Parametric resonance voltage response of electrostatically actuated Micro-Electro-Mechanical Systems cantilever resonators

Abstract

This paper investigates the parametric resonance voltage response of nonlinear parametrically actuated Micro-Electro-Mechanical Systems (MEMS) cantilever resonators. A soft AC voltage of frequency near natural frequency is applied between the resonator and a parallel ground plate. This produces an electrostatic force that leads the structure into parametric resonance. The model consists of an Euler–Bernoulli thin cantilever under the actuation of electrostatic force to include fringe effect, and damping force. Two methods of investigation are used, namely the Method of Multiple Scales (MMS) and Reduced Order Model (ROM) method. ROM convergence of the voltage response and the limitation of MMS to small to moderate amplitudes with respect to the gap (gap-amplitudes) are reported. MMS predicts accurately both Hopf supercritical and supercritical bifurcation voltages. However, MMS overestimates the large gap-amplitudes of the resonator, and. misses completely or overestimates the saddle-node bifurcation occurring at large gap-amplitudes. ROM produces valid results for small and/or large gap-amplitudes for a sufficient number of terms (vibration modes). As the voltage is swept up at constant frequency, the resonator maintains zero amplitude until reaches the subcritical Hopf bifurcation voltage where it loses stability and jumps up to large gap-amplitudes, next the gap-amplitude decreases until it reaches the supercritical Hopf bifurcation point, and after that the gap-amplitude remains zero, for the voltage range considered in this work. As the voltage is swept down at constant frequency, the zero gap-amplitude of the resonator starts increasing continuously after reaching the supercritical Hopf bifurcation voltage until it reaches the saddle-node bifurcation voltage when a sudden jump to zero gap-amplitude occurs. Effects of frequency, damping and fringe parameters on the voltage response show that (1) the supercritical Hopf bifurcation is shifted to lower voltage values with the increase of any of the mentioned parameters, (2) the subcritical Hopf bifurcation is shifted to larger voltage values with the increase of damping, shifted to lower voltage values with the increase of the fringe parameter, and not significantly altered by the change in frequency, (3) the saddle-node bifurcation voltage decreases with the increase of frequency and damping, and decrease of fringe parameter, and (4) the saddle-node bifurcation gap-amplitude decreases with the increase of frequency and damping, and it is not significantly altered by the change of the fringe parameter.