A new method for determination of natural frequency in bending vibration mode of single-walled carbon nanotubes

This paper investigates the bending vibration of single-walled carbon nanotubes (SWCNTs) based on a new theory called doublet mechanics (DM) with a scale parameter. A sixth order partial differential equation that governs the bending vibration for such nanotubes is derived. It is the first time that DM is used to model the bending vibration of carbon nanotube. Euler-Bernoulli beam model is used in this paper. Using DM, the relation between natural frequency and scale parameter is derived in the bending vibration mode. It is proven that scale parameter plays significant role in the vibration behavior of such nanotubes in lateral direction. Such effect decreases the natural frequency compared to the predictions of the classical continuum mechanics (CCM) models. However, with increasing the tube length, the scale effect on the natural frequency decreases. To show the accuracy and capability of this method, the results obtained herein are compared with the existing nonlocal and molecular dynamics (MD) results and good agreement is observed. It is notable that the results generated herein are new and can be used as a benchmark for future works.