Int J Adv Manuf Technol-Khalid

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ORIGINAL ARTICLE

Rolling element bearing faults diagnosis based

on autocorrelation of optimized: wavelet de-noising

technique

Khalid F. Al-Raheem & Asok Roy &

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

Received: 18 February 2007 /Accepted: 20 November 2007# Springer-Verlag London Limited 2007

Abstract Machinery failure diagnosis is an important

component of the condition based maintenance (CBM)activities for most engineering systems. Rolling element

bearings are the most common cause of rotating machinery

failure. The existence of the amplitude modulation and

noises in the faulty bearing vibration signal present

challenges to effective fault detection method. The wavelet

transform has been widely used in signal de-noising, due to

its extraordinary time-frequency representation capability.

In this paper, a new technique for rolling element bearing

fault diagnosis based on the autocorrelation of wavelet de-

noised vibration signal is applied. The wavelet base

function has been derived from the bearing impulse

response. To enhance the fault detection process the

wavelet shape parameters (damping factor and center

frequency) are optimized based on kurtosis maximization

criteria. The results show the effectiveness of the proposed

technique in revealing the bearing fault impulses and itsperiodicity for both simulated and real rolling bearing

vibration 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 turn

excites the bearing system resonance frequencies. There-

fore, the overall vibration signal measured on the bearing

housing shows a pattern consisting of succession of

oscillating bursts dominated by the major bearing system

resonance frequency. The duration of the impulse is

extremely short compared with the interval between

impulses, and so its energy is distributed at a very low

level over a wide range of frequency and, hence, can be

easily masked by noise and low frequency effects. Theses

impulses will occur with a frequency determined by the

velocity of the rolling element, the location of the defect

and the bearing geometry and denoted as bearing charac-

teristic frequencies (BCF); see the appendix.

The rolling elements experience some slippage as the

rolling elements enter and leave the bearing load zone. As a

consequence, 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 with

respect to the load distribution of the bearing, the series of

impulses is modulated in amplitude. However, the period-

icity and the amplitude of the impulses experience a certain

Int J Adv Manuf Technol

DOI 10.1007/s00170-007-1330-3

K. F. Al-Raheem (*) :K. P. RamachandranDepartment of Mechanical and Industrial Eng.,

Caledonian College of Eng.,

Muscat, Oman

e-mail: [email protected]

K. P. Ramachandran


A. Roy :D. K. Harrison : S. GraingerSchool of Engineering Science and Design,

Glasgow Caledonian University,

Glasgow, Scotland, UK

A. Roy


D. K. Harrison


S. Grainger



2/10

degree of randomness [14]. In such case, the signal is not

strictly periodic, but can be considered as cyclo-stationary

(periodically time-varying statistics), then the cyclic sec-

ond-order statistics (such as cyclic-autocorrelation and

cyclic spectral density) are suited to demodulate the signal

and extract the fault feature [57]. All these make the

bearing defects very difficult to detect by conventional fast

Fourier transform (FFT) spectrum analysis, which assumesthat the analyzed signal to be strictly periodic.

The wavelet transform provides powerful multi-resolution

analysis in both time and frequency domain, thereby

becoming a favored tool to extract the transitory features of

non-stationary vibration signal produced by the faulty bearing

[814]. The wavelet analysis results in a series of wavelet

coefficients, which indicate how close the signal is to the

particular wavelet. In order to extract the fault features of the

signal more effectively appropriate wavelet base function

should be selected [1521].

The wavelet de-noising technique included of decom-

poses the signal using wavelet transform, threshold the

resulting coefficients to eliminate the redundant information

and further enhance the interested spectral features of the

signal, then reconstruct the signal from the threshold

wavelet coefficients using inverse wavelet transform.

Wavelet de-noising using a Morlet wavelet as a base

function has been used to extract the impulses for bearingand gear faults detection by J. Lin et al. in [ 22]. Y. Shao and

K. Nezu [23] combined the wavelet de-noising with

adaptive noise canceling filter to improve the signal-to-

noise ratio when the signal is contaminated by noise for

incipient bearing fault detection. H. Qiu et al. [24]

optimized the Morlet wavelet shape factor using the

minimal Shannon entropy criteria when applied as a base

function in wavelet de-noising for bearing fault diagnosis.

