 | | 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 |
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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. |
| 5. | K. Y. R. Billah and M. Shinozuka, "Numerical method for colored
noise generation and its application to a bistable system," Phys. Rev.
A., Vol. 42, No. 12, pp. 7492-7495, 1990. |
| 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. |
| 18. | K. Karhunen, "Über die Struktur stationären zufälligen Functionen," Ark. Mat.,
No. 1, pp. 141-160, 1950. |
| 19. | F. Kozin, "Autoregressive moving average models of earthquake
records," Probab. Engng Mech., Vol. 3, No. 2, pp. 58-63, 1988. |
| 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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