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動(dòng)力傳動(dòng)圓錐漸開線齒輪的設(shè)計(jì)、制造和應(yīng)用
Dr.J.Borner,K.Humm,Dr.F.Joachim,Dr.H.akaria,
ZF Friedrichshafen AG,88038Friedrichshafen,Germany;
[摘要]圓錐漸開線齒輪（斜面體齒輪）被用于交叉或傾斜軸變速器和平行軸自由側(cè)隙變速器中。圓錐齒輪是在齒寬橫斷面上具有不同齒頂高修正（齒厚）的直齒或斜齒圓柱齒輪。這類齒輪的集合形狀是已知的，但應(yīng)用在動(dòng)力傳動(dòng)上則多少是個(gè)例外。ZF公司已將該斜面體齒輪裝置應(yīng)用于各種場(chǎng)合：4W D轎車傳動(dòng)裝置、船用變速器（主要用于快艇）機(jī)器人齒輪箱和工業(yè)傳動(dòng)等領(lǐng)域。斜面體齒輪的模數(shù)在0.7mm-8mm之間，交叉?zhèn)鲃?dòng)角0°-25°之間。這些邊界條件需要對(duì)斜面體齒輪的設(shè)計(jì)、制造和質(zhì)量有一個(gè)深入的理解。在錐齒輪傳動(dòng)中為獲得高承載能力和低噪音所必須進(jìn)行的齒側(cè)修形課采用范成法磨削工藝制造。為降低制造成本，機(jī)床設(shè)定和由于磨削加工造成的齒側(cè)偏差可在設(shè)計(jì)階段利用仿真制造進(jìn)行計(jì)算。本文從總體上介紹了動(dòng)力傳動(dòng)變速器斜面體齒輪的研發(fā)，包括：基本幾何形狀、宏觀及微觀幾何形狀的設(shè)計(jì)、仿真、制造、齒輪測(cè)量和試驗(yàn)。
1 前言
在變速器中如果各軸軸線不平行的話。轉(zhuǎn)矩傳遞課采用多種設(shè)計(jì)，例如：傘齒輪或冠齒輪、萬向節(jié)軸或圓錐漸開線齒輪（斜面體齒輪）。圓錐漸開線齒輪特別適用于小軸線角度（小于15°），該齒輪的優(yōu)點(diǎn)是在制造、結(jié)構(gòu)、特點(diǎn)和輸入多樣性等方面的簡(jiǎn)易。圓錐漸開線齒輪被用于直角或交叉軸傳動(dòng)的變速器或被用于平行軸自由側(cè)隙工況的變速器。由于錐角的選擇并不取決于軸線交角，配對(duì)的齒輪也可能采用圓柱齒輪。斜面體齒輪可制成外嚙合和內(nèi)嚙合齒輪，整個(gè)可選齒輪副矩陣見表1，它為設(shè)計(jì)者提供了高度的靈活性。
圓錐齒輪是在齒寬橫截面上具有不同齒頂高修正（齒厚）量的直齒或斜齒輪。它們能與各種用同一把基準(zhǔn)齒條刀具切制成的齒輪相嚙合。斜面體齒輪的集合形狀是已知的，但它們能與各種用同一把基準(zhǔn)齒條刀具切制成的齒輪相嚙合。斜面體齒輪的幾何形狀是已知的，但它們很少應(yīng)用在動(dòng)力傳動(dòng)上。過去，未曾對(duì)斜面體齒輪的承載能力和噪聲進(jìn)行過任何大范圍的試驗(yàn)研究。標(biāo)準(zhǔn)（諸如適用于圓柱齒輪的ISO06336）、計(jì)算方法和強(qiáng)度值都是未知的。因此，必須開發(fā)計(jì)算方法、獲得承載能力數(shù)值和算出用于生產(chǎn)和質(zhì)量保證的規(guī)范。在過去的15年中，ZF公司已為錐齒輪開發(fā)了多種應(yīng)用：
1、 輸出軸具有下傾角的船用變速[1、3]
圖.1
2.轉(zhuǎn)向器[1]
3、機(jī)器人用小齒隙行星齒輪裝置（交叉軸角度1° —3°）[2]
4、用車輛的輸送齒輪箱（垃圾傾倒車）
5、AWD用自動(dòng)變速箱[4],圖2
2齒輪幾何形狀
2.1宏觀幾何形狀
簡(jiǎn)而言之，斜面體齒輪可看成是一個(gè)在齒寬橫截面上連續(xù)改變齒頂高修正的圓柱齒輪，如圖3.為此，根據(jù)齒根錐角δ刀具向齒輪軸線傾斜[1]。結(jié)果形成了齒輪基圓尺寸。
螺旋角，左/右
(1)
橫向壓力角 左/右
（2）
基圓直徑 左右
(3)
左右側(cè)不同的基圓導(dǎo)致斜齒輪齒廓形狀的不均勻，圖3.采用齒條類刀具加工將使得齒根錐具有相應(yīng)的根錐角δ。齒頂角設(shè)計(jì)成這樣以使得頂端避免與被嚙合齒輪發(fā)生干涉，并獲得最大接觸區(qū)域。由此導(dǎo)致在齒寬橫截面上具有不同的齒高。由于幾何設(shè)計(jì)限制了根切和齒頂形狀，實(shí)際齒寬隨錐角增加而減小，錐齒輪傳動(dòng)合適的錐角最大=約為15°。
2.2微觀幾何形狀
