Shock and Vibration

Shock and Vibration / 2018 / Article
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Nonlinear Vibration of Continuous Systems

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Research Article | Open Access

Volume 2018 |Article ID 1382726 | https://doi.org/10.1155/2018/1382726

Chenguang Huang, Jianhui Lin, Jianming Ding, Yan Huang, "A Novel Wheelset Bearing Fault Diagnosis Method Integrated CEEMDAN, Periodic Segment Matrix, and SVD", Shock and Vibration, vol. 2018, Article ID 1382726, 18 pages, 2018. https://doi.org/10.1155/2018/1382726

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

Academic Editor: Konstantin Avramov
Received13 Jun 2018
Accepted28 Aug 2018
Published05 Nov 2018

Abstract

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. Thirdly, 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. The effectiveness of the proposed CPS has been verified by simulations and experiments. Compared to the well-known Hankel-based SVD, the proposed CPS performs better at extracting the weak periodic impulse responses from the measured signals with strong noise and interferences.

1. Introduction

The wheelset bearing, a key rotating component in high-speed 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. Therefore, 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]. Therefore, 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). The decomposition procedure of EMD is fully automatic and adaptive and does not require predetermined kernels, as does wavelet decomposition [6]. Therefore, EMD is fairly suitable for processing nonlinear and nonstationary signals. EMD has also been widely applied in the field of rotation machinery diagnosis [711], 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, 1719].

The essential step of SVD is constructing the trajectory matrix. In SVD with IMFs [11, 1719], 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. The embedding dimension of a Hankel matrix cannot be easily determined [2022]. 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 [2325]. Therefore, a novel trajectory matrix, the periodic segment matrix (PSM), is constructed instead of a Hankel matrix. The rank of the PSM of periodic signals is equal to one. Thus, 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. Thirdly, 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.

The remainder of this paper is organized as follows. CEEMDAN and its limitations are introduced in Section 2. Section 3 illustrates PSM-based SVD. The novel fault diagnosis method, CPS, is proposed in Section 4. The 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.

2. CEEMDAN and Its Limitations

2.1. Theory 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]. The calculation steps are summarized as follows:

Step 1: let and initialize .

Step 2: for every , add the mode denoted by decomposed with each by EMD to , i.e.,

Step 3: calculate the first mode of , denoted by , to obtain the residue:

Step 4: calculate the kth IMF:

Step 5: go to Step 2 with until the stoppage criterion is met, where is white Gaussian noise with zero mean and unit variance and is the standard deviation of the noise to add. , is a normalization of . , and the ensemble number is in this paper. In the CEEMDAN approach, signal x is decomposed in terms of IMFs, i.e.,where is the residue of signal x after K IMFs are extracted.

2.2. Limitations of CEEMDAN

When CEEMDAN is used to analyse the simulation signals in Section 5, the fault characteristic frequency  = 90.09 Hz and its harmonics are successfully demodulated in the squared envelope spectrum of IMF3 in Figure 1(g), but the fault characteristic frequencies  = 100 Hz and  = 113.64 Hz are not discovered. Although the fault characteristic frequency is extracted, it is still disturbed by strong background noise, as shown in Figure 1(g). There is only one second harmonic of in the spectrum, and 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.

The energy and amplitude of the fault signals associated with and are less than in other fault signals related to . As a result, the low-amplitude vibration signal is submerged in the IMFs decomposed by CEEMDAN, preventing the fault features from being effectively extracted. Therefore, a method for enhancing periodic impact components in an IMF is expected.

3. 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, 1719]. For an arbitrary real matrix , SVD can be expressed as [27]where , , and , , , and diagonal elements are the singular values of Y, and .

3.2. Periodic Segment Matrix

In Equation (5), the previous step of SVD is constructing trajectory matrix 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 [2022]. 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 [2022]. 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 [2325]. Therefore, the novel trajectory matrix, PSM, without accumulative error [28], is used as the trajectory matrix of SVD in this paper. The trajectory matrix with PSM properties can be expressed aswhere is any periodic impulse component of the IMFs obtained by CEEMDAN, is the number of periods, and l is the embedded dimension. , , , , , , and , where is a rounding operator, is times , and . 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]. Therefore, the embedding dimension can be expressed as

Naturally, the row number of is also determined as

For the optimization of noise reduction, the noise energies must be distributed into more singular values. This requirement is proven in Appendix. Therefore, h can be obtained by maximizing the rank of matrix , i.e.,where is the rank of matrix and is equal to .

3.3. Matrix Reconstruction and Signal Recovery

Because the rank of trajectory matrix of a pure periodic signal is [29], the trajectory matrix can be reconstructed by using the first maximal singular value, i.e.,

Finally, the periodic impact components are extracted by the inverse process of in Equation (6).

4. Proposed CPS

A novel fault diagnosis method, CPS, is proposed based on CEEMDAN and PSM-based SVD. The 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. Then, 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 is set equal to 1,000, the normalized standard deviation 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. The embedding dimension of the trajectory matrix can be determined using Equations (7)–(9). The PSM related to an IMF is constructed using Equation (6). PSM-based SVD is conducted. The trajectory matrix is reconstructed by Equation (10). The periodic impact components 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.

