Mechanical Systems and Signal Processing (2002) 16(2–3), 447–457

doi:10.1006/mssp.2002.1482, available online at http://www.idealibrary.com on

088

GEAR FAULTDIAGNOSIS BASEDONCONTINUOUSWAVELET TRANSFORM

H. Zheng, Z. Li and X. Chen

Dynamic Measuring Center, P.O. Box 02, Hefei University of Technology, Hefei,Anhui 230009, People’s Republic of China. E-mail: hbobzheng@yahoo.com.cn

(Received 8 June 2001, accepted 12 February 2002)

A new approach of gear fault diagnosis based on continuous wavelet transform ispresented. Continuous wavelet transform can provide a finer scale resolution thanorthogonal wavelet transform. It is more suitable for extracting mechanical faultinformation. In this paper, the concept of time-averaged wavelet spectrum (TAWS) basedon Morlet continuous wavelet transform is proposed. Two fault diagnosis methods namedspectrum comparison method (SCM) and feature energy method (FEM) based on TAWSare established. The results of the application to gearbox gear fault diagnosis show thatTAWS can effectively extract gear fault information. The feature energy of the TAWSfeatures the gear fault advancement very well and is conically proportional to the gear faultadvancement.

# 2002 Elsevier Science Ltd. All rights reserved.

1. INTRODUCTION

Vibration signal analysis has been widely used in the rotating machinery conditionmonitoring and fault diagnosis. A multitude of methods have been developed.Conventional techniques, such as power spectrum, cepstrum, time-domain averaging,adaptive noise cancellation, demodulation analysis, time-series analysis, are wellestablished and have proved to be very effective in machinery diagnostics. However, theyhave difficulties with certain applications such as the detection of cracking teeth in gearsystems and cylinder wear in reciprocating engines. This is because conventionaltechniques are based on the assumption of stationarity of the vibration signals. To dealwith non-stationary signals, a number of new techniques have been proposed, e.g. time–frequency distribution [1], wavelet [2] and higher-order statistics [3].

Among these new techniques, wavelet analysis possesses particular advantages forcharacterising signals at different localisation levels in time as well as frequency domains.It has a wide variety of applications in many engineering fields such as signal processing,image processing, pattern recognition, seismology, machine visualisation, etc. In the fieldof mechanical fault diagnosis, wavelet analysis has been used in gear diagnosis [4–7],rolling bearing diagnosis [7–9], compressor diagnosis [10] and diesel engine diagnosis [11].Wang and McFadden [4, 5], for example, used orthogonal wavelets such as the Daubechies4 and harmonic wavelets to disclose abnormal transients generated by early gear damagefrom gearbox casing vibration signal. Due to the limited number of scales, a single waveletamplitude map has not enough scales to describe all details of the signal. To discriminatethe fault advancement directly from the amplitude map is also a difficulty especially for anon-trained technician. Staszewski and Tomlinson [6] applied wavelet transform to faultdetection in a spur gear using similarity analysis of patterns featured by the wavelettransform coefficients. It was shown that Mahalanobis distance between contour plots of

8–3270/02/+$35.00/0 # 2002 Elsevier Science Ltd. All rights reserved.

H. ZHENG ET AL.448

the modulus of the wavelet transform increases monotonically with gear faultadvancement, which can be used as a fault detection symptom, but there is not anyregularity for the modulus itself with the increase of gear fault advancement. Orthogonalwavelet transform offers faster algorithms and no redundancy in the decomposition. Theoriginal signal is decomposed into a compact wavelet series, which is advantageous insignal reconstruction. However, they have three disadvantages making it unsuitable forextracting mechanical fault information. First, they demand orthogonal wavelet bases,which make it rather difficult to choose a proper wavelet function. Second, the samplinggrid in the scale direction is rather sparse. Some of the fault information will be leakedunder such sparse grid. Third, they are time translation variant. That is, the sametransients at different time may be shown as different patterns.

