Statistical Index Development from Time Domain for Rolling Element Bearings

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1 Iteratioal Joural of Performability Egieerig Vol.0, No.3, May 04, pp RAMS Cosultats Prited i Idia Statistical Idex Developmet from Time Domai for Rollig Elemet Bearigs YUAN FUQING * ad UDAY KUMAR Divisio of Operatio ad Maiteace, Luleå Uiversity of Techology, SE Lulea, SWEDEN (Received o November 30, 03, revised o March 9, ad March 30, 04) Abstract: Feature extractio is crucial to efficietly diagose fault. This paper discusses a umber of time-domai statistical features, icludig Kurtosis or the Crest Factor, the Mea by Deviatio Ratio (MDR), ad Symbolized Sequece Shao Etropy (SSSE). The SSSE reflects the spatial distributio of the sigal which is complemetary with the statistical features. A ew feature, Normalized Normal Negative Likelihood (NNNL), is used to improve the Normal Negative Likelihood (NNL). A Separatio Idex (SI) called the Exteded SI (ESI) evaluates the performace of each feature ad to remove oise feature. The Multi-Class Support Vector Machie (MSVM) recogizes bearig defect patters. A umerical case is preseted to demostrate these features, their feature subset selectio method ad the patter recogitio method. The MSVM is used to detect three differet types of bearig defects: defects i the ier race, outer race ad bearig ball. Keywords: Fault diagosis, time domai, feature extractio, exteded separatio idex (ESI); multi-class support vector machie (MSVM).. Itroductio Rollig elemet bearigs are critical compoets i idustrial machiery ad thus play a importat role i idustry []. Because uexpected failures i the bearigs lead to costly dowtime [], much research has cosidered their prevetio. Fault diagosis aims to fid the icipiet failure so that the catastrophic failure ca be preveted. Withi this paper, fault diagosis refers to determiig the state of the system (faulty or ormal) as well as the type of faults [3]. The fault diagosis of rollig elemet bearigs is geerally doe maually based o practical experiece. This is a ituitive ad subjective method; therefore, whe umerous patters are ivolved, this method is iefficiet. More recetly, patter recogitio has bee used to diagose fault automatically. There are four steps i automatic fault diagosis: sigal acquisitio, feature extractio, feature selectio ad patter recogitio. Sigal acquisitio istalls sesors ad collects the sigal for aalysis. Feature extractio is to extract the umerical measuremets of the sigal that is more iterpretable tha the raw sigal [4]. Feature extractio geerates measuremets, ad feature selectio chooses a subset of optimal features based o those measuremets. Fially, patter recogitio techiques such as classificatio or cluster algorithms ca be used to discrimiate amog differet bearig states, for example discrimiatig failure patters from ormal patters [4]. Sigal acquisitio i the case of the rollig elemet bearig focuses o collectig the vibratio sigal. The presece of a defect i a bearig produces impulses from the cotact betwee metal surfaces whe the bearig is ruig. These impulses are periodically produced with a frequecy determied by the locatio of the defect ad its size [5]. Correspodig author s xiaohagji@yahoo.com 33

