sensors Article
Optimal Resonant Band Demodulation Based on an Improved Correlated Kurtosis and Its Application in Bearing Fault Diagnosis Xianglong Chen 1 , Bingzhi Zhang 2 , Fuzhou Feng 1, * and Pengcheng Jiang 1 1 2
*
Department of Mechanical Engineering, Academy of Armored Forces Engineering, Beijing 100072, China;
[email protected] (X.C.);
[email protected] (P.J.) Beijing Special Vehicle Research Institute, Beijing 100072, China;
[email protected] Correspondence:
[email protected]; Tel.: +86-10-6671-8514
Academic Editor: Vittorio M. N. Passaro Received: 8 January 2017; Accepted: 6 February 2017; Published: 13 February 2017
Abstract: The kurtosis-based indexes are usually used to identify the optimal resonant frequency band. However, kurtosis can only describe the strength of transient impulses, which cannot differentiate impulse noises and repetitive transient impulses cyclically generated in bearing vibration signals. As a result, it may lead to inaccurate results in identifying resonant frequency bands, in demodulating fault features and hence in fault diagnosis. In view of those drawbacks, this manuscript redefines the correlated kurtosis based on kurtosis and auto-correlative function, puts forward an improved correlated kurtosis based on squared envelope spectrum of bearing vibration signals. Meanwhile, this manuscript proposes an optimal resonant band demodulation method, which can adaptively determine the optimal resonant frequency band and accurately demodulate transient fault features of rolling bearings, by combining the complex Morlet wavelet filter and the Particle Swarm Optimization algorithm. Analysis of both simulation data and experimental data reveal that the improved correlated kurtosis can effectively remedy the drawbacks of kurtosis-based indexes and the proposed optimal resonant band demodulation is more accurate in identifying the optimal central frequencies and bandwidth of resonant bands. Improved fault diagnosis results in experiment verified the validity and advantage of the proposed method over the traditional kurtosis-based indexes. Keywords: fault diagnosis; squared envelope spectrum; optimal resonant band demodulation; correlated kurtosis; rolling bearing
1. Introduction Rolling bearings are one of the most common but the most vulnerable parts in rotating mechanical systems. In order to ensure uninterrupted operation and avoid unexpected failures, research attention has been focused on the extraction of weak fault features of rolling bearings which constitute a key factor to condition monitoring and fault diagnosis of rotating mechanical systems [1]. Bearings usually generate wide-band impulses according to their fault frequency and force the bearing system to render an impulse attenuation response when a local fault failure occurs in the inner ring, outer ring, rolling element or cage of a rolling bearing,. As a result, transient impulses in vibration signals of rolling bearings occur cyclically. A popular consensus among the researchers is that the resonant band demodulation can be an effective method in extracting fault features and diagnosing faults of rolling bearings. The key factor of resonant band demodulation is to accurately achieve the central frequency and the bandwith of the optimal resonant frequency band. However, transient features of rolling bearings can be greatly compromised due to the heavy background noise and signal transmission
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paths. Consequently, it becomes difficult to identify resonant frequency bands, not to mention the accurate diagnosis of such faults by resonant frequency band demodulation [2]. Antoni [3,4] put forward the kurtogram based on spectral kurtosis for the detection of non-stationary transients and their frequency locations. By combining resonant frequency band demodulation, the kurtogram can effectively diagnose faults of rolling bearings. In order to improve the computation performance of the kurtogram, Antoni [5] further built the fast kurtogram by combining iterative segmentation of frequency range such as binary tree and band-pass filters such as short-time Fourier Transform. Taking advantage of the superiority of the fast kurtogram, several in-depth studies have been done in the area of bearing vibration monitoring and fault diagnosis [6–10]. However, the fast kurtogram has two deficiencies. Firstly, kurtosis can only characterize the strength of transient impulses, but it cannot differentiate impulse noises and transient impulses, which are cyclically generated in rolling bearing vibration signals. As a result, it may lead to the inaccurate resonance band identification results, unsatisfactory fault feature demodulation results and misleading rolling bearing fault diagnosis results [11–13]. Secondly, the fast kurtogram cannot accurately perfectly determine the central frequencies and bandwidth of rolling bearings’ resonant frequency bands by roughly segmenting frequency ranges [14,15], which may also lead the unsatisfactory demodulation results of fault features. Meanwhile, some artificial intelligence algorithms are also proposed to detect bearing faults [16–18]. In view of these deficiencies, Wang [10] proposed an enhanced kurtogram by calculating kurtosis based on the envelope spectrum of the wavelet package transform filtered signal, which performs well in determining resonance bands, and in demodulating the fault features of rolling bearings. Barszcz [11] proposed the protrugram method based on the kurtosis of the envelope spectrum and narrowband demodulation to select optimal resonant frequency band and to detect transient impulses with smaller rolling bearing vibration signal signal-to-noise ratios. The protrugram shows a superior detection ability of modulated signals in the presence of higher noise than in the case of the fast kurtogram. Enlightened by thermodynamics and entropic uncertainty principle. Antoni et al. [13] proposed the squared envelope infogram (SE infogram) and the squared envelope spectrum infogram (SES infogram) by measuring the negentropy of the squared envelope and the squared envelope spectrum of the short-time Fourier Transform filtered signal, respectively. The SES infogram can effectively provide new information about repetitive transients and locate the resonant frequency bands. Experimental results demonstrate the superiority of SE infogram and SES infogram in selecting resonant bands and demodulating bearing fault features in comparison to the fast kurtogram. Furthermore, Li et al. [14] proposed an optimal demodulation of bearing vibration signals by combining criterion fusion and bottom-up segmentation of the spectral sequence, which effectively determined the optimum resonant frequency band of rolling bearings and improved the robustness in identifying the optimal resonant central frequency. Tse et al. [12] proposed the sparsogram, which is constructed using the sparsity measurements of wavelet packet coefficients’ envelope power spectra at different decomposition depths. The sparsogram effectively locates the resonant bands and shows the superior capability in bearing fault diagnosis by comparison studies with kurtosis, smoothness index and Shannon entropy. Furthermore, Tse et al. [15] further proposed the automatic selection of a perfectly resonant frequency band by combining a complex Morlet wavelet filter and a genetic algorithm to enhance the sparsogram for fault feature demodulation of rolling bearings. The aforementioned review of related studies indicates that the identification quality of resonant frequency bands depends on the construction of criterion used to detect transient features and on the construction of a band-pass filter used to segment the frequency band. Among them, the construction of criteria used to detect transient features is the key factor to determine the quality of identified resonant frequency bands. Taking advantage of the periodical features of transient impulses, McDonald et al. [19] proposed the correlated kurtosis to detect repetitive transient impulses. As the kurtosis can only characterize the strength of transient impulses, which cannot differentiate impulse noises and repetitive transient
