Special Issue Article

Modeling on thermally induced coupled micro-motions of satellite with complex flexible appendages

Advances in Mechanical Engineering 2015, Vol. 7(6) 1–7 Ó The Author(s) 2015 DOI: 10.1177/1687814015590310 aime.sagepub.com

Zhicheng Zhou, Zhengshan Liu and Guangji Qu

Abstract To describe the characteristics of thermally induced coupled micro-motions more exactly, a numerical model is proposed for a satellite system consisting of a rigid body and the complex appendages. The coupled governing equations including the effects of transient temperature differences are formulated within the framework of the Lagrangian Method based on the finite element models of flexible structures. Meanwhile, the problem of coupling between attitude motions of rigid body and vibrations of flexible attachments are addressed with explicit expressions. Thermally induced micro-motions are examined in detail for a simple satellite with a large solar panel under the disturbance of thermal environment from earth shadow to sunlight area in the earth orbit. The results show that the thermal–mechanical performances of an on-orbit satellite can be well predicted by the proposed finite element model. Keywords Thermally coupled dynamics, Lagrangian method, flexible appendages, thermal disturbance

Date received: 10 December 2014; accepted: 29 April 2015 Academic Editor: Yan Jin

Introduction Thermally induced motions of a satellite with complex flexible appendages, which have a potential reverse effect on system performance, have attracted worldwide attention in engineering fields. For the on-orbit flexible appendages, the temperature differences, which are induced by the change of surrounding thermal environment, can result in elastic deformations due to different expansions or shrinkages. Additionally, these thermal responses can usually be classified into quasistatic deformations caused by slowly developing temperature differences, and dynamic structural motions caused by rapidly changing temperature differences. Meanwhile, the attitude rotations of the entire satellite may be induced by the thermal motions of flexible appendages. Therefore, to obtain the more accurate thermal response, it is necessary to investigate the modeling technologies on the coupled dynamics of entire

satellite system disturbed by the change of thermal environment. Since the basic concept of thermally induced microvibrations was first addressed to describe the motions of a cantilever beam by Boley,1 a series of research works have been proposed to analyze the thermal– mechanical characteristics. Most of the investigations focused on the thermal responses of fixed appendage structures. Levine2 described the Interferometry Program Experiments to investigate the potential thermal snapping when space structures undergo rapid thermal variations. Xue et al.3 gave a finite element China Academy of Space Technology, Beijing, China Corresponding author: Zhengshan Liu, China Academy of Space Technology, Beijing 100094, China. Email: [email protected]

Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 3.0 License (http://www.creativecommons.org/licenses/by/3.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (http://www.uk.sagepub.com/aboutus/ openaccess.htm).

2 (FE) scheme to solve the problem of thermally induced bending-torsion coupling vibration of large space structures. Heuer4 introduced alternative formulations of various higher order theories to offer complete analogies between the corresponding initial boundary value problems and those of homogenized single layer structures of effective parameters. Gupta and Sharma5 proposed the method to solve the orthotropic trapezoidal plate of variable thickness with clamped-simply supported-clamped-simply (C-S-C-S) boundary conditions. Iwata et al.6 performed a laboratory experiment with a scale model of the paddle and derived a FE analytical method to analyze the response for the Advanced Land Observing Satellite (ALOS). Rath and Sahu7 gave the numerical investigation on the free vibration behavior of laminated composite plates subjected to varying temperature and moisture. Mohammadreza et al.8 investigated the thermally induced vibration of a functionally graded (FG) cantilever micro-beam subjected to a moving laser beam via simulating the equivalent third-order dynamic system. Fan and Xiang9 proposed a novel optimal design to minimize an index determined by the system characteristic thermal time and natural frequency by using a kind of Fourier-finite element method (FEM) technique and the Lanczos method. On the other hand, the attitude rotation induced by the thermal deformations of flexible structures has also been taken into account to make the further error budget of mission pointing accuracy. Based on the thermal quasi-static motions of flexible appendages, Dennehy et al.,10 Foster et al.11 and Hail and De Kramer12 given the coupled models to analyze the change of attitude motion for different spacecrafts. Based on the effect of thermal dynamic response, a coupled method was proposed by Johnston and Thornton13,14 to demonstrate the characteristics of the simple spacecraft consisting of a rigid body and a cantilevered beam. The formulated governing equations are suitable for the thermal response analysis of the satellite system with simple flexible structures. Due to the difficulty in obtaining the analytic resolutions, it is necessary to develop an efficient computation method to deal with the analysis problem of the satellite system with complex flexible appendages. Meanwhile, the total angular momentum of the system is conserved, which implies that the thermal response for complex flexible structures and attitude dynamics should be considered simultaneously to obtain simulation results more exactly. Therefore, a numerical model is presented to analyze the characteristics of thermally induced coupled micromotions of a satellite system consisting of a rigid body and complex flexible appendages. The coupled thermal governing equations, including the transient thermal effects, are formulated with explicit expressions of coupling items between attitude rotation and appendage

Advances in Mechanical Engineering vibration by utilizing the Generalized Lagrangian formulas. An application is shown to predict the thermal micro-vibrations of a simple satellite with a large solar panel under the disturbance of thermal environment from earth shadow to sunlight. The computational results are obtained by studying an application example to demonstrate the thermally induced dynamic response.

