 | | 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: | | 11262 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8011
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| In English (Mech. Solids): | | 3251 |
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| M.V. Belichenko, "On the Orbital Stability of Pendulum-Type Motions in the Approximate Problem of Kovalevskaya Top Dynamics with a Vibrating Suspension Point," Mech. Solids. 57 (7), 1607-1619 (2022) |
| Year |
2022 |
Volume |
57 |
Number |
7 |
Pages |
1607-1619 |
| DOI |
10.3103/S0025654422070056 |
| Title |
On the Orbital Stability of Pendulum-Type Motions in the Approximate Problem of Kovalevskaya Top Dynamics with a Vibrating Suspension Point |
| Author(s) |
M.V. Belichenko (Moscow Aviation Institute (National Research University), Moscow, Russia, tuzemec1@rambler.ru) |
| Abstract |
The motion of a heavy rigid body is studied in the case when one of its points (the suspension point) undergoes high-frequency horizontal vibrations and the body mass geometry for this point
corresponds to the Kovalevskaya case. The problem is examined using an approximate autonomous
system of differential equations represented in the Hamiltonian form. Particular body motions are
studied: pendulum-type motions and rotations around the principle axis of inertia oriented horizontally, which is either a dynamic-symmetry axis or an axis located in the equatorial inertia plane.
The radius vector of the center of mass of the body relative to the suspension point performs pendulum-type motions in a vertical plane containing the vibration axis (longitudinal motions) or perpendicular to this axis (transverse motions). This publication completes the linear analysis of the orbital
stability of the described pendulum-type motions with consideration for spatial perturbations, which
was commenced earlier. The problem has been reduced to an equivalent problem of the stability of the
trivial equilibrium of a reduced non-autonomous system with two degrees of freedom. In the regions
of linear-approximation stability, a detailed nonlinear analysis of the orbital stability has been carried
out. Criteria of the stability for most (in the sense of the Lebesgue measure) initial conditions and the
formal stability criteria have been checked and cases of fourth-order resonances have been explored. |
| Keywords |
Kovalevskaya top, high-frequency vibrations, pendulum-type motions, orbital stability |
| Received |
25 April 2021 | Revised |
07 December 2021 | Accepted |
23 December 2021 |
| Link to Fulltext |
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