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impulses, the correlated kurtosis then becomes a potential substitution criterion for kurtosis-based indexes in detecting repetitive transients. However, the calculation of correlated kurtosis is performed on periodical signals, and previous research results have proved that repetitive transients of rolling bearing are not periodic, but rather are cyclostationary [13]. As a result, the calculation of correlated kurtosis on bearing vibration signals may lack an essential theoretical basis. Besides, the correlated kurtosis proposed by McDonald is only a construction form, which calls for a clear physical definition. To address these issues, the manuscript redefines the correlated kurtosis, on the basis of kurtosis and auto-correlative function, and puts forward an improved correlated kurtosis based on the squared envelope spectrum of rolling bearing vibration signals. Meanwhile, this manuscript proposes an optimal resonant frequency band demodulation by combining the complex Morlet wavelet band-pass filter and the Particle Swarm Optimization algorithm, which can adaptively identify the optimal resonant frequency bands and demodulate transient features of rolling bearings. The rest of the manuscript is organized as follows: the related theoretical background is presented in Section 2, the methods proposed by the authors are discussed in Section 3, the simulation analysis and test applications are presented in Section 4, and our conclusions in Section 5. 2. Kurtosis and Correlated Kurtosis 2.1. Kurtosis As we know, kurtosis can represent the characteristics of signals around their mean value and characterize the strength of transient impulses. Given a zero mean signals, kurtosis can be defined as the ratio among the fourth central moment of the signal to the squared second central moment of the e l,h as follows: signal [20]. Given a zero-mean filtered signal Xl,h , one can obtain its analytic signal X e l,h = Xl,h + j · Hilbert(Xl,h ), X
(1)
where j is the imaginary number, Hilbert(·) the Hilbert Transform function, h and l are respectively the upper and lower cut-off frequency of the band-pass filter f lh . Then the kurtosis can be expressed as: e 4 E ( X l,h ) e l,h ) = Kurtosis(X 2 , e 2 E( Xl,h )
(2)
where E(·) is the mathematical expectation operator. Taking a step further, the corresponding transient impulses can lead to an instantaneous energy fluctuation in vibration signals when a rolling bearing catches local faults. The squared envelope signal can represent the instantaneous energy fluctuation of the signal [13]. Given a zero-mean filtered signal e l,h , one can express its squared envelope signal SE(X e l,h ) as follows: X 2 e l,h ) = X e l,h = Xl,h + j · Hilbert(Xl,h ) 2 , SE(X
(3)
e l,h ) can be expressed as: the variance of SE(X h i h i2 n h io2 e l,h ) = E SE(X e l,h ) − E SE(X e l,h ) D SE(X ,
(4)
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h i e l,h ) can be expressed as: the normalized version of D SE(X h i2 e l,h ) E SE(X e l,h ) = n h d SE(X io2 − 1. e l,h ) E SE(X h
i
(5)
Substituting Equation (5) into Equation (1), a new expression of kurtosis can be expressed as: h i e l,h ) = d SE(X e l,h ) + 1. Kurtosis(X
(6)
As we know, the resonant frequency band filtered signal contains prominent repetitive transient fault impulses when a rolling bearing records local faults. Those repetitive transient fault impulses can lead to instantaneous energy fluctuations in the filtered signal and amplify the normalized variance in Equation (6). As a result, the resonant frequency band filtered signals usually have a high kurtosis value, and resonant frequency band demodulation based on kurtosis citation can be effective in detecting transient fault impulses and diagnosing faults of rolling bearings, such as the fast kurtogram. However, the vibration signals of rolling bearings always contain many other components, which may be useless for bearing diagnosis, such as discrete impulsive noises. Discrete impulsive noises also can give rise to an instantaneous energy fluctuation of signals. The filtered signal, which may not contain repetitive transient impulses but rather contain discrete impulsive noises, can also have a high kurtosis value. As a result, the kurtosis can only characterize the strength of impulses, but it cannot effectively differentiate impulsive noise and repetitive transient fault impulses. As a result, the kurtosis may lead to the inaccurate identification results of resonant bands, the unsatisfactory demodulation results of fault features and the misleading fault diagnosis results of rolling bearings. 2.2. Correlated Kurtosis Thanks to the superiority of kurtosis in characterizing transient impulses, it has been widely used in fault diagnosis of rolling bearings [6]. However, McDonald et al. [19] observed that when compared with a signal containing repetitive periodicity of impulses, a signal with a single impulse would generate a higher kurtosis value. He then concluded that the kurtosis is more effective in detecting a single impulse than repetitive impulses. In reality, rotating machines’ vibration signals usually contain repetitive impulses produced by mechanical faults, rather than a single impulse possibly produced by heavy background noise. Consequently, the kurtosis interfered by heavy noise may correspond with incorrect transient fault features. To address the problem, McDonald et al. [19] proposed the correlated kurtosis (CK) with the help of the impulse periodicity existing in signals. e l,h can be formally constructed as [19]: The CK of zero-mean filtered signal X N h i2 ∗ (n) · x el,h (n − T ) ∑ xel,h
e l,h , T ) = CK(X
n =1
N
2 ∑ xel,h (n)
2
,
(7)
n =1
e l,h . where T is the periodicity of impulses, e xl,h (n) the nth element of the filtered signal X McDonald et al. [19] later verified that CK hit a maximum at a certain period as opposed to the kurtosis which is likely to reach a maximum with a single impulse. CK takes advantage of the periodical features of transient impulses as well as the impulse-like vibration behavior associated with most types of faults. Therefore, it is more effective in detecting signal repetitive transients. Owing to its claimed superiority, some further studies have been done in bearing fault diagnosis and degradation analysis [21,22].
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3. The Proposed Optimal Resonant Band Demodulation Method Based on an Improved Correlated Kurtosis In this section, the authors firstly redefine the correlated kurtosis and then propose the improved correlated kurtosis based on squared envelope spectrum; then propose an optimal resonant band demodulation, by combining the complex Morlet wavelet filter and the Particle Swarm Optimization algorithm, to adaptively identify the optimal resonant frequency bands and demodulate the transient features of rolling bearings. 3.1. Redefinition of Correlated Kurtosis Taking advantage of the periodical features of transient impulses, the correlated kurtosis is a potential substitution criterion for kurtosis-based indexes in detecting repetitive transient impulses. However, the correlated kurtosis proposed by McDonald is only a construction form, which calls for a clear physical definition. In this manuscript, the authors redefine the correlated kurtosis as follows: e can be expressed as: As the auto-correlative function of the filtered signal X rXe (m) = E[ xe∗ (n) × xe(n + m)],
(8)
where m is the time delay coefficient. Substituting Equations (3) and (8) into Equation (2), a new expression of the kurtosis can be obtained as: e l,h ) = h rSE (0)i , Kurtosis(X 2 rXe (0)
(9)
l,h
e l,h when m equals to zero. And rSE (0) is the where rXe (0) is the auto-correlative function of X l,h e l,h ) when m equals to zero. auto-correlative function of squared envelope signal SE(X The auto-correlative function can be used to detect hidden periodicity of signals. If the periodicity e is T, the auto-correlative function r e (m) of X e will also present the periodicity of a finite length signal X X of T, and rXe (m) contains gradual attenuation peak values only when m = 0, T, 2T.... Enlightened by the characteristic of auto-correlation function and combining Equation (9), the authors redefine the correlated kurtosis as the ratio between rSE ( T ) to rXe (0)2 . Given a zero-mean l,h e l,h , the redefined correlated kurtosis can be expressed as: filtered signal X N
e l,h , T ) = h ReCK(X
rSE ( T ) i2 = N × rXe (0) l,h
2 ∑ [| xe(n)| × | xe(n + T )|]
n =1
N
∑ | xe(n)|
2
2
,
(10)
n =1
where N is the length of the filtered signal, T the periodicity of transient impulses, rSE ( T ) the auto-correlative function of the squared envelope signal when m equals to T, and rXe (0) the l,h auto-correlative function of the filtered signal when m equals to zero. In comparison with the correlated kurtosis proposed by McDonald in Equation (7), the redefined correlated kurtosis in Equation (10) has two distinctions. One is that the redefined correlated kurtosis utilizes signal length N as its correction factor and the other is that the computation of the redefined correlated kurtosis moves T points of the signal to right, instead of to left. Moreover, inheriting characteristics of auto-correlative function and kurtosis, the correlated kurtosis redefined in this manuscript can not only reduce impacts of impulsive noises and uncorrelated transient impulses on the detection of repetitive transient fault impulses, but also overcome the drawbacks of the kurtosis-based indexes. As a result, the redefined correlated kurtosis can be a potential substitution criterion for kurtosis-based indexes in detecting repetitive transient fault features.