Mathematical model The objective considered is to predict the coupled dynamic response of the satellite system (Figure 1) consisting of a rigid body and complex flexible appendages. The responses of flexible structures, which are assumed to be small deformations, are analyzed by the universal FEM. Three coordinate systems are utilized as follows: the orbit reference coordinate {r}, the satellite reference coordinate {b}, and the appendage reference coordinate {a}. As shown in Figure 1, symbols ‘‘B’’ and ‘‘Ai’’ represent the rigid body and the ith flexible structure, respectively. The original point of coordinate Ob in the coordinate {b} represents the center of the mass of the satellite system, and the original point of coordinate Oa in the coordinate {a} represents the joint node between the rigid body and the appendages. The symbols ‘‘xi, yi, zi (i = r, b, ai)’’ denote the three orthogonal axis in the coordinates {r}, {b}, and {a}. X defined in the coordinate {r} is the perturbation of Ob relative to the nominal position; dbi defined in the coordinate {b} is the radius vector from Oai to Ob; rai defined in the coordinate {a} is the radius vector from dmai to Oai; dai defined in the coordinate {a} is the thermal deformation for dmai. rb defined in the coordinate {b} is the radius vector for dmb to Ob. For simplicity, this article is only to address the thermal motions of a satellite in which the flexible structures are fixed onto

B dmb

xb

zb

rb

x

a ob d bi yb oai

Rb

ya

zr

X

Ra or xr

za

rai dmai

δ ai Ai

yr

Figure 1. Simplified satellite model including a rigid body and universal flexible appendages.

Zhou et al.

3

the rigid body. It is assumed that there are no external forces acting on the satellite system.

The rigid body. Assume that dmb is one arbitrary node in the rigid body. The radius vector from dmb to or in the coordinate {r} can be given as

System kinetic energy

Rb = X + Crb rb

In this section, the system kinetic energy will be obtained by utilizing the complex analytic methods in mechanics. Due to the scalar functions, the kinetic energy for flexible appendage and the rigid body will be shown in the reference coordinates {a} and {b} for the further formulas. The main steps are shown as follows. The universal flexible appendage. Assume that dmai is one arbitrary node in the ith appendage structures, the radius vector from dmai to or in the coordinate {r} can be given as Rai = X + Crb dbi + Crai ðrai + dai Þ Crai

ð1Þ

Crb

where and are the transformation matrices. Differentiating equation (1) with time once in the coordinate {r} and mapping the equation to the coordinate {a}, the velocity of dma can be obtained as b _ ai ~T T ai ~T Cai vb + d_ ai ~ vai = Cai C + r C + d d X + C ai b ai b b r b bi

ð2Þ

b where Cbr and Cai b are the transformation matrices, Cr is assumed to be the unit matrix for simplicity; the overhead symbols ‘‘;’’ and ‘‘.’’ represent the third-order symmetric matrix and the first derivative of time, and vb is the angle velocity of satellites. The kinetic energy of the ith flexible appendage in the reference coordinate {a} can be obtained by the following formula as

TAi =

ð 1 T v vai dm 2 ai A

’

X 1 mai X_ T X_ + X_ T Pai wb + X_ T dmai CbaT d_ ai 2 A

1 T w Iai wb 2 b X T + wTb dmai Cbai d~Tbi + r~Tai Cbai d_ ai +

ð3Þ

A

1X + dmai d_ Tai d_ ai 2 A where mai is the mass of the appendage; P ~ai )Cai rTai + d and Iai = Pai = A dmai ½d~Tbi + CaiT b (~ b P ai ~T ai T ai ~T T T ai rai Cb ) (Cb dbi + ~rai Cb ) are the static A dmai (Cb dbi + ~ moment and the moment of inertia for the flexible appendage relative to the center of mass of the entire system, respectively.

ð4Þ

The velocity of dmb in the coordinate {b} can be obtained as vb = X_ + r~Tb vb

ð5Þ

Hence, the kinetic energy of the rigid body in the coordinate {b} can be obtained by the following formula as ð 1 T 1 1 v vb dm = X_ T mb X_ + X_ T Pb vb + wTb Ib vb TB = 2 b 2 2 B

ð6Þ P

where mb isPthe mass of the rigid body; Pb = B dmb ~rTb and Ib = B dmb r~b ~rTb denote the static moment and the moment of inertia for the rigid body relative to the system center of mass of the entire system, respectively. The entire satellite system. By combining the equations (3) and (6), the kinetic energy of the entire satellite system can be calculated as T = TB +

X

TAi

i

X 1 _T _ X MX + X_ T Pwb + X_ T Ri d_ i 2 i X 1 1X _T + wTb Iwb + wTb Fi d_ i + d Mai d_ i 2 2 i i i =

ð7Þ

where Ma is the mass matrix of the appendage; di is the generalized P displacement vectors of the ith appendage; M = mb + i mP ai is the mass of the entire satellite system; P = Pb + iP Pai = 0 is the static moment of the satellite; I = Ib + i Iai , is the moment of inertia of E 0 E 0 E 0 is the satellite; Dai = r~ai1 E ~rai2 E r~ain E the rigid mode of the ith appendage; Ri = ½ CbaiT 0 Dai Mai = R11i Dai Mai is the coefficient Dai Mai = R22i Dai Mai is matrix; and Fi = ½ d~bi CaiT CaiT b b the coefficient matrix.

System potential energy For the universal flexible appendage structures, Boley and Weiner15 described the strain energy of the element including thermal effects as

4

Advances in Mechanical Engineering ððð Ue ’

1 T e De eT De0 dV 2

ð8Þ

V

zb

where e is the element strain vector, D is the constant flexible matrix, and e0 is the element thermal strain vector which can be expressed as e0 = f aDT aDT aDT 0 0 0 gT , in which a is the thermal expansion coefficient. Meanwhile, the relation between the strain field and the displacement vector of FE output nodes can be defined as e = Bue

ð9Þ

where B is the strain displacement transformation matrix. Substituting equation (9) into equation (8), the strain energy of the element, according to the FE theory, can be transformed into

ya

za

ob

yb

oa xa

xb

Figure 2. A simple satellite with a single solar panel.

coordinate {a}, the coupled governing equations for a satellite composed of complex flexible appendages attached to a rigid body can be obtained as X €i = 0 MX€ + Ri d ð14aÞ i

_ b + Ivb + Iv e T

e e

U e = ðu Þ K u

ðue ÞT reT

ððð

BT DBdV ,

reT =

V

ððð

BT De0 dV

ð11Þ

V

Therefore, after integrating all the flexible elements of the satellite system, the potential energy of the entire satellite system in the generalized coordinate {a}, including thermal effects, can be obtained as U=

X 1X T di Kai di dTi rTi 2 i i

ð12Þ

where Kai and rTi are the stiffness matrix and the thermal load of the ith appendage, respectively.

