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IssuesArchive of Issues2011-3pp.444-454

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V.D. Potapov, "Nonlinear Vibrations and Stability of Elastic and Viscoelastic Systems under Random Stationary Loads," Mech. Solids. 46 (3), 444-454 (2011)
Year 2011 Volume 46 Number 3 Pages 444-454
DOI 10.3103/S0025654411030113
Title Nonlinear Vibrations and Stability of Elastic and Viscoelastic Systems under Random Stationary Loads
Author(s) V.D. Potapov (Moscow State University of Railway Engineering, Obraztsova 15, GSP-4, Moscow, 127994 Russia, potapov@micnmic.msk.ru)
Abstract The paper deals with numerical analysis of nonlinear vibrations of viscoelastic systems under a stochastic action in the form of a Gaussian stationary process with rational spectral density. The analysis is based on numerical simulation of the original stationary process, numerical solution of the differential equations describing the motion of the system, and computation of the maximum Lyapunov exponent if the stability of this motion is studied. An example of a plate subjected to a random stationary load applied in its plane is used to consider specific issues concerning the application of the proposed method and the peculiarities of the behavior of geometrically nonlinear elastic and viscoelastic stochastic systems. Special attention is paid to the interaction of a deterministic periodic action and a stochastic action from the viewpoint of stability of the system motion. It is shown that in some cases imposing a "colored" noise may stabilize an unstable system subjected to a periodic load.
Keywords viscoelasticity, nonlinear vibrations, stability, Lyapunov exponent, random stationary process
References
1.  M. F. Dimentberg, Nonlinear Stochastic Problems of Mechanical Vibrations (Nauka, Moscow, 1980) [in Russian].
2.  M. F. Dimentberg, Statistical Dynamics and Time-Varying Systems (Taunton, Research Studies Press, New York, 1988).
3.  E. Simiu, Chaotic Transitions in Deterministic and Stochastic Systems. Application of Melnikov Processes in Engineering, Physics, and Neuroscience (Princeton Univ. Press, Princeton, Oxford, 2002; Fizmatlit, Moscow, 2007).
4.  V. D. Potapov, Stability of Stochastic Elastic and Viscoelastic Systems (Wiley, Chichester, 1999).
5.  A. S. Shalygin and Yu. I. Palagin, Applied Methods of Statistical Modelling (Mashinostroenie, Leningrad, 1986) [in Russian].
6.  V. D. Potapov, "Stability of Elastic and Viscoelastic Systems under Stochastic Parametric Excitation," Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 3, 123-136 (2005) [Mech. Solids (Engl. Transl.) 40 (3), 98-108 (2005)].
7.  G. Benettin, L. Galgani, A. Giorgilly, and J.-M. Strelcyn, "Liapunov Characteristics Exponent for Smooth Dynamical Systems and for Hamiltonian Systems; A Method for Computing All of Them. Pt. 1, 2," Meccanica 15 (1), 9-20; 21-30 (1980).
8.  J. Guckenheimer and Ph. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer-Verlag, New York-Berlin-Heidelberg, 1983; In-t Komp. Issled., Moscow-Izhevsk, 2002).
9.  M. A. Leibowits, "Statistical Behavior of Linear Systems with Randomly Varying Parameters," J. Mat. Phys. 4 (6), 852-858 (1963).
10.  R. Z. Khasminskii, Stability of Systems of Differential Equations under Stochastic Perturbations of Their Parameters (Nauka, Moscow, 1969) [in Russian].
Received 25 November 2008
Link to Fulltext http://www.springerlink.com/content/w68454t000j22886/
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