Application of generalized quadrature rule to vibration of a curved carbon nanotube on elastic foundation

There are several studies on modelling and simulation of vibrating CNTs in an elastic medium to predict the natural frequencies and associated mechanical properties. In all of these models, the nanotube was assumed to be perfectly straight. However, photomicrographs of nanocomposites indicate that CNTs may exhibit significant waviness. Significance of waviness effects on the elastic modulus of CNT-reinforced polymer composites has been shown in a few papers and it seems that no work has been thoroughly done on the effects of waviness on the vibration characteristics of CNTs. In this research, the governing equation for free vibration of a curved single-walled carbon nanotube (SWNT) on Winkler foundation is derived considering the effects of rotary inertia (RI) and shear deformation (SD). The nanotube waviness is described by a continuum circular model; and the differential equations of it are solved using generalized differential quadrature rule (GDQR). For validation, the results are compared with those for finite difference solution of this problem. In addition, as the radius of curvature approaches infinity, the GDQR model shows a good agreement with the results for a straight nanotube. The natural frequencies and associated mode shapes are calculated with clamped-clamped, hinged-hinged and clamped-hinged ends conditions. The model is solved for various curvatures and elastic mediums, which shows that the curvature of a CNT has a strong effect on modal frequencies, especially when the stiffness of foundation and aspect ratio of CNT are relatively small.