Expert Systems with Applications 37 (2010) 4568–4579

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Expert Systems with Applications

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Application of mother wavelet functions for automatic gear and bearingfault diagnosis

J. Rafiee a,*, M.A. Rafiee a, P.W. Tse b

a Department of Mechanical, Aerospace and Nuclear Engineering, Jonsson Engineering Center, 110 8th Street, Rensselaer Polytechnic Institute, NY 12180-3590, USAb Smart Engineering Asset Management Laboratory, Department of Manufacturing Engineering and Engineering Management, 83 Tat Chee Ave., City University of Hong Kong,Kowloon Tong, Hong Kong

a r t i c l e i n f o a b s t r a c t

Keywords:Condition monitoringFault detection and diagnosisFeature extractionMother waveletDaubechies 44 (db44)GearBearingVibration signalFourth central moments

0957-4174/$ - see front matter � 2009 Elsevier Ltd. Adoi:10.1016/j.eswa.2009.12.051

* Corresponding author. Tel.: +1 518 276 6351; faxE-mail addresses: rafiee@rpi.edu, krafiee81@gmail

This paper introduces an automatic feature extraction system for gear and bearing fault diagnosis usingwavelet-based signal processing. Vibration signals recorded from two experimental set-ups were pro-cessed for gears and bearing conditions. Four statistical features were selected: standard deviation, var-iance, kurtosis, and fourth central moment of continuous wavelet coefficients of synchronized vibrationsignals (CWC-SVS). In this research, the mother wavelet selection is broadly discussed. 324 mother wave-let candidates were studied, and results show that Daubechies 44 (db44) has the most similar shapeacross both gear and bearing vibration signals. Next, an automatic feature extraction algorithm is intro-duced for gear and bearing defects. It also shows that the fourth central moment of CWC-SVS is a properfeature for both bearing and gear failure diagnosis. Standard deviation and variance of CWC-SVS demon-strated more appropriate outcome for bearings than gears. Kurtosis of CWC-SVS illustrated the accept-able performance for gears only. Results also show that although db44 is the most similar motherwavelet function across the vibration signals, it is not the proper function for all wavelet-basedprocessing.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Gears and bearings are important machine components in con-dition monitoring. In condition monitoring between vibration(Boulahbal, Golnaraghi, & Ismail, 1999; Dalpiaz, Rivola, & Rubini,2000) and acoustic measures (Apostoloudia, Douka, Hadjileontia-dis, Rekanos, & Trochidis, 2007; Lin, 2001), vibration signals haveshown satisfactory results and are applicable in noisy industrialfactories. Intelligent vibration monitoring is on-going research forautomatic fault detection and diagnosis of mechanical systems,particularly to identify the incipient failures because of the com-plexity of the vibration signals.

To inspect raw vibration signals, a wide variety of techniqueshave been introduced that may be categorized into two maingroups: classic signal processing (McFadden & Smith, 1984) andintelligent systems (Paya, Esat, & Badi, 1997). To make mentionof a few, FFT (Kar & Mohanty, 2006), Wigner–Ville distribution(Baydar & Ball, 2001; Zou & Chen, 2004), wavelet (Newland,1994; Wang & Gao, 2003), Hilbert–Huang transform (Peng & Chu,2004), blind source separation (Tse, Zhang, & Wang, 2006),statistical signal analysis (Jardine, Lin, & Benjevic, 2006), and their

ll rights reserved.

: +1 518 276 6025..com (J. Rafiee).

combinations (Fan & Zuo, 2006; Farina, Lucas, & Doncarli, 2008) areclassic signal processing methods. ANN-based (Paya et al., 1997).GA-based (Samanta, 2004), FL-based (Saravanan, Cholairajan, &Ramachandran, 2009), various similar classifiers (Saravanan, Ku-mar Siddabattuni, and Ramachandran (2008)), expert systems(Ebersbach & Peng, 2008), and combined algorithms (Rafiee, Tse,Harifi, & Sadeghi, 2009; Wu, Hsu, & Wu, 2009) could be classifiedas intelligent systems. Currently, industrial applications of intelli-gent monitoring systems have been increased by the progress ofintelligent systems.

One of the most significant issues in intelligent monitoring isrelated to feature extraction. This research mainly focuses on find-ing applicable features for bearing and gear fault detection anddiagnosis. Wavelet transform (WT), capable of processing station-ary and non-stationary signals in time and frequency domainssimultaneously, was used for feature extraction (Daubechies,1991). Despite the amount of previous research on wavelet trans-form, the selection of the mother wavelet function, which is a sig-nificant topic in signal analysis, is open to question. WT can bemainly divided into discrete and continuous forms. The former isfaster with lower CPU time, but continuous forms are more effi-cient since there is good resolution throughout the signals.

