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機械原理 基于局部平均分解的階次跟蹤分析及其在齒輪故障診斷中的應用 Junsheng Cheng, Kang Zhang, Yu Yang 關鍵詞： 階次跟蹤分析 局部平均分解 解調 齒輪 故障診斷 摘要： 局部平均分解(LMD)是一種新的自適應時頻分析方法,這種方法特別適合處理多分量的調幅信號和調頻(AM-FM)信號。通過使用LMD方法,可以將任何復雜的信號分解為一系列的產(chǎn)品功能PF分量(PFs),每個PF分量都是純調頻信號和包絡信號的乘積，且通過純調頻信號可以獲得具有物理意義的瞬時頻率。從理論上講,每個PF分量都是一個單分量的AM-FM信號。 因此,可以將LMD的過程看作是信號解調的過程。齒輪發(fā)生故障時,振動信號呈現(xiàn)明顯的AM-FM特征。因此，針對齒輪升降速過程中故障振動信號為多分量的調制信號，以及故障特征頻率隨轉速變化的特點，提出了一種基于LMD和階次跟蹤分析的齒輪故障診斷方法。齒輪箱的故障診斷實驗表明本文提出的方法能有效地提出齒輪故障診斷特征。 1 引言 齒輪傳動是機械設備中常見的傳動方式, 故對齒輪進行故障診斷具有重要意義。 齒輪故障診斷的關鍵一步是故障特征的提取。一方面，傳統(tǒng)的齒輪故障診斷方法的重點在一個固定的旋轉速度檢測振動信號的頻譜分析。 而齒輪作為一種旋轉部件, 其升降速過程的振動信號往往包含了豐富的狀態(tài)信息, 一些在平穩(wěn)運行時不易反映的故障特征在升降速過程中可能會充分地表現(xiàn)出來[1],此外，來自齒輪振動信號的暫態(tài)過程中，速度依賴性總是顯示非平穩(wěn)特征。如果頻譜分析直接應用于非平穩(wěn)振動信號，混頻將不可避免的發(fā)生，這將對故障特征提取帶來不良影響。在以往的研究中，為了跟蹤技術，通常利用振動信號中添加旋轉機械軸轉速信息，已經(jīng)成為一個在旋轉機械故障診斷[2,3]的重要途徑。從本質上講，階次跟蹤分析技術可以在時域非平穩(wěn)信號轉換成角域靜止，可以突出的旋轉速度相關的振動信息和抑制無關的信息。因此，階次跟蹤分析是在助跑過程中齒輪的故障特征提取和運行了一個可取的方法 另一方面，當發(fā)生故障的齒輪振動信號，拿起在運行和運行過程中始終存在的振幅特性調制和頻率調制（AM–FM）。為了提取齒輪故障振動信號的調制特征，解調分析是最流行的方法之一[ 4,5 ]。然而，傳統(tǒng)的解調方法，如希爾伯特變換解調和傳統(tǒng)包絡分析有其自身的局限性[ 6 ]。這些缺點包括兩個方面：（1）在實踐中大多數(shù)的齒輪故障振動信號都是多組分是–調頻信號。這些信號，在傳統(tǒng)的解調方法，他們通常是通過帶通濾波器分解成單組分是–調頻信號的解調，然后提取的頻率和振幅信息。然而，這兩個數(shù)載波頻率的載波頻率成分和幅值都難以在實踐中被確定，所以帶通濾波器的中心頻率的選擇具有主體性，將解調誤差和使它提取機械故障振動信號的特征是無效的；（2）由于希爾伯特不可避免的窗口效應變換，當使用希爾伯特變換提取調制信息，目前的非瞬時響應特性，即，在調制信號被解調以及打破中間部分的兩端會再次產(chǎn)生調制，使振幅指數(shù)衰減的方式得到的波動，然后解調誤差將增加[ 7 ]。為了克服第一個缺點，一個合適的分解方法應尋找獨立的多分量信號為多個單組分是–調頻信號的包絡分析之前。由于EMD（經(jīng)驗模態(tài)分解）自適應復雜多分量信號分解為一系列固有模態(tài)函數(shù)（IMF）的瞬時頻率的物理意義[ 8,9 ]，基于EMD的階比跟蹤方法已廣泛應用于齒輪故障診斷[ 13 ]。然而，仍然存在許多不足之處[ 14 ]，如在EMD的端點效應和模態(tài)混 [ 15 ]，仍在進行。此外，對原信號通過EMD分解，產(chǎn)生了由希爾伯特變換（上面提到的）缺點是不可避免的在IMF進行希爾伯特變換的包絡分析。此外，有時無法解釋的負瞬態(tài)頻率時會出現(xiàn)瞬時頻率計算每個IMF進行希爾伯特變換[ 16 ] 局部均值分解（LMD）是一種新型的解調分析方法，特別適合于處理多組分的幅度調制和頻率調制（AM–調頻）信號[ 16 ]。用LMD，任何復雜的信號可以分解成許多產(chǎn)品功能（PFS），每一種產(chǎn)品的包絡線信號（獲得直接由分解）的PF瞬時振幅可以得到一個純粹的頻率調制信號從一個良好定義的瞬時頻率可以計算。在本質上，每個PF正是一種單組分我–調頻信號。因此，LMD的程序可以，事實上，作為解調過程。調制信息可以通過頻譜分析的瞬時振幅（包絡信號，直接獲得通過分解）每個PF分量進行希爾伯特變換，而不是由PF分量。因此，當LMD和EMD方法分別應用到解調分析，與EMD，LMD的突出優(yōu)點是避免希爾伯特變換。此外，LMD迭代過程中所采用的手段和當?shù)氐姆炔黄交牡胤接肊MD的三次樣條的方法，這可能帶來的包絡的誤差和影響的精度瞬時頻率和振幅。此外，與EMD端點效應相比并不明顯，因為在LMD方法更快的速度和算法的迭代次數(shù)更少[ 17 ]。 基于以上分析，階次跟蹤和解調技術，LMD最近的發(fā)展，科學相結合，并應用于齒輪故障診斷過程中各軸速度。首先，訂單跟蹤技術被用于將從時間域的齒輪振動信號角域。其次，分解角域重采樣信號的PF系列LMD，因此組件和相應的瞬時振幅和瞬時頻率可以得到的。最后，進行頻譜分析的故障信息含有顯性PF分量的瞬時幅值。從實驗的振動信號，表明該方法能有效地提取故障特征和分類準確齒輪工作狀態(tài)的分析結果。 本文的組織如下。第2節(jié)是一個給定的LMD方法理論。在第3節(jié)中的齒輪故障診斷方法中，以技術和LMD跟蹤相結合的提出和實踐應用表明，提出的方法。此外，LMD和基于EMD的比較也在第3節(jié)提到了基礎的方法。最后，我們得出了第4部分的結論。 2 LMD 方法 LMD方法的本質是通過迭代從原始信號中分離出純調頻信號和包絡信號,然后將純調頻信號和包絡信號相乘便可以得到一個瞬時頻率具有物理意義的PF分量,循環(huán)處理直至所有的PF分量分離出來對任意信號x(t),其分解過程如[16]: ( 1) 確定原始信號第i個局部極值及其對應的時刻,計算相鄰兩個局部極值和的平均值 (1) 將所有平均值點mi在其對應的時間段[,]內伸一線段,然后用滑動平均法進行0平滑處理,得到局均值m11(t) 。 ( 2) 采用局部極值點計算局部幅值 : =| -|/2 (2) 將所有局部幅值點ai在其對應的時間段[,]內伸成一條線段,然后采用滑動平均法進行平滑處理,得到包估計函數(shù)a11(t) 。 ( 3) 將局部均值函數(shù)m11(t)從原始信號x(t)中分離來, 即去掉一個低頻成分,得到 h11(t)=x(t)-m11(t) (3) ( 4)用h11(t)除以包絡估計函數(shù)A11( t)以對h11(t)進行解調,得到 s11(t)=h11(t)/A11(t) (4) 對s11( t)重復上述步驟便能得到s11(t)的包絡估計函數(shù)A12(t),若A12(t)不等于1,則s11( t)不是一個純調頻信號需要重復上述迭代過程n次,直至s1n(t)為一個純調頻信號,即 s1n(t)的包絡估計函數(shù) A1(n+1)(t)=1,所以,有 (5) (6) 為理論上, 迭代終止的條件 (7) 在實踐中,一種變體δ會提前確定。如果1?