Nonlinear dynamics of nano-resonators: an analytical approach

Prior to the design and fabrication of MEMS/NEMS devices, analysis of static and dynamic behaviors of these systems is necessary. In the present study, the nonlinear dynamic behavior of micro- and nano-mechanical resonators is investigated and classified based on the resonator’s physical parameters for first time. The Galerkin method is used to convert the distributed-parameter model to a nonlinear ordinary differential equation where mid-plane stretching, axial stress, DC electrostatic and AC harmonic voltages are taken into account. To obtain the analytical frequency response of the micro resonator near its primary resonance, the second order multiple scales method is applied to the general equation of motion with cubic, quadratic and parametric nonlinearities. It is demonstrated that variation of the micro resonator’s physical parameters strongly affects its dynamic behavior by changing equilibrium points and their stability properties and complex behaviors appear in its frequency and phase responses. The global dynamics of the resonator is classified into four different categories in terms of the system parameters in this paper. The dynamic characteristics and frequency response of each class are analyzed numerically as well as analytically. Comparison of the obtained closed-form solution with the numerical simulation results confirms its validity. A striking point of the obtained closed-form solution is that it predicts some complex nonlinear behaviors of the resonator. This paper presents a quick and efficient method for determining the global dynamics of the micro-resonators and can be useful in design and analyses of these devices.