Analytical solution of nonlinear differential equations two oscillators mechanism using Akbari-Ganji method

In the last decade, many potent analytical methods have been utilized to find the approximate solution of nonlinear differential equations. Some of these methods are energy balance method (EBM), homotopy perturbation method (HPM), variational iteration method (VIM), amplitude frequency formulation (AFF), and max-min approach (MMA). Besides the methods mentioned above, the Akbari-Ganji method (AGM) is a highly efficient analytical method to solve a wide range of nonlinear equations, including heat transfer, mass transfer, and vibration problems. In this study, it was constructed the approximate analytic solution for movement of two mechanical oscillators by employing the AGM. In the derived analytical method, both oscillator motion equations and the sensitivity analysis of the frequency were included. The AGM was validated through comparison against Runge-Kutta fourth-order numerical method and an excellent agreement was achieved. Based on the results, the highest sensitivity of the oscillation frequency is related to the mass. As alpha and beta increase, the slope of the system velocity and acceleration will increase.

Automated summary

This study introduced several potent analytical methods used in the past decade to solve nonlinear differential equations and demonstrated the application of the Akbari-Ganji method in solving mechanical oscillator movement problems. The sensitivity analysis showed that the mass has the highest sensitivity on oscillation frequency, and increasing alpha and beta will lead to an increase in the system velocity and acceleration slope.