一對(duì)傘齒輪通常形成點(diǎn)狀接觸。除接觸外，在齒側(cè)還存在間隙，如圖7.齒輪修行設(shè)計(jì)的目的是減小這些間隙以形成平坦二均勻的接觸。通過逐步應(yīng)用嚙合定律有可能對(duì)齒側(cè)進(jìn)行精確的計(jì)算[5],圖4.最后，在原始側(cè)生成半徑為rp1和法向矢量為n1的P1點(diǎn)。這生成速度矢量VP1 及對(duì)于在嚙合一側(cè)所生的點(diǎn)，有半徑矢量rp2:
3傳動(dòng)裝置設(shè)計(jì)
3.1根切和齒頂形狀
斜面體齒輪的可用齒寬受到大端齒頂形狀和小端根切的限制，見圖3.齒高越高（為獲得較大的齒高變位量）理論可用齒寬越窄。小端根切和大端齒頂形狀導(dǎo)致齒高變位量沿齒寬方向發(fā)生變化。當(dāng)一對(duì)齒輪的錐角大致相同時(shí)可獲得最大的可用齒寬。若齒輪副中小齒輪愈小，則該小齒輪必須采用更小的錐角。齒頂錐角小于齒根錐角時(shí)，通常能在小端獲得有用的漸開線，而在大端處有足夠齒頂間隙，這時(shí)大端的齒頂形狀并不太嚴(yán)重。
3.2工作區(qū)域和滑動(dòng)速度
斜面體齒輪工作區(qū)域產(chǎn)生扭歪的原因是圓錐半徑有形成平行四邊形趨勢(shì)。另外，工作壓力角在齒寬橫截面方向的改變也造成工作區(qū)域的扭曲。圖5是一個(gè)例子。在交叉軸傳動(dòng)的斜面體齒輪上存在一滾動(dòng)軸；如同圓柱齒輪副的滾動(dòng)點(diǎn)一樣，在該軸上不存在滑動(dòng)。對(duì)于傾斜軸布置而言，在輪齒嚙合處總存在另外的軸向滑動(dòng)。由于工作壓力角在齒寬橫截面上變化，從小端到大端的接觸區(qū)內(nèi)的接觸軌跡有很大的變化。因此，沿齒寬方向在齒頂和齒根處具有明顯不同的滑動(dòng)速度。在齒輪中部，齒頂高修正的選擇是基于圓柱齒輪副的規(guī)范；在主動(dòng)齒輪根部的接觸軌跡將小于齒頂?shù)慕佑|軌跡。圖6給出了斜面體齒輪副主動(dòng)齒輪滑動(dòng)速度的分布。
4接觸分析和修形
4.1點(diǎn)接觸和間隙
在未修正齒輪傳動(dòng)中，由于軸線傾斜，通常僅有一點(diǎn)接觸。沿可能接觸線出現(xiàn)的間隙可大致解釋為螺旋凸起和齒側(cè)廓線角度的偏差所致。圓柱齒輪左右側(cè)間隙與軸線交叉無關(guān)。對(duì)于螺旋齒輪而言，當(dāng)兩斜面體齒輪錐角大致相同時(shí)，其產(chǎn)生的間隙也幾乎相等。隨兩齒輪錐角和螺旋角不一致的增加，左右側(cè)間隙的不同程度也增加。
在工作壓力角較小時(shí)將導(dǎo)致更大的間隙。圖7給出了具體相同錐角交叉軸傳動(dòng)的斜面體齒輪副所出現(xiàn)的間隙。圖8顯示了具體相同10°交叉軸線和30°螺旋角齒輪在左右側(cè)間隙方面的差異。兩側(cè)平均間隙的數(shù)值在很大程度上與螺旋角無關(guān)，但與兩齒輪的錐角相關(guān)。
螺旋角和錐角的選擇決定了齒輪左右側(cè)平均間隙的分布。傾斜軸線布置對(duì)接觸間隙產(chǎn)生額外影響。這將有效減少齒輪一側(cè)的螺旋凸形。如果垂直軸線與總基圓半徑相同，并且基圓柱螺旋角之差等于交叉軸角的話，間隙減少到零并出現(xiàn)線接觸。然而，在另一側(cè)將出現(xiàn)明顯的間隙。如果正交的軸線進(jìn)一步擴(kuò)大直至變成圓柱交叉軸螺旋齒輪副的話，其兩側(cè)間隙等同于較小的螺旋凸形。除螺旋凸形外，明顯的齒廓扭曲（見圖8）也是斜面體齒輪的間隙特征。隨螺旋角增加齒廓扭曲也隨之增加。圖9表明圖7所示齒輪裝置的齒廓是如何扭曲。為補(bǔ)償齒輪嚙合中所存在的間隙，必須采用齒側(cè)拓?fù)湫扌?，該類修形可明顯補(bǔ)償螺旋凸形和輪廓扭曲。未對(duì)齒廓扭曲作補(bǔ)償?shù)脑?，在工作區(qū)域僅有一個(gè)對(duì)角線狀的接觸帶，見圖10.
4.2齒側(cè)修形
對(duì)于一定程度的補(bǔ)償而言，必需的齒面形狀可由實(shí)際間隙所決定。圖11給出了這些樣品的齒形幾何特征。采用修正后的接觸率得到了很大改善如圖12所示。為應(yīng)用在系列生產(chǎn)中，其目標(biāo)總是能使用磨床加工這類齒面，對(duì)此的選擇在第6節(jié)論述。除間隙補(bǔ)償外，齒頂修形也是有益的。修形減少了嚙合開始和結(jié)束階段負(fù)荷，并能提供一較低的噪聲激勵(lì)源。然而，斜面體齒輪的齒頂修形在齒寬橫截面上的加工總量和長(zhǎng)度上是不同的。問題主要出現(xiàn)在具有一個(gè)大根錐角但頂錐角與根錐角存在偏差的齒輪上。因此齒頂修形在小端明顯大于大端。如齒輪需要在嚙合開始和結(jié)束處修形，則必須接受這種不均勻的齒頂修形。利用其它錐角如根錐角進(jìn)行齒頂修形加工也是可以的。但是，這樣需要專門用于齒頂卸載的專用磨削設(shè)備。與范成法磨削方法無關(guān)，齒側(cè)修正可采用諸如珩磨等手段；但在斜面體齒輪上應(yīng)用這些方法尚處在早起開發(fā)階段。
