Vibration stability and bifurcation analysis of spur gear systems based on time-varying backlash and random load

Gear backlash and random load affect the stability of gear transmission, which are the main factors causing meshing impact. This study presents the establishment of a six-degree-of-freedom model for the spur gear system, employing the lumped parameter theory and D'Alembert's principle. The model comprehensively incorporates the effects of time-varying meshing stiffness (TVMS), static transmission error (STE), time-varying gear backlash, and random load. The system's equations of motion are solved using the fourth-order Runge-Kutta method. Subsequently, the dynamic response of the constant backlash constant load model is compared with that of the time-varying backlash constant load model and the constant backlash random load model. This research quantitatively dissects the impact of the nonlinear characteristics of backlash and load on the dynamic response of the gear system. This analysis is conducted utilizing a variety of methods, including bifurcation diagrams, Floquet characteristic multipliers, Lyapunov exponents, phase diagrams, and Poincaré sections. The study demonstrates that fluctuations in backlash and load during meshing compromise the stability of the periodic solution, thereby escalating the complexity of predicting system motion. Moreover, observations include the jump phenomena, quasi-periodic and periodic motions, and the trajectory of the system transitioning into a chaotic state via bifurcation. This investigation serves as a valuable reference for future explorations into spur gear systems.