# Interdisciplinary Applied Mathematics

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The partitioning problem can be particularly troublesome, because unlike purely electric circuits, general mechanical structures do not offer a clean mapping between geometry and the corresponding network topology. For example, when considering the electrostatic pull-in of the beam, the moving beam simultaneously serves as the moving boundary of a capacitor used for actuation and sets a moving boundary condition at the bounding surface of a gas film. The lumped parameter values for the physical elements are typically determined from a combination of analysis, numerical simulation, and constitutive properties obtained from test structures. An alternative approach to determine the lumped parameters is to use energy methods. In energy methods, a reasonable shape function with one or more undetermined parameters is assumed, the total stored energy is calculated with that shape function, and the stored energy is then minimized with respect to the parameters. The accuracy of energy methods depends on the quality of the shape function employed and on how well the stored energy

(a)    (b)

FIGURE 17.2. (a) A parallel plate electrostatic actuator. (b) Schematic representation of the parallel plate electrostatic actuator.

calculation is implemented. For a first-order analysis of device behavior, accounting for the dependence of device behavior on the geometry and material properties, energy methods are unmatched in terms of simplicity and speed. A generalized procedure for lumped parameter modeling is as follows:

1. The device is decomposed into a combination of rigid bodies, ideal springs, and ideal variable capacitors. The damping is considered external to the transducer.

2. The governing equations are linearized about an equilibrium signal.

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