減少偏離齒輪傳動裝載和卸載時的噪音 Faydor L. Litvina, Daniele Vecchiatoa, Kenji Yukishimaa, Alfonso Fuentesb, Ignacio Gonzalez-Perezb and Kenichi Hayasakac 芝加哥伊利諾州大學機械部門和工業(yè)工程齒輪研究中心， 842 W. IL 60607-7022, 泰勒圣，芝加哥， 美國 喀他赫納工藝綜合大學機械工程部博士， Murcia， 30202，喀他赫納，西班牙 山葉電動機股份有限公司齒輪半 徑研究發(fā)展中心， 2500 Shingai, Iwata, 靜岡 438-8501, 日本 2005 年二月 22 日定為標準； 2005 年五月 6 日校訂了； 2005 年五月 17 日被公認； 2006 年一月 25 日可在線應用 . 摘要 齒輪傳動時產(chǎn)生震動和噪音的主要原因是傳輸誤差。有關影響噪音傳輸誤差的兩個主要函數(shù)已被查明：（ 1）一個是線性的對應誤差；（ 2）一個是初步設計使用傳輸誤差以減少噪音而引起的。它顯示了傳輸誤差的線性關系，在一個周期內(nèi)形成了混合的循環(huán)嚙合：（ 1）如點對點接觸；（ 2）當從表面以曲線形 式移動到起始點時就產(chǎn)生嚙合。使用初步設計傳輸誤差能夠減少因為線性對應函數(shù)而引起的傳輸誤差，減少噪音和避免移動接觸。引起傳輸誤差的負載函數(shù)已被研究。齒牙的損壞能夠使在裝載的齒輪傳動中減少最大的傳輸誤差。用計算機處理的模擬齒輪嚙合，且齒輪傳動裝載和卸貨技術已發(fā)展相當水平。 關鍵詞：齒輪傳動；傳輸誤差；齒牙嚙合分析（ YCA）；限定的元素分析；噪音的減少。 文章概要 1. 緒論 2. 齒牙表面的修正 2.1. 螺旋狀的齒輪傳動 2.2. 螺旋狀的斜齒輪 2.3. 圓柱形的蝸桿齒輪傳動 3. 嚙合的類型和基本功能的 傳輸誤差 4. 裝載齒輪驅(qū)動的傳動誤差 4.1. 初步的考慮 4.2. 裝載的齒輪傳動果斷的運行應用限定的元素分析是為了傳輸誤差的函數(shù) 5. 數(shù)字例證 6. 噪音的兩個傳輸誤差函數(shù)的有力對比 6.1. 應用方式概念上的考慮 6.2. 線性函數(shù)的分段插補 7.結論 參考文獻 1. 緒論 模擬的齒輪傳動嚙合執(zhí)行應用齒牙接觸分析（ TCA）和測試齒輪傳動已被證實傳輸誤差的主要原因是齒輪箱的震動，這樣的震動引起齒輪傳動的噪音 1，2， 3， 4， 5， 6， 7， 10和 11。傳輸誤差函數(shù)的類型 依賴對應錯誤的類型且齒輪齒牙表面為了進一步的傳動在進行改善。（見第二節(jié)） 為減少噪音而依下列的計劃進行： （ 1） 齒牙接觸表面被局部化 （ 2） 提供一個傳輸誤差的函數(shù)。這種傳輸錯誤是由未對準的一次函數(shù)所引起的7。 （ 3） 對雙層表面之一進行最高倍數(shù)的修正。 見第 2 節(jié) 這通常是避免表面摩擦。 見第 5 節(jié) 已經(jīng)對裝載和卸載齒輪傳動應用 TCA 進行了比較，它顯示裝載的齒輪傳動的傳輸誤差較少。其發(fā)展的方式與數(shù)字進行一起舉例。 見第 5 節(jié) 2. 齒牙表面的修正 減少齒輪傳動的噪音需要修正接觸的雙表面之一。要修正齒輪傳動接觸表面有三種類型： 螺旋狀的齒輪，螺旋狀的斜齒輪，蝸桿齒輪。 2.1 螺旋狀的齒輪傳動 螺旋狀的齒輪最高剖面可能相交而表面產(chǎn)生兩個齒條刀形成錯誤的輪廓 5和 7。 完美輪廓允許接觸方向的局部化。最完美的輪廓比較是允許的：（ 1）避免邊緣接觸（交叉角和不同形狀角的相交齒輪）（ 2）提供一個傳輸誤差的拋物線函數(shù)。