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系 机械设计 系 主 任 批准日期 毕 业 设 计（论 文）任 务 书 机械设计 系 机械设计制造及其自动化 专业 班 学生 一、毕业设计(论文)课题 DF12型手扶拖拉机变速驱动系统设计 二、毕业设计(论文)工作自2012年 月 日起至 2012 年 12 月 日止三、毕业设计(论文)进行地点 西安理工大学 四、毕业设计(论文)的内容要求 拖拉机是一种牵引、驱动兼用型的农业机械，一般具有结构紧凑，耐用可靠，操纵灵活、马力大、重量轻、油耗低通用性好等特点，并具有乘坐装置，适用于水田、旱地、果园和丘陵地的农田耕作，配备相应的农用机具，可以完成犁地、旋耕、耙地、开沟、播种、收割、运输和其其它作业，还可以为排灌、喷灌、脱粒、磨粉、饲料加工等固定作业提供动力。 本次设计主要针对DF12型拖拉机变速驱动系统，通过对其进行传动箱和驱动转向机构的结构设计以及各主要零部件的受力分析与计算，强化机械设计综合应用能力的培养。 其设计所需原始数据如下： 发动机额定功率（马力）：12 最大扭矩：（公斤米）5 空转转速（最大/最小）；2300/600 驱动轮轮胎规格6.0012/2 理论行驶速度（公里/小时）：、倒、倒档：1.40、2.50、4.10、5.30、9.40、15.30、1.00、3.80 传动箱：前进档6个，倒档2个。 驱动桥中央传动型式：直齿圆柱齿轮 本次设计的主要任务及要求： 1.拟定DF12型手扶拖拉机传动箱的传动方案，要求所选的方案要能够实现拖拉机各种理论档位所要求的速度，传动方案设计合理、而且应该易于实现、操纵方便成本低廉； 2.在拟定的方案中合理选取满足设计要求的传动系统总成方案，合理地分配各传动比。 3.确定DF12型手扶拖拉机变速箱和驱动转向机构的相关结构参数和几何参数，对各主要零件进行运动和动力相关设计，对所选的传动方案主要零部件进行强度和刚度、寿命、润滑等必要的计算、校核； 4.完成DF12型手扶拖拉机传动箱装配图一份，完成DF12型手扶拖拉机主要零件的设计、计算、校核以及零件工作图的绘制，要求总图量0#2.5张以上）； 5.完成设计计算说明书一份，其内容应包括总体传动方案的拟定，方案的可行性以及所选方案优缺点分析，相关计算以及说明应全面、详细、公式出处等应该详细标明，注明各个参数含义，论文书写认真、规范； 6.完成文献综述一份，文献综述应该细致全面描述拖拉机行业的发展概况、发展趋势、发展中存在问题、可能存在的解决方案等； 7.完成翻译相关资料一份（大于2000单词量），题材必须和机械结构相关； 8.参考文献15篇以上，其中外文参考文献不少于3篇。 参考资料： 1 杨成康工程机械发动机与底盘构造北京：机械工业出版社，1989。 2 王 健工程机械构造北京：中国铁道出版社，1995。 3 黄声显重型汽车构造与维修北京：人民交通出版社，1992。 4 何挺继，展朝勇现代公路施工机械北京：人民交通出版社，1999。 5 吉林工业大学汽车教研室汽车构造北京：人民交通出版社，2005。 负责指导教师 指 导 教 师 接受设计论文任务开始执行日期 学生签名 毕业设计（论文）进度表 机械设计 系月日周次任务阶段名称及详细项目检查日期检 查 结 果布置任务、熟悉设计内容、要求，查阅，收集，整理和拖拉机传动系统相关的资料，尤其是传动系统总体布置、分配方面；初步拟定传动方案,其初始方案的选取应该在分析比较的基础上并要阐明原因以及所搜集方案的优缺点；1.拟定总体传动方案；2.绘制相关结构原理图，最终拟定传动方案；1.绘制相关结构原理图，最终拟定传动方案；2.DF12型手扶拖拉机传动箱结构和参数设计；1.DF12型手扶拖拉机传动箱结构和参数设计；2.DF12型手扶拖拉机传动箱主要传动部件的设计与计算；DF12型手扶拖拉机传动箱主要传动部件的设计与计算；1.DF12型手扶拖拉机传动箱主要传动部件的设计与计算；2.DF12型手扶拖拉机传动箱装配图绘制；1.DF12型手扶拖拉机传动箱装配图绘制；2.主要零件、部件零件图绘制；1.DF12型手扶拖拉机传动箱装配图绘制；2.主要零件、部件零件图绘制；月日周次任务阶段名称及详细项目检查日期检 查 结 果1.继续以上工作；2.完成文献综述；1.主要零件、部件零件图绘制；2.开始外文翻译和设计说明书的初步编写；1.完成外文资料的翻译；2.完成设计说明书的编写；准备答辩。指导教师(签名) 学生(签名) 年 月 日 年 月 日Gear crack level identification based on weighted K nearest neighborclassification algorithmYaguo Lei, Ming J. Zuo?Department of Mechanical Engineering, University of Alberta, Edmonton, Alberta, Canada T6G2G8a r t i c l e i n f oArticle history:Received 17 August 2008Received in revised form7 December 2008Accepted 19 January 2009Available online 31 January 2009Keywords:Feature extractionTwo-stage feature selection and weightingtechniqueWeighted K nearest neighbor algorithmGear crack level identificationFault diagnosisa b s t r a c tA crack fault is one of the damage modes most frequently occurring in gears. 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: /locate/jnlabr/ymsspMechanical 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 conditionDc;jffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1Mc? Mc? 1XMcl;m1pm;c;j? pl;c;j2vuut; l; m 1; 2; .; Mc; lam;(12)then getting the average distance of C gear conditionsDwj1CXCc1Dc;j.(13)(2) Defining and calculating the variance factor with the same gear condition as follows:VwjmaxDc;jminDc;j.(14)(3) Calculating the average feature value of all samples under the same gear conditionac;j1McXMcm1pm;c;j;(15)ARTICLE IN PRESSY. Lei, M.J. Zuo / Mechanical Systems and Signal Processing 23 (2009) 153515471540then obtaining the average distance between samples of different gear conditionsDbjffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1C ? C ? 1XCc;e1ae;j? ac;j2vuut; c;e 1; 2; .; C; cae.(16)(4) Defining and calculating the variance factor between different gear conditions as follows:Vbjmaxjae;j? ac;jjminjae;j? ac;jj; c; e 1; 2; .; C; cae.(17)(5) Defining and calculating the variance factor as follows:ljVwjmaxVwjVbjmaxVbj01A?1.(18)(6) Calculating the ratio Dj(b)and Dj(w)and assigning the variance factorEjljDbjDwj;(19)then normalizing Ejby its maximum value and getting the evaluation criteriaEjEjmaxEj.