Research ArticleFault Detection Enhancement in Rolling ElementBearings via Peak-Based Multiscale Decompositionand Envelope Demodulation

Hua-Qing Wang1 Wei Hou1 Gang Tang1 Hong-Fang Yuan2

Qing-Liang Zhao1 and Xi Cao2

1 School of Mechanical and Electrical Engineering Beijing University of Chemical Technology Beijing 100029 China2 School of Information Science and Technology Beijing University of Chemical Technology Beijing 100029 China

Correspondence should be addressed to Gang Tang tanggangmailbucteducn

Received 31 March 2014 Accepted 7 May 2014 Published 27 May 2014

Academic Editor Ruqiang Yan

Copyright copy 2014 Hua-Qing Wang et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Vibration signals of rolling element bearings faults are usually immersed in background noise which makes it difficult to detectthe faults Wavelet-based methods being used commonly can reduce some types of noise but there is still plenty of room forimprovement due to the insufficient sparseness of vibration signals in wavelet domain In this work in order to eliminate noiseand enhance the weak fault detection a new kind of peak-based approach combined with multiscale decomposition and envelopedemodulation is developed First to preserve effective middle-low frequency signals while making high frequency noise moresignificant a peak-based piecewise recombination is utilized to convert middle frequency components into low frequency onesThenewly generated signal becomes so smoother that it will have a sparser representation inwavelet domainThen a noise thresholdis applied after wavelet multiscale decomposition followed by inverse wavelet transform and backward peak-based piecewisetransform Finally the amplitude of fault characteristic frequency is enhanced bymeans of envelope demodulationThe effectivenessof the proposed method is validated by rolling bearings faults experiments Compared with traditional wavelet-based analysisexperimental results show that fault features can be enhanced significantly and detected easily by the proposed method

1 Introduction

A rolling bearing is one of the most widely used elementsin rotating machinery As a critical component it carriesmost of the load during the running of rotating machineryIf rolling bearing fails serious problems arise which will inturn result in the decrease of production efficiency and largeeconomic loss Records show that faulty bearings contributeto about thirty percent of failures in rotatingmachineryThusit is of great importance to study the effective fault diagnosisapproaches for rolling bearings

Various methods have been developed for bearing faultdiagnosis and condition monitoring such as vibration mon-itoring temperature monitoring chemical analysis acousticemission monitoring sound pressure monitoring and laser

monitoring [1] Vibration signal analysis is one of the mostefficient methods thanks to the useful information ofmachinersquos work status carried by vibration signals [2ndash4]To extract fault information for example partial defectsof bearings vibration signals are usually processed in timedomain frequency domain or both [5 6] In time domainsome statistical parameters for example probability densitykurtosis root mean square or skewness have been intro-duced into bearing defect detection [7] However these faultindicators are not effective in all cases especially for weakfault signals Therefore analysis approaches in frequencydomain are developed to detect bearing defect fault Based onthe efficiency of modern fast Fourier transform (FFT) spec-tral analysis of vibration signals is widely used to extract char-acteristic defect frequencies In fact bearing fault signal is

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 329458 11 pageshttpdxdoiorg1011552014329458

2 Mathematical Problems in Engineering

amplitude modulated at its characteristic defect frequenciesso a preprocessing of demodulation should be performedbefore FFT applied Traditionally envelope demodulationis used to detect the fault frequency [5 8] For obviousfault features traditional envelope demodulation has notableadvantages in fault type recognition However it does notwork well in many cases for the detection of weak faultfeatures To solve this problem various approaches have beendeveloped to denoise vibration signals and to enhance thecharacteristic defect frequency Cyclostationary analysis [9]for example spectral autocorrelation analysis [10] is such away In recent years some methods based on time-frequencyanalysis such as empirical mode decomposition (EMD) [11ndash13] local mean decomposition (LMD) [14 15] and wavelettransform [16 17] are also developed to enhance fault featuresand detect failures Both EMDand LMDare sensitive to noiselevel but they are so hard to implement properly since thesealgorithms contain imperfections In addition to the men-tioned signal processing approaches some other strategiesare also introduced for fault features extraction as well aspattern recognition for example symptom parameter waves[18] and fuzzy diagnosis method [19]

As mentioned above signalrsquos decomposition or sparserepresentation is the basis of those methods Howeverbecause of the existing high frequency components vibrationsignals are not always sparse enough after decomposition Asa popular strategy for bearing vibration signalsrsquo processingwavelet-based analysis is effective for feature extraction fromsmooth signals signal denoising and fault feature enhance-ment [20ndash22] But it cannot always represent vibration signalssparsely enough especially for thosemixed by plenty of inter-ferential signals generated by other related rolling elementsThis will be illustrated further with a typical example inthe next section In related research areas a so-called peaktransform based on piecewise curve recombination has beendeveloped The idea is to adjust piecewise curves to improvethe smoothness of the target signals or images and it has beenused in 2D image representation and coding [23 24] To ourknowledge there is no report in the literature by far on itsapplications to bearing vibration signal analysis

In order to overcome those disadvantages of wavelet-based methods for weak signals processing of bearing faultsthis paper developed a new multiscale decomposition strat-egy with sparsity-promoting by recombination of piece-wise signals under different frequencies The new strategyimproves the sparsity of original vibration signals and is ofgreat significance in fault signals denoising and enhance-ment Then the wavelet coefficients of vibration signals willbe sparse enough and a threshold method based onwavelet decomposition can eliminate most noise Finally theFFT-based Hilbert transform is conducted for signal demod-ulation and extracting fault signature of characteristic defectfrequency Fault features will be significantly enhanced andeasily detected through the proposed method

The paper is organized as follows Section 2 illustrates thesparseness of vibration signals in wavelet domain which isalso the motivation of this work The proposed method ofpeak-based multiscale decomposition and envelope demod-ulation is introduced in Section 3 Some experimental results

0 200 400 600 800 1000 1200 1400 1600 1800 2000

0

1

2

3

4

Data number

Am

plitu

de

minus3

minus2

minus1

Figure 1 Vibration waveform of a bearing with a point defect on theouter race

to show the effectiveness of the method are presented inSection 4 Finally discussions and conclusions are presentedin Section 5

2 Vibration Signal of Rolling Bearing Faultsand Its Sparsity in Wavelet Domain

Vibration signal is always acquired by sensors related toacceleration velocity or displacementmeasurement A seriesof signal processing steps are then applied to make the faultfeatures easy to be distinguished Basically the key of vibra-tion signal analysis is to find out appropriate parametersdescribing the running conditions of bearings clearly Thecommonly used characteristic defect frequency is one of themost effective indicators for rolling bearingrsquos fault If localdefects exist in a bearing the measured vibration signal willbe amplitude modulated Its carrier wave is a kind of thebearingrsquos inherent vibration with high frequency while itsmodulating wave is a pass vibration corresponding to thelocal defects whose frequency is called ball pass frequency orcharacteristic defect frequencyThe latter is usually utilized todetermine whether a fault exists and it is also adopted by thispaper

In order to reduce the potential impact of signal attenua-tion through complex paths sensors are often placed as closeto the bearings as possible for example bearing housingHowever transmission path still exists between the impactpoint and the vibration measurement point which wouldmake signals more complex and hard to be demodulated Ifdefects exist a fault in one surface of bearings would strikeanother then a force impulse will be generated and flowingresonances in the bearing and machine will also be excited[25] A series of impulse responses will be generated and lastcontinuously which may be amplitude modulated caused bythe fault passing through the load zone or the mentionedvarying transmission paths Therefore the ideal bearingdefect signal is a kind of periodical impulse-like signal withhigh frequency nonsmooth components A typical vibrationsignal of a bearing is shown in Figure 1 the vibration wave-form is obviously impacted by an existing point defect on theouter race In practice in addition to bearing condition

Mathematical Problems in Engineering 3

0 1 2 3 4 5 6 7 8 9 10

012345

times104Data number

Am

plitu

de

minus3

minus4

minus5

minus2

minus1

Figure 2 Wavelet coefficients of the bearing fault signal in Figure 1

information the measured vibration signals are often mixedwith lots of interfering information from other componentsMeanwhile various types of noise also make the signal morecomplicated In addition to the noise introduced by mea-surement instruments interfering signals can also be gen-erated accompanied with the existence of transmission pathbetween local impact point and vibration measurementpoint In a word bearing vibration signal is nonstationarywith wide frequency band which makes the bearing faultsignature extremely weak to be detected

Existing wavelet-based denoisingmethodsmainly rely onthe sparseness of wavelet coefficients [17] If a signal is notsparse enough in wavelet domain it becomes challengingfor feature extraction by wavelet analysis due to the difficul-ties to distinguish required features from large number ofwavelet coefficients Common vibration signals from faultybearings are often composed of impulse components Atypical vibration signal is shown in Figure 1 and its waveletcoefficients are shown in Figure 2 from which we can seethat most coefficients are nonzero that is not sparse enoughExperiments show that this is applicable to fault vibrationsignals of inner race outer race and ball element Thus it isimportant to find out a proper way to improve the sparsenessas well as reducing noise and enhancing the weak faultfeatures

3 The Proposed Method

31 The Basic Principle of Peak-Based Piecewise Recombi-nation Peak-based piecewise recombination (PPR) or so-called peak transform is a kind of nonlinear geometric trans-form tomake the signal smoother through transforming highfrequency subsections to low frequency ones Here a briefdescription of PPR is given as follows for more informationplease refer to [23]

For a continuous function 119891(119909) that has 119873-point peaks(or break points) 119909

119894(119894 = 1 sdot sdot sdot 119873) is defined over [119886 119887]

In fact 119891(119909) can be viewed as a piecewise function with119873+1 combined curves where the119873 peaksrsquo positions are theconnection pointsThen the 119894th piecewise curve can be calledsubfunction119891

119894(119909) over interval [119886 119887] The 119873-point forward

1

5

6

4

3

2Forward

PPRBackward

PPR

f1(x) f2(x)

f3(x) f4(x)

f5(x)f6(x)

x

x

a b

x0 = ax1 x2 x3 x4 x5 x6 = b

Figure 3 Schematic diagram of five-point peak-based piecewiserecombination

transform of peak-based piecewise recombination can bedefined as

119875119865 [119891 (119909)] = 119892119900 (119909) oplus 119892119890 (119909) (1)

where

119892119900 (119909) = 1198911 (119909) oplus 1198913 (119909) oplus sdot sdot sdot oplus 1198912lfloor(119873minus1)2rfloor+1 (119909)

119892119890 (119909) = 1198912 (119909) oplus 1198914 (119909) oplus sdot sdot sdot oplus 1198912lfloor1198732rfloor (119909)

(2)

are the cascades of all odd- and even-numbered curve seg-ments respectively Here 2lfloor(119873 minus 1)2rfloor + 1 and 2lfloor1198732rfloor arerespectively the largest odd and even integers that are lessthan or equal to119873

Geometrically speaking as illustrated in Figure 3 all pie-cewise curves with similar positive slopes are grouped to-gether and cascaded to be a new curve while all negativeones are grouped into another new curve finally these twonew subcurves are recombined to be a new curve It is appar-ent that the recombination only changes the order of the pie-cewise curves and is reversible The backward transformcan be done by simply recascading them according to theiroriginal orders which is symbolized as119875119865minus1[119891(119909)] accordingto (1)

It should be noted that although the forward and back-ward transforms discussed above are defined for curve func-tions they can also be defined in similarmanners for discrete-time signals of rolling bearingsrsquo vibration

32 Wavelet Decomposition Wavelet 120595(119886119887)(119905) is obtained by

translation and dilation based on a defined single function120595(119905) as

120595(119886119887) (119905) =

1

radic119886120595(119905 minus 119887

119886) (3)

4 Mathematical Problems in Engineering

where 119886 gt 0 is the so-called scaling parameter and 119887 isin 119877 isthe parameter denoting time localization which can be con-tinuous or discrete120595(119905) is called ldquomotherwaveletrdquo to generatewavelets 120595

(119886119887)(119905)

Given a signal 119909(119905) with finite energy the wavelet trans-form with analytic wavelet 120595(119905) can be viewed as the convo-lution of 119909(119905) with a scaled and conjugated wavelet

119882(119886 119887) =1

radic119886intinfin

minusinfin

119909 (119905) 120595lowast (119905 minus 119887

119886)119889119905 (4)

where 120595lowast(119905) stands for the complex conjugation of 120595(119905)The wavelet transform 119882(119886 119887) can be considered as a

function of translation 119887 at each scale 119886 From (4) we can seethat wavelet transform is a kind of time-frequency analysis ora time-scaled analysis Different from analysis either in timedomain or in frequency domain or even different from thatin time-frequency domain but with fixed-length windows byshort time Fourier transform (SFFT) vibration signals canbe decomposed bywavelet transformwithmultiscale analysisthrough dilation and translation so that the time-frequencyfeatures of vibration signals can bemore effectively extracted

Wavelet transform is also reversible as follows

119909 (119905) = 119862minus1

120595∬119882(119886 119887) 120595(119886119887) (119905)

119889119886

1198862119889119887 (5)

where

119862120595= intinfin

minusinfin

10038161003816100381610038161003816120595(120596)100381610038161003816100381610038162

|120596|119889120596 lt infin

(119908) = int120595 (119905) exp (minusj119908119905) 119889119905

(6)

which provides the possibility to reconstruct the originalvibration signals

33 Envelope Demodulation As mentioned in the previoussections if bearing defects exist the measured vibration sig-nal would be amplitude modulated at its characteristic defectfrequency The modulating wave is a pass vibration signalcorresponding to local defects and various demodulationtechniques have been developed for the separation In thiswork an envelope demodulation method based on Hilberttransform is employed

For a continuous time signal 119909(119905) the Hilbert transform119909(119905) is defined as

119909(119905) =1

120587int+infin

minusinfin

119909 (120591)

(119905 minus 120591)119889120591 (7)

Then combine 119909(119905) and 119909(119905) to form a new analytic signal

119892 (119905) = 119909 (119905) + 119895119909(119905) (8)

The envelope of 119909(119905) is defined as the amplitude of 119892(119905)

119860 (119905) = radic[119909 (119905)]2+ [119909 (119905)]2 (9)

After Hilbert transform envelope spectra could be obtainedby FFT to extract characteristic defect frequencies

34 Enhancement Frame with Peak-Based Multiscale Decom-position and Envelope Demodulation Wavelet transform is akind of time-frequency analysis method for signal denoisingby adjusting the discrete detail coefficients and approxima-tion coefficients obtained from multilevel decompositionThis scheme is called threshold processing one of whosepremises is the sparse representation of signals in waveletdomain However different from that of perfect bearing thevibration signal energy of bearings with defects increases athigh frequenciesThus vibration signals are not always sparsein wavelet domain Piecewise curve recombination can con-vert energies at high frequencies to ones at low frequenciesIn this way signal energies are focused in low frequency sub-bands so that the piecewise recombination can improve sig-nalrsquos smoothness as well as its sparseness in wavelet domain

The present paper proposed an enhancement frame forvibration signals by the combination of peak-based multi-scale decomposition and envelope demodulationThe proce-dure is described as follows

(1) Peak-based piecewise recombination of vibration sig-nal let 119909(119905) denote the measured vibration signalfirstly then transform the original signal 119909(119905) tosmooth signal 119875119865[119891(119909)] through piecewise recombi-nation according to Equation (1) The key is to deter-mine the peaks and to recombine different segmentsIn the present work a cascading strategy related tolocal minimum andmaximum is employedThe end-point of a segment is chosen as a peak If two adjacentpeaks are equal one of them is defined as the peak

(2) Wavelet decomposition of the new recombined sig-nal choose a wavelet basis (eg Morlet wavelet orDb4 wavelet) then decompose the new signal119875119865[119891(119909)] at level N

119882(119886 119887) =1

radic119886intinfin

minusinfin

119875119865 [119909 (119905)] 120595lowast (119905 minus 119887

119886)119889119905 (10)

(3) Coefficients threshold set a threshold value and applyit to the approximation coefficients at level119873 and thedetail coefficients at levels 1 to119873 The hard thresholdcan be expressed as

119910119904= 11991010038161003816100381610038161199101003816100381610038161003816 gt 119905

010038161003816100381610038161199101003816100381610038161003816 lt 119905

(11)

where 119905 is the selected threshold 119910 is the originalwavelet coefficients and 119910

119904is the wavelet coefficients

after thresholding In this work the value which cov-ers about 95 of the smallest wavelet coefficients is setas the threshold value Then 5 of the largest waveletcoefficients are kept as nonzero ones to reserve thesignal feature

(4) Signal recovery reconstruct the vibration signalbased on the modified approximation coefficients atlevel119873 and the modified detail coefficients at levels 1to119873

1198751198651015840 [119909 (119905)] = 119862minus1

120595∬1198821015840 (119886 119887) 120595(119886119887) (119905)

119889119886

1198862119889119887 (12)

Mathematical Problems in Engineering 5

Original signal

Peak-based piecewise transform

Newly generated smooth signal

Wavelet decomposition

Coefficients threshold Inverse wavelet transform

Peak-based piecewise transform backward

Enhanced signal

Envelope demodulation

Spectral analysis and detection

Figure 4 Flowchart of the proposed enhancement scheme for weakfault signal

CH 1

Figure 5 Fault rig of a roller element bearing

(5) Piecewise signal backward transform put the pro-cessed signal segments back to their original posi-tions then a new denoised vibration signal 1199091015840(119905) isgenerated

(6) Envelope demodulation according to (9) use Hilberttransform to obtain the envelope 119860(119905) of signal 1199091015840(119905)

(7) Extraction of characteristic defect frequency get theenvelope spectra via Fourier analysis and distinguishdifferent fault types of rolling bearings

In summary these steps are shown as in Figure 4

4 Experimental Results and Discussions

41 Experiment Setup To verify the effectiveness of theproposed method experiments are designed for fault rigs ofroller element bearing as shown in Figure 5 The rig is com-posed of a motor a coupling a rotor and a shaft with tworoller bearings Basically the sensors are located at the posi-tion near the bearings to mitigate the effects of signal atten-uation Usually bearing housing is one of the best locationsfor bearing arrangement Therefore here an accelerometeris located on the bearing housing to acquire the vibrationsignals

In the experiments a single point defect is introduced ininner raceway outer raceway and ball element of differ-ent bearings using electron-discharge machining with faultwidths of 3mm 7mm and 7mm and depths of 5mm25mm and 25mm respectively

Vibration signals are measured by an accelerometer lo-cated at the top of the bearing house (CH1 as shownin Figure 5) In all experiments the sample frequency is100 kHz and the shaft speed is finite 1300 rpm Four commonconditions are studied in the present paper including aperfect healthy bearing and three other bearings with pointdefect on the outer race on the inner race bearing and onthe ball element

In theory elements of roller bearings have their own spe-cific rotational frequencies which may appear in envelopespectra if defects existThese defect frequencies are called ballpass frequencies For a bearing with a stationary outer racethese frequencies can be given by the following formulas

outer race defect frequency 119908od =119885119908119904

2119889(1 minus

119889

119863cos120572)

inner race defect frequency 119908id =119885119908119904

2(1 +

119889

119863cos120572)

rolling element defect frequency 119908re = 119908119904119863

2119889

times (1 minus1198892

1198632cos2120572)

(13)

where 119908119904is the shaft rotation frequency in rads 119889 is the

diameter of rolling element 119863 is the pitch diameter 119885 is thenumber of rolling elements and 120572 is the contact angle

In the present experiments outer race defect frequencyinner race defect frequency and rolling element defect fre-quency are 8632Hz 14584Hz and 5113Hz respectively

42 Vibration Signals of Typical Bearing Faults and TheirEnvelop Spectra The time-domain waveforms of bearingvibration signals obtained from accelerometers are shown inFigure 6 from which we can see that different from that ofperfect bearings as shown in Figure 6(a) obvious impulsivephenomenon exists in faulty vibration signals of fault bear-ings as shown in Figures 6(b) 6(c) and 6(d)

To investigate the differences of the signals and todistinguish different fault types envelope demodulation isapplied first to vibration signals of the three typical bearingfaults that is outer race fault inner race fault and rollerelement fault The results are shown in Figures 7 8 and 9In Figure 7 the frequency magnitude of outer race defectis lower than the second harmonic frequency of outer racedefect In Figure 8 although the inner race defect frequencyslightly sticks out it is still possibly confused by the side-lobes with almost similarmagnitudes In Figure 9 it is almostimpossible to figure out the calculated ball pass frequencyof 5113Hz From the figures we can see that simply basedon envelope demodulation it is difficult to identify the faultfrequencies from those of other components especially forweak fault signals

43 Wavelet Coefficients Distributions of the Vibration Sig-nals with and without Peak-Based Piecewise RecombinationBefore envelope demodulation wavelet decomposition is

6 Mathematical Problems in Engineering

0 02

02

04

04

06 08 1

0

Am

plitu

de (V

)

Time (s)

minus04

minus02

(a)

0 02 04 06 08 1

0

2

Am

plitu

de (V

)

Time (s)

minus4

minus2

(b)

0 02 04 06 08 1

0

5

10

Time (s)

Am

plitu

de (V

)

minus10

minus5

(c)

Am

plitu

de (V

)

0 02 04 06 08 1

0

5

Time (s)

minus5

(d)

Figure 6 Vibration signals of rolling bearings (a) a normal bearing and bearings with a point defect on (b) outer race (c) inner race bearingand (d) ball element

0 50 100 150 200 250 300 350 4000

50

100

150

200

250

Am

plitu

de

Frequency (Hz)

X 8621Y 1714

Figure 7 Envelope spectra of vibration signal in Figure 6(b)induced by outer race fault

always applied to denoise and enhance the weak fault signalsHowever vibration signals are not always sparse enough inwavelet domain which would affect the effectiveness of theenhancement As mentioned above peak-based piecewiserecombination can help to improve the smoothness of vibra-tion signals and make their energies more concentrated in

Am

plitu

de

0 50 100 150 200 250 300 350 4000

50

100

150

200

250

Frequency (Hz)

X 145Y 1816

Figure 8 Envelope spectra of vibration signal in Figure 6(c)induced by inner race fault

wavelet domain Comparisons between wavelet decomposi-tions with and without peak-based piecewise recombinationare illustrated in this section

Taking outer race fault as an example the time-domainvibration signal of Figure 6(b) is converted to that of Figure 10after applying piecewise recombination The signal becomesmuch smoother than the original signal after piecewise

Mathematical Problems in Engineering 7

0 50 100 150 200 250 300 350 4000

10

20

30

40

50

60

Frequency (Hz)

Am

plitu

de

X 4959Y 2026

Figure 9 Envelope spectra of vibration signal in Figure 6(d)induced by roller element fault

0 1 2 3 4 5 6 7 8 9 100

50010001500200025003000350040004500

Data number

Am

plitu

de

times104

Figure 10 Vibration signal induced by outer race fault inFigure 6(b) after peak-based piecewise recombination

Table 1 Energy distributions of the outer race fault signalrsquos waveletcoefficients

Frequency band Low frequency High frequencyEnergies without PPR 4931 5069Energies with PPR 999 01

recombination that is high frequency components have beenconverted to low frequency ones

For comparisons then wavelet transform is applied tosignals in Figures 6(b) and 10 respectively where 6-leveldecomposition with db4 wavelet basis is employed Both dis-tributions of the wavelet coefficients are shown in Figure 11Without piecewise recombination most wavelet coefficientsare nonzero that is not sparse enough as depicted inFigure 11(a) However it is improved significantly withpreprocessing of peak-based piecewise recombination asdepicted in Figure 11(b) Energies of the signal in waveletdomain are concentrated in only a few low frequency inter-vals and almost all detail coefficients at scales 1 to 6 are closeto zero It implies that peak-based piecewise recombinationcan significantly improve the sparseness of vibration signalsin wavelet domain

Statistically energy distributions of wavelet coefficientsobtained by the two methods are listed in Table 1 from

0 1 2 3 4 5 6 7 8 9 10

012345

Data number

Am

plitu

de

minus1

minus2

minus3

minus4

minus5

times104

(a)

Data number0 1 2 3 4 5 6 7 8 9

0

05

1

15

2

25

3

35

Am

plitu

de

times104

times104

(b)

