A Rolling Bearing Fault Diagnosis-Optimized Scale-Space Representation for the Empirical Wavelet Transform

Rolling element bearings have been widely used in mechanical systems, such as electric motors, generators, pumps, gearboxes, railway axles, and turbines, etc. /erefore, the detection of rolling bearing faults has been a hot research topic in engineering practices. Envelope demodulation represents a fundamental method for extracting effective fault information from measured vibration signals. However, the performance of envelope demodulation depends heavily on the selection of the filter band and central frequencies. /e empirical wavelet transform (EWT), a new signal decomposition method, provides a framework for arbitrarily segmenting the Fourier spectrum of an analysed signal. Scale-space representation (SSR) can adaptively detect the boundaries of the EWT; however, it has two shortcomings: slow calculation speeds and invalid boundary detection results. Accordingly, an EWTmethod based on optimized scale-space representation (OSSR), namely, the EWTOSSR, is proposed. /e effectiveness of the EWTOSSR is verified by comparisons between the simulation and the experimental signals. /e results show that the EWTOSSR can automatically and effectively segment the EWT spectrum to extract fault information. Compared with three well-knownmethods (the traditional EWT, ensemble empirical mode decomposition (EEMD), and the fast kurtogram), the EWTOSSR exhibits a much better fault detection performance.


Introduction
Rolling element bearings have been widely used in mechanical systems, such as electric motors, generators, pumps, gearboxes, railways, and turbines [1].ey are closely associated with the reliability of a mechanical system [2], which is therefore a necessity to monitor their condition.Many methods are used to monitor bearing faults, such as vibration, acoustic, and temperature measurement techniques [3][4][5][6][7].However, the use of temperature measurements can delay the detection of faults, and acoustic measurements are easily obscured by noise.As a result, vibration monitoring has become a commonly used method.
Bearing defects can evoke a sharp impulse with a wide frequency range, and such evoked impulses modulate other vibration signals.erefore, to obtain the modulated fault information, a high-frequency resonance technique, also known as envelope analysis, was proposed [8]; the key of envelope analysis is to determine two filter parameters: the bandwidth and the centre frequency.In addition, the concept of the spectral kurtosis was proposed to select one sub-band with a maximal kurtosis to detect bearing faults [9].However, the computational cost required for a shorttime Fourier transform in spectral kurtosis is high [10].Consequently, a fast kurtogram algorithm was developed [11].Unfortunately, the bandwidths of the spectral kurtosis and the fast kurtogram are fixed, being incapable of covering the entire sideband surrounding the resonance frequency.To resolve the fixed bandwidth problem associated with the spectral kurtosis and fast kurtogram approaches, the adaptive spectral kurtosis (ASK) technique was proposed [12].However, the performance of the ASK method heavily depends on the initial superposition window width [13].
e empirical wavelet transform (EWT) [14], which possesses the fast computational properties of the wavelet transform [15,16] and the adaptive characteristic of empirical mode decomposition (EMD) [17,18], provides a framework for arbitrarily determining the bandwidth and centre frequency of a filter.e EWT has been widely applied in the field of rotational mechanical fault detection, and it obtains excellent fault detection results [19][20][21][22][23]. e key of the EWT is to determine the proper boundaries of spectrum segments associated with arbitrary bandwidths.Many methods for determining these boundaries have been developed, including operational modal analysis [24], order statistics filters [25], sparsity guides [26], Pearson's correlation coefficient [27], and SSR [28].However, with the exception of SSR, all boundary-determining methods require a priori knowledge, such as the fault characteristic frequency and order information.In contrast, SSR-based EWT is more adaptive and can therefore extract bearing fault information.However, it is computationally expensive, and some overly noisy spectrum segments exist that are subject to the presence of strong noise [29].
To solve the two problems mentioned above, an optimized scale-space representation (OSSR) and an effective scale-space curve (SSC) selection method are proposed in this paper.e calculation cost of the SSR is closely related to the initial scale; the larger the initial scale value, the faster the calculation efficiency of the SSR, and vice versa.However, with an increase in the initial scale value, low-scale information will be lost, resulting in the failure of the boundary detection process.It has been discovered that the minimum number of the SSR conforms to a logarithmic distribution; therefore, a function of this minimum number is used to adaptively adjust the scale value to achieve a faster calculation of the SSR.e differences in the lengths of the SSCs are used to determine the significant boundaries, and a novel EWT (EWTOSSR) fault detection method based on OSSR is proposed.
e contents followed are organized as follows: the EWT and SSR techniques are introduced in Section 2; Section 3 introduces an OSSR and the boundary selection method; in Section 4, the procedure of the proposed EWTOSSR method is introduced, and the proposed method is verified by a noisy mechanical fault simulation signal in Section 5, and then an experimental verification of the proposed method is conducted in Section 6; finally, the conclusions of this study are drawn in Section 7.

