机械毕业设计英文外文翻译546圆锥渐开线齿轮(斜面体齿轮).docx
机械毕业设计英文外文翻译546圆锥渐开线齿轮(斜面体齿轮)
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附录 外文资料翻译译文 摘要 圆锥渐开线齿轮 (斜面体齿轮 )被用于交叉或倾斜轴变速器和平行轴自由侧隙变速器中。圆锥齿轮是在齿宽横断面上具有不同齿顶高修正 (齿厚 )的直齿或斜齿圆柱齿轮。这类齿轮的几何形状是已知的，但应用在动力传动上则多少是个例外。 ZF 公司已将该斜面体齿轮装置应用于各种场合 :4W D 轿车传动装置、船用变速器 (主要用于快艇 )机器人齿轮箱和工业传动等领域。斜面体齿轮的模数在 0. 7 mm-8 mm 之间， 交叉传动角在 0 - 25。之间。这些边界条件需要对斜面体齿轮的设计、制造和质量有一个深入的理解。在锥齿轮传动中为获得高承载能力 和低噪声所必须进行的齿侧修形可采用范成法磨削工艺制造。为降低制造成本，机床设定和由于磨削加工造成的齿侧偏差可在设计阶段利用仿真制造进行计算。本文从总体上介绍了动力传动变速器斜面体齿轮的研发，包括 :基本几何形状、宏观及微观几何形状的设计、仿真、制造、齿轮测量和试验。 1 前言 在变速器中如果各轴轴线不平行的话，转矩传递可采用多种设计，例如 :伞齿轮或冠齿轮、万向节轴或圆锥渐开线齿轮 (斜面体齿轮 )。圆锥渐开线齿轮特别适用于小轴线角度 (小于 15 )，该齿轮的优点是在制造、结构特点和输入多样性等方而的简易。圆锥渐开线齿 轮被用于直角或交叉轴传动的变速器或被用于平行轴自由侧隙工况的变速器。由于锥角的选择并不取决于轴线交角，配对的齿轮也可能采用圆柱齿轮。斜面体齿轮可制成外啮合和内齿轮，整个可选齿轮副矩阵见表 1，它为设计者提供了高度的灵活性。 圆锥齿轮是在齿宽横截面上具有不同齿顶高修正 (齿厚 )量的直齿轮或斜齿轮。它们能与各种用同一把基准齿条刀具切制成的齿轮相啮合。斜面体齿轮的几何形状是已知的，但它们很少应用在动力传动上。过去，未曾对斜面体齿轮的承载能力和噪声进行过任何大范围的试验研究。标准 (诸如适用于圆柱齿轮的 IS06336)、计算方法和强度值都是未知的。因此，必须开发计算方法、获得承载能力数值和算出用于生产和质量保证nts的规范。在过去的 15年中， ZF 公司已为锥齿轮开发了多种应用 : 1、输出轴具有下倾角的船用变速1、 3 图 .1 2、转向器 1 3、机器人用小齿隙行星齿轮装置(交叉轴角度 1一 3 )2 4、商用车辆的输送齿轮箱 (垃圾倾倒车 ) 5、 AWD 用自动变速器 4，图 2 2 齿轮几何形状 2. 1 宏观几何形状 简而言之，斜面体齿轮可看成是一个在齿宽横截面上连续改变齿顶高修正的圆柱齿轮，如图 3。为此，根据齿根锥角刀具向齿轮轴线倾斜 1。结果形成了齿轮基圆尺寸。 螺旋角，左 /右 tanLR,=tan cos cossintan n (l) 横向压力角 左 /右 ta nc o s c o sta nta n , noLtR （ 2） 基圆直径 左 /右 LRLtRnLdR Zimd, c o sc o s (3) 左右侧不同的基圆导致斜齿轮齿廓形状的不均匀，图 3。采用齿条类刀具加工将使得齿根锥具有相应的根锥角。齿顶角设计成这样以使得nts顶端避免与被啮合齿轮发生干涉，并获得最大接触区域。由此导致在齿宽横截面上具有不同的齿高。由于几何设计限制了根切和齿顶形状，实际齿宽随锥角增加而减小。锥齿轮传动合适的锥角最大约为 15。 2. 2 微观几何形状 一对伞齿轮通常形成点状接触。除接触外，在齿侧还存在间隙，如图 7。齿轮修形设计的目的 是减小这些间隙以形成平坦而均匀的接触。通过逐步应用啮合定律有可能对齿侧进行精确的计算 5,图 4。最后，在原始侧生成半径为 rp1 和法向矢量为 n1 的 P1点。这生成速度矢量 V 1P 及对于在啮合一侧所生成的点，有半径矢量 rp2 : 0c o ss in1111 PPP rrv (4) 12 PP rar (5) 和速度矢量 2PV 0c o ss in212122 PPP rrV （ 6） 角速度根据齿轮速比确定： 1221 zz （ 7） 角度被反复迭代直至满足下代。 0121 PP vvn (8) 啮合点 Pa 偏转 2 角度 2112 zz （ 9） nts绕齿轮轴转动，形成共轭点 P2。 3 传动装置设计 3. 1 根切和齿顶形状 斜面体齿轮的可用齿宽受到大端齿顶形状和小端根切的限制，见图 3。齿高愈高 (为获得较大的齿高变位量 )，理论可用齿宽愈窄。小端根切和大端齿顶形状导致齿高变位量沿齿宽方向发生变化。当一对齿轮的锥角大致相同时可获得最大的可用齿宽。若齿轮副中小齿轮愈小，则该小齿轮必须采用更小的锥角。齿顶锥角小于齿根锥角时，通常能在小端获得有用的渐开线，而在大端处有足够齿顶间隙，这时大端的齿顶形状并不太严重。 3. 2 工作区域和滑动速度 斜面体齿轮工作区域产生扭歪的原因是圆锥半径有形成平行四边形趋势。另外，工作压力角在齿宽横截面方向的改变也造成工作区域的扭曲。图 5 是一个例子。在交叉轴传动的斜面体齿轮上存在一滚动轴 ;如同圆柱齿轮副的滚动点一样，在该轴上不存在滑动。对于倾斜轴布置而言，在轮齿啮合处总存在另外的轴向滑动。由于工作压力角在齿宽横截面上变化，从小端到大端的接触区内的接触轨迹有很大的变化。因此，沿齿宽方向在齿顶和齿根处具有明显不同的滑动速度。在齿轮中部，nts齿顶高修正的选择是基于圆柱齿轮副的规范 ;在主动齿轮根部的接触轨迹将小于齿顶的接触轨迹。图 6 给出了斜面体齿轮副主动齿轮滑动速度的分布。 4 接触分析和修形 4. 1 点接 触和间隙 在未修正齿轮传动中，由于轴线倾斜，通常仅有一点接触。沿可能接触线出现的间隙可大致解释为螺旋凸起和齿侧廓线角度的偏差所致。圆柱齿轮左右侧间隙与轴线交叉无关。对于螺旋齿轮而言，当两斜面体齿轮锥角大致相同时，其产生的间隙也几乎相等。随两齿轮锥角和螺旋角不一致的增加，左右侧间隙的不同程度也增加 。 在工作压力角较小时将导致更大的间隙。图 7给出了具有相同锥角交叉轴传动的斜面体齿轮副所出现的间隙。图 8显示了具有相同 10交叉轴线和 30螺旋角 齿轮在左右侧间隙方而的差异。两侧平均间隙的数值在很大程度上与螺旋角无关，但与两齿轮的锥角相关。 螺旋角和锥角的选择决定了齿轮左右侧平均间隙的分布。倾斜轴线布置对接触间隙产生额外影响。这将有效减少齿轮一侧的nts螺旋凸形。如果垂直轴线与总基圆半径相同，并且基圆柱螺旋角之差等于交叉轴角的话 ,间隙减小到零并出现线接触。然而 ,在另一侧将出现明显的间隙。如果正交的轴线进一步扩大直至变成圆柱交叉轴螺旋齿轮副的话 ,其两侧间隙等同于较小的螺旋凸形。除螺旋凸形外 ,明显的齿廓扭曲 (见图 8)也是斜面体齿轮的间隙特征。随螺旋角增加 齿廓扭曲也随之增加。图 9 表明图 7所示齿轮装置的齿廓是如何扭曲。为补偿齿轮啮合中所存在的间隙 ,必须采用齿侧拓扑修形 ,该类修形可明显补偿螺旋凸形和轮廓扭曲。