# Interdisciplinary Applied Mathematics

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Similarly, the micromotors rotating on a substrate or microcomb drives (discussed in Section 1.1) oscillating with very high frequencies also experience gas damping effects. Although the primary motion in such devices is lateral, depending on the allowable degrees of freedom, up and down motion and tilting in the pitch and yaw angles are also possible (Freeman et al., 1998). Also, MEMS torsion mirrors with high frequency pitch oscillations experience significant squeezed film effects (Pan et al., 1998). In this section, we will address the fluid-mechanical aspects of various types of motion experienced by moving MEMS components, which can be summarized as one or a combination of the following:

1. Lateral motion between two surfaces: oscillatory Couette flow (see Section 3.3).

2. Normal oscillations between two surfaces: squeezed film.

FIGURE 6.2. Dynamic response of the DMD mirrors subject to a step pulse under various air pressures. (The data were obtained by Dr. Larry Hornbeck (1988); Courtesy of Texas Instruments.)

3. Lateral motion with slight inclination: slider bearing.

4. Pitch-angle oscillatory motion between two plates: squeezed film.

All of these situations, with the exception of the first case, can be analyzed using the Reynolds equation for thin-film lubrication presented below.

The first case corresponds to fluid motion between oscillating parallel plates. Detailed analysis of this motion and the associated damping characteristics in the entire Knudsen regime were presented in Section 3.3. In oscillatory    Couette    flows,    the    lateral    plate    motion    does    not    generate

streamwise pressure variations. Hence, this case cannot be described by the Reynolds equation. In contrast to the squeezed film and the slider bearing problems, there will be no lift. We identify this case as the oscillatory Couette flow, which is a variation of the classical Stokes second problem (Batchelor, 1998) (flow over a laterally oscillating surface) with inclusion of a wall    over    a    thin fluid gap    of    ho.    The    lateral vibrations    of    comb-drives

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