Research ArticleApplication of Self-Adaptive Wavelet Ridge DemodulationMethod Based on LCD to Incipient Fault Diagnosis

Songrong Luo12 Junsheng Cheng2 and Jianping Fu2

1College of Mechanical Engineering Hunan University of Arts and Science Changde 415003 China2China College of Mechanical and Vehicle Engineering Hunan University Changsha 410082 China

Correspondence should be addressed to Songrong Luo luosr7351sinacom

Received 19 November 2014 Revised 20 March 2015 Accepted 5 April 2015

Academic Editor Lei Zuo

Copyright copy 2015 Songrong Luo et al 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

When a local defect occurs in gearbox the vibration signals present as the form of multicomponent amplitude modulation andfrequency modulation (AM-FM) Demodulation analysis is an effective way for this kind of signal A self-adaptive wavelet ridgedemodulation method based on LCD is proposed in this paper Firstly multicomponent AM-FM signal is decomposed intoseries of intrinsic scale components (ISCs) and the special intrinsic scale component is selected in order to decrease the lowerfrequency background noise Secondly the genetic algorithm is employed to optimize wavelet parameters according to the inherentcharacteristics of signal thirdly self-adaptive wavelet ridge demodulation wavelet for the selected ISC component is performedto get instantaneous amplitude (IA) or instantaneous frequency (IF) Lastly the characteristics frequency can be obtained toidentify the working state or failure information from its spectrum By two simulation signals the proposed method was comparedwith various existing demodulation methods the simulation results show that it has higher accuracy and higher noise tolerantperformance than others Furthermore the proposed method was applied to incipient fault diagnosis for gearbox and the resultsshow that it is simple and effective

1 Introduction

Fault diagnosis technique is of great significance to guar-antee the normal operation of mechanical and electricalequipment When a localized defect occurs in gearbox thevibration signals present as the form of multicomponentamplitude modulation and frequency modulation (AM-FM)[1] expressed as a frequency family on the spectrum whichcontains the center frequency and its harmonic frequencyFor this kind of signals some demodulation techniqueshave been used to find the fault feature information Hilbertdemodulation method is widely used in machinery faultdiagnosis [2 3] but there exists window effect and endeffect of Hilbert transforms inevitably resulting in greaterdemodulation error The energy separation algorithm (ESA)appears much popular in recent years for the applicationto machinery fault diagnosis [4ndash7] because it is suitable toextract the local dynamic characteristics of nonstationarysignal However ESA requires that the processed signal

should be narrow-band monocomponent [4 5] In additionESA is sensitive to noise [8] Compared with the above timedomain demodulation methods the wavelet ridge demod-ulation technique is time-frequency domain demodulationmethod which uses continuous wavelet transform (CWT)to obtain instantaneous amplitude (IA) information andinstantaneous frequency (IF) information [8 9] In generalthe analytic Morlet wavelet is used as the basic waveletdue to its similarity to the fault associated impacts [10ndash13] But the analytic Morlet wavelet parameters which arecenter frequency and shape factor would exert a greatimpact on the results of wavelet ridge demodulation Inorder to select the proper parameters some techniques havebeen employed [10ndash12] Unfortunately there is no maturetheory to tell us how to choose them In addition there arefew methods which can select both center frequency andshape factor of Morlet wavelet to obtain the optimal time-scale resolution Here genetic algorithm (GA) which notonly has better ability to search the optimal solution but

Hindawi Publishing CorporationShock and VibrationVolume 2015 Article ID 735853 13 pageshttpdxdoiorg1011552015735853

2 Shock and Vibration

also has fast convergence is introduced to obtain the twooptimal parameters according to the analyzed signal localcharacteristics and Morlet wavelet with optimal parametersusing GA is called self-adaptive wavelet Therefore we willutilize self-adaptive wavelet ridge demodulation approach toobtain better demodulation results in this paper

On the other hand to greatly eliminate the backgroundnoise and improve the demodulation accuracy multicompo-nent AM-FM signals should be decomposed into monocom-ponent AM-FM signals before using self-adaptive waveletridge demodulation approach Empirical mode decomposi-tion (EMD)method [3 4 14 15] or localmean decomposition(LMD) method [16ndash18] is widely employed to decomposemulticomponent AM-FM signal into monocomponent AM-FM signals in general However EMD method still hastheoretical limitations such as frequency confusion over-shooting undershooting end effect and the emergence ofnegative frequency components of nonphysical meaningCompared to the EMD method LMD method avoids theseproblems to some extent but its computing speed is muchslower than EMD Local characteristic-scale decomposition(LCD) is a new data-driving signal analysis method Basedon the inherent characteristics of the signal itself the LCDmethod can decompose a complex multicomponent AM-FM into several intrinsic scale components (ISC) Simul-taneously each ISC component is a monocomponent AM-FM signal which has obvious physical meaning Our teamrsquosresearch works show that compared with the LMD andEMD method LCD not only avoids the shortcomings ofEMD and LMD but also owns much faster computing speed[19ndash21] Therefore LCD method is used to decompose themulticomponent gearbox fault vibration signal to a numberof ISCs at first Subsequently one or several interesting ISCsare selected as analyzed component After that noise wouldbe greatly removed to clearly demodulate fault-associatedfeatures component from the selected ISCs

In summary targeting the demodulation solution of themulticomponent AM-FM vibration signal with low signal-noise ratio produced by gearbox failures we present a self-adaptive wavelet ridge demodulation method based on LCDfor fault diagnosis The rest of the paper is organized as fol-lows In Section 2 the wavelet ridge demodulation principlebased on LCD is introducedTheprocess to get a self-adaptivewavelet based on GA is described We describe the proposedmethod and the simulation study is provided in Section 4The proposed method is applied for incipient fault diagnosisof gearbox in Section 5 Finally we offer conclusions inSection 6

2 Wavelet Ridge Demodulation PrincipleBased on LCD

A real signal of monocomponent can be expressed as119904(119905) = 119860(119905)cos(120593(119905)) When the instantaneous frequencyof the signal is much larger than the amplitude modulationfrequency expressed as |119889120593(119905)119889119905| ≫ |(1119860(119905))(119889119860(119905)119889119905)|

this real signal monocomponent is called a progressive singlefrequency signal Then its analytical signal can be written as

119885119904(119905) = 119860

119904(119905) exp (119895120593

119904(119905)) (1)

The expression of the instantaneous frequency is

119891119904=

1

2120587

119889120593119904(119905)

119889119905

(2)

After selecting a mother wavelet 120595(119905) whose analyticalwavelet is expressed as (119905) = 119860

120595(119905)exp(119895120593

120595(119905)) 119885

119904(119905)

can be transformed by continuous wavelet transformation asfollows

119882119911(119886 119887) =

1

radic119886

int

infin

minusinfin

119860119886119887

(119905) exp (119895120593119886119887

(119905)) 119889119905 (3)

where 119886 is the scale parameter 119887 is the translation parameter

119860119886119887(119905) = 119860

119904(119905) 119860120595(

119905 minus 119887

119886

)

120593119886119887(119905) = 120593

119904(119905) minus 120593

120595(

119905 minus 119887

119886

)

(4)

As to any scale parameter 119886 and translation parameter 119887suppose 120593

119886119887(119905) only has a first-order stagnation 119905

119904= 119905119904(119886 119887)

and then the first-order stagnation satisfies 1205931015840119886119887(119905119904) = 0 and

12059310158401015840

119886119887(119905119904) = 0 that is

1205931015840

119904(119905119904) =

1

119886

1205931015840

(

119905119904minus 119887

119886

) (5)

Wavelet ridge is defined as a collection of all the pointswhich meet 119905

119904(119886 119887) = 119887 on a phase plane The expression of

the collection is 119877 = (119886 119887) isin 1198672(119877) 119905119904(119886 119887) = 119887 where

1198672(119877) is a Hardy real space A point (119886

119903(119887) 119887) on the wavelet

ridge line is called wavelet ridge point Obviously accordingto formula (5) there is

119886 = 119886119903(119887) =

1205931015840

120595(0)

1205931015840

119904(119887)

(6)

Here can be seen that instantaneous frequency can beextracted from the wavelet ridge points

Also the wavelet coefficients of signal 119904(119905) about (119905) canbe expressed as

119882119904(119886 119887) asymp

radic119886

2

119860119904(119887) exp (119895120593 (119887))

times ( (119886 [

1205960

119886

minus1205931015840

119904(119887)]))

(7)

where (120596) is the Fourier transform of 120595(119905) And the waveletcoefficients modulus is initially defined as

1003816100381610038161003816119882119904(119886 119887)

1003816100381610038161003816asymp

radic119886

2

119860119904(119887)

1003816100381610038161003816100381610038161003816

(119886 [

1205960

119886

minus1205931015840

119904(119887)])

1003816100381610038161003816100381610038161003816

(8)

Shock and Vibration 3

As to wavelet ridge point it can be seen that 1205960119886119903 minus1205931015840

119904(119887) = 0 from formula (6) So the wavelet coefficients

modulus of wavelet ridge point is further expressed as

1003816100381610038161003816119882119904(119886119903 119887)

1003816100381610038161003816asymp

radic119886

2

119860119904(119887)

1003816100381610038161003816 (0)

1003816100381610038161003816 (9)

So it can be seen that signal instantaneous frequencycan be gained after the wavelet ridge is extracted which isexpressed as

119891119904(119905) =

1

2120587

1205960

119886119903(119905)

(10)

where 1205960is the center frequency of (119905) that is 120596

0= 1205931015840

120595(0)

At the same time the signal instantaneous amplitude canbe expressed as

119860119904(119905) asymp

21003816100381610038161003816119882119904(119886119903(119905) 119905)

1003816100381610038161003816

radic119886119903(119905)

1003816100381610038161003816 (0)

1003816100381610038161003816

(11)

As presented above it is clear that the demodulationanalysis of monocomponent AM-FM signal based onwaveletridge is feasible Howevermost vibration signals produced bygearbox failures are generally multicomponent AM-FM sig-nalsThey should be decomposed intomonocomponentAM-FM signals by appropriate time-frequency signal processingmethod before demodulation In this paper LCD method isemployed to accomplish the signal decomposition

The LCD method has the assumptions that a complexsignal consists of a number of ISCs (Intrinsic Scale Compo-nent ISC) and any two ISCs are independent of each otherIn the entire data segment ISC must meet the following twoconditions

(I) The maximal value is positive the minimum value isnegative and the data set are monotonic between anytwo adjacent extreme points

(II) Let all the extreme points be written as (119883119896 120591119896)

119896 = 1 2 119872 the line 119897119896determined by any two

adjacent extreme points (119883119896 120591119896) and (119883

119896+2 120591119896+2) canbe expressed as

119897119896=

119883119896+2

minus 119883119896

120591119896+2

minus 120591119896

(119905 minus 120591119896) +119883119896 (12)

Remembering the value of 119897119896at the 120591

119896+1as 119860119896+1

therelation between 119860

119896+1and119883

119896+1should meet the following

120572119860119896+1

+ (1 minus 120572)119883119896+1

= 0 (13)

when 120572 = 05 119860119896+1

= minus119883119896+1

Based on this definition a complexmulticomponentAM-

FM signal can be decomposed into the sumof a finite numberof ISCs and a residual signal Each ISC is a monocomponentAM-FM signal whose instantaneous frequency has specificphysical meaning That is

119909 (119905) =

119873

sum

119894=1

119888119894(119905) + 119906 (119905) (14)

where 119888119894(119905) is the 119894th ISC component and 119906(119905) is the residual

signal

3 Self-Adaptive Wavelet

31 MorletWavelet Frequency Resolution When there is localfailure for gear the fault gear teeth will stimulate systemto produce a convergent impact response and the vibrationsignal collected by the acceleration sensor shows the obviousmulticomponent modulation characteristic Therefore asmentioned above we can adopt wavelet ridge demodulationbased on LCD to extract the fault feature In order to matchthis kind of signal analytic Morlet wavelet with impactfeature is chosen which is defined as

(119905) =

1

radic119891119887120587

exp(minus 1199052

119891119887

) exp (1198952120587119891119888119905) (15)

The Fourier transform of 120595(119905) is represented as

(2120587119891) = exp[(minus2120587119891 minus 2120587119891

119888

2radic119891119887

)

2

] (16)

where 119891119887is the shape factor and 119891

119888is the center frequency

whose numerical values determine the speed of thewaveformvibration damping respectively from formula (16) Morletwavelet quality factor is 119876 = radic2120587119891

119888radic119891119887radicln2 So the best

frequency resolution can be gained by adjusting 119891119887and 119891

119888

which can result in a good time-scale accumulation

32The Procedure of Obtaining Self-AdaptiveWavelet Sparsedegree of wavelet coefficients can characterize the degreeof similarity between the basic wavelet function and signalThe energy entropy of wavelet can indicate this sparsedegree which shows accumulation performance of waveletcoefficients As to each specific scale 119895 the wavelet energyentropy is defined as

119882119864= minussum

119895

119875119895ln119875119895 (17)

where 119875119895= 119864119895119864119879is the probability of energy distribution

(sum119875119895= 1) 119864

119895= int119882

2

(119895 120591)119889120591 is the wavelet energy 119882(119895 120591)

is the wavelet coefficient and 119864119879= sum119895119864119895is the total wavelet

energy within the time scale planeAccordingly the wavelet energy entropy is taken as

the objective function during selecting the optimal waveletparameters To optimize wavelet parameters genetic algo-rithm (GA) and particle swarm optimization (PSO) [2223] are two widely utilized approaches In this paper GAis employed to optimize either envelope factor or centerfrequency with wavelet energy entropy as the fitness func-tion That wavelet with optimal parameters is called self-adaptive wavelet The procedure to get self-adaptive waveletis described as follows

Step 1 Set search prime range and population size of param-eters 119891

119887and 119891

119888 and randomly generate initial population

In this paper the population size is set to 100 and theparameters are respectively encoded as 10-bit binary stringchromosomes by binary coding method

4 Shock and Vibration

Step 2 Make wavelet decomposition of the signal and calcu-late the fitness value of each individual according to formula(17) Then sort the fitness values by size

Step 3 Based on individual fitness value in the search spaceindividuals are screened and evolved by a series of geneticmanipulation selection reproduction crossover mutationand so forth to constantly update and select populations

Step 4 In this step it is determined whether iterationsatisfies the termination condition or not If satisfied theoptimal solution is finished If not satisfied go to Step 2 untilthe optimal solution is got The optimization procedure ofwavelet parameters via GA is shown in Figure 1

4 The Proposed Method and Simulation

Since demodulation technique is an effectiveway to reveal thefault characteristic frequency for fault diagnosis of gearboxa self-adaptive wavelet ridge demodulation based on LCDfor fault diagnosis is proposed in this paper Firstly LCDmethod is adopted to decompose the original signal intoa number of ISC components and lower frequency noiseis decreased by selecting special ISC component whichcontains rich fault feature Secondly the genetic algorithm isused to optimize wavelet parameters to obtain self-adaptivewavelet based on wavelet energy entropy which is servedas the objective function Thirdly the self-adaptive waveletridge demodulation for the selected ISC component is usedto extract dynamic information Finally the characteristicsfrequency can be obtained to identify the working state orfailure information from the spectrum The flowchart of theproposed fault diagnosis method was illustrated in Figure 2

To verify the validity of the proposed method let usconsider the following signal

119909 (119905) = 1199091(119905) + 119909

2(119905)

1199091(119905) = (1 + 05 cos 20120587119905) sin (200120587119905 + 2 cos 20120587119905)

1199092(119905) = sin120587119905 sin 20120587119905 119905 isin [0 1]

(18)

Obviously 119909(119905) is a complexmulticomponentAM-FM signalcontaining AM-FM component 1199091(119905) and AM component1199092(119905) The sampling frequency is 1000Hz Time domainwaveforms of simulation signal 119909(119905) and its LCD decompo-sition results are shown in Figure 3 where the two ISC com-ponents ISC1 and ISC2 correspond to the two components1199091(119905) and 1199092(119905)

Then the wavelet parameters are optimized accordingto the signal itself using genetic algorithms For ISC1 theoptimal parameters are determined as 119891

119887= 22316 and

119891119888= 10532 Figure 4 shows the evolution curve of GA For

ISC2 the optimal parameters are 119891119887

= 41203 and 119891119888

=

10012 Ultimately the signal is transformed self-adaptivelyusing Morlet wavelet to extract wavelet ridge and ISC1 andISC2 are demodulated based on the formula (10) and (11)The demodulation results are shown in Figures 5 and 6respectively

Genetic manipulation

crossover and mutation)

Optimal solution

Yes

No

Starting

Wavelet decomposition

Fitness value calculation and evaluation of individual fitness

Termination conditions

End

Population initialization

and energy entropy

(selection reproduction

Figure 1 Flowchart of wavelet parameters optimization by GA

For the instantaneous amplitude of ISC1 to eliminate theborder effect of the wavelet transformation the boundary isprocessed by symmetric extensionmethodThe result is givenin Figure 7

From the above analysis results it can be seen that self-adaptive wavelet ridge demodulation method based on LCDcan precisely demodulate a complex multicomponent AM-FMsignal In order tomake comparative analysis the demod-ulation analysis results of ISC1 using Hilbert demodulationmethod are given in Figure 8 fromwhich it can be found thatthe demodulation curve is not smooth and the demodulationerror is bigger due to inevitable window effects by Hilbertdemodulation method [4] The above comparison results

Shock and Vibration 5

Collect vibration signals

Apply LCD decomposition

Extract the fault characteristic

Select useful ISC component

Apply self-adaptive wavelet ridge

demodulation to the selected ISC

Optimize wavelet parameters by GA

Identify fault type

Figure 2 Flowchart of the proposed method

indicate the proposed method is superior to the Hilbertdemodulation method

Let us consider another multicomponent AM-FM signal119910(119905)which is a simulated faulty signal of gearbox and definedas follows

119910 (119905) = 1199101(119905) + 119910

2(119905)

1199101(119905) = [04 + 02 cos (2120587 times 10119905)]

sdot sin [2120587 times 600119905 + cos (2120587 times 20119905)]

1199102(119905) = [1 + 05 cos (2120587 times 20119905)]

sdot sin [2120587 times 300119905 + cos (2120587 times 15119905)]

(19)

Time (s)

Am

plitu

deA

mpl

itude

Am

plitu

deA

mpl

itude

0

0

5

0

2

0

1

005

0

01 02 03 04 05 06 07 08 09 1

Time (s) 0 01 02 03 04 05 06 07 08 09 1

Time (s) 0 01 02 03 04 05 06 07 08 09 1

Time (s) 0 01 02 03 04 05 06 07 08 09 1

minus5

minus2

x(t

)IS

C 1IS

C 2

minus1

minus005

u(t

)Figure 3 Time domain waveform of simulation signal 119909(119905) and itsLCD decomposition result

Wav

elet

ener

gy en

tropy

val

ue

0 5 10 15 20 25 30

14

145

15

155

16

165

135

Optimal valueMean value

times10minus3

Figure 4 Evolution curve of GA

The time domain waveform is shown in Figure 9 Thesampling frequency is 2000Hz Through LCD method twoISC components are presented in Figure 10 Obviously thetwo ISC components are consistent with the two components1199101(119905) and 1199102(119905) By optimizing wavelet energy entropy self-adaptive wavelet parameters for the first ISC signal aredetermined as 119891

119887= 97752 and 119891

119888= 07428 the parameters

for the second ISC signal as 119891119887

= 40585 and 119891119888

=

10678 The self-adaptive wavelet transform results for thetwo ISCs are provided in Figures 11 and 12 It can be foundthat the IArsquos frequency of the first ISC signal is 10Hz and

6 Shock and Vibration

0 01 02 03 04 05 06 07 08 09 1040608

11214

80

90

100

110

120

IA waveform

IF waveform

Am

plitu

deFr

eque

ncy

Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

Figure 5 Self-adaptive wavelet ridge demodulation results of the 1stISC

0 01 02 03 04 05 06 07 08 09 10

02040608

1

859

9510

10511

Freq

uenc

yA

mpl

itude

IF waveform

IA waveform

Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

Figure 6 Self-adaptive wavelet ridge demodulation results of the2nd ISC of simulation signal

IA waveform

0 01 02 03 04 05 06 07 08 09 104

06

08

1

12

14

Am

plitu

de

Time (s)

Figure 7 Processed instantaneous amplitude of the 1st ISC

0 01 02 03 04 05 06 07 08 09 10

05

1

15

2

8090

100110120130

IA waveform

IF waveform

Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

Am

plitu

deFr

eque

ncy

Figure 8 Hilbert demodulation results of the 1st ISC of simulationsignal

0

2

4

Am

plitu

de

minus2

minus40 01 02 03 04 05 06 07 08 09 1

Time (s)

Figure 9 Waveform of simulation signal 119910(119905)

Am

plitu

de

0 01 02 03 04 05 06Time (s)

07 08 09 1

0 01 02 03 04 05 06Time (s)

07 08 09 1

0

05

1

0

1

2

Am

plitu

de

minus05

minus1

minus1

minus2

ISC 1

ISC 2

Figure 10 LCD results of simulation signal 119910(119905)

Shock and Vibration 7

200

400

600

Scal

e

Time-scale distribution

Am

plitu

de

Time (s)

Freq

uenc

y (H

z)

IA waveform

IF waveform

020304050607

0 01 02 03 04 05 06 07 08 09 1

Time (s)0 01 02 03 04 05 06 07 08 09 1

Time (s)0 01 02 03 04 05 06 07 08 09

550

600

650

700

Figure 11 Demodulation results of the 1st ISC of simulation signal119910(119905) by the proposed method

the IFrsquos frequency is 20Hz and that IArsquos frequency of thesecond ISC signal is 20Hz and the IFrsquos frequency is 15HzThese results are consistent with the two components in theoriginal simulation signal Therefore the proposed methodcan effectively demodulate multicomponent AM-FM signaland is suitable for fault diagnosis of gearbox

For comparison we use EMD method to make signaldecomposition and use self-adaptive wavelet ridge demodu-lation approach to demodulate We obtained eight IMFs intotal After analysis we noted the first two IMFs reflected themodulation characteristics hence they are considered as thetwo components The demodulation results of the first twoIMFs by self-adaptive wavelet ridge demodulation approachare provided in Figures 13 and 14 It can be seen that theFM component of the first IMF failed to be demodulatedand the energy of the spectra of the second IMFs is lowerThen energy operator demodulation based on EMD [4] isused and the results are shown in Figure 15 From thesefigures it is shown that the serious mode mixing exists inthe EMDdecomposition which influences the demodulationaccuracy However from Figures 11 and 12 it is noted thatLCD approach may diminish this problem and be superiorto EMD approach

Finally we add a Gaussian white noise with deviation of02 to 119910(119905) LCD decomposes the noisy signal into five ISCsBy the method introduced in Section 32 the optimal Morlet

0 01 02 03 04 05 06 07 08 09

100

200

300

400

500

Time (s)

Scal

e

Time-scale distribution

0 01 02 03 04 05 06 07 08 09Time (s)

1

0 01 02 03 04 05 06 07 08 09Time (s)

1

05

1

15

280

290

300

310

320

Am

plitu

deFr

eque

ncy

IA waveform

IF waveform

Figure 12 Demodulation results of the 2nd ISC of simulation signal119910(119905) by the proposed method

wavelet parameters for the first two ISCs which containmodulation information are determined The demodulationresults of the first two ISCs are shown in Figure 16The resultsby energy operator demodulation based on LCD to the firsttwo ISCs are given in Figure 17 It can be obviously found thatself-adaptive wavelet demodulation approach based on LCDhas better noise tolerant performance than energy operatordemodulation approach based on LCD

5 Application to Incipient Fault Diagnosis

51 Gear Crack Fault Diagnosis A gear crack fault diagnosisexperiment is carried out on bearing-gear test rig as shown inFigure 18 In this test themotor power is 600W both drivinggear and driven gear are standard spur gear whose modulusis 25mm and the number of teeth is 37The input and outputshafts are arranged in parallel They are supported by tworoller bearings A crack with 015mm width and 1mm depthat the root of the driving gear tooth is set by wire cuttingmachining to simulate the gear incipient crack failure Thevibration signals were collected by an accelerometer attachedto the bearing housingThe shaft speed is 360 revmin that isthe drive shaft rotation frequency is 119891

119903= 6Hz The sampling

frequency is 1024Hz and the length of sampling data is 1024pointThedomainwaveformof the vibration signalmeasured

8 Shock and Vibration

0 01 02 03 04 05 06 07 08 09 10

01

02

03

04IA component of the 1st IMF

Time (s)

Am

plitu

de

0 01 02 03 04 05 06 07 08 09 1100

150

200

250IF component of the 1st IMF

Time (s)

Am

plitu

de

0 20 40 60 80 100 120 140 160 180 2000

0102030405

Spectrum

Am

plitu

de

01020304050 Spectrum

Am

plitu

de

Frequency (Hz)

0 20 40 60 80 100 120 140 160 180 200Frequency (Hz)

X 10Y 006344

X 15Y 2169

Figure 13 Demodulation results of the 1st IMF of simulation signal119910(119905) by self-adaptive wavelet ridge demodulation approach based onEMD

and envelope spectrum are shown in Figure 19 From whichit can be seen that fault characteristics are submerged by thebackground noise and the characteristics frequency fail to beidentified

Since the vibration signal with gear crack is a multi-component AM-FM signal (AM-FM) we used the proposedmethod for fault diagnosis Firstly the vibration accelerationsignal was decomposed into four ISC components ISC

1simISC4

and a residual component 119903 by LCD method as shown inFigure 20 Because the carrier frequency of gear vibration sig-nals is generally gear meshing frequency and its harmonicswe select the first ISC with the highest frequency for analysisFigure 21 shows the Hilbert demodulation results of thefirst ISC showing instantaneous amplitude contains complexhigh-frequency interference and some negative frequency

0 01 02 03 04 05 06 07 08 09 1040506070809

0 20 40 60 80 120100 140 160 180 2000

0102030405

Spectrum

IA component of the 2nd IMF

IF component of the 2nd IMF

Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

Am

plitu

deA

mpl

itude

Frequency (Hz)

0 20 40 60 80 120100 140 160 180 200Frequency (Hz)

Spectrum

767778798081

0

5

10

15

20

Am

plitu

deA

mpl

itude

X 20Y 003938

X 15Y 1266

Figure 14Demodulation results of the 2nd IMFof simulation signal119910(119905) by self-adaptive wavelet ridge demodulation approach based onEMD

exists in the instantaneous frequency waveform which isrelated to the measured signal

Then the optimum wavelet parameters 119891119887= 48574 and

119891119888= 07832 were selected Lastly the self-adaptive wavelet

ridge demodulation was carried out The IA waveform andits frequency spectrum obtained are shown in Figure 22from which the fault characteristics frequency 119891

119903= 6Hz

and its harmonics 3119891119903 4119891119903 and 5119891

119903are clearly found These

demonstrate that a local defect has occurred in the gear onthe drive shaft which is consistent with the drive gear stateThat is the proposed method is effective for gear crack faultdiagnosis

In addition we use the self-adaptive wavelet ridgedemodulation approach to analyse the original signal asexhibited in Figure 23 where the spectrum contains the faultcharacteristics frequency and its harmonic as well which

Shock and Vibration 9

0 01 02 03 04 05 06 07 08 09 10

051

152

25

0

200

400

600

800

Freq

uenc

yA

mpl

itude

Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

IA waveform of 1st IMF

IA waveform of 2nd IMF

IF waveform of 2nd IMF

IF waveform of 1st IMF

0

05

1

15

2

0100200300400500

Am

plitu

deFr

eque

ncy

Figure 15 Demodulation results of simulation signal 119910(119905) by energyoperator demodulation based on EMD

outperform the excellent time-frequency localization abilityof self-adaptive wavelet But some unknown frequency isinvolved because of background noise Moreover it can benoted that the harmonic in Figure 22 is richer and clearerthan that in Figure 23 In fact through signal decompositionby LCDand ISC selectionmost of noise can be removed fromanalysis signal Therefore the proposed method is effectiveand superior in application to weak fault diagnosis for gear

52 Bearing Inner-Race Fault Diagnosis The vibration signalof roller bearing with inner-race fault is complex and weakit is difficult to identify the fault state To further verifythe effectiveness of the proposed method we made bearinginner-race fault diagnosis experiment

The data was downloaded from the website of theCase Western Reserve University Bearing Center [24] Thetest stand consists of a 2 hp motor a torque transducera dynamometer and control electronics The test bearing

0 01 02 03 04 05 06 07 08 09 1010203040506

222224226228230232

IA waveform of 1st ISC

IF waveform of 1st ISC

IF waveform of 2nd ISC

IA waveform of 2nd ISC

Am

plitu

de

Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

Freq

uenc

y

040608

11214

155

160

165

170

Am

plitu

deFr

eque

ncy

Figure 16 Demodulation results of simulation signal with noise bythe proposed method

is at the drive end a single point defect was introducedinto the inner raceway of the test bearing The size of thesingle point defect is 0178mm in diameter and 0279mm indepth using electrodischarge machining An accelerometerattached to the bearing housing collected vibration data withthe sampling frequency as 12 kHz The shaft rotating speedof the bearing inner-race is 1750 revmin the characteristicfrequency of the roller bearing with inner-race fault is 119891

119900119894=

158HzFigure 24 presents the time domain waveform and spec-

trum of a bearing vibration signal From the spectrum it canbe seen that there are three center frequency bands and theirside frequency bands which show that main modulationcharacteristics exhibits in the high frequency band Howeverthe fault characteristic frequency is not clear in the spectrumHere we used the proposed method to demodulate Thebearing vibration signal was decomposed into thirteen ISCs

10 Shock and Vibration

0

1

2

3

0

200

400

600

800

IA waveform of 1st ISC

IF waveform of 1st ISC

Am

plitu

de

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

Freq

uenc

y

005

115

225

0

200

400

600

IA waveform of 2nd ISC

IF waveform of 2nd ISC

Am

plitu

deFr

eque

ncy

Figure 17 Demodulation results of simulation signal with noise byenergy operator demodulation approach based on LCD

and the first ISC with the highest frequency as shown inFigure 25 was selected for analysis The optimal Morletwavelet parameters were determined as 119891

119887= 128576 and

119891119888= 06862The IA waveform and its spectrum are shown in

Figure 26 from which we can clearly see the spectrum linesat the fault characteristic frequency 119891

119900119894and its harmonics

which shows that a local failure occurs in the inner racewayof bearing This is consistent with the fact state

Besides through comparison of Figure 24 with Figure 25we can find that applying LCD decomposition equals todesign adaptive band-pass filter whose center frequency isautomatically determined with the inherent characteristicsof analyzed signal Simultaneously the selection of specialISC with higher frequency for analysis leads to decrease ofthe influence of lower frequency noise to suit weak faultfeature extraction In addition due to concern of main

1 2 3 4 5 6 7 8 9 10 11

(1) Induction motor(2) Speed controller(3) Coupling(4) Bearing number 1 and bearing housing(5) Bearing number 2 and bearing housing(6) Drive gear(7) Driven gear

(9) Vibration acceleration sensor(10) Speed sensor

(8) Bearing number 3 and bearing housing

(11) Bearing number 4 and bearing housing

Figure 18 Bearing-gear test rig for vibration signal acquisition

0 01 02 03 04 05 06 07 08 09 1

0

50

100

Time (s)

Time domain waveform

0 50 100 150 200 250 300 350 4000

5

10

15

20 Envelope spectrum

minus50

minus100

Frequency (Hz)

X 21Y 914

Am

plitu

de (m

middotsminus2)

Acce

lera

tion

(mmiddotsminus

2)

Figure 19 Time domain waveform and envelope spectrum of afaulty gear vibration signal

energy on the time-scale distribution self-adaptive waveletridge modulation has excellent time-frequency localizationproperty as seen in Figure 23 Therefore the applicationresults show the proposed method is simple and effective forweak bearing inner-race fault diagnosis

6 Conclusion

LCD method is a new signal decomposition approachwhich is suitable to preprocess the multicomponent AM-FM signal Wavelet ridge demodulation method is based onwavelet transform in the time-frequency domain focusing

Shock and Vibration 11

0 01 02 03 04 05 06 07 08 09 1

200

100

50

010

Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

minus50

minus20

minus10

minus10

minus5

0

50

ISC 1

ISC 2

ISC 3

ISC 4

u(t

)

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

Figure 20 LCD decomposition results of a faulty gear vibration signal

IA waveform

0

20

40

60

80

0

500

1000IF waveform

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

Acce

lera

tion

(mmiddotsminus

2)

minus500

minus1000

Freq

uenc

y (H

z)

Figure 21 Demodulation results of the 1st ISC by Hilbert approach

on main energy on the time-scale distribution It is lesssensitive to signal bandwidth and noise especially usingthe wavelet transform with optimal parameters called self-adaptive wavelet Therefore with combination of the abovetwo approaches a self-adaptive wavelet ridge demodulationmethod based on LCD for fault diagnosis is presented

Using two simulation signals we compare the proposedmethod with the following four approaches Hilbert demod-ulation energy operator demodulation based on EMD self-adaptive wavelet ridge demodulation based on EMD andenergy operator demodulation based on LCD Analysis

