Response of Microbeam-Based Devices Accounting for the Electrostatic Force and Geometric Nonlinearities

A model of electrically actuated microbeam-based MEMS devices incorporating the nonlinearities associated with moderately large displacements and electrostatic forces is presented. The parallel-plate restriction used to calculate the electrostatic force between the two electrodes is relaxed. Boundary-element method is used to solve for the integral equation describing the capacitance established between the moving and the stationary electrodes and hence the electrostatic force is numerically cmputed. This gives us the ability to change the fixed electrode configuration and to have gaps between its segments. Therefore, the model can handle any capacitor configuration disposing of the complete electrode-overlapping, parallel-plate theory, restriction. The model accounts for the geometric nonlinearity arising from the midplane stretching of the microbeam. The boundary-value problem describing the static deflection of the microbeam under the electrostatic loading is solved numerically. The eigenvalue problem describing the vibration of the microbeam around its statically deflected position is solved numerically for the natural frequencies and mode shapes. Results generated by our model for the parallel-plate case are in agreement with published results. Our results show that the underlying assumption of the closed-form formula of the parallel-plate case underestimates the electrostatic force and leads to an overestimation of the pull-in voltage. The model provides an analytical tool to predict the static and dynamic response of any electrically actuated MEMS device based on clamped-clamped microbeams.