Research Article  Open Access
Fengtao Wang, Shouhai Chen, Jian Sun, Dawen Yan, Lei Wang, Lihua Zhang, "TimeFrequency Fault Feature Extraction for Rolling Bearing Based on the Tensor Manifold Method", Mathematical Problems in Engineering, vol. 2014, Article ID 198362, 15 pages, 2014. https://doi.org/10.1155/2014/198362
TimeFrequency Fault Feature Extraction for Rolling Bearing Based on the Tensor Manifold Method
Abstract
Rollingbearing faults can be effectively reflected using timefrequency characteristics. However, there are inevitable interference and redundancy components in the conventional timefrequency characteristics. Therefore, it is critical to extract the sensitive parameters that reflect the rollingbearing state from the timefrequency characteristics to accurately classify rollingbearing faults. Thus, a new tensor manifold method is proposed. First, we apply the HilbertHuang transform (HHT) to rollingbearing vibration signals to obtain the HHT timefrequency spectrum, which can be transformed into the HHT timefrequency energy histogram. Then, the tensor manifold timefrequency energy histogram is extracted from the traditional HHT timefrequency spectrum using the tensor manifold method. Five timefrequency characteristic parameters are defined to quantitatively depict the failure characteristics. Finally, the tensor manifold timefrequency characteristic parameters and probabilistic neural network (PNN) are combined to effectively classify the rollingbearing failure samples. Engineering data are used to validate the proposed method. Compared with traditional HHT timefrequency characteristic parameters, the information redundancy of the timefrequency characteristics is greatly reduced using the tensor manifold timefrequency characteristic parameters and different rollingbearing fault states are more effectively distinguished when combined with the PNN.
1. Introduction
Rolling bearings are widely used in modern rotating machinery, and their failure is one of the most common causes of machine breakdowns and accidents [1–3]. Therefore, fault diagnosis of rolling bearings is necessary to ensure the safe and efficient operation of machines in engineering applications. The main aspects of bearing fault diagnosis are classification and pattern recognition, where feature extraction directly affects the accuracy and reliability of the fault diagnosis [4]. Rollingbearing fault features can be generally divided into three categories: timedomain characteristics, frequencydomain characteristics, and time and frequencydomain characteristics [5, 6].
Timedomain characteristics are fairly intuitive; however, they fluctuate significantly and lack quantitative judging criteria. Thus, they cannot be directly used to diagnose bearing faults. In contrast, frequencydomain characteristics can be used to diagnose bearing fault conditions more accurately because different bearing faults correspond to different characteristic frequencies. However, there are typically noise and modulation components in the bearing fault signals. Thus, direct application of the frequencydomain method will submerge the fault characteristic frequency in noise or false frequency components because of improper selection of the demodulation parameters. Furthermore, signal denoising and demodulation must be conducted before extracting the bearing fault characteristic frequencies. In the process of signal denoising and demodulation, parameters such as the denoising parameters, demodulation center, and filter bandwidth should be properly selected based on experience, and a satisfactory selection is only obtained after numerous adjustments.
The time and frequencydomain characteristics, which have the intuitive feature of the timedomain characteristics and good timefrequency aggregation, can simultaneously reflect the timedomain and frequencydomain characteristics of a signal [7–10]. Therefore, extracting the timefrequency fault characteristics is important for fault diagnosis. Wang and Hu used the principle of timefrequency image analysis to diagnose gearbox faults in 1993 [11]; this effort was the first application of timefrequency image for the fault diagnosis of machinery and equipment. Zhang et al. subsequently used timefrequency images to classify diesel engine faults under complex vibration conditions [12]. Zhu et al. used shorttime Fourier transform to extract timefrequency features for fault diagnosis [13], and satisfactory results were achieved. However, the aforementioned timefrequency characteristics are not adaptive and can only be used for reciprocating machinery. To overcome the limitations of the above methods, Huang et al. proposed the HHT timefrequency spectrum, which is selfadaptive [14]. The HHT timefrequency spectrum is suitable for analyzing nonstationary signals because of its frequency instantaneity [15]. However, mode mixing is inevitable for signals with instantaneous frequency trajectory crossings [16, 17]. Li et al. used the geometric center of the HHT timefrequency spectrum as a feature vector [18, 19] in combination with SVM and classified rollingbearing fault signals. However, because the geometric center requires a considerable amount of calculations and lacks corresponding physical meaning, it can only provide qualitative classification criteria. Manifold learning has recently emerged in nonlinearfeature extraction because of its capability of effectively identifying hidden lowdimensional nonlinear structures in highdimensional data. He [20] proposed a timefrequency manifold feature by combining the timefrequency distribution and the nonlinear manifold for an effective quantitative representation of machinery health pattern.
