Determining Equations for a Wave Equation Arising Due to Collapse of Shafts in Power Transmission System

Abstract

The study of wave propagation in mechanical systems plays a crucial role in understanding the dynamic behavior of components under stress or failure. One such scenario is the collapse of shafts in power transmission systems, where the sudden failure of structural components can trigger wave-like disturbances that propagate through the system, potentially causing further damage and system-wide disruptions. The ability to predict and analyze these wave phenomena is essential for ensuring the stability, safety, and efficiency of power transmission networks. In this paper, we study a fourth-order nonlinear wave equation which arises due to collapse of shafts in power transmission systems. Lie symmetry analysis is used to derive the corresponding determining equations that describe the symmetries of the system, which can then be used to reduce the problem to lower-order equations, find exact solutions, or gain insights into the underlying dynamics. By applying Lie group analysis, we systematically prolonged the corresponding infinitesimal generator and applied the symmetry condition to the given wave equation to yield the required determining equations. These equations enable a pathway to exact solutions and system simplification. The work reinforces the value of symmetry-based methods in understanding the dynamic behavior of mechanical systems under collapse-induced wave propagation.