外文翻译--减少偏离齿轮传动装载和卸载时的噪音

减少偏离齿轮传动装载和卸载时的噪音 Faydor L. Litvina, Daniele Vecchiatoa, Kenji Yukishimaa, Alfonso Fuentesb, Ignacio Gonzalez-Perezb and Kenichi Hayasakac 芝加哥伊利诺州大学机械部门和工业工程齿轮研究中心， 842 W. IL 60607-7022, 泰勒圣，芝加哥， 美国 喀他赫纳工艺综合大学机械工程部博士， Murcia， 30202，喀他赫纳，西班牙 山叶电动机股份有限公司齿轮半 径研究发展中心， 2500 Shingai, Iwata, 静冈 438-8501, 日本 2005 年二月 22 日定为标准； 2005 年五月 6 日校订了； 2005 年五月 17 日被公认； 2006 年一月 25 日可在线应用 . 摘要 齿轮传动时产生震动和噪音的主要原因是传输误差。有关影响噪音传输误差的两个主要函数已被查明：（ 1）一个是线性的对应误差；（ 2）一个是初步设计使用传输误差以减少噪音而引起的。它显示了传输误差的线性关系，在一个周期内形成了混合的循环啮合：（ 1）如点对点接触；（ 2）当从表面以曲线形 式移动到起始点时就产生啮合。使用初步设计传输误差能够减少因为线性对应函数而引起的传输误差，减少噪音和避免移动接触。引起传输误差的负载函数已被研究。齿牙的损坏能够使在装载的齿轮传动中减少最大的传输误差。用计算机处理的模拟齿轮啮合，且齿轮传动装载和卸货技术已发展相当水平。 关键词：齿轮传动；传输误差；齿牙啮合分析（ YCA）；限定的元素分析；噪音的减少。 文章概要 1. 绪论 2. 齿牙表面的修正 2.1. 螺旋状的齿轮传动 2.2. 螺旋状的斜齿轮 2.3. 圆柱形的蜗杆齿轮传动 3. 啮合的类型和基本功能的 传输误差 4. 装载齿轮驱动的传动误差 4.1. 初步的考虑 4.2. 装载的齿轮传动果断的运行应用限定的元素分析是为了传输误差的函数 5. 数字例证 6. 噪音的两个传输误差函数的有力对比 6.1. 应用方式概念上的考虑 6.2. 线性函数的分段插补 7.结论 参考文献 1. 绪论 模拟的齿轮传动啮合执行应用齿牙接触分析（ TCA）和测试齿轮传动已被证实传输误差的主要原因是齿轮箱的震动，这样的震动引起齿轮传动的噪音 1，2， 3， 4， 5， 6， 7， 10和 11。传输误差函数的类型 依赖对应错误的类型且齿轮齿牙表面为了进一步的传动在进行改善。（见第二节） 为减少噪音而依下列的计划进行： （ 1） 齿牙接触表面被局部化 （ 2） 提供一个传输误差的函数。这种传输错误是由未对准的一次函数所引起的7。 （ 3） 对双层表面之一进行最高倍数的修正。 见第 2 节 这通常是避免表面摩擦。 见第 5 节 已经对装载和卸载齿轮传动应用 TCA 进行了比较，它显示装载的齿轮传动的传输误差较少。其发展的方式与数字进行一起举例。 见第 5 节 2. 齿牙表面的修正 减少齿轮传动的噪音需要修正接触的双表面之一。要修正齿轮传动接触表面有三种类型： 螺旋状的齿轮，螺旋状的斜齿轮，蜗杆齿轮。 2.1 螺旋状的齿轮传动 螺旋状的齿轮最高剖面可能相交而表面产生两个齿条刀形成错误的轮廓 5和 7。 完美轮廓允许接触方向的局部化。最完美的轮廓比较是允许的：（ 1）避免边缘接触（交叉角和不同形状角的相交齿轮）（ 2）提供一个传输误差的抛物线函数。双倍完美的执行突进的圆盘而产生小齿轮（见 REF 的第 15 章资料。 7）。 2.2 螺旋状的斜齿轮 应用提供两个有误差的刀尖 p 和 g 而有局部接触会产生 螺旋状的斜齿轮： p 和 g 二者是分别用来产生小齿轮和齿轮的 7。 俩个刀尖 p 和 g再齿呀的表面产生一个共同线 C。（当提供外层轮廓的情况下）再加倍的情况下产生配合误差表面 p 和 g 刀尖 只有接触的通常单一点，但不是一条接触的线。加倍可能产生齿轮而形成有斜齿的刀尖，或者是刀尖特有的部分。她是近代科技生产的齿轮当中教授欢迎的齿轮之一，通常小齿轮都被改良为滚动的 7。 2.3 圆柱型蜗杆齿轮传动 通常蜗轮制造工艺是以下列的方式为基础。蜗轮的生产和蜗杆齿轮传动一样都是由一个滚刀运行的。应用的机床设置模拟蜗杆和蜗轮啮合而形成齿轮传动。然而，观察发现在这些条件下的制造引起不宜的轴接触 ，和高度传动误差。为把这些误差减少到最低限度可用以下不同的方法完成： （ 1） 长期在齿轮箱中研磨加工而使齿轮传动畸形； （ 2） 齿轮传动在长期的运转下产生负载，近而达到最大负载； （ 3） 蜗轮在蜗轮箱中被刨且传动装置利用刨削蜗杆部分背离减少到最小化，等等。 制造者的方法是应用接触局限为基础的：（ a）一个特大号的滚齿刀，和（ b）几何学的修正。（见下面）。 有蜗轮传动几何学的各种不同类型 7，但是一个较好的是有 Klingelnberg类型的蜗杆。 这种蜗杆是由圆盘轮廓和锥形圆作成的 7。有关蜗杆传动要考虑圆盘的一个螺纹的产生（在生产 的方法中）。 通常，在蜗轮传动局限接触中以达成应用滚刀且是比较特大号的蜗轮传动。 3. 啮合的类型和传动误差的基本函数 它假定齿牙表面任何点相切是正当的局限定位。此后，我们考虑两种啮合：（ 1）面与面，（ 2）面与曲线。面与面相切是平等观察表面的位置向量和表面单位提供 7。面与曲线啮合是曲线边缘实在接触的结果 7。 面与面相切的 TCA 运算法则是以下列的矢量为基础的方程 7： (1) (2) 在固定的同等系统 Sf位置矢量 和表面常态 中表现。这里， (ui, i)是表面的参数而且 ( 1, 2)决定表 面的角位置。 面与曲线的运算法则是用 Sf方程来表现的 7： (3) (4) 在这里 描述表面的啮合曲线 是边缘曲线的切线。 TCA 的允许应用而发现两种啮合的类型，面与面和面与曲线。计算机处理的啮合模拟是以一个反复的程序为基础的非线性方程的数字解决方案 8 应用最高的相交表面之一，它可能变成：（ 1）避免边缘接触，（ 2）获得一个初步设计的抛物线函数 7（图 1）。初步设计的抛物线函数功能的应用是减少噪音的先决条件。 图 1 例证：（ a）齿轮驱动的一个不成直线的传动函数 1 和没有欠对准的理的线 性函数 2；（ b）周期函数抛物线形成的传动误差 2( 1)。 应用最高的允许向前分配传动的误差函数的是一个抛物线，而且允许分配同样最大误差值的 6-8 。初步设计预期大小的传动误差抛物线函数和投入大量生产的工具是有关联的。图 2表示在何处由于欠对准的误差的大小，传动误差函数形成两个支流： 面对面接触和 面与曲线接触。 图 2 一个螺旋状齿轮的最大 TCA 误差结果 = 10: （ a）传动误差函数在何处 符合面与面相切和何处 符合面与曲线相切；（ b）在小齿轮齿面上相切的路径：（ c）在齿轮表面的接触路径。 4.装 载齿轮传动的传动误差 这一部分内容覆盖了一般用途 FEM 电脑程序应用装载齿轮驱动的传动误差果断程序 3。 TCA 决定直接应用卸载齿轮驱动的传动误差。描述比较装载和卸载时齿轮驱动的传动误差在第 5节。 4.1 初步的考虑 （ 1）由于载入齿轮驱动的结果，最大的传动误差被减少，而且接触比增加了。 （ 2）创造者的方式允许在有限机械要素模型的自动生产之前的时候减少模型的准备 对于应用结构组的每个结构 1。 （ 3）图 3举例说明在负载之下被调查的一个结构。 TAC允许确定齿面 1 和 2 的接触的点 M,在负载被应用（图 3（ a）之前， N2 和 N1 是表面的法线。（图 3（ b）和（ c）获得小齿轮和传动机构的齿面的柔性变形应用扭距到传动机构的结果。图 3(b)的例证和（ c）以接触表面的不连续介绍为基础的。 图 3 说明了：（ a）一个单一接触结构（ b）和（ c）描述了不连续的接触表面及表面法线 N1 和 N2 （ 4） 图 4 概要的表示了 2D 空间的结构组。 TCA 决定了每个结构（将应用于柔性变形之前）的位置。 图 4 说明了装载齿轮驱动啮合组的模拟模型。 4.2 装载的齿轮驱动果断的运行应用限定的元素分析是为了传输误差的函数 描述的程序是可适用 于任何型的齿轮驱动。下列各项描述的是必须的阶段： （ 1）因为工作机的设定应用而决定分析并生产新的小齿轮和齿轮表面（包括内圆）。 （ 2） TCA 决定了相关角位置对 NF 结构（ a）（ Nf=8-16）和（ b）的观察关系。 (5) （ 3）一个预处理程序应用于生产 NF 结构的模型：（ a）小齿轮完全被强制放置，且 (b)传动机构有开关而使形成一个旋转的表面。且规定扭距被应用于这个表面。（图 5） 图 5不成型结构和弹性变形的度量 (4)从个方面获得一个装载齿轮驱动 的传动误差的总功能：（ 1）误差 引起受热面的配合误差， （ 2）有弹性的误差 。 (6) 5.数字例证 表 1是设计一个螺旋齿轮传动的设计参数。考虑下列啮合状态和传动接触： （ 1） 对于生产传动机构和小齿轮齿条，它们分别地有如横断面的一个直线和抛物线轮廓。所谓的高的配合误差是由生产齿条刀轮廓产生的。 （ 2） 齿轮驱动的欠对准是由轴角 0 的误差引起的。 （ 3） 给由 0 所引起的传动误差提供了一个初步设计的抛物线函数。 （ 4） TCA（齿接触分析）决心应用由 引起的卸货和装载齿轮驱动的传动误差。这种调查能够影响传动误差大小方面的负载。 （ 5） 电脑程式的应用能分析有限的机械要素而决定装载的 齿轮驱动的应力。 （ 6） 调查轴向接触的成型。 表 1 设计参数 小齿轮的齿牙数目， N1 21 传动机构的齿牙数目， N22 77 常态组件， mn 5.08 mm 正压力角 , n n 25 小齿轮螺旋线的方向 左手方 螺旋角 , 30 齿面宽 , b 70 mm 小齿轮齿条刀抛物线系数 , aca 0.002 mm1 圆柱蜗杆的定位半径 , rwa 98 mm 滚动小齿轮的修正系数 , amrb 0.00008 rad/mm2 小齿轮的应用扭距 c 250 N m 用下列的一个例子来描述： 例 1：考虑一个排列的齿轮驱动（ =0 ）卸下齿轮驱动。抛物线功能提供一个最大值的传动误差 2(1)=8 （图 6（ a）。循环啮合 .把小齿轮和轮齿方面的轴向接触定位纵向（图（ b）和（ c）。 图 6一个欠对准卸货齿轮传动的计算结果：（ a）传动误差函数（ b）和（ c）在小齿轮和轮齿表面上的接触路径。 6. 噪音的两个传输误差函数的有力对比 6.1 应用方式概念上的考虑 噪音信号的源动力是以假定 为基础的，声波发生的摆动的速度与传动机构的速度瞬时值成比例的变动。这一假定（即使大体上不是很正确）是很好的第一个猜测，因为它避免了齿轮驱动的一个复杂动模型的应用。 我们提议并强调应用下列的状态方式： （ a）目标信号的动力是不同的，但并不是肯定的绝对值信号。 （ b）不同的信号动力大体上引出一个不同结果为两个不同的光滑传动误差函数。 提议应用的传动误差函数引起的功能信号是以基部平均数角尺比较为基础的 9。定义如此的比较信号模拟强度 (7) 这里描述了传动机构的角速度偏差的平均值，而且 rms 描述了 rms需要的值 2(1) 。传动误差回收功率定义为 2= m 211+ 2(1), m 21 是齿数比。 区别考虑计时，我们获得传动机构的角速度 (8) 其中 假定为常数。在第二个术语的右边，（ 8）表现了对于速度的变动 (9) 上面的定义假设传输错误函数是连续可微的。在用有限元方法模拟负载齿轮启动器计算的情况下，这个函数是用有限个给定的点（ 1） i，（ 2） i（ i=1,2 ）来定义的。为了 Eq 的使用，各点的给定值必须用连续函数进行插值计算。 6.2. 分段函数的插补 在这种情况下（图 7）， 用一条直线将两个连续的数据点连接起来。在 i 和i-1 点之间的速度是不变的，并且由下式确定： (10) 图 7 插补函数传输误差分段的应用于线性函数 数据点的选择如下：（ i）增量 ( 1)i ( 1)i1在每个区间 i 内被认为是不变的。基于这种假设，两个功率量的比值式如下所示： (11) 7. 结论 通过先前的讨论，计算和数字的例子能够得到下列的结论： （ 1）齿轮驱动（如果没有提供充足的表面修正）的对准误差可能引起混合啮合：（ a）面与面和（ b）边缘接触（如表面与曲线）边缘接触可通过初步设计的抛 物线函数（ PPF）来避免。 （ 2）调查发现传动误差抛物线函数的应用可减少齿轮驱动的噪音和震动。应用 PPF 最少要修正生产齿轮驱动的一个构件，通常为小齿轮。（或者是蜗杆驱动的蜗杆） （ 3）负荷齿轮启动器的传输错误的确定需要运用一个一般用途的有限元电脑程序。负荷齿轮启动器配有弹性可变的轮齿，这样接触率增加，由于启动器的未对准而产生的传输错误将减少。由于使用了作者设计的有限元模块的自动产生方法使得模块的准备时间大大的缩短了。这种方法是专门为确定负荷齿轮传输错误而设计的。 致谢 作者对格林森基金会和日本雅马哈公司在财 政上的支持表示深切地感谢。 参考文献 1 J. Argyris, A. Fuentes and F.L. Litvin, Computerized integrated approach for design and stress analysis of spiral bevel gears, Comput. Methods Appl. Mech. Engrg. 191 (2002), pp. 1057 1095. SummaryPlus | Full Text + Links | PDF (1983 K) 2 Gleason Works, Understanding Tooth Contact Analysis, Rochester, New York, 1970. 3 Hibbit, Karlsson & Sirensen, Inc., ABAQUS/Standard Users Manual, 1800 Main Street, Pawtucket, RI 20860-4847, 1998. 4 Klingelnberg und Shne, Ettlingen, Kimos: Zahnkontakt-Analyse fr Kegelrder, 1996. 5 F.L. Litvin et al., Helical and spur gear drive with double crowned pinion tooth surfaces and conjugated gear tooth surfaces, USA Patent 6,205,879, 2001. 