A Novel Wheelset Bearing Fault Diagnosis Method Integrated CEEMDAN , Periodic Segment Matrix , and SVD

A novel fault diagnosis method, named CPS, is proposed based on the combination of CEEMDAN (complete ensemble empirical mode decomposition with adaptive noise), PSM (periodic segment matrix), and SVD (singular value decomposition). Firstly, the collected vibration signals are decomposed into a set of IMFs using CEEMDAN. Secondly, the PSM of the selected IMFs is constructed. )irdly, singular values are obtained by SVD conducted on the space of PSM. Fourthly, the impulse components are enhanced by the singular value reconstruction with the first maximal singular value. Finally, the squared envelope spectra of the reconstructed signals are used to diagnose the wheelset bearing faults. )e effectiveness of the proposed CPS has been verified by simulations and experiments. Compared to the well-known Hankelbased SVD, the proposed CPS performs better at extracting the weak periodic impulse responses from the measured signals with strong noise and interferences.


Introduction
e wheelset bearing, a key rotating component in highspeed trains, not only supports the weight of a train but also bears various dynamic loads [1].During the long-term operation of high-speed trains, dynamic loads will aggravate fault production and further expand.As a result, wheelset bearing faults will inevitably affect the quality of high-speed train service and endanger its running safety.erefore, conducting wheelset bearing fault diagnosis is of great significance.
Vibration-based fault diagnosis is a feasible solution due to low cost and convenient installation [2].However, the vibration signals induced by bearing faults are a type of nonlinear and nonstationary signal [3,4].In addition, the energies excited by the early faults are fairly weak and often submerged by strong measured noise and other vibration interferences [5].
erefore, extracting the weak fault information from measured vibration signals has been challenging.
Many advanced signal processing techniques for analysing nonstationary signals with strong noise have been developed.A well-known technique is empirical mode decomposition (EMD).EMD is a powerful time-frequency analysis technique [6] that can decompose a signal to be analysed into a set of intrinsic mode functions (IMFs).e decomposition procedure of EMD is fully automatic and adaptive and does not require predetermined kernels, as does wavelet decomposition [6].erefore, EMD is fairly suitable for processing nonlinear and nonstationary signals.EMD has also been widely applied in the field of rotation machinery diagnosis [7][8][9][10][11], as overviewed in Ref. [12].However, the main shortcoming of EMD is mode mixing.As a result, an IMF can include several resonant frequencies, or a resonant frequency can be decomposed into different IMFs.To reduce the adverse influences of mode mixing on extracting fault information, various variations of EMD have been proposed in succession, such as ensemble empirical mode decomposition (EEMD) [13], complementary EEMD (CEEMD) [14], and complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN) [15,16].CEEMDAN is the enhanced version of CEEMD.
Two main improvements of CEEMDAN are the avoidance of spurious modes and the reduction of the amount of noise in the modes [16].Because CEEMDAN has excellent performance for decomposing nonstationary signals, it is used to analyse signals collected from a fault wheelset bearing system.However, the research studies show that IMFs obtained by CEEMDAN cannot increase the periodic impulse responses caused by bearing faults considerably to extract fault features effectively.To enhance the periodic impact components in an IMF, singular value decomposition (SVD), as an effective method, has been widely utilized [11,[17][18][19].
e essential step of SVD is constructing the trajectory matrix.In SVD with IMFs [11,[17][18][19], a Hankel matrix is often constructed as the trajectory matrix.However, a Hankel matrix, as a trajectory matrix, has been shown to be unsuitable for strengthening periodic impulse responses induced by bearing faults.
e embedding dimension of a Hankel matrix cannot be easily determined [20][21][22].Since the agglomeration of singular values in a Hankel matrix is extremely low, the signal reconstructed by using the previous larger singular values usually contains a large amount of noise, and effective singular values to reconstruct the signal cannot be easily selected [23][24][25].erefore, a novel trajectory matrix, the periodic segment matrix (PSM), is constructed instead of a Hankel matrix.e rank of the PSM of periodic signals is equal to one.us, a PSM with excellent singular value agglomeration can accurately isolate the periodic impact components from the IMF with noise.
Here, based on the combination of CEEMDAN, PSM, and SVD, a novel fault diagnosis method, CPS, is proposed.Firstly, the collected vibration signals are decomposed into a set of IMFs using CEEMDAN.Secondly, the PSM of the selected IMFs is constructed.
irdly, singular values are obtained using SVD on the space of the PSM.Fourthly, the impulse components are enhanced by singular value reconstruction with the first maximal singular value.Finally, the envelope spectra of the reconstructed signals are used to diagnose the wheelset bearing faults [26].Compared to another combination-namely, CEEMDAN, a Hankel matrix, and SVD-the proposed method, CPS, has better performance for extracting periodic impulse responses caused by bearing faults.e remainder of this paper is organized as follows.CEEMDAN and its limitations are introduced in Section 2. Section 3 illustrates PSM-based SVD. e novel fault diagnosis method, CPS, is proposed in Section 4. e proposed method, CPS, is verified by simulation in Section 5.An experiment validation of CPS is conducted in Section 6.Finally, Section 7 concludes the paper.