Coupled dynamic equations Combining the system kinetic energy with the potential energy, the Lagrangian function of the system can be obtained as L=T U X 1 1 Ri d_ i + wTb Iwb = X_ T MX_ + X_ T 2 2 i X X ð13Þ 1 1X T d_ Ti Mai d_ i + wTb Fi d_ i + di Kai di 2 i 2 i i X + dTi rTi i

By utilizing the Lagrangian formula for the attitude motions in the satellite reference coordinate {b} and the thermal vibrations in the appendage reference

€i = 0 Fi d

ð14bÞ

i

ð10Þ

where the symbols Ke and rTe , which are the stiffness matrix and thermal load matrix, respectively, have the following expressions as Ke =

X

T€ €i + Cai d_ i + Kai di + FT v Mai d i _ b + Ri X = rTi

ð14cÞ

Numerical application To testify the validity of the proposed coupled model, a numerical study is employed to investigate the thermally induced dynamics of a satellite with a single solar panel as shown in Figure 2. The geometry of solar panel consisting of three parts is presented in Figure 3. The orbit reference coordinate {r} and the satellite reference coordinate {b} are overlapped, and the transformation relation from appendage reference coordinate {a} to satellite reference coordinate {b} is one unit matrix. Meanwhile, to make the demonstration for the further practical engineering application, the coupled equations are solved in the platform MATLAB, while the temperature data are obtained by IDEAS TMG to simulate the environment of the orbit. Furthermore, the dynamic response of the satellite is studied by the FE modal proposed involving the temperature data.

Transient temperature data of the solar panel To obtain the thermal data of the solar panel more accurately and efficiently, the commercial software IDEAS TMG is used to simulate the orbit environment and calculate the temperature gradient. The physical and orbit parameters are adopted here as shown in Tables 1 and 2. To simplify the calculation, one thermal coupled channel is constructed to simulate the heat conductivity from the lighted side panel to the shadowed side panel. Here, the conductivity is to be 1.2 W/ m °C. The eclipse transitions from earth shadow to sunlight area in the earth orbit are chosen for the further analysis. Figure 4 shows the time history of temperature for

Zhou et al.

5

P

P

P

P

P

P

P

P

P

P

P

P

Figure 3. Geometry size of the solar panel. Table 1. Heat physical parameters.

Sun side panel Dark side panel Beam

Mass density (kg/m3)

Heat conductivity (W/m °C)

Specific heat (J/kg °C)

Emissivity

Absorptivity

40 40 1630

0.4 0.4 0.1

1600 1600 1600

0.84 0.95 0.80

0.8 0.8 0.7

Table 2. Orbit parameters.

Values

Values

Solar flux (W/m2)

IR flux (W/m2)

Orbit altitude (km)

Earth radius (km)

Eccentricity

1410.94

236

350

6371

0.00117

Orbit inclination (°)

Argument of perigee (°)

Local time at the ascending node

Period(s)

99.34

90

6:00

5496

IR: infrared.

the solar radiation, which is the maximum thermal resources, induces the slowly developing temperature differences due to the normal direction of solar panel to sun. Meanwhile, the big temperature leap occurs when the solar radiation is to affect the solar panel suddenly.

50

Temperature /°C

Temperature difference

0

Satellite dynamics response

Sunside temperature

−50 Darkside temperature

−100 0

50

100

150

200 Time /s

250

300

350

400

Figure 4. Time history of temperature for the middle point.

the middle point. It can be obviously seen that the temperature differences in the earth shadow area and in sunlight area are close to be constant, and the numerical values change rapidly at the time of exiting from earth shadow. During earth shadow or sunlight area,

In this part, the solution for the dynamic response of the satellite system with the disturbance of thermal environment will be discussed by using the proposed model. An FE model based on the isotropy material is first constructed for the flexible solar panel. And the structural parameters are given in Table 3. For the case of the solar panel fixed on the left, the fundamental frequency can be obtained as 0.1158 Hz. For the further dynamic analysis, five flexible modes are used with a time step of 0.2 s. The thermal load is achieved by combining the transient temperature and the FE model. With the assumption for the initial values to be zero, Figure 5 shows the time traces of bending moment for the middle point. It

6

Advances in Mechanical Engineering

Table 3. Structural parameters.

