ = iY |X /iX|Y , 0 ≤ γ < ∞, and to deﬁne that cause Y and effect X are the processes for which γ > 1. If γ < 1, then inversely, X is cause and Y is effect. The case γ = 1 corresponds to the adiabatic (causeless) relationship of X and Y . Claim of retardation of effect relative to cause is introduced then as necessary condition of local connection of X and Y . It has been shown by theoretical and multiple experimental examples that such formal deﬁnition of causality does not contradict its intuitive understanding in obvious situations and can be used in non-obvious ones [17–19]. So, the above-mentioned claim: γ > 1 ⇒ τ = tY − tX < 0 corresponds to the principle of local or strong causality [14]. Phenomenon of quantum nonlocality, however, satisﬁes only weak causality principle according to which X must be retarded relative to Y , if Y is a process controlled (initiated) by an observer. If Y is non-controlled (natural) process, then advancement of X relative to Y is allowed [14]. VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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Thus calculating iX|Y and iY |X as a function of time shift τ , it is possible, by their minima, to ﬁnd optimal time shifts corresponding to transaction X and Y . Then, by value of γ relative to 1, it is possible to establish direction of causal connection. In the case if γ is known to be cause (e.g. Y is index of solar activity), while X — to be effect (e.g. X is detector signal), then for any classical interaction min iX|Y will be observed only at τ < 0, and this minimum would correspond to max γ > 1. Only for nonlocal transaction of X and Y it is possible that γ > 1 at τ > 0. In processing data of observations with the electrode detectors (self-potentials differences U ) was used, which, because of the technical gaps, was divided in two standards: (1) time series, one-hour sampled, with duration no less than 3 months, and (2) time series of daily averaged data with duration no less than a year. It provided an excerption error of i and γ about 1 %. In the spectral analysis the only preprocessing procedure was removing of trend, approximated by a cubic polynomial. In the causal and correlation analysis (as experience of the previous study [6–13] had shown that for detector reaction on the geomagnetic activity the signal/noise ratio was maximal in a range from 1 day to about 2 months, while for the reaction on the solar activity it increased monotonously along the period) the corresponding wide-band ﬁltration was applied. Series of standard (1) were used only for study of relationship of the signal and geomagnetic activity with using the previous low-pass ﬁltration (periods T > 1d ) and removing the cubic trend restricting the maximal period by half of the series length. Series of standard (2) were used for study of the solar and geomagnetic activity with the low-pass ﬁltration (T > 1d ). Nonlocal inﬂuence of other large-scale natural processes, (ﬁrst of all the meteorological activity [10, 12, 13] conﬁnes i below and γ above. At a given stage we will put up with this circumstance, although in future more detailed research this noise inﬂuence can be taken into account by employment of the generalized causal analysis. Results. Examples of synchronous amplitude spectra of solar radio wave ﬂux R at three frequencies, Dst-index and detector signal U are shown in Figs. 1 and 2. All the spectra have two main maxima — at period of solar rotation and its second harmonic. The Dst spectrum is more complicated due to the fact that Dst-variation is formed by multiplex and nonlinear inﬂuence of the Sun on the current, exciting the magnetic ﬁeld (both via source emf variations and via plasma conductivity variations). For realization presented in Fig. 1 the spectrum U is most similar to one R610 — by width of both spectral maxima and by general spectrum shape, including a rise in the most long period domain. Amplitude ratio of the ﬁrst and second harmonics of solar rotation period for U equals 0.95, for Dst — 0.69, R610 — 0.99, R1415 — 0.87, R2800 — 0.69 and thus by this parameter U is also closest to R610 . In the other realization (Fig. 2) the same ratio for U equals 1.2, for Dst — 0.74 and for all R — 1.1, that is in this case in U the ﬁrst harmonic is almost as more than the second one as it is in R, while in Dst, on the contrary, the ﬁrst harmonic essentially less than the second one. By width of the main maxima, and general spectrum shape, U is most similar to R2800 in this case. 176

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Fig. 1. Amplitude spectra of the solar radio wave ﬂux at frequencies 2800 MHz R2800 , 1415 MHz R1415 and 610 MHz R610 , the geomagnetic activity index Dst and detector signal U in the period range T from 10 days to 243 days (ff = 1/T ). Realization 10/26/1994–02/11/1996

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Fig. 2. Amplitude spectra of R2800 , R11415 , Dst and U in the period range T from 10 days to 274 days. Realization 03/16/1996–07/23/1997

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Fig. 3. Independence iX|Y and causality γ function of the detector signal and geomagnetic activity. τ is time shift, days; X = U ; Y = Dst Dst. Realization U 03/16/1994– 06/19/1994 (realization Dst begins 75 days before and ﬁnishes 75 days after realization U)

So spectral analysis points out the certainly closer relation of the detector signal with the solar activity proper than with its effect — geomagnetic activity. However optimal frequency of the solar radio wave ﬂux, reﬂecting level of the sourceprocesses in the solar atmosphere, may change in time. Turn now to the causal analysis of data. In [13] it was established that the deepest minima of both independence function and the highest maxima of the causality function (as well as the main extrema of the correlation one) were observed at advancement of the detector signal relative to Dst by 1–2 month. This conclusion was made using data of both GEMRI detectors and two time series data of non-protected detector and was illustrated by an example of Dst forecast with advancement of 33 days. In the present study all the remained series satisﬁng the standard (1) were processed. Such series proved to be 6, and the conclusion made before has not changed. Figure 3 illustrates it. The deepest minimum of iX|Y and single maximum of γ are observed at advancement of the detector signal U = X relative to Dst = Y by 34 days. Thus detector registers the advanced signal from the process of dissipation of the currents, exciting the Dst-variation. This process has essentially less power as compared with the solar (and meteorological) dissipative processes competing by inﬂuence on the detector, therefore due to a small signal/noise ratio, iX|Y and γ sligtly differ from 1. It should be noted that in data processed by the standard (2), due to the growing low-frequency direct inﬂuence of the solar activity, effect of inﬂuence Dst on U turned out insigniﬁcant. How much stronger the detector reaction is on the processes of solar activity can be seen in Fig. 4–6, where results of causal analysis for the year corresponding VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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to beginning of the present cycle are shown. In the advanced domain (τ > 0) values of the independence function of the detector signal U = X relative to R = Y at the all three frequencies are much lower than in the retarded domain , and the causality function is much more than 1. The deepest minimum iX|Y = 0.47 and the highest maximum γ = 1.58 are observed for Y = R2800 at τ = 42d . The latter value is close to estimation (about month) obtained in [10–12] by a simpler way. A big τ -interval corresponding to signiﬁcant γ > 1 is explained by a big volume of the solar atmosphere occupied by the source-processes with diffusion propagation. A common feature of reﬂected in Fig. 4–6 advanced connection of the detector signal with the radio ﬂux is availability of three γ pikes. For the frequency 2800 MHz these pikes correspond to advancement: 42, 119 and 280 days; for 1415 MHz: 63, 133 and 280 days; for 610 MHz: 98, 217 and 280 days. The most distant, third peak (τ = 280d ) is common for the all three levels of radio ﬂux generation and therefore it corresponds to the process of the most spatial scale. From comparison of the results of causal and correlation analysis the nonlinearity of relation of U and R follows. The main maxima of correlation function coincide by position with minima of iX|Y (maxima of γ) but their relative values are different. So at frequency 2800 MHz there is maximum of correlation r corresponding to the main γ pike: at τ = 42d r = 0.50 ± 0.03 (correlation is signiﬁcant with reliability not less than 0.999 [12]), but the biggest is maximum corresponding to the third γ peak: at τ = 280d r = 0.72 ± 0.02.

Fig. 4. Independence and causality functions of the detector signal and solar radio ﬂux at frequency 2800 MHz. X = U , Y = R2800 . Realization U 12/11/1996–12/10/1997 (realization R2800 begins 1 year before and ﬁnishes 1 year after realization U )

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Y = R1415 ) Fig. 5. The same as in Fig. 4 for frequency 1415 MHz (Y

Y = R610 ) Fig. 6. The same as in Fig. 4 for frequency 610 MHz (Y

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If instead of observed radio ﬂux data, we take thatadjusted to 1 A.U. (published in “Solar-Geophysical Data”), the dependence of U on R slightly decreases. So for frequency 2800 MHz at τ = 42d γ decreases 1.03 times, r decreases 1.11 times, iX|Y increases 1.01 times. Thus the weak affect of the Earth orbit ellipsisity also reﬂects in detector reaction. In Figs. 4–6 it is seen that the ﬁrst and second γ pikes move to larger τ as the frequency falls of the frequency (rise of the source level). Further, it is seen that with the fall of the frequency the larger γ values shift to the domain of larger τ (with a small decrease of γ and iX|Y extremal deﬂections) — for Y = R2800 : γ = 1.58, iX|Y = 0.47 (τ = 42d ); for Y = R1415 : γ = 1.56, iX|Y = 0.48 (τ = 63d ); for Y = R610 : γ = 1.56, iX|Y = 0.49 (τ = 280d ). These peculiarities indicate that processes at upper levels are activated later than at lower ones (diffusion of activity in the solar atmosphere goes upwards). Thus in spite of the unusual method of solar activity registration — by nonlocal insulated lab probe-process reaction, the result is quite natural. Availability of such essential advanced detector reaction on the solar activity gives a sense to an attempt of performance of the forecast problem. As it is seen in Figs. 4–6, relation of U and R is far from δ-correlated (the domain of advanced dependence is spread in wide τ range). Therefore a forecast algorithm must be based on plural regression — one forecasted value R is calculated as convolution of impulse transition characteristic with multitude of the preceding U values. In addition, the problem of optimal ﬁltration, suppressing interference from nonlocal inﬂuence of other natural processes, ﬁrst of all, the meteorological activity must

Fig. 7. U variation forecasting R2800 one with advancement of 42 days. The origin of time count (days) corresponds to 12/11/1996 182

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be previously solved. Elaboration of such algorithm is a complicated though quite standard task. For the present we will conﬁne ourselves by wittingly primitive simplest demonstration of the possibility of forecast. We select the highest observed γ pike corresponding to Y = R2800 , τ = 42d and shift by this τ the series X(= U ) and Y . The result is shown in Fig. 7. It is seen that detector signal really forecasts the solar activity. Conclusion. Effect of macroscopic nonlocality is reliably registered for the processes of solar and geomagnetic activity. Nonlocal detector reaction directly on the process of solar activity essentially prevails. The most prominent peculiarity of nonlocal correlations is their advanced character. Time of advancement is large: of the order of 10–100 days. There is a sense to put forward the problem of employment of macroscopic nonlocality effect for solar activity forecasting. Acknowledgement. This work was supported by RBRF and Moscow Region Government (grants 02-05-64006 and 01-05-97015).

REFERENCES 1. Mermin, N.D. Extreme quantum entanglement in a superposition of macroscopically distinct states // Phys. Rev. Lett. – 1990. V. 65. No 15. P. 1838–1840. 2. Home, D. & Majumdar, A.S. Incompatibility between quantum mechanics and classical realism in the strong macroscopic limit // Phys. Rev. A. 1995. V. 52. No 6. P. 4959–4962. 3. Kozyrev, N.A. On the possibility of experimental investigation of the properties of time // Time in Science and Philosophy / Ed. Zenam J. Prague: Academia, 1971. P. 111– 132. 4. Korotaev, S.M. Logic of causal mechanics: observations-theory-experiments // On the Way to Understanding the Time Phenomenon, Part 2 / Ed. Levich A.P. World Scientiﬁc, 1996, P. 60–74. 5. Zhirblis, V.E. Starts and koltsars // On the Way to Understanding the Time Phenomenon, Part 2 / Ed. Levich A.P. Word Scientiﬁc, 1996. P. 135–173. 6. Korotaev, S.M., Sorokin, M.O., Serdyuk, V.O. & Abramov, J.M. Experimental study of the nonlocal interaction of the macroscopic dissipative processes // Physical Though of Russia. 1998. No 2. P. 1–17. 7. Dvoruk, S.K., Korotaev, S.M., Morozov, A.N., Nazolin, A.L., Serdyuk, V.O., Solovyov, A.V., Sorokin, M.O., Tabalin, S.E. & Shishkin, G.V. Experimental studies of the irreversible processes in the electrolytes // Applied Mechanics and MachineBuilding Technologies. 1998. V. 1. No 4. P. 61–66. 8. Korotaev, S.M., Sorokin, M.O., Serdyuk, V.O. & Abramov, J.M. Experimental study of macroscopic nonlocality // Science & Technology in Russia. 1999. No 1. P. 16–19. 9. Korotaev, S.M., Sorokin, M.O., Serdyuk, V.O. & Abramov, J.M. Geophisical manifestation of interaction of the processes through the active properties of time // Physics and Chemistry of the Earth. A. 1999. V. 24. No 8. P. 735–740. 10. Korotaev, S.M., Serdyuk, V.O. & Sorokin, M.O. Experimental veriﬁcation of Kozyrev’s interaction of natural processes// Galilean Electrodynamics. 2000. V. 11. No 2. P. 23– 28. 11. Korotaev, S.M., Serdyuk, V.O. & Sorokin, M.O. Effect of macroscopic nonlocality on geomagnetic and solar-ionospheric processes // Geomagnetism and Aeronomy. 2000. V. 40. No 3. P. 323–330. VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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12. Korotaev, S.M., Morozov, A.N., Serdyuk, V.O. & Sorokin, M.O. Manifestation of macroscopic nonlocality in some natural dissipative processes // Russian Phisics Journal. 2002. No 5. P. 3–1413. 13. Korotaev, S.M., Morozov, A.N., Gorohov, J.V., Nalivayko, V.I., Novysh, A.V., Pulinets, S.A. & Serdyuk, V.O. Experimental study of macroscopic nonlocality of some magnetospheric-ionospheric and tropospheric processes. 14. Cramer, J.G. Generalized absorber theory and Enstain–Podolsky–Rosen paradox // Phys. Rev. D. V. 22. P. 362–376. 15. Korotaev, S.M. The force of time // Galilean Electrodynamics. 2000. V. 11. No 2. P. 29–33. 16. Korotaev, S.M. On the possibility of causal analysis of the geophysical processes // Geomagnetism and Aeronomy. 1993. V. 33. No 2. P. 128–133. 17. Korotaev, S.M. Role of different deﬁnitions of the entropy in the causal analysis and its employment to electromagnetic induction of the sea currents // Geomagnetism and Aeronomy. 1995. V. 35. No 3. P. 116–125. 18. Hachay, O.A., Korotaev, S.M. & Troyanov, A.K. Results of application of the causal analysis for bore-hole data on seismo-acoustic and electromagnetic emission // Volcalonogy and Seismology. 1992. No 3. P. 92–100. 19. Korotaev, S.M., Hachay, O.A. & Shabelyansky, S.V. Causal analysis of the process of horizontal informational diffusion of electromagnetic ﬁeld in the ocean // Geomagnetism and Aeronomy. 1993. V. 33. P. 116–125. 20. Korotaev, S.M., Shabelynsky, S.V. & Serdyuk, V.O. Generalized causal analysis and its employment for study of electromagnetic ﬁeld in the ocean // Izvestia Physics of the Solid Earth. 1992. No 6. P. 77–86. Korotaev Sergey Maratovich, born in 1950, graduated from LHMI in 1972, D. Sc. (Phys.-Math.), Head of Lab of GEMRI RAS, Professor of BMSTU. Author of more than 130 publications, including 3 monographs, in the area of electrodynamics, causal mechanics and informational-statistical methods of analysis of physical experimental data. A.N. Morozov, born in 1959, graduated from the Bauman Moscow Higher Technical School in 1981. D. Sc. (Phys.-Math.), professor, head of “Physics” Department of the Bauman Moscow State Technical University. Author of more than 100 publications in the ﬁeld of high precision measuring systems and physical kinetics theory. Serdyuk Vyacheslav Olegovich, born in 1969, graduated from the Lomonosov Moscow State University in 1992. Researcher of GEMRI RAS. Author of about 30 publications in the area of geophysics, statistical electrodynamics and causal analysis. Gorokhov Yury Vasilyevich, born in 1950, graduated from the Lomonosov Moscow State University in 1974, Ph. D. (Phys.-Math.), senior researcher of IZMIRAN RAS, author of about 40 publications in the area of radiophysics and solar-terrestrial physics. Pulinets Sergey Alexandrovich, born in 1949, graduated from the Lomonosov Moscow State University in 1972, D. Sc. (Phys.-Math.), Professor, Vice-director of IZMIRAN RAS. Author of about 180 publications in the area of solar-terrestrial physics. Nalivayko Valentin Iosifovich, born in 1928, graduated from OEIC in 1953, senior researcher of GEMRI RAS, author of about 50 publications in the area of techniques of weak electromagnetic ﬁelds measurements and solar-terrestrial physics. Novysh Andrew Vsevolodovich, born in 1947, graduated from Moscow Institute for Fine Chemical Technology, leading engineer-electronicist of GEMRI RAS. Author of 7 publications in the area of techniques of physical experiment.

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Gaidash Sergey Petrovich, born in 1947, graduated from Radiophysical Faculty of KhIRE, senior researcher of GEMRI RAS, author of about 50 publications in the area of techniques of weak electromagnetic ﬁelds measurements and solar- terrestrial physics. Kanonidi Harlampiy Dmitrievich, born in 1932, graduated from the Lomonosov Moscow State University in 1956, Ph. D. (Phys.-Math.), Director of Geophysical Observation Center of IZMIRAN RAS, author of about 70 publications, including 6 monographs, in the area of solar-terrestrial physics.

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185

M.A. Yakovlev (Bauman Moscow State Technical University)

GENERATION OF THE NEAR-SURFACE ELECTRON LAYER BY PICOSECOND LASER PULSES The inﬂuence of the self-consistent electric ﬁeld of a near-surface electron layer on the energy spectrum of photoelectrons emitted under the action of a picosecond laser pulse on a metallic target is determined.

Near-Surface Electron Layer. The main reason for the formation of a nearsurface electron layer under the action of ultranarrow laser pulses on a metallic target is a substantial gap (of the order of 1 eV) between the temperature of the electronic component and that of the lattice [5]. This causes a sharp increase in thermoemission current and the formation of a fairly extended region of the volume negative charge near the surface. The double electric layer near the surface of a metal is known to exist also under the conditions of thermodynamic equilibrium between the electronic and lattice subsystems. In this case, the electron gas is degenerate, and the concentration of electrons ne decreases at a very high rate as the distance from the surface z increases: ne ∝ z −2 exp(−βz), where β −1 is a value of the order of the mean interelectronic distance in the metal [7]. The near-surface layer of degenerate electrons can therefore be considered arbitrarily thin, and its inﬂuence on laser beam penetration inside the metal can be ignored. The degree of degeneracy, however, decreases sharply as the concentration of electrons lowers, and electrons experience the transition to the classical state when the Fermi energy EF (ne ) becomes of the order of kT . A further decrease in the concentration of electrons follows a much smoother law [6], µ ¶−2 z ne (z) = n0 1 + √ , (1) 2Ld where Ld = (ε0 kT /e2 n0 )1/2 is the Debye screening length and n0 is the concentration boundary determined from the degeneracy condition EF (n0 ) ≈ kT, that is, n0 ∝ T 3/2 . It follows that the size of the near-surface electron layer region, where the concentration decreases according to Eq. (1), increases as the electron temperature 186

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rises. Simultaneously, concentration boundary n0 also increases, and, at temperatures of the order of EF , all electrons in the near-surface electron layer experience the transition to the classical state. Their distribution then obeys Eq. (1), and n0 becomes of the order of the concentration of electrons in the metal. Thus, intense nonequilibrium heating of the electronic component of a metal can cause the formation of a fairly extended layer of electrons with a high concentration near the surface. The formation of a near-surface electron layer, whose electrophysical characteristics enable it to substantially inﬂuence near-surface processes under the action of ultranarrow laser pulses, is only possible within limited laser pulse intensity Iem and width τp intervals, min max Iem ≤ Iem ≤ Iem ,

τpmin ≤ τp ≤ τpmax .

min The Iem lower intensity boundary is determined from the condition that a fairly effective gap between the electronic component temperature Te and the lattice temperature Tl should be attained during the pulse duration,

∆T = Te − Tl ∼ TF ∼ 104 . . . 105 K, where TF is the temperature of the degenerate electronic subsystem; that is [8], min ≈ αl l∆T, Iem

where αl ∼ 1016 W/(m3 K) is the energy exchange rate between the electronic and lattice subsystems and l is the depth of the metal layer heated during the pulse √ duration, l ∼ max[δ, χτp ] (δ is the skin depth, and χ is the electron thermal min diffusivity). Picosecond pulses are characterized by l ∼ 10−5 cm, that is, Iem ∼ 9 2 ∼ 10 W/cm . According to [5], when picosecond laser pulses of intensity Iem ∼ 3 × × 109 W/cm2 act on the surface of typical metals, two competing processes of electron escape from the surface, namely, thermoemission and photoemission, produce current of the same order of magnitude. We can therefore expect that the collective thermoemission process should most noticeably inﬂuence the oneparticle photoemission process under these conditions. Thermoemission begins to predominate over photoemission as the intensity of radiation increases; that is, because of thermoemission, the formation of the near-surface electron layer at the speciﬁed radiation intensities occurs in time τs ≈ Ld /vT ∼ 10−14 . . . 10−13 s. Here, vT the thermal velocity of electrons [6]. Note in addition that the nature of thermoemission is then substantially different from that of thermoemission from an electrode in a closed circuit, because thermoemission that we are considering occurs from an insulated metallic surface, on which an uncompensated positive charge remains, and a volume negative charge is formed near the surface, which in turn inﬂuences the thermoemission current. This considerably complicates the VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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description of the formation of the near-surface electron layer. Nevertheless, the time τs , of formation of the spatial distribution is much shorter than the picosecond laser pulse width τp . The spatial distribution of the near-surface electron layer can therefore be considered stationary with the corresponding layer temperature virtually over the whole pulse width. max corresponds to the prethreshold region of The upper intensity boundary Iem the beginning of melting and ablation of the target material. Such processes are observed when the density of the energy accumulated in the electronic subsystem under the action of an ultranarrow laser pulse exceeds a certain threshold, max Iem τp ≤ Fabl ,

where the threshold energy density of laser ablation Fabl ≈ 0.2 . . . 0.5 J/cm2 max [9, 10]; that is, at the pulse width of τp ∼ 1 ps, we have Iem ∼ 1012 W/cm2 . The limitations imposed on the width of laser pulses τp are determined from the condition τpmin ≥ τs , where τs ∼ 10−13 s is the time of formation of the near-surface electron layer with a nondegenerate electronic component, and τpmax ≤ τel , where τel is the characteristic time of energy transfer from electrons to the lattice. The second condition guarantees that the lattice is not heated during a laser pulse and the pulse width is insufﬁcient for creating conditions of developed vaporization of the target material, when the role played by the near-surface electron layer loses signiﬁcance. To summarize, it follows from the results described above that the near-surface electron layer can exert substantial inﬂuence on surface processes within limited laser pulse intensity and width ranges, which are, however, fairly important for technical applications, namely, 1010 W/cm2 ≤ Iem ≤ 1012 W/cm2 and 10−13 ≤ ≤ τp ≤ 10−11 s respectively. The Electric Field of the Near-surface Electron Layer. As in [11], the selfconsistent electric ﬁeld of the near-surface electron layer Ez created by the positive charge of the conductor and electrons under surface irradiation by picosecond laser pulses was found by solving the set of equations that described the electronic component temperature variations with time and electron layer formation. The ionization of the neutral gas taken into account in [11] could, however, be ignored at pressures and picosecond laser pulse intensities under consideration (of about 1 atm and ∼1010 W/cm2 , respectively). Let us turn to the system of equations that describe the near-surface electron layer formation near the surface of a condensed substance. First consider the heat equation. At z < 0 (a conducting condensed substance), we have · ¸ ∂Te ∂Te ∂ Cm (2) = χm − α(Te − Tl ) + q(z, t). ∂t ∂z ∂z 188

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Here, Cm and χm are the heat capacity and the heat conductivity of the conducting condensed substance, respectively, and α is the coefﬁcient of heat exchange between electrons and the lattice, which is virtually independent of the electron temperature [12]: π 2 m c2s νeff nm α= , 6 Tl where νef f = νef f (Tl ) and cs is the velocity of sound in the conducting condensed substance. For typical metals with nm ∼ 1022 cm−3 , we have α∼1010 W/(cm3 K). The heat capacity and heat conductivity of the condensed substance were approximated by the asymptotic equations [12]

Cm

χm

µ ¶ 2 π kTe k n , m 2 EF = 3 n k, m 2 Te χm0 T , l = 1 nm k v 2e , νef f 2

kTe ¿ EF ,

kTe ≥ EF ;

kTe ¿ EF ,

kTe ≥ EF ,

where v e is the mean thermal velocity of electrons and χm0 is the equilibrium thermal conductivity coefﬁcient of the condensed conducting substance at initial temperature T0 . The q(z, t) volume energy release function has the form q(z, t) = κr κi k0 ε0 c|E0 |2 exp(2k0 κi z), where κr and κi are the refractive indices of the condensed conducting substance, k0 is a wavenumber (k0 = ω/c) and E0 is the amplitude of the wave ﬁeld at z = 0. The heat equation for a gas (z > 0) has the form · ¸ 1 ∂ ∂Te 3 ∂Te e2 |E|2 νe 3m k = k(Te − Ta )νe + , (3) χe − 2 ∂t ne ∂z ∂z M 2m(ω 2 + νe2 ) where M is the atomic weight of the gas, νe = σa na v e is the frequency of electronic collisions in the gas, and σa = σa (v e ) is the transport cross section of electron scattering by atoms. The σa (v e ) dependence is well known for rare gases. We used the data from [13] in our calculations. The equations describing variations in Tl — lattice temperature and Ta — temperature of the heavy gas component (atoms) — have the form Cl

∂Tl = α(Te − Tl ); ∂t

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where Cl is the heat capacity of the lattice: Cl ≈ 3nl k. Problem conditions correspond to very fast processes (t ¿ τl ∼10−10 s), and, according to the equations given above, the temperatures of the lattice and the heavy gas component change very insigniﬁcantly during such time intervals. For instance, even at the mean electron temperature Te ∼10 eV, the temperature of the lattice changes by a value of the order of 102 K in time t∼10−13 s. For this reason, we assumed in our calculations that Tl = Ta = T0 . The boundary conditions for (2) and (3) have the form ∂Te = 0, ∂z

z = −lm , la , (4)

·

¸ ∂Te χ = 0, ∂z

z = 0,

where lm and la are the boundaries of the region under consideration in the condensed conducting substance and gas, respectively (formally, lm , la → ∞). Next, consider the equations that describe the kinetics of the near-surface electron layer formation. The continuity equation for the concentration of electrons should be solved simultaneously with the Maxwell equations in the z < 0 and z > 0 regions. It is, however, sufﬁcient to consider the continuity equation in the z > 0 region rather than in the whole space. The boundary condition at z = 0 should then correspond to distributions of electrons of the near-surface electron layer Eq. (1) as functions of temperature (see above), ne |z=0 = n0 , where n0 is the boundary concentration of the near-surface electron layer at a given time moment (note that electrons of the near-surface electron layer form an ideal Coulomb system at T ≥ 5 eV). There is sufﬁcient time for the concentration proﬁle of the near-surface electron layer near the surface to keep track of temperature variations, because the characteristic time of the formation of this layer is about ωp−1 . Consider the continuity equation for the electronic component, ¸ · ∂ ∂ne ∂ne = + µ e E z ne , (5) De ∂t ∂z ∂z where De and µe , are the diffusion and mobility coefﬁcients deﬁned as µe =

e ; m e νe

De =

v 2e . 3νe

The boundary conditions for Eq. (5) have the form z=0:

ne = n 0 ;

z = la :

∂ne + µe Ez ne = 0. De ∂z

(6)

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The initial condition for ne corresponds to the distribution of near-surface layer electrons at the initial temperature T0 . The distribution of the longitudinal electric ﬁeld Ez is found from the equation ∂Ez ene . =− ∂z ε0

(7)

The boundary condition that corresponds to this equation at each time step has the form Zla e Ez |z=0 = ne) dz. ε0 0

REFERENCES 1. Farkas, G. & Toth, C. // Phys. Rev. A, 41, 4123 (1990). 2. Logothetis, E.M. & Hartman, P.L. // Phys. Rev. 187, 460 (1969). 3. Petite, G., Agostini, P., Trainham, R., et al. // Phys. Rev. B 45, 12210 (1992). 4. Varro, S. & Ehlotzky, F. // Phys. Rev. A 57, 663 (1998). 5. Anisimov, S.I., Kapelovich B.L. & Perel’man, T.L. “Electron Emission from Metal ´ Surface Under Ultranarrow Pulse Laser Radiation” (in Russian) // Zh. Eksp. Teor. Fiz., 66, 776(1974) [Sov. Phys. JETP 39, 375 (1974)]. 6. Ivlev, A.V., Pavlov, K.B. & Yakovlev, M.A. “Interaction of Radiation with Near-Surface Electron Layer” (in Russian) // Zh. Tekh. Fiz., 64 (9), 50(1994) [Tekh. Phys. 39, 888 (1994)]. 7. Gupta, A.K. & Singwi, K.S. “Gradient Corrections to the Exchange-Correlation Energy of Electrons at Metal Surfaces” // Phys. Rev. B 15, 1801(1977). 8. Anisimov, S.I. & Retfel’d, B. ”To Theory of Interaction of Ultranarrow Laser Pulse with Metal” (in Russian) // Izv. Akad. Nauk, Ser. Fiz. 61, 1642 (1997). 9. Afanas’ev, Yu. V., Demchenko, N.N., Zavestkovskaya, I.N., et al. “Simulation of Metal Ablation by Ultranarrow Laser Pulses” (in Russian) // Izv. Akad Nauk Ser. Fis. 63, 667 (1999) 10. Riley, D., Langley, A.J., Taday, P.F., et al. “Reﬂectivity Experiments with 60 Femtosecond Laser Pulses” // J.Phys. D. 31, 515 (1998). 11. Ivlev, A.V., Yakovlev, M.A., Bordenyuk, A.N. “Simulation of Gaz Breakdown by Boundary Layer Electrons while Irradiating Metallic Target by Picosecond Laser Pulses” (in Russian) // Zh. Tekh. Fiz., 68(8), 42(1998) [Tech. Phys. 43, 921(1998)]. 12. Anisimov, S.I., Makshantsev, B.I., Barsukov A.V. Opt. Acoust. Rev. 1,251(1990). 13. Kieffer, L.J. At. Data 2(4), 293(1971). M.A. Yakovlev (b. 1946), graduated from the Moscow State University n.a. M.V. Lomonosov in 1971. D. Sc. (Phys.-Math.), professor of “Physics” department of the Bauman Moscow State Technical University. Author of more than 100 publications in the ﬁeld of weakly-ionized plasma and interaction of concentrated energy ﬂuxes with substance. Author of the scientiﬁc discovery “Regularity of reduction of atom ionization potential in dense plasma”. VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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V.O. Gladyshev, T.M. Gladysheva, A.N. Morozov, V.Ye. Zubarev (Bauman Moscow State Technical University)

THE RESULTS OF THE EXPERIMENT ON REGISTRATING LIGHT DRAGGING OBSERVED IN AN INTERFEROMETER WITH A ROTATING OPTICAL DISC The interferometer, in which light beams propagate in a rotating optical disk, is considered in the work. Interference pattern shift (∆ = = 0.0076 of a bandwidth) has been registrated experimentally for beams, which passed through the rotating optical disk with a diameter of 120 mm and thickness of 30 mm in two opposite directions. It is shown that the shift can be explained with the longitudinal effect of light dragging and the effect of deviation from Snell’s law, when tangential break of velocity takes place. The value of the interference pattern shift as a result of light trajectory inclination due to a deviation from Snell’s law in a rotating medium is about 25 % due to the longitudinal dragging of light. The results represent the ﬁrst experimental test of the solution of the dispersion equation in the 3-dimensional case of medium motion.

1. Introduction. Effect of light dragging by a moving medium consists in dependence of the phase velocity of light in a moving medium upon the velocity of the medium [1]. This phenomenon is of a fundamental nature in moving medium optics, and provides a classical example of the relativistic velocity composition law and is a basis for creation of moving medium electrodynamics. When an electromagnetic wave is incident onto the moving medium with the tangential velocity break, a deviation of the refraction angle value in a moving medium from the one in a static medium occurs. As a result, light beam trajectories curve in a homogeneous medium with a complex motion [2]. Experimental registration of these effects is a new method of check of the relativity theory and the existing understanding of light interactions with a moving medium in a general case. The goal of this work is to develop an interferometry method for investigating the spatial effect of the light dragging in a rotating medium, to create an interferometer scheme and also to detect a phase shift for beams propagated in a rotating medium. 2. Estimation of the Expected Shift of an Interference Pattern. For achieving this goal the double-beam two-pass disk interferometer was designed and built with inputting beams into the ﬂat surface of an optical disk (Fig. 1). In this scheme, the light beam from laser L incident on a beam divider BD was split into two beams. These beams entered the optical disk OD (one of them after reﬂecting from the mirror M) to be reﬂected from ﬂat mirror surfaces of the OD. The exit beams reﬂected from angle prism AP changed paths, passed through the

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Fig. 1. The optical scheme of the interferometer with a rotating optical disk

optical disk in the reverse direction, and entered the divider again. Mixed on the divider mirror, the beams passed through an optical system OS to display the interference pattern on a screen S. The light intensity was measured by a photodetector PD. The signal from a photodetector came through a resistor system into oscillograph. The optical disk rotation in two opposite directions was provided by a motor. The digital camera Kodak DC240 with high resolution and a recording element took photos of oscillograms (25 tests for each combination of rotation direction and an adjustment) and further they were processed and analyzed with the help of a personal computer. Compensating difference features of the scheme provided a high protection against mechanical disturbances. The interferometer is tuned to the interference fringes of equal thickness and the shift of the light interference pattern is determined using the time signal from the photodetector, when the optical disk OD rotates. The direction of the interference pattern (IP) shift in the plane of the IP analysis depends on the rotation direction. This method of picking out the time signal is preferred as compared to the method of measuring the intensity variations, because the time of IP motion is measured by the equipment, which has a higher relative resolution and stability of parameters. After adjusting the OD, the spots of beams moved as ellipses on a screen, when we slowly rotated the OD. Ellipsisity of the curves is explained with the photoelastic effect. After adjusting the optical system, checking stability of the motor operation, and achieving a stable IP, the experiment was carried out. VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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Fig. 2. The optical disk OD. The upper surface of OD has the partial reﬂective layer with the radius R1 , the lower has full reﬂective cover. Due to the deviation from Snell’s law, the exit point of a beam B for the stationary disk shifts from its position to point B 0 for the rotating disk

When a light beam propagates in a rotating optical disk, the vector of a linear velocity changes its direction in space from point to point, therefore the longitudinal effect of light dragging and the transverse deviation of a beam trajectory should be observed. To estimate the light dragging effect, let us consider the solution of the dispersion equation in the plane XOZ and in the plane of incident and reﬂected beams. In our experiment (Fig. 2) a beam is incident (out of a medium with n0 = 1) onto the ﬂat surface of the optical disk with the refractive index n2 at the angle ϑ0 . In the plane XOZ the electromagnetic wave vector projections on the axis OX and OZ in the moving medium are determined as follows [3] k2x = k0x = (ω0 /c) sin ϑ0 , k2z =

¢1/2 i ¡ ω0 h −κ2 γ22 β2z ξ2 η2 ± η2 cos2 ϑ0 + κ2 γ22 ξ22 η22 , c

(1) (2)

where 2 η2−1 = 1 − κ2 γ22 β2z , κ2 = ε2 µ2 − 1, u2x u2z 2 2 = 1 − β2x − β2z , β2x = , β2z = , c c

ξ2 = 1 − β2x sin ϑ0 , γ2−2

ω0 — electromagnetic wave frequency, c — phase light velocity in vacuum, ε2 , µ2 — dielectric and magnetic permeabilities of the medium, u2x , u2z — components of the medium velocity, ϑ0 — is the incident angle.

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As the longitudinal light dragging effect is determined by the longitudinal component of a linear velocity of the medium, so we can consider (2) when β2x = 0: q ¡ ¢ 2 2 −κ γ β ± 1 + κ2 γ22 1 − β2z 2 2 2z ω0 k2z = , (3) 2 c 1 − κ2 γ22 β2z where cos ϑ0 = 1. 2 Considering β2z À β2z , we obtain the expression for the light phase velocity in a rotating medium µ ¶ ω0 c 1 0 ∼ ± u2 1 − 2 , (4) c =p 2 = 2 n2 n2 k2x + k2z √ where n2 = ε2 µ2 — index of refraction. Orientation of ~k2 and ~u2 varies along a beam trajectory, therefore for a spatial case the projection of the medium velocity vector onto the wave vector should be used instead of ~u2 in (4): pr~k ~u2 = u2z cos ϑ sin ϑ2 + u2x sin ϑ sin ϑ2 ,

(5)

where u2x = ω(R − z), u2z = ωx, R = OA, ϑ2 — angle of refraction. Further we will take into account that the phase accumulation due to trajectory curving in the plane XOZ is not large in comparison with the longitudinal effect [2]. Then we can show that pr~k ~u2 = ωR sin ϑ sin ϑ2 in any point of the trajectory. When beams pass through the rotating OD once, the IP shift is equal to ∆0 =

µ

t1 = cos ϑ2

c (t2 − t1 ) , λ

d µ ¶ ¶, 1 c + 1 − 2 ωR sin ϑ sin ϑ2 n2 n2

(6)

t2 =

n2 d , c cos ϑ2

(7)

where λ — wave length of light; t1 , t2 — time of passing through the rotating and static OD, respectively; ω — OD angular velocity and d — OD thickness. After substituting t1 , t2 into (6), we obtain in the limit ωR ¿ c: √ κ2 ωr R2 − r 2 . (8) ∆0 = λc Here r is the distance between projection of beam path in OD and the center of rotation, r = OA sin ϑ (Fig. 2). By specifying the value R we can ﬁnd an optimal value r, for which ∆0 will be √ ∂∆0 = 0 we obtain the optimal value r = R/ 2. maximum. From the equation ∂r The determined value of IP shift ∆ in the interferometer should be equal to 32∆0 . First of all, the disk edge surfaces are mirror coated, which increases the optical path twice as much. After reﬂecting from a prismatic reﬂector the beams VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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repeatedly pass through OD, which also increases the resulting IP shift by a factor of 2. In the scheme we use two opposite beams and two directions of rotation which increases the resulting IP shift by a factor of 4. Carrying out the experiments for two different adjustments allows another increase of the resulting IP shift by a factor of 2. As a result of this we can write ∆=

16lu2n (n22 − 1) , λc

(9)

where l = AB 0 is the projection of beam path in OD onto the ﬂat surface of the disk (see Fig. 2), u2n = ωr is the medium linear velocity along the light beam trajectory. In reality, the IP shift can be increased due to the refraction angle ϑ˜2 differing from the one calculated by Snell’s law (ϑ2 ). As the trajectory curvature is slight, we consider that a beam propagates in the plane POY (see Fig. 2). The projection of a wave vector on the axis P can be ﬁnd from the solution of 2 : the dispersion equation, supposing β2y = 0 and β2z À β2z ´ p ω0 ³ k2pz = −κ2 β2z + cos2 ϑ0 + κ2 , (10) c where ϑ0 — the angle of refraction. The refraction angle in a moving medium tgϑ˜2 = −κ2 β2z

sin ϑ0 √ . + cos2 ϑ0 + κ2

(11)

An increase of the optical path length in OD is equal to the difference between the equivalent optical path le in a rotation medium and the one in a static medium (l0e ): µ ¶ 1 1 ˜ ∆le = le − l0e = 4dn2 − . (12) cos ϑ˜2 cos ϑ2 If OD is stationary, a beam passes the path in air. The increase of the optical path length is equal to µ ¶ 1 ˜ . (13) ∆le = ∆le 1 − n2 This value determine the additional difference of the beam path, for its passing through the OD once in one direction, due to a deviation from Snell’s law. The resulting value of the additional shift due to this effect with account of all passes of two beams for all directions and adjustments in our experiment is equal to ∆le ∆S = 16 . (14) λ Also, we can estimate the value of the additional IP shift as a result of the curving of a beam trajectory in a rotating medium. However for our scheme the 196

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opposite beams have curved trajectories, so the resulting value of the IP shift will tend to zero. The total shift due to the light dragging effect and the deviation from Snell’s law, is (15) ∆Σ = ∆ + ∆ S . The expected shift was calculated with the following interferometer parameters used in our experiment: l = 0.0892 m, r = 0.024 m, ω ∼ = 25.26 Hz, λ = 0.632991 µm, n2 = 1.48, ϑ0 = 62◦ : ∆ = 0.0054 of a bandwidth, ∆S = 0.0013 of a bandwidth,

(16)

∆Σ = 0.0067 of a bandwidth. Therefore, in the experiment it should be measured ∆Σ = 0.0067 of a bandwidth. Registration of Spatial Effect of Light Dragging in a Rotating Medium. In the experiment the gas-atomic stabilised laser LGN-302, operating in continuous mode, was as a light source. The optical disk was a standard plate of a diameter 120 mm and thickness of 30 mm and was made of LK5 grade glass. One edge surface of a disk as a whole and a part of its another edge surface (a circle with a diameter of 80 mm) were mirror-coated to provide the reﬂection coefﬁcient ρ = 0.9. In the experiment we used the asynchronous three-phase motor. The average period of the OD rotation is T¯ = 0.0396 s which corresponds to the rotation frequency ν = 25.26 Hz. Deviation of the rotation speed was within ±1 %. To decrease the inﬂuence of vibrations, the optical system and the motor with OD were placed on different platforms and tables (Fig. 1). When we used the ﬁrst adjustment and the counterclockwise direction of motor rotation, interference fringes inclined to the right and were reset to the original position (Fig. 3, a). When the OD was slowly rotating, it was observed that three bands passed across the photodetector aperture in the forward and reverse directions. Between the two motions the stop point was observed. The photodetector was set in such a way that in the stop position its intensity value was between minimum and maximum of the band (Fig. 3). The main scale for the experiment was 1 ms per a division so that points 1–7 could be accommodated on the oscillograph screen. The readings tkj , where k is a number of the point, were taken from oscillograms. Thereupon the values were calculated, using δtj = (t5j − t3j )/2,

j = 1, 25;

(17)

∆tj = (t3j − t2j + t6j − t5j )/2,

(18)

where δtj corresponds to the time distance value between the IP stop point and the nearest interference band and ∆tj corresponds to the time width of the interference fringe. VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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After this, we processed the oscillograms, using the clockwise direction of motor rotation, in the same way (Fig. 3, b). The next adjustment was achieved with inclining interference fringes to the left. Two directions of motor rotation were also used for the second adjustment. In the experiment we detected time coordinates of the points tk while IP moving, so it is necessary to convert them to the spatial ones. Processing the Experimental Data. The relation ∆t and δt determines the relative position of an interference fringe as fractions of the time bandwidth. It was noticed that beams in a screen moved in ellipses. An ellipse arc length is determined using the elliptic integral of the second kind. We can change the values ∆t and δt for the ellipse arc lengths δL and ∆LΣ , respectively, expressed in radians:

δL = π

δt , T

∆LΣ = δL + ∆L/2 = π

δt + ∆t , T

(19)

Fig. 3. The dependence of relative voltage of the photodetector PD on time when the ﬁrst adjustment for the right (aa ) and left (bb ) directions of rotation was used 198

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Zϕ5 q 1 − sin2 α sin ϕdϕ = aE(ϕ5 \α),

δL = a

(20)

0

Zϕ6 q 1 − sin2 α sin ϕ dϕ = aE(ϕ6 \α),

∆LΣ = a

(21)

0

where sin2 α = 1 − b2 /a2 , a, b — major and minor ellipse half-axes, T — period of IP vibrations, E(ϕ5 \α), E(ϕ6 \α) — elliptic integral of the second kind, ϕ — parameter of the elliptic integral, ϕ5 , ϕ6 — parameters referred to the points 5 and 6. Substituting the values ∆t and δt, which were measured in the experiment, into (16), we can get ∆L and δLΣ . It follows from (16)–(18) that a = 1 because the values ∆t and δt are normalized to the period T . Then, using (17), (18), relation b/a, taken from the experiment, and tables of elliptic integrals of the second kind, we can determine ϕ5 and ϕ6 . The relative position of a band from the point 4 is determined as

∆=

δy 1 − cos ϕ5 = , ∆y 2(cos ϕ5 − cos ϕ6 )

(22)

where δy is the spatial position of a band from the point 4 and ∆y is the spatial bandwidth δy = b(1 − cos ϕ5 ), ∆y = 2b(1 − cos ϕ6 − δy). For two directions of rotation and two adjustments of interfering beams we can obtain the resulting measured value of IP shift: ∆Σ = (∆1 − ∆2 ) − (∆3 − ∆4 ).

(23)

Let us notice that calculating the IP shift is carried out for the values, which are normalized to a rotation period and interference band width. Thus the results of the calculation don’t depend on period vibrations and bandwidth for one measurement or another. For the series of the experimental data ∆i the conﬁdence intervals were calculated with the conﬁdence probability p = 0.9. Then the resulting value of shift was equal to = 0.0076 ± 0.0030. ∆Exp Σ

(24)

The theoretical magnitude ∆Σ = 0.0067 appears in the conﬁdence interval (21), = 0.0076 is about 13 %. moreover the relative error for the average value ∆Exp Σ Conclution. The results of theoretical calculations of the expected IP shift for the used parameters in the experiment are in a good agreement with the experimental results of the IP shift. It should be noted that the rotation velocity in the experiment was not high. But the spatial light dragging effect had an inﬂuence on the IP shift. VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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Therefore, electromagnetic wave dragging by a rotating medium should be taken into account for using non-relativistic velocities of a medium, and the effect may have an impact on the results of different measurement procedures. In order to make the ﬁnal conclusion that the experimental results are in agreement with predictions of the relativity, it is necessary to increase the experiment accuracy. It may be achieved by using a more stable motor with the higher frequency of rotation and more sophisticated system of transforming the signal from a photodetector. This work was supported by Grants Council of the President of the Russian Federation (grant No MD-170.2003.08).

REFERENCES 1. Fizeau, D’H. // 1859. Ann. de Chimie et de Phys. 57, p. 385. 2. Gladyshev, V.O. // 1993. JETP Lett. 58, pp. 569–572. 3. Bolotovsky, B.M., Stolyarov, S.N. (in Russian). // 1988. Uspekhi Fiz. Nauk 156, No 1, pp. 153–176. Gladyshev Vladimir (b. 1966) graduated from Bauman Moscow Higher Technical School in 1989. D. Sc. (Phys.-Math.), Prof. of Physics Department of Bauman Moscow Technical University. The main ﬁeld of scientiﬁc interests is electrodynamics of moving medium and the theory of relativity effects. Author of 80 publications, including 2 monographs, in the ﬁeld of theoretical physics. Gladysheva Tatyana Mikhailovna (b. 1966) Ph.D. (Eng.), assoc. professor of Physics Department of Bauman Moscow State Technical University. Author of 25 scientiﬁc publications in the ﬁeld of optical interferometry. The main ﬁeld of scientiﬁc interests is the precise optical interferometry, the moving medium optics. Morozov Andrei (b. 1959) graduated from Bauman Moscow Higher Technical School in 1981. Dr. Sc. (Phys.-Math.), Prof. and Head of Physics Department of Bauman Moscow Technical University, General Director of Applied Physics Center of Bauman Moscow Technical University. Main Editor of series “Natural Sciences” of journal “Vestnik MGTU”. Chairman of Organizing Committee Conference “Irreversible processes in Nature and Science”. The main ﬁeld of scientiﬁc interests is precision measurements and physical kinetics. Author of 150 publications and inventions, including 4 monographs. Zubarev Vyatcheslav (b. 1934) graduated from Bauman Moscow Higher Technical School. Ph. D. (Eng.), Assoc. Prof. of “Optoelectronic devices for Scientiﬁc Research” Department of BMSTU. Assoc. professor of “Optical and optoelectronic systems” Department; assoc professor of “Optoelectronic devices for Scientiﬁc Research” Department of the Bauman Moscow State Technical University. Author of about 60 publications, 6 inventions and patents and 4 monographs. The main ﬁeld of scientiﬁc interests is interferometers, spectral and polarizable instruments.

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M.C. Duffy (Liverpool University, Sunderland University), V.O. Gladyshev, A.N. Morozov (Bauman Moscow State Technical University), P. Rowlands (Department of Physics, University of Liverpool) PHYSICAL INTERPRETATIONS OF RELATIVITY THEORY (On the materials of International Scientiﬁc Conference in Bauman Moscow State Technical University, 30th June – 3rd July, 2003)

The international scientiﬁc conference “Physical Interpretations of Relativity Theory” (PIRT) was held in Moscow from 30th June to 3rd July, 2003. A program of the conference included more than 80 reports of representatives of leading scientiﬁc schools of Russia and other countries. The presented papers dealt with the following major themes: • Cosmology, Gravitation and Space-Time Structure. • Time, Reference Frames and the Fundamentals of Relativity. • Nature and Models of the Physical Vacuum. • Formal Structures and Physical Interpretations of Relativity Theory. • Experimental Aspects of Relativity. • The Poincare-Lorentz and the Einstein-Minkowski Expositions of the Relativity Principle. • The Relativistic World Ether: The Ether Geometrized; Dirac’s Ether and Ether Formulations of Relativity. • Analogues of Relativity and Quantum Mechanics. The Vortex Sponge. • Historical and Philosophical Aspects of Relativity. The main problems, which were discussed in the meeting, were theoretical and experimental examinations of physical theories, being based on Relativity principles, and also on interpretation of Relativity and its consequences. The considerable progress, linked with development and practical applications of the Relativity, was noted by participants. It was also discussed that at present the existing experimental grounds do not give a synonymous answer to some principle questions, including the question on existence of gravitational radiation, which was predicted by A. Einstein in his work [1], devoted to the solution of the equations of General Relativity and the calculation of gravitational radiation power. During past decades it was proposed a number of theories of gravity alternative to the General Relativity, and more general theories such as supergravity theory and theory of superstrings, multidimensional gravity theories, which in parameterized VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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post-Newtonian formalism give the same predictions for results of such classical experiments as veriﬁcation of the equivalence principle for massive bodies, comparison of indications of moving electromagnetic clocks, deviation of light rays, signal delay, perihelion precession, time variation of gravitational constant, Lense and Thirring effect etc. In this connection progress in Relativity development is often linked with newgeneration experiments — detection of gravitational waves (GW). In approximation of a weak ﬁeld, when the metric gµν slightly differs from Minkowski’s metric ηµν , A. Einstein obtained the equation ¤h0µν = 0, which contains a solution for a plane monochromatic gravitational wave in vacuum. It is an important circumstance that the equation and also quadruple formulae for power of gravitational radiation are derived in linear approximation, where as the powerful gravitational radiation can only arise in the ﬁeld of strong nonlinearity when it is needed to consider together the moving bodies and gravitation ﬁeld. In this connection the results of experimental works of T.M. Taylor, J.M. Weisberg and others [2, 3] give conﬁdence. In these works the investigation of the delay effect for a period of the binary stellar system PSR 1913+16 due to losses of energy in gravitational radiation was carried out. Hence, investigators consider that a direct test of GW detection should become its direction with ground-based or space gravitational antennas, most likely, with interferometers. Using the interferometer with coherent optical pumping for gravitation waves was proposed at ﬁrst in a work of the participant of the conference M.E. Gertsenshtein (M.V. Lomonosov Moscow State University) and V.N. Pustovoit [4]. An analysis of the supposed methods of GW detection by different authors leads to the conclusion that just the interferometer detection of gravitation radiation is promising [5]. Moreover, it leads to the conclusion that up-to-date test of Relativity and alternative theories is possible for only limiting acceptable accuracies of measurements, demanding to widely apply laws of general and applied physics. As a veriﬁcation is the fact that at present the success in implementing the similar ground-based GW experiment is achieved in Californian and Massachusetts Technological Universities, having the present-day technical base and high-qualiﬁcation specialists. On the whole, we can conclude that recent achievements in the ﬁeld of gravity, astrophysics, laser physics, computing, cosmonautics have led to possibility to create the ground-based and spacecraft-borne GW observatories, the global satellitenavigation systems [6–11]. ”High-tech” technological level, realized in the projects, allows one to fulﬁll measuring physical parameters such as distance, time, energy, impulse with a precision close to standard quantum limits [12], it is a needed condition for testing modern theories of gravity (report of V.N. Melnikov, Russian Gravity Society, Moscow), multi-dimensional unique theories (report of Yu.V. Vladimirov, Lomonosov Moscow State University), cosmological models of global evolution of the Universe (report of S.V. Chervon, Ulyanovsk State University).

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Operating the GW observatories and satellite systems on the level of precision demands the high stability of parameters, connection to world-wide time service and taking into account the motion of celestial bodies, activity of the Sun, seismic activity of the Earth, overfalls of temperature, pressure, electrical and magnetic ﬁelds, inﬂuence of cosmic rays etc. [13, 14]. The creation of the systems promoted developing new methods of limiting measurements and stimulated originating metrological procedures, which would more fully represent possibilities of modern devices. Now a demand has arisen for a detail description of electromagnetic processes with account of effects of moving media optics [15], for a precise space-time description of all elements of measuring channel [16], including astrophysical sources of radiation [17], for developing new mathematical methods of picking out and processing signals, for creating numerical methods. In the report of C.F. Levin (Moscow Institute of Expert Appraisal and Tests of State Standards of Russia) it was shown that experimental testing the Relativity principles requests to use the appropriate methods of mathematical processing of measurement results, when not only the estimation of the measured magnitude is used but also its distribution. New technological level requires to take into account dynamic features of all elements of measurement systems, including motion parameters of individual optical elements, effect of light pressure on the elements [18], transformation of ﬁne specter structure of a radiation source due to relative motion of the source and the detector [19], inﬂuence of medium of propagation of electromagnetic wave on its parameters [20, 21]. Here we can take into consideration the deep relation in a description of particles or bodies, free moving in curved space-time continuum, and light quanta, whose motion equations are the equations of geodesic lines: j k d 2 xi i dx dx + Γ = 0, jk dψ 2 dψ dψ

where Γijk are the Christoffel’s symbols of the second rank. The analogy between the descriptions allows us to simulate an inﬂuence of gravitational ﬁeld on moving macroscopic bodies by including optical nonhomogeneity, which distorts ray trajectories. Let us notice that the description of optical experiments, being accompanied, for example, with propagation of light beams in an interferometer, is usually made on the assumption that the gravitation ﬁeld of the Earth is absent. Hence, in the case, light also propagates along geodesic lines in a gravitation ﬁeld of a rotating mass. Therefore, applying a model of a free-falling interferometer is but an approximation to the reality. Together with that in a real interferometer for GW detection the mirrors represent a pendulum, being in the ﬁeld of light wave pressure. The motion equations have a form N X ˆ i ; ω0i \xi ) = 1 G(β Fni (t), Mi n=1

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2 1 0 ˆ i ; ω0i \xi ) = d xi + 2βi d xi + ω 2 d xi is a linear operator, acting on a where G(β 0i dt2 dt1 dt0 coordinate xi ; index i corresponds to the equation for the mirror Si ; βi is a damping factor; ω0i is a fundamental frequency of a test body with the mirror Si mass Mi , Fni (t) represents external actions, N is a natural number. The function Fni (t) contains different actions, including light pressure on mirrors and GW perturbation. Evidently, in the reality the situation is more complex, as we should take into account the non-homogeneous distribution of dielectric and magnetic features of light propagation media that reﬂects the real technology of optics manufacturing and inﬂuence of temperature and other inherent factors. And at last, in a general case, we should take into consideration that the velocity of motion of the medium, which can be characterized with a three-dimensional velocity ﬁeld, varying in time, inﬂuences a shift of the electromagnetic wave plane due to phenomena, which are similar to the Fizeau’s effect. The factors, taken separately, can lead to distortion of lights rays, and more possible for its combined inﬂuence. Therefore, by describing electromagnetic radiation in tasks of gravitation radiation detection, location and navigation, it is needed to use solutions of the dispersion equation for each local ﬁeld of a moving medium. In the case, projections of wave vectors ~ki for refracted and reﬂected waves in a plane case with tangential break of a velocity on boundaries between two ﬁelds for i-ﬁelds of the medium will be expressed as ´ ³ 1/2 2 ~ ~ γ (β − β ) 1 − d β β + κ i i it ± Qi in I1 , (kin )1,2 = c 1 − β 2 − κi γi2 (β − βin )2

~k0t = ~k1,1t = ~k2,1t = ~k2,2t = I~t , ω0 − ~k0~v = ω1,1 − ~k1,1~v = ω2,1 − ~k2,1~v = ω2,2 − ~k2,2~v = I1 , ui β = v/c, κi = εi µi − 1, βi = , c h i ¡ ¢ 2 2 2 2 2 2 Qi = 1 + κi γi 1 − βin − d 1 − β − κi γi (β − βin ) − h i ¢ ¡ −κi γ 2 d~β~it 2 (1 − ββin ) − 1 − β 2 d~β~it .

γi−2 = 1 − βi2 ,

d~ = −cI~t /I1 ,

i

The invariant I~t reﬂects an equivalence of the tangential components of wave vectors for all waves, and I1 reﬂects an equivalence of frequencies for all waves, conjugated in the reference frame, where the boundary of the medium rests. The magnitudes ~ui , εi , and µi are velocities, dielectric and magnetic permittivities, respectively; ω0 , ωi are frequencies of electromagnetic waves in vacuum and in the i-th medium ﬁeld, respectively. Here, a model with a plane boundary between media, which moves with the velocity ~v , perpendicularly to the boundary, is considered. Evidently, using the given approach is justiﬁed, if the wave length λ is much more than a distinctive size of non-homogeneity of the medium a, i.e. λ À a.

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But the main parameters, inﬂuencing the additional accumulation of phase, are the medium velocity and the velocity of a boundary between media. It should be noted that even a hardly detected velocity of the boundary can lead to the considerable contribution to the wave phase difference. For example, after n reﬂecting cycles the ray, having passed in Fabry-Perot interferometer, in which mirrors oscillate with velocities of β1 , β2 under the action of GW splash, will have a frequency µ ¶ 1 + β 1 1 + β2 n ω n = ω0 . 1 − β 1 1 − β2 The additional phase difference, appearing due to a frequency shift of an electromagnetic wave after n reﬂections in the limit β1 ∼ = β2 = β ¿ 1, is equal to ∆ϕω = 2 (n + 1) ω0 βt, where t = nt0 ; t0 is time for one passage of a ray along the resonator length. It is clear that the value can have an order of the phase difference, stipulated by the sought shift for mirrors ∆ϕx = 2nk0 ∆x, where k0 = 2π/λ0 . The arguments lead to the conclusion that Relativity development is possible with the composite approach to considering its consequences which was reﬂected in some reports. For example, in the Marmet’s report an analysis of Radio-Doppler and ranging data from distant Pioneer 10 and 11 spacecrafts indicated an apparent anomalous acceleration. And it is needed to take into account the availability of dust nearby satellites trajectories, when accelerations are calculated. Namely, the composite approach required for the exact prediction of the experimental results in the interferometers of Michelson — Morley, Fizeau, Sagnac and their modern analogues in the relativistic approximation. Together with this, the solutions of the dispersion equation should be tested for a general case of 3-dimensional ﬁeld of medium velocities and with strong nonhomogeneities, appearing, for example, in a ﬁeld of considerable light absorption. These works are carried out at Physics Department of BMSTU under support of the Council of the Russian President’s grants and collaboration with investigations of Liverpool and Sunderland Universities of Great Britain. Thus, the dispersion equation in the considered tasks with high-precision measurements, despite the additional difﬁculties, will allow us to take into account non-linear effects, which can decidedly further progress of the Relativity. Moreover, the analysis suggests that the non-linear effects of interactions between electromagnetic radiation and moving media can considerably inﬂuence a solution of the tasks for deﬁning radiation parameters on both cosmological and microcosmic scales. In the report of V.S. Gorelik (Lebedev Physics Institute of Russian Academy of Sciences, Moscow) regularities of an interaction between material media and electromagnetic waves, for example, in virtual models of crystal lattices of dielectrics and semiconductors were analyzed. Dynamics of elementary excitations for the system of “a medium — an electromagnetic ﬁeld”-polaritons was investigated; and conditions, under which a group velocity of polaritons is small in limit, that is the light “stops” in the material medium, were found. Features of a “crystal” model of physical vacuum were considered. VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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As is known, forms p of dependences of a dielectric permittivity ε (ω) and a refractive index n = ε (ω) can be found from the motion equation for a charged oscillator in the electromagnetic wave ﬁeld s ωl2 − ω 2 n(ω) = , ω02 − ω 2 where ωl2 = ω02 + ωp2 ; ω0 is a fundamental frequency of a charged oscillator; ωp is a plasma frequency. It is easily to notice that for high frequencies (ω À ω0 ) the refractive index tends to unity, i.e. the light phase velocity tends to c0 . On the contrary, for low frequencies (ω → 0) the following takes place:

ε(0) =

ωl2 ω02

and

n(0) =

ωl . ω0

If ω = ωl , the refractive index and dielectric permittivity are equal to zero: ε (ωl ) = 0. The result leads to a conclusion on possibility of existence of longitudinal electromagnetic waves with a frequency ω = ωl . The frequency of a longitudinal electromagnetic wave for the considered model of substance does not depend on a wave vector ~k, i.e. on a wave length. And on the contrary, if ω ∼ = ω0 , the refractive index and the dielectric permittivity sharply increase and considerably differ from unity. As an example, we can consider far infra-red part of the spectrum, which corresponds to low frequencies of polariton waves. In the case, the refractive index is ω0 √ more than unity, n = ε0 = > 1. Therefore, phase and group velocities of ωl propagation of millimeter electromagnetic waves in a material medium can be essentially (by some orders) less than the light velocity in vacuum. Accordingly, the dielectric constant ε0 can be substantially more than unity. The especial situation takes place, when a fundamental frequency of vibration for Lorentz’s oscillators becomes very low. The dielectric constant highly increases: for some substances its magnitude achieves a value of 104 . In this connection we can notice the experimentally established fact of inﬂuence of an instellar medium motion on a velocity of electromagnetic wave propagation. In a model of the expanding Universe the radial propagation of an electromagnetic wave in IRF of an observer with account of a medium motion effect will be described with the equation: Zt0

Zr1 dt = R(t)

t1

1 dr √ . βe (r, t) 1 − kr 2 0

Here t1 , t0 are moments of emission and detection, respectively, counted from the singular state; R (t) is the cosmological scale factor; k is a spatial curvature; r1 is the dimensionless distance to space source of radiation in the Earth IRF. A sign before the integral in the right part is taken as positive which corresponds to a period 206

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of expanding the Universe, βe = c/c0 , c is a group velocity of an electromagnetic wave, c0 is the light velocity in vacuum. As cosmological extension can delay with time, so the group velocity of light depends on time t. Moreover, the light velocity in the i-th layer of a medium along the radiation direction is deﬁned by features of the instellar medium and velocity of its motion in the local ﬁeld of space, where a wave propagates: ci = ω0 /kin . Thus, by using the results of observation astronomy for the instellar medium distribution along a beam of observation, time of light beam propagation from distant parts of the Universe to an observer can be calculated. Fields in the Universe with refractive indices, which are more than unity, essentially inﬂuence the velocity and, accordingly, time of electromagnetic signal propagation. If such ﬁelds are very distant ﬁelds of the Universe, estimation of light propagation time and Universe dimensions can strongly depend on the velocity law of medium motion along a ray. For a general case of 3-dimensional medium motion there are effects of longitudinal dragging and distortion of an electromagnetic wave trajectory, which are analogous to distortion in non-homogeneous gravitation ﬁelds. Observational estimations of distortion, magnitudes for electromagnetic waves with account of a medium motion velocity in the distant ﬁelds of the Universe are a basis for new testing of the expanding space hypothesis. Similar effects should be expected in the tasks of navigation, including space navigation and location. In this connection the works, in which attempts of experimental detection of non-local features of physical interactions are discussed, can have an especial meaning (the report of S.M. Korotaev, Geo-electromagnetic Research Institute of Russian Academy, Troitsk). A part of reports was devoted to methodical aspects of the Relativity. In the report of G. Keswani (New Delhi, India) the historical aspects of the experimental basis — Fizeau’s effect and the stellar aberration phenomenon — were discussed. In the report of S. Hajra (Calcutta, India) it was shown that electrodynamics equations contain features of space-time continuum, which in an explicit form can be written as the Lorentz’s transformations, and recommendations on training the question in electrodynamics course were made. A deep analysis of interpretations of the Poincare-Lorentz’s relativity principles was carried out in the work of M.C. Duffy (Sunderland and Liverpool Universities, Great Britain). The reports of P. Rowlands (Liverpool University, Great Britain) and E.Trell (Linkoping, Sweden) are devoted to profound links between Relativity and Physics of Elementary Particles and also are examples of researching physical interpretations of observational features of space and time. Despite a wide spectrum of the presented reports, all participants were united with one generic task of searching the formalism which would be adequate to physical reality, and interpretations of observable phenomena and properties of spacetime continuum corresponding to it.

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In his speech, V.N. Melnikov (the president of Russian Gravity Society) noticed that the Relativity is not only a powerful tool for researchers, but it has become a base for engineering calculations long ago and the fact should be taken into account in training specialists. Durability of the educational base is nonseparably linked with breadth of natural sciences outlook, therefore, engineering education should be based on Gravity Theory, Astrophysics, Cosmology and Physics of Elementary Particles. Together with this, a solution of up-to-date experimental tasks, as time history of modern sciences shows, is possible only where polytechnic base is created, and it allows one to carry out new precise experiments, using the composite approach, by applying nowadays technologies and the accumulated world experience. The participants of the conference concluded, by noticing that BMSTU was the center of engineering culture more than 170 years, that the University was chosen as the right place to hold the conference. All information, concerning the conference, is on the web-sites of Physics Department of BMSTU [22] and Sunderland University [23].

REFERENCES 1. Einstein, A. Naherungsweise Integration der Fildgleichungen der Gravitation. – Berlin: Preuss. Akad. Wiss., 1916. – P. 688–696. 2. Taylor, J.H., Weisberg, J.M. A New Test of General Relativity: Gravitational Radiation and the Binary Pulsar PSR 1913+16 // Astrophysical Journal. – 1982. – V. 253, No 2, Pt. 1. – P. 908–920. 3. Taylor, J.H. Gravitational Radiation and the Binary Pulsar//Proc. 2-th Marcel Gross. Meeting on Gen. Relativity, 1979, – Amsterdam e.a., – 1982. – P. 15–19. 4. Gertsenshtein, M.Ye., Pustovoit, V.I. On the Question of Low Frequency Gravitational Waves Detection // JETP (Rus.). – 1962. – V. 43. – P. 605–627. 5. Gladyshev, V.O., Morozov, A.N. Gravitational Waves Classiﬁcation on Method Gravitation Radiation Detection (in Russian). // Measurement Technique – 2000. – V. 9. – P. 21–25. 6. Giazotto, A. Interferometric Detection of Gravitational Waves//Physics Reports. – 1989. – V. 182, No 6. – P. 365–424. 7. Rudenko, V.N. Prospects of Gravitational Astronomy (in Russian). //Einsteinovski sbornik, 1986–1990. – M.:Nauka, 1990. – P. 351–374. 8. Bradaskia, C. et al. A VIRGO Project. A Wide Band Antenna for Gravitational Wave Detection // Nucl. Instr. and Meth. Phys. Res. A. – 1990. – V. 289, No 3. – P. 518–525. 9. Anderson, A.J. The Space Multi-arm Interferometer and the Search for Cosmic Background Gravitational Wave Radiation (SMILE) // Proc. Int. Assoc. Geod. Symp., Vancouver, 1987. – V. 1. – P. 83–90. 10. Ashby, N., Spilker, J.J. (Jr.) Introduction to Relativistic Effects on the Global Positioning System // Global Positioning System: Theory and Applications. Washington, 1997. – P. 623–697. 11. Fomichev, A.A., Dmitriev, V.G., Kolchev, A.B. et al. Complex Inertial Satellite Navigational System on the Base of Laser Gyroscopes for Civil Aircrafts (in Russian). // SPIE/RUS Bulletin. – 1995. – No 1(5). – P. 28–31.

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12. Braginski, V.B. Resolution in Macroscopic Measurements: Achievements and Prospective (in Russian). // Uspekhi ﬁzicheskih fauk. – 1988. – V. 156, No. 1. – . 93–115. 13. Brillet, A., Damour, T., Tourrenc, Ph. Introduction to Gravitational Research // Annales de Physique (Fr). – 1985. – V. 10, No 3. – P. 210–218. 14. Brillet, A. Interferometric Gravitational Wave Antennae// Annales de Physique (Fr). – 1985. – V. 10, No 3. – P. 219–226. 15. Bolotovsky, B.M., Stolyarov, S.N. Reﬂection from Moving Mirror (in Russian). // Uspekhi ﬁzicheskikh nauk. – 1989. – V. 159, No. 1. – P. 155–180. 16. Izmailov, G.N. et al. Stable Interferometer for Precise Physical Experiments (in Russian) // Tekhnicheskaya ﬁzika. – 1987. – V. 57, No 6. – P. 1194–1197. 17. Lipunov, V.M., Postnov, K.A., Prokhorov, M.E. The Sources of Gravitational Waves with Continuous and Discrete Spectra // Astron Astrophysics. – 1987. – V. 176. – P. L1–L4. 18. Tourrenc, P., Deruelle, N. Effects of the Time Delays in a Nonlinear Pendular FabryPerot // Annales de Physique (Fr). – 1985. – V. 10. – P. 241–252. 19. Zagorodnov, O.G., Fainberg, Ya.B., Egorov, A.M. Multiplication Frequency with Plasma Collapse (in Russian). // JETP. – 1960. – V. 38, No. 1. – P. 7–9. 20. Welsh, B.Y., Vedder, P.W., Vallerga, J.V. High-resolution Sodium Absorption-line Observations of the Local Interstellar Medium. // Astrophys.J. – 1990. – V. 358. – P. 473–484. 21. Kimata Fumiaki, Mannodzi Nobutaka // Kisho Kenkyu noto. – 1998. – No 192. – P. 49–59. 22. http://fn.bmstu.ru/phys/nov/konf/pirt/pirt main.html. 23. http://www.cet.sunderland.ac.uk/webedit/allweb/news/ Philosophy of Science/PIRFORM.htm.

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RADIO-PHYSICS & RADIOLOCATION B. Rozanov1 , S. Solomonov2 , A. Zrazhevsky3 , V. Nagnibeda4 , Ye. Kropotkina2 , S. Rozanov2 , N. Zharkova1 , T. Lebedjuk∗ , I. Fetisov1 , M. Loukicheva4 1 Bauman Moscow State Technical University (BMSTU), 2 Lebedev Physics Institute of the Russian Academy of Science (LPI), 3 Institute of Radio Engineering and Electronics of the Russian Academy of Science, (IRE) 4 Saint-Petersburg State University (SPbSU)

THE ATMOSPHERIC AND SOLAR INVESTIGATIONS AT MILLIMETER WAVES In three last decades the short millimeter wave range was mastered for solution of both scientiﬁc and practical problems. Investigations of the Sun and the Earth’s atmosphere by means of special unique radio astronomical and radio physical instruments played a considerable part in the progress. Some results of the investigations obtained by joining efforts and experience of researches from several academic institutes and universities are presented in the paper. Results of the stratospheric and mesospheric ozone observations at frequency of 142 GHz at the Lebedev Physics Institute of the Russian Academy of Sciences are considered in the ﬁrst part. It is shown that millimeter-wave radiometry is very effective techniques for investigation of dynamical and chemical processes in the atmosphere affecting depletion of the ozone layer. Results of experimental investigations of refraction, ﬂuctuations of vertical and horizontal arrival angles and intensity of the received waves in the 2. . . 3 mm waveband for near-surface communication lines about 10 km long are presented in the second part. The data was obtained at the Bauman Moscow State Technical University (BMSTU) and Institute of Radio Engineering and Electronics of the Russian Academy of Sciences as the result of long-term series of observations at various meteorological conditions. The third part of the paper is devoted to analysis of results of solar investigations at wavelengths of 2.2 and 3.3 mm at the radio telescope RT-7.5 in Moscow region. The research was done by BMSTU and SaintPetersburg State University (SPbSU).

1. MILLIMETER-WAVE MONITORING THE ATMOSPHERIC OZONE Introduction. Radio physical methods of the Earth’s atmosphere remote sensing based on measurements of thermal emission of atmospheric gases at millimeter (MM) and submillimeter (subMM) waves opened new possibilities for studying the atmosphere and its ozone layer [1–3]. Both ﬁrst investigations of the atmospheric subMM emission from balloons and satellites [1] and initial ground-based research of the ozone layer over Moscow at MM waves were done at the Lebedev 210

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Physics Institute of the Russian Academy of Sciences (LPI) [4]. Now the measurements of the vertical ozone distribution (VOD) at 2-mm transparency window of the atmosphere are carried out regularly. Some problems of remote sensing the Earth’s ozonosphere at microwaves, design of the ozone spectrometer operating at 142 GHz and main results of the MM-wave monitoring the ozone layer at the LPI are considered in the paper. It is shown that millimeter-wave radiometry is very effective techniques for investigation of dynamical and chemical processes in the atmosphere affecting depletion of the ozone layer. Problems of radio physical investigations of the ozonosphere. The purpose of radio physical investigations of the ozone layer is to answer questions on changes in the ozonosphere and their reasons. Important problems of ground-based research of the ozone layer at MM waves are studying the VOD variations of different time scales resulted from natural reasons: atmospheric circulation, emission, chemical and thermal processes in the atmosphere and its interactions with space. Studying the human activity inﬂuence on the ozone layer is a topical problem as well. Knowledge of features of changes in ozone over high-populated mid-latitude territories such as Moscow region is of special importance because possible ozone depletion may result in injury to health of a large number of people. The impacts of the natural and human factors on the ozonosphere depend on altitude and may have various time scales (for example, diurnal, seasonal, quasibiannual changes in the atmosphere are known as well as inﬂuence of 11-year cycle of the Sun activity). So the ozone variations have to be studied for broad temporal and altitude ranges. The MM-wave ground-based remote sensing is the techniques to study the variations of different time scales at altitudes from the lower stratosphere to the mesosphere. The new data on the VOD is important to create modern model of the ozonosphere taking into account inﬂuence of the above mentioned natural and human factors on the atmospheric ozone. The well-known ozone reference models for the COSPAR international reference atmosphere [5] were based on satellite data from late 70s and early 80s when last anomalous phenomena in the ozonosphere were not known yet. Else one important practical problem is carrying out synchronous ground-based suborbit measurements of the VOD at MM waves for validation of satellite ozone data. To solve all the problems both high-sensitive MM-wave spectrometer and higheffective methods of ozone observations and data processing were done at the LPI [4, 6, 7] providing reliable and uniform in quality series of data on the ozone layer over Moscow for many years. Instruments and methods. The VOD in the stratosphere and mesosphere may be retrieved from shape of pressure-broadened rotational spectral line of ozone thermal emission. The pressure in the atmosphere rapidly falls with altitude, so the line measured at the ground level contains variously broadened contributions from ozone molecules located at different altitudes. Doppler broadening prevails over pressure broadening for altitudes above 70 km. The line contour may be expressed as an integral resulted from the radiation transfer equation. There exist special methods of the VOD retrieval from the line measured [3, 7]. The ozone line centered at 142.175 GHz is used for the spectral measurements at the LPI. The LPI MM-wave ozone spectrometer (Fig. 1) consists of low-noise heterodyne Dicker radiometer, two ﬁlter-bank spectrum analyzers and a PC with special VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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Fig. 1. General view of receiver of the LPI spectrometer

interfaces and software. The analyzers were done at Institute of Applied Physics (Nizhny Novgorod) with participation of LPI. The spectrometer was mounted in a laboratory room before the radio transparent window. The receiver antenna has Gaussian pattern with beamwidth of 1.5 degrees at – 3 dB level (the input lens diameter is 120 mm). The antenna zenith angle can be optimized depending on meteorological conditions. Typically the angle is of 60–70 degrees. Special design of the receiver quasi-optical input units and methods of calibration and observations (see details in [6]) allowed one to minimize distortions of the ozone spectra by standing waves. Planar Schottky diode of AA138V-3 type from Institute of Semiconductor Devices (Tomsk) is used in the ﬁrst mixer of the receiver [8]. The mixer has simple design, low conversion loss and noise temperature together with low required local oscillator power. It may be cooled at least down to liquid nitrogen temperature. Reliability of the mixer was conﬁrmed by its steady operation in the ozone spectrometer for many years. The spectrometer single sideband noise temperature is of about 700 K under cooling the receiver down to 80 K and of about 1500 K without cooling. The spectrometer has a broadband channel to measure total power in the signal band correspondent to background atmospheric emission. Brightness temperature of the emission is used to determine tropospheric attenuation of the ozone line.

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Typical calibration time of the receiver is 100 s, and signal integration time is 200 s. The unit spectra are stored in the controlling PC before further processing. Retrieval of the VOD from the integral spectrum obtained after averaging the unit experimental spectra for 1–2 hours is done using the method [7]. An altitude range for which the VOD may be retrieved is from 15 to 75 km for day-time and to about 95 km for night-time measurements of mesospheric ozone [4, 7, 9]. Errors of the ozone proﬁle retrieval are no more than 5–7 % for 20–50 km and no more than 20– 30 % for 15–20 and 50–75 km. For night mesospheric ozone the errors are no more than 30 % for 80–95 km. For altitudes 75–80 km where minimum of relative ozone content is located the errors may reach 50 %. The LPI ozone spectrometer was incorporated in to the global network for ozone measurements in the framework of international campaigns DYANA (1990), CRISTA/MAHRSI (1994 and 1997), SOLVE 2000 (1999-2000). Results of remote sensing the ozone layer over Moscow coincide well with data from ozonosondes and satellites [4]. Results of investigations. Regular observations of the atmospheric ozone over Moscow region at MM waves have been done at the LPI since 1987. Altitudetemporal distribution of the ozonemixing ratio was obtained. An example of the distribution for cold months of 1999–2000 is presented at the outside of the journal back cover. Ozone variations of different time scales were detected. Firstly it is necessary to note seasonal ozone variations. The variations for altitude of 35 km are shown in Fig. 2. The reasons of the seasonal ozone variations are seasonal changes in dynamical and photochemical atmospheric processes affecting the ozone [10]. The observed seasonal variations of the ozone content qualitatively correspond to the model [5]. However noticeable systematic deviation of MM-wave data from that of the model was found, especially for cold seasons. For some periods ozone content at altitudes between 25 and 45 km decreased by 30–45 % from the model values. The anomalous events usually coincided in time with some decrease of total ozone content in the atmosphere (typically by 10–15 %) measured over Moscow region by optical methods in Central Aerological Observatory (Dolgoprudny). More fast noise-like ozone variations with time scales from several days to several weeks were detected besides the seasonal ones. Magnitude of the short-term variations increases from summer to winter. For example, at 35 km the magnitude is ±(15 − 20) % around the mean value in summer and rises up to approximately ±40 % in winter. Data of aerological sounding from Russian and foreign observatories as well as satellite data (maps of potential vorticity, geopotential altitudes in the stratosphere, etc.) were analyzed to understand reasons of the fast changes in VOD. The analysis shown that the VOD variations resulted from the atmospheric dynamics, from transport of air masses with different ozone content. Correlation of the ozone content at 15–45 km and dynamical parameters of the atmosphere was revealed. For example, absolute values of the correlation coefﬁcients between the ozone content at the altitudes and potential vorticity at isoentropic surface near 30 km were up to 0.9 in cold half-years. The fact shows strong inﬂuence of the

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Fig. 2. Relative ozone content over Moscow at altitude of 35 km for 1996-2000. Monthly averaged values are shown too

atmospheric dynamics on the stratospheric ozone al altitudes lower than 45 km. Large planetary wave activity in cold months result in the above mentioned shortterm ozone variation in this time of year. It is well known that the polar vortex appears and develops in the stratosphere and mesosphere in a cold halfyear, and destruction of the ozone takes place in the central isolated part of the vortex [10]. Correlation of the stratospheric ozone content over Moscow with movements of the polar vortex was established. Changes in the VOD when the vortex approaches Moscow are shown in Fig. 3. On November 30, 1999 the stratosphere over Moscow occurred inside the vortex and was notable for high potential Fig. 3. Changes in the VOD resulted vorticity and decreased ozone content. Monitoring the ozone layer at MM from approaching the polar vortex to Moscow for 24, 26, and 30 November waves allowed the study of the ozone pro1999 by MM-wave observations ﬁle evolution over Moscow under inﬂuence of the atmospheric processes and the investigation of peculiarities of the altitude-temporal ozone distribution for every year of observations. It was found that the distribution changes noticeably from year to year as a result of variations in the atmospheric dynamics course. In particular it was discovered that the most long and deep ozone depletion (especially

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at 25–45 km where the ozone content decreased by 40–45 % in comparison to the model [5]) was in cold seasons of 1995–1996 and 1999–2000. In winter 1999–2000 the decreased ozone content was kept with short breaks almost 3 months from middle of November 1999 to middle of March 2000. These cold seasons were notable for the very deep and stable polar vortex. Diurnal variations of the mesospheric ozone were measured and studied as well [9]. Considerable increase in the mesospheric ozone at night corresponds to photochemical theory. Variations of the night ozone at altitudes of 65–75 km with periods of 2 and 4 hours were discovered [4, 9]. The variations may be connected with internal gravity waves in the atmosphere. Inﬂuence of planetary waves on the night ozone at mesopause altitudes (80– 95 km) was found, the ozone content at the altitudes being varied from 1.5 to 8 ppm (1 ppm= 10−6 ). Correlations of the ozone mixing ratio in the region of the upper atmosphere and parameters of the low and middle stratosphere (potential vorticity, geopotential altitudes, total ozone content, etc.) were determined. The correlations revealed when the polar vortex moving over Moscow region at time of large planetary wave activity. It is known [11, 12] that the processes envelope layers from the lower stratosphere to thermosphere and may affect the mesospheric ozone by change in vertical transport of atomic oxygen and hydrogen, the species involved together with ozone in to chemical reactions in the upper atmosphere. So, the ozone may be an indicator of transport both in the stratosphere up to 45 km and in the upper atmosphere. The above-mentioned difference of the VOD obtained at MM waves from the model [5] may be connected with long-term ozone decrease for almost two decades from time when the satellite measurements used in [5] were done. The ozone trend value estimated from comparison of the LPI MM-wave data with model [5] has maximum at 40 km and conforms to data of other experiments [10]. It is believed that the reason of ozone depletion at altitudes of about 40 km is ozone destruction in catalytic chemical reactions with chlorine compounds of anthropogenic origin. Conclusion. The MM-wave ozone measurements allow studying processes in the ozone layer of different time scales. The results obtained show importance of further monitoring the ozonosphere at MM waves to investigate long-term changes in the ozone layer. It is desirable to create a network of ground-based observational points in Russia and other countries of Union of Independent States for monitoring the ozonosphere at MM waves. Simultaneous microwave measurements of ozone, chlorine monoxide, nitric compounds and other minor gas species of the atmosphere are important too. The measurements are in preparation at the LPI [4]. So, remote sensing the atmosphere at MM waves gives essential contribution to investigations of the ozone layer and our knowledge about its changes it. 1

Authors are grateful to Correspondent Member of the RAS I.I.Sobel’man for support of the work, and to researchers of the LPI A.N. Lukin and V.N. Leonov for their contributions to the investigations. The work was supported by Council of “Integration” Program, by RFBR, grants No 99-02-18132, 0302-17436, and 00-05-64976, and by “Leading Scientiﬁc Schools” Program, grants No 00-15-96586 and SS-1254.2003.2. VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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2. THE SHORT MILLIMETER WAVE PROPAGATION THROUGH THE NEAR- SURFACE LAYER OF THE EARTH ATMOSPHERE Introduction. The investigations of short millimeter wave propagation through the near-surface layer of the Earth atmosphere (altitudes H = 0 . . . 100 m) are essential for use of this band for communications, radars, radio astronomy and so on. The investigations were carried out in 1970–1990 by BMSTU and Institute of Radio Engineering and Electronics of the Russian Academy of Sciences (IRE) in Dmitrov district of the Moscow region. Theoretical models and computation methods for deriving the propagation characteristics [13] are not applicable for this layer of the atmosphere because of signiﬁcant inﬂuence of the Earth surface and fast variability of statistical estimations of meteorological parameters in the layer. The wavelength range λ = 2.2 . . . 3.3 mm is optimal for studying the propagation characteristics in near-surface layer of the atmosphere (structure factor of the refraction index, its effective vertical gradient, turbulence anisotropy). At longer wavelengths it is difﬁcult to obtain narrow antenna beam for exclusion of the Earth surface inﬂuence. At shorter wavelengths the atmospheric losses increase. The propagation characteristics deﬁned in the range of 2.2 . . . 3.3 mm can be applied for any longer wavelength because at λ > 2 mm temperature dependence of the refraction index is negligible [14]. Instruments and Methods. The measurements of the refraction, intensity, and arrival angle ﬂuctuations were carried out at λ = 2.2 . . . 3.3 mm using two communication lines of 7.7 and 14 km long. The radio beam propagated through the atmosphere at the altitude of 20 . . . 50 m above bushes and arable terrain. Suppression of the surface reﬂections was realized by high angular resolution of the receiving antenna RT-7.5 of the BMSTU Radio telescope with the dish diameter of 7.8 m, having beamwidth Θ0.5 = 1.50 . . . 2.40 at λ = 2.2 . . . 3.3 mm and near-zone of approximately 50 km long [15]. The transmitter was a BW-oscillator of 10−2 W power with antenna dish of 0.6 m diameter. During the observations we measured simultaneously vertical and horizontal arrival angles Ψv , Ψh and intensity of the received wave I. Fluctuations of the parameters were studied in the frequency interval of 0.01 . . . 1 Hz. Angle sensitivity of the system equal to 0.100 was reached at 30 dB signal to noise ratio by conical scanning of the narrow receiving antenna beam around direction of the transmitter. The observations of the ﬂuctuations were possible under conditions of both clear weather and precipitation. Observations of the ﬂuctuations. The ﬂuctuations rms value σ for the parameters I, Ψv , and Ψh as well as the mean value I of the intensity I were calculated and rms deviation of both the total arrival angle σt2 = σv2 + σh2 and the chaotic modulation depth mI = σI /I were determined for experimental records of approximately 10 min long. Two long-term observation cycles were done at λ = 2.2 mm:

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1) of the intensity and arrival angles for the 7.7 km line during 24 hours in spring (45 records each of 10 min long); 2) of the intensity and arrival angles (partly) for the 14 km line during 19 days in different seasons at day-time (76 records each 10 min long). Some conclusions were obtained from the results. The observed variations of the propagation characteristics have the following limits: — for the 24-hour cycle σh = 0.5 . . . 300 ,

σv = 0.3 . . . 500 ,

σt = 0.6 . . . 700 ,

mI = 0.5 . . . 6 %.

— for the annual cycle mI = 1 . . . 25 % . In summer the σ values were several times higher than during cold period of the year; the mean values of mI being of 18 % for June, 6 % for November and 3.5 % for months from December to April. The rms σ values were 3 . . . 5 times lower at night than at day-time. The ﬂuctuations reduced by 3 times during weak rain. Fog gave no changes in the σ. Fall of the ﬂuctuations may be observed for some time before and after the sunset during anticyclone. Anisotropy of the atmospheric turbulence was detected as well. In Fig. 4, a variations of the arrival angles at hot summer midday are shown. It is seen clearly that the vertical ﬂuctuations are 2.5 times more than horizontal ones. At night ﬂuctuations in both directions are approximately equal (Fig. 4, b). Correlation between σh and σv can be seen at Fig. 5, a. In Fig. 5, b the correlation between σt and mI (%) is presented. The results of the mI measurements provided the possibility to

Fig. 4. The visible transmitter moving during 30s time interval: a — in the daytime; b — at night VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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Fig. 5. The results simultaneous measurements of: a — σv and σh ; b — σt and mI , %; 1 — in the daytime, 2 — at night, 3 — in the daytime; — the regression line, — calculated dependence

derive the important parameter of the atmosphere: the structure constant Cn of the refraction index n of the air. The experimental dependence of the Cn on the mI is in good agreement with theoretical one [16]: σt = 1.78mI (Fig. 5, b). Investigations of the Refraction. Investigations of the refraction were done using angle data units of the radio telescope for measurements of direction corresponding to maximum intensity of the arrival wave. The rms error of a single measurement was of about 1000 . For 10 measurements the mean error decreased to 300 . Two measurement series were realized for the 14-km line. In the ﬁrst series (1979), 138 measurements including two 24-hour cycles were done at λ = 2.2 mm. In the second series (1987-1989) 13 vertical refraction measurements of 24-hour cycles were done with 30-min intervals between the data at λ = 3.3 mm. At the same period 50 measurements were made simultaneously in millimeter and optical regions. Processing of the data allowed us to make some conclusions. The 24-hour variations ∆αv of the vertical arrival angle at millimeter wave region is 2 . . . 3 times higher at warm period of the year than in winter because of increased gradient of the air humidity and correspondingly increased gradient of the refraction index. The horizontal arrival angle variations were no more than the measurement errors (300 . . . 1000 ). The upper limit of slow horizontal angle variations was about 102 times lower than for the ∆αv . The greatest refraction was correlated with sharp changes in the weather. The sharp variations of the refraction may be observed at sunrise or sunset. At the anticyclones the night refraction values were higher than the day values. A variation by 50 was observed with the top angular speed of 30 per hour. A sudden variation by 60 for 30 min was observed at the cold front after thunderstorm. Such rare phenomenon as appearance of several maxima of intensity within a 1’ interval of the 218

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vertical arrival angles was observed during several minutes. The minimum variations of the refraction were observed in windy weather under cyclonic conditions. Inﬂuence of rain and snowfall on the refraction was not detected. The correlation of mm- and optical wave refraction is high (K = 0.97) in winter and low (K = 0.4) in summer. This results from low humidity of air in winter when refraction indices for mm- and optical waves are approximately equal. The data on the refraction variations were within the limit ∆αv = 1 . . . 50 all over the year. The corresponding 24-hour variations of gradient of the air refraction index were calculated: ∆gn = 2∆αv /L = (0.4 . . . 2) · 10−7 m−1 . The result is in agreement with data of [17]. 3. INVESTIGATION OF THE SOLAR EMISSION AT MILLIMETER WAVELENGTHS WITH RT-7.5 Introduction. Studies of the Sun’s radio emission at millimeter wavelengths provides a simple diagnostics of physical conditions in the solar chromosphere and the lower part of the transition region. Such studies offer several distinct advantages over those in visible and ultraviolet light. Non-burst millimeter-wave emission is generated under conditions of local thermodynamic equilibrium (LTE), it has a thermal nature and is formed due to bremsstrahlung, other mechanisms of emission do not make any signiﬁcant contribution into mm emission. The intensity (or brightness temperature) of mm-wave emission depends linearly on temperature of chromosphere plasma and therefore is able to provide a suitable and independent diagnostics of the solar atmosphere. Solar observations with RT-7.5 were begun in 1982 in collaboration with solar radio astronomers from St.-Petersburg State University. Since 1987 regular solar observations made possible to obtain a long set of solar maps at 3.4 mm wavelength (the frequency of 85 GHz) with 2.50 spatial resolution. Two years ago the second radiometer operating at 2 mm wavelength (150 GHz) was set in RT-7.5. Solar mapping at the telescope is carried out by means of raster, radial or circular scanning of the solar disc. The reader is referred to Solovyov et al. [32] for a detailed description of the telescope observations. Solar mapping at short millimeter wavelengths has large difﬁculties due to strong inﬂuence of terrestrial atmosphere and small contrast of sources on the solar disk. Nevertheless solar maps at millimeter wavelengths has shown the existence of various spatial structures in brightness distribution on the disc — “quiet Sun”, large-scale structures and sources of the S-component connected with the active regions. “Quiet Sun” and chromosphere models. After more than 50 years of theoretical and observational research the solar chromosphere still is poorly understood. At the present time there are some standard static and dynamic chromosphere models. All these semi-empirical models VAL (Vernazza et al. [34]), FAL (Fontenla et al. [21]), dynamic models CS (Carlsson and Stein [20]) are based on the chromosphere UV and visible continua observations. Thus the millimeter continua, which are formed in LTE-condition, may be able to provide an independent test of the models. VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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Fig. 6. The observed millimeter and submillimeter brightness temperatures of the quiet Sun at disc center.

The observed millimeter and submillimeter brightness temperatures of the quiet Sun at disc center, taken from original literature, are plotted vs. wavelength in Fig. 6, including data, obtained with RT-7.5 during 15 years. For measurements of center disc brightness temperature we used radio emission of the New Moon as a standard signal. At such a period the Sun and the Moon are placed close to each other on the sky, thus the observational conditions are the same for both objects. All observational data used for the current investigation were obtained during solar cycles 19, 20 and 21 and, thus, corresponded to the chromosphere at different activity phases — from activity minimum to activity maximum. We tried to look for solar cycle variations of the quiet Sun but at the moment the dispersion of the experimental points was too large to ﬁnd such variations. We compute the spectrum of the brightness temperature for millimeter wavelengths range using two sets of model atmospheres. The ﬁrst set consists of three models FAL corresponding to different components of the quiet Sun: A (a faint cell center area), C (an average intensity area), and F (a bright area representing the network). The other set of models considered is simulation of the photosphere and chromosphere due to Carlsson and Stein (CS). They studied [21] the dynamics of acoustic waves in the solar atmosphere using a one-dimensional, non-LTE, radiation hydrodynamic code. We calculated the submillimeter and millimeter radiation emerging from the above static and dynamic models. The radiation transfer was computed under the assumption that radio emission for these wavelengths occurred due to bremsstrahlung (free-free emission). The 220

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calculated spectra of radiation for the quiet Sun FAL models A, C and F are depicted in Fig. 6 together with the observational data. Among the considered models, FAL A, corresponding to the internetwork, provides the best ﬁt to the data. But all in all, the FAL models indicate that the millimeter-wave data is not compatible with UV spectra within the context of single-component time-independent planeparallel models: the UV data require a warmer model than the microwave data (Loukitcheva et al. [26]). A combination of a cell interior (FAL A) and a network model (FAL F) with ﬁlling factor about 0.13, looks more promising. The analysis of the dynamic simulations of CS reveals that radio emission (temperature contrbution functions and brightness spectrum) at millimeter wavelengths is extremely sensitive to the dynamic processes in the chromosphere. The most striking result is that the dynamic picture of the solar internetwork chromosphere is consistent with millimeter and submillimeter brightness observations. The average spectrum obtained by averaging over spectra from all time-steps of CS dynamic simulations provides a good ﬁt to observed temporally and spatially averaged data. Millimeter observations indicate that using radio technique it is possible to extend observations of the solar oscillatory component to the heights above those previously observed in the optical and UV spectral lines. Large-scale structures. Solar millimeter-wave maps allow us to ﬁnd out socalled “belts of activity” — enhanced emission clearly distinguished between heliolatitudes of 10◦ and 50◦ in both solar hemispheres. In these belts of activity S-component sources appear from time to time, but enhanced brightness exists all time though the evident activity in optics is absent. The brightness temperature difference between equatorial and latitudinal belts is not large and is equal to about 100–200 K. Comparison of our maps with solar images in UV spectral lines (mainly, in HeII 304 MM line) shows a close similarity. Nevertheless an estimation of HeII emitted layers contribution to the brightness temperature at 3 mm wavelengths shows unimportant role of these layers (Nagnibeda and Piotrovitch, [28]). Indeed, HeII emission is generated in the regions with T ∼ 100000 K that is too high for MM wave emission. Another conspicuous type of large-scale structures is related to dark regions or depressions or low-temperature regions. These depressions are observed during many years mainly at long mm waves and as a rule they are connected with Hα ﬁlaments or channels of ﬁlaments on the solar disk. But on our solar maps at 2–3 mm wavelengths we do not practically ﬁnd the depressions connected with ﬁlaments. But on the other hand on our maps we have found many regions with low brightness, lower than the quiet Sun level. And most of them seem to be related to coronal holes. Usually the coronal holes are deﬁned as dark regions on the space images of the Sun in UV spectral lines (HeII 304, for example), or in X-ray continuum images, or (as bright regions) from the ground-based observations in the IR HeI line at 10830. Statistical investigation of low radio brightness showed that about 75 % of equatorial coronal holes are associated with such regions (Nagnibeda, [29]). It is well-known that coronal holes are connected with open magnetic structures and are clearly visible on the disk as dark regions at decimeter and long centimeter VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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(CM) wavelengths also. One of the important conclusions of that investigation is: coronal holes are not seen at short CM wavelengths. Indeed, it is widely believed that coronal-hole-associated depressions become indistinguishable at short wavelengths and at millimeter wavelengths all the more. But in literature one can ﬁnd very poor data from this domain. There are some observations by Kundu (1976) at 3.5 and 9 mm wavelengths — coronal hole as a dark region, and by Kosugi et al.(1986) at 3 and 8 mm wavelengths — no evidence at 3 mm and brightening at 8 mm. So we had very poor and contrary observations and data for mm-wave domain (Moran et al. 2001). Model calculations corroborate the absence of evidences of coronal holes in MM-wave range (Papagiannis and Baker, [30]; Borovik et al. [18]). As to our short mm-wave solar maps, ﬁrstly, we ﬁnd out many dark regions on it, and secondly, we ﬁnd out high correlation of brightness depression with X-ray dark regions as well as with HeI and HeII spectral lines depressions. Thus one can deﬁnitely afﬁrm that in many cases at short MM wavelengths there is coronalhole-associated darkening. Various possible sources of the discrepancy between observations in CM and MM-wave ranges may exist, in particular, difﬁculties of calibration of data. But we are sure that there is a physical reason for the discrepancy: at long wavelengths coronal holes emission is observed but at short wavelengths under coronal hole chromosphere emission is observed. So we need to look for the distinction of chromosphere under coronal hole and outside it in a quiet region. Indeed, there is the set of observational and theoretical results which show that chromospheric spicule structures are different in the active, quiet and coronal hole regions. In particular, spicules are longer under coronal holes than in quiet regions (Bray and Loughhead, [19]; Rabin and Moore, [31]). Sources of S -component. Studies of the Sun’s active region radio emission at millimeter wavelengths provide a simple diagnostic of physical conditions in the active solar atmosphere and the lower part of the transition region as well as in quiet Sun atmosphere. Nevertheless, there are some disadvantages in mapping the Sun in the short MM wavelength regime: the angular resolution available is relatively modest at present, it is limited by strong inﬂuence of terrestrial atmosphere. As a result, conclusions about MM emission based on data obtained by instruments with different angular resolution turn out to be contradictory. Furthermore, observational results are in conﬂict with prevailing homogeneous chromospheric models due to the fact that inhomogenities are present in chromosphere structure at MM wavelengths. Two-dimensional solar maps at millimeter wavelengths are compared with solar maps obtained at Japan heliograph Nobeyama (NoRH) at 17.6 mm wavelength (the frequency of 17 GHz). On NoRH maps obtained with high spatial resolution of 10 arc seconds a ﬁne structure of an active region can be derived. Meanwhile, spatial resolution of 2.5 arc minutes that corresponds to RT-7.5 is not high enough to resolve the active region into individual components, even if they are present. Thus, it is impossible to study spectra of sources in detail. Nevertheless some interesting spectral features were found from this investigation.

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First, for different local sources the spectral behavior of millimeter emission is different. The investigation has shown that this fact can be explained by the considering of active regions of various types: active regions with highly developed sunspot groups, with weak sunspots or without them. Second, it can be seen that in general there is an increase in ﬂux density toward short wavelengths in MM domain. The existence of ﬁne structure in sources radio emission consisting of 2 components: sunspot emission and plage emission, leads to the signiﬁcance of the resolving sunspot and plage contributions into the total ﬂux of S-component sources. It can be easily done for NoRH maps, where a bright source over a sunspot group can be resolved. It contribution into the total ﬂux can be estimated and it is considered that the remaining emission is due to plage complexes. For source radio emission at 3.4 mm wavelength contributions of different components of ﬁne structures was estimated under assumption that 3.4 mm emission of these sources occurred mainly due to emission of plages, and sunspot contribution into the total ﬂux at 3.4 mm wavelength was out of consideration. A statistical investigation of the dependence between the observed radio source ﬂux density and the size of corresponding active region was carried out. It was found that source ﬂux density at 17.6 mm is correlated with the sunspot area, as for the ﬂux at 3.4 mm, it showed a linear correlation with the total active region area, where plage area dominates. Thus, the basic contribution into radio emission at 3.4 mm wavelength is made by plages and moreover, it comes from our observations that the brightness temperature of a source over sunspot group does not differ much from the brightness temperature of the whole active region. And at 17.6 mm wavelength sunspot component dominates in the total source ﬂux. There was an attempt to place the observational results in the context provided by existing models of the solar chromosphere (Loukitcheva and Nagnibeda [25]). The distributions of electron temperature and electron concentration for sunspots and plages were considered from various chromospheric models : VAL and FAL for the upper chromosphere and transition region, sunspot models by Lites and Skumanich [24] and by Staude et al. [33]. Radiation transfer was estimated under the assumption that radio emission at MM wavelengths occurred mainly due to bremsstrahlung (for < 3 cm). From the results of these calculations it is obvious that theoretical values of brightness temperature are systematically higher than observed ones for all considered components of radio emission: the quiet Sun, sunspot emission and plage emission. Clearly, chromospheric models of different radio emission components based on the UV and optic observations and computed under assumption of hydrostatic equilibrium fail to account for the observed spectral characteristics of sources of millimeter radio emission. To obtain correspondence between calculated and observed values of brightness temperature some authors introduce a ﬁlling factor α. The estimated values of the ﬁlling factor α were found to be about 0.15. These values can serve as an estimation of relative contribution of cold and hot structures into the solar chromosphere.

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REFERENCES 1. Development of radio astronomy in the USSR (in Russian) / Ed. by A.E. Salomonovich. Moscow: ”Nauka”, 1988. 2. Spectral Investigations of Cosmic and Atmospheric Radiation (in Russian) / Ed. by A.G. Kislyakov. Gorky: IAP RAS, 1979. 3. Atmospheric Remote Sensing by Microwave Radiometry / Ed. by M.A. Janssen. New York: J.Willey& Sons, Inc., 1993. 4. Solomonov, S.V. (in Russian). // Rus. Telecommunications and Radio Engineering. – 2003. – T. 1, 2003. P. 9–25. 5. Keating, G.M., Chiou, L.S., Hsu, N.C. // Adv. Space Res. 1996. – V. 18. – No 9/10. – P. 11–58. 6. Solomonov, S.V., Rozanov, S.B., Kropotkina, E.P., Lukin A.N. (in Russian) // Radiotekhnika i elektronika. – 2000. – V. 45. – No. 12. – P. 1519–1525. 7. Gaikovich, K.P., Kropotkina, E.P., Solomonov S.V. (in Russian) // Isvestia Akad. Nauk, Phys. atmosphery and okeana. – 1999. – V. 35. – No 1. – P. 86–95. 8. Rozanov, S.B. (in Russian) // Radiotekhnika i elektronika. – 1996. – V. 41. – No. 3. – P. 362–369. 9. Solomonov, S.V., Kropotkina, E.P., Semyonov, A.I. // Bulletin of the Lebedev Physics Institute (RAS). – 2001. – No. 10. – P. 30–38. 10. Scientiﬁc Assessment of Ozone Depletion: 1998. World Meteorological Organization, Global Ozone Research and Monitoring Project. — Report No. 44, Geneva, 1999. 11. Allen, M., Lunine, J.I., Yung, Y.L. // J. Geophys. Res. – 1984. – V. 89. – No. D3. – P. 4841–4872. 12. Perminov, V.I., Kropotkina, E.P., Bakanas, V.V. et al. (in Russian) // Geomagnetism and aeronomy. – 2002. – V. 42. – No. 6. – P. 814–820. 13. Tatarsky, V. Rasprostranenie voln v turbulentnoi atmosfere (in Russian). Moscow: Nauka, 1967. P. 548. 14. Issledovanij rasprostranenij millimetrovyh voln v prizemnom sloe atmosfery (in Russian) / B. Rozanov, I. Fetisov, A. Zrazhevsky at al // Vestnik MGTU. Priborostroenie. – 1990. – No. 1. – P. 60–66. 15. Millimetrovyi radioteleskop RT-7.5 of MGTU (in Russian) / B. Rozanov // Isvestija vuzov SSSR Radioelektronika. – 1981. – T. 24. – No. 3. – P. 3–8. 16. Andreev, G., Chernaj, L. Fluctuacii puchka MMV pri rasprostranenii v turbulentnoi atmosfere zemli (in Russian) // Radiotechnika. – 1978. – V. 33. – No. 1. – P. 16–29. 17. Badulin, N., Tatarinov, V. Ovlijanii sutochnih variacii parametrov sredy na statistiku uglov refrakcii (in Russian) // Radiotekhnika i elektronika. – 1980. – V. 25. – No. 12. – P. 2498–2503. 18. Borovik, V.N., Kurbanov, M., Livshits, M.A. and Ryabov, B.I. (in Russian) // Astron. Zh., 1993, Vol. 70, 403–414 (in Russian). 19. Bray, R.J. and Loughhead, R.E. (1974): “The Solar Chromosphere”, L.,384 pp. 20. Carlsson, M. and Stein, R.F. (1995): ApJ, 440. L29. 21. Fontenla, J.M., Avrett, E.H., Loeser, R. (1993): Ap. J. 406. P. 319. 22. Kosugi, T., Ishiguro, M. and Shibasaki, K. (1986): Publ. Astron. Soc. Japan, Vol. 38, 1–11. 23. Kundu, M.R. and Liu, S.Y. (1976): Solar Phys., Vol. 49, 267–269. 24. Lites, B.W., Skumanich, A. (1982): Ap. J., 49, 293.

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25. Loukitcheva, M.A. and Nagnibeda, V.G. (1999): Proc.8-th SOHO Workshop “Plasma Dynamics and Diagnostics in the Solar Transition Region and Corona”, ESA SP-446, 451. 26. Loukitcheva, M.A., Solanki, S.K., Carlsson, M. (2003): A& A, (in press). 27. Moran, T.; Gopalswamy, N.; Dammasch, I. E.; Wilhelm, K. (2001): A & A, v. 378, p. 1037. 28. Nagnibeda, V.G. and Piotrovitch, V.V. (1994): Solar Phys., Vol. 152, 175. 29. Nagnibeda, V.G. (1997): Proc. 31-st ESLAB Symp., ESA SP-415, 353. 30. Papagiannis, M.D. and Baker K.B. (1982): Solar Phys., Vol. 79, 365. 31. Rabin, D. and Moore, R.L. (1980): ApJ., 241, 394. 32. Solovyov, G.N., Rozanov, B.A., Ivanov, V.N., Nagnibeda, V.G., Piotrovitch, V.V. (1992): in: Fedorov (ed.) “Topics in Radioelectronics and Laser System Design”. CRC Press, Boca Raton, Mayami, 167. 33. Staude, J., Kruger, A., Hildebrandt, J., Furstenberg, F. (1985): A & A., 143, 72. 34. Vernazza, J.E., Avrett, E.H., Loeser, R. (1981): Ap. J. Supp., 46, 635. B.A. Rozanov (b. 1933) graduated from Bauman Moscow Higher Technical School University in 1956 and Lomonosov Moscow State University in I960. D.Sc. (Eng.), professor of “Radio and Electronic Devices” Department of the Bauman Moscow State Technical University. Author of more than 150 publications in the ﬁeld of radio astronomy radio physics signal processing & MM-wave engineering. S.V. Solomonov (b. 1941) graduated from Lomonosov Moscow State University in 1964. Ph.D. (Phys.-Math.). Leading researcher of the Lebcdev Physics Institute of the Russian Academy of Science. Author of over 100 publications in ﬁelds of radio physics. spectroscopy, atmospheric physics. A.Yu. Zrazhevsky (b.1936) graduated from the Bauman Moscow Higher Technical School in 1959. D.Sc. (Eng.), Head of laboratory in Institute of Radio Engineering and Electronics of Russian Academy of Sciences (IRE RAS, Moscow). Author over 100 publications in ﬁelds of radio phisycs, remote sensing, MM-waves propagation in atmosphere. V.G. Nagnibeda (b. 1937) graduated from Astronomical Department of Leningrad State University in 1960. Ph.D. (Phys.-Math.), professor of astrophysics and radio astronomy in St. Petersburg State University. Author of more than 85 publications in the ﬁelds of Solar physics and astrophysics. Ye.P. Kropotkina (b. 1940) graduated from the Lenin Moscow Teachers Training Institute in 1963. Ph.D. (Phys.-Math.), senior research worker of the Lebedev Physics Institute of the Russian Academy of Sciences. Author of about 50 publications in ﬁelds of atmospheric physics spectroscopy, and radio physics. S.B. Rozanov (b. 1958) graduated from Lomonosov Moscow State University in 1980. Ph.D. (Phys.-Math.), head of Laboratory of millimeter-wave spectroscopy of the Lebedev Physics Institute of the Russian Academy of Sciences. Author of about 80 publications in ﬁelds of radio physics, spectroscopy and atmospheric physics. T.S. Lebediuk (b. 1944) graduated from the Bauman Moscow Higher Technical School in 1968. Research worker of Bauman Moscow Slate Technical University. Author of 17 publications in the ﬁelds of radio astronomy and radio physics. VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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N.A. Zharkova. (b. 1947) graduated from Moscow Institute for Mountain Engineering in 1970. Ph.D. (Eng.), assoc. professor of the Bauman Moscow State Technical University. Author of over 40 publications in ﬁelds of radio physics, low-noise receivers & MM-wave engineering. I.N. Fetisov (b. 1931) graduated from Lomonosov Moscow State University in 1955. Ph.D. (Phys.-Math.), assoc. professor of the Bauman Moscow State Technical University. Author of over 90 publications in ﬁelds of radio physics, atmospheric physics, MM-wave propagation. M.A. Loukictheva (b. 1975) graduated from Astronomical Department of St. Petersburg State University in 1992. Researcher of Astronomical Institute of St. Petersburg State University. Author of 10 publications in the ﬁelds of solar physics and helioseismology.

BMSTU Press has published the book: Information Technologies in Radio Technical Systems. – 2nd edition, revised and supplemented / V.A. Vasin, I.B. Vlasov, Yu.M. Yegorov et al.; edited by I.B. Fedorov. – M.: Izd-vo MGTU im. N.E. Baumana, 2004. – 768 p. Fundamentals of the statistic theory of radio systems are set forth. Problems of the signal detection, identiﬁcation and resolution against the noise background and problems of measuring signal parameters including those varying for the observation time are considered. An attempt is made to generalize the methodical approach to analysis and synthesis of basic information technologies for various radio technical systems: radar, satellite, radio navigational and for data transfer systems. The 2nd edition (1st in 2003) is revised and supplemented. Some chapters, devoted to information characteristics of radio technical systems and advanced data transfer systems, in particular, to cellular ones, are added. The textbook content corresponds to the course of lectures delivered by the authors in the Bauman Moscow State Technical University. For students of higher educational establishments, learning the trade “Radio Technology”. Can be useful for post-graduates, workers of research and industrial institutions, practicing in the ﬁeld of development of multi-purpose radio technical systems.

