Fault diagnosis of roller bearings based on Laplacian energy feature extraction of path graphs

Accepted Manuscript Fault diagnosis of roller bearings based on Laplacian energy feature extraction of path graphs Lu Ou, Dejie Yu PII: DOI: Reference:

S0263-2241(16)30218-4 http://dx.doi.org/10.1016/j.measurement.2016.05.061 MEASUR 4077

To appear in:

Measurement

Received Date: Revised Date: Accepted Date:

11 September 2015 13 May 2016 14 May 2016

Please cite this article as: L. Ou, D. Yu, Fault diagnosis of roller bearings based on Laplacian energy feature extraction of path graphs, Measurement (2016), doi: http://dx.doi.org/10.1016/j.measurement.2016.05.061

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Fault diagnosis of roller bearings based on Laplacian energy feature extraction of path graphs 

Lu Ou, Dejie Yu

State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha

410082, PR China

Abstract: Feature extraction of roller bearing is always an intractable problem and has attracted considerable attention for a long time. The vibration signal of roller bearing can be treated as the path graph in a manifold perspective. Generally, vibration signals of roller bearings with different faults have different correlation matrices of path graphs which including different adjacency matrices and Laplacian matrices. Therefore, as a complexity feature of the path graph, the Laplacian energy (LE) can be employed to analyze the roller bearing vibration signals. In this paper, LE is introduced as the fault feature of bearing vibration signals from graph spectrum domain and then a new fault diagnosis method based on the LE single feature extraction and Mahalanobis distance (MD) criterion function is proposed and applied to the analysis of roller bearing vibration signals. Experimental analysis results show that the proposed method can identify the roller bearing faults accurately and effectively only with a small amount of sampling points and training samples. Keywords: Path graph, Laplacian energy, Mahalanobis distance, Feature extraction, Roller bearing, Fault diagnosis

1. Introduction Roller bearings are key parts of mechanical systems and play a crucial role in the modern ﹡Corresponding author. Tel.: +86 731 88821915; fax: +86 731 88823946. E-mail address: [email protected] (D. Yu). 1

manufacturing industry, and their failure is one of the most common causes of the mechanical breakdowns in engineering applications. Usually, vibration signals are used to detect the faults of machine components and reduce the damage of machinery by applying fault diagnosis methods [1–4]. The essence of fault diagnosis is the pattern recognition and classification. Naturally, feature extraction is a critical section in pattern recognition. The conventional feature extraction methods include time-domain methods, frequency-domain methods, and time-frequency methods [5]. Time-domain methods are based on the time waveform index, e.g. peak amplitude, root-mean-square amplitude, variance, kurtosis and entropy [6–8]. Frequency-domain methods are based on the transformed signals in frequency domain, i.e. Fourier spectrum, cepstrum analysis, and envelope spectrum [9–11]. Time-frequency methods investigate signals in both the time and the frequency domains, such as the wavelet transform (WT), the empirical mode decomposition (EMD) [12–15] etc. These methods extracted the features either from the time domain, the frequency domain or the time–frequency domain. Nonetheless, these methods often produce unsatisfactory results because of their respective drawbacks when used to analyze complex roller bearing vibration data [16–18]. Accordingly, there is a need to develop new methods for feature extraction from other domains, such as from graph spectrum domain. In the last few years, techniques based on the spectrum graph theory [19,20] provide a “frequency” interpretation of graph data and have been proven to be quite popular in some application areas [21–23]. Additionally, a growing amount of research works have been dedicated to extending and complementing the spectrum graph techniques, leading to the emergence of the Graph Signal Processing (GSP) [24]. David I Shuman [24,25] outlined the main challenges of the area, discussed different ways to define the graph spectrum domain, which is the analogue to the

