This paper investigates the coupled axial-radial (CAR) vibration of single-walled carbon nanotubes (SWCNTs) based on doublet mechanics (DM) with a scale parameter. Two coupled forth order partial differential equations that govern the CAR vibration of SWCNTs are derived. It is the first time that DM is used to model the CAR vibration of SWCNTs. To obtain the natural frequency and dynamic response of the CAR vibration, the equations of motion are solved and the relation between natural frequencies and scale parameter is derived. It is found that there are two frequencies in the frequency spectrum and these CAR vibrational frequencies are complicated due to coupling between two vibration modes. The advantage of these analytical formulas is that they are explicitly dependent to scale parameter and chirality effect. The influence of changing some geometrical and mechanical parameters of SWCNT on its CAR frequencies has been investigated, too. It is shown that the chirality and scale parameter play significant role in the CAR vibration response of SWCNTs. The scale parameter decreases the higher band CAR frequency compared to the predictions of the classical continuum models. However, with increase in tube radius and length, the effect of the scale parameter on the natural frequencies decreases. The lower band CAR frequency is nearly independent to scale effect and tube diameter. The CAR frequencies of SWCNTs decrease as the length of the tube increases. This decreasing is higher for higher band CAR frequency. To show the accuracy and ability of this method, the results obtained herein are compared with the existing theoretical and experimental results and good agreement is observed. تفاصيل المقالة

This paper investigates the nonlinear coupled radial-axial vibration of single-walled carbon nanotubes (SWCNTs) based on numerical methods. Two coupled partial differential equations that govern the nonlinear coupled radial-axial vibration for such nanotube are derived using nonlocal doublet mechanics (DM) theory. To obtain the nonlinear natural frequencies in coupled radial-axial vibration mode, these equations are solved using Homotopy perturbation method (HPM). It is found that the coupled radial-axial vibrational frequencies are complicated due to coupling between two vibration modes. The influences of some commonly used boundary conditions, changes in vibration modes and variations of the nanotubes geometrical parameters on the nonlinear coupled radial-axial vibration characteristics of SWCNTs are discussed. It was shown that boundary conditions and maximum vibration velocity play significant roles in the nonlinear coupled radial-axial vibration response of SWCNTs. It was shown that unlike the linear one, the nonlinear natural frequencies are dependent to maximum vibration velocity. Increasing the maximum vibration velocity increases the natural frequency of vibration compared to the prediction of the linear model. However, with increase in tube length, the effect of the maximum vibration velocity on the natural frequencies decreases. It was also shown that the amount and variation of nonlinear natural frequencies are more apparent in higher vibration modes and two clamped boundary conditions. To show the accuracy and capability of this method, the results obtained herein are compared with the fourth order Runge-Kuta numerical results and also with the other available results and good agreement is observed. It is notable that the results generated herein are new and can be served as a benchmark for future works. تفاصيل المقالة