S. Abbasion et al. [25] proposed discrete Meyer wavelet as

base 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)

Acceleration(m.s

-2)

Acceleration(m.s

-2)

Acceleration(m.s

-2)

Acceleration(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: 0.04908

Y: 0.2558

X: 0.05883

Y: 0.208

X: 0.03958

Y: -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

0

0.1

0.2

0.3

X: 0.09392

Y: -0.1905 X: 0.1

Y: -0.2162

X: 0.1306

Y: 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: -0.00975

Y: 0.5478

X: 0.00975

Y: 0.5478

Delay (s)

Correlationcoefficient

e

b

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16-6

-4

-2

0

2

4

6

Time (s)

f

-0 .05 -0 .04 -0 .03 -0.0 2 - 0.01 0 0. 01 0.0 2 0 .03 0. 04 0 .0 5-0.5

0

0.5

1

X: -0.006167

Y: 0.1692

X: 0.006167

Y: 0.1692

X: -0.03083

Y: 0.4573

Delay (s)

Correlationcoefficient

Fig. 1 The simulated vibrationsignal, the corresponding wave-

let de-noised signal and, the

auto-correlation function Rx()

for bearing with outer-race fault

(a, b, and c) , inner-race fault (d,

e and f)



3/10

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 fault

detection. Z. K. Peng and F. L. Chu gave a comprehensive

overview to the wavelet de-noising for mechanical fault

diagnosis. A number of threshold methods to eliminate the

effects of the signal noise from the resulting wavelet

coefficients are applied in [28, 29].

a

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

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Time (s)

Amplitude

b

0 5 10 15 20 25 30 35 40 45 500

0.2

0.4

0.6

0.8

1

1.2x 10

-3

Frequency (Hz)

PowerSpectrum

Fig. 2 (a) the impulse wavelet

time waveform, (b) its FFT-

spectrum

a

b

c

Fig. 3 (a) The simulated noise

signal (kurtosis=3.0843),

(b) The overall vibration signal

(noise and impulses) (kurtosis=

7.7644), and (c) The pure fault

impulses (kurtosis=8.5312),

with the corresponding intensity

distribution curve for bearing

with outer-race fault



4/10

In this paper, a new technique based on the wavelet

de-noising method for rolling bearing fault detection has

been developed and tested on both simulated and real

bearing vibration signals. To enhance the generated

wavelet coefficients related to the bearing fault impulses,

the wavelet base function has been constructed based onthe impulse response of the bearing system. Moreover, the

wavelet shape parameters are optimized using maximum

kurtosis criteria.

The remaining sections of the paper are as follows: In

the next section the vibration model for rolling bearing with

outer and inner-races fault is derived. In Sect. 3 the

procedures of the proposed approach is set up. The

implementations of the proposed approach for detection of

localized 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 localized

defects

Every time the rolling element strikes a defect in the

raceway or every time a defect in the rolling element hits

the raceway, a force impulse of short duration is produced,

which in turn excites the natural frequencies of the bearing

parts and housing structure. The structure resonance in the

system acts as an amplifier of low energy impacts.

Therefore, the overall vibration signal measured on the

bearing 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) the

experimentally collected signal, and (c) the CWRU signal, for outer-race fault bearing



5/10

oscillating bursts dominated by the major resonance

frequency of the structure.

The response of the bearing structure as an under-

damped second-order mass-spring-damper system to a

single impulse force is given by the following [30]:

S t

Ce

x

ffiffiffiffiffiffi1x2

p wdtsin wdt

1

where is the damping ratio, d is the damped naturalfrequency of the bearing structure, and C is an amplitude

scaling factor.

As the shaft rotates, this process occurs periodically

every time a defect hits another part of the bearing, and its

rate of occurrence is equal to one of the BCF. In reality,

there is a slight random fluctuation in the spacing between

impulses because the load angle on each rolling element

changes as the rolling element passes through the load

zone. Furthermore, the amplitude of the impulse response

will be modulated as a result of the passage of the fault

through the load zone:

x t X

i

AiS t T n t 2

where S(t-Ti) is the waveform generated by the ith impact at

the time Ti, and Ti =iT+i, where T is the average time

between two impacts, and i describe the random slips of

the rolling elements. Ai is the time varying amplitude-

demodulation, and n(t) is an additive background noise

which takes into account the effects of the other vibrations

in the bearing structure.

