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0301-679X/$ - s

doi:10.1016/j.tr

�CorrespondE-mail addr

(J. Poshtan).

Tribology International 40 (2007) 763–769

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Bearing fault detection using wavelet packet transformof induction motor stator current

Jafar Zarei, Javad Poshtan�

Department of Electrical Engineering, Iran University of Science and Technology, Narmak 16846, Tehran, Iran

Received 27 November 2005; received in revised form 10 April 2006; accepted 19 July 2006

Available online 20 September 2006

Abstract

Induction motor vibrations, caused by bearing defects, result in the modulation of the stator current. In this research, bearing defect is

detected using the stator current analysis via Meyer wavelet in the wavelet packet structure, with energy comparison as the fault index.

The advantage of this method is in the detection of incipient faults. The presented method is evaluated using experimental signals. Sets of

data are gathered before and after using defective bearings. Compared to conventional methods, the superiority of the proposed method

is shown in the success of fault detection.

r 2006 Elsevier Ltd. All rights reserved.

Keywords: Wavelet packet transform; Induction motor; Bearing; Fault detection

1. Introduction

Electrical motors are the majority of the industry primemovers and are the most popular for their reliability andsimplicity of construction [1].

Since the apparatus driven by induction motors hasimportant role in industry, their safety, reliability, effi-ciency and performance are highly considered by engineers.Although induction motors are reliable, they are subjectedto some failures. Therefore in the past two decades, therehas been substantial amount of research to provide newcondition monitoring techniques for induction motorsmostly based on analyzing vibration signals, and hencemany commercial tools are available in this area [1–3].

Recent studies show that more than 40% of inductionmotor failures are related to bearings. Therefore, this typeof fault must be detected as soon as possible to avoid fatalbreakdowns of machines that may lead to loss ofproduction. Bearing defects may be categorized as ‘‘dis-tributed’’ or ‘‘local’’ [4]. Distributed defects include surfaceroughness, waviness, misaligned races and off-size rolling

ee front matter r 2006 Elsevier Ltd. All rights reserved.

iboint.2006.07.002

ing author. Tel.: +98 21 772 40492; fax: +98 21 772 40490.

esses: jzarei@iust.ac.ir (J. Zarei), jposhtan@iust.ac.ir

elements. Localized defects include cracks, pits and spallson the rolling surfaces. The dominant mode of failure inrolling element bearings is spalling of the races or therolling elements. Localized defects generate a series ofimpact vibrations every time a running roller passes overthe surface of a defect. Therefore, vibration analysis is aconventional method for bearing fault detection. Althoughvibration analysis has been used for mechanical faultdetection for many decades, more recent studies oninduction motors concentrate on monitoring electricalsignals such as stator current [5]. Because the vibrationproduced by defect is also modulated on the stator current,and this signal can be easily measured for conditionmonitoring and control purposes, Motor Current Signa-ture Analysis provides a non-intrusive approach to obtaininformation about bearing health using already availableline current.Time, frequency, and time–frequency domain analysis

methods are used to analyze vibration signals [4]. However,frequency and time–frequency domain analyses are oftenused to analyze stator current signals [5,6]. Time domainmethods such as RMS and Crest Factor have achievedlimited success for the detection of localized defects [4].Some statistical properties, such as Kurtosis also has beenused in the time domain. Kurtosis is the fourth moment,

ARTICLE IN PRESS

↓2

f

L

H ↓2

cA1

cD1

↑2 L

H↑2

cA1

cD1

f

Fig. 1. Basic steps of decomposition and reconstruction of the wavelet

transform filter bank.

0 8 4 2

v0 w0 w1 w2

H (�

)

� � � �

�

Fig. 2. Frequency bands for two-band wavelet transform.

J. Zarei, J. Poshtan / Tribology International 40 (2007) 763–769764

normalized with respect to the fourth power of standarddeviation. Kurtosis has not become a very popular methodin industry for the condition monitoring of bearings;however, it has been suggested to measure kurtosis inselected frequency bands [4].

Frequency analysis methods using vibration and currentsignals are developed extensively for bearing fault detec-tion. FFT is the simplest frequency domain analysismethod [5]. The Combination of FFT and Envelopemethods using vibration analysis has shown successfulapplication in industry [4]. This method is the examinationof high-frequency resonances caused by fault in thespectrum of the signal. Because the impact vibrationgenerated by a bearing fault has relatively low energy, itis often overwhelmed by noise with higher energy andvibration generated from other macrostructural compo-nents. Therefore, it is difficult to identify the bearing faultin the spectra using conventional FFT methods.

