后支架零件沖壓模具設(shè)計(jì)
后支架零件沖壓模具設(shè)計(jì),支架,零件,沖壓,模具設(shè)計(jì)
畢業(yè)設(shè)計(jì)（外文翻譯）
題 目： 輪輻柔性變形效果和滾動(dòng)接觸的潛變力追蹤
輪輻的柔性變形結(jié)構(gòu)的效果和在滾動(dòng)接觸的輪/ 軌道的潛變力的追蹤
摘錄：
在這一篇論文中，對(duì)滾動(dòng)接觸機(jī)械裝置上的滾動(dòng)接觸體結(jié)構(gòu)柔性變形的效果簡(jiǎn)短地分析。輪副和軌道對(duì)輪的潛變力的結(jié)構(gòu)變形的效果和軌條詳細(xì)地被分析研究。輪副的一般結(jié)構(gòu)柔性變形和軌道首先分別用有限元的機(jī)械要素方法和關(guān)系一起分析,從而獲得表達(dá)滾動(dòng)方向和輪副的橫方向的結(jié)構(gòu)柔性變形和對(duì)應(yīng)的負(fù)載。按照它們之間的關(guān)系,我們計(jì)算輪和軌條的在一點(diǎn)相接接觸的影響力系數(shù)。影響力系數(shù)代表發(fā)生在輪/軌道接觸的一個(gè)小的矩形面積上的單位面積的牽引力引起的結(jié)構(gòu)柔性變形。他們習(xí)慣校訂一些與Kalker的無(wú)赫茲的形狀滾動(dòng)接觸的三維空間的有柔性體的理論 Bossinesq 和 Cerruti 的公式一起獲得的影響力系數(shù)。在潛變力的分析中, 利用了修正的 Kalker 的理論。從輪副和軌道的結(jié)構(gòu)柔性變形中獲得的數(shù)字結(jié)果表明潛變力發(fā)揮的很大影響力。
2002 Elsevier 科學(xué)出版社版權(quán)所有。
關(guān)鍵字: 輪/軌條; 滾動(dòng)接觸;潛變力;柔性變形結(jié)構(gòu)
1.介紹
由于火車輪副和軌道之間的很大相對(duì)運(yùn)動(dòng)作用力引起輪副和軌道的結(jié)構(gòu)較大的柔性變形。大的結(jié)構(gòu)變形極大影輪和軌條響滾動(dòng)接觸的性能,如潛變力，波形 [1 – 3] ，黏著，滾動(dòng)接觸疲勞, 噪音 [4,5] 和脫軌[6]等等. 到現(xiàn)在為止在輪/ 軌道的潛變力的分析中廣泛應(yīng)用的滾動(dòng)接觸理論是以柔性一半的空間假定為基礎(chǔ)的 [7 – 12]. 換句話說(shuō)，輪/ 軌道的一個(gè)接觸的柔性變形和牽引之間的關(guān)系可以用Bossinesq 和 Cerruti 的理論公式表達(dá)。實(shí)際, 當(dāng)輪副在軌道上持續(xù)運(yùn)動(dòng),接觸的柔性變形是比那些以滾動(dòng)接觸的現(xiàn)在理論公式計(jì)算的更大。因?yàn)檩喐? 軌道的撓性是比柔性一半的空間更加大 。由對(duì)應(yīng)的負(fù)荷所引起的輪副/ 軌道柔性變形結(jié)構(gòu)在圖中被顯示。如 1 和 2. 在圖中輪副彎曲變形被顯示出來(lái)。在圖 1a 中被顯示的輪副彎曲變形主要由車輛和輪副/軌條的垂直動(dòng)載荷所引起。在圖 1 b 中描述的輪副扭轉(zhuǎn)的變形是由于輪和軌道之間的縱潛變力的作用生產(chǎn)的。在圖 1 c 中顯示的輪副斜角彎曲變形和在圖 2 中顯示的軌道翻折變形主要地由交通工具和輪副/軌道的橫動(dòng)態(tài)負(fù)荷所引起。在輪副 (圖 1 d) 的軸周圍的和旋轉(zhuǎn)裝置相同方向的扭轉(zhuǎn)變形,火車可以使用的,主要在電動(dòng)機(jī)的輪/ 軌條和驅(qū)動(dòng)扭矩的接觸補(bǔ)綴上的牽引所引起。到目前為止很少的出版物討論滾動(dòng)接觸的輪副和軌道之間的爬動(dòng)和潛變力的效果。
