fault diagnosis of ball bearings using continuous wavelet transform

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Applied Soft Computing 11 (2011) 2300–2312

Contents lists available at ScienceDirect

Applied Soft Computing

journa l homepage: www.e lsev ier .com/ locate /asoc

ault diagnosis of ball bearings using continuous wavelet transform

.K. Kankar, Satish C. Sharma, S.P. Harsha ∗

ibration and Noise Control Laboratory, Mechanical & Industrial Engineering Department, Indian Institute of Technology Roorkee, Roorkee 247667, Uttaranchal, India

r t i c l e i n f o

rticle history:eceived 4 January 2010eceived in revised form 12 July 2010ccepted 1 August 2010vailable online 26 November 2010

eywords:nergy to Shannon Entropy ratioelative Wavelet Energyupport vector machinertificial neural networkelf-organizing maps

a b s t r a c t

Bearing failure is one of the foremost causes of breakdown in rotating machines, resulting in costly sys-tems downtime. This paper presents a methodology for rolling element bearings fault diagnosis usingcontinuous wavelet transform (CWT). The fault diagnosis method consists of three steps, firstly the sixdifferent base wavelets are considered in which three are from real valued and other three from com-plex valued. Out of these six wavelets, the base wavelet is selected based on wavelet selection criterionto extract statistical features from wavelet coefficients of raw vibration signals. Two wavelet selectioncriteria Maximum Energy to Shannon Entropy ratio and Maximum Relative Wavelet Energy are used andcompared to select an appropriate wavelet for feature extraction. Finally, the bearing faults are classifiedusing these statistical features as input to machine learning techniques. Three machine learning tech-niques are used for faults classifications, out of which two are supervised machine learning techniques,i.e. support vector machine (SVM), artificial neural network (ANN) and other one is an unsupervised
machine learning technique, i.e. self-organizing maps (SOM). The methodology presented in the paper isapplied to the rolling element bearings fault diagnosis. The Meyer wavelet is selected based on MaximumEnergy to Shannon Entropy ratio and the Complex Morlet wavelet is selected using Maximum RelativeWavelet Energy criterion. The test result showed that the SVM identified the fault categories of rollingelement bearing more accurately for both Meyer wavelet and Complex Morlet wavelet and has a better
comn effi

diagnosis performance ashigher faults classificatio

. Introduction

Rolling element bearings are used in a wide variety of rotatingachineries from small hand-held devices to heavy duty industrial

ystems and are primary cause of breakdowns in these machines.here are several vibration and acoustic measurement methodsave been used for the detection of defects in rolling elementearings [1]. However, the complex and non-stationary vibrationignals with a large amount of noise make the challenging inault detection of rolling element bearings, especially at the earlytage. Therefore, development of effective maintenance strategiesnd novel diagnosis procedures are needed using different signalnalyzing procedures for features extraction and soft computingechniques in order to avoid the system shutdowns, and even catas-
rophes involving human fatalities and material damage.
To analyze vibration signals, different techniques such asime, frequency and time–frequency domain are extensively used.amantha and Balushi [2] have presented a procedure for fault diag-

∗ Corresponding author. Tel.: +91 9917489849/01332 286602;ax: +91 1332 285665.

E-mail addresses: [email protected] (P.K. Kankar),[email protected] (S.C. Sharma), [email protected] (S.P. Harsha).

568-4946/$ – see front matter © 2010 Elsevier B.V. All rights reserved.oi:10.1016/j.asoc.2010.08.011

pared to the ANN and SOM. Features selected using Meyer wavelet givesciency with SVM classifier.

© 2010 Elsevier B.V. All rights reserved.

nosis of rolling element bearings through artificial neural network(ANN). The characteristic features of time-domain vibration signalsof the rotating machinery with normal and defective bearings haveused as inputs to the ANN. Lei et al. [3] have proposed a method forintelligent fault diagnosis of rotating machinery based on waveletpacket transform (WPT), empirical mode decomposition (EMD),dimensionless parameters, a distance evaluation technique andradial basis function (RBF) network. The effectiveness of wavelet-based features for fault diagnosis of gears using support vectormachines (SVM) and proximal support vector machines (PSVM) hasbeen revealed by Saravanan et al. [4]. Yang et al. [5] have proposeda method of fault feature extraction for roller bearings based onintrinsic mode function (IMF) envelope spectrum. Li et al. [6] haveshown that the feature vectors obtained by the FFT, wavelet trans-form, bi-spectrum, etc., can be used as fault features and the HMMsas the classifiers to recognize the faults of the speed-up and speed-down process in rotating machinery. Fault diagnosis of turbo-pumprotor based on support vector machines with parameter optimiza-tion by artificial immunization algorithm has been done by Yuan
and Chu [7]. Various artificial intelligence techniques are used withwavelet transforms for fault detection in rotating machines [8–14].
An extensive comparative study concerning the performance ofSVM against 16 other popular classifiers, using 21 different datasets, is carried out by Meyer et al. [15]. For classification, simple