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3.2. An Improved Correlated Kurtosis Unlike the kurtosis, the correlated kurtosis both in Equations (7) and (10) is defined on the basis of periodic signals. However, according to cyclostationary analysis, bearing vibration signals have typically some second-order cyclostationary characteristics, repetitive transient fault features of bearing vibration signals are not periodic, but rather cyclostationary, which means the instantaneous energy fluctuates rather than their waveform being periodic [13]. As a result, applying the correlated kurtosis directly on bearing vibration signal may lack in necessary theoretical foundations. To overcome this weakness of correlated kurtosis, the authors observe the frequency spectrum of instantaneous energy fluctuations signal, namely the squared envelope spectrum (SES) of bearing vibration signal, which can be expressed as: h i e l,h ) = SES( f ) = DFT SE(X
∑ i∈Z SES( f , k)δ( f − k × ffault ),
(11)
where f fault is the fault feature frequency of rolling bearing, SES( f , k) the kth spectrum line of SES, and δ the unit-impulse function. It can be seen from Equation (11) that the fault feature frequency f fault and its harmonics are periodically distributed on frequency axis of SES, and the periodicity of those harmonics corresponds with the fault feature frequency. By performing the computation of the redefined correlated kurtosis on SES, the correlated kurtosis may remedy the weakness. Then, this manuscript proposes an improved correlated kurtosis on the basis of the redefined correlated kurtosis in Equation (10) and SES, which can be expressed as: h−l
2 ∑ [SES( f ) × SES( f + f fault )]
e l,h , f fault ) = (h − l + 1) × SESCK(X
f =0
"
h−l
#2
(12)
2 ∑ SES( f )
f =0
where h and l are upper and lower cut-off frequency of band-pass filter f lh . Comparing Equation (10) with Equation (12), one can learn that Equation (12) is the implementation of Equation (10) on SES. The improved correlated kurtosis can approach a higher value when the SES of filtered signal contains higher peak values of the fault feature frequency, more harmonics, and wider filtered bandwidth. On the contrary, the improved correlated kurtosis approaches an especially small value when the SES contains lower peak values of the fault feature frequency, less harmonics, and narrower filtered bandwidth, or even doesn’t contain any fault feature frequency, nor its harmonics. As a result, it also can overcome the drawbacks of the kurtosis-based indexes. 3.3. Performance Comparison In previous subsections, the authors have redefined the correlated kurtosis and proposed an improved correlated kurtosis based on SES. In order to verify the superiority of the redefined correlated kurtosis and the improved correlated kurtosis against the kurtosis, spectral kurtosis and the correlated kurtosis originated by McDonald, several discrete testing signals, shown in Figure 1, are constructed as follows: Signal Y1 : a unit-impulse function δ(20) and its length equals to 40. Signal Y2 : a Dirac comb function composed by unit-impulse functions δ(20), δ(40) and δ(60), its length equals to 80. Signal Y3 : a Dirac comb function composed by unit-impulse functions δ(20), δ(40), δ(60), δ(80) and δ(100), its length equals to 120. Signal Y4 : a sine function and its length equals to 200.
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As the kurtosis, the original correlated kurtosis and the redefined correlated kurtosis are all 7 of 19 defined in the time domain, in order to compare their performances in detecting repetitive transient impulses, firstly, treating testing signals in Figure 1 as time singles. The results calculated by the kurtosis, theYcorrelated kurtosis originated by McDonald and the redefined correlated kurtosis are Signal 5 : a Gaussian noise function and its length equals to 200. shown in Table 1. Sensors 2017, 17, 360
Y
1
1 0.5
Y
2
0 1 0.5
Y
3
0 1 0.5
Y
5
Y
4
0 1 0 -1 2 0 -2 0
20
40
60
80
100 120 Point [n]
200
Figure Figure 1. 1. Testing Testing signals. signals. Table 1. The results of testing signals in time domain calculated by the kurtosis (K), the correlated As is mentioned in references [11,13], a Dirac comb stands as an idealization of a series of repetitive kurtosis originated by McDonald (CK) and the redefined correlated kurtosis (ReCK).
transients and a single impulse is a particular instance of a Dirac comb, though they are unlikely to occur in practice. TheSignal characteristicKof repetitive CK transients in timeReCK is also a series harmonic in SES. Fortunately, a Dirac comb same structure0in both time domain Y1 has the 15.0526 0 and SES. Therefore, Dirac comb functions can beYutilized herein to compare performances of different indexes in detecting 2 15.0526 0.2222 13.3333 repetitive transients. Y3 15.0526 0.16 16 As the kurtosis, the original correlated kurtosis and the redefined correlated kurtosis are all 0.2928 0.0048 0.9694 Y4 defined in the time domain, in order to compare their performances in detecting repetitive transient −1.5 0.0068 1.35 Y5 impulses, firstly, treating testing signals in Figure 1 as time singles. The results calculated by the kurtosis, the correlated kurtosis originated by McDonald and the redefined correlated kurtosis are From Table 1, one can learn that, the results of Y1, Y2 and Y3 calculated by the kurtosis have the shown in Table 1. same value, even if Y1, Y2 and Y3 contain different quantities of repetitive transient impulses. As a result, the kurtosis in theoftime domain a single and(K), a series of repetitive testing signalscannot in timedifferentiate domain calculated by impulse the kurtosis the correlated Table 1. The results impulses, theoriginated latter of which is usually to be more helpfulkurtosis in fault(ReCK). diagnosis. The result of Y2 kurtosis by McDonald (CK)believed and the redefined correlated calculated by the correlated kurtosis originated by McDonald is larger than other signals, especially Signal K CK more obvious ReCK the optimal repetitive impulses with periodicity than Y2. Such larger than Y3, which is is the case, the correlated might fail in0 identifying the optimal Y1 kurtosis originated 15.0526 by McDonald 0 Y2 15.0526 impulses. 0.2222 13.3333 filtered signal which contains more repetitive Finally, the result of Y3 calculated by the Y 15.0526 0.16 16 3 redefined correlated kurtosis is larger than other signals. Thus, the redefined correlated kurtosis Y4 0.2928 0.0048 0.9694 claims the superiority over other methods in identifying the optimal filtered signal which contains Y5 −1.5 0.0068 1.35 more repetitive impulses. As a Dirac comb has the same structure in both time domain and SES, treating testing signals Y1, From Table 1, one that, the results Y1 , Y2 and Yresults have the 3 calculated Y2 and Y3 in Figure 1 ascan SESlearn of themselves. Thenof computation of Y1, by Y2 the andkurtosis Y3 calculated by same value, even if Y , Y and Y contain different quantities of repetitive transient impulses. As a 3 spectral kurtosis and1 the2 improved correlated kurtosis are shown in Table 2. Due to the same result, the kurtosis in the and time spectral domain cannot differentiate a single impulse andredefined a series ofcorrelated repetitive construction of kurtosis kurtosis, the same construction of the impulses, the latter of which is usually believed to be more helpful in fault diagnosis. The result Y2 kurtosis and the improved correlated kurtosis based on SES. The computation results in Table 2 of have calculated byasthe kurtosis originated bytesting McDonald is larger than other signals, especially same values is correlated shown in Table 1 when treating signals as their SES. Then, one can come a larger than Y , which is the optimal repetitive impulses with more obvious periodicity than Y2 .and Such 3 same conclusion that, the spectral kurtosis cannot differentiate a single impulsive frequency a is the case, the correlated kurtosis originated by McDonald might fail in identifying the optimal series of repetitive impulse harmonics, the improved correlated kurtosis can identify the optimal filtered frequency signal which contains repetitive theharmonics result of Yin by the 3 calculated filtered band, which more contains optimalimpulses. repetitive Finally, impulsive SES. redefined correlated kurtosis is larger than other signals. Thus, the redefined correlated kurtosis claims the superiority over other methods in identifying the optimal filtered signal which contains more repetitive impulses.
