Mechanics of Solids (about journal) Mechanics of Solids
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in January 1966
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IssuesArchive of Issues2002-6pp.42-47

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V. V. Lyubimov, "External stability of a resonance in a nonlinear system with slowly changing variables," Mech. Solids. 37 (6), 42-47 (2002)
Year 2002 Volume 37 Number 6 Pages 42-47
Title External stability of a resonance in a nonlinear system with slowly changing variables
Author(s) V. V. Lyubimov (Samara)
Abstract The external stability of resonances in nonlinear systems with several slowly changing variables is analyzed. The external stability is determined by the behavior of the trajectories of the system on nonresonant segments of motion apart from a small resonance zone [1]. For the motion along the nonresonant segments, the resonant terms appear in the second approximation of the averaging method and, if the evolution does not occur in the first approximation, manifest themselves in secondary resonant effects. Conditions for the external stability of a resonance are stated in nonlinear formulation. The method is illustrated by the case investigation of the external stability of a resonance during the motion of an asymmetric rigid body about a fixed point. A phase portrait constructed for nonresonant segments is presented to illustrate the evolution of the slow variables in the cases of externally stable and externally unstable resonances.
1.  E. A. Grebennikov and Yu. A. Ryabov, Constructive Methods of Analysis of Nonlinear Systems [in Russian], Nauka, Moscow, 1979.
2.  M. M. Khapaev, Asymptotic Methods and Stability in the Theory of Nonlinear Oscillations [in Russian], Vysshaya Shkola, Moscow, 1988.
3.  F. L. Chernous'ko, "On a resonance in an essentially nonlinear system," Zh. Vychislit. Matemematiki i Matem. Fiziki [Journal of Computational Mathematics and Mathematical Physics], Vol. 3, No. 1, pp. 131-144, 1963.
4.  Yu. A. Sadov, "Secondary resonant effects in mechanical systems," Izv. AN SSSR. MTT [Mechanics of Solids], No. 4, pp. 20-24, 1990.
5.  V. I. Arnol'd, V. V. Kozlov, and A. I. Neishtadt, "Mathematical aspects of the classical and celestial mechanics," in Achievements in Science and Technology. Modern Problems of Mathematics. Fundamental Aspects. Volume 3 [in Russian], VINITI, Moscow, 1985.
6.  Yu. M. Zabolotnov, "A method for analyzing the resonance motion in a nonlinear oscillatory system," Izv. AN. MTT [Mechanics of Solids], No. 1, pp. 33-45, 1999.
7.  N. N. Moiseev, Asymptotic Methods of Nonlinear Mechanics [in Russian], Nauka, Moscow, 1981.
Received 20 June 2000
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