Chaotic and periodic vibration of a carbon nanotube supported by nonlinear foundation

The nonlinear vibrations and chaotic motion of a carbon nanotube subjected to axial loads and rested on a Winkler and Pasternak foundation are studied. It is assumed that the carbon nanotube is simply-supported at both ends. The governing equation of motion for the nanotube and foundation system using the Euler-Bernoulli beam theory is developed. The Galerkin method is considered for the solution of the governing equation to obtain the nonlinear ordinary differential equation of the system. The multiple time scale method is utilized to find the frequency response of the system. The bifurcation diagram, phase-plane trajectories and Poincaré section are considered to identify the chaotic region of oscillations of carbon nanotube.