Solar panel

Mass density (kg/m3)

Poisson ratio

E (N/m2)

Damping coefficient

Expansion parameters (1/°C)

118.281

0.32

7.727 3 109

1 3 1023

3.35 3 1026

0

0

−5

−0.1 −0.2

−10

Along the axis x,y Displacement /mm

Bending moment /N.mm

x y z

Around the axis x,z

−15 −20 −25 −30

Around the axis y

−0.3 −0.4 −0.5 Along the axis z

−0.6 −0.7

−35

−0.8

−40 −0.9

−45 0

100

150

200 Time /s

250

300

350

50

100

150

400

Figure 5. Time traces of the bending moment for the middle point.

is obvious that the values of the bending moment around the axis y remain consistent with the change trend of transient temperature differences as shown in Figure 4, while the bending moment around the axes x and z belonged to the small level is not significant in Figure 5. The potential of thermally induced dynamic response can be analyzed by utilizing the key parameter proposed by Boley.16 The thermal response time tT can be obtained to be 70 s according to the change of temperature as presented in Figure 4. The structural response time ts from the fundamental frequency can be calculated as 2.855 s. Then, the key parameter B isﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ to be 2.85 through the following formula: B = tT =ts . Due to the numerical data close to 1, it can be inferred that the dynamic micro-vibration of the solar panel may probably occur. Finally, the thermally induced motions of entire satellite system disturbed by thermal moment are analyzed in brief by utilizing the proposed equations. With the structural damping parameters being 5& and the mass moments of inertia of the three parts being 22,000 kg m2, 22,000 kg m2, and 20,000 kg m2, respectively. The thermal deformations of the middle point in the flexible solar panel and the induced attitude motions of the satellite can be obtained, as shown in Figures 6 and 7. It is obvious that the values of thermal motions are small: the maximum displacement of flexible panel can reach to the millimeter level, and the attitude angular

200 Time /s

250

300

350

400

Figure 6. Time history of the thermal displacements along the axis direction for the middle point.

45 40 35 Attitude angular /milliarcsec

50

30 25

Around the axis y

20 15 10 Around the axis x,z 5 0 −5

0

50

100

150

200 Time /s

250

300

350

400

Figure 7. Time history of the thermal angular displacement around the axis direction for the satellite’s attitude.

displacement can reach to the milliarcsec level. The thermally induced motions of the flexible solar panel and the satellite’s attitude belong to the quasi-static deformations composed of small dynamic deformations.

Conclusion A numerical FEM for investigating the coupled mechanical characteristics of thermally induced

Zhou et al. motions of a satellite with large flexible appendages has been proposed in this article. The governing equations, taking the transient thermal effect into consideration, are obtained with the interactive items between attitude rotation and appendage vibration by utilizing the generalized form of Lagrangian equations. And the correctness of the proposed mechanical model has been demonstrated with the results of a numerical example. The conclusion can be summarized as follows: 1.

2.

3.

The formulation of thermal disturbance (shown in equations (11) and (12)) is defined by combining the structure’s temperature gradient with the flexible characteristics. This means that how much the budget for the payload performance will be disturbed by the thermal environment can be calculated with a qualitative analysis. The thermal quasi-static/dynamic deformations of the large-scale flexible appendages can lead to the attitude motions of a satellite. Through the coupled equations, the satellite’s flexible characteristics including the changes of the attitude and deformations of the appendages can be analyzed. The results of the presented numerical example show that the thermal response consists of the quasi-static deformations composed of small dynamic deformations for a simple satellite with a large flexible solar panel.

7

3.

4.

5.

6.

7.

8.

9.

10.

11.

Declaration of conflicting interests The authors declare that there is no conflict of interest.

12.

Funding

13.

This work was financially supported by the National Natural Science Foundation of China (Nos 11302244 and 11272334). 14.

References 1. Boley BA. Thermally induced vibrations of beams. J Aeronaut Sci 1956; 23: 179–181. 2. Levine MB. The interferometry program flight experiments: IPEX I and II. In: Proceedings of the SPIE— astronomical telescopes and instrumentation conference,

15. 16.

Kona, HI, 20 March 1998, paper 3350-3314. Bellingham, WA: SPIE. Xue MD, Jin D and Xiang ZH. Thermally-induced bending-torsion coupling vibrations of large scale space structures. Comput Mech 2007; 40: 707–723. Heuer R. Equivalences in the analysis of thermally induced vibrations of sandwich structures. J Therm Stresses 2007; 30: 605–621. Gupta AK and Sharma S. Thermally induced vibration of orthotropic trapezoidal plate of linearly varying thickness. J Vib Contr 2011; 17: 1591–1598. Iwata T, Matsumoto K and Hoshino H. Thermally induced dynamics of large solar array paddle: from laboratory experiment to flight data analysis. In: AIAA guidance, navigation, and control conference, Portland, OR, 8–11 August 2011. Reston, VA: AIAA. Rath MK and Sahu SK. Vibration of woven fiber laminated composite plates in hygrothermal environment. J Vib Contr 2012; 18: 1957–1970. Mohammadreza Z, Ghader R, Ilgar JP, et al. Thermally induced vibration of a functionally graded micro-beam subjected to a moving laser beam. Int J Appl Mech 2014; 6: 1450066-1–1450066-16. Fan LJ and Xiang ZH. Suppressing the thermally induced vibration of large-scale space structures via structural optimization. J Therm Stresses 2015; 38: 1–21. Dennehy CJ, Zimbelman DF and Welch RV. Sunrise/ sunset thermal shock disturbance analysis and simulation for the TOPEXE satellite. In: 28th aerospace sciences meeting, Reno, NV, 8–11 January 1990, AIAA paper 90-0470. Reston, VA: AIAA. Foster CL, Tinker ML and Nurre GS. The solar arrayinduced disturbance of the Hubble space telescope pointing system. J Spacecraft Rockets 1995; 32: 634–644. Hail W and De Kramer C. Thermal jitter analysis of SA3. SAI-TM-0798, Revision A, 1998. Beltsville, MD: Swales Aerospace. Johnston JD and Thornton EA. Thermally induced attitude dynamics of a spacecraft with a flexible appendage. J Guid Contr Dynam 1998; 21: 581–587. Johnston JD and Thornton EA. Thermally induced dynamics of satellite solar panels. J Spacecraft Rockets 2000; 37: 604–613. Boley BA and Weiner JH. Theory of thermal stresses. New York: John Wiley & Sons, 1960. Boley BA. Approximate analysis of thermally induced vibrations of beams and plates. J Appl Mech Trans ASME 1972; 39: 212–216.

