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Mechanical Systems and Signal Processing

Mechanical Systems and Signal Processing 40 (2013) 499–519

0888-32http://d

n CorrE-m

journal homepage: www.elsevier.com/locate/ymssp

The design of a new sparsogram for fast bearing faultdiagnosis: Part 1 of the two related manuscripts that havea joint title as “Two automatic vibration-based fault diagnosticmethods using the novel sparsity measurement – Parts 1 and 2”

Peter W. Tse n, Dong WangThe Smart Engineering Asset Management Laboratory (SEAM) and the Croucher Optical Non-destructive Testing and Quality InspectionLaboratory (CNDT), Department of Systems Engineering & Engineering Management, City University of Hong Kong, Tat Chee Avenue, HongKong, China

a r t i c l e i n f o

Article history:Received 8 February 2012Received in revised form24 May 2013Accepted 29 May 2013Available online 5 July 2013

Keywords:Wavelet packet decompositionResonant frequency bandBearing fault diagnosisSparsogramSparsity measurement

70/$ - see front matter & 2013 Elsevier Ltd.x.doi.org/10.1016/j.ymssp.2013.05.024

esponding author. Tel.: +852 92660207; faxail addresses: meptse@cityu.edu.hk, mepwts

a b s t r a c t

Rolling element bearings are widely used in rotating machines. An early warning ofbearing faults helps to prevent machinery breakdown and economic loss. Vibration-basedenvelope analysis has been proven to be one of the most effective methods for bearingfault diagnosis. The core of an envelope analysis is to find a resonant frequency band for aband-pass filtering for the enhancement of weak bearing fault signals. A new conceptcalled a sparsogram is proposed in Part 1 paper. The aim of the sparsogram is to quicklydetermine the resonant frequency bands. The sparsogram is constructed usingthe sparsity measurements of the power spectra from the envelopes of wavelet packetcoefficients at different wavelet packet decomposition depths. The optimal wavelet packetnode can be selected by visually inspecting the largest sparsity value of the waveletpacket coefficients obtained from all wavelet packet nodes. Then, the wavelet packetcoefficients extracted from the selected wavelet packet node is demodulated for envelopeanalysis. Several case studies including a simulated bearing fault signal mixed with heavynoise and real bearing fault signals collected from a rotary motor were used to validate thesparsogram. The results show that the sparsogram effectively locates the resonantfrequency bands, where the bearing fault signature has been magnified in these bands.Several comparison studies with three popular wavelet packet decomposition basedmethods were conducted to show the superior capability of sparsogram in bearing faultdiagnosis.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Rolling element bearings are widely used to support rotating components. Bearing failures may cause machinerybreakdown and economic loss. Consequently, it is crucial to identify bearing faults at their early developing stages toprevent severe machinery failures. Because vibration signals are easily collected by attached transducers, vibration signal-based analysis is the dominant way to identify machine faults. A rolling element bearing usually consists of an inner race, an

All rights reserved.

: +852 34420173.e@cityu.edu.hk (P.W. Tse), dongwang4-c@my.cityu.edu.hk (D. Wang).

P.W. Tse, D. Wang / Mechanical Systems and Signal Processing 40 (2013) 499–519500

outer race, rollers and a cage. Once localized faults develop on the surface of any of these components, the strikes of rollerson the fault surfaces excite the resonant frequencies of structures between the bearing and the transducers, triggering themodulation phenomenon. It has been found that envelope signals obtained by amplitude demodulation contain many morebearing fault-related signatures than the original signals. The core of amplitude demodulation is the appropriate selection ofthe resonant frequency band. Many previous studies have been conducted to solve the resonant frequency band selectionproblem [1,2].

In recent years, Antoni and Randall [3] showed that spectral kurtosis was a solution for designing an optimal band-passfilter that maximized the signal to noise ratio. It should be noted that the performance of short-time Fourier transform(STFT) depends on the selections of different window lengths. Hence, a new concept called a kurtogram [3] was developedby considering a series of different window lengths. However, the original kurtogram was time-consuming and not suitablefor on-line health monitoring. To overcome the disadvantage of a long computing time, the 1/3-binary tree fast kurtogramestimator was devised [4]. This estimator made on-line condition monitoring a reality. Barszcz and Jabłoński [5] found thatthe kurtosis of the envelope spectrum amplitudes of demodulated signals was more effective than the kurtosis of thefiltered temporal signal when the signal to noise ratio was low. Therefore, they proposed the concept of a protrugram toselect the optimal frequency band for amplitude demodulation.

From the previous studies [5–7], it is found that the bearing fault characteristic frequency and its harmonics shown in theenvelope spectrum can be regarded as a few spikes with large amplitudes that reflect bearing fault signatures. In this paper,we define these spikes in the envelope spectrum as the sparse representation of a bearing fault signal in a frequencydomain. The envelope spectrum analysis was applied to the signals captured from a normal bearing and a bearing sufferedfrom an outer race fault. The results are shown in Fig. 1(a) and (b) for the purpose to verify the aforementioned definition.From the envelope spectrum as shown in Fig. 1(b), the signal, which was generated by the bearing suffered from an outerrace defect, contains a few outstanding amplitudes at the bearing outer race fault characteristic frequency and its harmonics.This result proves that the definition of the sparse representation is useful to indicate that a bearing is defective. Therefore, asparsity measurement may be a potential approach for quantifying the amplitudes of bearing fault characteristic frequencyand its harmonics in an envelope spectrum.

