Fault Detection of a Wheelset Bearing Based on Appropriately Sparse Impulse Extraction

Convolution sparse representation (CSR) is a novel compressive sensing technique proposed in 2016 and provides an excellent framework for extracting the impulses induced by bearing faults and the unevenness of wheel tread. However, its sparsity performance on extracting impulses is sensitive to the improper penalty parameter. So, a novel fault detectionmethod, appropriately sparse impulse extraction, is proposed based on the combination of CSR, estimating the number of atom types (ENA), and crest factor. The type of atoms embedded in vibration signals is estimated by ENA. Aiming at the different types of atoms, the impulses with different sparse characteristic are spanned by CSR with different penalty parameters. The appropriately sparse impulses are selected for fault detection based on the maximal crest factor. The simulation validation, experiment verification, and practical application are conducted to validate the effectiveness of the proposed appropriately sparse impulses extraction.These results show that the proposed appropriately sparse impulse extraction not only can obtain fault-characteristic frequency and its harmonics for fault judgment but also describes the dynamic behaviour between elementary defects and their matching surfaces. In addition, the proposed appropriately sparse impulse extraction can isolate the impulses with different types of atoms and is very suitable for detecting the wheelset bearing faults.


Introduction
A wheelset bearing is one of crucial mechanical components in a high-speed train and plays an important role in load bearing, power transmission, and motion transform.During the long-term running process of high-speed train, the sustainable effects of the heavy loads and alternating stresses on the components of wheelset bearing easily lead to the bearing fault initiation (such as spalls, pitting, and burningoff).Once these faults exist in the wheelset bearing, they become increasing worse and finally endanger the service safety of high-speed train [1][2][3].Therefore, it is of great significance to detect wheelset bearing faults for ensuring the operational safety and realizing the condition-based maintenance of high-speed train.
Acceleration-based vibration monitoring is a feasible and effective route to detect wheelset bearing faults because of the convenient installation of the measured sensors and the low cost of collecting vibrational signals [4,5].The vibrational signals carry the fruitful information about the working status of the bearing.Once the bearing fault appears, the vibrational signals contain impulse sequences with the rotation of wheelset bearing [6].The impulses are the essential symptom for bearing faults [7].So, how to extract the impulses from the vibrational signals using an advanced signal proceeding technique becomes a key of detecting bearing faults.Many advanced signal processing methods are proposed, such as enhanced wavelet transforms [8,9] and modified empirical model decomposition [10].
Compressed sensing (CS) that emerged in 2006 is a new method to extract impulses from the measured vibration signals [11].The core theory about CS consists of sparse representation and dictionary design.Sparse representation models the analyzed signals as the linear combination of basic elements or atoms in a redundant dictionary.Dictionary design is adapted to the self-feature of the analyzed signals to well match high-level structures embedded in the analyzed signals, as much as possible.CS has been widely applied and generates the state-of-the-art results in learning, neuroscience, signal processing, image and audio processing, classification, and statistics [12][13][14].
In terms of sparse representation, its extract resolution is an NP-hard problem [15].Thus, the approximate solutions based on the pursuit algorithms are considered instead.The pursuit algorithms mainly include the greed-based matching pursuit and the convex optimization-based basis pursuit.Matching pursuits are very simple via selecting the dictionary atoms sequentially by the similar measure between the analyzed signal and atoms [8], mainly including matching pursuit (MP) [16], orthogonal matching pursuit (OMP) [17,18], block orthogonal matching pursuit (BOMP) [19], compressive sampling matching pursuit (COSAMP) [20], and subspace pursuit (SSMP) [21].The resolution methods of basis pursuit have been developed in recent years, mainly including the gradient projection method (GP) [22], sparse reconstruction by separable approximation (SPARSA) [23], iterative thresholding method (ITS) [24], fast iterative thresholding method (FIST) [25], coordinate gradient descent method (CGD) [26], and Bregman iterative algorithm (BIT) [27].In fact, matching pursuit and basis pursuit share the common the sparse solutions under some favorable conditions [28].In general, basis pursuit is computationally tractable and not combinational algorithm compared to matching pursuit [29].So basis pursuit is more suitable for processing large-scale and high-dimension engineering data.
