Vibration of multi-walled carbon nanotubes with initial axial force and radial pressure

A study is performed for the vibrational characteristics of multi-walled carbon nanotubes (MWCNTs) with initial axial force and radial pressure based on the Donnell model. In particular, explicit expressions are obtained for double-walled carbon nanotubes. Numerical results show that the intertube frequencies are insensitive to the initial axial stress (axial tensile stress or axial compressive stress), internal pressure or external pressure in MWCNTs, while the natural frequency is sensitive to the initial axial stress, internal pressure or external pressure. Further, the natural frequency increases with increasing axial tensile stress and decreases with increasing axial compressive stress. The frequencies increase with increasing internal pressure and decrease with increasing external pressure. The associated modes are insensitive to the initial axial stress, internal pressure or external pressure in MWCNTs.