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.
Free vibrations of a double-walled carbon nanotube (DWNT) are studied. The inner and outer carbon nanotubes are modeled as two individual elastic beams interacting each other by van der Waals forces. An original method is proposed to calculate the first seven order resonant frequencies and relative vibrational modes. Detailed results are demonstrated for DWNTs according to the different boundary conditions between inner and outer tubes, such as fixed-free, cantilever-free, fixed-simple and fixed-fixed (reduced form) supported ends. Our results indicate that there is a special invariable frequency for a DWNT that is not affected by different combinations of boundary conditions. All vibrational modes of the DWNT must be coaxial when the resonant frequency is smaller than this frequency. Some noncoaxial vibrations will occur when their resonant frequencies exceed the frequency. Especially, the first noncoaxial resonant frequency is still invariable for all different boundary conditions. A change of resonant frequency for various lengths of DWNTs is discussed in detail. In addition, our model predicts a new coaxial-noncoaxial vibrational mode in fixed-simple supports for inner and outer tubes of a DWNT.
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