Phase-lag predicts nonlinear response maxima in liquid-sloshing experiments

TL;DR

This study demonstrates that the phase-lag between the liquid's center of mass and the forcing predicts the maximum nonlinear response in liquid sloshing, revealing a consistent 90-degree lag at resonance regardless of wave pattern complexity.

Contribution

It introduces phase-lag as a key predictor for nonlinear response maxima in liquid sloshing, extending understanding beyond traditional amplitude-based models.

Findings

Phase-lag predicts nonlinear response maxima.

At resonance, sloshing lags forcing by 90 degrees.

Multimodal model overestimates amplitudes and hysteresis.

Abstract

Mass-spring models are essential for the description of sloshing resonances in engineering. By experimentally measuring the liquid's centre of mass in a horizontally oscillated rectangular tank, we show that low-amplitude sloshing obeys the Duffing equation. A bending of the response curve in analogy to a softening spring is observed, with growing hysteresis as the driving amplitude increases. At large amplitudes, complex wave patterns emerge (including wave-breaking and run up at the tank walls), competition between flow states is observed and the dynamics departs progressively from Duffing. We also provide a quantitative comparison of wave shapes and response curves to the predictions of a multimodal model based on potential flow theory (Faltinsen & Timokha 2009) and show that it systematically overestimates the sloshing amplitudes and the hysteresis. We find that the phase-lag between the liquid's centre of mass and the forcing is the key predictor of the nonlinear response maxima. The phase-lag reflects precisely the onset of deviations from Duffing dynamics and - most importantly - at resonance the sloshing motion always lags the driving by 90{\deg} (independently of the wave pattern). This confirms the theoretical 90{\deg}-phase-lag criterion (Cenedese & Haller 2020).