Instabilites in Arch Shaped MEMS

Arch shaped microelectromechanical systems (MEMS) have been used as mechanical memories, micro-relays, micro-valves, optical switches, and digital micro-mirrors. A bi-stable structure is characterized by a multivalued load deflection curve. Here, the symmetry breaking, the snap-through instability, and the pullin instability of a sinusoidal shaped MEMS under static and dynamic electric loads have been studied. The electric load is a nonlinear function of the a priori unknown deformed shape of the arch, and is thus a follower type load. The nonlinear partial differential equation governing transient deformations of the arch is solved numerically using the Galerkin method and the resulting ordinary differential equations are integrated by using the Livermore solver for ordinary differential equations. For the static problem, the displacement control and the pseudo-arc length continuation methods are used to obtain the bifurcation curve of the MEMS displacement versus a load parameter. The displacement control method fails to compute asymmetric deformations of the MEMS, which are found by the pseudo-arc-length continuation method. Two distinct mechanisms of the snap-through instability for the dynamic problem are demonstrated. It is found that critical loads and geometric parameters for instabilities of an arch under an electric load with and without the consideration of mechanical inertia effects are quite different.