This paper deals with the amplitude voltage response of electrostatically actuated MEMS cantilever resonators undergoing superharmonic resonance of fourth order. This can be used as sensing mechanism. The system consists of a MEMS cantilever beam held parallel to a ground plate with an applied voltage of alternating current (AC) causing the cantilever to vibrate. The driving frequency of the excitation voltage is near one eighth of the first natural frequency of the cantilever. This causes the cantilever to experience superharmonic resonance of order four. In order for this resonance to occur hard excitations are required wherein the magnitude of the excitation voltage must be large enough. This work models the electrostatic force to include fringe effect. The fringe effect is modeled using Palmer’s formula. Reduced order models (ROMs) are used in this work. The methods used to solve these models are 1) the method of multiple scales (MMS), 2) homotopy analysis method (HAM), and 3) numerical integration for ROM with 2 modes of vibration. The amplitude voltage response shows a softening. The response consists of three branches: two stable and one unstable. As the voltage is increased the system is stable until the first saddle-node bifurcation point is reached. Here the system experiences instability and it jumps to higher amplitude on the stable branch. As the voltage is swept down the system is stable until the second saddle-node bifurcation point in high amplitudes is reached and the system jumps down to lower amplitudes on the first stable branch. This is the biosensing mechanism proposed in this work. All three methods show excellent agreement with one another for detuning frequency values up to σ = −0.025. As the magnitude of the detuning frequency increases the MMS and HAM begin to disagree with the time responses obtained from the numerical integration of the ROM with 2 modes of vibration (or terms).
This demonstrates the limitations of MMS and HAM to accurately predict the behavior for hard excitations where the voltage is very high.