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ECOLOGY PROBLEMS N.P. Demenkov, V.A. Matveev (Bauman Moscow State Technical University)

FUZZY SYSTEMS OF ECOLOGICAL MONITORING AND CONTROL It is offered that systems of ecological monitoring and control be regarded as intellectual systems, i.e. systems based on human knowledge. The approach to designing systems of ecological monitoring and control, based on the T. Saaty method of an analytic hierarchy process (AHP) and the L. Zadeh theory of fuzzy sets, is investigated. Some investigation results are given.

Introduction. The improvement of ecological environment implies solving a large number of rather complex, nonlinear problems, difﬁcult to formalize and requiring knowledge on many ecology aspects, including the environmental monitoring. As a whole, a monitoring system should be regarded as an expert system that carries out the monitoring of the environment condition and helps people to inﬂuence it. At present, solving ecological problems is a difﬁcult task for all countries of the world. First of all, it is necessary to mention economic difﬁculties. To develop, implement and employ special ecological systems (information, research systems etc.) is very costly, therefore not every country in the world can afford it. But, if there are no economic difﬁculties, instead of them others come — difﬁculties of construction, production, etc. Data is collected in databanks while arriving from observation platforms: mobile laboratories, balloon probes, planes and helicopters, rockets, geostationary monitoring satellites or low-altitude viewing satellites. The data passes check in databanks. The current condition data of a given object is compared to its past condition data, and information on its change and a quantitative estimation of this change are transferred to the decision-making person. The appropriate control action can also be recommended. Both modern means of ecological monitoring and information-and-control systems, supporting them, are complex multifunctional multimode distributed systems. In such systems the joint processing of the data and knowledge, having complicated organization, is carried out. They should be developed on the basis of modern information technologies, which would ensure the essential rise of a level of the information and intellectual support. The informatization problems in solving ecological tasks become fundamental in connection with a widespread application of local and global computer networks. The efﬁciency of predicting the development of the ecological situation (including destruction) in this or that area, at an enterprise or facility depends on solving these problems.

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Using information technologies, based on knowledge, is a way out for the created situation. The representation of the domain knowledge and organization of its processing are provided in such systems at various hierarchical levels to increase the efﬁciency of control and decision-making. The urgency of consideration of such class systems is caused by their ability to accumulate and generalize knowledge, to develop hypotheses and forecast, to make decisions and explain them, etc. An intellectual system is understood as an aggregate of hardware and software, united by the information process and working independently or interacting with a person (group of people), capable to synthesize a goal, to make decisions on action and to ﬁnd rational ways for achieving the goal on the basis of items of information and knowledge with the available motivation [1]. Solving Multi-Criteria Optimization Tasks under Condition of Qualitative Uncertainty. While solving a number of ecological tasks, the situations occur when either there are no necessary sensors of the primary information or the existing measuring means do not provide the real-time reception of the required information or the qualitative information on the control object is only available. In such situations information technologies are required that allow getting the required control information with the help of computer processing of the qualitative (or fuzzy) information about the control object and goals. Let us consider a multi-criteria optimization problem in the formulation, when the goals, outcomes (criteria values) and alternatives (actions) are given fuzzily, but the preference relations, i.e. function of utility, are given crisply. For solving the problem it is necessary to deﬁne and maximize the utility function U for alternatives under consideration: (1) max{U (z1 , z2 , . . . , zk )}, where zi = fi (x ∈ S ⊂ Rn ), i = 1 . . . k are components of the criteria vector; x is a point in the decision space; S is a set of the allowable decisions. The basic difﬁculty, arising in solving the problem, is to derive the mathematical description of the utility function U [2]. In the utility theory the utility function U is considered and calculated as a probabilistic value, however, in many complex non-formalized decision-making problems it is difﬁcult and even impossible to estimate the multi-dimensional distribution of probability. In this paper the utility function is not considered as a probabilistic value, but as a fuzzy one, with membership functions of fuzzy sets being regarded as subjective measurements of the decision-making persons (DMP). The multi-criteria optimization problem can be presented in a form of the hierarchical decomposition (Fig. 1), where the set, formalizing the global goal, is denoted M . If the goals are deﬁned by too sophisticated concepts, they can be presented as a hierarchy of simpler ones. The elements of the hierarchy, i.e. goals and alternatives, are fuzzy sets denoted in Fig. 1, respectively, Gj , j = 1, . . . , m, where m is a number of goals, and Xr , r = 1, . . . , l, where l is a number of alternatives. The hierarchy levels are denoted Ln−1 , Ln , Ln+1 . The utility function is considered as a membership function of the global goal M on the set of alternatives X and is denoted µM (X). So, what is wanted is to get µM (X). 228

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Fig. 1. Hierarchical decomposition of multi-criteria optimization problem

The method of an analytic hierarchy process (AHP) [3], being a method to solve multi-criteria problems in a difﬁcult situation with hierarchical structures including non-formalized elements, is used in this paper as an indirect method to determine membership functions of fuzzy sets [4]. Let it be required to determine a membership function µD0 (x) of a fuzzy set D0 , deﬁning some qualitative concept. For this purpose, a DMP is offered to compare quantitative elements of a universal set X to one another by a degree of their conformity to this qualitative concept and to ﬁll in a matrix of pairwise comparisons A = {aij }, whose elements aij are estimates of a degree of membership of elements ai ∈ X to the fuzzy set D0 in comparison with elements aj ∈ X. The membership function is found as an eigenvector ω of the matrix A, corresponding to its maximal eigenvalue λmax : Aω = λmax ω. When complex properties, being represented as a hierarchical system, are analyzed, the above-described approach is used to compare the property constituents by a degree of their conformity to this complex property. For the hierarchical case, T. Saaty [3] proved the following theorem: If H is a complete hierarchy, having h levels and the greatest element b on the top (zero) level, Bn is a matrix of priorities of the n-th level, n = 1, . . . , h, and W 0 is a vector of priorities of the p-th level with respect to some element z in the (p − 1)-th level, then the vector of priorities W of the q-th level (p < q) with respect to z is deﬁned through the Cartesian product W = Bq × Bq−1 . . . Bp+1 × W 0 .

(2)

According to this theorem, a vector of priorities of the lowermost level with reference to the element b is deﬁned by the following expression: W = Bh × Bh−1 . . . B2 × W 0 .

(3)

Unfortunately, Eq. (2) and, hence, Eq. (3) can not be considered as the solution of the problem (1), i.e. the obtained ranking can not be considered as a function of utility. Besides, engineering problems, requiring formalization of qualitative VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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characteristics of physical values, can not be solved by the AHP method in its classical statement. The validity of the T. Saaty theorem is proved in [5, 6] for a case, when not all the hierarchy is considered as a fuzzy set (with its universal set at the lower level), but the hierarchy individual elements are fuzzy sets. The utility function for alternatives can be considered as a function of membership of the global goal on the set of alternatives, with the membership function being believed as a subjective value, instead of a probabilistic one. The membership functions of fuzzy sets formalizing each criterion (complex, represented as a hierarchy, or simple) are deﬁned on the set of alternatives, thus the set of alternatives is a base set for the criteria fuzzy sets. At every level we have various crisp (non-fuzzy) ordered sets that consist of elements, being fuzzy (qualitative), and each is deﬁned by the membership function of its own. Elements of each level are fuzzy subsets of the crisp ordered set and are deﬁned just in this sense. Let the hierarchy be an aggregate of levels Ln , n = 1, . . . , h. Any n-th level of the hierarchy is an array of the individual elements lnj , j = 1, . . . , m where m is a number of elements of the given level. The array of elements lnj represents “fuzzy properties” for elements of the lower, (n + 1)-th level, i.e. lnj is an array of fuzzy sets; universal sets for these fuzzy sets are sets of elements of the lower levels. Let us write out elements of each level: m

level Ln−1 : level Ln

:

n−1 1 2 3 ln−1 ln−1 ln−1 . . . ln−1 ;

ln1

ln2

ln3

...

lnmn ; m

level Ln+1 :

n+1 1 2 3 ln+1 ln+1 ln+1 . . . ln+1 .

The connection among the neighboring levels of hierarchy is deﬁned by a matrix of eigenvectors, i.e. by the matrix of priorities B: ω12 ... ω1mn ω11 ω21 ω22 ... ω2mn , (4) Bn+1 = ω31 ω32 ... ω3mn ωmn+1 1 ωmn+1 2 . . . ωmn+1 mn where Bn+1 is a matrix of eigenvectors of the level Ln+1 , mn+1 is a number of elements of the level Ln+1 , mn is a number of elements of the level Ln . 1 , A membership degree of the element ln1 to a fuzzy set, being the element ln+1 is determined as 1 µln+1 (ln1 ) = ω11 = µln1 (ln+1 ). 1 1 A membership function of the fuzzy set, being the element ln+1 , 1 (Ln ) = [ω11 ω12 . . . ω1mn ] ≡ Wn+1,n µln+1 1

is a membership function of the ﬁrst element of the (n + 1)-th level, deﬁned on the base set of the level Ln (row of the matrix from Eq. (4)). A membership function of a fuzzy set, being the element ln1 ,

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1 µln+1 (Ln+1 ) = [ω11 ω21 ω31 . . . ωmn+1 1 ] ≡ Wn,n+1 1

is a membership function of the ﬁrst element of the n-th level, deﬁned on the base set of the level Ln+1 (column of the matrix from Eq. (4)). It is obvious, that the matrix Bn+1 speciﬁes a binary fuzzy relation between fuzzy sets belonging to levels n and n + 1: Bn+1 ≡ Rn+1,n : Ln+1 ◦ Ln → [0, 1].

(5)

The problem of the hierarchies analysis is reduced to ﬁnding µl1 (Ln+1 ), i.e. to ﬁnding a membership function of the fuzzy set of the element of the ﬁrst level of the hierarchy determined on the base set of the (n + 1)-th level. We consider, that the level L1 consists of one element and there are n + 1 levels in the hierarchy. It is necessary to ﬁnd the relation between the fuzzy set l1 of the top level and the fuzzy sets belonging to the level Ln+1 : R(Ln+1 , L1 ) : Ln+1 ◦ L1 → [0, 1]. From Eq. (5), Bn+1 is a matrix of fuzzy relations of fuzzy sets in levels Ln and Ln+1 Bn+1 = R(Ln+1 , Ln ). So, the membership function µln (Ln+1 ) can be deﬁned as a composition of fuzzy relations: R(Ln−1 ) = µl1 (Ln+1 ) = R(Ln+1 , Ln ) ◦ µl1 (Ln ). The composition can be rewritten in a different way: mn+1

m \n

[ µl1 (Ln+1 ) =

µl j

(Ln )

µli (Ln ),

(6)

1 n+1

j=1

i=1

where mn is a number of elements of the n-th level; mn+1 is a number of elements of the (n + 1)-th level. Equation (6) is equivalent to Eq. (2). Hence, the validity is proved of considering the global vector of alternative priorities (vector of priorities) of the last level as the membership function of the global goal of the problem solution which, in turn, can be considered as the utility function in solving multi-criteria problems in a fuzzy statement. The latter assertion allows the AHP method to be applied to solving such multicriteria problems, as the estimation of fuzzy models of complex systems, where qualitative estimates of physical values should be formalized on quantitative scales. Thus, the AHP method can effectively be used not only for making a decision in non-formalized domains and solving tasks of ranking the ﬁnite set of complex objects presented as a hierarchical structure, but also for making decisions in engineering tasks, where the formalization of all fuzzy concepts is required with the VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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help of fuzzy sets. It opens additional opportunities for introduction of new information technologies, for example, to replace human operators, who manage technological processes, to improve the quality of management. It will allow building the intellectual systems capable to make a decision and adequately react on changes of a situation on the basis of human knowledge. Fuzzy Logic Controller. A technique and software, based on the suggested approach to solving non-formalized optimization problems, have been developed for creating a model of the fuzzy logic controller (FLC) destined for managing complex engineering objects. The main part of information, required for the model construction is qualitative or fuzzy. The data is presented in a form of rules (where the physical concepts are expressed in the natural language), called the linguistic rules of control (LRC). The operation of minimum is used as implication, if the rule is formulated as IF-THEN, or the operation of maximum — for the rule IF–THEN–OTHERWISE. The composition rule is used as a rule of inference, with the composition being calculated as maximal-minimal. The weight method is applied in choosing the crisp value of the management function, according to which the value, having the maximal membership function, gets out as the only value for management. The model of receiving data for control, i.e. the model of a controller, is developed as a hierarchical structure, whose elements are the input linguistic variables (LV) [4], inﬂuencing the process, the input linguistic rules of control, in which these variables are involved, and qualitative estimations of these linguistic variables. The main goal of solving the problem — to get the optimal control — is arranged at the ﬁrst level. As control only depends on the corresponding input variables, the input LVs, inﬂuencing the above-mentioned control, are arranged at the second level and the LRCs, used for getting the above-mentioned control with the help of the LVs, are arranged at the third level. Qualitative values of the input LVs are arranged at the fourth level. The LRC validity is taken into account on the basis of estimation of the qualitative values of the current observation data. Elements of the ﬁfth level are quantitative values of the input LVs (i.e. universal sets for fuzzy sets of the LV terms) used for estimation of membership function of their qualitative values. The technique makes possible to estimate the signiﬁcance of the input LVs, the relative importance of LRC in technological process and the LRC validity with the help of the obtained qualitative data and also to take into account a degree of each variable inﬂuence on control. If several output values are required for control, then hierarchical structures are deﬁned and estimated for each variable. The ad hoc software, developed according to the suggested technique, uses the MATLAB package and can also be implemented on any freely programmed controller, with the compiler for the programming language C++ available. Figure 2 illustrates the situation when there are three input LVs with two LRCs and, consequently, two qualitative values for each variable; V, W, Z are the universal sets with base variables v, w, z, respectively. The qualitative estimation of values of input variables, used in LRC, is formalized with fuzzy subsets A, C and D of sets of values of parameters V, W and Z, respectively; A0 , C 0 , D 0 are qual-

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Fig. 2. Hierarchical decomposition in designing FLC

itative values of observation for LV 1, LV 2, LV 3, respectively. The qualitative estimation of values of control action u is formalized with the fuzzy subset B of the set of control actions U . An assertion of the type “if A then A × B” is a binary fuzzy relation in V × U which presents a Cartesian product of A and B and is deﬁned as follows: if A then B = A × B. For conversion of fuzzy inferences at the linguistic level, a method is used on the basis of the compositional rule of inference, which is interpreted as a process of solving a system of equations of assignment in relations where linguistic values are assigned to fuzzy restrictions. If V and U are two universal sets with base variables v and u, respectively, R(v), R(v, u) and R(u) denote restrictions for v, (v, u) and u, respectively, and present fuzzy relations in V , V × U and U , and A and F are fuzzy subsets of sets V and V × U , then the compositional rule of inference states that the solution to the system of assignment equations R(v) = A, R(v, u) = F is of the form R(u) = A ◦ F, VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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where A ◦ F is a composition of A and F . In this sense we can make a conclusion that R(u) = A ◦ F from R(v) = A and R(v, u) = F . Deﬁnition. Let A0 , A and B be fuzzy subsets of the sets V , U and U , respectively. We suppose that the value A0 is assigned to the restriction R(v) and the relation A ⇒ B is assigned to the restriction R(v, u), i.e. R(v) = A0 , R(v, u) = A ⇒ B. These equations of assignment in relations can be resolved with respect to the restriction for u in the following way R(u) = A ◦ (A ⇒ B). The expression of this inference in the form A0

premise

A ⇒ B implication A0 ◦ (A ⇒ B) inf erence presents the formulation of the generalized rule modus ponens. Basic operations applied in getting the fuzzy inference with the help of the compositional rule of inference are described as follows. A union of fuzzy sets A and B in V denotes the fuzzy set A + B with the membership function of the form µA+B (v) = max{µA (v), µB (v)},

v ∈ V.

A Cartesian product A1 ×. . .×An of fuzzy sets Ai in Vi , i = 1, . . . , n is deﬁned as a fuzzy set in the Cartesian product V = V1 × . . . × Vn with the membership function of the form µA (v) = min{µA1 (v1), . . . , µAn (vn)}, v = (v1 , . . . , vn ) ∈ V. A complement (negation) of a fuzzy set A in V denotes the fuzzy set A¯ with the membership function of the form µA¯ (v) = 1 − µA (v),

v ∈ V.

A maximin product A ◦ B, in case of the ﬁnite set V , is equal to the maximin product of relation matrices A and B, i.e. µA◦B (v, u) = max min{µA (v), µB (u)}. v∈V

A fuzzy inference presents an application of the maximin composition as a compositional rule of the fuzzy inference and operation of ﬁnding minimum as a fuzzy implication (if the rule IF–THEN is used):

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µB 0 (u) = max min{µA0 (v), µR (v, u)} = v∈V

= max min{µA0 (v) min[µA (v), µB (u)]}, (7) v∈V

where B 0 is a fuzzy control action and A0 is a speciﬁed value of current data of observations. Then the defazziﬁcation follows, with which on the basis of the membership function µB 0 (u) for B the exact value u (to be used for the control operation) for each point of U is obtained. As a defuzziﬁcation method the weight method is applied which is based on using the variable u that has a maximal value of the membership function. In the FLC implementation several rules are commonly used whose structures may be different. While performing operations with the help of the maximin composition accompanied with the calculation of the maximal membership function value for u the inference result for each rule is obtained according to Eq. (7). As a ﬁnal result of the inferences, a sum is assumed of fuzzy sets which are inferences resulting from each rule (operation of summing or ﬁnding maximum). In other words, the rules are processed in parallel. The ad hoc software, implementing the suggested technique in MATLAB package, features the following: — a maximal number of control actions, i.e. estimated hierarchies, is 4; — a maximal number of input variables is 12; — a maximal number of LRCs for a single LV is 8, i.e. a maximal total number of LRCs is 384. At ﬁrst, a user estimates membership functions for all qualitative values, comparing quantitative elements in pairs by degree of their conformity with the qualitative concept on a 9-point verbal scale. Results are issued in the numerical and graphical form. A conformity relation is calculated for each estimated membership function. Then priorities of LVs, LRCs and the LRCs validity are estimated. The estimation is conducted in the same manner as in the AHP software [5], destined for solving multi-criteria problems, in case of 5-level incomplete hierarchy. Results of Research. The scientiﬁc-educational complex “Computer Sciences and Control Systems” of the Bauman Moscow State Technical University has been engaged in problems of development of theoretical bases of construction of intellectual information-and- control systems to solve ecological tasks for a number of years [6–9]. Some results of the complex activities are as follows. The intellectual system of environment monitoring is investigated. The algorithms of the environment condition estimation and forecast are formulated. They might be used for management of society activities to improve the quality of life. The ecological processes are investigated on the basis of the parametrical analysis of inﬂuence of internal and external environment on optimum indemnities in gerontological and toxicological models of natural technologies of human organism.

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The program of computer thermodynamic modeling of the kinetics of chemical transformation of gaseous mixtures is developed. The general basic description of a method to calculate the composition and properties of an equilibrium condition of multi-component heterogeneous systems is given. The methods, algorithms and software to design the intellectual regulators for ecological systems are investigated on the basis of the fuzzy combined strategy of management and the relevant approach to generate fuzzy rules. The strategy of the fuzzy PID management is implemented in the automatic system of decreasing toxicity of diesels on the basis of recycling soot ﬁlters. The combination of PI- and PD-algorithms with the help of two input variables considerably reduces a number of managing rules without deteriorating the quality of regulation. The problems of organization of data processing and management on the basis of technology of systems based on knowledge, named S-technology, are investigated. The offered architecture of information support system for making decisions provides for the base of knowledge the functioning of subsystems of management and of knowledge output as uniﬁed system. It allows the system to be quickly adapted to various software at the expense of updating the user interface of the subsystem of knowledge output. The subsystem of management of base of knowledge remains unchanged in this case. The partitioned architecture allows using and modifying components of the subsystems independently. The procedure mechanism makes possible the data exchange with external devices and expands functionalities of the system. The algorithms of cosmonaut’s actions in performing typical tasks of viewing the emergency areas with the use of the cosmonaut helmet information and control system (“NIUSK”) are developed. The generalized and partial criteria of an estimation of efﬁciency of the algorithms are offered. They are substantiated with the system approach and method of parallel analogy. The algorithms to detect outﬂows and breaks are produced for the system of detection of outﬂows and localization of breaks in urban networks of water supply. The software for the reception, processing and displaying of the information is offered. The research results are used in the educational process during the laboratory training, development of the course and degree projects. They are, on the one hand, objects of research, and, on the other hand, they form the ecological thinking of the students. Conclusion. We have discussed some results of development of fuzzy systems for ecological monitoring and control. The general problem of optimization is understood as a problem with fuzzily expressed criteria and alternatives. As the research results have shown, the fuzzy logic approach can improve our ability to ﬁnd out solutions for ecological dynamical systems and speed up computations. The level of solving ecological problems in any world country can serve the most authentic and complex criterion of an estimation of a level of stability of this state and its economic development, and also moral condition of a society in the modern world.

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The formation of the ecological culture of an engineer is therefore an important aspect of the engineering education. The formation of the ecological thinking based on the comprehension of the main principle — “absoluteness” of ecological priorities while solving scientiﬁc and technical tasks is the major purpose of the ecological education. The ecological education has the special importance in technical higher educational establishments, as it, in many respects, deﬁnes efﬁciency of solving ecological problems directly at site of their occurrence.

REFERENCES 1. Pupkov, K.A., Konkov, V.G. Intellectual systems (in Russian). M.: Izdatelstvo MGTU im. N.Ye. Baumana, 2003. 2. Shtoer, R. Multicriteria Optimization. Theory, Calculation and Applications / Translated from English. M.: Radio i Svyaz, 1992. 3. Saaty, T.L. Making Decisions. Method of Analysis of Hierarchies / Translated from English. M.: Radio i Svyaz, 1993. 4. Zadeh, L.A. Concept of Linguistic Variable and its Application to Making Approximated Decisions / Translated from English. M.: Mir, 1976. 5. Grunina, G.S., Demenkov, N.P., Yevlampiev, A.A. Solution of Multicriterial Optimization Problem Under Conditions of Qualitative Uncertainty (in Russian) / Vestnik MGTU, 1998, No 1. 6. Demenkov, N.P. Solution of Multicriterial Optimization Problems and Making Decisions in Fuzzy Statement (in Russian) // Trudy III Mezhdunarodnoi konferentsii po myagkim vychisleniyam i izmereniyam, S-Pb, 2000. 7. Demenkov, N.P., Matveev, V.A. Computer Technologies of Decrease of Intensity of Allocation of Harmful Engines of Power (in Russian) / Ekologicheskie sistemy i pribory, 2001, No 11. 8. Demenkov, N.P., Matveev, V.A. Intellectual Information and Control Systems for Solution of Ecological Tasks (in Russian) / Konversiya v mashinostroenii, 2002, No 1. 9. Demenkov, N.P., Matveev, V.A. Logical-linguistical Models in Ecological Tasks of Municipal Economy (in Russian) / Promyshlennye ASU i kontrollery, 2003, No 2. V.A. Matveev (b. 1939), D. Sc. (Eng.), Professor, the Head of the Research and Education Unit “Information Technology and Control Systems” (NUK IU) of the Bauman MSTU. Scientiﬁc interests: gyroscopic systems, navigational complexes, control systems, optimization of dynamic system at indeﬁnite external parameters of the environment. N.P. Demenkov (b. 1944) graduated from Bauman Moscow Higher Technical School in 1968. Ph. D. (Eng.), assoc. professor of “Automatic Control Systems” Department of Bauman Moscov State Technical University. Winner of Komsomol Prize in the ﬁeld of science and technology. Author of more than 150 publications in the ﬁeld of optimization of dynamic system at indeﬁnite external parameters of the environment, designing the automatic control systems for space vehicles, automatic and automatized control systems for technological processes and objects.

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FUNDAMENTAL PROBLEMS of MECHANICAL ENGINEERING Yu.N. Drozdov (Mechanical Engineering Research Institute, Russian Academy of Sciences) Ye.G. Yudin (Bauman Moscow State Technical University)

WEAR PREDICTION CONSIDERING MECHANICAL, PHYSICO-CHEMICAL AND GEOMETRICAL FACTORS Wear of solid bodies depends on physical-chemical and mechanical processes in the contact zone. The processes are greatly inﬂuenced by kinematics of mating surfaces motion (sliding, rolling, rolling with sliding, spinning and etc.), by structure and composition of material layers in the surface and surﬁcial region, by formation of surface compounds, by geometric properties of contacting surfaces and their change in time. A technique of obtaining relations to calculate a wear-rate is based on the synthesis of experimental data and mathematical models. A theoretical and invariant method, developed by the authors, to calculate the rate of the friction-induced surface destruction of solid bodies is based on equations of elastic-hydrodynamic lubrication theory, chemical kinetics, contact problem of elasticity theory, theory of strength, analysis of thermalphysical, adsorption and diffusion processes. Theoretical studies were closely connected with experimental ones. A heuristic role of physically informative tribological invariants is shown.

Introduction. After known calculated dependences for deﬁnition of magnitude of a wear — papers by Holm, Khrushchov, Kuznetsov, Archard, Rabinovich, Kragelsky, Flyaisher — the new basic equations have not appeared in literature within several decades. Only empirical approaches concerning separate units and concrete situations in maintenance predominate. The known calculated equations attract with the simplicity, but, unfortunately, do not take into account many rather important factors. For example, inﬂuence of a lubricant layer (boundary and elastichydrodynamic), thermalphysic conditions of tribocontact, adsorption and diffusive phenomena, modiﬁcation of rubbing surfaces. In practice only rather approximated, comparative and estimated calculations therefore are used. Prediction of service life at a design stage is acutely necessary for modern engineering, especially for the friction units operating in extreme conditions (aerospace, atomicenergetic, high-speed transport, technique for mastering Ocean, deep bowels of the Earth etc.). The urgent problem consists in generalization of experimental data, prediction of reliability and service life in a wide range of voltage, temperatures, and environment change, of effect of ﬁelds of a different physical nature (acoustic, electromagnetic, radiant etc.).

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The proposal of the authors consists in use of physically informative criteria obtained from theoretical dependences, describing process, and experimental data. The structure of obtained calculated dependences contains the generalized variables, allowing understanding of dominating factors during wear process. The quantitative response is achieved with the use of the calculated dependences obtained on the basis of special experimental research or from practical generalizations. In a basis of research the theory of similarity and simulation lies. The real possibility of the development of calculated models of wear process is shown in works [1–5]. dh , where We shall use the concept of intensity of linear wear process Jh = dS h — magnitude of a linear wear, S — a path of wear. Speed of wear process or dh intensity of wear process with respect to time is deﬁned as Jhτ = = Jh v, where dτ τ — time of wear process, v — speed of tribo-surfaces migration. Having installed the regularity for deﬁnition of intensity of wear process in different conditions, it does not represent difﬁculties to deﬁne a service life by criterion of a wear of real friction units [5]. Thickness of Elastic-Hydrodynamic Lubrication Layer. For bodies rolling with sliding the process of shaping of lubricant layers is described by the equations of elastic-hydrodynamic theory of lubrication. For simplicity the isothermal equation of Reynolds is used: ¤ £ dp/dx = 6µ (v1 + v2 ) (h − h0 ) /h3 , where p = 0 at x = −∞; p = dp/dx = 0 at x = x0 . We shall write down the equation for deﬁnition of a clearance as ¶ Zx0

µ x2 − x20 2 h = h0 + − 2r π

1 − v12 1 − v22 + E1 E2

¯ ¯ ¯ ζ −x ¯ ¯ dζ. ¯ p (ζ) ln ¯ ζ − x0 ¯

−∞

The equation, expressing conservation of energy, will be as follows µ ¶2 dt d2 t dv dp ρlub club = λlub 2 + µ + kvt , dx dy dy dx where t = t0 at x = −∞; ¶1/2 Zx

µ 1 πρ1 c1 λ1 v1

t (x, 0) =

¯ dt ¯¯ dω λ1 ¯ + t0 , dy y=0 (x − ω)1/2

−∞

¶1/2 Zx

µ t (x, h) =

1 πρ2 c2 λ2 v2

¯ dt ¯¯ dω −λ2 ¯ + t0 . dy y=h (x − ω)1/2

−∞

Dependence of viscosity of a lubricant on pressure and temperature is: µ = µ0 exp (βp − ψ∆t) , VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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where p — pressure; µ — a dynamic viscosity of a lubricant; v1 , v2 — peripheral speeds; h — a clearance; h0 — a minimum clearance; r — a reduced radius of curvature 1/r = 1/r1 ± 1/r2 ; r1 , r2 — radii of cylinders; “+” sign corresponds to external contact, “−” sign — to internal contact of bodies; E1 , E2 — coefﬁcients of elasticity of contacting bodies; ν1 , ν2 — factors of Poisson of materials of bodies; t, ∆t — temperature and an increment of temperature; ρlub , club , λlub , ρ1 , C1 , λ1 , ρ2 , C2 , λ2 — densities, speciﬁc thermal capacities, factors of a thermal conduction of a lubricant and materials of bodies, respectively; β — piezo-coefﬁcient of viscositiy; ψ — factor of dependence of viscosity on temperature in the formula of Reynolds; k — factor of a thermal expansion of a lubricant; x, y — coordinates up and down a ﬁlm; ζ, ω — additional variables; x0 — an abscissa of the point where pressure and a gradient of pressure are equal to zero. On the basis of a method of integrated analogs, initial set of the equations, boundary conditions and conditions of unambiguity, criteria of a similarity, the principal dimensionless complexes describing process of friction and lubrication were found. The criterion describing a carrying capacity of contact µv/Pn , where v = vsl in conditions of sliding and v = v1 + v2 = vΣk in conditions of rolling friction. Deformation of the contact is ¢being expressed ¢by the criterion Pn η/r, where Pn — ¡ ¡ running loading, η = 1 − v12 /E1 + 1 − v22 /E2 — elastic constant of touching bodies. The temperature processes ¢ which are ﬂowing inside a lubricant layer are ¡ 2 deﬁned by criterion λlub / ψµv , where v = vsl at the analysis of the temperatures, happening as a result of energy dissipations at sliding, v = vΣk — at rolling. Pekle numbers Pe1 = bv1 /a1 and Pe2 = bv2 /a2 characterize a ratio of heat content of a stream in axial and transversal directions, where b — a halfwidth of a strip of contact by Hertz; a1 , a2 — factors of temperature conductivity of materials. The numerical solution of reduced system of essentially nonlinear integrodifferential equations may not give an acceptable quantitative response as the mathematical model with all its completeness and complexity does not reﬂect many latent and complex-connected phenomena accompanying processes of friction, lubrication, destruction of a layer of lubrication and real solid bodies. Drawing up the calculated equations on the basis of experimental results and obtained physically informative performances allows one to compensate some latent operating factors to some extent and to discover the best values of factors. In result the equations appear which have the physical contents, expressing qualitative inﬂuence of major factors and giving quantitative results necessary in engineering practice. Elastic-hydrodynamic theory of lubrication has been developed for more than 60 years. However, at rolling with sliding of bodies (cogged and friction gears, rolling bearings, cam mechanisms etc.) inﬂuence of speed of sliding on the width of a lubricant layer is not established. It is difﬁcult to construct the theoretical model, which would take into account simultaneously the inﬂuence of speed of rolling and sliding on forming and destruction of a lubricant layer. Observations in practice and laboratory research of real mechanisms and physical models display essential inﬂuence of speed of sliding on the wear of rubbing bodies in the lubricant environment, and, hence, on the width of a lubricant layer. Thus, well-known calcu-

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lated dependences have the restricted application for real processes (small speeds, cinematically pure rolling). On the basis of the generalized criteria (dimensionless complexes) and experimental researches the authors have developed the following dependences, which take into account the inﬂuence of speeds of rolling and sliding. For the boundary conditions of friction appropriate to the jam origination the equation is derived: Ã !µ ¶ µ ¶e ¶ µ r µvΣk 0.7 Pn β 0.6 λ K3 p 2 61 Pn r αµ0 vS2 Pe0.5 Ra1 + Ra22 1.2 where K3 and e — the experimentally deﬁned magnitudes depending on physical and chemical processes (magnitude K3 varies depending on materials of surfaces of cog-wheels and used oils: for gears K3 = 1 . . . 3, for samples — rollers K3 = = 0.5 . . . 5; magnitude e = 0.23 . . . 0.35); r + Ra22

p

Ra21

— a complex describing micro- and macro-geometry of contacting bodies; Ph β r — a complex describing deformability of contacting bodies and piezo-viscosity properties of a lubricant; λ 2 µ0 vS Pe0.5 1.2 — thermophysical complex;

Pe1.2 =

Pe1 + Pe2 2

— average by Pekle. Using the obtained dimensionless complexes and results of experimental research of the lubricated ﬁlm width between cylindrical bodies rolling with sliding, it is possible to get calculated dependences for the deﬁnition of the width of a lubricant layer considering heat release as a result of a viscous dissipation at rolling with sliding [6]:

µ h = 1.65 r

¶0.1 "

¶0.75 µ βµvΣk 2r

E p

1 + 0.18

2 ψµvΣk 4λlub

¶0.8 #−1

µ

¶0.7

µ

+ 0.45

2 ψµvsl λ

.

The analysis of regularities of change of elastic- hydrodynamic layer of lubrication displays an essential roleµof speed ¶ of sliding. At rolling with sliding with the λlub , the width of a lubricant layer diminishes. increase of speed of sliding ψµVsl2 VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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At rather thin layers the inﬂuence of temperature processes in discrete points of contingence is revealed. Destruction of a lubricant layer results in intensive wear process, jam of rubbing surfaces, growth of a factor of sliding friction f . Appearance of signiﬁcant thermal sources in spots of a contingence fσ vsl results in burn-off of rubbing bodies, thus the factor of friction does not increase [7]. Mechanical and Geometrical Factors of Wear Process. In case of dominant mechanical destruction of friction surfaces the determining criteria are the following: p fp ; HB HB — the criterion, describing a tension of contact, the dimensionless area of an actual contingence of bodies. This criterion is most representative and is applied in calculations with different types of wear process; here p — direct stress of compresh sion; HB — hardness of a wearing material; — the criterion, deﬁning a relative ζ width of a lubricant layer; h — a width of a lubricant layer; ζ — the characteristic size p of the abrasive particle, cutting roughness or the reduced size of roughness Ra21 + Ra22 . The criterion is applied for estimation of conditions of lubrication and fatigue surface destruction. ν1 p σ0 — the criterion of fatigue strength of rubbing surfaces, where p — a direct stress of compression; ν1 — the factor depending on factor of sliding friction and tension in contact; σ0 — fatigue limit of a material in the considered conditions of friction. ´ ³ 1/v Rmax / rkk bkk kk — the criterion of a roughness (factor by Kragelsky–Kombalov); Rmax — the greatest height of irregularities of the proﬁle; rkk — reduced radius of irregularities; bkk , vkk — parameters of a basic curve. Physico-Chemical Factors of Wear Process. In case of a physical and chemical effect of environment (lubricant) it is necessary to apply additional criteria ratio: RT /Q — the criterion describing stability of boundary lubricant layers, where R — universal gas constant; T — absolute temperature; Q — heat of adsorption. RT /Echem — the criterion describing chemical modiﬁcation of surfaces of friction, where Echem — energy of activation of disintegration of interatomic links of surface connections. Kinetic processes of forming and destruction of surface layers (connections) depend on the time factor. Characteristic time for bodies rolling with sliding is: τr.av. = dav /v — average time of solitary actual contact; dav — average diameter of an actual tooth contact; v — speed of relative driving of bodies; τr.f ree = lav /v — an average free time; lav — average distance between microprotuberances or τr.f ree = πHBdav /(8pv); time of contact on the nominal area τnom = 2b/v — for a case of contact of cylinders or a cylinder with a plane (where 2b — length of

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a strip of the contact, calculated by the method of Hertz); τout = Sτ /v — time of free stay of a surface outside the zone of contact (where Sτ — a path, ran by a body before the next nominal contact). Thermalphysic Factors of Wear Process. Inﬂuence of thermalphysic processes on the intensity of wear process is deﬁned by the following: tcon /tcr — criterion, deﬁning inﬂuence of contact temperature, where tcon — temperature in contact; tcr — critical temperature (for example, temperature of fusion, homological temperature, temperature of characteristic physicochemical and structural transformations in the materials of rubbing bodies); qδt /λ1,2 tcr — the criterion, deﬁning inﬂuence of temperature gradient and thermal boundary layer, where q — the speciﬁc thermal stream operating on the given body (a density of a thermal stream), λ1,2 — factor of a thermal conduction of materials, δt — width of a thermal boundary layer in a rubbing body. For approximated deﬁnition of a width of the warmed up (thermal) boundary layer appearing in a body at friction, it is possible to use the following equation: √ √ tδ /tcon π = ierfc (δT /2 aT τ ) , where tδ — temperature at a distance δT from a surface of friction; tcon — contact temperature on the surface; aT — factor of temperature conductivity; τ — time of an operation of a thermal source. Thermostress of surface layer is characterized by a criterion of the type: Eαt ∆t , (1 − ν) σlim where αt — factor of linear temperature expansion, ∆t — an increment of temperature in the surface layer; ν — factor by Poisson; σlim — limiting stress for the surface layer. ρ (tnom − t) σs — the thermomechanical criterion, where ρ — density of a material, c — a speciﬁc thermal capacity, t — a body temperature, tcon.f us. — temperature of contact fusion, σs — a limit of ﬂuctuation of a material. The criterion characterizes propensity of metals to seizing and factor of friction. Minimum values of a ratio of heat content of a material during its heating up to temperature of contact fusion [ρc(tcon.f us. − t)] to a limit of ﬂuctuation correspond to smaller values of a factor of friction and propensity to seizing [8]. Use of the speciﬁed criteria ratios in the calculated equation allows one to connect intensity of destruction with temperature in a place of contact, a temperature gradient, thermostresses, a thermal boundary layer, temperature constancy of a material, the characteristic temperatures, inﬂuencing content and structure of contacting materials. Load-Carrying Capability of Tribocontact. General ﬁeld of stresses on a surface of tribocontact σij can be presented as a ﬁeld consisting of mechanical VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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(power) and temperature stresses [9–11]: £ i ¤ t ¯ij (x, y, ν, f ) + Φt σ σij = σm σ ¯ij (x, y, v, Fo) . Here the dimensionless functions are as follows: σij depends on coordinates of a t depends on point x, y, ν (factor by Poisson), f (factor of a sliding friction); σij x, y, v and numbers Fourier Fo: Fo =

2aT , vract

where aT — factor of thermal conductivity, v — speed, ract — radius of an area of actual contact. The dimensionless parameter is determined by the formula:

Φt =

2 (1 + ν) αt G γf vract , (1 − ν) λ

where G — the module of shift, αt — factor of linear temperature expansion, γ — factor of distribution of thermal streams between rubbing bodies. The operation of mechanical and temperature stresses results in lowering the load-carrying capacity of tribocontact, and also in the beginning of plastic deformations and wear process of contacting bodies. Temperature factor Φt was accepted by authors as thermomechanical criterion of the beginning of wear process [9–11]. Examples of use of obtained calculated dependences for different friction units are presented in papers [5, 12–15]. More detailed research of obtained criteria is carried out nowadays on the basis of the consideration of dislocation processes in contacting materials . Conclusion. Designed on the basis of theoretical models, generalized dependences allow us to take into account mechanical, physicochemical and geometrical factors of wear process in the calculation equations. Interpretation of experimental results in the criteria form is important. The profound study of physics, chemistry and mechanics of the wear process will allow us to understand and take into account the phenomena taking place under the microlevel consideration in calculated dependences of wear process.

REFERENCES 1. Drozdov, Yu.N. Determination of Wear Rate of Machine Parts [In Russian]. Vestnik Mashinostroeniya, 1980, No 6, pp. 12–15. 2. Drozdov, Yu.N., Frolov, K.V. Theoretical-invariant Method for Calculating the Intensity of Solid Body Surface Wear due to Friction [In Russian]. Poverkhnost. Fizika, Khimiya, Mekhanika, 1982, No 5, pp. 138–146. 3. Drozdov, Yu.N. Transmission Mechanisms [In Russian]. Trenie, iznashivanie i smazka. Ed. by Kragelsky, I. V., Alisin, V. V. Mashinostroenie. Book 2, 1979, pp. 113–147. Transmissions Tribology-Lubrication, Frictionand Wear, 2001, pp. 597– 639. 244

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4. Drozdov, Yu.N. Key Invariants in Calculations of Wear Process Intensity at Friction [In Russian]. Mashinovedenie. 1980, No 2, pp. 93–98. 5. Kogaev, V.P., Drozdov, Yu.N. Strength and Wear resistance of Machine Parts [in Russian]. Moscow, Vysshaya Shkola, 1991, 320 p. 6. Shirobokov, V.V., Drozdov, Yu.N. Width of Lubrication Layer at Rolling with Sliding of Bodies Considering Thermal Processes [In Russian]. Mashinovedenie. 1979, No 4, pp. 90–93. 7. Drozdov, Yu.N., Archegov, V.G., Smirnov, V.I. Antiscuff Stability of Rubbing Bodies [In Russian]. Moscow, Nauka, 1981, 139 p. 8. Drozdov, Yu.N. Tribological Contact of Sliding in Space [In Russian]. Problemy mashinostroeniya i nadyozhnosti mashin. 2001, No 1, pp. 69–76. 9. Hamilton G.M., Goodman L.E. Trans ASME. Ser E Journal of Applied Mechanics. 1966, v. 33, No 1, pp. 371–376. 10. Ting B.Y., Winer W.O. Journal of Tribology. 1989, p. 315. 11. Yevtushenko, A.A., Uhanskaya, O.M. Thermomechanical Criterion of Wear Process [In Russian]. Trenie i iznos. 1994, Vol. 15, No 3, pp. 379–388. 12. Drozdov, Yu.N., Pavlov, V.G. and Puchkov, V.N. Friction and Wear in Extreme Conditions [in Russian], Mashinostroenie, Moscow, 1986, 224 p. 13. Drozdov, Yu.N., Tumabishvili G.I. Calculation of Seizing by Limiting Width of Lubricant Layer [In Russian]. Vestnik mashinostroeniya, 1982, No 4, pp. 19–22. 14. Drozdov, Yu.N., Mudryak, V.I. and Dyntu, S.I. Generalized Characteristics for Predicting the Wear of Rubbing Surfaces [In Russian]. Trenie i Iznos, Vol. 18, No 6, 1997, pp. 715–721. 15. Drozdov, Yu.N. Development of Tribology for Extreme Conditions [In Russian]. Tribologiya. Issledovaniya i prilozheniya. Opyt SSHA i stran SNG. Ed. by Belyi, V. A., Ludema, K., Myshkin, N. K. Moskva: Mashinostroenie. 1993, pp. 296–311. Ye.G. Yudin (b. 1940) graduated from the Bauman Moscow Higher Technical School in 1964. Ph. D. (Eng.), assoc. professor of “Multipurpose Caterpillar Vehicles and Mobile Robots”, First Vice-Rector – Vice-Rector for Educational Activity of the Bauman Moscow State Technical University, member of the Russian Academy of Natural Sciences. Author of more than 70 publications in the ﬁeld of theory and design of caterpillar machines, tribological reliability of transmissions. Yu.N. Drozdov (b. 1936) graduated from the Bauman Moscow Higher Technical School in 1959. Honored Worker of Science of the Russian Federation, Winner of the USSR State Prize, D. Sc. (Eng.), professor. Head of “Friction, Wear and Lubrication, Tribology” department of the Institute for Machine Science n.a. A.A. Blagonravov of the Russian Academy of Sciences. Author of more than 500 publications in the ﬁeld of complicated mechanical systems.

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Yu.N. Drozdov (Mechanical Engineering Research Institute, Russian Academy of Sciences) Ye.G. Yudin (Bauman Moscow State Technical University)

TRIBOLOGICAL PROBLEMS OF MECHANICAL SYSTEM CREATION FOR THE MOON A renewal and evolution of activities on the Moon development should be expected in XXI century. Building the manned lunar bases having production, astrophysical, extractive and maintenance equipment and services are in sight. Interplanetary ﬂights from the Moon require less power expenditures than those from the Earth, so it is proﬁtable to use Moon’s surface for erecting cosmodromes. An experience of creating technical systems for the Moon-application shows the bafﬂing complexity in providing the workability and reliability of tribotechnical systems. Some problems and results of experimental and theoretical studies on tribology for application in the Moon environment are discussed in the paper.

Introduction. Automatic interplanetary stations (AIS) like “Luna” (USSR), “Surveyor”, “Apollo” (USA) series were used in research of the Moon. The stations executed functions of ﬂying, landing, of artiﬁcial satellites of the Moon, of moonrovers and manned spacecrafts such as “Apollo” and they have allowed us to better understand the conditions of the Moon in XX century [1–3]. Automatic interplanetary stations “Luna-1” and “Luna-2” consisted of the thin spherical coverings formed by the connection of two hemispheres with a radius of 400 mm, a length of the station was 1300 mm. They were made of an aluminummagnesium alloy, hermetically incorporated with each other through sealing laying, which was made of the special rubber. Hemispheres were fastened by 48 bolts through a frame, with the diameter of 850 mm. The container was ﬁlled with gas (nitrogen) at pressure 0.13 · 105 Pa , temperature 20 ◦ C. The gross weight was 361.3 kg. “Luna-1” was launched on 2nd January 1959. The station has ﬂown close to the Moon and has left into its own orbit around the Sun. On 12th September 1959 the AIS “Luna-2” was launched, it reached a surface of the Moon on 14th September 1959 in the western part of the Sea of Rains, in 800 km from the centre of a visible disk. “Luna-3” has photographed the reverse of the Moon for the ﬁrst time. The station liftoff was realized on 4th October 1959. “Luna-3” represented a thin-walled hermetical cylinder with the spherical bottoms. The maximal transversal size was 1200 mm. “Luna-9” made soft landing to the Moon (1966), “Luna-16” supplied the Moon’s ground to the Earth (1970). The ﬁrst ﬂight of a spacecraft around the Moon lasted several days since 21st December 1968 (“Apollo-8”, astronauts Borman, Lovell, Anders). Expedition to the Moon came true on 16th July 1969 by the spacecraft “Apollo-11” with the astronauts Armstrong, Aldrin and Collins. The lunar module has executed necessary functions — landing, takeoff, joining the command module. In this ﬂight people

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landed on the Moon for the ﬁrst time, and in XX century the Moon was visited by 12 people. On 17th November 1970 the AIS “Luna-17” landed on the Moon with the ﬁrst automatic planet research vehicle “Lunokhod-1”. In total 5 transport machines were delivered to the Moon and they covered the following distances: “Lunokhod1” — 10 km, “Lunokhod-2” — 37 km, “Apollo-15” — 28 km, “Apollo-16” — 27 km, “Apollo-17” — 37 km. Three American moon-rovers (“Rover”) were used for astronauts’ moving along the Moon’s surface — “Apollo-15, 16, 17”. The Moon and Libration Points. Let us notice some geometrical and physical characteristics of the Moon: the diameter (equatorial) is 3476 km, medium density of the planet is 3.33 g/cm3 , acceleration of force of gravitation on a surface is 1.62 m/s2 , gravitation parameter 4.89 · 103 km3 /s2 , sidereal cycle time (relative to stars) is 27.32 days, synodical cycle time (the period of full change of phases) 29.5 days, circular speed is 1.65 km/s, a ratio of densities of an atmosphere at the surface of the Moon and the Earth is ≈ 2 · 10−13 , a ground basically porous (70 . . . 80 % of surface), the density of ground on depth from 0 up to 10 cm — 0.6 . . . 0.7 g/cm3 , from 10 up to 20 cm — 1 g/cm3 , from 1 up to 10 m — 2 . . . 3 g/cm3 . Elements of lunar rock are mostly oxides. A surface stratum (regolith) has generated as a result of fragmentation, sintering, hashing of rock at the falling of meteorites. A lot of dust appeared on a surface. Table 1 Main elements of Moon’s rock Rock

Stones Dust

Si 20 20

Fe 14 12

Ca 7 8

Elements, % Al Mg Ti 6 5 5 6 6 5

Cr 0,4 0,3

Mn 0,3 0,2

K 0,2 0,1

Physical-mechanical properties of the ground of the Moon were deﬁned thanks to supplies to the Earth of samples of lunar ground and to their detailed research in the earth laboratories. Samples of ground were selected, supplied to the Earth, investigated by crews of spacecrafts “Apollo” and automatic stations “Luna16”, “Luna-20”, “Luna-24”, “Surveyor”, planet research vehicles “Lunokhod-1”, “Lunokhod-2”. In the system “Earth—Moon” there are unique points, in which total effect of forces is equal to zero. They are referred to as points of libration. Existence of such points was predicted by Lagrange in 1772. In libration point a space body goes under inﬂuence of the gravity of two bodies, it can be in a condition of relative equilibrium in relation to these two bodies, for example, the Earth and the Moon. Let us remark that under libration of the Moon (from the latin word libratio — swinging, oscillation) periodic penduliform oscillations of the Moon relative to its center-of-mass are understood. The system “Earth—Moon” has ﬁve libration points: three of them are on the straight line connecting the centres of the Earth VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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and the Moon, and are unstable; two points, located on a conditional triangle, are steady [4]. The last two points are in a plane of an orbit of driving of the Moon relative to the Earth. Spacecrafts, started up in a steady libration point, can serve intermediate stations for the ﬂights to the Moon and to the long-distance planets. Flights of crews to libration points are considered for the second decade of XXI century. Table 2 Distance between the cen- Distance from the surface of Position of the Moon relaters of the Moon and the the Moon to the libration tive to the Earth Earth, km point, km The Moon in apogee The Moon at the middle distance from the Earth The Moon in perigee

402000

61791

384400

59024

356414

54784

It is supposed to place space telescopes, assembly of factories for manufacturing the equipment for the further mastering the Moon in libration points. Friction Units on the Moon. Open friction units on the Moon are to work under following conditions: pressure of environment — vacuum, p < 10−10 Pa; wide range of temperature change on surface: +150 · · · − 170 ◦ C, high evaporability of lubricants; inﬂuence of temperature gradients and warping of constructions; sublimation of elements of constructional materials; irradiation of different physical nature; effect of micrometeorites; reduced gravity; inﬂuence of abrasive particles of lunar ground; requirements for minimization of size characteristics of a construction (high tension); undesirability (impossibility) of application of liquid lubricants and grease; vibration, shock, acoustic loadings during takeoffs and landings to the Earth; difﬁculties in repair-regenerative operations in conditions of the Moon etc. Adhesive interaction of conjugated surfaces is a principal danger for rubbing pairs on the Moon. Forces of adhesive interaction depend on a degree of clearing of surfaces from adsorbed stratums, pollution, presence of lubricating material, oxides. On the areas of actual contact of rubbing elements under effect of normal and tangential efforts (strains), temperatures and wear, destruction of screening layers occurs, there are contact atom-pure surfaces [5, 6]. Temperature, vacuum and irradiation essentially inﬂuence the evaporability and decomposition of lubrication. Evaporability is reﬂected on operating characteristics of a lubricant and allowed period of storage. Speed of ageing of lubricant depends on evaporability and thermo-chemical disintegration (polymerization). Thermal stability is the major characteristic of a lubricant. The increasing of service life of friction units is achieved due to the use of a complex of additives and physicochemical technologies of surface modiﬁcations of rubbing bodies. Effects of vacuum, temperatures, different aspects of an irradiation (infrared, x-ray etc.), a stream of electrons on a surface of materials inﬂuence the thermo-kinetic processes which are ﬂowing in the volume — diffusion, degassing,

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sublimation etc. become more active. Thus the chemical composition, structure and physical-mechanical properties, dimensional stability, relaxation chracteristics vary. Ultimate conditions for a friction unit are characterized by: heightened value of forces (rate) of friction as a result of intensive adhesive interaction of rubbing surfaces, wear process of antifrictional wear-resistant coatings, exhaustion of a lubricant, change of clearances in process of a wear at interface. There is a probability of jamming from abrasive’s droppings and from temperature deformations. Process of seizing (grip) of rubbing parts is dangerous. Typical friction units of mechanical systems on the Moon: cogged and wave transmissions, bearings of rolling and sliding, guiders, cam mechanisms, screw transmissions, sealing elements, demountable and carving connections, robot mechanisms and manipulators, joints, locks etc. Violation of rules of tribological design, manufacturing and operation of friction units results in emergencies and catastrophes in Space [6]. Provision of Tribological Reliability. In the sixtieth of the last century in the USSR wide researches of tribological reliability of mechanical systems under conditions similar to that of the Moon were conducted by integrated efforts of institutes of Academy of Science of the USSR, branch institutes, higher educational institutions. New laboratories, experimental stands, testing ranges were created. Model experiments in conditions of Space were carried out. For example, on AIS “Luna10” (31.03.1966) and “Luna-11” (24.08.1966) the experimental reducer “R-1” was tested which worked in a free Space and transmitted data from lunar orbit about serviceability, temperature conditions and losses of energy on friction. Calculationtheoretical methods on the determination of the efﬁciency of geared systems of moon-rovers were developed. Naturally, at the ﬁrst stage of friction unit creation there was the desire to use traditional methods of calculation and technique of friction unit design accepted in general machine industry. Experiments, modeling the Moon condition in laboratory conditions resulted in a full disappointment. The mechanism of destruction of conjugated surfaces essentially differed from the one realized under earth conditions. Methods of calculation appeared unusable, conventional rubbing materials were unsuitable. The attention of scientists and engineers was attracted to emergency and catastrophe, happened in Space. Owing to friction in space conditions the mechanism of the plotter of infrared vertical broke down during the ﬂight of the ﬁrst unmanned spaceship “Vostok” (on 15th May 1960). Cosmonaut V.M. Komarov was killed (on 24th April 1967, “Soyuz-1”) — efforts of a parachute exhaust system did not sufﬁced to overcome forces of friction and to draw out the basic parachute. As a result of a rigid landing there was a strong impact, explosion, ﬁre. For the friction units to be maintained on the Moon, special composite, selflubricating materials, solid-lubricating coatings, surface modiﬁcations, methods of lubrication etc. were created. In connection with absence of a solid lubricant in open friction units there is a continuous wear during maintenance of dry rubbing

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parts. The design procedure of a service life calculation by a wear criterion, essentially distinguished from the conventional one [7–12], was created. Magnet-active powder methods of lubrication which essentially increase service life of gear systems in conditions of the Moon were created for the ﬁrst time [11–13]. Method of research of magnet-active lubrications is scientiﬁcally justiﬁed and the complex of the experimental equipment for the research of heavy-loaded gears is created [11–14]. On the basis of ﬁber optics in enclosed volume the visual and photo-observations were carried on with the purpose of research of wear and performance of lubrication in the magnetic ﬁeld during the operation. Application of magnet-active lubricants allows (10 times) increasing the durability of heavy-loaded transmissions, as contrasted to durability of the transmissions made of self-lubricating composite materials and with solid-lubricating coatings. Use of the determined and stochastic criteria of a similarity in the analysis of processes of abrasion, wear process and seizing allows one to take into account physical processes and to evaluate reliability at effect of random factors. Friction units on the Moon till now are unique articles and there is no yet sufﬁcient statistical information on their load and real condition. The statistical exposition of probability events in friction units will develop in process of the mastering the Moon. In friction units of “Lunokhod-1” and “Lunokhod-2” the following materials have found application as a result of cooperative research of many organizations [14]: thermally processed ceramic-metal iron-glass material C-5 for gears and sliding bearings; titanic alloy BT-14 with nitriding for epicycle of planetary gear-box; solid-lubricating coating STM-1 on a basis of molybdenum disulﬁde for open friction pairs; grease VNIINP-246 on a basis of ploychlorosiloxane for lubricating of gears and rolling bearings; self-lubricating polymeric material FN-3 on the basis of nickel and ﬂuoroplastic which was applied to sealing elements. The composite antifrictional material, consisting of rust-resistant steel with a working surface modiﬁed by a copper-silver alloy, a material was applied in the sliding bearings working in pair with titanic alloy VT-14, having a nitrated surface. For the ﬁrst time gravitation forces were taken into account in the calculations of efﬁciency of gear-boxes on the Moon [11]. The accepted scientiﬁcally-based approach to the methodology of ground research in laboratory conditions, in model tests in Space, the designed cycle of trials and techniques of design were the basis of successful functioning of friction units on the Moon. Conclusion. Provision of reliability and service life of friction units on the Moon represents a difﬁcult and complex scientiﬁc and technical problem. Carried out system research of the friction units in the second half of XX century have created base for the future mastering the Moon and other planets. To the achievements of science and technology of the Soviet Union which originated the space age — the

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ﬁrst artiﬁcial satellite of the Earth was put into orbit, the ﬁrst ﬂight of a man in space was realized, an extra vehicular activity of a man in outer space was undertaken, lunar devices with soft landing and mobile laboratory “Lunokhod” and others were created — it should be added the following: the experimental-theoretical methods of provision and prediction of friction unit reliability for extreme conditions of the maintenance on the Moon were created for the ﬁrst time. Subsequently these achievements have found their application in general machine industry, aerospace machinery, atomic engineering and others. In future, for creation of heavy-loaded friction units for a long service life for engineering systems of the Moon not only accumulated experience and designed techniques, but also wide-scale research and development activities in tribological ﬁeld will be required.

REFERENCES 1. Raushenbakh, B.V. ed.: Korolyov and His Undertaking [In Russian]. Moscow: Nauka 1998. 2. Feoktistov, K.P.: Space Technology. Prospects of Development [In Russian]. Moscow: Izd-vo MGTU imeni N.E. Baumana, 1997. 3. Planet-rovers (In Russian). ed.: A.L. Kemurdzhian. Moscow: Mashinostroenie 1993. 4. Meshcheryakov, I.V. In the World of Cosmonautics [In Russian]. Nizhny Novgorod, 1996. 5. Kragelskiy, I.V., Lubarskiy, I.M., Guslyakov, A.A. Friction and Wear in Vacuum (In Russian). Moscow: Mashinostroenie 1973. 6. Drozdov, Yu.N. Tribological Aspect of Reliability Problem of Space Technique (In Russian). Problems of engineering and machine reliability. 2000. No 1. 7. Drozdov, Yu.N. To the Calculation of Gears for Wear (In Russian). Machine Science. 1969. No 2. 8. Drozdov, Yu.N. Principal Scheme of Calculation of Machine Parts for Wear (In Russian). Proceedings of Perm Polytechnic Institute. 1970. No 62. 9. Drozdov, Yu.N. To the Development of Calculation Method for Wear Process and Friction Simulation (In Russian). Endurance Collection. Ed. A.A. Blagonravov, R.M. Matveevsky. Moscow: Nauka. 10. Drozdov, Yu.N. Development of a Method for the Calculation of the Life of Dry Frictional Contacts under Extreme Operating Conditions. Wear, 38 (1976), 217–223. 11. Drozdov, Yu.N. Transmissions. Tribology — Lubrication, Friction, and Wear. Edited by J.V. Kragelsky and V.V. Alisin. Professional Engineering Publishing Limited (2001) pp. 597–639. 12. Drozdov, Yu.N., Pavlov, V.G. and Puchkov, V.N. Friction and Wear in Extreme Condition (In Russian). Moscow 1986. 13. Vaisefeld, L.O., Drozdov, Yu.N., Kemurdzhian, A.L. Efﬁciency of Gear Drives with Magnetic-Powder Lubrication in the Different Environmental Conditions (In Russian). Engineering Bulletin. 1979, No 3. 14. Vaisefeld, L.O., Mitskevich, A.V., Tarasov, V.M. Scientiﬁc-technical Solutions on Friction Units of Space Apparatus (In Russian). Transport Engineering Bulletin. Moscow, 1995, No 3–4.

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PRODUCTION PROCESS PROCEDURES & MACHINES A.M. Dmitriev (Bauman Moscow State Technical University), A.L. Vorontsov (Moscow State Academy of Instrument Engineering & Informatics)

ANALYSIS OF EXTRUSION PROCESS OF CORELESS CYLINDRICAL PRODUCTS WITH AN EXTERNAL CORE IN THE BOTTOM The stress condition of a part is determined for the operation of extruding coreless cylindrical products with an external core. Formulas for calculating of the basic technological parameters of the combined extrusion process have been obtained. Accuracy of the obtained theoretical results is conﬁrmed by comparing them with experimental data.

In machine-building engineering the parts schematically shown in Fig. 1 are used. The most effective way to manufacture them is the extrusion with two simultaneous metal ﬂow directions that results in reduction of speciﬁc force as opposed to the traditional extrusion. For calculation of the basic technological parameters of such extrusion there were no formulas derived on the basis of a uniﬁed method. The extrusion of products with an external core can look as it is shown in Fig. 2. For a small relative size of the matrix aperture (Fig. 2, a), the process will begin with the traditional extrusion of walls with growth of their height. For a product with the dimensions ratio as shown in Fig. 2, b, on the most part of a working travel there may be a direct extrusion of a core, and formation of a cavity of a product will only begin at a ﬁnal stage.

Fig. 1. Combined extrusion of coreless cylindrical products with an external core in the bottom 252

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Fig. 2. Variants of forms during the extrusion of coreless cylindrical products with an external core

If the product has the dimensions ratio as shown in Fig. 2, c, the cavity may be formed due to the shift of the metal located under a punch, in the matrix aperture. If the values of speciﬁc forces for extrusion of the cavity and core are close to each other, formation of these elements will begin practically simultaneously (Fig. 2, d). On the matrix-part contact surface friction forces compensate each other; therefore the constant of friction on this surface is equal to 0. Substituting this value into the formula (2.9) from [2], we determine the relative speciﬁc extrusion force by the formula ¸ · hy 0.5 + µ1 (1) + + qtr q = 1,1 2 + ln R + 2(R2 − 1) 4hy where µ1 is the constant of friction between the part and punch. In this and subsequent formulas, we use relative values. Forces are divided by σs — the average pressure of ﬂuidity of the material of parts in the plastic deformation zone. The values of the parameters are divided by the radius of a working site of the punch: r = 1, R — relative radius of a matrix. The height of the top zone of plastic deformation hy is determined using the following formulas. If deformation hardening is insigniﬁcant (hy = h), s (R2 − 1)(0.5 + µ1 ) h= . (2) 2(1 + 2µR) When the high quality greasing is used, µ, µ1 and µ0 (Fig. 1) equal to 0.1, with lower quality greasing — µ, µ1 and µ0 = 0.3. In the presence of deformation hardening hy = h[1 + ky (1 − 0.2e−s − 0.8e−5s )],

(3)

where the factor of hardening is µ ¶ σs2 /σs1 − 1 ky = 1 − exp −10 , e2 − e1

(4)

s is the relative length of a working travel for which we determine cumulative deformations, e is the natural logarithm basis, and σs1 and σs2 are the pressures VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

253

of the ﬂuidity taken from the hardening curve of the given material at cumulative deformations e1 = 0.2 . . . 0.4 and e2 = 1 . . . 1.2. Coefﬁcients of hardening of various materials that have been obtained are presented in table 1. Table 1 Coefﬁcients of hardening of materials

Material

e1

σs1 , MPa

e2

σs2 , MPa

ky

Aluminium alloy AB

0.4

240

1.2

300

0.956

Aluminium alloy D16

0.2

290

1.2

405

0.981

Steel 10

0.2

470

1.2

705

0.993

Steel 20

0.2

610

1.0

980

0.999

Steel 12X18H9T

0.1

600

0.6

1000

1.000

Steel 12X18H9T(training)

0.1

560

0.6

960

1.000

Calculations demonstrate that the values of coefﬁcient of hardening are in between 0.9 and 1. Therefore it is possible to accept ky = 0.95. Inﬂuence of the matrix deﬂection on the speciﬁc force of extrusion in the formula (1) according to [3] is described by the formula

qtr = 1.1

µR3 s. (R2 − 1)2

(5)

The extrusion force will grow until the working travel will reach the value

str =

(R2 − 1)2 , R2 (1 + µR)

(6)

which corresponds to its maximum value:

qtr = 1.1

µR . 1 + µR

(7)

If the current value is s < str , then it is necessary to use expression (5), and if it is s ≥ str , then it is necessary to use the formula (7). When the thickness of the bottom of the part decreases up to the size s (R2 − 1)(µ0 + µ1 ) Hc = , (8) 2(1 + 2µR) there will be a merge of the top and the bottom plastic deformation zones (Fig. 1). The total height of the plastic deformation zone becomes equal to the thickness of the bottom of the part H. 254

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

In the theoretical model shown in Fig. 1, the cylindrical coordinate system ρ, θ, z, is used, and the plastic deformation zone is represented as four areas numbered: 1 and 2, 3 and 4. The following assumptions are made: (1) the material is hard plastic, and hardening is described by average pressure of ﬂuidity in the zone of plastic deformation, σs ; (2) contact friction forces are determined from Zibel0 s law as τk = µβσs , where β — is Lode0 s coefﬁcient. Area 1. Axial speed of ﬂow is accepted as: vz = A[z − ϕ(ρ)],

(9)

where A — the factor determined from the volume stability condition, and ϕ(ρ) is an arbitrary function. Function ϕ(ρ) may be determined by an estimation procedure presented in [4]. From the volume stability condition, which looks like ∂vρ vρ ∂vz + + = 0, ∂z ∂ρ ρ or ¸ · 1 ∂ ∂vz (vρ ρ) = − , ρ ∂ρ ∂z

(10)

it follows, that A f (z) vρ = − ρ + . 2 ρ Taking into account the boundary condition vρ = 0 at ρ = r0 f (z) =

A 2 r . 2 0

we obtain the radial speed: µ ¶ A r02 −ρ . vρ = 2 ρ Deformations speeds: ¶ µ ∂vρ A r02 +1 , =− ξρ = ∂ρ 2 ρ2 ¶ µ vρ A r02 ξθ = −1 , = ρ 2 ρ2 ∂vz ξz = = A, ∂z ∂v ∂v ηρz = ρ + z = −Aϕ0 (ρ), ∂z ∂ρ

(11)

(12)

and deformation speed intensities: ξi = β|ξmax | = βξz = βA. VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

(13) 255

The equations of balance look like: ∂τ σ − σθ ∂σ ρ + ρz + ρ = 0, ∂ρ ∂z ρ ∂σ ∂τ τ z + zρ + zρ = 0. ∂z ∂ρ ρ

(14)

From Levy–Mizes equations, taking into account formula (13) and system (12) it follows, that τρz depends only from ρ, and 2 2r2 (ξρ − ξθ ) = − 02 , 3ξi 3βρ

σρ − σ θ =

(15)

From the ﬁrst equation of balance (14)

σρ = −

r02 + f1 (z) + C1 . 3βρ2

(16)

Taking into account this expression and the condition of plasticity σz − σρ = β

(17)

we can write the second equation of system (14) in a form: ∂τρz τρz ∂f1 (z) =− − . ∂z ∂ρ ρ

(18)

Since the left-hand part of the equation (18) depends only on z, and the righthand part only on ρ, both of these parts are equal to constant C2 . The right-hand part of the equation (18) can be presented as −

1 ∂ [τρz ρ] = C2 , ρ ∂ρ

then C2 ρ C 3 + . (19) 2 ρ As the left-hand part of the formula (18) is equated with the constant C2 , then τρz = −

f1 (z) = C2 z.

(20)

From the boundary conditions τρz = −µ2 β

at

ρ = r0 ,

τρz z = 0.5β

at

ρ = r1 ,

it follows that r1 + 2µ2 r0 C2 = −β r 2 − r2 , 1 0 r 0 + 2µ2 r1 . C3 = −0.5βr0 r1 2 r1 − r02 256

(21)

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

The solution to equations (17), (16) and (20) gives:

σz = β −

r02 + C2 z + C1 . 3βρ2

(22)

Using the condition σz = 0 at z = −H and ρ = r1 , we determine the constant C1 = −β +

r02 + C2 H. 3βr12

(23)

The solution to equation (17), (22) and (23) at z = −H/2 and ρ = r1 gives an average radial pressure from area 1 to area 2: σρ1 = −β + 0.5C2 H.

(24)

Area 2. The axial speed looks like: vz = −f2 (z). Taking into account vρ = 0 at ρ = Rg we determine: µ ¶ 1 ∂f2 (z) ρ2 − Rg2 vρ = . 2 ∂z ρ From the common parts of equation (12) speeds of deformations are determined: ∂f2 (z) , ξz = − ∂z µ ¶ R2g 1 ∂f2 (z) ξρ = 1+ , 2 ∂z ρ2 ¶ µ (25) R2g 1 ∂f2 (z) ξθ = 1− 2 , 2 ∂z ρ ¶ µ 2 2 1 ∂ f2 (z) ρ − Rg2 . ηρz = 2 ∂z 2 ρ Similarly to (13), ∂f2 (z) . ∂z From (25) and (26) and of Levy–Mizes equations: ξi = β

τρz

∂ 2 f2 (z) µ µ 2 ¶ ¶ ρ − Rg2 ρ2 − Rg2 1 2 ∂z = = f3 (z) . 6β ∂f2 (z) ρ ρ ∂z

Substituting (27) into the second equation of system (14) gives: Z σz = −2 f3 (z) dz + f (ρ) + C. VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

(26)

(27)

(28) 257

Substituting (28) into the condition of plasticity σρ − σz = β, we obtain

Z σρ = β − 2

f3 (z) dz + f (ρ) + C.

(29)

Further the condition of plasticity is used in the form σp − σθ = β.

(30)

Substitution of formulas (27), (28) and (20) in the ﬁrst equation of system (14) gives: · ¸ ρ ∂f (ρ) β ∂f3 (z) + . =− 2 2 ρ − Rg ∂ρ ρ ∂z Since the left-hand part of this equation depends only on ρ, and right-hand part only on z, then both of these parts should be equal to constant C4 , whence it follows that: f3 (z) = −C4 z + C5 , f (ρ) =

C4 ρ2 − (β + C4 Tg2 ) ln ρ. 2

(31) (32)

Taking into account (31) from the formula (27) we determine

τρz

µ 2 ¶ Rg − ρ2 = (C4 z − C5 ) . ρ

(33)

From the boundary conditions τρz = −βµ0 τρz = 0.5β

at at

ρ = r1 and z = −H, ρ = r1 and z = 0,

it follows that arbitrary constants in formula (33) are equal to (34): β(0.5 + µ0 )r1 C4 = H(R2 − r 2 ) , g 1 βr 1 C5 = − 2(R2 − r 2 ) . g 1

(34)

After substituting formulas (31) and (32) into formulas (28) and (29) we obtain: ( σz = (C4 z − 2C5 )z + C4 (0.5ρ2 − R2g ln ρ) − β ln ρ + C, (35) σρ = β + (C4 z − 2C5 )z + C4 (0.5ρ2 − Rg2 ln ρ) − β ln ρ + C.

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VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

To determine the arbitrary constant C we use the average value σρ1 on the border between areas 1 and 2, which is determined by equation (24). Equating it to the value σp from the system (35) ρ = r1 and z = 0 gives its value: C = β(ln r1 − 2) + 0.5C2 H − C4 (0.5r12 − Rg2 ln r1 ).

(36)

Force of extrusion is determined by the sum of forces in areas 1 and 2: P = P 1 + P2 .

(37)

From (22) and (23) at z = 0 Zr1

·

µ

2r02 r1 |σz |ρ dρ = π − ln 3β r0

P1 = 2π

¶ ¸ r02 2 2 C2 H (r1 − r0 ) . 3βr12

r0

It is possible to neglect the following terms of equations 2r02 r1 ln 3β r0

and

−

r02 (r2 − r02 ), 3βr12 1

obtaining P1 = −πC2 H(r12 − r02 ).

(38)

From formulas (35) and (36) at z = 0 ZRg |σz |ρ dρ = π[(2β − 0.5C2 H + 0.5C4 )(Rg2 − r12 )−

P2 = 2π r1

− 0.25C4 (Rg4 − r14 ) + (β + C4 R2g )(R2g ln Rg − r12 ln r1 − 0.5R2g + 0.5r12 )]. (39) From formulas (21), (34), (38), (39) and the average value of Lode0 s coefﬁcient β = 1.1 the speciﬁc force working on a punch from areas 1 and 2 is determined: q1 =

P 1 + P2 1.1 = 2 1,5(Rg2 − r12 )+ 2 2 π(R − r0 ) Rg − r02 ·

¸ (0.5 + µ02 r0 )(R2g + r12 − 2r02 ) − (µ − µ )r 1 0 1 H+ r12 − r02 ¸ · µ ¶ Rg 4 2 2 4 − 0.75 + Rg r1 − 0.25r1 r1 (µ0 + µ1 ) Rg ln R r g 1 2 + Rg ln + . (40) r1 Rg2 − r12 )H VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

259

In the formula (40) a value of the calculated friction factor µ02 is chosen according to the type of the mandrel used in a process. If the mandrel is rigidly attached to the punch, then µ02 = 0.5µ2 . If the mandrel is rigidly attached to the basis of the stamp, then µ02 = µ2 . If the mandrel moves forcedly in the direction of extrusion of a tubular core, then µ02 = −µ2 . Area 3. The axial speed is similar to the one in formula (9). In this case, as shown in [4] for the traditional extrusion, the stress-stain condition looks like: C6 ρ C7 + , τρz = − 2 ρ R (41) σρ = −β − β ln + C6 z − qtr , ρ σz = σρ + β, where qtr is determined by formula (7), and 1 + 2µR , C6 = β 2 R −1 (42) C7 = 0.5βR R + 2µ . R2 − 1 From the second formula of system (41) at ρ = R and z = −H the maximum pressure upon a wall of the matrix is determined: ¸ · 1 + 2µR H + qtr . (43) p = 1.1 1 + 2 R −1 At ρ = 1 an average value of a radial pressure on the border between areas 3 and 4 is equal to: ¸ · 1 + 2µR H − qtr . (44) σρ2 = −β 1 + ln R + 2(R2 − 1) Area 4. The axial speed looks like: vz = −f4 (z). Using the boundary condition vρ = 0 at ρ = Rg , we get: µ ¶ 1 ∂f4 (z) ρ2 − R2g vρ = . 2 ∂z ρ According to the formula (12), speeds of deformations are equal to: ∂f4 (z) , ξz = − ∂z ¶ µ Rg2 1 ∂f4 (z) 1+ 2 , ξρ = 2 ∂z ρ µ ¶ (45) Rg2 1 ∂f4 (z) 1− 2 , ξ = θ 2 ∂z ρ ¶ µ 2 2 1 ∂ f4 (z) ρ − R2g . ηρz = 2 ∂z 2 ρ 260

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

According to (13), the intensity of deformation speeds is equal to: ξi = β

∂f4 (z) . ∂z

(46)

From formulas (45) and (46) the formula similar to (27) is obtained: ¶ µ R2g τρz = f5 (z) ρ − . ρ

σρ − σθ =

2 2 Rg2 (ξρ − ξθ ) = . 3ξi 3β ρ2

(47)

(48)

From the second equation of system (14), being solved together with (47), it follows that Z (49) σz = −2 f5 (z) dz + f1 (ρ) + C8 . The joint solution of formulas (47), (48) and conditions of plasticity σρ − σz = β,

(50)

and also (49), and the ﬁrst equation of system (14) results in formula: µ ¶ ρ 2 Rg2 ∂f (ρ) ∂f5 (z) + . =− 2 2 3 ρ − Rg ∂ρ 3β ρ ∂z As the left-hand part of this equation depends only on ρ, and the right-hand part one only on z both of these parts should be equal to constant C9 , and thus: f5 (z) = −C9 z + C10 ,

(51)

ρ2 1 Rg2 + C − C9 Rg2 ln ρ. 9 3β ρ2 2 By substituting (51) in the formula (47) we determine, that µ 2 ¶ ρ − Rg2 τρz = (C10 − C9 z . ρ f1 (rho) =

(52)

(53)

From the boundary conditions τρz = −βµ1 at ρ = 1, z = 0 and τρz = βµ0 at ρ = 1 and z = −H, it follows, that constants in (53) are equal: β(µ0 + µ1 ) C9 = H (1 − R2 ) , g (54) βµ 1 C10 = − 1 − R2 . g As a result of substituting (51) and (52) in (49), we determine that: σz = (C9 z − 2C10 )z + C9 (0.5ρ2 − R2g ln ρ) + VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

1 Rg2 + C8 . 3β ρ2

(55) 261

The radial pressure determined by substituting (55) into (50) looks as follows: σρ = β + (C9 z − 2C10 )z + C9 (0.5ρ2 − Rg2 ln ρ) +

1 R2g + C8 . 3β ρ2

(56)

To determine the constant C8 we use the average value σρ2 from formula (44), on the border between areas 3 and 4, which is determined on the assumption of at σρ = 1 and z = 0. As a result: C8 = −

R2g 1 + 2µR C9 − − 2β − β ln R − H − qtr . 2 3β 2(R2 − 1)

(57)

Using formulas (7), (54), (55) and (57) at z = 0, and also an average value of Lode0 s coefﬁcient β = 1.1 we can determine the speciﬁc force applied to the punch from area 4: Z1 2 q2 = 1 − Rg2

½ |σz |ρ dρ = 1.1 2 + ln R +

1 + 2µR H+ 2(R2 − 1)

Rg

¾ (µ0 + µ1 )[1 − 4Rg2 + Rg4 (3 − 4 ln Rg )] µR + + . (58) 4(1 − R2g )2 H 1 + µR The relative speciﬁc force of the combined extrusion: q=

q1 (1 − R2g ) + q2 (R2g − r02 ) . 1 − r02

(59)

The Rg value is determined from the condition of a minimum of formula (59). With the accuracy sufﬁcient for practical calculations of speciﬁc force, the value of Rg is determined using the following approximated formula: Rg =

Rr1 − r0 . R − 1 + r1 − r0

(60)

To verify the theoretical equations (40), (58), (59) and (60) experiments on extrusion of products made of lead C00 were performed. When comparing experimental and theoretical values of relative speciﬁc forces it was assumed, that the relative thickness of the bottom of a product H = Hc . The results of comparison of calculations with experimental data are given in Table 2. Table 2 Comparison of theoretical and experimental values of the relative speciﬁc force of extrusion of products with an external core made of lead C00 at r0 = 0 R

r1

µ

µ1

µ0

H

Rg

q1

q2

q

qe

δ, %

1.07

0.49

0.3

0.3

0.1

0.133

0.936

3.440

2.623

3.584

3.47

3.2

1.07

0.41

0.1

0.1

0.3

0.155

0.914

3.174

3.179

3.820

3.77

1.3

1.22

0.57

0.3

0.3

0.3

0.291

0.880

3.371

2.268

3.728

3.50

6.1

1.22

0.49

0.3

0.3

0.1

0.237

0.842

3.274

2.518

3.603

3.58

0.6

262

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Since comparison of theoretical and experimental results shows that they are in a good agreement, we believe that the formulas obtained in this work can be recommended for use in practical calculations.

REFERENCES 1. Cold die forging. The directory / Edited by G.A.Navrotskiy. Moscow: Publishing House “Machine building”. – 1973. 496 p. 2. Dmitriev, A.M., Vorontsov, A.L. Technology of forging and die forging. Part 1. Die forging by extrusion. The textbook for high schools recommended by the Ministry of Education of the Russian Federation. Moscow, Publishing House “The Higher School”. – 2002. 400 p. 3. Dmitriev, A.M., Vorontsov, A.L. Deﬁnition of technological parameters of die forging by extrusion with an allowance of elastic deformation of a matrix // Vestnik of Moscow State Technical University named after Bauman. – 2002. No 2. PP. 76–93. 4. Osadchij, V.J., Vorontsov, A.L., Beznosikov, I.I. The theory and calculations of technological parameters of die forging by extrusion. The manual for high schools recommended by the Ministry of Education of the Russian Federation. Moscow, Moscow State Academy of Informatics. – 2001. 307 p. 5. Dmitriev, A.M., Vorontsov, A.L. Estimation of loading on the extruding tool for parts with a through step cavity. Part 1 // Forging-stamping manufacturing Journal. 2002. No 10. PP. 21–28. 6. Dmitriev, A.M., Vorontsov, A.L. Estimation of loading on the extruding tool for parts with a through step cavity. Part 2 // Forging-stamping manufacturing Journal. 2002. No 11. PP. 21–28. A.M. Dmitriev (b. 1948) graduated from the Bauman Moscow Higher Technical School in 1972. Corresponding member of the Russian Academy of Sciences, D. Sc. (Eng.), professor, head of “Technology of Metal Forming” Department of the Bauman Moscow State Technical University. Author of 201 publications in the ﬁeld of plastic metal working. A. L. Vorontsov (b. 1955) graduated from the Bauman Moscow Higher Technical School in 1978. D. Sc. (Eng.), professor of “Applied Mechanics” Department of the Moscow State Academy of Instrument Engineering and Informatics (MGAPI). Author of 151 publications in the ﬁelds of plastic metal working and applied mechanics.

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

263

I.N. Shiganov (Bauman Moscow State Technical University)

FUSION WELDING OF METALLIC COMPOSITE MATERIALS Physical and metallurgical features of fusion welding of metallic composite materials, strengthened by particles, are considered. Different materials are classiﬁed by a weldability, which allows the possible defects during a welding to be predicted. Physical effects, accompanying the forming of a weld during the argon-arc, electron-beam and laser welding, are considered. Typical defects of welds are revealed and a mechanism of their formation is determined. On the basis of conducted researches we propose some technological techniques for these types of welding process which enable avoiding the welding defects formation. Our researches and developed technologies have made possible to get the good quality of welds for different groups of metallic composite materials, such as Al–Be, Al–Be–Mg, Fe–Cu, Fe–Cu–Pb, Fe–Cu–Pb–Sn, Al–Pb, with a thickness from 2 to 20 mm.

Classiﬁcation of Composites by Weldability. The application of materials with speciﬁed properties — metallic composite materials (MCM) — results in completely new technical characteristics of products. In developing structures made of MCM, it is necessary to use the fusion welding, especially welding by highproductivity methods, for example, using a laser [1]. It is well known [2] that melting of a composite material is accompanied in the majority of cases by the disruption of its initial structure, resulting in a deterioration or large change of the physical and mechanical properties. On the basis of the available data on welding dissimilar and composite materials [3], it is possible to deﬁne four groups of composites considering the interfacial interaction of their components in fusion welding. Group 1 (Fig. 1, a) includes MCM whose components form a homogeneous liquid during melting and are almost insoluble in each other in the solid state: Fe–Cu, Al–Be, Ti–Mg, etc. Melting and solidiﬁcation of these alloys result in the formation of a uniform heterogeneous structure with alternating particles of the matrix and the hardening agent. Fusion welding of these materials is most promising because the structure of the composite is distorted to a lesser degree, and, consequently, the properties of the weld metal differ only slightly from those of the parent metal. Group 2 (Fig. 1, b) includes MCM whose components in melting and solidiﬁcation have limited or unlimited solubility: Nb–W, Ni–W, Al–Si, Co–Cr, etc. Melting of these composites is accompanied by the formation of solid solutions with a smoothly changing concentration. The strength can be close to that of the parent metal. In welding MCM with limited solubility of the components the weld can contain eutectics and peritectics, in addition to solid solutions, depending on the equilibrium diagram.

264

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

Fig. 1. Structural state of welded composites: a — group 1, b — group 2, c — group 3, d — group 4

Group 3 (Fig. 1, c) includes MCM whose components do not interact during welding. This is associated with the relation between the melting points of the refractory component and the boiling point or the low-melting component. The two liquid phases can not be in equilibrium and, consequently, no superheating would increase the mutual solubility of the components in the liquid state. In most cases, this group includes not two-component but three-component materials and materials with a larger number of components: Al–Be–Mg, Fe–Mg, Fe–Cu–Pb, W–Ag, etc. Welding of these materials is difﬁcult without using special measures suppressing evaporation of the low-melting component. Group 4 (Fig. 1, d) includes MCM whose components are not miscible in the liquid state: Fe–Pb, Fe–Cu–Pb, Al–Pb, etc. In fusion welding, the joint easily fails through the fusion area. This classiﬁcation is rather conventional because, under the effect of the welding thermal-diffusion cycle, MCM can belong to several groups at the same time. However, the classiﬁcation does make it possible to specify the main groups of MCM and the criteria that can be used to evaluate their weldability VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

265

In this work, investigations were carried out into materials of the Al–Be–Mg, Fe–Cu–Pb and Al–Pb systems characterized by the formation of the largest number of defects in welding. Research of Materials of Al–Be–Mg System. The composites of the Al– Be–Mg system have an aluminum matrix with beryllium particles distributed in it. Magnesium is in the solution with aluminum. The mostly used alloys of this type are: Al–30Be–5Mg, Al–40Be–3Mg.The unique feature of these materials is that they combine low density (2.1 − 2.3 g/cm3 ) with the high melting point (1150 − 1160 ◦ C), strength (450 MPa) and elasticity modulus (140 − 150 MPa). A combination of high strength and rigidity of these materials gives them advantages in comparison with the structural alloys based on aluminum, titanium and magnesium. These materials are used to produce large-sized components for the aerospace, aviation and ship building industries, with all component structures are mostly welded (girders, casings, antennae, fuel tanks, platforms, etc.). The thickness of the welded materials varies from 2 to 20 mm. Usually they are applied for butts welded with continuous penetration. Argon-arc welding is used to weld MCM up to 3 mm thick. At larger thickness there are defects in the form of the weld lowering below the joint level, i.e. sagging associated with the existence of the solid-liquid state in the weld zone for a long period of time which does not guarantee the required strength of the metal (the aluminum matrix is still melted, whereas the beryllium grains are already solidiﬁed). The results of examination of the effect of the welding conditions showed special features of the welded materials up to 3 mm thick. The results show that, over the entire range of the conditions, beryllium concentrates in the heat affected zone (HAZ) in the form of an almost continuous band (Fig. 2, a). This zone is weakened under impact loading, is subjected to corrosion and does not withstand vibration loads, i.e. many requirements imposed on the important components are not fulﬁlled in this case. The results of modeling of the process, physical investigations and calculations show that the enrichment of the weld zone with Be is caused by its thermal diffusion in the aluminum melt. The criterion for the start of this process is the time (0.9–1.5 s) of holding the metal at the weld boundary at a temperature higher than 1500 ◦ C. The changes of the welding conditions did not make it possible to overcome this criterion. The holding time of the metal at the given temperature was reduced by using copper heat-removing straps of the appropriate weight and design. The structure of the weld after welding with heat-removing devices is shown in Fig. 2, b. Another method is the use of concentrated heat sources. Composite materials 3–20 mm thick were joined by electron beam welding. The vacuum chamber protected the operator against harmful beryllium fumes. The results of initial experiments with melting the material 5–10 mm thick showed that there are no defects in the HAZ as a result of the rigid thermal cycle of the concentrated heat source. However, it was not possible to produce welded joints due to speciﬁc defects in the form of a longitudinal cavity (Fig. 3) — the metal was ejected from the melt to the surface. Defects of this type were not detected in any of the materials used in electron beam welding.

266

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Fig. 2. Microstructure of welded joints in Al–30Be–5Mg composite with (a) and without (b) heat removal

Application of ﬁlming for the examination of physical processes in the penetration channel: evaporation dynamics, vapor composition, pressure in the channel, etc., has made it possible to determine the mechanism of formation of an unusual defect. Its formation is caused by the special feature of the structure of the material evident under the vacuum and high temperature conditions: at the melting point of the material (1150 ◦ C), the magnesium, present in it, already has reached the boiling point (1060 ◦ C). This combination of temperatures under vacuum conditions at a relatively high magnesium content (up to 5 %) leads to the development of the processes of volume boiling of the melt and its removal from the penetration channel to the surface. The main criterion for the start of boiling under the welding conditions is that the pressure in the gas bubbles should be

Fig. 3. Macrostructure of welded joints in Al-30Be-5Mg composite after welding with electron beam (without oscillations)

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higher than the external pressure (p0 > p) [4]. The defect formation criterion is the critical size of the bubble at which the metal is ejected. These conditions are described by the following equation: µ ¶ 2Vm σ 2σ . (1) p0 exp − >p+ rcT r where p0 and p are pressures of the vapors in the bubble and above the melt surface, respectively; Vm is a speciﬁc volume of the liquid; σ is a surface tension coefﬁcient; r is a critical radius of the bubble; c is the heat capacity; T is the temperature. The experiments and calculations show that for the Al–Be–Mg system there are limiting permissible amounts of easily evaporated elements, present in the material. For example, an equation was derived for determining the critical molar fraction of magnesium in relation to the beryllium content:

NMgcr =

0.38 4CBe , 0.11 − 2800 2457CBe

(2)

where CBe is the beryllium content in the alloy. The experiments show that in the alloys with NMg < NMgcr , the effects were not found, whereas at NMg > NMgcr , there are defects in the form of a longitudinal cavity. For Al–30Be–5Mg alloy used often in structures, NMg = 0.0348 and NMgcr = 0.0188, i.e. this alloy cannot be welded by electron beam welding. Analysis of the physical mechanism of the process shows that the defect can be eliminated only by reducing the magnesium content of the welding zone and by suppressing the boiling processes. Boiling can be suppressed by applying longitudinal transverse oscillations of a beam. The magnesium content in the welding zone was reduced by placing interlayer of aluminum alloy with no magnesium in the joint between the edges of the components to be welded. The relation between the fraction of the parent metal and the interlayer material in the weld was determined using the equation µ ¶ δ ∆Pb = 1 − · 100 %, (3) a+b where ∆Pb is a fraction of the parent metal; δ is a thickness of the interlayer; a and b are widths of the weld in the upper and central parts, respectively. The best results for a metal thickness of 10 mm were obtained at an interlayer thickness of 1 mm. This technology was used to weld a number of important components. Highquality joints were produced in the metal 5–20 mm thick (Fig. 4), with the mechanical properties of the joint close to those of the parent metal. No defects in the form of pores, cracks or cavities were found. Research of Materials of Fe–Cu–Pb System. Another main type of composites is the Fe–Cu–Pb system, which components are not miscible in the liquid and solid states. When they are melted together, it is not possible to produce a homogeneous structure of the ingot because of phase separation of lead with iron and their 268

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liquation. Copper with lead also have a phase separation region with a temperature range 1500 − 1536 ◦ C [5]. The structure consists of a copper-lead matrix with iron intrusions. Similar materials have been produced in sufﬁciently large amounts quite recently by contact alloying [6]. They have high damping properties and an exceptionally high friction factor with sufﬁciently high strength. Iron-copper composites substitute for many types of bronze in producing spring elements, roller bearings, compressor blades, piston rings, toothed wheels, etc. All these components require fusion welding. Since the welded thickness varies from 0.5 to 5 mm, it is convenient to examine the application of argon-arc welding and laser welding for joining these materials. The experiments, carried out with variation of the conditions of argon-arc welding of materials of the 50Fe–40Cu–l0Pb and 50Fe–45Cu–5Pb systems, did not give positive results. In all cases, the weld contained the same phase separation defect, namely the precipitation of a copper-lead mixture and separation of its individual conglomerates. These compounds failed at low loads. To produce a strong welded joint, it was necessary to use a concentrated heat source in the form of a powerful laser beam. Materials with a thickness of 1.5 to 4 mm were welded in a CO2 laser with a power of up to 5 kW. Argon was used as the shielding gas. The welding speed was 20–30 mm/s and at an energy concentration of 105 W/cm2 resulted in a cooling rate of 103 ◦ C/s. A priori these conditions should prevent phase separation. However, as shown by the results of a large number of experiments, under the above conditions the copper-lead mixture managed to precipitate in large conglomerates (Fig. 4). To explain the physical nature of this defect formation, the theory of breakdown and phase separation of liquid-metal systems was used [7] according to which the criteria for the retention of the structure of phase-separation systems in Fig. 4. Macrostructure of Al– fusion welding are the possibility to obtain in 30Be–5Mg composite welded by heating with the welding power source the mix- electron beam welding with beam ing temperature Tcr and the duration no longer oscillations and interlayers: than (10 − 50) · 103 s of existence of the liquid- thickness of material 10 mm metal systems in the dissociation and phase separation temperature range. The ﬁrst criterion can be evaluated using the dependence derived by Hildenbrandt and Scott [8] : m Tcr >

0.25(V1 + V2 )(δ1 − δ2 ) , R

(4)

where V1 and V2 are atomic volumes of the separating components; δ1 , δ2 are paVESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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rameters of solubility of the components equal to H/V (H is the sublimation heat); R is the gas constant. The right hand part of [4] reﬂects the heat of formation of the liquid solution. If it is higher than the value Tcr , the phase separation takes place in the liquid state. As shown by the calculations, the laser welding conditions do not allow these conditions to be satisﬁed, because this requires solidiﬁcation rates of the order of 105 − 106 ◦ C/s, which is unrealistic under the welding conditions. Another method of preventing this type dem of the melt fect formation is the reduction of Tcr by adding additional alloying elements to the melt zone. Taking into account the conditions of chemical compatibility and the effect on Tcr , experiments and calculations showed that nickel is a suitable alloying element. The interlayer of this material in the form of a strip was placed between the specimens. The thickness of the interlayer B1 in laser welding of the 50Fe–40Cu– 10Pb composite was selected in relation to the lead content CPb , of the material: B1 = 0.27BW (CPb − 2.5),

(5)

where BW is a width of the weld, mm. The last equation was derived by analyzFig. 5. Macrostructure of laser beam welded joints in 50Fe- ing the experimental dependence of the required 40Cu-10Pb composite without nickel content of the weld on the lead content of nickel interlayer the material. Welding with the nickel interlayer has made it possible to prevent completely phase separation and obtain welded joints with a strength of 0.8 ∼ 0.9 of that of the parent metal (Fig. 5). Research of Materials of Al–Pb System. The Al–Pb system in accordance with the equilibrium diagram is a composite insoluble in the liquid state. In the solid state, the mutual miscibility is restricted and equals 0.15 − 0.12 %. The composite materials based on this system have appeared relatively recently and are also produced by contact alloying. These materials are applied where it is necessary to have the radiation protection using light materials: under space ﬂight conditions, in nuclear power stations and other objects associated with radiation. Technology of welding the materials of the Al–Pb composite type is based on the experience obtained in previous applications, especially laser welding using interlayer. On the basis of the calculations with the use the procedures proposed previously, a tin interlayer was recommended. The experiments with laser welding with interlayers made of tin gave positive results. The criteria and the methods of removing defects can be applied to larger groups of materials and also used when developing the fusion welding technology for dissimilar materials. Conclusions. 1. A classiﬁcation of MCM, based on the interfacial interaction of the components in fusion welding, is proposed. 270

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2. Speciﬁc defects when fusion welding MCM are considered and criteria for their formation are determined. 3. Technological techniques are proposed for producing high-quality welded joints by argon-arc, electron beam and laser welding of Al–Be–Mg, Fe–Cu, Fe– Cu–Pb and Al–Pb composites.

REFERENCES 1. Grigoryants, A.G. and Shiganov I.N: Laser welding of metals (in Russian). – M.: Vysshaya shkola, 1988. 2. Edited by Kreider N. Composite materials with a metallic matrix (in Russian). Vol 4. – M.: Mashinostroenie, 1978. 3. Komarov, M.A. Speciﬁc features of structural transformations in the fusion zone of welded joints in aluminum–beryllium alloys (in Russian). Aut. Svarka 1985 No 10. 4. Kokh, B.A. Fundamentals of thermodynamics of metallic processes (in Russian). – M.: Sudostroenie, 1975. 5. Binary and multicomponent systems based on copper (in Russian). Handbook. – M.: Nauka, 1979. 6. Tuchinskii, Ya.I. Composite materials produced by impregnation (in Russian). – M.: Metallurgia, 1986. 7. Vozdvizhenskii, V.M. Formation of delamination in the liquid state in binary systems (in Russian). In “Equilibrium diagrams of metallic systems”. – M.: Nauka, 1974. 8. Wilson, D.R. Structure of liquid metals and alloys (in Russian). – M.: Metallurgia, 1972. Shiganov Igor Nikolaevich graduated from the Bauman Moscow Higher Technical School in 1971, D. Sc. (Eng.), professor, Director of the Materials & Technological Processes Research Institute. Author of more than 120 publications in the ﬁeld of the welding & laser technologies.

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Yu. Bocharov, Yu. Gladkov (Bauman Moscow State Technical University)

INTEGRATED CONTROL-MONITORIUNGDIAGNOSIS SYSTEM DEVELOPMENT IN TECHNOLOGY OF PLASTICITY Problems and results of integrated control system development for closed-die hot metal forming are discussed in the paper. Process and Maxipress mechanisms simulation models are employed for an adaptive control system with goal of a deformational force stabilization. Operational parameters monitoring and press mechanisms diagnostics are realized with a help of two level module subsystem for a disc friction clutch and the closed gap regulation mechanisms. An adaptive algorithm for this purpose has been developed.

Introduction. Monitoring operational metal forming process parameters and diagnostics of forging machine-tool mechanisms are necessary for reliable atomized forging complexes and lines operation and product quality assurance. It involves computer process and machine-tool modeling, computer numerical control (CNC) operational algorithms and programs development, monitoring parameters selection, diagnostic mechanisms and algorithms development, set of proper sensors and hardware selection and dozens of other problems to be solved. A general methodology approach to the problems of integrated CNC development in technology of plasticity has been worked out. A pilot integrated CNC system incorporating control-monitoring-diagnosis subsystems for the forging line based on the 25 MN nominal force Maxipress has been designed. According to CNC strategy analysis [1] the billet control strategy has been selected, since the mechanical crank operating Maxipress demonstrates a low degree of machine parameters controllability. To test and verify the system an experimental set-up, based on the 6.3 MN crank-operated press, electric furnace, personal computer (PC) and set of sensors, has been constructed. Physical process has been simulated with experimental dies and steel billets heated to the forging temperature. Process and Machine-tool Modeling and CNC. Process and Maxipress machine-tool simulation models are employed for obtaining nominal cycle parameters to compare with real time variables in adaptive control, monitoring and diagnosis subsystems. Besides they allow exploring acceptable successfully processed process cycles. Product model includes geometry and height values, material structure (microstructure, macrostructure) as main variables; forging model considers billet mass (or volume) and time dependant functions of temperature, deformation force, whose absolute values are constrained between upper and lower limits. Process dynamics model usually presents the simpliﬁed mathematical description for relationship between directly actuated process variables and product geometry or microstructure quantities [3] The computer simulation model helps to test, 272

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Fig. 1. Hot-Die Forging complex with 25 MN mechanical press: 1 — inductor, 2 — forming rollers, 3 — conveyer, 4 — mechanical press, 5 — feeder, 6 — conveyer, 7 — press-cutter

adjust, improve integrated CNC for hot-die forging complex in the course of its design and to control and monitor production cycle parameters in the operational mode. Main variables are volume (mass) and temperature of the billet. Random variation of those parameters causes the corresponding variations in the value of deformation force and elastic. Forging complexes based on hot-die forging mechanical presses (Maxi-presses) are widely employed in automotive, agricultural, road construction machinery and other industries. The precision of a product height depends upon the press-die stiffness; die inserts temperature and deformation force variables, resulting in a variable gap between die-inserts. Hot-die forging complex of TMP Company (Voronezh, Russia) (Fig. 1) serves for die-forging of pitman type components and consists of a preheater (450−500 ◦ C), press-cutter, induction heater (1230−1260 ◦ C), forging rolls (initial preforming), mechanical press 25 MH (preforming, forming, ﬂash cutting). The same complex is employed for gear and ﬂange type components forging in four stages (upsetting, preforming, ﬁnal forming and burr cutting). The closed loop adaptive CNC system able to stabilize deformation force with a billet temperature as correcting factor of the billet volume (mass) variations has been developed (Fig. 2) [3]. Adaptive algorithms of the independent billet mass, billet temperature and integrated mass and temperature control have been designed and executed with the help of the original program “PC-Swage Master” [4]. Linear regression model expressing the billet temperature and mass inﬂuence on deformation force has been used in a die forging complex based in Maxipress of 25 MN nominal force CNC

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Fig. 2. Adaptive CNC system structure

system. “PC Swage-Master” program [4] has been used as a process simulator Pi = P0 + KI(Mi − M0 ) − K2(Ti − T0 ),

(1)

where P0 , M0 , T0 — optimal calculated values of the deformation force, billet mass and temperature; Mi , Ti — current values of a billet mass and temperature; K1 and K2 — correlation coefﬁcients. Adaptation of the regression model (1) is executed with a periodic calculation of K1 and K2 coefﬁcients. Billet temperature after inductor is considered as a random value in the course of the forging process. The billet mass (volume) could be slightly different from a cut value due to a billet material oxidation. Both uncontrollable factors are considered as random values having normal distribution mode in the simulation model. The following devices are to be involved in the control signals generation: — billet cutting device (press-cutter) and heating device (inductor) as control objects executing control signals; — transfers and other performing devices (like forging rollers) acting in the way from inductor to the ﬁrst die position as a system time-delay (inertia) factors. (Time-delay factors are considered with a help of decreasing coefﬁcients K1 and K2 programmable calculation); — billet mass (volume) and temperature transducers as feed-back information source; — deformation force transducers (tensometers) as an integrated feed-back information means of a billet mass (volume) and temperature measurements; — die insert temperature sensors;

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— closed gap between die inserts transducer as an integrated control system feed-back information source. According to integrated algorithm the initial billet volume (mass) value is compared with calculated one and a billet temperature needed to compensate the volume (mass) deﬂection is calculated. The deformation force at every process cycle versus press slide travel serves as the feed-back information and is compared with the calculated diagram. Three control and diagnosis levels are presented in the control system structure (Fig. 2). Informational system tasks are divided into three blocks: forging complex drive and servomechanisms monitoring, production time statistics, diagnostics. Diagnostics subsystem solves the problems of the following groups in real time: 1) monitoring complex main parameters within permissible value corridor and realization of the mechanisms preventing stoppage in the case of parameters critical values; 2) diagnosis the main complex mechanisms operation and warning signals generation in the case of any forecasted malfunction; 3) diagnosis of the main complex mechanisms temperature and wear conditions and maintenance and repair planning. The ﬁrst group of problems is solved with the help of PLC with corresponding software, while the second group requires execution of calculation algorithms. In the case of low probability of critical parameter value the diagnosis is executed by PLC. The third group of problems serves for information monitoring to the control terminal. Forging process simulation model. The forging process computer simulation model of hot-die and semi-hot-die forging process is based on the original software with the following function blocks: process variables calculation, process simulation core, operator interface, remote control interface [4]. Simulation program is employed for the following tasks: (1) computer testing and adjusting adaptive control algorithms; (2) executing the adaptive CNC and data terminal control in real time; 3) collecting and storing operational data. The program can operate in the following modes: operation, tuning, control algorithm analysis and recall mode. In operation mode input data are received from sensors. Output data are the control commands to complex devices. Tuning mode serves for analysis and adjusting of complex operation with the help of simulation model. The operation mode permits to analyze different complex processes and monitor operational parameters. Control algorithm analysis mode serves for calculation the system responding time to single exposure and comparing the algorithm efﬁciency. Recall mode allows regenerating complex operation parameters with a help of a protocol ﬁle. This function permits the multiple analysis of any process on the base of input data form the protocol ﬁle. The simulation program also provides facilities for calculation the optimal parameter values for control algorithms (optimal values of decreasing coefﬁcients and others).

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System requirements: Pentium 166 MHz, 32 Mb RAM, 50 Mb HDD, SVGA High Color (16 bit), Windows sound card, Microsoft Windows 0 95/98. The program has been developed in Borland C++ 5.02 and OWL [10]. Each billet is supplied with a data passport and is served in the queue. Queue elements record current parameters of mass and temperature and the queue itself simulates the process logistics. The simulation program executes the following tasks. 1. Compensation of a billet mass (volume) deﬂection dM of the calculated value (2) Mi = Mi−1 + dM ; It is realized in the billet press-cutter control net. Mean deviation in the batch of n billets ¶ µX n ± (3) Mk n, Mm = k=1

is compared with the calculated value M0 dM = (M0 − Mm )K,

(4)

where K — decreasing coefﬁcient, calculated in simulation model at the condition of regulation time minimization. 2. Compensation of a billet temperature in inductor in the billet heater control subsystem Ti = Ti−1 + dT ; dT = (T0 − Tm )K; µX ¶ n ± Tk n. Tm =

(5)

k=1

3) Additional compensation of a billet mass (volume) inaccuracy by a billet temperature Ti = Ti−1 + dTmi . (6) Program interface displays the system information to a human operator in real time. There are facilities for the main window customizing, changing parameters of the hot, semi-hot or cold die-forging process and simulation model, operational and critical situation monitoring and others. Button area of the main window includes buttons: “Scissors”, “Inductor”, “Rejecter” and “Press” allowing an appropriate device for operational diagnosis to be selected. List of displayed parameters can be selected with buttons: button ”All” for all devices, ”Selection” for an appropriate composition of the die-forging complex. Window area consists of four parts: “Star”, “message” and two “diagram” windows. Each ray of the star displays one parameter as a point between ray ends that denote maximum and minimum values. Red, yellow and green areas of the ray visually demonstrate a state of parameter that can be not admissible, critical or normal. 276

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In the message window the current values of parameters, system components state and warnings are displayed in text messages. Two diagram windows display process parameters versus time. Red, yellow and green lines on the background of the windows denote areas of not admissible, critical and normal values of parameters. There is also a facility for customizing diagram windows (diagram width; link/unlink points; x-line step changing). Main menu items realize access to dialog boxes for customizing program and changing parameters of the die-forging process, control algorithms and others. The program also offers a suitable interface for manual adjustment of process parameters. Operator can select necessary parameters and they will be displayed as a number of scrollbars. Scrolling will bring about changes in the corresponding parameter value. The list of parameters depends on the program operation mode. Press mechanisms simulation models. For clutch-brake diagnosis a doublelevel imitation model has been developed: module structure for a lower level and probability mode of operation (with weight coefﬁcients) for an upper level. Each of several lower level modules sends press mechanisms diagnosis signals to the upper level. The upper level calculates reliability degree of each diagnosis signal. If a reliability degree exceeds a limit value the system may monitor diagnostic values and correct process-machine parameters regulating Maxipress closed gap at the time interval between the cycles, correcting billet temperature etc. Lower level modules for clutch and press closed gap regulation mechanism are presented here. a) Clutch module. Diagnosis parameters are angular velocities of active and passive friction discs. Clutch cycle (Fig. 3) is analyzed and diagnosis parameters are monitored. For example, in the case of t exceeds permissible level diagnosis of control valves, pneumatic cylinder, seals, or friction discs is suggested. b) Press closed gap regulating mechanism module. Diagnosis parameter are a slide linear displacement, press closed gap value, deformation force, billet mass and temperature. An adaptive relation between parameters (Fig. 4 ), diagnosis algorithm for a forging product height variation and operations for a programmable

Fig. 3. Tachogram of crank press clutch cycle

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Fig. 4. Adaptive control algorithm

press closed gap correction are developed [6] with adaptive constants recalculation depending on a press closed gap variations. Information processing is executed with a program and experimental imitation model and control algorithms veriﬁcation and correction are possible. The same methodology could be used for other mechanisms imitation models and relations between lower and upper levels development. Conclusion. The developed simulation program and the control algorithms could be after certain modiﬁcation employed also for semi-hot and cold die-forging processes in automated complexes and lines on the base of other types of forging processes and machine-tools. The requirements to the processes could be speciﬁed as follows. The process parameters could be presented in the linear regression model. Endogenous parameters (like the billet mass and the temperature) should permit the calculation of the only exogenous parameter (like the maximal deformation force value). The closed loop in the integrated CNC System should be based on the exogenous parameters. Veriﬁcation of the computer simulation program and model has been done in computer and physical experiments. It has been demonstrated that accuracy of the height size of the forged components in hot-die forging complexes based in mechanical forging presses could be increased by 60–70 % with the help of suggested adaptive CNC. The computer simulation model helps to test, adjust, and improve the adaptive CNC for hot-die forging complex in the course of its design, saving time and resources. In operational mode it helps to control and monitor production cycle parameters and to regenerate statistical process data in the cases of unscheduled stopovers and change of production program. Computer simulation program with incorporated monitoring and diagnosis functions helps to increase reliability of the production lines and accuracy of the forged items.

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REFERENCES 1. Bocharov, Yu. A. CNC Strategy in Technology of Plasticity // Advanced Technology of Plasticity, 1999, 6 ICTP, 1999, v. 1, p. 195–200. 2. Bocharov, Yu.A. CNC Systems in Technology of Plasticity (in Russian) // KuznechnoShtampovochnoye Proizvodstvo (Technology of Plasticity), No 7, 2000. – PP. 39–46. 3. Balagansky, V.I., Bocharov, Yu.A., Gladkov, Yu.A. (in Russian) System of Adaptive Control for Hot-Die Complex // Kuznechno-Shtampovochnoye Proizvodstvo (Technology of Plasticity), No 6, 2001. – PP. 26–30. 4. Gladkov, Yu., Khvostenko, A. Interface module of CNC for hot-die forging complex - PC Swage Master (in Russian)// RosPatent Certiﬁcate, No 990746, 1999. 5. Prutckov, R., Balagansky, V., Bocharov, Yu. Maxipress Cosed Gap Operational Regulation (in Russian) // Kuznechno-Shtampovochnoye Proizvodstvo (Technology of Plasticity), No 6, 2001. – PP. 26–30 (Russian). Yury A. Bocharov, graduated in Mechanical Engineering from Bauman Moscow Higher Technical School in 1953. Professor, Technology of Metal Forming department of Bauman Moscow State Technical University. D.Sc. (Eng.), RANS Full member, Honored Scientist of RF. Has published more than 300 papers, textbooks and books; has a number of invention certiﬁcates and patents in the ﬁeld of Metal Forming Technology, Machine Theory and Design, CNC and diagnosis. Yury A. Gladkov, graduated in Mechanical Engineering (MSc) in 1998 and Computer Control Engineering (BSc) in 2000 from Bauman Moscow State Technical University. Assistant-Professor, Technology of Metal Forming department. Ph.D. (Eng.) Has published more than 30 papers; has a number of Software Development Certiﬁcates related to Metal Forming Machinery and processes CNC.

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S. Golovashchenko (Ford Motor Co, USA), V. Kondratenko, A. Vlasov (Bauman Moscow State Technical University)

EXPERIMENTAL INVESTIGATIONS OF FLANGING ALUMINIUM PANELS AS THE FIRST STAGE OF HEMMING PROCESS Hemming is often used as the last stage of stamping operations in automobile industry. Mostly it is used to attach one sheet metal part to another for instance outer vehicle door panel and inner door panel. The main problem associated with the hemming of aluminium body panels is cracks on the outside hemmed radius. We offer and study a number of new ways of ﬂanging (the ﬁrst stage of hemming), based on idea of increasing of hydrostatic pressure in the center of plastic deformation. Most investigations were carried out in Ford Research Laboratory.

Background. We develop technological process of ﬂat hemming thin sheet aluminium alloy 6111-T4 panels (closely related to Russian AD33). Traditional hemming process has three stages: ﬂanging, pre-hemming and ﬁnal hemming. Finally, after hemming, the design should look as follows (Fig. 1): The key factor for using of hemming for aluminium automobile body panels is a lack of any cracks on the outside hemmed radius as well as lack of any roughening on A surface of outer panel. Thus, inside radius should not exceed half of thickness of a material. The thickness of a material is 0,0400 (approximately 1 mm). The resource of plasticity of 6111-T4 does not sufﬁce for traditional hemming process. It is known (Bridgman and others), that the increase of hydrostatic pressure in a zone of deformation raises a resource of plasticity. We tried to suggest some schemes, which used the idea. The review of new technologies. Let0 s consider the suggested schemes of new technologies: 1. First offered the scheme, in which ﬂanging is divided into two stages: a usual bending on large radius and then ﬂanging with compressive force applying to

Fig. 1

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Fig. 2

Fig. 3

outer panel trim edge (Fig. 2). In this schemes the force causes the high hydrostatic pressure, and polyurethane prevents buckling failure. 2. The technology, shown on Fig. 3, based on the idea of using of elastic punch for creating the pressure in bending zone. We also used the proﬁled elastic punch. 3. The technology, shown on Fig. 4, has two steps. First (stage A — not shown) — bending with large radius and then (stage B — shown on Fig. 4) ﬂanging by means of rigid punch moving horizontally. In this scheme the rigid punch was used for creating active friction forces, which create compressive force in bending zone. Experimental results. We used two sets of material in experiments. The ﬁrst set is the old strips of 6111, which have more 2 years old. So due to aging treatment the old strips lost much of their ductility. Rolling direction at a bend of old strips placed along a line of a bend. In further these samples we shall designate as material # 1. The second type of strips was 6111 with 7 % prestrain. Rolling direction at a bend of new strips placed along a line of a bend. In further these samples we shall designate material # 2. Polyurethane - (made in Russia) used for elastic punches during the experiments.

