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Mechanical Systems and Signal Processing
Mechanical Systems and Signal Processing 35 (2013) 176–199
0888-32
http://d
n Corr
E-m
journal homepage: www.elsevier.com/locate/ymssp
An enhanced Kurtogram method for fault diagnosis of rollingelement bearings
Dong Wang a,n, Peter W. Tse a,b, Kwok Leung Tsui b
a Smart Engineering Asset Management Laboratory (SEAM), and Croucher Optical Non-destructive Testing and Quality Inspection Laboratory (CNDT),
Department of Systems Engineering & Engineering Management, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong, Chinab Center for System Informatics and Quality Engineering, Department of Systems Engineering & Engineering Management, City University of Hong Kong,
Tat Chee Avenue, Kowloon, Hong Kong, China
a r t i c l e i n f o
Article history:
Received 13 May 2011
Received in revised form
19 September 2012
Accepted 1 October 2012Available online 25 October 2012
Keywords:
Kurtogram
Rolling element bearing
Fault diagnosis
Wavelet packet transform
Low signal-to-noise ratio
70/$ - see front matter & 2012 Elsevier Ltd. A
x.doi.org/10.1016/j.ymssp.2012.10.003
esponding author. Tel.: þ852 34422648; fax
ail addresses: [email protected] (D. W
a b s t r a c t
The Kurtogram is based on the kurtosis of temporal signals that are filtered by the
short-time Fourier transform (STFT), and has proved useful in the diagnosis of bearing
faults. To extract transient impulsive signals more effectively, wavelet packet transform
is regarded as an alternative method to STFT for signal decomposition. Although
kurtosis based on temporal signals is effective under some conditions, its performance
is low in the presence of a low signal-to-noise ratio and non-Gaussian noise. This paper
proposes an enhanced Kurtogram, the major innovation of which is kurtosis values
calculated based on the power spectrum of the envelope of the signals extracted from
wavelet packet nodes at different depths. The power spectrum of the envelope of the
signals defines the sparse representation of the signals and kurtosis measures the
protrusion of the sparse representation. This enhanced Kurtogram helps to determine
the location of resonant frequency bands for further demodulation with envelope
analysis. The frequency signatures of the envelope signal can then be used to determine
the type of fault that has affected a bearing by identifying its characteristic frequency. In
many cases, discrete frequency noise always exists and may mask the weak bearing
faults. It is usually preferable to remove such discrete frequency noise by using
autoregressive filtering before the enhanced Kurtogram is performed. At last, we used
a number of simulated bearing fault signals and three real bearing fault signals obtained
from an experimental motor to validate the efficiency of these proposed modifications.
The results show that both the proposed method and the enhanced Kurtogram are
effective in the detection of various bearing faults.
& 2012 Elsevier Ltd. All rights reserved.
1. Introduction
Rolling element bearings are widely used in rotating machinery to support rotating shafts, and the major cause ofmachinery breakdowns is bearing failure. Hence, it is necessary to detect bearing faults at an early stage. Rolling elementbearings usually consist of an inner race, an outer race, several rollers and a cage. When the surface of one of thesecomponents develops a localised fault, the impacts generated excite the resonant frequencies of the bearing and adjacentcomponents, and induce a modulating phenomenon [1]. Demodulation of the original signal with envelope analysis can
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: þ852 34420415.
ang), [email protected] (P.W. Tse), [email protected] (K.L. Tsui).
D. Wang et al. / Mechanical Systems and Signal Processing 35 (2013) 176–199 177
reveal additional fault-related signatures [2]. To enhance the signal-to-noise ratio of the original signal, a band-pass filteris usually set manually to maintain the desired resonance frequency band before demodulation is performed.
Antoni [3] recently analysed spectral kurtosis thoroughly. A short-time Fourier transform (STFT)-based spectral kurtosis wasthen investigated for the diagnosis of rotating machine faults [4]. To reduce computing time, a 1/3-binary tree Kurtogramestimator was proposed to perform fast on-line fault detection [5]. Since that time, improvements of both spectral kurtosis andthe Kurtogram have attracted a great deal of attention. To clarify the impulses, Sawalhi et al. [6] proposed a method thatcombined minimum entropy deconvolution with spectral kurtosis. Their results showed that this method can hone impulsesand increase the values of kurtosis. Combet and Gelman [7] presented spectral kurtosis-based optimal filtering for the residualsignal of gears. The results showed that their method could enhance small transients in gear vibration signals. Zhang andRandall [8] considered that the fast Kurtogram was only an approximate estimation and therefore combined it with a geneticalgorithm to determine the optimal centre frequency and bandwidth for resonance demodulation with envelope analysis.Barszcz and Randall [9] reported that a spectral kurtosis-based method was more effective than others for the early detection oftooth crack. Wang and Liang [10] proposed an adaptive spectral kurtosis method that could adaptively determine thebandwidth and centre frequency of a filter by merging right-expanded windows to make the kurtosis of the filtered signalmaximum. Lei et al. [11] thought that wavelet packet transform (WPT) filters could process non-stationary transient vibrationsignals more efficiently than STFT. Therefore, they replaced STFT with WPT to improve the original Kurtogram. Barszcz andJab"onski [12] found that temporal signal-based kurtosis can be considerably affected by noise, and proposed a novel methodcalled the Protrugram, which calculated the kurtosis of envelope spectrum amplitudes. It is logical to measure kurtosis in thefrequency domain. When a bearing is healthy, its envelope spectrum is randomly distributed over that of the whole frequency.However, when it has localised faults, bearing fault characteristic frequencies dominate the envelope spectrum [13,14]. Peakvalues of bearing fault characteristic frequency and its harmonics can be measured by kurtosis with higher values becausekurtosis can be used to measure the protrusion of a signal. Moreover, other faults, such as misalignment, eccentric fault and soon, can also be measured by kurtosis in the frequency domain because the envelope spectrum actually is a sparserepresentation of the envelope. In contrast to the study of Lei et al. [11], we replaced kurtosis of the temporal signalsextracted from wavelet packet nodes with that of the power spectrum of the envelope of the signals extracted from waveletpacket nodes. Moreover, the power spectrum was used because it can enhance fault frequency more effectively than theFourier spectrum used by Barszcz and Jab"onski [12].
The remainder of paper is organised as follows. Section 2 describes the basic concept of spectral kurtosis and the binaryWPT for the decomposition of the frequency support of the original signal. In Section 3, a new detection method and theenhanced Kurtogram are proposed for the detection of rolling element bearing faults. In Section 4, we used simulated dataand real data obtained in a laboratory to validate the proposed method and the enhanced Kurtogram. In Section 4, thecomparison of the proposed method with the fast Kurtogram proposed by Antoni [5] and the improved Kurtogramproposed by Lei et al. [11] are conducted to analyse the same signals. Our conclusions are summarised in Section 5.