As for dictionary design, the route to designing a dictionary mainly includes manually predefined dictionary (also called the implicit dictionary) and dictionary learning (also called explicit dictionary).The predefined dictionary is designed via the signal transform techniques (such as Dirac, Heaviside, Fourier transform, wavelet transform, or merging mentioned dictionaries to create mega-dictionary) [30].Dictionary learning is that a dictionary is directly inferred from the measured signals by machine learning technique, mainly including regular dictionary learning (such as optimal directions (MOD, RSDL) [31], -SVD [32], online dictionary learning [33], and iterative LS-based dictionary learning (ILS-DLA) [34]) and shift-invariant dictionary learning (Lagrange dual method and shift-invariant -SVD, explicit shift-invariant dictionary learning) [35][36][37].Shift-invariant dictionary learning (SIDL) is an extension of regular dictionary learning (RDL) and makes each atom replicated and shifted at each time offset within the signal and more suitable for designing the dictionary to process the signals with the circular modes.Dictionary learning can adaptively fit high-level structures of the analyzed signals and generates more sparse resolution compared to the predefined dictionary.
CS-based impulse extraction techniques become increasing popular by virtue of the excellent performances (such as flexibility, sparsity, and superresolution).The CS based on the combination of matching pursuit and predefined dictionary is used to extract impulses induced by the rotational machine faults for bearing and gear fault detection [38,39].The CS based on the combination of basis pursuit and predefined dictionary is employed to capture the impulse patterns for rotational machine fault diagnosis [40][41][42].The performance of CS on detecting fault outperforms the wavelet transform and empirical model decomposition [38][39][40][41][42].Because dictionary learning has the advantages and potential on mining high-level structures embedded in vibration signals over the predefined dictionary, SIDL-based CS and basis pursuit is used to extract the circular impulses submerged in the vibration signals of rotating machine system [43][44][45][46].
Convolution sparse representation (CSR) based on alternating direction method of multiplier (ADMM) is proposed in 2016 as a novel research production of CS [47].It realizes not only SIDL but also shift-invariant sparse representation (SISR) of analyzed signals.The strategy of the interleaved optimization between SIDL and SISR, instead of the alternating optimization between SIDL and basis pursuit in the traditional SIDL, leads to higher computational efficiency and accuracy.CSR is introduced into the field of wheelset bearing fault detection in this paper.Although CSR has the potential on extracting the impulses embedded in vibration signals, its performance on extracting impulses is sensitive to the improper penalty parameter.When the penalty parameter is very large, the extracted impulses are so sparse that the number of the extracted impulses is far less than the true number of the impulses induced by bearing faults.As a result, this easily leads to the loss of the impulses containing fruitful fault-information.Conversely, the number of the extracted impulses is far more than the true number of the impulses.Then, the true impulses induced by faults are contaminated by the extra false impulses which have adverse influence on detecting faults.The impulses with different sparse characteristic are extracted by CSR with different penalty parameters.An impulse indictor is regarded as a criterion to select the impulses with the properly sparse characteristic for fault judgment.The impulse indictors mainly include kurtosis [48], sparsity [49], permutation entropy [50], infogram [51], and crest factor [52].Those indictors (e.g., kurtosis, sparsity, and permutation entropy) can only indicate impulse existence but cannot characterize the repetitive characteristic of the impulses [52].However, the repetitive characteristic of impulses is closely related to their sparse characteristic.So, crest factor of the impulse envelope spectra is regarded as a measured criterion to select the impulses with the properly sparse characteristic in this paper.Meanwhile, CSR cannot learn the number of atoms embedded in vibration signals.So a novel fault detection method, appropriately sparse impulse extraction based on the combination of CSR, crest factor, and ENA, is proposed in this paper.
This paper is organized as follows.The basis theory of the appropriately sparse impulse extraction is introduced in Section 2. Simulation validation is executed in Section 3. Experiment verifications are conducted in Section 4. Section 5 represents the practical application of the proposed method.Section 6 concludes the paper.

Basis Theory of the Appropriately
Sparse Impulse Extraction The one inverse problem is that the sparse coefficients x , are resolved by the shift-invariance sparse coding (SISC) of ( 1), when the signals a  and atoms d  are given.The SISC can be expressed as In ( 2), the mutual coupling of the different sparse coefficients x , does not exist with respect to different analyzed signals s  and the sparse coefficients x  corresponding to signals a can be individually solved by The atoms d  are obtained by SIDL of (1), when the signal a  and sparse coefficients x , are known.SIDL is described as For the details of the algorithms of SISC and SIDL in CSR based on alternating direction method of multipliers (ADMM), one can refer to [47].Given the analyzed signals a, the interval signals a  ∈ R  ( = 1, 2, . . ., ) are obtained so that a = [a 1  , . . ., a   ]  ∈ R  ( = 1024,  = 8 in this paper).The procedure of extracting impulses from the signals a based on the SISR and SIDL is proposed and shown in Figure 1.