The dominant contributions to optimum mother waveletselection can be found in literature (e.g. Ahuja, Lertrattanapanich,

Normal condition

Ball fault

Cage fault

800 1600

Inner race fault

Fig. 1. Three segmented signals recorded in four bearing conditions (X-axis: number of datapoints–time, Y-axis: amplitude).

J. Rafiee et al. / Expert Systems with Applications 37 (2010) 4568–4579 4569

& Bose, 2005; Brechet, Lucas, Doncarli, & Farina, 2007; Flanders,2002; Singh & Tiwari, 2006; Tse, Yang, & Tam, 2004). Althoughapplication of Daubechies wavelets (db) has been referred to inseveral papers, low-order db (db1–db20) (Wu & Liu, 2007) aregenerally used, and the application of high-order db is rare(Antonino-Daviu, Riera-Guasp, Folch, & Palomares, 2006). In thisresearch, we have focused on 324 candidate mother functionsfrom different families to understand their behaviors for gearand bearing fault diagnosis. Then, a wavelet-based automatic fea-ture extraction system, applicable for intelligent systems (Rafiee,

-200

0

200

-200

0

200Slight

-200

0

200Medium

100 200 300-400

-200

0

200

400Broke

Nor

Fig. 2. Non-synchronous gearbox vibrations in one sample sig

Arvani, Harifi, & Sadeghi, 2007), is presented for gears and bear-ings faults.

2. Developed algorithm

The research follows these steps:

(1) For gear defects, raw vibration signals were recorded from amotorcycle gearbox in four conditions including: normalgearbox (NG), broken-tooth gear (BT), slight-worn gear

-worn

-worn

400 500 600 700

n-teeth

mal

nal (X-axis: number of samples–time, Y-axis: amplitude).

Fig. 3. Machine fault simulator TM.

Table 1Machine fault simulator™.

Specification Explanation

Motor ½ HP Marathon Electric� Three Phase AC motorController Delta VFD-S inverter or equivalentMax. motor

RPM10,000 (short duration) (around 166.67 Hz)

Range 0–10,000 rpm variableVoltage Drive input 120 or 230 VAC, single phase, 50/60 HzShaft

diameter5/8 in diameter, steel (around 1.5875 cm)

Bearings Two each, sealed rolling element with shaft centering featureBearing

housingTwo each, aluminum horizontally split bracket for simple andeasy changes,tapped to accept transducer mount bearing housing

No. of balls 8Ball diameter 0.2813 inPitch

diameter1.1228 in

BPFO 2.998BPFI 5.002BSF 1.871

4570 J. Rafiee et al. / Expert Systems with Applications 37 (2010) 4568–4579

(SW), and medium-worn gear (MW). For bearing defects,vibrations were recorded from a common machine faultsimulator for inner race fault (IR), ball fault (BL), and cagefault (CG).

(2) Recorded vibration signals were segmented into smallerunits: fifty segmented signals for each condition of the gear-box (e.g. NG and BT), and fifty segmented signals for bearingconditions (e.g. IR and CG) in duration of one complete rev-olution of the driven shaft (see Fig. 1).

(3) Piecewise cubic spline interpolation (PCSI) (Rafiee et al.,2009) was used to only synchronize the gearbox segmentedsignals because gearbox signals were not synchronous (e.g.see Fig. 2).

(4) Continuous wavelet coefficients of synchronized vibrationsignals (CWC-SVS) were calculated for gear data, and CWCwere calculated for bearing vibrations, both in the third levelof decomposition (23 series of wavelet coefficients for eachcondition). Decomposing the signals into higher levels maynot be efficient in this case as the coefficients did not showconsiderable difference in high scales for our case studies.

(5) Standard deviations, variances, kurtoses, and fourth centralmoments of CWC-SVS were calculated by 324 mother wave-let candidates. The most similar mother wavelet functionwas selected based on variance of CWC-SVS for gearboxand variance of CWC for bearing vibrations. Note that thefourth decomposition level was considered to find the mostsimilar mother wavelet across our experimental signalssince it is an off-line process.

(6) An automatic system is presented using the aforementionedstatistical features for gear and bearing fault identification.

2.1. Experimental bearing and gear dataset

A machine fault simulator (MFS) was used to collect the bearingdataset (see Fig. 3). In MFS, accelerometers can be installed on anumber of measurement points, which are labeled from ‘A’ to ‘G’for different components. The bearing dataset was recorded fromthe left-side bearing housing using the ‘C’ point. Motor rotationalfrequency and sampling frequency were fixed at 50 Hz and40 kHz, respectively. More detail can be found in Table 1. Vibrationsignals were recorded from a machine with four bearing conditionsincluding ball, cage, and outer race defects, and the normalcondition.

