Nonlinear Vibrations of Elastic Structures Containing a Cylindrical Liquid Tank under Vertical Excitation

Abstract

This study investigates the nonlinear vibrations of an elastic structure, with a liquid-filled cylindrical tank, which is subjected to a vertical sinusoidal excitation. This structure-tank system behaves as an autoparametric system. Modal equations governing the coupled motions of the structure and liquid sloshing are derived when the natural frequency of the structure is equal to twice the natural frequency of the first axisymmetric sloshing mode. Van der Pol's method is applied to the modal equations to determine the theoretical resonance curves. The theoretical results can be concluded as follows: (1) As the liquid level decreases, the resonance curve for the liquid sloshing changes from a soft spring type to a hard spring type. (2) The structure's resonance curve flattens out at small amplitude when the liquid level is appropriate. (3) Amplitude-modulated motions appear for the negative and positive values of the internal resonance ratio's deviation (the detuning parameter) in the high and low liquid levels, respectively. (4) Furthermore, the results of the bifurcation analysis, Poincare maps and Lyapunov exponents reveal that amplitude-modulated motions and chaotic oscillations can occur in the system. In experiments, the theoretical resonance curves were quantitatively in agreement with the experimental data.