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Identifyingdifferent crack levels, especially for early cracks is a challenge in gear fault diagnosis.This paper aims to propose a method to classify the different levels of gear cracksautomatically and reliably. In this method, feature parameters in time domain, speciallydesigned for gear damage detection and in frequency domain are extracted tocharacterize the gear conditions. A two-stage feature selection and weighting technique(TFSWT) via Euclidean distance evaluation technique (EDET) is presented and adoptedto select sensitive features and remove fault-unrelated features. A weighted K nearestneighbor (WKNN) classification algorithm is utilized to identify the gear crack levels.The gear crack experiments were conducted and the vibration signals were capturedfrom the gears under different loads and motor speeds. The proposed method is appliedto identifying the gear crack levels and the applied results demonstrate its effectiveness.& 2009 Elsevier Ltd. All rights reserved.1. IntroductionGearboxes are one of the fundamental and most important parts of rotating machinery employed in industries. Theirfunction is to transfer torque and power from one shaft to another. Typical applications include airplanes, automobiles,power turbines, and steel mills. If faults occur in any gears of these machines during operating conditions, seriousconsequences may occur. Therefore, the fault diagnosis of the gearboxes is crucial to prevent the mechanical system frommalfunction that could cause damage or the entire system to halt.Gear faults may be classified as distributed faults (e.g., wear and misalignment) and localized faults (e.g., cracks andchipping). The former may reduce transmission accuracy, and increase the vibration level of rotating machinery. The lattermay not only increase transmission errors, but also cause catastrophic accidents in machines such as airplanes andhelicopters. Furthermore, the distributed faults are usually initiated from the localized faults 1. Therefore, diagnosing thelocalized faults of the gears is more significant and we will focus on the localized fault diagnosis in this paper. Up to nowthe fault diagnosis of the gears has received intensive study and many investigations have been carried out. One of theprincipal tools for diagnosing the gear faults is the vibration-based analysis because of its ease of measurement. It ispossible to obtain vital diagnosis information from the vibration signals through the use of signal processing methodsbased on the vibration signals, which include statistical methods 2, cepstrum estimation 3, time-domain averaging 4,demodulation 5, WignerViller distribution 6, wavelet transform 7, independent or principal component analysis8,9, cyclostationarity analysis 10, and empirical mode decomposition 11. Among these methods, the statisticalContents lists available at ScienceDirectjournal homepage: Systems and Signal ProcessingARTICLE IN PRESS0888-3270/$-see front matter & 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.ymssp.2009.01.009?Corresponding author.E-mail address: ming.zuoualberta.ca (M.J. Zuo).Mechanical Systems and Signal Processing 23 (2009) 15351547methods using root mean square, kurtosis, crest factor, etc., have been proved to be relatively simple but effective in thefault diagnosis of the gears and also widely reported in the literature 1215. The reported results show different statisticalfeatures display different sensitivity to fault advancements. Thus if multiple statistical features are combined to diagnosethe gear faults, more accurate results can be obtained. However, when multiple features are applied, a diagnostician canhardly pay attention to all the features and deal with the contradictive features. Especially for an early fault, difficulty maybe encountered in diagnosing it only using visual observation shown by the statistical features for the diagnostician. Thealternative is to adopt an automated diagnosis scheme.Multiple statistical features when combined with pattern recognition techniques is an effective solution to overcomethe above difficulty and is able to provide an automated, convenient and reliable fault diagnosis method for the gear faults.Samanta used genetic algorithm to select optimal features from the statistical features of both the raw and preprocessedvibration signals, and adopted artificial neural networks and support vector machines for gear fault detection 12. Kanget al. implemented gear fault category identification of tooth breakage and wear by using Bayesian networks and the time-domain statistical parameters of vibration signals 13. Abumahfouz presented a gear fault diagnosis method which appliedthe time-domain statistical features and the neural networks for the classification of the worn and missing