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中國地質(zhì)大學(xué)長城學(xué)院
本科畢業(yè)設(shè)計外文資料翻譯
系 別： 工程技術(shù)系
專 業(yè)：機械設(shè)計制造及其自動化
姓 名： 商殿尊
學(xué) 號： 05208332
2011 年12 月 30 日
外文資料翻譯譯文
1.1轉(zhuǎn)子動態(tài)特性的軸承
在每種類型的旋轉(zhuǎn)機械中，轉(zhuǎn)子振動的因素是很重要的設(shè)計。在最常見的應(yīng)用中，主要的重點是適當(dāng)?shù)钠胶?，盡量減少轉(zhuǎn)子剩余不平衡振動水平。在更先進的設(shè)計轉(zhuǎn)速中，功率密度和性能受到實際的限制，那么將大大增加對轉(zhuǎn)子動力學(xué)設(shè)計的水平和復(fù)雜程度的考慮。高性能渦輪機械技術(shù)要求最苛刻的地方分別在旋轉(zhuǎn)機械方面的轉(zhuǎn)子動力學(xué)以及其他許多重要的工程材料學(xué)方面。
先進的渦輪機械需要用廣泛的計算研究和預(yù)測的轉(zhuǎn)子動力學(xué)特性的成分進行設(shè)計，i.e., (i) 關(guān)鍵（共振）的速度，(ii) 響應(yīng)和靈敏度轉(zhuǎn)子質(zhì)量不平衡分布，(iii)不穩(wěn)定（自激）閾值的速度。標(biāo)準(zhǔn)治療這種分析涉及到數(shù)學(xué)建模的轉(zhuǎn)子和支持系統(tǒng)的范圍內(nèi)假定非線性動力學(xué)模型[ 1,2 ] 。在專門的情況下（如刀片損失的事件）切合實際的預(yù)測不能沒有包括占主導(dǎo)地位非線性[ 3 ] 。然而，這是假設(shè)的動態(tài)線性，還有大多數(shù)轉(zhuǎn)子動力學(xué)設(shè)計分析的工作要做。
在數(shù)學(xué)表述中的線性模型橫向轉(zhuǎn)子振動很簡單，并體現(xiàn)在標(biāo)準(zhǔn)線性振動模型的任何多自由度系統(tǒng)，表現(xiàn)在以下緊湊型矩陣形式：
[M](q) + [C]{q} + [K]{q} = {F(t)} (1.1)
在模式
[M], [C], [K] =質(zhì)量，阻尼和剛度的速度取決于系數(shù)矩陣
{q}, {q}, {q}= 位移，速度和加速度矢量廣義坐標(biāo)
（函數(shù)f （噸） ） =廣義力載體
轉(zhuǎn)子動力系統(tǒng)一個有趣的特點是，運動方程通常有非對稱矩陣，尤其是剛度[ K ]和阻尼[C]矩陣。在[ K ]矩陣通常是非對稱由于動態(tài)特性的軸承，密封和其他轉(zhuǎn)子定子流體動力學(xué)相互作用力量。非對稱的[C]矩陣源自轉(zhuǎn)子陀螺效應(yīng)和流體的慣性影響密封和程度較輕的軸承。一些研究提出了數(shù)學(xué)模型，使群眾矩陣，[ M ]檔，也將非對稱類似的原因， [ K ]和[C]矩陣的非對稱。然而，令人信服的理由，如在[ 2 ] 中所描述，已說服了轉(zhuǎn)子動力學(xué)說放棄這一想法，取而代之的是對稱的質(zhì)量矩陣。
雖然理論上正式聲明的線性轉(zhuǎn)子動力學(xué)分析模型是明確的，即均衡器。 （ 1.1 ） ，雖然計算算法，充分利用這一分析模型現(xiàn)在很標(biāo)準(zhǔn)，但事實仍然是想要可靠和準(zhǔn)確的轉(zhuǎn)子振動的預(yù)測仍然是一個相當(dāng)大的挑戰(zhàn)。為什么？眾所周知的，是由于一些重要的“投入”不夠好。因此，雖然存在許多有效的計算機代碼，使轉(zhuǎn)子振動分析， “產(chǎn)出”這種代碼是唯一，好“投入” 。但最不確定的投入轉(zhuǎn)子動態(tài)系數(shù)為軸承轉(zhuǎn)子定子，密封件和其他轉(zhuǎn)子定子流體動力學(xué)的相互作用。
液體膜動力軸承，是最常用的模型，為小擾動的雜質(zhì)靜態(tài)平衡立場，就是所謂的8 -系數(shù)剛度和阻尼模型，并具有下列形式：
