Phase lag predicts nonlinear response maxima in liquid-sloshing experiments

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, Sloshing. Cambridge University Press, 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 $90^{\circ }$ (independently of the wave pattern). This confirms the theoretical $90^{\circ }$-phase-lag criterion (Cenedese & Haller, Proc. R. Soc. A, vol. 476, no. 2234, 2020, 20190494).