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P.K. Kankar et al. / Applied Soft Com

Nomenclature

||w||−2 Geometrical marginC Error penalty�i The distance between the margin and the examples

xi that lying on the wrong side of the marginB Bias or threshold�i Lagrange multipliersW(�) Lagrange functionxpi ith input of the pth input vectorwjk Synaptic weight between hidden and output layerN Scale numberE(n) Energy of nth scaleCn,i ith wavelet coefficient of nth scalem Number of wavelet coefficientsEtotal Total energy of all scalesPn Relative Wavelet Energy of nth scale�(n) Energy to Shannon Entropy ratio of nth scaleSentropy(n) Shannon Entropy of nth scale

sidFsrclafaaSinahfi

wwadsliasMaai((eo

2

si

Pi Energy probability distribution of the wavelet coef-ficients

tatistical procedures and ensemble methods proved very compet-tive, mostly producing good results without the inconvenience ofelicate and computationally expensive hyper parameter tuning.or regression tasks, neural networks, projection pursuit regres-ion and random forests often yielded better results than SVMs. Theesults verify that SVM classifiers rank at the very top among theselassifiers, although there are cases for which other classifiers gaveower error rates [15]. Kankar et al. [16] have conducted a compar-tive experimental study for the effectiveness of ANN and SVM inault diagnosis of ball bearings and concluded that the classificationccuracy for SVM is better than of ANN. Based on the comparisonnd recommendation of previous studies, authors have employedVM and ANN for bearing faults classification. In many situations,t is not easy to collect training data set because of routine mainte-ance and periodically repairs. To solve this problem, authors havelso used SOM because unlike SVM and ANN, SOM-based approachas the practical advantage of learning and producing fault classi-cations without any supervision.

Previous research articles have highlighted the advantages ofavelet transforms when applied to fault diagnosis. In presentork, a methodology is proposed for selection of most appropri-

te wavelet and to determine scale corresponding to characteristicefect frequency based on wavelet selection criterion. These rawignals are divided into 27 sub-signals, i.e. 128 scales in seventhevel of decomposition to convert the complex vibration signalsnto simplified signals with more resolution. Six different waveletsre considered with each 27 sub-signals, i.e. 128 scales. Two waveletelection criteria Maximum Energy to Shannon Entropy ratio andaximum Relative Wavelet Energy are used and compared to select

n appropriate wavelet for feature extraction. Statistical featuresre calculated from continuous wavelet coefficients and are fed asnput to machine learning techniques, i.e. support vector machineSVM), artificial neural network (ANN) and self-organizing mapsSOM). The results showed that the proposed methodology canxtract useful features from the original data and dimension ofriginal data can also be reduced by removing irrelevant features.

. Review of machine learning techniques

Machine learning is an approach of using examples (data) toynthesize programs. In the particular case when the examples arenput/output pairs, it is called supervised learning. In a case, where

puting 11 (2011) 2300–2312 2301

there are no output values and the learning task is to gain someunderstanding of the process that generated the data, this type oflearning is said to be unsupervised. In the present study, the twosupervised machine learning techniques, i.e. SVM and ANN are con-sidered and the unsupervised machine learning technique like SOMis considered. Pattern recognition and classification using machinelearning techniques are described here [17].

2.1. Support vector machine (SVM)

SVM is a supervised machine learning method based on the sta-tistical learning theory. It is a useful method for classification andregression in small-sample cases such as fault diagnosis. Meyeret al. [15] have carried out an extensive comparative study concern-ing the performance of SVM against 16 other popular classifiers,using 21 different data sets. Their results verified that SVM clas-sifiers rank at the top among other classifiers, although there aresome cases for which other classifiers gave lower error rates. In thismethod, a boundary is placed between the two different classesand orients it, in such a way that the margin is maximized, whichresults in least generalization error. The nearest data points thathave been used to define the margin are called Support Vectors. Thisis implemented by reducing it to a convex optimization problem:minimizing a quadratic function under linear inequality constraints[18]. A training sample set {(xi,yi)}; i = 1–N is considered, whereN is total number of samples. The hyperplane f(x) = 0 that sepa-rates the given data can be obtained as a solution to the followingoptimization problem.