Sparsity measurement is frequently used in ultrasonic non-destructive testing to improve the performance of ultrasonicsignal processing algorithms. An ultrasonic echo signal is a convolution between an impulse response and a series ofreflection sequences (spikes). After deconvolution, a series of reflection sequences are useful for accurately estimating thelocations of media discontinuations. These reflection sequences can be seen as the sparse representation of the originalultrasonic echo signal. Based on sparsity measurement, Liang et al. [8] constructed a non-linear function to improve theconvergence of a blind deconvolution algorithm. Further, they [9] added sparsity measurement to the Moore-penroseinverse to obtain sparse ultrasonic reflection sequences. Chen et al. [10] used the same sparsity measurement to pick themost useful intrinsic mode functions decomposed by empirical mode decomposition for magnetic flux leakage signal-based

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4x 104

Am

plitu

de

Am

plitu

de

Frequency (Hz)

Frequency (Hz)

Bearingfault characterisicfrequency and its harmonics

Sparse representation

Fig. 1. The envelope spectra of (a) the signal from a normal bearing and (b) the signal from the bearing with an outer race defect.

P.W. Tse, D. Wang / Mechanical Systems and Signal Processing 40 (2013) 499–519 501

non-destructive inspection. The sparsity measurement used in non-destructive testing [8–10] was employed in this paper toquantify bearing characteristic frequencies in an envelope spectrum.

Before sparsity measurement is performed on an envelope spectrum, a band-pass filter can be used to retain one of theresonant frequency bands for enhancing the signal to noise ratio of a weak bearing fault signal. Compared with discretewavelet transform [11] and continuous wavelet transform [12], binary wavelet packet transform (WPT) is used todecompose an original signal into some wavelet packet coefficients, which have orthogonal sub-frequency bands with anequal bandwidth at a wavelet packet decomposition depth. Moreover, the high computing efficiency and sufficient time–frequency resolution of binary WPT is more attractive. WPT as a precise filter bank has been demonstrated to be an effectiveway to extract bearing fault signatures [13–17]. As a result, a new concept called a sparsogram is proposed in Part 1 paper forthe fast selection of a resonant frequency band that contains many bearing fault signatures. Additionally, the sparsogram hasability to give initial center frequencies and bandwidths for the optimization of a complex Morlet wavelet filter which havebeen developed in Part 2 paper.

The rest of Part 1 paper is organized as follow. Section 2 introduces binary WPT and proposes the new sparsogram and itscorresponding bearing fault diagnosis method. In Section 3, simulated and real bearing fault signals are used to validate theeffectiveness of the proposed method. Comparison studies with three binary WPT based popular methods are conducted.Section 4 concludes Part 1 paper.

2. Binary WPT and the proposed fast detection method for bearing fault diagnosis

2.1. The basic theory of binary WPT [18]

A binary WPT that has good local properties in both time and frequency spaces is an extension of discrete wavelettransform. The binary WPT uses more filters than the discrete wavelet transform to analyze a signal. It means that the WPTdecomposes the high frequency bands that are not split by discrete wavelet transform. Assume a space Wp

j and itsorthonormal basis ψp

j ðt−2jnÞn∈Z , where j is the wavelet packet decomposition depth and p is the pth wavelet packet basis.The above orthogonal basis of the space at wavelet packet node (j, p) can be decomposed into two new orthogonal bases asfollow:

ψ2pjþ1ðtÞ ¼ ∑

þ∞

n ¼ −∞hðnÞψp

j ðt−2jnÞ; ð1Þ

ψ2pþ1jþ1 ðtÞ ¼ ∑

þ∞

n ¼ −∞gðnÞψp

j ðt−2jnÞ; ð2Þ

hðnÞ ¼ ⟨ψ2pjþ1ðuÞ;ψ

pj ðt−2jnÞ⟩; ð3Þ

gðnÞ ¼ ⟨ψ2pþ1jþ1 ðuÞ;ψp

j ðt−2jnÞ⟩: ð4Þ

here, hðnÞ and gðnÞ are a pair of conjugate mirror filters and ⟨; ⟩ is the inner product.Take the Fourier transform of Eqs. (1) and (2) as follow:

ϕ2pjþ1ðωÞ ¼Hð2jωÞϕp

j ðωÞ; ð5Þ

ϕ2pþ1jþ1 ðωÞ ¼ Gð2jωÞϕp

j ðωÞ; ð6Þ

where HðωÞ, GðωÞ and ϕðωÞ are the Fourier transform of hðnÞ, gðnÞ and ψðtÞ respectively. Therefore, Eqs. (5) and (6) explainthat the frequency support of ϕp

j ðωÞ is divided into two frequency bands whose energies concentrate on Hð2jωÞ and Gð2jωÞ,respectively. For any node (j, p), wavelet packet coefficients of an original signal can be calculated by taking the innerproduct of the original signal with every wavelet packet basis. In the fast binary wavelet packet decomposition algorithm,wavelet packet coefficients d2pjþ1ðnÞ and d2pþ1

jþ1 ðnÞ are obtained by

d2pjþ1ðnÞ ¼ dpj nhð−2nÞ; ð7Þ

d2pþ1jþ1 ðnÞ ¼ dpj ngð−2nÞ; ð8Þ

where n is the convolution operator. In the reconstruction algorithm of wavelet packet decomposition, wavelet packetcoefficients are recalculated by

dpj ðnÞ ¼D2pjþ1nhðnÞ þ D2pþ1

jþ1 ngðnÞ; ð9Þ

where D means inserting a zero between each sample of d.The frequency band of wavelet packet coefficients at a specific wavelet packet node ðj;pÞ is located in

½p� Fs=2Jþ1; ðpþ 1Þ � Fs=2

Jþ1�, 0≤p≤2J−1, where Fs is the sampling frequency. This indicates that a specific frequency bandcan be established for a specific wavelet packet node (j, p). In other words, binary WPT has the ability to extract frequency

P.W. Tse, D. Wang / Mechanical Systems and Signal Processing 40 (2013) 499–519502

components at a desired frequency band. However, it is possible to reconstruct wavelet packet coefficients at a specificwavelet packet node with the same temporal length as the original signal. This step can be realized by setting the waveletpacket coefficients at all other wavelet packet nodes to zero and then reconstructing the specific wavelet packet coefficientsusing Eq. (9). For convenient notation, the reconstructed wavelet packet coefficients are also represented by dpj ðnÞ.

