Research Article Application of Reassigned Wavelet...

Research ArticleApplication of Reassigned Wavelet Scalogram inWind Turbine Planetary Gearbox Fault Diagnosis underNonstationary Conditions

Xiaowang Chen and Zhipeng Feng

School of Mechanical Engineering University of Science and Technology Beijing Beijing 100083 China

Correspondence should be addressed to Zhipeng Feng fengzpustbeducn

Received 27 June 2015 Accepted 31 August 2015

Academic Editor Chuan Li

Copyright copy 2016 X Chen and Z FengThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Wind turbine planetary gearboxes often run under nonstationary conditions due to volatile wind conditions thus resulting innonstationary vibration signals Time-frequency analysis gives insight into the structure of an arbitrary nonstationary signal in jointtime-frequency domain but conventional time-frequency representations suffer from either time-frequency smearing or cross-term interferences Reassigned wavelet scalogram has merits of fine time-frequency resolution and cross-term free nature but hasvery limited applications in machinery fault diagnosis In this paper we use reassigned wavelet scalogram to extract fault featurefrom wind turbine planetary gearbox vibration signals Both experimental and in situ vibration signals are used to evaluate theeffectiveness of reassigned wavelet scalogram in fault diagnosis of wind turbine planetary gearbox For experimental evaluationthe gear characteristic instantaneous frequency curves on time-frequency plane are clearly pinpointed in both local and distributedsun gear fault cases For in situ evaluation the periodical impulses due to planet gear fault are also clearly identified The resultsverify the feasibility and effectiveness of reassigned wavelet scalogram in planetary gearbox fault diagnosis under nonstationaryconditions

1 Introduction

Wind turbines are playing an increasingly significant rolein energy strategy However harsh working conditions forexample wind gust dust and unpredictable heavy loadmake the power transmission system prone to fault whichmay lead to catastrophic breakdown or even productivityand economic losses Planetary gearbox is one of the keycomponents of wind turbine powertrain for itsmerits of highgear transmission ratio and large load bearing capacity in acompact structure Therefore planetary gearbox fault diag-nosis has been an important topic and has drawn increasingattentions recently [1ndash3]

Planetary gearbox exhibits quite unique dynamic behav-ior due to its special transmission structure and kineticsMcFadden [4] recognized that the varying phase angle of theplanet gear vibration is responsible for spectra asymmetryInalpolat and Kahraman [5 6] investigated the mechanismsof sideband harmonics in the vicinity of meshing frequency

and its integer multiples using a general model and furtherdeveloped a dynamicmodel to predict modulation sidebandsdue to manufacturing errors Feng and Zuo [7] proposedplanetary gearbox vibration signal models and deprivedequations to calculate both local and distributed fault fre-quencies These researches show that planetary gearboxvibration signals feature complexity and modulation andtherefore have quite different spectral characteristics fromthat of fixed-shaft gearboxes

In order to extract potential fault feature from planetarygearbox vibration signals researchers proposed differentmethods McNames [8] used continuous-time Fourier series(CTFS) analysis to invest the frequency characteristics Chenet al [9] applied ensemble multiwavelet transform withadaptive multiwavelet basis function to extract weak faultfeature of a planetary gearbox Lei et al [10] proposedan adaptive stochastic resonance method to extract weakfault feature characteristics from noisy vibration signals ofplanetary gearbox Considering the spectral complexity of

Hindawi Publishing CorporationShock and VibrationVolume 2016 Article ID 6723267 12 pageshttpdxdoiorg10115520166723267

2 Shock and Vibration

planetary gearbox vibration signals Feng et al [11] presenteda joint amplitude and frequency demodulation analysis basedon Teager energy operator and ensemble empirical modedecomposition (EEMD) to diagnose planetary gearbox faultBarszcz and Randall [12] employed the spectral kurtosis todetect gear tooth crack in a wind turbine planetary gearboxSun et al [13] customized the multiwavelets based on theredundant symmetric lifting scheme to detect the incipientpitting faults in planetary gearbox Li et al [14] extracted thefault characteristics frequencies of planetary gearbox basedon empirical mode decomposition and adaptive multiscalemorphological gradient filter

