# Alpha-Stable Distribution and Multifractal Detrended Fluctuation Analysis-Based Fault Diagnosis Method Application for Axle Box Bearings

Authors: Qing Xiong; Weihua Zhang; Yanhai Xu; Yiqiang Peng; Pengyi Deng
Journal: Shock and Vibration
 (2018)

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## Abstract

A railway vehicle’s key components, such as wheelset treads and axle box bearings, often suffer from fatigue failures. If these faults are not detected and dealt with in time, the running safety of the railway vehicle will be seriously affected. To detect these components’ early failure and extend their fatigue life, a regular maintenance becomes critical. Currently, the regular maintenance of axle box bearings mainly depends on manual off-line inspection, which has low working efficiency and precision of fault diagnosis. In order to improve the maintenance efficiency and effectiveness of railway vehicles, this study proposes a method of integrating the vibration monitoring system of the axle box bearing in the underfloor wheelset lathe, where the integration scheme and work flow of the system are introduced followed by the detailed fault diagnosis method and application examples. Firstly, the band-pass filter and envelope analysis is successively performed on the original signal acquired by an accelerometer. Secondly, the alpha-stable distribution (ASD) and multifractal detrended fluctuation analysis (MFDFA) analysis of the envelope signal are performed, and five characteristic parameters with significant stability and sensitivity are extracted and then brought into the least squares support vectors machine based on particle swarm optimization to determine the state of the bearing quantitatively. Finally, the effectiveness of the method is validated by bench test data. The results demonstrated that the proposed method can accomplish effective diagnosis of axle box bearings’ fault location and fault degree and can yield better diagnosis accuracy than the single method of ASD or MFDFA.


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## Body

## 1. Introduction

A railway vehicle is usually subjected to complex conditions with variable speeds, loading, and temperatures, which unavoidably lead to wear, pitting, and even peeling fault of its core components, such as wheel treads and axle box bearings (Figure1) [1–3]. These failures will not only reduce its ride comfort and service life but also impair its safety that may cause serious economic losses or catastrophic casualties, as well as the formation of negative social influence. In August 2007, there was a train derailment accident in Zhengzhou City, Henan Province, China. After investigation, it was concluded that the accident was due to a failure of axle box bearing extending to the axle fracture, eventually leading to the train derailment. In July 2008, an ICE3 high-speed train rated for 330 km/h service speed derailed during departure from Cologne, Central Station, Germany, due to fatigue failure of one of the driving axles. In November 2012, an axle box bearing of a train suddenly broke down whose temperature rose to about 200°C in a very short time, and a lot of burning smell came out. This accident detained the train for a long time and caused a huge economic loss [4, 5]. Therefore, it is necessary to diagnose and process these faults in time.Figure 1
Common localized faults of wheel tread and axle box bearing: (a) failure on wheel tread; (b) failure on the outer race of bearing; (c) failure on the inner race of bearing; (d) failure on the ball of bearing.Currently, the reparation of wheel tread and the fault detection of the axle box bearing are carried out separately. An under-floor wheelset lathe (UWL) is usually applied to regularly maintain railway vehicles and repair its wheel tread failure, since UWL can profile the wheel tread with faults precisely when the wheel set is not broken down [6]. However, the regular maintenance of axle box bearings mainly depends on manual off-line inspection, which has low working efficiency and precision of fault diagnosis due to heavy weight and small assembly space along with several steps of process [7].The maintenance period is usually formulated by the manufacturer and related to the type and speed level of the railway vehicles. Generally, the profile period of wheel treads is about 200 thousand kilometers [8], and the maintenance period of axle box bearings is generally from one million kilometers [9]. If bearing maintenance and tread repair happen at the same time, the maintenance period of axle box bearings can be shortened and thus the cost can be reduced. Secondly, when the UWL works, the wheel will rotate at a low speed that provides objective conditions of bearing fault diagnosis by monitoring its vibration. Also, this method can reduce the manual intervention which is expected to further improve the diagnostic accuracy and efficiency.This study proposes the applicable method of integrating the fault diagnosis system of axle box bearings based on vibration monitoring with UWL. Without changing the original structure and function of UWL, this method can diagnose the common faults of axle box bearings. It is different from the current working mode that UWL can only deal with tread damage, and it is of great significance to improve and enrich the existing vehicle maintenance strategy. Meanwhile, the installation of the fault diagnosis system of axle box bearings on UWL has definite engineering background and wide application prospect. It not only increases a maintenance opportunity of bearings which is beneficial to the early detection and prediction of the bearings’ faults more accurately but also improves the operation efficiency of the railway vehicle and the economic benefits of the railway business.


## 2. Integrated Strategy

### 2.1. System Workflow

An UWL, as illustrated in Figure2, is composed of a frame, a track system, a positioning device, a parameter measuring device, a supporting device, a clamping device, a profiling device, and a numerical control system [10]. The system workflow that integrates the bearings’ fault diagnosis with UWL is shown in Figure 3.Figure 2
Underfloor wheelset lathe.Figure 3
System workflow.The proposed system workflow has specific steps as follows:(1)
In the interaction of the clamping device, supporting device, and driving friction wheels, the measured wheel is firmly fixed in the center position of UWL.(2)
The parameter measuring device measures the size parameters of the wheelset, such as wheel diameter, wheel inner distance, wheel flange thickness, and wheel flange height. Then, the numerical control system calculates these parameters to form the optimum profiling strategy.(3)
Under the action of two driving friction wheels, the wheelset begins to rotate and reaches a predetermined speed. Then, the profiling device is raised and repair starts.(4)
After one profiling work, the device measures the relevant parameters again. If the parameters are still unqualified, go back to step (2) until all parameters are qualified.(5)
When all parameters are qualified, put down the profiling tool. At the same time, let the wheelset continue to rotate and enter the fault diagnosis stage of axle box bearings.


