Dynamic analysis of a micro-resonator driven by electrostatic combs

Abstract

The dynamic response of a micro-resonator driven by electrostatic combs is investigated in this work. The micro-resonator is assumed to consist of eight flexible beams and three rigid bodies. The nonlinear partial differential equations that govern the motions of the flexible beams are obtained, as well as their boundary and matching conditions. The natural matching conditions for the flexible beams are the governing equations for the rigid bodies. The undamped natural frequencies and mode shapes of the linearized model of the micro-resonator are determined, and the orthogonality relation of the undamped global mode shapes is established. The modified Newton iterative method is used to simultaneously solve for the frequency equation and identify repeated natural frequencies that can occur in the micro-resonator and their multiplicities. The Gram-Schmidt orthogonalization method is extended to orthogonalize the mode shapes of the continuous system corresponding to the repeated natural frequencies. The undamped global mode shapes are used to spatially discretize the nonlinear partial differential equations of the micro-resonator. The simulation results show that the geometric nonlinearities of the flexible beams can have a significant effect on the dynamic response of the micro-resonator.