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Home » Issues » Archive of Issues » 2019 - 2 issue » pp. 245-260

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Russia Moscow
Year
2019
Volume
29
Issue
2
Pages
245-260
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Section Mechanics
Title On periodic motions of a rigid body suspended on a thread in a uniform gravity field
Author(-s) Markeev A.P.ab
Affiliations Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciencea, Moscow Aviation Instituteb
Abstract The planar motion of a rigid body in a uniform gravity field is considered. The body is suspended on a weightless inextensible thread. The thread is assumed to remain taut during the motion of the body. Nonlinear periodic oscillations of the body in the vicinity of its stable equilibrium position on the vertical are studied. These motions are generated by small (linear) normal body vibrations. The question of the existence of such motions is solved with the Lyapunov theorem on a holomorphic integral. An algorithm for constructing these motions using the canonical transformation method is proposed. The corresponding solutions are represented in the form of series in a small parameter characterizing the amplitude of the generating normal oscillations. A rigorous solution is given to the nonlinear problem of orbital stability of the motions obtained. Possible regions of parametric resonance (instability regions) are indicated. The third and fourth order resonance cases, as well as a nonresonant case, are considered. The study is based on the Lyapunov and Poincaré methods and KAM-theory.
Keywords periodic motions, Hamiltonian system, resonance, stability
UDC 531.36, 531.53
MSC 70E20, 70H14, 70K28
DOI 10.20537/vm190208
Received 22 May 2019
Language Russian
Citation Markeev A.P. On periodic motions of a rigid body suspended on a thread in a uniform gravity field, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2019, vol. 29, issue 2, pp. 245-260.
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