Clustering methods have been widely applied to the fault diagnosis of mechanical system, but the characteristic that the number of cluster needs to be determined in advance limits the application range of the method. In this paper, a novel clustering method combining the adaptive resonance theory (ART) with the similarity measure based on the Yu’s norm is presented and applied to the fault diagnosis of rolling element bearings, which can be adaptive to generate the number of cluster by the vigilance parameter test. Timedomain features, frequencydomain features, and time series model parameters are extracted to demonstrate the faultrelated information about the bearings, and then considering the irrelevance or redundancy of some features many salient features are selected by an improved distance discriminant technique and input into the proposed clustering method to diagnose the faults of bearings. The experiment results confirmed that the proposed clustering method can diagnose the fault categories accurately and has better diagnosis performance compared with fuzzy ART and SelfOrganizing Feature Map (SOFM).
In order to decrease the downtime on production machinery and to increase reliability against possible failures, some important machinery is equipped with condition monitoring systems, but how to be intelligent to classify the data samples collected by the condition monitoring system is challenging. Artificial neural networks (ANNs) used as an intelligent classification tool have been widely applied in the fault diagnosis field of machine conditions which are treated as classification problems based on learning pattern from empirical data modeling in complex mechanical processes and systems [
Currently clustering methods, owing to their superiority in independency of supervisors, have been widely studied and applied to the field of fault diagnosis. According to the principle that similar objects are within the same cluster and dissimilar objects are in different clusters, most of clustering methods mainly employ the similarity and distance measure to partition a dataset into several clusters, such as
To the best of our knowledge, the clustering method using similarity measure based on the Yu’s norm is seldom applied in the fault diagnosis of mechanical system. And in the fault diagnosis application when the samples of different fault classes overlap in some regions in the feature space, the traditional hard (crisp) clustering method mainly uses distance to compare data sample to fault classes and classify the data sample into one and only one cluster [
The rest of the paper is organized as follows. The review of adaptive resonance theory and clustering method using similarity measure based on Yu’s norm is introduced in Section
Adaptive resonance theory (ART) which is treated as a theory of human cognitive information processing was designed by Carpenter and Grossberg in 1976. ART network is an online learning system, and it mainly utilizes the selforganization to develop the stable and plastic clustering of input samples; namely, the ART network uses the vigilance test to resolve the stabilityplasticity dilemma. In the learning process when a new sample is input into the ART network, it can attempt to categorize the sample by comparing it with the stored weight vectors of existing cluster node which represented a category. If the sample is similar to the existing categories and the match degree is greater than or equal to the vigilance value, the sample is classified into the specified category and the weight vectors of the corresponding cluster node are modified. Otherwise, a new category is created without affecting the existing memory. Its detailed dynamic character and algorithm can be seen in [
Because the fuzzy relation can be called a similarity relation [
The clustering algorithm is described as follows. Assume that a set
The ART neural network and the clustering method using similarity measure based on Yu’s norm have their respective advantages as has been noted. The proposed ARTsimilarity clustering method is the synthesized product of their respective advantages. Its architecture which is shown in Figure
The architecture of ARTsimilarity classifier based on Yu’s norm.
The fault diagnosis system is shown in Figure
Architecture of fault diagnosis system.
The schematic diagram of experiment rig is shown in Figure
The schematic diagram of the experimental setup.
The test bearing type is 62052RS JEM SKF, deep groove ball bearing. The single point defects are introduced into the driveend bearing of the motor by the electrodischarge machining. Four different defect diameters (0.007, 0.014, 0.021, and 0.028 inch) are introduced into the balls to simulate different fault severity of bearings; 0.014 inch defect diameter is introduced into the inner race and outer race, respectively, to simulate the different fault categories of bearings, and these defects’ depth is all 0.011 inch; each bearing is tested under four different loads (0, 1, 2, and 3 hp) and rpm ≈ 1800. Thus, the bearing data sets can be obtained from the experimental system under different operation loads and seven different fault conditions: (1) in normal condition; (2) with outer race fault; (3) with inner race fault; (4) with four different severity levels of ball faults.
Feature parameters are mainly utilized to depict the faultrelate information about the bearings. To acquire more information many different feature parameters are extracted from the vibration signals.
Statistical feature parameters in time domain and frequency domain are often used to characterize the shape of vibration signal from different perspectives. In this study nine timedomain feature parameters and seven frequencydomain feature parameters are extracted and used as the basis for the fault diagnosis of bearings, which are listed in Table
Timedomain and frequencydomain feature parameters.
Timedomain feature parameters  Frequencydomain feature parameters  

Feature  Equation  Feature  Equation 
Mean ( 

Mean frequency 

Root mean square ( 

Average frequency that is wave shape of signal crosses the mean of timedomain signal 

Standard deviation ( 

Stabilization factor of wave shape 

Skewness ( 

Coefficient of variability 

Kurtosis ( 

Frequencydomain skewness 

Crest factor (CF) 

Frequencydomain kurtosis 

Latitude factor (LF) 

