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

TL;DR

This paper numerically investigates the bifurcation and stability thresholds in MEMS models, revealing how different equations and device parameters influence the pull-in instability and operational regimes.

Contribution

It provides the first numerical bifurcation analysis of stationary solutions and dynamical thresholds for MEMS models with free boundaries, including heat and damped wave equations.

Findings

Dynamical thresholds match static values for heat equations.

Damped wave equations have smaller dynamical thresholds than static.

Aspect ratio significantly influences pull-in stability more than inertia.

Abstract

In this work we numerically compute the bifurcation curve of stationary solutions for 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 given by either a heat or a damped wave equation. This threshold is what we term the dynamical pull-in value: it separates the stable operation 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 agreement with the observations reported for a mass-spring system 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.