Finite-amplitude motions of beam resonators and their stability

We study the motions of a beam resonator using a nonlinear beam model encompassing a capacitive electrostatic force, the restoring force of the beam, and an axial load applied to the beam. A perturbation method, the method of multiple scales, is applied to this distributed-parameter system to produce analytical expressions describing small but finite-amplitude motions of the resonator under primary, superharmonic, and subharmonic excitations. In each case, we obtain two first-order nonlinear ordinary-differential equations that describe the modulation of the amplitude and phase of the response and its stability, and hence the bifurcations of the response. The resulting expressions provide an analytical tool to predict the resonator response to primary, superharmonic, and subharmonic excitations, including the locations of sudden jumps and regions of hysteretic behavior and hence allowing designers of resonant micro- and nano-sensors and RF filters to study the sensitivity and optimize the design parameters of these devices.