Abstract
Rolling element bearings are widely used in high-speed rotating machinery; thus proper monitoring and fault diagnosis procedure to avoid major machine failures is necessary. As feature extraction and classification based on vibration signals are important in condition monitoring technique, and superfluous features may degrade the classification performance, it is needed to extract independent features, so LSSVM (least square support vector machine) based on hybrid KICA-GDA (kernel independent component analysis-generalized discriminate analysis) is presented in this study. A new method named sensitive subband feature set design (SSFD) based on wavelet packet is also presented; using proposed variance differential spectrum method, the sensitive subbands are selected. Firstly, independent features are obtained by KICA; the feature redundancy is reduced. Secondly, feature dimension is reduced by GDA. Finally, the projected feature is classified by LSSVM. The whole paper aims to classify the feature vectors extracted from the time series and magnitude of spectral analysis and to discriminate the state of the rolling element bearings by virtue of multiclass LSSVM. Experimental results from two different fault-seeded bearing tests show good performance of the proposed method.
1. Introduction
Rotor machinery condition monitoring is gaining importance as the need to increase reliability and to decrease possible loss of production due to machine breakdown. Predictive maintenance based on vibration signal is efficient [1]. Bearings serve as the best location for measuring machinery vibration since this is where the basic loads and forces of machine are applied [2]; bearing faults are common problems in high speed rotating machinery such as aero-engine, and bearing faults recognition has been an important research topic in pattern recognition over the last decade. Many reasons cause rolling bearing faults, for example, unbalance, critical speed, improper mounting, rubbing, bad lubrication, and plastic deformation. Some fault vibration phenomena can be interpreted as an amplitude modulation of the characteristic vibration frequency of the machine. Statistical data [3] shows that 90% of faults which occur in rolling bearings are due to cracks in inner and outer race; the rest are cracks in balls or cage.
SVMs are widely used classifiers in machine diagnosis of rolling element bearings applications [4, 5]. Data preprocessing is important for good classification; it will reduce the data dimension and retain as much information as possible.
Feature selection is necessary to remove garbage features and reduce the correlation [6], which may degrade classification performance. Raymer et al. employ GAs for automatic feature selection in machine condition monitoring [7]. Hou et al. extract feature based on genetic programming and linear discriminant analysis with aims to eliminate correlations between features and reduce dimensions of the high-dimensional data [8]. Hu et al. combine rough set and kernel Fisher discriminant analysis for aero engine fault feature extraction by virtue of optimizing between-class and within-class scatter [9]. Lee et al. apply KNMF to extract discriminative spectral feature from EEG data for classification [10]. Cao et al. propose a hybrid scheme which is composed of independent component analysis (ICA) and support vector machine (SVM) to determine the fault quality variables when a step-change disturbance existed in a multivariate process; it is concluded that, among the three methods including PCA, KPCA, and ICA, there is the best performance in KPCA feature extraction, followed by ICA feature extraction [11]. Keskes et al. utilize SWPT for feature extraction under lower sampling rate [12]. Totally, almost all the traditional methods usually remove the redundant and irrelevant features and extract independent features from the original feature set to obtain better performance. Consequently, it is worth conducting dimension reduction before further works. And the sensitive feature design method is rarely developed.
The purpose of this investigation is to present a useful method for fault feature extraction of rolling bearing based on the KICA-GDA, and sensitive feature selection method is proposed here. Particularly, we extract independent features from original selected features by virtue of KICA and reduce dimensionality of the input feature space to select useful discriminative features with the help of GDA. Then we classify the preprocessed data by LSSVM. Finally, we compare the effectiveness of LSSVM classifiers on the original and preprocessed data to examine the effectiveness of the proposed method.
2. Rolling Element Bearing Vibration Features
2.1. Traditional Features
The bearing mainly consists of inner ring, out ring, retainer, and ball. When there exists defect in rolling element, the impulse-forced response is produced. Generally, most of the bearing vibration is comprised of periodic motion.
A lot of works have studied the vibration characteristics of rolling bearing [13, 14]; this paper select features from time and frequency domain. Assume observation samples , the feature selected shown in Table 1.
Some detail expressions in Table 1 are as follows.
Mean value is
Root mean square value is
Peak value is
Root square value is
Standard deviation is
2.2. Sensitive Feature Set Design
A new method named sensitive subband feature set design (SSFD) based on Wavelet packet is presented here. The features listed in Table 1 are calculated in each subband, and the information entropy is substituted by wavelet packet Shannon entropy, since they both reflect the time series complexity.