2

classical frequency domain, and highlighted the importance of incorporating the irregular structures of graph data domains when processing signals on graph. Ameya Agaskar [26,27] justified the use of the graph Laplacian’s eigenbasis as the surrogate of the Fourier basis for graphs, defined the notions of “spread” in the graph and spectrum domains, and investigated the uncertainty principle and localization of a graph signal in different domains. Xiaofan Zhu [28] provided a detailed theoretical analysis on why the graph Laplacian eigenbasis can be regarded as the Fourier transform of graphs and discussed whether the Laplacian eigenvectors are meaningful basis vectors for all graphs. Aliaksei Sandryhaila [29–31] extended Discrete Signal Processing (DSP) to “DSP on graphs”, including transform, impulse response, spectrum representation, Fourier transform, frequency response, and illustrated DSP on graphs by classifying blogs, linear predicting and compressing data from irregularly located weather stations and so on. Obviously, the main targets of these studies are generally the case for all class graphs, but the study of a particular class graph is ignored, namely the path graph. Generally, a path graph is a sequence of vertices which is connected by edges in sequence, while a path graph signal is a function defined on the set of vertices of the path graph. Path graph is one type of graph with the simplest and most intuitive structure, such as harmonic signals, mechanical vibration signals, and ECG signals, which are time series signals with the structure of path graph. As the time series signal is one class of path graph signal, it is of important significance to introduce the path graph signal analysis into the time series analysis. Meanwhile, there are structural corresponding relationships between the time-series and the path graph, which are the sequence structure of time-series signal correspond to the graph structure of path graph and the function value of time-series signal correspond to the graph signal value of path graph. The roller bearing vibration

3

signals are time-series signals and the signal processing in fault diagnosis is actually the signal processing in time-series [32], hence the path graph can be used to analyze the roller bearing vibration signals. In fact, vibration signals of roller bearings with different faults have different path graph structures, and thus the characteristics of its Laplacian matrix are different. The eigenvalues of the Laplacian matrix contain the most important characteristics of a graph in graph spectrum domain, and then these eigenvalues could reflect the internal structure information of the path graph. Consequently, the Laplacian energy (LE) [33], which is calculated from the eigenvalues of the Laplacian matrix, can be used as a measure of graph complexity in many areas. LE of a graph has a clear connection to chemical problems [34] and there are some known results in the mathematical literature [35,36]. Song [37–39] introduced component-wise LE to filter image description hierarchies. Livi [40–42] applied LE in the characterization of graphs for protein structure modeling and recognition of solubility. Zhao [43] proposed a new high-resolution satellite image classification and segmentation method which applies LE as a generic measure to reduce the number of levels and regions in the hierarchy. Zhang [44] proposed a new hierarchical segmentation method that applies graph LE as a generic measure for segmentation in hierarchical remote sensing image analysis. Ayyalasomayajula [45] used topological clustering of LE segmentation algorithm in text binarization. In general, LE is a powerful tool for uncovering the structural characteristics of graphs. With the present of relationship between the time-series and the path graph, LE can be used to discover the intrinsic structure of path graph from roller bearing vibration signals. In this paper, LE [33,46] and Laplacian-energy-like invariant (LEL) [47,48] are employed to

4

represent the fault features of roller bearing vibration signals from graph spectrum domain. Since LE is a single feature that only contains one value, Mahalanobis distance (MD) criterion function [49] is employed to achieve fault classification. Accordingly, a fault diagnosis method of roller bearings based on the LE single feature extraction and MD classification is proposed. In the proposed method, the adjacency matrices and Laplacian matrices of the roller bearing vibration signals are firstly constructed. Then, LE is calculated by using eigenvalues which come from the standard orthogonal decomposition of Laplacian matrices. Finally, the fault characteristic vectors consisting of LE are input to the MD classifiers and the work condition and fault patterns of roller bearings can be identified. The analysis results from experimental signals with normal and defective roller bearings indicate that the proposed approach shows the effectiveness and availability for the fault diagnosis of roller bearings and have the characteristics of requiring only minute amounts of sampling points and training samples. This paper is organized as follows. In Section 2, path graph and spectrum graph theory are introduced. In Section 3, the definitions of LE and MD are given briefly. In Section 4, the new approach for the fault diagnosis of roller bearings is proposed. In Section 5, the effectiveness of the proposed method is verified by the test results and the conclusions are given in Section 6.

2. Path graph and spectrum graph theory A path graph is a sequence of vertices such that from each of its vertex there is an edge to the next vertex in sequence. A path graph P10 with 10 vertices is shown in Fig. 1. In Fig.1, the vertex set is {v1, v2, v3, v4, v5, v6, v7, v8, v9, v10}, and the edge set is {v1v2, v2v3, v3v4, v4v5, v5v6, v6v7, v7v8, v8v9, v9v10}.