Figure 11 Wavelet coefficients of the vibration signal induced byouter race fault (a) without PPR preprocessing and (b) with PPRpreprocessing

which we can see that after employing peak-based piecewiserecombination (PPR) preprocessing energies of the signal areconcentrated into only a few coefficients in wavelet domain999 in low frequency bands comparedwith 4931withoutpeak-based piecewise recombination

The experimental results of inner race fault and rollerelement fault are similar to those of the outer race fault whichare shown in Figures 12 and 13

44 Comparisons between Traditional Wavelet-Based Detec-tion Method and the Proposed Strategy with Envelop Demod-ulation Envelop demodulation is always combined withwavelet-based denoising and enhancement methods to ana-lyze vibration signals Comparisons of the results after en-velop demodulation based on the results of Section 43 arepresented in this section

Envelop demodulation is applied to the signals recon-structed from the thresholding of wavelet coefficients and abackward peak-based piecewise recombination is imple-mented for the proposed strategyThe demodulated envelopespectra are shown in Figures 14ndash16 In Figure 14(a) althoughthe fault frequency can be figured out the magnitude of

8 Mathematical Problems in Engineering

0 2 4 6 8 10

0

5

10

15

Data number

Am

plitu

de

minus20

minus15

minus10

minus5

times104

(a)

0Data number

2 4 6 8 10

0

05

1

15

2

25

3

35

Am

plitu

detimes104

times104

(b)

Figure 12 Wavelet coefficients of the vibration signal induced by inner race fault (a) without PPR preprocessing and (b) with PPRpreprocessing

Data number0 2 4 6 8 10

0

2

4

6

8

10

Am

plitu

de

minus8

minus6

minus4

minus2

times104

(a)

Data number0 2 4 6 8 10

0

05

1

15

2

25

Am

plitu

de

minus05

times104

times104

(b)

Figure 13 Wavelet coefficients of the vibration signal induced by roller element fault (a) without PPR preprocessing and (b) with PPR pre-processing

the outer race defect frequency is not the most remarkableone among all frequency components thus it is not alwayscertain that a bearing defect exists in outer race With theproposed method presented in this paper the magnitudeof outer race defect frequency becomes more prominent

where the magnitudes of outer race defect frequency havebeen enhanced and are more easily detected as Figure 14(b)depicted The proposed strategy is also more effective forinner race and roller bearing fault detection of rolling bear-ings and the results are shown as in Figures 15 and 16

Mathematical Problems in Engineering 9

0 100 200 300 4000

50

100

150

200

250

300

Frequency (Hz)

Am

plitu

de

X 8621Y 149

(a)

0 100 200 300 4000

200

400

600

800

1000

1200

Frequency (Hz)

Am

plitu

de

X 8621Y 9952

(b)

Figure 14 Envelope spectra of the vibration signal induced by outer race fault (a) wavelet analysis method and (b) the proposed method

0 100 200 300 4000

50

100

150

200

250

300

Frequency (Hz)

Am

plitu

de

Y 1777X 145

(a)

0 100 200 300 4000

50

100

150

200

250

300

350

400

Frequency (Hz)

Am

plitu

de

X 145Y 2525

(b)

Figure 15 Envelope spectra of the vibration signal induced by inner race fault (a) wavelet analysis method and (b) the proposed method

If defect exists in the bearing the magnitudes of faultsfrequencieswill increase in the envelope spectrumThereforethey are often used to describe the fault severity and alsoemployed to judge whether the fault features are enhancedIn this work magnitudes of faults frequencies are chosen asthe major indictors to reflect the fault features As recordedand listed in Table 2 with the proposed method magnitudesof faults frequencies are all enhanced and they become more

easily distinguished from interfered noise for the experimen-tal faults

5 Conclusions

In order to improve the current wavelet-based method forfault detection of rolling bearings the present paper devel-oped a new way to improve vibration signalsrsquo smoothness

10 Mathematical Problems in Engineering

0 100 200 300 4000

50

100

150

200

250

300

Frequency (Hz)

Am

plitu

de

X 4959Y 2026

(a)

0 100 200 300 4000

50

100

150

200

250

300

Frequency (Hz)

Am

plitu

de

X 4959Y 2792

(b)

Figure 16 Envelope spectra of the vibration signal induced by roller element fault (a) wavelet analysis method and (b) the proposedmethod

Table 2 Major indictors for bearing fault features

Defect frequencymagnitude of

Traditionalwavelet-based

methodThe proposed method

outer race 149 9952inner race 1777 2525roller element 2026 2792

as well as their sparseness in wavelet domain by employingpeak-based piecewise recombination and envelope demodu-lation Through the proposed method most noise mixed invibration signals can be eliminated and the weak fault signalsof rolling bearings can be enhanced and become more easilydetectableTherationality and validity of themethods are alsoproved by various experiments It shows that compared withtraditional wavelet-based method the proposed method hasmany advantages in bearing fault signals enhancement anddetection and is very simple and easy to implement

There is still room to improve the present work Thesensitivity of the method to noise and methods to makeenergy more concentrated are what will be studied in futurework

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This project is partially supported by the National NaturalScience Foundation of China (Grant no 51375037) and theNational Program on Key Basic Research Project (Grant no2012CB026000) The third author would also like to thankthe support of China Fundamental Research Funds for theCentral Universities (ZY1410) and the Public Hatching Plat-form for Recruited Talents of Beijing University of ChemicalTechnology

References

[1] W Zhou T G Habetler and R G Harley ldquoBearing conditionmonitoring methods for electric machines a general reviewrdquo inProceedings of the IEEE International Symposium on Diagnosticsfor Electric Machines Power Electronics and Drives (SDEMPEDrsquo07) pp 3ndash6 September 2007

[2] H Q Wang and P Chen ldquoIntelligent diagnosis method for acentrifugal pump using features of vibration signalsrdquo NeuralComputing and Applications vol 18 no 4 pp 397ndash405 2009

[3] H Wang and P Chen ldquoSequential diagnosis for rolling bearingusing fuzzy neural networkrdquo in Proceedings of the IEEEASMEInternational Conference on Advanced Intelligent Mechatronics(AIM rsquo08) pp 56ndash61 Xirsquoan China August 2008

[4] T Mitoma H Wang and P Chen ldquoFault diagnosis and condi-tion surveillance for plant rotating machinery using partially-linearized neural networkrdquo Computers and Industrial Engineer-ing vol 55 no 4 pp 783ndash794 2008

[5] P D McFadden and J D Smith ldquoVibration monitoring of roll-ing element bearings by the high-frequency resonance tech-niquemdasha reviewrdquo Tribology International vol 17 no 1 pp 3ndash101984

Mathematical Problems in Engineering 11

[6] N Tandon and A Choudhury ldquoReview of vibration and acous-tic measurement methods for the detection of defects in rollingelement bearingsrdquo Tribology International vol 32 no 8 pp469ndash480 1999

[7] R B W Heng and M J M Nor ldquoStatistical analysis of soundand vibration signals for monitoring rolling element bearingconditionrdquo Applied Acoustics vol 53 no 1ndash3 pp 211ndash226 1998

[8] C Junsheng Y Dejie and Y Yu ldquoThe application of energyoperator demodulation approach based on EMD in machineryfault diagnosisrdquo Mechanical Systems and Signal Processing vol21 no 2 pp 668ndash677 2007

[9] R B Randall J Antoni and S Chobsaard ldquoThe relationship be-tween spectral correlation and envelope analysis in the diagnos-tics of bearing faults and other cyclostationarymachine signalsrdquoMechanical Systems and Signal Processing vol 15 no 5 pp 945ndash962 2001

[10] A B Ming Z Y Qin W Zhang et al ldquoSpectrum auto-cor-relation analysis and its application to fault diagnosis of rollingelement bearingsrdquo Mechanical Systems and Signal Processingvol 41 no 1 pp 141ndash154 2013

[11] D Yu J Cheng and Y Yang ldquoApplication of EMDmethod andHilbert spectrum to the fault diagnosis of roller bearingsrdquo Me-chanical Systems and Signal Processing vol 19 no 2 pp 259ndash270 2005

[12] V K Rai and A RMohanty ldquoBearing fault diagnosis using FFTof intrinsic mode functions in Hilbert-Huang transformrdquo Me-chanical Systems and Signal Processing vol 21 no 6 pp 2607ndash2615 2007

[13] Y Lei N Li J Lin and SWang ldquoFault diagnosis of rotatingma-chinery based on an adaptive ensemble empirical mode decom-positionrdquo Sensors vol 13 no 12 pp 16950ndash16964 2013

[14] J Cheng Y Yang and Y Yang ldquoA rotatingmachinery fault diag-nosis method based on local mean decompositionrdquo Digital Sig-nal Processing vol 22 no 2 pp 356ndash366 2012

[15] B Chen Z He X Chen H Cao G Cai and Y Zi ldquoA demod-ulating approach based on local mean decomposition and itsapplications in mechanical fault diagnosisrdquo Measurement Sci-ence and Technology vol 22 no 5 Article ID 055704 2011

[16] S Prabhakar A R Mohanty and A S Sekhar ldquoApplication ofdiscrete wavelet transform for detection of ball bearing racefaultsrdquoTribology International vol 35 no 12 pp 793ndash800 2002

[17] R Yan R X Gao and X Chen ldquoWavelets for fault diagnosis ofrotary machines a review with applicationsrdquo Signal Processingvol 96 part A pp 1ndash15 2014

[18] H Wang and P Chen ldquoA feature extraction method based oninformation theory for fault diagnosis of reciprocating machin-eryrdquo Sensors vol 9 no 4 pp 2415ndash2436 2009

[19] K Li X L Ping H Q Wang and P Chen ldquoSequential fuzzydiagnosis method for motor roller bearing in variable operatingconditions based on vibration analysisrdquo Sensors vol 13 no 6pp 8013ndash8041 2013

[20] R Rubini and U Meneghetti ldquoApplication of the envelope andwavelet transform analyses for the diagnosis of incipient faultsin ball bearingsrdquoMechanical Systems and Signal Processing vol15 no 2 pp 287ndash302 2001

[21] L Gao Z Yang L Cai H Wang and P Chen ldquoRoller bearingfault diagnosis based on nonlinear redundant lifting waveletpacket analysisrdquo Sensors vol 11 no 1 pp 260ndash277 2011

[22] Z Yang L Cai L Gao and H Wang ldquoAdaptive redundant lift-ing wavelet transform based on fitting for fault feature extrac-tion of roller bearingsrdquo Sensors vol 12 no 4 pp 4381ndash43982012

[23] Z He ldquoPeak transform for efficient image representation andcodingrdquo IEEE Transactions on Image Processing vol 16 no 7pp 1741ndash1754 2007

[24] S Anila and N Devarajan ldquoThe usage of peak transform forimage compressionrdquo International Journal of Computer SystemsScience amp Engineering vol 2 no 11 pp 6308ndash6316 2010

[25] D Ho and R B Randall ldquoOptimization of bearing diagnostictechniques using simulated and actual bearing fault signalsrdquoMechanical Systems and Signal Processing vol 14 no 5 pp 763ndash788 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

amplitude modulated at its characteristic defect frequenciesso a preprocessing of demodulation should be performedbefore FFT applied Traditionally envelope demodulationis used to detect the fault frequency [5 8] For obviousfault features traditional envelope demodulation has notableadvantages in fault type recognition However it does notwork well in many cases for the detection of weak faultfeatures To solve this problem various approaches have beendeveloped to denoise vibration signals and to enhance thecharacteristic defect frequency Cyclostationary analysis [9]for example spectral autocorrelation analysis [10] is such away In recent years some methods based on time-frequencyanalysis such as empirical mode decomposition (EMD) [11ndash13] local mean decomposition (LMD) [14 15] and wavelettransform [16 17] are also developed to enhance fault featuresand detect failures Both EMDand LMDare sensitive to noiselevel but they are so hard to implement properly since thesealgorithms contain imperfections In addition to the men-tioned signal processing approaches some other strategiesare also introduced for fault features extraction as well aspattern recognition for example symptom parameter waves[18] and fuzzy diagnosis method [19]

As mentioned above signalrsquos decomposition or sparserepresentation is the basis of those methods Howeverbecause of the existing high frequency components vibrationsignals are not always sparse enough after decomposition Asa popular strategy for bearing vibration signalsrsquo processingwavelet-based analysis is effective for feature extraction fromsmooth signals signal denoising and fault feature enhance-ment [20ndash22] But it cannot always represent vibration signalssparsely enough especially for thosemixed by plenty of inter-ferential signals generated by other related rolling elementsThis will be illustrated further with a typical example inthe next section In related research areas a so-called peaktransform based on piecewise curve recombination has beendeveloped The idea is to adjust piecewise curves to improvethe smoothness of the target signals or images and it has beenused in 2D image representation and coding [23 24] To ourknowledge there is no report in the literature by far on itsapplications to bearing vibration signal analysis

In order to overcome those disadvantages of wavelet-based methods for weak signals processing of bearing faultsthis paper developed a new multiscale decomposition strat-egy with sparsity-promoting by recombination of piece-wise signals under different frequencies The new strategyimproves the sparsity of original vibration signals and is ofgreat significance in fault signals denoising and enhance-ment Then the wavelet coefficients of vibration signals willbe sparse enough and a threshold method based onwavelet decomposition can eliminate most noise Finally theFFT-based Hilbert transform is conducted for signal demod-ulation and extracting fault signature of characteristic defectfrequency Fault features will be significantly enhanced andeasily detected through the proposed method

The paper is organized as follows Section 2 illustrates thesparseness of vibration signals in wavelet domain which isalso the motivation of this work The proposed method ofpeak-based multiscale decomposition and envelope demod-ulation is introduced in Section 3 Some experimental results

0 200 400 600 800 1000 1200 1400 1600 1800 2000

0

1

2

3

4

Data number

Am

plitu

de

minus3

minus2

minus1

Figure 1 Vibration waveform of a bearing with a point defect on theouter race

to show the effectiveness of the method are presented inSection 4 Finally discussions and conclusions are presentedin Section 5

2 Vibration Signal of Rolling Bearing Faultsand Its Sparsity in Wavelet Domain

Vibration signal is always acquired by sensors related toacceleration velocity or displacementmeasurement A seriesof signal processing steps are then applied to make the faultfeatures easy to be distinguished Basically the key of vibra-tion signal analysis is to find out appropriate parametersdescribing the running conditions of bearings clearly Thecommonly used characteristic defect frequency is one of themost effective indicators for rolling bearingrsquos fault If localdefects exist in a bearing the measured vibration signal willbe amplitude modulated Its carrier wave is a kind of thebearingrsquos inherent vibration with high frequency while itsmodulating wave is a pass vibration corresponding to thelocal defects whose frequency is called ball pass frequency orcharacteristic defect frequencyThe latter is usually utilized todetermine whether a fault exists and it is also adopted by thispaper

In order to reduce the potential impact of signal attenua-tion through complex paths sensors are often placed as closeto the bearings as possible for example bearing housingHowever transmission path still exists between the impactpoint and the vibration measurement point which wouldmake signals more complex and hard to be demodulated Ifdefects exist a fault in one surface of bearings would strikeanother then a force impulse will be generated and flowingresonances in the bearing and machine will also be excited[25] A series of impulse responses will be generated and lastcontinuously which may be amplitude modulated caused bythe fault passing through the load zone or the mentionedvarying transmission paths Therefore the ideal bearingdefect signal is a kind of periodical impulse-like signal withhigh frequency nonsmooth components A typical vibrationsignal of a bearing is shown in Figure 1 the vibration wave-form is obviously impacted by an existing point defect on theouter race In practice in addition to bearing condition

Mathematical Problems in Engineering 3

0 1 2 3 4 5 6 7 8 9 10

012345

times104Data number

Am

plitu

de

minus3

minus4

minus5

minus2

minus1

Figure 2 Wavelet coefficients of the bearing fault signal in Figure 1

information the measured vibration signals are often mixedwith lots of interfering information from other componentsMeanwhile various types of noise also make the signal morecomplicated In addition to the noise introduced by mea-surement instruments interfering signals can also be gen-erated accompanied with the existence of transmission pathbetween local impact point and vibration measurementpoint In a word bearing vibration signal is nonstationarywith wide frequency band which makes the bearing faultsignature extremely weak to be detected

Existing wavelet-based denoisingmethodsmainly rely onthe sparseness of wavelet coefficients [17] If a signal is notsparse enough in wavelet domain it becomes challengingfor feature extraction by wavelet analysis due to the difficul-ties to distinguish required features from large number ofwavelet coefficients Common vibration signals from faultybearings are often composed of impulse components Atypical vibration signal is shown in Figure 1 and its waveletcoefficients are shown in Figure 2 from which we can seethat most coefficients are nonzero that is not sparse enoughExperiments show that this is applicable to fault vibrationsignals of inner race outer race and ball element Thus it isimportant to find out a proper way to improve the sparsenessas well as reducing noise and enhancing the weak faultfeatures

3 The Proposed Method

31 The Basic Principle of Peak-Based Piecewise Recombi-nation Peak-based piecewise recombination (PPR) or so-called peak transform is a kind of nonlinear geometric trans-form tomake the signal smoother through transforming highfrequency subsections to low frequency ones Here a briefdescription of PPR is given as follows for more informationplease refer to [23]

For a continuous function 119891(119909) that has 119873-point peaks(or break points) 119909

119894(119894 = 1 sdot sdot sdot 119873) is defined over [119886 119887]

In fact 119891(119909) can be viewed as a piecewise function with119873+1 combined curves where the119873 peaksrsquo positions are theconnection pointsThen the 119894th piecewise curve can be calledsubfunction119891

119894(119909) over interval [119886 119887] The 119873-point forward

1

5

6

4

3

2Forward

PPRBackward

PPR

f1(x) f2(x)

f3(x) f4(x)

f5(x)f6(x)

x

x

a b

x0 = ax1 x2 x3 x4 x5 x6 = b

Figure 3 Schematic diagram of five-point peak-based piecewiserecombination

transform of peak-based piecewise recombination can bedefined as

119875119865 [119891 (119909)] = 119892119900 (119909) oplus 119892119890 (119909) (1)

where

119892119900 (119909) = 1198911 (119909) oplus 1198913 (119909) oplus sdot sdot sdot oplus 1198912lfloor(119873minus1)2rfloor+1 (119909)

119892119890 (119909) = 1198912 (119909) oplus 1198914 (119909) oplus sdot sdot sdot oplus 1198912lfloor1198732rfloor (119909)

(2)

are the cascades of all odd- and even-numbered curve seg-ments respectively Here 2lfloor(119873 minus 1)2rfloor + 1 and 2lfloor1198732rfloor arerespectively the largest odd and even integers that are lessthan or equal to119873

Geometrically speaking as illustrated in Figure 3 all pie-cewise curves with similar positive slopes are grouped to-gether and cascaded to be a new curve while all negativeones are grouped into another new curve finally these twonew subcurves are recombined to be a new curve It is appar-ent that the recombination only changes the order of the pie-cewise curves and is reversible The backward transformcan be done by simply recascading them according to theiroriginal orders which is symbolized as119875119865minus1[119891(119909)] accordingto (1)

It should be noted that although the forward and back-ward transforms discussed above are defined for curve func-tions they can also be defined in similarmanners for discrete-time signals of rolling bearingsrsquo vibration

32 Wavelet Decomposition Wavelet 120595(119886119887)(119905) is obtained by

translation and dilation based on a defined single function120595(119905) as

120595(119886119887) (119905) =

1

radic119886120595(119905 minus 119887

119886) (3)

4 Mathematical Problems in Engineering

where 119886 gt 0 is the so-called scaling parameter and 119887 isin 119877 isthe parameter denoting time localization which can be con-tinuous or discrete120595(119905) is called ldquomotherwaveletrdquo to generatewavelets 120595

(119886119887)(119905)

Given a signal 119909(119905) with finite energy the wavelet trans-form with analytic wavelet 120595(119905) can be viewed as the convo-lution of 119909(119905) with a scaled and conjugated wavelet

119882(119886 119887) =1

radic119886intinfin

minusinfin

119909 (119905) 120595lowast (119905 minus 119887

119886)119889119905 (4)

where 120595lowast(119905) stands for the complex conjugation of 120595(119905)The wavelet transform 119882(119886 119887) can be considered as a

function of translation 119887 at each scale 119886 From (4) we can seethat wavelet transform is a kind of time-frequency analysis ora time-scaled analysis Different from analysis either in timedomain or in frequency domain or even different from thatin time-frequency domain but with fixed-length windows byshort time Fourier transform (SFFT) vibration signals canbe decomposed bywavelet transformwithmultiscale analysisthrough dilation and translation so that the time-frequencyfeatures of vibration signals can bemore effectively extracted

Wavelet transform is also reversible as follows

119909 (119905) = 119862minus1

120595∬119882(119886 119887) 120595(119886119887) (119905)

119889119886

1198862119889119887 (5)

where

119862120595= intinfin

minusinfin

10038161003816100381610038161003816120595(120596)100381610038161003816100381610038162

|120596|119889120596 lt infin

(119908) = int120595 (119905) exp (minusj119908119905) 119889119905

(6)

which provides the possibility to reconstruct the originalvibration signals

33 Envelope Demodulation As mentioned in the previoussections if bearing defects exist the measured vibration sig-nal would be amplitude modulated at its characteristic defectfrequency The modulating wave is a pass vibration signalcorresponding to local defects and various demodulationtechniques have been developed for the separation In thiswork an envelope demodulation method based on Hilberttransform is employed

For a continuous time signal 119909(119905) the Hilbert transform119909(119905) is defined as

119909(119905) =1

120587int+infin

minusinfin

119909 (120591)

(119905 minus 120591)119889120591 (7)

Then combine 119909(119905) and 119909(119905) to form a new analytic signal

119892 (119905) = 119909 (119905) + 119895119909(119905) (8)

The envelope of 119909(119905) is defined as the amplitude of 119892(119905)

119860 (119905) = radic[119909 (119905)]2+ [119909 (119905)]2 (9)

After Hilbert transform envelope spectra could be obtainedby FFT to extract characteristic defect frequencies

34 Enhancement Frame with Peak-Based Multiscale Decom-position and Envelope Demodulation Wavelet transform is akind of time-frequency analysis method for signal denoisingby adjusting the discrete detail coefficients and approxima-tion coefficients obtained from multilevel decompositionThis scheme is called threshold processing one of whosepremises is the sparse representation of signals in waveletdomain However different from that of perfect bearing thevibration signal energy of bearings with defects increases athigh frequenciesThus vibration signals are not always sparsein wavelet domain Piecewise curve recombination can con-vert energies at high frequencies to ones at low frequenciesIn this way signal energies are focused in low frequency sub-bands so that the piecewise recombination can improve sig-nalrsquos smoothness as well as its sparseness in wavelet domain

The present paper proposed an enhancement frame forvibration signals by the combination of peak-based multi-scale decomposition and envelope demodulationThe proce-dure is described as follows

(1) Peak-based piecewise recombination of vibration sig-nal let 119909(119905) denote the measured vibration signalfirstly then transform the original signal 119909(119905) tosmooth signal 119875119865[119891(119909)] through piecewise recombi-nation according to Equation (1) The key is to deter-mine the peaks and to recombine different segmentsIn the present work a cascading strategy related tolocal minimum andmaximum is employedThe end-point of a segment is chosen as a peak If two adjacentpeaks are equal one of them is defined as the peak

(2) Wavelet decomposition of the new recombined sig-nal choose a wavelet basis (eg Morlet wavelet orDb4 wavelet) then decompose the new signal119875119865[119891(119909)] at level N

119882(119886 119887) =1

radic119886intinfin

minusinfin

119875119865 [119909 (119905)] 120595lowast (119905 minus 119887

119886)119889119905 (10)

(3) Coefficients threshold set a threshold value and applyit to the approximation coefficients at level119873 and thedetail coefficients at levels 1 to119873 The hard thresholdcan be expressed as

119910119904= 11991010038161003816100381610038161199101003816100381610038161003816 gt 119905

010038161003816100381610038161199101003816100381610038161003816 lt 119905

(11)

where 119905 is the selected threshold 119910 is the originalwavelet coefficients and 119910

