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Dynamic analysis of a micro-cantilever beam in non-contact mode: Classic and Strain Gradient theories
Microscopy Research and Technique ( IF 2.5 ) Pub Date : 2021-08-25 , DOI: 10.1002/jemt.23909
Mohammad Ali Mohammadi 1 , Meisam Farajollahi 2 , Aghil Yousefi-Koma 1
Affiliation  

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.

中文翻译:

非接触模式下微悬臂梁的动力学分析:经典和应变梯度理论

本文基于经典和应变梯度连续介质理论,研究了非接触式原子力显微镜中微悬臂梁的动态行为。系统的控制方程是使用 Euler-Bernoulli 和应变梯度理论作为非线性偏微分方程导出的。然后,利用伽辽金法将方程转化为非线性常微分方程,推导出集总模型。研究了范德瓦尔斯斥力项的影响,证明其斥力项与其吸引项相比对系统的动力学没有显着影响。系统的稳定区域和频率响应是通过分析和数值计算和验证的。得到的解析方程用于比较经典和应变梯度系统的频率响应。观察到经典理论预测系统的非线性行为,包括特定维度的软化,而从应变梯度理论的角度来看,系统的动态行为对于相同的维度是线性的。这种差异可能源于经典理论忽略了亚微米尺度的尺寸效应。
更新日期:2021-08-25
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