Analyzing Time-varying Meshing Stiffness with Different Spalling Shapes Considered
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摘要: 考虑斜齿轮副端面重合度大于轴向重合度时的单齿啮合接触线表达式, 针对行星轮系统中啮合齿轮中基圆大于齿根圆的情况, 建立斜齿轮变截面悬臂梁模型, 采用势能法、切片法和自适应递推复合Lobatto数值积分法求解斜齿轮时变啮合刚度, 通过与有限元方法及经验法进行对比, 验证所建模型的可行性。在此基础上, 分析了不同长度、宽度、径向位置(齿根到齿顶)的剥落故障及不同剥落形状对时变啮合刚度的影响, 研究结果表明: 不同剥落长度对斜齿轮副在剥落区域的啮合位置影响较为明显; 随着剥落宽度的增加, 时变啮合刚度线性降低, 不同径向位置剥落, 在越靠近齿根的位置对时变啮合刚度影响越大; 不同剥落形状下, 三角形和圆形剥落引起时变啮合刚度非线性降低, 四边形剥落使时变啮合刚度线性降低。Abstract: Taking into consideration the single-tooth meshing pair contact line expression when the helical gear pair transverse contact ratio is greater than the overlap ratio and when the base circle of the helical gear is larger than the root circle, a helical gear variable cross-section cantilever beam model is established. The slicing method and the adaptive recursive compound Lobatto numerical integration method are used to solve the time-varying mesh stiffness (TVMS) of the helical gear. The feasibility of the model is verified by comparing it with the FEM and the empirical method. Based on the model, the effects of spalling fault and different spalling shapes on the TVMS of different spalling lengths, widths, radial positions are analyzed. The results show that different spalling lengths have obvious influences on the meshing position of the helical gear pair in the spalling area. The TVMS decreases linearly with the increase of the spalling width; the influence of spalls at different radial positions is greater than that on the TVMS at the positions closer to the root. With different spalling shapes, triangular and circular spalls cause the non-linear reduction of TVMS; quadrilateral spalls cause their uniform reduction.
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Key words:
- potential energy method /
- helical gear /
- slicing method /
- time-varying mesh stiffness
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表 1 多项式中各参数的值
系数 L* M* P* Q* Ai -5.574×10-5 60.111×10-5 -50.952×10-5 -6.2042×10-5 Bi -1.9986×10-3 28.100×10-3 185.50×10-3 9.0889×10-3 Ci -2.3015×10-4 -83.431×10-4 0.0538×10-4 -4.0964×10-4 Di 4.7702×10-3 -9.9256×10-3 53.300×10-3 7.8297×10-3 Ei 0.0271 0.1624 0.2895 -0.1472 Fi 6.8045 0.9086 0.9236 0.6904 
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表 2 基体变形中各参数的值
齿轮 L* M* P* Q* 太阳轮 6.882 1 1.085 8 3.415 5 0.531 5 行星轮 6.877 4 1.037 2 3.165 3 0.567 2
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表 3 齿轮的基本参数
参数 太阳轮(被动轮) 行星轮(主动轮) 模数mn/mm 15 15 压力角α0/(°) 20 20 螺旋角β/(°) 7.5 7.5 齿数Z 18 34 齿宽/mm 345 345 总切片数N 120 120 总重合度ε 2.546 2.546 半径比h 1.69 1.4
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表 4 不同方式得到的时变啮合刚度最大值比较
方法 啮合刚度最大值/1010 (N·m-1) 误差/% 经验法 - - 有限元法 2.123 0 本文 2.079 2.1 仅考虑齿宽 2.193 25 3.2 重合度建立接触线 2.236 79 5.0
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表 5 不同方式得到的时变啮合刚度最小值比较
方法 啮合刚度最小值/1010 (N·m-1) 误差/% 经验法 - - 有限元法 1.998 0 本文 1.976 1.1 仅考虑齿宽 2.101 43 4.9 重合度建立接触线 2.124 15 5.9
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表 6 不同方式得到的平均啮合刚度比较
方法 平均啮合刚度/1010 (N·m-1) 误差/% 经验法 2.030 5 0 有限元法 2.059 5 1.4 本文 2.027 5 0.15 仅考虑齿宽 2.147 34 5.4 重合度建立接触线 2.180 47 7.3
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