This paper proposes a new tensor manifold timefrequency feature extraction method to overcome the weakness of traditional HHT timefrequency characteristics. The HHT timefrequency spectrum, which contains a considerable amount of failure information, is used as the research object. The tensor manifold learning method is applied to extract the tensor manifold timefrequency characteristics of the HHT timefrequency spectrum. The twodimensional timefrequency information does not need to be converted into a onedimensional vector when calculating the tensor manifold, and the information loss is significantly reduced. On this basis, five timefrequency characteristic parameters are defined. The tensor manifold timefrequency characteristic parameters can distinguish different rollingbearing fault states more effectively than traditional HHT timefrequency characteristic parameters. Combined with PNN, the tensor manifold timefrequency characteristic parameters can effectively distinguish different rollingbearing fault states. Engineering vibration signals were used to evaluate the efficiency of the proposed method.
The remainder of this paper is organized as follows. The theory basis is introduced in Section 2, and the tensor manifold timefrequency fault feature extraction method is described in Section 3. Section 4 presents the adaption of the proposed method to rollingbearing fault classification. Rollingbearing fault classification is implemented in Section 5. Finally, conclusions are drawn in Section 6.
2. Theory Basis
2.1. HHT TimeFrequency Spectrum
Based on the definition of instantaneous frequency and EMD, the HHT timefrequency spectrum is analytically derived as follows.
Apply the EMD to signal to obtain the IMFs of . Then, the analytical form of can be expressed as where Re is the real part of the selected signal, is the instantaneous amplitude of the ith IMF, and is the corresponding instantaneous frequency.
The time, frequency, and amplitude of the signals can be combined to form the threedimensional timefrequency space. Then, the amplitude distribution on timefrequency plane is referred to as the HHT timefrequency spectrum, which is expressed as where Re is the real part of the selected signal and is the indicator variable. When , , and when , .
The HHT timefrequency analysis is a decomposition method based on signal local characteristics, which provides a physical basis for the concept of instantaneous frequency and sets this method apart from conventional methods through its use of numerous harmonic components to describe complex nonlinear and nonstationary signals. Therefore, from the concept definition and the nature of signal analysis, the HHT timefrequency spectrum eliminates the limitations of Fourier transform and can accurately describe nonstationary signal characteristics.
2.2. Tensor Manifold Algorithm
2.2.1. Locality Preserving Projection (LPP) Manifold Learning Algorithm
The LPP manifold learning algorithm aims at finding the linear transformation matrix to reduce the dimensionality of highdimensional data. There are training samples , and can be obtained by minimizing the following objective function: where is the similarity measure among objects and can be defined using the knearestneighbor method: where denotes the nearest neighbor of and is a positive constant. Both and can be determined empirically.
Equation (3) demonstrates the feature space after dimension reduction can maintain the local structure of the original highdimensional space. We apply an algebraic transformation to (3) as follows: where , denotes an diagonal matrix, where the diagonal element , , and .
Then, the problem of solving for the optimal vector can be transformed into the following eigenvalue problem:
2.2.2. Tensor LPP Manifold Learning Algorithm
The LPP manifold learning algorithm [21] can only be regarded as a onedimensional manifold feature extraction algorithm. However, the number of training images in the twodimensional (e.g., timefrequency spectrum) image feature extraction process is notably small compared to the dimensions of the image vectors, which results in a singularity of and failure of the LPP algorithm. To alleviate the drawback of the LPP, this paper uses a new tensor LPP manifold learning algorithm (TenLoPP) [22] to extract the timefrequency spectrum fault characteristics.
There are twodimensional training images , where denotes an dimensional unitization column vector. The main objective of the tensor manifold algorithm is to make each image matrix project onto using a linear transformation . In this manner, an dimensional column vector can be obtained and considered a projection feature vector of image . The objective function of the tensor manifold algorithm is expressed as follows: where , the definitions of and are identical to those in the LPP manifold learning method, and denotes the Kronecker product.