In order to overcome FFT problems, recent advancedsignal processing methods such as Short-Time-Fourier-Transform, Wigner–Ville and wavelet analysis have beenused. In [7], wavelet transform using vibration signal isused to detect inner and outer race defects independentlyand in conjunction with each other. In [8], using waveletpacket transform and Daubechies 12 wavelet analysis ofvibration signal, faults have been detected on the inner andouter races of bearing.

In this study, stator current analysis via Meyer motherwavelet function, because of high resolution, in waveletpacket structure is used in stator current analysis forinduction motor bearing fault detection. In this regard, thepaper will appear as follows: a brief review of wavelet andwavelet packet transform theory is explained in Sections 2and 3, respectively. Fault detection algorithm via WPT isdescribed in Section 4, and mother wavelet selection forthis application is discussed in Section 5. The experimentalresults, given in Section 6, show the success of the proposedalgorithm in detecting rings defect.

2. A review of wavelet transform theory

The continuous wavelet transform of finite-energysignals ðf ðtÞ 2 L2ðrÞÞ with the analyzing wavelet c(t), isthe convolution of f(t) with a scaled and conjugatedwavelet [9]:

W f ða; bÞ ¼

Z þ1�1

f ðtÞ1ffiffiffiap c�

t� a

b

� �dt, (1)

where c(t) is the wavelet function, a and b are the dilationand translation respectively. The factor 1=

ffiffiffiap

is used forenergy preservation. Eq. (1) indicates that wavelet analysisis a time–frequency analysis, or more properly termed atime-scaled analysis.

Since continuous a and b cause computational complex-ity, usually the discrete forms are used. A useful selectionis a ¼ 2�j and b ¼ k2�j, where k and j are integer values.This wavelet system is called dyadic wavelet transform.

Therefore, wavelet transform is obtained by:

W f ðj; kÞ ¼

Z 1�1

f ðtÞ2j=2c�ð2j t� kÞdt. (2)

Discrete wavelet analysis can be implemented by scalingfilter h(n), which is a low-pass filter related to the scalingfunction f(t), and the wavelet filter g(n), which is a high-pass filter related to the wavelet function c(t):

fjðtÞ ¼X

k

hðkÞ2ðjþ1Þ=2fð2jþ1t� kÞ, (3)

cjðtÞ ¼X

k

gðkÞ2ðjþ1Þ=2fð2jþ1t� kÞ. (4)

The basic step of a fast wavelet algorithm is illustrated inFig. 1 which can be implemented in two oppositedirections: decomposition and reconstruction. In thedecomposition step, the discrete signal f is convolved witha low-pass filter L and a high-pass filter H, resulting in twovectors cA1 and cD1. The elements of the vector cA1 arecalled approximation coefficients, and the elements ofvector cD1 are called detailed coefficients. The symbol k2denotes down sampling.

3. Wavelet packet transform

The classical two-band wavelet transform results in alogarithmic frequency resolution [10]. The low frequencieshave narrow bandwidths and the high frequencies havewide bandwidths, as illustrated in Fig. 2. Therefore, the lowfrequencies are investigated with finer resolution, whilewide bandwidth at high frequencies results in a poorresolution. The wavelet packet system is a generalizationof wavelet transform, in which at all stages both the low-pass and high-pass bands are split. Therefore, it canallow a finer adjustable resolution of frequencies at high

ARTICLE IN PRESSJ. Zarei, J. Poshtan / Tribology International 40 (2007) 763–769 765

frequencies. It also gives a rich structure that allowsadaptation to particular signals or signal classes. The costof this richer structure is a computational complexity ofO(N log2(N)), similar to the FFT, in contrast to theclassical wavelet transform which is O(N) [10].

If the conjugate mirror filters h and g have finite impulseresponses of size K, it has been proved that f ¼ c0

0 has acompact support of size K�1, hence:

c2pjþ1 ¼

X1n¼�1

hðnÞcpj ðt� 2�jnÞ; 0opo2j � 1,

c2pþ1jþ1 ¼

X1n¼�1

gðnÞcpj ðt� 2�jnÞ; 0opo2j � 1. ð5Þ

The frequency localization of wavelet packets is morecomplicated to analyze. The Fourier transform of (5)proves that the Fourier transforms of wavelet packetchildren are related to their parent by:

c2p

jþ1 ¼ hð2joÞcp

j ðoÞ,

c2pþ1

jþ1 ¼ gð2joÞcp

j ðoÞ. ð6Þ

The energy of cp

j is mostly concentrated over a frequencyband and the two filters hð2joÞ and gð2joÞ select the loweror higher frequency components within this band.