5承載能力和噪聲激勵(lì)
5.1計(jì)算標(biāo)準(zhǔn)的應(yīng)用
斜面體齒輪齒側(cè)和根部承載能力僅可用圓柱齒輪的計(jì)算標(biāo)準(zhǔn)（ISO 6336，DIN 3990，AGMAC95）作近似估算。具體計(jì)算時(shí)用圓柱齒輪副替代斜面體齒輪，用斜面體齒輪中部的齒寬來定義圓柱齒輪的參數(shù)。雖然斜面體齒輪齒寬是非對(duì)稱的，但在替代齒輪中可不予考慮。替代齒輪中心距由斜面體齒輪中部齒寬處的工作節(jié)圓半徑確定。當(dāng)計(jì)及齒寬橫截面時(shí)，各項(xiàng)獨(dú)立的參數(shù)都會(huì)變化，這將明顯影響承載能力。
表2給出了影響齒根和齒側(cè)承載能力的主要因素。由于沿大端方向減少輪齒齒根圓角半徑所產(chǎn)生較大的凹口效應(yīng)阻止了根部齒厚的增加。另外，在大端處，較大的節(jié)圓直徑可獲得較小的切向力；然而，大端處的齒高變位量也隨之變小。由于主要影響得到很好的平衡，因此可用替代齒輪副獲得十分近似的承載能力計(jì)算結(jié)果。齒寬橫截面上的載荷分布可用齒寬系數(shù)（例如DIN/ISO標(biāo)準(zhǔn)中的）表示和利用補(bǔ)充的負(fù)載曲線圖分析來確定。
5.2輪齒接觸分析
如圖在圓柱齒輪副中那樣，更精確的承載能力計(jì)算可采用三維輪齒接觸分析。同樣采用替代齒輪，而且齒側(cè)處接觸狀況被認(rèn)為非常理想。該齒側(cè)形狀通過疊加經(jīng)齒側(cè)修正的無負(fù)載接觸間隙而獲得。在這里，接觸線由替代齒輪所確定，它們和斜面體齒輪的接觸狀況稍有不同。圖13給出了以這方法獲得的載荷分布，并與已有的負(fù)載曲線圖作對(duì)比，兩者的相關(guān)性非常好。
輪齒接觸分析也將生成一個(gè)作為激振源的由輪齒嚙合產(chǎn)生的傳動(dòng)誤差。然而這僅能作為一個(gè)粗略的引導(dǎo)。在傳動(dòng)誤差方面，斜面體齒輪接觸計(jì)算的不精確性是一個(gè)比載荷分布更大的影響因素。
5.3采用有限元法的精確建模
斜面體齒輪的應(yīng)力也能利用有限元法計(jì)算。圖14是齒輪橫截面建模的實(shí)例。圖15給出了使用PERMAS軟件由計(jì)算機(jī)生產(chǎn)的主動(dòng)齒輪在嚙合位置的輪齒嚙合區(qū)模型和應(yīng)力分布計(jì)算值.可對(duì)多個(gè)嚙合位置進(jìn)行計(jì)算，并能求出齒輪旋裝產(chǎn)生的傳動(dòng)誤差。
5.4承載能力和噪聲試驗(yàn)
在交叉軸背靠背試驗(yàn)臺(tái)上對(duì)AWD變速器進(jìn)行試驗(yàn)以測(cè)量其承載能力，圖16。試驗(yàn)齒輪采用不同的修正，以確定它們對(duì)承載能力的影響。承載能力的試驗(yàn)與有限元計(jì)算結(jié)果相當(dāng)吻合。值得注意的是，由于大端硬度提高使得載荷曲線圖朝大端由一個(gè)額外的移動(dòng)。這種移動(dòng)在替代的圓柱齒輪副計(jì)算中不能被辨別。在進(jìn)行承載能力試驗(yàn)的同時(shí)，傳動(dòng)誤差和旋轉(zhuǎn)加速度的測(cè)量在通用噪聲試驗(yàn)臺(tái)上進(jìn)行，圖17。除了載荷影響外，這些試驗(yàn)還要測(cè)量了附加軸線傾斜所引起的噪聲激勵(lì)，關(guān)于軸線附加傾斜，試驗(yàn)中未發(fā)現(xiàn)有明顯的影響。
6仿真制造
借助于仿真制造，可獲得機(jī)床設(shè)置及連續(xù)范成磨削和產(chǎn)生齒廓扭曲的運(yùn)動(dòng)。齒廓瘦迫扭曲現(xiàn)象可在變速器設(shè)計(jì)階段就被認(rèn)識(shí)到并與承載能力及噪聲一并進(jìn)行分析。斜面體齒輪制造仿真軟件由ZF公司開發(fā)，詳見「9」。
6.1適用于斜面體齒輪的制造方法
