Frequency-Amplitude Response of Subharmonic Resonance of One-Third Order of Electrostatically Actuated MEMS Circular Plates

Abstract

Abstract This work deals with the subharmonic resonance of one-third order of electrostatically actuated clamped MEMS circular plate resonators. The system consists of flexible MEMS circular plate parallel to a ground plate actuated only by AC voltage. Hard excitations due to large enough AC voltage of frequency near three-halves of the natural frequency of the MEMS plate resonator lead it into a subharmonic resonance. The partial differential equation describing the motion of the resonator is nondimensionalized and two reduced order models are developed. The first one consists of a one mode of vibration model which is solved using the Method of Multiple Scales (MMS). The frequency-amplitude response (bifurcation diagram) is predicted. Hard excitations were modeled by keeping the first term of the Taylor polynomial of the electrostatic force as a large term and the rest of them as small terms. The second model uses two modes of vibration, and it is solved through numerical integration. This produces time responses of the resonator. Both methods show a zero-amplitude steady-state stable branch for the entire range of resonant frequencies. Also, two branches, one unstable and one stable, with a saddle node bifurcation point are predicted for non-zero steady state amplitudes. One can notice that non-zero steady state amplitudes can be reached only from large enough initial amplitudes. Both methods are in agreement for amplitudes up to 0.7 of the gap. The effect of damping and voltage on the frequency response are reported. As the damping increases, the saddle-node bifurcation point and consequently the non-zero steady-state branches are shifted to larger amplitudes. As the voltage increases, the saddle-node bifurcation point and the non-zero branches are shifted to lower amplitudes and lower frequency.