0 01 02 03 04 05 06 07 08 09 10

1020304050

0 10 20 30 40 706050 80 90 10002468

10

Time (s)

IA waveform

Spectrum

Frequency (Hz)

fr

3fr4fr 5fr

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

Figure 22 Instantaneous amplitude and its spectrum of the 1st ISCby the proposed method

results show that the proposedmethod has higher demodula-tion accuracy and higher noise tolerant performance than theothers Finally we applied the proposed method to incipientfault diagnosis of gearbox The application results show theproposed method is simple and effective for incipient faultdiagnosis of gearbox

Conflict of Interests

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

12 Shock and Vibration

0 01 02 03 04 05 06 07 08 09 10

20

40

60

80

0 10 20 30 40 50 60 70 80 90 10002468

10

Time (s)

IA waveform

Spectrum

5fr

fr

Frequency (Hz)

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

Figure 23 Instantaneous amplitude and its spectrum of the orig-inal vibration signal by self-adaptive wavelet ridge demodulationapproach

0 01 02 03 04 05 06 07 08 09 1

0

1

2

0 1000 2000 3000 4000 5000 60000

005

01

015

02

Waveform

Spectrum

Time (s)

Frequency (Hz)

minus2

minus1

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

Figure 24 Time domain waveform and spectrum of a roller bearingvibration signal with inner-race defect

Acknowledgments

The support from Chinese National Science FoundationGrant (no 51075131) the construct program of the key dis-cipline in Hunan Province (Mechanical Design and Theory)(XJF2011 [76]) cooperative Demonstration Base of Universi-ties in Hunan ldquoR amp D and Industrialization of Rock DrillingMachinesrdquo (XJT [2014] 239) Hunan Major Special Projectsof Science and Technology (2014GK1043) and scientificresearch project of Hunan Province Education Department(no 14C0789) is greatly acknowledged The authors wouldlike to express their appreciation to Case Western ReserveUniversity for offering the free bearing data

0 1000 2000 3000 4000 5000 60000

005

01

015

02

Waveform

Spectrum

0

1

2

0 01 02 03 04 05 06 07 08 09 1Time (s)

minus2

minus1

Frequency (Hz)

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

Figure 25 Time domain waveform and spectrum of the 1st ISC ofa roller bearing vibration signal with inner-race defect

0 01 02 03 04 05 06 07 08 09 10

02040608

1

0 100 200 300 400 500 600 700 800 900 10000

005

01

015

02

IA waveform

Spectrum

Time (s)

Frequency (Hz)

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

foi

2foi

3foi

Figure 26 Instantaneous amplitude waveform and spectrum of the1st ISC of a roller bearing vibration signal with inner-race defect

References

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

[2] M Feldman ldquoNonparametric identification of asymmetricnonlinear vibration systems with the Hilbert transformrdquo Jour-nal of Sound and Vibration vol 331 no 14 pp 3386ndash3396 2012

[3] N E Huang Z Shen S R Long et al ldquoThe empiricalmode decomposition and the Hilbert spectrum for nonlinearand non-stationary time series analysisrdquo The Royal Societyof London Proceedings Series A Mathematical Physical andEngineering Sciences vol 454 no 1971 pp 903ndash995 1998

[4] J Cheng D Yu and Y Yang ldquoThe application of energyoperator demodulation approach based on EMD in machinery

Shock and Vibration 13

fault diagnosisrdquo Mechanical Systems and Signal Processing vol21 no 2 pp 668ndash677 2007

[5] Z Feng M Liang Y Zhang and S Hou ldquoFault diagnosis forwind turbine planetary gearboxes via demodulation analysisbased on ensemble empirical mode decomposition and energyseparationrdquo Renewable Energy vol 47 pp 112ndash126 2012

[6] Y Qin ldquoMulticomponent AM-FM demodulation based onenergy separation and adaptive filteringrdquo Mechanical Systemsand Signal Processing vol 38 no 2 pp 440ndash459 2013

[7] M Liang and H Faghidi ldquoIntelligent bearing fault detection byenhanced energy operatorrdquo Expert Systems with Applicationsvol 41 no 16 pp 7223ndash7234 2014

[8] J Wang and Q He ldquoExchanged ridge demodulation of time-scale manifold for enhanced fault diagnosis of rotating machin-eryrdquo Journal of Sound and Vibration vol 333 no 11 pp 2450ndash2464 2014

[9] Y Qin S Qin and Y Mao ldquoDemodulation approach based onwavelet ridge and its application in fault diagnosis of rotatingmachineryrdquo Chinese Journal of Mechanical Engineering vol 45no 2 pp 231ndash237 2009

[10] W He Z Jiang and Q Qin ldquoA joint adaptive wavelet filter andmorphological signal processing method for weak mechanicalimpulse extractionrdquo Journal of Mechanical Science and Technol-ogy vol 24 no 8 pp 1709ndash1716 2010

[11] Y Jiang B Tang YQin andW Liu ldquoFeature extractionmethodof wind turbine based on adaptive Morlet wavelet and SVDrdquoRenewable Energy vol 36 no 8 pp 2146ndash2153 2011

[12] W Su FWang H Zhu Z Zhang and Z Guo ldquoRolling elementbearing faults diagnosis based on optimal Morlet wavelet filterand autocorrelation enhancementrdquo Mechanical Systems andSignal Processing vol 24 no 5 pp 1458ndash1472 2010

[13] H Qiu J Lee J Lin and G Yu ldquoWavelet filter-based weaksignature detection method and its application on rollingelement bearing prognosticsrdquo Journal of Sound and Vibrationvol 289 no 4-5 pp 1066ndash1090 2006

[14] M Amarnath and I Praveen Krishna ldquoLocal fault detection inhelical gears via vibration and acoustic signals using EMDbasedstatistical parameter analysisrdquo Measurement vol 58 pp 154ndash164 2014

[15] Y Lei J Lin Z He and M J Zuo ldquoA review on empiricalmode decomposition in fault diagnosis of rotating machineryrdquoMechanical Systems and Signal Processing vol 35 no 1-2 pp108ndash126 2013

[16] J S Smith ldquoThe local mean decomposition and its applicationto EEG perception datardquo Journal of the Royal Society Interfacevol 2 no 5 pp 443ndash454 2005

[17] H Liu and M Han ldquoA fault diagnosis method based onlocal mean decomposition and multi-scale entropy for rollerbearingsrdquo Mechanism and Machine Theory vol 75 pp 67ndash782014

[18] J Cheng and Y Yang ldquoA rotating machinery fault diagnosismethod based on local mean decompositionrdquo Digital SignalProcessing vol 22 no 2 pp 356ndash366 2012

[19] J Zheng J Cheng and Y Yang ldquoA rolling bearing fault diagno-sis approach based on LCD and fuzzy entropyrdquoMechanism andMachine Theory vol 70 pp 441ndash453 2013

[20] J Zheng J Cheng Y Nie and S Luo ldquoA signal pro-cessing method for fault diagnosis-complete ensemble localcharacteristic-scale decompositionrdquo Journal of Vibration Engi-neering vol 27 no 4 pp 637ndash646 2014

[21] H L Ao J Cheng K Li andTK Truong ldquoA roller bearing faultdiagnosis method based on LCD energy entropy and ACROA-SVMrdquo Shock and Vibration vol 2014 Article ID 825825 12pages 2014

[22] MUnalMOnatMDemetgul andHKucuk ldquoFault diagnosisof rolling bearings using a genetic algorithm optimized neuralnetworkrdquoMeasurement vol 58 pp 187ndash196 2014

[23] B Long W Xian M Li and H Wang ldquoImproved diagnosticsfor the incipient faults in analog circuits using LSSVM based onPSO algorithm with Mahalanobis distancerdquo Neurocomputingvol 133 no 10 pp 237ndash248 2014

[24] CWRU Bearing Data Center Seeded Fault Test Data 2013httpcsegroupscaseedubearingdatacenterhome

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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

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Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

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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

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Navigation and Observation

International Journal of

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DistributedSensor Networks

International Journal of

2 Shock and Vibration

also has fast convergence is introduced to obtain the twooptimal parameters according to the analyzed signal localcharacteristics and Morlet wavelet with optimal parametersusing GA is called self-adaptive wavelet Therefore we willutilize self-adaptive wavelet ridge demodulation approach toobtain better demodulation results in this paper

On the other hand to greatly eliminate the backgroundnoise and improve the demodulation accuracy multicompo-nent AM-FM signals should be decomposed into monocom-ponent AM-FM signals before using self-adaptive waveletridge demodulation approach Empirical mode decomposi-tion (EMD)method [3 4 14 15] or localmean decomposition(LMD) method [16ndash18] is widely employed to decomposemulticomponent AM-FM signal into monocomponent AM-FM signals in general However EMD method still hastheoretical limitations such as frequency confusion over-shooting undershooting end effect and the emergence ofnegative frequency components of nonphysical meaningCompared to the EMD method LMD method avoids theseproblems to some extent but its computing speed is muchslower than EMD Local characteristic-scale decomposition(LCD) is a new data-driving signal analysis method Basedon the inherent characteristics of the signal itself the LCDmethod can decompose a complex multicomponent AM-FM into several intrinsic scale components (ISC) Simul-taneously each ISC component is a monocomponent AM-FM signal which has obvious physical meaning Our teamrsquosresearch works show that compared with the LMD andEMD method LCD not only avoids the shortcomings ofEMD and LMD but also owns much faster computing speed[19ndash21] Therefore LCD method is used to decompose themulticomponent gearbox fault vibration signal to a numberof ISCs at first Subsequently one or several interesting ISCsare selected as analyzed component After that noise wouldbe greatly removed to clearly demodulate fault-associatedfeatures component from the selected ISCs

In summary targeting the demodulation solution of themulticomponent AM-FM vibration signal with low signal-noise ratio produced by gearbox failures we present a self-adaptive wavelet ridge demodulation method based on LCDfor fault diagnosis The rest of the paper is organized as fol-lows In Section 2 the wavelet ridge demodulation principlebased on LCD is introducedTheprocess to get a self-adaptivewavelet based on GA is described We describe the proposedmethod and the simulation study is provided in Section 4The proposed method is applied for incipient fault diagnosisof gearbox in Section 5 Finally we offer conclusions inSection 6

2 Wavelet Ridge Demodulation PrincipleBased on LCD

A real signal of monocomponent can be expressed as119904(119905) = 119860(119905)cos(120593(119905)) When the instantaneous frequencyof the signal is much larger than the amplitude modulationfrequency expressed as |119889120593(119905)119889119905| ≫ |(1119860(119905))(119889119860(119905)119889119905)|

this real signal monocomponent is called a progressive singlefrequency signal Then its analytical signal can be written as

119885119904(119905) = 119860

119904(119905) exp (119895120593

119904(119905)) (1)

The expression of the instantaneous frequency is

119891119904=

1

2120587

119889120593119904(119905)

119889119905

(2)

After selecting a mother wavelet 120595(119905) whose analyticalwavelet is expressed as (119905) = 119860

120595(119905)exp(119895120593

120595(119905)) 119885

119904(119905)

can be transformed by continuous wavelet transformation asfollows

119882119911(119886 119887) =

1

radic119886

int

infin

minusinfin

119860119886119887

(119905) exp (119895120593119886119887

(119905)) 119889119905 (3)

where 119886 is the scale parameter 119887 is the translation parameter

119860119886119887(119905) = 119860

119904(119905) 119860120595(

119905 minus 119887

119886

)

120593119886119887(119905) = 120593

119904(119905) minus 120593

120595(

119905 minus 119887

119886

)

(4)

As to any scale parameter 119886 and translation parameter 119887suppose 120593

119886119887(119905) only has a first-order stagnation 119905

119904= 119905119904(119886 119887)

and then the first-order stagnation satisfies 1205931015840119886119887(119905119904) = 0 and

12059310158401015840

119886119887(119905119904) = 0 that is

1205931015840

119904(119905119904) =

1

119886

1205931015840

(

119905119904minus 119887

119886

) (5)

Wavelet ridge is defined as a collection of all the pointswhich meet 119905

119904(119886 119887) = 119887 on a phase plane The expression of

the collection is 119877 = (119886 119887) isin 1198672(119877) 119905119904(119886 119887) = 119887 where

1198672(119877) is a Hardy real space A point (119886

119903(119887) 119887) on the wavelet

ridge line is called wavelet ridge point Obviously accordingto formula (5) there is

119886 = 119886119903(119887) =

1205931015840

120595(0)

1205931015840

119904(119887)

(6)

Here can be seen that instantaneous frequency can beextracted from the wavelet ridge points

Also the wavelet coefficients of signal 119904(119905) about (119905) canbe expressed as

119882119904(119886 119887) asymp

radic119886

2

119860119904(119887) exp (119895120593 (119887))

times ( (119886 [

1205960

119886

minus1205931015840

119904(119887)]))

(7)

where (120596) is the Fourier transform of 120595(119905) And the waveletcoefficients modulus is initially defined as

1003816100381610038161003816119882119904(119886 119887)

1003816100381610038161003816asymp

radic119886

2

119860119904(119887)

1003816100381610038161003816100381610038161003816

(119886 [

1205960

119886

minus1205931015840

119904(119887)])

1003816100381610038161003816100381610038161003816

(8)

Shock and Vibration 3

As to wavelet ridge point it can be seen that 1205960119886119903 minus1205931015840

119904(119887) = 0 from formula (6) So the wavelet coefficients

modulus of wavelet ridge point is further expressed as

1003816100381610038161003816119882119904(119886119903 119887)

1003816100381610038161003816asymp

radic119886

2

119860119904(119887)

1003816100381610038161003816 (0)

1003816100381610038161003816 (9)

So it can be seen that signal instantaneous frequencycan be gained after the wavelet ridge is extracted which isexpressed as

119891119904(119905) =

1

2120587

1205960

119886119903(119905)

(10)

where 1205960is the center frequency of (119905) that is 120596

0= 1205931015840

120595(0)

At the same time the signal instantaneous amplitude canbe expressed as

119860119904(119905) asymp

21003816100381610038161003816119882119904(119886119903(119905) 119905)

1003816100381610038161003816

radic119886119903(119905)

1003816100381610038161003816 (0)

1003816100381610038161003816

(11)

As presented above it is clear that the demodulationanalysis of monocomponent AM-FM signal based onwaveletridge is feasible Howevermost vibration signals produced bygearbox failures are generally multicomponent AM-FM sig-nalsThey should be decomposed intomonocomponentAM-FM signals by appropriate time-frequency signal processingmethod before demodulation In this paper LCD method isemployed to accomplish the signal decomposition

The LCD method has the assumptions that a complexsignal consists of a number of ISCs (Intrinsic Scale Compo-nent ISC) and any two ISCs are independent of each otherIn the entire data segment ISC must meet the following twoconditions

(I) The maximal value is positive the minimum value isnegative and the data set are monotonic between anytwo adjacent extreme points

(II) Let all the extreme points be written as (119883119896 120591119896)

119896 = 1 2 119872 the line 119897119896determined by any two

adjacent extreme points (119883119896 120591119896) and (119883

119896+2 120591119896+2) canbe expressed as

119897119896=

119883119896+2

minus 119883119896

120591119896+2

minus 120591119896

(119905 minus 120591119896) +119883119896 (12)

Remembering the value of 119897119896at the 120591

119896+1as 119860119896+1

therelation between 119860

119896+1and119883

119896+1should meet the following

120572119860119896+1

+ (1 minus 120572)119883119896+1

= 0 (13)

when 120572 = 05 119860119896+1

= minus119883119896+1

Based on this definition a complexmulticomponentAM-

FM signal can be decomposed into the sumof a finite numberof ISCs and a residual signal Each ISC is a monocomponentAM-FM signal whose instantaneous frequency has specificphysical meaning That is

119909 (119905) =

119873

sum

119894=1

119888119894(119905) + 119906 (119905) (14)

where 119888119894(119905) is the 119894th ISC component and 119906(119905) is the residual

signal

3 Self-Adaptive Wavelet

31 MorletWavelet Frequency Resolution When there is localfailure for gear the fault gear teeth will stimulate systemto produce a convergent impact response and the vibrationsignal collected by the acceleration sensor shows the obviousmulticomponent modulation characteristic Therefore asmentioned above we can adopt wavelet ridge demodulationbased on LCD to extract the fault feature In order to matchthis kind of signal analytic Morlet wavelet with impactfeature is chosen which is defined as

(119905) =

1

radic119891119887120587

exp(minus 1199052

119891119887

) exp (1198952120587119891119888119905) (15)

The Fourier transform of 120595(119905) is represented as

(2120587119891) = exp[(minus2120587119891 minus 2120587119891

119888

2radic119891119887

)

2

] (16)

where 119891119887is the shape factor and 119891

119888is the center frequency

whose numerical values determine the speed of thewaveformvibration damping respectively from formula (16) Morletwavelet quality factor is 119876 = radic2120587119891

119888radic119891119887radicln2 So the best

frequency resolution can be gained by adjusting 119891119887and 119891

119888

which can result in a good time-scale accumulation

32The Procedure of Obtaining Self-AdaptiveWavelet Sparsedegree of wavelet coefficients can characterize the degreeof similarity between the basic wavelet function and signalThe energy entropy of wavelet can indicate this sparsedegree which shows accumulation performance of waveletcoefficients As to each specific scale 119895 the wavelet energyentropy is defined as

119882119864= minussum

119895

119875119895ln119875119895 (17)

where 119875119895= 119864119895119864119879is the probability of energy distribution

(sum119875119895= 1) 119864

119895= int119882

2

(119895 120591)119889120591 is the wavelet energy 119882(119895 120591)

is the wavelet coefficient and 119864119879= sum119895119864119895is the total wavelet

energy within the time scale planeAccordingly the wavelet energy entropy is taken as

the objective function during selecting the optimal waveletparameters To optimize wavelet parameters genetic algo-rithm (GA) and particle swarm optimization (PSO) [2223] are two widely utilized approaches In this paper GAis employed to optimize either envelope factor or centerfrequency with wavelet energy entropy as the fitness func-tion That wavelet with optimal parameters is called self-adaptive wavelet The procedure to get self-adaptive waveletis described as follows

Step 1 Set search prime range and population size of param-eters 119891

119887and 119891

119888 and randomly generate initial population

In this paper the population size is set to 100 and theparameters are respectively encoded as 10-bit binary stringchromosomes by binary coding method

4 Shock and Vibration

Step 2 Make wavelet decomposition of the signal and calcu-late the fitness value of each individual according to formula(17) Then sort the fitness values by size

Step 3 Based on individual fitness value in the search spaceindividuals are screened and evolved by a series of geneticmanipulation selection reproduction crossover mutationand so forth to constantly update and select populations

Step 4 In this step it is determined whether iterationsatisfies the termination condition or not If satisfied theoptimal solution is finished If not satisfied go to Step 2 untilthe optimal solution is got The optimization procedure ofwavelet parameters via GA is shown in Figure 1

4 The Proposed Method and Simulation

Since demodulation technique is an effectiveway to reveal thefault characteristic frequency for fault diagnosis of gearboxa self-adaptive wavelet ridge demodulation based on LCDfor fault diagnosis is proposed in this paper Firstly LCDmethod is adopted to decompose the original signal intoa number of ISC components and lower frequency noiseis decreased by selecting special ISC component whichcontains rich fault feature Secondly the genetic algorithm isused to optimize wavelet parameters to obtain self-adaptivewavelet based on wavelet energy entropy which is servedas the objective function Thirdly the self-adaptive waveletridge demodulation for the selected ISC component is usedto extract dynamic information Finally the characteristicsfrequency can be obtained to identify the working state orfailure information from the spectrum The flowchart of theproposed fault diagnosis method was illustrated in Figure 2

To verify the validity of the proposed method let usconsider the following signal

119909 (119905) = 1199091(119905) + 119909

2(119905)

1199091(119905) = (1 + 05 cos 20120587119905) sin (200120587119905 + 2 cos 20120587119905)

1199092(119905) = sin120587119905 sin 20120587119905 119905 isin [0 1]

(18)

Obviously 119909(119905) is a complexmulticomponentAM-FM signalcontaining AM-FM component 1199091(119905) and AM component1199092(119905) The sampling frequency is 1000Hz Time domainwaveforms of simulation signal 119909(119905) and its LCD decompo-sition results are shown in Figure 3 where the two ISC com-ponents ISC1 and ISC2 correspond to the two components1199091(119905) and 1199092(119905)

Then the wavelet parameters are optimized accordingto the signal itself using genetic algorithms For ISC1 theoptimal parameters are determined as 119891

119887= 22316 and

119891119888= 10532 Figure 4 shows the evolution curve of GA For

ISC2 the optimal parameters are 119891119887

= 41203 and 119891119888

=

10012 Ultimately the signal is transformed self-adaptivelyusing Morlet wavelet to extract wavelet ridge and ISC1 andISC2 are demodulated based on the formula (10) and (11)The demodulation results are shown in Figures 5 and 6respectively

Genetic manipulation

crossover and mutation)

Optimal solution

Yes

No

Starting

Wavelet decomposition

Fitness value calculation and evaluation of individual fitness

Termination conditions

End

Population initialization

and energy entropy

(selection reproduction

Figure 1 Flowchart of wavelet parameters optimization by GA

For the instantaneous amplitude of ISC1 to eliminate theborder effect of the wavelet transformation the boundary isprocessed by symmetric extensionmethodThe result is givenin Figure 7

From the above analysis results it can be seen that self-adaptive wavelet ridge demodulation method based on LCDcan precisely demodulate a complex multicomponent AM-FMsignal In order tomake comparative analysis the demod-ulation analysis results of ISC1 using Hilbert demodulationmethod are given in Figure 8 fromwhich it can be found thatthe demodulation curve is not smooth and the demodulationerror is bigger due to inevitable window effects by Hilbertdemodulation method [4] The above comparison results

Shock and Vibration 5

Collect vibration signals

Apply LCD decomposition

Extract the fault characteristic

Select useful ISC component

Apply self-adaptive wavelet ridge

demodulation to the selected ISC

Optimize wavelet parameters by GA

Identify fault type

Figure 2 Flowchart of the proposed method

indicate the proposed method is superior to the Hilbertdemodulation method

Let us consider another multicomponent AM-FM signal119910(119905)which is a simulated faulty signal of gearbox and definedas follows

119910 (119905) = 1199101(119905) + 119910

2(119905)

1199101(119905) = [04 + 02 cos (2120587 times 10119905)]

sdot sin [2120587 times 600119905 + cos (2120587 times 20119905)]

1199102(119905) = [1 + 05 cos (2120587 times 20119905)]

sdot sin [2120587 times 300119905 + cos (2120587 times 15119905)]

(19)

Time (s)

Am

plitu

deA

mpl

itude

Am

plitu

deA

mpl

itude

0

0

5

0

2

0

1

005

0

01 02 03 04 05 06 07 08 09 1

Time (s) 0 01 02 03 04 05 06 07 08 09 1

Time (s) 0 01 02 03 04 05 06 07 08 09 1

Time (s) 0 01 02 03 04 05 06 07 08 09 1

minus5

minus2

x(t

)IS

C 1IS

C 2

minus1

minus005

u(t

)Figure 3 Time domain waveform of simulation signal 119909(119905) and itsLCD decomposition result

Wav

elet

ener

gy en

tropy

val

ue

0 5 10 15 20 25 30

14

145

15

155

16

165

135

Optimal valueMean value

times10minus3

Figure 4 Evolution curve of GA

The time domain waveform is shown in Figure 9 Thesampling frequency is 2000Hz Through LCD method twoISC components are presented in Figure 10 Obviously thetwo ISC components are consistent with the two components1199101(119905) and 1199102(119905) By optimizing wavelet energy entropy self-adaptive wavelet parameters for the first ISC signal aredetermined as 119891

119887= 97752 and 119891

119888= 07428 the parameters

for the second ISC signal as 119891119887

= 40585 and 119891119888

=

10678 The self-adaptive wavelet transform results for thetwo ISCs are provided in Figures 11 and 12 It can be foundthat the IArsquos frequency of the first ISC signal is 10Hz and

6 Shock and Vibration

0 01 02 03 04 05 06 07 08 09 1040608

11214

80

90

100

110

120

IA waveform

IF waveform

Am

plitu

deFr

eque

ncy

Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

Figure 5 Self-adaptive wavelet ridge demodulation results of the 1stISC

0 01 02 03 04 05 06 07 08 09 10

02040608

1

859

9510

10511

Freq

uenc

yA

mpl

itude

IF waveform

IA waveform

Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

Figure 6 Self-adaptive wavelet ridge demodulation results of the2nd ISC of simulation signal

IA waveform

0 01 02 03 04 05 06 07 08 09 104

06

08

1

12

14

Am

plitu

de

Time (s)

Figure 7 Processed instantaneous amplitude of the 1st ISC

0 01 02 03 04 05 06 07 08 09 10

05

1

15

2

8090

100110120130

IA waveform

IF waveform

Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

Am

plitu

deFr

eque

ncy

Figure 8 Hilbert demodulation results of the 1st ISC of simulationsignal

0

2

4

Am

plitu

de

minus2

minus40 01 02 03 04 05 06 07 08 09 1

Time (s)

Figure 9 Waveform of simulation signal 119910(119905)

Am

plitu

de

0 01 02 03 04 05 06Time (s)

07 08 09 1

0 01 02 03 04 05 06Time (s)

07 08 09 1

0

05

1

0

1

2

Am

plitu

de

minus05

minus1

minus1

minus2

ISC 1

ISC 2

Figure 10 LCD results of simulation signal 119910(119905)

Shock and Vibration 7

200

400

600

Scal

e

Time-scale distribution

Am

plitu

de

Time (s)

Freq

uenc

y (H

z)

IA waveform

IF waveform

020304050607

0 01 02 03 04 05 06 07 08 09 1

Time (s)0 01 02 03 04 05 06 07 08 09 1

Time (s)0 01 02 03 04 05 06 07 08 09

550

600

650

700

Figure 11 Demodulation results of the 1st ISC of simulation signal119910(119905) by the proposed method

the IFrsquos frequency is 20Hz and that IArsquos frequency of thesecond ISC signal is 20Hz and the IFrsquos frequency is 15HzThese results are consistent with the two components in theoriginal simulation signal Therefore the proposed methodcan effectively demodulate multicomponent AM-FM signaland is suitable for fault diagnosis of gearbox

For comparison we use EMD method to make signaldecomposition and use self-adaptive wavelet ridge demodu-lation approach to demodulate We obtained eight IMFs intotal After analysis we noted the first two IMFs reflected themodulation characteristics hence they are considered as thetwo components The demodulation results of the first twoIMFs by self-adaptive wavelet ridge demodulation approachare provided in Figures 13 and 14 It can be seen that theFM component of the first IMF failed to be demodulatedand the energy of the spectra of the second IMFs is lowerThen energy operator demodulation based on EMD [4] isused and the results are shown in Figure 15 From thesefigures it is shown that the serious mode mixing exists inthe EMDdecomposition which influences the demodulationaccuracy However from Figures 11 and 12 it is noted thatLCD approach may diminish this problem and be superiorto EMD approach

Finally we add a Gaussian white noise with deviation of02 to 119910(119905) LCD decomposes the noisy signal into five ISCsBy the method introduced in Section 32 the optimal Morlet

0 01 02 03 04 05 06 07 08 09

100

200

300

400

500

Time (s)

Scal

e

Time-scale distribution

0 01 02 03 04 05 06 07 08 09Time (s)

1

0 01 02 03 04 05 06 07 08 09Time (s)

1

05

1

15

280

290

300

310

320

Am

plitu

deFr

eque

ncy

IA waveform

IF waveform

Figure 12 Demodulation results of the 2nd ISC of simulation signal119910(119905) by the proposed method

wavelet parameters for the first two ISCs which containmodulation information are determined The demodulationresults of the first two ISCs are shown in Figure 16The resultsby energy operator demodulation based on LCD to the firsttwo ISCs are given in Figure 17 It can be obviously found thatself-adaptive wavelet demodulation approach based on LCDhas better noise tolerant performance than energy operatordemodulation approach based on LCD

5 Application to Incipient Fault Diagnosis

51 Gear Crack Fault Diagnosis A gear crack fault diagnosisexperiment is carried out on bearing-gear test rig as shown inFigure 18 In this test themotor power is 600W both drivinggear and driven gear are standard spur gear whose modulusis 25mm and the number of teeth is 37The input and outputshafts are arranged in parallel They are supported by tworoller bearings A crack with 015mm width and 1mm depthat the root of the driving gear tooth is set by wire cuttingmachining to simulate the gear incipient crack failure Thevibration signals were collected by an accelerometer attachedto the bearing housingThe shaft speed is 360 revmin that isthe drive shaft rotation frequency is 119891

119903= 6Hz The sampling

frequency is 1024Hz and the length of sampling data is 1024pointThedomainwaveformof the vibration signalmeasured

8 Shock and Vibration

0 01 02 03 04 05 06 07 08 09 10

01

02

03

04IA component of the 1st IMF

Time (s)

Am

plitu

de

0 01 02 03 04 05 06 07 08 09 1100

150

200

250IF component of the 1st IMF

Time (s)

Am

plitu

de

0 20 40 60 80 100 120 140 160 180 2000

0102030405

Spectrum

Am

plitu

de

01020304050 Spectrum

Am

plitu

de

Frequency (Hz)

0 20 40 60 80 100 120 140 160 180 200Frequency (Hz)

X 10Y 006344

X 15Y 2169

Figure 13 Demodulation results of the 1st IMF of simulation signal119910(119905) by self-adaptive wavelet ridge demodulation approach based onEMD

and envelope spectrum are shown in Figure 19 From whichit can be seen that fault characteristics are submerged by thebackground noise and the characteristics frequency fail to beidentified

Since the vibration signal with gear crack is a multi-component AM-FM signal (AM-FM) we used the proposedmethod for fault diagnosis Firstly the vibration accelerationsignal was decomposed into four ISC components ISC

1simISC4

and a residual component 119903 by LCD method as shown inFigure 20 Because the carrier frequency of gear vibration sig-nals is generally gear meshing frequency and its harmonicswe select the first ISC with the highest frequency for analysisFigure 21 shows the Hilbert demodulation results of thefirst ISC showing instantaneous amplitude contains complexhigh-frequency interference and some negative frequency

0 01 02 03 04 05 06 07 08 09 1040506070809

0 20 40 60 80 120100 140 160 180 2000

0102030405

Spectrum

IA component of the 2nd IMF

IF component of the 2nd IMF

Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

Am

plitu

deA

mpl

itude

Frequency (Hz)

0 20 40 60 80 120100 140 160 180 200Frequency (Hz)

Spectrum

767778798081

0

5

10

15

20

Am

plitu

deA

mpl

itude

X 20Y 003938

X 15Y 1266

Figure 14Demodulation results of the 2nd IMFof simulation signal119910(119905) by self-adaptive wavelet ridge demodulation approach based onEMD

exists in the instantaneous frequency waveform which isrelated to the measured signal

Then the optimum wavelet parameters 119891119887= 48574 and

119891119888= 07832 were selected Lastly the self-adaptive wavelet

ridge demodulation was carried out The IA waveform andits frequency spectrum obtained are shown in Figure 22from which the fault characteristics frequency 119891

119903= 6Hz

and its harmonics 3119891119903 4119891119903 and 5119891

119903are clearly found These

demonstrate that a local defect has occurred in the gear onthe drive shaft which is consistent with the drive gear stateThat is the proposed method is effective for gear crack faultdiagnosis

In addition we use the self-adaptive wavelet ridgedemodulation approach to analyse the original signal asexhibited in Figure 23 where the spectrum contains the faultcharacteristics frequency and its harmonic as well which

Shock and Vibration 9

0 01 02 03 04 05 06 07 08 09 10

051

152

25

0

200

400

600

800

Freq

uenc

yA

mpl

itude

Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

IA waveform of 1st IMF

IA waveform of 2nd IMF

IF waveform of 2nd IMF

IF waveform of 1st IMF

0

05

1

15

2

0100200300400500

Am

plitu

deFr

eque

ncy

Figure 15 Demodulation results of simulation signal 119910(119905) by energyoperator demodulation based on EMD

outperform the excellent time-frequency localization abilityof self-adaptive wavelet But some unknown frequency isinvolved because of background noise Moreover it can benoted that the harmonic in Figure 22 is richer and clearerthan that in Figure 23 In fact through signal decompositionby LCDand ISC selectionmost of noise can be removed fromanalysis signal Therefore the proposed method is effectiveand superior in application to weak fault diagnosis for gear

52 Bearing Inner-Race Fault Diagnosis The vibration signalof roller bearing with inner-race fault is complex and weakit is difficult to identify the fault state To further verifythe effectiveness of the proposed method we made bearinginner-race fault diagnosis experiment

The data was downloaded from the website of theCase Western Reserve University Bearing Center [24] Thetest stand consists of a 2 hp motor a torque transducera dynamometer and control electronics The test bearing

