On the influence of vertical vibrations on the stability of permanent rotations of a rigid body about axes lying in the main plane of inertia

The motion of a rigid body in a uniform gravity field is considered for the case of high-frequency vertical harmonic small-amplitude oscillations of one of its points (the suspension point). The center of mass of the body is assumed to lie on one of the principal axes of inertia for the suspension point. In the framework of an approximate autonomous system of differential equations of motion written in the canonical Hamiltonian form the special motions of the body are studied, which are permanent rotations about the axes directed vertically and lying in the principal planes of inertia containing the above-mentioned principal axis. Analogous permanent rotations exist for the body with a fixed suspension point. The influence of the fast vibrations on the stability of these rotations is examined. For all admissible values of the four-dimensional parameter space (two inertial parameters, and parameters characterizing the vibration frequency and the rotation angular velocity) the necessary and in some cases sufficient conditions for stability are written and illustrated. They are considered as the stability conditions of the corresponding equilibrium positions of the reduced (in the sense of Routh) autonomous Hamiltonian two-degree-of-freedom system. Nonlinear stability analysis is carried out for two special cases of the inertial parameter corresponding to the dynamically symmetric body and the body with the geometry of the mass for the Bobylev–Steklov case. The nonresonant and resonant cases are considered as well as the degeneration cases. A comparison is made between the results obtained and the corresponding results for the body with the fixed suspension point.