### 2.2. The Process of Fault Diagnosis for Axle Box Bearings

In order to ensure the profiling quality, the profiling speed is usually between 30 and 120 m/min [11] when the wheel tread is repairing. According to the standard wheel diameter of 0.84 m [12], the corresponding wheel speed can be calculated as about 12 to 46 r/min. At such a low speed, it is very difficult to identify the bearing faults stably and efficiently.In practical application, the envelope spectrum analysis is the most common method for fault diagnosis of rolling bearing. But this method has the following drawbacks [13, 14]. First, at low-speed conditions, the fault frequency of the bearing is very small that can easily be disturbed by environmental noise. Especially for minor faults, the results of spectrum analysis are poor and the omission judgment occurs sometimes; second, this method can only distinguish fault locations qualitatively but cannot analyze the fault degrees quantitatively.Therefore, in order to achieve the accurate diagnosis of axle box bearings on UWL, in addition to the envelope spectrum method, this paper also adopts a quantitative method based on alpha-stable distribution (ASD) and multifractal detrended fluctuation analysis (MFDFA).The widely used ASD has good robustness in the modeling of pulse shape in non-Gaussian signals [15]; MFDFA is able to characterize the internal dynamics mechanism of the fault signal and to detect slight changes in the complex environment of rotating machinery [16]. In References [17, 18], we have studied the sensitivity and stability on fault degrees of the characteristic parameters extracted by ASD and MFDFA, respectively. The results show that the three ASD parameters (the characteristic exponent α, the scale factor γ, and the peak value of the probability density function H) and the two MFDFA parameters (the singularity exponent of the pole hq0 and the minimum singularity exponent hqmin) have excellent sensitivity and stability and can be used as the fault characteristics to distinguish the rolling bearings’ faults with different locations and degrees.In summary, this paper will further explore the practicality of the five characteristic parameters (shown in Figure4) at low speed and aims to realize the intelligent diagnosis of axle box bearings’ faults with different locations and degrees on UWL.Figure 4
The five characteristic parameters extracted by ASD and MFDFA.The fault diagnosis system of axle box bearings consists of an accelerometer, a speed sensor, a data acquisition device, a communication line, and a high-performance computer (including analysis software). The specific process of the fault diagnosis shown in Figure5 can be described as follows:(1)
Signal Acquisition. Connect the devices as shown in Figure 5. The two acceleration sensors are affixed to the left and right clamping devices of the UWL by two magnetic bases. The speed sensor we selected is a noncontact photoelectric one which must be arranged close to the wheel and need to face the reflective label on the wheel. Subsequently, the output signals of all sensors are collected and processed by the data acquisition device and then communicated to the high-performance computer through the OPC protocol. It is worth noting that the acceleration sensors are mounted on the clamping devices rather than on the axle boxes. Such arrangement can improve the maintenance efficiency because there is no need to rearrange the sensors when replacing wheel sets. Furthermore, this arrangement is easy to install because the material of the clamping device is steel and the upper surface of clamping device is smooth enough to arrange the magnetic base conveniently.(2)
Signal Processing. The band-pass filter is performed on the original signal obtained by acquisition, and then the envelope analysis is carried out.(3)
Fault Diagnosis. The ASD and MFDFA analyses of the envelope signal are carried out, and five fault characteristic parameters (α, γ, H, hq0, and hqmin) are extracted into the PSO-LSSVM model (least squares support vectors machine based on particle swarm optimization) which has been well trained to determine the state of the bearing quantitatively.Figure 5
The process of fault diagnosis for axle box bearings.It should be noted that the PSO-LSSVM model in Step (3) above needs to be established and trained before the bearing fault diagnosis begins. The detailed process of building the PSO-LSSVM model (shown in Figure6) is as follows:(1)
Under the working speed of UWL,n status of vibration signals of axle box bearings are obtained as training samples, including normal, inner-race faults (different fault degrees), outer-race faults (different fault degrees), and ball faults (different fault degrees), and labeled as S1, S2, S3, … Sn, respectively(2)
All signals are band-pass filtered and analyzed by envelope analysis to extract envelope signals(3)
Take ASD and MFDFA analyses on all the envelope signals and extract five fault characteristic parameters (α, γ, H, hq0, and hqmin)(4)
Consider the five parameters (α, γ, H, hq0, and hqmin) as input samples and optimize the regularization parameter λ and kernel parameter σ of LSSVM by PSO(5)
Establish the PSO-LSSVM model based on the optimal parametersλ and σFigure 6
The detailed process of building the PSO-LSSVM model.


## 2.1. System Workflow

An UWL, as illustrated in Figure2, is composed of a frame, a track system, a positioning device, a parameter measuring device, a supporting device, a clamping device, a profiling device, and a numerical control system [10]. The system workflow that integrates the bearings’ fault diagnosis with UWL is shown in Figure 3.Figure 2
Underfloor wheelset lathe.Figure 3
System workflow.The proposed system workflow has specific steps as follows:(1)
In the interaction of the clamping device, supporting device, and driving friction wheels, the measured wheel is firmly fixed in the center position of UWL.(2)
The parameter measuring device measures the size parameters of the wheelset, such as wheel diameter, wheel inner distance, wheel flange thickness, and wheel flange height. Then, the numerical control system calculates these parameters to form the optimum profiling strategy.(3)
Under the action of two driving friction wheels, the wheelset begins to rotate and reaches a predetermined speed. Then, the profiling device is raised and repair starts.(4)
After one profiling work, the device measures the relevant parameters again. If the parameters are still unqualified, go back to step (2) until all parameters are qualified.(5)
When all parameters are qualified, put down the profiling tool. At the same time, let the wheelset continue to rotate and enter the fault diagnosis stage of axle box bearings.