Signals to be analyzed
Estimating the number of atom types Extracting impulses with different sparsities Selecting appropriately sparse impulses In this paper,   =  1 + 0.5( − 1) ( = 1, 2, . . ., ),  1 = 1, and  is determined when the number of the nonzero elements of sparse coefficients x  +0.5  is equal to zero.

Selection of the Appropriately Sparse Impulses. The Hilbert transform of the extracted impulses
The envelope spectra E    of the extracted impulses I    are written as where  denotes Fourier transform.Crest factor is a measure of a waveform and defined as the ratio of peak values to the effective value.When the sparsity of the extracted impulses is almost identical to the true sparsity of the impulses induced by bearing fault, the fault-characteristic frequency and its harmonics exist in the envelope spectra of the extracted impulses.The higher the number of the periodic impulses is, the larger the peak value of the amplitude about the fault-characteristic frequency  is.The crest factor of envelope spectra is suitable for characterizing the periodic impulses and expressed as So the maximal crest factor  of envelope spectra is regarded as an indicator to select the approximately sparse impulses.

Estimating the Number of the Atom Types (ENA).
The number of atoms  is not determined from (1) to (3).According to the prior knowledge about the vibration mechanism of impulses induced by bearing faults [5], an impulse can be completely described by two parameters of resonance frequency and damping coefficient.The resonance frequency can identify different atoms.So the dominant frequencies of the learned atoms are used to estimate the number of atoms .The main steps about ENA are expressed as follows.
Step 1.The number  is preset as 4 in wheelset bearing fault detection considering the defects of outer race, inner race, roller, and wheel tread.
Step 2. The atoms of  are obtained by CSR; the dominant frequencies   ( = 1, 2, . . ., ) of each atom are extracted by Fourier transformation.
Step 3. Judge whether the difference of any two dominant frequencies   ( = 1, 2, . . ., ) is more than the frequency resolution    −1 (  denotes the sampling frequency).If so,  is output as the estimated value of the number of atoms; if not,  =  − 1.
Step 4. Circularly execute Steps 2 and 3 until the difference of any two dominant frequencies is more than frequency resolution.

Fault Detection Procedure of Appropriately Sparse Impulse
Extraction.The fault detection procedure based on the appropriately sparse impulse extraction is proposed and shown in Figure 2. The main steps include the following.Step 1.The signals to be analyzed a = [a 1  , . . ., a   ]  ∈ R  and resulting interval signals a  are given.

Shock and Vibration
Step 2. Estimate the number of the atom types based on the dominant frequency analysis.
Step 3. Extract the impulses with the different sparse characteristic through CSR with the different penalty parameters.
Step 4. Select the appropriately sparse impulses for fault judgment.

Simulation Validation
To illustrate the effectiveness of the proposed method, the acceleration responses a() induced by a bearing fault are simulated by the impulse function of a mass-spring-damper system [5] and expressed as (11) where   denotes the amplitude of the th fault impulse and is set as 0.000000005 in this paper,   denotes the reciprocal of fault-character frequency   and is set as 49.1 Hz in this paper,  means structural damping coefficient and is set as 1200 Ns/m,  denotes nature frequency and is equal to 2000 Hz, and  means the number of the simulated impulses and is equal to 39.The fault simulation signals at a −7 dB SNR shown in Figure 3 are generated by mixing the fault response in (11) with white noises.The impulses with different sparse characteristic are extracted from the signals in Figure 3(b) and shown in Figure 4.When the penalty parameter increases, the impulses become increasingly sparse.In the range from penalty parameter 1 to penalty parameter 4, there are false impulses or noise between the true simulation impulses.In the scope of both the penalty parameter 5 and penalty parameter 8, a part of true impulses are discarded.The impulses with the penalty parameter 4.5 are almost identical to the true impulses of simulation signal in Figure 3(a); there are 39 impulses.The proposed maximal crest factor of envelope spectra is used to select the appropriately sparse impulse.For comparison, the indicators (such as kurtosis value, sparsity value, and permutation entropy value) are computed.These four indicators are shown in Figure 5. Based on the maximal kurtosis value, maximal sparsity value, and minimal permutation entropy value of impulse and impulse envelope, the appropriately sparse impulse should be the impulses with penalty parameter 8. Based on the proposed maximal crest factor of impulse envelope spectra, the appropriately sparse impulse should be the impulses with penalty parameter 4.5.This is identical to the previous analysis in

