EDP​​ SCIENCES徽标
Open Access
Issue
金博宝
体积22,2021
文章编号 33.
Number of page(s) 11
迪伊 https://doi.org/10.1051/meca/2021032
网上发布 2021年4月30日

© L. Geng et al., Published by EDP Sciences 2021

Licence Creative CommonsThis is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1介绍

螺旋锥齿轮广泛应用于航空,航空航天,船用和机床的优点,其优点是顺畅,传输效率高,负载能力优异等,因此,螺旋锥齿轮的制造一直是一个重要的话题。许多专家和学者都进行了深入研究。主要的螺旋锥齿轮加工方法是面部铣削和面部滚动。

For face milling, according to machine method of pinion can be divided into five-cut and two-cut. Generalized theory and methods of spiral bevel and hypoid gears manufactured by the five-cut method have been comprehensively presented by several gear scientists [13]。Shtipelman [4] introduced the generalized theory of the five-cut method and calculated machine settings parameters of spiral bevel and hypoid gears by the five-cut method in Gleason Works. Litvin [5,6] proposed local synthesis and applied for tooth contact analysis (TCA), and determining the optimal machine settings parameters. For five-cut, the concave side and convex side adopts different machine setting parameters which the mesh performance can effectively control and correction.

在Gleason technology, two-cut method includes duplex spread blade method and duplex helical method. Traditionally, duplex spread blade method is used to process small module of spiral bevel gear. In book of spiral bevel gears published by Beijing gear factory proposed the calculation of machine setting parameters by duplex spread blade method, but its principle is not revealed [7]。K. Kawasaki and H. Tamura [8] proposed a method to manufacture gear with a large radius of curvature cutting edge and a modified tooth surface is obtained. Recently, Deng et al. [9[提出了一种通过双工扩散叶片方法制造螺旋锥齿轮的方法,并通过齿面修饰优化网状性能。

Duplex helix method is fist put forward by Gleason [10], but principle and calculation of machine setting parameters are not revealed. Tsay and Lin [11] developed a mathematical model can be applied to simulate tooth surface geometry machined by duplex spread blade and duplex helical method. Gonzalez-Perez [12] approached conversion of the specific machine settings parameters of a given generator to neutral machine-tool setting by duplex heliacal method, and parabolic profile on the blades of the head cutter was applied to adjust the contact pattern. Zhang et al. [13,14[揭示了通过定义三个参考点的非生成齿轮的螺旋方法的广义理论,并通过定义三个参考点来计算基本机器设置参数。然而,通过双工螺旋方法计算机器设置参数复杂,并且机器的要求是苛刻的。难以控制和正确的网格性能对于凹面,凸面由相同的机器设置参数处理。

In this paper, a double–side milling method to machine spiral bevel gear is proposed which the calculation of machine setting parameters is simple. Through inclination of root line and considering cut parameters, geometrical parameters are designed by double–side milling method. In order to guarantee the normal tooth meshing, a modified mean point is selected, and machine setting parameters are calculated in modified mean point. Aiming at optimize the bias in contact caused by cut number; a helical motion is introduced to modify the pressure angle on the pitch line. Contact performance is controlled by adjusting coefficient of helical motion.

2 Geometric parameter design

2.1 Tooth taper

The taper of tooth blank is different from single-side method to double-side method. Dedendum angle will affect tooth height in the direction of tooth length from toe to heel, and the influence of dedendum angle on tooth taper are analyzed as follow.

In single-side method, tooth height and tooth thickness are proportional to cone distance, as shown in图1a. A tangent angle formed at mean pointPa两侧的牙齿空间Pa1Pa2can be expressed as(1)

Heres是the mean point arc tooth thickness of paired gear.

虽然对于双面方法,通过双侧切割同时处理牙齿表面的两侧。两侧的齿线是同心弧,如图所示图1b. Then the tangent angle formed at mean pointPb两侧的牙齿空间Pb1Pb2can be expressed as(2)

HereO是the center of cut, andr是the nominal radius of cut,β是mean spiral angle.

Comparing图1, a difference of tangent angle occurs and can be expressed as(3)

HereRm是mean cone distance.