tooth 14. Lai etal. utilized the radial basis function network based classifier and the high-order cumulants of vibration signals toidentify gear spalling and worn teeth 15. Refs. 1215 presented the automated and effective methods for gear damagedetection and fault categories classification. In gear fault diagnosis, however, the damage level identification is moredifficult than the damage detection and category classification. Few papers reported this research topic about the levelidentification of gear damage. Oztu rk et al. presented a method in which a scalogram and its mean frequency variationwere used to detect and recognize the pitting levels in gears 16. Loutridis utilized instantaneous energy density and localscaling exponent algorithm to detect the gear crack and identify the crack levels effectively 17,18. However, the abovemethods require the expertise of a diagnostician to apply them successfully and can not distinguish the damage levels ofgears automatically.To approach this challenge of the level identification for the gear faults, the objective of this paper is to develop a faultdiagnosis method to automatically and accurately identify the levels of the gear cracks which is one of the faultmodes most frequently occurring in gears. Not only time- and frequency-domain statistical features but also featureparameters specially designed for gear damage detection are adopted in this paper to improve the diagnosis accuracy of thegear cracks. Unfortunately, too many features will have large dimensionality, which may increase the computationalburden of a subsequent classifier, and degrade the generalization capability of the classifier. Thus, to overcome theseshortcomings, a few features obviously characterizing the gear conditions need to be selected from all the features. Here, asimple but reliable two-stage feature selection and weighting technique (TFSWT) via Euclidean distance evaluationtechnique (EDET) is developed, and the first stage feature selection is utilized to select sensitive features closely related tothe gear faults.The K nearest neighbor (KNN) algorithm 19,20, as a pattern recognition technique, has been proved to be simpler andmore stable than neural networks, classification trees, etc., and has good classification performance on a wide range of real-world data sets 21. It has been studied extensively and used successfully in many pattern recognition applications. But theKNN algorithm faces a serious problem when samples of different classes overlap in some regions in the feature space. Tosolve this problem, the second stage feature weighting of TFSWT is used to improve the performance of the KNN algorithm.The improved KNN is referred as the weighted K nearest neighbor (WKNN) algorithm in this paper.In view of the above analysis, a new method for gear crack level identification is presented in this paper. The method iscreated by adopting TFSWT based on EDET to select sensitive features and compute feature weights, and using WKNN toidentify the crack levels. In comparison with the existing methods reported in pattern recognition applications 2224, theproposed method harnesses the merits that the computation of feature weights is simpler and the weights are easier to beunderstood. Gear experiments on a test rig were carried out to test the performance of the proposed method. Vibrationsignals were measured from the gears under various loads and speeds as well as different crack levels. The diagnosis resultsvalidate that the method is able to recognize the gear crack levels effectively.2. Experimental setup and data acquisitionFig. 1(a) shows the experimental system used in this paper to verify the performance of the proposed method. Thediagram of the system is displayed in Fig.1(b). The system includes a gearbox, a 3-hp ac motor for driving the gearbox, anda magnetic brake for loading. The motor rotating speed is controlled by a speed controller, which allows the tested gear tooperate under various speeds. The load is provided by the magnetic brake connected to the output shaft and the torque canbe adjusted by a brake controller. As shown in Fig.1(b), the gearbox is driven by the motor through a timing belt and thereare three shafts inside the gearbox, which are mounted to the gearbox housing by rolling element bearings. Gear 1 on shaft1 has 48 teeth and meshes with gear 2 with 16 teeth. Gear 3 on shaft 2 has 24 teeth and meshes with gear 4, which is on theoutput shaft (shaft 3) and has 40 teeth. Gear 3 is the tested gear.Crack is a very common fault mode studied in gear fault diagnosis. For this reason, crack faults are simulated in ourgearbox experiments. Letabe the crack angle, a one half of the chordal tooth thickness, and b the face width, as shown inFig. 