Experiment Verifications
The wheelset bearing test bench used for a practical test of the proposed method is shown in Figure 8.The test bench consists of a motor, driving wheel, loading device, wheelset, and axle box.The motor delivers driving power with different motor speeds.The driving power is conveyed to the driving wheel though rubber belts.The traction power of the driving wheel is then transmitted to the wheelset.The artificial faults on outer race and a roller, shown in Figure 9, are introduced to the roller bearing which is installed in the axle box in Figure 10 on which the accelerometer is mounted.The parameters of the fault bearing are listed in Table 1.shown in Figure 11 when the outer race of the wheelset bearing has the fault in Figure 9(a).

The Fault Detection on Outer
The 122 impulses series extracted from the vibration signals in Figure 11 by CSR with different penalty parameters cannot be displayed due to large number (similar to Figure 4).The kurtosis value, sparsity value, permutation entropy value, and crest factor of these 122 impulses series are calculated and shown in Figure 12.So, the impulses with penalty parameters 20 and 122 are selected to judge fault.The impulse with penalty parameters 22 is the one extracted by the proposed appropriately sparse impulse method shown in Figure 13(a).Figure 13(b) shows the outer-race fault-characteristic frequency  BPFO and its 8-order harmonics.The outer-race faultcharacteristic frequency  BPFO is expressed as where   denotes roller number,   denotes roller diameter,   means pitch diameter,  means contact angle, and   denotes the rotation speed.
The impulses with penalty parameter 122 are the impulses selected by three other indicators and shown in Figure 13(c),  The kurtosis value, sparsity value, permutation entropy value, and crest factor of the extracted 96 impulse series with different sparse characteristic are computed and shown in Figure 15.The three impulses series with penalty parameters 47, 47.5, and 11 are selected.The impulses with penalty parameter 11 are the ones extracted by the proposed appropriately sparse impulse extraction and shown in Figure 16(a).In order to understand the vibrational phenomenon of the extracted impulses in Figure 16(a), the interaction relations of the roller defect and its matching races are described and illustrated in Figure 17.When a defective roller leaves the no-load zone (NLZ) of the wheelset bearing from position A1 and enters the load zone (LZ) of the wheelset bearing at location B1 after a half-rotation of the roller, the interactions between the roller defect and inner race generate impulse B1.After a half-rotation of the roller, the impulse C1 induced by the interactions between the roller defect and outer race emerges.
Similarly, the impulse sequences (e.g., D1, E1, F1, G1, and H1) are produced through the alternating interactions of the roller defect with the outer race and inner race.There are 7 impulses in the LZ of wheelset bearing.At position J1, the defective roller enters the NLZ of the bearing, and the dynamic interactions between the roller and races are much smaller than those in the LZ of the bearing because the forces applied on the roller only include small self-gravity and centrifugal forces when the defective roller lies in the NLZ of the bearing.After passing over the entire NLZ (K1, L1, M1, N1, and A2), the defective roller enters the LZ.When the defective roller alternatively enters and leaves the LZ of wheelset bearing, the position and direction of the forces excited by the defect applied on races and roller vary.As a result, the amplitudes of the impulses induced by roller defect are modulated by the cage rotation frequency  FTF .The time interval between both the adjacent impulses at a constant rotational speed is properly equal to the roller faultcharacteristic frequency  BSF which is expressed as In Figure 16(a), there are properly 7 impulses (e.g., from impulse B1 to impulse B7) in the LZ of wheelset bearing, and there are almost impulses in NLZ of the wheelset bearing due to the small forces applied on defective roller.These show that the proposed appropriately sparse impulse extraction can accurately characterize the dynamic behaviour of defective roller; Figure 16