在制造期间,切口轴垂直于牙齿坯料的根锥。因此,牙齿坯料外部将比内部和在牙齿间隙两侧的平均点形成的切线切割,这将弥补Δψ1

为了研究Dependum角度对δ的影响ψ1, another pointp′ was taken near the tooth line, and the position of point was: ΔRin the direction of cone distance, Δhin the direction of tooth height, and Δs牙齿空间宽度的差异。两点之间的距离是δL,见图2.。The increment of tangent angle on tooth line can be expressed as:(4)

Hereα是标称压力角。

如果δ.ψ1 ≠ Δψ2, it may cause abnormal tooth taper using double-side method, which will seriously affect the strength and cut life. So this situation should be avoided.

There are many factors affect Δψ1and Δψ2from equations(3)and(4)。但确定后不应更改一些参数。所以专注角度θf是changed to meet the requirements Δψ1 = Δψ2。Finally, an ideal dedendum angle can be calculated as(5)

The above analysis is suitable for pinion and gear. If both machined by double-side method, the following relationship can be obtained(6)

Hereθf1,θf2are dedendum angle of pinion and gear respectively;s1,s2are the mean point arc tooth thickness, and meetz0(s1 + s2) = 2πRm,z0是等价数量的牙齿。

Then equation(6)can be expressed as(7)

假设标准深度专用角度σθs满足条件δψ1 = Δψ2, then a nominal radius of cut can be obtained from Equations(3)and(4)(8)

然后是一个标称半径rc可以根据等式选择(8), Substitutingrcinto equation(7), the corresponding sum of dedendum angle ∑θTcan be expressed as(9)

The nominal radiusrcwill affect sum of dedendum angle ∑θT和侧翼锥度。为了避免在计算和选择之间的标称半径差异引起的过度齿坯,推荐的标称半径范围为1.1Rmsinβ ≤ rc ≤ Rm

thumbnail Fig. 1

Flank lines on the pitch cone. (a) Tooth blank in single-side method. (b) Tooth blank in double-side method.

thumbnail Fig. 2

专职角度对牙齿锥度的影响。

2.2 Geometric parameter design

Different from the traditional method of calculating geometric parameters by outer transverse module in heel, the double-side geometric parameter design of spiral bevel gear is completed by mean normal module. In order to ensure the proper meshing depth of tooth at the mean point, mean working depthhmwcan be calculated as(10)

Hereh是工作牙齿高度因素,mn是normal modulus of mean point,mt是outer transverse module, andR是outer cone distance.

根据平均点计算螺旋锥齿轮的间隙,并在齿长的方向上保持恒定。可以根据设计要求调整间隙并计算为(11)

Herec1是clearance factor.

Then mean whole depth can be expressed as(12)

根据平均附录因素c火腿, the mean addendum and mean dedendum can be expressed as(13)

Dedendum angle is distributed according to the ratio of mean dedendum to mean whole depth(14)

Then theoretical cut number can be calculated as(15)

由于截止数字已经标准化和序列化,因此接近截止数量N0是selected. In order to make up for the difference of real and theoretical cut number, root line is tilted around mean point and the amount of tilt can be expressed as(16)

Finally dedendum angle can be obtained according to equation(14)and(16)。附录角可以表示为(17)

以上参数在平均点计算。为了便于生产测量,应将上述数据转换为脚跟。

面对锥角度δa1,δa2can be expressed as(18)

Root cone angleδf1,δf2can be expressed as(19)

然后外职职业和外表可以表达为(20)

Outer working depth can be expressed as(21)

外部深度可以表示为(22)

外径可以表示为(23)

节锥顶皇冠可以表示为(24)

Then geometric parameters of spiral bevel gear can be calculated

3计算机器设置参数

3.1 Initial machining setting parameters

设计后,标称压力角α, mean spiral angleβ, equivalent tooth numberz0are unchangeable. As the root line tilted, the sum of dedendum angle will not meet equation(7)。The spiral angle varies along the length of the tooth. So a pointm是selected to ensure the normal taper and a well meshing performance. The point is named as modified mean point can calculated as(25)