CEEMDAN and Its Limitations
2.1.eory of CEEMDAN.CEEMDAN was proposed to solve the averaging problem of CEEMD [15].CEEMDAN was further improved to address the presence of residual noise in the modes and the existence of spurious modes [16].
e calculation steps are summarized as follows: Step 1: let k � 1 and initialize r k−1 � x.
Step 2: for every i � 1, . . ., I, add the kth mode denoted by M k (w (i) ) decomposed with each w (i) by EMD to r k−1 , i.e., Step 3: calculate the first mode of X (i) , denoted by M 1 (X (i) ), to obtain the kth residue: Step 4: calculate the kth IMF: Step 5: go to Step 2 with k � k + 1 until the stoppage criterion is met, where w (i) is white Gaussian noise with zero mean and unit variance and β k−1 is the standard deviation of the noise to add.
and the ensemble number is I � 1000 in this paper.In the CEEMDAN approach, signal x is decomposed in terms of IMFs, i.e., where r K+1 is the residue of signal x after K IMFs are extracted.

Limitations of CEEMDAN.
When CEEMDAN is used to analyse the simulation signals in Section 5, the fault characteristic frequency f p1 � 90.09Hz and its harmonics are successfully demodulated in the squared envelope spectrum of IMF 3 in Figure 1(g), but the fault characteristic frequencies f p2 � 100 Hz and f p3 � 113.64 Hz are not discovered.Although the fault characteristic frequency f p1 is extracted, it is still disturbed by strong background noise, as shown in Figure 1(g).ere is only one second harmonic of f p1 in the spectrum, and f p2 does not have an obvious harmonic.Meanwhile, significant interference frequencies are generated due to the differences in the two fault frequencies in the demodulation process.e energy and amplitude of the fault signals associated with f p2 and f p3 are less than in other fault signals related to f p1.As a result, the low-amplitude vibration signal is submerged in the IMFs decomposed by CEEMDAN, preventing the fault features from being effectively extracted.erefore, a method for enhancing periodic impact components in an IMF is expected.

