An accurate spring-mass finite element model for vibration analysis of single-walled carbon nanotubes

An accurate spring-mass model, in the context of a three-dimensional finite element formulation, is developed for investigating the vibrational characteristics of single-walled carbon nanotubes (SWCNTs). Atoms are replaced by lumped mass elements at their locations and appropriate spring-type elements are defined as interconnections between the atoms in order to simulate the interatomic interactions. The effect of out of plane angle variation energy is incorporated into the model. The obtained results for the fundamental frequency of single-walled carbon nanotubes of various kinds are graphically illustrated. The influences of some commonly-used boundary conditions and changes in the nanotube geometrical parameters on vibration frequencies are examined. The numerical results show good agreement with other published results in the literature. Also, some novel relations are deduced which can be more useful in predicting the fundamental frequency of SWCNTs with great number of atoms.