δ≤a1(n + 1)(t)≤1 +δand?1≤s1n(t)≤1,然后迭代過程將停止 ( 5) 把迭代過程中產(chǎn)生的所有包絡估計函數(shù)相乘便可以得到包絡信號( 瞬時幅值函數(shù)) : （8） ( 6) 將包絡信號A1(t)和純調頻信號s1n(t)相乘便可以得到原始信號的第一個PF分量: PF1(t）=a1(t)s1n(t) ( 9) PF1(t)包含了原始信號中頻率值最高的成分,是一個單分量的調幅-調頻信號,PF1(t)的瞬時幅值就是包絡信號A1(t),PF1(t)的瞬時頻率f1(t)則可由純調頻信號s1n(t)求出,即: (10) ( 7)將第一個PF分量PF1(t)從原始信號x(t)中分離出來, 得到一個新的信號u1(t),將u1( t)作為原始數(shù)據(jù)重復以上步驟,循環(huán)k次,直到 uk為一個單調函數(shù)為止,即: (11) 原始信號x(t)能夠被所有的PF分量和uk重構,即: (12) 產(chǎn)品功能p的數(shù)量在哪里.此外,相應的完整的時頻分布可以通過組裝瞬時幅度和瞬時頻率的PF組件。 3 基于階次跟蹤分析與 L M D 的齒輪故障診斷 3.1 階次跟蹤分析 階次跟蹤分析首先根據(jù)參考軸的轉速信息對時域信號進行等角度重采樣, 將時域非平穩(wěn)信號轉換為角域平穩(wěn)信號, 再對角域平穩(wěn)信號進行譜分析得到階次譜。階次跟蹤分析能夠提取信號中與參考軸轉速有關的信息, 同時抑制與轉速無關的信號, 因此非常適合分析旋轉機械在變轉速過程下的振動信號。實現(xiàn)階次跟蹤分析技術的關鍵在于, 如何實現(xiàn)被分析信號相對于參考軸的等角度重采樣, 即階次重采樣。常用的階次重采樣方法有硬件階次跟蹤法[ 6]、計算階次跟蹤法[ 7]和基于瞬時頻率估計的階次跟蹤法[ 8]等。硬件階次跟蹤法直接通過專用的模擬設備實現(xiàn)信號的等角度重采樣,實時性好,但只適用于軸轉速較穩(wěn)定的情況,且成本很高;基于瞬時頻率估計的階次跟蹤法不需要專門的硬件設備,無需考慮硬件安裝問題,且成本較低, 但是不適用于分析多分量信號,而實際工程信號大多為多分量信號, 因此其實際應用意義不大;COT法通過軟件的形式實現(xiàn)等角度重采樣,分析精度高, 對被分析的信號沒有特別的要求,并且無需特定的硬件, 因此是一種應用廣泛的階次跟蹤分析方法。 根據(jù)試驗條件采用COT法實現(xiàn)信號的階次重采樣,其具體步驟如下: 1. 對振動信號和轉速信號分兩路同時進行等時間間隔(間隔為$t)采樣,得到異步采樣信號; 2. 通過轉速信號計算等角度增量 $H 所對應的時間序列ti ; 3. 根據(jù)時間序列ti的值,對振動信號進行插值,求出其對應的幅值,得到振動信號的同步采樣信號,即角域平穩(wěn)信號; 4.使用LMD分解平衡角重采樣信號,因此sPF系列組件和相應的瞬間振幅和瞬時頻率可以獲得 5.光譜分析應用于每個PF的瞬時振幅組件,然后我們有訂單譜 3.2 齒輪故障診斷實例 升降速過程中的齒輪故障振動信號通常是多分量的調幅-調頻信號,并且故障特征頻率會隨著轉速的變化而改變。針對升降速過程齒輪故障振動信號的這些特點, 提出了基于階次跟蹤分析和 LM D 的齒輪故障診斷方法。首先采用階次跟蹤分析將齒輪升降速過程的時域振動信號轉換成角域平穩(wěn)信號;然后對角域信號進行LMD分解,得到一系列PF分量,以及各個PF分量的瞬時幅值和瞬時頻率; 最后對各個PF分量的瞬時幅值進行頻譜分析,便可以有效地提取出齒輪故障特征。為了驗證方法的正確性,在旋轉機械試驗臺上進行了齒輪正常和齒根裂紋兩種工況的試驗。該系統(tǒng)中, 電機輸入軸齒輪齒數(shù)z1=55, 輸出軸齒輪齒數(shù)z2 = 75。在輸入軸齒輪齒根上加工出小槽,以模擬齒根紋故 障, 因此齒輪嚙合階次xm=55,故障特征階次xc=1。圖1和圖2所示分別為由轉速傳感器測得的輸入軸瞬時轉速n(t),以及由振動傳感器測得的齒輪故障 振動加速度a(t),其中采樣頻率為8192H z,采樣時間為20s從圖1可以看出,輸入軸轉速首先從150r/min逐漸加速至1410r/min, 然后再減速到820r/min,而加速度信號的幅值也隨著作出了相應的變化。不失一般性,截取圖2中5~ 7s升速過程的信號 a1(t)進行分析。 圖 1 輸 入軸的瞬時轉速 n ( t ) 圖 2 齒輪故障振動加速度信號 a( t ) 值在秩序O=55和O=110相應的齒輪嚙合秩序和雙。因此這意味著頻率混淆現(xiàn)象已經(jīng)在很大程度上消除。然而,為j1(θ)仍然是一個多個組件MA-MF信號。因此,一邊頻帶反映故障特征頻率模糊。有效地提取故障特征,應用LMD j - 1(θ),因此七PF組件和殘渣可以得到圖6所示,這意味著LMD解調的進展。因此,它是可以提取齒輪故障特性,利用頻譜分析的瞬時振幅PF組件包含主要故障信息。通過分析,我們知道失敗的主要信息包括在第一個PF組件。因此,無花果。7和8給瞬時振幅a1(θ)的第一個PF組件PF 1(θ)和相應的秩序光譜的a1(θ),很明顯,有不同的光譜峰值在第一順序(O = 1)對應齒輪階次跟蹤功能,符合齒輪的實際工況。 圖9和圖10顯示轉速信號的n(t)和振動加速度信號的時域波形s(t)齒輪分別與破碎的牙齒,采樣率為8192 Hz和總樣品時間是20年代。斷齒故障引入輸入軸上的齒輪與激光切割槽的牙根。首先,一段信號s1(t)5 s-7年代為進一步分析的進步是攔截;其次,假設樣本點每旋轉400;第三,角域信號為j1(θ)圖11所示可以通過執(zhí)行命令重采樣s1(t);第四,LMD適用于j-1(θ);最后,相應的秩序頻譜圖12所示的瞬時振幅首先PF組件PF 1(θ)可以了,很明顯,有不同的光譜峰值(比在圖8)在第一順序(O = 1)階次跟蹤分析對應于齒輪故障功能,符合齒輪的實際工況。 同樣的,我們同樣可以做正常的齒輪。轉速信號n(t)和振動的時域波形加速度信號s(t)的正常齒輪分別列在無花果。13和14,采樣率為8192 Hz和總樣品時間是20多歲。在上述相同的方法應用于原始信號圖14所示,結果無花果所示。15和16。圖15顯示了角域j - 1(θ)執(zhí)行順序重采樣后的信號部分(5s-7年代在籌備進展)的原始信號。圖16顯示了相應的瞬時振幅譜第一個PF組件,很難找到齒輪故障特征,也符合實際的工作狀態(tài)的裝備。 目前,多組分的另一個競爭解調方法AM-FM信號,即經(jīng)驗模式分解(EMD)存在,已經(jīng)被廣泛應用于信號解調分析(7、22)。為了比較兩個EMD方法,取代LMD,我們能做的同樣使用EMD進行重采樣信號無花果所示。圖4、11和15 圖 3 齒輪故障振動加速度信號的頻譜 圖 4 階次重采樣后的齒輪故障振動 加速度信號 圖5 j1(θ)的階次譜 分別,因此可以獲得一系列國際貨幣基金組織(IMF)組件。此外,相應的瞬時振幅和國際貨幣基金組織每個組件的瞬時頻率可以通過希爾伯特變換計算。通過分析,我們知道,IMF主要特征信息包含在第一個組件。因此,只有應用于瞬時頻譜分析第一個國際貨幣基金組織(IMF)組件的振幅。無花果。17日至19日給訂單頻譜對應三種振動信號的破解斷層、斷齒故障和正常的齒輪,分別,很明顯,訂單跟蹤分析基于EMD也可以提取齒輪故障特性,確定齒輪的工作狀態(tài)。盡管EMD和LMD都可以分解原始信號實際上,兩種方法之間的差異仍然存在。EMD方法比較,如第一節(jié)中所述,LMD有更多迭代次數(shù)少等優(yōu)點,不明顯的效果和更少的瞬時頻率的虛假成分,可以使用更多的應用在實踐中。 圖 6 角域信號j1( θ )的LMD分解結果 圖 7 PF1(θ)的瞬時幅值A1(θ) 圖 8 第1個PF分量的幅值譜 圖 9 輸入軸的瞬時轉速 n(t) 圖 1 0 正常齒輪的振動加速度信號 a(t) 圖11 階次重采樣后的正常齒輪振動加速度信號j1(θ） 圖 12 第一個PF分量的幅值譜 圖13 輸入軸轉速r(t)正常齒輪前和過程中 圖圖14 齒輪的振動加速度信號(t)在正常狀態(tài) 圖15 相應的振動加速度信號為j1(θ)角域通過應用順序重采樣tos(t)圖14所示。 圖17 第一個IMF分量的幅值譜 圖 18 第一個IMF分量的幅值譜 3 結論 在齒輪故障診斷技術、階次跟蹤是一個著名的技術,可用于故障檢測的旋轉機器采用振動信號。