Finite Element Simulation of Dynamic Stability of Plane Free-Surface of a Liquid under Vertical Excitation

Abstract

When partially filled liquid containers are excited vertically, the plane free-surface of the liquid can be stable or unstable depending on the amplitude and frequency of the external excitation. For some combinations of amplitude and frequency, the free-surface undergoes unbounded motion leading to instability called parametric instability or parametric resonance, and, for few other combinations, the free-surface undergoes bounded stable motion. In parametric resonance, a small initial perturbation on the free-surface can build up unboundedly even for small external excitation, if the excitation acts on the tank for sufficiently long time. In this paper, the stability of the plane free-surface is investigated by numerical simulation. Stability chart for the governing Mathieu equation is plotted analytically using linear equations. Applying fully nonlinear finite element method based on nonlinear potential theory, the response of the plane free-surface is simulated for various cases.