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

Abstract

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