未对齿廓扭曲作补偿的话 ,在工作区域仅有一个对角线状的接触带 ,见图 10。 4.2 齿侧修形 对于一定程度的补偿而言 ,必需的齿面形状可由实际间隙所决定。图11 给出了这些样品的齿形几何特征。采用修正后的接触率得到了很大改善如图 12 所示。为应用在系列生产中 ,其目标总是能使用磨床加工这类齿面 ,对此的选择在第 6 节论述。除间隙补偿外 ,齿顶修形也是有益的。修形减少了啮合开 始和结束阶段的负荷 ,并能提供一较低的噪声激励源。然而 ,斜面体齿轮的齿顶修形在齿宽横截面上的加工总量上和长度上是不同的。问题主要出现在具有一个大根锥角但顶锥角与根锥角存在偏差的齿轮上。因此齿顶修形在小端明显大于大端。如齿轮需要在啮合开始和结束处修形 ,则必须接受这种不均匀的齿顶修形。利用其它锥角如根锥角进行齿顶修形加工也是可行的。但是 ,这样需要专门用于齿顶卸载的专用磨削设备。与范成法磨削方法无关 ,nts齿侧修正可采用诸如珩磨等手段 ;但在斜面体齿轮上应用这些方法尚处在早期开发阶段。 5 承载能力和噪声激励 5.1 计 算标准的应用 斜面体齿轮齿侧和根部承载能力仅可用圆柱齿轮的计 算标准 (ISO 6336, DIN 3990, AGMAC95) 作近似估算。具体计算时用圆柱齿轮副替代斜面体齿轮 ,用斜面体齿轮中部的齿宽来定义圆柱齿轮的参数。虽然斜面体齿轮齿廓是非对称的 ,但在替代齿轮中可不予考虑。替代齿轮的中心距由斜面体齿轮中部齿宽处的工作节圆半径确定。当计及齿宽横截面时 ,各项独立的参数都会变化 ,这将明显影响承载能力。 表 2 给出了影响齿根和齿侧承载能力的主要因素。由于沿大端方向减小轮齿齿根圆角半径所产生较大的凹口效应阻 止了根部齿厚的增加。另外 ,在大端处 ,较大的节圆直径可获得较小的切向力 ;然而 ,大端处的齿高变位nts量也随之变小。由于主要影响得到很好的平衡 ,因此可用替代齿轮副获得十分近似的承载能力计算结果。齿宽横截面上的载荷分布可用齿宽系数 (例如 DIN/ISO 标准中的 KH和 KF)表示和利用补充的负载曲线图分析来确定。 5.2 轮齿接触分析 如同在圆柱齿轮副中那样 ,更精确的承载能力计算可采用三维轮齿接触分析。同样采用替代齿轮 ,而且齿侧处接触状况被认为非常理想。该齿侧形状通过叠加经齿侧修正的无负载接触间隙而获得。在这里 ,接触线由替代齿轮所确定 ,它们和斜面体齿轮的接触状况稍有不同。图 13 给出了以这方法获得的载荷分布 ,并与已有的负载曲线图作对比 ,两者的相关性非常好。 轮齿接触分析也将生成一个作为激振源的由轮齿啮合产生的传动误差。然而这仅能作为一个粗略的引导。在传动误差方面 ,斜面体齿轮接触计算的不精确性是一个比载荷分布更大的影响因素。 5.3 采用 有限元法的精确建模 斜面体齿轮的应力也能利用有限元法计算。图 14 是齿轮横断面建模的实例。图 15给出了使用 PERMAS软件由计算机生成的主动齿轮在啮合位置的轮齿啮合区模型和应力分布计算值 7。可对多个啮 nts合位置进行计算 ,并能求出齿轮旋转产生的传动误差。 5.4 承载能力和噪声试验 在交叉轴背靠背试验台上对 AWD 变速器进行试验以测量其承载能力 ,图 16。试验齿轮采用不同的修正 ,以确定它们对承载能力的影响。承载能力的试验与有限元计算结果相当吻合。值得注意的是 ,由于大端硬度提高使得载荷曲线图朝大端由一个额外 的移动。这种移动在替代的圆柱齿轮副计算中不能被辨别。在进行承载能力试验的同时 ,传动误差和旋转加速度的测量在通用噪声试验台上进行 ,图 17。除了载荷影响外 ,这些试验还测量了附加轴线倾斜所引起的噪声激励 ,关于轴线附加倾斜 ,试验中未发现有明显的影响。 nts6 仿真制造 借助于仿真制造 ,可获得机床设置及连续范成磨削和产生齿廓扭曲的运动。齿廓受迫扭曲现象可在变速器设计阶段就被认识到并与承载能力及噪声一并进行分析。斜面体齿轮制造仿真软件由 ZF 公司开发 ,详见9。 6.1 适用于斜面体齿轮的制造方法 斜面体齿轮仅可用范成法加工 ,因为齿廓形状沿齿宽方向有明显的变化。尽管是锥角非常小的斜面体齿轮 ,必须承认在修整处理中仍然会出现齿廓角度偏差。滚刀最方便用于预切削。理论上也可采用刨削 ,但是 ,所需的运动在现有机床上很难实现。内齿圆锥齿轮仅能用类似小齿轮的刀具精确制造 ,如果刀具轴线和工具轴线平行并且锥角是通过改变中心距生成的。如果内齿轮利用轴线倾斜的小齿轮刀具如同加工差速器锥齿轮那样来制造的话 ,将导致齿沟凸起和无修正运动的齿廓扭曲。对于小锥角而言这些偏差足够小 ,可以被忽略。对于终加工 ,范成法螺旋磨削是一个最佳选择。 如果工件或机床夹具能被另外倾斜 ,也可采用部分范成法。如果齿轮锥角处于机床控制范围内 ,拓扑磨削工艺也是可能的 (例如 5轴机床 ),但是会耗费巨大的努力。原则上 ,珩磨等方法也能被用于加工 ,但是 ,在斜面体齿轮应用这些方法仍需大量的开发工作。双齿侧范成法磨削工艺并利用中心距弧形减少方法可实现齿沟凸起的目标。该方法所得到的齿廓扭曲与造成啮合间隙的齿廓扭曲相反。因此该方法可在很大程度上补偿齿廓扭曲并可承受比圆柱齿轮更大的载荷。 6.2 工件表面形状 以下的关于工件描述被应用在仿真中 : 原始齿轮 (留有磨削所需的余量 ) 理想齿 轮 (来自齿轮数据 ,无齿侧修形 ) 完成的齿轮 (具有制造偏差和齿侧修形 ) nts 参考文献： 1. J. A. MacBain, J. J. Conover, and A. D. Brooker, “Full -vehicle simulation for series hybrid vehicles,” presented at the SAE Tech. Paper, Future Transportation Technology Conf., Costa Mesa, CA, Jun. 2003, Paper 2003-01-2301. 2. X. He and I. Hodgson,“Hybrid electric vehicle simulation and evaluation for UT-HEV,”prmented at the SAE Tech. Paper Series, Future Transpotation Technology Conf., Costa Mesa, CA, Aug. 2000, Paper 2000-01-3105. 3. K. E. Bailey and B. K. Powell,“A hybrid electric vehicle powertrain dynamic model,”inProc. Amer. Control Conf., Jun. 21 -23, 1995, vol. 3, pp. 1677-1682. 