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外文翻譯
減少偏離齒輪傳動(dòng)裝載和卸載時(shí)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(xué)機(jī)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(guó)
喀他赫納工藝
綜合大學(xué)機(jī)械工程部博士，Murcia，30202，喀他赫納，西班牙
山葉電動(dòng)機(jī)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(biāo)準(zhǔn)；2005 年五月 6 日校訂了；2005 年五月 17 日被公認(rèn)；2006 年一月 25 日可在線應(yīng)用.
摘要
齒輪傳動(dòng)時(shí)產(chǎn)生震動(dòng)和噪音的主要原因是傳輸誤差。有關(guān)影響噪音傳輸誤差的兩個(gè)主要函數(shù)已被查明：（1）一個(gè)是線性的對(duì)應(yīng)誤差；（2）一個(gè)是初步設(shè)計(jì)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(guān)系，在一個(gè)周期內(nèi)形成了混合的循環(huán)嚙合：（1）如點(diǎn)對(duì)點(diǎn)接觸；（2）當(dāng)從表面以曲線形式移動(dòng)到起始點(diǎn)時(shí)就產(chǎn)生嚙合。使用初步設(shè)計(jì)傳輸誤差能夠減少因?yàn)榫€性對(duì)應(yīng)函數(shù)而引起的傳輸誤差，減少噪音和避免移動(dòng)接觸。引起傳輸誤差的負(fù)載函數(shù)已被研究。齒牙的損壞能夠使在裝載的齒輪傳動(dòng)中減少最大的傳輸誤差。用計(jì)算機(jī)處理的模擬齒輪嚙合，且齒輪傳動(dòng)裝載和卸貨技術(shù)已發(fā)展相當(dāng)水平。
關(guān)鍵字：齒輪傳動(dòng)；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(dòng)
2.2. 螺旋狀的斜齒輪
2.3. 圓柱形的蝸桿齒輪傳動(dòng)
3. 嚙合的類型和基本功能的傳輸誤差
4. 裝載齒輪驅(qū)動(dòng)的傳動(dòng)誤差
4.1. 初步的考慮
4.2. 裝載的齒輪傳動(dòng)果斷的運(yùn)行應(yīng)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(gè)傳輸誤差函數(shù)的有力對(duì)比
6.1. 應(yīng)用方式概念上的考慮
6.2. 線性函數(shù)的分段插補(bǔ)
7.結(jié)論
1. 緒論
模擬的齒輪傳動(dòng)嚙合執(zhí)行應(yīng)用齒牙接觸分析（TCA）和測(cè)試齒輪傳動(dòng)已被證實(shí)傳輸誤差的主要原因是齒輪箱的震動(dòng)，這樣的震動(dòng)引起齒輪傳動(dòng)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(duì)應(yīng)錯(cuò)誤的類型且齒輪齒牙表面為了進(jìn)一步的傳動(dòng)在進(jìn)行改善。（見(jiàn)第二節(jié)）
為減少噪音而依下列的計(jì)劃進(jìn)行：
（1） 牙齒接觸表面被局部化
（2） 提供一個(gè)傳輸誤差的函數(shù)。這種傳輸錯(cuò)誤是由未對(duì)準(zhǔn)的一次函數(shù)所引起的[7]。
（3） 對(duì)雙層表面之一進(jìn)行最高倍數(shù)的修正。[見(jiàn)第2節(jié)]這通常是避免表面摩擦。[見(jiàn)第5節(jié)]
已經(jīng)對(duì)裝載和卸載齒輪傳動(dòng)應(yīng)用TCA進(jìn)行了比較，它顯示裝載的齒輪傳動(dòng)的傳輸誤差較少。其發(fā)展的方式與數(shù)字進(jìn)行一起舉例。[見(jiàn)第5節(jié)]
2. 齒牙表面的修正
減少齒輪傳動(dòng)的噪音需要修正接觸的雙表面之一。要修正齒輪傳動(dòng)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(dòng)
螺旋狀的齒輪最高剖面可能相交而表面產(chǎn)生兩個(gè)齒條刀形成錯(cuò)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(gè)傳輸誤差的拋物線函數(shù)。雙倍完美的執(zhí)行突進(jìn)的圓盤而產(chǎn)生小齒輪（見(jiàn)REF的第15章資料。[7]）。
2.2 螺旋狀的斜齒輪
應(yīng)用提供兩個(gè)有誤差的刀尖Σp 和Σg 而有局部接觸會(huì)產(chǎn)生螺旋狀的斜齒輪：Σp和Σg二者是分別用來(lái)產(chǎn)生小齒輪和齒輪的[7]。倆個(gè)刀尖Σp和Σg再齒呀的表面產(chǎn)生一個(gè)共同線C。（當(dāng)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Σp和Σg刀尖只有接觸的通常單一點(diǎn)，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(dāng)中教授歡迎的齒輪之一，通常小齒輪都被改良為滾動(dòng)的[7]。