(20)It is clear that a larger Ej(j 1, 2,y,J) suggests that the corresponding features are better to distinguish the C gearconditions. Therefore, the sensitive features may be selected from the feature set when their evaluation criteria EjXf,wherefis a predefined threshold for feature selection.3.2.2. Stage 2: feature weightingAlthough the sensitive features have been selected from the original feature set via stage 1 of TFSWT, the selectedfeatures have different sensitivities in the identification of gear crack levels. Thus, feature weighting is necessary to achievea more reliable diagnosis result. Feature weighting is a general method in which each feature is multiplied by a numberwithin 0, 1 and proportional to the ability of the feature to distinguish different classes. In the Euclidean space, featureweighting is to extend the axes corresponding to the sensitive features and shrink the axes corresponding to the featuresunrelated to the fault.Following the feature selection procedure outlined in Section 3.2.1, we have reduced the number of featuresto be further considered from 25 down to, say, D, where Dp25. In order to find the weights of these remaining features, weapply the same EDET procedure on these D features. Going from Eqs. (12)(20), we have obtained the new evaluationcriterion values of these remaining features and they are used as their respective weights wf (wf (wf1,y,wfd,y,wfD.They are assigned to each of the remaining features to point out their sensitivities in the identification of the gear cracklevels.4. Level identification method of the gear cracks using WKNNIn KNN classification algorithm, each training sample is represented in a D-dimensional space according to the value ofeach of its D features. The testing sample is then represented in the same space, and its K nearest neighbors are selected.The class of each of these K neighbors is then tallied, and the class with the largest number of votes is selected as theclassification of the testing sample. The K nearest neighbors are usually determined by computing the Euclidean distancebetween the testing sample and each of the training samples 19,20. The Euclidean distance between the testing sampleTEdand the mth training sample TRm,dis defined asDmXDd1TEd? TRm;d2#1=2; d 1; 2; .; D; m 1; 2; .; M(21)where D and M are the numbers of features and training samples, respectively.The simplicity of the KNN classification algorithm makes it easy to implement. However, it suffers from poorclassification performance when samples of different classes overlap in some regions in the feature space. Due to the use ofthe Euclidean distance, the KNN algorithm is sensitive to scaling of the feature values. Therefore, a feature weightingtechnique is useful for overcoming the shortcomings of KNN. The weighted Euclidean distance between the testing sampleARTICLE IN PRESSY. Lei, M.J. Zuo / Mechanical Systems and Signal Processing 23 (2009) 153515471541Tedand the mth training sample Trmdcan be expressed as:DwfmXDd1wfdTEd? TRmd2#1=2,(22)where wfddenotes the weight of the dth feature.As mentioned in Section 3.2.2, feature weights wf are computed using the feature weighting stage of TFSWT.Substituting wf into Eq. (26), the KNN algorithm using the weighted Euclidean distance metric is developed and referred asthe weighted K nearest neighbor (WKNN) algorithm in this paper.Adopting WKNN as a classifier, a level identification method for gear cracks is proposed and shown in Fig. 4. First,vibration signals captured from the gears are preprocessed with Hilbert transform and Fourier transform, etc. to obtain thedifference and residual signals and frequency spectrums. Second, the 25 feature parameters are extracted from the rawvibration signals or the preprocessed signals. Third, TFSWT based on EDET is proposed. The feature selection stage is usedto select the sensitive features according to the evaluation criteria and the threshold. And the feature weighting stage is tocompute the weights of the selected sensitive features. Finally, the WKNN classification algorithm is applied to the levelidentification of the gear cracks and final diagnosis result can be obtained.5. Experimental results and discussion5.1. Experiments and resultsThe vibration data acquired from the experimental system of the gears are used to demonstrate the effectiveness of theproposed diagnosis method for the gear faults. The evaluation result of the 25 feature parameters using the featureselection stage of TFSWT is shown in Fig. 5(a). The threshold valuej(in the range from 0 to 1) must be properly selected inorder to keep only the important features. If it is large, only a few really important features will be kept. If it is small, mostof the features will be kept. This means that if most features are relatively unimportant, a larger threshold value should beused; while if most features are pretty