0 01 02 03 04 05 06 07 08 09 1010203040506

222224226228230232

IA waveform of 1st ISC

IF waveform of 1st ISC

IF waveform of 2nd ISC

IA waveform of 2nd ISC

Am

plitu

de

Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

Freq

uenc

y

040608

11214

155

160

165

170

Am

plitu

deFr

eque

ncy

Figure 16 Demodulation results of simulation signal with noise bythe proposed method

is at the drive end a single point defect was introducedinto the inner raceway of the test bearing The size of thesingle point defect is 0178mm in diameter and 0279mm indepth using electrodischarge machining An accelerometerattached to the bearing housing collected vibration data withthe sampling frequency as 12 kHz The shaft rotating speedof the bearing inner-race is 1750 revmin the characteristicfrequency of the roller bearing with inner-race fault is 119891

119900119894=

158HzFigure 24 presents the time domain waveform and spec-

trum of a bearing vibration signal From the spectrum it canbe seen that there are three center frequency bands and theirside frequency bands which show that main modulationcharacteristics exhibits in the high frequency band Howeverthe fault characteristic frequency is not clear in the spectrumHere we used the proposed method to demodulate Thebearing vibration signal was decomposed into thirteen ISCs

10 Shock and Vibration

0

1

2

3

0

200

400

600

800

IA waveform of 1st ISC

IF waveform of 1st ISC

Am

plitu

de

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

Freq

uenc

y

005

115

225

0

200

400

600

IA waveform of 2nd ISC

IF waveform of 2nd ISC

Am

plitu

deFr

eque

ncy

Figure 17 Demodulation results of simulation signal with noise byenergy operator demodulation approach based on LCD

and the first ISC with the highest frequency as shown inFigure 25 was selected for analysis The optimal Morletwavelet parameters were determined as 119891

119887= 128576 and

119891119888= 06862The IA waveform and its spectrum are shown in

Figure 26 from which we can clearly see the spectrum linesat the fault characteristic frequency 119891

119900119894and its harmonics

which shows that a local failure occurs in the inner racewayof bearing This is consistent with the fact state

Besides through comparison of Figure 24 with Figure 25we can find that applying LCD decomposition equals todesign adaptive band-pass filter whose center frequency isautomatically determined with the inherent characteristicsof analyzed signal Simultaneously the selection of specialISC with higher frequency for analysis leads to decrease ofthe influence of lower frequency noise to suit weak faultfeature extraction In addition due to concern of main

1 2 3 4 5 6 7 8 9 10 11

(1) Induction motor(2) Speed controller(3) Coupling(4) Bearing number 1 and bearing housing(5) Bearing number 2 and bearing housing(6) Drive gear(7) Driven gear

(9) Vibration acceleration sensor(10) Speed sensor

(8) Bearing number 3 and bearing housing

(11) Bearing number 4 and bearing housing

Figure 18 Bearing-gear test rig for vibration signal acquisition

0 01 02 03 04 05 06 07 08 09 1

0

50

100

Time (s)

Time domain waveform

0 50 100 150 200 250 300 350 4000

5

10

15

20 Envelope spectrum

minus50

minus100

Frequency (Hz)

X 21Y 914

Am

plitu

de (m

middotsminus2)

Acce

lera

tion

(mmiddotsminus

2)

Figure 19 Time domain waveform and envelope spectrum of afaulty gear vibration signal

energy on the time-scale distribution self-adaptive waveletridge modulation has excellent time-frequency localizationproperty as seen in Figure 23 Therefore the applicationresults show the proposed method is simple and effective forweak bearing inner-race fault diagnosis

6 Conclusion

LCD method is a new signal decomposition approachwhich is suitable to preprocess the multicomponent AM-FM signal Wavelet ridge demodulation method is based onwavelet transform in the time-frequency domain focusing

Shock and Vibration 11

0 01 02 03 04 05 06 07 08 09 1

200

100

50

010

Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

minus50

minus20

minus10

minus10

minus5

0

50

ISC 1

ISC 2

ISC 3

ISC 4

u(t

)

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

Figure 20 LCD decomposition results of a faulty gear vibration signal

IA waveform

0

20

40

60

80

0

500

1000IF waveform

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

Acce

lera

tion

(mmiddotsminus

2)

minus500

minus1000

Freq

uenc

y (H

z)

Figure 21 Demodulation results of the 1st ISC by Hilbert approach

on main energy on the time-scale distribution It is lesssensitive to signal bandwidth and noise especially usingthe wavelet transform with optimal parameters called self-adaptive wavelet Therefore with combination of the abovetwo approaches a self-adaptive wavelet ridge demodulationmethod based on LCD for fault diagnosis is presented

Using two simulation signals we compare the proposedmethod with the following four approaches Hilbert demod-ulation energy operator demodulation based on EMD self-adaptive wavelet ridge demodulation based on EMD andenergy operator demodulation based on LCD Analysis

0 01 02 03 04 05 06 07 08 09 10

1020304050

0 10 20 30 40 706050 80 90 10002468

10

Time (s)

IA waveform

Spectrum

Frequency (Hz)

fr

3fr4fr 5fr

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

Figure 22 Instantaneous amplitude and its spectrum of the 1st ISCby the proposed method

results show that the proposedmethod has higher demodula-tion accuracy and higher noise tolerant performance than theothers Finally we applied the proposed method to incipientfault diagnosis of gearbox The application results show theproposed method is simple and effective for incipient faultdiagnosis of gearbox

Conflict of Interests

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

12 Shock and Vibration

0 01 02 03 04 05 06 07 08 09 10

20

40

60

80

0 10 20 30 40 50 60 70 80 90 10002468

10

Time (s)

IA waveform

Spectrum

5fr

fr

Frequency (Hz)

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

Figure 23 Instantaneous amplitude and its spectrum of the orig-inal vibration signal by self-adaptive wavelet ridge demodulationapproach

0 01 02 03 04 05 06 07 08 09 1

0

1

2

0 1000 2000 3000 4000 5000 60000

005

01

015

02

Waveform

Spectrum

Time (s)

Frequency (Hz)

minus2

minus1

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

Figure 24 Time domain waveform and spectrum of a roller bearingvibration signal with inner-race defect

Acknowledgments

The support from Chinese National Science FoundationGrant (no 51075131) the construct program of the key dis-cipline in Hunan Province (Mechanical Design and Theory)(XJF2011 [76]) cooperative Demonstration Base of Universi-ties in Hunan ldquoR amp D and Industrialization of Rock DrillingMachinesrdquo (XJT [2014] 239) Hunan Major Special Projectsof Science and Technology (2014GK1043) and scientificresearch project of Hunan Province Education Department(no 14C0789) is greatly acknowledged The authors wouldlike to express their appreciation to Case Western ReserveUniversity for offering the free bearing data

0 1000 2000 3000 4000 5000 60000

005

01

015

02

Waveform

Spectrum

0

1

2

0 01 02 03 04 05 06 07 08 09 1Time (s)

minus2

minus1

Frequency (Hz)

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

Figure 25 Time domain waveform and spectrum of the 1st ISC ofa roller bearing vibration signal with inner-race defect

0 01 02 03 04 05 06 07 08 09 10

02040608

1

0 100 200 300 400 500 600 700 800 900 10000

005

01

015

02

IA waveform

Spectrum

Time (s)

Frequency (Hz)

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

foi

2foi

3foi

Figure 26 Instantaneous amplitude waveform and spectrum of the1st ISC of a roller bearing vibration signal with inner-race defect

References

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

[2] M Feldman ldquoNonparametric identification of asymmetricnonlinear vibration systems with the Hilbert transformrdquo Jour-nal of Sound and Vibration vol 331 no 14 pp 3386ndash3396 2012

[3] N E Huang Z Shen S R Long et al ldquoThe empiricalmode decomposition and the Hilbert spectrum for nonlinearand non-stationary time series analysisrdquo The Royal Societyof London Proceedings Series A Mathematical Physical andEngineering Sciences vol 454 no 1971 pp 903ndash995 1998

[4] J Cheng D Yu and Y Yang ldquoThe application of energyoperator demodulation approach based on EMD in machinery

Shock and Vibration 13

fault diagnosisrdquo Mechanical Systems and Signal Processing vol21 no 2 pp 668ndash677 2007

[5] Z Feng M Liang Y Zhang and S Hou ldquoFault diagnosis forwind turbine planetary gearboxes via demodulation analysisbased on ensemble empirical mode decomposition and energyseparationrdquo Renewable Energy vol 47 pp 112ndash126 2012

[6] Y Qin ldquoMulticomponent AM-FM demodulation based onenergy separation and adaptive filteringrdquo Mechanical Systemsand Signal Processing vol 38 no 2 pp 440ndash459 2013

[7] M Liang and H Faghidi ldquoIntelligent bearing fault detection byenhanced energy operatorrdquo Expert Systems with Applicationsvol 41 no 16 pp 7223ndash7234 2014

[8] J Wang and Q He ldquoExchanged ridge demodulation of time-scale manifold for enhanced fault diagnosis of rotating machin-eryrdquo Journal of Sound and Vibration vol 333 no 11 pp 2450ndash2464 2014

[9] Y Qin S Qin and Y Mao ldquoDemodulation approach based onwavelet ridge and its application in fault diagnosis of rotatingmachineryrdquo Chinese Journal of Mechanical Engineering vol 45no 2 pp 231ndash237 2009

[10] W He Z Jiang and Q Qin ldquoA joint adaptive wavelet filter andmorphological signal processing method for weak mechanicalimpulse extractionrdquo Journal of Mechanical Science and Technol-ogy vol 24 no 8 pp 1709ndash1716 2010

[11] Y Jiang B Tang YQin andW Liu ldquoFeature extractionmethodof wind turbine based on adaptive Morlet wavelet and SVDrdquoRenewable Energy vol 36 no 8 pp 2146ndash2153 2011

[12] W Su FWang H Zhu Z Zhang and Z Guo ldquoRolling elementbearing faults diagnosis based on optimal Morlet wavelet filterand autocorrelation enhancementrdquo Mechanical Systems andSignal Processing vol 24 no 5 pp 1458ndash1472 2010

[13] H Qiu J Lee J Lin and G Yu ldquoWavelet filter-based weaksignature detection method and its application on rollingelement bearing prognosticsrdquo Journal of Sound and Vibrationvol 289 no 4-5 pp 1066ndash1090 2006

[14] M Amarnath and I Praveen Krishna ldquoLocal fault detection inhelical gears via vibration and acoustic signals using EMDbasedstatistical parameter analysisrdquo Measurement vol 58 pp 154ndash164 2014

[15] Y Lei J Lin Z He and M J Zuo ldquoA review on empiricalmode decomposition in fault diagnosis of rotating machineryrdquoMechanical Systems and Signal Processing vol 35 no 1-2 pp108ndash126 2013

[16] J S Smith ldquoThe local mean decomposition and its applicationto EEG perception datardquo Journal of the Royal Society Interfacevol 2 no 5 pp 443ndash454 2005

[17] H Liu and M Han ldquoA fault diagnosis method based onlocal mean decomposition and multi-scale entropy for rollerbearingsrdquo Mechanism and Machine Theory vol 75 pp 67ndash782014

[18] J Cheng and Y Yang ldquoA rotating machinery fault diagnosismethod based on local mean decompositionrdquo Digital SignalProcessing vol 22 no 2 pp 356ndash366 2012

[19] J Zheng J Cheng and Y Yang ldquoA rolling bearing fault diagno-sis approach based on LCD and fuzzy entropyrdquoMechanism andMachine Theory vol 70 pp 441ndash453 2013

[20] J Zheng J Cheng Y Nie and S Luo ldquoA signal pro-cessing method for fault diagnosis-complete ensemble localcharacteristic-scale decompositionrdquo Journal of Vibration Engi-neering vol 27 no 4 pp 637ndash646 2014

[21] H L Ao J Cheng K Li andTK Truong ldquoA roller bearing faultdiagnosis method based on LCD energy entropy and ACROA-SVMrdquo Shock and Vibration vol 2014 Article ID 825825 12pages 2014

[22] MUnalMOnatMDemetgul andHKucuk ldquoFault diagnosisof rolling bearings using a genetic algorithm optimized neuralnetworkrdquoMeasurement vol 58 pp 187ndash196 2014

[23] B Long W Xian M Li and H Wang ldquoImproved diagnosticsfor the incipient faults in analog circuits using LSSVM based onPSO algorithm with Mahalanobis distancerdquo Neurocomputingvol 133 no 10 pp 237ndash248 2014

[24] CWRU Bearing Data Center Seeded Fault Test Data 2013httpcsegroupscaseedubearingdatacenterhome

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Shock and Vibration 3

As to wavelet ridge point it can be seen that 1205960119886119903 minus1205931015840

119904(119887) = 0 from formula (6) So the wavelet coefficients

modulus of wavelet ridge point is further expressed as

1003816100381610038161003816119882119904(119886119903 119887)

1003816100381610038161003816asymp

radic119886

2

119860119904(119887)

1003816100381610038161003816 (0)

1003816100381610038161003816 (9)

So it can be seen that signal instantaneous frequencycan be gained after the wavelet ridge is extracted which isexpressed as

119891119904(119905) =

1

2120587

1205960

119886119903(119905)

(10)

where 1205960is the center frequency of (119905) that is 120596

0= 1205931015840

120595(0)

At the same time the signal instantaneous amplitude canbe expressed as

119860119904(119905) asymp

21003816100381610038161003816119882119904(119886119903(119905) 119905)

1003816100381610038161003816

radic119886119903(119905)

1003816100381610038161003816 (0)

1003816100381610038161003816

(11)

As presented above it is clear that the demodulationanalysis of monocomponent AM-FM signal based onwaveletridge is feasible Howevermost vibration signals produced bygearbox failures are generally multicomponent AM-FM sig-nalsThey should be decomposed intomonocomponentAM-FM signals by appropriate time-frequency signal processingmethod before demodulation In this paper LCD method isemployed to accomplish the signal decomposition

The LCD method has the assumptions that a complexsignal consists of a number of ISCs (Intrinsic Scale Compo-nent ISC) and any two ISCs are independent of each otherIn the entire data segment ISC must meet the following twoconditions

(I) The maximal value is positive the minimum value isnegative and the data set are monotonic between anytwo adjacent extreme points

(II) Let all the extreme points be written as (119883119896 120591119896)

119896 = 1 2 119872 the line 119897119896determined by any two

adjacent extreme points (119883119896 120591119896) and (119883

119896+2 120591119896+2) canbe expressed as

119897119896=

119883119896+2

minus 119883119896

120591119896+2

minus 120591119896

(119905 minus 120591119896) +119883119896 (12)

Remembering the value of 119897119896at the 120591

119896+1as 119860119896+1

therelation between 119860

119896+1and119883

119896+1should meet the following

120572119860119896+1

+ (1 minus 120572)119883119896+1

= 0 (13)

when 120572 = 05 119860119896+1

= minus119883119896+1

Based on this definition a complexmulticomponentAM-

FM signal can be decomposed into the sumof a finite numberof ISCs and a residual signal Each ISC is a monocomponentAM-FM signal whose instantaneous frequency has specificphysical meaning That is

119909 (119905) =

119873

sum

119894=1

119888119894(119905) + 119906 (119905) (14)

where 119888119894(119905) is the 119894th ISC component and 119906(119905) is the residual

signal

3 Self-Adaptive Wavelet

31 MorletWavelet Frequency Resolution When there is localfailure for gear the fault gear teeth will stimulate systemto produce a convergent impact response and the vibrationsignal collected by the acceleration sensor shows the obviousmulticomponent modulation characteristic Therefore asmentioned above we can adopt wavelet ridge demodulationbased on LCD to extract the fault feature In order to matchthis kind of signal analytic Morlet wavelet with impactfeature is chosen which is defined as

(119905) =

1

radic119891119887120587

exp(minus 1199052

119891119887

) exp (1198952120587119891119888119905) (15)

The Fourier transform of 120595(119905) is represented as

(2120587119891) = exp[(minus2120587119891 minus 2120587119891

119888

2radic119891119887

)

2

] (16)

where 119891119887is the shape factor and 119891

119888is the center frequency

whose numerical values determine the speed of thewaveformvibration damping respectively from formula (16) Morletwavelet quality factor is 119876 = radic2120587119891

119888radic119891119887radicln2 So the best

frequency resolution can be gained by adjusting 119891119887and 119891

119888

which can result in a good time-scale accumulation

32The Procedure of Obtaining Self-AdaptiveWavelet Sparsedegree of wavelet coefficients can characterize the degreeof similarity between the basic wavelet function and signalThe energy entropy of wavelet can indicate this sparsedegree which shows accumulation performance of waveletcoefficients As to each specific scale 119895 the wavelet energyentropy is defined as

119882119864= minussum

119895

119875119895ln119875119895 (17)

where 119875119895= 119864119895119864119879is the probability of energy distribution

(sum119875119895= 1) 119864

119895= int119882

2

(119895 120591)119889120591 is the wavelet energy 119882(119895 120591)

is the wavelet coefficient and 119864119879= sum119895119864119895is the total wavelet

energy within the time scale planeAccordingly the wavelet energy entropy is taken as

the objective function during selecting the optimal waveletparameters To optimize wavelet parameters genetic algo-rithm (GA) and particle swarm optimization (PSO) [2223] are two widely utilized approaches In this paper GAis employed to optimize either envelope factor or centerfrequency with wavelet energy entropy as the fitness func-tion That wavelet with optimal parameters is called self-adaptive wavelet The procedure to get self-adaptive waveletis described as follows

Step 1 Set search prime range and population size of param-eters 119891

119887and 119891

119888 and randomly generate initial population

In this paper the population size is set to 100 and theparameters are respectively encoded as 10-bit binary stringchromosomes by binary coding method

4 Shock and Vibration

Step 2 Make wavelet decomposition of the signal and calcu-late the fitness value of each individual according to formula(17) Then sort the fitness values by size

Step 3 Based on individual fitness value in the search spaceindividuals are screened and evolved by a series of geneticmanipulation selection reproduction crossover mutationand so forth to constantly update and select populations

Step 4 In this step it is determined whether iterationsatisfies the termination condition or not If satisfied theoptimal solution is finished If not satisfied go to Step 2 untilthe optimal solution is got The optimization procedure ofwavelet parameters via GA is shown in Figure 1

4 The Proposed Method and Simulation

Since demodulation technique is an effectiveway to reveal thefault characteristic frequency for fault diagnosis of gearboxa self-adaptive wavelet ridge demodulation based on LCDfor fault diagnosis is proposed in this paper Firstly LCDmethod is adopted to decompose the original signal intoa number of ISC components and lower frequency noiseis decreased by selecting special ISC component whichcontains rich fault feature Secondly the genetic algorithm isused to optimize wavelet parameters to obtain self-adaptivewavelet based on wavelet energy entropy which is servedas the objective function Thirdly the self-adaptive waveletridge demodulation for the selected ISC component is usedto extract dynamic information Finally the characteristicsfrequency can be obtained to identify the working state orfailure information from the spectrum The flowchart of theproposed fault diagnosis method was illustrated in Figure 2

To verify the validity of the proposed method let usconsider the following signal

119909 (119905) = 1199091(119905) + 119909

2(119905)

1199091(119905) = (1 + 05 cos 20120587119905) sin (200120587119905 + 2 cos 20120587119905)

1199092(119905) = sin120587119905 sin 20120587119905 119905 isin [0 1]

(18)

Obviously 119909(119905) is a complexmulticomponentAM-FM signalcontaining AM-FM component 1199091(119905) and AM component1199092(119905) The sampling frequency is 1000Hz Time domainwaveforms of simulation signal 119909(119905) and its LCD decompo-sition results are shown in Figure 3 where the two ISC com-ponents ISC1 and ISC2 correspond to the two components1199091(119905) and 1199092(119905)

Then the wavelet parameters are optimized accordingto the signal itself using genetic algorithms For ISC1 theoptimal parameters are determined as 119891

119887= 22316 and

119891119888= 10532 Figure 4 shows the evolution curve of GA For

ISC2 the optimal parameters are 119891119887

= 41203 and 119891119888

=

10012 Ultimately the signal is transformed self-adaptivelyusing Morlet wavelet to extract wavelet ridge and ISC1 andISC2 are demodulated based on the formula (10) and (11)The demodulation results are shown in Figures 5 and 6respectively

Genetic manipulation

crossover and mutation)

Optimal solution

Yes

No

Starting

Wavelet decomposition

Fitness value calculation and evaluation of individual fitness

Termination conditions

End

Population initialization

and energy entropy

(selection reproduction

Figure 1 Flowchart of wavelet parameters optimization by GA

For the instantaneous amplitude of ISC1 to eliminate theborder effect of the wavelet transformation the boundary isprocessed by symmetric extensionmethodThe result is givenin Figure 7

From the above analysis results it can be seen that self-adaptive wavelet ridge demodulation method based on LCDcan precisely demodulate a complex multicomponent AM-FMsignal In order tomake comparative analysis the demod-ulation analysis results of ISC1 using Hilbert demodulationmethod are given in Figure 8 fromwhich it can be found thatthe demodulation curve is not smooth and the demodulationerror is bigger due to inevitable window effects by Hilbertdemodulation method [4] The above comparison results

Shock and Vibration 5

Collect vibration signals

Apply LCD decomposition

Extract the fault characteristic

Select useful ISC component

Apply self-adaptive wavelet ridge

demodulation to the selected ISC

Optimize wavelet parameters by GA

Identify fault type

Figure 2 Flowchart of the proposed method

indicate the proposed method is superior to the Hilbertdemodulation method

Let us consider another multicomponent AM-FM signal119910(119905)which is a simulated faulty signal of gearbox and definedas follows

119910 (119905) = 1199101(119905) + 119910

2(119905)

1199101(119905) = [04 + 02 cos (2120587 times 10119905)]

sdot sin [2120587 times 600119905 + cos (2120587 times 20119905)]

1199102(119905) = [1 + 05 cos (2120587 times 20119905)]

sdot sin [2120587 times 300119905 + cos (2120587 times 15119905)]

(19)

Time (s)

Am

plitu

deA

mpl

itude

Am

plitu

deA

mpl

itude

0

0

5

0

2

0

1

005

0

01 02 03 04 05 06 07 08 09 1

Time (s) 0 01 02 03 04 05 06 07 08 09 1

Time (s) 0 01 02 03 04 05 06 07 08 09 1

Time (s) 0 01 02 03 04 05 06 07 08 09 1

minus5

minus2

x(t

)IS

C 1IS

C 2

minus1

minus005

u(t

)Figure 3 Time domain waveform of simulation signal 119909(119905) and itsLCD decomposition result

Wav

elet

ener

gy en

tropy

val

ue

0 5 10 15 20 25 30

14

145

15

155

16

165

135

Optimal valueMean value

times10minus3

Figure 4 Evolution curve of GA

The time domain waveform is shown in Figure 9 Thesampling frequency is 2000Hz Through LCD method twoISC components are presented in Figure 10 Obviously thetwo ISC components are consistent with the two components1199101(119905) and 1199102(119905) By optimizing wavelet energy entropy self-adaptive wavelet parameters for the first ISC signal aredetermined as 119891

119887= 97752 and 119891

119888= 07428 the parameters

for the second ISC signal as 119891119887

= 40585 and 119891119888

=

10678 The self-adaptive wavelet transform results for thetwo ISCs are provided in Figures 11 and 12 It can be foundthat the IArsquos frequency of the first ISC signal is 10Hz and

6 Shock and Vibration

0 01 02 03 04 05 06 07 08 09 1040608

11214

80

90

100

110

120

IA waveform

IF waveform

Am

plitu

deFr

eque

ncy

Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

Figure 5 Self-adaptive wavelet ridge demodulation results of the 1stISC

0 01 02 03 04 05 06 07 08 09 10

02040608

1

859

9510

10511

Freq

uenc

yA

mpl

itude

IF waveform

IA waveform

Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

Figure 6 Self-adaptive wavelet ridge demodulation results of the2nd ISC of simulation signal

IA waveform

0 01 02 03 04 05 06 07 08 09 104

06

08

1

12

14

Am

plitu

de

Time (s)

Figure 7 Processed instantaneous amplitude of the 1st ISC

0 01 02 03 04 05 06 07 08 09 10

05

1

15

2

8090

100110120130

IA waveform

IF waveform

Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

Am

plitu

deFr

eque

ncy

Figure 8 Hilbert demodulation results of the 1st ISC of simulationsignal

0

2

4

Am

plitu

de

minus2

minus40 01 02 03 04 05 06 07 08 09 1

Time (s)

Figure 9 Waveform of simulation signal 119910(119905)

Am

plitu

de

0 01 02 03 04 05 06Time (s)

07 08 09 1

0 01 02 03 04 05 06Time (s)

07 08 09 1

0

05

1

0

1

2

Am

plitu

de

minus05

minus1

minus1

minus2

ISC 1

ISC 2

Figure 10 LCD results of simulation signal 119910(119905)

Shock and Vibration 7

200

400

600

Scal

e

Time-scale distribution

Am

plitu

de

Time (s)

Freq

uenc

y (H

z)

IA waveform

IF waveform

020304050607

0 01 02 03 04 05 06 07 08 09 1

Time (s)0 01 02 03 04 05 06 07 08 09 1

Time (s)0 01 02 03 04 05 06 07 08 09

550

600

650

700

Figure 11 Demodulation results of the 1st ISC of simulation signal119910(119905) by the proposed method

the IFrsquos frequency is 20Hz and that IArsquos frequency of thesecond ISC signal is 20Hz and the IFrsquos frequency is 15HzThese results are consistent with the two components in theoriginal simulation signal Therefore the proposed methodcan effectively demodulate multicomponent AM-FM signaland is suitable for fault diagnosis of gearbox

For comparison we use EMD method to make signaldecomposition and use self-adaptive wavelet ridge demodu-lation approach to demodulate We obtained eight IMFs intotal After analysis we noted the first two IMFs reflected themodulation characteristics hence they are considered as thetwo components The demodulation results of the first twoIMFs by self-adaptive wavelet ridge demodulation approachare provided in Figures 13 and 14 It can be seen that theFM component of the first IMF failed to be demodulatedand the energy of the spectra of the second IMFs is lowerThen energy operator demodulation based on EMD [4] isused and the results are shown in Figure 15 From thesefigures it is shown that the serious mode mixing exists inthe EMDdecomposition which influences the demodulationaccuracy However from Figures 11 and 12 it is noted thatLCD approach may diminish this problem and be superiorto EMD approach

Finally we add a Gaussian white noise with deviation of02 to 119910(119905) LCD decomposes the noisy signal into five ISCsBy the method introduced in Section 32 the optimal Morlet

0 01 02 03 04 05 06 07 08 09

100

200

300

400

500

Time (s)

Scal

e

Time-scale distribution

0 01 02 03 04 05 06 07 08 09Time (s)

1

0 01 02 03 04 05 06 07 08 09Time (s)

1

05

1

15

280

290

300

310

320

Am

plitu

deFr

eque

ncy

IA waveform

IF waveform

Figure 12 Demodulation results of the 2nd ISC of simulation signal119910(119905) by the proposed method

wavelet parameters for the first two ISCs which containmodulation information are determined The demodulationresults of the first two ISCs are shown in Figure 16The resultsby energy operator demodulation based on LCD to the firsttwo ISCs are given in Figure 17 It can be obviously found thatself-adaptive wavelet demodulation approach based on LCDhas better noise tolerant performance than energy operatordemodulation approach based on LCD

5 Application to Incipient Fault Diagnosis

51 Gear Crack Fault Diagnosis A gear crack fault diagnosisexperiment is carried out on bearing-gear test rig as shown inFigure 18 In this test themotor power is 600W both drivinggear and driven gear are standard spur gear whose modulusis 25mm and the number of teeth is 37The input and outputshafts are arranged in parallel They are supported by tworoller bearings A crack with 015mm width and 1mm depthat the root of the driving gear tooth is set by wire cuttingmachining to simulate the gear incipient crack failure Thevibration signals were collected by an accelerometer attachedto the bearing housingThe shaft speed is 360 revmin that isthe drive shaft rotation frequency is 119891

119903= 6Hz The sampling

frequency is 1024Hz and the length of sampling data is 1024pointThedomainwaveformof the vibration signalmeasured

8 Shock and Vibration

0 01 02 03 04 05 06 07 08 09 10

01

02

03

04IA component of the 1st IMF

Time (s)

Am

plitu

de

0 01 02 03 04 05 06 07 08 09 1100

150

200

250IF component of the 1st IMF

Time (s)

Am

plitu

de

0 20 40 60 80 100 120 140 160 180 2000

0102030405

Spectrum

Am

plitu

de

01020304050 Spectrum

Am

plitu

de

Frequency (Hz)

0 20 40 60 80 100 120 140 160 180 200Frequency (Hz)

X 10Y 006344

X 15Y 2169

Figure 13 Demodulation results of the 1st IMF of simulation signal119910(119905) by self-adaptive wavelet ridge demodulation approach based onEMD

and envelope spectrum are shown in Figure 19 From whichit can be seen that fault characteristics are submerged by thebackground noise and the characteristics frequency fail to beidentified

Since the vibration signal with gear crack is a multi-component AM-FM signal (AM-FM) we used the proposedmethod for fault diagnosis Firstly the vibration accelerationsignal was decomposed into four ISC components ISC

1simISC4

and a residual component 119903 by LCD method as shown inFigure 20 Because the carrier frequency of gear vibration sig-nals is generally gear meshing frequency and its harmonicswe select the first ISC with the highest frequency for analysisFigure 21 shows the Hilbert demodulation results of thefirst ISC showing instantaneous amplitude contains complexhigh-frequency interference and some negative frequency

0 01 02 03 04 05 06 07 08 09 1040506070809

0 20 40 60 80 120100 140 160 180 2000

0102030405

Spectrum

IA component of the 2nd IMF

IF component of the 2nd IMF

Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

Am

plitu

deA

mpl

itude

Frequency (Hz)

0 20 40 60 80 120100 140 160 180 200Frequency (Hz)

Spectrum

767778798081

0

5

10

15

20

Am

plitu

deA

mpl

itude

X 20Y 003938

X 15Y 1266

Figure 14Demodulation results of the 2nd IMFof simulation signal119910(119905) by self-adaptive wavelet ridge demodulation approach based onEMD

exists in the instantaneous frequency waveform which isrelated to the measured signal

Then the optimum wavelet parameters 119891119887= 48574 and

119891119888= 07832 were selected Lastly the self-adaptive wavelet

ridge demodulation was carried out The IA waveform andits frequency spectrum obtained are shown in Figure 22from which the fault characteristics frequency 119891

119903= 6Hz

and its harmonics 3119891119903 4119891119903 and 5119891

119903are clearly found These

demonstrate that a local defect has occurred in the gear onthe drive shaft which is consistent with the drive gear stateThat is the proposed method is effective for gear crack faultdiagnosis

In addition we use the self-adaptive wavelet ridgedemodulation approach to analyse the original signal asexhibited in Figure 23 where the spectrum contains the faultcharacteristics frequency and its harmonic as well which

Shock and Vibration 9

0 01 02 03 04 05 06 07 08 09 10

051

152

25

0

200

400

600

800

Freq

uenc

yA

mpl

itude

Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

IA waveform of 1st IMF

IA waveform of 2nd IMF

IF waveform of 2nd IMF

IF waveform of 1st IMF

0

05

1

15

2

0100200300400500

Am

plitu

deFr

eque

ncy

Figure 15 Demodulation results of simulation signal 119910(119905) by energyoperator demodulation based on EMD

outperform the excellent time-frequency localization abilityof self-adaptive wavelet But some unknown frequency isinvolved because of background noise Moreover it can benoted that the harmonic in Figure 22 is richer and clearerthan that in Figure 23 In fact through signal decompositionby LCDand ISC selectionmost of noise can be removed fromanalysis signal Therefore the proposed method is effectiveand superior in application to weak fault diagnosis for gear

52 Bearing Inner-Race Fault Diagnosis The vibration signalof roller bearing with inner-race fault is complex and weakit is difficult to identify the fault state To further verifythe effectiveness of the proposed method we made bearinginner-race fault diagnosis experiment

The data was downloaded from the website of theCase Western Reserve University Bearing Center [24] Thetest stand consists of a 2 hp motor a torque transducera dynamometer and control electronics The test bearing

0 01 02 03 04 05 06 07 08 09 1010203040506

222224226228230232

IA waveform of 1st ISC

IF waveform of 1st ISC

IF waveform of 2nd ISC

IA waveform of 2nd ISC

Am

plitu

de

Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

Freq

uenc

y

040608

11214

155

160

165

170

Am

plitu

deFr

eque

ncy

Figure 16 Demodulation results of simulation signal with noise bythe proposed method

is at the drive end a single point defect was introducedinto the inner raceway of the test bearing The size of thesingle point defect is 0178mm in diameter and 0279mm indepth using electrodischarge machining An accelerometerattached to the bearing housing collected vibration data withthe sampling frequency as 12 kHz The shaft rotating speedof the bearing inner-race is 1750 revmin the characteristicfrequency of the roller bearing with inner-race fault is 119891

