MATEC Web of Conferences
Volume 362, 2022XXII International Conference on Computational Mechanics and Modern Applied Software Systems (CMMASS 2021)
|Number of page(s)||9|
|Published online||14 September 2022|
Analysis of the orbital stability of periodic pendulum motions of a heavy rigid body with a fixed point under the Goryachev–Chaplygin condition
1 Department of Mechatronics and Theoretical Mechanics, Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, 125993, Russia
2 Department of Higher Mathematics, Moscow Automobile and Road Construction State Technical University, 64, Leningradsky prospect, Moscow, 125319, Russia
3 Mechanical Engineering Research Institute of the Russian Academy of Sciences, 4, M. Kharitonyevskiy pereulok, Moscow, 101990, Russia
* e-mail: firstname.lastname@example.org
In this paper the motion of a rigid body with a fixed point in a uniform gravity field is considered. It is assumed that the main moments of inertia of the body correspond to the case of Goryachev–Chaplygin, i.e., they are in the ratio of 1:1:4. In contrast to the integrable case of Goryachev–Chaplygin, there are no restrictions imposed on the position of the center of mass of the body. The problem of the orbital stability of periodic pendulum motions of a body (oscillations and rotations) is investigated. The equations of perturbed motion were obtained and the problem of orbital stability was reduced to the stability problem of the equilibrium position of a second-order linear system with 2π-periodic coefficients, the right-hand sides of which depend on two parameters. Based on the analysis of the linearized system, it has been established that the rotations are orbitally unstable for all possible values of the parameters. Moreover, a diagram of stability of pendulum oscillations has been constructed, on which the regions of orbital instability (parametric resonance) and regions of orbital stability in the linear approximation are indicated. At small values of oscillations amplitudes, a nonlinear analysis was performed. Based on the analysis of the coefficients of the normalized Hamilton function using theorems of the KAM theory, rigorous conclusions on the orbital stability of pendulum oscillations with small amplitudes were obtained.
© The Authors, published by EDP Sciences, 2022
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