Chaotic region of elastically restrained single-walled carbon nanotube

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.