Dynamic analysis of a micro-cantilever beam in non-contact mode: Classic and Strain Gradient theories.

Abstract

This article investigates the dynamic behavior of the micro-cantilever beam in a non-contact atomic force microscope based on classic and Strain Gradient continuum theories. Governing equation of the system is derived using Euler-Bernoulli and Strain Gradient theories as a nonlinear partial differential equation. Then, the equation is converted to a nonlinear ordinary differential equation using the Galerkin method, and the lumped model is derived. The effect of van der Waals repulsive term is investigated, and it is demonstrated that its repulsive term has no considerable effect on the dynamic of the system compared with its attractive term. The stability region of the system and the frequency response are computed and validated analytically and numerically. The obtained analytical equation is used for comparing the frequency response of the classic and Strain Gradient systems. It was observed that the classic theory predicts the nonlinear behavior of the system including softening for a specific dimension, while the dynamic behavior of the system is linear for the same dimensions from the Strain Gradient theory point of view. This difference probably roots in ignoring the size effect in submicron scales by the classic theory. HIGHLIGHTS: The influence of the repulsive term of van der Waals force on the dynamic behavior of micro-beam and this term is investigated. A general closed-form equation is derived for the stability region of AFM by an analytical approach. Effect of size on the stability region is investigated. The frequency response of the system is derived using the multiscale method (MSM) approach.