﻿ 环面蜗轮滚刀刃带宽受周向定位误差影响分析<sup>*</sup>
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1. 中国农业大学 工学院, 北京 100083;
2. 现代农业装备优化设计北京市重点实验室, 北京 100083

Influence of circumferential positioning error of an hourglass worm wheel hob on land width
RUI Chengjie1, LI Haitao1,2, YANG Jie1, LONG Xinjiani1, TAI Jianjian1, DING Ning1
1. College of Engineering, China Agricultural University, Beijing 100083, China;
2. Key Laboratory of Optimal Design of Modern Agricultural Equipment in Beijing, Beijing 100083, China
Received: 2018-10-22; Accepted: 2018-11-08; Published online: 2019-02-20 15:52
Foundation item: National Natural Science Foundation of China (51475460)
Corresponding author. LI Haitao, E-mail: h89533@cau.edu.cn
Abstract: Land surfaces of an hourglass worm wheel hob needs to be controlled within a range to ensure the strength and sharpness of cutting edges. Circumference positioning errors lead to the changes of the land width. In order to grind out the required land surfaces, this paper studies the influence of the circumferential positioning errors on the land width. A mathematical model for solving the land width with the circumferential positioning error is established. The variation of the land width along the axial direction and the tooth height direction of the hob is studied. Effects of the circumferential positioning errors on the land width are simulated based on a four-axis linkage virtual hourglass worm grinding machine in VERICUT. The results show that the circumferential positioning errors have the greatest influence on the land width of the side teeth of the hob. There is a strong correlation between the land width and the errors, showing a linear relationship. In an example, the allowable range of the circumferential positioning error is between -0.122° and 0.054° when the land width e=1.0 mm varies in the range of ±10%.
Keywords: hourglass worm wheel hob     grinding     land width     circumferential positioning error     error limitation

1 环面蜗轮滚刀的特征

 图 1 整体式环面蜗轮滚刀 Fig. 1 Integral hourglass worm wheel hob
 图 2 环面蜗轮滚刀刀齿结构示意图 Fig. 2 Schematic of tooth structure of hourglass worm wheel hob

2 磨削侧后角面的数学模型 2.1 双锥面包络环面蜗杆磨床及其加工坐标系

 图 3 环面蜗杆磨床 Fig. 3 An hourglass worm grinding machine
 图 4 双锥磨头 Fig. 4 A dual-cone grinding head

 图 5 加工坐标系 Fig. 5 Machining coordinate systems

 图 6 双锥砂轮 Fig. 6 A dual-cone grinding wheel
2.2 双锥面包络环面蜗杆螺旋面的数学方程

 图 7 双锥面包络环面蜗杆螺旋面成形原理 Fig. 7 Forming principle of helical surface of dual-cone enveloping hourglass worm

 (1)

 (2)

 (3)

2.3 侧后角面磨削运动的求解

 图 8 侧后角面磨削 Fig. 8 Relief surface grinding

 (4)

3 含周向定位误差的刃口线求解

 图 9 展成螺旋槽前刀面 Fig. 9 Generating spiral flute rake face

γm为环面蜗轮滚刀的喉部导程角，d1为环面蜗轮滚刀分度圆直径。在坐标系σ1中，前刀面的位置向量为rc。根据齿轮啮合原理，基准螺旋槽前刀面的方程如式(5)所示，其他前刀面是基准前刀面沿滚刀轴线的圆周阵列。

 (5)

 图 10 周向定位误差与前刀面 Fig. 10 Circumferential positioning errors and rake faces

 (6)

 (7)

4 典型算例分析

 参数 数值 中心距a/mm 260 传动比i1d 8 蜗杆头数z1 5 蜗轮齿数z2 40 滚刀分度圆直径d1/mm 110 蜗轮分度圆直径d2/mm 410 滚刀齿顶环面半径Ra/mm 193 滚刀齿根环面半径Rfd/mm 214 侧后角面磨削半径Rf/mm 220 滚刀工作长度L/mm 135 滚刀喉部导程角γm/(°) 24.98 砂轮安装倾角β/(°) 30 砂轮齿形角αd/(°) 20 砂轮半径rd/mm 50 砂轮距a0/mm 164 砂轮齿顶宽sa/mm 6