針對齒輪故障振動信號的調制特點在助跑和破敗的和缺點在齒輪經(jīng)常可以發(fā)相關軸轉速在瞬態(tài)過程中,階次跟蹤和技術LMD相結合用于齒輪故障診斷。從理論分析和實驗結果以下幾點得出結論: ( 1) 在分析齒輪變轉速狀態(tài)下的振動信號時,轉速波動會引起頻譜圖出現(xiàn)頻率混疊, 而階次跟蹤分析通過對信號進行階次重采樣能夠在很大程度上消除頻率混疊, 使頻譜圖的譜線清晰可讀。 ( 2) 齒輪故障時的振動信號為一多分量的調幅- 調頻信號, 采用LMD方法能將其分解為若干個PF分量之和,同得到各個PF分量的瞬時幅值和瞬時頻率, 實現(xiàn)了原信號的解調。對含有齒輪故障特征的PF分量的瞬時幅值進行頻譜分析, 能夠準確地提取出齒輪故障特征信息。 圖19 階次的第一個國際貨幣基金組織(IMF)組件的正常使用EMD齒輪 ( 3) 對齒輪正常和齒根裂紋兩種工況的振動信號進行了分析,分析結果表明, 本文方法能夠準確地反映出齒輪的實際工況。 References [1] S.K. 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An order tracking technique for the gear fault diagnosis using local mean decomposition method Junsheng Cheng, Kang Zhang, Yu Yang State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, 410082, PR China College of Mechanical and Vehicle Engineering, Hunan University, Changsha, 410082, PR China article info abstract Article history: Received 17 November 2010 Received in revised form 13 December 2011 Accepted 30 April 2012 Available online 28 May 2012 Local mean decomposition (LMD) is a new self-adaptive timefrequency analysis method, which is particularly suitable for the processing of multi-component amplitude-modulated and frequency-modulated (AMFM) signals. By using LMD, any complicated signal can be decomposed into a number of product functions (PFs), each of which is the product of an envelope signal and a purely frequency modulated signal from which physically meaningful instantaneous frequencies can be obtained. Theoretically, each PF is exactly a mono-component AMFMsignal. Therefore, the procedure of LMD can beregardedastheprocess of demodulation. While fault occurs in gear, the vibration signals would exactly present AMFM characteristics. Therefore, targeting the modulation feature of gear fault vibration signal in run-ups and run- downs and the fact that fault characteristics found in gear vibration signal could often be related to revolution of the shaft in the transient process, a gear fault diagnosis method in which order tracking technique and local mean decomposition is put forward. The analysis results from the practical gearbox vibration signal demonstrate that the proposed algorithm is effective in gear fault feature extraction. 2012 Elsevier Ltd. All rights reserved. Keywords: Order tracking technique Local mean decomposition Demodulation Gear Fault diagnosis 1. Introduction Gears are the important and frequently encountered components in the rotating machines that find widespread industrial applications. Therefore, the corresponding gear fault diagnosis has been the subject of extensive research. The key step of gear fault diagnosis is the extraction of fault feature. On the one hand, the conventional gear fault diagnosis methods focus on examining the frequency spectrum analysis of vibration signal at a fixed rotation speed. Unfortunately, the information obtained thus is only partial because some faults maybe do not respond significantly at the fixed operation speed. Since faults commonly found in gear could often be related to revolution of the shaft, more comprehensive information may be acquired by measuring the gear vibration signal in the process of run-up and run-down 1. In addition, vibration signals derived fromgearinthetransientprocessthatarespeed-dependentalwaysdisplaynon-stationaryfeature.Iffrequencyspectrumanalysis is directly applied to the non-stationary vibration signal, frequency mixing would occur inevitably, which will