The sensitivity is measured by variance based on a basic fact that the more disperse the feature distributed, the larger the variance. The schematic table of one feature value of different faults under different operating conditions is shown in Table 2. The value in each band is mean value of times divided time series.
The variance is calculated as follows: where means value of .
In order to select the sensitive features in the whole bands, the variance differential spectrum is defined as follows: where .
The advantage of variance differential spectrum is that it can clearly show the dominate variances; the first big value index indicates the number. Figure 1 shows its advantage, from which it can be seen clearly that there are 4 dominant components.
Aimed at different operating condition, the union set is employed to define sensitive features; it can be expressed as follows: where indicates the sensitive subband set. The new features are summarized as features of .
3. Hybrid KICA-GDA-LSSVM Method
The task of the ICA algorithms is to estimate the independent features from original features which can be modeled by finding a separating or demixing matrix such that where is feature vector and is the estimated feature vector .
KICA is a nonlinear feature extraction technique which defines contrast functions based on canonical correlations in a reproducing kernel Hilbert space. The key issue is employing a “kernelized” version of CCA to compute a flexible contrast function for ICA. The kernel idea is to define a new contrast function for ICA in the reproducing kernel Hilbert space.
Firstly, we map the original feature into feature space by introducing a kernel function , where is nonlinear map function, which projects features from Euclidean space to Hilbert space . Then we formulate and optimize the contrast function to obtain the demixing matrix based on CCA. More detailed procedure is referenced in paper [15].
Secondly, we classify the extracted independent features by virtue of GDA with the aim to improve the separability.
The features are mapped into higher dimension space, then Fisher discriminate analysis is employed, and then the features’ nonlinear discrimination in original space is realized.
The criterion function in mapped space can be expressed as follows: where . Assume data has been centered in ; and are within class scatter matrix and total scatter matrix which are expressed as follows: where .
It has been proved that the optimal solution of (2) can be obtained by solving the following generalized eigenvalue and eigenvector problem: Furthermore, can be expressed by the linear combination of ; that is, where is expansion coefficient.
With the kernel trick, the projection of test data can be simply expressed as follows: where are the new projected vectors in dimension space.
Finally, are classified by LSSVM which is the evolution of SVM. The most important difference is that the loss function of LSSVM is least square linear system rather than quadratic optimization method. As an improvement of Vapnik’s standard SVM, the objective function of LSSVM is formulated as
The dual problems are built as follows:
Then we can get support vectors by KKT condition; the detailed procedure is in [2].
The whole procedure of our proposed method can be decomposed as the following several steps.(1)Calculate features in time and frequency domain listed in Table 1.(2)Calculate subband features and select the sensitivity band using proposed method.(3)Extract independent features in mapping space by virtue of KICA.(4)Reduce the independent features dimension with the help of GDA.(5)Classify the low-dimensional features by LSSVM.
4. The Experiment Results
4.1. Induction Motor Ball Bearing Fault Recognition
The fault deep groove ball bearing experimental data used in this paper is obtained from Case Western Reverse Lab [16]. The test stand consists of a 2 hp, three-phase induction motor and it is connected to a dynamometer; the test bearings support the motor shaft. Single point faults are introduced to the test bearings using electro discharge machining with fault diameters of 0.5334 mm, depth 0.2794 mm. This paper considers inner raceway fault, ball fault, and outer raceway fault. The data set includes 240 samples with ten features and three output labels. Each class includes 80 samples. Training data set includes 210 samples (70 samples from each class). Testing data set includes 30 samples (10 samples from each class), the test sets are selected randomly in 60 times, and the subsequent KICA-GDA-LSSVM calculation is also repeated 60 times.
The features listed in Table 2 are calculated, the original experimental data are filtered by wavelet packet, using Symlets wavelet function, and the signal is decomposed into 3 layers. Its correlation matrix is shown in Figure 2, from which we can find that the features are not independent separately. And the 3 fault states features in feature space are shown in Figure 3; it shows that 3 fault states features are correlated to each other.
Then the data is classified by LSSVM classifier directly, a radial basis function (RBF) is used, and the resulting optimum values were 0.02 and 2 for and . The average classification accuracy result over 60 runs is 90.13%. The schematic diagram is shown in Figure 4; the feature relativity and high feature dimensionality degrade the performance of classifier.
