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On the Nonlinear Oscillations of a Triaxial Ellipsoid on a Smooth Horizontal Plane

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Abstract

The motion of a homogeneous ellipsoid on a fixed horizontal plane in a uniform gravity field is considered. The plane is considered to be perfectly smooth and the semiaxes of the ellipsoid are different. There is a position of stable equilibrium, when the ellipsoid rests on the plane with the lowest point of its surface. The nonlinear oscillations of the ellipsoid in the vicinity of this equilibrium are studied. Analysis is performed using the methods of the classical perturbation theory, the Kolmogorov–Arnold–Moser (KAM) theory, and computer algebra algorithms. The normal form of the Hamiltonian function of the perturbed motion is obtained including the terms of the sixth degree relative to deviations from the equilibrium position. An approximate analytical representation of the Kolmogorov set of conditionally periodic oscillations and an estimate of the measure of this set are presented. The problem of the existence and orbital stability of periodic motions occurring from the stable equilibrium in the resonant and nonresonant cases is studied.

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Funding

The study was supported by the Russian Science Foundation (project no. 19-11-00116) at the Moscow Aviation Institute (National Research University) and Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences.

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Authors and Affiliations

  1. Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, 119526, Moscow, Russia

    A. P. Markeev

  2. Moscow Aviation Institute (National Research University), 125993, Moscow, Russia

    A. P. Markeev

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  1. A. P. Markeev
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Correspondence to A. P. Markeev.

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Translated by N. Podymova

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Markeev, A.P. On the Nonlinear Oscillations of a Triaxial Ellipsoid on a Smooth Horizontal Plane. Mech. Solids 57, 1805–1818 (2022). https://doi.org/10.3103/S0025654422080209

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  • DOI: https://doi.org/10.3103/S0025654422080209

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