On the dynamics of a spinning top under high-frequency excitation

Abstract

We investigate the dynamics of a spinning top whose pivot point undergoes a small amplitude high-frequency vertical vibration. The method of Direct Partition of Motion is used to obtain an autonomous equation governing the leading order slow dynamics of the top’s nutation and to derive an approximate closed form solution for the forced spinning top problem. We show that the fast vibration can lead to the stabilization of the “sleeping top” state and an expression for the minimum amplitude required is given in terms of system parameters. We also show the existence of a degenerate family of special solutions in which the spinning top is locked at constant nutation and precession angles; we refer to those as “skewed sleeping top” states. We derive the conditions under which these states exist and are stable. The results are verified through numerical integration of the full non-autonomous system