事實(shí)上，上面提到輪副/ 軌道的柔性變形結(jié)構(gòu)是在輪/軌道的常態(tài)和切線的接觸剛性以下運(yùn)動(dòng)。輪/ 軌道的正常的接觸點(diǎn)的剛性通常低于軌道的下沉位置。
低于正常接觸點(diǎn)的剛性很少的影響接觸面積上的正常壓力。那低于切線的接觸剛性很大影響接觸面積的黏結(jié)/ 滑移面積狀態(tài)和牽引力。如果滾動(dòng)接觸的柔性變形結(jié)構(gòu)的影響被對(duì)于輪/軌道的分析考慮進(jìn)去，一對(duì)接觸面積的全體微?；婆c用現(xiàn)在滾動(dòng)接觸理論計(jì)算的結(jié)果不同。所有的連絡(luò)顆粒和摩擦功的總的滑移比那在分析輪/軌道淺動(dòng)力的時(shí)候，被忽略的柔性變形結(jié)構(gòu)更小。同樣一個(gè)接觸面積的根/ 轉(zhuǎn)差面積的比率比沒(méi)有考慮的柔性變形結(jié)構(gòu)的效果更大。在這一篇論文中，在滾動(dòng)接觸性能上的滾動(dòng)接觸的車體柔性變形機(jī)構(gòu)的裝置被簡(jiǎn)短地分析，而且和Kalker''''s 無(wú)赫茲的形狀滾動(dòng)接觸的三度空間的有柔性車體的理論模型用來(lái)分析在輪副和軌道之間的潛變力。在數(shù)值分析中挑選的輪副和軌條分別地,是貨車輪副的錐形輪廓，中國(guó) "兆位元組" 和鋼軌條的質(zhì)量是60 公斤/m 。有限元分析方法用來(lái)決定他們的柔性變形結(jié)構(gòu)。依照柔性變形結(jié)構(gòu)的關(guān)系和對(duì)應(yīng)的由于 FEM 獲得負(fù)荷, 表示輪副的柔性變位的影響系數(shù)是由輪/ 軌條的接觸單位面積密度有所反應(yīng)的牽引生產(chǎn)的軌條所決定。這些影響系數(shù)用來(lái)代替一些與 Kalker''''s 的理論 Bossinesq 和 Cerruti 的公式一起計(jì)算的影響系數(shù)。在圖 1a 中被顯示的輪副彎曲變形的效果和在輪副軌道的柔性變形結(jié)構(gòu)之中的橫斷的影響力在研究中被疏忽。獲得的數(shù)字結(jié)果表明在輪副/軌道柔性變形結(jié)構(gòu)的潛變力效果考慮和疏忽的條件之間的顯著差別。
2. 減少連絡(luò)剛性機(jī)構(gòu)增加接觸面積的根粘滯/滑動(dòng)比
為了要使輪副/ 軌道關(guān)于滾動(dòng)接觸的輪/ 軌的的柔性變形結(jié)構(gòu)的效果較好的理解, 我們必需簡(jiǎn)短地解釋減少的接觸剛性的機(jī)構(gòu)增加在沒(méi)有飽和的潛變力的狀態(tài)下面的接觸面積的粘滯/ 滑移面積的比。通常在一個(gè)接觸面積的一對(duì)接觸顆粒之間的總的滑移含有剛性的滑移,局部一個(gè)接觸面積和柔性變形結(jié)構(gòu)的柔性變形。圖 3 a一描述一對(duì)滾動(dòng)接觸車體①和沒(méi)有柔性變形②接觸顆粒， A1 和 A2 的狀態(tài) 。在圖 3 a中的線A1A 1 和 A2A 2, 為了要作描述的讓大家接受而被作記號(hào)。在車體的形變發(fā)生之后,線的位和形變,A1A 1 和 A2A2,在圖 3 b 中被顯示。