Research ArticleCyclostationary Analysis for Gearboxand Bearing Fault Diagnosis

Zhipeng Feng1 and Fulei Chu2

1School of Mechanical Engineering University of Science and Technology Beijing Beijing 100083 China2Department of Mechanical Engineering Tsinghua University Beijing 100084 China

Correspondence should be addressed to Zhipeng Feng fengzpustbeducn

Received 5 May 2015 Revised 24 July 2015 Accepted 27 July 2015

Academic Editor Dong Wang

Copyright copy 2015 Z Feng and F Chu This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Gearbox and rolling element bearing vibration signals feature modulation thus being cyclostationary Therefore the cycliccorrelation and cyclic spectrum are suited to analyze their modulation characteristics and thereby extract gearbox and bearingfault symptoms In order to thoroughly understand the cyclostationarity of gearbox and bearing vibrations the explicitexpressions of cyclic correlation and cyclic spectrum for amplitude modulation and frequency modulation (AM-FM) signalsare derived and their properties are summarized The theoretical derivations are illustrated and validated by gearbox andbearing experimental signal analyses The modulation characteristics caused by gearbox and bearing faults are extracted Infaulty gearbox and bearing cases more peaks appear in cyclic correlation slice of 0 lag and cyclic spectrum than in healthycases The gear and bearing faults are detected by checking the presence or monitoring the magnitude change of peaks incyclic correlation and cyclic spectrum and are located according to the peak cyclic frequency locations or sideband frequencyspacing

1 Introduction

Gearboxes and rolling element bearings are critical mechani-cal components and widely used in many types of machinery[1ndash3] Gear and bearing faults will result in deficiency oftransmission or even shut-down of the entire machineryTherefore gearbox and bearing fault diagnosis play an impor-tant role

The vibration signals of gearboxes and rolling elementbearings are usually cyclostationary since their statistics (interms of ensemble average) change periodically with time dueto their rotation Hence cyclostationary analysis is suitable toprocess gearbox and bearing vibration signals Dalpiaz et al[4] made a comparison study between various vibration sig-nal analysismethods (including cepstrum time-synchronousaverage wavelet transform and cyclostationary analysis)for gear localized fault detection and they found spectralcorrelation density is effective inmonitoring gear crack devel-opment Sidahmed et al [5 6] showed that time-synchronous

average can be considered as a first order cyclostationarityand spectral correlation as a second order cyclostationarityfound that gear vibration signals have second order cyclo-stationarity and early detected gear tooth spalling usingspectral correlation analysis Zhu et al [7] investigated theeffectiveness of cyclostationarity from the first order to thethird order that is spectrum of time-synchronous averagecyclic spectrum and cyclic bispectrum in gearbox conditionmonitoring Bi et al [8] proposed to extract the amplitudemodulation and phase modulation information from gearvibration signals using slice spectral correlation density soas to detect gear defects Li and Qu [9] deduced the cycliccorrelation and cyclic spectrum of amplitude modulationsignals and applied them to rolling element bearing faultdiagnosis Recently Antoni et al [10ndash15] conducted a seriesof researches on cyclostationary signal analysis and appliedit to fault diagnosis of rotating machinery To reduce thecomputational complexity of cyclic energy indicator basedon cyclic spectral density Wang and Shen [16] proposed

Hindawi Publishing CorporationShock and VibrationVolume 2015 Article ID 542472 12 pageshttpdxdoiorg1011552015542472

2 Shock and Vibration

an equivalent cyclic energy indicator for rolling elementbearing degradation evaluation These researches illustratethe effectiveness of cyclostationary analysis in gearbox andbearing fault diagnosis However the explicit relationshipbetween the cyclostationary features of vibration signalsand the gearbox and bearing dynamic nature still needsfurther investigation in order for thoroughly understandingthe vibration signal characteristics and thereby effectivelydiagnosing fault

Gearbox and rolling element bearing vibration signalsusually feature amplitude modulation and frequency modu-lation (AM-FM) and the modulation characteristics containtheir health status information [17ndash21] Cyclic correlationand cyclic spectrum are effective in extracting modulationfeatures from amplitudemodulation (AM) frequencymodu-lation (FM) and AM-FM signals Feng and his collaborators[22ndash24] derived the expressions of cyclic correlation andcyclic spectrum for gear AM-FM vibration signals andproposed indicators based on cyclic correlation and cyclicspectrum for detection and assessment of gearbox faultNevertheless the carrier frequency of rolling element bear-ing vibration signals (resonance frequency) is completelydifferent from that of gear vibration signals (gear meshingfrequency and its harmonics) Therefore it is importantto investigate the cyclic correlation and cyclic spectrum ofbearing vibration signals in depth considering both the AMand the FM effects due to bearing fault Meanwhile how toexplain the cyclostationary features displayed by the cycliccorrelation and cyclic spectrum and to map the modulationcharacteristics to gear and bearing fault are still importantissues for application of cyclostationary analysis in gearboxand bearing fault diagnosis In this paper we derive theexplicit expressions of cyclic correlation and cyclic spectrumfor general AM-FM signals summarize their propertiesand further extend the theoretical derivations to modulationanalysis of both gear and rolling element bearing vibrationsignals thus enabling cyclostationary analysis to detect andlocate both gearbox and bearing fault

2 Cyclic Correlation

21 Definition The statistics of cyclostationary signals haveperiodicity ormultiperiodicitywith respect to time evolutionCyclic statistics are suitable to process such signals Amongthose second order cyclic statistics that is cyclic correlationand cyclic spectrum are effective in extracting the modula-tion features of cyclostationary signals

For a signal 119909(119905) the cyclic autocorrelation function isdefined as [25]

119877120572

119909(120591) = int

infin

minusinfin

119909(119905 +

120591

2)

lowast

119909(119905 minus

120591

2) exp (minus1198952120587120572119905) 119889119905 (1)

where 120591 is time lag and 120572 is cyclic frequency

22 Cyclic Correlation of AM-FMSignal During the constantspeed running of gearboxes and rolling element bearings theexistence of fault machining defect and assembling erroroften leads to periodical changes in vibration signals Forgearboxes such periodical changes modulate both the ampli-tude envelope and instantaneous frequency of gear meshingvibration For bearings such repeated changes excite reso-nance periodically The excited resonance vanishes rapidlydue to damping before next resonance comes resulting inAM featureMeanwhile in one repeating cycle the resonanceexists in early period and the instantaneous frequency equalsapproximately the resonance frequency while in later periodthe resonance vanishes due to damping and the instanta-neous frequency becomes 0 That means the instantaneousfrequency changes periodically resulting in frequency mod-ulation (FM) Hence the vibration signals of both gearboxesand rolling element bearings can be modeled as an AM-FMprocess [17 21] plus a random noise

119909 (119905) =

119870

sum

119896=0119886119896(119905) cos [2120587119891

119888119905 + 119887119896(119905) + 120579

119896] + 120576 (119905) (2)

where

119886119896(119905) =

119873

sum

119899=0(1+119860

119896119899) cos (2120587119899119891

119898119905 + 120601119896119899) (3)

119887119896(119905) =

119871

sum

119897=0119861119896119894sin (2120587119897119891

119898119905 + 120593119896119897) (4)

are the AM and FM functions respectively 119860 gt 0 and 119861 gt 0

are themagnitude ofAMandFMrespectively119891119888is the carrier

frequency (for gearboxes it is the gear meshing frequencyor its 119896th harmonics for rolling element bearings it is theresonance frequency of bearing system)119891

119898is themodulating

frequency equal to the gear or bearing fault characteristicfrequency 120579 120601 and 120593 are the initial phase of AM and FMrespectively and 120576(119905) is a white Gaussian noise due to randombackground interferences

Without loss of generality consider only the fundamentalfrequency of the AM and FM terms then (2) becomes

119909 (119905) = [1+119860 cos (2120587119891119898119905 + 120601)]

sdot cos [2120587119891119888119905 + 119861 sin (2120587119891

119898119905 + 120593) + 120579] + 120576 (119905)

(5)

According to the identity [26]

exp [119895119911 sin (120573)] =infin

sum

119896=minusinfin

119869119896(119911) exp (119895119896120573) (6)

where 119869119896(119911) is Bessel function of the first kind with integer

order 119896 and argument 119911 the FM term in (5) can be expandedas a Bessel series and then (5) becomes

119909 (119905) = [1+119860 cos (2120587119891119898119905 + 120601)]

sdot

infin

sum

119899=minusinfin

119869119899(119861) cos (2120587119891

119888119905 + 2120587119899119891

119898119905 + 119899120593+ 120579)

+ 120576 (119905)

(7)

Shock and Vibration 3

For such a signal expressed as (7) the time-varying fea-ture of its autocorrelation function is mainly determined bythe AM-FM part since the autocorrelation function of awhite Gaussian noise 119877

120576(120591) = 120575(120591) (ie its Fourier transform

has peak at 0 only and therefore the noise 120576(119905) does not affectidentification of modulating frequency via cyclic correlationand cyclic spectrum analysis) In addition it is the AM and

FM effects on the carrier signal that leads to the cyclo-stationarity of bearing and gearbox vibration signals and werely on detection of the modulating frequency of such AMand FM effects to diagnose bearing and gearbox fault There-fore we neglect the noise 120576(119905) and focus on the deterministicAM-FM part only in the following analysis Then the cyclicautocorrelation function of (7) can be derived as [22]

119877120572

119909(120591) =

1

2

[ℎ119871(0 0) +

1

4

1198602

ℎ119871(minus2119891119898 0) +

1

4

1198602

ℎ119871(2119891119898 0)] 120572 = plusmn (119899 minus 119899

1015840

) 119891119898

1

4

119860 [ℎ119871(∓119891119898 plusmn120601) + ℎ

119871(plusmn119891119898 plusmn120601)] 120572 = plusmn (119899 minus 119899

1015840

plusmn 1)119891119898

1

8

1198602

ℎ119871(0 plusmn2120601) 120572 = plusmn (119899 minus 119899

1015840

plusmn 2)119891119898

(8a)

in lower cyclic frequency domain and

119877120572

119909(120591) =

1

2

[ℎ119867(0 0) +

1

4

1198602

ℎ119867(minus2119891119898 0) +

1

4

1198602

ℎ119867(2119891119898 0)] 120572 = plusmn [2119891

119888+ (119899 + 119899

1015840

) 119891119898]

1

4

119860 [ℎ119867(∓119891119898 plusmn120601) + ℎ

119867(plusmn119891119898 plusmn120601)] 120572 = plusmn [2119891

119888+ (119899 + 119899

1015840

plusmn 1)119891119898]

1

8

1198602

ℎ119867(0 plusmn2120601) 120572 = plusmn [2119891

119888+ (119899 + 119899

1015840

plusmn 2)119891119898]

(8b)

in higher cyclic frequency domain where the intermediatefunctions

ℎ119871(120578 120595) =

infin

sum

119899=minusinfin

infin

sum

1198991015840=minusinfin

119869119899(119861) 1198691198991015840 (119861)

sdot exp (plusmn119895 120587 [2119891119888+ (119899 + 119899

1015840

) 119891119898+ 120578] 120591 + (119899 minus 119899

1015840

) 120593 +120595)

(9a)

ℎ119867(120578 120595) =

infin

sum

119899=minusinfin

infin

sum

1198991015840=minusinfin

119869119899(119861) 1198691198991015840 (119861)

sdot exp (plusmn119895 120587 [(119899 minus 1198991015840) 119891119898+ 120578] 120591 + (119899 + 119899

1015840

) 120593 + 2120579 +120595)

(9b)

Set the time lag to 0 yielding the slice of cyclic autocorre-lation function

119877120572

119909(0) =

(

1

2

+

1

4

1198602

) ℎ1198710(0) 120572 = plusmn (119899 minus 119899

1015840

) 119891119898

1

2

119860ℎ1198710(plusmn120601) 120572 = plusmn (119899 minus 119899

1015840

plusmn 1)119891119898

1

8

1198602

ℎ1198710(plusmn2120601) 120572 = plusmn (119899 minus 119899

1015840

plusmn 2)119891119898

(10a)

119877120572

119909(0)

=

1

2

(1 +

1

2

1198602

) ℎ1198670(0) 120572 = plusmn [2119891

119888+ (119899 + 119899

1015840

) 119891119898]

1

2

119860ℎ1198670(plusmn120601) 120572 = plusmn [2119891

119888+ (119899 + 119899

1015840

plusmn 1)119891119898]

1

8

1198602

ℎ1198670(plusmn2120601) 120572 = plusmn [2119891

119888+ (119899 + 119899

1015840

plusmn 2)119891119898]

(10b)

in lower and higher cyclic frequency domains respectivelywhere the intermediate functions

ℎ1198710(120595) =

infin

sum

119899=minusinfin

infin

sum

1198991015840=minusinfin

119869119899(119861) 1198691198991015840 (119861)

sdot exp plusmn119895 [(119899 minus 1198991015840) 120593 +120595]

(11a)

ℎ1198670(120595) =

infin

sum

119899=minusinfin

infin

sum

1198991015840=minusinfin

119869119899(119861) 1198691198991015840 (119861)

sdot exp plusmn119895 [(119899 + 1198991015840) 120593 + 2120579 +120595]

(11b)

According to (10a) and (10b) and (11a) and (11b) theslice of cyclic autocorrelation function has two clusters ofcyclic frequencies one cluster concentrates around the cyclicfrequency of 0Hz separated by the modulating frequency119891119898 and the other cluster spreads around twice the carrier

4 Shock and Vibration

frequency 2119891119888with a spacing equal to the modulating fre-

quency 119891119898

In lower cyclic frequency domain the cyclic frequencylocations of present peaks are dependent on the difference ofthe two Bessel function orders 119899minus1198991015840 For any peak at a specificcyclic frequency 119899 minus 119899

1015840

= constant This leads to constantcomplex exponentials in (11a) Furthermore according to theaddition theorem of Bessel functions [26]

infin

sum

119899=minusinfin

infin

sum

1198991015840=minusinfin

119869119899(119911) 1198691198991015840 (119911) =

1 119899 = 1198991015840

0 119899 = 1198991015840

(12)

peaks appear in the slice of cyclic autocorrelation functionin lower cyclic frequency domain if and only if the ordersof the two Bessel functions are equal to each other that is119899 = 119899

1015840 Therefore the cyclic autocorrelation slice in lowercyclic frequency domain can be further simplified as

119877120572

119909(0) =

1

2

+

1

4

1198602

120572 = 0

1

2

119860 exp (plusmn119895120601) 120572 = plusmn119891119898

1

8

1198602 exp (plusmn1198952120601) 120572 = plusmn2119891

119898

(13)

According to (13) in lower cyclic frequency domain ofthe cyclic autocorrelation slice peaks appear at the cyclicfrequencies of 0Hz the modulating frequency 119891

119898 and its

twice 2119891119898 If higher order harmonics of the modulating

frequency are taken into account then peaks also appear atthe modulating frequency harmonics 119896119891

119898

The addition theorem of Bessel functions does not applyto the slice of cyclic autocorrelation function in higher cyclicfrequency domain The cyclic frequency locations of presentpeaks are dependent on the sum of the two Bessel functionorders 119899+1198991015840 For any specific peak 119899+1198991015840 = constantThis doesnot mean 119899 minus 1198991015840 = constant According to (10b) and (11b) inhigher cyclic frequency domain of the cyclic autocorrelationslice sidebands appear at both sides of twice the carrierfrequency 2119891

119888 with a spacing equal to the modulating freq-

uency 119891119898

According to the above derivations we can detect gearboxand bearing fault by monitoring the presence or magnitudechange of sidebands around the cyclic frequency of 0 or twicethe carrier frequency that is twice themeshing frequency forgearboxes and twice the resonance frequency for bearingsin the 0 lag slice of cyclic autocorrelation function We canfurther locate the gearbox and bearing fault by matching thesideband spacing with the fault characteristic frequencies

3 Cyclic Spectrum

31 Definition For a signal 119909(119905) the cyclic spectral densityis defined as the Fourier transform of the cyclic correlationfunction [25]

119878120572

119909(119891) = int

infin

minusinfin

119877120572

119909(120591) exp (minus1198952120587119891120591) 119889120591 (14)

where 119891 is frequency

32 Cyclic Spectrum of AM-FM Signal Without loss of gen-erality we still consider the simplified gearbox and bearingvibration signal model (5) by focusing on the fundamentalfrequency of the AM and FM terms Its cyclic spectral densitycan be derived as [23 24]

119878120572

119909(119891) =

1

2

119867 (0) 120572 = (119899 minus 1198991015840

) 119891119898 119891 = 119891

119888+

1

2

(119899 + 1198991015840

) 119891119898

1

8

1198602

119867(0) 120572 = (119899 minus 1198991015840

) 119891119898 119891 = 119891

119888+

1

2

(119899 + 1198991015840

∓ 2)119891119898

1

4

119860119867 (plusmn120601) 120572 = (119899 minus 1198991015840

plusmn 1)119891119898 119891 = 119891

119888+

1

2

(119899 + 1198991015840

∓ 1)119891119898

1

8

1198602

119867(2120601) 120572 = (119899 minus 1198991015840

plusmn 2)119891119898 119891 = 119891

119888+

1

2

(119899 + 1198991015840

) 119891119898

(15a)

119878120572

119909(119891) =

1

2

119867 (2120579) 120572 = 2119891119888+ (119899 + 119899

1015840

) 119891119898 119891 =

1

2

(119899 minus 1198991015840

) 119891119898

1

8

1198602

119867(2120579) 120572 = 2119891119888+ (119899 + 119899

1015840

) 119891119898 119891 =

1

2

(119899 minus 1198991015840

∓ 2)119891119898

1

4

119860119867 (2120579 plusmn 120601) 120572 = 2119891119888+ (119899 + 119899

1015840

plusmn 1)119891119898 119891 =

1

2

(119899 minus 1198991015840

∓ 1)119891119898

1

8

1198602

119867(2120579 plusmn 2120601) 120572 = 2119891119888+ (119899 + 119899

1015840

plusmn 2)119891119898 119891 =

1

2

(119899 minus 1198991015840

) 119891119898

(15b)

Shock and Vibration 5

in lower and higher cyclic frequency domains respectivelywhere the intermediate function

119867(120595) =

infin

sum

119899=minusinfin

infin

sum

1198991015840=minusinfin

119869119899(119861) 1198691198991015840 (119861) exp (plusmn119895120595) (16)

According to (15a) and (15b) peaks appear at specificlocations only on the cyclic frequencyminusfrequency planeTheircyclic frequency locations in lower cyclic frequency domainas well as their frequency locations in higher cyclic frequencydomain are dependent on the difference of the two Besselfunction orders 119899minus1198991015840 For any specific peaks 119899minus1198991015840 = constantThen the addition theorem of Bessel functions [26] (12)applies to the cyclic spectral density (15a) and (15b) Henceon the cyclic frequencyminusfrequency plane of the cyclic spectraldensity peaks appear if and only if the orders of the twoBessel

functions are equal to each other that is 119899 = 1198991015840 Then cyclic

spectral density can be simplified as

119878120572

119909(119891) = 119869

2

119899(119861)

sdot

1

2

120572 = 0 119891 = 119891119888+ 119899119891119898

1

8

1198602

120572 = 0 119891 = 119891119888+ (119899 ∓ 1) 119891

119898

1

4

119860 exp (plusmn119895120601) 120572 = plusmn119891119898 119891 = 119891

119888+ (119899 ∓

1

2

)119891119898

1

8

1198602 exp (plusmn1198952120601) 120572 = plusmn2119891

119898 119891 = 119891

119888+ 119899119891119898

(17a)

in lower cyclic frequency domain and

119878120572

119909(119891) = 119869

2119899(119861)

1

2

exp (plusmn1198952120579) 120572 = 2119891119888+ 2119899119891119898 119891 = 0

1

8

1198602 exp (plusmn1198952120579) 120572 = 2119891

119888+ 2119899119891119898 119891 = ∓119891

119898

1

4

119860 exp [plusmn119895 (2120579 plusmn 120601)] 120572 = 2119891119888+ (2119899 plusmn 1) 119891

119898 119891 = ∓

1

2

119891119898

1

8

1198602 exp [plusmn119895 (2120579 plusmn 2120601)] 120572 = 2119891

119888+ (2119899 plusmn 2) 119891

119898 119891 = 0

(17b)

in higher cyclic frequency domainAccording to (17a) and (17b) for any peak on the cyclic

frequencyminusfrequency plane its frequency location in lower

cyclic frequency domain as well as its cyclic frequencylocation in higher cyclic frequency domain is dependent onthe Bessel function order 119899 Meanwhile the peak magnitudeonly involves a few Bessel functionsThus (17a) and (17b) canbe further simplified as

119878120572

119909(119891) =

1

2

1198692

119899(119861) +

1

8

1198602

[1198692

119899minus1(119861) + 119869

2

119899+1(119861)] 120572 = 0 119891 = 119891

119888+ 119899119891119898

1

4

119860 [1198692

119899(119861) + 119869

2

119899+1(119861)] exp (plusmn119895120601) 120572 = plusmn119891

119898 119891 = 119891

119888+ (119899 +

1

2

)119891119898

1

8

1198602

1198692

119899(119861) exp (plusmn1198952120601) 120572 = plusmn2119891

119898 119891 = 119891

119888+ 119899119891119898

(18a)

119878120572

119909(119891) =

1

2

1198692

119899(119861) +

1

8

1198602

[1198692

119899minus1(119861) + 119869

2

119899+1(119861)] exp (plusmn1198952120579) 120572 = 2119891

119888+ 2119899119891119898 119891 = 0

1

4

119860 [1198692

119899(119861) + 119869

2

119899+1(119861)] exp [plusmn119895 (2120579 plusmn 120601)] 120572 = 2119891

119888+ (2119899 + 1) 119891

119898 119891 = ∓

1

2

119891119898

1

8

1198602

1198692

119899(119861) exp [plusmn119895 (2120579 plusmn 2120601)] 120572 = 2119891

119888+ 2119899119891119898 119891 = ∓119891

119898

(18b)

in lower and higher cyclic frequency domains respectivelyAccording to (18a) in lower cyclic frequency domain

peaks appear at the cyclic frequencies of 0Hz modulatingfrequency119891

119898 and its twice 2119891

119898 If higher order harmonics of

modulating frequency are taken into account then peaks alsoappear at the cyclic frequencies of the modulating frequencyharmonics 119896119891

119898 Along the frequency axis these peaks center

around the carrier frequency 119891119888 with a spacing equal to the

modulating frequency 119891119898

According to (18b) in higher cyclic frequency domainpeaks appear at the frequencies of 0Hz modulating fre-quency 119891

119898 and its half 12119891

119898 If higher order harmonics

of modulating frequency are taken into account then peaksalso appear at the modulating frequency harmonics 119896119891

119898and

6 Shock and Vibration

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 50 100 150 200150

200

250

300

350

5

10

15

20

25

30

35

Figure 1 Cyclic spectrumdistribution characteristics of anAM-FMsignal

their half 1198962119891119898 Along the cyclic frequency axis these peaks

center around twice the carrier frequency 2119891119888 with a spacing

equal to twice the modulating frequency 2119891119898

Observing the peak locations in (18a) and (18b) the peaksin cyclic spectrum distribute along the lines

119891 = plusmn

120572

2

plusmn (119891119888plusmn 119899119891119898) (19)

These lines form diamonds on the cyclic frequencyminusfre-quency plane as illustrated by Figure 1 which is the cyclicspectrum of an AM-FM signal with a carrier frequency of256Hz and a modulating frequency of 16Hz

According to the above derivations we can detect gearboxand bearing fault by monitoring the presence or magnitudechange of peaks in the cyclic spectrum For example in lowercyclic frequency band focus on the points corresponding tothe cyclic frequency locations of modulating frequency 119891

119898

and its multiples 119896119891119898and to the frequency locations of the

carrier frequency plus the modulating frequency multiples119891119888+ 119899119891119898 We can further locate the gearbox and bearing fault

by matching the cyclic frequency spacing of peaks with thefault characteristic frequencies

4 Gearbox Signal Analysis

41 Specification of Experiment Figure 2 shows the experi-mental system The gearbox has one gear pair Table 1 liststhe configuration and running condition of the gearbox Twostatuses of the gearbox are simulated Under the healthystatus the gear pair is perfect While under the faulty statusone of the pinion teeth is spalled whereas the gear is perfectDuring the experiment accelerometer signals are collectedat a sampling frequency of 20000Hz and each record lastsfor 3 s (ie 60000 samples) which covers more than 51and 48 revolutions for the drive pinion and the driven

Coupling

Motor

Accelerometer

Brake

Gear

Pinion

Bearing Amplifier

Computer

Figure 2 Gearbox experimental setup

Table 1 Gearbox configuration and running condition

Drive pinion Driven gearNumber of teeth 20 21Rotating frequency (Hz) 17285 16462Meshing frequency (Hz) 3457Brake torque (Nm) mdash 200

gear respectively and is long enough to reveal the cyclosta-tionarity

42 Signal Analysis Figure 3 shows the waveform Fourierspectrum cyclic correlation 0 lag slice and cyclic spectrum(in lower cyclic frequency band) of the healthy gearbox In thefollowing analysis the peak frequencies do not correspondexactly to those in Table 1 This is reasonable because theactual speed is inevitably different from the set one in realexperiments In the cyclic correlation slice Figure 3(c) mostof the present peaks correspond to the driven gear rotatingfrequency harmonics Although a few peaks appear at thepinion and gear characteristic frequencies and their harmon-ics their magnitudes are not strong In the cyclic spectrum(the colorbar on the right shows the magnitude the samesetting in the following cyclic spectra) Figure 3(d) alongthe frequency axis the present peaks center around the gearmeshing frequency 3457Hz and its harmonics and alongthe cyclic axis they appear at the pinion and gear character-istic frequencies and their harmonics The presence of thesepeaks is reasonable since gear manufacturing errors andminor defects are inevitable they will result in these peaks

Figure 4 shows the waveform Fourier spectrum cycliccorrelation 0 lag slice and cyclic spectrum (in lower cyclicfrequency band) of the faulty gearbox In the cyclic correla-tion slice Figure 4(c) peaks appear at the pinion and gearcharacteristic frequencies and their harmonics Except thefirst two peaks nearly all the other peaks appear at the cyclicfrequency locations of the drive pinion rotating frequencyharmonics and they are higher than those in that of thehealthy gearbox In the cyclic spectrum Figure 4(d) moreand stronger peaks appear than in that of the healthy gearboxFigure 4(e) shows the zoomed-in cyclic spectrum for reveal-ing the peak distribution details Along the cyclic frequencyaxis peaks appear along lines corresponding to the pinion

Shock and Vibration 7

0 01 02 03 04 05 06 07 08 09 1minus04

minus02

0

02

04

Time (s)

Am

plitu

de (V

)

(a) Waveform

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

0005

001

0015

002

0025

003

Frequency (Hz)

Am

plitu

de (V

)

(b) Fourier spectrum

0 20 40 60 80 100 120 140 160 180 2000

05

1

15

2

Cyclic frequency (Hz)

17

323

333

483

333

643

333

806

667

118

6667

135

3333

169

3333

times10minus3

Mag

nitu

de

(c) Cyclic correlation 0 lag slice

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 50 100 150 2000

500

1000

1500

2000

2500

3000

3500

4000

05

1

15

2

25

3

35

4

(d) Cyclic spectrum

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 50 100 150 2001200

1250

1300

1350

1400

1450

1500

1550

1600

05

1

15

2

25

3

35

4

(e) Zoomed-in cyclic spectrum

Figure 3 Healthy gearbox signal

characteristic frequency and its multiples While along thefrequency axis sidebands appear around 1400Hz about fourtimes the gear meshing frequency with a spacing equal tothe pinion characteristic frequency Based on the theoreticalderivations in Sections 2 and 3 the cyclic frequency locationsof peaks in cyclic correlation slice and cyclic spectrum aswell as the sideband spacing along frequency axis in cyclicspectrum relate to the modulating frequency of AM-FMsignalsThese features imply thatmore harmonics of the drivepinion rotating frequency get involved inmodulating the gear

meshing vibration and that the drive pinion has a strongermodulation effect than the driven gear indicating the pinionfault These findings are consistent with the actual conditionof the faulty gearbox

5 Bearing Signal Analysis

51 Specification of Experiment Figure 5 shows the experi-mental setup of the test [27] A shaft is supported by four

8 Shock and Vibration

0 01 02 03 04 05 06 07 08 09 1minus04

minus02

0

02

04

Time (s)

Am

plitu

de (V

)

(a) Waveform

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

001

002

003

004

Frequency (Hz)

Am

plitu

de (V

)

(b) Fourier spectrum

0 20 40 60 80 100 120 140 160 180 2000

02

04

06

08

1

Cyclic frequency (Hz)

Mag

nitu

de

173

333

33

52 69

863

333 10

366

67

121

138

3333 15

566

67

172

6667 190

times10minus3

(c) Cyclic correlation 0 lag slice

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 50 100 150 2000

500

1000

1500

2000

2500

3000

3500

4000

051152253354455

(d) Cyclic spectrum

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 50 100 150 2001200

1250

1300

1350

1400

1450

1500

1550

1600

051152253354455

(e) Zoomed-in cyclic spectrum

Figure 4 Faulty gearbox signal

Rexnord ZA-2115 double row rolling element bearings andis driven by an AC motor through rub belts at a speed of2000 rpm A radial load of 6000 lbs is applied to the shaftand bearings by a spring mechanism The bearing test waskept running for eight days until damage occurs to the outerrace of bearing 1 In the normal case all the four bearings arehealthy While in the faulty case the outer race of bearing1 has damage as shown in Figure 6 Accelerometers aremounted on the bearing housings and the vibration signals

are collected at a sampling frequency of 20480Hz and 20480data points are recorded for both the healthy and faultycases The main parameters of the four bearings are listed inTable 2 The characteristic frequency of each element fault iscalculated and listed in Table 3

52 Signal Analysis Figure 7 shows the waveform Fourierspectrum cyclic correlation 0 lag slice and cyclic spectrum

Shock and Vibration 9

Accelerometers Radial load Thermocouples

Bearing 1 Bearing 2 Bearing 3 Bearing 4

Motor

Figure 5 Bearing experimental setup [27]

Figure 6 Outer race damage of bearing 1 [27]

Table 2 Parameters of bearing Rexnord ZA-2115

Number ofrollers Roller diameter Pitch diameter Contact angle

16 0331 [in] 2815 [in] 1517 [∘]