2.3 圓柱型蝸桿齒輪傳動(dòng)
通常蝸輪制造工藝是以下列的方式為基礎(chǔ)。蝸輪的生產(chǎn)和蝸桿齒輪傳動(dòng)一樣都是由一個(gè)滾刀運(yùn)行的。應(yīng)用的機(jī)床設(shè)置模擬蝸桿和蝸輪嚙合而形成齒輪傳動(dòng)。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(dòng)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(zhǎng)期在齒輪箱中研磨加工而使齒輪傳動(dòng)畸形；
（2） 齒輪傳動(dòng)在長(zhǎng)期的運(yùn)轉(zhuǎn)下產(chǎn)生負(fù)載，近而達(dá)到最大負(fù)載；
（3） 蝸輪在蝸輪箱中被刨且傳動(dòng)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(yīng)用接觸局限為基礎(chǔ)的：（a）一個(gè)特大號(hào)的滾齒刀，和（b）幾何學(xué)的修正。（見(jiàn)下面）。
有蝸輪傳動(dòng)幾何學(xué)的各種不同類型[7]，但是一個(gè)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(guān)蝸桿傳動(dòng)要考慮圓盤的一個(gè)螺紋的產(chǎn)生（在生產(chǎn)的方法中）。
時(shí)常，再蝸輪傳動(dòng)局限接觸中以達(dá)成應(yīng)用滾刀且是比較特大號(hào)的蝸輪傳動(dòng)。
3. 嚙合的類型和傳動(dòng)誤差的基本函數(shù)
它假定齒牙表面任何點(diǎn)相切是正當(dāng)?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(shí)在接觸的結(jié)果[7]。
面與面相切的TCA運(yùn)算法則是以下列的矢量為基礎(chǔ)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, θi)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(yùn)算法則是用Sf方程來(lái)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(yīng)用而發(fā)現(xiàn)兩種嚙合的類型，面與面和面與曲線。計(jì)算機(jī)處理的嚙合模擬是以一個(gè)反復(fù)的程序?yàn)榛A(chǔ)的非線性方程的數(shù)字解決方案[8]
應(yīng)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(gè)初步設(shè)計(jì)的拋物線函數(shù)[7]（圖1）。初步設(shè)計(jì)的拋物線函數(shù)功能的應(yīng)用是減少噪音的先決條件。
圖1例證：（a）齒輪驅(qū)動(dòng)的一個(gè)不成直線的傳動(dòng)函數(shù)1和沒(méi)有欠對(duì)準(zhǔn)的理想的線性函數(shù)2；（b）周期函數(shù)拋物線形成的傳動(dòng)誤差Δ2(1)。
應(yīng)用最高的允許向前分配傳動(dòng)的誤差函數(shù)的是一個(gè)拋物線，而且允許分配同樣最大誤差值的6-8 ″。初步設(shè)計(jì)預(yù)期大小的傳動(dòng)誤差拋物線函數(shù)和投入大量生產(chǎn)的工具是有關(guān)聯(lián)的。圖2表示在何處由于欠對(duì)準(zhǔn)的誤差的大小，傳動(dòng)誤差函數(shù)形成兩個(gè)支流：面對(duì)面接觸和面與曲線接觸。
圖2一個(gè)螺旋狀齒輪的最大TCA誤差結(jié)果Δγ?=?10′:（a）傳動(dòng)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(dòng)的傳動(dòng)誤差
這一部分內(nèi)容覆蓋了一般用途FEM電腦程序應(yīng)用裝載齒輪驅(qū)動(dòng)的傳動(dòng)誤差果斷程序[3]。TCA決定直接應(yīng)用卸載齒輪驅(qū)動(dòng)的傳動(dòng)誤差。描述比較裝載和卸載時(shí)齒輪驅(qū)動(dòng)的傳動(dòng)誤差在第5節(jié)。
4.1 初步的考慮
（1）由于載入齒輪驅(qū)動(dòng)的結(jié)果，最大的傳動(dòng)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(jī)械要素模型的自動(dòng)生產(chǎn)之前的時(shí)候減少模型的準(zhǔn)備[對(duì)于應(yīng)用結(jié)構(gòu)組的每個(gè)結(jié)構(gòu)1]。
（3）圖3舉例說(shuō)明在負(fù)載之下被調(diào)查的一個(gè)結(jié)構(gòu)。TAC允許確定齒面Σ 1 和Σ 2 的接觸的點(diǎn) M,在負(fù)載被應(yīng)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(dòng)機(jī)構(gòu)的齒面的柔性變形應(yīng)用扭距到傳動(dòng)機(jī)構(gòu)的結(jié)果。圖3(b)的例證和（c）以接觸表面的不連續(xù)介紹為基礎(chǔ)的。
