Numerical Bifurcation Analysis of Gear Transmission in Non-stationary Regimes Under Angular Formalism

Numerical continuation methods, a powerful tool for studying the dynamics of nonlinear systems, are typically limited to steady-state regimes such as static or periodic solutions. Moreover, they cannot be directly applied to rotating mechanisms due to the presence of rigid-body modes. In this contribution, we propose a method to extend such methods to the aforementioned applications, such that they may provide meaningful information about the bifurcations to be encountered under non-stationary conditions. The key idea is to use an angular transformation of the equations of motion, followed by changes of variables and the use of the Harmonic Balance Method (HBM). Our method is tested on an academic gear transmission system with one gear mesh. Our model clearly shows how nonlinearity is introduced by the cyclic periodic stiffness of gear mesh without any assumption on shaft speed. The stiffness variation will be described by a Fourier series and its waveform will be explored. Based on this simulation data set, we propose to investigate the differences induced by a generic defect by analyzing the different nonlinear characteristics of the signals of interest, such as Instantaneous Angular Speed (IAS) or vibration. These simulations will also offer opportunities to better understand nonlinear expression of gear faults in non-stationary conditions of gear transmission systems.