2. Because the thinnest knife of the machine tools in our lab is 0.4mm, the crack thickness is 0.4mm in theARTICLE IN PRESSY. Lei, M.J. Zuo / Mechanical Systems and Signal Processing 23 (2009) 153515471536experiments. The gears with different crack levels are summarized in Table 1. As a result, three gears with differentconditions (F0F2) including one normal gear and two faulty gears are tested in the experiment. These faulty gears areshown in Fig. 3.ARTICLE IN PRESS#1#2#3#4BrakeMotorTested gear Shaft 1Shaft 2Shaft 3Speed controller Brake controllerTiming beltGearboxLaptop Accelerometers Gearbox system Siglab analyzer Fig. 1. (a) Experimental system, (b) the diagram of the system.Face width, bChordal tooth thickness, 2aCrack angleCrackFig. 2. Crack angle, face width and chordal tooth thickness of a gear.Table 1Geometry of the crack faults.Crack fault modeGeometry of faultDepth (mm)Width (mm)Thickness (mm)Crack angleF0000F1(1/4)a(1/4)b0.4451F2(1/2)a(1/2)b0.4451Fig. 3. Different level cracks in the gears.Y. Lei, M.J. Zuo / Mechanical Systems and Signal Processing 23 (2009) 153515471537The vibration was measured for each of the three gears using two acceleration sensors, which were produced by PCBElectronics with the model number 352C67. They were mounted on the gearbox casing in both the vertical and horizontaldirections. A DSP Siglab analyzer 2042 and a laptop with the data acquisition software were used to collect the vibrationdata for further processing. The speed of the driving motor and the load of the magnetic brake were varied to simulate thegeneral gearbox operating conditions. The vibration data were acquired under three different loads and four differentmotor speeds from 1200 to 1800rpm with an increment of 200rpm. The three different load levels are labeled as loads 0,1and 2, respectively. Load 0 denotes that there was no load applied to the gears. Load 1 denotes that half the maximum loadwas applied to the gears and load 2 denotes that the maximum load was applied to the gears. The maximum load wascalculated when the maximum stress was less than the allowable stress. The maximum loads and the meshing frequenciesare summarized in Table 2. The sampling frequency is 5120Hz and sampling points 8192. Because the vibration signals ofthe vertical direction were more sensitive to the crack levels, they were considered and analyzed in this paper. Under anidentical operating condition, two data samples were collected. Therefore, 24 data samples were obtained for each cracklevel and there are altogether 72 data samples for F0F2.3. Feature extraction, selection and weighting3.1. Feature definition and extractionIn this work, 25 statistical feature parameters are extracted and used to recognize the gear conditions. They can bedivided into three parts: time-domain feature parameters, feature parameters specially developed for gear damagedetection, and frequency-domain feature parameters.(1) The first part includes ten time-domain feature parameters commonly used in literature 1215. They are mean,standard deviation, root mean square, peak, skewness, kurtosis, crest factor, clearance factor, shape factor, and impulsefactor. The definitions of these features can be found in Refs. 1215.(2) The second part contains 11 statistical feature parameters which were specially developed to serve for gear damagedetection and presented frequently in NASA technical reports 25,26 but seldom in published paper 27. Thesefeatures are defined as follows 2528.FM0 is a robust indicator of major faults in a gear mesh and given asFM0 PPxPHh0Ph,(1)where PPxis the maximum peak-to-peak value of time record xi(i 1, 2,y,n), Phis the amplitude of the nth harmonicof the meshing frequency, and H is the total number of harmonics considered. Generally, the complete data seriescollected is also called a run ensemble. It is further divided into M time records each including n data points. For each ofthe time records, FM0 can be calculated.FM4 is designed based on the difference signal 28. The kurtosis of the difference signal is defined as FM4.FM4 nPni1di?d4Pni1di?d22,(2)where n is the total number of points in each time record, diis the ith measurement of the difference signal in a timerecord, andd is the average of the difference signal.FM4* is developed for monitoring the progression of a gear fault instead of detection of the initial fault. Among the Mtime records in the run ensemble, the first M0time records are classified as the data collected when gears are runningunder normal condition. The remaining records contain information on the growth of the fault. FM4* is expressed asFM4?1=nPni1di?d41=M0PM0j11=nPnk1djk?dj22,(3)where djkis the kth measurement in the ith time record captured under the normal condition of gears.ARTICLE IN PRESSTable 2Motor speeds, maximum loads and characteristic frequencies of the gearbox.Motor speed (rpm)Maximum torque (Nm)f1 (Hz)f12 (Hz)f2 (Hz)f34 (Hz)f3 (Hz)1200.0034.514.76228.5714.29342.868.571400.0029.585.56266.6716.67400.0010.001600.0025.896.35304.7619.05457.1411.431800.0023.017.14342.8621.43514.2912.86Note: f1, f2 and f3 are the rotating frequencies of shaft 1, shaft 2 and shaft 3, respectively. f12 and f34 are the meshing frequencies of gears 1 and 2, andgears 3 and 4, respectively.Y. Lei, M.J. Zuo / Mechanical Systems and Signal Processing 23 (2009) 153515471538MA6 is defined asMA6 1=nPni1di?d61=nPni1di?d23.