Timoshenko beam model for vibration analysis of multi-walled carbon nanotubes

This paper is concerned with the use of the Timoshenko beam model for free vibration analysis of multi-walled carbon nanotubes (CNTs). Unlike the Euler beam model, the Timoshenko beam model allows for the effects of transverse shear deformation and rotary inertia. These effects become significant for CNTs with small length-to-diameter ratios that are normally encountered in applications such as nanoprobes. By using the differential quadrature (DQ) method, the governing Timoshenko equations are solved for CNTs of different length-to-diameter ratios and boundary conditions. By comparing results based on the Timoshenko and the Euler beam theories, we show that the frequencies are significantly overpredicted by the Euler beam theory when the length-to-diameter ratios are small and when considering high vibration modes. For such situations, the Timoshenko beam model should be used for a better prediction of the frequencies.