The Non-Perturbative Approach in Examining the Motion of a Simple Pendulum Associated with a Rolling Wheel with a Time-Delay

The present study aims to examine the movement of a simple pendulum (SP) that is connected by a lightweight spring and connected with a rotating wheel. The motivation behind this topic is to gain a comprehensive understanding of intricate dynamic systems that involve interconnected mechanical components and feedback with temporal delays. This system is not only theoretically attractive but also practically applicable in domains such as robotics, engineering, and control systems. As well known, all classical perturbation methods exploit Taylor expansion to simplify the reality of restoring forces. In contrast, the non-perturbative approach (NPA), as a novel methodology, transforms any nonlinear ordinary differential equation (ODE) into a linear one. It scrutinizes the restoring forces, away from employing Taylor expansion; hence it eliminates the previous weakness. The concept of the NPA is based mainly on the He’s frequency formula (HFF). The confidence of the NPA comes from the numerical compatibility between the nonlinear and linear ODEs via the Mathematica Software (MS). Therefore, instead of handling the nonlinear ODE, we investigate the linear one. The achieved response is plotted over time to show the impact of the acted parameters during a specified time interval. Moreover, the phase plane curves that correspond to the plotted solution are presented and examined. The stability criteria of the analogous linear ODE are provided and drawn to explore the stability/instability zones. The performance is applicable in engineering and other fields due to its ease of adaptation to different nonlinear systems. Therefore, the NPA can be regarded as substantial, successful, and interesting and can extended to be applied in further categories within the field of couples dynamical systems.