A numerical study of vibrational properties of single-walled carbon nanotubes

The vibrational response of the two well-known single-walled carbon nanotubes, zig-zag and arm-chair are investigated in this work by numerical simulation using finite element method. The simulations are carried out for three types of zig-zag nanotubes (10, 0), (15, 0) and (20, 0) and arm-chair nanotubes (10, 10), (15, 15) and (20, 20). Three different lengths of 15, 20 and 25 nm are considered for each nanotube. The natural frequencies and their corresponding modes of deformation versus the change of the geometry are determined for the nanotubes. The finite element models of nanotubes consist of beam elements in the form of a mechanical structure. The nanostructure is assumed to behave like mechanical structures in which the links are the carbon-carbon bond covalent force and the carbon themselves are the joints of the structure. The results obtained for this structure (as model) is then extended to the nanotubes (real structure) under investigation using the geometrically similar structures with a scale factor of β. The results indicate that the natural frequency of the nanotubes with the same diameter decreases with the increase of length. The increase of diameter gives rise to the increase of frequency. The increase, however, is not regular for arm-chair nanotubes. The frequencies of the two nanotubes (m, 0) and (m, m) are close for the first two modes and are significantly different at higher modes of vibration. The curves of natural frequencies versus the vibrating modes converge periodically nearly to the same point for both types of nanotubes of different diameters, impling that the frequency of the nanotubes is independent of the diameter at those periodic modes of vibration.