5

Fig. 1. A path graph P10 with 10 vertices.

A signal or a function f defined on the vertices of a graph can be represented as a vector

f  R N , where the ith component of the vector f represents the signal value at the ith vertex. As can be seen from the definition, the vector and the graph signal is the one-to-one correspondence, and this means that the sort order of graph signal is determined by the sort order of vertices. Consider undirected, connected, weighted graph G = (V, E, W), where V is a finite set of vertices with V  N , E is a set of edges with E  M , N and M are the number of vertices and edges respectively, and W is a weighted adjacency matrix. Let wij be the element of W at the ith row and jth column, if there is an edge eij connecting vertices i and j, then the element

wij that represents the weight of eij is nonzero; otherwise, wij  0 . General definitions of nonzero wij are the following three ways [24]

wij  1

(1)

wij  xi  x j

(2)

wij  e



xi  x j 2t

2

2

.

(3)

where t is a suitable constant that indicates the thermonuclear width and xi and xj are the values of vertex i and vertex j, respectively. The degree matrix D is a diagonal matrix with the ith diagonal element di 



j i

w ji . The graph Laplacian matrix is defined as L  D W

(4)

As the graph Laplacian matrix L is a real symmetric matrix, it has a complete set of

6

orthonormal eigenvectors

X  [ x0 , x1 , x2

xl , {l  0,1

N 1} and we denote eigenvector matrix by

xN 1 ] . Without loss of generality, we assume that the associated real,

nonnegative Laplacian eigenvalues are ordered as 0  0  1  2 denote the graph Laplacian spectrum by

 ( L)  {0 , 1 , 2

 N 1  max , and we

N 1} .

The Laplacian eigenvalues l and Laplacian eigenvectors xl can be obtained by the standard orthogonal decomposition of the graph Laplacian L, which meets

Lxl  l xl , l  0,1,

N  1.

(5)

3. Laplacian energy (LE) and Mahalanobis distance (MD) 3.1. Definition of LE and LEL Graph complexity feature is of important value to applications, such as: embedding, classification, and the construction of prototypes. The graph feature we use is LE defined by Gutman and Zhou [33]. Consider undirected, connected graph G = (V, E), where V is a finite set of vertices with

V  N , E is a set of edges with E  M . The corresponding Laplacian spectrum is

 ( L)  {0 , 1 , 2

N 1} . The LE [33,46] of graph G is defined as N 1

LE (G )   i  i 0

2M N

(6)

The Laplacian-energy-like invariant (LEL) [47,48] of graph G is defined as N 1

LEL(G )   i

(7)

i 1

3.2. Description of MD Since LE is a single feature with one value, in this paper, a simple classifier MD is employed to achieve roller bearing fault classification. Mathematically, MD is a statistical

7

method for measuring the similarities of two sets of data [49]. Different from Euclidean distance,

MD considers the correlations between data and is scale-invariant, which is defined as

MDi 

LEx ( n )  mean( S LE i ) var ( S LE i )

, i  1, 2, 3, 4

(8)

Where LEx ( n ) represents the fault feature of test sample x(n) , S LE i represents the fault feature set of i th state of training samples, mean( S LE i ) and var( S LE i ) are the mean and variance of S LE i , respectively, MDi is the MD discriminate distance of LEx ( n ) to S LE i . The subscript i ( i  1, 2, 3, 4 ) corresponds to the normal, roller element fault, inner race fault and outer race fault conditions of roller bearings, respectively. The MD approach can provide a number for gauging similarity of an unknown sample set to a known one. Generally, the samples are more similar and more possible to belong to the same fault class if their MD value is smaller. Thus, MD can be used to classify samples from different fault classes

