 | | 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: | | 12854 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8044
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| In English (Mech. Solids): | | 4810 |
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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. |
| References |
| 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 |
| << Previous article | Volume 37, Issue 6 / 2002 | Next article >> |
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