2.2. The proposed fast detection method based on the sparsogram for bearing fault diagnosis

When rolling element bearings suffer from localized faults, the strikes of rollers on the fault surfaces excite the resonantfrequencies of structures, which are accompanied by fault characteristic frequencies. Therefore, the bearing fault signaturesmust be located within some frequency bands in the high frequency region. In terms of the principle of binary waveletpacket decomposition algorithm, the bearing fault signatures can be reflected by wavelet packet coefficients located at oneor several wavelet packet nodes at different wavelet packet decomposition depths. One of these wavelet packet nodes thatcontain more fault signatures than any other wavelet packet node is then chosen for further envelope analysis. Consideringall wavelet packet nodes at different wavelet packet decomposition depths, a new concept called a sparsogram is proposedfor the fast selection of the optimal wavelet packet node and the resonant frequency band. The new sparsogram is formedby calculating the sparsity values of the power spectra from the envelopes of binary wavelet packet coefficients at differentwavelet packet decomposition depths and different wavelet packet nodes. Depending on the different sparsity values, thenew sparsogram can reflect the optimal wavelet packet node that contains many bearing fault signatures. Then, based onthe selected wavelet packet node, an envelope analysis can be used to detect bearing faults. The proposed bearing faultdiagnosis method based on the new sparsogram is shown in Fig. 2, where the details of the flowchart are given in thefollowing steps.

Step 1: Load an original bearing fault signal. Assume the maximum wavelet packet decomposition depth is equal to J.Perform binary WPT on the original fault signal at different wavelet packet decomposition depths equal to 1,2,…, J.To establish the upper limit of the maximum wavelet packet decomposition depth, it is suggested that the minimumbandwidth of the wavelet packet coefficients obtained by WPT at the maximum decomposition depth J should be largerthan three times the outer race fault characteristic frequency [16].

Generate a new sparsogram using the

sparsity measurements

Load an original vibration signal

Perform binary wavelet packet transform on

the original signal at different depths and

reconstruct signals with the same temporal

length as the original signal

Select a valuable wavelet packet node from

all wavelet packet nodes

Perform power spectrum on the envelope

signal and identify bearing fault

characteristic frequency and its harmonics

Start

End

Demodulate the selected wavelet packet

node signal by Hilbert transform to obtain an

envelope signal

Fig. 2. The flowchart of the proposed method.

P.W. Tse, D. Wang / Mechanical Systems and Signal Processing 40 (2013) 499–519 503

The most important step for the use of binary WPT is the selection of a proper wavelet packet basis function becausewavelet packet coefficients are calculated according to the principle of inner product operation. Mathematically, it meansthat the inner product operation is used to measure the similarity between two signals. Therefore, if the wavelet basisfunction is very similar to the analyzed signal, the result obtained by the inner product operation will have a large value.For bearing fault diagnosis, the Daubechies mother wavelet provides the fast binary WPT and is widely used in bearing faultdiagnosis [16,17,19,20] because its shape is similar to the impulse generated by bearing localized faults. Hence, a Daubechies10 wavelet [17] was employed to implement binary WPT.

Step 2: After wavelet packet coefficients at different wavelet packet nodes and different wavelet packet decompositiondepths are obtained, the sparsity values measured from the power spectra of the envelopes of wavelet packet coefficientsare calculated. In order to get the envelopes, first, it is necessary to construct an analytical signal. The real part of theanalytical signal is the wavelet packet coefficients at each wavelet packet node and the imaginary part of the analyticalsignal is the Hilbert transform of wavelet packet coefficients at each wavelet packet node. The envelope signal for eachwavelet packet node is obtained by taking the modulus of the analytical signal. After that, autocorrelation with the Fouriertransform, namely the power spectrum, is used to map the temporal envelope signals of wavelet packet coefficients into afrequency domain. Here, assume the envelope signal has a mean of zero and its corresponding power spectrum is denotedas dpj ðf Þ. The sparsity value can be calculated using Eq. (10) given as follow [8–10]:

Sðj; pÞ ¼ Spj ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi∑

Fs=2−1

f ¼ 0ðdpj ðf ÞÞ2

s

∑Fs=2−1

f ¼ 0jdpj ðf Þj

¼∥dpj ðf Þ∥2∥dpj ðf Þ∥1

; 1≤j≤J; 0≤p≤2J−1; ð10Þ

where ∥dpj ðf Þ∥2 and ∥dpj ðf Þ∥1 are L2 norm and L1 norm, respectively. Once all sparsity values for all wavelet packet nodes atdifferent wavelet packet decomposition depths are calculated, the paving of the new sparsogram is shown in Fig. 3, where aone-dimensional signal is depicted in a two-dimensional sparsity value based figure. Here, assume the maximum waveletpacket decomposition depth is 4.

In the case of only bearing faults, the largest sparsity value is selected to indicate the optimal wavelet packet node thatcontains the most useful resonant frequency band. However, it should be pointed out that the frequency band of the waveletpacket coefficients extracted from the optimal wavelet packet node may be not proper to cover a whole resonant frequencyband because the frequency band of the wavelet packet coefficients is mathematically fixed prior to the signal analysis andmay only cover the part of the resonant frequency band. Therefore, the sparsogram just reports a fast detection scheme tochoose the coarse resonant frequency band.

