optimising dynamic characteristics of machinery

5 downloads 0 Views 591KB Size Report
the systems incorporating a dynamic vibration eliminator. The investigations were ... carried out on the example of a dynamic system with a vibration eliminator.
European Congress on Computational Methods in Applied Sciences and Engineering ECCOMAS 2000 Barcelona, 11-14 September 2000  ECCOMAS

OPTIMISING DYNAMIC CHARACTERISTICS OF MACHINERY Arkadiusz Mężyk, Eugeniusz Świtoński Department of Applied Mechanics Silesian University of Technology Konarskiego 18a, 44-100 Gliwice, Poland e-mail: [email protected], web page: http://www.kms.polsl.gliwice.pl/

Key words: Dynamic characteristics, Sensitivity Analysis, Dynamic Model, Optimisation, Vibration. Abstract. A methodology for shaping the dynamic characteristics of machines has been presented in the paper. The objective of the performed investigations was to minimize dynamic reactions in kinematic pairs of the system under analysis. In the formulated algorithm for selecting dynamic parameters of the systems, numerical methods of the sensitivity analysis and optimisation found application. Time-dependent objective functions, describing time courses of dynamic reactions of the systems in unsteady states, and frequencydependent objective functions, describing resonance curves, are taken into account in the presented algorithm. The so formulated problems served for minimizing the peak values of time courses of the dynamic forces as well as for optimisation of resonance characteristics of the systems incorporating a dynamic vibration eliminator. The investigations were performed when making use of discrete models of dynamic electromechanical machinery systems. A method of sequential linear programming was applied to search for optimum dynamic parameters for all types of the objective function. The method was used to minimize electromechanical vibration amplitudes of drive systems in unsteady states (start-up, sudden change in load). The paper deals also with shaping of dynamic responses to the system with the aid of resonance curves in the frequency domain. In this case the optimisation was oriented at minimizing the peak values corresponding to selected natural frequencies. An optimisation of the objective function describing a coefficient of amplitude amplification was carried out on the example of a dynamic system with a vibration eliminator. The algorithm enables to optimise dynamic characteristics of machines for different objective functions and constraints. Calculations were made when taking the optimisation of dynamic characteristics of an electromechanical drive system of a working machine as an example.

1

Arkadiusz Mężyk, Eugeniusz Świtoński

1 INTRODUCTION Practically, all contemporary branches of technology are faced with questions involved in selecting the dynamic characteristics of physical systems. Optimisation is still rarely used as a designing method applicable for solving the technical problems of dynamic character. Providing proper shapes of dynamic characteristics of machines can minimize amplitudes of dynamic reactions. Numerical methods of the sensitivity analysis and optimisation found application in the formulated algorithm of selecting the dynamic parameters of systems. In the presented algorithm of the sensitivity analysis and optimisation the following objective functions are taken into account: • objective function, dependent on a set of design variables and time, that describes time courses of dynamic reactions of vibration for mechanical and electromechanical systems in unsteady states; • objective function, dependent on a set of design variables and frequency that describes resonance curves. 2. MODELLING OF A SYSTEM Drive systems of machines are complex dynamic systems with many degrees of freedom. One of the methods for determining the dynamic state of a machine consists in modelling its form as a complex dynamic system composed of a number of simple subsystems, which interact one on another. Modelling of an electromechanical drive system as a system with feedback between its electric part and mechanical one is an example of such an approach. A model of the mechanical part, taken as a discrete form, describes the following system of ordinary differential equations expressed as a matrix:  + CV q + Kq = Q Mq

(1)

where: M, Cv and K - matrices of inertia, damping and stiffness, q -column matrix of generalized coordinates, Q - column matrix of generalized forces. The model of an electric motor describes electromagnetic phenomena connected with the magnetic field action in a machine. The character of time courses of dynamic quantities is, to a considerable degree, conditioned by characteristics and power of a driving motor. Thus, the accuracy of the obtained solutions depends on the exactness of the assumptions made in a physical model of the mechanical system and in a model of the electric motor. Dynamic phenomena taking place in asynchronous machines can be described by the following differential equations: d Li + Ri = U dt ∂ 1 Li M el = i T 2 ∂ϕ 1

2

(2)