bring undesirable effect to the fault feature extraction. In past research, order-tracking technique, which normally exploits a vibration signal supplemented with information of shaft speed of rotating machinery, has become one of the significant approaches for fault diagnosis in rotating machinery 2,3. Essentially, order-tracking technique can transform a non-stationary signal in time domain into stationary one in angular domain, which can highlight the vibration information related to rotation speed and restrain the unrelated information. Therefore, order tracking is a desirable method to extract gear fault feature in the process of run-up and run-down. Mechanism and Machine Theory 55 (2012) 6776 Corresponding author at: State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, 410082, PR China. Tel.: +86 731 88664008; fax: +86 731 88711911. E-mail address: (J. Cheng). 0094-114X/$ see front matter 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2012.04.008 Contents lists available at SciVerse ScienceDirect Mechanism and Machine Theory journal homepage: On the other hand, while faults occur in gears, the vibration signal picked up in run-up and run-down process always present the characteristics of amplitude-modulated and frequency-modulated (AMFM). In order to extract the modulation feature of gear fault vibration signals, demodulation analysis is one of the most popular methods 4,5. However, conventional demodulation approaches such as Hilbert transform demodulation and traditional envelope analysis have their own limitations 6. These drawbacks include two aspects: (1) in practice most gear fault vibration signals are all multi-component AMFM signals. For these signals, in conventional demodulation approaches, they are usually decomposed into single component AMFM signals by band-pass filter and then demodulated to extract frequencies and amplitudes information. However, both the number of the carrier frequency components and the magnitude of the carrier frequency are hard to be determined in practice, so the selection of central frequency of band-pass filter carries great subjectivity that would bring demodulation error and make it ineffective to extract the characteristic of machinery fault vibration signal; (2) owing to the inevitable window effect of Hilbert transform, when Hilbert transform is used to extract the modulate information, the demodulation results present non- instantaneous response characteristic, that is, at the two ends of the modulated signal which has been demodulated as well as the middle part with break would produce modulation again, which makes the amplitude get fluctuation in an exponential attenuation way, and then the demodulation error would increase 7. In order to overcome the first drawback, an appropriate decomposition method should be looked for to separate multi-component signal into a number of single component AMFM signals before the envelope analysis. Since EMD (Empirical mode decomposition) could adaptively decompose a complicated multi-component signal into a sum of intrinsic mode functions (IMFs) whose instantaneous frequencies have physical significance 8,9, order tracking method based on EMD has been widely used in the gear fault diagnosis 1013. However, there still exist many deficiencies in EMD such as the end effects 14 and modes mixing 15 that are still underway. In addition, after