On the motion of a heavy dynamically symmetric rigid body with vibrating suspension point

The motion of a dynamically symmetric rigid body in a homogeneous field of gravity is studied. One point lying on the symmetry axis of the body (the suspension point) performs high-frequency periodic or conditionally periodic vibrations of small amplitude. In the framework of approximate equations of motion obtained earlier, we find necessary and sufficient conditions for the stability of the body rotation about the vertical symmetry axis and study the existence and stability of regular precessions of the body in the coordinate system translationally moving together with the suspension point.