斜面體齒輪僅可用范成法加工，因?yàn)辇X寬形狀沿齒寬方向有明顯的變化。盡管是錐角非常小的斜面體齒輪，必須承認(rèn)在修整處理中任然會(huì)出現(xiàn)齒寬角度偏差。滾刀最方便用于預(yù)切削。理論上也可采用刨削，但是，所需的運(yùn)動(dòng)在現(xiàn)有機(jī)床上很難實(shí)現(xiàn)。內(nèi)齒圓錐齒輪僅能用類似小齒輪的刀具精確制造，如果刀具軸線和和工具軸線平行并且錐角是通過改變中心距生成的。如果內(nèi)齒輪利用軸線傾斜的小齒輪刀具如同加工差速器錐齒輪那樣來制造的話，將導(dǎo)致齒溝凸起和修正運(yùn)動(dòng)的齒廓扭曲。對(duì)于小錐角而言這些偏差足夠小，可以被忽略。對(duì)于終加工，范成法螺旋磨削是一個(gè)最佳選擇。如果工件或機(jī)床夾具能被另外傾斜，可可以采用部分范成法。如果齒輪錐角處于機(jī)床控制范圍內(nèi)，拓?fù)淠ハ鞴に囈彩强赡艿模ɡ?軸機(jī)床），但是會(huì)消耗巨大的努力。原則上，珩磨等方法也能被用于加工，但是，在斜面體齒輪應(yīng)用這些方法任需大量的開發(fā)工作。雙齒側(cè)范成法磨削工藝并利用中心距弧形減少方法可實(shí)現(xiàn)齒溝凸起的目標(biāo)。該方法所得到的齒廓扭曲與造成嚙合間隙的齒廓扭曲相反。因此該方法可在很大程度上補(bǔ)償齒廓扭曲并可承受比圓柱齒輪更大的載荷。
6.2工作表面形狀
以下關(guān)于工作描述被應(yīng)用在仿真中：
原始齒輪（留有磨削所需的余量）
理想齒輪（來自齒輪數(shù)據(jù)，無齒側(cè)修形）
完成的齒輪（具有制造偏差和齒側(cè)修形）
參考文獻(xiàn)
1.J. A. macbain, J. J. Conover, and A. D. brooker，“Full-vehicle simulation for series hybrid vehicles,” presented at SAE Tech. paper, Future Transportation technology Conf., Costa Mesa, CA,jun.2003, Paper2003-01-2301.
2.X.Heand I. Hodgson, “Hybrid electric vehicle simulation and evaluation for UT-HEV,”pemented at the SAE Tech. Paper Series,Future Transpotation Technology Cong., Costa Mesa, CA,Aug,2000,Paper 2000-01-3105.
3.K.E.Bailey and B. K. Powell, “ A hybira electric vehicle powertrain dynamic model,”in proc.Amer. Control Conf.,Jun.21-23,1995,3,pp.1667-1682
4.B. K. Powell, K. E. Bailey,and S. R. Cikanek,”Dynamic modeling and control of hybird 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 hybird electric vehicle design,”IEEE Trans, Veh.Technol., vol. 48, no. 6, pp. 1770-1778, Nov. 1999.
6.K. L. 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,” in Proc. 16th Annu Battery Conf. Appl. And Adv., Jan. 9-12.2001. pp. 23-39.
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,”PSLM-based modeling of automotive power systema: Conventional, electric, and hybrid electric vehicles,”IEEE Trans. Veh. Technol., vol. 53, no. 2, pp. 390-400, Mar. 2001.
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.