在這里，互動式的動態(tài)徑向力組件（fx, fy）造成的徑向位移（ X，Y ）的相對靜態(tài)平衡態(tài)和徑向速度（ X，Y ）這一位移。這個概念是畫報圖所示。1 。請注意，在該模型所描述的方程。 （ 1.2 ） ，該動力是一個功能只有位置和速度，而不是加速。這是符合古典雷諾潤滑方程，其中忽略流體的慣性作用。此外，惰性少流，軸承剛度矩陣可非對稱（即Kxy ！ = Kyx ） ，但同時阻尼矩陣應(yīng)假定為對稱（即Cxy = Cyx ） ，因為任何斜對稱添加劑軸承阻尼矩陣必須有一個后果，流體的慣性作用[ 1 ] 。在高雷諾數(shù)轉(zhuǎn)子定子液環(huán)，如海豹和一些軸承（例如，水潤滑軸承） ，這是不恰當(dāng)?shù)暮鲆暳黧w的慣性的影響，因此，另外一組系數(shù)矩陣的需要，包括轉(zhuǎn)子軌道振動加速度的影響后的總轉(zhuǎn)子定子的互動動力。這導(dǎo)致一般各向異性模型顯示如下：
在這里，Dxy = Dyx時應(yīng)實行。而
有11個完全轉(zhuǎn)子動態(tài)系數(shù)來確定上述各向異性線性模型。這些系數(shù)一般職能的軸轉(zhuǎn)速和軌道頻率。其中幾個重要的獨特功能CWRU轉(zhuǎn)子動力學(xué)試驗設(shè)施是它設(shè)定為允許提取的所有系數(shù)的各向異性模型與慣性，所描繪均衡器。 （ 1.3 ） 。目前大部分業(yè)務(wù)測試平臺的基礎(chǔ)之上更近似各向同性模型，這是嚴(yán)格只適用于旋轉(zhuǎn)對稱流場。對于各向同性的模式，均衡器。 （ 1.3 ）降低以下。
原因減少版本的均衡器。 （ 1.4 ） （不包括慣性矩陣）不能用于流體軸承，是因為這種軸承，其基本功能，支持靜態(tài)徑向負(fù)荷，必須運行在相當(dāng)大的靜態(tài)偏心率，因此，是眾所周知的轉(zhuǎn)子動態(tài)相當(dāng)各向異性。
靜和混合（合并靜水和動水）軸承，同時還有各向異性靜態(tài)偏心，也受到頻率特性的依賴慣性的影響，即使在其中的一部分，同時占主導(dǎo)地位的是粘性的影響。這是由于一些原因： （一）雄厚的財力（相比，薄膜厚度）的概念所固有的靜/混合軸承， （二）流量的急劇區(qū)之間的過渡口袋，薄膜部分的軸承， （三）流體的慣性影響，流動供應(yīng)線， （四）可能流體的慣性的影響，即使在薄膜部分。
考慮到所有上述考慮，很明顯，轉(zhuǎn)子動態(tài)上講，混合軸承結(jié)合了最復(fù)雜的特點，既流體軸承和密封。也就是說，妥善處理混合軸承，需要考慮到雙方的各向異性和慣性的影響，合并。
因此，線性模型體現(xiàn)在均衡器。 （ 1.3 ）是必要的。這并不排除潛在的有用的均衡器。 （ 1.2 ）或（ 1.4 ） ，甚至以下。（1.5 ） ，在某些特殊情況下，在特意的實驗，并分析將證明這種簡化。
各向異性模型的慣性。 （ 1.3 ） ，當(dāng)然是最好的辦法，提供了一個已提供了足夠的試驗裝置通用允許提取的所有系數(shù)的方程模型。 （ 1.3 ） 。
最近，動態(tài)特性，靜水和混合軸承已引起特別注意，因為它們在高速渦輪機械中增加應(yīng)用負(fù)載的支持要素。結(jié)合靜水行動的動力效應(yīng)許可證的混合軸承納入轉(zhuǎn)子設(shè)計在外部提供的潤滑是不切實際或不可能的。在地方汽輪機油或其他外部潤滑油，工作液中的轉(zhuǎn)子可作為一種潤滑劑。葉輪軸承指導(dǎo)中發(fā)現(xiàn)的核冷卻劑泵和火箭發(fā)動機軸承液態(tài)氫或氧氣泵是兩個例子這種類型的應(yīng)用程序。大承載能力的可能性，很長的壽命和更多的支持阻尼的反摩擦軸承使混合軸承更具吸引力。正是由于這些原因，美國宇航局目前正在大力推行混合軸承用于航天飛機和其他先進的發(fā)射系統(tǒng)。
靜壓軸承可以設(shè)計各種各樣的配置如上 圖2和 圖3 說明更詳細(xì)的幾種不同的設(shè)計，有可能為實驗，主旨，并結(jié)合實驗和推力軸承。
1.2 綜 述
影響流體軸承性能的轉(zhuǎn)子軸承系統(tǒng)已經(jīng)被認(rèn)識到了許多年。最早的一個嘗試模型實驗由斯托多拉[ 4 ] 報道于1925年，他調(diào)查了油膜剛度對臨界轉(zhuǎn)速的骨干支持軸承的影響。進一步開展工作的建模與線性軸承因為它們影響到轉(zhuǎn)子的動態(tài)特性，由哈格和?；鵞 5 ]斯塔雷特[ 6 ] 報道。