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As a Dirac comb has the same structure in both time domain and SES, treating testing signals Y1 , Y2 and Y3 in Figure 1 as SES of themselves. Then computation results of Y1 , Y2 and Y3 calculated by spectral kurtosis and the improved correlated kurtosis are shown in Table 2. Due to the same construction of kurtosis and spectral kurtosis, the same construction of the redefined correlated kurtosis and the improved correlated kurtosis based on SES. The computation results in Table 2 have same values as is shown in Table 1 when treating testing signals as their SES. Then, one can come a same conclusion that, the spectral kurtosis cannot differentiate a single impulsive frequency and a series of repetitive impulse harmonics, the improved correlated kurtosis can identify the optimal filtered frequency band, which contains optimal repetitive impulsive harmonics in SES. Table 2. The results of testing signals in SES calculated by the spectral kurtosis (SK), and the improved correlated kurtosis (SESCK). Signal
SK
SESCK
Y1 Y2 Y3
15.0526 15.0526 15.0526
0 13.3333 16
As is mentioned above, the redefined correlated kurtosis can detect the optimal repetitive transient impulses in time domain and the improved correlated kurtosis can detect the optimal repetitive impulsive harmonics in SES. Consequently, the performance comparison can verify the superiority of the redefined correlated kurtosis and the improved correlated kurtosis. 3.4. The Optimal Resonant Band Demodulation The resonant frequency band demodulation is an effective method for extracting fault features and diagnosing faults of rolling bearings, and the key factor of resonant frequency band demodulation is to accurately identify the central frequency and the bandwith of the optimal resonant frequency band. However, the identification quality of resonant band depends on the construction of criterion used in detecting transient features and on the construction of band-pass filter used in segmenting frequency band. Firstly, as is demonstrated in previous subsections, the improved correlated kurtosis on the basis of SES can not only reduce the impacts of impulse noises in detecting repetitive transient fault features, but also can remedy the drawbacks of the kurtosis-based indexes. Thus, the improved correlated kurtosis can be a substitution criterion for kurtosis-based indexes in detecting repetitive transient fault features. Secondly, the complex Morlet wavelet is a kind of band-pass filter. Its wavelet function contains exponential attenuation components, which correspond with transient characteristics of rolling bearing faults. The wavelet function of complex Morlet wavelet ϕ(t) can be expressed as [23–26]: α ϕ(t) = √ exp(−α2 π 2 ) exp(j2π f c t), π
(13)
its Fourier Transform ψ( f ) can be expressed as: π2 ψ( f ) = ψ∗ ( f ) = exp − 2 ( f − f c )2 , δ
(14)
where α is the envelope factor and f c the central frequency of the complex Morlet wavelet. Taking normalized central frequency f c = 0.25 Hz and envelope factor α = 0.06 Hz as an example, the Morlet wavelet waveform both in time and frequency domain are shown in Figure 2. The filtered bandwidth is defined as β, and it equals to the frequency range √ between the upper cut-off frequency and lower cut-off frequency, which correspond with the 2/2 times maximum Morlet
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wavelet amplitude in frequency domain [26], as shown in Figure 2b.√As a result, the filtered range of Sensors 2017, 17,Morlet 360 9 of 19 the complex wavelet is from f c − β/2 to f c + β/2, and β = α 2 ln 2/π.
