New frequency predictions for a simple pendulum: Application of Harmonic Balance and Akbari-Ganji methods

In the analysis of nonlinear dynamical systems, developing an accurate understanding of simple mechanical models—such as the pendulum—is of fundamental importance in both engineering and physics. Although the simple pendulum is often introduced in its linearized form for small oscillations, its true behavior becomes highly nonlinear at larger amplitudes. The nonlinear pendulum, therefore, serves as a classical yet powerful example for exploring the rich dynamics that emerge in real-world systems where linear approximations fail. In this study, the non-linear dynamic analysis of a simple pendulum is revisited. Two new formulas for the period and frequency are proposed based on the Harmonic Balance Method and the Akbari-Ganji Method. Furthermore, to obtain more accurate results, improvements are made to the formulas of the harmonic balance method and the Akbari-Ganji method. These improvements provide more reliable outcomes, especially in systems requiring high accuracy. Two of the most prominent formulas in the literature are derived using the Akbari-Ganji Method. As a result of this, the frequencies obtained by the present method and the other methods are compared. The obtained results emphasize the accuracy and efficiency of the proposed approaches. Consequently, this study encourages the use of alternative methods in the analysis of non-linear dynamic systems.