Crack Fault Classification for Planetary Gearbox Based on Feature Selection Technique and K-means Clustering Method

Chinese Journal of Mechanical Engineering, Feb 2018

During the condition monitoring of a planetary gearbox, features are extracted from raw data for a fault diagnosis. However, different features have different sensitivity for identifying different fault types, and thus, the selection of a sensitive feature subset from an entire feature set and retaining as much of the class discriminatory information as possible has a directly effect on the accuracy of the classification results. In this paper, an improved hybrid feature selection technique (IHFST) that combines a distance evaluation technique (DET), Pearson’s correlation analysis, and an ad hoc technique is proposed. In IHFST, a temporary feature subset without irrelevant features is first selected according to the distance evaluation criterion of DET, and the Pearson’s correlation analysis and ad hoc technique are then employed to find and remove redundant features in the temporary feature subset, respectively, and hence, a sensitive feature subset without irrelevant or redundant features is selected from the entire feature set. Further, the k-means clustering method is applied to classify the different kinds of health conditions. The effectiveness of the proposed method was validated through several experiments carried out on a planetary gearbox with incipient cracks seeded in the tooth root of the sun gear, planet gear, and ring gear. The results show that the proposed method can successfully distinguish the different health conditions of a planetary gearbox, and achieves a better classification performance than other methods. This study proposes a sensitive feature subset selection method that achieves an obvious improvement in terms of the accuracy of the fault classification.

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Crack Fault Classification for Planetary Gearbox Based on Feature Selection Technique and K-means Clustering Method

Wang and Shao Chin. J. Mech. Eng. Crack Fault Classification for Planetary Gearbox Based on Feature Selection Technique and K-means Clustering Method Li‑Ming Wang 0 Yi‑Min Shao 0 0 State Key Laboratory of Mechanical Transmission, Chongqing University , Chongqing 400044 , China During the condition monitoring of a planetary gearbox, features are extracted from raw data for a fault diagnosis. However, different features have different sensitivity for identifying different fault types, and thus, the selection of a sensitive feature subset from an entire feature set and retaining as much of the class discriminatory information as possible has a directly effect on the accuracy of the classification results. In this paper, an improved hybrid feature selection technique (IHFST) that combines a distance evaluation technique (DET), Pearson's correlation analysis, and an ad hoc technique is proposed. In IHFST, a temporary feature subset without irrelevant features is first selected according to the distance evaluation criterion of DET, and the Pearson's correlation analysis and ad hoc technique are then employed to find and remove redundant features in the temporary feature subset, respectively, and hence, a sensitive feature subset without irrelevant or redundant features is selected from the entire feature set. Further, the k‑ means clustering method is applied to classify the different kinds of health conditions. The effectiveness of the proposed method was validated through several experiments carried out on a planetary gearbox with incipient cracks seeded in the tooth root of the sun gear, planet gear, and ring gear. The results show that the proposed method can successfully distinguish the different health conditions of a planetary gearbox, and achieves a better classification performance than other methods. This study proposes a sensitive feature subset selection method that achieves an obvious improvement in terms of the accuracy of the fault classification. Planetary gearbox; Gear crack; Feature selection technique; K‑ means classification 1 Introduction Owing to its advantages of a compact structure, large transmission ratio, and high load capacity, a planetary gear transmission system is widely used in large-scale and complex mechanical equipment [ 1, 2 ], e.g., wind turbines, helicopters, and automobiles. A planetary gearbox typically consists of some key components: a sun gear, planet gear, ring gear, carrier, and bearing, and faults may occur in these components