 | | 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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| O. V. Kholostova, "On the periodic motion of Lagrange's top with vibrating suspension," Mech. Solids. 37 (1), 26-38 (2002) |
| Year |
2002 |
Volume |
37 |
Number |
1 |
Pages |
26-38 |
| Title |
On the periodic motion of Lagrange's top with vibrating suspension |
| Author(s) |
O. V. Kholostova (Moscow) |
| Abstract |
The motion of Lagrange's top the suspension point of which
harmonically vibrates with a small amplitude along the vertical
is considered. The periodic motions (with the period coinciding
with that of the vibration of the suspension point) generated
from regular precessions of the top with a fixed point
are constructed both in the presence and in the absence
of the forced vibration resonance. The stability
of these motions is analyzed. Special cases
of the parametric resonance and the third-order resonance
are considered. The existence and stability of periodic motions
of the top with the period equal to the double and triple period
of vibration of the suspension point are investigated. We consider
also the issue of the boundedness of the motions of the top starting
in a sufficiently small neighborhood of its regular precession
in the indicated resonance cases and give an estimate
of the width of this neighborhood.
This study is based on the general results of [1-3] obtained for
single-degree-of-freedom Hamiltonian systems. In Section 5,
an estimate is obtained for the width of the neighborhood
in which the motion of single-degree-of-freedom Hamiltonian
systems is bounded in the case of forced vibration resonance. |
| References |
| 1. | O. V. Kholostova, "Resonant forced vibrations of a
Hamiltonian system with one degree of freedom," Izv. AN. MTT [Mechanics of
Solids], No. 3, pp. 167-175, 1996. |
| 2. | A. P. Markeev, "Parametric resonance and nonlinear vibrations
of a heavy rigid body in the neighborhood of its plane rotations,"
Izv. AN. MTT [Mechanics of Solids], No. 5, pp. 34-44, 1995. |
| 3. | O. V. Kholostova, "On the nonlinear vibrations of a satellite
in the case of the third-order resonance,"
PMM [Applied Mathematics and Mechanics], Vol. 61, No. 4, pp. 556-565, 1997. |
| 4. | N. V. Roze, Dynamics of a Rigid Body [in Russian],
KUBUCH, Leningrad, 1932. |
| 5. | R. Grammel, Gyroscope Theory and Applications. Volume 1
[Russian translation], Izd-vo Inostr. Lit-ry, Moscow, 1952. |
| 6. | K. Magnus, Gyroscope: Theory and Application
[Russian translation], Mir, Moscow, 1974. |
| 7. | O. V. Kholostova, "On the dynamics of Lagrange's top
with vibrating suspension point,"
PMM [Applied Mathematics and Mechanics], Vol. 63, No. 5, pp. 786-797, 1999. |
| 8. | J. Moser, Lectures on Hamiltonian Systems
[Russian translation], Mir, Moscow, 1973. |
| 9. | A. P. Markeev. Libration Points in Celestial Mechanics and
Space Flight Dynamics [in Russian], Nauka, Moscow, 1978. |
| 10. | M. Born, Lectures on Atomic Mechanics [Russian translation],
Gos. Nauch.-tekhn. Izd-vo Ukrainy, Kharkov,
Kiev, 1934. |
|
| Received |
05 October 1999 |
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