Inclination Effects on the Frequency Tuning of Comb-Driven Resonators

The comb fingers of high aspect ratio structures fabricated by micromachining technology are usually not parallel. Effects of the inclination of the fingers and edge effect on the capacitance, driving electrostatic force, and electrostatic spring constant are studied. The complex nonlinear air damping in the 3-D resonators is also determined accurately. The governing equations are presented to describe the complex dynamic problem by taking both linear and nonlinear mechanical spring stiffness constants into account. The dynamic responses of the micro-resonator driven by electrostatic combs are investigated using the multiscale method. Stability analysis is presented using the maximum Lyapunov index map, and effects of vacuum pressure on the frequency tuning and stability are also discussed. The comparisons show that the numerical results agree well with the experimental data reported in the literature, and it verified the validity of the presented dynamic model. The results also demonstrate that the inclination of the fingers causes the resonance frequency to increase and the electrostatic spring to harden under applied dc voltage. Therefore, it can provide an effective approach to balance the traditional resonance frequency decreasing and stiffness softening from driving electrostatic force. The inclination of the fingers can be helpful for strengthening the stability of the MEMS resonators, and avoiding the occurrence of pull-in.