Minimize12

‖w‖2 + C

N∑i=1

�i (1)

Subject to

{yi(wT xi + b) ≥ 1 − �i

�i ≥ 0, i = 1, 2, . . . , N(2)

where C is a constant representing error penalty. Rewriting theabove optimization problem in terms of Lagrange multipliers, leadsto the following problem,

Maximize W(�) =N∑

i=1

�i − 12

N∑i,j=1

yiyj�i�j(xixj) (3)

Subject to

⎧⎪⎨⎪⎩

0 ≤ �i ≤ CN∑

i=1

�iyi = 0, i = 1, 2, . . . , N(4)

The Sequential Minimal Optimization (SMO) algorithm givesan efficient way of solving the dual problem arising from thederivation of the SVM. SMO decomposes the overall quadratic pro-gramming problem into quadratic programming sub-problems.

2.2. Artificial neural network (ANN)

Artificial neural network is an interconnected group of artifi-cial neurons. These neurons use a mathematical or computationalmodel for information processing. ANN is an adaptive system thatchanges its structure based on information that flows through thenetwork [17].

A single neuron consists of synapses, adder and activation func-tion. Bias is an external parameter of neural network. Model of a
neuron can be represented by following mathematical model.
yk = �

(p∑

i=1

wkixi + wk0

)(5)

2302 P.K. Kankar et al. / Applied Soft Computing 11 (2011) 2300–2312

t(en(w

2

ortiinsf

gncs

3

tipprTDwrlh

Table 1Parameters of ball bearings.

Parameter Value

Outer race diameter 28.262 mmInner race diameter 18.738 mm

Fig. 1. Kohonen model of SOM.

Input vector comprises of ‘p’ inputs multiplied by their respec-ive synaptic weights, and sum off all weighted inputs. A thresholdbias) is used with constant input. Activation function converts gen-ral output into a limited range of output. Intelligence of neuraletwork lies in the weights between neurons. Back PropagationBP) algorithm is used as learning algorithm for calculating synapticeights.

.3. Self-organizing maps (SOM)

Self-organizing maps are special class of ANN and are basedn competitive learning [17]. In self-organizing maps, the neu-ons are placed at the nodes of a lattice that is usually one- orwo-dimensional. The neurons become selectively tuned to variousnput patterns or classes of input patterns in the course of a compet-tive learning process. The location of neuron is so tuned (winningeurons) that it becomes ordered with respect to each other inuch a way that a meaningful co-ordinate system for different inputeatures is created over the lattice.

A SOM is therefore characterized by the formation of a topo-raphic map of the input patterns. The spatial locations of theeurons in the lattice are indicative of intrinsic statistical featuresontained in the input patterns. The type of SOM used in the presenttudy is Kohonen model as shown in Fig. 1.

. Experimental setup

The problem of predicting the degradation of working condi-ions of bearings before they reach to the alarm or failure thresholds extremely important in industries to fully utilize the machineroduction capacity and to reduce the plant downtime. In theresent study, an experimental test rig is used and vibrationesponses for healthy bearing and bearing with faults are obtained.able 1 shows dimensions of the ball bearings used for the study.
ata acquisition and analysis system consists of Vibra-Quest soft-are which is designed in LabVIEW for quick data acquisition,
eview and storage. Data acquisition hardware consists of 16 ana-og input channels for simultaneous sampling. PCI bus providedigh-speed data acquisition (102.4k samples/s). A remote optical

Fig. 2. Bearing components with faults induced in them. (a) Outer ra

Ball diameter 4.762 mmBall number 8Contact angle 0◦

Radial clearance 10 �m

sensor with a visible red LED light source is used to measure rotorspeed. Piezoelectric accelerometers (IMI 603C01) are used for pick-ing up the vibration signals from various stations on the rig. Theseaccelerometers are having measurement range as ±490 m/s2.

The test rig is used to run with healthy bearing initially to estab-lish the base-line data. Then the data are collected for different faultconditions of bearings. Various defects considered in bearing com-ponents are as shown in Fig. 2 [19]. A variety of faults on bearings aresimulated on the rig at different rotor speed 250, 500, 1000, 1500and 2000 rpm. Following five bearing conditions are considered forthe study:

1. Healthy bearings (HB)2. Bearing with spall on inner race (BSIR)3. Bearing with spall on outer race (BSOR)4. Bearing with spall on ball (BSB)5. Combined bearing component defects (CCD)

