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Accepted 19 January 2009 Available online 31 January 2009 designed for gear damage detection and in frequency domain are extracted to characterize the gear conditions. A two-stage feature selection and weighting technique wer from faults the now the It is possible to obtain vital diagnosis information from the vibration signals through the use of signal processing methods 8,9, cyclostationarity analysis 10, and empirical mode decomposition 11. Among these methods, the statistical Contents lists available at ScienceDirect Mechanical Systems and Signal Processing ARTICLE IN PRESS Mechanical Systems and Signal Processing 23 (2009) 15351547 0888-3270/$-see front matter j C0 p l;c;j 2 ; l; m 1; 2; .; M c ; lam; (12) C gear conditions (13) factor with the same gear condition as follows: (14) of all samples under the same gear condition (15) ARTICLE IN PRESS Y. Lei, M.J. Zuo / Mechanical Systems and Signal Processing 23 (2009) 153515471540 a c;j 1 M c m1 p m;c;j ; X Mc (3) Calculating the average feature value V j minD c;j . w maxD c;j (2) Defining and calculating the variance D j C c1 D c;j . w 1 X C then getting the average distance of M c C2M c C0 1 l;m1 D c;j 1 p m;c t X Mc v u u (1) Calculating the average distance of the Each of the vibration signals collected from the gears is processed to extract the above 25 feature parameters. Therefore, a feature set p m,c,j , m 1, 2,y,M c ; c 1, 2,y,C; j 1, 2,y,J can be acquired, which is an M c -by-C-by-J matrix, where p m,c,j is the jth feature value of the mth sample under the cth condition, M c is 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, M c equals 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 classifier, sensitive features providing gear fault-related information must be selected and highlighted and irrelevant features discarded or weakened to improve the classification 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 gearboxexperiment, 72 data samples were obtained for the three gear conditions (F0F2). Foreach sample, the 25 features are extracted to represent the characteristic information contained in the sample. Thus, a feature set p m,c,j with 24C23C225 feature values is obtained. Then the first stage feature selection procedure based on EDET can be described as follows: 24. R 25. S 6A* 4 4* 4 B4* ratio operator frequency (MF) requency centre (FC) oot mean square frequency (RMSF) tandard deviation frequency (STDF) 14. M 15. M factor 0 4 4* 6A 9. Shape 10. Impulse t factor ance factor factor to be apply criteri They levels 4. Lev In each betw TE d and where The classification the Euclidean techniq ARTICLE IN PRESS Y. Lei, M.J. Zuo / Mechanical Systems and Signal Processing 23 (2009) 15351547 1541 d1 d m;d D and M are the numbers of features and training samples, respectively. simplicity of the KNN classification algorithm makes it easy to implement. However, it suffers from poor performance when samples of different classes overlap in some regions in the feature space. Due to the use of distance, the KNN algorithm is sensitive to scaling of the feature values. Therefore, a feature weighting ue is useful for overcoming the shortcomings of KNN. The weighted Euclidean distance between the testing sample of the testing sample. The K nearest neighbors are usually determined by computing the Euclidean distance een the testing sample and each of the training samples 19,20. The Euclidean distance between the testing sample the mth training sample TR m,d is defined as D m X D TE C0 TR 2 # 1=2 ; d 1; 2; .; D; m 1; 2; .; M (21) The class classification el identification method of the gear cracks using WKNN KNN classification algorithm, each training sample is represented in a D-dimensional space according to the value of of its D features. The testing sample is then represented in the same space, and its K nearest neighbors are selected. of each of these K neighbors is then tallied, and the class with the largest number of votes is selected as the further considered from 25 down to, say, D, where Dp25. In order to find the weights of these remaining features, we the same EDET procedure on these D features. Going from Eqs. (12)(20), we have obtained the new evaluation on values of these remaining features and they are used as their respective weights wf (wf (wf 1 ,y,wf d ,y,wf D . are assigned to each of the remaining features to point out their sensitivities in the identification of the gear crack . then obtaining the average distance between samples of different gear conditions D b j 1 C C2C C0 1 X C c;e1 a e;j C0 a c;j 2 v u u t ; c;e 1; 2; .; C; cae. (16) (4) Defining and calculating the variance factor between different gear conditions as follows: V b j maxja e;j C0 a c;j j minja e;j C0 a c;j j ; c; e 1; 2; .; C; cae. (17) (5) Defining and calculating the variance factor as follows: l j V w j maxV w j V b j maxV b j 0 1 A C01 . (18) (6) Calculating the ratio D j (b) and D j (w) and assigning the variance factor E j l j D b j D w j ; (19) then normalizing E j by its maximum value and getting the evaluation criteria E j E j maxE j . (20) It is clear that a larger E j (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 E j Xf, where f is a predefined 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 identification 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 ARTICLE IN PRESS Y. Lei, M.J. Zuo / Mechanical Systems and Signal Processing 23 (2009) 153515471542 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 Te d and the mth training sample Tr md can be expressed as: D wf m X D d1 wf d TE d C0 TR md 2 # 1=2 , (22) where wf d denotes 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 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 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 classification algorithm is applied to the level identification of the gear cracks and final 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 value j (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 Fig. 4. Flow chart of the proposed method. ARTICLE IN PRESS Table 3 Y. Lei, M.J. Zuo / Mechanical Systems and Signal Processing 23 (2009) 15351547 1543 Data description of the three experiments. Experiment Number of training / testing samples Fault modes of training/testing Motor speeds of training/ testing samples (rpm) Loads of training/ testing Samples Label of classification 1 12/12 F0/F0 12001800/12001800 0, 1, 2/0, 1, 2 1 12/12 F1/F1 12001800/12001800 0, 1, 2/0, 1, 2 2 12/12 F2/F2 12001800/12001800 0, 1, 2/0, 1, 2 3 2 12/12 F0/F0 1200, 1600/1400, 1800 0, 1, 2/0, 1, 2 1 12/12 F1/F1 1200, 1600/1400, 1800 0, 1, 2/0, 1, 2 2 12/12 F2/F2 1200, 1600/1400, 1800 0, 1, 2/0, 1, 2 3 3 8/16 F0/F0 12001800/12001800 0/1, 2 1 8/16 F1/F1 12001800/12001800 0/1, 2 2 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. 