119900119894=

158HzFigure 24 presents the time domain waveform and spec-

trum of a bearing vibration signal From the spectrum it canbe seen that there are three center frequency bands and theirside frequency bands which show that main modulationcharacteristics exhibits in the high frequency band Howeverthe fault characteristic frequency is not clear in the spectrumHere we used the proposed method to demodulate Thebearing vibration signal was decomposed into thirteen ISCs

10 Shock and Vibration

0

1

2

3

0

200

400

600

800

IA waveform of 1st ISC

IF waveform of 1st ISC

Am

plitu

de

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

Freq

uenc

y

005

115

225

0

200

400

600

IA waveform of 2nd ISC

IF waveform of 2nd ISC

Am

plitu

deFr

eque

ncy

Figure 17 Demodulation results of simulation signal with noise byenergy operator demodulation approach based on LCD

and the first ISC with the highest frequency as shown inFigure 25 was selected for analysis The optimal Morletwavelet parameters were determined as 119891

119887= 128576 and

119891119888= 06862The IA waveform and its spectrum are shown in

Figure 26 from which we can clearly see the spectrum linesat the fault characteristic frequency 119891

119900119894and its harmonics

which shows that a local failure occurs in the inner racewayof bearing This is consistent with the fact state

Besides through comparison of Figure 24 with Figure 25we can find that applying LCD decomposition equals todesign adaptive band-pass filter whose center frequency isautomatically determined with the inherent characteristicsof analyzed signal Simultaneously the selection of specialISC with higher frequency for analysis leads to decrease ofthe influence of lower frequency noise to suit weak faultfeature extraction In addition due to concern of main

1 2 3 4 5 6 7 8 9 10 11

(1) Induction motor(2) Speed controller(3) Coupling(4) Bearing number 1 and bearing housing(5) Bearing number 2 and bearing housing(6) Drive gear(7) Driven gear

(9) Vibration acceleration sensor(10) Speed sensor

(8) Bearing number 3 and bearing housing

(11) Bearing number 4 and bearing housing

Figure 18 Bearing-gear test rig for vibration signal acquisition

0 01 02 03 04 05 06 07 08 09 1

0

50

100

Time (s)

Time domain waveform

0 50 100 150 200 250 300 350 4000

5

10

15

20 Envelope spectrum

minus50

minus100

Frequency (Hz)

X 21Y 914

Am

plitu

de (m

middotsminus2)

Acce

lera

tion

(mmiddotsminus

2)

Figure 19 Time domain waveform and envelope spectrum of afaulty gear vibration signal

energy on the time-scale distribution self-adaptive waveletridge modulation has excellent time-frequency localizationproperty as seen in Figure 23 Therefore the applicationresults show the proposed method is simple and effective forweak bearing inner-race fault diagnosis

6 Conclusion

LCD method is a new signal decomposition approachwhich is suitable to preprocess the multicomponent AM-FM signal Wavelet ridge demodulation method is based onwavelet transform in the time-frequency domain focusing

Shock and Vibration 11

0 01 02 03 04 05 06 07 08 09 1

200

100

50

010

Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

minus50

minus20

minus10

minus10

minus5

0

50

ISC 1

ISC 2

ISC 3

ISC 4

u(t

)

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

Figure 20 LCD decomposition results of a faulty gear vibration signal

IA waveform

0

20

40

60

80

0

500

1000IF waveform

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

Acce

lera

tion

(mmiddotsminus

2)

minus500

minus1000

Freq

uenc

y (H

z)

Figure 21 Demodulation results of the 1st ISC by Hilbert approach

on main energy on the time-scale distribution It is lesssensitive to signal bandwidth and noise especially usingthe wavelet transform with optimal parameters called self-adaptive wavelet Therefore with combination of the abovetwo approaches a self-adaptive wavelet ridge demodulationmethod based on LCD for fault diagnosis is presented

Using two simulation signals we compare the proposedmethod with the following four approaches Hilbert demod-ulation energy operator demodulation based on EMD self-adaptive wavelet ridge demodulation based on EMD andenergy operator demodulation based on LCD Analysis

0 01 02 03 04 05 06 07 08 09 10

1020304050

0 10 20 30 40 706050 80 90 10002468

10

Time (s)

IA waveform

Spectrum

Frequency (Hz)

fr

3fr4fr 5fr

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

Figure 22 Instantaneous amplitude and its spectrum of the 1st ISCby the proposed method

results show that the proposedmethod has higher demodula-tion accuracy and higher noise tolerant performance than theothers Finally we applied the proposed method to incipientfault diagnosis of gearbox The application results show theproposed method is simple and effective for incipient faultdiagnosis of gearbox

Conflict of Interests

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

12 Shock and Vibration

0 01 02 03 04 05 06 07 08 09 10

20

40

60

80

0 10 20 30 40 50 60 70 80 90 10002468

10

Time (s)

IA waveform

Spectrum

5fr

fr

Frequency (Hz)

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

Figure 23 Instantaneous amplitude and its spectrum of the orig-inal vibration signal by self-adaptive wavelet ridge demodulationapproach

0 01 02 03 04 05 06 07 08 09 1

0

1

2

0 1000 2000 3000 4000 5000 60000

005

01

015

02

Waveform

Spectrum

Time (s)

Frequency (Hz)

minus2

minus1

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

Figure 24 Time domain waveform and spectrum of a roller bearingvibration signal with inner-race defect

Acknowledgments

The support from Chinese National Science FoundationGrant (no 51075131) the construct program of the key dis-cipline in Hunan Province (Mechanical Design and Theory)(XJF2011 [76]) cooperative Demonstration Base of Universi-ties in Hunan ldquoR amp D and Industrialization of Rock DrillingMachinesrdquo (XJT [2014] 239) Hunan Major Special Projectsof Science and Technology (2014GK1043) and scientificresearch project of Hunan Province Education Department(no 14C0789) is greatly acknowledged The authors wouldlike to express their appreciation to Case Western ReserveUniversity for offering the free bearing data

0 1000 2000 3000 4000 5000 60000

005

01

015

02

Waveform

Spectrum

0

1

2

0 01 02 03 04 05 06 07 08 09 1Time (s)

minus2

minus1

Frequency (Hz)

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

Figure 25 Time domain waveform and spectrum of the 1st ISC ofa roller bearing vibration signal with inner-race defect

0 01 02 03 04 05 06 07 08 09 10

02040608

1

0 100 200 300 400 500 600 700 800 900 10000

005

01

015

02

IA waveform

Spectrum

Time (s)

Frequency (Hz)

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

foi

2foi

3foi

Figure 26 Instantaneous amplitude waveform and spectrum of the1st ISC of a roller bearing vibration signal with inner-race defect

References

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

[2] M Feldman ldquoNonparametric identification of asymmetricnonlinear vibration systems with the Hilbert transformrdquo Jour-nal of Sound and Vibration vol 331 no 14 pp 3386ndash3396 2012

[3] N E Huang Z Shen S R Long et al ldquoThe empiricalmode decomposition and the Hilbert spectrum for nonlinearand non-stationary time series analysisrdquo The Royal Societyof London Proceedings Series A Mathematical Physical andEngineering Sciences vol 454 no 1971 pp 903ndash995 1998

[4] J Cheng D Yu and Y Yang ldquoThe application of energyoperator demodulation approach based on EMD in machinery

Shock and Vibration 13

fault diagnosisrdquo Mechanical Systems and Signal Processing vol21 no 2 pp 668ndash677 2007

[5] Z Feng M Liang Y Zhang and S Hou ldquoFault diagnosis forwind turbine planetary gearboxes via demodulation analysisbased on ensemble empirical mode decomposition and energyseparationrdquo Renewable Energy vol 47 pp 112ndash126 2012

[6] Y Qin ldquoMulticomponent AM-FM demodulation based onenergy separation and adaptive filteringrdquo Mechanical Systemsand Signal Processing vol 38 no 2 pp 440ndash459 2013

[7] M Liang and H Faghidi ldquoIntelligent bearing fault detection byenhanced energy operatorrdquo Expert Systems with Applicationsvol 41 no 16 pp 7223ndash7234 2014

[8] J Wang and Q He ldquoExchanged ridge demodulation of time-scale manifold for enhanced fault diagnosis of rotating machin-eryrdquo Journal of Sound and Vibration vol 333 no 11 pp 2450ndash2464 2014

[9] Y Qin S Qin and Y Mao ldquoDemodulation approach based onwavelet ridge and its application in fault diagnosis of rotatingmachineryrdquo Chinese Journal of Mechanical Engineering vol 45no 2 pp 231ndash237 2009

[10] W He Z Jiang and Q Qin ldquoA joint adaptive wavelet filter andmorphological signal processing method for weak mechanicalimpulse extractionrdquo Journal of Mechanical Science and Technol-ogy vol 24 no 8 pp 1709ndash1716 2010

[11] Y Jiang B Tang YQin andW Liu ldquoFeature extractionmethodof wind turbine based on adaptive Morlet wavelet and SVDrdquoRenewable Energy vol 36 no 8 pp 2146ndash2153 2011

[12] W Su FWang H Zhu Z Zhang and Z Guo ldquoRolling elementbearing faults diagnosis based on optimal Morlet wavelet filterand autocorrelation enhancementrdquo Mechanical Systems andSignal Processing vol 24 no 5 pp 1458ndash1472 2010

[13] H Qiu J Lee J Lin and G Yu ldquoWavelet filter-based weaksignature detection method and its application on rollingelement bearing prognosticsrdquo Journal of Sound and Vibrationvol 289 no 4-5 pp 1066ndash1090 2006

[14] M Amarnath and I Praveen Krishna ldquoLocal fault detection inhelical gears via vibration and acoustic signals using EMDbasedstatistical parameter analysisrdquo Measurement vol 58 pp 154ndash164 2014

[15] Y Lei J Lin Z He and M J Zuo ldquoA review on empiricalmode decomposition in fault diagnosis of rotating machineryrdquoMechanical Systems and Signal Processing vol 35 no 1-2 pp108ndash126 2013

[16] J S Smith ldquoThe local mean decomposition and its applicationto EEG perception datardquo Journal of the Royal Society Interfacevol 2 no 5 pp 443ndash454 2005

[17] H Liu and M Han ldquoA fault diagnosis method based onlocal mean decomposition and multi-scale entropy for rollerbearingsrdquo Mechanism and Machine Theory vol 75 pp 67ndash782014

[18] J Cheng and Y Yang ldquoA rotating machinery fault diagnosismethod based on local mean decompositionrdquo Digital SignalProcessing vol 22 no 2 pp 356ndash366 2012

[19] J Zheng J Cheng and Y Yang ldquoA rolling bearing fault diagno-sis approach based on LCD and fuzzy entropyrdquoMechanism andMachine Theory vol 70 pp 441ndash453 2013

[20] J Zheng J Cheng Y Nie and S Luo ldquoA signal pro-cessing method for fault diagnosis-complete ensemble localcharacteristic-scale decompositionrdquo Journal of Vibration Engi-neering vol 27 no 4 pp 637ndash646 2014

[21] H L Ao J Cheng K Li andTK Truong ldquoA roller bearing faultdiagnosis method based on LCD energy entropy and ACROA-SVMrdquo Shock and Vibration vol 2014 Article ID 825825 12pages 2014

[22] MUnalMOnatMDemetgul andHKucuk ldquoFault diagnosisof rolling bearings using a genetic algorithm optimized neuralnetworkrdquoMeasurement vol 58 pp 187ndash196 2014

[23] B Long W Xian M Li and H Wang ldquoImproved diagnosticsfor the incipient faults in analog circuits using LSSVM based onPSO algorithm with Mahalanobis distancerdquo Neurocomputingvol 133 no 10 pp 237ndash248 2014

[24] CWRU Bearing Data Center Seeded Fault Test Data 2013httpcsegroupscaseedubearingdatacenterhome

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

4 Shock and Vibration

Step 2 Make wavelet decomposition of the signal and calcu-late the fitness value of each individual according to formula(17) Then sort the fitness values by size

Step 3 Based on individual fitness value in the search spaceindividuals are screened and evolved by a series of geneticmanipulation selection reproduction crossover mutationand so forth to constantly update and select populations

Step 4 In this step it is determined whether iterationsatisfies the termination condition or not If satisfied theoptimal solution is finished If not satisfied go to Step 2 untilthe optimal solution is got The optimization procedure ofwavelet parameters via GA is shown in Figure 1

4 The Proposed Method and Simulation

Since demodulation technique is an effectiveway to reveal thefault characteristic frequency for fault diagnosis of gearboxa self-adaptive wavelet ridge demodulation based on LCDfor fault diagnosis is proposed in this paper Firstly LCDmethod is adopted to decompose the original signal intoa number of ISC components and lower frequency noiseis decreased by selecting special ISC component whichcontains rich fault feature Secondly the genetic algorithm isused to optimize wavelet parameters to obtain self-adaptivewavelet based on wavelet energy entropy which is servedas the objective function Thirdly the self-adaptive waveletridge demodulation for the selected ISC component is usedto extract dynamic information Finally the characteristicsfrequency can be obtained to identify the working state orfailure information from the spectrum The flowchart of theproposed fault diagnosis method was illustrated in Figure 2

To verify the validity of the proposed method let usconsider the following signal

119909 (119905) = 1199091(119905) + 119909

2(119905)

1199091(119905) = (1 + 05 cos 20120587119905) sin (200120587119905 + 2 cos 20120587119905)

1199092(119905) = sin120587119905 sin 20120587119905 119905 isin [0 1]

(18)

Obviously 119909(119905) is a complexmulticomponentAM-FM signalcontaining AM-FM component 1199091(119905) and AM component1199092(119905) The sampling frequency is 1000Hz Time domainwaveforms of simulation signal 119909(119905) and its LCD decompo-sition results are shown in Figure 3 where the two ISC com-ponents ISC1 and ISC2 correspond to the two components1199091(119905) and 1199092(119905)

Then the wavelet parameters are optimized accordingto the signal itself using genetic algorithms For ISC1 theoptimal parameters are determined as 119891

119887= 22316 and

119891119888= 10532 Figure 4 shows the evolution curve of GA For

ISC2 the optimal parameters are 119891119887

= 41203 and 119891119888

=

10012 Ultimately the signal is transformed self-adaptivelyusing Morlet wavelet to extract wavelet ridge and ISC1 andISC2 are demodulated based on the formula (10) and (11)The demodulation results are shown in Figures 5 and 6respectively

Genetic manipulation

crossover and mutation)

Optimal solution

Yes

No

Starting

Wavelet decomposition

Fitness value calculation and evaluation of individual fitness

Termination conditions

End

Population initialization

and energy entropy

(selection reproduction

Figure 1 Flowchart of wavelet parameters optimization by GA

For the instantaneous amplitude of ISC1 to eliminate theborder effect of the wavelet transformation the boundary isprocessed by symmetric extensionmethodThe result is givenin Figure 7

From the above analysis results it can be seen that self-adaptive wavelet ridge demodulation method based on LCDcan precisely demodulate a complex multicomponent AM-FMsignal In order tomake comparative analysis the demod-ulation analysis results of ISC1 using Hilbert demodulationmethod are given in Figure 8 fromwhich it can be found thatthe demodulation curve is not smooth and the demodulationerror is bigger due to inevitable window effects by Hilbertdemodulation method [4] The above comparison results

Shock and Vibration 5

Collect vibration signals

Apply LCD decomposition

Extract the fault characteristic

Select useful ISC component

Apply self-adaptive wavelet ridge

demodulation to the selected ISC

Optimize wavelet parameters by GA

Identify fault type

Figure 2 Flowchart of the proposed method

indicate the proposed method is superior to the Hilbertdemodulation method

Let us consider another multicomponent AM-FM signal119910(119905)which is a simulated faulty signal of gearbox and definedas follows

119910 (119905) = 1199101(119905) + 119910

2(119905)

1199101(119905) = [04 + 02 cos (2120587 times 10119905)]

sdot sin [2120587 times 600119905 + cos (2120587 times 20119905)]

1199102(119905) = [1 + 05 cos (2120587 times 20119905)]

sdot sin [2120587 times 300119905 + cos (2120587 times 15119905)]

(19)

Time (s)

Am

plitu

deA

mpl

itude

Am

plitu

deA

mpl

itude

0

0

5

0

2

0

1

005

0

01 02 03 04 05 06 07 08 09 1

Time (s) 0 01 02 03 04 05 06 07 08 09 1

Time (s) 0 01 02 03 04 05 06 07 08 09 1

Time (s) 0 01 02 03 04 05 06 07 08 09 1

minus5

minus2

x(t

)IS

C 1IS

C 2

minus1

minus005

u(t

)Figure 3 Time domain waveform of simulation signal 119909(119905) and itsLCD decomposition result

Wav

elet

ener

gy en

tropy

val

ue

0 5 10 15 20 25 30

14

145

15

155

16

165

135

Optimal valueMean value

times10minus3

Figure 4 Evolution curve of GA

The time domain waveform is shown in Figure 9 Thesampling frequency is 2000Hz Through LCD method twoISC components are presented in Figure 10 Obviously thetwo ISC components are consistent with the two components1199101(119905) and 1199102(119905) By optimizing wavelet energy entropy self-adaptive wavelet parameters for the first ISC signal aredetermined as 119891

119887= 97752 and 119891

119888= 07428 the parameters

for the second ISC signal as 119891119887

= 40585 and 119891119888

=

10678 The self-adaptive wavelet transform results for thetwo ISCs are provided in Figures 11 and 12 It can be foundthat the IArsquos frequency of the first ISC signal is 10Hz and

6 Shock and Vibration

0 01 02 03 04 05 06 07 08 09 1040608

11214

80

90

100

110

120

IA waveform

IF waveform

Am

plitu

deFr

eque

ncy

Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

Figure 5 Self-adaptive wavelet ridge demodulation results of the 1stISC

0 01 02 03 04 05 06 07 08 09 10

02040608

1

859

9510

10511

Freq

uenc

yA

mpl

itude

IF waveform

IA waveform

Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

Figure 6 Self-adaptive wavelet ridge demodulation results of the2nd ISC of simulation signal

IA waveform

0 01 02 03 04 05 06 07 08 09 104

06

08

1

12

14

Am

plitu

de

Time (s)

Figure 7 Processed instantaneous amplitude of the 1st ISC

0 01 02 03 04 05 06 07 08 09 10

05

1

15

2

8090

100110120130

IA waveform

IF waveform

Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

Am

plitu

deFr

eque

ncy

Figure 8 Hilbert demodulation results of the 1st ISC of simulationsignal

0

2

4

Am

plitu

de

minus2

minus40 01 02 03 04 05 06 07 08 09 1

Time (s)

Figure 9 Waveform of simulation signal 119910(119905)

Am

plitu

de

0 01 02 03 04 05 06Time (s)

07 08 09 1

0 01 02 03 04 05 06Time (s)

07 08 09 1

0

05

1

0

1

2

Am

plitu

de

minus05

minus1

minus1

minus2

ISC 1

ISC 2

Figure 10 LCD results of simulation signal 119910(119905)

Shock and Vibration 7

200

400

600

Scal

e

Time-scale distribution

Am

plitu

de

Time (s)

Freq

uenc

y (H

z)

IA waveform

IF waveform

020304050607

0 01 02 03 04 05 06 07 08 09 1

Time (s)0 01 02 03 04 05 06 07 08 09 1

Time (s)0 01 02 03 04 05 06 07 08 09

550

600

650

700

Figure 11 Demodulation results of the 1st ISC of simulation signal119910(119905) by the proposed method

the IFrsquos frequency is 20Hz and that IArsquos frequency of thesecond ISC signal is 20Hz and the IFrsquos frequency is 15HzThese results are consistent with the two components in theoriginal simulation signal Therefore the proposed methodcan effectively demodulate multicomponent AM-FM signaland is suitable for fault diagnosis of gearbox

For comparison we use EMD method to make signaldecomposition and use self-adaptive wavelet ridge demodu-lation approach to demodulate We obtained eight IMFs intotal After analysis we noted the first two IMFs reflected themodulation characteristics hence they are considered as thetwo components The demodulation results of the first twoIMFs by self-adaptive wavelet ridge demodulation approachare provided in Figures 13 and 14 It can be seen that theFM component of the first IMF failed to be demodulatedand the energy of the spectra of the second IMFs is lowerThen energy operator demodulation based on EMD [4] isused and the results are shown in Figure 15 From thesefigures it is shown that the serious mode mixing exists inthe EMDdecomposition which influences the demodulationaccuracy However from Figures 11 and 12 it is noted thatLCD approach may diminish this problem and be superiorto EMD approach

Finally we add a Gaussian white noise with deviation of02 to 119910(119905) LCD decomposes the noisy signal into five ISCsBy the method introduced in Section 32 the optimal Morlet

0 01 02 03 04 05 06 07 08 09

100

200

300

400

500

Time (s)

Scal

e

Time-scale distribution

0 01 02 03 04 05 06 07 08 09Time (s)

1

0 01 02 03 04 05 06 07 08 09Time (s)

1

05

1

15

280

290

300

310

320

Am

plitu

deFr

eque

ncy

IA waveform

IF waveform

Figure 12 Demodulation results of the 2nd ISC of simulation signal119910(119905) by the proposed method

wavelet parameters for the first two ISCs which containmodulation information are determined The demodulationresults of the first two ISCs are shown in Figure 16The resultsby energy operator demodulation based on LCD to the firsttwo ISCs are given in Figure 17 It can be obviously found thatself-adaptive wavelet demodulation approach based on LCDhas better noise tolerant performance than energy operatordemodulation approach based on LCD

5 Application to Incipient Fault Diagnosis

51 Gear Crack Fault Diagnosis A gear crack fault diagnosisexperiment is carried out on bearing-gear test rig as shown inFigure 18 In this test themotor power is 600W both drivinggear and driven gear are standard spur gear whose modulusis 25mm and the number of teeth is 37The input and outputshafts are arranged in parallel They are supported by tworoller bearings A crack with 015mm width and 1mm depthat the root of the driving gear tooth is set by wire cuttingmachining to simulate the gear incipient crack failure Thevibration signals were collected by an accelerometer attachedto the bearing housingThe shaft speed is 360 revmin that isthe drive shaft rotation frequency is 119891

119903= 6Hz The sampling

frequency is 1024Hz and the length of sampling data is 1024pointThedomainwaveformof the vibration signalmeasured

8 Shock and Vibration

0 01 02 03 04 05 06 07 08 09 10

01

02

03

04IA component of the 1st IMF

Time (s)

Am

plitu

de

0 01 02 03 04 05 06 07 08 09 1100

150

200

250IF component of the 1st IMF

Time (s)

Am

plitu

de

0 20 40 60 80 100 120 140 160 180 2000

0102030405

Spectrum

Am

plitu

de

01020304050 Spectrum

Am

plitu

de

Frequency (Hz)

0 20 40 60 80 100 120 140 160 180 200Frequency (Hz)

X 10Y 006344

X 15Y 2169

Figure 13 Demodulation results of the 1st IMF of simulation signal119910(119905) by self-adaptive wavelet ridge demodulation approach based onEMD

and envelope spectrum are shown in Figure 19 From whichit can be seen that fault characteristics are submerged by thebackground noise and the characteristics frequency fail to beidentified

Since the vibration signal with gear crack is a multi-component AM-FM signal (AM-FM) we used the proposedmethod for fault diagnosis Firstly the vibration accelerationsignal was decomposed into four ISC components ISC

1simISC4

and a residual component 119903 by LCD method as shown inFigure 20 Because the carrier frequency of gear vibration sig-nals is generally gear meshing frequency and its harmonicswe select the first ISC with the highest frequency for analysisFigure 21 shows the Hilbert demodulation results of thefirst ISC showing instantaneous amplitude contains complexhigh-frequency interference and some negative frequency

0 01 02 03 04 05 06 07 08 09 1040506070809

0 20 40 60 80 120100 140 160 180 2000

0102030405

Spectrum

IA component of the 2nd IMF

IF component of the 2nd IMF

Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

Am

plitu

deA

mpl

itude

Frequency (Hz)

0 20 40 60 80 120100 140 160 180 200Frequency (Hz)

Spectrum

767778798081

0

5

10

15

20

Am

plitu

deA

mpl

itude

X 20Y 003938

X 15Y 1266

Figure 14Demodulation results of the 2nd IMFof simulation signal119910(119905) by self-adaptive wavelet ridge demodulation approach based onEMD

exists in the instantaneous frequency waveform which isrelated to the measured signal

Then the optimum wavelet parameters 119891119887= 48574 and

119891119888= 07832 were selected Lastly the self-adaptive wavelet

ridge demodulation was carried out The IA waveform andits frequency spectrum obtained are shown in Figure 22from which the fault characteristics frequency 119891

119903= 6Hz

and its harmonics 3119891119903 4119891119903 and 5119891

119903are clearly found These

demonstrate that a local defect has occurred in the gear onthe drive shaft which is consistent with the drive gear stateThat is the proposed method is effective for gear crack faultdiagnosis

In addition we use the self-adaptive wavelet ridgedemodulation approach to analyse the original signal asexhibited in Figure 23 where the spectrum contains the faultcharacteristics frequency and its harmonic as well which

Shock and Vibration 9

0 01 02 03 04 05 06 07 08 09 10

051

152

25

0

200

400

600

800

Freq

uenc

yA

mpl

itude

Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

IA waveform of 1st IMF

IA waveform of 2nd IMF

IF waveform of 2nd IMF

IF waveform of 1st IMF

0

05

1

15

2

0100200300400500

Am

plitu

deFr

eque

ncy

Figure 15 Demodulation results of simulation signal 119910(119905) by energyoperator demodulation based on EMD

outperform the excellent time-frequency localization abilityof self-adaptive wavelet But some unknown frequency isinvolved because of background noise Moreover it can benoted that the harmonic in Figure 22 is richer and clearerthan that in Figure 23 In fact through signal decompositionby LCDand ISC selectionmost of noise can be removed fromanalysis signal Therefore the proposed method is effectiveand superior in application to weak fault diagnosis for gear

52 Bearing Inner-Race Fault Diagnosis The vibration signalof roller bearing with inner-race fault is complex and weakit is difficult to identify the fault state To further verifythe effectiveness of the proposed method we made bearinginner-race fault diagnosis experiment

The data was downloaded from the website of theCase Western Reserve University Bearing Center [24] Thetest stand consists of a 2 hp motor a torque transducera dynamometer and control electronics The test bearing

0 01 02 03 04 05 06 07 08 09 1010203040506

222224226228230232

IA waveform of 1st ISC

IF waveform of 1st ISC

IF waveform of 2nd ISC

IA waveform of 2nd ISC

Am

plitu

de

Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

Freq

uenc

y

040608

11214

155

160

165

170

Am

plitu

deFr

eque

ncy

Figure 16 Demodulation results of simulation signal with noise bythe proposed method

is at the drive end a single point defect was introducedinto the inner raceway of the test bearing The size of thesingle point defect is 0178mm in diameter and 0279mm indepth using electrodischarge machining An accelerometerattached to the bearing housing collected vibration data withthe sampling frequency as 12 kHz The shaft rotating speedof the bearing inner-race is 1750 revmin the characteristicfrequency of the roller bearing with inner-race fault is 119891

119900119894=

158HzFigure 24 presents the time domain waveform and spec-

trum of a bearing vibration signal From the spectrum it canbe seen that there are three center frequency bands and theirside frequency bands which show that main modulationcharacteristics exhibits in the high frequency band Howeverthe fault characteristic frequency is not clear in the spectrumHere we used the proposed method to demodulate Thebearing vibration signal was decomposed into thirteen ISCs

10 Shock and Vibration

0

1

2

3

0

200

400

600

800

IA waveform of 1st ISC

IF waveform of 1st ISC

Am

plitu

de

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

Freq

uenc

y

005

115

225

0

200

400

600

IA waveform of 2nd ISC

IF waveform of 2nd ISC

Am

plitu

deFr

eque

ncy

Figure 17 Demodulation results of simulation signal with noise byenergy operator demodulation approach based on LCD

and the first ISC with the highest frequency as shown inFigure 25 was selected for analysis The optimal Morletwavelet parameters were determined as 119891

119887= 128576 and

119891119888= 06862The IA waveform and its spectrum are shown in

Figure 26 from which we can clearly see the spectrum linesat the fault characteristic frequency 119891

119900119894and its harmonics

which shows that a local failure occurs in the inner racewayof bearing This is consistent with the fact state

Besides through comparison of Figure 24 with Figure 25we can find that applying LCD decomposition equals todesign adaptive band-pass filter whose center frequency isautomatically determined with the inherent characteristicsof analyzed signal Simultaneously the selection of specialISC with higher frequency for analysis leads to decrease ofthe influence of lower frequency noise to suit weak faultfeature extraction In addition due to concern of main

1 2 3 4 5 6 7 8 9 10 11

(1) Induction motor(2) Speed controller(3) Coupling(4) Bearing number 1 and bearing housing(5) Bearing number 2 and bearing housing(6) Drive gear(7) Driven gear

(9) Vibration acceleration sensor(10) Speed sensor

(8) Bearing number 3 and bearing housing

(11) Bearing number 4 and bearing housing

Figure 18 Bearing-gear test rig for vibration signal acquisition

0 01 02 03 04 05 06 07 08 09 1

0

50

100

Time (s)

Time domain waveform

0 50 100 150 200 250 300 350 4000

5

10

15

20 Envelope spectrum

minus50

minus100

Frequency (Hz)

X 21Y 914

Am

plitu

de (m

middotsminus2)

Acce

lera

tion

(mmiddotsminus

2)

Figure 19 Time domain waveform and envelope spectrum of afaulty gear vibration signal

energy on the time-scale distribution self-adaptive waveletridge modulation has excellent time-frequency localizationproperty as seen in Figure 23 Therefore the applicationresults show the proposed method is simple and effective forweak bearing inner-race fault diagnosis

6 Conclusion

LCD method is a new signal decomposition approachwhich is suitable to preprocess the multicomponent AM-FM signal Wavelet ridge demodulation method is based onwavelet transform in the time-frequency domain focusing

Shock and Vibration 11

0 01 02 03 04 05 06 07 08 09 1

200

100

50

010

Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

minus50

minus20

minus10

minus10

minus5

0

50

ISC 1

ISC 2

ISC 3

ISC 4

u(t

)

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

Figure 20 LCD decomposition results of a faulty gear vibration signal

IA waveform

0

20

40

60

80

0

500

1000IF waveform

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

Acce

lera

tion

(mmiddotsminus

2)

minus500

minus1000

Freq

uenc

y (H

z)

Figure 21 Demodulation results of the 1st ISC by Hilbert approach

on main energy on the time-scale distribution It is lesssensitive to signal bandwidth and noise especially usingthe wavelet transform with optimal parameters called self-adaptive wavelet Therefore with combination of the abovetwo approaches a self-adaptive wavelet ridge demodulationmethod based on LCD for fault diagnosis is presented

Using two simulation signals we compare the proposedmethod with the following four approaches Hilbert demod-ulation energy operator demodulation based on EMD self-adaptive wavelet ridge demodulation based on EMD andenergy operator demodulation based on LCD Analysis

0 01 02 03 04 05 06 07 08 09 10

1020304050

0 10 20 30 40 706050 80 90 10002468

10

Time (s)

IA waveform

Spectrum

Frequency (Hz)

fr

3fr4fr 5fr

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

Figure 22 Instantaneous amplitude and its spectrum of the 1st ISCby the proposed method

results show that the proposedmethod has higher demodula-tion accuracy and higher noise tolerant performance than theothers Finally we applied the proposed method to incipientfault diagnosis of gearbox The application results show theproposed method is simple and effective for incipient faultdiagnosis of gearbox

Conflict of Interests

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

12 Shock and Vibration

0 01 02 03 04 05 06 07 08 09 10

20

40

60

80

0 10 20 30 40 50 60 70 80 90 10002468

10

Time (s)

IA waveform

Spectrum

5fr

fr

Frequency (Hz)

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

Figure 23 Instantaneous amplitude and its spectrum of the orig-inal vibration signal by self-adaptive wavelet ridge demodulationapproach

0 01 02 03 04 05 06 07 08 09 1

0

1

2

0 1000 2000 3000 4000 5000 60000

005

01

015

02

Waveform

Spectrum

Time (s)

Frequency (Hz)

minus2

minus1

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

Figure 24 Time domain waveform and spectrum of a roller bearingvibration signal with inner-race defect

Acknowledgments

The support from Chinese National Science FoundationGrant (no 51075131) the construct program of the key dis-cipline in Hunan Province (Mechanical Design and Theory)(XJF2011 [76]) cooperative Demonstration Base of Universi-ties in Hunan ldquoR amp D and Industrialization of Rock DrillingMachinesrdquo (XJT [2014] 239) Hunan Major Special Projectsof Science and Technology (2014GK1043) and scientificresearch project of Hunan Province Education Department(no 14C0789) is greatly acknowledged The authors wouldlike to express their appreciation to Case Western ReserveUniversity for offering the free bearing data

0 1000 2000 3000 4000 5000 60000

005

01

015

02

Waveform

Spectrum

0

1

2

0 01 02 03 04 05 06 07 08 09 1Time (s)

minus2

minus1

Frequency (Hz)

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

Figure 25 Time domain waveform and spectrum of the 1st ISC ofa roller bearing vibration signal with inner-race defect

