基于解析解的人字齿轮齿廓修形动态分析

Dynamic analysis of tooth profile modification to herringbone gears based on analytical solutions

根据人字齿轮传动的结构,考虑时变啮合刚度、齿侧间隙,并把齿廓修形作为一种时变齿侧间隙计入,建立了单级人字齿轮传动的弯-扭-轴耦合动力学模型和相应的非线性动力学方程.对方程的数值解进行分析,发现合理的修形使人字齿轮传动不发生齿面的完全分离.基于多尺度法对不发生齿面完全分离时的人字齿轮传动系统进行了摄动分析,得到了系统在主共振频率附近幅频响应的近似解析解.相比数值方法,解析法具有很高的求解效率.最后,根据近似解析解研究了修形参数对动载系数幅值的影响.研究发现:螺旋角越大使动载系数最小所需的修形量越大,而所需的修形长度越小.另外,使动载系数最小的最优修形参数解域呈新月形,其随着主动轮支承刚度的增大而先增大后减小;随着被动轮支承刚度的增大而持续减小.

Based on the structure of a single-stage herringbone gear transmission,a nonlinear lateral-torsional-axial coupling dynamic model and corresponding dynamic equations considering the time-varying mesh stiffness,tooth profile modification and tooth backlash were proposed,for which the tooth profile modification was considered as a time-varying tooth backlash.The numerical solutions were analyzed and the results show that the total tooth separation can be suppressed with rational tooth profile modification.Then,according to the method of multiple scales,perturbation analysis of the dynamic equations finds that appropriate analytical frequency response solutions for the case cause no total tooth separation at primary resonance.Compared to the numerical method,the analytical method is more efficient.At last,the influence of the modification parameters on the amplitude of the dynamic load factor was analyzed on the basis of the analytical solutions.The analysis results present that the magnitude of tip relief decreases as the helical angle increases to reduce the dynamic load factor,while the length of modification increases in this case.In addition,the optimal solution domain of the minimum dynamic load factor is a meniscoid zone,it increases firstly as the support stiffness of the driving gear increases to a value,after that value,it decreases when the support stiffness of the driving gear continues to increase;the optimal solution domain decreases as the support stiffness of the driven gear continuously increases.