Time responses are obtained at various speeds and at differ-ent loader conditions with considerations of all cases in a phasedmanner. The time responses at 1000 rpm with no loader conditionare shown in Fig. 3. The five different bearing conditions are usedas (a) Healthy bearings, (b) bearing with spall on inner race, (c)bearing with spall on outer race, (d) bearing with spall on ball,and (e) combined bearing components defects. For healthy bear-ings, the periodic responses are observed with low magnitude ofpeak of excitation as shown in Fig. 3(a). The more severe vibrationsare appeared in the vibration spectra with spall on inner race andon the ball as shown in Fig. 3(b) and (d). The nature of solutionobserved is aperiodic in nature with intermittent behavior. Whenthe spall has been introduced in the outer race of bearings, the timeresponse is shown aperiodic nature with type-I intermittency (withvertical response) as shown in Fig. 3(c). With the combined defects,system shows chaos and beat like structure as shown in Fig. 3(e).Numbers of responses are obtained as 75 for each training and test-ing of classifiers by considering different bearing conditions, loaderconditions and various rotor speeds.

4. Wavelet based feature extraction methodology

The effectiveness of signal processing techniques to handle alarge quantity of data, present a bottleneck for timely and accurateassessment of the bearing conditions. The underlying simplifica-tions and idealization of signal (e.g. assuming signal stationary orsystem linearity) can lead to inaccurate and improper assessment

ce with spall, (b) inner race with spall, and (c) ball with spall.

P.K. Kankar et al. / Applied Soft Computing 11 (2011) 2300–2312 2303

F bearinr ent de

oaeum

ig. 3. Vibration signatures for various bearing conditions at 1000 rpm. (a) Healthyace (BSOR), (d) bearing with spall on ball (BSB), and (e) combined bearing compon

f the realistic bearing conditions, thus reducing the overall reli-bility of the health diagnosis techniques. Statistical parametersxtracted from vibration signals in time and frequency domain aresed as features for fault diagnosis. In this paper, to get more infor-ation and identify faults more accurately, wavelet based feature

gs (HB), (b) bearing with spall on inner race (BSIR), (c) bearing with spall on outerfects (CCD).

extraction methodology is used. Rotor bearing system is a non-linear dynamical system and is sensitive to different parameterslike radial internal clearance, type of defect (distributed or local-ized), defect severity, number of rolling elements carrying load, etc.Therefore, total six different wavelets (three complex valued and


wavel

tdswp

4

sfwiro

E

ww

E

W[

p

Fig. 4. Flowchart for

hree real valued) have been considered for the present study toemonstrate the methodology proposed for bearing fault diagno-is, which is capable of extracting most appropriate features fromavelet selected. Two wavelet selection criteria are used and com-ared to select an appropriate wavelet for feature extraction.

.1. Maximum Relative Wavelet Energy criterion

Relative Wavelet Energy (RWE) is considered as time-scale den-ity that can be used to detect a specific phenomenon in time andrequency planes. RWE gives information about relative energyith associated frequency bands and can detect the degree of sim-

larity between segments of a signal [20,21]. The energy at eachesolution level n, will be the energy content of signal at each res-lution is given by

(n) =m∑

i=1

|Cn,i|2 (6)

here ‘m’ is the number of wavelet coefficients and Cn,i is the ithavelet coefficient of nth scale.

The total energy can be obtained by

total =∑

n

∑i

|Cn,i|2 =∑

n

E(n) (7)

Then, the normalized values, which represent the Relative
avelet Energy, is the energy probability distribution, defined as
21]

n = E(n)Etotal

(8)

et selection criterion.

where∑

n

pn = 1, and the distribution, pn, is considered as a time-

scale density.

4.2. Maximum Energy to Shannon Entropy ratio criterion

An appropriate wavelet is selected as the base wavelet, whichcan extract the maximum amount of Energy while minimizing theShannon entropy of the corresponding wavelet coefficients. A com-bination of the Energy and Shannon entropy content of a signal’swavelet coefficients is denoted by Energy to Shannon Entropy ratio[14] and is given as

�(n) = E(n)Sentropy(n)

(9)

where entropy of signal wavelet coefficients is given by

Sentropy(n) = −m∑

i=1

pi · logzpi (10)

where pi is the energy probability distribution of the wavelet coef-ficients, defined as

2
pi = |Cn,i|
E(n)(11)

With∑m

i=1pi = 1, and in the case of pi = 0 for some i, the valueof pi · logzpi is taken as zero.

ft Com

4

w

1

Fo1

P.K. Kankar et al. / Applied So

.3. Wavelet selection methodology

The following steps explain the methodology for selecting a baseavelet for the vibration signals under study:

. In this study, healthy bearings, bearings with spall in outer race,inner race, ball and bearing with combined component defects(CCD) are considered. Total 150 vibration signals in time domainare obtained both in horizontal and vertical directions for each

ig. 5. Plot between Energy to Shannon Entropy ratio vs. Scale number for Meyer wavelef 1000 rpm with one loader, (c) rotor running at speed of 1000 rpm with two loader, (d500 rpm with one loader, and (f) rotor running at speed of 1500 rpm with two loader.

puting 11 (2011) 2300–2312 2305

bearing condition at different rotor speed 250, 500, 1000, 1500and 2000 rpm under loader and no loader conditions.