## 2.2. The Process of Fault Diagnosis for Axle Box Bearings

In order to ensure the profiling quality, the profiling speed is usually between 30 and 120 m/min [11] when the wheel tread is repairing. According to the standard wheel diameter of 0.84 m [12], the corresponding wheel speed can be calculated as about 12 to 46 r/min. At such a low speed, it is very difficult to identify the bearing faults stably and efficiently.In practical application, the envelope spectrum analysis is the most common method for fault diagnosis of rolling bearing. But this method has the following drawbacks [13, 14]. First, at low-speed conditions, the fault frequency of the bearing is very small that can easily be disturbed by environmental noise. Especially for minor faults, the results of spectrum analysis are poor and the omission judgment occurs sometimes; second, this method can only distinguish fault locations qualitatively but cannot analyze the fault degrees quantitatively.Therefore, in order to achieve the accurate diagnosis of axle box bearings on UWL, in addition to the envelope spectrum method, this paper also adopts a quantitative method based on alpha-stable distribution (ASD) and multifractal detrended fluctuation analysis (MFDFA).The widely used ASD has good robustness in the modeling of pulse shape in non-Gaussian signals [15]; MFDFA is able to characterize the internal dynamics mechanism of the fault signal and to detect slight changes in the complex environment of rotating machinery [16]. In References [17, 18], we have studied the sensitivity and stability on fault degrees of the characteristic parameters extracted by ASD and MFDFA, respectively. The results show that the three ASD parameters (the characteristic exponent α, the scale factor γ, and the peak value of the probability density function H) and the two MFDFA parameters (the singularity exponent of the pole hq0 and the minimum singularity exponent hqmin) have excellent sensitivity and stability and can be used as the fault characteristics to distinguish the rolling bearings’ faults with different locations and degrees.In summary, this paper will further explore the practicality of the five characteristic parameters (shown in Figure4) at low speed and aims to realize the intelligent diagnosis of axle box bearings’ faults with different locations and degrees on UWL.Figure 4
The five characteristic parameters extracted by ASD and MFDFA.The fault diagnosis system of axle box bearings consists of an accelerometer, a speed sensor, a data acquisition device, a communication line, and a high-performance computer (including analysis software). The specific process of the fault diagnosis shown in Figure5 can be described as follows:(1)
Signal Acquisition. Connect the devices as shown in Figure 5. The two acceleration sensors are affixed to the left and right clamping devices of the UWL by two magnetic bases. The speed sensor we selected is a noncontact photoelectric one which must be arranged close to the wheel and need to face the reflective label on the wheel. Subsequently, the output signals of all sensors are collected and processed by the data acquisition device and then communicated to the high-performance computer through the OPC protocol. It is worth noting that the acceleration sensors are mounted on the clamping devices rather than on the axle boxes. Such arrangement can improve the maintenance efficiency because there is no need to rearrange the sensors when replacing wheel sets. Furthermore, this arrangement is easy to install because the material of the clamping device is steel and the upper surface of clamping device is smooth enough to arrange the magnetic base conveniently.(2)
Signal Processing. The band-pass filter is performed on the original signal obtained by acquisition, and then the envelope analysis is carried out.(3)
Fault Diagnosis. The ASD and MFDFA analyses of the envelope signal are carried out, and five fault characteristic parameters (α, γ, H, hq0, and hqmin) are extracted into the PSO-LSSVM model (least squares support vectors machine based on particle swarm optimization) which has been well trained to determine the state of the bearing quantitatively.Figure 5
The process of fault diagnosis for axle box bearings.It should be noted that the PSO-LSSVM model in Step (3) above needs to be established and trained before the bearing fault diagnosis begins. The detailed process of building the PSO-LSSVM model (shown in Figure6) is as follows:(1)
Under the working speed of UWL,n status of vibration signals of axle box bearings are obtained as training samples, including normal, inner-race faults (different fault degrees), outer-race faults (different fault degrees), and ball faults (different fault degrees), and labeled as S1, S2, S3, … Sn, respectively(2)
All signals are band-pass filtered and analyzed by envelope analysis to extract envelope signals(3)
Take ASD and MFDFA analyses on all the envelope signals and extract five fault characteristic parameters (α, γ, H, hq0, and hqmin)(4)
Consider the five parameters (α, γ, H, hq0, and hqmin) as input samples and optimize the regularization parameter λ and kernel parameter σ of LSSVM by PSO(5)
Establish the PSO-LSSVM model based on the optimal parametersλ and σFigure 6
The detailed process of building the PSO-LSSVM model.


## 3. Applications

### 3.1. Data Acquisition and Processing

In order to further prove the effectiveness of the method proposed in this study, we have carried out the fault setting test of axle box bearings on UGL-type UWL (Figure7(a)) in Hefei Rolling Stock Depot of Shanghai Railway Administration.Figure 7
Experimental setup. (a) UGL-type underfloor wheelset lathe; (b) the installation location of the acceleration sensor.The test bearings are double row cylindrical roller bearings (NJ(P)3226X1) which are widely used in the axle box of railway vehicles. The geometrical parameters of the bearings are shown in Table1.Table 1
Geometrical parameters of the bearings.
Axle box bearing
Outside diameter (mm)
Inside diameter (mm)
Pitch diameter (mm)
Thickness (mm)
Ball diameter (mm)
Ball thickness (mm)
NJ(P)3226X1
250
130
190
80
32
52Jining mould company in Shandong Province provided the test bearings with the precise machining to achieve seven faults with different locations and degrees, as shown in Figure8. Every fault location consists of two degrees: slight faults (the width is 5 mm and the depth is 1.5 mm) and serious faults (the width is 10 mm and the depth is 2.0 mm).Figure 8
Artificial faults on the components of the axle box bearing: (a) normal; (b) slight outer-race faults; (c) serious outer-race faults; (d) slight inner-race faults; (e) serious inner-race faults; (f) slight ball faults; (g) serious ball faults.When the UWL is running, the wheel speed is 45 r/min. In other words, the outer race of the axle box bearing is fixed, while the inner race of the axle box bearing rotates with the rotational frequency of 0.75 Hz; and the load of the axle box is 80 kN. The acceleration sensor affixed to the clamping devices of the UWL is instrumented to collect vibration signals (Figure7(b)), with the sampling frequency of 25.6 kHz.Through experiments, we obtain seven kinds of vibration acceleration signals with different fault locations and different fault degrees of axle box bearings. The fault labels and sizes of seven signals are shown in Table2.Table 2
The detail descriptions of seven vibration acceleration signals.
Label
Running state
Fault width (mm)
Speed (r/min)
Sampling frequency (kHZ)
Load (kN)
S1
Normal
0
45
25.6
80
S2
Slight inner-race faults
5
45
25.6
80
S3
Serious inner-race faults
10
45
25.6
80
S4
Slight outer-race faults
5
45
25.6
80
S5
Serious outer-race faults
10
45
25.6
80
S6
Slight ball faults
5
45
25.6
80
S7
Serious ball faults
10
45
25.6
80Figure9(a) illustrates the gathered first half-second acceleration time histories associated with the seven states of axle box bearing. As we can see from the graph, the seven states cannot be distinguished only by original waveforms.Figure 9
The waveforms of seven vibration acceleration signals. (a) Original signal. (b) Envelope signals.
(a)
(b)In order to remove low-frequency noise and enhance the periodic signals which contain fault information, the band-pass filtering is performed on the original signals. According to the document [19] and combined with our repeated experimental experience, we select a 500 to 10 kHz band-pass filter. Then, all filtered signals are subjected to the envelope detector, and seven kinds of envelope signals are obtained as shown in Figure 9(b). Since the Hilbert transformation is used to demodulate the signal in the envelope analysis, the amplitude unit of the envelope signal is dimensionless [20].Compare the two figures shown in Figures9(a) and 9(b), we can find that, after band-pass filtering and envelope analysis, the impact characteristics hidden in the signals appear. However, it is still impossible to judge the states of the bearings directly through these signals.