 图 11 侧后角面磨削过程 Fig. 11 Relief surface grinding process
 图 12 环面蜗轮滚刀 Fig. 12 Hourglass worm wheel hob
4.1 误差影响规律及误差最大位置的分析

 图 13 刃带宽变化情况(DC=0°) Fig. 13 Variation of land width with DC=0°

 图 14 刃带宽变化情况(DC=-0.5°) Fig. 14 Variation of land width with DC=-0.5°

 图 15 刃带宽变化情况(DC=0.5°) Fig. 15 Variation of land width with DC=0.5°

 DC/(°) 齿顶刃带宽/mm 刀齿1 刀齿2 刀齿3 刀齿4 刀齿5 -0.5 0.4693 0.4903 0.4987 0.4927 0.4730 -0.35 0.6402 0.6495 0.6535 0.6503 0.6406 -0.15 0.8681 0.8619 0.8599 0.8604 0.8642 0 1.0389 1.0211 1.0147 1.0180 1.0319 0.15 1.2097 1.1803 1.1696 1.1756 1.1996 0.35 1.4374 1.3925 1.3760 1.3858 1.4233 0.5 1.6082 1.5516 1.5308 1.5435 1.5911

 图 16 周向定位误差对刀齿齿顶刃带宽的影响 Fig. 16 Influence of circumferential positioning error on top land width of each tooth
4.2 刃带宽与周向定位误差的相关性分析

 DC/(°) 刃带宽/mm 齿顶 分度 齿根 -0.5 0.4693 0.5182 0.5623 -0.35 0.6402 0.6673 0.6964 -0.15 0.8681 0.8660 0.8751 0 1.0389 1.0150 1.0091 0.15 1.2097 1.1639 1.1430 0.35 1.4374 1.3625 1.3215 0.5 1.6082 1.5113 1.4554

 图 17 刀齿1刃带宽变化趋势 Fig. 17 Variation of land width of tooth No.1

DC=0°时，刃带宽为e0。直线的斜率k表示周向定位误差对刃带宽的影响程度，刃带宽与误差之间的关系式可表达为

 (8)

4.3 周向定位误差允许误差限

 图 18 刀齿1齿顶刃带宽与周向定位误差的变化关系 Fig. 18 Relationship between top land width of tooth No.1 and circumferential positioning error

5 结论

1) 螺旋面和侧后角面是在同一台四轴联动双锥面包络环面蜗杆磨床上连续加工而成的，因此，无论周向定位误差是否存在，磨削出来的侧后角面相对于螺旋面的位置不变，刃带线在螺旋面上的位置不发生变化。当存在周向定位误差时，侧后角面和螺旋面相对于螺旋槽前刀面的位置发生了变化，这种变化导致刃口线的位置发生了变化，最终引起刃带面的刃带宽发生变化。

2) 通过使无误差的前刀面绕滚刀轴线旋转一个周向定位误差角，可得到一个新的前刀面。新的前刀面能正确表达含有周向定位误差时，前刀面、刃带面和侧后角面三者之间的位置关系。通过联立含新前刀面的数学方程和螺旋面的数学方程，可得到新刃口线的坐标点。通过计算新刃口线和刃带线之间的距离，即可得到含有周向定位误差时的刃带宽。

3) 当周向定位误差DC < 0时，刃带宽es小于给定值e，当DC>0时，刃带宽es大于给定的刃带宽e。周向定位误差对滚刀边齿齿顶的刃带宽影响最大。

4) 刃带宽与周向定位误差存在线性关系，斜率k表示误差对该处的影响程度。对于特定的滚刀，当给定刃带宽e时，可以获得边齿齿顶的k值，并且能够获得表达式es=kDC+e0。根据此表达式，可计算出在刃带宽允许的变化范围内，周向定位误差DC允许的变化范围，为实际生产中机床的调整提供参考。

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#### 文章信息

RUI Chengjie, LI Haitao, YANG Jie, LONG Xinjiani, TAI Jianjian, DING Ning

Influence of circumferential positioning error of an hourglass worm wheel hob on land width

Journal of Beijing University of Aeronautics and Astronsutics, 2019, 45(6): 1096-1105
http://dx.doi.org/10.13700/j.bh.1001-5965.2018.0607