Dynamics of nonlinear pendulum equations: Modified homotopy perturbation method
Nazmul Sharif, M. S. Alam, Md. Helal Uddin Molla
Abstract
This paper aims to deepen the understanding of the dynamic behavior of a pendulum attached to a rotating rigid frame, where the frame rotates with a constant angular velocity around a vertical axis passing through the pendulum’s pivot point. To provide a basis for comparison, the motion of a simple pendulum is also examined. The motivation for this study arises from the need for accurate and efficient analytical methods to predict the complex dynamics of such systems under varying conditions. A key contribution is the development of a modified version of He’s homotopy perturbation method (HPM), achieved through a novel time scaling and a reformulated power series expansion for the frequency. This approach allows the infinite series solution to be truncated at the first-order approximation, with iterative refinements introduced into higher-order terms of the governing differential equation. The validity and accuracy of the modified HPM are demonstrated by comparing its analytical solutions with numerical results obtained via the fourth-order Runge–Kutta method (RK4) and with established findings from the literature. Additionally, graphical analyses of the pendulum’s time histories are presented to explore how system parameters such as angular velocity and initial conditions influence its dynamic behavior. These findings provide a robust modified homotopy perturbation method (MHPM) for investigating similar nonlinear dynamical systems, addressing practical challenges in predicting their motion and contributing to the broader understanding of their complex interactions.