6 F.L. Litvin, A. Fuentes and K. Hayasaka, Design, manufacture, stress analysis, and experimental tests of low-noise high endurance spiral bevel gears, Mech. Mach. Theory 41 (2006), pp. 83 118. SummaryPlus | Full Text + Links | PDF (1234 K) 7 F.L. Litvin and A. Fuentes, Gear Geometry and Applied Theory (second ed.), Cambridge University Press, New York (2004). 8 J.J. Mor, B.S. Garbow, K.E. Hillstrom, User Guide for MINPACK-1, Argonne National Laboratory Report ANL-80-74, Argonne, Illinois, 1980. 9 A.D. Pierce, Acoustics. An Introduction to Its Physical Principles and Applications, Acoustical Society of America (1994). 10 J.D. Smith, Gears and Their Vibration, Marcel Dekker, New York (1983). 11 H.J. Stadtfeld, Gleason Bevel Gear Technology Manufacturing, Inspection and Optimization, Collected Publications, The Gleason Works, Rochester, New York (1995). 12 O.C. Zienkiewicz and Reduction of noise of loaded and unloaded misaligned gear drives Faydor L. Litvina, Daniele Vecchiatoa, Kenji Yukishimaa, Alfonso Fuentesb, , , Ignacio Gonzalez-Perezb and Kenichi Hayasakac aGear Research Center, Department of Mechanical and Industrial Engineering, University of Illinois at Chicago, 842 W. Taylor St., Chicago, IL 60607-7022, USA bDepartment of Mechanical Engineering, Polytechnic University of Cartagena, C/Doctor Fleming, s/n, 30202, Cartagena, Murcia, Spain cGear R&D Group, Research and Development Center, Yamaha Motor Co., Ltd., 2500 Shingai, Iwata, Shizuoka 438-8501, Japan Received 22 February 2005； revised 6 May 2005； accepted 17 May 2005. Available online 25 January 2006. Abstract Transmission errors are considered as the main source of vibration and noise of gear drives. The impact of two main functions of transmission errors on noise is investigated: (i) a linear one, caused by errors of alignment, and (ii) a predesigned parabolic function of transmission errors, applied for reduction of noise. It is shown that a linear function of transmission errors is accompanied with edge contact, and then inside the cycle of meshing, the meshing becomes a mixed one: (i) as surface-to-surface tangency, and (ii) surface-to-curve meshing when edge contact starts. Application of a predesigned parabolic function of transmission errors enables to absorb the linear functions of transmission errors caused by errors of alignment, reduce noise, and avoid edge contact. The influence of the load on the function of transmission errors is investigated. Elastic deformations of teeth enable to reduce the maximal transmission errors in loaded gear drives. Computerized simulation of meshing and contact is developed for loaded and unloaded gear drives. Numerical examples for illustration of the developed theory are provided. Keywords: Gear drives ； Transmission errors； Tooth contact analysis (TCA)； Finite element analysis； Reduction of noise Article Outline 1. Introduction 2. Modification of tooth surfaces 2.1. Helical gear drives 2.2. Spiral bevel