Analytical approaches for vibration analysis of multi-walled carbon nanotubes modeled as multiple nonlocal Euler beams

This paper concerns with the effect of small scale on the vibrational characteristics of multi-walled carbon nanotubes (MWCNTs) modeled as multiple nonlocal Euler beams. In this model, each nanotube interacts with its neighbors through the van der Waals force. Analytical approaches are expressed to solve coupled governing equations of the motion. Results for double- and five-walled carbon nanotubes (DWCNTs and FWCNTs), as two specific examples of MWCNTs, are presented for various boundary conditions. Then, effect of small scale on the natural and intertube resonant frequencies and their associated amplitude ratios are discussed. Besides the effect of small scale, the effect of end conditions on the vibrational properties and a comparison between the methods are provided. Natural and intertube frequencies reduce with the introduction of nonlocal parameter. However, reduction of intertube frequencies is less than the natural frequencies. Moreover, it is provided that the effect of small scale stiffens the van der Waals force and causes MWCNTs to behave similar to a single beam in high values of nonlocal parameter. Also, this study reveals that in high mode numbers, natural frequencies of a multiple classical Euler beams system tend to frequencies of its constituent beams.