119904is the wavelet coefficients

after thresholding In this work the value which cov-ers about 95 of the smallest wavelet coefficients is setas the threshold value Then 5 of the largest waveletcoefficients are kept as nonzero ones to reserve thesignal feature

(4) Signal recovery reconstruct the vibration signalbased on the modified approximation coefficients atlevel119873 and the modified detail coefficients at levels 1to119873

1198751198651015840 [119909 (119905)] = 119862minus1

120595∬1198821015840 (119886 119887) 120595(119886119887) (119905)

119889119886

1198862119889119887 (12)

Mathematical Problems in Engineering 5

Original signal

Peak-based piecewise transform

Newly generated smooth signal

Wavelet decomposition

Coefficients threshold Inverse wavelet transform

Peak-based piecewise transform backward

Enhanced signal

Envelope demodulation

Spectral analysis and detection

Figure 4 Flowchart of the proposed enhancement scheme for weakfault signal

CH 1

Figure 5 Fault rig of a roller element bearing

(5) Piecewise signal backward transform put the pro-cessed signal segments back to their original posi-tions then a new denoised vibration signal 1199091015840(119905) isgenerated

(6) Envelope demodulation according to (9) use Hilberttransform to obtain the envelope 119860(119905) of signal 1199091015840(119905)

(7) Extraction of characteristic defect frequency get theenvelope spectra via Fourier analysis and distinguishdifferent fault types of rolling bearings

In summary these steps are shown as in Figure 4

4 Experimental Results and Discussions

41 Experiment Setup To verify the effectiveness of theproposed method experiments are designed for fault rigs ofroller element bearing as shown in Figure 5 The rig is com-posed of a motor a coupling a rotor and a shaft with tworoller bearings Basically the sensors are located at the posi-tion near the bearings to mitigate the effects of signal atten-uation Usually bearing housing is one of the best locationsfor bearing arrangement Therefore here an accelerometeris located on the bearing housing to acquire the vibrationsignals

In the experiments a single point defect is introduced ininner raceway outer raceway and ball element of differ-ent bearings using electron-discharge machining with faultwidths of 3mm 7mm and 7mm and depths of 5mm25mm and 25mm respectively

Vibration signals are measured by an accelerometer lo-cated at the top of the bearing house (CH1 as shownin Figure 5) In all experiments the sample frequency is100 kHz and the shaft speed is finite 1300 rpm Four commonconditions are studied in the present paper including aperfect healthy bearing and three other bearings with pointdefect on the outer race on the inner race bearing and onthe ball element

In theory elements of roller bearings have their own spe-cific rotational frequencies which may appear in envelopespectra if defects existThese defect frequencies are called ballpass frequencies For a bearing with a stationary outer racethese frequencies can be given by the following formulas

outer race defect frequency 119908od =119885119908119904

2119889(1 minus

119889

119863cos120572)

inner race defect frequency 119908id =119885119908119904

2(1 +

119889

119863cos120572)

rolling element defect frequency 119908re = 119908119904119863

2119889

times (1 minus1198892

1198632cos2120572)

(13)

where 119908119904is the shaft rotation frequency in rads 119889 is the

diameter of rolling element 119863 is the pitch diameter 119885 is thenumber of rolling elements and 120572 is the contact angle

In the present experiments outer race defect frequencyinner race defect frequency and rolling element defect fre-quency are 8632Hz 14584Hz and 5113Hz respectively

42 Vibration Signals of Typical Bearing Faults and TheirEnvelop Spectra The time-domain waveforms of bearingvibration signals obtained from accelerometers are shown inFigure 6 from which we can see that different from that ofperfect bearings as shown in Figure 6(a) obvious impulsivephenomenon exists in faulty vibration signals of fault bear-ings as shown in Figures 6(b) 6(c) and 6(d)

To investigate the differences of the signals and todistinguish different fault types envelope demodulation isapplied first to vibration signals of the three typical bearingfaults that is outer race fault inner race fault and rollerelement fault The results are shown in Figures 7 8 and 9In Figure 7 the frequency magnitude of outer race defectis lower than the second harmonic frequency of outer racedefect In Figure 8 although the inner race defect frequencyslightly sticks out it is still possibly confused by the side-lobes with almost similarmagnitudes In Figure 9 it is almostimpossible to figure out the calculated ball pass frequencyof 5113Hz From the figures we can see that simply basedon envelope demodulation it is difficult to identify the faultfrequencies from those of other components especially forweak fault signals

43 Wavelet Coefficients Distributions of the Vibration Sig-nals with and without Peak-Based Piecewise RecombinationBefore envelope demodulation wavelet decomposition is

6 Mathematical Problems in Engineering

0 02

02

04

04

06 08 1

0

Am

plitu

de (V

)

Time (s)

minus04

minus02

(a)

0 02 04 06 08 1

0

2

Am

plitu

de (V

)

Time (s)

minus4

minus2

(b)

0 02 04 06 08 1

0

5

10

Time (s)

Am

plitu

de (V

)

minus10

minus5

(c)

Am

plitu

de (V

)

0 02 04 06 08 1

0

5

Time (s)

minus5

(d)

Figure 6 Vibration signals of rolling bearings (a) a normal bearing and bearings with a point defect on (b) outer race (c) inner race bearingand (d) ball element

0 50 100 150 200 250 300 350 4000

50

100

150

200

250

Am

plitu

de

Frequency (Hz)

X 8621Y 1714

Figure 7 Envelope spectra of vibration signal in Figure 6(b)induced by outer race fault

always applied to denoise and enhance the weak fault signalsHowever vibration signals are not always sparse enough inwavelet domain which would affect the effectiveness of theenhancement As mentioned above peak-based piecewiserecombination can help to improve the smoothness of vibra-tion signals and make their energies more concentrated in

Am

plitu

de

0 50 100 150 200 250 300 350 4000

50

100

150

200

250

Frequency (Hz)

X 145Y 1816

Figure 8 Envelope spectra of vibration signal in Figure 6(c)induced by inner race fault

wavelet domain Comparisons between wavelet decomposi-tions with and without peak-based piecewise recombinationare illustrated in this section

Taking outer race fault as an example the time-domainvibration signal of Figure 6(b) is converted to that of Figure 10after applying piecewise recombination The signal becomesmuch smoother than the original signal after piecewise

Mathematical Problems in Engineering 7

0 50 100 150 200 250 300 350 4000

10

20

30

40

50

60

Frequency (Hz)

Am

plitu

de

X 4959Y 2026

Figure 9 Envelope spectra of vibration signal in Figure 6(d)induced by roller element fault

0 1 2 3 4 5 6 7 8 9 100

50010001500200025003000350040004500

Data number

Am

plitu

de

times104

Figure 10 Vibration signal induced by outer race fault inFigure 6(b) after peak-based piecewise recombination

Table 1 Energy distributions of the outer race fault signalrsquos waveletcoefficients

Frequency band Low frequency High frequencyEnergies without PPR 4931 5069Energies with PPR 999 01

recombination that is high frequency components have beenconverted to low frequency ones

For comparisons then wavelet transform is applied tosignals in Figures 6(b) and 10 respectively where 6-leveldecomposition with db4 wavelet basis is employed Both dis-tributions of the wavelet coefficients are shown in Figure 11Without piecewise recombination most wavelet coefficientsare nonzero that is not sparse enough as depicted inFigure 11(a) However it is improved significantly withpreprocessing of peak-based piecewise recombination asdepicted in Figure 11(b) Energies of the signal in waveletdomain are concentrated in only a few low frequency inter-vals and almost all detail coefficients at scales 1 to 6 are closeto zero It implies that peak-based piecewise recombinationcan significantly improve the sparseness of vibration signalsin wavelet domain

Statistically energy distributions of wavelet coefficientsobtained by the two methods are listed in Table 1 from

0 1 2 3 4 5 6 7 8 9 10

012345

Data number

Am

plitu

de

minus1

minus2

minus3

minus4

minus5

times104

(a)

Data number0 1 2 3 4 5 6 7 8 9

0

05

1

15

2

25

3

35

Am

plitu

de

times104

times104

(b)

Figure 11 Wavelet coefficients of the vibration signal induced byouter race fault (a) without PPR preprocessing and (b) with PPRpreprocessing

which we can see that after employing peak-based piecewiserecombination (PPR) preprocessing energies of the signal areconcentrated into only a few coefficients in wavelet domain999 in low frequency bands comparedwith 4931withoutpeak-based piecewise recombination

The experimental results of inner race fault and rollerelement fault are similar to those of the outer race fault whichare shown in Figures 12 and 13

44 Comparisons between Traditional Wavelet-Based Detec-tion Method and the Proposed Strategy with Envelop Demod-ulation Envelop demodulation is always combined withwavelet-based denoising and enhancement methods to ana-lyze vibration signals Comparisons of the results after en-velop demodulation based on the results of Section 43 arepresented in this section

Envelop demodulation is applied to the signals recon-structed from the thresholding of wavelet coefficients and abackward peak-based piecewise recombination is imple-mented for the proposed strategyThe demodulated envelopespectra are shown in Figures 14ndash16 In Figure 14(a) althoughthe fault frequency can be figured out the magnitude of

8 Mathematical Problems in Engineering

0 2 4 6 8 10

0

5

10

15

Data number

Am

plitu

de

minus20

minus15

minus10

minus5

times104

(a)

0Data number

2 4 6 8 10

0

05

1

15

2

25

3

35

Am

plitu

detimes104

times104

(b)

Figure 12 Wavelet coefficients of the vibration signal induced by inner race fault (a) without PPR preprocessing and (b) with PPRpreprocessing

Data number0 2 4 6 8 10

0

2

4

6

8

10

Am

plitu

de

minus8

minus6

minus4

minus2

times104

(a)

Data number0 2 4 6 8 10

0

05

1

15

2

25

Am

plitu

de

minus05

times104

times104

(b)

Figure 13 Wavelet coefficients of the vibration signal induced by roller element fault (a) without PPR preprocessing and (b) with PPR pre-processing

the outer race defect frequency is not the most remarkableone among all frequency components thus it is not alwayscertain that a bearing defect exists in outer race With theproposed method presented in this paper the magnitudeof outer race defect frequency becomes more prominent

where the magnitudes of outer race defect frequency havebeen enhanced and are more easily detected as Figure 14(b)depicted The proposed strategy is also more effective forinner race and roller bearing fault detection of rolling bear-ings and the results are shown as in Figures 15 and 16

Mathematical Problems in Engineering 9

0 100 200 300 4000

50

100

150

200

250

300

Frequency (Hz)

Am

plitu

de

X 8621Y 149

(a)

0 100 200 300 4000

200

400

600

800

1000

1200

Frequency (Hz)

Am

plitu

de

X 8621Y 9952

(b)

Figure 14 Envelope spectra of the vibration signal induced by outer race fault (a) wavelet analysis method and (b) the proposed method

0 100 200 300 4000

50

100

150

200

250

300

Frequency (Hz)

Am

plitu

de

Y 1777X 145

(a)

0 100 200 300 4000

50

100

150

200

250

300

350

400

Frequency (Hz)

Am

plitu

de

X 145Y 2525

(b)

Figure 15 Envelope spectra of the vibration signal induced by inner race fault (a) wavelet analysis method and (b) the proposed method

If defect exists in the bearing the magnitudes of faultsfrequencieswill increase in the envelope spectrumThereforethey are often used to describe the fault severity and alsoemployed to judge whether the fault features are enhancedIn this work magnitudes of faults frequencies are chosen asthe major indictors to reflect the fault features As recordedand listed in Table 2 with the proposed method magnitudesof faults frequencies are all enhanced and they become more

easily distinguished from interfered noise for the experimen-tal faults

5 Conclusions

In order to improve the current wavelet-based method forfault detection of rolling bearings the present paper devel-oped a new way to improve vibration signalsrsquo smoothness

10 Mathematical Problems in Engineering

0 100 200 300 4000

50

100

150

200

250

300

Frequency (Hz)

Am

plitu

de

X 4959Y 2026

(a)

0 100 200 300 4000

50

100

150

200

250

300

Frequency (Hz)

Am

plitu

de

X 4959Y 2792

(b)

Figure 16 Envelope spectra of the vibration signal induced by roller element fault (a) wavelet analysis method and (b) the proposedmethod

Table 2 Major indictors for bearing fault features

Defect frequencymagnitude of

Traditionalwavelet-based

methodThe proposed method

outer race 149 9952inner race 1777 2525roller element 2026 2792

as well as their sparseness in wavelet domain by employingpeak-based piecewise recombination and envelope demodu-lation Through the proposed method most noise mixed invibration signals can be eliminated and the weak fault signalsof rolling bearings can be enhanced and become more easilydetectableTherationality and validity of themethods are alsoproved by various experiments It shows that compared withtraditional wavelet-based method the proposed method hasmany advantages in bearing fault signals enhancement anddetection and is very simple and easy to implement

There is still room to improve the present work Thesensitivity of the method to noise and methods to makeenergy more concentrated are what will be studied in futurework

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This project is partially supported by the National NaturalScience Foundation of China (Grant no 51375037) and theNational Program on Key Basic Research Project (Grant no2012CB026000) The third author would also like to thankthe support of China Fundamental Research Funds for theCentral Universities (ZY1410) and the Public Hatching Plat-form for Recruited Talents of Beijing University of ChemicalTechnology

References

[1] W Zhou T G Habetler and R G Harley ldquoBearing conditionmonitoring methods for electric machines a general reviewrdquo inProceedings of the IEEE International Symposium on Diagnosticsfor Electric Machines Power Electronics and Drives (SDEMPEDrsquo07) pp 3ndash6 September 2007

[2] H Q Wang and P Chen ldquoIntelligent diagnosis method for acentrifugal pump using features of vibration signalsrdquo NeuralComputing and Applications vol 18 no 4 pp 397ndash405 2009

[3] H Wang and P Chen ldquoSequential diagnosis for rolling bearingusing fuzzy neural networkrdquo in Proceedings of the IEEEASMEInternational Conference on Advanced Intelligent Mechatronics(AIM rsquo08) pp 56ndash61 Xirsquoan China August 2008

[4] T Mitoma H Wang and P Chen ldquoFault diagnosis and condi-tion surveillance for plant rotating machinery using partially-linearized neural networkrdquo Computers and Industrial Engineer-ing vol 55 no 4 pp 783ndash794 2008

[5] P D McFadden and J D Smith ldquoVibration monitoring of roll-ing element bearings by the high-frequency resonance tech-niquemdasha reviewrdquo Tribology International vol 17 no 1 pp 3ndash101984

Mathematical Problems in Engineering 11

[6] N Tandon and A Choudhury ldquoReview of vibration and acous-tic measurement methods for the detection of defects in rollingelement bearingsrdquo Tribology International vol 32 no 8 pp469ndash480 1999

[7] R B W Heng and M J M Nor ldquoStatistical analysis of soundand vibration signals for monitoring rolling element bearingconditionrdquo Applied Acoustics vol 53 no 1ndash3 pp 211ndash226 1998

[8] C Junsheng Y Dejie and Y Yu ldquoThe application of energyoperator demodulation approach based on EMD in machineryfault diagnosisrdquo Mechanical Systems and Signal Processing vol21 no 2 pp 668ndash677 2007

[9] R B Randall J Antoni and S Chobsaard ldquoThe relationship be-tween spectral correlation and envelope analysis in the diagnos-tics of bearing faults and other cyclostationarymachine signalsrdquoMechanical Systems and Signal Processing vol 15 no 5 pp 945ndash962 2001

[10] A B Ming Z Y Qin W Zhang et al ldquoSpectrum auto-cor-relation analysis and its application to fault diagnosis of rollingelement bearingsrdquo Mechanical Systems and Signal Processingvol 41 no 1 pp 141ndash154 2013

[11] D Yu J Cheng and Y Yang ldquoApplication of EMDmethod andHilbert spectrum to the fault diagnosis of roller bearingsrdquo Me-chanical Systems and Signal Processing vol 19 no 2 pp 259ndash270 2005

[12] V K Rai and A RMohanty ldquoBearing fault diagnosis using FFTof intrinsic mode functions in Hilbert-Huang transformrdquo Me-chanical Systems and Signal Processing vol 21 no 6 pp 2607ndash2615 2007

[13] Y Lei N Li J Lin and SWang ldquoFault diagnosis of rotatingma-chinery based on an adaptive ensemble empirical mode decom-positionrdquo Sensors vol 13 no 12 pp 16950ndash16964 2013

[14] J Cheng Y Yang and Y Yang ldquoA rotatingmachinery fault diag-nosis method based on local mean decompositionrdquo Digital Sig-nal Processing vol 22 no 2 pp 356ndash366 2012

[15] B Chen Z He X Chen H Cao G Cai and Y Zi ldquoA demod-ulating approach based on local mean decomposition and itsapplications in mechanical fault diagnosisrdquo Measurement Sci-ence and Technology vol 22 no 5 Article ID 055704 2011

[16] S Prabhakar A R Mohanty and A S Sekhar ldquoApplication ofdiscrete wavelet transform for detection of ball bearing racefaultsrdquoTribology International vol 35 no 12 pp 793ndash800 2002

[17] R Yan R X Gao and X Chen ldquoWavelets for fault diagnosis ofrotary machines a review with applicationsrdquo Signal Processingvol 96 part A pp 1ndash15 2014

[18] H Wang and P Chen ldquoA feature extraction method based oninformation theory for fault diagnosis of reciprocating machin-eryrdquo Sensors vol 9 no 4 pp 2415ndash2436 2009

[19] K Li X L Ping H Q Wang and P Chen ldquoSequential fuzzydiagnosis method for motor roller bearing in variable operatingconditions based on vibration analysisrdquo Sensors vol 13 no 6pp 8013ndash8041 2013

[20] R Rubini and U Meneghetti ldquoApplication of the envelope andwavelet transform analyses for the diagnosis of incipient faultsin ball bearingsrdquoMechanical Systems and Signal Processing vol15 no 2 pp 287ndash302 2001

[21] L Gao Z Yang L Cai H Wang and P Chen ldquoRoller bearingfault diagnosis based on nonlinear redundant lifting waveletpacket analysisrdquo Sensors vol 11 no 1 pp 260ndash277 2011

[22] Z Yang L Cai L Gao and H Wang ldquoAdaptive redundant lift-ing wavelet transform based on fitting for fault feature extrac-tion of roller bearingsrdquo Sensors vol 12 no 4 pp 4381ndash43982012

[23] Z He ldquoPeak transform for efficient image representation andcodingrdquo IEEE Transactions on Image Processing vol 16 no 7pp 1741ndash1754 2007

[24] S Anila and N Devarajan ldquoThe usage of peak transform forimage compressionrdquo International Journal of Computer SystemsScience amp Engineering vol 2 no 11 pp 6308ndash6316 2010

[25] D Ho and R B Randall ldquoOptimization of bearing diagnostictechniques using simulated and actual bearing fault signalsrdquoMechanical Systems and Signal Processing vol 14 no 5 pp 763ndash788 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

0 1 2 3 4 5 6 7 8 9 10

012345

times104Data number

Am

plitu

de

minus3

minus4

minus5

minus2

minus1

Figure 2 Wavelet coefficients of the bearing fault signal in Figure 1

information the measured vibration signals are often mixedwith lots of interfering information from other componentsMeanwhile various types of noise also make the signal morecomplicated In addition to the noise introduced by mea-surement instruments interfering signals can also be gen-erated accompanied with the existence of transmission pathbetween local impact point and vibration measurementpoint In a word bearing vibration signal is nonstationarywith wide frequency band which makes the bearing faultsignature extremely weak to be detected

Existing wavelet-based denoisingmethodsmainly rely onthe sparseness of wavelet coefficients [17] If a signal is notsparse enough in wavelet domain it becomes challengingfor feature extraction by wavelet analysis due to the difficul-ties to distinguish required features from large number ofwavelet coefficients Common vibration signals from faultybearings are often composed of impulse components Atypical vibration signal is shown in Figure 1 and its waveletcoefficients are shown in Figure 2 from which we can seethat most coefficients are nonzero that is not sparse enoughExperiments show that this is applicable to fault vibrationsignals of inner race outer race and ball element Thus it isimportant to find out a proper way to improve the sparsenessas well as reducing noise and enhancing the weak faultfeatures

3 The Proposed Method

31 The Basic Principle of Peak-Based Piecewise Recombi-nation Peak-based piecewise recombination (PPR) or so-called peak transform is a kind of nonlinear geometric trans-form tomake the signal smoother through transforming highfrequency subsections to low frequency ones Here a briefdescription of PPR is given as follows for more informationplease refer to [23]

For a continuous function 119891(119909) that has 119873-point peaks(or break points) 119909

119894(119894 = 1 sdot sdot sdot 119873) is defined over [119886 119887]

In fact 119891(119909) can be viewed as a piecewise function with119873+1 combined curves where the119873 peaksrsquo positions are theconnection pointsThen the 119894th piecewise curve can be calledsubfunction119891

119894(119909) over interval [119886 119887] The 119873-point forward

1

5

6

4

3

2Forward

PPRBackward

PPR

f1(x) f2(x)

f3(x) f4(x)

f5(x)f6(x)

x

x

a b

x0 = ax1 x2 x3 x4 x5 x6 = b

Figure 3 Schematic diagram of five-point peak-based piecewiserecombination

transform of peak-based piecewise recombination can bedefined as

119875119865 [119891 (119909)] = 119892119900 (119909) oplus 119892119890 (119909) (1)

where

119892119900 (119909) = 1198911 (119909) oplus 1198913 (119909) oplus sdot sdot sdot oplus 1198912lfloor(119873minus1)2rfloor+1 (119909)

119892119890 (119909) = 1198912 (119909) oplus 1198914 (119909) oplus sdot sdot sdot oplus 1198912lfloor1198732rfloor (119909)

(2)

are the cascades of all odd- and even-numbered curve seg-ments respectively Here 2lfloor(119873 minus 1)2rfloor + 1 and 2lfloor1198732rfloor arerespectively the largest odd and even integers that are lessthan or equal to119873

Geometrically speaking as illustrated in Figure 3 all pie-cewise curves with similar positive slopes are grouped to-gether and cascaded to be a new curve while all negativeones are grouped into another new curve finally these twonew subcurves are recombined to be a new curve It is appar-ent that the recombination only changes the order of the pie-cewise curves and is reversible The backward transformcan be done by simply recascading them according to theiroriginal orders which is symbolized as119875119865minus1[119891(119909)] accordingto (1)

It should be noted that although the forward and back-ward transforms discussed above are defined for curve func-tions they can also be defined in similarmanners for discrete-time signals of rolling bearingsrsquo vibration

32 Wavelet Decomposition Wavelet 120595(119886119887)(119905) is obtained by

translation and dilation based on a defined single function120595(119905) as

120595(119886119887) (119905) =

1

radic119886120595(119905 minus 119887

119886) (3)

4 Mathematical Problems in Engineering

where 119886 gt 0 is the so-called scaling parameter and 119887 isin 119877 isthe parameter denoting time localization which can be con-tinuous or discrete120595(119905) is called ldquomotherwaveletrdquo to generatewavelets 120595

(119886119887)(119905)

Given a signal 119909(119905) with finite energy the wavelet trans-form with analytic wavelet 120595(119905) can be viewed as the convo-lution of 119909(119905) with a scaled and conjugated wavelet

119882(119886 119887) =1

radic119886intinfin

minusinfin

119909 (119905) 120595lowast (119905 minus 119887

119886)119889119905 (4)

where 120595lowast(119905) stands for the complex conjugation of 120595(119905)The wavelet transform 119882(119886 119887) can be considered as a

function of translation 119887 at each scale 119886 From (4) we can seethat wavelet transform is a kind of time-frequency analysis ora time-scaled analysis Different from analysis either in timedomain or in frequency domain or even different from thatin time-frequency domain but with fixed-length windows byshort time Fourier transform (SFFT) vibration signals canbe decomposed bywavelet transformwithmultiscale analysisthrough dilation and translation so that the time-frequencyfeatures of vibration signals can bemore effectively extracted

Wavelet transform is also reversible as follows

119909 (119905) = 119862minus1

120595∬119882(119886 119887) 120595(119886119887) (119905)