To make up for the deficiency of orthogonal wavelet transform in the applicationof mechanical fault diagnosis, continuous wavelet transform (CWT) was introducedinto the field of mechanical fault diagnosis. Lin and Qu [7] developed a denoising methodbased on Morlet CWT, which can completely extract the feature periodic impulses ofgearbox immersed in the noise. Unfortunately, they had not presented a sound gear faultdetection method, just to remove the noise. Dalpiaz et al. [14] presented applications ofCWT by using residual radial signals of gear vibration. Wang et al. [15] found CWT canprovide a good visual inspection especially when residual signals of gear vibration areused. Most of the applications just make use of the visual advantage of CWT to localisethe fault.

This paper presents a new CWT-based approach to deal with gear fault diagnosis.Similar to other CWT-based applications, visual advantage of CWT is also employed tolocalise the fault. A new concept of time-averaged wavelet spectrum (TAWS) based onCWT is proposed. As seen later, TAWS can features gear fault advancement very well.Two gearbox gear fault diagnosis methods named spectrum comparison method (SCM)and feature energy method (FEM) based on TAWS are established. Both of the methodsare reliable and easy to be carried out.

The structure of the paper is as follows. In Section 2, the theoretical backgroundof CWT and TAWS are given. In Section 3, the test rig, results of traditionalspectrum analysis and results of TAWS are presented. In Section 4, two gearboxgear fault diagnosis methods, SCM and FEM, are introduced in detail, respectively.Some discussions are given in Section 5. The conclusion of this paper is given in Section 6.

2. THEORETICAL BACKGROUND

2.1. CONTINUOUS WAVELET TRANSFORM

If cðtÞ 2 L2ðRÞ and its Fourier transform #ccðoÞ satisfies the admissibility condition

Cc ¼Z 1

�1j #ccðoÞj2=joj do51 ð1Þ

we call cðtÞ mother wavelet or wavelet function, where L2ðRÞ is the space of squareintegrable complex functions. The corresponding family of wavelets consists of a series ofson wavelets, which are generated by dilation and translation from the mother wavelet cðtÞshown as follows:

ca;bðtÞ ¼ jaj�1=2ct� b

a

� �ð2Þ

where a is scale factor and b is time location, the jaj�1=2 is used to ensure energypreservation.

GEAR FAULT DIAGNOSIS 449

The CWT of the signal xðtÞ is defined as the inner product in the Hilbert space of the L2

norm as follows:

WbðaÞ ¼ 5ca;bðtÞ; xðtÞ >¼ jaj�1=2

ZxðtÞc*

a;b dt: ð3Þ

Here the asterisk stands for complex conjugate. Scale factor a and time location b varycontinuously.

For a discrete sequence xm, let t ¼ mdt and b ¼ ndt, where m; n ¼ 0; 1; 2; :::; N � 1, N isthe sampling point number and dt is the sampling interval. The CWT of xm is defined asfollows:

WnðajÞ ¼XN�1

m¼0

xmc� ðm� nÞdt

aj

� �: ð4Þ

By varying the index j and n corresponding to scale factor a and time location b,respectively, one can construct a picture showing both the amplitude of any features vs thescale and how this amplitude varies with time. The details of the implementation of CWTcan be found in reference [12].

2.2. THE CHOICE OF WAVELET FUNCTION AND SCALE FACTOR

There are two kinds of wavelet functions, orthogonal and non-orthogonal. Among thewidely used orthogonal wavelet functions are Haar, Daubechies, Coiflets, Symlets andMeyer, etc, while the non-orthogonal ones include Morlet, Mexican hat and DOG, etc. Indyadic discrete wavelet transform and wavelet packet transform, one must chooseorthogonal wavelet function, while in CWT, one can choose either orthogonal ornon-orthogonal wavelet function, which provides much bigger freedom of choice. In fact,wavelet coefficients measure the similarity between the signal and each of its son wavelets.The more the son wavelet is similar to the feature component, the larger is thecorresponding wavelet coefficient [7]. So it is important to choose a proper waveletfunction for the mechanical fault diagnosis. The basic principle is to choose a waveletfunction whose shape is similar to the vibration signal caused by the mechanical fault. Aswe know, when a fault occurs, the vibration signal of the machine includes periodicimpulses, whose shape is like Morlet wavelet. So we choose Morlet wavelet as the waveletfunction used in the CWT.