雙倍完美的執(zhí)行突進的圓盤而產(chǎn)生小齒輪（見 REF 的第 15 章資料。 7）。 2.2 螺旋狀的斜齒輪 應用提供兩個有誤差的刀尖 p 和 g 而有局部接觸會產(chǎn)生 螺旋狀的斜齒輪： p 和 g 二者是分別用來產(chǎn)生小齒輪和齒輪的 7。 倆個刀尖 p 和 g再齒呀的表面產(chǎn)生一個共同線 C。（當提供外層輪廓的情況下）再加倍的情況下產(chǎn)生配合誤差表面 p 和 g 刀尖 只有接觸的通常單一點，但不是一條接觸的線。加倍可能產(chǎn)生齒輪而形成有斜齒的刀尖，或者是刀尖特有的部分。她是近代科技生產(chǎn)的齒輪當中教授歡迎的齒輪之一，通常小齒輪都被改良為滾動的 7。 2.3 圓柱型蝸桿齒輪傳動 通常蝸輪制造工藝是以下列的方式為基礎。蝸輪的生產(chǎn)和蝸桿齒輪傳動一樣都是由一個滾刀運行的。應用的機床設置模擬蝸桿和蝸輪嚙合而形成齒輪傳動。然而，觀察發(fā)現(xiàn)在這些條件下的制造引起不宜的軸接觸 ，和高度傳動誤差。為把這些誤差減少到最低限度可用以下不同的方法完成： （ 1） 長期在齒輪箱中研磨加工而使齒輪傳動畸形； （ 2） 齒輪傳動在長期的運轉(zhuǎn)下產(chǎn)生負載，近而達到最大負載； （ 3） 蝸輪在蝸輪箱中被刨且傳動裝置利用刨削蝸桿部分背離減少到最小化，等等。 制造者的方法是應用接觸局限為基礎的：（ a）一個特大號的滾齒刀，和（ b）幾何學的修正。（見下面）。 有蝸輪傳動幾何學的各種不同類型 7，但是一個較好的是有 Klingelnberg類型的蝸桿。 這種蝸桿是由圓盤輪廓和錐形圓作成的 7。有關蝸桿傳動要考慮圓盤的一個螺紋的產(chǎn)生（在生產(chǎn) 的方法中）。 通常，在蝸輪傳動局限接觸中以達成應用滾刀且是比較特大號的蝸輪傳動。 3. 嚙合的類型和傳動誤差的基本函數(shù) 它假定齒牙表面任何點相切是正當?shù)木窒薅ㄎ?。此后，我們考慮兩種嚙合：（ 1）面與面，（ 2）面與曲線。面與面相切是平等觀察表面的位置向量和表面單位提供 7。面與曲線嚙合是曲線邊緣實在接觸的結果 7。 面與面相切的 TCA 運算法則是以下列的矢量為基礎的方程 7： (1) (2) 在固定的同等系統(tǒng) Sf位置矢量 和表面常態(tài) 中表現(xiàn)。這里， (ui, i)是表面的參數(shù)而且 ( 1, 2)決定表 面的角位置。 面與曲線的運算法則是用 Sf方程來表現(xiàn)的 7： (3) (4) 在這里 描述表面的嚙合曲線 是邊緣曲線的切線。 TCA 的允許應用而發(fā)現(xiàn)兩種嚙合的類型，面與面和面與曲線。計算機處理的嚙合模擬是以一個反復的程序為基礎的非線性方程的數(shù)字解決方案 8 應用最高的相交表面之一，它可能變成：（ 1）避免邊緣接觸，（ 2）獲得一個初步設計的拋物線函數(shù) 7（圖 1）。初步設計的拋物線函數(shù)功能的應用是減少噪音的先決條件。 圖 1 例證：（ a）齒輪驅(qū)動的一個不成直線的傳動函數(shù) 1 和沒有欠對準的理的線 性函數(shù) 2；（ b）周期函數(shù)拋物線形成的傳動誤差 2( 1)。 應用最高的允許向前分配傳動的誤差函數(shù)的是一個拋物線，而且允許分配同樣最大誤差值的 6-8 。初步設計預期大小的傳動誤差拋物線函數(shù)和投入大量生產(chǎn)的工具是有關聯(lián)的。圖 2表示在何處由于欠對準的誤差的大小，傳動誤差函數(shù)形成兩個支流： 面對面接觸和 面與曲線接觸。 