Fig. 4

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During the experiments we used a scale for evaluation the quality of bend line. Following Baartman1 we used the scale of hem quality given in Table 1. Table 1 Score

Classiﬁcation

10

No visual surface defect

8

Mild surface roughening

6

Ridge-like surface roughening (“orange peel”)

4

Small surface cracks

2

Continuous surface cracks

0

Complete cracks through material

Experiments for scheme # 1 was produced with material # 1. The quality of bend line of the blanks being bended such way had score 10 (the best quality). Besides we obtained the sharp inner radius (a lot of less than half of thickness). But the A-surface had some roughening. Due to bad A-surface quality of A-surface we rejected this scheme. On the other hand the best hem quality shows that the increasing of hydrostatic pressure raised the bend ductility dramatically. Experiments for scheme # 2 was produced with material # 1 too. We varied the following parameters at this investigation (see Fig. 3): A — The depth of a cavity in rigid punch; B, C — The sizes of elastic punch; D — The interference between punch and die. The transparency ﬁlm placed atop of blank was also used in some experiments. At ﬁrst step we used following elastic punch sizes: 10 × 17, 10 × 15, 10 × 13, 10 × 10, 10 × 6, 13 × 13, 13 × 10, 13 × 6, and varied interference in the limits: 0 . . . 6 mm. The best results was for punch with sizes B × C = 10 × 6 mm. We obtained quality score 8 with using a ﬁlm and 6 without ﬁlm (for material # 1) and 5 for material # 2. But the durability of elastic punch was moderate. The wear-out of elastic punches was less for interference 1 . . . 2 mm. A larger interference invokes a fast wear-out. The second step was for the investigations of form of elastic punch. The numbers of punches, their forms and sizes are given in Fig. 5. The best results was for punches ## 12,18 with a quality score 6 without ﬁlm. The third step was for the inﬂuence of lubrication. We used soap, graphite lubrication as well as ﬁlm and two blanks ﬂanged together. It is found out, that the friction is not the major factor inﬂuencing quality of hem. The positive inﬂuence of a ﬁlm consists in increasing thickness of blanks. 1

R.Baartman, E.H.Atzema and J.Bottema. Optimization of the hemming process for AA6016-T4 aluminium body sheet. 98NM056.

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Fig. 5

This hypothesis proved by improvement of quality at use of a ﬁlm of the greater thickness, and at joint bending of two blanks placed one atop other. Both experiments have given the best results in comparison with other, received at normal conditions and at use of lubrications. The analysis of experimental results with schemes # 2 shows, that form and sizes of elastic punch that obtained stable good quality without using ﬁlm are not found out. The main reason is that the schemes do not create hydrostatic pressure. And what is more — they increase tensile stress in bending zone due to inﬂuence of contact friction. The best form of the elastic punch obtained from experiments listed above is the punch # 12 which has complicated form. Its form we varied at the next stage. The varying parameters are explained in Fig. 6. The values are given in Table 2. All experiments were performed without any lubrication or ﬁlm.

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Table 2 Punch No

C, mm

F , mm

E, mm

R, mm

I

26

9

6,5

3.5

II

22

7

6.5

3.5

III

18

8

6.5

3.5

IV

15

7

6.5

3.5

V

20

8

6

4

We achieve the quality score 6 without ﬁlm. The analysis of experimental results with schemes # 3 shows, that form and sizes of elastic punch that obtained stable good quality for prestrained material (matherial # 2) are not found out. The best results are for following: 1. The creating of a direct corner occurs by turning of lug of elastic punch (size E in Fig. 6) at the end of process. Width E should be as greatest as possible. 2. The deformation of elastic punch should be maximal. If the deformation is insufﬁcient, elasticity of the lug does not sufﬁce for obtaining of a direct corner. The experiments with the scheme # 3 were carried out in the following sequence: 1. Preliminary ﬂanging: a) preliminary bend on large radius with use of laboratory installation for bending by the moment; b) preliminary bend in a die with the big gap between punch and die. 2. Final ﬂanging with use of rigid punch. At ﬁnal stage we varied the following sizes (see Fig. 7, 8): • Inner preliminary blank radius — Rb = 2 . . . 7 mm;

Fig. 7 284

Fig. 8

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• Preliminary gap — ∆ = 2 . . . 10 mm; • Die radius — Rm = 0, 1; 0, 5 mm. The obtained experimental scores of quality are given in Tables 3, 4. We used the material # 2 (7 % prestrained) for this experiments. Table 3 Rb, mm Rm = 0,1 mm Rm = 0,5 mm 2 6 (# 1) 6 (# 7) 4 (# 2) 4 (# 8) 3 4 (# 3) 6 (# 9) 4 2 (# 4) 2 (# 10) 5 2 (# 5) 2 (# 11) 6 2 (# 6) 4 (# 12) 7

Table 4 ∆, mm Rm = 0,1 mm Rm = 0,5 mm 2 2 (# 13) 8 (# 20) 4 (# 14) 6 (# 21) 3 2 (# 15) 8 (# 22) 4 2 (# 16) 8 (# 23) 5 4 (# 17) 8 (# 24) 6 4 (# 18) 6 (# 25) 7 2 (# 19) 6 (# 26) 8 no data 4 (# 27) 9 no data no data 10

In Fig. 9 we show the result # 7 from Table 3 and # 24 from Table 4. You can see the sharp inner radius and the negative springback obtained after the ﬁnal ﬂanging. Analyzing the results of experiments for schemes # 3 it is possible to make the following conclusions: 1. Using the preliminary bend with large radius or large gaps ensure the good quality for ﬁnal bending by rigid-elastic punch and with some sizes for rigid punch. 2. The key factor in two-step ﬂanging is the precision mounting of preliminary bended blank into the ﬁnal die set. The preliminary bended zone must be out of contact with ﬁnal die (see Fig. 10). In the case of contact the results was worst.

Fig. 9

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285

Fig. 10

Conclusions. The key idea, consisting in increasing the hydrostatic pressure in the bending zone for increasing ductility of material being bended, is proved by experimental way. The way of realization of the approach is found. The ﬂanged blank with high quality of the A-surface and of the bending zone is obtained.

S.F. Golovashchenko graduated from Bauman Moscow Higher Technical Scholl in 1985. D. Sc. (Eng.), Senior Technical Specialist of Scientiﬁc Research Laboratory, Ford Motor Company. Author of more than 50 publications in the ﬁeld of theory of plasticity and metal forming technologies. V.G. Kondratenko graduated from Bauman Moscow Higher Technical School in 1959. Ph. D. (Eng.), professor. Author of more than 150 publications in the ﬁeld of theory of plasticity and metal forming technologies. A.V. Vlasov graduated from Bauman Moscow Higher Technical School in 1978. D. Sc. (Eng.), professor of “Metal forming technologies” Department of Bauman Moscow State Technical University. Author of more than 50 publications in the ﬁeld of forging equipment design and metal forming technologies.

286

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CRIOGENIC ENGINEERING & TECHNOLOGY A.M. Arkharov, S.D. Glukhov, L.V. Grekhov, A.A. Zherdev, N.A. Ivashchenko, D.N. Kalinin, A.V. Sharaburin, A.A. Aleksandrov (Bauman Moscow State Technical University)

CREATION OF HARMLESS FOR ENVIRONMENT ENGINES AND INSTALLATIONS USING DIMETHYL ETHER The article deals with the problem of replacing pollutant substances with ecologically safe ones used in the urban transport. Dimethyl ether is proposed as it has ozone depletion potential and global worming potential of zero value. It can be used as a fuel and as a refrigerant at the same time. The article cites the advantages of dimethyl ether use as well as it analyses experimental researches.

Ecological problems for such megapolicies like Moscow are becoming more and more acute. The Moscow government realizes such an unhealthy situation; it undertakes concrete measures to reduce pollution of environment including water ozonization, conservation of different gas discharges — for example at ﬁlling/service stations; avoidance of soot use during winter period, planting of greenery, amelioration of transport schedule, development of electric traction etc. Among such measures the creation of harmless diesel engines as well as replacement of ozone depletion refrigerants used in low temperature installations have a signiﬁcant place because diesel engines and refrigerants are involved in urban economic development. The problem of energy and cold regeneration in vehicles is of great importance for megapolicies, large cities and densely populated regions, where major technical and technological advances of 20th century are concentrated: propulsion engineering, motor-car construction, chemical engineering, atomic and power engineering, refrigerating and cryogenic engineering, food industry, aerospace and chemical complexes etc. In a number of biggest megapolicies (for example, Los-Angeles, Tokyo, London, Shanghai) rigid constraint has been adopted, ﬁrst of all with respect to vehicles, refrigerating, heating, ﬁltration, and puriﬁcation systems. Properties of working substances of many energy-power refrigerating installations as well as properties of different consumer goods are regulated too [3]. In 1987 the Montreal Protocol was signed that limited the production of ozone depletion refrigerants; in 1997 the Kyoto Protocol was drafted that would require to limit use of chemicals stimulating greenhouse effect of our planet. Besides, in Russia, since 2003 the Euro-III standards have been adopted that regulate the toxic substances content in used gas of internal combustion engines. Today two criteria exist: ODP (ozone depletion potential) and GWP (global worming potential). Ideally these potentials must be of zero value for working substances and fuels used. Such a substance is dimethyl ether (DME) CH3 −O−CH3 VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

287

that can be considered as a fuel and as a refrigerant at the same time. DME was used in the ﬁrst vapor compression machines, and then it was forced out by ammonia, later — by refrigerants. Last few years DME has been considered as a perspective fuel for diesels used in transport, ﬁrst of all thanks to signiﬁcant reduction of toxic substances issue (Euro III standards without use of neutralizers). These qualities of DME are objective prerequisites for creating combined energetic plants with engine and refrigerating unit for the urban transport. The Moscow State Technical University named after N.E. Bauman (BMSTU) has carried out researches of DME as a diesel fuel as well as a refrigerant component [4]. Table 1 The Characteristics of Motor Fuels Properties

DF

DME

835–850

668

Combustion heat, MJ/kg

43.5

28.8

Stoichiometric ratio

14.5

9

170

46.07

240–310

235

◦

3

Density at 20 C, kg/m

Molecular mass ◦

Temperature of self-inﬂammability, C ◦

Heat of vapor formation at 20 C, kJ/(kg·K)

210

410

Cetane number

45

> 55

0.4

34.8

< 0.001

0.51

—

5.37 (400)

Oxygen content, % ◦

Saturated vapors pressure at 20 C, MPa Critical pressure, MPa ◦

Compressibility factor at 20 C and 0,1 MPa, 1/Pa

−11

86 · 10

210 · 10−11

DME as a motor fuel. The DME characteristics as a motor fuel [1, 2] differ much from those of a diesel fuel (DF) (Table 1). For example, volume cycle feedings are needed 1.7–1.9 times as much to keep diesel engine capacity, taking into account lower density and caloriﬁc efﬁciency. While designing a high pressure fuel pump it is necessary to take into account the fact that at typical operation the volumetric efﬁciency must be increased 2.4–2.7 times due to the higher DME compressibility. When changing over from DF to DME the fuel feeding and the diesel working cycle undergo great modiﬁcations. Because of a high compressibility the fuel feeding begins later. So, when using the diesel D245.12 in the motor car “Bychok”, that has a modernized fuel equipment, injection lead decreases by 10.7 degrees of crankshaft turn when changing over to the DME at typical operation (Table 2). The optimal angle of lead increases when a rotational speed of the shaft increases; on the contrary the real angle decreases. The greater is compressibility of the fuel, the stronger is this contradiction. That’s why an automatic lead coupling is to be designed specially for operation on DME. The operation with a standard coupling diesel is not optimum, all the more without it (diesel D245.12 288

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case). This deﬁciency is compensated by quicker DME inﬂammation. The results cited in Table 2 have been obtained with the program complex “Vprysk” [5] that was adapted for the studied object. This program complex was used for optimizing and designing a modernized fuel system for DME feeding. Table 2 Indices of Fuel Injection in Diesel D245.12 Properties Cycle feeding gc , mg Active running of plunger Value of additional injection, % of g Maximum pressure, Mpa: in front of a fuel nozzle pressure of injection Average pressure of injection, MPa Characteristics of feeding, degree of crankshaft turn: duration retard of start Maximum moment on camshaft H·m

DF 79 2.15 0

DME 118 5.56 1.2

90% DF + 10% DME 82.7 2.56 0

55.09 44.36 23.95

38.38 28.79 19.56

55.08 42.66 23.93

23.12 11.60 92.00

44.12 22.32 66.56

24.40 12.90 85.18

The decrease of the injection pressure (in the case of diesel D245.12 1.5 the pressure decreases 1.5 times) is another signiﬁcant modiﬁcation of the fuel feeding index. The decrease of the injection pressure takes place in spite of the increase of the cycle feeding. Such conditions are not optimal for mixture formation but at least it is not a limiting factor: spraying of DME liquid jet is better due to less surface tension and viscosity; ether is vaporized without delay at charge temperatures in the cylinder. More important is the fact that the duration of injection (feeding) at the operating regime increases by 11 degrees of the crankshaft turning (Table 2). Thanks to special properties of DME a smoke of used gases may be avoided but the delay of injection doesn’t allow increasing the motor effectiveness. The proposed increase of a sectional area of nozzles doesn’t correspond to the possibilities of a universal dual-fuel diesel engine. Other speciﬁc problems occur while feeding a diesel with DME. Filling of the plunger cavity becomes difﬁcult. It results in non-stability of feeding. When using DME without the Lubrizol fuel additive imported there are problems to start the engine due to low viscosity; leaks occur and the life period of precision couples decreases; there are problems with fuel pumping in the low pressure lines. A shorter and larger ﬂame of DME leads to redistribution of thermal loads on elements of the cylinder-piston assembly as well as to overheat of central part elements of the combustion chamber etc. It’s evident that it is necessary to optimize working process to obtain an effective work of the diesel. Even if the diesel is working effectively the creation of a universal dual-fuel diesel engine on the basis of existent technical decisions VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

289

is rather problematical. The results of such an approach are well known thanks to 50 years experience of creating a universal engine at forced ignition working on gasoline and gas. Besides, modern motor fuel market and structure of modern service/ﬁlling stations should need serious modiﬁcations to exploit specialized cars using DME. Creation of new generation transport diesels using engine mechanical elements as well as electronic control systems optimizing the working process may be more acceptable [6]. A fuel feeding system must be changed a little according to new working conditions. In such a case the most perspective is application of an accumulator system with Common-Rail electronic control. We may certify that in the beginning of 21st century Russia doesn’t produce diesels with an electronic control system (except for small-serial diesel GAZ-560). And what is more there is no basis to expect for a quick start of Common-Rail systems’ production. The fact of the matter is that the hydrodynamic and thermal and physical peculiar properties of DME behavior under conditions of fast-acting physical and chemical processes of mixture formation and combustion (at high pressure drop) are not well studied, retards the creation of a diesel with effective working process. At the same time ecological situation of large cities demands to look for improvement of this situation based on the equipment produced as well as on the equipment in operation. Thereupon the suggestion to change fuel market structure in favor of DME is rather constructive (Moscow Government Decree #170-ΠΠ of 12.03.2002). The positive results have been obtained in diesels of 4ch10,5/12 type. Liquid gases have been supplied into a cylinder through a fuel supplying equipment improved. We have the experience of additionally makeup fuel supply of high pressure lines with other alternative fuels: hydrogen, gases, dense carbon suspension and others [7, 8]. In particular, the results of experience with a ZIL-645 diesel, when gases were supplied into the high pressure line showed other interesting things. While supplying hydrogen, synthesis-gas and pure air, the 20–45% decrease of CH, CO, NOx emission has been registered. In virtue of above-stated and taking into account the MSTU experience [9] we suggest to implement the ecological fuel DME in mixture with traditional diesel fuel. Its share is to be from 10 to 40% for different types of diesels and different operating conditions. This will allow smoothing away or eliminating many speciﬁc problems of diesel feeding with the DME. The passing of working processes (dispersion, evaporation, inﬂation, and combustion) will be rather improved. The decrease of toxic substances emission will permit to meet legislative standards on toxic emission (even while using rather an old model of diesels) using accessible technical means. Thereto a system of fueling of a transport diesel engine was designed and ﬁnished off (Fig. 1). The use of a mix fuel with DME addition in diesel engines exploitable now gives the possibility to obtain the following results: •Due to a high cetane number the low noisiness of operation, easier emission of used gases, decrease of nitrogen oxides emission are assured.

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•Smoke of used gases decreases due to a high content of oxygen in DME as well as to enhancement of spraying of the mix fuel, which contains a low-boiling component. •High pressure, minor duration of spraying, and low consumption of DME are kept due to supply of the DF as a main component of the mix fuel. Also there is a possibility to use the base fuel equipment Fig. 1. System of fuel supply of a transand to secure passing to operation with a port diesel: 1 — tank with DME; 2 — tank for DF; 3 pure DF. •Non-expensive retroﬁt of fuel equip- — electric control valve; 4, 8 — pumps for DME and DF pumping respectively; 5 — ment doesn’t inﬂuence the car cost and al- high pressure fuel pump ; 6 — fuel nozzle; low accelerating the renovation of a ﬂeet 7 — pressure regulator; 9 — DME feed valve of cars. •Use of expensive fuel additives imported is not needed, low consumption of the DME makes easy the cars exploitation as well as it makes possible creating combined installations with DME feeding of a refrigerating plant and a diesel engine. The top ﬁgure at the inside of the journal back cover demonstrates a car operating on DME. In our opinion the proposed design differs from others by low expenses, simplicity, universality, efﬁciency. You needn’t resolve speciﬁc problems when using the pure DME. But it is necessary to continue researches and to look for rational technologies. DME as a refrigerant. Owing to the fact that there are no performance attributes of the ﬁrst refrigerating machines operating on DME, a computerized program for determining thermodynamic properties of DME has been created, the diagrams have been plotted, and theoretical characteristics of vapor-compressing refrigerating machines [10] operating at DME have been calculated (Fig. 2 and 3). Later the refrigerating facilities using DME as a working refrigerant were installed. The necessary experimental studies were carried out. The obtained characteristics were correlated with the characteristics of the machines operating on R12 and R134a. The analysis of functions calculated allowed us to conclude that DME was a real alternative to well-known refrigerants but de plus it was of unique ecological purity for environment. The parameters of the R12 refrigerating cycle were compared with the experimental data according to main values, which characterized the given cycle: Q0 — cycle refrigerating capacity; L — compressor work; cop = Q0 /L — experimental coefﬁcient of performance. The results of the experiments have shown (Fig. 4) that use of a recuperative heat exchanger in DME cycles makes the refrigerating machine construction more complicated, and doesn’t provide efﬁciency and refrigerating capacity needed.

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291

Fig. 2. Calculated dependences of volume speciﬁc refrigerating capacity qv on boiling temperature t0 for DME, R12 and R134a: a — non-regenerative cycle; b — regenerative cycle

Fig. 3. Dependence (a) of the theoretical coefﬁcient of performance copt , (b) of the ratio of delivery pressure and suction p2 /p1 on boiling temperature t0 for DME, R12 and R134a

292

Fig. 4. Experimental dependences (a) of refrigerating capacity Q0 , (b) compressor work L, and (c) experimental coefﬁcient of performance cope on boiling t0 and condensation tc temperatures for DME and R12

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

The small refrigerating machines laboratory of the BMSTU carried out the DME tests with the Stinol 106 refrigerator of serial production. Without any construction conversion and oil replacement, DME was used instead of R12; an optimal mass of its charging was determined. Variations in energy consumption at different DME mass charging are shown in Fig. 5. The dotted line corresponds to a daily energy consumption of the Stinol 106 refrigerator when operating on R12 (nameplate charging mass is 240 g). During tests it was found that the Fig. 5. Daily electric energy consumpoptimal DME mass charging for such a tion N of the Stinol 106 refrigerator on model of the refrigerator (without chang- DME depending on charge mass m at ◦ ing a capillary tube dimensions) was 60– tm = −24 C in the freezing room: a, b — at ambient temperature +25 and 70 g. A daily energy consumption reduce +32◦ C respectively was noted (by 14–16 %). Other parameters were kept unaltered. These experimental researches have demonstrated that DME is adequate to replace R12 and R134a in the existent equipment as well as in a new designed one without degradation of its characteristics. In some cases DME may be considered as a preferable component of the motor fuel due to its low cost and availability. That’s why it is recommended to conduct long term resource tests with DME, to examine its interaction with different sealing materials which are used now in the refrigerating equipment. It is necessary to study the processes of DME boiling and condensation at different temperatures and pressures, to obtain data on dynamic and kinematic viscosity, gas and liquid heat conduction in the range of working temperatures and pressures of the refrigerating equipment. DME as a fuel and as a refrigerant for diesel transport using refrigeration and conditioning systems. Refrigerated lorries, buses, cars, ships, cutters, yachts Fig. 6. Grouping of a diesel refrigerator have to do with such hauling units. In when using DME as a fuel and a refrigcontrast to liquid propane-butane gas the erant: 1 — vehicle engine; 2 — compressor; DME is supplied under 3–5 MPa pressure. 3 — ventilator and condenser; 4 — ventiThat’s why the application of an open cy- lator and air cooler; 5 — temperature excle refrigerating or conditioning system is pansion valve; 6 — DF Tank; 7 — DME not feasible. But when the same substance tank; 8 — accumulator; 9 — fuel supply system is applied as a fuel as well as a refrigerant, VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

293

fueling up with a refrigerant is not needed because a fuel tank is a receiver of a low temperature system at the same time. With this, the cost of the refrigerating equipment and its maintenance is reduced. Figure 6 shows the grouping of a refrigerated lorry. Approximate costs of a refrigerating installation for the refrigerated lorry on the basis of the “Bychok” ZIL car are shown in Table 3. The main advantage of this lorry is that it is not expensive and doesn’t pollute environment. An experimental model of such a car was created in the BMSTU and demonstrated in Moscow City Administration (the bottom ﬁgure at the inside of the journal back cover. Table 3 Initial Expenses

Refrigerating installation costs Cost of car conversion using DME Cost of refrigerating installation assembling Consumed material cost (tubing, refrigerant, oils, assembling elements, isolation etc.)

Refrigerant R314a DME $3,000 $3,000 — $450 $1,000 $900 $500 $400

A payback depends on many factors: market conditions, transport service costs, operation conditions, climate etc. A preliminary analysis shows that costs of the refrigerating and conditioning equipment don’t exceed costs of traditional installations operating with R134a. REFERENCES 1. A.M. Arkharov, S.D. Glukhov, L.V. Grekhov, N.A. Ivashchenko, D.N. Kalinin, A.A. Alexandrov. Technology of Dimethyl Ether Use for Urban Transport //MSTU Vestnik, 2002, p. 162–167. 2. A.M. Arkharov, S.D. Glukhov, L.V. Grekhov, A.A. Zherdev, N.A. Ivashchenko, D.N. Kalinin, A.V. Sharaburin, A.A. Alexandrov. Dimethyl Ether Use as a Motor Fuel as Well as a Refrigerant // Khimicheskoe i neftegazovoe mashinostroenie, 2203, No 6, p. 17–21. 3. I.M. Kalinin, A.I. Smyslov, K.N. Fadekov. Perspectives Application of Safe for Environment Refrigerants in Household Equipment. Kholodilnaya Tekhnika // 2001. No 12. P. 4–8. 4. V.N. Bogachenko,, S.D. Glukhov, A.A. Zherdev et al. Dimethyl Ether — a Fuel and a Refrigerant for Diesel Refrigerator Trucks // MSTU Vestnik. Special Issue. Ser. Mashinostroenie, 2000. 5. Description of PK “Injection”. MSTU INTERNET site: http://www.bmstu.ru/facult/em/em2/ inject/illrus.htm. 6. Christisen R., Sorenson S.C., Jensen M.G. Hansen K.F. Engine Operation on Dimethyl Ether in a Naturally Aspirated DI Diesel Engine // SAE Paper, No 971665, 1997, p. 101–110. 7. N.N. Patrakhaltsev. Facilities for Gas-Diesel Process // Avtomobilnaya Promyshlennost, 1988, NO 7, p. 16–17.

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8. V.A. Vagner, D.D. Matievsky. Additive of Hydrogen to Fuel and Its Inﬂuence on Diesel Operation Characteristics // Dvigatelestroenie, 1985, No 2, p. 11–13. 9. L.V. Grekhov. Fundamentals of Development of Systems of Fuel Feed into Cylinders of Explosion Engine. Abstract to the Doctor Dissertation, M. 1999, 32 pages. 10. A.A. Zherdev, B.A. Makarov. Cycles Calculation of a Vapor Compression Refrigerating Machine with Redlih-Quong Equation of State // MSTU Vestnik, 2002, No 11, p. 71–80. A. M. Arkharov (b. 1931) graduated from the Bauman Moscow Higher Technical School in 1954. D. Sc. (Eng.), professor, head of ”Refrigeration, Cryogenics, Air Conditioning and Life Support Systems” Department of the Bauman Moscow State Technical University. USSR State prize winner, merited scientist of Russia, corresponding member of the Russian Academy of Technological Sciences. Vice-president of the Scientiﬁc Council of International Institute of Refrigeration. Author of 14 books, more than 50 inventions and more than 200 publications in the ﬁeld of cryogenic technology. S. D. Glukhov (b. 1938) graduated from the Bauman Moscow Higher Technical School in 1962. Ph. D. (Eng.), assoc. professor, section head of “Power Engineering” Research Institute of the Bauman Moscow State Technical University. Author of about 60 publications in the ﬁeld of cryogenic and refrigeration technology. A. A. Zherdev (b. 1955) graduated from the Bauman Moscow Higher Technical School in 1977. Ph. D. (Eng.), senior researcher, section head of “Power Engineering” Research Institute of the Bauman Moscow State Technical University. Author of more than 30 publications in the ﬁeld of cryogenic technology. N.A. Ivashchenko (b. 1940) graduated from the Bauman Moscow Higher Technical School in 1968. Ph. D. (Eng.), Honoured Science Worker of the Russian Federation, professor, head of “Reciprocator Engines” department of the Bauman Moscow State Technical University. Author of about 200 publications including 11 educational books and monographs in the ﬁeld of engine engineering. S. A. Kalinin (b. 1951) graduated from the Lomonosov Moscow State University in 1973. Ph. D. (Biology), assoc. professor of “Biomedical Technical Systems and Devices” Department of the Bauman Moscow State Technical University. Author of 35 publications in the ﬁeld of mathematical modelling and control in biological, medical, and bioengineering systems. A.V. Sharaburin (b. 1976) graduated from the Bauman Moscow State Technical University in 2000. Ph. D. (Eng.), senior researcher of “Refrigeration, Cryogenics, Air Conditioning and Life Support Systems” department of the Bauman Moscow State Technical University. Author of 4 publications in the ﬁeld of refrigeration technology. A.A. Aleksandrov (b. 1951) graduated from the Bauman Moscow Higher Technical School in 1975. Ph. D. (Eng.), assoc. Professor, director of the Experimental Factory of the Bauman Moscow State Technical University. Winner of the Prize of Government of the Russian Federation. Author of 2 monographs, 2 patents and 10 publications in the ﬁeld of safety of life support.

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INTERNATIONAL COLLABORATION OF BMSTU WITH UNIVERSITIES

BMSTU teaches about 500 international students, post-graduates and trainees from 25 countries throughout the world. So it makes special arrangements to provide support and assistance for them. Actually, BMSTU has agreements for cooperation with other one hundred universities from Argentina, Azerbaijan, America, UK, Hungary, Vietnam, Germany, Greece, Egypt, Italy, Canada, The People’s Republic of China, Columbia, Malaysia, Mexico, Mongolia, The Netherlands, Norway, Peru, The Republic of Korea, Singapore, Syria, the USA, Turkey, Ukraine, Finland, France, Chile, and Switzerland. International activities such as student and staff exchanges, joint education and research projects, conferences and seminars, joint development and publication of cooperative articles, textbooks and monographs play a signiﬁcant role. Membership in international associations of universities 1. T.I.M.E. (Top Industrial Managers for Europe) is the Association of engineering universities of Europe aimed at obtaining a double-degree by some selected students who are highly motivated to study at one of 43 partner universities of this Association for two years. On having completed a study programme at BMSTU, participants are awarded the engineering degrees of both home university and host one. 2. IGIP (International Geselschaft fur Ingenieurpadagogik) is an International Society for Engineering Education. Its objective is to elaborate teaching, scientiﬁc and organisational recommendations to train teaching staff for engineering universities to meet IGIP criteria for the title ”Teacher of European dimension for engineering higher education institution”. 3. IACE is an International Association of Counting Engineering Education aimed at developing and implementing international projects in distance learning. 4. INKORVUZ XXI is an International Co-ordinating Council of graduates of the education establishments. Its objectives are to select international citizens, while promoting study programmes of Russian universities to implement research projects in co-operation with international partners. 5. EAIE is a professional European Association for International education. Its goal is to stimulate and facilitate the internationalisation of education within the global perspective, in particular in higher education. EAIE organisers annual conferences, and training programmes for employees being in charge of international education, produces a variety of publications reﬂecting the interests and activities of its members.

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Editorial board: A.M. ARKHAROV, D. Sc. (Eng.), Professor V.O. GLADYSHEV, D. Sc. (Phys.-Math.), Professor K.Ye. DEMIKHOV , D. Sc. (Eng.), Professor V.V. DEVYATKOV, D. Sc. (Eng.), Professor Yu.I. DIMITRIENKO, D. Sc. (Eng.), Professor S.F. KONOVALOV, D. Sc. (Eng.), Professor V.A. MATVEEV, D. Sc. (Eng.), Professor A.N. MOROZOV, D. Sc. (Phys.-Math.), Professor B.P. NAZARENKO, D. Sc. (Eng.), Professor I.P. NORENKOV, D. Sc. (Eng.), Professor M.I. OSIPOV, D. Sc. (Eng.), Professor B.A. ROZANOV, D. Sc. (Eng.), Professor S.T. SURZHIKOV, D. Sc. (Phys.-Math.), Professor A.S. YUSHCHENKO, D. Sc. (Eng.), Professor

Manuscripts have been submitted in English and papers are published without any redaction

Proof-reader M.V. SAMOKHINA Art design S.S. VODCHITS, N.G. STOLYAROVA

Address of the Editorial Ofﬁce: Vestnik MGTU, 5, Vtoraya Baumanskaya ul., Moscow, 105005, RUSSIA Tel: (095) 263-62-60, 263-67-98, 267-63-49 Fax: (095) 265-42-98 E-mail: [email protected], [email protected] c Bauman Moscow State Technical University ° VESTNIK. Journal of the Bauman Moscow State Technical University, 2005

Wide-section journal for publications in theoretical and applied sciences

Printed by the Bauman MSTU Press

Natural Sciences & Engineering

CONTENTS Scientiﬁc Schools of the Bauman Moscow State Technical University (Devoted to the 175th anniversary of the BMSTU foundation) . . . . . . . . . . . . . .

3

Simulation of Processes & Systems H. Krier, S.T. Surzhikov. Computing Model of Burning of Bi-modal AP/HTPB Composite Solid Rockets Propellants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

V.T. Kalugin, A. Yu. Lutsenko, Ye.G. Stolyarova. Hysteresis in Aerodynamics of Flying Vehicles under Condition of Unsteady Movement . . . . . . . . . . . . . . . . .

31

Yu.I. Dimitrienko, M.L. Glazikov. Local Transport Phenomena in Porous Periodic Structures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

Control Systems A. Golubev, R. Johansson, A. Robertsson, S. Tkachev. Output Tracking for a Class of Nonlinear Nonminimum-phase Systems Using Observer Backstepping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

Laser & Optic-Electronic Systems V.N. Rozhdestvin, O.A. Smirnova. Combined Q-switch and Mode-lock in Laser Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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V.Ye. Karasik. Laser Range-gated Imaging System . . . . . . . . . . . . . . . . . . . . . . . .

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Navigational & Gyroscopic Systems V.A. Matveev and M.A. Basarab. Numerical Modeling of Heat Diffusion Processes in the Solid-state Wave Gyro Resonator by the R-function Method . . . .

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B.S. Konovalov, S.F. Konovalov, A.V. Kuleshov, D.V. Mayorov, V.P. Podtchezertsev, V.V. Fateev. New Types of Vibrating Gyro for Rotating Carrier . . . . . . .

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Information & Computation Technology V.V. Devyatkov. Multiagent Hierarchical Recognition on the Basis of Fuzzy Situational Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I. Balepin, S. Maltsev, J. Rowe, K. Levitt. Using Speciﬁcation-based Intrusion Detection for Automated Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Physics S.M. Korotaev, A.N. Morozov, V.O. Serdyuk, Yu.V. Gorokhov, S.A. Pulinets, V.I. Nalivayko, A.V. Novysh, S.P. Gaidash, H.D. Kanonidy. Manifestation of Macroscopic Nonlocality in the Processes of Solar and Geomagnetic Activity M.A. Yakovlev. Generation of the Near-surface Electron Layer by Picosecond Laser Pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V.O. Gladyshev, T.M. Gladysheva, A.N. Morozov, V.E. Zubarev. The Results of the Experiment on Registrating Light Dragging Observed in an Interferometer with a Rotating Optical Disc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M.C. Duffy, V.O. Gladyshev, A.N. Morozov, P. Rowlands Physical Interpretations of Relativity Theory (On the materials of International Scientiﬁc Conference in Bauman Moscow State Technical University, 30th June – 3rd July, 2003) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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186

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Radio-Physics & Radiolocation B. Rozanov, S. Solomonov, A. Zrazhevsky, V. Nagnibeda, Ye. Kropotkina, S. Rozanov, N. Zharkova, T. Lebedjuk, I. Fetisov, M. Loukicheva. The Atmospheric and Solar Investigations at Millimeter Waves . . . . . . . . . . . . . . . . . . . . . . .

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Ecology Problems N.P. Demenkov, V.A. Matveev. Fuzzy Systems of Ecological Monitoring and Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Fundamental Problems of Mechanical Engineering Yu.N. Drozdov, Ye.G. Yudin. Wear Prediction Considering Mechanical Physical-chemical and Geometrical Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Yu.N. Drozdov, Ye.G. Yudin. Tribological Problems of Mechanical System Creation for the Moon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Production Process Procedures & Machines A.M. Dmitriev, A.L. Vorontsov. Analysis of Extrusion Process of Coreless Cylindrical Products with an External Core in the Bottom . . . . . . . . . . . . . . . . . .

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I.N. Shiganov. Fusion Welding of Metallic Composite Materials . . . . . . . . . . . .

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Yu. Bocharov, Yu. Gladkov. Integrated Control-Monitoring-Diagnosis System Development in Technology of Plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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S. Golovashchenko, V. Kondratenko, A. Vlasov. Experimental Investigations of Flanging Aluminium Panels as the First Stage of Hemming Process . . . . . . .

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Cryogenic Engineering & Technology A.M. Arkharov, S.D. Glukhov, L.V. Grekhov, A.A. Zherdev, N.A. Ivashchenko, D.N. Kalinin, A.V. Sharaburin, A.A. Aleksandrov. Creation of Harmless for Environment Engines and Installations Using Dimethyl Ether . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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SCIENTIFIC SCHOOLS OF THE BAUMAN MOSCOW STATE TECHNICAL UNIVERSITY (Devoted to the 175th anniversary of the BMSTU foundation)

In 2005 the Bauman Moscow State Technical University (former Bauman Moscow Higher Technical School) will mark by celebration the jubilee — 175th anniversary of its foundation. The history of BMSTU as an oldest technical higher educational establishment of the country is closely connected with the history of Russia, the development of its economy, science, technology, engineering, the Russian Army. The University has sprung from the vocational school. In 1826 the widowed empress Mariya Fyodorovna “deigned to royally enjoin that large workshops of various handicrafts be established” for orphan boys of the Foundling Hospital. With this purpose, the Slobodskoy Palace in the German Suburb of Moscow (Nemetskaya Sloboda), having suffered from ﬁre in 1812, was reconstructed by the famous Moscow architect D.I. Zhilyardi. The building has acquired an appearance inherent in the late Moscow Empire-style as we can see it today. It has been decorated with an attic, representing the multi-ﬁgure sculptural group “Minerva” (by sculptor I.P. Vitali), that symbolizes scientiﬁc achievements and practical craftsman skills. On 1st July 1830, the emperor Nicholas I signed the Edict to open the Moscow Vocational School (MVS) for training “craftsmen in theoretical knowledge, serving for modernization of crafts and factory works” and approved the “Statute on the Vocational School”. (The existing BMSTU counts off its chronology since that year.) At the same time, two main trends in teaching alumni were determined: mechanical and chemical crafts. From the very beginning MVS became a center of the technical education in Russia, and the incipient home industry at once recognized a high level of training specialists in the new educational establishment. In 1868 the Regulations of the School were approved, in the ﬁrst section of which the following was written: “The Emperor’s Moscow Technical School (EMTS) is a higher special educational establishment having its main goal to educate civil engineers, mechanical engineers and technological engineers”. Early chairs in the school were those of higher mathematics, general mechanics, general and applied physics, machine building, construction art, technology of ﬁber substances, general chemistry, chemical technology. “Russian method” of training engineers. In XIX century, technical sciences and the higher education as a whole experienced the process of formation, which could not but have an immediate impact on shaping the learning process in the School. Everything depended, to a large measure, on people who worked within its precincts. The professorate and teaching staff of EMTS included such distinguished scientists as D.I. Mendeleev, N.Ye. Zhukovsky, P.L. Chebyshev, S.A. Chaplygin, A.S. Yershov, D.K. Sovetkin, A.V. Letnikov. An immense prestige of the professorate and teaching staff and a high level of training specialists put EMTS into a line of leading European polytechnic schools. An orderly system of professional teaching of future specialists that was accepted in EMTS — so-called “Russian method VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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of training engineers” — gained recognition all over the world. The method became well-known, especially after its successful demonstrations at international exhibitions: in Vienna (1873), where it was awarded the Big Golden Medal, and also in Philadelphia (1878, 1900), where the method was declared the best. With advances in engineering the “Russian method” of the nineteen seventies was progressively upgraded. By 1903 EMTS was generally recognized as the best higher educational establishment for mechanical engineering in Russia. It had become clear by that time that only scientiﬁcally formulated experiments conducted in a laboratory equipped with up-to-date apparatus could promote the technology development. Later, some world higher educational establishments were created according to that pattern. So, the President of the famous Massachusetts Institute of Technology J. Rounkle, having received from EMTS the collection of models, specially made for the American Institute, to teach engineers by the “Russian method”, wrote to the EMTS rector V.K. Della-Vos: “Russia is recognized to be completely successful in solving so important problem of the technical education. . . . After this, no other system will be applied in America”. He also published a small brochure titled “Russian Method of Professional Teaching of Engineers and Mechanics”. Development of scientiﬁc ties. Mutual enrichment of different systems of the technical education proceeded. International exhibitions, congresses, meetings attended by EMTS representatives contributed to expansion of scientiﬁc ties, enchanced the prestige of EMTS. Business-like relations with the largest foreign companies began taking shape; no less important were ties with Russian business circles, whose representatives were aware of higher educational establishments being centers of the contemporary science and technology. Scientiﬁc societies, proceedings and other mutual initiatives strengthened and enriched relations among scientists. Creating the Polytechnic Exhibition in 1872 became one of such initiatives, which gave birth to the known-to-everyone Polytechnic Museum. The Polytechnic Society, established by EMTS in 1877, only consisted of EMTS alumni and held a proper position in the line of Russian scientiﬁc and technical institutions. “Society for Contribution to Successes of Experimental Sciences and their Practical Applications” named after Kh.S. Ledentsov was created by the EMTS professorate and teaching staff and yielded inestimable fruit. It subsidized talented inventors and provided funds for equipping laboratories, including those having become early research institutes in Russia: the physical laboratory headed by P.N. Lebedev, I.P. Pavlov’s laboratory for psycho-physiological research, the aerodynamic laboratory headed by N.Ye. Zhukovsky, etc. This was the only society in Russia, performing the work similar to activity of Carnegie Fund or the German Society n.a. Caesar William. Turn for polytechnic aspects. In early XX century the world situation sharpened problems of the technical progress and as a consequence — problems of the Russian technical education. In 1915 the then EMTS rector V.I. Grinevetsky, wellknown scientist in thermal engineering, suggested the “Draft on EMTS Transformation to School of a Polytechnic Type”. He believed that “engineering education should develop in two ways. On the one hand, the specialization of training should

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increase, on the other hand — the interaction and close collaboration of various professions should be strengthened. Only a polytechnic type school can satisfy both requirements given it has a fairy ﬂexible management.” With this, V.I. Grinevetsky was sure that in a contemporary higher educational establishment the “encyclopedic learning — broad engineering scope” had to be connected with a “certain specialization”. Attention should be also paid to another thought, uttered by him, about the necessity of the economical education of an engineer. Today it is especially important to appreciate it. Such a profound, conﬁrmed with examples and statistics, view on the further development of the School was supported by the EMTS professorate and teaching staff, and then put into practice. Scientiﬁc schools in early XX century. At the XIX–XX century border the signiﬁcance of EMTS in the scientiﬁc and industrial life of the country was increasing. There were formed in EMTS fundamental scientiﬁc schools, called into being by the intensive industry growth and by enlistment in the School of top scientists, mainly, Moscow University alumni. The ﬁeld of theoretical mechanics and aeromechanics was presented by “Father of the Russian Aviation” — N.Ye. Zhukovsky with his disciples S.A. Chaplygin, B.N. Yuriev, V.P. Vetchinkin. They developed fundamentals of aerodynamics, laid a scientiﬁc base for engineering design in aircraft building. The Central Aerodynamic and Hydrodynamic Institute and Air Force Engineering Academy were established with their active participation. The ﬁeld of physics was presented by P.N. Lebedev, P.P. Lazarev, V.S. Shcheglyaev, S.I. Vavilov. The scientists highly contributed to the development of photometry, luminescence, roentgenoscopy, to the research in wireless telegraphy. Owing to work of A.S. Yershov, P.L. Chebyshev, N.Ye. Zhukovsky, N.I. Mertsalov the home science — theory of mechanisms and machines — was born. Studies of D.N. Lebedev, A.I. Sidorov, P.K. Khudyakov laid a foundation for the development of computations in strength of materials and parts of machines. M.M. Cherepashinsky, N.S. Streletsky, P.A. Velikhov were founders of methods of computation and design of building structures. Thermal engineering was formed by activities of V.I. Grinevetsky who developed methods for designing boilers and thermal processes in engines of internal combustion. He also became an initiator of creating diesel locomotives. In the ﬁeld of electrical engineering it is necessary to highlight works performed by K.I. Shenfer, K.A. Krug, B.I. Ugrimov and devoted to high voltages of electric systems, multi-phase motors, facilities for transferring energy over large distances. Later, the School’s scientists took the most active part in the development of GOELRO plan. Studies, conducted by A.Ye. Chichibabin, S.A. Fyodorov, Ya.Ya. Nikitinsky, N.A. Shilov and others in the ﬁeld of chemistry and chemical technology, technology of ﬁber, food, explosive and pharmaceutical substances, were of great importance. Transformations from EMTS to BMHTS. After 1917 the School experienced a number of transformations. Its name changed for Moscow Higher Technical School (MHTS). During the period of 1920–1930 some individual higher educational establishments and research institutes were extracted from

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it. Those were: aerohydrodynamic, automotive, chemical-technological laboratories, on which basis were established later the following institutes: the Central Aerodynamic and Hydrodynamic Institute (“TsAGI”), All-union Institute for Aviation Materials (“VIAM”), Central Institute for Aviation Engines (“TsIAM”), Scientiﬁc Institute for Automotive Engines (“NAMI”) and a number of others. The most serious transformations of the School occurred in 1930, when on its base some technical higher educational establishments were organized which later became the largest educational institutions: Moscow Aviation Institute (“MAI”), Moscow Institute for Power Engineering (“MEI”), Moscow Institute for Building Engineering (“MISI”), Academy for Chemical Protection and some others. The mechanical faculty, remained in the original MHTS building, was given a new denomination — Moscow Mechanical Machine-building Institute (MMMI). In December 1930 it was also named after N.E. Bauman (MMMI n.a. N.E. Bauman). Then, in 1943, its former denomination — MHTS — was restored. The renewed name signiﬁed not only the recognition of the Moscow Higher Technical School n.a. N.E. Bauman (BMHTS) as a descendant of the famous Russian engineering school but also its peculiar status in the system of the higher technical education of the country. As early as before the 1917 Revolution the Industrial School was established on the initiative of the EMTS teachers; after the Revolution its name became Moscow Chemical-Technological Institute (“MkhTI”) n.a. D.I. Mendeleev. In 1918 the MHTS-based Aviation Technical Secondary School was transformed into the Academy for Air Force Engineers. Based on the EMTS laboratory of ﬁlament technology, some scientiﬁc trends in Textile Institute began to progress. A number of the MHTS chairs moved into the Institute of Food Industry Technology, Institute of Chemical Engineering, Electric Engineering Institute for Communications, Building Engineering Academy. In 1921, the All-Union Thermal-Technical Institute n.a. V.I. Grinevetsky and K.V. Kirsh was established around the thermal-technical laboratory. The All-Union Electrical Engineering Institute was founded the same year on the initiative of the MHTS professors. In the nineteen thirties the Central Research Institute for Ferrous Metallurgy (“TsNIIChERMET”) n.a. I.P. Bardin was created on the basis of the MHTS laboratory of rolling. In the nineteenth forties, the physical engineering faculty moved into the just established Moscow Engineering Physical Institute (“MIFI”). The present-day Vladimir Polytechnic Institute was founded in XIX century on the initiative of EMTS teachers in town Vladimir as the Vocational Maltsev’s school. The Izhevsk Mechanical Institute emerged in 1943 (during the World War II) due to activities of BMHTS professors and teachers, evacuated to town Izhevsk. Those multiple transformations had no essential effect on the BMHTS scientiﬁc schools life. General science and technology chairs even though suffered from leaving of a number of leading scientists, but rapidly gained strength thanks to the talented young people who came to succeed their teachers and inherited all the

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best from them. New chairs became headed by MHTS alumni, keepers of its traditions, acute for the novelty, directed towards the future. Physical and mathematical sciences made further progress through activities of S.I. Vavilov, S.P. Finikov, A.S. Predvoditelev, A.P. Kotelnikov et al. Studies, performed by N.R. Briling, Ye.K. Mazing, A.S. Orlin, G.G. Kalish, V.Ye. Tsydzik, S.Ya. Gersh, G.F. Knorre, L.K. Ramzin, V.V. Uvarov, promoted the scientiﬁc schools development in the ﬁeld of power engineering. I.I. Kukolevsky and S.S. Rudnev created the school of hydromechanical engineering; Ye.A. Chudakov, M.K. Kristi — the school of wheeled and caterpillar vehicles; L.G. Kifer — the school of hoisting transportation machines. In the ﬁeld of technology of mechanical engineering and standardization the following scientists fruitfully worked: E.A. Satel, M.A. Saverin, V.M. Kovan. The professors K.K. Khrenov, N.N. Rykalin, G.A. Nikolaev founded the school of welding practice; A.M. Bochvar, I.I. Sidorin — the school of science of materials; N.N. Rubtsov and N.P. Aksyonov — the school of casting practice; A.I. Tselikov, A.I. Zimin, M.V. Storozhev — the school of non-cutting shaping of materials. Instrumental engineering schools developed with fast paces: precise instrumental engineering (F.V. Drozdov, S.O. Dobrogursky); optical engineering (S.I. Freiberg, I.A. Turygin); gyroscopy (S.S. Tikhmenyov, B.V. Bulgakov); radioelectronics (M.V. Shuleikin, A.M. Kugushev); automatic control device engineering (V.V. Solodovnikov). In 1938 three new faculties were set up: tank, artillery and ammunition engineering. Among the ﬁrst professors — the chairs organizers — there were M.K. Kristi, A.G. Gorst, K.P. Stanyukovich, A.A. Tolochkov, V.Ye. Slukhotsky. In 1948 a faculty of rocket technology was created, with which the activities in BMHTS of the rocket engineering founders — S.P. Korolyov, Yu.A. Pobedonostsev, V.I. Feodosiev, V.N. Chelomey, V.P. Barmin — were connected. BMHTS museum. It is impossible to enumerate here all noteworthy scientists names, to describe their contribution to the formation and development of the present-day Bauman Moscow State Technical University. One can become familiar in detail with the history of the university scientiﬁc schools development in the BMHTS museum, that was set up in 1967 and for years of its existence it became one of the most known museums of higher educational establishments in Russia. The main fund of museum objects consists of more than 1000 items. The museum book storage includes more than 3000 volumes: from XVIII century folios to ﬁrst home monographs and textbooks on thermal engineering, automotive engineering, cybernetics, rocket engineering, robotics. The museum closely collaborates with the Museum of History of Moscow and Polytechnic Museum and also with many museums of higher educational establishments in our country; it is engaged in many enlightenment activities. Educational-experimental center. No less unique exposition — engineering exhibition is presented in the BMSTU Branch — Educational-Experimental Center, situated at 87 km from Moscow, in the settlement Orevo, Dmitrovsky region of Moscow oblast. The Center construction was initiated in 1960 and it was dictated by the necessity to withdraw some laboratories from Moscow, because they could

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not be accomodated within the main BMHTS building. The Center has become the ﬁrst of such structural divisions not only in BMHTS, but in the whole Higher School of the country as well. Today it remains single of the kind due to the unique equipment it has and the research being conducted there. In one of its laboratories, for example, the unique exposition of samples of the rocket and space engineering was managed to collect, that has no analog not only in higher educational establishments but also in rocket and space centers of the country. However, the most valuable is the fact that these articles serve not only as museum exhibits, but also as acting laboratory test-benches, which are permanently used for educational purposes. State appreciation. The state highly appreciated the contribution of the BMHTS collective to formation and development of new trends in science, engineering and contemporary technologies, to creation of scientiﬁc and pedagogical schools, to training of highly skilled specialists: — in XIX century the status of Emperor’s was conferred on the School; — in 1933, according to the Decree of the Central Executive Committee of the USSR, the Bauman Moscow Mechanical and Machine-building Institute was awarded the Order of Labor Red Banner for great merits in socialism construction; — in 1955, according to the Edict of The Presidium of the Supreme Soviet of the USSR, the Bauman Moscow Higher Technical School was awarded the Order of Lenin in connection with the 125th anniversary of its foundation and for merits in development of science and technology and in training of highly skilled specialists; — in 1980, according to the Edict of The Presidium of the Supreme Soviet of the USSR, the Bauman Moscow Higher Technical School was awarded the Order of October Revolution in connection with the 150th anniversary of its foundation and for merits in scientiﬁc activity and in training of highly skilled specialists; By mutual decision of the Central Committee of the CPSU and Council of Ministers of the USSR No. 452 of 17th April 1987, the necessity was recognized to set up the higher educational establishment of a new type on the basis of BMHTS, entrusting it with training specialists according to the principles, which combine the advanced forms of the fundamental university’s education with the engineeringand- technological one. Since then a structure of the higher educational establishment has changed and remains the same up till now. By the Order of the USSR State Committee on People’s Education No. 617 of 27th July 1989 the Bauman Moscow Higher Technical School was transformed into the Bauman Moscow State Technical University (BMSTU). Thus, the organization of technical universities, a number of which nowadays exceeds 100, has been initiated in Russia. 8

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On 5th April 1993 the Decree of the Russian Federation Government was signed on the further BMSTU development, which provided some measures on sustaining and advancing the University’s intellectual potential. On 24th January 1995 by the Edict No. 64 of the Russian Federation President the BMSTU was involved in the State Fund of Greatly Valuable Objects of Cultural Heritage of Russian Federation Peoples. In 2000 the gratitude of the Russian Federation President was conferred on the University for the great contribution to the development of the higher education and to training of highly skilled specialists. The BMSTU has trained and graduated many thousands of high class engineers. It is practically impossible to name a ﬁeld in engineering and scientiﬁc activity where the BMSTU alumni would not do justice to themselves. It is also impossible within the framework of this paper to enumerate all outstanding BMSTU graduates. The following eminent organizers of industry and science are among them: S.A. Afanasiev, B.L. Vannikov, V.A. Malyshev; prominent scientists and designers: V.P. Barmin, A.A. Bochvar, N.A. Dollezhal, V.Ya. Klimov, S.P. Korolyov, S.A. Lebedev, V.M. Myasishchev, P.O. Sukhoy, A.N. Tupolev, Ye.A. Chudakov, A.I. Tselikov, V.G. Shukhov and many others. BMSTU today. In today’s BMSTU there are 18 thousand students of exclusively full form of learning and more than 4500 professors and teachers including 450 doctors of science (D. Sc.) and about 3000 candidates of science (Ph. D.). Eminent scientists from research institutes of the Russian Academy of Sciences and other Russian academies, from branch research institutions and international visiting scientists deliver lectures in BMSTU. About 1000 post-graduates of 86 scientiﬁc professions are taught and 26 dissertation councils, covering 62 scientiﬁc professions, function here. BMSTU has licenses for training bachelors and masters by 26 professions and certiﬁed specialists by 75 professions of a practically full set of ﬁelds of the contemporary mechanical and instrumental engineering. The BMSTU structure has no analog among technical universities of Russia and, in our opinion, it approximates most closely to that of a university of the future — the research university. It is destined to provide the integration of the fundamental theoretical training with a practical activity of future specialists that is connected with solving urgent scientiﬁc and engineering problems, with performing design and experimental work in research institutes and in BMSTU departments. The BMSTU structure also offers an opportunity to the professorate and teaching staff to accomplish the research work inside the university in a close cooperation with specialists of industrial institutions. The professional training of BMSTU students is based on the principles, combining advanced methods of fundamental university’s and engineering education, and is realized in seven research and education units (REU), which include educational faculties and the proper research institutes with design bureaus: • REU “Manufacturing Engineering” (Faculty “Manufacturing Engineering” and Research Institute “Structural Materials and Manufacturing Processes”);

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• REU “Information Technology and Control Systems” (Faculty “Information Technology and Control Systems”, “Instrumental Engineering” Faculty and Research Institute “Information Technology and Control Systems”); • REU “Radio-electronics, Laser and Medical Technology” (Faculty “Radioelectronics and Laser Technology”, Faculty “Biomedical Engineering”, Faculty “Optoelectronic Device Engineering”, “Radio Engineering” Faculty and Research Institute “Radio-electronics and Laser Technology”, Research Institute “Biomedical Engineering”); • REU “Robotics and Integrated Automation” (Faculty “Robotics and Integrated Automation” and Research Institute “Manufacturing Processes Automation”); • REU “Power Engineering” (Faculty “Power Engineering”, Faculty “Rocket and Space Engineering” and Research Institute “Power Engineering”); • REU “Special-purpose Mechanical Engineering” (Faculty “Special-purpose Mechanical Engineering”, “Aerospace Engineering” Faculty and Research Institute “Special-purpose Mechanical Engineering”); • REU “Fundamental Sciences” (Faculty “Fundamental Sciences”, “Linguistic” Faculty, Research Institute “Applied Mathematics and Mechanics”). A REU provides the theoretical education of future specialists in fundamental and professional disciplines as well as the practical training of students by their involvement in solving concrete engineering problems. Research institutes, included in REUs, are under the methodical supervision of the proper divisions of the Russian Academy of Sciences. The economical instruction of all students is performed by the Faculty “Engineering Business and Management” which also trains engineers (managers) in the ﬁeld of high technologies. The Faculty “Humanitarian and Social Sciences” provides education of students in humanitarian and social sciences. Besides, BMSTU has Faculty of Sports and Rehabilitation and Faculty of Military Training. BMSTU is one of fewer higher educational establishments in Russia with a six year term of training of certiﬁed specialists. This term allows the more profound fundamental knowledge and professional skills to be acquired by future graduates. Those students, who have successfully completed the basic (six year) course of training, may get the more thorough instruction (seventh year of training) to improve their professional skills or to prepare for the further study of the post-graduate course. Altogether in the University there are more than 100 chairs, annually graduating more than 2000 highly skilled specialists. The full form training by 16 basic professions is provided by BMSTU Kaluga Branch which has commemorated the 45th anniversary of the foundation. Students of the University have an opportunity to get additional educational services, e.g. another profession or specialization

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(conﬁrmed by a diploma), on the contract basis. The BMSTU structure also includes the Institute for Improvement and Change of Professional Skills of Skilled Personnel and the Experimental Institute for Improvement of Skills of Teachers. Since 1934 BMSTU has been engaged in teaching of hard-of-hearing students and is the world’s ﬁrst educational establishment of a general type, where professional programs of the engineering education for this category of disabled people are successfully implemented and efﬁciently keep developing. It has become the largest education institution in Russia where training of highly skilled specialists is realized for hard-of-hearing citizens; nowadays many of them are prominent engineers and scientists. The unique system, developed in BMSTU, includes both the integrated continuous multi-level professional education, variative in a form, and the complex rehabilitation, accompanying the learning process. The suggested model of the innovation education, being integrated with the social policy and meeting the best world standards, may serve the basis for creating the federal system of the higher engineering education for disabled people. For many years BMSTU has been working systematically to organize a broad net of educational institutions within the framework of under-graduate activity. The net includes: more than 200 schools (secondary, lyceum and gymnasium type) of Moscow and Moscow oblast and other regions of Russia; Russian youth programs “Step into the Future” and “Cosmonautics”; multiple preparatory courses, various olympiads etc. For 12 years the Physical and Mathematical Lyceum has been successfully functioning at BMSTU. The Lyceum annually graduates about 300 future students for the University. The worked-through system is based on the all-year cycle and includes various forms of professional orientation and knowledge check of university entrants. Its main goals are as follows: creating optimal conditions to future BMSTU students for the high-quality completion of the secondary education process, preparation for entering the University and adaptation for further study there; attracting the talented youth; selecting the more prepared and professionally oriented university entrants. BMSTU takes an active part in accomplishing many scientiﬁc and technical programs of the Russian Federation Ministry for Education and Science and other ministries and organizations and BMSTU is frequently a leading executor of them. On University’s base the Teaching and Methodical Association on Polytechnic University’s Education functions, as well as 12 products-certiﬁcation centers, many other divisions, supporting close ties between the University and industrial and scientiﬁc institutions. The purposeful and fruitful work to create a strategy of developing the engineering education, a methodical support of the teaching process is carried out in BMSTU. Late in the nineteen nineties the Association of Technical Universities, headed by BMSTU, developed a concept of a contemporary technical university, the proper classiﬁer and educational standards. The BMSTU Press successfully functions in BMSTU. It publishes a periodical bulletin “Vestnik MGTU imeni N.E. Baumana” (in three series: Mechanical Engineering, Instrumental Engineering and Natural Sciences). Publishing series

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of textbooks, for example, “Mechanics in Technical University”, “Informatics in Technical University”, etc. is characteristic of BMSTU. The BMSTU technical library, having more than 3 million volumes in its scientiﬁc book-stock, is unique. The fund of the rare books storage is of exclusive value and includes volumes of XVIII–XX centuries, e.g. dissertation by D.I. Mendeleev, scientiﬁc papers by N.Ye. Zhukovsky and many others. BMSTU maintains and further develops international relations with practically all leading world technical universities. About 250 BMSTU students and postgraduates annually learn courses and work on probation in leading world universities. BMSTU scientiﬁc developments are successfully implemented by the largest companies of the USA, China, Republic of Korea, Japan, Canada, Great Britain and other countries. Russian engineers, who have graduated from BMSTU, are considered among the best in the world. Rationally combining the fundamental and special technical education and having a high level of the practical professional training, they successfully withstand a severe competition in intellectual labor markets of our country and abroad. The BMSTU development is inseparably linked with an inexhaustible “supply of strength” — above all, with an intellectual potential, owing to which for many dozens of years BMSTU has been holding its positions of a leader of our higher technical school in science, engineering and manufacturing procedures and process. A combination of an accurate engineering design with an engineer’s intuition, careful observance of traditions of graduating engineers in accordance with the “Russian method”, acute feeling for a novelty of scientiﬁc trends, complex social and economical approach to solving the most complicated technical problems, fundamental and humanitarian aspects of training specialists — all allow BMSTU to keep remaining on the edge of the world scientiﬁc and engineering progress, to head the Community of Russian Technical Universities and the whole higher engineering education in Russia. Commemorating the important jubilee — the 175th anniversary of the BMSTU foundation, we may wish the University to continue and develop the many years glorious traditions of leading scientiﬁc schools, arisen within its precincts and recognized by the world; to improve the training of highly skilled specialists — BMSTU alumni; to keep for oneself the high status of a leader of the Russian university’s engineering education. Editor-in-chief, Rector of BMSTU, Corresponding Member of the Russian Academy of Sciences I.B. Fedorov

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VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

SIMULATION OF PROCESSES & SYSTEMS Herman Krier (Center for Simulation of Advanced Rockets University of Illinois at Urbana-Champaign Urbana, IL 61801, USA), Sergey T. Surzhikov (Institute for Problems in Mechanics of Russian Academy of Sciences — Bauman Moscow State Technical University)

COMPUTING MODEL OF BURNING OF BI-MODAL AP/HTPB COMPOSITE SOLID ROCKETS PROPELLANTS A numerical simulation model of thermo-gas-dynamic processes near to a burning surface of heterogeneous solid rocket propellants on the base of AP and HTPB is represented, including the heat transfer processes inside a solid phase of these materials. The model takes into account the gas phase chemical reactions within the framework of the concept of the global chemical reactions. Two global chemical reactions for ﬁve effective chemical components are accepted in the model.

Introduction. Composite solid propellants are a heterogeneous mix of crystalline oxidizer and polymer binder, plus other burning rate additives. A long-range goal is to eventually model the combustion of such heterogeneous material to resolve ﬂame characteristics that range from 5 to 500 µm, at rocket motor pressures ranged from 40 to 100 atm. A two-dimensional model for the unsteady burning of idealized mixtures of oxidizer (ammonium perchlorate, AP) and binder (hydroxyl-terminated polybutadiene, HTPB) was developed previously [1]. That study included the modeling by a reduced scheme of the chemistry of AP and binder combustion. Both the gas and solid phase concentration relations were solved, assuming a 2D planar geometry. The energy balances between phases and a pyrolysis rate relation allowed for the coupling and the prediction of the AP, binder, and mixture burning rates. The work reported in [1] showed how the gas phase micro-ﬂame structure supplied the local heat ﬂuxes which, in turn, determined to local regression rate, but always for a periodic “sandwich” of AP and binder. Figures 1–3 are typical results from the 2D model [1]. Since composite propellants are heterogeneous in all three dimensions, the work presented below is the next-step toward such simulations. Instead of regular 2D “sandwich” conﬁguration, we now consider the mixture of AP/Binder to be un-even, as depicted in Figure 4, a, to simulate a bi-modal oxidizer mixture. Mathematical Formulation of the Problem. The numerical simulation model includes groups of equations for describing mass, energy and species concentration transfer processes in the gas and solid phases of heterogeneous propellants.

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Fig. 1. Gas-phase temperature and velocity distributions at p = 40 atm and ϕ = 0.75 0.75. BDP-pyrolysis model, 8-component kinetics model of gas phase. Full 2D solid phase heat conduction model

Fig. 2. Distributions of convective heat ﬂux along burning surface at p = 40 atm and ϕ = 0.75 0.75. BDP-pyrolysis model, 8component kinetics model of gas phase. Full 2D solid phase heat conduction model

Fig. 3. Burning rate distribution at p = 60 atm and ϕ = 0.75 0.75. BDPpyrolysis model, 8-th-component kinetics model of gas phase. Full 2D solid phase heat conduction model

For solving these equations, a Cartesian coordinate system (Fig. 4) is used. The calculated domain (a − b − c − d) is shown here as a physical region which includes solid oxidizer (AP; a−e−y1 −g; y2 −f −d−h), solid binder (HTPB; y1 −y2 −h−g), gas phase region (e − b − c − f ), and also the symmetry planes (a − b; c − d).

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VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

Fig. 4. Schematic of the problem where two different rectangular AP oxidizers bounded the HTPB binder (aa ). Calculation domain (bb). According to Fig. 4, a pieces of the AP (aa − e − y1 − g and h − y2 − f − dd) have different depths

Gas phase. The gas phase description is based on the system of mass conservation, the Navier-Stokes equations, species conservation equations for gas mixture and for each chemical component, and the energy transfer equation for a chemically reacting mixture. The non-dimensional gas-phase relations were presented in [1], and are: ∂ρ + div(ρV) = 0, (1) ∂t ∂ρu ∂p + div(ρuV) = − + ∂t½ ∂x · µ ¶¸ µ ¶¾ 2 ∂ ∂ ∂u ∂v ∂ ∂u 1 ρ + − (µdivV) + µ + +2 µ − 2 , (2) 3 ∂x ∂y ∂y ∂x ∂x ∂x Re Fr ∂ρv ∂p + div(ρvV) = − + ∂t ∂y ¾ ½ ∂ ∂u ∂v ∂ ∂v 2 ∂ 1 (µdivV) + [µ( + )] + 2 (µ ) , (3) + − 3 ∂y ∂x ∂y ∂x ∂y ∂y Re ∂Yk + ρVgradYk = ∂t L 1 = , div(ρDk gradYk ) − $k Re Sc ρ0 u 0

ρ

ρcp

k = 1, 2, . . . , Nc

(4)

∂T + ρcp VgradT = ∂t N

=

c 1 1 X cp,k ρDk (gradYk · gradT ) + div(λgradT ) + Pr Re Re Sc k µ ¶ Nc X L + , hk $ k ρ0 cp,0 u0 T0

(5)

k

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where x, y are the longitudinal and transversal coordinates; u and v are the projections of the ﬂow speed V on the x and y axes; ρ, p, cp , µ, λ are the density, pressure, speciﬁc heat at constant pressure, viscosity, and thermal conductivity respectively; T is the temperature; cp,k is the speciﬁc heat capacity at constant pressure for the k-th component; Yk = ρk /ρ is the mass fraction of the k-th component; ρk is the density of the k-th component; Dk is the effective diffusion coefﬁcient of the k-th component; hk , $k are the enthalpy and reaction rate of the k-th species (Here: reaction rate means value of mass consumed or produced per unit volume per unit time, e.g. g/(cm3 ·sec)); Nc is the number of chemical components. The variables (x, y, t), (u, v), (ρ, p, T, cp , µ, λ) in the system of Eqs. (1)–(5) are nondimensional. To non-dimensionalize the variables, the following parameters were used: ρ0 , µ0 , L0 , u0 , where the “0” index is used from here on to denote the parameters at known ﬁxed conditions: (x, r) =

T =

T˜ , T0

µ=

(˜ x, r˜) , L0 µ ˜ , µ0

t=

λ=

t˜u0 , L0

˜ λ , λ0

(u, v) =

D=

˜ D , D0

(˜ u, v˜) , u0 cp =

ρ=

c˜p , cp,0

ρ˜ , ρ0 p=

p˜ . ρ0 u20

ρ 0 u0 L 0 is the Following criteria of the heat and mass transfer are used: Re = µ0 µ0 cp,0 µ0 Reynolds number, Pr = is the Prandtl number, Sc = is the Schmidt λ0 ρ 0 D0 uo number, Fr = √ is the Froude number, where g is the gravitational acceleragL0 tion. The following normalization parameters were used: u0 = 10 cm/sec, L0 = = 0.01 cm, ρ0 = 1.177 · 10−3 g/cm3 , µ0 = 1.983 · 10−4 g/(cm·sec), λ0 = 2.624 · λ0 µ0 · 10−4 W/(cm·K), T0 = 300 K, cp,0 = , D0 = . µ0 ρ0 The chemical kinetics model. A concept of the global chemical reactions for burning of heterogeneous solid propellants was discussed in [1, 2]. A 5-component model for two global chemical reactions is used for describing heat release processes in the gas phase k 1 a1,1 [FA ] + a1,2 [Ox] −→ b1,1 [P1 ],

(6a)

k 2 b2,1 [P2 ]. a2,1 [FB ] + a2,2 [P1 ] −→

(6b)

Correspondence between the conventional and real names of the chemical components is listed in Table 1. The global chemical reactions constants are expressed by the Arrhenius law as µ ¶ El kl (T ) = Al exp − , l = 1, 2, (7) R0 T where El is the activation energy; Al is the pre-exponent factor (see Table 2); R0 is the universal gas constant, R0 = 1.9858 cal/(mol·K). 16

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

Table 1 Chemical components taken into account in the 5-component numerical simulation model 1

2

3

4

5

Conventional names of the effective chemical components

FA

Ox

P1

FB

P2

Names of the effective chemical components

NH3

HClO4

P1

C4 H6

P2

(Yk,w )AP

0.05

0.04

0.91

0.0

0.0

(Yk,w )HTPB

0.0

0.0

0.0

0.3

0.7

Number of species

Table 2 Suggested input for the global chemical reactions constants

Number of the reaction

Al , (cm3 · sec)/mol

El , cal/mol

13

1

3.6 · 10

15236 11

1.8 · 10

2

14955

In this case the global reactions rates are expressed as, ¶µ ¶ µ ρ2 ρ1 , ω˙ 1 = k1 (T ) W1 W2 µ ¶µ ¶ ρ4 ρ3 ω˙ 2 = k2 (T ) , W4 W3

(8a)

(8b)

where Wk is the molecular weight of the species k. Here, the dimension of the ω˙ k is mol/(cm3 ·sec). The species reaction rates are calculated using, $1 = −a1,1 W1 ω˙ 1 ,

$2 = −a1,2 W2 ω˙ 1 ,

$4 = −a2,1 W4 ω˙ 2 ,

$5 = +b2,1 W5 ω˙ 2 .

$3 = +b1,1 W3 ω˙ 1 − a2,2 W3 ω˙ 2 , (9)

where a dimension of all reaction rates is g/(cm3 ·sec). Thermodynamical and Transport Properties of the Gas Phase. For calculating the thermophysical and transport properties of the gas mixture, the following relations were used: R0 p=ρ T, (10) WΣ Nc Nc X X ρk , h = h k Yk , (11) ρ= k

k

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Nc X

1 cp =

cp,k Yk ,

WΣ =

.

(12)

Nc X

Yk Wk

k k

Average characteristics for the momentum, heat and mass transfer coefﬁcient assumed that sµ ¶ T 3 273 + 117 µ(T ) = 1.983 · 10−4 , g/(cm · sec), (13) T + 117 273

λ(T ) =

µcp , Pr

Dk = D(T ) =

µ . ρSc

(14)

To approximate the thermodynamic properties the following approximations are used for the speciﬁc heat and the enthalpy [3]: cp,k = A + Bt + Ct2 + Dt3 + E/t2 , J/(mol · K); hk − hk,T =298.15K = 2

(15) 3

4

= At + Bt /2 + Ct /3 + Dt /4 − E/t + F −

0 ∆Hf,298.15 ,

kJ/mol,

0 is the enthalpy of formation at 298.15 K; where t = T /1000; ∆Hf,298.15 A, B, C, D, E, F are the approximation coefﬁcients given in [3]. It is assumed that average molecular weight of the global chemical reactions products and approximations for the enthalpy are WP1 = WP2 = 29 g/mol, hPk = APk + BPk T , k = 1, 2, where approximation coefﬁcients are presented in Table 3.

Table 3 Approximation coefﬁcients for products enthalpy of the global chemical reactions

Number of the reaction

APk , cal/mol

BPk , cal/(mol·K)

1

–59500

10

2

–41000

12

Boundary conditions. The boundary conditions for solving Eqs. (1)–(7) are formulated in relation to the velocity, temperature, and mass fractions of the chemical components. Boundary conditions at xg = 0 (surface of the heterogeneous material). Mass fractions distributions along the surface: at y < y1 , y > y2 (AP surface): Yk = Yk,w ,

18

k = 1, 2, 3;

Yk = 0,

k = 4, 5;

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

at y1 < y < y2 (binder surface): Yk = 0,

k = 1, 2, 3;

Yk = Yk,w ,

k = 4, 5.

The mass fractions Yk,w are presented in Table 1. It was assumed that the mass fractions for the chemical components considered (including effective product species P1 , P2 ) do not depend on external burning conditions or pressure. Note that because we assume the values for the interface mass fractions, we do not utilize the species mass ﬂux boundary condition which includes the diffusional velocity, related to gradYk,w . Temperature and blowing velocity distributions along the surface: at y < y1 (AP surface): Tw = Tw,1 ,

uw = uw,1 ;

(16)

Tw = Tw,3 ,

uw = uw,3 ;

(17)

uw = uw,2 .

(18)

at y > y2 (AP surface):

at y1 < y < y2 (binder surface): Tw = Tw,2 ,

Note that Tw,1 , Tw,2 , Tw,3 and uw,1 , uw,2 , uw,3 are determined after solving heat transfer equations for the heterogeneous material solid phases. Boundary conditions at the symmetry axis (plane) (y = 0).

y = 0 : v = 0,

∂T ∂Yk ∂u = = = 0, ∂y ∂y ∂y

k = 1, 2, . . . , Nc .

(19)

Boundary conditions at the symmetry plane (y = L).

y = L : v = 0,

∂T ∂Yk ∂u = = = 0, ∂y ∂y ∂y

k = 1, 2, . . . , Nc .

(20)

Boundary conditions at the upper boundary of the calculated domain (x → ∞).

x → ∞,

∂ 2v ∂Yk ∂u ∂T = = = 0, = 2 ∂x ∂x ∂x ∂x

k = 1, 2, . . . , Nc .

(21)

The downstream dimensions for the computational domain to approximate the inﬁnity conditions are varied and range from 100 to 300 µm, as was done in [1]. Initial Conditions. To obtain steady-state solutions, temperature, velocity and species concentration distributions inside the calculated domain are the same as at surface. Pressure is assumed to be constant. To obtain other unsteady solutions, known steady-state values were used to provide initial conditions, as discussed in [1].

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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Heat Transfer Model for the Condensed Phases. The calculated domain for solving the solid phase heat transfer equations was shown in Fig. 4. Heat transfer equations are formulated for each of three sub-domains (see Fig. 4, b) µ ¶ α1,2 ∂T1 ∂T1 ∂T1 ∂ C 1 ρ1 ˙1 = (22) + C1 m (T1 − T2 ), λ1 −2 ∂t ∂xs ∂xs ∂xs ϕ1 L µ ¶ ∂ ∂T2 ∂T2 ∂T2 + C2 m ˙2 = λ2 + C 2 ρ2 ∂t ∂xs ∂xs ∂xs α2,3 α1,2 (T1 − T2 ) + 2 (T3 − T2 ), (23) +2 ϕ2 L ϕ2 L µ ¶ α2,3 ∂T3 ∂T3 ∂T3 ∂ λ3 −2 + C3 m (T3 − T2 ), C 3 ρ3 ˙3 = (24) ∂t ∂xs ∂xs ∂xs ϕ3 L where T1 , T2 , T3 are the temperatures in the ﬁrst, second, and third regions; C1 , C2 , C3 = C1 are the molar heat capacity of corresponding materials; ρ1 , ρ2 , ρ3 = ρ1 , λ1 , λ2 , λ3 = λ1 are the densities and heat conductivities of the materials. Note that α1,2 , α2,3 are the heat transfer coefﬁcients (between the 1st , 2nd, and 3rd materials). Also ϕ1 , ϕ2 , ϕ3 are deﬁned as the volume fraction of the 1st, 2nd, and 3rd materials in composite material (see Fig. 4). Coefﬁcients “2” in the right-hand side parts of these equations take into account 3D-character of the heat transfer processes. The temperature dependencies for the heat capacity and heat conductivity were borrowed from [6]. Thus, 0 00 C1,2 = C1,2 + C1,2 T,

(25a)

λ1,2 = λ01,2 + λ001,2 T,

(25b)

where the coefﬁcients are listed in Table 4. This table also contains values for the material densities. Table 4 Physical properties for the solid AP and HTPB

0 C1,2

C1,2 = cal/(g·K)

+

00 C1,2 T,

λ1,2 = λ01,2 + λ001,2 T , cal/(cm·sec·K) ρ1,2 , g/cm3

Solid AP

Solid HTPB

C10 = 0.15, C100 = 0.00041

C20 = 0.25, C200 = 0.00085

λ01 = 1.5 · 10−3 , λ001 = −9.2 · 10−7

λ02 = 4.4 · 10−4 , λ002 = 1.3 · 10−7

1.76

0.88

A signiﬁcant part of the coupled model are the boundary conditions on the burning surface of the solid material. Here xs = Hs and 20

x = xg = 0, y ∈ [0, y1 ] :

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

¶

µ λ1

∂T1 ∂x

= qw,1 + m ˙ 1 Q1,w ,

(T1 )w,s = (T1 )w,g ;

(26)

w,s

xs = Hs and x = xg = 0, y ∈ [y1 , y2 ] : ¶ µ ∂T2 λ2 = qw,2 + m ˙ 2 Q2,w , (T2 )w,s = (T2 )w,g ; ∂x w,s xs = Hs and x = xg = 0, y ∈ [y2 , L] : µ ¶ ∂T3 = qw,3 + m ˙ 3 Q3,w , (T3 )w,s = (T3 )w,g , λ3 ∂x w,s

(27)

(28)

where: Zy1 qw,1

1 = y1

1 qw dr = y1 0

¶ Zy1 µ ∂T λ dy, ∂x x=0 0

Zy2 qw,2

1 = y2 − y1

1 qw dr = y2 − y1 y1

y1

ZL qw,3

1 = L − y2

1 qw dr = L − y2 y2

¶ Zy2 µ ∂T λ dy, ∂x x=0

¶ ZL µ ∂T λ dy. ∂x x=0 y2

Q1,w , Q2,w , Q3,w are the latent heat terms for the surface phase transformations (J/g) (Table 5); m ˙ 1, m ˙ 2, m ˙ 3 are the mass burning rates for oxidizer and binder (g/(cm2 ·sec)). These mass burning rates are used in Arrhenius form: ¶ µ Eox ; (29) m ˙1=m ˙ oxidizer = Aox exp − R0 Tw,1 ¶ µ Ebin ; (30) ˙ binder = Abin exp − m ˙2=m R0 Tw,2 µ ¶ Eox ˙ oxidizer = Aox exp − m ˙3=m , (31) R0 Tw,3 where Aox , Abin are the pre-exponential factors; Eox , Ebin are the activation energies (see Table 5). Numerical Simulation Results. This can be used to calculate the burning rate of solid propellant based on AP/HTPB at different gas pressures. The calculations were performed for one model of the AP pyrolysis which was offered in [4] (see Eqs. (29), (31), and Table 5), and for two models of the HTPB pyrolysis. The ﬁrst model was offered in [4] (further we will call this model as BDP-pyrolysis model) and the second model was offered in [5] (CFD-pyrolysis model) (see Eq. (30), and Table 4). To demonstrate the inﬂuence of the heat transfer conditions in a solid phase of the composite material on a burning rate, factors of the heat exchange α1,2 , α2,3 VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

21

Table 5 Input values for calculation of the pyrolysis rates (BDP-pyrolysis model) A, g/(cm2 · sec)

E, cal/mol

Qw , cal/g

5.0 · 105

22000

100

15000

–50

Oxidizer

3

2.5 · 10

Binder

Constants for the pyrolysis mass burning rates (CFD-pyrolysis model) A, g/(cm2 · sec)

E, cal/mol

Qw , cal/g

22000

100

16900

–433

5

Oxidizer

5.0 · 10 2

2.9 · 10

Binder

were also varied. A base series of calculation was performed for α1,2 = 0.1 and α2,3 = 1. The input data for these calculations are presented in Tables 1–5. The following sizes of calculation domain (see Fig. 4, b) were set: Hg = 0.01 cm, L = 0.01 cm, Hs = 0.2 cm, y1 = 0.005 cm, y2 = 0.008 cm. Numerical simulation results corresponding to the BDP-pyrolysis model [4] are presented in Table 6 and in Figs. 5–7. Numerical simulations corresponding to the CFD-pyrolysis model [5] are presented in Table 7 and in Figs. 8–10. The following values are presented in those tables: r˙1 , r˙2 , r˙3 are the burning rates for AP (y < y1 , ϕ1 = 0.5), HTPB (y1 < y < y2 , ϕ2 = 0.3) and AP (y > y2 , ϕ3 = 0.2) respectively; r˙av is the average burning rate of the composite material, which is calculated by the following formula

r˙av =

ϕ1 ρ1 r˙1 + ϕ2 ρ2 r˙2 + ϕ3 ρ3 r˙3 , ϕ1 ρ1 + ϕ2 ρ2 + ϕ3 ρ3

(32)

Table 6 Calculated data for the BDP-pyrolysis model

r˙1 , cm/sec r˙2 , cm/sec r˙3 , cm/sec r˙av , cm/sec uw,1 , cm/sec uw,2 , cm/sec uw,3 , cm/sec Tw,1 , K Tw,2 , K Tw,3 , K

22

p = 40 atm 0.584 0.890 0.55 0.63 58.2 1.89 54.9 880 936 877

p = 60 atm 0.953 1.37 0.96 1.02 65.9 2.05 66.1 916 989 916

p = 80 atm 1.23 1.84 1.32 1.35 65.4 2.14 70.3 936 941 1030

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

Table 7 Calculated data for the CFD-pyrolysis model p = 40 atm, α1,2 = 0.1, α2,3 = 1.0

p = 60 atm, α1,2 = 0.1, α2,3 = 1.0

p = 80 atm, α1,2 = 0.1, α2,3 = 1.0

p = 40 atm, α1,2 = 0.01, α2,3 = 1.0

r˙1 , cm/sec

0.65

0.99

1.32

0.59

r˙2 , cm/sec

0.45

0.51

0.67

0.38

r˙3 , cm/sec

1.12

1.42

1.82

0.78

r˙av , cm/sec

0.72

1.00

1.32

0.60

uw,1 , cm/sec

65.3

68.4

70.6

59.1

uw,2 , cm/sec

1.31

1.01

1.03

1.07

uw,3 , cm/sec

117.0

101.0

99.7

79.9

Tw,1 , K

888

919

942

881

Tw,2 , K

1280

1310

1370

1250

Tw,3 , K

928

947

968

901

or with regard to ρ1 = ρ3 = 2ρ2 r˙av =

2ϕ1 r˙1 + ϕ2 r˙2 + 2ϕ3 r˙3 ; 2ϕ1 + ϕ2 ρ2 + 2ϕ3

(33)

uw,1 , uw,2 , uw,3 are the velocities of gas injection from the AP surface (y < y1 ), the HTPB surface (y1 < y < y2 ) and the AP surface at y > y2 respectively; Tw,1 , Tw,2 , Tw,3 are the temperatures of appropriate surfaces. Key interpretations of the quasi-3D simulation results (Figs. 5–10) include the following: 1. The burning rates of components (AP and HTPB) of the solid propellant, and also average burning rate will increase with a gas pressure buildup. This regularity is predicted by both HTPB pyrolysis models (compare Tables 6 and 7). 2. The burning rates obtained for two HTPB-pyrolysis models are similar. This can be explained, ﬁrst of all, by the low binder mass fraction. The largest temperature of a ﬂame (in local zones near to the burning surface), which grows with a pressure buildup, is achieved for the CFD-pyrolysis model. 3. A dependence of average burning rate upon pressure can be approximated ³ p ´0.86 by the following formula (for 40 < p < 80 atm) r˙av = 0.72 , cm/sec, (for 40 the CFD-pyrolysis model; pressure in atm), but using the BDP-pyrolysis model. 4. Comparison of the calculated burning rates with data obtained in [1] (where burning rate of a “sandwich” composite material with a characteristic size of AP layers — 150 µm and HTPB layers — 30 µm is studied) conﬁrms the experimentally established fact that bi-modal composite materials of a matrix type have greater burning rates [7].

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

23

Fig. 5. a : Temperature and velocity distribution above the burning surface at p = 40 atm. The BDP-pyrolysis model is used. Magnitudes of the gas velocities above the surfaces are uw,1 = 58.2 cm/sec, uw,2 = 1.89 cm/sec, uw,3 = 54.9 cm/sec; the surface temperatures are Tw,1 = 880 K, Tw,2 = 936 K, Tw,3 = 877 K; b : Temperature distribution above the burning surface at p = 40 atm. The BDPpyrolysis model is used. Note that maximum x = 50 µ µm; ϕ1 = 0.5, ϕ2 = 0.3 0.3, ϕ3 = 0.2 0.2; c: Heat ﬂux near the burning surface at p = 40 atm. The BDP-pyrolysis model is used; ϕ1 = 0.5 0.5, ϕ2 = 0.3 0.3, ϕ3 = 0.2

5. Gas velocities from the burning AP surface are in the region from 50 to 100 cm/sec (for different pressures and different HTPB-pyrolysis models). The gas velocities of the burning HTPB surface are predicted to be approximately ∼ 1 . . . 2 cm/sec, relatively small compared to the AP. 6. In the calculations cases, one can observe penetration of the gases (injected from the burning surfaces) through a high-temperature zone above the HTPB surface. However, for other conditions (not included here), for lower gas velocities a by-pass of the high-temperature area was predicted.

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Fig. 6. a : Temperature and velocity distribution above the burning surface at p = = 60 atm. The BDP-pyrolysis model is used. Magnitudes of the gas velocities at the surfaces are uw,1 = 65.9 cm/sec, uw,2 = 2.05 cm/sec, uw,3 = 66.1 cm/sec; the surface temperatures are Tw,1 = 916 K, Tw,2 = 989 K, Tw,3 = 916 K; b : Temperature distribution above the burning surface at p = 60 atm. The BDPpyrolysis model is used; ϕ1 = 0.5 0.5, ϕ2 = 0.3 0.3, ϕ3 = 0.2 0.2; c : Heat ﬂux near the burning surface at p = 60 atm. The BDP-pyrolysis model is used; ϕ1 = 0.5 0.5, ϕ2 = 0.3 0.3, ϕ3 = 0.2

7. The temperature of the HTPB burning surface exceeds the temperature of the AP burning surface by about 200–300 K in the CFD-pyrolysis model. For the BDP-pyrolysis model the temperatures of the AP and HTPB surfaces are almost equal. 8. The density of isotherms near to the burning surfaces increases with pressure (compare Figs. 5, b–6, b–7, b). One can see that as this takes place, the hightemperature zone of the ﬂame approaches to the burning surface. This explains the increase of the heat ﬂuxes to the burning surface with increasing pressure (see sequentially Figs. 5, c–6, c–7, c). 9. In all prediction cases the heat ﬂuxes at the burning surface are maximum near to the boundaries of the AP and the HTPB. VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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Fig. 7. a : Temperature and velocity distribution above the burning surface at p = 80 atm. The BDP-pyrolysis model is used. Magnitudes of the gas velocities above the surfaces are uw,1 = 65.4 cm/sec, uw,2 = 2.14 cm/sec, u w,3 = 70.3 cm/sec; the surface temperatures are Tw,1 = 936 K, Tw,2 = 941 K, Tw,3 = 1030 K; b : Temperature distribution above the burning surface at p = 80 atm. The BDPpyrolysis model is used; ϕ1 = 0.5 0.5, ϕ2 = 0.3 0.3, ϕ3 = 0.2 0.2; c : Heat ﬂux near the burning surface at p = 80 atm. The BDP-pyrolysis model is used; ϕ1 = 0.5 0.5, ϕ2 = 0.3 0.3, ϕ3 = 0.2

10. The variation of the heat exchange factors (α1,2 , α2,3 ) between the components of the investigated material in the solid phase has shown that burning rates of the AP and HTPB decrease with decreasing of the α1,2 and α2,3 (compare the second and last columns in Table 7). Conclusions. A numerical simulation model of thermo-gas-dynamic processes near to a burning surface of heterogeneous solid propellants on the base of AP and HTPB is represented, including the heat transfer processes inside a solid phase of these materials. The model takes into account the gas phase chemical reactions within the framework of the concept of the global chemical reactions.

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Fig. 8. a : Temperature and velocity distribution above the burning surface at p = 40 atm. The CFD-pyrolysis model is used. Magnitudes of the gas velocities above the surfaces are uw,1 = 65.3 cm/sec, uw,2 = 1.31 cm/sec, uw,3 = 117.0 cm/sec; the surface temperatures are Tw,1 = 888 K, Tw,2 = 1280 K, Tw,3 = 928 K; b : Temperature distribution above the burning surface at p = 40 atm. The CFDpyrolysis model is used; ϕ1 = 0.5 0.5, ϕ2 = 0.3 0.3, ϕ3 = 0.2 0.2; c : Heat ﬂux near the burning surface at p = 40 atm. The CFD-pyrolysis model is used; ϕ1 = 0.5 0.5, ϕ2 = 0.3 0.3, ϕ3 = 0.2

Two global chemical reactions for ﬁve effective chemical components are accepted in the model. A joint coupling of the heat-mass-transfer processes in a gas phase and thermal processes in a solid phase allows us to predict burning rates of the separate components of the material and the mass averaged burning rate of the heterogeneous propellant, including the surface temperatures and the gas (injection) velocities of the burning process gas products.

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Fig. 9. a : Temperature and velocity distribution above the burning surface at p = 60 atm. The CFD-pyrolysis model is used. Magnitudes of the gas velocities above the surfaces are uw,1 = 68.4 cm/sec, uw,2 = 1.01 cm/sec, u w,3 = 101.0 cm/sec; the surface temperatures are Tw,1 = 919 K, Tw,2 = 1310 K, Tw,3 = 947 K; b : Temperature distribution above the burning surface at p = 60 atm. The CFDpyrolysis model is used; ϕ1 = 0.5 0.5, ϕ2 = 0.3 0.3, ϕ3 = 0.2 0.2; c : Heat ﬂux near the burning surface at p = 60 atm. The CFD-pyrolysis model is used; ϕ1 = 0.5 0.5, ϕ2 = 0.3 0.3, ϕ3 = 0.2

The gas dynamics of the process was studied within the framework of the full unsteady Navier-Stokes equations. Actual properties for gas and condensed phases were also taken into account. The systematic numerical research of the burning process of bi-modal composite material of a matrix structure for various binder pyrolysis models is fulﬁlled, although the next step is to model the “spherical” shaped oxidizer. Based on the calculations the approximate dependence of an average burning ³ p ´0.86 rate upon on pressure is offered r˙av = 0.72 , cm/sec (with p in atm). 40 28

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Fig. 10. a : Temperature and velocity distribution above the burning surface at p = 80 atm. The CFD-pyrolysis model. Magnitudes of the gas velocities above the surfaces are uw,1 = 70.6 cm/sec, uw,2 = 1.03 cm/sec, uw,3 = 99.7 cm/sec; the surface temperatures are Tw,1 = 942 K, Tw,2 = 1370 K, T w,3 = 968 K; b : Temperature and velocity distribution above the burning surface at p = 80 atm. The CFD-pyrolysis model is used; ϕ1 = 0.5 0.5, ϕ2 = 0.3 0.3, ϕ3 = 0.2 0.2; c : Heat ﬂux near the burning surface at p = 80 atm. The CFD-pyrolysis model is used; ϕ1 = 0.5 0.5, ϕ2 = 0.3 0.3, ϕ3 = 0.2

It was shown that the burning rate of a bi-modal composite material of a matrix type exceeds the burning rate of materials with the “sandwich” structure (see [1]) for similar characteristic sizes of AP/binder. The study was partially supported by Russian Foundation for Basic Research grant No. 01-02-17532.

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REFERENCES 1. Surzhikov, S.T., Murphy, J.J., Krier, H. 2D Model for Unsteady Burning Heterogeneous AP/Binder Solid Propellants // AIAA Paper 2000-3573, 36-th AIAA/ASME/SAE/ASEE Joint Propulsion Conference 16–19 July, 2000/Huntsville, AL. 2. Ermolin, N.E., Korobeinichev, O.P., Fomin, V.M., et al. Study of Flame Structure For Mixed Solid Fuels Based on Ammonium Perchlorate and Polybutadiene Rubber // Combustion, Explosion, and Shock Waves, 1992, V. 28, No. 4, pp. 59–65. 3. Chemistry Web Book, http://webbook.nist.gov, 2000. 4. Beckstead, M.W., Derr, R.L., Price, C.F. A model of Composite Solid-Propellant Combustion Based on Multiple Flames // AIAA Journal, 1970, V. 8, No. 12, pp. 2200– 2207. 5. Cohen, N.S., Fleming, R.W., Derr, R.L. Role of Binders in Solid Propellant Combustion // AIAA Journal, 1974, No. 2, pp. 212–218. 6. Jeppson, M.B., Beckstead, M.W., Jing, Q. Model for the Premixed Combustion of a Fine AP/HTPB Composite Propellant. 35-th JANNAF Combustion Meeting, Tucson Arizona, 1998. 7. Price, E.W., Chakravarthy, S.R., Sigman, R.K., Freeman, J.M. Pressure Dependence of Burning Rate of Ammonium Perchlorate-Hydrocarbon Binder Solid Propellants // AIAA 97-3106, 33rd AIAA/ ASME/ SAE/ ASEE Joint Propulsion Conference & Exhibit, July 6–9, 1997, Seattle, WA.

S.T. Surzhikov graduated from the Bauman Moscow Higher Technical School in 1975. D.Sc. (Phys.-Math.), Head of the Computational Physical-Chemical and Radiative Gas Dynamics Laboratory of the Institute for Problems in Mechanics of Russian Academy of Sciences. Author of more than 300 publications in radiative gas dynamics and theory of heat and mass transfer.

BMSTU Press has published the book: Surzhikov S.T. Radiation of Heat in Gases and Plasma (in Russian) – M.: Izdatelstvo MGTU imeni N.E. Baumana, 2004. 544 p. Basic concepts of the radiant energy transfer theory in hot gases and low-temperature plasma are introduced. Phenomenological coefﬁcients and functions of the transfer theory are worded as well as their association with quantum properties. Main laws of the theory of transfer of radiation of heat are presented. The transfer equation is formulated and its most-used particular forms are given. Peculiarities of application of models of elementary radiation processes to construction of phenomenological models of radiation transfer are discussed. Methods of integrating the radiation transfer equation over a frequency and 3D-coordinates are presented. The book is intended for researchers and engineers practising heat exchange by radiation, physical gas dynamics and physics of low-temperature plasma as well as for students and post-graduates learning physical and technical trades in universities.

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V.T. Kalugin, A. Yu. Lutsenko, Ye.G. Stolyarova (Bauman Moscow State Technical University)

HYSTERESIS IN AERODYNAMICS OF FLYING VEHICLES UNDER CONDITION OF UNSTEADY MOVEMENT On the basis of the experimental research of aerodynamic hysteresis in a wide range of the governing parameters M∞ , α under condition of unsteady movement, some regularities are found in transformation of structures of a ﬂow nearby blunt bodies. Inﬂuence of unsteady effects on moments of transformation of a ﬂow from separated to attached one is established. Unsteady aerodynamic characteristics (in view of movement background) and their derivatives for various frequencies and amplitudes of body oscillations are obtained.

