纳米圆轴在弹性介质中的扭转振动分析

Torsional Vibration Analysis of Round Nanoshaft in Elastic Medium

基于非局部应变梯度理论,结合Winkler模型,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.讨论了细长度、两个小尺度参数及其比值、弹性介质刚度对振动频率的影响.研究结果表明,扭转自由振动频率随细长度、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.

The torsional vibration of a nanoshaft embedded in an elastic medium is investigated based on the nonlocal strain gradient theory. First, the governing equations and boundary conditions are derived by Hamilton’s principle. Then the discrete forms of governing equations and boundary conditions are obtained using the differential quadrature method. Finally, the torsional vibration characteristics are analyzed through numerical results to evaluate the effects of small-scale parameters and the stiffness of elastic medium on vibration frequency. The coupling effect of two small-scale parameters on vibration frequency is reflected through the influence of the scale parameter ratio. It is found that the frequency of free torsional vibration decreases with the increases of nonlocal parameters, but increases with the increase of strain gradient scale parameter or elastic medium stiffness. When the nonlocal parameter is large, the scale utility is reflected as a nonlocal effect;otherwise, it is reflected as a strain gradient effect.