0 01 02 03 04 05 06 07 08 09 10

02040608

1

0 100 200 300 400 500 600 700 800 900 10000

005

01

015

02

IA waveform

Spectrum

Time (s)

Frequency (Hz)

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

foi

2foi

3foi

Figure 26 Instantaneous amplitude waveform and spectrum of the1st ISC of a roller bearing vibration signal with inner-race defect

References

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

[2] M Feldman ldquoNonparametric identification of asymmetricnonlinear vibration systems with the Hilbert transformrdquo Jour-nal of Sound and Vibration vol 331 no 14 pp 3386ndash3396 2012

[3] N E Huang Z Shen S R Long et al ldquoThe empiricalmode decomposition and the Hilbert spectrum for nonlinearand non-stationary time series analysisrdquo The Royal Societyof London Proceedings Series A Mathematical Physical andEngineering Sciences vol 454 no 1971 pp 903ndash995 1998

[4] J Cheng D Yu and Y Yang ldquoThe application of energyoperator demodulation approach based on EMD in machinery

Shock and Vibration 13

fault diagnosisrdquo Mechanical Systems and Signal Processing vol21 no 2 pp 668ndash677 2007

[5] Z Feng M Liang Y Zhang and S Hou ldquoFault diagnosis forwind turbine planetary gearboxes via demodulation analysisbased on ensemble empirical mode decomposition and energyseparationrdquo Renewable Energy vol 47 pp 112ndash126 2012

[6] Y Qin ldquoMulticomponent AM-FM demodulation based onenergy separation and adaptive filteringrdquo Mechanical Systemsand Signal Processing vol 38 no 2 pp 440ndash459 2013

[7] M Liang and H Faghidi ldquoIntelligent bearing fault detection byenhanced energy operatorrdquo Expert Systems with Applicationsvol 41 no 16 pp 7223ndash7234 2014

[8] J Wang and Q He ldquoExchanged ridge demodulation of time-scale manifold for enhanced fault diagnosis of rotating machin-eryrdquo Journal of Sound and Vibration vol 333 no 11 pp 2450ndash2464 2014

[9] Y Qin S Qin and Y Mao ldquoDemodulation approach based onwavelet ridge and its application in fault diagnosis of rotatingmachineryrdquo Chinese Journal of Mechanical Engineering vol 45no 2 pp 231ndash237 2009

[10] W He Z Jiang and Q Qin ldquoA joint adaptive wavelet filter andmorphological signal processing method for weak mechanicalimpulse extractionrdquo Journal of Mechanical Science and Technol-ogy vol 24 no 8 pp 1709ndash1716 2010

[11] Y Jiang B Tang YQin andW Liu ldquoFeature extractionmethodof wind turbine based on adaptive Morlet wavelet and SVDrdquoRenewable Energy vol 36 no 8 pp 2146ndash2153 2011

[12] W Su FWang H Zhu Z Zhang and Z Guo ldquoRolling elementbearing faults diagnosis based on optimal Morlet wavelet filterand autocorrelation enhancementrdquo Mechanical Systems andSignal Processing vol 24 no 5 pp 1458ndash1472 2010

[13] H Qiu J Lee J Lin and G Yu ldquoWavelet filter-based weaksignature detection method and its application on rollingelement bearing prognosticsrdquo Journal of Sound and Vibrationvol 289 no 4-5 pp 1066ndash1090 2006

[14] M Amarnath and I Praveen Krishna ldquoLocal fault detection inhelical gears via vibration and acoustic signals using EMDbasedstatistical parameter analysisrdquo Measurement vol 58 pp 154ndash164 2014

[15] Y Lei J Lin Z He and M J Zuo ldquoA review on empiricalmode decomposition in fault diagnosis of rotating machineryrdquoMechanical Systems and Signal Processing vol 35 no 1-2 pp108ndash126 2013

[16] J S Smith ldquoThe local mean decomposition and its applicationto EEG perception datardquo Journal of the Royal Society Interfacevol 2 no 5 pp 443ndash454 2005

[17] H Liu and M Han ldquoA fault diagnosis method based onlocal mean decomposition and multi-scale entropy for rollerbearingsrdquo Mechanism and Machine Theory vol 75 pp 67ndash782014

[18] J Cheng and Y Yang ldquoA rotating machinery fault diagnosismethod based on local mean decompositionrdquo Digital SignalProcessing vol 22 no 2 pp 356ndash366 2012

[19] J Zheng J Cheng and Y Yang ldquoA rolling bearing fault diagno-sis approach based on LCD and fuzzy entropyrdquoMechanism andMachine Theory vol 70 pp 441ndash453 2013

[20] J Zheng J Cheng Y Nie and S Luo ldquoA signal pro-cessing method for fault diagnosis-complete ensemble localcharacteristic-scale decompositionrdquo Journal of Vibration Engi-neering vol 27 no 4 pp 637ndash646 2014

[21] H L Ao J Cheng K Li andTK Truong ldquoA roller bearing faultdiagnosis method based on LCD energy entropy and ACROA-SVMrdquo Shock and Vibration vol 2014 Article ID 825825 12pages 2014

[22] MUnalMOnatMDemetgul andHKucuk ldquoFault diagnosisof rolling bearings using a genetic algorithm optimized neuralnetworkrdquoMeasurement vol 58 pp 187ndash196 2014

[23] B Long W Xian M Li and H Wang ldquoImproved diagnosticsfor the incipient faults in analog circuits using LSSVM based onPSO algorithm with Mahalanobis distancerdquo Neurocomputingvol 133 no 10 pp 237ndash248 2014

[24] CWRU Bearing Data Center Seeded Fault Test Data 2013httpcsegroupscaseedubearingdatacenterhome

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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

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Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 5

Collect vibration signals

Apply LCD decomposition

Extract the fault characteristic

Select useful ISC component

Apply self-adaptive wavelet ridge

demodulation to the selected ISC

Optimize wavelet parameters by GA

Identify fault type

Figure 2 Flowchart of the proposed method

indicate the proposed method is superior to the Hilbertdemodulation method

Let us consider another multicomponent AM-FM signal119910(119905)which is a simulated faulty signal of gearbox and definedas follows

119910 (119905) = 1199101(119905) + 119910

2(119905)

1199101(119905) = [04 + 02 cos (2120587 times 10119905)]

sdot sin [2120587 times 600119905 + cos (2120587 times 20119905)]

1199102(119905) = [1 + 05 cos (2120587 times 20119905)]

sdot sin [2120587 times 300119905 + cos (2120587 times 15119905)]

(19)

Time (s)

Am

plitu

deA

mpl

itude

Am

plitu

deA

mpl

itude

0

0

5

0

2

0

1

005

0

01 02 03 04 05 06 07 08 09 1

Time (s) 0 01 02 03 04 05 06 07 08 09 1

Time (s) 0 01 02 03 04 05 06 07 08 09 1

Time (s) 0 01 02 03 04 05 06 07 08 09 1

minus5

minus2

x(t

)IS

C 1IS

C 2

minus1

minus005

u(t

)Figure 3 Time domain waveform of simulation signal 119909(119905) and itsLCD decomposition result

Wav

elet

ener

gy en

tropy

val

ue

0 5 10 15 20 25 30

14

145

15

155

16

165

135

Optimal valueMean value

times10minus3

Figure 4 Evolution curve of GA

The time domain waveform is shown in Figure 9 Thesampling frequency is 2000Hz Through LCD method twoISC components are presented in Figure 10 Obviously thetwo ISC components are consistent with the two components1199101(119905) and 1199102(119905) By optimizing wavelet energy entropy self-adaptive wavelet parameters for the first ISC signal aredetermined as 119891

119887= 97752 and 119891

119888= 07428 the parameters

for the second ISC signal as 119891119887

= 40585 and 119891119888

=

10678 The self-adaptive wavelet transform results for thetwo ISCs are provided in Figures 11 and 12 It can be foundthat the IArsquos frequency of the first ISC signal is 10Hz and

6 Shock and Vibration

0 01 02 03 04 05 06 07 08 09 1040608

11214

80

90

100

110

120

IA waveform

IF waveform

Am

plitu

deFr

eque

ncy

Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

Figure 5 Self-adaptive wavelet ridge demodulation results of the 1stISC

0 01 02 03 04 05 06 07 08 09 10

02040608

1

859

9510

10511

Freq

uenc

yA

mpl

itude

IF waveform

IA waveform

Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

Figure 6 Self-adaptive wavelet ridge demodulation results of the2nd ISC of simulation signal

IA waveform

0 01 02 03 04 05 06 07 08 09 104

06

08

1

12

14

Am

plitu

de

Time (s)

Figure 7 Processed instantaneous amplitude of the 1st ISC

0 01 02 03 04 05 06 07 08 09 10

05

1

15

2

8090

100110120130

IA waveform

IF waveform

Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

Am

plitu

deFr

eque

ncy

Figure 8 Hilbert demodulation results of the 1st ISC of simulationsignal

0

2

4

Am

plitu

de

minus2

minus40 01 02 03 04 05 06 07 08 09 1

Time (s)

Figure 9 Waveform of simulation signal 119910(119905)

Am

plitu

de

0 01 02 03 04 05 06Time (s)

07 08 09 1

0 01 02 03 04 05 06Time (s)

07 08 09 1

0

05

1

0

1

2

Am

plitu

de

minus05

minus1

minus1

minus2

ISC 1

ISC 2

Figure 10 LCD results of simulation signal 119910(119905)

Shock and Vibration 7

200

400

600

Scal

e

Time-scale distribution

Am

plitu

de

Time (s)

Freq

uenc

y (H

z)

IA waveform

IF waveform

020304050607

0 01 02 03 04 05 06 07 08 09 1

Time (s)0 01 02 03 04 05 06 07 08 09 1

Time (s)0 01 02 03 04 05 06 07 08 09

550

600

650

700

Figure 11 Demodulation results of the 1st ISC of simulation signal119910(119905) by the proposed method

the IFrsquos frequency is 20Hz and that IArsquos frequency of thesecond ISC signal is 20Hz and the IFrsquos frequency is 15HzThese results are consistent with the two components in theoriginal simulation signal Therefore the proposed methodcan effectively demodulate multicomponent AM-FM signaland is suitable for fault diagnosis of gearbox

For comparison we use EMD method to make signaldecomposition and use self-adaptive wavelet ridge demodu-lation approach to demodulate We obtained eight IMFs intotal After analysis we noted the first two IMFs reflected themodulation characteristics hence they are considered as thetwo components The demodulation results of the first twoIMFs by self-adaptive wavelet ridge demodulation approachare provided in Figures 13 and 14 It can be seen that theFM component of the first IMF failed to be demodulatedand the energy of the spectra of the second IMFs is lowerThen energy operator demodulation based on EMD [4] isused and the results are shown in Figure 15 From thesefigures it is shown that the serious mode mixing exists inthe EMDdecomposition which influences the demodulationaccuracy However from Figures 11 and 12 it is noted thatLCD approach may diminish this problem and be superiorto EMD approach

Finally we add a Gaussian white noise with deviation of02 to 119910(119905) LCD decomposes the noisy signal into five ISCsBy the method introduced in Section 32 the optimal Morlet

0 01 02 03 04 05 06 07 08 09

100

200

300

400

500

Time (s)

Scal

e

Time-scale distribution

0 01 02 03 04 05 06 07 08 09Time (s)

1

0 01 02 03 04 05 06 07 08 09Time (s)

1

05

1

15

280

290

300

310

320

Am

plitu

deFr

eque

ncy

IA waveform

IF waveform

Figure 12 Demodulation results of the 2nd ISC of simulation signal119910(119905) by the proposed method

wavelet parameters for the first two ISCs which containmodulation information are determined The demodulationresults of the first two ISCs are shown in Figure 16The resultsby energy operator demodulation based on LCD to the firsttwo ISCs are given in Figure 17 It can be obviously found thatself-adaptive wavelet demodulation approach based on LCDhas better noise tolerant performance than energy operatordemodulation approach based on LCD

5 Application to Incipient Fault Diagnosis

51 Gear Crack Fault Diagnosis A gear crack fault diagnosisexperiment is carried out on bearing-gear test rig as shown inFigure 18 In this test themotor power is 600W both drivinggear and driven gear are standard spur gear whose modulusis 25mm and the number of teeth is 37The input and outputshafts are arranged in parallel They are supported by tworoller bearings A crack with 015mm width and 1mm depthat the root of the driving gear tooth is set by wire cuttingmachining to simulate the gear incipient crack failure Thevibration signals were collected by an accelerometer attachedto the bearing housingThe shaft speed is 360 revmin that isthe drive shaft rotation frequency is 119891

119903= 6Hz The sampling

frequency is 1024Hz and the length of sampling data is 1024pointThedomainwaveformof the vibration signalmeasured

8 Shock and Vibration

0 01 02 03 04 05 06 07 08 09 10

01

02

03

04IA component of the 1st IMF

Time (s)

Am

plitu

de

0 01 02 03 04 05 06 07 08 09 1100

150

200

250IF component of the 1st IMF

Time (s)

Am

plitu

de

0 20 40 60 80 100 120 140 160 180 2000

0102030405

Spectrum

Am

plitu

de

01020304050 Spectrum

Am

plitu

de

Frequency (Hz)

0 20 40 60 80 100 120 140 160 180 200Frequency (Hz)

X 10Y 006344

X 15Y 2169

Figure 13 Demodulation results of the 1st IMF of simulation signal119910(119905) by self-adaptive wavelet ridge demodulation approach based onEMD

and envelope spectrum are shown in Figure 19 From whichit can be seen that fault characteristics are submerged by thebackground noise and the characteristics frequency fail to beidentified

Since the vibration signal with gear crack is a multi-component AM-FM signal (AM-FM) we used the proposedmethod for fault diagnosis Firstly the vibration accelerationsignal was decomposed into four ISC components ISC

1simISC4

and a residual component 119903 by LCD method as shown inFigure 20 Because the carrier frequency of gear vibration sig-nals is generally gear meshing frequency and its harmonicswe select the first ISC with the highest frequency for analysisFigure 21 shows the Hilbert demodulation results of thefirst ISC showing instantaneous amplitude contains complexhigh-frequency interference and some negative frequency

0 01 02 03 04 05 06 07 08 09 1040506070809

0 20 40 60 80 120100 140 160 180 2000

0102030405

Spectrum

IA component of the 2nd IMF

IF component of the 2nd IMF

Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

Am

plitu

deA

mpl

itude

Frequency (Hz)

0 20 40 60 80 120100 140 160 180 200Frequency (Hz)

Spectrum

767778798081

0

5

10

15

20

Am

plitu

deA

mpl

itude

X 20Y 003938

X 15Y 1266

Figure 14Demodulation results of the 2nd IMFof simulation signal119910(119905) by self-adaptive wavelet ridge demodulation approach based onEMD

exists in the instantaneous frequency waveform which isrelated to the measured signal

Then the optimum wavelet parameters 119891119887= 48574 and

119891119888= 07832 were selected Lastly the self-adaptive wavelet

ridge demodulation was carried out The IA waveform andits frequency spectrum obtained are shown in Figure 22from which the fault characteristics frequency 119891

119903= 6Hz

and its harmonics 3119891119903 4119891119903 and 5119891

119903are clearly found These

demonstrate that a local defect has occurred in the gear onthe drive shaft which is consistent with the drive gear stateThat is the proposed method is effective for gear crack faultdiagnosis

In addition we use the self-adaptive wavelet ridgedemodulation approach to analyse the original signal asexhibited in Figure 23 where the spectrum contains the faultcharacteristics frequency and its harmonic as well which

Shock and Vibration 9

0 01 02 03 04 05 06 07 08 09 10

051

152

25

0

200

400

600

800

Freq

uenc

yA

mpl

itude

Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

IA waveform of 1st IMF

IA waveform of 2nd IMF

IF waveform of 2nd IMF

IF waveform of 1st IMF

0

05

1

15

2

0100200300400500

Am

plitu

deFr

eque

ncy

Figure 15 Demodulation results of simulation signal 119910(119905) by energyoperator demodulation based on EMD

outperform the excellent time-frequency localization abilityof self-adaptive wavelet But some unknown frequency isinvolved because of background noise Moreover it can benoted that the harmonic in Figure 22 is richer and clearerthan that in Figure 23 In fact through signal decompositionby LCDand ISC selectionmost of noise can be removed fromanalysis signal Therefore the proposed method is effectiveand superior in application to weak fault diagnosis for gear

52 Bearing Inner-Race Fault Diagnosis The vibration signalof roller bearing with inner-race fault is complex and weakit is difficult to identify the fault state To further verifythe effectiveness of the proposed method we made bearinginner-race fault diagnosis experiment

The data was downloaded from the website of theCase Western Reserve University Bearing Center [24] Thetest stand consists of a 2 hp motor a torque transducera dynamometer and control electronics The test bearing

0 01 02 03 04 05 06 07 08 09 1010203040506

222224226228230232

IA waveform of 1st ISC

IF waveform of 1st ISC

IF waveform of 2nd ISC

IA waveform of 2nd ISC

Am

plitu

de

Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

Freq

uenc

y

040608

11214

155

160

165

170

Am

plitu

deFr

eque

ncy

Figure 16 Demodulation results of simulation signal with noise bythe proposed method

is at the drive end a single point defect was introducedinto the inner raceway of the test bearing The size of thesingle point defect is 0178mm in diameter and 0279mm indepth using electrodischarge machining An accelerometerattached to the bearing housing collected vibration data withthe sampling frequency as 12 kHz The shaft rotating speedof the bearing inner-race is 1750 revmin the characteristicfrequency of the roller bearing with inner-race fault is 119891

119900119894=

158HzFigure 24 presents the time domain waveform and spec-

trum of a bearing vibration signal From the spectrum it canbe seen that there are three center frequency bands and theirside frequency bands which show that main modulationcharacteristics exhibits in the high frequency band Howeverthe fault characteristic frequency is not clear in the spectrumHere we used the proposed method to demodulate Thebearing vibration signal was decomposed into thirteen ISCs

10 Shock and Vibration

0

1

2

3

0

200

400

600

800

IA waveform of 1st ISC

IF waveform of 1st ISC

Am

plitu

de

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

Freq

uenc

y

005

115

225

0

200

400

600

IA waveform of 2nd ISC

IF waveform of 2nd ISC

Am

plitu

deFr

eque

ncy

Figure 17 Demodulation results of simulation signal with noise byenergy operator demodulation approach based on LCD

and the first ISC with the highest frequency as shown inFigure 25 was selected for analysis The optimal Morletwavelet parameters were determined as 119891

119887= 128576 and

119891119888= 06862The IA waveform and its spectrum are shown in

Figure 26 from which we can clearly see the spectrum linesat the fault characteristic frequency 119891

119900119894and its harmonics

which shows that a local failure occurs in the inner racewayof bearing This is consistent with the fact state

Besides through comparison of Figure 24 with Figure 25we can find that applying LCD decomposition equals todesign adaptive band-pass filter whose center frequency isautomatically determined with the inherent characteristicsof analyzed signal Simultaneously the selection of specialISC with higher frequency for analysis leads to decrease ofthe influence of lower frequency noise to suit weak faultfeature extraction In addition due to concern of main

1 2 3 4 5 6 7 8 9 10 11

(1) Induction motor(2) Speed controller(3) Coupling(4) Bearing number 1 and bearing housing(5) Bearing number 2 and bearing housing(6) Drive gear(7) Driven gear

(9) Vibration acceleration sensor(10) Speed sensor

(8) Bearing number 3 and bearing housing

(11) Bearing number 4 and bearing housing

Figure 18 Bearing-gear test rig for vibration signal acquisition

0 01 02 03 04 05 06 07 08 09 1

0

50

100

Time (s)

Time domain waveform

0 50 100 150 200 250 300 350 4000

5

10

15

20 Envelope spectrum

minus50

minus100

Frequency (Hz)

X 21Y 914

Am

plitu

de (m

middotsminus2)

Acce

lera

tion

(mmiddotsminus

2)

Figure 19 Time domain waveform and envelope spectrum of afaulty gear vibration signal

energy on the time-scale distribution self-adaptive waveletridge modulation has excellent time-frequency localizationproperty as seen in Figure 23 Therefore the applicationresults show the proposed method is simple and effective forweak bearing inner-race fault diagnosis

6 Conclusion

LCD method is a new signal decomposition approachwhich is suitable to preprocess the multicomponent AM-FM signal Wavelet ridge demodulation method is based onwavelet transform in the time-frequency domain focusing

Shock and Vibration 11

0 01 02 03 04 05 06 07 08 09 1

200

100

50

010

Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

minus50

minus20

minus10

minus10

minus5

0

50

ISC 1

ISC 2

ISC 3

ISC 4

u(t

)

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

Figure 20 LCD decomposition results of a faulty gear vibration signal

IA waveform

0

20

40

60

80

0

500

1000IF waveform

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

Acce

lera

tion

(mmiddotsminus

2)

minus500

minus1000

Freq

uenc

y (H

z)

Figure 21 Demodulation results of the 1st ISC by Hilbert approach

on main energy on the time-scale distribution It is lesssensitive to signal bandwidth and noise especially usingthe wavelet transform with optimal parameters called self-adaptive wavelet Therefore with combination of the abovetwo approaches a self-adaptive wavelet ridge demodulationmethod based on LCD for fault diagnosis is presented

Using two simulation signals we compare the proposedmethod with the following four approaches Hilbert demod-ulation energy operator demodulation based on EMD self-adaptive wavelet ridge demodulation based on EMD andenergy operator demodulation based on LCD Analysis

0 01 02 03 04 05 06 07 08 09 10

1020304050

0 10 20 30 40 706050 80 90 10002468

10

Time (s)

IA waveform

Spectrum

Frequency (Hz)

fr

3fr4fr 5fr

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

Figure 22 Instantaneous amplitude and its spectrum of the 1st ISCby the proposed method

results show that the proposedmethod has higher demodula-tion accuracy and higher noise tolerant performance than theothers Finally we applied the proposed method to incipientfault diagnosis of gearbox The application results show theproposed method is simple and effective for incipient faultdiagnosis of gearbox

Conflict of Interests

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

12 Shock and Vibration

0 01 02 03 04 05 06 07 08 09 10

20

40

60

80

0 10 20 30 40 50 60 70 80 90 10002468

10

Time (s)

IA waveform

Spectrum

5fr

fr

Frequency (Hz)

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

Figure 23 Instantaneous amplitude and its spectrum of the orig-inal vibration signal by self-adaptive wavelet ridge demodulationapproach

0 01 02 03 04 05 06 07 08 09 1

0

1

2

0 1000 2000 3000 4000 5000 60000

005

01

015

02

Waveform

Spectrum

Time (s)

Frequency (Hz)

minus2

minus1

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

Figure 24 Time domain waveform and spectrum of a roller bearingvibration signal with inner-race defect

Acknowledgments

The support from Chinese National Science FoundationGrant (no 51075131) the construct program of the key dis-cipline in Hunan Province (Mechanical Design and Theory)(XJF2011 [76]) cooperative Demonstration Base of Universi-ties in Hunan ldquoR amp D and Industrialization of Rock DrillingMachinesrdquo (XJT [2014] 239) Hunan Major Special Projectsof Science and Technology (2014GK1043) and scientificresearch project of Hunan Province Education Department(no 14C0789) is greatly acknowledged The authors wouldlike to express their appreciation to Case Western ReserveUniversity for offering the free bearing data

0 1000 2000 3000 4000 5000 60000

005

01

015

02

Waveform

Spectrum

0

1

2

0 01 02 03 04 05 06 07 08 09 1Time (s)

minus2

minus1

Frequency (Hz)

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

Figure 25 Time domain waveform and spectrum of the 1st ISC ofa roller bearing vibration signal with inner-race defect

0 01 02 03 04 05 06 07 08 09 10

02040608

1

0 100 200 300 400 500 600 700 800 900 10000

005

01

015

02

IA waveform

Spectrum

Time (s)

Frequency (Hz)

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

foi

2foi

3foi

Figure 26 Instantaneous amplitude waveform and spectrum of the1st ISC of a roller bearing vibration signal with inner-race defect

References

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

[2] M Feldman ldquoNonparametric identification of asymmetricnonlinear vibration systems with the Hilbert transformrdquo Jour-nal of Sound and Vibration vol 331 no 14 pp 3386ndash3396 2012

[3] N E Huang Z Shen S R Long et al ldquoThe empiricalmode decomposition and the Hilbert spectrum for nonlinearand non-stationary time series analysisrdquo The Royal Societyof London Proceedings Series A Mathematical Physical andEngineering Sciences vol 454 no 1971 pp 903ndash995 1998

[4] J Cheng D Yu and Y Yang ldquoThe application of energyoperator demodulation approach based on EMD in machinery

Shock and Vibration 13

fault diagnosisrdquo Mechanical Systems and Signal Processing vol21 no 2 pp 668ndash677 2007

[5] Z Feng M Liang Y Zhang and S Hou ldquoFault diagnosis forwind turbine planetary gearboxes via demodulation analysisbased on ensemble empirical mode decomposition and energyseparationrdquo Renewable Energy vol 47 pp 112ndash126 2012

[6] Y Qin ldquoMulticomponent AM-FM demodulation based onenergy separation and adaptive filteringrdquo Mechanical Systemsand Signal Processing vol 38 no 2 pp 440ndash459 2013

[7] M Liang and H Faghidi ldquoIntelligent bearing fault detection byenhanced energy operatorrdquo Expert Systems with Applicationsvol 41 no 16 pp 7223ndash7234 2014

[8] J Wang and Q He ldquoExchanged ridge demodulation of time-scale manifold for enhanced fault diagnosis of rotating machin-eryrdquo Journal of Sound and Vibration vol 333 no 11 pp 2450ndash2464 2014

[9] Y Qin S Qin and Y Mao ldquoDemodulation approach based onwavelet ridge and its application in fault diagnosis of rotatingmachineryrdquo Chinese Journal of Mechanical Engineering vol 45no 2 pp 231ndash237 2009

[10] W He Z Jiang and Q Qin ldquoA joint adaptive wavelet filter andmorphological signal processing method for weak mechanicalimpulse extractionrdquo Journal of Mechanical Science and Technol-ogy vol 24 no 8 pp 1709ndash1716 2010

[11] Y Jiang B Tang YQin andW Liu ldquoFeature extractionmethodof wind turbine based on adaptive Morlet wavelet and SVDrdquoRenewable Energy vol 36 no 8 pp 2146ndash2153 2011

[12] W Su FWang H Zhu Z Zhang and Z Guo ldquoRolling elementbearing faults diagnosis based on optimal Morlet wavelet filterand autocorrelation enhancementrdquo Mechanical Systems andSignal Processing vol 24 no 5 pp 1458ndash1472 2010

[13] H Qiu J Lee J Lin and G Yu ldquoWavelet filter-based weaksignature detection method and its application on rollingelement bearing prognosticsrdquo Journal of Sound and Vibrationvol 289 no 4-5 pp 1066ndash1090 2006

[14] M Amarnath and I Praveen Krishna ldquoLocal fault detection inhelical gears via vibration and acoustic signals using EMDbasedstatistical parameter analysisrdquo Measurement vol 58 pp 154ndash164 2014

[15] Y Lei J Lin Z He and M J Zuo ldquoA review on empiricalmode decomposition in fault diagnosis of rotating machineryrdquoMechanical Systems and Signal Processing vol 35 no 1-2 pp108ndash126 2013

[16] J S Smith ldquoThe local mean decomposition and its applicationto EEG perception datardquo Journal of the Royal Society Interfacevol 2 no 5 pp 443ndash454 2005

[17] H Liu and M Han ldquoA fault diagnosis method based onlocal mean decomposition and multi-scale entropy for rollerbearingsrdquo Mechanism and Machine Theory vol 75 pp 67ndash782014

[18] J Cheng and Y Yang ldquoA rotating machinery fault diagnosismethod based on local mean decompositionrdquo Digital SignalProcessing vol 22 no 2 pp 356ndash366 2012

[19] J Zheng J Cheng and Y Yang ldquoA rolling bearing fault diagno-sis approach based on LCD and fuzzy entropyrdquoMechanism andMachine Theory vol 70 pp 441ndash453 2013

[20] J Zheng J Cheng Y Nie and S Luo ldquoA signal pro-cessing method for fault diagnosis-complete ensemble localcharacteristic-scale decompositionrdquo Journal of Vibration Engi-neering vol 27 no 4 pp 637ndash646 2014

[21] H L Ao J Cheng K Li andTK Truong ldquoA roller bearing faultdiagnosis method based on LCD energy entropy and ACROA-SVMrdquo Shock and Vibration vol 2014 Article ID 825825 12pages 2014

[22] MUnalMOnatMDemetgul andHKucuk ldquoFault diagnosisof rolling bearings using a genetic algorithm optimized neuralnetworkrdquoMeasurement vol 58 pp 187ndash196 2014

[23] B Long W Xian M Li and H Wang ldquoImproved diagnosticsfor the incipient faults in analog circuits using LSSVM based onPSO algorithm with Mahalanobis distancerdquo Neurocomputingvol 133 no 10 pp 237ndash248 2014

[24] CWRU Bearing Data Center Seeded Fault Test Data 2013httpcsegroupscaseedubearingdatacenterhome

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

0 01 02 03 04 05 06 07 08 09 1040608

11214

80

90

100

110

120

IA waveform

IF waveform

Am

plitu

deFr

eque

ncy

Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

Figure 5 Self-adaptive wavelet ridge demodulation results of the 1stISC

0 01 02 03 04 05 06 07 08 09 10

02040608

1

859

9510

10511

Freq

uenc

yA

mpl

itude

IF waveform

IA waveform

Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

Figure 6 Self-adaptive wavelet ridge demodulation results of the2nd ISC of simulation signal

IA waveform

0 01 02 03 04 05 06 07 08 09 104

06

08

1

12

14

Am

plitu

de

Time (s)

Figure 7 Processed instantaneous amplitude of the 1st ISC

0 01 02 03 04 05 06 07 08 09 10

05

1

15

2

8090

100110120130

IA waveform

IF waveform

Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

Am

plitu

deFr

eque

ncy

Figure 8 Hilbert demodulation results of the 1st ISC of simulationsignal

0

2

4

Am

plitu

de

minus2

minus40 01 02 03 04 05 06 07 08 09 1

Time (s)

Figure 9 Waveform of simulation signal 119910(119905)

Am

plitu

de

0 01 02 03 04 05 06Time (s)

07 08 09 1

0 01 02 03 04 05 06Time (s)

07 08 09 1

0

05

1

0

1

2

Am

plitu

de

minus05

minus1

minus1

minus2

ISC 1

ISC 2

Figure 10 LCD results of simulation signal 119910(119905)

Shock and Vibration 7

200

400

600

Scal

e

Time-scale distribution

Am

plitu

de

Time (s)

Freq

uenc

y (H

z)

IA waveform

IF waveform

020304050607

0 01 02 03 04 05 06 07 08 09 1

Time (s)0 01 02 03 04 05 06 07 08 09 1

Time (s)0 01 02 03 04 05 06 07 08 09

550

600

650

700

Figure 11 Demodulation results of the 1st ISC of simulation signal119910(119905) by the proposed method

the IFrsquos frequency is 20Hz and that IArsquos frequency of thesecond ISC signal is 20Hz and the IFrsquos frequency is 15HzThese results are consistent with the two components in theoriginal simulation signal Therefore the proposed methodcan effectively demodulate multicomponent AM-FM signaland is suitable for fault diagnosis of gearbox

For comparison we use EMD method to make signaldecomposition and use self-adaptive wavelet ridge demodu-lation approach to demodulate We obtained eight IMFs intotal After analysis we noted the first two IMFs reflected themodulation characteristics hence they are considered as thetwo components The demodulation results of the first twoIMFs by self-adaptive wavelet ridge demodulation approachare provided in Figures 13 and 14 It can be seen that theFM component of the first IMF failed to be demodulatedand the energy of the spectra of the second IMFs is lowerThen energy operator demodulation based on EMD [4] isused and the results are shown in Figure 15 From thesefigures it is shown that the serious mode mixing exists inthe EMDdecomposition which influences the demodulationaccuracy However from Figures 11 and 12 it is noted thatLCD approach may diminish this problem and be superiorto EMD approach

Finally we add a Gaussian white noise with deviation of02 to 119910(119905) LCD decomposes the noisy signal into five ISCsBy the method introduced in Section 32 the optimal Morlet

0 01 02 03 04 05 06 07 08 09

100

200

300

400

500

Time (s)

Scal

e

Time-scale distribution

0 01 02 03 04 05 06 07 08 09Time (s)

1

0 01 02 03 04 05 06 07 08 09Time (s)

1

05

1

15

280

290

300

310

320

Am

plitu

deFr

eque

ncy

IA waveform

IF waveform

Figure 12 Demodulation results of the 2nd ISC of simulation signal119910(119905) by the proposed method

wavelet parameters for the first two ISCs which containmodulation information are determined The demodulationresults of the first two ISCs are shown in Figure 16The resultsby energy operator demodulation based on LCD to the firsttwo ISCs are given in Figure 17 It can be obviously found thatself-adaptive wavelet demodulation approach based on LCDhas better noise tolerant performance than energy operatordemodulation approach based on LCD