The data set is then processed by KICA-GDA method; Figure 5 shows the correlation coefficient matrix between each state processed by KICA. Compared with Figure 2, it is easy to show that when the dataset is processed by KICA, the correlation between each feature is reduced. Then the independent features’ dimension is reduced by GDA; the features projected in the feature space are shown in Figure 6(b). Compared with Figure 6(a), it can be see that, in the feature space, the different fault classes are located close to each other and obviously separated from the other classes; the box-plots of new features are shown in Figure 7. The distribution difference between each feature is obvious. The processed data set is then classified by LSSVM; the average classification accuracy reaches 99.54%. Figure 8 is a schematic of classification accuracy calculation; it can be seen that eliminating redundant information and reducing the sample dimension can improve classification performance.
(a) Features processed by KICA
(b) Features processed by KICA-GDA
(a) First feature
(b) Second feature
According to the results presented in Table 3, KICA has no improvement in the classification performance of LSSVM classifier obviously, GDA has better performance than LDA, and the hybrid KICA-GDA-LSSVM method demonstrates the best classification performance.
The sensitive subband features are taken into consideration; the features’ distribution under whole operation status considered in this paper is shown in Figure 9. By extracting several dominant subbands features, the classification performance improved as Table 4 shows.
4.2. Gear Box Bearing Fault Recognition
The transfer-box type is JZQ250, its test stand is shown in Figure 10(a), it is driven by three-phase induction motor, and the driving force is transferred from gearbox to Load wheel, so the core apparatus is gear box, the inner structure schematic is shown in Figure 10(b), the gear box bears torque and pressure changes, so it inclines to failure, and it is important to take research in gear-box monitoring and diagnosis. The data sampling rate is 8 k Hz, the rotation speed varied from 150 rpm to 1800 rpm, and it is kept constant in each sample. The faults include out race fault, inner race fault, and bearing retainer fault. The time domain and frequency domain methods are used to calculate features as described in Section 2. The original experimental data are filtered by wavelet packet, using Symlets wavelet function; the signal is decomposed into 3 layers. Three class faults include 390 feature samples, each class includes 130 samples, training data set includes 330 samples (110 samples from each class), testing data set includes 30 samples (10 samples from each class), and the test samples are selected randomly. All data related to the bearing and the system under investigation is used with kind permission of Dr. Meng [17]. The calculation is also run 60 times.
(a) Test rig and experiment
(b) The gear box schematic diagram
The original features are classified by LSSVM, one of the calculation results is shown in Figure 11, and the average accuracy is 63.72%.
The new features are gained by KICA-GDA. Its box-like plots are shown in Figure 12, associated with Figure 13. It can be found that the processed features’ distribution is better than original features. Within the low-dimensional space, the same classes are concentrated, and the different classes are separated from the other classes.
(a) First feature
(b) Second feature
(a) Features processed by KICA
(b) Features processed by KICA-GDA
The new reduced feature classification schematic is shown in Figure 14, the detail classification results of several common methods are shown in Table 5, it can be seen that there has large improvement in the classification performance of proposed hybrid KICA-GDA-LSSVM method.
Using the proposed new features design method, the classification results are shown in Table 6, from which we can find that the hybrid new method gains better performance.
5. Conclusion
Encouraged by the success of KICA and GDA, we present a hybrid procedure for detection of bearing fault. The experimental results classified by LSSVM show that our proposed method has the best performance compared to other four combinations.
(1) The proposed method named sensitive subband feature set design (SSFD) is an improved method compared to traditional features calculation method; the proposed variance differential spectrum can get the sensitive subbands easily. The experiment’s results show its advantage in feature extraction.
(2) Correlation between original features may degrade the classification performance, while KICA removes redundancy to select independent feature, consequently avoiding the degradation. Furthermore, GDA reduces the feature dimensionality to improve the separability of different fault classes. Finally, classifying the features with the classical classifier LSSVM, the prediction performance is better.
(3) Two different rolling bearing fault experiments are taken into consideration. Our method gets a better result with best classification accuracy compared with other methods. It validates our motivation that effective and reasonable data preprocess is important for classification.
(4) Different fault states under different rotor speed are considered. The classification results show that the classification accuracy becomes higher, which indicates that the proposed hybrid KICA-GDA-LSSVM method has stronger robustness.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported by the basic research of Nanjing University of Aeronautics & Astronautics business integration of science and engineering project funds (NZ2015103). These supports are gratefully acknowledged.