Most of above works were based on an assumption ofsignal stationarity that is constant running speed and loadHowever in real-world wind turbine applications volatilewind conditions result in nonstationary vibration signalsFor example the running speed varies with respect to theunpredictable wind power and transient strong wind maylead to a shock Under nonstationary conditions planetarygearbox vibration signals always feature intricate frequencycomponent structure [15 16] manifesting as time-varyingamplitude modulation (AM) and frequency modulation(FM) Due to the nonstationary nature conventional Fouriertransform most likely fails to thoroughly reveal the truefrequency structure To the best of our knowledge theliterature reported onplanetary gearbox fault diagnosis undernonstationary conditions has been very limited [17] Bartel-mus and Zimroz [18] proposed an indicator that reflects thelinear dependence between themeshing frequency amplitudeand the operating condition for condition monitoring ofplanetary gearboxes Chaari et al [19] developed amathemat-ical model to understand the dynamic behavior of planetarygear under variable load condition Yang and Zhang [20]studied the running characteristics of wind turbine planetarygearbox considering time-varying speed and load

Time-frequency analysis (TFA) is capable of revealingtime-frequency structure despite time variation of frequencycomponents and thus is suitable for analyzing nonstation-ary signals [21] Linear time-frequency analysis methodslike short time Fourier transform (STFT) and continuouswavelet transform (CWT) essentially represent signals byweighted sum of a series of bases localized in both time andfrequency domains But their time-frequency resolution isgoverned by Heisenberg uncertainty principle and resolutionin one domain can only be improved at expense of theother resulting in time-frequency smearing [22] Bilineartime-frequency distribution presents the signal energy ontime-frequency plane [23 24] The Wigner-Ville distribution(WVD) is the basis of almost all bilinear TFA methodswith merit of finest time-frequency resolution Howeverit is subjected to inevitable cross-term interferences whenanalyzing multicomponent signals Even for monocompo-nent signals it still suffers from inner interference if the IFcurve is nonlinear Although improved methods smoothedpseudo Wigner-Ville distribution (SPWVD) for examplecan effectively suppress the cross-term interferences they inturn deteriorate time-frequency resolution

Fortunately reassigned wavelet scalogram has merits ofexcellent time-frequency resolution and good suppression of

cross-terms and thus can clearly exhibit the time-frequencystructure of nonstationary signals However its applicationsinmachinery fault diagnosis have been very limited and onlya few published works focus on this topic Peng et al [25]studied the characteristics of rotor-stator rubbing impactsand applied both scalogram and reassigned scalogram forimpact detection Ma et al [26] summarized the symptomsof rotor-stator rubbing fault based on reassigned scalogram

In this paper we further extend the application of reas-signed wavelet scalogram to planetary gearbox fault diag-nosis under nonstationary conditions [27] and validate itseffectiveness in extracting gear fault features The remainderof this paper is structured as follows In Section 2 thedetailed algorithm of scalogram and reassigned scalogramis introduced In Section 3 a numerical simulated nonsta-tionary signal encompassing a time-varying frequency andperiodical impulses is analyzed to illustrate the performanceof reassigned scalogram In Sections 4 and 5 the effectivenessof reassigned scalogram is further validated by analysis oflab experimental planetary gearbox vibration signals in bothlocal and distributed sun gear fault cases as well as in situwind turbine planetary gearbox vibration signals measuredfrom awind farm Finally conclusions are drawn in Section 6

2 Reassigned Wavelet Scalogram

21 ContinuousWavelet Transform Wavelet transform (WT)has advantage of better time-frequency resolution over shorttime Fourier transform (STFT) Rather than using a constantwindow function for whole time-frequency plane analysis itemploys different window functions for analyzing differentfrequency bands of the signal Therefore WT is popularamong linear TFA methods

For an arbitrary finite-energy signal 119909(119905) isin 1198712(119877) its

CWT is defined as

119882119909(119886 119887 120595) = ⟨119909 (119905) 120595

119886119887(119905)⟩

= 119886minus12

int

infin

minusinfin

119909 (119905) 120595lowast

119886119887(119905) 119889119905

(1)

where asterisk lowast stands for complex conjugate 119886 gt 0 119886 and119887 respectively represent the scale and time offset and 120595

119886119887(119905)

is the family of wavelets generated by dilation and translationfrom the mother wavelet 120595(119905)

120595119886119887

(119905) = 119886minus12

120595(

119905 minus 119887

119886

) (2)

Morlet wavelet is most commonly used in CWT and it isdefined as

120595 (119905) = 120587minus14

(119890minus1198941205960119905

minus 119890minus1205962

02) 119890minus11990522 (3)

