Nonlinear free vibration of single-walled carbon nanotubes using nonlocal Timoshenko beam theory

Nonlinear free vibration of single-walled carbon nanotubes (SWCNTs) is studied in this paper based on von Kármán geometric nonlinearity and Eringenʹs nonlocal elasticity theory. The SWCNTs are modeled as nanobeams where the effects of transverse shear deformation and rotary inertia are considered within the framework of Timoshenko beam theory. The governing equations and boundary conditions are derived by using the Hamiltonʹs principle. The differential quadrature (DQ) method is employed to discretize the nonlinear governing equations which are then solved by a direct iterative method to obtain the nonlinear vibration frequencies of SWCNTs with different boundary conditions. Zigzag (5, 0), (8, 0), (9, 0) and (11, 0) SWCNTs are considered in numerical calculations and the elastic modulus is obtained through molecular mechanics (MM) simulation. A detailed parametric study is conducted to study the influences of nonlocal parameter, length and radius of the SWCNTs and end supports on the nonlinear free vibration characteristics of SWCNTs.