Nonlinear dynamic analysis of a V-shaped microcantilever of an atomic force microscope

Abstract

This paper is devoted to investigate the nonlinear behaviors of a V-shaped microcantilever of an atomic force microscope (AFM) operating in its two major modes: amplitude modulation and frequency modulation. The nonlinear behavior of the AFM is due to the nonlinear nature of the AFM tip–sample interaction caused by the Van der Waals attraction/repulsion force. Considering the V-shaped microcantilever as a flexible continuous system, the resonant frequencies, mode shapes, governing nonlinear partial and ordinary differential equations (PDE and ODE) of motion, boundary conditions, frequency and time responses, potential function and phase-plane of the system are obtained analytically. The governing PDE is determined by employing the Hamilton principle. Subsequently, the Galerkin method is utilized to gain the governing nonlinear ODE. Afterward, the resulting ODE is analytically solved by means of some perturbation techniques including the method of multiple scales and the Lindsted–Poincare method. In addition, the effects of different parameters including geometrical one on the frequency response of the system are assessed.