Empirical Wavelet Transform.
e fundamental principle of EWT, which was introduced by Gilles in 2013 to achieve the adaptive decomposition of signals, is to establish a set of filters covering all resonant frequency bands.e EWT includes a low-pass filter representation approximation and N-1 bandpass filters to describe signals in greater detail.If (N+1) divided boundaries are obtained, the Fourier spectrum of the analysed signal will be divided into N corresponding intervals.
e empirical wavelet functions associated with each divided interval are generated using Littlewood-Paley and Meyer's wavelets [14].As a result, the empirical scale function  ϕ n (ω) and empirical wavelet function  ψ n (ω) are written as follows [30]: where ω n denotes a detected boundary; a transition phase T n with a width 2τ n is defined for each ω n .η(x) is an arbitrary C k ([0, 1]) function which satisfies [13]: and the most used function η(x) is e transition phase between two adjacent wavelets can be calculated from the linear relationship τ n � cω n , where 0 < c < 1.An empirical wavelet with a divided pattern is shown in Figure 1.
erefore, a key step of the EWT is to determine the divided boundaries.To effectively obtain these boundaries, as mentioned above, many methods have been proposed,

Scale-Space Representation.
e SSR L(f, s) of the discrete Fourier transform (DFT) X(f) of a discrete signal x(n) can be expressed as [31]: ) is a kernel function, s denotes the scale parameter, and M denotes the width of the kernel function.To ensure the approximation error of the kernel function during the discretization, the value of M is set as M � C � s √ + 1, where 3 ≤ C ≤ 6 (with an increase in the scale parameter, the width of the kernel function should also increase) [32].
e SSR can be interpreted as a global trend of the Fourier spectrum under different scale parameters.e scale parameter will update by � s √ � N �� s 0 √ , where �� s 0 √ � 0.5 is the initial scale parameter corresponding to half of the distance between two Fourier spectrum points, and N � 1, . . .N max are integers.To perform a finite maximum number of steps, N max is defined as twice the length of the signal.Detailed local information of characteristic lengths smaller than � s √ will be removed by the SSR L(f, s).W N denotes the number of local minima in the N-th scale step; the number of local minima with respect to f of L(f, s) is a decreasing function of the step N (the relationship of N and s has been defined previously), although no new minima will appear as the step N increases.e number of initial local minima of L(f, s) (with an initial scale �� s 0 √ ) will be denoted as W0, and each of these local minima defines a scale-space curve Ci (i ∈ [1, W0]) with a length Li (i ∈ [1, W0]), and Li can be regarded as the life span of the i-th minimum [33].erefore, Li is defined as e supports of meaningful modes in Fourier spectra are achievable by delimiting two long SSCs.erefore, the selection of the SSC is the fundamental step of the SSR, and it is solved by finding a threshold T such that the SSCs with lengths larger than T are the curves delimiting the supports of meaningful modes. is is a two-class clustering problem.Many methods have been proposed to solve this problem, including k-Means clustering [34] and probabilistic approaches [35] in addition to the mean value method and Otsu's method [36].