2 34 Yua Fuqig, ad Uday Kumar Accelerometer sesors ca be mouted o the bearig house to measure the vibratio sigal which cotais the defect iformatio. As the vibratio sigal ca be collected olie, by aalyzig this sigal the status of the bearig, e.g., the locatio of the defect ad possibly the severity of the problem, ca be diagosed without stoppig the machie. This is the advatage of vibratio aalysis. Feature extractio is very importat as the raw vibratio sigal is ormally too chaotic to fid defect directly. A feature ca be cosidered to represet the sigal ad ca therefore be used as the idicator of the bearig state. Features ca be extracted either from time domais, such as Kurtosis, Crest Factor, etc., or from frequecy or timefrequecy domais. Most aalyses of vibratio sigals use the frequecy domai or timefrequecy domais. Nevertheless, the use of time domais has bee argued to be computatioally efficiet [6], uderstadable ad more easily implemeted. Additioally, as argued by Tao et al., the time domai feature is less sesitive to bearig load ad speed variace [7]. Fially, i the early stages of the defect, the frequecy features are ot sigificat, ad frequecy domai aalysis is iefficiet. Extractig a large umber of features may provide more iformatio o the sigal, but it adversely affects the computatioal cost iversely ad could degrade the performace of diagosis. I feature selectio, a set of optimal features are chose. State-of-the-art feature selectio methods are idividual feature selectio ad subset feature selectio. The Separatio Idex (SI) ca be used to measure the classifiability of a idividual feature or a subset feature ad thus ca be used to perform feature selectio [8, 9]. For example, Zio et al. propose a SI called the classifiability evaluatio fuctio to select a subset of features. Qiu et al. propose a method based o the geometry distace. Kim et al. develop oe SI to evaluate the sigificace of a idividual feature. This paper exteds Kim s SI [0] by improvig the statistical property ad usig it to remove oise feature. This paper discusses state-of the-art time domai features ad proposes some ew oes to represet vibratio sigal more accurately; these iclude features adopted from aother field ad improvemets o existig oes. A separatio idex is proposed to remove oise feature, ad a commoly used subset selectio method chooses the optimal subset of features. The optimal subset features are used as Support Vector Machie (SVM) iput to diagose fault. The multi-class SVM is selected to discrimiate fault patters, as the SVM ca be liear or oliear depedig o the kerel fuctio chose. Sectio of the paper discusses state-of-the-art time domai features ad the proposed ew feature. Sectio 3 presets a feature selectio method. Sectio 4 discusses the multiclass SVM as a classifier of time domai features for diagosis ad progosis purposes. Sectio 5 presets a umerical example usig public test-rig data. Sectio 6 discusses the fidigs.. Feature Extractio from Time Domai. State-of-the-art Features i the Time Domai I the early stage of fault developmet, the bearig is ot sigificatly damaged ad the defective sigal is masked by the oise. As the periodicity of the occurrece of the sigal is ot sigificat, the spectral aalysis is ieffective. Eve whe the periodicity is sigificat, usig the time domai feature is still recommeded because ormal ad defective sigals differ i their statistical characteristics. Kurtosis is a importat ad popular feature used i rollig elemet machies. It defies the peakedess of the amplitude of the sigal. Beta parameters are the shape ad scale parameters i the Beta distributio whe the amplitude of the sigal is assumed to follow a Beta distributio. This is a flexible distributio ad most sigals ca fit it. Sice

3 Statistical Idex Developmet from Time Domai for Rollig Elemet Bearigs 35 the parameters i Beta distributio for a ormal vibratio sigal (bearigs without defects) ad a defective sigal (bearigs with defects) differ, they ca be used to differetiate betwee types of defects []. However, some critics, such as Heg ad Nor, argue that the Beta method has o sigificat advatage over usig the Kurtosis ad Crest factor for rollig elemet bearigs []. The Kurtosis, Crest ad Impulse factors are o-dimesioal features ad are idepedet of the magitude of the sigal power. RMS, Peak value, stadard deviatio, ad Normal Negative Likelihood (NNL) value are fully depedat o the sigal power. Some uisace factors such as the quality of the sesors ad the locatio where they are mouted ca ifluece the power of the sigal. The mai advatage of o-dimesioal features is that they are more immue from uisace factors tha dimesioal features. RMS is a importat feature i sigal processig. It measures the power of the sigal ad ca be used to ormalize the sigal. Therefore, some features are ormalized by RMS. Certai other features used i the past are ormalized by RMS, as for example, Beta- Kurtosis [], Weibull egative likelihood value [3], Kurtosis Ratio [4], etc. They are ot discussed here as the focus is o commoly used time domai features.. Normalized NNL Normal Negative Likelihood (NNL) has bee used by some researchers to diagose fault [3]. I NNL, the amplitudes of the sigal are assumed to follow Normal distributio. The parameters u adσ are calculated usig the maximum likelihood estimator method. This paper proves that the performace of NNL is equivalet to a much simpler feature. Let the amplitudes of the sigal deoted by a series x, x i,.. x discretely. Whe parameters u adσ are ukow, the egative likelihood fuctio of this series is: ( ) / x µ i= i f ( x, x,..., x µ, σ ) = f ( x µ, σ ) = ( ) exp( ) () i i= πσ σ The maximum likelihood estimator of u adσ is: u = x = x i / ad x i = ( i ˆ µ ) ˆ σ = () i= Substitutig Formula () ito Formula () ad simplifyig it, the followig equatio is obtaied: / f ( x, x,..., x µ, σ ) = ( ) exp( / ). σ (3) π Thus the egative likelihood is: / L = l ( ) exp( / ) lσ (4) π It ca be cocluded from the above that NNL is essetially equivalet to l σ ad obviously, lσ is ot o-dimesioal. I order to make the feature idepedet of power, as Kurtosis does, we ormalize it by usig RMS i the followig way: σ NNNL = l (5) RMS This ew feature is called Normalized Normal Negative Likelihood Value (NNNL). Essetially, the old NNL is ot a stable feature as it o-ecessarily depeds o the umber of sample size as show i Formula (4). For a sceario where two sigals are idetical but differ i legth, the NNL values will differ. This is evidetly ot reasoable

