ORIGINAL ARTICLE

Rolling element bearing faults diagnosis basedon autocorrelation of optimized: wavelet de-noisingtechnique

Khalid F. Al-Raheem & Asok Roy &

K. P. Ramachandran & D. K. Harrison & Steven Grainger

Received: 18 February 2007 /Accepted: 20 November 2007 / Published online: 17 January 2008# Springer-Verlag London Limited 2007

Abstract Machinery failure diagnosis is an importantcomponent of the condition based maintenance (CBM)activities for most engineering systems. Rolling elementbearings are the most common cause of rotating machineryfailure. The existence of the amplitude modulation andnoises in the faulty bearing vibration signal presentchallenges to effective fault detection method. The wavelettransform has been widely used in signal de-noising, due toits extraordinary time-frequency representation capability.In this paper, a new technique for rolling element bearingfault diagnosis based on the autocorrelation of wavelet de-noised vibration signal is applied. The wavelet basefunction has been derived from the bearing impulseresponse. To enhance the fault detection process thewavelet shape parameters (damping factor and centerfrequency) are optimized based on kurtosis maximization

criteria. The results show the effectiveness of the proposedtechnique in revealing the bearing fault impulses and itsperiodicity for both simulated and real rolling bearingvibration signals.

Keywords Bearing fault detection .Wavelet de-noising .

Impulse-response wavelet . Kurtosis maximization .

Autocorrelation

1 Introduction

Every time the rolling element hits a defect in the raceway,an impulse of short duration is generated, which in turnexcites the bearing system resonance frequencies. There-fore, the overall vibration signal measured on the bearinghousing shows a pattern consisting of succession ofoscillating bursts dominated by the major bearing systemresonance frequency. The duration of the impulse isextremely short compared with the interval betweenimpulses, and so its energy is distributed at a very lowlevel over a wide range of frequency and, hence, can beeasily masked by noise and low frequency effects. Thesesimpulses will occur with a frequency determined by thevelocity of the rolling element, the location of the defectand the bearing geometry and denoted as bearing charac-teristic frequencies (BCF); see the appendix.

The rolling elements experience some slippage as therolling elements enter and leave the bearing load zone. As aconsequence, the occurrence of the impacts never repro-duce exactly at the same position from one cycle to another.Moreover when the position of the defect is moving withrespect to the load distribution of the bearing, the series ofimpulses is modulated in amplitude. However, the period-icity and the amplitude of the impulses experience a certain

Int J Adv Manuf Technol (2009) 40:393–402DOI 10.1007/s00170-007-1330-3

K. F. Al-Raheem (*) :K. P. RamachandranDepartment of Mechanical and Industrial Eng.,Caledonian College of Eng.,Muscat, Omane-mail: khalid@caledonian.edu.om

K. P. Ramachandrane-mail: ramkp@caledonian.edu.om

A. Roy :D. K. Harrison : S. GraingerSchool of Engineering Science and Design,Glasgow Caledonian University,Glasgow, Scotland, UK

A. Roye-mail: A.Roy@gcal.ac.uk

D. K. Harrisone-mail: dha2@gcal.ac.uk

S. Graingere-mail: S.grainger@gcal.ac.uk

degree of randomness [1–4]. In such case, the signal is notstrictly periodic, but can be considered as cyclo-stationary(periodically time-varying statistics), then the cyclic sec-ond-order statistics (such as cyclic-autocorrelation andcyclic spectral density) are suited to demodulate the signaland extract the fault feature [5–7]. All these make thebearing defects very difficult to detect by conventional fastFourier transform (FFT) spectrum analysis, which assumesthat the analyzed signal to be strictly periodic.

The wavelet transform provides powerful multi-resolutionanalysis in both time and frequency domain, therebybecoming a favored tool to extract the transitory features ofnon-stationary vibration signal produced by the faulty bearing[8–14]. The wavelet analysis results in a series of waveletcoefficients, which indicate how close the signal is to theparticular wavelet. In order to extract the fault features of thesignal more effectively appropriate wavelet base functionshould be selected [15–21].

