Nonlinear stability analysis of a spinning top with an interior liquid-filled cavity

¹ Department of Mechanical Engineering and Materials Science, University of Pittsburgh, Pittsburgh, USA.
² Department of Mathematics and Statistics, Queen’s University, Kingston, Canada.

^* Corresponding author: galdi@pitt.edu

Received: 5 February 2020
Accepted: 30 November 2020

Abstract

Consider the motion of the coupled system, 𝒮, constituted by a (non-necessarily symmetric) top, ℬ, with an interior cavity, 𝒞, filled up with a Navier-Stokes liquid, ℒ. A particular steady-state motion (say) of 𝒮, is when ℒ is at rest with respect to ℬ, and 𝒮, as a whole rigid body, spins with a constant angular velocity around a vertical axis passing through its center of mass G in its highest position (upright spinning top). We then provide a complete characterization of the nonlinear stability of by showing, roughly speaking, that is stable if and only if is sufficiently large, all other physical parameters being fixed. Moreover we show that, unlike the case when 𝒞 is empty, under the above stability conditions, the top will eventually return to the unperturbed upright configuration.