The wavelet scalogram is defined as

119878119909(119886 119887 120595) =

1003816100381610038161003816119882119909(119886 119887 120595)

1003816100381610038161003816

2

(4)

22 Reassigned Wavelet Scalogram Although wavelet scalo-gram can reveal the constituent frequency components of

Shock and Vibration 3

0

Am

plitu

de (V

)

minus06

minus04

minus02

02

04

06

08

1

02 04 06 08 10Time (s)

(a) Waveform

Am

plitu

de (V

)

0

0002

0004

0006

0008

001

200 400 600 800 10000Frequency (Hz)

(b) Fourier spectrum

Figure 1 Simulated signal

a signal and show how the energy of each component varieswith time and frequency its time-frequency resolution is stillsubject to the Heisenberg-Gabor inequality that is fine timelocalization and frequency resolution can hardly be obtainedsimultaneously For example it has fine time localization butcoarse frequency resolution in high-frequency regions whileit has coarse time localization but fine frequency resolution inlow-frequency regions Therefore for nonstationary signalswith instantaneous frequency curves changing in a widefrequency region the time-frequency blurring phenomenonmay arise thus hindering an accurate fault feature extraction

In order to improve the readability of conventionalwavelet scalograms Auger and Flandrin [28] proposed thereassignmentmethodThewavelet scalogram 119878

119909(119886 119887 120595) does

not represent the signal energy at the location (119886 119887) onthe time-scalefrequency plane but the mean energy in thevicinity of location (119886 119887) Therefore it is reasonable toreallocate the signal energy 119878

119909(119886 119887 120595) to the center of gravity

in the vicinity of location (119886 119887) The new coordinates become

1205960

1198861015840(119887 119886)

=

1205960

119886

+ Im

CWT119909(119886 119887 )CWTlowast

119909(119886 119887 120595)

21205871198861003816100381610038161003816CWT

119909(119886 119887 120595)

1003816100381610038161003816

2

1198871015840(119886 119887)

= 119887 minus Re119886CWT

119909(119886 119887 120595

1015840)CWTlowast

119909(119886 119887 120595)

1003816100381610038161003816CWT

119909(119886 119887 120595)

1003816100381610038161003816

2

(5)

where

1205951015840(119905) = 119905120595 (119905)

(119905) =

119889120595

119889119905

(119905)

(6)

Thus the reassigned wavelet scalogram is cast as

119877119878119909(1198861015840 1198871015840) = int

+infin

minusinfin

int

+infin

minusinfin

(

119886

119886

)

2

119878119909(119886 119887) 120575 [

119887 minus 1198871015840(119886 119887)]

sdot 120575 [119886 minus 1198861015840(119886 119887)] 119889119886 119889119887

(7)

3 Simulation Evaluation

In this section we examine the performance of reassignedwavelet scalogram in identifying time-varying frequencycomponents and transient impulses characteristic of typicalgear fault under nonstationary conditions A numericalsimulated signal is generated

119904 (119905) = 119860 sin(2120587(int800119905

chirp

+ int 50 cos (212058710119905)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

FM

))

+ 120575 (119905)

impulse+ 119899 (119905)

(8)

where 119905 = 0 00005 1 s 119860 = 01 is the amplitude of time-varying frequency component and 120575(119905) stands for impulsesat time instant of 025 05 and 075 s To mimic backgroundnoise a Gaussian white noise 119899(119905) at signal to noise ratio(SNR) of 8 dB is added This signal encompasses threenonstationary conditions that is linear frequency modu-lation (chirp) frequency fluctuations (FM) and periodicalimpulses Its waveform and Fourier spectrum are plotted inFigures 1(a) and 1(b)

From the signal waveform Figure 1(a) we can clearly seethe time instant of the three impulses but cannot identify theperiod of time-varying frequency components Neither canwe do from the Fourier spectrum Figure 1(b) due to thenonstationarity of the simulated signal

Figure 2 shows the time-frequency analysis results ofthe signal In the wavelet scalogram Figure 2(a) the majorsignal components and their time variation are roughly


0

200

400

600

800

1000Fr

eque

ncy

(Hz)

02 04 06 08 10Time (s)

(a) Wavelet scalogram

0

200

400

600

800

1000

Freq

uenc

y (H

z)

02 04 06 08 10Time (s)

(b) Reassigned scalogram

0

200

400

600

800

1000

Freq

uenc

y (H

z)

