Akbari–Ganji Method for Solving Equations of Euler–Bernoulli Beam with Quintic Nonlinearity

In many real word applications, beam has nonlinear transversely vibrations. Solving nonlinear beam systems is complicated because of the high dependency of the system variables and boundary conditions. It is important to have an accurate parametric analysis for understanding the nonlinear vibration characteristics. This paper presents an approximate solution of a nonlinear transversely vibrating beam with odd and even nonlinear terms using the Akbari–Ganji Method (AGM). This method is an effective approach to solve nonlinear differential equations. AGM is already used in the heat transfer science for solving differential equations, and in this research for the first time, it is applied to find the approximate solution of a nonlinear transversely vibrating beam. The advantage of creating new boundary conditions in this method in additional to predefined boundary conditions is checked for the proposed nonlinear case. To illustrate the applicability and accuracy of the AGM, the governing equation of transversely vibrating nonlinear beams is treated with different initial conditions. Since simply supported and clamped-clamped structures can be encountered in many engineering applications, these two boundary conditions are considered. The periodic response curves and the natural frequency are obtained by AGM and contrasted with the energy balance method (EBM) and the numerical solution. The results show that the present method has excellent agreements in contrast with numerical and EBM calculations. In most cases, AGM is applied straightforwardly to obtain the nonlinear frequency– amplitude relationship for dynamic behaviour of vibrating beams. The natural frequencies tested for various values of amplitude are clearly stated the AGM is an applicable method for the proposed nonlinear system. It is demonstrated that this technique saves computational time without compromising the accuracy of the solution. This approach can be easily extended to other nonlinear systems and is therefore widely applicable in engineering and other sciences.