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圓錐漸開(kāi)線(xiàn)齒輪
[摘要]圓錐漸開(kāi)線(xiàn)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(lèi)齒輪的幾何形狀是已知的，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(chē)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. 7 mm-8 mm之間，
交叉?zhèn)鲃?dòng)角在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聲所必須進(jìn)行的齒側(cè)修形可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(xiàn)不平行的話(huà)，轉(zhuǎn)矩傳遞可采用多種設(shè)計(jì)，例如:傘齒輪或冠齒輪、萬(wàn)向節(jié)軸或圓錐漸開(kāi)線(xiàn)齒輪(斜面體齒輪)。圓錐漸開(kāi)線(xiàn)齒輪特別適用于小軸線(xiàn)角度(小于15°)，該齒輪的優(yōu)點(diǎn)是在制造、結(jié)構(gòu)特點(diǎn)和輸入多樣性等方而的簡(jiǎn)易。圓錐漸開(kāi)線(xiàn)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(xiàn)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)齒輪，整個(gè)可選齒輪副矩陣見(jiàn)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)量的直齒輪或斜齒輪。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(guò)去，未曾對(duì)斜面體齒輪的承載能力和噪聲進(jìn)行過(guò)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IS06336)、計(jì)算方法和強(qiáng)度值都是未知的。因此，必須開(kāi)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(guò)去的15年中，ZF公司已為錐齒輪開(kāi)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°一3°)[2]
4、商用車(chē)輛的輸送齒輪箱(垃圾傾倒車(chē))
5、AWD用自動(dòng)變速器[ 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(xiàn)傾斜[ 1]。結(jié)果形成了齒輪基圓尺寸。
螺旋角，左/右
tanβ=tanβ·cosδ (l)
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(lèi)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。齒輪修形設(shè)計(jì)的目的是減小這些間隙以形成平坦而均勻的接觸。通過(guò)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rp和法向矢量為n的P1點(diǎn)。這生成速度矢量V及對(duì)于在嚙合一側(cè)所生成的點(diǎn)，有半徑矢量rp:
(4)
(5)
和速度矢量
（6）
角速度根據(jù)齒輪速比確定：
（7）
角度γ被反復(fù)迭代直至滿(mǎn)足下代。
(8)
嚙合點(diǎn)Pa偏轉(zhuǎn)角度
（9）
繞齒輪軸轉(zhuǎn)動(dòng)，形成共軛點(diǎn)P。
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(jiàn)圖3。齒高愈高(為獲得較大的齒高變位量)，理論可用齒寬愈窄。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(kāi)線(xiàn)，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(xiàn)傾斜，通常僅有一點(diǎn)接觸。沿可能接觸線(xiàn)出現(xiàn)的間隙可大致解釋為螺旋凸起和齒側(cè)廓線(xiàn)角度的偏差所致。圓柱齒輪左右側(cè)間隙與軸線(xiàn)交叉無(wú)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給出了具有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顯示了具有相同10°交叉軸線(xiàn)和30°螺旋角齒輪在左右側(cè)間隙方而的差異。兩側(cè)平均間隙的數(shù)值在很大程度上與螺旋角無(wú)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(xiàn)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(xiàn)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(huà),間隙減小到零并出現(xiàn)線(xiàn)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(xiàn)進(jìn)一步擴(kuò)大直至變成圓柱交叉軸螺旋齒輪副的話(huà),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(jiàn)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(lèi)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?huà),在工作區(qū)域僅有一個(gè)對(duì)角線(xiàn)狀的接觸帶,見(jiàn)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(lèi)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(kāi)始和結(jié)束階段的負(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上和長(zhǎng)度上是不同的。問(wèn)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(kāi)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行的。但是,這樣需要專(zhuān)門(mén)用于齒頂卸載的專(zhuān)用磨削設(shè)備。與范成法磨削方法無(wú)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期開(kāi)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(lái)定義圓柱齒輪的參數(shù)。雖然斜面體齒輪齒廓是非對(duì)稱(chēng)的,但在替代齒輪中可不予考慮。替代齒輪的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。由于沿大端方向減小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K和K)表示和利用補(bǔ)充的負(fù)載曲線(xiàn)圖分析來(lái)確定。
5.2 輪齒接觸分析
如同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(guò)疊加經(jīng)齒側(cè)修正的無(wú)負(fù)載接觸間隙而獲得。在這里,接觸線(xiàn)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(xiàn)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是齒輪橫斷面建模的實(shí)例。圖15給出了使用PERMAS軟件由計(jì)算機(jī)生成的主動(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[7]。可對(duì)多個(gè)嚙
合位置進(jìn)行計(jì)算,并能求出齒輪旋轉(zhuǎn)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(xiàn)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)還測(cè)量了附加軸線(xiàn)傾斜所引起的噪聲激勵(lì),關(guān)于軸線(xiàn)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)。齒廓受迫扭曲現(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(kāi)發(fā),詳見(jiàn)[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廓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)在修整處理中仍然會(huì)出現(xiàn)齒廓角度偏差。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(lèi)似小齒輪的刀具精確制造,如果刀具軸線(xiàn)和工具軸線(xiàn)平行并且錐角是通過(guò)改變中心距生成的。如果內(nèi)齒輪利用軸線(xiàn)傾斜的小齒輪刀具如同加工差速器錐齒輪那樣來(lái)制造的話(huà),將導(dǎo)致齒溝凸起和無(wú)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,也可采用部分范成法。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?例如5軸機(jī)床),但是會(huì)耗費(fèi)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)用這些方法仍需大量的開(kāi)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 工件表面形狀
以下的關(guān)于工件描述被應(yīng)用在仿真中:
? 原始齒輪(留有磨削所需的余量)
?理想齒輪(來(lái)自齒輪數(shù)據(jù),無(wú)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 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.