2. The review of spectral kurtosis and binary wavelet packet transform
2.1. Spectral kurtosis
According to Wold–Cramer representation, a zero-mean non-stationary random process x(n) can be decomposed into [5]:
xðnÞ ¼
Z þ1=2
�1=2H n,fð Þej2pf ndZxðf Þ ð1Þ
where dZx(f) is a spectral increment and H(n,f) is the complex envelope of x(n) (the time varying transfer function of the system)at frequency f. Therefore, the spectral kurtosis can be represented by the fourth-order normalized cumulant [5]:
Kx fð Þ ¼/9H n,fð Þ94S
/9H n,fð Þ92S2�2 ð2Þ
where /dS stands for the temporal averaging operator. �2 is used in Eq. (2) because H(n, f) is complex. Considering thepresence of stationary additive noise, spectral kurtosis of the signal y(n) is described by [5]:
Ky fð Þ ¼Kxðf Þ
½1þrðf Þ�2ð3Þ
where r(f) is the noise to signal ratio at frequency f. Therefore, the spectral kurtosis is able to detect and localize the presence ofnon-stationarities represented by a signal. Antoni [4,5] proposed two methods to calculate spectral kurtosis. One is based onShort Time Fourier Transform (the called Kurtogram for finding the optimal filter) [4] and the other is based on 1/3 binaryfilter banks (fast Kurtogram for on-line condition monitoring and fault diagnosis) [5]. More details can be found in the originalpapers [4,5]. Spectral kurtosis has been successfully used in many cases [1,4–9]. It can be one of the solutions toblind component separation that is a useful concept proposed by Antoni [15], for decomposing a vibration signal intoperiodic components, random transient components and random stationary components. Recently, Lei et al. [11] indicated thata short-time Fourier transform (STFT)-based or filters-based spectral kurtosis was not as precise as wavelet packet transform is.
D. Wang et al. / Mechanical Systems and Signal Processing 35 (2013) 176–199178
Therefore, they used WPT to replace the STFT in extracting transient characteristics, and measured the kurtosis of the temporalsignal filtered by WPT. In another work, Barszcz and Jab"onski [12] proposed a novel Protugram to measure the kurtosis of theenvelope spectrum in the case of lower signal to noise ratio. They also pointed out that Kurtogram could find the frequencyband with high kurtosis as the optimal band, but sometimes being incorrect [12].
2.2. Binary wavelet packet transform (WPT) [16]
Assume a space Vj is decomposed into a lower resolution or coarser space Vjþ1 and a detail space Wjþ1 bymultiresolution approximation. In the above process, the orthogonal basis fj(t�2jn)nAZ of Vj should be divided intofjþ1(t�2jþ1n)nAZ of Vjþ1 and Wjþ1(t�2jþ1n)nAZ of Wjþ1. It is obvious that only the approximated spaces Vj are recursivelyused to construct the detail spaces and wavelet bases. For further division of the detail spaces, it must consider orthogonalsplitting (of detail spaces). Wavelet packets offset the disadvantage of wavelets with multiresolution approximations.Assume a space Wp
j and its orthonormal basis cpj ðt�2jnÞn2Z , where j is its depth and p is the number of nodes. Therefore,
the orthogonal basis of the space at node (j, p) can be divided into two new orthogonal bases as follows:
c2pjþ1ðtÞ ¼
Xþ1n ¼ �1
hðnÞcpj ðt�2jnÞ, ð4Þ
c2pþ1jþ1 ðtÞ ¼
Xþ1n ¼ �1
gðnÞcpj ðt�2jnÞ, ð5Þ
hðnÞ ¼/c2pjþ1ðuÞ,c
pj ðt�2jnÞS, ð6Þ
gðnÞ ¼/c2pþ1jþ1 ðuÞ,c
pj ðt�2jnÞS ð7Þ
here, h(n) and g(n) is a pair of conjugate mirror filters, and /,S is the inner product. In terms of c2pjþ1ðt�2jþ1nÞn2Z and
c2pþ1jþ1 ðt�2jþ1nÞn2Z , orthonormal bases of two orthogonal spaces W2p
jþ1 and W2pþ1jþ1 , respectively, it is deduced that:
W2pjþ1 �W2pþ1
jþ1 ¼Wpj , ð8Þ
where � is the direct sum. Consequently, the iterative splitting steps result in two orthogonal spaces for each node (j, p).For example, the maximum depth J is set to 3. According to Eq. (5), W0
0can be recursively decomposed as:
W00 ¼ �
1p ¼ 0 Wp
1 ¼ �3p ¼ 0 Wp
2 ¼ �7p ¼ 0 Wp
3 ð9Þ
To generalize Eq. (9), W00 can be recursively decomposed at the maximum depth J by Eq. (8) as:
W00 ¼ �
1p ¼ 0 Wp
1 ¼ �3p ¼ 0 Wp
2 ¼ � � � ¼ �2J�1
p ¼ 0 WpJ ð10Þ
On the other hand, considering the Fourier transform of Eqs. (4) and (5), the orthogonal basis at note (j, p) can be shown as:
c 2p
jþ1ðoÞ ¼ h½2jo�cp
j ðoÞ, ð11Þ
c 2pþ1
jþ1 ðoÞ ¼ g½2jo�cp
j ðoÞ, ð12Þ
Due to the energy concentration of h ½2jo� and g ½2jo� in their own orthogonal frequency bands, Eqs. (11) and (12) canbe interpreted as a division of frequency support of c p
j ðoÞ. Therefore, it is concluded that division of frequency support ofc 0
0ðoÞ at maximum depth J can be given as follows:
c 0
0ðoÞ ¼ �1p ¼ 0
c p
1ðoÞ ¼ �3p ¼ 0
c p
2ðoÞ ¼ � � � ¼ �2J�1
p ¼ 0c p
J ðoÞ: ð13Þ
Moreover, it should be pointed out that frequency support c p
j ðoÞ at the same depth j has the same bandwidth. For anynode (j, p), wavelet packet coefficients of the original signal xðtÞ 2W0
0 can be calculated by taking the inner product of theoriginal signal with every wavelet packet basis:
dpj ½n� ¼/xðtÞ,cp
j ðt�2jnÞS ð14Þ
For the fast wavelet transform algorithm, wavelet packet coefficients are calculated in the decomposition:
d2pjþ1½n� ¼ dp
j nh �2nð Þ, ð15Þ
d2pþ1jþ1 ½n� ¼ dp
j ng �2nð Þ, ð16Þ
where * is convolution operator. By applying the recursive splitting of Eqs. (13)–(16), wavelet packet coefficients arecalculated in the reconstruction:
dpj ½n� ¼
ed2p
jþ1nhðnÞþed2pþ1
jþ1 ngðnÞ, ð17Þ
x(t)
Node (0,0)
0-Fs/2 Hz Node (1,0)
0-Fs/4 Hz
Node (1,1)
Fs/4-Fs/2 Hz
Depth 1
Node (2,0)
0-Fs/8 Hz
Node (2,1)
Fs/8-Fs/4 Hz
Node (2,2)
Fs/4-3Fs/8 Hz
Node (2,3)
3Fs/8-Fs/2 Hz Depth 2
Node (3,0)
0-Fs/16 Hz
Depth 3 Node (3,1)
Fs/16- Fs/8 Hz
Node (3,2)
Fs/8-3Fs/16 Hz
Node (3,3)
3Fs/16- Fs/4 Hz
Node (3,4)
Fs/4- 5Fs/16 Hz
Node (3,5)
5Fs/16-3Fs/8 Hz
Node (3,6)
3Fs/8-7Fs/16 Hz
Node (3,7)
7Fs/16-Fs/2 Hz
Fig. 1. Three-depth binary wavelet packet decomposition tree.