### 3.2. The Quantitative Diagnosis Method Based on ASD and MFDFA

In order to reduce the randomness of operation, we collected the data length of 3 minutes in each state and divided each data set evenly into 20 segments. Since the wheel speed is 45 r/min, the axle box bearing rotating a revolution needs 1.333 (60/45) seconds and has 34125 (1.333 × 25600) points. The length of total data is 4608000 (25600 × 60 × 3) points, and each segment has 230400 (4608000/20) data points, so there are about 7 (230400/34125) circles that can be rotated for bearing. According to our experience, the fault information in each segment is enough for further analysis.Then, 10 segments in each state were randomly selected for training samples and the remaining 10 segments as validation samples. Hence, the total number of training and validation samples is both 70, respectively.Take ASD analysis and MFDFA analysis on the 70 training samples, the values of five fault characteristic parameters (α, γ, H, hq0, and hqmin) are extracted as shown in Table 3. After averaging, the probability density functions (PDFs) and the multifractal spectrums of each state can be gained, as shown in Figures 10 and 11, respectively.Table 3
The detailed descriptions of seven vibration acceleration signals.
Operational status
α
γ
H
h
q0
h
qmin
Normal (S1)
1.8993
0.0096
29.4120
1.8457
1.4556
1.9067
0.0087
29.0582
1.8591
1.5075
1.9151
0.0089
30.4335
1.8383
1.4541
1.9094
0.0092
30.9254
1.8247
1.5057
1.9195
0.0088
30.0025
1.8480
1.4841
1.9034
0.0095
29.6416
1.8510
1.4693
1.9052
0.0095
29.7845
1.8449
1.4516
1.9132
0.0094
30.0323
1.8395
1.4932
1.9077
0.0093
30.3889
1.8541
1.5014
1.9023
0.0090
31.1737
1.8451
1.4525
Slight inner-race faults (S2)
1.3320
0.0107
26.1155
1.7160
1.0720
1.3154
0.0102
25.5469
1.7137
1.0633
1.3296
0.0101
25.8140
1.7135
1.0638
1.3230
0.0112
26.1135
1.7087
1.0595
1.3287
0.0109
26.1692
1.7065
1.0607
1.3292
0.0104
25.9836
1.7066
1.0590
1.3172
0.0104
27.0754
1.7176
1.0719
1.3133
0.0109
26.2662
1.7060
1.0611
1.3212
0.0111
26.0422
1.7197
1.0644
1.3164
0.0115
26.3823
1.7269
1.0789
Serious inner-race faults (S3)
0.7018
0.0151
19.4513
1.6388
0.9678
0.6925
0.0144
18.1511
1.6445
0.9641
0.6956
0.0149
18.9960
1.6459
0.9738
0.7049
0.0152
18.8235
1.6483
0.9577
0.6841
0.0146
19.9549
1.6394
0.9867
0.6898
0.0144
19.9818
1.6560
0.9787
0.6985
0.0152
20.1924
1.6455
0.9669
0.6819
0.0153
19.1857
1.6498
0.9629
0.6900
0.0150
19.8159
1.6552
0.9838
0.6961
0.0156
19.4017
1.6496
0.9741
Slight outer-race faults (S4)
1.5162
0.0158
17.6917
1.5515
1.2857
1.5307
0.0158
17.7755
1.5296
1.2828
1.5223
0.0164
17.7314
1.5451
1.3100
1.5254
0.0158
17.7576
1.5231
1.3149
1.5047
0.0159
17.7512
1.5361
1.2986
1.5282
0.0161
17.5879
1.5306
1.2912
1.5093
0.0158
17.8053
1.5216
1.2982
1.5190
0.0159
17.7555
1.5507
1.3040
1.5242
0.0161
17.6028
1.5440
1.3017
1.5221
0.0162
17.5019
1.5411
1.2853
Serious outer-race faults (S5)
1.4205
0.0188
15.1236
1.4037
1.1938
1.4250
0.0186
15.2996
1.3930
1.2019
1.4322
0.0181
15.0908
1.3960
1.2091
1.4158
0.0186
15.1885
1.3908
1.1998
1.4307
0.0187
15.1936
1.3810
1.2067
1.4247
0.0185
15.3594
1.3926
1.2138
1.4193
0.0177
16.1091
1.3934
1.1992
1.4096
0.0193
15.7148
1.3824
1.2013
1.4166
0.0193
15.7449
1.3909
1.2060
1.4202
0.0191
15.8696
1.3758
1.1869
Slight ball faults (S6)
1.5929
0.0121
23.1508
1.6817
1.3872
1.6082
0.0119
23.4960
1.6839
1.3865
1.6002
0.0121
23.2410
1.6746
1.3822
1.5919
0.0119
23.6557
1.6863
1.3884
1.5844
0.0123
22.9531
1.6848
1.3910
1.5880
0.0120
23.4184
1.6831
1.3833
1.5999
0.0121
23.1717
1.6728
1.3847
1.5890
0.0117
24.0167
1.6733
1.3857
1.6057
0.0120
23.3415
1.6825
1.3832
1.5960
0.0123
22.8547
1.6835
1.3765
Serious ball faults (S7)
1.3643
0.0203
12.7830
1.4739
1.2476
1.3555
0.0199
12.9226
1.4870
1.2584
1.3634
0.0203
13.3643
1.4676
1.2451
1.3552
0.0210
13.2624
1.4715
1.2599
1.3583
0.0197
13.4797
1.4755
1.2608
1.3419
0.0199
14.0001
1.4803
1.2462
1.3540
0.0205
13.6946
1.4822
1.2461
1.3492
0.0211
13.3486
1.4821
1.2570
1.3509
0.0207
13.5415
1.4813
1.2592
1.3679
0.0215
13.1382
1.4982
1.2529Figure 10
The PDFs of seven signals.Figure 11
The multifractal spectrums of seven signals.From Figure10, we can see the size, shape, and location of PDFs of the seven signals are different, and also there are clear distinction in parameter H. It illustrates that the seven signals have different statistic characteristics. Similarly, from Figure 11, we can see the size, shape, and location of multifractal spectrums of seven signals are different, and also there are clear distinction in parameters hq0 and hqmin. It illustrates that the seven signals have different potential dynamics mechanism and multifractal characteristics.It can be seen from Table3 that, in different states, the values of the five parameters for each state are inequable. It is worth noting that the four fault characteristic parameters α, H, hq0, and hqmin have the same regularity which can be applied to discern whether there is a fault in the axle box bearing (failure < normal) and also to depict the fault degree under the same fault location (serious faults < slight faults). It is also important to note that the value of γ can be applied to discern whether there is a fault in the axle box bearing (failure > normal) and also to depict the fault degree under the same fault location (serious faults > slight faults).In order to show the stability and sensitivity for fault quantitative expression of these parameters more clearly, the data of each column in Table3 graphically expressed are shown in Figures 12(a)–12(e).Figure 12
The stability and sensitivity of the five fault characteristic parameters of seven signals. (a)α. (b) γ. (c) H. (d) hq0. (e) hqmin.
(a)
(b)
(c)
(d)
(e)Figures12(a) and 12(d) demonstrate that the values of α and hq0 do not change significantly which may indicate that the two parameters can be stably determined from each divided sample data set for all considered bearing states. However, in Figures 12(b), 12(c), and 12(e), the parameter fluctuations of some states are relatively large, especially for the serious outer-race fault states in Figure 12(b) and normal states in Figures 12(c) and 12(e), which may indicate that the stability of parameters γ, H, and hqmin is relatively poor compared with that of the other two parameters. Regarding the parameter sensitivity, significant value differences can be seen between each other for all state curves in all Figures 12(a)–12(e), and such value differences are enough to distinguish the seven states, which indicates the sensitivity of all the five fault characteristic parameters (α, γ, H, hq0, and hqmin) are significant in every state.The above analysis proves that the five characteristic parameters (α, γ, H, hq0, and hqmin) selected in this paper can quantitatively reflect the fault degree of the axle box bearings. If these parameters are combined with an appropriate classification method, the intelligent diagnosis of the axle box bearings on UWL may be realized.Therefore, we put the 5D fault characteristic parameters (α, γ, H, hq0, hqmin) = [(α1, γ1, H1, hq01, hqmin1), (α2, γ2, H2, hq02, hqmin2), …, (α10, γ10, H10, hq010, hqmin10)] obtained in Table 3 into the PSO-LSSVM model for training. The primary parameters of PSO are shown in Table 4. After iterated operation, we got the optimal parameters for LSSVM: σ = 0.0973 and λ = 4.1701.Table 4
The primary parameters of PSO.
Parameter
Value
Primary position and its increment vector
Generate randomly
Population size
40
Maximum iteration number
200
Accelerating coefficientsc1 and c2
2
Optimal range ofλ and σ
0.01–100Meanwhile, we used the remaining ten segments data as the validation samples to extract the five fault characteristic parameters (α, γ, H, hq0, hqmin)′ = [(α11, γ11, H11, hq011, hqmin11), (α12, γ12, H12, hq012, hqmin12), …, (α20, γ20, H20, hq020, hqmin20)] and lastly put it into the PSO-LSSVM model which had been trained well to classify. In order to reduce the amount of calculation, we chose the coding method of minimum output coding and adopt three classifiers to meet the requirement of diagnosis.In order to investigate the advantages of the proposed method, we also built the PSO-LSSVM model based on the 3D ASD characteristic parameters (α, γ, and H) and the 2D MFDFA characteristic parameters (hq0 and hqmin), respectively. After decoding, the classification results with three methods are shown in Figures 13–15.Figure 13
Classification results of the PSO-LSSVM model based on ASD and MFDFA parameters.Figure 14
Classification results of the PSO-LSSVM model based on ASD parameters.Figure 15
Classification results of the PSO-LSSVM model based on MFDFA parameters.It can be seen from Figures13–15, under the condition of training samples and validation samples are both 70, all the above three methods can distinguish the states of the validation samples, though their diagnostic accuracies are different. This once again shows that the five parameters (α, γ, H, hq0, and hqmin) can be used as the fault characteristics to distinguish different fault locations and fault degrees of axle box bearings on UWL.Regarding the generalization ability, we can see from Figure13 that the method that combined five parameters (α, γ, H, hq0, and hqmin) with the PSO-LSSVM model has only one mistake; that is, a sample originally belonged to S4 but was classified under S3 (see the green arrow), and the diagnostic accuracy is 98.6%. Figure 14 shows that the method that combined only three ASD parameters (α, γ, and H) with the PSO-LSSVM model has five green arrows which indicates that there are five mistakes: a sample in S2 is assigned to S7, a sample in S3 is assigned to S4, two samples in S4 are assigned to S3, and a sample in S6 is assigned to S2, and the diagnostic accuracy is 92.9%. Similarly, it also can be seen from Figure 15 that the method that combined only two MFDFA parameters (hq0 and hqmin) with the PSO-LSSVM model has six mistakes, and the diagnostic accuracy is 91.4%.So, the PSO-LSSVM model based on five parameters (α, γ, H, hq0, and hqmin) which is proposed in this paper has the highest diagnostic accuracy. It demonstrated that the combination of the characteristic parameters extracted by ASD and MFDFA can improve their effectiveness and enhance the recognition ability of the classifier; hence, the diagnosis accuracy of axle box bearings on UWL can be improved.