02 04 06 08 10Time (s)

(c) True frequency constitution

Figure 2 TFRs of the simulated signal

shown a nearly sinusoidal fluctuating frequency rides on alinear trend together with three impulses However time-frequency blurring occurs because the instantaneous fre-quency of the time-varying frequency component is nonlin-earThis results in a lower time-frequency resolution andmayhinder an accurate time-frequency localization of each signalcomponent

In the reassigned scalogram Figure 2(b) both the time-varying frequency component and the three impulses areclearly displayed owing to the much better time-frequencyresolution More importantly the identified instantaneousfrequency curve of the time-varying component and thetime-frequency distribution of the three impulses matchwell with the true time-frequency constitution shown inFigure 2(c) This result demonstrates the capability of reas-signed scalogram in identifying both time-varying frequencycomponent and transient impulses Therefore reassignedscalogram has the potential to diagnose planetary gearboxfault under nonstationary conditions

4 Experimental Evaluations

Experiments were carried out on a wind turbine drivetraintest rig in University of Ottawa lab to examine the effective-ness of reassigned wavelet scalogram on real-world planetarygearbox fault diagnosis

Table 1 Fixed-shaft gearbox configuration

Gear Number of gear teethStage 1 Stage 2

Input 32 mdashIntermediate 80 40Output mdash 72

41 Experimental Settings In order to simulate the workingcondition of real-world planetary gearbox in wind farms awind turbine drivetrain composed of a two-stage planetarygearbox and a two-stage fixed-shaft gearbox is set up asshown in Figure 3 The power flow path is as followsdrive motorrarr fixed-shaft gearboxrarr planetary gearboxrarrmain shaftrarrmagnetic powder brake The configurationparameters of the two gearboxes are listed in Tables 1 and 2respectively

Accelerometers are mounted on the top of planetarygearbox casing tomeasure planetary gearbox vibration signalat a sampling frequency of 20000Hz A load of 163Nsdotmis applied at the output end of planetary gearbox stage 2by loading device In consideration of the wind turbineblade rated speed of 25 rpm that is 0417Hz we collectthe vibration data during specific speed-down processes


Table 2 Planetary gearbox configuration


Ring 100 100Planet 40 (4) 36 (4)Sun 20 28Note the number of planet gears is in the parenthesis

1 3 4 5 6 72

Figure 3 WT drivetrain test rig (1) motor (2) tachometer (3)fixed-shaft gearbox (4) planetary gearbox stage 1 (5) planetarygearbox stage 2 (6) accelerometers and (7) loading device

The drive motor speed decreases from 60Hz to 50Hz andthe corresponding blade speed varies continuously from0486Hz to 0405Hz within the typical rated speed rangeof wind turbine blades in real world The motor speed ismeanwhile recorded at a sampling frequency of 20Hz

To simulate gear faults two types of gear damage wereintroduced to sun gears of planetary gearbox stage 1 andstage 2 respectively while other gears are healthy Stage 1 sungear has wear on every tooth and stage 2 sun gear has anindividual chipped tooth as illustrated in Figures 4(a) and4(b) Therefore three sets of experiments are carried out asbaseline wear and chipping

Given the drive motor speed 119891119889(119905) we can calculate the

characteristic frequencies of both stages of planetary gearboxaccording to the aforementioned gearbox configuration aslisted in Table 3 For example at the motor speed 119891

119889(119905) =

60Hz the sun gear related characteristic frequencies arecalculated as 119891

1198981(119905) = 2222Hz 119891(119903)

1199041(119905) = 133Hz and

1198911199041(119905) = 444Hz for stage 1 and 119891

1198982(119905) = 486Hz 119891(119903)

1199042(119905) =

22Hz and 1198911199041(119905) = 69Hz for stage 2

42 Baseline Signal Analysis Firstly we analyze the baselinesignal for comparison study with the gear fault cases Inbaseline case all gears are healthy Figures 5(a) and 5(b)exhibit the waveform and Fourier spectrum of the baselinevibration signal and Figure 5(c) shows the correspondingspeedThemaximummeshing frequency is 2222Hz for stage1 and 486Hz for stage 2 Since gear fault usually results insidebands around meshing frequency and its harmonics wefocus on two frequency ranges of 0ndash400Hz and 0ndash80Hz