119889119886

1198862119889119887 (5)

where

119862120595= intinfin

minusinfin

10038161003816100381610038161003816120595(120596)100381610038161003816100381610038162

|120596|119889120596 lt infin

(119908) = int120595 (119905) exp (minusj119908119905) 119889119905

(6)

which provides the possibility to reconstruct the originalvibration signals

33 Envelope Demodulation As mentioned in the previoussections if bearing defects exist the measured vibration sig-nal would be amplitude modulated at its characteristic defectfrequency The modulating wave is a pass vibration signalcorresponding to local defects and various demodulationtechniques have been developed for the separation In thiswork an envelope demodulation method based on Hilberttransform is employed

For a continuous time signal 119909(119905) the Hilbert transform119909(119905) is defined as

119909(119905) =1

120587int+infin

minusinfin

119909 (120591)

(119905 minus 120591)119889120591 (7)

Then combine 119909(119905) and 119909(119905) to form a new analytic signal

119892 (119905) = 119909 (119905) + 119895119909(119905) (8)

The envelope of 119909(119905) is defined as the amplitude of 119892(119905)

119860 (119905) = radic[119909 (119905)]2+ [119909 (119905)]2 (9)

After Hilbert transform envelope spectra could be obtainedby FFT to extract characteristic defect frequencies

34 Enhancement Frame with Peak-Based Multiscale Decom-position and Envelope Demodulation Wavelet transform is akind of time-frequency analysis method for signal denoisingby adjusting the discrete detail coefficients and approxima-tion coefficients obtained from multilevel decompositionThis scheme is called threshold processing one of whosepremises is the sparse representation of signals in waveletdomain However different from that of perfect bearing thevibration signal energy of bearings with defects increases athigh frequenciesThus vibration signals are not always sparsein wavelet domain Piecewise curve recombination can con-vert energies at high frequencies to ones at low frequenciesIn this way signal energies are focused in low frequency sub-bands so that the piecewise recombination can improve sig-nalrsquos smoothness as well as its sparseness in wavelet domain

The present paper proposed an enhancement frame forvibration signals by the combination of peak-based multi-scale decomposition and envelope demodulationThe proce-dure is described as follows

(1) Peak-based piecewise recombination of vibration sig-nal let 119909(119905) denote the measured vibration signalfirstly then transform the original signal 119909(119905) tosmooth signal 119875119865[119891(119909)] through piecewise recombi-nation according to Equation (1) The key is to deter-mine the peaks and to recombine different segmentsIn the present work a cascading strategy related tolocal minimum andmaximum is employedThe end-point of a segment is chosen as a peak If two adjacentpeaks are equal one of them is defined as the peak

(2) Wavelet decomposition of the new recombined sig-nal choose a wavelet basis (eg Morlet wavelet orDb4 wavelet) then decompose the new signal119875119865[119891(119909)] at level N

119882(119886 119887) =1

radic119886intinfin

minusinfin

119875119865 [119909 (119905)] 120595lowast (119905 minus 119887

119886)119889119905 (10)

(3) Coefficients threshold set a threshold value and applyit to the approximation coefficients at level119873 and thedetail coefficients at levels 1 to119873 The hard thresholdcan be expressed as

119910119904= 11991010038161003816100381610038161199101003816100381610038161003816 gt 119905

010038161003816100381610038161199101003816100381610038161003816 lt 119905

(11)

where 119905 is the selected threshold 119910 is the originalwavelet coefficients and 119910

119904is the wavelet coefficients

after thresholding In this work the value which cov-ers about 95 of the smallest wavelet coefficients is setas the threshold value Then 5 of the largest waveletcoefficients are kept as nonzero ones to reserve thesignal feature

(4) Signal recovery reconstruct the vibration signalbased on the modified approximation coefficients atlevel119873 and the modified detail coefficients at levels 1to119873

1198751198651015840 [119909 (119905)] = 119862minus1

120595∬1198821015840 (119886 119887) 120595(119886119887) (119905)

119889119886

1198862119889119887 (12)

Mathematical Problems in Engineering 5

Original signal

Peak-based piecewise transform

Newly generated smooth signal

Wavelet decomposition

Coefficients threshold Inverse wavelet transform

Peak-based piecewise transform backward

Enhanced signal

Envelope demodulation

Spectral analysis and detection

Figure 4 Flowchart of the proposed enhancement scheme for weakfault signal

CH 1

Figure 5 Fault rig of a roller element bearing

(5) Piecewise signal backward transform put the pro-cessed signal segments back to their original posi-tions then a new denoised vibration signal 1199091015840(119905) isgenerated

(6) Envelope demodulation according to (9) use Hilberttransform to obtain the envelope 119860(119905) of signal 1199091015840(119905)

(7) Extraction of characteristic defect frequency get theenvelope spectra via Fourier analysis and distinguishdifferent fault types of rolling bearings

In summary these steps are shown as in Figure 4

4 Experimental Results and Discussions

41 Experiment Setup To verify the effectiveness of theproposed method experiments are designed for fault rigs ofroller element bearing as shown in Figure 5 The rig is com-posed of a motor a coupling a rotor and a shaft with tworoller bearings Basically the sensors are located at the posi-tion near the bearings to mitigate the effects of signal atten-uation Usually bearing housing is one of the best locationsfor bearing arrangement Therefore here an accelerometeris located on the bearing housing to acquire the vibrationsignals

In the experiments a single point defect is introduced ininner raceway outer raceway and ball element of differ-ent bearings using electron-discharge machining with faultwidths of 3mm 7mm and 7mm and depths of 5mm25mm and 25mm respectively

Vibration signals are measured by an accelerometer lo-cated at the top of the bearing house (CH1 as shownin Figure 5) In all experiments the sample frequency is100 kHz and the shaft speed is finite 1300 rpm Four commonconditions are studied in the present paper including aperfect healthy bearing and three other bearings with pointdefect on the outer race on the inner race bearing and onthe ball element

In theory elements of roller bearings have their own spe-cific rotational frequencies which may appear in envelopespectra if defects existThese defect frequencies are called ballpass frequencies For a bearing with a stationary outer racethese frequencies can be given by the following formulas

outer race defect frequency 119908od =119885119908119904

2119889(1 minus

119889

119863cos120572)

inner race defect frequency 119908id =119885119908119904

2(1 +

119889

119863cos120572)

rolling element defect frequency 119908re = 119908119904119863

2119889

times (1 minus1198892

1198632cos2120572)

(13)

where 119908119904is the shaft rotation frequency in rads 119889 is the

diameter of rolling element 119863 is the pitch diameter 119885 is thenumber of rolling elements and 120572 is the contact angle

In the present experiments outer race defect frequencyinner race defect frequency and rolling element defect fre-quency are 8632Hz 14584Hz and 5113Hz respectively

42 Vibration Signals of Typical Bearing Faults and TheirEnvelop Spectra The time-domain waveforms of bearingvibration signals obtained from accelerometers are shown inFigure 6 from which we can see that different from that ofperfect bearings as shown in Figure 6(a) obvious impulsivephenomenon exists in faulty vibration signals of fault bear-ings as shown in Figures 6(b) 6(c) and 6(d)

To investigate the differences of the signals and todistinguish different fault types envelope demodulation isapplied first to vibration signals of the three typical bearingfaults that is outer race fault inner race fault and rollerelement fault The results are shown in Figures 7 8 and 9In Figure 7 the frequency magnitude of outer race defectis lower than the second harmonic frequency of outer racedefect In Figure 8 although the inner race defect frequencyslightly sticks out it is still possibly confused by the side-lobes with almost similarmagnitudes In Figure 9 it is almostimpossible to figure out the calculated ball pass frequencyof 5113Hz From the figures we can see that simply basedon envelope demodulation it is difficult to identify the faultfrequencies from those of other components especially forweak fault signals

43 Wavelet Coefficients Distributions of the Vibration Sig-nals with and without Peak-Based Piecewise RecombinationBefore envelope demodulation wavelet decomposition is

6 Mathematical Problems in Engineering

0 02

02

04

04

06 08 1

0

Am

plitu

de (V

)

Time (s)

minus04

minus02

(a)

0 02 04 06 08 1

0

2

Am

plitu

de (V

)

Time (s)

minus4

minus2

(b)

0 02 04 06 08 1

0

5

10

Time (s)

Am

plitu

de (V

)

minus10

minus5

(c)

Am

plitu

de (V

)

0 02 04 06 08 1

0

5

Time (s)

minus5

(d)

Figure 6 Vibration signals of rolling bearings (a) a normal bearing and bearings with a point defect on (b) outer race (c) inner race bearingand (d) ball element

0 50 100 150 200 250 300 350 4000

50

100

150

200

250

Am

plitu

de

Frequency (Hz)

X 8621Y 1714

Figure 7 Envelope spectra of vibration signal in Figure 6(b)induced by outer race fault

always applied to denoise and enhance the weak fault signalsHowever vibration signals are not always sparse enough inwavelet domain which would affect the effectiveness of theenhancement As mentioned above peak-based piecewiserecombination can help to improve the smoothness of vibra-tion signals and make their energies more concentrated in

Am

plitu

de

0 50 100 150 200 250 300 350 4000

50

100

150

200

250

Frequency (Hz)

X 145Y 1816

Figure 8 Envelope spectra of vibration signal in Figure 6(c)induced by inner race fault

wavelet domain Comparisons between wavelet decomposi-tions with and without peak-based piecewise recombinationare illustrated in this section

Taking outer race fault as an example the time-domainvibration signal of Figure 6(b) is converted to that of Figure 10after applying piecewise recombination The signal becomesmuch smoother than the original signal after piecewise

Mathematical Problems in Engineering 7

0 50 100 150 200 250 300 350 4000

10

20

30

40

50

60

Frequency (Hz)

Am

plitu

de

X 4959Y 2026

Figure 9 Envelope spectra of vibration signal in Figure 6(d)induced by roller element fault

0 1 2 3 4 5 6 7 8 9 100

50010001500200025003000350040004500

Data number

Am

plitu

de

times104

Figure 10 Vibration signal induced by outer race fault inFigure 6(b) after peak-based piecewise recombination

Table 1 Energy distributions of the outer race fault signalrsquos waveletcoefficients

Frequency band Low frequency High frequencyEnergies without PPR 4931 5069Energies with PPR 999 01

recombination that is high frequency components have beenconverted to low frequency ones

For comparisons then wavelet transform is applied tosignals in Figures 6(b) and 10 respectively where 6-leveldecomposition with db4 wavelet basis is employed Both dis-tributions of the wavelet coefficients are shown in Figure 11Without piecewise recombination most wavelet coefficientsare nonzero that is not sparse enough as depicted inFigure 11(a) However it is improved significantly withpreprocessing of peak-based piecewise recombination asdepicted in Figure 11(b) Energies of the signal in waveletdomain are concentrated in only a few low frequency inter-vals and almost all detail coefficients at scales 1 to 6 are closeto zero It implies that peak-based piecewise recombinationcan significantly improve the sparseness of vibration signalsin wavelet domain

Statistically energy distributions of wavelet coefficientsobtained by the two methods are listed in Table 1 from

0 1 2 3 4 5 6 7 8 9 10

012345

Data number

Am

plitu

de

minus1

minus2

minus3

minus4

minus5

times104

(a)

Data number0 1 2 3 4 5 6 7 8 9

0

05

1

15

2

25

3

35

Am

plitu

de

times104

times104

(b)

Figure 11 Wavelet coefficients of the vibration signal induced byouter race fault (a) without PPR preprocessing and (b) with PPRpreprocessing

which we can see that after employing peak-based piecewiserecombination (PPR) preprocessing energies of the signal areconcentrated into only a few coefficients in wavelet domain999 in low frequency bands comparedwith 4931withoutpeak-based piecewise recombination

The experimental results of inner race fault and rollerelement fault are similar to those of the outer race fault whichare shown in Figures 12 and 13

44 Comparisons between Traditional Wavelet-Based Detec-tion Method and the Proposed Strategy with Envelop Demod-ulation Envelop demodulation is always combined withwavelet-based denoising and enhancement methods to ana-lyze vibration signals Comparisons of the results after en-velop demodulation based on the results of Section 43 arepresented in this section

Envelop demodulation is applied to the signals recon-structed from the thresholding of wavelet coefficients and abackward peak-based piecewise recombination is imple-mented for the proposed strategyThe demodulated envelopespectra are shown in Figures 14ndash16 In Figure 14(a) althoughthe fault frequency can be figured out the magnitude of

8 Mathematical Problems in Engineering

0 2 4 6 8 10

0

5

10

15

Data number

Am

plitu

de

minus20

minus15

minus10

minus5

times104

(a)

0Data number

2 4 6 8 10

0

05

1

15

2

25

3

35

Am

plitu

detimes104

times104

(b)

Figure 12 Wavelet coefficients of the vibration signal induced by inner race fault (a) without PPR preprocessing and (b) with PPRpreprocessing

Data number0 2 4 6 8 10

0

2

4

6

8

10

Am

plitu

de

minus8

minus6

minus4

minus2

times104

(a)

Data number0 2 4 6 8 10

0

05

1

15

2

25

Am

plitu

de

minus05

times104

times104

(b)

Figure 13 Wavelet coefficients of the vibration signal induced by roller element fault (a) without PPR preprocessing and (b) with PPR pre-processing

the outer race defect frequency is not the most remarkableone among all frequency components thus it is not alwayscertain that a bearing defect exists in outer race With theproposed method presented in this paper the magnitudeof outer race defect frequency becomes more prominent

where the magnitudes of outer race defect frequency havebeen enhanced and are more easily detected as Figure 14(b)depicted The proposed strategy is also more effective forinner race and roller bearing fault detection of rolling bear-ings and the results are shown as in Figures 15 and 16

Mathematical Problems in Engineering 9

0 100 200 300 4000

50

100

150

200

250

300

Frequency (Hz)

Am

plitu

de

X 8621Y 149

(a)

0 100 200 300 4000

200

400

600

800

1000

1200

Frequency (Hz)

Am

plitu

de

X 8621Y 9952

(b)

Figure 14 Envelope spectra of the vibration signal induced by outer race fault (a) wavelet analysis method and (b) the proposed method

0 100 200 300 4000

50

100

150

200

250

300

Frequency (Hz)

Am

plitu

de

Y 1777X 145

(a)

0 100 200 300 4000

50

100

150

200

250

300

350

400

Frequency (Hz)

Am

plitu

de

X 145Y 2525

(b)

Figure 15 Envelope spectra of the vibration signal induced by inner race fault (a) wavelet analysis method and (b) the proposed method

If defect exists in the bearing the magnitudes of faultsfrequencieswill increase in the envelope spectrumThereforethey are often used to describe the fault severity and alsoemployed to judge whether the fault features are enhancedIn this work magnitudes of faults frequencies are chosen asthe major indictors to reflect the fault features As recordedand listed in Table 2 with the proposed method magnitudesof faults frequencies are all enhanced and they become more

easily distinguished from interfered noise for the experimen-tal faults

5 Conclusions

In order to improve the current wavelet-based method forfault detection of rolling bearings the present paper devel-oped a new way to improve vibration signalsrsquo smoothness

10 Mathematical Problems in Engineering

0 100 200 300 4000

50

100

150

200

250

300

Frequency (Hz)

Am

plitu

de

X 4959Y 2026

(a)

0 100 200 300 4000

50

100

150

200

250

300

Frequency (Hz)

Am

plitu

de

X 4959Y 2792

(b)

Figure 16 Envelope spectra of the vibration signal induced by roller element fault (a) wavelet analysis method and (b) the proposedmethod

Table 2 Major indictors for bearing fault features

Defect frequencymagnitude of

Traditionalwavelet-based

methodThe proposed method

outer race 149 9952inner race 1777 2525roller element 2026 2792

as well as their sparseness in wavelet domain by employingpeak-based piecewise recombination and envelope demodu-lation Through the proposed method most noise mixed invibration signals can be eliminated and the weak fault signalsof rolling bearings can be enhanced and become more easilydetectableTherationality and validity of themethods are alsoproved by various experiments It shows that compared withtraditional wavelet-based method the proposed method hasmany advantages in bearing fault signals enhancement anddetection and is very simple and easy to implement

There is still room to improve the present work Thesensitivity of the method to noise and methods to makeenergy more concentrated are what will be studied in futurework

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This project is partially supported by the National NaturalScience Foundation of China (Grant no 51375037) and theNational Program on Key Basic Research Project (Grant no2012CB026000) The third author would also like to thankthe support of China Fundamental Research Funds for theCentral Universities (ZY1410) and the Public Hatching Plat-form for Recruited Talents of Beijing University of ChemicalTechnology

References

[1] W Zhou T G Habetler and R G Harley ldquoBearing conditionmonitoring methods for electric machines a general reviewrdquo inProceedings of the IEEE International Symposium on Diagnosticsfor Electric Machines Power Electronics and Drives (SDEMPEDrsquo07) pp 3ndash6 September 2007

[2] H Q Wang and P Chen ldquoIntelligent diagnosis method for acentrifugal pump using features of vibration signalsrdquo NeuralComputing and Applications vol 18 no 4 pp 397ndash405 2009

[3] H Wang and P Chen ldquoSequential diagnosis for rolling bearingusing fuzzy neural networkrdquo in Proceedings of the IEEEASMEInternational Conference on Advanced Intelligent Mechatronics(AIM rsquo08) pp 56ndash61 Xirsquoan China August 2008

[4] T Mitoma H Wang and P Chen ldquoFault diagnosis and condi-tion surveillance for plant rotating machinery using partially-linearized neural networkrdquo Computers and Industrial Engineer-ing vol 55 no 4 pp 783ndash794 2008

[5] P D McFadden and J D Smith ldquoVibration monitoring of roll-ing element bearings by the high-frequency resonance tech-niquemdasha reviewrdquo Tribology International vol 17 no 1 pp 3ndash101984

Mathematical Problems in Engineering 11

[6] N Tandon and A Choudhury ldquoReview of vibration and acous-tic measurement methods for the detection of defects in rollingelement bearingsrdquo Tribology International vol 32 no 8 pp469ndash480 1999

[7] R B W Heng and M J M Nor ldquoStatistical analysis of soundand vibration signals for monitoring rolling element bearingconditionrdquo Applied Acoustics vol 53 no 1ndash3 pp 211ndash226 1998

[8] C Junsheng Y Dejie and Y Yu ldquoThe application of energyoperator demodulation approach based on EMD in machineryfault diagnosisrdquo Mechanical Systems and Signal Processing vol21 no 2 pp 668ndash677 2007

[9] R B Randall J Antoni and S Chobsaard ldquoThe relationship be-tween spectral correlation and envelope analysis in the diagnos-tics of bearing faults and other cyclostationarymachine signalsrdquoMechanical Systems and Signal Processing vol 15 no 5 pp 945ndash962 2001

[10] A B Ming Z Y Qin W Zhang et al ldquoSpectrum auto-cor-relation analysis and its application to fault diagnosis of rollingelement bearingsrdquo Mechanical Systems and Signal Processingvol 41 no 1 pp 141ndash154 2013

[11] D Yu J Cheng and Y Yang ldquoApplication of EMDmethod andHilbert spectrum to the fault diagnosis of roller bearingsrdquo Me-chanical Systems and Signal Processing vol 19 no 2 pp 259ndash270 2005

[12] V K Rai and A RMohanty ldquoBearing fault diagnosis using FFTof intrinsic mode functions in Hilbert-Huang transformrdquo Me-chanical Systems and Signal Processing vol 21 no 6 pp 2607ndash2615 2007

[13] Y Lei N Li J Lin and SWang ldquoFault diagnosis of rotatingma-chinery based on an adaptive ensemble empirical mode decom-positionrdquo Sensors vol 13 no 12 pp 16950ndash16964 2013

[14] J Cheng Y Yang and Y Yang ldquoA rotatingmachinery fault diag-nosis method based on local mean decompositionrdquo Digital Sig-nal Processing vol 22 no 2 pp 356ndash366 2012

[15] B Chen Z He X Chen H Cao G Cai and Y Zi ldquoA demod-ulating approach based on local mean decomposition and itsapplications in mechanical fault diagnosisrdquo Measurement Sci-ence and Technology vol 22 no 5 Article ID 055704 2011

[16] S Prabhakar A R Mohanty and A S Sekhar ldquoApplication ofdiscrete wavelet transform for detection of ball bearing racefaultsrdquoTribology International vol 35 no 12 pp 793ndash800 2002

[17] R Yan R X Gao and X Chen ldquoWavelets for fault diagnosis ofrotary machines a review with applicationsrdquo Signal Processingvol 96 part A pp 1ndash15 2014

[18] H Wang and P Chen ldquoA feature extraction method based oninformation theory for fault diagnosis of reciprocating machin-eryrdquo Sensors vol 9 no 4 pp 2415ndash2436 2009

[19] K Li X L Ping H Q Wang and P Chen ldquoSequential fuzzydiagnosis method for motor roller bearing in variable operatingconditions based on vibration analysisrdquo Sensors vol 13 no 6pp 8013ndash8041 2013

[20] R Rubini and U Meneghetti ldquoApplication of the envelope andwavelet transform analyses for the diagnosis of incipient faultsin ball bearingsrdquoMechanical Systems and Signal Processing vol15 no 2 pp 287ndash302 2001

[21] L Gao Z Yang L Cai H Wang and P Chen ldquoRoller bearingfault diagnosis based on nonlinear redundant lifting waveletpacket analysisrdquo Sensors vol 11 no 1 pp 260ndash277 2011

[22] Z Yang L Cai L Gao and H Wang ldquoAdaptive redundant lift-ing wavelet transform based on fitting for fault feature extrac-tion of roller bearingsrdquo Sensors vol 12 no 4 pp 4381ndash43982012

[23] Z He ldquoPeak transform for efficient image representation andcodingrdquo IEEE Transactions on Image Processing vol 16 no 7pp 1741ndash1754 2007

[24] S Anila and N Devarajan ldquoThe usage of peak transform forimage compressionrdquo International Journal of Computer SystemsScience amp Engineering vol 2 no 11 pp 6308ndash6316 2010

[25] D Ho and R B Randall ldquoOptimization of bearing diagnostictechniques using simulated and actual bearing fault signalsrdquoMechanical Systems and Signal Processing vol 14 no 5 pp 763ndash788 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

where 119886 gt 0 is the so-called scaling parameter and 119887 isin 119877 isthe parameter denoting time localization which can be con-tinuous or discrete120595(119905) is called ldquomotherwaveletrdquo to generatewavelets 120595

(119886119887)(119905)

Given a signal 119909(119905) with finite energy the wavelet trans-form with analytic wavelet 120595(119905) can be viewed as the convo-lution of 119909(119905) with a scaled and conjugated wavelet

119882(119886 119887) =1

radic119886intinfin

minusinfin

119909 (119905) 120595lowast (119905 minus 119887

119886)119889119905 (4)

where 120595lowast(119905) stands for the complex conjugation of 120595(119905)The wavelet transform 119882(119886 119887) can be considered as a

function of translation 119887 at each scale 119886 From (4) we can seethat wavelet transform is a kind of time-frequency analysis ora time-scaled analysis Different from analysis either in timedomain or in frequency domain or even different from thatin time-frequency domain but with fixed-length windows byshort time Fourier transform (SFFT) vibration signals canbe decomposed bywavelet transformwithmultiscale analysisthrough dilation and translation so that the time-frequencyfeatures of vibration signals can bemore effectively extracted

Wavelet transform is also reversible as follows

119909 (119905) = 119862minus1

120595∬119882(119886 119887) 120595(119886119887) (119905)

119889119886

1198862119889119887 (5)

where

119862120595= intinfin

minusinfin

10038161003816100381610038161003816120595(120596)100381610038161003816100381610038162

|120596|119889120596 lt infin

(119908) = int120595 (119905) exp (minusj119908119905) 119889119905

(6)

which provides the possibility to reconstruct the originalvibration signals

33 Envelope Demodulation As mentioned in the previoussections if bearing defects exist the measured vibration sig-nal would be amplitude modulated at its characteristic defectfrequency The modulating wave is a pass vibration signalcorresponding to local defects and various demodulationtechniques have been developed for the separation In thiswork an envelope demodulation method based on Hilberttransform is employed

For a continuous time signal 119909(119905) the Hilbert transform119909(119905) is defined as

119909(119905) =1

120587int+infin

minusinfin

119909 (120591)

(119905 minus 120591)119889120591 (7)

