 | | 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: | | 11223 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8011
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| In English (Mech. Solids): | | 3212 |
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| A. G. Petrov, "Nonlinear vibrations of a swinging spring at resonance," Mech. Solids. 41 (5), 13-22 (2006) |
| Year |
2006 |
Volume |
41 |
Number |
5 |
Pages |
13-22 |
| Title |
Nonlinear vibrations of a swinging spring at resonance |
| Author(s) |
A. G. Petrov (Moscow) |
| Abstract |
We propose to study nonlinear vibrations of a swinging spring by the
Poincaré-Birkhoff normal form method. In this method [1, 2], the
Hamiltonian of the system is represented as the sum of the quadratic part,
which is said to be unperturbed, and terms of order higher than 2. Using
canonical transformations, one can simplify the Hamiltonian system so that
it becomes integrable up to fourth- and higher-order terms. Thus we obtain
an asymptotic solution of the nonlinear problem. Traditional normalization
methods for a system with two degrees of freedom are rather cumbersome and
require a lot of computations [2-6]. The change of variables is sought
with the use of generating functions or a generating Hamiltonian.
In the present paper, we use the definition of invariant normal form given
in [7, 8], which does not require separation into the autonomous and
nonautonomous or resonance and nonresonance cases and can be performed in
the framework of a unified approach. The asymptotics of the normal form is
obtained by successive computations of the quadratures. In contrast to the
Zhuravlev method [7, 8], the generating function [9-11] is used instead of
the generating Hamiltonian. In the present paper, we continue the study,
initiated in [12], of nonlinear vibrations of a swinging spring in the
resonance case. |
| References |
| 1. | H. Poincaré, Selected Works. Volume 2 [Russian translation],
Nauka, Moscow, 1972. |
| 2. | D. D. Birkhoff, Dynamical Systems [Russian translation],
Gostekhizdat, Moscow-Leningrad, 1941. |
| 3. | A. D. Bryuno, Restricted Three-Body Problem [in Russian], Nauka,
Moscow, 1990. |
| 4. | V. I. Arnold, Mathematical Aspects of Classical Mechanics [in
Russian], Editorial URSS, Moscow, 2000. |
| 5. | V. I. Arnold, Additional Chapters of Ordinary Differential
Equations [in Russian], Nauka, Moscow, 1978. |
| 6. | V. I. Arnold, V. V. Kozlov, and A. I. Neishtadt, Mathematical
Aspects of Classical and Celestial Mechanics [in Russian], VINITI, Moscow,
1985. |
| 7. | V. Ph. Zhuravlev, Foundations of Theoretical Mechanics [in Russian],
Nauka, Fizmatlit, Moscow, 1997. |
| 8. | V. Ph. Zhuravlev,
"Invariant normalization of nonautonomous Hamiltonian systems,"
PMM [Applied Mathematics and Mechanics], Vol. 66, No. 3,
pp. 356-365, 2002. |
| | |
| 9. | A. G. Petrov, "A parametric method for constructing Poincaré
mappings in hydrodynamical systems," PMM [Applied Mathematics and
Mechanics], Vol. 66, No. 6, pp. 948-967, 2002. |
| 10. | A. G. Petrov,
"A modification of the invariant normalization method for Hamiltonians
using the parametrization of canonical transformations,"
Doklady RAN, Vol. 386, No. 4, pp. 482-486, 2002. |
| 11. | A. G. Petrov,
"On invariant normalization of nonautonomous Hamiltonian systems,"
PMM [Applied Mathematics and Mechanics], Vol. 68, No. 3,
pp. 402-413, 1970. |
| 12. | M. N. Zaripov and A. G. Petrov, "Nonlinear vibrations of a
swinging string," Doklady RAN, Vol. 399, No. 3, pp. 347-352, 2004. |
| 13. | A. H. Nayfeh,
Perturbation Methods,
Wiley, New York, 1973. |
| 14. | V. M. Starzhinskii, Applied Methods in Nonlinear Vibrations [in
Russian], Nauka, Moscow, 1977. |
| 15. | V. N. Bogaevskii and A. Ya. Povzner, Algebraic Methods in
Nonlinear Perturbation Theory [in Russian], Nauka, Moscow, 1987. |
| 16. | W. A. Mersman, "A new algorithm for the Lie transformation,"
Celest. Mech., Vol. 3, Mo. 1, pp. 81-89, 1970. |
| 17. | A. P. Markeev and A. G. Sokol'skii,
Several Calculational Algorithms
for Normalization of Hamiltonian Systems.
Preprint No. 31 [in Russian], In-t Problem Mekhaniki RAN,
Moscow, 1976. |
|
| Received |
10 June 2005 |
| << Previous article | Volume 41, Issue 5 / 2006 | Next article >> |
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