Autoparametric Resonances of Elastic Structures Coupled With Two Sloshing Modes in a Square Liquid Tank

Nonlinear vibrations of an elastic structure coupled with liquid sloshing in a square tank subjected to vertical sinusoidal excitation are investigated. In the theoretical analysis, the ratios of the natural frequencies of the structure and two sloshing modes satisfy 2:1:1. The equations of motion for the structure and seven sloshing modes are derived using Galerkin’s method while considering the nonlinearity of sloshing. The linear damping terms are then incorporated into the modal equations to consider the damping effect of sloshing. The frequency response curves are determined using van der Pol’s method. The influences of the liquid level, the aspect ratio of the tank cross-section, the deviation of the tuning condition, and the excitation amplitude are investigated. When the liquid level is high, and depending on the excitation frequency, there are three patterns of sloshing: (i) both (1,0) and (0,1) sloshing modes appear at identical amplitudes; (ii) these two modes appear at different amplitudes; and (iii) either (1,0) or (0,1) mode appears. Small deviations of the tuning condition may cause Hopf bifurcations to occur followed by amplitude modulated motion including chaotic vibrations. Bifurcation sets are also calculated to illustrate the influence of the system parameters on the response of the system. It is found that for low liquid levels, square tuned liquid dampers (TLDs) work more effectively than rectangular TLDs. Experiments were also conducted in order to confirm the validity of the theoretical results and were in good agreement with the experimental data.