important, a smaller threshold value should be used. Experience is helpful inselection of this parameter. When there is no a prior knowledge of setting the threshold, one may start with the median ofthe range of the evaluation criteria. In this paper, for illustration of the proposed methodology, we have selected 0.5 to beARTICLE IN PRESSData acquisition Difference and residual signals Hilbert envelope spectrum 11 feature parameters specially for gear damage detection Arrange all features from large evaluation criteria to small using TFSWT based on EDET Crack level identification with the WKNN algorithm Frequency spectrum Gears with accelerometers Diagnosis result 10 time-domain feature parameters Select sensitive features according to the predefined threshold Compute feature weights using TFSWT based on EDET 4 frequency-domain feature parameters Fig. 4. Flow chart of the proposed method.Y. Lei, M.J. Zuo / Mechanical Systems and Signal Processing 23 (2009) 153515471542the threshold value for feature selection. As illustrated in Fig. 5(a), features #7, #8, #9, #10, #16, #23, and #24 have beenselected to be the remaining features. These features are crest factor, clearance factor, shape factor, impulse factor, NA4,frequency centre (FC), and root mean square frequency (RMSF).Following the procedure for calculation of feature weights, the weights of these selected features have been calculatedand given in Fig. 5(b). The weights of these features are 0.915, 0.950, 0.743, 0.930, 0.640, 1.000, and 0.700, respectively.Three experiments are conducted over the three different organizations of the training and testing data. For comparison,the KNN without feature selection (method 1), the KNN with feature selection randomly (method 2), the KNN with theproposed feature selection and no weighting (method 3) are also employed to analyse the same data sets, respectively. Forconvenience, the proposed method is referred as method 4 in the following section. For all the four method, theneighborhood parameter K is changed from 1 to the number of the training samples.5.1.1. Experiment 1As mentioned in Section 2, under the same gearbox operating condition (identical motor speed, load and fault mode),two data samples were collected. For each of the three fault modes F0, F1 and F2, 24 samples are acquired, and thereforethe whole data set corresponding to the three gear conditions includes altogether 72 samples. Thirty-six data samples areselected for training and the remaining 36 samples under the identical operating condition are used to test. The trainingand testing data in this experiment are listed in Table 3, respectively.The proposed method based on WKNN is used to identify the three levels of the gear cracks. The seven features selectedwith the first stage of TFSWT are adopted as the input of the WKNN classifier. The evaluation criteria of the selected sevenfeatures using the second stage of TFSWT are used as the weights of the WKNN classifier. The identification accuracies ofthe proposed method with the different values of the neighborhood parameter K are shown in Fig. 6. Table 4 gives thestatistical results of the identification accuracies.For method 1, all the 25 features are used and fed into the KNN classifier. The results are shown in Fig. 6 and Table 4,respectively. In method 2, seven features, the same number of the selected feature as the proposed method, are selectedfrom the 25 features randomly. The KNN classifier is used to recognize the three gear conditions. This method is repeatedfifteen times and the average results are also given in Fig. 6 and Table 4, respectively. For method 3, the feature selection ofTFSWT is utilized and the selected seven sensitive features are input the KNN classifier to distinguish the different gearconditions. Fig. 6 and Table 4 give its diagnosis results, respectively.ARTICLE IN PRESSFeature weightsNumber of selected features 00.51#8#9#10#16#24#23#7Number of all features Evaluation criteria00.51Threshold = 0.5#5#10#15#20#25Fig. 5. (a) Evaluation criteria of all 25 features, (b) feature weights of the selected 7 features.Table 3Data description of the three experiments.ExperimentNumber of training /testing samplesFault modes oftraining/testingMotor speeds of training/testing samples (rpm)Loads of training/testing SamplesLabel