During the movement of ﬂying vehicles their kinematic parameters (speed of ﬂight, angles of attack and sliding) and parameters of a ﬂow (pressure, density, temperature) are continuously changing. Not only the values of the above-mentioned parameters but also a sense of their change, i.e. decrease or increase, have a vital importance for the structure and characteristics of separated ﬂows. It means, that with the same set of parameters a vehicle may have various aerodynamic characteristics. One of the reasons of it is the aerodynamic hysteresis caused by transformation of a ﬂow structure. Because of rather small time of the transformation process, accompanied with a sharp change of pressure upon surfaces, there always occur shock loads. This should be taken into account in design of ﬂying vehicles. Aerodynamic hysteresis is most pronounced at transonic speeds of ﬂow around the ﬂying vehicles with generatrix breaks (combinations of cylindrical and conical surfaces; bodies of rotation with segmental and butt bluntness, with rod superstructures) when at least one of the following parameters has changed: a speed of an oncoming ﬂow, angle of attack or Reynolds number. Experimental complex investigations of a ﬂow nearby the segmental bluntnosed bodies were conducted in the transonic speed range. They included viewing ﬂow patterns and measuring aerodynamic forces and moments both under the condition of the model oscillations and with ﬁxed angles of attack. Experiments were conducted in the wind tunnel with the closed test portion at Mach numbers of 0.7. . . 1.3, Reynolds numbers of (1.2 . . . 4.1) · 106 , the undisturbed ﬂow pressure and stagnation temperature were equal to 12.6 Pa and 290 K, respectively. Models represented low-lengthening bodies with ﬂat side surfaces, the square cross section l × l = 64 · 10−4 m2 and the lengthening λ = l/b = 0.422, where b is a body length. Unsteady aerodynamic characteristics were determined with the method of forced oscillations. A principle of harmonic analysis was at the heart of experimental studies of aerodynamic derivatives of forces and longitudinal moment [1]. In these studies the models oscillated relative to the ﬁxed angles of attack with the VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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small amplitude θα = ±4◦ . The estimation of inﬂuence of the ﬂow separation under the unsteady ﬂow condition on total aerodynamic characteristics and acquisition of experimental data about the angular speed inﬂuence on the ﬂow pattern transformation were conducted for oscillations of the models with various amplitudes θ1α ≤ ±16◦ , when an oscillation span exceeded the angle of attack at which the ﬂow pattern transformation occurred. Structures of a ﬂow around a blunt body. An axisymmetric ﬂow around a blunt body, both at a subsonic and supersonic speed, is accompanied with formation of separated ﬂow zones behind a frontal part of the body. From all variety of ﬂow structures under condition of unsteady movement we can highlight the three basic ones, shown in Fig. 1 [2]. With change of Mach number M∞ of the free stream the ﬂow structure transforms. At small subsonic speeds near the point of a generatrix break the ﬂow separates with formation of an open zone of the separated ﬂow 1 on the body surface (see Fig. 1, ). In the process of M∞ increase within the range of subsonic speeds the separated ﬂow zone nestles up to the streamlined surface; the zone cross-section decreases and ﬂow pressure in the zone decreases too. The further speed increase causes a change in the exterior of the ﬂow structure. In front of the body and near the external border of the separated ﬂow zone the shock

Fig. 1. Typical structures of a separated ﬂow around blunt bodies

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waves appear, but an open type of the separated ﬂow is kept. The pressure behind the generatrix break point levels off at a constant value. The ﬂow structure transformation occurs almost instantly at some ﬁrst critical Mach number (Mcr1 ) when instead of the advanced separated ﬂow, closed for a wake, a local separated zone 2 appears in the frontal part of the body (see Fig. 1, b). It is accompanied with forming nearby the streamlined surface a system of shock waves 3 due to the local separated ﬂow attachment and external ﬂow braking. The above-described ﬂow structure is referred to as a supersonic one. With the following reduction of Mach number down to 1, the type of the ﬂow structure remains the same, as there is no disturbing factor to cause the ﬂow structure transformation. However, at Mach numbers, smaller than 1, such a factor is available. It is known that the braking of a supersonic stream occurs through a shock wave. If a ﬂow speed becomes below the sound speed, a local supersonic ﬂow transforms to a subsonic one through a normal shock wave. The shock wave interaction with a boundary layer on the body surface results in its separation, the starting point of which comes nearer to the frontal part as the speed reduces. At the moment of merger of a circulating zone and a separated one, caused by the shock wave, the ﬂow structure is transforming, pressure is sharply growing, and separation, beginning in the generatrix break point, extends for the whole body surface. Such transformation of the ﬂow is achieved with the second critical Mach number, smaller than the ﬁrst critical one. Thus, at a various sense of the change of the free stream Mach number a ﬂow around the body has different structures, because transformation of ﬂow structures at the direct and reverse change of Mach number occurs under a different initial condition of the system and its reconstruction from this condition needs the additional energy which is taken away from the free stream. The variety of ﬂow structures within a range of small supersonic speeds also takes place with change of the body angle of attack. For zero and small values of the angle of attack there is a ﬂow structure with a forward shock wave, a forward local separated zone and a shock of the secondary compression of the ﬂow. Up to certain values of angle of attack the ﬂow structure varies insigniﬁcantly. On achievement of some critical value of angle of attack αtr1 the ﬂow structure transformation occurs almost instantly. It is explained by the following reasons. The increased pressure from a windward side propagates over the whole surface including a leeward side. For angles of attack, smaller than the ﬁrst critical angle, the propagating disturbances are not capable to cause separation from the leeward surface. If the angle of attack is more than the ﬁrst critical angle or equal to it, a merger of the local separated zone with that on the leeward surface occurs. Due to the ring-type separated ﬂow available the disturbances extend to a signiﬁcant part of the body surface which results in formation of the advanced separated ﬂow 4 (see Fig. 1, c). With the further increase of the value of angle of attack over the ﬁrst critical angle, only geometrical parameters of separated zones change. During the following reduction of angle of attack within the range from the initial value (corresponding to the separated ﬂow structure) to some second critical

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angle of attack, the type of a ﬂow structure remains constant. When the angle of attack becomes equal to the second critical angle, a reverse transformation of the separated ﬂow structure to that of a ﬂow with local separated zones occurs. The range of angles, where there is a variety of ﬂow structures, depends on design parameters of the body, Mach and Reynolds numbers in the free stream, but the reason for the hysteresis occurrence is the same, as in case of increase and reduction of the free stream speed [3]. Mathematical simulation. For construction of the mathematical model, determining the moment of transformation of a steady ﬂow structure into a separated one, we suppose, that in each section the ﬂow is ﬂat, steady, without heat exchange, the constant mass of gas circulates in the separated zone, and there are no longitudinal and transverse gradients of pressure in the zone. We believe, that the angle of transformation αtr1 corresponds to such ﬂow conditions when the angle of an inclination of a leeward surface is equal to the ﬂow separation angle βs (Fig. 2). If αtr1 is less than βs , the ﬂow structure is continuous; if αtr1 appears more than βs , then a separated zone arises on the leeward surface. Let us calculate the ﬂow separation angle βs with the use of a method of the ﬂow-dividing line (f dl). For the calculation of parameters of a ﬂow with separation we assume the direction β = βsd + α, corresponding to the critical ﬂow speed (M=1), as initial one. Here βsd is an angle of the ﬂow turn behind the forward shock wave, at which the speed is equal to the critical one; it is determined by the solution of the following system of equations [4]: 2M2∞ (1 − sin2 θsd ) 2 + (k − 1)M2∞ + = 1, 2 2kM2∞ sin θsd − (k − 1) 2 + (k − 1)M2∞ sin2 θsd · βsd = θsd − arctg

¸ 2 + (k − 1)M2∞ sin2 θsd tgθ sd , (k + 1)M2∞ sin2 θsd

Fig. 2. The scheme of a ﬂow for determination of αtr1 34

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where θsd is an angle of shock wave inclination at Mach number M=1; k is the ratio of speciﬁc thermal capacities. The task of ﬁnding an angle, at which the ﬂow structure transforms, is solved with a method of iterations. We assume pp = 0,9p∞ as initial value of the pressure in the separated zone on the leeward surface (p∞ is the static pressure in the free stream) and determine with the help of gasdynamic functions the Mach and Crokko numbers (Mlw and Crlw ) behind the frontal surface generatrix break. Using the solution of the integral equation [5] +∞ Z

+∞ Z

ϕ2 dη = 1 − Cr2lw ϕ2 −∞

ϕ dη , 1 − Cr2lw ϕ2 ηf dl

the dimensionless speed, Crokko and Mach numbers on the ﬂow-dividing line are determined as Ã !0,5 Cr2f dl 2 ϕf dl = 0,5(1 + erf ηf dl ), Crf dl = ϕf dl Crlw , Mf dl = , k − 1 1 − Cr2f dl where η is a dimensionless coordinate. Then the angle of the stream turn is calculated: r r q k+1 k−1 2 arctg (Mlw − 1) − acrtg (M2lw − 1) − β. βb = k−1 k+1 Further, on the basis of the theory of shock waves, an inclination angle θsh for a shock wave, created by attachment of the separated ﬂow in the base area, and a value of the static pressure psh behind shock wave are estimated. Using the attachment criterion psh = p0fdl , the value plw is then reﬁned. If the calculated angle βb > 0, we have a continuous structure of ﬂow, if βb < 0 — a separated one. Equality βb = 0 corresponds to the transformation condition α = αtr1 . The unsteady ﬂow condition has an essential inﬂuence on aerodynamics of similar bodies. In the unsteady movement, e.g. the body oscillations (even with a small amplitude), there can be a transformation of areas of the aerodynamic characteristics ambiguity, which should be taken into consideration in aerodynamic calculations. The most widespread method of the unsteady aerodynamic characteristics determination is a method of “small amplitudes” with the governing parameters change according to the harmonic law. As is known, in this case the coefﬁcients of the aerodynamic normal force and aerodynamic longitudinal moment can be presented as follows: ¯ ¯ ¯ ¯˙ z ; Cy = Cyα · α + Cyα˙ · α ˙ + Cyω¯ z · ω ¯ z + Cyω˙ z · ω ¯

¯

¯˙ + mωz¯ z · ω ¯˙ z , ¯ z + mωz˙ z · ω mz = mαz · α + mαz˙ · α ¯

¯

¯

¯

where Cyα , Cyα˙ , Cyω¯ z , Cyω˙ z , mαz , mαz˙ , mωz¯ z , mzω˙ z are derivatives of corresponding ¯˙ z , where α ¯˙ = ¯˙ ω aerodynamic coefﬁcients with respect to parameters α, α, ¯z , ω VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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= α˙ · b/V∞ is a dimensionless derivative of an angle of attack with respect to 2 ¯˙ z = ω˙ z · b2 /V∞ is time; ω ¯ z = ωz · b/V∞ is a dimensionless pitch angular speed; ω a dimensionless derivative of the pitch angular speed with respect to time; V∞ is a speed of the free stream; b is a length of the body. Such representation of aerodynamic characteristics is based on the assumption of a possibility to use a principle of superposition of independent private movements for which the structure of ﬂow around ﬂying vehicles remains unchanged — with or without separation. In this case it appears possible to choose independently the main parameters inﬂuencing the aerodynamic characteristics of a body. Experimental results. Let us discuss the given below unsteady aerodynamic characteristics, obtained experimentally for a body of small lengthening with segmental bluntness for speciﬁc values of amplitude, frequency and Struhal number. These experimental results can form a basis for calculations of transition processes of ﬂying vehicles with any law of movement. Unsteady aerodynamic characteristics at a zero angle of attack. Figure 3 presents the experimental unsteady aerodynamic characteristics — derivatives Cyα , ´ ³ ´ ³ ¯ ¯ mαz , Cyω¯ z + Cyα˙ , mzω¯ z + mαz˙ for Mach numbers M∞ = 0.6 . . . 1.2 at a zero angle of attack and various positions of the conventional center-of-gravity x ¯cg (¯ xcg = xcg /b where xcg is a distance from the conventional center-of-gravity to ¯cg2 = 0.428; the body frontal surface; b is a length of the body): x ¯cg1 = 0.262; x ¯cg1 ) shows the following: x ¯cg3 = 0.76). The analysis of these curves (for x in the subsonic range of speeds (M∞ = 0.6 . . . 0.9) with the M∞ increase the value Cyα = f (M∞ ) decreases (Fig. 3, a), remaining positive; in the interval of Mach numbers M∞ = 0.93 . . . 1.1 the derivative changes its sign, with the minimal value Cyα = −0.5 achieved at M∞ = 1.0; the further increase of Mach numbers (M∞ > 1,0) results in growth of the derivative Cyα , and at M∞ > 1.1 the value Cyα > 0. It means, that with the body oscillations (relative to the zero angle of attack) in the range of M∞ = 0.9 . . . 1.1 the normal force direction has an opposite sense to change of an angle of attack (Cyα < 0). The experimental research of a steady ﬂow [3] has shown, that only with a ﬂow separation there can occur a negative aerodynamic normal force (α > 0), acting on a body, and a value of the number M∞ , at which the sign of Cyα changes from negative to positive, corresponds to the moment of transformation of the separated ﬂow (see Fig. 1, a) to the continuous one (see Fig. 1, b). In body oscillations with respect to the zero angle of attack, the ﬂow structure transformation occurs at M∞ = 1.15, while in conditions of a steady ﬂow it occurs at M∞ = 1.05. Thus, the pitch angular speed, which is characteristic of the oscillatory movement of the ﬂying vehicles, leads to some prolongation of existence of the separated ﬂow structure for the larger values of M∞ . The important practical task in investigations of a ﬂow around the oscillating bodies is to determine the inﬂuence of a position of the conventional center-ofgravity x ¯cg on their aerodynamic characteristics. The analysis of the obtained data shows that a character of change of the derivative Cyα = f (M∞ ) at various positions of the rotation axis is the same for all investigated values of x ¯cg . At subsonic 36

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Fig. 3. Derivatives of aerodynamic force and moment at α0 = 0 (•, •0 — increase and reduction of M∞ , respectively)

speeds of the free stream (M∞ < 0,9) the aft displacement of the conventional center-of-gravity results in a small increase of the normal force derivative. It is evident most essentially in a range of numbers M∞ = 1.0 . . . 1.15 where the ﬂow has an unstable pattern. When a ﬂow around the body becomes attached, the derivative

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Cyα remains constant for various positions of the conventional center-of-gravity and practically becomes equal to the value Cyα obtained for a steady ﬂow. An area of negative values of Cyα and, consequently, of resulting negative normal forces de¯cg1 ) creases with the aft displacement of the conventional center-of-gravity (¯ xcg > x and almost disappears with its maximal approach to a base cut. As this force only occurs in the separated ﬂow mode, the range of Mach numbers, associated with the force, also reduces. Compared to the case of a forward position of the conventional center-of-gravity x ¯cg , the ﬂow structure transformation from ﬂow with separation to ﬂow without separation occurs at smaller numbers M∞ , practically equal to the Mach number of transformation under steady conditions (ωz = 0). It is connected with the inﬂuence of an angular speed of oscillations. With the forward position of the conventional center-of-gravity, e.g. x ¯cg1 , the aft linear speed distribution is such that promotes (at positive angular speeds) the separated zone expansion on the top surface of a body and its reduction on the bottom one. Therefore, the separated ﬂow exists in a wider range of M∞ . At the signiﬁcant aft displacement of the conventional center-of-gravity the distribution of linear speeds, caused by movement of a body, practically does not inﬂuence the ﬂow structure formation in this area, and the forebody ﬂow is rather steady. The analysis of dependence mαz = f (M∞ ) in the investigated range of ﬂow speeds has shown, that at M∞ < 1 the value mαz < 0, and at M∞ > 1 the derivative sign is determined by a position of the conventional center-of-gravity (see Fig. 3, b). For separated ﬂow modes at numbers M∞ < 1.15 the values of mαz , obtained under conditions of forced oscillations and a steady ﬂow, essentially differ from each other. It allows us to make a conclusion that the body oscillations result in the more appreciable transformation of the separated ﬂow structure at subsonic and transonic speeds of free stream and, consequently, in the change of body’s aerodynamic characteristics as compared to those obtained for a steady ﬂow. Figure 3, c presents the Mach number dependence of rotary derivatives (Cyω¯ z + ¯ + Cyα˙ ) with the zero angle of attack and various positions of the body’s axis of rotation (the conventional center-of-gravity). With the M∞ change in the subsonic range of speeds (M∞ < 0.9) a combination of derivatives of the normal force ¯ coefﬁcient (Cyω¯ z +Cyα˙ ) reduces continuously, and at transonic speeds the derivatives ¯ at ﬁrst increase and then decrease (the maximal value (Cyω¯ z + Cyα˙ ) is reached at M∞ = 1). Such a behavior of the M∞ -dependence of the derivatives combination can be explained by estimating individually each derivative in the combination. At speeds of the oncoming ﬂow corresponding to 0.6 6 M∞ 6 0.9 the structure of the separated ﬂow nearby the body (see Fig. 1, ) is transformed insigniﬁ¯ cantly. It is known that a value of the derivative Cyα˙ , basically caused by delay of the ﬂow structure transformation with respect to the angle of oscillations, remains practically constant. According to this, a change of a combination of derivatives ¯ (Cyω¯ z + Cyα˙ ) is determined completely by the law of the Cyω¯ z change. The rotary derivative Cyω¯ z depends on the local angles of attack with the body oscillations in the pitch plane. In the body rotation with angular speed ωz , at various points of the body surface the additional speed components of the ﬂow appear which are

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directed, in case of the inverted movement, opposite to the body circular speed. For the investigated position of a rotation axis (¯ xcg1 ) at ωz > 0 these speeds result in a negative angle of attack for the frontal part of a body and positive — for the rear part. The direction of the aerodynamic force, depending on ωz , will be determined, ﬁrstly, by the position of a rotation axis, secondly, by the size of a region of action of positive and negative normal forces, respectively, that depends for the investigated body on the speed of the free stream. So, for example, at x ¯cg = 0.262 ω ¯z (M∞ < 0.75) the value Cy > 0, i.e. normal forces in frontal and rear parts of the body have the same direction — positive. With increasing of the free stream speed the derivative Cyω¯ z decreases due to the extension of the body surface area (where the negative normal force Cyα < 0 is created) behind an axis of oscillations and to the occurrence of a negative force with the point of its application located downstream, behind the axis of oscillations. In a speed range corresponding to 0.9 < M∞ ≤ 1.0 the great increase in value ¯ of a combination of derivatives (Cyω¯ z + Cyα˙ ) is observed. However, it is necessary to note, that at M∞ = 1 the derivative Cyα is negative and reaches its minimum. ¯ Hence, the sharp increase of the combination of unsteady derivatives (Cyω¯ z + Cyα˙ ) ¯ is basically determined by the value Cyα˙ > 0. The subsequent growth of the free stream speed (M∞ > 1) results in reduction of the derivatives combination. Such stepwise change of unsteady derivatives corresponds to conditions under which an unstable separated ﬂow arises. In this case it occurs at M∞ = 1. With increase of the free stream speed (M∞ > 1.15) the ﬂow structure transformation to the ¯ ﬂow without separation is observed. At the same time the derivative Cyα˙ remains positive. In conditions of ﬂow without separation the pressure distribution on the body surface provides only the positive force creation (Cyα > 0). Hence, the sign and value of the derivative Cyω¯ z is determined by the sum of two differently directed forces in the frontal and rear parts of the body. The M∞ -dependence of derivatives of the aerodynamic longitudinal moment ¯ coefﬁcients (mωz¯ z + mαz˙ ) is presented in Fig. 3, d. The behavior of curves is similar ¯ to the specular reﬂection of the dependence Cyω¯ z + Cyα˙ = f (M∞ ). Position of the dynamic focus at various speeds of a ﬂow. In the analysis of unsteady aerodynamic characteristics it is interesting to investigate a position of dynamic focus at various speeds of a ﬂow. A coordinate of the dynamic focus is determined by the following formula: ¯

¯cg − x ¯Fω = xFω /b = x

mzω¯ z + mαz˙ . Cyω¯ z + Cyα¯˙

¯cg = 0.262 (α0 = 0) are presented Results of calculation of values x ¯Fω at x ¯ ¯˙ in a range of in Fig. 4. The point of applying the unsteady force (Cyω¯ z + Cyα˙ ) · α small subsonic speeds (M∞ = 0.6) is situated in a frontal part of the model, i.e. the coefﬁcient of the anti- damping moment is basically determined by the value Cyω¯ z of the frontal part of the model. With the subsequent growth of the free stream speed (M∞ 6 0.7) the point of this force application transfers outside the body. ¯ This means that an absolute value of the aerodynamic derivative Cyα˙ is greater than VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

39

Fig. 4. Change of the dynamic focus position at various ﬂow speeds

Cyω¯ z of a rear part of the model, and the point of application of the unsteady force ¯ ¯ ¯ ˙ is behind the axis of oscillations. When (Cyω¯ z + Cyα˙ ) = 0, the curve (Cyα˙ · α) x ¯Fω = f (M∞ ) has a discontinuity. At transonic speeds of the free stream (M∞ = 0.9 . . . 1.2) the combination ¯ ω ¯z (mz + mαz˙ ) >> 0. Anti-damping appears in the following cases: for (Cyω¯ z + ¯ + Cyα˙ ) < 0, if a coordinate of the unsteady force application point is more than that ¯ ¯cg , and for (Cyω¯ z + Cyα˙ ) > 0 if, of the conventional center-of-gravity, i.e. x ¯ Fω > x on the contrary, x ¯ Fω < x ¯cg . At the ﬂow speed, corresponding to the moment of the ﬂow structure transformation, x ¯Fω asymptotically tends to inﬁnity. If the structure of ﬂow remains separated, the dynamic focus is situated downstream behind the conventional center¯cg at a ﬂow without separation. With a supersonic ﬂow of-gravity, but x ¯ Fω < x ¯ (M∞ > 1.15) the combination (mzω¯ z + mαz˙ ) < 0. In this case the point of ap¯ ¯˙ which is positive, is plication of the aerodynamic unsteady force (Cyω¯ z + Cyα˙ ) · α, situated behind the axis of oscillations. When the point of x ¯Fω coincides with the position of the axis of oscillations (conventional center-of-gravity) a value of the longitudinal damping moment is equal to zero. ¯ The conducted analysis of the Mach number dependencies of Cyω¯ z +Cyα˙ , mωz¯ z + ¯ + mαz˙ and the dynamic focus position has revealed rather a complex structure of a ﬂow nearby a body under condition of the body oscillations at subsonic and transonic numbers M∞ . Inﬂuence of an initial angle of the model installation. For an estimation of inﬂuence of an initial angle of the model installation on its aerodynamic derivatives ¯ ¯ Cyα , mαz , Cyω¯ z + Cyα˙ , mzω¯ z + mαz˙ an experimental research has been carried out at x ¯cg = 0.262 and α0 = 5◦ , 10◦ . If α0 = 5◦ , the general character of the dependence Cyα = f (M∞ ) practically remains the same, as at a zero angle of the model installation (Fig. 5, ). In this case, the less intensive reduction of the derivative Cyα

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Fig. 5. Derivatives of aerodynamic force and moment at α0 6= 0

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with the free stream speed increase is due to the insigniﬁcant transformation of the separated ﬂow on a leeward side of the body. The derivative of the aerodynamic moment coefﬁcient mαz (Fig. 5,b) as against a case of α0 = 0 changes insigniﬁcantly in the whole range of the investigated Mach numbers. The sign and magni¯ tude of the derivatives combination Cyω¯ z + Cyα˙ depend on the distribution of normal forces over the body surface (Fig. 5, c). The change of Cyα is the supreme reason for the fact that in some range of transonic numbers M∞ the absolute magnitude ¯ of the combination Cyω¯ z + Cyα˙ decreases in comparison with that in the case of the zero angle of the body installation. The increase of the installation angle up to α0 = 10◦ leads to essential transformation of a ﬂow structure and, consequently, to the change of behavior of the ¯ curve Cyω¯ z + Cyα˙ = f (M∞ ) in the whole investigated range of Mach numbers. At such angle of installation a negative force does not arise in the body frontal part under conditions of the separated ﬂow, i.e. Cyα > 0; therefore a change of the value ¯ Cyω¯ z + Cyα˙ will be caused by redistribution of positive normal forces with respect to the rotation axis. As the ﬂow structure remains rather steady (α0 = 10◦ ) up to numbers M∞ 6 1.0, the sign and magnitude of the combination of derivatives ¯ ¯ Cyω¯ z + Cyα˙ are determined by the component Cyω¯ z . The sharp increase of Cyω¯ z + Cyα˙ ¯ at M∞ ≈ 1.0 is connected with a change of Cyα . The greatest increase of Cyω¯ z + Cyα˙ is caused by the fact that in a rear part of the body a system of oblique shock waves, closed in the separated zone, appears. The experimentally obtained dependence of a combination of derivatives of the ¯ longitudinal moment coefﬁcient (mzω¯ z + mαz˙ ) on various speeds of an free stream is presented in Fig. 5, d for α0 = 5, 10◦ . The redistribution of forces on the body surface, with its oscillatory movement, results in the essential change of the damping properties in comparison with a case of the zero angle of installation. We shall note that a frontal part of the body is practically neutral in creation of the damping ¯ moment. At an unsteady ﬂow the sign of the combination mωz¯ z + mαz˙ is determined by the action of forces in a rear part of the body. As distinction from the case of small initial angles of the body installation, with α0 = 10◦ the small damping is observed in a range of M∞ = 0.6 . . . 0.7, then for larger M∞ the anti-damping properties increase gradually. In the vicinity of the Mach number equal to 1.0 the sharp increase of the damping coefﬁcient is observed. With the following growth of Mach numbers anti-damping suddenly ¯ appears (mωz¯ z + mαz˙ ) > 0. There should be noted a peculiarity in recalculation of rotary derivatives from one position of the conventional center-of-gravity (axis of rotation) to another. Ac¯ cording to the linear theory, the value Cyα˙ does not depend on a position of the rotation axis and is to be constant for the given conditions of a ﬂow. Hence, the ¯ xcg ) is determined by the value Cyω¯ z = f (¯ xcg ) sum of derivatives Cyω¯ z + Cyα˙ = f (¯ and can be presented as follows: ´ ´ ³ ³ ¯ ¯ = Cyω¯ z + Cyα˙ + ∆¯ xcg · Cyα , Cyω¯ z + Cyα˙ x ¯cg

42

x ¯cg0

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´ ³ ¯ where Cyω¯ z + Cyα˙

´ ³ ¯ , Cyω¯ z + Cyα˙ x ¯cg0

are values of derivatives for the starting x ¯cg

¯cg , respectively; ∆¯ xcg0 = position of the rotation axis x ¯cg0 and the chosen one x ¯cg . =x ¯cg0 − x ¯ However, the unsteady aerodynamic characteristics of the body Cyω¯ z + Cyα˙ , obtained in experimental way, at various positions of the conventional center-ofgravity x ¯cg do not coincide with the calculation results, obtained with the help of the above-mentioned equation. This fact can be explained by the ﬂow structure transformation with the x ¯cg change for the examined range of speeds. The similar conclusion can be also made while analyzing the damping characteristics of the ¯ body mzω¯ z + mαz˙ = f (¯ xcg ), obtained at various positions of the rotation axis. The conducted analysis of experimental results, obtained with the usage of the “small amplitudes” method, has allowed us to make an important conclusion, that the representation of aerodynamic characteristics of bodies with a generatrix break within the framework of the harmonicity hypothesis is only possible at the certain ﬁxed position of the conventional center-of-gravity for some ﬁnite range of change of the determining parameters — an angle of attack, Mach number, amplitude and frequency of oscillations which have paramount inﬂuence on a ﬂow structure nearby such bodies. Using the “small amplitudes” method for determination of the aerodynamic coefﬁcients, required for transient process calculations, allows one to receive them only for the ﬂow structure of the same type (within the limits of oscillation amplitude) and does not enable to establish true moments of the ﬂow transformation. At oscillatory movement of a body with the amplitude exceeding angles of attack, corresponding to the ﬂow separation in steady conditions, the essential change of aerodynamic characteristics is observed. Total aerodynamic characteristics. To determine the aerodynamic coefﬁcients of bodies under consideration in conditions of oscillatory movement with various amplitudes, the experimental research has been carried out providing registration of aerodynamic forces and moments, corresponding to the instant structure of ﬂow. The angle-of-attack dependence of coefﬁcients of the body total aerodynamic normal force Cyt (α) and longitudinal moment mzt (α) in the unsteady movement with parameters M∞ = 0.8, θα = ±16◦ , f = 0.2 . . . 2.5 Hz are presented in Fig. 6. Here, for comparison, the research results Cy0 (α) and m0z (α), obtained for steady conditions, are shown by dashed lines (the same designations will be used in all subsequent ﬁgures). The structure of ﬂow corresponded to that of the separated one. The analysis of the obtained data shows, that the oscillatory movement results in change of the characteristics inherent to a steady ﬂow. In a range of angles of attack 0 ≤ α ≤ 3◦ (f = 0.2 Hz, θα = 16◦ ) the coefﬁcient of aerodynamic normal force is practically equal to zero. With an unsteady ﬂow the negative force occurs in the frontal part of a body for a little bit larger range of angles of attack, than under steady conditions, hence, the area of angles α extends, where Cyt is close to zero. An increase of an angle of attack (α > 3◦ ) results in an increase of Cyt (α, ωz , ω˙ z ), and it is necessary to note that a change of the oscillation frequency VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

43

Fig. 6. Total aerodynamic characteristics (•, •0 — increase and reduction of an angle of attack α , respectively)

in the investigated range of values (f = 0.2 . . . 2.5 Hz) do not inﬂuence essentially the behavior of the α-dependence of Cyt . The reason for this is a small increase of ¯ ¯ the unsteady component (Cyω¯ z + Cyα˙ ) · α ˙ at oscillation frequencies up to f = 2.5 Hz for the investigated kind of a ﬂow structure. The same reason can explain the fact 44

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that the values of Cyt at a direct and reverse change of angles of attack during the oscillatory movement are equal. Within the limits of the oscillation amplitude the total normal force coefﬁcient is a little bit less, than the Cyt (α) values for a steady ﬂow. However, the gradient of Cyt change remains the same, as for Cy0 (α). Some reduction of carrying properties of the body at the unsteady ﬂow is due to the less intensive growth of the positive normal force in the rear part of a body with the change of α. In a range of angles of attack 0 ≤ α ≤ 3◦ where t = 0, values of the total aerodynamic longitudinal moment coefﬁcient mzt (see Fig. 6, b) are close to the corresponding values at the steady ﬂow m0z . With increase of α (α > 3◦ ) within the limits of the oscillation amplitude the values of aerodynamic coefﬁcients mzt (α) and m0z (α) are slightly different, which is caused by transformation of the subsonic separated ﬂow structure at the unsteady movement of the body in comparison with a steady ﬂow. The analysis of the obtained results shows, that inﬂuence of a position of the conventional center-of-gravity xcg on aerodynamic characteristics is different for the steady and unsteady ﬂow. For the case of the unsteady movement under research, it is impossible to recalculate the moment characteristics based on some position of xcg to those based on another xcg . In the vicinity of a zero angle of attack at transonic speeds M∞ > 1 under steady conditions the supersonic ﬂow without separation is observed which is characterized by a local λ-shaped shock wave and the closed circulating zone within the body frontal part. At some angle of attack α = αtr1 (during a direct course — increasing α) the λ-shaped shock wave on a leeward side is breaking, the circulating zone is uniting with the separated area having developed by this moment on the body surface and the ﬂow structure is transforming from continuous to separated one. With reduction of an angle of attack (during a reverse course — reducing α) the transformation of the ﬂow structure to the continuous one occurs at the smaller value of this angle (α = αtr2 ) in comparison to that during the direct course. It is possible to consider that this pattern of the ﬂow structure change remains the same for an oscillatory movement with a large amplitude. Curves of the angle of attack dependence of the longitudinal moment coefﬁcient ¯cg1 = 0.262 are presented mzt for M∞ = 1.12, f = 0.2 . . . 1.0 Hz, θα = ±16◦ , x in Fig. 6, d, e. The ﬂow is without separation at the body oscillation frequency f = = 0.2 Hz in a range of angles of attack 0 ≤ α ≤ 8◦ . The increase of α within the limits of the oscillation amplitude results in the ﬂow transformation to a separated ﬂow. During the reduction of angle of attack (reverse course) the transformation of the separated ﬂow to a continuous one occurs at smaller α, so there is a ﬁeld of ambiguity of body’s aerodynamic characteristics. A change of the oscillation frequency within the limits from 0.2 z up to 0,5 Hz does not inﬂuence essentially the behavior of the dependence mzt = f (α). The increase of the oscillation frequency up to f = 1.0 Hz causes the essential change of aerodynamic characteristics and transformation of the ﬁeld of ambiguity of the longitudinal moment coefﬁcient. At such oscillation frequencies and angles

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45

of attack up to α ≤ 8◦ the absolute value of the aerodynamic longitudinal moment coefﬁcient grows (mzt < 0). No continuous ﬂow of the kind that was observed at smaller frequencies occurs now on the body. In the vicinity of a zero angle of attack the value of coefﬁcient mzt is less than that inherent to the steady ﬂow with a continuous structure. Such distinction in aerodynamic characteristics of the body is explained by formation of the separated ﬂow at its surface. However, if we compare the longitudinal moment coefﬁcients at the unsteady movement and at a stationary ﬂow for the separated ﬂow structure, it is visible that mzt > m0z . With reduction of angles of attack in a range of 3◦ ≤ α ≤ 9◦ the mzt values coincide with the results obtained for a steady ﬂow at the same Mach number (M∞ = 1.12). The analysis of the conducted research allows one to make a conclusion that in conditions of the body oscillations with the frequency f ≥ 1 the separated ﬂow may occur which remains for all angles of attack while in steady conditions there are two types of a ﬂow — with and without separation. In a steady ﬂow the ambiguity of aerodynamic characteristics (presence of hysteresis areas) is explained by the ﬂow transformation delay during a direct and reverse course of changing some governing parameter (Mach number M∞ , an angle of attack or sliding, etc.). In conditions of the body oscillations the ambiguity of its aerodynamic characteristics can be observed at the same ﬂow structure type, for example, at the separated one, which is explained by various changes of geometry of the separated zones and ﬂow parameters in them. With increase of the oscillation frequency (f ≥ 1 Hz) the behavior of the dependence mzt = f (α) within the limits of the investigated oscillation amplitudes does not change, but the area of the aerodynamic characteristics ambiguity extends to somewhat wider range of angles of attack. A change of a position of the conventional center-of-gravity of a body (its aft displacement, for example, x ¯cg2 = 0.428) under the continuous ﬂow and at the oscillation frequencies f ≤ 0.5 z does not result in the change of the total aerodynamic longitudinal moment coefﬁcients with respect to those obtained for a steady ﬂow (see Fig. 6, d). The transformation of the ﬂow to the separated one occurs in the vicinity of the angle of attack α ≈ 9◦ . At values α > 9◦ there is a ﬂow with separation on windward and leeward sides. During the reduction of an angle of attack, starting from the maximal value α = θα = +16◦ , values of the total longitudinal moment coefﬁcient mzt differ essentially from those obtained under steady conditions. For a considered range of frequencies f < 0.5 Hz in the vicinity of α ≈ 5.5◦ (reverse motion) a stepwise reduction of the aerodynamic longitudinal moment coefﬁcient mzt occurs. Such a behavior of the dependence is kept for all investigated oscillation frequencies (f ≤ 2.5 Hz). Analyzing changes of aerodynamic characteristics during the reduction of an angle of attack α, it is possible to make a conclusion, that the continuous structure ﬂow on the bottom and top surfaces of the body occurs at various angles of attack (in the vicinity of α = 5.5◦ and α = 0, respectively). The increase of the body oscillation frequency up to f = 1.0 Hz results in appearance of peculiarities of the aerodynamic longitudinal moment coefﬁcient

46

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change in the vicinity of a zero angle of attack (see Fig. 6, f). Both at transition from positive angles of attack to negative ones (α˙ < 0), and at their reverse change (α˙ > 0) the aerodynamic longitudinal moment coefﬁcient is not equal to zero and grows with the increase of the oscillations frequency. It can’t be only explained by existence of the moment due to the maximal angular speed, since the value of ¯ ¯ ˙ max is less by an order than mzt . Hence, the abovethe component (mωz¯ z + mαz˙ )α mentioned result can be only caused by the asymmetrical ﬂow structure owing to the delay of the ﬂow transformation from one ﬂow type to another. For example, at reduction of a positive angle of attack (α˙ < 0) the transformation of a ﬂow type from separated to continuous on the top body surface is delayed up to the reach of the region of negative angles of attack. Thus, at the unsteady movement there is no pronounced moment of the reconstruction of a continuous ﬂow on the top surface, characteristic of the steady ﬂow conditions. As a result, the behavior of the dependence mzt = f (α) at α = 0◦ has the above-mentioned peculiarity. The additional proof of existence of an asymmetrical ﬂow around a symmetric body was obtained during the unsteady movement research in which an artiﬁcial braking was applied in the vicinity of a zero angle of attack so that α˙ would change the sign. In this case the transformation duration appreciably reduces and the total aerodynamic longitudinal moment coefﬁcient mzt at a zero angle of attack becomes close to zero. The amplitude of oscillations θα is an important factor inﬂuencing the total aerodynamic characteristics of a body. Research was carried out at various values of θα in the range of 8 . . . 16◦ . The choice of oscillation amplitudes provided ﬂow regimes in which the ﬂow structure transformation inherent to the steady conditions occurred, which allowed the inﬂuence of the body oscillatory movement on the ﬂow structure transformation and the body aerodynamic characteristics to be estimated. In particular, at oscillatory movement with θα = 8◦ (¯ xcg1 = 0.262, f = 1.0 Hz), for which θα < αtr1 , there is no transformation of the ﬂow structure to the separated one. The longitudinal moment coefﬁcients mzt and mz at the unsteady movement and under steady conditions are approximately equal. The oscillation frequency variation in the range of f = 0.2 . . . 2.5 Hz does not inﬂuence the behavior of the dependence mzt = f (α). The analysis of the research results shows that in case of the unsteady movement with the amplitude θα = 10◦ (θα > αtr1 , x ¯cg1 = 0.262, f = 0.5 Hz) at a change of the current angle of attack α < αtr1 , i.e. with a continuous ﬂow structure, the absolute values of the longitudinal moment and normal force coefﬁcients, obtained both at the body oscillations and for steady conditions, are practically equal. The oscillatory process causes the ﬂow structure transformation delay, so a change of the longitudinal moment coefﬁcient mzt , corresponding to appearance of separated ﬂow, occurs at larger angles of attack than it was observed in steady conditions. Besides, parameters and sizes of separated zones on the top and bottom surfaces of the investigated body have changed which is conﬁrmed by essential distinction of the curves of the total normal force coefﬁcients obtained for ﬂow conditions under investigation. VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

47

At reduction of an angle of attack with the body oscillations the transformation of a ﬂow structure from separated to a continuous one occurs at smaller angles of attack. An increase of the oscillation frequency up to f = 2.5 Hz does not bring essential changes in behavior of curves Cyt (α), mzt (α). In case of θα = ±16◦ the character of the angle of attack dependence of the investigated aerodynamic characteristics is the same as at θα = ±10◦ and only has insigniﬁcant distinctions in the area of critical regimes. The comparison of experimental results obtained for various positions of the conventional center-of-gravity x ¯cg shows, that with the aft displacement of the rotation axis position the signiﬁcant transformation of areas of the aerodynamic characteristics ambiguity is observed, at the same time the difference between normal force coefﬁcients at the increase and reduction of the angle of attack is smaller. On the basis of the carried out analysis it is possible to make a conclusion that at the unsteady movement there is an essential transformation of both the ambiguity areas of aerodynamic characteristics and their values in comparison with a steady ﬂow; it is caused by changing of critical angles of attack at which a ﬂow transformation occurs both for a ﬂow with separation and without one. Thus, in the unsteady supersonic ﬂow such factors as frequency, amplitude of oscillations and position of the rotation axis (position of the conventional centerof-gravity) exert an inﬂuence on the aerodynamic characteristics of the body. The degree of this inﬂuence depends on the speed of the free stream. At M∞ values, close to the speeds corresponding to the moment of the ﬂow structure transformation from separated to continuous (with local separated zones), we have the greatest change of aerodynamic characteristics in comparison with those under steady conditions. Work is performed with ﬁnancial support of the Ministry of Education of the Russian Federation: grant 2-06.8-1134.

REFERENCES 1. Belotserkovsky, S.M., Skripach, B.K., Tabachnikov, V.G. Wing in Nonstationary Gas Stream (in Russian). – Moscow: “Nauka”, 1971. – 757 p. 2. Krasnov, N.F., Koshevoy, V.N., Kalugin, V.T. Aerodynamics of Separated Flows (in Russian). – Moscow: “Vysshaya Shkola”, 1988. – 349 p. 3. Kalugin, V.T., Lutsenko, A.Yu., Stolyarova, Ye.G. Experimental Research of Aerodynamic Hysteresis in Transonic Flow Around Segment-Blunted Bodies with Flat Sides (in Russian) // “Vestnik MGTU”, series “Mashinostroenie”. – 2002. – No 3, pp. 15– 30. 4. Krasnov, N.F. Aerodynamics of Body of Revolution. – New York, 10017, 52 Vanderbuilt Avenue: American Elsevier Publishing Company, Inc., 1970. – 867 p. 5. Korst, H.H. A theory for Base Pressure in Transonic and Supersonic Flow // Journal of Applied Mechanics. Vol. 23, 1956, pp. 593–600.

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V.T. Kalugin (b. 1949) graduated from the Bauman Moscow Higher Technical School in 1972. D. Sc. (Eng.), professor of “Ballistics and Aerodynamics” department of the Bauman Moscow State Technical University. Author of over 200 publications in the ﬁeld of aero-gas dynamics of jet and detached ﬂows, design of ﬂight controls. A.Yu. Lutsenko graduated from the Bauman Moscow Higher Technical School in 1987. Ph. D. (Eng.), assoc. professor of “Ballistics and Aerodynamics” department of the Bauman Moscow State Technical University. Author of more than 40 publications in the ﬁeld of aero-gas dynamics of continuous and separated ﬂows. Ye.G. Stolyarova graduated from the Moscow Aviation Institute n. a. S. Ordzhonikidze in 1971. Ph. D. (Eng.), assoc. professor of “Ballistics and Aerodynamics” department of the Bauman Moscow State Technical University. Author of more than 40 publications in the ﬁeld of non-stationary aerodynamics of continuous and separated ﬂows.

BMSTU Press has published the book: Kalugin V.T. Aero-Gas Dynamics of Flying Vehicle Controls (In Russian). – M.: Izdatelstvo MGTU imeni N.E. Baumana, 2004. – 648 p. Results of study of various ways to control the aerodynamic characteristics of rockets, rocket stages and descending spacecrafts are presented. Methods of mathematical and physical simulation of ﬂows around ﬂight controls are given. The book material on aerodynamic, jet and gas dynamical controls is classiﬁed, which makes possible the creation of the data base for aero-gas dynamical design of controlling and braking devices for ﬂying vehicles. The book is intended for under-graduate and post-graduate students of aviation and rocket and spacecraft trades and technical universities. It may be also useful for engineers and researchers specializing in the ﬁeld of aero-gas dynamics and design of ﬂying vehicles.

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Yu.I. Dimitrienko, M.L. Glazikov (Bauman Moscow State Technical University)

LOCAL TRANSPORT PHENOMENA IN POROUS PERIODIC STRUCTURES The present paper continues investigations devoted to application of the asymptotic averaging method to the transport theory in porous media. The method of solving so-called local problems of gas dynamics over a periodic cell has been developed there. Solution of the problems allows us to ﬁnd accurate local distributions of velocities and pressure inside a separate pore, and also to evaluate the gas-permeability coefﬁcient when only geometric shape of pores is known. Computed results are shown and they are compared with experimental data.

1. Introduction. At present in many works, heat and mass transport processes in porous media are investigated only as macroscopic phenomena: Darcy’s law or its modiﬁcations are postulated for relatively slow processes, and dynamical equations of gas/ﬂuid motion through a porous structure are considered for rapidly running processes. Considerable local effects occuring in an individual pore are taken into account only integrally (Darcy’s law includes them in the form of average speed of gas motion and the permeability coefﬁcient determined in experiments). For dynamical problems, speciﬁc hypotheses on local functions are usually introduced (for example, the hypothesis on uniform distribution of gas speeds in a pore etc.). These macroscopic approaches maintain comparative simplicity of the mathematical problem of transport (the set usually includes parabolic equations of ﬁltration or dynamical equations of gas dynamics), but they can not take into account the actual transport processes running in porous structures with complex internal geometry of pores. But the role of local processes (i.e. occuring in an individual pore) is of great importance: character of internal geometry of pores and processes of gas (ﬂuid) motion in a pore determine permeability coefﬁcients of the whole porous system, values of which are known to change within the limits of several orders of the magnitude. Moreover, with the help of reasonable construction of geometry of porous structures we can achieve desired permeability macrocharacteristics. This problem is especially actual for designing of special technical ﬁlters. Taking account of the above-mentioned local transport effects in porous systems can be efﬁciently realized with the help of the asymptotic averaging method for porous periodic structures which has been suggested in our preceding works [3–5]. This method is based on the idea of homogenization of periodic media [2, 7, 1, 4] and introduction of two scale coordinates. Works by Dimitrienko [3–5] formulated so-called local gasodynamics problem over a periodic cell within the frames of “slow” ﬂows, and work by Dimitrienko [5] presented the similar problem for dynamical rapidly running processes. 50

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

These local problems have some speciﬁc peculiarities: they are of integrodifferential type and include periodic boundary conditions. There is no numerical investigation of these problems in scientiﬁc works up to now (works [3, 4]) suggested only approximate solution). The purpose of the present work is to investigate the local gasodynamics problem over a periodic cell and to develop computation methods for solving the problem for some typical geometric structures of pores. It should be noted once again that solving these problems is conducted for the ﬁrst time here. 2. Main Assumptions. Let us consider a porous medium ﬁlled with linearly viscous gas (or compressible ﬂuid). The medium V is assumed to have a periodic structure (Fig. 1) so that a periodic cell (PC) can be separated. Domains Vg and Vs occupied by gas and solid phase, respectively, are oneconnected in the three-dimensional case. All the assumptions in detail have been given in works [3, 4]. The present paper considers the model two-dimensional case, where pores are longitudinal channels along the coordinate axis Ox3 , and domain Vs stops keeping one-connected. This scheme simulates to certain degree, for example, a set of long plane cracks ﬁlled with gas in rocks or a set of longitudinal cracks in composite ﬁbrous unidirectional materials which appear at the ﬁber-matrix interface in solidiﬁcation. Let l0 be the characteristic size of periodic cell Vξ , and x0 be the characteristic global size of the whole porous medium V where the following relationship is satisﬁed: κ = l0 /x0 ¿ 1 (κ is the small parameter). The parameter κ has been introduced in detail in works [3, 4]. Let us also introduce local coordinates ξ = x ¯/κ, where x ¯ = x/ξ0 are the dimensionless global Cartesian coordinates. Then all main functions Ω describing a gas ﬂow in pores can be considered to be quasiperiodic (i.e. depending on ξ and x ¯) and periodic in ξ. Differentiation of the functions is realized by the following

Fig. 1. A scheme of periodic structure and periodic cell (PC)

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51

rule: ¯i ) + (1/κ)(∂Ω/∂ξi ). ∂Ω(x, ξ)/∂xi −→ (∂Ω/∂ x Below the following notation for derivatives with respect to local and global coordinates will be also used: ∂Ω/∂ x ¯i ≡ Ω,i ,

∂Ω/∂ξi ≡ Ω/i .

Index i takes the values i = 1, 2, 3 for three-dimensional case, and i = 1, 2 for two-dimensional periodic structure. The gas ﬂow process in pores is assumed to be sufﬁciently slow so that inert terms in the momentum equation can be neglected, and we will consider only quasistatic equations of equilibrium. The solid phase is considered to be non-deformable, and gas (ﬂuid) is compressible, linearly viscous, perfect. Thermal effects in this work are not considered (i.e. the ﬂow is isothermal). 3. Statement of the Local Problem. Gas motion through porous medium V in the frames of the above-mentioned assumptions is described by the set of dimensionless equations (see [3, 4]) in the Cartesian coordinate system: ρv¯i ),i = 0, (∂ ρ¯/∂t) + (¯ (1) ¯ v¯i,jj = 0, x ¯i ∈ Vg , −p¯,i + µ v¯i |Σsg = 0, p¯ = Aρ¯. Here p¯, ρ¯, v¯ are the dimensionless pressure, density and velocity of gas, respectively; p, ρ, v are their dimensional values; p0 , ρ0 , v0 are their typical magnitudes so that: p¯ = p/p0 , ρ¯ = ρ/ρ0 , v¯ = v/v0 , and t¯ = tv0 /x0 is the dimensionless time, A = Rρ0 θ0 /p0 . Dimensionless viscous coefﬁcient µ ¯ is introduced as follows: µ ¯ = µv0 /(p0 x0 ), and µ is the true value of viscosity. The magnitude µ ¯ is connected to the Reynolds and Euler numbers: µ ¯ = 1/(Re Eu), where Re = ρ0 v0 x0 /µ and Eu = p0 /ρ0 v02 . Let us make one more assumption. Gas viscosity is considered to be sufﬁciently small so that the number µ ¯ can be written in the form ¯0 , µ ¯ = κ2 µ

(2)

where µ ¯0 has the order O(1). This assumption is practically always satisﬁed for real gases: so if p0 = 106 Pa, v0 = 10 m/s, x0 = 1 m, l0 = 10−5 m and µ = 1.5 · 10−5 Pa · s (gas CO2 ), then κ = 10−5 ,

µ ¯ = 1.5 · 10−10

and we obtain µ ¯ 0 = 1.5. 52

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

Since domain Vg has a complex porous structure (Figure 1), immediate computation of the problem (1) is practically impossible. However, due to periodicity of the structure, the solution may be represented in the form of asymptotic expansion in terms of parameter κ [3, 4]: (0)

(1)

v¯i = vi (x, ξ, t) + κvi (x, ξ, t) + O(κ2 ), ρ¯ = ρ(0) (x, t) + κρ(1) (x, ξ, t) + O(κ2 ),

(3)

p¯ = p(0) (x, t) + κp(1) (x, ξ, t) + O(κ2 ). Substitute the expansions (3) into set (1), taking account of the above-mentioned differentiation rules and assumptions (2) collect terms with the same powers of κ and equate them to zero. On collecting terms with lowest powers of κ, we obtain the desired local problem over periodic cell Vξg [3, 4]: (0) vi/i = 0, (1) (0) (0) (4) ¯0 vi/jj = p,i ξi ∈ Vξg , −p/i + µ (0) vi |Σξsg = 0. This problem statement includes the additional condition that average pulsations of pressure hp(1) i are equal to zero, where Z Z 1 (1) (1) p dVξg , ϕg = dVξg (5) hp i = |ϕg | Vξg Vξg is the integral over domain Vξg occupied by gas in PC. The necessity of this condition is explained in works (Dimitrienko, 1997, 1998). Moreover, the set (4) is (0) complemented by the condition on periodicity of functions p(1) and vi which can be written, for example, for 2-D case as follows: [[Ω]] = 0 =⇒ Ω(ξ1 , −1/2) = Ω(ξ1 , 1/2), Ω(1/2, ξ2 ) = Ω(−1/2, ξ2 ),

Ω = {v (0) , p(1) }.

(6)

Periodic cell Vξ in this case is a square with unit side, and the local coordinate system ξi is placed into its center. (0) Pressure gradient p,i in (4) depends only on global coordinates, and this is considered in the problem (4)–(6) as known value (input data of the problem). Due to linearity of local problem (4)–(6), its solution can be always written (0) formally as a linear function of input data, i.e. of p,i : k X (1)

p

(α)

=

P

(ξ)p(0) ,α ,

α=1

(0) vi

k 1 X (α) = W (ξ)p(0) ,α , µ ¯ 0 α=1 i

(7)

(α)

where functions P (α) (ξ) and Wi

(ξ) depend only on coordinate ξ.

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53

On substituting expression (7) into the local problem (4) - (6), after elimination of p(0) we obtain: (α) Wi/i = 0, (8) (α)

(α)

−P/i + ∆Wi

(α)

= hi , ξi ∈ Vξg ,

(9)

(α)

Wi

|Σξg = 0,

hP (α) i = 0,

(10) (11)

(α)

[[Wi

]] = 0,

[[P (α) ]] = 0

(12) (α)

— the collection of k local problems to determine functions P (α) (ξ), Wi (ξ). (α) (α) Here ∆Wi = Wi/jj is Laplacian operator. Unlike problem (4)–(6), the problems (8) do not include any properties of gas phase and do not depend explicitly on input data. Solution of problems (8) is determined only by internal geometric structure of pores, therefore their solution may be applied (taking account of the assumptions made in paragraph 1) for computations of transport processes for any gaseous and ﬂuid media. (α) The object hi is introduced as follows: ( 0, i 6= α or (i = α and p(0) ,α ≡ 0), (α) hi = (13) (0) 1, i = α and p,α 6= 0, that allows us to take into account the singular case of the structure where there are no through pore channels along coordinate direction xk . Figure 2 shows examples of singular and nonsingular (branching) porous structures for 2-D case. Singular structures for 2-D case can be 1-channel or 0-channel (completely closed porosity). (0) (k) For the closed porosity we have: p,k ≡ 0, hi = 0 ∀k. It follows from (8) that (k)

Wi ≡ 0, p(k) =0, i.e. there is no ﬂow. Therefore, this case will not be considered below.

Fig. 2. Examples of nonsingular (aa) and singular (1-channel (bb) and closed porosity (cc) porous structures

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In general, the number of through channels in a porous structure is equal to number k in formula (7). 4. Solving Method of Local Problem (8) for 2-D Structures. For 2-D porous structures in (8)–(12) index i takes values 1 and 2. For every ﬁxed value of k, the problem (8)–(12) is a stationary problem of ﬂowing some imaginary linear-viscous incompressible ﬂuid but under speciﬁc additional conditions. In the ﬁrst place, since function p(1) in (7) physically means a pulsation of gas pressure p¯ with respect to the average value p(0) : it follows from (3) that κp(1) = p¯ − p¯(0) + O(κ2 ), and hp(0) i = p¯ due to hp(1) i = 0, so sign of p(1) as well as the sign of function P (k) may be both positive and negative. In the second place, since there is condition hP (k) i = 0 there, the problem (8)– (12) proves to be integrodifferential, that together with periodicity conditions for (k) functions Wi and P (k) considerably complicates its solving. The following theorem allows us to simplify solving the problem. Theorem. Let 2-D porous structure (1-channel or branching) have a mirror symmetry about planes Oξ1 ξ2 and Oξ2 ξ3 , then solution of the problem (8)–(12) (α) Wi , P (α) can be constructed with the help of symmetric or antisymmetric extenf (k) , Pe(k) which are deﬁned in a quarter sion (according to Figure 3) of functions W i of PC (in the ﬁrst quadrant Veξg ), and they are solutions of the following problems: (α)

Wi/i = 0, (α) f (α) = h(α) , −Pe/i + ∆W i i

(14) ξi ∈ Veξg ,

f (α) |Σ = 0, W ξsg i

(15) (16)

Fig. 3. Symmetric and antisymmetric extensions of solution for the problem (8)–(12) VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

55

ξα = 0 :

f (α) = 0, W α/α

f (α) = 0, Pe(α) = 0, W β

ξα = 1/2 : ξβ = 0 :

f (α) = 0, W α/β

fα(α) = 0, W

ξβ = 1/2 :

f (α) = 0, Pe (α) = 0, W β

(17) (18)

f (α) = 0, Pe (α) = 0, W β /β

Pe (α) = 0, α, β = 1, 2, α 6= β.

(19) (20)

Sign “+” in Figure 3 means that a function at passage from the ﬁrst quadrant into the considered one does not change its sign, and sign “−” means the change. Proof of the theorem will be given in Appendix. Local problems (14)–(20) are considerably easier for numerical solving because they do not include integral and periodic conditions. To solve the problems we can apply the method of curl and stream functions [6]. Introduce functions of curl ω (α) and stream ψ (α) for every of problems (14)–(20) as follows: f (α) − W f (α) , ω (α) = W (21) α/β β/α f (α) = ψ (α) , W α /β

f (α) = ψ (α) α 6= β, α, β = 1, 2, W β /α

(22)

then equations (14) are satisﬁed identically, and for ω (α) , ψ (α) and Pe (α) we obtain the following equation set in the standard way [6]: (α) ∆ω = 0, (23) ∆ψ (α) = ω (α) , ∆Pe (α) = 0, ξ ∈ Ve . i ξg However, the boundary conditions for these functions following from (16)–(20) differ from the well-known conditions for curl and stream functions [6]. On differentiating (20) with respect to ξα and then with respect to ξβ , we get: (α)

(α)

(α)

(α)

f f −W , ω/α = W α/βα β/αα

(α)

(α)

f f ω/β = W −W . α/ββ β/βα

Having substituted these expressions into equation (15) with account of (14) and (13), we ﬁnd the relationships (α) (α) ω/α = −Pe/β ,

(α) (α) ω/β = hαα + Pe/α ,

α 6= β.

(24)

On applying these relations for boundaries ξα = 0, ξα = 1/2 and ξβ = 1/2, we ﬁnd additional boundary conditions on the surfaces. Then with taking account of f (α) = 0 at ξα = 0, f (α) = 0 at ξα = 0 is always satisﬁed if W that the condition W β α/α we obtain systems of three equations (23): there are three boundary conditions at each boundary of domain Veξg : Z ξβ (α) (α) (α) α ξα = 0 and ξα = 1/2 : ψ/α = 0, ω = hα ξβ + Pe/α dξ, Pe (α) = 0, 0 (α)

ξβ = 0 :

ψ

(α)

= 0,

ω

= 0,

(α) Pe/β Z ξα

(α)

ξβ = 1/2 :

ψ/β = 0,

(α) Pe/β dξ,

ω (α) = −

(25)

= 0, Pe (α) = 0.

0

56

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

To ﬁnd boundary conditions for functions ψ (α) , ω (α) and Pe(α) on surface Σsg of contact with solid phase, we apply the expansion of the stream function in the vicinity of the surface (Roache, 1976). For example, let equation of the surface (or of its part) have the form ξβ = S(ξα ), where S(ξα ) is the one-valued differentiable function. Then, on passing to curvilinear coordinates ξ¯α connected to this surface as follows: ξ¯α = ξα , ξ¯β = aξβ /S(ξα ), 0 < a < 0.5, we obtain the desired boundary condition on surface Σξg : ξ¯2 = const :

ω (α) =

2[ψ (α) (ξ¯α , ξ¯β − ∆ξ¯β ) − 2ψ (α) (ξ¯α , ξ¯β )]S 2 , ∆ξ22 )a2 + (ξ¯β ∂S/∂ ξ¯β )2 )

(26)

ψ (α) = const. The third condition at the boundary for function Pe (α) we obtain by combining both conditions (24): (α) (α) ∂ Pe (α) /∂ ξ¯β = (ω/β − hαα )(∂ξ α /∂ ξ¯β ) − ω/α (∂ξ β /∂ ξ¯β ),

α 6= β.

(27)

In the similar way, we may ﬁnd conditions on the part of surface Σsg where there is no one-to-one connection ξα = S(ξβ ) (at vertical and horizontal walls of pores). 5. Numerical Solution of the Local Problem. The problem (23) for every of values α = 1, 2 with boundary conditions (25)–(27) is the set of three classical elliptic equations with boundary conditions of the standard type, when either the function itself or its derivative at the boundary is given. To solve the problem we may apply corresponding numerical methods for similar equations [6]. The present work demonstrates the iterative method combined with the method of rectifying coordinates ξ¯α transforming domain Veξg to the domain with rectangular boundaries. We have also applied the scheme by Pismen–Reckford with the help of the longitudinal-cross sweep method. Consider some results of computations. The computations have been conducted for one-channel porous medium (Figure 2, c). For this case, we have the only funcf (1) ≡ W f1 , tions’ set not equal to zero: ψ (1) ≡ ψ, ω (1) ≡ ω, Pe (1) ≡ Pe and W 1 (1) f2 . f ≡W W 2 The surface shape Σξg has been chosen cosine: ξ2 = S(ξ1 ) = (b + a + +(b−a)(cos 2πξ1 )1+n )/2, where n, a and b are the parameters: a = 0.04, b = 0.4, n ≥ 0. f2 depending on coordinates f1 and W Figures 4, a–e show functions ψ, ω, Pe , W ξ1 , ξ2 for n = 0. Typical values of these dimensionless functions are indicted in parentheses there. The stream function (−ψ) has a maximum on surface Σξg of solid phase, and the curl function ω — on symmetry surface ξ1 = 0 in the zone of the largest pore diameter. Pressure pulsation function Pe takes both negative values in a wide part of the pore and positive values in a narrow section of the pore. This picture may be explained physically by that in the narrow section the total gas pressure

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57

f2 (e) depending on coordinates ξ1 , f1 (dd ) and W Fig. 4. Functions ψ (aa), ω (bb ), Pe (cc), W ξ2 for n = 0

58

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

exceeds the average pressure in PC, and in the wide section the total gas pressure is smaller than p(0) . f1 takes negative values, and its absolute magnitude has maximum Velocity W equal to 0.47 in the most narrow section of the pore near the solid wall. Transf2 has a maximum, being sufﬁciently high and equal to 0.1, in the verse velocity W zone of maximum bending the surface Σξg (from the side of narrow pore section). f2 in comparison with velocity These considerable values of transverse velocity W f1 demostrate the brightly expressed not one-dimensional character of the local W transport process in the porous medium. 6. Comparison with Poiseuille’s Flow. At a = b, for the considered example of one-channel structure we have a porous medium with plane straight channels with a constant cross-section. For this case, the problem (20)–(23) has the following exact solution: ψ = −0.5ξ2 (a2 − ξ 2 /3),

ω = ξ2 ,

w = −0.5(a2 − ξ22 ),

(28)

that precisely corresponds to Poiseuille’s ﬂow of linearly viscous ﬂuid. Table 1 shows testing results computed by formulas (28) and obtained by immediate calculation from equations (23) for a = b. So the numerical method is seen to maintain a quite satisfactory accuracy of computations. Table 1 Distributions of stream ψ, curl ω functions and speed W along coordinate ξ2 of the pore at a = b, where numerator is exact solution (28), and denominator is computed solution of system (23)

ξ2

0

0.04

0.08

0.12

0.16

0.2

0.24

0.28

0.32

0.36

0.4

−ψ · 102

0 0

0.32 0.3

0.63 0.6

0.93 0.88

1.21 1.15

1.47 1.39

1.69 1.59

1.87 1.76

2.01 1.89

2.1 1.96

2.13 1.97

ω

0 0

0.04 0.04

0.08 0.08

0.12 0.12

0.16 0.16

0.2 0.2

0.24 0.24

0.28 0.28

0.32 0.32

0.36 0.36

0.4 0.4

W · 103

8 7.6

7.92 7.52

7.68 7.28

7.28 6.88

6.72 6.32

6.0 5.6

5.12 4.7

4.08 3.68

2.88 2.48

1.52 1.12

0 0

7. Determination of Permeability Coefﬁcient. After the local transport paf (k) , Pe (α) , ψ (α) and ω (α) in Veξ have been determined, in accordance rameters W β with Theorem we may ﬁnd corresponding functions in the whole PC. Substitute the functions into relatioships (7) and average the obtained expressions over the PC. Then we ﬁnd: (0) (0) µ0 )p,j , (29) hvi i = −(K ji /¯ where (j)

K ji = −hWi i.

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(30)

59

The relationship (29) is the ﬁltration law by Darcy works (see [3, 4]), and coefﬁcients K ji are gas permeability coefﬁcients of the porous medium. On using (α) Theorem, we obtain that matrix K ji is diagonal, because hWβ i = 0 at α 6= β due (α)

to antisymmetry of functions Wβ at α 6= β. For the special case when porosity is one-channel (for example, along axis (1) (2) Oξ1 ), then, as it has been shown above, Wi 6= 0 and Wi = 0, thus (1)

all K ji = 0 except K 11 = K = −hW1 i.

(31)

On passing to the integral over Veξ , we obtain Z1/2Za 4 K=− a

f (1) (ξ1 , ξ2 )dξ2 S(ξ1 )dξ1 . W 1 0

(32)

0

For the example considered in paragraph 5, magnitudes of dimensionless gas¯ for different values of parameter n characterizing the permeability coefﬁcient K pore shape and different porosities are given in Table 2. Table 2 Values of gas permeability coefﬁcient for different porosities and different shapes of pores

ϕg

0.44

0.48

0.52

0.60

0.80

K · 10 at n = 0

5.19

5.75

6.25

7.19

9.22

K · 103 at n = 2

3.53

4.15

5.12

6.08

9.22

3

The computed results allow us to make the following important conclusion. At n = 0 the pore has such shape that passage from maximum size at ξ1 = 0 to minimum at ξ1 = 1/2 is more smooth than at n = 2. This favours increasing f (1) and thus increasing gas permeability coefﬁcient K. Therefore, gas velocity W 1 we may change gas permeability of porous media with varying only pores’ shape without changing the total porosity. 8. Comparison with Experiments. Work by Dimitrienko [4] shows data on gas permeability of composite materials with different porosities ϕg . Pores’ shape may be considered appoximately as two-dimensional due to the presence of a set of parallel ﬁbers in a composite which promote the appearance of sufﬁciently large longitudinal cracks and pores. A typical size of these pores is close to monoﬁbers’ diameter and is equal to l0 ≈ 10−5 m. Experimental value of dimensionless permeability coefﬁcient K at porosity ϕg = 0.4 is K = 4·10−3 . Computed results shown in Table 2 give K = 5.19·10−3 . With taking account of rather undeterminate shape of pores in experimental data and statistic scattering, the obtained computed results should be considered as sufﬁciently accurate. 60

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

9. Conclusions. 1. The present paper continues investigations of works [3–5]. As a result, the local not one-dimensional problem of transport in porous periodic medium has been solved numerically here for the ﬁrst time in frames of the asymptotic averaging method. 2. Sufﬁciently general approach to solving the local problem has been developed, which allows us to compute a faithful (in the asymptotic sense: at κ −→ 0) local distributions of speeds and pressure inside a separate pore for 2-D case. 3. The calculation method has been developed for gas permeability coefﬁcient of porous medium when only internal pores’ geometry is known. 4. Computations conducted as the example have shown that a local gas ﬂow in pores is considerably nonuniform, and pores’ shape (at the same total porosity) has an essential effect on the value of the total gas-permeability coefﬁcient. 5. Comparison of computed results of gas permeability with experimental data for composite materials allows us to tell about sufﬁciently high accuracy of computations by the developed method. Appendix. Prove Theorem from paragraph 4. Consider the case when α = 1. Introduce new coordinates: ξ¯1 = −ξ1 , ξ¯2 = ξ2 and extend the problem (10)–(16) solution from the ﬁrst to the second quadrant according to Figure 2, a, i.e. assume (1) f (ξ1 , ξ2 ), W (1) (ξ¯1 , ξ¯2 ) = −W f (1) (ξ1 , ξ2 ), P (1) (ξ¯1 , ξ¯2 ) = that W1 (ξ¯1 , ξ¯2 ) = W 2 2 −Pe (1) (ξ1 , ξ2 ). For this substitution, differential operators in the second quadrant are connected to corresponding operators of the ﬁrst quadrant as follows: (1) f (1) , Wi//i = −W i/i

(1) (1) P//1 = Pe/1 ,

(1) (1) P//2 = −Pe/2 ,

and (1) (1) f (1) , W1//11 + W1//22 = ∆W 1

(1) (1) f (1) , W2//11 + W2//22 = −∆W 2

where //i = ∂/∂ ξ¯i . This means that in equations (4), (5) (when i = 2) at passage to the second quadrant all summands change in their signs (here h12 = 0), and in equation (5) (when i = 1) they do not change. Therefore, in the second quadrant equations (8)–(10) are satisﬁed. Boundary conditions (16)–(20) are evident to maintain the satisfaction of conditions (10), (12) at passage to the second quadrant. Moreover, due to antisymmetry of function P (1) , integral condition (11) is satisﬁed as well. In the similar way, we can prove that extension of the solution to the third and fourth quadrants realized according to Figure 3 maintains the satisfaction of all equations (8)–(12) in these quadrants. The above is also valid when α = 2 with taking account of symmetric and antisymmetric extensions of the functions by data of Figure 3, c.

REFERENCES 1. Bakhvalov, N. & Panasenko, G.: 1989, Homogenisation: Averaging Processes in Periodic Media (Mathematical Problems in Mechanics of Composite Materials), Kluwer Academic Publishers.

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2. Bensoussan, A., Lions, J.L. & Papanicolaou, G.: 1978, Asymptotic Analysis for Periodic Structures, North Holland, Amsterdam. 3. Dimitrienko, Yu.I.: 1997, Heat-mass transport and thermal stresses in porous charring materials, Transport in Porous Media, 27 (2), 143–170. 4. Dimitrienko, Yu.I.: 1998, Thermomechanics of Composites under High Temperatures, Kluwer Academic Publishers, Dordrecht. 5. Dimitrienko, Yu.I.: 1999, Dynamic transport phenomena in porous polymer materials under impulse thermal effects, Transport in Porous Media, 35, 299–326. 6. Roache, P.: 1976, Computational Fluid Dynamics, Hermosa Publishers, Albuquerque. 7. Sanchez-Palencia, E.: 1980, Non-homogeneous media and vibration theory. New– York, Springer-Verlag.

Yu. I. Dimitrienko graduated fom the Lomonosov Moscow State University in 1984. D.Sc (Phys.-Math.), professor, head of “Computational Mathematics and Mathematical Physics” department of the Bauman Moscow State Technical University, academician of the Russian Academy of Engineering Sciences. Author of over 100 publications in the ﬁeld of thermal mechanics, composite mechanics, multiphase media mechanics, mathematical modelling and numerical methods for problems of mathematical physics. M.L. Glazikov graduated from the Bauman Moscow State Technical University in 1999. Assistant of “Computational Mathematics and Mathematical Physics” department of the Bauman Moscow State Technical University. Author of a number of publications in the ﬁeld of numerical methods of the ﬁltration theory.

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VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

CONTROL SYSTEMS A. Golubev∗ , R. Johansson∗∗ , A. Robertsson∗∗ , S. Tkachev∗ (∗ Bauman Moscow State Technical University), (∗∗ Lund Institute of Technology, Sweden)

OUTPUT TRACKING FOR A CLASS OF NONLINEAR NONMINIMUM-PHASE SYSTEMS USING OBSERVER BACKSTEPPING In this paper we address the problem of output tracking for a class of nonlinear nonminimum-phase systems with linear zero dynamics. Using observer backstepping a globally stabilizing control law is designed to force the tracking error to converge asymptotically to zero. The observer backstepping procedure is extended to global stabilization of a class of nonlinear dynamical systems in the output-feedback form with linear unstable zero dynamics. An output feedback controller is designed for a ﬂexible one-link robot arm as an illustrative example.

Introduction. The integrator backstepping [1, 2] is an effective control design tool for stabilization of nonlinear dynamical systems that can be transformed into the strict feedback form x˙ 1 = x2 + f1 (x1 ), x˙ 2 = x3 + f2 (x1 , x2 ), .. .

(1)

x˙ n−1 = xn + fn−1 (x1 , . . . , xn−1 ), x˙ n = u + fn (x1 , . . . , xn ), where u ∈ R is the control input, under the assumption that the full state x = = (x1 , . . . , xn )T ∈ Rn of the system is measured. In more realistic problems, where only part of the state is available for measurement, ﬁrst an observer system is designed, that is a dynamical system which provides sufﬁciently fast (for instance, exponentially) convergent estimates of the unmeasured states. Then the so called observer backstepping procedure [1, 2] can be used for stabilizing control design. It consists in the following. The integrator backstepping is applied to a new system in which the equations of the unmeasured states have been replaced by the corresponding equations of their estimates from the observer. At each step of the procedure observation errors are treated as disturbances and accounted for using the so called nonlinear “damping” [2]. Further, we consider the output tracking problem for nonlinear dynamical systems written in the output-feedback form

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

63

x˙ 1 = x2 + ϕ1 (y), .. . x˙ r−1 = xr + ϕr−1 (y), x˙ r = xr+1 + β(y)u + ϕr (y), x˙ r+1 = xr+2 + b1 β(y)u + ϕr+1 (y), .. .

(2)

x˙ n−1 = xn + bm−1 β(y)u + ϕn−1 (y), x˙ n = bm β(y)u + ϕn (y), y = x1 , where the nonlinearities ϕi (·) ∈ C ∞ (R), i = 1, n depend only on the output y = x1 of the system; bj , j = 1, m are some constants; n − r = m; β(y) 6= 0 for all y ∈ R. In (2) only output of the system is supposed to be available for measurement. For the considered class of systems (2) our purpose is to design a stabilizing feedback control law that uses information only about the measured output y, guarantees global boundedness of any solution x(t) of the closed-loop system (2) and forces the tracking error y(t) − yd (t) to converge to zero as t → ∞ for all x(0), where yd (t) is a sufﬁciently smooth reference output trajectory, with its derivatives (n) y˙ d , . . . , yd being known and bounded on [0, ∞). Note that a global exponential observer for (2) is x ˆ˙ = Aˆ x + GC(ˆ x − x) + ϕ(y) + Bβ(y)u,

(3)

where x ˆ ∈ Rn ; the matrices A, B correspond to the system (2) written in vector form; ϕ(y) = (ϕ1 (y), . . . , ϕn (y))T ; C = (1, 0, . . . , 0); G = (g1 , . . . , gn )T is chosen so that A + GC is Hurwitz. Therefore, state estimation error e = x ˆ−x equation is e˙ = (A + GC)e (4) and its equilibrium e = 0 is globally exponentially stable [3]. By the analysis of the system (2) the notion of zero dynamics [4] is very useful. Note that for the inital conditions x1 (0) = 0, x2 (0) = −ϕ1 (0), . . ., xr (0) = = −ϕr−1 (0) and any xr+1 (0) = xr+1,0 , . . . , xn (0) = xn,0 the input u(t) that keeps the output y(t) indentically equal to zero for all t > 0 is uniquely deﬁned and given by xr+1 (t) + ϕr (0) . (5) u(t) = − β(0) The zero dynamics [4] of the system (2), that is dynamics of (2) with the control law (5) and the above initial conditions x1 (0), . . . , xn (0), have the following form: x˙ r+1 = xr+2 − b1 xr+1 + ϕr+1 (0) − b1 ϕr (0), .. . (6) x˙ n−1 = xn − bm−1 xr+1 + ϕn−1 (0) − bm−1 ϕr (0), x˙ n = −bm xr+1 + ϕn (0) − bm ϕr (0). 64

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

If the system (2) is minimum-phase, that is the equilibrium point of the zero dynamics (6) is globally asymptotically stable, then one of the globally stabilizing feedback control laws obtained via the observer backstepping procedure that keep any solution x(t) of the closed-loop system (2) globally bounded and force the tracking error y(t) − yd (t) to converge asymptotically to zero is [2] u=

1 (r) (αr − x ˆr+1 − yd ), β(y)

(7)

with αi , zi , i = 1, r given by z 1 = x1 − y d , (i−2)

zi = x ˆi − αi−1 (y, x ˆ1 , . . . , x ˆi−1 , yd , . . . , yd α1 = −c1 z1 − d1 z1 − ϕ1 (y),

i−1 X ∂αi−1

i−2 X ∂αi−1

x ˆ˙ j + ∂x ˆj j=1

(j+1)

yd

−(

(j) j=1 ∂yd

,

i = 2, r,

c1 > 0, d1 > 0,

x1 − y) − ϕi (y) + αi = −ci zi − zi−1 − gi (ˆ

+

(i−1)

) − yd

∂αi−1 (ˆ x2 + ϕ1 (y))+ ∂y

∂αi−1 2 ) d i zi , ∂y

ci > 0, di > 0,

i = 2, r.

In case when (2) is nonminimum-phase, that is the equilibrium point of the zero dynamics (6) is unstable, the observer backstepping control (7) as it is given in [2] fails to achieve global boundedness of x(t). Hence, solutions x(t) of the system (2) with control (7) may be not well-deﬁned for all t > 0 and asymptotic convergence of the tracking error to zero may not take place. In this paper we solve the output tracking problem for nonlinear nonminimumphase systems of the form (2). According to the ideas presented in [5] the observer backstepping control design procedure is extended to global stabilization of nonlinear dynamical systems written in the output-feedback form (2) with linear unstable zero dynamics. As an illustrative example an output feedback controller is designed to stabilize angular position of a ﬂexible one-link robot arm. Output tracking in case of one-dimensional zero dynamics. Consider ﬁrst the system (2) with n − r = 1: x˙ 1 = x2 + ϕ1 (y), .. . (8) x˙ n−2 = xn−1 + ϕn−2 (y), x˙ n−1 = xn + β(y)u + ϕn−1 (y), x˙ n = b1 β(y)u + ϕn (y), y = x1 . The linear change of variables given by k1 k2 χ1 1 0 χ2 .. = 0 1 . .. . . . . χn 0 0

. . . kn x1 ... 0 x2 ... 0 .. . . .. . .. xn ... 1 0 k3 0 0 .. .

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

(9)

65

with ki = (−1)i+1 b1 −i , i = 1, n transforms (8) into the form χ˙ 1 = −b1 χ1 + χ2 + ϕn (y), χ˙ 2 = χ3 + ϕ1 (y), .. . (10) χ˙ n−1 = χn + ϕn−2 (y), χ˙ n = LT χ + β(y)u + ϕn−1 (y), y = χ2 , n X

1 k1 kn−1 T ,− ,...,− ) ∈ Rn . kn kn kn i=1 An exponential observer for the system (10) is written as

where ϕn (y) =

ki ϕi (y); L = (

χˆ˙ = Aχˆ + GC(χˆ − χ) + φ(y) + Bβ(y)u,

(11)

where χˆ ∈ Rn ; φ(y) = (ϕn (y), ϕ1 (y), . . . , ϕn−1 (y))T ; the matrices A, B refer to the vector form of (10); C = (0, 1, 0, . . . , 0). In (11) the gain vector G = = (g1 , g2 , . . . , gn )T is chosen so that the matrix A + GC is Hurwitz. The state estimation error e = χˆ − χ equation is of the form (4), where the matrices A, G, C refer to the matrices in (11). Hence, its equilibrium e = 0 is globally exponentially stable [3]. A Lyapunov function for the system (4) is taken [3] as Vo (e) = eT Po e with its derivative satisfying V˙o (e)|(4) = −eT Qo e ≤ def

≤ −λmin (Qo )| e |2 = −l| e |2 . Here | · | denotes the Euclidean norm on Rn , the matrices Po = Po T > 0 and Qo = Qo T > 0 satisfy the Lyapunov equation (A + GC)T Po + Po (A + GC) = −Qo .

(12)

Note that, when solving the output tracking problem in question for the system (10), we cannot directly use state feedback control laws of the form u = u(χ) since only χ2 is available for measurement. Therefore, we will ﬁnd stabilizing control laws as u = u(χ2 , χ) ˆ = u(χ2 , χ + e), where the state estimate χˆ is generated by the observer (11). Further, consider the system χ˙ 1 = −b1 χ1 + χ2 + φ1 (y), χ˙ 2 = χ3 + φ2 (y), .. . (13) χ˙ n−1 = χn + φn−1 (y), χ˙ n = LT χ + β(y)u + φn (y), e˙ = (A + GC)e, y = χ2 composed of equations of the system (10) and the state estimation error e = χˆ − χ equation. Note that, if the function f (χ2 ) = χ2 + φ1 (χ2 ) deﬁnes a diffeomorphism, then def

for the system (13) the reference output trajectory y = yd (t) = χ2d (t) tracking 66

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

problem is equivalent to that of the tracking of a globally bounded reference trajectory χ1 = χ1d (t) that satisﬁes the differential equation χ˙ 1d (t) + b1 χ1d (t) = χ2d (t) + φ1 (χ2d (t)). Suppose that the reference output trajectory χ2d (·) is smooth and belongs to L1 ∩ L∞ . The conditions on the right-hand side of of the ﬁrst equation in (13) under which for any globally bounded output trajectory χ2d (·) ∈ L1 ∩ L∞ there exists a unique solution χ1d (·) ∈ L1 ∩ L∞ ∩ C 0 , χ1d (±∞) = 0 of the differential equation χ˙ 1 + b1 χ1 = χ2d (t) + φ1 (χ2d (t)) are given in [6, 7]. Further we will assume that these conditions are satisﬁed and the smooth reference trajectory χ1d (t) that corresponds to the output reference trajectory χ2d (t) is found, for instance, via the procedure given in [6, 7]. Using the fact that e = χˆ − χ, the system (13) can be written in the following form: χ˙ 1 = −b1 χ1 + χˆ2 − e2 + φ1 (y), ˆ3 + g2 e2 + φ2 (y), χˆ˙ 2 = χ .. . (14) χˆ˙ n−1 = χˆn + gn−1 e2 + φn−1 (y), χˆ˙ n = LT χˆ + gn e2 + β(y)u + φn (y), e˙ = (A + GC)e, y = χˆ2 − e2 , ˆn )T is the state of the observer (11). where χˆ = (χˆ1 , . . . , χ The standard observer backstepping design as it is given in [1, 2] cannot be directly applied to stabilization of the system (14) since the ﬁrst state χ1 in the integrator chain of (14) is not measured but estimated only. To overcome this difﬁculty an extra “damping” term is used similar to what is done in the standard observer backstepping. To solve the output tracking problem for the system (14) we use the following extension of the standard observer backstepping procedure. Step 1. Consider the ﬁrst equation of the system (14) χ˙ 1 = −b1 χ1 + χˆ2 − e2 + φ1 (y) = −b1 χˆ1 + χˆ2 + φ1 (y) + b1 e1 − e2 , where e1 (t) = χˆ1 (t) − χ1 (t), e2 (t) = χˆ2 (t) − χ2 (t). If χˆ2 were the control input, then, using a control law χˆ2 = α1 (χˆ1 , χ1d , χ˙ 1d , y), one could compensate for the uncertainties e1 , e2 in the ﬁrst equation of the system (14) and force the tracking error χ1 (t) − χ1d (t) to converge to zero as t → ∞. To ˆ1 , χ1d , χ˙ 1d , y) consider the following design the “virtual” control law χ ˆ 2 = α1 ( χ positive deﬁnite function:

V1 (z1 , e) =

1 2 z + Vo (e) > 0, (z1 , e) 6= 0, 2 1

where z1 = χ1 − χ1d , Vo (e) is the Lyapunov function for the state estimation error e = χˆ − χ equation. For convenience sake deﬁne also the error variable z2 = = χˆ2 − α1 .

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The time derivative of V1 along the solutions of (14) is expressed as V˙ 1 |(14) = z1 z˙1 + V˙ o (e)|(14) ≤ z1 (−b1 χˆ1 + χ ˆ2 + φ1 (y)+ + b1 e1 − e2 − χ˙ 1d ) − l| e |2 ≤ z1 (z2 + α1 − b1 χˆ1 + φ1 (y)+ + b1 e1 − e2 − χ˙ 1d ) − le21 − le22 . By choosing the “stabilizing function” α1 as α1 = (b1 −c1 −d11 −d12 )χˆ1 −φ1 (y)+(c1 +d11 +d12 )χ1d + χ˙ 1d ,

c1 , d11 , d12 > 0

and using the fact that χˆ1 = e1 + χ1 = e1 + χ1d + z1 , one gets V˙ 1 |(14) ≤ z1 (−c1 z1 + z2 − d11 z1 − d12 z1 + (b1 − c1 − d11 − d12 )e1 − e2 )− − le21 − le22 = −c1 z12 + z1 z2 − (d11 z12 − (b1 − c1 − d11 − − d12 )z1 e1 + le21 ) − (d12 z12 + z1 e2 + le22 ) = −c1 z12 + z1 z2 − S1 , (15) µ √

¶2

κ1 d11 z1 − √ e1 2 d11 ¶

¶ ¶2 µ √ κ21 1 2 √ e + + l− d12 z1 + e2 + 4d11 1 2 d12 µ

with S1 = µ 1 + l− e2 ≥ 0, where κ1 = b1 − c1 − d11 − d12 and d11 , d12 are chosen 4d12 2 ½ ¾ κ21 1 such that l > max , . 4d11 4d12 Note that in the stabilizing function α1 (χˆ1 , χ1d , χ˙ 1d , y) the “damping” term −(d11 + d12 )χˆ1 is introduced to complete the squares in the right-hand side of the inequality (15) and in this way to compensate for the esimation errors e1 and e2 . Hence, if χˆ2 were the control input, then by the choice χˆ2 = α1 (χˆ1 , χ1d , χ˙ 1d , y) and appropriately taken positive constants d11 , d12 the derivative V˙ 1 along the solutions of the system (14) would be less or equal to the expression that contains a negative deﬁnite part plus indeﬁnite term z1 z2 . Step 2. Consider the subsystem composed of the ﬁrst two equations of (14). It can be expressed as ˆ 2 − α1 ) − e 2 , χ˙ 1 = −b1 χ1 + α1 + φ1 (y) + (χ ˙ χˆ2 = χˆ3 + g2 (χˆ2 − y) + φ2 (y). In this step χ ˆ3 is used as virtual control input instead of χˆ2 . By using an appropriately taken virtual control law χ ˆ 3 = α2 ( χ ˆ1 , χˆ2 , χ1d , χ˙ 1d , χ¨1d , y) one can force the errors z1 = χ1 − χ1d and z2 = χˆ2 − α1 to converge to zero as t → ∞. To design this virtual control law consider the following positive deﬁnite function: 1 V2 (z1 , z2 , e) = V1 (z1 , e) + z2 2 + Vo (e) > 0, (z1 , z2 , e) 6= 0. 2 After introducing the error variable z3 = χˆ3 − α2 , the time derivative of V2 along the solutions of (14) is computed as 68

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

V˙ 2 |(14) = V˙ 1 |(14) + z2 z˙2 + V˙ o |(14) = z1 z˙1 + z2 z˙2 + 2V˙ o |(14) = = z1 (−b1 χ1 + α1 + φ1 (y) − e2 − χ˙ 1d ) + z1 (χ ˆ2 − α1 )+ + z2 ( χ ˆ˙ 2 − α˙ 1 ) + 2V˙ o |(14) ≤ −c1 z12 + z1 z2 − S1 + z2 (χˆ˙ 2 − α˙ 1 )+ + V˙ o |(14) ≤ −c1 z12 − S1 + z2 (z1 + χ ˆ3 + g2 (χˆ2 − y) + φ2 (y)− ∂α1 ˙ ∂α1 ∂α1 ∂α1 (χˆ3 + φ2 (y) − e3 ) − χ˙ 1d − χ¨1d ) − le21 − le23 . − χˆ1 − ∂y ∂ χˆ1 ∂χ1d ∂ χ˙ 1d

V˙ 2 |(14) ≤ −c1 z12 − S1 + z2 (χˆ1 − e1 − χ1d + µ ¶ ∂α1 ∂α1 + (z3 + α2 ) 1 − + g2 (χˆ2 − y) + φ2 (y) − (φ2 (y) − e3 )− ∂y ∂y ∂α1 ˙ ∂α1 ∂α1 − χ˙ 1d − χ ¨1d ) − le21 − le23 . (16) χˆ − ∂ χˆ1 1 ∂χ1d ∂ χ˙ 1d ¶ µ ∂α1 6= 0 for all y ∈ R. This assumption is equivalent to Suppose that 1 − ∂y µ ¶ ∂φ1 1+ 6= 0 for all y ∈ R. Then by the choice ∂y µ 1 α2 = (1 − ∂α1 /∂y)

− χˆ1 + χ1d − c2 z2 −

∂α1 ∂α1 ˙ ∂α1 φ2 (y) + χ˙ 1d + χˆ + ∂y ∂χ ˆ1 1 ∂χ1d µ ¶ ¶ ∂α1 ∂α1 2 + χ ¨1d − d21 z2 − d23 z2 , c2 , d21 , d23 > 0 ∂ χ˙ 1d ∂y

− g2 (χˆ2 − y) − φ2 (y) +

one gets ∂α1 )z2 z3 − S1 − S2 , V˙ 2 |(14) ≤ −c1 z12 − c2 z22 + (1 − ∂y µp

¶2 µp ¶2 µ ¶ ∂α1 1 1 √ with S2 = z2 − + d23 e3 + l− e21 + ∂y 4d 2 d23 21 ½ ¾ µ 1 1 1 2 , e ≥ 0, where d21 , d23 are taken such that l > max . + l− 4d23 3 4d21 4d23 Note that in this step the uncertainties e1 and e3 in the right-hand side of the ¶ µ ∂α1 2 z2 in inequality (16) are compensated by the damping term −d21 z2 − d23 ∂y the stabilizing function α2 (χˆ1 , χˆ2 , χ1d , χ˙ 1d , χ¨1d , y). Therefore, if χˆ3 were the control input, then, by choosing χˆ3 = α2 (χˆ1 , χˆ2 , χ1d , ¨1d , y) with the appropriately taken positive constants d11 , d12 , d21 , d23 , the χ˙ 1d , χ 1 d21 z2 + √ e1 2 d21 ¶

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

69

derivative V˙ 2 along the solutions of the system (14) would be less µ or equal to ¶ the ex∂α1 z2 z3 . pression that contains a negative deﬁnite part plus indeﬁnite term 1 − ∂y Step 3. Consider the subsystem that is composed of the ﬁrst three equations of the system (14) and is written as ˆ 2 − α1 ) − e 2 , χ˙ 1 = −b1 χ1 + α1 + φ1 (y) + (χ ˆ2 − y) + φ2 (y) + (χˆ3 − α2 ), χˆ˙ 2 = α2 + g2 (χ χˆ˙ 3 = χˆ4 + g3 (χˆ2 − y) + φ3 (y). Now analogously to the previous steps of the design use the positive deﬁnite function 1 V3 (z1 , z2 , z3 , e) = V2 (z1 , z2 , e) + z3 2 + Vo (e) > 0, (z1 , z2 , z3 , e) 6= 0 2 (3)

ˆ2 , χ ˆ3 , χ1d , χ˙ 1d , χ ¨1d , χ1d , y). in order to ﬁnd a virtual control law χˆ4 = α3 (χˆ1 , χ Using the error variable z4 = χˆ4 − α3 , the time derivative of V3 along the solutions of (14) can be evaluated as follows: V˙ 3 |(14) = z1 z˙1 + z2 z˙2 + z3 z˙3 + 3V˙ o |(14) = µ = z1 (−b1 χ1 +α1 +φ1 (y)−e2 − χ˙ 1d )+z1 (χˆ2 −α1 )+z2 α2 +g2 (χˆ2 −y)+φ2 (y)− ¶ ∂α1 ∂α1 ˙ ∂α1 ∂α1 (α2 + φ2 (y) − e3 ) − − χ˙ 1d − χ¨1d + z2 (χˆ3 − α2 )− χˆ − ∂y ∂ χˆ1 1 ∂χ1d ∂ χ˙ 1d µ ¶ ∂α1 ∂α1 z2 (χˆ3 − α2 ) + z3 (χˆ˙ 3 − α˙ 2 ) + 3V˙ o |(14) ≤ −c1 z12 − c2 z22 + 1 − − z2 z3 − ∂y ∂y − S1 − S2 + z3 (χˆ˙ 3 − α˙ 2 ) + V˙ o |(14) ≤ −c1 z12 − c2 z22 − S1 − S2 + ¶ µµ ∂α2 ∂α1 + z3 ˆ2 − y) + φ3 (y) − z2 + z 4 + α 3 + g 3 ( χ (χˆ3 + φ2 (y) − e3 )− 1− ∂y ∂y ¶ ∂α2 ˙ ∂α2 ˙ ∂α2 ∂α2 ∂α2 (3) − le23 . (17) − χ˙ 1d − χ¨1d − χ χˆ − χˆ − ∂ χˆ1 1 ∂ χˆ2 2 ∂χ1d ∂ χ˙ 1d ∂ χ¨1d 1d By choosing ¶ ∂α1 z2 − g 3 ( χ ˆ2 − y)− α3 = −c3 z3 − 1 − ∂y ∂α2 ∂α2 ˙ ∂α2 ˙ (χˆ3 + φ2 (y) + − φ3 (y) + χˆ1 + χˆ + ∂y ∂χ ˆ1 ∂ χˆ2 2 ¶ µ ∂α2 ∂α2 (3) ∂α2 2 ∂α2 χ˙ 1d + χ¨1d + χ − d33 z3 , c3 , d33 > 0, + ∂χ1d ∂ χ˙ 1d ∂ χ¨1d 1d ∂y µ

one gets V˙ 3 |(14) ≤ −c1 z12 − c2 z22 − c3 z32 + z3 z4 − S1 − S2 − S3 , 70

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

¶2

µp ∂α2 1 with S3 = d33 z3 − √ e3 ∂y 2 d33 1 such that l > . 4d33

¶ 1 e2 ≥ 0, where d33 is taken + l− 4d33 3 µ

In this step the only uncertainty that is contained in the right-hand side of the ¶ µ ∂α2 2 z3 inequality (17) is e3 and it is compensated by the damping term −d33 ∂y (3) in the stabilizing function α3 (χˆ1 , χ ˆ2 , χˆ3 , χ1d , χ˙ 1d , χ ¨1d , χ1d , y). Thus, if χˆ4 were the control input, then, by choosing χ ˆ4 = α3 (χˆ1 , χˆ2 , χˆ3 , χ1d , (3) χ˙ 1d , χ ¨1d , χ1d , y) with the appropriately taken positive constants d11 , d12 , d21 , d23 , d33 , the derivative V˙ 3 along the solutions of the system (14) would be less or equal to the expression that contains a negative deﬁnite part plus indeﬁnite term z3 z4 . Further, the steps 4 – (n − 1) of our control design procedure are analogous to the step 3 and in each step the corresponding virtual control law χ ˆi = (i−1) = αi−1 (χˆ1 , . . . , χˆi−1 , χ1d , χ˙ 1d , . . . , χ1d , y), i = 5, n is designed. Step n. To solve the reference trajectory χ1 = χ1d (t) tracking problem and, hence, the considered output tracking problem by exploiting the virtual control laws, the control input u can be used to force the errors z1 = χ1 − χ1d (t), z2 = χˆ2 − α1 , . . . , zn = χˆn − αn−1 to converge asymptotically to zero. Indeed, consider the following Lyapunov function candidate for the system (14): 1 Vn (z, e) = Vn−1 (z1 , . . . , zn−1 , e)+ zn 2 +Vo (e) > 0, (z, e) 6= 0, 2

Vn (0, 0) = 0.

The time derivative of Vn along the solutions of (14) is given by V˙ n |(14) = V˙ n−1 |(14) + zn (χˆ˙ n − α˙ n−1 ) + V˙ o |(14) ≤ n−1 X

n−1 X

ci zi2 −

≤− i=1

Si + zn (zn−1 + LT χˆ + gn (χˆ2 − y) + β(y)u + φn (y)− i=1

n−1 n−1 n X ∂αn−1 X ∂αn−1 (i+1) X ∂αn−1 2 ˙ (χˆ3 +φ2 (y)−e3 )− − χ )−le3 −l e2i . χ ˆ − (i) 1d ∂y ∂ χˆi i i=1 i=0 ∂χ1d i=4

Hence, by the choice of control law

u(χ, ˆ y) =

1 (−cn zn − zn−1 − LT χˆ − gn (χˆ2 − y)− β(y) n−1 X ∂αn−1 ∂αn−1 (χˆ3 + φ2 (y)) + − φn (y) + χˆ˙ + ∂y ∂ χˆi i i=1 n−2 X

∂αn−1

(i+1)

χ1d

+ (i) i=0 ∂χ1d

−(

∂αn−1 2 ) dn3 zn ), ∂y

cn , dn3 > 0 (18)

for the derivative V˙ n along the solutions of (14) holds the following: VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

71

n X

n X

ci zi2

V˙ n |(14) ≤ − i=1

µp

−

n X

e2i =

Si − l i=1

i=4

µ µp ¶ ¶2 κ1 κ21 1 2 = d11 z1 − √ e1 − l − d12 z1 + √ e2 − e − 4d11 1 2 d11 2 d12 i=1 µ µp ¶ ¶2 1 1 − l− d21 z2 + √ e1 − e22 − 4d12 2 d21 µp ¶2 µ ¶ ¶ µ 1 1 1 ∂α1 2 − z2 − √ e 3 − l − e −− l− e2 − d23 ∂y 4d21 1 4d23 3 2 d23 ¶2 X n µ n n X X p ∂αj−1 1 1 − dj3 (l − )e23 − l e2i . (19) z j − p e3 − ∂y 4dj3 2 dj3 n X

¶2

ci zi2 −

j=3

j=3

i=4

While taking the positive constants dij , j = 1, 3, i = 1, n such that ¾ ½ 2 1 1 1 1 1 κ1 def l > max , , , , ,..., = lmax , 4d11 4d12 4d21 4d23 4d33 4dn3 one gets V˙ n |(14) < 0,

(z, e) 6= 0.

Note that the condition l > lmax can be satisﬁed, for instance, in the following way. One can ﬁrst choose the coefﬁcients dij > 0, j = 1, 3; i = 1, n and then ﬁnd such matrix Qo that satisﬁes the Lyapunov equation (12) and the inequality l > lmax . In the error variables z1 , . . . , zn , e the system (14) in closed-loop form with the control law (18) is written as follows: z˙1 = −c1 z1 + z2 − d11 z1 − d12 z1 + (b1 − c1 − d11 − d12 )e1 − e2 , ∂α1 ∂α1 2 ∂α1 )z3 − d21 z2 − d23 ( ) z2 + e3 , z˙2 = −c2 z2 − z1 + (1 − ∂y ∂y ∂y ∂α1 ∂α2 2 ∂α2 )z2 + z4 − d33 ( ) z3 + e3 , z˙3 = −c3 z3 − (1 − ∂y ∂y ∂y ................................................................. ∂αn−1 2 ∂αn−1 ) zn + e3 , e˙ = (A + GC)e. z˙n = −cn zn − zn−1 − dn3 ( ∂y ∂y

(20)

Now if the condition l > lmax is satisﬁed, then global asymptotic stability of the equilibrium point z = 0, e = 0 of the system (20) follows from the inequality (19) and the main Lyapunov theorem [2]. Further, from the expressions z1 = χ1 − χ1d , z2 = χˆ2 − α1 , . . ., zn = χˆn − −αn−1 , e = e and global asymptotic stability of the equilibrium point z = 0, e = 0 of the system (20) it follows that any trajectory χ(t), e(t) of the closed-loop system (14) is globally bounded and the tracking error χ1 (t) − χ1d (t) converges asymptotically to zero for any initial conditions χ(0), e(0) of the closed-loop system (14). T T Then from the inequality |χ(t) − χd (t)| ≤ |(χT (t), eT (t))T − (χT d (t), 0 ) |, where χd (t) is the appropriate reference trajectory, it follows that for the closedloop system (10) the feedback control law u(χ + e, y) (18) also forces the tracking 72

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

error χ1 (t) − χ1d (t) to converge asymptotically to zero for any initial conditions χ(0). Finally, since the change of coordinates (9) is linear we conclude that for any initial conditions x(0) the trajectories x(t) of the closed-loop system (8) are globally bounded and the output reference trajectory tracking error y(t) − yd (t) converges asymptotically to zero. Output tracking in case of m -dimensional zero dynamics. Now consider the system (2) with n − r = m. The linear change of coordinates x1 χ1 · ¸ K1 K2 .. .. (21) . , . = Ir×r 0r×m χn xn 1 2 where K1 = [ kij ], i = 1, m, j = 1, r and K2 = [ kij ], i = 1, m, j = 1, m are appropriate matrices, with their elements being functions of bi , i = 1, m; Ir×r is the identity matrix of dimension r × r; 0r×m is the zero matrix of dimension r × m, transforms the system (2) into the form

χ˙ 1 = −b1 χ1 + χ2 + ϕr+1 (y), .. . χ˙ m−1 = −bm−1 χ1 + χm + ϕn−1 (y), χ˙ m = −bm χ1 + χm+1 + ϕn (y), χ˙ m+1 = χm+2 + ϕ1 (y), .. .

(22)

χ˙ n−1 = χn + ϕr−1 (y), χ˙ n = LT χ + β(y)u + ϕr (y), y = χm+1 . Here ϕi , i = r + 1, n are linear combinations of the nonlinearities ϕk , k = 1, n 1 , i = 1, m, and L ∈ Rn is a vector, with its components being functions of kij 2 j = 1, r and kij , i = 1, m, j = 1, m. A global exponential observer for the system (22) is written as follows: χˆ˙ = Aχˆ + GC(χˆ − χ) + φ(y) + Bβ(y)u,

(23)

where χˆ ∈ Rn ; φ(y) = (ϕr+1 (y), . . . , ϕn (y), ϕ1 (y), . . . , ϕr (y))T ; the matrices A, B refer to the vector form of (22); C = (0, . . . , 0, 1, 0, . . . , 0) has a one in the (m + 1)st entry. In (23) the gain vector G = (g1 , g2 , . . . , gn )T is chosen so that the matrix A + GC is Hurwitz. The state estimation error e = χ ˆ − χ equation is of the form (4) with the matrices A, G, C taken from (23) and its equilibrium e = 0 is globally exponentially stable, since the matrix A + GC is Hurwitz [3]. Note that in general case the backstepping procedure cannot be applied to stabilization of the system (22), since it is not in the strict feedback form (1). Further we will assume the folowing. Assumption 1. Let the following equations hold for the right-hand side of the system (2) with m > 1: σm (k)T (K1 ϕ(1:r) (y) + K2 ϕ(r+1:n) (y)) = 0,

k = 1, . . . , m − 1,

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

(24) 73

where σm (k) = (0, . . . , 0, 1, 0, . . . , 0)T ∈ Rm has a one in the kth position and ϕ(i:j) (y) = (ϕi (y), ϕi+1 (y), . . . , ϕj−1 (y), ϕj (y))T , j > i. The conditions (24) imply that in (22) the nonlinerities ϕi (y), i = r + 1, n − 1 satisfy ϕi (y) ≡ 0, i = r + 1, n − 1 and the system (22) preserves the structure similar to the strict feedback form (1), except that the nonlinearity fm (χ1 , . . . , χm , χm+1 ) = ϕn (χm+1 ) − bm χ1 depends also on the (m + 1)st variable. However, the observer backstepping procedure as it is applied to the stabilization of the system (10) cannot be directly used for stabilization of the system (22). Indeed, using the fact that e = χˆ − χ the system (22) is written as follows, the conditions (24) being satisﬁed: χ˙ 1 = −b1 χ1 + χˆ2 − e2 , χˆ˙ 2 = −b2 χˆ1 + g2 em+1 + χˆ3 , .. . χˆ˙ m−1 = −bm−1 χˆ1 + gm−1 em+1 + χˆm , χˆ˙ m = −bm χˆ1 + gm em+1 + χˆm+1 + ϕn (y), ˆm+2 + gm+1 em+1 + ϕ1 (y), χˆ˙ m+1 = χ .. . ˙ χˆn−1 = χˆn + gn−1 em+1 + ϕr−1 (y), χˆ˙ n = LT χˆ + gn em+1 + β(y)u + ϕr (y), e˙ = (A + GC)e, y = χˆm+1 − em+1 .

(25)

The presence of the terms gi em+1 = gi (χˆm+1 − χm+1 ), i = 2, m − 1 in the ﬁrst m − 1 equations of the system (25) implies that the system is not in the strict feedback form and the observer backstepping as it is considered in the previous case of one-dimensional zero dynamics is not applicable to the considered system. Now consider the following subsystem composed of the ﬁrst m equations of (22): (26) χ˙ (1:m) = Am χ(1:m) + Bm (χm+1 + ϕn (y)), where χ(1:m) = (χ1 , . . . , χm )T ; Am = (aij ) is a matrix of dimension m × m with its elements satisfying aij = 1 if j − i = 1, aij = −bi if j = 1 and aij = 0 if j − i 6= 1 and j 6= 1; Bm = (0, . . . , 0, 1)T ∈ Rm . Note that, if the function f (χm+1 ) = χm+1 + ϕn (χm+1 )) deﬁnes a diffeodef

morphism, then for the system (22) the reference output trajectory yd (t) = def

= χ(m+1)d (t) tracking problem is equivalent to that of the tracking of a globally bounded reference trajectory χ(1:m) = χ(1:m)d (t) that satisﬁes the system χ˙ (1:m)d = Am χ(1:m)d + Bm (χ(m+1)d + ϕn (χ(m+1)d )). The conditions on the right-hand side of (26) under which for any globally bounded reference trajectory χ(m+1)d (·) ∈ L1 ∩ L∞ there exists a unique solution χ(1:m)d (·) ∈ L1 ∩ L∞ ∩ C 0 , χ(1:m)d (±∞) = 0 of the system χ˙ (1:m)d = = Am χ(1:m)d + Bm (χ(m+1)d + ϕn (χ(m+1)d )) are given in [6, 7]. Further we will suppose that these conditions are satisﬁed and the smooth reference trajectory χ(1:m)d (t) that corresponds to the considered reference output trajectory χ(m+1)d (t) is found, for instance, following the procedure given in [6, 7]. 74

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

By using the fact that em+1 = χˆm+1 − χm+1 , the system (26) can be written as follows: χ˙ (1:m) = Am χ(1:m) + Bm (χˆm+1 − em+1 + ϕn (y)),

(27)

where χˆm+1 is the (m + 1)st component of the state vector estimate χˆ generated by the observer (23). Note that the linear subsystem χ˙ (1:m) = Am χ(1:m) +Bm χ ˆm+1 with χˆm+1 as its control input is completely controllable. Let us use χˆm+1 as a virtual control input to solve the reference trajectory χ1 (t) = χ1d (t), . . ., χm (t) = χmd (t) tracking problem for the system (27) by exploiting the notions of block backstepping [2] and vectorial observer backstepping [8]. Consider the following positive deﬁnite function: T Pm z(1:m) + 2V0 (e) > 0, Vm (z(1:m) , e) = z(1:m)

(z(1:m) , e) 6= 0,

T there z(1:m) = (z1 , . . . , zm )T = (χ1 − χ1d , . . . , χm − χmd )T , Pm = Pm > 0. ˆm+1 − αm , the time derivative of Vm along the Using the error variable zm+1 = χ solutions of (27) can be expressed as follows:

T V˙ m |(27) = 2z(1:m) Pm [Am χ(1:m) +

+ Bm (zm+1 + αm − em+1 + ϕn (y)) − χ˙ (1:m)d ] − 2eT Qo e. Then, having written the system (27) in the error variables (z1 , . . . , zm )T = = (χ1 − χ1d , . . . , χm − χmd )T , one can ﬁnd a control law αm that solves the considered reference trajectory tracking problem for the system χ˙ (1:m) = = Am χ(1:m) + Bm (αm + ϕn (y)). One of such control laws is given by m X

αm = − m X

−

ci (χi − χid )− i=1 i−1 X (i−j−1) (−bj χ1d

ci i=2

m X

+ χid −

(i−1) χ1d )

(m−i)

− ϕn (y) +

j=1

bi χ1d

(m)

+ χ1d =

i=1 i−1 X

m X

(i−j−1)

= −Cm (χ(1:m) − χ(1:m)d ) −

ci i=2

(−bj χ1d

(i−1)

+ χid − χ1d

)−

j=1 m X (m−i)

− ϕn (y) +

bi χ1d

(m)

+ χ1d ,

i=1

where Cm is chosen so that the matrix Ac = Am − Bm Cm is Hurwitz. Hence, T > 0 and Qm = QT there exist the matrices Pm = Pm m > 0 satisfying the Lyapunov equation [3] AT c Pm + Pm Ac = −Qm − 2Im , where Im is the identity matrix of dimension m × m. Note that here the matrix −Qm represents the effect of stabilization without taking care of the uncertainties VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

75

and the two identity matrices −2Im represent the effect of the extra damping to take care of the estimation errors e(1:m) = χˆ(1:m) − χ(1:m) and em+1 = χˆm+1 − χm+1 . This could be compared with the role of the constants c1 and d11 , d22 in the ﬁrst step of the one-dimensional case in the previous section. Substitute in αm the state estimate χˆ(1:m) instead of the true state χ(1:m) . Then the time derivative of Vm along the solutions of (27) can be written in the following way:

T T (AT V˙ m |(27) = z(1:m) c Pm + Pm Ac )z(1:m) − 2z(1:m) Pm Bm Cm e(1:m) − T T − 2z(1:m) Pm Bm em+1 + 2z(1:m) Pm Bm zm+1 − eT Qo e − eT Qo e ≤ T Qm z(1:m) − (z(1:m) + Pm Bm Cm e(1:m) )T (z(1:m) + Pm Bm Cm e(1:m) )− ≤ −z(1:m)

− eT (Qo − Λ)e − (z(1:m) + Pm Bm em+1 )T (z(1:m) + Pm Bm em+1 )− T T T − (l − Bm Pm Pm Bm )e2m+1 + 2z(1:m) Pm Bm zm+1 ,

where ¸

· Λ=

(Pm Bm Cm )T (Pm Bm Cm ) 0m×r 0r×m 0r×r

.

Here by the appropriate choice of the matrices Pm and Cm one can satisfy the T T inequalities Qo − Λ > 0 and l − Bm Pm Pm Bm > 0. Thus, if χˆm+1 were the control input, then, by choosing χˆm+1 = αm (χˆ(1:m) , (m)

χ(1:m)d , χ˙ 1d , . . . , χ1d , y) with the appropriately taken matrices Pm and Cm , the derivative V˙ m along the solutions of the system (27) would be less or equal to the expression that contains a negative deﬁnite part plus indeﬁnite term T 2z(1:m) Pm Bm zm+1 . Finally, the further steps of our stabilizing control design are analogous to the above case of one-dimensional zero dynamics and consist in the subsequent adding of the rest (n − m) equations of the system (22) to the considered subsystem (26). In each step the appropriate virtual control law is designed untill the whole system (22) is considered and the stabilizing control law is found. Remark 1. The assumption 1 can be relaxed to allow for linear terms of the form Bχm+1 , B ∈ Rm in the ﬁst m equations of the system (22) as long as the system χ˙ (1:m) = Am χ(1:m) + (Bm + B)χm+1 is stabilizable. Illustrative example. Consider the model for a ﬂexible one-link robot arm [9]. The dynamics are given by χ˙ 1 = χ2 , χ˙ 2 = −M1 sin χ1 − k1 (χ1 − χ3 ), χ˙ 3 = χ4 , χ˙ 4 = −b1 χ4 + k2 (χ1 − χ3 ) + u/J, y = χ3 ,

(28)

where χ1 is the angle of the arm, χ2 is the angular velocity of the arm, χ3 is the angle on the motor side, χ4 is the angular velocity on the motor side, the input signal u is the driving torque from the motor, the constants M, b1 , k1 , k2 , J are all positive. 76

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

It is supposed that only angular rotation χ3 on the motor side is available for measurements. For this system our purpose is to design a feedback control law that uses only values of angular coordinate χ3 and forces the tracking error χ3 (t) − χ3d to converge to zero as t → ∞ for all χ(0), where χ3d = const is a constant reference signal. First, let us design a global exponential observer for the system (28). To this end, use the new variables x 1 = χ 3 , x 2 = χ 4 , x 3 = k2 χ 1 − k 2 χ 3 − b 1 χ 4 , x4 = −k2 b1 χ1 + k2 χ2 + k2 b1 χ3 + (b21 − k2 )χ4 .

(29)

The linear change of coordinates χ = Φ(x), Φ(0) = 0 given by (29) deﬁnes a diffeomorphism R4 = {x} onto R4 = {χ}. Furthermore, in the new variables x the system (28) is written as follows: x˙ 1 = x2 , x˙ 2 = x3 + u/J, x˙ 3 = x4 − b1 u/J, x˙ 4 = a4 (x) + (b21 − k2 )u/J, y = x1 ,

(30)

where a4 (x) = −b1 k1 x2 −(k1 +k2 )x3 −b1 x4 −k2 M1 sin ((k2 x1 + b1 x2 + x3 )/k2 ). Note that both the system (28) and the dynamical part of the system (30) are in the strict feedback form (1). However, neither of these systems is in the outputfeedback form (2). The structure of (30) is similar to the output-feedback form (2), except that the nonlinearity a4 (x) also depends on unmeasured states. Hence, one cannot design observers for the system (30) directly in the form of (3). However, since the system (30) is written in the uniformly observable canonical form [10] and the right-hand side of the system is globally Lipschitz in x uniformly on u, a global exponential observer for the system (30) is readily constructed as [10] x ˆ˙ = Aˆ x + a(ˆ x) + GC(ˆ x − x) + Bu,

(31)

where A = (aij ), i = 1, 4, j = 1, 4 is a square matrix, with its elements satisfying x) = (0, 0, 0, a4 (ˆ x))T ; aij = 1, if j − i = 1 and aij = 0, if j − i 6= 1; a(ˆ B = (0, 1/J, −b1 /J, (b21 − k2 )/J)T ; C = (1, 0, 0, 0). In (31) the gain vector G = = (g1 , g2 , g3 , g4 )T has the following structure G = (θl1 , θ 2 l2 , θ 3 l3 , θ 4 l4 )T , where θ > 1 is some positive constant, L = (l1 , l2 , l3 , l4 )T is chosen so that the matrix A + LC is Hurwitz. In the variables χˆ = Φ(ˆ x) the system (31) is written as ˆ3 − χ3 ), χ ˆ˙ 1 = χˆ2 + (θ 3 l3 /k2 + θl1 + b1 θ 2 l2 /k2 )(χ ˆ1 − k1 (χˆ1 − χ ˆ3 )+ χ ˆ˙ 2 = −M1 sin χ +(θ 4 l4 /k2 + θ 2 l2 + b1 θ 3 l3 /k2 )(χˆ3 − χ3 ), ˙ ˆ3 − χ3 ), χ ˆ3 = χˆ4 + θl1 (χ ˙ ˆ1 − χˆ3 ) + θ 2 l2 (χ ˆ3 − χ3 ) + u/J. χ ˆ4 = −b1 χˆ4 + k2 (χ

(32)

Once the change of coordinates χ = Φ(x) given by (29) is linear, the system (32) is a global exponential observer for the system (28). VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

77

Further, to ﬁnd a stabilizing control law that is solution to the tracking problem in question via the introduced in the previous sections backstepping procedure, the following system is used: ˆ 2 − e2 , χ˙ 1 = χ χ ˆ˙ 2 = (θ 4 l4 /k2 + θ 2 l2 + b1 θ 3 l3 /k2 + k1 )χˆ3 − M1 sin χˆ1 − k1 χˆ1 − − (θ 4 l4 /k2 + θ 2 l2 + b1 θ 3 l3 /k2 )χ3 , ˆ4 + θl1 (χˆ3 − χ3 ), χ ˆ˙ 3 = χ ˙χ ˆ4 = −b1 χˆ4 + k2 (χˆ1 − χˆ3 ) + θ 2 l2 (χˆ3 − χ3 ) + u/J, ˜ + ϕ(χ) e˙ = Ae ˆ − ϕ(χ ˆ − e), y = χ ˆ 3 − e3 ,

(33)

where e = χˆ − χ, the matrix A˜ refers to the linear part of the system (32) with u ≡ 0 written in vector form, ϕ(χ) = (0, −M1 sin χ1 , 0, 0)T . Finally, the feedback control law that when applied to the system (28) forces the tracking error χ3 (t) − χ3d to converge to zero as t → ∞ for all χ(0) is given by µ µ ¶ ∂α2 + b1 χˆ4 − u(χ, ˆ χ3 ) = J −c4 z4 − z3 1 − ∂χ3 ∂α3 − k2 (χˆ1 − χˆ3 ) − θ 2 l2 (χˆ3 − χ3 ) + χˆ4 + ∂χ3 ¶ 3 X ∂α3 ˙ ∂α3 2 χˆ − d44 ( ) z4 , c4 , d44 > 0, (34) + ∂ χˆi i ∂χ3 i=1

where αi , zj , i = 1, 3, j = 1, 4 are calculated as follows: z1 = χ1 − χ1d , zi = χˆi − αi−1 (χ3 , χˆ1 , . . . , χˆi−1 , χ1d ), α1 = −(c1 + d11 + d12 )χˆ1 + (c1 + d11 + d12 )χ1d ,

i = 2, 4,

c1 > 0, d11 > 0, d12 > 0,

∂α1 ˜ ³ α2 = 1/(k1 + θ˜2 − ˆ1 + χ1d + θ˜2 χ3 + θ1 ) −c2 z2 − χ ∂ χˆ1 ´ ∂α1 (χ ˆ2 − θ˜1 χ3 ) − d21 z2 , + M1 sin χˆ1 + k1 χˆ1 + ∂ χˆ1

.³ α3 = 1

78

c2 > 0, d21 > 0,

µ µ ¶ ∂α2 ´ ∂α1 ˜ ˜ −c3 z3 − z2 k1 + θ2 − 1− θ1 − θl1 (χˆ3 − χ3 )+ ∂χ3 ∂ χˆ1 µ ¶ ¶ ∂α2 ˙ ∂α2 ˙ ∂α2 2 + z3 , c3 > 0, d34 > 0, χˆ + χˆ − d34 ∂χ ˆ1 1 ∂ χˆ2 2 ∂χ3

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

with θ˜1 = θ 3 l3 /k2 + θl1 + b1 θ 2 l2 /k2 , θ˜2 = θ 4 l4 /k2 + θ 2 l2 + b1 θ 3 l3 /k2 and the constants d11 , d12 , d21 , d34 , d44 being appropriately chosen. Conclusions. In this paper the output feedback control problem for a class of nonlinear nonminimum-phase systems is considered. The standard observer backstepping method is extended to stabilization of a class of nonlinear nonminimumphase systems in the output-feedback form with linear unstable zero dynamics. The proposed design achieves global stabilization and allows tracking of reference output trajectories. However, the main restriction on the class of systems to which the design is applicable is assumption 1, which is a condition for preserving strict feedback form in the state transformations used for the backstepping design. This work has been supported by Grant 02-01-00704 from the Russian Foundation for Basic Research, by Grant HU-2094.2003.1 of Support of Leading Scientiﬁc Schools and by the National Board for Industrial and Technical Development (NUTEK).

REFERENCES 1. Kanellakopoulos, I., Kokotovi´c, P., Morse, S. A toolkit for nonlinear feedback control // System and Control Letters. 1992. V. 18. P. 83–91. 2. Krsti´c, M., Kanellakopoulos, I., Kokotovi´c, P. Nonlinear and Adaptive Control Design. – New York: John Wiley and Sons, 1995. 3. Krasovskiy, N.N. Some problems of stability of motion theory (in Russian). – Moscow, 1959. 4. Isidori, A. Nonlinear Control Systems. 3rd edition. – Springer-Verlag, 1995. 5. Robertsson, A., Johansson, R. Observer backstepping for a class of nonminimumphase systems // Proceedings of IEEE Conference on Decision and Control. Phoenix, Arizona. 1999. P. 4866–4871. 6. Degang, C., Paden, B. Stable inversion of nonlinear nonminimum-phase systems // International Journal of Control. 1996. V. 64. No 1. P. 81–97. 7. Devasia, S., Degang, C., Paden, B. Nonlinear inversion-based output tracking // IEEE Transactions on Automatic Control. 1996. V. 41. No 7. P. 930–942. 8. Fossen, T.I., Grovlen, A. Nonlinear output feedback control of dynamically positioned ships using vectorial observer backstepping // IEEE Transactions on Control System and Technology. 1998. V. 6. P. 121–128. 9. Marino, R., Tomei, P. Nonlinear Control Design: Geometric, Adaptive and Robust. Prentice Hall information and system sciences series. – London: Prentice-Hall, 1995. 10. Gauthier, J.P., Hammouri, H., Othman, S. A simple observer for nonlinear systems. Applications to bioreactors // IEEE Trans. Autom. Contr. – 1992. – V. 37, No 6. – P. 875 – 880.

Alexey Evgenievich Golubev (b. 1978) graduated from Bauman Moscow State Technical University (BMSTU) in 2002. Currently, he is post graduate student at the “Mathematical modeling” Department, BMSTU. He specializes in the ﬁeld of mathematical control theory and has published 3 papers on output feedback control of nonlinear dynamical systems.

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

79

Rolf Johansson received the Master of Science degree in Technical Physics in 1977, the Bachelor of Medicine degree in 1980, the doctorate in control theory in 1983, was appointed Docent in 1985, and received the Doctor of Medicine degree in 1986, all from Lund University, Lund, Scandinavia. He is member of SIAM and IEEE and is a fellow of the Swedish Society of Medicine. Currently, he is Professor of control theory at the Lund Institute of Technology and a coordinating director in robotics research with participants from several departments of Lund University. In his scientiﬁc work, he has been involved in research in adaptive system theory, mathematical modeling, system identiﬁcation, robotics and signal processing. Anders Robertsson (b. 1967) received the M.Sc. degree in Electrical Engineering and the Ph.D. degree in Automatic Control from Lund Institute of Technology, Lund, Sweden, in 1992 and 1999, respectively. During 2000 he worked as a Research Fellow and he is currently appointed as an Assistent Professor at the Department of Automatic Control, Lund. His research interests are in nonlinear control, real-time systems and robotics with particular attention to the output feedback control problem and force feedback control for robot manipulators. Sergey Borisovich Tkachev (b. 1961) graduated from Bauman Moscow Higher Technical School in 1984 Ph. D. (Phys.-Math.), Associate Professor at the “Mathematical modeling” Department of the Bauman Moscow State Technical University. He is author of more then 35 publications in the ﬁeld of mathematical control theory. His research interests include nonlinear control, nonlinear observation theory, geometric stabilization methods and the output feedback control problem.

BMSTU Press has published the book: Optical and Electronic Systems for Ecological Monitoring of Environment (In Russian) / V.I. Kozintsev, V.M. Orlov, M.L. Belov et al.; edited by V.N. Rozhdestvin. – M.: Izdatelstvo MGTU imeni N.E. Baumana, 2002. – 528 p. The book has two parts titled: “Laser Optical and Electronic Systems for Ecological Monitoring of Environment” and “Passive Optical and Electronic Systems for Ecological Monitoring of Environment”. Part I sets forth the physical background of laser sounding, concepts of building lidar systems of ecological monitoring and contains some examples of the systems. Part II is devoted to the physical background of passive optical monitoring, to concepts of organization of satellite systems of ecological monitoring and to building of passive optical and electronic devices for remote monitoring of environment. Examples of the satellite optical and electronic apparatus for ecological monitoring of environment are given. The book incorporates material of lectures delivered by the authors in the Bauman Moscow State Technical University. The book is intended for technical university students trained for “Optotechnology” profession and also for scientiﬁc and engineering workers specializing in instrumental engineering.

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LASER & OPTIC-ELECTRONIC SYSTEMS V.N. Rozhdestvin, O.A. Smirnova (Bauman Moscow State Technical University)

COMBINED Q-SWITCH AND MODE-LOCK IN LASER SYSTEMS Various laser system conﬁgurations are discussed which are based on the combined usage of Q-switch and mode-lock and allow generating ultrashort pulses with high peak power. Three different schemes of their realization are considered. The particular attention is given to the saturable absorber Cr+4 :YAG and its possible operation regimes. Rate equations describing the dynamics of processes inside the Cr+4 :YAG crystal are shown for each regime. Experimental results supporting theoretical expectations are presented.

Introduction. Pulse laser systems, generating short pulses (nanoseconds– picoseconds) of high peak power (kW0 s—GW0 s), have found their application in many ﬁelds of science and technology. Spectroscopy and dynamic studies in physics, nonlinear optics, chemistry, biology, optoelectronics were historically the ﬁrst areas of applications of ultrashort pulsed laser systems. Ultrafast lasers (pulse repetition rate up to 100 MHz) can be used in location systems, communication systems, systems for remote measurement of material properties, remote sensing, material processing, environmental studies, and monitoring production processes. Medical and industrial applications, metrology, astronomy of gravitational waves, optical databases, high deﬁnition laser television, optical computers — are only a few of their applications already proposed [1–3]. There is a number of known methods of obtaining the pulse generation in lasers. By means of pulse pumping the pulses of 10−6 . . . 10−3 s can be obtained. When signal short pulses are generated using Q-switch modulators, it is possible to achieve pulses with duration of about 10−9 . . . 10−8 s with peak power up to several GW. To obtain even shorter pulses (10−13 . . . 10−12 s) the mode-locking regime should be used [4, 5]. It is possible to generate short high power pulses by combining the Q-switch and mode-lock in one laser system. The mode-locking allows to obtain the very short pulse duration, while Q-switching provides the high output power [5–7]. In laser system design it is also important to consider the fact that the system should be inexpensive, simple (reduced number of components, diminished in size and weight), and have the high stability of output pulses parameters. There are some different system conﬁgurations based on the combined usage of Q-switch and mode-lock regimes. Methods of Combined Q-switching and Mode-Locking and their Realization Schemes. An injection seeding method for generating high peak power pulses at high repetition rates from Q-switched Nd:glass laser is described in [6]. VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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Fig. 1. Injection mode-locked and Q-switched laser system (two-cavity laser system)

A schematic diagram of the injection mode-locked and Q-switched laser system is shown in Fig. 1. This laser system was proposed by S. Basu and R. Byer from Stanford university in 1990. The system includes two cavities with a Q-switch and a mode-lock in separate cavities. The mode-locked laser output is injected into the Q-switched cavity through an optical isolator and a partial beam splitter. A full width of the Q-switch pulse is 64 ns. When 50 ps pulses from the mode-locked laser are injected, the output becomes a series of short pulses under a Q-switched envelope. As a result of the experiment, with 11 picosecond pulses they are able to extract a maximum of 6.3 mJ per pulse, which corresponds to approximate of 500 MW pulse peak power. Combined Q-switch and mode-lock in one cavity is the most common conﬁguration of combined Q-switch and mode-lock laser system (Fig. 2). In closed state the Q-switch is a bit transparent to the lasing radiation. This allows enough intensity through, for mode-locking process to reach a steady state. And when the Q-switch opens, an envelope of very high peak power mode-locked pulses will be radiated.