Arkadiusz Mężyk, Eugeniusz Świtoński

where: L, R. i, U - matrices of inductance, resistance, supply currents and voltages, Mel motor torque, ϕ1 - angular displacement of a rotor. The electromechanical coupling is realized through the angular displacement of the rotor ϕ1 as determined from the model of the mechanical system and through the torque Mel calculated from the electric motor model. The solution of the above model coupled with the model of the mechanical system allows dynamic characteristics of the electromechanical drive system to be obtained. In consequence of describing of the physical model of the system in state coordinates a system of differential equations of the second order is put to a system of differential equations of the first order. x (t ) = Ax(t ) + Bu(t ) y (t ) = Cx(t ) + Du(t )

(3)

where: A - matrix of the system, B - matrix of inputs, C - matrix of outputs, D - matrix of direct effect of a column matrix of inputs u(t) on a column matrix of outputs y(t). Such a notation makes the problem more convenient for solving by most numerical methods. 3. SENSITIVITY ANALYSIS OF DYNAMIC CHARACTERISTICS The sensitivity analysis constitutes an indispensable source of information about the influence of design features on dynamic phenomena occurring in kinematic pairs of the system. However, the necessity of determining the derivatives of the objective function in relation to design variables is the basic difficulty encountered in the sensitivity analysis. Among methods of the analysis there are distinguished: • direct methods • adjoined methods • approximate methods. The paper deals with the application of direct and approximate methods only. 3.1 Investigation of the sensitivity of eigenvalues Natural frequencies are basic quantities characterizing the dynamic characteristics of physical systems. The knowledge of the influence exerted by design features on values of the vibration frequency is of vital importance for the obtainment of required dynamic characteristics. Such knowledge can be acquired through the sensitivity analysis. In case of a system, in which there is not any damping and multiple eigenvalues are absent, it is possible to determine a derivative of the selected natural frequency in relation to parameters of the model from the relationship: ∂ω j  ∂K ∂M   X j = X Tj  −ω (4) ∂ bi ∂ b ∂ b i   i

3

Arkadiusz Mężyk, Eugeniusz Świtoński

where: ωj - eigenvalue j, Xj - eigenvector corresponding to j -th eigenvalue that is so normalized that the condition XTj MXj = 1 is fulfilled. 3.2 Investigation of the sensitivity of time courses When aiming at investigation of the sensitivity of time courses, it is necessary to solve a mathematical model of the system being modelled. Such a model described in state coordinates is, in addition, a function of selected design variables entered in the matrix b and can be formulated as follows: x = f(x, b) x( t 0) = h(b)

(5)

where: x(t) - column matrix of state variables, b - column matrix of design variables, to , tk initial time and end time. The selected objective function describes the state of the system at the defined moment of time stated as time tk:

ψ = g( x( t k ),b )

(6)

The so formulated objective function makes it possible to investigate the influence of dissipation of energy in the system, external loading of the system and of the coupling between a driving motor and the machinery on vibration amplitudes in unsteady states (startup, sudden change in load) as well as assures that steady state is obtained in minimum time. By way of differentiation of the functional (6) in relation to b and after transformations we obtain : T

dψ  ∂g  ∂x ∂g = + db  ∂x  ∂b ∂b

(7)

Partial derivatives, that are present in the above expression, can be determined analytically, except for ∂x(tk)/∂b, that cannot be determined by analytic methods. This results from the fact that x(t) is determined numerically. It is possible to determine this expression or to avoid its determination by using a direct differentiation method or a adjoined variables method4,7 . The direct differentiation method consists in solving an additional equation obtained by means of differentiation of (5) in relation to b. ∂x ∂f ∂x ∂f + = ⋅ ∂b ∂x ∂b ∂b ∂x( t 0) ∂h = ∂b ∂b t0 ≤ t ≤ tk

(8)

In consequence of solving the initial problem (8) the wanted values of the derivative ∂x(t)/∂b

4

Arkadiusz Mężyk, Eugeniusz Świtoński

are obtained. A very simple algorithm and possibilities of calculating ∂x(t)/∂b and x(t) at the same time are the advantage of this method. Its disadvantage lies in that it is necessary to solve a great number of additional differential equations. 3.3 Investigation of the sensitivity of spectral transfer function A spectral transfer function is one of the basic characteristics used to describe dynamic linear systems. The relationship describing the spectral transfer function of the system is obtained when differential equations of motion put in state coordinates (3) are subjected to the Laplace transform. In case of a stationary linear system and on the assumption that initial conditions are zero it is possible to obtain an expression describing the transmittance as a frequency function. H ( jω ) = C( jωI − A ) B + D −1

