Vibrations of a Free-Surface Fluid in an Ellipsoidal Tank in the Vicinity Of Resonance

The problem of the motion of a free-surface fluid in an ellipsoidal tank under a harmonic force with a frequency close to the resonant one is considered. A nonlinear coupled problem statement is used. Additional attention is paid to the solvability condition for the Neumann problem for the Laplace equation, which is auxiliary to solving the problem using asymptotic methods. A mathematical model that allows us to analyze the behavior of the system over a sufficiently long period of time is developed. The evolution of surface waves is analyzed for an elongated and vertically compressed ellipsoid. It is found out that the oscillation does not become steady-state in all the cases. The system shows high sensitivity to changes in excitation frequencies, and modulation is strongly manifested in almost all modes. In addition, it is shown that the mean value of vibration drifts at low excitation frequencies. The manifestation of nonlinearity becomes stronger with increasing vertical deviations of the side wall of the tank near the unperturbed free surface. Under oscillations at frequencies exceeding the resonance and under strong manifestations of nonlinearities, the regularity of oscillations is disturbed, and antiresonance is observed in some modes. The results are consistent with known theoretical and experimental data.