Vibrations and pull-in instabilities of microelectromechanical von Kármán elliptic plates incorporating the Casimir force

We consider the von Kármán nonlinearity and the Casimir force to first develop a reduced-order model for a prestressed clamped elliptic electrostatically actuated microplate, and then use it to study vibrations and pull-in instability. The reduced-order model is derived by taking a family of linearly independent kinematically admissible functions as basis functions for the transverse displacement. The in-plane displacement vector is expressed as the sum of displacements for irrotational and isochoric waves in a two-dimensional medium. The potentials of these two displacement vector fields satisfy an eigenvalue problem analogous to that of transverse vibrations of a linear elastic membrane. Basis functions for the transverse and the in-plane displacements are related by using the nonlinear equation governing the plate's in-plane motion. The reduced-order model is derived from the equation governing the transverse deflection of the plate. Pull-in parameters are found using the displacement iteration pull-in extraction method and by studying small vibrations of the plate about its predeformed configuration. However, the effect of inertia forces on pull-in parameters has not been analyzed. The reduced-order model for a linear elliptic micromembrane is derived as a special case of that for an elliptic plate.