早期對于靜壓軸承的興趣出現(xiàn)在1940年年底，并且重點是他們的高負(fù)荷和剛度的能力是在沒有滑動速度的要求和幾乎為零脫離摩擦的條件下。早期出版物的靜態(tài)負(fù)荷計算方法和設(shè)計曲線也提供了基本的靜態(tài)剛度的信息，因為負(fù)荷計算可以作為一個功能的位移（即薄膜厚度） 。在20世紀(jì)60年代末，需要擁有一個全面的會計轉(zhuǎn)子動態(tài)性能造成更完整的處理靜水和混合軸承動態(tài)特性的東西。
戴維斯[ 7,8 ]利用貧瘠的土地，集中參數(shù)近似類型的研究動態(tài)行為的靜壓軸承。這種類型的分析可以公開表達(dá)形式來寫的潤滑油流量的軸承;這些問題都可以用來計算近似壓力分布和力量。送達(dá)后，以確定性能特點層流靜壓軸承的優(yōu)點和局限性的各種方法，包括薄土地的方法，這種方法也適用于由倫納德和羅[ 9 ]羅[ 10 ] ，已由奧多諾霍等人獲得。 [ 11 ] 。
1969年，亞當(dāng)斯和夏皮羅[ 12 ]利用計算機分析，以確定擠壓油膜墊性能的各種賠償類型。也包含在參考是有見地的說明阻尼效應(yīng)的內(nèi)在軸承和靜壓之間的關(guān)系阻尼墊一個單位到一個普通平板具有相同的比例。
羅德和伊扎特[ 13 ]計算表明，效果明顯的潤滑油壓縮在凹槽和補給線，動態(tài)行為特點是“打破頻率”上面剛度急劇增加和阻尼急劇下降以及。這些結(jié)果也得到了韋斯納斯[ 14 ]和高斯[ 15 ] 和戈什[ 16 ] 等人的分析，使用一階攝動法，確定剛度和阻尼性能的同時旋轉(zhuǎn)是受到飛機諧波的激發(fā)。影響可壓縮流體在休會量被忽視。結(jié)果表明，改善動態(tài)特性有可能通過適當(dāng)?shù)倪x擇壓力和偏心率和供應(yīng)的壓力。羅[ 10 ]羅和鄭[ 17 ]本理論剛度和阻尼的結(jié)果混合軸承，包括非對稱部分的剛度矩陣抓住了其中的潛力自激轉(zhuǎn)子振動。戈什[ 18 ]研究了流體慣性的影片土地層流，毛細(xì)管補償混合軸承。他表明，流體的慣性的影響減少動剛度系數(shù)。
最近的理論治療爭取獲得更多的流體力學(xué)，特別是動蕩和流體的慣性作用，逐步成為更重要的高轉(zhuǎn)速雜志正在成為至關(guān)重要的各種航空航天應(yīng)用。羅德里夫和沃爾 [ 19 ]分析了軸承設(shè)計的低溫火箭渦輪泵使用液體氣體作為潤滑劑。有限差分格式包括動蕩的影響，慣性和可壓縮流體中的發(fā)展。幾何包括一些凹降息的影響，但不包括軸向溝槽。流量，壓力分布，和剛度進行了計算，但是，沒有阻尼。實驗計劃是發(fā)達(dá)國家比較結(jié)果與理論模型。協(xié)定被認(rèn)為是良好的比較特點。有限差分方法也采用海[ 20 ]的混合軸承有關(guān)的渦輪泵。該分析模型包括動蕩和入口慣性的影響，但不可液體流動的同時，摩擦損失，承載能力和動態(tài)系數(shù)計算。實驗驗證了六個容器里，水潤滑軸承雜交表明，流體的慣性在凹嚴(yán)重影響流量的預(yù)測相比，不包括此類慣性的影響。其他性能因素，并不是有很多影響。
阿瑞帝雷絲 ，瓦倫懷特，和夏皮羅[ 21 ]提出了一個數(shù)學(xué)模型預(yù)測的靜態(tài)和動態(tài)性能特點湍流混合軸承。矩陣柱法[ 22 ]適用于可變規(guī)模有限差分網(wǎng)格是用來解決執(zhí)政潤滑方程在內(nèi)部外地點。迭代計劃之間的雷諾方程和流動連續(xù)性方程被啟用。慣性效應(yīng)在休會邊緣，但被列入可壓縮流體作用忽視。軸承審議了直徑和審批相類似的可能是部署在火箭渦輪泵。液態(tài)氦和氧被用作潤滑劑。
有限元技術(shù)已被布賽義德和曹莫利費爾[ 23 ]用來分析湍流混合軸承 。分析結(jié)果相比，得到了曹莫利費爾和尼古拉[ 24 ]的認(rèn)可 。一般來說，協(xié)議被認(rèn)為是良好的預(yù)測與實驗的特點。
最近由圣安德烈斯[ 25,26 ]發(fā)表的一本出版物完整記載了慣性和高效率的數(shù)值分析準(zhǔn)確預(yù)測的動態(tài)性能湍流混合軸承。散流的勢頭方程來描述湍流慣性流動軸承，特別考慮到下游的壓力，發(fā)展的孔板補給線和隱窩邊緣入口處的影響。這些分析都帶進觀點的重要性，流體力學(xué)和液體可壓縮性的影響動態(tài)特性的混合軸承和生產(chǎn)標(biāo)準(zhǔn)，以確保穩(wěn)定運行。數(shù)值結(jié)果預(yù)測的性能特點湍流混合軸承運行在任意中心。