Table 3 Planetary gearbox characteristic frequencies

Frequency Stage 1 Stage 2Meshing 119891

119898(119905) (10027) 119891

119889(119905) (175216) 119891

119889(119905)

Sun rotating 119891(119903)119904(119905) (29) 119891

119889(119905) (127) 119891

119889(119905)

Planet carrier rotating 119891119888(119905) (127) 119891

119889(119905) (7864) 119891

119889(119905)

Sun fault 119891119904(119905) (2027) 119891

119889(119905) (1751512) 119891

119889(119905)

Planet fault 119891119901(119905) (554) 119891

119889(119905) (1757776) 119891

119889(119905)

Ring fault 119891119903(119905) (427) 119891

119889(119905) (1755400) 119891

119889(119905)

which cover 32 times stage 1 and stage 2meshing frequenciesrespectively

Figures 6(a) and 6(b) show the reassigned wavelet scalo-gram of the baseline signal It clearly displays the constituentfrequency components and their variations along with timeThe value of each characteristic frequency at any time canbe calculated according to the equations in Table 3 thus theinstantaneous frequency curves in the time-frequency analy-sis results can be identified In Figure 6(a) the gear meshingfrequency minus the sun gear fault frequency 119891

1198981(119905) minus 119891

1199041(119905)

and the gear meshing frequency minus four times the sunrotating frequency 119891

1198981(119905) minus 4119891

(119903)

1199041(119905) are presented but they

are not dominant in comparison with the other componentsalmost uniformly distributed in higher frequency region[200 400]Hz Even for healthy gearboxes manufacturingerrors and minor defects are inevitable Hence the presenceof these frequency components does not imply the sun gearfault In Figure 6(b) only a few weak components are visibleshowing that the gearbox is healthy which is consistent withthe experimental setting

43 Detection of Sun Gear Wear In this section we replacedthe healthy stage 1 sun gear with a worn one while othersettings are unchanged and detect the sun gear wear bycomparing worn sun gear signal with baseline signal injoint time and frequency domain The waveform Fourierspectrum and corresponding motor speed are plotted inFigures 7(a)ndash7(c) Since gear damage locates on stage 1 sungear in this case we focus on 0ndash400Hz to look for faultsymptom around stage 1 meshing frequency

Figure 8 presents the reassigned wavelet scalogram offaulty sun signal Compared to the baseline signal case inFigure 6(a) the gear meshing frequency minus the sun faultcharacteristic frequency 119891

1198981(119905) minus 119891

1199041(119905) and the gear meshing

frequency minus four times the sun rotating frequency1198911198981(119905) minus 4119891

(119903)

1199041(119905) still remain Besides more characteristic

frequencies appear including the gear meshing frequency1198911198981(119905) the gear meshing frequency minus 74 times the

sun fault characteristic frequency 1198911198981(119905) minus (74)119891

1199041(119905) and

the gear meshing frequency plus three times the sun faultcharacteristic frequency 119891

1198981(119905) + 3119891

1199041(119905) All these frequency

components have a pronounced magnitude and they areall related to the sun gear This finding shows that the 1ststage sun gear is faulty in consistence with the experimentalsetting


(a) Stage 1 sun gear wear (b) Stage 2 sun gear chipping

Figure 4 Sun gear damage

0 1 2 3 4 5Time (s)

minus8

minus6

minus4

minus2

0

Am

plitu

de (m

s2)

2

4

6

8

(a) Waveform

50 100 150 200 250 300 350 4000Frequency (Hz)

0

001

002

003

Am

plitu

de (m

s2)

004

005

006


45

50

55

60

65

Spee

d (H

z)

1 2 3 4 50Time (s)

(c) Motor speed

Figure 5 Baseline signal

44 Detection of Sun Gear Chipping Another type of gearfault that is gear chipping is brought in by replacing thehealthy stage 2 sun gear with a chipped one while othersettings remain the same with baseline case Still we detectthe fault by comparing chipped sun gear signal with baseline

signal in joint time and frequency domain but anotherfrequency range of 0ndash80Hz is focused on since themaximumstage 2 meshing frequency is 486HzThe waveform Fourierspectrum and corresponding motor speed of chipped sungear signal are presented in Figures 9(a)ndash9(c)


1 2 3 4 50Time (s)

0

100

200

300

400Fr

eque

ncy

(Hz)

fm1 minus fs1 fm1 minus 4f(r)s1

(a) Frequency band of 0ndash400Hz

1 2 3 4 50Time (s)