The resulting three-scale analysis tree (three-stage tree) isillustrated in Fig. 3. The delicate point is to realize thath(2jo) does not always play the role of a low-pass filterbecause of the side lobes that are brought into the interval[�p, p] by the dilation. Therefore, unlike Wavelet Trans-form, frequency order is not the same as the nodeorder. Coifman and Wickerhauser have a proposition to

f (t)

(0-125 Hz) (125-250 Hz) (375-500 Hz) (250-375 Hz)

Level 2

Level 3

(3, 0) (3, 1) (3, 2) (3, 3)

(2, 0)

(1, 0)

(2, 1)

(0-250 Hz) (250-500 Hz)

(0-500 Hz)

(0, 0

(0-1000Level 1

Fig. 3. Three-level wavelet pa

arrange nodes at depth j, proportional to the frequencyintervals [9].WPT coefficients at each stage are computed by:

d2kjþ1ðnÞ ¼ dk

j ðnÞ � hð�2nÞ; 0oko2j � 1, (7)

d2kþ1jþ1 ðnÞ ¼ dk

j ðnÞ � gð�2nÞ; 0oko2j � 1. (8)

In wavelet tree, scale parameter (depth) is demonstratedby j, and frequency parameter (nodes) by 2k and 2k+1.Coefficient energy at each node is computed by:

E ¼

PMk¼1ðd

pj ðkÞÞ

2

M

!1=2

. (9)

where M is the number of samples at the node.

4. Fault detection algorithm

Local defects or wear defects cause periodic impulses invibration signal. Amplitude and period of these impulsesare determined by shaft rotational speed, fault location,and bearing dimensions. The frequency of these impulses,considering different fault locations as in Fig. 4 areobtained by (10)–(13) [4].Fundamental cage frequency is given by:

f c ¼f s

21�

d

DcosðaÞ

� �. (10)

Ball defect frequency is two times the ball spin frequencyand can be calculated as:

f bd ¼D

df s 1�

d2

D2cos2ðaÞ

� �. (11)

(750-875 Hz)(875-1000 Hz) (500-625 Hz)

(3, 4) (3, 5) (3, 6) (3, 7)

(625-750 Hz)

(1, 1)

(2, 2)(2, 3)

(750- 1000 Hz) (500- 750 Hz)

(500-1000 Hz)

)

Hz)

cket decomposition tree.

ARTICLE IN PRESS

Decision

Input Data

Calculating WP

Coefficients

Comparison with

Healthy Condition

Zoom on defect

frequency range

Calculating energy

Fig. 5. Fault detection algorithm procedure.

0.5

1

1.5Meyer wavelet

J. Zarei, J. Poshtan / Tribology International 40 (2007) 763–769766

Inner race defect frequency is given by:

f id ¼ nðf s � f cÞ ¼nf s

21þ

d

DcosðaÞ

� �. (12)

Outer race defect frequency is given by:

f od ¼ nf c ¼nf s

21�

d

DcosðaÞ

� �, (13)

where fs is the shaft rotation frequency, n is the number ofrollers, d is the roller diameter, and D is the pitch diameterof the bearing.

Since these mechanical vibrations produce anomalies inthe air gap flux density, they result in the modulation ofstator current. These frequencies can be calculated by [11]:

f bng ¼ f e �m � f v

�� ��, (14)

where m ¼ 1; 2; 3; . . . and fe is the electrical power supplyfrequency and fv is one of the characteristic vibrationfrequencies which are calculated by (10)–(13). As theimpact vibration generated by a bearing fault has relativelylow energy, it is often overwhelmed by noise with higherenergy and vibration generated from other macrostructuralcomponents. Therefore, it is difficult to identify the bearingfault in the spectra using the conventional FFT method,and hence advanced signal processing techniques areneeded. Since wavelet packet transform is able toconcentrate on a frequency range, characteristic frequen-cies of a signal can be achieved carefully.

In this study, stator current measured in variousconditions is first decomposed in sub-bands at predeter-mined levels using wavelet packet algorithm. Then, defectfrequency region is determined, and coefficient energies inrelated nodes are calculated. In comparison with a healthycondition, energy is increased in the nodes related to defectfrequency regions, therefore it can be used as a fault index.Fig. 5 shows the fault detection procedure using thisalgorithm. Bearing defects of an induction motor aresuccessfully detected using the proposed method in alaboratory experiment.

fc

fod

fod

fid

fr

d

�

D

Fig. 4. Bearing dimension and characteristic defect frequencies.