格利尼克橋[ 27 ] ， （ 66年至1967年）發(fā)布了有關(guān)測量工作和鑒定軸承轉(zhuǎn)子動力學(xué)系數(shù)的工作，從頻域方程中得出軸承部分的120毫米模型同時或兩個相互正交方向同時測量振幅和相位之間的相對運動軸承的剛度和阻尼計算系數(shù)。莫頓[ 28 ]通過這一技術(shù)更全面的計算了308毫米工業(yè)軸承的剛度和阻尼系數(shù)，并隨后開發(fā)出一種技術(shù)，讓一個步驟改變生效，適用于旋轉(zhuǎn)軸。據(jù)估計由散射實驗產(chǎn)生的剛度和阻尼會表現(xiàn)出相當(dāng)?shù)南禂?shù)?？缱枘釛l件特別差界定和莫頓將此歸因于條件不足的矩陣，但他并不追求這一點。
帕金斯[ 30,31 ]采用了剛性轉(zhuǎn)子與兩個外部的獨立的正弦負(fù)載。他調(diào)整了相對幅值和相位，使軸承的動議最早是純粹的橫向，然后是純粹的縱向。因此，他能夠簡化運動方程。他評價系數(shù)為平原軸承的360 °環(huán)狀溝。當(dāng)他相比，預(yù)測和衡量系數(shù)，他常常發(fā)現(xiàn)超過百分之百的分歧。
在1977年伯羅斯和斯坦韋[ 32 ]提出了利用偽隨機二進制序列（ PRBS ）的時域方法進行數(shù)據(jù)分析。阿多元回歸估計是在離散域或從時域性差別轉(zhuǎn)子軸承模型。然而，這一估計可能會產(chǎn)生偏見[ 33 ] ，這項工作產(chǎn)生了明顯的成果。與其優(yōu)勢相比，這種技術(shù)與其他方法的測試轉(zhuǎn)子軸承系統(tǒng)已經(jīng)經(jīng)過了討論[ 34 ] 。使用多頻測試[ 35,36,37 ]可以克服一些缺點與時域算法[ 32 ]。這種方法具有若干優(yōu)點，其中包括所有的系統(tǒng)模式內(nèi)訂明的頻率范圍，高噪聲抑制。該方法涉及迫使該系統(tǒng)在x和y方向，在所有頻率范圍內(nèi)的頻息，同時進行。然而，隨著使用PRBS迫使類型的信號，有一種危險的飽和的系統(tǒng)，以便一些振幅頻率是如此之大，非遇到非線性和測試變得無效。在[ 37 ， 38 ]布羅斯等人利用施羅德相諧波信號（ SPHS ） 克服了這一問題，他們希望在一個特定的頻率范圍內(nèi)[ 39 ] 激發(fā)由平等的振幅正弦信號的頻率。
安田等人[ 40 ]同時轉(zhuǎn)子系統(tǒng)由兩個獨立的統(tǒng)計學(xué)隨機輸入信號的測量頻率響應(yīng)函數(shù)水泵水封。 12動態(tài)系數(shù)分別提取采用最小二乘技術(shù)。該方法在較短的時間內(nèi)獲得的數(shù)據(jù)比席卷正弦的方法更激動人心。此外，數(shù)據(jù)的X方向和Y方向也同時獲得。
諾德曼和斯克里 [ 41 ]適用于沖擊力的投入轉(zhuǎn)子軸承系統(tǒng)，并采用了曲線擬合技術(shù)的頻率響應(yīng)函數(shù)獲得的實驗研究。該試驗臺的對稱配置是剛性轉(zhuǎn)子運行的兩個“相同的”軸承。瞬態(tài)振動轉(zhuǎn)子而引起的運用武力沖動轉(zhuǎn)子（由轉(zhuǎn)子突出一個“校準(zhǔn)錘” ） 。輸入信號（動力）和輸出信號（位移轉(zhuǎn)子）轉(zhuǎn)化為頻域和復(fù)雜的頻率響應(yīng)函數(shù)從而計算。分析頻率響應(yīng)函數(shù)，這取決于軸承系數(shù)，被安裝在測量功能。剛度和阻尼系數(shù)結(jié)果擬合過程迭代最小二乘錯誤作為一個標(biāo)準(zhǔn)。同樣的技術(shù)也適用于測定動態(tài)系數(shù)的環(huán)形湍流密封渦輪泵[ 42 ] 。
神吉和川[ 43 ]構(gòu)建了一個試驗臺的水密封對稱配置和流動的支持下套管空氣波紋管和液壓傳動。雙方向前和向后圓形武力激發(fā)適用。解決后的未知阻抗的職能， 8剛度和阻尼系數(shù)的各向異性模型，得到了曲線擬合在廣泛的頻率范圍。
潛在的航空航天應(yīng)用了一些新的實驗旨在測試水壓和混合軸承轉(zhuǎn)子動力學(xué)特點，包括在本論文和[ 44]墨菲和瓦格納[ 45 ]最近提出的剛度和阻尼系數(shù)提取同步軌道偏心雜志與制冷劑- 113的工作液。他們限制于無法從同步軌道提取慣性系數(shù)以及其他數(shù)據(jù)，來確定頻率和所有的系數(shù)。