Amplitude [m·s-2]
(a)
0.1 0.05 0 -0.05 -0.1 -1
Amplitude [m·s-2]
(b)
-0.5
0 T ime [s]
0.5
1
1 fc - /2
fc + /2
0.5
0
0
0.1
0.2 0.3 Frequency [Hz]
0.4
0.5
Figure 2. (a) waveform of of the the Morlet Morlet wavelet; wavelet; (b) (b) frequency frequency waveform waveform of of the the Morlet Morlet wavelet. wavelet. Figure 2. (a) Time Time waveform
Finally, an optimal resonant band demodulation method, by combining the improved correlated Then the complex Morlet wavelet filtered signal can be expressed as: kurtosis and the complex Morlet wavelet filter, is proposed for the purpose of diagnosing faults of rolling bearings in this manuscript.WCoupled with Swarm Optimization algorithm,(15) the −1 the Particle ∗ x ( f c , β ) = F [ X ( f ) ψ ( f )], proposed method can adaptively identify the optimal resonant bands and demodulate transient 1 is features rolling bearings. where F −of the inverse Fourier Transform function, X ( f ) the Fourier Transform of signal x (t) and flowchart of the proposed method is shown in Figure 3, and the detailed steps are as follows: Wx ( fThe , β ) the filtered signal of x (t) by the complex Morlet wavelet filter. c an optimal resonant band demodulation method, combiningparameters the improved correlated (1) Finally, Calculating the fault feature frequency according to thebygeometric of the rolling kurtosis and the complex Morlet wavelet filter, is proposed for the purpose of diagnosing faults bearing and operation conditions of the mechanical system. of bearings the in this manuscript. with the Particle Swarm algorithm, (2)rolling Determining searching rangeCoupled of the resonant bandwidth. AsOptimization the band-pass filtered the proposed method can adaptively identify the optimal resonant bands and demodulate transient bandwidth should be greater than three times of the fault feature frequency, the searching range features of rolling bearings. of the resonant bandwidth can be set as 3 ffault 10 ffault . The flowchart of the proposed method is shown in Figure 3, and the detailed steps are as follows: (3) Determining the searching range of the resonant central frequency. As the filtered range of the fc 2,frequency fc 2 , theaccording searching to range the resonant central frequency can Morlet wavelet Calculating thefilter faultisfeature the of geometric parameters of the rolling min /2 fc fs /2 conditions /2, whereoffsthe bearing operation mechanical represents the system. sampling frequency. be set asand min (4) Initializing the Particle Swarm Optimization algorithm. TheAs dimension of thefiltered particle swarm is (2) Determining the searching range of the resonant bandwidth. the band-pass bandwidth set as 2, be thegreater size of than the particle swarm setfault as 30.feature The initial particlethe swarm can berange achieved in should three times ofisthe frequency, searching of the line with bandwidth the searching in 3step (2) and resonant cansettings be set as × f fault ≤ βstep ≤ 10(3). × f fault . (5) Performing onrange the initial swarm, calculating values of the improved (3) DeterminingSES theanalysis searching of theparticle resonant central frequency.their As the filtered range of the correlated kurtosis and achieving the individual maximum and global maximum. Morlet wavelet filter is [ f c − β/2, f c + β/2], the searching range of the resonant central frequency (6) Updating of − theβ min particle swarm iterativelythe until the maximum iterations. can be set speed as β minand /2 ≤location f c ≤ f s /2 /2, where f s represents sampling frequency. f . particle swarm is central frequency c_opt and (4) Achieving Initializingthe theoptimal Particleresonant Swarm Optimization algorithm. Thebandwidth dimension ofoptthe bandwidth (7) Setting of swarm the optimal Morlet wavelet filter as fcan set as 2,the thecentral size offrequency the particle is setcomplex as 30. The initial particle swarm be achieved in c_opt , the line with the searching settings in step (2) and step (3). of the optimal complex Morlet wavelet filter as opt . Achieving the optimal resonant band (5) filtered Performing SES analysis on the initial particle swarm, calculating their values of the improved signal. correlated kurtosis achieving individual and and global maximum. (8) Performing the SESand of the optimalthe resonant bandmaximum filtered signal diagnosing faults of rolling (6) bearings. Updating speed and location of the particle swarm iteratively until the maximum iterations. Achieving the optimal resonant central frequency f c_opt and bandwidth β opt .
(1)
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(7)
(8)
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Setting the central frequency of the optimal complex Morlet wavelet filter as f c_opt , the bandwidth of the optimal complex Morlet wavelet filter as β opt . Achieving the optimal resonant band filtered signal. Performing the SES of the optimal resonant band filtered signal and diagnosing faults of rolling bearings.
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Vibration signal
Calculating fault feature frequency f fault Determining the searching range of the resonant central frequency fc and bandwidth β Initializing the dimension and size of Particle Swarm Optimization algorithm Achieving the optimal central frequency and bandwidth of resonant band Constructing the optimal Morlet wavelet filter and filtering rolling bearing signals
Achieving the initial particle swarm
Performing SES analysis on the current particle swarm Calculating the improved correlated kurtosis of the current particle swarm
Achieving the individual maximum and global maximum
Yes
Maximum iterations?
Performing SES analysis on the filtered signal No Diagnosing fault state of rolling bearings
Updating speed and location
End Figure 3. Flowchart of the proposed optimal resonant band demodulation method. Figure 3. Flowchart of the proposed optimal resonant band demodulation method.
4. Analysis 4. Analysis
Simulation Analysis 4.1.4.1. Simulation Analysis numerical simulation model rolling bearing vibration is used construct bearing failure AA numerical simulation model of of rolling bearing vibration is used to to construct bearing failure simulation signals verify validity proposed method. The numerical simulation model simulation signals to to verify thethe validity of of thethe proposed method. The numerical simulation model is is expressed as follows: expressed as follows: x(t ) s(t ) n(t ) x (t) =s(t) + n(t) h(t j T )+p(t t )+p(t t ) s(t) =s(t )h(t − jj × T − τi ) +i p(t + t11) + p(t +2 t2 ) ∑ j h(t ) A0 exp( Ct()2π cos(2 Ct) cos f n t)f n t ) h(t) =A0 exp(− p(t ) M exp( Dt ) cos(2 f t ) p(t) = M0 exp(− 0 Dt ) cos(2π f m t ) m
(16) (16)
where rolling bearing simulation signal impulse and signal noise signal (t) includes s(t) noise where thethe rolling bearing simulation signal x (t) xincludes impulse seriesseries s(t) and n(t). T nis( t ) f . T is the average period of impulse series; is the impulsive feature frequency which equals to the i the average period of impulse series; f i is the impulsive feature frequency which equals to the reciprocal is the tiny random fluctuation impulse and T Hz; andτi set to tiny 100 random Hz; i fluctuation of reciprocal T and set toof100 is the of the i-th impulse and τ of ∼ the N (0,i-th 0.05T ); C is the damping coefficient which equalscoefficient to 900; f n is the resonant frequency bearing damping which equals to 900; fn ofisthe thesimulated resonant rolling frequency of the N(0,0.05 T) ; C is the system which equals to 4000 Hz; A is the vibration amplitude which equals to 1; p ( t ) is additional 0 simulated rolling bearing system which equals to 4000 Hz; A0 is the vibration amplitude which impulsive noise; D is the damping coefficient of the impulsive noise which equals to 600; f m is the equals to 1; p(t) is additional impulsive noise; D is the damping coefficient of the impulsive noise which equals to 600; fm is the resonant frequency of the impulsive noise which equals to 2500 Hz; and M0 is the vibration amplitude of the impulsive noise which equals to 5. Two impulse noise components are added to s(t) at the moment of t1 0.125 s and t1 0.225 s , respectively. The noise signal n( t )
is white Gaussian noise with noise variation equaling to 1. The sampling frequency of
