Static response and free vibration of MEMS arches assuming out-of-plane actuation pattern

In this research work, the nonlinear structural behavior of a non-parallel plates micro-actuator design is investigated via a reduced-order modeling. The micro-actuator is considered to be made of an initially flexible curved doubly-clamped microbeam and two evenly arranged stationary rectangular shaped out-of-plane electrodes of different lengths. The subsequent actuating attractive electrostatic force is mainly formed by the unevenness of the electric fringing-fields (non-parallel) electrodes arrangement. Results of this numerical investigation show that by increasing the actuation voltage the shallow arch behavior tend to reach a softening like behavior, mainly due to its dominant flexibility effect. With a further increase in the actuation voltage, the micro-arch starts to stretch and thus develops a hardening like behavior. Furthermore, and for certain values of the design parameters (mainly the shape and the length of the stationary electrodes), the shallow arch presented a snap-through like bi-stability behavior. Indeed, when the voltage approaches a certain critical value, the micro-arch static profile alters from a symmetrical shape to an asymmetric one, thus showing a symmetry breaking like behavior that depends a lot on the shape and length of the stationary non-parallel actuating electrodes. It is also demonstrated that by varying few of the micro-actuator design parameters, the variation of the normalized fundamental frequency showed rises and declines for certain ranges of the applied bias voltage. This approves that with such electrostatically actuated micro-actuator design, and with a smart tuning of its geometrical design parameters, one can obtain softening as well as hardening like behaviors. Modes frequency variations curves showed also that for certain actuation DC loads, it is possible to achieve a one-to-one internal resonance state involving both the first symmetric and the first antisymmetric modes of the shallow micro-arch.