5. 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 aswhere is Gaussian white noise whose variance is determined by the signal-to-noise ratio (SNR) and is a periodical impact signal written as [31]where is the amplitude of the periodic impulse component, is the ith resonance frequency, is the fault frequency, is the damping coefficient, and is the remainder of divided by . The fourth signal component is a sinusoidal interference signal, which is expressed as

The parameters related to the simulation signal are listed in Table 1.



11.20.180090.09
20.80.1800100.00
310.13500113.64
40.13450

When the sampling frequency is 10 kHz, the sequence length is 40,000, and the SNR is set equal to −10 dB. The time-domain waveforms of the simulation signals are shown in Figure 3. The Fourier and squared envelope spectra of the simulation signal in Figure 3(f) are shown in Figure 4. Only one fault characteristic frequency and its three harmonics are clearly extracted. The other two fault characteristic frequencies, and , 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. The 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. The SVR spectrum of the first IMF is calculated and shown in Figure 6. The kurtosis of the obtained SVR spectrum is 9.50, which is greater than 4. Therefore, the first IMF is selected to extract the fault information. The 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. The 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. Therefore, the processing results of the fourth IMF are not shown to save space. The third IMF is further analysed. The kurtosis of the SVR spectra of the third IMF is 51.11. The number of rows and columns in the trajectory matrix PSM is 180 and 222, respectively. The first periodic impact signals are extracted and shown in Figure 7(d). The kurtosis of the SVR spectra of the resulting residue is 50.7. The 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). The kurtosis of the SVR spectra of the final obtained residue is less than 4. Thus, 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 that the proposed CPS is fairly effective for bearing fault diagnosis.

In order to illustrate the advancement of the proposed CPS, the well-known Hankel matrix [11, 1719] 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. The results obtained by Hankel-based SVD are shown in Figure 8. Although the fault characteristic frequency and its fifth harmonics are extracted in Figure 8(c), they are disturbed by other complex spectra lines. The other two characteristic fault frequencies and are confused and are not discovered. Therefore, the performance of the proposed CPS for diagnosing bearing faults is superior compared to the results obtained by Hankel-based SVD in Figure 8.

6. Experiment Validation

To further verify the effectiveness of the proposed CPS, a fault experiment rig was built, as shown in Figure 9(a). Three 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. The parameters of the tested bearing are listed in Table 2.


Roller diameter, (mm)Pitch diameter, (mm)Number of rollers, Contact angle, (radian)

26.918019

The fault-related characteristic frequencies are expressed as follows:

Ball pass frequency over outer race, :

Fundamental train frequency, :

Ball spin frequency, :where 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.


Fault typeCharacteristic frequency (Hz)

Ball pass frequency over outer race ()83.23
Ball spin frequency ()33.93
Fundamental train frequency ()4.39

6.1. Outer Race Fault Experiment

A wheelset bearing with three defects on the surface of the outer race, as shown in Figure 9(b), was tested. The fault signals were collected at a sampling rate of 10 kHz and are shown in Figure 10(a). Its squared envelope spectra are shown in Figure 10(b). The outer race fault frequency and its harmonics cannot be discovered in Figure 10(b).

The proposed CPS is used to analyse the experiment signals in Figure 10(a). First, the measured signals are decomposed into sixteen IMFs using CEEMDAN. The KSVR of each IMF obtained by CEEMDAN is shown in Figure 11, also the first seven of whose KSVR are greater than 4. Thus, these IMFs are executed by the proposed CPS. The IMFs from the 8th to 16th fail to meet the KSVR criteria proposed in Section 3; they are not processed further. The 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.

The 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. The third IMF is processed further. However, the kurtosis of the SVR spectrum of the first IMF is 6.60, greater than 4. The first IMF must be processed further. The period determined by its SVR spectrum is 969. The embedding dimension and the number of rows of the trajectory matrix PSM computed using Equations (7)–(9) are 969 and 41, respectively. Then, the resulting PSM is constructed using Equation (6), and PSM-based 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. Then, the second periodic impact signal is recovered and shown in Figure 12(c). Its squared envelope spectrum is counted in Figure 12(f). The 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. The embedding dimension of the Hankel matrix and the selection of the effective singular values are based on the conclusion in Ref. [23].

The results obtained by Hankel-based SVD are shown in Figure 13. The rotation frequency and its harmonics are 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.

6.2. Roller Fault Experiment

A wheelset bearing with two defects on the surface of two rolling elements in Figure 9(c) was tested. The fault signals are collected at a sampling rate of 10 kHz and shown in Figure 14(a). The envelope spectra are shown in Figure 14(b). The roller fault signatures, including the double ball spin frequency () and the fundamental train frequency () and their harmonics, must be detected. Moreover, sidebands with a spacing of fundamental train frequency around the double ball spin frequency and their harmonics are also expected to be detected. However, only can be discovered in Figure 14(b), and its harmonics cannot be detected.