5 Application to Incipient Fault Diagnosis

51 Gear Crack Fault Diagnosis A gear crack fault diagnosisexperiment is carried out on bearing-gear test rig as shown inFigure 18 In this test themotor power is 600W both drivinggear and driven gear are standard spur gear whose modulusis 25mm and the number of teeth is 37The input and outputshafts are arranged in parallel They are supported by tworoller bearings A crack with 015mm width and 1mm depthat the root of the driving gear tooth is set by wire cuttingmachining to simulate the gear incipient crack failure Thevibration signals were collected by an accelerometer attachedto the bearing housingThe shaft speed is 360 revmin that isthe drive shaft rotation frequency is 119891

119903= 6Hz The sampling

frequency is 1024Hz and the length of sampling data is 1024pointThedomainwaveformof the vibration signalmeasured

8 Shock and Vibration

0 01 02 03 04 05 06 07 08 09 10

01

02

03

04IA component of the 1st IMF

Time (s)

Am

plitu

de

0 01 02 03 04 05 06 07 08 09 1100

150

200

250IF component of the 1st IMF

Time (s)

Am

plitu

de

0 20 40 60 80 100 120 140 160 180 2000

0102030405

Spectrum

Am

plitu

de

01020304050 Spectrum

Am

plitu

de

Frequency (Hz)

0 20 40 60 80 100 120 140 160 180 200Frequency (Hz)

X 10Y 006344

X 15Y 2169

Figure 13 Demodulation results of the 1st IMF of simulation signal119910(119905) by self-adaptive wavelet ridge demodulation approach based onEMD

and envelope spectrum are shown in Figure 19 From whichit can be seen that fault characteristics are submerged by thebackground noise and the characteristics frequency fail to beidentified

Since the vibration signal with gear crack is a multi-component AM-FM signal (AM-FM) we used the proposedmethod for fault diagnosis Firstly the vibration accelerationsignal was decomposed into four ISC components ISC

1simISC4

and a residual component 119903 by LCD method as shown inFigure 20 Because the carrier frequency of gear vibration sig-nals is generally gear meshing frequency and its harmonicswe select the first ISC with the highest frequency for analysisFigure 21 shows the Hilbert demodulation results of thefirst ISC showing instantaneous amplitude contains complexhigh-frequency interference and some negative frequency

0 01 02 03 04 05 06 07 08 09 1040506070809

0 20 40 60 80 120100 140 160 180 2000

0102030405

Spectrum

IA component of the 2nd IMF

IF component of the 2nd IMF

Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

Am

plitu

deA

mpl

itude

Frequency (Hz)

0 20 40 60 80 120100 140 160 180 200Frequency (Hz)

Spectrum

767778798081

0

5

10

15

20

Am

plitu

deA

mpl

itude

X 20Y 003938

X 15Y 1266

Figure 14Demodulation results of the 2nd IMFof simulation signal119910(119905) by self-adaptive wavelet ridge demodulation approach based onEMD

exists in the instantaneous frequency waveform which isrelated to the measured signal

Then the optimum wavelet parameters 119891119887= 48574 and

119891119888= 07832 were selected Lastly the self-adaptive wavelet

ridge demodulation was carried out The IA waveform andits frequency spectrum obtained are shown in Figure 22from which the fault characteristics frequency 119891

119903= 6Hz

and its harmonics 3119891119903 4119891119903 and 5119891

119903are clearly found These

demonstrate that a local defect has occurred in the gear onthe drive shaft which is consistent with the drive gear stateThat is the proposed method is effective for gear crack faultdiagnosis

In addition we use the self-adaptive wavelet ridgedemodulation approach to analyse the original signal asexhibited in Figure 23 where the spectrum contains the faultcharacteristics frequency and its harmonic as well which

Shock and Vibration 9

0 01 02 03 04 05 06 07 08 09 10

051

152

25

0

200

400

600

800

Freq

uenc

yA

mpl

itude

Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

IA waveform of 1st IMF

IA waveform of 2nd IMF

IF waveform of 2nd IMF

IF waveform of 1st IMF

0

05

1

15

2

0100200300400500

Am

plitu

deFr

eque

ncy

Figure 15 Demodulation results of simulation signal 119910(119905) by energyoperator demodulation based on EMD

outperform the excellent time-frequency localization abilityof self-adaptive wavelet But some unknown frequency isinvolved because of background noise Moreover it can benoted that the harmonic in Figure 22 is richer and clearerthan that in Figure 23 In fact through signal decompositionby LCDand ISC selectionmost of noise can be removed fromanalysis signal Therefore the proposed method is effectiveand superior in application to weak fault diagnosis for gear

52 Bearing Inner-Race Fault Diagnosis The vibration signalof roller bearing with inner-race fault is complex and weakit is difficult to identify the fault state To further verifythe effectiveness of the proposed method we made bearinginner-race fault diagnosis experiment

The data was downloaded from the website of theCase Western Reserve University Bearing Center [24] Thetest stand consists of a 2 hp motor a torque transducera dynamometer and control electronics The test bearing

0 01 02 03 04 05 06 07 08 09 1010203040506

222224226228230232

IA waveform of 1st ISC

IF waveform of 1st ISC

IF waveform of 2nd ISC

IA waveform of 2nd ISC

Am

plitu

de

Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

Freq

uenc

y

040608

11214

155

160

165

170

Am

plitu

deFr

eque

ncy

Figure 16 Demodulation results of simulation signal with noise bythe proposed method

is at the drive end a single point defect was introducedinto the inner raceway of the test bearing The size of thesingle point defect is 0178mm in diameter and 0279mm indepth using electrodischarge machining An accelerometerattached to the bearing housing collected vibration data withthe sampling frequency as 12 kHz The shaft rotating speedof the bearing inner-race is 1750 revmin the characteristicfrequency of the roller bearing with inner-race fault is 119891

119900119894=

158HzFigure 24 presents the time domain waveform and spec-

trum of a bearing vibration signal From the spectrum it canbe seen that there are three center frequency bands and theirside frequency bands which show that main modulationcharacteristics exhibits in the high frequency band Howeverthe fault characteristic frequency is not clear in the spectrumHere we used the proposed method to demodulate Thebearing vibration signal was decomposed into thirteen ISCs

10 Shock and Vibration

0

1

2

3

0

200

400

600

800

IA waveform of 1st ISC

IF waveform of 1st ISC

Am

plitu

de

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

Freq

uenc

y

005

115

225

0

200

400

600

IA waveform of 2nd ISC

IF waveform of 2nd ISC

Am

plitu

deFr

eque

ncy

Figure 17 Demodulation results of simulation signal with noise byenergy operator demodulation approach based on LCD

and the first ISC with the highest frequency as shown inFigure 25 was selected for analysis The optimal Morletwavelet parameters were determined as 119891

119887= 128576 and

119891119888= 06862The IA waveform and its spectrum are shown in

Figure 26 from which we can clearly see the spectrum linesat the fault characteristic frequency 119891

119900119894and its harmonics

which shows that a local failure occurs in the inner racewayof bearing This is consistent with the fact state

Besides through comparison of Figure 24 with Figure 25we can find that applying LCD decomposition equals todesign adaptive band-pass filter whose center frequency isautomatically determined with the inherent characteristicsof analyzed signal Simultaneously the selection of specialISC with higher frequency for analysis leads to decrease ofthe influence of lower frequency noise to suit weak faultfeature extraction In addition due to concern of main

1 2 3 4 5 6 7 8 9 10 11

(1) Induction motor(2) Speed controller(3) Coupling(4) Bearing number 1 and bearing housing(5) Bearing number 2 and bearing housing(6) Drive gear(7) Driven gear

(9) Vibration acceleration sensor(10) Speed sensor

(8) Bearing number 3 and bearing housing

(11) Bearing number 4 and bearing housing

Figure 18 Bearing-gear test rig for vibration signal acquisition

0 01 02 03 04 05 06 07 08 09 1

0

50

100

Time (s)

Time domain waveform

0 50 100 150 200 250 300 350 4000

5

10

15

20 Envelope spectrum

minus50

minus100

Frequency (Hz)

X 21Y 914

Am

plitu

de (m

middotsminus2)

Acce

lera

tion

(mmiddotsminus

2)

Figure 19 Time domain waveform and envelope spectrum of afaulty gear vibration signal

energy on the time-scale distribution self-adaptive waveletridge modulation has excellent time-frequency localizationproperty as seen in Figure 23 Therefore the applicationresults show the proposed method is simple and effective forweak bearing inner-race fault diagnosis

6 Conclusion

LCD method is a new signal decomposition approachwhich is suitable to preprocess the multicomponent AM-FM signal Wavelet ridge demodulation method is based onwavelet transform in the time-frequency domain focusing

Shock and Vibration 11

0 01 02 03 04 05 06 07 08 09 1

200

100

50

010

Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

minus50

minus20

minus10

minus10

minus5

0

50

ISC 1

ISC 2

ISC 3

ISC 4

u(t

)

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

Figure 20 LCD decomposition results of a faulty gear vibration signal

IA waveform

0

20

40

60

80

0

500

1000IF waveform

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

Acce

lera

tion

(mmiddotsminus

2)

minus500

minus1000

Freq

uenc

y (H

z)

Figure 21 Demodulation results of the 1st ISC by Hilbert approach

on main energy on the time-scale distribution It is lesssensitive to signal bandwidth and noise especially usingthe wavelet transform with optimal parameters called self-adaptive wavelet Therefore with combination of the abovetwo approaches a self-adaptive wavelet ridge demodulationmethod based on LCD for fault diagnosis is presented

Using two simulation signals we compare the proposedmethod with the following four approaches Hilbert demod-ulation energy operator demodulation based on EMD self-adaptive wavelet ridge demodulation based on EMD andenergy operator demodulation based on LCD Analysis

0 01 02 03 04 05 06 07 08 09 10

1020304050

0 10 20 30 40 706050 80 90 10002468

10

Time (s)

IA waveform

Spectrum

Frequency (Hz)

fr

3fr4fr 5fr

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

Figure 22 Instantaneous amplitude and its spectrum of the 1st ISCby the proposed method

results show that the proposedmethod has higher demodula-tion accuracy and higher noise tolerant performance than theothers Finally we applied the proposed method to incipientfault diagnosis of gearbox The application results show theproposed method is simple and effective for incipient faultdiagnosis of gearbox

Conflict of Interests

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

12 Shock and Vibration

0 01 02 03 04 05 06 07 08 09 10

20

40

60

80

0 10 20 30 40 50 60 70 80 90 10002468

10

Time (s)

IA waveform

Spectrum

5fr

fr

Frequency (Hz)

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

Figure 23 Instantaneous amplitude and its spectrum of the orig-inal vibration signal by self-adaptive wavelet ridge demodulationapproach

0 01 02 03 04 05 06 07 08 09 1

0

1

2

0 1000 2000 3000 4000 5000 60000

005

01

015

02

Waveform

Spectrum

Time (s)

Frequency (Hz)

minus2

minus1

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

Figure 24 Time domain waveform and spectrum of a roller bearingvibration signal with inner-race defect

Acknowledgments

The support from Chinese National Science FoundationGrant (no 51075131) the construct program of the key dis-cipline in Hunan Province (Mechanical Design and Theory)(XJF2011 [76]) cooperative Demonstration Base of Universi-ties in Hunan ldquoR amp D and Industrialization of Rock DrillingMachinesrdquo (XJT [2014] 239) Hunan Major Special Projectsof Science and Technology (2014GK1043) and scientificresearch project of Hunan Province Education Department(no 14C0789) is greatly acknowledged The authors wouldlike to express their appreciation to Case Western ReserveUniversity for offering the free bearing data

0 1000 2000 3000 4000 5000 60000

005

01

015

02

Waveform

Spectrum

0

1

2

0 01 02 03 04 05 06 07 08 09 1Time (s)

minus2

minus1

Frequency (Hz)

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

Figure 25 Time domain waveform and spectrum of the 1st ISC ofa roller bearing vibration signal with inner-race defect

0 01 02 03 04 05 06 07 08 09 10

02040608

1

0 100 200 300 400 500 600 700 800 900 10000

005

01

015

02

IA waveform

Spectrum

Time (s)

Frequency (Hz)

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

foi

2foi

3foi

Figure 26 Instantaneous amplitude waveform and spectrum of the1st ISC of a roller bearing vibration signal with inner-race defect

References

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

[2] M Feldman ldquoNonparametric identification of asymmetricnonlinear vibration systems with the Hilbert transformrdquo Jour-nal of Sound and Vibration vol 331 no 14 pp 3386ndash3396 2012

[3] N E Huang Z Shen S R Long et al ldquoThe empiricalmode decomposition and the Hilbert spectrum for nonlinearand non-stationary time series analysisrdquo The Royal Societyof London Proceedings Series A Mathematical Physical andEngineering Sciences vol 454 no 1971 pp 903ndash995 1998

[4] J Cheng D Yu and Y Yang ldquoThe application of energyoperator demodulation approach based on EMD in machinery

Shock and Vibration 13

fault diagnosisrdquo Mechanical Systems and Signal Processing vol21 no 2 pp 668ndash677 2007

[5] Z Feng M Liang Y Zhang and S Hou ldquoFault diagnosis forwind turbine planetary gearboxes via demodulation analysisbased on ensemble empirical mode decomposition and energyseparationrdquo Renewable Energy vol 47 pp 112ndash126 2012

[6] Y Qin ldquoMulticomponent AM-FM demodulation based onenergy separation and adaptive filteringrdquo Mechanical Systemsand Signal Processing vol 38 no 2 pp 440ndash459 2013

[7] M Liang and H Faghidi ldquoIntelligent bearing fault detection byenhanced energy operatorrdquo Expert Systems with Applicationsvol 41 no 16 pp 7223ndash7234 2014

[8] J Wang and Q He ldquoExchanged ridge demodulation of time-scale manifold for enhanced fault diagnosis of rotating machin-eryrdquo Journal of Sound and Vibration vol 333 no 11 pp 2450ndash2464 2014

[9] Y Qin S Qin and Y Mao ldquoDemodulation approach based onwavelet ridge and its application in fault diagnosis of rotatingmachineryrdquo Chinese Journal of Mechanical Engineering vol 45no 2 pp 231ndash237 2009

[10] W He Z Jiang and Q Qin ldquoA joint adaptive wavelet filter andmorphological signal processing method for weak mechanicalimpulse extractionrdquo Journal of Mechanical Science and Technol-ogy vol 24 no 8 pp 1709ndash1716 2010

[11] Y Jiang B Tang YQin andW Liu ldquoFeature extractionmethodof wind turbine based on adaptive Morlet wavelet and SVDrdquoRenewable Energy vol 36 no 8 pp 2146ndash2153 2011

[12] W Su FWang H Zhu Z Zhang and Z Guo ldquoRolling elementbearing faults diagnosis based on optimal Morlet wavelet filterand autocorrelation enhancementrdquo Mechanical Systems andSignal Processing vol 24 no 5 pp 1458ndash1472 2010

[13] H Qiu J Lee J Lin and G Yu ldquoWavelet filter-based weaksignature detection method and its application on rollingelement bearing prognosticsrdquo Journal of Sound and Vibrationvol 289 no 4-5 pp 1066ndash1090 2006

[14] M Amarnath and I Praveen Krishna ldquoLocal fault detection inhelical gears via vibration and acoustic signals using EMDbasedstatistical parameter analysisrdquo Measurement vol 58 pp 154ndash164 2014

[15] Y Lei J Lin Z He and M J Zuo ldquoA review on empiricalmode decomposition in fault diagnosis of rotating machineryrdquoMechanical Systems and Signal Processing vol 35 no 1-2 pp108ndash126 2013

[16] J S Smith ldquoThe local mean decomposition and its applicationto EEG perception datardquo Journal of the Royal Society Interfacevol 2 no 5 pp 443ndash454 2005

[17] H Liu and M Han ldquoA fault diagnosis method based onlocal mean decomposition and multi-scale entropy for rollerbearingsrdquo Mechanism and Machine Theory vol 75 pp 67ndash782014

[18] J Cheng and Y Yang ldquoA rotating machinery fault diagnosismethod based on local mean decompositionrdquo Digital SignalProcessing vol 22 no 2 pp 356ndash366 2012

[19] J Zheng J Cheng and Y Yang ldquoA rolling bearing fault diagno-sis approach based on LCD and fuzzy entropyrdquoMechanism andMachine Theory vol 70 pp 441ndash453 2013

[20] J Zheng J Cheng Y Nie and S Luo ldquoA signal pro-cessing method for fault diagnosis-complete ensemble localcharacteristic-scale decompositionrdquo Journal of Vibration Engi-neering vol 27 no 4 pp 637ndash646 2014

[21] H L Ao J Cheng K Li andTK Truong ldquoA roller bearing faultdiagnosis method based on LCD energy entropy and ACROA-SVMrdquo Shock and Vibration vol 2014 Article ID 825825 12pages 2014

[22] MUnalMOnatMDemetgul andHKucuk ldquoFault diagnosisof rolling bearings using a genetic algorithm optimized neuralnetworkrdquoMeasurement vol 58 pp 187ndash196 2014

[23] B Long W Xian M Li and H Wang ldquoImproved diagnosticsfor the incipient faults in analog circuits using LSSVM based onPSO algorithm with Mahalanobis distancerdquo Neurocomputingvol 133 no 10 pp 237ndash248 2014

[24] CWRU Bearing Data Center Seeded Fault Test Data 2013httpcsegroupscaseedubearingdatacenterhome

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

200

400

600

Scal

e

Time-scale distribution

Am

plitu

de

Time (s)

Freq

uenc

y (H

z)

IA waveform

IF waveform

020304050607

0 01 02 03 04 05 06 07 08 09 1

Time (s)0 01 02 03 04 05 06 07 08 09 1

Time (s)0 01 02 03 04 05 06 07 08 09

550

600

650

700

Figure 11 Demodulation results of the 1st ISC of simulation signal119910(119905) by the proposed method

the IFrsquos frequency is 20Hz and that IArsquos frequency of thesecond ISC signal is 20Hz and the IFrsquos frequency is 15HzThese results are consistent with the two components in theoriginal simulation signal Therefore the proposed methodcan effectively demodulate multicomponent AM-FM signaland is suitable for fault diagnosis of gearbox

For comparison we use EMD method to make signaldecomposition and use self-adaptive wavelet ridge demodu-lation approach to demodulate We obtained eight IMFs intotal After analysis we noted the first two IMFs reflected themodulation characteristics hence they are considered as thetwo components The demodulation results of the first twoIMFs by self-adaptive wavelet ridge demodulation approachare provided in Figures 13 and 14 It can be seen that theFM component of the first IMF failed to be demodulatedand the energy of the spectra of the second IMFs is lowerThen energy operator demodulation based on EMD [4] isused and the results are shown in Figure 15 From thesefigures it is shown that the serious mode mixing exists inthe EMDdecomposition which influences the demodulationaccuracy However from Figures 11 and 12 it is noted thatLCD approach may diminish this problem and be superiorto EMD approach

Finally we add a Gaussian white noise with deviation of02 to 119910(119905) LCD decomposes the noisy signal into five ISCsBy the method introduced in Section 32 the optimal Morlet

0 01 02 03 04 05 06 07 08 09

100

200

300

400

500

Time (s)

Scal

e

Time-scale distribution

0 01 02 03 04 05 06 07 08 09Time (s)

1

0 01 02 03 04 05 06 07 08 09Time (s)

1

05

1

15

280

290

300

310

320

Am

plitu

deFr

eque

ncy

IA waveform

IF waveform

Figure 12 Demodulation results of the 2nd ISC of simulation signal119910(119905) by the proposed method

wavelet parameters for the first two ISCs which containmodulation information are determined The demodulationresults of the first two ISCs are shown in Figure 16The resultsby energy operator demodulation based on LCD to the firsttwo ISCs are given in Figure 17 It can be obviously found thatself-adaptive wavelet demodulation approach based on LCDhas better noise tolerant performance than energy operatordemodulation approach based on LCD

5 Application to Incipient Fault Diagnosis

51 Gear Crack Fault Diagnosis A gear crack fault diagnosisexperiment is carried out on bearing-gear test rig as shown inFigure 18 In this test themotor power is 600W both drivinggear and driven gear are standard spur gear whose modulusis 25mm and the number of teeth is 37The input and outputshafts are arranged in parallel They are supported by tworoller bearings A crack with 015mm width and 1mm depthat the root of the driving gear tooth is set by wire cuttingmachining to simulate the gear incipient crack failure Thevibration signals were collected by an accelerometer attachedto the bearing housingThe shaft speed is 360 revmin that isthe drive shaft rotation frequency is 119891

119903= 6Hz The sampling

frequency is 1024Hz and the length of sampling data is 1024pointThedomainwaveformof the vibration signalmeasured

8 Shock and Vibration

0 01 02 03 04 05 06 07 08 09 10

01

02

03

04IA component of the 1st IMF

Time (s)

Am

plitu

de

0 01 02 03 04 05 06 07 08 09 1100

150

200

250IF component of the 1st IMF

Time (s)

Am

plitu

de

0 20 40 60 80 100 120 140 160 180 2000

0102030405

Spectrum

Am

plitu

de

01020304050 Spectrum

Am

plitu

de

Frequency (Hz)

0 20 40 60 80 100 120 140 160 180 200Frequency (Hz)

X 10Y 006344

X 15Y 2169

Figure 13 Demodulation results of the 1st IMF of simulation signal119910(119905) by self-adaptive wavelet ridge demodulation approach based onEMD

and envelope spectrum are shown in Figure 19 From whichit can be seen that fault characteristics are submerged by thebackground noise and the characteristics frequency fail to beidentified

Since the vibration signal with gear crack is a multi-component AM-FM signal (AM-FM) we used the proposedmethod for fault diagnosis Firstly the vibration accelerationsignal was decomposed into four ISC components ISC

1simISC4

and a residual component 119903 by LCD method as shown inFigure 20 Because the carrier frequency of gear vibration sig-nals is generally gear meshing frequency and its harmonicswe select the first ISC with the highest frequency for analysisFigure 21 shows the Hilbert demodulation results of thefirst ISC showing instantaneous amplitude contains complexhigh-frequency interference and some negative frequency

0 01 02 03 04 05 06 07 08 09 1040506070809

0 20 40 60 80 120100 140 160 180 2000

0102030405

Spectrum

IA component of the 2nd IMF

IF component of the 2nd IMF

Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

Am

plitu

deA

mpl

itude

Frequency (Hz)

0 20 40 60 80 120100 140 160 180 200Frequency (Hz)

Spectrum

767778798081

0

5

10

15

20

Am

plitu

deA

mpl

itude

X 20Y 003938

X 15Y 1266

Figure 14Demodulation results of the 2nd IMFof simulation signal119910(119905) by self-adaptive wavelet ridge demodulation approach based onEMD

exists in the instantaneous frequency waveform which isrelated to the measured signal

Then the optimum wavelet parameters 119891119887= 48574 and

119891119888= 07832 were selected Lastly the self-adaptive wavelet

ridge demodulation was carried out The IA waveform andits frequency spectrum obtained are shown in Figure 22from which the fault characteristics frequency 119891

119903= 6Hz

and its harmonics 3119891119903 4119891119903 and 5119891

119903are clearly found These

demonstrate that a local defect has occurred in the gear onthe drive shaft which is consistent with the drive gear stateThat is the proposed method is effective for gear crack faultdiagnosis

In addition we use the self-adaptive wavelet ridgedemodulation approach to analyse the original signal asexhibited in Figure 23 where the spectrum contains the faultcharacteristics frequency and its harmonic as well which

Shock and Vibration 9

0 01 02 03 04 05 06 07 08 09 10

051

152

25

0

200

400

600

800

Freq

uenc

yA

mpl

itude

Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

IA waveform of 1st IMF

IA waveform of 2nd IMF

IF waveform of 2nd IMF

IF waveform of 1st IMF

0

05

1

15

2

0100200300400500

Am

plitu

deFr

eque

ncy

Figure 15 Demodulation results of simulation signal 119910(119905) by energyoperator demodulation based on EMD

outperform the excellent time-frequency localization abilityof self-adaptive wavelet But some unknown frequency isinvolved because of background noise Moreover it can benoted that the harmonic in Figure 22 is richer and clearerthan that in Figure 23 In fact through signal decompositionby LCDand ISC selectionmost of noise can be removed fromanalysis signal Therefore the proposed method is effectiveand superior in application to weak fault diagnosis for gear

52 Bearing Inner-Race Fault Diagnosis The vibration signalof roller bearing with inner-race fault is complex and weakit is difficult to identify the fault state To further verifythe effectiveness of the proposed method we made bearinginner-race fault diagnosis experiment

The data was downloaded from the website of theCase Western Reserve University Bearing Center [24] Thetest stand consists of a 2 hp motor a torque transducera dynamometer and control electronics The test bearing

0 01 02 03 04 05 06 07 08 09 1010203040506

222224226228230232

IA waveform of 1st ISC

IF waveform of 1st ISC

IF waveform of 2nd ISC

IA waveform of 2nd ISC

Am

plitu

de

Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

Freq

uenc

y

040608

11214

155

160

165

170

Am

plitu

deFr

eque

ncy

Figure 16 Demodulation results of simulation signal with noise bythe proposed method

is at the drive end a single point defect was introducedinto the inner raceway of the test bearing The size of thesingle point defect is 0178mm in diameter and 0279mm indepth using electrodischarge machining An accelerometerattached to the bearing housing collected vibration data withthe sampling frequency as 12 kHz The shaft rotating speedof the bearing inner-race is 1750 revmin the characteristicfrequency of the roller bearing with inner-race fault is 119891

119900119894=

158HzFigure 24 presents the time domain waveform and spec-

trum of a bearing vibration signal From the spectrum it canbe seen that there are three center frequency bands and theirside frequency bands which show that main modulationcharacteristics exhibits in the high frequency band Howeverthe fault characteristic frequency is not clear in the spectrumHere we used the proposed method to demodulate Thebearing vibration signal was decomposed into thirteen ISCs

10 Shock and Vibration

0

1

2

3

0

200

400

600

800

IA waveform of 1st ISC

IF waveform of 1st ISC

Am

plitu

de

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

Freq

uenc

y

005

115

225

0

200

400

600

IA waveform of 2nd ISC

IF waveform of 2nd ISC

Am

plitu

deFr

eque

ncy

Figure 17 Demodulation results of simulation signal with noise byenergy operator demodulation approach based on LCD

and the first ISC with the highest frequency as shown inFigure 25 was selected for analysis The optimal Morletwavelet parameters were determined as 119891

119887= 128576 and

119891119888= 06862The IA waveform and its spectrum are shown in

Figure 26 from which we can clearly see the spectrum linesat the fault characteristic frequency 119891

119900119894and its harmonics

which shows that a local failure occurs in the inner racewayof bearing This is consistent with the fact state

Besides through comparison of Figure 24 with Figure 25we can find that applying LCD decomposition equals todesign adaptive band-pass filter whose center frequency isautomatically determined with the inherent characteristicsof analyzed signal Simultaneously the selection of specialISC with higher frequency for analysis leads to decrease ofthe influence of lower frequency noise to suit weak faultfeature extraction In addition due to concern of main

1 2 3 4 5 6 7 8 9 10 11

(1) Induction motor(2) Speed controller(3) Coupling(4) Bearing number 1 and bearing housing(5) Bearing number 2 and bearing housing(6) Drive gear(7) Driven gear

(9) Vibration acceleration sensor(10) Speed sensor

(8) Bearing number 3 and bearing housing

(11) Bearing number 4 and bearing housing

Figure 18 Bearing-gear test rig for vibration signal acquisition

0 01 02 03 04 05 06 07 08 09 1

0

50

100

Time (s)

Time domain waveform

0 50 100 150 200 250 300 350 4000

5

10

15

20 Envelope spectrum

minus50

minus100

Frequency (Hz)

X 21Y 914

Am

plitu

de (m

middotsminus2)

Acce

lera

tion

(mmiddotsminus

2)

Figure 19 Time domain waveform and envelope spectrum of afaulty gear vibration signal

energy on the time-scale distribution self-adaptive waveletridge modulation has excellent time-frequency localizationproperty as seen in Figure 23 Therefore the applicationresults show the proposed method is simple and effective forweak bearing inner-race fault diagnosis

6 Conclusion

LCD method is a new signal decomposition approachwhich is suitable to preprocess the multicomponent AM-FM signal Wavelet ridge demodulation method is based onwavelet transform in the time-frequency domain focusing

Shock and Vibration 11

0 01 02 03 04 05 06 07 08 09 1

200

100

50

010

Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

minus50

minus20

minus10

minus10

minus5

0

50

ISC 1

ISC 2

ISC 3

ISC 4

u(t

)

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

Figure 20 LCD decomposition results of a faulty gear vibration signal

IA waveform

0

20

40

60

80

0

500

1000IF waveform

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

Acce

lera

tion

(mmiddotsminus

2)

minus500

minus1000

Freq

uenc

y (H

z)

Figure 21 Demodulation results of the 1st ISC by Hilbert approach

on main energy on the time-scale distribution It is lesssensitive to signal bandwidth and noise especially usingthe wavelet transform with optimal parameters called self-adaptive wavelet Therefore with combination of the abovetwo approaches a self-adaptive wavelet ridge demodulationmethod based on LCD for fault diagnosis is presented

Using two simulation signals we compare the proposedmethod with the following four approaches Hilbert demod-ulation energy operator demodulation based on EMD self-adaptive wavelet ridge demodulation based on EMD andenergy operator demodulation based on LCD Analysis

0 01 02 03 04 05 06 07 08 09 10

1020304050

0 10 20 30 40 706050 80 90 10002468

10

Time (s)

IA waveform

Spectrum

Frequency (Hz)

fr

3fr4fr 5fr

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

Figure 22 Instantaneous amplitude and its spectrum of the 1st ISCby the proposed method

results show that the proposedmethod has higher demodula-tion accuracy and higher noise tolerant performance than theothers Finally we applied the proposed method to incipientfault diagnosis of gearbox The application results show theproposed method is simple and effective for incipient faultdiagnosis of gearbox

Conflict of Interests

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

12 Shock and Vibration

0 01 02 03 04 05 06 07 08 09 10

20

40

60

80

0 10 20 30 40 50 60 70 80 90 10002468

10

Time (s)

IA waveform

Spectrum

5fr

fr

Frequency (Hz)

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

Figure 23 Instantaneous amplitude and its spectrum of the orig-inal vibration signal by self-adaptive wavelet ridge demodulationapproach

0 01 02 03 04 05 06 07 08 09 1

0

1

2

0 1000 2000 3000 4000 5000 60000

005

01

015

02

Waveform

Spectrum

Time (s)

Frequency (Hz)

minus2

minus1

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

Figure 24 Time domain waveform and spectrum of a roller bearingvibration signal with inner-race defect

Acknowledgments

The support from Chinese National Science FoundationGrant (no 51075131) the construct program of the key dis-cipline in Hunan Province (Mechanical Design and Theory)(XJF2011 [76]) cooperative Demonstration Base of Universi-ties in Hunan ldquoR amp D and Industrialization of Rock DrillingMachinesrdquo (XJT [2014] 239) Hunan Major Special Projectsof Science and Technology (2014GK1043) and scientificresearch project of Hunan Province Education Department(no 14C0789) is greatly acknowledged The authors wouldlike to express their appreciation to Case Western ReserveUniversity for offering the free bearing data

0 1000 2000 3000 4000 5000 60000

005

01

015

02

Waveform

Spectrum

0

1

2

0 01 02 03 04 05 06 07 08 09 1Time (s)

minus2

minus1

Frequency (Hz)

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

Figure 25 Time domain waveform and spectrum of the 1st ISC ofa roller bearing vibration signal with inner-race defect

0 01 02 03 04 05 06 07 08 09 10

02040608

1

0 100 200 300 400 500 600 700 800 900 10000

005

01

015

02

IA waveform

Spectrum

Time (s)

Frequency (Hz)

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

foi

2foi

3foi

Figure 26 Instantaneous amplitude waveform and spectrum of the1st ISC of a roller bearing vibration signal with inner-race defect

References

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

[2] M Feldman ldquoNonparametric identification of asymmetricnonlinear vibration systems with the Hilbert transformrdquo Jour-nal of Sound and Vibration vol 331 no 14 pp 3386ndash3396 2012

[3] N E Huang Z Shen S R Long et al ldquoThe empiricalmode decomposition and the Hilbert spectrum for nonlinearand non-stationary time series analysisrdquo The Royal Societyof London Proceedings Series A Mathematical Physical andEngineering Sciences vol 454 no 1971 pp 903ndash995 1998

[4] J Cheng D Yu and Y Yang ldquoThe application of energyoperator demodulation approach based on EMD in machinery

Shock and Vibration 13

fault diagnosisrdquo Mechanical Systems and Signal Processing vol21 no 2 pp 668ndash677 2007

[5] Z Feng M Liang Y Zhang and S Hou ldquoFault diagnosis forwind turbine planetary gearboxes via demodulation analysisbased on ensemble empirical mode decomposition and energyseparationrdquo Renewable Energy vol 47 pp 112ndash126 2012