Then, the problem of solving for the optimal vector is transformed into the following eigenvalue problem: where is comprised of feature vectors that correspond to the smallest nonzero eigenvalues; that is, there are optimal projection vectors , which can form the projection matrix . For any image , there is where are the projection feature vectors of the sample image and , which is comprised of projection feature vectors, is the characteristic matrix of the sample image .
2.3. Probabilistic Neural Network (PNN)
The neural composition structure and elements of the PNN are shown in Figure 1
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In the PNN, characteristic parameters were transported into each node on the pattern layer through the input layer. Then, we apply layer nonlinear mapping to the input parameters in each node of the PNN pattern and complete the comparison between an unknown type with a known type. Finally, the characteristic parameters that represent the types are input to the next layer for processing. The node structure of the layers is used to be called the RBF center, and the node output is expressed as follows: where the center vector is , which is the same size as the input vector. denotes the radial basis function, which is typically a Gaussian function; that is, where denotes the shape parameter that corresponds to the component of the radial basis function.
To facilitate the calculation, and are processed with mathematical regularization and unit. Assume that . Then, the above expression is expressed as follows:
Finally, through the output layer (decisionmaking layer), the characteristic parameters, which are derived from the pattern layer, are accumulated to provide the category feature vector, which is
The PNN has the following characteristics: the training convergence speed is high, making the PNN suitable for the realtime processing of various data types; the pattern unit can form any nonlinear mapping judgment surface, which is closest to the optimal judgment surface bayes; the selection of the RBF center kernel function has diversity, and the form of the kernel function has a small effect on the recognition results; and the number of neuron nodes in each PNN layer is relatively stable, the hardware processing is convenient, and the fault tolerance is high. The PNN has been widely used in pattern recognition, prediction estimation, and filtering denoising.
3. Tensor Manifold TimeFrequency Fault Feature Extraction Method
3.1. Description of the Proposed Method
The manifold learning method is a nonlinear dimension reduction method to extract lowdimensional nonlinear characteristics from highdimensional data. Unlike the conventional linear dimension reduction methods, such as multidimensional scaling (MDS), principal component analysis (PCA), and linear discriminant analysis (LDA), this method is a nonlinear method to address the part before the whole. By satisfying the entire optimization, the manifold learning method can preserve the partial manifold characteristics and effectively extract the nonlinear manifold characteristics that are inherent in the highdimensional characteristic set. However, the manifold learning algorithm suffers from information loss and error that are caused by the transformation from a set of twodimensional timefrequency characteristics to a onedimensional vector.
To alleviate the drawback of information loss and error, this section presents a tensor manifold timefrequency fault feature extraction method based on the tensor manifold algorithm to extract the set of lowdimensional timefrequency characteristics from the set of highdimensional timefrequency characteristics. Then, five tensor manifold timefrequency characteristic parameters were defined and combined with the PNN to classify the rollingbearing failure samples.
The tensor manifold timefrequency fault feature extraction method is described as follows, and Figure 2 presents its flow chart.(1)Group the rollingbearing vibration signal samples to be classified and for training and then calculate the HHT timefrequency spectrum. To hasten the calculation of the tensor manifold algorithm, grid the timefrequency regions, integrate the energy value of the HHT timefrequency spectrum of each mesh, and convert the HHT timefrequency spectrum into HHT timefrequency energy histograms.(2)The HHT timefrequency energy histograms are essentially twodimensional matrices. Use the HHT timefrequency energy histograms that correspond to signal samples to form a set of highdimensional timefrequency characteristics.(3)Apply the tensor manifold algorithm to extract the set of lowdimensional timefrequency characteristics from the set of highdimensional timefrequency characteristics. In this manner, the tensor manifold timefrequency energy histograms are obtained.(4)Based on the result of step , define different tensor manifold timefrequency characteristic parameters. Input the defined parameters of the training signal samples into the PNN for the rollingbearing fault classification.(5)Input the tensor manifold timefrequency characteristic parameters of the tobeclassified signal samples into the trained PNN to classify the rollingbearing faults.
3.2. Definition of the TimeFrequency Characteristic Parameters
The tensor manifold timefrequency energy histogram is a nonlinear timefrequency fault feature and can effectively differentiate different rollingbearing fault signals. However, it is equal to a twodimensional matrix, which makes it unsuitable for direct application in fault classification. In this section, several parameters are presented to quantitatively measure the difference among the tensor manifold timefrequency energy histograms. Their definitions are provided as follows.