2. To convert the complex vibration signals into simplified signalswith more resolution in time and frequency domain, these raw

7
signals are divided into 2 sub-signals, i.e. 128 scales in seventhlevel of decomposition.
3. For healthy and faulty bearings, continuous wavelet coefficients(CWC) of vibration signals are calculated using six differentmother wavelets in which three are real valued as Meyer,

t. (a) Rotor running at speed of 1000 rpm with no loader, (b) rotor running at speed) rotor running at speed of 1500 rpm with no loader, (e) rotor running at speed of


(Cont

4

each scale by taking ratio of Total Energy of nth scale and sumof Total Energy of all scale. Maximum value of RWE is selectedfor each wavelet as shown in Table 2.

Table 2Comparison of parameters for wavelet selection.

Wavelet type Energy to ShannonEntropy ratio

Maximum RelativeWavelet Energy

Meyer 123.569 0.014079

Fig. 5.

Coiflet5, Symlet2 wavelets and other three are complex valuedas complex Gaussian, complex Morlet and Shannon wavelets.

. Wavelet selection criterion is used to select an appropriatewavelet using either Step 4(a) or 4(b) as:a. The Total Energy and Total Shannon Entropy of CWC is calcu-

lated for vibration signals at different rotor speed 250, 500,1000, 1500 and 2000 rpm and for different loading conditionsusing healthy and faulty bearings. The Total Energy to TotalShannon Entropy ratio for each wavelet is calculated as shown
in Table 2.
b. The Total Energy is calculated for each scale and for vibra-tion signals at different rotor speed 250, 500, 1000, 1500 and2000 rpm and for different loading conditions using healthyand faulty bearings. Relative Wavelet Energy is calculated for

inued ).

Coiflet5 119.019 0.014132Symlet2 110.039 0.013703Complex Gaussian 118.898 0.014009Shannon 88.126 0.031747Complex Morlet 22.909 0.049158

ft Com

5

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Fo1

P.K. Kankar et al. / Applied So

. The wavelet having Maximum Energy to Shannon Entropyratio/Maximum Relative Wavelet Energy is considered for faultdiagnosis of rolling element bearings.

The flowchart for above-mentioned procedure is shown in Fig. 4.he Energy to Shannon Entropy ratio is obtained maximum for
eyer wavelet. Hence, Meyer wavelet is considered as the baseavelet to extract features for fault diagnosis. For healthy and
aulty bearings, the plots between Energy to Shannon Entropy rationd Scale number at rotor speed 1000 and 1500 rpm and differ-nt loading conditions using Meyer wavelet are shown in Fig. 5. At

ig. 6. Plot between Relative Wavelet Energy vs. Scale number for complex Morlet wavelef 1000 rpm with one loader, (c) rotor running at speed of 1000 rpm with two loader, (d500 rpm with one loader, and (f) rotor running at speed of 1500 rpm with two loader.

puting 11 (2011) 2300–2312 2307

rotor speed 1000 rpm, that outer race defect in ball bearing giveshigh Energy to Shannon Entropy ratio as shown in Fig. 5(a)–(c). Atthe rotor speed 1500 rpm, inner race defect in ball bearing giveshigh Energy to Shannon Entropy ratio as shown in Fig. 5(d)–(f). Forhealthy bearings, it is observed that Energy to Shannon Entropyratio comes out to be less than as compared to other faults in
bearing as shown in Fig. 5. Similarly, the Complex Morlet waveletis selected based on Maximum Relative Wavelet Energy criterion.The plots between Relative Wavelet Energy and Scale number usingcomplex Morlet wavelet at rotor speed 1000 and 1500 rpm are asshown in Fig. 6.
t. (a) Rotor running at speed of 1000 rpm with no loader, (b) rotor running at speed) rotor running at speed of 1500 rpm with no loader, (e) rotor running at speed of


(Cont

4

canwWcttrotcts

Fig. 6.

