Simplified Single-Stage Planetary Gearbox and Rolling Element Bearings Dynamic Analysis Using Lagrange’s Theorem and Comparison of Vulnerable Frequencies of Vibration

Introduction Planetary gearboxes are widely used in the aerospace, automotive and renewable energy sectors. These gearboxes are the critical component in a system's reliability; hence, monitoring this component is essential. Methods To study the vibration characteristics, a novel dynamic modelling approach using Lagrange's equations is presented in this paper. Newtonian equations of motion have been developed and solved using Lagrange's theorem, and modal analysis has been performed to estimate the dynamic characteristics of a single-stage planetary gearbox. The torsional stiffness and time-varying mesh stiffness have been considered in this paper for dynamic modelling. This study applies the perturbation method to solve the conditions of the system. The significant resonance frequencies of the individual components in the gearbox have been identified using coordinates based on eigenvectors. The finite-element analysis results have been considered for validation purposes and compared to the numerical model. The frequency range of the gearbox components' resonances and dynamic characteristics have been obtained from the FEA. Conclusions The study shows that Lagrange's dynamic modes match with modes obtained from the finite-element modal analysis. In addition, the resonance frequencies produced by the sun and planet gears and the bearings are detected. Therefore, the results show a positive potential in gearbox fault diagnostics and characteristics frequencies studies for individual components in the gearbox.