D. Wang et al. / Mechanical Systems and Signal Processing 35 (2013) 176–199 179
where ed means inserting a zero between each sample of d. The process of the frequency support decomposition usingwavelet packet transform is illustrated in Fig. 1, where maximum depth is equal to 3 and Fs is the sampling frequency.
3. The proposed method for bearing fault diagnosis
Kurtosis for the measurement of temporal signals in the diagnosis of machinery faults has attracted a great deal ofattention [4–11]. However, when the signal-to-noise ratio is low, the use of kurtosis to detect impulses hidden in the signalis difficult because potential periodic peak values are overwhelmed by unexpected heavy noises. In addition, kurtosis mayhave greater values for non-Gaussian noises. Recently, Barszcz and Jab"onski [12] proposed a novel method called theProtrugram to measure kurtosis in the frequency domain. Their results showed that this is an effective method ofdiagnosing fault signals with a low signal-to-noise ratio, and the use of kurtosis to measure the envelope spectrum islogical. When a bearing is healthy, the envelope spectrum is randomly distributed over that of the whole frequency, butonce it develops a localised fault, the bearing fault characteristic frequency and its harmonics become the majorcomponents in the envelope spectrum [13,14]. Therefore, the envelope signal or its counterpart in the frequency domainusually contains much more fault information than the original signal [1,2]. Lei et al. [11] believed that WPT had good localproperties in the domains of both time and frequency, and could extract transient characteristics more efficiently thanSTFT. Thus, they proposed a method based on kurtosis of the temporal signals extracted from wavelet packet nodes.
As mentioned above, it is difficult to identify fault-related signatures hidden in a signal with a low signal-to-noise ratiousing kurtosis of the temporal signal. This paper proposes a new method, which is based on binary WPT and kurtosismeasurements in the frequency domain. A flowchart of the proposed method is shown in Fig. 2 and the details aredescribed below.
Step 1. The original vibration signal measured by an accelerometer is first loaded. In many cases, the discrete frequencynoise caused by low frequency periodic components, such as shaft rotating frequency, misalignment and eccentric fault,always exists and may mask the weak bearing signals. Refs. [1] and [15] suggest that it is preferable to separate thebearing fault signal from the discrete frequency noise before the bearing fault signal is analysed. Some other options [1]for removing discrete frequency noise are linear prediction, adaptive noise cancellation, self-adaptive noise cancella-tion, discrete/random separation, time synchronous averaging and eigenvector algorithm [17]. In this paper, pre-whitening processing could be performed prior to the analysis of bearing fault signal if there is a disturbance caused bydiscrete frequency noise. Autoregressive model (AR) is a popular method to establish the deterministic periodiccomponents. Considering an additive noise term v(n), the mathematical equation of AR for an actual signal x(n) isexpressed as follows [18]:
xðnÞ ¼ �Xp ¼ q
p ¼ 1aðpÞx n�pð ÞþvðnÞ, ð18Þ
The coefficients a(p) are the parameters of AR model, which could be obtained by the solution of the Yule–Walkerequations through the Levinson–Durbin recursion algorithm (LDR) or Burg’s method (BM) [18]. The order of model q
could be decided by minimizing Akaike information criterion (AIC) given as [18]:
AIC qð Þ ¼ ln v nð Þð Þþ2 qþ1ð Þ
N, ð19Þ
Generate the enhanced Kurtogram
Load the original vibration signal and perform pre-whitening processing
Perform binary wavelet packet transform on the original signal at different depths and
reconstruct the obtained signals as the same temporal length with the original signal
Select the valuable node signals from all of the node signals for further investigation
Perform power spectrum on the envelope signal and identify the bearing fault
characteristics
Start
End
Demodulate the selected node signal by Hilbert transform to obtain the envelope
Fig. 2. Flowchart of the proposed method.
D. Wang et al. / Mechanical Systems and Signal Processing 35 (2013) 176–199180
where N is the length of the signal x(n). Because the Fourier transform of the convolution of two signals is equal to theproduct of the Fourier transform of the two signals, the Fourier transform of Eq. (18) is given as follows:
vðf Þ ¼ xðf Þaðf Þ: ð20Þ
Eq. (20) shows that only random components are retained, which means that transient impulses and stationary noiseare left. The temporal signal v(n) of v(f) is said to be pre-whitened. The above process is usually called pre-whiteningprocessing.Binary WPT was then conducted on random components v(n) at different depths for the enhancement of bearing faultsignal. The minimum bandwidth decomposed by binary WPT at maximum depth needed to be three times longer thanthe inner race fault frequency so that sufficient bearing fault-related signatures could be retained in the desiredfrequency band, and thus provided an alternative method of determining the maximum depth for binary WPT. Forwavelet packet coefficients at a specific node to have the same temporal length as the original signal, those obtained ata specific node were reconstructed by setting those at the other nodes to zero. For simplification, the same notationdp
j ðnÞ is used to express these reconstructed wavelet packet coefficients.Step 2. It should be noted that the proposed method differs from both of those in Refs. [11] and [12]. The kurtosis of theenvelope spectrum was used to improve that of the temporal signal in Ref. [11] to determine fault-related signatures ina signal with a low signal-to-noise ratio. The power spectrum was used to map the envelope signal into the frequency-dependent signal because the Fourier transform used in Ref. [12] is affected by residual noise in the filtered signal. Thecore of the proposed method, the enhanced Kurtogram, was prepared as follows. First, it was necessary to obtain adesired envelope signal that contains many fault-related signatures. The envelope signal ep
j ðnÞ for each node at differentdepths was obtained by taking the modulus of an analytical signal zp
j ðnÞ ¼ zpj realðnÞþ izp
j imagðnÞ. An alternative approach[19] for calculating the analytic signal zp
j ðnÞ of a discrete signal dpj ðnÞ is introduced in the following. Assume the length of
D. Wang et al. / Mechanical Systems and Signal Processing 35 (2013) 176–199 181
the discrete-time signal is N and even. The N-point discrete Fourier transform (DFT) of the discrete signal is given as:
Dpj mð Þ ¼
XN�1
n ¼ 0
dpj nð Þexp
�i2pmn
N
� �, m¼ 0,1,. . .,N�1: ð21Þ
In order to satisfy the analytic-like properties, two constraints are used:
zpj realðnÞ ¼ dp
j ðnÞ, n¼ 0,1,2,. . .,N�1, ð22Þ
XN�1
n ¼ 0
zpj realðnÞz
pj imagðnÞ ¼ 0: ð23Þ
Construct N-point one-side discrete analytic signal transform:
Zpj mð Þ
Dpj ð0Þ m¼ 0
2Dpj ðmÞ m¼ 1,2:. . .,N�1:
Dpj
N2
� �, m¼ N
2 ,
0, m¼ N2 þ1, N
2 þ2,. . .,N�1:
8>>>>><>>>>>:ð24Þ
Consider an N-point inverse DFT of Zpj ðmÞ, then the analytic signal zp
j ðnÞ is obtained by:
zpj nð Þ ¼
PN�1n ¼ 0 Zp
j ðmÞexp i2pmn=N� �
N¼ zp
j realðnÞþ izpj imagðnÞ, m¼ 0,1,. . .,N�1: ð25Þ
After the envelope signal epj ðnÞ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðzp
j realðnÞÞ2þðzp
j imagðnÞÞ2
qis obtained by taking the modulus of Eq. (25), the power
spectrum Epj ðmÞ can be calculated by either parametric or non-parametric approaches. In the non-parametric approach,
Fourier transform of the autocorrelation function and the segment averaging method are two popular methods toestimate the power spectrum. The bias and variance considerations of the power spectra obtained by different methodsare discussed in Ref. [20].When a bearing suffers from a localized fault (in this case it is the bearing outer race fault), a few coefficients (bearing fault
characteristic frequency and its harmonics) of Epj ðmÞ dominate the envelope spectrum. Then, kurtosis is employed to quantify
the protrusion of Epj ðmÞ . Therefore, the value of kurtosis is high in the envelope spectrum of bearing fault signal. The same
approach can be applied to inner race fault signals and rolling element fault signals. On the other hand, if there is no bearing
fault, the protrusion of Epj ðmÞ will not be so observed. The value of kurtosis in the power spectrum should be relatively low.
