Unified Model of Gear Characteristic Lines and Its Application in the Evaluation of Gears 3D Deviation
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摘要: 表征渐开螺旋齿轮的特征线有多种，广为熟知的是几何意义明确的渐开线和螺旋线。其实齿面上还有法向啮合齿形、接触线等工程价值突出的其他特征线。但特征线增多带来了两个问题，一是复杂的特征线方程彼此不关联，数学上缺乏统一性；二是除了渐开线和螺旋线，其他特征线没有测量手段，缺乏可测性。据渐开线齿轮传动的特点，将齿面特征线映射到啮合平面里，发现齿面上各条特征线在啮合平面里都有各自对应的二维直线，以此建立直线模型统一表达了齿面各种特征线；基于齿轮三维误差测量数据和特征线统一模型，提出了各种特征线偏差的提取方法，应用于测量实践，通过与通用齿轮仪器测量的渐开线偏差和螺旋线偏差作比对，证明了特征线统一模型及特征线偏差提取方法的有效性和实用性，解决了齿面复杂特征线的可测性问题。同时，齿轮特征线统一模型在齿轮工艺误差溯源、传动性能预报等方面也有重要应用价值。
表征渐开螺旋齿轮的特征线有多种，广为熟知的是几何意义明确的渐开线和螺旋线。其实齿面上还有法向啮合齿形、接触线等工程价值突出的其他特征线。但特征线增多带来了两个问题，一是复杂的特征线方程彼此不关联，数学上缺乏统一性；二是除了渐开线和螺旋线，其他特征线没有测量手段，缺乏可测性。据渐开线齿轮传动的特点，将齿面特征线映射到啮合平面里，发现齿面上各条特征线在啮合平面里都有各自对应的二维直线，以此建立直线模型统一表达了齿面各种特征线；基于齿轮三维误差测量数据和特征线统一模型，提出了各种特征线偏差的提取方法，应用于测量实践，通过与通用齿轮仪器测量的渐开线偏差和螺旋线偏差作比对，证明了特征线统一模型及特征线偏差提取方法的有效性和实用性，解决了齿面复杂特征线的可测性问题。同时，齿轮特征线统一模型在齿轮工艺误差溯源、传动性能预报等方面也有重要应用价值。
Abstract: There are many kinds of characteristic curves that characterize involute helical gears. The most well-known are involute and helix with clear geometrical definition. In fact, there are other characteristic curves with outstanding engineering value such as profile of path of contact and contact line on the tooth surface. However, the addition of characteristic lines brings two problems. One is that the complex equations of the characteristic curves are not related to each other and lack mathematical unity. The other is that except for involute and helix, other characteristic curves have no measuring method, which results in a lack of measurability. According to the characteristics of involute gear transmission, the characteristic curves on the tooth surface are mapped to the meshing plane. It is found that each characteristic curves on the tooth surface has its corresponding 2-D straight line in the meshing plane. Based on this, a straight-line model is established to uniformly express various characteristic curves on the tooth surface. On basis of the 3-D deviation measurement data of the gear and the unified model of the characteristic curves, the extraction method of the deviations of various characteristic curves is proposed, which is applied to the measurement practice. By comparing with the profile deviation and helix deviation measured by the general gear measuring instrument, the effectiveness and practicability of the unified model of the characteristic curves and the extraction method of the deviations of the characteristic curves are proved, and the measurability of the complex characteristic curves on the tooth surface is solved. Meanwhile, the unified model of the gear characteristic curves also has important application value in the traceability of gear process error and the prediction of transmission performance. -
表 1 测量结果对比
名称 序号 统一模型评定数值/μm P26测量值/ μm 渐开线倾斜偏差fHα 1 -18.5 -18.7 2 -2.5 -3.0 3 13.9 14.1 渐开线形状偏差ffα 1 12.0 11.7 2 12.4 13.3 3 12.3 12.0 渐开线总偏差Fα 1 19.8 20.8 2 11.7 12.2 3 18.8 21.2 螺旋线倾斜偏差fHβ 1 40.3 43.5 2 25.8 24.2 3 4.5 4.4 螺旋线形状偏差ffβ 1 12.0 11.3 2 12.0 12.1 3 11.5 12.1 螺旋线总偏差Fβ 1 34.8 35.9 2 24.3 23.9 3 15.9 14.0 法向齿形倾斜偏差fHan 1 5.8 — 法向齿形形状偏差ffan 1 20.1 — 法向齿形总偏差Fan 1 22.8 — 接触线倾斜偏差fHc 1 16.5 — 接触线形状偏差ffc 1 16.9 — 接触线总偏差Fc 1 26.5 — 
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