克魯?shù)俨痰热薣 46 ]最近建立了一個高速試驗臺研究水潤滑軸承，并用它來測試靜態(tài)孔口補償混合軸承速度可達(dá)25000轉(zhuǎn)。在他們的平臺，測試軸承是自由暫停高速軸在中間立場的支持軸承。對照靜態(tài)負(fù)荷適用于通過輪換耐鋼絲繩。雖然只有靜態(tài)負(fù)荷結(jié)果[ 46 ] ，動態(tài)測試指出了目前的進展情況。
墨菲斯科爾等 [ 47 ]描述了多功能測試儀器來衡量轉(zhuǎn)子動力學(xué)系數(shù)的軸承和密封件。徑向磁軸承將用于靜態(tài)和動態(tài)負(fù)載軸。徑向磁軸承將用于靜態(tài)和動態(tài)負(fù)載軸。迅速正弦波掃描是作為動力激發(fā)瞬態(tài)投入測試。測試環(huán)境密切關(guān)系到真正的火箭發(fā)動機渦輪泵的運行條件。
1.3 問 題 的 聲 明
這項研究在本論文集中的實驗和理論確定轉(zhuǎn)子動力學(xué)系數(shù)為石油喂靜壓口補償軸承，其中包括混合效應(yīng)輪換。當(dāng)前利益和缺乏實驗數(shù)據(jù)和充分的分析工具，強調(diào)需要這種類型的研究。在凱斯西儲大學(xué)使用試驗設(shè)施實驗研究已完成。計算機代碼的基礎(chǔ)上有限差分模型被用來進行理論預(yù)測，同樣的配置和經(jīng)營狀況的測試矩陣。代碼確定靜態(tài)和動態(tài)性能的動力，在制度層狀沒有流體的慣性，靜水或混合軸承是不可使用潤滑劑的。動態(tài)性能受到了擾動的位置和速度，解決了靜態(tài)特性。改變力向量（綜合壓力）可以判斷剛度和阻尼矩陣系數(shù)的混合軸承。
這一論斷的安排如下：第二章中，測試平臺和儀器儀表的描述。此外，實驗過程中已列入第二章。第三章，位置所用的方法在衡量和確定轉(zhuǎn)子動力學(xué)系數(shù)的軸承和密封件。特別是發(fā)達(dá)國家的方法是此工作的重點。發(fā)展功能混合/靜壓軸承靜態(tài)和動態(tài)分析（ HBSADA ）計算機代碼顯示在第四章。之前，討論方法的數(shù)值解，簡要概述了一個典型的混合軸承的運作。實驗和計算結(jié)果公布在第五章，誤差分析也載于附錄第五章。有關(guān)工作列入年底。
外文原文
1.1 ROTORDYNAMIC PROPERTIES OF BEARINGS
Rotor vibration considerations are important to the design of nearly every type of rotating machinery. In the least demanding applications, the primary focus is on adequate balancing of the rotor to minimize residual unbalance vibration levels. In more advanced designs, as rotational speed, power density and performance are pushed to practical limits, invariably the level and sophistication of required attention to rotor dynamical design considerations increase considerably. High performance turbo machinery is technologically the most demanding branch of rotating machinery in regards to rotor dynamics as well as many other critical engineering ingredients.