## 3.1. Data Acquisition and Processing

In order to further prove the effectiveness of the method proposed in this study, we have carried out the fault setting test of axle box bearings on UGL-type UWL (Figure7(a)) in Hefei Rolling Stock Depot of Shanghai Railway Administration.Figure 7
Experimental setup. (a) UGL-type underfloor wheelset lathe; (b) the installation location of the acceleration sensor.The test bearings are double row cylindrical roller bearings (NJ(P)3226X1) which are widely used in the axle box of railway vehicles. The geometrical parameters of the bearings are shown in Table1.Table 1
Geometrical parameters of the bearings.
Axle box bearing
Outside diameter (mm)
Inside diameter (mm)
Pitch diameter (mm)
Thickness (mm)
Ball diameter (mm)
Ball thickness (mm)
NJ(P)3226X1
250
130
190
80
32
52Jining mould company in Shandong Province provided the test bearings with the precise machining to achieve seven faults with different locations and degrees, as shown in Figure8. Every fault location consists of two degrees: slight faults (the width is 5 mm and the depth is 1.5 mm) and serious faults (the width is 10 mm and the depth is 2.0 mm).Figure 8
Artificial faults on the components of the axle box bearing: (a) normal; (b) slight outer-race faults; (c) serious outer-race faults; (d) slight inner-race faults; (e) serious inner-race faults; (f) slight ball faults; (g) serious ball faults.When the UWL is running, the wheel speed is 45 r/min. In other words, the outer race of the axle box bearing is fixed, while the inner race of the axle box bearing rotates with the rotational frequency of 0.75 Hz; and the load of the axle box is 80 kN. The acceleration sensor affixed to the clamping devices of the UWL is instrumented to collect vibration signals (Figure7(b)), with the sampling frequency of 25.6 kHz.Through experiments, we obtain seven kinds of vibration acceleration signals with different fault locations and different fault degrees of axle box bearings. The fault labels and sizes of seven signals are shown in Table2.Table 2
The detail descriptions of seven vibration acceleration signals.
Label
Running state
Fault width (mm)
Speed (r/min)
Sampling frequency (kHZ)
Load (kN)
S1
Normal
0
45
25.6
80
S2
Slight inner-race faults
5
45
25.6
80
S3
Serious inner-race faults
10
45
25.6
80
S4
Slight outer-race faults
5
45
25.6
80
S5
Serious outer-race faults
10
45
25.6
80
S6
Slight ball faults
5
45
25.6
80
S7
Serious ball faults
10
45
25.6
80Figure9(a) illustrates the gathered first half-second acceleration time histories associated with the seven states of axle box bearing. As we can see from the graph, the seven states cannot be distinguished only by original waveforms.Figure 9
The waveforms of seven vibration acceleration signals. (a) Original signal. (b) Envelope signals.
(a)
(b)In order to remove low-frequency noise and enhance the periodic signals which contain fault information, the band-pass filtering is performed on the original signals. According to the document [19] and combined with our repeated experimental experience, we select a 500 to 10 kHz band-pass filter. Then, all filtered signals are subjected to the envelope detector, and seven kinds of envelope signals are obtained as shown in Figure 9(b). Since the Hilbert transformation is used to demodulate the signal in the envelope analysis, the amplitude unit of the envelope signal is dimensionless [20].Compare the two figures shown in Figures9(a) and 9(b), we can find that, after band-pass filtering and envelope analysis, the impact characteristics hidden in the signals appear. However, it is still impossible to judge the states of the bearings directly through these signals.