Figure 1a and b show the acceleration signals (d2x(t)/

dt2) generated by the model in Eq. 2 with random slip

() of 10% of the period T and signal-to-noise ratio of

0.6 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 time

domain signal with the translated and dialed wavelet-base

function. The wavelet transform resulting coefficients

reflect the correlation between the signal and the selected

wavelet-base function. Therefore, to increase the amplitudeof the generated wavelet coefficients related to the fault

impulses, and to enhance the fault detection process, the

selected wavelet-base function should be similar in charac-

teristics to the bearing impulse response generated by

presence of bearing incipient fault, Eq. 1. Based on that,

the proposed wavelet-base function is denoted as impulse-

response wavelet and given by

y t A ebffiffiffiffiffiffi

1b2p wct

sin wct 3

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



6/10

where is the damping factor that control the decay rate of

the exponential envelop in the time, hence regulating the

resolution of the wavelet, simultaneously corresponds to the

frequency bandwidth of the wavelet in the frequency

domain, c determining the number of significant oscil-lations of the wavelet in the time domain and correspond to

the wavelet centre frequency in frequency domain, and A is

an arbitrary scaling factor. Figure 2 shows the proposed

wavelet and its power spectrum.

The proposed wavelet satisfy the admissibility condition,

Cg Z1

1

b= f j j2f

df< 1 4

where Cg is the admissibility constant and, y^(f) is the

Fourier transform ofy (t). This implies that the wavelet has

no zero frequency component, y^

(0)=0 or, the wavelety(t)must have a zero mean [31].

The proposed wavelet de-noising technique consists of

the following steps:

1 Optimize the wavelet shape parameters ( and c)

based on maximization kurtosis of the signal-wavelet

inner product.

It is possible to find optimal values of and c for a

given vibration signal by adjusting the time-frequency

resolution of the Impulse wavelet to the decay rate and

frequency of impulses to be extracted. Kurtosis is an

indicator that reflects thepeakiness

of a signal, which is

a property of the impulses and also it measures the

divergence from a fundamental Gaussian distribution. A

high kurtosis value indicates high impulsive content of the

signal with more sharpness in the signal intensity distribu-

tion. Figure 3 shows the kurtosis value and the intensity

distribution for a white noise signal, pure impulsive signal,

and impulsive signal mixed with noise.

The objective of the impulse wavelet shape optimization

process is to find out the wavelet shape parameters ( and

c), which maximize the kurtosis of the wavelet transform

output:

Optimal b;wc max PNn1

WT4 x t ;yb;wc t

PNn1

WT2 x t ;yb;wc t !2

2666437775 5

a b cFig. 6 (a) The collected vibration signal, (b) The corresponding wavelet de-noised signal, and (c) The auto-correlation function, for bearing with

outer-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



7/10

2 Apply the wavelet de-noising technique:

a- Perform a wavelet transform for the bearing

vibration signal x(t) using the optimized wavelet,

W T x t ; a; bf g =a;b x t 1ffiffiffi

ap

Zx t *a;b t dt

6

where indicates the inner product, the super-

script asterisk* stands for the complex conjugate.

The ya, b is a family of daughter wavelets derived

from the mother wavelet y(t) by continuously

varying the scale factor a and the translation

parameter b. The factor 1

ffiffia

p is used to ensure energypreservation.

b- Shrink the wavelet coefficients expressed in Eq.5by soft thresholding:

WTsoft 0sign WT WT thr

WTj j < thrWTj j > thr

&7

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

yields

thr e Max WT a;b j j x e Max WT a;b j j x 8

where > 0 is parameter governing the shape of

the threshold function.

c- Perform the inverse wavelet transform to recon-

struct the signal using the shrunken wavelet

coefficients.

x*

t C1gZ1

1WTsoft a; t da

a3=29

3 Evaluate the auto-correlation function Rx () for the de-

noised signal x*(t) to estimate the periodicity of the

extracted impulses,

Rx C E x*

t x*

t C ! 10where is the time lag, and E [ ] denotes ensemble

average value of the quantity in square brackets.

4 Wavelet de-noising technique for rolling bearing fault

detection

To demonstrate the performance of the proposed approach,

this section presents several application examples for the

detection of localized bearing defects. In all the examples,

the impulse wavelet has been used as a wavelet base-

function. The wavelet parameters (damping factor and

centre frequency) are optimized based on maximizing the

kurtosis value for the wavelet coefficients as shown in

Fig. 4.

