On Stability of Relative Equilibriums of a Rigid Body with a Vibrating Point of Support

In this paper motions of a heavy rigid body, of which point of support performs vertical harmonical vibrations of a high frequency and a small amplitude, are studied. Using an approximate autonomous system of differential equations of the motion, an analysis of existence, bifurcations and stability of relative equilibrium is carried out, for which the mass center and the point of support of the body don't lie at the same vertical. It is shown that all of these relative equilibria are unstable.

Euler-Poisson equations, high-frequency vibration, vibratory moment, relative equilibrium, stability