High-order contact analysis method of spiral bevel gear tooth surface based on ease-off
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摘要:
为应对弧齿锥齿轮二阶接触分析方法的不足与其高阶接触理论实现的复杂问题，基于ease-off拓扑曲面方程与弧齿锥齿轮齿面方程的结合以及传动误差与接触迹线和ease-off之间的解析关系，提出以传动比高阶导数和接触迹线短程曲率为高阶接触参数的离散齿面的高阶接触分析方法，并建立基于有限差分的简便计算方法。结果表明，高阶齿面的传动比高阶导数波动值分别为0.0031、0.0019与0.001，数值反映齿面形貌的全局特性；接触线短程曲率波动值分别为0.0000769、0.000586和0.000127，说明沿接触迹线的齿面接触过程的复杂性。结果不仅验证了离散齿面高阶接触分析方法的正确性与有效性，而且说明该方法降低了高阶接触参数的计算难度，为齿面全局设计提供了可能。
Abstract:In order to deal with the shortcomings of the second-order contact analysis method of spiral bevel gears and the complex problems of its high-order contact theory realization, based on the combination of ease-off topological surface equation and tooth surface equation of spiral bevel gears and the analytical relationship between transmission error and contact trace and ease-off, a high-order contact analysis method for discrete tooth surfaces with high-order contact parameters of high-order derivative of transmission ratio and short-range curvature of contact trace was proposed, and a simple calculation method based on finite difference was established. The results showed that the fluctuation values of the high-order derivative of the transmission ratio of the high-order tooth surface were 0.0031, 0.0019 and 0.001, respectively, which reflected the global characteristics of the tooth surface morphology. The short-range curvature fluctuation values of the contact line were 0.0000769, 0.000586 and 0.000127, respectively, indicating the complexity of the tooth surface contact process along the contact trace. The results not only verified the correctness and effectiveness of the discrete tooth surface high-order contact analysis method, but also showed that the method reduced the calculation difficulty of high-order contact parameters, providing the possibility for the global design of tooth surface.
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图 3 Ease-off模型与坐标投影关系
Figure 3. Relationship between ease-off model and coordinate projection
图 6 离散齿面高阶接触参数求解网格
Figure 6. A mesh for solving high-order contact parameters of discrete tooth surfaces
图 8 三组改进齿面的ease-off拓扑曲面（单位：μm）
Figure 8. Three groups modified tooth surface ease-off topological surfaces（unit：μm）
图 10 三组改进齿面的传动误差曲线
Figure 10. Three groups modified tooth surface transmission error curves
图 11 三组改进齿面的传动比高阶导数曲线
Figure 11. Three groups modified tooth surfaces higher order derivative curves of transmission ratio
图 12 三组改进齿面的接触迹线短程曲率曲线
Figure 12. Three groups modified tooth surface contact trace short-range curvature curves
图 13 初始齿面与三组修形齿面传动比高阶导数结果
Figure 13. Higher order derivative results of the transmission ratio between the initial tooth surface and the three groups modified tooth surfaces
图 14 初始齿面与三组修形齿面接触迹线结果
Figure 14. Contact trace results between initial tooth surface and three groups modified tooth surface
图 15 初始齿面与三组修形齿面接触迹线短程曲率结果
Figure 15. Short-range curvature results of contact trace between initial tooth surface and three groups modified tooth surface
表 1 齿轮副基本参数
Table 1. Basic parameters of gear
参数 数值或说明 小轮 大轮 齿数 34 43 大端模数/mm 2.0 齿宽/mm 15 压力角/（°） 20 螺旋角/（°） 20 轴交角/（°） 108 节锥角/（°） 44.8608 63.1392 面锥角/（°） 49.4637 66.0355 根锥角/（°） 41.9645 58.5362 旋向 右旋 左旋 
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表 2 小轮工作面切齿参数
Table 2. Pinion cutting parameters of working surface
参数 数值 刀尖半径$ {R_{\text{c}}} $/mm 44.4713 角向刀位$ {q_0} $/rad 1.0827 径向刀位$ {S_{\text{k}}} $/mm 51.5697 滚比$ {R_{\text{a}}} $ 1.4391 垂直轮位$ E $/mm −3.2839 轴向轮位$ {X_{\text{p}}} $/mm −0.2389 床位$ {X_{\text{b}}} $/mm 0.1598 安装角$ {\delta _{\text{m}}} $/rad 0.7324
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表 3 预置ease-off修形参数
Table 3. Preset ease-off trim parameters
参数 a11 a12 a13 a14 a15 1 0 0.000005 0 0.0000012 0.0002063 2 0 −0.000005 0 0.0000014 0.0002063 3 0.00001 0 0.000006 0.0000014 0.0002063
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