Nonlinear vibration of embedded single-walled carbon nanotube with geometrical imperfection under harmonic load based on nonlocal Timoshenko beam theory

Based on the nonlocal continuum theory, the nonlinear vibration of an embedded single-walled carbon nanotube （SWCNT） subjected to a harmonic load is in- vestigated. In the present study, the SWCNT is assumed to be a curved beam, which is unlike previous similar work. Firstly, the governing equations of motion are derived by the Hamilton principle, meanwhile, the Galerkin approach is carried out to convert the nonlinear integral-differential equation into a second-order nonlinear ordinary differ- ential equation. Then, the precise integration method based on the local linearzation is appropriately designed for solving the above dynamic equations. Besides, the numerical example is presented, the effects of the nonlocal parameters, the elastic medium constants, the waviness ratios, and the material lengths on the dynamic response are analyzed. The results show that the above mentioned effects have influences on the dynamic behavior of the SWCNT.

On the limitations of linear beams for the problems of moving mass-beam interaction using a meshfree method

This paper deals with the capabilities of linear and nonlinear beam theories in predicting the dynamic response of an elastically supported thin beam traversed by a moving mass. To this end, the discrete equations of motion are developed based on Lagrange＇s equations via reproducing kernel particle method （RKPM）. For a particular case of a simply supported beam, Galerkin method is also employed to verify the results obtained by RKPM, and a reasonably good agreement is achieved. Variations of the maximum dynamic deflection and bending moment associated with the linear and nonlinear beam theories are investigated in terms of moving mass weight and velocity for various beam boundary conditions. It is demonstrated that for majority of the moving mass velocities, the differences between the results of linear and nonlinear analyses become remarkable as the moving mass weight increases, particularly for high levels of moving mass velocity. Except for the cantilever beam, the nonlinear beam theory predicts higher possibility of moving mass separation from the base beam compared to the linear one. Furthermore, the accuracy levels of the linear beam theory are determined for thin beams under large deflections and small rotations as a function of moving mass weight and velocity in various boundary conditions.