The wavelet de-noising technique included of decom-poses the signal using wavelet transform, threshold theresulting coefficients to eliminate the redundant informationand further enhance the interested spectral features of thesignal, then reconstruct the signal from the thresholdwavelet coefficients using inverse wavelet transform.Wavelet de-noising using a Morlet wavelet as a basefunction has been used to extract the impulses for bearingand gear faults detection by J. Lin et al. in [22]. Y. Shao andK. Nezu [23] combined the wavelet de-noising withadaptive noise canceling filter to improve the signal-to-noise ratio when the signal is contaminated by noise forincipient bearing fault detection. H. Qiu et al. [24]optimized the Morlet wavelet shape factor using theminimal Shannon entropy criteria when applied as a basefunction in wavelet de-noising for bearing fault diagnosis.S. Abbasion et al. [25] proposed discrete Meyer wavelet asbase function for signal de-noising and bearing fault

a

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16-5

-4

-3

-2

-1

0

1

2

3

4

Time (s)

Acc

eler

atio

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m.s-2)

Acc

eler

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n (

m.s-2)

Acc

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m.s-2)

Acc

eler

atio

n (

m.s-2)

c

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

X: 0.04908Y: 0.2558

X: 0.05883Y: 0.208

X: 0.03958Y: -0.09365

Time (s)

d

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16-0.4

-0.3

-0.2

-0.1

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0.1

0.2

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X: 0.09392Y: -0.1905 X: 0.1

Y: -0.2162

X: 0.1306Y: 0.1915

Time (s)

-0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

X: -0.00975Y: 0.5478

X: 0.00975Y: 0.5478

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Co

rrel

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b

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16-6

-4

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f

-0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05-0.5

0

0.5

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X: -0.006167Y: 0.1692

X: 0.006167Y: 0.1692

X: -0.03083Y: 0.4573

Delay (s)

Co

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atio

n c

oef

fici

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Fig. 1 The simulated vibrationsignal, the corresponding wave-let de-noised signal and, theauto-correlation function Rx(τ)for bearing with outer-race fault(a, b, and c) , inner-race fault (d,e and f)

394 Int J Adv Manuf Technol (2009) 40:393–402

classification using a support vector machine (SVM). D.Giaouris and J.W. Finch [26] applied the wavelet de-noising of the electrical motor current signal for faultdetection. Z. K. Peng and F. L. Chu gave a comprehensive

overview to the wavelet de-noising for mechanical faultdiagnosis. A number of threshold methods to eliminate theeffects of the signal noise from the resulting waveletcoefficients are applied in [28, 29].

a

-5 -4 -3 -2 -1 0 1 2 3 4 5-0.8

-0.6

-0.4

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Am

plit

ud

e

b

0 5 10 15 20 25 30 35 40 45 500

0.2

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0.6

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1

1.2x 10

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Frequency (Hz)

Po

wer

Sp

ectr

um

Fig. 2 (a) the impulse wavelettime waveform, (b) its FFT-spectrum

a

b

c

Fig. 3 (a) The simulated noisesignal (kurtosis=3.0843),(b) The overall vibration signal(noise and impulses) (kurtosis=7.7644), and (c) The pure faultimpulses (kurtosis=8.5312),with the corresponding intensitydistribution curve for bearingwith outer-race fault

Int J Adv Manuf Technol (2009) 40:393–402 395

In this paper, a new technique based on the waveletde-noising method for rolling bearing fault detection hasbeen developed and tested on both simulated and realbearing vibration signals. To enhance the generatedwavelet coefficients related to the bearing fault impulses,the wavelet base function has been constructed based onthe impulse response of the bearing system. Moreover, thewavelet shape parameters are optimized using maximumkurtosis criteria.

The remaining sections of the paper are as follows: Inthe next section the vibration model for rolling bearing withouter and inner-races fault is derived. In Sect. 3 theprocedures of the proposed approach is set up. Theimplementations of the proposed approach for detection oflocalized ball bearing defects for both simulated and actual

bearing vibration signals are presented in Sect. 4. Finally,the conclusions are given in Sect. 5.

2 Vibration model for rolling element bearing localizeddefects

Every time the rolling element strikes a defect in theraceway or every time a defect in the rolling element hitsthe raceway, a force impulse of short duration is produced,which in turn excites the natural frequencies of the bearingparts and housing structure. The structure resonance in thesystem acts as an amplifier of low energy impacts.Therefore, the overall vibration signal measured on thebearing shows a pattern consisting of a succession of

a b

cFig. 4 The optimization of the wavelet parameters based on maximization of the kurtosis value for (a) simulated vibration signal, (b) theexperimentally collected signal, and (c) the CWRU signal, for outer-race fault bearing

396 Int J Adv Manuf Technol (2009) 40:393–402

oscillating bursts dominated by the major resonancefrequency of the structure.

