Analytical Expression of Nonlinear Vibration Analysis of Triple-Walled Carbon Nanotubes

This article employs the homotopy perturbation method (HPM) to derive analytical solutions for the nonlinear vibrations of triple-walled carbon nanotubes (TWCNTs) embedded in an elastic medium. A triple-beam model is used, where the governing equations for each layer are coupled with those of adjacent layers through van der Waals interlayer forces. The study examines the amplitude-frequency response of TWCNTs under large-amplitude vibrations, analysing the effects of variations in the elastic medium's material properties as well as changes in the nanotube's geometric parameters. Using the homotopy perturbation method (HPM), a nonlinear system of equations can be reformulated into an approximate analytical expression. The numerical results demonstrate the rapid convergence of the derived series solutions toward the exact solution. Additionally, the analytical findings are validated against simulation results obtained using a MATLAB program, showing strong agreement between the two approaches.