In the case of multiple faults, it is possible to consider the first several largest sparsity values obtained from differentwavelet packet nodes because different fault signatures may be located in different frequency bands (such as, low frequencybands for periodic fault components being caused by unbalance, misalignments, eccentricities, etc., and high frequencybands for random transient fault components being caused by bearing faults). In the case of multiple vibration componentsthat result from a single vibration mixture, it is a blind component separation problem. Different separation algorithms [1],such as linear prediction, adaptive noise cancellation, self-adaptive cancellation, and discrete/random separation, could beused to separate low-frequency periodic fault components from the single vibration mixture before the new sparsogram isused for bearing fault diagnosis.

Step 3: After the useful wavelet packet node is indicated by the sparsogram, the demodulation (the Hilbert transformdemodulation method stated in Step 2) is performed to get the envelope of the wavelet packet coefficients extracted fromthe useful wavelet packet node, which shows more obvious fault signatures than the original fault signal.

Step 4: Autocorrelation [21,22] with the Fourier analysis, namely the power spectrum, is useful for identifying thepotential periodic intervals. In the time domain, the autocorrelation of the envelope signal has the ability to exhibit potentialbearing fault characteristic periods, which are approximately the reciprocal of bearing fault characteristic frequencies. In thefrequency domain, the power spectrum directly shows the bearing fault characteristic frequencies themselves. The outerrace fault characteristic frequency fo, the inner race fault characteristic frequency fI and the ball spinning frequency fBS are

Fig. 3. The paving of the sparsogram at the maximum wavelet packet decomposition depth of 4.

P.W. Tse, D. Wang / Mechanical Systems and Signal Processing 40 (2013) 499–519504

given as follow [1]:

f O ¼ Z � f s2

1−dD

cos β� �

; ð11Þ

f I ¼Z � f s

21þ d

Dcos β

� �; ð12Þ

f BS ¼D� f s2d

1−dD

cos β� �2

!; ð13Þ

where fs is the shaft rotating frequency in Hz, d and D are the diameters of the rolling element and pitch diameter,respectively. Z is the number of rolling elements and β is the contact angle. Eq. (13) is the frequency calculated byconsidering that a defective roller strikes both inner and outer races.

3. Validation of the proposed method

3.1. A simulated bearing fault signal with two resonant frequency bands

The similar simulated bearing fault signal used in Ref. [23] was produced as

yðkÞ ¼∑rexpð−α� ðk−r � Fs=f m−τrÞ=FsÞ � sin ð2πf 1 � ðk−r � Fs=f m−τrÞ=FsÞ

þ∑rexpð−α� ðk−r � Fs=f m−τrÞ=FsÞ � sin ð2πf 2 � ðk−r � Fs=f m−τrÞ=FsÞ; ð14Þ

where α is equal to 900, fm is the fault frequency (equal to 100 Hz), Fs is the sampling frequency (set to 12,000 Hz), f1 is theresonant frequency (equal to 1700 Hz) and f2 is equal to 4200 Hz. τr is subject to a discrete uniform distribution, which isused to simulate the randomness caused by roller slippage. Here, two resonance frequencies were embedded in thesimulated bearing fault signal because Ref. [24] reported that the first several resonant frequencies were distributed indifferent frequency bands. Although 24,000 samples were used, to display the transient components clearly, only 2500samples were displayed in the simulated case. A normally distributed random heavy noise signal with a mean of 0 and a

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24

Am

plitu

deA

mpl

itude

Am

plitu

de

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Am

plitu

de

Samples

Samples

Samples

Fig. 4. The temporal signals: (a) the simulated signal with two resonant frequency bands; (b) the intentionally added normally distributed noise withvariance of 0.6 (heavy noise); (c) the simulated signal mixed with the heavy noise and (d) the final signal obtained by the fast sparsogram.

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1

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4

1

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2

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3

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4

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5

x 10-4

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th

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Node (4,11)Node (4,4)

Fig. 5. The sparsogram of the mixed signal with two resonant frequency bands.

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plitu

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plitu

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Fig. 6. The signals in frequency domain: (a) the original mixed signal with two resonant frequency bands; (b) the low frequency components of Fig. 6(a);(c) the signal extracted from wavelet packet node (4, 4) by binary WPT.

P.W. Tse, D. Wang / Mechanical Systems and Signal Processing 40 (2013) 499–519 505

P.W. Tse, D. Wang / Mechanical Systems and Signal Processing 40 (2013) 499–519506

variance of 0.6 was added to Eq. (14). The simulated signal, the noisy signal and the mixed signal with noise are shown inFig. 4(a), (b) and (c), respectively.

In Fig. 4(c), it is difficult to distinguish the potential periodic intervals of the signal mixed with noise. The newsparsogram was applied to the mixed signal and the results are shown in Fig. 5. Through the analysis of Fig. 5, waveletpacket node (4, 4) with the largest sparsity value was used for further analysis. It should be noted that the sparsity atwavelet packet node (4, 11) has the second largest sparsity value among all of the wavelet packet nodes. This illustrates that

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200

400

600

Am

plitu

de

Frequency (Hz)

fm

2fm

3fm

Fig. 7. The power spectrum of the envelope of the signal extracted from wavelet packet node (4, 4) using the fast sparsogram.

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se v

aria

nce

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Fig. 8. The mixed signals with different noise variances (The mixed signal has two resonant frequency bands.).

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Fig. 9. The autocorrelation of the envelopes of the wavelet packet coefficients obtained by WPT in the case of the simulated signal mixed with heavy noise.

P.W. Tse, D. Wang / Mechanical Systems and Signal Processing 40 (2013) 499–519 507

the sparsogram is able to detect two resonant frequency bands at the same time. The frequency spectrum of the waveletpacket coefficients extracted from wavelet packet node (4, 4) is shown in Fig. 6(c). For comparison, the frequency spectrumof the mixed signal is plotted in Fig. 6(a), which indicates that the new sparsogram can provide the correct wavelet packetnode for further envelope analysis. Fig. 6(b) is the low frequency range of Fig. 6(a). Fig. 6(b) shows that the fault frequency of100 Hz is difficult to be directly identified by the fast Fourier transform (FFT) spectrum because the fault frequency of 100 Hzis overwhelmed by heavy noise.