PSM-Based Singular Value Decomposition and Reconstruction
3.1.Singular Value Decomposition.SVD has been widely utilized as an effective method for enhancing periodic impact components [11,[17][18][19].For an arbitrary real m × n matrix Y, SVD can be expressed as [27] Y m×n � U m×m D m×n V T n×n , (5) where and Shock and Vibration q � min(m, n), and diagonal elements σ i (i � 1, • • • , q) are the singular values of Y, and 3.2.Periodic Segment Matrix.In Equation ( 5), the previous step of SVD is constructing trajectory matrix Y with the IMFs decomposed by CEEMDAN.Many methods can be used to construct the matrix.One of the most famous methods is the Hankel matrix [20][21][22].However, the Hankel matrix is unsuitable for strengthening the periodic impulse responses induced by bearing faults in this paper.First, the embedding dimension of a Hankel matrix cannot be easily determined [20][21][22].Second, since the agglomeration of singular values of a Hankel matrix is extremely low, the reconstructed signal by using the previous larger singular values usually contains a large amount of noise.Furthermore, effective singular values for reconstructing the signal cannot be easily selected [23][24][25].erefore, the novel trajectory matrix, PSM, without accumulative error [28], is used as the trajectory matrix of SVD in this paper.e trajectory matrix Y with PSM properties can be expressed as where s is any periodic impulse component of the IMFs obtained by CEEMDAN, a is the number of periods, and l is the embedded dimension.l � <p> , , and c a + l ≤ N, where <•> is a rounding operator, p is h times T, and h ∈ N * .T is the period length of the periodic impulse component and can be determined by the singular value ratio (SVR) spectrum [29,30].Peaks at higher multiples of this length must be monitored [29].erefore, the embedding dimension l can be expressed as l � <p> � <hT>, Naturally, the row number a of Y is also determined as a � arg max a <(a − 1)hT> + <hT>.

Shock and Vibration
For the optimization of noise reduction, the noise energies must be distributed into more singular values.is requirement is proven in Appendix.
erefore, h can be obtained by maximizing the rank of matrix Y, i.e., where rank(Y) is the rank of matrix Y and is equal to min(a, l).

Matrix Reconstruction and Signal Recovery.
Because the rank of trajectory matrix Y of a pure periodic signal is rank(Y) � 1 [29], the trajectory matrix Y can be reconstructed by using the first maximal singular value, i.e., Finally, the periodic impact components  s are extracted by the inverse process of  Y in Equation ( 6).

Proposed CPS
A novel fault diagnosis method, CPS, is proposed based on CEEMDAN and PSM-based SVD. e flowchart of CPS is shown in Figure 2. CPS is comprised of four main steps: Step 1: the vibration signals collected from an acceleration sensor installed on an axle box are decomposed into IMFs by CEEMDAN.en, the effective IMFs containing fault information are selected based on the rule that the kurtosis of the SVR (KSVR) spectrum of the IMF is greater than 4. In this paper, the ensemble number I is set equal to 1,000, the normalized standard deviation ε k of the added noise is set equal to 0.2, and the range of the SVR spectrum is set from 10 to 1,000 with a step size of 0.1.
Step 2: the period lengths of each IMF are obtained by calculating the SVR spectrum.e embedding dimension of the trajectory matrix Y can be determined using Equations ( 7)-( 9).e PSM related to an IMF is constructed using

Shock and Vibration
Equation (6).PSM-based SVD is conducted.e trajectory matrix  Y is reconstructed by Equation ( 10).e periodic impact components  s are extracted.
Step 3: the periodic impact components are subtracted from the resulting IMF.If the kurtosis of the SVR (KSVR) spectrum of the residue is greater than a constant value (4 in this paper), continually execute Step 2, and the periodic impact components contained in an IMF can be extracted; otherwise, go to Step 4.
Step 4: the squared envelope spectra of the extracted impulse components are used to evaluate the wheelset bearing faults.

