Modeling and Nonlinear vibration analysis of Euler-Bernoulli beam under finite deformation

According to the wide presence of beams in engineering structures, it is very useful to understand how the beams vibrate nonlinearly in conditions where they oscillate with a large amplitude. In this paper, the nonlinear vibrations of an Euler-Bernoulli beam under finite strain are investigated. In this study, unlike other papers, in order to obtain the governing equations of beam, the field-displacement relationship has been done without approximation. Based on this, the strain-displacement relations are calculated using the Green Lagrange strain and the nonlinear form of the equations is obtained by using the Hamilton method. In order to solve the partial differential equation, using the Galerkin method, the equation has been converted to an Ordinary differential equation and finally solved using the multiple scale method and compared with the Rung-Kutta numerical method. To evaluate the accuracy of the method and the validity of the modeling, the obtained results are compared with the Euler–Bernoulli beam theory and the Von-Karman nonlinear model. The results show that the present method in low vibration amplitudes is consistent with the model of Euler-Bernoulli and Von-Karman, but with increasing amplitude of oscillations, the results of these models will be significantly different from each other, which is as expected.