Prediction of chaos in electrostatically actuated arch micro-nano resonators: Analytical approach

Chaotic vibrations of a bistable resonator comprised of doubly clamped shallow arch under simultaneous harmonic AC and static DC distributed electrostatic actuation are investigated. A single degree of freedom model obtained by application of the Galerkin decomposition method to the Euler-Bernoulli shallow arch equation is used for the studies. The bistable arch possessing an asymmetric double-well potential is vulnerable to homoclinic chaos at certain values of parameters. We have implemented the Melnikov's method to derive a necessary condition for the initiation of chaos in this type of resonators. Moreover, we have used a heuristic chaos prediction criterion, together with an analytical solution for the arch's frequency response given by the multiple scales method, to derive an analytical expression for prediction of chaos in the forced vibrations of the arch micro/nano-resonators. Numerical simulations confirm validity of the obtained analytical expressions.