# Interdisciplinary Applied Mathematics

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Fe =

ehV 2

d

where V is the voltage, h is the height of the fingers (direction perpendicular to the page in Figure 1.2), d is the gap, and e = 8.85 x 10~12 C2/Nm2. This formula    does    not    include    the    effect    of    the    width    of    the fingers,    but

accurate simulations performed in (Shi, 1995) show that the variation is almost linear.    The    magnitude    of    this    force is    very    small;    for    example,    for

h =1 p,m; d = 2.5 p,m, and V = 40 V the above formula gives Fe = 5.7 nN, which is smaller than the more accurate value 6.3 nN obtained with a boundary element simulation in (Shi, 1995).

We now turn to the flow analysis of the comb-drive shown in Figure 1.2, for which detailed measurements were obtained by Freeman using computer microvision (Freeman et al., 1998). Specifically, stroboscopic illumination was used to obtain images at evenly spaced phases during a sinusoidal excitation. The displacements between the images were obtained using algo-

FIGURE 1.3. Magnitude and phase measurements of the comb-drive using computer microvision techniques. (Courtesy of D. Freeman.)

rithms originally developed for machine vision, and they were subsequently integrated to produce a time series for which the magnitude and phase of the motion was determined (see Figure 1.3). The frequency response was fit by a second-order system with mass m, dashpot damping coefficient C, and spring stiffness k. The quality factor defined as

Q =

……..I ……..

In-plane motion

Out-of-plane motion

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