.4. Features extraction and faults classification

Based on two wavelet selection criteria, Meyer wavelet andomplex Morlet wavelet are selected as the best base waveletmong the other wavelets considered. The CWC of all the 150 sig-als with Meyer wavelet and complex Morlet wavelet as a baseavelet are calculated at seventh level of decomposition (27 scales).hen apply the wavelet transform to a signal, if a major frequency

omponent corresponding to a particular scale exists in the signal,he corresponding wavelet coefficients at that scale will have rela-ively high magnitudes. As a result, the Energy to Shannon Entropyatio and RWE of that particular scale is more. In the present study,
ut of 27 scales considered, the scale having the maximum Energyo Shannon Entropy ratio/maximum RWE are selected for Meyer andomplex Morlet wavelet, respectively. The statistical features ofhe continuous wavelet coefficients (CWC) corresponding to theelected scale are calculated.
inued ).

Root mean square (RMS) value, crest factor, kurtosis, skew-ness, standard deviation, etc., are most commonly used statisticalmeasures used for fault diagnosis of rolling element bearings [1].Statistical moments like kurtosis, skewness and standard deviationare descriptors of the shape of the amplitude distribution of vibra-tion data collected from a bearing, and have some advantages overtraditional time and frequency analysis, such as its lower sensitiv-ity to the variations of load and speed, the analysis of the conditionmonitoring results is easy and convenient, and no precious historyof the bearing life is required for assessing the bearing condition[22]. When selecting certain normalized statistical moments tomonitor the bearing condition, we usually need to consider two
most essential characteristics, i.e. sensitivity and robustness. Byrectifying the signal, Honarvar and Martin [22] compared the thirdmoment, skewness, of the rectified data to kurtosis, and foundthat this third moment has better characteristics than kurtosis.In present paper, author’s use statistical moments like kurtosis,


Table 3Sample input vector for SVM/ANN/SOM.

Features

Horizontal response Vertical response Loader Speed Class

Kurtosis Skewness Standard deviation Kurtosis Skewness Standard deviation

Amplitudeof features

2.7255872 −0.001213 0.021858 12.80515 0.00163772 0.011718 0 1000 BSB2.7860973 −0.001071 0.029106 11.81161 0.0122907 0.022992 0 1500 BSB1.8246985 0.001291 0.056189 3.233427 0.00395157 0.04545 0 2000 BSB2.017569 0.1479638 0.007982 4.825214 0.01391472 0.008245 1 1000 BSIR2.0085471 −0.000241 0.026249 2.892508 0.007512 0.022523 1 1500 BSIR2.149799 0.0006869 0.069483 3.618462 −0.0287274 0.038616 1 2000 BSIR2.2313071 0.0004708 0.024906 2.84579 −0.0160622 0.01495 1 1000 CCD2.7805259 −0.11345 0.020858 3.108108 −0.0065279 0.027154 1 1500 CCD1.6082328 0.0025935 0.164289 1.582226 0.00466979 0.086661 1 2000 CCD2.6645751 −0.003117 0.031612 2.615904 0.03257202 0.015022 2 1000 HB2.5214287 −0.002396 0.059733 3.32959 0.00329883 0.025953 2 1500 HB

6574865784466

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with each technique for faults classification. The correctly classi-fied instances using test set for SVM, ANN and SOM are 98.6667%,94.6667% and 76% respectively. For 10-fold cross validation, aver-age classification accuracies for SVM, ANN and SOM are 96%, 92%

1.8002664 −0.00014 0.115862 2.032.5300902 −0.006135 0.14601 2.692.5976952 −0.003402 0.108927 12.22.9964526 0.0210332 0.141186 7.53

kewness and standard deviation as features to effectively indicatearly faults occurring in rolling element bearing. These statisticaleatures are briefly described as follows:

Kurtosis: A statistical measure used to describe the distributionf observed data around the mean. Kurtosis is defined as the degreeo which a statistical frequency curve is peaked.

urtosis ={

n(n + 1)(n − 1)(n − 2)(n − 3)

∑(xi − x̄

s

)4}

− 3(n − 1)2

(n − 2)(n − 3)(12)

Skewness: Skewness characterizes the degree of asymmetry ofistribution around its mean. Skewness can be negative or positive.

kewness = n

(n − 1)(n − 2)

∑(xi − x̄

s

)3

(13)

Standard deviation: Standard deviation is measure of energyontent in the vibration signal.

tandard deviation =

√n∑

x2 −(∑

x)2

n(n − 1)(14)

These statistical features are fed as input to the machine learningechniques like ANN, SVM and SOM for faults classification. Fig. 7hows an overview of the methodology presented in this study forearing faults diagnosis.

. Results and discussion

In the present study, training and testing of the classifiers asVM, ANN and SOM has been carried out. The results on a test set inmulti-class prediction are displayed as a two dimensional confu-

ion matrix with a row and column for each class [23]. Each matrixlement is shown the number of test examples for which the actuallass is the row and the predicted class is the column.

