Vibration analysis of carbon nanotubes (CNTs) is very essential field owing to their many promising applications in tiny instruments. The unique and interesting properties of CNTs, particularly their mechanical and electrical features, have fascinated industries and researchers to implement CNTs for production of different electromechanical devices. Research on vibration behavior of CNTs has been done for a few decades. Accordingly, a number of mathematical theories have been developed to analyze the vibration of CNTs. Most of the articles on the vibration of CNTs are focused on the linear and free vibrations and also, CNTs are supported by linear foundations. The main goal of this research is to develop nonlinear vibration analysis of CNTs having geometrical nonlinearity and being rested on nonlinear Winkler and Pasternak foundations. The nonlinear vibrations of the curved CNTs are developed using the Eringen theory in conjunction with the Euler-Bernoulli beam model and the thermal effect is also considered for the vibration of CNTs. The effects of fluid conveying flow, geometrical nonlinearities and Winkler and Pasternak foundations are studied. Nonlinear equations of CNTs vibration postulating the longitudinal electromagnetic effect is also modeled. Furthermore, a new method is developed for the nonlinear vibrations of freeform carbon nanotube based on the Euler-Bernoulli beam theory in conjunction with Non-Uniform Rational B-Spline (NURBS). The proposed model is quite straightforward for the vibrational analysis of the curved structures. Moreover, the chaotic motions of the fluid conveying carbon nanotube rested on a Winkler and Pasternak foundation are studied. The bifurcation and time history diagrams, phase-plane trajectories and Poincaré sections are presented to examine the dynamical behavior of carbon nanotubes. The Galerkin approach in conjunction with the multiple time scales method is chosen to study the nonlinear vibrations of the proposed models. The effects of different parameters on the primary and secondary resonant cases and also the nonlinear and linear frequencies of CNTs are examined.