Simulation Validation
In order to verify the effectiveness of the proposed CPS, a compound of simulation signals containing multiple faults is constructed and is expressed as where ξ(t) is Gaussian white noise whose variance is determined by the signal-to-noise ratio (SNR) and s i (t), i � 1, 2, 3 is a periodical impact signal written as [31] where A i is the amplitude of the ith periodic impulse component, f ci is the ith resonance frequency, f pi is the ith fault frequency, ς i is the damping coefficient, and mod(t, f −1 p i ) is the remainder of t divided by f −1 p i .e fourth signal component is a sinusoidal interference signal, which is expressed as e parameters related to the simulation signal are listed in Table 1.
When the sampling frequency is 10 kHz, the sequence length is 40,000, and the SNR is set equal to −10 dB. e time-domain waveforms of the simulation signals are shown in Figure 3.
e Fourier and squared envelope spectra of the simulation signal in Figure 3(f ) are shown in Figure 4.Only one fault characteristic frequency f p1 and its three harmonics are clearly extracted.e other two fault characteristic frequencies, f p2 and f p3 , are not extracted.
To extract all the fault characteristic frequencies, the proposed CPS is used to analyse the same simulation signals in Figure 3(f ).According to the flowchart of the proposed CPS in Figure 2, the simulation signals in Figure 3(f ) are decomposed into seventeen IMFs by CEEMDAN.For brevity, the first four IMFs and their envelope spectra are shown in Figure 1. e KSVRs of the IMFs are shown in Figure 5. From the KSVR in Figure 5, the first, third, and fourth IMFs meet the KSVR criterion.e SVR spectrum of the first IMF is calculated and shown in Figure 6. e kurtosis of the obtained SVR spectrum is 9.50, which is greater than 4. erefore, the first IMF is selected to extract the fault information.
e period length of the periodic impulse component in the first IMF is 88 points in Figure 6.Next, the embedding dimension and number of rows of the trajectory matrix PSM are computed using Equations ( 7)- (9) and are 176 and 227, respectively.e resulting PSM is constructed in Equation ( 6), and the PSM-based SVD is conducted in Equation (5).Finally, the first periodic impact signals are recovered from the trajectory matrix PSM reconstructed in Equation (10) and are shown in Figure 7(a).Because the kurtosis of the SVR spectra of the resulting residue is less than 4, IMF processing is completed.Similarly, the SVR spectra of the other fourteen IMFs are less than 4, except for the third and fourth IMFs, which are not shown to save space.From the processing results, the periodic impact components in the third and fourth IMFs are the same.erefore, the processing results of the fourth IMF are not shown to save space.e third IMF is further analysed.e kurtosis of the SVR spectra of the third IMF is 51.11.e number of rows and columns in the trajectory matrix PSM is 180 and 222, respectively.e first periodic impact signals are extracted and shown in Figure 7(d).e kurtosis of the SVR spectra of the resulting residue is 50.7.e numbers of rows and columns in the trajectory matrix PSM are 200 and 200, respectively.As a result, the second periodic impact signals are extracted from the residue and shown in Figure 7(g).e kurtosis of the SVR spectra of the final obtained residue is less than 4. us, the processing of the third IMF is completed.In Figures 7(c), 7(f), and 7(i), the three fault characteristic frequencies and their harmonics are clearly extracted for diagnosing bearing faults, demonstrating In order to illustrate the advancement of the proposed CPS, the well-known Hankel matrix [11,[17][18][19] is used as the trajectory matrix of SVD to analyse the first and third IMFs obtained by CEEMDAN.According to the rule [23], the numbers of rows and columns of the Hankel matrix are set equal to 10,001 and 10,000, respectively, and the effective singular values are selected by the coordinate of the maximum peak of difference spectrum.e results obtained by Hankel-based SVD are shown in Figure 8.Although the fault characteristic frequency f p3 and its fifth harmonics are extracted in Figure 8(c), they are disturbed by other complex spectra lines.e other two characteristic fault frequencies f p1 and f p2 are confused and are not discovered.erefore, the performance of the proposed CPS for diagnosing bearing faults is superior compared to the results obtained by Hankel-based SVD in Figure 8.