Practical Application
To test the fault detection performance of the proposed method during the practical running condition of high-speed train, an accelerometer is installed on axle box of high-speed train and shown in Figure 18.The vibration acceleration signals are collected from the axle box of running high-speed train at the speed of 200 kmh −1 when the roller of wheelset bearing has a defect and shown in Figure 19.
The two types of atoms are embedded in vibration signals in Figure 19 through estimating the number of the atom types.Aiming at atom 1, the kurtosis value, sparsity value, permutation entropy value, and crest factor of the extracted 96 impulse series with different sparsity characteristic are computed and shown in Figure 20.The impulse series with penalties 48, 46.5, and 5.5 are selected and shown in Figure 21.The impulses with penalty 5.5 in Figure 21(a) are the ones extracted by the proposed appropriately sparse impulse extraction.There are 7 impulses in the LZ of wheelset bearing (e.g., from impulse B1 to impulse B7); when the defective roller enters the NLZ of wheelset bearing, there are almost significant impulses with a large shock amplitude compared to the LZ.This is identical to the conclusion drawn by the analysis of the interaction relations between the roller defect and inner-outer races.The envelope spectra of Figure 21(a) are shown in Figure 21(c) and can well capture the roller fault-characteristic frequency  BSF and its 3-order harmonics for fault judgment.The number of the impulses selected by other three measure indicators in Figure 21(b) obviously is less than the true one induced by roller faults, so that these impulses cannot provide enough information about judging the roller fault.These illustrate that the proposed appropriately sparse impulse extraction can effectively capture the impulses induced by roller faults.Aiming at atom 2, the kurtosis value, sparsity value, permutation entropy value, and crest factor of the extracted 96 impulse series with different sparsity characteristic are computed and shown in Figure 22.
The impulse series with penalties 45.5 and 5.5 are selected and shown in Figure 23.The impulse series with penalty parameter 5.5 are the ones obtained by the proposed appropriately sparse impulse extraction in Figure 23(a); its envelope spectra shown in Figure 23(c) capture 2, 4, 6, and 7.5 times the wheelset rotational frequency   .This illustrates that the wheel tread is out of roundness.The impulse series in Figure 23(b) are selected by other three indicators; only one shock exists in the impulse series.So the proposed appropriately sparse impulse extraction makes full use of the capability of CSR of extracting impulses embedded vibration signals and the capability of crest factor of determining the sparsity of impulses induced by the roller faults and unevenness of wheel tread.

Conclusion
In this paper, a novel fault detection method, the appropriately sparse impulse extraction, is proposed based on the combination of CSR, ENA, and crest factor.(2) The appropriately sparse impulse extraction not only can well capture the fault-characteristic frequencies and their harmonics for fault judgments but also analyzes the dynamic behaviour process of defective elements and their matching surfaces.
(3) The appropriately sparse impulse extraction can isolate the impulses induced by the bearing faults and wheel tread unevenness.It is every suitable for detecting fault of wheelset bearing.
In this paper, only one impulse series is selected and used to judge faults based on the maximal crest factor.The combined application of multi-impulse series with different penalty parameters should be further studied.

a 1 Interval signal a 2 Figure 1 :
Figure 1: The procedure of extracting the impulses.

Figure 4 :
Figure 4: The impulses with different sparse characteristic.

Figure 9 :Figure 10 :Figure 11 :
Figure 9: Photo of the artificial faults on the tapered roller and outer race.

Figure 13 :Figure 14 :
Figure 13: The extracted impulse: (a) by proposed appropriately sparse impulse extraction; (b) the envelope spectra of (a); (c) three other indicators; (d) the envelope spectra of (c).
Race.The vibration signals measured from the wheelset bearing system test bench are

Figure 21 :
Figure 21: The extracted impulse related to atom 1: (a) by proposed appropriately sparse impulse extraction; (b) three other indicators; (c) the envelope spectra of (a).

Figure 13 (
Figure 13(d) shows the Hilbert envelope spectra of Figure 13(c), and they fail to capture the useful feature information for fault detection.

4. 2 .
The Fault Detection on Roller.The collected vibration signals from the wheelset bearing system test bench are shown in Figure14, when the roller of the wheelset bearing has the fault shown in Figure9(b).

Figure 23 :
Figure 23: The extracted impulse related to atom 2: (a) by proposed appropriately sparse impulse extraction; (b) three other indicators; (c) the envelope spectra of (a).

( 1 )
CSR provides the excellent framework of extracting the impulses induced by the bearing faults and wheelrail interactions with the virtues of sparsity, flexibility, and superresolution.ENA can estimate the number of atoms embedded in vibration signals.Crest factor can select the impulse series with the most reasonable sparsity characteristic which makes the number of extracted impulses almost equal to the true one of impulses induced by bearing faults.
∈ R  denotes the th analyzed vibration acceleration signals,  means the length of analyzed signal, d  ∈ R  denotes a set of atoms,  is the length of atom, * means the convolution operation, x , ∈ R −+1 denotes sparse coefficients related to given signal s  and atom d  ,  is the number of interval signals s  ,  is the number of atoms d  , and  ∈ R + means penalty parameter which determines the sparse characteristic.

Table 1 :
The parameters of the test rolling bearing.