Table 3 Characteristic frequency of bearing Rexnord ZA-2115 [Hz]

Outer race Inner race Roller236404 296930 139917 279833

(in lower cyclic frequency band) of the normal case Inthe cyclic correlation slice Figure 7(c) a few small peaksappear but they do not correspond to any bearing componentfault characteristic frequency or its multiples Although somepeaks appear in the cyclic spectrum Figure 7(d) they do notdistribute along lines corresponding to the cyclic frequenciesof any bearing component fault characteristic frequency orits multiples These features imply that no significant mod-ulation on the resonance vibration and the bearing is healthy

Figure 8 shows the waveform Fourier spectrum cycliccorrelation 0 lag slice and cyclic spectrum (in lower cyclicfrequency band) of the faulty case In the close-up view of theFourier spectrum some sidebands appear but the frequencyspacing is not identically equal to the outer race fault charac-teristic frequency In the cyclic correlation slice Figure 8(d)more peaks appear and their magnitudes are higher than inthe normal case Moreover their cyclic frequency locationscorrespond to the outer race fault characteristic frequencyand multiples up to the 25th order In the cyclic spectrumFigure 8(e) many peaks are present More importantly theydistribute along lines associated with the outer race faultcharacteristic frequency and multiples At any cyclic fre-quency of the outer race fault characteristic frequency or itsmultiple these peaks form sidebands along the frequencyaxis with a spacing equal to the outer race fault characteristicfrequency For example in the zoomed-in cyclic spectrumFigure 8(f) the four peaks labeled by capitals A B C and Ddistribute along a line at the cyclic frequency of 710Hz (aboutthree times the outer race fault characteristic frequency) andthey have a spacing of 236Hz (approximately equal to theouter race fault characteristic frequency) along the frequencyaxis These findings imply that the outer race characteristics

10 Shock and Vibration

0 01 02 03 04 05 06 07 08 09 1minus04

minus02

0

02

04

Time (s)

Am

plitu

de (V

)

(a) Waveform

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

0004

0008

0012

0016

002

Frequency (Hz)

Am

plitu

de (V

)

(b) Fourier spectrum

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

02

04

06

08

1

Cyclic frequency (Hz)

Mag

nitu

de

times10minus3

(c) Cyclic correlation 0 lag slice

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

005

01

015

02

025

03

(d) Cyclic spectrum

Figure 7 Normal bearing signal

frequency has strong modulation effect on the resonanceindicating fault existence on the bearing outer race in accord-ance with the actual settings

6 Discussion and Conclusions

Since gearbox and rolling element bearing vibration signalsare characterized by AM-FM feature their Fourier spectrahave complex sideband structure due to the convolutionbetween the Fourier spectra ofAMandFMparts aswell as the

infinite Bessel series expansion of an FM term Althoughgearbox and bearing fault can be identified via conventionalFourier spectrum it involves complex sideband analysis andrelies on the sideband spacing to extract the fault character-istic frequency

Considering the cyclostationarity of gearbox and bearingvibration signals due to modulation and the suitability ofcyclostationary analysis to extract modulation features ofsuch signals the explicit expressions of cyclic correlation andcyclic spectrum of general AM-FM signals are derived and

Shock and Vibration 11

0 01 02 03 04 05 06 07 08 09 1minus4

minus2

0

2

4

Time (s)

Am

plitu

de (V

)

(a) Waveform

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

001

002

003

004

005

Frequency (Hz)

Am

plitu

de (V

)

(b) Fourier spectrum

2000 2500 3000 3500 4000 4500 5000 5500 60000

001

002

003

004

005

006

Frequency (Hz)

4965

3544

4033

3322

3086

4491

2612

2839

3796

4743

4325

5202

5428

5675A

mpl

itude

(V)

(c) Zoomed-in Fourier spectrum

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

002

004

006

008

01

012

Cyclic frequency (Hz)

237

473 71

094

711

83 1420

1657

1894

2130 23

6725

96 2833

3084

3306 3543

3780

4016 4253 4490

4726 49

6352

0054

5156

7359

10

Mag

nitu

de

(d) Cyclic correlation 0 lag slice

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

05

1

15

2

25

3

35

4

(e) Cyclic spectrum

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

660 680 700 720 740 7603000

3100

3200

3300

3400

3500

3600

3700

3800

3900

4000

05

1

15

2

25

3

35

4D

C

B

A

(f) Zoomed-in cyclic spectrum

Figure 8 Faulty bearing signal

validated via gearbox and bearing lab experimental signalanalysis According to the theoretical derivations the cyclicfrequency locations of peaks in cyclic correlation and cyclicspectrum directly correspond to the modulating frequency(ie the gear or bearing fault characteristic frequency) andharmonicsTherefore cyclostationary analysis offers an effec-tive approach to gearbox and bearing fault feature extraction

Conflict of Interests

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

Acknowledgments

This work is supported by National Natural Science Founda-tion ofChina (11272047 51475038) Program forNewCenturyExcellent Talents in University (NCET-12-0775) and CETIM(Centre des Etudes Techniques des Industries Mecaniques deSenlis) FranceThe authors would like to thank the reviewersfor their valuable comments and suggestions

References

[1] R Shao W Hu and J Li ldquoMulti-fault feature extraction anddiagnosis of gear transmission system using time-frequency

12 Shock and Vibration

analysis and wavelet threshold de-noising based on EMDrdquoShock and Vibration vol 20 no 4 pp 763ndash780 2013

[2] S Luo J Cheng and H Ao ldquoApplication of LCD-SVD tech-nique and CRO-SVMmethod to fault diagnosis for roller bear-ingrdquo Shock and Vibration vol 2015 Article ID 847802 8 pages2015

[3] R Yan M Shan J Cui and Y Wu ldquoMutual information-assisted wavelet function selection for enhanced rolling bearingfault diagnosisrdquo Shock and Vibration In press

[4] G Dalpiaz A Rivola and R Rubini ldquoEffectiveness and sensi-tivity of vibration processing techniques for local fault detectionin gearsrdquo Mechanical Systems and Signal Processing vol 14 no3 pp 387ndash412 2000

[5] C Capdessus M Sidahmed and J L Lacoume ldquoCyclostation-ary processes application in gear faults early diagnosisrdquoMechanical Systems and Signal Processing vol 14 no 3 pp 371ndash385 2000

[6] L Bouillaut and M Sidahmed ldquoCyclostationary approach andbilinear approach comparison applications to early diagnosisfor helicopter gearbox and classificationmethod based on hocsrdquoMechanical Systems and Signal Processing vol 15 no 5 pp 923ndash943 2001

[7] Z K Zhu Z H Feng and F R Kong ldquoCyclostationarity anal-ysis for gearbox condition monitoring approaches and effec-tivenessrdquo Mechanical Systems and Signal Processing vol 19 no3 pp 467ndash482 2005

[8] G Bi J Chen F C Zhou and J He ldquoApplication of slice spectralcorrelation density to gear defect detectionrdquo Proceedings ofthe Institution of Mechanical Engineers Part C Journal ofMechanical Engineering Science vol 220 no 9 pp 1385ndash13922006

[9] L Li and L Qu ldquoCyclic statistics in rolling bearing diagnosisrdquoJournal of Sound and Vibration vol 267 no 2 pp 253ndash2652003

[10] R B Randall J Antoni and S Chobsaard ldquoThe relationshipbetween spectral correlation and envelope analysis in the diag-nostics of bearing faults and other cyclostationary machinesignalsrdquoMechanical Systems and Signal Processing vol 15 no 5pp 945ndash962 2001

[11] J Antoni ldquoCyclic spectral analysis in practicerdquoMechanical Sys-tems and Signal Processing vol 21 no 2 pp 597ndash630 2007

[12] J Antoni ldquoCyclic spectral analysis of rolling-element bearingsignals facts and fictionsrdquo Journal of Sound and Vibration vol304 no 3-5 pp 497ndash529 2007

[13] A Raad J Antoni and M Sidahmed ldquoIndicators of cyclosta-tionarity theory and application to gear fault monitoringrdquoMechanical Systems and Signal Processing vol 22 no 3 pp 574ndash587 2008

[14] J Antoni ldquoCyclostationarity by examplesrdquo Mechanical Systemsand Signal Processing vol 23 no 4 pp 987ndash1036 2009

[15] J Urbanek T Barszcz and J Antoni ldquoTime-frequencyapproach to extraction of selected second-order cyclostation-ary vibration components for varying operational conditionsrdquoMeasurement vol 46 no 4 pp 1454ndash1463 2013

[16] DWang and C Shen ldquoAn equivalent cyclic energy indicator forbearing performance degradation assessmentrdquo Journal of Vibra-tion and Control 2014

[17] R B Randall ldquoA new method of modeling gear faultsrdquo Journalof Mechanical Design vol 104 no 2 pp 259ndash267 1982

[18] P DMcFadden ldquoDetecting fatigue cracks in gears by amplitudeand phase demodulation of the meshing vibrationrdquo Journal of

Vibration Acoustics Stress and Reliability in Design vol 108 no2 pp 165ndash170 1986

[19] P D McFadden and J D Smith ldquoModel for the vibrationproduced by a single point defect in a rolling element bearingrdquoJournal of Sound and Vibration vol 96 no 1 pp 69ndash82 1984

[20] P D McFadden and J D Smith ldquoThe vibration produced bymultiple point defects in a rolling element bearingrdquo Journal ofSound and Vibration vol 98 no 2 pp 263ndash273 1985

[21] M Liang and I S Bozchalooi ldquoAn energy operator approachto joint application of amplitude and frequency-demodulationsfor bearing fault detectionrdquo Mechanical Systems and SignalProcessing vol 24 no 5 pp 1473ndash1494 2010

[22] Z Feng M J Zuo R Hao and F Chu ldquoGear crack assessmentbased on cyclic correlation analysisrdquo in Proceedings of the8th International Conference on Reliability Maintainability andSafety (ICRMS rsquo09) pp 1071ndash1076 Chengdu China July 2009

[23] Z Feng R Hao F Chu M J Zuo andM El Badaoui ldquoApplica-tion of cyclic spectral analysis to gear damage assessmentrdquo inProceedings of the Prognostics and Health Management Confer-ence p 3058 Macau China January 2010

[24] Z Feng andM J Zuo ldquoGearbox diagnosis based on cyclic spec-tral analysisrdquo in Proceedings of the 3rd Annual IEEE Prognosticsand System Health Management Conference (PHM rsquo12) pp 1ndash5Beijing China May 2012

[25] W A Gardner Cyclostationarity in Communications and SignalProcessing IEEE Press New York NY USA 1994

[26] M Abramowitz and I A Stegun Handbook of MathematicalFunctions With Formulas Graphs and Mathematical TablesDover Publications New York NY USA 1972

[27] H Qiu J Lee J Lin and G Yu ldquoWavelet filter-based weak sig-nature detection method and its application on rolling elementbearing prognosticsrdquo Journal of Sound and Vibration vol 289no 4-5 pp 1066ndash1090 2006

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Submit your manuscripts athttpwwwhindawicom

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

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2 Shock and Vibration

an equivalent cyclic energy indicator for rolling elementbearing degradation evaluation These researches illustratethe effectiveness of cyclostationary analysis in gearbox andbearing fault diagnosis However the explicit relationshipbetween the cyclostationary features of vibration signalsand the gearbox and bearing dynamic nature still needsfurther investigation in order for thoroughly understandingthe vibration signal characteristics and thereby effectivelydiagnosing fault

Gearbox and rolling element bearing vibration signalsusually feature amplitude modulation and frequency modu-lation (AM-FM) and the modulation characteristics containtheir health status information [17ndash21] Cyclic correlationand cyclic spectrum are effective in extracting modulationfeatures from amplitudemodulation (AM) frequencymodu-lation (FM) and AM-FM signals Feng and his collaborators[22ndash24] derived the expressions of cyclic correlation andcyclic spectrum for gear AM-FM vibration signals andproposed indicators based on cyclic correlation and cyclicspectrum for detection and assessment of gearbox faultNevertheless the carrier frequency of rolling element bear-ing vibration signals (resonance frequency) is completelydifferent from that of gear vibration signals (gear meshingfrequency and its harmonics) Therefore it is importantto investigate the cyclic correlation and cyclic spectrum ofbearing vibration signals in depth considering both the AMand the FM effects due to bearing fault Meanwhile how toexplain the cyclostationary features displayed by the cycliccorrelation and cyclic spectrum and to map the modulationcharacteristics to gear and bearing fault are still importantissues for application of cyclostationary analysis in gearboxand bearing fault diagnosis In this paper we derive theexplicit expressions of cyclic correlation and cyclic spectrumfor general AM-FM signals summarize their propertiesand further extend the theoretical derivations to modulationanalysis of both gear and rolling element bearing vibrationsignals thus enabling cyclostationary analysis to detect andlocate both gearbox and bearing fault

2 Cyclic Correlation

21 Definition The statistics of cyclostationary signals haveperiodicity ormultiperiodicitywith respect to time evolutionCyclic statistics are suitable to process such signals Amongthose second order cyclic statistics that is cyclic correlationand cyclic spectrum are effective in extracting the modula-tion features of cyclostationary signals

For a signal 119909(119905) the cyclic autocorrelation function isdefined as [25]

119877120572

119909(120591) = int

infin

minusinfin

119909(119905 +

120591

2)

lowast

119909(119905 minus

120591

2) exp (minus1198952120587120572119905) 119889119905 (1)

where 120591 is time lag and 120572 is cyclic frequency

22 Cyclic Correlation of AM-FMSignal During the constantspeed running of gearboxes and rolling element bearings theexistence of fault machining defect and assembling erroroften leads to periodical changes in vibration signals Forgearboxes such periodical changes modulate both the ampli-tude envelope and instantaneous frequency of gear meshingvibration For bearings such repeated changes excite reso-nance periodically The excited resonance vanishes rapidlydue to damping before next resonance comes resulting inAM featureMeanwhile in one repeating cycle the resonanceexists in early period and the instantaneous frequency equalsapproximately the resonance frequency while in later periodthe resonance vanishes due to damping and the instanta-neous frequency becomes 0 That means the instantaneousfrequency changes periodically resulting in frequency mod-ulation (FM) Hence the vibration signals of both gearboxesand rolling element bearings can be modeled as an AM-FMprocess [17 21] plus a random noise

119909 (119905) =

119870

sum

119896=0119886119896(119905) cos [2120587119891

119888119905 + 119887119896(119905) + 120579

119896] + 120576 (119905) (2)

where

119886119896(119905) =

119873

sum

119899=0(1+119860

119896119899) cos (2120587119899119891

119898119905 + 120601119896119899) (3)

119887119896(119905) =

119871

sum

119897=0119861119896119894sin (2120587119897119891

119898119905 + 120593119896119897) (4)

are the AM and FM functions respectively 119860 gt 0 and 119861 gt 0

are themagnitude ofAMandFMrespectively119891119888is the carrier

frequency (for gearboxes it is the gear meshing frequencyor its 119896th harmonics for rolling element bearings it is theresonance frequency of bearing system)119891

119898is themodulating

frequency equal to the gear or bearing fault characteristicfrequency 120579 120601 and 120593 are the initial phase of AM and FMrespectively and 120576(119905) is a white Gaussian noise due to randombackground interferences

Without loss of generality consider only the fundamentalfrequency of the AM and FM terms then (2) becomes

119909 (119905) = [1+119860 cos (2120587119891119898119905 + 120601)]

sdot cos [2120587119891119888119905 + 119861 sin (2120587119891

119898119905 + 120593) + 120579] + 120576 (119905)

(5)

According to the identity [26]

exp [119895119911 sin (120573)] =infin

sum

119896=minusinfin

119869119896(119911) exp (119895119896120573) (6)

where 119869119896(119911) is Bessel function of the first kind with integer

order 119896 and argument 119911 the FM term in (5) can be expandedas a Bessel series and then (5) becomes

119909 (119905) = [1+119860 cos (2120587119891119898119905 + 120601)]

sdot

infin

sum

119899=minusinfin

119869119899(119861) cos (2120587119891

119888119905 + 2120587119899119891

119898119905 + 119899120593+ 120579)

+ 120576 (119905)

(7)

Shock and Vibration 3

For such a signal expressed as (7) the time-varying fea-ture of its autocorrelation function is mainly determined bythe AM-FM part since the autocorrelation function of awhite Gaussian noise 119877

120576(120591) = 120575(120591) (ie its Fourier transform

has peak at 0 only and therefore the noise 120576(119905) does not affectidentification of modulating frequency via cyclic correlationand cyclic spectrum analysis) In addition it is the AM and

FM effects on the carrier signal that leads to the cyclo-stationarity of bearing and gearbox vibration signals and werely on detection of the modulating frequency of such AMand FM effects to diagnose bearing and gearbox fault There-fore we neglect the noise 120576(119905) and focus on the deterministicAM-FM part only in the following analysis Then the cyclicautocorrelation function of (7) can be derived as [22]

119877120572

119909(120591) =

1

2

[ℎ119871(0 0) +

1

4

1198602

ℎ119871(minus2119891119898 0) +

1

4

1198602

ℎ119871(2119891119898 0)] 120572 = plusmn (119899 minus 119899

1015840

) 119891119898

1

4

119860 [ℎ119871(∓119891119898 plusmn120601) + ℎ

119871(plusmn119891119898 plusmn120601)] 120572 = plusmn (119899 minus 119899

1015840

plusmn 1)119891119898

1

8

1198602

ℎ119871(0 plusmn2120601) 120572 = plusmn (119899 minus 119899

1015840

plusmn 2)119891119898

(8a)

in lower cyclic frequency domain and

119877120572

119909(120591) =

1

2

[ℎ119867(0 0) +

1

4

1198602

ℎ119867(minus2119891119898 0) +

1

4

1198602

ℎ119867(2119891119898 0)] 120572 = plusmn [2119891

119888+ (119899 + 119899

1015840

) 119891119898]

1

4

119860 [ℎ119867(∓119891119898 plusmn120601) + ℎ

119867(plusmn119891119898 plusmn120601)] 120572 = plusmn [2119891

119888+ (119899 + 119899

1015840

plusmn 1)119891119898]

1

8

1198602

ℎ119867(0 plusmn2120601) 120572 = plusmn [2119891

119888+ (119899 + 119899

1015840

plusmn 2)119891119898]

(8b)

in higher cyclic frequency domain where the intermediatefunctions

ℎ119871(120578 120595) =

infin

sum

119899=minusinfin

infin

sum

1198991015840=minusinfin

119869119899(119861) 1198691198991015840 (119861)

sdot exp (plusmn119895 120587 [2119891119888+ (119899 + 119899

1015840

) 119891119898+ 120578] 120591 + (119899 minus 119899

1015840

) 120593 +120595)

(9a)

ℎ119867(120578 120595) =

infin

sum

119899=minusinfin

infin

sum

1198991015840=minusinfin

119869119899(119861) 1198691198991015840 (119861)

sdot exp (plusmn119895 120587 [(119899 minus 1198991015840) 119891119898+ 120578] 120591 + (119899 + 119899

1015840

) 120593 + 2120579 +120595)

(9b)

Set the time lag to 0 yielding the slice of cyclic autocorre-lation function

119877120572

119909(0) =

(

1

2

+

1

4

1198602

) ℎ1198710(0) 120572 = plusmn (119899 minus 119899

1015840

) 119891119898

1

2

119860ℎ1198710(plusmn120601) 120572 = plusmn (119899 minus 119899

1015840

plusmn 1)119891119898

1

8

1198602

ℎ1198710(plusmn2120601) 120572 = plusmn (119899 minus 119899

1015840

plusmn 2)119891119898

(10a)

119877120572

119909(0)

=

1

2

(1 +

1

2

1198602

) ℎ1198670(0) 120572 = plusmn [2119891

119888+ (119899 + 119899

1015840

) 119891119898]

1

2

119860ℎ1198670(plusmn120601) 120572 = plusmn [2119891

119888+ (119899 + 119899

1015840

plusmn 1)119891119898]

1

8

1198602

ℎ1198670(plusmn2120601) 120572 = plusmn [2119891

119888+ (119899 + 119899

1015840

plusmn 2)119891119898]

(10b)

in lower and higher cyclic frequency domains respectivelywhere the intermediate functions

ℎ1198710(120595) =

infin

sum

119899=minusinfin

infin

sum

1198991015840=minusinfin

119869119899(119861) 1198691198991015840 (119861)

sdot exp plusmn119895 [(119899 minus 1198991015840) 120593 +120595]

(11a)

ℎ1198670(120595) =

infin

sum

119899=minusinfin

infin

sum

1198991015840=minusinfin

119869119899(119861) 1198691198991015840 (119861)

sdot exp plusmn119895 [(119899 + 1198991015840) 120593 + 2120579 +120595]

(11b)

According to (10a) and (10b) and (11a) and (11b) theslice of cyclic autocorrelation function has two clusters ofcyclic frequencies one cluster concentrates around the cyclicfrequency of 0Hz separated by the modulating frequency119891119898 and the other cluster spreads around twice the carrier

4 Shock and Vibration

frequency 2119891119888with a spacing equal to the modulating fre-

quency 119891119898

In lower cyclic frequency domain the cyclic frequencylocations of present peaks are dependent on the difference ofthe two Bessel function orders 119899minus1198991015840 For any peak at a specificcyclic frequency 119899 minus 119899

1015840

= constant This leads to constantcomplex exponentials in (11a) Furthermore according to theaddition theorem of Bessel functions [26]

infin

sum

119899=minusinfin

infin

sum

1198991015840=minusinfin

119869119899(119911) 1198691198991015840 (119911) =

1 119899 = 1198991015840

0 119899 = 1198991015840

(12)

peaks appear in the slice of cyclic autocorrelation functionin lower cyclic frequency domain if and only if the ordersof the two Bessel functions are equal to each other that is119899 = 119899

1015840 Therefore the cyclic autocorrelation slice in lowercyclic frequency domain can be further simplified as

119877120572

119909(0) =

1

2

+

1

4

1198602

120572 = 0

1

2

119860 exp (plusmn119895120601) 120572 = plusmn119891119898

1

8

1198602 exp (plusmn1198952120601) 120572 = plusmn2119891

119898

(13)

According to (13) in lower cyclic frequency domain ofthe cyclic autocorrelation slice peaks appear at the cyclicfrequencies of 0Hz the modulating frequency 119891

119898 and its

twice 2119891119898 If higher order harmonics of the modulating

frequency are taken into account then peaks also appear atthe modulating frequency harmonics 119896119891

119898

The addition theorem of Bessel functions does not applyto the slice of cyclic autocorrelation function in higher cyclicfrequency domain The cyclic frequency locations of presentpeaks are dependent on the sum of the two Bessel functionorders 119899+1198991015840 For any specific peak 119899+1198991015840 = constantThis doesnot mean 119899 minus 1198991015840 = constant According to (10b) and (11b) inhigher cyclic frequency domain of the cyclic autocorrelationslice sidebands appear at both sides of twice the carrierfrequency 2119891

119888 with a spacing equal to the modulating freq-

uency 119891119898

According to the above derivations we can detect gearboxand bearing fault by monitoring the presence or magnitudechange of sidebands around the cyclic frequency of 0 or twicethe carrier frequency that is twice themeshing frequency forgearboxes and twice the resonance frequency for bearingsin the 0 lag slice of cyclic autocorrelation function We canfurther locate the gearbox and bearing fault by matching thesideband spacing with the fault characteristic frequencies

3 Cyclic Spectrum

31 Definition For a signal 119909(119905) the cyclic spectral densityis defined as the Fourier transform of the cyclic correlationfunction [25]

119878120572

119909(119891) = int

infin

minusinfin

119877120572

119909(120591) exp (minus1198952120587119891120591) 119889120591 (14)

where 119891 is frequency

32 Cyclic Spectrum of AM-FM Signal Without loss of gen-erality we still consider the simplified gearbox and bearingvibration signal model (5) by focusing on the fundamentalfrequency of the AM and FM terms Its cyclic spectral densitycan be derived as [23 24]

119878120572

119909(119891) =

1

2

119867 (0) 120572 = (119899 minus 1198991015840

) 119891119898 119891 = 119891

119888+

1

2

(119899 + 1198991015840

) 119891119898

1

8

1198602

119867(0) 120572 = (119899 minus 1198991015840

) 119891119898 119891 = 119891

119888+

1

2

(119899 + 1198991015840

∓ 2)119891119898

1

4

119860119867 (plusmn120601) 120572 = (119899 minus 1198991015840

plusmn 1)119891119898 119891 = 119891

119888+

1

2

(119899 + 1198991015840

∓ 1)119891119898

1

8

1198602

119867(2120601) 120572 = (119899 minus 1198991015840

plusmn 2)119891119898 119891 = 119891

119888+

1

2

(119899 + 1198991015840

) 119891119898

(15a)

119878120572

119909(119891) =

1

2

119867 (2120579) 120572 = 2119891119888+ (119899 + 119899

1015840

) 119891119898 119891 =

1

2

(119899 minus 1198991015840

) 119891119898

1

8

1198602

119867(2120579) 120572 = 2119891119888+ (119899 + 119899

1015840

) 119891119898 119891 =

1

2

(119899 minus 1198991015840

∓ 2)119891119898

1

4

119860119867 (2120579 plusmn 120601) 120572 = 2119891119888+ (119899 + 119899

1015840

plusmn 1)119891119898 119891 =

1

2

(119899 minus 1198991015840

∓ 1)119891119898

1

8

1198602

119867(2120579 plusmn 2120601) 120572 = 2119891119888+ (119899 + 119899

1015840

plusmn 2)119891119898 119891 =

1

2

(119899 minus 1198991015840

) 119891119898

(15b)

Shock and Vibration 5

in lower and higher cyclic frequency domains respectivelywhere the intermediate function

119867(120595) =

infin

sum

119899=minusinfin

infin

sum

1198991015840=minusinfin

119869119899(119861) 1198691198991015840 (119861) exp (plusmn119895120595) (16)

According to (15a) and (15b) peaks appear at specificlocations only on the cyclic frequencyminusfrequency planeTheircyclic frequency locations in lower cyclic frequency domainas well as their frequency locations in higher cyclic frequencydomain are dependent on the difference of the two Besselfunction orders 119899minus1198991015840 For any specific peaks 119899minus1198991015840 = constantThen the addition theorem of Bessel functions [26] (12)applies to the cyclic spectral density (15a) and (15b) Henceon the cyclic frequencyminusfrequency plane of the cyclic spectraldensity peaks appear if and only if the orders of the twoBessel

functions are equal to each other that is 119899 = 1198991015840 Then cyclic

spectral density can be simplified as

119878120572

119909(119891) = 119869

2

119899(119861)

sdot

1

2

120572 = 0 119891 = 119891119888+ 119899119891119898

1

8

1198602

120572 = 0 119891 = 119891119888+ (119899 ∓ 1) 119891

119898

1

4

119860 exp (plusmn119895120601) 120572 = plusmn119891119898 119891 = 119891

119888+ (119899 ∓

1

2

)119891119898

1

8

1198602 exp (plusmn1198952120601) 120572 = plusmn2119891

119898 119891 = 119891

119888+ 119899119891119898

(17a)

in lower cyclic frequency domain and

119878120572

119909(119891) = 119869

2119899(119861)

1

2

exp (plusmn1198952120579) 120572 = 2119891119888+ 2119899119891119898 119891 = 0

1

8

1198602 exp (plusmn1198952120579) 120572 = 2119891

119888+ 2119899119891119898 119891 = ∓119891

119898

1

4

119860 exp [plusmn119895 (2120579 plusmn 120601)] 120572 = 2119891119888+ (2119899 plusmn 1) 119891

119898 119891 = ∓

1

2

119891119898

1

8

1198602 exp [plusmn119895 (2120579 plusmn 2120601)] 120572 = 2119891

119888+ (2119899 plusmn 2) 119891

119898 119891 = 0

(17b)

in higher cyclic frequency domainAccording to (17a) and (17b) for any peak on the cyclic

frequencyminusfrequency plane its frequency location in lower

cyclic frequency domain as well as its cyclic frequencylocation in higher cyclic frequency domain is dependent onthe Bessel function order 119899 Meanwhile the peak magnitudeonly involves a few Bessel functionsThus (17a) and (17b) canbe further simplified as

119878120572

119909(119891) =

1

2

1198692

119899(119861) +

1

8

1198602

[1198692

119899minus1(119861) + 119869

2

119899+1(119861)] 120572 = 0 119891 = 119891

119888+ 119899119891119898

1

4

119860 [1198692

119899(119861) + 119869

2

119899+1(119861)] exp (plusmn119895120601) 120572 = plusmn119891

119898 119891 = 119891

119888+ (119899 +

1

2

)119891119898

1

8

1198602

1198692

119899(119861) exp (plusmn1198952120601) 120572 = plusmn2119891

119898 119891 = 119891

119888+ 119899119891119898

(18a)

119878120572

119909(119891) =

1

2

1198692

119899(119861) +

1

8

1198602

[1198692

119899minus1(119861) + 119869

2

119899+1(119861)] exp (plusmn1198952120579) 120572 = 2119891

119888+ 2119899119891119898 119891 = 0

1

4

119860 [1198692

119899(119861) + 119869

2

119899+1(119861)] exp [plusmn119895 (2120579 plusmn 120601)] 120572 = 2119891

119888+ (2119899 + 1) 119891

119898 119891 = ∓

1

2

119891119898

1

8

1198602

1198692

119899(119861) exp [plusmn119895 (2120579 plusmn 2120601)] 120572 = 2119891

119888+ 2119899119891119898 119891 = ∓119891

119898

(18b)

in lower and higher cyclic frequency domains respectivelyAccording to (18a) in lower cyclic frequency domain

peaks appear at the cyclic frequencies of 0Hz modulatingfrequency119891

119898 and its twice 2119891

119898 If higher order harmonics of

modulating frequency are taken into account then peaks alsoappear at the cyclic frequencies of the modulating frequencyharmonics 119896119891

119898 Along the frequency axis these peaks center

around the carrier frequency 119891119888 with a spacing equal to the

modulating frequency 119891119898

According to (18b) in higher cyclic frequency domainpeaks appear at the frequencies of 0Hz modulating fre-quency 119891

119898 and its half 12119891

119898 If higher order harmonics

of modulating frequency are taken into account then peaksalso appear at the modulating frequency harmonics 119896119891