The proposed CPS is used to analyse the experiment signals in Figure 14(a). First, the measured signals are decomposed into seventeen IMFs using CEEMDAN. The KSVR of each IMF obtained by CEEMDAN is shown in Figure 15. The first eight IMFs meet the KSVR criteria; thus, they are selected to be processed. The processing results show that the double ball spin frequency 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. The double ball spin frequency and quadruple ball spin frequency 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]. The 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 , the fundamental train frequency , and their harmonics are clearly detected. Moreover, sidebands with a spacing of the fundamental train frequency around the double ball spin frequency and their harmonics are also observed. These 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. Therefore, 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 66.7 Hz, respectively, which is consistent with the results shown in Figures 16(h) and 16(i).

The results obtained by Hankel-based SVD are shown in Figure 17. The roller fault signatures cannot be detected. The performance of the proposed CPS for diagnosing roller faults is superior to the results obtained by Hankel-based SVD in Figure 17.

6.3. 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. The 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 and its harmonics can be discovered in the envelope plotted in Figure 18(b).

The 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. The KSVR of each IMF obtained by CEEMDAN is shown in Figure 19. 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 proposed in Section 3, they are not processed further. The processing results show that the outer race ball pass frequency 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 and its harmonics can be found in the squared envelope spectrum of the periodic impact components extracted from the second IMF. The 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. The second IMF and its squared envelope spectrum are shown in Figures 20(a) and 20(d). The outer race fault frequency 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. The 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.

The results obtained by Hankel-based SVD are shown in Figure 21. The 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. Therefore, compared to the results obtained by Hankel-based SVD in Figure 21, the proposed CPS is more effective in diagnosing compound fault signals.

7. Conclusions

In this paper, a novel fault diagnosis method, CPS, is proposed based on the combination of CEEMDAN, PSM, and SVD. CEEMDAN is used to avoid spurious modes and improve the signal-to-noise ratio. PSM-based SVD is used to enhance the periodic impact components and isolate different impulse responses with different resonance frequencies. The details of determining the embedded dimension and number of rows of the PSM by SVR are discussed. Simulations and experiments are used to verify the effectiveness of the proposed CPS. Compared to the Hankel-based SVD, CPS is more effective in extracting weak periodic impulse responses from the measured complex vibration signals with strong noise and interferences. Therefore, CPS is suitable for extracting the fault features of a wheelset bearing.

Appendix

Logarithmic Relationship between the Matrix Rank and Enhancement SNR. The SNR enhancement equation can be written aswhere R is the noise in the raw signal and r is the residual noise in the denoised signal.

The singular values for the PSM can be obtained by SVD, which is constructed by Gaussian white noise with zero mean and unit variance. The energy of the recovered signal for each singular value can be calculated using the square of the singular value [29]. Therefore, the energies of R and r can be written as

For convenience, the singular values are approximately expressed as a linear equation:

Let . Clearly, . By substituting Equation (19) into Equation (20), can be expressed as

Considering Equations (17), (18), and (21)

When PSM is a square matrix, the last singular value is very small or approximately zero. Simplified with  = 0, the previous equation is expressed as

As q approaches infinity, can be further simplified with an operation of sequence limit, which is expressed aswhere is the coefficient for the rank and  = 1/3. The singular values do not obey strict linear relationships. Therefore, the theoretical value ( = 1/3) is not identical to the simulated value. Figure 22 illustrates the simulation results and a fitting curve with a slope of 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.

Nomenclature

x(t):Signal decomposed by CEEMDAN
I:Number of iterations
(i = 1, ... , I):A zero mean unit variance Gaussian white noise
:kth residue
:Residue of signal x
:Standard deviation of added noise
:Normalization of
IMFk:kth IMF
Mk(·):Operator to calculate the kth mode by EMD
:Arbitrary real matrix: left singular matrix, for Y
:Left singular vectors, for Y
:Right singular matrix, for Y
:Right singular vectors, for Y
:Characteristic singular matrix, which is rectangular with the same dimensions as Y
:Symmetric matrix, the diagonal entries of which are the singular values, for Y
:Number of singular values, for Y
:Singular values, for Y
s:Any periodic impulse component of IMFs obtained by CEEMDAN
a:Number of periods l: embedded dimension
T:Period length of the periodic impulse component
:Rounding operator
:Rank of matrix Y
:Reconstructed trajectory matrix
:Signal recovered from
:Gaussian white noise with zero mean
s(t):Compound of simulation signals
si(t):ith simulation signal component of s(t)
Ai:Amplitude of si(t)
fci:ith resonance frequency of si(t)
fpi:ith fault frequency of si(t)
ith damping coefficient of si(t)
d:Rolling element diameter
D:Bearing pitch diameter
Z:Number of rolling elements
:Angle of load from the radial plane
fr:Rotation frequency
fBPFO:Ball pass frequency, outer race
fBSF:Ball (roller) spin frequency
fFTF:Fundamental train frequency (cage speed).

Data Availability

The 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. The bearing test is conducted in ORRC, and the data are confidential.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The work was supported by the National Natural Science Foundation of China (Nos. 51305358, 51875481, and 51775456) and the China National Key Research and Development Plan for Advanced Rail Transit (No. 2017YFB1201004).

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Copyright © 2018 Chenguang Huang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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