Based on elastic continuum mechanics, the nonlinear free and force vibrational analysis of an embedded single walled carbon nanotube (SWCNT) with waviness along its axis is analyzed. The single-walled carbon nanotube embeds in a Pasternak elastic foundation. Two analytical approaches utilize to obtain the frequency-amplitude relationship of the free vibrational model, and another analytical approach propose to obtain the forced vibrations of the curved single walled carbon nanotube on the Pasternak elastic foundation. Subsequently, a parametric study is performed to study the importance of different parameters, such as the amplitude of oscillation and the curvature radius, on the nonlinear behavior of the system. Finally, numerical simulation is carried out to obtain the results and investigate the accuracy of the analytical solution methods. Comparison of the results obtained by proposed methods show excellent agreement with those obtained by numerical solution.