Fig. 3 shows the power spectra of envelope of a normal bearing signal and fault bearing signals (The data are provided byBearing Data Centre [21]). The protrusion of the power spectra of the envelope of these signals is quantified by kurtosis.
Assume the envelope signal has zero mean. Their corresponding kurtosis values are shown in Fig. 3. Kurtosis of Epj ðmÞ for each
node was calculated as:
K j,pð Þ ¼
PNm ¼ 1 ðE
pj ðmÞÞ
4=N
ðPN
m ¼ 1 ðEpj ðmÞÞ
2=NÞ
2
,1r jr J,0rpr2J�1, ð26Þ
Since the power spectrum is a real and positive function, it is not necessary to subtract the mean value from the powerspectrum. Once the kurtosis values of all nodes had been calculated, the enhanced Kurtogram is paved. For example, theenhanced Kurtogram at maximum depth 3 is plotted in Fig. 4.Step 3. A colour map is used to represent kurtosis of all nodes in Fig. 4. The depth of the colour values was proportional to thelevel of the kurtosis values. In the case of one resonant frequency band, the node with the maximum kurtosis was deemedworthy of further analysis (a simulation with one resonant frequency band is analysed in Section 4.1). When more than oneresonant frequency band can be seen, the first few maximum values at different frequency bands can be considered for furtheranalysis (a simulation with two resonant frequency bands is analysed in Section 4.2). After the nodes that could be assessed forbearing fault diagnosis had been selected, demodulation with envelope analysis was performed on the selected nodes tochange the high-frequency signal to a low-frequency signal (bearing fault characteristic frequency-related signal).Step 4. Autocorrelation was performed on the envelope of the desired signal filtered by WPT to extract potentialperiodic characteristics in the time domain. Its Fourier transform, namely the power spectrum, was used to map thetime-dependent signal into the frequency-dependent signal to identify bearing fault characteristic frequencies, whichare usually calculated by Eqs. (27)–(31). The outer-race fault characteristic frequency fO, the inner-race fault
500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000
200400600800
1000
500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000
200400600800
1000
500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000
200400600800
1000
Am
plitu
deA
mpl
itude
Frequency (Hz)
Frequency (Hz)
Am
plitu
de
Frequency (Hz)
Kurtosis=46
Kurtosis=2170
Kurtosis=4309
Sparse representation of outerrace fault signal
Sparse representation of innerrace fault signal
Fig. 3. Power spectra of the envelope of signals. (a) Normal bearing signal; (b) bearing outer race fault signal; (c) bearing inner race fault signal
(Note: the resolution of all spectra is equal to 0.5 Hz).
Depth K(j,p)3 K (3,0) K (3,1) K (3,2) K (3,3) K (3,4) K (3,5) K (3,6) K (3,7)
2 K (2,0) K (2,1) K (2,2) K (2,3)
1 K (1,0) K (1,1)
Fs/2Fs/40 Fs/8 3Fs/8
Frequency (Hz)
Fig. 4. The paving of the enhanced Kurtogram.
D. Wang et al. / Mechanical Systems and Signal Processing 35 (2013) 176–199182
characteristic frequency fI, the rolling element fault characteristic frequency fB, the fundamental cage frequency fC andthe ball spinning frequency fBS were formulated as follows [1]:
f O ¼Z � f s
21�
d
Dcos a
� �, ð27Þ
f I ¼Z � f s
21þ
d
Dcos a
� �, ð28Þ
f B ¼D� f s
d1�
d
Dcos a
� �2 !
, ð29Þ
f C ¼f s
21�
d
Dcos a
� �ð30Þ
f BS ¼D� f s
2d1�
d
Dcos a
� �2 !
, ð31Þ
D. Wang et al. / Mechanical Systems and Signal Processing 35 (2013) 176–199 183
where fs is the shaft rotating frequency in Hz, d and D are diameters of the rolling element and pitch diameter,respectively. Z is the number of rolling elements and a is the contact angle.
4. The proposed method validated by both simulated and real case studies
4.1. Case 1: Simulated bearing fault signals with single resonant frequency
We used the similar simulated signal as that given in Refs. [10,22], because a bearing fault signal consists of periodicbursts of exponentially decaying sinusoidal vibration [23]. Signals with single resonant frequency enhanced by differentsignal-to-noise ratios were considered initially. The selection of mother wavelet function for wavelet analysis is a hottopic. Among the wavelet families, Daubechies family (db M) is the most attractive one for discrete wavelet analysisbecause their daughter wavelets are orthogonal, biorthogonal and compact supported. Associated scaling filters areminimum-phase filters. Its support width, filter length and the number of vanishing moments are 2M-1, 2M and M. Forbearing fault diagnosis, the low-order Daubechies wavelets, such as db2 [24], db5 [25], db10 [11,26] and db12 [27] aremore preferable to be used in discrete wavelet analysis. A Daubechies 10 wavelet used in Lei’s work [11] was employed inour method to implement binary WPT. The simulated signal with a resonant frequency is given as:
yðkÞ ¼X
r
e�b� k�r�Fs=f m�trð Þ=Fs � sin 2pf 1 � k�r � Fs=f m�tr
� �=FsÞ,
�ð32Þ
where b is equal to 900, fm is the fault characteristic frequency (equal to 100 Hz), Fs is the sampling frequency set to12,000 Hz, tr, which is subject to a discrete uniform distribution, is used to simulate the randomness caused by slippageand f1 is the resonant frequency, equal to 1700 Hz. A total of 24,000 samplings were used for each simulated signal.To display the temporal signal clearly, only 2500 samplings are shown in both Sections 4.1 and 4.2. A normally distributedrandom signal with a mean of 0 and a variance of 0.5 were added to Eq. (32). It is noted that the simulated bearing faultsignal is an ideal one that is not interrupted by low frequency periodic components. Therefore, in the simulated cases ofSections 4.1 and 4.2, pre-whitening processing is not performed on the simulated signal.
