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

Key Laboratory of Fluid and Power Machinery, Ministry of Education, Xihua University, Chengdu, Sichuan 610039, China Key Laboratory of Automotive Measurement, Control and Safety, Xihua University, Chengdu, Sichuan 610039, China School of Automobile & Transportation, Xihua University, Chengdu, Sichuan 610039, China State Key Laboratory of Traction Power, Southwest Jiaotong University, Chengdu, Sichuan 610031, China


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 (Figure 1) [1][2][3].ese 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. is accident detained the train for a long time and caused a huge economic loss [4,5].erefore, it is necessary to diagnose and process these faults in time.
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].
e 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.
is 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.

System
Workflow.An UWL, as illustrated in Figure 2, 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].e system workflow that integrates the bearings' fault diagnosis with UWL is shown in Figure 3.
e 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) e parameter measuring device measures the size parameters of the wheelset, such as wheel diameter, wheel inner distance, wheel flange thickness, and wheel flange height.en, 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.en, 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

e 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.erefore, 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).
e 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.e results show that the three ASD parameters (the characteristic exponent α, the scale factor c, and the peak value of the probability density function H) and the two MFDFA parameters (the singularity exponent of the pole h q0 and the minimum singularity exponent h qmin ) 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 Figure 4) at low speed and aims to realize the intelligent diagnosis of axle box bearings' faults with different locations and degrees on UWL. e 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).e specific process of the fault diagnosis shown in Figure 5 can be described as follows: (1) Signal Acquisition.Connect the devices as shown in Figure 5. e two acceleration sensors are affixed to the left and right clamping devices of the UWL by two magnetic bases.e 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 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.
e detailed process of building the PSO-LSSVM model (shown in Figure 6) is as follows: (  1. Jining 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 Figure 8.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).
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.  2.
Figure 9(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.
In order to remove low-frequency noise and enhance the periodic signals which contain fault information, the bandpass 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 bandpass filter.
en, 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 Figures 9(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.

e 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.e 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.en, 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, c, H, h q0 , and h qmin ) 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 It can be seen from Table 3 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, h q0 , and h qmin 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 c can be applied to discern whether there is a fault in the axle box bearing

Shock and Vibration
(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 Table 3 graphically expressed are shown in Figures 12(a)-12(e).
Figures 12(a) and 12(d) demonstrate that the values of α and h q0 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 c, H, and h qmin 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, c, H, h q0 , and h qmin ) are significant in every state.
Meanwhile, we used the remaining ten segments data as the validation samples to extract the five fault characteristic parameters (α, c, H, h q0 , h qmin )′ � [(α 11 , c 11 , H 11 , h q011 , h qmin11 ), (α 12 , c 12 , H 12 , h q012 , h qmin12 ), . .., (α 20 , c 20 , H 20 , h q020 , h qmin20 )] 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  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, c, and H) and the 2D MFDFA characteristic parameters (h q0 and h qmin ), respectively.After decoding, the classification results with three methods are shown in Figures 13-15

Shock and Vibration
It can be seen from Figures 13-15, 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.is once again shows that the five parameters (α, c, H, h q0 , and h qmin ) 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 Figure 13 that the method that combined five parameters (α, c, H, h q0 , and h qmin ) with the PSO-LSSVM model has only one mistake; that is, a sample originally belonged to S4 but 10 Shock and Vibration 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, c, 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 (h q0 and h qmin ) 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, c, H, h q0 , and h qmin ) 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.

Conclusions and Future Works
Based on the study, the following conclusions can be drawn: (1) rough 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) e method that combined the five ASD and MFDFA parameters (α, c, H, h q0 , and h qmin ) 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.
ere 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. en, 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).
en, the signals after separation could be fed to the proposed method in this paper to realize the diagnosis of compound faults.

Data Availability
e vibration acceleration data of axle box bearing used to support the findings of this study are obtained by our own experiments.In order to obtain these data which contain different fault locations and fault degrees of axle box bearings, our two units (Xihua University and Southwest Jiaotong University) experienced the process of connecting manufacturers (Hefei Rolling Stock Depot of Shanghai Railway Administration and Jining mould company in Shandong Province), purchasing bearings, setting up faults, upgrading test rig, hiring testers, bench test, and so on.Especially in the bench test, because the manufacturers have their own production and maintenance tasks, we can only take experiments in spare time.Moreover, the dismantling, replacement, cleaning, marking, and other work of axle box bearing must comply with the manufacturer's working time flow, so the whole test took more than half a year, and we had spent a lot of time, money, and labor.erefore, the data in this paper cannot be open to the public free of charge.If there is any need, contact the corresponding author via e-mail.If our two units (Xihua University and Southwest Jiaotong University) are licensed, these data can be provided.

Figure 1 :
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.

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, c, H, h q0 , and h qmin ) (4) Consider the five parameters (α, c, H, h q0 , and h qmin ) 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 parameters λ and σ 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 (Figure 7(a)) in Hefei Rolling Stock Depot of Shanghai Railway Administration.e test bearings are double row cylindrical roller bearings (NJ(P)3226X1) which are widely used in the axle box of railway vehicles.e geometrical parameters of the bearings are shown in Table e acceleration sensor affixed to the clamping devices of the UWL is instrumented to collect vibration signals (Figure 7(b)), with the sampling frequency of 25.6 kHz.rough experiments, we obtain seven kinds of vibration acceleration signals with different fault locations and different fault degrees of axle box bearings.e fault labels and sizes of seven signals are shown in Table

Figure 4 :
Figure 4: e five characteristic parameters extracted by ASD and MFDFA.
, respectively.From Figure 10, 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 h q0 and h qmin .It illustrates that the seven signals have different potential dynamics mechanism and multifractal characteristics.

Figure 5 :
Figure 5: e process of fault diagnosis for axle box bearings.

Figure 6 :
Figure 6: e process of building the PSO-LSSVM model.
Shock and Vibration primary parameters of PSO are shown in Table

Figure 13 :Figure 14 :Figure 15 :
Figure 13: Classification results of the PSO-LSSVM model based on ASD and MFDFA parameters.

Table 1 :
Geometrical parameters of the bearings.

Table 3 :
e detailed descriptions of seven vibration acceleration signals.

Table 4 :
e primary parameters of PSO.