Nonlinear free and forced vibration analysis of a single-walled carbon nanotube using shell model

In this Paper, the nonlinear free and force vibration of a single-walled carbon nanotube (SWCNT) with simply supported ends is investigated based on von Karman’s geometric nonlinearity. The SWCNT described as an individual shell and the Donnell’s equations of cylindrical shells are used to obtain the governing equations. The Galerkin's procedure is used to discretized partial differential equations of the governing into the ordinary differential equations of motion. The method of averaging is applied to analyze the nonlinear vibration of (10, 0), (20, 0) and (30, 0) zigzag SWCNTs in the analytical calculations. The effects of the nonlinear parameters, different aspect ratios, different circumferential wave numbers and longitudinal half-wave numbers are investigated. Both free and forced motions (due to harmonic excitation) are considered. It is shown that (30, 0) zigzag SWCNT has less nonlinear behavior than the other CNTs for a constant aspect ratio. The type of nonlinearity is determined by the aspect ratio. It is seen from the results that for Small values of aspect ratios, the vibration behavior is softening type for the low amplitudes, and it is hardening type for the large amplitudes. And for large value of the aspect ratio, the vibration behavior is hardening type for all amplitudes.