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 0°and 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 benefits 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. 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 results in the basic gear dimensions:
Helix angle, right/left
tanβ=tanβ·cosδ (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 asymmetrical tooth profiles at helical gears, Fig. 3. Manufacturing with a rack-type cutter results in a tooth root cone with root cone angle δ. 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 possible with the step-by-step application of the gearing law /5/, Fig. 4. To that end , a point (P) with the radiusrP1and normal vectorn1is generated on the original flank. This generates the speed vector V with
(4)
For the point created on the mating flank, the radial vector rp:
(5)
and the speed vector apply
(6)
The angular velocities are generated from the gear ratio:
(7)
The angle γ 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 P.
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 tooth 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 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 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 perpendicular 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
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 ratios have improved greatly with these corrections as can be seen in Fig.12. For use in series production, 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 beneficial. This relief 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 large root cone angle and a tip cone angle deviating from this angle. The tip relief at the toe is significantly larger than that 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 only for the tip relief. Independently of the generating grinding process, targeted flank topography 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 CAPACITY AND NOISE EXCITATION
5. 1 APPLICATION OF THE CALCULATION STANDARDS
The flank and root load capacity of beveloid gearing can only 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 parameters 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 larger 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. Kand K in DIN/ISO) 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 pairs. The substitute cylindrical gear pair can be used in this analysis and the contact conditions are considered very well with flank topography. This topography is obtained from the superimposition 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 transmission 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 does on the load distribution.
5.3 EXACT MODELING USING THE FINITE-ELEMENT METHOD
The stress at the beveloid gears can 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 calculation was carried out for 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 gears in order to ascertain their influence on the load capacity. There was good correlation between the load capacity in the test and the FE (finite element) results. Particularly noteworthy is an additional shift of the load pattern towards the heel due to the increased 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 addition to the load influence, the influence of additional axis tilt on the noise excitation was also examined in these tests. With regard to this axis tilt, no large amount of sensitivity in the tested gear sets was found.
6 MANUFACTURING SIMULATION
With the assistance of the manufacturing simulation, machine settings and movements 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 CAN BE USED FOR 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 beveloids 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. Gear planning would theoretically be useable as well; however, the kinematics required makes this not really feasible on existing machines. Internal conical gears can then only be precisely manufactured with pinion-type cutters if the 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 tilted pinion cutter axis such as used for crown gears, this results in a hollow crowning and a profile twist without corrective movements. These deviations are small enough to be ignored for minor cone angles. For final processing, continuous generation grinding with a grinding worm appears to be the best option. If the workpiece or tool fixture can be additionally tilted， 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 crowning can be created in the generation grinding process in the dual-flank grinding process via a bowshaped reduction in the center distance. This method results in a profile twist, that is the reverse of the profile twist from the contact gaping. Thus, this 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).
References
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 ec