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IssuesArchive of Issues2004-2pp.10-17

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A. V. Shatina, "Fast and slow dissipative evolution in mechanical systems containing viscoelastic elements," Mech. Solids. 39 (2), 10-17 (2004)
Year 2004 Volume 39 Number 2 Pages 10-17
Title Fast and slow dissipative evolution in mechanical systems containing viscoelastic elements
Author(s) A. V. Shatina (Moscow)
Abstract An asymptotic method is proposed for the analysis of the evolution of the motion of mechanical systems containing viscoelastic elements. This method combines the separation of motion for systems with infinite number of degrees of freedom and the generalized Krylov-Bogolyubov method for systems with fast and slow variables. The proposed method is a development and generalization of the asymptotic method of [1]. It can be applied to systems in which, apart from the perturbation due to the elastic compliance and dissipation, there are small periodic perturbations due to the external force field. As an example, the motion of a satellite with flexible viscoelastic beams in a circular orbit is considered.
References
1.  V. G. Vil'ke, "Separation of motions and averaging in the mechanics of systems with infinite number of degrees of freedom," Vestnik MGU [Bulletin of Moscow State University], Ser. 1. Matematika. Mekhanika, No. 5, pp. 54-59, 1983.
2.  V. G. Vil'ke, Analytical and Qualitative Methods in the Dynamics of Systems with Infinite Number of Degrees of Freedom [in Russian], Izd-vo MGU, Moscow, 1986.
3.  F. L. Chernousko, "On the motion of a rigid body with elastic and dissipative elements," PMM [Applied Mathematics and Mechanics], Vol. 42, No. 1, pp. 34-42, 1978.
4.  F. L. Chernousko, "On the motion of a viscoelastic solid about the center of mass," Izv. AN SSSR. MTT [Mechanics of Solids], No. 1, pp. 22-26, 1980.
5.  F. L. Chernousko and A. S. Shamaev, "Singular perturbation asymptotics in dynamics of a rigid body with elastic and dissipative elements," Izv. AN SSSR. MTT [Mechanics of Solids], No. 3, pp. 33-42, 1983.
6.  E. V. Sinitsyn, "Singular perturbation asymptotics in the study of the translational-rotational motion of a viscoelastic body," Izv. AN SSSR. MTT [Mechanics of Solids], No. 1, pp. 104-110, 1991.
7.  A. B. Vasil'eva and V. F. Butuzov, Asymptotic Expansions of the Solutions of Singularly Perturbed Equations [in Russian], Nauka, Moscow, 1973.
8.  V. V. Sidorenko, "On the evolution of the motion of a mechanical system with a linear damper having high rigidity," PMM [Applied Mathematics and Mechanics], Vol. 59, No. 4, pp. 562-568, 1995.
9.  Yu. A. Mitropol'skii and O. B. Lykova, Integral Manifolds in Nonlinear Mechanics [in Russian], Nauka, Moscow, 1973.
10.  V. V. Strygin and V. A. Sobolev, Separation of Motion by the Integral Manifold Method [in Russian], Nauka, Moscow, 1988.
11.  V. I. Arnold, V. V. Kozlov, and A. I. Neishtadt, "Mathematical aspects of the classical and celestial mechanics," in Modern Issues of Mathematics. Fundamental Directions. Volume 3 [in Russian], VINITI, Moscow, 1985.
Received 16 June 2001
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