gears 2.3. Worm gear drives with cylindrical worm 3. Types of meshing and basic functions of transmission errors 4. Transmission errors of a loaded gear drive 4.1. Preliminary considerations 4.2. Application of finite element analysis for determination of function of transmission errors of a loaded gear drive 5. Numerical examples 6. Comparison of the power of noise for two functions of transmission errors 6.1. Conceptual consideration of applied approach 6.2. Interpolation by a piecewise linear function 7. Conclusion Acknowledgements References 1. Introduction Simulation of meshing of gear drives performed by application of tooth contact analysis (TCA) and test of gear drives have confirmed that transmission errors are the main source of vibrations of the gear box and such vibrations cause the noise of gear drive 1, 2, 4, 5, 6, 7, 10 and 11. The shape of functions of transmission errors depends on the type of errors of alignment and on the way of modification of gear tooth surfaces performed for improvement of the drive (see Section 2). The reduction of noise proposed by the authors is achieved as follows: (1) The bearing contact of tooth surfaces is localized. (2) A parabolic function of transmission errors is provided. This allows to absorb linear functions of transmission errors caused by misalignment 7. (3) One of the pair of mating surfaces is modified by double-crowning (see Section 2). This allows usually to avoid edge contact (see Section 5). The authors have compared the results of application of TCA for loaded and unloaded gear drives. It is shown that transmission errors of a loaded gear drive are reduced. The developed approach is illustrated with numerical examples (see Section 5). 2. Modification of tooth surfaces Reduction of noise of a gear drive requires modification of one of the pair of contacting surfaces. The surface modification is illustrated for three types of gear drives: helical gears, spiral bevel gears, and worm gear drives. 2.1. Helical gear drives Profile crowning of helical gears may be illustrated considering that the mating surfaces are generated by two rack-cutters with mismatched profiles 5 and 7. Profile crowning allows to localize the bearing contact. Double-crowning in comparison with profile crowning allows to: (i) avoid edge contact (caused by errors of crossing angle and different helix angles of mating gears), and (ii) provide a parabolic function of transmission errors. Double-crowning is performed by plunging of the disk that generates the pinion (see details in Chapter 15 of Ref. 7). 2.2. Spiral bevel gears Localization of contact of generated spiral bevel gears is provided by application of two mismatched head-cutters p and g used for generation of the pinion and the gear, respectively 7. Two head-cutters p and g have a common line C of generating tooth surfaces (in the case when profile crowning is provided). In the case of double-crowning, the mismatched generating surfaces p and g of the head -cutters have only a common single point of tangency, but not a line of tangency. Double-crowning of a generated gear may be achieved by tilting of one of the pair of generating head-cutters, or by proper installment of one of the head-cutters. It is very popular for the modern technology that during the generation of one of the mating gears, usually of the pinion, modified roll is provided 7. 