0

20

40

60

80

Freq

uenc

y (H

z)

(b) Frequency band of 0ndash80Hz

Figure 6 Reassigned scalogram of baseline signal

Am

plitu

de (m

s2)

minus8

minus6

minus4

minus2

0

2

4

6

8

1 2 3 4 50Time (s)

(a) Waveform

Am

plitu

de (m

s2)

50 100 150 200 250 300 350 4000Frequency (Hz)

0

001

002

003

004

005

006


Spee

d (H

z)

45

50

55

60

65

41 2 3 50Time (s)

(c) Motor speed

Figure 7 Sun gear wear signal

In the reassigned scalogram Figure 10 except for themotor rotating frequency component 119891

119889 two new charac-

teristic frequency components are also evident One corre-sponds to the meshing frequency minus the third harmonicof the sum of the sun gear fault characteristic frequency andthe sun gear rotating frequency of the 2nd stage119891

1198982minus3[119891(119903)

1199042+

1198911199042] The other is very close to motor rotating frequency but

by virtue of the fine time-frequency resolution it is identifiedto be the sum of the meshing frequency and the secondharmonic of sun gear fault characteristic frequency of the 2ndstage 119891

1198982+ 21198911199042(as labeled with a black dashed line) These

two extra frequency components correlate to the 2nd stagesun gear fault characteristic frequency and have prominentmagnitude thus implying the 2nd stage sun gear fault


0

100

200

300

400

Freq

uenc

y (H

z)

1 2 3 4 50Time (s)

fm1


fm1 + 3fs1

fm1 + (74)fs1

Figure 8 Reassigned scalogram of sun gear wear signal

minus1

minus05

0

05

1

Am

plitu

de (m

s2)

1 2 3 4 50Time (s)

(a) Waveform

0

005

01

015

02

025

03

035

Am

plitu

de (m

s2)

10 20 30 40 50 60 70 800Frequency (Hz)


0 1 2 3 4 5Time (s)

Spee

d (H

z)

45

50

55

60

65

(c) Motor speed

Figure 9 Chipped sun gear signal

The sun gear fault usually exhibits weak symptom inthe vibration signal because the sun gear is relatively farfrom the accelerometer mounted on the gearbox casing Butthe time-varying sun gear fault characteristic frequencies arestill identified owing to the fine time-frequency resolutionand capability in suppressing cross-term interferences ofreassigned scalogram This shows its effectiveness in extract-ing planetary gearbox fault features under nonstationaryconditions

5 In Situ Signal Analysis

In this section we analyze the in situ collected signalsof a wind turbine planetary gearbox in a wind farm tofurther validate the reassigned scalogram for engineeringapplications

51 Gearbox Configuration The wind turbine drivetrain ina wind farm to be analyzed is composed of a one-stage


fm2 + 2fs2 fd fm2 minus 3[f(r)s2 + fs2]

1 2 3 4 50Time (s)

0

20

40

60

80

Freq

uenc

y (H

z)

Figure 10 Reassigned scalogram of sun gear chipping signal

Table 4 Planetary gearbox configuration parameters

Gear Number of gear teethRing 96Planet 37 (3)Sun 21Note the number of planet gears is in the parenthesis

Table 5 Fixed-shaft gearbox configuration parameters

Gear Number of gear teethInput stage Output stage

Input gear 71 mdashIntermediate shaft gear 19 89Output gear mdash 30

planetary gearbox and a two-stage fixed-shaft gearbox Thewind power flow path is as follows bladesrarrmain shaftrarrplanetary gearboxrarr fixed-shaft gearboxrarr generator Con-sidering the demand of wind farm information protectionphotos of the wind turbine drivetrain are not attached in thepaper Tables 4 and 5 list the configuration parameters of thefixed-shaft gearbox and the planetary gearbox respectively

The rated rotating speed of the wind turbine blades is25 rpm that is 0416Hz At the rated speed the characteristicfrequencies of the drivetrain gearboxes are calculated as listedin Table 6 They will be used to estimate the frequencycomponents of vibration signals for the wind turbine speedusually fluctuates around the rated speed

The vibration signals from an accelerometer mounted onthe planetary gearbox casing of both a normal and a faultywind turbine are collected Since the gear fault always resultsin abnormal sidebands or impulses centered around gearmeshing frequency and its harmonics a frequency band of0ndash70Hz which covers over 32 times the planetary gearboxmeshing frequency (4203Hz) will be focused on