Among the several required ingredients in the design of advanced turbo machinery are extensive computational studies and predictions of rotor dynamical characteristics, i.e., (i) critical (resonance) speeds, (ii) response and sensitivity to rotor mass unbalance distribution, and (iii) instability (self-excited) threshold speeds. Standard treatments of such analyses involve the mathematical modeling of the rotor and support system within the context of an assumed linear dynamics model [1,2]. In specialized situations (e.g., blade loss events) realistic predictions can not be made without including dominant nonlinearities [3]. However, it is with the assumption of dynamic linearity that most rotor dynamic design analyses are done.
The mathematical formulation of the linear model for lateral rotor vibrations is quite straightforward, and is embodied within the standard linear vibration model for any multi-degree-of-freedom system, as shown in the following compact matrix form:
[M](q) + [C]{q} + [K]{q} = {F(t)} (1.1)
where
[M], [C], [K] = mass, damping, and stiffness speed dependent coefficient matrices
{q}, {q}, {q}= a displacement, velocity, and acceleration vectors of the generalized coordinates
{F（t）}=generalized force vector
An interesting characteristic of rotor dynamical systems is that the equations of motion typically have non-symmetric matrices, especially the stiffness [K] and damping [C] matrices. The [K] matrix is typically non-symmetric because of dynamic characteristics of bearings, seals and other rotor-stator fluid dynamical interaction forces. Non-symmetry of the [C] matrix arises from the rotor's gyroscopic effects and fluid inertia effects in seals and to a lesser degree in bearings. Some researches have proposed mathematical models which allow the mass matrix, [M], also to be non-symmetric for similar reasons that the [K] and [C] matrices are non-symmetric. However, compelling arguments, such as made in [2], have convinced serious rotordynamicists to drop this idea in favor of a symmetric mass matrix.
