On Nonlinear Resonant Oscillations of a Rigid Body Generated by Its Conical Precession
2018, Vol. 14, no. 4, pp. 503-518
Author(s): Markeev A. P.
This work is licensed under a Creative Commons Attribution-NoDerivs 3.0 Unported License
Author(s): Markeev A. P.
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.
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