Experimental observation of Duffing dynamics in liquid sloshing

The nonlinear resonances of sloshing liquids in ships (e.g. liquified natural gas) and of liquid fuel in rockets pose serious risks. Their dynamics resembles that Duffing of oscillators and it has been predicted with potential theory that sloshing in a rectangular container obeys the Duffing equation. However, potential theory does not include dissipation and even for modern sloshing models the description of dissipation remains a challenge, which prevents predictions of the nonlinear response maxima. \newline We show that low-amplitude sloshing in a horizontally oscillated rectangular tank obeys Duffing dynamics with linear damping. The motion of the liquid's centre of mass is used to characterize the amplitude and phase-lag of the sloshing unambiguously and globally. As the driving amplitude increases deviations from Duffing dynamics are first seen in the phase-lag, well before complex wave patterns emerge. We observe that at resonance the sloshing motion is in quadrature with the driving independently of the observed flow state. This confirms the theoretical $90^\circ$-phase-lag criterion and highlights the phase-lag (so far rarely measured in experiments) as a key indicator of transitions in sloshing liquids.