The experimental set-up to collect the gear dataset consists of afour-speed motorcycle gearbox holding the oil during data extrac-

tion, a driven motor with a constant nominal rotation speed of1420 RPM, a load mechanism, a multi-channel pulse analyzersystem, a triaxial accelerometer, a tachometer and four shockabsorbers under the bases of a test-bed. Gearbox vibration signalswere sampled at 16,384 Hz. More detail can be found in reference(Rafiee & Tse, 2009; Rafiee et al., 2009).

2.2. Feature extraction

In intelligent systems, feature extraction is of paramountimportance. Since there is no clear-cut rule for practical vibrationsignals, a reliable feature is the main factor. We applied continuouswavelet transform (CWT) in this research. The basic theory of CWThas been discussed in several papers (e.g. Peng & Chu, 2004; Rubini& Meneghetti, 2001; Staszewski & Tomlinson, 1994; Wang & McF-adden, 1996).

Continuous wavelet coefficients (CWC) show how well awavelet function correlates with the signal, supposing signal en-ergy and wavelet function energy are equal to one. In this re-search, CWC-SVS and CWC were calculated for gear and bearingsegmented signals with 324 mother wavelet candidates from dif-ferent wavelet families: Haar, Daubechies, Symlet, Coiflet, Gauss-ian, Morlet, complex Morlet, Mexican hat, bio-orthogonal, reversebio-orthogonal, Meyer, discrete approximation of Meyer, complexGaussian, Shannon, and frequency B-spline wavelets (see Tables2a–3b). To select the most similar mother wavelet for each set-up, 4 � 50 sample signals were processed for faulty and normalconditions. The similar mother wavelet function was selectedbased on a statistical algorithm, called SUMVAR for simplicity, re-ported in Rafiee et al. (2009). In that paper, the most similar func-tion to gear vibration signals was introduced. In current research,the same algorithm is being tested for bearing vibration signals.The relation between mother wavelet functions and SUMVAR isshown in Fig. 4 for bearings. In this figure, the relation has a peakrepresenting the db44 function. One of the reasons for this peak isthat the shape difference is slight between two mother waveletfunctions in the low-order db family (e.g. db5 and db6). However,the shape difference between two adjacent high-order db func-tions is obvious. Furthermore, the shape of db44 shows near-sym-metrical properties that the other high-order db functions lack. Byzooming in step by step on the relation in Fig. 4, the ranking ofthe less similar functions across our signals is revealed (seeFig. 4a and b).

Table 2aStudied wavelet families in this research.

No. Family (short form) Order Best order

1 Haar (db1) db 1 db 12-45 Daubechies(db) db 2 to db 45 db4446-50 Coiflet (coif) coif 1 to coif 5 Coif 551 Morlet (Morl) morl Morl52-147 Complex morlet (cmor Fb-Fc)a Included Table 2b Cmor 1-0.1148 Discrete Meyer (dmey) dmey Dmey149 Meyer (meyr) meyr Meyr150 Mexican Hat (mexh) mexh Mexh151-200 Shannon (Shan Fb-Fc) a Included Table 2b Shan 1-0.1201-260 Frequency B-spline (fbsp M–Fb–Fc) a Included Table 2b Fbsp 2-1-0.1261-267 Gaussian (gaus) gaus 1 to gaus7 Gaus 7268-275 Complex Gaussian (cgau) cgau 1 to cgau8 Cgau 8276-290 Biorthogonal (bior Nr.Nd) b Included Table 2b Bior 3.1291-305 Reverse Biorthogonal (rbio Nr.Nd) b Included Table 2b Rbio 5.5306-324 Symlet (sym) Sym 2 to sym 20 Sym 15

a Fb is a bandwidth parameter, Fc is a wavelet center frequency, M is an integer order parameter.b Nr and Nd are orders: r for reconstruction/d for decomposition.

Table 2bStudied wavelet families in detail.

