Simulating vibration of single-walled carbon nanotube using Rayleigh-Ritz's method

In this paper, a new method based on the Sander theory is developed for SWCNTs to predict the vibrational behavior of length and ratio of thickness-to-radius according to various end conditions. The motion equation for this system is developed using Rayleigh-Ritz's method. The proposed model shows the vibration frequencies of armchair (5, 5), (7, 7), (9, 9), zigzag (12, 0), (14, 0), (19, 0) and chiral (8, 3), (10, 2), (14, 5) under different support conditions namely; SS-SS, C-F, C-C, and C-SS. The solutions of frequency equations have been given for different boundary condition, which have been given in several graphs. Several parameters of nanotubes with characteristic frequencies are given and vary continuously in length and ratio of thickness-to-radius. It has been illustrated that an enhancing the length of SWCNTs results in decreasing of the frequency range. It was demonstrated by increasing of the height-to-radius ratio of CNTs, the fundamental natural frequency would increase. Moreover, effects of length and ratio of height-to-radius with different boundary conditions have been investigated in detail. It was found that the fundamental frequencies of C-F are always lower than that of other conditions, respectively. In addition, the existence of boundary conditions has a significant impact on the vibration of SWCNTs. To generate the fundamental natural frequencies of SWCNTs, computer software MATLAB engaged. The numerical results are validated with existing open text. Since the percentage of error is negligible, the model has been concluded as valid.