Analysis equation(25), the real cut numberN0将影响修改平均点的位置,因此可以通过选择切割数来控制接触图案的近似位置。

The machine setting parameters are calculated in modified mean point. The installment of cut is shown in图3.。返回和垂直偏移的机器中心都是0.其他计算公式如下所示

The radial setting position is calculated as(26)

Initial cradle angle setting is calculated as(27)

根倾斜后,机器中心回来被改变。变化被确定为(28)

And other parameters are calculated as(29)

thumbnail Fig. 3

分期付款的。

3.2 Influence of cutter number on pressure angle

After real cutter number is determined, profile angle can be determined as(30)

The difference of pressure angle on concave side and convex side in modified mean point is(31)

Only spiral angle at modified mean point is considered and varies along the tooth length. So the difference of pressure angle on concave side and convex side in the any point can be expressed(32)

Hereβy,Ry是分别对应于该点的螺旋角和锥形距离。

The difference of pressure angle between modified mean point and any point can be expressed as(33)

Then equation(33)can be arranged as(34)

In modified mean point of tooth lineβ = βy, so Δ′α = 0. As far away from the modified mean point, the difference of pressure angle becomes large which will affect mesh performance and leads to bias in contact pattern.

3.3螺旋校正运动对压力角的影响

Helical motion is a modified movement. In addition to generating motion during manufacture, the work piece and cut have a linear motion relation in direction of cradle axis. The helical motion has an effect on correction of pressure angle; the basic principle of the modification is shown in图4.

产生由工件旋转和切割速度组成vt(沿着齿轮的切线方向)Asposition1所示,以及网点中的压力角m1α1; considering helical motion (position 2), then a radial motionvr是added, resultant motion cut can be expressed asv,见图4.。The pressure angle in mesh pointm2α2, andα2 = α1+ Δ α

压力角的变化取决于径向速度的比率vrand tangential velocityvt(35)

这里,ω是the rotational angular velocity of cradle;p是the lead of helical motion.

压力角δ的变化αyat any point in tooth line can be expressed as(36)

The change in normal pressure angle can be expressed as(37)

Then in modified mean point(38)

The difference Δ″αbetween at any point and modified mean point in tooth line can be expressed as(39)

The difference is proportional to the distance from modified mean point.

如果设置δ.α + Δα = 0, the bias in generated by cut number will be eliminated by the difference of pressure angle generated by helical motion. Then the helical motion coefficient can be expressed as(40)

While for some special working conditions, a bias in contact is more suitable. Then coefficient of helical motion can be adjusted to get an ideal contact pattern.

thumbnail Fig. 4

Influence of helical motion on pressure angle.

4 Technological process of double-side milling method

According to the method proposed in this paper, the flow chart is shown in图5.

As shown in图5.根据基本参数,可以计算标准锥度的专职角度和理论标称半径的总和。然后可以获得双工锥度中的专职角度和理论切割数。由于实际切割数可能不等于理论,因此由根倾斜修改专职角度的总和。确定专职角度后,几何参数可以计数。选择修改的平均点,并在修改的平均点计算机器设置参数。仅保证修改平均点中的压力角。切割编号,压力角误差会导致齿长的方向。引入螺旋运动以在齿长方向上修改压力角。然后牙齿接触分析检查机器设置参数。调整螺旋运动系数和实际切割数以优化接触性能。

thumbnail Fig. 5

技术过程的流程图。

5 Numerical examples

A pair of spiral bevel gear was taken as an example for experimental verification. The geometric parameters were shown in表格1并且和机器设置参数显示在Table 2

The helical motion coefficient of pinion was set as −1, 0, 1 respectively to calculate the tooth surface. The comparison of tooth surface is shown in图6.。蓝色对应于小齿轮理论齿表面,黑色是对应的p =1, and red is the corresponding top = −1

WhenH = 1, for concave side, the correction is −0.10789 mm in toe of topland, −0.035967 mm in toe of root, 0.049979 mm in heel of top land, 0.13008 mm in heel of root; for convex side, the correction is 0.04147 mm in toe of topland, 0.10691 mm in toe of root, −0.15265 mm in heel of top land, −0.024998 mm in heel of root. Compared with theoretical tooth surface, for concave side, the pressure angle become small comparing to theory in toe of top land, and become large in heel of root, while the change of pressure angle are bigger comparing to the change in the direction from toe of root to heel of top land.

