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Gear crack level identifi cation based on weighted K nearest neighbor classifi cation algorithm Yaguo Lei, Ming J. Zuo? Department of Mechanical Engineering, University of Alberta, Edmonton, Alberta, Canada T6G2G8 a r t i c l e i n f o Article history: Received 17 August 2008 Received in revised form 7 December 2008 Accepted 19 January 2009 Available online 31 January 2009 Keywords: Feature extraction Two-stage feature selection and weighting technique Weighted K nearest neighbor algorithm Gear crack level identifi cation Fault diagnosis a b s t r a c t A crack fault is one of the damage modes most frequently occurring in gears. Identifying different 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 cracks automatically and reliably. In this method, feature parameters in time domain, specially designed for gear damage detection and in frequency domain are extracted to characterize the gear conditions. A two-stage feature selection and weighting technique (TFSWT) via Euclidean distance evaluation technique (EDET) is presented and adopted to select sensitive features and remove fault-unrelated features. A weighted K nearest neighbor (WKNN) classifi cation algorithm is utilized to identify the gear crack levels. The gear crack experiments were conducted and the vibration signals were captured from the gears under different loads and motor speeds. The proposed method is applied to identifying the gear crack levels and the applied results demonstrate its effectiveness. c 1, 2,y,C; j 1, 2,y,J can be acquired, which is an Mc-by-C-by-J matrix, where pm,c,jis the jth feature value of the mth sample under the cth condition, Mcis the number of samples under the cth gear condition, C is the number of the gear conditions, and J is the number of features. In this paper, Mcequals 24, C equals 3, and J equals 25. 3.2. TFSWT based on EDET The 25 features listed above may identify the crack levels of the gears from different aspects, but they have varying potential 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 classifi er, sensitive features providing gear fault-related information must be selected and highlighted and irrelevant features discarded or weakened to improve the classifi cation performance and avoid the curse of dimensionality. In this paper, a TFSWT based on EDET is presented, which consists of two stages: feature selection and feature weighting. 3.2.1. Stage 1: feature selection In the gearbox experiment, 72 data samples were obtained for the three gear conditions (F0F2). For each sample, the 25 features are extracted to represent the characteristic information contained in the sample. Thus, a feature set pm,c,j with 24?3?25 feature values is obtained. Then the fi rst stage feature selection procedure based on EDET can be described as follows: (1) Calculating the average distance of the samples of the same gear condition Dc;j ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi 1 Mc? Mc? 1 X Mc l;m1 pm;c;j? pl;c;j2 v u u t ; l; m 1; 2; .; Mc; lam;(12) then getting the average distance of C gear conditions Dw j 1 C X C c1 Dc;j.(13) (2) Defi ning and calculating the variance factor with the same gear condition as follows: Vw j maxDc;j minDc;j .(14) (3) Calculating the average feature value of all samples under the same gear condition ac;j 1 Mc X Mc m1 pm;c;j;(15) ARTICLE IN PRESS Y. Lei, M.J. Zuo / Mechanical Systems and Signal Processing 23 (2009) 153515471540 then obtaining the average distance between samples of different gear conditions Db j ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi 1 C ? C ? 1 X C c;e1 ae;j? ac;j2 v u u t ; c;e 1; 2; .; C; cae.(16) (4) Defi ning and calculating the variance factor between different gear conditions as follows: Vb j maxjae;j? ac;jj minjae;j? ac;jj ; c; e 1; 2; .; C; cae.(17) (5) Defi ning and calculating the variance factor as follows: lj Vw j maxVw j Vb j maxVb j 0 1 A ?1 .(18) (6) Calculating the ratio Dj(b)and Dj(w)and assigning the variance factor Ejlj Db j Dw j ;(19) then normalizing Ejby its maximum value and getting the evaluation criteria Ej Ej maxEj .(20) It is clear that a larger Ej(j 1, 2,y,J) suggests that the corresponding features are better to distinguish the C gear conditions. Therefore, the sensitive features may be selected from the feature set when their evaluation criteria EjXf, wheref is a predefi ned threshold for feature selection. 