Geodesic
Vibrational Stability of Periodic Solutions of the Liouville Equations
Russian journal of nonlinear dynamics, Tome 15 (2019) no. 3, pp. 351-363

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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.
Mots-clés : Liouville equations, Euler – Poisson equations, Mathieu equation
Keywords: Hill’s equation, parametric resonance, vibrostabilization, Euler – Poinsot case, Joukowski – Volterra case. 
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     author = {E. V. Vetchanin and E. A. Mikishanina},
     title = {Vibrational {Stability} of {Periodic} {Solutions} of the {Liouville} {Equations}},
     journal = {Russian journal of nonlinear dynamics},
     pages = {351--363},
     publisher = {mathdoc},
     volume = {15},
     number = {3},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ND_2019_15_3_a11/}
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E. V. Vetchanin; E. A. Mikishanina. Vibrational Stability of Periodic Solutions of the Liouville Equations. Russian journal of nonlinear dynamics, Tome 15 (2019) no. 3, pp. 351-363. http://geodesic.mathdoc.fr/item/ND_2019_15_3_a11/