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 horizontal vibration. The method of direct partition of motion is used to obtain an autonomous two-degree-of-freedom system governing the leading-order slow dynamics of the top’s nutation and precession angles. We show that the fast vibration leads to loss of stability of the upright “sleeping top” equilibrium state of the spinning top. We also show the existence of two new apparent equilibrium states that correspond to special solutions in which the spinning top is locked at constant nutation with the precession angle being aligned with the direction of excitation. We refer to those as “skewed sleeping top” states. We derive the condition for which these states exist and show that they are stable. The results are verified through numerical integration of the full non-autonomous system. In addition, we illustrate how these states allow the control of the attitude of the top through slow variation of the amplitude of fast excitation.