owing to fatigue or tough working conditions. According to a condition-monitoring report on wind turbines, a gearbox failure is the leading contributor to all wind turbine failures [ 3 ]. A vibration-based method was proven to be one of the most popular techniques in the fault diagnosis of rotating machinery, and it has been determined that certain changes to the vibration signals can be seen when a fault occurs, e.g., crack or spalling [ 4–6 ]. The commonly used vibration signal processing methods can be divided into three categories: time domain methods, frequency domain methods, and time–frequency domain methods. Time domain methods refer to the analysis of a signal with respect to time, and are relatively easy and direct compared to both frequency and time–frequency domain methods. Statistical indicators, the time synchronous averaging (TSA) method, and an autoregressive (AR) model are typically used in the fault diagnosis of rotating machinery [ 4, 7–9 ]. Frequency domain methods refer to an analysis of the signals with respect to the frequency, and a periodic signal in the time domain can be converted into a frequency component through a Fourier transformation. In this way, researchers can identify the difference between the spectrum of a normal vibration signal and a fault vibration signal with commonly used methods that include a spectrum-based analysis, resonance demodulation technique, and cepstrum analysis [ 10–12 ]. In contrast, time–frequency domain methods are used to study a signal in both the time and frequency domains simultaneously, allowing both the constituent frequency components and their time variation features to be revealed and analyzed. Researchers have developed various time–frequency domain methods including a short-time Fourier transformation, Wigner–Ville distribution, continuous wavelet transform, and HilbertHuang transformation [ 13–16 ]. Although vibrationbased methods have been successfully used in the fault diagnosis and condition monitoring of rotating machinery, the appearance of faults in the analysis results has to be identified artificially, e.g., the identification of a fault characteristic frequency in the spectrum, the determination of a filter sub-band in a demodulation analysis, or the determination of a wavelet type, all of which require considerable of experience and expertise [ 17–19 ]. Therefore, it is necessary to develop some intelligent techniques that can automatically determine the health conditions of a planetary gearbox. Feature extraction is commonly the first important step in an intelligent technique. However, different features display different sensitivity to fault advancements, and some of the features are redundant or irrelevant to the fault diagnosis or classification result [ 19–21 ]. Selecting a sensitive feature subset and retaining as much of the class discriminatory information as possible has a direct effect on the accuracy of the results. Kang et  al. [ 20 ] developed an outlier-insensitive hybrid feature selection methodology to reduce diagnostic performance deterioration caused by outliers in data-driven diagnostics. Peng et  al. [ 22 ] studied how to select good features according to the maximal statistical dependency criterion based on mutual information. Yang et al. [ 23 ] demonstrated an approach to the multi-criteria optimization problem of feature subset selection using a genetic algorithm. Li et al. [ 24 ] presented a two-stage feature selection approach combining filter and wrapper techniques to obtain a more compact feature subset for accurate classification of the hybrid faults of a gearbox. Liu et  al. [ 25 ] introduced a hybrid dimension reduction method that combines kernel feature selection and a kernel Fisher discriminant analysis for a fault-level diagnosis of planetary gearboxes. Lei et al. [ 19, 26 ] and Shen et al. [27] proposed a feature subset selection method based on a distance evaluation technique (DET) for fault classification of a roller bearing and gear reducer. Among these feature subset selection methods, DET is relatively simpler and more efficient than the other methods, and has thus been widely used in fault diagnosis [ 28–30 ]. However, DET tends to select redundant features because it does not consider the relationships between features; one relevant feature selected by DET may be redundant in the presence of another relevant feature with which it is strongly correlated. To address this problem, an improved hybrid feature selection technique (IHFST) combining DET, Pearson’s