the original signal is decomposed by EMD, the drawback produced by Hilbert transform (above mentioned) is inevitable when IMF is performed envelope analysis by Hilbert transform. Moreover, sometimes the unexplainable negative instantaneous frequency would appear when calculating instantaneous frequency by performing Hilbert transform to each IMF 16. Local mean decomposition (LMD) is a novel demodulation analysis method,which is particularlysuitable for the processing of multi-component amplitude-modulated and frequency-modulated (AMFM) signals 16. By using LMD, any complicated signal can be decomposed into a number of product functions (PFs), each of which is the product of an envelope signal (obtained directly by the decomposition) from which instantaneous amplitude of the PF can be obtained and a purely frequency modulated signal from which a well-defined instantaneous frequency could be calculated. In essence, each PF is exactly a mono-component AMFM signal. Therefore, the procedure of LMD could be, in fact, regarded as the process of demodulation. Modulation informationcan be extracted by performing spectrum analysis to the instantaneous amplitude (envelope signal, obtained directly by the decomposition) of each PF component rather than by performing Hilbert transform to the PF components. Hence, when LMD and EMD are applied to the demodulation analysis respectively, compared with EMD, the prominent advantage of LMD is to avoid the Hilberttransform. In addition, the LMD iterationprocess which uses smoothed local means and local magnitudes avoids the cubic spline approach used in EMD, which maybe bring the envelope errors and influence on the precision of the instantaneous frequency and amplitude.Moreover, compared with EMD the end effect is not obvious in LMD approachbecause of faster algorithm speed and less iterative times 17. Based upon the above analysis, order-tracking analysis and the recent development of demodulation techniques, LMD, are combined and applied to the gear fault diagnosis of various shaft speeds process. Firstly, order tracking technique is used to transformthe gear vibration signals from time domainto angular domain. Secondly,decompose the re-sampling signal of angular domain by LMD, thus s series PF components and corresponding instantaneous amplitudes and instantaneous frequencies can be obtained.Finally,spectrumanalysisiscarriedoutto theinstantaneousamplitudesof the PFcomponentcontainingdominant fault information. The analysis results from the experimental vibration signal show that the proposed method can extract fault feature of the gear effectively and classify working condition accurately. This paper is organized as follows. A theory of the LMD approach is given in Section 2. In Section 3 a gear fault diagnosis approach in which order tracking technique and LMD are combined is put forward and the practice applications of proposed method are demonstrated. In addition, the comparison between LMD-based and EMD-based method is also given in Section 3. Finally, we offer the conclusion in Section 4. 