Free transverse vibrations of double-walled carbon nanotubes using a theory of nonlocal elasticity

Based on theory of nonlocal elasticity, a nonlocal double-elastic beam model is developed for the free transverse vibrations of double-walled carbon nanotubes, The effect of small length scale is incorporated in the formulation. With this nonlocal double-elastic beam model, explicit expressions are derived for natural frequencies and associated amplitude ratios of the inner to the outer tubes for the case of simply supported double-walled carbon nanotubes. The effect of small length scale on the properties of vibrations is discussed. It is demonstrated that the natural frequencies and the associated amplitude ratios of the inner to the outer tubes are dependent upon the small length scale. The effect of small length scale is related to the vibrational mode and the aspect ratio.