Fig. 2. Q-switched and mode-locked one cavity laser system

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Fig. 3. Laser system based on corner cube resonator with saturable absorber modulator

It should be speciﬁed, that in such a laser cavity conﬁguration, the interaction between two passive modulators is rather complicated and hardly compatible. As a rule, a combination of an active mode-lock and a passive Q-switch is used. For now, solid state lasers with the highest output characteristics are based on the conﬁguration with Q-switch and mode-lock in one cavity [4]. Combined Q-switch and mode-lock in one nonlinear element (Fig. 3) is the next laser conﬁguration [7, 8]. Several kinds of the passive nonlinear saturable absorbers that played double role of the Q-switch as well as the Q-switch and mode-lock at the same time are investigated. (The corner cube in the resonator conﬁguration was designed for practical purposes to be insensitive to the resonator misalignment.) Good example of such absorbers is Cr+4 :YAG crystal. Saturable Absorber Cr+4 :YAG. Solid-state laser engineers were eager to have at their disposal a crystal that would include in itself all main optical cavity parts (modulators, gain medium, etc.). This became possible with introducing saturable absorbers based on the same matrix as the active medium (such as YAG — yttrium aluminum garnet). This allowed to grow up crystals doped partially with saturable and partially with gain particles (though, ideally, we would like to have gain and saturable element spread over the whole cavity length). With mirrors added at the sides of such a crystal, and a small efﬁcient diode as a pump, we can have a solid-state high power pulsed laser that can be ﬁt in a volume smaller then a pencil. Let us consider the 5-level saturable absorber based on the Cr+4 :YAG crystal. The latter has at least two transitions at the same frequency, which can be used to absorb the laser radiation coming through the modulator. The schematic energy level diagram of the crystal [7, 8] is shown in Fig. 4. VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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Fig. 4. Schematic energy level diagram of Cr+4 :YAG crystal

The following designations are used: N1 , N2 , N3 are population densities; σg , σe are cross sections of the ground state and exciting state absorption respectively, with σg = 8.7·10−19 cm2 ; σe = 2.2·10−19 cm2 . At room temperature, τ1 , relaxation time from the state 3 T2 to the ground state 3 A2 , is 3.4 µs, and τ2 , relaxation time from the state 3 T1 to 3 T2 , is 50 ps. The rate equations that describe the dynamics of populations of the passive crystal Cr+4 :YAG energy levels are as follows: ³ σ ´ ∂N1 N2 g , = −N1 I+ ∂t 2π~ν τ1 ³ σ ´ ³ σ ´ ∂N2 N2 N3 g e − N2 , = N1 I− I+ ∂t 2π~ν τ1 2π~ν τ2 ³ σ ´ ∂N3 N3 e , = N2 I− ∂t 2π~ν τ2 ∂I = −I(σg N1 + σe N2 ), ∂z N0 = N1 + N2 + N3 , where I is intensity of laser radiation inside the cavity; z is the horizontal axis along the crystal length; N0 is concentration of the Cr+4 particles in the crystal. At the starting point for the small signal case we have N2 = N3 = 0, N1 = N0 . Let us consider the Q-switch regime ﬁrst [7, 9]. As photons start to go through the ﬁlter, the atoms jump to the lower excited level and stay there for the relaxation time τ1 . When all atoms are pumped up from the ground energy level, the absorption on the σg σe transition does not occur any more, and photons come through the ﬁlter without absorption. Since σg is 4 times larger then σe , then the probability of σe transition is small, but it occurs once in a while. As a result of this process, the maximum ﬁlter transparency in Q-switch regime cannot reach 100 %. The lifetime of the upper excited state τ2 is so small, that the population density 84

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N3 almost does not change, and can be approximated by zero. Hence, the rate equations can be rewritten in the following way: ³ σ ´ ∂N1 N2 g = −N1 I+ , ∂t 2π~ν τ1 ³ σ ´ N2 ∂N2 g = N1 I− . ∂t 2π~ν τ1 For the saturation case, N2 = N3 and the saturation intensity for Q-switch regime is W 2π~ν Isg = = 6.3 · 104 2 . σg τ1 cm Now, let us see what happens if we dramatically increase the intensity inside the laser cavity to turn it into the combined Q-switch and mode-lock regime [7, 8]. The atoms are quickly pumped up to the lower excited level but do not stay there. The intensity is high enough now to pump atoms even farther, to the upper excited level, where they will ﬁnally stay for the short period of time τ2 , allowing a group of photons to come through without absorption. After atoms relax back to the second level, they are met by the returning and ampliﬁed group photons that pump them up, again making the ﬁlter transparent for photons. In an actual experiment there can be as many as 100 of such loops or even more, which depends both on the combination of characteristics of the active and saturable elements and on the pumping rates. When the signal ceases, atoms slowly relax back, to the ground level. The very small lifetime τ2 , about 50 ps, is what makes the mode-locking process possible. During this process atoms spend most of the time in excited states, leaving N1 practically equal to zero. Therefore, the rate equations have the following representation: ³ σ ´ ∂N2 N3 e = −N2 I+ , ∂t 2π~ν τ2 ³ σ ´ N3 ∂N3 e = N2 I− . ∂t 2π~ν τ2 Again saturation intensity can be calculated as Ise =

2π~ν W = 7.5 · 109 2 , σe τ2 cm

giving us the result by ﬁve orders greater then for the Q-switch regime, which proves the necessity of much higher intensity for the combined Q-switch and modelock regime. The experimental results of the above-mentioned investigations [7, 8] for both the Q-switch and the combined Q-switch and mode-lock regimes are shown in Figs. 5 and 6. The horizontal time scale is 50 ns per division. It can be easily seen, that in combined regime the output becomes a train of short pulses under a Q-switched envelope [7]. VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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Fig. 5. Output for Nd:YAG laser passively Q-switched by Cr+4 :YAG crystal

Fig. 6. Output for Nd:YAG laser passively Q-switched and mode-locked by Cr+4 :YAG crystal

The performance of the Cr+4 :YAG crystal, that passively modulated the Nd:YAG laser, and the summarized experimental results are as follows: Q-switched pulsewidth (FWHM) . . . . . . . . . . . . . . 140 ns Q-switched pulsewidth (Full-width) . . . . . . . . . . . . 430 ns Mode-locked pulses number . . . . . . . . . . . . . . . . . . . 108 Mode-locked pulses separation. . . . . . . . . . . . . . . . . 4 ns Mode-locked pulsewidth . . . . . . . . . . . . . . . . . . . . . . 197 ps Q-switched pulse output power . . . . . . . . . . . . . . . . 0.07 MW Mode-locked pulses output power . . . . . . . . . . . . . . 0.47 MW

The achieved results might be not close to the limits, but they demonstrate the advanced capabilities of such systems, and suggest many possible applications with further improvements. Conclusions. In our age of simpliﬁcation, miniaturization and improved reliability, the described laser systems will surely be in great demand. Passive elements are relatively cheap and reliable, do not need external control systems, and can be made very small in size. Communication, medical application, high deﬁnition laser television, and optical data storage, are only a few examples of their possible applications. In this work there was presented the overview of combined Q-switching and mode-locking methods of generating ultrashort pulses with high peak power. Three different realization schemes of laser systems using the combined Q-switch and mode-lock regime were considered. In greater detail the saturable absorber on the basis of the Cr+4 :YAG crystal was discussed — rate equations, describing the dynamics of processes inside the Cr+4 :YAG crystal, were derived for each regime. Experimental results supporting theoretical expectations were also considered.

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REFERENCES 1. Clark, M., Sharples, S.D. & Somekh, M.G. Fast laser generated Rayleih wave scanning microscope, in Proceedings of the IEEE Ultrasonics Symposium, vol. 2, 1998, pp. 1317–1320. 2. Wayne H. Knox, Practical lasers will spawn varied ultra-fast applications, Laser Focus World, June 1996. 3. Brunner, F., Keller, U., Kuleshov, N.V., et. al., 240-fs pulses with 22-W average power from a mode-locked thin-disk Yb:KY(WO4 )2 laser, Optics Letters, vol. 27, no 13, pp. 1162–1164, 2002. 4. Malcolm, G.P.A., Ferguson, A.I. Diode-pumped solid-state lasers, Contemporary Physics, vol. 32, no 5, pp. 305–319, 1991. 5. Honninger, C., Paschotta, R., Morier-Genoud, F. & Keller, U. Q-switching stability limits of continuous-wave passive mode-locking, Journal of Optical Society of America, Vol. 16, no .1, pp. 46–56, 1999. 6. Basu, S. & Byer, R.L. Short pulse injection seeding of Q-switched Nd:glass laser oscillators theory and experiment, IEEE Journal of Quantum Electronics, vol. 26, no 1, pp. 149–157, 1990. 7. Chen, J., Yau, H.F., Liu, H.P., Chen, T.C., Cheng, C.C. & Liu, F.M. Passive Q-switch and mode-locking modulators for Nd: hosted lasers, Optics & Laser Technology, vol. 32, pp. 215–219, 2000. 8. Chen, J., Chen, D.K., Chen, R.S., Chen, Y.F. & Tsai, S.W. Passive Q-switch and modelocking modulators as well as ﬂashlamp-pump Nd: hosted lasers, in IEEE Paciﬁc Rim Conference on Lasers and Electro-Optics, CLEO, vol. 2, 2001, pp. II470–II471. 9. Shimony, Y., Burshtein, Z. & Kalisky, Y. Cr+4 :YAG as passive Q-switch and Brewster plate in a pulsed Nd:YAG laser, IEEE Journal of Quantum Electronics, vol. 31, no 10, pp. 1738–1741, 1995.

V.N. Rozhdestvin (b. 1940) graduated from the Bauman Moscow Higher Technical School in 1963. D. Sc. (Eng.), professor of “Radio Electronic Systems and Devices” department of the Bauman Moscow State Technical University, Winner of the State Prize of the Russian Federation, Honored Worker of Science and Technology of the Russian Federation, full member of the Russian Academy of Natural Sciences and Russian Academy of Military Sciences. Author of more than 200 publications in the ﬁeld of microwave and laser technology. O.A. Smirnova (b. 1947) graduated from the Bauman Moscow Higher Technical School in 1971. Ph. D. (Eng.), assoc. professor of “Radio Electronic Systems and Devices” department of the Bauman Moscow State Technical University, corresponding member of the Russian Academy of Informatization. Author of more than 80 publications in the ﬁeld of microwave and laser technology and physics of lasers.

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V.Ye. Karasik (Bauman Moscow State Technical University) LASER RANGE-GATED IMAGING SYSTEM Imaging systems of a new type — laser range-gated imaging systems are considered, in which as a ﬁnal link of decision making process an operator acts. The acquisition characteristics of such systems are introduced which take into account properties of the human visual system of an observer. A technique of their calculation based on determination of one of the most important parameters — the perceptible signal-to-noise ratio — is offered. For check of correctness of the offered computational relations another technique for an experimental estimation of the acquisition characteristics is developed and a prototype of laser imaging system is made with semiconductor laser pulse source and the gated receiving optical-electronic device containing the third-generation microchannelplate image intensiﬁer and the TV-camera on the basis of a photosensitive CCD-array. Experimental results of measuring the imaging system characteristics are discussed and compared with the calculated values.

Introduction. Laser imaging systems (LIS) are intended for formation an image of remote objects to detect and identify them under bad conditions of observation caused by low natural light and a various sort of clutters. Despite the recent intensive development of vision systems with automatic image processing, it should be admitted, that the man-operator, given a sufﬁcient resource of time available for decision making, copes with tasks of the object detection and identiﬁcation by its image better than any automatic system. Therefore, in overwhelming majority of LIS types the operator acts as a decision-maker, and the LIS characteristics should be determined in view of the observer0 s human visual system (HVS) properties. The laser imaging systems with pulse illumination are most popular now. Their operation principle is based on periodic illumination of an object by short pulses of laser radiation followed by photo-registration of the image in the receiving channel based on the staring CCD-array. For higher acquisition characteristics of LIS the microchannel-plate image intensiﬁer (MPII) is incorporated in the receiving channel, which increases the image brightness and carries out the gating of the received signal for clutters ﬁltering. At present, MPII of the third generation with gallium-arsenide (GaAs) photo-cathode has the greatest sensitivity. The application of such MPII makes possible to use as an illuminating laser not only pulse solid-state lasers, but also semiconductor pulse sources, which radiation spectrum falls within the maximal sensitivity area of GaAs photo-cathode. It allows creating the compact LIS with low weight and energy consumption. The main difference of LIS with pulse illumination from night vision devices with continuous laser illumination consists in the coordinated work of receiving and transmitting channels: MPII is open only for the time when the radiation pulse, reﬂected from the object, is expected to arrive. This results in the following:

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— effective time and spectrum gating of the signal is achieved with simultaneous suppression of the image background component caused by natural radiation; — contrast image still can be formed under the strong radiation dispersion caused by bad conditions of observation (smoke, fog, atmospheric precipitates); — high noise-immunity and range-gating of objects are ensured. At the same time, it is expected, that the overall dimensions and weight of LIS with semiconductor sources will not exceed essentially the similar parameters for night vision devices. This allows the competitiveness of similar systems to be predicted. Theoretical Analysis. The LIS performance is deﬁned by its acquisition characteristics, which are understood as its capabilities of detection, recognition and identiﬁcation of objects. Acquisition characteristics of LIS can be a priori estimated on the basis of its perceptible resolution with using the known Johnson0 s chart [1]. For this purpose, the equivalent mire (bar pattern, as a rule, four-bar pattern) is assigned to each level of the real object recognition. It is considered, that the probability of the real object recognition will be no less than the given one, if the perceptible signal-to-noise ratio (PSNR) µp∆ for an image of the equivalent p pt bar pattern is no less than the threshold value µpt ∆ , i.e. the condition µ∆ > µ∆ is satisﬁed. By threshold is meant such a value of PSNR, at which the probability of the bar pattern resolution is equal to 0.5; usually µpt ∆ is set equal to 2.5–3. If we know how PSNR depends on internal g1 , g2 , . . . gm and external q1 , q2 , . . . . . . qn parameters of the system, we are able to estimate a priori its acquisition characteristics, to carry out the light and energy calculations, to obtain such important system characteristics, as the minimum perceptible contrast and maximum distance of vision, to choose correctly the component base of LIS and to optimize its parameters. For this purpose, we only need to solve the equation µp∆ {(g) , (q)} = µpt ∆

(1)

for the unknown parameter. A technique of PSNR calculation is offered in [2, 3], where peculiarities of the image formation in LIS are deﬁned and, considering them, the mathematical model of LIS is developed in the form of a sequence of linear spatially-invariant parts and generators of additive homogeneous clutters. The functional model of the observer0 s human visual system in the form of a quasi-optimum ﬁltering receiver is proved. In the accepted model the object image on the monitor screen is described by the brightness distribution L00 (x, y) or its space-frequency spectrum ˜ 0 (νx , νy ), and image of a background or underlying surface — by average (SFS) L 0 brightness L0b . A signal component of the image is understood as a deviation of brightness value in the object image from the average background brightness, i.e. ∆L0 (x, y) = |L00 (x, y) − L0b |. Except for the object image, there is on the monitor screen a random additive clutter of the image characterized by the spectral density ˜ im (νx , νy ). Kr The HVS model presents an optimal ﬁlter receiver with noise and contains some linear parts with the appropriate transfer characteristics and a nonlinear part, VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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Fig. 1. Model of human visual system

which takes into account a nonlinear (as a rule, logarithmic) dependence of subjective (perceptible) brightness upon the real one. The inherent noise of HVS is taken into account by introduction of noise generators (photon noise, etc.) in the model. As a result, the sufﬁciently simple HVS model (Fig. 1) is synthesized that consists of only three parts: noise generator 3, optimal ﬁlter 4 with transfer func˜ f (νx , νy ) and the threshold device 5, which threshold of coming into action tion H depends on the chosen criterion of detection. Within the framework of the accepted model an expression for the perceptible signal-to-noise ratio will be as follows [2]: 1

Z+∞ Z

¯ ¯2 ¯ ˜0 ¯ ¯∆L (νx , νy )¯

µp∆ = K1 K2

2

dνx dνy ,

(2)

˜ im (νx , νy ) + Kr ˜ 0 Kr HV S (νx , νy ) −∞

˜ 0 (νx , νy ) is SFS of the image signal component on the monitor screen; where ∆L ˜ 0 ˜ = Krim (νx , νy ) is a spectral density of image noise; Kr HV S (νx , νy ) ˜ KrHV S (νx , νy ) = ¯ ¯2 is a spectral density of HVS noise, reduced to the monitor ¯ ¯˜ ¯Heye (νx , νy )¯ ˜ eye (νx , νy ) is an eye transfer function; K1 is a factor which takes account screen; H of the non-ideality of HVS, it is accepted equal to 0.6 on the basis of experimental data [4]; K2 is a factor which takes into account the ability of HVS for integrating the image over time [5]; νx , νy are spatial frequencies in two mutually perpendicular directions. To calculate the K2 value as applied to LIS with television reception at frame sequence frequency fs > 20, we can use the relation K2 = (fs τeye )1/2 , where τeye ≈ 0.1 is an “eye lag” (persistence of vision). The advantage of the offered model is that the spectral density of HVS noise, reduced to the monitor screen, has the same dimensions as that of image noise [cd2 ·m−4 ·rad2 ]. SFS of the image signal component is determined by the expression: ˜ 0 (νx , νy ) = ∆˜ ˜ LIS (νx , νy ), ∆L ρ(νx , νy )H

(3)

where ∆˜ ρ(νx , νy ) is a spectral density of the function ∆ρ(x, y), describing a devia˜ LIS (νx , νy ) is a transfer function of tion of reﬂection factor ρ(x, y) from average; H 90

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LIS. Believing clutters in imaging system to be additive, we can present a spectral density of the total image noise as ¯ ¯2 X ˜ i−e (νx , νy )¯¯ , ˜ im (νx , νy ) = ˜ i (νx , νy ) · ¯¯H (4) Kr Kr i

˜ i (νx , νy ) is a spectral density of noise created by i-th noise source; where Kr ˜ i−e (νx , νy ) is a transfer function of an optical-electronic path from the i-th noise H source to the monitor screen. In what follows we believe that the operator works under normal illumination condition, i.e. the brightness of adaptation is approximately equal to average of the background brightness and the image contrast can be adjusted. As the capability of low-contrast objects detection is limited by HVS noise, it is possible to assume, that this noise will not exert an appreciable effect on the contrast image identiﬁcation. It means that probability of detection of an object with using its image of sufﬁcient contrast is limited basically by image noise. Thus, when the condition ˜ im (νx , νy ) > Kr ˜ 0 Kr HV S (νx , νy ) is satisﬁed, the contrast may be increased only to a certain level, and the further increase becomes senseless and even harmful because the detection probability is decreased due to nonlinear transformations of the HVS signal component. According to the above-stated, the Eq. (2) can be put in the following form: 1 ¯2 ¯ 2 Z+∞ Z ¯¯∆L ˜ 0 (νx , νy )¯¯ . dν dν µp∆ = K1 K2 ˜ im (νx , νy ) x y Kr

(5)

−∞

Though the total noise components can have various distribution laws, the probability density of the perceptible brightness is close to the Gaussian function. It is caused by the HVS ability to integrate the image over space and time. As a result, the perceptible brightness is formed by summation of multitude of independent random values — brightness determined in various points of the image area and at various moments of time. According to the central limiting theorem, a probability density of the sum of a large number of such independent random values tends to the Gaussian function. Hence, the probability of the bar pattern resolution Pres , which is equal to the probability that the perceptible brightness will exceed the threshold value, is determined as ³ ´i 1h pt 1 + Φ µp∆ − µ∆ , (6) Pres = 2 r Zz µ 2¶ 2 x where Φ (z) = exp − dx is an integral of probabilities. π 2 0

The value µpt ∆ is determined by an acting rule of decision making and generally can change depending on a PSNR value, on a priori available information about the object, and on a level of training and experience of the operator. If a priori VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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probabilities of object occurrence and remunerations for the correct and erroneous decisions are not given, and the signal level is not known, then the decisionmaking rule is determined by the Neumann–Pearson criterion, and the threshold signal-to-noise ratio does not depend on a PSNR value. The dependence of object detection probability upon a PSNR value is shown in Fig. 2 on a perFig. 2. Dependence of bar pattern resolution unit basis for µpt = 2.5. ∆ probability upon PSNR Thus, knowing transfer functions of separate parts and the system as a whole, and also the noise spectral density of individual sources, we can calculate the image SFS and the image noise spectral density. However this calculation is rather labor-consuming and implies a knowledge of all characteristics of the system by the developer, which is not always possible. Therefore the calculation of PSNR is made with certain assumptions and approximations, resulting in signiﬁcant mistakes for a number of cases. In this connection the development of a technique for the experimental estimation of LIS acquisition characteristics and PSNR values represents the scientiﬁc and practical interest. Experimental Research. For experimental check of the above-described computational relations the prototype of LIS with a semiconductor source has been developed in the Bauman Moscow State Technical University (Fig. 3). The transmitting channel generates illuminating pulses with duration 100–200 nsec and repetition frequency up to 2 kHz. The image of object generated by receiving lens is ampliﬁed by photoreceiving device and then transferred by reproduction lens into the plane of sensitive elements of the CCD-camera. An output video signal of the CCD-camera is processed in the electronic unit of the videocontrol device and comes to the monitor. For preliminary pointing at an object under conditions of sufﬁcient natural light the optical sight is applied which is set co-axially with transmitting and receiving channels on a mobile platform. The latter can turn through some azimuth and elevation angle. To maintain LIS high acquisition characteristics and extend its functions the third-generation MPII is applied in the receiving channel. It operates in a pulse mode and carries out the gating of the signal coming at its input. The usage of gating allows to ﬁlter off effectively a back scattering clutter and background component. Besides, in the passive operational mode the gating by the “photocathode — microchannel plate” interval makes possible to limit an average current of the photo-cathode and to avoid its premature degradation. In the LIS transmitting channel the semiconductor laser source IDLP-100M-810E is applied, having a radiation wave length of 0.81 µm and pulse power of 70 W.

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Fig. 3. Structural and functional scheme of LIS experimental sample

The laser source luminescence body represents a lattice of elementary radiating p−n-junctions. The radiating source is located close to the lens focus, so that defocused images of elementary p−n-junctions are partially blocked from falling on object image. Thus the object appears to be lighted practically uniformly. The selection of distances between components of the lens and radiating source allows for the change of the target radiation angular divergence within a range of 0.600 × × 0.400 to 200 × 1.400 . The pumping of laser source is carried out by the power supply containing the initial generator G1, the shaper of pulses of the given duration G2 and the gate, through which a discharge of the integrating capacitor C to radiator D is provided by a ﬁxed value current. The univibrator G3 forms pulses synchronized with current pulses through radiating source (and, hence, and with pulses of radiation) and required for operation of the photodetecting device of the receiving channel. VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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The basic purpose of the LIS experimental research was to check the modulation transfer function (MTF), spectral density of a clutter in the image, PSNR, acquisition characteristics and to compare the measured values with calculated ones. During the LIS ﬁeld tests the generated image of the test-object — set of ten bar patterns Fig. 4. Image of test-object with the period from 16 to 160 mm (denoted in ﬁgures as No. 1, No. 2,. . . , No. 10) — was recorded using the standard VHS tape-recorder. Then separate pictures were put into the personal computer as bmp-ﬁles. ¯ pictures is shown in Fig. 4. ¯ One of the ¯ ¯˜ To obtain the experimental MTF ¯H (νx , νy )¯, the contrast k of the image of each bar pattern is determined as k = (Lmax − Lmin )/(Lmax + Lmin ), where Lmax and Lmin are maximal and minimal brightness of the bar pattern image respectively. The system MTF with frequency νx0 can be determined by the known Koltman formula [7] ¸ · ¯ ¯ π k(3νx0 ) k(5νx0 ) k(7νx0 ) ¯ ¯˜ − + + ... , k(νx0 ) + ¯H (νx0 , 0)¯ = 4kp 3 5 7

(7)

where kp is a contrast of the bar pattern; k(νx0 ) is a depth of modulation of the bar pattern image with spatial frequency νx0 . The calculated MTF of the LIS prototype and its elements as well as experimentally determined MTF values and conﬁdential intervals for the 90 % probability for several angular spatial frequencies are given in Fig. 5. The calculated curves are drawn according to the developed mathematical LIS model. From this ﬁgure we can notice a good agreement of the calculated MTF and experimentally determined values. Small discrepancies of the calculated and measured values within the frequency range of 4 . . . 6 mrad−1 can be explained by the fact that in this area the divergence between the real MTF of MPII and its approximation function, used in calculations, is observed. Spectral noise density is the second major LIS characteristic inﬂuencing the PSNR. It was determined by statistical processing of brightness for the image area, containing no bar patterns. This made possible to consider a clutter as an ergodic one and, according to the Wiener–Khinchin theorem, to ﬁnd its Fourier transformation of the correlation function Kr. For each of ten initial images of bar patterns ˜ was generated as arithmetic mean of arrays Kri . After Fourier transthe array Kr ˜ exp (νx , 0) of spectral density of image formation the one-dimensional function Kr noise was found on the basis of experimental data. It should be noted, that the automatic brightness adjustment system, available in the CCD-camera, makes impossible to measure absolute values of spectral densities. Therefore the measured and calculated spectral densities were normalized on the basis of the average image brightness square. The normalized measured and 94

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Fig. 5. Results of MTF measurements for LIS prototype: calculated curves (1 — MTF of atmosphere, 2 — MTF of receiving lens, 3 — MTF of MPII, 4 — MTF of reproduction lens, 5 — MTF of staring arrays, 6 — MTF of LIS prototype) with experimental estimates by bar pattern images marked with dots (with corresponding numbers) and vertical line segments which denote conﬁdential intervals of MTF estimates

calculated spectral densities of image clutters are given in Fig. 6. The good concurrence of the calculated and experimental curves is seen. Distinctions between them in the frequency region of 0 . . . 3 mrad−1 can be explained by presence of the transmitting channel clutter which was not taken into account during calculations. This clutter arises as a result of re-superposition of defocused images of p−n-junctions of the laser luminescence body, which spatial frequency is equal to 1.4 mrad−1 . The recession of the experimental curve of noise spectral density at frequencies above 9 mrad−1 can be explained by suppression of high-frequency components in the electronic path of the video-recorder (9 mrad−1 corresponds to 280 TV lines, which is a resolution limit for VHS video-recorders). The experimental value of PSNR was calculated by the approximated formula derived from Eq. (2): r (Lmax − Lmin )3.5Tp π p q µ∆ ≈ K1 , (8) 4 ˜ (νx0 , 0) 2 Kr ˜ where Tp is a period of the bar pattern image; Kr(ν x0 , 0) is a spectral density of image noise at the bar pattern frequency. Experimental values of PSNR measured in the image of each bar pattern and also the dependence of PSNR on spatial frequency of bar pattern, calculated in accordance with the above-mentioned technique, are given in Fig. 7, where a good concurrence of the calculated curve and experimental data, denoted by dots, can be seen. The deviation of some dots from the calculated curve is explained by nonuniformity of the test-object illumination and by difference between the real MTF of MPII and that of approximating function. Images of the test bar patterns reproduced by LIS on the monitor screen, allow for experimental estimates of the system acquisition characteristic — dependence VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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Fig. 6. Spectral density of image noises: 1 — calculated shot noise of MPII; 2 — calculated internal noise of CCD-camera; 3 — calculated total noise; 4 — total noise, determined experimentally

Fig. 7. Dependence of PSNR upon the bar pattern frequency

of the object detection probability upon PSNR. Ten images of bar patterns similar to those shown in Fig. 2 were demonstrated to ﬁve observers, and the cases were ﬁxed when the bar patterns were recognized. By results of the experiment (Fig. 8) the estimates of resolution probability for each bar pattern and conﬁdential intervals for the 90 % probability are found. The dependence of the bar pattern resolution probability upon PSNR, calculated according to Eq. (6), is also shown in Fig. 8. The PSNR threshold is accepted equal to 2.8 on a condition of the best conformity of the calculated curve to the experimental data. The obtained value µpt ∆ agrees well with the data of other researches. Let us note, that during the prototype experimental research the high efﬁciency was conﬁrmed when using the receiving channel gating for ﬁltration of a back scattering clutter and for the range selection of objects. The limiting range of LIS vision was estimated: for identiﬁcation of a car it was 800 m (Fig. 9), for its detection — 1200 m. Thus, the results of experimental research of LIS with semiconductor pulse radiating source and gated receiving channel on the basis of the third-generation

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Fig. 8. Dependence of probability of the bar pattern resolution upon PSNR

Fig. 9. The image of the car formed by LIS at a distance of 800 m

MPII and CCD-camera have shown, that these systems, at least, compare well with night vision devices, having active illumination, by efﬁciency, functionality and quality of formed images. The correctness of the offered simulation insight and computational relations is conﬁrmed experimentally with acceptable reliability. Acknowledgements. The author is grateful to the staff of the Bauman Moscow State Technical University, V. Popov and Ye. Mukhina, for their help in realization of experiments and preparation of this material for publication.

REFERENCES 1. Holst, Ger. “CCD Arrays, Cameras, and Displays”. JCD Publishing and SPIE Optical Engineering Press. Bellingham, WA, 1996. 2. Gusev, M.A., Karasik, V.E., Shestov, S.N. Investigation of acquisition characteristics of laser imaging systems (in Russian). Vestnik MGTU. Ser. Priborostroyenie. No 3, 1995, pp. 27–35, BMSTU Press, Moscow. 3. Karasik, V.E., Orlov V.M. Laser Imaging Systems (in Russian). Moscow. Izd-vo MGTU imeni N/E/ Baumana, 2001. P. 352. 4. Pavlov, N.I., Voronin, Yu. M. Probability of objects detection on the monitor screen in optical-electronic system, of supervision (in Russian). Optical magazine. 1994. No 7, pp. 3–11, Moscow. 5. Vaﬁadi, A.V. Analytical models of scanning thermo-vision devices (in Russian). Moscow. Optical magazine. 1997. No 1, pp. 32–36. 6. Mosyagin, G.M., Nemtinov, V.B., Lebedev, E.N. Theory of optical-electronic systems (in Russian). Moscow. Mashinostroenie, 1990. P. 432. 7. Kreopalova, G.V., Lazareva, N.L., Puryaev, D.T. Optical measurements (in Russian)/ Edited by D.T. Puryaev. Moscow. Mashinostroenie, 1987. P. 266. V.Ye. Karasik (b. 1939) graduated from the Bauman Moscow Higher Technical School in 1964. D. Sc. (Eng.), professor of “Laser & Optical-and-Electronic Systems” department of the Bauman Moscow State Technical University. Author of over 160 publications in the ﬁeld of the laser location and laser imaging.

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NAVIGATIONAL & GYROSCOPIC SYSTEMS V.A. Matveev and M.A. Basarab (Bauman Moscow State Technical University)

NUMERICAL MODELING OF HEAT DIFFUSION PROCESSES IN THE SOLID-STATE WAVE GYRO RESONATOR BY THE R-FUNCTION METHOD Stationary and non-stationary heat-transfer processes in a solid-state wave gyro are studied. The proposed numerical method is based on the theory of R-functions.

Introduction. The solid-state wave gyro (SWG) [1–3] belongs to the class of non-traditional gyro devices and is used as a sensitive element in navigational systems. One of the sources of SWG errors is a heat component of its drift [2, 4]. To identify its parameters, one must evaluate stationary and non-stationary thermal ﬁelds in the SWG resonator. The work is devoted to the use of the numericalanalytical method of R-functions (RFM) [5, 6] for solving the aforementioned problem. Mathematical Model of Heat-Transfer Processes in the SWG Resonator. The sensitive element of a real SWG is a hemisphere thin-shelled resonator made of melted quartz glass ensuring high isotropy of physical-mechanical properties and a small linear expansion coefﬁcient. Internal and external surfaces of the resonator are covered with a thin conducting ﬁlm and are not connected with each other. The resonator is ﬁxed in the device body with the help of the cylindrical leg. Due to the symmetry, we shall consider only the part of this system shown in Fig. 1, a. To identify parameters of the SWG drift rate model, the following model of an output signal was proposed [2]: ϑ˙ = −KΩ + Ω0 + Ωm sin 4(ϑ − ϕ0 ) + ΩT + W,

(1)

where ϑ = ϑ(t) is the standing wave rotation angle; K ≈ 0.3 is the SWG scaling factor; Ω is the angular velocity of SWG rotation around the symmetry axis; Ω0 is a constant component; Ωm is the amplitude, ϕ0 is an angle of orientation of the maximum quality factor axis; ΩT is a heat component of the drift rate, and W is a random component (white noise) of the drift rate. The constant parameters Ω0 , Ωm , ϕ0 , and the component ΩT are unknown and must be identiﬁed. The time constant ˙ T = −ΩT /T is taken in the form of a set of ﬁxed values T of the heat component Ω 0 T1 , T2 , . . . , TN . Its best approximation is determined on the set {Ti }N i=1 as a result of the most adequate identiﬁcation reconstructing the original realization ϑ = ϑ(t). The drift rate model (1) is not complete and may be considered as the basic one for an output signal and, after the stepwise complication, can be represented in the form providing high accuracy of the SWG characteristics determination.

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Fig. 1

SWG accuracy and its time of readiness are mostly deﬁned by heat exchange processes during heat radiation by internal sources and variations of external temperature. Inﬂuence of these factors yields the change of the SWG sensitive element temperature and, therefore, appearance of the thermal drift. To analyze the time of readiness and accuracy characteristics of the SWG, and to choose an appropriate thermal control system, one must know the temperature distribution in SWG elements. To investigate thermal processes in the SWG, it is necessary to — construct a mathematical model of heat diffusion processes in the SWG; — determine temperature gradients inside the SWG resonator; — develop the method for identiﬁcation of the heat model parameters with the use of experimental results; — choose a thermostat system. The main reason for appearance of the thermal drift is a shift of the SWG resonator center-of-mass caused by nonuniform linear deformations of its elements due to the nonzero temperature gradient. This drift induces radial and axial components of reactive forces in resonator supports during its main form of oscillations A part of the resonator energy is transferred to its support through the connecting leg and dissipates due to the damping effect. Damping in the support reduces the resonator quality factor Q and causes its dependence on the standing wave orientation. The latter effect yields the SWG drift proportional to the value −1 (Q−1 min − Qmax ), where Qmin and Qmax are the minimum and maximum values of the resonator quality factor. As in the dynamically adjusted gyros [4], one can distinguish the following main ways for decreasing the resonator center-of-mass shift caused by the temperature gradient: — decreasing the heat ﬂow from the device body to the sensitive element; VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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— increasing the resonator thermal conductivity; — decreasing the resonator linear expansion coefﬁcient. The problem of ensuring the mass center stability depends on the stability of holding up deep vacuum in the device cavity. Consider the following simpliﬁed model of heat diffusion processes in the SWG resonator (see Fig. 1, a). Due to the fact that the resonator is under vacuum, we shall not consider the convective heat exchange. We shall neglect also the radiation from the sensors. In places of the cylindrical leg ﬁxing, we assume the temperature is given (thermostatic conditions). Thus, the stationary temperature ﬁeld T (r, z) in the resonator must satisfy the following two-dimensional Laplace equation in cylindrical coordinates r, z [7, 8]: ∂ 2T 1 ∂T ∂2T + = 0; + ∂r2 r ∂r ∂z 2

(2)

the boundary conditions are T (R1 , z) = T1

at

z ∈ [0, d1 ],

∂T /∂n = 0

T (R1 , z) = T2

at

z ∈ [l − d2 , l],

on the remaining part of the boundary.

(3) (4)

In Eq. (4), n denotes the external normal to the domain boundary. Heat diffusion boundary-value problem (Eqs. (2)–(4)) can be solved by one of numerical or numerical-analytical methods [7–15]. In this work we shall consider an approach based on the use of the RFM. The R-function Method and General Solution Structures. The R-function method allows all prescribed boundary conditions to be satisﬁed exactly at all boundary points. The original idea underlying the RFM is due to Kantorovich [9]. He proposed that the homogeneous Dirichlet conditions may be satisﬁed exactly by representing the solution as the product of two functions: (1) a real-valued function that takes on zero values at the boundary points; and (2) an unknown function that allows one to satisfy (exactly or approximately) the differential equation of the problem. The idea appeared to have a limited use because it was not clear how such functions could be constructed for complex shapes and because the method did not seem to be generalized to other types of boundary value problems. Rvachev [5, 6] suggested that both of these obstacles may be overcome using R-functions — the real-valued functions that behave as continuous analogs of logical Boolean functions. With R-functions, it became possible to construct functions with prescribed values and derivatives at speciﬁed locations. Furthermore, the constructed functions possess desired differential properties and may be assembled into a solution structure that is guaranteed to contain solutions to the posed boundary value problems. A function z = f (x), x = (x1 , . . . , xN ), is called an R-function if its sign is completely determined by the signs (but not magnitudes) of its arguments. In this work, we shall consider the two-dimensional case (N = 2) when z = f (x, y).

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The most popular system of R-functions is the system > 0. If f is an R-function corresponding to the Boolean function F , then the implicit function for the resulting geometric domain is immediately given by Ω=(f (ω1 , . . . , ωn )>0). The function f (ω1 , . . . , ωn ) is negative outside Ω and the equation f (ω1 , . . . , ωn ) = 0 deﬁnes the boundary ∂Ω of the region Ω. It is known that the equation of the boundary ∂Ω = (f = 0) is called normal if the value of f (p) is equal to the Euclidean distance from the point p to the boundary ∂Ω. Similarly a function f that coincides with the normal function only on the boundary ∂Ω is called normalized and has a property that ¯ ∂f ¯¯ = −1, (7) ∂n ¯∂Ω where n is a vector of the external normal to ∂Ω. A normalized equation ω ˜ (x, y) = 0 of the boundary can be obtained from an ordinary one (ω(x, y) = 0) in the following way:

ω ˜ (x, y) =

ω(x, y) . |∂ω/∂n|∂Ω

(8)

To avoid indeterminacy due to division by zero, another normalizing formula can be used instead of Eq. (8): ω(x, y) ω ˜ (x, y) = q

. ω2

+

(9)

|∂ω/∂n|2∂Ω

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If every primitive implicit function ωi is normalized at the primitive boundaries ∂Ωi , then all of the above R-functions preserve this property, and the function f (ω1 , . . . , ωn ) is normalized at the boundary ∂Ω. Let us ﬁnd the solution to the operator equation Au = f

(10)

inside a bounded domain Ω ⊂ R2 with the given boundary conditions Lu|∂Ω = ϕ

(11)

on the boundary ∂Ω. The general solution structure (GSS) of a boundary-value problem is determined by the expression (12) u = B(Φ, ω, ωi , f, ϕ) satisfying the boundary conditions exactly under an arbitrary choice of the undetermined component Φ. Here, B is the operator depending on the geometry of the domain Ω and parts of its boundary ∂Ωi . The undetermined component Φ of the GSS in Eq. (12) is usually expressed by the series M X cn ψ n , (13) Φ= n=1

where ψn (x, y) are elements of a full system of basic functions (algebraic and trigonometric polynomials, splines, etc.), and cn are unknown coefﬁcients found with the help of one of variational or projection methods [9–12]. Some frequently encountered types of boundary conditions and GSSs of corresponding boundary-value problems are as follows. (i) The Dirichlet condition u|∂Ω = ϕ (14) is satisﬁed by the GSS u = ωΦ + ϕ.

(15)

∂u/∂n|∂Ω = ϕ

(16)

u = (1 − ωD)Φ − ωϕ,

(17)

(ii) The Neumann condition

is satisﬁed by the GSS

where µ D≡

¶ ∂ω ∂ ∂ω ∂ + ∂x ∂x ∂y ∂y

;

D|∂Ω

¯ ∂ ¯¯ = . ∂n ¯∂Ω

(18)

(iii) The GSS for the 3rd kind boundary condition (∂u/∂n + hu)|∂Ω = ϕ 102

(19)

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

is u = [1 + ω(h − D)] Φ − ωϕ.

(20)

(iv) The GSS ¸ · ω1 ω2 ω1 ω2 (2) (h − D ) (ω1 Φ + ϕ1 ) − ϕ2 , u= 1+ ω1 + ω2 ω1 + ω2 where µ (2)

D

≡

¶ ∂ω2 ∂ ∂ω2 ∂ + ∂x ∂x ∂y ∂y

(2)

;

D

|∂Ω2

¯ ∂ ¯¯ = , ∂n ¯∂Ω2

(21)

(22)

corresponds to the mixed boundary conditions ( u|∂Ω1 = ϕ1 , (23) (∂u/∂n + hu)|∂Ω2 = ϕ2 . It should be noted that the functions ω(x, y) and ω2 (x, y) in GSSs for conditions of a differential type (ii)–(iv) are normalized. Besides, an arbitrary combination of different conditions on boundary parts ∂Ωk is possible. Here, the “glue” operation for boundary conditions must be used. Let, for example, u(x, y)|∂Ωk = ϕk (x, y), k = 1, . . . , K.

(24)

Then the GSS is represented as , K X

K X

ωk−1 ϕk

u = ωΦ + k=1

ωk−1 ,

(25)

k=1

where ωk are the equations of boundary parts ∂Ωk , such that ωk > 0 in (Ω ∪ ∂Ω) ∩ ∂Ωk . The latter expression is called the generalized Lagrange formula. In a general case, only a part of boundary conditions may be satisﬁed by GSSs, or the structures may obey known conditions on given manifolds inside the domain. The Rote Method for Solving Non-stationary Problems. Consider the parabolic equation of heat transfer ∆T =

1 ∂T − f λ−1 . κ ∂t

(26)

Here, t is time, κ = λ/(cρ) is the temperature conductance [m2 /s], λ is the thermal conductivity [W/(m × K)], c is the speciﬁc heat [J /(kg × K)], and ρ is the volume density [kg /m3 ]. Besides for Eq. (26), the initial temperature distribution must be given: (27) T |t=0 = ψ. The variational statement of boundary-value problems for Eq. (26) and methods of their solution are based on the combined use of the Rote method [16–18] and VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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variational principle for elliptic-type problems [11]. Here, the time digitization of nonstationary boundary-value problem (26) is carried out and then the obtained stationary problems are solved by variational methods, i.e. the parabolic-type problem is reduced to a set of elliptic-type problems. Consider the boundary-value problems for Eq. (26) in the domain Ω at t ∈ (0, T ]. Introduce the partition {∆pj }, j = 1, 2, . . . , p, of the time interval [0, T ] as p X ∆tpj = T, ∆tpj = tpj − tpj−1 . (1) j=1

Then replace the time derivative in Eq. (26) by the approximate difference relation ∂T (x, y, tpj )

T (x, y, tpj ) − T (x, y, tpj−1 ) ≈

.

(2)

∆tpj

∂t

Thus, the original problem is reduced to the set of the following problems: p

∆T (x, y, tpj ) −

p

1 T (x, y, tj ) 1 T (x, y, tj−1 ) =− − f λ−1 p κ κ ∆tj ∆tpj

(29)

which must be solved sequentially. At j = 1, initial condition (27) must be used. In [17] it is proven that a limit of the solutions sequence exists. This limit is called the solution of the non-stationary problem, coinciding with the classical solution if the latter exists. Equation (29) approximates Eq. (26) with an error of order of max ∆tpj according to the use of approximate expression (28). Other schemes of j

the Rote method with higher orders of accuracy with respect to the time variable also exist [16]. The Lumped Capacity Condition. A thin coating of a body may be considered as the lumped capacity adjoining to the body [19]. Assuming the uniform temperature distribution in a thin subsurface layer and integrating both parts of the heat transfer equation along the layer depth (averaging), we can pass from the original complicated model to the model with lumped capacity. From the mathematical point of view, the time derivative in the boundary conditions is characteristic of the latter model. Such assumption allows one to simplify essentially solutions of heat exchange problems in compound bodies with sharp contrast of geometrical and thermal-physical characteristics. However, unlike the heat transfer boundary-value problems with conventional types of boundary conditions, the well-posedness of the problem with lumped capacity conditions depends on a ratio between coefﬁcients at derivatives in boundary conditions. At deﬁnite values of this ratio, the stated problem becomes conditionally well-posed or well-posed by Tychonoff. Since a resonator surface is covered with a thin metal layer, then on the part of the boundary, corresponding to the hemisphere, the lumped capacity boundary condition must be given [19] as

k 104

∂T ∂T =C , ∂ν ∂t

(30)

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where k is the heat conduction; C = cρS is the lumped capacity; ρS = ρm hm is the surface density of the layer material [kg /m2 ]; ρm and hm are the volume density and thickness of the layer. In practice, the lumped capacity as well as mechanical factors is one of the reasons for sharp decrease of the SWG resonator quality factor. ∂T → 0 as t → ∞, and in the stationary case, the condition (30) is Obviously, ∂t reduced to the ordinary Neumann boundary condition (4). Evaluation of Stationary and Non-stationary Thermal Fields in the SWG resonator. According to the R-function method, let us write the general solution structure of the problem (Eqs. (2)–(4)) as

u = ωI P −

ω (1) T2 + ωI(2) T1 ωI ωII (II) D1 (ωI P ) + I (1) − ωI + ωII ωI + ωI(2) −

(1) (2) ωI ωII (II) ωI T2 + ωI T1 D1 , (31) ωI + ωII ωI(1) + ωI(2)

where u is the approximate temperature distribution, P is the unknown component; ωI is the equation of the boundary part with the Dirichlet conditions,

ωI(1)

ωI = ωI(1) ∧0 ωI(2) , µq ¶ 2 2 = (R1 − r) ∨0 −circle R1 + d1 , r, z , ωII(1) = (R1 − r) ∨0 (l − d2 − z),

∂ωII ∂ 1 ∂ωII ∂ (II) (R2 − x2 − y 2 ), D1 ≡ + 2R ∂r ∂r ∂z ∂r and ωII is the equation of the boundary part with the Neumann conditions (see Fig. 1, b), ωII = ωII(1) ∧0 (ωII(2) ∨0 ωII(3) ), circle(R, x, y) ≡

¢ 1¡ (l/2)2 − (z − l/2)2 , l (2) ωII = −circle(R2 − h/2, r, z) ∧0 circle(R2 + h/2, r, z), µq ¶ (3) 2 2 ωII = [(R1 − r) ∨0 (z − l + d2 )] ∨0 circle R1 + d1 , r, z . ωII(1) = r ∧0

The boundary-value problem is solved by the variational-difference method [5]. Instead of the differential operator in the left-hand side of Eq. (2), its ﬁnitedifference approximation of the second order on the uniform mesh is considered [15] · µ ¶ µ ¶¸ 1 1 1 r r + − u(r + ∆r, z) + u(r − ∆r, z) + r∆r ∆r 2 ∆r 2 (32) µ ¶ ¸ · 1 1 u(r, z + ∆z) + u(r, z − ∆z) −2 + u(r, z) = 0. + ∆z 2 ∆r 2 ∆z 2 VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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Approximation of derivatives in the expression for the differential operator is also realized by means of ﬁnite-difference relations. To increase the rate of convergence, the undetermined component P is represented by the following series with respect to trigonometric basic functions, a priori satisfying boundary conditions at deﬁnite parts of the boundary:

(II) D1

µ

¶

M X

Cq,s cos

P =

πqr R2 + h/2

³ πsz ´ .

cos

(33)

l

q+s=0

Substituting the structure Eq. (31) in Eq. (32), we obtain the system of linear algebraic equations with respect to unknown coefﬁcients Cq,s of expansion (33). Let us calculate the temperature distribution for the resonator with parameters R1 = 5 mm, R2 = 25 mm, h = 2.5 mm, l = 50 mm, d1 = d2 = 0.1l, T1 − T0 = = 0.5 K, T2 − T0 = 1 K. Digitization is performed on the mesh ri = i

R2 + h/2 (i = 0, 1, . . . , 25), 25

zj = j

l (j = 0, 1, . . . , 50); 50

parameters of ﬁnite difference analogs of derivatives are ∆r = ∆z = 0.001(R2 + + h/2). In Eq. (33), M = 2 is chosen that corresponds to 6 terms of the expansion. Figure 2 illustrates contour and surface plots of the stationary temperature ﬁeld in the SWG resonator. It should be noted that, in spite of the small number of expansion terms, the solution obtained adequately describes the thermal ﬁeld distribution and, in particular, satisﬁes given boundary conditions (3)–(4) exactly. Non-stationary overheat may be evaluated with the help of the combined use of the R-function and Rote methods. Calculate the thermal time constant of the SWG resonator under the following conditions. Let at the initial moment t0 = 0 the whole resonator, uniformly heated

Fig. 2

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to the temperature T0 , be placed into the medium with the given temperature Tc (t). On the boundary we have the non-stationary 3rd kind boundary condition λ

∂u + α(u − Tc ) = 0, ∂ν

(34)

where α is the thermal exchange coefﬁcient [W/(m2 · K)]. An approximate solution can be written in the form of the structure for the 3rd kind boundary condition u = −ωϕ0 + Φ − ω (D1 Φ + gΦ) , where g = α/λ, ϕ0 = αTc /λ, and µ ω=

[−circle(R2 − h/2, r, z) ∧0 circle(R2 + ¶ R21 − r2 (l/2)2 − (z − l/2)2 +h/2)] ∨0 . (35) ∧0 2R1 l Contour lines of the function (35) are shown in Fig. 3. The solution of the stated problem can be found by the Rote method according to Eq. (29). Here, we neglect the lumped capacity boundary condition (30). Consider another numerical technique for evaluation of non-stationary heat transfer processes in the SWG, based on averaged parameters [13]. By V and S denote the volume and surface area of the resonator, respectively. By deﬁnition, the average volume temperature of a body is Z 1 udV . u ¯= V

Fig. 3

V

Suppose the thermal parameters of the resonator are time-independent. Then, using the Green formula and boundary condition (34), we obtain the equation d¯ u + m¯ u = mTc (t), dt

(36)

αS . Taking into account the initial condition, the general integral of cρV Eq. (36) takes the form where m =

Zt −mt

u ¯ = T0 e

−mt

Tc (τ )emτ dτ .

+ me

(37)

0

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Consider the following three cases. 1. Tc = const. The relative overheat is (¯ u − Tc )/(T0 − Tc ) = e−mt ,

(38)

i.e. the difference between the ambient temperature and the temperature of the body varies according to the exponential law. 2. Tc = bt + Tc0 . The relative overheat is expressed as u ¯ − Tc = (T0 − Tc0 + b/m)e−mt − b/m.

(39)

As time elapses, the ﬁrst term in Eq. (39) becomes negligibly small in comparison with the second one and the difference between the temperatures tends to the constant value u ¯ − Tc = −b/m. 3. Tc = T¯c +A cos ωt, where T¯c is the average value of the ambient temperature, A and ω are the amplitude and frequency of oscillations. From Eq. (36) for the difference between the temperatures, we have u ¯ − Tc = [(T0 − Tc (0)) + A∗ sin β] e−mt + A∗ sin(ωt − β), √ where β = arctg(ω/m) and A∗ = Aω/ m2 + ω 2 = A sin β. For large time values, the regular regime is achieved when u ¯ − Tc = A∗ sin(ωt − β), or u ¯ − T¯c = A cos β cos(ωt − β). From the analysis of the aforementioned formulas, the important role played by the parameter m is seen in all typical cases. This parameter is called the cooling (heating) rate of a body and is determined either experimentally or numerically. The inverse value ε = 1/m is called the exponent of thermal inertia characterizing the rate of attainment of the thermal equilibrium. In fact, this exponent is the thermal time constant if the ambient temperature is constant and equals the time when the relative temperature increases (decreases) e times. In cases 2 (linearly changing ambient temperature) and 3 the thermal inertia is characterized by more complicated parameters bε and ωε, respectively. Another important parameter expressing inertial properties of a body is the time of the system stabilization t∗ corresponding to the moment when the difference between the ambient temperature and the temperature of the system becomes less than a deﬁnite value ∆ = u ¯ − Tc . For the constant ambient temperature ¯ ¯ ¯ T 0 − Tc ¯ ∗ ¯ ¯. t = ε ln ¯ ∆ ¯ 108

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

To calculate thermal parameters of the SWG resonator, we assume λ = = 1.5 W /(m × K), = 900 J /(kg × K), ρ = 2.2 kg / m3 . It is easily seen that the area of the resonator surface area is equal to

h S = 2π R12 + R1 (l − δ) + R2 h+ i + 2(R22 + h2 /4) − (R2 + h/2)g1 − (R2 − h/2)g2 , and its volume is ·q V = πR12 (l − δ) +

π 6

¡ ¢ (R2 + h/2)2 − R21 4(R2 + h/2)2 + 2R21 − ¸ q ¡ −

(R2 −

h/2)2

−

R12

¢ 2

4(R2 − h/2) +

2R12

.

Here, q g1 = R2 + h/2 −

(R2 + h/2)2 − R12 ,

q g2 = R2 − h/2 − (R2 − h/2)2 − R12 , q q δ = (R2 + h/2)2 − R12 − (R2 − h/2)2 − R21 . Suppose α = 1 W /(m2 K), Tc − T0 = 1 K, and ∆ = 0.1 K. Then, for the aforementioned geometrical parameters we get m = 3.65 · 10−4 s−1 , ε = 2.74 · 103 s, t∗ = 6.31 · 103 s ≈ 1 hour and 45 minutes. Now, we shall restrict ourselves to the case when heat exchange occurs only on the side of the resonator leg, i.e., z ∈ [0, d1 ] ∪ [l − d2 , l], r = R1 . On the other parts of the surface we assume α = 0 in Eq. (34) (thermal insulation conditions). Instead of S in the aforementioned expressions we must take S1 = 2πR1 (d1 + d2 ). In this case, the following values of thermal parameters are obtained: m1 = = 1.18 · 10−5 s−1 , ε = 8.51 · 104 s, t∗ = 1.96 · 105 s ≈ 54 hours and 24 minutes, i.e. due to decrease of the heat exchange surface, the thermal time constant of the SWG resonator is increased almost 30 times. Conclusion. Results of numerical experiments conﬁrm efﬁciency of the R-function method for solving heat-transfer problems in the SWG resonator. The use of solution structures allows one to take a small number of expansion terms to achieve the appropriate accuracy of the approximation solution satisfying the given boundary conditions exactly. Application of the R-function method gives the possibility to identify main thermal characteristics of the resonator, to take into account the thermal component in the drift model, and to improve the device accuracy. VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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REFERENCES 1. Zhuravlev, V.F. & Klimov, D.M. The Solid-State Wave Gyro (in Russian), Moscow: Nauka, 1985. 2. Matveev, V.A., Lipatnikov, V.I., & Alekhin, A.V. Design of the Solid-State Wave Gyro (in Russian), Moscow: Izd-vo MGTU imeni N.E. Baumana, 1997. 3. Basarab, M.A., Kravchenko,V.F., Matveev, V.A., & Pustovoit, V.I. Atomic Functions in the Problem of Evaluating the Rayleigh Functions and the Precession Coefﬁcient for the Resonator of a Wave Solid-State Gyroscope, Doklady Physics, 2001, vol. 46, no. 2, pp. 113–118. 4. Pelpor, D.S., Matveev, V.A., & Arsenyev, V.D. Dynamically Adjusted Gyroscopes (in Russian), Moscow: Mashinostroenie, 1988. 5. Rvachev, V.L. & Sheiko, T.I. R-functions in Boundary Value Problems in Mechanics, Appl. Mech. Rev., 1995, vol. 48, no. 4, pp. 151–188. 6. Rvachev, V.L., Sheiko, T.I., Shapiro, V., & Uicker, J.J. Implicit Function Modeling of Solidiﬁcation in Metal Casting, ASME Transaction, Journal of Mechanical Design, 1997, vol. 119, no. 4, pp. 466–473. 7. Luikov, A.V. Heat-Conductivity Theory (in Russian), Moscow: Nauka, 1986. 8. Kartashov, E.M. Analytical Methods in the Theory of Heat Transfer in Solid State Bodies (in Russian), Moscow: Vysshaya Shkola, 2001. 9. Kantorovich, L.V. & Krylov, V.I. Approximate Methods of Higher Analysis, Noordhof, Groningen, 1958. 10. Mikhlin, S.G. Variational Methods in Mathematical Physics. New-York: MacMillan, 1964. 11. Mikhlin, S.G. The Numerical Performance of Variational Methods. Groningen: Noordhoff, 1971. 12. Marchuk, G.I. & Agoshkov, V.I. Introduction to Projective-Difference Methods (in Russian). Moscow: Nauka, 1981. (in Russian). 13. Dulnev, G.N. Heat- and Mass-Exchange in Radio Electronic Equipment (in Russian). Moscow: Vysshaya Shkola, 1984. 14. Kotlyar, Ya. M., Sovershennyi, V.D., & Strizhenov, D.S. Methods and Problems of Heat- and Mass-Exchange (in Russian). Moscow: Mashinostroenie, 1987. 15. Samarskii, A.A. The Theory of Finite Difference Schemes (in Russian). Moscow: Nauka, 1989. 16. Liskovets, O.A. The Rote Method (Survey). Differential Equations. 1965, vol. 4, no. 12, pp. 1662–1678. 17. Mosolov, P.P. Variational Methods in Non-stationary Problems (Parabolic Case), (in Russian). Izvestiya AN SSSR, ser. Mat., 1970, vol. 34, no. 2, pp. 425–457. 18. Kalinichenko, V.I., Koshchii, A.F., & Ropavka, A.I. On A Posteriori Evaluations of Boundary-Value Problems for Some Elliptic-Type Equations and Their Applications for the Nonstationary Case (in Russian). Preprint of the Siberia Branch of Russ. Acad. Sci., 1982, no. 96. 19. Erzhanov, R.Zh., Matsevityi, Yu.M., Sultangazin, U.M., & Sheryshev, V.P. Lumped Capacity in Problems of Thermal Physics and Microelectronics (in Russian). Kiev: Naukova Dumka, 1992. Matveev Valerii Aleksandrovich (born 1939) is a Dr. Sc. (Eng.), Professor, the Head of the Research and Education Unit “Information Technology and Control Systems” (NUK IU) of the Bauman MSTU. Scientiﬁc interests: gyroscopic systems, navigational complexes, control systems. Basarab Mikhail Alekseevich (born 1970) is a Cand. Sci. (Eng.), post-doctoral student of the Bauman MSTU. Scientiﬁc interests: applied mathematics, numerical methods. 110

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B.S. Konovalov, S.F. Konovalov, A.V. Kuleshov, D.V. Mayorov, V.P. Podtchezertsev, V.V. Fateev (Bauman Moscow State Technical University)

NEW TYPES OF VIBRATING GYRO FOR ROTATING CARRIER The paper presents the information of design and experimental study of prototypes of three types of vibrating rate sensors developed by BMSTU. The sensors are constructed using the design concept of a vibrating gyro (VG) which is to be used on a fast-rotating object for measuring angular velocity components ωξ , ωη of its turning around axes perpendicular to the axis of fast rotation. Being installed on the rotating carrier the sensitive element of VG gets an angular moment and becomes capable to respond to initiation of angular rate ωξ , ωη , by making oscillations, which amplitudes are proportional to ωξ , ωη . A prototype of the given sensor was developed by A.I. Suchkov more than 50 years ago, and now similar gyro sensors are in large scale production, but they show low accuracy characteristics. The complex of activities on creation of more precise VG has been conducted at BMSTU. The experimental batch of the VG and test equipment for their testing in conditions of batch production have been made.

Introduction. Two of the main directions of gyroscope technology evolution are simpliﬁcation and miniaturization of sensors designed for measuring parameters of spatial moving objects with a sufﬁcient accuracy. Vibrating gyroscopes are best to satisfy these requirements as they allow one to construct on their basis simple in design, reliable and cheap sensors. In particular, vibrating gyroscopes (VG) are very suitable for using on a rotating carrier. To solve mentioned above tasks of modern gyroscopic technology for building miniature, cheap and reliable sensors of primary information, the most simple and reliable among rotary vibrating gyroscopes is a single-channel, two-dimensional measuring device. Real possibilities of similar device construction have been proposed and proved in publications of A.I. Suchkov [1] and evolved by L.I. Brozgul [2], U.B. Vlasov [3], M.N. Lyuty [4] and others. The angular rate of the carrier along the longitudinal axis (typically from 10 to 25 rpm) can be used by such gyroscopes installed on the rotating carrier to obtain the angular momentum of the sensing element. This allows denying a rotary engine in the device, which substantially decreases dimensions, mass and value of the gyroscope, simpliﬁes its construction and manufacturing method and raises its reliability. But, together with the mentioned advantages, a large instability of a carrier angular rate along the longitudinal axis (10 . . . 20 rps) and its small value create large difﬁculties for developing the device and guaranteeing a given accuracy of its parameters. Sensor without feedback. The now known VG for a rotating carrier and new developed devices of this class represent a sensing element in a one-axis suspension, whose plane is perpendicular to the longitudinal axis of a rotating carrier VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

111

Fig. 1. To derivation of the equation of VG sensing element motion: ξηζ — absolute frame of reference; X1 Y1 Z1 — frame of reference, bound to carrier; XY Z — frame of reference, bound to gyroscope sensing element

(Fig. 1). When the carrier turns about the cross axis the sensing element is producing harmonic vibration in suspension support with a frequency equal to the carrier rotary rate and an amplitude proportional to angular rate about the cross axis. We can get the equation of motion of such gyroscope in a frame of reference, bound to the carrier, using dynamic non-generalized Euler’s equations (see Fig. 1): B1 Ω˙ Y − (C1 − A1 ) ΩX ΩZ = −Dα α˙ − Kα + MYEx , where OX — sensing axis, OY — sensor axis, OZ — rotation axis, A1 , B1 , C1 — inertia moments of the sensing element relatively to its three axis, Dα α˙ — damping moment, Dα — damping coefﬁcient, Kα — elastic moment, K — elastic coefﬁcient (angular stiffness of suspension), α — sensing element angular deﬂection, MYEx — disturbing (harmful) moment relatively to sensor axis, caused by the sensing element misbalance or some other reasons. Using the following equations for Ωx , Ωy , Ωz , deﬁned through the projections of the carrier angular speed on the axes of an absolute frame of reference Ωx ≈ Ωξ sin ϕ − Ωη cos ϕ − αϕ Ωy ≈ α + Ωξ cos ϕ + Ωη sin ϕ , Ωz ≈ ϕ˙ 0 where ϕ˙ 0 — an angular rate of the carrier rotation along the longitudinal axis, we derive the equation of motion of the sensing element: α ¨ + 2ξω0 α˙ + ω02 α =

H ˙ Ωsin (ϕ˙ 0 t − δ1 ) − Ωsin (ϕ˙ 0 t + δ2 ) , B1

q Ω2ξ + Ω2η is a projection of the carrier angular rate on the plane, q perpendicular to the carrier own rotation vector; Ω˙ = Ω˙ 2ξ + Ω˙ 2η is a projection

where Ω =

112

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

of the angular acceleration to the plane, perpendicular to the carrier own rotation vector, and phase shifts δ1 and δ2 are angles, deﬁning appropriate positions of the angular rate vector and angular acceleration vector in the sensing plane: Ã ! µ ¶ Ω˙ ξ Ωη δ1 = arctg ; , δ2 = arctg Ωξ Ω˙ η s ω0 =

ξ=

(C1 − A1 ) ϕ˙ 20 + K ; B1

Dα Dα = q ¡ ¢; 2B1 ω0 2 (C 2 B1 ˙0 + K 1 − A1 ) ϕ H0 = (C1 − A1 + B1 ) ϕ˙ 0 .

The derived equation of the sensing element motion and its solution are common for all VGs of this type, used on a rotating carrier. Angular deﬂection of the gyroscope sensing element in the presence of a constant measurable angular rate is deﬁned by the following equation:

α = q¡

(C1 − A1 + B1 ) ϕ˙ 0 × ¢2 2 2 (C1 − A1 − B1 ) ϕ˙ 0 + K + (Dα ϕ˙ 0 ) µ µ ¶¶ Dα ϕ˙ 0 × Ω sin ϕ˙ 0 t − arctg . (C1 − A1 − B1 ) ϕ˙ 0 + K

Such devices with different construction shemes can operate in either beforeresonance or resonance mode, using dynamic tuning. Measuring the angular deﬂection or angular rate of the sensing element gives information about its position. A construction scheme of one of the ﬁrst gyroscopes is shown in Fig. 2. A sensing element of the device is made as a frame, with a signal winding round the

Fig. 2. Construction scheme of VG VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

113

brass skeleton. On the two sides of the frame two holders with semi-axis are glued, on which the frame is suspended in the centering screw with corundum bearing. Two other sides of the frame are placed in the gap of the magnetic system, which consists of a sensor foundation and magnet. When the frame is moving in the magnetic system gaps an output signal is induced in the sensing winding by the law of electromagnetic induction, and through the ﬂexible conduction this signal arrives at the sensor output. In case of free suspension of the sensing element, accuracy and stability of the resonance tuning do not depend on the unstable carrier angular rate along the longitudinal axis, and are deﬁned only by the relationship of the frame moments of inertia: C1 − A1 = B1 , as it follows from the equation of motion of the sensing element. But such relationship of inertia moments is not possible in practice, because it requires a zero thickness of the sensing element. That’s why a resonance tuning in the device can only be realized for C1 − A1 = (0.9 . . . 0.95)B1 . As a result, a device operates in a post-resonant zone in the immediate vicinity to resonance. In such work mode, the frame angular deﬂection and output sensor signal are: µ µ ¶¶ (C1 − A1 + B1 ) Dα α= q Ω sin ϕ˙ 0 t − arctg , (C1 − A1 − B1 ) ϕ˙ 0 (C1 − A1 − B1 )2 ϕ˙ 20 + Dα2

Ksw (C1 − A1 + B1 ) ϕ˙ 0 U = Ksw α˙ = q

× (C1 − A1 − B1 )2 ϕ˙ 20 + Dα2 µ µ × Ω cos ϕ˙ 0 t − arctg

¶¶ Dα (C1 − A1 − B1 ) ϕ˙ 0

.

r (C1 − A1 ) Dα Dα p ; H0 = ; ξ = = B1 2B1 ω0 2ϕ˙ 0 B1 (C1 − A1 ) = (C1 − A1 + B1 ) ϕ˙ 0 ; Ksw — transfer coefﬁcient of the signal winding. As is seen, the scale factor of the sensor depends on ϕ˙ 0 . In this case, an error is up to 30% in the range of the carrier angular rate from 12 to 22 rps. This error might be corrected by algorithmic compensation with the use of measuring the carrier angular rate ϕ˙ 0 . Information about ϕ˙ 0 is contained in the output signal. If there were a resonant mode the output signal would be in phase with the measured angular rate. The output signal phase error is caused by the inaccurate resonance tuning and is deﬁned by: r 1 − µ2 C1 − A 1 ∆χ = . , where µ = 2ξµ B1 where ω0 = ϕ˙ 0

The scale factor of the sensor depends on damping moment. This moment is created by Laurence force appearing in the short-circuit frame skeleton due to eddy currents. In this case the value of damping moment is proportional to electrical resistance of the frame skeleton, which is strongly effected by ambient temperature changes. Brass was used as a skeleton material, because it provides the best 114

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

Fig. 3. Construction scheme of VG with DF

correlation between electrical conductivity and temperature coefﬁcient of speciﬁc electric resistance. Errors of the scale factor can reach 30% in ambient temperature range of −40 · · · + 75 o C. Sensor with feedback. The errors have essentially reduced in a new scheme of VG for a rotating carrier with degenerative speed feedback (VG with DF). The sensing element of this gyroscope (Fig. 3) is an electrically non-conducting frame. On the frame two electrical coils are placed. One of them — signal coil — is a means for measuring the angular rate α˙ of the sensing element vibration. Other coil provides the moment effect on the frame. The coils connected with each other by an ampliﬁer. Thus a sufﬁcient value of damping in the device is provided using feedback. Current in the torque motor coil with its following integration provides readout of the output signal. An amplitude of the device output signal is proportional to that of the sensing element vibration and, consequently, to the measured angular rate. VG with DF works in a mode close to resonance. Bringing the feedback in the device sets a problem of providing stability in the system. In Fig. 4 there is a sensor block scheme. The scheme uses the following designations: M Ex — inertia moment, caused by presence of an angular rate of the carrier along the cross axis; Md — damping moment, created by feedback; s.e. — sensing element with transfer function Wf (s),

Wf (s) =

1 α(s) 1 1 = ; = · ∆M (s) B1 s2 +D0 s + B0 B0 T0 s2 + 2ξ0 T0 s + 1

D0 — air damping coefﬁcient; α — angular deﬂection of the sensing element vibration; SC — signal coil of the sensing element with transfer coefﬁcient Ksc ; VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

115

Fig. 4. Structural scheme of VG with generative feedback

esc — electromotive force directing in the signal coil; Ka — gain of the preampliﬁer; Kc — transfer coefﬁcient of the current ampliﬁer; Kmi — mutual induction coefﬁcient; emi — electromotive force, as a result of mutual induction of coils; TMC — torque moment coil of the sensing element with transfer coefﬁcient Ktmc ; Wi (s) — transfer function of the integrator:

Wi (s) =

(1 + T1 s) . + 2ξi T2 s + 1

T22 s2

During the research of stability and accuracy of the gyroscope in the transducerampliﬁer, consisting of pre-ampliﬁer and current ampliﬁer, the mutual induction of coils is taken into consideration. Examination of the system stability has shown that the system is structurally stable in all range of the carrier angular rate along the longitudinal axis. By corresponding choice of the device parameters we can provide phase stability with a margin, reaching tens degrees. In VG for a rotating carrier it is necessary to have information about the instantaneous angular rate value, measured by the device. This raises requirements to the phase characteristics of the device. To reduce inﬂuence from the carrier rotating rate on the phase of the output signal, a correcting unit is brought in. This provides the phase stability, not exceeding 1 deg. Considering a coefﬁcient n, which describes the coverage of mutual induction, the transfer function of a self-contained system is:

Φ(s) =

(1 + T1 s) KK i · 2 2 × Ktmc Kc T2 s + 2ξi T2 s + 1 s (1 + Td s) ¢ ¡ 2 , × 2 T0 nT c s3 + T0 + 2ξ0 T0 nTc s2 + (nT c + 2ξ0 T0 + K) s + 1

where Td — time constant of the correcting unit; Tc — time constant of the transducer-ampliﬁer: Tc = Kc Ka Kmi ; K = Ktmc Ksc Ka Kc /B0 .

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In compliance with !

Ã

Uout = |Φ(jω)| H0 Ω sin ϕ˙ 0 t + arg (Φ(jω)) , ω=ϕ˙ 0

ω=ϕ˙ 0

in case of ideal integration the values T1 and T2 are big, and with admission n = 0, ξ0 = 0 while measuring the constant angular rate we receive the amplitude of output signal: Uampl = (C1 − A1 + B1 ) ϕ˙ 20 Ω

KKi · Ktmc Kc

q¡ T22 ϕ˙ 0

T1 ϕ˙ 0 . ¢2 1 − T02 ϕ˙ 20 + (2ξT0 ϕ˙ 0 )2

If the device operated in a resonance mode the output signal amplitude would be detrmined only by feedback parameters, integrator characteristics and moments of inertia of the sensing element, whose stability is no more than some fractions of a percent and the phase is 90 deg: Uampl =

2B1 Ki T1 Ω. Ktmc Kc T22

Inaccuracy of the resonance tuning creates errors in both amplitude and phase of the output signal. Besides that, imperfection of the integrator creates an additional dependence on the carrier rotation rate, and the correcting unit adds amplitude errors. That’s why it is necessary to choose the time constant Td as a compromise between the accurate compensation of phase errors and amplitude small errors. In the developed sensor the amplitude errors are not exceeding 1% in the range of change of the carrier rotation rate along the longitudinal axis from 12 to 22 rps. That allows us to use VG with DF without measuring ϕ˙ 0 and without algorithmic ways of compensation. Temperature instability of the output signal parameters are mostly deﬁned by temperature instability of induction in gaps of the magnetic system. For magnetic materials (Nd15 Fe77 B8 ) used in the sensor the values of output signal errors do not exceed 3% by amplitude and 2 deg by phase in the temperature range from −40 to +75 o C. If necessary, the error can be decreased using thermomagnetic shunts. Investigation of inﬂuence of the carrier angular vibration on accuracy of angular rate measurements was performed by comparing two amplitude-manipulated values in the frame of reference bound with the carrier. These values are the device output signal and external inﬂuence on the sensing element. The investigation of measurement accuracy for a variable angular rate is difﬁcult because values of the sensing element rotation rate ϕ˙ 0 and measurable angular rate ν are close to each other: carrier angular vibration frequency reaches half the sensing element own rotation rate. Let’s consider a case of angular vibration of the carrier. Here both external inﬂuence on the sensing element and the sensor output signal are sums of two harmonic values with frequencies equal to a sum and difference of previously mentioned frequencies ϕ˙ 0 and ν. VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

117

µ α ¨ + 2ξω0 α˙ +

ω02 α

¶ H0 ν + 2B1 2

=

Ω sin ((ϕ˙ 0 + ν) t) + ¶ µ ν H0 Ω sin ((ϕ˙ 0 − ν) t) . + − 2B1 2

An output signal of the device can be presented as: Uout = Um sin (ϕ˙ 0 t + χm ) , where q Um =

U+ 2 + U−2 + 2U+ U− cos (2νt − χ− + χ+ ),

q χm

U+ 2 + U−2 − 2U+ U− cos (χ+ + χ− ) = arctg × q 2 U+ + U−2 + 2U+ U− cos (χ+ + χ− ) ¶¶ µ µ U+ sin χ+ + U− sin χ− sin νt + arctg U cos χ+ − U− cos χ− µ µ + ¶¶ × , U+ sin χ+ − U− sin χ− cos νt + arctg U+ cos χ+ + U− cos χ− KKi ((C1 − A1 + B1 ) ϕ˙ 0 + νB1 ) Ω r ³

U+ =

´2 ,

2

2Ktm Kc Ti

2

(2ξT0 (ϕ˙ 0 + ν)) + 1 − T02 (ϕ˙ 0 + ν)

KKi ((C1 − A1 + B1 ) ϕ˙ 0 − νB1 ) Ω r ³

U− =

´2 ,

2

2Ktm Kc Ti

(2ξT0 (ϕ˙ 0 − ν)) + 1 −

2

T02 (ϕ˙ 0

− ν)

Ã π χ+ = + arctg (Td (ϕ˙ 0 + ν)) − arctg (i (ϕ˙ 0 + ν)) − arctg 2

! 2ξ0 (ϕ˙ 0 + ν) , 1 −20 (ϕ˙ 0 + ν)2 !

Ã π χ− = +arctg (Td (ϕ˙ 0 − ν))−arctg (Ti (ϕ˙ 0 − ν))−arctg 2

˙ 0 − ν) 2ξ0 (ϕT . 1 − T02 (ϕ˙ 0 − ν)2

By analogy with the device output signal the input inﬂuence on the sensing element can be presented like that: M Ex = Mm sin (ϕ˙ 0 t + χinm ) , where s µ Mm = Ω

118

H02 ν 2 B12 + + 2 2

¶ H02 ν 2 B12 − 2 2

cos (2νt);

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

µ χinm = arctg

¶ νB1 tg (νt) . H0

At angular oscillations of the carrier, phase and amplitude depend on time. That’s why it was offered to estimate accuracy of measuring the harmonic angular rate by comparing phases and amplitudes of envelopes of the output signal and external inﬂuence on the sensing element, and also their carrier phases at the moment of envelope maximum, when useful signal is maximal. In the result the relative frequency dependence of the device output signal is deﬁned by the following equation: q m (µ2 − 1)2 + (2ξµ)2 max 2 + 1) ∆Um 1 (µ r = − max ´2 ∆Mm 2 ³ 2 2 2 + (2ξµ (1 + m)) µ − (1 + m) q m (µ2 − 1)2 + (2ξµ)2 2 (µ + 1) − r³ + ´ 2

µ2 − (1 − m)2 + (2ξµ (1 − m))2 q (µ2 − 1)2 + (2ξµ)2 + r³ + ´2 2 2 2 + (2ξµ (1 + m)) µ − (1 + m) q + r³

max ν Mm ;m= . 0) (ν = 0) ϕ˙ 0 Ratios of amplitudes of the output signal and input moment in relative values are shown in Fig. 5. max where ∆Um =

max Um max Um (ν =

(µ2 − 1)2 + (2ξµ)2 ; ´2 2 2 2 + (2ξµ (1 − m)) µ − (1 − m) max ; ∆Mm =

max Mm

Fig. 5. Relative frequency dependence of amplitude of output signal envelope

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119

This ratio is deﬁned only by coefﬁcients of detuning and vibration damping of the sensing element of the gyroscope. To reduce amplitude errors it is necessary to increase the damping coefﬁcient and improve the resonance tuning accuracy. On account of dependence of the vibration damping coefﬁcient on the rotation speed of a carrier the real error values can vary in the range between curves shown in the ﬁgure. The same conclusion can be made regarding phase differences of envelopes and carriers of the output signal and external inﬂuence on the sensing element. The difference of phases of envelopes for the output signal and external inﬂuence on the sensing element will become µ ¶ 1 2ξµ (1 + m) − χ0 = (arctg (Td ϕ˙ 0 (1+m)) −arctg (Ti ϕ˙ 0 (1 + m)) −arctg 2 µ2 − (1 + m)2 µ ¶¶ 2ξµ (1 − m) −arctg (Td ϕ˙ 0 (1 − m)) + arctg (Ti ϕ˙ 0 (1 − m)) + arctg . µ2 − (1 − m)2 The frequency dependence of the envelopes phase difference is shown in Fig. 6. To determine the angular vibration frequency inﬂuence on the output signal carrier it is necessary to take into consideration a frequency dependence of phase differences on the carrier frequency of the output signal and external inﬂuence on the sensing element of the device in moments of the envelope maximum, when useful signal is maximal. This dependence is deﬁned by the following equation: v ³ ´ u ((µ2 +1) +m)2 λ+ 2 + ((µ2 +1) −m)2 λ2− −2 (µ2 +1)2 −m2 λ+ λ− cos(χ+ +χ− ) u u ´ ³ × χc = arctg t ((µ2 +1) +m)2 λ+ 2 + ((µ2 +1) −m)2 λ2− +2 (µ2 +1)2 − m2 λ+ λ− cos(χ+ +χ− )

Ã sin

− Ã

× cos

−

Ã ¡¡ !! ¢ ¢ ¡¡ ¢ ¢ µ2 + 1 + m λ+ sin χ+ + µ2 + 1 − m λ− sin χ− ((µ2 + 1) + m) λ+ cos χ+ − ((µ2 + 1) − m) λ− cos χ− Ã ¡¡ !! ¢ ¢ ¡¡ ¢ ¢ , 2 2 µ + 1 + m λ+ sin χ+ − µ + 1 − m λ− sin χ− + arctg ((µ2 + 1) + m) λ+ cos χ+ + ((µ2 + 1) − m) λ− cos χ−

χ+ − χ− + arctg 2 χ+ − χ− 2

Fig. 6. Envelope phase difference

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Fig. 7. Carrier phase difference at the moments of maximum envelope amplitude

where 1 λ+ = v ! ; Ã uµ ¶ 2 2 u 2ξ (ϕ˙ 0 + ν) 2 (ϕ˙ 0 + ν) t + 1− ω0 ω02 1 λ− = v Ã ! . uµ 2 2 u 2ξ (ϕ˙ 0 − ν) ¶2 (ϕ˙ 0 − ν) t + 1− ω0 ω02

The frequency dependence of carriers phase difference is shown in Fig. 7. In VG with DF the errors, appearing during angular and circular oscillations of a carrier, are essentially less (more than 10 times), than those of the device shown in Fig. 2. Micromechanical sensor. Let us consider one more design of VG for a rotating carrier. The device (Fig. 8) consists of a silicon plate 1, suspended in the body on ﬂexible torsions 2. Relatively to the body, the plate has one degree of freedom, deﬁned by the relative angular deﬂection α. The α angle is measured by the differential capacitive pick-off 3, whose moving electrode is the silicon plate, and motionless electrode is put on the isolated substrate. The sensing element comprises a plate, torsions and foundation and represents a uniﬁed construction, made of silicon using anisotropic etching. The photo of the created device is shown in Fig. 9. In micromechanical sensors a ﬂexible suspension of the sensing element is used. In compliance with common equations of the VG motion for a rotating carrier the equation of motion of a micromechanical gyroscope and its solution with a constant measurable angular rate are: ¢ ¡ B1 α ¨ + D α˙ + (C1 − A1 ) ϕ˙ 20 + KT α = (C1 + B1 − A1 ) ϕ˙ 0 Ω sin (ϕ˙ 0 t) ,

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Fig. 8. Construction scheme of micromechanical sensor

Fig. 9. Micromechanical sensor

α = q¡

µ (C1 + B1 − A1 ) ϕ˙ 0 Ω sin ϕ˙ 0 t− ¢2 2 2 (C1 − A1 − B1 ) ϕ˙ 0 + KT + (D ϕ˙ 0 ) µ ¶¶ D ϕ˙ 0 −arctg , (C1 − A1 − B1 ) ϕ˙ 20 + KT

where KT — angular stiffness of suspension. Conditions of resonance tuning with angular stiffness depend on the carrier rotation rate along the longitudinal axis, which doesn’t allow one to ensure the tuning stability in all range of carrier rotation rate changes. This causes output characteristics to be dependent of the carrier rotation speed along the longitudinal axis. In such design of the sensor because of a small thickness (C1 − A1 − B1 ) ϕ˙ 20 ¿ KT . If during the choice of device parameters we will try to insure KT > DΩ, then the moment being measured will be balanced mostly by stiffness moment of torsions, which makes the output signal stable enough during temperature changes. 122

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But in this case the output signal will be dependent of the carrier rotation rate along the longitudinal axis which requires to use the algorithmic compensation. In the case when DΩ À KT and the moment being measured is balanced mostly by damping moment, the dependence of output signal on the carrier rotation rate, as in previous devices, is much less. The created micromechanical sensor also works in the mode, close to resonance. Equation for an amplitude Um of the output signal is: Um =

KE (C1 + B1 − A1 ) ϕ˙ 0 Ω q . KT2 + (D ϕ˙ 0 )2

where KE — transfer coefﬁcient of the electric part of the device. Because of presence of torsion angular stiffness in the last equation an inﬂuence of the carrier rotation rate along the longitudinal axis on the output signal amplitude is slightly larger than for the case of VG with DF. According to the results of theoretical calculations this inﬂuence is no more than 12% in the range of the carrier rotation rate from 12 to 22 rps. Test results. A special equipment to test sensors was created, representing two rotary stands with a device, placed on them, which simulates the carrier rotation along the longitudinal axis, where a sensor is attached. The simulator of own carrier rotation is controlled using a personal computer and contains the pulse sensor of the simulator rotation rate, whose output signal is compared to a frequency that is set in the computer. Digital control system provides the rotation rate to be constant in limits from 10 up to 25 rps. Technical parameters of the rotation simulator 1. The range of changing rotation rate . . . . . . . . . . . . . . . . . . . . . . . 2. Stability of speciﬁed angular rate . . . . . . . . . . . . . . . . . . . . . . . . . 3. Mass (with holder for mounting sensor to the stand) . . . . . . . . 4. Dimension (with holder) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Power source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Operation conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Technical parameters of the test stand 1. The range of changing stand table constant angular rate . . . . . 2. Maximum frequency of oscillations . . . . . . . . . . . . . . . . . . . . . . . 3. Maximum angular rate of oscillations at 7 Hz frequency . . . . 4. The form of table oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Measurement accuracy of table angular rate . . . . . . . . . . . . . . . . 6. Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Power source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. Operation conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10 . . . 25 rps 1.5% 3.7 kg 150 × 150 × 220 mm 45 V; 1,5 A laboratory ±300 deg/s 7 Hz 50 deg/s sinusoidal 1% 40 kg ∅450 × 245 mm 2 × 40 V; 10 A laboratory

The stand for dynamic tests (Fig. 10) provides a constant rate of angular motion velocity and vibratory motion with controllable frequency and amplitude. The stand refers to a single-axis indicator gyrostabilizer with a ﬁber-optic angular rate sensor served as a sensing element. The stand is controlled by a personal computer. VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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Fig. 10. Kinematic scheme of test equipment

The computer receives data on an angular rate of the stand table, an imitator rotation rate and output signal of the sensor under test. The stand has its own thermostat mounted on the simulator rotary table (temperature within the range from +20 to 80 o C). Photo of the stand for dynamic tests with rotation simulator and ﬁxed testing sensor is shown in Fig. 11. Forms of the output signal on the computers display at constant and variable measured rates are shown in Fig. 12. The accuracy of reproduction of an angular rate by the dynamic stand is not high because of a mechanical reducer used in the stand drive. To perform accuracy tests with a constant angular rate in a large temperature range, including negative temperatures, the second rotary stand with a mechanical variator, Fig. 11. Test stand allowing rotation with the constant angular speed within the range from 0.01 deg/s to 1100 deg/s was used.

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Fig. 12. Output signal of VG with DF on computer’s display: a — when measuring a constant angular rate; b — at a harmonic angular rate

Since commonly used thermal chambers of “Tabai” type are not suitable for testing at negative temperatures because of large errors due to temperature change when the rotary stand with a simulator of rotation is placed into the chamber, we have developed the special thermal chamber (with a volume of 1 litre), mounted on the simulator holder. The chamber can house only the object table of the simulator with VG installed on it for testing and it has a nipple for supply of cooled gas. The nipple axis is to be aligned with the rotation axis of the stand table. The scheme of the thermostatic control system is shown in Fig. 13.

Fig. 13. Thermostabilization system

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Photos of the stand in the process of its adjusting, and the simulator together with thermal chamber, located on the stand table, are shown in Figs. 14, 15 respectively. Parameters of the stand 1. Range of angular rate speciﬁed . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Minimum rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Stability of angular rate speciﬁed . . . . . . . . . . . . . . . . . . . . . . . . . Parameters of thermostatic control system 1. Range of supported temperature . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Absolute accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Irregularity of temperature distribution on an internal surface of the chamber . . . . . . . . . . . . . . . . . . . . . 4. Amplitude of temperature ﬂuctuation in adjusting . . . . . . . . . . 5. Operating time of one ﬁlled Dewar ﬂask . . . . . . . . . . . . . . . . . . . 6. Time of temperature changing from room temperature to limit of temperature range . . . . . . . . . . . . . . . . . .

1300 deg/s 0.1 deg/s 0.5% −60 · · · + 90 o C ≤ 1.0 o C 0.5 o C 0.05 o C 12 hours less than 15 min

All three schemes of the sensor were investigated using the indicated equipment. The sensors show normal performance at the following operating conditions 1. Linear acceleration of arbitrary orientation . . . . . . . . . . . . . . . . . 2. Linear acceleration on rotation axis . . . . . . . . . . . . . . . . . . . . . . . 3. Impact in the form of half sine wave . . . . . . . . . . . . . . . . . . . . . . 4. Random vibration of arbitrary orientation . . . . . . . . . . . . . . . . . . 5. Environmental temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12 g 70 g 150 g, 6 ms 5 g, f = 20 . . . 2000 Hz (−40 · · · + 75) o C

Technical parameters of the devices are summarized in Table 1. Table 1 Sensors parameters

Range of measurable speeds Threshold of sensitivity

VG scheme Fig. 2 Fig. 3 300 deg/s 300 deg/s 1 deg/s 0.3 deg/s 30 1 mV/(deg/s) mV/(deg/s) 1 deg/s 0.4 deg/s

Scale factor Zero signal (3σ) Temperature changing of scale factor (3σ) 30% 3% in the range from −40 . . . 75 o C Changing of scale factor (3σ) with carrier rotation rate change in the range from 12 to 22 rps (without algorithmic compensation) 30% 1% ∅30 × 15 Dimension (mm) without ∅30×15 with electronics electronics 40 g 40 g Mass

126

Fig. 8 300 deg/s 0.15 deg/s 30 mV/(deg/s) 0.2 deg/s 6%

12% ∅22 × 8 with electronics 12 g

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Fig. 14. Thermo-chamber

Fig. 15. Simulator with thermo-chamber

Conclusion. VGs for the rotating carrier have been developed, made and tested. The accuracy of VG with DF is signiﬁcantly (more than 10 times) exceeds the accuracy of the existing devices of the given class. The application of technology of etching of silicon for manufacturing a sensing element has resulted in creation of the micromechanical sensor of angular rate and allowed reducing weight and dimensions of the device while maintaining the satisfactory accuracy, which can be improved by algorithmic compensation of inﬂuence of the carrier rotation rate along the longitudinal axis.

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REFERENCES 1. Author’s certiﬁcate. No 108731 USSR. Gyroscopic Rate Sensor (in Russian) / A.I. Suchkov // Bulleten izobretenii. – 1957. – No 12. 2. Brozgul, L.I., Smirnov, E.L. Vibrating Gyroscopes (in Russian). – M.: Mashinostroenie, 1970. 3. Vlasov, Y.B., Filonov, O.M. Rotor Vibrating Gyroscopes in Navigation Systems. – Leningrad.: Sudostroenie, 1980. 4. Fateev, V.V., Podtchezertsev, V.P., Lyuty, M.N. Vibrating Rate Sensor (in Russian) // Vestnik MGTU. – 1999. – No 1. Boris S. Konovalov (b. 1972) graduated from Bauman Moscow State Technical University in 1996, a post-graduate student of BMSTU. Sergei F. Konovalov (b. 1941) graduated from Bauman Moscow Higher Technical School in 1964. D. Sc.(Eng.), Professor, a Head of “Orientation, Stabilization and Navigation Devices & Systems” Department of BMSTU. Author of more than 150 publications in the ﬁeld of development of navigation accelerometers, gyros, test equipment, inclinometering of gas and oil wells. Alexander V. Kuleshov (b. 1972) graduated from Bauman Moscow State Technical University in 1998, a lecturer of “Orientation, Stabilization and Navigation Devices & Systems” Department of BMSTU. Author of 6 publications in the ﬁeld of gyroscopic technology development. Denis V. Mayorov (b. 1972) graduated from Bauman Moscow State Technical University in 1996, a post-graduate student of “Orientation, Stabilization and Navigation Devices & Systems” Department of BMSTU. Victor P. Podtchezertsev (b. 1945) graduated from Bauman Moscow Higher Technical School in 1969, Ph. D., Assoc. Professor of “Orientation, Stabilization and Navigation Devices & Systems” Department of BMSTU. Author of more than 30 publications in the ﬁeld of gyroscopic technology development. Vladimir V. Fateev (b. 1940) graduated from Bauman Moscow Higher Technical School in 1963, Ph. D., Professor of “Orientation, Stabilization and Navigation Devices & Systems” Department of BMSTU. Author of more 60 publications in the ﬁeld of gyroscopic technology development.

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INFORMATION & COMPUTATION TECHNOLOGY V.V. Devyatkov (Bauman Moscow State Technical University)

MULTIAGENT HIERARCHICAL RECOGNITION ON THE BASIS OF FUZZY SITUATIONAL CALCULUS The effectiveness of images (signals, images, scenes) recognition system depends on ability to reveal properties of separate segments of images, relations between them in space and in time. It is known, that recognition can be considered as the syntactic analysis in some language. It is also known that languages in which fuzzy sets underlie semantics, can serve as the convenient tool for recognition of fuzzy and irregular images. The languages based on fuzzy rules are popular. This paper presents the approach to recognition of signals and images, based on the fuzzy situational calculus used by hierarchically organized community of intellectual agents. Each agent deals with the volume of knowledge represented in fuzzy situational calculus as ontology of this agent. In the paper, within the framework of the accepted concept of fuzzy hierarchical multiagent recognition, the essence of the hierarchical approach, the concrete language of fuzzy situational calculus, principles of solving problems of recognition by a collective of agents are considered. Illustrative examples are given.

1. Introduction. Various kinds of fuzzy calculi for modeling those or other ﬁelds of knowledge [4, 6–8, 9–15] are known. In the paper the concrete kind of situational calculus and its use by community of intellectual agents for multiagent hierarchical recognition (MHR) are considered. The basic attention in paper is given to principles of formation of ontology for recognition of signals, images and scenes (SIS) by communities of intellectual agents. The term “ontology” is widely applied in works on artiﬁcial intellect [1, 2, 25–27]. In the present paper under ontology the formalized description of any properties of images (signals, images, scenes) in language of fuzzy situational calculus (FSC) is understood [28]. The situation is understood as some property of images [3] deduced in FSC in process of multiagent hierarchical recognition (MHR) which they get in certain time or in the certain place. The situation is deﬁned by special parameters (variables). The property also can be independent of time and space. There is no uniﬁed methodology for ontology construction though on this theme there is a set of publications. In the paper rather universal methodology of the ontology creation and use on the basis of FSC for MHR is offered, including (1) principles of hierarchical multiagent recognition; (2) foundation of FSC; (3) consideration of formation ontology principles for a separate agent; (4) processes of recognition by a separate agent. Any calculus uses some language with its own syntax and semantics. The analysis or reasoning in calculus can be considered as syntactic recognition [15, VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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19]. Last years various syntactic methods of recognition were developed. Some authors offer to create a hierarchy of small, but stable knowledge bases (ontologies) with the description of pattern properties with which the image properties are compared [13–15] to achieve fast and reliable recognition,. The drawbacks of known approaches are the certain expressive restrictions of languages, impossibility to use modern hierarchical multiagent approaches when agents can solve problems of recognition, carrying out the communications with each other in a manner “question – answer” in the distributed environment, impossibility to query the recognition system using rather complex questions, demanding the modal and situational reasoning during the recognition. Irrespective of as far as good the ontology is there can be always an image with some unexpected features to which existing descriptions of pattern images do not correspond. To overcome this problem some mechanism of ontology adaptation can be used. The most obvious one is updating from outside, for example by a user. There are some solutions to this problem using neuron or genetic algorithms [16– 18]. In the paper this problem is not mentioned, but our point of view is that FSC should be ﬂexible enough to allow realization of correction, and the organization and interaction in MHR should be such that adaptation could be realized by community of agents using the certain criteria, known beforehand or developed during functioning. Further in the paper some data from area of fuzzy linear segmentation of signals and images is entered, principles of ontology creation for agents of the bottom level are shown. Then the basic concepts of MHR and FSC are introduced, the illustrative example of inference at the bottom level of the MHR is considered. In the conclusion, problems of the near future are formulated. 2. Fuzzy syntactic recognition. An image is representation of some physical reality (object, phenomenon), having at least one parameter which contains the information and is referred to as informative. This information can be measured for each parameter from a determined set of values. Values of informative parameters of images are used for reasoning about properties of the image, for example, for the purpose of diagnostics of heart diseases, if the image is a cardiogram, or for the purpose of management of the pumps rotation speed if images represent diagrams of the liquid pressure change in pipelines. Parameters of images can be continuous and discrete. The continuous parameter is given in all range of deﬁnition X. The discrete parameter is given only in separate points of a range of deﬁnition, for example, through the same step of digitization ∆x , since some initial value: x0 = = x, x1 = x + ∆x , x2 = x + 2∆x , x3 = x + 3∆x , . . . Steps of digitization can be various. Digital images, as a rule, have only discrete parameters. For simplicity we shall consider a case when the image has only two parameters x ∈ X and y ∈ Y . Parameters are in the certain relation R(x, y) ∈ X × ×Y. In Fig. 1 the example of the digital image (signal) having two parameters x and y is shown. The steps of digitization of parameters x and y are ∆x , ∆y . We shall put in conformity to each relation R(xi , yj ) the node of the graph signed with a symbol Ni , i = 0, 1, . . . , 12. (We shall designate nodes by a circle). The most left node we shall also sign with a symbol S = N0 . We connect each pair of the neighboring nodes i, i + 1 by an arch directed from node i to node i + 1. An arch directed from node i to node i + 1 we shall sign with a sym130

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Fig. 1. A digital signal (a step of digitization on abscissa axis is ∆x )

Fig. 2. Segmentation of a digital image

Fig. 3. Transitions graph of the automation M

bol ai in alphabet A = {a0 , a1 , a2 , a3 , . . . , am } (Fig. 2). If to write out designations of all arches from left to right we shall obtain a sequence of symbols a0 a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 . We denote the arches, having the same direction and length in a sequence a0 a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 with the same symbol, that appears ﬁrst at viewing the sequence from left to right. In the result we shall obtain a sequence a0 a1 a2 a3 a4 a3 a6 a7 a8 a9 a6 a0 and a graph depicted in Fig. 3. This sequence can be considered as a word of some language L = L(G), generated by automaton grammar G = {V, A, P, S}, V = {N1 , N2 , N3 , N4 , N5 , A = {a0 , a1 , a2 , a3 , a4 , a6 , a7 , a8 , a9 }, P = N6 , N7 , N8 , N9 , N10 , N11 }, = {S → a0 N1 , N1 → a1 N2 , N2 →a2 N3 , N3 → a3 N4 , N4 → a4 N5 , N5 → → a3 N6 , N6 → a6 N7 , N7 → a7 N8 , N8 → a8 N9 , N9 → a9 N10 , N10 → a6 N11 , N11 → a0 }. VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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To this grammar there corresponds the recognizing automaton device M = = (A, B, f, b0 , F ), where A is the input alphabet, B is the output alphabet, f is a function of transitions, b0 is an initial state, F is a set of ﬁnal states, and B = {b1 , b2 , b3 , b4 , b5 , b6 , b7 , b8 , b9 , b10 , b11 , b12 }, f : f (a0 , b0 ) = b1 , f (a1 , b1 ) = b2 , f (a2 , b2 ) = b3 , f (a3 , b3 ) = b4 , f (a4 , b4 ) = b5 , f(a3 , b5 ) = b6 , f (a6 , b6 ) = b7 , f (a7 , b7 ) = b8 , f (a8 , b8 ) = b9 , f (a9 , b9 ) = b10 , f(a6 , b10 ) = b11 , f (a0 , b11 ) = = b12 , F = b12 . Thus, to an arch in Fig. 2 there corresponds a transition of the transitions graph of the automation (see Fig. 3). Our task is ascription of an image to one of pattern classes of images. For these purposes we shall construct the fuzzy grammar GF using the crisp grammar G on the basis of the following principles. To each arch of graph transitions there corresponds two neighboring nodes bi and bj (Fig. 4). Coordinates of node bi are xi and yj , and coordinates of node bj are xj and yj . We allow, that coordinates of both nodes can change within the range of some rectangular areas, for example, the squares given in Fig. 4 with steps of digitization accordingly δx and δy . Such assumption means, that instead of one node bi with coordinates xi , yi we shall have a set of nodes bni ∈ B(bi ) with the coordinates varied within the range, outlining the node (in Fig. 4 it is a square), and

n=

+ − + (x− i − xi ) (yi − yi ) · . δx δy

Instead of one node bj with coordinates xj , yj we shall have a set of nodes bsj ∈ B(bj ) with the coordinates varied within the range of a square, outlined

Fig. 4. Membership functions of transitions graph nodes

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around node bj , and instead of one arch ai , connecting node bi with node bj we shall have a set of all arches ali ∈ A(ai ) = {(bri , bsj )|bri ∈ B(bi ), bsj ∈ B(bj ), l ≤ n2 , r ≤ n, s ≤ n}, connecting each node of the set B(bi ) with each node of the set B(bj ). Projections of sides of the squares, outlining sets B(bi ) and B(bj ), to axis X + − + − + − + will be, respectively, x− i , xi and xj , xj , and to axis Y : yi , yi and yj , yj . We shall set four triangular membership functions µi (x), µi (y), µj (x), µj (y) which + are given on universes X and Y by triplets of values 0, 1, 0 in points {x− i , xi , xi }, − + − + − + {yi , yi , yi }, {xj , xj , xj }, {yj , yj , yj }, respectively (see Fig. 4). These functions deﬁne a measure of nearness of the graph node coordinates of to “the best coordinates” to which the membership function value, equal to 1, corresponds. The membership function is expressed by the formula u− u − k u − u− − u − u− , if uk ≤ u ≤ uk , k k k k µk (x) = + u −u + + k , if uk ≤ u ≤ u+ + k, uk − uk uk − uk where u = x, y; k = i, j. Let us believe, that membership function of each arch ali ∈ A(ai ), incidental to nodes bri ∈ B(bi ) and bsj ∈ B(bj ), is deﬁned as follows µA(ai ) (ali ) = min{µi (xri ), µi (yri ), µj (xsj ), µj (ysj )}. The fuzzy automaton grammar GF = {V, AF , PF , SF } is derived from the crisp automaton grammar G = {V, A, P, S} as follows [ A(ai ). AF = i

The unique initial nonterminal symbol of the crisp grammar is replaced with a set of initial nonterminal symbols SF = B(b0 ). The set of rules PF of the fuzzy grammar GF is deﬁned by an expression PF = {(Ni → ali Ni+1 , µ(Ni → ali Ni+1 ) = µA(ai ) (ali )), ali ∈ A(ai ), i = 0, 1, . . . , m, l ≤ n2 }. The fuzzy automaton grammar GF generates the fuzzy language {L(GF ), RL(GF ) } (in automaton designations it will be the language {L(MF ), RL(MF ) }). L(GF ) = {l∗ |l∗ = al0 al1 al2 . . . alm , ali ∈ A(ai ), l ≤ n2 }, RL(GF ) = {µL(GF ) (l∗ )/l∗ |l∗ = al0 al1 al2 . . . alm , ali ∈ A(ai ), µL(GF ) (l∗ ) =

min

{µA(ai ) (ali )}.

a=al0 ,al1 ,...,alm

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Thus, recognition by pattern images in two-dimentional space can be organized as follows. For each pattern image the construction of pattern fuzzy grammar is carried out. The number of grammars will be equal to that of pattern images. If to be limited to a syntactic level of recognition on fuzzy pattern grammars then the process of recognition of an image consists of the following steps. 1. One of pattern images (the fuzzy pattern grammar appropriate to it) is chosen. 2. The image to be recognized is processed with the same steps of digitization, as the chosen pattern. 3. Syntactic analysis of the image to be recognized with the help of the fuzzy grammar of the chosen pattern image is carried out. 4. If syntactic analysis appears unsuccessful, next pattern image is chosen out of still unconsidered ones and all repeats from item 2. 5. If syntactic analysis appears successful then membership function of the image being recognized is calculated and memorized. 6. If all pattern images are not yet considered, one more is chosen out of them and all repeats from item 2. 7. If no pattern images remain unconsidered, and there has been no successful syntactic analysis the recognition of the image comes to an end with a failure (the image is not recognized). 8. If there have been some successful syntactic analyses then recognition ﬁnishes successfully, and the recognized image can be identiﬁed with a class of that pattern image, whose grammar provided analysis having returned the maximal membership function for the recognized image. It is known [3] that rules of any grammar are easy to represent by rules of adequate calculus. The initial construction of the grammar is a convenient form of a well-formalizable way of transition from graphic representation of pattern images or their elements (given in two-dimensional space and having the form of pieces, lines and curves) to describing the pattern ontology in fuzzy calculus. If it is difﬁcult to allocate pieces or curves in an image, the considered method of ontology formation is inapplicable. Fortunately, the allocation of pieces and curves for the practical majority of images can be always made and it is an important part of recognition process. In case of the automaton grammar, it is easy to organize inference on the basis of calculus with the help of the automaton. Now we shall proceed to considering the possible organization of multiagent hierarchical recognition. 3. Multiagent hierarchical recognition. In the present paper the problem of research and substantiations of a certain original architecture of multiagent hierarchical recognition is not put. There are many works devoted to multiagent architectures. It is enough to mention one of the latest work [22]. Intellectual agents are united in an agency for solving problems. In our case it is a problem of recognition of images. Each of agents in agency solves some individual problem according to its role. Together they solve the common problem of the agency. Each agent has, as a minimum, input and output ports, base of knowledge (ontology) and the machine of inference. Through an input port an agent receives messages and inquiries from other agents, and also the information from other sources which are not agents in

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Fig. 5. Interrelation between semantic and syntactic levels

the speciﬁed sense. Through an output port an agent gives out messages and inquiries to other agents, and also the information to the addressees, which are not agents. In a basis of the approach considered in the given paper the idea lies about the differentiation of the syntactic and semantic description of image properties in the language of situational modal fuzzy calculus [29] (Fig. 5). The syntactic description serves as a means of data exchange between agents. The semantic description is used by agents for calculations. In this connection there are two levels of data representation. The ﬁrst (syntactic) level represents the ontology in the formal language of situational modal fuzzy calculus. The second (semantic) level represents the ontology in the language of fuzzy relations [22–25]. It can be considered as an original formal system with the language, rules of inference and interpretation. The inference is made at a semantic level. Communication between these two levels is carried out by means of two procedures: (1) semantic translation which carries out translation of the text presented at a syntactic level into the language of fuzzy relations, that refers to a semantic level; (2) linguistic approximation which carries out translation of the conclusions derived at a semantic level, into the language referring to a syntactic level (see Fig. 5). Agents communicate at a syntactic level. A diagram of multiagent hierarchy for recognition of images is shown in Fig. 6. A number of levels of agents hierarchy can be anyone. Agents of the ﬁrst level are engaged in segmentation of images. An example of segmentation by considering principles of syntactic recognition is given further. Other agents calculate relations by results of linguistic approximation of relations executed by agents of the previous levels. 4. Fuzzy relations. Let us remind some concepts from area of fuzzy relations. An expression ai (Xi , Ri ), where Xi = {xi1 , xi2 , . . . , xiri }, is a set of term values of a parameter x; Ri is a fuzzy set appropriate to the attribute ai and given on Xi . The attribute ai unambiguously corresponds to a term. A special case of a term, which we shall name the elementary one, is an expression ai (xij , µ(xij )), where xij is a single objective variable of the universe and µ(xij ) is the membership function determined for this variable. An expression α(a1 (X1 , R1 ), . . . , aN (XN , RN )) is referred to as the scheme of relations or the fuzzy scheme of relations where α is VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

135

Fig. 6. Multiagent hierarchy of images recognition

a name or an attribute of the scheme of relations; ai (Xi , Ri )) is a term; Ri is the fuzzy set appropriate to the attribute ai . Generally, if we have some attributes we shall believe, that we can always index these attributes in some order by integers from 1 up to N , where N is a quantity of attributes. Then a subset of attributes from all such sets, consisting of i attributes whose indices run a set of values I (i) , we shall designate A(I (i) ). Attributes are always ordered according to the order of indices in the set I (i) , and I (i) is arranged in increasing order. Thus, A(I (i) ) = {aj |j ∈ I (i) }. For example, images can be given by a matrix of dimension m×n. Coordinates (i, j) of elements of this matrix, where i = 1, . . . , m; j = 1, . . . , n, form universe Umn on which the set of values of some membership function µ(i, j) is given. Each element of the matrix with coordinates (i, j) can be represented by the elementary term aij (xij , µ(xij )), where xij = (i, j), i = 1, . . . , m; j = 1, . . . , n, and all matrix is the scheme of relations initial (a11 (x11 , µ(x11 )), a12 (x12 , µ(x12 )), . . . . . . , amn (xmn , µ(xmn ))), named as initial. We shall name the set A(I (i) ) as a basis of the scheme of relations. Conformity between the set of attributes ai and fuzzy sets Ri can be expressed as a function dom(ai ) = Ri . A set of all various subsets of indices I (i) , where 1 ≤ i ≤ N , is (N ) (i) i i }, where CN is a quantity of combinations designated Ji = {Ij |j = 1, . . . , CN from N by i. In view of the entered designations the scheme of relations can be designated as α[A(I (m) )], R[A(I (m) )] or R[i |i ∈ I (m) ]: dom(A(I (i) )) = X dom(aj ), j∈I (i)

where X is a sign of operation of the Cartesian product. j∈I (i)

136

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The scheme of relations R[A(I (i) )] V is referred to as normal if R is a normal fuzzy set, i.e. (∃u)[u ∈ dom(A(I (i) )) (µR (u) = 1)]. The scheme of relations R[A(I(i))] is referred to as separable, if for any partition A = {ξj |j = 1, . . . , m} of sets I (i) such, that m [

I (i) =

ξj ,

j 6= i,

ξj ∩ ξi = ∅,

j=1

the following takes place: m

R[A(I (i) )] = X Projξj R[A(I (i) )], j=1

where Projξj R[A(I (i) )] is a projection of fuzzy set R[A(I (i) )] onto ξj . The separable scheme of relations R[A(I (i) )] has the remarkable properties: (1) it can be derived as the Cartesian product of schemes of relations R[ξj ], being projections of the scheme of relations R[A(I (i) )] appropriate to disjoint subsets ξj ∈ A of the set I (i) , forming the partition A; (2) schemes of relations R[ξj ] can be derived as a result of a projection of the scheme of relations R[A(I (i) )]. Let us add the following designations: a set of all bases of schemes of relations {A(I (i) )}: n [

[ A(I (i) );

(i)

{A(I )} = i=1 I (i) ∈J (N ) i

a set of all schemes of relations {R[A(I (i) )]}, having the basis A(I (i) ): {R[A(I (i) )]} = {R[A(I (i) )]|A(I (i) ) ∈ {A(I (i) )}}. Above schemes of relations the following operations can be carried out. The supplementation, designated ¬:

R0 [A(I (i) )] = ¬R[A(I (i) )],

where ∀u ∈ dom(A(I (i) )),

µR0 (u) = 1−µR (u).

The union, designated ∪: P [A(I (i) )] = R[A(I (i) )] ∪ Q[A(I (i) )], where ∀u ∈ dom(A(I (i) )), µP (u) = max(µR (u), µQ (u)). The intersection, designated ∩: P [A(I (i) )] = R[A(I (i) )] ∩ Q[A(I (i) )], where ∀u ∈ dom(A(I (i) )), µP (u) = min(µR (u), µQ (u)). The implication, designated ⊃ [23]: g

P [A(I (i) )] = [R[A(I (i) )] ⊃ Q[A(I (i) )]], g

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137

where ∀u ∈ dom(A(I (i) )), µP (u) = [µR (u) ⊃ µQ (u)], and g

( 1, a 6 b, a⊃b =

∀a, b ∈ [0, 1].

g

b, a > b, The cylindrical continuation, designated CA(K (j) ) : R0 [A(I (i) ∪ K (j) )] = CA(K (j) ) R[A(I (i) )], where ∀u ∈ dom(A(I (i) )), ∀v ∈ dom(A(K (j) )), µR0 (u, v) = µR (u). (j) The projection designated ProjA(K) : R0 [A(K (j) )] = ProjA(K (j) ) R[A(K (j) ∪ I (i) )], where ∀u ∈ dom(A(K (j) )), µR0 (u) = sup{µR (u, v)|v ∈ dom(A(I (i) ))}. The linear transformation, designated Λ(k,b) : R0 [A(I (i) )] = Λ(k,b) R[A(I (i) )], where ∀u ∈ dom(A(I (i) )), µR0 (u) = max(0, min(1, k · µR (u) + b)). A result of linear transformation operation depends on a ratio of its parameters. In particular, if k > 1 and (−k) < b ≤ (1 − k), or if 0 < k < 1 and (−k) < < b < 0 then membership function values of a result of linear transformation do not exceed membership function values of its argument in all range of deﬁnition of these functions. In this sense, linear transformation can be considered as a version of an operation of concentration [24]. On the contrary, if 0 < k < 1 and (1 − k) < < b < 1, or if k > 1 and 0 ≤ b < 1 then membership function values of an argument of the operation do not exceed membership function values of its result, that is linear transformation in these cases can be considered as a version of an operation of stretching [24]. At values of the parameters satisfying a condition k > 1 and (1 − k) < b < 0, an operation of linear transformation can be considered as a version of an operation of the contrast intensiﬁcation [24] as it increases those values of the membership function which are more than (−b)/(1 + k) and reduces those less than (−b)/(1 + k). The comparison, designated =: R[A(I (i) )] = P [A(K (j) )], and

if

I (i) = K (j)

∀u ∈ dom(A(I (i) )), µR (u) = µP (u).

The inclusion, designated ⊆: R[A(I (i) )] ⊆ P [A(K (j) )], and

138

if

I (i) = K (j)

∀u ∈ dom(A(I (i) )), µR (u) ≤ µP (u).