ofclassification112/12F0/F012001800/120018000, 1, 2/0, 1, 2112/12F1/F112001800/120018000, 1, 2/0, 1, 2212/12F2/F212001800/120018000, 1, 2/0, 1, 23212/12F0/F01200, 1600/1400, 18000, 1, 2/0, 1, 2112/12F1/F11200, 1600/1400, 18000, 1, 2/0, 1, 2212/12F2/F21200, 1600/1400, 18000, 1, 2/0, 1, 2338/16F0/F012001800/120018000/1, 218/16F1/F112001800/120018000/1, 228/16F2/F212001800/120018000/1, 23Y. Lei, M.J. Zuo / Mechanical Systems and Signal Processing 23 (2009) 153515471543From Fig. 6 and Table 4, it can be seen that using the feature selection stage of TFSWT, methods 3 and 4 obtain thehigher identification accuracies. The accuracy ranges of these two methods are from 86.11% to 100.00%, respectively.Method 2 selects the input features randomly and produces the worst result (65.9396.67%). Method 1 uses not only thesensitive features but also the other fault-unrelated features to recognize the crack levels, which lead to the middleclassification result (75100%). The CPU times taken to carry out these four methods in this experiment are 0.2344, 0.1719,0.2500 and 0.2656s, respectively. They are listed in Table 4.5.1.2. Experiment 2In this experiment, the training and testing data are reorganized as depicted in Table 3. The 36 training samples werecollected under the motor speeds 1200 and 1600rpm, while the 36 testing samples were collected under the motor speeds1400 and 1800rpm. The experiment for these training and testing data is carried out to further investigate thegeneralization when the proposed method is tested by the data with different motor speeds.The seven features are selected with the first stage of TFSWT as the diagnosis features and their weights computed viathe second stage of TFSWTare used as the weights. Applying the method based on WKNN to the three level identification ofthe gear cracks, the identification correct rates with the neighborhood parameter K are shown in Fig. 7 and Table 4,respectively. The testing results of methods 13 in this experiment are also given in Fig. 7 and Table 4 for comparison.ARTICLE IN PRESSMethod 1 Method 2 Method 3 Method 4 510152025303560708090100Identification accuracy % KFig. 6. Accuracy comparison of the four methods for experiment 1.Table 4Diagnosis results of the four methods in the three experiments.Exper-imentMethod 1Method 2Method 3Method 4Accuracy (%)CPUtimes (s)Accuracy (%)CPUtimes (s)Accuracy (%)CPUtimes (s)Accuracy (%)CPUtimes (s)Max.MeanMin.Max.MeanMin.Max.MeanMin.Max.MeanMin.1100.0089.5875.000.234496.6780.1765.930.1719100.0096.5386.110.2500100.0096.6886.110.2656297.2287.4277.780.234483.3377.2966.300.1719100.0099.1597.220.2500100.0099.6197.220.2656397.9287.6777.080.140690.2878.0865.280.109497.9292.1087.500.1875100.0092.6287.500.2031Method 1 Method 2 Method 3 Method 4 510152025303560708090100Identification accuracy % KFig. 7. Accuracy comparison of the four methods for experiment 2.Y. Lei, M.J. Zuo / Mechanical Systems and Signal Processing 23 (2009) 153515471544It is observed from Fig. 7 and Table 4 that the same ranges of the correct rates (97.22100.00%) are achieved by bothmethods 3 and 4. The diagnosis accuracies of method 1 (77.7897.22%) are lower than those of methods 3 and 4. Method 2obtains the lowest accuracies (66.3083.33%). This observation is similar to that of experiment 1, which indicates that thegeneralization of methods 3 and 4 is superior to those of the others, method 1 is inferior and method 2 is the worst one.Because experiment 2 has the same computational burden as experiment 1, the CPU times are the same betweenexperiments 1 and 2 for each of the four methods. The CPU times of the four methods in experiment 2 are 0.2344, 0.1719,0.2500 and 0.2656s, respectively. They are listed in Table 4.5.1.3. Experiment 3To clarify the generalization of the proposed method with the various loads, the testing data with the different loadsfrom those of the training data are used in this experiment, which are described in Table 3. The 24 training samples wereacquired under load 0 (without torque) and the remaining 48 testing samples under loads 1 and 2, respectively.The mentioned four methods are employed again and the corresponding diagnosis results are given in Fig. 8 and Table 4,respectively. It is found from the diagnosis results that the identification accuracies of methods 1, 2 and 3 are from 77.08%to 97.92%, 65.28% to 