Vibration of short carbon nanotubes using generalized differential quadrature rule

In this paper, the transversely vibrating governing equations of a multi-wall carbon nanotube (MWNT) on a Winkler elastic foundation have been derived. The model includes the effects of shear deformation (SD) and rotary inertia (RI) using Timoshenko beam theory for short nanotubes. The van der Waals interaction between the tubes is taken into account. The model supports not only traditional boundary conditions such as free-free, clamped-clamped and cantilever; but also all possible end conditions. This makes the model more compatible for real applications, especially in nanocomposites. By using generalized differential quadrature method (GDQM), the equations are solved for carbon nanotubes (CNTs) with different aspect ratios, various boundary conditions and different foundation stiffness to obtain resonant frequencies. Numerical results for double-wall and multi-wall carbon nanotubes showed that by decreasing aspect ratio of the CNT, the effects of SD and RI increase. Moreover, the effects of ends stiffness play an important role to determine resonant frequencies, especially for small aspect ratios when the stiffness of elastic medium is relatively small.