Rootmeansquare ratio 

Shape factor (SF) 

where 

Impulse factor (IF) 


where 
Time series model can characterize the dynamic process of mechanical system. Because of sensitiveness of these model parameters to the shape of vibration data, these parameters are also used as feature parameters to demonstrate the faultrelated information about the bearing. Autoregression (AR) model which is the basis time series model can work as predictor; its basic expression can be written as follows:
Thus, an original feature set containing 32 feature parameters is obtained, which can preserve faultrelated information that cover timedomain, frequencydomain and waveletdomain.
When the above 32 features are used as the input of the proposed ARTsimilarity clustering method to diagnose the bearings, there is a possibility that the diagnosis accuracy decreases and the computation time is increased because of the redundancy or irrelevance of some features. In order to improve the diagnosis performance, some sensitive features providing characteristic information for the diagnosis system need to be selected, and irrelevant or redundant features must be removed. Here, the distance discriminant technique [
Assume that a feature set of
Calculate the standard deviation and the mean of all samples in the
Calculate the standard deviation and the mean of the sample in the
Calculate the weighted standard deviation of the class center
Calculate the distance of the
Define and calculate the variance factor of
Calculate the distance of the
Define and calculate the variance factor of
The compensation factor of the
Calculate the modified distance discriminant factor of the
Rank
Set a threshold value
Further, in order to demonstrate the superiority and character of the improved distance discriminant technique, a numerical example to compare the improved distance discriminant technique with the original distance discriminant technique is presented in the appendix.
In the phase of fault diagnosis some data samples of bearings are utilized to evaluate the performance of the proposed method, the data samples contain seven different fault conditions, the fault conditions are labeled by Arabic numerals
Statistics of each fault condition of bearing.
Condition  Normal  Inner race fault  Outer race fault  Different severity level of ball fault  

Defect diameter (inch)  0  0.014  0.014  0.007  0.014  0.021  0.028 


Label  1  2  3  4  5  6  7 


Number of data samples  25  25  25  25  25  25  25 
The detailed fault diagnosis flow chart of the proposed method is shown in Figure
Fault diagnosis flowchart of the proposed method.
Salient feature parameters selection.
Finally, the proposed ARTsimilarity clustering method based on Yu’s norm is applied to the fault diagnosis of bearings. Its characteristics are training and test together. The 175 data samples are used for training and test. In the beginning of the training of the cluster model, it is empty. When the first data sample is input into the cluster model, the first cluster node is produced which is considered as one fault category. When the next input sample enters the model, it is compared with the first cluster node. If the similarity degree is bigger than the set vigilance parameter
Generally, one fault class is needed to use many cluster nodes to learn because of the complex fault mechanism. To evaluate the performance of the proposed Yu’s norm based on ARTsimilarity clustering method and understand the relationship of classification accuracy, the number of cluster nodes, and vigilance parameter, a series of fault diagnosis experiments with the increasing vigilance parameters are conducted. For convenience of computation, the classification accuracy can be obtained by the following formula [
Figure
Classification accuracy with the different vigilance parameters.
Relationship of cluster nodes number and vigilance parameters.
For convenience of understanding, Figure
Number of neurons presenting each condition.
Label  1  2  3  4  5  6  7  Total 

Number of neurons  1  2  1  1  1  1  8  15 
Classification of all data samples.
It is well known that the initial conditions affect the performance of ARTsimilarity clustering method. To study the stability and generalization of the proposed method, the bootstrap method is used to compute the estimated mean, standard deviation, and confidence interval for the classification accuracy, which is useful for estimating a parameter when the underlying distribution function of parameter is unknown [
Statistical performance of ARTsimilarity clustering method.
Clustering method  Mean (%)  Standard deviation (%)  Confidence interval (%) 

ARTsimilarity  98.95  0.54  [97.87, 99.58] 
In order to validate the superiority of the proposed ARTsimilarity clustering method, the classification result produced by ARTsimilarity classifier is compared with that produced by other conventional unsupervised neural networks, such as the fuzzy ART and SOFM network. Same data samples are utilized to evaluate these methods: 100 of these samples are used for the training of these networks, and the rest, for test. The classification results of the ARTsimilarity clustering method versus other classification methods with the same salient feature parameters are shown in Table
Comparison of classification with different neural networks.
Classification method  Fuzzy ART  SOFM  ARTsimilarity 

Number of cluster nodes  79  76  15 


Classification accuracy (%)  96.57  94.36  100 
In this paper a new clustering method that combines the adaptive resonance theory (ART) with the similarity measure based on Yu’s norm is presented to diagnose the faults of rolling element bearings, which can generate the cluster nodes dynamically. Before application of the proposed clustering method to the fault diagnosis of bearings, timedomain statistical characteristics features, frequencydomain statistical characteristics features, and AR time series model parameters are extracted to characterize the faultrelated information of bearing.
Owing to the redundancy and irrelevance of some features the improved distance discriminant techniques are used to select the sensitive features, and then they are input into the proposed clustering method to diagnose the fault categories of bearings. The experiment result showed that the proposed ARTsimilarity clustering method can diagnose the faults of bearings successfully, and its diagnosis accuracy is higher than fuzzy ART and SOFM. And because the initial conditions affect the performance of the proposed clustering method, the bootstrap method is utilized to analyze the diagnosis results. The statistical analysis result shows that the proposed clustering method is stable and generalized. All these indicate that the proposed method has better diagnosis ability and performance and further demonstrate that the proposed clustering method has a good promise in the field of fault diagnosis of mechanical system.
Suppose the simulated date set is comprised of four classes. Each class consists of 35 samples, and each sample is depicted by six features. Figure
The distribution of six features under four classes, where
To ascertain classification ability of each feature, the distance discriminant technique is adopted to evaluate the sensitivities of the six features. The evaluation results are shown in Figure
The distance discriminant factors of the six features (a) produced by the distance discriminant technique and (b) produced by the modified distance discriminant technique.
The authors declare that they have no competing interests.
The work is supported financially by the National Natural Science Foundation of China (Grant No. 51405353), and the Open Fund of the Key Laboratory for Metallurgical Equipment and Control of Ministry of Education in Wuhan University of Science and Technology (2014B01), and National Natural Science Foundation of China (Grant nos. 51475339 and 51575202).