梁成成(1994—), 男, 重庆大学硕士研究生, 主要从事准双曲面齿轮啮合理论研究, (E-mail)
基于奥利康制准双曲面齿轮切齿原理和加工方法,分析了三面刀刀头的结构和安装位置,提出了基于刀具NS(neutral surface)平面法向基准下刀盘数学模型的建立方法。在此基础上,推导了奥利康制准双曲面齿轮的加工机床坐标系,建立了成形法大轮和展成法小轮的齿面数学模型,整理了一套基于三面刀盘奥利康制准双曲面齿轮精确化建模流程。通过齿面模型得到的数学齿面与通过KIMOS软件得到的45点齿面进行对比和实际接触印痕与理论接触印痕对比两种方法进行齿面验证。结果表明:大小轮推导齿面与实际齿面齿线和几何形貌基本一致,小大轮齿面基本重合;小轮凹面最大误差位于小端偏齿顶处,其值为0.007 5 mm,大轮凸面最大为0.002 3 mm;KIMOS计算的理论轮齿接触分析(TCA)、轮齿承载接触分析(LTCA)印痕与有限元计算印痕的位置方向基本一致,验证了齿面的正确性。
Based on the tooth cutting principle and manufacturing method of Oerlikon's hypoid gear, the structure and assembly location of three-faced blade is analyzed and the method to establish the mathematical model of three-face cutter head and plate (SPIRON) by the normal benchmark of the NS plane of tool is proposed. According to the manufacturing coordinate systems through HFT methods, the mathematical models of pinion with generating method and gear with shaping method is established on the basis of the NS benchmark. Furthermore, the technological process of Oerlikon's hypoid gear modeling based on the three-face cutter is proposed. The two compare methods were made between the mathematical tooth flank by tooth surface mathematical model and 45 point tooth surface by the KIMOS of both pinion and gear and the contact pattern with actual and theory. Results reveal that the geometry and tooth line of theoretical tooth surface of gear and pinion corresponds well with the actual tooth surface, the maximum the concave error of pinion was 0.007 5 mm, the maximum convex error of gear about 0.002 3 mm. The contact pattern is consistent between the contact patterns of tooth contact analysis (TCA) and loaded tooth contact analysis (LTCA) by KIMOS and calculated by the initial element method, it verified the precision of the mathematical model.
准双曲面齿轮具有重合度大、容差能力强、啮合稳定性高,且适用于大交角、可偏置安装等优点,广泛运用于航空、汽车、船舶等传动系统之中。与格里森制齿轮相比,奥利康制齿轮具有加工效率高、承载能力强及噪声低等优点。由于奥利康制齿轮加工为连续分度切削的过程, 刀头的结构、安装位置方向及刀盘摇台加工齿轮三者间的协调运动要求十分严格;同时,基于不同的参考基准,刀盘和切削参数是不相同的,造成奥利康制准双曲面齿轮齿面成形理论变得更加复杂。在现有研究中通常采用切屑方向基准进行齿面建模,而对于基于NS(neutral surface)法向基准的齿面建模几乎没有,因此有必要研究此基准下齿面数学模型的推导。
针对准双曲面齿轮齿面成形理论,国内外相关学者进行了一定的研究。Litvin[
奥利康制准双曲面齿轮采用端面滚齿法加工。
奥利康准双曲面齿轮切齿原理
Cutting principle of Oerlikon's hypoid gear
准双曲面齿轮建模基准一般分为两种:切削方向基准和NS法向基准。NS法向基准建立在NS平面的法向方向,切削方向基准建立在切削曲线的相切方向,如