圖3說(shuō)明了：（a）一個(gè)單一接觸結(jié)構(gòu)（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(jié)構(gòu)組。TCA決定了每個(gè)結(jié)構(gòu)（將應(yīng)用于柔性變形之前）的位置。
圖4說(shuō)明了裝載齒輪驅(qū)動(dòng)嚙合組的模擬模型。
4.2 裝載的齒輪驅(qū)動(dòng)果斷的運(yùn)行應(yīng)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(dòng)。下列各項(xiàng)描述的是必須的階段：
（1） 因?yàn)楣ぷ鳈C(jī)的設(shè)定應(yīng)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(guān)角位置對(duì)NF結(jié)構(gòu)（a）（Nf=8-16）和（b）的觀察關(guān)系。
(5)
（3）一個(gè)預(yù)處理程序應(yīng)用于生產(chǎn)NF結(jié)構(gòu)的模型：（a）小齒輪完全被強(qiáng)制放置，且(b)傳動(dòng)機(jī)構(gòu)有開(kāi)關(guān)而使形成一個(gè)旋轉(zhuǎn)的表面。且規(guī)定扭距被應(yīng)用于這個(gè)表面。（圖5）
圖5不成型結(jié)構(gòu)和彈性變形的度量
(4)從個(gè)方面獲得一個(gè)裝載齒輪驅(qū)動(dòng)的傳動(dòng)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(shè)計(jì)一個(gè)螺旋齒輪傳動(dòng)的設(shè)計(jì)參數(shù)?？紤]下列嚙合狀態(tài)和傳動(dòng)接觸：
（1） 對(duì)于生產(chǎn)傳動(dòng)機(jī)構(gòu)和小齒輪齒條，它們分別地有如橫斷面的一個(gè)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(dòng)的欠對(duì)準(zhǔn)是由軸角Δγ≠ 0的誤差引起的。
（3） 給由Δγ≠ 0所引起的傳動(dòng)誤差提供了一個(gè)初步設(shè)計(jì)的拋物線函數(shù)。
（4） TCA（齒接觸分析）決心應(yīng)用由Δγ引起的卸貨和裝載齒輪驅(qū)動(dòng)的傳動(dòng)誤差。這種調(diào)查能夠影響傳動(dòng)誤差大小方面的負(fù)載。
（5） 電腦程式的應(yīng)用能分析有限的機(jī)械要素而決定裝載的齒輪驅(qū)動(dòng)的應(yīng)力。
（6） 調(diào)查軸向接觸的成型。
表1
設(shè)計(jì)參數(shù)
小齒輪的齒牙數(shù)目，N1
21
傳動(dòng)機(jī)構(gòu)的齒牙數(shù)目， N22
77
常態(tài)組件，mn
5.08?mm
正壓力角, α nn
25°
小齒輪螺旋線的方向
左手方
螺旋角, β
30°
齒面寬, b
70?mm
小齒輪齒條刀拋物線系數(shù), aca
0.002?mm?1
圓柱蝸桿的定位半徑, rwa
98?mm
滾動(dòng)小齒輪的修正系數(shù), amrb
0.00008?rad/mm2
小齒輪的應(yīng)用扭距c
250?N?m
用下列的一個(gè)例子來(lái)描述：
例1：考慮一個(gè)排列的齒輪驅(qū)動(dòng)（Δγ =0）卸下齒輪驅(qū)動(dòng)。拋物線功能提供一個(gè)最大值的傳動(dòng)誤差Δ 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(gè)欠對(duì)準(zhǔn)卸貨齒輪傳動(dòng)的計(jì)算結(jié)果：（a）傳動(dòng)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(gè)傳輸誤差函數(shù)的有力對(duì)比
6.1 應(yīng)用方式概念上的考慮
噪音信號(hào)的源動(dòng)力是以假定為基礎(chǔ)的，聲波發(fā)生的擺動(dòng)的速度與傳動(dòng)機(jī)構(gòu)的速度瞬時(shí)值成比例的變動(dòng)。這一假定（即使大體上不是很正確）是很好的第一個(gè)猜測(cè)，因?yàn)樗苊饬她X輪驅(qū)動(dòng)的一個(gè)復(fù)雜動(dòng)模型的應(yīng)用。
我們提議并強(qiáng)調(diào)應(yīng)用下列的狀態(tài)方式：
（a）目標(biāo)信號(hào)的動(dòng)力是不同的，但并不是肯定的絕對(duì)值信號(hào)。
（b）不同的信號(hào)動(dòng)力大體上引出一個(gè)不同結(jié)果為兩個(gè)不同的光滑傳動(dòng)誤差函數(shù)。
提議應(yīng)用的傳動(dòng)誤差函數(shù)引起的功能信號(hào)是以基部平均數(shù)角尺比較為基礎(chǔ)的[9]。定義如此的比較信號(hào)模擬強(qiáng)度
(7)
