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

Abstract

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.