Interdisciplinary Applied Mathematics

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18.2.2 Galerkin Methods

Linear modes of vibration of a system have been used for reduced-order modeling of MEM dynamics as discussed in Chapter 17. In this section, squeezed film damping has been considered in such a model-order reduction. The fluid (film) in between the plates typically undergoes Stokes flow (low Reynolds number, hence negligible inertia effects) and thus does not have any “normal modes” of its own that could be used for basis functions in combination with the elastic modes. One obvious approach is to linearize the dissipative effect under an assumption of small motion. Once linearized, frequency-domain analysis can be used, converting the time-dependent dissipation problem in the time domain into a time-independent frequency-domain calculation of amplitude and phase response. This approach was the basis for the early squeezed film damping work involving rigid body motion and has been widely used in the MEMS field. Even when the moving body is flexible, it is possible to use the modal amplitude to create a

moving boundary condition for the fluidic system and calculate the reaction force. This has been done for small-amplitude damped resonant motions of flexible microbeams and resonators. When the amplitudes are large, such as for the electrostatic pull-in of a beam, linearized modal solutions are not accurate. In (Mehner et al., 2003), an approach has been presented to add dissipative effects of squeezed film damping (Reynolds equation) in the transient and harmonic analysis of MEMS. The macromodels are automatically generated by a modal projection technique based on the harmonic transfer functions of the fluidic domain. The transfer functions are either obtained at    the    initial    position    (small    signal    case)    or    at    various    deflec

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