5. Wavelet selection

In order to study the frequency characteristic of a signal,a high-frequency-resolution wavelet is required. Amongorthogonal wavelets, Shannon wavelet has the mostresolution theoretically [9]. High- and low-frequency filterof Shannon wavelet select the frequency range ½�p=2;p=2�and ½�p;p=2� [ ½p=2; p�, respectively. Sharp edges of thesefilters make them non-causal, and hence in practice theirapproximation is used. Meyer wavelet is an approximationof Shannon wavelet. This wavelet is a frequency band-limited function whose Fourier transform is smooth, unlikethat of the Shannon wavelet, and cause a faster decay ofwavelet coefficient in the time domain. However, the timedecay of this wavelet in time domain is high. It is fasterthan Shannon wavelet, but the supporting area in timedomain is not limited and an approximation of it is used as

-6 -4 -2 0 2 4 6-1

-0.5

0

-6 -4 -2 0 2 4 6-0.5

0

0.5

1

1.5Meyer scaling function

Fig. 6. Discrete Meyer wavelet and scaling functions.

ARTICLE IN PRESS

Sym 4

Sample

Nod

e

2000 4000 6000 8000 10000 12000 14000 16000

4748504953545251596062615758565539404241454644433536383733343231

20

40

60

80

100

120

db4

Sample

Nod

e

2000 4000 6000 8000 10000 12000 14000 16000

4748504953545251596062615758565539404241454644433536383733343231

20

40

60

80

100

120

dmey

Sample

Nod

e

2000 4000 6000 8000 10000 12000 14000 16000

4748504953545251596062615758565539404241454644433536383733343231

20

40

60

80

100

120

Coif4

Sample

Nod

e

2000 4000 6000 8000 10000 12000 14000 16000

4748504953545251596062615758565539404241454644433536383733343231

20

40

60

80

100

120

Fig. 7. Time–frequency space of decomposed signal using different mother wavelets.

Table 1

Characteristic vibration frequency and modulation effect on the stator current

Condition Characteristic defect in vibration (Hz) Modulation effect on stator current and related nodes

Defective outer race 89.2 39.2Hz (8–15), 128.4Hz (8–48), 139.2Hz (8–50)

Defective inner race 135.5 85.5Hz (8–31), 185.5Hz (8–56), 221Hz (8–36)

Distributed defect 117.5 66Hz (8–24), 166Hz (8–63), 182Hz (8–57)

J. Zarei, J. Poshtan / Tribology International 40 (2007) 763–769 767

‘‘Discrete Meyer’’. Fig. 6 shows the Discrete Meyer waveletand the scaling function.

In order to compare the resolution of the DMeyermother wavelet function with other wavelets such asDaubiches, Symlet and Coiflet, WPT coefficients of asinusoid signal with two components (5 and 25Hz) arecomputed, and the resulting time–frequency space in eachcase is shown in Fig. 7. The signal was defined as:

f ðtÞ ¼ sinð2p5tÞ þ sinð2p25tÞ. (15)

This signal was sampled at 64Hz, and was decomposedinto five levels via WPT algorithm using dB4, coif4, sym4

and dmey wavelets. In this figure the amplitude of eachfrequency is shown by a color bar. It is observed that whendmey wavelet is used, the separation of frequency band isobtained with a better resolution. Therefore, in thisresearch, dmey is used as the mother wavelet function.

6. Experimental result

In this study a three-phase, 1.2KW, 380V, 50Hz,1400 rpm, four pole induction motor was used. Bothshaft-end and fan-end bearings are 6205-2Z. From the6205-2Z bearing data sheet, the outside diameter is 52mm

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0 20 40 60 80 100-80

-60

-40

-20

0

20

Frequency (Hz)

Po

wer

/Fre

qu

ency

(d

B/H

z)

Power Spectral Density Estimste via Welch Algorithm

Healthy Bearing

Single Defect on Outer race

Fig. 9. Stator current spectrum: solid line (—) of healthy bearing; dotted

line (y) with a hole in the outer race of the shaft-end bearing.

Table 2

Energy comparison around 39.18Hz (outer race defect frequency)

Condition Frequency range

Node (8–12) Node (8–13) Node (8–15)

(31.25–35.15Hz) (35.15–39.06Hz) (39.06–42.96Hz)

A 108.1 37.75 31.9

B 200.3 69.72 42.29

C 187.3 63.74 44.03

D 126.6 39.69 28.48

E 32.06 22.64 32.06

Table 3

Energy comparison around 85.5Hz (inner race defect frequency)

Condition Frequency range

Node (8–30) Node (8–31) Node (8–29)

(78.12–82.03Hz) (82.03–85.93Hz) (85.93–89.84Hz)