The response of the bearing structure as an under-damped second-order mass-spring-damper system to asingle impulse force is given by the following [30]:

S tð Þ ¼ Ce� xffiffiffiffiffiffi

1�x2p wd t

sin wdtð Þ ð1Þwhere ξ is the damping ratio, ωd is the damped naturalfrequency of the bearing structure, and C is an amplitudescaling factor.

As the shaft rotates, this process occurs periodicallyevery time a defect hits another part of the bearing, and itsrate of occurrence is equal to one of the BCF. In reality,there is a slight random fluctuation in the spacing betweenimpulses because the load angle on each rolling elementchanges as the rolling element passes through the loadzone. Furthermore, the amplitude of the impulse responsewill be modulated as a result of the passage of the faultthrough the load zone:

x tð Þ ¼Xi

AiS t � Tð Þ þ n tð Þ ð2Þ

where S(t-Ti) is the waveform generated by the ith impact atthe time Ti, and Ti=iT+τi, where T is the average timebetween two impacts, and τi describe the random slips ofthe rolling elements. Ai is the time varying amplitude-demodulation, and n(t) is an additive background noisewhich takes into account the effects of the other vibrationsin the bearing structure.

Figure 1a and b show the acceleration signals (d2x(t)/dt2) generated by the model in Eq. 2 with random slip(τ) of 10% of the period T and signal-to-noise ratio of0.6 dB for outer-race and inner-race bearing faults,respectively.

3 Wavelet de-noising technique

The wavelet transform (WT) is the inner product of a timedomain signal with the translated and dialed wavelet-basefunction. The wavelet transform resulting coefficientsreflect the correlation between the signal and the selectedwavelet-base function. Therefore, to increase the amplitudeof the generated wavelet coefficients related to the faultimpulses, and to enhance the fault detection process, theselected wavelet-base function should be similar in charac-teristics to the bearing impulse response generated bypresence of bearing incipient fault, Eq. 1. Based on that,the proposed wavelet-base function is denoted as impulse-response wavelet and given by

y tð Þ ¼ A e� bffiffiffiffiffiffi

1�b2p wct

sin wctð Þ ð3Þ

a

b

c

Delay (s)

Delay (s)

Delay (s)

impulsenon

morcor1

Fig. 5 The autocorrelation function of the wavelet de-noised outer-race fault signal using (a) optimized impulse-wavelet, (b) non-optimized impulse-wavelet, and (c) Morlet-wavelet

Int J Adv Manuf Technol (2009) 40:393–402 397

where β is the damping factor that control the decay rate ofthe exponential envelop in the time, hence regulating theresolution of the wavelet, simultaneously corresponds to thefrequency bandwidth of the wavelet in the frequencydomain, ωc determining the number of significant oscil-lations of the wavelet in the time domain and correspond tothe wavelet centre frequency in frequency domain, and A isan arbitrary scaling factor. Figure 2 shows the proposedwavelet and its power spectrum.

The proposed wavelet satisfy the admissibility condition,

Cg ¼Z1�1

b= fð Þj j2f

df < 1 ð4Þ

where Cg is the admissibility constant and, y^(f) is theFourier transform of y (t). This implies that the wavelet hasno zero frequency component, y^(0)=0 or, the wavelet y(t)must have a zero mean [31].

The proposed wavelet de-noising technique consists ofthe following steps:

1 Optimize the wavelet shape parameters (β and ωc)based on maximization kurtosis of the signal-waveletinner product.

It is possible to find optimal values of β and ωc for agiven vibration signal by adjusting the time-frequencyresolution of the Impulse wavelet to the decay rate andfrequency of impulses to be extracted. Kurtosis is anindicator that reflects the “peakiness” of a signal, which isa property of the impulses and also it measures thedivergence from a fundamental Gaussian distribution. Ahigh kurtosis value indicates high impulsive content of thesignal with more sharpness in the signal intensity distribu-tion. Figure 3 shows the kurtosis value and the intensitydistribution for a white noise signal, pure impulsive signal,and impulsive signal mixed with noise.

