Interdisciplinary Applied Mathematics

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This procedure is fairly accurate only for conservative energy domains. For a more accurate analysis, the inertia and damping terms must be considered. Forces are expressed as appropriate gradients of suitably constructed potential energy or coenergy functions, and these functions are calculated quasi-statically. If one has knowledge of mass distribution, one can assess accelerations and kinetic energy in response to these forces and hence can    construct    complete    dynamic    models    of    the    device    using    only

quasi-static simulations in the potential energy domain.

The steps followed in the quasi-static reduced-order modeling are as follows (Senturia et al., 1997):

1. Select an idealized structure that is close to the desired model.

2. Model the idealized problem analytically, either by solving the governing differential equation, or by approximating the solution with Rayleigh-Ritz energy minimization methods.

3. Identify a set of nondimensionalized numerical constants that can be varied within the analytical form of the solution.

4. Perform meshed numerical simulations of the desired structure over the design space of interest, and adjust the nondimensionalized numerical quantities in the macromodel for agreement with the numerical simulations.

The method has some advantages: (i) simple to use and easy to implement; (ii) reasonably accurate for conservative energy systems when mass distribution is known; (iii) can be used to determine material constants; (iv) even if analytical functions exist for nonlinear behavior, in most cases nonlinearities can be taken care of by a simple fit function. However, the major disadvantage of this method is that it cannot be used in a nonconservative energy system, i.e., when dissipation is involved.

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