位移差別 , w1, 在圖 3 b 的二個(gè)劃線之間由車體的剛性運(yùn)動(dòng)①和②所引起(滾動(dòng)或變化). 局部點(diǎn) A1 和 A2 的柔性變形,被 u11 和 u21 指示,與基于有柔性- 半份空間的假設(shè)滾動(dòng)接觸的一些現(xiàn)代的理論一起決定,他們有差別在于點(diǎn) A1 和點(diǎn)A2之間的有柔性位移 u1= u11- u21。如果車體的結(jié)構(gòu)柔性變形的效果和被忽視的A1 和 A2點(diǎn)之間的總轉(zhuǎn)差 , 能用公式: S1 = w1 ? u1 = w1 ? (u11 ? u21)
表示。柔性變形結(jié)構(gòu)車體 1 和 2 主要地由牽引力所引起,p 和 p 代表接觸插線和車體的其他邊界條件1和 2,他們做線,A1A 1 和 A2A 2 產(chǎn)生與接觸面積的局部的坐標(biāo) (ox1x3,圖 3 a) 無(wú)關(guān)的剛性運(yùn)動(dòng)。u10 和 u20 用來(lái)表達(dá)點(diǎn) A1 和點(diǎn)A2的位移,各自歸于結(jié)構(gòu)柔性變形。在任何的荷載階段他們?yōu)橐?guī)定的邊界條件和車體 1 和 2 的幾何學(xué)可能被當(dāng)做有不防礙局部的坐標(biāo)常數(shù)。在點(diǎn) A1 和點(diǎn) A2 之間的位移差別取決于 u10 和 u20, 應(yīng)該是 u0= u10-u20。如此在考慮車體 1 和 2的柔性變形結(jié)構(gòu)的條件之下,在點(diǎn)之間的總滑移 , A1 和 A2,同樣地用公式:S*1 = w1 - u1 - u0表示。明顯的 S1 和 S?1 是不同的。在一對(duì)接觸顆粒之間的牽引 ( 或潛變力)非常仰賴 S1( 或 S?1) 。當(dāng) |S1|>0(或 |S?1|>0)那對(duì)接觸顆粒是在滑移中和牽引力進(jìn)入飽和。在進(jìn)入飽和的情形中, 依照庫(kù)倫摩擦定律的如果一樣的磨擦力系數(shù)而且正常的壓力被假定的二個(gè)條件，牽引是相同的。如此對(duì) u1 的牽引影響在二個(gè)條件之下也是相同的。如果 |S1|=|S?1|>0,|w1| 在 (2) 必須是比在(1)更大。即沒(méi)有 u0 的影響的那對(duì)接觸顆粒比有 u0 的影響的滑移更快。相應(yīng)地沒(méi)有 u0 的影響整個(gè)的接觸面積進(jìn)入滑移情況快于有 u0 的影響。因此，在接觸面積上的粘滯/ 滑移面積的比率和在上面被討論的二個(gè)類型的總牽引是不同的,他們只是被圖 4a 和 b一起被簡(jiǎn)單描述。圖 4a表明粘滯/ 滑移面積的情況。圖 4a 的號(hào)訊 1 表明不考慮 u0 和 2的效果而指示外殼 即用 u0 的效果指示。圖 4 b表示在接觸面積上總的接觸牽引力F1和車體的滑動(dòng)關(guān)系的一種規(guī)律。在圖 4 b 中的號(hào)訊 1 和 2 和圖 4 中的意義相同。從圖 4 b 中已知 , 在一點(diǎn)相接牽引力 F1 在 w1=w 時(shí)到達(dá)它的最大值 F1max 不考慮 u0 和 F1 接觸的效果在 w1=w 它的最大 F1max 僅由于 u0 的效果來(lái)看w1< w 1. u0 主要仰賴于車體的柔性變形結(jié)構(gòu)和在接觸面積上的牽引力。大的柔性變形結(jié)構(gòu)引起滾動(dòng)接觸的在二個(gè)車體之間的大 u0 和小的接觸剛性。那是為什么增加一個(gè)接觸面積的根/ 滑移面積的比率和減少?zèng)]有全滑移的在接觸面積的條件下面的全體的牽引力而減少的接觸剛性。