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

The consistency, designated CP: τ = CP(R[A(I (i) )], P [A(I (i) )]), µτ (t) = sup{µP (u)|u ∈

if

∀t ∈ [0, 1],

µ−1 R (t)},

where µ−1 R (t) = {u|x = µR (u), u ∈ U }. 5. Syntax of fuzzy situational calculus language. The alphabet of a language of fuzzy situational calculus, on which the formal ontology of agents is described, comprises the following groups of symbols: names of linguistic variables {X1 , X2 , . . . , Xn }; linguistic values n [

θ=

θ(Xi ) i=1

of these variables; modalities Mod = {m}; logic symbols {¬, &, ∨, ⊃}; the dyadic predicate “is” and also brackets “(”, “)”. According to [23] a linguistic variable is characterized by the quintuple < X, θB (X), U , G, M > in which X is a linguistic variable name, θB (X) is a base term-set, U is the universe, the symbol G designates a syntactic rule, and the symbol M designates a semantic rule of a linguistic variable. A base term-set is ﬁnite and usually contains a small number of linguistic values. It is given and directly appears in the linguistic variable deﬁnition. All other values are derived by the syntactic rule G and form a term-set which is designated by the symbol θ(X). Elements of this term-set are constructed of elements of θB (X) with the help of a ﬁnite set of linguistic modiﬁers and connectives. Fuzzy sets R given on the universe U are put in conformity to the linguistic values. Thus each fuzzy set expresses a sense of the appropriate concrete linguistic value. Conformity between linguistic values of the set θ(X) and the fuzzy sets is established by the semantic rule M . The modalities included in the language under consideration are expressed as a kind of conformity of the semantic contents of modal propositions with the real world, and force or gradation of this conformity. Symbols of a modality include a preﬁx, expressing a modality type, and a natural number, expressing a gradation value. Let us consider modalities: “impossibility”, “probability”, “possibility” and “necessity”. The preﬁxes, corresponding to them, are: {Impossible , Probable , Possible , Necessary }. A number after a preﬁx, expressing the gradation of the appropriate modality, should not exceed hundred. The symbols of modalities are read as follows: I mpossible N −”with the N % impossibility”; P robable N — ”with the N % probability”; P ossible N — ”with the N % possibility”; N ecessary N — ”with the N % necessity”, where N is a gradation value expressed in percentage. All 400 symbols of modalities form the set M od. Thus, any well-formed formula (wff) of a language in which a formal ontology is described, represents a chain of symbols, satisfying the following BNF: atom ::= Y is X VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

139

wﬀ ::=< atom > |¬ < wﬀ > |(< wﬀ > & < wﬀ >)|(< wﬀ > ∨ < wﬀ >)| (< wﬀ >⊃< wﬀ >)|m < wﬀ >, where Y ∈ {X1 , X2 , . . . , Xn }; X ∈ θ(Y ); m ∈ Mod . A procedure of semantic translation establishes conformity between expressions of a language and their semantic contents expressed by fuzzy relations. Thus it is considered, that each linguistic variable one-to-one corresponds to an attribute, and the universe of linguistic variables one-to-one corresponds to a domain. The atom a =”Y is X” corresponds to the scheme Rα [A(I (1) )] = α(a(d, Rα )), where a = Y , Rα = M (X), d = U . Conformity between other syntactic constructions and fuzzy schemes of relations is shown in Table 1 where N means gradation of the appropriate modality, parameter n is calculated under the formula n = N/100, and i, j are indices of linguistic values in formulas. Table 1 The formula of a language

The fuzzy relation

wﬀ i

Rwﬀ i [A(I (i) )]

wﬀ j

Rwﬀ j [A(K (j) )]

¬wﬀ i

¬Rwﬀ i [A(I (i) )]

(wﬀ i &wﬀ j )

CK Rwﬀ i [A(I (i) )] ∩ CI Rwﬀ j [A(K (j) )]

(wﬀ i ∨ wﬀ j )

CK Rwﬀ i [A(I (i) )] ∪ CI Rwﬀ j [A(K (j) )]

(wﬀ i → wﬀ j )

CK Rwﬀ i [A(I (i) )] ⊃ CI Rwﬀ j [A(K (j) )]

(j)

(i)

(j)

(i)

(j)

(i) g

Impossible N wﬀ i

¬Λ(1/(1−n),0) Rwﬀ i [A(I (i) )]

Probable N wﬀ i

¬Λ(1/n,−(1−n)/n) Rwﬀ i [A(I (i) )]

Possible N wﬀ i

Λ(1/n,0) Rwﬀ i [A(I (i) )]

Necessary N wﬀ i

Λ(1/(1−n),−n/(1−n)) Rwﬀ i [A(I (i) )]

6. Inference and linguistic approximation. Using the considered operations for fuzzy relations we shall enter the following rules for realization of inference. Successor rule. If fuzzy relation Rα [A(I (i) )] is speciﬁed and it is known, that Rα [A(I (i) )] ⊆ Rβ [A(I (i) )], then it is possible to deduce fuzzy relation Rβ [A(I (i) )]. To put it differently, if the formula α has the semantic contents Rα [A(I (i) )], that we shall write down as < α >= Rα [A(I (i) )], and the fuzzy relation Rβ [A(I (i) )] exists, such that Rα [A(I (i) )] ⊆ Rβ [A(I (i) )], then the formula β with semantic contents Rβ [A(I (i) )] is deduced from the formula α: α, < α >= Rα [A(I (i) )], Rβ [A(I (i) )], Rα [A(I (i) )] ⊆ Rβ [A(I (i) )] ` β, < β >= Rβ [A(I (i) )]. For example, let it be known, that the person is very young. This fact can be expressed by the formula “the person is very young”, where the person is a 140

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name of a linguistic variable “person”, young is its base linguistic value, very is a linguistic modiﬁer. The semantic contents of this formula is expressed by the fuzzy relation the person (very young (U, M (very young)). On the other hand, for example, it is known, that the fuzzy set M (young) includes M (very young), i.e. M (very young) ⊂ M (young). On this basis it is possible to establish, that the given person is young. The formula “the person is young” corresponds to this conclusion. Generalization rule. If the fuzzy relation Rα [A(I (i) )] is known then it is possible to deduce the fuzzy relation Rβ [A(I (i) ∪K (j) )] such, that Rβ [A(I (i) ∪ K (j) )] = = CA(K (j) ) Rα [A(I (i) )], where K (j) ∩ I (i) = ∅. In other words, if the formula α has semantic contents Rα [A(I (i) )] and some attributes which are not included in the scheme A(I (i) ) whose indices make the set K (j) , then the formula β, having semantic contents Rβ [A(I (i) ∪ K (j) )] = = CA(K (j) ) R[A( I (i) )], is deduced from the formula α: α, < α >= Rα [A(I (i) )], K (j) ∩ I (i) = ∅ ` β, < β >= CA(K)( j) R[A(I (i) )]. For example, if it is known, that “the person is young”, and nothing is known about the person’s weight, it is possible by a generalization rule to infer: “the person is young and weight is any”. Projection rule. If fuzzy relation Rα [A(I (i) )] is known then it is possible to deduce the fuzzy relation Rβ [A(K (j) )] such, that Rβ [A(K (j) )] = = ProjA(K (j) ) Rα [A(I (i) )], where K (j) ⊂ I (i) . That is, if the formula α has the semantic contents Rα [A(I (i) )] and some subset of attributes is allocated from the scheme A(I (i) ), whose indices make the K (j) , then the formula β with the value Rβ [A(K (j) )] = ProjA(K (j) ) Rα [A(I (i) )] is deduced from the formula α: α, < α >= Rα [A(I (i) )], K (j) ⊂ I (i) ` β, < β >= ProjA(K (j) ) Rα [A(I (i) )]. For example, if it is known, that “the person is young and weight is big” by a projection rule then it is possible to deduce formulas: “the person is young” and “weight is big”. Intersection rule. If two fuzzy relations Rα [A(I (i) )] and Rβ [A(I (i) )] are known then it is possible to deduce the third fuzzy relation Rγ [A(I (i) )] such, that Rγ [A(I (i) )] = Rα [A(I (i) )] ∩ Rβ [A(I (i) )]. To put it differently, if the formula α has the semantic contents Rα [A(I (i) )] and the formula β has the semantic contents Rβ [A(I (i) )], then the formula γ, which is coordinated to semantics of formulas α and β is deduced from them: α, < α >= Rα [A(I (i) )], β, < β >= Rβ [A(I (i) )] ` γ, < γ >= Rα [A(I (i) )] ∩ Rβ [A(I (i) )]. As an inference process we shall understand such a sequence of application of the above-listed rules that provides deducing the required output formula with the appropriate semantic contents. VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

141

An accountable set of formulas at a syntactic level and a continuum of the semantic contents expressed by fuzzy relations at a semantic level can be presented. In the process of inference, which is carried out at a semantic level, a fuzzy relation, having no expression at a syntactic level, can be derived. In this connection there appears a problem of linguistic approximation. Let the fuzzy relation Rβ [A(I (i) )], where A(I (i) ) = {X1 , X2 , . . . , Xi }, be obtained as a result of inference. Linguistic approximation of the relation Rβ [A(I (i) )] consists in ﬁnding such a formula α that its semantic contents < α >= Rα [A(I (i) )] would coincide with Rβ [A(I (i) )] or be close to it. For solving the problem we shall search among the formulas having the form (α1 &(. . . &(αi−1 &αi ) . . .)), in which every subformula αk (k ∈ 1, 2, . . . , i}) has the structure: (mk,1 Xk is Xk,1 &(. . . &(mk,q−1 Xk is Xk,q−1 &mk,q Xk is Xk,q ) . . .)), where {Xk,1 , . . . , Xk,q } = θB (Xk ) is the base term-set of the linguistic variable Xk ∈ A(I (i) ); mk,1 , . . . , mk,q are symbols of modalities. As the structure of the formula α is ﬁxed, linguistic approximation is reduced to ﬁnding all symbols of the modalities which are included in the subformula of a kind: mk,j Xk is Xk,j (Xk ∈ A(I (i) ), Xk,j ∈ θB (Xk )). For ﬁnding every concrete mk,j we shall carry out semantic translation of atom Xk is Xk,j and we shall construct projection Rβ [A(I (i) )] on the scheme {Xk }. We shall designate these fuzzy relations Rα [Xk ] and Rβ [Xk ]. We shall ﬁnd the degree of consistency τ = CP(Rα [Xk ], Rβ [Xk ]) of the semantic contents of the atom Xk is Xk,j , and the semantic contents of the inference concerning a value of the variable Xk . By deﬁnition of a degree of consistency, τ is a fuzzy set in [0, 1] and expresses an estimation of conﬁdence that the given inference can be expressed by the formula Xk is Xk,j . On the other hand, the modality is also an estimation of the conﬁdence, but only presented at a syntactic level. We shall note, that the modal semantics, expressed in a procedure of semantic translation through operations of linear transformation with the appropriate parameters, can be also deﬁned in another way. We shall consider the linguistic variable < confidence, Mod, [0, 1], G, M > whose base term-set contains all earlier considered symbols of modalities, syntactic rule G generates only elements of Mod, and semantic rule M establishes conformity between symbols of modalities and fuzzy sets in an interval [0, 1]. These fuzzy sets have trapezoid membership functions which can be constructed proceeding from intuitive representations about a nature of the modal concepts expressing feeling of conﬁdence. In particular, for modalities Impossible N , Probable N , Possible N and Necessary N the membership functions of the fuzzy sets generated by the semantic procedure have the following form: ∀x ∈ [0, 1]: µM (Impossible

N ) (x)

= 1 − max(0, min(1, x/(1 − n)));

µM (Probable

N ) (x)

= 1 − max(0, min(1, (x + n − 1)/n));

µM (Possible

N ) (x)

= max(0, min(1, x/n));

µM (Necessary

N ) (x)

= max(0, min(1, (x − n)/(1 − n))),

where n = N/100. 142

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

Proceeding from this, linguistic approximation consists in a choice of such symbols of the modality mk,j , that difference of values µM (mk,j ) and appropriate values µτ would be minimal and ∀t ∈ [0, 1], µM (mk,j ) (t) > µτ (t). Formulas in the language of fuzzy situational calculus grow out of works of separate agents and are transferred to other agents for the subsequent use during recognition. 7. Example. Let us consider an example of evaluation of the membership function and linguistic approximation by the agent of the ﬁrst level according to results of segmentation after which in two-dimensional space the coordinate of a point (node) on the abscissa axis is determined. Let the arrangement of a node be characterized by two linguistic variables: < X, θB (X), U, G, M > and < Y, θB (Y ), V, G, M >, where X and Y are linguistic variables, describing the node position on the abscissa and ordinate axes, respectively; θB (X) = = θB (Y ) = {At the left , At the center , At the right } are base sets of linguistic values; U = V = [−1, 1] is the universe of values of coordinates on axes X and Y ; G — a syntactic rule, M — a semantic rule. Base sets of linguistic values of both linguistic variables are chosen conterminous for convenience of a statement. Identical dimensions universes values of coordinates from −1 up to 1 on both axes are chosen conterminous also for convenience of a statement. Let us believe, that U = V = {−1, −0.8, −0.6, −0.4, −0.2, 0, 0.2, 0.4, 0.6, 0.8, 1}. Semantic contents of linguistic values At the left, At the center, At the right are deﬁned by semantic rules: M (At the left ) = µL , M (At thecenter ) = µC , M (At the right ) = µR . Diagrams of membership functions µL , µC , µR are presented in Fig. 7. The ontology contains the following axioms expressing restrictions on an arrangement of nodes of the given segment, caused by its properties. (X is At the left ⊃ Y is At the right )

(1)

(X is At the center ⊃ Y is At the center )

(2)

(X is At the right ⊃ Y is At the left )

(3)

(Poss 50 X is At the center and Poss 50 X is At the right ).

(4)

It is required to ﬁnd the answer (linguistic approximation) to a question, which represents the following essence: “If the ontology, represented by axioms (1)–(4),

Fig. 7. Membership functions of node coordinates VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

143

is known, what is the unknown coordinate of a node?”. The axiom 4 determines a coordinate of the node on axis X. Therefore only a coordinate of the node on axis Y is unknown. In our calculus the question is put by the following expression (m1 Y is At the left and (m2 Y is At the center and m3 Y is At the right )) (5) In this expression the agent should deﬁne values of modalities m1 , m2 , m3 , describing an arrangement of nodes on axis Y . Let us carry out semantic translation of the ontology (1)–(4), and then linguistic approximation of the expression (5). We have two linguistic variables X, Y . For this set of linguistic variables three bases of schemes of relations {X}, {Y }, {X, Y } are possible. We shall illustrate semantic translation of the ﬁrst axiom of the ontology in detail: (X is At the left ⊃ ⊃ Y is At the right ). The given formula includes atoms X is At the left and Y is At the right. Results of semantic translation of these atoms are fuzzy relations RL [X] =< {X}, M (At the left ) > and RR [Y ] =< {Y }, M (At the right ) >, where RL = M (At the left ) is fuzzy set in universe U = [−1, 1], RR = M (At the right ) is fuzzy set in universe V = [−1, 1]. Fuzzy sets RL and RR (Fig. 7) have the membership functions presented in Tables 2 and 3. Table 2 x

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1

0.8

0.6

0.4

0.2

0

0

0

0

0

0

µRL (x)

Table 3 y

µRR (y)

−1

0

−0.8

0

The result of semantic translation of the formula (X is At the left ⊃ Y is At the right ) = is the fuzzy relation R1 [X, Y ] C{Y } RL [X] ⊃ C{X} RR [Y ], where R1 is a fuzzy g

−0.6

0

−0.4

0

−0.2

0

0

0

0.2

0.2

0.4

0.4

0.6

0.6

0.8

0.8

1

1

144

set in the universe U × V = [−1, 1] × [−1, 1]. For calculation of its membership function it is necessary to receive cylindrical continuation C{X} RR [Y ] of the fuzzy relation RL [X] on a basis of the scheme {Y } and cylindrical continuation C{X} RR [Y ] of the fuzzy relation RR [Y ] on a basis of the scheme {X}. Membership functions of these cylindrical continuations are given in Tables 4 and 5 respectively. They are represented as matrices in which rows are signed with elements of the universe V , and columns — with elements of the universe U . On crossing of columns x and rows y there are values of the membership function µ(x, y).

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

Table 4 x y −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1

1

0.8

0.6

0.4

0.2

0

0

0

0

0

0

−0.8

1

0.8

0.6

0.4

0.2

0

0

0

0

0

0

−0.6

1

0.8

0.6

0.4

0.2

0

0

0

0

0

0

−0.4

1

0.8

0.6

0.4

0.2

0

0

0

0

0

0

−0.2

1

0.8

0.6

0.4

0.2

0

0

0

0

0

0

0

1

0.8

0.6

0.4

0.2

0

0

0

0

0

0

0.2

1

0.8

0.6

0.4

0.2

0

0

0

0

0

0

0.4

1

0.8

0.6

0.4

0.2

0

0

0

0

0

0

0.6

1

0.8

0.6

0.4

0.2

0

0

0

0

0

0

0.8

1

0.8

0.6

0.4

0.2

0

0

0

0

0

0

1

1

0.8

0.6

0.4

0.2

0

0

0

0

0

0

Table 5 x y −1 −0.8 −0.6 −0.4 −0.2

0

0.2 0.4 0.6 0.8

1

−1

0

0

0

0

0

0

0

0

0

0

0

−0.8

0

0

0

0

0

0

0

0

0

0

0

−0.6

0

0

0

0

0

0

0

0

0

0

0

−0.4

0

0

0

0

0

0

0

0

0

0

0

−0.2

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0.2

0.2

0.2

0.2

0.2

0.2

0.2 0.2 0.2 0.2 0.2 0.2

0.4

0.4

0.4

0.4

0.4

0.4

0.4 0.4 0.4 0.4 0.4 0.4

0.6

0.6

0.6

0.6

0.6

0.6

0.6 0.6 0.6 0.6 0.6 0.6

0.8

0.8

0.8

0.8

0.8

0.8

0.8 0.8 0.8 0.8 0.8 0.8

1

1

1

1

1

1

1

1

1

1

1

1

The next step in evaluation of the membership function of the fuzzy relation R1 [X, Y ] is application of implication to the obtained cylindrical continuations. The membership function µR1 (x, y) is given in the matrix form in Table 6.

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Table 6 x y −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 −0.8 0 0 0 0 0 1 1 1 1 1 1 −0.6 0 0 0 0 0 1 1 1 1 1 1 −0.4 0 0 0 0 0 1 1 1 1 1 1 −0.2 0 0 0 0 0 1 1 1 1 1 1 0 0.2 0.2 0.2 0.2 1 1 1 1 1 1 1 0.2 0.4 0.4 0.4 1 1 1 1 1 1 1 1 0.4 0.6 0.6 1 1 1 1 1 1 1 1 1 0.6 0.8 1 1 1 1 1 1 1 1 1 1 0.8 1 1 1 1 1 1 1 1 1 1 1 1

We shall not carry out semantic translation of other axioms of the ontology in detail. We shall only show formulas for semantic translation (left column of Tables 7, 8) and fuzzy relations which are used for calculation (right column of Tables 7, 8). Table 7 The formula of a language X is At the left Y is At the right X is At the center Y is At the center X is At the right Y is At the left

The fuzzy relation RL [X] =< {X}, M (At the left) > RR [Y ] =< {Y }, M (At the right ) > RC [X] =< {X}, M (At the left) > RC [Y ] =< {Y }, M (At the center ) > RR [X] =< {X}, M (At the right) > RL [Y ] =< {Y }, M (At the left ) > Table 8

The formula of language

The fuzzy relation

(X is At the left ⊃ Y is At the right )

R1 [X, Y ] = C{Y } RL [X] ⊃ C{X} RR [Y ]

(X is At the center ⊃ Y is At the center )

R2 [X, Y ] = C{Y } RC [X] ⊃ C{X} RC [Y ]

(Xis At the right ⊃ Y is At the left)

R3 [X, Y ] = C{Y } RR [X] ⊃ C{X } RL [Y ]

g

g

g

(Possible 50 X is At the center and Possible 50 X is At the right)

R4 [X] = λ(2,0) RC [X] ∩ λ(2,0) RR [X]

Final results of semantic translation of the second and third axiom are given in Tables 9 and 10 (membership functions of fuzzy relations R2 [X, Y ] and R3 [X, Y ] in the matrix form). 146

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

Table 9 x y −1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

−1

1

0

0

0

0

0

0

0

0

0

1

−0.8

1

1

0.2

0.2

0.2

0.2

0.2

0.2

0.2

1

1

−0.6

1

1

1

0.4

0.4

0.4

0.4

0.4

1

1

1

−0.4

1

1

1

1

0.6

0.6

0.6

1

1

1

1

−0.2

1

1

1

1

1

0.8

1

1

1

1

1

0

1

1

1

1

1

1

1

1

1

1

1

0.2

1

1

1

1

1

0.8

1

1

1

1

1

0.4

1

1

1

1

0.6

0.6

0.6

1

1

1

1

0.6

1

1

1

0.4

0.4

0.4

0.4

0.4

1

1

1

0.8

1

1

0.2

0.2

0.2

0.2

0.2

0.2

0.2

1

1

1

1

0

0

0

0

0

0

0

0

0

1

Table 10 x y −1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

−1

1

1

1

1

1

1

1

1

1

1

1

−0.8

1

1

1

1

1

1

1

1

1

1

0.8

−0.6

1

1

1

1

1

1

1

1

1

0.6

0.6

−0.4

1

1

1

1

1

1

1

1

0.4

0.4

0.4

−0.2

1

1

1

1

1

1

0.2

0.2

0.2

0.2

0

1

1

1

1

1

1

0

0

0

0

0

0.2

1

1

1

1

1

1

0

0

0

0

0

0.4

1

1

1

1

1

1

0

0

0

0

0

0.6

1

1

1

1

1

1

0

0

0

0

0

0.8

1

1

1

1

1

1

0

0

0

0

0

1

1

1

1

1

1

1

0

0

0

0

0

It is especially needed to emphasize semantic translation of the last axiom of the ontology as it illustrates application of an operation of linear transformation. In Table 11 results of semantic translation of axiom Possible 50 X is At the center and Possible 50 X is At the right are given.

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Table 11

x −1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

µR C l(2,0) µRC (x)

0

0.2

0.4

0.6

0.8

1

0.8

0.6

0.4

0.2

0

0

0.4

0.8

1

1

1

1

1

0.8

0.4

0

µRR (x) l(2,0) µRR (x)

0

0

0

0

0

0

0.2

0.4

0.6

0.8

1

0

0

0

0

0

0

0.4

0.8

1

1

1

µR4 (x)

0

0

0

0

0

0

0.4

0.8

0.8

0.4

0

It should be noted, that the membership function µR4 (x) takes on a value of unity at x = 0.5, however this value is not given in Table 11, as value x = 0.5 drops out of consideration at the accepted digitization of the universe. Thus, semantic translation of axioms is completed and it is possible to pass to inference.

1) R1 [X, Y ],

an axiom,

2) R2 [X, Y ],

an axiom,

3) R3 [X, Y ],

an axiom,

4) R4 [X],

an axiom,

5) R5 [X, Y ] = C{Y } R4 [X],

from 4) by a generalization rule,

6) R6 [X, Y ] = R1 [X, Y ] ∩ R2 [X, Y ],

from 1) and 2) by an intersection rule,

7) R7 [X, Y ] = R6 [X, Y ] ∩ R3 [X, Y ],

from 6) and 3) by an intersection rule,

8) R8 [X, Y ] = R7 [X, Y ] ∩ R5 [X, Y ],

from 7) and 5) by an intersection rule,

9) R9 [Y ] = Proj{Y } R8 [X, Y ],

from 8) by a projection rule.

Each value of the membership function of the fuzzy relation R8 [X, Y ] represents minimum of the appropriate values of membership functions of fuzzy relations R1 [X, Y ], R2 [X, Y ], R3 [X, Y ] and cylindrical continuation onto a basis of the scheme {Y } of the fuzzy relation R4 [X]. The membership function of the fuzzy relation R4 [X] is given in the matrix form in Table 12. The conclusion concerning a value of the coordinate Y represents the fuzzy relation R9 [Y ], which is the projection R8 [X, Y ] onto a basis of the scheme {Y }. The membership function of this fuzzy relation is given in Table 13.

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Table 12 x y

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

−1 0 0 0 0 0 0 0 0 0 0 0

−0.8 0 0 0 0 0 0 0 0 0 0 0

−0.6 0 0 0 0 0 0 0 0 0 0 0

−0.4 0 0 0 0 0 0 0 0 0 0 0

−0.2 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

The result of linguistic approximation of the fuzzy relation R9 [Y ] is the formula (m1 Y is At the left &(m2 Y is At the center & m3 Y is At the right )) in which we should deﬁne modality symbols: m1 , m2 and m3 . We shall ﬁnd degrees of consistency τ1 = = CP(RL [Y ], R9 [Y ]), τ2 = CP(RC [Y ], R9 [Y ]), τ3 = CP(RR [Y ], R9 [Y ]). Their membership functions are calculated as follows: µτ1 (x) = sup{µR9 (y)|x = µL (y)}, µτ2 (x) = sup{µR9 (y)|x = µC (y)}, µτ3 (x) = sup{µR9 (y)|x = µR (y)}.

0.2 0 0.2 0.4 0.4 0.4 0 0 0 0 0 0

0.4 0 0.2 0.4 0.8 0.2 0 0 0 0 0 0

0.6 0 0.2 0.8 0.4 0.2 0 0 0 0 0 0

0.8 0 0.4 0.4 0.4 0.2 0 0 0 0 0 0

1 0 0 0 0 0 0 0 0 0 0 0

Table 13 y –1 –0.8 –0.6 –0.4 –0.2 0 0.2 0.4 0.6 0.8 1

µR9 (y) 0 0.4 0.8 0.8 0.4 0 0 0 0 0 0

The data on the basis of which the calculations are made is given in Table 14, and results of the calculations are given in Table 15. For determination of membership functions of a degree of consistency it is possible to take advantage of the following technique. For example we want to deﬁne membership function of a degree of consistency τ2 = CP(RC [Y ], R9 [Y ]) at x = 0.4. For this purpose it is necessary to determine at what values of an argument µC (y) = 0.4. In this case we have y ∈ {−0.6, 0.6}. Further, it is necessary to choose the greatest value from values µR9 (−0.6) = 0.8 and µR9 (0.6) = 0 (in this case 0.8). Consequently we have µτ2 (0.4) = 0.8. Diagrams of membership functions µτ1 , µτ2 and µτ3 are given in Fig. 8. Diagrams of membership functions approximating semantics of modalities Possible 50 and Impossible 95 are given in the same ﬁgure. VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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Table 15

Table 14 y x = µL (y) x = µC (y) x = µR (y) µR9 (y) –1 1 0 0 0 0.8 0.2 0 0.4 –0.8 0.6 0.4 0 0.8 –0.6 0.4 0.6 0 0.8 –0.4 0.2 0.8 0 0.4 –0.2 0 1 0 0 0 0 0.8 0.2 0 0.2 0 0.6 0.4 0 0.4 0 0.4 0.6 0 0.6 0 0.2 0.8 0 0.8 0 0 1 0 1

x µτ 1 (x) µτ 2 (x) µτ 3 (x) 0

0

0

1

0.2

0.4

0.4

0

0.4

0.8

0.8

0

0.6

0.8

0.8

0

0.8

0.4

0.4

0

1

0

0

0

Fig. 8. Approximation of membership functions

Thus, the result of linguistic approximation of the fuzzy relation R9 [Y ] is the formula (Possible 50 Y is At the left and (Possible 50 Y is At the center and Impossible 95 Y is At the right)) which can be interpreted as “Y is more to the left of the centre”.

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8. Conclusion. In the paper the multiagent hierarchical approach to recognition of images is stated on the basis of use of fuzzy situational calculus. The offered calculus is based on using relations between segments of images. Due to application of modalities the external language, in which the agents ontology is described, is expanded and translation rules for the modal statement are realized. The original solution to the problem of linguistic approximation of inference is also found. Further development of the present approach is supposed in the following directions. 1. Realization of the offered approach in one of the known tools designed for multiagents technology. 2. Approbation of the approach for recognition of images in various areas of applied sciences. 3. Development and substantiation of new formulas of the linguistic approximation adequate to inquiries of users. 4. Development of situational fuzzy calculus towards the insertion of sorts of parameters (situations, actions etc.). 5. Estimation of complexity of solving the recognition problems of various classes. 6. Development of procedures of adaptation and correction of ontologies. 7. Development of procedures of segmentation and preliminary processing of images.

REFERENCES 1. Ushold, M., Gruninger, M. Ontologies: Principles, Methods and Application Knowledge Engineering Review. Vol. 11, No 2, 1996. 2. Fadel, F.G., Fox, M.S., Gruninger, M. A Generic Enterprise Resource Ontology, Proceeding of the third IEEE Workshop on Enabling Technologies: Infrastructure for Collaborative Enterprises, April 1994, Morgantown, West Virginia (WET ICE ’ 94) 3. Russel, S.J., Norvig, P. Artiﬁcial Intelligence. A Modern Approach. NJ, Prentice-Hall, 1998. 4. Baldwin, J.F., Pilsworth, B. W. Axiomatic Approach to Implication for Approximate Reasoning with Fuzzy Logic, Fuzzy Sets and Systems, 1980. – Vol. 3. – PP.193–219. 5. Fu, K.S. Syntactic Pattern Recognition and applications, NJ, Prentice-Hall, 1982. 6. Gary, M.T. et al. A Fuzzy-Attributed Graph Approach to Handwritten Character Recognition, FUZZ- IEEE-93, pp. 570–575, 1993. 7. Guyon et al. UNIPEN Project of On-line Data Exchange and Recognition Benchmarks, 13th IEEE-ICPR, pp. 29–33, Israel, 1994. 8. Jorge, J.A. Fuzzy Relational Grammars for Interactive Gesture Recognition, 2nd International Conf. on Fuzzy Set Theory and Technology, Durham, NC, Oct. 13–16, 1993. 9. Keller, J.M. et al. Evidence Aggregation networks for fuzzy logic inference, IEEE T. on Neural Networks, vol.3, No. 5, pp. 761–769, Sept, 1992. 10. Lan, M.-S. et al. Character Recognition using Fuzzy Rules Extracted from Data, FUZZ-IEEE-94, pp.415–420, Oriando, June, 1994. 11. Lee, E.T., Zadeh, L.A. Note on Fuzzy Languages, Information Sciences-1, pp. 421– 434,1969.

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12. KIette, R., Zamperoni, P. Handbook of Image Processing Operators, Wiley, Chichester, 1996. 13. Malaviya et al. FOHDEL — A Fuzzy Handwriting Description Language, FUZZIEEE, June 1994. 14. Malaviya et al. Automaton Generation of Fuzzy Rule Base for Online Handwriting Recognition, EU FIT-94, Aachen, 1994. 15. Malaviya and Peters, L. Extracting Meaningful Handwriting Features with Fuzzy Aggregation Method, Conf. on Document Analysis and Recognition, Montreal Canada, 1995. 16. Malaviya and Klette, R. A Fuzzy Syntactic Method for On-line Handwriting Recognition, Lecture notes in Computer Science 1121, Springer, Advances in structural syntactical pattern recognition, SSPR 96, pp. 381–392, 1996. 17. Oram, M.W. and Perrett, D.I. Modeling Visual Recognition from Neurobiological Constraints, Neural Networks, vol. 7, No. 6, 7, pp. 945–972, 1994. 18. Shaw, A.C. A Formal Picture Description Scheme as a Basis for Picture Processing Systems, Information and Control–14, pp. 9–52, 1969. 19. Sloman On Designing a Visual System; J.Exp. Theor. A.l., No. pp.289-337,1989. 20. Yau, K.C. and Fu, K.S. A Syntactic Approach to Shape Recognition Using Attributed Grammars, IEEE-SMC-9, No. 6, pp. 334–345, 1979. 21. Tarasov, V.B. From Multiagent Systems to Intellectual Organizations (in Russian). Moscow: URSS, 2002, 348 p. 22. Zadeh, L.A. Outline of New Approach to the Analysis of complex Systems and Decision Processes // IEEE Trans. Syst. – 1973. – Vol. SMC-3. – P. 28–44. 23. Zadeh, L.A. A Theory of Approximate Reasoning // Machine Intelligence. – 1979. – Vol. 9. – P. 149–194. 24. Mizumoto, M., Fukami, S., Tanaka, K. Some Methods for Fuzzy Reasoning // Advances in Fuzzy Set. Theory and Applications / Ed. by M.M. Gupta, R.K. Ragade, R.R. – Amsterdam: North Holland, 1977. – P. 117–136. 25. Zemankova-Leech, M., Kandel, A. Fuzzy Relational Databases: A Key to Expert Systems. – Koln: Verlag TUV Rheinland, 1984. – 180 p. 26. Devyatkov, V.V., Gaidulov, V.A. Complex Ontology of Heat Supply for Analysis of Heat Supply Efﬁciency in Housing Design (in Russian). /Trudy instituta problem upravleniya (Proceeding of Institute for Control Problems). 1999, vol. 1. PP. 80–89. 27. Devyatkov, V.V. Ontologies and Design of Systems (in Russian). /Pribory i sistemy. Upravlenie, kontrol, diagnostika (devices and Systems). Vol. 1, 2000 28. Devyatkov, V.V. Ontologies and their Application (in Russian). /Programnye produkty i sistemy (Program Products and Systems). Vol. 3, 2000. 29. Devyatkov, V.V., Rumbesht, V.V. Fuzzy Modal Situational Calculus for Analysis of Complex Objects (in Russian). Vestnik MGTU. No. 3 (44), 2001, pp. 3–21. V.V. Devyatkov (b. 1939) graduated from the Leningrad Institute for Precise Mechanics and Optics in 1963. D. Sc. (Eng.), Professor, Head of “Information Systems and Telecommunications” department of the Bauman State Technical University. Academician of International Academy of Informatization. Author of over 80 publications in the ﬁeld of logical control, computer systems and complexes, technical cybernetics.

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Ivan Balepin∗ , Sergei Maltsev∗∗ , Jeff Rowe∗ and Karl Levitt∗ (∗ University of California, Davis; ∗∗ Bauman Moscow State Technical University)

USING SPECIFICATION-BASED INTRUSION DETECTION FOR AUTOMATED RESPONSE One of the most controversial issues in intrusion detection is automating responses to intrusions, which can provide an efﬁcient way to react to a progressing attack much quicker and with more precision than a human. However, it comes with several disadvantages that can lead to a waste of resources, and have so far prevented wide acceptance of automated response-enabled systems. We feel that a structured approach to the problem is needed, that will account for the above mentioned disadvantages. In this work, we brieﬂy describe what has been done in the area before. Then we start addressing the problem by coupling automated response with speciﬁcation-based, host-based intrusion detection. We describe the system map, and the map-based action cost model that give us the basis for deciding on response. We also show the process of suspending the attack, and designing the optimal response strategy, even in the presence of uncertainty. Finally, we discuss the implementation issues, our experience with the early automated response agent prototype, the Automated Response Broker (ARB), and suggest topics for further research.

1. Introduction. Automated response to intrusions is an exciting area of research in intrusion detection. A chance to develop a system that would resist attacks carried out or programmed by another human being can be approached in many ways, including the one in which we teach the machine to beat an attacker in the game of intrusion and response. Let us begin by formulating the objectives of our work. Objectives. With growing speed intensity of computer attacks [14] grows the need for quick and well-planned responses. Currently, some of the most intense intrusions are automated. A reliable automated response system, with the right approach, could certainly provide an efﬁcient protection, or a degree of tolerance for all kinds of attacks. However, automated response remains mostly an area of research due to the following issues: • primitive response systems that do not take the cost of intrusion and response into account apply response actions that cause more harm than the intrusion itself; • a large part of commercially available Intrusion Detection Systems (IDS) produces an extensive number of false positive alerts, causing numerous costly response actions [13]. Both cases lead to a denial of service to legitimate users of the system. VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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The objective of this work is to develop a consistent, organized, cost-based approach to automated response that would address these issues. An optimal response would stop the progressing intrusion at early stages, and clean up after it as much as feasible. The scheme that we describe in this work is geared to produce such response. We start addressing the problem by considering the host-based automated response. The key parts of our approach are the basis for response decisions — the system map, the cost model, and the process — response selection even in presence of uncertainty. Let us brieﬂy summarize the work previously done in the area. Related Work. Primitive automated response actions are implemented in some Intrusion Detection Systems (IDS) commercially available today. [13] However, these actions are rather simple and reﬂexive by their nature. Even with a limited response arsenal, many practitioners report that they disable the systems’ intrusion prevention capabilities due to a high number of false positives from IDS’s which give a false basis for response, and also denial of service caused by not very sophisticated response decision strategies. An interesting research work on Survivable Autonomic Response Architecture (SARA) [2] uses the term autonomic response by drawing an analogy with the autonomic nervous system, which automatically controls certain functions of an organism without any conscious input. The authors also propose to have two separate “loops” of response, a local autonomic response, and a global response carried out by the hosts in a system in co-operation. The primary focus of the work is a network with multiple hosts. Alphatech’s Light Autonomic Defense System (αLADS) attempts to utilize control theory when selecting a response [3]. The authors describe it as a part Autonomic Computing, which, according to them, is an emerging area of study of design and construction of self-managed computing systems with a minimum of human interference. Alphatech’s work was most likely oriented towards known systems and system tasks, and it is not applicable to general-purpose computer systems. The focus of this work was most likely on developing a full-scale solution that would have its own proﬁle-based intrusion detection components and it is intended to defend a very speciﬁc range of systems. The issue of compatibility with existing intrusion detection systems has not received much attention in published descriptions of αLADS. However, Alphatech’s work is of interest to further automated response research, since it is one of the few early organized approaches to the problem of quick automated responses. Also, another interesting work on network-oriented automated response that relies on the Control Theory is currently done at UC Davis [4]. Toth, et.al., [5] propose yet another interesting model for automating intrusion response. They suggest approaching the problem of response to network intrusions by constructing dependency trees that model conﬁguration of the network, and they suggest an outline of a cost model for estimating the effect of a response.

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Other signiﬁcant response works include a thorough consideration of some intrusion detection and response cost modeling aspects by Lee, et.al. [6], a response taxonomy by C. Carver and U. Pooch [7], and Fred Cohen’s work on deception [8], which is another very interesting perspective on countering malicious activity. The analysis of related work leads us to conclusion that the primary area of interest, and, consequently, the primary system to model, so far has been a computer network that consists of multiple hosts. The idea of responding at a level of a single host has received relatively little attention. Also, we note that despite the efforts to produce a working cost model for a set of protected resources, no welldeveloped and well-tested model currently exists, that guarantees a consistent and fair representation of protected resources, and their true value. This Work. This paper has the following remaining sections: Section 2 in which we describe the basis for constructing a response chain, Section 3, in which we discuss an implementation of our model, Section 4, which lists possible directions for future work, and, ﬁnally, we sum our work up in Section 5. The reason why we decided to separate the basis for response decisions from implementation notes on our prototype is to attempt to describe a model for host-based response in Section 2 that would not be tied to any particular operating system, and potentially could be used even for applications other than host-based response. 2. Basis for Automated Response. Several pieces of information are necessary in order to plan a sequence of response actions. For the system we are protecting, we need a clear representation of the most valuable resources and also the underlying resources that provide the basic functionality. The true value of some resources (for example, the TCP/IP network service) is heavily inﬂuenced by other resources that depend on them (network is needed by httpd, etc.), and we need a clear way to reﬂect these dependencies before we can decide how to deal with a compromised entity. We also need an organized way to store information about malicious and compromised entities, and to decide how they relate to our key resources. Part of this representation will be highly dynamic, since some entities reﬂected (processes, etc.) are dynamic; however, a large part of it, such as ﬁle structure, program conﬁguration (dependence on ﬁles, sockets, etc.), and system conﬁguration, can be determined statically. We narrow the scope of the problem by noting that transferring an entire computer system to a safe state is a challenging task, and limiting the scope of the problem to returning a set of critical system resources to a reasonably safe and working state. Resources we will model are anything of value in our system — system subjects and objects, ﬁles and running processes, sockets, ﬁle systems, etc. We arrange them in two different ways — the resource type hierarchy and the system map. Resource Type Hierarchy. We found it convenient to group resources by their type, since every such group most likely will have common response actions associated with it. Also, resource types can be arranged in a hierarchy similar to the one in Fig. 1.

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Fig. 1. An example of a response type hierarchy

On a Linux system, for example, we can subdivide resources into ﬁles, sockets, processes, etc., in advance. However, in certain situations, a more speciﬁc category would be appropriate. Consider a ﬁle that contains keys for automatic encryption/decryption of emails using the GNU Privacy Guard software (GPG, [9]). In addition to response action that applies to a more general node in the hierarchy (conﬁguration ﬁles, “restore the conﬁguration ﬁle from backup and restart the corresponding service”), we deﬁne a response action speciﬁc to this sub-category (“also revoke and re-issue the keys”). Since these more speciﬁc categories, and their corresponding response actions, depend highly on the system conﬁguration and nature of the software installed on the system, we cannot deﬁne all of them in advance. We should provide a way for the users of the response system to deﬁne new custom categories with response actions tailored to speciﬁc resources on the target systems. System Map. Although the response type hierarchy is useful for storing general response actions that might apply to a resource, it carries no information about the speciﬁc instances of resources on a system, merely their types. Therefore, we need an additional data structure to satisfy our requirements for a decision basis mentioned above. We suggest representing the necessary information as a directed graph, which we will refer to as the system map. The vertices of the graph, which we will refer to as map nodes, represent the resources in our system. Besides nodes, our map has node templates and edges. Map Nodes. The system map contains important nodes of all types — the system’s priorities. By “important nodes” we mean “all nodes with a non-zero cost”, with cost assigned according to our cost model described in the corresponding subsection. In addition to the priorities, our map also reﬂects the underlying basic resources that these priorities need for proper operation. For example, most applications need a working mounted ﬁle system with read/write access right in order to operate properly. Therefore, if we have applications A, B, and C that are our priorities, we place them on the map along with the node that represents the ﬁle system. We also note that the ﬁle system, an underlying basic resource, does not need to be explicitly speciﬁed as a priority itself, since in this simple example it

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does not have any value of its own. It costs only as much as the priority nodes that depend on it. Each node holds information about the resource it represents. Namely, we need to know the type of the resource according to our type hierarchy. We also need to know some type-speciﬁc information such as path, ﬁlename and inode for type and sub-types of “ﬁle”; PID, name and owner for type and sub-types of “process”, etc. Also, a node also has a cost value associated with it. Some static nodes might have several node templates associated with them in order to later construct dynamic dependent nodes. Finally, every node has a list of applicable response actions associated with it. Node Responses. Every node has a list of basic response actions that restore its functionality. Currently, we require that this list contain only the actions that completely restore the node to a working state. The node’s list of responses is constructed from response actions that are listed for this type of node and its parent types in the type hierarchy. Each such response action has an activation condition associated with it. Referring to the example we have used before, type “conﬁguration ﬁle” would inherit a response “restore from backup” from the parent type “ﬁle”. The activation condition would be, “the node of this type was a target of an illegal write system call” or “the node of this type was a target of an illegal unlink system call”. Another important property of a simple response action is what nodes it affects. Currently, we assume that an action either damages several resources, or does not. If the technique we rely upon for intrusion detection is oriented towards system calls, activation conditions for each response action will be also oriented towards system calls. Since the number of system calls is ﬁnite, and the number of node types is ﬁnite, it seems very feasible to pre-deﬁne response lists for every combination. We will discuss details of creating a response list in the implementation section. We also complement the node’s response list with a response “take no action”. That is an essential response alternative that has a certain cost, just like other responses, and by including it, we will ensure that any response action we take is not more expensive than the intrusion itself. Therefore, an entry in a node’s response list has three ﬁelds: — the action itself (a Linux command, etc.); — the activation criteria; — the list of nodes the action damages. Map Edges. Edges on our map represent dependencies between the resources. If an edge is directed from node A to node B, it means that A provides some service to B, B depends on A, and, most likely, A produces information that B consumes. However, it does not seem feasible to attempt to trace information ﬂow through our map, since it contains nodes that are often times not comparable (for example, ﬁle systems and sockets), and also nodes that obscure information ﬂow (if node A reads from node “ﬁle system”, node B writes to node “ﬁle system”, there is not necessarily an information ﬂow from B to A). Therefore, we do not use our map VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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for intrusion detection. For all information about the intrusion we rely on some detection technique. The true value of the map edges is that they allow us to properly carry out single response actions that involve several nodes (“restart the service that corresponds to this conﬁguration ﬁle”. What service? The one that consumes information from the ﬁle). Also, the edges allow us to collect information about the nodes that depend on a certain node, therefore allowing us to calculate the dynamic cost of the node in our system. Of course, relationships between the nodes can be speciﬁed with greater detail, such as “node A writes to node B that often”, or “node A writes to node B with probability N”. But for our purposes, it is sufﬁcient to only reﬂect the fact that one node provides services to another node, and therefore, the latter depends on the former. Also, some authors model dependency alternatives (node A depends on node B or node C) [5]. From a standpoint of resources of a single host, this is a relatively rare situation, so we will not consider it here. Constructing the Map. As we have mentioned before, the map will have a static part, which will consist of nodes that can be produced by static analysis of our priority resources when no processes are running. The static part of the map will have information about objects of the system, but not subjects. We begin operating just with the static part of the map. Therefore, as the system runs, we will need to add dynamic nodes to the map. In our design, there are ﬁve ways we can add a node to the map. Static nodes are added to the map upon upgrades/reconﬁgurations of the protected system. For the dynamic part, we propose to add new nodes for every subject or object mentioned in the incident alert from the IDS that was not previously on the map. Such nodes would be assigned cost 0, since they were not included in the list of priorities, and they get assigned the most speciﬁc type from the type hierarchy that we are able to determine automatically. Consequently, the node will have a response list that corresponds to its assigned type. Also, as we will describe in later sections, sometimes we will be able to classify a whole group of subjects as malicious, whereas only a few of them might have been explicitly mentioned in alerts. Such situation can occur, for example, when a malicious process caused an alert, and immediately produced a number of children processes that have not yet done anything illegal themselves. We will put the whole related group of subjects on the map, despite the fact that only a few were mentioned in the alert, and they will be treated just like the nodes mentioned in the alerts. We mark the nodes that were mentioned in the alerts, or the ones with their relatives mentioned in the alert, as suspicious, or “contaminated”. Finally, we will have some dynamic nodes that will represent our priority resources. Often times, a resource in general can be mapped to several nodes on our map. For example, a “web server” resource encompasses the executable ﬁle, a number of running processes, and dependent resources (conﬁguration ﬁles, sockets, etc.). At the time of static analysis, we will not have a running instance of a

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web server; however, we can get most of information about the web server process node at that time. By analyzing the static executable ﬁle, we can predict the cost of a running instance of a web server, build the response list, determine the type of the node that will correspond to this process, and also get information about the resources web server depends on. Therefore, at the time of static analysis, with every important executable we create a set of templates, that will characterize the subjects and objects that will later be produced by running the executable ﬁle. A node template is a prototype for building nodes that has all information in place except for the type-speciﬁc information (like PID or ﬁlename) that get ﬁlled upon use of the template. Properties and Beneﬁts of the Map. We are not proposing to build a complete kernel map, or a map of an entire system. Our map has only a few static and dynamic nodes that are critical to the system’s operation. They are not updated periodically; rather, we update them only when signiﬁcant events happen (alerts for dynamic nodes and system re-conﬁguration for static ones). Therefore, if our system runs for a long time without getting attacked, the map will not be updated in order to minimize the overhead. The nodes on the map can be of very different nature, so not always they can be compared directly (for example, ﬁle systems and processes). Let us illustrate some properties of the map with a small example. Suppose, we have a Linux system equipped with System Health and Intrusion Monitoring IDS (SHIM, [11]) that has been compromised, and now has an active malicious process A that was produced by a program B that is not supposed to make system calls from the exec family. Process A has its parent’s speciﬁcations imposed on it by SHIM. Suppose then that process A produces process C, and process C writes to a ﬁle. Since SHIM would promptly alert our response system about A, B and C being involved in several illegal exec system calls, the whole family would appear on the map, and would be marked as malicious. As far as the ﬁle that C has written to, if speciﬁcations for A allowed such behavior, then we would not get an alert about the ﬁle write, and, therefore, would not reﬂect that fact on the map. However, if it was not legal according to A’s speciﬁcations and the system policy, we would get an alert about a possibly contaminated ﬁle, place the corresponding node on the map, and plan our response strategy with that alert in mind. As we have shown above, our map contains all necessary information about our priorities, and resources they need to operate. The map also will reﬂect information about malicious entities, and their relation to our priorities. The map, as we have described it, gives us a solid basis for designing an intrusion response strategy. Cost Model. When deciding on response to a malicious action, most of the time, we will have several alternatives. As we mentioned, one response alternative is always present in any system — take no action. Most of the time, we will ﬁnd some more response actions whose activation criteria matches the current situation. We solve the problem of comparing these alternatives and selecting the optimal one VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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by introducing an action cost model. The cost model helps us pick the best response and also ensure that we don’t cause denial of service to ourselves by performing responses that are more harmful (i.e. more costly to us) than the intrusion itself. Our cost model is based on numerical cost values associated with every map node. Designing a cost model that allows us to quickly associate a number with a resource that precisely reﬂects the value of that resource, is a difﬁcult task. Most of the attempts to produce such a model left it up to the system administrators to determine cost values for their resources. Although it is true that only the system’s owner, familiar with its conﬁguration and primary functions, can point out the true value of the resources, it is very hard to assign the cost values in a consistent manner that would always guarantee optimal response, without exhaustive testing of the system. In our implementation, we rely on ordering the resources by their importance to help producing a cost conﬁguration that would yield optimal response. There are only a few priority nodes that have an actual cost values in our model. For example, let us consider a system with only one such priority — the web server. In the static part of the map, it is represented with the executable ﬁle of the web server. There will be a static node for the ﬁle itself, and it will have a cost of 0. The static node for the executable ﬁle will, however, have a template for web server processes to be created, and that template will have a cost value associated with it. In our model, all process nodes that get created according to that template, will share an equal fraction of the template’s cost with existing processes. For example, suppose the system’s owner has estimated that the web server has a cost x. When there is no web server running, according to our model, the executable ﬁle will have no cost value. If one instance of httpd gets started, its node will get assigned a cost value x. If y nodes of httpd get created, each will get a cost value of x/y. A static node can also get an explicit cost value assigned to its, and not to its templates; or it might not even have any templates. For example, on a certain system, some ﬁles might be indicated as a priority, even though they are not used by any subjects of that system. Cost-wise, another category of nodes on our map is the underlying service nodes. Most likely, these nodes will have a zero cost. However, any harmful action to these nodes will also affect the costly resources that rely on them, and by reﬂecting these dependencies on the map and accounting for them in our response actions, we will take into account the true value of the services. For example, let us consider a system with only two priorities — a web server and an ftp server. These resources depend heavily on the network service to operate. However, the network service does not have any value of its own; its true value is providing part of functionality of our priority resources; and without any dependents, the network service would be useless. Finally, we have all the resources that were not put on our map as a priority resource or its dependency. We assign all such resources cost 0; if they become

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malicious or get involved in an incident, they do get on the map, and a response action that affects these 0-cost nodes even in the most dramatic way will not be harmful for the whole system. Once we determine the cost values for our map nodes based on these factors, we then can associate a cost value with any action that an intruder or the response system takes. We deﬁne the cost of an intrusion action as the sum of costs of map nodes, previously in safe state, that get negatively affected by the action. We deﬁne the beneﬁt of a response action as the sum of costs of nodes, previously in the set of affected nodes, that this response action restores to a working state. Finally, we deﬁne the cost of a response action in terms of costs of the nodes that get negatively affected by the response action (“lost to the intruder”). The goal of a response system is to carry out the response sequence that yields the maximum beneﬁt at the minimum cost. We note that such an approach does not emphasize transferring the system to the ultimate safe state, or completely recovering from an intrusion, since there are situations in which these goals would be much more costly than the intrusion itself. With our approach, we are, however, guaranteed to come up with response strategy that is optimal for the current situation. Response Selection. Once we have the whole picture of the intrusion, our goal is then to “win” the resources on the “contaminated” side back. We start by listing all response alternatives at every contaminated node whose activation condition matches the intrusion. The goal of response selection is to build a response action sequence that will have one action out of a list of every contaminated node. That way, we ensure that every contaminated node is addressed. As mentioned before, an optimal response action is the one that that yields the maximum beneﬁt at the minimum cost. We then assume that a response sequence (response strategy) is optimal if it consists of response actions that are optimal for every node. Therefore, if we have the complete picture of the intrusion, we build the response chain from optimal responses at every node, and carry it out. Managing Uncertainty. Sometimes we might encounter situations where we do not know for certain what exactly the intruder has done. For example, suppose the capabilities allowed the intruder to perform a write call on a ﬁle, which is illegal according to the current system policy. The ﬁle could have been overwritten, appended to, or erased completely (overwritten with an empty string). In certain situations, response actions, and their cost, may vary depending on what has really happened. Then we turn to decision theory, which provides well-deﬁned ways to construct the response plan, for different requirements in presence of uncertainty. The possible results of a write call would be over-written data in the ﬁle, data appended to the ﬁle, or data completely erased from the ﬁle (the latter being a special case of the ﬁrst one). This allows us to list the possible system states. Every one of these states will have a potential damage value and a probability associated with it. Now, using the decision theory convention [10], we can describe VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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the situation with the following “gain matrix”:

A1 ... AN Q

Π1 a11

Π2 a12

aN 1 q1

aN 2 q2

Π3 a13 ... aN 3 q3

Π4 A14

Π5 a15

aN 4 q4

aN 5 q5

where Πi are the possible states, qi are the probabilities, and Ai ’s represent the response alternatives. aij in this matrix, again, represents the usefulness, or beneﬁt, of using the ith decision in case of a jth sub-state. This value can be estimated as: aij = (−εij )γ+1 Bj − ci .

(1)

where Bj is the potential damage of a sub-state; ci — response cost; εij — efﬁciency (beneﬁt) of response i in sub-state j, and γ is 0 if εij = 0, or 1 — otherwise. Considering the above parameters, we observe that the greater the value of ai j, the more useful the corresponding response alternative will be in the corresponding state. We deﬁne the risk of losing in a particular game situation (rij ) as the difference between the player’s gain for strategy Ai for conditions of Πj , and the player’s gain for the strategy he would have chosen, had he known the conditions of Πj . It is clear that had the player known the system state and its conditions in advance, he would have chosen the strategy that yields the maximum gain in its matrix column (mj ). According to our deﬁnition, rij = mj − aij ,

where

mj = max aij .

(2)

i

One important property of risk that follows is that it is always greater than zero. Deﬁned in this way, the concept of risk also reﬂects how favorable a given nature state is to us. Consequently, a risk matrix constructed similarly to the gain matrix, gives us a more complete picture than the gain matrix. Relying on probability signiﬁcantly simpliﬁes the decision making process, especially if we can produce relatively accurate probability estimates using the system history, general knowledge, anomaly analysis tools, etc. A promising way to eliminate the uncertainty, or at least, estimate the values of probability of a certain intrusion sequence, is monitoring the system for a long period of time, and building a proﬁle for important resources. For that, machine learning techniques can be used; also, much can be drawn from the anomaly- based and misuse-based intrusion detection techniques [1]. We will discuss our suggestions in more details in the Section 4. Let us take mathematical expectation of probability-based gain ai to be the effectiveness criterion W that we obviously would like to maximize ai = q1 ai1 + q2 ai2 + . . . + qn ain . 162

(3)

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The optimal strategy is the one that yields the maximum ai in the gain matrix. It would also yield a minimum average risk based on the risk matrix. As we mentioned above, special care must be taken to accurately estimate probability. Pure probability, as a statistics-based value, might not always be available. In that case, it can be subjectively estimated. Certain events might be more likely than others according to the system logs. There are several techniques available that help us quantify these subjective estimates. For cases in which we have no statistical information for the system states, we can assign equal probabilities to each possible state, i.e.: q1 = q2 = . . . = qn = 1/n.

(4)

This approach is called Laplace insufﬁcient reason criterion [10]. For another approach, we assume that we can order possible system states by their likeliness. In order to represent the probabilities in this case, we can use a converging arithmetic series: q1 : q2 : . . . : qn = n : (n − 1) : . . . : 1.

(5)

where: qi =

2(n − i + 1) . n(n + 1)

(6)

We can also rely on expert estimates. If we manage to completely eliminate uncertainty in some situations, the probability values for the determined system state becomes 1, probabilities of all other states become 0, the matrix turns into a single column, and decision making becomes trivial. The Optimal Decision Criteria. There are several methods for selecting the decision criteria in the decision theory [10]. In the Minimax risk criterion (Savage criterion) we select the strategy from the risk matrix that provides us with the minimal risk value under the most unfavorable conditions. The efﬁciency W is then estimated as W = min max rij . The Minimax risk approach allows us to i

j

avoid making the high-risk decisions. The Maximin criterion (Waldt criterion) favors strategies with the largest minimal gain (with W deﬁned differently, see [10]). The Hurwitz criterion is neither pessimistic nor optimistic. Risk-based criterion is analogous to Hurwitz. As we mentioned before, selection of criterion and its parameters is subjective. In any case, however, it is useful to analyze the situation using various approaches. If several criterions indicate that a certain strategy is optimal, it should certainly be selected. If several different criteria suggest different strategies, in the game theory, it is up to the system owner to select (or pre-select) the right strategy based on the fact that some criterion might be preferred over others. 3. Implementation. We have implemented several concepts mentioned in the previous section in a prototype response system, the Automated Response Broker VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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(ARB). ARB is developed for Linux, and it relies on a speciﬁcation-based intrusion detection system, namely, SHIM, for detection. Let us brieﬂy mention why we chose SHIM for that role. Intrusion Detection: SHIM. SHIM is speciﬁcation-based. It relies on the Generic Software Wrapper Toolkit (GSWTK,[12]) for all information about the system calls. SHIM does not try to recognize an attack as a whole. Instead, it relies on a set of speciﬁcations (for programs, or protocols, etc.) that reﬂect the system policy. An interesting feature of SHIM implementation is that it addresses a large part of intrusions by enforcing speciﬁcations for privileged Linux programs. For the majority of current speciﬁcations, system calls of interest are reported by the GSWTK, and then classiﬁed as legal or illegal according to the speciﬁcations, with an alert being issued for the latter. SHIM is a great vehicle for testing our automated response scheme. Such a ﬁne event granularity allows us to catch the exact system call that started the intrusion. Also, the fact that SHIM does not need the whole intrusion to recognize its signature, allows it to catch unknown intrusions, and intrusions that are still in progress. The last feature also gives us a chance to stop an intrusion in progress by responding to the ﬁrst few steps of it that have been detected. The underlying assumption about SHIM that we make is that it always promptly detects and reports all intrusions. Also, SHIM and GSWTK give us a capability to check if a system call is legal according to the system policy before it is executed. However, such mode of operation causes a large overhead for every system call, and does not seem feasible. Map Implementation. We build the map starting with a set of nodes we want to protect. It is the set of all programs that are constrained by SHIM (regardless of whether they are among our priorities; the cost will reﬂect that fact), and several nodes for resources that might not be constrained by SHIM, but the system owner wants to protect as well. The type hierarchy is constructed once upon installation of a system. It does not have a dynamic part and does not change, since it simply contains information about the types of nodes, not the nodes themselves. In ARB, the type hierarchy is constructed in the source code in C++. While it might be sufﬁcient for experiments and testing, obviously, a more convenient interface for conﬁguring the type hierarchy is needed. Currently, we experiment with XML for type hierarchy deﬁnitions. XML so far has proved to be powerful enough to express all the information necessary, and there is an abundance of tools for parsing the type hierarchy deﬁned in XML into our program. Below is an example of a response list of a conﬁguration ﬁle node. Event name and target constitute the activation condition for the action. The victim tag marks the damaged nodes.

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restore attributes; kill offender; offender delete self; kill offender; self

dependents offender restore from backup; ﬁre event(”httpd”,”restart”,true); ... Should XML fail to be descriptive enough for the task, we will design a new domain-speciﬁc language (DSL) for describing the type hierarchy. Similarly to the type hierarchy, the map itself is constructed manually as a collection of C++ data structures. We are also currently experimenting with more ﬂexible ways to deﬁne a map, such as, again, XML or a new DSL. As mentioned above, all components of a system in our prototype are determined manually. However, some of them can be pre-deﬁned for most systems; some can be determined by automated analysis upon installation or reconﬁguration. Eventually, our goal is to let the user of ARB specify just the custom types, responses to custom types, and the system’s priorities. The remainder of the map, such as the basic types, underlying service nodes, all dependencies and templates can be determined automatically. We list the requirements for automating the map construction, and some problems with it, in the future work section. In ARB, the map gets updated only when malicious activity happens, and node’s cost is calculated only when the actual response decision is made, otherwise, much computing resources can be wasted by keeping the map and the node costs up-to-date unnecessarily. Node’s response lists are constructed from the type hierarchy. The set of response actions that are implemented, or will be implemented in the prototype include: delete a ﬁle, restore a ﬁle from backup, restart a service, change permissions, kill process(es), reboot the system, block a connection, re-conﬁgure a ﬁrewall rule, unmount a ﬁle system, change the owner of the process(es), start checkpointing,

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slow down the process(es), roll back to a checkpoint, return a random result, perform a random action, operate on a fake ﬁle, tunnel the process(es) to a sandbox, operate on a fake socket. Node Costs. The most difﬁcult task of any implementation of a response system would be to perform a consistent cost assignment that would reﬂect the true value of resources. This part of map construction cannot be completed in advance, or even automated, since it needs input from the owner of the system. Currently, we approach the problem by ﬁrst manually ordering the key resources of the system, so that the resources (Ri ) are listed in the form, similar to the one below: R0 < R1 = R2 = R . . . < Ri−1 < Ri .

(7)

Then, the least important resource gets assigned priority 1, and the priority of all other (more important) resources is approximated as N times the priority of the next less important one: P riority(Rj ) = N ∗ P riority(Rj−1 ),

(8)

where N is an approximate value and is determined experimentally. Finally, for convenience, we obtain a cost value Ci for a resource Rj from priority values according to the following formula: X Ci = 100∗ P riority(Ri )/ P riority(Ri ), (9) X where P riority(Ri ) is a sum of priorities of all resources. Currently, the process of assignment is completely manual, and all information is reﬂected in the source code of ARB, in C++. However, we plan to add a script that would guide a user of ARB through the cost assignment process. As a result, the script would update the map description, in the language of choice, with the cost values. Also, the map customization interface naturally will provide ways to directly specify the node cost bypassing the script. The cost assignment method described above is certainly only an approximation of the real resource costs. We are working on improving the method to ensure consistency of the cost assignment. Damage Assessment and Response Selection. Let us say ARB received an alert about some malicious actions involving several nodes on our map. Currently, ARB reads alerts from a socket that SHIM writes to. Later, a closer form of integration with SHIM will be developed, since the current implementation scheme might be sufﬁcient for evaluation of response, but it is rather vulnerable to attacks. First, we need to stop the intrusion, if it is still in progress. We partition the map into two sets: a set of nodes that are affected (or might have been affected) by the incident, and a set of nodes that are not dependent on any in the ﬁrst set, and therefore, not affected by the incident at all. After such a partition is done, we then will know what resources are possibly under the attacker’s control. In order to stop the intrusion in progress, we propose to temporarily suspend the subjects on the map involved in the incident, and also all subjects reported in the alert as 166

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suspicious, their children and parents (excluding, of course, PID 1). We freeze the intrusion in progress by going through the hierarchy of suspicious processes and suspending them one by one (by sending them a kill-19 message). We also note that some of the nodes that we put on the map as contaminated were not necessarily explicitly speciﬁed in the alert. Upon an alert that, say, mentions only one subject, the damage assessment procedure of ARB puts the all children this subject may have on the map independently of further alerts. That is done in order to freeze the intrusion in progress before the children attempt to perform further intrusive action, since having a suspicious process as a parent already gives us a right to mark a child process as suspicious as well, without waiting for further alerts. ARB operates with a concept of an incident. Alerts are grouped to form a single incident if they report subjects from the same family as suspicious. ARB considers the damage assessment procedure as complete when it has constructed and has frozen the entire family of suspicious processes. All new alerts are treated as a new incident. The testing of ARB that we have done so far indicates that such approach allows us to clearly separate individual incidents, freeze an incident, assess the damage, and carry out response actions. Upon completion of the damage assessment procedure, we have the suspended intrusion, the frozen suspicious processes, and the complete picture of an intrusion in form of the partitioned map. Finally, the response strategy is built and carried out, as described in the previous sections. Example. Let us demonstrate how ARB carries out the entire process of response selection with an actual example. We will consider a classic vulnerability in the RedHat Linux 6.2 dump utility [15], which examines the ﬁles on a ﬁle system, and determines the ﬁles that need to be backed up. These ﬁles are copied to a disk, tape, or other storage medium. The dump utility depends on the environment variables TAPE and RSH. The goal of the dump exploit is to set the RSH environment variable to an executable ﬁle that will be executed with suid root privileges. In the particular case we will consider, the executable ﬁle will copy ﬁle /bin/bash to /tmp/rootshell, creating a root shell in the /tmp directory. This program will then execute the root shell. The complete speciﬁcations for the dump utility can be found with any distribution of SHIM source code. According to them, dump is allowed to make a few system calls; namely, open and read certain ﬁles, fork, and connect. Consequently, when this intrusion happens on a system that runs SHIM, but is not protected with ARB, the system administrator will get several alerts. Namely, there will be an alert about dump copying the shell executable to the /tmp directory. Another key alert is issued when dump executes ﬁle /tmp/rootshell. Finally, the last alert will be issued when the attacker uses the obtained root shell to issue the open and write system calls to the target. The target in our example will be the ﬁle secring.gpg, which contains the keys the GPG software uses for encryption/decryption.

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Let us ﬁrst show the few relevant parts of the map before the intrusion begins (Fig. 2).

Fig. 2. The map of a part of a computer system before an intrusion. Only a few essential nodes are shown

The map was built according to the type hierarchy in Fig. 1. According to the map, our only priority in the entire system is the gpg program that encrypts and decrypts email messages. However, we also put dump node on the map, since it has SHIM speciﬁcations. Experiments with the ARB prototype showed that it takes a variable period of time for SHIM to issue an alert, and for ARB to receive it and process it. For certain test cases with favorable conditions, that period was not too long, and ARB was actually able to freeze the entire attack right after the ﬁrst alert. For test cases under the least favorable conditions, however, ARB completed the damage assessment procedure only after the attacker already had access to the root shell. Regardless of the current conditions, our goal is to stop the intrusion and clean up after the actions that already happened. Let us re-draw the system map for the worst case that has been observed (Fig. 3). According to the new map (Fig. 3), there are four contaminated nodes as the result of the intrusion. A node for /tmp/rootshell appeared on the map because the ﬁle was involved in an illegal ﬁle copy by cp. However, the cp process itself is gone by the time we ﬁnish damage assessment, so it is not reﬂected on the map. ARB starts building the response sequence addressing node by node, in arbitrary order. The dump process node is an issuer of an illegal exec system call, so ARB chooses the most efﬁcient response — killing the process — since the value of nodes affected by the response is 0. The /tmp/rootshell node was created as a 168

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result of an illegal create system call, and it does not have any cost or dependencies. The matching response would be to remove the ﬁle. Finally, the response for the secring.gpg ﬁle is selected as follows. Several response alternatives apply to the ﬁle, including deleting it, and restoring it from the backup. Deleting the ﬁle would certainly damage it. By using the map, we detect that the gpg process depends on the ﬁle; therefore, deleting the key ﬁle would damage the ﬁle and the process, and the cost of such response would equal to the sum of affected nodes — namely, 100 points. Another alternative with an activation criterion that matches a write system call is restoring the ﬁle from backup, with a cost of 0. We select the second alternative as the least expensive one. Another matching response action is “restart the corresponding service(s)”, and it was inherited by the custom type “key conﬁguration ﬁle” from general type “conﬁguration ﬁle”. By using the map, we determine the corresponding service to be the gpg in this case, and we restart it with a restored key ﬁle. We also note that in this case we do not re-issue the keys, since SHIM has not indicated that the content of the ﬁle has been read. Therefore, the response alternative “re-issue all keys” does not apply, and we do not re-issue the keys. Experience with ARB. The ARB prototype was tested for several well-known attack scripts. Work is in progress to extend it and test it with the broadest range of other intrusions. ARB can only be run on Linux kernel 2.2.14, since the current version of GSWTK relies on that kernel version, and the current version of SHIM relies heavily on GSWTK. As we mentioned before, the map in ARB is built manually for only a subset of all resources that really should be on the map. The current version of ARB does not handle uncertainty in intrusions. It does, however, successfully freeze the set of test attacks, stop them, and respond to them. The attacks we handle include the two examples from this paper. The prototype so far has forced us to re-design our original approach to automated response greatly, and posed several new problems, which were not obvious

Fig. 3. The map of a part of a computer system after ARB has stopped an attack

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before. For example, one such problem was, as we mentioned before, the fact that we did not deﬁne when the damage assessment procedure is complete, and we can actually start deciding and carrying out response. In order to resolve the issue, the concept of an incident was introduced in the prototype. We currently continue to work on the prototype, and we expect promising results from the future work with ARB. 4. Future Work. Automating the Map Construction. First, since with SHIM all malicious actions that involve a map node can be expressed as Linux system calls, and the number of Linux system calls is relatively small, we can automate the generation of nodes’ response lists. For a new type of a node, we list all applicable system calls that this node can make as an activation criteria. Then, we either borrow the corresponding response actions from the type above in the hierarchy, or ask the system user to deﬁne a response action and the damaged nodes. Then we construct a list of applicable system calls that target this type of a node as activation criteria, and obtain the corresponding response and damaged node information in the similar manner. Therefore, we can simplify the task of constructing response lists by guiding the user through the process and producing the output in some convenient format like XML. Also, construction of the map itself and analysis of node dependencies can be done in part automatically. When constructing a map, we can rely, for example, on the program installation package (Linux RedHat Package Manager information, for instance), the program’s source code if available, documentation (man pages), etc., for dependency information about opened ﬁles, sockets, pipes, inter-process communication, etc. Designing a tool that would assist a system’s user with map construction presents an interesting implementation task. Other Directions. As we mentioned before, introducing nodes of type CPU, or memory, or user may allow us to model and respond to denial-of-service attacks. We did not consider the topic in this work, but it seems promising; especially when the intrusion detection technology will provide us with ways to clearly identify denial-of-service intrusions. Storing information about past intrusions and incorporating that knowledge in response also is promising. For example, a large number of attacks in a small period of time might cause the system to take extra response measures targeted at preventing future intrusions rather than responding to ones already in effect. Also, we might design a set of stricter speciﬁcations for the privileged programs that would reﬂect a stricter system policy in response to a large number of intrusions. Another option would be to implement a “pre-emptive” mode as a wrapper in GSWTK: all system calls would be checked in advance, and not carried out if illegal. This mode of operation would cause a large overhead for every single system call; however, it might be useful when trying to counter particularly severe types of intrusions. We would, again, have to carefully weight the cost of switching to such mode. Also, currently, our response model does not consider actions that partially restore a node, and it assumes that an action either damages resources, or does not.

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Considering actions that only partially restore resources and introducing a degree of damage also deserve consideration for further work. Another interesting research direction would be to attempt to combine our hostbased approach to response and network-oriented response mentioned in Section 1, to design a network-wide response system that possibly might be based on single host components, such as ARB, cooperating with each other to protect the entire network. Finally, in our opinion, the most exciting future work option is combining a speciﬁcation-based IDS, features of anomaly- and misuse-based IDS’s and the requires/provides model of intrusions [12] to form basis for response decisions. With SHIM being a “low-level”, system-call oriented IDS that ignores the intrusion as a whole, and focuses on individual constraint violations instead, it is able to catch violations that have never been seen before, and cannot be detected with signaturebased detection systems; whereas signature-based systems can see farther ahead than SHIM, since they have a signature of the entire intrusion. In a situation where we receive several SHIM alerts (which represent the ﬁrst few steps of an intrusion), we can use our system map to calculate the capabilities of the attacker, describing them in JIGSAW [12], and also browse the signature database for all signatures that match the current intrusion at least partially. By using some historical data from an anomaly-based system we can determine probability of each intrusion path (signature), and initiate a game with the intruder. By winning such a game, we will be able to prevent complex intrusions instead of responding to the ones that are already in full progress. 5. Conclusion. In this paper, we gave brief overview of related works, stated the problems associated with automated response, and began addressing them. First, we outlined the basis for our response system, and then described the current implementation, the ARB. The central parts of our approach are the system map, the action cost model, and the game approach to response, which also open up plenty of opportunity for further research.

REFERENCES 1. Amoroso, E: Intrusion Detection: introduction to Internet surveillance, correlation, trace back, traps, and response, Intrusion.net Books, New Jersey. (1999). 2. S. Lewandowski, D. Hook, O. O’Leary, J. Haines, L. Rosse, SARA: Survivable Autonomic Response Architecnire, DISCEX II’01, Anaheim, CA. (June 2001). 3. Alphatech: ALPHATECH Light Autonomic Defense System, http://www.alphatech.com/secondary/techpro/alads.html (last accessed April 6, 2003). 4. Tylutki, M.: Optimal Intrusion Recovery and Response Through Resource and Attack Modeling, Ph.D. Thesis, Davis, CA. (September 2003). 5. Toth, T., Kruegel, C.: Evaluating the impact of automated intrusion response mechanisms, 18th Annual Computer Security Applications Conference, Las Vegas, Nevada. (December 9–13, 2002).

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6. Lee, W., Fan, W., Miller, M., Stolfo, R., Zadok, E. Toward Cost-Sensitive Modeling for Intrusion Detection and Response, Joumal of Computer Security, Vol. 10, Numbers 1, 2 (2002). 7. Carver, C.A, Jr. and Pooch, U.W.: An Intrusion Response Taxonomy and its Role in Automatic Intrusion Response, Proceedings of the 2000 IEEE Workshop n Information Assurance and Security, United States Military Academy, West Point, NY. (6–7 June, 2000). 8. Fred Cohen & Associates, Deception for Protection, http://all.net/journal/deception/index.html (last accessed April 6, 2003). 9. Free Software Foundation, Inc., The GNU Privacy Guard, http://www.gnupg.org (last accessed April 6, 2003). 10. Raiffa, H.: Decision Analysis: Introductory Lectures n Choices under Uncertainty, Addison-Wesley, Reading, . (1968). 11. Ko, C.C.W.: Execution Monitoring of Security-Critical Programs in Distributed System: Specitication-Based Approach, Ph.D. Thesis, Davis, CA. (August 1996). 12. Templeton, S., Levitt, K.: A requires/provides model for computer attacks. In Proceedings of the New Security Paradigms Workshop, Cork, Ireland. (September 2000). 13. Security Focus, Mailing List: FOCUS-DS, http://www.secuitfocus./hiv/9/310579/200302-03/2003-02-09/1 (last accessed April 6, 2003). 14. Staniford, S., Paxson, V., Weaver, N.: How to Own the Internet in Your Spare Time, Proceedings of the 11th USENI.X Security Symposium (2002). 15. Red Hat, Inc.: Red Hat Security Advisory RHS-2000:100-02, http://rhn.rdht.com/rrat/RS-2000-100.html (lst accessed April 6, 2003).

BMSTU Press has published the book: Heat Technology (in Russian) / A.M. Arkharov, V.L. Bondarenko, B.P. Borisov et al. Under general editorship by A.M. Arkharov, V.I. Afanasiev. – 2nd edition, revised and supplemented. – M.: Izd-vo MGTU imeni N.E. Baumana. 2004. – 712 p.

The textbook (1st edition in 1986, edited by V.I. Krutov) has an encyclopedic nature and deals with bases of thermodynamics and heat exchange theory; fuel and its burning; schemes and elements for designing boilers, industrial furnaces, vapor and gas turbine facilities, refrigeration units and compressors, engines of internal and external combustion, rocket and aircraft engines, atom and plasma power units. Calculations of heater systems, ventilation and air conditioning systems are given. Besides, some important materials are included concerning space power facilities; heat exchange apparatus; hydro-machines; photon power systems; cryogenic systems for liqueﬁed gases, air separation, production of neon, krypton, xenon; thermo-electronic and thermo-magnetic low-temperature facilities and control systems as well. Well-known specialists were involved into the activity for writing new parts of the textbook. Great attention was paid to problems of ecology and environment protection. The textbook content corresponds to courses of lectures being delivered by the authors in the Bauman Moscow State Technical University and some other Russian and Foreign universities. For students of the higher educational establishments, learning the “Power Engineering” trade.

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PHYSICS S.M. Korotaev1,2 , A.N. Morozov1 , V.O. Serdyuk2 , Yu.V. Gorokhov3 , S.A. Pulinets3 , V.I. Nalivayko2 , A.V. Novysh2 , S.P. Gaidash2 , H.D. Kanonidy3 (1 Bauman Moscow State Technical University, 2 Geoelectromagnetic Research Institute, Russian Academy of Sciences, 3 Institute of Terrestrial Magnetism, Ionosphere and Radio Wave Propagation, Russian Academy of Sciences)

MANIFESTATION OF MACROSCOPIC NONLOCALITY IN THE PROCESSES OF SOLAR AND GEOMAGNETIC ACTIVITY The results of long-term experiments on study of macroscopic nonlocality of the solar and geomagnetic dissipative processes are presented. Advanced nonlocal detector reaction on these processes has been revealed. Inﬂuence of the solar activity prevails over geomagnetic one. Advancement proved to be of order of value 10–100 days. Level of advanced signal allows us to put forward a problem of employment of the macroscopic nonlocality effect for solar activity forecast.

Introduction. Macroscopic nonlocality consists in correlation of the dissipative processes without any local interaction carriers. At the micro-level quantum nonlocality was studied for a long time, but only recently theoretical reasoning on persisting of nonlocality in the macro-limit has been appeared [1, 2]. Experimentally macroscopic nonlocality was discovered in the early works on causal mechanics, although it was interpreted in another terms [3]. It was suggested that number of statistically reliable but classically impossible correlations between some astrophysical and geophysical processes might be explained by macroscopic nonlocality [3–5]. The examples are: inﬂuence of the solar activity on the meteorological processes, on the physical-chemical insulated lab processes and so on. These two examples point out two ways for study of macroscopic nonlocality. The ﬁrst way consists in study of correlation of the natural processes which, by energetic reasons, must be considered as local-independent. The second way is study of reaction of the insulated probe-process (detector) of well known nature on the external largescale source-processes. Taking into account complexity of the natural processes and embryo of macroscopic nonlocality theory, the ﬁrst way can not be realized so rigorously as it requires for veriﬁcation of a new physical hypothesis. The second way is more perspective and exactly it has been recently realized [6–13]. Bell-type VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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inequalities violation was established and nonlocal detectors reaction on the processes of solar, meteorological, geomagnetic and ionospheric activity was revealed. The most prominent property of these phenomena was (theoretically predicted) availability, side by side with retarded correlation, of advanced one [14, 15]. Moreover a level of the advanced correlation according to the theoretical ideas [6, 8, 9] and the experiment [6–13] proved to be higher than the retarded one (due to the lesser efﬁciency of absorption of the advanced direct particle ﬁeld by the intermediate medium). This work is devoted to more detailed study of effect of the solar activity and is most closely related with it the global geomagnetic activity. Statement of the Problem. The problem consists in study of dependence of the probe-process entropy production on the source-process one, which are related by the macroscopic nonlocality equation [6, 8, 9, 12]. Although any dissipative process can be taken as the probe one, its choice is dictated by relative value of effect and theoretical transparency, allowing us to relate the measured signal with the entropy production and to consciously take exhaustive steps on screening and/or control of all possible local noise factors (temperature, pressure, electromagnetic ﬁeld etc.). From several types of the detectors employed in the experiments [6–12], the most long-term observation series have been obtained with the detector based on the spontaneous variations of self-potentials of weakly polarized electrodes in the electrolyte. Such detector is an element of GEMRI setup for study of macroscopic nonlocality with which the ﬁrst purpose-directed experiment of annual duration was conducted in 1996–97. At present a new synchronous experiment with GEMRI setup (electrode and photocathode detectors) and CAP one (ion mobility detector) is under way. It turned out also that for study of macroscopic nonlocality the other, more prolonged measurements of the self-potentials of weakly polarized electrodes could be used. These measurements were conducted in 1993-97 with technical aim, without exhaustive steps on protection against local noise factors, but nevertheless allowing statistical detection of the macroscopic nonlocality effect [6, 8–13]. In this work all available electrode detectors data is used. As the main source-process we shall consider the solar activity. In the previous works [10–12], the question on the indices of the solar activity most adequate to the given problem was investigated. The detector signals proved to be most correlated with the solar radio wave ﬂux in the middle of the standard frequency range of 245 . . . 15400 MHz — about 1415 MHz, corresponding to radiation from the level of upper chromosphere — lower corona, that is just from the level of maximal dissipation of the sound energy. Therefore in this work we use radio ﬂux data at frequencies 610, 1415 and 2800 MHz. In works [6, 8–13] the detectors reaction on various ionospheric-magnetospheric processes and their indices was also studied. As a result [13] it has been found that the most correlated with detectors signal was Dst — index of geomagnetic activity, which therefore is used in the present work (standard international hourly data).

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The ﬁnal task is revealing the advanced detector reaction on the solar geomagnetic processes. It should be stressed that the electrode detector is not sensitive to the solar radio waves or the geomagnetic ﬁeld. They are only qualitative indices of the source entropy production ( quantitative estimation is also possible, e.g. it is possible to relate geomagnetic variation amplitude via medium impedance and temperature with the entropy production of the source electric current dissipation [6, 8–12]). Special investigation was taken, which shown absence of direct or indirect (for example, via the cosmic rays variations) inﬂuence of the solar-terrestrial electromagnetic ﬁelds on the electrode detector within its sensitivity [10–12]. Data Processing. Data was processed by both the standard methods (spectral and correlational analysis) and by the modern informational-statistical methods (causal analysis). Taking into account the comparative novelty of the causal analysis [16–17], we present here the essence of its formalism. For any processes X and Y via conditional H(X|Y ), H(Y |X) and marginal H(X), H (Y ) Shannon entropies the independence functions i are introduced: iY |X = H(Y |X)/H(Y ),

iX|Y = H(X|Y )/H(X),

0 ≤ i ≤ 1.

Values of i characterize one-sided independence of the processes. If e.g. iX|Y = 0 then X is a single-valued function of Y , if iX|Y = 1 then X is independent of Y . Roughly speaking, values of i behave inversely to module of the correlation coefﬁcient (more exactly such analogue is (1 − iY |X )(1 − iX|Y )). However in contrast to the correlation function, the independence ones equally ﬁt to any (nonlinear) type of dependence of X and Y , but the main thing is that they reﬂect the asymmetry, characteristic of causal-effect relationship. It allows us to introduce the causality function γ: γ = iY |X /iX|Y , 0 ≤ γ < ∞, and to deﬁne that cause Y and effect X are the processes for which γ > 1. If γ < 1, then inversely, X is cause and Y is effect. The case γ = 1 corresponds to the adiabatic (causeless) relationship of X and Y . Claim of retardation of effect relative to cause is introduced then as necessary condition of local connection of X and Y . It has been shown by theoretical and multiple experimental examples that such formal deﬁnition of causality does not contradict its intuitive understanding in obvious situations and can be used in non-obvious ones [17–19]. So, the above-mentioned claim: γ > 1 ⇒ τ = tY − tX < 0 corresponds to the principle of local or strong causality [14]. Phenomenon of quantum nonlocality, however, satisﬁes only weak causality principle according to which X must be retarded relative to Y , if Y is a process controlled (initiated) by an observer. If Y is non-controlled (natural) process, then advancement of X relative to Y is allowed [14]. VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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Thus calculating iX|Y and iY |X as a function of time shift τ , it is possible, by their minima, to ﬁnd optimal time shifts corresponding to transaction X and Y . Then, by value of γ relative to 1, it is possible to establish direction of causal connection. In the case if γ is known to be cause (e.g. Y is index of solar activity), while X — to be effect (e.g. X is detector signal), then for any classical interaction min iX|Y will be observed only at τ < 0, and this minimum would correspond to max γ > 1. Only for nonlocal transaction of X and Y it is possible that γ > 1 at τ > 0. In processing data of observations with the electrode detectors (self-potentials differences U ) was used, which, because of the technical gaps, was divided in two standards: (1) time series, one-hour sampled, with duration no less than 3 months, and (2) time series of daily averaged data with duration no less than a year. It provided an excerption error of i and γ about 1 %. In the spectral analysis the only preprocessing procedure was removing of trend, approximated by a cubic polynomial. In the causal and correlation analysis (as experience of the previous study [6–13] had shown that for detector reaction on the geomagnetic activity the signal/noise ratio was maximal in a range from 1 day to about 2 months, while for the reaction on the solar activity it increased monotonously along the period) the corresponding wide-band ﬁltration was applied. Series of standard (1) were used only for study of relationship of the signal and geomagnetic activity with using the previous low-pass ﬁltration (periods T > 1d ) and removing the cubic trend restricting the maximal period by half of the series length. Series of standard (2) were used for study of the solar and geomagnetic activity with the low-pass ﬁltration (T > 1d ). Nonlocal inﬂuence of other large-scale natural processes, (ﬁrst of all the meteorological activity [10, 12, 13] conﬁnes i below and γ above. At a given stage we will put up with this circumstance, although in future more detailed research this noise inﬂuence can be taken into account by employment of the generalized causal analysis. Results. Examples of synchronous amplitude spectra of solar radio wave ﬂux R at three frequencies, Dst-index and detector signal U are shown in Figs. 1 and 2. All the spectra have two main maxima — at period of solar rotation and its second harmonic. The Dst spectrum is more complicated due to the fact that Dst-variation is formed by multiplex and nonlinear inﬂuence of the Sun on the current, exciting the magnetic ﬁeld (both via source emf variations and via plasma conductivity variations). For realization presented in Fig. 1 the spectrum U is most similar to one R610 — by width of both spectral maxima and by general spectrum shape, including a rise in the most long period domain. Amplitude ratio of the ﬁrst and second harmonics of solar rotation period for U equals 0.95, for Dst — 0.69, R610 — 0.99, R1415 — 0.87, R2800 — 0.69 and thus by this parameter U is also closest to R610 . In the other realization (Fig. 2) the same ratio for U equals 1.2, for Dst — 0.74 and for all R — 1.1, that is in this case in U the ﬁrst harmonic is almost as more than the second one as it is in R, while in Dst, on the contrary, the ﬁrst harmonic essentially less than the second one. By width of the main maxima, and general spectrum shape, U is most similar to R2800 in this case. 176

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Fig. 1. Amplitude spectra of the solar radio wave ﬂux at frequencies 2800 MHz R2800 , 1415 MHz R1415 and 610 MHz R610 , the geomagnetic activity index Dst and detector signal U in the period range T from 10 days to 243 days (ff = 1/T ). Realization 10/26/1994–02/11/1996

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Fig. 2. Amplitude spectra of R2800 , R11415 , Dst and U in the period range T from 10 days to 274 days. Realization 03/16/1996–07/23/1997

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Fig. 3. Independence iX|Y and causality γ function of the detector signal and geomagnetic activity. τ is time shift, days; X = U ; Y = Dst Dst. Realization U 03/16/1994– 06/19/1994 (realization Dst begins 75 days before and ﬁnishes 75 days after realization U)

So spectral analysis points out the certainly closer relation of the detector signal with the solar activity proper than with its effect — geomagnetic activity. However optimal frequency of the solar radio wave ﬂux, reﬂecting level of the sourceprocesses in the solar atmosphere, may change in time. Turn now to the causal analysis of data. In [13] it was established that the deepest minima of both independence function and the highest maxima of the causality function (as well as the main extrema of the correlation one) were observed at advancement of the detector signal relative to Dst by 1–2 month. This conclusion was made using data of both GEMRI detectors and two time series data of non-protected detector and was illustrated by an example of Dst forecast with advancement of 33 days. In the present study all the remained series satisﬁng the standard (1) were processed. Such series proved to be 6, and the conclusion made before has not changed. Figure 3 illustrates it. The deepest minimum of iX|Y and single maximum of γ are observed at advancement of the detector signal U = X relative to Dst = Y by 34 days. Thus detector registers the advanced signal from the process of dissipation of the currents, exciting the Dst-variation. This process has essentially less power as compared with the solar (and meteorological) dissipative processes competing by inﬂuence on the detector, therefore due to a small signal/noise ratio, iX|Y and γ sligtly differ from 1. It should be noted that in data processed by the standard (2), due to the growing low-frequency direct inﬂuence of the solar activity, effect of inﬂuence Dst on U turned out insigniﬁcant. How much stronger the detector reaction is on the processes of solar activity can be seen in Fig. 4–6, where results of causal analysis for the year corresponding VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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to beginning of the present cycle are shown. In the advanced domain (τ > 0) values of the independence function of the detector signal U = X relative to R = Y at the all three frequencies are much lower than in the retarded domain , and the causality function is much more than 1. The deepest minimum iX|Y = 0.47 and the highest maximum γ = 1.58 are observed for Y = R2800 at τ = 42d . The latter value is close to estimation (about month) obtained in [10–12] by a simpler way. A big τ -interval corresponding to signiﬁcant γ > 1 is explained by a big volume of the solar atmosphere occupied by the source-processes with diffusion propagation. A common feature of reﬂected in Fig. 4–6 advanced connection of the detector signal with the radio ﬂux is availability of three γ pikes. For the frequency 2800 MHz these pikes correspond to advancement: 42, 119 and 280 days; for 1415 MHz: 63, 133 and 280 days; for 610 MHz: 98, 217 and 280 days. The most distant, third peak (τ = 280d ) is common for the all three levels of radio ﬂux generation and therefore it corresponds to the process of the most spatial scale. From comparison of the results of causal and correlation analysis the nonlinearity of relation of U and R follows. The main maxima of correlation function coincide by position with minima of iX|Y (maxima of γ) but their relative values are different. So at frequency 2800 MHz there is maximum of correlation r corresponding to the main γ pike: at τ = 42d r = 0.50 ± 0.03 (correlation is signiﬁcant with reliability not less than 0.999 [12]), but the biggest is maximum corresponding to the third γ peak: at τ = 280d r = 0.72 ± 0.02.

Fig. 4. Independence and causality functions of the detector signal and solar radio ﬂux at frequency 2800 MHz. X = U , Y = R2800 . Realization U 12/11/1996–12/10/1997 (realization R2800 begins 1 year before and ﬁnishes 1 year after realization U )

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Y = R1415 ) Fig. 5. The same as in Fig. 4 for frequency 1415 MHz (Y

Y = R610 ) Fig. 6. The same as in Fig. 4 for frequency 610 MHz (Y

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If instead of observed radio ﬂux data, we take thatadjusted to 1 A.U. (published in “Solar-Geophysical Data”), the dependence of U on R slightly decreases. So for frequency 2800 MHz at τ = 42d γ decreases 1.03 times, r decreases 1.11 times, iX|Y increases 1.01 times. Thus the weak affect of the Earth orbit ellipsisity also reﬂects in detector reaction. In Figs. 4–6 it is seen that the ﬁrst and second γ pikes move to larger τ as the frequency falls of the frequency (rise of the source level). Further, it is seen that with the fall of the frequency the larger γ values shift to the domain of larger τ (with a small decrease of γ and iX|Y extremal deﬂections) — for Y = R2800 : γ = 1.58, iX|Y = 0.47 (τ = 42d ); for Y = R1415 : γ = 1.56, iX|Y = 0.48 (τ = 63d ); for Y = R610 : γ = 1.56, iX|Y = 0.49 (τ = 280d ). These peculiarities indicate that processes at upper levels are activated later than at lower ones (diffusion of activity in the solar atmosphere goes upwards). Thus in spite of the unusual method of solar activity registration — by nonlocal insulated lab probe-process reaction, the result is quite natural. Availability of such essential advanced detector reaction on the solar activity gives a sense to an attempt of performance of the forecast problem. As it is seen in Figs. 4–6, relation of U and R is far from δ-correlated (the domain of advanced dependence is spread in wide τ range). Therefore a forecast algorithm must be based on plural regression — one forecasted value R is calculated as convolution of impulse transition characteristic with multitude of the preceding U values. In addition, the problem of optimal ﬁltration, suppressing interference from nonlocal inﬂuence of other natural processes, ﬁrst of all, the meteorological activity must

Fig. 7. U variation forecasting R2800 one with advancement of 42 days. The origin of time count (days) corresponds to 12/11/1996 182

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be previously solved. Elaboration of such algorithm is a complicated though quite standard task. For the present we will conﬁne ourselves by wittingly primitive simplest demonstration of the possibility of forecast. We select the highest observed γ pike corresponding to Y = R2800 , τ = 42d and shift by this τ the series X(= U ) and Y . The result is shown in Fig. 7. It is seen that detector signal really forecasts the solar activity. Conclusion. Effect of macroscopic nonlocality is reliably registered for the processes of solar and geomagnetic activity. Nonlocal detector reaction directly on the process of solar activity essentially prevails. The most prominent peculiarity of nonlocal correlations is their advanced character. Time of advancement is large: of the order of 10–100 days. There is a sense to put forward the problem of employment of macroscopic nonlocality effect for solar activity forecasting. Acknowledgement. This work was supported by RBRF and Moscow Region Government (grants 02-05-64006 and 01-05-97015).

REFERENCES 1. Mermin, N.D. Extreme quantum entanglement in a superposition of macroscopically distinct states // Phys. Rev. Lett. – 1990. V. 65. No 15. P. 1838–1840. 2. Home, D. & Majumdar, A.S. Incompatibility between quantum mechanics and classical realism in the strong macroscopic limit // Phys. Rev. A. 1995. V. 52. No 6. P. 4959–4962. 3. Kozyrev, N.A. On the possibility of experimental investigation of the properties of time // Time in Science and Philosophy / Ed. Zenam J. Prague: Academia, 1971. P. 111– 132. 4. Korotaev, S.M. Logic of causal mechanics: observations-theory-experiments // On the Way to Understanding the Time Phenomenon, Part 2 / Ed. Levich A.P. World Scientiﬁc, 1996, P. 60–74. 5. Zhirblis, V.E. Starts and koltsars // On the Way to Understanding the Time Phenomenon, Part 2 / Ed. Levich A.P. Word Scientiﬁc, 1996. P. 135–173. 6. Korotaev, S.M., Sorokin, M.O., Serdyuk, V.O. & Abramov, J.M. Experimental study of the nonlocal interaction of the macroscopic dissipative processes // Physical Though of Russia. 1998. No 2. P. 1–17. 7. Dvoruk, S.K., Korotaev, S.M., Morozov, A.N., Nazolin, A.L., Serdyuk, V.O., Solovyov, A.V., Sorokin, M.O., Tabalin, S.E. & Shishkin, G.V. Experimental studies of the irreversible processes in the electrolytes // Applied Mechanics and MachineBuilding Technologies. 1998. V. 1. No 4. P. 61–66. 8. Korotaev, S.M., Sorokin, M.O., Serdyuk, V.O. & Abramov, J.M. Experimental study of macroscopic nonlocality // Science & Technology in Russia. 1999. No 1. P. 16–19. 9. Korotaev, S.M., Sorokin, M.O., Serdyuk, V.O. & Abramov, J.M. Geophisical manifestation of interaction of the processes through the active properties of time // Physics and Chemistry of the Earth. A. 1999. V. 24. No 8. P. 735–740. 10. Korotaev, S.M., Serdyuk, V.O. & Sorokin, M.O. Experimental veriﬁcation of Kozyrev’s interaction of natural processes// Galilean Electrodynamics. 2000. V. 11. No 2. P. 23– 28. 11. Korotaev, S.M., Serdyuk, V.O. & Sorokin, M.O. Effect of macroscopic nonlocality on geomagnetic and solar-ionospheric processes // Geomagnetism and Aeronomy. 2000. V. 40. No 3. P. 323–330. VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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12. Korotaev, S.M., Morozov, A.N., Serdyuk, V.O. & Sorokin, M.O. Manifestation of macroscopic nonlocality in some natural dissipative processes // Russian Phisics Journal. 2002. No 5. P. 3–1413. 13. Korotaev, S.M., Morozov, A.N., Gorohov, J.V., Nalivayko, V.I., Novysh, A.V., Pulinets, S.A. & Serdyuk, V.O. Experimental study of macroscopic nonlocality of some magnetospheric-ionospheric and tropospheric processes. 14. Cramer, J.G. Generalized absorber theory and Enstain–Podolsky–Rosen paradox // Phys. Rev. D. V. 22. P. 362–376. 15. Korotaev, S.M. The force of time // Galilean Electrodynamics. 2000. V. 11. No 2. P. 29–33. 16. Korotaev, S.M. On the possibility of causal analysis of the geophysical processes // Geomagnetism and Aeronomy. 1993. V. 33. No 2. P. 128–133. 17. Korotaev, S.M. Role of different deﬁnitions of the entropy in the causal analysis and its employment to electromagnetic induction of the sea currents // Geomagnetism and Aeronomy. 1995. V. 35. No 3. P. 116–125. 18. Hachay, O.A., Korotaev, S.M. & Troyanov, A.K. Results of application of the causal analysis for bore-hole data on seismo-acoustic and electromagnetic emission // Volcalonogy and Seismology. 1992. No 3. P. 92–100. 19. Korotaev, S.M., Hachay, O.A. & Shabelyansky, S.V. Causal analysis of the process of horizontal informational diffusion of electromagnetic ﬁeld in the ocean // Geomagnetism and Aeronomy. 1993. V. 33. P. 116–125. 20. Korotaev, S.M., Shabelynsky, S.V. & Serdyuk, V.O. Generalized causal analysis and its employment for study of electromagnetic ﬁeld in the ocean // Izvestia Physics of the Solid Earth. 1992. No 6. P. 77–86. Korotaev Sergey Maratovich, born in 1950, graduated from LHMI in 1972, D. Sc. (Phys.-Math.), Head of Lab of GEMRI RAS, Professor of BMSTU. Author of more than 130 publications, including 3 monographs, in the area of electrodynamics, causal mechanics and informational-statistical methods of analysis of physical experimental data. A.N. Morozov, born in 1959, graduated from the Bauman Moscow Higher Technical School in 1981. D. Sc. (Phys.-Math.), professor, head of “Physics” Department of the Bauman Moscow State Technical University. Author of more than 100 publications in the ﬁeld of high precision measuring systems and physical kinetics theory. Serdyuk Vyacheslav Olegovich, born in 1969, graduated from the Lomonosov Moscow State University in 1992. Researcher of GEMRI RAS. Author of about 30 publications in the area of geophysics, statistical electrodynamics and causal analysis. Gorokhov Yury Vasilyevich, born in 1950, graduated from the Lomonosov Moscow State University in 1974, Ph. D. (Phys.-Math.), senior researcher of IZMIRAN RAS, author of about 40 publications in the area of radiophysics and solar-terrestrial physics. Pulinets Sergey Alexandrovich, born in 1949, graduated from the Lomonosov Moscow State University in 1972, D. Sc. (Phys.-Math.), Professor, Vice-director of IZMIRAN RAS. Author of about 180 publications in the area of solar-terrestrial physics. Nalivayko Valentin Iosifovich, born in 1928, graduated from OEIC in 1953, senior researcher of GEMRI RAS, author of about 50 publications in the area of techniques of weak electromagnetic ﬁelds measurements and solar-terrestrial physics. Novysh Andrew Vsevolodovich, born in 1947, graduated from Moscow Institute for Fine Chemical Technology, leading engineer-electronicist of GEMRI RAS. Author of 7 publications in the area of techniques of physical experiment.

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Gaidash Sergey Petrovich, born in 1947, graduated from Radiophysical Faculty of KhIRE, senior researcher of GEMRI RAS, author of about 50 publications in the area of techniques of weak electromagnetic ﬁelds measurements and solar- terrestrial physics. Kanonidi Harlampiy Dmitrievich, born in 1932, graduated from the Lomonosov Moscow State University in 1956, Ph. D. (Phys.-Math.), Director of Geophysical Observation Center of IZMIRAN RAS, author of about 70 publications, including 6 monographs, in the area of solar-terrestrial physics.

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185

M.A. Yakovlev (Bauman Moscow State Technical University)

GENERATION OF THE NEAR-SURFACE ELECTRON LAYER BY PICOSECOND LASER PULSES The inﬂuence of the self-consistent electric ﬁeld of a near-surface electron layer on the energy spectrum of photoelectrons emitted under the action of a picosecond laser pulse on a metallic target is determined.

Near-Surface Electron Layer. The main reason for the formation of a nearsurface electron layer under the action of ultranarrow laser pulses on a metallic target is a substantial gap (of the order of 1 eV) between the temperature of the electronic component and that of the lattice [5]. This causes a sharp increase in thermoemission current and the formation of a fairly extended region of the volume negative charge near the surface. The double electric layer near the surface of a metal is known to exist also under the conditions of thermodynamic equilibrium between the electronic and lattice subsystems. In this case, the electron gas is degenerate, and the concentration of electrons ne decreases at a very high rate as the distance from the surface z increases: ne ∝ z −2 exp(−βz), where β −1 is a value of the order of the mean interelectronic distance in the metal [7]. The near-surface layer of degenerate electrons can therefore be considered arbitrarily thin, and its inﬂuence on laser beam penetration inside the metal can be ignored. The degree of degeneracy, however, decreases sharply as the concentration of electrons lowers, and electrons experience the transition to the classical state when the Fermi energy EF (ne ) becomes of the order of kT . A further decrease in the concentration of electrons follows a much smoother law [6], µ ¶−2 z ne (z) = n0 1 + √ , (1) 2Ld where Ld = (ε0 kT /e2 n0 )1/2 is the Debye screening length and n0 is the concentration boundary determined from the degeneracy condition EF (n0 ) ≈ kT, that is, n0 ∝ T 3/2 . It follows that the size of the near-surface electron layer region, where the concentration decreases according to Eq. (1), increases as the electron temperature 186

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rises. Simultaneously, concentration boundary n0 also increases, and, at temperatures of the order of EF , all electrons in the near-surface electron layer experience the transition to the classical state. Their distribution then obeys Eq. (1), and n0 becomes of the order of the concentration of electrons in the metal. Thus, intense nonequilibrium heating of the electronic component of a metal can cause the formation of a fairly extended layer of electrons with a high concentration near the surface. The formation of a near-surface electron layer, whose electrophysical characteristics enable it to substantially inﬂuence near-surface processes under the action of ultranarrow laser pulses, is only possible within limited laser pulse intensity Iem and width τp intervals, min max Iem ≤ Iem ≤ Iem ,

τpmin ≤ τp ≤ τpmax .

min The Iem lower intensity boundary is determined from the condition that a fairly effective gap between the electronic component temperature Te and the lattice temperature Tl should be attained during the pulse duration,

∆T = Te − Tl ∼ TF ∼ 104 . . . 105 K, where TF is the temperature of the degenerate electronic subsystem; that is [8], min ≈ αl l∆T, Iem

where αl ∼ 1016 W/(m3 K) is the energy exchange rate between the electronic and lattice subsystems and l is the depth of the metal layer heated during the pulse √ duration, l ∼ max[δ, χτp ] (δ is the skin depth, and χ is the electron thermal min diffusivity). Picosecond pulses are characterized by l ∼ 10−5 cm, that is, Iem ∼ 9 2 ∼ 10 W/cm . According to [5], when picosecond laser pulses of intensity Iem ∼ 3 × × 109 W/cm2 act on the surface of typical metals, two competing processes of electron escape from the surface, namely, thermoemission and photoemission, produce current of the same order of magnitude. We can therefore expect that the collective thermoemission process should most noticeably inﬂuence the oneparticle photoemission process under these conditions. Thermoemission begins to predominate over photoemission as the intensity of radiation increases; that is, because of thermoemission, the formation of the near-surface electron layer at the speciﬁed radiation intensities occurs in time τs ≈ Ld /vT ∼ 10−14 . . . 10−13 s. Here, vT the thermal velocity of electrons [6]. Note in addition that the nature of thermoemission is then substantially different from that of thermoemission from an electrode in a closed circuit, because thermoemission that we are considering occurs from an insulated metallic surface, on which an uncompensated positive charge remains, and a volume negative charge is formed near the surface, which in turn inﬂuences the thermoemission current. This considerably complicates the VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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description of the formation of the near-surface electron layer. Nevertheless, the time τs , of formation of the spatial distribution is much shorter than the picosecond laser pulse width τp . The spatial distribution of the near-surface electron layer can therefore be considered stationary with the corresponding layer temperature virtually over the whole pulse width. max corresponds to the prethreshold region of The upper intensity boundary Iem the beginning of melting and ablation of the target material. Such processes are observed when the density of the energy accumulated in the electronic subsystem under the action of an ultranarrow laser pulse exceeds a certain threshold, max Iem τp ≤ Fabl ,

where the threshold energy density of laser ablation Fabl ≈ 0.2 . . . 0.5 J/cm2 max [9, 10]; that is, at the pulse width of τp ∼ 1 ps, we have Iem ∼ 1012 W/cm2 . The limitations imposed on the width of laser pulses τp are determined from the condition τpmin ≥ τs , where τs ∼ 10−13 s is the time of formation of the near-surface electron layer with a nondegenerate electronic component, and τpmax ≤ τel , where τel is the characteristic time of energy transfer from electrons to the lattice. The second condition guarantees that the lattice is not heated during a laser pulse and the pulse width is insufﬁcient for creating conditions of developed vaporization of the target material, when the role played by the near-surface electron layer loses signiﬁcance. To summarize, it follows from the results described above that the near-surface electron layer can exert substantial inﬂuence on surface processes within limited laser pulse intensity and width ranges, which are, however, fairly important for technical applications, namely, 1010 W/cm2 ≤ Iem ≤ 1012 W/cm2 and 10−13 ≤ ≤ τp ≤ 10−11 s respectively. The Electric Field of the Near-surface Electron Layer. As in [11], the selfconsistent electric ﬁeld of the near-surface electron layer Ez created by the positive charge of the conductor and electrons under surface irradiation by picosecond laser pulses was found by solving the set of equations that described the electronic component temperature variations with time and electron layer formation. The ionization of the neutral gas taken into account in [11] could, however, be ignored at pressures and picosecond laser pulse intensities under consideration (of about 1 atm and ∼1010 W/cm2 , respectively). Let us turn to the system of equations that describe the near-surface electron layer formation near the surface of a condensed substance. First consider the heat equation. At z < 0 (a conducting condensed substance), we have · ¸ ∂Te ∂Te ∂ Cm (2) = χm − α(Te − Tl ) + q(z, t). ∂t ∂z ∂z 188

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Here, Cm and χm are the heat capacity and the heat conductivity of the conducting condensed substance, respectively, and α is the coefﬁcient of heat exchange between electrons and the lattice, which is virtually independent of the electron temperature [12]: π 2 m c2s νeff nm α= , 6 Tl where νef f = νef f (Tl ) and cs is the velocity of sound in the conducting condensed substance. For typical metals with nm ∼ 1022 cm−3 , we have α∼1010 W/(cm3 K). The heat capacity and heat conductivity of the condensed substance were approximated by the asymptotic equations [12]

Cm

χm

µ ¶ 2 π kTe k n , m 2 EF = 3 n k, m 2 Te χm0 T , l = 1 nm k v 2e , νef f 2

kTe ¿ EF ,

kTe ≥ EF ;

kTe ¿ EF ,

kTe ≥ EF ,

where v e is the mean thermal velocity of electrons and χm0 is the equilibrium thermal conductivity coefﬁcient of the condensed conducting substance at initial temperature T0 . The q(z, t) volume energy release function has the form q(z, t) = κr κi k0 ε0 c|E0 |2 exp(2k0 κi z), where κr and κi are the refractive indices of the condensed conducting substance, k0 is a wavenumber (k0 = ω/c) and E0 is the amplitude of the wave ﬁeld at z = 0. The heat equation for a gas (z > 0) has the form · ¸ 1 ∂ ∂Te 3 ∂Te e2 |E|2 νe 3m k = k(Te − Ta )νe + , (3) χe − 2 ∂t ne ∂z ∂z M 2m(ω 2 + νe2 ) where M is the atomic weight of the gas, νe = σa na v e is the frequency of electronic collisions in the gas, and σa = σa (v e ) is the transport cross section of electron scattering by atoms. The σa (v e ) dependence is well known for rare gases. We used the data from [13] in our calculations. The equations describing variations in Tl — lattice temperature and Ta — temperature of the heavy gas component (atoms) — have the form Cl

∂Tl = α(Te − Tl ); ∂t

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where Cl is the heat capacity of the lattice: Cl ≈ 3nl k. Problem conditions correspond to very fast processes (t ¿ τl ∼10−10 s), and, according to the equations given above, the temperatures of the lattice and the heavy gas component change very insigniﬁcantly during such time intervals. For instance, even at the mean electron temperature Te ∼10 eV, the temperature of the lattice changes by a value of the order of 102 K in time t∼10−13 s. For this reason, we assumed in our calculations that Tl = Ta = T0 . The boundary conditions for (2) and (3) have the form ∂Te = 0, ∂z

z = −lm , la , (4)

·

¸ ∂Te χ = 0, ∂z

z = 0,

where lm and la are the boundaries of the region under consideration in the condensed conducting substance and gas, respectively (formally, lm , la → ∞). Next, consider the equations that describe the kinetics of the near-surface electron layer formation. The continuity equation for the concentration of electrons should be solved simultaneously with the Maxwell equations in the z < 0 and z > 0 regions. It is, however, sufﬁcient to consider the continuity equation in the z > 0 region rather than in the whole space. The boundary condition at z = 0 should then correspond to distributions of electrons of the near-surface electron layer Eq. (1) as functions of temperature (see above), ne |z=0 = n0 , where n0 is the boundary concentration of the near-surface electron layer at a given time moment (note that electrons of the near-surface electron layer form an ideal Coulomb system at T ≥ 5 eV). There is sufﬁcient time for the concentration proﬁle of the near-surface electron layer near the surface to keep track of temperature variations, because the characteristic time of the formation of this layer is about ωp−1 . Consider the continuity equation for the electronic component, ¸ · ∂ ∂ne ∂ne = + µ e E z ne , (5) De ∂t ∂z ∂z where De and µe , are the diffusion and mobility coefﬁcients deﬁned as µe =

e ; m e νe

De =

v 2e . 3νe

The boundary conditions for Eq. (5) have the form z=0:

ne = n 0 ;

z = la :

∂ne + µe Ez ne = 0. De ∂z

(6)

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The initial condition for ne corresponds to the distribution of near-surface layer electrons at the initial temperature T0 . The distribution of the longitudinal electric ﬁeld Ez is found from the equation ∂Ez ene . =− ∂z ε0

(7)

The boundary condition that corresponds to this equation at each time step has the form Zla e Ez |z=0 = ne) dz. ε0 0

REFERENCES 1. Farkas, G. & Toth, C. // Phys. Rev. A, 41, 4123 (1990). 2. Logothetis, E.M. & Hartman, P.L. // Phys. Rev. 187, 460 (1969). 3. Petite, G., Agostini, P., Trainham, R., et al. // Phys. Rev. B 45, 12210 (1992). 4. Varro, S. & Ehlotzky, F. // Phys. Rev. A 57, 663 (1998). 5. Anisimov, S.I., Kapelovich B.L. & Perel’man, T.L. “Electron Emission from Metal ´ Surface Under Ultranarrow Pulse Laser Radiation” (in Russian) // Zh. Eksp. Teor. Fiz., 66, 776(1974) [Sov. Phys. JETP 39, 375 (1974)]. 6. Ivlev, A.V., Pavlov, K.B. & Yakovlev, M.A. “Interaction of Radiation with Near-Surface Electron Layer” (in Russian) // Zh. Tekh. Fiz., 64 (9), 50(1994) [Tekh. Phys. 39, 888 (1994)]. 7. Gupta, A.K. & Singwi, K.S. “Gradient Corrections to the Exchange-Correlation Energy of Electrons at Metal Surfaces” // Phys. Rev. B 15, 1801(1977). 8. Anisimov, S.I. & Retfel’d, B. ”To Theory of Interaction of Ultranarrow Laser Pulse with Metal” (in Russian) // Izv. Akad. Nauk, Ser. Fiz. 61, 1642 (1997). 9. Afanas’ev, Yu. V., Demchenko, N.N., Zavestkovskaya, I.N., et al. “Simulation of Metal Ablation by Ultranarrow Laser Pulses” (in Russian) // Izv. Akad Nauk Ser. Fis. 63, 667 (1999) 10. Riley, D., Langley, A.J., Taday, P.F., et al. “Reﬂectivity Experiments with 60 Femtosecond Laser Pulses” // J.Phys. D. 31, 515 (1998). 11. Ivlev, A.V., Yakovlev, M.A., Bordenyuk, A.N. “Simulation of Gaz Breakdown by Boundary Layer Electrons while Irradiating Metallic Target by Picosecond Laser Pulses” (in Russian) // Zh. Tekh. Fiz., 68(8), 42(1998) [Tech. Phys. 43, 921(1998)]. 12. Anisimov, S.I., Makshantsev, B.I., Barsukov A.V. Opt. Acoust. Rev. 1,251(1990). 13. Kieffer, L.J. At. Data 2(4), 293(1971). M.A. Yakovlev (b. 1946), graduated from the Moscow State University n.a. M.V. Lomonosov in 1971. D. Sc. (Phys.-Math.), professor of “Physics” department of the Bauman Moscow State Technical University. Author of more than 100 publications in the ﬁeld of weakly-ionized plasma and interaction of concentrated energy ﬂuxes with substance. Author of the scientiﬁc discovery “Regularity of reduction of atom ionization potential in dense plasma”. VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

191

V.O. Gladyshev, T.M. Gladysheva, A.N. Morozov, V.Ye. Zubarev (Bauman Moscow State Technical University)

THE RESULTS OF THE EXPERIMENT ON REGISTRATING LIGHT DRAGGING OBSERVED IN AN INTERFEROMETER WITH A ROTATING OPTICAL DISC The interferometer, in which light beams propagate in a rotating optical disk, is considered in the work. Interference pattern shift (∆ = = 0.0076 of a bandwidth) has been registrated experimentally for beams, which passed through the rotating optical disk with a diameter of 120 mm and thickness of 30 mm in two opposite directions. It is shown that the shift can be explained with the longitudinal effect of light dragging and the effect of deviation from Snell’s law, when tangential break of velocity takes place. The value of the interference pattern shift as a result of light trajectory inclination due to a deviation from Snell’s law in a rotating medium is about 25 % due to the longitudinal dragging of light. The results represent the ﬁrst experimental test of the solution of the dispersion equation in the 3-dimensional case of medium motion.

