﻿ 基于双锥砂轮的环面蜗轮滚刀前刀面成形方法<sup>*</sup>
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1. 中国农业大学 工学院, 北京 100083;
2. 现代农装优化设计北京市重点实验室, 北京 100083

Forming method of hourglass worm gear hob rake faces based on double-cone grinding wheel
YANG Jie1, LI Haitao1,2, RUI Chengjie1, LONG Xinjiani1, WEI Wenjun1
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: 2018-02-08; Accepted: 2018-03-23; Published online: 2018-05-21 15:11
Foundation item: National Natural Science Foundation of China (51475460)
Corresponding author. LI Haitao, E-mail:h.li@cau.edu.cn
Abstract: For multi-thread hourglass worm gear hobs, spiral flutes are typically used to reduce the absolute value of the negative rake angle and to equalize the rake angles on the left and right sides of each tooth, which will improve the cutting performance of the hob. Because of the different spiral angle on the basic worm helical surface, the rake angle on both sides of each tooth is more difficult to balance. However, there are few researches on the precise forming method of the spiral flute rake face. Based on the NC machine tool for hourglass worm, a method for grinding the rake face of spiral flute with double-cone grinding wheel is proposed. According to the gear meshing theory, a mathematical model was established for the rake face of the planar double enveloping hourglass worm gear hob spiral flute formed by the double-cone grinding surface. The formula for calculating the rake angle on both sides of the tooth is given. The calculation results show that if the hob with straight flute is used, the rake angles are between -19.530 3° and 19.530 4°. Through proper parameter selection, the rake angles can be between -8.1° and 7.3° when the hob with spiral flute is used, which effectively reduces the absolute value of the negative rake angle of the tooth. The spiral flute of hourglass worm gear hob was simulated and the rake angle was measured in the simulation software. The measured results coincide with the calculated results and this proves the correctness of the method.
Keywords: hourglass worm gear hob     chip flute     rake face     rake angles     double-cone grinding surface

1 前刀面的数学模型

1.1 加工前刀面的坐标系及相对运动关系

 图 1 加工前刀面的坐标系 Fig. 1 Coordinate systems of rake face machining

 (1)

1.2 坐标变换

1.3 滚刀基本蜗杆与刀座速比iqd1的求解

 (2)

 (3)

 (4)

 (5)
1.4 滚刀基本蜗杆螺旋面的矢量表达式

 (6)

 (7)

1.5 双锥产形面的矢量表达式

 图 2 双锥产形面 Fig. 2 Double-cone grinding surface

 (8)

 (9)

 图 3 砂轮在刀座静坐标系中的位置 Fig. 3 Position of grinding wheel in tool holder's static coordinate system
 (10)

 (11)

1.6 在σq1中滚刀前刀面的方程

σqod坐标系中，双锥产形面与前刀面的相对速度为(vqd1)qod，由齿轮啮合原理[19-21]得到双锥产形面与前刀面共轭条件函数为

 (12)

Фqod=0得到双锥产形面的一次包络共轭条件方程为

 (13)

 (14)
 (15)

 (16)
1.7 前角Vq的表达式

σq1坐标系中，由刃口线上任一点在前刀面上的法向量(nqd)q1和在螺旋面上的法向量(nd)q1求得该点处的前角Vq表达式为[10-11, 22]

 (17)

2 算例及结果分析

 参数 数值 中心距a/mm 160 速比i12 10 滚刀头数z1 4 蜗轮分度圆压力角α/(°) 21.8667 蜗轮分度圆直径d2/mm 255 蜗杆分度圆直径d1/mm 65 滚刀齿顶圆弧半径Ra1/mm 122.24 滚刀齿根圆弧半径Rf1/mm 133.24 滚刀工作长度Lw/mm 90 滚刀喉部分度圆导程角γm/(°) 21.4205 产形平面轴线倾角β/(°) 22.5 主基圆直径db/mm 95

 齿号 左侧 右侧 前角/(°) 轴向位置/mm 前角/(°) 轴向位置/mm 1号 -15.7159 -43.5 16.9998 -35.8 2号 -18.1932 -24.5 19.0122 -15.4 3号 -19.5303 -4.71 19.5304 4.71 4号 -19.0126 15.4 18.1937 24.5 5号 -17.0004 35.8 15.7165 43.5

 齿号 左侧 右侧 前角/(°) 轴向位置/mm 前角/(°) 轴向位置/mm 1号 5.5645 -37.6659 -5.4788 -30.6111 2号 6.6612 -21.2130 3.0636 -13.3234 3号 7.2522 -4.1854 6.2043 3.9941 4号 1.6208 13.3526 0.7470 21.4250 5号 -1.3445 31.1470 -8.1071 38.3567

 图 4 不同容屑槽两侧前角变化规律 Fig. 4 Change law of rake angles on both sides of different spiral flutes

3 VERICUT加工滚刀螺旋槽

 图 5 仿真加工滚刀过程 Fig. 5 Simulation of processing hob process

 图 6 测量分度环面上刀齿两侧的夹角 Fig. 6 Measured angle between sides of the teeth on indexing tours

Vq=Wq-90°，计算出刀齿两侧的前角Vq，如表 4所示。

 齿号 左侧 右侧 前角/(°) 轴向位置/mm 前角/(°) 轴向位置/mm 1号 5.6286 -37.6659 -5.3759 -30.6111 2号 6.3011 -21.2130 2.5098 -13.3234 3号 7.1336 -4.1854 5.6398 3.9941 4号 1.6094 13.3526 0.1771 21.4250 5号 -1.0385 31.1470 -8.3156 38.3567

4 结论

1) 针对直槽滚刀负前角绝对值较大、刀齿左右两侧切削条件不均衡的问题，提出了采用双锥砂轮磨削加工环面蜗轮滚刀螺旋槽前刀面的方法。

2) 通过建立磨削加工坐标系，推导由双锥产形面展成平面二次包络环面蜗轮滚刀螺旋容屑槽前刀面的数学模型，完成每个刀齿在分度环面螺旋线上的前角数值的计算。计算结果表明，相较于直容屑槽滚刀，螺旋容屑槽滚刀对应的前角下降了60%，有效减小了负前角的绝对值，有利于均衡刀齿两侧的切削条件。

3) 对滚刀的螺旋容屑槽进行仿真加工并且在仿真软件中对前角进行测量，测量结果与理论计算误差小于0.6°，验证了本文方法的正确性。

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#### 文章信息

YANG Jie, LI Haitao, RUI Chengjie, LONG Xinjiani, WEI Wenjun

Forming method of hourglass worm gear hob rake faces based on double-cone grinding wheel

Journal of Beijing University of Aeronautics and Astronsutics, 2018, 44(11): 2423-2429
http://dx.doi.org/10.13700/j.bh.1001-5965.2018.0090