3.2.1. Energy Entropy
Entropy is proposed to measure the data complexity and the probability to generate the new signal model. Here, the energy entropy is defined as follows: where is the energy entropy, is the value of the time frequency energy histogram, and is the proportion of each in the total . In addition, the energy entropy can reflect the uncertainty in the energy distribution.
3.2.2. Energy Correlation Coefficient
Divide the timefrequency energy histogram into sections by frequency and mark and . Because each varies with different timefrequency energy histograms, we can analyze the relevance of and to measure the difference in the timefrequency energy histogram. The energy correlation coefficient vector is defined as follows: where and is the crosscorrelation function.
3.2.3. Energy Sparsity
The signal energy distribution of the timefrequency energy histogram varies more significantly as it approaches zero. The sparsity expresses the sparse distribution of energy, and the purpose of estimating the sparsity is to obtain a function ,. If is sparse, then is relatively large, and vice versa. Generally, the norm of vector is used to quantitatively estimate the sparsity. Here, we define the norm of the standardized form of vector as follows: where , and we select such that can accurately reflect the energy distribution of the histogram.
3.2.4. Energy Mutual Information
Mutual information is proposed to measure the degree of independence among random variables. The mutual information of multiple variables is defined as the divergence of the multivariate joint probability density and its marginal probability density product: where is the divergence, , is the multivariate joint probability density function, and , is the marginal probability density function. Then is defined as follows: where , are two different probability density functions of a random vector x. The energy mutual information can be calculated for each histogram according to (18).
3.2.5. Energy Kurtosis
Kurtosis is a physical parameter that is proposed to measure the degree of Gaussian distribution of a random variable. A larger energy kurtosis in the timefrequency energy histogram corresponds to weaker Gaussianity of the energy distribution, whereas a smaller kurtosis indicates stronger Gaussianity. If the Gaussianity of the energy distribution is strong, the energy distribution presents the “middle big, two sides small” phenomenon. The energy values are mainly within the middle range, and larger or smaller values are less likely to occur. For the sequence of energy values, the overall kurtosis is defined as
4. Application of the Proposed Method to RollingBearing Fault Classification
4.1. RollingBearing Fault Data
To verify the effectiveness of the proposed method, the new method was used to analyze bearing fault data from the Bearing Data Center of Case Western Reserve University [23].
The test rig, which is shown in Figure 3, was constructed for the runtofailure testing of the rolling bearing. A 1.5 kW 3phase induction motor was connected to a power meter and torque sensor by selfcalibration coupling, which drove the fan. The load was adjusted using the fan. Data were collected using a vibration acceleration sensor, which was vertically fixed above the chassis of the drive end bearings of the induction motor. The bearings are deepgroove ball bearings of the type SKF62052RS JEM. There is a single point of failure in the inner ball and outer surface of the machining spark; the failure sizes are 0.18 mm in diameter and 0.28 mm deep. The experimental data were collected with a sample frequency of 12,000 Hz and a shaft running speed of 29.53 Hz (1,772 rpm). The corresponding ball pass frequency innerrace (BPFI), ball rotation frequency (BS), and ball pass frequency outerrace (BPFO) were estimated to be 159.93, 139.19, and 105.87 Hz, respectively.
Figures 4(a)–4(d) depict a group of normal, innerrace fault, ball fault, and outerrace fault timedomain signals. Although there are several differences among the four types of signals in the timedomain wave nature, it is difficult to distinguish the rollingbearing fault conditions using these intuitive qualitative differences. Therefore, the fault features that quantitatively represent the differences of different rollingbearing fault statuses must be studied.
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4.2. HHT TimeFrequency Characteristics of Rolling Bearings
First, the HHT timefrequency spectra of four states were calculated using the HHT method. Then, the HHT timefrequency spectrum was divided into 64 regions of identical size. The histogram of the HHT timefrequency spectrum was obtained via the integral of the energy amplitude for each region. Different types of signal HHT timefrequency spectra and their histograms are shown in Figure 5.
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Figure 5 describes the HHT timefrequency spectrum and corresponding timefrequency energy histograms of the normal rollingbearing vibration signal. As shown in Figure 5, the timefrequency energy is mainly distributed in the lowfrequency region and decreases with increasing frequency. The amplitude ranges from 0 to 20 g^{2}.