119898and

6 Shock and Vibration

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 50 100 150 200150

200

250

300

350

5

10

15

20

25

30

35

Figure 1 Cyclic spectrumdistribution characteristics of anAM-FMsignal

their half 1198962119891119898 Along the cyclic frequency axis these peaks

center around twice the carrier frequency 2119891119888 with a spacing

equal to twice the modulating frequency 2119891119898

Observing the peak locations in (18a) and (18b) the peaksin cyclic spectrum distribute along the lines

119891 = plusmn

120572

2

plusmn (119891119888plusmn 119899119891119898) (19)

These lines form diamonds on the cyclic frequencyminusfre-quency plane as illustrated by Figure 1 which is the cyclicspectrum of an AM-FM signal with a carrier frequency of256Hz and a modulating frequency of 16Hz

According to the above derivations we can detect gearboxand bearing fault by monitoring the presence or magnitudechange of peaks in the cyclic spectrum For example in lowercyclic frequency band focus on the points corresponding tothe cyclic frequency locations of modulating frequency 119891

119898

and its multiples 119896119891119898and to the frequency locations of the

carrier frequency plus the modulating frequency multiples119891119888+ 119899119891119898 We can further locate the gearbox and bearing fault

by matching the cyclic frequency spacing of peaks with thefault characteristic frequencies

4 Gearbox Signal Analysis

41 Specification of Experiment Figure 2 shows the experi-mental system The gearbox has one gear pair Table 1 liststhe configuration and running condition of the gearbox Twostatuses of the gearbox are simulated Under the healthystatus the gear pair is perfect While under the faulty statusone of the pinion teeth is spalled whereas the gear is perfectDuring the experiment accelerometer signals are collectedat a sampling frequency of 20000Hz and each record lastsfor 3 s (ie 60000 samples) which covers more than 51and 48 revolutions for the drive pinion and the driven

Coupling

Motor

Accelerometer

Brake

Gear

Pinion

Bearing Amplifier

Computer

Figure 2 Gearbox experimental setup

Table 1 Gearbox configuration and running condition

Drive pinion Driven gearNumber of teeth 20 21Rotating frequency (Hz) 17285 16462Meshing frequency (Hz) 3457Brake torque (Nm) mdash 200

gear respectively and is long enough to reveal the cyclosta-tionarity

42 Signal Analysis Figure 3 shows the waveform Fourierspectrum cyclic correlation 0 lag slice and cyclic spectrum(in lower cyclic frequency band) of the healthy gearbox In thefollowing analysis the peak frequencies do not correspondexactly to those in Table 1 This is reasonable because theactual speed is inevitably different from the set one in realexperiments In the cyclic correlation slice Figure 3(c) mostof the present peaks correspond to the driven gear rotatingfrequency harmonics Although a few peaks appear at thepinion and gear characteristic frequencies and their harmon-ics their magnitudes are not strong In the cyclic spectrum(the colorbar on the right shows the magnitude the samesetting in the following cyclic spectra) Figure 3(d) alongthe frequency axis the present peaks center around the gearmeshing frequency 3457Hz and its harmonics and alongthe cyclic axis they appear at the pinion and gear character-istic frequencies and their harmonics The presence of thesepeaks is reasonable since gear manufacturing errors andminor defects are inevitable they will result in these peaks

Figure 4 shows the waveform Fourier spectrum cycliccorrelation 0 lag slice and cyclic spectrum (in lower cyclicfrequency band) of the faulty gearbox In the cyclic correla-tion slice Figure 4(c) peaks appear at the pinion and gearcharacteristic frequencies and their harmonics Except thefirst two peaks nearly all the other peaks appear at the cyclicfrequency locations of the drive pinion rotating frequencyharmonics and they are higher than those in that of thehealthy gearbox In the cyclic spectrum Figure 4(d) moreand stronger peaks appear than in that of the healthy gearboxFigure 4(e) shows the zoomed-in cyclic spectrum for reveal-ing the peak distribution details Along the cyclic frequencyaxis peaks appear along lines corresponding to the pinion

Shock and Vibration 7

0 01 02 03 04 05 06 07 08 09 1minus04

minus02

0

02

04

Time (s)

Am

plitu

de (V

)

(a) Waveform

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

0005

001

0015

002

0025

003

Frequency (Hz)

Am

plitu

de (V

)

(b) Fourier spectrum

0 20 40 60 80 100 120 140 160 180 2000

05

1

15

2

Cyclic frequency (Hz)

17

323

333

483

333

643

333

806

667

118

6667

135

3333

169

3333

times10minus3

Mag

nitu

de

(c) Cyclic correlation 0 lag slice

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 50 100 150 2000

500

1000

1500

2000

2500

3000

3500

4000

05

1

15

2

25

3

35

4

(d) Cyclic spectrum

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 50 100 150 2001200

1250

1300

1350

1400

1450

1500

1550

1600

05

1

15

2

25

3

35

4

(e) Zoomed-in cyclic spectrum

Figure 3 Healthy gearbox signal

characteristic frequency and its multiples While along thefrequency axis sidebands appear around 1400Hz about fourtimes the gear meshing frequency with a spacing equal tothe pinion characteristic frequency Based on the theoreticalderivations in Sections 2 and 3 the cyclic frequency locationsof peaks in cyclic correlation slice and cyclic spectrum aswell as the sideband spacing along frequency axis in cyclicspectrum relate to the modulating frequency of AM-FMsignalsThese features imply thatmore harmonics of the drivepinion rotating frequency get involved inmodulating the gear

meshing vibration and that the drive pinion has a strongermodulation effect than the driven gear indicating the pinionfault These findings are consistent with the actual conditionof the faulty gearbox

5 Bearing Signal Analysis

51 Specification of Experiment Figure 5 shows the experi-mental setup of the test [27] A shaft is supported by four

8 Shock and Vibration

0 01 02 03 04 05 06 07 08 09 1minus04

minus02

0

02

04

Time (s)

Am

plitu

de (V

)

(a) Waveform

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

001

002

003

004

Frequency (Hz)

Am

plitu

de (V

)

(b) Fourier spectrum

0 20 40 60 80 100 120 140 160 180 2000

02

04

06

08

1

Cyclic frequency (Hz)

Mag

nitu

de

173

333

33

52 69

863

333 10

366

67

121

138

3333 15

566

67

172

6667 190

times10minus3

(c) Cyclic correlation 0 lag slice

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 50 100 150 2000

500

1000

1500

2000

2500

3000

3500

4000

051152253354455

(d) Cyclic spectrum

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 50 100 150 2001200

1250

1300

1350

1400

1450

1500

1550

1600

051152253354455

(e) Zoomed-in cyclic spectrum

Figure 4 Faulty gearbox signal

Rexnord ZA-2115 double row rolling element bearings andis driven by an AC motor through rub belts at a speed of2000 rpm A radial load of 6000 lbs is applied to the shaftand bearings by a spring mechanism The bearing test waskept running for eight days until damage occurs to the outerrace of bearing 1 In the normal case all the four bearings arehealthy While in the faulty case the outer race of bearing1 has damage as shown in Figure 6 Accelerometers aremounted on the bearing housings and the vibration signals

are collected at a sampling frequency of 20480Hz and 20480data points are recorded for both the healthy and faultycases The main parameters of the four bearings are listed inTable 2 The characteristic frequency of each element fault iscalculated and listed in Table 3

52 Signal Analysis Figure 7 shows the waveform Fourierspectrum cyclic correlation 0 lag slice and cyclic spectrum

Shock and Vibration 9

Accelerometers Radial load Thermocouples

Bearing 1 Bearing 2 Bearing 3 Bearing 4

Motor

Figure 5 Bearing experimental setup [27]

Figure 6 Outer race damage of bearing 1 [27]

Table 2 Parameters of bearing Rexnord ZA-2115

Number ofrollers Roller diameter Pitch diameter Contact angle

16 0331 [in] 2815 [in] 1517 [∘]

Table 3 Characteristic frequency of bearing Rexnord ZA-2115 [Hz]

Outer race Inner race Roller236404 296930 139917 279833

(in lower cyclic frequency band) of the normal case Inthe cyclic correlation slice Figure 7(c) a few small peaksappear but they do not correspond to any bearing componentfault characteristic frequency or its multiples Although somepeaks appear in the cyclic spectrum Figure 7(d) they do notdistribute along lines corresponding to the cyclic frequenciesof any bearing component fault characteristic frequency orits multiples These features imply that no significant mod-ulation on the resonance vibration and the bearing is healthy

Figure 8 shows the waveform Fourier spectrum cycliccorrelation 0 lag slice and cyclic spectrum (in lower cyclicfrequency band) of the faulty case In the close-up view of theFourier spectrum some sidebands appear but the frequencyspacing is not identically equal to the outer race fault charac-teristic frequency In the cyclic correlation slice Figure 8(d)more peaks appear and their magnitudes are higher than inthe normal case Moreover their cyclic frequency locationscorrespond to the outer race fault characteristic frequencyand multiples up to the 25th order In the cyclic spectrumFigure 8(e) many peaks are present More importantly theydistribute along lines associated with the outer race faultcharacteristic frequency and multiples At any cyclic fre-quency of the outer race fault characteristic frequency or itsmultiple these peaks form sidebands along the frequencyaxis with a spacing equal to the outer race fault characteristicfrequency For example in the zoomed-in cyclic spectrumFigure 8(f) the four peaks labeled by capitals A B C and Ddistribute along a line at the cyclic frequency of 710Hz (aboutthree times the outer race fault characteristic frequency) andthey have a spacing of 236Hz (approximately equal to theouter race fault characteristic frequency) along the frequencyaxis These findings imply that the outer race characteristics

10 Shock and Vibration

0 01 02 03 04 05 06 07 08 09 1minus04

minus02

0

02

04

Time (s)

Am

plitu

de (V

)

(a) Waveform

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

0004

0008

0012

0016

002

Frequency (Hz)

Am

plitu

de (V

)

(b) Fourier spectrum

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

02

04

06

08

1

Cyclic frequency (Hz)

Mag

nitu

de

times10minus3

(c) Cyclic correlation 0 lag slice

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

005

01

015

02

025

03

(d) Cyclic spectrum

Figure 7 Normal bearing signal

frequency has strong modulation effect on the resonanceindicating fault existence on the bearing outer race in accord-ance with the actual settings

6 Discussion and Conclusions

Since gearbox and rolling element bearing vibration signalsare characterized by AM-FM feature their Fourier spectrahave complex sideband structure due to the convolutionbetween the Fourier spectra ofAMandFMparts aswell as the

infinite Bessel series expansion of an FM term Althoughgearbox and bearing fault can be identified via conventionalFourier spectrum it involves complex sideband analysis andrelies on the sideband spacing to extract the fault character-istic frequency

Considering the cyclostationarity of gearbox and bearingvibration signals due to modulation and the suitability ofcyclostationary analysis to extract modulation features ofsuch signals the explicit expressions of cyclic correlation andcyclic spectrum of general AM-FM signals are derived and

Shock and Vibration 11

0 01 02 03 04 05 06 07 08 09 1minus4

minus2

0

2

4

Time (s)

Am

plitu

de (V

)

(a) Waveform

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

001

002

003

004

005

Frequency (Hz)

Am

plitu

de (V

)

(b) Fourier spectrum

2000 2500 3000 3500 4000 4500 5000 5500 60000

001

002

003

004

005

006

Frequency (Hz)

4965

3544

4033

3322

3086

4491

2612

2839

3796

4743

4325

5202

5428

5675A

mpl

itude

(V)

(c) Zoomed-in Fourier spectrum

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

002

004

006

008

01

012

Cyclic frequency (Hz)

237

473 71

094

711

83 1420

1657

1894

2130 23

6725

96 2833

3084

3306 3543

3780

4016 4253 4490

4726 49

6352

0054

5156

7359

10

Mag

nitu

de

(d) Cyclic correlation 0 lag slice

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

05

1

15

2

25

3

35

4

(e) Cyclic spectrum

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

660 680 700 720 740 7603000

3100

3200

3300

3400

3500

3600

3700

3800

3900

4000

05

1

15

2

25

3

35

4D

C

B

A

(f) Zoomed-in cyclic spectrum

Figure 8 Faulty bearing signal

validated via gearbox and bearing lab experimental signalanalysis According to the theoretical derivations the cyclicfrequency locations of peaks in cyclic correlation and cyclicspectrum directly correspond to the modulating frequency(ie the gear or bearing fault characteristic frequency) andharmonicsTherefore cyclostationary analysis offers an effec-tive approach to gearbox and bearing fault feature extraction

Conflict of Interests

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

Acknowledgments

This work is supported by National Natural Science Founda-tion ofChina (11272047 51475038) Program forNewCenturyExcellent Talents in University (NCET-12-0775) and CETIM(Centre des Etudes Techniques des Industries Mecaniques deSenlis) FranceThe authors would like to thank the reviewersfor their valuable comments and suggestions

References

[1] R Shao W Hu and J Li ldquoMulti-fault feature extraction anddiagnosis of gear transmission system using time-frequency

12 Shock and Vibration

analysis and wavelet threshold de-noising based on EMDrdquoShock and Vibration vol 20 no 4 pp 763ndash780 2013

[2] S Luo J Cheng and H Ao ldquoApplication of LCD-SVD tech-nique and CRO-SVMmethod to fault diagnosis for roller bear-ingrdquo Shock and Vibration vol 2015 Article ID 847802 8 pages2015

[3] R Yan M Shan J Cui and Y Wu ldquoMutual information-assisted wavelet function selection for enhanced rolling bearingfault diagnosisrdquo Shock and Vibration In press

[4] G Dalpiaz A Rivola and R Rubini ldquoEffectiveness and sensi-tivity of vibration processing techniques for local fault detectionin gearsrdquo Mechanical Systems and Signal Processing vol 14 no3 pp 387ndash412 2000

[5] C Capdessus M Sidahmed and J L Lacoume ldquoCyclostation-ary processes application in gear faults early diagnosisrdquoMechanical Systems and Signal Processing vol 14 no 3 pp 371ndash385 2000

[6] L Bouillaut and M Sidahmed ldquoCyclostationary approach andbilinear approach comparison applications to early diagnosisfor helicopter gearbox and classificationmethod based on hocsrdquoMechanical Systems and Signal Processing vol 15 no 5 pp 923ndash943 2001

[7] Z K Zhu Z H Feng and F R Kong ldquoCyclostationarity anal-ysis for gearbox condition monitoring approaches and effec-tivenessrdquo Mechanical Systems and Signal Processing vol 19 no3 pp 467ndash482 2005

[8] G Bi J Chen F C Zhou and J He ldquoApplication of slice spectralcorrelation density to gear defect detectionrdquo Proceedings ofthe Institution of Mechanical Engineers Part C Journal ofMechanical Engineering Science vol 220 no 9 pp 1385ndash13922006

[9] L Li and L Qu ldquoCyclic statistics in rolling bearing diagnosisrdquoJournal of Sound and Vibration vol 267 no 2 pp 253ndash2652003

[10] R B Randall J Antoni and S Chobsaard ldquoThe relationshipbetween spectral correlation and envelope analysis in the diag-nostics of bearing faults and other cyclostationary machinesignalsrdquoMechanical Systems and Signal Processing vol 15 no 5pp 945ndash962 2001

[11] J Antoni ldquoCyclic spectral analysis in practicerdquoMechanical Sys-tems and Signal Processing vol 21 no 2 pp 597ndash630 2007

[12] J Antoni ldquoCyclic spectral analysis of rolling-element bearingsignals facts and fictionsrdquo Journal of Sound and Vibration vol304 no 3-5 pp 497ndash529 2007

[13] A Raad J Antoni and M Sidahmed ldquoIndicators of cyclosta-tionarity theory and application to gear fault monitoringrdquoMechanical Systems and Signal Processing vol 22 no 3 pp 574ndash587 2008

[14] J Antoni ldquoCyclostationarity by examplesrdquo Mechanical Systemsand Signal Processing vol 23 no 4 pp 987ndash1036 2009

[15] J Urbanek T Barszcz and J Antoni ldquoTime-frequencyapproach to extraction of selected second-order cyclostation-ary vibration components for varying operational conditionsrdquoMeasurement vol 46 no 4 pp 1454ndash1463 2013

[16] DWang and C Shen ldquoAn equivalent cyclic energy indicator forbearing performance degradation assessmentrdquo Journal of Vibra-tion and Control 2014

[17] R B Randall ldquoA new method of modeling gear faultsrdquo Journalof Mechanical Design vol 104 no 2 pp 259ndash267 1982

[18] P DMcFadden ldquoDetecting fatigue cracks in gears by amplitudeand phase demodulation of the meshing vibrationrdquo Journal of

Vibration Acoustics Stress and Reliability in Design vol 108 no2 pp 165ndash170 1986

[19] P D McFadden and J D Smith ldquoModel for the vibrationproduced by a single point defect in a rolling element bearingrdquoJournal of Sound and Vibration vol 96 no 1 pp 69ndash82 1984

[20] P D McFadden and J D Smith ldquoThe vibration produced bymultiple point defects in a rolling element bearingrdquo Journal ofSound and Vibration vol 98 no 2 pp 263ndash273 1985

[21] M Liang and I S Bozchalooi ldquoAn energy operator approachto joint application of amplitude and frequency-demodulationsfor bearing fault detectionrdquo Mechanical Systems and SignalProcessing vol 24 no 5 pp 1473ndash1494 2010

[22] Z Feng M J Zuo R Hao and F Chu ldquoGear crack assessmentbased on cyclic correlation analysisrdquo in Proceedings of the8th International Conference on Reliability Maintainability andSafety (ICRMS rsquo09) pp 1071ndash1076 Chengdu China July 2009

[23] Z Feng R Hao F Chu M J Zuo andM El Badaoui ldquoApplica-tion of cyclic spectral analysis to gear damage assessmentrdquo inProceedings of the Prognostics and Health Management Confer-ence p 3058 Macau China January 2010

[24] Z Feng andM J Zuo ldquoGearbox diagnosis based on cyclic spec-tral analysisrdquo in Proceedings of the 3rd Annual IEEE Prognosticsand System Health Management Conference (PHM rsquo12) pp 1ndash5Beijing China May 2012

[25] W A Gardner Cyclostationarity in Communications and SignalProcessing IEEE Press New York NY USA 1994

[26] M Abramowitz and I A Stegun Handbook of MathematicalFunctions With Formulas Graphs and Mathematical TablesDover Publications New York NY USA 1972

[27] H Qiu J Lee J Lin and G Yu ldquoWavelet filter-based weak sig-nature detection method and its application on rolling elementbearing prognosticsrdquo Journal of Sound and Vibration vol 289no 4-5 pp 1066ndash1090 2006

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Shock and Vibration 3

For such a signal expressed as (7) the time-varying fea-ture of its autocorrelation function is mainly determined bythe AM-FM part since the autocorrelation function of awhite Gaussian noise 119877

120576(120591) = 120575(120591) (ie its Fourier transform

has peak at 0 only and therefore the noise 120576(119905) does not affectidentification of modulating frequency via cyclic correlationand cyclic spectrum analysis) In addition it is the AM and

FM effects on the carrier signal that leads to the cyclo-stationarity of bearing and gearbox vibration signals and werely on detection of the modulating frequency of such AMand FM effects to diagnose bearing and gearbox fault There-fore we neglect the noise 120576(119905) and focus on the deterministicAM-FM part only in the following analysis Then the cyclicautocorrelation function of (7) can be derived as [22]

119877120572

119909(120591) =

1

2

[ℎ119871(0 0) +

1

4

1198602

ℎ119871(minus2119891119898 0) +

1

4

1198602

ℎ119871(2119891119898 0)] 120572 = plusmn (119899 minus 119899

1015840

) 119891119898

1

4

119860 [ℎ119871(∓119891119898 plusmn120601) + ℎ

119871(plusmn119891119898 plusmn120601)] 120572 = plusmn (119899 minus 119899

1015840

plusmn 1)119891119898

1

8

1198602

ℎ119871(0 plusmn2120601) 120572 = plusmn (119899 minus 119899

1015840

plusmn 2)119891119898

(8a)

in lower cyclic frequency domain and

119877120572

119909(120591) =

1

2

[ℎ119867(0 0) +

1

4

1198602

ℎ119867(minus2119891119898 0) +

1

4

1198602

ℎ119867(2119891119898 0)] 120572 = plusmn [2119891

119888+ (119899 + 119899

1015840

) 119891119898]

1

4

119860 [ℎ119867(∓119891119898 plusmn120601) + ℎ

119867(plusmn119891119898 plusmn120601)] 120572 = plusmn [2119891

119888+ (119899 + 119899

1015840

plusmn 1)119891119898]

1

8

1198602

ℎ119867(0 plusmn2120601) 120572 = plusmn [2119891

119888+ (119899 + 119899

1015840

plusmn 2)119891119898]

(8b)

in higher cyclic frequency domain where the intermediatefunctions

ℎ119871(120578 120595) =

infin

sum

119899=minusinfin

infin

sum

1198991015840=minusinfin

119869119899(119861) 1198691198991015840 (119861)

sdot exp (plusmn119895 120587 [2119891119888+ (119899 + 119899

1015840

) 119891119898+ 120578] 120591 + (119899 minus 119899

1015840

) 120593 +120595)

(9a)

ℎ119867(120578 120595) =

infin

sum

119899=minusinfin

infin

sum

1198991015840=minusinfin

119869119899(119861) 1198691198991015840 (119861)

sdot exp (plusmn119895 120587 [(119899 minus 1198991015840) 119891119898+ 120578] 120591 + (119899 + 119899

1015840

) 120593 + 2120579 +120595)

(9b)

Set the time lag to 0 yielding the slice of cyclic autocorre-lation function

119877120572

119909(0) =

(

1

2

+

1

4

1198602

) ℎ1198710(0) 120572 = plusmn (119899 minus 119899

1015840

) 119891119898

1

2

119860ℎ1198710(plusmn120601) 120572 = plusmn (119899 minus 119899

1015840

plusmn 1)119891119898

1

8

1198602

ℎ1198710(plusmn2120601) 120572 = plusmn (119899 minus 119899

1015840

plusmn 2)119891119898

(10a)

119877120572

119909(0)

=

1

2

(1 +

1

2

1198602

) ℎ1198670(0) 120572 = plusmn [2119891

119888+ (119899 + 119899

1015840

) 119891119898]

1

2

119860ℎ1198670(plusmn120601) 120572 = plusmn [2119891

119888+ (119899 + 119899

1015840

plusmn 1)119891119898]

1

8

1198602

ℎ1198670(plusmn2120601) 120572 = plusmn [2119891

119888+ (119899 + 119899

1015840

plusmn 2)119891119898]

(10b)

in lower and higher cyclic frequency domains respectivelywhere the intermediate functions

ℎ1198710(120595) =

infin

sum

119899=minusinfin

infin

sum

1198991015840=minusinfin

119869119899(119861) 1198691198991015840 (119861)

sdot exp plusmn119895 [(119899 minus 1198991015840) 120593 +120595]

(11a)

ℎ1198670(120595) =

infin

sum

119899=minusinfin

infin

sum

1198991015840=minusinfin

119869119899(119861) 1198691198991015840 (119861)

sdot exp plusmn119895 [(119899 + 1198991015840) 120593 + 2120579 +120595]

(11b)

According to (10a) and (10b) and (11a) and (11b) theslice of cyclic autocorrelation function has two clusters ofcyclic frequencies one cluster concentrates around the cyclicfrequency of 0Hz separated by the modulating frequency119891119898 and the other cluster spreads around twice the carrier

4 Shock and Vibration

frequency 2119891119888with a spacing equal to the modulating fre-

quency 119891119898

In lower cyclic frequency domain the cyclic frequencylocations of present peaks are dependent on the difference ofthe two Bessel function orders 119899minus1198991015840 For any peak at a specificcyclic frequency 119899 minus 119899

1015840

= constant This leads to constantcomplex exponentials in (11a) Furthermore according to theaddition theorem of Bessel functions [26]

infin

sum

119899=minusinfin

infin

sum

1198991015840=minusinfin

119869119899(119911) 1198691198991015840 (119911) =

1 119899 = 1198991015840

0 119899 = 1198991015840

(12)

peaks appear in the slice of cyclic autocorrelation functionin lower cyclic frequency domain if and only if the ordersof the two Bessel functions are equal to each other that is119899 = 119899

1015840 Therefore the cyclic autocorrelation slice in lowercyclic frequency domain can be further simplified as

119877120572

119909(0) =

1

2

+

1

4

1198602

120572 = 0

1

2

119860 exp (plusmn119895120601) 120572 = plusmn119891119898

1

8

1198602 exp (plusmn1198952120601) 120572 = plusmn2119891

119898

(13)

According to (13) in lower cyclic frequency domain ofthe cyclic autocorrelation slice peaks appear at the cyclicfrequencies of 0Hz the modulating frequency 119891

119898 and its

twice 2119891119898 If higher order harmonics of the modulating

frequency are taken into account then peaks also appear atthe modulating frequency harmonics 119896119891

119898

The addition theorem of Bessel functions does not applyto the slice of cyclic autocorrelation function in higher cyclicfrequency domain The cyclic frequency locations of presentpeaks are dependent on the sum of the two Bessel functionorders 119899+1198991015840 For any specific peak 119899+1198991015840 = constantThis doesnot mean 119899 minus 1198991015840 = constant According to (10b) and (11b) inhigher cyclic frequency domain of the cyclic autocorrelationslice sidebands appear at both sides of twice the carrierfrequency 2119891

119888 with a spacing equal to the modulating freq-

uency 119891119898

According to the above derivations we can detect gearboxand bearing fault by monitoring the presence or magnitudechange of sidebands around the cyclic frequency of 0 or twicethe carrier frequency that is twice themeshing frequency forgearboxes and twice the resonance frequency for bearingsin the 0 lag slice of cyclic autocorrelation function We canfurther locate the gearbox and bearing fault by matching thesideband spacing with the fault characteristic frequencies

3 Cyclic Spectrum

31 Definition For a signal 119909(119905) the cyclic spectral densityis defined as the Fourier transform of the cyclic correlationfunction [25]

119878120572

119909(119891) = int

infin

minusinfin

119877120572

119909(120591) exp (minus1198952120587119891120591) 119889120591 (14)

where 119891 is frequency

32 Cyclic Spectrum of AM-FM Signal Without loss of gen-erality we still consider the simplified gearbox and bearingvibration signal model (5) by focusing on the fundamentalfrequency of the AM and FM terms Its cyclic spectral densitycan be derived as [23 24]

119878120572

119909(119891) =

1

2

119867 (0) 120572 = (119899 minus 1198991015840

) 119891119898 119891 = 119891

119888+

1

2

(119899 + 1198991015840

) 119891119898

1

8

1198602

119867(0) 120572 = (119899 minus 1198991015840

) 119891119898 119891 = 119891

119888+

1

2

(119899 + 1198991015840

∓ 2)119891119898

1

4

119860119867 (plusmn120601) 120572 = (119899 minus 1198991015840

plusmn 1)119891119898 119891 = 119891

119888+

1

2

(119899 + 1198991015840

∓ 1)119891119898

1

8

1198602

119867(2120601) 120572 = (119899 minus 1198991015840

plusmn 2)119891119898 119891 = 119891

119888+

1

2

(119899 + 1198991015840

) 119891119898

(15a)

119878120572

119909(119891) =

1

2

119867 (2120579) 120572 = 2119891119888+ (119899 + 119899

1015840

) 119891119898 119891 =

1

2

(119899 minus 1198991015840

) 119891119898

1

8

1198602

119867(2120579) 120572 = 2119891119888+ (119899 + 119899

1015840

) 119891119898 119891 =

1

2

(119899 minus 1198991015840

∓ 2)119891119898

1

4

119860119867 (2120579 plusmn 120601) 120572 = 2119891119888+ (119899 + 119899

1015840

plusmn 1)119891119898 119891 =

1

2

(119899 minus 1198991015840

∓ 1)119891119898

1

8

1198602

119867(2120579 plusmn 2120601) 120572 = 2119891119888+ (119899 + 119899

1015840

plusmn 2)119891119898 119891 =

1

2

(119899 minus 1198991015840

) 119891119898

(15b)

Shock and Vibration 5

in lower and higher cyclic frequency domains respectivelywhere the intermediate function

119867(120595) =

infin

sum

119899=minusinfin

infin

sum

1198991015840=minusinfin

119869119899(119861) 1198691198991015840 (119861) exp (plusmn119895120595) (16)

According to (15a) and (15b) peaks appear at specificlocations only on the cyclic frequencyminusfrequency planeTheircyclic frequency locations in lower cyclic frequency domainas well as their frequency locations in higher cyclic frequencydomain are dependent on the difference of the two Besselfunction orders 119899minus1198991015840 For any specific peaks 119899minus1198991015840 = constantThen the addition theorem of Bessel functions [26] (12)applies to the cyclic spectral density (15a) and (15b) Henceon the cyclic frequencyminusfrequency plane of the cyclic spectraldensity peaks appear if and only if the orders of the twoBessel

functions are equal to each other that is 119899 = 1198991015840 Then cyclic

spectral density can be simplified as

119878120572

119909(119891) = 119869

2

119899(119861)

sdot

1

2

120572 = 0 119891 = 119891119888+ 119899119891119898

1

8

1198602

120572 = 0 119891 = 119891119888+ (119899 ∓ 1) 119891

119898

1

4

119860 exp (plusmn119895120601) 120572 = plusmn119891119898 119891 = 119891

119888+ (119899 ∓

1

2

)119891119898

1

8

1198602 exp (plusmn1198952120601) 120572 = plusmn2119891

119898 119891 = 119891

119888+ 119899119891119898

(17a)

in lower cyclic frequency domain and

119878120572

119909(119891) = 119869

2119899(119861)

1

2

exp (plusmn1198952120579) 120572 = 2119891119888+ 2119899119891119898 119891 = 0

1

8

1198602 exp (plusmn1198952120579) 120572 = 2119891

119888+ 2119899119891119898 119891 = ∓119891

119898

1

4

119860 exp [plusmn119895 (2120579 plusmn 120601)] 120572 = 2119891119888+ (2119899 plusmn 1) 119891

119898 119891 = ∓

1

2

119891119898

1

8

1198602 exp [plusmn119895 (2120579 plusmn 2120601)] 120572 = 2119891

119888+ (2119899 plusmn 2) 119891

119898 119891 = 0

(17b)

in higher cyclic frequency domainAccording to (17a) and (17b) for any peak on the cyclic

frequencyminusfrequency plane its frequency location in lower

cyclic frequency domain as well as its cyclic frequencylocation in higher cyclic frequency domain is dependent onthe Bessel function order 119899 Meanwhile the peak magnitudeonly involves a few Bessel functionsThus (17a) and (17b) canbe further simplified as

