Nonlinear free vibration of embedded double-walled carbon nanotubes based on nonlocal Timoshenko beam theory

Nonlinear free vibration of embedded double-walled carbon nanotubes (DWNTs) is studied in this paper based on Eringen’s nonlocal elasticity theory and von Kármán geometric nonlinearity. The effects of the transverse shear deformation and rotary inertia are considered within the framework of Timoshenko beam theory. The surrounding elastic medium is described as the Winkler model characterized by the spring. The governing equations and boundary conditions are derived by using the Hamilton’s 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 nonlocal DWNTs with different boundary conditions. A detailed parametric study is conducted to investigate the influences of nonlocal parameter, length of the tubes, spring constant and end supports on the nonlinear free vibration characteristics of DWNTs.