On the vibrations of single-walled carbon nanotubes

In this paper, a detailed numerical study on the free and forced vibrations of single walled carbon nanotubes is presented. A simple and straightforward method developed such that the proximity of the mathematical model to the actual atomic structure of the nanotube is significantly retained, is used for this purpose. Both zigzag and armchair chiralities of the carbon nanotubes for clamped-free and clamped-clamped boundary conditions are analyzed and their natural frequencies and corresponding mode shapes are obtained. Results pertaining to axial, bending, and torsional modes of vibration are reported with discussions. These modes of vibration appear in the eigen-values and eigen-vectors without any distinction. The direct integration method by Newmark is used extensively along with the fast Fourier transform to identify different types of vibrational modes. In the case of zigzag nanotubes, the axial, bending, and torsional modes appear to be decoupled, whereas the armchair nanotubes show coupling between such modes.