[6] Y Qin ldquoMulticomponent AM-FM demodulation based onenergy separation and adaptive filteringrdquo Mechanical Systemsand Signal Processing vol 38 no 2 pp 440ndash459 2013

[7] M Liang and H Faghidi ldquoIntelligent bearing fault detection byenhanced energy operatorrdquo Expert Systems with Applicationsvol 41 no 16 pp 7223ndash7234 2014

[8] J Wang and Q He ldquoExchanged ridge demodulation of time-scale manifold for enhanced fault diagnosis of rotating machin-eryrdquo Journal of Sound and Vibration vol 333 no 11 pp 2450ndash2464 2014

[9] Y Qin S Qin and Y Mao ldquoDemodulation approach based onwavelet ridge and its application in fault diagnosis of rotatingmachineryrdquo Chinese Journal of Mechanical Engineering vol 45no 2 pp 231ndash237 2009

[10] W He Z Jiang and Q Qin ldquoA joint adaptive wavelet filter andmorphological signal processing method for weak mechanicalimpulse extractionrdquo Journal of Mechanical Science and Technol-ogy vol 24 no 8 pp 1709ndash1716 2010

[11] Y Jiang B Tang YQin andW Liu ldquoFeature extractionmethodof wind turbine based on adaptive Morlet wavelet and SVDrdquoRenewable Energy vol 36 no 8 pp 2146ndash2153 2011

[12] W Su FWang H Zhu Z Zhang and Z Guo ldquoRolling elementbearing faults diagnosis based on optimal Morlet wavelet filterand autocorrelation enhancementrdquo Mechanical Systems andSignal Processing vol 24 no 5 pp 1458ndash1472 2010

[13] H Qiu J Lee J Lin and G Yu ldquoWavelet filter-based weaksignature detection method and its application on rollingelement bearing prognosticsrdquo Journal of Sound and Vibrationvol 289 no 4-5 pp 1066ndash1090 2006

[14] M Amarnath and I Praveen Krishna ldquoLocal fault detection inhelical gears via vibration and acoustic signals using EMDbasedstatistical parameter analysisrdquo Measurement vol 58 pp 154ndash164 2014

[15] Y Lei J Lin Z He and M J Zuo ldquoA review on empiricalmode decomposition in fault diagnosis of rotating machineryrdquoMechanical Systems and Signal Processing vol 35 no 1-2 pp108ndash126 2013

[16] J S Smith ldquoThe local mean decomposition and its applicationto EEG perception datardquo Journal of the Royal Society Interfacevol 2 no 5 pp 443ndash454 2005

[17] H Liu and M Han ldquoA fault diagnosis method based onlocal mean decomposition and multi-scale entropy for rollerbearingsrdquo Mechanism and Machine Theory vol 75 pp 67ndash782014

[18] J Cheng and Y Yang ldquoA rotating machinery fault diagnosismethod based on local mean decompositionrdquo Digital SignalProcessing vol 22 no 2 pp 356ndash366 2012

[19] J Zheng J Cheng and Y Yang ldquoA rolling bearing fault diagno-sis approach based on LCD and fuzzy entropyrdquoMechanism andMachine Theory vol 70 pp 441ndash453 2013

[20] J Zheng J Cheng Y Nie and S Luo ldquoA signal pro-cessing method for fault diagnosis-complete ensemble localcharacteristic-scale decompositionrdquo Journal of Vibration Engi-neering vol 27 no 4 pp 637ndash646 2014

[21] H L Ao J Cheng K Li andTK Truong ldquoA roller bearing faultdiagnosis method based on LCD energy entropy and ACROA-SVMrdquo Shock and Vibration vol 2014 Article ID 825825 12pages 2014

[22] MUnalMOnatMDemetgul andHKucuk ldquoFault diagnosisof rolling bearings using a genetic algorithm optimized neuralnetworkrdquoMeasurement vol 58 pp 187ndash196 2014

[23] B Long W Xian M Li and H Wang ldquoImproved diagnosticsfor the incipient faults in analog circuits using LSSVM based onPSO algorithm with Mahalanobis distancerdquo Neurocomputingvol 133 no 10 pp 237ndash248 2014

[24] CWRU Bearing Data Center Seeded Fault Test Data 2013httpcsegroupscaseedubearingdatacenterhome

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 10

01

02

03

04IA component of the 1st IMF

Time (s)

Am

plitu

de

0 01 02 03 04 05 06 07 08 09 1100

150

200

250IF component of the 1st IMF

Time (s)

Am

plitu

de

0 20 40 60 80 100 120 140 160 180 2000

0102030405

Spectrum

Am

plitu

de

01020304050 Spectrum

Am

plitu

de

Frequency (Hz)

0 20 40 60 80 100 120 140 160 180 200Frequency (Hz)

X 10Y 006344

X 15Y 2169

Figure 13 Demodulation results of the 1st IMF of simulation signal119910(119905) by self-adaptive wavelet ridge demodulation approach based onEMD

and envelope spectrum are shown in Figure 19 From whichit can be seen that fault characteristics are submerged by thebackground noise and the characteristics frequency fail to beidentified

Since the vibration signal with gear crack is a multi-component AM-FM signal (AM-FM) we used the proposedmethod for fault diagnosis Firstly the vibration accelerationsignal was decomposed into four ISC components ISC

1simISC4

and a residual component 119903 by LCD method as shown inFigure 20 Because the carrier frequency of gear vibration sig-nals is generally gear meshing frequency and its harmonicswe select the first ISC with the highest frequency for analysisFigure 21 shows the Hilbert demodulation results of thefirst ISC showing instantaneous amplitude contains complexhigh-frequency interference and some negative frequency

0 01 02 03 04 05 06 07 08 09 1040506070809

0 20 40 60 80 120100 140 160 180 2000

0102030405

Spectrum

IA component of the 2nd IMF

IF component of the 2nd IMF

Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

Am

plitu

deA

mpl

itude

Frequency (Hz)

0 20 40 60 80 120100 140 160 180 200Frequency (Hz)

Spectrum

767778798081

0

5

10

15

20

Am

plitu

deA

mpl

itude

X 20Y 003938

X 15Y 1266

Figure 14Demodulation results of the 2nd IMFof simulation signal119910(119905) by self-adaptive wavelet ridge demodulation approach based onEMD

exists in the instantaneous frequency waveform which isrelated to the measured signal

Then the optimum wavelet parameters 119891119887= 48574 and

119891119888= 07832 were selected Lastly the self-adaptive wavelet

ridge demodulation was carried out The IA waveform andits frequency spectrum obtained are shown in Figure 22from which the fault characteristics frequency 119891

119903= 6Hz

and its harmonics 3119891119903 4119891119903 and 5119891

119903are clearly found These

demonstrate that a local defect has occurred in the gear onthe drive shaft which is consistent with the drive gear stateThat is the proposed method is effective for gear crack faultdiagnosis

In addition we use the self-adaptive wavelet ridgedemodulation approach to analyse the original signal asexhibited in Figure 23 where the spectrum contains the faultcharacteristics frequency and its harmonic as well which

Shock and Vibration 9

0 01 02 03 04 05 06 07 08 09 10

051

152

25

0

200

400

600

800

Freq

uenc

yA

mpl

itude

Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

IA waveform of 1st IMF

IA waveform of 2nd IMF

IF waveform of 2nd IMF

IF waveform of 1st IMF

0

05

1

15

2

0100200300400500

Am

plitu

deFr

eque

ncy

Figure 15 Demodulation results of simulation signal 119910(119905) by energyoperator demodulation based on EMD

outperform the excellent time-frequency localization abilityof self-adaptive wavelet But some unknown frequency isinvolved because of background noise Moreover it can benoted that the harmonic in Figure 22 is richer and clearerthan that in Figure 23 In fact through signal decompositionby LCDand ISC selectionmost of noise can be removed fromanalysis signal Therefore the proposed method is effectiveand superior in application to weak fault diagnosis for gear

52 Bearing Inner-Race Fault Diagnosis The vibration signalof roller bearing with inner-race fault is complex and weakit is difficult to identify the fault state To further verifythe effectiveness of the proposed method we made bearinginner-race fault diagnosis experiment

The data was downloaded from the website of theCase Western Reserve University Bearing Center [24] Thetest stand consists of a 2 hp motor a torque transducera dynamometer and control electronics The test bearing

0 01 02 03 04 05 06 07 08 09 1010203040506

222224226228230232

IA waveform of 1st ISC

IF waveform of 1st ISC

IF waveform of 2nd ISC

IA waveform of 2nd ISC

Am

plitu

de

Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

Freq

uenc

y

040608

11214

155

160

165

170

Am

plitu

deFr

eque

ncy

Figure 16 Demodulation results of simulation signal with noise bythe proposed method

is at the drive end a single point defect was introducedinto the inner raceway of the test bearing The size of thesingle point defect is 0178mm in diameter and 0279mm indepth using electrodischarge machining An accelerometerattached to the bearing housing collected vibration data withthe sampling frequency as 12 kHz The shaft rotating speedof the bearing inner-race is 1750 revmin the characteristicfrequency of the roller bearing with inner-race fault is 119891

119900119894=

158HzFigure 24 presents the time domain waveform and spec-

trum of a bearing vibration signal From the spectrum it canbe seen that there are three center frequency bands and theirside frequency bands which show that main modulationcharacteristics exhibits in the high frequency band Howeverthe fault characteristic frequency is not clear in the spectrumHere we used the proposed method to demodulate Thebearing vibration signal was decomposed into thirteen ISCs

10 Shock and Vibration

0

1

2

3

0

200

400

600

800

IA waveform of 1st ISC

IF waveform of 1st ISC

Am

plitu

de

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

Freq

uenc

y

005

115

225

0

200

400

600

IA waveform of 2nd ISC

IF waveform of 2nd ISC

Am

plitu

deFr

eque

ncy

Figure 17 Demodulation results of simulation signal with noise byenergy operator demodulation approach based on LCD

and the first ISC with the highest frequency as shown inFigure 25 was selected for analysis The optimal Morletwavelet parameters were determined as 119891

119887= 128576 and

119891119888= 06862The IA waveform and its spectrum are shown in

Figure 26 from which we can clearly see the spectrum linesat the fault characteristic frequency 119891

119900119894and its harmonics

which shows that a local failure occurs in the inner racewayof bearing This is consistent with the fact state

Besides through comparison of Figure 24 with Figure 25we can find that applying LCD decomposition equals todesign adaptive band-pass filter whose center frequency isautomatically determined with the inherent characteristicsof analyzed signal Simultaneously the selection of specialISC with higher frequency for analysis leads to decrease ofthe influence of lower frequency noise to suit weak faultfeature extraction In addition due to concern of main

1 2 3 4 5 6 7 8 9 10 11

(1) Induction motor(2) Speed controller(3) Coupling(4) Bearing number 1 and bearing housing(5) Bearing number 2 and bearing housing(6) Drive gear(7) Driven gear

(9) Vibration acceleration sensor(10) Speed sensor

(8) Bearing number 3 and bearing housing

(11) Bearing number 4 and bearing housing

Figure 18 Bearing-gear test rig for vibration signal acquisition

0 01 02 03 04 05 06 07 08 09 1

0

50

100

Time (s)

Time domain waveform

0 50 100 150 200 250 300 350 4000

5

10

15

20 Envelope spectrum

minus50

minus100

Frequency (Hz)

X 21Y 914

Am

plitu

de (m

middotsminus2)

Acce

lera

tion

(mmiddotsminus

2)

Figure 19 Time domain waveform and envelope spectrum of afaulty gear vibration signal

energy on the time-scale distribution self-adaptive waveletridge modulation has excellent time-frequency localizationproperty as seen in Figure 23 Therefore the applicationresults show the proposed method is simple and effective forweak bearing inner-race fault diagnosis

6 Conclusion

LCD method is a new signal decomposition approachwhich is suitable to preprocess the multicomponent AM-FM signal Wavelet ridge demodulation method is based onwavelet transform in the time-frequency domain focusing

Shock and Vibration 11

0 01 02 03 04 05 06 07 08 09 1

200

100

50

010

Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

minus50

minus20

minus10

minus10

minus5

0

50

ISC 1

ISC 2

ISC 3

ISC 4

u(t

)

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

Figure 20 LCD decomposition results of a faulty gear vibration signal

IA waveform

0

20

40

60

80

0

500

1000IF waveform

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

Acce

lera

tion

(mmiddotsminus

2)

minus500

minus1000

Freq

uenc

y (H

z)

Figure 21 Demodulation results of the 1st ISC by Hilbert approach

on main energy on the time-scale distribution It is lesssensitive to signal bandwidth and noise especially usingthe wavelet transform with optimal parameters called self-adaptive wavelet Therefore with combination of the abovetwo approaches a self-adaptive wavelet ridge demodulationmethod based on LCD for fault diagnosis is presented

Using two simulation signals we compare the proposedmethod with the following four approaches Hilbert demod-ulation energy operator demodulation based on EMD self-adaptive wavelet ridge demodulation based on EMD andenergy operator demodulation based on LCD Analysis

0 01 02 03 04 05 06 07 08 09 10

1020304050

0 10 20 30 40 706050 80 90 10002468

10

Time (s)

IA waveform

Spectrum

Frequency (Hz)

fr

3fr4fr 5fr

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

Figure 22 Instantaneous amplitude and its spectrum of the 1st ISCby the proposed method

results show that the proposedmethod has higher demodula-tion accuracy and higher noise tolerant performance than theothers Finally we applied the proposed method to incipientfault diagnosis of gearbox The application results show theproposed method is simple and effective for incipient faultdiagnosis of gearbox

Conflict of Interests

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

12 Shock and Vibration

0 01 02 03 04 05 06 07 08 09 10

20

40

60

80

0 10 20 30 40 50 60 70 80 90 10002468

10

Time (s)

IA waveform

Spectrum

5fr

fr

Frequency (Hz)

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

Figure 23 Instantaneous amplitude and its spectrum of the orig-inal vibration signal by self-adaptive wavelet ridge demodulationapproach

0 01 02 03 04 05 06 07 08 09 1

0

1

2

0 1000 2000 3000 4000 5000 60000

005

01

015

02

Waveform

Spectrum

Time (s)

Frequency (Hz)

minus2

minus1

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

Figure 24 Time domain waveform and spectrum of a roller bearingvibration signal with inner-race defect

Acknowledgments

The support from Chinese National Science FoundationGrant (no 51075131) the construct program of the key dis-cipline in Hunan Province (Mechanical Design and Theory)(XJF2011 [76]) cooperative Demonstration Base of Universi-ties in Hunan ldquoR amp D and Industrialization of Rock DrillingMachinesrdquo (XJT [2014] 239) Hunan Major Special Projectsof Science and Technology (2014GK1043) and scientificresearch project of Hunan Province Education Department(no 14C0789) is greatly acknowledged The authors wouldlike to express their appreciation to Case Western ReserveUniversity for offering the free bearing data

0 1000 2000 3000 4000 5000 60000

005

01

015

02

Waveform

Spectrum

0

1

2

0 01 02 03 04 05 06 07 08 09 1Time (s)

minus2

minus1

Frequency (Hz)

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

Figure 25 Time domain waveform and spectrum of the 1st ISC ofa roller bearing vibration signal with inner-race defect

0 01 02 03 04 05 06 07 08 09 10

02040608

1

0 100 200 300 400 500 600 700 800 900 10000

005

01

015

02

IA waveform

Spectrum

Time (s)

Frequency (Hz)

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

foi

2foi

3foi

Figure 26 Instantaneous amplitude waveform and spectrum of the1st ISC of a roller bearing vibration signal with inner-race defect

References

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

[2] M Feldman ldquoNonparametric identification of asymmetricnonlinear vibration systems with the Hilbert transformrdquo Jour-nal of Sound and Vibration vol 331 no 14 pp 3386ndash3396 2012

[3] N E Huang Z Shen S R Long et al ldquoThe empiricalmode decomposition and the Hilbert spectrum for nonlinearand non-stationary time series analysisrdquo The Royal Societyof London Proceedings Series A Mathematical Physical andEngineering Sciences vol 454 no 1971 pp 903ndash995 1998

[4] J Cheng D Yu and Y Yang ldquoThe application of energyoperator demodulation approach based on EMD in machinery

Shock and Vibration 13

fault diagnosisrdquo Mechanical Systems and Signal Processing vol21 no 2 pp 668ndash677 2007

[5] Z Feng M Liang Y Zhang and S Hou ldquoFault diagnosis forwind turbine planetary gearboxes via demodulation analysisbased on ensemble empirical mode decomposition and energyseparationrdquo Renewable Energy vol 47 pp 112ndash126 2012

[6] Y Qin ldquoMulticomponent AM-FM demodulation based onenergy separation and adaptive filteringrdquo Mechanical Systemsand Signal Processing vol 38 no 2 pp 440ndash459 2013

[7] M Liang and H Faghidi ldquoIntelligent bearing fault detection byenhanced energy operatorrdquo Expert Systems with Applicationsvol 41 no 16 pp 7223ndash7234 2014

[8] J Wang and Q He ldquoExchanged ridge demodulation of time-scale manifold for enhanced fault diagnosis of rotating machin-eryrdquo Journal of Sound and Vibration vol 333 no 11 pp 2450ndash2464 2014

[9] Y Qin S Qin and Y Mao ldquoDemodulation approach based onwavelet ridge and its application in fault diagnosis of rotatingmachineryrdquo Chinese Journal of Mechanical Engineering vol 45no 2 pp 231ndash237 2009

[10] W He Z Jiang and Q Qin ldquoA joint adaptive wavelet filter andmorphological signal processing method for weak mechanicalimpulse extractionrdquo Journal of Mechanical Science and Technol-ogy vol 24 no 8 pp 1709ndash1716 2010

[11] Y Jiang B Tang YQin andW Liu ldquoFeature extractionmethodof wind turbine based on adaptive Morlet wavelet and SVDrdquoRenewable Energy vol 36 no 8 pp 2146ndash2153 2011

[12] W Su FWang H Zhu Z Zhang and Z Guo ldquoRolling elementbearing faults diagnosis based on optimal Morlet wavelet filterand autocorrelation enhancementrdquo Mechanical Systems andSignal Processing vol 24 no 5 pp 1458ndash1472 2010

[13] H Qiu J Lee J Lin and G Yu ldquoWavelet filter-based weaksignature detection method and its application on rollingelement bearing prognosticsrdquo Journal of Sound and Vibrationvol 289 no 4-5 pp 1066ndash1090 2006

[14] M Amarnath and I Praveen Krishna ldquoLocal fault detection inhelical gears via vibration and acoustic signals using EMDbasedstatistical parameter analysisrdquo Measurement vol 58 pp 154ndash164 2014

[15] Y Lei J Lin Z He and M J Zuo ldquoA review on empiricalmode decomposition in fault diagnosis of rotating machineryrdquoMechanical Systems and Signal Processing vol 35 no 1-2 pp108ndash126 2013

[16] J S Smith ldquoThe local mean decomposition and its applicationto EEG perception datardquo Journal of the Royal Society Interfacevol 2 no 5 pp 443ndash454 2005

[17] H Liu and M Han ldquoA fault diagnosis method based onlocal mean decomposition and multi-scale entropy for rollerbearingsrdquo Mechanism and Machine Theory vol 75 pp 67ndash782014

[18] J Cheng and Y Yang ldquoA rotating machinery fault diagnosismethod based on local mean decompositionrdquo Digital SignalProcessing vol 22 no 2 pp 356ndash366 2012

[19] J Zheng J Cheng and Y Yang ldquoA rolling bearing fault diagno-sis approach based on LCD and fuzzy entropyrdquoMechanism andMachine Theory vol 70 pp 441ndash453 2013

[20] J Zheng J Cheng Y Nie and S Luo ldquoA signal pro-cessing method for fault diagnosis-complete ensemble localcharacteristic-scale decompositionrdquo Journal of Vibration Engi-neering vol 27 no 4 pp 637ndash646 2014

[21] H L Ao J Cheng K Li andTK Truong ldquoA roller bearing faultdiagnosis method based on LCD energy entropy and ACROA-SVMrdquo Shock and Vibration vol 2014 Article ID 825825 12pages 2014

[22] MUnalMOnatMDemetgul andHKucuk ldquoFault diagnosisof rolling bearings using a genetic algorithm optimized neuralnetworkrdquoMeasurement vol 58 pp 187ndash196 2014

[23] B Long W Xian M Li and H Wang ldquoImproved diagnosticsfor the incipient faults in analog circuits using LSSVM based onPSO algorithm with Mahalanobis distancerdquo Neurocomputingvol 133 no 10 pp 237ndash248 2014

[24] CWRU Bearing Data Center Seeded Fault Test Data 2013httpcsegroupscaseedubearingdatacenterhome

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

0 01 02 03 04 05 06 07 08 09 10

051

152

25

0

200

400

600

800

Freq

uenc

yA

mpl

itude

Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

IA waveform of 1st IMF

IA waveform of 2nd IMF

IF waveform of 2nd IMF

IF waveform of 1st IMF

0

05

1

15

2

0100200300400500

Am

plitu

deFr

eque

ncy

Figure 15 Demodulation results of simulation signal 119910(119905) by energyoperator demodulation based on EMD

outperform the excellent time-frequency localization abilityof self-adaptive wavelet But some unknown frequency isinvolved because of background noise Moreover it can benoted that the harmonic in Figure 22 is richer and clearerthan that in Figure 23 In fact through signal decompositionby LCDand ISC selectionmost of noise can be removed fromanalysis signal Therefore the proposed method is effectiveand superior in application to weak fault diagnosis for gear

52 Bearing Inner-Race Fault Diagnosis The vibration signalof roller bearing with inner-race fault is complex and weakit is difficult to identify the fault state To further verifythe effectiveness of the proposed method we made bearinginner-race fault diagnosis experiment

The data was downloaded from the website of theCase Western Reserve University Bearing Center [24] Thetest stand consists of a 2 hp motor a torque transducera dynamometer and control electronics The test bearing

0 01 02 03 04 05 06 07 08 09 1010203040506

222224226228230232

IA waveform of 1st ISC

IF waveform of 1st ISC

IF waveform of 2nd ISC

IA waveform of 2nd ISC

Am

plitu

de

Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

Freq

uenc

y

040608

11214

155

160

165

170

Am

plitu

deFr

eque

ncy

Figure 16 Demodulation results of simulation signal with noise bythe proposed method

is at the drive end a single point defect was introducedinto the inner raceway of the test bearing The size of thesingle point defect is 0178mm in diameter and 0279mm indepth using electrodischarge machining An accelerometerattached to the bearing housing collected vibration data withthe sampling frequency as 12 kHz The shaft rotating speedof the bearing inner-race is 1750 revmin the characteristicfrequency of the roller bearing with inner-race fault is 119891

119900119894=

158HzFigure 24 presents the time domain waveform and spec-

trum of a bearing vibration signal From the spectrum it canbe seen that there are three center frequency bands and theirside frequency bands which show that main modulationcharacteristics exhibits in the high frequency band Howeverthe fault characteristic frequency is not clear in the spectrumHere we used the proposed method to demodulate Thebearing vibration signal was decomposed into thirteen ISCs

10 Shock and Vibration

0

1

2

3

0

200

400

600

800

IA waveform of 1st ISC

IF waveform of 1st ISC

Am

plitu

de

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

Freq

uenc

y

005

115

225

0

200

400

600

IA waveform of 2nd ISC

IF waveform of 2nd ISC

Am

plitu

deFr

eque

ncy

Figure 17 Demodulation results of simulation signal with noise byenergy operator demodulation approach based on LCD

and the first ISC with the highest frequency as shown inFigure 25 was selected for analysis The optimal Morletwavelet parameters were determined as 119891

119887= 128576 and

119891119888= 06862The IA waveform and its spectrum are shown in

Figure 26 from which we can clearly see the spectrum linesat the fault characteristic frequency 119891

119900119894and its harmonics

which shows that a local failure occurs in the inner racewayof bearing This is consistent with the fact state

Besides through comparison of Figure 24 with Figure 25we can find that applying LCD decomposition equals todesign adaptive band-pass filter whose center frequency isautomatically determined with the inherent characteristicsof analyzed signal Simultaneously the selection of specialISC with higher frequency for analysis leads to decrease ofthe influence of lower frequency noise to suit weak faultfeature extraction In addition due to concern of main

1 2 3 4 5 6 7 8 9 10 11

(1) Induction motor(2) Speed controller(3) Coupling(4) Bearing number 1 and bearing housing(5) Bearing number 2 and bearing housing(6) Drive gear(7) Driven gear

(9) Vibration acceleration sensor(10) Speed sensor

(8) Bearing number 3 and bearing housing

(11) Bearing number 4 and bearing housing

Figure 18 Bearing-gear test rig for vibration signal acquisition

0 01 02 03 04 05 06 07 08 09 1

0

50

100

Time (s)

Time domain waveform

0 50 100 150 200 250 300 350 4000

5

10

15

20 Envelope spectrum

minus50

minus100

Frequency (Hz)

X 21Y 914

Am

plitu

de (m

middotsminus2)

Acce

lera

tion

(mmiddotsminus

2)

Figure 19 Time domain waveform and envelope spectrum of afaulty gear vibration signal

energy on the time-scale distribution self-adaptive waveletridge modulation has excellent time-frequency localizationproperty as seen in Figure 23 Therefore the applicationresults show the proposed method is simple and effective forweak bearing inner-race fault diagnosis

6 Conclusion

LCD method is a new signal decomposition approachwhich is suitable to preprocess the multicomponent AM-FM signal Wavelet ridge demodulation method is based onwavelet transform in the time-frequency domain focusing

Shock and Vibration 11

0 01 02 03 04 05 06 07 08 09 1

200

100

50

010

Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

minus50

minus20

minus10

minus10

minus5

0

50

ISC 1

ISC 2

ISC 3

ISC 4

u(t

)

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

Figure 20 LCD decomposition results of a faulty gear vibration signal

IA waveform

0

20

40

60

80

0

500

1000IF waveform

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

Acce

lera

tion

(mmiddotsminus

2)

minus500

minus1000

Freq

uenc

y (H

z)

Figure 21 Demodulation results of the 1st ISC by Hilbert approach

on main energy on the time-scale distribution It is lesssensitive to signal bandwidth and noise especially usingthe wavelet transform with optimal parameters called self-adaptive wavelet Therefore with combination of the abovetwo approaches a self-adaptive wavelet ridge demodulationmethod based on LCD for fault diagnosis is presented

Using two simulation signals we compare the proposedmethod with the following four approaches Hilbert demod-ulation energy operator demodulation based on EMD self-adaptive wavelet ridge demodulation based on EMD andenergy operator demodulation based on LCD Analysis

0 01 02 03 04 05 06 07 08 09 10

1020304050

0 10 20 30 40 706050 80 90 10002468

10

Time (s)

IA waveform

Spectrum

Frequency (Hz)

fr

3fr4fr 5fr

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

Figure 22 Instantaneous amplitude and its spectrum of the 1st ISCby the proposed method

results show that the proposedmethod has higher demodula-tion accuracy and higher noise tolerant performance than theothers Finally we applied the proposed method to incipientfault diagnosis of gearbox The application results show theproposed method is simple and effective for incipient faultdiagnosis of gearbox

Conflict of Interests

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

12 Shock and Vibration

0 01 02 03 04 05 06 07 08 09 10

20

40

60

80

0 10 20 30 40 50 60 70 80 90 10002468

10

Time (s)

IA waveform

Spectrum

5fr

fr

Frequency (Hz)

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

Figure 23 Instantaneous amplitude and its spectrum of the orig-inal vibration signal by self-adaptive wavelet ridge demodulationapproach

0 01 02 03 04 05 06 07 08 09 1

0

1

2

0 1000 2000 3000 4000 5000 60000

005

01

015

02

Waveform

Spectrum

Time (s)

Frequency (Hz)

minus2

minus1

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

Figure 24 Time domain waveform and spectrum of a roller bearingvibration signal with inner-race defect

Acknowledgments

The support from Chinese National Science FoundationGrant (no 51075131) the construct program of the key dis-cipline in Hunan Province (Mechanical Design and Theory)(XJF2011 [76]) cooperative Demonstration Base of Universi-ties in Hunan ldquoR amp D and Industrialization of Rock DrillingMachinesrdquo (XJT [2014] 239) Hunan Major Special Projectsof Science and Technology (2014GK1043) and scientificresearch project of Hunan Province Education Department(no 14C0789) is greatly acknowledged The authors wouldlike to express their appreciation to Case Western ReserveUniversity for offering the free bearing data

0 1000 2000 3000 4000 5000 60000

005

01

015

02

Waveform

Spectrum

0

1

2

0 01 02 03 04 05 06 07 08 09 1Time (s)

minus2

minus1

Frequency (Hz)

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

Figure 25 Time domain waveform and spectrum of the 1st ISC ofa roller bearing vibration signal with inner-race defect

0 01 02 03 04 05 06 07 08 09 10

02040608

1

0 100 200 300 400 500 600 700 800 900 10000

005

01

015

02

IA waveform

Spectrum

Time (s)

Frequency (Hz)

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

foi

2foi

3foi

Figure 26 Instantaneous amplitude waveform and spectrum of the1st ISC of a roller bearing vibration signal with inner-race defect

References

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

[2] M Feldman ldquoNonparametric identification of asymmetricnonlinear vibration systems with the Hilbert transformrdquo Jour-nal of Sound and Vibration vol 331 no 14 pp 3386ndash3396 2012

[3] N E Huang Z Shen S R Long et al ldquoThe empiricalmode decomposition and the Hilbert spectrum for nonlinearand non-stationary time series analysisrdquo The Royal Societyof London Proceedings Series A Mathematical Physical andEngineering Sciences vol 454 no 1971 pp 903ndash995 1998

[4] J Cheng D Yu and Y Yang ldquoThe application of energyoperator demodulation approach based on EMD in machinery

Shock and Vibration 13

fault diagnosisrdquo Mechanical Systems and Signal Processing vol21 no 2 pp 668ndash677 2007

[5] Z Feng M Liang Y Zhang and S Hou ldquoFault diagnosis forwind turbine planetary gearboxes via demodulation analysisbased on ensemble empirical mode decomposition and energyseparationrdquo Renewable Energy vol 47 pp 112ndash126 2012

[6] Y Qin ldquoMulticomponent AM-FM demodulation based onenergy separation and adaptive filteringrdquo Mechanical Systemsand Signal Processing vol 38 no 2 pp 440ndash459 2013

[7] M Liang and H Faghidi ldquoIntelligent bearing fault detection byenhanced energy operatorrdquo Expert Systems with Applicationsvol 41 no 16 pp 7223ndash7234 2014

[8] J Wang and Q He ldquoExchanged ridge demodulation of time-scale manifold for enhanced fault diagnosis of rotating machin-eryrdquo Journal of Sound and Vibration vol 333 no 11 pp 2450ndash2464 2014

[9] Y Qin S Qin and Y Mao ldquoDemodulation approach based onwavelet ridge and its application in fault diagnosis of rotatingmachineryrdquo Chinese Journal of Mechanical Engineering vol 45no 2 pp 231ndash237 2009

[10] W He Z Jiang and Q Qin ldquoA joint adaptive wavelet filter andmorphological signal processing method for weak mechanicalimpulse extractionrdquo Journal of Mechanical Science and Technol-ogy vol 24 no 8 pp 1709ndash1716 2010

[11] Y Jiang B Tang YQin andW Liu ldquoFeature extractionmethodof wind turbine based on adaptive Morlet wavelet and SVDrdquoRenewable Energy vol 36 no 8 pp 2146ndash2153 2011

[12] W Su FWang H Zhu Z Zhang and Z Guo ldquoRolling elementbearing faults diagnosis based on optimal Morlet wavelet filterand autocorrelation enhancementrdquo Mechanical Systems andSignal Processing vol 24 no 5 pp 1458ndash1472 2010

[13] H Qiu J Lee J Lin and G Yu ldquoWavelet filter-based weaksignature detection method and its application on rollingelement bearing prognosticsrdquo Journal of Sound and Vibrationvol 289 no 4-5 pp 1066ndash1090 2006

[14] M Amarnath and I Praveen Krishna ldquoLocal fault detection inhelical gears via vibration and acoustic signals using EMDbasedstatistical parameter analysisrdquo Measurement vol 58 pp 154ndash164 2014

[15] Y Lei J Lin Z He and M J Zuo ldquoA review on empiricalmode decomposition in fault diagnosis of rotating machineryrdquoMechanical Systems and Signal Processing vol 35 no 1-2 pp108ndash126 2013

[16] J S Smith ldquoThe local mean decomposition and its applicationto EEG perception datardquo Journal of the Royal Society Interfacevol 2 no 5 pp 443ndash454 2005

[17] H Liu and M Han ldquoA fault diagnosis method based onlocal mean decomposition and multi-scale entropy for rollerbearingsrdquo Mechanism and Machine Theory vol 75 pp 67ndash782014

[18] J Cheng and Y Yang ldquoA rotating machinery fault diagnosismethod based on local mean decompositionrdquo Digital SignalProcessing vol 22 no 2 pp 356ndash366 2012

