Effects of ring supports on vibration of armchair and zigzag FGM rotating carbon nanotubes using Galerkin's method

Abstract

This paper aims to investigate analytically the vibrational characteristics of functionally graded (FG) single-walled carbon nanotube (SWCNTs) based on modified Love shell theory. By using Galerkin's technique, the governing carbon nanotube (CNT) dynamics equation of the motion of the considered system is developed. For theoretical point of view, a nanotube problem is framed by amending some physical terms to accommodate the nature of the problem. For the present problem, vibrations of rotating carbon nanotubes with ring supports are examined. The effects of angular velocity, length- and height-to-radius ratios of rotating armchairs and zigzag SWCNTs with ring supports are fully investigated. Rings supports are attached to CNT in the radial direction. Our investigations show that increasing of length- and thickness-to-radius ratios yields decreasing and increasing frequency behaviors for rotating FG-CNT with clamped-clamped (C-C) and clamped-free (C-F) boundary conditions, respectively. It is found that with increasing the angular speed, frequencies of forward waves decrease and backward waves increase. Moreover, the effect of frequency is also performed to place the ring support at different locations of the CNT. Computational software MATLAB is engaged to characterize the frequencies based on CNT model to estimate the accuracy and effectiveness of this model. A few comparisons of analytical results for non-rotating and rotating CNTs are performed to confirm the validity, efficiency and accuracy of the numerical methodology.