Bernoulli-Euler微梁振动特性的尺寸效应

Size effect of vibration characteristics of Bernoulli-Euler microbeam

基于修正偶应力理论推导任意截面形状Bernoulli-Euler微梁的变形能、弯曲刚度、外力功、动能等基本变量的偶应力理论表达式,进而通过Hamilton原理建立Bernoulli-Euler微梁的偶应力理论动力微分方程。根据偶应力理论弯曲刚度表达式,研究任意截面形状Bernoulli-Euler微梁弯曲刚度的尺寸效应,分析泊松系数和截面形状对Bernoulli-Euler微梁弯曲刚度及其尺寸效应的影响。基于偶应力理论动力微分方程,求解Bernoulli-Euler简支微梁的偶应力理论固有频率,据此研究任意面形状Bernoulli-Euler微梁固有频率的尺寸效应,分析泊松系数和截面形状对Bernoulli-Euler微梁固有频率及其尺寸效应的影响。结果表明,建立的由基本变量偶应力理论表达式和偶应力理论动力微分方程构成的动力学模型能有效地描述任意截面形状Bernoulli-Euler微梁动力学特性的尺寸效应。

Based on the modified couple stress theory,the couple stress theoretical expressions of the basic variables such as deformation energy,bending stiffness,external force work and kinetic energy of Bernoulli-Euler microbeam with arbitrary cross-section shape were derived.Then the couple stress theoretical dynamic differential equation of Bernoulli-Euler microbeam was established according to Hamilton principle.The scale effect of bending stiffness of Bernoulli-Euler microbeam with arbitrary section shape was assessed using the couple stress theoretical expression of the bending stiffness.And the influences of Poisson's coefficient and section shape on the bending stiffness and its scale effect of Bernoulli-Euler microbeam were analyzed.Based on the couple stress theoretical dynamic differential equation,the natural frequency of Bernoulli-Euler simply supported beam was obtained.The scale effect of the natural frequency of Bernoulli-Euler beams with arbitrary cross-section shape was assessed.And the influences of Poisson’s coefficient and cross-section shape on the natural frequency and its scale effect of Bernoulli-Euler beams were analyzed.The results show that the developed dynamic model,which consists of the couple stress theoretical expressions of the aforementioned basic variables and the couple stress theoretical dynamic differential equation,can effectively describe the scale effect of the dynamic characteristics of the Bernoulli-Euler microbeam.