基于非局部理论的黏弹性纳米杆轴向振动与波传播研究

根据非局部理论和Kelvin黏弹性理论,针对黏弹性纳米杆自由振动和波传播的轴向动力学问题进行研究.首先,推导了黏弹性纳米杆的轴向动力学微分控制方程.然后,通过无量纲化讨论了3种典型边界纳米杆的前三阶振动特性.最后,研究黏弹性纳米杆波的传播问题,导出了圆频率、波速与波数之间的关系.数值结果表明,非局部效应使第一、二阶固有频率持续减小,第三阶频率先增大再减小,出现结构刚度削弱和增强两种趋势.特别地,对于自由端存在集中质量的情形,第二阶频率随着黏性系数增大出现了多值情况,易导致杆件失稳.数值算例还说明了非局部效应的增强可有效降低黏性材料的阻尼效应,产生逃逸频率,使得纵波能够在高波数段传播.另外,黏性系数在低波数段对阻尼比影响可忽略不计,而在高波数段下,黏性系数越大则阻尼比越大.

Longitudinal Vibration and Wave Propagation of Viscoelastic Nanorods Based on the Nonlocal Theory

The longitudinal dynamics of viscoelastic nanorods was investigated based on the nonlocal theory and the Kelvin viscoelastic theory, including axial free vibration and wave propagation. Firstly, the partial differential governing equations were derived and then the 1st 3 vibration properties were discussed under 3 kinds of typical boundary conditions with the dimensionless method. Finally, the relationships between the circular frequency, the wave speed and the wave number were obtained in the problem of wave propagation. The numerical results show that, the small-scale effect makes the 1st and 2nd frequencies decrease persistently and the 3rd frequency increase first and decrease later, which indicates that the nanostructural stiffness is weakened or strengthened. In particular, for a concentrated mass at the free end of the nanorod, the 2nd frequency has multiple values when the viscoelastic coefficient increases, which may cause instability. The numerical examples also prove that stronger nonlocal effect brings lower damping effect of viscoelastic materials. The longitudinal wave can propagate at high wave numbers due to occurrence of the escape frequency. The effects of viscoelastic coefficients on the damping ratio may be ignored at low wave numbers, however, be significant at high wave numbers.