The main objective of this research is to study the nonlinear vibrations of a single walled carbon nanotube. For this purpose, the lattice structure of carbon nanotube is replaced with a continuum structure using nanoscale continuum mechanics. Firstly, each carbon-carbon bond is replaced with an equivalent beam element and then the whole discrete structure of carbon nanotube is replaced with a virtual continuum medium representing hollow cylinder. Then, governing equations for vibrations is obtained taking into account geometric nonlinearity arisen from stretching of a mid-plane due to bending. Perturbation technique is used to analyze the nonlinear vibrations of carbon nanotubes. Frequency responses of carbon nanotubes for free vibrations and force vibrations in both primary and secondary resonance cases are studied. Obtained results are in a very good agreement with numerical integration technique. The results imply on hardening behavior of carbon nanotube. Moreover, nonlinear bifurcation and nonlinear jump phenomena are observed.