This article is cited in 9 papers

Mathematical problems of nonlinearity

Vibrational Stability of Periodic Solutions of the Liouville Equations

E. V. Vetchanin^a, E. A. Mikishanina<sup>b

a</sup> Udmurt State University, ul. Universitetskaya 1, Izhevsk, 426034 Russia
^b Chuvash State University, Moskovskii prosp. 15, Cheboksary, 428015 Russia

Abstract: The dynamics of a body with a fixed point, variable moments of inertia and internal rotors are considered. A stability analysis of permanent rotations and periodic solutions of the system is carried out. In some simplest cases the stability analysis is reduced to investigating the stability of the zero solution of Hill’s equation. It is shown that by periodically changing the moments of inertia it is possible to stabilize unstable permanent rotations of the system. In addition, stable dynamical regimes can lose stability due to a parametric resonance. It is shown that, as the oscillation frequency of the moments of inertia increases, the dynamics of the system becomes close to an integrable one.

Keywords: Liouville equations, Euler – Poisson equations, Hill’s equation, Mathieu equation, parametric resonance, vibrostabilization, Euler – Poinsot case, Joukowski – Volterra case.

MSC: 70E17, 70J40

Received: 17.07.2019
Accepted: 23.09.2019

DOI: 10.20537/nd190312

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