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As indicated by (3.19), the spring-mass system embodied by an unbiased clamped-clamped beam resonator exhibits a different frequency when subjected to an applied bias VP. This implies that its equivalent mass-spring- damper is bias-dependent.

The corresponding bias-dependent mechanical equivalent was derived by Bannon et al. [46] in two steps: first, the unbiased parameters were derived; and second, the effect of bias was applied. With respect to Figure 3.34, the equivalent resonator mass at the location y along its length is given by the ratio of the peak kinetic energy to one-half the square of the velocity, both at y [49]: Equations (3.20) to (3.23) give the mechanical parameters for the unbiased resonator.

Since the overall effect on resonance frequency of an applied dc bias is captured by (3.20), it suffices to determine the electrical spring stiffness, or the function g, in order to obtain the characterization of the biased resonator. Bannon et al. [46] defined the electrical spring stiffness ke as resulting from the nonlinear dependence of the beam-to-electrode capacitance Cn(x) on the displacement x. where d(y¢) denotes the beam-to-electrode gap resulting from the application of the dc bias VP .

By assuming that the beam deformation resulting from the polarization voltage is identical to the fundamental mode, the gap distance is approximated as [46] where d0 is the beam-to-electrode gap at VP = 0. An examination of (3.25) reveals that it contains d(y) on both sides, so it must be solved iteratively, beginning by assuming d(y) = d0 on the right-hand side until convergence is achieved [46].

Finally, the quantity g(d,VP), which embodies the effect of the electrical stiffness, is given by An excited beam resonator may exhibit nonlinear response due to two aspects: the nonlinear nature of its gap capacitance, and material nonlinearity due to the violation of Hooke?s law. Thus, it is important to have an idea of the maximum displacement at resonance. Accordingly, at resonance the displacement at a point y along the beam is Q times the displacement under nonresonance conditions;

it is given by where keff (y) is the effective stiffness at location y, given by [46] To calibrate our intuition, it should be noted that a MEM resonator of 8-mm-width, 40.8-mm-length, 1.98-mm-thickness, Young?s modulus of 150 GPa, mass density of 2,300 kg/m3, gap 1,300Ã…, and an ac input voltage of ni = 3mV together with a dc bias voltage nP = 10V, will exhibit a vibration amplitude of 49Ã… at the beam center. It shows the transmission characteristic of a clamped-clamped resonator.