4 36 Yua Fuqig, ad Uday Kumar ad is the major disadvatage of this feature. The ew feature, NNNL, is ot oly idepedet of the sample size but idepedet of the sigal power..3 Mea Deviatio Ratio The distributio of amplitude i the sigal samples differs from ormal ad defective rollig elemet bearig sigal. The ormal sigal without defect is comprised by some oise sigals ad the shape of the sigal thus teds to be peak. The distributio of defective sigal has more wide amplitude so the variace is bigger tha that of ormal. It ca be show from Figure. The left figure i Figure is from ormal sigal of a bearig ad the other oe from defective sigal from the same bearig. It is evidet that the defective sigal differs from ormal sigal ad it has wider variace. Therefore, it is straightforward to be remided that the Mea ad Deviatio Ratio (MDR) could be a feature to discrimiate both defective ad ormal coditio sigals. The defiitio of MDR is: xi (6) i= MDR = ( xi x) i= Obviously, MDR is also a o-dimesioal feature idepedet of sigal power. MDR implies the degree of scatter for the distributio of sigal amplitude. Figure : Normal ad Defective Sigal.4 Symbolized Sequece Shao Etropy Most statistical features cosider statistical characteristics of the amplitude distributio; however, i all of these features, the iformatio o the spacious patter of the amplitude is lost. For example, for rollig elemet bearig, whe defect exists, the amplitude teds to be periodic ad however this periodicity caot be reflected i the statistical features. Figure shows a simple example to verify this argumet. This figure is comprised by 00 samples. The amplitude of each sample is comprised by {,,3,4} ad each value has idetical probability of appearace. The upper figure ad the lower figure i Figure are plotted by same samples, but with differet spacious distributio. I the upper figure, the sigal is periodic where the amplitude is distributed determiistically with a sequece of 34, 34, iteratively. I the lower figure, the amplitudes are radomly distributed.

5 Statistical Idex Developmet from Time Domai for Rollig Elemet Bearigs 37 Figure : Periodic Sigal ad Radom Sigal From the Figure, it is evidet that both sigals are differet. However their statistical features are the same, i.e., the statistical features are ot able to discrimiate them. The Shao etropy has bee kow as a parameter capable to measure the ucertaity of a radom process. Rollig elemet bearig without defect teds to geerate a more radom sigal, while the machie with existig defect usually teds to have more determiistic sigal, i.e., their Shao etropy will be differet. To extract the periodicity i the sigal, a feature amed Symbolized Sequece Shao Etropy (SSSE) is used. I this feature, the sigal is firstly symbolized ad the the Shao etropy is used. This SSSE has bee used to detect weak sigal i other research fields [5,6]. This paper uses the SSSE to diagose fault for vibratio sigal for rollig elemet bearig. The procedure for SSSE calculatio is:. Discretize the sigal. A threshold is predefied. The amplitude below the threshold is coded as 0 ad the above is coded as. Thus the sigal is discretized ito a biary sequece, which is deoted by b b, b,... b,..., 3 i. Segmet the biary sigal with equal legth L. For example, segmet the biary sequece ito 0, 00, 00 with legth L=3. Calculate the decimal value of each segmet. It is 6,, i this example. 3. Calculate the probability of each segmet. The probability is cosidered as the frequecy. For 6 i this example, it is /3 ad for it is /3. 4. Calculate the etropy usig the followig Shao etropy formula: H = p i log p i (7) log N i where, N is the total umber of uique segmeted biary sequece. The pi is the probability of the i th kid of the uique sequece. I a periodic sigal, some sequeces will occur frequetly so the Shao etropy will be lower. Therefore, the Shao etropy values vary with differet acquired data so it ca be used as feature to measure the characteristics of a sigal. For a pure radom data, the Shao etropy value is. For determiistic sigal, the etropy is betwee 0 ad [5]. The more determiistic is the sigal, the lower its SSSE value. Usig the above procedure, for the example of Figure, the SSSE of periodic sigal (upper i Figure ) is 0 ad that of the radom sigal is (lower i Figure ). These two sigals ca be sigificatly discrimiated. Similarly differet defect i rollig elemet bearig ca be discrimiated attedig to the radomess existig i the acquired sigals.