The simulated signal, the noise signal and the mixed signal are shown in Fig. 5(a–c). The proposed method was used to analysethe mixed signal shown in Fig. 5(c). The enhanced Kurtogram is paved in Fig. 6(a). Node (4, 4) has the highest kurtosis of all thenodes. Moreover, it should be noted that the most useful node at each depth can be clearly identified, such as node (1, 0) at depthone, node (2, 1) at depth two, node (3, 2) at depth three and node (4, 4) at depth four. The temporal signal corresponding to node
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-2
0
2
Am
plitu
de
Am
plitu
de
Am
plitu
de
Samplings
Fig. 5. Signals in the time domain: (a) the simulated signal with one resonance frequency; (b) the noise signal with a normal distribution and a variance
of 0.5 and (c) the mixed signal.
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4
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Dep
ths
Frequency (Hz)
Frequency (Hz)
Am
plitu
de fm
2fm
3fm4fm
Fig. 6. The results obtained by the proposed method for processing the mixed signal with one resonant frequency. (a) The enhanced Kurtogram used in
this paper and (b) power spectrum of the envelope of the signal extracted from node (4, 4) by wavelet packet transform.
D. Wang et al. / Mechanical Systems and Signal Processing 35 (2013) 176–199184
(4, 4) was then extracted by WPT. The power spectrum of the envelope of the signal extracted from node (4, 4) by WPT is given inFig. 6(b), in which the fault frequency, 100 Hz, and its harmonics are evident.
The original fast Kurtogram proposed by Antoni [5] and the improved Kurtogram proposed by Lei et al. [11] are appliedto analyse the same mixed signal with one resonant frequency in the case of heavy noise. The common point of these twomethods is that the kurtosis is computed from the envelopes of the temporal signals obtained by a set of filters. One oftheir limitations is that a high kurtosis value may be caused by other components rather than by bearing faults [12]. Theirdiagrams are shown in Figs. 7 and 8(a), where it is found that both the fast Kurtogram and the improved Kurtogram fail todetect the single resonant frequency band. The fault characteristic frequency 100 Hz and its harmonics are difficult toidentify from their envelope spectra shown in Figs. 7 and 8(b). As a result, it is illustrated that the fast Kurtogram and theimproved Kurtogram based on kurtosis measurement from the temporal signal is difficult to indicate the resonantfrequency band in the case of a low signal-to-noise ratio.
Finally, a group of normally distributed random signals with a mean of 0 and different variances (from 0.1 to 0.9 withstep length¼0.05) were added to Eq. (32). Fig. 9 shows 17 mixed signals with different noise variances. The potentialperiodic characteristics of these mixed signals cannot be detected even though the simulated signal was mixed with anoise variance of 0.1 (the top row of Fig. 9). The results obtained by the proposed method are shown in Figs. 10 and 11.Fig. 10 shows temporal signals (autocorrelation of the envelope of the desired signals filtered by WPT) obtained. It is clearthat this method is effective in extracting the potential periodic characteristics of the fault even when large noise variancesare added to the original simulated signal.
Fig. 11 shows the frequency-dependent results (power spectra of the envelope of the desired signals filtered by WPT)obtained by the proposed method, from which the fault frequency, 100 Hz, and its harmonics are clearly visible. Thisillustrates that the proposed method can be applied to the detection of early bearing fault signals that are usuallyoverwhelmed by heavy noise.
4.2. Case 2: Simulated bearing fault signals with two resonant frequencies
In the second case study, another resonant frequency f2 (3900 Hz) was added to Eq. (32). Its final formation was:
yðkÞ ¼X
r
e�b� k�r�Fs=f m�trð Þ=Fs � ½sinð2pf 1 � ðk�r � Fs=f m�trÞ=FsÞþsinð2pf 2 � k�r � Fs=f m�tr
� �=FsÞ�, ð33Þ
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2
4
6 x 10-3
Frequency (Hz)
Am
plitu
de
Frequency (Hz)
Leve
l k
Fig. 7. The results obtained by the fast Kurtogram for processing the mixed signal with one resonant frequency. (a) The fast Kurtogram and (b) frequency
spectrum of the envelope of the signal obtained by a filter (the center frequency of 4500 Hz and the bandwidth of 3000 Hz).
D. Wang et al. / Mechanical Systems and Signal Processing 35 (2013) 176–199 185
Owing to the increased amplitudes of the overlapping temporal signal shown in Fig. 12(a), normally distributed randomsignals with a mean of 0 and a variance of 0.6 were introduced into Eq. (33). The simulated signal, the noise signal and themixed signal are shown in Fig. 12(a–c).
The enhanced Kurtogram was first applied to the mixed signal with two resonant frequencies. The paving of theenhanced Kurtogram shown in Fig. 13(a) illustrates that the node with the highest kurtosis is located at (4, 4). At the samedepth of 4, the second highest kurtosis corresponds to node (4, 10), which indicates the existence of another resonantfrequency band and provides an alternative node for further analysis. Fig. 13(a) also clearly shows the locations of the tworesonant frequencies at each depth. The power spectrum of the envelope of the signal extracted from node (4, 4) by WPT isplotted in Fig. 13(b), where it can be seen that the proposed method successfully detects the fault frequency, 100 Hz, andits harmonics.
The fast Kurtogram and the improved Kurtogram are used to analyse the same mixed signal with two resonantfrequencies, respectively. The paving of the fast Kurtogram is shown in Fig. 14(a), where it is indicated that the optimalfilter has a centre frequency of 5500 Hz and a bandwidth of 1000 Hz. The frequency spectrum of the envelope of the signalis plotted in Fig. 14(b), where the fault characteristic frequency 100 Hz is difficult to be detected. The paving of theimproved Kurtogram is given in Fig. 15(a). Different from the simulated resonant frequencies, the improved Kurtogramindicates a wrong location for selecting the most useful node. The frequency spectrum of the envelope of the signalextracted from node (4, 8) by wavelet packet transform is shown in Fig. 15(b), in which it is found that the frequencyspectrum cannot provide any fault information to indicate the existence of the fault characteristic frequency of 100 Hz.Consequently, in the case of a low signal-to-noise ratio, the proposed method is better than the fast Kurtogram and theimproved Kurtogram for indicating the existence of the resonant frequency bands. Besides, the enhanced Kurtogram cansimultaneously indicate multiple resonant frequency bands in this case.
Finally, a group of normally distributed random signals with a mean of 0 and different variances (from 0.1 to 0.9 with astep length of 0.05) were added to Eq. (33). Fig. 16 shows 17 mixed signals with 17 different noise variances in the case oftwo resonant frequencies. The results (autocorrelation of the envelope of the desired signals filtered by WPT) obtained bythe proposed method are shown in Fig. 17. Although strong noise variances were added to Eq. (33), the proposed method isstill effectively able to detect fault frequency-related signals at different noise variances. The fault frequency of 100 Hz and
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5
5.5
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Dep
ths
Frequency (Hz)
Frequency (Hz)
Am
plitu
de
Fig. 8. The results obtained by the improved Kurtogram proposed by Lei et al. for processing the mixed signal with one resonant frequency.