WhenH = −1, for concave side, the correction is 0.10929 mm in toe of topland, 0.038207 mm in toe of root, 0.038207 mm in heel of topland, 0.13002 mm in heel of root; for convex side, the correction is −0.04147 mm in toe of topland, −0.10775 mm in toe of root, 0.15276 mm in heel of topland, 0.024574 mm in heel of root. Compared with theoretical tooth surface, for concave side, the pressure angle become large comparing to theory in toe of topland, and become small in heel of root, while the change of pressure angle are bigger comparing to the change in the direction from toe of topland to heel of root.

Analysis图6., the influence trend of helical motion coefficient on concave side and convex side are opposite, and helical motion coefficient has no effect on modified mean point. As far away from modified mean point, the change of pressure angle is large. Analysis results are consistent with equation(40)。So a well performance can be obtained by adjust helical motion coefficient.

综合考虑接触模式和传输错误曲线,呈螺旋运动系数p = −0.3, and the corresponding results of tooth contact analysis (TCA) are shown in the图7.

有限元分析也是分析牙齿表面触点的有效方法[15]。不仅可以直观地观察到在相应的负载下的齿面的接触时刻,也可以通过有限元分析来看接触应力。基于文献建立了一体的3D模型[9]。Then the finite element model was established by preprocess as shown in图8.。The results of finite element analysis by Abaqus with a load of 500N are shown in the图9.

As图9.shown, (a) is the stress distribution for gear convex side and (b) is the stress distribution for gear concave side. The biggest contact stress is 242.8 and 303.4 MPa under the load of 500N. The contact pattern is in the middle of tooth surface and there is no edge contact occurs, which is consistent with the results of TCA.

The contact performance is verified by tooth contact analysis and finite element analysis. The method proposed in this paper is proved to be effective in theory.

表格1

Geometric parameters.

Table 2

机器设置参数。

thumbnail Fig. 6

Influence of helical motion on tooth surface topology.

thumbnail Fig. 7

Result of TCA. (a) Convex side. (b) Concave side.

thumbnail Fig. 8

Model of finite element analysis.

thumbnail Fig. 9

Results of finite element analysis. (a) Convex side of gear. (b) Concave side of gear.

5.1 Simulation

According to the parameters listed inTable 2,剪切模型如图所示图10。通过vericut进行模拟来检查和调试处理程序。模拟过程和小齿轮的产品显示在图11。完成了具有理论的仿真产品的比较,结果显示在图12。只有模拟小齿轮处理。

As the results showed in图12, the simulated product and established model are basically the same. In tooth surface, the biggest error in tooth surface is 0.02 mm for concave side and convex side. While the main error is in the root, the overcut and residue error is 0.05 mm. The tooth surface error comes from automatic approximated of parameters during model establishment for the precision of parameters are reserved to three decimal places or even more. The error in root mainly comes from the error caused by approximation of machine root angle. The comparison error meet engineering requirement. The correctness of parameters and machining procedures are verified.

thumbnail Fig. 10

刀具模型。

thumbnail Fig. 11

Simulation process and product.

thumbnail Fig. 12

比较的结果。

5.2切割实验

牙齿切割实验是在YK2260X上进行的,是由洛阳Keda Yuege CNC Machinetool Co.,Ltd的五轴四连杆CNC铣床。工件安装在0.01毫米的直径跳动和面部跳动中。切割过程中没有颤抖。牙齿切割的场景显示在图13

In order to check quality of process tooth surface, pinion and gear were measured after chamfering, burring and cleaning. The scenes of tooth surface measurement are shown in图14。The results of measurement are shown in图15