這里描述了傳動(dòng)機(jī)構(gòu)的角速度偏差的平均值，而且ω rms描述了rms需要的值ω 2(1)。傳動(dòng)誤差回收功率定義為2= m 211+ Δ 2(1), m 21 是齒數(shù)比。
區(qū)別考慮計(jì)時(shí)，我們獲得傳動(dòng)機(jī)構(gòu)的角速度
(8)
其中假定為常數(shù)。在第二個(gè)術(shù)語(yǔ)的右邊，（8）表現(xiàn)了對(duì)于速度的變動(dòng)
(9)
上面的定義假設(shè)傳輸錯(cuò)誤函數(shù)是連續(xù)可微的。在用有限元方法模擬負(fù)載齒輪啟動(dòng)器計(jì)算的情況下，這個(gè)函數(shù)是用有限個(gè)給定的點(diǎn)（（φ1）i，（△φ2）i（i=1,2……）來(lái)定義的。為了Eq的使用，各點(diǎn)的給定值必須用連續(xù)函數(shù)進(jìn)行插值計(jì)算。
6.2. 分段函數(shù)的插補(bǔ)
在這種情況下（圖7），用一條直線將兩個(gè)連續(xù)的數(shù)據(jù)點(diǎn)連接起來(lái)。在i和i-1點(diǎn)之間的速度是不變的，并且由下式確定：
(10)
圖7插補(bǔ)函數(shù)傳輸誤差分段的應(yīng)用于線性函數(shù)
數(shù)據(jù)點(diǎn)的選擇如下：（i）增量(1)i???(1)i?1在每個(gè)區(qū)間i 內(nèi)被認(rèn)為是不變的?；谶@種假設(shè)，兩個(gè)功率量的比值式如下所示：
(11)
7. 結(jié)論
通過(guò)先前的討論，計(jì)算和數(shù)字的例子能夠得到下列的結(jié)論：
（1）齒輪驅(qū)動(dòng)（如果沒(méi)有提供充足的表面修正）的對(duì)準(zhǔn)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(guò)初步設(shè)計(jì)的拋物線函數(shù)（PPF）來(lái)避免。
（2）調(diào)查發(fā)現(xiàn)傳動(dòng)誤差拋物線函數(shù)的應(yīng)用可減少齒輪驅(qū)動(dòng)的噪音和震動(dòng)。應(yīng)用PPF最少要修正生產(chǎn)齒輪驅(qū)動(dòng)的一個(gè)構(gòu)件，通常為小齒輪。（或者是蝸桿驅(qū)動(dòng)的蝸桿）
（3）負(fù)荷齒輪啟動(dòng)器的傳輸錯(cuò)誤的確定需要運(yùn)用一個(gè)一般用途的有限元電腦程序。負(fù)荷齒輪啟動(dòng)器配有彈性可變的輪齒，這樣接觸率增加，由于啟動(dòng)器的未對(duì)準(zhǔn)而產(chǎn)生的傳輸錯(cuò)誤將減少。由于使用了作者設(shè)計(jì)的有限元模塊的自動(dòng)產(chǎn)生方法使得模塊的準(zhǔn)備時(shí)間大大的縮短了。這種方法是專門為確定負(fù)荷齒輪傳輸錯(cuò)誤而設(shè)計(jì)的。
致謝
作者對(duì)格林森基金會(huì)和日本雅馬哈公司在財(cái)政上的支持表示深切地感謝。
參考文獻(xiàn)
[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 User’s Manual, 1800 Main Street, Pawtucket, RI 20860-4847, 1998.
[4] Klingelnberg und S?hne, Ettlingen, Kimos: Zahnkontakt-Analyse für Kegelr?der, 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. Moré, 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
外文原文
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 Σp and Σg used for generation of the pinion and the gear, respectively [7]. Two head-cutters Σp and Σg 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 Σp and Σg 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 Klingelnberg’s 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, θi) 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 Δ2(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 Δγ?=?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 Σ1 and Σ2, before the load will 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 gear’s 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 p
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