4. Fault diagnosis of roller bearings based on LE and MD 4.1. The relationship between a path graph and a vibration signal A path graph is a sequence of vertices connected by edges. However, a vibration signal can also be seen as a sequence of sampling points connected by edges. In addition, there are structural corresponding relationships between the vibration signal and the path graph, which are the sampling points of a vibration signal correspond to the vertices of a path graph and the values of sampling points in a vibration signal correspond to the values of vertices in a path graph. That is to say, no need of special transform, a roller bearing vibration signal can be directly seen as a path graph. The adjacency relation of vertices, which is obtained by the definition of weight, can be represented by a path graph, while it can not be represented by a vibration signal. Further, the 8

relationship between the sampling points and the internal structure characteristics of a vibration signal can be reflected by a path graph. In this study, roller bearing vibration signals are transformed to corresponding path graphs, and then the spectrum graph theory can be used to extract fault features of vibration signals. 4.2. LE feature extraction For the LE feature extraction in this study, we denote W1 and W2 as the adjacency matrix defined in Eq. (2) and Eq. (3), respectively, while the parameter in W2 is set as t  1 . Meanwhile, the corresponding LE and LEL defined in Eq. (6) and Eq. (7) are denoted as LE1 and LE2 , respectively. Then, four feature extraction methods can be given in the following ways:

W1 and LE1 , W1 and LE2 , W2 and LE1 , W2 and LE2 . In the procedures of fault diagnosis method which are detailed in Section 4.2, the LE single feature extraction is calculated by W1 and LE1 for instance. 4.3. Fault diagnosis method based on LE and MD When the roller bearing works with fault, the corresponding structures of path graph will be changed, then the structure of the Laplacian matrix of path graph will be changed too. Naturally, LE can be used to reflect the change of Laplacian energy of roller bearings in different operating conditions. In the proposed method, LE is first developed to extract the fault feature of roller bearings from graph spectrum domain. Since the extracted feature (LE) of oscillation signals only contains one value, MD is then used to identify the work condition and fault patterns of the roller bearings. The fault diagnosis method based on LE

and MD is given as the following steps:

(1) Collect vibration signals under the four states, i.e. normal roller bearing, roller bearing with

9

roller element fault, roller bearing with inner race fault and roller bearing with outer race fault. A total number of m samples is divided into two groups: the training sample set and the test sample set, wherein, the number of test samples of each state is K and the number of data points of each sample is N; (2) Construct the adjacency matrix W1 of each sample by Eq. (2); (3) Calculate the Laplacian matrix L1 according to Eq. (3); (4) Obtain the Laplacian spectrum

 ( L1 ) of each sample according to Eq. (5);

(5) Calculate the fault feature LE1 for all training and test samples according to Eq. (6); (6) Calculate the mean and variance of fault feature set S LE i for training samples of each state; (7) Calculate the MD discriminant distance MDi (i  1, 2, 3, 4), for test samples according to Eq. (8); (8) Choose the test samples correspond to K minimum MD values as the corresponding fault state of MD1 , MD2 , MD3 or MD4 . Then the fault type of roller bearing can be determined. Fig. 2 shows the process of the proposed method.

10

Start

Input vibration signal

Calculate the fault feature

Calculate the MD distance

Identify the condition

Output the result

End

Fig. 2. The flow chart of the proposed method.

5. Experimental validation 5.1. Description of the test data The data utilized in this study is obtained from the Case Western Reserve University Bearing Data Center [50]. The tested bearing is 6205-2RS JEM SKF deep groove ball bearing, with motor load about 2206.50 watts and motor speed 1772 rpm (round per minute). Here the roller bearings with roller element fault, outer race fault and inner race fault are under our consideration and single point faults with defect sizes 0.1778 mm in diameter and 0.2794 mm in depth are set into the tested bearing using electro-discharge machining. The data collection system consists of a high-bandwidth amplifier particularly designed for vibration signals and a data recorder with a sampling frequency of 12000 Hz per channel. 5.2. Implementation description

11

The proposed method contains the standard orthogonal decomposition of matrix and too much data size will increase the cost of calculation and reduce the efficiency of the method. Therefore, according to the motor speed and the sampling frequency, each signal sample is acquired with 400 signal points. There are 20 vibration signals for each type and totally 80 samples are chosen randomly from the data sets. The typical vibration signals of normal bearing and bearings with faults (roller element fault, inner race fault and outer race fault) are depicted in Fig. 3. 0.2

a

0

-0.2

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Amplitude

0.5

b

0

-0.5 2

c

0 -2 5

d

0 -5

Time(s)

Fig. 3. The time domain waveforms of typical vibration signals of roller bearings: (a) normal roller bearing; (b) roller bearing with roller element fault, (c) roller bearing with inner race fault and (d) roller bearing with outer race fault.