52 Signal Analysis

521 Normal Case The waveform and Fourier spectrumof the normal wind turbine planetary gearbox vibration

signal are plotted in Figures 11(a) and 11(b) Neverthelessdetails of nonstationary characteristics are still veiled dueto information loss Figure 12 shows the reassigned waveletscalogram of the normal wind turbine planetary gearboxvibration signal The generator rotating frequency 27Hz andthe planetary gearbox meshing frequency 42Hz are domi-nant The second harmonic of generator rotating frequency54Hz is also visible but in a relatively low magnitude Allthese frequencies are time-varying due to the unsteady speedof wind turbine blades caused by the nonstationary windpower Thus it is difficult to identify frequency componentsvia traditional Fourier transform based spectral analysis

522 Faulty Planet Gear Case In the faulty case similarly thewaveform and Fourier spectrum of faulty planetary gearboxvibration signal are presented in Figures 13(a) and 13(b) Dueto the unsteady motor speed and nonstationary conditionsit is difficult to identify frequency components via traditionalspectral analysis Figure 14 shows the reassigned scalogramof the faulty wind turbine planetary gearbox vibration signalIt has a time-frequency structure far different from thatin the normal case Only the planetary gearbox meshingfrequency is dominant In addition some big impulses appearalmost periodically as labeled by evenly spaced vertical lineswith capitals AndashG in Figure 14(a) The interval betweenadjacent impulses corresponds approximately to the planetcarrier rotating frequency 044Hz Furthermore if we zoomin Figure 14(a) and focus on the interval between twoconsecutive primary impulses for example B and C somesmall almost periodic impulses also appear as indicated bylower-case letters andashf in Figure 14(b) The interval betweenthese adjacent small impulses corresponds approximately tothe planet gear fault characteristic frequency that is 227HzThese findings imply the planet gear fault in accordance withthe check result after opening the planetary gearbox

The planet gear fault symptom is complicated becausethe planet gear not only spins around its own axis but alsorevolves around the sun gear with the planet carrier andmeanwhile it meshes with both the sun gear and the ring gearBut the gear fault induced impulses are still identified illus-trating the effectiveness of reassigned scalogram in extractingcomplicated fault features under nonstationary conditions


Table 6 Characteristic frequencies of drivetrain gearbox

Planetary Fixed-shaft stage 1 Fixed-shaft stage 2Mesh Ring Planet Sun Mesh Drive Driven Mesh Drive Driven4203 044 227 201 17318 244 911 81120 911 2704

minus10

minus5

0

5

10

Am

plitu

de (m

s2)

5 10 15 20 25 300Time (s)

(a) Waveform

0

0002

0004

0006

0008

001

0012

0014

0016

0018

Am

plitu

de (m

s2)

10 20 30 40 50 60 700Frequency (Hz)


Figure 11 Normal signal

0 5 10 15 20 25 30

Time (s)

0

10

20

30

40

50

60

70

Freq

uenc

y (H

z)

Figure 12 Reassigned scalogram of normal signal

Am

plitu

de (m

s2)

minus40

minus30

minus20

minus10

0

10

20

30

40

5 10 15 20 25 300Time (s)

(a) Waveform

0 10 20 30 40 50 60 70Frequency (Hz)

Am

plitu

de (m

s2)

0

001

002

003

004

005


Figure 13 Faulty signal


0 5 10 15 20 25 30

A B C D E F G

Freq

uenc

y (H

z)

0

10

20

30

40

50

60

70

Time (s)

(a) Full time span view

Freq

uenc

y (H

z)

B C

a b c d e f

1 2 3 4 5 6 7 8 90 10

Time (s)

0

10

20

30

40

50

60

70

(b) Close-up view of the interval from 0 to 10 s

Figure 14 Reassigned scalogram of faulty signal

6 Conclusions

The reassigned wavelet scalogram has excellent merits offine time-frequency resolution and capability in suppressingcross-term interferences and thus is suited to extract thetime-frequency features of nonstationary signals We havevalidated the effectiveness of this method for planetary gear-box fault diagnosis under nonstationary conditions usingboth lab experimental signals and in situ measured datasetsBoth the time-varying gear characteristic frequencies andgear fault induced impulses are identified and thereby thelocalized and distributed sun gear fault as well as naturallydeveloped planet gear fault are diagnosed even thoughthey have weak and complicated fault symptoms undernonstationary conditions