A Discussion of Scale-Space Representation Shortcomings.
A noisy simulated signal is employed to illustrate the shortcomings of SSR when diagnosing a faulty bearing signal.e simulated signal is expressed as follows [37]: where A j is the amplitude of the j-th fault impulse, T p is the time period corresponding to the characteristic fault frequency, f R denotes the excited resonance frequency, β is the structural damping coefficient, J is the number of impulses, u(t) is a unit step function, and s(t) denotes the simulated signal.e fixed parameters are listed in Table 1.e sampling frequency is 4096 Hz, and the length of the simulated signal is 8192.e simulated signal and the noised signal (SNR � −14 dB) are shown in Figures 2(a) and 2(b), respectively.e Fourier spectrum of the noised signal and the Fourier spectrum of the squared envelope (i.e., the squared envelope spectrum) of the noised signal are shown in Figures 2(c) and 2(d), respectively.
Because of the large noise in the simulated signal, the characteristic fault frequency cannot be identified from the squared envelope spectrum of the noised signal.e partial discrete SSRs L(f, s) with different scale parameters s are computed using Equation (3) and shown in Figure 3. Obviously, with the s increase, the SSRs become smoother.
e SSP and threshold T obtained by different two-class   16) and the detected boundaries are shown in Figure 4(b), from which too many false boundaries are evident, thereby providing an incorrect determination of the practical resonance frequency band.
e number of minima declines with an increase in the step according to a similar logarithmic law of attenuation, as shown in Figure 5. ere are two attenuation stages with different slopes.However, during the calculation of L(f, s), the scale parameter � s √ (i.e., the smoothing effect is related to � s √ ) increases linearly.As a result, Gaussian functions with an excessive number of steps are used to smooth the Fourier spectra of the analysed signal; consequently, the resulting L(f, s) is so redundant that numerous error boundaries without any information partitions are found.When the step reaches a maximum number, the number of minimum of L(f, s) is still 11, and it is greater than 2 (as expected).
erefore, a fixed number of steps is not suitable for diagnosing fault bearing signals.In addition, the cost of generating the SSR is fairly expensive.erefore, traditional SSRs have two main shortcomings:   Shock and Vibration (1) Too many false boundaries are found, preventing a determination of the actual boundaries (2) e cost of generating the SSR is fairly high

Optimized Scale-Space Representation.
To overcome the abovementioned shortcomings, a novel scale-updating mechanism is proposed as follows: where W N−1 denotes the number of local minima of L(f, s) at the N-1 th step, and W T is the minimum threshold number set as ten percent of the initial minimum number (W 0 ) according to experimental analysis.When the number of local minima is less than the set threshold W T , the scale parameter is updated linearly ( erefore, upon reaching the stage of minimum rapid change (Figure 5), the novel updating mechanism captures the crucial information for the boundary partitions.Meanwhile, in the slow stage, it not only eliminates the scale redundancy but also minimizes the calculation cost with fewer steps.
Meanwhile, the traditional maximum number of steps N max is replaced by N e corresponding to the no minimum in L(f, N 2 e �� s 0 √ ).erefore, the dynamic determination of the maximum step can expand the scale values to perform effectively with relatively few boundary divisions.