Then combine 119909(119905) and 119909(119905) to form a new analytic signal

119892 (119905) = 119909 (119905) + 119895119909(119905) (8)

The envelope of 119909(119905) is defined as the amplitude of 119892(119905)

119860 (119905) = radic[119909 (119905)]2+ [119909 (119905)]2 (9)

After Hilbert transform envelope spectra could be obtainedby FFT to extract characteristic defect frequencies

34 Enhancement Frame with Peak-Based Multiscale Decom-position and Envelope Demodulation Wavelet transform is akind of time-frequency analysis method for signal denoisingby adjusting the discrete detail coefficients and approxima-tion coefficients obtained from multilevel decompositionThis scheme is called threshold processing one of whosepremises is the sparse representation of signals in waveletdomain However different from that of perfect bearing thevibration signal energy of bearings with defects increases athigh frequenciesThus vibration signals are not always sparsein wavelet domain Piecewise curve recombination can con-vert energies at high frequencies to ones at low frequenciesIn this way signal energies are focused in low frequency sub-bands so that the piecewise recombination can improve sig-nalrsquos smoothness as well as its sparseness in wavelet domain

The present paper proposed an enhancement frame forvibration signals by the combination of peak-based multi-scale decomposition and envelope demodulationThe proce-dure is described as follows

(1) Peak-based piecewise recombination of vibration sig-nal let 119909(119905) denote the measured vibration signalfirstly then transform the original signal 119909(119905) tosmooth signal 119875119865[119891(119909)] through piecewise recombi-nation according to Equation (1) The key is to deter-mine the peaks and to recombine different segmentsIn the present work a cascading strategy related tolocal minimum andmaximum is employedThe end-point of a segment is chosen as a peak If two adjacentpeaks are equal one of them is defined as the peak

(2) Wavelet decomposition of the new recombined sig-nal choose a wavelet basis (eg Morlet wavelet orDb4 wavelet) then decompose the new signal119875119865[119891(119909)] at level N

119882(119886 119887) =1

radic119886intinfin

minusinfin

119875119865 [119909 (119905)] 120595lowast (119905 minus 119887

119886)119889119905 (10)

(3) Coefficients threshold set a threshold value and applyit to the approximation coefficients at level119873 and thedetail coefficients at levels 1 to119873 The hard thresholdcan be expressed as

119910119904= 11991010038161003816100381610038161199101003816100381610038161003816 gt 119905

010038161003816100381610038161199101003816100381610038161003816 lt 119905

(11)

where 119905 is the selected threshold 119910 is the originalwavelet coefficients and 119910

119904is the wavelet coefficients

after thresholding In this work the value which cov-ers about 95 of the smallest wavelet coefficients is setas the threshold value Then 5 of the largest waveletcoefficients are kept as nonzero ones to reserve thesignal feature

(4) Signal recovery reconstruct the vibration signalbased on the modified approximation coefficients atlevel119873 and the modified detail coefficients at levels 1to119873

1198751198651015840 [119909 (119905)] = 119862minus1

120595∬1198821015840 (119886 119887) 120595(119886119887) (119905)

119889119886

1198862119889119887 (12)

Mathematical Problems in Engineering 5

Original signal

Peak-based piecewise transform

Newly generated smooth signal

Wavelet decomposition

Coefficients threshold Inverse wavelet transform

Peak-based piecewise transform backward

Enhanced signal

Envelope demodulation

Spectral analysis and detection

Figure 4 Flowchart of the proposed enhancement scheme for weakfault signal

CH 1

Figure 5 Fault rig of a roller element bearing

(5) Piecewise signal backward transform put the pro-cessed signal segments back to their original posi-tions then a new denoised vibration signal 1199091015840(119905) isgenerated

(6) Envelope demodulation according to (9) use Hilberttransform to obtain the envelope 119860(119905) of signal 1199091015840(119905)

(7) Extraction of characteristic defect frequency get theenvelope spectra via Fourier analysis and distinguishdifferent fault types of rolling bearings

In summary these steps are shown as in Figure 4

4 Experimental Results and Discussions

41 Experiment Setup To verify the effectiveness of theproposed method experiments are designed for fault rigs ofroller element bearing as shown in Figure 5 The rig is com-posed of a motor a coupling a rotor and a shaft with tworoller bearings Basically the sensors are located at the posi-tion near the bearings to mitigate the effects of signal atten-uation Usually bearing housing is one of the best locationsfor bearing arrangement Therefore here an accelerometeris located on the bearing housing to acquire the vibrationsignals

In the experiments a single point defect is introduced ininner raceway outer raceway and ball element of differ-ent bearings using electron-discharge machining with faultwidths of 3mm 7mm and 7mm and depths of 5mm25mm and 25mm respectively

Vibration signals are measured by an accelerometer lo-cated at the top of the bearing house (CH1 as shownin Figure 5) In all experiments the sample frequency is100 kHz and the shaft speed is finite 1300 rpm Four commonconditions are studied in the present paper including aperfect healthy bearing and three other bearings with pointdefect on the outer race on the inner race bearing and onthe ball element

In theory elements of roller bearings have their own spe-cific rotational frequencies which may appear in envelopespectra if defects existThese defect frequencies are called ballpass frequencies For a bearing with a stationary outer racethese frequencies can be given by the following formulas

outer race defect frequency 119908od =119885119908119904

2119889(1 minus

119889

119863cos120572)

inner race defect frequency 119908id =119885119908119904

2(1 +

119889

119863cos120572)

rolling element defect frequency 119908re = 119908119904119863

2119889

times (1 minus1198892

1198632cos2120572)

(13)

where 119908119904is the shaft rotation frequency in rads 119889 is the

diameter of rolling element 119863 is the pitch diameter 119885 is thenumber of rolling elements and 120572 is the contact angle

In the present experiments outer race defect frequencyinner race defect frequency and rolling element defect fre-quency are 8632Hz 14584Hz and 5113Hz respectively

42 Vibration Signals of Typical Bearing Faults and TheirEnvelop Spectra The time-domain waveforms of bearingvibration signals obtained from accelerometers are shown inFigure 6 from which we can see that different from that ofperfect bearings as shown in Figure 6(a) obvious impulsivephenomenon exists in faulty vibration signals of fault bear-ings as shown in Figures 6(b) 6(c) and 6(d)

To investigate the differences of the signals and todistinguish different fault types envelope demodulation isapplied first to vibration signals of the three typical bearingfaults that is outer race fault inner race fault and rollerelement fault The results are shown in Figures 7 8 and 9In Figure 7 the frequency magnitude of outer race defectis lower than the second harmonic frequency of outer racedefect In Figure 8 although the inner race defect frequencyslightly sticks out it is still possibly confused by the side-lobes with almost similarmagnitudes In Figure 9 it is almostimpossible to figure out the calculated ball pass frequencyof 5113Hz From the figures we can see that simply basedon envelope demodulation it is difficult to identify the faultfrequencies from those of other components especially forweak fault signals

43 Wavelet Coefficients Distributions of the Vibration Sig-nals with and without Peak-Based Piecewise RecombinationBefore envelope demodulation wavelet decomposition is

6 Mathematical Problems in Engineering

0 02

02

04

04

06 08 1

0

Am

plitu

de (V

)

Time (s)

minus04

minus02

(a)

0 02 04 06 08 1

0

2

Am

plitu

de (V

)

Time (s)

minus4

minus2

(b)

0 02 04 06 08 1

0

5

10

Time (s)

Am

plitu

de (V

)

minus10

minus5

(c)

Am

plitu

de (V

)

0 02 04 06 08 1

0

5

Time (s)

minus5

(d)

Figure 6 Vibration signals of rolling bearings (a) a normal bearing and bearings with a point defect on (b) outer race (c) inner race bearingand (d) ball element

0 50 100 150 200 250 300 350 4000

50

100

150

200

250

Am

plitu

de

Frequency (Hz)

X 8621Y 1714

Figure 7 Envelope spectra of vibration signal in Figure 6(b)induced by outer race fault

always applied to denoise and enhance the weak fault signalsHowever vibration signals are not always sparse enough inwavelet domain which would affect the effectiveness of theenhancement As mentioned above peak-based piecewiserecombination can help to improve the smoothness of vibra-tion signals and make their energies more concentrated in

Am

plitu

de

0 50 100 150 200 250 300 350 4000

50

100

150

200

250

Frequency (Hz)

X 145Y 1816

Figure 8 Envelope spectra of vibration signal in Figure 6(c)induced by inner race fault

wavelet domain Comparisons between wavelet decomposi-tions with and without peak-based piecewise recombinationare illustrated in this section

Taking outer race fault as an example the time-domainvibration signal of Figure 6(b) is converted to that of Figure 10after applying piecewise recombination The signal becomesmuch smoother than the original signal after piecewise

Mathematical Problems in Engineering 7

0 50 100 150 200 250 300 350 4000

10

20

30

40

50

60

Frequency (Hz)

Am

plitu

de

X 4959Y 2026

Figure 9 Envelope spectra of vibration signal in Figure 6(d)induced by roller element fault

0 1 2 3 4 5 6 7 8 9 100

50010001500200025003000350040004500

Data number

Am

plitu

de

times104

Figure 10 Vibration signal induced by outer race fault inFigure 6(b) after peak-based piecewise recombination

Table 1 Energy distributions of the outer race fault signalrsquos waveletcoefficients

Frequency band Low frequency High frequencyEnergies without PPR 4931 5069Energies with PPR 999 01

recombination that is high frequency components have beenconverted to low frequency ones

For comparisons then wavelet transform is applied tosignals in Figures 6(b) and 10 respectively where 6-leveldecomposition with db4 wavelet basis is employed Both dis-tributions of the wavelet coefficients are shown in Figure 11Without piecewise recombination most wavelet coefficientsare nonzero that is not sparse enough as depicted inFigure 11(a) However it is improved significantly withpreprocessing of peak-based piecewise recombination asdepicted in Figure 11(b) Energies of the signal in waveletdomain are concentrated in only a few low frequency inter-vals and almost all detail coefficients at scales 1 to 6 are closeto zero It implies that peak-based piecewise recombinationcan significantly improve the sparseness of vibration signalsin wavelet domain

Statistically energy distributions of wavelet coefficientsobtained by the two methods are listed in Table 1 from

0 1 2 3 4 5 6 7 8 9 10

012345

Data number

Am

plitu

de

minus1

minus2

minus3

minus4

minus5

times104

(a)

Data number0 1 2 3 4 5 6 7 8 9

0

05

1

15

2

25

3

35

Am

plitu

de

times104

times104

(b)

Figure 11 Wavelet coefficients of the vibration signal induced byouter race fault (a) without PPR preprocessing and (b) with PPRpreprocessing

which we can see that after employing peak-based piecewiserecombination (PPR) preprocessing energies of the signal areconcentrated into only a few coefficients in wavelet domain999 in low frequency bands comparedwith 4931withoutpeak-based piecewise recombination

The experimental results of inner race fault and rollerelement fault are similar to those of the outer race fault whichare shown in Figures 12 and 13

44 Comparisons between Traditional Wavelet-Based Detec-tion Method and the Proposed Strategy with Envelop Demod-ulation Envelop demodulation is always combined withwavelet-based denoising and enhancement methods to ana-lyze vibration signals Comparisons of the results after en-velop demodulation based on the results of Section 43 arepresented in this section

Envelop demodulation is applied to the signals recon-structed from the thresholding of wavelet coefficients and abackward peak-based piecewise recombination is imple-mented for the proposed strategyThe demodulated envelopespectra are shown in Figures 14ndash16 In Figure 14(a) althoughthe fault frequency can be figured out the magnitude of

8 Mathematical Problems in Engineering

0 2 4 6 8 10

0

5

10

15

Data number

Am

plitu

de

minus20

minus15

minus10

minus5

times104

(a)

0Data number

2 4 6 8 10

0

05

1

15

2

25

3

35

Am

plitu

detimes104

times104

(b)

Figure 12 Wavelet coefficients of the vibration signal induced by inner race fault (a) without PPR preprocessing and (b) with PPRpreprocessing

Data number0 2 4 6 8 10

0

2

4

6

8

10

Am

plitu

de

minus8

minus6

minus4

minus2

times104

(a)

Data number0 2 4 6 8 10

0

05

1

15

2

25

Am

plitu

de

minus05

times104

times104

(b)

Figure 13 Wavelet coefficients of the vibration signal induced by roller element fault (a) without PPR preprocessing and (b) with PPR pre-processing

the outer race defect frequency is not the most remarkableone among all frequency components thus it is not alwayscertain that a bearing defect exists in outer race With theproposed method presented in this paper the magnitudeof outer race defect frequency becomes more prominent

where the magnitudes of outer race defect frequency havebeen enhanced and are more easily detected as Figure 14(b)depicted The proposed strategy is also more effective forinner race and roller bearing fault detection of rolling bear-ings and the results are shown as in Figures 15 and 16

Mathematical Problems in Engineering 9

0 100 200 300 4000

50

100

150

200

250

300

Frequency (Hz)

Am

plitu

de

X 8621Y 149

(a)

0 100 200 300 4000

200

400

600

800

1000

1200

Frequency (Hz)

Am

plitu

de

X 8621Y 9952

(b)

Figure 14 Envelope spectra of the vibration signal induced by outer race fault (a) wavelet analysis method and (b) the proposed method

0 100 200 300 4000

50

100

150

200

250

300

Frequency (Hz)

Am

plitu

de

Y 1777X 145

(a)

0 100 200 300 4000

50

100

150

200

250

300

350

400

Frequency (Hz)

Am

plitu

de

X 145Y 2525

(b)

Figure 15 Envelope spectra of the vibration signal induced by inner race fault (a) wavelet analysis method and (b) the proposed method

If defect exists in the bearing the magnitudes of faultsfrequencieswill increase in the envelope spectrumThereforethey are often used to describe the fault severity and alsoemployed to judge whether the fault features are enhancedIn this work magnitudes of faults frequencies are chosen asthe major indictors to reflect the fault features As recordedand listed in Table 2 with the proposed method magnitudesof faults frequencies are all enhanced and they become more

easily distinguished from interfered noise for the experimen-tal faults

5 Conclusions

In order to improve the current wavelet-based method forfault detection of rolling bearings the present paper devel-oped a new way to improve vibration signalsrsquo smoothness

10 Mathematical Problems in Engineering

0 100 200 300 4000

50

100

150

200

250

300

Frequency (Hz)

Am

plitu

de

X 4959Y 2026

(a)

0 100 200 300 4000

50

100

150

200

250

300

Frequency (Hz)

Am

plitu

de

X 4959Y 2792

(b)

Figure 16 Envelope spectra of the vibration signal induced by roller element fault (a) wavelet analysis method and (b) the proposedmethod

Table 2 Major indictors for bearing fault features

Defect frequencymagnitude of

Traditionalwavelet-based

methodThe proposed method

outer race 149 9952inner race 1777 2525roller element 2026 2792

as well as their sparseness in wavelet domain by employingpeak-based piecewise recombination and envelope demodu-lation Through the proposed method most noise mixed invibration signals can be eliminated and the weak fault signalsof rolling bearings can be enhanced and become more easilydetectableTherationality and validity of themethods are alsoproved by various experiments It shows that compared withtraditional wavelet-based method the proposed method hasmany advantages in bearing fault signals enhancement anddetection and is very simple and easy to implement

There is still room to improve the present work Thesensitivity of the method to noise and methods to makeenergy more concentrated are what will be studied in futurework

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This project is partially supported by the National NaturalScience Foundation of China (Grant no 51375037) and theNational Program on Key Basic Research Project (Grant no2012CB026000) The third author would also like to thankthe support of China Fundamental Research Funds for theCentral Universities (ZY1410) and the Public Hatching Plat-form for Recruited Talents of Beijing University of ChemicalTechnology

References

[1] W Zhou T G Habetler and R G Harley ldquoBearing conditionmonitoring methods for electric machines a general reviewrdquo inProceedings of the IEEE International Symposium on Diagnosticsfor Electric Machines Power Electronics and Drives (SDEMPEDrsquo07) pp 3ndash6 September 2007

[2] H Q Wang and P Chen ldquoIntelligent diagnosis method for acentrifugal pump using features of vibration signalsrdquo NeuralComputing and Applications vol 18 no 4 pp 397ndash405 2009

[3] H Wang and P Chen ldquoSequential diagnosis for rolling bearingusing fuzzy neural networkrdquo in Proceedings of the IEEEASMEInternational Conference on Advanced Intelligent Mechatronics(AIM rsquo08) pp 56ndash61 Xirsquoan China August 2008

[4] T Mitoma H Wang and P Chen ldquoFault diagnosis and condi-tion surveillance for plant rotating machinery using partially-linearized neural networkrdquo Computers and Industrial Engineer-ing vol 55 no 4 pp 783ndash794 2008

[5] P D McFadden and J D Smith ldquoVibration monitoring of roll-ing element bearings by the high-frequency resonance tech-niquemdasha reviewrdquo Tribology International vol 17 no 1 pp 3ndash101984

Mathematical Problems in Engineering 11

[6] N Tandon and A Choudhury ldquoReview of vibration and acous-tic measurement methods for the detection of defects in rollingelement bearingsrdquo Tribology International vol 32 no 8 pp469ndash480 1999

[7] R B W Heng and M J M Nor ldquoStatistical analysis of soundand vibration signals for monitoring rolling element bearingconditionrdquo Applied Acoustics vol 53 no 1ndash3 pp 211ndash226 1998

[8] C Junsheng Y Dejie and Y Yu ldquoThe application of energyoperator demodulation approach based on EMD in machineryfault diagnosisrdquo Mechanical Systems and Signal Processing vol21 no 2 pp 668ndash677 2007

[9] R B Randall J Antoni and S Chobsaard ldquoThe relationship be-tween spectral correlation and envelope analysis in the diagnos-tics of bearing faults and other cyclostationarymachine signalsrdquoMechanical Systems and Signal Processing vol 15 no 5 pp 945ndash962 2001

[10] A B Ming Z Y Qin W Zhang et al ldquoSpectrum auto-cor-relation analysis and its application to fault diagnosis of rollingelement bearingsrdquo Mechanical Systems and Signal Processingvol 41 no 1 pp 141ndash154 2013

[11] D Yu J Cheng and Y Yang ldquoApplication of EMDmethod andHilbert spectrum to the fault diagnosis of roller bearingsrdquo Me-chanical Systems and Signal Processing vol 19 no 2 pp 259ndash270 2005

[12] V K Rai and A RMohanty ldquoBearing fault diagnosis using FFTof intrinsic mode functions in Hilbert-Huang transformrdquo Me-chanical Systems and Signal Processing vol 21 no 6 pp 2607ndash2615 2007

[13] Y Lei N Li J Lin and SWang ldquoFault diagnosis of rotatingma-chinery based on an adaptive ensemble empirical mode decom-positionrdquo Sensors vol 13 no 12 pp 16950ndash16964 2013

[14] J Cheng Y Yang and Y Yang ldquoA rotatingmachinery fault diag-nosis method based on local mean decompositionrdquo Digital Sig-nal Processing vol 22 no 2 pp 356ndash366 2012

[15] B Chen Z He X Chen H Cao G Cai and Y Zi ldquoA demod-ulating approach based on local mean decomposition and itsapplications in mechanical fault diagnosisrdquo Measurement Sci-ence and Technology vol 22 no 5 Article ID 055704 2011

[16] S Prabhakar A R Mohanty and A S Sekhar ldquoApplication ofdiscrete wavelet transform for detection of ball bearing racefaultsrdquoTribology International vol 35 no 12 pp 793ndash800 2002

[17] R Yan R X Gao and X Chen ldquoWavelets for fault diagnosis ofrotary machines a review with applicationsrdquo Signal Processingvol 96 part A pp 1ndash15 2014

[18] H Wang and P Chen ldquoA feature extraction method based oninformation theory for fault diagnosis of reciprocating machin-eryrdquo Sensors vol 9 no 4 pp 2415ndash2436 2009

[19] K Li X L Ping H Q Wang and P Chen ldquoSequential fuzzydiagnosis method for motor roller bearing in variable operatingconditions based on vibration analysisrdquo Sensors vol 13 no 6pp 8013ndash8041 2013

[20] R Rubini and U Meneghetti ldquoApplication of the envelope andwavelet transform analyses for the diagnosis of incipient faultsin ball bearingsrdquoMechanical Systems and Signal Processing vol15 no 2 pp 287ndash302 2001

[21] L Gao Z Yang L Cai H Wang and P Chen ldquoRoller bearingfault diagnosis based on nonlinear redundant lifting waveletpacket analysisrdquo Sensors vol 11 no 1 pp 260ndash277 2011

[22] Z Yang L Cai L Gao and H Wang ldquoAdaptive redundant lift-ing wavelet transform based on fitting for fault feature extrac-tion of roller bearingsrdquo Sensors vol 12 no 4 pp 4381ndash43982012

[23] Z He ldquoPeak transform for efficient image representation andcodingrdquo IEEE Transactions on Image Processing vol 16 no 7pp 1741ndash1754 2007

[24] S Anila and N Devarajan ldquoThe usage of peak transform forimage compressionrdquo International Journal of Computer SystemsScience amp Engineering vol 2 no 11 pp 6308ndash6316 2010

[25] D Ho and R B Randall ldquoOptimization of bearing diagnostictechniques using simulated and actual bearing fault signalsrdquoMechanical Systems and Signal Processing vol 14 no 5 pp 763ndash788 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

Original signal

Peak-based piecewise transform

Newly generated smooth signal

Wavelet decomposition

Coefficients threshold Inverse wavelet transform

Peak-based piecewise transform backward

Enhanced signal

Envelope demodulation

Spectral analysis and detection

Figure 4 Flowchart of the proposed enhancement scheme for weakfault signal

CH 1

Figure 5 Fault rig of a roller element bearing

(5) Piecewise signal backward transform put the pro-cessed signal segments back to their original posi-tions then a new denoised vibration signal 1199091015840(119905) isgenerated

(6) Envelope demodulation according to (9) use Hilberttransform to obtain the envelope 119860(119905) of signal 1199091015840(119905)

(7) Extraction of characteristic defect frequency get theenvelope spectra via Fourier analysis and distinguishdifferent fault types of rolling bearings

In summary these steps are shown as in Figure 4

4 Experimental Results and Discussions

41 Experiment Setup To verify the effectiveness of theproposed method experiments are designed for fault rigs ofroller element bearing as shown in Figure 5 The rig is com-posed of a motor a coupling a rotor and a shaft with tworoller bearings Basically the sensors are located at the posi-tion near the bearings to mitigate the effects of signal atten-uation Usually bearing housing is one of the best locationsfor bearing arrangement Therefore here an accelerometeris located on the bearing housing to acquire the vibrationsignals

In the experiments a single point defect is introduced ininner raceway outer raceway and ball element of differ-ent bearings using electron-discharge machining with faultwidths of 3mm 7mm and 7mm and depths of 5mm25mm and 25mm respectively

Vibration signals are measured by an accelerometer lo-cated at the top of the bearing house (CH1 as shownin Figure 5) In all experiments the sample frequency is100 kHz and the shaft speed is finite 1300 rpm Four commonconditions are studied in the present paper including aperfect healthy bearing and three other bearings with pointdefect on the outer race on the inner race bearing and onthe ball element

In theory elements of roller bearings have their own spe-cific rotational frequencies which may appear in envelopespectra if defects existThese defect frequencies are called ballpass frequencies For a bearing with a stationary outer racethese frequencies can be given by the following formulas

outer race defect frequency 119908od =119885119908119904

2119889(1 minus

119889

119863cos120572)

inner race defect frequency 119908id =119885119908119904

2(1 +

119889

119863cos120572)

rolling element defect frequency 119908re = 119908119904119863

2119889

times (1 minus1198892

1198632cos2120572)

(13)

where 119908119904is the shaft rotation frequency in rads 119889 is the

diameter of rolling element 119863 is the pitch diameter 119885 is thenumber of rolling elements and 120572 is the contact angle

In the present experiments outer race defect frequencyinner race defect frequency and rolling element defect fre-quency are 8632Hz 14584Hz and 5113Hz respectively