Modeling on thermally induced coupled micro-motions of satellite with complex flexible appendages

Advances in Mechanical Engineering 2015, Vol. 7(6) 1–7 Ó The Author(s) 2015 DOI: 10.1177/1687814015590310 aime.sagepub.com

Zhicheng Zhou, Zhengshan Liu and Guangji Qu

Abstract To describe the characteristics of thermally induced coupled micro-motions more exactly, a numerical model is proposed for a satellite system consisting of a rigid body and the complex appendages. The coupled governing equations including the effects of transient temperature differences are formulated within the framework of the Lagrangian Method based on the finite element models of flexible structures. Meanwhile, the problem of coupling between attitude motions of rigid body and vibrations of flexible attachments are addressed with explicit expressions. Thermally induced micro-motions are examined in detail for a simple satellite with a large solar panel under the disturbance of thermal environment from earth shadow to sunlight area in the earth orbit. The results show that the thermal–mechanical performances of an on-orbit satellite can be well predicted by the proposed finite element model. Keywords Thermally coupled dynamics, Lagrangian method, flexible appendages, thermal disturbance

Date received: 10 December 2014; accepted: 29 April 2015 Academic Editor: Yan Jin

Introduction Thermally induced motions of a satellite with complex flexible appendages, which have a potential reverse effect on system performance, have attracted worldwide attention in engineering fields. For the on-orbit flexible appendages, the temperature differences, which are induced by the change of surrounding thermal environment, can result in elastic deformations due to different expansions or shrinkages. Additionally, these thermal responses can usually be classified into quasistatic deformations caused by slowly developing temperature differences, and dynamic structural motions caused by rapidly changing temperature differences. Meanwhile, the attitude rotations of the entire satellite may be induced by the thermal motions of flexible appendages. Therefore, to obtain the more accurate thermal response, it is necessary to investigate the modeling technologies on the coupled dynamics of entire

satellite system disturbed by the change of thermal environment. Since the basic concept of thermally induced microvibrations was first addressed to describe the motions of a cantilever beam by Boley,1 a series of research works have been proposed to analyze the thermal– mechanical characteristics. Most of the investigations focused on the thermal responses of fixed appendage structures. Levine2 described the Interferometry Program Experiments to investigate the potential thermal snapping when space structures undergo rapid thermal variations. Xue et al.3 gave a finite element China Academy of Space Technology, Beijing, China Corresponding author: Zhengshan Liu, China Academy of Space Technology, Beijing 100094, China. Email: [email protected]

Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 3.0 License (http://www.creativecommons.org/licenses/by/3.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (http://www.uk.sagepub.com/aboutus/ openaccess.htm).

2 (FE) scheme to solve the problem of thermally induced bending-torsion coupling vibration of large space structures. Heuer4 introduced alternative formulations of various higher order theories to offer complete analogies between the corresponding initial boundary value problems and those of homogenized single layer structures of effective parameters. Gupta and Sharma5 proposed the method to solve the orthotropic trapezoidal plate of variable thickness with clamped-simply supported-clamped-simply (C-S-C-S) boundary conditions. Iwata et al.6 performed a laboratory experiment with a scale model of the paddle and derived a FE analytical method to analyze the response for the Advanced Land Observing Satellite (ALOS). Rath and Sahu7 gave the numerical investigation on the free vibration behavior of laminated composite plates subjected to varying temperature and moisture. Mohammadreza et al.8 investigated the thermally induced vibration of a functionally graded (FG) cantilever micro-beam subjected to a moving laser beam via simulating the equivalent third-order dynamic system. Fan and Xiang9 proposed a novel optimal design to minimize an index determined by the system characteristic thermal time and natural frequency by using a kind of Fourier-finite element method (FEM) technique and the Lanczos method. On the other hand, the attitude rotation induced by the thermal deformations of flexible structures has also been taken into account to make the further error budget of mission pointing accuracy. Based on the thermal quasi-static motions of flexible appendages, Dennehy et al.,10 Foster et al.11 and Hail and De Kramer12 given the coupled models to analyze the change of attitude motion for different spacecrafts. Based on the effect of thermal dynamic response, a coupled method was proposed by Johnston and Thornton13,14 to demonstrate the characteristics of the simple spacecraft consisting of a rigid body and a cantilevered beam. The formulated governing equations are suitable for the thermal response analysis of the satellite system with simple flexible structures. Due to the difficulty in obtaining the analytic resolutions, it is necessary to develop an efficient computation method to deal with the analysis problem of the satellite system with complex flexible appendages. Meanwhile, the total angular momentum of the system is conserved, which implies that the thermal response for complex flexible structures and attitude dynamics should be considered simultaneously to obtain simulation results more exactly. Therefore, a numerical model is presented to analyze the characteristics of thermally induced coupled micromotions of a satellite system consisting of a rigid body and complex flexible appendages. The coupled thermal governing equations, including the transient thermal effects, are formulated with explicit expressions of coupling items between attitude rotation and appendage

Advances in Mechanical Engineering vibration by utilizing the Generalized Lagrangian formulas. An application is shown to predict the thermal micro-vibrations of a simple satellite with a large solar panel under the disturbance of thermal environment from earth shadow to sunlight. The computational results are obtained by studying an application example to demonstrate the thermally induced dynamic response.

