Free-Vibration Analysis Of Helically Coiled Carbon Nanotubes Considering Nonlocal Effect Using Curved-Beam Elements

In this paper a numerical method is proposed to study the free vibration of helically coiled carbon nanotubes considering the nonlocal effects. In this method, the governing equations are obtained using the spatial curved-beam theory based on Washizu's static model. In the equations of motion all displacement functions are defined at the centroid axis and the effects of rotary inertia and transverse shear deformation are included. Moreover, nonlocal theory of elasticity is used in 3D curved-beam modeling. Therefore, six coupled equations including stresses and their second derivatives are obtained which should be combined with six coupled partial differential equations of motion of the system. Finite element method is used to solve the resulting equations, numerically. Curved elements with three nodes and six degrees of freedom per node are used in this method. In order to verify the developed MATLAB code, the results obtained from the proposed method by neglecting nonlocal effects are compared with those of ANSYS simulation. Besides, the effects of different boundary conditions and various parameters including helix radius, pitch, number of turns, and nonlocal parameter are studied.