Analysis of the vibration characteristics of double-walled carbon nanotubes

The induced electric-field has been applied to measure the elastic modulus of carbon nanotubes. However, the vibrational modes of the multi-walled carbon nanotubes are quite different from those of the single-walled carbon nanotubes. Analysis of the vibration characteristics of double-walled carbon nanotubes (DWCNTs) with simply supported boundary condition are carried out based on Euler–Bernoulli beam theory. The DWCNTs are considered as two nanotube shells coupled through the van der Waals interaction between them. It is found that the vibrational modes of DWCNTs are noncoaxial intertube vibrations, and the deflections of the inner and outer nanotubes can occur in the same or in opposite deflections. In the same vibrational mode, the resonant frequencies of DWCNTs with deflections between the inner and outer nanotubes in the same direction are smaller than those of DWCNTs with the opposite deflections.