Stability Of A Liquid-filled Spinning Top: A Numerical Approach

Abstract

The stability of an axisymmetric liquid-filled spinning top is studied numerically using the finite element method. The top contains a cylindrical cavity. Its size and axial distance from the pivot and the dry top's center of mass distance from the pivot can be varied as parameters in addition to the top's inertia property. The stability is examined for various liquid fill-fractions. The goal of this paper is (i) to benchmark a novel numerical approach using a cavity geometry that already has known analytical solutions; and (ii) to extend some of the existing results of the problem to a wider parameter region, which appears only to be accessible numerically. Our results show that for a free top (pivot and cavity center located at the dry top center of mass), all the odd modes destabilize nutation for a dry body's axial-to-spin inertia ratio, σ, to be less than one. For a top under the action of gravity, nutation instability can occur even for σ greater than one. The classical degenerate condition for a rigid top can manifest into a triple resonance, by the interaction between the two rigid modes and a two-dimensional liquid mode. For an inverted top (a spinning compound pendulum), nutation instability can still occur when σ is less than one. Both the two-dimensional modes (no axial variations) and even modes become destabilizing to nutation if the axial cavity location is offset from a balanced position.