圖 2 一個螺旋狀齒輪的最大 TCA 誤差結果 = 10: （ a）傳動誤差函數(shù)在何處 符合面與面相切和何處 符合面與曲線相切；（ b）在小齒輪齒面上相切的路徑：（ c）在齒輪表面的接觸路徑。 4.裝 載齒輪傳動的傳動誤差 這一部分內(nèi)容覆蓋了一般用途 FEM 電腦程序應用裝載齒輪驅(qū)動的傳動誤差果斷程序 3。 TCA 決定直接應用卸載齒輪驅(qū)動的傳動誤差。描述比較裝載和卸載時齒輪驅(qū)動的傳動誤差在第 5節(jié)。 4.1 初步的考慮 （ 1）由于載入齒輪驅(qū)動的結果，最大的傳動誤差被減少，而且接觸比增加了。 （ 2）創(chuàng)造者的方式允許在有限機械要素模型的自動生產(chǎn)之前的時候減少模型的準備 對于應用結構組的每個結構 1。 （ 3）圖 3舉例說明在負載之下被調(diào)查的一個結構。 TAC允許確定齒面 1 和 2 的接觸的點 M,在負載被應用（圖 3（ a）之前， N2 和 N1 是表面的法線。（圖 3（ b）和（ c）獲得小齒輪和傳動機構的齒面的柔性變形應用扭距到傳動機構的結果。圖 3(b)的例證和（ c）以接觸表面的不連續(xù)介紹為基礎的。 圖 3 說明了：（ a）一個單一接觸結構（ b）和（ c）描述了不連續(xù)的接觸表面及表面法線 N1 和 N2 （ 4） 圖 4 概要的表示了 2D 空間的結構組。 TCA 決定了每個結構（將應用于柔性變形之前）的位置。 圖 4 說明了裝載齒輪驅(qū)動嚙合組的模擬模型。 4.2 裝載的齒輪驅(qū)動果斷的運行應用限定的元素分析是為了傳輸誤差的函數(shù) 描述的程序是可適用 于任何型的齒輪驅(qū)動。下列各項描述的是必須的階段： （ 1）因為工作機的設定應用而決定分析并生產(chǎn)新的小齒輪和齒輪表面（包括內(nèi)圓）。 （ 2） TCA 決定了相關角位置對 NF 結構（ a）（ Nf=8-16）和（ b）的觀察關系。 (5) （ 3）一個預處理程序應用于生產(chǎn) NF 結構的模型：（ a）小齒輪完全被強制放置，且 (b)傳動機構有開關而使形成一個旋轉(zhuǎn)的表面。且規(guī)定扭距被應用于這個表面。（圖 5） 圖 5不成型結構和彈性變形的度量 (4)從個方面獲得一個裝載齒輪驅(qū)動 的傳動誤差的總功能：（ 1）誤差 引起受熱面的配合誤差， （ 2）有彈性的誤差 。 (6) 5.數(shù)字例證 表 1是設計一個螺旋齒輪傳動的設計參數(shù)。考慮下列嚙合狀態(tài)和傳動接觸： （ 1） 對于生產(chǎn)傳動機構和小齒輪齒條，它們分別地有如橫斷面的一個直線和拋物線輪廓。所謂的高的配合誤差是由生產(chǎn)齒條刀輪廓產(chǎn)生的。 （ 2） 齒輪驅(qū)動的欠對準是由軸角 0 的誤差引起的。 （ 3） 給由 0 所引起的傳動誤差提供了一個初步設計的拋物線函數(shù)。 （ 4） TCA（齒接觸分析）決心應用由 引起的卸貨和裝載齒輪驅(qū)動的傳動誤差。這種調(diào)查能夠影響傳動誤差大小方面的負載。 （ 5） 電腦程式的應用能分析有限的機械要素而決定裝載的 齒輪驅(qū)動的應力。 （ 6） 調(diào)查軸向接觸的成型。 表 1 設計參數(shù) 小齒輪的齒牙數(shù)目， N1 21 傳動機構的齒牙數(shù)目， N22 77 常態(tài)組件， mn 5.08 mm 正壓力角 , n n 25 小齒輪螺旋線的方向 左手方 螺旋角 , 30 齒面寬 , b 70 mm 小齒輪齒條刀拋物線系數(shù) , aca 0.002 mm1 圓柱蝸桿的定位半徑 , rwa 98 mm 滾動小齒輪的修正系數(shù) , amrb 0.00008 rad/mm2 小齒輪的應用扭距 c 250 N m 用下列的一個例子來描述： 例 1：考慮一個排列的齒輪驅(qū)動（ =0 ）卸下齒輪驅(qū)動。