90.28% and 87.50% to 97.92%, respectively. The highest accuracies are obtained by method 4 and theyare from 87.50% to 100.00%. The CPU times taken by the four methods in this experiment are 0.1406, 0.1094, 0.1875 and0.2031s, which are listed in Table 4. They are smaller than those of experiments 1 and 2 because there are fewer trainingsamples in experiment 3, which lead to the lower computational burden.5.2. Discussion(1) In the three experiments, the lowest identification accuracies are yielded by method 2 and medium accuracies areobtained by method 1. However, the best diagnosis results are provided by methods 3 and 4. It is because differentfeatures have varying potential in distinguishing crack levels of the gears. Some features are sensitive and closelyrelated to the crack levels, but others are not. Methods 3 and 4 select these sensitive features from the original featureset using TFSWT to identify the crack levels, and therefore the diagnosis accuracies are improved. However, Method 2randomly selects the same number of the diagnosis features as methods 3 and 4. It means that it could select and usefeatures which contain too much gear crack fault-unrelated information and there is a high degree of overlap betweenthe values of these features between the different crack levels. These features would confuse the classification process,and when they are used to distinguish different crack levels, the identification success rate will decline clearly. Thus,method 2 produces the worst diagnosis results. For method 1, it uses not only the sensitive features but also the otherfault-unrelated features to recognize the crack levels and therefore it leads to a middle classification result.(2) Another comparison between methods 4 and 3 indicates that method 4 is generally superior to method 3 in the light ofthe diagnosis accuracies. Although they have the same ranges of identification accuracies in experiments 1 and 2, themaximum diagnosis accuracy (100%) of method 4 outperforms that of method 3 (97.92%) in experiment 3. Thissuggests that method 4 has better classification performance than method 3 when the loads of gears are changed.Moreover, for each of the three experiments, the higher mean accuracies are achieved by method 4 (96.53% forexperiment 1, 99.15% for experiment 2 and 92.10% for experiment 3) than method 3 (96.68% for experiment 1, 99.61%for experiment 2 and 92.62% for experiment 3). These confirm our idea that the proposed WKNN method based onTFSWT may not only select the sensitive features from the original feature set but also consider their differentimportance in the level identification of the gear cracks. Thus, it may produce the best diagnosis result among the fourdiagnosis methods.(3) In view of the above result analyses, it is appropriate to conclude that the proposed method achieves the higherclassification accuracies and provides a better generalization capability compared to the other three methods in thispaper. The success obtained by the proposed method may be attributed to the following three points. (1) The commonARTICLE IN PRESSMethod 1 Method 2 Method 3 Method 4 510152060708090100Identification accuracy % KFig. 8. Accuracy comparison of the four methods for experiment 3.Y. Lei, M.J. Zuo / Mechanical Systems and Signal Processing 23 (2009) 153515471545feature parameters in time domain, specially developed parameters for gear damage detection as well as featureparameters in frequency domain are all extracted and utilized, which may provide more characteristic information ofthe crack levels. (2) A few sensitive features are selected and other fault-unrelated features, which would confuse theclassifier and therefore cause the classification success rate to decline, are discarded. (3) It is believed that in the WKNNclassification algorithm the sensitive features still have varying potential in distinguishing the different levels of thegear cracks. It is able to perform feature weighting in the classification process to reflect the different importantdegrees of the different sensitive features.