2.3. Worm gear drives with cylindrical worm Very often the technology of manufacturing of a worm-gear is based on the following approach. The generation of the worm-gear is performed by a hob that is identical to the worm of the gear drive. The applied machine-tool settings simulate the meshing of the worm and worm-gear of the drive. However, manufacture with observation of these conditions causes an unfavorable bearing contact, and high level of transmission errors. Minimization of such disadvantages may be achieved by various ways: (i) by long-time lapping of the produced gear drive in the box of the drive； (ii) by running of the gear drive under gradually increased load, up to the maximal load； (iii) by shaving of the worm-gear in the box of the drive by using a shaver with minimized deviations of the worm-member, etc. The authors approach is based on localization of bearing contact by application of: (a) an oversized hob, and (b) modification of geometry (see below). There are various types of geometry of worm gear drives 7, but the preferable one is the drive with Klingelnbergs type of worm. Such a worm is generated by a disk with profiles of a circular cone 7. The relative motion of the worm with respect to the generating disk is a screw one (in the process of generation). Very often localization of bearing contact in a worm gear drive is achieved by application of a hob that is oversized in comparison with the worm of the drive. 3. Types of meshing and basic functions of transmission errors It is assumed that the tooth surfaces are at any instant in point tangency due to the localization of contact. Henceforth, we will consider two types of meshing: (i) surface-to-surface, and (ii) surface-to-curve. Surface-to-surface tangency is provided by the observation of equality of position vectors and surface unit normals 7. Surface-to-curve meshing is the result of existence of edge contact 7. The algorithm of TCA for surface-to-surface tangency is based on the following vector equations 7: (1) (2) that represent in fixed coordinate system Sf position vectors and surface unit normals . Here, (ui, i) are the surface parameters and ( 1, 2) determine the angular positions of surfaces. The algorithm for surface-to-curve tangency is represented in Sf by equations 7 (3) (4) Here, represents the surface that is in mesh with curve is the tangent to the curve of the edge. Application of TCA allows to discover both types of meshing, surface-to-surface and surface-to-curve. Computerized simulation of meshing is an iterative process based on numerical solution of nonlinear equations 8. By applying double-crowning to one of the mating surfaces, it