Mathematical Analysis of Electrostatically Actuated MEMS Devices

We perform a rigorous mathematical analysis of a simple membrane based model of an electrostatically ac-tuated MEMS device. Using both analytical and numerical techniques, we prove the existence of a fold in the solution space of the displacement, implying the existence of a critical voltage beyond which there are no solutions of the equation. This critical voltage corresponds to the pull-in voltage observed in simpler lumped models, the numerical solution of three dimensional models, and also in experimental devices. We show h o w pseudo-arclength continuation may be used to eeciently compute the solutions on both sides of the fold.