Analytical Solution for the Forced Vibrations of a Nano-Resonator with Cubic Nonlinearities Using Homotopy Analysis Method

Document Type : Research Paper


1 School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, I. R. Iran

2 Mechanical Engineering Department, Sharif University of Technology, Tehran, I. R. Iran.

3 Department of Mechanical Engineering, King Fahd University of Petroleum and Minerals, 31261, Dhahran, Kingdom of Saudi Arabia, Saudi Arabia


Many of nonlinear systems in the field of engineering such as nano-resonator and atomic force microscope can be modeled based on Duffing equation. Analytical frequency response of this system helps us analyze different interesting nonlinear behaviors appearing in its response due to its rich dynamics. In this paper, the general form of Duffing equation with cubic nonlinearity as well as parametric excitations is considered and its frequency response is derived utilizing Homotopy Analysis Method (HAM) for the first time. Although time response of different Duffing systems has been analyzed using HAM, derivation of its frequency response equation by applying this powerful method has not been presented. The main advantage of proposed simple closed-form solution is that it is not restricted to weakly nonlinear systems in contrast with perturbation methods. Because of numerous applications of Micro-electro-mechanical resonator and its rich and nonlinear dynamics, it is considered as a case study in this paper and the obtained analytical equation is applied to find its frequency response. The validation of analytical method is verified by comparing the results with numerical simulations. It is also shown that proposed closed-form equation for nano-resonator frequency response can capture both hardening and softening behavior of nano-resonator as well as jump phenomenon. The results of this paper can be useful in analysis of different engineering systems modeled by general Duffing equation.