The final result obtained by the proposed method is shown in Fig. 7, which demonstrates that the proposed methodeffectively detects fault frequency 100 Hz and its harmonic frequencies. The corresponding temporal signal of the signalshown in Fig. 7 is shown in Fig. 4(d), compared to the temporal mixed signal in Fig. 4(c). The intervals of 120 samples couldbe clearly observed.

Finally, a group of normally distributed random signals with a mean of 0 and different variances (from 0 to 1 with a steplength of 0.05) were added to Eq. (14). The mixed signals with different noise variances are plotted in Fig. 8. Obviously, thepotential periodic intervals are difficult to be identified when noise variances increase. The results obtained by the proposedmethod are shown in Figs. 9 and 10. In Fig. 9, it is easy to identify the periodic intervals, even though the mixed signals areoverwhelmed by heavy noises. The results in Fig. 10 show that the proposed method can detect the fault frequency 100 Hzand its harmonics at different noise variances.

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aria

nce

Frequency (Hz)

Fig. 10. The power spectra of the envelopes of the wavelet packet coefficients obtained by WPT for the simulated signal mixed with heavy noise(two resonant frequencies).

Tested bearing

Accelerometer

Fig. 11. An experiment motor and the faulty components of the tested bearings. (a) An experimental motor, (b) a tested bearing (SKF 1206 EKTN9), (c) adefect on an outer race, (d) An defect on an inner race and (e) a defect in a ball.

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Fig. 12. The collected bearing signals caused by (a) an outer race defect; (b) an inner race defect; (c) a ball defect; (d) outer race and inner race defects; andthe frequency spectra of the signals generated from (e) an outer race defect; (f) an inner race defect; (g) a ball defect and (h) outer race and inner racedefects.

P.W. Tse, D. Wang / Mechanical Systems and Signal Processing 40 (2013) 499–519508

3.2. Experimental validation through real bearing data

Several experiments were conducted in the Smart Engineering Asset Management Laboratory using an AC inductionmotor driven system. The speed of the motor was set to around 1400 rpm. The motor, the monitored bearing and each typeof artificially introduced fault are shown in Fig. 11. Four kinds of bearing defects, including an outer race defect, an inner racedefect, a rolling element defect and the combination of the outer race and inner race defects were introduced to thebearings. The sampling frequency was set to 80 kHz. The length of each sampled fault signal was 16,000 samples. Accordingto Eqs. (11)–(13), the bearing outer race fault characteristic frequency, the bearing inner race fault characteristic frequency,and the bearing ball spinning frequency were calculated as 136 Hz, 192 Hz, and 64 Hz, respectively.

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600

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plitu

deA

mpl

itude

Frequency (Hz)

Samples

x 104

Fig. 13. The collected bearing signal caused by a bearing with multiple faults, which included the outer race and inner race defects: (a) the temporal signaland (b) the frequency spectra.

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1.5

2

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x 10-3

Dep

th

Frequency (Hz)

Node (4, 3)

Fig. 14. The sparsogram for detecting an outer race defect.

P.W. Tse, D. Wang / Mechanical Systems and Signal Processing 40 (2013) 499–519 509

The bearing outer race fault signal, the bearing inner race fault signal and the bearing ball fault signal and theircorresponding frequency spectra are plotted in Fig. 12. An extra bearing vibration signal from a bearing that sufferedmultiple faults which included the outer race and inner race defects was collected. The temporal waveform of the raw multi-fault vibration signal is plotted in Fig. 13(a) and its corresponding frequency spectrum is plotted in Fig. 13(b). Note that theimpacts caused by outer race and inner race defects are hardly distinguishable from the temporal waveform. The expectedouter race fault characteristic frequency and inner race fault characteristic frequency are also difficult to be identified fromthe frequency spectrum.

The proposed method was applied to the bearing outer race fault signal shown in Fig. 12(a). The result obtained by thesparsogram is given in Fig. 14, where it is clear that wavelet packet node (4, 3) has the largest sparsity value. Therefore, it isoptimal to choose wavelet packet node (4, 3) for envelope analysis. The envelope of the wavelet packet coefficients extractedfromwavelet packet node (4, 3) is shown in Fig. 15(a). The potential periodic interval is equal to 591 samples (approximatelyequal to 80,000/136¼588 samples). Furthermore, its counterpart in the frequency domain as displayed in Fig. 15(b) shows

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2000

4000

6000

Ampl

itude

Am

plitu

de

Frequency (Hz)

Samples

fO

2fO

3fO4fO

5fO 6fO

Fig. 15. The results obtained by the sparsogram for detecting an outer race defect: (a) the envelope of the wavelet packet coefficients extracted fromwavelet packet node (4, 3); (b) the power spectrum of the envelope of the wavelet packet coefficients extracted from wavelet packet node (4, 3).

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1

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1

1.2

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1.6

1.8

2

2.2

x 10-3

Dep

th

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Node (4,2) Node (4,6)

Fig. 16. The sparsogram in this paper for detecting an inner race defect.

P.W. Tse, D. Wang / Mechanical Systems and Signal Processing 40 (2013) 499–519510

that the bearing outer race characteristic frequency and its harmonics are easily detected. The results shown in Fig. 15demonstrate that the sparsogram effectively extracts the most useful frequency band for bearing outer race fault diagnosis.