correlation analysis, and an ad hoc technique is proposed. Using IHFST, the relevant features are selected according to the distance evaluation criterion of DET, and the redundant features are then further suppressed based on the Pearson’s correlation analysis and ad hoc technique. Hence, not only irrelevant features, but also redundant features, are removed from the entire feature set based on the proposed IHFST. Once a sensitive feature subset is selected, it can be classified into several classifications based on some machine learning (ML) methods. The k-means clustering method is a widely used unsupervised pattern recognition algorithm that aims to partition n observations into k clusters, where each observation belongs to the cluster with the nearest mean [ 31–34 ]. Therefore, the k-means clustering method is further employed to validate the effectiveness of the proposed IHFST method. The rest of this paper is organized as follows: Section 2 briefly introduces the feature-subset selection based on DET. Section  3 presents the fault classification process based on the proposed IHFST and k-means method. Section  4 describes the experiment system and data acquisition. The effectiveness of the proposed method as validated using various datasets is then described in Section 5. Finally, some concluding remarks are provided in Section 6. 2 Brief Review of DET In fault diagnosis, different features can reflect different aspects of the vibration properties of the machinery, and certain features are sensitive to certain changes in machine conditions [ 19–21, 25 ]. If all features are applied to a fault diagnosis without a careful selection, the computation complexity of the algorithm will increase with little gain [26]. Therefore, a feature selection process is necessary to reduce the computation complexity and improve the accuracy of the fault diagnosis. The feature selection process based on DET is briefly introduced. The basic concept of DET is to measure the ratio of the between-class distance to the within-class distance in the feature vector space, as illustrated in Figure  1, where a high DET value indicates high sensitivity or class separability of the feature. The detailed process of DET can be expressed as follows [ 19, 26, 27 ]. {qi,k,j, i = 1, 2, . . . , Ik ; k = 1, 2, . . . , K ; j = 1, 2, . . . , J }, (1) where qi,k,j is the feature value of the jth feature from the ith sample of the kth health condition, Ik is the total sample number of the kth condition, K represents the condition number, and J is the feature number in the feature vector of each sample. The process of informative feature selection based on DET is as follows: Step 1: Calculate the within-class average distance of the same condition samples, dk,j = 1 Ik (Ik − 1) Ik l,j=1 qi,k,j − ql,k,j , l, i = 1, 2, . . . , Ik , l = i, and then obtain the average distance of K conditions, d(w) = 1 j K K k=1 dk,j. 1 Ik Ik i=1 qi,j,k . µ k,j = d(b) = j Step 2: Calculate the average feature value of all samples under the same condition, Then, obtain the average between-class distance between different condition samples, 1 K K (K − 1) k,e=1 ue,j − uk,j , k, e = 1, 2, . . . , K , k = e. Step 3: Calculate the ratio between d(b) and dj(w), assigning the compensation factor εj = dj(b)/dj(w), and j then normalize it based on the maximum value and obtain the distance evaluation criterion αj = εj/max(εj). Hence, a feature subset can be selected when the distance evaluation criterion αj is greater than a given threshold. However, the DET method only removes irrelevant features from the entire feature set. Although the (2) (3) (4) (5) features selected by DET carry good classification information when treated separately, there may be little gain if they are combined into a feature vector because of a high mutual correlation [ 21–25 ]. 3 Fault Classification Based on the Proposed IHFST and K‑means Figure  2 shows a flow chart of fault classification based on the proposed IHFST and k-means classification method. The entire feature set is first extracted from time domain signal, frequency domain signal, and difference signal. A sensitive feature subset without irrelevant features or redundant features is then selected based on the proposed IHFST. The feature subset is then classified using the k-means clustering method. 