Simultaneous Parametric and Internal Resonances in Systems Simulating Liquid Sloshing Impact

Abstract

The nonlinear dynamic interaction between liquid sloshing impact and elastic structural dynamics is examined in the neighborhood of simultaneous parametric and internal resonance conditions. The analytical modeling of the liquid impact forces is represented by a simple pendulum. The impact forces with the tank walls are phenomenologically described by a power function. In the absence of internal resonance, Pilipchuk and Ibrahim (1997) analyzed the system to describe the in-phase and out-of-phase strongly non-linear periodic regimes. The present work considers both weakly and strongly nonlinear forces of interaction. The method of multiple scales is used to determine the system response in the neighborhood of combination parametric resonance and internal resonance conditions. Under combination parametric resonance, and in the absence of impact forces, the response is found to sensitive to initial conditions. Depending on the initial conditions and internal detuning parameter, the response can be quasi-periodic or chaotic with irregular jumps between two unstable equilibria. In the presence of impact forces, the system preserves fixed response amplitude response within a small range of internal detuning parameter. Beyond that range, the response exhibits quasi-periodic and snap-through regimes. Each regime is governed by the initial conditions, internal detuning parameter, damping ratios and excitation level. The paper briefly describes the response characteristics under fist and second mode parametric excitations.