Vibration Identification of Folded-MEMS Comb Drive Resonators

Abstract

Natural frequency and frequency response are two important indicators for the performances of resonant microelectromechanical systems (MEMS) devices. This paper analytically and numerically investigates the vibration identification of the primary resonance of one type of folded-MEMS comb drive resonator. The governing equation of motion, considering structure and electrostatic nonlinearities, is firstly introduced. To overcome the shortcoming of frequency assumption in the literature, an improved theoretical solution procedure combined with the method of multiple scales and the homotopy concept is applied for primary resonance solutions in which frequency shift due to DC voltage is thoroughly considered. Through theoretical predictions and numerical results via the finite difference method and fourth-order Runge-Kutta simulation, we find that the primary frequency response actually includes low and high-energy branches when AC excitation is small enough. As AC excitation increases to a certain value, both branches intersect with each other. Then, based on the variation properties of frequency response branches, hardening and softening bending, and the ideal estimation of dynamic pull-in instability, a zoning diagram depicting extreme vibration amplitude versus DC voltage is then obtained that separates the dynamic response into five regions. Excellent agreements between the theoretical predictions and simulation results illustrate the effectiveness of the analyses.

Bifurcation Analysis on the Periodic Response of a Comb Drive MEMS Resonator

“…V D is the DC voltage, V A is the amplitude of the AC voltage, ω is the driving frequency. Equation ( 1) can be rewritten as [5,26].…”

“…The stability of the periodic solution is determined by the eigenvalues of the Jacobian matrix of Equation ( 8) [26]. Figure 2 shows the response curves given by Equation (17) corresponding to different DC and AC voltages.…”

“…We also present the results of ref. [26] and the numerical results of Equation ( 4) obtained by the fourth-order Runge-Kutta method. To illustrate the results clearly, we give two points P 1 and P 2 in Figure 2b-d, which are the bifurcation points near the peak of the solution curves of this paper and ref.…”

“…Elshurafa et al [5] studied analytically and numerically the spring softening and hardening phenomena of a resonator considering both the transverse and longitudinal capacitance of the combs. According to the variation of softening/hardening characteristics of the resonator, Han et al [26] classified the system response on the plane of the extremely amplitude and DC voltage. Zhong et al [27] investigated the effects of the inclination of the fingers and edge effect on the capacitance, driving electrostatic force, and electrostatic spring constant of a resonator considering nonlinear air damping.…”

“…Of most analytical investigations, the electrostatic force is fitted to a polynomial of the displacement, which inevitably leads to solution errors. In references [5,26,27], the dynamic equation with fractional nonlinearity is solved directly to give a more accurate solution. However, the harmonic function term including the ratio of the AC voltage to DC voltage is ignored in their calculations, so that the conclusions obtained may not reflect the influence of the AC voltage on the response very well.…”