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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.

An analytical model and a numerical procedure to simulate coupled energy domains in capacitive microdevices are presented. The model consists of a flexible microplate under the effect of electrostatic forces and squeeze film damping. The coupled system is described by the linear plate equation and a modified compressible Reynolds equation to account for the rarefied gas in the narrow gap between the microplate and substrate. A numerical method based on Differential Quadrature Method (DQM) is employed to discretize and solve the coupled differential equations of motion for complex eigenvalues, mode shapes, and quality factors. The simulation results are compared to the experimental data available in the literature. The analysis highlights the effect of air pressure on quality factors and natural frequencies of the coupled system.