Application, Design, and Manufacturing of Conical Involute Gears for Power Transmissions
Dr. J. Borner, K. Humm, Dr. F. Joachim, Dr. H. Yakaria,
ZF Friedrichshafen AG , 88038Friedrichshafen, Germany:
[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 gecometry of such gears is generally known, but applications in power transmissions are more or less exceptional. ZF has implemented beveloid gear sets in varioue applications: 4WD gear units for passenger cars, marine transmissions ( mostly used in yachts ), gear boxes for robotics, and indusrtial drives. The module of these beveloids varies between 0.7 mm and 8 mm in size, and the crossed axes angle varies between 0°and 250°. 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
P ossible 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 benefits with regard to ease of production, design features, and overall input. Conical involute gaars can be used in transmissions with intersecting or overall input. Conical involute gears can be uese in transimissions 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 rarele 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. In 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 1…3°) 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 result s in the basic gear dimensions:
Helix angle, right/left
(1) Transverse pressure angle right/left
(2) Base circle diameter right/left
(3) The differing base circles for the left and right flanks lead to asymmertrical tooth profiles at helical gears, Fig. 3. Manufacturing with a rack- type cutter results in a tooth root cone with root cone angle q. The addendum angle is designed so that tip edge interferences with the mating gear are avoided and a maximally large tooth height results across the face width. Due to the geometric design limits r for undercut and tip formation, the possible face width decreases as the cone angle increase. Sufficiently well-proportioned gearing is possible up to a cone angle of approx.15°.