No Wave No Wave No Wave No Wave No Wave

52 1-1.5 100 3-1.1 148 dmey 196 2-0.6 244 2-0.2-0.453 1-1 101 3-1.2 149 meyr 197 2-0.7 245 2-0.2-0.554 1-0.5 102 3-1.3 150 mexh 198 2-0.8 246 2-0.2-0.655 1-0.3 103 3-1.4 151 0.1-0.1 199 2-0.9 247 2-0.2-0.756 1-0.2 104 3-1.5 152 0.1-0.2 200 1-1 248 2-0.2-0.857 1-0.1 105 3-1.6 153 0.1-0.3 201 1-0.1-0.1 249 2-0.2-0.958 1-0.05 106 3-1.8 154 0.1-0.4 202 1-0.1-0.2 250 2-0.2-159 1-0.02 107 3-1.9 155 0.1-0.5 203 1-0.1-0.3 251 3-0.2-0.160 1-0.01 108 3-2 156 0.1-0.6 204 1-0.1-0.4 252 3-0.2-0.261 2-0.1 109 3-2.1 157 0.1-0.7 205 1-0.1-0.5 253 3-0.2-0.362 2-0.2 110 3-2.2 158 0.1-0.8 206 1-0.1-0.6 254 3-0.2-0.463 2-0.3 111 3-2.3 159 0.1-0.9 207 1-0.1-0.7 255 3-0.2-0.564 2-0.4 112 3-2.4 160 0.1-1 208 1-0.1-0.8 256 3-0.2-0.665 2-0.5 113 3-2.5 161 0.2-0.1 209 1-0.1-0.9 257 3-0.2-0.766 2-0.6 114 3-2.6 162 0.2-0.2 210 1-0.1-1 258 3-0.2-0.867 2-0.7 115 3-2.7 163 0.2-0.3 211 2-0.1-0.1 259 3-0.2-0.968 2-0.8 116 3-2.8 164 0.2-0.4 212 2-0.1-0.2 260 3-0.2-169 2-0.9 117 3-2.9 165 0.2-0.5 213 2-0.1-0.3 276 1.170 2-1 118 3-3 166 0.2-0.6 214 2-0.1-0.4 277 1.371 2-1.1 119 4-0.1 167 0.2-0.7 215 2-0.1-0.5 278 1.572 2-1.2 120 4-0.2 168 0.2-0.8 216 2-0.1-0.6 279 2.273 2-1.3 121 4-0.3 169 0.2-0.9 217 2-0.1-0.7 280 2.474 2-1.4 122 4-0.4 170 0.2-1 218 2-0.1-0.8 281 2.675 2-1.5 123 4-0.5 171 0.5-0.1 219 2-0.1-0.9 282 2.876 2-1.6 124 4-0.6 172 0.5-0.2 220 2-0.1-1 283 3.177 2-1.8 125 4-0.7 173 0.5-0.3 221 3-0.1-0.1 284 3.378 2-1.9 126 4-0.8 174 0.5-0.4 222 3-0.1-0.2 285 3.579 2-2 127 4-0.9 175 0.5-0.5 223 3-0.1-0.3 286 3.780 2-2.1 128 4-1 176 0.5-0.6 224 3-0.1-0.4 287 3.981 2-2.2 129 4-1.1 177 0.5-0.7 225 3-0.1-0.5 288 4.482 2-2.3 130 4-1.2 178 0.5-0.8 226 3-0.1-0.6 289 5.583 2-2.4 131 4-1.3 179 0.5-0.9 227 3-0.1-0.7 290 6.884 2-2.5 132 4-1.4 180 0.5-1 228 3-0.1-0.8 291 1.185 2-2.6 133 4-1.5 181 1-0.1 229 3-0.1-0.9 292 1.386 2-2.7 134 4-1.6 182 1-0.2 230 3-0.1-1 293 1.587 2-2.8 135 4-1.8 183 1-0.3 231 1-0.2-0.1 294 2.288 2-2.9 136 4-1.9 184 1-0.4 232 1-0.2-0.2 295 2.489 2-3 137 4-2 185 1-0.5 233 1-0.2-0.3 296 2.690 3-0.1 138 4-2.1 186 1-0.6 234 1-0.2-0.4 297 2.891 3-0.2 139 4-2.2 187 1-0.7 235 1-0.2-0.5 298 3.192 3-0.3 140 4-2.3 188 1-0.8 236 1-0.2-0.6 299 3.393 3-0.4 141 4-2.4 189 1-0.9 237 1-0.2-0.7 300 3.594 3-0.5 142 4-2.5 190 1-1 238 1-0.2-0.8 301 3.795 3-0.6 143 4-2.6 191 2-0.1 239 1-0.2-0.9 302 3.996 3-0.7 144 4-2.7 192 2-0.2 240 1-0.2-1 303 4.497 3-0.8 145 4-2.8 193 2-0.3 241 2-0.2-0.1 304 5.598 3-0.9 146 4-2.9 194 2-0.4 242 2-0.2-0.2 305 6.899 3-1 147 4-3 195 2-0.5 243 2-0.2-0.3

J. Rafiee et al. / Expert Systems with Applications 37 (2010) 4568–4579 4571

Table 3aAppropriate mother wavelets for gear fault identification system.