In the experiments, W1 is first calculated for 80 samples and L1 is calculated by W1 . Then, the Laplacian spectrum

 ( L1 ) , which is used to calculate fault features LE1 for the samples,

are obtained by the standard orthogonal decomposition of L1 . For each fault type, 4 samples are randomly chosen for training and the remaining 16 samples for test, namely a total of 64 test samples are obtained. All the data used to test is different from the data used to training. Then, the mean and variance of fault feature set for training samples of each state are calculated for training and the MD discriminate distance of test samples, which are sequentially arranged by the order of four fault states, are calculated according to Eq. (8). The MD values from the test samples to the different state of training samples are represented

12

by different MDi (i  1, 2, 3, 4) . MD1 represents the MD values from the test samples to the normal state; MD2 represents the MD values from the test samples to the state of roller element fault; MD3 represents the MD values from the test samples to the state of inner race fault; MD4 represents the MD values from the test samples to the state of outer race fault. The state of test sample sequences is arranged as follows: 1-16 are the test samples of normal bearing, 17-32 are the test samples of bearing with roller element fault, 33-48 are the test samples of bearing with inner race fault and 49-64 are the test samples of bearing with outer race fault. The identified results based on W1 and LE1 are shown in Fig. 4. As can be seen from Fig. 4(a), MD1 of 1-16 test samples are significantly less than the remainder samples and thus 1-16 samples belong to the normal state. From Fig. 4(b), we can see that MD2 of 17-32 test samples are obviously less than the remainder samples and thus 17-32 samples belong to the state of roller element fault. From Fig. 4(c), we can see that MD3 of 33-48 test samples are distinctly less than the remainder of the samples and thus 33-48 samples belong to the state of inner race fault. From Fig. 4(d), we can see that MD4 of 49-64 test samples are clearly less than the remainder of the samples and thus 49-64 samples belong to the state of outer race fault. It can be seen that the fault diagnosis method based on W1 and LE1 fault feature extraction and MD classification can be used to identify the fault patterns of roller bearings accurately.

13

a

800

b

0.6

MD2

MD1

600 400 200 0

0.8

0.4 0.2

0

16

32

48

0

64

0

Test sample numbers 5

c

16

32

48

64

Test sample numbers

d

1.5

1

3

MD4

MD3

4

2

0.5

1 0

0

16

32

48

0

64

0

Test sample numbers

16

32

48

64

Test sample numbers

Fig. 4. The MD classification results based on W1 and LE1 feature extraction: (a), (b), (c) and (d) represent the MD discriminat distance of test samples to training samples of normal bearing, bearings with roller element fault, inner race fault and outer race fault, respectively.

In order to demonstrate the extensibility of the proposed method, another three single feature extraction methods are investigated, which are the combination of W1 and LE2 , W2 and LE1 ,

W2 and LE2 . The associated analysis process is the same as the fault feature extraction method based on W1 and LE1 as shown in Fig. 2. The corresponding identification results are shown in Figs. 5, 6 and 7, respectively. The identification results demonstrate that these feature extraction methods can be used to extract the fault features of roller bearings effectively and accurately. Therefore, these feature extraction methods can be effectively used to diagnose the roller bearing faults under the auxiliary of MD criterion function. a

150

b

1

MD2

MD1

100 0.5

50

0

0

16

32

48

0

64

0

Test sample numbers

c

16

32

48

64

Test sample numbers

15

d MD4

1

MD3

10

1.5

5

0

0.5

0

16

32

48

0

64

Test sample numbers

0

16

32

48

Test sample numbers

14

64

Fig. 5. The MD classification results based on the W2 and LE1 feature extraction: (a), (b), (c) and (d) represent the MD discriminat distance of test samples to training samples of normal bearing, bearings with roller element fault, inner race fault and outer race fault, respectively. 5

6

a

x 10

b MD2

10

MD1

4

15

2

0

5

0

16

32

48

0

64

0

Test sample numbers

d

15

c

16

32

48

64

Test sample numbers 1 0.8

MD4

MD3

10

5

0.6 0.4 0.2

0

0

16

32

48

0

64

0

Test sample numbers

16

32

48

64

Test sample numbers

Fig. 6. The MD classification results based on the W1 and LE2 feature extraction: (a), (b), (c) and (d) represent the MD discriminat distance of test samples to training samples of normal bearing, bearings with roller element fault, inner race fault and outer race fault, respectively. a

200

400

b

300

MD2

MD1

150 100 50 0

200 100

0

16

32

48

0

64

0

Test sample numbers

c

400

d MD4

MD3

32

48

64

10 8

300 200 100 0

16

Test sample numbers

6 4 2

0

16

32

48

0

64

Test sample numbers

0

16

32

48

64

Test sample numbers

Fig. 7. The MD classification results based on the W2 and LE2 feature extraction: (a), (b), (c) and (d) represent the MD discriminat distance of test samples to training samples of normal bearing, bearings with roller element fault, inner race fault and outer race fault, respectively.

