Comprehensive Reduced-Order Models of Electrostatically Actuated MEMS Switches and Their Dynamics Including Impact and Bounce

Abstract

As MEMS technology develops it is becoming better understood that MEMS designers must account for the large uncertainties characteristic of the relevant manufacturing processes. Uncertainty quantification tasks the designer with evaluating many different possible outcomes from the manufacturing process which creates a demand for models that are accurate and comprehensive, yet fast to evaluate. This work presents a comprehensive reduced-order model of electrostatically actuated switches incorporating a range of effects that are typically included only in FE modeling codes. Specifically, the model accounts for variable electrode geometry, stretching of centerline or large displacement effects, fringing field, squeeze film and rarefied gas damping, and allows for elastic contact with the dielectric substrate. Individual compact models for each of these effects are taken from literature and included in the model for the system. The dielectric substrate is modeled as an elastic foundation. The resulting partial differential equation for the switch modeled as a beam is discritized via a Galerkin method into ordinary differential equations for modal amplitudes. The Galerkin method uses the linear un-damped mode shapes of the beam to approximate the solution. Both cantilever and fixed-fixed type switches are analyzed. Static equilibrium solutions as a function of the applied voltage are developed along with their stability. Static pull-in voltages, first time of switch closure, and voltage for lift-off are studied with the model.

A comparative study of reduced-order modeling techniques for nonlinear MEMS beams

“…As the TS approach, this increasingly popular technique (see [6][7][8] for example) has the advantage of turning a non-polynomial nonlinearity into a polynomial one and has full-gap range of validity, provided N is sufficiently large. However, for a number of reasons, most authors using this technique limit themselves to the N=1 case, as in [7].…”

An equation-based nonlinear model for non-flat MEMS fixed–fixed beams with non-vertical anchoring supports

“…The approach based on solving the entire beam and support structure by FEM simulations [20] can take into account the shape of the support, but it is computationally expensive. Another approach used a torsional stiffness as boundary conditions to model the rotational compliance but neglected the translational and axial compliance of the supports [6,18,[21][22][23]. The boundary torsional stiffness was assumed constant and independent of load and residual stress.…”

Residual Stress Extraction of Surface-Micromachined Fixed-Fixed Nickel Beams Using a Wafer-Scale Technique

“…With a residual stress σ 0 , the total axial force is P x = P w + σ 0 bh. To adapt to the Galerkin method for solving the beam model [33], [39], a boundary torsional stiffness k T is defined to present the rotational compliance of the supports which may vary with the beam displacement and residual stress:…”