No. of properfeatures

Mother wavelet functions

8 cmor2-0.1, cmor3-0.1, cmor4-0.17 cmor1-0.1, cmor1-0.05, cmor1-0.02, cmor1-0.01, cmor2-

0.2, shan0.2-0.1, shan0.5-0.2, fbsp1-0.2-0.16 Haar, cmor1-0.2, cmor3-0.1, shan 0.5-0.1, shan 1-0.1,

bior1.1, rbio1.1

Table 3bAppropriate mother wavelets for bearing fault identification system.

No. of properfeatures

Mother wavelet functions

22 rbio3.120 Haar, bior3.1, rbio1.1, cmor1-0.1, cmor1-0.2, shan2-0.1,

shan2-0.2, shan2-0.3, shan2-0.418 cmor1-0.05, cmor1-0.02, cmor1-0.1, cmor2-0.1, cmor3-0.1,

shan2-0.5, shan2-0.6

50 100 150 200 250 3000

20

40

60

80

Decomposition level=4

Mother wavelet function

SU

MV

AR

Crit

erio

n Db44

Db45

Fig. 4. SUMVAR vs. 324 mother wavelets for bearing signals.

50 100 1500

2

4

6

8

10

12x 1010

Mother wave

SU

MV

AR

crit

erio

n

100 1500

2

4

6

8x 109

Db 44

Cmor3-0.1

Cmor1-0.01

Shan0.2-0.1

Cmor4-0.1

Shan0.5-0.1

Sha

Fig. 4a. The complex mother w

4572 J. Rafiee et al. / Expert Systems with Applications 37 (2010) 4568–4579

3. Results of mother wavelet selection

The behaviors of wavelet functions are almost the same for bothexperimental set-ups with a few minor differences. Results showthat complex mother wavelet functions have proper similarity tothe gearbox and bearing signals after db44, db45, and bior 3.1.The minor differences in wavelet functions fitted for bearing andgear signals are mostly related to center frequencies in complexmother wavelets, which are dependent on the frequency contentsof signals.

In literature, the Morlet function is a common function in ma-chine condition monitoring (Lin & Qu, 2000; Nikolaou & Antonia-dis, 2002; Saravanan et al., 2008). The research verifies thatMorlet has better similarity to both vibration signals in comparisonto many other functions such as Daubechies (1–43), Coiflet, Symlet,complex Morlet, Gaussian, complex Gaussian, and Meyer (seeFig. 4c) for both set-ups.

Among the studied mother wavelets, results also show thatdb44 is the most similar function across both gear and bearingvibration signals. Using this function, the periodic behavior ofvibration signals can be interpreted from Fig. 5. In this figure,CWC-SVS calculated by db44 and calculated by Haar were com-pared in a broken-tooth gear signal. It can be seen that periodicbehaviors of vibration signals are revealed using db44 as well ashigher values in wavelet coefficients.

The drawback of the db44 function is that the high-order dbfunctions take more CPU time than most others (see Fig. 6).

4. Automatic gear fault identification system

In this section, there are two goals. The primary goal is featureextraction for detection and identification of gears and bearings.The secondary goal is to find that the most similar function isnot the proper function for any wavelet-based processingtechniques.

As mentioned, the third decomposition level was used for signaldecomposition in this section. Four features vectors were selected

200 250 300 324let functions

200 250

Bior3.1

n0.5-0.2

Fbsp3-0.1-0.5

Fbsp2-0.1-0.5

Fbsp3-0.1-1

Fbsp2-0.1-1

Fbsp3-0.2-1

Fbsp2-0.2-1

avelets vs. SUMVAR (Gear).

Fig. 4b. The complex Morlet vs. SUMVAR (Gear).

20 40 60 80 100 120 1400

1

2

3

4x 10-4

Morlet

Db43Cmor4-0.1

Db1Coiflet

Mexh

Meyr

Dmey

Cmor1-1.5

Cmor3-0.1

Cmor1-0.2

Cmor2-0.1

Fig. 4c. The other mother wavelets vs. SUMVAR (bearing).