Mathematical model The objective considered is to predict the coupled dynamic response of the satellite system (Figure 1) consisting of a rigid body and complex flexible appendages. The responses of flexible structures, which are assumed to be small deformations, are analyzed by the universal FEM. Three coordinate systems are utilized as follows: the orbit reference coordinate {r}, the satellite reference coordinate {b}, and the appendage reference coordinate {a}. As shown in Figure 1, symbols ‘‘B’’ and ‘‘Ai’’ represent the rigid body and the ith flexible structure, respectively. The original point of coordinate Ob in the coordinate {b} represents the center of the mass of the satellite system, and the original point of coordinate Oa in the coordinate {a} represents the joint node between the rigid body and the appendages. The symbols ‘‘xi, yi, zi (i = r, b, ai)’’ denote the three orthogonal axis in the coordinates {r}, {b}, and {a}. X defined in the coordinate {r} is the perturbation of Ob relative to the nominal position; dbi defined in the coordinate {b} is the radius vector from Oai to Ob; rai defined in the coordinate {a} is the radius vector from dmai to Oai; dai defined in the coordinate {a} is the thermal deformation for dmai. rb defined in the coordinate {b} is the radius vector for dmb to Ob. For simplicity, this article is only to address the thermal motions of a satellite in which the flexible structures are fixed onto

B dmb

xb

zb

rb

x

a ob d bi yb oai

Rb

ya

zr

X

Ra or xr

za

rai dmai

δ ai Ai

yr

Figure 1. Simplified satellite model including a rigid body and universal flexible appendages.

Zhou et al.

3

the rigid body. It is assumed that there are no external forces acting on the satellite system.

The rigid body. Assume that dmb is one arbitrary node in the rigid body. The radius vector from dmb to or in the coordinate {r} can be given as

System kinetic energy

Rb = X + Crb rb

In this section, the system kinetic energy will be obtained by utilizing the complex analytic methods in mechanics. Due to the scalar functions, the kinetic energy for flexible appendage and the rigid body will be shown in the reference coordinates {a} and {b} for the further formulas. The main steps are shown as follows. The universal flexible appendage. Assume that dmai is one arbitrary node in the ith appendage structures, the radius vector from dmai to or in the coordinate {r} can be given as Rai = X + Crb dbi + Crai ðrai + dai Þ Crai

ð1Þ

Crb

where and are the transformation matrices. Differentiating equation (1) with time once in the coordinate {r} and mapping the equation to the coordinate {a}, the velocity of dma can be obtained as b _ ai ~T T ai ~T Cai vb + d_ ai ~ vai = Cai C + r C + d d X + C ai b ai b b r b bi

ð2Þ

b where Cbr and Cai b are the transformation matrices, Cr is assumed to be the unit matrix for simplicity; the overhead symbols ‘‘;’’ and ‘‘.’’ represent the third-order symmetric matrix and the first derivative of time, and vb is the angle velocity of satellites. The kinetic energy of the ith flexible appendage in the reference coordinate {a} can be obtained by the following formula as

TAi =

ð 1 T v vai dm 2 ai A

’

X 1 mai X_ T X_ + X_ T Pai wb + X_ T dmai CbaT d_ ai 2 A

1 T w Iai wb 2 b X T + wTb dmai Cbai d~Tbi + r~Tai Cbai d_ ai +

ð3Þ

A

1X + dmai d_ Tai d_ ai 2 A where mai is the mass of the appendage; P ~ai )Cai rTai + d and Iai = Pai = A dmai ½d~Tbi + CaiT b (~ b P ai ~T ai T ai ~T T T ai rai Cb ) (Cb dbi + ~rai Cb ) are the static A dmai (Cb dbi + ~ moment and the moment of inertia for the flexible appendage relative to the center of mass of the entire system, respectively.

ð4Þ

The velocity of dmb in the coordinate {b} can be obtained as vb = X_ + r~Tb vb

ð5Þ

Hence, the kinetic energy of the rigid body in the coordinate {b} can be obtained by the following formula as ð 1 T 1 1 v vb dm = X_ T mb X_ + X_ T Pb vb + wTb Ib vb TB = 2 b 2 2 B

ð6Þ P

where mb isPthe mass of the rigid body; Pb = B dmb ~rTb and Ib = B dmb r~b ~rTb denote the static moment and the moment of inertia for the rigid body relative to the system center of mass of the entire system, respectively. The entire satellite system. By combining the equations (3) and (6), the kinetic energy of the entire satellite system can be calculated as T = TB +

X

TAi

i

X 1 _T _ X MX + X_ T Pwb + X_ T Ri d_ i 2 i X 1 1X _T + wTb Iwb + wTb Fi d_ i + d Mai d_ i 2 2 i i i =

ð7Þ

where Ma is the mass matrix of the appendage; di is the generalized P displacement vectors of the ith appendage; M = mb + i mP ai is the mass of the entire satellite system; P = Pb + iP Pai = 0 is the static moment of the satellite; I = Ib + i Iai , is the moment of inertia of E 0 E 0 E 0 is the satellite; Dai = r~ai1 E ~rai2 E r~ain E the rigid mode of the ith appendage; Ri = ½ CbaiT 0 Dai Mai = R11i Dai Mai is the coefficient Dai Mai = R22i Dai Mai is matrix; and Fi = ½ d~bi CaiT CaiT b b the coefficient matrix.

