This paper explores the nonlinear characteristics of coupled radial-tangential vibrations in single-walled carbon nanotubes (SWCNTs) using numerical methods. The authors derive two coupled partial differential equations governing these vibrations through the application of doublet mechanics (DM). It is the first time that coupled nonlinear radial-tangential vibration of SWCNTs studied using DM. On the other hands; the tangential vibration mode has been less studied in scientific researches due to the complexity of the governing equation. To calculate the nonlinear natural frequencies, they utilize the Homotopy Perturbation Method (HPM) to solve the derived equations. The analysis reveals that the coupling between the two vibration modes leads to complex natural frequency behaviors. The study examines how different boundary conditions, vibration modes, and geometrical parameters affect the nonlinear dynamics of these vibrations. Significantly, the research finds that the maximum vibration velocity has a considerable impact on the nonlinear coupled radial-tangential vibration response of SWCNTs. In contrast to linear models, the nonlinear natural frequencies are influenced by the maximum vibration velocity; specifically, an increase in this velocity results in higher natural frequencies. However, as the tube diameter increases, the influence of the maximum vibration velocity decreases. The variations in nonlinear natural frequencies become more pronounced in higher vibration modes. To demonstrate the effectiveness of this method, we compare their findings with results from the fourth-order Runge-Kutta method, observing a strong correlation between both sets of results. The outcomes presented in this paper are original and offer a valuable benchmark for future investigations in this field.