becomes possible to: (i) avoid edge contact, and (ii) obtain a predesigned parabolic function 7 (Fig. 1). Application of a predesigned parabolic function is the precondition of reduction of noise. Fig. 1. Illustration of: (a) transmission functions 1 of a misaligned gear drive and linear function 2 of an ideal gear drive without misalignment； (b) periodic functions 2( 1) of transmission errors formed by parabolas. Application of double-crowning allows to assign ahead that function of transmission errors is a parabolic one, and allows to assign as well the maximal value of transmission errors as of 6 8. The expected magnitude of the predesign parabolic function of transmission errors and the magnitude of the parabolic plunge of the generating tool have to be correlated. Fig. 2 shows the case wherein due to a large magnitude of error of misalignment, the function of transmission errors is formed by two branches: of surface-to-surface contact and of surface-to-curve contact. Fig. 2. Results of TCA of a case of double-crowned helical gear drive with a large error = 10: (a) function of transmission errors wherein corresponds to surface-to-surface tangency and correspond to surface-to-curve tangency； (b) path of contact on pinion tooth surface； (c) path of contact on gear tooth surface. 4. Transmission errors of a loaded gear drive The contents of this section cover the procedure of determination of transmission errors of a loaded gear drive by application of a general purpose FEM computer program 3. Transmission errors of an unloaded gear drive are directly determined by application of TCA. Comparison of transmission errors for unloaded and loaded gear drives is represented in Section 5. 4.1. Preliminary considerations (i) Due to the effect of loading of the gear drive, the maximal transmission errors are reduced and the contact ratio is increased (ii) The authors approach allows to reduce the time of preparation of the model by the automatic generation of the finite element model 1 for each configuration of the set of applied configurations. (iii) Fig. 3 illustrates a configuration that is investigated under the load. TCA allows to determine point M of tangency of tooth surfaces 1 and 2, before the load wil l be applied (Fig. 3(a), where N2 and N1 are the surface normals (Fig. 3(b) and (c). The elastic deformations of tooth surfaces of the pinion and the gear are obtained as the result of applying the torque to the gear. The illustrations of Fig. 3(b) and (c) are based on discrete presentations of the contacting surfaces. Fig. 3. Illustration of: (a) a single configuration； (b) and (c) discrete presentations of contacting surfaces and surface normals N1 and N2. (iv) Fig. 4 shows schematically the set of configurations in 2D space. The location of each configuration (before the elastic deformation will be applied) is determined by TCA. Fig. 4. Illustration of set of models for simulation of meshing of a loaded gear drive. 