Experiment Validation
To further verify the effectiveness of the proposed CPS, a fault experiment rig was built, as shown in Figure 9(a).ree defects on the surface of the outer race and two defects on the surface of the rolling elements are shown in Figures 9(b) and 9(c), respectively.e parameters of the tested bearing are listed in Table 2.
e fault-related characteristic frequencies are expressed as follows: Ball pass frequency over outer race, f BPFO : Fundamental train frequency, f FTF : Ball spin frequency, f BSF : where f r is the rotational frequency of the wheelset.When the rotation frequency of the tested bearing is equal to 10.28 Hz, the fault characteristic frequencies are calculated as shown in Table 3.    e KSVR of each IMF obtained by CEEMDAN is shown in Figure 11, also the first seven of whose KSVR are greater than 4. us, these IMFs are executed by the proposed CPS. e IMFs from the 8th to 16th fail to meet the KSVR criteria proposed in Section 3; they are not processed further.e processing results show that only the rotation frequency and 50 Hz interferences are extracted from the 1st to 2nd IMFs and from the 4th to 7th IMFs.To save space, the processing results of these six IMFs are not shown.Finally, the 3rd IMF and the resulting squared envelope spectra are shown in Figure 12.
e outer race frequency cannot be detected in the squared envelope spectra of the outer race fault signal and its IMFs.However, the kurtosis of the SVR spectrum of the third IMF is 259.2, greater than 4. e third IMF is processed further.However, the kurtosis of the SVR spectrum of the first IMF is 6.60, greater than 4.
e first IMF must be processed further.
e period determined by its SVR spectrum is 969.e embedding dimension and the number of rows of the trajectory matrix PSM computed using Equations ( 7)-( 9) are 969 and 41, respectively.en, the resulting PSM is constructed using Equation ( 6), and PSMbased SVD is conducted using Equation (5).Finally, the first periodic impact signal is recovered from the trajectory matrix PSM reconstructed using Equation (10) and is shown in Figure 12(b).Its squared envelope spectrum is counted in Figure 12(e).Similar to the process mentioned above, the size of the trajectory matrix PSM constructed by the residual signal is 835.1 and 47, respectively.en, the second periodic impact signal is recovered and shown in Figure 12(c).Its squared envelope spectrum is counted in Figure 12(f ).e rotation frequency and its harmonics are clearly observed in Figure 12(e), and the outer race fault characteristic frequency, which was overwhelmed by heavy noise and rotation frequency, is observed in Figure 12(f ).In summary, although the energy of the measured signals is mainly concentrated in the rotation frequency and its harmonics, the weak outer race fault characteristic frequency can be extracted by the proposed CPS.
For illustrating the advancement of the proposed CPS, the Hankel matrix is used as the trajectory matrix of SVD to analyse the same IMFs obtained by CEEMDAN.e embedding dimension of the Hankel matrix and the selection of the effective singular values are based on the conclusion in Ref. [23].
e results obtained by Hankel-based SVD are shown in Figure 13.
e rotation frequency and its harmonics are  Shock and Vibration discovered.However, the outer race fault characteristic frequency and its harmonics are not extracted.Compared to the results obtained by Hankel-based SVD in Figure 13, the performance of the proposed CPS for diagnosing outer race faults is superior to that of Hankel-based SVD.