3. 輪副/軌條的結(jié)構(gòu)形變的計(jì)算
為了要計(jì)算在圖 1 b – d, 和圖 2 中被描述的柔性變形結(jié)構(gòu)，輪副的離散化而且軌條被虛構(gòu)。他們的 FEM 網(wǎng)目的方案
在圖 5,7 和 9中被顯示。假定輪副和軌條有相同的物理性質(zhì)。剪[切]模量:G=82,000個(gè)牛頓/mm2,泊松比: μ =0.28. 圖 5 用來(lái)決定輪副的扭轉(zhuǎn)形變。因?yàn)?它是關(guān)于輪副 (圖 1 b) 的中心對(duì)稱,一個(gè)一半的輪副被選擇來(lái)分析。輪副的切斷橫斷面被安裝,如圖 5 所示一。負(fù)荷被應(yīng)用到圓周方向輪副的胎面,在輪的不同的母圓上。荷載的分布分別位于從輪的內(nèi)部邊測(cè)量31.6,40.8 和 60.0毫米, 圖 6 表示縱方向扭轉(zhuǎn)的形變相對(duì)于負(fù)荷位置。他們都是線性載荷，載荷的不同點(diǎn)都非常接近。在圖 5 中被顯示Y軸方向形變的的負(fù)荷被忽略。
i=1,2 分別代表左邊和右邊的邊輪/軌條。叁數(shù) (3) 在現(xiàn)在的論文命名中被詳細(xì)地義。如果它被傾斜，當(dāng)輪副向軌道和ψ >0 的左邊變檔的時(shí)候 ,在順時(shí)針?lè)较?在輪副的軸線和左邊的軌道的橫向方向之間，我們定義那 y>0。叁數(shù)仰賴y 和ψ，輪和軌條的輪廓。但是如果輪和軌條的輪廓被指定他們主要地仰賴 y[16]. 詳細(xì)的討論用數(shù)字的方法被屈服[16,17] 和輪/軌條的接觸幾何學(xué)的結(jié)果。
當(dāng)一個(gè)輪副移動(dòng)到一個(gè)正切追蹤剛性蠕動(dòng)輪副和軌條的時(shí)候當(dāng)做 [17]:
i=1,2時(shí)它有如同寫在底下在(3)的 i 一樣的意義。在 (4)的不明確的叁數(shù)能在命名法中看到。很明顯蠕動(dòng)不僅與接觸幾何學(xué)的叁數(shù)有關(guān), 而且也與輪副運(yùn)動(dòng)的狀態(tài)有關(guān)。因?yàn)榻佑|幾何學(xué)的叁數(shù)變化主要依靠一些他們的導(dǎo)出于計(jì)時(shí)輪/ 軌條的規(guī)定輪廓y的變化有關(guān)被記做：
把(5)放進(jìn)(4)之內(nèi)我們獲得:
在輪/軌條的接觸幾何學(xué)和滑移的計(jì)算，大范圍的偏角和輪副的橫向位移被選用以便輪輻的滑移和接觸角含盡可能完全地在磁場(chǎng)中被產(chǎn)生的情況被獲得。因此我們選擇 毫米 和 與中央的用不同的方法和ri, φ和 ?i 和 y l0=746.5mm ， r0=420mm比較的數(shù)字結(jié)果一起計(jì)算。使用選擇的y ，ψ，˙ y/ v 和 r0 ˙ψ/ v 的范圍在我們獲得上面ξ i 1個(gè)范圍從 ?0.0034 到 0.0034,ξ i 2個(gè)范圍從 ?0.03 到 0.03, ξ i 3 排列從 ?0.00013 到 0.00013(毫米?1), 和接觸角δ i 是從到 2.88 到 55.83度。由于論文的長(zhǎng)度限制滑動(dòng)和接觸幾何學(xué)的詳細(xì)數(shù)字的結(jié)果不被在這篇論文中顯示。
4．結(jié)論