## 3.2. The Quantitative Diagnosis Method Based on ASD and MFDFA

In order to reduce the randomness of operation, we collected the data length of 3 minutes in each state and divided each data set evenly into 20 segments. Since the wheel speed is 45 r/min, the axle box bearing rotating a revolution needs 1.333 (60/45) seconds and has 34125 (1.333 × 25600) points. The length of total data is 4608000 (25600 × 60 × 3) points, and each segment has 230400 (4608000/20) data points, so there are about 7 (230400/34125) circles that can be rotated for bearing. According to our experience, the fault information in each segment is enough for further analysis.Then, 10 segments in each state were randomly selected for training samples and the remaining 10 segments as validation samples. Hence, the total number of training and validation samples is both 70, respectively.Take ASD analysis and MFDFA analysis on the 70 training samples, the values of five fault characteristic parameters (α, γ, H, hq0, and hqmin) are extracted as shown in Table 3. After averaging, the probability density functions (PDFs) and the multifractal spectrums of each state can be gained, as shown in Figures 10 and 11, respectively.Table 3
The detailed descriptions of seven vibration acceleration signals.
Operational status
α
γ
H
h
q0
h
qmin
Normal (S1)
1.8993
0.0096
29.4120
1.8457
1.4556
1.9067
0.0087
29.0582
1.8591
1.5075
1.9151
0.0089
30.4335
1.8383
1.4541
1.9094
0.0092
30.9254
1.8247
1.5057
1.9195
0.0088
30.0025
1.8480
1.4841
1.9034
0.0095
29.6416
1.8510
1.4693
1.9052
0.0095
29.7845
1.8449
1.4516
1.9132
0.0094
30.0323
1.8395
1.4932
1.9077
0.0093
30.3889
1.8541
1.5014
1.9023
0.0090
31.1737
1.8451
1.4525
Slight inner-race faults (S2)
1.3320
0.0107
26.1155
1.7160
1.0720
1.3154
0.0102
25.5469
1.7137
1.0633
1.3296
0.0101
25.8140
1.7135
1.0638
1.3230
0.0112
26.1135
1.7087
1.0595
1.3287
0.0109
26.1692
1.7065
1.0607
1.3292
0.0104
25.9836
1.7066
1.0590
1.3172
0.0104
27.0754
1.7176
1.0719
1.3133
0.0109
26.2662
1.7060
1.0611
1.3212
0.0111
26.0422
1.7197
1.0644
1.3164
0.0115
26.3823
1.7269
1.0789
Serious inner-race faults (S3)
0.7018
0.0151
19.4513
1.6388
0.9678
0.6925
0.0144
18.1511
1.6445
0.9641
0.6956
0.0149
18.9960
1.6459
0.9738
0.7049
0.0152
18.8235
1.6483
0.9577
0.6841
0.0146
19.9549
1.6394
0.9867
0.6898
0.0144
19.9818
1.6560
0.9787
0.6985
0.0152
20.1924
1.6455
0.9669
0.6819
0.0153
19.1857
1.6498
0.9629
0.6900
0.0150
19.8159
1.6552
0.9838
0.6961
0.0156
19.4017
1.6496
0.9741
Slight outer-race faults (S4)
1.5162
0.0158
17.6917
1.5515
1.2857
1.5307
0.0158
17.7755
1.5296
1.2828
1.5223
0.0164
17.7314
1.5451
1.3100
1.5254
0.0158
17.7576
1.5231
1.3149
1.5047
0.0159
17.7512
1.5361
1.2986
1.5282
0.0161
17.5879
1.5306
1.2912
1.5093
0.0158
17.8053
1.5216
1.2982
1.5190
0.0159
17.7555
1.5507
1.3040
1.5242
0.0161
17.6028
1.5440
1.3017
1.5221
0.0162
17.5019
1.5411
1.2853
Serious outer-race faults (S5)
1.4205
0.0188
15.1236
1.4037
1.1938
1.4250
0.0186
15.2996
1.3930
1.2019
1.4322
0.0181
15.0908
1.3960
1.2091
1.4158
0.0186
15.1885
1.3908
1.1998
1.4307
0.0187
15.1936
1.3810
1.2067
1.4247
0.0185
15.3594
1.3926
1.2138
1.4193
0.0177
16.1091
1.3934
1.1992
1.4096
0.0193
15.7148
1.3824
1.2013
1.4166
0.0193
15.7449
1.3909
1.2060
1.4202
0.0191
15.8696
1.3758
1.1869
Slight ball faults (S6)
1.5929
0.0121
23.1508
1.6817
1.3872
1.6082
0.0119
23.4960
1.6839
1.3865
1.6002
0.0121
23.2410
1.6746
1.3822
1.5919
0.0119
23.6557
1.6863
1.3884
1.5844
0.0123
22.9531
1.6848
1.3910
1.5880
0.0120
23.4184
1.6831
1.3833
1.5999
0.0121
23.1717
1.6728
1.3847
1.5890
0.0117
24.0167
1.6733
1.3857
1.6057
0.0120
23.3415
1.6825
1.3832
1.5960
0.0123
22.8547
1.6835
1.3765
Serious ball faults (S7)
1.3643
0.0203
12.7830
1.4739
1.2476
1.3555
0.0199
12.9226
1.4870
1.2584
1.3634
0.0203
13.3643
1.4676
1.2451
1.3552
0.0210
13.2624
1.4715
1.2599
1.3583
0.0197
13.4797
1.4755
1.2608
1.3419
0.0199
14.0001
1.4803
1.2462
1.3540
0.0205
13.6946
1.4822
1.2461
1.3492
0.0211
13.3486
1.4821
1.2570
1.3509
0.0207
13.5415
1.4813
1.2592
1.3679
0.0215
13.1382
1.4982
1.2529Figure 10
The PDFs of seven signals.Figure 11
The multifractal spectrums of seven signals.From Figure10, we can see the size, shape, and location of PDFs of the seven signals are different, and also there are clear distinction in parameter H. It illustrates that the seven signals have different statistic characteristics. Similarly, from Figure 11, we can see the size, shape, and location of multifractal spectrums of seven signals are different, and also there are clear distinction in parameters hq0 and hqmin. It illustrates that the seven signals have different potential dynamics mechanism and multifractal characteristics.It can be seen from Table3 that, in different states, the values of the five parameters for each state are inequable. It is worth noting that the four fault characteristic parameters α, H, hq0, and hqmin have the same regularity which can be applied to discern whether there is a fault in the axle box bearing (failure < normal) and also to depict the fault degree under the same fault location (serious faults < slight faults). It is also important to note that the value of γ can be applied to discern whether there is a fault in the axle box bearing (failure > normal) and also to depict the fault degree under the same fault location (serious faults > slight faults).In order to show the stability and sensitivity for fault quantitative expression of these parameters more clearly, the data of each column in Table3 graphically expressed are shown in Figures 12(a)–12(e).Figure 12
