Autoparametric resonances in elastic structures carrying two rectangular tanks partially filled with liquid

Nonlinear vibrations of elastic structures carrying two liquid-filled tanks under vertical harmonic excitation are investigated. A 2:1:1 ratio of internal resonance is satisfied between the natural frequency of the structure and the lowest natural frequencies of sloshing in two rectangular liquid tanks. The theoretical analysis here is restricted to two-dimensional flow and three-dimensional Faraday waves are not considered. The finite fluid depths are also assumed. Modal equations of motion are derived by using Galerkin's method, taking into account the nonlinearity of the hydrodynamic force. Then, frequency response curves are determined by using van der Pol's method. The respective influences of the liquid levels and the deviation of the internal resonance ratio (i.e., the mistuning) on the frequency response curves are demonstrated. As a result, we have obtained results different from those obtained in the case of a structure with a single liquid-filled tank. It is found that two types of vibrations can occur depending on the liquid levels and the excitation frequency: a single-mode solution (sloshing in either tank) and a double-mode solution (sloshing in both tanks). Amplitude- and phase-modulated motion including chaotic vibrations appears on the branches of the frequency response curves, which correspond to the single mode solution, when some deviation of the internal resonance ratio exists. The validity of the theoretical results was verified by the experiments.