# Interdisciplinary Applied Mathematics

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drag is caused by the lower surface. However, that analysis did not include slip effects, which may modify the viscous drag contribution.

A similar type of gas microflow occurs in another classic MEMS device, the electrostatic comb microdrive (Tang et al., 1989), which is shown in Figure 1.2. Electrostatic comb-drives are excellent resonant actuators that produce large motions at low drive voltage. For typical operating conditions with a resonance frequency of 75 kHz, the dimensions shown in the figure, and with the gap between the stationary and movable comb arms of order 1 pm, we calculate that Re = 0.74 and M = 0.014. Both the micromotor and the comb-drive flows are sustained due to the motion of a thin layer of polysilicon across the silicon substrate. In the simplest form, these flows can be modeled by a shear-driven flow (see Chapter 3).

An electrostatic comb-drive is one of the most important first-generation

4

80 [4 m

Stationary

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FIGURE 1.2. Left: Diagram of a typical electrostatic comb-drive mechanism with typical dimensions. Right: Actual electrostatic comb-drive. The resonator consists of a central    shuttle    that    is suspended by a    cantilever    beam    so    that it    can    move

horizontally. (Courtesy of D. Freeman.)

Movable

Resonance Frequency 75 kHz

MEMS devices. It is driven by interdigitated capacitors, the electrostatic combs. In a standard comb-drive, the capacitance varies linearly with the displacement, resulting in an electrostatic force that is independent of the position of the moving fingers, except at the end of the travel, where the force becomes large (full overlap). An approximate equation for the driving electrostatic force Fe on each of the moving fingers is

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