 | | 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: | | 12804 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8044
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| In English (Mech. Solids): | | 4760 |
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| A. P. Markeev, "On the identity resonance in a particular case of the problem of stability of periodic motions of a rigid body," Mech. Solids. 38 (3), 22-26 (2003) |
| Year |
2003 |
Volume |
38 |
Number |
3 |
Pages |
22-26 |
| Title |
On the identity resonance in a particular case of the problem of stability of periodic motions of a rigid body |
| Author(s) |
A. P. Markeev (Moscow) |
| Abstract |
The motion of a rigid body with one fixed point is considered.
The mass geometry and the initial conditions of motion of the body
are assumed to correspond to Goryachev-Chaplygin integrable case [1, 2].
In this case, there exist families of periodic motions corresponding
to vibrations or rotations of the body about a principal axis
of inertia that occupies an invariable horizontal position.
In the present paper, we assume that this axis is the equatorial
axis of the ellipsoid of inertia and study the orbital stability
of the aforementioned periodic motions. It is established that
the problem in question is always a resonant problem. More precisely,
it is established that
for any vibration amplitude (or any angular velocity of rotation)
of the body in the unperturbed motion, a parametric resonance
occurs in the perturbed motion (all multipliers are equal to unity).
It is shown that the periodic motions under study are orbitally
unstable in the first approximation. |
| References |
| 1. | D. N. Goryachev, "On the motion of a heavy rigid body about
a fixed point in the case A=B=4C," Matematicheskii Sbornik
Kruzhka Lyubitelei Matematicheskikh Nauk, Vol. 21, No. 3,
pp. 431-438, 1900. |
| 2. | S. A. Chaplygin, "New case of rotation of a heavy rigid body
supported at one point," Trudy Otdeleniya Fizicheskikh Nauk
Obshchestva Lyubitelei Estestvoznaniya, Vol. 10, No. 2,
pp. 32-34, 1901. |
| 3. | G. V. Gorr, L. V. Kudryashova, and L. A. Stepanova,
Classical Problems of Dynamics of a Rigid Body: Development
and State-of-the-art [in Russian],
Naukova Dumka, Kiev, 1978. |
| 4. | V. V. Kozlov, Qualitative Analysis Methods in Dynamics
of a Rigid Body [in Russian], NITs "Regulyarnaya i Khaoticheskaya
Dinamika," Izhevsk, 2001. |
| 5. | A. I. Dashkevich, Closed-form Solutions of Euler-Poisson
Equations [in Russian],
Naukova Dumka, Kiev, 1992. |
| 6. | A. V. Borisov and I. S. Mamaev, Dynamics of a Rigid Body
[in Russian], NITs "Regulyarnaya i Khaoticheskaya
Dinamika," Izhevsk, 2001. |
| 7. | A. P. Markeev, Theoretical Mechanics
[in Russian], NITs "Regulyarnaya i Khaoticheskaya
Dinamika," Izhevsk, 2001. |
| 8. | A. P. Markeev, "On the stability of plane motions of
a rigid body in the Kovalevskaya case,"
PMM [Applied Mathematics and Mechanics], Vol. 65, No. 1,
pp. 51-58, 2001. |
| 9. | A. M. Zhuravskii, A Handbook on Elliptic Functions [in Russian],
Izd-vo AN SSSR, Moscow, Leningrad, 1941. |
| 10. | I. G. Malkin, Theory of Stability of Motion [in Russian],
Nauka, Moscow, 1966. |
| 11. | A. P. Markeev, "Stability analysis of periodic motions
of an autonomous Hamiltonian system in one critical case,"
PMM [Applied Mathematics and Mechanics], Vol. 64, No. 5,
pp. 833-847, 2000. |
|
| Received |
25 November 2002 |
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