Chaotic dynamics of spiral bevel gears

Abstract

This study investigated the nonlinear dynamics of a Spiral Bevel Gear (SBG) system used in helicopter transmissions. Two approaches were employed to determine the Mesh Stiffness (MS): the average slope method (average-MS) and a hybrid analytical-computational method (adaptive-MS), which accounts for Hertzian contact nonlinearity. The dynamic mesh force was computed using polynomial interpolation techniques and loaded tooth contact analysis. The accuracy of the adaptive-MS approach was validated through comparison with a verified nonlinear finite element method (Nonlinear-FEM) simulation, demonstrating strong agreement. The GearDyns-SBG program was used to solve the system's dynamic model under varying working conditions. Backward and forward simulations were conducted to track stable branches, providing insights into the system's behavior. The study evaluated the dynamic responses based on both mesh stiffness approaches using tools such as FFT spectrum, amplitude-frequency analysis, nonlinear time series analysis, Poincaré maps, phase diagrams, and recurrence plots. The results revealed complex behaviors, including tooth separation, backside contact, boundary crises, and period-doubling cascades. Additionally, the largest Lyapunov exponent and fractal dimension were used to characterize the dynamics and 3D bifurcation analysis captured transitions between regular and chaotic regimes. The periodicity characteristics of the system were evaluated by recurrence quantification analyses. These findings enhance the understanding of nonlinear gear dynamics of SBG and provide reliable methods for predicting and analyzing their behavior.