[19] J Zheng J Cheng and Y Yang ldquoA rolling bearing fault diagno-sis approach based on LCD and fuzzy entropyrdquoMechanism andMachine Theory vol 70 pp 441ndash453 2013

[20] J Zheng J Cheng Y Nie and S Luo ldquoA signal pro-cessing method for fault diagnosis-complete ensemble localcharacteristic-scale decompositionrdquo Journal of Vibration Engi-neering vol 27 no 4 pp 637ndash646 2014

[21] H L Ao J Cheng K Li andTK Truong ldquoA roller bearing faultdiagnosis method based on LCD energy entropy and ACROA-SVMrdquo Shock and Vibration vol 2014 Article ID 825825 12pages 2014

[22] MUnalMOnatMDemetgul andHKucuk ldquoFault diagnosisof rolling bearings using a genetic algorithm optimized neuralnetworkrdquoMeasurement vol 58 pp 187ndash196 2014

[23] B Long W Xian M Li and H Wang ldquoImproved diagnosticsfor the incipient faults in analog circuits using LSSVM based onPSO algorithm with Mahalanobis distancerdquo Neurocomputingvol 133 no 10 pp 237ndash248 2014

[24] CWRU Bearing Data Center Seeded Fault Test Data 2013httpcsegroupscaseedubearingdatacenterhome

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

1

2

3

0

200

400

600

800

IA waveform of 1st ISC

IF waveform of 1st ISC

Am

plitu

de

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

Freq

uenc

y

005

115

225

0

200

400

600

IA waveform of 2nd ISC

IF waveform of 2nd ISC

Am

plitu

deFr

eque

ncy

Figure 17 Demodulation results of simulation signal with noise byenergy operator demodulation approach based on LCD

and the first ISC with the highest frequency as shown inFigure 25 was selected for analysis The optimal Morletwavelet parameters were determined as 119891

119887= 128576 and

119891119888= 06862The IA waveform and its spectrum are shown in

Figure 26 from which we can clearly see the spectrum linesat the fault characteristic frequency 119891

119900119894and its harmonics

which shows that a local failure occurs in the inner racewayof bearing This is consistent with the fact state

Besides through comparison of Figure 24 with Figure 25we can find that applying LCD decomposition equals todesign adaptive band-pass filter whose center frequency isautomatically determined with the inherent characteristicsof analyzed signal Simultaneously the selection of specialISC with higher frequency for analysis leads to decrease ofthe influence of lower frequency noise to suit weak faultfeature extraction In addition due to concern of main

1 2 3 4 5 6 7 8 9 10 11

(1) Induction motor(2) Speed controller(3) Coupling(4) Bearing number 1 and bearing housing(5) Bearing number 2 and bearing housing(6) Drive gear(7) Driven gear

(9) Vibration acceleration sensor(10) Speed sensor

(8) Bearing number 3 and bearing housing

(11) Bearing number 4 and bearing housing

Figure 18 Bearing-gear test rig for vibration signal acquisition

0 01 02 03 04 05 06 07 08 09 1

0

50

100

Time (s)

Time domain waveform

0 50 100 150 200 250 300 350 4000

5

10

15

20 Envelope spectrum

minus50

minus100

Frequency (Hz)

X 21Y 914

Am

plitu

de (m

middotsminus2)

Acce

lera

tion

(mmiddotsminus

2)

Figure 19 Time domain waveform and envelope spectrum of afaulty gear vibration signal

energy on the time-scale distribution self-adaptive waveletridge modulation has excellent time-frequency localizationproperty as seen in Figure 23 Therefore the applicationresults show the proposed method is simple and effective forweak bearing inner-race fault diagnosis

6 Conclusion

LCD method is a new signal decomposition approachwhich is suitable to preprocess the multicomponent AM-FM signal Wavelet ridge demodulation method is based onwavelet transform in the time-frequency domain focusing

Shock and Vibration 11

0 01 02 03 04 05 06 07 08 09 1

200

100

50

010

Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

minus50

minus20

minus10

minus10

minus5

0

50

ISC 1

ISC 2

ISC 3

ISC 4

u(t

)

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

Figure 20 LCD decomposition results of a faulty gear vibration signal

IA waveform

0

20

40

60

80

0

500

1000IF waveform

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

Acce

lera

tion

(mmiddotsminus

2)

minus500

minus1000

Freq

uenc

y (H

z)

Figure 21 Demodulation results of the 1st ISC by Hilbert approach

on main energy on the time-scale distribution It is lesssensitive to signal bandwidth and noise especially usingthe wavelet transform with optimal parameters called self-adaptive wavelet Therefore with combination of the abovetwo approaches a self-adaptive wavelet ridge demodulationmethod based on LCD for fault diagnosis is presented

Using two simulation signals we compare the proposedmethod with the following four approaches Hilbert demod-ulation energy operator demodulation based on EMD self-adaptive wavelet ridge demodulation based on EMD andenergy operator demodulation based on LCD Analysis

0 01 02 03 04 05 06 07 08 09 10

1020304050

0 10 20 30 40 706050 80 90 10002468

10

Time (s)

IA waveform

Spectrum

Frequency (Hz)

fr

3fr4fr 5fr

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

Figure 22 Instantaneous amplitude and its spectrum of the 1st ISCby the proposed method

results show that the proposedmethod has higher demodula-tion accuracy and higher noise tolerant performance than theothers Finally we applied the proposed method to incipientfault diagnosis of gearbox The application results show theproposed method is simple and effective for incipient faultdiagnosis of gearbox

Conflict of Interests

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

12 Shock and Vibration

0 01 02 03 04 05 06 07 08 09 10

20

40

60

80

0 10 20 30 40 50 60 70 80 90 10002468

10

Time (s)

IA waveform

Spectrum

5fr

fr

Frequency (Hz)

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

Figure 23 Instantaneous amplitude and its spectrum of the orig-inal vibration signal by self-adaptive wavelet ridge demodulationapproach

0 01 02 03 04 05 06 07 08 09 1

0

1

2

0 1000 2000 3000 4000 5000 60000

005

01

015

02

Waveform

Spectrum

Time (s)

Frequency (Hz)

minus2

minus1

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

Figure 24 Time domain waveform and spectrum of a roller bearingvibration signal with inner-race defect

Acknowledgments

The support from Chinese National Science FoundationGrant (no 51075131) the construct program of the key dis-cipline in Hunan Province (Mechanical Design and Theory)(XJF2011 [76]) cooperative Demonstration Base of Universi-ties in Hunan ldquoR amp D and Industrialization of Rock DrillingMachinesrdquo (XJT [2014] 239) Hunan Major Special Projectsof Science and Technology (2014GK1043) and scientificresearch project of Hunan Province Education Department(no 14C0789) is greatly acknowledged The authors wouldlike to express their appreciation to Case Western ReserveUniversity for offering the free bearing data

0 1000 2000 3000 4000 5000 60000

005

01

015

02

Waveform

Spectrum

0

1

2

0 01 02 03 04 05 06 07 08 09 1Time (s)

minus2

minus1

Frequency (Hz)

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

Figure 25 Time domain waveform and spectrum of the 1st ISC ofa roller bearing vibration signal with inner-race defect

0 01 02 03 04 05 06 07 08 09 10

02040608

1

0 100 200 300 400 500 600 700 800 900 10000

005

01

015

02

IA waveform

Spectrum

Time (s)

Frequency (Hz)

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

foi

2foi

3foi

Figure 26 Instantaneous amplitude waveform and spectrum of the1st ISC of a roller bearing vibration signal with inner-race defect

References

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

[2] M Feldman ldquoNonparametric identification of asymmetricnonlinear vibration systems with the Hilbert transformrdquo Jour-nal of Sound and Vibration vol 331 no 14 pp 3386ndash3396 2012

[3] N E Huang Z Shen S R Long et al ldquoThe empiricalmode decomposition and the Hilbert spectrum for nonlinearand non-stationary time series analysisrdquo The Royal Societyof London Proceedings Series A Mathematical Physical andEngineering Sciences vol 454 no 1971 pp 903ndash995 1998

[4] J Cheng D Yu and Y Yang ldquoThe application of energyoperator demodulation approach based on EMD in machinery

Shock and Vibration 13

fault diagnosisrdquo Mechanical Systems and Signal Processing vol21 no 2 pp 668ndash677 2007

[5] Z Feng M Liang Y Zhang and S Hou ldquoFault diagnosis forwind turbine planetary gearboxes via demodulation analysisbased on ensemble empirical mode decomposition and energyseparationrdquo Renewable Energy vol 47 pp 112ndash126 2012

[6] Y Qin ldquoMulticomponent AM-FM demodulation based onenergy separation and adaptive filteringrdquo Mechanical Systemsand Signal Processing vol 38 no 2 pp 440ndash459 2013

[7] M Liang and H Faghidi ldquoIntelligent bearing fault detection byenhanced energy operatorrdquo Expert Systems with Applicationsvol 41 no 16 pp 7223ndash7234 2014

[8] J Wang and Q He ldquoExchanged ridge demodulation of time-scale manifold for enhanced fault diagnosis of rotating machin-eryrdquo Journal of Sound and Vibration vol 333 no 11 pp 2450ndash2464 2014

[9] Y Qin S Qin and Y Mao ldquoDemodulation approach based onwavelet ridge and its application in fault diagnosis of rotatingmachineryrdquo Chinese Journal of Mechanical Engineering vol 45no 2 pp 231ndash237 2009

[10] W He Z Jiang and Q Qin ldquoA joint adaptive wavelet filter andmorphological signal processing method for weak mechanicalimpulse extractionrdquo Journal of Mechanical Science and Technol-ogy vol 24 no 8 pp 1709ndash1716 2010

[11] Y Jiang B Tang YQin andW Liu ldquoFeature extractionmethodof wind turbine based on adaptive Morlet wavelet and SVDrdquoRenewable Energy vol 36 no 8 pp 2146ndash2153 2011

[12] W Su FWang H Zhu Z Zhang and Z Guo ldquoRolling elementbearing faults diagnosis based on optimal Morlet wavelet filterand autocorrelation enhancementrdquo Mechanical Systems andSignal Processing vol 24 no 5 pp 1458ndash1472 2010

[13] H Qiu J Lee J Lin and G Yu ldquoWavelet filter-based weaksignature detection method and its application on rollingelement bearing prognosticsrdquo Journal of Sound and Vibrationvol 289 no 4-5 pp 1066ndash1090 2006

[14] M Amarnath and I Praveen Krishna ldquoLocal fault detection inhelical gears via vibration and acoustic signals using EMDbasedstatistical parameter analysisrdquo Measurement vol 58 pp 154ndash164 2014

[15] Y Lei J Lin Z He and M J Zuo ldquoA review on empiricalmode decomposition in fault diagnosis of rotating machineryrdquoMechanical Systems and Signal Processing vol 35 no 1-2 pp108ndash126 2013

[16] J S Smith ldquoThe local mean decomposition and its applicationto EEG perception datardquo Journal of the Royal Society Interfacevol 2 no 5 pp 443ndash454 2005

[17] H Liu and M Han ldquoA fault diagnosis method based onlocal mean decomposition and multi-scale entropy for rollerbearingsrdquo Mechanism and Machine Theory vol 75 pp 67ndash782014

[18] J Cheng and Y Yang ldquoA rotating machinery fault diagnosismethod based on local mean decompositionrdquo Digital SignalProcessing vol 22 no 2 pp 356ndash366 2012

[19] J Zheng J Cheng and Y Yang ldquoA rolling bearing fault diagno-sis approach based on LCD and fuzzy entropyrdquoMechanism andMachine Theory vol 70 pp 441ndash453 2013

[20] J Zheng J Cheng Y Nie and S Luo ldquoA signal pro-cessing method for fault diagnosis-complete ensemble localcharacteristic-scale decompositionrdquo Journal of Vibration Engi-neering vol 27 no 4 pp 637ndash646 2014

[21] H L Ao J Cheng K Li andTK Truong ldquoA roller bearing faultdiagnosis method based on LCD energy entropy and ACROA-SVMrdquo Shock and Vibration vol 2014 Article ID 825825 12pages 2014

[22] MUnalMOnatMDemetgul andHKucuk ldquoFault diagnosisof rolling bearings using a genetic algorithm optimized neuralnetworkrdquoMeasurement vol 58 pp 187ndash196 2014

[23] B Long W Xian M Li and H Wang ldquoImproved diagnosticsfor the incipient faults in analog circuits using LSSVM based onPSO algorithm with Mahalanobis distancerdquo Neurocomputingvol 133 no 10 pp 237ndash248 2014

[24] CWRU Bearing Data Center Seeded Fault Test Data 2013httpcsegroupscaseedubearingdatacenterhome

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 1

200

100

50

010

Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

minus50

minus20

minus10

minus10

minus5

0

50

ISC 1

ISC 2

ISC 3

ISC 4

u(t

)

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

Figure 20 LCD decomposition results of a faulty gear vibration signal

IA waveform

0

20

40

60

80

0

500

1000IF waveform

0 01 02 03 04 05 06 07 08 09 1Time (s)

0 01 02 03 04 05 06 07 08 09 1Time (s)

Acce

lera

tion

(mmiddotsminus

2)

minus500

minus1000

Freq

uenc

y (H

z)

Figure 21 Demodulation results of the 1st ISC by Hilbert approach

on main energy on the time-scale distribution It is lesssensitive to signal bandwidth and noise especially usingthe wavelet transform with optimal parameters called self-adaptive wavelet Therefore with combination of the abovetwo approaches a self-adaptive wavelet ridge demodulationmethod based on LCD for fault diagnosis is presented

Using two simulation signals we compare the proposedmethod with the following four approaches Hilbert demod-ulation energy operator demodulation based on EMD self-adaptive wavelet ridge demodulation based on EMD andenergy operator demodulation based on LCD Analysis

0 01 02 03 04 05 06 07 08 09 10

1020304050

0 10 20 30 40 706050 80 90 10002468

10

Time (s)

IA waveform

Spectrum

Frequency (Hz)

fr

3fr4fr 5fr

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

Figure 22 Instantaneous amplitude and its spectrum of the 1st ISCby the proposed method

results show that the proposedmethod has higher demodula-tion accuracy and higher noise tolerant performance than theothers Finally we applied the proposed method to incipientfault diagnosis of gearbox The application results show theproposed method is simple and effective for incipient faultdiagnosis of gearbox

Conflict of Interests

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

12 Shock and Vibration

0 01 02 03 04 05 06 07 08 09 10

20

40

60

80

0 10 20 30 40 50 60 70 80 90 10002468

10

Time (s)

IA waveform

Spectrum

5fr

fr

Frequency (Hz)

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

Figure 23 Instantaneous amplitude and its spectrum of the orig-inal vibration signal by self-adaptive wavelet ridge demodulationapproach

0 01 02 03 04 05 06 07 08 09 1

0

1

2

0 1000 2000 3000 4000 5000 60000

005

01

015

02

Waveform

Spectrum

Time (s)

Frequency (Hz)

minus2

minus1

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

Figure 24 Time domain waveform and spectrum of a roller bearingvibration signal with inner-race defect

Acknowledgments

The support from Chinese National Science FoundationGrant (no 51075131) the construct program of the key dis-cipline in Hunan Province (Mechanical Design and Theory)(XJF2011 [76]) cooperative Demonstration Base of Universi-ties in Hunan ldquoR amp D and Industrialization of Rock DrillingMachinesrdquo (XJT [2014] 239) Hunan Major Special Projectsof Science and Technology (2014GK1043) and scientificresearch project of Hunan Province Education Department(no 14C0789) is greatly acknowledged The authors wouldlike to express their appreciation to Case Western ReserveUniversity for offering the free bearing data

0 1000 2000 3000 4000 5000 60000

005

01

015

02

Waveform

Spectrum

0

1

2

0 01 02 03 04 05 06 07 08 09 1Time (s)

minus2

minus1

Frequency (Hz)

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

Figure 25 Time domain waveform and spectrum of the 1st ISC ofa roller bearing vibration signal with inner-race defect

0 01 02 03 04 05 06 07 08 09 10

02040608

1

0 100 200 300 400 500 600 700 800 900 10000

005

01

015

02

IA waveform

Spectrum

Time (s)

Frequency (Hz)

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

foi

2foi

3foi

Figure 26 Instantaneous amplitude waveform and spectrum of the1st ISC of a roller bearing vibration signal with inner-race defect

References

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

[2] M Feldman ldquoNonparametric identification of asymmetricnonlinear vibration systems with the Hilbert transformrdquo Jour-nal of Sound and Vibration vol 331 no 14 pp 3386ndash3396 2012

[3] N E Huang Z Shen S R Long et al ldquoThe empiricalmode decomposition and the Hilbert spectrum for nonlinearand non-stationary time series analysisrdquo The Royal Societyof London Proceedings Series A Mathematical Physical andEngineering Sciences vol 454 no 1971 pp 903ndash995 1998

[4] J Cheng D Yu and Y Yang ldquoThe application of energyoperator demodulation approach based on EMD in machinery

Shock and Vibration 13

fault diagnosisrdquo Mechanical Systems and Signal Processing vol21 no 2 pp 668ndash677 2007

[5] Z Feng M Liang Y Zhang and S Hou ldquoFault diagnosis forwind turbine planetary gearboxes via demodulation analysisbased on ensemble empirical mode decomposition and energyseparationrdquo Renewable Energy vol 47 pp 112ndash126 2012

[6] Y Qin ldquoMulticomponent AM-FM demodulation based onenergy separation and adaptive filteringrdquo Mechanical Systemsand Signal Processing vol 38 no 2 pp 440ndash459 2013

[7] M Liang and H Faghidi ldquoIntelligent bearing fault detection byenhanced energy operatorrdquo Expert Systems with Applicationsvol 41 no 16 pp 7223ndash7234 2014

[8] J Wang and Q He ldquoExchanged ridge demodulation of time-scale manifold for enhanced fault diagnosis of rotating machin-eryrdquo Journal of Sound and Vibration vol 333 no 11 pp 2450ndash2464 2014

[9] Y Qin S Qin and Y Mao ldquoDemodulation approach based onwavelet ridge and its application in fault diagnosis of rotatingmachineryrdquo Chinese Journal of Mechanical Engineering vol 45no 2 pp 231ndash237 2009

[10] W He Z Jiang and Q Qin ldquoA joint adaptive wavelet filter andmorphological signal processing method for weak mechanicalimpulse extractionrdquo Journal of Mechanical Science and Technol-ogy vol 24 no 8 pp 1709ndash1716 2010

[11] Y Jiang B Tang YQin andW Liu ldquoFeature extractionmethodof wind turbine based on adaptive Morlet wavelet and SVDrdquoRenewable Energy vol 36 no 8 pp 2146ndash2153 2011

[12] W Su FWang H Zhu Z Zhang and Z Guo ldquoRolling elementbearing faults diagnosis based on optimal Morlet wavelet filterand autocorrelation enhancementrdquo Mechanical Systems andSignal Processing vol 24 no 5 pp 1458ndash1472 2010

[13] H Qiu J Lee J Lin and G Yu ldquoWavelet filter-based weaksignature detection method and its application on rollingelement bearing prognosticsrdquo Journal of Sound and Vibrationvol 289 no 4-5 pp 1066ndash1090 2006

[14] M Amarnath and I Praveen Krishna ldquoLocal fault detection inhelical gears via vibration and acoustic signals using EMDbasedstatistical parameter analysisrdquo Measurement vol 58 pp 154ndash164 2014

[15] Y Lei J Lin Z He and M J Zuo ldquoA review on empiricalmode decomposition in fault diagnosis of rotating machineryrdquoMechanical Systems and Signal Processing vol 35 no 1-2 pp108ndash126 2013

[16] J S Smith ldquoThe local mean decomposition and its applicationto EEG perception datardquo Journal of the Royal Society Interfacevol 2 no 5 pp 443ndash454 2005

[17] H Liu and M Han ldquoA fault diagnosis method based onlocal mean decomposition and multi-scale entropy for rollerbearingsrdquo Mechanism and Machine Theory vol 75 pp 67ndash782014

[18] J Cheng and Y Yang ldquoA rotating machinery fault diagnosismethod based on local mean decompositionrdquo Digital SignalProcessing vol 22 no 2 pp 356ndash366 2012

[19] J Zheng J Cheng and Y Yang ldquoA rolling bearing fault diagno-sis approach based on LCD and fuzzy entropyrdquoMechanism andMachine Theory vol 70 pp 441ndash453 2013

[20] J Zheng J Cheng Y Nie and S Luo ldquoA signal pro-cessing method for fault diagnosis-complete ensemble localcharacteristic-scale decompositionrdquo Journal of Vibration Engi-neering vol 27 no 4 pp 637ndash646 2014

[21] H L Ao J Cheng K Li andTK Truong ldquoA roller bearing faultdiagnosis method based on LCD energy entropy and ACROA-SVMrdquo Shock and Vibration vol 2014 Article ID 825825 12pages 2014

[22] MUnalMOnatMDemetgul andHKucuk ldquoFault diagnosisof rolling bearings using a genetic algorithm optimized neuralnetworkrdquoMeasurement vol 58 pp 187ndash196 2014

[23] B Long W Xian M Li and H Wang ldquoImproved diagnosticsfor the incipient faults in analog circuits using LSSVM based onPSO algorithm with Mahalanobis distancerdquo Neurocomputingvol 133 no 10 pp 237ndash248 2014

[24] CWRU Bearing Data Center Seeded Fault Test Data 2013httpcsegroupscaseedubearingdatacenterhome

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

0 01 02 03 04 05 06 07 08 09 10

20

40

60

80

0 10 20 30 40 50 60 70 80 90 10002468

10

Time (s)

IA waveform

Spectrum

5fr

fr

Frequency (Hz)

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

Figure 23 Instantaneous amplitude and its spectrum of the orig-inal vibration signal by self-adaptive wavelet ridge demodulationapproach

0 01 02 03 04 05 06 07 08 09 1

0

1

2

0 1000 2000 3000 4000 5000 60000

005

01

015

02

Waveform

Spectrum

Time (s)

Frequency (Hz)

minus2

minus1

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

Figure 24 Time domain waveform and spectrum of a roller bearingvibration signal with inner-race defect

Acknowledgments

The support from Chinese National Science FoundationGrant (no 51075131) the construct program of the key dis-cipline in Hunan Province (Mechanical Design and Theory)(XJF2011 [76]) cooperative Demonstration Base of Universi-ties in Hunan ldquoR amp D and Industrialization of Rock DrillingMachinesrdquo (XJT [2014] 239) Hunan Major Special Projectsof Science and Technology (2014GK1043) and scientificresearch project of Hunan Province Education Department(no 14C0789) is greatly acknowledged The authors wouldlike to express their appreciation to Case Western ReserveUniversity for offering the free bearing data

0 1000 2000 3000 4000 5000 60000

005

01

015

02

Waveform

Spectrum

0

1

2

0 01 02 03 04 05 06 07 08 09 1Time (s)

minus2

minus1

Frequency (Hz)

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

Figure 25 Time domain waveform and spectrum of the 1st ISC ofa roller bearing vibration signal with inner-race defect

0 01 02 03 04 05 06 07 08 09 10

02040608

1

0 100 200 300 400 500 600 700 800 900 10000

005

01

015

02

IA waveform

Spectrum

Time (s)

Frequency (Hz)

Acce

lera

tion

(mmiddotsminus

2)

Am

plitu

de (m

middotsminus2)

foi

2foi

3foi

Figure 26 Instantaneous amplitude waveform and spectrum of the1st ISC of a roller bearing vibration signal with inner-race defect

References

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

[2] M Feldman ldquoNonparametric identification of asymmetricnonlinear vibration systems with the Hilbert transformrdquo Jour-nal of Sound and Vibration vol 331 no 14 pp 3386ndash3396 2012

[3] N E Huang Z Shen S R Long et al ldquoThe empiricalmode decomposition and the Hilbert spectrum for nonlinearand non-stationary time series analysisrdquo The Royal Societyof London Proceedings Series A Mathematical Physical andEngineering Sciences vol 454 no 1971 pp 903ndash995 1998

[4] J Cheng D Yu and Y Yang ldquoThe application of energyoperator demodulation approach based on EMD in machinery

Shock and Vibration 13

fault diagnosisrdquo Mechanical Systems and Signal Processing vol21 no 2 pp 668ndash677 2007

[5] Z Feng M Liang Y Zhang and S Hou ldquoFault diagnosis forwind turbine planetary gearboxes via demodulation analysisbased on ensemble empirical mode decomposition and energyseparationrdquo Renewable Energy vol 47 pp 112ndash126 2012

[6] Y Qin ldquoMulticomponent AM-FM demodulation based onenergy separation and adaptive filteringrdquo Mechanical Systemsand Signal Processing vol 38 no 2 pp 440ndash459 2013

[7] M Liang and H Faghidi ldquoIntelligent bearing fault detection byenhanced energy operatorrdquo Expert Systems with Applicationsvol 41 no 16 pp 7223ndash7234 2014

[8] J Wang and Q He ldquoExchanged ridge demodulation of time-scale manifold for enhanced fault diagnosis of rotating machin-eryrdquo Journal of Sound and Vibration vol 333 no 11 pp 2450ndash2464 2014

[9] Y Qin S Qin and Y Mao ldquoDemodulation approach based onwavelet ridge and its application in fault diagnosis of rotatingmachineryrdquo Chinese Journal of Mechanical Engineering vol 45no 2 pp 231ndash237 2009

[10] W He Z Jiang and Q Qin ldquoA joint adaptive wavelet filter andmorphological signal processing method for weak mechanicalimpulse extractionrdquo Journal of Mechanical Science and Technol-ogy vol 24 no 8 pp 1709ndash1716 2010

[11] Y Jiang B Tang YQin andW Liu ldquoFeature extractionmethodof wind turbine based on adaptive Morlet wavelet and SVDrdquoRenewable Energy vol 36 no 8 pp 2146ndash2153 2011

[12] W Su FWang H Zhu Z Zhang and Z Guo ldquoRolling elementbearing faults diagnosis based on optimal Morlet wavelet filterand autocorrelation enhancementrdquo Mechanical Systems andSignal Processing vol 24 no 5 pp 1458ndash1472 2010

[13] H Qiu J Lee J Lin and G Yu ldquoWavelet filter-based weaksignature detection method and its application on rollingelement bearing prognosticsrdquo Journal of Sound and Vibrationvol 289 no 4-5 pp 1066ndash1090 2006

[14] M Amarnath and I Praveen Krishna ldquoLocal fault detection inhelical gears via vibration and acoustic signals using EMDbasedstatistical parameter analysisrdquo Measurement vol 58 pp 154ndash164 2014

[15] Y Lei J Lin Z He and M J Zuo ldquoA review on empiricalmode decomposition in fault diagnosis of rotating machineryrdquoMechanical Systems and Signal Processing vol 35 no 1-2 pp108ndash126 2013

[16] J S Smith ldquoThe local mean decomposition and its applicationto EEG perception datardquo Journal of the Royal Society Interfacevol 2 no 5 pp 443ndash454 2005

[17] H Liu and M Han ldquoA fault diagnosis method based onlocal mean decomposition and multi-scale entropy for rollerbearingsrdquo Mechanism and Machine Theory vol 75 pp 67ndash782014

[18] J Cheng and Y Yang ldquoA rotating machinery fault diagnosismethod based on local mean decompositionrdquo Digital SignalProcessing vol 22 no 2 pp 356ndash366 2012

[19] J Zheng J Cheng and Y Yang ldquoA rolling bearing fault diagno-sis approach based on LCD and fuzzy entropyrdquoMechanism andMachine Theory vol 70 pp 441ndash453 2013

[20] J Zheng J Cheng Y Nie and S Luo ldquoA signal pro-cessing method for fault diagnosis-complete ensemble localcharacteristic-scale decompositionrdquo Journal of Vibration Engi-neering vol 27 no 4 pp 637ndash646 2014

[21] H L Ao J Cheng K Li andTK Truong ldquoA roller bearing faultdiagnosis method based on LCD energy entropy and ACROA-SVMrdquo Shock and Vibration vol 2014 Article ID 825825 12pages 2014

[22] MUnalMOnatMDemetgul andHKucuk ldquoFault diagnosisof rolling bearings using a genetic algorithm optimized neuralnetworkrdquoMeasurement vol 58 pp 187ndash196 2014

[23] B Long W Xian M Li and H Wang ldquoImproved diagnosticsfor the incipient faults in analog circuits using LSSVM based onPSO algorithm with Mahalanobis distancerdquo Neurocomputingvol 133 no 10 pp 237ndash248 2014

[24] CWRU Bearing Data Center Seeded Fault Test Data 2013httpcsegroupscaseedubearingdatacenterhome

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 13

fault diagnosisrdquo Mechanical Systems and Signal Processing vol21 no 2 pp 668ndash677 2007

[5] Z Feng M Liang Y Zhang and S Hou ldquoFault diagnosis forwind turbine planetary gearboxes via demodulation analysisbased on ensemble empirical mode decomposition and energyseparationrdquo Renewable Energy vol 47 pp 112ndash126 2012

[6] Y Qin ldquoMulticomponent AM-FM demodulation based onenergy separation and adaptive filteringrdquo Mechanical Systemsand Signal Processing vol 38 no 2 pp 440ndash459 2013

[7] M Liang and H Faghidi ldquoIntelligent bearing fault detection byenhanced energy operatorrdquo Expert Systems with Applicationsvol 41 no 16 pp 7223ndash7234 2014

[8] J Wang and Q He ldquoExchanged ridge demodulation of time-scale manifold for enhanced fault diagnosis of rotating machin-eryrdquo Journal of Sound and Vibration vol 333 no 11 pp 2450ndash2464 2014

[9] Y Qin S Qin and Y Mao ldquoDemodulation approach based onwavelet ridge and its application in fault diagnosis of rotatingmachineryrdquo Chinese Journal of Mechanical Engineering vol 45no 2 pp 231ndash237 2009

[10] W He Z Jiang and Q Qin ldquoA joint adaptive wavelet filter andmorphological signal processing method for weak mechanicalimpulse extractionrdquo Journal of Mechanical Science and Technol-ogy vol 24 no 8 pp 1709ndash1716 2010

[11] Y Jiang B Tang YQin andW Liu ldquoFeature extractionmethodof wind turbine based on adaptive Morlet wavelet and SVDrdquoRenewable Energy vol 36 no 8 pp 2146ndash2153 2011

[12] W Su FWang H Zhu Z Zhang and Z Guo ldquoRolling elementbearing faults diagnosis based on optimal Morlet wavelet filterand autocorrelation enhancementrdquo Mechanical Systems andSignal Processing vol 24 no 5 pp 1458ndash1472 2010

[13] H Qiu J Lee J Lin and G Yu ldquoWavelet filter-based weaksignature detection method and its application on rollingelement bearing prognosticsrdquo Journal of Sound and Vibrationvol 289 no 4-5 pp 1066ndash1090 2006

[14] M Amarnath and I Praveen Krishna ldquoLocal fault detection inhelical gears via vibration and acoustic signals using EMDbasedstatistical parameter analysisrdquo Measurement vol 58 pp 154ndash164 2014

[15] Y Lei J Lin Z He and M J Zuo ldquoA review on empiricalmode decomposition in fault diagnosis of rotating machineryrdquoMechanical Systems and Signal Processing vol 35 no 1-2 pp108ndash126 2013

[16] J S Smith ldquoThe local mean decomposition and its applicationto EEG perception datardquo Journal of the Royal Society Interfacevol 2 no 5 pp 443ndash454 2005

[17] H Liu and M Han ldquoA fault diagnosis method based onlocal mean decomposition and multi-scale entropy for rollerbearingsrdquo Mechanism and Machine Theory vol 75 pp 67ndash782014

[18] J Cheng and Y Yang ldquoA rotating machinery fault diagnosismethod based on local mean decompositionrdquo Digital SignalProcessing vol 22 no 2 pp 356ndash366 2012

[19] J Zheng J Cheng and Y Yang ldquoA rolling bearing fault diagno-sis approach based on LCD and fuzzy entropyrdquoMechanism andMachine Theory vol 70 pp 441ndash453 2013

[20] J Zheng J Cheng Y Nie and S Luo ldquoA signal pro-cessing method for fault diagnosis-complete ensemble localcharacteristic-scale decompositionrdquo Journal of Vibration Engi-neering vol 27 no 4 pp 637ndash646 2014

[21] H L Ao J Cheng K Li andTK Truong ldquoA roller bearing faultdiagnosis method based on LCD energy entropy and ACROA-SVMrdquo Shock and Vibration vol 2014 Article ID 825825 12pages 2014

[22] MUnalMOnatMDemetgul andHKucuk ldquoFault diagnosisof rolling bearings using a genetic algorithm optimized neuralnetworkrdquoMeasurement vol 58 pp 187ndash196 2014

[23] B Long W Xian M Li and H Wang ldquoImproved diagnosticsfor the incipient faults in analog circuits using LSSVM based onPSO algorithm with Mahalanobis distancerdquo Neurocomputingvol 133 no 10 pp 237ndash248 2014

[24] CWRU Bearing Data Center Seeded Fault Test Data 2013httpcsegroupscaseedubearingdatacenterhome

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