3.2.2. Stage 2: feature weighting Although the sensitive features have been selected from the original feature set via stage 1 of TFSWT, the selected features have different sensitivities in the identifi cation of gear crack levels. Thus, feature weighting is necessary to achieve a more reliable diagnosis result. Feature weighting is a general method in which each feature is multiplied by a number within 0, 1 and proportional to the ability of the feature to distinguish different classes. In the Euclidean space, feature weighting is to extend the axes corresponding to the sensitive features and shrink the axes corresponding to the features unrelated to the fault. Following the feature selection procedure outlined in Section 3.2.1, we have reduced the number of features to be further considered from 25 down to, say, D, where Dp25. In order to fi nd the weights of these remaining features, we apply the same EDET procedure on these D features. Going from Eqs. (12)(20), we have obtained the new evaluation criterion 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 identifi cation of the gear crack levels. 4. Level identifi cation method of the gear cracks using WKNN In KNN classifi cation algorithm, each training sample is represented in a D-dimensional space according to the value of each 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 the classifi cation of the testing sample. The K nearest neighbors are usually determined by computing the Euclidean distance between the testing sample and each of the training samples 19,20. The Euclidean distance between the testing sample TEdand the mth training sample TRm,d is defi ned as Dm X D d1 TEd? 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 classifi cation algorithm makes it easy to implement. However, it suffers from poor classifi cation performance when samples of different classes overlap in some regions in the feature space. Due to the use of the Euclidean distance, the KNN algorithm is sensitive to scaling of the feature values. Therefore, a feature weighting technique is useful for overcoming the shortcomings of KNN. The weighted Euclidean distance between the testing sample ARTICLE IN PRESS Y. Lei, M.J. Zuo / Mechanical Systems and Signal Processing 23 (2009) 153515471541 Tedand the mth training sample Trmdcan be expressed as: Dwf m X D d1 wfdTEd? 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 as the weighted K nearest neighbor (WKNN) algorithm in this paper. Adopting WKNN as a classifi er, a level identifi cation 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 the difference and residual signals and frequency spectrums. Second, the 25 feature parameters are extracted from the raw vibration signals or the preprocessed signals. Third, TFSWT based on EDET is proposed. The feature selection stage is used to select the sensitive features according to the evaluation criteria and the threshold. And the feature weighting stage is to compute the weights of the selected sensitive features. Finally, the WKNN classifi cation algorithm is applied to the level identifi cation of the gear cracks and fi nal diagnosis result can be obtained. 5. Experimental results and discussion 5.1. Experiments and results The vibration data acquired from the experimental system of the gears are used to demonstrate the effectiveness of the proposed diagnosis method for the gear faults. The evaluation result of the 25 feature parameters using the feature selection stage of TFSWT is shown in Fig. 5(a). The threshold valuej(in the range from 0 to 1) must be properly selected in order to keep only the important features. If it is large, only a few really important features will be kept. If it is small, most of the features will be kept. This means that if most features are relatively unimportant, a larger threshold value should be used; while if most features are pretty important, a smaller threshold value should be used. Experience is helpful in selection of this parameter. When there is no a prior knowledge of setting the threshold, one may start with the median of the range of the evaluation criteria. In this paper, for illustration of the proposed methodology, we have selected 0.5 to be ARTICLE IN PRESS Data 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) 153515471542 the threshold value for feature selection. As illustrated in Fig. 5(a), features #7, #8, #9, #10, #16, #23, and #24 have been selected 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 calculated and 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 the proposed feature selection and no weighting (method 3) are also employed to analyse the same data sets, respectively. For convenience, the proposed method is referred as method 4 in the following section. For all the four method, the neighborhood parameter K is changed from 1 to the number of the training samples. 