1. Introduction. Effect of light dragging by a moving medium consists in dependence of the phase velocity of light in a moving medium upon the velocity of the medium [1]. This phenomenon is of a fundamental nature in moving medium optics, and provides a classical example of the relativistic velocity composition law and is a basis for creation of moving medium electrodynamics. When an electromagnetic wave is incident onto the moving medium with the tangential velocity break, a deviation of the refraction angle value in a moving medium from the one in a static medium occurs. As a result, light beam trajectories curve in a homogeneous medium with a complex motion [2]. Experimental registration of these effects is a new method of check of the relativity theory and the existing understanding of light interactions with a moving medium in a general case. The goal of this work is to develop an interferometry method for investigating the spatial effect of the light dragging in a rotating medium, to create an interferometer scheme and also to detect a phase shift for beams propagated in a rotating medium. 2. Estimation of the Expected Shift of an Interference Pattern. For achieving this goal the double-beam two-pass disk interferometer was designed and built with inputting beams into the ﬂat surface of an optical disk (Fig. 1). In this scheme, the light beam from laser L incident on a beam divider BD was split into two beams. These beams entered the optical disk OD (one of them after reﬂecting from the mirror M) to be reﬂected from ﬂat mirror surfaces of the OD. The exit beams reﬂected from angle prism AP changed paths, passed through the

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Fig. 1. The optical scheme of the interferometer with a rotating optical disk

optical disk in the reverse direction, and entered the divider again. Mixed on the divider mirror, the beams passed through an optical system OS to display the interference pattern on a screen S. The light intensity was measured by a photodetector PD. The signal from a photodetector came through a resistor system into oscillograph. The optical disk rotation in two opposite directions was provided by a motor. The digital camera Kodak DC240 with high resolution and a recording element took photos of oscillograms (25 tests for each combination of rotation direction and an adjustment) and further they were processed and analyzed with the help of a personal computer. Compensating difference features of the scheme provided a high protection against mechanical disturbances. The interferometer is tuned to the interference fringes of equal thickness and the shift of the light interference pattern is determined using the time signal from the photodetector, when the optical disk OD rotates. The direction of the interference pattern (IP) shift in the plane of the IP analysis depends on the rotation direction. This method of picking out the time signal is preferred as compared to the method of measuring the intensity variations, because the time of IP motion is measured by the equipment, which has a higher relative resolution and stability of parameters. After adjusting the OD, the spots of beams moved as ellipses on a screen, when we slowly rotated the OD. Ellipsisity of the curves is explained with the photoelastic effect. After adjusting the optical system, checking stability of the motor operation, and achieving a stable IP, the experiment was carried out. VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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Fig. 2. The optical disk OD. The upper surface of OD has the partial reﬂective layer with the radius R1 , the lower has full reﬂective cover. Due to the deviation from Snell’s law, the exit point of a beam B for the stationary disk shifts from its position to point B 0 for the rotating disk

When a light beam propagates in a rotating optical disk, the vector of a linear velocity changes its direction in space from point to point, therefore the longitudinal effect of light dragging and the transverse deviation of a beam trajectory should be observed. To estimate the light dragging effect, let us consider the solution of the dispersion equation in the plane XOZ and in the plane of incident and reﬂected beams. In our experiment (Fig. 2) a beam is incident (out of a medium with n0 = 1) onto the ﬂat surface of the optical disk with the refractive index n2 at the angle ϑ0 . In the plane XOZ the electromagnetic wave vector projections on the axis OX and OZ in the moving medium are determined as follows [3] k2x = k0x = (ω0 /c) sin ϑ0 , k2z =

¢1/2 i ¡ ω0 h −κ2 γ22 β2z ξ2 η2 ± η2 cos2 ϑ0 + κ2 γ22 ξ22 η22 , c

(1) (2)

where 2 η2−1 = 1 − κ2 γ22 β2z , κ2 = ε2 µ2 − 1, u2x u2z 2 2 = 1 − β2x − β2z , β2x = , β2z = , c c

ξ2 = 1 − β2x sin ϑ0 , γ2−2

ω0 — electromagnetic wave frequency, c — phase light velocity in vacuum, ε2 , µ2 — dielectric and magnetic permeabilities of the medium, u2x , u2z — components of the medium velocity, ϑ0 — is the incident angle.

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As the longitudinal light dragging effect is determined by the longitudinal component of a linear velocity of the medium, so we can consider (2) when β2x = 0: q ¡ ¢ 2 2 −κ γ β ± 1 + κ2 γ22 1 − β2z 2 2 2z ω0 k2z = , (3) 2 c 1 − κ2 γ22 β2z where cos ϑ0 = 1. 2 Considering β2z À β2z , we obtain the expression for the light phase velocity in a rotating medium µ ¶ ω0 c 1 0 ∼ ± u2 1 − 2 , (4) c =p 2 = 2 n2 n2 k2x + k2z √ where n2 = ε2 µ2 — index of refraction. Orientation of ~k2 and ~u2 varies along a beam trajectory, therefore for a spatial case the projection of the medium velocity vector onto the wave vector should be used instead of ~u2 in (4): pr~k ~u2 = u2z cos ϑ sin ϑ2 + u2x sin ϑ sin ϑ2 ,

(5)

where u2x = ω(R − z), u2z = ωx, R = OA, ϑ2 — angle of refraction. Further we will take into account that the phase accumulation due to trajectory curving in the plane XOZ is not large in comparison with the longitudinal effect [2]. Then we can show that pr~k ~u2 = ωR sin ϑ sin ϑ2 in any point of the trajectory. When beams pass through the rotating OD once, the IP shift is equal to ∆0 =

µ

t1 = cos ϑ2

c (t2 − t1 ) , λ

d µ ¶ ¶, 1 c + 1 − 2 ωR sin ϑ sin ϑ2 n2 n2

(6)

t2 =

n2 d , c cos ϑ2

(7)

where λ — wave length of light; t1 , t2 — time of passing through the rotating and static OD, respectively; ω — OD angular velocity and d — OD thickness. After substituting t1 , t2 into (6), we obtain in the limit ωR ¿ c: √ κ2 ωr R2 − r 2 . (8) ∆0 = λc Here r is the distance between projection of beam path in OD and the center of rotation, r = OA sin ϑ (Fig. 2). By specifying the value R we can ﬁnd an optimal value r, for which ∆0 will be √ ∂∆0 = 0 we obtain the optimal value r = R/ 2. maximum. From the equation ∂r The determined value of IP shift ∆ in the interferometer should be equal to 32∆0 . First of all, the disk edge surfaces are mirror coated, which increases the optical path twice as much. After reﬂecting from a prismatic reﬂector the beams VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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repeatedly pass through OD, which also increases the resulting IP shift by a factor of 2. In the scheme we use two opposite beams and two directions of rotation which increases the resulting IP shift by a factor of 4. Carrying out the experiments for two different adjustments allows another increase of the resulting IP shift by a factor of 2. As a result of this we can write ∆=

16lu2n (n22 − 1) , λc

(9)

where l = AB 0 is the projection of beam path in OD onto the ﬂat surface of the disk (see Fig. 2), u2n = ωr is the medium linear velocity along the light beam trajectory. In reality, the IP shift can be increased due to the refraction angle ϑ˜2 differing from the one calculated by Snell’s law (ϑ2 ). As the trajectory curvature is slight, we consider that a beam propagates in the plane POY (see Fig. 2). The projection of a wave vector on the axis P can be ﬁnd from the solution of 2 : the dispersion equation, supposing β2y = 0 and β2z À β2z ´ p ω0 ³ k2pz = −κ2 β2z + cos2 ϑ0 + κ2 , (10) c where ϑ0 — the angle of refraction. The refraction angle in a moving medium tgϑ˜2 = −κ2 β2z

sin ϑ0 √ . + cos2 ϑ0 + κ2

(11)

An increase of the optical path length in OD is equal to the difference between the equivalent optical path le in a rotation medium and the one in a static medium (l0e ): µ ¶ 1 1 ˜ ∆le = le − l0e = 4dn2 − . (12) cos ϑ˜2 cos ϑ2 If OD is stationary, a beam passes the path in air. The increase of the optical path length is equal to µ ¶ 1 ˜ . (13) ∆le = ∆le 1 − n2 This value determine the additional difference of the beam path, for its passing through the OD once in one direction, due to a deviation from Snell’s law. The resulting value of the additional shift due to this effect with account of all passes of two beams for all directions and adjustments in our experiment is equal to ∆le ∆S = 16 . (14) λ Also, we can estimate the value of the additional IP shift as a result of the curving of a beam trajectory in a rotating medium. However for our scheme the 196

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opposite beams have curved trajectories, so the resulting value of the IP shift will tend to zero. The total shift due to the light dragging effect and the deviation from Snell’s law, is (15) ∆Σ = ∆ + ∆ S . The expected shift was calculated with the following interferometer parameters used in our experiment: l = 0.0892 m, r = 0.024 m, ω ∼ = 25.26 Hz, λ = 0.632991 µm, n2 = 1.48, ϑ0 = 62◦ : ∆ = 0.0054 of a bandwidth, ∆S = 0.0013 of a bandwidth,

(16)

∆Σ = 0.0067 of a bandwidth. Therefore, in the experiment it should be measured ∆Σ = 0.0067 of a bandwidth. Registration of Spatial Effect of Light Dragging in a Rotating Medium. In the experiment the gas-atomic stabilised laser LGN-302, operating in continuous mode, was as a light source. The optical disk was a standard plate of a diameter 120 mm and thickness of 30 mm and was made of LK5 grade glass. One edge surface of a disk as a whole and a part of its another edge surface (a circle with a diameter of 80 mm) were mirror-coated to provide the reﬂection coefﬁcient ρ = 0.9. In the experiment we used the asynchronous three-phase motor. The average period of the OD rotation is T¯ = 0.0396 s which corresponds to the rotation frequency ν = 25.26 Hz. Deviation of the rotation speed was within ±1 %. To decrease the inﬂuence of vibrations, the optical system and the motor with OD were placed on different platforms and tables (Fig. 1). When we used the ﬁrst adjustment and the counterclockwise direction of motor rotation, interference fringes inclined to the right and were reset to the original position (Fig. 3, a). When the OD was slowly rotating, it was observed that three bands passed across the photodetector aperture in the forward and reverse directions. Between the two motions the stop point was observed. The photodetector was set in such a way that in the stop position its intensity value was between minimum and maximum of the band (Fig. 3). The main scale for the experiment was 1 ms per a division so that points 1–7 could be accommodated on the oscillograph screen. The readings tkj , where k is a number of the point, were taken from oscillograms. Thereupon the values were calculated, using δtj = (t5j − t3j )/2,

j = 1, 25;

(17)

∆tj = (t3j − t2j + t6j − t5j )/2,

(18)

where δtj corresponds to the time distance value between the IP stop point and the nearest interference band and ∆tj corresponds to the time width of the interference fringe. VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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After this, we processed the oscillograms, using the clockwise direction of motor rotation, in the same way (Fig. 3, b). The next adjustment was achieved with inclining interference fringes to the left. Two directions of motor rotation were also used for the second adjustment. In the experiment we detected time coordinates of the points tk while IP moving, so it is necessary to convert them to the spatial ones. Processing the Experimental Data. The relation ∆t and δt determines the relative position of an interference fringe as fractions of the time bandwidth. It was noticed that beams in a screen moved in ellipses. An ellipse arc length is determined using the elliptic integral of the second kind. We can change the values ∆t and δt for the ellipse arc lengths δL and ∆LΣ , respectively, expressed in radians:

δL = π

δt , T

∆LΣ = δL + ∆L/2 = π

δt + ∆t , T

(19)

Fig. 3. The dependence of relative voltage of the photodetector PD on time when the ﬁrst adjustment for the right (aa ) and left (bb ) directions of rotation was used 198

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Zϕ5 q 1 − sin2 α sin ϕdϕ = aE(ϕ5 \α),

δL = a

(20)

0

Zϕ6 q 1 − sin2 α sin ϕ dϕ = aE(ϕ6 \α),

∆LΣ = a

(21)

0

where sin2 α = 1 − b2 /a2 , a, b — major and minor ellipse half-axes, T — period of IP vibrations, E(ϕ5 \α), E(ϕ6 \α) — elliptic integral of the second kind, ϕ — parameter of the elliptic integral, ϕ5 , ϕ6 — parameters referred to the points 5 and 6. Substituting the values ∆t and δt, which were measured in the experiment, into (16), we can get ∆L and δLΣ . It follows from (16)–(18) that a = 1 because the values ∆t and δt are normalized to the period T . Then, using (17), (18), relation b/a, taken from the experiment, and tables of elliptic integrals of the second kind, we can determine ϕ5 and ϕ6 . The relative position of a band from the point 4 is determined as

∆=

δy 1 − cos ϕ5 = , ∆y 2(cos ϕ5 − cos ϕ6 )

(22)

where δy is the spatial position of a band from the point 4 and ∆y is the spatial bandwidth δy = b(1 − cos ϕ5 ), ∆y = 2b(1 − cos ϕ6 − δy). For two directions of rotation and two adjustments of interfering beams we can obtain the resulting measured value of IP shift: ∆Σ = (∆1 − ∆2 ) − (∆3 − ∆4 ).

(23)

Let us notice that calculating the IP shift is carried out for the values, which are normalized to a rotation period and interference band width. Thus the results of the calculation don’t depend on period vibrations and bandwidth for one measurement or another. For the series of the experimental data ∆i the conﬁdence intervals were calculated with the conﬁdence probability p = 0.9. Then the resulting value of shift was equal to = 0.0076 ± 0.0030. ∆Exp Σ

(24)

The theoretical magnitude ∆Σ = 0.0067 appears in the conﬁdence interval (21), = 0.0076 is about 13 %. moreover the relative error for the average value ∆Exp Σ Conclution. The results of theoretical calculations of the expected IP shift for the used parameters in the experiment are in a good agreement with the experimental results of the IP shift. It should be noted that the rotation velocity in the experiment was not high. But the spatial light dragging effect had an inﬂuence on the IP shift. VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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Therefore, electromagnetic wave dragging by a rotating medium should be taken into account for using non-relativistic velocities of a medium, and the effect may have an impact on the results of different measurement procedures. In order to make the ﬁnal conclusion that the experimental results are in agreement with predictions of the relativity, it is necessary to increase the experiment accuracy. It may be achieved by using a more stable motor with the higher frequency of rotation and more sophisticated system of transforming the signal from a photodetector. This work was supported by Grants Council of the President of the Russian Federation (grant No MD-170.2003.08).

REFERENCES 1. Fizeau, D’H. // 1859. Ann. de Chimie et de Phys. 57, p. 385. 2. Gladyshev, V.O. // 1993. JETP Lett. 58, pp. 569–572. 3. Bolotovsky, B.M., Stolyarov, S.N. (in Russian). // 1988. Uspekhi Fiz. Nauk 156, No 1, pp. 153–176. Gladyshev Vladimir (b. 1966) graduated from Bauman Moscow Higher Technical School in 1989. D. Sc. (Phys.-Math.), Prof. of Physics Department of Bauman Moscow Technical University. The main ﬁeld of scientiﬁc interests is electrodynamics of moving medium and the theory of relativity effects. Author of 80 publications, including 2 monographs, in the ﬁeld of theoretical physics. Gladysheva Tatyana Mikhailovna (b. 1966) Ph.D. (Eng.), assoc. professor of Physics Department of Bauman Moscow State Technical University. Author of 25 scientiﬁc publications in the ﬁeld of optical interferometry. The main ﬁeld of scientiﬁc interests is the precise optical interferometry, the moving medium optics. Morozov Andrei (b. 1959) graduated from Bauman Moscow Higher Technical School in 1981. Dr. Sc. (Phys.-Math.), Prof. and Head of Physics Department of Bauman Moscow Technical University, General Director of Applied Physics Center of Bauman Moscow Technical University. Main Editor of series “Natural Sciences” of journal “Vestnik MGTU”. Chairman of Organizing Committee Conference “Irreversible processes in Nature and Science”. The main ﬁeld of scientiﬁc interests is precision measurements and physical kinetics. Author of 150 publications and inventions, including 4 monographs. Zubarev Vyatcheslav (b. 1934) graduated from Bauman Moscow Higher Technical School. Ph. D. (Eng.), Assoc. Prof. of “Optoelectronic devices for Scientiﬁc Research” Department of BMSTU. Assoc. professor of “Optical and optoelectronic systems” Department; assoc professor of “Optoelectronic devices for Scientiﬁc Research” Department of the Bauman Moscow State Technical University. Author of about 60 publications, 6 inventions and patents and 4 monographs. The main ﬁeld of scientiﬁc interests is interferometers, spectral and polarizable instruments.

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M.C. Duffy (Liverpool University, Sunderland University), V.O. Gladyshev, A.N. Morozov (Bauman Moscow State Technical University), P. Rowlands (Department of Physics, University of Liverpool) PHYSICAL INTERPRETATIONS OF RELATIVITY THEORY (On the materials of International Scientiﬁc Conference in Bauman Moscow State Technical University, 30th June – 3rd July, 2003)

The international scientiﬁc conference “Physical Interpretations of Relativity Theory” (PIRT) was held in Moscow from 30th June to 3rd July, 2003. A program of the conference included more than 80 reports of representatives of leading scientiﬁc schools of Russia and other countries. The presented papers dealt with the following major themes: • Cosmology, Gravitation and Space-Time Structure. • Time, Reference Frames and the Fundamentals of Relativity. • Nature and Models of the Physical Vacuum. • Formal Structures and Physical Interpretations of Relativity Theory. • Experimental Aspects of Relativity. • The Poincare-Lorentz and the Einstein-Minkowski Expositions of the Relativity Principle. • The Relativistic World Ether: The Ether Geometrized; Dirac’s Ether and Ether Formulations of Relativity. • Analogues of Relativity and Quantum Mechanics. The Vortex Sponge. • Historical and Philosophical Aspects of Relativity. The main problems, which were discussed in the meeting, were theoretical and experimental examinations of physical theories, being based on Relativity principles, and also on interpretation of Relativity and its consequences. The considerable progress, linked with development and practical applications of the Relativity, was noted by participants. It was also discussed that at present the existing experimental grounds do not give a synonymous answer to some principle questions, including the question on existence of gravitational radiation, which was predicted by A. Einstein in his work [1], devoted to the solution of the equations of General Relativity and the calculation of gravitational radiation power. During past decades it was proposed a number of theories of gravity alternative to the General Relativity, and more general theories such as supergravity theory and theory of superstrings, multidimensional gravity theories, which in parameterized VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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post-Newtonian formalism give the same predictions for results of such classical experiments as veriﬁcation of the equivalence principle for massive bodies, comparison of indications of moving electromagnetic clocks, deviation of light rays, signal delay, perihelion precession, time variation of gravitational constant, Lense and Thirring effect etc. In this connection progress in Relativity development is often linked with newgeneration experiments — detection of gravitational waves (GW). In approximation of a weak ﬁeld, when the metric gµν slightly differs from Minkowski’s metric ηµν , A. Einstein obtained the equation ¤h0µν = 0, which contains a solution for a plane monochromatic gravitational wave in vacuum. It is an important circumstance that the equation and also quadruple formulae for power of gravitational radiation are derived in linear approximation, where as the powerful gravitational radiation can only arise in the ﬁeld of strong nonlinearity when it is needed to consider together the moving bodies and gravitation ﬁeld. In this connection the results of experimental works of T.M. Taylor, J.M. Weisberg and others [2, 3] give conﬁdence. In these works the investigation of the delay effect for a period of the binary stellar system PSR 1913+16 due to losses of energy in gravitational radiation was carried out. Hence, investigators consider that a direct test of GW detection should become its direction with ground-based or space gravitational antennas, most likely, with interferometers. Using the interferometer with coherent optical pumping for gravitation waves was proposed at ﬁrst in a work of the participant of the conference M.E. Gertsenshtein (M.V. Lomonosov Moscow State University) and V.N. Pustovoit [4]. An analysis of the supposed methods of GW detection by different authors leads to the conclusion that just the interferometer detection of gravitation radiation is promising [5]. Moreover, it leads to the conclusion that up-to-date test of Relativity and alternative theories is possible for only limiting acceptable accuracies of measurements, demanding to widely apply laws of general and applied physics. As a veriﬁcation is the fact that at present the success in implementing the similar ground-based GW experiment is achieved in Californian and Massachusetts Technological Universities, having the present-day technical base and high-qualiﬁcation specialists. On the whole, we can conclude that recent achievements in the ﬁeld of gravity, astrophysics, laser physics, computing, cosmonautics have led to possibility to create the ground-based and spacecraft-borne GW observatories, the global satellitenavigation systems [6–11]. ”High-tech” technological level, realized in the projects, allows one to fulﬁll measuring physical parameters such as distance, time, energy, impulse with a precision close to standard quantum limits [12], it is a needed condition for testing modern theories of gravity (report of V.N. Melnikov, Russian Gravity Society, Moscow), multi-dimensional unique theories (report of Yu.V. Vladimirov, Lomonosov Moscow State University), cosmological models of global evolution of the Universe (report of S.V. Chervon, Ulyanovsk State University).

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Operating the GW observatories and satellite systems on the level of precision demands the high stability of parameters, connection to world-wide time service and taking into account the motion of celestial bodies, activity of the Sun, seismic activity of the Earth, overfalls of temperature, pressure, electrical and magnetic ﬁelds, inﬂuence of cosmic rays etc. [13, 14]. The creation of the systems promoted developing new methods of limiting measurements and stimulated originating metrological procedures, which would more fully represent possibilities of modern devices. Now a demand has arisen for a detail description of electromagnetic processes with account of effects of moving media optics [15], for a precise space-time description of all elements of measuring channel [16], including astrophysical sources of radiation [17], for developing new mathematical methods of picking out and processing signals, for creating numerical methods. In the report of C.F. Levin (Moscow Institute of Expert Appraisal and Tests of State Standards of Russia) it was shown that experimental testing the Relativity principles requests to use the appropriate methods of mathematical processing of measurement results, when not only the estimation of the measured magnitude is used but also its distribution. New technological level requires to take into account dynamic features of all elements of measurement systems, including motion parameters of individual optical elements, effect of light pressure on the elements [18], transformation of ﬁne specter structure of a radiation source due to relative motion of the source and the detector [19], inﬂuence of medium of propagation of electromagnetic wave on its parameters [20, 21]. Here we can take into consideration the deep relation in a description of particles or bodies, free moving in curved space-time continuum, and light quanta, whose motion equations are the equations of geodesic lines: j k d 2 xi i dx dx + Γ = 0, jk dψ 2 dψ dψ

where Γijk are the Christoffel’s symbols of the second rank. The analogy between the descriptions allows us to simulate an inﬂuence of gravitational ﬁeld on moving macroscopic bodies by including optical nonhomogeneity, which distorts ray trajectories. Let us notice that the description of optical experiments, being accompanied, for example, with propagation of light beams in an interferometer, is usually made on the assumption that the gravitation ﬁeld of the Earth is absent. Hence, in the case, light also propagates along geodesic lines in a gravitation ﬁeld of a rotating mass. Therefore, applying a model of a free-falling interferometer is but an approximation to the reality. Together with that in a real interferometer for GW detection the mirrors represent a pendulum, being in the ﬁeld of light wave pressure. The motion equations have a form N X ˆ i ; ω0i \xi ) = 1 G(β Fni (t), Mi n=1

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2 1 0 ˆ i ; ω0i \xi ) = d xi + 2βi d xi + ω 2 d xi is a linear operator, acting on a where G(β 0i dt2 dt1 dt0 coordinate xi ; index i corresponds to the equation for the mirror Si ; βi is a damping factor; ω0i is a fundamental frequency of a test body with the mirror Si mass Mi , Fni (t) represents external actions, N is a natural number. The function Fni (t) contains different actions, including light pressure on mirrors and GW perturbation. Evidently, in the reality the situation is more complex, as we should take into account the non-homogeneous distribution of dielectric and magnetic features of light propagation media that reﬂects the real technology of optics manufacturing and inﬂuence of temperature and other inherent factors. And at last, in a general case, we should take into consideration that the velocity of motion of the medium, which can be characterized with a three-dimensional velocity ﬁeld, varying in time, inﬂuences a shift of the electromagnetic wave plane due to phenomena, which are similar to the Fizeau’s effect. The factors, taken separately, can lead to distortion of lights rays, and more possible for its combined inﬂuence. Therefore, by describing electromagnetic radiation in tasks of gravitation radiation detection, location and navigation, it is needed to use solutions of the dispersion equation for each local ﬁeld of a moving medium. In the case, projections of wave vectors ~ki for refracted and reﬂected waves in a plane case with tangential break of a velocity on boundaries between two ﬁelds for i-ﬁelds of the medium will be expressed as ´ ³ 1/2 2 ~ ~ γ (β − β ) 1 − d β β + κ i i it ± Qi in I1 , (kin )1,2 = c 1 − β 2 − κi γi2 (β − βin )2

~k0t = ~k1,1t = ~k2,1t = ~k2,2t = I~t , ω0 − ~k0~v = ω1,1 − ~k1,1~v = ω2,1 − ~k2,1~v = ω2,2 − ~k2,2~v = I1 , ui β = v/c, κi = εi µi − 1, βi = , c h i ¡ ¢ 2 2 2 2 2 2 Qi = 1 + κi γi 1 − βin − d 1 − β − κi γi (β − βin ) − h i ¢ ¡ −κi γ 2 d~β~it 2 (1 − ββin ) − 1 − β 2 d~β~it .

γi−2 = 1 − βi2 ,

d~ = −cI~t /I1 ,

i

The invariant I~t reﬂects an equivalence of the tangential components of wave vectors for all waves, and I1 reﬂects an equivalence of frequencies for all waves, conjugated in the reference frame, where the boundary of the medium rests. The magnitudes ~ui , εi , and µi are velocities, dielectric and magnetic permittivities, respectively; ω0 , ωi are frequencies of electromagnetic waves in vacuum and in the i-th medium ﬁeld, respectively. Here, a model with a plane boundary between media, which moves with the velocity ~v , perpendicularly to the boundary, is considered. Evidently, using the given approach is justiﬁed, if the wave length λ is much more than a distinctive size of non-homogeneity of the medium a, i.e. λ À a.

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But the main parameters, inﬂuencing the additional accumulation of phase, are the medium velocity and the velocity of a boundary between media. It should be noted that even a hardly detected velocity of the boundary can lead to the considerable contribution to the wave phase difference. For example, after n reﬂecting cycles the ray, having passed in Fabry-Perot interferometer, in which mirrors oscillate with velocities of β1 , β2 under the action of GW splash, will have a frequency µ ¶ 1 + β 1 1 + β2 n ω n = ω0 . 1 − β 1 1 − β2 The additional phase difference, appearing due to a frequency shift of an electromagnetic wave after n reﬂections in the limit β1 ∼ = β2 = β ¿ 1, is equal to ∆ϕω = 2 (n + 1) ω0 βt, where t = nt0 ; t0 is time for one passage of a ray along the resonator length. It is clear that the value can have an order of the phase difference, stipulated by the sought shift for mirrors ∆ϕx = 2nk0 ∆x, where k0 = 2π/λ0 . The arguments lead to the conclusion that Relativity development is possible with the composite approach to considering its consequences which was reﬂected in some reports. For example, in the Marmet’s report an analysis of Radio-Doppler and ranging data from distant Pioneer 10 and 11 spacecrafts indicated an apparent anomalous acceleration. And it is needed to take into account the availability of dust nearby satellites trajectories, when accelerations are calculated. Namely, the composite approach required for the exact prediction of the experimental results in the interferometers of Michelson — Morley, Fizeau, Sagnac and their modern analogues in the relativistic approximation. Together with this, the solutions of the dispersion equation should be tested for a general case of 3-dimensional ﬁeld of medium velocities and with strong nonhomogeneities, appearing, for example, in a ﬁeld of considerable light absorption. These works are carried out at Physics Department of BMSTU under support of the Council of the Russian President’s grants and collaboration with investigations of Liverpool and Sunderland Universities of Great Britain. Thus, the dispersion equation in the considered tasks with high-precision measurements, despite the additional difﬁculties, will allow us to take into account non-linear effects, which can decidedly further progress of the Relativity. Moreover, the analysis suggests that the non-linear effects of interactions between electromagnetic radiation and moving media can considerably inﬂuence a solution of the tasks for deﬁning radiation parameters on both cosmological and microcosmic scales. In the report of V.S. Gorelik (Lebedev Physics Institute of Russian Academy of Sciences, Moscow) regularities of an interaction between material media and electromagnetic waves, for example, in virtual models of crystal lattices of dielectrics and semiconductors were analyzed. Dynamics of elementary excitations for the system of “a medium — an electromagnetic ﬁeld”-polaritons was investigated; and conditions, under which a group velocity of polaritons is small in limit, that is the light “stops” in the material medium, were found. Features of a “crystal” model of physical vacuum were considered. VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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As is known, forms p of dependences of a dielectric permittivity ε (ω) and a refractive index n = ε (ω) can be found from the motion equation for a charged oscillator in the electromagnetic wave ﬁeld s ωl2 − ω 2 n(ω) = , ω02 − ω 2 where ωl2 = ω02 + ωp2 ; ω0 is a fundamental frequency of a charged oscillator; ωp is a plasma frequency. It is easily to notice that for high frequencies (ω À ω0 ) the refractive index tends to unity, i.e. the light phase velocity tends to c0 . On the contrary, for low frequencies (ω → 0) the following takes place:

ε(0) =

ωl2 ω02

and

n(0) =

ωl . ω0

If ω = ωl , the refractive index and dielectric permittivity are equal to zero: ε (ωl ) = 0. The result leads to a conclusion on possibility of existence of longitudinal electromagnetic waves with a frequency ω = ωl . The frequency of a longitudinal electromagnetic wave for the considered model of substance does not depend on a wave vector ~k, i.e. on a wave length. And on the contrary, if ω ∼ = ω0 , the refractive index and the dielectric permittivity sharply increase and considerably differ from unity. As an example, we can consider far infra-red part of the spectrum, which corresponds to low frequencies of polariton waves. In the case, the refractive index is ω0 √ more than unity, n = ε0 = > 1. Therefore, phase and group velocities of ωl propagation of millimeter electromagnetic waves in a material medium can be essentially (by some orders) less than the light velocity in vacuum. Accordingly, the dielectric constant ε0 can be substantially more than unity. The especial situation takes place, when a fundamental frequency of vibration for Lorentz’s oscillators becomes very low. The dielectric constant highly increases: for some substances its magnitude achieves a value of 104 . In this connection we can notice the experimentally established fact of inﬂuence of an instellar medium motion on a velocity of electromagnetic wave propagation. In a model of the expanding Universe the radial propagation of an electromagnetic wave in IRF of an observer with account of a medium motion effect will be described with the equation: Zt0

Zr1 dt = R(t)

t1

1 dr √ . βe (r, t) 1 − kr 2 0

Here t1 , t0 are moments of emission and detection, respectively, counted from the singular state; R (t) is the cosmological scale factor; k is a spatial curvature; r1 is the dimensionless distance to space source of radiation in the Earth IRF. A sign before the integral in the right part is taken as positive which corresponds to a period 206

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of expanding the Universe, βe = c/c0 , c is a group velocity of an electromagnetic wave, c0 is the light velocity in vacuum. As cosmological extension can delay with time, so the group velocity of light depends on time t. Moreover, the light velocity in the i-th layer of a medium along the radiation direction is deﬁned by features of the instellar medium and velocity of its motion in the local ﬁeld of space, where a wave propagates: ci = ω0 /kin . Thus, by using the results of observation astronomy for the instellar medium distribution along a beam of observation, time of light beam propagation from distant parts of the Universe to an observer can be calculated. Fields in the Universe with refractive indices, which are more than unity, essentially inﬂuence the velocity and, accordingly, time of electromagnetic signal propagation. If such ﬁelds are very distant ﬁelds of the Universe, estimation of light propagation time and Universe dimensions can strongly depend on the velocity law of medium motion along a ray. For a general case of 3-dimensional medium motion there are effects of longitudinal dragging and distortion of an electromagnetic wave trajectory, which are analogous to distortion in non-homogeneous gravitation ﬁelds. Observational estimations of distortion, magnitudes for electromagnetic waves with account of a medium motion velocity in the distant ﬁelds of the Universe are a basis for new testing of the expanding space hypothesis. Similar effects should be expected in the tasks of navigation, including space navigation and location. In this connection the works, in which attempts of experimental detection of non-local features of physical interactions are discussed, can have an especial meaning (the report of S.M. Korotaev, Geo-electromagnetic Research Institute of Russian Academy, Troitsk). A part of reports was devoted to methodical aspects of the Relativity. In the report of G. Keswani (New Delhi, India) the historical aspects of the experimental basis — Fizeau’s effect and the stellar aberration phenomenon — were discussed. In the report of S. Hajra (Calcutta, India) it was shown that electrodynamics equations contain features of space-time continuum, which in an explicit form can be written as the Lorentz’s transformations, and recommendations on training the question in electrodynamics course were made. A deep analysis of interpretations of the Poincare-Lorentz’s relativity principles was carried out in the work of M.C. Duffy (Sunderland and Liverpool Universities, Great Britain). The reports of P. Rowlands (Liverpool University, Great Britain) and E.Trell (Linkoping, Sweden) are devoted to profound links between Relativity and Physics of Elementary Particles and also are examples of researching physical interpretations of observational features of space and time. Despite a wide spectrum of the presented reports, all participants were united with one generic task of searching the formalism which would be adequate to physical reality, and interpretations of observable phenomena and properties of spacetime continuum corresponding to it.

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In his speech, V.N. Melnikov (the president of Russian Gravity Society) noticed that the Relativity is not only a powerful tool for researchers, but it has become a base for engineering calculations long ago and the fact should be taken into account in training specialists. Durability of the educational base is nonseparably linked with breadth of natural sciences outlook, therefore, engineering education should be based on Gravity Theory, Astrophysics, Cosmology and Physics of Elementary Particles. Together with this, a solution of up-to-date experimental tasks, as time history of modern sciences shows, is possible only where polytechnic base is created, and it allows one to carry out new precise experiments, using the composite approach, by applying nowadays technologies and the accumulated world experience. The participants of the conference concluded, by noticing that BMSTU was the center of engineering culture more than 170 years, that the University was chosen as the right place to hold the conference. All information, concerning the conference, is on the web-sites of Physics Department of BMSTU [22] and Sunderland University [23].

REFERENCES 1. Einstein, A. Naherungsweise Integration der Fildgleichungen der Gravitation. – Berlin: Preuss. Akad. Wiss., 1916. – P. 688–696. 2. Taylor, J.H., Weisberg, J.M. A New Test of General Relativity: Gravitational Radiation and the Binary Pulsar PSR 1913+16 // Astrophysical Journal. – 1982. – V. 253, No 2, Pt. 1. – P. 908–920. 3. Taylor, J.H. Gravitational Radiation and the Binary Pulsar//Proc. 2-th Marcel Gross. Meeting on Gen. Relativity, 1979, – Amsterdam e.a., – 1982. – P. 15–19. 4. Gertsenshtein, M.Ye., Pustovoit, V.I. On the Question of Low Frequency Gravitational Waves Detection // JETP (Rus.). – 1962. – V. 43. – P. 605–627. 5. Gladyshev, V.O., Morozov, A.N. Gravitational Waves Classiﬁcation on Method Gravitation Radiation Detection (in Russian). // Measurement Technique – 2000. – V. 9. – P. 21–25. 6. Giazotto, A. Interferometric Detection of Gravitational Waves//Physics Reports. – 1989. – V. 182, No 6. – P. 365–424. 7. Rudenko, V.N. Prospects of Gravitational Astronomy (in Russian). //Einsteinovski sbornik, 1986–1990. – M.:Nauka, 1990. – P. 351–374. 8. Bradaskia, C. et al. A VIRGO Project. A Wide Band Antenna for Gravitational Wave Detection // Nucl. Instr. and Meth. Phys. Res. A. – 1990. – V. 289, No 3. – P. 518–525. 9. Anderson, A.J. The Space Multi-arm Interferometer and the Search for Cosmic Background Gravitational Wave Radiation (SMILE) // Proc. Int. Assoc. Geod. Symp., Vancouver, 1987. – V. 1. – P. 83–90. 10. Ashby, N., Spilker, J.J. (Jr.) Introduction to Relativistic Effects on the Global Positioning System // Global Positioning System: Theory and Applications. Washington, 1997. – P. 623–697. 11. Fomichev, A.A., Dmitriev, V.G., Kolchev, A.B. et al. Complex Inertial Satellite Navigational System on the Base of Laser Gyroscopes for Civil Aircrafts (in Russian). // SPIE/RUS Bulletin. – 1995. – No 1(5). – P. 28–31.

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12. Braginski, V.B. Resolution in Macroscopic Measurements: Achievements and Prospective (in Russian). // Uspekhi ﬁzicheskih fauk. – 1988. – V. 156, No. 1. – . 93–115. 13. Brillet, A., Damour, T., Tourrenc, Ph. Introduction to Gravitational Research // Annales de Physique (Fr). – 1985. – V. 10, No 3. – P. 210–218. 14. Brillet, A. Interferometric Gravitational Wave Antennae// Annales de Physique (Fr). – 1985. – V. 10, No 3. – P. 219–226. 15. Bolotovsky, B.M., Stolyarov, S.N. Reﬂection from Moving Mirror (in Russian). // Uspekhi ﬁzicheskikh nauk. – 1989. – V. 159, No. 1. – P. 155–180. 16. Izmailov, G.N. et al. Stable Interferometer for Precise Physical Experiments (in Russian) // Tekhnicheskaya ﬁzika. – 1987. – V. 57, No 6. – P. 1194–1197. 17. Lipunov, V.M., Postnov, K.A., Prokhorov, M.E. The Sources of Gravitational Waves with Continuous and Discrete Spectra // Astron Astrophysics. – 1987. – V. 176. – P. L1–L4. 18. Tourrenc, P., Deruelle, N. Effects of the Time Delays in a Nonlinear Pendular FabryPerot // Annales de Physique (Fr). – 1985. – V. 10. – P. 241–252. 19. Zagorodnov, O.G., Fainberg, Ya.B., Egorov, A.M. Multiplication Frequency with Plasma Collapse (in Russian). // JETP. – 1960. – V. 38, No. 1. – P. 7–9. 20. Welsh, B.Y., Vedder, P.W., Vallerga, J.V. High-resolution Sodium Absorption-line Observations of the Local Interstellar Medium. // Astrophys.J. – 1990. – V. 358. – P. 473–484. 21. Kimata Fumiaki, Mannodzi Nobutaka // Kisho Kenkyu noto. – 1998. – No 192. – P. 49–59. 22. http://fn.bmstu.ru/phys/nov/konf/pirt/pirt main.html. 23. http://www.cet.sunderland.ac.uk/webedit/allweb/news/ Philosophy of Science/PIRFORM.htm.

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RADIO-PHYSICS & RADIOLOCATION B. Rozanov1 , S. Solomonov2 , A. Zrazhevsky3 , V. Nagnibeda4 , Ye. Kropotkina2 , S. Rozanov2 , N. Zharkova1 , T. Lebedjuk∗ , I. Fetisov1 , M. Loukicheva4 1 Bauman Moscow State Technical University (BMSTU), 2 Lebedev Physics Institute of the Russian Academy of Science (LPI), 3 Institute of Radio Engineering and Electronics of the Russian Academy of Science, (IRE) 4 Saint-Petersburg State University (SPbSU)

THE ATMOSPHERIC AND SOLAR INVESTIGATIONS AT MILLIMETER WAVES In three last decades the short millimeter wave range was mastered for solution of both scientiﬁc and practical problems. Investigations of the Sun and the Earth’s atmosphere by means of special unique radio astronomical and radio physical instruments played a considerable part in the progress. Some results of the investigations obtained by joining efforts and experience of researches from several academic institutes and universities are presented in the paper. Results of the stratospheric and mesospheric ozone observations at frequency of 142 GHz at the Lebedev Physics Institute of the Russian Academy of Sciences are considered in the ﬁrst part. It is shown that millimeter-wave radiometry is very effective techniques for investigation of dynamical and chemical processes in the atmosphere affecting depletion of the ozone layer. Results of experimental investigations of refraction, ﬂuctuations of vertical and horizontal arrival angles and intensity of the received waves in the 2. . . 3 mm waveband for near-surface communication lines about 10 km long are presented in the second part. The data was obtained at the Bauman Moscow State Technical University (BMSTU) and Institute of Radio Engineering and Electronics of the Russian Academy of Sciences as the result of long-term series of observations at various meteorological conditions. The third part of the paper is devoted to analysis of results of solar investigations at wavelengths of 2.2 and 3.3 mm at the radio telescope RT-7.5 in Moscow region. The research was done by BMSTU and SaintPetersburg State University (SPbSU).

1. MILLIMETER-WAVE MONITORING THE ATMOSPHERIC OZONE Introduction. Radio physical methods of the Earth’s atmosphere remote sensing based on measurements of thermal emission of atmospheric gases at millimeter (MM) and submillimeter (subMM) waves opened new possibilities for studying the atmosphere and its ozone layer [1–3]. Both ﬁrst investigations of the atmospheric subMM emission from balloons and satellites [1] and initial ground-based research of the ozone layer over Moscow at MM waves were done at the Lebedev 210

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Physics Institute of the Russian Academy of Sciences (LPI) [4]. Now the measurements of the vertical ozone distribution (VOD) at 2-mm transparency window of the atmosphere are carried out regularly. Some problems of remote sensing the Earth’s ozonosphere at microwaves, design of the ozone spectrometer operating at 142 GHz and main results of the MM-wave monitoring the ozone layer at the LPI are considered in the paper. It is shown that millimeter-wave radiometry is very effective techniques for investigation of dynamical and chemical processes in the atmosphere affecting depletion of the ozone layer. Problems of radio physical investigations of the ozonosphere. The purpose of radio physical investigations of the ozone layer is to answer questions on changes in the ozonosphere and their reasons. Important problems of ground-based research of the ozone layer at MM waves are studying the VOD variations of different time scales resulted from natural reasons: atmospheric circulation, emission, chemical and thermal processes in the atmosphere and its interactions with space. Studying the human activity inﬂuence on the ozone layer is a topical problem as well. Knowledge of features of changes in ozone over high-populated mid-latitude territories such as Moscow region is of special importance because possible ozone depletion may result in injury to health of a large number of people. The impacts of the natural and human factors on the ozonosphere depend on altitude and may have various time scales (for example, diurnal, seasonal, quasibiannual changes in the atmosphere are known as well as inﬂuence of 11-year cycle of the Sun activity). So the ozone variations have to be studied for broad temporal and altitude ranges. The MM-wave ground-based remote sensing is the techniques to study the variations of different time scales at altitudes from the lower stratosphere to the mesosphere. The new data on the VOD is important to create modern model of the ozonosphere taking into account inﬂuence of the above mentioned natural and human factors on the atmospheric ozone. The well-known ozone reference models for the COSPAR international reference atmosphere [5] were based on satellite data from late 70s and early 80s when last anomalous phenomena in the ozonosphere were not known yet. Else one important practical problem is carrying out synchronous ground-based suborbit measurements of the VOD at MM waves for validation of satellite ozone data. To solve all the problems both high-sensitive MM-wave spectrometer and higheffective methods of ozone observations and data processing were done at the LPI [4, 6, 7] providing reliable and uniform in quality series of data on the ozone layer over Moscow for many years. Instruments and methods. The VOD in the stratosphere and mesosphere may be retrieved from shape of pressure-broadened rotational spectral line of ozone thermal emission. The pressure in the atmosphere rapidly falls with altitude, so the line measured at the ground level contains variously broadened contributions from ozone molecules located at different altitudes. Doppler broadening prevails over pressure broadening for altitudes above 70 km. The line contour may be expressed as an integral resulted from the radiation transfer equation. There exist special methods of the VOD retrieval from the line measured [3, 7]. The ozone line centered at 142.175 GHz is used for the spectral measurements at the LPI. The LPI MM-wave ozone spectrometer (Fig. 1) consists of low-noise heterodyne Dicker radiometer, two ﬁlter-bank spectrum analyzers and a PC with special VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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Fig. 1. General view of receiver of the LPI spectrometer

interfaces and software. The analyzers were done at Institute of Applied Physics (Nizhny Novgorod) with participation of LPI. The spectrometer was mounted in a laboratory room before the radio transparent window. The receiver antenna has Gaussian pattern with beamwidth of 1.5 degrees at – 3 dB level (the input lens diameter is 120 mm). The antenna zenith angle can be optimized depending on meteorological conditions. Typically the angle is of 60–70 degrees. Special design of the receiver quasi-optical input units and methods of calibration and observations (see details in [6]) allowed one to minimize distortions of the ozone spectra by standing waves. Planar Schottky diode of AA138V-3 type from Institute of Semiconductor Devices (Tomsk) is used in the ﬁrst mixer of the receiver [8]. The mixer has simple design, low conversion loss and noise temperature together with low required local oscillator power. It may be cooled at least down to liquid nitrogen temperature. Reliability of the mixer was conﬁrmed by its steady operation in the ozone spectrometer for many years. The spectrometer single sideband noise temperature is of about 700 K under cooling the receiver down to 80 K and of about 1500 K without cooling. The spectrometer has a broadband channel to measure total power in the signal band correspondent to background atmospheric emission. Brightness temperature of the emission is used to determine tropospheric attenuation of the ozone line.

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Typical calibration time of the receiver is 100 s, and signal integration time is 200 s. The unit spectra are stored in the controlling PC before further processing. Retrieval of the VOD from the integral spectrum obtained after averaging the unit experimental spectra for 1–2 hours is done using the method [7]. An altitude range for which the VOD may be retrieved is from 15 to 75 km for day-time and to about 95 km for night-time measurements of mesospheric ozone [4, 7, 9]. Errors of the ozone proﬁle retrieval are no more than 5–7 % for 20–50 km and no more than 20– 30 % for 15–20 and 50–75 km. For night mesospheric ozone the errors are no more than 30 % for 80–95 km. For altitudes 75–80 km where minimum of relative ozone content is located the errors may reach 50 %. The LPI ozone spectrometer was incorporated in to the global network for ozone measurements in the framework of international campaigns DYANA (1990), CRISTA/MAHRSI (1994 and 1997), SOLVE 2000 (1999-2000). Results of remote sensing the ozone layer over Moscow coincide well with data from ozonosondes and satellites [4]. Results of investigations. Regular observations of the atmospheric ozone over Moscow region at MM waves have been done at the LPI since 1987. Altitudetemporal distribution of the ozonemixing ratio was obtained. An example of the distribution for cold months of 1999–2000 is presented at the outside of the journal back cover. Ozone variations of different time scales were detected. Firstly it is necessary to note seasonal ozone variations. The variations for altitude of 35 km are shown in Fig. 2. The reasons of the seasonal ozone variations are seasonal changes in dynamical and photochemical atmospheric processes affecting the ozone [10]. The observed seasonal variations of the ozone content qualitatively correspond to the model [5]. However noticeable systematic deviation of MM-wave data from that of the model was found, especially for cold seasons. For some periods ozone content at altitudes between 25 and 45 km decreased by 30–45 % from the model values. The anomalous events usually coincided in time with some decrease of total ozone content in the atmosphere (typically by 10–15 %) measured over Moscow region by optical methods in Central Aerological Observatory (Dolgoprudny). More fast noise-like ozone variations with time scales from several days to several weeks were detected besides the seasonal ones. Magnitude of the short-term variations increases from summer to winter. For example, at 35 km the magnitude is ±(15 − 20) % around the mean value in summer and rises up to approximately ±40 % in winter. Data of aerological sounding from Russian and foreign observatories as well as satellite data (maps of potential vorticity, geopotential altitudes in the stratosphere, etc.) were analyzed to understand reasons of the fast changes in VOD. The analysis shown that the VOD variations resulted from the atmospheric dynamics, from transport of air masses with different ozone content. Correlation of the ozone content at 15–45 km and dynamical parameters of the atmosphere was revealed. For example, absolute values of the correlation coefﬁcients between the ozone content at the altitudes and potential vorticity at isoentropic surface near 30 km were up to 0.9 in cold half-years. The fact shows strong inﬂuence of the

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Fig. 2. Relative ozone content over Moscow at altitude of 35 km for 1996-2000. Monthly averaged values are shown too

atmospheric dynamics on the stratospheric ozone al altitudes lower than 45 km. Large planetary wave activity in cold months result in the above mentioned shortterm ozone variation in this time of year. It is well known that the polar vortex appears and develops in the stratosphere and mesosphere in a cold halfyear, and destruction of the ozone takes place in the central isolated part of the vortex [10]. Correlation of the stratospheric ozone content over Moscow with movements of the polar vortex was established. Changes in the VOD when the vortex approaches Moscow are shown in Fig. 3. On November 30, 1999 the stratosphere over Moscow occurred inside the vortex and was notable for high potential Fig. 3. Changes in the VOD resulted vorticity and decreased ozone content. Monitoring the ozone layer at MM from approaching the polar vortex to Moscow for 24, 26, and 30 November waves allowed the study of the ozone pro1999 by MM-wave observations ﬁle evolution over Moscow under inﬂuence of the atmospheric processes and the investigation of peculiarities of the altitude-temporal ozone distribution for every year of observations. It was found that the distribution changes noticeably from year to year as a result of variations in the atmospheric dynamics course. In particular it was discovered that the most long and deep ozone depletion (especially

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at 25–45 km where the ozone content decreased by 40–45 % in comparison to the model [5]) was in cold seasons of 1995–1996 and 1999–2000. In winter 1999–2000 the decreased ozone content was kept with short breaks almost 3 months from middle of November 1999 to middle of March 2000. These cold seasons were notable for the very deep and stable polar vortex. Diurnal variations of the mesospheric ozone were measured and studied as well [9]. Considerable increase in the mesospheric ozone at night corresponds to photochemical theory. Variations of the night ozone at altitudes of 65–75 km with periods of 2 and 4 hours were discovered [4, 9]. The variations may be connected with internal gravity waves in the atmosphere. Inﬂuence of planetary waves on the night ozone at mesopause altitudes (80– 95 km) was found, the ozone content at the altitudes being varied from 1.5 to 8 ppm (1 ppm= 10−6 ). Correlations of the ozone mixing ratio in the region of the upper atmosphere and parameters of the low and middle stratosphere (potential vorticity, geopotential altitudes, total ozone content, etc.) were determined. The correlations revealed when the polar vortex moving over Moscow region at time of large planetary wave activity. It is known [11, 12] that the processes envelope layers from the lower stratosphere to thermosphere and may affect the mesospheric ozone by change in vertical transport of atomic oxygen and hydrogen, the species involved together with ozone in to chemical reactions in the upper atmosphere. So, the ozone may be an indicator of transport both in the stratosphere up to 45 km and in the upper atmosphere. The above-mentioned difference of the VOD obtained at MM waves from the model [5] may be connected with long-term ozone decrease for almost two decades from time when the satellite measurements used in [5] were done. The ozone trend value estimated from comparison of the LPI MM-wave data with model [5] has maximum at 40 km and conforms to data of other experiments [10]. It is believed that the reason of ozone depletion at altitudes of about 40 km is ozone destruction in catalytic chemical reactions with chlorine compounds of anthropogenic origin. Conclusion. The MM-wave ozone measurements allow studying processes in the ozone layer of different time scales. The results obtained show importance of further monitoring the ozonosphere at MM waves to investigate long-term changes in the ozone layer. It is desirable to create a network of ground-based observational points in Russia and other countries of Union of Independent States for monitoring the ozonosphere at MM waves. Simultaneous microwave measurements of ozone, chlorine monoxide, nitric compounds and other minor gas species of the atmosphere are important too. The measurements are in preparation at the LPI [4]. So, remote sensing the atmosphere at MM waves gives essential contribution to investigations of the ozone layer and our knowledge about its changes it. 1

Authors are grateful to Correspondent Member of the RAS I.I.Sobel’man for support of the work, and to researchers of the LPI A.N. Lukin and V.N. Leonov for their contributions to the investigations. The work was supported by Council of “Integration” Program, by RFBR, grants No 99-02-18132, 0302-17436, and 00-05-64976, and by “Leading Scientiﬁc Schools” Program, grants No 00-15-96586 and SS-1254.2003.2. VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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2. THE SHORT MILLIMETER WAVE PROPAGATION THROUGH THE NEAR- SURFACE LAYER OF THE EARTH ATMOSPHERE Introduction. The investigations of short millimeter wave propagation through the near-surface layer of the Earth atmosphere (altitudes H = 0 . . . 100 m) are essential for use of this band for communications, radars, radio astronomy and so on. The investigations were carried out in 1970–1990 by BMSTU and Institute of Radio Engineering and Electronics of the Russian Academy of Sciences (IRE) in Dmitrov district of the Moscow region. Theoretical models and computation methods for deriving the propagation characteristics [13] are not applicable for this layer of the atmosphere because of signiﬁcant inﬂuence of the Earth surface and fast variability of statistical estimations of meteorological parameters in the layer. The wavelength range λ = 2.2 . . . 3.3 mm is optimal for studying the propagation characteristics in near-surface layer of the atmosphere (structure factor of the refraction index, its effective vertical gradient, turbulence anisotropy). At longer wavelengths it is difﬁcult to obtain narrow antenna beam for exclusion of the Earth surface inﬂuence. At shorter wavelengths the atmospheric losses increase. The propagation characteristics deﬁned in the range of 2.2 . . . 3.3 mm can be applied for any longer wavelength because at λ > 2 mm temperature dependence of the refraction index is negligible [14]. Instruments and Methods. The measurements of the refraction, intensity, and arrival angle ﬂuctuations were carried out at λ = 2.2 . . . 3.3 mm using two communication lines of 7.7 and 14 km long. The radio beam propagated through the atmosphere at the altitude of 20 . . . 50 m above bushes and arable terrain. Suppression of the surface reﬂections was realized by high angular resolution of the receiving antenna RT-7.5 of the BMSTU Radio telescope with the dish diameter of 7.8 m, having beamwidth Θ0.5 = 1.50 . . . 2.40 at λ = 2.2 . . . 3.3 mm and near-zone of approximately 50 km long [15]. The transmitter was a BW-oscillator of 10−2 W power with antenna dish of 0.6 m diameter. During the observations we measured simultaneously vertical and horizontal arrival angles Ψv , Ψh and intensity of the received wave I. Fluctuations of the parameters were studied in the frequency interval of 0.01 . . . 1 Hz. Angle sensitivity of the system equal to 0.100 was reached at 30 dB signal to noise ratio by conical scanning of the narrow receiving antenna beam around direction of the transmitter. The observations of the ﬂuctuations were possible under conditions of both clear weather and precipitation. Observations of the ﬂuctuations. The ﬂuctuations rms value σ for the parameters I, Ψv , and Ψh as well as the mean value I of the intensity I were calculated and rms deviation of both the total arrival angle σt2 = σv2 + σh2 and the chaotic modulation depth mI = σI /I were determined for experimental records of approximately 10 min long. Two long-term observation cycles were done at λ = 2.2 mm:

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1) of the intensity and arrival angles for the 7.7 km line during 24 hours in spring (45 records each of 10 min long); 2) of the intensity and arrival angles (partly) for the 14 km line during 19 days in different seasons at day-time (76 records each 10 min long). Some conclusions were obtained from the results. The observed variations of the propagation characteristics have the following limits: — for the 24-hour cycle σh = 0.5 . . . 300 ,

σv = 0.3 . . . 500 ,

σt = 0.6 . . . 700 ,

mI = 0.5 . . . 6 %.

— for the annual cycle mI = 1 . . . 25 % . In summer the σ values were several times higher than during cold period of the year; the mean values of mI being of 18 % for June, 6 % for November and 3.5 % for months from December to April. The rms σ values were 3 . . . 5 times lower at night than at day-time. The ﬂuctuations reduced by 3 times during weak rain. Fog gave no changes in the σ. Fall of the ﬂuctuations may be observed for some time before and after the sunset during anticyclone. Anisotropy of the atmospheric turbulence was detected as well. In Fig. 4, a variations of the arrival angles at hot summer midday are shown. It is seen clearly that the vertical ﬂuctuations are 2.5 times more than horizontal ones. At night ﬂuctuations in both directions are approximately equal (Fig. 4, b). Correlation between σh and σv can be seen at Fig. 5, a. In Fig. 5, b the correlation between σt and mI (%) is presented. The results of the mI measurements provided the possibility to

Fig. 4. The visible transmitter moving during 30s time interval: a — in the daytime; b — at night VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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Fig. 5. The results simultaneous measurements of: a — σv and σh ; b — σt and mI , %; 1 — in the daytime, 2 — at night, 3 — in the daytime; — the regression line, — calculated dependence

derive the important parameter of the atmosphere: the structure constant Cn of the refraction index n of the air. The experimental dependence of the Cn on the mI is in good agreement with theoretical one [16]: σt = 1.78mI (Fig. 5, b). Investigations of the Refraction. Investigations of the refraction were done using angle data units of the radio telescope for measurements of direction corresponding to maximum intensity of the arrival wave. The rms error of a single measurement was of about 1000 . For 10 measurements the mean error decreased to 300 . Two measurement series were realized for the 14-km line. In the ﬁrst series (1979), 138 measurements including two 24-hour cycles were done at λ = 2.2 mm. In the second series (1987-1989) 13 vertical refraction measurements of 24-hour cycles were done with 30-min intervals between the data at λ = 3.3 mm. At the same period 50 measurements were made simultaneously in millimeter and optical regions. Processing of the data allowed us to make some conclusions. The 24-hour variations ∆αv of the vertical arrival angle at millimeter wave region is 2 . . . 3 times higher at warm period of the year than in winter because of increased gradient of the air humidity and correspondingly increased gradient of the refraction index. The horizontal arrival angle variations were no more than the measurement errors (300 . . . 1000 ). The upper limit of slow horizontal angle variations was about 102 times lower than for the ∆αv . The greatest refraction was correlated with sharp changes in the weather. The sharp variations of the refraction may be observed at sunrise or sunset. At the anticyclones the night refraction values were higher than the day values. A variation by 50 was observed with the top angular speed of 30 per hour. A sudden variation by 60 for 30 min was observed at the cold front after thunderstorm. Such rare phenomenon as appearance of several maxima of intensity within a 1’ interval of the 218

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vertical arrival angles was observed during several minutes. The minimum variations of the refraction were observed in windy weather under cyclonic conditions. Inﬂuence of rain and snowfall on the refraction was not detected. The correlation of mm- and optical wave refraction is high (K = 0.97) in winter and low (K = 0.4) in summer. This results from low humidity of air in winter when refraction indices for mm- and optical waves are approximately equal. The data on the refraction variations were within the limit ∆αv = 1 . . . 50 all over the year. The corresponding 24-hour variations of gradient of the air refraction index were calculated: ∆gn = 2∆αv /L = (0.4 . . . 2) · 10−7 m−1 . The result is in agreement with data of [17]. 3. INVESTIGATION OF THE SOLAR EMISSION AT MILLIMETER WAVELENGTHS WITH RT-7.5 Introduction. Studies of the Sun’s radio emission at millimeter wavelengths provides a simple diagnostics of physical conditions in the solar chromosphere and the lower part of the transition region. Such studies offer several distinct advantages over those in visible and ultraviolet light. Non-burst millimeter-wave emission is generated under conditions of local thermodynamic equilibrium (LTE), it has a thermal nature and is formed due to bremsstrahlung, other mechanisms of emission do not make any signiﬁcant contribution into mm emission. The intensity (or brightness temperature) of mm-wave emission depends linearly on temperature of chromosphere plasma and therefore is able to provide a suitable and independent diagnostics of the solar atmosphere. Solar observations with RT-7.5 were begun in 1982 in collaboration with solar radio astronomers from St.-Petersburg State University. Since 1987 regular solar observations made possible to obtain a long set of solar maps at 3.4 mm wavelength (the frequency of 85 GHz) with 2.50 spatial resolution. Two years ago the second radiometer operating at 2 mm wavelength (150 GHz) was set in RT-7.5. Solar mapping at the telescope is carried out by means of raster, radial or circular scanning of the solar disc. The reader is referred to Solovyov et al. [32] for a detailed description of the telescope observations. Solar mapping at short millimeter wavelengths has large difﬁculties due to strong inﬂuence of terrestrial atmosphere and small contrast of sources on the solar disk. Nevertheless solar maps at millimeter wavelengths has shown the existence of various spatial structures in brightness distribution on the disc — “quiet Sun”, large-scale structures and sources of the S-component connected with the active regions. “Quiet Sun” and chromosphere models. After more than 50 years of theoretical and observational research the solar chromosphere still is poorly understood. At the present time there are some standard static and dynamic chromosphere models. All these semi-empirical models VAL (Vernazza et al. [34]), FAL (Fontenla et al. [21]), dynamic models CS (Carlsson and Stein [20]) are based on the chromosphere UV and visible continua observations. Thus the millimeter continua, which are formed in LTE-condition, may be able to provide an independent test of the models. VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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Fig. 6. The observed millimeter and submillimeter brightness temperatures of the quiet Sun at disc center.

The observed millimeter and submillimeter brightness temperatures of the quiet Sun at disc center, taken from original literature, are plotted vs. wavelength in Fig. 6, including data, obtained with RT-7.5 during 15 years. For measurements of center disc brightness temperature we used radio emission of the New Moon as a standard signal. At such a period the Sun and the Moon are placed close to each other on the sky, thus the observational conditions are the same for both objects. All observational data used for the current investigation were obtained during solar cycles 19, 20 and 21 and, thus, corresponded to the chromosphere at different activity phases — from activity minimum to activity maximum. We tried to look for solar cycle variations of the quiet Sun but at the moment the dispersion of the experimental points was too large to ﬁnd such variations. We compute the spectrum of the brightness temperature for millimeter wavelengths range using two sets of model atmospheres. The ﬁrst set consists of three models FAL corresponding to different components of the quiet Sun: A (a faint cell center area), C (an average intensity area), and F (a bright area representing the network). The other set of models considered is simulation of the photosphere and chromosphere due to Carlsson and Stein (CS). They studied [21] the dynamics of acoustic waves in the solar atmosphere using a one-dimensional, non-LTE, radiation hydrodynamic code. We calculated the submillimeter and millimeter radiation emerging from the above static and dynamic models. The radiation transfer was computed under the assumption that radio emission for these wavelengths occurred due to bremsstrahlung (free-free emission). The 220

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calculated spectra of radiation for the quiet Sun FAL models A, C and F are depicted in Fig. 6 together with the observational data. Among the considered models, FAL A, corresponding to the internetwork, provides the best ﬁt to the data. But all in all, the FAL models indicate that the millimeter-wave data is not compatible with UV spectra within the context of single-component time-independent planeparallel models: the UV data require a warmer model than the microwave data (Loukitcheva et al. [26]). A combination of a cell interior (FAL A) and a network model (FAL F) with ﬁlling factor about 0.13, looks more promising. The analysis of the dynamic simulations of CS reveals that radio emission (temperature contrbution functions and brightness spectrum) at millimeter wavelengths is extremely sensitive to the dynamic processes in the chromosphere. The most striking result is that the dynamic picture of the solar internetwork chromosphere is consistent with millimeter and submillimeter brightness observations. The average spectrum obtained by averaging over spectra from all time-steps of CS dynamic simulations provides a good ﬁt to observed temporally and spatially averaged data. Millimeter observations indicate that using radio technique it is possible to extend observations of the solar oscillatory component to the heights above those previously observed in the optical and UV spectral lines. Large-scale structures. Solar millimeter-wave maps allow us to ﬁnd out socalled “belts of activity” — enhanced emission clearly distinguished between heliolatitudes of 10◦ and 50◦ in both solar hemispheres. In these belts of activity S-component sources appear from time to time, but enhanced brightness exists all time though the evident activity in optics is absent. The brightness temperature difference between equatorial and latitudinal belts is not large and is equal to about 100–200 K. Comparison of our maps with solar images in UV spectral lines (mainly, in HeII 304 MM line) shows a close similarity. Nevertheless an estimation of HeII emitted layers contribution to the brightness temperature at 3 mm wavelengths shows unimportant role of these layers (Nagnibeda and Piotrovitch, [28]). Indeed, HeII emission is generated in the regions with T ∼ 100000 K that is too high for MM wave emission. Another conspicuous type of large-scale structures is related to dark regions or depressions or low-temperature regions. These depressions are observed during many years mainly at long mm waves and as a rule they are connected with Hα ﬁlaments or channels of ﬁlaments on the solar disk. But on our solar maps at 2–3 mm wavelengths we do not practically ﬁnd the depressions connected with ﬁlaments. But on the other hand on our maps we have found many regions with low brightness, lower than the quiet Sun level. And most of them seem to be related to coronal holes. Usually the coronal holes are deﬁned as dark regions on the space images of the Sun in UV spectral lines (HeII 304, for example), or in X-ray continuum images, or (as bright regions) from the ground-based observations in the IR HeI line at 10830. Statistical investigation of low radio brightness showed that about 75 % of equatorial coronal holes are associated with such regions (Nagnibeda, [29]). It is well-known that coronal holes are connected with open magnetic structures and are clearly visible on the disk as dark regions at decimeter and long centimeter VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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(CM) wavelengths also. One of the important conclusions of that investigation is: coronal holes are not seen at short CM wavelengths. Indeed, it is widely believed that coronal-hole-associated depressions become indistinguishable at short wavelengths and at millimeter wavelengths all the more. But in literature one can ﬁnd very poor data from this domain. There are some observations by Kundu (1976) at 3.5 and 9 mm wavelengths — coronal hole as a dark region, and by Kosugi et al.(1986) at 3 and 8 mm wavelengths — no evidence at 3 mm and brightening at 8 mm. So we had very poor and contrary observations and data for mm-wave domain (Moran et al. 2001). Model calculations corroborate the absence of evidences of coronal holes in MM-wave range (Papagiannis and Baker, [30]; Borovik et al. [18]). As to our short mm-wave solar maps, ﬁrstly, we ﬁnd out many dark regions on it, and secondly, we ﬁnd out high correlation of brightness depression with X-ray dark regions as well as with HeI and HeII spectral lines depressions. Thus one can deﬁnitely afﬁrm that in many cases at short MM wavelengths there is coronalhole-associated darkening. Various possible sources of the discrepancy between observations in CM and MM-wave ranges may exist, in particular, difﬁculties of calibration of data. But we are sure that there is a physical reason for the discrepancy: at long wavelengths coronal holes emission is observed but at short wavelengths under coronal hole chromosphere emission is observed. So we need to look for the distinction of chromosphere under coronal hole and outside it in a quiet region. Indeed, there is the set of observational and theoretical results which show that chromospheric spicule structures are different in the active, quiet and coronal hole regions. In particular, spicules are longer under coronal holes than in quiet regions (Bray and Loughhead, [19]; Rabin and Moore, [31]). Sources of S -component. Studies of the Sun’s active region radio emission at millimeter wavelengths provide a simple diagnostic of physical conditions in the active solar atmosphere and the lower part of the transition region as well as in quiet Sun atmosphere. Nevertheless, there are some disadvantages in mapping the Sun in the short MM wavelength regime: the angular resolution available is relatively modest at present, it is limited by strong inﬂuence of terrestrial atmosphere. As a result, conclusions about MM emission based on data obtained by instruments with different angular resolution turn out to be contradictory. Furthermore, observational results are in conﬂict with prevailing homogeneous chromospheric models due to the fact that inhomogenities are present in chromosphere structure at MM wavelengths. Two-dimensional solar maps at millimeter wavelengths are compared with solar maps obtained at Japan heliograph Nobeyama (NoRH) at 17.6 mm wavelength (the frequency of 17 GHz). On NoRH maps obtained with high spatial resolution of 10 arc seconds a ﬁne structure of an active region can be derived. Meanwhile, spatial resolution of 2.5 arc minutes that corresponds to RT-7.5 is not high enough to resolve the active region into individual components, even if they are present. Thus, it is impossible to study spectra of sources in detail. Nevertheless some interesting spectral features were found from this investigation.

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First, for different local sources the spectral behavior of millimeter emission is different. The investigation has shown that this fact can be explained by the considering of active regions of various types: active regions with highly developed sunspot groups, with weak sunspots or without them. Second, it can be seen that in general there is an increase in ﬂux density toward short wavelengths in MM domain. The existence of ﬁne structure in sources radio emission consisting of 2 components: sunspot emission and plage emission, leads to the signiﬁcance of the resolving sunspot and plage contributions into the total ﬂux of S-component sources. It can be easily done for NoRH maps, where a bright source over a sunspot group can be resolved. It contribution into the total ﬂux can be estimated and it is considered that the remaining emission is due to plage complexes. For source radio emission at 3.4 mm wavelength contributions of different components of ﬁne structures was estimated under assumption that 3.4 mm emission of these sources occurred mainly due to emission of plages, and sunspot contribution into the total ﬂux at 3.4 mm wavelength was out of consideration. A statistical investigation of the dependence between the observed radio source ﬂux density and the size of corresponding active region was carried out. It was found that source ﬂux density at 17.6 mm is correlated with the sunspot area, as for the ﬂux at 3.4 mm, it showed a linear correlation with the total active region area, where plage area dominates. Thus, the basic contribution into radio emission at 3.4 mm wavelength is made by plages and moreover, it comes from our observations that the brightness temperature of a source over sunspot group does not differ much from the brightness temperature of the whole active region. And at 17.6 mm wavelength sunspot component dominates in the total source ﬂux. There was an attempt to place the observational results in the context provided by existing models of the solar chromosphere (Loukitcheva and Nagnibeda [25]). The distributions of electron temperature and electron concentration for sunspots and plages were considered from various chromospheric models : VAL and FAL for the upper chromosphere and transition region, sunspot models by Lites and Skumanich [24] and by Staude et al. [33]. Radiation transfer was estimated under the assumption that radio emission at MM wavelengths occurred mainly due to bremsstrahlung (for < 3 cm). From the results of these calculations it is obvious that theoretical values of brightness temperature are systematically higher than observed ones for all considered components of radio emission: the quiet Sun, sunspot emission and plage emission. Clearly, chromospheric models of different radio emission components based on the UV and optic observations and computed under assumption of hydrostatic equilibrium fail to account for the observed spectral characteristics of sources of millimeter radio emission. To obtain correspondence between calculated and observed values of brightness temperature some authors introduce a ﬁlling factor α. The estimated values of the ﬁlling factor α were found to be about 0.15. These values can serve as an estimation of relative contribution of cold and hot structures into the solar chromosphere.

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REFERENCES 1. Development of radio astronomy in the USSR (in Russian) / Ed. by A.E. Salomonovich. Moscow: ”Nauka”, 1988. 2. Spectral Investigations of Cosmic and Atmospheric Radiation (in Russian) / Ed. by A.G. Kislyakov. Gorky: IAP RAS, 1979. 3. Atmospheric Remote Sensing by Microwave Radiometry / Ed. by M.A. Janssen. New York: J.Willey& Sons, Inc., 1993. 4. Solomonov, S.V. (in Russian). // Rus. Telecommunications and Radio Engineering. – 2003. – T. 1, 2003. P. 9–25. 5. Keating, G.M., Chiou, L.S., Hsu, N.C. // Adv. Space Res. 1996. – V. 18. – No 9/10. – P. 11–58. 6. Solomonov, S.V., Rozanov, S.B., Kropotkina, E.P., Lukin A.N. (in Russian) // Radiotekhnika i elektronika. – 2000. – V. 45. – No. 12. – P. 1519–1525. 7. Gaikovich, K.P., Kropotkina, E.P., Solomonov S.V. (in Russian) // Isvestia Akad. Nauk, Phys. atmosphery and okeana. – 1999. – V. 35. – No 1. – P. 86–95. 8. Rozanov, S.B. (in Russian) // Radiotekhnika i elektronika. – 1996. – V. 41. – No. 3. – P. 362–369. 9. Solomonov, S.V., Kropotkina, E.P., Semyonov, A.I. // Bulletin of the Lebedev Physics Institute (RAS). – 2001. – No. 10. – P. 30–38. 10. Scientiﬁc Assessment of Ozone Depletion: 1998. World Meteorological Organization, Global Ozone Research and Monitoring Project. — Report No. 44, Geneva, 1999. 11. Allen, M., Lunine, J.I., Yung, Y.L. // J. Geophys. Res. – 1984. – V. 89. – No. D3. – P. 4841–4872. 12. Perminov, V.I., Kropotkina, E.P., Bakanas, V.V. et al. (in Russian) // Geomagnetism and aeronomy. – 2002. – V. 42. – No. 6. – P. 814–820. 13. Tatarsky, V. Rasprostranenie voln v turbulentnoi atmosfere (in Russian). Moscow: Nauka, 1967. P. 548. 14. Issledovanij rasprostranenij millimetrovyh voln v prizemnom sloe atmosfery (in Russian) / B. Rozanov, I. Fetisov, A. Zrazhevsky at al // Vestnik MGTU. Priborostroenie. – 1990. – No. 1. – P. 60–66. 15. Millimetrovyi radioteleskop RT-7.5 of MGTU (in Russian) / B. Rozanov // Isvestija vuzov SSSR Radioelektronika. – 1981. – T. 24. – No. 3. – P. 3–8. 16. Andreev, G., Chernaj, L. Fluctuacii puchka MMV pri rasprostranenii v turbulentnoi atmosfere zemli (in Russian) // Radiotechnika. – 1978. – V. 33. – No. 1. – P. 16–29. 17. Badulin, N., Tatarinov, V. Ovlijanii sutochnih variacii parametrov sredy na statistiku uglov refrakcii (in Russian) // Radiotekhnika i elektronika. – 1980. – V. 25. – No. 12. – P. 2498–2503. 18. Borovik, V.N., Kurbanov, M., Livshits, M.A. and Ryabov, B.I. (in Russian) // Astron. Zh., 1993, Vol. 70, 403–414 (in Russian). 19. Bray, R.J. and Loughhead, R.E. (1974): “The Solar Chromosphere”, L.,384 pp. 20. Carlsson, M. and Stein, R.F. (1995): ApJ, 440. L29. 21. Fontenla, J.M., Avrett, E.H., Loeser, R. (1993): Ap. J. 406. P. 319. 22. Kosugi, T., Ishiguro, M. and Shibasaki, K. (1986): Publ. Astron. Soc. Japan, Vol. 38, 1–11. 23. Kundu, M.R. and Liu, S.Y. (1976): Solar Phys., Vol. 49, 267–269. 24. Lites, B.W., Skumanich, A. (1982): Ap. J., 49, 293.

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25. Loukitcheva, M.A. and Nagnibeda, V.G. (1999): Proc.8-th SOHO Workshop “Plasma Dynamics and Diagnostics in the Solar Transition Region and Corona”, ESA SP-446, 451. 26. Loukitcheva, M.A., Solanki, S.K., Carlsson, M. (2003): A& A, (in press). 27. Moran, T.; Gopalswamy, N.; Dammasch, I. E.; Wilhelm, K. (2001): A & A, v. 378, p. 1037. 28. Nagnibeda, V.G. and Piotrovitch, V.V. (1994): Solar Phys., Vol. 152, 175. 29. Nagnibeda, V.G. (1997): Proc. 31-st ESLAB Symp., ESA SP-415, 353. 30. Papagiannis, M.D. and Baker K.B. (1982): Solar Phys., Vol. 79, 365. 31. Rabin, D. and Moore, R.L. (1980): ApJ., 241, 394. 32. Solovyov, G.N., Rozanov, B.A., Ivanov, V.N., Nagnibeda, V.G., Piotrovitch, V.V. (1992): in: Fedorov (ed.) “Topics in Radioelectronics and Laser System Design”. CRC Press, Boca Raton, Mayami, 167. 33. Staude, J., Kruger, A., Hildebrandt, J., Furstenberg, F. (1985): A & A., 143, 72. 34. Vernazza, J.E., Avrett, E.H., Loeser, R. (1981): Ap. J. Supp., 46, 635. B.A. Rozanov (b. 1933) graduated from Bauman Moscow Higher Technical School University in 1956 and Lomonosov Moscow State University in I960. D.Sc. (Eng.), professor of “Radio and Electronic Devices” Department of the Bauman Moscow State Technical University. Author of more than 150 publications in the ﬁeld of radio astronomy radio physics signal processing & MM-wave engineering. S.V. Solomonov (b. 1941) graduated from Lomonosov Moscow State University in 1964. Ph.D. (Phys.-Math.). Leading researcher of the Lebcdev Physics Institute of the Russian Academy of Science. Author of over 100 publications in ﬁelds of radio physics. spectroscopy, atmospheric physics. A.Yu. Zrazhevsky (b.1936) graduated from the Bauman Moscow Higher Technical School in 1959. D.Sc. (Eng.), Head of laboratory in Institute of Radio Engineering and Electronics of Russian Academy of Sciences (IRE RAS, Moscow). Author over 100 publications in ﬁelds of radio phisycs, remote sensing, MM-waves propagation in atmosphere. V.G. Nagnibeda (b. 1937) graduated from Astronomical Department of Leningrad State University in 1960. Ph.D. (Phys.-Math.), professor of astrophysics and radio astronomy in St. Petersburg State University. Author of more than 85 publications in the ﬁelds of Solar physics and astrophysics. Ye.P. Kropotkina (b. 1940) graduated from the Lenin Moscow Teachers Training Institute in 1963. Ph.D. (Phys.-Math.), senior research worker of the Lebedev Physics Institute of the Russian Academy of Sciences. Author of about 50 publications in ﬁelds of atmospheric physics spectroscopy, and radio physics. S.B. Rozanov (b. 1958) graduated from Lomonosov Moscow State University in 1980. Ph.D. (Phys.-Math.), head of Laboratory of millimeter-wave spectroscopy of the Lebedev Physics Institute of the Russian Academy of Sciences. Author of about 80 publications in ﬁelds of radio physics, spectroscopy and atmospheric physics. T.S. Lebediuk (b. 1944) graduated from the Bauman Moscow Higher Technical School in 1968. Research worker of Bauman Moscow Slate Technical University. Author of 17 publications in the ﬁelds of radio astronomy and radio physics. VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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N.A. Zharkova. (b. 1947) graduated from Moscow Institute for Mountain Engineering in 1970. Ph.D. (Eng.), assoc. professor of the Bauman Moscow State Technical University. Author of over 40 publications in ﬁelds of radio physics, low-noise receivers & MM-wave engineering. I.N. Fetisov (b. 1931) graduated from Lomonosov Moscow State University in 1955. Ph.D. (Phys.-Math.), assoc. professor of the Bauman Moscow State Technical University. Author of over 90 publications in ﬁelds of radio physics, atmospheric physics, MM-wave propagation. M.A. Loukictheva (b. 1975) graduated from Astronomical Department of St. Petersburg State University in 1992. Researcher of Astronomical Institute of St. Petersburg State University. Author of 10 publications in the ﬁelds of solar physics and helioseismology.

BMSTU Press has published the book: Information Technologies in Radio Technical Systems. – 2nd edition, revised and supplemented / V.A. Vasin, I.B. Vlasov, Yu.M. Yegorov et al.; edited by I.B. Fedorov. – M.: Izd-vo MGTU im. N.E. Baumana, 2004. – 768 p. Fundamentals of the statistic theory of radio systems are set forth. Problems of the signal detection, identiﬁcation and resolution against the noise background and problems of measuring signal parameters including those varying for the observation time are considered. An attempt is made to generalize the methodical approach to analysis and synthesis of basic information technologies for various radio technical systems: radar, satellite, radio navigational and for data transfer systems. The 2nd edition (1st in 2003) is revised and supplemented. Some chapters, devoted to information characteristics of radio technical systems and advanced data transfer systems, in particular, to cellular ones, are added. The textbook content corresponds to the course of lectures delivered by the authors in the Bauman Moscow State Technical University. For students of higher educational establishments, learning the trade “Radio Technology”. Can be useful for post-graduates, workers of research and industrial institutions, practicing in the ﬁeld of development of multi-purpose radio technical systems.

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ECOLOGY PROBLEMS N.P. Demenkov, V.A. Matveev (Bauman Moscow State Technical University)

FUZZY SYSTEMS OF ECOLOGICAL MONITORING AND CONTROL It is offered that systems of ecological monitoring and control be regarded as intellectual systems, i.e. systems based on human knowledge. The approach to designing systems of ecological monitoring and control, based on the T. Saaty method of an analytic hierarchy process (AHP) and the L. Zadeh theory of fuzzy sets, is investigated. Some investigation results are given.

Introduction. The improvement of ecological environment implies solving a large number of rather complex, nonlinear problems, difﬁcult to formalize and requiring knowledge on many ecology aspects, including the environmental monitoring. As a whole, a monitoring system should be regarded as an expert system that carries out the monitoring of the environment condition and helps people to inﬂuence it. At present, solving ecological problems is a difﬁcult task for all countries of the world. First of all, it is necessary to mention economic difﬁculties. To develop, implement and employ special ecological systems (information, research systems etc.) is very costly, therefore not every country in the world can afford it. But, if there are no economic difﬁculties, instead of them others come — difﬁculties of construction, production, etc. Data is collected in databanks while arriving from observation platforms: mobile laboratories, balloon probes, planes and helicopters, rockets, geostationary monitoring satellites or low-altitude viewing satellites. The data passes check in databanks. The current condition data of a given object is compared to its past condition data, and information on its change and a quantitative estimation of this change are transferred to the decision-making person. The appropriate control action can also be recommended. Both modern means of ecological monitoring and information-and-control systems, supporting them, are complex multifunctional multimode distributed systems. In such systems the joint processing of the data and knowledge, having complicated organization, is carried out. They should be developed on the basis of modern information technologies, which would ensure the essential rise of a level of the information and intellectual support. The informatization problems in solving ecological tasks become fundamental in connection with a widespread application of local and global computer networks. The efﬁciency of predicting the development of the ecological situation (including destruction) in this or that area, at an enterprise or facility depends on solving these problems.

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Using information technologies, based on knowledge, is a way out for the created situation. The representation of the domain knowledge and organization of its processing are provided in such systems at various hierarchical levels to increase the efﬁciency of control and decision-making. The urgency of consideration of such class systems is caused by their ability to accumulate and generalize knowledge, to develop hypotheses and forecast, to make decisions and explain them, etc. An intellectual system is understood as an aggregate of hardware and software, united by the information process and working independently or interacting with a person (group of people), capable to synthesize a goal, to make decisions on action and to ﬁnd rational ways for achieving the goal on the basis of items of information and knowledge with the available motivation [1]. Solving Multi-Criteria Optimization Tasks under Condition of Qualitative Uncertainty. While solving a number of ecological tasks, the situations occur when either there are no necessary sensors of the primary information or the existing measuring means do not provide the real-time reception of the required information or the qualitative information on the control object is only available. In such situations information technologies are required that allow getting the required control information with the help of computer processing of the qualitative (or fuzzy) information about the control object and goals. Let us consider a multi-criteria optimization problem in the formulation, when the goals, outcomes (criteria values) and alternatives (actions) are given fuzzily, but the preference relations, i.e. function of utility, are given crisply. For solving the problem it is necessary to deﬁne and maximize the utility function U for alternatives under consideration: (1) max{U (z1 , z2 , . . . , zk )}, where zi = fi (x ∈ S ⊂ Rn ), i = 1 . . . k are components of the criteria vector; x is a point in the decision space; S is a set of the allowable decisions. The basic difﬁculty, arising in solving the problem, is to derive the mathematical description of the utility function U [2]. In the utility theory the utility function U is considered and calculated as a probabilistic value, however, in many complex non-formalized decision-making problems it is difﬁcult and even impossible to estimate the multi-dimensional distribution of probability. In this paper the utility function is not considered as a probabilistic value, but as a fuzzy one, with membership functions of fuzzy sets being regarded as subjective measurements of the decision-making persons (DMP). The multi-criteria optimization problem can be presented in a form of the hierarchical decomposition (Fig. 1), where the set, formalizing the global goal, is denoted M . If the goals are deﬁned by too sophisticated concepts, they can be presented as a hierarchy of simpler ones. The elements of the hierarchy, i.e. goals and alternatives, are fuzzy sets denoted in Fig. 1, respectively, Gj , j = 1, . . . , m, where m is a number of goals, and Xr , r = 1, . . . , l, where l is a number of alternatives. The hierarchy levels are denoted Ln−1 , Ln , Ln+1 . The utility function is considered as a membership function of the global goal M on the set of alternatives X and is denoted µM (X). So, what is wanted is to get µM (X). 228

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Fig. 1. Hierarchical decomposition of multi-criteria optimization problem

The method of an analytic hierarchy process (AHP) [3], being a method to solve multi-criteria problems in a difﬁcult situation with hierarchical structures including non-formalized elements, is used in this paper as an indirect method to determine membership functions of fuzzy sets [4]. Let it be required to determine a membership function µD0 (x) of a fuzzy set D0 , deﬁning some qualitative concept. For this purpose, a DMP is offered to compare quantitative elements of a universal set X to one another by a degree of their conformity to this qualitative concept and to ﬁll in a matrix of pairwise comparisons A = {aij }, whose elements aij are estimates of a degree of membership of elements ai ∈ X to the fuzzy set D0 in comparison with elements aj ∈ X. The membership function is found as an eigenvector ω of the matrix A, corresponding to its maximal eigenvalue λmax : Aω = λmax ω. When complex properties, being represented as a hierarchical system, are analyzed, the above-described approach is used to compare the property constituents by a degree of their conformity to this complex property. For the hierarchical case, T. Saaty [3] proved the following theorem: If H is a complete hierarchy, having h levels and the greatest element b on the top (zero) level, Bn is a matrix of priorities of the n-th level, n = 1, . . . , h, and W 0 is a vector of priorities of the p-th level with respect to some element z in the (p − 1)-th level, then the vector of priorities W of the q-th level (p < q) with respect to z is deﬁned through the Cartesian product W = Bq × Bq−1 . . . Bp+1 × W 0 .

(2)

According to this theorem, a vector of priorities of the lowermost level with reference to the element b is deﬁned by the following expression: W = Bh × Bh−1 . . . B2 × W 0 .

(3)

Unfortunately, Eq. (2) and, hence, Eq. (3) can not be considered as the solution of the problem (1), i.e. the obtained ranking can not be considered as a function of utility. Besides, engineering problems, requiring formalization of qualitative VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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characteristics of physical values, can not be solved by the AHP method in its classical statement. The validity of the T. Saaty theorem is proved in [5, 6] for a case, when not all the hierarchy is considered as a fuzzy set (with its universal set at the lower level), but the hierarchy individual elements are fuzzy sets. The utility function for alternatives can be considered as a function of membership of the global goal on the set of alternatives, with the membership function being believed as a subjective value, instead of a probabilistic one. The membership functions of fuzzy sets formalizing each criterion (complex, represented as a hierarchy, or simple) are deﬁned on the set of alternatives, thus the set of alternatives is a base set for the criteria fuzzy sets. At every level we have various crisp (non-fuzzy) ordered sets that consist of elements, being fuzzy (qualitative), and each is deﬁned by the membership function of its own. Elements of each level are fuzzy subsets of the crisp ordered set and are deﬁned just in this sense. Let the hierarchy be an aggregate of levels Ln , n = 1, . . . , h. Any n-th level of the hierarchy is an array of the individual elements lnj , j = 1, . . . , m where m is a number of elements of the given level. The array of elements lnj represents “fuzzy properties” for elements of the lower, (n + 1)-th level, i.e. lnj is an array of fuzzy sets; universal sets for these fuzzy sets are sets of elements of the lower levels. Let us write out elements of each level: m

level Ln−1 : level Ln

:

n−1 1 2 3 ln−1 ln−1 ln−1 . . . ln−1 ;

ln1

ln2

ln3

...

lnmn ; m

level Ln+1 :

n+1 1 2 3 ln+1 ln+1 ln+1 . . . ln+1 .

The connection among the neighboring levels of hierarchy is deﬁned by a matrix of eigenvectors, i.e. by the matrix of priorities B: ω12 ... ω1mn ω11 ω21 ω22 ... ω2mn , (4) Bn+1 = ω31 ω32 ... ω3mn ωmn+1 1 ωmn+1 2 . . . ωmn+1 mn where Bn+1 is a matrix of eigenvectors of the level Ln+1 , mn+1 is a number of elements of the level Ln+1 , mn is a number of elements of the level Ln . 1 , A membership degree of the element ln1 to a fuzzy set, being the element ln+1 is determined as 1 µln+1 (ln1 ) = ω11 = µln1 (ln+1 ). 1 1 A membership function of the fuzzy set, being the element ln+1 , 1 (Ln ) = [ω11 ω12 . . . ω1mn ] ≡ Wn+1,n µln+1 1

is a membership function of the ﬁrst element of the (n + 1)-th level, deﬁned on the base set of the level Ln (row of the matrix from Eq. (4)). A membership function of a fuzzy set, being the element ln1 ,

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1 µln+1 (Ln+1 ) = [ω11 ω21 ω31 . . . ωmn+1 1 ] ≡ Wn,n+1 1

is a membership function of the ﬁrst element of the n-th level, deﬁned on the base set of the level Ln+1 (column of the matrix from Eq. (4)). It is obvious, that the matrix Bn+1 speciﬁes a binary fuzzy relation between fuzzy sets belonging to levels n and n + 1: Bn+1 ≡ Rn+1,n : Ln+1 ◦ Ln → [0, 1].

(5)

The problem of the hierarchies analysis is reduced to ﬁnding µl1 (Ln+1 ), i.e. to ﬁnding a membership function of the fuzzy set of the element of the ﬁrst level of the hierarchy determined on the base set of the (n + 1)-th level. We consider, that the level L1 consists of one element and there are n + 1 levels in the hierarchy. It is necessary to ﬁnd the relation between the fuzzy set l1 of the top level and the fuzzy sets belonging to the level Ln+1 : R(Ln+1 , L1 ) : Ln+1 ◦ L1 → [0, 1]. From Eq. (5), Bn+1 is a matrix of fuzzy relations of fuzzy sets in levels Ln and Ln+1 Bn+1 = R(Ln+1 , Ln ). So, the membership function µln (Ln+1 ) can be deﬁned as a composition of fuzzy relations: R(Ln−1 ) = µl1 (Ln+1 ) = R(Ln+1 , Ln ) ◦ µl1 (Ln ). The composition can be rewritten in a different way: mn+1

m \n

[ µl1 (Ln+1 ) =

µl j

(Ln )

µli (Ln ),

(6)

1 n+1

j=1

i=1

where mn is a number of elements of the n-th level; mn+1 is a number of elements of the (n + 1)-th level. Equation (6) is equivalent to Eq. (2). Hence, the validity is proved of considering the global vector of alternative priorities (vector of priorities) of the last level as the membership function of the global goal of the problem solution which, in turn, can be considered as the utility function in solving multi-criteria problems in a fuzzy statement. The latter assertion allows the AHP method to be applied to solving such multicriteria problems, as the estimation of fuzzy models of complex systems, where qualitative estimates of physical values should be formalized on quantitative scales. Thus, the AHP method can effectively be used not only for making a decision in non-formalized domains and solving tasks of ranking the ﬁnite set of complex objects presented as a hierarchical structure, but also for making decisions in engineering tasks, where the formalization of all fuzzy concepts is required with the VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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help of fuzzy sets. It opens additional opportunities for introduction of new information technologies, for example, to replace human operators, who manage technological processes, to improve the quality of management. It will allow building the intellectual systems capable to make a decision and adequately react on changes of a situation on the basis of human knowledge. Fuzzy Logic Controller. A technique and software, based on the suggested approach to solving non-formalized optimization problems, have been developed for creating a model of the fuzzy logic controller (FLC) destined for managing complex engineering objects. The main part of information, required for the model construction is qualitative or fuzzy. The data is presented in a form of rules (where the physical concepts are expressed in the natural language), called the linguistic rules of control (LRC). The operation of minimum is used as implication, if the rule is formulated as IF-THEN, or the operation of maximum — for the rule IF–THEN–OTHERWISE. The composition rule is used as a rule of inference, with the composition being calculated as maximal-minimal. The weight method is applied in choosing the crisp value of the management function, according to which the value, having the maximal membership function, gets out as the only value for management. The model of receiving data for control, i.e. the model of a controller, is developed as a hierarchical structure, whose elements are the input linguistic variables (LV) [4], inﬂuencing the process, the input linguistic rules of control, in which these variables are involved, and qualitative estimations of these linguistic variables. The main goal of solving the problem — to get the optimal control — is arranged at the ﬁrst level. As control only depends on the corresponding input variables, the input LVs, inﬂuencing the above-mentioned control, are arranged at the second level and the LRCs, used for getting the above-mentioned control with the help of the LVs, are arranged at the third level. Qualitative values of the input LVs are arranged at the fourth level. The LRC validity is taken into account on the basis of estimation of the qualitative values of the current observation data. Elements of the ﬁfth level are quantitative values of the input LVs (i.e. universal sets for fuzzy sets of the LV terms) used for estimation of membership function of their qualitative values. The technique makes possible to estimate the signiﬁcance of the input LVs, the relative importance of LRC in technological process and the LRC validity with the help of the obtained qualitative data and also to take into account a degree of each variable inﬂuence on control. If several output values are required for control, then hierarchical structures are deﬁned and estimated for each variable. The ad hoc software, developed according to the suggested technique, uses the MATLAB package and can also be implemented on any freely programmed controller, with the compiler for the programming language C++ available. Figure 2 illustrates the situation when there are three input LVs with two LRCs and, consequently, two qualitative values for each variable; V, W, Z are the universal sets with base variables v, w, z, respectively. The qualitative estimation of values of input variables, used in LRC, is formalized with fuzzy subsets A, C and D of sets of values of parameters V, W and Z, respectively; A0 , C 0 , D 0 are qual-

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Fig. 2. Hierarchical decomposition in designing FLC

itative values of observation for LV 1, LV 2, LV 3, respectively. The qualitative estimation of values of control action u is formalized with the fuzzy subset B of the set of control actions U . An assertion of the type “if A then A × B” is a binary fuzzy relation in V × U which presents a Cartesian product of A and B and is deﬁned as follows: if A then B = A × B. For conversion of fuzzy inferences at the linguistic level, a method is used on the basis of the compositional rule of inference, which is interpreted as a process of solving a system of equations of assignment in relations where linguistic values are assigned to fuzzy restrictions. If V and U are two universal sets with base variables v and u, respectively, R(v), R(v, u) and R(u) denote restrictions for v, (v, u) and u, respectively, and present fuzzy relations in V , V × U and U , and A and F are fuzzy subsets of sets V and V × U , then the compositional rule of inference states that the solution to the system of assignment equations R(v) = A, R(v, u) = F is of the form R(u) = A ◦ F, VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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where A ◦ F is a composition of A and F . In this sense we can make a conclusion that R(u) = A ◦ F from R(v) = A and R(v, u) = F . Deﬁnition. Let A0 , A and B be fuzzy subsets of the sets V , U and U , respectively. We suppose that the value A0 is assigned to the restriction R(v) and the relation A ⇒ B is assigned to the restriction R(v, u), i.e. R(v) = A0 , R(v, u) = A ⇒ B. These equations of assignment in relations can be resolved with respect to the restriction for u in the following way R(u) = A ◦ (A ⇒ B). The expression of this inference in the form A0

premise

A ⇒ B implication A0 ◦ (A ⇒ B) inf erence presents the formulation of the generalized rule modus ponens. Basic operations applied in getting the fuzzy inference with the help of the compositional rule of inference are described as follows. A union of fuzzy sets A and B in V denotes the fuzzy set A + B with the membership function of the form µA+B (v) = max{µA (v), µB (v)},

v ∈ V.

A Cartesian product A1 ×. . .×An of fuzzy sets Ai in Vi , i = 1, . . . , n is deﬁned as a fuzzy set in the Cartesian product V = V1 × . . . × Vn with the membership function of the form µA (v) = min{µA1 (v1), . . . , µAn (vn)}, v = (v1 , . . . , vn ) ∈ V. A complement (negation) of a fuzzy set A in V denotes the fuzzy set A¯ with the membership function of the form µA¯ (v) = 1 − µA (v),

v ∈ V.

A maximin product A ◦ B, in case of the ﬁnite set V , is equal to the maximin product of relation matrices A and B, i.e. µA◦B (v, u) = max min{µA (v), µB (u)}. v∈V

A fuzzy inference presents an application of the maximin composition as a compositional rule of the fuzzy inference and operation of ﬁnding minimum as a fuzzy implication (if the rule IF–THEN is used):

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µB 0 (u) = max min{µA0 (v), µR (v, u)} = v∈V

= max min{µA0 (v) min[µA (v), µB (u)]}, (7) v∈V

where B 0 is a fuzzy control action and A0 is a speciﬁed value of current data of observations. Then the defazziﬁcation follows, with which on the basis of the membership function µB 0 (u) for B the exact value u (to be used for the control operation) for each point of U is obtained. As a defuzziﬁcation method the weight method is applied which is based on using the variable u that has a maximal value of the membership function. In the FLC implementation several rules are commonly used whose structures may be different. While performing operations with the help of the maximin composition accompanied with the calculation of the maximal membership function value for u the inference result for each rule is obtained according to Eq. (7). As a ﬁnal result of the inferences, a sum is assumed of fuzzy sets which are inferences resulting from each rule (operation of summing or ﬁnding maximum). In other words, the rules are processed in parallel. The ad hoc software, implementing the suggested technique in MATLAB package, features the following: — a maximal number of control actions, i.e. estimated hierarchies, is 4; — a maximal number of input variables is 12; — a maximal number of LRCs for a single LV is 8, i.e. a maximal total number of LRCs is 384. At ﬁrst, a user estimates membership functions for all qualitative values, comparing quantitative elements in pairs by degree of their conformity with the qualitative concept on a 9-point verbal scale. Results are issued in the numerical and graphical form. A conformity relation is calculated for each estimated membership function. Then priorities of LVs, LRCs and the LRCs validity are estimated. The estimation is conducted in the same manner as in the AHP software [5], destined for solving multi-criteria problems, in case of 5-level incomplete hierarchy. Results of Research. The scientiﬁc-educational complex “Computer Sciences and Control Systems” of the Bauman Moscow State Technical University has been engaged in problems of development of theoretical bases of construction of intellectual information-and- control systems to solve ecological tasks for a number of years [6–9]. Some results of the complex activities are as follows. The intellectual system of environment monitoring is investigated. The algorithms of the environment condition estimation and forecast are formulated. They might be used for management of society activities to improve the quality of life. The ecological processes are investigated on the basis of the parametrical analysis of inﬂuence of internal and external environment on optimum indemnities in gerontological and toxicological models of natural technologies of human organism.

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The program of computer thermodynamic modeling of the kinetics of chemical transformation of gaseous mixtures is developed. The general basic description of a method to calculate the composition and properties of an equilibrium condition of multi-component heterogeneous systems is given. The methods, algorithms and software to design the intellectual regulators for ecological systems are investigated on the basis of the fuzzy combined strategy of management and the relevant approach to generate fuzzy rules. The strategy of the fuzzy PID management is implemented in the automatic system of decreasing toxicity of diesels on the basis of recycling soot ﬁlters. The combination of PI- and PD-algorithms with the help of two input variables considerably reduces a number of managing rules without deteriorating the quality of regulation. The problems of organization of data processing and management on the basis of technology of systems based on knowledge, named S-technology, are investigated. The offered architecture of information support system for making decisions provides for the base of knowledge the functioning of subsystems of management and of knowledge output as uniﬁed system. It allows the system to be quickly adapted to various software at the expense of updating the user interface of the subsystem of knowledge output. The subsystem of management of base of knowledge remains unchanged in this case. The partitioned architecture allows using and modifying components of the subsystems independently. The procedure mechanism makes possible the data exchange with external devices and expands functionalities of the system. The algorithms of cosmonaut’s actions in performing typical tasks of viewing the emergency areas with the use of the cosmonaut helmet information and control system (“NIUSK”) are developed. The generalized and partial criteria of an estimation of efﬁciency of the algorithms are offered. They are substantiated with the system approach and method of parallel analogy. The algorithms to detect outﬂows and breaks are produced for the system of detection of outﬂows and localization of breaks in urban networks of water supply. The software for the reception, processing and displaying of the information is offered. The research results are used in the educational process during the laboratory training, development of the course and degree projects. They are, on the one hand, objects of research, and, on the other hand, they form the ecological thinking of the students. Conclusion. We have discussed some results of development of fuzzy systems for ecological monitoring and control. The general problem of optimization is understood as a problem with fuzzily expressed criteria and alternatives. As the research results have shown, the fuzzy logic approach can improve our ability to ﬁnd out solutions for ecological dynamical systems and speed up computations. The level of solving ecological problems in any world country can serve the most authentic and complex criterion of an estimation of a level of stability of this state and its economic development, and also moral condition of a society in the modern world.

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The formation of the ecological culture of an engineer is therefore an important aspect of the engineering education. The formation of the ecological thinking based on the comprehension of the main principle — “absoluteness” of ecological priorities while solving scientiﬁc and technical tasks is the major purpose of the ecological education. The ecological education has the special importance in technical higher educational establishments, as it, in many respects, deﬁnes efﬁciency of solving ecological problems directly at site of their occurrence.

REFERENCES 1. Pupkov, K.A., Konkov, V.G. Intellectual systems (in Russian). M.: Izdatelstvo MGTU im. N.Ye. Baumana, 2003. 2. Shtoer, R. Multicriteria Optimization. Theory, Calculation and Applications / Translated from English. M.: Radio i Svyaz, 1992. 3. Saaty, T.L. Making Decisions. Method of Analysis of Hierarchies / Translated from English. M.: Radio i Svyaz, 1993. 4. Zadeh, L.A. Concept of Linguistic Variable and its Application to Making Approximated Decisions / Translated from English. M.: Mir, 1976. 5. Grunina, G.S., Demenkov, N.P., Yevlampiev, A.A. Solution of Multicriterial Optimization Problem Under Conditions of Qualitative Uncertainty (in Russian) / Vestnik MGTU, 1998, No 1. 6. Demenkov, N.P. Solution of Multicriterial Optimization Problems and Making Decisions in Fuzzy Statement (in Russian) // Trudy III Mezhdunarodnoi konferentsii po myagkim vychisleniyam i izmereniyam, S-Pb, 2000. 7. Demenkov, N.P., Matveev, V.A. Computer Technologies of Decrease of Intensity of Allocation of Harmful Engines of Power (in Russian) / Ekologicheskie sistemy i pribory, 2001, No 11. 8. Demenkov, N.P., Matveev, V.A. Intellectual Information and Control Systems for Solution of Ecological Tasks (in Russian) / Konversiya v mashinostroenii, 2002, No 1. 9. Demenkov, N.P., Matveev, V.A. Logical-linguistical Models in Ecological Tasks of Municipal Economy (in Russian) / Promyshlennye ASU i kontrollery, 2003, No 2. V.A. Matveev (b. 1939), D. Sc. (Eng.), Professor, the Head of the Research and Education Unit “Information Technology and Control Systems” (NUK IU) of the Bauman MSTU. Scientiﬁc interests: gyroscopic systems, navigational complexes, control systems, optimization of dynamic system at indeﬁnite external parameters of the environment. N.P. Demenkov (b. 1944) graduated from Bauman Moscow Higher Technical School in 1968. Ph. D. (Eng.), assoc. professor of “Automatic Control Systems” Department of Bauman Moscov State Technical University. Winner of Komsomol Prize in the ﬁeld of science and technology. Author of more than 150 publications in the ﬁeld of optimization of dynamic system at indeﬁnite external parameters of the environment, designing the automatic control systems for space vehicles, automatic and automatized control systems for technological processes and objects.

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FUNDAMENTAL PROBLEMS of MECHANICAL ENGINEERING Yu.N. Drozdov (Mechanical Engineering Research Institute, Russian Academy of Sciences) Ye.G. Yudin (Bauman Moscow State Technical University)

WEAR PREDICTION CONSIDERING MECHANICAL, PHYSICO-CHEMICAL AND GEOMETRICAL FACTORS Wear of solid bodies depends on physical-chemical and mechanical processes in the contact zone. The processes are greatly inﬂuenced by kinematics of mating surfaces motion (sliding, rolling, rolling with sliding, spinning and etc.), by structure and composition of material layers in the surface and surﬁcial region, by formation of surface compounds, by geometric properties of contacting surfaces and their change in time. A technique of obtaining relations to calculate a wear-rate is based on the synthesis of experimental data and mathematical models. A theoretical and invariant method, developed by the authors, to calculate the rate of the friction-induced surface destruction of solid bodies is based on equations of elastic-hydrodynamic lubrication theory, chemical kinetics, contact problem of elasticity theory, theory of strength, analysis of thermalphysical, adsorption and diffusion processes. Theoretical studies were closely connected with experimental ones. A heuristic role of physically informative tribological invariants is shown.

Introduction. After known calculated dependences for deﬁnition of magnitude of a wear — papers by Holm, Khrushchov, Kuznetsov, Archard, Rabinovich, Kragelsky, Flyaisher — the new basic equations have not appeared in literature within several decades. Only empirical approaches concerning separate units and concrete situations in maintenance predominate. The known calculated equations attract with the simplicity, but, unfortunately, do not take into account many rather important factors. For example, inﬂuence of a lubricant layer (boundary and elastichydrodynamic), thermalphysic conditions of tribocontact, adsorption and diffusive phenomena, modiﬁcation of rubbing surfaces. In practice only rather approximated, comparative and estimated calculations therefore are used. Prediction of service life at a design stage is acutely necessary for modern engineering, especially for the friction units operating in extreme conditions (aerospace, atomicenergetic, high-speed transport, technique for mastering Ocean, deep bowels of the Earth etc.). The urgent problem consists in generalization of experimental data, prediction of reliability and service life in a wide range of voltage, temperatures, and environment change, of effect of ﬁelds of a different physical nature (acoustic, electromagnetic, radiant etc.).

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The proposal of the authors consists in use of physically informative criteria obtained from theoretical dependences, describing process, and experimental data. The structure of obtained calculated dependences contains the generalized variables, allowing understanding of dominating factors during wear process. The quantitative response is achieved with the use of the calculated dependences obtained on the basis of special experimental research or from practical generalizations. In a basis of research the theory of similarity and simulation lies. The real possibility of the development of calculated models of wear process is shown in works [1–5]. dh , where We shall use the concept of intensity of linear wear process Jh = dS h — magnitude of a linear wear, S — a path of wear. Speed of wear process or dh intensity of wear process with respect to time is deﬁned as Jhτ = = Jh v, where dτ τ — time of wear process, v — speed of tribo-surfaces migration. Having installed the regularity for deﬁnition of intensity of wear process in different conditions, it does not represent difﬁculties to deﬁne a service life by criterion of a wear of real friction units [5]. Thickness of Elastic-Hydrodynamic Lubrication Layer. For bodies rolling with sliding the process of shaping of lubricant layers is described by the equations of elastic-hydrodynamic theory of lubrication. For simplicity the isothermal equation of Reynolds is used: ¤ £ dp/dx = 6µ (v1 + v2 ) (h − h0 ) /h3 , where p = 0 at x = −∞; p = dp/dx = 0 at x = x0 . We shall write down the equation for deﬁnition of a clearance as ¶ Zx0

µ x2 − x20 2 h = h0 + − 2r π

1 − v12 1 − v22 + E1 E2

¯ ¯ ¯ ζ −x ¯ ¯ dζ. ¯ p (ζ) ln ¯ ζ − x0 ¯

−∞

The equation, expressing conservation of energy, will be as follows µ ¶2 dt d2 t dv dp ρlub club = λlub 2 + µ + kvt , dx dy dy dx where t = t0 at x = −∞; ¶1/2 Zx

µ 1 πρ1 c1 λ1 v1

t (x, 0) =

¯ dt ¯¯ dω λ1 ¯ + t0 , dy y=0 (x − ω)1/2

−∞

¶1/2 Zx

µ t (x, h) =

1 πρ2 c2 λ2 v2

¯ dt ¯¯ dω −λ2 ¯ + t0 . dy y=h (x − ω)1/2

−∞

Dependence of viscosity of a lubricant on pressure and temperature is: µ = µ0 exp (βp − ψ∆t) , VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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where p — pressure; µ — a dynamic viscosity of a lubricant; v1 , v2 — peripheral speeds; h — a clearance; h0 — a minimum clearance; r — a reduced radius of curvature 1/r = 1/r1 ± 1/r2 ; r1 , r2 — radii of cylinders; “+” sign corresponds to external contact, “−” sign — to internal contact of bodies; E1 , E2 — coefﬁcients of elasticity of contacting bodies; ν1 , ν2 — factors of Poisson of materials of bodies; t, ∆t — temperature and an increment of temperature; ρlub , club , λlub , ρ1 , C1 , λ1 , ρ2 , C2 , λ2 — densities, speciﬁc thermal capacities, factors of a thermal conduction of a lubricant and materials of bodies, respectively; β — piezo-coefﬁcient of viscositiy; ψ — factor of dependence of viscosity on temperature in the formula of Reynolds; k — factor of a thermal expansion of a lubricant; x, y — coordinates up and down a ﬁlm; ζ, ω — additional variables; x0 — an abscissa of the point where pressure and a gradient of pressure are equal to zero. On the basis of a method of integrated analogs, initial set of the equations, boundary conditions and conditions of unambiguity, criteria of a similarity, the principal dimensionless complexes describing process of friction and lubrication were found. The criterion describing a carrying capacity of contact µv/Pn , where v = vsl in conditions of sliding and v = v1 + v2 = vΣk in conditions of rolling friction. Deformation of the contact is ¢being expressed ¢by the criterion Pn η/r, where Pn — ¡ ¡ running loading, η = 1 − v12 /E1 + 1 − v22 /E2 — elastic constant of touching bodies. The temperature processes ¢ which are ﬂowing inside a lubricant layer are ¡ 2 deﬁned by criterion λlub / ψµv , where v = vsl at the analysis of the temperatures, happening as a result of energy dissipations at sliding, v = vΣk — at rolling. Pekle numbers Pe1 = bv1 /a1 and Pe2 = bv2 /a2 characterize a ratio of heat content of a stream in axial and transversal directions, where b — a halfwidth of a strip of contact by Hertz; a1 , a2 — factors of temperature conductivity of materials. The numerical solution of reduced system of essentially nonlinear integrodifferential equations may not give an acceptable quantitative response as the mathematical model with all its completeness and complexity does not reﬂect many latent and complex-connected phenomena accompanying processes of friction, lubrication, destruction of a layer of lubrication and real solid bodies. Drawing up the calculated equations on the basis of experimental results and obtained physically informative performances allows one to compensate some latent operating factors to some extent and to discover the best values of factors. In result the equations appear which have the physical contents, expressing qualitative inﬂuence of major factors and giving quantitative results necessary in engineering practice. Elastic-hydrodynamic theory of lubrication has been developed for more than 60 years. However, at rolling with sliding of bodies (cogged and friction gears, rolling bearings, cam mechanisms etc.) inﬂuence of speed of sliding on the width of a lubricant layer is not established. It is difﬁcult to construct the theoretical model, which would take into account simultaneously the inﬂuence of speed of rolling and sliding on forming and destruction of a lubricant layer. Observations in practice and laboratory research of real mechanisms and physical models display essential inﬂuence of speed of sliding on the wear of rubbing bodies in the lubricant environment, and, hence, on the width of a lubricant layer. Thus, well-known calcu-

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lated dependences have the restricted application for real processes (small speeds, cinematically pure rolling). On the basis of the generalized criteria (dimensionless complexes) and experimental researches the authors have developed the following dependences, which take into account the inﬂuence of speeds of rolling and sliding. For the boundary conditions of friction appropriate to the jam origination the equation is derived: Ã !µ ¶ µ ¶e ¶ µ r µvΣk 0.7 Pn β 0.6 λ K3 p 2 61 Pn r αµ0 vS2 Pe0.5 Ra1 + Ra22 1.2 where K3 and e — the experimentally deﬁned magnitudes depending on physical and chemical processes (magnitude K3 varies depending on materials of surfaces of cog-wheels and used oils: for gears K3 = 1 . . . 3, for samples — rollers K3 = = 0.5 . . . 5; magnitude e = 0.23 . . . 0.35); r + Ra22

p

Ra21

— a complex describing micro- and macro-geometry of contacting bodies; Ph β r — a complex describing deformability of contacting bodies and piezo-viscosity properties of a lubricant; λ 2 µ0 vS Pe0.5 1.2 — thermophysical complex;

Pe1.2 =

Pe1 + Pe2 2

— average by Pekle. Using the obtained dimensionless complexes and results of experimental research of the lubricated ﬁlm width between cylindrical bodies rolling with sliding, it is possible to get calculated dependences for the deﬁnition of the width of a lubricant layer considering heat release as a result of a viscous dissipation at rolling with sliding [6]:

µ h = 1.65 r

¶0.1 "

¶0.75 µ βµvΣk 2r

E p

1 + 0.18

2 ψµvΣk 4λlub

¶0.8 #−1

µ

¶0.7

µ

+ 0.45

2 ψµvsl λ

.