3.1 Feature Extraction Feature extraction is the first important step of the proposed method. Before calculating the features, the raw vibration signals are divided into several segments by multiplying a sliding Hanning window owing to its good properties, e.g., better side-lobe behavior shown in the spectrum [ 35 ], as indicated in Figure 3. The sliding Hanning window equation is given as follows:  0.5 1 − cos 2π(m−δ) , δ ≤ m ≤ M + δ − 1 ,  M−1 W (m) =  0, otherwise,   (6) where M = k × fs is the width of the sliding Hanning window, k is the time length of the window, fs is the sampling frequency, δ  =  j  ×  fs is the sliding distance, and j is the time length of the sliding step. Here, M and δ are both integers, and the values of k and j were chosen as 4 and 0.5, respectively. Three types of features are extracted from time domain signal, frequency domain signal, and difference signal, respectively, as given in Table 1. The difference signal is defined as Ref. [ 4 ] kept and act as feature weights of the selected where x is the raw vibration signal, yd is the signal containing the mesh frequencies, their harmonics, and their first-order sidebands. Thus, d is composed of higher-order sidebands and Gaussian noise. d = x − yd , (7) Here, x is the original time domain signal; N is the number of samples; Sk is the spectrum of x for k = 1, 2,…, K; K is the number of spectrum lines; y¯d and d¯ are the mean values of yd and d, respectively; PPx indicates the maximum peak-to-peak amplitude of signal x; Ph is the amplitude of the hth harmonic; H is the total number of harmonics within the frequency ranges. In this study, eight features are calculated from time domain signal, four features from frequency domain signal, and four from difference signal. In addition, because the rotating speed is low, the calculated gear mesh frequency [ 36 ] and its harmonics are concentrated at a low frequency range, and thus, a low-pass digital filter is used to cover the main gear mesh components of the vibration signals, and cut-off frequencies of 650 and 900 Hz are chosen for the low-pass digital filter at 300 and 400 r/min, respectively. Thus, another sixteen features are calculated from the filtered vibration signals, and a total of thirtytwo features are obtained within the entire feature set. 3.2 IHFST To remove both irrelevant and redundant features from the entire feature set, the IHFST method combining DET, Pearson’s correlation analysis, and the ad hoc technique is proposed, as shown in Figure  4. The distance evaluation criterion αj of each feature is obtained based on DET, and the entire feature set is then sorted in descending order according to αj. Next, the mean correlation coefficient of the sorted feature set can be computed based on the Pearson’s correlation analysis. Figure 4 Scheme of the proposed IHFST ρ¯j = 1 M − 1 ρ(qk,Jm , qk,j) = M−1 m=1 ρ(qk,Jm , qk,j) and iI=k1 (qi,k,Jm − q¯k,Jm )2 iI=k1 (qi,k,Jm − q¯k,Jm )(qi,k,j − q¯k,j) I i=k1 (qi,k,j − q¯k,j)2 , Jm = j where, ρ¯j is the mean correlation coefficient of the jth feature, M is the ranking index of the jth feature according to αj, Jm is the feature index of the top informative feature for m  =  1, 2,…, M  −  1, qk,Jm = [q1,k,Jm , q2,k,Jm , . . . , qIk ,k,Jm ] and qk,j = [q1,k,j, q2,k,j, . . . , qIk ,k,j] are two feature vectors, and q¯k,Jm and q¯k,j are the mean values of these two features, respectively. The correlation coefficient ranges from − 1 to + 1, which indicates a high degree of negative or positive linear correlation between qk,Jm and qk,j. Finally, the ad hoc technique associated with the mean correlation coefficient is applied to reevaluate the distance evaluation criteria αj, and a feature that is highly correlated with the top informative features will be suppressed or removed. The new distance evaluation criterion βj is obtained, i.e., βj = αj, when M = 1, θ1αj − θ2ρ¯j, when M ≥ 2, (10) where, θ1 and θ2 are two weighting factors that determine the relative importance of the informative and the correlation terms, respectively. A large θ1 factor emphasizes the informative term, and a relatively larger θ2 factor weights the correlation term more heavily, and can thus produce a feature subset with less redundancy [ 25 ]. In this paper, when the mean correlation coefficient is higher than 0.7, the feature will be recognized as a severe redundant feature, and will thus be removed. In another situation described herein, the two factors are chosen as θ1 = 1 and θ2 = 0.2. Hence, a feature subset without irrelevant or redundant features can be selected according to the value of the new distance evaluation criteria βj. 3.3 Normalization and Weighting Before classification, all features are normalized as follows: nFEj = FEj − mean(FEj) , j = 1, 2, . . . , J , std(FEj) (11) where FEj is the feature to be normalized, mean() is the mean function, and std() is the standard variance function. Although the feature subset is selected based on the IHFST, different features in the feature subset have different sensitivities during the fault diagnosis process. Thus, to emphasize the importance of sensitive features, (8) (9) a feature-weighting step is needed to guarantee a more accurate classification result. Supposing there are L features selected by IHFST, described in Section 3.2, the new distance evaluation criteria βj of these selected features are also kept and act as feature weights of the selected features. 