An efficient numerical model for vibration analysis of single-walled carbon nanotubes

This paper presents a linear spring-based element formulation for computation of vibrational characteristics of single-walled carbon nanotubes (CNTs). Three-dimensional nanoscale elements and corresponding elemental equations are developed for the numerical treatment of the dynamic behaviour of single-walled CNTs, including appropriate stiffness and mass characteristics. The atomistic microstructure of nanotubes is used to assemble the elemental equations and construct the dynamic equilibrium equation. The developed elements simulate the relative translations and rotations between atoms as well as the mass of the atoms. In this way, molecular mechanics theory can be applied directly because the atomic bonds are modelled by using exclusively physical variables such as bond stretching. The modelling is regenerative and can provide simulations for different geometric characteristics of the nanotubes. Numerical results are presented that illustrates new natural frequencies and mode shapes, going beyond the usual ones for various nanotubes under different support conditions and defects. Comparisons with corresponding numerical predictions from the literature, where they are possible, show very good agreement.