119878120572

119909(119891) =

1

2

1198692

119899(119861) +

1

8

1198602

[1198692

119899minus1(119861) + 119869

2

119899+1(119861)] 120572 = 0 119891 = 119891

119888+ 119899119891119898

1

4

119860 [1198692

119899(119861) + 119869

2

119899+1(119861)] exp (plusmn119895120601) 120572 = plusmn119891

119898 119891 = 119891

119888+ (119899 +

1

2

)119891119898

1

8

1198602

1198692

119899(119861) exp (plusmn1198952120601) 120572 = plusmn2119891

119898 119891 = 119891

119888+ 119899119891119898

(18a)

119878120572

119909(119891) =

1

2

1198692

119899(119861) +

1

8

1198602

[1198692

119899minus1(119861) + 119869

2

119899+1(119861)] exp (plusmn1198952120579) 120572 = 2119891

119888+ 2119899119891119898 119891 = 0

1

4

119860 [1198692

119899(119861) + 119869

2

119899+1(119861)] exp [plusmn119895 (2120579 plusmn 120601)] 120572 = 2119891

119888+ (2119899 + 1) 119891

119898 119891 = ∓

1

2

119891119898

1

8

1198602

1198692

119899(119861) exp [plusmn119895 (2120579 plusmn 2120601)] 120572 = 2119891

119888+ 2119899119891119898 119891 = ∓119891

119898

(18b)

in lower and higher cyclic frequency domains respectivelyAccording to (18a) in lower cyclic frequency domain

peaks appear at the cyclic frequencies of 0Hz modulatingfrequency119891

119898 and its twice 2119891

119898 If higher order harmonics of

modulating frequency are taken into account then peaks alsoappear at the cyclic frequencies of the modulating frequencyharmonics 119896119891

119898 Along the frequency axis these peaks center

around the carrier frequency 119891119888 with a spacing equal to the

modulating frequency 119891119898

According to (18b) in higher cyclic frequency domainpeaks appear at the frequencies of 0Hz modulating fre-quency 119891

119898 and its half 12119891

119898 If higher order harmonics

of modulating frequency are taken into account then peaksalso appear at the modulating frequency harmonics 119896119891

119898and

6 Shock and Vibration

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 50 100 150 200150

200

250

300

350

5

10

15

20

25

30

35

Figure 1 Cyclic spectrumdistribution characteristics of anAM-FMsignal

their half 1198962119891119898 Along the cyclic frequency axis these peaks

center around twice the carrier frequency 2119891119888 with a spacing

equal to twice the modulating frequency 2119891119898

Observing the peak locations in (18a) and (18b) the peaksin cyclic spectrum distribute along the lines

119891 = plusmn

120572

2

plusmn (119891119888plusmn 119899119891119898) (19)

These lines form diamonds on the cyclic frequencyminusfre-quency plane as illustrated by Figure 1 which is the cyclicspectrum of an AM-FM signal with a carrier frequency of256Hz and a modulating frequency of 16Hz

According to the above derivations we can detect gearboxand bearing fault by monitoring the presence or magnitudechange of peaks in the cyclic spectrum For example in lowercyclic frequency band focus on the points corresponding tothe cyclic frequency locations of modulating frequency 119891

119898

and its multiples 119896119891119898and to the frequency locations of the

carrier frequency plus the modulating frequency multiples119891119888+ 119899119891119898 We can further locate the gearbox and bearing fault

by matching the cyclic frequency spacing of peaks with thefault characteristic frequencies

4 Gearbox Signal Analysis

41 Specification of Experiment Figure 2 shows the experi-mental system The gearbox has one gear pair Table 1 liststhe configuration and running condition of the gearbox Twostatuses of the gearbox are simulated Under the healthystatus the gear pair is perfect While under the faulty statusone of the pinion teeth is spalled whereas the gear is perfectDuring the experiment accelerometer signals are collectedat a sampling frequency of 20000Hz and each record lastsfor 3 s (ie 60000 samples) which covers more than 51and 48 revolutions for the drive pinion and the driven

Coupling

Motor

Accelerometer

Brake

Gear

Pinion

Bearing Amplifier

Computer

Figure 2 Gearbox experimental setup

Table 1 Gearbox configuration and running condition

Drive pinion Driven gearNumber of teeth 20 21Rotating frequency (Hz) 17285 16462Meshing frequency (Hz) 3457Brake torque (Nm) mdash 200

gear respectively and is long enough to reveal the cyclosta-tionarity

42 Signal Analysis Figure 3 shows the waveform Fourierspectrum cyclic correlation 0 lag slice and cyclic spectrum(in lower cyclic frequency band) of the healthy gearbox In thefollowing analysis the peak frequencies do not correspondexactly to those in Table 1 This is reasonable because theactual speed is inevitably different from the set one in realexperiments In the cyclic correlation slice Figure 3(c) mostof the present peaks correspond to the driven gear rotatingfrequency harmonics Although a few peaks appear at thepinion and gear characteristic frequencies and their harmon-ics their magnitudes are not strong In the cyclic spectrum(the colorbar on the right shows the magnitude the samesetting in the following cyclic spectra) Figure 3(d) alongthe frequency axis the present peaks center around the gearmeshing frequency 3457Hz and its harmonics and alongthe cyclic axis they appear at the pinion and gear character-istic frequencies and their harmonics The presence of thesepeaks is reasonable since gear manufacturing errors andminor defects are inevitable they will result in these peaks

Figure 4 shows the waveform Fourier spectrum cycliccorrelation 0 lag slice and cyclic spectrum (in lower cyclicfrequency band) of the faulty gearbox In the cyclic correla-tion slice Figure 4(c) peaks appear at the pinion and gearcharacteristic frequencies and their harmonics Except thefirst two peaks nearly all the other peaks appear at the cyclicfrequency locations of the drive pinion rotating frequencyharmonics and they are higher than those in that of thehealthy gearbox In the cyclic spectrum Figure 4(d) moreand stronger peaks appear than in that of the healthy gearboxFigure 4(e) shows the zoomed-in cyclic spectrum for reveal-ing the peak distribution details Along the cyclic frequencyaxis peaks appear along lines corresponding to the pinion

Shock and Vibration 7

0 01 02 03 04 05 06 07 08 09 1minus04

minus02

0

02

04

Time (s)

Am

plitu

de (V

)

(a) Waveform

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

0005

001

0015

002

0025

003

Frequency (Hz)

Am

plitu

de (V

)

(b) Fourier spectrum

0 20 40 60 80 100 120 140 160 180 2000

05

1

15

2

Cyclic frequency (Hz)

17

323

333

483

333

643

333

806

667

118

6667

135

3333

169

3333

times10minus3

Mag

nitu

de

(c) Cyclic correlation 0 lag slice

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 50 100 150 2000

500

1000

1500

2000

2500

3000

3500

4000

05

1

15

2

25

3

35

4

(d) Cyclic spectrum

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 50 100 150 2001200

1250

1300

1350

1400

1450

1500

1550

1600

05

1

15

2

25

3

35

4

(e) Zoomed-in cyclic spectrum

Figure 3 Healthy gearbox signal

characteristic frequency and its multiples While along thefrequency axis sidebands appear around 1400Hz about fourtimes the gear meshing frequency with a spacing equal tothe pinion characteristic frequency Based on the theoreticalderivations in Sections 2 and 3 the cyclic frequency locationsof peaks in cyclic correlation slice and cyclic spectrum aswell as the sideband spacing along frequency axis in cyclicspectrum relate to the modulating frequency of AM-FMsignalsThese features imply thatmore harmonics of the drivepinion rotating frequency get involved inmodulating the gear

meshing vibration and that the drive pinion has a strongermodulation effect than the driven gear indicating the pinionfault These findings are consistent with the actual conditionof the faulty gearbox

5 Bearing Signal Analysis

51 Specification of Experiment Figure 5 shows the experi-mental setup of the test [27] A shaft is supported by four

8 Shock and Vibration

0 01 02 03 04 05 06 07 08 09 1minus04

minus02

0

02

04

Time (s)

Am

plitu

de (V

)

(a) Waveform

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

001

002

003

004

Frequency (Hz)

Am

plitu

de (V

)

(b) Fourier spectrum

0 20 40 60 80 100 120 140 160 180 2000

02

04

06

08

1

Cyclic frequency (Hz)

Mag

nitu

de

173

333

33

52 69

863

333 10

366

67

121

138

3333 15

566

67

172

6667 190

times10minus3

(c) Cyclic correlation 0 lag slice

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 50 100 150 2000

500

1000

1500

2000

2500

3000

3500

4000

051152253354455

(d) Cyclic spectrum

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 50 100 150 2001200

1250

1300

1350

1400

1450

1500

1550

1600

051152253354455

(e) Zoomed-in cyclic spectrum

Figure 4 Faulty gearbox signal

Rexnord ZA-2115 double row rolling element bearings andis driven by an AC motor through rub belts at a speed of2000 rpm A radial load of 6000 lbs is applied to the shaftand bearings by a spring mechanism The bearing test waskept running for eight days until damage occurs to the outerrace of bearing 1 In the normal case all the four bearings arehealthy While in the faulty case the outer race of bearing1 has damage as shown in Figure 6 Accelerometers aremounted on the bearing housings and the vibration signals

are collected at a sampling frequency of 20480Hz and 20480data points are recorded for both the healthy and faultycases The main parameters of the four bearings are listed inTable 2 The characteristic frequency of each element fault iscalculated and listed in Table 3

52 Signal Analysis Figure 7 shows the waveform Fourierspectrum cyclic correlation 0 lag slice and cyclic spectrum

Shock and Vibration 9

Accelerometers Radial load Thermocouples

Bearing 1 Bearing 2 Bearing 3 Bearing 4

Motor

Figure 5 Bearing experimental setup [27]

Figure 6 Outer race damage of bearing 1 [27]

Table 2 Parameters of bearing Rexnord ZA-2115

Number ofrollers Roller diameter Pitch diameter Contact angle

16 0331 [in] 2815 [in] 1517 [∘]

Table 3 Characteristic frequency of bearing Rexnord ZA-2115 [Hz]

Outer race Inner race Roller236404 296930 139917 279833

(in lower cyclic frequency band) of the normal case Inthe cyclic correlation slice Figure 7(c) a few small peaksappear but they do not correspond to any bearing componentfault characteristic frequency or its multiples Although somepeaks appear in the cyclic spectrum Figure 7(d) they do notdistribute along lines corresponding to the cyclic frequenciesof any bearing component fault characteristic frequency orits multiples These features imply that no significant mod-ulation on the resonance vibration and the bearing is healthy

Figure 8 shows the waveform Fourier spectrum cycliccorrelation 0 lag slice and cyclic spectrum (in lower cyclicfrequency band) of the faulty case In the close-up view of theFourier spectrum some sidebands appear but the frequencyspacing is not identically equal to the outer race fault charac-teristic frequency In the cyclic correlation slice Figure 8(d)more peaks appear and their magnitudes are higher than inthe normal case Moreover their cyclic frequency locationscorrespond to the outer race fault characteristic frequencyand multiples up to the 25th order In the cyclic spectrumFigure 8(e) many peaks are present More importantly theydistribute along lines associated with the outer race faultcharacteristic frequency and multiples At any cyclic fre-quency of the outer race fault characteristic frequency or itsmultiple these peaks form sidebands along the frequencyaxis with a spacing equal to the outer race fault characteristicfrequency For example in the zoomed-in cyclic spectrumFigure 8(f) the four peaks labeled by capitals A B C and Ddistribute along a line at the cyclic frequency of 710Hz (aboutthree times the outer race fault characteristic frequency) andthey have a spacing of 236Hz (approximately equal to theouter race fault characteristic frequency) along the frequencyaxis These findings imply that the outer race characteristics

10 Shock and Vibration

0 01 02 03 04 05 06 07 08 09 1minus04

minus02

0

02

04

Time (s)

Am

plitu

de (V

)

(a) Waveform

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

0004

0008

0012

0016

002

Frequency (Hz)

Am

plitu

de (V

)

(b) Fourier spectrum

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

02

04

06

08

1

Cyclic frequency (Hz)

Mag

nitu

de

times10minus3

(c) Cyclic correlation 0 lag slice

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

005

01

015

02

025

03

(d) Cyclic spectrum

Figure 7 Normal bearing signal

frequency has strong modulation effect on the resonanceindicating fault existence on the bearing outer race in accord-ance with the actual settings

6 Discussion and Conclusions

Since gearbox and rolling element bearing vibration signalsare characterized by AM-FM feature their Fourier spectrahave complex sideband structure due to the convolutionbetween the Fourier spectra ofAMandFMparts aswell as the

infinite Bessel series expansion of an FM term Althoughgearbox and bearing fault can be identified via conventionalFourier spectrum it involves complex sideband analysis andrelies on the sideband spacing to extract the fault character-istic frequency

Considering the cyclostationarity of gearbox and bearingvibration signals due to modulation and the suitability ofcyclostationary analysis to extract modulation features ofsuch signals the explicit expressions of cyclic correlation andcyclic spectrum of general AM-FM signals are derived and

Shock and Vibration 11

0 01 02 03 04 05 06 07 08 09 1minus4

minus2

0

2

4

Time (s)

Am

plitu

de (V

)

(a) Waveform

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

001

002

003

004

005

Frequency (Hz)

Am

plitu

de (V

)

(b) Fourier spectrum

2000 2500 3000 3500 4000 4500 5000 5500 60000

001

002

003

004

005

006

Frequency (Hz)

4965

3544

4033

3322

3086

4491

2612

2839

3796

4743

4325

5202

5428

5675A

mpl

itude

(V)

(c) Zoomed-in Fourier spectrum

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

002

004

006

008

01

012

Cyclic frequency (Hz)

237

473 71

094

711

83 1420

1657

1894

2130 23

6725

96 2833

3084

3306 3543

3780

4016 4253 4490

4726 49

6352

0054

5156

7359

10

Mag

nitu

de

(d) Cyclic correlation 0 lag slice

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

05

1

15

2

25

3

35

4

(e) Cyclic spectrum

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

660 680 700 720 740 7603000

3100

3200

3300

3400

3500

3600

3700

3800

3900

4000

05

1

15

2

25

3

35

4D

C

B

A

(f) Zoomed-in cyclic spectrum

Figure 8 Faulty bearing signal

validated via gearbox and bearing lab experimental signalanalysis According to the theoretical derivations the cyclicfrequency locations of peaks in cyclic correlation and cyclicspectrum directly correspond to the modulating frequency(ie the gear or bearing fault characteristic frequency) andharmonicsTherefore cyclostationary analysis offers an effec-tive approach to gearbox and bearing fault feature extraction

Conflict of Interests

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

Acknowledgments

This work is supported by National Natural Science Founda-tion ofChina (11272047 51475038) Program forNewCenturyExcellent Talents in University (NCET-12-0775) and CETIM(Centre des Etudes Techniques des Industries Mecaniques deSenlis) FranceThe authors would like to thank the reviewersfor their valuable comments and suggestions

References

[1] R Shao W Hu and J Li ldquoMulti-fault feature extraction anddiagnosis of gear transmission system using time-frequency

12 Shock and Vibration

analysis and wavelet threshold de-noising based on EMDrdquoShock and Vibration vol 20 no 4 pp 763ndash780 2013

[2] S Luo J Cheng and H Ao ldquoApplication of LCD-SVD tech-nique and CRO-SVMmethod to fault diagnosis for roller bear-ingrdquo Shock and Vibration vol 2015 Article ID 847802 8 pages2015

[3] R Yan M Shan J Cui and Y Wu ldquoMutual information-assisted wavelet function selection for enhanced rolling bearingfault diagnosisrdquo Shock and Vibration In press

[4] G Dalpiaz A Rivola and R Rubini ldquoEffectiveness and sensi-tivity of vibration processing techniques for local fault detectionin gearsrdquo Mechanical Systems and Signal Processing vol 14 no3 pp 387ndash412 2000

[5] C Capdessus M Sidahmed and J L Lacoume ldquoCyclostation-ary processes application in gear faults early diagnosisrdquoMechanical Systems and Signal Processing vol 14 no 3 pp 371ndash385 2000

[6] L Bouillaut and M Sidahmed ldquoCyclostationary approach andbilinear approach comparison applications to early diagnosisfor helicopter gearbox and classificationmethod based on hocsrdquoMechanical Systems and Signal Processing vol 15 no 5 pp 923ndash943 2001

[7] Z K Zhu Z H Feng and F R Kong ldquoCyclostationarity anal-ysis for gearbox condition monitoring approaches and effec-tivenessrdquo Mechanical Systems and Signal Processing vol 19 no3 pp 467ndash482 2005

[8] G Bi J Chen F C Zhou and J He ldquoApplication of slice spectralcorrelation density to gear defect detectionrdquo Proceedings ofthe Institution of Mechanical Engineers Part C Journal ofMechanical Engineering Science vol 220 no 9 pp 1385ndash13922006

[9] L Li and L Qu ldquoCyclic statistics in rolling bearing diagnosisrdquoJournal of Sound and Vibration vol 267 no 2 pp 253ndash2652003

[10] R B Randall J Antoni and S Chobsaard ldquoThe relationshipbetween spectral correlation and envelope analysis in the diag-nostics of bearing faults and other cyclostationary machinesignalsrdquoMechanical Systems and Signal Processing vol 15 no 5pp 945ndash962 2001

[11] J Antoni ldquoCyclic spectral analysis in practicerdquoMechanical Sys-tems and Signal Processing vol 21 no 2 pp 597ndash630 2007

[12] J Antoni ldquoCyclic spectral analysis of rolling-element bearingsignals facts and fictionsrdquo Journal of Sound and Vibration vol304 no 3-5 pp 497ndash529 2007

[13] A Raad J Antoni and M Sidahmed ldquoIndicators of cyclosta-tionarity theory and application to gear fault monitoringrdquoMechanical Systems and Signal Processing vol 22 no 3 pp 574ndash587 2008

[14] J Antoni ldquoCyclostationarity by examplesrdquo Mechanical Systemsand Signal Processing vol 23 no 4 pp 987ndash1036 2009

[15] J Urbanek T Barszcz and J Antoni ldquoTime-frequencyapproach to extraction of selected second-order cyclostation-ary vibration components for varying operational conditionsrdquoMeasurement vol 46 no 4 pp 1454ndash1463 2013

[16] DWang and C Shen ldquoAn equivalent cyclic energy indicator forbearing performance degradation assessmentrdquo Journal of Vibra-tion and Control 2014

[17] R B Randall ldquoA new method of modeling gear faultsrdquo Journalof Mechanical Design vol 104 no 2 pp 259ndash267 1982

[18] P DMcFadden ldquoDetecting fatigue cracks in gears by amplitudeand phase demodulation of the meshing vibrationrdquo Journal of

Vibration Acoustics Stress and Reliability in Design vol 108 no2 pp 165ndash170 1986

[19] P D McFadden and J D Smith ldquoModel for the vibrationproduced by a single point defect in a rolling element bearingrdquoJournal of Sound and Vibration vol 96 no 1 pp 69ndash82 1984

[20] P D McFadden and J D Smith ldquoThe vibration produced bymultiple point defects in a rolling element bearingrdquo Journal ofSound and Vibration vol 98 no 2 pp 263ndash273 1985

[21] M Liang and I S Bozchalooi ldquoAn energy operator approachto joint application of amplitude and frequency-demodulationsfor bearing fault detectionrdquo Mechanical Systems and SignalProcessing vol 24 no 5 pp 1473ndash1494 2010

[22] Z Feng M J Zuo R Hao and F Chu ldquoGear crack assessmentbased on cyclic correlation analysisrdquo in Proceedings of the8th International Conference on Reliability Maintainability andSafety (ICRMS rsquo09) pp 1071ndash1076 Chengdu China July 2009

[23] Z Feng R Hao F Chu M J Zuo andM El Badaoui ldquoApplica-tion of cyclic spectral analysis to gear damage assessmentrdquo inProceedings of the Prognostics and Health Management Confer-ence p 3058 Macau China January 2010

[24] Z Feng andM J Zuo ldquoGearbox diagnosis based on cyclic spec-tral analysisrdquo in Proceedings of the 3rd Annual IEEE Prognosticsand System Health Management Conference (PHM rsquo12) pp 1ndash5Beijing China May 2012

[25] W A Gardner Cyclostationarity in Communications and SignalProcessing IEEE Press New York NY USA 1994

[26] M Abramowitz and I A Stegun Handbook of MathematicalFunctions With Formulas Graphs and Mathematical TablesDover Publications New York NY USA 1972

[27] H Qiu J Lee J Lin and G Yu ldquoWavelet filter-based weak sig-nature detection method and its application on rolling elementbearing prognosticsrdquo Journal of Sound and Vibration vol 289no 4-5 pp 1066ndash1090 2006

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RotatingMachinery

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Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

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Shock and Vibration

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Advances inOptoElectronics

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

Propagation

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Navigation and Observation

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DistributedSensor Networks

International Journal of

4 Shock and Vibration

frequency 2119891119888with a spacing equal to the modulating fre-

quency 119891119898

In lower cyclic frequency domain the cyclic frequencylocations of present peaks are dependent on the difference ofthe two Bessel function orders 119899minus1198991015840 For any peak at a specificcyclic frequency 119899 minus 119899

1015840

= constant This leads to constantcomplex exponentials in (11a) Furthermore according to theaddition theorem of Bessel functions [26]

infin

sum

119899=minusinfin

infin

sum

1198991015840=minusinfin

119869119899(119911) 1198691198991015840 (119911) =

1 119899 = 1198991015840

0 119899 = 1198991015840

(12)

peaks appear in the slice of cyclic autocorrelation functionin lower cyclic frequency domain if and only if the ordersof the two Bessel functions are equal to each other that is119899 = 119899

1015840 Therefore the cyclic autocorrelation slice in lowercyclic frequency domain can be further simplified as

119877120572

119909(0) =

1

2

+

1

4

1198602

120572 = 0

1

2

119860 exp (plusmn119895120601) 120572 = plusmn119891119898

1

8

1198602 exp (plusmn1198952120601) 120572 = plusmn2119891

119898

(13)

According to (13) in lower cyclic frequency domain ofthe cyclic autocorrelation slice peaks appear at the cyclicfrequencies of 0Hz the modulating frequency 119891

119898 and its

twice 2119891119898 If higher order harmonics of the modulating

frequency are taken into account then peaks also appear atthe modulating frequency harmonics 119896119891

119898

The addition theorem of Bessel functions does not applyto the slice of cyclic autocorrelation function in higher cyclicfrequency domain The cyclic frequency locations of presentpeaks are dependent on the sum of the two Bessel functionorders 119899+1198991015840 For any specific peak 119899+1198991015840 = constantThis doesnot mean 119899 minus 1198991015840 = constant According to (10b) and (11b) inhigher cyclic frequency domain of the cyclic autocorrelationslice sidebands appear at both sides of twice the carrierfrequency 2119891

119888 with a spacing equal to the modulating freq-

uency 119891119898

According to the above derivations we can detect gearboxand bearing fault by monitoring the presence or magnitudechange of sidebands around the cyclic frequency of 0 or twicethe carrier frequency that is twice themeshing frequency forgearboxes and twice the resonance frequency for bearingsin the 0 lag slice of cyclic autocorrelation function We canfurther locate the gearbox and bearing fault by matching thesideband spacing with the fault characteristic frequencies

3 Cyclic Spectrum

31 Definition For a signal 119909(119905) the cyclic spectral densityis defined as the Fourier transform of the cyclic correlationfunction [25]

119878120572

119909(119891) = int

infin

minusinfin

119877120572

119909(120591) exp (minus1198952120587119891120591) 119889120591 (14)

where 119891 is frequency

32 Cyclic Spectrum of AM-FM Signal Without loss of gen-erality we still consider the simplified gearbox and bearingvibration signal model (5) by focusing on the fundamentalfrequency of the AM and FM terms Its cyclic spectral densitycan be derived as [23 24]

119878120572

119909(119891) =

1

2

119867 (0) 120572 = (119899 minus 1198991015840

) 119891119898 119891 = 119891

119888+

1

2

(119899 + 1198991015840

) 119891119898

1

8

1198602

119867(0) 120572 = (119899 minus 1198991015840

) 119891119898 119891 = 119891

119888+

1

2

(119899 + 1198991015840

∓ 2)119891119898

1

4

119860119867 (plusmn120601) 120572 = (119899 minus 1198991015840

plusmn 1)119891119898 119891 = 119891

119888+

1

2

(119899 + 1198991015840

∓ 1)119891119898

1

8

1198602

119867(2120601) 120572 = (119899 minus 1198991015840

plusmn 2)119891119898 119891 = 119891

119888+

1

2

(119899 + 1198991015840

) 119891119898

(15a)

119878120572

119909(119891) =

1

2

119867 (2120579) 120572 = 2119891119888+ (119899 + 119899

1015840

) 119891119898 119891 =

1

2

(119899 minus 1198991015840

) 119891119898

1

8

1198602

119867(2120579) 120572 = 2119891119888+ (119899 + 119899

1015840

) 119891119898 119891 =

1

2

(119899 minus 1198991015840

∓ 2)119891119898

1

4

119860119867 (2120579 plusmn 120601) 120572 = 2119891119888+ (119899 + 119899

1015840

plusmn 1)119891119898 119891 =

1

2

(119899 minus 1198991015840

∓ 1)119891119898

1

8

1198602

119867(2120579 plusmn 2120601) 120572 = 2119891119888+ (119899 + 119899

1015840

plusmn 2)119891119898 119891 =

1

2

(119899 minus 1198991015840

) 119891119898

(15b)

Shock and Vibration 5

in lower and higher cyclic frequency domains respectivelywhere the intermediate function

119867(120595) =

infin

sum

119899=minusinfin

infin

sum

1198991015840=minusinfin

119869119899(119861) 1198691198991015840 (119861) exp (plusmn119895120595) (16)

According to (15a) and (15b) peaks appear at specificlocations only on the cyclic frequencyminusfrequency planeTheircyclic frequency locations in lower cyclic frequency domainas well as their frequency locations in higher cyclic frequencydomain are dependent on the difference of the two Besselfunction orders 119899minus1198991015840 For any specific peaks 119899minus1198991015840 = constantThen the addition theorem of Bessel functions [26] (12)applies to the cyclic spectral density (15a) and (15b) Henceon the cyclic frequencyminusfrequency plane of the cyclic spectraldensity peaks appear if and only if the orders of the twoBessel

functions are equal to each other that is 119899 = 1198991015840 Then cyclic

spectral density can be simplified as

119878120572

119909(119891) = 119869

2

119899(119861)

sdot

1

2

120572 = 0 119891 = 119891119888+ 119899119891119898

1

8

1198602

120572 = 0 119891 = 119891119888+ (119899 ∓ 1) 119891

119898

1

4

119860 exp (plusmn119895120601) 120572 = plusmn119891119898 119891 = 119891

119888+ (119899 ∓

1

2

)119891119898

1

8

1198602 exp (plusmn1198952120601) 120572 = plusmn2119891

119898 119891 = 119891

119888+ 119899119891119898

(17a)

in lower cyclic frequency domain and

119878120572

119909(119891) = 119869

2119899(119861)

1

2

exp (plusmn1198952120579) 120572 = 2119891119888+ 2119899119891119898 119891 = 0

1

8

1198602 exp (plusmn1198952120579) 120572 = 2119891

119888+ 2119899119891119898 119891 = ∓119891

119898

1

4

119860 exp [plusmn119895 (2120579 plusmn 120601)] 120572 = 2119891119888+ (2119899 plusmn 1) 119891

119898 119891 = ∓

1

2

119891119898

1

8

1198602 exp [plusmn119895 (2120579 plusmn 2120601)] 120572 = 2119891

119888+ (2119899 plusmn 2) 119891

119898 119891 = 0

(17b)

in higher cyclic frequency domainAccording to (17a) and (17b) for any peak on the cyclic

frequencyminusfrequency plane its frequency location in lower

cyclic frequency domain as well as its cyclic frequencylocation in higher cyclic frequency domain is dependent onthe Bessel function order 119899 Meanwhile the peak magnitudeonly involves a few Bessel functionsThus (17a) and (17b) canbe further simplified as

119878120572

119909(119891) =

1

2

1198692

119899(119861) +

1

8

1198602

[1198692

119899minus1(119861) + 119869

2

119899+1(119861)] 120572 = 0 119891 = 119891

119888+ 119899119891119898

1

4

119860 [1198692

119899(119861) + 119869

2

119899+1(119861)] exp (plusmn119895120601) 120572 = plusmn119891

119898 119891 = 119891

119888+ (119899 +

1

2

)119891119898

1

8

1198602

1198692

119899(119861) exp (plusmn1198952120601) 120572 = plusmn2119891

119898 119891 = 119891

119888+ 119899119891119898

(18a)

119878120572

119909(119891) =

1

2

1198692

119899(119861) +

1

8

1198602

[1198692

119899minus1(119861) + 119869

2

119899+1(119861)] exp (plusmn1198952120579) 120572 = 2119891

119888+ 2119899119891119898 119891 = 0

1

4

119860 [1198692

119899(119861) + 119869

2

119899+1(119861)] exp [plusmn119895 (2120579 plusmn 120601)] 120572 = 2119891

119888+ (2119899 + 1) 119891

119898 119891 = ∓

1

2

119891119898

1

8

1198602

1198692

119899(119861) exp [plusmn119895 (2120579 plusmn 2120601)] 120572 = 2119891

119888+ 2119899119891119898 119891 = ∓119891

119898

(18b)

in lower and higher cyclic frequency domains respectivelyAccording to (18a) in lower cyclic frequency domain

peaks appear at the cyclic frequencies of 0Hz modulatingfrequency119891

119898 and its twice 2119891

119898 If higher order harmonics of

modulating frequency are taken into account then peaks alsoappear at the cyclic frequencies of the modulating frequencyharmonics 119896119891

119898 Along the frequency axis these peaks center

around the carrier frequency 119891119888 with a spacing equal to the

modulating frequency 119891119898

According to (18b) in higher cyclic frequency domainpeaks appear at the frequencies of 0Hz modulating fre-quency 119891

119898 and its half 12119891

119898 If higher order harmonics

of modulating frequency are taken into account then peaksalso appear at the modulating frequency harmonics 119896119891

