Static and Dynamic Analysis of Electrostatically Actuated Microcantilevers Using the Spectral Element Method

Abstract

The objective of this paper is to present the Spectral Element Method (SEM) as an accurate and efficient design tool for static and dynamic simulations of cantilever based MEMS devices. The microcantilever under consideration is modeled as a Timoshenko beam and discretized using the spectral element formulation that accounts for fringing field and the nonlinearity arising from the electrostatic driving force. The static analysis has been carried out using Picard’s iteration method and the static pull-in displacement and voltage have been calculated. An eigenvalue analysis of this beam is also carried out to determine its natural frequencies. In addition, the dynamics of this cantilever is studied using the explicit Newmark predictor-corrector method to generate the time history. In all cases, the results have been compared to the one-dimensional Finite Element Method and three-dimensional finite element method (implemented through the commercial package COMSOL Multiphysics) to examine the accuracy and computational speed of the proposed SEM. The results of the simulations were also compared to those obtained by experiments in the existing scientific literature. These comparisons lead to the inference that the SEM is able to reproduce the static and dynamic response of the beam to a high degree of accuracy. It was also found that several numerical features inherent in the SEM lead to a significantly faster computation than the corresponding finite element method for equivalent degrees of freedom. This advantage was verified by using the SEM to carry out static and dynamic simulations of variable width microcantilevers. We therefore propose that the SEM is a viable tool for the MEMS community to accurately and quickly determine the static and dynamic pull-in parameters, frequency eigenvalues, and static and dynamic behavior at the design stage.