Modeling of Micromachined Beams Subject to Nonlinear Restoring or Damping Forces

Electrostatic, Casimir, or squeeze-film damping forces are some of the many inverse power-law forces that may affect micromachined devices. The behavior of structures that are subject to such forces, even simple ones such as clamped-clamped or cantilever beams, is difficult to model in a way that is both computationally efficient and physically meaningful. The main contribution of this paper is a semianalytical modeling approach for microbeams that are subject to inverse power-law force densities, which is accurate, computationally efficient, and amenable to physical interpretation. This approach is based on the observation that, for large deformations, inverse power-law forces affect all mathematically equivalent shapes in the same way. Starting from reference shapes for which the analytical expressions of the Galerkin projection of the inverse power-law forces are known, semianalytical expressions are then established for all equivalent shapes, using a simple least squares fitting procedure. In a context of transient simulation, the resulting models are economical because they do not require costly deformation-dependent integral evaluations. Furthermore, they can be used to establish analytical expressions of relevant physical quantities. This approach is validated with simulated and experimental data and compared qualitatively and quantitatively to other reduced-order modeling methods.