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. 1. Introduction In many engineering applications, it is of great interest to study the oscillatory motion of a mechanical system under external excitations. In all of the vibrating systems, it is essential to know the response of the system and the nature of the oscillations. It is very important to get information about the different resonance cases of the system to avoid the harmful ones in its design. Under external periodic excitations, the system may undergo two kinds of oscillations: forced oscillations and parametric oscillations. Forced oscillations correspond to the oscillatory response of the system in the direction of external excitation, and the system undergoes resonance when external excitation is equal to natural frequency of the system. In resonance, systems amplitude increases linearly. Parametric oscillations refer to an oscillatory motion in a mechanical system due to time-varying changes in the system parameters caused due to external excitation. The system parameters can be inertia or stiffness. The response of the system is orthogonal to the direction of external excitation. System undergoes parametric resonance when the external excitation is equal to integral multiple of natural frequency of the system. In parametric resonance, systems amplitude increases exponentially and may grow without limit. This exponential unlimited increase of amplitude is potentially dangerous to the system. Parametric resonance is also known as parametric instability or dynamic instability. Although parametric instability is secondary, the system may undergo failure near
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