Nonlinear dynamics of MEMS/NEMS resonators: analytical solution by the homotopy analysis method

Abstract

Due to various sources of nonlinearities, micro/nano-electro-mechanical-system (MEMS/NEMS) resonators present highly nonlinear behaviors including softening- or hardening-type frequency responses, bistability, chaos, etc. The general Duffing equation with quadratic and cubic nonlinearities serves as a characterizing model for a wide class of MEMS/NEMS resonators as well as lots of other engineering and physical systems. In this paper, after brief reviewing of various sources of nonlinearities in micro/nano-resonators and discussing how they contribute to the Duffing-type nonlinearities, we propose a Homotopy Analysis Method (HAM) approach for derivation of analytical solutions for the frequency response of the resonators. Toward this aim, we first apply the HAM to the proposed Duffing equation, and through this procedure, we derive the first-order and second-order HAM-based analytical solutions for the frequency response of the resonator. As the main novelty, we show that the second-order solution benefits from a tunable parameter, known as the convergence-control parameter, which is a distinguishing aspect of the HAM and plays a key role in enhancing the accuracy of the obtained analytical expressions in strongly nonlinear problems. We use the obtained analytical solutions for the study of nonlinear dynamics in two types of electrostatically actuated MEMS resonators proposing hardening, softening or mixed behaviors near their primary resonance frequency. Numerical simulations are performed to validate the analytical results.