 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
Archive of Issues
| Total articles in the database: | | 12804 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8044
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| In English (Mech. Solids): | | 4760 |
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| << Previous article | Volume 46, Issue 3 / 2011 | Next article >> |
| 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 |
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