On the equilibriums stability in an approximate problem of the dynamics of a rigid body with a suspension point vibrating along an inclined straight line

Abstract

We consider the heavy rigid body dynamics under the assumption that one of the body points (the suspension point) performs the specified high-frequency vibrations of small amplitude along an inclined straight line, and the body mass centre lies on the principal axis of inertia for the suspension point. In the framework of an approximate autonomous system of canonical equations of motion, the question of existence, number, and stability of the body relative equilibrium positions is solved. It is shown that for all such equilibria, the mass centre radius-vector lies in the vertical plane containing the vibration axis. The number of equilibria is four, six or eight depending on the vibration intensity. Sufficient and necessary conditions for their stability are found. The existence of high-frequency periodic motions of the initial non-autonomous system, which are generated by the investigated equilibria, is justified using Poincare’s method. Conclusions about stability (in linear approximation) of these periodic motions are drawn.