6 38 Yua Fuqig, ad Uday Kumar 3. Feature Selectio 3. Noise Feature Removal usig Separatio Idex This paper extracts several features from time domai. Some features may ot cotribute to the fault diagosis ad eve degrade the performace of the diagosis. I order to remove these o-sigificat features or oise features, a idex amed Separatio Idex is used to defie the sigificace of features [3]. For the sake of simplicity, each feature value is called a sample i this sectio. If two sigals are preseted to be compared, let m d ad mh deotes the mea of samples, Sd ad S h deotes stadard deviatio. Kim et al. developed oe Separatio Idex (SI) as [0] SI m S d d m + S h h = (8) This separatio idex is used defiig the separability of features. This paper has improved this SI to gai a better statistical property. The ew SI, which is called Exteded SI (ESI), as: SI m d S m + d S h = (9) Assume the samples are ormal distributio ad the umber of sample size from each sigal is equal with. The quatity SI ca be test its sigificace usig t-test. That is, whe > t (0) SI, the md ad degree of freedom. The v i Iequality (0) is: ( + u) ( ) v = () + u where σ v h mh has sigificatly differece, where σ is sigificace level ad v is u = S d S. The statistical proof of this ca be foud i referece [7]. / h The t-test proposed above assumes that the sample size is equal from both sides. For a problem with uequal sample size, oe ca use the same Separatio Idex but the t-test is differet from Formula (). Oe ca refer to the referece [7] for more detail for the uequal sample size t-test. The sigificace of the differece betwee two sigals the has more statistical foudatio by usig ESI. This is the advatage of the ew proposed separator idex. This ESI ca be used to remove oise feature. For example, whe the feature values from two sigals are tested sigificatly differet, the feature will be retaied; otherwise they should be excluded from further cosideratio.

7 Statistical Idex Developmet from Time Domai for Rollig Elemet Bearigs Feature Subset Selectio The ESI ca be used to remove some oise feature i the iitial step of feature selectio. Feature subset selectio is to select the compact optimal feature set. The feature subset selectio is ecessary as features are possibly correlated or redudat. There are umerous methods available to perform subset selectio [8] however the simplest oe is the exhaustive method. This method eumerates all the subsets ad selects the oe with highest performace, for example the highest fault diagosis accuracy, as the optimal. Noise Feature Removal usig ESI Select oe from the subsets Perform diagosis usig SVM All subset selected? Y Select the subset with highest diagosis accuracy as optimal N Ed Figure 3: Feature Selectio Procedure This exhaustive method is computatioal cost, as for a umber of features, there have as much as subsets, therefore, the exhaustive method is oly suitable for small umber of features. The advatage of the exhaustive method is a global optimal solutio ca be obtaied. The detailed procedure of performig feature selectio i this paper is show i Figure Patter Recogitio usig SVM The stadard SVM is a biary classifier classifyig two classes of objects. To accommodate the multi-classes problems, oe has to exted the stadard SVM. Oe solutio is to combie several biary SVMs together. Oe-Agaist-All multi-class SVM is oe of them. The Oe-Agaist-All method trasforms the k-class problem ito a k sub biary classificatio problem. The i th sub biary classificatio problem labels the idicator of data sets belog to the i th class with ad label all the remaiig data sets with - [9]. SVM is a kerel method which depeds o the kerel fuctio [0, ]. Before oe uses the SVM for patter recogitio, oe should select a proper kerel fuctio. The discussio o kerel fuctio is omitted as it is ot the cocer of this paper. The motivatio to select the SVM for fault diagosis is because the decisio fuctio is flexible as it ca be liear or oliear depedig o the kerel fuctio. 5. Numerical Case 5. Data Descriptio This umerical case uses the public bearig data which is collected from a test rig i Case Wester Reverse Lab []. I this rig two bearigs are istalled which are located at the

