Vibration of a double-walled carbon nanotube aroused by nonlinear intertube van der Waals forces

Vibration of a double-walled carbon nanotube aroused by nonlinear interlayer van der Waals (vdW) forces is studied. The interlayer vdW forces as a nonlinear function are described by the interlayer spacing. The inner and outer carbon nanotubes are modeled as two individual elastic beams. Detailed results are demonstrated for double-walled carbon nanotubes (DWCNTs) with an aspect ratios of 10 and 20, based on the simply supported, fixed, or free end conditions, respectively. Harmonic balance method is used to analyze the relation between the amplitudes of deflection and the frequencies of coaxial and noncoaxial free vibrations. Our results indicate that the nonlinear factors of vdW forces have little effect on the coaxial free vibration, and that the deflection amplitudes increase rapidly with the increasing frequency, which are almost the same with those of the linear free vibration. On the other hand, the nonlinear factors of vdW forces have a great effect on noncoaxial free vibration. The relation between the deflection amplitudes and the frequencies shows nonlinear trend, which indicates that the aspect ratio and end condition almost have no affect on the noncoaxial amplitudes of DWCNTs.