Resonances in the heavy symmetrical top with vibrating pivot

Abstract.

In this work we consider a heavy symmetrical top whose pivot is subjected to small-amplitude, high-frequency vibrations in the vertical direction. For analytical simplicity we confine ourselves to positions of the top close to the vertical. We first apply the slow-fast separation method originally devised by Kapitza and Landau for analysing the vibrational stabilization of an inverted pendulum. This analysis yields the slow precession frequency but we see that the equations become undefined at a particular value of the vibration frequency. This breakdown is seen to correspond to a resonance and we use Euler's equations to write down the solution at the resonance. For vibration in the horizontal direction there is a resonance at more or less the same frequency as before but the dynamics at the resonance is different from the former case.