Because it is necessary to choose an orthogonal wavelet function in the dyadic discretewavelet transform and wavelet packet transform, the scale factor a is discretised as a ¼ 2j,j ¼ 0; 1; 2; :::; k. For such sparse sampling grid, missing fault information is inevitable. Asfor CWT, one can choose discrete interval discretionarily. In this paper, the scale factor isdiscretised as follows:

aj ¼ a02jdj ; j ¼ 0; 1; :::; J ð5Þ

where a0 ¼ 2dt is the minimum scale factor and dj ¼ 0:05, J ¼ 100. dt is the samplinginterval. If the sampling frequency is equal to 3000Hz, one can get 101 scales between0.6660 and 21.3120.

2.3. TIME-AVERAGED WAVELET SPECTRUM

A Morlet wavelet function is defined as

cðtÞ ¼ expð�t2=2Þ expðjo0tÞ; o055: ð6Þ

It is a complex wavelet function, so its CWT coefficient, WnðajÞ, is also complex. Thecoefficient can then be divided into the real part, RfWnðajÞg, and the imaginary part,IfWnðajÞg, or amplitude, jWnðajÞj, and phase, tan�1½IfWnðajÞg=RfWnðajÞg�. Finally, one

H. ZHENG ET AL.450

can define the wavelet power spectrum as jWnðajÞj2. In fact, it is a two-dimensionalmatrix, which has the number of rows and columns equal to that of the scales and thesampling points, respectively. We define time-averaged wavelet spectrum (TAWS) asfollows:

WðajÞ ¼1

N

XN�1

n¼0

jWnðajÞj2 ð7Þ

where N is the number of sampling points, j ¼ 0; 1; :::; J. It reflects the distribution ofenergy of wavelet power spectrum in the direction of scale.

3. EXPERIMENTAL ANALYSIS

3.1. TEST RIG

Gearboxes are very popular in industry applications. A broken gear tooth failuremay cause many fatal accidents, so the recognition of gear tooth cracks is veryimportant for the safety of a gearbox. Most of the gear fault diagnosis methodsare developed on the basis of seeded faults [6, 13]. It does have some profound valuesfor the gear fault detection, but the simulated faults have much difference from theactual ones. To study the gear failure progress lively, our experiment concerns with a lifetest of an automobile transmission box, which has five forward speeds and one backwardspeed.

In this paper, the signal was acquired by an accelerometer mounted on the outer case ofa gearbox, when it was loaded in the third speed. We recorded the vibration signal fromthe beginning to the end of the life test, when one of the gears of the third speed wasbroken, and then picked up five blocks at equal intervals. The first and last blockcorrespond to 0, i.e. beginning of the test, and 100%, i.e. end of the test, gear faultadvancement, so the second, the third and the fourth block correspond to 25, 50 and 75%gear fault advancement, respectively. It was sampled at 3 kHz randomly with 1024 pointseach time.

For gear transmission, the meshing frequency of the gear is calculated by

fz ¼ fr z ¼ nz=ð60iÞ ð8Þ

where z is the number of gears, n is the rotating speed of the input shaft and i is thetransmission ratio. In our experiment, z, n and i equals to 27, 1600 rpm and 1.44,respectively. It follows from equation (8) that the meshing frequency of the third speedgear is 500Hz.

3.2. POWER SPECTRUM

Figure 1 shows the time-domain vibration signals under 0, 50% and 100% gear faultadvancement. Although some evident impulses occur under 100% gear fault advancement,it is hardly possible to evaluate the gear fault condition only through such signals.

Figure 2 exhibits the power spectrum of the above signals. 500Hz frequencycomponent, which is the meshing frequency, can be clearly seen in all the conditions. Inthe case of 100% gear fault advancement, frequencies around 250Hz dominate the powerspectrum [Fig. 2(c)]. But there is no frequency component around 250Hz under 0 gearfault advancement [Fig. 2(a)]. Different gear fault advancement can be separatedaccording to the amplitude of frequency component around 250Hz. The bigger thefrequency amplitude is, the higher the fault advancement will be. So the featurefrequencies caused by the gear fault are around 250Hz. To some extent, classical Fourier

0 100 200 300

_4

_2

0

2

4

Time (ms)

Acc

eler

atio

n (m

/s2 )

(a)

0 100 200 300 _4

_2

0

2

4

Time (ms)

Acc

eler

atio

n (m

/s2 )

(c)

0 100 200 300_4

_2

0

2

4

Time (ms)

Acc

eler

atio

n (m

/s2 )

(b)

Figure 1. Time-domain vibration signals: (a) 0 gear fault advancement; (b) 50% gear fault advancement;(c) 100% gear fault advancement.