5.3. Method performance analysis A good feature extraction method should be generally immune to noise. Therefore, the performance of the proposed method under different noise levels is examined. The same processing is done to the above data set (4 training samples and 16 test samples for each state), which are added considerable Gaussian noise with different SNR. The results are listed in Table 1. 15

It can be seen that the recognition rates of the proposed methods are 100% with SNR  25 dB and SNR  0 dB . Although the recognition rates of W1 + LE1 and W2 + LE2 are reduced a little with the change of SNR, the recognition rates of W2 + LE1 and W1 + LE2 are still 100%. This means that the proposed methods are highly immune to noises and interferences. Table 1 Test results with different SNR. Feature

Recognition

Recognition

Recognition

Recognition

Recognition

Recognition

extraction

rate of

rate of

rate of

rate of

rate of

rate of

method

SNR=-25 dB

SNR=-20 dB

SNR=-15 dB

SNR=-10 dB

SNR=-5 dB

SNR=0 dB

W1+LE1

100%

96%

89%

87%

78%

100%

W2+LE1

100%

100%

100%

100%

100%

100%

W1+LE2

100%

100%

100%

100%

100%

100%

W2+LE2

100%

100%

98%

82%

100%

100%

It is clear that the computational efficiency of such single feature extraction methods are focused on the standard orthogonal decomposition of graph Laplacian matrix. In order to investigate its requirement on training samples and sample data points under the same sampling frequency, we select different training samples and different sampling points to identify the fault states. The number of training sampling is taken as 1, 2 and 3, respectively, and the sampling points are the continuous interception of sample data. The number of sampling points N is taken as 100, 150, 200, 250, 300 and 350, respectively. For different sampling points, the recognition rates of one training sample, two training samples and three training samples are shown in Tables 2, 3 and 4, respectively. Table 2 Test results with 1 training sample and 19 test samples. Feature

Recognition

Recognition

Recognition

Recognition

Recognition

Recognition

extraction

rate of

rate of

rate of

rate of

rate of

rate of

method

N=100

N=150

N=200

N=250

N=300

N=350

W1+LE1

68%

79%

95%

97%

97%

100%

W2+LE1

74%

83%

93%

97%

100%

100%

W1+LE2

71%

82%

97%

100%

100%

100%

W2+LE2

75%

84%

100%

100%

100%

100%

16

Table 3 Test results with 2 training samples and 18 test samples. Feature

Recognition

Recognition

Recognition

Recognition

Recognition

Recognition

extraction

rate of

rate of

rate of

rate of

rate of

rate of

method

N=100

N=150

N=200

N=250

N=300

N=350

W1+LE1

82%

89%

90%

97%

97%

100%

W2+LE1

81%

88%

93%

97%

100%

100%

W1+LE2

83%

92%

94%

99%

100%

100%

W2+LE2

89%

94%

99%

100%

100%

100%

Table 4 Test results with 3 training sample and 17 test samples. Feature

Recognition

Recognition

Recognition

Recognition

Recognition

Recognition

extraction

rate of

rate of

rate of

rate of

rate of

rate of

method

N=100

N=150

N=200

N=250

N=300

N=350

W1+LE1

75%

90%

94%

97%

97%

100%

W2+LE1

75%

87%

99%

100%

100%

100%

W1+LE2

79%

93%

96%

100%

100%

100%

W2+LE2

82%

94%

100%

100%

100%

100%

From the above results, we can see that the recognition rates of the proposed single feature extraction methods with sampling points above 350 are 100%. However, the number of training samples has little effect on the test results. In general, the recognition rates of the feature extraction methods based on W2 are higher than that based on W1 and the recognition rates of the feature extraction methods based on LE2 are higher than that based on LE1 . This is because the element values of W2 are between 0 and 1, which are more compact than the element values of W1 . At the same time, the defined values of LE2 are more concentrated, while the values of LE1 are quite dispersed. The more the data are concentrated, the higher the recognition rate of MD classifier. Since the feature extraction method based on W2 and LE2 only needs one training sample and sample points above 250 to ensure the 100% recognition rate in this experiment, it is more efficient than the other three single feature extraction methods. 5.4. Comparison to other techniques

