Nonlinear vibrations of an elastic structure coupled with liquid sloshing in a square tank subjected to vertical sinusoidal excitation are investigated. Previous studies examined the vibrations of a structure coupled with only one sloshing mode in a rectangular tank. However, square tanks are expected to work more efficiently as a vibration suppression device (Tuned Liquid Damper, TLD) because two sloshing modes, (1,0) and (0,1) modes, simultaneously appear when the internal resonance ratio 2:1:1 is satisfied. In reality, it is impossible to build a perfectly square tank. Therefore, a nearly square liquid tank is also considered when the tuning condition is slightly deviated. In the theoretical analysis, the fluid in the tank is assumed to be perfect. The modal equations of motion for seven sloshing modes are derived using Galerkin’s method, considering the nonlinear terms. 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 (based on the harmonic balance method). From these response curves, the influences of the liquid level, the aspect ratio of the tank cross section, and the deviation of the tuning condition are investigated. For a square tank it is found that (1,0) and (0,1) modes are nonlinearly coupled. When the liquid level is high, there are three patterns for 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. Compared with the performance of a rectangular TLD, a square TLD works more efficiently when the liquid level is low. Small deviations of the tuning condition may cause amplitude modulated motion to appear. Bifurcation sets are also calculated to illustrate the influence of the system parameters on the performance of the TLD. Experiments were also conducted in order to confirm the validity of the theoretical results. These results were in good agreement with the experimental data.

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