GEAR FAULT DIAGNOSIS 451

analysis can detect gear fault advancement. But it has also some limitations such as unableto process non-stationary signals, only time-domain representation.

3.3. WAVELET POWER SPECTRUM AND TAWS

Figure 3 is the contour plots of the CWT wavelet power spectrum. They present theenergy distribution of the vibration signals under different gear fault advancement in time–frequency domain. A time–frequency distribution describes simultaneously when a signalcomponent occurs and how its frequency spectrum develops with time. Most of the energyis around scale 2 when it is under 0 gear fault advancement [Fig. 3(a)]. There is no evidentperiodicity in the energy distribution. Some energy move to higher scales (around scale 4)when the gear fault advancement is 50% [Fig. 3(b)]. While in the case of 100% gear faultadvancement, most of the energy accumulated to the scales around scale 4 [Fig. 3(c)]. Theperiodicity is very clear and the period is equal to one revolution time of the third speedgear, i.e. 27/500=54 ms. It accords with the transmission box principle. Such figures canhelp localize gear faults. Considering the power spectrum features, scale 4 and scale 2correspond to gear fault feature frequency (250Hz) and gear meshing frequency (500Hz),respectively.

Figure 4 represents the time-averaged wavelet spectrum (TAWS). We find easily that thevalue of TAWS around scale 2 change not much in the three specified gear faultadvancement conditions. But it change violently around scale 4. So the value of TAWSaround scale 4 features the gear fault.

250 500 750 1000 1250 0

2

4

6

Frequency (Hz)

Am

plitu

de

(a)

250 500 750 1000 1250 0

5

10

15

Frequency (Hz)

Am

plitu

de

(c)

250 500 750 1000 12500

1

2

3

4

5

Frequency (Hz)

Am

plitu

de

(b)

Figure 2. Power spectrum of the vibration signals: (a) 0 gear fault advancement; (b) 50% gear faultadvancement; (c) 100% gear fault advancement.

H. ZHENG ET AL.452

4. FAULT DIAGNOSIS METHODS BASED ON TAWS

4.1. SPECTRUM COMPARISON METHOD

Because the signal was sampled randomly and the revolution of the input shaft of thetransmission box was not stable, the energy distribution of TAWS has some differenceeach other even in the same gear fault advancement condition. To eliminate such affection,we adopt the parameter average method. Concretely, for each gear fault advancementcondition, we get a standard TAWS after 10 times averages. The procedure of spectrumcomparison method (SCM) for the gear fault diagnosis is as follows:

(1) Sampling 10 times for each gear fault advancement and then computing the CWTcoefficients WnðajÞ using equation (4).

(2) Computing the TAWS WðajÞ using equation (7).(3) Computing the standard TAWS of each gear fault advancement condition after ten

times averaging.(4) Sampling 2 times randomly for each gear fault advancement condition and then

computing the TAWS WðajÞ using equation (7). We say these TAWSs are waitingmodel (WM) while the standard TAWSs are standard model (SM).

(5) Computing Euclid distance between each WM and each SM. The WM belongs to theSM if the Euclid distance is minimum.

Figure 5 shows the standard TAWS for each gear fault advancement condition. Figure5(a)–(e) correspond to the gear fault advancement 0, 25, 50, 75 and 100%, respectively.

Figure 3. Contour plots of wavelet power spectrum: (a) 0 gear fault advancement; (b) 50% gear faultadvancement; (c) 100% gear fault advancement.

GEAR FAULT DIAGNOSIS 453

Table 1 is the Euclid distance between each WM and each SM and the diagnosis result. Itis very clear that, when a WM belongs to a SM, the value of Euclid distance is muchsmaller than that between others. That is also to say, the TAWS shape of the WM is themost similar to that of the SM. So we can detect the gear fault advancement from the valueof the Euclid distance between the WM and SM based on TAWS.