Figure 6 presents the HHT timefrequency spectrum and corresponding timefrequency energy histogram with an innerrace fault. As shown in Figure 6, the timefrequency energy is widely distributed in both the low and highfrequency regions, with a lull in the midfrequency region. The amplitude ranges from 0 to 40 g^{2}, the maximum value of which is greater than that in the normal case.
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Figure 7 presents the HHT timefrequency spectrum and corresponding timefrequency energy histograms with a ball fault. As shown in Figure 7, the timefrequency energy is mainly distributed in the highfrequency region and exhibits a less significant yet stable distribution in the lowfrequency region. The amplitude of the energy histogram ranges from 0 to 30 g^{2}.
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Figure 8 presents the HHT timefrequency spectrum and corresponding timefrequency energy histograms with an outerrace fault. As shown in Figure 8, the timefrequency energy is centered in the highfrequency region and exhibits a lull in the lowfrequency region. The distribution trend begins at a rather low frequency and increases abruptly at a certain high frequency. The magnitude ranges from 0 to 100 g^{2}.
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4.3. Extraction of the Tensor Manifold TimeFrequency Characteristic Parameters
For convenience and conciseness, we only discuss the tensor manifold timefrequency characteristic parameters of four types of rollingbearing faults in this section; in other situations, such as different damage degrees in the same fault type or different fault types with different damage degrees, the classification result will also be discussed in subsequent Section 5.
We consider 20 normal innerrace fault, ball fault, and outerrace fault timedomain signals (each sample dataset has 1,024 points, and 80 samples are used for training). The HHT timefrequency spectrum and HHT timefrequency energy histogram of each signal are obtained using the aforementioned method. Then, a highdimensional timefrequency feature combination with 80 samples is obtained. The tensor manifold algorithm is applied to extract the lowdimensional tensor manifold features from the highdimensional characteristic set. The optimal projected vectors are obtained, and the parameter is defined as 6 because of the distribution of eigenvalues. Finally, we project the matrix energy histogram onto W and obtain the tensor manifold energy histograms.
According to the aforementioned definition of the five timefrequency characteristic parameters, we take the absolute value of the elements of the obtained tensor manifold energy histograms and calculate the tensor manifold timefrequency characteristic parameters of the tensor manifold energy histograms.
Five tensor manifold timefrequency characteristic parameters of the above 80 samples are obtained. Below, only 10 samples of the above four different types of rollingbearing signals are considered for clarity in the graphics.
The results are as follows.
4.3.1. Manifold Energy Entropy
Samples 1–10 correspond to the normal signals, samples 11–20 correspond to the innerrace fault signals, samples 21–30 correspond to the ball fault signals, and samples 31–40 correspond to the outerrace fault signals. The manifold energy entropy of each tensor manifold timefrequency energy histogram is calculated and depicted in Figure 9(a). The energy entropy of the HHT timefrequency energy histogram without a tensor manifold analysis is presented in Figure 9(b). Compared to Figure 9(a), the energy entropy shown in Figure 9(b) cannot provide a clear distinction between innerrace faults and ball faults. Thus, the manifold energy entropy is more appropriate for classifying rollingbearing faults.
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4.3.2. Manifold Energy Correlation Coefficient
The manifold energy correlation coefficient (manifold energy for short) is obtained by calculating the manifold energy between and . The results are shown in Figure 10.
As shown in Figure 10, the manifold energy can generally distinguish different fault signals, but different manifold energies have different abilities. First, can generally distinguish four rollingbearing failures. is also suitable for distinguishing failures, except for the normal and ball fault samples. failed to distinguish the ball fault and outerrace fault, failed to distinguish the innerrace fault and ball fault, and and failed to distinguish all faults. Thus, is accepted as the parameter that is best able to distinguish different rollingbearing failures.
Figure 11 presents the energy correlation coefficient (hereafter denoted as “energy ”), which is calculated using six large energy bands of the HHT timefrequency energy histogram without manifold analysis. As shown in Figure 11, the energy , where , cannot provide clear distinctions and thus is not suitable for classifying different rollingbearing faults.
