Pendulum between two springs using Ms- DTM

Abstract This study provides a comprehensive analysis of a pendulum system suspended between two horizontal springs, focusing on its dynamic behavior and the application of advanced mathematical methods. The research begins by formulating the Lagrangian of the system, which serves as the foundation for deriving the equations of motion, offering a rigorous mathematical framework for understanding the system’s dynamics. The nonlinear equations of motion are then solved using the multi-step differential transformation method (Ms-DTM), whose accuracy and effectiveness are evaluated through comparison with the well-established Runge–Kutta method. This comparative analysis highlights the strengths and limitations of each numerical approach in managing the complexity of the system. The study also includes a series of simulated plots that depict the system’s behavior under various initial conditions and parameter settings. These visualizations provide valuable insights into the system’s dynamic responses, revealing phenomena such as chaotic motion and periodic oscillations. The results underscore the utility of sophisticated mathematical methods in addressing complex physical problems and enhancing our understanding of nonlinear dynamic systems.