(a) The improved Kurtogram and (b) frequency spectrum of the envelope of the signal extracted from node (4, 7) by wavelet packet transform.
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se V
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se R
aito
-11.45
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-23.49
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-30.34
-28.18
-29.45
Fig. 9. The mixed signal with different noise variances in the case of one resonant frequency.
D. Wang et al. / Mechanical Systems and Signal Processing 35 (2013) 176–199186
its harmonics are also easy to identify in Fig. 18. These results demonstrate that the proposed method can effectivelydetect fault signatures with a low signal-to-noise ratio.
4.3. Different bearing fault signals obtained from an experimental motor and analysed by the proposed method
A small AC motor with a speed around 1400 rpm in the Smart Engineering Asset Management Laboratory was used asthe basic drive. The experimental setup and bearings are shown in Fig. 19(a) and (b). Three types of localised bearing faultswere seeded, including an outer race fault, an inner race fault and a rolling element fault. The locations of the bearingfaults are given in Fig. 19(c–e). The sampling frequency was set at 80 kHz. Each fault signal with the length of 75,000samplings was used and they are shown in Fig. 20(a–c), respectively. Bearing outer race, inner race, fundamental cagefrequency and ball spinning frequency were calculated as 136 Hz, 192 Hz, 9.7 Hz and 64 Hz by Eqs. (27)–(31).
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se R
aito
-11.45
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-20.75
-23.49
-25.24
-26.83
-30.34
-28.18
-29.45
Fig. 10. Results (autocorrelation signals of the envelope of the signal filtered by wavelet packet transform) obtained by the proposed method in the time
domain in the case of one resonant frequency.
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Rai
to
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-26.83
-30.34
-28.18
-29.45
Fig. 11. Results (power spectra of the envelope of the signal filtered by wavelet packet transform) obtained by the proposed method in the frequency
domain in the case of one resonant frequency.
D. Wang et al. / Mechanical Systems and Signal Processing 35 (2013) 176–199 187
In order to reduce the signal to noise ratio of the original bearing fault signals and remove the influence caused by theshaft rotating frequency, pre-whitening processing was conducted on all original fault signals. The values of the Akaikeinformation criterion for the different orders of the AR for processing the bearing outer race, inner race and ball faultsignals are plotted in Fig. 21(a–c), separately. The parameter of the AR filtering, namely the order of AR model, was chosenas 200 by minimizing the Akaike information criterion. In Fig. 22(a–c), temporal signals obtained by pre-whiteningprocessing contained the outer race, inner race and roller fault signatures, respectively. These pre-whitened signals wereequally analysed in the following sections by using the enhanced Kurtogram, the improved Kurtogram and the fastKurtogram, respectively.
4.3.1. Case 3: A real bearing outer race fault signal obtained from an experimental motor
To identify the bearing fault characteristic frequency correctly, the enhanced Kurtogram was applied to the signalshown in Fig. 22(a) to determine the most useful node of all. The colour map obtained is shown in Fig. 23(a), where node(4, 3) is the best among all nodes. This node was selected for further analysis by the proposed method. Power spectrum ofthe envelope of the signal extracted from node (4, 3) by WPT is shown in Fig. 23(b). From the result shown in Fig. 23(b),bearing outer race fault-related signatures are easily identified by inspecting the outer race fault characteristic frequencyand its harmonics. For comparison, the fast Kurtogram and the improved Kurtogram are applied to the same outer racefault signal shown in Fig. 22(a). The paving of the fast Kurtogram is plotted in Fig. 24(a), where an optimal filter with theoptimal centre frequency of 12,500 Hz and the bandwidth of 5000 Hz is found. In Fig. 24(b), envelope spectrum ofthe signal obtained by the optimal filter provides outer race fault signatures for bearing fault diagnosis. In Fig. 25(a), theimproved Kurtogram indicates that the most valuable node is (4, 7) which is obtained by maximizing the kurtosis of thetemporal signal filtered by wavelet packet transform. In Fig. 25(b), the frequency spectrum of the envelope of the signalextracted from node (4, 7) by wavelet packet transform reveals outer race fault signatures. In the case of outer race faultdiagnosis, although the three methods are effective in detecting the outer race localized faults, frequency spectrumobtained by the proposed method is clearest to show outer race fault characteristic frequency and its harmonics.Frequency spectra obtained by the other methods contain heavy noise which corrupts the visual inspection ability.
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2
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0
2
Am
plitu
de
Am
plitu
de
Am
plitu
de
Samplings
Fig. 12. Signals in the time domain: (a) the simulated signal with two resonance frequencies; (b) the noise signal with a normal distribution
and a variance of 0.6 and (c) the mixed signal.
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4
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2500
Dep
ths
Frequency (Hz)
Frequency (Hz)
Am
plitu
de
fm
2fm
3fm
4fm
Fig. 13. The results obtained by the proposed method for processing the mixed signal with two resonant frequencies. (a) The enhanced Kurtogram used
in this paper and (b) power spectrum of the envelope of the signal extracted from node (4, 4) by wavelet packet transform.
D. Wang et al. / Mechanical Systems and Signal Processing 35 (2013) 176–199188
0 1000 2000 3000 4000 5000 6000
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40
0.05
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0.25
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1
2
3x 10-3
Frequency (Hz)
Am
plitu
de
Frequency (Hz)
Leve
l k
Fig. 14. The results obtained by the fast Kurtogram for processing the mixed signal with two resonant frequencies. (a) The fast Kurtogram and
(b) frequency spectrum of the envelope of the signal obtained by a filter (the center frequency of 5500 Hz and the bandwidth of 1000 Hz).
D. Wang et al. / Mechanical Systems and Signal Processing 35 (2013) 176–199 189
4.3.2. Case 4: A real bearing inner race fault signal obtained from an experimental motor
The enhanced Kurtogram was then applied to the signal in Fig. 22(b). In Fig. 26(a), it can be seen that node (4, 8) has thehighest kurtosis and would thus be useful in extracting the bearing fault feature. The final result generated by theproposed method is plotted in Fig. 26(b). Fig. 26(b) displays an inner race fault characteristic frequency and its severalharmonics, which indicate that the bearing inner race had localised faults. For comparison, the fast Kurtogram is applied tothe same inner race fault signal. The paving of the fast Kurtogram is shown in Fig. 27(a), where an optimal filter with thecentre frequency of 18,333 Hz and the bandwidth of 3333 Hz is automatically chosen. The envelope spectrum of the signalfiltered by the optimal filter shows fault related signatures in Fig. 27(b). The improved Kurtogram is used to analyse thesame inner race fault signal. The paving of the improved Kurtogram is plotted in Fig. 28(a), in which the node (4, 8) is themost useful node for providing inner race fault signatures. The envelope spectrum of the signal extracted from node (4, 8)by WPT provides information about inner race faults in Fig. 28(b). The most value node (4, 8) obtained by the enhancedKurtogram coincides with that obtained by the improved Kurtogram. The results shown in Figs. 26 and 28(b) are powerspectrum and frequency spectrum of the envelope of the signal extracted from the same node (4, 8), respectively. It isclearly found that the power spectrum is capable of keeping inner race fault dominating frequency and depressing heavynoise. In the case of inner race fault diagnosis, the three methods are effective in detecting inner race fault characteristicfrequency. From the results shown in Figs. 26–28(b), envelope spectra obtained by the proposed method and the improvedKurtogram have better visual inspection ability than the envelope spectrum obtained by the fast Kurtogram.