2.2 MCRO GEOMETRY
The pairing of two conical gears generally leads to a point-shaped tooth contact. Out-side this 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 possible with the step-by-step application of the gearing law /5/, Fig. 4. To that end, a point (p) with the radiusrpland normal vectornlis generated on the original lank. This generates the speed vector V with
(4) For the point created on the mating flank, the radial vecor rp:
(5) And the speed vector apply
(6) The angular velocities are generated form the gear ratio:
(7) The angle Y is iterated until the gearing law in the form
(8) Is fulfilled. The meshing point Pa found is then rotated through the angle
(9) Around the gear axis, and this results in the conjugate flank point
3 GEARING DESIGN
3.1 UNDERCUT AND FORMTION
The usable face with on he beveloid gearing is limited by tip formation on the heel and undercut on the toe as shown in Fig. 3. The greater the selected tooth height (in the 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 form 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 pinion , a smaller cone angle must be used on this pinion. Top 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 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 on 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 slide 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 diver should be smaller than the tip contact path. Fig. 6 shows the distribution of the sliding velocity 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 in contact due to the tilting of the axes. The gaping that results along the potential contact line can be approximately described by helix crowing 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 differentce between the gap values on the left and right flanks increase. 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 shouws 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 largr 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 perpendicular is identical to the total of the base radii and the difference in the base helix angle is equivaklent to the 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 helical beveloids. This profile twist grows significantly as the helix angle increases. Fig.9 shows how the profile twist on the example gear set fron 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 compensation of the profil twist, only a diagonally patterned contact strip is obtained in the field of action, as shown in Fig.10.
4.2 FLANK MODIFICATIONS
For 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 prototypes. The contact rations have improved greatly with these corrections as can be seen in Fig.12. For use in series production, the targer is always to manufacture such topographies on commonly used grinding machines. The target is always to manufacture such topographies on commonly used grinding machines. The options for this are described in section 6. In addition to the gaping compensation , tip relief is also benefical. Thid reduces the load at the start and at the end of meshing and can also provide lower noise excitation. However, tip relief manufactured at beveloid gears is not constant in amount and length across the face width. The problem primarily occurs on gearing with a largr root cone angle and a tip cone angle deviating from this angle .The tip relief at the toe is significantly larger than at the heel. This uneven tip relief must be accepted if relief of the start and end of meshing is required. The production of tip relief using another cone angle as the root cone angle is possible; however , this requires an additional grinding step onle for the tip relief . Independently of the generating grinding process, targeted flank topograpjy can be manufactured by coroning or honing ; the application of this method on beveloids , however, is still in the early stages of development.
5 LOAD APACITY AND NOISE EXCITATION
5.1 APPLICATION OF THE CALCULATION STANDARDA
The flank and root load capacity of beveeloid gearing can onle approximately be deter-mined using the calculation standards (ISO6336, DIN3990,AGMA C95) for cylindrical gearing. A substitute cylindrical gear pair has to be used ,which is defined by the gear parameners at the center of the face width. The profile of the beveloid tooth is asymmetrical; that can , however, be ignored on the substitute gears. The substitute center distance is obtained by adding up the operating pitch radii at the center of the face width. When viewed across the face width, individual parameters will change , which significantly influence the load capacity. Table 2 shows the main influences on the root and flank load capacities. The large notch effect due to the decrease in the tooth root fillet radius towards the heel is in opposition to the increase in the root thickness .In addition, there is a smaller tangential force on the larger operating pitch circle at the heel; at the same time, however, the addendum modification on the heel is smaller. The primary influences are nearly well-balanced so that the load capacity can be calculated sufficiently approximate with the substitute gear pair . The load distribution across the face width can be considered with the width factors (e. g. k. and k. in DIN/ISIO) and should be determined from additional load pattern analyses.
5.2 USE OF THE TOOTH CONTACT ANALYSIS
A more precise calculation of the load capacity is possible with a three-dimensional tooth contact analysis , as used at cylindrical gear pair can be used in this analysis and the contact conditions are considered from the supermposition of of the load-free contact ease-off with the flank corrections used on the gear. In this process, the contact lines are determined on the substitute cylindrical gear and they differ slightly from the contact at the beveloid gear. Fig. 13 shows the load distributions calculated in this manner as compared to the load patterns recorded , and a very goodcorrelation can be seen.
This tooth contact analysis also generates the tansmission error resulting from the tooth mesh as vibrational excitation. It can, however, only be used as a rough guide . The impreciseness in the contact behavior calculated has a stronger effect on the transmission error than it douse on the load distribution.