42 Vibration Signals of Typical Bearing Faults and TheirEnvelop Spectra The time-domain waveforms of bearingvibration signals obtained from accelerometers are shown inFigure 6 from which we can see that different from that ofperfect bearings as shown in Figure 6(a) obvious impulsivephenomenon exists in faulty vibration signals of fault bear-ings as shown in Figures 6(b) 6(c) and 6(d)

To investigate the differences of the signals and todistinguish different fault types envelope demodulation isapplied first to vibration signals of the three typical bearingfaults that is outer race fault inner race fault and rollerelement fault The results are shown in Figures 7 8 and 9In Figure 7 the frequency magnitude of outer race defectis lower than the second harmonic frequency of outer racedefect In Figure 8 although the inner race defect frequencyslightly sticks out it is still possibly confused by the side-lobes with almost similarmagnitudes In Figure 9 it is almostimpossible to figure out the calculated ball pass frequencyof 5113Hz From the figures we can see that simply basedon envelope demodulation it is difficult to identify the faultfrequencies from those of other components especially forweak fault signals

43 Wavelet Coefficients Distributions of the Vibration Sig-nals with and without Peak-Based Piecewise RecombinationBefore envelope demodulation wavelet decomposition is

6 Mathematical Problems in Engineering

0 02

02

04

04

06 08 1

0

Am

plitu

de (V

)

Time (s)

minus04

minus02

(a)

0 02 04 06 08 1

0

2

Am

plitu

de (V

)

Time (s)

minus4

minus2

(b)

0 02 04 06 08 1

0

5

10

Time (s)

Am

plitu

de (V

)

minus10

minus5

(c)

Am

plitu

de (V

)

0 02 04 06 08 1

0

5

Time (s)

minus5

(d)

Figure 6 Vibration signals of rolling bearings (a) a normal bearing and bearings with a point defect on (b) outer race (c) inner race bearingand (d) ball element

0 50 100 150 200 250 300 350 4000

50

100

150

200

250

Am

plitu

de

Frequency (Hz)

X 8621Y 1714

Figure 7 Envelope spectra of vibration signal in Figure 6(b)induced by outer race fault

always applied to denoise and enhance the weak fault signalsHowever vibration signals are not always sparse enough inwavelet domain which would affect the effectiveness of theenhancement As mentioned above peak-based piecewiserecombination can help to improve the smoothness of vibra-tion signals and make their energies more concentrated in

Am

plitu

de

0 50 100 150 200 250 300 350 4000

50

100

150

200

250

Frequency (Hz)

X 145Y 1816

Figure 8 Envelope spectra of vibration signal in Figure 6(c)induced by inner race fault

wavelet domain Comparisons between wavelet decomposi-tions with and without peak-based piecewise recombinationare illustrated in this section

Taking outer race fault as an example the time-domainvibration signal of Figure 6(b) is converted to that of Figure 10after applying piecewise recombination The signal becomesmuch smoother than the original signal after piecewise

Mathematical Problems in Engineering 7

0 50 100 150 200 250 300 350 4000

10

20

30

40

50

60

Frequency (Hz)

Am

plitu

de

X 4959Y 2026

Figure 9 Envelope spectra of vibration signal in Figure 6(d)induced by roller element fault

0 1 2 3 4 5 6 7 8 9 100

50010001500200025003000350040004500

Data number

Am

plitu

de

times104

Figure 10 Vibration signal induced by outer race fault inFigure 6(b) after peak-based piecewise recombination

Table 1 Energy distributions of the outer race fault signalrsquos waveletcoefficients

Frequency band Low frequency High frequencyEnergies without PPR 4931 5069Energies with PPR 999 01

recombination that is high frequency components have beenconverted to low frequency ones

For comparisons then wavelet transform is applied tosignals in Figures 6(b) and 10 respectively where 6-leveldecomposition with db4 wavelet basis is employed Both dis-tributions of the wavelet coefficients are shown in Figure 11Without piecewise recombination most wavelet coefficientsare nonzero that is not sparse enough as depicted inFigure 11(a) However it is improved significantly withpreprocessing of peak-based piecewise recombination asdepicted in Figure 11(b) Energies of the signal in waveletdomain are concentrated in only a few low frequency inter-vals and almost all detail coefficients at scales 1 to 6 are closeto zero It implies that peak-based piecewise recombinationcan significantly improve the sparseness of vibration signalsin wavelet domain

Statistically energy distributions of wavelet coefficientsobtained by the two methods are listed in Table 1 from

0 1 2 3 4 5 6 7 8 9 10

012345

Data number

Am

plitu

de

minus1

minus2

minus3

minus4

minus5

times104

(a)

Data number0 1 2 3 4 5 6 7 8 9

0

05

1

15

2

25

3

35

Am

plitu

de

times104

times104

(b)

Figure 11 Wavelet coefficients of the vibration signal induced byouter race fault (a) without PPR preprocessing and (b) with PPRpreprocessing

which we can see that after employing peak-based piecewiserecombination (PPR) preprocessing energies of the signal areconcentrated into only a few coefficients in wavelet domain999 in low frequency bands comparedwith 4931withoutpeak-based piecewise recombination

The experimental results of inner race fault and rollerelement fault are similar to those of the outer race fault whichare shown in Figures 12 and 13

44 Comparisons between Traditional Wavelet-Based Detec-tion Method and the Proposed Strategy with Envelop Demod-ulation Envelop demodulation is always combined withwavelet-based denoising and enhancement methods to ana-lyze vibration signals Comparisons of the results after en-velop demodulation based on the results of Section 43 arepresented in this section

Envelop demodulation is applied to the signals recon-structed from the thresholding of wavelet coefficients and abackward peak-based piecewise recombination is imple-mented for the proposed strategyThe demodulated envelopespectra are shown in Figures 14ndash16 In Figure 14(a) althoughthe fault frequency can be figured out the magnitude of

8 Mathematical Problems in Engineering

0 2 4 6 8 10

0

5

10

15

Data number

Am

plitu

de

minus20

minus15

minus10

minus5

times104

(a)

0Data number

2 4 6 8 10

0

05

1

15

2

25

3

35

Am

plitu

detimes104

times104

(b)

Figure 12 Wavelet coefficients of the vibration signal induced by inner race fault (a) without PPR preprocessing and (b) with PPRpreprocessing

Data number0 2 4 6 8 10

0

2

4

6

8

10

Am

plitu

de

minus8

minus6

minus4

minus2

times104

(a)

Data number0 2 4 6 8 10

0

05

1

15

2

25

Am

plitu

de

minus05

times104

times104

(b)

Figure 13 Wavelet coefficients of the vibration signal induced by roller element fault (a) without PPR preprocessing and (b) with PPR pre-processing

the outer race defect frequency is not the most remarkableone among all frequency components thus it is not alwayscertain that a bearing defect exists in outer race With theproposed method presented in this paper the magnitudeof outer race defect frequency becomes more prominent

where the magnitudes of outer race defect frequency havebeen enhanced and are more easily detected as Figure 14(b)depicted The proposed strategy is also more effective forinner race and roller bearing fault detection of rolling bear-ings and the results are shown as in Figures 15 and 16

Mathematical Problems in Engineering 9

0 100 200 300 4000

50

100

150

200

250

300

Frequency (Hz)

Am

plitu

de

X 8621Y 149

(a)

0 100 200 300 4000

200

400

600

800

1000

1200

Frequency (Hz)

Am

plitu

de

X 8621Y 9952

(b)

Figure 14 Envelope spectra of the vibration signal induced by outer race fault (a) wavelet analysis method and (b) the proposed method

0 100 200 300 4000

50

100

150

200

250

300

Frequency (Hz)

Am

plitu

de

Y 1777X 145

(a)

0 100 200 300 4000

50

100

150

200

250

300

350

400

Frequency (Hz)

Am

plitu

de

X 145Y 2525

(b)

Figure 15 Envelope spectra of the vibration signal induced by inner race fault (a) wavelet analysis method and (b) the proposed method

If defect exists in the bearing the magnitudes of faultsfrequencieswill increase in the envelope spectrumThereforethey are often used to describe the fault severity and alsoemployed to judge whether the fault features are enhancedIn this work magnitudes of faults frequencies are chosen asthe major indictors to reflect the fault features As recordedand listed in Table 2 with the proposed method magnitudesof faults frequencies are all enhanced and they become more

easily distinguished from interfered noise for the experimen-tal faults

5 Conclusions

In order to improve the current wavelet-based method forfault detection of rolling bearings the present paper devel-oped a new way to improve vibration signalsrsquo smoothness

10 Mathematical Problems in Engineering

0 100 200 300 4000

50

100

150

200

250

300

Frequency (Hz)

Am

plitu

de

X 4959Y 2026

(a)

0 100 200 300 4000

50

100

150

200

250

300

Frequency (Hz)

Am

plitu

de

X 4959Y 2792

(b)

Figure 16 Envelope spectra of the vibration signal induced by roller element fault (a) wavelet analysis method and (b) the proposedmethod

Table 2 Major indictors for bearing fault features

Defect frequencymagnitude of

Traditionalwavelet-based

methodThe proposed method

outer race 149 9952inner race 1777 2525roller element 2026 2792

as well as their sparseness in wavelet domain by employingpeak-based piecewise recombination and envelope demodu-lation Through the proposed method most noise mixed invibration signals can be eliminated and the weak fault signalsof rolling bearings can be enhanced and become more easilydetectableTherationality and validity of themethods are alsoproved by various experiments It shows that compared withtraditional wavelet-based method the proposed method hasmany advantages in bearing fault signals enhancement anddetection and is very simple and easy to implement

There is still room to improve the present work Thesensitivity of the method to noise and methods to makeenergy more concentrated are what will be studied in futurework

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This project is partially supported by the National NaturalScience Foundation of China (Grant no 51375037) and theNational Program on Key Basic Research Project (Grant no2012CB026000) The third author would also like to thankthe support of China Fundamental Research Funds for theCentral Universities (ZY1410) and the Public Hatching Plat-form for Recruited Talents of Beijing University of ChemicalTechnology

References

[1] W Zhou T G Habetler and R G Harley ldquoBearing conditionmonitoring methods for electric machines a general reviewrdquo inProceedings of the IEEE International Symposium on Diagnosticsfor Electric Machines Power Electronics and Drives (SDEMPEDrsquo07) pp 3ndash6 September 2007

[2] H Q Wang and P Chen ldquoIntelligent diagnosis method for acentrifugal pump using features of vibration signalsrdquo NeuralComputing and Applications vol 18 no 4 pp 397ndash405 2009

[3] H Wang and P Chen ldquoSequential diagnosis for rolling bearingusing fuzzy neural networkrdquo in Proceedings of the IEEEASMEInternational Conference on Advanced Intelligent Mechatronics(AIM rsquo08) pp 56ndash61 Xirsquoan China August 2008

[4] T Mitoma H Wang and P Chen ldquoFault diagnosis and condi-tion surveillance for plant rotating machinery using partially-linearized neural networkrdquo Computers and Industrial Engineer-ing vol 55 no 4 pp 783ndash794 2008

[5] P D McFadden and J D Smith ldquoVibration monitoring of roll-ing element bearings by the high-frequency resonance tech-niquemdasha reviewrdquo Tribology International vol 17 no 1 pp 3ndash101984

Mathematical Problems in Engineering 11

[6] N Tandon and A Choudhury ldquoReview of vibration and acous-tic measurement methods for the detection of defects in rollingelement bearingsrdquo Tribology International vol 32 no 8 pp469ndash480 1999

[7] R B W Heng and M J M Nor ldquoStatistical analysis of soundand vibration signals for monitoring rolling element bearingconditionrdquo Applied Acoustics vol 53 no 1ndash3 pp 211ndash226 1998

[8] C Junsheng Y Dejie and Y Yu ldquoThe application of energyoperator demodulation approach based on EMD in machineryfault diagnosisrdquo Mechanical Systems and Signal Processing vol21 no 2 pp 668ndash677 2007

[9] R B Randall J Antoni and S Chobsaard ldquoThe relationship be-tween spectral correlation and envelope analysis in the diagnos-tics of bearing faults and other cyclostationarymachine signalsrdquoMechanical Systems and Signal Processing vol 15 no 5 pp 945ndash962 2001

[10] A B Ming Z Y Qin W Zhang et al ldquoSpectrum auto-cor-relation analysis and its application to fault diagnosis of rollingelement bearingsrdquo Mechanical Systems and Signal Processingvol 41 no 1 pp 141ndash154 2013

[11] D Yu J Cheng and Y Yang ldquoApplication of EMDmethod andHilbert spectrum to the fault diagnosis of roller bearingsrdquo Me-chanical Systems and Signal Processing vol 19 no 2 pp 259ndash270 2005

[12] V K Rai and A RMohanty ldquoBearing fault diagnosis using FFTof intrinsic mode functions in Hilbert-Huang transformrdquo Me-chanical Systems and Signal Processing vol 21 no 6 pp 2607ndash2615 2007

[13] Y Lei N Li J Lin and SWang ldquoFault diagnosis of rotatingma-chinery based on an adaptive ensemble empirical mode decom-positionrdquo Sensors vol 13 no 12 pp 16950ndash16964 2013

[14] J Cheng Y Yang and Y Yang ldquoA rotatingmachinery fault diag-nosis method based on local mean decompositionrdquo Digital Sig-nal Processing vol 22 no 2 pp 356ndash366 2012

[15] B Chen Z He X Chen H Cao G Cai and Y Zi ldquoA demod-ulating approach based on local mean decomposition and itsapplications in mechanical fault diagnosisrdquo Measurement Sci-ence and Technology vol 22 no 5 Article ID 055704 2011

[16] S Prabhakar A R Mohanty and A S Sekhar ldquoApplication ofdiscrete wavelet transform for detection of ball bearing racefaultsrdquoTribology International vol 35 no 12 pp 793ndash800 2002

[17] R Yan R X Gao and X Chen ldquoWavelets for fault diagnosis ofrotary machines a review with applicationsrdquo Signal Processingvol 96 part A pp 1ndash15 2014

[18] H Wang and P Chen ldquoA feature extraction method based oninformation theory for fault diagnosis of reciprocating machin-eryrdquo Sensors vol 9 no 4 pp 2415ndash2436 2009

[19] K Li X L Ping H Q Wang and P Chen ldquoSequential fuzzydiagnosis method for motor roller bearing in variable operatingconditions based on vibration analysisrdquo Sensors vol 13 no 6pp 8013ndash8041 2013

[20] R Rubini and U Meneghetti ldquoApplication of the envelope andwavelet transform analyses for the diagnosis of incipient faultsin ball bearingsrdquoMechanical Systems and Signal Processing vol15 no 2 pp 287ndash302 2001

[21] L Gao Z Yang L Cai H Wang and P Chen ldquoRoller bearingfault diagnosis based on nonlinear redundant lifting waveletpacket analysisrdquo Sensors vol 11 no 1 pp 260ndash277 2011

[22] Z Yang L Cai L Gao and H Wang ldquoAdaptive redundant lift-ing wavelet transform based on fitting for fault feature extrac-tion of roller bearingsrdquo Sensors vol 12 no 4 pp 4381ndash43982012

[23] Z He ldquoPeak transform for efficient image representation andcodingrdquo IEEE Transactions on Image Processing vol 16 no 7pp 1741ndash1754 2007

[24] S Anila and N Devarajan ldquoThe usage of peak transform forimage compressionrdquo International Journal of Computer SystemsScience amp Engineering vol 2 no 11 pp 6308ndash6316 2010

[25] D Ho and R B Randall ldquoOptimization of bearing diagnostictechniques using simulated and actual bearing fault signalsrdquoMechanical Systems and Signal Processing vol 14 no 5 pp 763ndash788 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

0 02

02

04

04

06 08 1

0

Am

plitu

de (V

)

Time (s)

minus04

minus02

(a)

0 02 04 06 08 1

0

2

Am

plitu

de (V

)

Time (s)

minus4

minus2

(b)

0 02 04 06 08 1

0

5

10

Time (s)

Am

plitu

de (V

)

minus10

minus5

(c)

Am

plitu

de (V

)

0 02 04 06 08 1

0

5

Time (s)

minus5

(d)

Figure 6 Vibration signals of rolling bearings (a) a normal bearing and bearings with a point defect on (b) outer race (c) inner race bearingand (d) ball element

0 50 100 150 200 250 300 350 4000

50

100

150

200

250

Am

plitu

de

Frequency (Hz)

X 8621Y 1714

Figure 7 Envelope spectra of vibration signal in Figure 6(b)induced by outer race fault

always applied to denoise and enhance the weak fault signalsHowever vibration signals are not always sparse enough inwavelet domain which would affect the effectiveness of theenhancement As mentioned above peak-based piecewiserecombination can help to improve the smoothness of vibra-tion signals and make their energies more concentrated in

Am

plitu

de

0 50 100 150 200 250 300 350 4000

50

100

150

200

250

Frequency (Hz)

X 145Y 1816

Figure 8 Envelope spectra of vibration signal in Figure 6(c)induced by inner race fault

wavelet domain Comparisons between wavelet decomposi-tions with and without peak-based piecewise recombinationare illustrated in this section

Taking outer race fault as an example the time-domainvibration signal of Figure 6(b) is converted to that of Figure 10after applying piecewise recombination The signal becomesmuch smoother than the original signal after piecewise

Mathematical Problems in Engineering 7

0 50 100 150 200 250 300 350 4000

10

20

30

40

50

60

Frequency (Hz)

Am

plitu

de

X 4959Y 2026

Figure 9 Envelope spectra of vibration signal in Figure 6(d)induced by roller element fault

0 1 2 3 4 5 6 7 8 9 100

50010001500200025003000350040004500

Data number

Am

plitu

de

times104

Figure 10 Vibration signal induced by outer race fault inFigure 6(b) after peak-based piecewise recombination

Table 1 Energy distributions of the outer race fault signalrsquos waveletcoefficients

Frequency band Low frequency High frequencyEnergies without PPR 4931 5069Energies with PPR 999 01

recombination that is high frequency components have beenconverted to low frequency ones

For comparisons then wavelet transform is applied tosignals in Figures 6(b) and 10 respectively where 6-leveldecomposition with db4 wavelet basis is employed Both dis-tributions of the wavelet coefficients are shown in Figure 11Without piecewise recombination most wavelet coefficientsare nonzero that is not sparse enough as depicted inFigure 11(a) However it is improved significantly withpreprocessing of peak-based piecewise recombination asdepicted in Figure 11(b) Energies of the signal in waveletdomain are concentrated in only a few low frequency inter-vals and almost all detail coefficients at scales 1 to 6 are closeto zero It implies that peak-based piecewise recombinationcan significantly improve the sparseness of vibration signalsin wavelet domain

Statistically energy distributions of wavelet coefficientsobtained by the two methods are listed in Table 1 from

0 1 2 3 4 5 6 7 8 9 10

012345

Data number

Am

plitu

de

minus1

minus2

minus3

minus4

minus5

times104

(a)

Data number0 1 2 3 4 5 6 7 8 9

0

05

1

15

2

25

3

35

Am

plitu

de

times104

times104

(b)

Figure 11 Wavelet coefficients of the vibration signal induced byouter race fault (a) without PPR preprocessing and (b) with PPRpreprocessing

which we can see that after employing peak-based piecewiserecombination (PPR) preprocessing energies of the signal areconcentrated into only a few coefficients in wavelet domain999 in low frequency bands comparedwith 4931withoutpeak-based piecewise recombination

The experimental results of inner race fault and rollerelement fault are similar to those of the outer race fault whichare shown in Figures 12 and 13

44 Comparisons between Traditional Wavelet-Based Detec-tion Method and the Proposed Strategy with Envelop Demod-ulation Envelop demodulation is always combined withwavelet-based denoising and enhancement methods to ana-lyze vibration signals Comparisons of the results after en-velop demodulation based on the results of Section 43 arepresented in this section

Envelop demodulation is applied to the signals recon-structed from the thresholding of wavelet coefficients and abackward peak-based piecewise recombination is imple-mented for the proposed strategyThe demodulated envelopespectra are shown in Figures 14ndash16 In Figure 14(a) althoughthe fault frequency can be figured out the magnitude of

8 Mathematical Problems in Engineering

0 2 4 6 8 10

0

5

10

15

Data number

Am

plitu

de

minus20

minus15

minus10

minus5

times104

(a)

0Data number

2 4 6 8 10

0

05

1

15

2

25

3

35

Am

plitu

detimes104

times104

(b)

Figure 12 Wavelet coefficients of the vibration signal induced by inner race fault (a) without PPR preprocessing and (b) with PPRpreprocessing

Data number0 2 4 6 8 10

0

2

4

6

8

10

Am

plitu

de

minus8

minus6

minus4

minus2

times104

(a)

Data number0 2 4 6 8 10

0

05

1

15

2

25

Am

plitu

de

minus05

times104

times104

(b)

Figure 13 Wavelet coefficients of the vibration signal induced by roller element fault (a) without PPR preprocessing and (b) with PPR pre-processing

the outer race defect frequency is not the most remarkableone among all frequency components thus it is not alwayscertain that a bearing defect exists in outer race With theproposed method presented in this paper the magnitudeof outer race defect frequency becomes more prominent

where the magnitudes of outer race defect frequency havebeen enhanced and are more easily detected as Figure 14(b)depicted The proposed strategy is also more effective forinner race and roller bearing fault detection of rolling bear-ings and the results are shown as in Figures 15 and 16

Mathematical Problems in Engineering 9

0 100 200 300 4000

50

100

150

200

250

300

Frequency (Hz)

Am

plitu

de

X 8621Y 149

(a)

0 100 200 300 4000

200

400

600

800

1000

1200

Frequency (Hz)

Am

plitu

de

X 8621Y 9952

(b)

Figure 14 Envelope spectra of the vibration signal induced by outer race fault (a) wavelet analysis method and (b) the proposed method

0 100 200 300 4000

50

100

150

200

250

300

Frequency (Hz)

Am

plitu

de

Y 1777X 145

(a)

0 100 200 300 4000

50

100

150

200

250

300

350

400

Frequency (Hz)

Am

plitu

de

X 145Y 2525

(b)

Figure 15 Envelope spectra of the vibration signal induced by inner race fault (a) wavelet analysis method and (b) the proposed method

If defect exists in the bearing the magnitudes of faultsfrequencieswill increase in the envelope spectrumThereforethey are often used to describe the fault severity and alsoemployed to judge whether the fault features are enhancedIn this work magnitudes of faults frequencies are chosen asthe major indictors to reflect the fault features As recordedand listed in Table 2 with the proposed method magnitudesof faults frequencies are all enhanced and they become more

easily distinguished from interfered noise for the experimen-tal faults

5 Conclusions

In order to improve the current wavelet-based method forfault detection of rolling bearings the present paper devel-oped a new way to improve vibration signalsrsquo smoothness

10 Mathematical Problems in Engineering

0 100 200 300 4000

50

100

150

200

250

300

Frequency (Hz)

Am

plitu

de

X 4959Y 2026

(a)

0 100 200 300 4000

50

100

150

200

250

300

Frequency (Hz)

Am

plitu

de

X 4959Y 2792

(b)

Figure 16 Envelope spectra of the vibration signal induced by roller element fault (a) wavelet analysis method and (b) the proposedmethod

Table 2 Major indictors for bearing fault features

Defect frequencymagnitude of

Traditionalwavelet-based

methodThe proposed method

outer race 149 9952inner race 1777 2525roller element 2026 2792

as well as their sparseness in wavelet domain by employingpeak-based piecewise recombination and envelope demodu-lation Through the proposed method most noise mixed invibration signals can be eliminated and the weak fault signalsof rolling bearings can be enhanced and become more easilydetectableTherationality and validity of themethods are alsoproved by various experiments It shows that compared withtraditional wavelet-based method the proposed method hasmany advantages in bearing fault signals enhancement anddetection and is very simple and easy to implement

There is still room to improve the present work Thesensitivity of the method to noise and methods to makeenergy more concentrated are what will be studied in futurework

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This project is partially supported by the National NaturalScience Foundation of China (Grant no 51375037) and theNational Program on Key Basic Research Project (Grant no2012CB026000) The third author would also like to thankthe support of China Fundamental Research Funds for theCentral Universities (ZY1410) and the Public Hatching Plat-form for Recruited Talents of Beijing University of ChemicalTechnology