(1). 在滾動(dòng)接觸性能上的滾動(dòng)接觸車體的柔性變形結(jié)構(gòu)的效果機(jī)構(gòu)被簡(jiǎn)短地分析。一般了解連絡(luò)車體的接觸剛性減少則接觸面積在不全滑移情形中的粘滯/ 滑移面積增加。
(2). Kalker''''s 的和無(wú)赫茲的形狀滾動(dòng)接觸的三度空間的彈性體的理論模型被用來(lái)分析在輪副和軌道之間的潛動(dòng)力。在分析中，有限元法被用決定作用于每個(gè)矩形元件單位牽引生產(chǎn)的輪副/軌道有柔性位移表達(dá)的影響系數(shù),用來(lái)代替一些與 Kalker''''s 的理論 Bossinesq 和 Cerruti 的公式一起計(jì)算的影響系數(shù)。被獲得的數(shù)字結(jié)果表明在輪副/ 軌條結(jié)構(gòu)柔性變形的效果被考慮和忽略的兩種情況之下輪副/ 軌條類型的潛動(dòng)力的差別。
(3). 輪副和軌道的柔性變形結(jié)構(gòu)低于運(yùn)行輪副和軌道的接觸剛性, 而且在沒(méi)有飽和的潛動(dòng)力的條件之下顯著地減少在輪副和軌道之間的潛動(dòng)力。因此，這種情況有利于減少磨損和輪與軌條的滾動(dòng)接觸疲勞。
(4).在研究中，在圖 1 中顯示的輪副彎曲形變的因素被忽略,而橫斷的影響系數(shù) 不被修正。因此，獲得數(shù)字結(jié)果的精確度很低。除此之外, 當(dāng)輪副中心的橫向位移, y>10 mm,凸圓作用發(fā)生。在如此的情形中，接觸角非常大，而且橫的方向正常負(fù)載的元件也非常大。大的橫力引起軌道和輪副產(chǎn)生大的結(jié)構(gòu)形變,影響輪/ 軌條的接觸幾何學(xué)的叁數(shù)和剛性的滑動(dòng)。因此,剛性滑動(dòng),潛動(dòng)力, 接觸幾何學(xué)的叁數(shù)，柔性變形結(jié)構(gòu)和輪副的運(yùn)動(dòng)彼此有很大的影響。他們必需綜合地分析考慮。他們的數(shù)字結(jié)果能與一個(gè)其它可能的迭代法一起獲得
。或許共形的接觸或輪和軌條之間的點(diǎn)接觸在凸圓的作用期間發(fā)生。滾動(dòng)接觸的輪副和軌條的現(xiàn)象是非常復(fù)雜的, 而且可能與可能是包括結(jié)構(gòu)形變和包括輪副和軌道的所有邊界條件在不久的將來(lái)內(nèi)的效果 FEM 模型的滾動(dòng)接觸的一個(gè)新的理論被分析。
這一個(gè)工作被研究計(jì)劃的中國(guó)自然的科學(xué)基礎(chǔ)委員會(huì)支持了: 輪和軌條和滾動(dòng)接觸疲勞的接觸表面的波形。(59935100)國(guó)家牽引動(dòng)力實(shí)驗(yàn)室,西南交通大學(xué)。
它也被中國(guó)的教育部鍵老師大學(xué)也提供基金支持。
參考文獻(xiàn)
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[3] W.F. Hayes, H.G. Tucker, Wheelset–track resonance as a possible source of corrugation wear, Wear 144 (1991) 211–226.
[4] P.J. Remington, Wheel–rail noise. Part IV. Rolling noise, J. Sound Vibrat. 46 (1975) 419–436.
