Sloshing in coffee as a pumped pendulum

TL;DR

This paper models the dynamics of a coffee cup during walking as a nonlinear pendulum with parametric and direct excitation, analyzing stability and resonances using averaging and multiple scales methods.

Contribution

It introduces a lumped nonlinear pendulum model for coffee sloshing with analytical stability and resonance analysis methods.

Findings

Identified stability regions for steady-state solutions.

Determined system resonances through analytical methods.

Validated results with exact numerical integration.

Abstract

We present a study of the dynamics of a cup of coffee while walking with it in terms of a lumped model. The model considered is a planar (nonlinear) pendulum with parametric and direct excitation. The position of the pivot is changed with time, which leads to parametric excitation in the vertical direction; and parametric/direct excitation in the horizontal one. For the former case, the Method of Averaging is used to determine the regions of stability for two steady-state solutions. In the vertical/horizontal excitation, we determined the resonances of the system using the Method of Multiple Scales and show the results computed with \emph{exact} numerical integration.