A new physically realisable internal 1:1 resonance in the coupled pendulum-slosh system

The problem of dynamic coupling between a rectangular container undergoing planar pendular oscillations and its interior potential fluid sloshing is studied. The Neumann boundary-value problem for the fluid motion inside the container is deduced from the Bateman-Luke variational principle. The governing integro-differential equation for the motion of the suspended container, from a single rigid pivoting rod, is the Euler-Lagrange equation for a forced pendulum. The fluid and rigid-body partial differential equations are linearised, and the characteristic equation for the natural and resonant frequencies of the coupled dynamical system are presented. It is found that internal 1:1 resonances exist for an experimentally realistic setup, which has important physical implications. In addition, a new instability is found in the linearised coupled problem whereby instability occurs when the rod length is shorter than a critical length, and an explicit formula is given.& COPY; 2022 The Author(s). Published by Elsevier Masson SAS. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

Automated summary

This study investigates the dynamic coupling between a rectangular container undergoing planar pendular oscillations and the potential fluid sloshing inside it. The Neumann boundary-value problem for the fluid motion is derived using the Bateman-Luke variational principle. The suspended container's motion is described by the Euler-Lagrange equation for a forced pendulum. The linearized fluid and rigid-body partial differential equations yield the characteristic equation for the coupled dynamical system's natural and resonant frequencies. It is noteworthy that internal 1:1 resonances exist in a realistic experimental setup, which has significant physical implications. Additionally, a new instability has been discovered in the linearized coupled problem, where instability arises when the rod length is shorter than a critical length, and an explicit formula is provided.