Improved Nonlinear Dynamic Model of Helical Gears Considering Frictional Excitation and Fractal Effects in Backlash

Surface roughness and sliding friction are pivotal in determining the dynamic meshing performance of helical gears, especially under conditions of flexible support. In addition, the meshing parameters influenced by gear vibrations exhibit time-varying characteristics under flexible support stiffness, which is disregarded by many scholars. Based on this, a nonlinear dynamic model of a helical cylindrical gear system under flexible support conditions is developed, considering the coupling effects of dynamic friction and backlash influenced by fractal surface roughness. The motion differential equations of the system are derived using the Lagrange method, and numerical solutions are obtained through the Runge–Kutta method. The effect of several control parameters (driving speed, surface roughness and fractal dimension) on the dynamic response of gear system is studied, and the proposed dynamic model is compared with the traditional model under different support stiffness to demonstrate its adaptability to highly flexible support scenarios. The results indicate that the proposed dynamic model is better suited for flexible support structures. Moreover, the coupling effects of sliding friction and fractal backlash amplify the dynamic response of the gear system and introduce complex spectrum characteristics. This study provides theoretical guidance for the optimization of vibration and noise reduction designs in helical gear systems.