Vibration of multi-walled carbon nanotubes with initial axial loading

This paper studies the vibrational characteristics of multi-walled carbon nanotubes (MWNTs) with initial axial loading based on Donnell equations. Both the effect of initial axial stress and the effect of the van der Waals forces are discussed in detail. It is shown that the resonant frequencies is related to the tension or compression forms of initial axial stress, which increase with the increase of axial tensile stress, and decrease with the increase of axial compressive stress. Further, the intertube resonant frequencies are insensitive to the initial axial stress in MWNTs, while the natural resonant frequency is sensitive to the initial axial stress, especially for the lower ones. However, the associated vibrational modes are independent of the initial axial stress. In particular, for double-walled carbon nanotubes (DWNTs), the natural resonant frequency of free vibration or vibration with initial axial stress is always between the corresponding one of the isolated inner tube and that of the isolated outer tube. In addition, some of the results obtained are compared with previous results.