Axial vibration analysis of nanocones based on nonlocal elasticity theory

Carbon nanocones have quite fascinating electronic and structural properties, whose axial vibration is seldom investigated in previous studies. In this paper, based on a nonlocal elasticity theory, a nonuniform rod model is applied to investigate the small-scale effect and the nonuniform effect on axial vibration of nanocones. Using the modified Wentzel-Brillouin-Kramers (WBK) method, an asymptotic solution is obtained for the axial vibration of general nonuniform nanorods. Then, using similar procedure, the axial vibration of nanocones is analyzed for nonuniform parameters, mode number and nonlocal parameters. Explicit expressions are derived for mode frequencies of clamped-clamped and clamped-free boundary conditions. It is found that axial vibration frequencies are highly overestimated by the classical rod model because of ignorance of the effect of small length scale.