A 6.28 5.72 4.37

B 9.18 3.59 10.14

C 4.4 5.74 8.33

D 7.42 8.45 6.25

E 8.89 7.07 3.88

J. Zarei, J. Poshtan / Tribology International 40 (2007) 763–769768

and the inside diameter is 25mm. Assuming equalthickness for the inner and outer races leads to a pitchdiameter equal to 38.5mm (D ¼ 38.5mm). The bearing hasnine balls (n ¼ 9) with an approximated diameter of7.938mm (d ¼ 7.938mm). Assuming a contact angle, a,of 01, and motor operation at the measured shaft speed of1498 rpm (frm ¼ 24.96Hz), the characteristic vibrationfrequencies are calculated from (10)–(13), and the modu-lated frequency on stator current is derived by (14), asshown in Table 1. Five tests were conducted to evaluate theability of the proposed method. At first, while bearing wasin a healthy condition (named A), the stored stator currentwas used as a baseline. In the second experiment as shownin Fig. 8, a 1-mm hole was drilled on the outer race (namedB), while in the third experiment a similar hole was drilledon the inner race (named C). Because inner race damagehas more transfer segments when transmitting to the outerrace surface of the case, usually the impulse componentsare rather weak in the vibration signals. Hence, diagnosisfor inner race damages is very difficult [12]. Therefore, a3-mm hole was drilled on the inner race to exaggerate thecase (named D). At last, in order to study a distributeddefect, two holes were drilled on the outer race (named E).

In all tests, the machine is connected to a line directly,and stator current was sampled at Fs ¼ 2KHz before andafter defects were made. Fig. 9 shows the average period-ogram of signals with a sample number of N ¼ 80; 000,Hanning window with length L ¼ 4000 and 50% overlap.As seen in this Fig. 9, conventional spectral analysis couldnot successfully detect the defect. For example, outer racecomponent at 39.18Hz in the healthy condition is greaterthan that in the faulty condition.

Now the stator current is analyzed with the proposedalgorithm in Section 4. In the first step, the stator current isdecomposed via DMeyer wavelet at 8 levels. Therefore, wehave 256 nodes at ending level of which the frequencyresolution is 3.9Hz. Afterwards energy is computed at allnodes.

In Table 2 energy is compared around the outer racedefect frequency (39.18Hz) for all cases. The largest energyis related to bearing B with a 1-mm hole on the outer race.As can be seen from this table, energy is increased for othercases. This is caused by the modulation effect of sidebandsthat have been attributed to time-related changes in thedefect position relative to the vibration measuring positionin cases when defect is on the moving elements such as

Fig. 8. Defective bearing used in the second experiment.

Table 4

Energy comparison around 67Hz

Condition Frequency range

Node (8–24) Node (8–25) Node (8–27)

(62.5–66.4Hz) (66.4–70.31Hz) (70.31–74.21Hz)

A 77.03 80.13 33.68

B 17.79 13.32 10.51

C 51.61 55.55 21.7

D 62.58 74.4 38.49

E 86.69 91.2 42.01

ARTICLE IN PRESSJ. Zarei, J. Poshtan / Tribology International 40 (2007) 763–769 769

inner race or a rolling element. The result shows that thesesidebands are located at 85Hz whose modulation effect isapproximately near in the same range as 89.2Hz.

In Table 3 energy is compared around 85.5Hz. Sincebearing C has an incipient inner race defect, energy isslightly increased. In case D with a 3-mm hole on the innerrace, the increase of energy is better viewed as expected.

Energy variations around 67Hz related to distributeddefect (two 1-mm holes in the outer race) are shown inTable 4. In Comparison with Table 2, it is observed thatthe energy increase has happened around 67Hz, and thereis no increase around 39.18Hz. This fact is interpreted asthe modulation of stator current by vibration componentas follows: vibration signals of the motor housing wereevaluated using combination of envelope and WPTmethods as described in [8]. It was noticed again that inthe vibration spectrum, instead of a component in 89.2Hz,a component in 117.5Hz was observed which wasmodulated in 67.5Hz in the stator current.

7. Conclusions

In this paper, Wavelet Packet Analysis is used as apowerful diagnostic method for the detection of incipientbearing failures via stator current analysis. An advancedcurrent signal processing algorithm applied to electriccurrent signals was proposed as a suitable alternative tovibration signal to detect bearing faults.

The proposed method has several advantages overFourier analysis. Stator current in nature is non-stationary;therefore, wavelet packet transform can provide betteranalysis under various conditions. Moreover, the frequencybands in defect detection are more tolerant due to the factthat the actual bearing-defect induced vibration frequencymay vary slightly from the predicted values due to slippage

that occurs within bearing. Wavelet packet transform cancover this range of frequency band. The proposed methodwas successfully verified through five laboratory tests.

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