4.3.3. Manifold Energy Sparsity
The energy distributions of different fault signals are different, as are the energy distributions of different regions in the timefrequency energy histogram. Figure 12(a) presents the manifold energy sparsity of four rollingbearing signals. As shown in Figure 12(a), the manifold energy sparsity can effectively distinguish different fault samples and can be used to classify different rollingbearing faults. Figure 12(b) presents the energy sparsity of four types of signal samples. Although the energy sparsity can distinguish different rollingbearing samples, the energy sparsity within the sample fluctuations, and the difference in energy sparsity of the interclass sample is not obvious. Therefore, the energy sparsity is not a rollingbearing fault parameter.
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4.3.4. Manifold Energy Mutual Information
We divide the tensor manifold timefrequency energy histogram and HHT timefrequency energy histogram of the four types of signal samples into 6 regions based on the frequency. Then, we calculate the corresponding mutual manifold energy information and mutual energy information.
As described in Figure 13(a), different rollingbearing faults can be accurately distinguished using the mutual manifold energy information, which is clearly different among the fault samples; thus, the mutual manifold energy information can be used as the rollingbearing fault characteristic parameter. Figure 13(b) illustrates that the mutual energy information of normal and ball fault samples is similar, and, thus, these two fault types cannot be distinguished. Therefore, the energy mutual information is not suitable for use as the rollingbearing fault characteristic parameter.
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4.3.5. Manifold Energy Kurtosis
We calculate the manifold energy kurtosis and energy kurtosis based on the corresponding energy histograms. Figure 14(a) presents the manifold energy kurtosis of the four different samples; the manifold energy kurtosis varies significantly, and, thus, the mutual energy manifold information can be used to classify different rollingbearing faults. Figure 14(b) presents the energy kurtosis of the four different samples; as shown, all samples exhibit highly similar energy kurtosis values. In particular, the values of the normal fault, ball fault, and outerrace fault are extremely similar. Therefore, we cannot distinguish different rollingbearing faults using energy kurtosis.
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4.4. Discussion
The merits of the proposed method for extracting the tensor manifold timefrequency characteristic parameters are mainly based on the fact that the tensor manifold timefrequency feature explores the timevarying characteristic of the nonstationary fault signals. The tensor manifold utilizes the HHT timefrequency fault feature of the rollingbearing fault vibration signals. Thus, the advanced feature is suitable for nonstationary vibration signals. Moreover, there are simple features that are widely used for classification in rollingbearing fault diagnosis, such as timedomain features (e.g., kurtosis, variance), frequencydomain features (e.g., subband energy), and timefrequency domain features (e.g., HHT timefrequency spectrum). These simple features are not as advantageous as the tensor manifold timefrequency features for capturing synthetic signal information. Thus, we use the tensor manifold timefrequency parameters for rollingbearing fault diagnosis in this paper.
To demonstrate the benefit of the proposed parameters, simple features based on the HHT timefrequency spectrum are also conducted to analyze the four types of rollingbearing fault signals. The test results are presented in Figures 9–13. These simple features perform worse in classification than do the tensormanifoldbased features. To avoid possible mistakes in pattern identification, it is necessary to improve the classification capability for reliable pattern diagnosis by exploring advanced features, which is the purpose of this paper.
5. RollingBearing Fault Classification
The preceding analysis demonstrates that the five parameters (manifold energy entropy, manifold energy correlation coefficient, manifold energy sparsity, manifold energy mutual information, and manifold energy kurtosis) can efficiently distinguish the rollingbearing fault states. Thus, they are used as the PNN input parameters for the bearing fault classification.
To verify the effectiveness of the manifold feature for identifying the four bearing faults, 20 samples of four types (normal, innerrace faults, ball faults, and outerrace faults) were used as training samples. The other 20 samples of each type were used for classification purposes. Each sample was extracted for the five aforementioned manifold feature parameters. The characteristic parameters of 80 training samples were used to train the PNN, and the numbers of nodes in the four PNNs were 5, 30, 4, and 4. Finally, the characteristic parameters of the 80 tobeclassified samples were input into the PNN for classification. The PNN classification results of the four bearing faults are shown in Table 1.

Table 1 illustrates that, when the tensor manifold timefrequency characteristic parameters are used as inputs for the PNN, four types of rollingbearing fault samples can be effectively distinguished and each of the 20 tobeclassified samples for each type of fault can be correctly classified. The normal sample classification exhibits the best results, whereas the minimum components of the category vectors of the innerrace fault samples, ball fault samples, and outerrace fault samples are 0.92, 0.93, and 0.92, respectively. The classification results of the PNN indicate that the rollingbearing fault condition can be effectively described using the tensor manifold timefrequency characteristic parameters and that the rollingbearing fault type can be accurately identified with the PNN.