A sample training/testing vector is shown in Table 3. Total 75nstances and 8 features are used for the study. These eight featuresre used as an input to train and test machine learning techniques.
hese features include 6 statistical features, i.e. kurtosis, skewnessnd standard deviation, each for horizontal and vertical response.umber of loader used and rotor speed are also considered as the
eatures to train and test classifiers as shown in Table 3. For Meyeravelet, the test results for three machine learning techniques,

−0.0268434 0.039708 2 2000 HB0.0547617 0.036282 2 1000 BSOR0.01326271 0.041251 2 1500 BSOR−0.0019256 0.069958 2 2000 BSOR

i.e. SVM, ANN and SOM, using test set and 10-fold cross valida-tion are shown in Tables 4–6. Total 75 numbers of instances areobtained in which 15 cases are considered with each of BSB, BSIR,CCD, HB and BSOR, respectively. Table 7 shows accuracy associated

Fig. 7. Flow chart of methodology proposed for bearing fault diagnosis.


Table 4Confusion matrix for SVM using Meyer wavelet (Maximum Energy to Shannon Entropy ratio criterion).

Using test set Using 10-fold cross validation

BSB BSIR CCD HB BSOR Classified as BSB BSIR CCD HB BSOR Classified as

15 0 0 0 0 BSB 14 0 1 0 0 BSB0 15 0 0 0 BSIR 0 15 0 0 0 BSIR0 0 14 1 0 CCD 0 0 14 1 0 CCD0 0 0 15 0 HB 0 0 0 15 0 HB0 0 0 0 15 BSOR 0 0 0 1 14 BSOR

Table 5Confusion matrix for ANN using Meyer wavelet (Maximum Energy to Shannon Entropy ratio criterion).




Table 6Confusion matrix for SOM using Meyer wavelet (Maximum Energy to Shannon Entropy ratio criterion).




Table 7Evaluation of the success of the numeric prediction Meyer wavelet (Maximum Energy to Shannon Entropy ratio criterion).

Parameters SVM ANN SOM

Test set 10-Fold cross validation Test set 10-Fold cross validation Test set 10-Fold cross validation

Correctly classified instances 74 (98.667%) 72 (96%) 71 (94.667%) 69 (92%) 57 (76%) 55 (73.333%)Incorrectly classified instances 1 (1.333%) 3 (4%) 4 (5.333%) 6 (8%) 18 (24%) 20 (26.667)Kappa statistic 0.9833 0.95 0.9333 0.9 0.7 0.6667Total number of instances 75 75 75 75 75 75

Table 8Confusion matrix for SVM using complex Morlet wavelet (Relative Wavelet Energy criterion).



15 0 0 0 0 BSB 15 0 0 0 0 BSB

aclw

TC

0 13 0 2 0 BSIR0 0 15 0 0 CCD0 1 1 13 0 HB0 0 0 1 14 BSOR

nd 73.333% respectively, which is slightly less than the previousase. Similarly, Tables 8–10 show the test results for three machineearning techniques, i.e. SVM, ANN and SOM using complex Morletavelet. Classification accuracies associated with each technique

able 9onfusion matrix for ANN using complex Morlet wavelet (Relative Wavelet Energy criter

Using test set

BSB BSIR CCD HB BSOR Classified as

14 0 1 0 0 BSB0 13 0 2 0 BSIR0 0 14 0 1 CCD0 0 0 15 0 HB0 0 1 0 14 BSOR

0 12 0 3 0 BSIR0 0 14 1 0 CCD0 0 1 14 0 HB0 1 0 1 13 BSOR

are shown in Table 11. For test set, correctly classified instancesfor SVM, ANN and SOM are 93.333%, 93.333% and 72% respectively.While using 10-fold cross validation average classification accu-racies are 90.667%, 89.333% and 70.667% for SVM, ANN and SOM

ion).

Using 10-fold cross validation

BSB BSIR CCD HB BSOR Classified as

13 0 2 0 0 BSB0 14 0 1 0 BSIR0 0 12 0 3 CCD0 0 1 14 0 HB0 0 1 0 14 BSOR


Table 10Confusion matrix for SOM using complex Morlet wavelet (Relative Wavelet Energy criterion).




Table 11Evaluation of the success of the numeric prediction using complex Morlet wavelet (Relative Wavelet Energy criterion).