(9)

The above function describes a response of the system to sinusoidal excitation. The response can be presented in a dimensionless form that determines an amplitude amplification factor. 3.3.1 Determining of a derivative by a direct method Investigation of the sensitivity of spectral transfer function needs determining the derivatives H(jω) in relation to design variables b. When taking into consideration the principles of differentiation we obtain ∂H ( jω ) ∂A −1  ∂B ( jωI − A )−1 B + ∂D = C( jωI − A )  + ∂b  ∂b ∂b  ∂b

(10)

Seeing that the argument of the function (9) is an imaginary number, we plot the characteristic curve as the relationship H ( jω ) . The equation (10) represents also a function of the imaginary variable. Thus, the wanted function of the sensitivity is found as 1  ∂  ∂H ( jω )   ∂H ( jω )  H ( jω ) = Re(H ( jω ))Re  + Im(H ( jω ))Im   H ( jω )  ∂b  ∂b   ∂b  3.3.2

(11)

Determining of a derivative by an approximate method

A derivative of the spectral transfer function can be determined approximately by means of the finite differences method. In such a case we make use of the relationship: H ( jω ,b + ∆b ) − H ( jω ,b ) ∂ H ( j ω ,b ) = ∂b Δb

(12)

This method of determining the derivatives provides a very simple algorithm, that, however, may prove to be low- accurate and time-consuming in case of some applications especially

5

Arkadiusz Mężyk, Eugeniusz Świtoński

when there is a great number of design variables. 4. OPTIMISATION OF DYNAMIC CHARACTERISTICS Optimisation of dynamic characteristics of dynamic drive systems is one of the methods leading to increased reliability and longer service life of machines. The selection of design variables is conditioned by the purpose of optimisation and by technical possibilities9,10. Objective functions describing the eigenvalue of the system are used most often. It is much more difficult to optimise maximum values of time- varying functions. A set of design variables can contain any parameters of the system such as: parameters of an electric motor model, masses and mass moments of inertia of elements of the system, coefficients of stiffness and damping, state variables and the like. Most of the problems aimed at the optimisation of dynamic characteristics consist in minimizing the maximum values Min Max ψ (t)

(13)

The difficulty lies in that when making calculations for definite values of design variables it is not known what value of the independent variable involves the peak value of the objective function. Problems of this type are strongly nonlinear. Moreover, several peak values are likely to occur. From results obtained when applying the methods of optimisation for solving problems of this type it appears that sequential methods are highly effective1,2,3. A method of sequential linear programming is one of them6,8. The essence of this method consists in changing of the problem of nonlinear optimisation of the objective function into a sequence of linear optimisation problems based on gradients. In such a case we consider the variable ψmax for an additional unknown subjected to additional constraints8. The most important constraint is that value of the objective function should not exceed ψmax at each step of calculations. In this way the problem of optimisation will be changed into a sequence of problems of linear optimisation b

(i +1)

= b(i) + Δ b(i)

(14)

where: b(i) - column matrix of parameters after i optimisations, Δb(i) - increase of b in i -th sequence of optimisation. In case of a very little increase of the parameter, a linear increment of the objective function in the neighbourhood b can be assumed. It is possible to express the optimisation problem (13) in a form: Min ψ max ψ j + ∇ψ j ∆b - ψ max ≤ 0

(15)

j = 1..k where: k - number of the peak value of the objective function that is nearing ψmax in the actual analysis, ψj – value of the objective function for j-th peak,∇ψj - gradient ψj in relation to

6

Arkadiusz Mężyk, Eugeniusz Świtoński

design variables. The so formulated problem of optimisation can be solved by the SIMPLEX method. The optimisation was carried out for objective functions described in the domain of time and frequency. In case of such problems a set of design variables comprises design parameters of the system such as masses, mass moments of inertia, coefficients of damping and stiffness. Because of technical constraints it is not always possible to change freely parameters of the system (e.g. coefficient of internal damping for the given material can be changed within a small range only). In such cases changing a structure of the system can effect the changes of dynamic characteristic. M(t)