J. Rafiee et al. / Expert Systems with Applications 37 (2010) 4568–4579 4573

for the automatic failures identification system. Therefore, thereare 4 � 23 feature elements for each condition. The range of thefeature vectors (upper and lower limits) is defined for each classof faults and for each feature as Rlk

ðiÞ:

RlkðiÞ ¼ ½minflkðiÞg;maxflkðiÞg�: ð1Þ

Let lk ¼ fl1;l2;l3;l4g, where l1, l2, l3 and l4 are standarddeviation, variance, kurtosis, and the fourth central moment ofCWC-SVS (e.g. see Fig. 7 for gearbox system), respectively, andi = {1, 2, . . . , 2l}, where l is the decomposition level. 4 � 50 samplesignals were used to calculate Rlk

ðiÞ. (e.g. see Fig. 8, where RlkðiÞ

shown by blue dash-lines). In this figure, there is a fair classifica-tion between the faults and the normal gearbox. However, a fewsamples are not in the determined ranges (shown with red1 dashline) in slight- and medium-worn gears, because these failures aresimilar to each other. Therefore, this feature may not be an appropri-ate feature for fault identification of the gear failures.

We also optimized the feature ranges using the followingequation:

ORlkðiÞ ¼ Rlk

ðiÞ � adc; ð2Þ

1 For interpretation of color in Figs. 1–14, the reader is referred to the web versionof this article.

where a is a ratio of the corresponding condition range to therange of nearest condition, and dc is the distance between theupper/lower limit of one condition and the lower/upper limit ofthe closest condition. In Fig. 9, Rl4

ð8Þ, the range of failures(SW, MW), the range of the defects and the optimized range aredepicted based on the fourth central moment of CWC-SVS. If amonitoring system is to be designed using the algorithm basedon the optimized range, then all related failures should be takeninto consideration. For example, if the system is designed for bear-ing fault identification, then all related defects (e.g. inner race, out-er race, ball, and cage faults) should be considered in the design ofthe monitoring system.

For simplicity, statistical variables are often used in non-dimen-sionalized form, and normalization is a simple tool to meet thisobjective. For any statistical variable, x, its standardized normalvariable, b, is defined as:

b ¼ x� x0

l1ð3Þ

in which x0

is the mean value of the population, and l1 is its stan-dard deviation. After normalizing the statistical features (e.g. seeFig. 10), a Boolean-based diagonal matrix is defined to comparethe feature ranges of the four conditions for each set up as follows:

100 200 300 400 500 600 700-1.5

-1

-0.5

0

0.5

1

1.5x 104 (4,2), variance=3873934.66

100 200 300 400 500 600 700-500

-400

-300

-200

-100

0

100

200

300

400

500(4,2), variance=3254.14

Db 1Db44

100 200 300 400 500 600 700-800

-600

-400

-200

0

200

400

600

800(4,13), variance=1269.29

100 200 300 400 500 600 700-800

-600

-400

-200

0

200

400

600

800(4,13), variance=41926.39

Db 1Db44

Fig. 5. CWC-SVS of a sample signal of broken-tooth gear calculated by db44 and Haar (variance of CWC-SVS and tree decomposition in title).

50 100 150 200 250 3000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Mother wavelet functions

Com

puta

tiona

l tim

e of

CW

C-S

VS

(se

cond

)

10 20 30 40

0.1

0.2

0.3

0.4

Db1 to Db45310 315 320

0

0.5

1

1.5

2

Sym2 to Sym20

Db 44

Dmey

Fig. 6. CPU time calculated for CWC-SVS of one sample signal using a common P4 PC.

4574 J. Rafiee et al. / Expert Systems with Applications 37 (2010) 4568–4579

BT

MW

SW

NG

Lm

BTMWSWNG

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

1/01/01/01/0

1/01/01/01/0

1/01/01/01/0

1/01/01/01/0

μ

IR

CG

BL

NB

Lm

IRCGBLNB

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

1/01/01/01/0

1/01/01/01/0

1/01/01/01/0

1/01/01/01/0

μð4Þ

Fig. 7. Statistical features extracted from gear signals (db44, third decomposition level).

50 100 1500

0.05

0.1

0.15

0.2

0.25

60 70 80 90 100 110 120 130 140 1500.015

0.016

0.017

0.018

0.019

0.02

0.021

0.022

0.023

0.024

0.025 Medium worn

Slight worn

Broken tooth

Normal gearbox

Fig. 8. Feature ranges in four gearbox conditions.

J. Rafiee et al. / Expert Systems with Applications 37 (2010) 4568–4579 4575

for gear and bearing, respectively. To determine the values for Lml:

Lmlði; jÞ ¼ 0() FaðliÞ \ FbðliÞ ¼ /; ð5Þ

where Fa(li) and Fb(li) stand for the set of samples in feature li, fortwo different conditions.