4. B. K. Powell, K. E. Bailey, and S. R. Cikanek,“Dynamic modeling and control of hybrid electrie vehicle powertrain system,”IEEE Control Syst. Mag., vol, 18, no. 5. pp. 17-33, Oct. 1998. 5. K. L. Butler, M. Ehsani, and P. Kamath,“A Matlabbared modeling and simulation package for electric and hybrid electric vehicle design,”IEEE Trans. Veh.Technol., vol. 48, no. 6, pp. 1770-1778, Nov. 1999. 6. K. B. Wipke, M. R. Cuddy, and S. D. Burch,“ADVISOR 2.1: A user -friendly advanced powertrain simulation using a combined backward/forward approach,” IEEE Trans. Veh. Technol., vol. 48. no. 6, pp.1751-1761, Nov. 1999. 7. T. Markel and K. Wipke,“Modeling grid -connected hybrid electric vehicles using ADVISOR,”inProc.16th Annu. Battery Conf. Appl. and Adv.,Jan. 9 -12.2001. pp. 23-29. 8. S. M. Lukic and A. Emadi,“Effects of drivetrain hybridization on fuel economy and dynamic performance of parallel hybrid electric vehicles,”IEEE Trans. Veh.Technol., vol. 53, no. 2, pp. 385-389, Mar. 2004. 9. A. Emadi and S. Onoda,“PSIM -based modeling of automotive power systems: Conventional, electric, and hybrid electric vehicles,”IEEE Trans. Veh. Technol.,vol. 53, no. 2, pp. 390-400, Mar. 2004. 10. J. M. Tyrus, R. M. Long, M. Kramskaya, Y. Fertman, and A. Emadi,“Hybrid electric sport utility vehicles,”IEEE Trans. Veh. Technol., vol. 53, no. 5,pp. 1607 -1622, Sep. 2004. nts 附件 2：外文原文 ABSTRACT Conical involute gears (beveloids) are used in transmissions with intersecting or skew axes and for backlash-free transmissions with parallel axes. Conical gears are spur or helical gears with variable addendum modification (tooth thickness) across the face width. The geometry of such gears is generally known, but applications in power transmissions are more or less exceptional. ZF has implemented beveloid gear sets in various applications: 4WD gear units for passenger cars, marine transmissions (mostly used in yachts), gear boxes for robotics, and industrial drives. The module of these beveloids varies between 0.7 mm and 8 mm in size, and the crossed axes angle varies between 0and 25. These boundary conditions require a deep understanding of the design, manufacturing, and quality assurance of beveloid gears. Flank modifications, which are necessary for achieving a high load capacity and a low noise emission in the conical gears, can be produced with the continuous generation grinding process. In order to reduce the manufacturing costs, the machine settings as well as the flank deviations caused by the grinding process can be calculated in the design phase using a manufacturing simulation. This presentation gives an overview of the development of conical gears for power transmissions: Basic geometry, design of macro and micro geometry, simulation, manufacturing, gear measurement, and testing. 