4.3.3. Case 5: A real bearing ball fault signal obtained from an experimental motor
Finally, the proposed method was applied to the signal in Fig. 22(c). The result provided by the enhanced Kurtogram inFig. 29(a) shows that node (4, 3) has the highest kurtosis. The final power spectrum of the envelope of the signal extractedfrom node (4, 3) by WPT is shown in Fig. 29(b). A ball defect is successfully diagnosed through the identification of a rollingelement fault characteristic frequency. For comparison, both the fast Kurtogram and the improved Kurtogram are appliedto the same ball localized fault signal. In Fig. 30(a), the optimal filter indicated by the fast Kurtogram has the centrefrequency of 23,333 Hz and the bandwidth of 6666 Hz. In Fig. 30(b), the frequency spectrum of the envelope of the signalfiltered by the optimal filter indicates the existence of ball localized faults. However, the fault signatures are not obviousenough. Fault related fault frequencies are nearly overwhelmed by heavy noise. In Fig. 31(a), the paving of the improvedKurtogram indicates that the node (3, 3) is the most useful node among all nodes for bearing ball localized fault diagnosis.
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5.5
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50100150200250
Dep
ths
Frequency (Hz)
Frequency (Hz)
Am
plitu
de
Fig. 15. The results obtained by the improved Kurtogram proposed by Lei et al. for processing the mixed signal with two resonant frequencies.
(a) The improved Kurtogram and (b) frequency spectrum of the envelope of the signal extracted from node (4, 8) by wavelet packet transform.
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-20.39
-24.05
-21.78
-22.96
Fig. 16. The mixed signal with different noise variances in the case of two resonant frequencies.
D. Wang et al. / Mechanical Systems and Signal Processing 35 (2013) 176–199190
Frequency spectrum of the envelope of the signal extracted from node (3, 3) by WPT is effective in detecting the ball faultcharacteristic frequency. The visual inspection ability of the result obtained by the improved Kurtogram is not as good asthat of the result obtained by the proposed method. In the case of ball localized fault diagnosis, although the threemethods are effective in detecting ball localized faults, the frequency spectrum obtained by the proposed method providesthe best visual inspection ability for bearing ball localized fault diagnosis.
In Section 4, the simulated bearing fault signals and the real bearing fault signals have been simultaneously analysed bythe enhanced Kurtogram, the fast Kurtogram and the improved Kurtogram. The results of performance comparison of theenhanced Kurtogram, the fast Kurtogram and the improved Kurtogram are summarized in Table 1, where Case 1 concernsthe mixed signal with one resonant frequency; Case 2 concerns the mixed signal with two resonant frequencies; Case 3concerns the real experimental outer race fault signal; Case 4 concerns the real experimental inner race fault signal andCase 5 concerns the real experimental ball fault signal. From the results shown in Table 1, it is found that the enhanced
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Fig. 17. Results (autocorrelation signals of the envelope of the signal filtered by wavelet packet transform) obtained by the proposed method in the time
domain in the case of two resonant frequencies.
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Fig. 18. Results (power spectra of the envelope of the signal filtered by wavelet packet transform) obtained by the proposed method in the frequency
domain in the case of two resonant frequencies.
Fig. 19. Experiment equipment and faulty elements of the tested bearings. (a) An AC motor with speed 1400 rpm, (b) tested bearing (SKF 1206 EKTN9),
(c) an outer race fault, (d) an inner race fault and (e) a ball fault.
D. Wang et al. / Mechanical Systems and Signal Processing 35 (2013) 176–199 191
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x 104
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plitu
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plitu
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Samplings
Am
plitu
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Samplings
Fig. 20. The original signals in time domain (a) bearing with localized outer race fault; (b) bearing with localized inner race fault and (c) bearing with
localized rolling element fault.
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AIC
A
IC
The order of AR
The order of AR
AIC
The order of AR
Fig. 21. The values of AIC for the different orders of AR for processing (a) the bearing with localized outer race fault; (b) the bearing with localized inner
race fault and (c) the bearing with localized rolling element fault.
D. Wang et al. / Mechanical Systems and Signal Processing 35 (2013) 176–199192
1 2 3 4 5 6 7
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plitu
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plitu
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x 104 Samplings
x 104Samplings
Am
plitu
de
x 104Samplings
Fig. 22. Signals obtained by pre-whitening processing (a) bearing with localized outer race fault; (b) bearing with localized inner race fault and
(c) bearing with localized rolling element fault.
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ths
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Am
plitu
de fO
2fO
3fO
Fig. 23. The results obtained by the enhanced Kurtogram in this paper for detecting an outer race fault. (a) The enhanced Kurtogram and (b) power
spectrum of the envelope of the signal extracted from node (4, 3) by wavelet packet transform.
D. Wang et al. / Mechanical Systems and Signal Processing 35 (2013) 176–199 193
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plitu
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Frequency (Hz)
fO 2fO 3fO 4fO 5fO
×10 4
Leve
l k
Fig. 24. The results obtained by the fast Kurtogram for detecting an outer race fault. (a) The fast Kurtogram and (b) frequency spectrum of the envelope
of the signal obtained by a filter (the center frequency of 12,500 Hz and the bandwidth of 5000 Hz).
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ths
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plitu
de fO
2fO3fO
Fig. 25. The results obtained by the improved Kurtogram proposed by Lei et al. for detecting an outer race fault. (a) The improved Kurtogram and
(b) frequency spectrum of the envelope of the signal extracted from node (4, 7) by wavelet packet transform.
D. Wang et al. / Mechanical Systems and Signal Processing 35 (2013) 176–199194
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plitu
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2fI
fs 3fI 4fI
Fig. 26. The results obtained by the enhanced Kurtogram in this paper for detecting an inner race fault. (a) The enhanced Kurtogram and (b) power
spectrum of the envelope of the signal extracted from node (4, 8) by wavelet packet transform.
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plitu
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Frequency (Hz) ×104
Leve
l k
fI 2fI 3fI
Fig. 27. The results obtained by the fast Kurtogram for detecting an inner race fault. (a) The fast Kurtogram and (b) frequency spectrum of the envelope
of the signal obtained by a filter (the center frequency of 18,333 Hz and the bandwidth of 3333 Hz).