2. LMD analysis method As mentioned above, the nature of LMD is to demodulate AMFM signals. By using LMD a complicated signal can be decomposed into a set of product functions, each of which is the product of an envelope signal and a purely frequency modulated signal.Furthermore,the completedtimefrequencydistributionoftheoriginalsignalcanbeobtained. Foranysignal x(t),it canbe decomposed as follows 16: (1) Determine all local extrema n i of the original signal x(t), and then the mean value m i of two successive extrema n i and n i+1 can be calculated by m i n i n i1 2 1 All mean value m i of two successive extreme are connected by straight lines, and then local mean function m 11 (t) can be formed by using moving averaging to smooth the local means m i . 68 J. Cheng et al. / Mechanism and Machine Theory 55 (2012) 6776 (2) A corresponding envelope estimate a i is given by a i n i n i1 C12 C12 C12 C12 2 2 Similarly, the envelope estimate a i is smoothed in the same way and the corresponding envelope function a 11 (t)is formed. (3) The local mean function m 11 (t) is subtracted from the original signal x(t) and the resulting signal h 11 (t) is given by h 11 txtm 11 t 3 (4) h 11 (t) can be amplitude demodulated by dividing it by envelope function a 11 (t) s 11 th 11 t=a 11 t 4 Ideally, s 11 (t) is a purely frequency modulated signal, namely, the envelope function a 12 (t)ofs 11 (t)shouldsatisfy a 12 (t)=1.Ifa 12 (t)1, then s 11 (t) is regarded as the original signal and the above procedure needs to be repeated until a purely frequency modulated signal s 1n (t) that meets 1s 1n (t)1 is derived. In other words, envelope function a 1(n+1) (t) of the resulting s 1n (t)shouldsatisfya 1(n+1) (t)=1. Therefore h 11 txtm 11 t h 12 s 11 tm 12 t h 1n ts 1 n1 tm 1n t 8 : 5 in which, s 11 th 11 t=a 11 t s 12 th 12 t=a 12 t s 1n th 1n t=a 1n t 8 : 6 where the objective is that lim n a 1n t1 7 In practice, a variation can be determined in advance. If 1a 1(n+1) (t)1+ and 1s 1n (t)1, then iterative process would be stopped. (5) Envelope signal a 1 (t), namely, instantaneous amplitude function, can be derived by multiplying together the successive envelope estimate functions that are acquired during the iterative process described above. a 1 ta 11 ta 12 ta 1n t n q1 a 1q t 8 where q is the times of the iterative process. (6) Multiplying envelope signal a 1 (t) by the purely frequency modulated signal s 1n (t) the first product function PF 1 of the original signal can be obtained. PF 1 ta 1 ts 1n t 9 PF 1 contains the highest frequency oscillations of the original signal. Meantime, it is a mono-component AMFM signal, whose instantaneous amplitude is exactly the envelope signal a 1 (t) and instantaneous frequency is defined from the purely frequency modulated signal s 1n (t)as f 1 t 1 2 d arccos s 1n tC138 dt 10 (7) Subtract the first PF component PF 1 (t) from the original signal x(t) and we have a new signal u 1 (t), which becomes the new originalsignalandthewholeoftheaboveprocedureisrepeated,i.e.uptoktimes,untilu k becomesmonotonicfunction u 1 txtPF 1 t u 2 tu 1 tPF 2 t u k tu k1 tPF k t 8 : 11 69J. Cheng et al. / Mechanism and Machine Theory 55 (2012) 6776 Thus, the original signal x(t) was decomposed into k-product and a monotonic function u k xt X k p1 PF p t u k t 12 where p is the number of the product function. Furthermore, the corresponding complete timefrequency distribution could be obtained by assembling the instantaneous amplitude and instantaneous frequency of all PF components. 