# Interdisciplinary Applied Mathematics

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(6.16)

_ ffP = dh 12/xe    dt dt

where /ae    is    the    effective    viscosity    coefficient    that models    the    local    (lin

earized) rarefaction effects on the volumetric flowrate QP.

For rectangular surfaces and normal-only motion, equation (6.16) is solved analytically, and the result is formulated as a mechanical admittance. This admittance is approximately implemented as an electric equivalent circuit. The accelerometer model is a combination of this damping circuit and the equivalent circuit of the mass-spring system. Efficient time- and frequency-domain accelerometer simulations can be performed with the circuit simulation program APLAC, similar to SPICE, discussed in Chapter 18. More details of the lumped parameterization and the equivalent electric circuit model of the mechanical structure, including the squeezed gas

cantilever beams electric feedthrough sealed gas chamber

seismic mass

borosilicate glass

FIGURE 6.7. Cross-section of a micromechanical accelerometer. The cantilever beams support the moving mass, which forms two varying capacitances with the thin film electrodes.    The    cross-section    is    not    drawn    to    scale.    (Courtesy    of    T.

Ve ijola.)

film effects are given in Chapter 17. The simulated steady-state frequency-domain response of the system was compared with experimental measurements, and a transient time-domain response to a step acceleration was also obtained. The results point to the following behavior in the frequency domain as a function of the actuation frequency:

• Below a certain frequency identified as the cut-off frequency, the gas has enough time to flow away from the closing gap, inducing dissipation to the system. However, above this frequency the gas is trapped in the gap, and it is squeezed between the moving plates, introducing a springlike behavior with low dissipation.

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