Energy equipartition and fluctuation-dissipation theorems for damped flexible structures
Authors:
Ronald K. Pearson and Timothy L. Johnson
Journal:
Quart. Appl. Math. 45 (1987), 223-238
MSC:
Primary 73K35; Secondary 34F05
DOI:
https://doi.org/10.1090/qam/895095
MathSciNet review:
895095
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Abstract: Dynamics and control of flexible mechanical structures has been the topic of much recent research. Here, we examine the energy distribution in finite-dimensional flexible structure models of the type obtained through finite element analysis. Modeling external disturbance forces as zero-mean white noise, we establish that symmetry of the damping matrix is a sufficient condition for the equipartition of potential and kinetic energy in the structure. In addition, we develop upper and lower bounds on the total energy stored in symmetrically damped structures in terms of the strength of the stochastic driving term and the Euclidean norms of the damping matrix and its inverse. In two special cases, explicit solutions for the total energy are obtained and may be viewed as fluctuation-dissipation theorems for the structure models. Convergence conditions for modal expansions of distributed parameter flexible structure models are then developed from these finite-dimensional results. These conditions are interpreted physically as interrelations between assumed damping mechanisms and disturbance force actuator models that must exist in formulating well-posed stochastic distributed-parameter flexible structure models.
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M. J. Balas, Modal control of certain flexible dynamic systems, SIAM J. Control Optim. 16, 450–462 (1978)
M. J. Balas, Direct velocity feedback control of large space structures, AIAA J. Guidance and Control 2, 252–253 (1979)
G. Chen and D. L. Russell, A mathematical model for linear elastic systems with structural damping, Quart. Appl. Math. 39, 433–454 (1978)
R. W. Clough and J. Penzien, Dynamics of structures, McGraw-Hill, New York, 1975
R. F. Curtain and A. J. Pritchard, Infinite dimensional linear systems theory, Springer-Verlag, New York, 1978
R. F. Fox and G. E. Uhlenbeck, Contributions to non-equilibrium thermodynamics. I. Theory of hydrodynamical fluctuations, Phys. Fluids 13, 1893–1902 (1970)
J. C. Geromel and J. Bernussou, On bounds of Lyapunov’s matrix equation, IEEE Trans. Automat. Control 24, 482–487 (1979)
J. S. Gibson, A note on stabilization of infinite dimensional oscillators by compact linear feedback, SIAM J. Control Optim. 18, 311–316 (1980)
J. S. Gibson, An analysis of optimal modal regulation: convergence and stability, SIAM J. Control Optim. 19, 686–707 (1981)
P. C. Hughes and R. E. Skelton, Controllability and observability for flexible spacecraft, J. Guidance and Control 3, 452–460 (1980)
T. Kato, Perturbation theory for linear operators, Springer-Verlag, New York, 1980
W. H. Kwon and A. E. Pearson, A note on the algebraic matrix Riccati equation, IEEE Trans. Automat. Control 22, 143-144 (1977)
L. D. Landau and E. M. Lifschitz, Statistical physics, Pergamon Press, New York, 1980
B. G. Levich, Theoretical physics, vol. 2: Statistical physics, electromagnetic processes in matter, John Wiley & Sons, New York, 1971
Industry workshop on large space structures: Compilation of company presentations and responses to a key issue questionnaire, NASA CR-144997, McDonnell-Douglas Astronautics Co., May 1976
A. H. Nayfeh and D. T. Mook, Nonlinear oscillations, Wiley, New York, 1979
R. V. Patel and M. Toda, On norm bounds for algebraic Riccatti and Lyapunov equations, IEEE Trans. Automat. Control 23, 87–88 (1978)
D. L. Russell, Decay rates for weakly damped systems in Hilbert space obtained with control-theoretic methods, J. Differential Equations 19, 344–370 (1975)
Y. Sakawa, Feedback control of second order evolution equations with damping, SIAM J. Control Optim. 22, 343–361 (1984)
G. Shilov, Elementary functional analysis, MIT Press, Cambridge, Massachusetts, 1974
Proceedings of VPI & SU/AIAA symposium on dynamics and control of large flexible spacecraft, Blacksburg, Virginia, June 1979
Dynamics and control of large flexible spacecraft: proceedings of the third VPI & SU/AIAA symposium, Blacksburg, Virginia, June 1981
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Article copyright:
© Copyright 1987
American Mathematical Society