I1 k1

Ie

ke

ce

ϕ1

ϕe

Figure 1: Dynamic model of the system with vibration eliminator

Application of a dynamic vibration eliminator (Fig.1), the dynamic features of which are conditioned by parameters of the drive system, serves as an example of such approach. In order to obtain the best effects in the field of vibration elimination, it is indispensably necessary to adjust the eliminator to the system. Results of numerical simulations have indicated that a system with an eliminator selected by means of optimisation methods is characterized by a low value of the coefficient of vibration amplification within a wide range of frequency. 5. MINIMIZATION OF THE DYNAMIC REACTIONS IN KINEMATIC PAIRS OF A DRIVE SYSTEM An electromechanical discrete model of a drive system composed of an asynchronous double squirrel-cage motor, a series toothed gear and of a two-stage planetary gear was assumed for the purpose of oriented-oriented investigations (Fig.2). The effect of clearances and varying meshing stiffness was left out of account during the investigations. The carried out numerical calculations allowed the influence of design parameters of the system on maximum values of forces in kinematic pairs to be determined.

7

Arkadiusz Mężyk, Eugeniusz Świtoński

k n-3 I2

I1 k1

I3

cn-3

In-4

cn-1

k n-1

k2

kn-4

c2

cn-4

k n-2

kn

kn+1 Mo

M el

c1

In-3

c n-2

In-2

cn

In-1

c n+1

In

mn-2

mn-3

Figure 2: Dynamic model of the drive system

The force in first kinematic pair was selected to be the objective function. Three first peaks of the objective function were taken into account when making calculations (Fig.3). When analysing the influence of particular parameters of the system on maximum values of forces (Fig.4) five parameters of a physical model of the mechanical part of the drive system (two moments of inertia, coefficient of stiffness and two coefficients of damping) were selected for optimising at the first stage of investigations. It was assumed that the permissible range of changes in parameter values should be ± 20 %. Peak 1

1.6e+0

Peak 2

1.4e+0 1.2e+0 1e+0 80000 Peak 3

60000 40000 20000 0

0.00

0.0

0.01

0.0

0.02

0.0

0.03

0.0

0.04

[s]

Figure 3: Objective function ψ = P12 describing dynamical forces in the first kinematic pair of the drive system

8

Arkadiusz Mężyk, Eugeniusz Świtoński

Change of the objective function 0,4 0,2 0 -0,2 -0,4 -0,6 -0,8 -1 -1,2 -1,4 -1,6

I1

I3

I5

I7

m8

I10

I11

k1

k3

k5

k7

k9

k11 k13

c2

c4

c6

c8

c10 c12

Parameter peak 1

peak 2

peak 3

Figure 4: Change of value of the objective function ψ = P12 due to small change of each parameter of the model

[Nm] a) b)

5000 4000 3000 2000 1000 0 -1000 -2000 -3000 -4000 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

[s]

Figure 5: Dynamic torque M13 for the system before a) and after oriented b)

Moreover, calculations were made in order to minimize amplitudes by changing the structure of a vibrating system. A dynamic vibration eliminator was attached to the system. Parameters of the torsional vibration eliminator of the system were selected as a result of the process of sensitivity analysis and optimisation carried out when making use of a model of the system with one degree of freedom (Fig.1). Results of sensitivity analysis applying direct method and

9

Arkadiusz Mężyk, Eugeniusz Świtoński

finite differences method are presented in the figure 6. Resonance curve can be described by a dimensionless amplitude amplification factor. The resonance curve for a system with an eliminator, the parameters of which have been selected by way of optimisation, is shown in the figure 7. The obtained characteristic indicates a low value of the coefficient of amplification of the vibration amplitude corresponding to the first natural frequency of the system within a wide range of frequency. The effectiveness of functioning of the eliminator will depend on its location. When making use of the sensitivity analysis of natural frequencies and when taking the technical conditions into consideration the eliminator was attached to the first disk of the model corresponding to the moment of inertia of a rotor of the driving motor. This was caused by technical possibilities of making design alterations in the system and by great influence of the moment of inertia of this element on the first natural frequency of the system. The fact that this mass is under direct influence of time-varying torque of the driving motor gave additional reasons for such location of the eliminator. Numerical calculations were carried out for the system incorporating the eliminator and the results obtained were compared with those relating to a system without the eliminator (Fig.8,9). 4

