# Interdisciplinary Applied Mathematics

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The rarefaction effects in (Veijola et al., 1995a), were incorporated into the linearized Reynolds equation (6.16) using an effective viscosity model defined as

pe

И’о

1 + /(Kn) ’

(6.17)

where po    is    the    tabulated    viscosity    coefficient    of    the    gas, and f (Kn)    is

a function    that    models    the    flowrate    changes    with    respect    to the    no-slip

Poiseuille flow. The function f (Kn) is obtained either by asymptotic analytical solutions at a certain Knudsen regime or via a curve fit to the volumetric flowrate of linearized Boltzmann solutions. The flowrate relations used by    Veijola    have    similarities    to    our    unified    flowrate    model    presented

in Section 4.2. The effective viscosity model adopted by Veijola employing data in (Fukui and Kaneko, 1988), is also similar to the model introduced in Section 4.2.2 using the concept of rarefaction coefficient.

Accelerometer transient response APLAC 6.21 User: HUT Circuit Theory Lab. Oct 25 1994

FIGURE 6.9. Simulated accelerometer displacement step response at pressures 30, 300, and 3000 Pa. The parameters are the same as in the frequency domain simulation in Figure 6.8. (Courtesy of T. Veijola.)

Complex geometries are a challenge for accurate numerical modeling of squeezed film damping effects in various microsystems. A simplified rectangular capacitive accelerometer with holes in the middle portion was simulated by Veijola et al., who solved the two-dimensional Reynolds equation with a finite difference method (Veijola et al., 1995b). The ambient pressure conditions were specified in the hole region, where air is free to escape. This approach could be used for engineering modeling purposes, and it gives very reasonable results for thin structures. In the case of thick seismic mass, the hole(s) will act as finite-length flow suction/ejection channels (depending on the motion) with considerable pressure variations through the thick hole region. Figure 6.10 shows the simulated pressure distribution in the air gap of a tilting rectangular accelerometer with a hole in its middle. The tilting of the plate creates asymmetric pressure distribution in the system. The results are obtained by finite differences on a mesh of 22 x 22 grid points.    The effect    of    the    number    of    holes    on    the    steady-state    frequency

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