8 30 Yua Fuqig, ad Uday Kumar ed of the driver ad fa respectively. Artificial defect are itroduced to the ier race, outer race ad ball i the bearigs. The vibratio data is collected by accelerometers attached to the housig with magetic bases. The data usig i this case is from the bearig located i the driver ed which is a sigal sampled at khz as show i Figure Performace of Idividual Feature Figure 4: Normal ad Ball Defect Sigal The raw sigal is divided ito several o-overlappig segmets usig a fixed widow size. The widow size used i the case study is 3000 for each segmet which is more tha the miimal requiremet w le = 96 suggested by followig Iequality [4]: 4. fs w le f () where f s BDF deotes the samplig frequecy ad f BDF deotes the fault frequecy, for example the ball pass frequecy o the ier race. The sigals used are a ormal bearig sigal ad ball defective sigal show i Figure 4. Features extracted from time domai are: MDR, SSSE, NNNL, Kurtosis, Crest Factor, Clear Factor, Impulse Factor, Shape factor. All these features are o-dimesioal features. By usig the t-test described i Sectio 3., the feature SSSE is sigificatly equal for the two sigals so it is removed for cosideratio i this compariso. The ESI value is computed by Eq. (9) ad is compared with the accuracy obtaied by simplest liear polyomial fuctio SVM, as show i Table, where the 0.8, 0.6, 0.5 meas the ratio selected as traiig data, the remaiig data sets are test datasets. The selectio of the data sets is radom ad each ratio is repeated 30 times. The umber listed i Table is the mea of the 30 rus. Table : Accuracy usig Sigle Feature Features Kurtosis Shape Clear NNNL MDR Impulse Crest SI Accuracy(%) for Ratio Accuracy(%) for Ratio Accuracy(%) for Ratio As show i Table, the higher ESI implies the accuracy teds to be higher. The Kurtosis has the highest ESI, so the accuracy is also the highest. The ESI of the Crest is the lowest, the correspodig accuracy also lowest. This aligmet implies the efficiecy of ESI as a feature performace idicator.

9 Statistical Idex Developmet from Time Domai for Rollig Elemet Bearigs Fault Diagosis usig MSVM There are three types of defects itroduced i the ier race, out race, ball of the bearig. Therefore, icludig ormal state (bearig without defect), four patters are eeded to be discrimiated: ormal, ier race defect, outer racer defect, ad ball bearig defect. Multi- SVM is used to recogize these patters. Segmet the sigal obtaied from accelerometer sesors ad use the t-test proposed i Sectio 3. to filter oise feature. The test results show that except feature SSSE, all the other features are sigificatly differece. The SSSE fails to pass the t-test betwee ormal ad ball defect. However, the SSSE is sigificat differet betwee other sigals. Therefore this feature is still kept i the feature selectio. As a cosequece, the iput of SVM is the vector of feature values: MDR, NNNL, Crest Factor, Clear Factor, Impulse Factor, Shape factor, Kurtosis, SSSE. The output of the SVM is the status of the bearig: Normal, Ier Race defect, Ball defect, Outer race defect Patter Recogitio usig MSVM The polyomial fuctio as writte follows is selected as kerel fuctio : ' ' d K ( x, x ) = ( < x, x > + ) (3) The parameter d above is a predefied parameter. Whe d=, the SVM is a liear classifier; Whe d > it is oliear. The higher of d, the more flexible of the SVM classifies the data; whereas a too flexible classifier is proe to occur overfittig pheomeo [3], which meas a classifier has low trai error but have a high predictio error. The Figure 5 shows the overfittig pheomeo. I this figure, the traiig accuracy is always icreasig with d. This meas icreasig the order ca always improve the traiig accuracy as the classifier becomes more flexible. After d>4, the traiig accuracy reaches 00%. However, the test accuracy is ot icreasig with d, iversely it deceases with the order whe d 4. The higher order d turs out to have lower test accuracy. This is the overfittig pheomeo. The highest test accuracy i the figure is order d=. Therefore, i this case, d= is selected for the polyomial kerel fuctio. Figure 5: Performace of Various Kerel Parameters Divide the data extracted from sigals at ratio 0.8, 0.6, 0.5 respectively ad the first part is used for traiig, the secod for test. To reduce radomess, the divisio of data for traiig ad test are also radom ad each subset has bee ra 30 times. Utilizig the approach metioed i Sectio 3. selects the optimal subset features. The resultig optimal subset ad its diagosis accuracy, that is the mea of the 30 rus, are show i Table.

