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

Abstract

Mass-spring models are essential for the description of sloshing resonances\nin engineering. By experimentally measuring the liquid's centre of mass in a\nhorizontally oscillated rectangular tank, we show that low-amplitude sloshing\nobeys the Duffing equation. A bending of the response curve in analogy to a\nsoftening spring is observed, with growing hysteresis as the driving amplitude\nincreases. At large amplitudes, complex wave patterns emerge (including\nwave-breaking and run up at the tank walls), competition between flow states is\nobserved and the dynamics departs progressively from Duffing. We also provide a\nquantitative comparison of wave shapes and response curves to the predictions\nof a multimodal model based on potential flow theory (Faltinsen & Timokha 2009)\nand show that it systematically overestimates the sloshing amplitudes and the\nhysteresis. We find that the phase-lag between the liquid's centre of mass and\nthe forcing is the key predictor of the nonlinear response maxima. The\nphase-lag reflects precisely the onset of deviations from Duffing dynamics and\n- most importantly - at resonance the sloshing motion always lags the driving\nby 90{\\deg} (independently of the wave pattern). This confirms the theoretical\n90{\\deg}-phase-lag criterion (Cenedese & Haller 2020).\n