4.2. FEATURE ENERGY METHOD

Figure 5 shows clearly that the energy of TAWS between scale 2.31 and 6.11, which iscaused by the gear fault, is becoming larger with the increase of gear fault advancement. Itfeatures the evolution of the gear fault advancement very well. So we define feature energy(FE) as follows:

FE ¼Z 6:11

2:31WðaÞ da ¼

X63j¼36

WðajÞðajþ1 � ajÞ ð9Þ

where j ¼ 36; 37; 38; :::; 63, a36 ¼ 2:31, a64 ¼ 6:11. It is the area under the TAWS linebetween 2.31–6.11.

Figure 6(a) shows the values of standard TAWS feature energy and the conic fittingcurve based on least mean square algorithm vs gear fault advancement. The mathematicalexpression can be concluded as

FEðxÞ ¼ 0:0038x2 þ 0:1251xþ 3:7767 ð10Þ

where x is the gear fault advancement, FE(x) is the standard TAWS feature energy.

1 2 4 8 16 0

10

20

30

40

50

Scale

TAW

S

(a)

1 2 4 8 16 0

10

20

30

40

50

Scale

TAW

S

(c)

1 2 4 8 160

10

20

30

40

50

Scale

TAW

S

(b)

Figure 4. TAWS: (a) 0 gear fault advancement; (b) 50% gear fault advancement; (c) 100% gear faultadvancement.

0

50

0

50

0

50

0

50

1 2 4 8 16

0

50

(a)

(b)

(c)

(d)

(e)

Scale

Stan

dard

TA

WS

2.31 6.11

Figure 5. Standard TAWS: (a) 0 gear fault advancement; (b) 25% gear fault advancement; (c) 50% gear faultadvancement; (d) 75% gear fault advancement; (e) 100% gear fault advancement.

H. ZHENG ET AL.454

Table 1

Euclid distance between WM and SM, and the diagnosis result

SM (a) SM (b) SM (c) SM (d) SM (e) Result

WM1 1.41 11.95 30.88 63.36 95.25 0WM2 1.48 12.48 31.37 63.85 95.63 0WM3 14.85 6.75 20.85 51.15 84.18 25%WM4 10.50 5.37 21.71 54.20 86.27 25%WM5 33.16 21.61 3.76 32.79 69.33 50%WM6 33.76 22.38 4.64 32.24 67.63 50%WM7 72.36 60.31 43.32 9.15 32.27 75%WM8 69.14 57.03 40.43 6.49 35.57 75%WM9 105.99 94.60 80.53 48.46 10.55 100%WM10 97.52 86.06 72.19 40.04 7.01 100%

FE

0 25 50 75 1000

20

40

60

80

0 25 50 75 1000

20

40

60

80

Fault advancement (%) Fault advancement (%) (a) (b)

FE

Figure 6. (a) Value of the standard TAWS feature energy and the conic fitting curve’: }}, fitting curve;*, standard FE; (b) results of the gear fault diagnosis: }}, fitting curve; +, diagnosis results.

GEAR FAULT DIAGNOSIS 455

To improve the fault diagnosis accuracy, we form a WM after 5 times averages for thevalue of TAWS feature energy. Following is the procedure using feature energy method(FEM) for gear fault diagnosis:

(1) Same as steps (1)–(3) in SCM.(2) Computing the values of standard TAWS feature energy for each gear fault

advancement condition using equation (9).(3) Getting the conic fitting curve and function (10).(4) Getting five WMs for each gear fault advancement condition and then computing the

corresponding fault advancement using equation (10).

Figure 6(b) shows the fault diagnosis results. Different fault advancement WMs can beeasily separated. So it is very easy to detect the fault advancement using function (10).

5. DISCUSSION

As we have pointed out in Section 3 in this paper, many gear fault diagnosis methodsare developed on the basis of seeded faults in laboratory. They do have useful referencevalues for real applications, but seeded faults have much difference from actual ones. Theexperiment carried out here concerns with a life test of an automobile transmission box. It

H. ZHENG ET AL.456

was performed on a test rig designed by ourselves. The experiment lasted for several daysuntil one of the third speed gears was broken, which is more similar to the damage inindustrial applications. So the experimental data acquired here is more reliable than thatfrom the seeded fault experiment. Fault diagnosis method developed on the basis of suchdata is more valuable.