1 Introduction In transmissions with shafts that are not arranged parallel to the axis, torque transmission is possible by means of various designs such as bevel or crown gears , universal shafts , or conical involute gears (beveloids). The use of conical involute gears is particularly ideal for small shaft angles (less than 15), as they offer ntsbenefits with regard to ease of production, design features, and overall input. Conical involute gears can be used in transmissions with intersecting or skew axes or in transmissions with parallel axes for backlash-free operation. Due to the fact that selection of the cone angle does not depend on the crossed axes angle, pairing is also possible with cylindrical gears. As beveloids can be produced as external and internal gears, a whole matrix of pairing options results and the designer is provided with a high degree of flexibility; Table 1. Conical gears are spur or helical gears with variable addendum correction (tooth thickness) across the face width. They can mesh with all gears made with a tool with the same basic rack. The geometry of beveloids is generally known, but they have so far rarely been used in power transmissions. Neither the load capacity nor the noise behavior of beveloids has been examined to any great extent in the past. Standards (such as ISO 6336 for cylindrical gears ), calculation methods, and strength values are not available. Therefore, it was necessary to develop the calculation method, obtain the load capacity values, and calculate specifications for production and quality assurance. ntsIn the last 15 years, ZF has developed various applications with conical gears: Marine transmissions with down-angle output shafts /1, 3/, Fig. 1 Steering transmissions /1/ Low-backlash planetary gears (crossed axes angle 13) for robots /2/ Transfer gears for commercial vehicles (dumper) Automatic car transmissions for AWD /4/, Fig. 2 2 GEAR GEOMETRY 2.1 MACRO GEOMETRY To put it simply, a beveloid is a spur gear with continuously changing addendum modification across the face width, as shown in Fig. 3. To accomplish this, the tool is tilted towards the gear axis by the root cone angle ? /1/. This results in the basic gear dimensions: Helix angle, right/left tanLR,=tan cos cossintan n(1) Transverse pressure angle right/left ta nc o s c o sta nta n , noLtR (2) Base circle diameter right/left ntsLRLtRnLdR Zimd, c o sc o s (3) The differing base circles for the left and right flanks lead to asymmetrical tooth profiles at helical gears, Fig. 3. Manufacturing with a rack-type cutter results in a tooth root cone with root cone angle . The addendum angle is designed so that tip edge interferences with the mating gear are avoided and a maximally large contact ratio is obtained. Thus, a differing tooth height results across the face width.Due to the geometric design limits for undercut and tip formation, the possible face width decreases as the cone angle increases. Sufficiently well-proportioned gearing is possible up to a cone angle of approx. 15. 