Lmlði; jÞ ¼ 0() a ¼ b: ð6Þ

It means that all diagonal terms in Lml are 0.In Lml, when all terms are 1 excluding diagonal terms, it means

that the system can identify the related condition. Simply, the fea-

ture range in that condition is not overlapping with the other con-ditions and then Lml is:

Lml ¼

0 1 1 11 0 1 11 1 0 11 1 1 0

26664

37775: ð7Þ

After calculation of Lml for all feature vectors, Lml matrix entriesare added and we have BFz:

50 100 1500

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

Range of SW Optimized range

Optimized range

Range of MW

Fig. 9. Optimized feature range for gear fault identification [Feature: fourth centralmoment of CWC/scale: (3, 7)].

50 1000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2(3,3), Normalized variance

50 100.01

0.015

0.02

0.025

0.03

NG

SW

Fig. 10. A sample selected feature

Table 4aBFz for statistical features calculated by Haar vs. decomposition tree (Gearbox).

Statistical criteria (3, 0) (3, 1) (3, 2) (3, 3)

StD 10 10 10 10Variance 10 10 10 10Fourth central moment 12 12 12 12Kurtosis 10 10 10 10Rt

Table 4bBFz for statistical features calculated by db44 vs. decomposition tree (Gearbox).

Statistical criteria (3, 0) (3, 1) (3, 2) (3, 3)

StD 10 8 8 2Variance 10 8 8 2Fourth central moment 10 8 8 6Kurtosis 10 6 6 6Rt

Table 4cBFz for statistical features calculated by rbio3.1 vs. decomposition tree (bearing).

Statistical criteria (3, 0) (3, 1) (3, 2) (3, 3)

StD 12 12 12 12Variance 12 12 12 12Fourth central moment 10 10 12 12Rt

4576 J. Rafiee et al. / Expert Systems with Applications 37 (2010) 4568–4579

BFz ¼Xn

i;j¼1

Lmlði; jÞ; ð8Þ

where n is the number of conditions (n = 4 in this research for bothset-ups). In Tables 4a–4d, BFz has been calculated for proposed sta-tistical criteria using Haar and db44, respectively:

if BFz ¼ nðn� 1Þ; ð9Þ

then the proper feature is selected for identification purposes (e.g.see Fig. 10). Among 4 � 23 features, those features having the largerentries for the condition explained by Eq. (9) are the proper fea-tures. Therefore, the first goal is fulfilled, which was to design anautomatic feature selection system for gears and bearings.

For automatic fault identification, mother wavelet functions areselected such that the failures are properly identified. In otherwords, suitable mother wavelets are those which can magnify thefaulty components of the signals for the sake of easy identification.

150 200

of CWC-SVS by cmor1-1.5

0 150

MW

BT

for gear fault identification.

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

6 8 8 10 72 756 8 8 10 72 7510 8 12 12 90 93.7510 10 6 8 74 77.08

80.2

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

8 8 4 0 48 508 8 4 0 48 508 8 2 0 50 52.086 6 4 4 48 50

50.52

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

12 12 12 12 96 10012 12 12 12 96 10012 12 12 12 96 95.83

98.61

10 20 30 4084

86

88

90

92

94

96

98

Decomposition level=3

Db family

Perc

ent o

f cla

ssifi

catio

n

Db45

Haar

Db44

Fig. 11. Classification rate vs. db family for bearings.

Table 4dBFz for statistical features calculated by db44 vs. decomposition tree (bearing).

Statistical criteria (3, 0) (3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6) (3, 7) Total Total (%)

StD 10 10 12 12 10 8 10 12 84 87.5%Variance 10 10 12 12 10 8 10 12 84 87.5%Fourth central moment 10 10 10 10 10 8 10 10 78 81.25%Rt 85.41

50 100 1500

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016(3,0)

50 100 1500

0.005

0.01

0.015

0.02(3,2)

Normalized 4th central moment of CWC-SVS

NG NG

SWSW

MW MW

Fig. 12. Normalized fourth central moment of CWC-SVS for worn gears calculatedwith Haar.

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

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

NBBL CG

IR

IRNB BL CG

Fig. 13. Normalized variance of CWC for 4 � 50 sample signals calculated by bior3.1 (bearing).

J. Rafiee et al. / Expert Systems with Applications 37 (2010) 4568–4579 4577

4578 J. Rafiee et al. / Expert Systems with Applications 37 (2010) 4568–4579

For 324 mother wavelet functions, the summation of features ful-filling the condition defined by Eq. (9) for each function is calcu-lated. The largest of these calculations represents the most propermother function for the fault identification algorithm.

5. Results and discussion of part II

In this section, the classification rate between two conditions isdefined as follows:

Rt ¼P4

i¼1

P2l

li¼1BFz

4� nðn� 1Þ � 100; ð10Þ

where i is the number of feature vectors (three and four for bearingand gear, respectively). l is the decomposition level. Note that allfeatures were used to calculate classification rate for gear signals.However, the kurtosis feature was not considered for bearing sig-nals. The most proper mother wavelet was selected based on theclassification rate that optimally classifies the signals. Therefore,the appropriate features could automatically be selected for gearfault identification purposes using Rt.