(4)M6A* is based on the M6A parameter with the exception that the normal records are differentiated from the faultyrecords. It is shown asM6A?1=nPni1di?d61=M0PM0j11=nPnk1d0jk? d0j23.(5)NA4 is developed based on a residual signal 28 of a time record. NA4 is expressed asNA4 1=nPni1ri? r41=MPMj11=nPnk1rjk? rj22,(6)where riis the ith measurement of the residual signal in a time record, r is the average of the residual signal in that timerecord.NA4* is considered as an enhancement to NA4. The calculation method is similar to the one used in FM4* and it isgiven byNA4?1=nPni1ri? r41=M0PM0j11=nPnk1rjk? rj22.(7)NB4 is defined on the envelope of the obtained bandpass filtered signal as an indicator of localized gear toothdamage. NB4 is given asNB4 1=nPni1si? s41=MPMj11=nPnk1sjk? sj22,(8)where s(t) is the envelope expressed as s(t) |b(t)+iHb(t)|, b(t) is the signal bandpass filtered about the meshingfrequency, and Hb(t) is the Hilbert transform of b(t).NB4* aims to improve the performance of NB4 in tracking damage progression and is defined asNB4?1=nPni1si? s41=M0PM0j11=nPnk1sjk? sj22.(9)Energy ratio (ER) is expressed asER ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=nPni1di21=nPni1d0i2s,(10)where d0iis the ith measurement of the regular meshing components which include the shaft frequencies and theirharmonics, the meshing frequencies and their harmonics, and all first-order sidebands.Energy operator (EOP) is developed by first calculating the value xi2?xi?1?xi+1for every point xi(i 1, 2,y,n) of thesignal. The energy operator is then computed by taking the kurtosis of the resulting signal and shown asEOP nPni1rei? re4Pni1rei? re2?2,(11)where reiequals xi2?xi?1?xi+1and is the ith measurement of the resulting signal, and re is the average of the resultingsignal.(3) The third part covers four statistical feature parameters based on the frequency spectrum of a vibration signal. Thesefour frequency-domain parameters may reveal some information that cannot be found using the time-domain featureparameters. They are mean frequency (MF), frequency centre (FC), root mean square frequency (RMSF), and standarddeviation frequency (STDF), as introduced in Refs. 29,30.These 25 features considered in this study as summarized above are listed below:1. Mean2. Standard deviation3. Root mean square4. Peak5. Skewness6. KurtosisARTICLE IN PRESSY. Lei, M.J. Zuo / Mechanical Systems and Signal Processing 23 (2009) 1535154715397. Crest factor8. Clearance factor9. Shape factor10. Impulse factor11. FM012. FM413. FM4*14. M6A15. M6A*16. NA417. NA4*18. NB419. NB4*20. Energy ratio21. Energy operator22. Mean frequency (MF)23. Frequency centre (FC)24. Root mean square frequency (RMSF)25. Standard deviation frequency (STDF)Each of the vibration signals collected from the gears is processed to extract the above 25 feature parameters. Therefore,a feature set pm,c,j, m 1, 2,y,Mc; c 1, 2,y,C; j 1, 2,y,J can be acquired, which is an Mc-by-C-by-J matrix, wherepm,c,jis the jth feature value of the mth sample under the cth condition, Mcis the number of samples under the cth gearcondition, C is the number of the gear conditions, and J is the number of features. In this paper, Mcequals 24, C equals 3, andJ equals 25.3.2. TFSWT based on EDETThe 25 features listed above may identify the crack levels of the gears from different aspects, but they have varyingpotential in distinguishing the crack faults. Some features are sensitive and closely related to the fault, but others are not.Thus, before the whole feature set is fed into a classifier, sensitive features providing gear fault-related information must beselected and highlighted and irrelevant features discarded or weakened to improve the classification performance andavoid the curse of dimensionality. In this paper, a TFSWT based on EDET is presented, which consists of two stages: featureselection and feature weighting.3.2.1. Stage 1: feature selectionIn the gearbox experiment, 72 data samples were obtained for the three gear conditions (F0F2). For each sample, the 25features are extracted to represent the characteristic information contained in the sample. Thus, a feature set pm,c,j with24?3?25 feature values is obtained. Then the first stage feature selection procedure based on EDET can be described asfollows:(1) Calculating the average distance of the samples of the same gear c