Second, to detect the fault signal shown in Fig. 12(b), the sparsogramwas used to select the optimal wavelet packet nodefor further analysis. The result obtained by the sparsogram is shown in Fig. 16. Wavelet packet node (4, 2) has the largestsparsity value and wavelet packet node (4, 6) has the second largest sparsity value. Thus, the wavelet packet coefficientsextracted from wavelet packet node (4, 2) using WPT was used for further envelope analysis. The final results are shown inFig. 17(a) and (b). Although it is not easy to find the periodic characteristics of the temporal signal shown in Fig. 17(a), thebearing inner race fault characteristic frequency and its first harmonic in Fig. 17(b) clearly report the bearing suffered frominner race localized faults. Therefore, the proposed method based on the sparsogram effectively extracts the bearing innerrace fault features.

Thirdly, the sparsogram was employed to analyze the bearing fault signal plotted in Fig. 12(c). The result obtained by thesparsogram is shown in Fig. 18, indicating that wavelet packet node (4, 5) has the largest sparsity value and that wavelet

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0

2

4

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2000

4000

6000

8000

Am

plitu

deA

mpl

itude

Frequency (Hz)

Samples

fI

2fI

Fig. 17. The results obtained by the proposed method for detecting an inner race defect: (a) the envelope of the wavelet packet coefficients extracted fromwavelet packet node (4, 2); (b) the power spectrum of the envelope of the wavelet packet coefficients extracted from wavelet packet node (4, 2).

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1

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6

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x 10-4

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th

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x 104

Fig. 18. The sparsogram in this paper for detecting a ball defect.

P.W. Tse, D. Wang / Mechanical Systems and Signal Processing 40 (2013) 499–519 511

packet node (4, 3) has the second largest sparsity value. The envelope of the signal extracted fromwavelet packet node (4, 5)is shown in Fig. 19(a), where the potential periodic intervals are found. In contrast, the power spectrum from the envelope ofthe signal extracted from wavelet packet node (4, 5) is shown in Fig. 19(b), where the ball spinning frequency and itsharmonics are detected.

Finally, the raw multi-fault signal plotted in Fig. 13(a) was analyzed by the sparsogram. The result obtained by thesparsogram is given in Fig. 20, where the most useful wavelet packet node (4, 9) has the largest sparsity value among allwavelet packet nodes on the sparsogram. In order to find fault signatures in a lower frequency band, the wavelet packetcoefficients extracted from wavelet packet node (4, 9) was demodulated for envelope analysis. The envelope signal fromwavelet packet node (4, 9) and its corresponding power spectrum are plotted in Fig. 21(a) and (b). Both outer race faultcharacteristic frequency and inner race fault characteristic frequency can be detected visually. The results illustrate that theproposed method is able to detect the multi-fault signatures.

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0

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6

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1000

2000

3000

4000

5000

Ampl

itude

Ampl

itude

Samples

fBS

2fBS

3fBS

5fBS

6fBS

Frequency (Hz)

4fBS

Fig. 19. The results obtained by the fast sparsogram for detecting a ball defect: (a) the envelope of the wavelet packet coefficients extracted from waveletpacket node (4, 5); (b) the power spectrum of the wavelet packet coefficients in Fig. 19(a).

0.5 1 1.5 2 2.5 3 3.5 4

1

2

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4

4

6

8

10

12

14x 10-4

Dep

th

Frequency (Hz)

Node (4,9)

x 104

Fig. 20. The sparsogram in this paper for detecting outer race and inner race defects.

P.W. Tse, D. Wang / Mechanical Systems and Signal Processing 40 (2013) 499–519512

In conclusion, the sparsogram-based bearing fault diagnosis method effectively detects different bearing defects,including an outer race defect, an inner race defect, a ball defect and multiple defects which included outer race and innerrace defects. Additionally, these significant wavelet packet nodes selected for the envelope analysis are able to fasten theconvergence of an optimal complex Morlet wavelet filter, which will be reported in Part 2 paper. In other words, thesesignificant wavelet packet nodes can be used to provide the proper initial center frequencies and bandwidths for the use ofgenetic algorithm that searches the best solution over a narrow solution zone. The details about these initial centerfrequencies and bandwidths provided by the selected wavelet packet nodes are illustrated as follow. Recalling thatthe frequency band of a specific wavelet packet node (j, p) is located in the frequency range ½p� Fs=2

Jþ1; ðpþ 1Þ � Fs=2Jþ1�,

the center frequency and bandwidth for the specific wavelet packet node are ½ð2pþ 1Þ � Fs=2Jþ2; Fs=2Jþ1�. Therefore, for thesimulated bearing fault signal, the initial center frequencies and bandwidths provided by wavelet packet nodes (4, 4) and(4, 11) are (1687.5 Hz, 375 Hz) and (4312.5 Hz, 375 Hz). The initial center frequency and bandwidth provided by waveletpacket node (4, 3) for the bearing outer race fault signal is (8750 Hz, 2500 Hz). For the bearing inner race fault signal, the

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100

200

300

400

Ampl

itude

Ampl

itude

Frequency (Hz)

Samples

fO2fO

2fIfI

3fO4fO

5fO 6fO

Fig. 21. The results obtained by the fast sparsogram for detecting outer race and inner race defects: (a) the envelope of the wavelet packet coefficientsextracted from wavelet packet node (4, 9); (b) the power spectrum of the wavelet packet coefficients shown in Fig. 21(a).

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th

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Am

plitu

de

Frequency (Hz)

Frequency (Hz)

Fig. 22. The results obtained by the kurtosis based wavelet packet decomposition method: (a) the paving of the kurtosis values of the envelopes of thewavelet packet coefficients for the simulated signal mixed with heavy noise; (b) the power spectrum of the envelope of the wavelet packet coefficientsextracted from wavelet packet node (4, 8).

P.W. Tse, D. Wang / Mechanical Systems and Signal Processing 40 (2013) 499–519 513

1000 2000 3000 4000 5000 6000

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0.84

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1000

2000

Dep

th

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Node (4,7)

Frequency (Hz)

Am

plitu

de

Fig. 23. The results obtained by the smoothness index based wavelet packet decomposition method: (a) the paving of the smoothness index values of theenvelopes of the wavelet packet coefficients for the simulated signal mixed with heavy noise; (b) the power spectrum of the envelope of the wavelet packetcoefficients extracted from wavelet packet node (4, 7).