System potential energy For the universal flexible appendage structures, Boley and Weiner15 described the strain energy of the element including thermal effects as

4

Advances in Mechanical Engineering ððð Ue ’

1 T e De eT De0 dV 2

ð8Þ

V

zb

where e is the element strain vector, D is the constant flexible matrix, and e0 is the element thermal strain vector which can be expressed as e0 = f aDT aDT aDT 0 0 0 gT , in which a is the thermal expansion coefficient. Meanwhile, the relation between the strain field and the displacement vector of FE output nodes can be defined as e = Bue

ð9Þ

where B is the strain displacement transformation matrix. Substituting equation (9) into equation (8), the strain energy of the element, according to the FE theory, can be transformed into

ya

za

ob

yb

oa xa

xb

Figure 2. A simple satellite with a single solar panel.

coordinate {a}, the coupled governing equations for a satellite composed of complex flexible appendages attached to a rigid body can be obtained as X €i = 0 MX€ + Ri d ð14aÞ i

_ b + Ivb + Iv e T

e e

U e = ðu Þ K u

ðue ÞT reT

ððð

BT DBdV ,

reT =

V

ððð

BT De0 dV

ð11Þ

V

Therefore, after integrating all the flexible elements of the satellite system, the potential energy of the entire satellite system in the generalized coordinate {a}, including thermal effects, can be obtained as U=

X 1X T di Kai di dTi rTi 2 i i

ð12Þ

where Kai and rTi are the stiffness matrix and the thermal load of the ith appendage, respectively.

Coupled dynamic equations Combining the system kinetic energy with the potential energy, the Lagrangian function of the system can be obtained as L=T U X 1 1 Ri d_ i + wTb Iwb = X_ T MX_ + X_ T 2 2 i X X ð13Þ 1 1X T d_ Ti Mai d_ i + wTb Fi d_ i + di Kai di 2 i 2 i i X + dTi rTi i

By utilizing the Lagrangian formula for the attitude motions in the satellite reference coordinate {b} and the thermal vibrations in the appendage reference

€i = 0 Fi d

ð14bÞ

i

ð10Þ

where the symbols Ke and rTe , which are the stiffness matrix and thermal load matrix, respectively, have the following expressions as Ke =

X

T€ €i + Cai d_ i + Kai di + FT v Mai d i _ b + Ri X = rTi

ð14cÞ

Numerical application To testify the validity of the proposed coupled model, a numerical study is employed to investigate the thermally induced dynamics of a satellite with a single solar panel as shown in Figure 2. The geometry of solar panel consisting of three parts is presented in Figure 3. The orbit reference coordinate {r} and the satellite reference coordinate {b} are overlapped, and the transformation relation from appendage reference coordinate {a} to satellite reference coordinate {b} is one unit matrix. Meanwhile, to make the demonstration for the further practical engineering application, the coupled equations are solved in the platform MATLAB, while the temperature data are obtained by IDEAS TMG to simulate the environment of the orbit. Furthermore, the dynamic response of the satellite is studied by the FE modal proposed involving the temperature data.

Transient temperature data of the solar panel To obtain the thermal data of the solar panel more accurately and efficiently, the commercial software IDEAS TMG is used to simulate the orbit environment and calculate the temperature gradient. The physical and orbit parameters are adopted here as shown in Tables 1 and 2. To simplify the calculation, one thermal coupled channel is constructed to simulate the heat conductivity from the lighted side panel to the shadowed side panel. Here, the conductivity is to be 1.2 W/ m °C. The eclipse transitions from earth shadow to sunlight area in the earth orbit are chosen for the further analysis. Figure 4 shows the time history of temperature for

Zhou et al.

5

P

P

P

P

P

P

P

P

P

P

P

P

Figure 3. Geometry size of the solar panel. Table 1. Heat physical parameters.

Sun side panel Dark side panel Beam

Mass density (kg/m3)

Heat conductivity (W/m °C)

Specific heat (J/kg °C)

Emissivity

Absorptivity

40 40 1630

0.4 0.4 0.1

1600 1600 1600

0.84 0.95 0.80

0.8 0.8 0.7

Table 2. Orbit parameters.

Values

Values

Solar flux (W/m2)

IR flux (W/m2)

Orbit altitude (km)

Earth radius (km)

Eccentricity

1410.94

236

350

6371

0.00117

Orbit inclination (°)

Argument of perigee (°)

Local time at the ascending node

Period(s)

99.34

90

6:00

5496

IR: infrared.

the solar radiation, which is the maximum thermal resources, induces the slowly developing temperature differences due to the normal direction of solar panel to sun. Meanwhile, the big temperature leap occurs when the solar radiation is to affect the solar panel suddenly.

50

Temperature /°C

Temperature difference

0

Satellite dynamics response

Sunside temperature

−50 Darkside temperature

−100 0

50

100

150

200 Time /s

250

300

350

400

Figure 4. Time history of temperature for the middle point.

the middle point. It can be obviously seen that the temperature differences in the earth shadow area and in sunlight area are close to be constant, and the numerical values change rapidly at the time of exiting from earth shadow. During earth shadow or sunlight area,

In this part, the solution for the dynamic response of the satellite system with the disturbance of thermal environment will be discussed by using the proposed model. An FE model based on the isotropy material is first constructed for the flexible solar panel. And the structural parameters are given in Table 3. For the case of the solar panel fixed on the left, the fundamental frequency can be obtained as 0.1158 Hz. For the further dynamic analysis, five flexible modes are used with a time step of 0.2 s. The thermal load is achieved by combining the transient temperature and the FE model. With the assumption for the initial values to be zero, Figure 5 shows the time traces of bending moment for the middle point. It

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Advances in Mechanical Engineering

Table 3. Structural parameters.

