Numerical investigation on effect of different parameters on nonlinear vibration response of fully geometrically exact Timoshenko beams

In this paper, a FE formulation for nonlinear vibration of fully geometrically exact Timoshenko beams is derived. Firstly, the strong form of the governing equation of motion is obtained without any approximation. Next, the weak relations are used to derive the FE formulation. To obtain the vibration response, a direct integration method is employed to solve highly nonlinear formulation for geometrically exact beams. Finally, some examples are investigated, and very good results with analytical solutions available in the literature are achieved. The results show that the formulation presented in this paper can predict the vibration response of the Timoshenko beam with high accuracy. Moreover, investigating the effect of parameters shows that increasing the length parameter leads to increasing natural frequencies. Besides, the frequency value increases when the amplitude of vibration increases. Furthermore, adding an axial spring can lead to asymptotic behavior in vibration response of the fully exact Timoshenko beams.