Mechanical behaviors of electrostatic microresonators with initial offset imperfection: qualitative analysis via time-varying capacitors

Abstract

This paper investigates analytically and numerically the effect of initial offset imperfection on the mechanical behaviors of microbeam-based resonators. Symmetry breaking of DC actuation, due to different initial offset distances of microbeam to lower and upper electrodes, is concerned. For qualitative analysis, time-varying capacitors are introduced and a lumped parameter model, considering nonlinear electrostatic force and midplane stretching of microbeam, is adopted to examine the system statics and dynamics. The Method of Multiple Scales (MMS) is applied to determine the primary resonance solution under small vibration assumption. Meanwhile, the Finite Difference Method (FDM) combined with Floquet theory is utilized to generate frequency response curves for medium- and large-amplitude vibration simulations. Static bifurcation, phase portrait and Hamiltonian function are firstly investigated to examine the system inherent behaviors. Besides, basins of attraction are briefly depicted to grasp the effects of initial offset and AC excitation on the system global dynamics. Then, variation of equivalent natural frequency versus DC voltage is analyzed. Results show that initial offset may induce complex frequency rebound phenomenon as well as a separate frequency branch under secondary pull-in condition. In what follows, emergences of softening, linear and hardening vibration are classified through discussing a key parameter obtained from the frequency response equation. New linear behavior induced by initial offset imperfection is found, which exhibits much higher sensitivity to DC voltage. Medium- and large-amplitude in-well motions are also investigated, indicating the existence of alternations of softening and hardening behaviors. Finally, lumped parameters are deduced via Galerkin procedure, and case studies are provided to illustrate the effectiveness of the whole analysis.