3.4 K‑means Clustering Method As an unsupervised clustering method, k-means is commonly used to automatically partition samples into k clusters [ 31 ]. The purpose of the k-means clustering method is to assign all N samples into k clusters by minimizing the sum of point-to-centroid distances as follows: J = arg min C x − μi , k i=1 x∈Ci where C = {C1, C2, . . . , Ck } indicates k clusters, x is an N × R feature matrix, R represents the dimensions of the matrix, each row is a single observation or sample, and μi indicates the cluster centroid of the ith cluster. The detailed process of the k-means clustering method is given as follows. Step 1: Initialization Randomly choose k cluster centroids for all feature samples. Step 2: Assignation Assign each sample to the nearest cluster centroid by measuring the distance between the sample and each centroid as follows: Cit = {xp : xp − μit 2 ≤ xp − μjt 2 ∀j, 1 ≤ j ≤ k}. Step 3: Update Find all samples in each cluster, and determine the new cluster centroid using (12) (13) (14) μt+1 = i 1 N t i xj∈Cit xj, where Nit is the sample number of the ith cluster at the tth iteration. Step 4: Repeat Steps 2 and 3 until the cluster centroid remains unchanged or the function achieves convergence. 4 Experimental System Figure  5 illustrates the “back-to-back” planetary gearbox experiment setup, which consists of two planetary gearboxes (PGB), two motors, an industrial computer, a control box, and a set of vibration acquisition systems. The test bed is symmetrically arranged, PGB 1 and PGB 2 have the same structure and two motors each, and the four components are connected using three teeth-shaft couplings. The design parameters of the PGB are given in Table 2, where the PGB is comprised of an inner sun gear surrounded by four planet gears and a standstill ring gear, and the gear ratio of the PGB is 6.25. It should be noted that the simulated faults are located in PGB 1. The two motors are SIEMENS three-phase 15  kW induction motors, where motor 1 is for driving and motor 2 is for loading; in addition, the range of the motor speed is 0–1450 r/min, and the two motors are controlled using an industrial computer through a control box. To measure the vibration signals, five Kistler integrated circuit piezoelectric accelerometers, denoted as #1, #2, #3, #4, and #5, are placed in the test bed. Here, #1 is placed in the motor base, #2 is placed in the housing of the input bearing, #3 is placed in the housing of the planetary gearbox, #4 is placed in the ring gear of the planetary gearbox, and #5 is placed in the housing of the output bearing, as illustrated in Figure  5a. An LMS SCADAS system and a computer are used for data acquisition, the sampling frequency of which is 20,480 Hz, and the sampling length is 11 s. Figure  6a–c illustrate seeded incipient cracks located at the tooth root of the sun gear, planet gear, and ring, respectively. The parameters of the seeded crack are defined as (q0, q1, Wc, αc), as shown in Figure 7. As is well known, it may be more difficult to detect gear faults at low speed because low-frequency characteristics are easily masked by heavy noise [ 18 ]. To verify the effectiveness of the proposed method, vibration datasets were collected under two relatively low rotating speeds of 300/400  r/min and four loading conditions of 0, 20, 40, and 60  Nm, as listed in Table  3. Altogether, there were eight vibration datasets collected, each of which includes four types of health conditions: normal, cracked sun gear (CS), cracked planet gear (CP), and cracked ring gear (CR). In addition, the vibration signals of accelerometer #4 mounted on the outer ring gear of PGB 1 were analyzed in this study. In addition, Figure 8a, b show an example of vibration signals for each health condition in dataset 1 and their spectrum, respectively. It can be observed there are few differences in the raw vibration signals and their spectrum for the four health conditions: normal, CS, CP, and CR, and it is difficult to distinguish the three types of faults from the normal condition. 