Dynamics of Electrostatic MEMS Actuators

TL;DR

This paper provides rigorous theorems describing the dynamics of undamped electrostatic MEMS actuators with one degree of freedom, revealing the conditions for stable oscillations and collapse based on voltage levels and elastic forces.

Contribution

It introduces two sharp theorems that characterize the nonlinear and linear dynamics of electrostatic MEMS actuators, including explicit and complete descriptions of key parameters.

Findings

Existence of one-stagnation-point periodic response below pull-in voltage

Finite-time collapse of the actuator above pull-in voltage

Explicit formulas for stagnation level, pull-in voltage, and coordinate in linear case

Abstract

Electrostatic actuators are simple but important switching devices for MEMS applications. Due to the difficulties associated with the electrostatic nonlinearity, precise mathematical description is often hard to obtain for the dynamics of these actuators. Here we present two sharp theorems concerning the dynamics of an undamped electrostatic actuator with one-degree of freedom, subject to linear and nonlinear elastic forces, respectively. We prove that both situations are characterized by the onset of one-stagnation-point periodic response below a well-defined pull-in voltage and a finite-time touch-down or collapse of the actuator above this pull-in voltage. In the linear-force situation, the stagnation level, pull-in voltage, and pull-in coordinate of the movable electrode may all be determined explicitly, following the recent work of Leus and Elata based on numerics. Furthermore, in the nonlinear-force situation, the stagnation level, pull-in voltage, and pull-in coordinate may be described completely in terms of the electrostatic and mechanical parameters of the model so that they approach those in the linear-force situation monotonically in the zero nonlinear-force limit.