In the last part of this thesis, the overall conclusions of single-walled CNTs for vibrational analysis have been presented and validated with existing open text. Since the percentage of error is negligible, the models have been concluded as valid. The vibrations of the carbon nanotube that have been investigated are based on shell theory of Donnell, the theory of modified Flügge shell and nonlocal theory of elasticity. To generate the fundamental natural frequencies of SWCNTs, computer software MATLAB engaged. Free vibrations of the CNTs has been developed using Donnell‟s shell theory and wave propagation approach. In our new study, vibration spectra of various physical parameters such as ratio of length-to-diameter/aspect ratio were obtained with the effects of various parameters and in-plane rigidity for the vibrational behavior of different sorts of SWCNTs. Also the effects of mass density are originally examined for these types of tubes. It has been demonstrated that the fundamental natural frequencies depend on the material parameters of the nanotube and the frequencies for three sorts of SWCNTs under C-C and C-F edge conditions increase with the decrease of length-to diameter ratio. In addition, results indicated that frequencies of three sorts of SWCNTs increasing in increases the frequencies and increasing in decreases the frequencies. Throughout the computation from this model, the frequencies of zigzag and chiral under end conditions are higher than that of armchair SWCNTs. We double-checked our results with the frequency value versus ratio of length-to-diameter arising from the MD simulation and continuum shell model showing the same divergences that those have been previously shown. Also, the resonant frequency peaks in the present results show excellent stability across a range of length/diameter ratio. Furthermore, a new method based on the Flügge 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 a novel wave propagation 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 of three different radii 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-toradius 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. A comprehensive estimation regarding nonlocal theory has been considered for vibrational behavior of the SWCNTs with distinct nonlocal parameters (x = 0.5, 1, 1.5, 2). Vibrations of SWCNTs for armchair indices (5, 5), (7, 7), (9, 9), zigzag indices (8, 0), (15, 0), (20, 0) and chiral indices (12, 5), (22, 7), (25, 10) have been analyzed. We developed a new model from the combination of the nonlocal theory of elasticity with wave propagation approach. It is noted that the frequencies of C-C is higher than that of C-F. Also, WPA has been utilized for first time to consider the effects of bending rigidity on SWCNTs vibration. This modified model has less complication and has been compared with the earlier methods. The computational results indicated that there is inverse relation of nonlocal parameters and frequencies. The frequency curves of clamped-free are lower throughout the computation than the clamped-clamped carbon nanotube. The obtained results show that by increasing aspect ratio of carbon nanotubes, frequency value increases at all boundary conditions. In our measurement we indicated that with higher aspect ratio, the BCs have a momentous influence on vibration of CNT. It can be concluded that frequencies would increase by increasing of the bending rigidity. This means that smaller effects play an important role in predicting SWCNT frequencies, which local theory cannot capture. In summary, we obtained a complete set of three form of SWCNTs by systematically comparing some of the most important simulation (molecular dynamics) with continuum theory. The result obtained by these models shows the correlation between the simulations of MD (Zhang et al., 2009), shell model (Wang et al., 2006), nonlocal theories (Bocko and Lengvarský, 2014; Amirian et al., 2013). We developed a new model from the combination of using shell theories based on novel wave propagation approach. Our calculation for armchair, zigzag and chiral SWCNTs frequencies have been investigated with different indices based on shell theories and are in agreement with other studies. This study of different models based on wave propagation approach might lead to very novel results with different boundary conditions and simplify the complicated vibrational behaviors of SWCNTs. These boundary conditions seem to play an important role on the vibration of SWCNTs. Calculating the frequencies of SWCNTs with various parameters of vibration may contribute to understand this physical phenomenon.