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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: wangdonguestc@yahoo.cn (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

ll rights reserved.

: þ852 34420415.

ang), meptse@cityu.edu.hk (P.W. Tse), kltsui@cityu.edu.hk (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

0 500 1000 1500 2000 2500-1

0

1

0 500 1000 1500 2000 2500

-2

0

2

0 500 1000 1500 2000 2500

-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.

0 100 200 300 400 500 600 700 800 900 10000

100

200

300

400

1000 2000 3000 4000 5000 6000

1

2

3

4

500

1000

1500

2000

2500

3000

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

0 1000 2000 3000 4000 5000 6000

0

1

1.6

2

2.6

3

3.6

4 0.02

0.04

0.06

0.08

0.1

0.12

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0.16

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0.2

0.22

100 200 300 400 500 600 700 800 900 10000

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

1000 2000 3000 4000 5000 6000

1

2

3

4

3

3.5

4

4.5

5

5.5

0 100 200 300 400 500 600 700 800 900 10000

50

100

150

200

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.

500 1000 1500 2000 2500

0.1

0.2

0.3

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0.7

0.8

0.9 -3

-2

-1

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1

2

3

Noi

se V

aria

nces

Samplings

Sign

al to

Noi

se R

aito

-11.45

-17.31

-20.75

-23.49

-25.24

-26.83

-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).

500 1000 1500 2000 2500

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0.2

0.3

0.4

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0.6

0.7

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0

0.2

0.4

0.6

0.8

Noi

se V

aria

nces

Samplings

Sign

al to

Noi

se R

aito

-11.45

-17.31

-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.

0 100 200 300 400 500 600 700 800 900

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

50

100

150

200

250

Noi

se V

aria

nces

Frequency(Hz)

Sig

nal t

o N

oise

Rai

to

-11.45

-17.31

-20.75

-23.49

-25.24

-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.

0 500 1000 1500 2000 2500-2

0

2

0 500 1000 1500 2000 2500

-2

0

2

0 500 1000 1500 2000 2500

-2

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.

0 100 200 300 400 500 600 700 800 900 10000

200

400

600

1000 2000 3000 4000 5000 6000

1

2

3

4

500

1000

1500

2000

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

0

1

1.6

2

2.6

3

3.6

40

0.05

0.1

0.15

0.2

0.25

100 200 300 400 500 600 700 800 900 10000

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.

1000 2000 3000 4000 5000 60000.5

1

1.5

2

2.5

3

3.5

4

4.5

3

3.5

4

4.5

5

5.5

0 100 200 300 400 500 600 700 800 900 10000

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.

500 1000 1500 2000 2500

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0.5

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0.9 -3

-2

-1

0

1

2

3

4

Noi

se V

aria

nces

Samplings

Sig

nal t

o N

oise

Rai

to

-5.07

-10.98

-14.35

-17.09

-18.96

-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

500 1000 1500 2000 2500

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se V

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Sig

nal t

o N

oise

Rai

to

-5.07

-10.98

-14.35

-17.09

-18.96

-20.39

-24.05

-21.78

-22.96

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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Sig

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o N

oise

Rai

to

-5.07

-10.98

-14.35

-17.09

-18.96

-20.39

-24.05

-21.78

-22.96

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

1 2 3 4 5 6 7x 104

-1

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1

1 2 3 4 5 6 7x 104

x 104

-4

-2

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2

1 2 3 4 5 6 7

-1

0

1

Am

plitu

de

Am

plitu

de

Samplings

Samplings

Am

plitu

de

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.

0 50 100 150 200 250 300-10

-8

-6

-4

0 50 100 150 200 250 300-8

-6

-4

0 50 100 150 200 250 300-10

-8

-6

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

-0.1-0.05

00.050.1

1 2 3 4 5 6 7

-0.2

0

0.2

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-0.2-0.1

00.10.2

Am

plitu

de

Am

plitu

de

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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x 104

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6000

7000

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10000

11000

Dep

ths

Frequency (Hz)

Frequency (Hz)

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

0 0.5 1 1.5 2 2.5 3 3.5 4

0

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10

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1

2

3

4x 10-6

Frequency (Hz)

Am

plitu

de

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).

0.5 1 1.5 2 2.5 3 3.5 4

x 104

0.5

1

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24

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10

20

30

Dep

ths

Frequency (Hz)

Frequency (Hz)

Am

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

0 100 200 300 400 500 600 700 800 900 10000

500

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1500

2000

0.5 1 1.5 2 2.5 3 3.5 4

x 104

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4000

6000

8000

10000

12000

Dep

ths

Frequency (Hz)

Frequency (Hz)

Am

plitu

de fI

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.

0 0.5 1 1.5 2 2.5 3 3.5 4

0

1

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4

6

8x 10-6

Frequency (Hz)

Am

plitu

de

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

0.5 1 1.5 2 2.5 3 3.5 4

x 104

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80

Frequency (Hz)

Frequency (Hz)

Am

plitu

de fI

2fI

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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x 104

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ths

Frequency (Hz)

Frequency (Hz)

Am

plitu

de

2fBS

fBS

fC3fBS

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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