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resonant frequency of the impulsive noise which equals to 2500 Hz; and M0 is the vibration amplitude of the impulsive noise which equals to 5. Two impulse noise components are added to s(t) at the moment of t1 = 0.125 s and t1 = 0.225 s, respectively. The noise signal n(t) is white Gaussian noise with noise variation equaling to 1. The sampling frequency of simulation signal is set to 12,800 Hz while the sampling points are set to 6400. In line with Equation (16), a simulation signal is generated as is shown in Figure 4. The simulation signal only17,includes impulsive series and impulsive noise components is shown in Figure Sensors 2017, 360 11 of4a. 19 The repetitive transient impulsive series, additional impulsive noise components and their transient features distinctare in the signal.inWhen the simulation signal includessignal noise includes signal and noisesignal variation transientare features distinct the signal. When the simulation noise and equals to 1, the signal-to-noise ratio of the simulation signal is − 14.9 dB. And the simulation signal noise variation equals to 1, the signal-to-noise ratio of the simulation signal is −14.9 dB. And the waveform inwaveform time domain in frequency domain are respectively shown in Figure 4b,c. simulationboth signal bothand in time domain and in frequency domain are respectively shown One can learn the influence heavy background noise, none obvious feature in Figure 4b,c.that, One because can learnofthat, because ofofthe influence of heavy background noise,signal none obvious can be captured both time domain and in frequency domain. The squared envelope spectrum (SES) signal feature can be in captured both in time domain and in frequency domain. The squared envelope of the simulation by conducting envelope analysis envelope directly on the simulation shown in spectrum (SES) ofsignal the simulation signal by conducting analysis directly signal on theissimulation Figure 4d,shown whereinwe cannot any transient fault features in thefault SES. features in the SES. signal is Figure 4d,recognize where we cannot recognize any transient 5 0 -5 0.1
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Firstly, utilizing the proposed optimal resonant band demodulation method to analyze the Firstly, utilizing the proposed optimal resonant band demodulation method to analyze the simulation signal, which is shown in Figure 4b. According to the algorithm flow in Figure 3, the simulation signal, which is shown in Figure 4b. According to the algorithm flow in Figure 3, impulsive feature frequency f equals 100 Hz; the central frequency searching range of the optimal the impulsive feature frequencyi f i equals 100 Hz; the central frequency searching range of the optimal resonant band is set between 50 Hz and 6350 Hz; and the bandwidth searching range of the optimal resonant band is set between 50 Hz and 6350 Hz; and the bandwidth searching range of the optimal resonant band is set between 100 Hz and 1000 Hz. By initializing the Particle Swarm Optimization resonant band is set between 100 Hz and 1000 Hz. By initializing the Particle Swarm Optimization algorithm, the calculation result after 30 times’ interactive computation is shown in Figure 5a, where algorithm, the calculation result after 30 times’ interactive computation is shown in Figure 5a, where the maximum of the improved correlated kurtosis occurs at 26th calculation, which equals 2.617, and the maximum of the improved correlated kurtosis occurs at 26th calculation, which equals 2.617, and the acquired optimal center frequency and bandwidth of resonant frequency band are 4015 Hz and the acquired optimal center frequency and bandwidth of resonant frequency band are 4015 Hz and 760 Hz, respectively. 760 Hz, respectively. Constructing the complex Morlet wavelet filter based on the optimal resonant frequency band parameters, the filtered simulation signal both in time domain and in frequency domain are shown in Figure 5b,c, respectively. The signal filtered by the optimal complex Morlet filter roughly shows transient impulse feature. The filtering central frequency of the optimal complex Morlet filter is almost identical to the resonant central frequency of the simulation signal, which equals to 4000 Hz, and the filtering frequency range can properly cover the resonant frequency band. The SES of the simulation signal performed by the proposed method is shown in Figure 5d, where the fault feature
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Figure5.5.Analysis Analysis results of proposed the proposed optimal resonant band demodulation based on the Figure results of the optimal resonant band demodulation based on the improved improvedkurtosis. correlated (a) The iterative computation of the (b)signal the filtered signal correlated (a)kurtosis. The iterative computation results of theresults PSO; (b) the PSO; filtered by optimal by optimal Morlet wavelet (c) the of signal; the filtered signal; (d)the thefiltered SES of the filtered signal. Morlet wavelet filter; (c) thefilter; spectrum ofspectrum the filtered (d) the SES of signal.
To compare the correlated kurtosis and spectral kurtosis, the kurtosis Constructing the improved complex Morlet wavelet filter with basedkurtosis on the optimal resonant frequency band and spectral kurtosis of squared envelope signal are used to substitute the improved correlated parameters, the filtered simulation signal both in time domain and in frequency domain are shown respectively. According the same algorithm flow incomplex Figure 3Morlet and the same initialization inkurtosis, Figure 5b,c, respectively. The signal filtered by the optimal filter roughly shows parameters used in Figure 5, analysis results of the same simulation signal obtained by kurtosis and transient impulse feature. The filtering central frequency of the optimal complex Morlet filter is almost spectral kurtosis are shown in Figures 6 and 7, respectively. identical to the resonant central frequency of the simulation signal, which equals to 4000 Hz, and the One can learnrange from can Figure 6a thatcover the maximal kurtosis of the squared envelope occurs at filtering frequency properly the resonant frequency band. The SES of signal the simulation the 26th iterative calculation and the acquired optimal center frequency and bandwidth of resonant signal performed by the proposed method is shown in Figure 5d, where the fault feature frequency band are 3115 Hz and 723 Hz, respectively. Constructing complex Morlet wavelet filter based onare the f i and its harmonic frequencies are easy to identify. In the meanwhile, fault feature frequencies acquired optimal resonant frequency parameters, filtered correlated signal is shown in Figure 6b andthe its dominant in frequency domain. Consequently, thethe improved kurtosis can conquer frequency spectrum shown in Figure 6c. From Figure 6b, it can be learned that, there are two disturbance of impulsive noise, identify the repetitive transient impulse series. Combining complex prominent the filteredthe signal, whichoptimal are consistent with the impulsive noise Morlet filterimpulsive and PSO components optimizationinalgorithm, proposed resonant band demodulation components. From Figure 6c, the filtering central frequency of the optimal complex Morlet filter is method can adaptively identify the optimal resonant band and demodulate fault features. approaching to the resonant central frequency of the impulsive noise components, which equals to To compare the improved correlated kurtosis with kurtosis and spectral kurtosis, the kurtosis and 2500 Hz.kurtosis As a result, the filtered signalsignal contains impulsive components. From Figure spectral of squared envelope are prominent used to substitute thenoise improved correlated kurtosis, 6d, the SES of the filtered signal cannot extract any distinct fault features. respectively. According the same algorithm flow in Figure 3 and the same initialization parameters used in Figure 5, analysis results of the same simulation signal obtained by kurtosis and spectral kurtosis are shown in Figures 6 and 7, respectively. One can learn from Figure 6a that the maximal kurtosis of the squared envelope signal occurs at the 26th iterative calculation and the acquired optimal center frequency and bandwidth of resonant band are 3115 Hz and 723 Hz, respectively. Constructing complex Morlet wavelet filter based on the acquired optimal resonant frequency parameters, the filtered signal is shown in Figure 6b and its frequency spectrum shown in Figure 6c. From Figure 6b, it can be learned that, there are two prominent impulsive components in the filtered signal, which are consistent with the impulsive noise components.
