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IssuesArchive of Issues2006-2pp.134-143

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L. P. Dzubak, G. V. Manucharyan, Yu. V. Mikhlin, and T. V. Shmatko, "Stability of regular and chaotic vibration modes in systems with several equilibrium states," Mech. Solids. 41 (2), 134-143 (2006)
Year 2006 Volume 41 Number 2 Pages 134-143
Title Stability of regular and chaotic vibration modes in systems with several equilibrium states
Author(s) L. P. Dzubak (Kharkov)
G. V. Manucharyan (Kharkov)
Yu. V. Mikhlin (Kharkov)
T. V. Shmatko (Kharkov)
Abstract Forced vibrations of a system with two degrees of freedom and several equilibrium states are considered. Such systems can be obtained by discretizing elastic systems in a post-critical state. The modes of vibrations that are periodic if the amplitude of the external periodic excitation is small and become chaotic as this amplitude increases are studied. To investigate the stability of such vibration modes, computational procedures based on the classical definition of the Lyapunov stability are utilized. The stability of vibration modes of nonlinear rods, shells, and arches is analyzed.

The mutual instability of the phase trajectories is utilized as a criterion of the appearance of the chaotic behavior in a nonlinear system. Trajectories with very close initial conditions are compared. The computational procedures based on the classical definitions of the Lyapunov stability enable one to judge the stability or instability of these trajectories. Specific calculations for the time-varying Duffing equation and the von Mises truss allow one to observe the appearance and expansion of the chaotic behavior domains.
References
1.  P. J. Holmes, "A nonlinear oscillator with a strange attractor," Philos. Trans. Royal Soc. London, Ser. A, Vol. 292, pp. 419-448, 1979.
2.  F. C. Moon, Chaotic Vibrations, Wiley, New York, 1987.
3.  A. M. Lyapunov, General Problem of the Stability of Motion [in Russian], ONTI, Leningrad, Moscow, 1935.
4.  E. L. Incze, Ordinary Differential Equations [in Russian], Gosudarstvennoe Nauchno-tekhnicheskoe Izdatel'stvo Ukrainy, Kharkov, 1939.
5.  A. A. Andronov, A. A. Vitt, and S. E. Khaikin, Theory of Oscillations [in Russian], Fizmatgiz, Moscow, 1959.
6.  I. G. Malkin, Theory of Stability of Motion [in Russian], Nauka, Moscow, 1966.
7.  N. Minirsky, Nonlinear Oscillatons, Van Nostrand, Princeton, 1962.
8.  C. L. Siegel and J. K. Moser, Lectures on Celestial Mechanics, Springer, New York, 1971.
9.  H. Poincaré, Les Méthodes Nouvelles de la Méchanique Céleste. 3 Vols, Gauthier-Villars, Paris, Vol. 1, 1982; Vol. 2, 1983; Vol. 3, 1897-1899.
10.  E. N. Lorenz, "Deterministic non-periodic flow," J. Atmos. Sci., Vol. 20, pp. 130-141, 1963.
11.  D. Ruelle and F. Takens, "On the nature of turbulence," Communs. Math. Phys., Vol. 20, pp. 167-172, 1971.
12.  V. K. Mel'nikov, "On the stability of the center under time-periodic excitations," in Transactions of the Moscow Mathematical Society [in Russian], Vol. 12, pp. 1-57, 1963.
13.  J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York, 1983.
14.  S. W. Wiggins, Introducation to Applied Nonlinear Dynamical Systems and Chaos, Springer, New York, 1990.
15.  A. J. Lichtenberg and M. A. Lieberman, Regular and Stochastic Motion, Springer, New York, 1983.
16.  Y. Ueda, "Randomly transitional phenomena in the system governed by Duffing's equation," J. Stat. Phys., Vol. 20, pp. 181-196, 1979.
17.  A. Wolf, "Quantifying chaos with Lyapunov exponents," in A. V. Holden (Editor), Chaos Nonlinear Science: Theory and Applications. Volume 1, Univ. Press, Manchester, 1986.
18.  H. Kauderer, Nonlinear Mechanics [Russian translation], Izd-vo Inostr. Lit-ry, Moscow, 1961.
19.  Thin-walled Shell Structures [in Russian], Mashinostroenie, Moscow, 1980.
20.  V. D. Kubenko, P. S. Koval'chuk, and T. S. Krasnopol'skaya, Nonlinear Interaction of Flexural Vibration Modes of Cylindrical Shells [in Russian], Naukova Dumka, Kiev, 1984.
21.  L. Perko, Differential Equations and Dynamical Systems, Springer, New York, 1991.
23.  G. Pecelli and E. S. Thomas, "Normal modes, incoupling, and stability for a class of nonlinear oscillators," Quart. of Appl. Math., Vol. 37, pp. 281-301, 1979.
24.  L. I. Manevich, Yu. V. Mikhlin, and V. N. Pilipchuk, Normal Oscillation Method for Essentially Nonlinear Systems [in Russian], Nauka, Moscow, 1989.
25.  K. V. Avramov and Yu. V. Mikhlin, "Forced oscillations of a system containing a snap-through truss close to its equilibrium position," Nonlinear Dynamics, Vol. 35, pp. 361-379.
26.  K. V. Avramov and Yu. V. Mikhlin, "Snap-through truss as a vibration absorber," Journal of Vibration and Control, Vol. 10, pp. 291-308, 2004.
Received 20 August 2002
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