On the stability of the specific motions of a heavy rigid body due to fast vertical vibrations of one of its points

We consider the motion of a heavy rigid body with one point performing the specified highfrequency harmonic oscillations along the vertical. In the framework of an approximate autonomous system of differential equations of motion two new types of permanent rotations of the body about the vertical are found. These motions are affected by presence of fast vibrations and do not exist in the case of a body with a fixed point. The problem of stability of the motions is investigated.