A numerical study of the pull-in instability in some free boundary models for MEMS

In this work we numerically compute the bifurcation curve for stationary solutions of the free boundary problem for MEMS in one space dimension. It has a single turning point, as in the case of the small aspect ratio limit. We also find a threshold for the existence of global in time solutions of the evolution equation for the MEMS in the form of either a heat or a damped wave equation. This threshold is what we term the dynamical pull-in value: it separates the stable operating regime from the touchdown regime. The numerical calculations show that the dynamical threshold values for the heat equation coincide with the static values. For the damped wave equation the dynamical threshold values are smaller than the static values. This result is in qualitative agreement with those reported for the model of the MEMS based on a simplified mass-spring system, as studied in the engineering literature. In the case of the damped wave equation, we also show that the aspect ratio of the device is more important than the inertia in the determination of the pull-in value.