The objective of the impulse wavelet shape optimizationprocess is to find out the wavelet shape parameters (β andωc), which maximize the kurtosis of the wavelet transformoutput:

Optimal b;wcð Þ ¼ max �PNn¼1

WT 4 x tð Þ;yb;wctð Þ� �

PNn¼1

WT 2 x tð Þ;yb;wctð Þ� �� �2

2666437775 ð5Þ

a b cFig. 6 (a) The collected vibration signal, (b) The corresponding wavelet de-noised signal, and (c) The auto-correlation function, for bearing withouter-race fault at shaft rotational speed of 983.887 rpm

a b cFig. 7 (a) The collected vibration signal, (b) Corresponding wavelet de-noised signal, and (c) Auto-correlation function, for bearing with outer-race fault at shaft rotational speed of 2080.28 rpm

398 Int J Adv Manuf Technol (2009) 40:393–402

2 Apply the wavelet de-noising technique:

a- Perform a wavelet transform for the bearingvibration signal x(t) using the optimized wavelet,

WT x tð Þ; a; bf g ¼ =a;b � x tð Þ� �¼ 1ffiffiffi

ap

Zx tð ÞΨ*a;b tð Þdt ð6Þ

where <.,.> indicates the inner product, the super-script asterisk ‘*’ stands for the complex conjugate.The ya, b is a family of daughter wavelets derivedfrom the mother wavelet y(t) by continuouslyvarying the scale factor a and the translationparameter b. The factor 1ffiffi

ap is used to ensure energy

preservation.b- Shrink the wavelet coefficients expressed in Eq.5

by soft thresholding:

WTsoft ¼ 0sign ðWTÞ ðWT � thrÞ

WTj j < thrWTj j > thr

�ð7Þ

using soft-threshold function (thr) proposed by [29]yields

thr ¼ e �Max WT a;bð Þj jð Þx½ � � e� Max WT a;bð Þj jð Þx½ � ð8Þ

where ξ > 0 is parameter governing the shape ofthe threshold function.

c- Perform the inverse wavelet transform to recon-struct the signal using the shrunken waveletcoefficients.

x*tð Þ ¼ C�1

g

Z1�1

WTsoft a; tð Þ da

a3=2ð9Þ

3 Evaluate the auto-correlation function Rx (τ) for the de-noised signal x*(t) to estimate the periodicity of theextracted impulses,

Rx Cð Þ ¼ E x*tð Þ � x* t þ Cð Þ

� �ð10Þ

where τ is the time lag, and E [ ] denotes ensembleaverage value of the quantity in square brackets.

4 Wavelet de-noising technique for rolling bearing faultdetection

To demonstrate the performance of the proposed approach,this section presents several application examples for thedetection of localized bearing defects. In all the examples,the impulse wavelet has been used as a wavelet base-function. The wavelet parameters (damping factor andcentre frequency) are optimized based on maximizing thekurtosis value for the wavelet coefficients as shown inFig. 4.

To evaluate the performance of the proposed method,the autocorrelation functions of the optimized impulsewavelet, impulse wavelet with non-optimized parameters,and the widely used Morlet wavelet are carried out andshown in Fig. 5. The comparison of Fig. 5a,b and c shows

a b cFig. 8 (a) The collected vibration signal, (b) Corresponding wavelet de-noised signal, and (c) Auto-correlation function, for rolling with outer-race fault at shaft rotational speed of 3541.11 rpm

Table 1 The calculated and extracted BCFs at different shaftrotational speed

Shaft speed(rpm)

Calculated BCF(Hz)

Periodextracted (s)

BCF extracted(Hz)

983.887 50.32 0.020310 49.2362080.28 106.4 0.009297 107.5613541.11 181.12 0.005391 185.493

Int J Adv Manuf Technol (2009) 40:393–402 399

the increased effectiveness of the optimized impulsewavelet over non-optimized impulse and Morlet waveletfor extraction of the bearing fault impulses and period-icity. Consequently the performance of the bearing faultdiagnosis process has been increased using the proposedtechnique.

