SIMULTANEOUS PARAMETRIC AND INTERNAL RESONANCES IN SYSTEMS INVOLVING STRONG NON-LINEARITIES

Abstract

The non-linear dynamic interaction between the impact of the first asymmetric liquid sloshing mode, represented by an equivalent pendulum, and the elastic structural dynamics is examined in the neighborhood of simultaneous occurrence of parametric and internal resonance conditions. The analytical modelling of the impact force is described by a power function. The present work considers both weak and strong non-linear forces on interaction. The method of multiple scales is used to determine the system response in the neighborhood of three sets of different resonance conditions. Under first- or mixed-mode parametric excitation, the normal modes interact through internal resonance condition. The system response is found to be strongly dependent on 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 motion mainly governed by the initial conditions, internal detuning parameter, damping ratios and excitation level. Under second- mode parametric excitation, the second mode reaches fixed response, depending on initial conditions, with no energy sharing with the first mode. However, the phase angles are found to vary with time. Under combination parametric resonance, and in the absence of impact forces, the response is found to be sensitive to initial conditions.