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In this article, the homotopy analysis method (HAM) is implemented for obtaining semi-analytical solutions to the problem of the nonlinear vibrations of multiwalled carbon nanotubes embedded in an elastic medium. A multiple-beam model is utilized in which the governing equations of each layer are coupled with those of its adjacent ones via the Van der Waals inter layer forces. The amplitude-frequency curves for large-amplitude vibrations of single-walled, double-walled and triple-walled carbon nanotubes are obtained. The influence of changes in material constants of the surrounding elastic medium and the effect of changes in nanotube geometrical parameters on the vibration characteristics are studied by comparing the results with those from the previous work. Series solutions of the problem under consideration are developed by means of HAM and the recurrence relations are given explicitly. The obtained numerical results show the rapid convergence of the series constructed by the proposed method to the exact solution. Test problems have been considered to ensure that HAM is accurate and efficient compared with the Adomian decomposition method.

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