Free Vibration Analysis of Carbon Nanotubes by Using Finite Element Method

Abstract

In the present study, an efficient and accurate finite element model for vibration analysis of carbon nanotubes (CNTs) with both Euler–Bernoulli and Timoshenko beam theory has been presented. For this purpose, an analytical solution for the exact dynamic shape functions of CNTs based on both Euler–Bernoulli and Timoshenko beam theories has been derived. The solution is general and is not restricted to a particular range of magnitudes of the nonlocal parameters. The exact dynamic shape functions have been utilized to derive analytic expressions for the coefficients of the exact dynamic (frequency-dependent) element stiffness matrix. Numerical results are presented to figure out the effects of nonlocal parameter, mode number and slenderness ratio on the vibration characteristics of CNTs. It is shown that these results are in good agreement with those reported in the literature. Present element formulation will be useful for structural analyses of nanostructures with complex geometries, loadings, material properties and boundary conditions.