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Onecan canlearn learnfrom fromFigure Figure7a 7athat thatthe themaximal maximalspectral spectralkurtosis kurtosisofofthe thesquared squaredenvelope envelopesignal signal One occursatat24th 24thiterative iterativecalculation calculationand andthe theacquired acquiredoptimal optimalcenter centerfrequency frequencyand andbandwidth bandwidthofof occurs resonant band are 4560 Hz and 1000 Hz, respectively. Constructing complex Morlet waveletfilter filter resonant band are 4560 Hz and 1000 Hz, respectively. Constructing complex Morlet wavelet based on the acquired optimal resonant frequency parameters, the filtered signal is shown in Figure based on the acquired optimal resonant frequency parameters, the filtered signal is shown in Figure 7b 7b and its frequency spectrum is shown in Figure 7c. From Figure 7c, the filtering central frequency and its frequency spectrum is shown in Figure 7c. From Figure 7c, the filtering central frequency of of the optimal complex Morlet filter is not identical to the resonant central frequency of the simulation the optimal complex Morlet filter is not identical to the resonant central frequency of the simulation signal,but butthe thefiltering filteringfrequency frequencyband bandstill stillcan canproperly properlycover coverthe theresonant resonantcentral centralfrequency frequencyofofthe the signal, simulationsignal. signal. Consequently, Consequently, from from Figure Figure7d, 7d,the theSES SESof ofthe thefiltered filteredsignal signalcan canextract extractdistinct distinct simulation fault features. fault features. Asmentioned mentioned in this subsection, kurtosis cancharacterize only characterize the of strength ofimpulses, transient As in this subsection, kurtosis can only the strength transient impulses, it cannot differentiate impulse noises and repetitive transient impulses, the SES of the it cannot differentiate impulse noises and repetitive transient impulses, the SES of the filtered signal filtered signal in Figure 6d cannot extract any useful fault features. The SES of the filtered signal in Figure 6d cannot extract any useful fault features. The SES of the filtered signal in Figure 7din Figure 7d can extract fault butsmaller it contains featureand frequency and less can extract correct faultcorrect features, butfeatures, it contains faultsmaller featurefault frequency less harmonics. harmonics. The SES of the filtered signal in Figure 5d can extract optimal fault features, The SES of the filtered signal in Figure 5d can extract optimal fault features, which contains which clear contains clear fault feature frequency andAs its aharmonics. a result,correlated the improved correlated kurtosis fault feature frequency and its harmonics. result, the As improved kurtosis can overcome can overcomeofshortcomings kurtosis-based indexes, and the resonant proposedband optimal resonant band shortcomings kurtosis-basedofindexes, and the proposed optimal demodulation can demodulation can identify the natural resonance frequency of rolling bearing vibration signals. Analysis results of the proposed method can clearly extract fault features and correctly diagnose rolling bearing faults, that is, the analysis results verify the validity and superiority of the proposed method.
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4.2. Experimental Experimental Analysis Analysis 4.2. Vibration data data from from the the Case Case Western Western Reserve Reserve University University (CWRU) (CWRU) Bearing Bearing Data Data Center Center are are Vibration utilized to verify the validity of the proposed method. The bearing test rig of CWRU [27] is shown in utilized to verify the validity of the proposed method. The bearing test rig of CWRU [27] is shown in Figure 8, which consists of a two horsepower Reliance Electric motors, a torque transducer/encoder, Figure 8, which consists of a two horsepower Reliance Electric motors, a torque transducer/encoder, a a dynamometer control electronics. bearings are seeded with faults using electrodynamometer andand control electronics. MotorMotor bearings are seeded with faults using electro-discharge discharge machining. Faults ranging from 0.007 inches to 0.021 inches in diameter are introduced machining. Faults ranging from 0.007 inches to 0.021 inches in diameter are introduced separately on separately on the inner ring. Faulty bearings are then reinstalled into the test rig with motor the inner ring. Faulty bearings are then reinstalled into the test rig with motor loads ranging fromloads zero ranging from zero toand three horsepower and motor speeds rotating at and a rate between 1797 and data 1720 to three horsepower motor speeds rotating at a rate between 1797 1720 RPM. Vibration RPM. Vibration data are collected using accelerometers, which are mounted at the 12 o’clock position are collected using accelerometers, which are mounted at the 12 o’clock position at both the drive end at both andhousing. fan end Vibration of the motor data sets areacquisition recorded using a data and fan the enddrive of theend motor datahousing. sets are Vibration recorded using a data system and acquisition system and the sampling frequency is set to 12 KHz. According to the geometric the sampling frequency is set to 12 KHz. According to the geometric parameters of the rolling bearing, parameters of thefrequency rolling bearing, the fault feature of inner ring=faults calculated the fault feature of inner ring faults canfrequency be calculated as f bpfi 5.415can × f rbe , where f bpfi as is fr is frequency and of rotating of the motor fbpfi =fault 5.415 feature fr , where fbpfi is the the frequency andfault f is feature the rotating frequency thethe motor drivefrequency shaft. Data sampling r
drive shaft.and Data sampling conditions and their fault feature frequencies aredetails listed in Table 3. the Further conditions their fault feature frequencies are listed in Table 3. Further regarding test detailscan regarding can be found onCenter the CWRU Bearing setup be foundthe ontest thesetup CWRU Bearing Data Website [27]. Data Center Website [27].
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Figure 8. The bearing test rig of the CWRU Bearing Data Center [27].
Figure 8. The bearing test rig of the CWRU Bearing Data Center [27].
In line with the algorithm flow in Figure 3, three kinds of indexes, the improved correlated 3. Data sampling conditions fault(SE-K) feature frequencies. kurtosis (SE-SCK),Table the kurtosis of squared envelopeand signal and the spectral kurtosis of the squared envelope signal (SE-SK), are utilized to identify the optimal resonant frequency band of Fault Fault Motor Rotating Fault Feature CWRU No. data. According to data sampling conditions and fault feature frequencies, the searching Depth/mm Diameter/mm Load/kW Speed/rpm Frequency/Hz range of the optimal resonant frequency bandwidth is set as fbpfi 10 fbpfi ; the searching range of IR007-0 1797 the Particle 162 Swarm the optimal resonant central frequency is set as min /2 fc 0 fs /2 min /2 . Initializing IR007-1 0.7355 1772 159 2.7940 and performing 0.177830 times’ interactive computation. The identified resonant Optimization algorithm IR007-2 1.4710 1748 157 central frequency and bandwidth are shown in Figure 2.2065 9a,b, respectively. 1721 IR007-3 155 One can learn from Figure 9a,b that for the improved correlated kurtosis, the identified resonant IR014-0 0 1796 162 central frequencies are centering around 3000 Hz, and the identified resonant frequency bandwidth IR014-1 0.7355 1774 160 2.7940 0.3556 is centering around 1600 Hz. For the kurtosis of squared envelope signal, the identified resonant IR014-2 1.4710 1752 158 central frequencies are around 5000 Hz, but the identified resonant 1728 frequency bandwidth IR014-3 2.2065 156 is dispersed. For the spectral kurtosis of squared envelope signal, the identified resonant central IR021-0 0 1797 162 frequencies are around 1500 Hz, and the identified resonant frequency bandwidth is around 160 Hz. IR021-1 0.7355 1774 160 Data IR007-0, which represents the 0.5334 minimal fault level and minimal motor load, and data IR021-3, 2.7940 IR021-2 1.4710 1752 158 which represents the maximal fault level and maximal2.2065 motor load, are utilized IR021-3 1728 to perform optimal 156 resonant frequency band demodulation to verify the effectiveness of the identified resonant frequency bands in Figure 9.
