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

Abstract

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

On the free rotations of rigid bodies with a liquid-filled gap

“…If this happens, from the boundary conditions (4), also ω is expected to decay, and the system would then move as a whole rigid body. This kind of behavior has been rigorously proved to be true in other problems in liquid-solid interactions without the inner core for inertial motions and motions driven by gravity (see [9,13,14,22,23,25,24,15]).…”

“…To this aim, we will prove the existence of a special basis of H 2 (V) and of a special basis of H 2 2 (V). We start by noticing that, taking (15) with q = 2, the norm • 1,2 is induced by the following inner product…”

On the Motion of a Pendulum with a Cavity Filled with a Compressible Fluid

“…In spite of its relevance, a rigorous and systematic mathematical analysis of the motion of a body with a fluid-filled cavity has started only a few years ago [5,12,[15][16][17][18][19][24][25][26]. These works have, on the one hand, produced a full explanation of experimental observations and, on the other hand, hinted at other, new interesting features that might be supported by numerical or lab tests.…”