Nonlocal analysis of chaotic vibration, primary and super-harmonic resonance of single walled carbon nanotube in thermal environment

In this article, a nonlinear elastic Bernoulli–Euler beam model is presented to investigate the chaotic behavior and primary and superharminic resonance of single walled carbon nanotubes embedded in a visco-elastic medium at an elevated temperature. Using the Galerkin method and fourth-order Runge-Kutta method the governing equation is solved. The bifurcation diagram and largest Lyapunov exponent are employed to detect the critical amplitude of external force of periodic and chaotic response of single walled carbon. Having known the critical values, phase portrait and Poincare maps are presented to observe the periodic and chaotic behavior of the system. Moreover, the amplitude–frequency response for the primary superharmonic resonance of system is derived with the multiple scale method to investigate the feasibility of jump phenomenon. The sensitivity of jump phenomenon are studied for the selected viscoelastic foundation parameters, detuning parameter and external amplitude load. The results show that the amplitude of external force, viscoelastic foundation parameters, detuning parameter and temperature change in the cases of high and low temperature have a significant effect on the frequency response with jump phenomenon of system. In addition, the chaotic vibration of CNT can be controlled by changing of amplitude of external force.