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Nonlocal Elasticity Theory for Free Vibration of Single-Walled Carbon Nanotubes

This paper concerns with free vibration analysis of single-walled carbon nanotubes including the effect of small length scale based on the nonlocal elasticity theory. The governing equation of nanotube is derived from Euler beam theory including a nonlocal parameter in the function of masses. Classical solutions are obtained and then compared with the numerical solutions provided by finite element models. Effect of tube chirality and various geometrically boundary conditions are considered. The finite element models of nanotubes are assumed as the virtually analogous frame structures. In the numerical technique, the atomic masses existing on the both ends of beams are assigned by physical and chemical properties of carbon element. The results show that the natural frequencies significantly increase when the nonlocal parameters decrease. The numerical results are in good agreement with the classical solutions for the nanotubes with low aspect ratios and are acceptable for high aspect ratios. Furthermore, the first-ten mode shapes are demonstrated for various aspect ratios and boundary conditions, and the repeated natural frequencies are also highlighted in this study.