Pull-in and snap-through instabilities in transient deformations of microelectromechanical systems

We analyze transient finite electroelastodynamic deformations of a perfect electrically conducting undamped clamped-clamped beam, a clamped-clamped parabolic arch and a clamped-clamped bell-shaped arch suspended over a flat rigid semi-infinite perfect conductor. The pull-in instability in a beam and the pull-in and the snap-through instabilities in the two arches due to time-dependent potential difference between the two electrodes have been studied. The potential difference is applied either suddenly or is increased linearly in time. Since the time scale of the transient electric forces is very small as compared to that of the mechanical forces, inertia effects only in the mechanical deformations are considered. Effects of both material and geometric nonlinearities are incorporated in the problem formulation and solution; however, damping due to the interaction of the structure with the surrounding medium is neglected. The coupled nonlinear partial differential equations for mechanical deformations are solved numerically by the finite element method and those for the electrical problem by the boundary element method. The Coulomb pressure due to the potential difference between the two electrodes is a nonlinear function of the a priori unknown distance between them. The potential difference that induces either the pull-in instability in a beam or the snap-through followed by the pull-in instabilities in an arch has been computed. Wherever possible these results are compared with those available in the literature. With a decrease in the rate of the applied potential difference, the pull-in and the snap-through parameters approach those for a static problem. Also, for large rates of increase in the potential difference between the two electrodes, the snap-through instability in an arch is suppressed and only the pull-in instability occurs.