10 3 Yua Fuqig, ad Uday Kumar Table : Optimal Subsets ad Their Accuracy Features Kurtosis Crest Clear Impulse Shape NNNL MDR SSSE Accuracy% Ratio 0.6 Ratio 0.5 / / 98.7 / / / / / / / / / / Ratio 0.8 / 99.5 Note: deote the feature i the subset, / deote ot i the subset All the optimal subsets iclude feature Kurtosis, SSSE, Impulse ad Shape. These four features are commo features i the subsets havig highest accuracy. The feature MDR is also i the optimal subset of Ratio 0.6, which meas it is a good cadidate feature for time domai fault diagosis. Moreover, the proposed ew NNNL is also importat. The secod row for each ratio i Table list the subsets cotaiig NNNL, which have accuracy very close to the highest accuracy. Noetheless, the Kurtosis ad SSSE are the most icredible i this case. All the statistical features cotai oly the amplitude distributio of the sigal. The iformatio about the spacious distributio of the sigal has lost. This lost ca be proved i the simulated example i Sectio.4. However, the feature SSSE cosiders the spacious distributio iformatio but it loses the amplitude distributio iformatio. I this sese, the SSSE ad the statistical features are complemetary. Therefore, combiig the statistical features ad the SSSE could achieve higher diagosis accuracy, which ca be verified from the results i Table. 6. Coclusio The fault diagosis o rollig elemet bearig is mostly o frequecy domai. This paper proposes a approach o time domai. The feature o time domai is less depedet o the machie load ad rotatio speed, ad whe the defect is i its early stage, the defect frequecy is isigificat ad diagosis o time domai is ecessary. The umerical case shows the diagosis o time domai is feasible. The feature SSSE shows icredible i the umerical case thus it shows its effectiveess used for fault diagosis. The ew feature NNNL has also show its importace as usig this feature the diagosis accuracy is high as well. The proposed MDR is i the optimal subset so it is a feature of importace for the diagosis. The SSSE is complemetary with the other statistical features ad thus combiig them ca have a good fault diagosis result. This ca be verified from the bearig case. The proposed Exteded Separatio Idex shows its efficiecy i the bearig case. I the bearig case, the higher ESI implies the higher diagosis accuracy. Moreover, the MSVM used i this paper shows its flexibility by adaptig itself to data. By tuig the order i the Polyomial kerel fuctio, the SVM ca be liear or oliear. I the bearig case, it fially uses the simplest liear Polyomial kerel fuctio that shows better performace tha the more complex oliear Polyomial kerel fuctio.

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12 34 Yua Fuqig, ad Uday Kumar Yua Fuqig obtaied his M.Tech. i System Egieerig at Beijig Uiversity of Aeroautics ad Astroautics, Chia, i the year 007. He joied the Divisio of Operatio ad Maiteace Egieerig, Luleå Uiversity of Techology, Swede, i September 007 to work for the degree of Ph.D. His area of research deals with reliability data aalysis ad statistical learig theory. Uday Kumar obtaied his B.Tech. i Idia durig the year 979. After workig for 6 years i Idia miig compaies, he joied the postgraduate programme of Luleå Uiversity of Techology, Luleå, Swede, ad obtaied the degree of PhD i the field of Reliability ad Maiteace durig 990. Presetly, he is Professor of Operatio ad Maiteace Egieerig at Luleå Uiversity of Techology, Luleå, Swede. His research iterests are equipmet maiteace, equipmet selectio, reliability ad maitaiability aalysis, system aalysis, etc. He has published more tha 70 papers i iteratioal jourals ad coferece proceedigs.