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Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2019, Volume 29, Issue 2, Pages 245–260 (Mi vuu679)  

MECHANICS

On periodic motions of a rigid body suspended on a thread in a uniform gravity field

A. P. Markeevab

a Ishlinsky Institute for Problems in Mechanics, Russian Academy of Science, pr. Vernadskogo, 101/1, Moscow, 119526, Russia
b Moscow Aviation Institute (National Research University), Volokolamskoe shosse, 4, Moscow, 125080, Russia

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.

Funding Agency Grant Number
Russian Academy of Sciences - Federal Agency for Scientific Organizations АААА-А17-117021310382-5
Russian Foundation for Basic Research 17-01-00123_а
The study was performed on the state task (registration no. АААА-А17-117021310382-5), was supported in part by the Russian Foundation for Basic Research (project no. 17-01-00123) at the Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Science and at the Moscow Aviation Institute (National Research University).


DOI: https://doi.org/10.20537/vm190208

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Bibliographic databases:

UDC: 531.36, 531.53
MSC: 70E20, 70H14, 70K28
Received: 22.05.2019

Citation: A. P. Markeev, “On periodic motions of a rigid body suspended on a thread in a uniform gravity field”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 29:2 (2019), 245–260

Citation in format AMSBIB
\Bibitem{Mar19}
\by A.~P.~Markeev
\paper On periodic motions of a rigid body suspended on a thread in a uniform gravity field
\jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki
\yr 2019
\vol 29
\issue 2
\pages 245--260
\mathnet{http://mi.mathnet.ru/vuu679}
\crossref{https://doi.org/10.20537/vm190208}
\elib{https://elibrary.ru/item.asp?id=39136249}


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