Roller Fault Experiment.
A wheelset bearing with two defects on the surface of two rolling elements in Figure 9(c) was tested.e fault signals are collected at a sampling rate of 10 kHz and shown in Figure 14(a).e envelope spectra are shown in Figure 14(b).e roller fault signatures, including the double ball spin frequency (2f BSF ) and the fundamental train frequency (f FTF ) and their harmonics, must be detected.Moreover, sidebands with a spacing of fundamental train frequency f FTF around the double ball spin frequency 2f BSF and their harmonics are also expected to be detected.However, only 2f BSF can be discovered in Figure 14(b), and its harmonics cannot be detected.
e proposed CPS is used to analyse the experiment signals in Figure 14(a).First, the measured signals are decomposed into seventeen IMFs using CEEMDAN.e KSVR of each IMF obtained by CEEMDAN is shown in Figure 15.e first eight IMFs meet the KSVR criteria; thus, they are selected to be processed.
e processing results show that the double ball spin frequency 2f BSF and its harmonics can be found in the squared envelope spectra of the periodic impact components extracted from the first three IMFs.From the processing results of the third to eighth IMF, the rotation frequency information and 50 Hz interferences can be extracted.Because of the limitation of space, taking the second IMF as an example, the processing results are shown in Figure 16.
e double ball spin frequency 2f BSF and quadruple ball spin frequency 4f BSF are detected in the squared envelope spectrum of IMF 2 shown in Figure 16(g), but the higher harmonics tend to smear over each other due to the random speed fluctuations of the rolling elements [32,33].e first and second periodic impact signals separated from IMF 2 are plotted in Figures 16(b) and 16(c), and their squared envelope spectra are shown in Figures 16(h) and 16(i).From the squared envelope spectra in Figures 16(h) and 16(i), the double ball spin frequency 2f BSF , the fundamental train frequency f FTF , and their harmonics are clearly detected.Moreover, sidebands with a spacing of the fundamental train frequency f FTF around the double ball spin frequency 2f BSF and their harmonics are also observed.10 Shock and Vibration ese roller fault signatures are more abundant and clearer than in Figure 16(g).From the locally zoomed figures shown in Figures 16(d)-16(f ), the two groups of impacts in IMF 2 which are excited by two rolling elements defects, respectively, are separated.Due to the random sliding of the rolling elements and the cage, there is a slight difference in the average periods of the impulses excited by the two rolling element defects.erefore, the characteristic frequencies detected in the envelope spectra shown in Figures 16(h) and 16(i) are different, i.e., 66.25 Hz and 66.75 Hz, respectively.In fact, a further analysis is executed using higher frequency resolution data (0.1 Hz frequency resolution), and it is found that the more accurate characteristic frequencies are 66.4 Hz and  Shock and Vibration 66.7 Hz, respectively, which is consistent with the results shown in Figures 16(h) and 16(i).
e results obtained by Hankel-based SVD are shown in Figure 17. e roller fault signatures cannot be detected.e performance of the proposed CPS for diagnosing roller faults is superior to the results obtained by Hankel-based SVD in Figure 17.

Compound Fault Experiment.
A wheelset bearing with three defects on the surface of the outer race (shown in Figure 9(b)) and with two defects on the surface of two rolling elements (shown in Figure 9(c)) was tested.e fault signals were collected at a sampling rate of 10 kHz and are shown in Figure 18(a).Its envelope spectra are shown in Figure 18(b).However, only the outer race fault frequency f BPFO and its harmonics can be discovered in the envelope plotted in Figure 18(b).
e proposed CPS is used to analyse the compound fault signals in Figure 18(a).First, the measured signals are decomposed into sixteen IMFs using CEEMDAN.e KSVR of each IMF obtained by CEEMDAN is shown in Figure 19.e first eight IMFs meet the KSVR criteria; thus, they are selected to be processed.Since, the 9th to 16th IMFs do not meet the KSVR criteria proposed in Section 3, they are not processed further.e processing results show that the outer race ball pass frequency f BPFO and its harmonics can be detected in the squared envelope spectra of the periodic impact components extracted from the first four IMFs and that the double ball spin frequency 2f BSF and its harmonics can be found in the squared envelope spectrum of the periodic impact components extracted from the second IMF.e IMFs from the third to the eighth contain the shaft rotation information and 50 Hz interference.Since the second IMF contains both types of fault information, it is exemplified, and its results are shown in Figure 20.Because of the limitation of length, the results of the other IMFs are not displayed.e second IMF and its squared envelope spectrum are shown in Figures 20(a) and 20(d).
e outer race fault frequency f BPFO and its harmonics are easily detected.However, the roller fault signatures are not detected.By employing the proposed CPS method, two periodic impact components are separated from IMF 2, which are shown in Figures 20(b) and 20(c), respectively.e outer race fault signatures and the roller fault signatures are detected from the envelope spectra of the two components, as shown in Figures 20(e) and 20(f ), respectively.
e results obtained by Hankel-based SVD are shown in Figure 21.
e outer race fault signatures can be clearly observed.However, the roller fault signatures in IMF 2 cannot be detected in the squared envelope spectra in Figure 21(b).Because the outer race fault component has a dominant role in the compound fault signal, it can be successfully extracted by Hankel-SVD.However, the roller fault signatures cannot be extracted by Hankel-SVD due to the disturbance of the outer race fault information.erefore, compared to the results obtained by Hankel-based SVD in Figure 21, the proposed CPS is more effective in diagnosing compound fault signals.