拋物線功能提供一個最大值的傳動誤差 2(1)=8 （圖 6（ a）。循環(huán)嚙合 .把小齒輪和輪齒方面的軸向接觸定位縱向（圖（ b）和（ c）。 圖 6一個欠對準卸貨齒輪傳動的計算結果：（ a）傳動誤差函數(shù)（ b）和（ c）在小齒輪和輪齒表面上的接觸路徑。 6. 噪音的兩個傳輸誤差函數(shù)的有力對比 6.1 應用方式概念上的考慮 噪音信號的源動力是以假定 為基礎的，聲波發(fā)生的擺動的速度與傳動機構的速度瞬時值成比例的變動。這一假定（即使大體上不是很正確）是很好的第一個猜測，因為它避免了齒輪驅(qū)動的一個復雜動模型的應用。 我們提議并強調(diào)應用下列的狀態(tài)方式： （ a）目標信號的動力是不同的，但并不是肯定的絕對值信號。 （ b）不同的信號動力大體上引出一個不同結果為兩個不同的光滑傳動誤差函數(shù)。 提議應用的傳動誤差函數(shù)引起的功能信號是以基部平均數(shù)角尺比較為基礎的 9。定義如此的比較信號模擬強度 (7) 這里描述了傳動機構的角速度偏差的平均值，而且 rms 描述了 rms需要的值 2(1) 。傳動誤差回收功率定義為 2= m 211+ 2(1), m 21 是齒數(shù)比。 區(qū)別考慮計時，我們獲得傳動機構的角速度 (8) 其中 假定為常數(shù)。在第二個術語的右邊，（ 8）表現(xiàn)了對于速度的變動 (9) 上面的定義假設傳輸錯誤函數(shù)是連續(xù)可微的。在用有限元方法模擬負載齒輪啟動器計算的情況下，這個函數(shù)是用有限個給定的點（ 1） i，（ 2） i（ i=1,2 ）來定義的。為了 Eq 的使用，各點的給定值必須用連續(xù)函數(shù)進行插值計算。 6.2. 分段函數(shù)的插補 在這種情況下（圖 7）， 用一條直線將兩個連續(xù)的數(shù)據(jù)點連接起來。在 i 和i-1 點之間的速度是不變的，并且由下式確定： (10) 圖 7 插補函數(shù)傳輸誤差分段的應用于線性函數(shù) 數(shù)據(jù)點的選擇如下：（ i）增量 ( 1)i ( 1)i1在每個區(qū)間 i 內(nèi)被認為是不變的?；谶@種假設，兩個功率量的比值式如下所示： (11) 7. 結論 通過先前的討論，計算和數(shù)字的例子能夠得到下列的結論： （ 1）齒輪驅(qū)動（如果沒有提供充足的表面修正）的對準誤差可能引起混合嚙合：（ a）面與面和（ b）邊緣接觸（如表面與曲線）邊緣接觸可通過初步設計的拋 物線函數(shù)（ PPF）來避免。 （ 2）調(diào)查發(fā)現(xiàn)傳動誤差拋物線函數(shù)的應用可減少齒輪驅(qū)動的噪音和震動。應用 PPF 最少要修正生產(chǎn)齒輪驅(qū)動的一個構件，通常為小齒輪。（或者是蝸桿驅(qū)動的蝸桿） （ 3）負荷齒輪啟動器的傳輸錯誤的確定需要運用一個一般用途的有限元電腦程序。負荷齒輪啟動器配有彈性可變的輪齒，這樣接觸率增加，由于啟動器的未對準而產(chǎn)生的傳輸錯誤將減少。由于使用了作者設計的有限元模塊的自動產(chǎn)生方法使得模塊的準備時間大大的縮短了。這種方法是專門為確定負荷齒輪傳輸錯誤而設計的。 致謝 作者對格林森基金會和日本雅馬哈公司在財 政上的支持表示深切地感謝。 參考文獻 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