(4) Besides, comparing the CPU times taken to perform the four methods in the three experiments, it can be seen thatmethod 2 takes the shortest CPU times (0.1719s for experiment 1, 0.1719s for experiment 2 and 0.1094s for experiment3) because only seven features are used in method 2. In method 1, all features are used and therefore longer CPU timesare taken for each of the experiments (0.2344s for experiment 1, 0.2344s for experiment 2 and 0.1406s for experiment3). A feature selection process is introduced into method 3 and a feature selection and weighting process in method 4.Therefore the corresponding CPU times rise in methods 3 and 4. However, the CPU times taken by method 4 (0.2656sfor experiment 1, 0.2656s for experiment 2 and 0.2031s for experiment 3) are acceptable. Actually, only for the firsttime of gear fault diagnosis, sensitive features need to be selected and feature weights computed. Once the sensitivefeatures are selected and the feature weights computed, it is unnecessary to do them again in subsequent tests andapplications. Thus, the CPU times taken by method 4 will be the same as those of method 2 because they have the samenumber of diagnosis features. Thus, it can be seen that method 4 is able to achieve the best identification accuracies aswell as decrease the computational burden.(5) In the experiments, the selected seven features are crest factor, clearance factor, shape factor, impulse factor, NA4,frequency centre, and root mean square frequency. From Fig. 5, it can be seen that FC plays the most important role andmost of the selected features are the time-domain feature parameters for crack level identification. Among the 11feature parameters specially defined for gear damage detection, only NA4 is selected and used for the experiment data.As its definition, NA4 can detect not only the onset of fault but also fault severity. Theoretically, the 11 featureparameters should be effective to the gear fault diagnosis because they are specially designed to diagnose gear faultsand their calculations are related to the characteristic frequencies of the gear faults. But in our experiments, althoughthe vibration signals were captured under the specified constant speed of the motor, there were still speed fluctuationswithin a signal sample. However, these 11 feature parameters can be used only if the collected vibration data arestationary because they require the identification or removal of certain frequency elements from the collected data. Ifthe data are non-stationary, it is impossible to remove such frequency data 28. Thus, there are errors in thecalculations of these feature parameters in our gearbox experiment, which make these features suffer on diagnosissensitivity. This analysis shows that the 11 feature parameters are not always the most appropriate features fordiagnosing gear faults when the rotating speed of gears is fluctuant or unstable.(6) The collected vibration data covered the different gear crack levels under several operating loads and speeds. Thus, theproblem studied in this paper is a typical diagnosis case of gear faults. The satisfied experimental results demonstratethe effectiveness and generalization of the proposed method.6. ConclusionA new method based on a weighted K nearest neighbor (WKNN) classification algorithm is proposed for crack levelidentification of gears in this paper. Feature parameters in time domain, specially developed for gear damage detection andin frequency domain are extracted to reflect the gear conditions. In order to remove the irrelevant information, a two-stagefeature selection and weighting technique (TFSWT) via Euclidean distance evaluation technique (EDET) is developed andused to select the sensitive features. The WKNN classification algorithm is presented to overcome the shortcoming of KNN.Gear crack experiments were conducted on a real gearbox. The vibration signals were measured from the gears underdifferent loads and motor speeds. The proposed method is applied to recognizing the gear crack levels. The results showthat the proposed method achieves higher identification accuracies and therefore it is a promising method to gear faultdiagnosis.AcknowledgementsThis research is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). Valuablecomments on the paper from anonymous reviewers are very much appreciated.References1 W. Wang, An enhanced diagnostic system for gear system monitoring, IEEE Transactions on Systems, Man, and Cybernetics 38 (2008) 102112.2 S.J. Loutridis, Gear failure prediction using multiscale local statistics, Engineering Structur