119898and

6 Shock and Vibration

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 50 100 150 200150

200

250

300

350

5

10

15

20

25

30

35

Figure 1 Cyclic spectrumdistribution characteristics of anAM-FMsignal

their half 1198962119891119898 Along the cyclic frequency axis these peaks

center around twice the carrier frequency 2119891119888 with a spacing

equal to twice the modulating frequency 2119891119898

Observing the peak locations in (18a) and (18b) the peaksin cyclic spectrum distribute along the lines

119891 = plusmn

120572

2

plusmn (119891119888plusmn 119899119891119898) (19)

These lines form diamonds on the cyclic frequencyminusfre-quency plane as illustrated by Figure 1 which is the cyclicspectrum of an AM-FM signal with a carrier frequency of256Hz and a modulating frequency of 16Hz

According to the above derivations we can detect gearboxand bearing fault by monitoring the presence or magnitudechange of peaks in the cyclic spectrum For example in lowercyclic frequency band focus on the points corresponding tothe cyclic frequency locations of modulating frequency 119891

119898

and its multiples 119896119891119898and to the frequency locations of the

carrier frequency plus the modulating frequency multiples119891119888+ 119899119891119898 We can further locate the gearbox and bearing fault

by matching the cyclic frequency spacing of peaks with thefault characteristic frequencies

4 Gearbox Signal Analysis

41 Specification of Experiment Figure 2 shows the experi-mental system The gearbox has one gear pair Table 1 liststhe configuration and running condition of the gearbox Twostatuses of the gearbox are simulated Under the healthystatus the gear pair is perfect While under the faulty statusone of the pinion teeth is spalled whereas the gear is perfectDuring the experiment accelerometer signals are collectedat a sampling frequency of 20000Hz and each record lastsfor 3 s (ie 60000 samples) which covers more than 51and 48 revolutions for the drive pinion and the driven

Coupling

Motor

Accelerometer

Brake

Gear

Pinion

Bearing Amplifier

Computer

Figure 2 Gearbox experimental setup

Table 1 Gearbox configuration and running condition

Drive pinion Driven gearNumber of teeth 20 21Rotating frequency (Hz) 17285 16462Meshing frequency (Hz) 3457Brake torque (Nm) mdash 200

gear respectively and is long enough to reveal the cyclosta-tionarity

42 Signal Analysis Figure 3 shows the waveform Fourierspectrum cyclic correlation 0 lag slice and cyclic spectrum(in lower cyclic frequency band) of the healthy gearbox In thefollowing analysis the peak frequencies do not correspondexactly to those in Table 1 This is reasonable because theactual speed is inevitably different from the set one in realexperiments In the cyclic correlation slice Figure 3(c) mostof the present peaks correspond to the driven gear rotatingfrequency harmonics Although a few peaks appear at thepinion and gear characteristic frequencies and their harmon-ics their magnitudes are not strong In the cyclic spectrum(the colorbar on the right shows the magnitude the samesetting in the following cyclic spectra) Figure 3(d) alongthe frequency axis the present peaks center around the gearmeshing frequency 3457Hz and its harmonics and alongthe cyclic axis they appear at the pinion and gear character-istic frequencies and their harmonics The presence of thesepeaks is reasonable since gear manufacturing errors andminor defects are inevitable they will result in these peaks

Figure 4 shows the waveform Fourier spectrum cycliccorrelation 0 lag slice and cyclic spectrum (in lower cyclicfrequency band) of the faulty gearbox In the cyclic correla-tion slice Figure 4(c) peaks appear at the pinion and gearcharacteristic frequencies and their harmonics Except thefirst two peaks nearly all the other peaks appear at the cyclicfrequency locations of the drive pinion rotating frequencyharmonics and they are higher than those in that of thehealthy gearbox In the cyclic spectrum Figure 4(d) moreand stronger peaks appear than in that of the healthy gearboxFigure 4(e) shows the zoomed-in cyclic spectrum for reveal-ing the peak distribution details Along the cyclic frequencyaxis peaks appear along lines corresponding to the pinion

Shock and Vibration 7

0 01 02 03 04 05 06 07 08 09 1minus04

minus02

0

02

04

Time (s)

Am

plitu

de (V

)

(a) Waveform

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

0005

001

0015

002

0025

003

Frequency (Hz)

Am

plitu

de (V

)

(b) Fourier spectrum

0 20 40 60 80 100 120 140 160 180 2000

05

1

15

2

Cyclic frequency (Hz)

17

323

333

483

333

643

333

806

667

118

6667

135

3333

169

3333

times10minus3

Mag

nitu

de

(c) Cyclic correlation 0 lag slice

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 50 100 150 2000

500

1000

1500

2000

2500

3000

3500

4000

05

1

15

2

25

3

35

4

(d) Cyclic spectrum

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 50 100 150 2001200

1250

1300

1350

1400

1450

1500

1550

1600

05

1

15

2

25

3

35

4

(e) Zoomed-in cyclic spectrum

Figure 3 Healthy gearbox signal

characteristic frequency and its multiples While along thefrequency axis sidebands appear around 1400Hz about fourtimes the gear meshing frequency with a spacing equal tothe pinion characteristic frequency Based on the theoreticalderivations in Sections 2 and 3 the cyclic frequency locationsof peaks in cyclic correlation slice and cyclic spectrum aswell as the sideband spacing along frequency axis in cyclicspectrum relate to the modulating frequency of AM-FMsignalsThese features imply thatmore harmonics of the drivepinion rotating frequency get involved inmodulating the gear

meshing vibration and that the drive pinion has a strongermodulation effect than the driven gear indicating the pinionfault These findings are consistent with the actual conditionof the faulty gearbox

5 Bearing Signal Analysis

51 Specification of Experiment Figure 5 shows the experi-mental setup of the test [27] A shaft is supported by four

8 Shock and Vibration

0 01 02 03 04 05 06 07 08 09 1minus04

minus02

0

02

04

Time (s)

Am

plitu

de (V

)

(a) Waveform

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

001

002

003

004

Frequency (Hz)

Am

plitu

de (V

)

(b) Fourier spectrum

0 20 40 60 80 100 120 140 160 180 2000

02

04

06

08

1

Cyclic frequency (Hz)

Mag

nitu

de

173

333

33

52 69

863

333 10

366

67

121

138

3333 15

566

67

172

6667 190

times10minus3

(c) Cyclic correlation 0 lag slice

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 50 100 150 2000

500

1000

1500

2000

2500

3000

3500

4000

051152253354455

(d) Cyclic spectrum

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 50 100 150 2001200

1250

1300

1350

1400

1450

1500

1550

1600

051152253354455

(e) Zoomed-in cyclic spectrum

Figure 4 Faulty gearbox signal

Rexnord ZA-2115 double row rolling element bearings andis driven by an AC motor through rub belts at a speed of2000 rpm A radial load of 6000 lbs is applied to the shaftand bearings by a spring mechanism The bearing test waskept running for eight days until damage occurs to the outerrace of bearing 1 In the normal case all the four bearings arehealthy While in the faulty case the outer race of bearing1 has damage as shown in Figure 6 Accelerometers aremounted on the bearing housings and the vibration signals

are collected at a sampling frequency of 20480Hz and 20480data points are recorded for both the healthy and faultycases The main parameters of the four bearings are listed inTable 2 The characteristic frequency of each element fault iscalculated and listed in Table 3

52 Signal Analysis Figure 7 shows the waveform Fourierspectrum cyclic correlation 0 lag slice and cyclic spectrum

Shock and Vibration 9

Accelerometers Radial load Thermocouples

Bearing 1 Bearing 2 Bearing 3 Bearing 4

Motor

Figure 5 Bearing experimental setup [27]

Figure 6 Outer race damage of bearing 1 [27]

Table 2 Parameters of bearing Rexnord ZA-2115

Number ofrollers Roller diameter Pitch diameter Contact angle

16 0331 [in] 2815 [in] 1517 [∘]

Table 3 Characteristic frequency of bearing Rexnord ZA-2115 [Hz]

Outer race Inner race Roller236404 296930 139917 279833

(in lower cyclic frequency band) of the normal case Inthe cyclic correlation slice Figure 7(c) a few small peaksappear but they do not correspond to any bearing componentfault characteristic frequency or its multiples Although somepeaks appear in the cyclic spectrum Figure 7(d) they do notdistribute along lines corresponding to the cyclic frequenciesof any bearing component fault characteristic frequency orits multiples These features imply that no significant mod-ulation on the resonance vibration and the bearing is healthy

Figure 8 shows the waveform Fourier spectrum cycliccorrelation 0 lag slice and cyclic spectrum (in lower cyclicfrequency band) of the faulty case In the close-up view of theFourier spectrum some sidebands appear but the frequencyspacing is not identically equal to the outer race fault charac-teristic frequency In the cyclic correlation slice Figure 8(d)more peaks appear and their magnitudes are higher than inthe normal case Moreover their cyclic frequency locationscorrespond to the outer race fault characteristic frequencyand multiples up to the 25th order In the cyclic spectrumFigure 8(e) many peaks are present More importantly theydistribute along lines associated with the outer race faultcharacteristic frequency and multiples At any cyclic fre-quency of the outer race fault characteristic frequency or itsmultiple these peaks form sidebands along the frequencyaxis with a spacing equal to the outer race fault characteristicfrequency For example in the zoomed-in cyclic spectrumFigure 8(f) the four peaks labeled by capitals A B C and Ddistribute along a line at the cyclic frequency of 710Hz (aboutthree times the outer race fault characteristic frequency) andthey have a spacing of 236Hz (approximately equal to theouter race fault characteristic frequency) along the frequencyaxis These findings imply that the outer race characteristics

10 Shock and Vibration

0 01 02 03 04 05 06 07 08 09 1minus04

minus02

0

02

04

Time (s)

Am

plitu

de (V

)

(a) Waveform

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

0004

0008

0012

0016

002

Frequency (Hz)

Am

plitu

de (V

)

(b) Fourier spectrum

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

02

04

06

08

1

Cyclic frequency (Hz)

Mag

nitu

de

times10minus3

(c) Cyclic correlation 0 lag slice

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

005

01

015

02

025

03

(d) Cyclic spectrum

Figure 7 Normal bearing signal

frequency has strong modulation effect on the resonanceindicating fault existence on the bearing outer race in accord-ance with the actual settings

6 Discussion and Conclusions

Since gearbox and rolling element bearing vibration signalsare characterized by AM-FM feature their Fourier spectrahave complex sideband structure due to the convolutionbetween the Fourier spectra ofAMandFMparts aswell as the

infinite Bessel series expansion of an FM term Althoughgearbox and bearing fault can be identified via conventionalFourier spectrum it involves complex sideband analysis andrelies on the sideband spacing to extract the fault character-istic frequency

Considering the cyclostationarity of gearbox and bearingvibration signals due to modulation and the suitability ofcyclostationary analysis to extract modulation features ofsuch signals the explicit expressions of cyclic correlation andcyclic spectrum of general AM-FM signals are derived and

Shock and Vibration 11

0 01 02 03 04 05 06 07 08 09 1minus4

minus2

0

2

4

Time (s)

Am

plitu

de (V

)

(a) Waveform

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

001

002

003

004

005

Frequency (Hz)

Am

plitu

de (V

)

(b) Fourier spectrum

2000 2500 3000 3500 4000 4500 5000 5500 60000

001

002

003

004

005

006

Frequency (Hz)

4965

3544

4033

3322

3086

4491

2612

2839

3796

4743

4325

5202

5428

5675A

mpl

itude

(V)

(c) Zoomed-in Fourier spectrum

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

002

004

006

008

01

012

Cyclic frequency (Hz)

237

473 71

094

711

83 1420

1657

1894

2130 23

6725

96 2833

3084

3306 3543

3780

4016 4253 4490

4726 49

6352

0054

5156

7359

10

Mag

nitu

de

(d) Cyclic correlation 0 lag slice

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

05

1

15

2

25

3

35

4

(e) Cyclic spectrum

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

660 680 700 720 740 7603000

3100

3200

3300

3400

3500

3600

3700

3800

3900

4000

05

1

15

2

25

3

35

4D

C

B

A

(f) Zoomed-in cyclic spectrum

Figure 8 Faulty bearing signal

validated via gearbox and bearing lab experimental signalanalysis According to the theoretical derivations the cyclicfrequency locations of peaks in cyclic correlation and cyclicspectrum directly correspond to the modulating frequency(ie the gear or bearing fault characteristic frequency) andharmonicsTherefore cyclostationary analysis offers an effec-tive approach to gearbox and bearing fault feature extraction

Conflict of Interests

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

Acknowledgments

This work is supported by National Natural Science Founda-tion ofChina (11272047 51475038) Program forNewCenturyExcellent Talents in University (NCET-12-0775) and CETIM(Centre des Etudes Techniques des Industries Mecaniques deSenlis) FranceThe authors would like to thank the reviewersfor their valuable comments and suggestions

References

[1] R Shao W Hu and J Li ldquoMulti-fault feature extraction anddiagnosis of gear transmission system using time-frequency

12 Shock and Vibration

analysis and wavelet threshold de-noising based on EMDrdquoShock and Vibration vol 20 no 4 pp 763ndash780 2013

[2] S Luo J Cheng and H Ao ldquoApplication of LCD-SVD tech-nique and CRO-SVMmethod to fault diagnosis for roller bear-ingrdquo Shock and Vibration vol 2015 Article ID 847802 8 pages2015

[3] R Yan M Shan J Cui and Y Wu ldquoMutual information-assisted wavelet function selection for enhanced rolling bearingfault diagnosisrdquo Shock and Vibration In press

[4] G Dalpiaz A Rivola and R Rubini ldquoEffectiveness and sensi-tivity of vibration processing techniques for local fault detectionin gearsrdquo Mechanical Systems and Signal Processing vol 14 no3 pp 387ndash412 2000

[5] C Capdessus M Sidahmed and J L Lacoume ldquoCyclostation-ary processes application in gear faults early diagnosisrdquoMechanical Systems and Signal Processing vol 14 no 3 pp 371ndash385 2000

[6] L Bouillaut and M Sidahmed ldquoCyclostationary approach andbilinear approach comparison applications to early diagnosisfor helicopter gearbox and classificationmethod based on hocsrdquoMechanical Systems and Signal Processing vol 15 no 5 pp 923ndash943 2001

[7] Z K Zhu Z H Feng and F R Kong ldquoCyclostationarity anal-ysis for gearbox condition monitoring approaches and effec-tivenessrdquo Mechanical Systems and Signal Processing vol 19 no3 pp 467ndash482 2005

[8] G Bi J Chen F C Zhou and J He ldquoApplication of slice spectralcorrelation density to gear defect detectionrdquo Proceedings ofthe Institution of Mechanical Engineers Part C Journal ofMechanical Engineering Science vol 220 no 9 pp 1385ndash13922006

[9] L Li and L Qu ldquoCyclic statistics in rolling bearing diagnosisrdquoJournal of Sound and Vibration vol 267 no 2 pp 253ndash2652003

[10] R B Randall J Antoni and S Chobsaard ldquoThe relationshipbetween spectral correlation and envelope analysis in the diag-nostics of bearing faults and other cyclostationary machinesignalsrdquoMechanical Systems and Signal Processing vol 15 no 5pp 945ndash962 2001

[11] J Antoni ldquoCyclic spectral analysis in practicerdquoMechanical Sys-tems and Signal Processing vol 21 no 2 pp 597ndash630 2007

[12] J Antoni ldquoCyclic spectral analysis of rolling-element bearingsignals facts and fictionsrdquo Journal of Sound and Vibration vol304 no 3-5 pp 497ndash529 2007

[13] A Raad J Antoni and M Sidahmed ldquoIndicators of cyclosta-tionarity theory and application to gear fault monitoringrdquoMechanical Systems and Signal Processing vol 22 no 3 pp 574ndash587 2008

[14] J Antoni ldquoCyclostationarity by examplesrdquo Mechanical Systemsand Signal Processing vol 23 no 4 pp 987ndash1036 2009

[15] J Urbanek T Barszcz and J Antoni ldquoTime-frequencyapproach to extraction of selected second-order cyclostation-ary vibration components for varying operational conditionsrdquoMeasurement vol 46 no 4 pp 1454ndash1463 2013

[16] DWang and C Shen ldquoAn equivalent cyclic energy indicator forbearing performance degradation assessmentrdquo Journal of Vibra-tion and Control 2014

[17] R B Randall ldquoA new method of modeling gear faultsrdquo Journalof Mechanical Design vol 104 no 2 pp 259ndash267 1982

[18] P DMcFadden ldquoDetecting fatigue cracks in gears by amplitudeand phase demodulation of the meshing vibrationrdquo Journal of

Vibration Acoustics Stress and Reliability in Design vol 108 no2 pp 165ndash170 1986

[19] P D McFadden and J D Smith ldquoModel for the vibrationproduced by a single point defect in a rolling element bearingrdquoJournal of Sound and Vibration vol 96 no 1 pp 69ndash82 1984

[20] P D McFadden and J D Smith ldquoThe vibration produced bymultiple point defects in a rolling element bearingrdquo Journal ofSound and Vibration vol 98 no 2 pp 263ndash273 1985

[21] M Liang and I S Bozchalooi ldquoAn energy operator approachto joint application of amplitude and frequency-demodulationsfor bearing fault detectionrdquo Mechanical Systems and SignalProcessing vol 24 no 5 pp 1473ndash1494 2010

[22] Z Feng M J Zuo R Hao and F Chu ldquoGear crack assessmentbased on cyclic correlation analysisrdquo in Proceedings of the8th International Conference on Reliability Maintainability andSafety (ICRMS rsquo09) pp 1071ndash1076 Chengdu China July 2009

[23] Z Feng R Hao F Chu M J Zuo andM El Badaoui ldquoApplica-tion of cyclic spectral analysis to gear damage assessmentrdquo inProceedings of the Prognostics and Health Management Confer-ence p 3058 Macau China January 2010

[24] Z Feng andM J Zuo ldquoGearbox diagnosis based on cyclic spec-tral analysisrdquo in Proceedings of the 3rd Annual IEEE Prognosticsand System Health Management Conference (PHM rsquo12) pp 1ndash5Beijing China May 2012

[25] W A Gardner Cyclostationarity in Communications and SignalProcessing IEEE Press New York NY USA 1994

[26] M Abramowitz and I A Stegun Handbook of MathematicalFunctions With Formulas Graphs and Mathematical TablesDover Publications New York NY USA 1972

[27] H Qiu J Lee J Lin and G Yu ldquoWavelet filter-based weak sig-nature detection method and its application on rolling elementbearing prognosticsrdquo Journal of Sound and Vibration vol 289no 4-5 pp 1066ndash1090 2006

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Active and Passive Electronic Components

Control Scienceand Engineering

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

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Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

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Navigation and Observation

International Journal of

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DistributedSensor Networks

International Journal of

Shock and Vibration 5

in lower and higher cyclic frequency domains respectivelywhere the intermediate function

119867(120595) =

infin

sum

119899=minusinfin

infin

sum

1198991015840=minusinfin

119869119899(119861) 1198691198991015840 (119861) exp (plusmn119895120595) (16)

According to (15a) and (15b) peaks appear at specificlocations only on the cyclic frequencyminusfrequency planeTheircyclic frequency locations in lower cyclic frequency domainas well as their frequency locations in higher cyclic frequencydomain are dependent on the difference of the two Besselfunction orders 119899minus1198991015840 For any specific peaks 119899minus1198991015840 = constantThen the addition theorem of Bessel functions [26] (12)applies to the cyclic spectral density (15a) and (15b) Henceon the cyclic frequencyminusfrequency plane of the cyclic spectraldensity peaks appear if and only if the orders of the twoBessel

functions are equal to each other that is 119899 = 1198991015840 Then cyclic

spectral density can be simplified as

119878120572

119909(119891) = 119869

2

119899(119861)

sdot

1

2

120572 = 0 119891 = 119891119888+ 119899119891119898

1

8

1198602

120572 = 0 119891 = 119891119888+ (119899 ∓ 1) 119891

119898

1

4

119860 exp (plusmn119895120601) 120572 = plusmn119891119898 119891 = 119891

119888+ (119899 ∓

1

2

)119891119898

1

8

1198602 exp (plusmn1198952120601) 120572 = plusmn2119891

119898 119891 = 119891

119888+ 119899119891119898

(17a)

in lower cyclic frequency domain and

119878120572

119909(119891) = 119869

2119899(119861)

1

2

exp (plusmn1198952120579) 120572 = 2119891119888+ 2119899119891119898 119891 = 0

1

8

1198602 exp (plusmn1198952120579) 120572 = 2119891

119888+ 2119899119891119898 119891 = ∓119891

119898

1

4

119860 exp [plusmn119895 (2120579 plusmn 120601)] 120572 = 2119891119888+ (2119899 plusmn 1) 119891

119898 119891 = ∓

1

2

119891119898

1

8

1198602 exp [plusmn119895 (2120579 plusmn 2120601)] 120572 = 2119891

119888+ (2119899 plusmn 2) 119891

119898 119891 = 0

(17b)

in higher cyclic frequency domainAccording to (17a) and (17b) for any peak on the cyclic

frequencyminusfrequency plane its frequency location in lower

cyclic frequency domain as well as its cyclic frequencylocation in higher cyclic frequency domain is dependent onthe Bessel function order 119899 Meanwhile the peak magnitudeonly involves a few Bessel functionsThus (17a) and (17b) canbe further simplified as

119878120572

119909(119891) =

1

2

1198692

119899(119861) +

1

8

1198602

[1198692

119899minus1(119861) + 119869

2

119899+1(119861)] 120572 = 0 119891 = 119891

119888+ 119899119891119898

1

4

119860 [1198692

119899(119861) + 119869

2

119899+1(119861)] exp (plusmn119895120601) 120572 = plusmn119891

119898 119891 = 119891

119888+ (119899 +

1

2

)119891119898

1

8

1198602

1198692

119899(119861) exp (plusmn1198952120601) 120572 = plusmn2119891

119898 119891 = 119891

119888+ 119899119891119898

(18a)

119878120572

119909(119891) =

1

2

1198692

119899(119861) +

1

8

1198602

[1198692

119899minus1(119861) + 119869

2

119899+1(119861)] exp (plusmn1198952120579) 120572 = 2119891

119888+ 2119899119891119898 119891 = 0

1

4

119860 [1198692

119899(119861) + 119869

2

119899+1(119861)] exp [plusmn119895 (2120579 plusmn 120601)] 120572 = 2119891

119888+ (2119899 + 1) 119891

119898 119891 = ∓

1

2

119891119898

1

8

1198602

1198692

119899(119861) exp [plusmn119895 (2120579 plusmn 2120601)] 120572 = 2119891

119888+ 2119899119891119898 119891 = ∓119891

119898

(18b)

in lower and higher cyclic frequency domains respectivelyAccording to (18a) in lower cyclic frequency domain

peaks appear at the cyclic frequencies of 0Hz modulatingfrequency119891

119898 and its twice 2119891

119898 If higher order harmonics of

modulating frequency are taken into account then peaks alsoappear at the cyclic frequencies of the modulating frequencyharmonics 119896119891

119898 Along the frequency axis these peaks center

around the carrier frequency 119891119888 with a spacing equal to the

modulating frequency 119891119898

According to (18b) in higher cyclic frequency domainpeaks appear at the frequencies of 0Hz modulating fre-quency 119891

119898 and its half 12119891

119898 If higher order harmonics

of modulating frequency are taken into account then peaksalso appear at the modulating frequency harmonics 119896119891

119898and

6 Shock and Vibration

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 50 100 150 200150

200

250

300

350

5

10

15

20

25

30

35

Figure 1 Cyclic spectrumdistribution characteristics of anAM-FMsignal

their half 1198962119891119898 Along the cyclic frequency axis these peaks

center around twice the carrier frequency 2119891119888 with a spacing

equal to twice the modulating frequency 2119891119898

Observing the peak locations in (18a) and (18b) the peaksin cyclic spectrum distribute along the lines

119891 = plusmn

120572

2

plusmn (119891119888plusmn 119899119891119898) (19)

These lines form diamonds on the cyclic frequencyminusfre-quency plane as illustrated by Figure 1 which is the cyclicspectrum of an AM-FM signal with a carrier frequency of256Hz and a modulating frequency of 16Hz

According to the above derivations we can detect gearboxand bearing fault by monitoring the presence or magnitudechange of peaks in the cyclic spectrum For example in lowercyclic frequency band focus on the points corresponding tothe cyclic frequency locations of modulating frequency 119891

119898

and its multiples 119896119891119898and to the frequency locations of the

carrier frequency plus the modulating frequency multiples119891119888+ 119899119891119898 We can further locate the gearbox and bearing fault

by matching the cyclic frequency spacing of peaks with thefault characteristic frequencies

4 Gearbox Signal Analysis

41 Specification of Experiment Figure 2 shows the experi-mental system The gearbox has one gear pair Table 1 liststhe configuration and running condition of the gearbox Twostatuses of the gearbox are simulated Under the healthystatus the gear pair is perfect While under the faulty statusone of the pinion teeth is spalled whereas the gear is perfectDuring the experiment accelerometer signals are collectedat a sampling frequency of 20000Hz and each record lastsfor 3 s (ie 60000 samples) which covers more than 51and 48 revolutions for the drive pinion and the driven

Coupling

Motor

Accelerometer

Brake

Gear

Pinion

Bearing Amplifier

Computer

Figure 2 Gearbox experimental setup

Table 1 Gearbox configuration and running condition

Drive pinion Driven gearNumber of teeth 20 21Rotating frequency (Hz) 17285 16462Meshing frequency (Hz) 3457Brake torque (Nm) mdash 200

gear respectively and is long enough to reveal the cyclosta-tionarity

42 Signal Analysis Figure 3 shows the waveform Fourierspectrum cyclic correlation 0 lag slice and cyclic spectrum(in lower cyclic frequency band) of the healthy gearbox In thefollowing analysis the peak frequencies do not correspondexactly to those in Table 1 This is reasonable because theactual speed is inevitably different from the set one in realexperiments In the cyclic correlation slice Figure 3(c) mostof the present peaks correspond to the driven gear rotatingfrequency harmonics Although a few peaks appear at thepinion and gear characteristic frequencies and their harmon-ics their magnitudes are not strong In the cyclic spectrum(the colorbar on the right shows the magnitude the samesetting in the following cyclic spectra) Figure 3(d) alongthe frequency axis the present peaks center around the gearmeshing frequency 3457Hz and its harmonics and alongthe cyclic axis they appear at the pinion and gear character-istic frequencies and their harmonics The presence of thesepeaks is reasonable since gear manufacturing errors andminor defects are inevitable they will result in these peaks

Figure 4 shows the waveform Fourier spectrum cycliccorrelation 0 lag slice and cyclic spectrum (in lower cyclicfrequency band) of the faulty gearbox In the cyclic correla-tion slice Figure 4(c) peaks appear at the pinion and gearcharacteristic frequencies and their harmonics Except thefirst two peaks nearly all the other peaks appear at the cyclicfrequency locations of the drive pinion rotating frequencyharmonics and they are higher than those in that of thehealthy gearbox In the cyclic spectrum Figure 4(d) moreand stronger peaks appear than in that of the healthy gearboxFigure 4(e) shows the zoomed-in cyclic spectrum for reveal-ing the peak distribution details Along the cyclic frequencyaxis peaks appear along lines corresponding to the pinion

Shock and Vibration 7

0 01 02 03 04 05 06 07 08 09 1minus04

minus02

0

02

04

Time (s)

Am

plitu

de (V

)

(a) Waveform

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

0005

001

0015

002

0025

003

Frequency (Hz)

Am

plitu

de (V

)

(b) Fourier spectrum

0 20 40 60 80 100 120 140 160 180 2000

05

1

15

2

Cyclic frequency (Hz)

17

323

333

483

333

643

333

806

667

118

6667

135

3333

169

3333

times10minus3

Mag

nitu

de

(c) Cyclic correlation 0 lag slice

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 50 100 150 2000

500

1000

1500

2000

2500

3000

3500

4000

05

1

15

2

25

3

35

4

(d) Cyclic spectrum

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 50 100 150 2001200

1250

1300

1350

1400

1450

1500

1550

1600

05

1

15

2

25

3

35

4

(e) Zoomed-in cyclic spectrum

Figure 3 Healthy gearbox signal

characteristic frequency and its multiples While along thefrequency axis sidebands appear around 1400Hz about fourtimes the gear meshing frequency with a spacing equal tothe pinion characteristic frequency Based on the theoreticalderivations in Sections 2 and 3 the cyclic frequency locationsof peaks in cyclic correlation slice and cyclic spectrum aswell as the sideband spacing along frequency axis in cyclicspectrum relate to the modulating frequency of AM-FMsignalsThese features imply thatmore harmonics of the drivepinion rotating frequency get involved inmodulating the gear

meshing vibration and that the drive pinion has a strongermodulation effect than the driven gear indicating the pinionfault These findings are consistent with the actual conditionof the faulty gearbox

5 Bearing Signal Analysis

51 Specification of Experiment Figure 5 shows the experi-mental setup of the test [27] A shaft is supported by four

8 Shock and Vibration

0 01 02 03 04 05 06 07 08 09 1minus04

minus02

0

02

04

Time (s)

Am

plitu

de (V

)

(a) Waveform

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

001

002

003

004

Frequency (Hz)

Am

plitu

de (V

)

(b) Fourier spectrum

0 20 40 60 80 100 120 140 160 180 2000

02

04

06

08

1

Cyclic frequency (Hz)

Mag

nitu

de

173

333

33

52 69

863

333 10

366

67

121

138

3333 15

566

67

172

6667 190

times10minus3

(c) Cyclic correlation 0 lag slice

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 50 100 150 2000

500

1000

1500

2000

2500

3000

3500

4000

051152253354455

(d) Cyclic spectrum

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 50 100 150 2001200

1250

1300

1350

1400

1450

1500

1550

1600

051152253354455

(e) Zoomed-in cyclic spectrum

Figure 4 Faulty gearbox signal

Rexnord ZA-2115 double row rolling element bearings andis driven by an AC motor through rub belts at a speed of2000 rpm A radial load of 6000 lbs is applied to the shaftand bearings by a spring mechanism The bearing test waskept running for eight days until damage occurs to the outerrace of bearing 1 In the normal case all the four bearings arehealthy While in the faulty case the outer race of bearing1 has damage as shown in Figure 6 Accelerometers aremounted on the bearing housings and the vibration signals

are collected at a sampling frequency of 20480Hz and 20480data points are recorded for both the healthy and faultycases The main parameters of the four bearings are listed inTable 2 The characteristic frequency of each element fault iscalculated and listed in Table 3