Parameters SVM ANN SOM

Test set 10-Fold cross validation Test set 10-Fold cross validation Test set 10-Fold cross validation

Correctly classified instances 70 (93.333%) 68 (90.667%) 70 (93.333%) 67 (89.333%) 54 (72%) 53 (70.667)Incorrectly classified instances 5 (6.667%) 7 (9.333%) 5 (6.667%) 8 (10.667%) 21 (28%) 22 (29.333%)Kappa statistic 0.9167 0.8833 0.9167 0.8666 0.65 0.6334Total number of instances 75 75 75 75 75 75

Table 12A compressive study between the present work and some recent publications.

References

Xian and Zeng [8] Paya et al. [9] Staszewski andTomlinson [10]

Abbasion et al. [11] Prabhakar et al.[12]

Present work

Objects Gears Bearings and gears Spur gears Rolling elementbearings

Rolling elementbearings

Rolling elementbearings

0Defectsconsidered

Tooth root crack,fatigue wear,surface pitting,surface scrape

Defects on innerrace of bearingand gear toothirregularity

Broken tooth Bearing looseness,defects in rollingelements andbearing raceways

One scratch markeach on innerrace (on thetrack) and outerrace (on thetrack), twoscratch marks onouter race (180◦

apart on thetrack), onescratch mark oneach of inner raceand outer race(on the track)

Spall in inner race,outer race,rolling elementand combinedcomponentdefects

Techniques usedfor vibrationsignature analysis

Wavelet packetanalysis(compared threedifferentwavelet, i.e. Db2,sym5 andDmeyer)

D4 wavelet Morlet wavelet Meyer wavelet Daubechies 4 Meyer, Coiflet5,Symlet2,Gaussian,complex Morletand Shannonwavelets

Featuresconsidered

Amplitudes of thespecificfrequency inpower spectrum,the specificrotating periodin cestrum andthe componentsenergy valuesfrom waveletpacketdecomposition

10 waveletnumbersindicating bothtime andfrequency andtheir 10correspondingamplitudes

Statisticalsimilarityanalysis is used(kurtosis andmahalanobisdistance)

Fundamental cagefrequency (Fc),ball pass innerracewayfrequency (FBPI),ball pass outerracewayfrequency (FBPO)and ballrotationalfrequency (FB)

RMS, kurtosis Statistical featuresnamely, kurtosis,skewness andstandarddeviation fromwaveletcoefficientscorresponding toscale maximizingEnergy toShannon entropyratio or RelativeWavelet Energy

Classifier used Back propagationneural networks,hybrid SVM

Artificial neuralnetworks

NA Support vectormachine

NA Support vectormachines,artificial neuralnetworks,self-organizingmaps

Classifierefficiencies

Best averageefficiencyobtained usingDmeyer waveletand Hybrid SVM– 92.5%

96% NA 100% NA Best efficiencyobtained usingtest set withMeyer waveletand SVM –98.6667%

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espectively. The prediction performance of SVM is coming outo be best mainly due to its good generalization capability whichs also reported by Meyer et al. [15] in an extensive comparativetudy. SVM and ANN are effective in predicting the bearing faultsell in advance of the impending catastrophic failure as their cor-

ectly classified instances are high. SOM which is an unsupervisedearning technique, is also given good classification with efficiencyf 76% in the present study. To show the efficiency of the selectedeatures and the methodology, a comparison between the currentork and some published literatures has been shown in Table 12.

n this table, comparison has been made on the basis of the objectssed, defects considered for the study, techniques used for vibra-ion signature analysis, features considered, classifier used and thelassifier efficiencies in each paper.

. Conclusions

This study presents, a methodology for detection of bearingaults by classifying them using three machine learning methods,ike SVM, ANN, and SOM. This methodology incorporates mostppropriate features, which are extracted from wavelet coefficientsf raw vibration signals. Two wavelet selection criteria Maximumnergy to Shannon Entropy ratio and Maximum Relative Waveletnergy are used and compared to select an appropriate wavelet foreature extraction. Results obtained from the two criteria are shownn Tables 7 and 11 and concluded that the wavelet selected using

aximum Energy to Shannon Entropy ratio criterion (Meyer wavelet)ives better classification efficiency. Though all three techniqueserformed well, but the results of faults classification with SVMnd ANN are better than SOM. The performance of SVM is foundo be the best due to its inherent generalization capability. Bysing proposed methodology, useful features can be extracted fromhe original data and dimension of original data can be reducedy removing irrelevant features, so that the classifier can achievehigher accuracy. The results show the potential application of

roposed methodology with machine learning techniques for theevelopment of on-line fault diagnosis systems for machine condi-ion.

eferences

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http://scholarworks.umass.edu/dissertations/AAI3275786

fault diagnosis of ball bearings using continuous wavelet transform

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