3. The gear fault diagnosis method based on order tracking technique and LMD 3.1. Order tracking analysis and the corresponding fault diagnosis method Order-trackingtechniquecouldtransformanon-stationarysignalintimedomainintoastationarysignalinangulardomainby applyingequi-angularre-samplingto vibrationsignalwithreference toshaftspeed.Furthermore, orderspectrumcan beobtained by using spectrum analysis to stationary signal in angular domain, thus the information related to rotation speed can be highlighted and the unrelated one could be restrained. Therefore, order-tracking is suitable for the vibration signal analysis of rotation machine. There are three popular techniques for producing synchronously sampled data: a traditional hardware solution, computed order tracking (COT) and order tracking based on estimation of instantaneous frequency 1820. The traditional hardware approach, which uses specialized hardware to dynamically adapt the sample rate, is only suitable for the case that rotating speed of shaft is relatively smooth, thus resulting to a high cost. The method of order tracking based on estimation of instantaneous frequency has no need for specialized hardware and thus cost is relatively low, however, it has failed to analyze multiple component signal. While in practice most gear fault vibration signals exactly present the characteristic of multi-component. Therefore, this technique has little practice significance. COT technique realized equi-angular re-sampling by software, therefore it not onlyrequires nospecialized hardware,but alsohave nolimitation for analysissignal thatmeans it is more flexibleand more accurate. Just for this reason, COT is introduced into the gear fault detection in this paper. The step of the gear fault diagnosis method based on order tracking technique and LMD can be listed as follows: (1) The vibration signals and a tachometer signal are asynchronously sampled, that is, they are sampled conventionally at equal time incrementst; (2) Calculate the time series t i corresponding to equi-angular increments by tachometer signals; (3) According to the time series t i , apply interpolation to the vibration signals, thus the synchronous sampling signal, namely, stationary signal in angular domain, can be obtained; (4) Use LMD to decompose the equi-angular re-sampling signal, thus s series PF componentsand corresponding instantaneous amplitudes and instantaneous frequencies can be acquired; (5) Apply spectrum analysis to the instantaneous amplitude of each PF component, and then we have the order spectrum. 3.2. Application Since the gear fault vibration signal in run-up and run-down process are always multiple component AMFM signals and fault feature frequency would vary with rotation speed, the fault diagnosis method in which order tracking technique and LMD are combined would be suitable for gear fault detection. To verify the effectiveness of the proposed method, the fault diagnosis method based on order tracking technique and LMD was applied to the experimental gear vibration signals analysis. An experiment has been carried out on the rotating machinery test rig that is used for modeling different gear faults 21. Here we consider three working conditions that are gear with normal condition, with cracked tooth and with broken tooth. Standard gears with teeth number z=55 and z=75 are used on input and output shafts respectively, in which the crack fault is introduced into the gear on the input shaft by cutting slot with laser in the root of tooth, and the width of the slot