2.2 MICRO GEOMETRY The pairing of two conical gears generally leads to a point-shaped tooth contact. Out-side this contact, there is gaping between the tooth flanks , Fig. 7. The goal of the gearing correction design is to reduce this gaping in order to create a flat and uniform contact. An exact calculation of the tooth flank is ntspossible with the step-by-step application of the gearing law /5/, Fig. 4. To that end , a point (P1) with the radiusrP1and normal vectorn1is generated on the original flank. This generates the speed vector V1Pwith 0c o ss in1111 PPP rrv (4) For the point created on the mating flank, the radial vector rp2 : 12 PP rar (5) and the speed vector 2PV apply 0c o ss in212122 PPP rrV (6) The angular velocities are generated from the gear ratio: 1221 zz (7) The angle is iterated until the gearing law in the form 0121 PP vvn (8) is fulfilled. The meshing point Pa found is then rotated through the angle2 2112 zz (9) around the gear axis, and this results in the conjugate flank point P2 . 3 GEARING DESIGN 3.1 UNDERCUT AND TIP FORMATION The usable face width on the beveloid gearing is limited by tip formation on the heel and undercut on the toe as shown in Fig. 3. The greater the selected ntstooth height (in order to obtain a larger addendum modification), the smaller the theoretically useable face width is. Undercut on the toe and tip formation on the heel result from changing the addendum modification along the face width. The maximum usable face width is achieved when the cone angle on both gears of the pairing is selected to be approximately the same size. With pairs having a significantly smaller pinion, a smaller cone angle must be used on this pinion. Tip formation on the heel is less critical if the tip cone angle is smaller than the root cone angle, which often provides good use of the available involute on the toe and for sufficient tip clearance in the heel. 3.2 FIELD OF ACTION AND SLIDING VELOCITY The field of action for the beveloid gearing is distorted by the radial conicity with a tendency towards the shape of a parallelogram. In addition, the field of action is twisted due to the working pressure angle change across the face width. Fig. 5 shows an example of this. There is a roll axis on the beveloid gearing with crossed axes; there is no sliding on this axis as there is on the roll point of cylindrical gear pairs. With a skewed axis arrangement, there is always yet another axial ntsslide in the tooth engagement. Due to the working pressure angle that changes across the face width, there is varying distribution of the contact path to the tip and root contact. Thus, significantly differing sliding velocities can result on the tooth tip and the tooth root along the face width. In the center section, the selection of the addendum modification should be based on the specifications for the cylindrical gear pairs; the root contact path at the driver should be smaller than the tip contact path. Fig. 6 shows the distribution of the sliding velocity on the driver of a beveloid gear