Solar panel

Mass density (kg/m3)

Poisson ratio

E (N/m2)

Damping coefficient

Expansion parameters (1/°C)

118.281

0.32

7.727 3 109

1 3 1023

3.35 3 1026

0

0

−5

−0.1 −0.2

−10

Along the axis x,y Displacement /mm

Bending moment /N.mm

x y z

Around the axis x,z

−15 −20 −25 −30

Around the axis y

−0.3 −0.4 −0.5 Along the axis z

−0.6 −0.7

−35

−0.8

−40 −0.9

−45 0

100

150

200 Time /s

250

300

350

50

100

150

400

Figure 5. Time traces of the bending moment for the middle point.

is obvious that the values of the bending moment around the axis y remain consistent with the change trend of transient temperature differences as shown in Figure 4, while the bending moment around the axes x and z belonged to the small level is not significant in Figure 5. The potential of thermally induced dynamic response can be analyzed by utilizing the key parameter proposed by Boley.16 The thermal response time tT can be obtained to be 70 s according to the change of temperature as presented in Figure 4. The structural response time ts from the fundamental frequency can be calculated as 2.855 s. Then, the key parameter B isﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ to be 2.85 through the following formula: B = tT =ts . Due to the numerical data close to 1, it can be inferred that the dynamic micro-vibration of the solar panel may probably occur. Finally, the thermally induced motions of entire satellite system disturbed by thermal moment are analyzed in brief by utilizing the proposed equations. With the structural damping parameters being 5& and the mass moments of inertia of the three parts being 22,000 kg m2, 22,000 kg m2, and 20,000 kg m2, respectively. The thermal deformations of the middle point in the flexible solar panel and the induced attitude motions of the satellite can be obtained, as shown in Figures 6 and 7. It is obvious that the values of thermal motions are small: the maximum displacement of flexible panel can reach to the millimeter level, and the attitude angular

200 Time /s

250

300

350

400

Figure 6. Time history of the thermal displacements along the axis direction for the middle point.

45 40 35 Attitude angular /milliarcsec

50

30 25

Around the axis y

20 15 10 Around the axis x,z 5 0 −5

0

50

100

150

200 Time /s

250

300

350

400

Figure 7. Time history of the thermal angular displacement around the axis direction for the satellite’s attitude.

displacement can reach to the milliarcsec level. The thermally induced motions of the flexible solar panel and the satellite’s attitude belong to the quasi-static deformations composed of small dynamic deformations.

Conclusion A numerical FEM for investigating the coupled mechanical characteristics of thermally induced

Zhou et al. motions of a satellite with large flexible appendages has been proposed in this article. The governing equations, taking the transient thermal effect into consideration, are obtained with the interactive items between attitude rotation and appendage vibration by utilizing the generalized form of Lagrangian equations. And the correctness of the proposed mechanical model has been demonstrated with the results of a numerical example. The conclusion can be summarized as follows: 1.

2.

3.

The formulation of thermal disturbance (shown in equations (11) and (12)) is defined by combining the structure’s temperature gradient with the flexible characteristics. This means that how much the budget for the payload performance will be disturbed by the thermal environment can be calculated with a qualitative analysis. The thermal quasi-static/dynamic deformations of the large-scale flexible appendages can lead to the attitude motions of a satellite. Through the coupled equations, the satellite’s flexible characteristics including the changes of the attitude and deformations of the appendages can be analyzed. The results of the presented numerical example show that the thermal response consists of the quasi-static deformations composed of small dynamic deformations for a simple satellite with a large flexible solar panel.

7

3.

4.

5.

6.

7.

8.

9.

10.

11.

Declaration of conflicting interests The authors declare that there is no conflict of interest.

12.

Funding

13.

This work was financially supported by the National Natural Science Foundation of China (Nos 11302244 and 11272334). 14.

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Kona, HI, 20 March 1998, paper 3350-3314. Bellingham, WA: SPIE. Xue MD, Jin D and Xiang ZH. Thermally-induced bending-torsion coupling vibrations of large scale space structures. Comput Mech 2007; 40: 707–723. Heuer R. Equivalences in the analysis of thermally induced vibrations of sandwich structures. J Therm Stresses 2007; 30: 605–621. Gupta AK and Sharma S. Thermally induced vibration of orthotropic trapezoidal plate of linearly varying thickness. J Vib Contr 2011; 17: 1591–1598. Iwata T, Matsumoto K and Hoshino H. Thermally induced dynamics of large solar array paddle: from laboratory experiment to flight data analysis. In: AIAA guidance, navigation, and control conference, Portland, OR, 8–11 August 2011. Reston, VA: AIAA. Rath MK and Sahu SK. Vibration of woven fiber laminated composite plates in hygrothermal environment. J Vib Contr 2012; 18: 1957–1970. Mohammadreza Z, Ghader R, Ilgar JP, et al. Thermally induced vibration of a functionally graded micro-beam subjected to a moving laser beam. Int J Appl Mech 2014; 6: 1450066-1–1450066-16. Fan LJ and Xiang ZH. Suppressing the thermally induced vibration of large-scale space structures via structural optimization. J Therm Stresses 2015; 38: 1–21. Dennehy CJ, Zimbelman DF and Welch RV. Sunrise/ sunset thermal shock disturbance analysis and simulation for the TOPEXE satellite. In: 28th aerospace sciences meeting, Reno, NV, 8–11 January 1990, AIAA paper 90-0470. Reston, VA: AIAA. Foster CL, Tinker ML and Nurre GS. The solar arrayinduced disturbance of the Hubble space telescope pointing system. J Spacecraft Rockets 1995; 32: 634–644. Hail W and De Kramer C. Thermal jitter analysis of SA3. SAI-TM-0798, Revision A, 1998. Beltsville, MD: Swales Aerospace. Johnston JD and Thornton EA. Thermally induced attitude dynamics of a spacecraft with a flexible appendage. J Guid Contr Dynam 1998; 21: 581–587. Johnston JD and Thornton EA. Thermally induced dynamics of satellite solar panels. J Spacecraft Rockets 2000; 37: 604–613. Boley BA and Weiner JH. Theory of thermal stresses. New York: John Wiley & Sons, 1960. Boley BA. Approximate analysis of thermally induced vibrations of beams and plates. J Appl Mech Trans ASME 1972; 39: 212–216.