Hu Weipeng, Song Mingzhe, Deng Zichen, Zou Hailin, Wei Bingqing
School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnical University, Xi'an, Shaanxi 710072, People's Republic of China.
State Key Laboratory of Solidification Processing, Northwestern Polytechnical University, Xi'an, Shaanxi 710072, People's Republic of China.
Chaos. 2017 Feb;27(2):023118. doi: 10.1063/1.4977193.
The occurrence of chaos in the transverse oscillation of the carbon nanotube in all of the precise micro-nano mechanical systems has a strong impact on the stability and the precision of the micro-nano systems, the conditions of which are related with the boundary restraints of the carbon nanotube. To generalize some transverse oscillation problems of the carbon nanotube studied in current references, the elastic restraints at both ends of the single-walled carbon nanotube are considered by means of rotational and translational springs to investigate the effects of the boundary restraints on the chaotic properties of the carbon nanotube in this paper. Based on the generalized multi-symplectic theory, both the generalized multi-symplectic formulations for the governing equation describing the transverse oscillation of the single-walled carbon nanotube subjected to the transverse load and the constraint equations resulting from the elastic restraints are presented firstly. Then, the structure-preserving scheme with discrete constraint equations is constructed to simulate the transverse oscillation process of the carbon nanotube. Finally, the chaotic region of the carbon nanotube is captured, and the oscillations of the two extreme cases (including simply supported and cantilever) are investigated in the numerical investigations. From the numerical results, it can be concluded that the relative bending stiffness coefficient and the absolute bending stiffness coefficients at both ends of the carbon nanotube are two important factors that affect the chaotic region of the carbon nanotube, which provides guidance on the design and manufacture of precise micro-nano mechanical systems. In addition, the different routes to the chaos of the carbon nanotube in two extreme cases are revealed.
在所有精密微纳机械系统中，碳纳米管横向振动中混沌现象的出现，对微纳系统的稳定性和精度有很大影响，其情况与碳纳米管的边界约束有关。为了归纳当前参考文献中研究的碳纳米管的一些横向振动问题，本文通过转动弹簧和平移弹簧来考虑单壁碳纳米管两端的弹性约束，以研究边界约束对碳纳米管混沌特性的影响。基于广义多辛理论，首先给出了描述受横向载荷作用的单壁碳纳米管横向振动的控制方程的广义多辛格式以及由弹性约束产生的约束方程。然后，构建了具有离散约束方程的结构保持格式来模拟碳纳米管的横向振动过程。最后，捕捉了碳纳米管的混沌区域，并在数值研究中研究了两种极端情况（包括简支和悬臂）的振动。从数值结果可以得出，碳纳米管两端的相对弯曲刚度系数和绝对弯曲刚度系数是影响碳纳米管混沌区域的两个重要因素，这为精密微纳机械系统的设计和制造提供了指导。此外，还揭示了碳纳米管在两种极端情况下通向混沌的不同路径。
