A closed-form approach for the resonant frequency analysis of clamped rectangular microplates under distributed electrostatic force

This paper proposes a fast-converged reduced-order model for small vibrations of electrostatically actuated rectangular microplates around their deformed states. On this foundation, one-mode analysis method and thus closed-form analytical expressions are developed for the fundamental resonant frequency analysis. Numerical multi-mode analysis is conducted to investigate the convergence. It is concluded that the one-mode analysis method can give a converged solution, which demonstrates that the proposed model has a faster convergence than previous models where multi modes are needed. The directly calculated resonant frequencies by the closed-form expressions are in good agreement with the numerical results by FEM simulations with less than 5% variation before pull-in. Parametric study shows the closed-form expressions are applicable to the cases in which the thickness is less than 1/20 of the length and width under a nominal length to width ratio of 1∼2.5, and the gap distance is less than or equal to the thickness of the microplate. Additionally, a kind of electrostatic configurations based on multilayer microplates, capacitive micromachined ultrasonic transducers (CMUTs), were used to experimentally validate the closed-form expressions and good agreement was observed.