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

Giovanni P Galdi,Giusy Mazzone

doi:10.1051/mmnp/2020053

Giovanni P Galdi, Giusy Mazzone

Open Access

https://doi.org/10.1051/mmnp/2020053

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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 [see formula in PDF] (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 [see formula in PDF] around a vertical axis passing through its center of massGin its highest position (upright spinning top). We then provide a complete characterization of the nonlinear stability of [see formula in PDF] by showing, roughly speaking, that [see formula in PDF] is stable if and only if [see formula in PDF] 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.

Highlights

The stability of an upright spinning top is a renowned, classical problem in rigid body dynamics
Among the many motions that B can execute, interesting is the one where B spins with constant angular velocity, ω, around the vertical axis, a, passing through O and its center of mass G at its highest position, and coinciding with one of the principal axes of inertia eB
In this paper we investigate the analogous question when the top has an interior cavity, C, entirely filled with a viscous liquid, L

Summary

Introduction

The stability of an upright spinning top is a renowned, classical problem in rigid body dynamics. By “asymptotic stability” we mean thats is stable in the classical sense of Liapunov and, the generic perturbed motion with initial data in an appropriate neighborhood ofs will converge, exponentially fast, to a steady-state where the top continues to spin in the upright position. This “stabilizing effect” is just due to the presence the liquid since, as we noted earlier on, if the cavity is empty the perturbed motion of the top is precession-nutation-like around the vertical axis through the fixed point O.

Formulation of the problem and preliminary considerations

Stability properties for an abstract evolution problem

Preliminary properties of the operator L

On the location of the spectrum of the operator L

Properties of the operator N

Nonlinear stability properties