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 first stage of TFSWT are adopted as the input of the WKNN classifier. The evaluation criteria of the selected seven features using the second stage of TFSWT are used as the weights of the WKNN classifier. The identification 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 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 selected from the 25 features randomly. The KNN classifier is used to recognize the three gear conditions. This method is repeated fifteen 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 classifier to distinguish the different gear conditions. Fig. 6 and Table 4 give its diagnosis results, respectively. 8/16 F2/F2 12001800/12001800 0/1, 2 3 ARTICLE IN PRESS Y. Lei, M.J. Zuo / Mechanical Systems and Signal Processing 23 (2009) 153515471544 Method 1 Method 2 Method 3 Method 4 5 101520253035 60 70 80 90 100 Identification accuracy % K Fig. 6. Accuracy comparison of the four methods for experiment 1. 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 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 the sensitive features but also the other fault-unrelated features to recognize the crack levels, which lead to the middle classification 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 first 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 tothe three level identification of the 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. Table 4 Diagnosis results of the four methods in the three experiments. Exper- iment Method 1 Method 2 Method 3 Method 4 Accuracy (%) CPU times (s) Accuracy (%) CPU times (s) Accuracy (%) CPU times (s) Accuracy (%) CPU times (s) Max. Mean Min. Max. Mean Min. Max. Mean Min. Max. Mean Min. 1 100.00 89.58 75.00 0.2344 96.67 80.17 65.93 0.1719 100.00 96.53 86.11 0.2500 100.00 96.68 86.11 0.2656 2 97.22 87.42 77.78 0.2344 83.33 77.29 66.30 0.1719 100.00 99.15 97.22 0.2500 100.00 99.61 97.22 0.2656 3 97.92 87.67 77.08 0.1406 90.28 78.08 65.28 0.1094 97.92 92.10 87.50 0.1875 100.00 92.62 87.50 0.2031 Method 1 Method 2 Method 3 Method 4 5 101520253035 60 70 80 90 100 Identification accuracy % K Fig. 7. Accuracy comparison of the four methods for experiment 2. It methods obtains gener Because experiments ARTICLE IN PRESS Y. Lei, M.J. Zuo / Mechanical Systems and Signal Processing 23 (2009) 15351547 1545 5.1.3. Experiment 3 To clarify the generalization of the proposed method with the various loads, the testing data with the different loads from those of the training data are used in this experiment, which are described in Table 3. The 24 training samples were acquired 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 they are from 87.50% to 100.00%. The CPU times taken by the four methods in this experiment are 0.1406, 0.1094, 0.1875 and 0.2031s, which are listed in Table 4. They are smaller than those of experiments 1 and 2 because there are fewer training samples 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 are obtained by method 1. However, the best diagnosis results are provided by methods 3 and 4. It is because different features have varying potential in distinguishing crack levels of the gears. Some features are sensitive and closely related to the crack levels, but others are not. Methods 3 and 4 select these sensitive features from the original feature set using TFSWT to identify the crack levels, and therefore the diagnosis accuracies are improved. However, Method 2 randomly selects the same number of the diagnosis features as methods 3 and 4. It means that it could select and use features which contain too much gear crack fault-unrelated information and there is a high degree of overlap between the values of these features between the different crack levels. These features would confuse the classification process, 0.250 (2) (3) is observed from Fig. 7 and Table 4 that the same ranges of the correct rates (97.22100.00%) are achieved by both 3 and 4. The diagnosis accuracies of method 1 (77.7897.22%) are lower than those of methods 3 and 4. Method 2 the lowest accuracies (66.3083.33%). This observation is similar to that of experiment 1, which indicates that the alization of methods 3 and 4 is superior to those of the others, method 1 is inferior and method 2 is the worst one. experiment 2 has the same computational burden as experiment 1, the CPU times are the same between 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 and 0.2656s, respectively. They are listed in Table 4. K Fig. 8. Accuracy comparison of the four methods for experiment 3. Method 1 Method 2 Method 3 Method 4 5101520 60 70 80 90 Identification accuracy % 100 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 other fault-unrelated features to recognize the crack levels and therefore it leads to a middle classification result. Another comparison between methods 4 and 3 indicates that method 4 is generally superior to method 3 in the light of the diagnosis accuracies. Although they have the same ranges of identification accuracies in experiments 1 and 2, the maximum diagnosis accuracy (100%) of method 4 outperforms that of method 3 (97.92%) in experiment 3. This suggests that method