切削基准的定义
Benchmark definition of cutting
刀具切削刃截面
Cutting edge section of cutting tool
以左弦内刀为例,在坐标系
刀尖圆弧(
其中
图中,
端面滚齿刀盘模型
Knife plant model of face-hobbed
切削内刀与外刀切削刃参考点间的夹角为
式中:
奥利康制准双曲面齿轮加工机床(如
左旋准双曲面齿轮机床加工坐标系
Machining coordinate system of left hypoid gear machine
式中:
成形法加工时,摇台固定不动,摇台增量角为零,即
式中:
展成法加工时,齿面方程通过式(11)依旧超静定,需要经过啮合方程求解出各个刀盘旋转角下的摇台增量角或者毛坯旋转角进行求解。在毛坯坐标系
基于齿轮啮合理论,齿轮的啮合方程为
式中:
对于右旋准双曲面齿轮齿面,采用右旋刀盘,加工刀盘逆时针旋转,刀具安装方向沿着切削方向;在机床坐标系中,刀盘安装于第1象限,毛坯安装于第4象限,摇台初始安装角、垂直轮位与左旋齿轮相反;毛坯旋转方向与左旋齿轮相反为顺时针旋转,以相同的求解方式可得到右旋齿轮在毛坯坐标系下的切削轨迹面方程。
齿轮基本参数
Basic parameters of gear
参数 | 偏置距 |
轴交角 |
头数 |
齿数 |
起始安装角 |
切削刃半径 |
齿廓压力角 |
刀具方向角 |
重磨角 |
主刃后角 |
前角 |
|
小轮 | 凹面 | 38.1 | 90 | 17 | 11 | -21.176 | 1 324.49 | 19.445 | 22.93 | 4.705 | 12.673 | 9.642 |
凸面 | -10.588 | 1 305.95 | 20.554 | 22.93 | 4.935 | 13.389 | 10.320 | |||||
大轮 | 凹面 | 47 | 10.588 | 1 030.74 | 17.485 | 22.93 | 4.304 | 12.678 | 10.290 | |||
凸面 | 21.176 | 1 276.45 | 22.515 | 22.93 | 5.319 | 13.382 | 9.601 |
机床参数
Machine parameters
参数 | 刀倾角 |
刀转角 |
径向刀位 |
初始摇台角 |
垂直轮位 |
床位 |
水平轮位 |
机床根锥角 |
滚比 |
小轮 | 3.332 5 | 146.940 5 | 119.639 1 | 52.667 3 | 39.835 2 | -12.248 6 | 3.115 6 | 23.445 3 | 4.722 168 |
大轮 | 0.000 0 | 0.000 0 | 122.009 0 | -29.704 8 | 0.000 0 | 0.000 0 | 0.000 0 | 64.799 5 | 0.000 000 |
根据上文推导的齿面数学方程,依据
啮合实体装配模型
The model of meshing solid assembly
奥利康准双曲面齿轮建模流程
Modeling process of Oerlikon's hypoid gear
根据
加工与数学齿轮模型
Machine and mathematics model of pinion and gear
通过KIMOS设计模块得到45点齿面点集,部分数据如
Kimos理论齿面45点齿面坐标点集(部分)
Theory tooth surface 45 points set of Kimos (Portion)
点集编号 | 小轮凹面 |
小轮凸面 |
大轮凹面 |
大轮凸面 |
1 | 27.137,27.489,106.206 | 30.170, 24.121, 106.206 | 3.338, 95.311, 36.380 | -1.157, 95.364, 36.679 |
2 | 28.876,27.103,106.206 | 31.015, 24.625, 106.206 | 4.120, 95.882, 35.413 | -1.729, 95.956, 35.413 |
3 | 30.650,26.593,106.206 | 31.840, 25.155, 106.206 | 4.862, 96.449, 34.147 | -2.304, 96.956, 34.147 |
4 | 32.454,25.952,106.206 | 32.648, 25.707, 106.206 | 5.608, 97.009, 32.881 | -2.883, 97.129, 32.881 |
5 | 34.278,25.176,106.206 | 33.441, 26.276, 106.206 | 6.358, 97.565, 31.615 | -3.465, 97.710, 31.615 |
6 | 19.051,31.198,101.472 | 22.614, 28.720, 101.472 | 5.459, 91.735, 35.034 | 1.094, 91.891, 35.033 |