is 0.15 mm, as well as its depth is 0.3 mm. Therefore, the mesh order is x m =55 and the fault feature order is x c =1.Figs. 1 and 2 give the rotation speed signal r(t) picked up by a tachometer and vibration acceleration signal s(t) of the gear with crack fault collected by a piezoelectric acceleration sensor respectively, in which the sample frequency is 8192 Hz and total sample time is 20 s, and from which we know the speed of input shaft increased gradually from 150 rpm to 1410 rpm, then decreased to 820 rpm. Meantime, the amplitude of vibration acceleration signal accordingly changed, from which a section of signal s 1 (t)of5s7 s in the run-up progress is intercepted for further analysis. Fig. 3 gives the spectrum of s 1 (t)by applying spectrum analysis directly to vibration signal. For the rotation speed changes with time, the frequency mixing arises. Therefore, it is impossible to find meshing frequency and fault feature frequency in Fig. 3. As a result, actual gear working condition cannot be identified. Replace direct spectrum analysis by the order tracking method. Firstly, assume sample point per rotation is 400, namely,the maximum analysis orderis 200. Secondly,angular domainsignal j 1 () shown in Fig. 4 can be obtained by performing order re-sampling to s 1 (t), in which horizontal ordinate has changed from time to radian. Thirdly, the corresponding order spectrum of j 1 () can be calculated that is illustrated in Fig. 5, from which we can find obvious spectral peak 70 J. Cheng et al. / Mechanism and Machine Theory 55 (2012) 6776 values at order O=55 and O=110 corresponding to gear meshing order and the double. Thus it means that frequency aliasing phenomenon has been eliminated to a large degree. However, j 1 () is still a multiple component MAMF signal. Therefore, side frequency band reflecting fault feature frequency is indistinct. To extract fault characteristic effectively, apply LMD to j 1 (), thus sevenPF componentsand aresiduecan beobtainedshowninFig.6, whichmeansLMDis a demodulation progress. Therefore,it is possible to extract gear fault feature by utilizing spectrum analysis to the instantaneous amplitude of PF component containing dominant fault information. By analysis, we know that the main failure information is included in the first PF component. Therefore, Figs. 7 and 8 give instantaneous amplitude a 1 () of the first PF component PF 1 () and the corresponding order spectrum of a 1 (), from which it is clear that there are distinct spectral peak value at the 1st order (O=1) corresponding to gear fault feature order x c , which accords with the actual working condition of the gear. Figs. 9 and 10 show the rotation speed signal n(t) and the time domain waveform of vibration acceleration signal s(t) of the gear with broken tooth respectively, in which the sample rate is 8192 Hz and total sample time is 20 s. The broken tooth fault is introduced into the gear on the input shaft by cutting slot with laser in the root of tooth. Firstly, a section of signal s 1 (t)of5s7s in the run-up progress is intercepted for further analysis; secondly, assume sample point per rotation is 400; thirdly, angular domain signal j 1 () shown in Fig. 11 can be obtained by performing order re-sampling to s 1 (t); fourthly, apply LMD to j 1 (); finally, the corresponding order spectrum shown in Fig. 12 of instantaneous amplitude of t