52 Signal Analysis Figure 7 shows the waveform Fourierspectrum cyclic correlation 0 lag slice and cyclic spectrum

Shock and Vibration 9

Accelerometers Radial load Thermocouples

Bearing 1 Bearing 2 Bearing 3 Bearing 4

Motor

Figure 5 Bearing experimental setup [27]

Figure 6 Outer race damage of bearing 1 [27]

Table 2 Parameters of bearing Rexnord ZA-2115

Number ofrollers Roller diameter Pitch diameter Contact angle

16 0331 [in] 2815 [in] 1517 [∘]

Table 3 Characteristic frequency of bearing Rexnord ZA-2115 [Hz]

Outer race Inner race Roller236404 296930 139917 279833

(in lower cyclic frequency band) of the normal case Inthe cyclic correlation slice Figure 7(c) a few small peaksappear but they do not correspond to any bearing componentfault characteristic frequency or its multiples Although somepeaks appear in the cyclic spectrum Figure 7(d) they do notdistribute along lines corresponding to the cyclic frequenciesof any bearing component fault characteristic frequency orits multiples These features imply that no significant mod-ulation on the resonance vibration and the bearing is healthy

Figure 8 shows the waveform Fourier spectrum cycliccorrelation 0 lag slice and cyclic spectrum (in lower cyclicfrequency band) of the faulty case In the close-up view of theFourier spectrum some sidebands appear but the frequencyspacing is not identically equal to the outer race fault charac-teristic frequency In the cyclic correlation slice Figure 8(d)more peaks appear and their magnitudes are higher than inthe normal case Moreover their cyclic frequency locationscorrespond to the outer race fault characteristic frequencyand multiples up to the 25th order In the cyclic spectrumFigure 8(e) many peaks are present More importantly theydistribute along lines associated with the outer race faultcharacteristic frequency and multiples At any cyclic fre-quency of the outer race fault characteristic frequency or itsmultiple these peaks form sidebands along the frequencyaxis with a spacing equal to the outer race fault characteristicfrequency For example in the zoomed-in cyclic spectrumFigure 8(f) the four peaks labeled by capitals A B C and Ddistribute along a line at the cyclic frequency of 710Hz (aboutthree times the outer race fault characteristic frequency) andthey have a spacing of 236Hz (approximately equal to theouter race fault characteristic frequency) along the frequencyaxis These findings imply that the outer race characteristics

10 Shock and Vibration

0 01 02 03 04 05 06 07 08 09 1minus04

minus02

0

02

04

Time (s)

Am

plitu

de (V

)

(a) Waveform

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

0004

0008

0012

0016

002

Frequency (Hz)

Am

plitu

de (V

)

(b) Fourier spectrum

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

02

04

06

08

1

Cyclic frequency (Hz)

Mag

nitu

de

times10minus3

(c) Cyclic correlation 0 lag slice

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

005

01

015

02

025

03

(d) Cyclic spectrum

Figure 7 Normal bearing signal

frequency has strong modulation effect on the resonanceindicating fault existence on the bearing outer race in accord-ance with the actual settings

6 Discussion and Conclusions

Since gearbox and rolling element bearing vibration signalsare characterized by AM-FM feature their Fourier spectrahave complex sideband structure due to the convolutionbetween the Fourier spectra ofAMandFMparts aswell as the

infinite Bessel series expansion of an FM term Althoughgearbox and bearing fault can be identified via conventionalFourier spectrum it involves complex sideband analysis andrelies on the sideband spacing to extract the fault character-istic frequency

Considering the cyclostationarity of gearbox and bearingvibration signals due to modulation and the suitability ofcyclostationary analysis to extract modulation features ofsuch signals the explicit expressions of cyclic correlation andcyclic spectrum of general AM-FM signals are derived and

Shock and Vibration 11

0 01 02 03 04 05 06 07 08 09 1minus4

minus2

0

2

4

Time (s)

Am

plitu

de (V

)

(a) Waveform

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

001

002

003

004

005

Frequency (Hz)

Am

plitu

de (V

)

(b) Fourier spectrum

2000 2500 3000 3500 4000 4500 5000 5500 60000

001

002

003

004

005

006

Frequency (Hz)

4965

3544

4033

3322

3086

4491

2612

2839

3796

4743

4325

5202

5428

5675A

mpl

itude

(V)

(c) Zoomed-in Fourier spectrum

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

002

004

006

008

01

012

Cyclic frequency (Hz)

237

473 71

094

711

83 1420

1657

1894

2130 23

6725

96 2833

3084

3306 3543

3780

4016 4253 4490

4726 49

6352

0054

5156

7359

10

Mag

nitu

de

(d) Cyclic correlation 0 lag slice

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

05

1

15

2

25

3

35

4

(e) Cyclic spectrum

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

660 680 700 720 740 7603000

3100

3200

3300

3400

3500

3600

3700

3800

3900

4000

05

1

15

2

25

3

35

4D

C

B

A

(f) Zoomed-in cyclic spectrum

Figure 8 Faulty bearing signal

validated via gearbox and bearing lab experimental signalanalysis According to the theoretical derivations the cyclicfrequency locations of peaks in cyclic correlation and cyclicspectrum directly correspond to the modulating frequency(ie the gear or bearing fault characteristic frequency) andharmonicsTherefore cyclostationary analysis offers an effec-tive approach to gearbox and bearing fault feature extraction

Conflict of Interests

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

Acknowledgments

This work is supported by National Natural Science Founda-tion ofChina (11272047 51475038) Program forNewCenturyExcellent Talents in University (NCET-12-0775) and CETIM(Centre des Etudes Techniques des Industries Mecaniques deSenlis) FranceThe authors would like to thank the reviewersfor their valuable comments and suggestions

References

[1] R Shao W Hu and J Li ldquoMulti-fault feature extraction anddiagnosis of gear transmission system using time-frequency

12 Shock and Vibration

analysis and wavelet threshold de-noising based on EMDrdquoShock and Vibration vol 20 no 4 pp 763ndash780 2013

[2] S Luo J Cheng and H Ao ldquoApplication of LCD-SVD tech-nique and CRO-SVMmethod to fault diagnosis for roller bear-ingrdquo Shock and Vibration vol 2015 Article ID 847802 8 pages2015

[3] R Yan M Shan J Cui and Y Wu ldquoMutual information-assisted wavelet function selection for enhanced rolling bearingfault diagnosisrdquo Shock and Vibration In press

[4] G Dalpiaz A Rivola and R Rubini ldquoEffectiveness and sensi-tivity of vibration processing techniques for local fault detectionin gearsrdquo Mechanical Systems and Signal Processing vol 14 no3 pp 387ndash412 2000

[5] C Capdessus M Sidahmed and J L Lacoume ldquoCyclostation-ary processes application in gear faults early diagnosisrdquoMechanical Systems and Signal Processing vol 14 no 3 pp 371ndash385 2000

[6] L Bouillaut and M Sidahmed ldquoCyclostationary approach andbilinear approach comparison applications to early diagnosisfor helicopter gearbox and classificationmethod based on hocsrdquoMechanical Systems and Signal Processing vol 15 no 5 pp 923ndash943 2001

[7] Z K Zhu Z H Feng and F R Kong ldquoCyclostationarity anal-ysis for gearbox condition monitoring approaches and effec-tivenessrdquo Mechanical Systems and Signal Processing vol 19 no3 pp 467ndash482 2005

[8] G Bi J Chen F C Zhou and J He ldquoApplication of slice spectralcorrelation density to gear defect detectionrdquo Proceedings ofthe Institution of Mechanical Engineers Part C Journal ofMechanical Engineering Science vol 220 no 9 pp 1385ndash13922006

[9] L Li and L Qu ldquoCyclic statistics in rolling bearing diagnosisrdquoJournal of Sound and Vibration vol 267 no 2 pp 253ndash2652003

[10] R B Randall J Antoni and S Chobsaard ldquoThe relationshipbetween spectral correlation and envelope analysis in the diag-nostics of bearing faults and other cyclostationary machinesignalsrdquoMechanical Systems and Signal Processing vol 15 no 5pp 945ndash962 2001

[11] J Antoni ldquoCyclic spectral analysis in practicerdquoMechanical Sys-tems and Signal Processing vol 21 no 2 pp 597ndash630 2007

[12] J Antoni ldquoCyclic spectral analysis of rolling-element bearingsignals facts and fictionsrdquo Journal of Sound and Vibration vol304 no 3-5 pp 497ndash529 2007

[13] A Raad J Antoni and M Sidahmed ldquoIndicators of cyclosta-tionarity theory and application to gear fault monitoringrdquoMechanical Systems and Signal Processing vol 22 no 3 pp 574ndash587 2008

[14] J Antoni ldquoCyclostationarity by examplesrdquo Mechanical Systemsand Signal Processing vol 23 no 4 pp 987ndash1036 2009

[15] J Urbanek T Barszcz and J Antoni ldquoTime-frequencyapproach to extraction of selected second-order cyclostation-ary vibration components for varying operational conditionsrdquoMeasurement vol 46 no 4 pp 1454ndash1463 2013

[16] DWang and C Shen ldquoAn equivalent cyclic energy indicator forbearing performance degradation assessmentrdquo Journal of Vibra-tion and Control 2014

[17] R B Randall ldquoA new method of modeling gear faultsrdquo Journalof Mechanical Design vol 104 no 2 pp 259ndash267 1982

[18] P DMcFadden ldquoDetecting fatigue cracks in gears by amplitudeand phase demodulation of the meshing vibrationrdquo Journal of

Vibration Acoustics Stress and Reliability in Design vol 108 no2 pp 165ndash170 1986

[19] P D McFadden and J D Smith ldquoModel for the vibrationproduced by a single point defect in a rolling element bearingrdquoJournal of Sound and Vibration vol 96 no 1 pp 69ndash82 1984

[20] P D McFadden and J D Smith ldquoThe vibration produced bymultiple point defects in a rolling element bearingrdquo Journal ofSound and Vibration vol 98 no 2 pp 263ndash273 1985

[21] M Liang and I S Bozchalooi ldquoAn energy operator approachto joint application of amplitude and frequency-demodulationsfor bearing fault detectionrdquo Mechanical Systems and SignalProcessing vol 24 no 5 pp 1473ndash1494 2010

[22] Z Feng M J Zuo R Hao and F Chu ldquoGear crack assessmentbased on cyclic correlation analysisrdquo in Proceedings of the8th International Conference on Reliability Maintainability andSafety (ICRMS rsquo09) pp 1071ndash1076 Chengdu China July 2009

[23] Z Feng R Hao F Chu M J Zuo andM El Badaoui ldquoApplica-tion of cyclic spectral analysis to gear damage assessmentrdquo inProceedings of the Prognostics and Health Management Confer-ence p 3058 Macau China January 2010

[24] Z Feng andM J Zuo ldquoGearbox diagnosis based on cyclic spec-tral analysisrdquo in Proceedings of the 3rd Annual IEEE Prognosticsand System Health Management Conference (PHM rsquo12) pp 1ndash5Beijing China May 2012

[25] W A Gardner Cyclostationarity in Communications and SignalProcessing IEEE Press New York NY USA 1994

[26] M Abramowitz and I A Stegun Handbook of MathematicalFunctions With Formulas Graphs and Mathematical TablesDover Publications New York NY USA 1972

[27] H Qiu J Lee J Lin and G Yu ldquoWavelet filter-based weak sig-nature detection method and its application on rolling elementbearing prognosticsrdquo Journal of Sound and Vibration vol 289no 4-5 pp 1066ndash1090 2006

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

6 Shock and Vibration

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 50 100 150 200150

200

250

300

350

5

10

15

20

25

30

35

Figure 1 Cyclic spectrumdistribution characteristics of anAM-FMsignal

their half 1198962119891119898 Along the cyclic frequency axis these peaks

center around twice the carrier frequency 2119891119888 with a spacing

equal to twice the modulating frequency 2119891119898

Observing the peak locations in (18a) and (18b) the peaksin cyclic spectrum distribute along the lines

119891 = plusmn

120572

2

plusmn (119891119888plusmn 119899119891119898) (19)

These lines form diamonds on the cyclic frequencyminusfre-quency plane as illustrated by Figure 1 which is the cyclicspectrum of an AM-FM signal with a carrier frequency of256Hz and a modulating frequency of 16Hz

According to the above derivations we can detect gearboxand bearing fault by monitoring the presence or magnitudechange of peaks in the cyclic spectrum For example in lowercyclic frequency band focus on the points corresponding tothe cyclic frequency locations of modulating frequency 119891

119898

and its multiples 119896119891119898and to the frequency locations of the

carrier frequency plus the modulating frequency multiples119891119888+ 119899119891119898 We can further locate the gearbox and bearing fault

by matching the cyclic frequency spacing of peaks with thefault characteristic frequencies

4 Gearbox Signal Analysis

41 Specification of Experiment Figure 2 shows the experi-mental system The gearbox has one gear pair Table 1 liststhe configuration and running condition of the gearbox Twostatuses of the gearbox are simulated Under the healthystatus the gear pair is perfect While under the faulty statusone of the pinion teeth is spalled whereas the gear is perfectDuring the experiment accelerometer signals are collectedat a sampling frequency of 20000Hz and each record lastsfor 3 s (ie 60000 samples) which covers more than 51and 48 revolutions for the drive pinion and the driven

Coupling

Motor

Accelerometer

Brake

Gear

Pinion

Bearing Amplifier

Computer

Figure 2 Gearbox experimental setup

Table 1 Gearbox configuration and running condition

Drive pinion Driven gearNumber of teeth 20 21Rotating frequency (Hz) 17285 16462Meshing frequency (Hz) 3457Brake torque (Nm) mdash 200

gear respectively and is long enough to reveal the cyclosta-tionarity

42 Signal Analysis Figure 3 shows the waveform Fourierspectrum cyclic correlation 0 lag slice and cyclic spectrum(in lower cyclic frequency band) of the healthy gearbox In thefollowing analysis the peak frequencies do not correspondexactly to those in Table 1 This is reasonable because theactual speed is inevitably different from the set one in realexperiments In the cyclic correlation slice Figure 3(c) mostof the present peaks correspond to the driven gear rotatingfrequency harmonics Although a few peaks appear at thepinion and gear characteristic frequencies and their harmon-ics their magnitudes are not strong In the cyclic spectrum(the colorbar on the right shows the magnitude the samesetting in the following cyclic spectra) Figure 3(d) alongthe frequency axis the present peaks center around the gearmeshing frequency 3457Hz and its harmonics and alongthe cyclic axis they appear at the pinion and gear character-istic frequencies and their harmonics The presence of thesepeaks is reasonable since gear manufacturing errors andminor defects are inevitable they will result in these peaks

Figure 4 shows the waveform Fourier spectrum cycliccorrelation 0 lag slice and cyclic spectrum (in lower cyclicfrequency band) of the faulty gearbox In the cyclic correla-tion slice Figure 4(c) peaks appear at the pinion and gearcharacteristic frequencies and their harmonics Except thefirst two peaks nearly all the other peaks appear at the cyclicfrequency locations of the drive pinion rotating frequencyharmonics and they are higher than those in that of thehealthy gearbox In the cyclic spectrum Figure 4(d) moreand stronger peaks appear than in that of the healthy gearboxFigure 4(e) shows the zoomed-in cyclic spectrum for reveal-ing the peak distribution details Along the cyclic frequencyaxis peaks appear along lines corresponding to the pinion

Shock and Vibration 7

0 01 02 03 04 05 06 07 08 09 1minus04

minus02

0

02

04

Time (s)

Am

plitu

de (V

)

(a) Waveform

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

0005

001

0015

002

0025

003

Frequency (Hz)

Am

plitu

de (V

)

(b) Fourier spectrum

0 20 40 60 80 100 120 140 160 180 2000

05

1

15

2

Cyclic frequency (Hz)

17

323

333

483

333

643

333

806

667

118

6667

135

3333

169

3333

times10minus3

Mag

nitu

de

(c) Cyclic correlation 0 lag slice

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 50 100 150 2000

500

1000

1500

2000

2500

3000

3500

4000

05

1

15

2

25

3

35

4

(d) Cyclic spectrum

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 50 100 150 2001200

1250

1300

1350

1400

1450

1500

1550

1600

05

1

15

2

25

3

35

4

(e) Zoomed-in cyclic spectrum

Figure 3 Healthy gearbox signal

characteristic frequency and its multiples While along thefrequency axis sidebands appear around 1400Hz about fourtimes the gear meshing frequency with a spacing equal tothe pinion characteristic frequency Based on the theoreticalderivations in Sections 2 and 3 the cyclic frequency locationsof peaks in cyclic correlation slice and cyclic spectrum aswell as the sideband spacing along frequency axis in cyclicspectrum relate to the modulating frequency of AM-FMsignalsThese features imply thatmore harmonics of the drivepinion rotating frequency get involved inmodulating the gear

meshing vibration and that the drive pinion has a strongermodulation effect than the driven gear indicating the pinionfault These findings are consistent with the actual conditionof the faulty gearbox

5 Bearing Signal Analysis

51 Specification of Experiment Figure 5 shows the experi-mental setup of the test [27] A shaft is supported by four

8 Shock and Vibration

0 01 02 03 04 05 06 07 08 09 1minus04

minus02

0

02

04

Time (s)

Am

plitu

de (V

)

(a) Waveform

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

001

002

003

004

Frequency (Hz)

Am

plitu

de (V

)

(b) Fourier spectrum

0 20 40 60 80 100 120 140 160 180 2000

02

04

06

08

1

Cyclic frequency (Hz)

Mag

nitu

de

173

333

33

52 69

863

333 10

366

67

121

138

3333 15

566

67

172

6667 190

times10minus3

(c) Cyclic correlation 0 lag slice

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 50 100 150 2000

500

1000

1500

2000

2500

3000

3500

4000

051152253354455

(d) Cyclic spectrum

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 50 100 150 2001200

1250

1300

1350

1400

1450

1500

1550

1600

051152253354455

(e) Zoomed-in cyclic spectrum

Figure 4 Faulty gearbox signal

Rexnord ZA-2115 double row rolling element bearings andis driven by an AC motor through rub belts at a speed of2000 rpm A radial load of 6000 lbs is applied to the shaftand bearings by a spring mechanism The bearing test waskept running for eight days until damage occurs to the outerrace of bearing 1 In the normal case all the four bearings arehealthy While in the faulty case the outer race of bearing1 has damage as shown in Figure 6 Accelerometers aremounted on the bearing housings and the vibration signals

are collected at a sampling frequency of 20480Hz and 20480data points are recorded for both the healthy and faultycases The main parameters of the four bearings are listed inTable 2 The characteristic frequency of each element fault iscalculated and listed in Table 3

52 Signal Analysis Figure 7 shows the waveform Fourierspectrum cyclic correlation 0 lag slice and cyclic spectrum

Shock and Vibration 9

Accelerometers Radial load Thermocouples

Bearing 1 Bearing 2 Bearing 3 Bearing 4

Motor

Figure 5 Bearing experimental setup [27]

Figure 6 Outer race damage of bearing 1 [27]

Table 2 Parameters of bearing Rexnord ZA-2115

Number ofrollers Roller diameter Pitch diameter Contact angle

16 0331 [in] 2815 [in] 1517 [∘]

Table 3 Characteristic frequency of bearing Rexnord ZA-2115 [Hz]

Outer race Inner race Roller236404 296930 139917 279833

(in lower cyclic frequency band) of the normal case Inthe cyclic correlation slice Figure 7(c) a few small peaksappear but they do not correspond to any bearing componentfault characteristic frequency or its multiples Although somepeaks appear in the cyclic spectrum Figure 7(d) they do notdistribute along lines corresponding to the cyclic frequenciesof any bearing component fault characteristic frequency orits multiples These features imply that no significant mod-ulation on the resonance vibration and the bearing is healthy

Figure 8 shows the waveform Fourier spectrum cycliccorrelation 0 lag slice and cyclic spectrum (in lower cyclicfrequency band) of the faulty case In the close-up view of theFourier spectrum some sidebands appear but the frequencyspacing is not identically equal to the outer race fault charac-teristic frequency In the cyclic correlation slice Figure 8(d)more peaks appear and their magnitudes are higher than inthe normal case Moreover their cyclic frequency locationscorrespond to the outer race fault characteristic frequencyand multiples up to the 25th order In the cyclic spectrumFigure 8(e) many peaks are present More importantly theydistribute along lines associated with the outer race faultcharacteristic frequency and multiples At any cyclic fre-quency of the outer race fault characteristic frequency or itsmultiple these peaks form sidebands along the frequencyaxis with a spacing equal to the outer race fault characteristicfrequency For example in the zoomed-in cyclic spectrumFigure 8(f) the four peaks labeled by capitals A B C and Ddistribute along a line at the cyclic frequency of 710Hz (aboutthree times the outer race fault characteristic frequency) andthey have a spacing of 236Hz (approximately equal to theouter race fault characteristic frequency) along the frequencyaxis These findings imply that the outer race characteristics

10 Shock and Vibration

0 01 02 03 04 05 06 07 08 09 1minus04

minus02

0

02

04

Time (s)

Am

plitu

de (V

)

(a) Waveform

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

0004

0008

0012

0016

002

Frequency (Hz)

Am

plitu

de (V

)

(b) Fourier spectrum

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

02

04

06

08

1

Cyclic frequency (Hz)

Mag

nitu

de

times10minus3

(c) Cyclic correlation 0 lag slice

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

005

01

015

02

025

03

(d) Cyclic spectrum

Figure 7 Normal bearing signal

frequency has strong modulation effect on the resonanceindicating fault existence on the bearing outer race in accord-ance with the actual settings

6 Discussion and Conclusions

Since gearbox and rolling element bearing vibration signalsare characterized by AM-FM feature their Fourier spectrahave complex sideband structure due to the convolutionbetween the Fourier spectra ofAMandFMparts aswell as the

infinite Bessel series expansion of an FM term Althoughgearbox and bearing fault can be identified via conventionalFourier spectrum it involves complex sideband analysis andrelies on the sideband spacing to extract the fault character-istic frequency

Considering the cyclostationarity of gearbox and bearingvibration signals due to modulation and the suitability ofcyclostationary analysis to extract modulation features ofsuch signals the explicit expressions of cyclic correlation andcyclic spectrum of general AM-FM signals are derived and

Shock and Vibration 11

0 01 02 03 04 05 06 07 08 09 1minus4

minus2

0

2

4

Time (s)

Am

plitu

de (V

)

(a) Waveform

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

001

002

003

004

005

Frequency (Hz)

Am

plitu

de (V

)

(b) Fourier spectrum

2000 2500 3000 3500 4000 4500 5000 5500 60000

001

002

003

004

005

006

Frequency (Hz)

4965

3544

4033

3322

3086

4491

2612

2839

3796

4743

4325

5202

5428

5675A

mpl

itude

(V)

(c) Zoomed-in Fourier spectrum

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

002

004

006

008

01

012

Cyclic frequency (Hz)

237

473 71

094

711

83 1420

1657

1894

2130 23

6725

96 2833

3084

3306 3543

3780

4016 4253 4490

4726 49

6352

0054

5156

7359

10

Mag

nitu

de

(d) Cyclic correlation 0 lag slice

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

05

1

15

2

25

3

35

4

(e) Cyclic spectrum

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

660 680 700 720 740 7603000

3100

3200

3300

3400

3500

3600

3700

3800

3900

4000

05

1

15

2

25

3

35

4D

C

B

A

(f) Zoomed-in cyclic spectrum

Figure 8 Faulty bearing signal

validated via gearbox and bearing lab experimental signalanalysis According to the theoretical derivations the cyclicfrequency locations of peaks in cyclic correlation and cyclicspectrum directly correspond to the modulating frequency(ie the gear or bearing fault characteristic frequency) andharmonicsTherefore cyclostationary analysis offers an effec-tive approach to gearbox and bearing fault feature extraction

Conflict of Interests

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

Acknowledgments

This work is supported by National Natural Science Founda-tion ofChina (11272047 51475038) Program forNewCenturyExcellent Talents in University (NCET-12-0775) and CETIM(Centre des Etudes Techniques des Industries Mecaniques deSenlis) FranceThe authors would like to thank the reviewersfor their valuable comments and suggestions

References

[1] R Shao W Hu and J Li ldquoMulti-fault feature extraction anddiagnosis of gear transmission system using time-frequency

12 Shock and Vibration

analysis and wavelet threshold de-noising based on EMDrdquoShock and Vibration vol 20 no 4 pp 763ndash780 2013

[2] S Luo J Cheng and H Ao ldquoApplication of LCD-SVD tech-nique and CRO-SVMmethod to fault diagnosis for roller bear-ingrdquo Shock and Vibration vol 2015 Article ID 847802 8 pages2015

[3] R Yan M Shan J Cui and Y Wu ldquoMutual information-assisted wavelet function selection for enhanced rolling bearingfault diagnosisrdquo Shock and Vibration In press

[4] G Dalpiaz A Rivola and R Rubini ldquoEffectiveness and sensi-tivity of vibration processing techniques for local fault detectionin gearsrdquo Mechanical Systems and Signal Processing vol 14 no3 pp 387ndash412 2000

[5] C Capdessus M Sidahmed and J L Lacoume ldquoCyclostation-ary processes application in gear faults early diagnosisrdquoMechanical Systems and Signal Processing vol 14 no 3 pp 371ndash385 2000

[6] L Bouillaut and M Sidahmed ldquoCyclostationary approach andbilinear approach comparison applications to early diagnosisfor helicopter gearbox and classificationmethod based on hocsrdquoMechanical Systems and Signal Processing vol 15 no 5 pp 923ndash943 2001

[7] Z K Zhu Z H Feng and F R Kong ldquoCyclostationarity anal-ysis for gearbox condition monitoring approaches and effec-tivenessrdquo Mechanical Systems and Signal Processing vol 19 no3 pp 467ndash482 2005

[8] G Bi J Chen F C Zhou and J He ldquoApplication of slice spectralcorrelation density to gear defect detectionrdquo Proceedings ofthe Institution of Mechanical Engineers Part C Journal ofMechanical Engineering Science vol 220 no 9 pp 1385ndash13922006

[9] L Li and L Qu ldquoCyclic statistics in rolling bearing diagnosisrdquoJournal of Sound and Vibration vol 267 no 2 pp 253ndash2652003

[10] R B Randall J Antoni and S Chobsaard ldquoThe relationshipbetween spectral correlation and envelope analysis in the diag-nostics of bearing faults and other cyclostationary machinesignalsrdquoMechanical Systems and Signal Processing vol 15 no 5pp 945ndash962 2001

[11] J Antoni ldquoCyclic spectral analysis in practicerdquoMechanical Sys-tems and Signal Processing vol 21 no 2 pp 597ndash630 2007

[12] J Antoni ldquoCyclic spectral analysis of rolling-element bearingsignals facts and fictionsrdquo Journal of Sound and Vibration vol304 no 3-5 pp 497ndash529 2007

[13] A Raad J Antoni and M Sidahmed ldquoIndicators of cyclosta-tionarity theory and application to gear fault monitoringrdquoMechanical Systems and Signal Processing vol 22 no 3 pp 574ndash587 2008

[14] J Antoni ldquoCyclostationarity by examplesrdquo Mechanical Systemsand Signal Processing vol 23 no 4 pp 987ndash1036 2009

[15] J Urbanek T Barszcz and J Antoni ldquoTime-frequencyapproach to extraction of selected second-order cyclostation-ary vibration components for varying operational conditionsrdquoMeasurement vol 46 no 4 pp 1454ndash1463 2013

[16] DWang and C Shen ldquoAn equivalent cyclic energy indicator forbearing performance degradation assessmentrdquo Journal of Vibra-tion and Control 2014

[17] R B Randall ldquoA new method of modeling gear faultsrdquo Journalof Mechanical Design vol 104 no 2 pp 259ndash267 1982

[18] P DMcFadden ldquoDetecting fatigue cracks in gears by amplitudeand phase demodulation of the meshing vibrationrdquo Journal of

Vibration Acoustics Stress and Reliability in Design vol 108 no2 pp 165ndash170 1986

[19] P D McFadden and J D Smith ldquoModel for the vibrationproduced by a single point defect in a rolling element bearingrdquoJournal of Sound and Vibration vol 96 no 1 pp 69ndash82 1984

[20] P D McFadden and J D Smith ldquoThe vibration produced bymultiple point defects in a rolling element bearingrdquo Journal ofSound and Vibration vol 98 no 2 pp 263ndash273 1985

[21] M Liang and I S Bozchalooi ldquoAn energy operator approachto joint application of amplitude and frequency-demodulationsfor bearing fault detectionrdquo Mechanical Systems and SignalProcessing vol 24 no 5 pp 1473ndash1494 2010

[22] Z Feng M J Zuo R Hao and F Chu ldquoGear crack assessmentbased on cyclic correlation analysisrdquo in Proceedings of the8th International Conference on Reliability Maintainability andSafety (ICRMS rsquo09) pp 1071ndash1076 Chengdu China July 2009

[23] Z Feng R Hao F Chu M J Zuo andM El Badaoui ldquoApplica-tion of cyclic spectral analysis to gear damage assessmentrdquo inProceedings of the Prognostics and Health Management Confer-ence p 3058 Macau China January 2010

[24] Z Feng andM J Zuo ldquoGearbox diagnosis based on cyclic spec-tral analysisrdquo in Proceedings of the 3rd Annual IEEE Prognosticsand System Health Management Conference (PHM rsquo12) pp 1ndash5Beijing China May 2012

[25] W A Gardner Cyclostationarity in Communications and SignalProcessing IEEE Press New York NY USA 1994

[26] M Abramowitz and I A Stegun Handbook of MathematicalFunctions With Formulas Graphs and Mathematical TablesDover Publications New York NY USA 1972

[27] H Qiu J Lee J Lin and G Yu ldquoWavelet filter-based weak sig-nature detection method and its application on rolling elementbearing prognosticsrdquo Journal of Sound and Vibration vol 289no 4-5 pp 1066ndash1090 2006

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 7

0 01 02 03 04 05 06 07 08 09 1minus04

minus02

0

02

04

Time (s)

Am

plitu

de (V

)

(a) Waveform

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

0005

001

0015

002

0025

003

Frequency (Hz)

Am

plitu

de (V

)

(b) Fourier spectrum

0 20 40 60 80 100 120 140 160 180 2000

05

1

15

2

Cyclic frequency (Hz)