7 | 21.551,31.347,101.472 | 23.491, 29.920, 101.472 | 6.186, 92.291, 33.767 | 0.548, 92.496, 33.767 |
8 | 24.205,31.247,101.472 | 24.292, 31.180, 101.472 | 6.917, 92.841, 32.501 | 0.000, 93.098, 32.501 |
9 | 26.992,30.877,101.472 | 25.028, 32.489, 101.472 | 7.651, 93.385, 31.235 | -0.552, 93.696, 31.235 |
10 | 29.888,30.211,101.472 | 25.706, 33.956, 101.472 | 8.390, 93.924, 29.969 | -1.107, 94.292, 29.969 |
数学齿面坐标点集(部分)
Mathematical tooth surface points set (Portion)
点集编号 | 小轮凹面 |
小轮凸面 |
大轮凹面 |
大轮凸面 |
1 | 138.050, 46.199, -55.395 | 137.517, 44.124, -26.463 | 107.746, 41.534, 28.305 | 106.350, 12.844, 44.532 |
2 | 138.131, 45.291, -54.983 | 137.610, 44.062, -26.388 | 108.122, 38.497, 29.754 | 107.210, 14.158, 42.135 |
3 | 138.255, 43.930, -54.365 | 137.713, 44.018, -26.312 | 108.823, 32.807, 31.656 | 108.119, 15.316, 39.813 |
4 | 138.381, 42.569, -53.747 | 137.782, 43.993, -26.236 | 109.791, 25.589, 32.548 | 109.078, 16.348, 37.635 |
5 | 138.509, 41.208, -53.130 | 137.934, 43.987, -26.162 | 110.209, 22.954, 32.405 | 110.343, 17.551, 35.236 |
6 | 137.310, 50.835, -47.901 | 134.770, 46.831, -18.258 | 101.755, 31.179, 36.677 | 102.619, 4.170, 4.460 |
7 | 137.424, 49.420, -47.415 | 134.862, 46.757, -18.193 | 102.283, 23.681, 37.736 | 103.777, 6.463, 4.192 |
8 | 137.541, 48.006, -46.929 | 134.962, 46.701, -18.123 | 102.642, 185.518, 37.556 | 104.762, 8.075, 3.984 |
9 | 137.660, 46.591, -46.443 | 135.070, 46.663, -18.050 | 103.009, 13.352, 36.498 | 105.796, 9.509, 3.789 |
10 | 137.822, 44.707, -45.795 | 135.180, 46.646, -17.975 | 103.340, 93.920, 34.930 | 106.876, 10.820, 3.620 |
数学齿面与45点齿面对比
Comparison of 45 points tooth surface
通过ABAQUS加载特性分析得到大轮凸面的接触斑点与KIMOS轮齿接触分析(TCA)及承载接触分析(LTCA)在450 NM时计算的大轮凸面接触印痕进行对比,进一步验证齿面模型推导的准确性。KIMOS计算结果如
KIMOS理论TCA与LTCA印痕(大轮凸面)
Contact patterns (KIMOS)
根据上文建立的齿轮实体模型,通过ABAQUS建立有限元啮合模型,如
有限元啮合模型
Finite element meshing model
瞬时接触线
Instantaneous contact line
有限元计算印痕
Contact patterns (ABAQUS)
在实际接触印痕的位置、大小设计过程中,热处理工艺以及轴系支撑变形等因素应当被考虑。通过调整刀具及机床参数,可以调整设计印痕的位置,确保在热处理变形及支撑变形之后,啮合印痕的区域位于齿中,提高齿轮的接触状态。
通过齿面对比和接触印痕的验证,基于刀具NS法向基准的准双曲面齿轮建模方法是准确可行的。
笔者从奥利康准双曲面齿轮的实际加工出发,研究了加工刀盘的结构和位置,建立了基于刀具NS法向基准的刀盘数学模型和奥利康制准双曲面齿轮实体模型,并进行了相关验证,提出了一套基于NS基准的建模流程。该方法对准双曲面齿轮的精确化建模提供了新的思路,为后续相关齿轮齿面的设计优化提供了基础,具有较强的实用价值。
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