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

TL;DR

This paper analyzes how a viscous, compressible fluid inside a pendulum's cavity influences its motion, demonstrating that the fluid acts as a damper and accelerates the system’s return to equilibrium, supported by theoretical proofs and numerical tests.

Contribution

The study proves that a compressible fluid inside a pendulum acts as a damper, leading to eventual rest, without restrictions on initial data, and highlights the effect of compressibility on damping efficiency.

Findings

Fluid acts as a damper, leading to rest state.

Large compressibility reduces damping time.

Results hold for weak solutions with finite energy.

Abstract

We study the motion of the coupled system, $\mathscr S$, constituted by a physical pendulum, $\mathscr B$, with an interior cavity entirely filled with a viscous, compressible fluid, $\mathscr F$. The presence of the fluid may strongly affect on the motion of $\mathscr B$. In fact, we prove that, under appropriate assumptions, the fluid acts as a damper, namely, $\mathscr S$ must eventually reach a rest-state. Such a state is characterized by a suitable time-independent density distribution of $\mathscr F$ and a corresponding equilibrium position of the center of mass of $\mathscr S$. These results are proved in the very general class of weak solutions and do not require any restriction on the initial data, other than having a finite energy. We complement our findings with some numerical tests. The latter show, among other things, the interesting property that ``large" compressibility favors the damping effect, since it drastically reduces the time that $\mathscr S$ takes to go to rest.