On Pendulum-Type Motions and Permanent Rotations in an Approximate Problem of the Dynamics of a Rigid Body with a Vibrating Suspension

The motion of a heavy rigid body with suspension point performing high-frequency periodic vibrations of small amplitude is considered. The study is carried out within the framework of an approximate autonomous system written in the form of the modified Euler – Poisson equations, to the right-hand sides of which the components of the vibration moment are added. The question of the existence of two particular motions of the body is resolved, they are permanent rotations and pendulum-type motions. It is shown that permanent rotations of the body can occur in the case of vibration symmetry relative to a vertically located axis. The search for pendulum-type motions is restricted to the case when the axis of these motions is one of the principal inertia axes of the body, as in the case of a heavy rigid body with a fixed point. Two basic variants of vibrations are considered, when the suspension point vibrates along a straight line and along an ellipse. To the latter variant any planar vibrations and a wide class of spatial vibrations of the suspension point are reduced. It is shown that for both basic cases of vibrations, pendulum-type motions are of two types. The motions of the first type are similar to the Mlodzeevsky’s pendulum-type motions of a heavy rigid body with a fixed point. For them, the body’s mass center lies in the principal plane of inertia, and the axis of the pendulum-type motions is perpendicular to this plane. Pendulum-type motions of the second type occur around the principal axis of inertia containing the body’s center of mass. Such motions are absent in the gravitational problem, they are caused by the presence of vibrations. To search for the pendulum-type motions, an approach is proposed that combines the results of the problem of gravitation (without vibration) and that of vibration (ignoring gravity). As an illustration, a number of examples of the interaction of the gravitational field and the vibration field corresponding to both basic variants of vibrations of the body’s suspension point are considered.