Static and Dynamic Instabilities of Electrostatic Actuated MEMS Devices

Fast and accurate characterization of stability regions and operational range with respect to pull-in voltage and displacement is critical in the design and development of MEMS resonators and switches. This paper presents a mathematical and computational procedure for modeling and analysis of static and dynamic instabilities of capacitive microdevices employing resonant microbeams. The mathematical model consists of a nonlinear microbeam under distributed electrostatic actuation and squeeze film damping. The coupled system is described by the nonlinear beam equation and a modified compressible Reynolds equation to account for the rarefied gas in the narrow gap between the microbeam and substrate. The Differential Quadrature Method (DQM) is used to discretize partial differential equations of motion and solve for static deflection, natural frequencies, static pull-in voltage, and quality factors for various encapsulation air pressures and applied DC voltages. A reduced order model is employed to study the dynamics of the microbeam under combined DC and AC voltages. It is shown that there is possible to trigger pull-in at a DC voltage much lower than static pull-in voltage. This combination can be used to minimize driving voltage of MEMS switches; however, it has to be avoided in the case of MEMS resonators to prevent the failure of the device.