The analysis of regularities of change of elastic- hydrodynamic layer of lubrication displays an essential roleµof speed ¶ of sliding. At rolling with sliding with the λlub , the width of a lubricant layer diminishes. increase of speed of sliding ψµVsl2 VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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At rather thin layers the inﬂuence of temperature processes in discrete points of contingence is revealed. Destruction of a lubricant layer results in intensive wear process, jam of rubbing surfaces, growth of a factor of sliding friction f . Appearance of signiﬁcant thermal sources in spots of a contingence fσ vsl results in burn-off of rubbing bodies, thus the factor of friction does not increase [7]. Mechanical and Geometrical Factors of Wear Process. In case of dominant mechanical destruction of friction surfaces the determining criteria are the following: p fp ; HB HB — the criterion, describing a tension of contact, the dimensionless area of an actual contingence of bodies. This criterion is most representative and is applied in calculations with different types of wear process; here p — direct stress of compresh sion; HB — hardness of a wearing material; — the criterion, deﬁning a relative ζ width of a lubricant layer; h — a width of a lubricant layer; ζ — the characteristic size p of the abrasive particle, cutting roughness or the reduced size of roughness Ra21 + Ra22 . The criterion is applied for estimation of conditions of lubrication and fatigue surface destruction. ν1 p σ0 — the criterion of fatigue strength of rubbing surfaces, where p — a direct stress of compression; ν1 — the factor depending on factor of sliding friction and tension in contact; σ0 — fatigue limit of a material in the considered conditions of friction. ´ ³ 1/v Rmax / rkk bkk kk — the criterion of a roughness (factor by Kragelsky–Kombalov); Rmax — the greatest height of irregularities of the proﬁle; rkk — reduced radius of irregularities; bkk , vkk — parameters of a basic curve. Physico-Chemical Factors of Wear Process. In case of a physical and chemical effect of environment (lubricant) it is necessary to apply additional criteria ratio: RT /Q — the criterion describing stability of boundary lubricant layers, where R — universal gas constant; T — absolute temperature; Q — heat of adsorption. RT /Echem — the criterion describing chemical modiﬁcation of surfaces of friction, where Echem — energy of activation of disintegration of interatomic links of surface connections. Kinetic processes of forming and destruction of surface layers (connections) depend on the time factor. Characteristic time for bodies rolling with sliding is: τr.av. = dav /v — average time of solitary actual contact; dav — average diameter of an actual tooth contact; v — speed of relative driving of bodies; τr.f ree = lav /v — an average free time; lav — average distance between microprotuberances or τr.f ree = πHBdav /(8pv); time of contact on the nominal area τnom = 2b/v — for a case of contact of cylinders or a cylinder with a plane (where 2b — length of

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a strip of the contact, calculated by the method of Hertz); τout = Sτ /v — time of free stay of a surface outside the zone of contact (where Sτ — a path, ran by a body before the next nominal contact). Thermalphysic Factors of Wear Process. Inﬂuence of thermalphysic processes on the intensity of wear process is deﬁned by the following: tcon /tcr — criterion, deﬁning inﬂuence of contact temperature, where tcon — temperature in contact; tcr — critical temperature (for example, temperature of fusion, homological temperature, temperature of characteristic physicochemical and structural transformations in the materials of rubbing bodies); qδt /λ1,2 tcr — the criterion, deﬁning inﬂuence of temperature gradient and thermal boundary layer, where q — the speciﬁc thermal stream operating on the given body (a density of a thermal stream), λ1,2 — factor of a thermal conduction of materials, δt — width of a thermal boundary layer in a rubbing body. For approximated deﬁnition of a width of the warmed up (thermal) boundary layer appearing in a body at friction, it is possible to use the following equation: √ √ tδ /tcon π = ierfc (δT /2 aT τ ) , where tδ — temperature at a distance δT from a surface of friction; tcon — contact temperature on the surface; aT — factor of temperature conductivity; τ — time of an operation of a thermal source. Thermostress of surface layer is characterized by a criterion of the type: Eαt ∆t , (1 − ν) σlim where αt — factor of linear temperature expansion, ∆t — an increment of temperature in the surface layer; ν — factor by Poisson; σlim — limiting stress for the surface layer. ρ (tnom − t) σs — the thermomechanical criterion, where ρ — density of a material, c — a speciﬁc thermal capacity, t — a body temperature, tcon.f us. — temperature of contact fusion, σs — a limit of ﬂuctuation of a material. The criterion characterizes propensity of metals to seizing and factor of friction. Minimum values of a ratio of heat content of a material during its heating up to temperature of contact fusion [ρc(tcon.f us. − t)] to a limit of ﬂuctuation correspond to smaller values of a factor of friction and propensity to seizing [8]. Use of the speciﬁed criteria ratios in the calculated equation allows one to connect intensity of destruction with temperature in a place of contact, a temperature gradient, thermostresses, a thermal boundary layer, temperature constancy of a material, the characteristic temperatures, inﬂuencing content and structure of contacting materials. Load-Carrying Capability of Tribocontact. General ﬁeld of stresses on a surface of tribocontact σij can be presented as a ﬁeld consisting of mechanical VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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(power) and temperature stresses [9–11]: £ i ¤ t ¯ij (x, y, ν, f ) + Φt σ σij = σm σ ¯ij (x, y, v, Fo) . Here the dimensionless functions are as follows: σij depends on coordinates of a t depends on point x, y, ν (factor by Poisson), f (factor of a sliding friction); σij x, y, v and numbers Fourier Fo: Fo =

2aT , vract

where aT — factor of thermal conductivity, v — speed, ract — radius of an area of actual contact. The dimensionless parameter is determined by the formula:

Φt =

2 (1 + ν) αt G γf vract , (1 − ν) λ

where G — the module of shift, αt — factor of linear temperature expansion, γ — factor of distribution of thermal streams between rubbing bodies. The operation of mechanical and temperature stresses results in lowering the load-carrying capacity of tribocontact, and also in the beginning of plastic deformations and wear process of contacting bodies. Temperature factor Φt was accepted by authors as thermomechanical criterion of the beginning of wear process [9–11]. Examples of use of obtained calculated dependences for different friction units are presented in papers [5, 12–15]. More detailed research of obtained criteria is carried out nowadays on the basis of the consideration of dislocation processes in contacting materials . Conclusion. Designed on the basis of theoretical models, generalized dependences allow us to take into account mechanical, physicochemical and geometrical factors of wear process in the calculation equations. Interpretation of experimental results in the criteria form is important. The profound study of physics, chemistry and mechanics of the wear process will allow us to understand and take into account the phenomena taking place under the microlevel consideration in calculated dependences of wear process.

REFERENCES 1. Drozdov, Yu.N. Determination of Wear Rate of Machine Parts [In Russian]. Vestnik Mashinostroeniya, 1980, No 6, pp. 12–15. 2. Drozdov, Yu.N., Frolov, K.V. Theoretical-invariant Method for Calculating the Intensity of Solid Body Surface Wear due to Friction [In Russian]. Poverkhnost. Fizika, Khimiya, Mekhanika, 1982, No 5, pp. 138–146. 3. Drozdov, Yu.N. Transmission Mechanisms [In Russian]. Trenie, iznashivanie i smazka. Ed. by Kragelsky, I. V., Alisin, V. V. Mashinostroenie. Book 2, 1979, pp. 113–147. Transmissions Tribology-Lubrication, Frictionand Wear, 2001, pp. 597– 639. 244

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4. Drozdov, Yu.N. Key Invariants in Calculations of Wear Process Intensity at Friction [In Russian]. Mashinovedenie. 1980, No 2, pp. 93–98. 5. Kogaev, V.P., Drozdov, Yu.N. Strength and Wear resistance of Machine Parts [in Russian]. Moscow, Vysshaya Shkola, 1991, 320 p. 6. Shirobokov, V.V., Drozdov, Yu.N. Width of Lubrication Layer at Rolling with Sliding of Bodies Considering Thermal Processes [In Russian]. Mashinovedenie. 1979, No 4, pp. 90–93. 7. Drozdov, Yu.N., Archegov, V.G., Smirnov, V.I. Antiscuff Stability of Rubbing Bodies [In Russian]. Moscow, Nauka, 1981, 139 p. 8. Drozdov, Yu.N. Tribological Contact of Sliding in Space [In Russian]. Problemy mashinostroeniya i nadyozhnosti mashin. 2001, No 1, pp. 69–76. 9. Hamilton G.M., Goodman L.E. Trans ASME. Ser E Journal of Applied Mechanics. 1966, v. 33, No 1, pp. 371–376. 10. Ting B.Y., Winer W.O. Journal of Tribology. 1989, p. 315. 11. Yevtushenko, A.A., Uhanskaya, O.M. Thermomechanical Criterion of Wear Process [In Russian]. Trenie i iznos. 1994, Vol. 15, No 3, pp. 379–388. 12. Drozdov, Yu.N., Pavlov, V.G. and Puchkov, V.N. Friction and Wear in Extreme Conditions [in Russian], Mashinostroenie, Moscow, 1986, 224 p. 13. Drozdov, Yu.N., Tumabishvili G.I. Calculation of Seizing by Limiting Width of Lubricant Layer [In Russian]. Vestnik mashinostroeniya, 1982, No 4, pp. 19–22. 14. Drozdov, Yu.N., Mudryak, V.I. and Dyntu, S.I. Generalized Characteristics for Predicting the Wear of Rubbing Surfaces [In Russian]. Trenie i Iznos, Vol. 18, No 6, 1997, pp. 715–721. 15. Drozdov, Yu.N. Development of Tribology for Extreme Conditions [In Russian]. Tribologiya. Issledovaniya i prilozheniya. Opyt SSHA i stran SNG. Ed. by Belyi, V. A., Ludema, K., Myshkin, N. K. Moskva: Mashinostroenie. 1993, pp. 296–311. Ye.G. Yudin (b. 1940) graduated from the Bauman Moscow Higher Technical School in 1964. Ph. D. (Eng.), assoc. professor of “Multipurpose Caterpillar Vehicles and Mobile Robots”, First Vice-Rector – Vice-Rector for Educational Activity of the Bauman Moscow State Technical University, member of the Russian Academy of Natural Sciences. Author of more than 70 publications in the ﬁeld of theory and design of caterpillar machines, tribological reliability of transmissions. Yu.N. Drozdov (b. 1936) graduated from the Bauman Moscow Higher Technical School in 1959. Honored Worker of Science of the Russian Federation, Winner of the USSR State Prize, D. Sc. (Eng.), professor. Head of “Friction, Wear and Lubrication, Tribology” department of the Institute for Machine Science n.a. A.A. Blagonravov of the Russian Academy of Sciences. Author of more than 500 publications in the ﬁeld of complicated mechanical systems.

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Yu.N. Drozdov (Mechanical Engineering Research Institute, Russian Academy of Sciences) Ye.G. Yudin (Bauman Moscow State Technical University)

TRIBOLOGICAL PROBLEMS OF MECHANICAL SYSTEM CREATION FOR THE MOON A renewal and evolution of activities on the Moon development should be expected in XXI century. Building the manned lunar bases having production, astrophysical, extractive and maintenance equipment and services are in sight. Interplanetary ﬂights from the Moon require less power expenditures than those from the Earth, so it is proﬁtable to use Moon’s surface for erecting cosmodromes. An experience of creating technical systems for the Moon-application shows the bafﬂing complexity in providing the workability and reliability of tribotechnical systems. Some problems and results of experimental and theoretical studies on tribology for application in the Moon environment are discussed in the paper.

Introduction. Automatic interplanetary stations (AIS) like “Luna” (USSR), “Surveyor”, “Apollo” (USA) series were used in research of the Moon. The stations executed functions of ﬂying, landing, of artiﬁcial satellites of the Moon, of moonrovers and manned spacecrafts such as “Apollo” and they have allowed us to better understand the conditions of the Moon in XX century [1–3]. Automatic interplanetary stations “Luna-1” and “Luna-2” consisted of the thin spherical coverings formed by the connection of two hemispheres with a radius of 400 mm, a length of the station was 1300 mm. They were made of an aluminummagnesium alloy, hermetically incorporated with each other through sealing laying, which was made of the special rubber. Hemispheres were fastened by 48 bolts through a frame, with the diameter of 850 mm. The container was ﬁlled with gas (nitrogen) at pressure 0.13 · 105 Pa , temperature 20 ◦ C. The gross weight was 361.3 kg. “Luna-1” was launched on 2nd January 1959. The station has ﬂown close to the Moon and has left into its own orbit around the Sun. On 12th September 1959 the AIS “Luna-2” was launched, it reached a surface of the Moon on 14th September 1959 in the western part of the Sea of Rains, in 800 km from the centre of a visible disk. “Luna-3” has photographed the reverse of the Moon for the ﬁrst time. The station liftoff was realized on 4th October 1959. “Luna-3” represented a thin-walled hermetical cylinder with the spherical bottoms. The maximal transversal size was 1200 mm. “Luna-9” made soft landing to the Moon (1966), “Luna-16” supplied the Moon’s ground to the Earth (1970). The ﬁrst ﬂight of a spacecraft around the Moon lasted several days since 21st December 1968 (“Apollo-8”, astronauts Borman, Lovell, Anders). Expedition to the Moon came true on 16th July 1969 by the spacecraft “Apollo-11” with the astronauts Armstrong, Aldrin and Collins. The lunar module has executed necessary functions — landing, takeoff, joining the command module. In this ﬂight people

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landed on the Moon for the ﬁrst time, and in XX century the Moon was visited by 12 people. On 17th November 1970 the AIS “Luna-17” landed on the Moon with the ﬁrst automatic planet research vehicle “Lunokhod-1”. In total 5 transport machines were delivered to the Moon and they covered the following distances: “Lunokhod1” — 10 km, “Lunokhod-2” — 37 km, “Apollo-15” — 28 km, “Apollo-16” — 27 km, “Apollo-17” — 37 km. Three American moon-rovers (“Rover”) were used for astronauts’ moving along the Moon’s surface — “Apollo-15, 16, 17”. The Moon and Libration Points. Let us notice some geometrical and physical characteristics of the Moon: the diameter (equatorial) is 3476 km, medium density of the planet is 3.33 g/cm3 , acceleration of force of gravitation on a surface is 1.62 m/s2 , gravitation parameter 4.89 · 103 km3 /s2 , sidereal cycle time (relative to stars) is 27.32 days, synodical cycle time (the period of full change of phases) 29.5 days, circular speed is 1.65 km/s, a ratio of densities of an atmosphere at the surface of the Moon and the Earth is ≈ 2 · 10−13 , a ground basically porous (70 . . . 80 % of surface), the density of ground on depth from 0 up to 10 cm — 0.6 . . . 0.7 g/cm3 , from 10 up to 20 cm — 1 g/cm3 , from 1 up to 10 m — 2 . . . 3 g/cm3 . Elements of lunar rock are mostly oxides. A surface stratum (regolith) has generated as a result of fragmentation, sintering, hashing of rock at the falling of meteorites. A lot of dust appeared on a surface. Table 1 Main elements of Moon’s rock Rock

Stones Dust

Si 20 20

Fe 14 12

Ca 7 8

Elements, % Al Mg Ti 6 5 5 6 6 5

Cr 0,4 0,3

Mn 0,3 0,2

K 0,2 0,1

Physical-mechanical properties of the ground of the Moon were deﬁned thanks to supplies to the Earth of samples of lunar ground and to their detailed research in the earth laboratories. Samples of ground were selected, supplied to the Earth, investigated by crews of spacecrafts “Apollo” and automatic stations “Luna16”, “Luna-20”, “Luna-24”, “Surveyor”, planet research vehicles “Lunokhod-1”, “Lunokhod-2”. In the system “Earth—Moon” there are unique points, in which total effect of forces is equal to zero. They are referred to as points of libration. Existence of such points was predicted by Lagrange in 1772. In libration point a space body goes under inﬂuence of the gravity of two bodies, it can be in a condition of relative equilibrium in relation to these two bodies, for example, the Earth and the Moon. Let us remark that under libration of the Moon (from the latin word libratio — swinging, oscillation) periodic penduliform oscillations of the Moon relative to its center-of-mass are understood. The system “Earth—Moon” has ﬁve libration points: three of them are on the straight line connecting the centres of the Earth VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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and the Moon, and are unstable; two points, located on a conditional triangle, are steady [4]. The last two points are in a plane of an orbit of driving of the Moon relative to the Earth. Spacecrafts, started up in a steady libration point, can serve intermediate stations for the ﬂights to the Moon and to the long-distance planets. Flights of crews to libration points are considered for the second decade of XXI century. Table 2 Distance between the cen- Distance from the surface of Position of the Moon relaters of the Moon and the the Moon to the libration tive to the Earth Earth, km point, km The Moon in apogee The Moon at the middle distance from the Earth The Moon in perigee

402000

61791

384400

59024

356414

54784

It is supposed to place space telescopes, assembly of factories for manufacturing the equipment for the further mastering the Moon in libration points. Friction Units on the Moon. Open friction units on the Moon are to work under following conditions: pressure of environment — vacuum, p < 10−10 Pa; wide range of temperature change on surface: +150 · · · − 170 ◦ C, high evaporability of lubricants; inﬂuence of temperature gradients and warping of constructions; sublimation of elements of constructional materials; irradiation of different physical nature; effect of micrometeorites; reduced gravity; inﬂuence of abrasive particles of lunar ground; requirements for minimization of size characteristics of a construction (high tension); undesirability (impossibility) of application of liquid lubricants and grease; vibration, shock, acoustic loadings during takeoffs and landings to the Earth; difﬁculties in repair-regenerative operations in conditions of the Moon etc. Adhesive interaction of conjugated surfaces is a principal danger for rubbing pairs on the Moon. Forces of adhesive interaction depend on a degree of clearing of surfaces from adsorbed stratums, pollution, presence of lubricating material, oxides. On the areas of actual contact of rubbing elements under effect of normal and tangential efforts (strains), temperatures and wear, destruction of screening layers occurs, there are contact atom-pure surfaces [5, 6]. Temperature, vacuum and irradiation essentially inﬂuence the evaporability and decomposition of lubrication. Evaporability is reﬂected on operating characteristics of a lubricant and allowed period of storage. Speed of ageing of lubricant depends on evaporability and thermo-chemical disintegration (polymerization). Thermal stability is the major characteristic of a lubricant. The increasing of service life of friction units is achieved due to the use of a complex of additives and physicochemical technologies of surface modiﬁcations of rubbing bodies. Effects of vacuum, temperatures, different aspects of an irradiation (infrared, x-ray etc.), a stream of electrons on a surface of materials inﬂuence the thermo-kinetic processes which are ﬂowing in the volume — diffusion, degassing,

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sublimation etc. become more active. Thus the chemical composition, structure and physical-mechanical properties, dimensional stability, relaxation chracteristics vary. Ultimate conditions for a friction unit are characterized by: heightened value of forces (rate) of friction as a result of intensive adhesive interaction of rubbing surfaces, wear process of antifrictional wear-resistant coatings, exhaustion of a lubricant, change of clearances in process of a wear at interface. There is a probability of jamming from abrasive’s droppings and from temperature deformations. Process of seizing (grip) of rubbing parts is dangerous. Typical friction units of mechanical systems on the Moon: cogged and wave transmissions, bearings of rolling and sliding, guiders, cam mechanisms, screw transmissions, sealing elements, demountable and carving connections, robot mechanisms and manipulators, joints, locks etc. Violation of rules of tribological design, manufacturing and operation of friction units results in emergencies and catastrophes in Space [6]. Provision of Tribological Reliability. In the sixtieth of the last century in the USSR wide researches of tribological reliability of mechanical systems under conditions similar to that of the Moon were conducted by integrated efforts of institutes of Academy of Science of the USSR, branch institutes, higher educational institutions. New laboratories, experimental stands, testing ranges were created. Model experiments in conditions of Space were carried out. For example, on AIS “Luna10” (31.03.1966) and “Luna-11” (24.08.1966) the experimental reducer “R-1” was tested which worked in a free Space and transmitted data from lunar orbit about serviceability, temperature conditions and losses of energy on friction. Calculationtheoretical methods on the determination of the efﬁciency of geared systems of moon-rovers were developed. Naturally, at the ﬁrst stage of friction unit creation there was the desire to use traditional methods of calculation and technique of friction unit design accepted in general machine industry. Experiments, modeling the Moon condition in laboratory conditions resulted in a full disappointment. The mechanism of destruction of conjugated surfaces essentially differed from the one realized under earth conditions. Methods of calculation appeared unusable, conventional rubbing materials were unsuitable. The attention of scientists and engineers was attracted to emergency and catastrophe, happened in Space. Owing to friction in space conditions the mechanism of the plotter of infrared vertical broke down during the ﬂight of the ﬁrst unmanned spaceship “Vostok” (on 15th May 1960). Cosmonaut V.M. Komarov was killed (on 24th April 1967, “Soyuz-1”) — efforts of a parachute exhaust system did not sufﬁced to overcome forces of friction and to draw out the basic parachute. As a result of a rigid landing there was a strong impact, explosion, ﬁre. For the friction units to be maintained on the Moon, special composite, selflubricating materials, solid-lubricating coatings, surface modiﬁcations, methods of lubrication etc. were created. In connection with absence of a solid lubricant in open friction units there is a continuous wear during maintenance of dry rubbing

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parts. The design procedure of a service life calculation by a wear criterion, essentially distinguished from the conventional one [7–12], was created. Magnet-active powder methods of lubrication which essentially increase service life of gear systems in conditions of the Moon were created for the ﬁrst time [11–13]. Method of research of magnet-active lubrications is scientiﬁcally justiﬁed and the complex of the experimental equipment for the research of heavy-loaded gears is created [11–14]. On the basis of ﬁber optics in enclosed volume the visual and photo-observations were carried on with the purpose of research of wear and performance of lubrication in the magnetic ﬁeld during the operation. Application of magnet-active lubricants allows (10 times) increasing the durability of heavy-loaded transmissions, as contrasted to durability of the transmissions made of self-lubricating composite materials and with solid-lubricating coatings. Use of the determined and stochastic criteria of a similarity in the analysis of processes of abrasion, wear process and seizing allows one to take into account physical processes and to evaluate reliability at effect of random factors. Friction units on the Moon till now are unique articles and there is no yet sufﬁcient statistical information on their load and real condition. The statistical exposition of probability events in friction units will develop in process of the mastering the Moon. In friction units of “Lunokhod-1” and “Lunokhod-2” the following materials have found application as a result of cooperative research of many organizations [14]: thermally processed ceramic-metal iron-glass material C-5 for gears and sliding bearings; titanic alloy BT-14 with nitriding for epicycle of planetary gear-box; solid-lubricating coating STM-1 on a basis of molybdenum disulﬁde for open friction pairs; grease VNIINP-246 on a basis of ploychlorosiloxane for lubricating of gears and rolling bearings; self-lubricating polymeric material FN-3 on the basis of nickel and ﬂuoroplastic which was applied to sealing elements. The composite antifrictional material, consisting of rust-resistant steel with a working surface modiﬁed by a copper-silver alloy, a material was applied in the sliding bearings working in pair with titanic alloy VT-14, having a nitrated surface. For the ﬁrst time gravitation forces were taken into account in the calculations of efﬁciency of gear-boxes on the Moon [11]. The accepted scientiﬁcally-based approach to the methodology of ground research in laboratory conditions, in model tests in Space, the designed cycle of trials and techniques of design were the basis of successful functioning of friction units on the Moon. Conclusion. Provision of reliability and service life of friction units on the Moon represents a difﬁcult and complex scientiﬁc and technical problem. Carried out system research of the friction units in the second half of XX century have created base for the future mastering the Moon and other planets. To the achievements of science and technology of the Soviet Union which originated the space age — the

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ﬁrst artiﬁcial satellite of the Earth was put into orbit, the ﬁrst ﬂight of a man in space was realized, an extra vehicular activity of a man in outer space was undertaken, lunar devices with soft landing and mobile laboratory “Lunokhod” and others were created — it should be added the following: the experimental-theoretical methods of provision and prediction of friction unit reliability for extreme conditions of the maintenance on the Moon were created for the ﬁrst time. Subsequently these achievements have found their application in general machine industry, aerospace machinery, atomic engineering and others. In future, for creation of heavy-loaded friction units for a long service life for engineering systems of the Moon not only accumulated experience and designed techniques, but also wide-scale research and development activities in tribological ﬁeld will be required.

REFERENCES 1. Raushenbakh, B.V. ed.: Korolyov and His Undertaking [In Russian]. Moscow: Nauka 1998. 2. Feoktistov, K.P.: Space Technology. Prospects of Development [In Russian]. Moscow: Izd-vo MGTU imeni N.E. Baumana, 1997. 3. Planet-rovers (In Russian). ed.: A.L. Kemurdzhian. Moscow: Mashinostroenie 1993. 4. Meshcheryakov, I.V. In the World of Cosmonautics [In Russian]. Nizhny Novgorod, 1996. 5. Kragelskiy, I.V., Lubarskiy, I.M., Guslyakov, A.A. Friction and Wear in Vacuum (In Russian). Moscow: Mashinostroenie 1973. 6. Drozdov, Yu.N. Tribological Aspect of Reliability Problem of Space Technique (In Russian). Problems of engineering and machine reliability. 2000. No 1. 7. Drozdov, Yu.N. To the Calculation of Gears for Wear (In Russian). Machine Science. 1969. No 2. 8. Drozdov, Yu.N. Principal Scheme of Calculation of Machine Parts for Wear (In Russian). Proceedings of Perm Polytechnic Institute. 1970. No 62. 9. Drozdov, Yu.N. To the Development of Calculation Method for Wear Process and Friction Simulation (In Russian). Endurance Collection. Ed. A.A. Blagonravov, R.M. Matveevsky. Moscow: Nauka. 10. Drozdov, Yu.N. Development of a Method for the Calculation of the Life of Dry Frictional Contacts under Extreme Operating Conditions. Wear, 38 (1976), 217–223. 11. Drozdov, Yu.N. Transmissions. Tribology — Lubrication, Friction, and Wear. Edited by J.V. Kragelsky and V.V. Alisin. Professional Engineering Publishing Limited (2001) pp. 597–639. 12. Drozdov, Yu.N., Pavlov, V.G. and Puchkov, V.N. Friction and Wear in Extreme Condition (In Russian). Moscow 1986. 13. Vaisefeld, L.O., Drozdov, Yu.N., Kemurdzhian, A.L. Efﬁciency of Gear Drives with Magnetic-Powder Lubrication in the Different Environmental Conditions (In Russian). Engineering Bulletin. 1979, No 3. 14. Vaisefeld, L.O., Mitskevich, A.V., Tarasov, V.M. Scientiﬁc-technical Solutions on Friction Units of Space Apparatus (In Russian). Transport Engineering Bulletin. Moscow, 1995, No 3–4.

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PRODUCTION PROCESS PROCEDURES & MACHINES A.M. Dmitriev (Bauman Moscow State Technical University), A.L. Vorontsov (Moscow State Academy of Instrument Engineering & Informatics)

ANALYSIS OF EXTRUSION PROCESS OF CORELESS CYLINDRICAL PRODUCTS WITH AN EXTERNAL CORE IN THE BOTTOM The stress condition of a part is determined for the operation of extruding coreless cylindrical products with an external core. Formulas for calculating of the basic technological parameters of the combined extrusion process have been obtained. Accuracy of the obtained theoretical results is conﬁrmed by comparing them with experimental data.

In machine-building engineering the parts schematically shown in Fig. 1 are used. The most effective way to manufacture them is the extrusion with two simultaneous metal ﬂow directions that results in reduction of speciﬁc force as opposed to the traditional extrusion. For calculation of the basic technological parameters of such extrusion there were no formulas derived on the basis of a uniﬁed method. The extrusion of products with an external core can look as it is shown in Fig. 2. For a small relative size of the matrix aperture (Fig. 2, a), the process will begin with the traditional extrusion of walls with growth of their height. For a product with the dimensions ratio as shown in Fig. 2, b, on the most part of a working travel there may be a direct extrusion of a core, and formation of a cavity of a product will only begin at a ﬁnal stage.

Fig. 1. Combined extrusion of coreless cylindrical products with an external core in the bottom 252

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Fig. 2. Variants of forms during the extrusion of coreless cylindrical products with an external core

If the product has the dimensions ratio as shown in Fig. 2, c, the cavity may be formed due to the shift of the metal located under a punch, in the matrix aperture. If the values of speciﬁc forces for extrusion of the cavity and core are close to each other, formation of these elements will begin practically simultaneously (Fig. 2, d). On the matrix-part contact surface friction forces compensate each other; therefore the constant of friction on this surface is equal to 0. Substituting this value into the formula (2.9) from [2], we determine the relative speciﬁc extrusion force by the formula ¸ · hy 0.5 + µ1 (1) + + qtr q = 1,1 2 + ln R + 2(R2 − 1) 4hy where µ1 is the constant of friction between the part and punch. In this and subsequent formulas, we use relative values. Forces are divided by σs — the average pressure of ﬂuidity of the material of parts in the plastic deformation zone. The values of the parameters are divided by the radius of a working site of the punch: r = 1, R — relative radius of a matrix. The height of the top zone of plastic deformation hy is determined using the following formulas. If deformation hardening is insigniﬁcant (hy = h), s (R2 − 1)(0.5 + µ1 ) h= . (2) 2(1 + 2µR) When the high quality greasing is used, µ, µ1 and µ0 (Fig. 1) equal to 0.1, with lower quality greasing — µ, µ1 and µ0 = 0.3. In the presence of deformation hardening hy = h[1 + ky (1 − 0.2e−s − 0.8e−5s )],

(3)

where the factor of hardening is µ ¶ σs2 /σs1 − 1 ky = 1 − exp −10 , e2 − e1

(4)

s is the relative length of a working travel for which we determine cumulative deformations, e is the natural logarithm basis, and σs1 and σs2 are the pressures VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

253

of the ﬂuidity taken from the hardening curve of the given material at cumulative deformations e1 = 0.2 . . . 0.4 and e2 = 1 . . . 1.2. Coefﬁcients of hardening of various materials that have been obtained are presented in table 1. Table 1 Coefﬁcients of hardening of materials

Material

e1

σs1 , MPa

e2

σs2 , MPa

ky

Aluminium alloy AB

0.4

240

1.2

300

0.956

Aluminium alloy D16

0.2

290

1.2

405

0.981

Steel 10

0.2

470

1.2

705

0.993

Steel 20

0.2

610

1.0

980

0.999

Steel 12X18H9T

0.1

600

0.6

1000

1.000

Steel 12X18H9T(training)

0.1

560

0.6

960

1.000

Calculations demonstrate that the values of coefﬁcient of hardening are in between 0.9 and 1. Therefore it is possible to accept ky = 0.95. Inﬂuence of the matrix deﬂection on the speciﬁc force of extrusion in the formula (1) according to [3] is described by the formula

qtr = 1.1

µR3 s. (R2 − 1)2

(5)

The extrusion force will grow until the working travel will reach the value

str =

(R2 − 1)2 , R2 (1 + µR)

(6)

which corresponds to its maximum value:

qtr = 1.1

µR . 1 + µR

(7)

If the current value is s < str , then it is necessary to use expression (5), and if it is s ≥ str , then it is necessary to use the formula (7). When the thickness of the bottom of the part decreases up to the size s (R2 − 1)(µ0 + µ1 ) Hc = , (8) 2(1 + 2µR) there will be a merge of the top and the bottom plastic deformation zones (Fig. 1). The total height of the plastic deformation zone becomes equal to the thickness of the bottom of the part H. 254

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

In the theoretical model shown in Fig. 1, the cylindrical coordinate system ρ, θ, z, is used, and the plastic deformation zone is represented as four areas numbered: 1 and 2, 3 and 4. The following assumptions are made: (1) the material is hard plastic, and hardening is described by average pressure of ﬂuidity in the zone of plastic deformation, σs ; (2) contact friction forces are determined from Zibel0 s law as τk = µβσs , where β — is Lode0 s coefﬁcient. Area 1. Axial speed of ﬂow is accepted as: vz = A[z − ϕ(ρ)],

(9)

where A — the factor determined from the volume stability condition, and ϕ(ρ) is an arbitrary function. Function ϕ(ρ) may be determined by an estimation procedure presented in [4]. From the volume stability condition, which looks like ∂vρ vρ ∂vz + + = 0, ∂z ∂ρ ρ or ¸ · 1 ∂ ∂vz (vρ ρ) = − , ρ ∂ρ ∂z

(10)

it follows, that A f (z) vρ = − ρ + . 2 ρ Taking into account the boundary condition vρ = 0 at ρ = r0 f (z) =

A 2 r . 2 0

we obtain the radial speed: µ ¶ A r02 −ρ . vρ = 2 ρ Deformations speeds: ¶ µ ∂vρ A r02 +1 , =− ξρ = ∂ρ 2 ρ2 ¶ µ vρ A r02 ξθ = −1 , = ρ 2 ρ2 ∂vz ξz = = A, ∂z ∂v ∂v ηρz = ρ + z = −Aϕ0 (ρ), ∂z ∂ρ

(11)

(12)

and deformation speed intensities: ξi = β|ξmax | = βξz = βA. VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

(13) 255

The equations of balance look like: ∂τ σ − σθ ∂σ ρ + ρz + ρ = 0, ∂ρ ∂z ρ ∂σ ∂τ τ z + zρ + zρ = 0. ∂z ∂ρ ρ

(14)

From Levy–Mizes equations, taking into account formula (13) and system (12) it follows, that τρz depends only from ρ, and 2 2r2 (ξρ − ξθ ) = − 02 , 3ξi 3βρ

σρ − σ θ =

(15)

From the ﬁrst equation of balance (14)

σρ = −

r02 + f1 (z) + C1 . 3βρ2

(16)

Taking into account this expression and the condition of plasticity σz − σρ = β

(17)

we can write the second equation of system (14) in a form: ∂τρz τρz ∂f1 (z) =− − . ∂z ∂ρ ρ

(18)

Since the left-hand part of the equation (18) depends only on z, and the righthand part only on ρ, both of these parts are equal to constant C2 . The right-hand part of the equation (18) can be presented as −

1 ∂ [τρz ρ] = C2 , ρ ∂ρ

then C2 ρ C 3 + . (19) 2 ρ As the left-hand part of the formula (18) is equated with the constant C2 , then τρz = −

f1 (z) = C2 z.

(20)

From the boundary conditions τρz = −µ2 β

at

ρ = r0 ,

τρz z = 0.5β

at

ρ = r1 ,

it follows that r1 + 2µ2 r0 C2 = −β r 2 − r2 , 1 0 r 0 + 2µ2 r1 . C3 = −0.5βr0 r1 2 r1 − r02 256

(21)

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

The solution to equations (17), (16) and (20) gives:

σz = β −

r02 + C2 z + C1 . 3βρ2

(22)

Using the condition σz = 0 at z = −H and ρ = r1 , we determine the constant C1 = −β +

r02 + C2 H. 3βr12

(23)

The solution to equation (17), (22) and (23) at z = −H/2 and ρ = r1 gives an average radial pressure from area 1 to area 2: σρ1 = −β + 0.5C2 H.

(24)

Area 2. The axial speed looks like: vz = −f2 (z). Taking into account vρ = 0 at ρ = Rg we determine: µ ¶ 1 ∂f2 (z) ρ2 − Rg2 vρ = . 2 ∂z ρ From the common parts of equation (12) speeds of deformations are determined: ∂f2 (z) , ξz = − ∂z µ ¶ R2g 1 ∂f2 (z) ξρ = 1+ , 2 ∂z ρ2 ¶ µ (25) R2g 1 ∂f2 (z) ξθ = 1− 2 , 2 ∂z ρ ¶ µ 2 2 1 ∂ f2 (z) ρ − Rg2 . ηρz = 2 ∂z 2 ρ Similarly to (13), ∂f2 (z) . ∂z From (25) and (26) and of Levy–Mizes equations: ξi = β

τρz

∂ 2 f2 (z) µ µ 2 ¶ ¶ ρ − Rg2 ρ2 − Rg2 1 2 ∂z = = f3 (z) . 6β ∂f2 (z) ρ ρ ∂z

Substituting (27) into the second equation of system (14) gives: Z σz = −2 f3 (z) dz + f (ρ) + C. VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

(26)

(27)

(28) 257

Substituting (28) into the condition of plasticity σρ − σz = β, we obtain

Z σρ = β − 2

f3 (z) dz + f (ρ) + C.

(29)

Further the condition of plasticity is used in the form σp − σθ = β.

(30)

Substitution of formulas (27), (28) and (20) in the ﬁrst equation of system (14) gives: · ¸ ρ ∂f (ρ) β ∂f3 (z) + . =− 2 2 ρ − Rg ∂ρ ρ ∂z Since the left-hand part of this equation depends only on ρ, and right-hand part only on z, then both of these parts should be equal to constant C4 , whence it follows that: f3 (z) = −C4 z + C5 , f (ρ) =

C4 ρ2 − (β + C4 Tg2 ) ln ρ. 2

(31) (32)

Taking into account (31) from the formula (27) we determine

τρz

µ 2 ¶ Rg − ρ2 = (C4 z − C5 ) . ρ

(33)

From the boundary conditions τρz = −βµ0 τρz = 0.5β

at at

ρ = r1 and z = −H, ρ = r1 and z = 0,

it follows that arbitrary constants in formula (33) are equal to (34): β(0.5 + µ0 )r1 C4 = H(R2 − r 2 ) , g 1 βr 1 C5 = − 2(R2 − r 2 ) . g 1

(34)

After substituting formulas (31) and (32) into formulas (28) and (29) we obtain: ( σz = (C4 z − 2C5 )z + C4 (0.5ρ2 − R2g ln ρ) − β ln ρ + C, (35) σρ = β + (C4 z − 2C5 )z + C4 (0.5ρ2 − Rg2 ln ρ) − β ln ρ + C.

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VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

To determine the arbitrary constant C we use the average value σρ1 on the border between areas 1 and 2, which is determined by equation (24). Equating it to the value σp from the system (35) ρ = r1 and z = 0 gives its value: C = β(ln r1 − 2) + 0.5C2 H − C4 (0.5r12 − Rg2 ln r1 ).

(36)

Force of extrusion is determined by the sum of forces in areas 1 and 2: P = P 1 + P2 .

(37)

From (22) and (23) at z = 0 Zr1

·

µ

2r02 r1 |σz |ρ dρ = π − ln 3β r0

P1 = 2π

¶ ¸ r02 2 2 C2 H (r1 − r0 ) . 3βr12

r0

It is possible to neglect the following terms of equations 2r02 r1 ln 3β r0

and

−

r02 (r2 − r02 ), 3βr12 1

obtaining P1 = −πC2 H(r12 − r02 ).

(38)

From formulas (35) and (36) at z = 0 ZRg |σz |ρ dρ = π[(2β − 0.5C2 H + 0.5C4 )(Rg2 − r12 )−

P2 = 2π r1

− 0.25C4 (Rg4 − r14 ) + (β + C4 R2g )(R2g ln Rg − r12 ln r1 − 0.5R2g + 0.5r12 )]. (39) From formulas (21), (34), (38), (39) and the average value of Lode0 s coefﬁcient β = 1.1 the speciﬁc force working on a punch from areas 1 and 2 is determined: q1 =

P 1 + P2 1.1 = 2 1,5(Rg2 − r12 )+ 2 2 π(R − r0 ) Rg − r02 ·

¸ (0.5 + µ02 r0 )(R2g + r12 − 2r02 ) − (µ − µ )r 1 0 1 H+ r12 − r02 ¸ · µ ¶ Rg 4 2 2 4 − 0.75 + Rg r1 − 0.25r1 r1 (µ0 + µ1 ) Rg ln R r g 1 2 + Rg ln + . (40) r1 Rg2 − r12 )H VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

259

In the formula (40) a value of the calculated friction factor µ02 is chosen according to the type of the mandrel used in a process. If the mandrel is rigidly attached to the punch, then µ02 = 0.5µ2 . If the mandrel is rigidly attached to the basis of the stamp, then µ02 = µ2 . If the mandrel moves forcedly in the direction of extrusion of a tubular core, then µ02 = −µ2 . Area 3. The axial speed is similar to the one in formula (9). In this case, as shown in [4] for the traditional extrusion, the stress-stain condition looks like: C6 ρ C7 + , τρz = − 2 ρ R (41) σρ = −β − β ln + C6 z − qtr , ρ σz = σρ + β, where qtr is determined by formula (7), and 1 + 2µR , C6 = β 2 R −1 (42) C7 = 0.5βR R + 2µ . R2 − 1 From the second formula of system (41) at ρ = R and z = −H the maximum pressure upon a wall of the matrix is determined: ¸ · 1 + 2µR H + qtr . (43) p = 1.1 1 + 2 R −1 At ρ = 1 an average value of a radial pressure on the border between areas 3 and 4 is equal to: ¸ · 1 + 2µR H − qtr . (44) σρ2 = −β 1 + ln R + 2(R2 − 1) Area 4. The axial speed looks like: vz = −f4 (z). Using the boundary condition vρ = 0 at ρ = Rg , we get: µ ¶ 1 ∂f4 (z) ρ2 − R2g vρ = . 2 ∂z ρ According to the formula (12), speeds of deformations are equal to: ∂f4 (z) , ξz = − ∂z ¶ µ Rg2 1 ∂f4 (z) 1+ 2 , ξρ = 2 ∂z ρ µ ¶ (45) Rg2 1 ∂f4 (z) 1− 2 , ξ = θ 2 ∂z ρ ¶ µ 2 2 1 ∂ f4 (z) ρ − R2g . ηρz = 2 ∂z 2 ρ 260

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

According to (13), the intensity of deformation speeds is equal to: ξi = β

∂f4 (z) . ∂z

(46)

From formulas (45) and (46) the formula similar to (27) is obtained: ¶ µ R2g τρz = f5 (z) ρ − . ρ

σρ − σθ =

2 2 Rg2 (ξρ − ξθ ) = . 3ξi 3β ρ2

(47)

(48)

From the second equation of system (14), being solved together with (47), it follows that Z (49) σz = −2 f5 (z) dz + f1 (ρ) + C8 . The joint solution of formulas (47), (48) and conditions of plasticity σρ − σz = β,

(50)

and also (49), and the ﬁrst equation of system (14) results in formula: µ ¶ ρ 2 Rg2 ∂f (ρ) ∂f5 (z) + . =− 2 2 3 ρ − Rg ∂ρ 3β ρ ∂z As the left-hand part of this equation depends only on ρ, and the right-hand part one only on z both of these parts should be equal to constant C9 , and thus: f5 (z) = −C9 z + C10 ,

(51)

ρ2 1 Rg2 + C − C9 Rg2 ln ρ. 9 3β ρ2 2 By substituting (51) in the formula (47) we determine, that µ 2 ¶ ρ − Rg2 τρz = (C10 − C9 z . ρ f1 (rho) =

(52)

(53)

From the boundary conditions τρz = −βµ1 at ρ = 1, z = 0 and τρz = βµ0 at ρ = 1 and z = −H, it follows, that constants in (53) are equal: β(µ0 + µ1 ) C9 = H (1 − R2 ) , g (54) βµ 1 C10 = − 1 − R2 . g As a result of substituting (51) and (52) in (49), we determine that: σz = (C9 z − 2C10 )z + C9 (0.5ρ2 − R2g ln ρ) + VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

1 Rg2 + C8 . 3β ρ2

(55) 261

The radial pressure determined by substituting (55) into (50) looks as follows: σρ = β + (C9 z − 2C10 )z + C9 (0.5ρ2 − Rg2 ln ρ) +

1 R2g + C8 . 3β ρ2

(56)

To determine the constant C8 we use the average value σρ2 from formula (44), on the border between areas 3 and 4, which is determined on the assumption of at σρ = 1 and z = 0. As a result: C8 = −

R2g 1 + 2µR C9 − − 2β − β ln R − H − qtr . 2 3β 2(R2 − 1)

(57)

Using formulas (7), (54), (55) and (57) at z = 0, and also an average value of Lode0 s coefﬁcient β = 1.1 we can determine the speciﬁc force applied to the punch from area 4: Z1 2 q2 = 1 − Rg2

½ |σz |ρ dρ = 1.1 2 + ln R +

1 + 2µR H+ 2(R2 − 1)

Rg

¾ (µ0 + µ1 )[1 − 4Rg2 + Rg4 (3 − 4 ln Rg )] µR + + . (58) 4(1 − R2g )2 H 1 + µR The relative speciﬁc force of the combined extrusion: q=

q1 (1 − R2g ) + q2 (R2g − r02 ) . 1 − r02

(59)

The Rg value is determined from the condition of a minimum of formula (59). With the accuracy sufﬁcient for practical calculations of speciﬁc force, the value of Rg is determined using the following approximated formula: Rg =

Rr1 − r0 . R − 1 + r1 − r0

(60)

To verify the theoretical equations (40), (58), (59) and (60) experiments on extrusion of products made of lead C00 were performed. When comparing experimental and theoretical values of relative speciﬁc forces it was assumed, that the relative thickness of the bottom of a product H = Hc . The results of comparison of calculations with experimental data are given in Table 2. Table 2 Comparison of theoretical and experimental values of the relative speciﬁc force of extrusion of products with an external core made of lead C00 at r0 = 0 R

r1

µ

µ1

µ0

H

Rg

q1

q2

q

qe

δ, %

1.07

0.49

0.3

0.3

0.1

0.133

0.936

3.440

2.623

3.584

3.47

3.2

1.07

0.41

0.1

0.1

0.3

0.155

0.914

3.174

3.179

3.820

3.77

1.3

1.22

0.57

0.3

0.3

0.3

0.291

0.880

3.371

2.268

3.728

3.50

6.1

1.22

0.49

0.3

0.3

0.1

0.237

0.842

3.274

2.518

3.603

3.58

0.6

262

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Since comparison of theoretical and experimental results shows that they are in a good agreement, we believe that the formulas obtained in this work can be recommended for use in practical calculations.

REFERENCES 1. Cold die forging. The directory / Edited by G.A.Navrotskiy. Moscow: Publishing House “Machine building”. – 1973. 496 p. 2. Dmitriev, A.M., Vorontsov, A.L. Technology of forging and die forging. Part 1. Die forging by extrusion. The textbook for high schools recommended by the Ministry of Education of the Russian Federation. Moscow, Publishing House “The Higher School”. – 2002. 400 p. 3. Dmitriev, A.M., Vorontsov, A.L. Deﬁnition of technological parameters of die forging by extrusion with an allowance of elastic deformation of a matrix // Vestnik of Moscow State Technical University named after Bauman. – 2002. No 2. PP. 76–93. 4. Osadchij, V.J., Vorontsov, A.L., Beznosikov, I.I. The theory and calculations of technological parameters of die forging by extrusion. The manual for high schools recommended by the Ministry of Education of the Russian Federation. Moscow, Moscow State Academy of Informatics. – 2001. 307 p. 5. Dmitriev, A.M., Vorontsov, A.L. Estimation of loading on the extruding tool for parts with a through step cavity. Part 1 // Forging-stamping manufacturing Journal. 2002. No 10. PP. 21–28. 6. Dmitriev, A.M., Vorontsov, A.L. Estimation of loading on the extruding tool for parts with a through step cavity. Part 2 // Forging-stamping manufacturing Journal. 2002. No 11. PP. 21–28. A.M. Dmitriev (b. 1948) graduated from the Bauman Moscow Higher Technical School in 1972. Corresponding member of the Russian Academy of Sciences, D. Sc. (Eng.), professor, head of “Technology of Metal Forming” Department of the Bauman Moscow State Technical University. Author of 201 publications in the ﬁeld of plastic metal working. A. L. Vorontsov (b. 1955) graduated from the Bauman Moscow Higher Technical School in 1978. D. Sc. (Eng.), professor of “Applied Mechanics” Department of the Moscow State Academy of Instrument Engineering and Informatics (MGAPI). Author of 151 publications in the ﬁelds of plastic metal working and applied mechanics.

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

263

I.N. Shiganov (Bauman Moscow State Technical University)

FUSION WELDING OF METALLIC COMPOSITE MATERIALS Physical and metallurgical features of fusion welding of metallic composite materials, strengthened by particles, are considered. Different materials are classiﬁed by a weldability, which allows the possible defects during a welding to be predicted. Physical effects, accompanying the forming of a weld during the argon-arc, electron-beam and laser welding, are considered. Typical defects of welds are revealed and a mechanism of their formation is determined. On the basis of conducted researches we propose some technological techniques for these types of welding process which enable avoiding the welding defects formation. Our researches and developed technologies have made possible to get the good quality of welds for different groups of metallic composite materials, such as Al–Be, Al–Be–Mg, Fe–Cu, Fe–Cu–Pb, Fe–Cu–Pb–Sn, Al–Pb, with a thickness from 2 to 20 mm.

Classiﬁcation of Composites by Weldability. The application of materials with speciﬁed properties — metallic composite materials (MCM) — results in completely new technical characteristics of products. In developing structures made of MCM, it is necessary to use the fusion welding, especially welding by highproductivity methods, for example, using a laser [1]. It is well known [2] that melting of a composite material is accompanied in the majority of cases by the disruption of its initial structure, resulting in a deterioration or large change of the physical and mechanical properties. On the basis of the available data on welding dissimilar and composite materials [3], it is possible to deﬁne four groups of composites considering the interfacial interaction of their components in fusion welding. Group 1 (Fig. 1, a) includes MCM whose components form a homogeneous liquid during melting and are almost insoluble in each other in the solid state: Fe–Cu, Al–Be, Ti–Mg, etc. Melting and solidiﬁcation of these alloys result in the formation of a uniform heterogeneous structure with alternating particles of the matrix and the hardening agent. Fusion welding of these materials is most promising because the structure of the composite is distorted to a lesser degree, and, consequently, the properties of the weld metal differ only slightly from those of the parent metal. Group 2 (Fig. 1, b) includes MCM whose components in melting and solidiﬁcation have limited or unlimited solubility: Nb–W, Ni–W, Al–Si, Co–Cr, etc. Melting of these composites is accompanied by the formation of solid solutions with a smoothly changing concentration. The strength can be close to that of the parent metal. In welding MCM with limited solubility of the components the weld can contain eutectics and peritectics, in addition to solid solutions, depending on the equilibrium diagram.

264

VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

Fig. 1. Structural state of welded composites: a — group 1, b — group 2, c — group 3, d — group 4

Group 3 (Fig. 1, c) includes MCM whose components do not interact during welding. This is associated with the relation between the melting points of the refractory component and the boiling point or the low-melting component. The two liquid phases can not be in equilibrium and, consequently, no superheating would increase the mutual solubility of the components in the liquid state. In most cases, this group includes not two-component but three-component materials and materials with a larger number of components: Al–Be–Mg, Fe–Mg, Fe–Cu–Pb, W–Ag, etc. Welding of these materials is difﬁcult without using special measures suppressing evaporation of the low-melting component. Group 4 (Fig. 1, d) includes MCM whose components are not miscible in the liquid state: Fe–Pb, Fe–Cu–Pb, Al–Pb, etc. In fusion welding, the joint easily fails through the fusion area. This classiﬁcation is rather conventional because, under the effect of the welding thermal-diffusion cycle, MCM can belong to several groups at the same time. However, the classiﬁcation does make it possible to specify the main groups of MCM and the criteria that can be used to evaluate their weldability VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

265

In this work, investigations were carried out into materials of the Al–Be–Mg, Fe–Cu–Pb and Al–Pb systems characterized by the formation of the largest number of defects in welding. Research of Materials of Al–Be–Mg System. The composites of the Al– Be–Mg system have an aluminum matrix with beryllium particles distributed in it. Magnesium is in the solution with aluminum. The mostly used alloys of this type are: Al–30Be–5Mg, Al–40Be–3Mg.The unique feature of these materials is that they combine low density (2.1 − 2.3 g/cm3 ) with the high melting point (1150 − 1160 ◦ C), strength (450 MPa) and elasticity modulus (140 − 150 MPa). A combination of high strength and rigidity of these materials gives them advantages in comparison with the structural alloys based on aluminum, titanium and magnesium. These materials are used to produce large-sized components for the aerospace, aviation and ship building industries, with all component structures are mostly welded (girders, casings, antennae, fuel tanks, platforms, etc.). The thickness of the welded materials varies from 2 to 20 mm. Usually they are applied for butts welded with continuous penetration. Argon-arc welding is used to weld MCM up to 3 mm thick. At larger thickness there are defects in the form of the weld lowering below the joint level, i.e. sagging associated with the existence of the solid-liquid state in the weld zone for a long period of time which does not guarantee the required strength of the metal (the aluminum matrix is still melted, whereas the beryllium grains are already solidiﬁed). The results of examination of the effect of the welding conditions showed special features of the welded materials up to 3 mm thick. The results show that, over the entire range of the conditions, beryllium concentrates in the heat affected zone (HAZ) in the form of an almost continuous band (Fig. 2, a). This zone is weakened under impact loading, is subjected to corrosion and does not withstand vibration loads, i.e. many requirements imposed on the important components are not fulﬁlled in this case. The results of modeling of the process, physical investigations and calculations show that the enrichment of the weld zone with Be is caused by its thermal diffusion in the aluminum melt. The criterion for the start of this process is the time (0.9–1.5 s) of holding the metal at the weld boundary at a temperature higher than 1500 ◦ C. The changes of the welding conditions did not make it possible to overcome this criterion. The holding time of the metal at the given temperature was reduced by using copper heat-removing straps of the appropriate weight and design. The structure of the weld after welding with heat-removing devices is shown in Fig. 2, b. Another method is the use of concentrated heat sources. Composite materials 3–20 mm thick were joined by electron beam welding. The vacuum chamber protected the operator against harmful beryllium fumes. The results of initial experiments with melting the material 5–10 mm thick showed that there are no defects in the HAZ as a result of the rigid thermal cycle of the concentrated heat source. However, it was not possible to produce welded joints due to speciﬁc defects in the form of a longitudinal cavity (Fig. 3) — the metal was ejected from the melt to the surface. Defects of this type were not detected in any of the materials used in electron beam welding.

266

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Fig. 2. Microstructure of welded joints in Al–30Be–5Mg composite with (a) and without (b) heat removal

Application of ﬁlming for the examination of physical processes in the penetration channel: evaporation dynamics, vapor composition, pressure in the channel, etc., has made it possible to determine the mechanism of formation of an unusual defect. Its formation is caused by the special feature of the structure of the material evident under the vacuum and high temperature conditions: at the melting point of the material (1150 ◦ C), the magnesium, present in it, already has reached the boiling point (1060 ◦ C). This combination of temperatures under vacuum conditions at a relatively high magnesium content (up to 5 %) leads to the development of the processes of volume boiling of the melt and its removal from the penetration channel to the surface. The main criterion for the start of boiling under the welding conditions is that the pressure in the gas bubbles should be

Fig. 3. Macrostructure of welded joints in Al-30Be-5Mg composite after welding with electron beam (without oscillations)

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higher than the external pressure (p0 > p) [4]. The defect formation criterion is the critical size of the bubble at which the metal is ejected. These conditions are described by the following equation: µ ¶ 2Vm σ 2σ . (1) p0 exp − >p+ rcT r where p0 and p are pressures of the vapors in the bubble and above the melt surface, respectively; Vm is a speciﬁc volume of the liquid; σ is a surface tension coefﬁcient; r is a critical radius of the bubble; c is the heat capacity; T is the temperature. The experiments and calculations show that for the Al–Be–Mg system there are limiting permissible amounts of easily evaporated elements, present in the material. For example, an equation was derived for determining the critical molar fraction of magnesium in relation to the beryllium content:

NMgcr =

0.38 4CBe , 0.11 − 2800 2457CBe

(2)

where CBe is the beryllium content in the alloy. The experiments show that in the alloys with NMg < NMgcr , the effects were not found, whereas at NMg > NMgcr , there are defects in the form of a longitudinal cavity. For Al–30Be–5Mg alloy used often in structures, NMg = 0.0348 and NMgcr = 0.0188, i.e. this alloy cannot be welded by electron beam welding. Analysis of the physical mechanism of the process shows that the defect can be eliminated only by reducing the magnesium content of the welding zone and by suppressing the boiling processes. Boiling can be suppressed by applying longitudinal transverse oscillations of a beam. The magnesium content in the welding zone was reduced by placing interlayer of aluminum alloy with no magnesium in the joint between the edges of the components to be welded. The relation between the fraction of the parent metal and the interlayer material in the weld was determined using the equation µ ¶ δ ∆Pb = 1 − · 100 %, (3) a+b where ∆Pb is a fraction of the parent metal; δ is a thickness of the interlayer; a and b are widths of the weld in the upper and central parts, respectively. The best results for a metal thickness of 10 mm were obtained at an interlayer thickness of 1 mm. This technology was used to weld a number of important components. Highquality joints were produced in the metal 5–20 mm thick (Fig. 4), with the mechanical properties of the joint close to those of the parent metal. No defects in the form of pores, cracks or cavities were found. Research of Materials of Fe–Cu–Pb System. Another main type of composites is the Fe–Cu–Pb system, which components are not miscible in the liquid and solid states. When they are melted together, it is not possible to produce a homogeneous structure of the ingot because of phase separation of lead with iron and their 268

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liquation. Copper with lead also have a phase separation region with a temperature range 1500 − 1536 ◦ C [5]. The structure consists of a copper-lead matrix with iron intrusions. Similar materials have been produced in sufﬁciently large amounts quite recently by contact alloying [6]. They have high damping properties and an exceptionally high friction factor with sufﬁciently high strength. Iron-copper composites substitute for many types of bronze in producing spring elements, roller bearings, compressor blades, piston rings, toothed wheels, etc. All these components require fusion welding. Since the welded thickness varies from 0.5 to 5 mm, it is convenient to examine the application of argon-arc welding and laser welding for joining these materials. The experiments, carried out with variation of the conditions of argon-arc welding of materials of the 50Fe–40Cu–l0Pb and 50Fe–45Cu–5Pb systems, did not give positive results. In all cases, the weld contained the same phase separation defect, namely the precipitation of a copper-lead mixture and separation of its individual conglomerates. These compounds failed at low loads. To produce a strong welded joint, it was necessary to use a concentrated heat source in the form of a powerful laser beam. Materials with a thickness of 1.5 to 4 mm were welded in a CO2 laser with a power of up to 5 kW. Argon was used as the shielding gas. The welding speed was 20–30 mm/s and at an energy concentration of 105 W/cm2 resulted in a cooling rate of 103 ◦ C/s. A priori these conditions should prevent phase separation. However, as shown by the results of a large number of experiments, under the above conditions the copper-lead mixture managed to precipitate in large conglomerates (Fig. 4). To explain the physical nature of this defect formation, the theory of breakdown and phase separation of liquid-metal systems was used [7] according to which the criteria for the retention of the structure of phase-separation systems in Fig. 4. Macrostructure of Al– fusion welding are the possibility to obtain in 30Be–5Mg composite welded by heating with the welding power source the mix- electron beam welding with beam ing temperature Tcr and the duration no longer oscillations and interlayers: than (10 − 50) · 103 s of existence of the liquid- thickness of material 10 mm metal systems in the dissociation and phase separation temperature range. The ﬁrst criterion can be evaluated using the dependence derived by Hildenbrandt and Scott [8] : m Tcr >

0.25(V1 + V2 )(δ1 − δ2 ) , R

(4)

where V1 and V2 are atomic volumes of the separating components; δ1 , δ2 are paVESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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rameters of solubility of the components equal to H/V (H is the sublimation heat); R is the gas constant. The right hand part of [4] reﬂects the heat of formation of the liquid solution. If it is higher than the value Tcr , the phase separation takes place in the liquid state. As shown by the calculations, the laser welding conditions do not allow these conditions to be satisﬁed, because this requires solidiﬁcation rates of the order of 105 − 106 ◦ C/s, which is unrealistic under the welding conditions. Another method of preventing this type dem of the melt fect formation is the reduction of Tcr by adding additional alloying elements to the melt zone. Taking into account the conditions of chemical compatibility and the effect on Tcr , experiments and calculations showed that nickel is a suitable alloying element. The interlayer of this material in the form of a strip was placed between the specimens. The thickness of the interlayer B1 in laser welding of the 50Fe–40Cu– 10Pb composite was selected in relation to the lead content CPb , of the material: B1 = 0.27BW (CPb − 2.5),

(5)

where BW is a width of the weld, mm. The last equation was derived by analyzFig. 5. Macrostructure of laser beam welded joints in 50Fe- ing the experimental dependence of the required 40Cu-10Pb composite without nickel content of the weld on the lead content of nickel interlayer the material. Welding with the nickel interlayer has made it possible to prevent completely phase separation and obtain welded joints with a strength of 0.8 ∼ 0.9 of that of the parent metal (Fig. 5). Research of Materials of Al–Pb System. The Al–Pb system in accordance with the equilibrium diagram is a composite insoluble in the liquid state. In the solid state, the mutual miscibility is restricted and equals 0.15 − 0.12 %. The composite materials based on this system have appeared relatively recently and are also produced by contact alloying. These materials are applied where it is necessary to have the radiation protection using light materials: under space ﬂight conditions, in nuclear power stations and other objects associated with radiation. Technology of welding the materials of the Al–Pb composite type is based on the experience obtained in previous applications, especially laser welding using interlayer. On the basis of the calculations with the use the procedures proposed previously, a tin interlayer was recommended. The experiments with laser welding with interlayers made of tin gave positive results. The criteria and the methods of removing defects can be applied to larger groups of materials and also used when developing the fusion welding technology for dissimilar materials. Conclusions. 1. A classiﬁcation of MCM, based on the interfacial interaction of the components in fusion welding, is proposed. 270

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2. Speciﬁc defects when fusion welding MCM are considered and criteria for their formation are determined. 3. Technological techniques are proposed for producing high-quality welded joints by argon-arc, electron beam and laser welding of Al–Be–Mg, Fe–Cu, Fe– Cu–Pb and Al–Pb composites.

REFERENCES 1. Grigoryants, A.G. and Shiganov I.N: Laser welding of metals (in Russian). – M.: Vysshaya shkola, 1988. 2. Edited by Kreider N. Composite materials with a metallic matrix (in Russian). Vol 4. – M.: Mashinostroenie, 1978. 3. Komarov, M.A. Speciﬁc features of structural transformations in the fusion zone of welded joints in aluminum–beryllium alloys (in Russian). Aut. Svarka 1985 No 10. 4. Kokh, B.A. Fundamentals of thermodynamics of metallic processes (in Russian). – M.: Sudostroenie, 1975. 5. Binary and multicomponent systems based on copper (in Russian). Handbook. – M.: Nauka, 1979. 6. Tuchinskii, Ya.I. Composite materials produced by impregnation (in Russian). – M.: Metallurgia, 1986. 7. Vozdvizhenskii, V.M. Formation of delamination in the liquid state in binary systems (in Russian). In “Equilibrium diagrams of metallic systems”. – M.: Nauka, 1974. 8. Wilson, D.R. Structure of liquid metals and alloys (in Russian). – M.: Metallurgia, 1972. Shiganov Igor Nikolaevich graduated from the Bauman Moscow Higher Technical School in 1971, D. Sc. (Eng.), professor, Director of the Materials & Technological Processes Research Institute. Author of more than 120 publications in the ﬁeld of the welding & laser technologies.

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Yu. Bocharov, Yu. Gladkov (Bauman Moscow State Technical University)

INTEGRATED CONTROL-MONITORIUNGDIAGNOSIS SYSTEM DEVELOPMENT IN TECHNOLOGY OF PLASTICITY Problems and results of integrated control system development for closed-die hot metal forming are discussed in the paper. Process and Maxipress mechanisms simulation models are employed for an adaptive control system with goal of a deformational force stabilization. Operational parameters monitoring and press mechanisms diagnostics are realized with a help of two level module subsystem for a disc friction clutch and the closed gap regulation mechanisms. An adaptive algorithm for this purpose has been developed.

Introduction. Monitoring operational metal forming process parameters and diagnostics of forging machine-tool mechanisms are necessary for reliable atomized forging complexes and lines operation and product quality assurance. It involves computer process and machine-tool modeling, computer numerical control (CNC) operational algorithms and programs development, monitoring parameters selection, diagnostic mechanisms and algorithms development, set of proper sensors and hardware selection and dozens of other problems to be solved. A general methodology approach to the problems of integrated CNC development in technology of plasticity has been worked out. A pilot integrated CNC system incorporating control-monitoring-diagnosis subsystems for the forging line based on the 25 MN nominal force Maxipress has been designed. According to CNC strategy analysis [1] the billet control strategy has been selected, since the mechanical crank operating Maxipress demonstrates a low degree of machine parameters controllability. To test and verify the system an experimental set-up, based on the 6.3 MN crank-operated press, electric furnace, personal computer (PC) and set of sensors, has been constructed. Physical process has been simulated with experimental dies and steel billets heated to the forging temperature. Process and Machine-tool Modeling and CNC. Process and Maxipress machine-tool simulation models are employed for obtaining nominal cycle parameters to compare with real time variables in adaptive control, monitoring and diagnosis subsystems. Besides they allow exploring acceptable successfully processed process cycles. Product model includes geometry and height values, material structure (microstructure, macrostructure) as main variables; forging model considers billet mass (or volume) and time dependant functions of temperature, deformation force, whose absolute values are constrained between upper and lower limits. Process dynamics model usually presents the simpliﬁed mathematical description for relationship between directly actuated process variables and product geometry or microstructure quantities [3] The computer simulation model helps to test, 272

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Fig. 1. Hot-Die Forging complex with 25 MN mechanical press: 1 — inductor, 2 — forming rollers, 3 — conveyer, 4 — mechanical press, 5 — feeder, 6 — conveyer, 7 — press-cutter

adjust, improve integrated CNC for hot-die forging complex in the course of its design and to control and monitor production cycle parameters in the operational mode. Main variables are volume (mass) and temperature of the billet. Random variation of those parameters causes the corresponding variations in the value of deformation force and elastic. Forging complexes based on hot-die forging mechanical presses (Maxi-presses) are widely employed in automotive, agricultural, road construction machinery and other industries. The precision of a product height depends upon the press-die stiffness; die inserts temperature and deformation force variables, resulting in a variable gap between die-inserts. Hot-die forging complex of TMP Company (Voronezh, Russia) (Fig. 1) serves for die-forging of pitman type components and consists of a preheater (450−500 ◦ C), press-cutter, induction heater (1230−1260 ◦ C), forging rolls (initial preforming), mechanical press 25 MH (preforming, forming, ﬂash cutting). The same complex is employed for gear and ﬂange type components forging in four stages (upsetting, preforming, ﬁnal forming and burr cutting). The closed loop adaptive CNC system able to stabilize deformation force with a billet temperature as correcting factor of the billet volume (mass) variations has been developed (Fig. 2) [3]. Adaptive algorithms of the independent billet mass, billet temperature and integrated mass and temperature control have been designed and executed with the help of the original program “PC-Swage Master” [4]. Linear regression model expressing the billet temperature and mass inﬂuence on deformation force has been used in a die forging complex based in Maxipress of 25 MN nominal force CNC

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Fig. 2. Adaptive CNC system structure

system. “PC Swage-Master” program [4] has been used as a process simulator Pi = P0 + KI(Mi − M0 ) − K2(Ti − T0 ),

(1)

where P0 , M0 , T0 — optimal calculated values of the deformation force, billet mass and temperature; Mi , Ti — current values of a billet mass and temperature; K1 and K2 — correlation coefﬁcients. Adaptation of the regression model (1) is executed with a periodic calculation of K1 and K2 coefﬁcients. Billet temperature after inductor is considered as a random value in the course of the forging process. The billet mass (volume) could be slightly different from a cut value due to a billet material oxidation. Both uncontrollable factors are considered as random values having normal distribution mode in the simulation model. The following devices are to be involved in the control signals generation: — billet cutting device (press-cutter) and heating device (inductor) as control objects executing control signals; — transfers and other performing devices (like forging rollers) acting in the way from inductor to the ﬁrst die position as a system time-delay (inertia) factors. (Time-delay factors are considered with a help of decreasing coefﬁcients K1 and K2 programmable calculation); — billet mass (volume) and temperature transducers as feed-back information source; — deformation force transducers (tensometers) as an integrated feed-back information means of a billet mass (volume) and temperature measurements; — die insert temperature sensors;

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— closed gap between die inserts transducer as an integrated control system feed-back information source. According to integrated algorithm the initial billet volume (mass) value is compared with calculated one and a billet temperature needed to compensate the volume (mass) deﬂection is calculated. The deformation force at every process cycle versus press slide travel serves as the feed-back information and is compared with the calculated diagram. Three control and diagnosis levels are presented in the control system structure (Fig. 2). Informational system tasks are divided into three blocks: forging complex drive and servomechanisms monitoring, production time statistics, diagnostics. Diagnostics subsystem solves the problems of the following groups in real time: 1) monitoring complex main parameters within permissible value corridor and realization of the mechanisms preventing stoppage in the case of parameters critical values; 2) diagnosis the main complex mechanisms operation and warning signals generation in the case of any forecasted malfunction; 3) diagnosis of the main complex mechanisms temperature and wear conditions and maintenance and repair planning. The ﬁrst group of problems is solved with the help of PLC with corresponding software, while the second group requires execution of calculation algorithms. In the case of low probability of critical parameter value the diagnosis is executed by PLC. The third group of problems serves for information monitoring to the control terminal. Forging process simulation model. The forging process computer simulation model of hot-die and semi-hot-die forging process is based on the original software with the following function blocks: process variables calculation, process simulation core, operator interface, remote control interface [4]. Simulation program is employed for the following tasks: (1) computer testing and adjusting adaptive control algorithms; (2) executing the adaptive CNC and data terminal control in real time; 3) collecting and storing operational data. The program can operate in the following modes: operation, tuning, control algorithm analysis and recall mode. In operation mode input data are received from sensors. Output data are the control commands to complex devices. Tuning mode serves for analysis and adjusting of complex operation with the help of simulation model. The operation mode permits to analyze different complex processes and monitor operational parameters. Control algorithm analysis mode serves for calculation the system responding time to single exposure and comparing the algorithm efﬁciency. Recall mode allows regenerating complex operation parameters with a help of a protocol ﬁle. This function permits the multiple analysis of any process on the base of input data form the protocol ﬁle. The simulation program also provides facilities for calculation the optimal parameter values for control algorithms (optimal values of decreasing coefﬁcients and others).

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System requirements: Pentium 166 MHz, 32 Mb RAM, 50 Mb HDD, SVGA High Color (16 bit), Windows sound card, Microsoft Windows 0 95/98. The program has been developed in Borland C++ 5.02 and OWL [10]. Each billet is supplied with a data passport and is served in the queue. Queue elements record current parameters of mass and temperature and the queue itself simulates the process logistics. The simulation program executes the following tasks. 1. Compensation of a billet mass (volume) deﬂection dM of the calculated value (2) Mi = Mi−1 + dM ; It is realized in the billet press-cutter control net. Mean deviation in the batch of n billets ¶ µX n ± (3) Mk n, Mm = k=1

is compared with the calculated value M0 dM = (M0 − Mm )K,

(4)

where K — decreasing coefﬁcient, calculated in simulation model at the condition of regulation time minimization. 2. Compensation of a billet temperature in inductor in the billet heater control subsystem Ti = Ti−1 + dT ; dT = (T0 − Tm )K; µX ¶ n ± Tk n. Tm =

(5)

k=1

3) Additional compensation of a billet mass (volume) inaccuracy by a billet temperature Ti = Ti−1 + dTmi . (6) Program interface displays the system information to a human operator in real time. There are facilities for the main window customizing, changing parameters of the hot, semi-hot or cold die-forging process and simulation model, operational and critical situation monitoring and others. Button area of the main window includes buttons: “Scissors”, “Inductor”, “Rejecter” and “Press” allowing an appropriate device for operational diagnosis to be selected. List of displayed parameters can be selected with buttons: button ”All” for all devices, ”Selection” for an appropriate composition of the die-forging complex. Window area consists of four parts: “Star”, “message” and two “diagram” windows. Each ray of the star displays one parameter as a point between ray ends that denote maximum and minimum values. Red, yellow and green areas of the ray visually demonstrate a state of parameter that can be not admissible, critical or normal. 276

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In the message window the current values of parameters, system components state and warnings are displayed in text messages. Two diagram windows display process parameters versus time. Red, yellow and green lines on the background of the windows denote areas of not admissible, critical and normal values of parameters. There is also a facility for customizing diagram windows (diagram width; link/unlink points; x-line step changing). Main menu items realize access to dialog boxes for customizing program and changing parameters of the die-forging process, control algorithms and others. The program also offers a suitable interface for manual adjustment of process parameters. Operator can select necessary parameters and they will be displayed as a number of scrollbars. Scrolling will bring about changes in the corresponding parameter value. The list of parameters depends on the program operation mode. Press mechanisms simulation models. For clutch-brake diagnosis a doublelevel imitation model has been developed: module structure for a lower level and probability mode of operation (with weight coefﬁcients) for an upper level. Each of several lower level modules sends press mechanisms diagnosis signals to the upper level. The upper level calculates reliability degree of each diagnosis signal. If a reliability degree exceeds a limit value the system may monitor diagnostic values and correct process-machine parameters regulating Maxipress closed gap at the time interval between the cycles, correcting billet temperature etc. Lower level modules for clutch and press closed gap regulation mechanism are presented here. a) Clutch module. Diagnosis parameters are angular velocities of active and passive friction discs. Clutch cycle (Fig. 3) is analyzed and diagnosis parameters are monitored. For example, in the case of t exceeds permissible level diagnosis of control valves, pneumatic cylinder, seals, or friction discs is suggested. b) Press closed gap regulating mechanism module. Diagnosis parameter are a slide linear displacement, press closed gap value, deformation force, billet mass and temperature. An adaptive relation between parameters (Fig. 4 ), diagnosis algorithm for a forging product height variation and operations for a programmable

Fig. 3. Tachogram of crank press clutch cycle

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Fig. 4. Adaptive control algorithm

press closed gap correction are developed [6] with adaptive constants recalculation depending on a press closed gap variations. Information processing is executed with a program and experimental imitation model and control algorithms veriﬁcation and correction are possible. The same methodology could be used for other mechanisms imitation models and relations between lower and upper levels development. Conclusion. The developed simulation program and the control algorithms could be after certain modiﬁcation employed also for semi-hot and cold die-forging processes in automated complexes and lines on the base of other types of forging processes and machine-tools. The requirements to the processes could be speciﬁed as follows. The process parameters could be presented in the linear regression model. Endogenous parameters (like the billet mass and the temperature) should permit the calculation of the only exogenous parameter (like the maximal deformation force value). The closed loop in the integrated CNC System should be based on the exogenous parameters. Veriﬁcation of the computer simulation program and model has been done in computer and physical experiments. It has been demonstrated that accuracy of the height size of the forged components in hot-die forging complexes based in mechanical forging presses could be increased by 60–70 % with the help of suggested adaptive CNC. The computer simulation model helps to test, adjust, and improve the adaptive CNC for hot-die forging complex in the course of its design, saving time and resources. In operational mode it helps to control and monitor production cycle parameters and to regenerate statistical process data in the cases of unscheduled stopovers and change of production program. Computer simulation program with incorporated monitoring and diagnosis functions helps to increase reliability of the production lines and accuracy of the forged items.

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REFERENCES 1. Bocharov, Yu. A. CNC Strategy in Technology of Plasticity // Advanced Technology of Plasticity, 1999, 6 ICTP, 1999, v. 1, p. 195–200. 2. Bocharov, Yu.A. CNC Systems in Technology of Plasticity (in Russian) // KuznechnoShtampovochnoye Proizvodstvo (Technology of Plasticity), No 7, 2000. – PP. 39–46. 3. Balagansky, V.I., Bocharov, Yu.A., Gladkov, Yu.A. (in Russian) System of Adaptive Control for Hot-Die Complex // Kuznechno-Shtampovochnoye Proizvodstvo (Technology of Plasticity), No 6, 2001. – PP. 26–30. 4. Gladkov, Yu., Khvostenko, A. Interface module of CNC for hot-die forging complex - PC Swage Master (in Russian)// RosPatent Certiﬁcate, No 990746, 1999. 5. Prutckov, R., Balagansky, V., Bocharov, Yu. Maxipress Cosed Gap Operational Regulation (in Russian) // Kuznechno-Shtampovochnoye Proizvodstvo (Technology of Plasticity), No 6, 2001. – PP. 26–30 (Russian). Yury A. Bocharov, graduated in Mechanical Engineering from Bauman Moscow Higher Technical School in 1953. Professor, Technology of Metal Forming department of Bauman Moscow State Technical University. D.Sc. (Eng.), RANS Full member, Honored Scientist of RF. Has published more than 300 papers, textbooks and books; has a number of invention certiﬁcates and patents in the ﬁeld of Metal Forming Technology, Machine Theory and Design, CNC and diagnosis. Yury A. Gladkov, graduated in Mechanical Engineering (MSc) in 1998 and Computer Control Engineering (BSc) in 2000 from Bauman Moscow State Technical University. Assistant-Professor, Technology of Metal Forming department. Ph.D. (Eng.) Has published more than 30 papers; has a number of Software Development Certiﬁcates related to Metal Forming Machinery and processes CNC.

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S. Golovashchenko (Ford Motor Co, USA), V. Kondratenko, A. Vlasov (Bauman Moscow State Technical University)

EXPERIMENTAL INVESTIGATIONS OF FLANGING ALUMINIUM PANELS AS THE FIRST STAGE OF HEMMING PROCESS Hemming is often used as the last stage of stamping operations in automobile industry. Mostly it is used to attach one sheet metal part to another for instance outer vehicle door panel and inner door panel. The main problem associated with the hemming of aluminium body panels is cracks on the outside hemmed radius. We offer and study a number of new ways of ﬂanging (the ﬁrst stage of hemming), based on idea of increasing of hydrostatic pressure in the center of plastic deformation. Most investigations were carried out in Ford Research Laboratory.

Background. We develop technological process of ﬂat hemming thin sheet aluminium alloy 6111-T4 panels (closely related to Russian AD33). Traditional hemming process has three stages: ﬂanging, pre-hemming and ﬁnal hemming. Finally, after hemming, the design should look as follows (Fig. 1): The key factor for using of hemming for aluminium automobile body panels is a lack of any cracks on the outside hemmed radius as well as lack of any roughening on A surface of outer panel. Thus, inside radius should not exceed half of thickness of a material. The thickness of a material is 0,0400 (approximately 1 mm). The resource of plasticity of 6111-T4 does not sufﬁce for traditional hemming process. It is known (Bridgman and others), that the increase of hydrostatic pressure in a zone of deformation raises a resource of plasticity. We tried to suggest some schemes, which used the idea. The review of new technologies. Let0 s consider the suggested schemes of new technologies: 1. First offered the scheme, in which ﬂanging is divided into two stages: a usual bending on large radius and then ﬂanging with compressive force applying to

Fig. 1

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Fig. 2

Fig. 3

outer panel trim edge (Fig. 2). In this schemes the force causes the high hydrostatic pressure, and polyurethane prevents buckling failure. 2. The technology, shown on Fig. 3, based on the idea of using of elastic punch for creating the pressure in bending zone. We also used the proﬁled elastic punch. 3. The technology, shown on Fig. 4, has two steps. First (stage A — not shown) — bending with large radius and then (stage B — shown on Fig. 4) ﬂanging by means of rigid punch moving horizontally. In this scheme the rigid punch was used for creating active friction forces, which create compressive force in bending zone. Experimental results. We used two sets of material in experiments. The ﬁrst set is the old strips of 6111, which have more 2 years old. So due to aging treatment the old strips lost much of their ductility. Rolling direction at a bend of old strips placed along a line of a bend. In further these samples we shall designate as material # 1. The second type of strips was 6111 with 7 % prestrain. Rolling direction at a bend of new strips placed along a line of a bend. In further these samples we shall designate material # 2. Polyurethane - (made in Russia) used for elastic punches during the experiments.

Fig. 4

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During the experiments we used a scale for evaluation the quality of bend line. Following Baartman1 we used the scale of hem quality given in Table 1. Table 1 Score

Classiﬁcation

10

No visual surface defect

8

Mild surface roughening

6

Ridge-like surface roughening (“orange peel”)

4

Small surface cracks

2

Continuous surface cracks

0

Complete cracks through material

Experiments for scheme # 1 was produced with material # 1. The quality of bend line of the blanks being bended such way had score 10 (the best quality). Besides we obtained the sharp inner radius (a lot of less than half of thickness). But the A-surface had some roughening. Due to bad A-surface quality of A-surface we rejected this scheme. On the other hand the best hem quality shows that the increasing of hydrostatic pressure raised the bend ductility dramatically. Experiments for scheme # 2 was produced with material # 1 too. We varied the following parameters at this investigation (see Fig. 3): A — The depth of a cavity in rigid punch; B, C — The sizes of elastic punch; D — The interference between punch and die. The transparency ﬁlm placed atop of blank was also used in some experiments. At ﬁrst step we used following elastic punch sizes: 10 × 17, 10 × 15, 10 × 13, 10 × 10, 10 × 6, 13 × 13, 13 × 10, 13 × 6, and varied interference in the limits: 0 . . . 6 mm. The best results was for punch with sizes B × C = 10 × 6 mm. We obtained quality score 8 with using a ﬁlm and 6 without ﬁlm (for material # 1) and 5 for material # 2. But the durability of elastic punch was moderate. The wear-out of elastic punches was less for interference 1 . . . 2 mm. A larger interference invokes a fast wear-out. The second step was for the investigations of form of elastic punch. The numbers of punches, their forms and sizes are given in Fig. 5. The best results was for punches ## 12,18 with a quality score 6 without ﬁlm. The third step was for the inﬂuence of lubrication. We used soap, graphite lubrication as well as ﬁlm and two blanks ﬂanged together. It is found out, that the friction is not the major factor inﬂuencing quality of hem. The positive inﬂuence of a ﬁlm consists in increasing thickness of blanks. 1

R.Baartman, E.H.Atzema and J.Bottema. Optimization of the hemming process for AA6016-T4 aluminium body sheet. 98NM056.

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Fig. 5

This hypothesis proved by improvement of quality at use of a ﬁlm of the greater thickness, and at joint bending of two blanks placed one atop other. Both experiments have given the best results in comparison with other, received at normal conditions and at use of lubrications. The analysis of experimental results with schemes # 2 shows, that form and sizes of elastic punch that obtained stable good quality without using ﬁlm are not found out. The main reason is that the schemes do not create hydrostatic pressure. And what is more — they increase tensile stress in bending zone due to inﬂuence of contact friction. The best form of the elastic punch obtained from experiments listed above is the punch # 12 which has complicated form. Its form we varied at the next stage. The varying parameters are explained in Fig. 6. The values are given in Table 2. All experiments were performed without any lubrication or ﬁlm.

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Table 2 Punch No

C, mm

F , mm

E, mm

R, mm

I

26

9

6,5

3.5

II

22

7

6.5

3.5

III

18

8

6.5

3.5

IV

15

7

6.5

3.5

V

20

8

6

4

We achieve the quality score 6 without ﬁlm. The analysis of experimental results with schemes # 3 shows, that form and sizes of elastic punch that obtained stable good quality for prestrained material (matherial # 2) are not found out. The best results are for following: 1. The creating of a direct corner occurs by turning of lug of elastic punch (size E in Fig. 6) at the end of process. Width E should be as greatest as possible. 2. The deformation of elastic punch should be maximal. If the deformation is insufﬁcient, elasticity of the lug does not sufﬁce for obtaining of a direct corner. The experiments with the scheme # 3 were carried out in the following sequence: 1. Preliminary ﬂanging: a) preliminary bend on large radius with use of laboratory installation for bending by the moment; b) preliminary bend in a die with the big gap between punch and die. 2. Final ﬂanging with use of rigid punch. At ﬁnal stage we varied the following sizes (see Fig. 7, 8): • Inner preliminary blank radius — Rb = 2 . . . 7 mm;

Fig. 7 284

Fig. 8

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• Preliminary gap — ∆ = 2 . . . 10 mm; • Die radius — Rm = 0, 1; 0, 5 mm. The obtained experimental scores of quality are given in Tables 3, 4. We used the material # 2 (7 % prestrained) for this experiments. Table 3 Rb, mm Rm = 0,1 mm Rm = 0,5 mm 2 6 (# 1) 6 (# 7) 4 (# 2) 4 (# 8) 3 4 (# 3) 6 (# 9) 4 2 (# 4) 2 (# 10) 5 2 (# 5) 2 (# 11) 6 2 (# 6) 4 (# 12) 7

Table 4 ∆, mm Rm = 0,1 mm Rm = 0,5 mm 2 2 (# 13) 8 (# 20) 4 (# 14) 6 (# 21) 3 2 (# 15) 8 (# 22) 4 2 (# 16) 8 (# 23) 5 4 (# 17) 8 (# 24) 6 4 (# 18) 6 (# 25) 7 2 (# 19) 6 (# 26) 8 no data 4 (# 27) 9 no data no data 10

In Fig. 9 we show the result # 7 from Table 3 and # 24 from Table 4. You can see the sharp inner radius and the negative springback obtained after the ﬁnal ﬂanging. Analyzing the results of experiments for schemes # 3 it is possible to make the following conclusions: 1. Using the preliminary bend with large radius or large gaps ensure the good quality for ﬁnal bending by rigid-elastic punch and with some sizes for rigid punch. 2. The key factor in two-step ﬂanging is the precision mounting of preliminary bended blank into the ﬁnal die set. The preliminary bended zone must be out of contact with ﬁnal die (see Fig. 10). In the case of contact the results was worst.

Fig. 9

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Fig. 10

Conclusions. The key idea, consisting in increasing the hydrostatic pressure in the bending zone for increasing ductility of material being bended, is proved by experimental way. The way of realization of the approach is found. The ﬂanged blank with high quality of the A-surface and of the bending zone is obtained.

S.F. Golovashchenko graduated from Bauman Moscow Higher Technical Scholl in 1985. D. Sc. (Eng.), Senior Technical Specialist of Scientiﬁc Research Laboratory, Ford Motor Company. Author of more than 50 publications in the ﬁeld of theory of plasticity and metal forming technologies. V.G. Kondratenko graduated from Bauman Moscow Higher Technical School in 1959. Ph. D. (Eng.), professor. Author of more than 150 publications in the ﬁeld of theory of plasticity and metal forming technologies. A.V. Vlasov graduated from Bauman Moscow Higher Technical School in 1978. D. Sc. (Eng.), professor of “Metal forming technologies” Department of Bauman Moscow State Technical University. Author of more than 50 publications in the ﬁeld of forging equipment design and metal forming technologies.

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CRIOGENIC ENGINEERING & TECHNOLOGY A.M. Arkharov, S.D. Glukhov, L.V. Grekhov, A.A. Zherdev, N.A. Ivashchenko, D.N. Kalinin, A.V. Sharaburin, A.A. Aleksandrov (Bauman Moscow State Technical University)

CREATION OF HARMLESS FOR ENVIRONMENT ENGINES AND INSTALLATIONS USING DIMETHYL ETHER The article deals with the problem of replacing pollutant substances with ecologically safe ones used in the urban transport. Dimethyl ether is proposed as it has ozone depletion potential and global worming potential of zero value. It can be used as a fuel and as a refrigerant at the same time. The article cites the advantages of dimethyl ether use as well as it analyses experimental researches.

Ecological problems for such megapolicies like Moscow are becoming more and more acute. The Moscow government realizes such an unhealthy situation; it undertakes concrete measures to reduce pollution of environment including water ozonization, conservation of different gas discharges — for example at ﬁlling/service stations; avoidance of soot use during winter period, planting of greenery, amelioration of transport schedule, development of electric traction etc. Among such measures the creation of harmless diesel engines as well as replacement of ozone depletion refrigerants used in low temperature installations have a signiﬁcant place because diesel engines and refrigerants are involved in urban economic development. The problem of energy and cold regeneration in vehicles is of great importance for megapolicies, large cities and densely populated regions, where major technical and technological advances of 20th century are concentrated: propulsion engineering, motor-car construction, chemical engineering, atomic and power engineering, refrigerating and cryogenic engineering, food industry, aerospace and chemical complexes etc. In a number of biggest megapolicies (for example, Los-Angeles, Tokyo, London, Shanghai) rigid constraint has been adopted, ﬁrst of all with respect to vehicles, refrigerating, heating, ﬁltration, and puriﬁcation systems. Properties of working substances of many energy-power refrigerating installations as well as properties of different consumer goods are regulated too [3]. In 1987 the Montreal Protocol was signed that limited the production of ozone depletion refrigerants; in 1997 the Kyoto Protocol was drafted that would require to limit use of chemicals stimulating greenhouse effect of our planet. Besides, in Russia, since 2003 the Euro-III standards have been adopted that regulate the toxic substances content in used gas of internal combustion engines. Today two criteria exist: ODP (ozone depletion potential) and GWP (global worming potential). Ideally these potentials must be of zero value for working substances and fuels used. Such a substance is dimethyl ether (DME) CH3 −O−CH3 VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

287

that can be considered as a fuel and as a refrigerant at the same time. DME was used in the ﬁrst vapor compression machines, and then it was forced out by ammonia, later — by refrigerants. Last few years DME has been considered as a perspective fuel for diesels used in transport, ﬁrst of all thanks to signiﬁcant reduction of toxic substances issue (Euro III standards without use of neutralizers). These qualities of DME are objective prerequisites for creating combined energetic plants with engine and refrigerating unit for the urban transport. The Moscow State Technical University named after N.E. Bauman (BMSTU) has carried out researches of DME as a diesel fuel as well as a refrigerant component [4]. Table 1 The Characteristics of Motor Fuels Properties

DF

DME

835–850

668

Combustion heat, MJ/kg

43.5

28.8

Stoichiometric ratio

14.5

9

170

46.07

240–310

235

◦

3

Density at 20 C, kg/m

Molecular mass ◦

Temperature of self-inﬂammability, C ◦

Heat of vapor formation at 20 C, kJ/(kg·K)

210

410

Cetane number

45

> 55

0.4

34.8

< 0.001

0.51

—

5.37 (400)

Oxygen content, % ◦

Saturated vapors pressure at 20 C, MPa Critical pressure, MPa ◦

Compressibility factor at 20 C and 0,1 MPa, 1/Pa

−11

86 · 10

210 · 10−11

DME as a motor fuel. The DME characteristics as a motor fuel [1, 2] differ much from those of a diesel fuel (DF) (Table 1). For example, volume cycle feedings are needed 1.7–1.9 times as much to keep diesel engine capacity, taking into account lower density and caloriﬁc efﬁciency. While designing a high pressure fuel pump it is necessary to take into account the fact that at typical operation the volumetric efﬁciency must be increased 2.4–2.7 times due to the higher DME compressibility. When changing over from DF to DME the fuel feeding and the diesel working cycle undergo great modiﬁcations. Because of a high compressibility the fuel feeding begins later. So, when using the diesel D245.12 in the motor car “Bychok”, that has a modernized fuel equipment, injection lead decreases by 10.7 degrees of crankshaft turn when changing over to the DME at typical operation (Table 2). The optimal angle of lead increases when a rotational speed of the shaft increases; on the contrary the real angle decreases. The greater is compressibility of the fuel, the stronger is this contradiction. That’s why an automatic lead coupling is to be designed specially for operation on DME. The operation with a standard coupling diesel is not optimum, all the more without it (diesel D245.12 288

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case). This deﬁciency is compensated by quicker DME inﬂammation. The results cited in Table 2 have been obtained with the program complex “Vprysk” [5] that was adapted for the studied object. This program complex was used for optimizing and designing a modernized fuel system for DME feeding. Table 2 Indices of Fuel Injection in Diesel D245.12 Properties Cycle feeding gc , mg Active running of plunger Value of additional injection, % of g Maximum pressure, Mpa: in front of a fuel nozzle pressure of injection Average pressure of injection, MPa Characteristics of feeding, degree of crankshaft turn: duration retard of start Maximum moment on camshaft H·m

DF 79 2.15 0

DME 118 5.56 1.2

90% DF + 10% DME 82.7 2.56 0

55.09 44.36 23.95

38.38 28.79 19.56

55.08 42.66 23.93

23.12 11.60 92.00

44.12 22.32 66.56

24.40 12.90 85.18

The decrease of the injection pressure (in the case of diesel D245.12 1.5 the pressure decreases 1.5 times) is another signiﬁcant modiﬁcation of the fuel feeding index. The decrease of the injection pressure takes place in spite of the increase of the cycle feeding. Such conditions are not optimal for mixture formation but at least it is not a limiting factor: spraying of DME liquid jet is better due to less surface tension and viscosity; ether is vaporized without delay at charge temperatures in the cylinder. More important is the fact that the duration of injection (feeding) at the operating regime increases by 11 degrees of the crankshaft turning (Table 2). Thanks to special properties of DME a smoke of used gases may be avoided but the delay of injection doesn’t allow increasing the motor effectiveness. The proposed increase of a sectional area of nozzles doesn’t correspond to the possibilities of a universal dual-fuel diesel engine. Other speciﬁc problems occur while feeding a diesel with DME. Filling of the plunger cavity becomes difﬁcult. It results in non-stability of feeding. When using DME without the Lubrizol fuel additive imported there are problems to start the engine due to low viscosity; leaks occur and the life period of precision couples decreases; there are problems with fuel pumping in the low pressure lines. A shorter and larger ﬂame of DME leads to redistribution of thermal loads on elements of the cylinder-piston assembly as well as to overheat of central part elements of the combustion chamber etc. It’s evident that it is necessary to optimize working process to obtain an effective work of the diesel. Even if the diesel is working effectively the creation of a universal dual-fuel diesel engine on the basis of existent technical decisions VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

289

is rather problematical. The results of such an approach are well known thanks to 50 years experience of creating a universal engine at forced ignition working on gasoline and gas. Besides, modern motor fuel market and structure of modern service/ﬁlling stations should need serious modiﬁcations to exploit specialized cars using DME. Creation of new generation transport diesels using engine mechanical elements as well as electronic control systems optimizing the working process may be more acceptable [6]. A fuel feeding system must be changed a little according to new working conditions. In such a case the most perspective is application of an accumulator system with Common-Rail electronic control. We may certify that in the beginning of 21st century Russia doesn’t produce diesels with an electronic control system (except for small-serial diesel GAZ-560). And what is more there is no basis to expect for a quick start of Common-Rail systems’ production. The fact of the matter is that the hydrodynamic and thermal and physical peculiar properties of DME behavior under conditions of fast-acting physical and chemical processes of mixture formation and combustion (at high pressure drop) are not well studied, retards the creation of a diesel with effective working process. At the same time ecological situation of large cities demands to look for improvement of this situation based on the equipment produced as well as on the equipment in operation. Thereupon the suggestion to change fuel market structure in favor of DME is rather constructive (Moscow Government Decree #170-ΠΠ of 12.03.2002). The positive results have been obtained in diesels of 4ch10,5/12 type. Liquid gases have been supplied into a cylinder through a fuel supplying equipment improved. We have the experience of additionally makeup fuel supply of high pressure lines with other alternative fuels: hydrogen, gases, dense carbon suspension and others [7, 8]. In particular, the results of experience with a ZIL-645 diesel, when gases were supplied into the high pressure line showed other interesting things. While supplying hydrogen, synthesis-gas and pure air, the 20–45% decrease of CH, CO, NOx emission has been registered. In virtue of above-stated and taking into account the MSTU experience [9] we suggest to implement the ecological fuel DME in mixture with traditional diesel fuel. Its share is to be from 10 to 40% for different types of diesels and different operating conditions. This will allow smoothing away or eliminating many speciﬁc problems of diesel feeding with the DME. The passing of working processes (dispersion, evaporation, inﬂation, and combustion) will be rather improved. The decrease of toxic substances emission will permit to meet legislative standards on toxic emission (even while using rather an old model of diesels) using accessible technical means. Thereto a system of fueling of a transport diesel engine was designed and ﬁnished off (Fig. 1). The use of a mix fuel with DME addition in diesel engines exploitable now gives the possibility to obtain the following results: •Due to a high cetane number the low noisiness of operation, easier emission of used gases, decrease of nitrogen oxides emission are assured.

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•Smoke of used gases decreases due to a high content of oxygen in DME as well as to enhancement of spraying of the mix fuel, which contains a low-boiling component. •High pressure, minor duration of spraying, and low consumption of DME are kept due to supply of the DF as a main component of the mix fuel. Also there is a possibility to use the base fuel equipment Fig. 1. System of fuel supply of a transand to secure passing to operation with a port diesel: 1 — tank with DME; 2 — tank for DF; 3 pure DF. •Non-expensive retroﬁt of fuel equip- — electric control valve; 4, 8 — pumps for DME and DF pumping respectively; 5 — ment doesn’t inﬂuence the car cost and al- high pressure fuel pump ; 6 — fuel nozzle; low accelerating the renovation of a ﬂeet 7 — pressure regulator; 9 — DME feed valve of cars. •Use of expensive fuel additives imported is not needed, low consumption of the DME makes easy the cars exploitation as well as it makes possible creating combined installations with DME feeding of a refrigerating plant and a diesel engine. The top ﬁgure at the inside of the journal back cover demonstrates a car operating on DME. In our opinion the proposed design differs from others by low expenses, simplicity, universality, efﬁciency. You needn’t resolve speciﬁc problems when using the pure DME. But it is necessary to continue researches and to look for rational technologies. DME as a refrigerant. Owing to the fact that there are no performance attributes of the ﬁrst refrigerating machines operating on DME, a computerized program for determining thermodynamic properties of DME has been created, the diagrams have been plotted, and theoretical characteristics of vapor-compressing refrigerating machines [10] operating at DME have been calculated (Fig. 2 and 3). Later the refrigerating facilities using DME as a working refrigerant were installed. The necessary experimental studies were carried out. The obtained characteristics were correlated with the characteristics of the machines operating on R12 and R134a. The analysis of functions calculated allowed us to conclude that DME was a real alternative to well-known refrigerants but de plus it was of unique ecological purity for environment. The parameters of the R12 refrigerating cycle were compared with the experimental data according to main values, which characterized the given cycle: Q0 — cycle refrigerating capacity; L — compressor work; cop = Q0 /L — experimental coefﬁcient of performance. The results of the experiments have shown (Fig. 4) that use of a recuperative heat exchanger in DME cycles makes the refrigerating machine construction more complicated, and doesn’t provide efﬁciency and refrigerating capacity needed.

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Fig. 2. Calculated dependences of volume speciﬁc refrigerating capacity qv on boiling temperature t0 for DME, R12 and R134a: a — non-regenerative cycle; b — regenerative cycle

Fig. 3. Dependence (a) of the theoretical coefﬁcient of performance copt , (b) of the ratio of delivery pressure and suction p2 /p1 on boiling temperature t0 for DME, R12 and R134a

292

Fig. 4. Experimental dependences (a) of refrigerating capacity Q0 , (b) compressor work L, and (c) experimental coefﬁcient of performance cope on boiling t0 and condensation tc temperatures for DME and R12

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The small refrigerating machines laboratory of the BMSTU carried out the DME tests with the Stinol 106 refrigerator of serial production. Without any construction conversion and oil replacement, DME was used instead of R12; an optimal mass of its charging was determined. Variations in energy consumption at different DME mass charging are shown in Fig. 5. The dotted line corresponds to a daily energy consumption of the Stinol 106 refrigerator when operating on R12 (nameplate charging mass is 240 g). During tests it was found that the Fig. 5. Daily electric energy consumpoptimal DME mass charging for such a tion N of the Stinol 106 refrigerator on model of the refrigerator (without chang- DME depending on charge mass m at ◦ ing a capillary tube dimensions) was 60– tm = −24 C in the freezing room: a, b — at ambient temperature +25 and 70 g. A daily energy consumption reduce +32◦ C respectively was noted (by 14–16 %). Other parameters were kept unaltered. These experimental researches have demonstrated that DME is adequate to replace R12 and R134a in the existent equipment as well as in a new designed one without degradation of its characteristics. In some cases DME may be considered as a preferable component of the motor fuel due to its low cost and availability. That’s why it is recommended to conduct long term resource tests with DME, to examine its interaction with different sealing materials which are used now in the refrigerating equipment. It is necessary to study the processes of DME boiling and condensation at different temperatures and pressures, to obtain data on dynamic and kinematic viscosity, gas and liquid heat conduction in the range of working temperatures and pressures of the refrigerating equipment. DME as a fuel and as a refrigerant for diesel transport using refrigeration and conditioning systems. Refrigerated lorries, buses, cars, ships, cutters, yachts Fig. 6. Grouping of a diesel refrigerator have to do with such hauling units. In when using DME as a fuel and a refrigcontrast to liquid propane-butane gas the erant: 1 — vehicle engine; 2 — compressor; DME is supplied under 3–5 MPa pressure. 3 — ventilator and condenser; 4 — ventiThat’s why the application of an open cy- lator and air cooler; 5 — temperature excle refrigerating or conditioning system is pansion valve; 6 — DF Tank; 7 — DME not feasible. But when the same substance tank; 8 — accumulator; 9 — fuel supply system is applied as a fuel as well as a refrigerant, VESTNIK. Journal of the Bauman Moscow State Technical University. Natural Sciences & Engineering. 2005

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fueling up with a refrigerant is not needed because a fuel tank is a receiver of a low temperature system at the same time. With this, the cost of the refrigerating equipment and its maintenance is reduced. Figure 6 shows the grouping of a refrigerated lorry. Approximate costs of a refrigerating installation for the refrigerated lorry on the basis of the “Bychok” ZIL car are shown in Table 3. The main advantage of this lorry is that it is not expensive and doesn’t pollute environment. An experimental model of such a car was created in the BMSTU and demonstrated in Moscow City Administration (the bottom ﬁgure at the inside of the journal back cover. Table 3 Initial Expenses

Refrigerating installation costs Cost of car conversion using DME Cost of refrigerating installation assembling Consumed material cost (tubing, refrigerant, oils, assembling elements, isolation etc.)

Refrigerant R314a DME $3,000 $3,000 — $450 $1,000 $900 $500 $400

A payback depends on many factors: market conditions, transport service costs, operation conditions, climate etc. A preliminary analysis shows that costs of the refrigerating and conditioning equipment don’t exceed costs of traditional installations operating with R134a. REFERENCES 1. A.M. Arkharov, S.D. Glukhov, L.V. Grekhov, N.A. Ivashchenko, D.N. Kalinin, A.A. Alexandrov. Technology of Dimethyl Ether Use for Urban Transport //MSTU Vestnik, 2002, p. 162–167. 2. A.M. Arkharov, S.D. Glukhov, L.V. Grekhov, A.A. Zherdev, N.A. Ivashchenko, D.N. Kalinin, A.V. Sharaburin, A.A. Alexandrov. Dimethyl Ether Use as a Motor Fuel as Well as a Refrigerant // Khimicheskoe i neftegazovoe mashinostroenie, 2203, No 6, p. 17–21. 3. I.M. Kalinin, A.I. Smyslov, K.N. Fadekov. Perspectives Application of Safe for Environment Refrigerants in Household Equipment. Kholodilnaya Tekhnika // 2001. No 12. P. 4–8. 4. V.N. Bogachenko,, S.D. Glukhov, A.A. Zherdev et al. Dimethyl Ether — a Fuel and a Refrigerant for Diesel Refrigerator Trucks // MSTU Vestnik. Special Issue. Ser. Mashinostroenie, 2000. 5. Description of PK “Injection”. MSTU INTERNET site: http://www.bmstu.ru/facult/em/em2/ inject/illrus.htm. 6. Christisen R., Sorenson S.C., Jensen M.G. Hansen K.F. Engine Operation on Dimethyl Ether in a Naturally Aspirated DI Diesel Engine // SAE Paper, No 971665, 1997, p. 101–110. 7. N.N. Patrakhaltsev. Facilities for Gas-Diesel Process // Avtomobilnaya Promyshlennost, 1988, NO 7, p. 16–17.

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8. V.A. Vagner, D.D. Matievsky. Additive of Hydrogen to Fuel and Its Inﬂuence on Diesel Operation Characteristics // Dvigatelestroenie, 1985, No 2, p. 11–13. 9. L.V. Grekhov. Fundamentals of Development of Systems of Fuel Feed into Cylinders of Explosion Engine. Abstract to the Doctor Dissertation, M. 1999, 32 pages. 10. A.A. Zherdev, B.A. Makarov. Cycles Calculation of a Vapor Compression Refrigerating Machine with Redlih-Quong Equation of State // MSTU Vestnik, 2002, No 11, p. 71–80. A. M. Arkharov (b. 1931) graduated from the Bauman Moscow Higher Technical School in 1954. D. Sc. (Eng.), professor, head of ”Refrigeration, Cryogenics, Air Conditioning and Life Support Systems” Department of the Bauman Moscow State Technical University. USSR State prize winner, merited scientist of Russia, corresponding member of the Russian Academy of Technological Sciences. Vice-president of the Scientiﬁc Council of International Institute of Refrigeration. Author of 14 books, more than 50 inventions and more than 200 publications in the ﬁeld of cryogenic technology. S. D. Glukhov (b. 1938) graduated from the Bauman Moscow Higher Technical School in 1962. Ph. D. (Eng.), assoc. professor, section head of “Power Engineering” Research Institute of the Bauman Moscow State Technical University. Author of about 60 publications in the ﬁeld of cryogenic and refrigeration technology. A. A. Zherdev (b. 1955) graduated from the Bauman Moscow Higher Technical School in 1977. Ph. D. (Eng.), senior researcher, section head of “Power Engineering” Research Institute of the Bauman Moscow State Technical University. Author of more than 30 publications in the ﬁeld of cryogenic technology. N.A. Ivashchenko (b. 1940) graduated from the Bauman Moscow Higher Technical School in 1968. Ph. D. (Eng.), Honoured Science Worker of the Russian Federation, professor, head of “Reciprocator Engines” department of the Bauman Moscow State Technical University. Author of about 200 publications including 11 educational books and monographs in the ﬁeld of engine engineering. S. A. Kalinin (b. 1951) graduated from the Lomonosov Moscow State University in 1973. Ph. D. (Biology), assoc. professor of “Biomedical Technical Systems and Devices” Department of the Bauman Moscow State Technical University. Author of 35 publications in the ﬁeld of mathematical modelling and control in biological, medical, and bioengineering systems. A.V. Sharaburin (b. 1976) graduated from the Bauman Moscow State Technical University in 2000. Ph. D. (Eng.), senior researcher of “Refrigeration, Cryogenics, Air Conditioning and Life Support Systems” department of the Bauman Moscow State Technical University. Author of 4 publications in the ﬁeld of refrigeration technology. A.A. Aleksandrov (b. 1951) graduated from the Bauman Moscow Higher Technical School in 1975. Ph. D. (Eng.), assoc. Professor, director of the Experimental Factory of the Bauman Moscow State Technical University. Winner of the Prize of Government of the Russian Federation. Author of 2 monographs, 2 patents and 10 publications in the ﬁeld of safety of life support.

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INTERNATIONAL COLLABORATION OF BMSTU WITH UNIVERSITIES

BMSTU teaches about 500 international students, post-graduates and trainees from 25 countries throughout the world. So it makes special arrangements to provide support and assistance for them. Actually, BMSTU has agreements for cooperation with other one hundred universities from Argentina, Azerbaijan, America, UK, Hungary, Vietnam, Germany, Greece, Egypt, Italy, Canada, The People’s Republic of China, Columbia, Malaysia, Mexico, Mongolia, The Netherlands, Norway, Peru, The Republic of Korea, Singapore, Syria, the USA, Turkey, Ukraine, Finland, France, Chile, and Switzerland. International activities such as student and staff exchanges, joint education and research projects, conferences and seminars, joint development and publication of cooperative articles, textbooks and monographs play a signiﬁcant role. Membership in international associations of universities 1. T.I.M.E. (Top Industrial Managers for Europe) is the Association of engineering universities of Europe aimed at obtaining a double-degree by some selected students who are highly motivated to study at one of 43 partner universities of this Association for two years. On having completed a study programme at BMSTU, participants are awarded the engineering degrees of both home university and host one. 2. IGIP (International Geselschaft fur Ingenieurpadagogik) is an International Society for Engineering Education. Its objective is to elaborate teaching, scientiﬁc and organisational recommendations to train teaching staff for engineering universities to meet IGIP criteria for the title ”Teacher of European dimension for engineering higher education institution”. 3. IACE is an International Association of Counting Engineering Education aimed at developing and implementing international projects in distance learning. 4. INKORVUZ XXI is an International Co-ordinating Council of graduates of the education establishments. Its objectives are to select international citizens, while promoting study programmes of Russian universities to implement research projects in co-operation with international partners. 5. EAIE is a professional European Association for International education. Its goal is to stimulate and facilitate the internationalisation of education within the global perspective, in particular in higher education. EAIE organisers annual conferences, and training programmes for employees being in charge of international education, produces a variety of publications reﬂecting the interests and activities of its members.

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