Nonlinear coupled axial torsional vibration of single-walled carbon nanotubes using homotopy perturbation method

This work analyses the nonlinear coupled axial torsional vibration of single -walled carbon nanotubes (SWCNTs) based on numerical methods. Two -second order partial differential equations that govern the nonlinear coupled axial torsional vibration for such nanotube are derived. First, these equations are reduced to ordinary differential equations using the Galerkin method and then solved using homotopy perturbation method (HPM) to obtain the nonlinear natural frequencies in coupled axial torsional vibration mode. It is found that the obtained frequencies are complicated due to coupling between two vibration modes. The dependence of boundary conditions, vibration modes and nanotubes geometry on the nonlinear coupled axial torsional vibration characteristics of SWCNTs are studied in detail. It was shown that boundary conditions and maximum initial vibration velocity have significant effects on the nonlinear coupled axial torsional vibration response of SWCNTs. It was also seen that unlike the linear model if the maximum vibration velocity increases, the natural frequencies of vibration increases too. To show the effectiveness and ability of this method, the results obtained with HPM are compared with the fourth-order Runge-Kutta numerical results and good agreement is observed. To the knowledge of authors, the results given herein are new and can be used as a foundation work for future work.