17

323

333

483

333

643

333

806

667

118

6667

135

3333

169

3333

times10minus3

Mag

nitu

de

(c) Cyclic correlation 0 lag slice

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 50 100 150 2000

500

1000

1500

2000

2500

3000

3500

4000

05

1

15

2

25

3

35

4

(d) Cyclic spectrum

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 50 100 150 2001200

1250

1300

1350

1400

1450

1500

1550

1600

05

1

15

2

25

3

35

4

(e) Zoomed-in cyclic spectrum

Figure 3 Healthy gearbox signal

characteristic frequency and its multiples While along thefrequency axis sidebands appear around 1400Hz about fourtimes the gear meshing frequency with a spacing equal tothe pinion characteristic frequency Based on the theoreticalderivations in Sections 2 and 3 the cyclic frequency locationsof peaks in cyclic correlation slice and cyclic spectrum aswell as the sideband spacing along frequency axis in cyclicspectrum relate to the modulating frequency of AM-FMsignalsThese features imply thatmore harmonics of the drivepinion rotating frequency get involved inmodulating the gear

meshing vibration and that the drive pinion has a strongermodulation effect than the driven gear indicating the pinionfault These findings are consistent with the actual conditionof the faulty gearbox

5 Bearing Signal Analysis

51 Specification of Experiment Figure 5 shows the experi-mental setup of the test [27] A shaft is supported by four

8 Shock and Vibration

0 01 02 03 04 05 06 07 08 09 1minus04

minus02

0

02

04

Time (s)

Am

plitu

de (V

)

(a) Waveform

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

001

002

003

004

Frequency (Hz)

Am

plitu

de (V

)

(b) Fourier spectrum

0 20 40 60 80 100 120 140 160 180 2000

02

04

06

08

1

Cyclic frequency (Hz)

Mag

nitu

de

173

333

33

52 69

863

333 10

366

67

121

138

3333 15

566

67

172

6667 190

times10minus3

(c) Cyclic correlation 0 lag slice

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 50 100 150 2000

500

1000

1500

2000

2500

3000

3500

4000

051152253354455

(d) Cyclic spectrum

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 50 100 150 2001200

1250

1300

1350

1400

1450

1500

1550

1600

051152253354455

(e) Zoomed-in cyclic spectrum

Figure 4 Faulty gearbox signal

Rexnord ZA-2115 double row rolling element bearings andis driven by an AC motor through rub belts at a speed of2000 rpm A radial load of 6000 lbs is applied to the shaftand bearings by a spring mechanism The bearing test waskept running for eight days until damage occurs to the outerrace of bearing 1 In the normal case all the four bearings arehealthy While in the faulty case the outer race of bearing1 has damage as shown in Figure 6 Accelerometers aremounted on the bearing housings and the vibration signals

are collected at a sampling frequency of 20480Hz and 20480data points are recorded for both the healthy and faultycases The main parameters of the four bearings are listed inTable 2 The characteristic frequency of each element fault iscalculated and listed in Table 3

52 Signal Analysis Figure 7 shows the waveform Fourierspectrum cyclic correlation 0 lag slice and cyclic spectrum

Shock and Vibration 9

Accelerometers Radial load Thermocouples

Bearing 1 Bearing 2 Bearing 3 Bearing 4

Motor

Figure 5 Bearing experimental setup [27]

Figure 6 Outer race damage of bearing 1 [27]

Table 2 Parameters of bearing Rexnord ZA-2115

Number ofrollers Roller diameter Pitch diameter Contact angle

16 0331 [in] 2815 [in] 1517 [∘]

Table 3 Characteristic frequency of bearing Rexnord ZA-2115 [Hz]

Outer race Inner race Roller236404 296930 139917 279833

(in lower cyclic frequency band) of the normal case Inthe cyclic correlation slice Figure 7(c) a few small peaksappear but they do not correspond to any bearing componentfault characteristic frequency or its multiples Although somepeaks appear in the cyclic spectrum Figure 7(d) they do notdistribute along lines corresponding to the cyclic frequenciesof any bearing component fault characteristic frequency orits multiples These features imply that no significant mod-ulation on the resonance vibration and the bearing is healthy

Figure 8 shows the waveform Fourier spectrum cycliccorrelation 0 lag slice and cyclic spectrum (in lower cyclicfrequency band) of the faulty case In the close-up view of theFourier spectrum some sidebands appear but the frequencyspacing is not identically equal to the outer race fault charac-teristic frequency In the cyclic correlation slice Figure 8(d)more peaks appear and their magnitudes are higher than inthe normal case Moreover their cyclic frequency locationscorrespond to the outer race fault characteristic frequencyand multiples up to the 25th order In the cyclic spectrumFigure 8(e) many peaks are present More importantly theydistribute along lines associated with the outer race faultcharacteristic frequency and multiples At any cyclic fre-quency of the outer race fault characteristic frequency or itsmultiple these peaks form sidebands along the frequencyaxis with a spacing equal to the outer race fault characteristicfrequency For example in the zoomed-in cyclic spectrumFigure 8(f) the four peaks labeled by capitals A B C and Ddistribute along a line at the cyclic frequency of 710Hz (aboutthree times the outer race fault characteristic frequency) andthey have a spacing of 236Hz (approximately equal to theouter race fault characteristic frequency) along the frequencyaxis These findings imply that the outer race characteristics

10 Shock and Vibration

0 01 02 03 04 05 06 07 08 09 1minus04

minus02

0

02

04

Time (s)

Am

plitu

de (V

)

(a) Waveform

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

0004

0008

0012

0016

002

Frequency (Hz)

Am

plitu

de (V

)

(b) Fourier spectrum

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

02

04

06

08

1

Cyclic frequency (Hz)

Mag

nitu

de

times10minus3

(c) Cyclic correlation 0 lag slice

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

005

01

015

02

025

03

(d) Cyclic spectrum

Figure 7 Normal bearing signal

frequency has strong modulation effect on the resonanceindicating fault existence on the bearing outer race in accord-ance with the actual settings

6 Discussion and Conclusions

Since gearbox and rolling element bearing vibration signalsare characterized by AM-FM feature their Fourier spectrahave complex sideband structure due to the convolutionbetween the Fourier spectra ofAMandFMparts aswell as the

infinite Bessel series expansion of an FM term Althoughgearbox and bearing fault can be identified via conventionalFourier spectrum it involves complex sideband analysis andrelies on the sideband spacing to extract the fault character-istic frequency

Considering the cyclostationarity of gearbox and bearingvibration signals due to modulation and the suitability ofcyclostationary analysis to extract modulation features ofsuch signals the explicit expressions of cyclic correlation andcyclic spectrum of general AM-FM signals are derived and

Shock and Vibration 11

0 01 02 03 04 05 06 07 08 09 1minus4

minus2

0

2

4

Time (s)

Am

plitu

de (V

)

(a) Waveform

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

001

002

003

004

005

Frequency (Hz)

Am

plitu

de (V

)

(b) Fourier spectrum

2000 2500 3000 3500 4000 4500 5000 5500 60000

001

002

003

004

005

006

Frequency (Hz)

4965

3544

4033

3322

3086

4491

2612

2839

3796

4743

4325

5202

5428

5675A

mpl

itude

(V)

(c) Zoomed-in Fourier spectrum

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

002

004

006

008

01

012

Cyclic frequency (Hz)

237

473 71

094

711

83 1420

1657

1894

2130 23

6725

96 2833

3084

3306 3543

3780

4016 4253 4490

4726 49

6352

0054

5156

7359

10

Mag

nitu

de

(d) Cyclic correlation 0 lag slice

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

05

1

15

2

25

3

35

4

(e) Cyclic spectrum

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

660 680 700 720 740 7603000

3100

3200

3300

3400

3500

3600

3700

3800

3900

4000

05

1

15

2

25

3

35

4D

C

B

A

(f) Zoomed-in cyclic spectrum

Figure 8 Faulty bearing signal

validated via gearbox and bearing lab experimental signalanalysis According to the theoretical derivations the cyclicfrequency locations of peaks in cyclic correlation and cyclicspectrum directly correspond to the modulating frequency(ie the gear or bearing fault characteristic frequency) andharmonicsTherefore cyclostationary analysis offers an effec-tive approach to gearbox and bearing fault feature extraction

Conflict of Interests

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

Acknowledgments

This work is supported by National Natural Science Founda-tion ofChina (11272047 51475038) Program forNewCenturyExcellent Talents in University (NCET-12-0775) and CETIM(Centre des Etudes Techniques des Industries Mecaniques deSenlis) FranceThe authors would like to thank the reviewersfor their valuable comments and suggestions

References

[1] R Shao W Hu and J Li ldquoMulti-fault feature extraction anddiagnosis of gear transmission system using time-frequency

12 Shock and Vibration

analysis and wavelet threshold de-noising based on EMDrdquoShock and Vibration vol 20 no 4 pp 763ndash780 2013

[2] S Luo J Cheng and H Ao ldquoApplication of LCD-SVD tech-nique and CRO-SVMmethod to fault diagnosis for roller bear-ingrdquo Shock and Vibration vol 2015 Article ID 847802 8 pages2015

[3] R Yan M Shan J Cui and Y Wu ldquoMutual information-assisted wavelet function selection for enhanced rolling bearingfault diagnosisrdquo Shock and Vibration In press

[4] G Dalpiaz A Rivola and R Rubini ldquoEffectiveness and sensi-tivity of vibration processing techniques for local fault detectionin gearsrdquo Mechanical Systems and Signal Processing vol 14 no3 pp 387ndash412 2000

[5] C Capdessus M Sidahmed and J L Lacoume ldquoCyclostation-ary processes application in gear faults early diagnosisrdquoMechanical Systems and Signal Processing vol 14 no 3 pp 371ndash385 2000

[6] L Bouillaut and M Sidahmed ldquoCyclostationary approach andbilinear approach comparison applications to early diagnosisfor helicopter gearbox and classificationmethod based on hocsrdquoMechanical Systems and Signal Processing vol 15 no 5 pp 923ndash943 2001

[7] Z K Zhu Z H Feng and F R Kong ldquoCyclostationarity anal-ysis for gearbox condition monitoring approaches and effec-tivenessrdquo Mechanical Systems and Signal Processing vol 19 no3 pp 467ndash482 2005

[8] G Bi J Chen F C Zhou and J He ldquoApplication of slice spectralcorrelation density to gear defect detectionrdquo Proceedings ofthe Institution of Mechanical Engineers Part C Journal ofMechanical Engineering Science vol 220 no 9 pp 1385ndash13922006

[9] L Li and L Qu ldquoCyclic statistics in rolling bearing diagnosisrdquoJournal of Sound and Vibration vol 267 no 2 pp 253ndash2652003

[10] R B Randall J Antoni and S Chobsaard ldquoThe relationshipbetween spectral correlation and envelope analysis in the diag-nostics of bearing faults and other cyclostationary machinesignalsrdquoMechanical Systems and Signal Processing vol 15 no 5pp 945ndash962 2001

[11] J Antoni ldquoCyclic spectral analysis in practicerdquoMechanical Sys-tems and Signal Processing vol 21 no 2 pp 597ndash630 2007

[12] J Antoni ldquoCyclic spectral analysis of rolling-element bearingsignals facts and fictionsrdquo Journal of Sound and Vibration vol304 no 3-5 pp 497ndash529 2007

[13] A Raad J Antoni and M Sidahmed ldquoIndicators of cyclosta-tionarity theory and application to gear fault monitoringrdquoMechanical Systems and Signal Processing vol 22 no 3 pp 574ndash587 2008

[14] J Antoni ldquoCyclostationarity by examplesrdquo Mechanical Systemsand Signal Processing vol 23 no 4 pp 987ndash1036 2009

[15] J Urbanek T Barszcz and J Antoni ldquoTime-frequencyapproach to extraction of selected second-order cyclostation-ary vibration components for varying operational conditionsrdquoMeasurement vol 46 no 4 pp 1454ndash1463 2013

[16] DWang and C Shen ldquoAn equivalent cyclic energy indicator forbearing performance degradation assessmentrdquo Journal of Vibra-tion and Control 2014

[17] R B Randall ldquoA new method of modeling gear faultsrdquo Journalof Mechanical Design vol 104 no 2 pp 259ndash267 1982

[18] P DMcFadden ldquoDetecting fatigue cracks in gears by amplitudeand phase demodulation of the meshing vibrationrdquo Journal of

Vibration Acoustics Stress and Reliability in Design vol 108 no2 pp 165ndash170 1986

[19] P D McFadden and J D Smith ldquoModel for the vibrationproduced by a single point defect in a rolling element bearingrdquoJournal of Sound and Vibration vol 96 no 1 pp 69ndash82 1984

[20] P D McFadden and J D Smith ldquoThe vibration produced bymultiple point defects in a rolling element bearingrdquo Journal ofSound and Vibration vol 98 no 2 pp 263ndash273 1985

[21] M Liang and I S Bozchalooi ldquoAn energy operator approachto joint application of amplitude and frequency-demodulationsfor bearing fault detectionrdquo Mechanical Systems and SignalProcessing vol 24 no 5 pp 1473ndash1494 2010

[22] Z Feng M J Zuo R Hao and F Chu ldquoGear crack assessmentbased on cyclic correlation analysisrdquo in Proceedings of the8th International Conference on Reliability Maintainability andSafety (ICRMS rsquo09) pp 1071ndash1076 Chengdu China July 2009

[23] Z Feng R Hao F Chu M J Zuo andM El Badaoui ldquoApplica-tion of cyclic spectral analysis to gear damage assessmentrdquo inProceedings of the Prognostics and Health Management Confer-ence p 3058 Macau China January 2010

[24] Z Feng andM J Zuo ldquoGearbox diagnosis based on cyclic spec-tral analysisrdquo in Proceedings of the 3rd Annual IEEE Prognosticsand System Health Management Conference (PHM rsquo12) pp 1ndash5Beijing China May 2012

[25] W A Gardner Cyclostationarity in Communications and SignalProcessing IEEE Press New York NY USA 1994

[26] M Abramowitz and I A Stegun Handbook of MathematicalFunctions With Formulas Graphs and Mathematical TablesDover Publications New York NY USA 1972

[27] H Qiu J Lee J Lin and G Yu ldquoWavelet filter-based weak sig-nature detection method and its application on rolling elementbearing prognosticsrdquo Journal of Sound and Vibration vol 289no 4-5 pp 1066ndash1090 2006

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 Shock and Vibration

0 01 02 03 04 05 06 07 08 09 1minus04

minus02

0

02

04

Time (s)

Am

plitu

de (V

)

(a) Waveform

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

001

002

003

004

Frequency (Hz)

Am

plitu

de (V

)

(b) Fourier spectrum

0 20 40 60 80 100 120 140 160 180 2000

02

04

06

08

1

Cyclic frequency (Hz)

Mag

nitu

de

173

333

33

52 69

863

333 10

366

67

121

138

3333 15

566

67

172

6667 190

times10minus3

(c) Cyclic correlation 0 lag slice

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 50 100 150 2000

500

1000

1500

2000

2500

3000

3500

4000

051152253354455

(d) Cyclic spectrum

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 50 100 150 2001200

1250

1300

1350

1400

1450

1500

1550

1600

051152253354455

(e) Zoomed-in cyclic spectrum

Figure 4 Faulty gearbox signal

Rexnord ZA-2115 double row rolling element bearings andis driven by an AC motor through rub belts at a speed of2000 rpm A radial load of 6000 lbs is applied to the shaftand bearings by a spring mechanism The bearing test waskept running for eight days until damage occurs to the outerrace of bearing 1 In the normal case all the four bearings arehealthy While in the faulty case the outer race of bearing1 has damage as shown in Figure 6 Accelerometers aremounted on the bearing housings and the vibration signals

are collected at a sampling frequency of 20480Hz and 20480data points are recorded for both the healthy and faultycases The main parameters of the four bearings are listed inTable 2 The characteristic frequency of each element fault iscalculated and listed in Table 3

52 Signal Analysis Figure 7 shows the waveform Fourierspectrum cyclic correlation 0 lag slice and cyclic spectrum

Shock and Vibration 9

Accelerometers Radial load Thermocouples

Bearing 1 Bearing 2 Bearing 3 Bearing 4

Motor

Figure 5 Bearing experimental setup [27]

Figure 6 Outer race damage of bearing 1 [27]

Table 2 Parameters of bearing Rexnord ZA-2115

Number ofrollers Roller diameter Pitch diameter Contact angle

16 0331 [in] 2815 [in] 1517 [∘]

Table 3 Characteristic frequency of bearing Rexnord ZA-2115 [Hz]

Outer race Inner race Roller236404 296930 139917 279833

(in lower cyclic frequency band) of the normal case Inthe cyclic correlation slice Figure 7(c) a few small peaksappear but they do not correspond to any bearing componentfault characteristic frequency or its multiples Although somepeaks appear in the cyclic spectrum Figure 7(d) they do notdistribute along lines corresponding to the cyclic frequenciesof any bearing component fault characteristic frequency orits multiples These features imply that no significant mod-ulation on the resonance vibration and the bearing is healthy

Figure 8 shows the waveform Fourier spectrum cycliccorrelation 0 lag slice and cyclic spectrum (in lower cyclicfrequency band) of the faulty case In the close-up view of theFourier spectrum some sidebands appear but the frequencyspacing is not identically equal to the outer race fault charac-teristic frequency In the cyclic correlation slice Figure 8(d)more peaks appear and their magnitudes are higher than inthe normal case Moreover their cyclic frequency locationscorrespond to the outer race fault characteristic frequencyand multiples up to the 25th order In the cyclic spectrumFigure 8(e) many peaks are present More importantly theydistribute along lines associated with the outer race faultcharacteristic frequency and multiples At any cyclic fre-quency of the outer race fault characteristic frequency or itsmultiple these peaks form sidebands along the frequencyaxis with a spacing equal to the outer race fault characteristicfrequency For example in the zoomed-in cyclic spectrumFigure 8(f) the four peaks labeled by capitals A B C and Ddistribute along a line at the cyclic frequency of 710Hz (aboutthree times the outer race fault characteristic frequency) andthey have a spacing of 236Hz (approximately equal to theouter race fault characteristic frequency) along the frequencyaxis These findings imply that the outer race characteristics

10 Shock and Vibration

0 01 02 03 04 05 06 07 08 09 1minus04

minus02

0

02

04

Time (s)

Am

plitu

de (V

)

(a) Waveform

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

0004

0008

0012

0016

002

Frequency (Hz)

Am

plitu

de (V

)

(b) Fourier spectrum

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

02

04

06

08

1

Cyclic frequency (Hz)

Mag

nitu

de

times10minus3

(c) Cyclic correlation 0 lag slice

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

005

01

015

02

025

03

(d) Cyclic spectrum

Figure 7 Normal bearing signal

frequency has strong modulation effect on the resonanceindicating fault existence on the bearing outer race in accord-ance with the actual settings

6 Discussion and Conclusions

Since gearbox and rolling element bearing vibration signalsare characterized by AM-FM feature their Fourier spectrahave complex sideband structure due to the convolutionbetween the Fourier spectra ofAMandFMparts aswell as the

infinite Bessel series expansion of an FM term Althoughgearbox and bearing fault can be identified via conventionalFourier spectrum it involves complex sideband analysis andrelies on the sideband spacing to extract the fault character-istic frequency

Considering the cyclostationarity of gearbox and bearingvibration signals due to modulation and the suitability ofcyclostationary analysis to extract modulation features ofsuch signals the explicit expressions of cyclic correlation andcyclic spectrum of general AM-FM signals are derived and

Shock and Vibration 11

0 01 02 03 04 05 06 07 08 09 1minus4

minus2

0

2

4

Time (s)

Am

plitu

de (V

)

(a) Waveform

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

001

002

003

004

005

Frequency (Hz)

Am

plitu

de (V

)

(b) Fourier spectrum

2000 2500 3000 3500 4000 4500 5000 5500 60000

001

002

003

004

005

006

Frequency (Hz)

4965

3544

4033

3322

3086

4491

2612

2839

3796

4743

4325

5202

5428

5675A

mpl

itude

(V)

(c) Zoomed-in Fourier spectrum

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

002

004

006

008

01

012

Cyclic frequency (Hz)

237

473 71

094

711

83 1420

1657

1894

2130 23

6725

96 2833

3084

3306 3543

3780

4016 4253 4490

4726 49

6352

0054

5156

7359

10

Mag

nitu

de

(d) Cyclic correlation 0 lag slice

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

05

1

15

2

25

3

35

4

(e) Cyclic spectrum

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

660 680 700 720 740 7603000

3100

3200

3300

3400

3500

3600

3700

3800

3900

4000

05

1

15

2

25

3

35

4D

C

B

A

(f) Zoomed-in cyclic spectrum

Figure 8 Faulty bearing signal

validated via gearbox and bearing lab experimental signalanalysis According to the theoretical derivations the cyclicfrequency locations of peaks in cyclic correlation and cyclicspectrum directly correspond to the modulating frequency(ie the gear or bearing fault characteristic frequency) andharmonicsTherefore cyclostationary analysis offers an effec-tive approach to gearbox and bearing fault feature extraction

Conflict of Interests

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

Acknowledgments

This work is supported by National Natural Science Founda-tion ofChina (11272047 51475038) Program forNewCenturyExcellent Talents in University (NCET-12-0775) and CETIM(Centre des Etudes Techniques des Industries Mecaniques deSenlis) FranceThe authors would like to thank the reviewersfor their valuable comments and suggestions

References

[1] R Shao W Hu and J Li ldquoMulti-fault feature extraction anddiagnosis of gear transmission system using time-frequency

12 Shock and Vibration

analysis and wavelet threshold de-noising based on EMDrdquoShock and Vibration vol 20 no 4 pp 763ndash780 2013

[2] S Luo J Cheng and H Ao ldquoApplication of LCD-SVD tech-nique and CRO-SVMmethod to fault diagnosis for roller bear-ingrdquo Shock and Vibration vol 2015 Article ID 847802 8 pages2015

[3] R Yan M Shan J Cui and Y Wu ldquoMutual information-assisted wavelet function selection for enhanced rolling bearingfault diagnosisrdquo Shock and Vibration In press

[4] G Dalpiaz A Rivola and R Rubini ldquoEffectiveness and sensi-tivity of vibration processing techniques for local fault detectionin gearsrdquo Mechanical Systems and Signal Processing vol 14 no3 pp 387ndash412 2000

[5] C Capdessus M Sidahmed and J L Lacoume ldquoCyclostation-ary processes application in gear faults early diagnosisrdquoMechanical Systems and Signal Processing vol 14 no 3 pp 371ndash385 2000

[6] L Bouillaut and M Sidahmed ldquoCyclostationary approach andbilinear approach comparison applications to early diagnosisfor helicopter gearbox and classificationmethod based on hocsrdquoMechanical Systems and Signal Processing vol 15 no 5 pp 923ndash943 2001

[7] Z K Zhu Z H Feng and F R Kong ldquoCyclostationarity anal-ysis for gearbox condition monitoring approaches and effec-tivenessrdquo Mechanical Systems and Signal Processing vol 19 no3 pp 467ndash482 2005

[8] G Bi J Chen F C Zhou and J He ldquoApplication of slice spectralcorrelation density to gear defect detectionrdquo Proceedings ofthe Institution of Mechanical Engineers Part C Journal ofMechanical Engineering Science vol 220 no 9 pp 1385ndash13922006

[9] L Li and L Qu ldquoCyclic statistics in rolling bearing diagnosisrdquoJournal of Sound and Vibration vol 267 no 2 pp 253ndash2652003

[10] R B Randall J Antoni and S Chobsaard ldquoThe relationshipbetween spectral correlation and envelope analysis in the diag-nostics of bearing faults and other cyclostationary machinesignalsrdquoMechanical Systems and Signal Processing vol 15 no 5pp 945ndash962 2001

[11] J Antoni ldquoCyclic spectral analysis in practicerdquoMechanical Sys-tems and Signal Processing vol 21 no 2 pp 597ndash630 2007

[12] J Antoni ldquoCyclic spectral analysis of rolling-element bearingsignals facts and fictionsrdquo Journal of Sound and Vibration vol304 no 3-5 pp 497ndash529 2007

[13] A Raad J Antoni and M Sidahmed ldquoIndicators of cyclosta-tionarity theory and application to gear fault monitoringrdquoMechanical Systems and Signal Processing vol 22 no 3 pp 574ndash587 2008

[14] J Antoni ldquoCyclostationarity by examplesrdquo Mechanical Systemsand Signal Processing vol 23 no 4 pp 987ndash1036 2009

[15] J Urbanek T Barszcz and J Antoni ldquoTime-frequencyapproach to extraction of selected second-order cyclostation-ary vibration components for varying operational conditionsrdquoMeasurement vol 46 no 4 pp 1454ndash1463 2013

[16] DWang and C Shen ldquoAn equivalent cyclic energy indicator forbearing performance degradation assessmentrdquo Journal of Vibra-tion and Control 2014

[17] R B Randall ldquoA new method of modeling gear faultsrdquo Journalof Mechanical Design vol 104 no 2 pp 259ndash267 1982

[18] P DMcFadden ldquoDetecting fatigue cracks in gears by amplitudeand phase demodulation of the meshing vibrationrdquo Journal of

Vibration Acoustics Stress and Reliability in Design vol 108 no2 pp 165ndash170 1986

[19] P D McFadden and J D Smith ldquoModel for the vibrationproduced by a single point defect in a rolling element bearingrdquoJournal of Sound and Vibration vol 96 no 1 pp 69ndash82 1984

[20] P D McFadden and J D Smith ldquoThe vibration produced bymultiple point defects in a rolling element bearingrdquo Journal ofSound and Vibration vol 98 no 2 pp 263ndash273 1985

[21] M Liang and I S Bozchalooi ldquoAn energy operator approachto joint application of amplitude and frequency-demodulationsfor bearing fault detectionrdquo Mechanical Systems and SignalProcessing vol 24 no 5 pp 1473ndash1494 2010

[22] Z Feng M J Zuo R Hao and F Chu ldquoGear crack assessmentbased on cyclic correlation analysisrdquo in Proceedings of the8th International Conference on Reliability Maintainability andSafety (ICRMS rsquo09) pp 1071ndash1076 Chengdu China July 2009

[23] Z Feng R Hao F Chu M J Zuo andM El Badaoui ldquoApplica-tion of cyclic spectral analysis to gear damage assessmentrdquo inProceedings of the Prognostics and Health Management Confer-ence p 3058 Macau China January 2010

[24] Z Feng andM J Zuo ldquoGearbox diagnosis based on cyclic spec-tral analysisrdquo in Proceedings of the 3rd Annual IEEE Prognosticsand System Health Management Conference (PHM rsquo12) pp 1ndash5Beijing China May 2012

[25] W A Gardner Cyclostationarity in Communications and SignalProcessing IEEE Press New York NY USA 1994

[26] M Abramowitz and I A Stegun Handbook of MathematicalFunctions With Formulas Graphs and Mathematical TablesDover Publications New York NY USA 1972

[27] H Qiu J Lee J Lin and G Yu ldquoWavelet filter-based weak sig-nature detection method and its application on rolling elementbearing prognosticsrdquo Journal of Sound and Vibration vol 289no 4-5 pp 1066ndash1090 2006

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 9

Accelerometers Radial load Thermocouples

Bearing 1 Bearing 2 Bearing 3 Bearing 4

Motor

Figure 5 Bearing experimental setup [27]

Figure 6 Outer race damage of bearing 1 [27]

Table 2 Parameters of bearing Rexnord ZA-2115

Number ofrollers Roller diameter Pitch diameter Contact angle

16 0331 [in] 2815 [in] 1517 [∘]

Table 3 Characteristic frequency of bearing Rexnord ZA-2115 [Hz]

Outer race Inner race Roller236404 296930 139917 279833

(in lower cyclic frequency band) of the normal case Inthe cyclic correlation slice Figure 7(c) a few small peaksappear but they do not correspond to any bearing componentfault characteristic frequency or its multiples Although somepeaks appear in the cyclic spectrum Figure 7(d) they do notdistribute along lines corresponding to the cyclic frequenciesof any bearing component fault characteristic frequency orits multiples These features imply that no significant mod-ulation on the resonance vibration and the bearing is healthy

Figure 8 shows the waveform Fourier spectrum cycliccorrelation 0 lag slice and cyclic spectrum (in lower cyclicfrequency band) of the faulty case In the close-up view of theFourier spectrum some sidebands appear but the frequencyspacing is not identically equal to the outer race fault charac-teristic frequency In the cyclic correlation slice Figure 8(d)more peaks appear and their magnitudes are higher than inthe normal case Moreover their cyclic frequency locationscorrespond to the outer race fault characteristic frequencyand multiples up to the 25th order In the cyclic spectrumFigure 8(e) many peaks are present More importantly theydistribute along lines associated with the outer race faultcharacteristic frequency and multiples At any cyclic fre-quency of the outer race fault characteristic frequency or itsmultiple these peaks form sidebands along the frequencyaxis with a spacing equal to the outer race fault characteristicfrequency For example in the zoomed-in cyclic spectrumFigure 8(f) the four peaks labeled by capitals A B C and Ddistribute along a line at the cyclic frequency of 710Hz (aboutthree times the outer race fault characteristic frequency) andthey have a spacing of 236Hz (approximately equal to theouter race fault characteristic frequency) along the frequencyaxis These findings imply that the outer race characteristics

10 Shock and Vibration

0 01 02 03 04 05 06 07 08 09 1minus04

minus02

0

02

04

Time (s)

Am

plitu

de (V

)

(a) Waveform

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

0004

0008

0012

0016

002

Frequency (Hz)

Am

plitu

de (V

)

(b) Fourier spectrum

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

02

04

06

08

1

Cyclic frequency (Hz)

Mag

nitu

de

times10minus3

(c) Cyclic correlation 0 lag slice

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

005

01

015

02

025

03

(d) Cyclic spectrum

Figure 7 Normal bearing signal

frequency has strong modulation effect on the resonanceindicating fault existence on the bearing outer race in accord-ance with the actual settings

6 Discussion and Conclusions

Since gearbox and rolling element bearing vibration signalsare characterized by AM-FM feature their Fourier spectrahave complex sideband structure due to the convolutionbetween the Fourier spectra ofAMandFMparts aswell as the

infinite Bessel series expansion of an FM term Althoughgearbox and bearing fault can be identified via conventionalFourier spectrum it involves complex sideband analysis andrelies on the sideband spacing to extract the fault character-istic frequency