As shown in图14(a)是齿轮过程的场景,(b)是小齿轮过程的场景。图15a and b is the measurement results of gear and pinion, respectively. For gear measurement as displayed in图15A,凸起侧和凹面的最大齿轮表面误差为0.004和0.006mm。用于小齿轮测量显示图15B,凸形侧和凹面的最大齿面误差为0.006和0.004mm。误差符合工程要求,对齿面的啮合性能几乎没有影响。

Finally a rolling test was carried out. The workpieces were installed in 0.01 mm of diameter runout and face runout. The rolling ran smoothly without obvious vibration noise. The results were shown in图16

As图16shows, (a) is the scene of rolling test, (b) is tooth contact pattern of gear convex side, and (c) is tooth contact pattern of gear concave side. The contact pattern is located in the middle of the tooth surface and there is no edge contact and other bad contacts. The rolling test results are basically consistent with图7.and9在位和形状。实验结果证明了本文提出的方法是有效可行的。

thumbnail Fig. 13

切割实验。(a)齿轮。(b)小齿轮。

thumbnail Fig. 14

Scene of measurement. (a) Gear. (b) Pinion.

thumbnail Fig. 15

Measurement results of tooth surface. (a) Gear. (b) Pinion.

thumbnail Fig. 16

滚动测试和结果。(a)滚动试验。(b)齿轮的凸面。(c)凹陷的凹陷侧。

6 Conclusions

Different from the traditional method of calculating geometric parameters by transverse module in heel, the double-side geometric parameter design of spiral bevel gear is completed by normal module in mean cone distance combined with the cut parameters.

根据双面方法的锥度,计算了一个名为改进的平均点的点满足啮合条件,并计算修改锥距离的加工设定参数。

分析了切割器数对齿长方向上的压力角的影响。研究了螺旋运动系数对牙齿表面的影响。结果表明,螺旋动作可以在齿长方向上校正压力角。实现了通过调整螺旋运动系数来优化接触性能。

The experimental results are consistent with the theoretical analysis results, which verify the effectiveness and feasibility of the double-side machining of spiral bevel gears proposed in this paper.

Acknowledgments

作者要感谢国家自然科学基金的经济援助和支持(授予第51975185号,授予第52005157号,并授予NO.51475141),河南省的主要科学和技术项目(授予号。191110213300-05)和国家重点研究和发展计划(授予NO。2020YFB1713505-4)。我们感谢审稿人和编辑的评价和建议。

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Cite this article as: L. Geng, X. Deng, H. Zhang, S. Nie, C. Jiang, Theory and experimental research on spiral bevel gear by double-side milling method, Mechanics & Industry22., 33 (2021)

All Tables

表格1

Geometric parameters.

Table 2

机器设置参数。

All Figures

thumbnail Fig. 1

Flank lines on the pitch cone. (a) Tooth blank in single-side method. (b) Tooth blank in double-side method.

In the text
thumbnail Fig. 2

专职角度对牙齿锥度的影响。

In the text
thumbnail Fig. 3

分期付款的。

In the text
thumbnail Fig. 4

Influence of helical motion on pressure angle.

In the text
thumbnail Fig. 5

技术过程的流程图。

In the text
thumbnail Fig. 6

Influence of helical motion on tooth surface topology.

In the text
thumbnail Fig. 7

Result of TCA. (a) Convex side. (b) Concave side.

In the text
thumbnail Fig. 8

Model of finite element analysis.

In the text
thumbnail Fig. 9

Results of finite element analysis. (a) Convex side of gear. (b) Concave side of gear.

In the text
thumbnail Fig. 10

刀具模型。

In the text
thumbnail Fig. 11

Simulation process and product.

In the text
thumbnail Fig. 12

比较的结果。

In the text
thumbnail Fig. 13

切割实验。(a)齿轮。(b)小齿轮。

In the text
thumbnail Fig. 14

Scene of measurement. (a) Gear. (b) Pinion.

In the text
thumbnail Fig. 15

Measurement results of tooth surface. (a) Gear. (b) Pinion.

In the text
thumbnail Fig. 16

滚动测试和结果。(a)滚动试验。(b)齿轮的凸面。(c)凹陷的凹陷侧。

In the text

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