Shock and Vibration
To illustrate the advantages of this approach, the optimized scale-space plane (OSSP) of a simulated signal is shown in Figure 6. e time required to calculate the OSSP using the OSSR on a laptop computer (i5-4460 processor, 3.20 GHz) is shown in Figure 7.
From Figure 7, it is obvious that the computational cost of the OSSR is much less than that of the SSR.rough a comparison between Figures 6 and 4(a), the number of false boundaries decreases obviously, and the SSCs (B1 and B2) that delineate the practical resonance band are clearer.erefore, the OSSR exhibits not only a better performance with regard to the determination of boundaries but also a faster calculation speed than the SSR approach.

e SSC Selection Method.
In the OSSR approach, as the length of the SSCs increase, the probability of becoming a valid boundary increases.
erefore, the problem of finding a meaningful mode can be solved by observing the change in the lengths of SSCs.e lengths L i of SSCs are sorted in descending order ( . en, the difference ΔL d i in the lengths L d i of the SSCs are calculated and expressed as e sequences of L d i and ΔL d i of the simulated signal are depicted in Figure 8. Obviously, the first peak of ΔL d i appears at an index of 2. Consequently, the two longest SSCs are selected to delimit the resonance frequency band, and the determined boundaries are shown in Figure 9.
Compared with Figure 5, the OSSR approach and the proposed boundary selection algorithm can more accurately determine the boundaries used to delineate the resonance band containing fruitful fault information, as shown in Figure 9.

Procedure of the Proposed Method
e procedure of the proposed EWTOSSR method is summarized in this section, and it is shown in Figure 10.
(1) Compute the Fourier spectrum of the analysed signal (2) Calculate the OSSP (3) Detect the boundaries using the differences in the lengths of scale-space curves (4) Decompose the analysed signals using the EWT (5) Extract the fault information from the squared envelope spectra of the decomposed signals.

Simulation Verification
To verify the effectiveness of the proposed method, the signal simulated using Equation ( 6) in Section 2 is analysed by the EWTOSSR technique.According the EWTOSSR procedure, the OSSP is shown in Figure 6. e differences in the length of the SSCs are shown in Figure 8. e peak of the difference appears at an index of 2. Consequently, the two longest SSCs are selected to delimit the resonance frequency band.e  To illustrate the advantages of the proposed EWTOSSR technique, three other well-known methods, namely, the traditional EWT [10], ensemble empirical mode decomposition (EEMD; the standard deviation set to 0.3, and the number of ensembles is set to 100) [38], and the fast kurtogram [6], are used to analyse the same signal simulated in Section 2. By the traditional EWT method, 17 subband signals are obtained.However, only the sub-band signals 7-10 approaching the resonant frequency band are depicted in Figure 12 because they have the greatest probability of extracting characteristic fault frequency from their squared envelope spectrum.e EEMD decomposes the noised signal into 13 intrinsic mode functions (IMFs).Because of the IMF5-13 in the low-frequency domain may not be modulated by fault characteristic frequency, only IMF1-4 and their resulting squared envelope spectra are shown in Figure 13.
e fast kurtogram are shown in Figure 14(a), and the squared envelope and squared envelope spectrum with the largest kurtosis have been shown in Figures 14(b) and 14(c).
e characteristic fault frequency 1/T p obtained by the traditional EWT can be observed only in Figure 12(g); no more harmonics could be detected.
e EEMD and fast kurtogram are unable to extract any useful fault information; instead, the fault characteristic frequency and its harmonics are still submerged in strong noise.
rough comparisons of Figures 12-14 with Figure 11, the EWTOSSR fault-detecting performance is superior to those of the three other methods.is is because the OSSR is capable of detecting relatively reasonable and effective frequency band boundaries containing maximal fault information and minimal noise, thereby highlighting the fault features.

