Stability of special motions (permanent rotations) of a heavy rigid body with a suspension point vibrating along the vertical

We consider a heavy rigid body with a point making the vertical high–frequency harmonic oscillations of small amplitude. The problem is considered in the framework of an approximate autonomous canonical system of differential equations of motion. The special motions are studied, which are permanent rotations of the body around the vertical principal axis of inertia containing its center of mass. Necessary and in some cases sufficient stability conditions for the corresponding equilibrium positions of the reduced two-degree-of-freedom system are found. The comparison of the results obtained with the corresponding results for a body with a fixed point is fulfilled. Nonlinear stability analysis is carried out for two special cases of mass geometry of the body.