Although the mathematically formal statement of the linear-analysis rotor dynamics model is well defined, i.e., Eq. (1.1), and although computational algorithms to fully utilize this analysis model are now quite standard, the fact remains that performing reliable and accurate rotor vibration predictions is still a considerable challenge. Why? Because some of the important "inputs" are not well enough known. Thus, while numerous valid computer codes exist to make rotor vibration analyses, the "outputs" of such codes are only as good as the "inputs". The most uncertain inputs are the rotor dynamic coefficients for the rotor-stator forces at bearings, seals and other rotor-stator fluid-dynamical interactions.
For fluid-film hydrodynamic journal bearings, the most commonly used model, for small perturbations of the journal from the static equilibrium position, is the so-called 8-coefficient stiffness and damping model, and has the following form:
Here, the dynamic interactive radial force components (fx, fy) are caused by the radial displacement (x, y) relative to the static equilibrium state and by the radial velocity (x, y) of this displacement. The concept is pictorially shown in Fig. 1. Note that in the model described by Eq. (1.2), the force is a function only position and velocity, but not acceleration. This is consistent with the classical Reynolds lubrication equation, which neglects fluid inertia effects. Also, for inertia less flow, the bearing stiffness matrix can be non-symmetric (i.e., Kxy!= Kyx ),but the bearing damping matrix should be postulated as symmetric (i.e., Cxy = Cyx ), since any skew-symmetric additive to the bearing damping matrix must be a consequence of fluid inertia effects [1]. In the case of higher Reynolds number rotor-stator fluid annuli, such as seals and some bearings (e.g., water lubricated bearings), it is not appropriate to neglect fluid inertia effects and, thus, an additional set of matrix coefficients are needed to include rotor orbital-vibration acceleration influences upon the total rotor-stator interaction dynamic force. This leads to the general anisotropic model shown as follows:
Where,
Dxy=Dyx shoud be imposed.
And,
There are totally 11 rotor dynamic coefficients to be determined in the above anisotropic linear model. These coefficients are generally functions of shaft spin speed and orbit frequency. One of the several important unique features of the CWRU rotor dynamics test facility is that it is configured to permit extractions of all the coefficients of the anisotropic model with inertia, as depicted in Eq. (1.3). Most currently operational test rigs are based upon the more approximate isotropic model, which is strictly valid only for rotationally symmetric flow fields. For the isotropic model, Eq. (1.3) reduces to the following.
The reason a reduced version of Eq. (1.4) (without the inertia matrix)is not used for hydrodynamic journal bearings is because such journal bearings, by their basic function to support static radial loads, must run at considerable static eccentricity and, thus, are well known to be rotor dynamically quite anisotropic.
Hydrostatic and hybrid (combined hydrostatic and hydrodynamic) journal bearings, while also anisotropic under static eccentricity, can also exhibit frequency dependence characteristic of inertia effects, even when the film part of the bearing is dominated by the viscous effects. This is so for a number of reasons: (i) the deep pockets (compared to film thickness) concept inherent in hydrostatic/hybrid bearings, (ii) the sharp flow-area transition between pockets and thin-film portions of bearing, (iii) fluid inertia effects in the flow-supply line, and (iv) possible fluid inertia effects even within the thin film portions.
Taking all of the above into account, it is apparent that rotor dynamically speaking, hybrid journal bearings combine the most complicated features of both hydrodynamic journal bearings and seals. That is, proper treatment of hybrid bearings requires taking account of both anisotropic and inertia effects, combined. Thus, the linear model embodied in Eq. (1.3) is required. This does not preclude potential usefulness of Eq. (1.2) or (1.4) or even the following, Eq. (1.5), in certain special situations, after bona fide experiments, and analyses would justify such simplifications.
The anisotropic model with inertia, Eq. (1.3), is certainly the best approach, provided one has available a test apparatus sufficiently versatile to permit extraction of all the coefficients of the model in Eq. (1.3).
Recently, dynamic characteristics of hydrostatic and hybrid journal bearings have attracted particular attention because of their increased application as load support elements in high speed turbo machinery. The combination of hydrostatic action with the hydrodynamic effects permits the hybrid bearing to be incorporated into rotor designs where externally supplied lubrication is impractical or just impossible. In place of turbine oil or other external lubricants, the working fluid in the rotor can be used as a lubricant. Impeller guide bearings found in nuclear coolant pumps and rocket motor bearings in liquid hydrogen or oxygen pumps are two examples of this type of application. Large load capacity, the possibility of very long life and increased support damping over anti-friction bearings make hybrid bearings attractive. It is for these reasons that NASA is now vigorously pursuing hybrid bearings for the Space Shuttle and other advanced launching systems.
