Electrically Actuated Microbeams: An Explicit Calculation of the Coulomb Integral in the Entire Stable and Unstable Regimes Using a Chebyshev-Edgeworth Approach

In this paper we present an analytical model for large displacements of electrostatically actuated elastic microbeams, including all their stable and unstable states. Such microbeams are the core element of microelectromechanical system-based resonators, transducers, and actuators. Their accurate modeling is essential to their development. Especially in the realm of system design, there is a need for lumped-parameter models with a single degree of freedom, where a large number of such subsystems can interact with each other. The state of the art suggests using zero-mode approximations far from the pull-in limit or using multi-degree-of-freedom Galerkin projection to get the results close to the pull-in limit. The challenges of finding a closed-form solution lie in the complicated structure of the Coulomb integral. Our approach to evaluating this integral is based on a Chebyshev-Edgeworth-type method common in analytic probability theory and allows the states to be accurately modeled where the kernel is close to the singularity. Although this approach to nonlinear dynamical systems is used here for a very specific purpose, where we report a user-friendly closed solution of a clamped-clamped microbeam, it has a much broader scope. It is apt to analyze different boundary conditions, electrostatic fringe field corrections, and squeeze film damping, to name a few applications. Moreover, the method presented here can generally be applied to the evaluation of singular integral kernels.