Frequency–amplitude response of superharmonic resonance of second order of electrostatically actuated MEMS cantilever resonators

Abstract

This paper deals with the frequency–amplitude response of superharmonic resonance of second order (order two) of electrostatically actuated Micro-Electro-Mechanical System (MEMS) cantilever resonators. The structure of MEMS resonators consists of a cantilever resonator over a parallel ground plate, with a given gap in between, and under AC voltage. This resonance results from hard excitations and AC voltage frequency near one-fourth of the natural frequency of the resonator. The forces acting on the resonator are the nonlinear electrostatic force to include fringe effect, and a linear damping force. In order to solve the dimensionless partial differential equation of motion along with boundary and initial conditions, two types of models are developed, namely Reduced Order Models (ROMs), and Boundary Value Problem (BVP) model. The BVP model is essentially a finite difference model with a discretization in time only. ROMs are developed using one through five modes of vibration. The Method of Multiple Scales (MMS), numerical integrations using MATLAB, as well as a continuation and bifurcation analysis are used to solve the ROMs. The BVP model, resulting from using finite differences for time derivatives, is also numerically integrated. Five modes of vibration ROM is found to make accurate predictions in all amplitudes. A softening effect of the response is predicted. The response consists of a bifurcation with a bifurcation point of amplitude one fourth of the gap, and a stable branch in larger frequencies with a pull-in instability end point at three fourths of the gap. The bifurcation point shifts to lower frequencies as the voltage and/or fringe effect increase, and/or damping decreases. If damping increases, the branches coalesce, peak amplitude decreases, and a linear behavior is experienced.