Analytical solution for nonlinear vibration of a new arch micro resonator model

Based on the nonlocal strain gradient theory, a new nonlinear dynamic model for an arch micro resonator is developed. The governing nonlinear partial differential equations of the doubly clamped Euler–Bernoulli arch micro resonator are derived by employing Hamilton's principle. Electrostatic actuation and the Casimir force are considered to provide the excitation of the micro resonator. By application of the Galerkin scheme, the nonlinear partial differential equation of dynamic motion of the micro resonator is discretized to the nonlinear ordinary differential equation with quadratic and cubic nonlinearities. Potential energy and bistability are investigated for the electrostatically actuated arch micro resonator. As an analytical approach, the perturbation method is used to examine the nonlinear forced vibration behavior of the resonator. The frequency–response curves are obtained to study the effect of size dependence on the amplitude of forced vibration of the arch micro resonator for the primary resonance. (paper)