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This paper aims to find all static states, including stable and unstable states, of electrostatically actuated microelectromechanical systems (MEMS) device models. We apply the numerical path-following technique to solve for the curve connecting the static states. We demonstrate that device models with 2 DOF can already exhibit symmetry-breaking bifurcations in the curve of static states and can have multiple disjoint solution paths. These features are also found in a finite-element method (FEM) model for a flexible beam suspended by a torsion spring. We have observed multiple hysteresis loops in measurements of a capacitive RF-MEMS device and have captured the qualitative features of these measurements in a model with 5 DOF. Numerical procedures for determining stability of solutions and finding bifurcation points are provided. Numerical path following is shown to be an efficient technique to find the curve of static states both in low-dimensional models and in FEM models.