P.W. Tse, D. Wang / Mechanical Systems and Signal Processing 40 (2013) 499–519514

initial center frequencies and bandwidths provided by wavelet packet nodes (4, 2) and (4, 6) are (6250 Hz, 2500 Hz) and(16,250 Hz, 2500 Hz). For the bearing ball fault signal, the initial center frequencies and bandwidths provided by waveletpacket nodes (4, 3) and (4, 5) are (8750 Hz, 2500 Hz) and (13,750 Hz, 2500 Hz). For the bearing outer race and inner racefault signal, the initial center frequency and bandwidth provided by wavelet packet node (4, 9) are (23,750 Hz, 2500 Hz).

3.3. Comparisons with other popular methods

For the purpose of comparing the effectiveness of the fast sparsogram against that contributed by other popular metricbased wavelet packet decomposition methods, such as the kurtosis [25], the smoothness index [26] and Shannon entropy [27],two previously used fault signals are used in this comparison study. The first signal is the simulated signal mixed byheavy noise as shown in Fig. 4(c). The second signal is obtained from the previously mentioned bearing ball faulty signal asshown in Fig. 12(c). The bearing ball fault signal is selected because it is a typical localized fault and contains morecomplexity than that generated by the outer race fault and the inner race fault. All compared methods are used to quantifythe envelopes of the wavelet packet coefficients extracted from wavelet packet nodes at different wavelet packetdecomposition depths. The use of the above three popular methods for the quantification of the wavelet packet coefficientsobtained by binary wavelet packet transform is capable of distinguishing different bearing health status. Kurtosis measuresthe peakedness of the probability distribution of a bearing fault signal. A large kurtosis value indicates the cyclicimpulsiveness of bearing fault signals. Some classic examples concerning the use of kurtosis for designing filters are thefast Kurtogram proposed by Antoni [4] and the improved Kurtogram proposed by Lei et al. [25]. The definition of thesmoothness index is the ratio of the geometric mean to the arithmetic mean of a positive signal [26]. A small smoothnessindex value indicates the occurrence of bearing fault impulses. Bozchalooi and Liang [26,28] used the smoothness index to

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th

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Node (4,8)

Frequency (Hz)

Am

plitu

de

Fig. 24. The results obtained by the Shannon entropy based wavelet packet decomposition method: (a) the paving of the Shannon entropy values of theenvelopes of the wavelet packet coefficients for the simulated signal mixed with heavy noise; (b) the power spectrum of the envelope of the wavelet packetcoefficients extracted from wavelet packet node (4, 8).

P.W. Tse, D. Wang / Mechanical Systems and Signal Processing 40 (2013) 499–519 515

quantify the envelope of the signal filtered by the complex Morlet wavelet for deciding the parameters of the optimalcomplex Morlet wavelet. Su et al. [27] suggested that Shannon entropy could be used to select the parameters of the Morletwavelet for bearing fault diagnosis. A small Shannon entropy value indicates the occurrence of bearing fault impulsivesignals.

The paving of the kurtosis values of the envelopes of the wavelet packet coefficients obtained by binary wavelet packettransform for the simulated signal mixed with heavy noise is plotted in Fig. 22(a), where wavelet packet node (4, 8) has thelargest kurtosis value which illustrates wavelet packet node (4, 8) contains the most useful bearing fault signatures.However, in Fig. 22(b), the power spectrum of the enveloped signal extracted from wavelet packet node (4, 8) does notprovide any signatures concerning the modulating frequency of 100 Hz. It means that this method fails to detect thesimulated bearing fault signatures. Actually, the paving of the kurtosis values calculated from the wavelet packet nodes isthe improved Kurtogram recently proposed by Lei et al. [25]. The paving of the smoothness index values of the envelopes ofthe wavelet packet coefficients obtained by binary wavelet packet transform for the same simulated signal mixed withheavy noise is depicted in Fig. 23(a). In this case, wavelet packet node (4, 7) is the optimal node among all wavelet packetnodes. However, in Fig. 23(b), the power spectrum of the envelope of the wavelet packet coefficients extracted fromwaveletpacket node (4, 7) does not exhibit the modulating frequency of 100 Hz and its harmonics. The paving of the Shannonentropy values of the envelopes of the wavelet packet coefficients for the simulated bearing fault signal accompanied withheavy noisy is plotted in Fig. 24(a), where wavelet packet node (4, 8) is selected as the optimal node. In Fig. 24(b), the powerspectrum of the envelope of the wavelet packet coefficients extracted from wavelet packet node (4, 8) illustrates that theShannon entropy based wavelet packet paving is ineffective in indicating one of the resonant frequency bands. Comparedwith the results shown in Figs. 22–24, the results from the fast sparsogram that are plotted in Figs. 5 and 7 illustrate that thefast sparsogram not only can detect the modulating frequency of 100 Hz but also can show the exact locations of two

P.W. Tse, D. Wang / Mechanical Systems and Signal Processing 40 (2013) 499–519516

resonant frequency bands. Hence, this comparison study demonstrates that the fast sparsogram is more effective indetecting weak simulated bearing fault signal that have been overwhelmed by heavy noise.

