纳米圆轴在弹性介质中的扭转振动分析

基于非局部应变梯度理论，考虑周围弹性介质的影响，研究纳米圆轴的扭转自由振动。首先通过Hamilton原理推导纳米圆轴扭转振动的控制方程及边界条件，接着采用微分求积法得到控制方程及三类边界条件的离散形式，最后由数值计算结果分析扭转振动特性。讨论了两个小尺度参数和弹性介质刚度的变化对振动频率的影响，并通过小尺度参数比对振动频率的影响分析两个尺度参数的耦合作用。研究结果表明，扭转自由振动频率随非局部参数增加而减小，随应变梯度尺度参数、弹性介质刚度增加而增大；当非局部参数大于应变梯度尺度参数时，小尺度效应体现为非局部效应，相反则体现为应变梯度效应。

Torsional Vibration Analysis of Nano-Round-Shaft in Elastic Medium

The torsional free vibration of the nano-shaft embedded in an elastic medium is investigated based on the nonlocal strain gradient theory. Firstly, the governing equations and boundary conditions are derived by Hamilton principle. Then the discrete form of the governing equations and boundary conditions are obtained by the differential quadrature method. Finally the torsional vibration characteristics are analyzed from the numerical results. The influence of two small-scale parameters and the stiffness of elastic media on vibration frequency are discussed. The coupling effect of the two scale parameters on vibration frequency is reflected through the influence of the scale parameter ratio. The results show that the torsional free vibration frequency decreases due to the increase of non-local parameters, and increases when the strain gradient scale parameter or elastic medium stiffness increase. When the non-local parameter is larger, the scale utility is reflected as a non-local effect , on the contrary, it is reflected as a strain gradient effect.