Experimental Validation
To validate the effectiveness of the proposed EWTOSSR method, a testing bench is constructed, as shown in Figure 15(a).e test bench consists of a motor, a drive wheel, a loading device, a wheelset, and an axle box.e power is conveyed to the driving wheel through rubber belts.e traction power of the driving wheel is then transmitted to the wheelset.Two types of artificial faults on the outer race and rollers are shown in Figures 15(b) and 15(c), respectively [39].e parameters of the tested wheelset bearing are listed in Table 2. e fault experiments are conducted at a rotation speed of 50 km/h, and the vibration signals are collected at a sampling frequency of 10 kHz.
where f r is the rotation frequency of the wheelset, d is the roller diameter, D denotes the pitch diameter, Z is the number of rollers, and α denotes the contact angle.e rotation frequency f r corresponding to a running speed of 50 km/h is 5.1 Hz.According to the parameters in Table 2, the outer-race fault characteristic frequency is 41.6 Hz. e sub-band signal filtered by a wavelet transform with Fourier support (811 Hz, 1333 Hz) can be used to extract the fault information.
e outer-race characteristic fault frequency f BPFO and its first five harmonics are clearly displayed in Figure 17(i).Consequently, it is determined that the bearing has suffered an outer-race fault.
Similarly, the three other methods are employed to analyse the same signal shown in Figure 16 Shock and Vibration about outer-race faults.e outer-race fault characteristic frequency f BPFO and its previous three harmonics are extracted and shown in Figures 18(e) and 18(f).However, both the amplitude and the number of its harmonics are less than those in Figure 17(i).is is because the second frequency band of EWTOSSR (from 811 Hz to 1333 Hz) in Figure 16(d) is split into two bandwidths determined by the traditional EWT (the first band is from 809 Hz to 992 Hz and the second band is from 992 Hz to 1211 Hz).As a result, the width of the two sub-band signals is too narrow to contain more useful fault information.en the signal is analysed by the EEMD, the signal is decomposed into 14 IMFs.Because of the IMF5-14 in the low-frequency domain may not be able to modulate the fault characteristic frequency, only IMF1-4 and their resulting squared envelope spectra are shown in Figure 19.e results obtained by fast kurtogram are shown in Figure 20.Obviously, the fault characteristic frequency and its harmonics cannot be identified from Figures 19 and  20.us, the EEMD and fast kurtogram could not extract the fault-related repetitive transients from the signal.By comparison, only the EWT and the proposed EWTOSSR have successfully extracted the fault impulses; however, the fault features obtained by the proposed method are more significant.us, the fault-detecting performance of the EWTOSSR is superior to the three comparison methods.

Rolling Fault Detection.
e measured signal of the rolling fault shown in Figure 15(c) is shown in Figure 21(a) under the test speed.Similarly, the OSSP is obtained and shown in Figure 21(f ), and the differences in the lengths of the SSCs are shown in Figure 21(b).e peak of the difference appears at an index of 2, and thus, the 2 longest SSCs are selected as the boundaries.
e two frequencies for determining the boundaries are 978 Hz and 2524 Hz, and the resulting boundaries are displayed in Figure 21(g).erefore, the measured signal is decomposed into three sub-band signals, and the sub-band signals and their squared envelope spectra are shown in Figure 21.
e rolling fault characteristic frequency f BSF is expressed as According to the rotation frequency of the wheelset and the wheelset bearing parameters listed in Table 2, f BSF is equal to 33.9 Hz.In both Figures 21(i) and 21(j), the rolling fault characteristic frequency f BSF and its harmonics are clearly extracted.
To further illustrate the advantages of the EWTOSSR method, the three other methods are used to analyse the same measured signal in Figure 21(a) for comparison.

Shock and Vibration
According to the process of the EWT, the SSP and threshold obtained by k-means are firstly calculated and shown in Figure 22(  Although these three methods can extract the characteristic rolling fault frequency f BSF and its harmonics, both the amplitudes of f BSF and the number of its harmonics are less than those extracted by the EWTOSSR method.erefore, the EWTOSSR method can be used to effectively determine the boundaries containing fruitful       To verify the superiority of the proposed EWTOSSR method, the three other methods are used to analyse the same