The results show that among a wide variety of mother wavelets,complex Morlet wavelets have satisfactory performances for bothbearing and gear fault identification. The point is to determinewhich mother wavelet has more distinctive wavelet coefficientsfor different conditions in fault identification. For example, asshown in Fig. 5, wavelet coefficients calculated by the Haar func-tion are more obvious than those calculated by db44 where thereare high-impact vibrations of the broken-tooth failure. Therefore,in this part, Haar has proven its ability for both gear and bearing

(3,1)

(3,4) (3,5)

(3,0)

CGBL

BLNB

CG

IR

NB

IR

Fig. 14. Normalized fourth central moment of CWC for bearing signals calculat

signals based on classification rate (see Fig. 11). The goal for satis-factory fault classification is to have the more limited range of ex-tracted features and therefore, the fluctuations of the features’values are not of interest. The normalized fourth central momentsof CWC, calculated by Haar, is demonstrated in Fig. 12 for gear sig-nals. The normalized features, calculated by bior3.1, are also dem-onstrated in Figs. 13 and 14 for bearings. Since a faulty cage is amore serious failure than the other faults, the fourth central mo-ment showed higher amplitudes compared to the other conditions.In Fig. 14, zooming step by step from (3, 0) to (3, 7) shows that thenormalized fourth central moments of CWC is a desired feature. Asdemonstrated in Tables 4a and 4b, the fourth central moment hadthe more accurate performance for both gear and bearing classifi-cation. Most importantly, it is a suitable feature for random excita-tion. Standard deviation and variance features have demonstratedproper results for gears and bearings, particularly for incipientfaults (e.g. worn gears). Kurtosis of CWC is only a mediocre featurefor gearbox signals, particularly serious failures such as a brokentooth gear.

Overall, the selected features showed accurate behaviors forboth bearings and gears. However, the performance of featuresfor bearings looks more satisfactory in this research due to thesophisticated gearbox signals and the similar worn gear failures.

In this part, the selected decomposition level was verified suchthat the number of appropriate features was compared in the lev-els three, four and five using the proposed algorithm. Results areapproximately the same when increasing the decomposition level.Therefore, the third decomposition level is enough for these casestudies. Results also show that db44 is not the appropriate functionfor the fault identification algorithm although db44 is significantly

(3,2) (3,3)

(3,6) (3,7)

ed with bior3.1-By zooming in step by step from scale (3, 0) to scale (3, 7).

J. Rafiee et al. / Expert Systems with Applications 37 (2010) 4568–4579 4579

similar to the signals. The Haar function (Li, Qu, & Liao, 2007),needing less CPU time, is a good function for the identification goal.

6. Conclusions

In summary:

1. db44 showed the most similarity across both faulty gears andbearings. It matches with randomly high impact signals (e.g.broken-tooth gear). It might also be effective for the otherwavelet-based applications which are based on the similaritybetween signals and the mother wavelet.

2. The fourth central moment of CWC is the appropriate featurefor gear and bearing defects, particularly uncorrelated randomexcitations. Standard deviation and variance of CWC are properfeatures for both set-ups, particularly bearings. Kurtosis of CWCdid not show any consequential relation towards the bearingdataset.

3. Proposed feature vectors are applicable for training purposes ofintelligent systems (e.g. ANNs) due to having enough value fluc-tuations in sample signals and small-size structures (Rafieeet al., 2009).

4. Symmetric mother wavelets had satisfactory results in theaforementioned case-studies. Although db functions are notsymmetric, db44 possesses a near-symmetric attribute.

5. Wavelet center frequencies are important factors in the selec-tion of complex mother functions. For the gearbox dataset,complex mother wavelets with the center frequency of 0.1had satisfactory results compared to those with other centerfrequencies.

6. In literature, Morlet function is one of the common wavelets inthe field. In this research, it has been shown that the Morlet is asimilar function across the vibration signals compared to sev-eral wavelet families.

7. Selection of the mother wavelet based on the similaritybetween signals and mother wavelets is not a proper methodfor all wavelet-based techniques. It is only appropriate for thosemethods based on the similarity.

Acknowledgements

The research is partially supported by Department of Mechani-cal, Aerospace and Nuclear Engineering at Rensselaer PolytechnicInstitute, NY, USA, and partially supported by Research GrantsCouncil of Hong Kong SAR, China (Project no. CityU 120506).

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