References

[1] W Zhou T G Habetler and R G Harley ldquoBearing conditionmonitoring methods for electric machines a general reviewrdquo inProceedings of the IEEE International Symposium on Diagnosticsfor Electric Machines Power Electronics and Drives (SDEMPEDrsquo07) pp 3ndash6 September 2007

[2] H Q Wang and P Chen ldquoIntelligent diagnosis method for acentrifugal pump using features of vibration signalsrdquo NeuralComputing and Applications vol 18 no 4 pp 397ndash405 2009

[3] H Wang and P Chen ldquoSequential diagnosis for rolling bearingusing fuzzy neural networkrdquo in Proceedings of the IEEEASMEInternational Conference on Advanced Intelligent Mechatronics(AIM rsquo08) pp 56ndash61 Xirsquoan China August 2008

[4] T Mitoma H Wang and P Chen ldquoFault diagnosis and condi-tion surveillance for plant rotating machinery using partially-linearized neural networkrdquo Computers and Industrial Engineer-ing vol 55 no 4 pp 783ndash794 2008

[5] P D McFadden and J D Smith ldquoVibration monitoring of roll-ing element bearings by the high-frequency resonance tech-niquemdasha reviewrdquo Tribology International vol 17 no 1 pp 3ndash101984

Mathematical Problems in Engineering 11

[6] N Tandon and A Choudhury ldquoReview of vibration and acous-tic measurement methods for the detection of defects in rollingelement bearingsrdquo Tribology International vol 32 no 8 pp469ndash480 1999

[7] R B W Heng and M J M Nor ldquoStatistical analysis of soundand vibration signals for monitoring rolling element bearingconditionrdquo Applied Acoustics vol 53 no 1ndash3 pp 211ndash226 1998

[8] C Junsheng Y Dejie and Y Yu ldquoThe application of energyoperator demodulation approach based on EMD in machineryfault diagnosisrdquo Mechanical Systems and Signal Processing vol21 no 2 pp 668ndash677 2007

[9] R B Randall J Antoni and S Chobsaard ldquoThe relationship be-tween spectral correlation and envelope analysis in the diagnos-tics of bearing faults and other cyclostationarymachine signalsrdquoMechanical Systems and Signal Processing vol 15 no 5 pp 945ndash962 2001

[10] A B Ming Z Y Qin W Zhang et al ldquoSpectrum auto-cor-relation analysis and its application to fault diagnosis of rollingelement bearingsrdquo Mechanical Systems and Signal Processingvol 41 no 1 pp 141ndash154 2013

[11] D Yu J Cheng and Y Yang ldquoApplication of EMDmethod andHilbert spectrum to the fault diagnosis of roller bearingsrdquo Me-chanical Systems and Signal Processing vol 19 no 2 pp 259ndash270 2005

[12] V K Rai and A RMohanty ldquoBearing fault diagnosis using FFTof intrinsic mode functions in Hilbert-Huang transformrdquo Me-chanical Systems and Signal Processing vol 21 no 6 pp 2607ndash2615 2007

[13] Y Lei N Li J Lin and SWang ldquoFault diagnosis of rotatingma-chinery based on an adaptive ensemble empirical mode decom-positionrdquo Sensors vol 13 no 12 pp 16950ndash16964 2013

[14] J Cheng Y Yang and Y Yang ldquoA rotatingmachinery fault diag-nosis method based on local mean decompositionrdquo Digital Sig-nal Processing vol 22 no 2 pp 356ndash366 2012

[15] B Chen Z He X Chen H Cao G Cai and Y Zi ldquoA demod-ulating approach based on local mean decomposition and itsapplications in mechanical fault diagnosisrdquo Measurement Sci-ence and Technology vol 22 no 5 Article ID 055704 2011

[16] S Prabhakar A R Mohanty and A S Sekhar ldquoApplication ofdiscrete wavelet transform for detection of ball bearing racefaultsrdquoTribology International vol 35 no 12 pp 793ndash800 2002

[17] R Yan R X Gao and X Chen ldquoWavelets for fault diagnosis ofrotary machines a review with applicationsrdquo Signal Processingvol 96 part A pp 1ndash15 2014

[18] H Wang and P Chen ldquoA feature extraction method based oninformation theory for fault diagnosis of reciprocating machin-eryrdquo Sensors vol 9 no 4 pp 2415ndash2436 2009

[19] K Li X L Ping H Q Wang and P Chen ldquoSequential fuzzydiagnosis method for motor roller bearing in variable operatingconditions based on vibration analysisrdquo Sensors vol 13 no 6pp 8013ndash8041 2013

[20] R Rubini and U Meneghetti ldquoApplication of the envelope andwavelet transform analyses for the diagnosis of incipient faultsin ball bearingsrdquoMechanical Systems and Signal Processing vol15 no 2 pp 287ndash302 2001

[21] L Gao Z Yang L Cai H Wang and P Chen ldquoRoller bearingfault diagnosis based on nonlinear redundant lifting waveletpacket analysisrdquo Sensors vol 11 no 1 pp 260ndash277 2011

[22] Z Yang L Cai L Gao and H Wang ldquoAdaptive redundant lift-ing wavelet transform based on fitting for fault feature extrac-tion of roller bearingsrdquo Sensors vol 12 no 4 pp 4381ndash43982012

[23] Z He ldquoPeak transform for efficient image representation andcodingrdquo IEEE Transactions on Image Processing vol 16 no 7pp 1741ndash1754 2007

[24] S Anila and N Devarajan ldquoThe usage of peak transform forimage compressionrdquo International Journal of Computer SystemsScience amp Engineering vol 2 no 11 pp 6308ndash6316 2010

[25] D Ho and R B Randall ldquoOptimization of bearing diagnostictechniques using simulated and actual bearing fault signalsrdquoMechanical Systems and Signal Processing vol 14 no 5 pp 763ndash788 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

0 50 100 150 200 250 300 350 4000

10

20

30

40

50

60

Frequency (Hz)

Am

plitu

de

X 4959Y 2026

Figure 9 Envelope spectra of vibration signal in Figure 6(d)induced by roller element fault

0 1 2 3 4 5 6 7 8 9 100

50010001500200025003000350040004500

Data number

Am

plitu

de

times104

Figure 10 Vibration signal induced by outer race fault inFigure 6(b) after peak-based piecewise recombination

Table 1 Energy distributions of the outer race fault signalrsquos waveletcoefficients

Frequency band Low frequency High frequencyEnergies without PPR 4931 5069Energies with PPR 999 01

recombination that is high frequency components have beenconverted to low frequency ones

For comparisons then wavelet transform is applied tosignals in Figures 6(b) and 10 respectively where 6-leveldecomposition with db4 wavelet basis is employed Both dis-tributions of the wavelet coefficients are shown in Figure 11Without piecewise recombination most wavelet coefficientsare nonzero that is not sparse enough as depicted inFigure 11(a) However it is improved significantly withpreprocessing of peak-based piecewise recombination asdepicted in Figure 11(b) Energies of the signal in waveletdomain are concentrated in only a few low frequency inter-vals and almost all detail coefficients at scales 1 to 6 are closeto zero It implies that peak-based piecewise recombinationcan significantly improve the sparseness of vibration signalsin wavelet domain

Statistically energy distributions of wavelet coefficientsobtained by the two methods are listed in Table 1 from

0 1 2 3 4 5 6 7 8 9 10

012345

Data number

Am

plitu

de

minus1

minus2

minus3

minus4

minus5

times104

(a)

Data number0 1 2 3 4 5 6 7 8 9

0

05

1

15

2

25

3

35

Am

plitu

de

times104

times104

(b)

Figure 11 Wavelet coefficients of the vibration signal induced byouter race fault (a) without PPR preprocessing and (b) with PPRpreprocessing

which we can see that after employing peak-based piecewiserecombination (PPR) preprocessing energies of the signal areconcentrated into only a few coefficients in wavelet domain999 in low frequency bands comparedwith 4931withoutpeak-based piecewise recombination

The experimental results of inner race fault and rollerelement fault are similar to those of the outer race fault whichare shown in Figures 12 and 13

44 Comparisons between Traditional Wavelet-Based Detec-tion Method and the Proposed Strategy with Envelop Demod-ulation Envelop demodulation is always combined withwavelet-based denoising and enhancement methods to ana-lyze vibration signals Comparisons of the results after en-velop demodulation based on the results of Section 43 arepresented in this section

Envelop demodulation is applied to the signals recon-structed from the thresholding of wavelet coefficients and abackward peak-based piecewise recombination is imple-mented for the proposed strategyThe demodulated envelopespectra are shown in Figures 14ndash16 In Figure 14(a) althoughthe fault frequency can be figured out the magnitude of

8 Mathematical Problems in Engineering

0 2 4 6 8 10

0

5

10

15

Data number

Am

plitu

de

minus20

minus15

minus10

minus5

times104

(a)

0Data number

2 4 6 8 10

0

05

1

15

2

25

3

35

Am

plitu

detimes104

times104

(b)

Figure 12 Wavelet coefficients of the vibration signal induced by inner race fault (a) without PPR preprocessing and (b) with PPRpreprocessing

Data number0 2 4 6 8 10

0

2

4

6

8

10

Am

plitu

de

minus8

minus6

minus4

minus2

times104

(a)

Data number0 2 4 6 8 10

0

05

1

15

2

25

Am

plitu

de

minus05

times104

times104

(b)

Figure 13 Wavelet coefficients of the vibration signal induced by roller element fault (a) without PPR preprocessing and (b) with PPR pre-processing

the outer race defect frequency is not the most remarkableone among all frequency components thus it is not alwayscertain that a bearing defect exists in outer race With theproposed method presented in this paper the magnitudeof outer race defect frequency becomes more prominent

where the magnitudes of outer race defect frequency havebeen enhanced and are more easily detected as Figure 14(b)depicted The proposed strategy is also more effective forinner race and roller bearing fault detection of rolling bear-ings and the results are shown as in Figures 15 and 16

Mathematical Problems in Engineering 9

0 100 200 300 4000

50

100

150

200

250

300

Frequency (Hz)

Am

plitu

de

X 8621Y 149

(a)

0 100 200 300 4000

200

400

600

800

1000

1200

Frequency (Hz)

Am

plitu

de

X 8621Y 9952

(b)

Figure 14 Envelope spectra of the vibration signal induced by outer race fault (a) wavelet analysis method and (b) the proposed method

0 100 200 300 4000

50

100

150

200

250

300

Frequency (Hz)

Am

plitu

de

Y 1777X 145

(a)

0 100 200 300 4000

50

100

150

200

250

300

350

400

Frequency (Hz)

Am

plitu

de

X 145Y 2525

(b)

Figure 15 Envelope spectra of the vibration signal induced by inner race fault (a) wavelet analysis method and (b) the proposed method

If defect exists in the bearing the magnitudes of faultsfrequencieswill increase in the envelope spectrumThereforethey are often used to describe the fault severity and alsoemployed to judge whether the fault features are enhancedIn this work magnitudes of faults frequencies are chosen asthe major indictors to reflect the fault features As recordedand listed in Table 2 with the proposed method magnitudesof faults frequencies are all enhanced and they become more

easily distinguished from interfered noise for the experimen-tal faults

5 Conclusions

In order to improve the current wavelet-based method forfault detection of rolling bearings the present paper devel-oped a new way to improve vibration signalsrsquo smoothness

10 Mathematical Problems in Engineering

0 100 200 300 4000

50

100

150

200

250

300

Frequency (Hz)

Am

plitu

de

X 4959Y 2026

(a)

0 100 200 300 4000

50

100

150

200

250

300

Frequency (Hz)

Am

plitu

de

X 4959Y 2792

(b)

Figure 16 Envelope spectra of the vibration signal induced by roller element fault (a) wavelet analysis method and (b) the proposedmethod

Table 2 Major indictors for bearing fault features

Defect frequencymagnitude of

Traditionalwavelet-based

methodThe proposed method

outer race 149 9952inner race 1777 2525roller element 2026 2792

as well as their sparseness in wavelet domain by employingpeak-based piecewise recombination and envelope demodu-lation Through the proposed method most noise mixed invibration signals can be eliminated and the weak fault signalsof rolling bearings can be enhanced and become more easilydetectableTherationality and validity of themethods are alsoproved by various experiments It shows that compared withtraditional wavelet-based method the proposed method hasmany advantages in bearing fault signals enhancement anddetection and is very simple and easy to implement

There is still room to improve the present work Thesensitivity of the method to noise and methods to makeenergy more concentrated are what will be studied in futurework

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This project is partially supported by the National NaturalScience Foundation of China (Grant no 51375037) and theNational Program on Key Basic Research Project (Grant no2012CB026000) The third author would also like to thankthe support of China Fundamental Research Funds for theCentral Universities (ZY1410) and the Public Hatching Plat-form for Recruited Talents of Beijing University of ChemicalTechnology

References

[1] W Zhou T G Habetler and R G Harley ldquoBearing conditionmonitoring methods for electric machines a general reviewrdquo inProceedings of the IEEE International Symposium on Diagnosticsfor Electric Machines Power Electronics and Drives (SDEMPEDrsquo07) pp 3ndash6 September 2007

[2] H Q Wang and P Chen ldquoIntelligent diagnosis method for acentrifugal pump using features of vibration signalsrdquo NeuralComputing and Applications vol 18 no 4 pp 397ndash405 2009

[3] H Wang and P Chen ldquoSequential diagnosis for rolling bearingusing fuzzy neural networkrdquo in Proceedings of the IEEEASMEInternational Conference on Advanced Intelligent Mechatronics(AIM rsquo08) pp 56ndash61 Xirsquoan China August 2008

[4] T Mitoma H Wang and P Chen ldquoFault diagnosis and condi-tion surveillance for plant rotating machinery using partially-linearized neural networkrdquo Computers and Industrial Engineer-ing vol 55 no 4 pp 783ndash794 2008

[5] P D McFadden and J D Smith ldquoVibration monitoring of roll-ing element bearings by the high-frequency resonance tech-niquemdasha reviewrdquo Tribology International vol 17 no 1 pp 3ndash101984

Mathematical Problems in Engineering 11

[6] N Tandon and A Choudhury ldquoReview of vibration and acous-tic measurement methods for the detection of defects in rollingelement bearingsrdquo Tribology International vol 32 no 8 pp469ndash480 1999

[7] R B W Heng and M J M Nor ldquoStatistical analysis of soundand vibration signals for monitoring rolling element bearingconditionrdquo Applied Acoustics vol 53 no 1ndash3 pp 211ndash226 1998

[8] C Junsheng Y Dejie and Y Yu ldquoThe application of energyoperator demodulation approach based on EMD in machineryfault diagnosisrdquo Mechanical Systems and Signal Processing vol21 no 2 pp 668ndash677 2007

[9] R B Randall J Antoni and S Chobsaard ldquoThe relationship be-tween spectral correlation and envelope analysis in the diagnos-tics of bearing faults and other cyclostationarymachine signalsrdquoMechanical Systems and Signal Processing vol 15 no 5 pp 945ndash962 2001

[10] A B Ming Z Y Qin W Zhang et al ldquoSpectrum auto-cor-relation analysis and its application to fault diagnosis of rollingelement bearingsrdquo Mechanical Systems and Signal Processingvol 41 no 1 pp 141ndash154 2013

[11] D Yu J Cheng and Y Yang ldquoApplication of EMDmethod andHilbert spectrum to the fault diagnosis of roller bearingsrdquo Me-chanical Systems and Signal Processing vol 19 no 2 pp 259ndash270 2005

[12] V K Rai and A RMohanty ldquoBearing fault diagnosis using FFTof intrinsic mode functions in Hilbert-Huang transformrdquo Me-chanical Systems and Signal Processing vol 21 no 6 pp 2607ndash2615 2007

[13] Y Lei N Li J Lin and SWang ldquoFault diagnosis of rotatingma-chinery based on an adaptive ensemble empirical mode decom-positionrdquo Sensors vol 13 no 12 pp 16950ndash16964 2013

[14] J Cheng Y Yang and Y Yang ldquoA rotatingmachinery fault diag-nosis method based on local mean decompositionrdquo Digital Sig-nal Processing vol 22 no 2 pp 356ndash366 2012

[15] B Chen Z He X Chen H Cao G Cai and Y Zi ldquoA demod-ulating approach based on local mean decomposition and itsapplications in mechanical fault diagnosisrdquo Measurement Sci-ence and Technology vol 22 no 5 Article ID 055704 2011

[16] S Prabhakar A R Mohanty and A S Sekhar ldquoApplication ofdiscrete wavelet transform for detection of ball bearing racefaultsrdquoTribology International vol 35 no 12 pp 793ndash800 2002

[17] R Yan R X Gao and X Chen ldquoWavelets for fault diagnosis ofrotary machines a review with applicationsrdquo Signal Processingvol 96 part A pp 1ndash15 2014

[18] H Wang and P Chen ldquoA feature extraction method based oninformation theory for fault diagnosis of reciprocating machin-eryrdquo Sensors vol 9 no 4 pp 2415ndash2436 2009

[19] K Li X L Ping H Q Wang and P Chen ldquoSequential fuzzydiagnosis method for motor roller bearing in variable operatingconditions based on vibration analysisrdquo Sensors vol 13 no 6pp 8013ndash8041 2013

[20] R Rubini and U Meneghetti ldquoApplication of the envelope andwavelet transform analyses for the diagnosis of incipient faultsin ball bearingsrdquoMechanical Systems and Signal Processing vol15 no 2 pp 287ndash302 2001

[21] L Gao Z Yang L Cai H Wang and P Chen ldquoRoller bearingfault diagnosis based on nonlinear redundant lifting waveletpacket analysisrdquo Sensors vol 11 no 1 pp 260ndash277 2011

[22] Z Yang L Cai L Gao and H Wang ldquoAdaptive redundant lift-ing wavelet transform based on fitting for fault feature extrac-tion of roller bearingsrdquo Sensors vol 12 no 4 pp 4381ndash43982012

[23] Z He ldquoPeak transform for efficient image representation andcodingrdquo IEEE Transactions on Image Processing vol 16 no 7pp 1741ndash1754 2007

[24] S Anila and N Devarajan ldquoThe usage of peak transform forimage compressionrdquo International Journal of Computer SystemsScience amp Engineering vol 2 no 11 pp 6308ndash6316 2010

[25] D Ho and R B Randall ldquoOptimization of bearing diagnostictechniques using simulated and actual bearing fault signalsrdquoMechanical Systems and Signal Processing vol 14 no 5 pp 763ndash788 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

0 2 4 6 8 10

0

5

10

15

Data number

Am

plitu

de

minus20

minus15

minus10

minus5

times104

(a)

0Data number

2 4 6 8 10

0

05

1

15

2

25

3

35

Am

plitu

detimes104

times104

(b)

Figure 12 Wavelet coefficients of the vibration signal induced by inner race fault (a) without PPR preprocessing and (b) with PPRpreprocessing

Data number0 2 4 6 8 10

0

2

4

6

8

10

Am

plitu

de

minus8

minus6

minus4

minus2

times104

(a)

Data number0 2 4 6 8 10

0

05

1

15

2

25

Am

plitu

de

minus05

times104

times104

(b)

Figure 13 Wavelet coefficients of the vibration signal induced by roller element fault (a) without PPR preprocessing and (b) with PPR pre-processing

the outer race defect frequency is not the most remarkableone among all frequency components thus it is not alwayscertain that a bearing defect exists in outer race With theproposed method presented in this paper the magnitudeof outer race defect frequency becomes more prominent

where the magnitudes of outer race defect frequency havebeen enhanced and are more easily detected as Figure 14(b)depicted The proposed strategy is also more effective forinner race and roller bearing fault detection of rolling bear-ings and the results are shown as in Figures 15 and 16

Mathematical Problems in Engineering 9

0 100 200 300 4000

50

100

150

200

250

300

Frequency (Hz)

Am

plitu

de

X 8621Y 149

(a)

0 100 200 300 4000

200

400

600

800

1000

1200

Frequency (Hz)

Am

plitu

de

X 8621Y 9952

(b)

Figure 14 Envelope spectra of the vibration signal induced by outer race fault (a) wavelet analysis method and (b) the proposed method

0 100 200 300 4000

50

100

150

200

250

300

Frequency (Hz)

Am

plitu

de

Y 1777X 145

(a)

0 100 200 300 4000

50

100

150

200

250

300

350

400

Frequency (Hz)

Am

plitu

de

X 145Y 2525

(b)

Figure 15 Envelope spectra of the vibration signal induced by inner race fault (a) wavelet analysis method and (b) the proposed method

If defect exists in the bearing the magnitudes of faultsfrequencieswill increase in the envelope spectrumThereforethey are often used to describe the fault severity and alsoemployed to judge whether the fault features are enhancedIn this work magnitudes of faults frequencies are chosen asthe major indictors to reflect the fault features As recordedand listed in Table 2 with the proposed method magnitudesof faults frequencies are all enhanced and they become more

easily distinguished from interfered noise for the experimen-tal faults

5 Conclusions

In order to improve the current wavelet-based method forfault detection of rolling bearings the present paper devel-oped a new way to improve vibration signalsrsquo smoothness

10 Mathematical Problems in Engineering

0 100 200 300 4000

50

100

150

200

250

300

Frequency (Hz)

Am

plitu

de

X 4959Y 2026

(a)

0 100 200 300 4000

50

100

150

200

250

300

Frequency (Hz)

Am

plitu

de

X 4959Y 2792

(b)

Figure 16 Envelope spectra of the vibration signal induced by roller element fault (a) wavelet analysis method and (b) the proposedmethod

Table 2 Major indictors for bearing fault features

Defect frequencymagnitude of

Traditionalwavelet-based

methodThe proposed method

outer race 149 9952inner race 1777 2525roller element 2026 2792

as well as their sparseness in wavelet domain by employingpeak-based piecewise recombination and envelope demodu-lation Through the proposed method most noise mixed invibration signals can be eliminated and the weak fault signalsof rolling bearings can be enhanced and become more easilydetectableTherationality and validity of themethods are alsoproved by various experiments It shows that compared withtraditional wavelet-based method the proposed method hasmany advantages in bearing fault signals enhancement anddetection and is very simple and easy to implement

There is still room to improve the present work Thesensitivity of the method to noise and methods to makeenergy more concentrated are what will be studied in futurework

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This project is partially supported by the National NaturalScience Foundation of China (Grant no 51375037) and theNational Program on Key Basic Research Project (Grant no2012CB026000) The third author would also like to thankthe support of China Fundamental Research Funds for theCentral Universities (ZY1410) and the Public Hatching Plat-form for Recruited Talents of Beijing University of ChemicalTechnology

References

[1] W Zhou T G Habetler and R G Harley ldquoBearing conditionmonitoring methods for electric machines a general reviewrdquo inProceedings of the IEEE International Symposium on Diagnosticsfor Electric Machines Power Electronics and Drives (SDEMPEDrsquo07) pp 3ndash6 September 2007

[2] H Q Wang and P Chen ldquoIntelligent diagnosis method for acentrifugal pump using features of vibration signalsrdquo NeuralComputing and Applications vol 18 no 4 pp 397ndash405 2009

[3] H Wang and P Chen ldquoSequential diagnosis for rolling bearingusing fuzzy neural networkrdquo in Proceedings of the IEEEASMEInternational Conference on Advanced Intelligent Mechatronics(AIM rsquo08) pp 56ndash61 Xirsquoan China August 2008

[4] T Mitoma H Wang and P Chen ldquoFault diagnosis and condi-tion surveillance for plant rotating machinery using partially-linearized neural networkrdquo Computers and Industrial Engineer-ing vol 55 no 4 pp 783ndash794 2008

[5] P D McFadden and J D Smith ldquoVibration monitoring of roll-ing element bearings by the high-frequency resonance tech-niquemdasha reviewrdquo Tribology International vol 17 no 1 pp 3ndash101984

Mathematical Problems in Engineering 11

[6] N Tandon and A Choudhury ldquoReview of vibration and acous-tic measurement methods for the detection of defects in rollingelement bearingsrdquo Tribology International vol 32 no 8 pp469ndash480 1999

[7] R B W Heng and M J M Nor ldquoStatistical analysis of soundand vibration signals for monitoring rolling element bearingconditionrdquo Applied Acoustics vol 53 no 1ndash3 pp 211ndash226 1998