4.2. Application of finite element analysis for determination of function of transmission errors of a loaded gear drive The described procedure is applicable for any type of a gear drive. The following is the description of the required steps: (i) The machine-tool settings applied for generation are known ahead, and then the pinion and gear tooth surfaces (including the fillet) may be determined analytically. (ii) Related angular positions are determined by (a) applying of TCA for Nf configurations (Nf = 8 16), and (b) observing the relation (5) (iii) A preprocessor is applied for generation of Nf models with the conditions: (a) the pinion is fully constrained to position , and (b) the gear has a rigid surface that can rotate about the gears axis ( Fig. 5). Prescribed torque is applied to this surface. (vi) The total function of transmission errors for a loaded gear drive is obtained considering: (i) the error caused due to the mismatched of generating surfaces, and (ii) the elastic approach . (6) 5. Numerical examples A helical gear drive with design parameters given in Table 1 is designed. The following conditions of meshing and contact of the drive are considered: (1) The gear and pinion rack-cutters are provided with a straight-line and parabolic profiles as cross-section profiles, respectively, for generation of the gear and the pinion. Mismatched rack-cutter profiles yield the so-called profile crowning. (2) The misalignment of gear drive is caused by an error of the shaft angle, 0. (3) A predesigned parabolic function for absorption of transmission errors caused by 0 is provided. （ Such a function for a double-crowned pinion tooth surface is obtained by plunging of the generating disk, or by modified roll of the grinding worm.） (4) TCA (tooth contact analysis) for unloaded and loaded gear drives are applied for determination of transmission errors caused by . This enables to investigate the influence of the load on the magnitude and shape of the function of transmission errors. (5) Application of a computer program for finite element analysis 3 enables to determine the stresses of a loaded gear drive. (6) Formation of bearing contact is investigated. Table 1. Design parameters Number of teeth of the pinion, N1 21 Number of teeth of the gear, N2 77 Normal module, mn 5.08 mm Normal pressure angle, n 25 Hand of helix of the pinion Left-hand Helix angle, 30 Face width, b 70 mm Parabolic coefficient of pinion rack-cutter, aca 0.002 mm1 Radius of the worm pitch cylinder, rwa 98 mm Parabolic coefficient of pinion modified roll, amrb 0.00008 rad/mm2 Applied torque to the pinionc 250 N m (i) Example 1: An aligned gear drive ( = 0) is considered. The gear drive is unloaded. A parabolic function with the maximal value of transmission errors 2( 1) = 8 is provided ( Fig. 6(a). The cycle of meshing is . The bearing contact on the pinion and gear tooth surfaces is oriented almost longitudinally (Fig. 6(b) and (c). Fig. 6. Results of computation for an unloaded gear drive without misalignment: (a) function of transmission errors； (b) and (c) paths of contact on pinion and gear tooth surfaces. 