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[7] F.W. Carter, On the action of a locomotive driving wheel, Proc. R. Soc. Lond. A 112 (1926) 151–157.
[8] J.K. Vermeulen, K.L. Johnson, Contact of non-spherical bodies transmitting tangential forces, J. Appl. Mech. 31 (1964) 338–340.
[9] J.J. Kalker. On the rolling contact of two elastic bodies in the presence of dry friction, Ph.D. thesis, Delft University, The Netherlands, 1967, pp. 64–100.
[10] J.J. Kalker. Simplified Theory of Rolling Contact, Delft Progress Report, Delft University Press, The Netherlands, 1973, pp. 1–10.
[11] J.J. Kalker. Three-Dimensional Elastic Bodies in Rolling Contact, Kluwer Academic Publishers, The Netherlands, 1990.
[12] Z.Y. Shen, J.K. Hedrick, J.A. Elkins. A comparison of alternative creep-force models for rail vehicles dynamic analysis, in: Proceedings of the Eighth IAVSD Symposium, Cambridge, MA, 1984, pp. 591–605.
[13] S. Guo, C. Cai, W. Zhai. A study of lateral coupling dynamics of vehicle/track system, J. China Railway Soc. Suppl. (1994) 91–98 (in Chinese).
[14] W. Wei, A model of rail track receptance analysis, J. Dalian Railway Inst. 19 (4) (1998) 33–38 (in Chinese).
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[17] X Jin. Study on creep theory of wheel and rail system and its experiment, Ph.D. thesis, Southwest Jiaotong University, Chengdu, China, 1999, pp. 39–53 (in Chinese).
Effects of structure elastic deformations of wheelset and track on creep forces of wheel/rail in rolling contact
Abstract：
In this paper the mechanism of effects of structure elastic deformations of bodies in rolling contact on rolling contact performance is briefly analyzed. Effects of structure deformations of wheelset and track on the creep forces of wheel and rail are investigated in detail. General structure elastic deformations of wheelset and track are previously analyzed with finite element method, and the relations, which express the structure elastic deformations and the corresponding loads in the rolling direction and the lateral direction of wheelset, respectively, are obtained. Using the relations, we calculate the influence coefficients of tangent contact of wheel and rail. The influence coefficients stand for the occurring of the structure elastic deformations due to the traction of unit density on a small rectangular area in thecontact area of wheel/rail. They are used to revise some of the influence coefficients obtained with the formula of Bossinesq and Cerruti in Kalker’s theory of three-dimensional elastic bodies in rolling contact with non-Hertzian form. In the analysis of the creep forces, the modified theory of Kalker is employed. The numerical results obtained show a great influence exerted by structure elastic deformations of wheelset and track upon the creep forces.
? 2002 Elsevier Science B.V. All rights reserved.
Keywords: Wheel/rail; Rolling contact; Creep force; Structure elastic deformation
1. Introduction
During running of a train on track the fierce action between wheelset and rails causes large elastic deformations of structure of wheelset and track. The large structure deformations greatly affect performances of wheels and rails in rolling contact, such as creep forces, corrugation [1–3], adhesion, rolling contact fatigue, noise [4,5] and derailment [6]. So far rolling contact theories widely used in the analysis of creep forces of wheel/rail are based on an assumption of elastic half space [7–12]. In other words, the relations between the elastic deformations and the traction in a contact patch of wheel/rail can be expressed with the formula of Bossinesq and Cerruti in the theories. In practice, when a wheelset is moving on track, the elastic deformations in the contact patch are larger than those calculated with the present theories of rolling contact. It is because the flexibility of wheelset/rail is much larger than that of elastic half space. Structure elastic deformations (SED) of wheelset/rail caused by the corresponding loads are shown in Figs. 1 and 2. The bending deformation of wheelset shown in Fig. 1a is mainly caused by vertical dynamic loads of vehicle and wheelset/rail. The torsional deformation of wheelset described in Fig. 1b is produced due to the action of longitudinal creep forces between wheels and rails. The oblique bending deformation of wheelset shown in Fig. 1c and the turnover deformation of rail shown in Fig. 2 are mainly caused by lateral dynamic loads of vehicle and wheelset/rail. The torsional deformations with the same direction of rotation around the axle of wheelset (see Fig. 1d), available for locomotive, are mainly caused by traction on the contact patch of wheel/rail and driving torque of motor. Up to now very few published papers have discussions on the effects of the SED on creepages and creep forces between wheelset and track in rolling contact.
