Dynamic analysis of embedded curved double-walled carbon nanotubes based on nonlocal Euler-Bernoulli Beam theory

The aim of this paper is to study the dynamic vibrations of embedded double-walled carbon nanotubes (DWCNTs) subjected to a moving harmonic load with simply supported boundary conditions. The model of DWCNTs is considered as an Euler-Bernoulli beam with waviness along the length, which is more accurate than the straight beam in previous works. Based on the nonlocal beam theory, the governing equations of motion are derived by using the Hamilton's principle, and then the separation of variables is carried out by the Galerkin approach, leading to two second-order ordinary differential equations (ODEs). The influences of the nonlocal parameter, the amplitude of the waviness, the surrounding elastic medium, the material length scale, load velocity and van der Waals force on the nonlinear vibration of DWCNTs are important. The dynamic responses of DWCNTs are obtained by using the precise integrator method to ordinary differential equations.