5 Results and Discussion The distance evaluation criteria α and β, and the corresponding feature weights of dataset 2, are shown in Figure  9a, b, respectively. It can be observed that three features (Nos. 19, 20, and 26) with higher mean correlation coefficients in the DET method are suppressed according to the new distance evaluation criteria β. Twelve sensitive features are selected from the entire feature set using the DET method with a given threshold, Thr(α) =  mean(αj), and eight sensitive features are selected using the IHFST method with a given threshold, Thr(β) = mean(βj), where j = 1, 2,…, 32 denotes the feature index. It should be noted that dataset 2 is taken as an example to show the effectiveness of the proposed method, and the other datasets are also analyzed. In this study, three types of feature selection methods are employed to analyze the same dataset using the proposed method for comparison, which are denoted as Method 1, Method 2, and Method 3. In Method 1, two commonly used features, the root mean square and kurtosis, are selected from the entire feature set. In Method 2, the dimensions of the entire feature set are directly reduced based on a principle component analysis (PCA) [ 37 ] without feature selection. In Method 3, features are selected based on DET. Finally, the proposed method is referred to as Method 4. The classification performances of these four methods are shown in Figure  10a–d. In addition, it should be noted that, for visualization, the PCA method is implemented based on the clustering results produced through Method 3 and Method 4, and the plots of the first two PCs are shown in Figure 10c, d. Many misclassification samples can be found in Figure  10a, b, which are based on Method 1 and Method 2, and the samples have a loose distribution. In Figure 10c, CS and CP can be clearly discriminated, but it is difficult to discriminate the CR from normal conditions, and there are five misclassification samples. In Figure  10d, the four health conditions can be clearly discriminated, and all four clusters have small within-class distances and large between-class distances. To compare the classification performance of the four methods, the silhouette value (SV) is adopted in this study. The SV is defined as a measure of how similar a point is to other points in its own cluster compared to points in other clusters, and ranges from −  1 to +  1, where a value close to + 1 indicates a better classification performance [ 38 ]. The SV can therefore be obtained: S(i) = min(b(i, :), 2) − a(i) max(a(i), min(b(i, :))) (15) where a(i) is the average distance from the ith point to other points in its cluster, and b(i, k) is the average distance from the ith point to points in another cluster k. The SV of the classification is obtained based on the average of all S(i). Table  4 presents the classification results of different datasets based on the four methods, along with their misclassification error and SV. There are 36 misclassification errors with Method 1, 44 misclassification errors with Method 2, 7 misclassification errors with Method 3, and zero misclassification errors with Method 4. In addition, the SV of Method 4 is greater than that of the other three methods for all datasets. The improvement of Method 4 over the other three methods is considered to be as follows: Sk = SVM4 − SVMk × 100%, k = 1, 2, 3, SVM4 (16) where SVM4 denotes the silhouette values of the classification using Method 4, and SVMk denotes these values for the other three methods, for k = 1, 2, and 3. Figure 11 shows a comparison of the classification performance between Method 4 and the other three methods. It can be observed that Method 4 consistently yields a better classification performance than the other three methods for the different datasets. Method 4 achieves a 30.51% to 51.74% improvement over Method 1, 30.92% to 42.43% improvement over Method 2, and 0.89% to 16.45% improvement over Method 3. As mentioned in Section  3.1, feature extraction is one of the most important steps in the proposed method, and the parameters of the sliding Hanning window have a SV direct influence on the classification performance. Therefore, the effects of the Hanning window width and sliding distance on the classification performance of the four methods are illustrated in Figures 12 and 13, respectively. Figure 