To evaluate the performance of the proposed method,

the autocorrelation functions of the optimized impulse

wavelet, impulse wavelet with non-optimized parameters,

and the widely used Morlet wavelet are carried out and

shown 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 shaft

rotational speed

Shaft speed

(rpm)

Calculated BCF

(Hz)

Period

extracted (s)

BCF extracted

(Hz)

983.887 50.32 0.020310 49.236

2080.28 106.4 0.009297 107.561

3541.11 181.12 0.005391 185.493



8/10

the increased effectiveness of the optimized impulse

wavelet over non-optimized impulse and Morlet wavelet

for extraction of the bearing fault impulses and period-

icity. Consequently the performance of the bearing fault

diagnosis process has been increased using the proposed

technique.

4.1 Simulated vibration data

For a rolling element bearing with pitch diameter of

51.16 mm, ball diameter of 11.9 mm, with eight rolling

elements and 0 contact angle, the calculated BCFs (see the

appendix) 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 bearing

with (a) outer-race fault, and (b) inner-race fault



9/10

107.36 Hz and 162.18 Hz for outer and inner-race fault,

respectively. Figure 1a and b show the time domain

waveform of the simulated signals for rolling bearing with

outer and inner-race faults based on the model described in

Sect. 2. The results of the wavelet de-noising method

(wavelet transform, shrink the wavelet coefficients and the

inverse wavelet transform) for rolling bearing with outer

and inner race faults using the optimized impulse waveletare displayed in Fig. 1c and d, respectively. The results

show that the signal noise has been diminished and the

impulses generated by the faulty bearing are easy to

identify in the wavelet de-noised signal. The impulse

periodicity of 0.00975 s (fo =102.564 Hz) for outer-race

fault and 0.006167 s (fi =162.153 Hz) for inner-race fault

are effectively extracted through the auto-correlation of the

de-noised signal, Fig. 1e and f, which are exactly matching

the theoretical estimation of the BCF.

4.2 Experimental vibration data

A B&K 752A12 piezoelectric accelerometer was used to

collect the vibration signals for an outer race defective,

deep groove, ball bearing (with same simulated specifica-

tions) at different shaft rotational speeds. The vibration

signals were transferred to the PC through a B&K

controller module type 7536 with data sampling frequency

of 12.8 kHz. Based on the bearing parameters, the

calculated outer race fault characteristic frequency is

0.05115 times the shaft rotational speed (rpm).

Figures 6, 7, 8 show the application of the proposed

wavelet de-noising technique for rolling bearing with outer-

race fault at different shaft rotational speed. The bearing

fault impulses and their periodicity are easily defined in the

wavelet de-noised signal and the de-noised autocorrelation

function, respectively. The comparison of Figs. 6c, 7c and

8c shows the sensitivity of the proposed de-noising

technique to the variation of the BCF as a result of

variation 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 with

outer and inner race faults using electro discharge machin-

ing (EDM). The calculated defect frequencies are 3.5848

and 5.4152 times the shaft rotational speed (Hz) for outer

and inner race fault, respectively. At shaft rotational speed

of 1797 rpm the calculated BCF are 107.36 Hz for outer-

race fault and, 162.185 Hz for inner-race fault. The time

course of the vibration signal for bearings with outer and

inner race faults, the corresponding wavelet de-noised

signal and the auto-correlation function are depicted in

Fig. 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 and

inner race fault, respectively, which are very close to the

calculated BCF.

5 Conclusions

A new approach for rolling bearing fault diagnosis based on

wavelet de-noising technique with wavelet-base function

derived from the impulse response of the bearing system is

presented. Wavelet shape parameters have been optimized

using maximum kurtosis criteria. The results for both

simulated as well as actual bearing vibration signals show

the effectiveness of the proposed approach to extract the

rolling bearing fault impulses buried in the noisy vibration

signal, and evaluate its periodicity using auto-correlation

function of the wavelet de-noised vibration signal.

Appendix

Fault bearing characteristic frequencies (BCF)

Each bearing element has its own characteristic frequency

of defect. Those frequencies can be calculated from the

kinematics relation, i.e., the geometry of the bearing and its

rotating speed. For a bearing with a stationary outer race,

the above defect characteristic frequencies can be obtained

as follows:

Characteristic frequency of the outer-race:

fo in:Hz 0:5zf 1 dD

cos

1

Characteristic frequency of the inner race:

fi in:Hz 0:5zf 1 dD

cos

2

Characteristic frequency of the rollers:

fr in:Hz fDd

1 dD

cos

2" #3

Characteristic frequency of the cage:

fC f2

1 dD

cos !

4



10/10

where z is the number of rollers, d is the diameter of the

rollers, D is the pitch diameter, is the contact angle, and f

is the rotating speed of shaft.

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