In fact, the SED of wheelset/rail mentioned above runs low the normal and tangential contact stiffness of wheel/rail. The normal contact stiffness of wheel/rail is mainly lowed by the subsidence of track. The normal contact stiffness lowed doesn’t affect the normal pressure on the contact area much. The lowed tangential contact stiffness affects the status of stick/slip areas and the traction in the contact area greatly. If the effects of the SED on the rolling contact are taken into account in analysis of rolling contact of wheel/rail, the total slip of a pair of contacting particles in a contact area is different from that calculated with the present rolling contact theories. The total slip of all the contacting particles and the friction work are smaller than those obtained under
condition that the SED is ignored in the analysis of creep forces of wheel/rail. Also the ratio of stick/slip areas in a contact area is larger than that without consideration of the effects of the SED.
In this paper the mechanism of effects of structure elastic deformations of bodies in rolling contact on rolling contact performance is briefly analyzed, and Kalker’s theoretical model of three-dimensional elastic bodies in rolling contact with non-Hertzian form is employed to analyze the creep forces between wheelset and track. In the numerical analysis the selected wheelset and rail are, respectively, a freight-car wheelset of conical profile, China “TB”, and steel rail of 60 kg/m. Finite element method is used to determine the SED of them. According to the relations of the SED and the corresponding loads obtained with FEM, the influence coefficients expressing elastic displacements of the wheelset and rail produced by unit density traction acting on the contact area of wheel/rail are determined. The influence coefficients are used to replace some of the influence coeffi- cients calculated with the formula of Bossinesq and Cerruti in Kalker’s theory. The effect of the bending deformation of wheelset shown in Fig. 1a and the crossed influences among the structure elastic deformations of wheelset and rail are neglected in the study. The numerical results obtained show marked differences between the creep forces of wheelset/rail under two kinds of the conditions that effects of the SED are taken into consideration and neglected.
2. Mechanism of reduced contact stiffness increasing the stick/slip ratio of contact area
In order to make better understanding of effects of the SED of wheelset/track on rolling contact of wheel/rail it is necessary that we briefly explain the mechanism of reduced contact stiffness increasing the ratio of stick/slip area in a contact area under the condition of unsaturated creep-force. Generally the total slip between a pair of contact particles in a contact area contains the rigid slip, the local elastic deformation in a contact area and the SED. Fig. 3a describes the status of a pair of the contact particles, A1 and A2, of rolling contact bodies and without elastic deformation. The lines, A1A_1 and A2A_2 in Fig. 3a, are marked in order to make a good understanding of the description. After the deformations of the bodies take place, the positions and deformations of lines, A1A_1 and A2A_2, are shown in Fig. 3b. The displacement difference, w1, between the two dash lines in Fig. 3b is caused by the rigid motions of the bodies and (rolling or shift). The local elastic deformations of points, A1 and A2, are indicated by u11 and u21, which are determined with some of the present theories of rolling contact based on the assumption of elastic-half space, they make the difference of elastic displacement between point A1 and point A2, u1 = u11 ? u21. If the effects of structure elastic
