On a Free Boundary Model for Three-Dimensional MEMS with a Hinged Top Plate I: Stationary Case

TL;DR

This paper analyzes a mathematical model of a 3D MEMS device with a hinged top plate, establishing conditions for the existence or nonexistence of stable stationary solutions based on voltage parameters.

Contribution

It introduces a coupled nonlocal fourth-order and Laplace equation model for MEMS with a hinged top plate and determines stability conditions for stationary solutions.

Findings

Stable solutions exist for small voltage differences.

No stationary solutions for large voltage differences.

The model couples deformation and electrostatic potential equations.

Abstract

A stationary free boundary problem modeling a three-dimensional electrostatic MEMS device is investigated. The device is made of a rigid ground plate and an elastic top plate which is hinged at its boundary, the plates being held at different voltages. The model couples a nonlocal fourth-order equation for the deformation of the top plate to a Laplace equation for the electrostatic potential in the free domain between the two plates. The strength of the coupling is tuned by a parameter $\lambda$ which is proportional to the square of the applied voltage difference. Existence of a stable stationary solution is established for small values of $\lambda$. Nonexistence of stationary solutions is obtained when $\lambda$ is large enough.