D. Wang et al. / Mechanical Systems and Signal Processing 35 (2013) 176–199 195
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plitu
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3fI 4fIfs
Dep
ths
Fig. 28. The results obtained by the improved Kurtogram proposed by Lei et al. for detecting an inner race fault. (a) The improved Kurtogram and
(b) frequency spectrum of the envelope of the signal extracted from node (4, 8) by wavelet packet transform.
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plitu
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4fBS
Fig. 29. The results obtained by the enhanced Kurtogram in this paper for detecting ball localized faults. (a) The enhanced Kurtogram and (b) power
spectrum of the envelope of the signal extracted from node (4, 3) by wavelet packet transform.
D. Wang et al. / Mechanical Systems and Signal Processing 35 (2013) 176–199196
0 0.5 1 1.5 2 2.5 3 3.5 4
0
1
1.6
2
2.6
3
3.6
4
20
40
60
80
100
120
100 200 300 400 500 600 700 800 900 10000
1
2
3
4 x 10-6
Frequency (Hz)
Am
plitu
de
Frequency (Hz)
4fBS2fBS
×104
Leve
l k
Fig. 30. The results obtained by the fast Kurtogram for detecting an ball localized fault. (a) The fast Kurtogram and (b) frequency spectrum of the
envelope of the signal obtained by a filter (the center frequency of 23,333 Hz and the bandwidth of 6666 Hz).
0.5 1 1.5 2 2.5 3 3.5 4
x 104
1
2
3
4
20
40
60
80
100
120
140
160
180
200
0 100 200 300 400 500 600 700 800 900 10000
5
10
15
20
Dep
ths
Frequency (Hz)
Frequency (Hz)
Am
plitu
de
2fBSfBS
3fBS 4fBS5fBS
fC
Fig. 31. The results obtained by the improved Kurtogram proposed by Lei et al. for detecting a ball localized fault. (a) The improved Kurtogram and
(b) frequency spectrum of the envelope of the signal extracted from node (3, 3) by wavelet packet transform.
D. Wang et al. / Mechanical Systems and Signal Processing 35 (2013) 176–199 197
Table 1Performance comparison of the enhanced Kurtogram, the fast Kurtogram and the improved Kurtogram.
Cases Is it effective in detecting bearing faults? Visual inspection ability
The enhanced
Kurtogram
The fast
Kurtogram
The improved
Kurtogram
The enhanced
Kurtogram
The fast
Kurtogram
The improved
Kurtogram
1 Yes No No High Null Null
2 Yes No No High Null Null
3 Yes Yes Yes High Medium Medium
4 Yes Yes Yes High Medium High
5 Yes Yes Yes High low Medium
Case 1 concerns the mixed signal with one resonant frequency; Case 2 concerns the mixed signal with two resonant frequencies; Case 3 concerns the real
laboratorial outer race fault signal; Case 4 concerns the real laboratorial inner race fault signal and Case 5 concerns the real laboratorial ball fault signal.
D. Wang et al. / Mechanical Systems and Signal Processing 35 (2013) 176–199198
Kurtogram is capable of detecting bearing faults in all cases. In the cases of 1 and 2, the enhanced Kurtogram not onlyestablished the most valuable wavelet packet node for further envelope spectrum analysis but also identified the most usefulnode at each depth. However, when the fast Kurtogram and the improved Kurtogram were applied to analyse the samesimulated signals, they failed to provide any bearing fault related signatures. In the cases of 3–5, although the three methodshad ability to successfully detect the three different kinds of bearing faults, their visual inspection ability varied with bearingfault signals. For bearing outer race fault diagnosis, the frequency spectrum obtained by the enhanced Kurtogram mostclearly showed the outer race fault frequency and its harmonics. The frequency spectra obtained by the fast Kurtogram andthe improved Kurtogram were corrupted by heavy noise. Therefore, in the case of real bearing outer race fault diagnosis, theenhanced Kurtogram has the best visual inspection ability because outer race fault frequency is the concerned topic. As forthe harmonics of outer race fault frequency, they are usually used to describe the regularity of a signal. The more regular (i.e.,less impulsive), the fast the decay of the amplitudes of the harmonics of the outer race fault frequency. For bearing inner racefault diagnosis, the same most valuable wavelet packet node was recommended by the enhanced Kurtogram and theimproved Kurtogram. The comparison between power spectrum and frequency spectrum of the envelope extracted from thesame node showed that the power spectrum suppresses heavy noise while the frequency spectrum retains the sidebandsaround the inner race fault frequency. Besides, the rotating frequency could be identified in both frequency spectra. Thefrequency spectrum obtained by the fast Kurtogram had difficulty in indicating the shaft rotating frequency. Moreover, thefrequency spectrum was corrupted by heavy noise. In the case of real inner race fault diagnosis, it was shown that the visualinspection ability of the enhanced Kurtogram and the improved Kurtogram was better than that of the fast Kurtogram. Forbearing ball fault diagnosis, it was easy to find that the frequency spectrum obtained by the proposed method had the bettervisual inspection ability than frequency spectra obtained by the fast Kurtogram and the improved Kurtogram. The reasonsare given as follows. First, heavy noise is suppressed by the proposed method. Second, the fundamental cage frequency canbe detected by the enhanced Kurtogram and the improved Kurtogram. Thirdly, sidebands extracted by the proposed methodare very clear. As a result, after the comprehensive comparison has been done, the enhanced Kurtogram has the bestperformance among the three methods for bearing fault diagnosis in our case studies.
5. Conclusions
This paper proposes a new method and an enhanced Kurtogram to detect bearing faults. The method uses the enhancedKurtogram to select the assessable WPT nodes from among all nodes. To construct the Kurtogram, the kurtosis of thepower spectrum of the envelope of the signals filtered by WPT at each node is calculated. The nodes corresponding to thehighest kurtosis can then be considered for further analysis. In the case of simulated signals mixed with noises, theproposed method has a good detection rate for fault frequency even though the signal-to-noise ratio is low. Moreover, it isable to locate the resonant frequency bands at different depths. However, the fast Kurtogram and the improved Kurtogramfail to diagnose the simulated signal mixed with heavy noise. In the case of the various real bearing fault signals obtainedin the laboratory, the proposed method is able to distinguish between different bearing faults, including outer race, innerrace and rolling element faults after AR filtering is used to remove the disturbance caused by discrete frequency noise, suchas the rotating frequency components. The pre-whitened bearing fault signals are equally analysed by the enhancedKurtogram, the improved Kurtogram and the fast Kurtogram, separately. Compared with the fast Kurtogram and theimproved Kurtogram, it is found that their visual inspection ability is not as good as that of the enhanced Kurtogram forshowing fault characteristic frequencies.
Acknowledgments
The work described in this paper was partly supported by a grant from the National Natural Science Foundation ofChina and Research Grants Council of the HKSAR Joint Research Scheme (Project No: N_CityU106/08) and a grant from
D. Wang et al. / Mechanical Systems and Signal Processing 35 (2013) 176–199 199
Croucher Foundation (Project No. 9220027). The authors wish to thank Professor K. A. Loparo for his permission to use thebearing data. At last, we would like to express our deepest appreciation for the valuable and constructive comments fromtwo anonymous referees. With their carefully reviews, this paper could be greatly improved.
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