deformations of bodies and are neglected the total slip between points, A1 and A2, can read as: S1 = w1 ? u1 = w1 ? (u11 ? u21) (1) The structure elastic deformations of bodies and are mainly caused by traction, p and p_ acting on the contact patch and the other boundary conditions of bodies and , they make lines, A1A_1 and A2A_2 generate rigid motions independent of the local coordinates (ox1x3, see Fig. 3a) in the contact area. The u10 and u20 are used to express the displacements of point A1 and point A2, respectively, due to the structure elastic deformations. At any loading step they can be treated as constants with respect to the local coordinates for prescribed boundary conditions and geometry of bodies and . The displacement difference between point A1 and point A2, due to u10 and u20, should be u0 = u10 ? u20. So under the condition of considering the structural elastic deformations of bodies and , the total slip between points, A1 and A2, can be written as: S?1 = w1 ? u1 ? u0 (2) It is obvious that S1 and S?1 are different. The traction (or creep-force) between a pair of contact particles depends on S1 (or S?1 ) greatly. When |S1| > 0 (or |S?1 | > 0) the pair of contact particles is in slip and the traction gets into saturation. In the situation, according to Coulomb’s friction law the tractions of the above two conditions are same if the
same frictional coefficients and the normal pressures are assumed. So the contribution of the traction to u1 is also same under the two conditions. If |S1| = |S?1 | > 0, |w1| in (2) has to be larger than that in (1). Namely the pairs of contact particles without the effect of u0 get into the slip situation faster than that with the effect of u0. Correspondingly the whole contact area without the effect of u0 gets into the slip situation fast than that with the effect of u0. Therefore, the ratios of stick/slip areas and the total traction on contact areas for two kinds of the conditions discussed above are different, they are simply described with Fig. 4a and b. Fig. 4a shows the situation of stick/slip areas. Sign in Fig. 4a indicates the case without considering the effect of u0 and indicates that with the effect of u0. Fig. 4b expresses a relationship law between the total tangent traction F1 of a contact area and the creepage w1 of the bodies. Signs and in Fig. 4b have the same meaning as those in Fig. 4a. From Fig. 4b it is known that the tangent traction F1 reaches its maximum F1max at w1 = w_1 without considering the effect of u0 and F1 reaches its maximum F1max at w1 = w_1 with considering the effect of u0, and w_1 < w__ 1 . u0 depends mainly on the SED of the bodies and the traction on the contact area. The large SED causes large u0 and the small contact stiffness between the two bodies in rolling contact. That is why the reduced contact stiffness increases the ratio of stick/slip area of a contact area and decreases the total tangent traction under the condition of the contact area without full-slip.
3. Calculation of structure deformation of wheelset/rail
In order to calculate the SED described in Fig. 1b–d, and Fig. 2, discretization of the wheelset and the rail is made. Their schemes of FEM mesh are shown in Figs. 5, 7 and 9. It is assumed that the materials of the wheelset and rail have the same physical properties. Shear modulus: G = 82,000 N/mm2, Poisson ratio: μ = 0.28. Fig. 5 is used to determine the torsional deformation of the wheelset. Since, it is symmetrical about the center of wheelset (see Fig. 1b), a half of the wheelset is selected for analysis. The cutting cross section of the wheelset is fixed, as shown in Fig. 5a. Loads are applied to the tread of the wheelset in the circumferential direction, on different rolling circles of the wheel. The positions of loading are, respectively, 31.6, 40.8 and 60.0 mm, measured from the inner side of the wheel. Fig. 6 indicates the torsional deformations versus loads in the longitudinal direction. They are all linear with loads, and very close for the different points of loading. The effect of the loads on the deformation of direction of y-axis, shown
in Fig. 5a, is neglected.
Parameters of contact geometry of wheelset/rail to be used in the latter analysis read as:
ri =ri(y,ψ)
δi = δi(y,ψ)
?i = ?i(y,ψ)
ai = ai(y,ψ)
hi = hi(y,ψ)
z = z(y,ψ)
φ = φ(y, ψ) (3)
where i = 1, 2 stand for the left and right side w
heels/rails, respectively. The parameters in (3) are defined in detail in the Nomenclature of the present paper.We define thaty > 0
when the wheelset shifts towards the left side of track and ψ > 0 if it is inclined, in the clockwise direction, between the axis of wheelset and the lateral direction of track pointing to the left side. The parameters depend on the profiles of wheel and rail, y and ψ. But if profiles of wheel and rail are prescribed they mainly depend on y [7]. Detailed discussion on the numerical method is given in [7,8] and
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