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IssuesArchive of Issues2005-3pp.98-108

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V. D. Potapov, "Stability of elastic and viscoelastic systems under the stochastic parametric excitation," Mech. Solids. 40 (3), 98-108 (2005)
Year 2005 Volume 40 Number 3 Pages 98-108
Title Stability of elastic and viscoelastic systems under the stochastic parametric excitation
Author(s) V. D. Potapov (Moscow)
Abstract Problems of stability of elastic and viscoelastic systems under the action of random loads in the form of steady stochastic processes were considered in a large number of papers. A reasonably detailed review of these papers is given in [1]. Major part of the results were obtained for the case where the steady process is assumed to be the Gaussian white noise. If the parametric loads are steady wideband processes, the solution of the stability problem for the system is much more difficult. In this case, only the sufficient conditions of the stability in probability were obtained. We note that the estimates of the stability region boundaries obtained using these criteria are usually very rough. More accurate estimates can be obtained using the methods of stochastic modeling in combination with numerical methods of structural design. The fundamental issues of this approach were discussed in [2-5]. A series of papers [1, 6-9] is devoted to the application of the method of canonical expansions in combination with the method of determining the maximum Lyapunov exponent to solving stability problems for elastic and viscoelastic systems. Within the problem formulation mentioned, the conditions of stability with respect to probability and statistic moments of various order (p-stability), in particular, the mean square stability, were obtained. However, the application of this method to the investigation of the p-stability indicates that this method is efficient only in the case where the maximum Lyapunov exponent is negative. In the case where this exponent is positive or negative but close to zero, the process of solving the problem meets certain difficulties.

In the present paper, we consider the stability with respect to statistic moments of elastic and viscoelastic systems under the action of random parametric loads in the form of steady processes with the fractional rational spectral density (colored noises). The method proposed in this paper is efficient in the stability analysis at arbitrary values of the maximum Lyapunov exponent.
References
1.  V. D. Potapov, Stability of Stochastic Elastic and Viscoelastic Systems, Wiley, Chichester, 1999.
2.  V. V. Bolotin, Random Vibrations in Elastic Systems [in Russian], Nauka, Moscow, 1979.
3.  J. R. Kloeden and E. Platen, The Numerical Solution of Stochastic Differential Equations. Volume 1, Springer, Berlin, 1992.
4.  G. N. Milstein, "Evaluation of moment Liapunov exponents for second order stochastic systems," Random and Comput. Dyn., Vol. 4, No. 4, pp. 301-315, 1996.
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6.  V. D. Potapov and P. Koirala, "Stability of elastic and viscoelastic system under action of random stationary narrow-band loads," Intern. J. Mech. Sci., Vol. 39, No. 8, pp. 935-942, 1997.
7.  V. D. Potapov, "Numerical method for investigation of stability of stochastic integro-differential equations," Appl. Numer. Math., Vol. 24, No. 2-3, pp. 191-201, 1997.
8.  V. D. Potapov and A. Y. Marasanov, "The investigation of the stability of elastic and viscoelastic rods under a stochastic excitation," Intern. J. Solids and Structures, Vol. 34, No. 11, pp. 1367-1377, 1997.
9.  V. D. Potapov, "On the stability of zero solution of a system of integro-differential equations under stochastic parametric excitation," Avtomatika i Telemekhanika [Automation and Remote Control], No. 4, pp. 45-53, 1997.
10.  A. S. Shalygin and Yu. I. Palagin, Applied Methods of Statistical Modeling [in Russian], Mashinostroenie, Leningrad, 1986.
11.  B. P. Demidovich, I. A. Maron, and E. Z. Shuvalova, Numerical Methods of Analysis [in Russian], Fizmatgiz, Moscow, 1963.
12.  G. Benettin, L. Galgani, A. Giorgolli, and J. M. Strelcyn, "Lyapunov characteristic exponent for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them, Pt. 1 and 2," Mechanica, Vol. 15, No. 1, pp. 9-20, 21-30, 1980.
13.  V. D. Potapov, "Analysis of the dynamic stability of viscoelastic systems by Lyapunov exponents," Izv. AN. MTT [Mechanics of Solids], No. 6, pp. 82-90, 2000.
14.  R. L. Stratonovich, Selected Issues of Theory of Fluctuations in Radio Engineering [in Russian], Sov. Radio, Moscow, 1961.
15.  S. T. Ariarathnam and D. S. F. Tam, "Moment stability of coupled linear systems under combined harmonic and stochastic excitation," in B. L. Clarkson (Editor), Stochastic Problems in Dynamics, Pitman, London, 1977.
16.  M. F. Dimentberg, Nonlinear Stochastic Problems of Mechanical Oscillations [in Russian], Nauka, Moscow, 1980.
17.  K. Karhunen, "Über lineare Methoden in der Wahrscheinlichkeitsrechnung," Ann. Acad. Sci. Fennicae. Ser. A-I. Math. Phys., Vol. 1, No. 37, pp. 3-79, 1947.
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20.  A. Tylikowski, "Stability and bounds on motion of viscoelastic column with imperfections and time-dependent forces," in M. Zyczkowski (Editor), Creep in Structures, Springer, Berlin, 1991.
Received 02 July 2003
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