Nonlinear and quasi-linear behavior of a curved carbon nanotube vibrating in an electric force field; an analytical approach

The nonlinear vibrational model of a slightly curved single-walled carbon nanotube (SWCNT) resting on a Winkler-type elastic foundation is developed using nonlocal Euler- Bernoulli elastic theory. The SWCNT is assumed to vibrate under an external harmonic electric force field and an analytical solution is proposed to obtain the nonlinear resonant frequencies. The results show good agreement with the numerical simulation and the obtained analytical frequency is completely related to the curvature of the nanotube. Our model predicts that although the model is nonlinear in nature, the curved SWCNT could behave linearly in a certain amount of curvatures and this quasi-linear vibrational behavior of curved SWCNT is a function of aspect ratio, nonlocal parameter, and stiffness of the foundation.