pair. 4 CONTACT ANALYSIS AND MODIFYCATIONS 4.1 POINT CONTACT AND EASE-OFF At the uncorrected gearing, there is only one point in contact due to the tilting of the axes. The gaping that results along the potential contact line can be approximately described by helix crowning and flank line angle deviation. Crossed axes result in no difference between the gaps on the left and right flanks on spur gears. With helical gearing, the resulting gaping is almost equivalent when both beveloid gears show approximately the same cone angle. The difference between the gap values on the left and right flanks increases as the difference between the cone angles increases and as the helix angle increases. This process results in larger gap values on the flank with the smaller working pressure angle. Fig.7 shows the resulting gaping (ease-off) for a beveloid gear pair with crossed axes and beveloid gears with an identical cone angle. Fig.8 shows the differences in the gaping that results for the left and right flanks for the same crossed axes angle of 10 and a helical angle of approx. 30. The mean gaping obtained from both flanks is, to a large extent, independent of the helix angle and the distribution of the cone angle to both gears. The selection of the helical and cone angles only determines the distribution of the mean gaping to the left and right flanks. A skewed axis arrangement results in additional influence on the contact gaping. There is a significant reduction in the effective helix crowning on one flank. If the axis ntsperpendicular is identical to the total of the base radii and the difference in the base helix angle is equivalent to the (projected) crossed axes angle, then the gaping decreases to zero and line contact appears. However, significant gaping remains on the opposite flank. If the axis perpendicular is further enlarged up to the point at which a cylindrical crossed helical gear pair is obtained, this results in equivalent minor helix crowning in the ease-off on both flanks. In addition to helix crowning, a notable profile twist (see Fig. 8) is also characteristic of the ease-off of helical beveloids. This profile twist grows significantly as the helix angle increases. Fig.9 shows how the profile twist on the example gear set from Fig.7 is changed depending on the helix angle. In order to compensate for the existing gaping in the tooth engagement, topological flank corrections are necessary; these corrections greatly compensate for the effective helix crowning as well as the profile twist. Without the compensation of the profile twist, only a diagonally patterned contact strip is obtained in the field of action, as shown in Fig. 10. 4.2 FLANK MODIFICATIONS ntsFor a given degree of compensation, the necessary topography can be determined from the existing ease-off. Fig. 11 shows these types of typographies, which were produced on prototyp
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