To compare the proposed method with traditional extraction methods, we extract the five defined parameters of the same training and tobeclassified samples using the traditional HHT timefrequency method as the PNN input parameters for the bearing fault classification. The results are shown in Table 2.

Table 2 illustrates that, when the HHT timefrequency characteristic parameters are used as inputs to the PNN, four types of rollingbearing fault samples can generally be distinguished, but the distinction is not adequate. The normal sample classification exhibits the best results, whereas the minimum components of the category vector of the innerrace fault samples, ball fault samples, and outerrace faults are 0.69, 0.76, and 0.73, respectively.
In Table 1, the minimum components of the category vectors of the four bearing faults are 0.92. In contrast, in Table 2, except the normalstate, the components of the category vectors of the innerrace fault, the ball fault, and the outerrace fault above 0.85 account for 65%, 80%, and 60%, respectively. Compared with the results of the PNN, which uses tensor manifold timefrequency characteristic parameters as inputs, the classification performance with traditional HHT timefrequency features is relatively poor.
To verify the effectiveness of the manifold feature for identifying different damage degrees in the same fault type of rollingbearing status, the proposed method was used to classify the innerrace fault samples with different damage degrees. 20 samples of four degrees (normal, mild damage, moderate damage, and severe damage) were used as training samples. The other 10 samples of each degree were used for classification purposes. The PNN classification results of the tobeclassified innerrace fault samples are shown in Table 3.

To compare the proposed method with traditional extraction methods, we extract the five defined parameters of the same training and tobeclassified innerrace fault samples with different damage degrees using the traditional HHT timefrequency method as the PNN input parameters for the bearing fault classification. The results are shown in Table 4.

Table 4 depicts that, when the HHT timefrequency characteristic parameters are used as inputs to the PNN, innerrace fault samples with different damage degrees can generally be distinguished, but the distinction is not adequate. The minimum components of the category vector of the innerrace milddamage fault samples, moderatedamage fault samples, and severedamage fault samples are 0.83, 0.86, and 0.78, respectively.
In Table 3, the minimum components of the category vectors of the four bearing faults are 0.93. In contrast, in Table 4, except the normalstate, the components of the category vectors of the innerrace fault, the ball fault, and the outerrace fault above 0.85 account for 90%, 100%, and 60%, respectively. Compared with the results in Table 3, the results of the PNN, which uses the traditional HHT timefrequency characteristic parameters as inputs, indicate a relatively poor classification performance.
6. Conclusion
This paper studies the problem of rollingbearing fault feature extraction. A timefrequency feature extraction method based on tensor manifolds for rolling bearings was proposed to overcome the deficiencies of the traditional HHT timefrequency feature extraction methods and to remove redundant timefrequency feature information. The HHT timefrequency energy histograms of the rollingbearing fault signal were used to compose highdimensional timefrequency fault feature sets. On this basis, the signal timefrequency characteristics were extracted using tensor manifold learning. Five tensor manifold timefrequency characteristic parameters were defined: manifold energy entropy, manifold energy correlation coefficient, manifold energy sparsity, manifold energy mutual information, and manifold energy kurtosis. These characteristic parameters and a PNN were combined to accurately classify rollingbearing fault samples. The tensor manifold method can realize the nonlinear fusion of the timefrequency information, which can effectively extract the intrinsic nonlinear characteristics of highdimensional timefrequency combination, and avoid the loss of information caused by traditional manifoldlearning methods. Compared with the HHT timefrequency characteristic parameters, the tensor manifold timefrequency characteristic parameters can more effectively distinguish the four bearing faults, different damage degrees in the same fault type, and different fault types with different damage degrees because of its strong nonlinearity and reduced information redundancy. The effectiveness of the proposed method was verified using real rollingbearing fault signals. Thus, this paper provides an important method to solve the rollingbearing feature extraction problems.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (no. 51375067), the Aviation Science Foundation of China (no. 20132163010), and the Fundamental Research Funds for the Central Universities of China (no. DUT13JS08).
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Copyright © 2014 Fengtao Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.