The stability and sensitivity of the five fault characteristic parameters of seven signals. (a)α. (b) γ. (c) H. (d) hq0. (e) hqmin.
(a)
(b)
(c)
(d)
(e)Figures12(a) and 12(d) demonstrate that the values of α and hq0 do not change significantly which may indicate that the two parameters can be stably determined from each divided sample data set for all considered bearing states. However, in Figures 12(b), 12(c), and 12(e), the parameter fluctuations of some states are relatively large, especially for the serious outer-race fault states in Figure 12(b) and normal states in Figures 12(c) and 12(e), which may indicate that the stability of parameters γ, H, and hqmin is relatively poor compared with that of the other two parameters. Regarding the parameter sensitivity, significant value differences can be seen between each other for all state curves in all Figures 12(a)–12(e), and such value differences are enough to distinguish the seven states, which indicates the sensitivity of all the five fault characteristic parameters (α, γ, H, hq0, and hqmin) are significant in every state.The above analysis proves that the five characteristic parameters (α, γ, H, hq0, and hqmin) selected in this paper can quantitatively reflect the fault degree of the axle box bearings. If these parameters are combined with an appropriate classification method, the intelligent diagnosis of the axle box bearings on UWL may be realized.Therefore, we put the 5D fault characteristic parameters (α, γ, H, hq0, hqmin) = [(α1, γ1, H1, hq01, hqmin1), (α2, γ2, H2, hq02, hqmin2), …, (α10, γ10, H10, hq010, hqmin10)] obtained in Table 3 into the PSO-LSSVM model for training. The primary parameters of PSO are shown in Table 4. After iterated operation, we got the optimal parameters for LSSVM: σ = 0.0973 and λ = 4.1701.Table 4
The primary parameters of PSO.
Parameter
Value
Primary position and its increment vector
Generate randomly
Population size
40
Maximum iteration number
200
Accelerating coefficientsc1 and c2
2
Optimal range ofλ and σ
0.01–100Meanwhile, we used the remaining ten segments data as the validation samples to extract the five fault characteristic parameters (α, γ, H, hq0, hqmin)′ = [(α11, γ11, H11, hq011, hqmin11), (α12, γ12, H12, hq012, hqmin12), …, (α20, γ20, H20, hq020, hqmin20)] and lastly put it into the PSO-LSSVM model which had been trained well to classify. In order to reduce the amount of calculation, we chose the coding method of minimum output coding and adopt three classifiers to meet the requirement of diagnosis.In order to investigate the advantages of the proposed method, we also built the PSO-LSSVM model based on the 3D ASD characteristic parameters (α, γ, and H) and the 2D MFDFA characteristic parameters (hq0 and hqmin), respectively. After decoding, the classification results with three methods are shown in Figures 13–15.Figure 13
Classification results of the PSO-LSSVM model based on ASD and MFDFA parameters.Figure 14
Classification results of the PSO-LSSVM model based on ASD parameters.Figure 15
Classification results of the PSO-LSSVM model based on MFDFA parameters.It can be seen from Figures13–15, under the condition of training samples and validation samples are both 70, all the above three methods can distinguish the states of the validation samples, though their diagnostic accuracies are different. This once again shows that the five parameters (α, γ, H, hq0, and hqmin) can be used as the fault characteristics to distinguish different fault locations and fault degrees of axle box bearings on UWL.Regarding the generalization ability, we can see from Figure13 that the method that combined five parameters (α, γ, H, hq0, and hqmin) with the PSO-LSSVM model has only one mistake; that is, a sample originally belonged to S4 but was classified under S3 (see the green arrow), and the diagnostic accuracy is 98.6%. Figure 14 shows that the method that combined only three ASD parameters (α, γ, and H) with the PSO-LSSVM model has five green arrows which indicates that there are five mistakes: a sample in S2 is assigned to S7, a sample in S3 is assigned to S4, two samples in S4 are assigned to S3, and a sample in S6 is assigned to S2, and the diagnostic accuracy is 92.9%. Similarly, it also can be seen from Figure 15 that the method that combined only two MFDFA parameters (hq0 and hqmin) with the PSO-LSSVM model has six mistakes, and the diagnostic accuracy is 91.4%.So, the PSO-LSSVM model based on five parameters (α, γ, H, hq0, and hqmin) which is proposed in this paper has the highest diagnostic accuracy. It demonstrated that the combination of the characteristic parameters extracted by ASD and MFDFA can improve their effectiveness and enhance the recognition ability of the classifier; hence, the diagnosis accuracy of axle box bearings on UWL can be improved.