In line with the algorithm flow in Figure 3, three kinds of indexes, the improved correlated kurtosis (SE-SCK), the kurtosis of squared envelope signal (SE-K) and the spectral kurtosis of the squared envelope signal (SE-SK), are utilized to identify the optimal resonant frequency band of CWRU data. According to data sampling conditions and fault feature frequencies, the searching range of the optimal resonant frequency bandwidth is set as f bpfi ≤ β ≤ 10 × f bpfi ; the searching range of the optimal resonant central frequency is set as β min /2 ≤ f c ≤ f s /2 − β min /2. Initializing the Particle Swarm Optimization algorithm and performing 30 times’ interactive computation. The identified resonant central frequency and bandwidth are shown in Figure 9a,b, respectively. One can learn from Figure 9a,b that for the improved correlated kurtosis, the identified resonant central frequencies are centering around 3000 Hz, and the identified resonant frequency bandwidth is centering around 1600 Hz. For the kurtosis of squared envelope signal, the identified resonant central frequencies are around 5000 Hz, but the identified resonant frequency bandwidth is dispersed. For the spectral kurtosis of squared envelope signal, the identified resonant central frequencies are around 1500 Hz, and the identified resonant frequency bandwidth is around 160 Hz. Data IR007-0, which represents the minimal fault level and minimal motor load, and data IR021-3, which represents the maximal fault level and maximal motor load, are utilized to perform optimal resonant frequency band demodulation to verify the effectiveness of the identified resonant frequency bands in Figure 9. Analysis results of IR007-0 and IR021-3 are shown in Figures 10 and 11, respectively.
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Figure 9. Identification results of optimal resonant frequency bands. (a) The optimal central Figure Identification results of optimal resonant frequency bands. (a)bands. The optimal central frequencies; Figure9. 9. Identification results of optimal resonant frequency (a) The optimal central frequencies; (b) the optimal bandwidth. (b) the optimal bandwidth. frequencies; (b) the optimal bandwidth.
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Figure Figure 11. 11. Analysis Analysis results results of of data data IR021-3. IR021-3. (a) (a)The Theresult result based basedon on the the improved improvedcorrelated correlatedkurtosis; kurtosis; (b) (b) the the result result based based on on the the kurtosis kurtosis of of squared squared envelope envelope signal; signal; (c) (c) the the result result based based on on the the spectral spectral kurtosis of squared envelope signal. kurtosis of squared envelope signal.
One that thethe identified resonant frequency bands based on One can can learn learnfrom fromFigures Figures10a 10aand and11a 11a that identified resonant frequency bands based the improved correlated kurtosis can can steady extract faultfault feature frequency and its The on the improved correlated kurtosis steady extract feature frequency andharmonics. its harmonics. SES of optimal resonant frequency band filtered signals contain highlighted fault feature frequency, The SES of optimal resonant frequency band filtered signals contain highlighted fault feature frequency, the the value value of of fault fault feature feature frequency frequency is is higher higher than than the the value value of of rotating rotating frequency, frequency, and and fault fault feature feature frequency frequency and and its its harmonics harmonics are are dominant dominant in in frequency frequency domain. domain. As As aa result, result, the the proposed proposed optimal optimal resonant frequencyband band demodulation based onimproved the improved correlated candiagnose clearly resonant frequency demodulation based on the correlated kurtosiskurtosis can clearly diagnose rollingfaults. bearing faults. From10b Figures 10b the andidentified 11b, the identified resonant frequency bands rolling bearing From Figures and 11b, resonant frequency bands based on based on the kurtosis of squared envelope signal can also recognize the fault feature frequency. the kurtosis of squared envelope signal can also recognize the fault feature frequency. However, for However, for harmonics data IR007-0, harmonics of thefrequency fault feature frequency is too small capture. For data data IR007-0, of the fault feature is too small to capture. For to data IR021-3, values IR021-3, values frequencies of fault feature andare itssmaller harmonics than the value of rotating of fault feature andfrequencies its harmonics thanare thesmaller value of rotating frequency, fault frequency, fault feature frequency and its harmonics are not dominant in frequency dominant. As a feature frequency and its harmonics are not dominant in frequency dominant. As a result, the optimal result, thefrequency optimal resonant frequency band based on the envelope kurtosis of squared envelope resonant band demodulation baseddemodulation on the kurtosis of squared signal cannot clearly signal cannot clearly diagnose rollingFigures bearing From 10cresonant and 11c, the identified diagnose rolling bearing faults. From 10cfaults. and 11c, the Figures identified frequency bands resonant frequency based on therecognize spectral the kurtosis can hardly recognize feature based on the spectralbands kurtosis can hardly fault feature frequency as forthe the fault extreme high frequency as for the extreme high value of rotating frequency. As a result, the optimal resonant value of rotating frequency. As a result, the optimal resonant frequency band demodulation based on frequency demodulation based on signal the spectral of squared envelope the spectralband kurtosis of squared envelope cannot kurtosis clearly diagnose rolling bearingsignal faults.cannot clearly diagnose rolling bearing faults. As is mentioned above, the proposed optimal resonant frequency band demodulation based on As is mentioned above, the proposed optimal resonant frequency band demodulation based on the improved correlated kurtosis can steady identify the optimal resonant bands, clearly demodulate the improved correlated kurtosis can steady identify the optimal resonant bands, clearly demodulate fault feature frequencies and their harmonics, and effectively diagnose rolling bearing faults. fault featureanalysis frequencies and their and effectively diagnose rolling of bearing faults. Comparison with kurtosis andharmonics, spectral kurtosis have verified the superiority the improved Comparison analysis with kurtosis and spectral kurtosis have verified the superiority of the correlated kurtosis. improved correlated kurtosis. 5. Conclusions 5. Conclusions As to the drawbacks of kurtosis in detecting repetitive transient impulses and diagnosing rolling As to the drawbacks of kurtosis detectingcorrelated repetitivekurtosis transient impulses and diagnosingfunction rolling bearing faults, this manuscript firstlyinredefines based on auto-correlation bearing thisThen, manuscript firstly correlated kurtosis based onbasis auto-correlation function and the faults, kurtosis. it proposes anredefines improved correlated kurtosis on the of squared envelope and the kurtosis. Then, it proposes an improved correlated kurtosis on the basis of squared envelope spectrum, which has been proved more effective in detecting repetitive transient impulses. Regarding spectrum, which has been proved more effective in detecting repetitive transient impulses. Regarding the improved correlated kurtosis, it can not only characterize the strength of repetitive transient
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the improved correlated kurtosis, it can not only characterize the strength of repetitive transient impulses, but also feature the cyclical occurrence of repetitive transient impulses, that is, the improved correlated kurtosis is effective in differentiating impulsive noises and repetitive transient impulses, and in reducing the noise impact on the detection of repetitive transient impulses as opposed to kurtosis-based indexes. Finally, this manuscript puts forward an optimal resonant band demodulation method based on improved correlated kurtosis by combining complex Morlet wavelet filter and the Particle Swarm Optimization algorithm. Analysis of both simulation signals and CWRU data demonstrate that, the proposed optimal demodulation method based on the improved correlated kurtosis can be more robust in identifying resonant frequency bands of rolling bearings and more accurate in demodulating transient fault features of rolling bearing. Analysis of data from experiment verify the validity and advantage of the proposed method over traditional kurtosis-based indexes. Acknowledgments: This research is supported by National Natural Science Foundation of China (51205427), Equipment Advanced Research Foundation (9140A27020115JB35071). Author Contributions: X.C. and B.Z. proposed the method. X.C. analyzed the data and wrote the draft manuscript, B.Z. provided some valuable advices, F.F. and P.J. participated in manuscript writing and revised the manuscript. Conflicts of Interest: The authors declare no conflicts of interest.
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