To further verify the effectiveness of sparsogram, the comparison study was extended to real bearing ball fault signal asplotted in Fig. 12(c). The results obtained from the sparsogram as shown in Fig. 19(b) are used to compare the resultsobtained from the three popular methods. In Fig. 25(a), the paving of the kurtosis values of the envelopes of the waveletpacket coefficients indicates that wavelet packet node (4, 2) contains the optimal bearing fault signatures. The powerspectrum of the envelope of the signal extracted from wavelet packet node (4, 2) is plotted in Fig. 25(b), where the ballspinning frequency and its harmonics can be observed. It should be noted that the ball spinning frequency and its harmonicsshown in Fig. 25(b) are not as remarkable as those shown in Fig. 19(b). The paving of the smoothness index values of theenvelopes of the wavelet packet coefficients at different wavelet packet nodes and different wavelet packet decompositiondepths for the bearing ball fault signal is shown in Fig. 26(a). In Fig. 26(b), the power spectrum of the envelope of the signalextracted from wavelet packet node (4, 3) indicates the existence of the bearing ball localized faults. Although the ballspinning frequency and its harmonics can be seen in Fig. 26(b), some of the harmonics, such as the 4fBS, is not as obvious asthat shown in Fig. 19(b). The paving of the Shannon entropy values of the envelopes of the wavelet packet coefficients forprocessing the bearing ball fault signal is plotted in Fig. 27(a), where wavelet packet node (4, 5) is selected as the optimalnode. In Fig. 27(b), the power spectrum of the envelope signal extracted from wavelet packet node (4, 5) is the same as thatshown in Fig. 19(b) because their results come from the same node.

Through the above analyses, it is concluded that the sparsogram can generate better visual images for the operatorsto detect bearing faults, especially for the complex ball fault signal and the signals that are overwhelmed with heavy noise.

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th

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Node (4, 2)

fBS 4fBS

5fBSAm

plitu

de

Frequency (Hz)

2fBS

3fBS

Fig. 25. The results obtained by the kurtosis based wavelet packet decomposition method: (a) the paving of the kurtosis values of the envelopes of thewavelet packet coefficients for processing the real bearing ball fault signal and (b) the power spectrum of the envelope of the wavelet packet coefficientsextracted from wavelet packet node (4, 2).

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th

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Frequency (Hz)

Am

plitu

de

5fBS

3fBS

2fBS

fBS

x 104Frequency (Hz)

Fig. 26. The results obtained by the smoothness index based wavelet packet decomposition method: (a) the paving of the smoothness index values of theenvelopes of the wavelet packet coefficients for processing the real bearing ball fault signal and (b) the power spectrum of the envelope of the waveletpacket coefficients extracted from wavelet packet node (4, 3).

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The comparisons of the performance for the sparsogram, the improved kurtogram, the smoothness index based WPT andthe Shannon entropy based WPT are summarized in Table 1. The results demonstrate that the sparsogram has betterperformance than that provided by the three popular methods, particularly for the case when the bearing fault signals havebeen corrupted by heavy noise.

4. Conclusion

This paper proposed a new and fast method, called sparsogram, for rolling element bearing fault detection. First, thewavelet packet coefficients at different wavelet packet depths and different wavelet packet nodes are obtained by usingbinary wavelet packet transform. Second, the sparsity values of the power spectra from the envelopes of these waveletpacket coefficients are measured. The sparsogram is then constructed by arranging these sparsity values in the form of atwo-dimensional diagram. From the sparsogram, the wavelet packet node that has the largest sparsity value can beidentified. This specific wavelet packet node should contain more bearing fault signatures that other nodes that have smallersparsity values. Hence, the sparsogram helps to find the optimal node for better bearing fault diagnosis. To validate theability of sparsogram, two studies, which included the simulated bearing fault signal mixed with heavy noise and the realbearing fault signals were investigated. From the results generated by the simulated case, they show that the sparsogram isa fast and effective method that is able to choose a useful fault frequency band that contains most bearing fault signatures.Moreover, the sparsogram is able to detect simulated fault signal even at a very low signal-to-noise ratio. In the cases oftesting with real bearings, the sparsogram could effectively detect bearings that had different types of fault as well as

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fBS2fBS

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3fBS

4fBS

5fBS6fBS

Fig. 27. The results obtained by the Shannon entropy based wavelet packet decomposition method: (a) the paving of the Shannon entropy values of theenvelopes of the wavelet packet coefficients for processing the real bearing ball fault signal and (b) the power spectrum of the envelope of the waveletpacket coefficients extracted from wavelet packet node (4, 5).

Table 1The performance comparisons of the sparsogram, the improved Kurtogram, the smoothness index based WPT and the Shannon entropy based WPT (Note:Case 1 is about the simulated signal (two resonant frequency bands) corrupted by heavy noise; Case 2 is about the real laboratorial ball fault signal.).

Effective in detecting bearing faults? Visual inspection ability Best wavelet packet node

Case 1 Case 2 Case 1 Case 2 Case 1 Case 2

Sparsogram Yes Yes High High (4, 4) (4, 5)Improved Kurtogram No Yes Null Medium (4, 8) (4, 2)Smoothness index based WPT No Yes Null Medium (4, 7) (4, 3)Shannon entropy based WPT No Yes Null High (4, 8) (4, 5)

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multiple faults. Comparison studies with three popular methods were also conducted. The results show that the sparsogramprovides better visual inspection than the other three methods commonly used for bearing fault detection.

The sparsogram may not work as expected if the bearing resonant frequency band is located in the overlappingfrequency band of a wavelet filter bank, the bearing resonant frequency band must be split into two adjacent frequencybands, resulting in the reduction of the bearing fault characteristics. Hence, an optimal wavelet filter based on the use of thesparsity measurement is required to be developed. A successive part of this Part 1 paper has been prepared to describe amethod in developing an optimal Morlet wavelet filter for bearing fault diagnosis. The detailed methodology is presented in

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a successive paper called ‘Part 2 – the automatic selection of an optimal wavelet filter and its enhancement by the newsparsogram for bearing fault detection’.

Acknowledgments

The work described in this paper was fully supported by a grant from the Research Grants Council of the Hong KongSpecial Administrative Region, China (Project no. CityU 122011) and a grant from City University of Hong Kong (Project no.7008187).

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