Frequency-Amplitude Response of Parametric Resonance of Electrostatically Actuated MEMS Cantilever Beams Driven by Fringe Effect

Abstract

Abstract This work deals with the frequency-amplitude response of parametric resonance of electrostatically actuated micro-electro-mechanical-system (MEMS) cantilever beam resonators driven by the fringe effect. The system involves a flexible MEMS cantilever beam parallel to a ground plate. The electrostatic force induced is due to the electric field between the cantilever beam and the ground plate (volume between cantilever and ground plate) and the electric field outside this volume which leads to the fringe effect. In this work, the cantilever is driven only by the fringe effect. The electrostatic force due to the electric field within the volume between cantilever and ground plate is neglected due to hole in the ground plate (size of cantilever). Excitations due to the fringe effect near the natural frequency of the MEMS cantilever beam lead the MEMS cantilever resonator into parametric resonance. The partial differential equation describing the motion of the cantilever resonator is nondimensionalized and a reduced order model is developed. This is a one-mode of vibration model which is solved using the Method of Multiple Scales (MMS). The frequency-amplitude response (bifurcation diagram) is predicted. The fringe effect was modeled by neglecting the electrostatic force term, and keeping the first four terms of the Taylor polynomial of the fringe effect (small terms due to small bookkeeping parameter). The method shows a zero-amplitude steady-state stable branch (trivial solution) for the entire range of resonant frequencies. In addition, two branches, one stable and one unstable, with two bifurcation points, subcritical and supercritical, are predicted. The response due to the fringe effect is compared to responses that involve the electrostatic force only as well as electrostatic and fringe combined. The influence of damping and voltage on the frequency-amplitude response are also investigated. As the damping increases, both non-zero steady-state branches, are shifted to higher frequencies and larger amplitudes. On the other hand, as the voltage increases the two non-zero branches are shifted to lower frequencies and smaller amplitudes.