One outstanding feature of TAWS defined in this paper is that it can vividly reveal theevolution of gear fault advancement (Fig. 5). The TAWS values around scale 2 and scale 4feature the gear meshing and gear fault respectively. The former keep fairly stable, whilethe latter vary violently. TAWS is a much more powerful tool in feature extraction thanothers such as power spectrum, time–frequency distribution etc. The two fault diagnosismethods developed are also easy to be carried out.

Nevertheless, it has also been demonstrated that parameter average method should beadopted in getting the waiting model and standard model in order to eliminate theaffection of random sampling and unstable input shaft revolution. The fault diagnosismethods developed here are both sensitive to these two facts. So before implementing weshould bear in mind this gist. But how many average times should be taken? It depends onthe specific experiment. The aim is to eliminate the affection as much as it could.

6. CONCLUSIONS

Based on the discussions above, the following conclusions can be drawn:

(1) The feature frequencies and scales of gear meshing and gear fault vibration signals aredifferent. They can be easily identified from the power spectrum and TAWS.

(2) The gear fault vibration signal has evident periodicity in the contour plot of thewavelet power spectrum, while the gear meshing vibration signal does not have.

(3) TAWS based on CWT using Morlet wavelet function can feature the gear faultadvancement effectively, especially the feature energy defined in this paper, which isconically proportional to the gear fault advancement.

(4) Spectrum comparison method and feature energy method based on TAWS can detectthe gear fault advancement accurately. They are simple in implementation and mayprovide practical utilities for fault diagnosis in other complex machines.

ACKNOWLEDGEMENTS

The authors wish to thank Professor Zhengmin Yang and Mr Zhongkui Zhu, the schoolof Mechanical and Automotive Engineering of Hefei University of Technology forsupplying the gearbox experiment data.

REFERENCES

1. Q. Meng and L. Qu 1991 Mechanical Systems and Signal Processing 3, 155–166. Rotatingmachinery fault diagnosis using Wigner distribution.

2. O. Riou and M. Vetterli 1991 IEEE Signal Processing Magazine 10, 14–18. Wavelets andsignal processing.

3. A. Swami, G. B. Giannakis and G. Zhou 1997 Signal Processing 60, 65–126. Bibliography onhigher-order statistics.

4. W. J. Wang and P. D. McFadden 1996 Journal of Sound and Vibration 192, 927–939.Application of wavelets to gearbox vibration signals for fault detection.

GEAR FAULT DIAGNOSIS 457

5. W. J. Wang and P. D. McFadden 1995 Mechanical System and Signal Processing 9, 497–507.Application of orthogonal wavelets to early gear damage detection.

6. W. J. Staszewski and G. R. Tomlinson 1994 Mechanical System and Signal Processing 8,289–307. Application of the wavelet transform to fault detection in a spur gear.

7. J. Lin and L. Qu 2000 Journal of Sound and Vibration 234, 135–148. Feature extraction based onMorlet wavelet and its application for mechanical fault diagnosis.

8. C. J. Li and J. Ma 1997 NDT & E International 30, 143–149. Wavelet decomposition ofvibrations for detection of bearing-localized defect.

9. Z. Geng and L. Qu 1994 The British Journal of NDT 36, 11–15. Vibrational diagnosis ofmachine parts using the wavelet packet technique.

10. J. Zhao 1997 Doctoral dissertation, Xi’an Jiaotong University. Study for practical diagnosistechnique based on wavelets and neural networks.

11. S. Liu, R. Du and S.Yang 2000 Journal of Vibration Engineering 13, 577–584. Fault diagnosisfor diesel engines by wavelet packet analysis of vibration signal measured on cylinder head.

12. O. Rioul and P. Duhamel 1992 IEEE Transactions on Information Theory 38, 569–586. Fastalgorithms for discrete and continuous wavelet transform.

13. P. D. McFadden 2000 Mechanical System and Signal Processing 5, 805–817. Detection of gearfaults by decomposition of matched differences of vibration signals.

14. G. Dalpiaz, A. Rivola and R. Rubini 2000 Mechanical System and Signal Processing 3,387–412. Effectiveness and sensitivity of vibration processing techniques for local fault detectionin gears.

15. W. Q. Wang, F. Ismail and M. F. Golnaraghi 2001Mechanical System and Signal Processing5, 905–922. Assessment of gear damage monitoring techniques using vibration measurements.