5.3EXACT MODELING USING THE FINTE-ELEMENT METHOD
The stress at the beveloid gears cab also be calculated using the finite-element method .Fig. 14 shows examples of the modeling of the transverse section on the gears. Fig. 15 shows the computer-generated model in the tooth mesh section and the stress distribution calculated with PERMAS/7/ on the driven gear in a mesh position. The multiple mesh positions and the transmission error can be determined from the rotation of the gears.
5.4 TESTS REGARDING LOAD CAPACITY AND NOISE
A back-to-back test bench with crossed axes, upon which gear pairs from AWD transmissions were tested, was used to determine the load capacity ,Fig.16. Different corrections were produced on the test geasrs in order to ascertain their influence on the load capacity. There was good correlation between the load capacity in the test and the FE results. Particularly stiffness in this area. This shift is not discernable in the calculation with the substitute cylindrical gear pair. Simultaneous to the load capacity tests, measurements of the transmission error and rotational acceleration were conducted in a universal noise test box, Fig . 17. In additional to the load influence, the influence of additional axis tilt on the noise excitation was also esamied in thse teasts .With regard to this axis tilt, no large amount of sensitivity in the tested gear sets was found .
6 MANUFACTURRIBF SIMULATION
With the assistance of the manufacturing simulation, machine settings and movemengts with continuous generation grinding as well as the produced profile twist can be obtained . Production-constrained profile twist can be considered as early as the design phase of a transmission and can be incorporated into the load capacity and noise analyses. Simulation software for the manufacturing of beveloids was specially developed at ZF, which is comparable to /9/.
6.1 PRODUCTION METHODS THAT CAB VW USED BEVELOIDS
Only generating methods can be used to produce the beveloid gearing, because the shape of the tooth profile changes significantly along the face width. Only very slightly conical bevelodis can be manufactured with the acknowledgment that there is profile angle deviation even with the shaping process. Hobs are the easiest to use for pre-cutting. G ear planning would theoretically be useable as well; however , the kinermatics required makes this not really feasible on sxisting machines. Internal conical gears can then only be precisely manufactured with pionion-type cutter axis is parallel to the tool axis and the cone is created by changing the center distance. If the internal gear is manufactured with a tilled pinion cutter without corrective movements. Theses deviations are small enough to be ignored for minor cone angles . For final processing , contiuous generation grinding with a grinding worm appears to be the best option. If the workpise or tool fixture can be additionally tilled ,then partial generation methods are also applicable . Processing in a topological grinding process is also possible (e. g. 5-axis machines ), but with great effort , when the cone angle of the gearing can be considered in the machine control. In principle , honing and coroning can also be used for the processing; however, the application of these methods in beveloids still needs extensive development. The targeted hollow crowing can be created in the generation grinding process in the dual-flank grinding process via a bowshaped reduction in the center distance . Thid method results in a profile twist , that is the reverse of the profile twist from the contact gaping. Thus , tisi method provides extensive compensation for the profile twist and a significantly more voluminous load pattern as is typical on cylindrical gears.
6.2 WORKPIECE GEOMETRY
The following workpiece descriptions are used in the simulation:
Initial gear ( with stock allowance for the grind processing)
Ideal gear (from the gear data , without flank corrections )
Finished gear ( with production-constrained deviations and flank corrections)
參考文獻(xiàn)
1.J. A. macbain, J. J. Conover, and A. D. brooker，“Full-vehicle simulation for series hybrid vehicles,” presented at SAE Tech. paper, Future Transportation technology Conf., Costa Mesa, CA,jun.2003, Paper2003-01-2301.
2.X.Heand I. Hodgson, “Hybrid electric vehicle simulation and evaluation for UT-HEV,”pemented at the SAE Tech. Paper Series,Future Transpotation Technology Cong., Costa Mesa, CA,Aug,2000,Paper 2000-01-3105.
3.K.E.Bailey and B. K. Powell, “ A hybira electric vehicle powertrain dynamic model,”in proc.Amer. Control Conf.,Jun.21-23,1995,3,pp.1667-1682
4.B. K. Powell, K. E. Bailey,and S. R. Cikanek,”Dynamic modeling and control of hybird 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 hybird electric vehicle design,”IEEE Trans, Veh.Technol., vol. 48, no. 6, pp. 1770-1778, Nov. 1999.
6.K. L. 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,” in Proc. 16th Annu Battery Conf. Appl. And Adv., Jan. 9-12.2001. pp. 23-39.
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,”PSLM-based modeling of automotive power systema: Conventional, electric, and
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