6. Comparison of the power of noise for two functions of transmission errors 6.1. Conceptual consideration of applied approach Determination of the power of the signal of noise is based on the assumption that the velocity of oscillation of the generated acoustic waves is proportional to the fluctuation of the instantaneous value of the velocity of the gears. This assumption (even if not accurate in general) is good as the first guess, since it allows to avoid application of a complex dynamic model of the gear drive. We emphasize that the proposed approach is applied for the following conditions: (a) The goal is the determination of difference of power of signals, but not the determination of absolute values of signals. (b) The difference of power of signals is the result mainly of the difference of first derivatives of two smooth functions of transmission errors. The proposed approach is based on the comparison of the root mean square of the signals (in rms) caused by two functions of transmission errors 9. Such comparison yields the simulation of the intensity (the power) of the signal defined as (7) Here 2( 1) represents the deviation of the angular velocity of the gear from the average value, and rms represents the desired rms value. The definition of function of transmission errors yields that 2 = m21 1 + 2( 1), where m21 is the gear ratio. By differentiation with respect to time, we obtain the angular velocity of the gear as (8) wherein is assumed as constant. The second term on the right side of Eq. (8) represents the sought-for fluctuation of velocity (9) The definition above assumes that the function of transmission errors (FTE) is a continuous and differentiable one. In the case of computation of a loaded gear drive simulated by FEM (finite element method), this function is defined by a finite number of given points ( 1)i, ( 2)i) (i = 1, , n). The given data of points have to be interpolated by continuous functions for application of Eq. (7).) 6.2. Interpolation by a piecewise linear function In this case (Fig. 7), two successive data points are connected by a straight line. The derivative (velocity) between point i and i 1 is constant and is determined as follows: (10) Fig. 7. Interpolation of function of transmission errors by application of a piecewise linear function. Data points have been chosen as follows: (i) an increment ( 1)i ( 1)i1 is considered as constant for each interval i, and (ii) as the same for the two functions (FTE) represented in Examples 2 and 3 (in Section 5). Based on this assumption, the ratio of two magnitudes of power by application of the mentioned functions is represented as (11) 7. Conclusion The previously presented discussions, computations, and numerical examples enable to draw the following conclusions: (1) Errors of alignment of a gear drive (if modification of surfaces is not provided enough) may cause a mixed meshing: (i) surface-to-surface and (ii) edge contact (as surface-to-curve). Edge contact may be usually