Hydrostatic bearings can be designed in a wide variety of configurations as indicated in Fig. 2. Fig. 3 illustrates in more detail the several different designs that are possible for journal, thrust, and combined journal and thrust bearings.
1.2 LITERATURE REVIEW
The influence of fluid bearings on the performance of rotor-bearing systems has been recognized for many years. One of the earliest attempts to model a journal bearing was reported in 1925 by Stodola [4], who investigated the effect of oil-film stiffness on the critical speed of a shaft supported in journal bearings. Further work on the modeling and linearization of bearings as they affect the rotor's dynamic behavior was reported by Hagg and Sankey [5] and Starlight [6].
Early interest in hydrostatic journal bearings emerged in the late 1940's and was focused on their high load and stiffness capability without a sliding velocity requirement and with virtually zero break-away friction. Early publications of static load calculation methods and design curves also provided the basic static stiffness information since load could be computed as a function of displacement (i.e., film thickness). In the late 1960's, the need for a fuller accounting of rotor dynamic performance resulted in more complete handling of hydrostatic and hybrid journal bearings dynamic properties.
Davies [7,8] employed the thin lands-lumped parameter typeof
approximation to study dynamic behavior of hydrostatic journal
bearings. This type of analysis enables closed form expression to be written for lubricant flow rates over the bearing lands; these can be used to calculate the approximate pressure distribution and forces. Such a method, applied also by Leonard and Rowe [9] and Rowe [10], served to determine the performance characteristics of laminar flow hydrostatic bearings. The advantages and limitations of various methods including thin-land methods have been given by O'Donoghue et al. [11].
In 1969 Adams and Shapiro [12] used computer analysis to determine squeeze film pad performance with the various compensation types. Also contained in that reference is an insightful description of the damping effect inherent in hydrostatic bearings and the relationship between the damping of a flat pad to that of a plain flat plate having the same proportions.
Rohde and Ezzat [13] computationally demonstrated the pronounced effects of lubricant compressibility in recesses and
supply line, with dynamic behavior characterized by a "break
frequency" above which stiffness increases sharply and damping
decreases sharply as well. These results also were supported by the analysis of Ghosh and Viswanath [14] and Ghosh et al. [15]. Ghosh [16], using a first order perturbation method, determined stiffness and damping properties for bearing with nonrotating journal subjected to plane harmonic excitation. The effect of fluid compressibility in the recess volume was neglected. The results show that an improvement of dynamic characteristics is possible by proper choice of pressure and eccentricity ratios and supply pressure. Rowe [10] and Rowe and Chong [17] present theoretical stiffness and damping results for hybrid bearings, including the non-symmetric portion of the stiffness matrix which captures the potential for self-excited rotor vibration. Ghosh [18] investigated the fluid inertia on the film lands of laminar flow, capillary compensated hybrid journal bearing. He showed that fluid inertia effects reduce the dynamic stiffness coefficients.
Recent theoretical treatments have sought to capture more of the fluid mechanics, specifically turbulence and fluid inertia effects which become progressively more important as high journal rotational speeds are becoming critical to various aerospace applications. Redecliff and Vohr [19] analyzed bearing designs for the cryogenic rocket turbopump using liquid gases as the lubricant. A finite difference scheme including the effects of turbulence, inertia, and compressibility in the fluid film was developed. The geometry included a number of recesses cut in the bearing, but did not include axial grooves. Flow rates, pressure distribution, and stiffness were calculated, however, damping was not. An experimental program was developed to compare results with the theoretical model. Agreement was found to be good for the compared characteristics.
Finite difference approach was also employed by Heller [20] for hybrid bearings related to turbopumps. The analytical model included turbulence and entrance inertia effects, but for incompressible fluid. The bearing flow, friction loss, load capacity, and dynamic coefficients were calculated. Experimental verification for a six pocket, water lubricated hybrid bearing, showed that fluid inertia at recesses grossly affected flow rates in comparison to predictions not including such inertia effects. Other performance factors were not nearly as much affect