17

The LE single feature extraction method can capture the fault information accurately and has an obvious advantage over the feature extraction method based on entropy theory. Shuen-De Wu et al investigate the feasibility of utilizing the multi-scale analysis and MD scheme to diagnose the roller bearing faults in rotating machinery [51]. The multiscale entropy (MSE) [52] and multiscale permutation entropy (MPE) [53] are employed to extract the fault-related features of roller bearing signals in different scales. To compare the LE feature extraction method with the feature extraction method based on the entropy theory, MSE and MPE are employed to identify the states of roller bearings in this experiment on the same dataset. The number of sampling points N is taken as 400. At first, the sampling points of roller bearing signals are processed through MSE and MPE for feature extraction. Then, referring to the fault diagnosis procedure described in Section 4, we take 20 vibration signals of each state, where 4 samples are chosen for training and the left 16 samples are chosen for test. Finally, the fault features in various scales are put into MD for fault classification. For convenience, we set the following parameters that seem to be the most suitable in the MSE calculations, the embedding dimension m  2 and the similar tolerance r  0.8SD (SD is the standard deviation of the original signal). In the MPE calculations, the embedding dimension is taken as m  6 and the time delay is set as

  2 . The scale factor in both

methods is taken as 4. The test results are shown in Table 5. Table 5 The MD classification results based on three kinds of feature extraction methods. Feature

MSE

MPE

extraction

Scale

Scale

Scale

Scale

Scale

Scale

Scale

Scale

method

factor 1

factor 2

factor 3

factor 4

factor 1

factor 2

factor 3

factor 4

Recognition

79.8%

89.1%

95.3%

93.8%

64.1%

56.3%

29.7%

23.4%

LE

100%

rate

As can be seen from Table 5, the recognition rate of MSE based feature extraction method is relatively high and the classification results under different scale factor are similar. However, the 18

recognition rate of MPE based feature extraction is fairly low, and the recognition rates under different scale factor vary greatly. By contrast, the LE based single feature extraction indicates satisfactory recognition rate, which is significantly higher than that of multiscale feature extraction method.

6. Conclusions In this paper, a new fault diagnosis method based on the LE single feature extraction and MD criterion function is proposed. In the proposed method, LE is first developed to measure the fault feature of roller bearings from graph spectrum domain. Then, MD is employed to achieve the fault classification. The proposed method is applied to the experiment signal analysis to testify the effectiveness and reliability. The main conclusions of this study are as following: (1) Since there are structural corresponding relationships between the time-series and the path graph, the vibration signals of roller bearing can be taken as the path graph in a manifold perspective. (2) Traditionally, features of roller bearing faults are extracted from the time domain, the frequency domain and the time-frequency domain. However, by considering the vibration signals of roller bearings as path graph signals, features of roller bearing faults can also be extracted from the graph spectrum domain and the LE and LEL can be employed as the fault features of roller bearings. Experimental results show that the proposed method can be able to identify the operational state of the roller bearings effectively. (3) Although the proposed method needs the standard orthogonal decomposition of the Laplacian matrix, it can be used to diagnose the roller bearing faults with only a small amount of sampling points and training samples. Furthermore, the identification success ratio of the single

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feature extraction method based on LE is far higher than that of the MSE and the MPE.

Acknowledgments This study was supported by the National Natural Science Foundation (51275161) and a project of the Advanced Design and Manufacturing of Automotive Body of Hunan University, which is an independent subject foundation (71375004) of the State Key Laboratory.

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Highlights

1.

The path graph is first introduced into the time series analysis in a manifold perspective.

2.

Laplacian energy (LE) is developed to measure the fault feature of rolling bearings from graph spectrum domain.

3.

A fault diagnosis method based on the LE single feature extraction and Mahalanobis distance (MD) criterion function is proposed.

36