4.1 Simulated vibration data

For a rolling element bearing with pitch diameter of51.16 mm, ball diameter of 11.9 mm, with eight rollingelements and 0° contact angle, the calculated BCFs (see theappendix) for shaft rotational speed of 1,797 rpm are

a bFig. 9 The CWRU collected vibration signal, corresponding wavelet de-noised signal and auto-correlation function, respectively, for a bearingwith (a) outer-race fault, and (b) inner-race fault

400 Int J Adv Manuf Technol (2009) 40:393–402

107.36 Hz and 162.18 Hz for outer and inner-race fault,respectively. Figure 1a and b show the time domainwaveform of the simulated signals for rolling bearing withouter and inner-race faults based on the model described inSect. 2. The results of the wavelet de-noising method(wavelet transform, shrink the wavelet coefficients and theinverse wavelet transform) for rolling bearing with outerand inner race faults using the optimized impulse waveletare displayed in Fig. 1c and d, respectively. The resultsshow that the signal noise has been diminished and theimpulses generated by the faulty bearing are easy toidentify in the wavelet de-noised signal. The impulseperiodicity of 0.00975 s (fo=102.564 Hz) for outer-racefault and 0.006167 s (fi=162.153 Hz) for inner-race faultare effectively extracted through the auto-correlation of thede-noised signal, Fig. 1e and f, which are exactly matchingthe theoretical estimation of the BCF.

4.2 Experimental vibration data

A B&K 752A12 piezoelectric accelerometer was used tocollect the vibration signals for an outer race defective,deep groove, ball bearing (with same simulated specifica-tions) at different shaft rotational speeds. The vibrationsignals were transferred to the PC through a B&Kcontroller module type 7536 with data sampling frequencyof 12.8 kHz. Based on the bearing parameters, thecalculated outer race fault characteristic frequency is0.05115 times the shaft rotational speed (rpm).

Figures 6, 7, 8 show the application of the proposedwavelet de-noising technique for rolling bearing with outer-race fault at different shaft rotational speed. The bearingfault impulses and their periodicity are easily defined in thewavelet de-noised signal and the de-noised autocorrelationfunction, respectively. The comparison of Figs. 6c, 7c and8c shows the sensitivity of the proposed de-noisingtechnique to the variation of the BCF as a result ofvariation in the of shaft rotational speed as listed in Table 1.

4.3 CWRU vibration data

We use data given by the Case Western Reserve University(CWRU) website [32] for rolling bearings seeded withouter and inner race faults using electro discharge machin-ing (EDM). The calculated defect frequencies are 3.5848and 5.4152 times the shaft rotational speed (Hz) for outerand inner race fault, respectively. At shaft rotational speedof 1797 rpm the calculated BCF are 107.36 Hz for outer-race fault and, 162.185 Hz for inner-race fault. The timecourse of the vibration signal for bearings with outer andinner race faults, the corresponding wavelet de-noisedsignal and the auto-correlation function are depicted inFig. 9a and b, respectively. The autocorrelation functions of

the de-noised signal reveal a periodicity of 0.009333 s (fo=107.14 Hz) and 0.006167 s (fi=162.153 Hz) for outer andinner race fault, respectively, which are very close to thecalculated BCF.

5 Conclusions

A new approach for rolling bearing fault diagnosis based onwavelet de-noising technique with wavelet-base functionderived from the impulse response of the bearing system ispresented. Wavelet shape parameters have been optimizedusing maximum kurtosis criteria. The results for bothsimulated as well as actual bearing vibration signals showthe effectiveness of the proposed approach to extract therolling bearing fault impulses buried in the noisy vibrationsignal, and evaluate its periodicity using auto-correlationfunction of the wavelet de-noised vibration signal.

Appendix

Fault bearing characteristic frequencies (BCF)

Each bearing element has its own characteristic frequencyof defect. Those frequencies can be calculated from thekinematics relation, i.e., the geometry of the bearing and itsrotating speed. For a bearing with a stationary outer race,the above defect characteristic frequencies can be obtainedas follows:

Characteristic frequency of the outer-race:

fo in:Hzð Þ ¼ 0:5zf 1� d

Dcosα

ð1Þ

Characteristic frequency of the inner race:

fi in:Hzð Þ ¼ 0:5zf 1þ d

Dcosα

ð2Þ

Characteristic frequency of the rollers:

fr in:Hzð Þ ¼ fD

d1� d

Dcosα

2" #

ð3Þ

Characteristic frequency of the cage:

fC ¼ f

21� d

Dcos αð Þ

� �ð4Þ

Int J Adv Manuf Technol (2009) 40:393–402 401

where z is the number of rollers, d is the diameter of therollers, D is the pitch diameter, α is the contact angle, and fis the rotating speed of shaft.

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