Considering the cyclostationarity of gearbox and bearingvibration signals due to modulation and the suitability ofcyclostationary analysis to extract modulation features ofsuch signals the explicit expressions of cyclic correlation andcyclic spectrum of general AM-FM signals are derived and

Shock and Vibration 11

0 01 02 03 04 05 06 07 08 09 1minus4

minus2

0

2

4

Time (s)

Am

plitu

de (V

)

(a) Waveform

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

001

002

003

004

005

Frequency (Hz)

Am

plitu

de (V

)

(b) Fourier spectrum

2000 2500 3000 3500 4000 4500 5000 5500 60000

001

002

003

004

005

006

Frequency (Hz)

4965

3544

4033

3322

3086

4491

2612

2839

3796

4743

4325

5202

5428

5675A

mpl

itude

(V)

(c) Zoomed-in Fourier spectrum

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

002

004

006

008

01

012

Cyclic frequency (Hz)

237

473 71

094

711

83 1420

1657

1894

2130 23

6725

96 2833

3084

3306 3543

3780

4016 4253 4490

4726 49

6352

0054

5156

7359

10

Mag

nitu

de

(d) Cyclic correlation 0 lag slice

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

05

1

15

2

25

3

35

4

(e) Cyclic spectrum

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

660 680 700 720 740 7603000

3100

3200

3300

3400

3500

3600

3700

3800

3900

4000

05

1

15

2

25

3

35

4D

C

B

A

(f) Zoomed-in cyclic spectrum

Figure 8 Faulty bearing signal

validated via gearbox and bearing lab experimental signalanalysis According to the theoretical derivations the cyclicfrequency locations of peaks in cyclic correlation and cyclicspectrum directly correspond to the modulating frequency(ie the gear or bearing fault characteristic frequency) andharmonicsTherefore cyclostationary analysis offers an effec-tive approach to gearbox and bearing fault feature extraction

Conflict of Interests

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

Acknowledgments

This work is supported by National Natural Science Founda-tion ofChina (11272047 51475038) Program forNewCenturyExcellent Talents in University (NCET-12-0775) and CETIM(Centre des Etudes Techniques des Industries Mecaniques deSenlis) FranceThe authors would like to thank the reviewersfor their valuable comments and suggestions

References

[1] R Shao W Hu and J Li ldquoMulti-fault feature extraction anddiagnosis of gear transmission system using time-frequency

12 Shock and Vibration

analysis and wavelet threshold de-noising based on EMDrdquoShock and Vibration vol 20 no 4 pp 763ndash780 2013

[2] S Luo J Cheng and H Ao ldquoApplication of LCD-SVD tech-nique and CRO-SVMmethod to fault diagnosis for roller bear-ingrdquo Shock and Vibration vol 2015 Article ID 847802 8 pages2015

[3] R Yan M Shan J Cui and Y Wu ldquoMutual information-assisted wavelet function selection for enhanced rolling bearingfault diagnosisrdquo Shock and Vibration In press

[4] G Dalpiaz A Rivola and R Rubini ldquoEffectiveness and sensi-tivity of vibration processing techniques for local fault detectionin gearsrdquo Mechanical Systems and Signal Processing vol 14 no3 pp 387ndash412 2000

[5] C Capdessus M Sidahmed and J L Lacoume ldquoCyclostation-ary processes application in gear faults early diagnosisrdquoMechanical Systems and Signal Processing vol 14 no 3 pp 371ndash385 2000

[6] L Bouillaut and M Sidahmed ldquoCyclostationary approach andbilinear approach comparison applications to early diagnosisfor helicopter gearbox and classificationmethod based on hocsrdquoMechanical Systems and Signal Processing vol 15 no 5 pp 923ndash943 2001

[7] Z K Zhu Z H Feng and F R Kong ldquoCyclostationarity anal-ysis for gearbox condition monitoring approaches and effec-tivenessrdquo Mechanical Systems and Signal Processing vol 19 no3 pp 467ndash482 2005

[8] G Bi J Chen F C Zhou and J He ldquoApplication of slice spectralcorrelation density to gear defect detectionrdquo Proceedings ofthe Institution of Mechanical Engineers Part C Journal ofMechanical Engineering Science vol 220 no 9 pp 1385ndash13922006

[9] L Li and L Qu ldquoCyclic statistics in rolling bearing diagnosisrdquoJournal of Sound and Vibration vol 267 no 2 pp 253ndash2652003

[10] R B Randall J Antoni and S Chobsaard ldquoThe relationshipbetween spectral correlation and envelope analysis in the diag-nostics of bearing faults and other cyclostationary machinesignalsrdquoMechanical Systems and Signal Processing vol 15 no 5pp 945ndash962 2001

[11] J Antoni ldquoCyclic spectral analysis in practicerdquoMechanical Sys-tems and Signal Processing vol 21 no 2 pp 597ndash630 2007

[12] J Antoni ldquoCyclic spectral analysis of rolling-element bearingsignals facts and fictionsrdquo Journal of Sound and Vibration vol304 no 3-5 pp 497ndash529 2007

[13] A Raad J Antoni and M Sidahmed ldquoIndicators of cyclosta-tionarity theory and application to gear fault monitoringrdquoMechanical Systems and Signal Processing vol 22 no 3 pp 574ndash587 2008

[14] J Antoni ldquoCyclostationarity by examplesrdquo Mechanical Systemsand Signal Processing vol 23 no 4 pp 987ndash1036 2009

[15] J Urbanek T Barszcz and J Antoni ldquoTime-frequencyapproach to extraction of selected second-order cyclostation-ary vibration components for varying operational conditionsrdquoMeasurement vol 46 no 4 pp 1454ndash1463 2013

[16] DWang and C Shen ldquoAn equivalent cyclic energy indicator forbearing performance degradation assessmentrdquo Journal of Vibra-tion and Control 2014

[17] R B Randall ldquoA new method of modeling gear faultsrdquo Journalof Mechanical Design vol 104 no 2 pp 259ndash267 1982

[18] P DMcFadden ldquoDetecting fatigue cracks in gears by amplitudeand phase demodulation of the meshing vibrationrdquo Journal of

Vibration Acoustics Stress and Reliability in Design vol 108 no2 pp 165ndash170 1986

[19] P D McFadden and J D Smith ldquoModel for the vibrationproduced by a single point defect in a rolling element bearingrdquoJournal of Sound and Vibration vol 96 no 1 pp 69ndash82 1984

[20] P D McFadden and J D Smith ldquoThe vibration produced bymultiple point defects in a rolling element bearingrdquo Journal ofSound and Vibration vol 98 no 2 pp 263ndash273 1985

[21] M Liang and I S Bozchalooi ldquoAn energy operator approachto joint application of amplitude and frequency-demodulationsfor bearing fault detectionrdquo Mechanical Systems and SignalProcessing vol 24 no 5 pp 1473ndash1494 2010

[22] Z Feng M J Zuo R Hao and F Chu ldquoGear crack assessmentbased on cyclic correlation analysisrdquo in Proceedings of the8th International Conference on Reliability Maintainability andSafety (ICRMS rsquo09) pp 1071ndash1076 Chengdu China July 2009

[23] Z Feng R Hao F Chu M J Zuo andM El Badaoui ldquoApplica-tion of cyclic spectral analysis to gear damage assessmentrdquo inProceedings of the Prognostics and Health Management Confer-ence p 3058 Macau China January 2010

[24] Z Feng andM J Zuo ldquoGearbox diagnosis based on cyclic spec-tral analysisrdquo in Proceedings of the 3rd Annual IEEE Prognosticsand System Health Management Conference (PHM rsquo12) pp 1ndash5Beijing China May 2012

[25] W A Gardner Cyclostationarity in Communications and SignalProcessing IEEE Press New York NY USA 1994

[26] M Abramowitz and I A Stegun Handbook of MathematicalFunctions With Formulas Graphs and Mathematical TablesDover Publications New York NY USA 1972

[27] H Qiu J Lee J Lin and G Yu ldquoWavelet filter-based weak sig-nature detection method and its application on rolling elementbearing prognosticsrdquo Journal of Sound and Vibration vol 289no 4-5 pp 1066ndash1090 2006

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 Shock and Vibration

0 01 02 03 04 05 06 07 08 09 1minus04

minus02

0

02

04

Time (s)

Am

plitu

de (V

)

(a) Waveform

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

0004

0008

0012

0016

002

Frequency (Hz)

Am

plitu

de (V

)

(b) Fourier spectrum

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

02

04

06

08

1

Cyclic frequency (Hz)

Mag

nitu

de

times10minus3

(c) Cyclic correlation 0 lag slice

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

005

01

015

02

025

03

(d) Cyclic spectrum

Figure 7 Normal bearing signal

frequency has strong modulation effect on the resonanceindicating fault existence on the bearing outer race in accord-ance with the actual settings

6 Discussion and Conclusions

Since gearbox and rolling element bearing vibration signalsare characterized by AM-FM feature their Fourier spectrahave complex sideband structure due to the convolutionbetween the Fourier spectra ofAMandFMparts aswell as the

infinite Bessel series expansion of an FM term Althoughgearbox and bearing fault can be identified via conventionalFourier spectrum it involves complex sideband analysis andrelies on the sideband spacing to extract the fault character-istic frequency

Considering the cyclostationarity of gearbox and bearingvibration signals due to modulation and the suitability ofcyclostationary analysis to extract modulation features ofsuch signals the explicit expressions of cyclic correlation andcyclic spectrum of general AM-FM signals are derived and

Shock and Vibration 11

0 01 02 03 04 05 06 07 08 09 1minus4

minus2

0

2

4

Time (s)

Am

plitu

de (V

)

(a) Waveform

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

001

002

003

004

005

Frequency (Hz)

Am

plitu

de (V

)

(b) Fourier spectrum

2000 2500 3000 3500 4000 4500 5000 5500 60000

001

002

003

004

005

006

Frequency (Hz)

4965

3544

4033

3322

3086

4491

2612

2839

3796

4743

4325

5202

5428

5675A

mpl

itude

(V)

(c) Zoomed-in Fourier spectrum

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

002

004

006

008

01

012

Cyclic frequency (Hz)

237

473 71

094

711

83 1420

1657

1894

2130 23

6725

96 2833

3084

3306 3543

3780

4016 4253 4490

4726 49

6352

0054

5156

7359

10

Mag

nitu

de

(d) Cyclic correlation 0 lag slice

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

05

1

15

2

25

3

35

4

(e) Cyclic spectrum

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

660 680 700 720 740 7603000

3100

3200

3300

3400

3500

3600

3700

3800

3900

4000

05

1

15

2

25

3

35

4D

C

B

A

(f) Zoomed-in cyclic spectrum

Figure 8 Faulty bearing signal

validated via gearbox and bearing lab experimental signalanalysis According to the theoretical derivations the cyclicfrequency locations of peaks in cyclic correlation and cyclicspectrum directly correspond to the modulating frequency(ie the gear or bearing fault characteristic frequency) andharmonicsTherefore cyclostationary analysis offers an effec-tive approach to gearbox and bearing fault feature extraction

Conflict of Interests

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

Acknowledgments

This work is supported by National Natural Science Founda-tion ofChina (11272047 51475038) Program forNewCenturyExcellent Talents in University (NCET-12-0775) and CETIM(Centre des Etudes Techniques des Industries Mecaniques deSenlis) FranceThe authors would like to thank the reviewersfor their valuable comments and suggestions

References

[1] R Shao W Hu and J Li ldquoMulti-fault feature extraction anddiagnosis of gear transmission system using time-frequency

12 Shock and Vibration

analysis and wavelet threshold de-noising based on EMDrdquoShock and Vibration vol 20 no 4 pp 763ndash780 2013

[2] S Luo J Cheng and H Ao ldquoApplication of LCD-SVD tech-nique and CRO-SVMmethod to fault diagnosis for roller bear-ingrdquo Shock and Vibration vol 2015 Article ID 847802 8 pages2015

[3] R Yan M Shan J Cui and Y Wu ldquoMutual information-assisted wavelet function selection for enhanced rolling bearingfault diagnosisrdquo Shock and Vibration In press

[4] G Dalpiaz A Rivola and R Rubini ldquoEffectiveness and sensi-tivity of vibration processing techniques for local fault detectionin gearsrdquo Mechanical Systems and Signal Processing vol 14 no3 pp 387ndash412 2000

[5] C Capdessus M Sidahmed and J L Lacoume ldquoCyclostation-ary processes application in gear faults early diagnosisrdquoMechanical Systems and Signal Processing vol 14 no 3 pp 371ndash385 2000

[6] L Bouillaut and M Sidahmed ldquoCyclostationary approach andbilinear approach comparison applications to early diagnosisfor helicopter gearbox and classificationmethod based on hocsrdquoMechanical Systems and Signal Processing vol 15 no 5 pp 923ndash943 2001

[7] Z K Zhu Z H Feng and F R Kong ldquoCyclostationarity anal-ysis for gearbox condition monitoring approaches and effec-tivenessrdquo Mechanical Systems and Signal Processing vol 19 no3 pp 467ndash482 2005

[8] G Bi J Chen F C Zhou and J He ldquoApplication of slice spectralcorrelation density to gear defect detectionrdquo Proceedings ofthe Institution of Mechanical Engineers Part C Journal ofMechanical Engineering Science vol 220 no 9 pp 1385ndash13922006

[9] L Li and L Qu ldquoCyclic statistics in rolling bearing diagnosisrdquoJournal of Sound and Vibration vol 267 no 2 pp 253ndash2652003

[10] R B Randall J Antoni and S Chobsaard ldquoThe relationshipbetween spectral correlation and envelope analysis in the diag-nostics of bearing faults and other cyclostationary machinesignalsrdquoMechanical Systems and Signal Processing vol 15 no 5pp 945ndash962 2001

[11] J Antoni ldquoCyclic spectral analysis in practicerdquoMechanical Sys-tems and Signal Processing vol 21 no 2 pp 597ndash630 2007

[12] J Antoni ldquoCyclic spectral analysis of rolling-element bearingsignals facts and fictionsrdquo Journal of Sound and Vibration vol304 no 3-5 pp 497ndash529 2007

[13] A Raad J Antoni and M Sidahmed ldquoIndicators of cyclosta-tionarity theory and application to gear fault monitoringrdquoMechanical Systems and Signal Processing vol 22 no 3 pp 574ndash587 2008

[14] J Antoni ldquoCyclostationarity by examplesrdquo Mechanical Systemsand Signal Processing vol 23 no 4 pp 987ndash1036 2009

[15] J Urbanek T Barszcz and J Antoni ldquoTime-frequencyapproach to extraction of selected second-order cyclostation-ary vibration components for varying operational conditionsrdquoMeasurement vol 46 no 4 pp 1454ndash1463 2013

[16] DWang and C Shen ldquoAn equivalent cyclic energy indicator forbearing performance degradation assessmentrdquo Journal of Vibra-tion and Control 2014

[17] R B Randall ldquoA new method of modeling gear faultsrdquo Journalof Mechanical Design vol 104 no 2 pp 259ndash267 1982

[18] P DMcFadden ldquoDetecting fatigue cracks in gears by amplitudeand phase demodulation of the meshing vibrationrdquo Journal of

Vibration Acoustics Stress and Reliability in Design vol 108 no2 pp 165ndash170 1986

[19] P D McFadden and J D Smith ldquoModel for the vibrationproduced by a single point defect in a rolling element bearingrdquoJournal of Sound and Vibration vol 96 no 1 pp 69ndash82 1984

[20] P D McFadden and J D Smith ldquoThe vibration produced bymultiple point defects in a rolling element bearingrdquo Journal ofSound and Vibration vol 98 no 2 pp 263ndash273 1985

[21] M Liang and I S Bozchalooi ldquoAn energy operator approachto joint application of amplitude and frequency-demodulationsfor bearing fault detectionrdquo Mechanical Systems and SignalProcessing vol 24 no 5 pp 1473ndash1494 2010

[22] Z Feng M J Zuo R Hao and F Chu ldquoGear crack assessmentbased on cyclic correlation analysisrdquo in Proceedings of the8th International Conference on Reliability Maintainability andSafety (ICRMS rsquo09) pp 1071ndash1076 Chengdu China July 2009

[23] Z Feng R Hao F Chu M J Zuo andM El Badaoui ldquoApplica-tion of cyclic spectral analysis to gear damage assessmentrdquo inProceedings of the Prognostics and Health Management Confer-ence p 3058 Macau China January 2010

[24] Z Feng andM J Zuo ldquoGearbox diagnosis based on cyclic spec-tral analysisrdquo in Proceedings of the 3rd Annual IEEE Prognosticsand System Health Management Conference (PHM rsquo12) pp 1ndash5Beijing China May 2012

[25] W A Gardner Cyclostationarity in Communications and SignalProcessing IEEE Press New York NY USA 1994

[26] M Abramowitz and I A Stegun Handbook of MathematicalFunctions With Formulas Graphs and Mathematical TablesDover Publications New York NY USA 1972

[27] H Qiu J Lee J Lin and G Yu ldquoWavelet filter-based weak sig-nature detection method and its application on rolling elementbearing prognosticsrdquo Journal of Sound and Vibration vol 289no 4-5 pp 1066ndash1090 2006

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 11

0 01 02 03 04 05 06 07 08 09 1minus4

minus2

0

2

4

Time (s)

Am

plitu

de (V

)

(a) Waveform

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

001

002

003

004

005

Frequency (Hz)

Am

plitu

de (V

)

(b) Fourier spectrum

2000 2500 3000 3500 4000 4500 5000 5500 60000

001

002

003

004

005

006

Frequency (Hz)

4965

3544

4033

3322

3086

4491

2612

2839

3796

4743

4325

5202

5428

5675A

mpl

itude

(V)

(c) Zoomed-in Fourier spectrum

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

002

004

006

008

01

012

Cyclic frequency (Hz)

237

473 71

094

711

83 1420

1657

1894

2130 23

6725

96 2833

3084

3306 3543

3780

4016 4253 4490

4726 49

6352

0054

5156

7359

10

Mag

nitu

de

(d) Cyclic correlation 0 lag slice

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

05

1

15

2

25

3

35

4

(e) Cyclic spectrum

Cyclic frequency (Hz)

Freq

uenc

y (H

z)

660 680 700 720 740 7603000

3100

3200

3300

3400

3500

3600

3700

3800

3900

4000

05

1

15

2

25

3

35

4D

C

B

A

(f) Zoomed-in cyclic spectrum

Figure 8 Faulty bearing signal

validated via gearbox and bearing lab experimental signalanalysis According to the theoretical derivations the cyclicfrequency locations of peaks in cyclic correlation and cyclicspectrum directly correspond to the modulating frequency(ie the gear or bearing fault characteristic frequency) andharmonicsTherefore cyclostationary analysis offers an effec-tive approach to gearbox and bearing fault feature extraction

Conflict of Interests

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

Acknowledgments

This work is supported by National Natural Science Founda-tion ofChina (11272047 51475038) Program forNewCenturyExcellent Talents in University (NCET-12-0775) and CETIM(Centre des Etudes Techniques des Industries Mecaniques deSenlis) FranceThe authors would like to thank the reviewersfor their valuable comments and suggestions

References

[1] R Shao W Hu and J Li ldquoMulti-fault feature extraction anddiagnosis of gear transmission system using time-frequency

12 Shock and Vibration

analysis and wavelet threshold de-noising based on EMDrdquoShock and Vibration vol 20 no 4 pp 763ndash780 2013

[2] S Luo J Cheng and H Ao ldquoApplication of LCD-SVD tech-nique and CRO-SVMmethod to fault diagnosis for roller bear-ingrdquo Shock and Vibration vol 2015 Article ID 847802 8 pages2015

[3] R Yan M Shan J Cui and Y Wu ldquoMutual information-assisted wavelet function selection for enhanced rolling bearingfault diagnosisrdquo Shock and Vibration In press

[4] G Dalpiaz A Rivola and R Rubini ldquoEffectiveness and sensi-tivity of vibration processing techniques for local fault detectionin gearsrdquo Mechanical Systems and Signal Processing vol 14 no3 pp 387ndash412 2000

[5] C Capdessus M Sidahmed and J L Lacoume ldquoCyclostation-ary processes application in gear faults early diagnosisrdquoMechanical Systems and Signal Processing vol 14 no 3 pp 371ndash385 2000

[6] L Bouillaut and M Sidahmed ldquoCyclostationary approach andbilinear approach comparison applications to early diagnosisfor helicopter gearbox and classificationmethod based on hocsrdquoMechanical Systems and Signal Processing vol 15 no 5 pp 923ndash943 2001

[7] Z K Zhu Z H Feng and F R Kong ldquoCyclostationarity anal-ysis for gearbox condition monitoring approaches and effec-tivenessrdquo Mechanical Systems and Signal Processing vol 19 no3 pp 467ndash482 2005

[8] G Bi J Chen F C Zhou and J He ldquoApplication of slice spectralcorrelation density to gear defect detectionrdquo Proceedings ofthe Institution of Mechanical Engineers Part C Journal ofMechanical Engineering Science vol 220 no 9 pp 1385ndash13922006

[9] L Li and L Qu ldquoCyclic statistics in rolling bearing diagnosisrdquoJournal of Sound and Vibration vol 267 no 2 pp 253ndash2652003

[10] R B Randall J Antoni and S Chobsaard ldquoThe relationshipbetween spectral correlation and envelope analysis in the diag-nostics of bearing faults and other cyclostationary machinesignalsrdquoMechanical Systems and Signal Processing vol 15 no 5pp 945ndash962 2001

[11] J Antoni ldquoCyclic spectral analysis in practicerdquoMechanical Sys-tems and Signal Processing vol 21 no 2 pp 597ndash630 2007

[12] J Antoni ldquoCyclic spectral analysis of rolling-element bearingsignals facts and fictionsrdquo Journal of Sound and Vibration vol304 no 3-5 pp 497ndash529 2007

[13] A Raad J Antoni and M Sidahmed ldquoIndicators of cyclosta-tionarity theory and application to gear fault monitoringrdquoMechanical Systems and Signal Processing vol 22 no 3 pp 574ndash587 2008

[14] J Antoni ldquoCyclostationarity by examplesrdquo Mechanical Systemsand Signal Processing vol 23 no 4 pp 987ndash1036 2009

[15] J Urbanek T Barszcz and J Antoni ldquoTime-frequencyapproach to extraction of selected second-order cyclostation-ary vibration components for varying operational conditionsrdquoMeasurement vol 46 no 4 pp 1454ndash1463 2013

[16] DWang and C Shen ldquoAn equivalent cyclic energy indicator forbearing performance degradation assessmentrdquo Journal of Vibra-tion and Control 2014

[17] R B Randall ldquoA new method of modeling gear faultsrdquo Journalof Mechanical Design vol 104 no 2 pp 259ndash267 1982

[18] P DMcFadden ldquoDetecting fatigue cracks in gears by amplitudeand phase demodulation of the meshing vibrationrdquo Journal of

Vibration Acoustics Stress and Reliability in Design vol 108 no2 pp 165ndash170 1986

[19] P D McFadden and J D Smith ldquoModel for the vibrationproduced by a single point defect in a rolling element bearingrdquoJournal of Sound and Vibration vol 96 no 1 pp 69ndash82 1984

[20] P D McFadden and J D Smith ldquoThe vibration produced bymultiple point defects in a rolling element bearingrdquo Journal ofSound and Vibration vol 98 no 2 pp 263ndash273 1985

[21] M Liang and I S Bozchalooi ldquoAn energy operator approachto joint application of amplitude and frequency-demodulationsfor bearing fault detectionrdquo Mechanical Systems and SignalProcessing vol 24 no 5 pp 1473ndash1494 2010

[22] Z Feng M J Zuo R Hao and F Chu ldquoGear crack assessmentbased on cyclic correlation analysisrdquo in Proceedings of the8th International Conference on Reliability Maintainability andSafety (ICRMS rsquo09) pp 1071ndash1076 Chengdu China July 2009

[23] Z Feng R Hao F Chu M J Zuo andM El Badaoui ldquoApplica-tion of cyclic spectral analysis to gear damage assessmentrdquo inProceedings of the Prognostics and Health Management Confer-ence p 3058 Macau China January 2010

[24] Z Feng andM J Zuo ldquoGearbox diagnosis based on cyclic spec-tral analysisrdquo in Proceedings of the 3rd Annual IEEE Prognosticsand System Health Management Conference (PHM rsquo12) pp 1ndash5Beijing China May 2012

[25] W A Gardner Cyclostationarity in Communications and SignalProcessing IEEE Press New York NY USA 1994

[26] M Abramowitz and I A Stegun Handbook of MathematicalFunctions With Formulas Graphs and Mathematical TablesDover Publications New York NY USA 1972

[27] H Qiu J Lee J Lin and G Yu ldquoWavelet filter-based weak sig-nature detection method and its application on rolling elementbearing prognosticsrdquo Journal of Sound and Vibration vol 289no 4-5 pp 1066ndash1090 2006

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

12 Shock and Vibration

analysis and wavelet threshold de-noising based on EMDrdquoShock and Vibration vol 20 no 4 pp 763ndash780 2013

[2] S Luo J Cheng and H Ao ldquoApplication of LCD-SVD tech-nique and CRO-SVMmethod to fault diagnosis for roller bear-ingrdquo Shock and Vibration vol 2015 Article ID 847802 8 pages2015

[3] R Yan M Shan J Cui and Y Wu ldquoMutual information-assisted wavelet function selection for enhanced rolling bearingfault diagnosisrdquo Shock and Vibration In press

[4] G Dalpiaz A Rivola and R Rubini ldquoEffectiveness and sensi-tivity of vibration processing techniques for local fault detectionin gearsrdquo Mechanical Systems and Signal Processing vol 14 no3 pp 387ndash412 2000

[5] C Capdessus M Sidahmed and J L Lacoume ldquoCyclostation-ary processes application in gear faults early diagnosisrdquoMechanical Systems and Signal Processing vol 14 no 3 pp 371ndash385 2000

[6] L Bouillaut and M Sidahmed ldquoCyclostationary approach andbilinear approach comparison applications to early diagnosisfor helicopter gearbox and classificationmethod based on hocsrdquoMechanical Systems and Signal Processing vol 15 no 5 pp 923ndash943 2001

[7] Z K Zhu Z H Feng and F R Kong ldquoCyclostationarity anal-ysis for gearbox condition monitoring approaches and effec-tivenessrdquo Mechanical Systems and Signal Processing vol 19 no3 pp 467ndash482 2005

[8] G Bi J Chen F C Zhou and J He ldquoApplication of slice spectralcorrelation density to gear defect detectionrdquo Proceedings ofthe Institution of Mechanical Engineers Part C Journal ofMechanical Engineering Science vol 220 no 9 pp 1385ndash13922006

[9] L Li and L Qu ldquoCyclic statistics in rolling bearing diagnosisrdquoJournal of Sound and Vibration vol 267 no 2 pp 253ndash2652003

[10] R B Randall J Antoni and S Chobsaard ldquoThe relationshipbetween spectral correlation and envelope analysis in the diag-nostics of bearing faults and other cyclostationary machinesignalsrdquoMechanical Systems and Signal Processing vol 15 no 5pp 945ndash962 2001

[11] J Antoni ldquoCyclic spectral analysis in practicerdquoMechanical Sys-tems and Signal Processing vol 21 no 2 pp 597ndash630 2007

[12] J Antoni ldquoCyclic spectral analysis of rolling-element bearingsignals facts and fictionsrdquo Journal of Sound and Vibration vol304 no 3-5 pp 497ndash529 2007

[13] A Raad J Antoni and M Sidahmed ldquoIndicators of cyclosta-tionarity theory and application to gear fault monitoringrdquoMechanical Systems and Signal Processing vol 22 no 3 pp 574ndash587 2008

[14] J Antoni ldquoCyclostationarity by examplesrdquo Mechanical Systemsand Signal Processing vol 23 no 4 pp 987ndash1036 2009

[15] J Urbanek T Barszcz and J Antoni ldquoTime-frequencyapproach to extraction of selected second-order cyclostation-ary vibration components for varying operational conditionsrdquoMeasurement vol 46 no 4 pp 1454ndash1463 2013

[16] DWang and C Shen ldquoAn equivalent cyclic energy indicator forbearing performance degradation assessmentrdquo Journal of Vibra-tion and Control 2014

[17] R B Randall ldquoA new method of modeling gear faultsrdquo Journalof Mechanical Design vol 104 no 2 pp 259ndash267 1982

[18] P DMcFadden ldquoDetecting fatigue cracks in gears by amplitudeand phase demodulation of the meshing vibrationrdquo Journal of

Vibration Acoustics Stress and Reliability in Design vol 108 no2 pp 165ndash170 1986

[19] P D McFadden and J D Smith ldquoModel for the vibrationproduced by a single point defect in a rolling element bearingrdquoJournal of Sound and Vibration vol 96 no 1 pp 69ndash82 1984

[20] P D McFadden and J D Smith ldquoThe vibration produced bymultiple point defects in a rolling element bearingrdquo Journal ofSound and Vibration vol 98 no 2 pp 263ndash273 1985

[21] M Liang and I S Bozchalooi ldquoAn energy operator approachto joint application of amplitude and frequency-demodulationsfor bearing fault detectionrdquo Mechanical Systems and SignalProcessing vol 24 no 5 pp 1473ndash1494 2010

[22] Z Feng M J Zuo R Hao and F Chu ldquoGear crack assessmentbased on cyclic correlation analysisrdquo in Proceedings of the8th International Conference on Reliability Maintainability andSafety (ICRMS rsquo09) pp 1071ndash1076 Chengdu China July 2009

[23] Z Feng R Hao F Chu M J Zuo andM El Badaoui ldquoApplica-tion of cyclic spectral analysis to gear damage assessmentrdquo inProceedings of the Prognostics and Health Management Confer-ence p 3058 Macau China January 2010

[24] Z Feng andM J Zuo ldquoGearbox diagnosis based on cyclic spec-tral analysisrdquo in Proceedings of the 3rd Annual IEEE Prognosticsand System Health Management Conference (PHM rsquo12) pp 1ndash5Beijing China May 2012

[25] W A Gardner Cyclostationarity in Communications and SignalProcessing IEEE Press New York NY USA 1994

[26] M Abramowitz and I A Stegun Handbook of MathematicalFunctions With Formulas Graphs and Mathematical TablesDover Publications New York NY USA 1972

[27] H Qiu J Lee J Lin and G Yu ldquoWavelet filter-based weak sig-nature detection method and its application on rolling elementbearing prognosticsrdquo Journal of Sound and Vibration vol 289no 4-5 pp 1066ndash1090 2006

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of