## 4. Conclusions and Future Works

Based on the study, the following conclusions can be drawn:(1)
Through the fault setting experiments, it is proved that the idea of integrating the fault diagnosis system of axle box bearings based on vibration monitoring with UWL is feasible and practically applicable.(2)
The method that combined the five ASD and MFDFA parameters (α, γ, H, hq0, and hqmin) with the PSO-LSSVM model can effectively distinguish the seven signals with different fault locations and fault degrees of axle box bearings on UWL, and its diagnostic accuracy is higher than the single method of ASD or MFDFA.There may be some achievable directions for next works. Firstly, this study has classified the states of axle box bearings into only seven classes under the circumstance of small samples, and thus more samples should be supplemented with various health states to further test its capability for multiclassifications. Then, the accelerated fatigue life tests of axle box bearings may be implemented to obtain the acceleration signals of all phases (normal, fatigue, and failure) of the bearings’ life cycle so as to form the failure feature sequences and set the failure thresholds of the bearings, which could be combined with the grey model, the support vector machine, or the random filter to predict the residual lives of the bearings.Secondly, the faults in this study were generated artificially which contains only a single fault in a bearing. However, axle box bearings often have faults at different locations at the same time in practice, which is called compound faults. To further enhance the engineering application value of the proposed method, enough samples with compound faults of axle box bearings should be supplemented to validate its effectiveness. On the basis of signal denoising, the different features of compound faults may be separated by the signal decomposition methods (such as wavelet transform, empirical mode decomposition, or their improved methods) combined with the blind source separation method (such as independent component analysis). Then, the signals after separation could be fed to the proposed method in this paper to realize the diagnosis of compound faults.



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*Source: 1737219-2018-11-12.xml*