Conclusions
In this paper, a novel fault diagnosis method, CPS, is proposed based on the combination of CEEMDAN, PSM, and SVD.The first eight IMFs meet the KSVR criteria; thus they are selected to be processed.
Since the 9th to 16th IMFs do not meet the KSVR criteria,they are not further processed

Appendix
Logarithmic Relationship between the Matrix Rank and Enhancement SNR. e SNR enhancement equation can be written as where R is the noise in the raw signal and r is the residual noise in the denoised signal.e singular values for the PSM can be obtained by SVD, which is constructed by Gaussian white noise with zero mean and unit variance.e energy of the recovered signal for each singular value can be calculated using the square of the singular value [29].erefore, the energies of R and r can be written as For convenience, the singular values are approximately expressed as a linear equation: Let α � σ q /σ 1 .Clearly, 0 < α < 1.By substituting Equation (19) into Equation ( 20), ‖R‖ 2  2 can be expressed as qσ 2 1 6(q − 1) 2 α 2 + 1  (2q − 1)(q − 1) + 2α(q + 2)(q + 1)  . ( Considering Equations ( 17), (18), and ( 21) � 10 log q 6(q − 1) 2 α 2 + 1  (2q − 1)(q − 1) + 2α(q + 2)(q + 1)  . ( When PSM is a square matrix, the last singular value σ q is very small or approximately zero.Simplified with α � 0, the previous equation is expressed as As q approaches infinity, Δ SNR can be further simplified with an operation of sequence limit, which is expressed as where λ is the coefficient for the rank and λ � 1/3.e singular values do not obey strict linear relationships.erefore, the theoretical value (λ �1/3) is not identical to the simulated value.Figure 22 illustrates the simulation results and a fitting curve with a slope of λ equals to 1/4.
Both the simulation and theoretical calculations indicate the logarithmic relationship between the rank of matrix and enhancement SNR, which provides a reference of the sequence length for analysis.Moreover, the noise energies must be distributed into more singular values to optimize the noise reduction.

Data Availability
e bearing test adopts the double-row tapered roller bearings with three types of faults, including three defects on the surface of outer race, two defects on the surface of two rolling elements and their combination.e bearing test is conducted in ORRC, and the data are confidential.

Figure 2 :
Figure 2: Flowchart of the proposed CPS.

Figure 7 :
Figure 7: Results obtained by the proposed CPS: (a) periodic impact components from the first IMF; (b) Fourier spectrum of (a); (c) envelope spectrum of (a); (d) first periodic impact components from the third IMF; (e) Fourier spectrum of (d); (f ) envelope spectrum of (d);(g) second periodic impact components from the third IMF; (h) Fourier spectrum of (g); (i) envelope spectrum of (g).

Figure 8 :Figure 9 :
Figure 8: Results obtained by Hankel-based SVD: (a) extracted signals from the first IMF; (b) Fourier spectrum of (a); (c) envelope spectrum of (a); (d) extracted signals from the third IMF; (e) Fourier spectrum of (d); (f ) envelope spectrum of (d).

Figure 12 :
Figure 12: Results obtained by the proposed CPS for a bearing outer race fault signal: (a) IMF 3; (b) first periodic impact component extracted from (a); (c) second periodic impact components extracted from (a); (d) squared envelope spectrum of (a); (e) squared envelope spectrum of (b); (f ) squared envelope spectrum of (c).

Figure 16 :Figure 17 :Figure 18 :
Figure 16: Results obtained by the proposed CPS for a bearing roller fault signal: (a) IMF 2; (b) first periodic impact component extracted from (a); (c) second periodic impact components extracted from (a); (d-f ) locally zoomed figures of the waveforms shown in (a-c), respectively; (g-i) squared envelope spectra of (a-c), respectively.

Figure 19 :Figure 20 :Figure 21 :
Figure 19: Kurtosis of the SVR spectra of the IMFs obtained by CEEMDAN for a compound fault signal.

Table 1 :
Parameters related to simulation signals.

Table 2 :
Specifications of the tested bearing.