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 Nelin. Dinam., 2018, Volume 14, Number 4, Pages 503–518 (Mi nd628)

Nonlinear physics and mechanics

On Nonlinear Resonant Oscillations of a Rigid Body Generated by Its Conical Precession

A. P. Markeevabc

a Moscow Institute of Physics and Technology (State University), Institutskii per. 9, Dolgoprudnyi, Moscow region, 141700 Russia
b Ishlinsky Institute for Problems in Mechanics RAS, pr. Vernadskogo 101-1, Moscow, 119526 Russia
c Moscow Aviation Institute (National Research University), Volokolamskoe shosse 4, Moscow, 125080 Russia

Abstract: The motion of a dynamically symmetric rigid body relative to its center of mass in the central Newtonian gravitational field in a circular orbit is investigated. This problem involves motion (called conical precession) where the dynamical symmetry axis of the body is located all the time in the plane perpendicular to the velocity vector of the center of mass of the body and makes a constant angle with the direction of the radius vector of the center of mass relative to the attracting center. This paper deals with a special case in which this angle is $\pi/4$ and the ratio between the polar and the equatorial principal central moments of inertia of the body is equal to the number $2/3$ or is close to it. In this case, the conical precession is stable with respect to the angles that define the position of the symmetry axis in an orbital coordinate system and with respect to the time derivatives of these angles, and the frequencies of small (linear) oscillations of the symmetry axis are equal or close to each other (that is, the 1:1 resonance takes place). Using classical perturbation theory and modern numerical and analytical methods of nonlinear dynamics, a solution is presented to the problem of the existence, bifurcations and stability of periodic motions of the symmetry axis of a body which are generated from its relative (in the orbital coordinate system) equilibrium corresponding to conical precession. The problem of the existence of conditionally periodic motions is also considered.

Keywords: resonance, stability, oscillations, canonical transformations

 Funding Agency Grant Number Russian Foundation for Basic Research 17-01-00123 Russian Academy of Sciences - Federal Agency for Scientific Organizations AAAA-A17-117021310382-5 This work was carried out within the framework of the State Assignment (registration No. AAAA-A17-117021310382-5) and was partially supported by the Russian Foundation for Basic Research (project No. 17-01-00123).

DOI: https://doi.org/10.20537/nd180406

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Accepted:25.08.2018
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Citation: A. P. Markeev, “On Nonlinear Resonant Oscillations of a Rigid Body Generated by Its Conical Precession”, Nelin. Dinam., 14:4 (2018), 503–518

Citation in format AMSBIB
\Bibitem{Mar18} \by A. P. Markeev \paper On Nonlinear Resonant Oscillations of a Rigid Body Generated by Its Conical Precession \jour Nelin. Dinam. \yr 2018 \vol 14 \issue 4 \pages 503--518 \mathnet{http://mi.mathnet.ru/nd628} \crossref{https://doi.org/10.20537/nd180406} \elib{https://elibrary.ru/item.asp?id=36686071}