Internal resonance of nonlinear sloshing in rectangular liquid tanks subjected to obliquely horizontal excitation

Nonlinear sloshing in rectangular tanks subjected to obliquely horizontal, harmonic excitation is investigated when the internal resonance condition 1:1 is satisfied between the natural frequencies of predominate modes (1, 0) and (0, 2). Galerkin’s method is employed to derive the nonlinear modal equations of motion for sloshing, considering nine sloshing modes. Then, van der Pol’s method is applied in order to obtain the expressions of the frequency response curves for amplitudes and phase angles of the predominate modes. The frequency response curves are calculated and reveal that (0, 2) mode may occur even though it is not directly excited because it is nonlinearly coupled with (1, 0) mode due to the autoparametric terms. In the numerical simulations, it is found that planar motions of (1, 0) mode, clockwise and counter-clockwise swirl motions, and translational motions may appear. Furthermore, Hopf bifurcation occurs, and amplitude modulated motions (AMMs), including chaotic motions, may appear depending on the value of the excitation frequency. Three-dimensional distribution charts of the maximum liquid surface elevation are calculated to show the risk of liquid overspill. The influence of the difference between the horizontal excitation direction and the tank side on the frequency response curves is also examined. Bifurcation sets are calculated to clarify this influence. Experimental data confirmed the validity of the theoretical results.