[8] C Junsheng Y Dejie and Y Yu ldquoThe application of energyoperator demodulation approach based on EMD in machineryfault diagnosisrdquo Mechanical Systems and Signal Processing vol21 no 2 pp 668ndash677 2007

[9] R B Randall J Antoni and S Chobsaard ldquoThe relationship be-tween spectral correlation and envelope analysis in the diagnos-tics of bearing faults and other cyclostationarymachine signalsrdquoMechanical Systems and Signal Processing vol 15 no 5 pp 945ndash962 2001

[10] A B Ming Z Y Qin W Zhang et al ldquoSpectrum auto-cor-relation analysis and its application to fault diagnosis of rollingelement bearingsrdquo Mechanical Systems and Signal Processingvol 41 no 1 pp 141ndash154 2013

[11] D Yu J Cheng and Y Yang ldquoApplication of EMDmethod andHilbert spectrum to the fault diagnosis of roller bearingsrdquo Me-chanical Systems and Signal Processing vol 19 no 2 pp 259ndash270 2005

[12] V K Rai and A RMohanty ldquoBearing fault diagnosis using FFTof intrinsic mode functions in Hilbert-Huang transformrdquo Me-chanical Systems and Signal Processing vol 21 no 6 pp 2607ndash2615 2007

[13] Y Lei N Li J Lin and SWang ldquoFault diagnosis of rotatingma-chinery based on an adaptive ensemble empirical mode decom-positionrdquo Sensors vol 13 no 12 pp 16950ndash16964 2013

[14] J Cheng Y Yang and Y Yang ldquoA rotatingmachinery fault diag-nosis method based on local mean decompositionrdquo Digital Sig-nal Processing vol 22 no 2 pp 356ndash366 2012

[15] B Chen Z He X Chen H Cao G Cai and Y Zi ldquoA demod-ulating approach based on local mean decomposition and itsapplications in mechanical fault diagnosisrdquo Measurement Sci-ence and Technology vol 22 no 5 Article ID 055704 2011

[16] S Prabhakar A R Mohanty and A S Sekhar ldquoApplication ofdiscrete wavelet transform for detection of ball bearing racefaultsrdquoTribology International vol 35 no 12 pp 793ndash800 2002

[17] R Yan R X Gao and X Chen ldquoWavelets for fault diagnosis ofrotary machines a review with applicationsrdquo Signal Processingvol 96 part A pp 1ndash15 2014

[18] H Wang and P Chen ldquoA feature extraction method based oninformation theory for fault diagnosis of reciprocating machin-eryrdquo Sensors vol 9 no 4 pp 2415ndash2436 2009

[19] K Li X L Ping H Q Wang and P Chen ldquoSequential fuzzydiagnosis method for motor roller bearing in variable operatingconditions based on vibration analysisrdquo Sensors vol 13 no 6pp 8013ndash8041 2013

[20] R Rubini and U Meneghetti ldquoApplication of the envelope andwavelet transform analyses for the diagnosis of incipient faultsin ball bearingsrdquoMechanical Systems and Signal Processing vol15 no 2 pp 287ndash302 2001

[21] L Gao Z Yang L Cai H Wang and P Chen ldquoRoller bearingfault diagnosis based on nonlinear redundant lifting waveletpacket analysisrdquo Sensors vol 11 no 1 pp 260ndash277 2011

[22] Z Yang L Cai L Gao and H Wang ldquoAdaptive redundant lift-ing wavelet transform based on fitting for fault feature extrac-tion of roller bearingsrdquo Sensors vol 12 no 4 pp 4381ndash43982012

[23] Z He ldquoPeak transform for efficient image representation andcodingrdquo IEEE Transactions on Image Processing vol 16 no 7pp 1741ndash1754 2007

[24] S Anila and N Devarajan ldquoThe usage of peak transform forimage compressionrdquo International Journal of Computer SystemsScience amp Engineering vol 2 no 11 pp 6308ndash6316 2010

[25] D Ho and R B Randall ldquoOptimization of bearing diagnostictechniques using simulated and actual bearing fault signalsrdquoMechanical Systems and Signal Processing vol 14 no 5 pp 763ndash788 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

0 100 200 300 4000

50

100

150

200

250

300

Frequency (Hz)

Am

plitu

de

X 8621Y 149

(a)

0 100 200 300 4000

200

400

600

800

1000

1200

Frequency (Hz)

Am

plitu

de

X 8621Y 9952

(b)

Figure 14 Envelope spectra of the vibration signal induced by outer race fault (a) wavelet analysis method and (b) the proposed method

0 100 200 300 4000

50

100

150

200

250

300

Frequency (Hz)

Am

plitu

de

Y 1777X 145

(a)

0 100 200 300 4000

50

100

150

200

250

300

350

400

Frequency (Hz)

Am

plitu

de

X 145Y 2525

(b)

Figure 15 Envelope spectra of the vibration signal induced by inner race fault (a) wavelet analysis method and (b) the proposed method

If defect exists in the bearing the magnitudes of faultsfrequencieswill increase in the envelope spectrumThereforethey are often used to describe the fault severity and alsoemployed to judge whether the fault features are enhancedIn this work magnitudes of faults frequencies are chosen asthe major indictors to reflect the fault features As recordedand listed in Table 2 with the proposed method magnitudesof faults frequencies are all enhanced and they become more

easily distinguished from interfered noise for the experimen-tal faults

5 Conclusions

In order to improve the current wavelet-based method forfault detection of rolling bearings the present paper devel-oped a new way to improve vibration signalsrsquo smoothness

10 Mathematical Problems in Engineering

0 100 200 300 4000

50

100

150

200

250

300

Frequency (Hz)

Am

plitu

de

X 4959Y 2026

(a)

0 100 200 300 4000

50

100

150

200

250

300

Frequency (Hz)

Am

plitu

de

X 4959Y 2792

(b)

Figure 16 Envelope spectra of the vibration signal induced by roller element fault (a) wavelet analysis method and (b) the proposedmethod

Table 2 Major indictors for bearing fault features

Defect frequencymagnitude of

Traditionalwavelet-based

methodThe proposed method

outer race 149 9952inner race 1777 2525roller element 2026 2792

as well as their sparseness in wavelet domain by employingpeak-based piecewise recombination and envelope demodu-lation Through the proposed method most noise mixed invibration signals can be eliminated and the weak fault signalsof rolling bearings can be enhanced and become more easilydetectableTherationality and validity of themethods are alsoproved by various experiments It shows that compared withtraditional wavelet-based method the proposed method hasmany advantages in bearing fault signals enhancement anddetection and is very simple and easy to implement

There is still room to improve the present work Thesensitivity of the method to noise and methods to makeenergy more concentrated are what will be studied in futurework

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This project is partially supported by the National NaturalScience Foundation of China (Grant no 51375037) and theNational Program on Key Basic Research Project (Grant no2012CB026000) The third author would also like to thankthe support of China Fundamental Research Funds for theCentral Universities (ZY1410) and the Public Hatching Plat-form for Recruited Talents of Beijing University of ChemicalTechnology

References

[1] W Zhou T G Habetler and R G Harley ldquoBearing conditionmonitoring methods for electric machines a general reviewrdquo inProceedings of the IEEE International Symposium on Diagnosticsfor Electric Machines Power Electronics and Drives (SDEMPEDrsquo07) pp 3ndash6 September 2007

[2] H Q Wang and P Chen ldquoIntelligent diagnosis method for acentrifugal pump using features of vibration signalsrdquo NeuralComputing and Applications vol 18 no 4 pp 397ndash405 2009

[3] H Wang and P Chen ldquoSequential diagnosis for rolling bearingusing fuzzy neural networkrdquo in Proceedings of the IEEEASMEInternational Conference on Advanced Intelligent Mechatronics(AIM rsquo08) pp 56ndash61 Xirsquoan China August 2008

[4] T Mitoma H Wang and P Chen ldquoFault diagnosis and condi-tion surveillance for plant rotating machinery using partially-linearized neural networkrdquo Computers and Industrial Engineer-ing vol 55 no 4 pp 783ndash794 2008

[5] P D McFadden and J D Smith ldquoVibration monitoring of roll-ing element bearings by the high-frequency resonance tech-niquemdasha reviewrdquo Tribology International vol 17 no 1 pp 3ndash101984

Mathematical Problems in Engineering 11

[6] N Tandon and A Choudhury ldquoReview of vibration and acous-tic measurement methods for the detection of defects in rollingelement bearingsrdquo Tribology International vol 32 no 8 pp469ndash480 1999

[7] R B W Heng and M J M Nor ldquoStatistical analysis of soundand vibration signals for monitoring rolling element bearingconditionrdquo Applied Acoustics vol 53 no 1ndash3 pp 211ndash226 1998

[8] C Junsheng Y Dejie and Y Yu ldquoThe application of energyoperator demodulation approach based on EMD in machineryfault diagnosisrdquo Mechanical Systems and Signal Processing vol21 no 2 pp 668ndash677 2007

[9] R B Randall J Antoni and S Chobsaard ldquoThe relationship be-tween spectral correlation and envelope analysis in the diagnos-tics of bearing faults and other cyclostationarymachine signalsrdquoMechanical Systems and Signal Processing vol 15 no 5 pp 945ndash962 2001

[10] A B Ming Z Y Qin W Zhang et al ldquoSpectrum auto-cor-relation analysis and its application to fault diagnosis of rollingelement bearingsrdquo Mechanical Systems and Signal Processingvol 41 no 1 pp 141ndash154 2013

[11] D Yu J Cheng and Y Yang ldquoApplication of EMDmethod andHilbert spectrum to the fault diagnosis of roller bearingsrdquo Me-chanical Systems and Signal Processing vol 19 no 2 pp 259ndash270 2005

[12] V K Rai and A RMohanty ldquoBearing fault diagnosis using FFTof intrinsic mode functions in Hilbert-Huang transformrdquo Me-chanical Systems and Signal Processing vol 21 no 6 pp 2607ndash2615 2007

[13] Y Lei N Li J Lin and SWang ldquoFault diagnosis of rotatingma-chinery based on an adaptive ensemble empirical mode decom-positionrdquo Sensors vol 13 no 12 pp 16950ndash16964 2013

[14] J Cheng Y Yang and Y Yang ldquoA rotatingmachinery fault diag-nosis method based on local mean decompositionrdquo Digital Sig-nal Processing vol 22 no 2 pp 356ndash366 2012

[15] B Chen Z He X Chen H Cao G Cai and Y Zi ldquoA demod-ulating approach based on local mean decomposition and itsapplications in mechanical fault diagnosisrdquo Measurement Sci-ence and Technology vol 22 no 5 Article ID 055704 2011

[16] S Prabhakar A R Mohanty and A S Sekhar ldquoApplication ofdiscrete wavelet transform for detection of ball bearing racefaultsrdquoTribology International vol 35 no 12 pp 793ndash800 2002

[17] R Yan R X Gao and X Chen ldquoWavelets for fault diagnosis ofrotary machines a review with applicationsrdquo Signal Processingvol 96 part A pp 1ndash15 2014

[18] H Wang and P Chen ldquoA feature extraction method based oninformation theory for fault diagnosis of reciprocating machin-eryrdquo Sensors vol 9 no 4 pp 2415ndash2436 2009

[19] K Li X L Ping H Q Wang and P Chen ldquoSequential fuzzydiagnosis method for motor roller bearing in variable operatingconditions based on vibration analysisrdquo Sensors vol 13 no 6pp 8013ndash8041 2013

[20] R Rubini and U Meneghetti ldquoApplication of the envelope andwavelet transform analyses for the diagnosis of incipient faultsin ball bearingsrdquoMechanical Systems and Signal Processing vol15 no 2 pp 287ndash302 2001

[21] L Gao Z Yang L Cai H Wang and P Chen ldquoRoller bearingfault diagnosis based on nonlinear redundant lifting waveletpacket analysisrdquo Sensors vol 11 no 1 pp 260ndash277 2011

[22] Z Yang L Cai L Gao and H Wang ldquoAdaptive redundant lift-ing wavelet transform based on fitting for fault feature extrac-tion of roller bearingsrdquo Sensors vol 12 no 4 pp 4381ndash43982012

[23] Z He ldquoPeak transform for efficient image representation andcodingrdquo IEEE Transactions on Image Processing vol 16 no 7pp 1741ndash1754 2007

[24] S Anila and N Devarajan ldquoThe usage of peak transform forimage compressionrdquo International Journal of Computer SystemsScience amp Engineering vol 2 no 11 pp 6308ndash6316 2010

[25] D Ho and R B Randall ldquoOptimization of bearing diagnostictechniques using simulated and actual bearing fault signalsrdquoMechanical Systems and Signal Processing vol 14 no 5 pp 763ndash788 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

0 100 200 300 4000

50

100

150

200

250

300

Frequency (Hz)

Am

plitu

de

X 4959Y 2026

(a)

0 100 200 300 4000

50

100

150

200

250

300

Frequency (Hz)

Am

plitu

de

X 4959Y 2792

(b)

Figure 16 Envelope spectra of the vibration signal induced by roller element fault (a) wavelet analysis method and (b) the proposedmethod

Table 2 Major indictors for bearing fault features

Defect frequencymagnitude of

Traditionalwavelet-based

methodThe proposed method

outer race 149 9952inner race 1777 2525roller element 2026 2792

as well as their sparseness in wavelet domain by employingpeak-based piecewise recombination and envelope demodu-lation Through the proposed method most noise mixed invibration signals can be eliminated and the weak fault signalsof rolling bearings can be enhanced and become more easilydetectableTherationality and validity of themethods are alsoproved by various experiments It shows that compared withtraditional wavelet-based method the proposed method hasmany advantages in bearing fault signals enhancement anddetection and is very simple and easy to implement

There is still room to improve the present work Thesensitivity of the method to noise and methods to makeenergy more concentrated are what will be studied in futurework

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This project is partially supported by the National NaturalScience Foundation of China (Grant no 51375037) and theNational Program on Key Basic Research Project (Grant no2012CB026000) The third author would also like to thankthe support of China Fundamental Research Funds for theCentral Universities (ZY1410) and the Public Hatching Plat-form for Recruited Talents of Beijing University of ChemicalTechnology

References

[1] W Zhou T G Habetler and R G Harley ldquoBearing conditionmonitoring methods for electric machines a general reviewrdquo inProceedings of the IEEE International Symposium on Diagnosticsfor Electric Machines Power Electronics and Drives (SDEMPEDrsquo07) pp 3ndash6 September 2007

[2] H Q Wang and P Chen ldquoIntelligent diagnosis method for acentrifugal pump using features of vibration signalsrdquo NeuralComputing and Applications vol 18 no 4 pp 397ndash405 2009

[3] H Wang and P Chen ldquoSequential diagnosis for rolling bearingusing fuzzy neural networkrdquo in Proceedings of the IEEEASMEInternational Conference on Advanced Intelligent Mechatronics(AIM rsquo08) pp 56ndash61 Xirsquoan China August 2008

[4] T Mitoma H Wang and P Chen ldquoFault diagnosis and condi-tion surveillance for plant rotating machinery using partially-linearized neural networkrdquo Computers and Industrial Engineer-ing vol 55 no 4 pp 783ndash794 2008

[5] P D McFadden and J D Smith ldquoVibration monitoring of roll-ing element bearings by the high-frequency resonance tech-niquemdasha reviewrdquo Tribology International vol 17 no 1 pp 3ndash101984

Mathematical Problems in Engineering 11

[6] N Tandon and A Choudhury ldquoReview of vibration and acous-tic measurement methods for the detection of defects in rollingelement bearingsrdquo Tribology International vol 32 no 8 pp469ndash480 1999

[7] R B W Heng and M J M Nor ldquoStatistical analysis of soundand vibration signals for monitoring rolling element bearingconditionrdquo Applied Acoustics vol 53 no 1ndash3 pp 211ndash226 1998

[8] C Junsheng Y Dejie and Y Yu ldquoThe application of energyoperator demodulation approach based on EMD in machineryfault diagnosisrdquo Mechanical Systems and Signal Processing vol21 no 2 pp 668ndash677 2007

[9] R B Randall J Antoni and S Chobsaard ldquoThe relationship be-tween spectral correlation and envelope analysis in the diagnos-tics of bearing faults and other cyclostationarymachine signalsrdquoMechanical Systems and Signal Processing vol 15 no 5 pp 945ndash962 2001

[10] A B Ming Z Y Qin W Zhang et al ldquoSpectrum auto-cor-relation analysis and its application to fault diagnosis of rollingelement bearingsrdquo Mechanical Systems and Signal Processingvol 41 no 1 pp 141ndash154 2013

[11] D Yu J Cheng and Y Yang ldquoApplication of EMDmethod andHilbert spectrum to the fault diagnosis of roller bearingsrdquo Me-chanical Systems and Signal Processing vol 19 no 2 pp 259ndash270 2005

[12] V K Rai and A RMohanty ldquoBearing fault diagnosis using FFTof intrinsic mode functions in Hilbert-Huang transformrdquo Me-chanical Systems and Signal Processing vol 21 no 6 pp 2607ndash2615 2007

[13] Y Lei N Li J Lin and SWang ldquoFault diagnosis of rotatingma-chinery based on an adaptive ensemble empirical mode decom-positionrdquo Sensors vol 13 no 12 pp 16950ndash16964 2013

[14] J Cheng Y Yang and Y Yang ldquoA rotatingmachinery fault diag-nosis method based on local mean decompositionrdquo Digital Sig-nal Processing vol 22 no 2 pp 356ndash366 2012

[15] B Chen Z He X Chen H Cao G Cai and Y Zi ldquoA demod-ulating approach based on local mean decomposition and itsapplications in mechanical fault diagnosisrdquo Measurement Sci-ence and Technology vol 22 no 5 Article ID 055704 2011

[16] S Prabhakar A R Mohanty and A S Sekhar ldquoApplication ofdiscrete wavelet transform for detection of ball bearing racefaultsrdquoTribology International vol 35 no 12 pp 793ndash800 2002

[17] R Yan R X Gao and X Chen ldquoWavelets for fault diagnosis ofrotary machines a review with applicationsrdquo Signal Processingvol 96 part A pp 1ndash15 2014

[18] H Wang and P Chen ldquoA feature extraction method based oninformation theory for fault diagnosis of reciprocating machin-eryrdquo Sensors vol 9 no 4 pp 2415ndash2436 2009

[19] K Li X L Ping H Q Wang and P Chen ldquoSequential fuzzydiagnosis method for motor roller bearing in variable operatingconditions based on vibration analysisrdquo Sensors vol 13 no 6pp 8013ndash8041 2013

[20] R Rubini and U Meneghetti ldquoApplication of the envelope andwavelet transform analyses for the diagnosis of incipient faultsin ball bearingsrdquoMechanical Systems and Signal Processing vol15 no 2 pp 287ndash302 2001

[21] L Gao Z Yang L Cai H Wang and P Chen ldquoRoller bearingfault diagnosis based on nonlinear redundant lifting waveletpacket analysisrdquo Sensors vol 11 no 1 pp 260ndash277 2011

[22] Z Yang L Cai L Gao and H Wang ldquoAdaptive redundant lift-ing wavelet transform based on fitting for fault feature extrac-tion of roller bearingsrdquo Sensors vol 12 no 4 pp 4381ndash43982012

[23] Z He ldquoPeak transform for efficient image representation andcodingrdquo IEEE Transactions on Image Processing vol 16 no 7pp 1741ndash1754 2007

[24] S Anila and N Devarajan ldquoThe usage of peak transform forimage compressionrdquo International Journal of Computer SystemsScience amp Engineering vol 2 no 11 pp 6308ndash6316 2010

[25] D Ho and R B Randall ldquoOptimization of bearing diagnostictechniques using simulated and actual bearing fault signalsrdquoMechanical Systems and Signal Processing vol 14 no 5 pp 763ndash788 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

[6] N Tandon and A Choudhury ldquoReview of vibration and acous-tic measurement methods for the detection of defects in rollingelement bearingsrdquo Tribology International vol 32 no 8 pp469ndash480 1999

[7] R B W Heng and M J M Nor ldquoStatistical analysis of soundand vibration signals for monitoring rolling element bearingconditionrdquo Applied Acoustics vol 53 no 1ndash3 pp 211ndash226 1998

[8] C Junsheng Y Dejie and Y Yu ldquoThe application of energyoperator demodulation approach based on EMD in machineryfault diagnosisrdquo Mechanical Systems and Signal Processing vol21 no 2 pp 668ndash677 2007

[9] R B Randall J Antoni and S Chobsaard ldquoThe relationship be-tween spectral correlation and envelope analysis in the diagnos-tics of bearing faults and other cyclostationarymachine signalsrdquoMechanical Systems and Signal Processing vol 15 no 5 pp 945ndash962 2001

[10] A B Ming Z Y Qin W Zhang et al ldquoSpectrum auto-cor-relation analysis and its application to fault diagnosis of rollingelement bearingsrdquo Mechanical Systems and Signal Processingvol 41 no 1 pp 141ndash154 2013

[11] D Yu J Cheng and Y Yang ldquoApplication of EMDmethod andHilbert spectrum to the fault diagnosis of roller bearingsrdquo Me-chanical Systems and Signal Processing vol 19 no 2 pp 259ndash270 2005

[12] V K Rai and A RMohanty ldquoBearing fault diagnosis using FFTof intrinsic mode functions in Hilbert-Huang transformrdquo Me-chanical Systems and Signal Processing vol 21 no 6 pp 2607ndash2615 2007

[13] Y Lei N Li J Lin and SWang ldquoFault diagnosis of rotatingma-chinery based on an adaptive ensemble empirical mode decom-positionrdquo Sensors vol 13 no 12 pp 16950ndash16964 2013

[14] J Cheng Y Yang and Y Yang ldquoA rotatingmachinery fault diag-nosis method based on local mean decompositionrdquo Digital Sig-nal Processing vol 22 no 2 pp 356ndash366 2012

[15] B Chen Z He X Chen H Cao G Cai and Y Zi ldquoA demod-ulating approach based on local mean decomposition and itsapplications in mechanical fault diagnosisrdquo Measurement Sci-ence and Technology vol 22 no 5 Article ID 055704 2011

[16] S Prabhakar A R Mohanty and A S Sekhar ldquoApplication ofdiscrete wavelet transform for detection of ball bearing racefaultsrdquoTribology International vol 35 no 12 pp 793ndash800 2002

[17] R Yan R X Gao and X Chen ldquoWavelets for fault diagnosis ofrotary machines a review with applicationsrdquo Signal Processingvol 96 part A pp 1ndash15 2014

[18] H Wang and P Chen ldquoA feature extraction method based oninformation theory for fault diagnosis of reciprocating machin-eryrdquo Sensors vol 9 no 4 pp 2415ndash2436 2009

[19] K Li X L Ping H Q Wang and P Chen ldquoSequential fuzzydiagnosis method for motor roller bearing in variable operatingconditions based on vibration analysisrdquo Sensors vol 13 no 6pp 8013ndash8041 2013

[20] R Rubini and U Meneghetti ldquoApplication of the envelope andwavelet transform analyses for the diagnosis of incipient faultsin ball bearingsrdquoMechanical Systems and Signal Processing vol15 no 2 pp 287ndash302 2001

[21] L Gao Z Yang L Cai H Wang and P Chen ldquoRoller bearingfault diagnosis based on nonlinear redundant lifting waveletpacket analysisrdquo Sensors vol 11 no 1 pp 260ndash277 2011

[22] Z Yang L Cai L Gao and H Wang ldquoAdaptive redundant lift-ing wavelet transform based on fitting for fault feature extrac-tion of roller bearingsrdquo Sensors vol 12 no 4 pp 4381ndash43982012

[23] Z He ldquoPeak transform for efficient image representation andcodingrdquo IEEE Transactions on Image Processing vol 16 no 7pp 1741ndash1754 2007

[24] S Anila and N Devarajan ldquoThe usage of peak transform forimage compressionrdquo International Journal of Computer SystemsScience amp Engineering vol 2 no 11 pp 6308ndash6316 2010

[25] D Ho and R B Randall ldquoOptimization of bearing diagnostictechniques using simulated and actual bearing fault signalsrdquoMechanical Systems and Signal Processing vol 14 no 5 pp 763ndash788 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of