5.1.1. Experiment 1 As 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 therefore the whole data set corresponding to the three gear conditions includes altogether 72 samples. Thirty-six data samples are selected for training and the remaining 36 samples under the identical operating condition are used to test. The training and 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 selected with the fi rst stage of TFSWT are adopted as the input of the WKNN classifi er. The evaluation criteria of the selected seven features using the second stage of TFSWT are used as the weights of the WKNN classifi er. The identifi cation accuracies of the proposed method with the different values of the neighborhood parameter K are shown in Fig. 6. Table 4 gives the statistical results of the identifi cation accuracies. For method 1, all the 25 features are used and fed into the KNN classifi er. 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 selected from the 25 features randomly. The KNN classifi er is used to recognize the three gear conditions. This method is repeated fi fteen times and the average results are also given in Fig. 6 and Table 4, respectively. For method 3, the feature selection of TFSWT is utilized and the selected seven sensitive features are input the KNN classifi er to distinguish the different gear conditions. Fig. 6 and Table 4 give its diagnosis results, respectively. ARTICLE IN PRESS Feature weights Number of selected features 0 0.5 1 #8#9#10#16#24#23#7 Number of all features Evaluation criteria 0 0.5 1 Threshold = 0.5 #5#10#15#20#25 Fig. 5. (a) Evaluation criteria of all 25 features, (b) feature weights of the selected 7 features. Table 3 Data description of the three experiments. ExperimentNumber of training / testing samples Fault modes of training/testing Motor speeds of training/ testing samples (rpm) Loads of training/ testing Samples Label of classifi cation 112/12F0/F012001800/120018000, 1, 2/0, 1, 21 12/12F1/F112001800/120018000, 1, 2/0, 1, 22 12/12F2/F212001800/120018000, 1, 2/0, 1, 23 212/12F0/F01200, 1600/1400, 18000, 1, 2/0, 1, 21 12/12F1/F11200, 1600/1400, 18000, 1, 2/0, 1, 22 12/12F2/F21200, 1600/1400, 18000, 1, 2/0, 1, 23 38/16F0/F012001800/120018000/1, 21 8/16F1/F112001800/120018000/1, 22 8/16F2/F212001800/120018000/1, 23 Y. Lei, M.J. Zuo / Mechanical Systems and Signal Processing 23 (2009) 153515471543 From Fig. 6 and Table 4, it can be seen that using the feature selection stage of TFSWT, methods 3 and 4 obtain the higher identifi cation 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 the sensitive features but also the other fault-unrelated features to recognize the crack levels, which lead to the middle classifi cation 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 2 In this experiment, the training and testing data are reorganized as depicted in Table 3. The 36 training samples were collected under the motor speeds 1200 and 1600rpm, while the 36 testing samples were collected under the motor speeds 1400 and 1800rpm. The experiment for these training and testing data is carried out to further investigate the generalization when the proposed method is tested by the data with different motor speeds. The seven features are selected with the fi rst stage of TFSWT as the diagnosis features and their weights computed via the second stage of TFSWTare used as the weights. Applying the method based on WKNN to the three level identifi cation of the gear cracks, the identifi cation 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 PRESS Method 1 Method 2 Method 3 Method 4 5101520253035 60 70 80 90 100 Identification accuracy % K Fig. 6. Accuracy comparison of the four methods for experiment 1. Table 4 Diagnosis results of the four methods in the three experiments. Exper- iment Method 1Method 2Method 3Method 4 Accuracy (%)CPU times (s) Accuracy (%)CPU times (s) Accuracy (%)CPU times (s) Accuracy (%)CPU times (s) Max.MeanMin.Max.MeanMin.Max.MeanMin.Max.MeanMin. 1100.0089.5875.000.234496.6780.1765.930.1719100.0096.5386.110.2500100.0096.6886.110.2656 297.2287.4277.780.234483.3377.2966.300.1719100.0099.1597.220.2500100.0099.6197.220.2656 397.9287.6777.080.140690.2878.0865.280.109497.9292.1087.500.1875100.0092.6287.500.2031 Method 1 Method 2 Method 3 Method 4 5101520253035 60 70 80 90 100 Identification accuracy % K Fig. 7. Accuracy comparison of the four methods for experiment 2. Y. Lei, M.J. Zuo / Mechanical Systems and Signal Processing 23 (2009) 153515471544 It is observed from Fig. 7 and Table 4 that the same ranges of the correct rates (97.22100.00%) are achieved by both methods 3 and 4. The diagnosis accuracies of method 1 (77.7897.22%) are lower than those of methods 3 and 4. Method 2 obtains the lowest accuracies (66.3083.33%). This observation is similar to that of experiment 1, which indicates that the generalization 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 between experiments 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. Experim