On the motion of a liquid-filled heavy body around a fixed point

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

“…If, however, T possess an interior cavity filled up with a viscous liquid, Theorem 6.1 tells us that under the same above circumstances, the axis a will eventually reposition itself in the vertical direction d, at an exponential rate. This fact provides a further example of the stabilizing influence of an interior liquid-filled cavity on the motion of a rigid body [4,9,11,20,21,22,23].…”

“…Also, by assumption and the spectral property of L, there is γ > 0 such that 11) which implies that the fractional powers L α 1 , α ∈ (0, 1), are well defined in X (1) . Thus, setting…”

“…Over the last few years, the present authors and their associates have started a systematic and stringent analysis of the motion of a rigid body with an interior cavity entirely filled with a viscous liquid [4,7,9,10,11,20,21,22,23]. In particular, in [11,Section 8], it has been shown in a rigorous way that condition (0.1) guarantees the full nonlinear stability of the steady-states in a very large class of perturbations (weak solutions). However, the instability counterpart of this result requires a condition that is more restrictive than that stated in (0.2).…”

Compressible fluid inside a linear oscillator

“…It is known that the presence of a fluid inside a freely moving body has a tremendous effect on the stability of the system. This has been investigated several times, most recently by Disser, Galdi, Mazzone and Zunino [2] for incompressible fluid inside a rotating body, Galdi and Mazzone [5] for incompressible fluid inside a pendulum (see also [6]). In both cases there appear some non-potential forces which cause the fluid to move unless the motion of the whole system is stabilized.…”

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

“…A few years ago, a rigorous and systematic mathematical analysis has eventually begun, under the assumption that the liquid filling the cavity is incompressible and governed by the Navier-Stokes equations [38,4,13,14,16,15,18,17,31]. 1 Besides the basic study of well-posedness of the relevant initial-boundary value problem, all these works mainly focus on the fundamental question of the "ultimate dynamics" of the coupled system, and the characterization of terminal states.…”