12a–h display the effect of the window width on eight datasets, 1 to 8. The window width ranges from 1 to 6 s, and the sliding distance is fixed to 0.5 s. It can be seen that, as the window width increases, the SVs of the four methods show an increasing trend, which may result from too little fault-related information in each segment when the window width is small. In addition, Method 4 shows a higher SV than Method 1 and Method 2 for each dataset, and shows a higher SV than Method 3 for datasets 1 to 4, and dataset 6. Figures 13a–h display the effect of the sliding distance on the eight datasets, 1 to 8. The sliding distance ranges from 0.1 to 0.9 s, and the window width is fixed to 4 s. It can be seen that, as the sliding distance increases, the SVs of Method 3 and Method 4 remain nearly unchanged, and the SVs of Method 1 and Method 2 have a small range of fluctuations. In addition, Method 4 shows a higher SV than Method 1 and Method 2 for each dataset, and a higher SV than Method 3 for most of the datasets. For Method 3 and Method 4, a feature subset is selected from the entire feature set with a given threshold, and it is important to clarify the relationship between the threshold and the classification performance. Hence, the effect of the feature selection threshold on the classification performance for Method 3 and Method 4 is also studied, as shown in Figure 14. The thresholds for Method 3 and Method 4 range from 0.5  ×  Thr(α) to 2.0 ×  Thr(α), and from 0.5 × Thr(β) to 2.0 × Thr(β), respectively. It can be seen that, as the threshold increases, the SVs of Method 3 and Method 4 have a small range of fluctuation, and Method 4 shows a higher SV than Method 3 for each dataset. 6 Conclusions In this paper, an IHFST method combining DET, Pearson’s correlation analysis, and an ad hoc technique was proposed. A sensitive feature subset without irrelevant or redundant features was selected from the entire feature set based on the proposed IHFST method. The k-means clustering method was further employed to automatically partition the vibration data acquired from a planetary gearbox with crack faults into several different classifications. To validate the effectiveness of the proposed method, three types of feature selection methods were employed to analyze the same dataset using the proposed method for comparison. The results indicate that the proposed method can discriminate the four types of health conditions of a planetary gearbox clearly for all the datasets used, and no misclassifications were found. The proposed method achieves a 30.51% to 51.74% improvement over Method 1, 30.92% to 42.43% improvement over Method 2, and 0.89% to 16.45% improvement over Method 3. In addition, the influence of the sliding Hanning window and feature selection threshold on the classification performance was investigated, and the proposed method again achieved a better classification performance than the other three methods. Authors’ contributions LW carried out the experiment studies and drafted the manuscript. YS carried out the sequence alignment and revision of the manuscript. All authors read and approved the final manuscript. Authors’ Information Li‑Ming, Wang born in 1987, is currently a PhD candidate at State Key Laboratory of Mechanical Transmission, Chongqing University, China. He received his bachelor degree from Chongqing University, China, in 2011. His main research interests include fault diagnosis of rotating machinery and signal processing. Yi‑Min Shao, born in 1963, is currently a professor at Chongqing University, China. He received his PhD degree from Gunma University, Japan, in 1997. His research interests include mechanical dynamics, signal processing, fault diagnosis of rotating machinery. Acknowledgements Supported by National Natural Science Foundation of China (Grant No. 51475053). Competing interests The authors declare that they have no competing interests. Ethics approval and consent to participate Not applicable. Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in pub‑ lished maps and institutional affiliations. 1. Y C Guo , R G Parker . Analytical determination of mesh phase relations in general compound planetary gears . Mechanism and Machine Theory , 2011 , 46 ( 12 ): 1869 - 1887 . 2. 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Li-Ming Wang, Yi-Min Shao. Crack Fault Classification for Planetary Gearbox Based on Feature Selection Technique and K-means Clustering Method, Chinese Journal of Mechanical Engineering, 2018, 4, DOI: 10.1186/s10033-018-0202-0