2

6

Sensitivity of the resonance curve

x 10

2.5

0

Sensitivity of the resonance curve

x 10

DDM FDM

2

-2

1.5

-4 1 -6 0.5 -8 0

-10 DDM FDM

-12 -14 0

0.02

0.04

0.06 0.08 Frequency [rad/s]

5000

0.1

-0.5

0.12

-1 0

a)

0.02

0.04

0.06 0.08 Frequency [rad/s]

0.1

0.12

b)

Sensitivity of the resonance curve

0

DDM FDM -5000

-10000 0

0.02

0.04

0.06 0.08 Frequency [rad/s]

0.1

0.12

c)

Figure 6: Sensitivity of the resonance curve calculated by direct method and finite differences method with respect to a) damping coefficient of eliminator, b) stiffness coefficient of eliminator, c) inertia moment of eliminator

10

Arkadiusz Mężyk, Eugeniusz Świtoński

a) b)

Amplitude amplification factor

12 10 8 6 4 2 0 800

850

900

950

1000

1050

1100

1150

Frequency [rad/s]

Figure 7: Resonance curves of the model with eliminator before a) and after optimisation [Nm] a) b)

5000 4000 3000 2000 1000 0 -1000 -2000 -3000 -4000 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

[s]

Figure 8: Dynamic torque M13 for the system without a) and with eliminator b) attached to the first disc of the model

6. RECAPITULATION The formulated algorithms of optimisation, which make use of dynamic models, allow dynamic characteristics of drive systems of machines to be shaped for any state of load. On the basis of results of the sensitivity analysis it is possible to assess the qualitative and quantitative effect of selected parameters of the model on values of the assumed objective

11

Arkadiusz Mężyk, Eugeniusz Świtoński

function. The numerical experiment carried out for the assumed drive system made it possible to determine optimum parameters of the system. These parameters bring about decreasing of peak values of forces in kinematic pairs by 20 - 40 % of initial amplitudes (Fig.5, 9). -6

5

Transfer function H(jω)

x 10

a) b) c)

4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 800

1000

1200

1400

1600 1800 2000 Frequency [rad/s]

2200

2400

2600

Figure 9: Transfer function for the dynamic model of the drive system a) before optimisation, b) with eliminator, c) after optimisation without eliminator

The presented algorithm of the optimisation of parameters of the system can be successfully applied for determining the design features of the system described by means of state variables as well as for assuring its long service life and high reliability. At the same time it provides an effective aid for the process involved in designing and constructing. The investigations have been carried out within the project no 7 T07A 023 17, sponsored by the Polish Committee of Scientific Research. REFERENCES [1]

[2] [3]

Abramson M.A., Chrissis J.W.: Sequential Quadratic programming and the ASTROS structural optimization system. Structural Optimization, Vol.15, pp.24-32, Springer Verlag 1998. Bestle D., Eberhard P.: Analyzing and Optimizing Multibody Systems. Mech. Struct. and Mach. 20(1), 67-92, 1992. Haftka R.T., Gürdal Z., Kamat M.P.: Elements of Structural Optimization. Kluwer, 396p., 1990.

12

Arkadiusz Mężyk, Eugeniusz Świtoński

[4]

Haug E.J., Choi K.K., Komkov V.: Design Sensitivity Analysis of Structural Systems. Academic Press, Inc. 1986. [5] Marchelek K.: Dynamics of machine tools. WNT Warszawa 1991. (In Polish). [6] Mężyk A.: Minimization of transient forces in an electromechanical system. Structural Optimization, Vol.8, pp.251-256, Springer Verlag 1994. [7] Mężyk A.: Sensitivity Analysis of Electromechanical Systems. Proceedings of the XI Polish Conference on Computer Methods in Mechanics. Kielce-Cedzyna, Poland, 11-14 May, 1993. [8] Pedersen P., Laursen C.L.: Design for Minimum Stress Concentration by Finite Elements and Linear Programming. J. Struct. Mech. 10, 243-271, 1982-83. [9] Rao S.S.: Multistage Multiobjective Optimization of Gearboxes. Journal of Mechanisms, Transmissions, and Automation in Design, Transactions of ASME vol.108, 1986. [10] Sobieszczański-Sobieski J., Haftka R.T.: Multidisciplinary aerospace design optimization: survey of recent developments. Structural Optimization, Vol.14, pp.1-23, Springer Verlag 1997.

13