# Interdisciplinary Applied Mathematics

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I_I_I…….I_

J_I…….

f (Hz)

f (Hz) …I

л/km

C was found to be Q = 27, and the best frequency for resonance was f0 = 19.2 kHz. The mass parameter was derived from the geometry with density 2300 kg/m3 resulting in m = 50.06 ng. The stiffness was obtained from the mass and measured resonant frequency resulting in k = 0.729 N/m.

In order for this system to be simulated accurately, the damping forces, which are primarily due to fluid motion, i.e., the viscous drag forces, should be computed accurately. A full three-dimensional simulation of this system was performed    for    the    first    time by    (Ye    et    al.,    1999)    using    the    FastStokes

program, which is based on boundary element methods and precorrected FFTs. A total of 23,424 panels were employed in their simulation, as shown in Figure 1.4. For kinematic viscosity of v = 0.145 cm2/sec and density p = 1.225 kg/m3, FastStokes predicted a drag force of 207.58 nN and cor-

FIGURE 1.4. Dimensions and boundary element discretization employed in the comb-drive three-dimensional simulation using the FastStokes program. (Courtesy of W. Ye and J. White.)

responding quality factor Q = 29.1. These predictions are very close to the experimental values. Simple steady or unsteady models based on Cou-ette flow (see Chapter 3) overpredict the Q factor by almost 100%. The estimated Knudsen number in this case was Kn « 0.03, and the Reynolds number was Re « 0.02, so rarefaction and nonlinearities were apparently second-order effects compared to very strong three-dimensional effects. The complete simulation of this problem requires models for the electrostatic driving force, flow models as above, and also mechanical models to investigate possible vibrational effects because of the very small width of the moving fingers. This mixed domain simulation requirement for the comb-drive is representative in the field of MEMS (see Section 1.7 and Chapters 17 and 18 for issues in full-system simulation of MEMS devices).

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