Conclusions
In this paper, a novel fault diagnosis method named the OSSR-based EWT, namely, the EWTOSSR method, was proposed.In the EWTOSSR technique, an optimized SSR replaces the traditional SSR to accelerate the calculation speed and increase the accuracy of the EWT with regard to boundary detection.e bandwidth determined by using these accurate boundaries can contain complete fault information with much less noise.
e EWTOSSR method was verified by simulated and experimental signals.e validation results show that the EWTOSSR approach is fairly suitable for extracting weak fault features from signals with strong noise.
Compared with three well-known fault diagnosis methods (i.e., the traditional EWT, EEMD, and the fast kurtogram), the EWTOSSR exhibits a much better fault detection performance.
Finally, although the EWTOSSR can effectively determine the resonant bands containing fruitful fault information, substantial in-band noise still exists and submerges the fault features to a certain extent.erefore, further research needs to be conducted to reduce or eliminate the in-band noise following the implementation of the EWTOSSR method.

Figure 2 :Figure 3 :
Figure 2: e simulated signal: (a) raw signal, (b) simulated signal with added noise, (c) Fourier spectrum of the noised signal, and (d) squared envelope spectrum of the noised signal.

Figure 8 :Figure 9 :Figure 6 :
Figure 8: e lengths of scale-space curves and the differences in the lengths of scale-space curves.

Figure 7 :
Figure 7: A comparison of the calculation costs.

6. 1 .
Outer-Race Fault Detection.e signals measured when the wheelset bearing in Figure 15(b) with three outer-race faults is installed on the tested bench shown in Figure 15(a) are shown in Figure 16(a).e EWTOSSR method is used to analyse the measured signal.e OSSP obtained according to the EWTOSSR calculation procedure is shown in Figure 16(b), and the calculated differences in the lengths of the SSCs are shown in Figure 16(c).e peak of the difference appears at an index of 6, and thus, the 6 longest scale-space curves are selected as the boundaries.e six frequencies for determining the boundaries are 811 Hz, 1333 Hz, 1615 Hz, 2168 Hz, 2417 Hz, and 3477 Hz, and the detected boundaries are depicted in Figure 16(d).As a result, the analysed signal in Figure 16(a) is decomposed into seven sub-band signals, and the sub-band signals are shown in the left column of Figure 17, while the resulting squared envelope spectra are shown in the right column of Figure17.e outer-race fault characteristic frequency is expressed as[39]

Figure 10 :
Figure 10: e procedure of the proposed method.

Figure 11 :
Figure 11: e proposed EWTOSSR method for diagnosing a simulated signal: (a-c) sub-band signals; (d-f ) squared envelope spectra (red dashed lines: the first three harmonics of the characteristic fault frequency 1/T p ).

Figure 12 :Figure 13
Figure 12: Results obtained by the traditional EWT and SSR: (a-d) sub-band signals; (e-h) squared envelope spectra (red dashed line: characteristic fault frequency 1/T p ).

10
Shock and Vibration spectra have been illustrated in Figure22.e same signal is also processed by the EEMD, and 14 IMFs are obtained.Because the IMF5-14 are low-frequency narrow-band signals which would not be modulated by defect-related frequency, only the first four IMFs and their squared envelope spectra are illustrated in Figure23.e result obtained by fast kurtogram is shown in Figure24.Obviously, the dominant components in Figures22(e) and 22(f ), Figure 23(e), and Figure 24(c) are fault characteristic frequency and its harmonics.

Figure 16 :
Figure 16: Results obtained by the EWTOSSR method: (a) measured signal; (b) OSSP; (c) differences in the lengths of the scale-space curves; (d) detected boundaries (green dash-dotted lines).

Figure 17 :Figure 18 :Figure 19 :
Figure 17: Sub-band signals and their envelope spectra: the left column shows the sub-band signals; the right column shows the squared envelope spectra corresponding to the left column (red dashed lines: f BPFO and its harmonics).

Table 1 :
Parameters of the simulated signal.

Table 2 :
e parameters of the tested wheelset bearing.
Width of the kernel function N max : Maximum number of scale step of scale-space representation WN: Number of local minimum of L(f, s) in N-th scale step