齿根安全条件下多状态啮合齿轮传动系统安全-吸引盆侵蚀与分岔

轮齿折断是齿轮主要失效形式之一。以包含单双齿交替啮合、轮齿脱啮和齿背接触等的多状态啮合直齿圆柱齿轮传动系统为研究对象,计算齿根危险截面上的弯曲应力,以齿根许用弯曲应力为边界条件,建立防止轮齿折断的齿根安全条件。基于胞映射方法,用四阶Runge-Kutta法数值计算考察区域内随负载和齿侧间隙变化时系统的吸引盆及其演变过程。判断齿根安全条件下吸引盆中不同运动类型吸引域的安全特性得到系统的安全-吸引盆。借助吸引子、多初值分岔图和最大Lyapunov指数图(TLE)分析安全-吸引盆的侵蚀与分岔机理。研究发现:吸引盆中不同运动类型吸引域在齿根安全条件下的安全特性不同;吸引子的出现和消失直接导致安全-吸引盆的分岔;多初值分岔图中出现周期跳跃和分岔是引发安全-吸引盆分岔的主要原因。研究结果可为直齿轮传动系统安全运行提供参考。

Erosion and bifurcation of safety-attraction basin for multi-state meshing gear transmission system under tooth root safety condition

Tooth fracture is one of the main failure modes of gears.Here,taking a meshing spur gear transmission system with multi-state of single and double teeth alternately meshing,tooth off meshing and tooth back contact as the research object,the bending stress on the dangerous section of tooth root was calculated,and the allowable bending stress of tooth root was taken as the boundary condition to establish the safety condition of tooth root to prevent tooth fracture.Based on the cell mapping method,the fourth-order Runge-Kutta method was used to investigate the system’s attraction basin and its evolution process with the variation of load and backlash.The safety characteristics of attraction field of different motion types in the attraction basin were determined under the condition of tooth root safety,and the safety-attraction basin of the system was obtained.The mechanism of erosion and bifurcation of the safety-attraction basin was analyzed using attractor,multi-initial value bifurcation diagram and top Lyapunov exponent(TLE).The results showed that safety characteristics of attraction field of different types of motion in the attraction basin are different under the condition of tooth root safety;appearance and disappearance of attractor directly lead to bifurcation of the safety-attraction basin;periodic jump and bifurcation occurring in the multi-initial value bifurcation diagram are the main cause for the safety-attraction basin bifurcation;the study results can provide a reference for the safe operation of spur gear transmission systems.