Conflict of Interests

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

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (11272047 and 51475038) the Programfor New Century Excellent Talents in University (NCET-12-0775) and the Natural Sciences and Engineering ResearchCouncil of Canada

References

[1] P D Samuel and D J Pines ldquoA review of vibration-basedtechniques for helicopter transmission diagnosticsrdquo Journal ofSound and Vibration vol 282 no 1-2 pp 475ndash508 2005

[2] Y Guo and R G Parker ldquoAnalytical determination of meshphase relations in general compound planetary gearsrdquo Mecha-nism and Machine Theory vol 46 no 12 pp 1869ndash1887 2011

[3] Y Lei J Lin M J Zuo and Z He ldquoCondition monitoring andfault diagnosis of planetary gearboxes a reviewrdquoMeasurementvol 48 no 1 pp 292ndash305 2014

[4] P D McFadden ldquoWindow functions for the calculation of thetime domain averages of the vibration of the individual planetgears and sun gear in an epicyclic gearboxrdquo Journal of Vibrationand Acoustics Transactions of the ASME vol 116 no 2 pp 179ndash187 1994

[5] M Inalpolat and A Kahraman ldquoA theoretical and experimentalinvestigation of modulation sidebands of planetary gear setsrdquoJournal of Sound and Vibration vol 323 no 3ndash5 pp 677ndash6962009

[6] M Inalpolat and A Kahraman ldquoA dynamic model to predictmodulation sidebands of a planetary gear set having manufac-turing errorsrdquo Journal of Sound and Vibration vol 329 no 4pp 371ndash393 2010

[7] Z Feng and M J Zuo ldquoVibration signal models for fault diag-nosis of planetary gearboxesrdquo Journal of Sound and Vibrationvol 331 no 22 pp 4919ndash4939 2012

[8] J McNames ldquoFourier series analysis of epicyclic gearbox vibra-tionrdquo Journal of Vibration and Acoustics vol 124 no 1 pp 150ndash160 2002

[9] J Chen C Zhang X Zhang Y Zi S He and Z YangldquoPlanetary gearbox conditionmonitoring of ship-based satellitecommunication antennas using ensemblemultiwavelet analysismethodrdquo Mechanical Systems and Signal Processing vol 54-55pp 277ndash292 2015

[10] Y Lei D Han J Lin and Z He ldquoPlanetary gearbox faultdiagnosis using an adaptive stochastic resonance methodrdquoMechanical Systems and Signal Processing vol 38 no 1 pp 113ndash124 2013

[11] 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

[12] T Barszcz and R B Randall ldquoApplication of spectral kurtosisfor detection of a tooth crack in the planetary gear of a windturbinerdquo Mechanical Systems and Signal Processing vol 23 no4 pp 1352ndash1365 2009

[13] H Sun Y Zi ZHe J Yuan XWang and L Chen ldquoCustomizedmultiwavelets for planetary gearbox fault detection based onvibration sensor signalsrdquo Sensors vol 13 no 1 pp 1183ndash12092013

[14] H Li J Zhao and W Song ldquoAn offline fault diagnosis methodfor planetary gearbox based on empirical mode decomposition


and adaptive multi-scale morphological gradient filterrdquo Journalof Vibroengineering vol 17 pp 705ndash719 2015

[15] Z Feng F Chu and M J Zuo ldquoTime-frequency analysis oftime-varying modulated signals based on improved energyseparation by iterative generalized demodulationrdquo Journal ofSound and Vibration vol 330 no 6 pp 1225ndash1243 2011

[16] Z Feng X Chen M Liang and F Ma ldquoTimendashfrequencydemodulation analysis based on iterative generalized demod-ulation for fault diagnosis of planetary gearbox under nonsta-tionary conditionsrdquo Mechanical Systems and Signal Processingvol 62-63 pp 54ndash74 2015

[17] Z Feng and M Liang ldquoFault diagnosis of wind turbineplanetary gearbox under nonstationary conditions via adaptiveoptimal kernel time-frequency analysisrdquoRenewable Energy vol66 pp 468ndash477 2014

[18] W Bartelmus and R Zimroz ldquoVibration condition monitoringof planetary gearbox under varying external loadrdquo MechanicalSystems and Signal Processing vol 23 no 1 pp 246ndash257 2009

[19] F Chaari M S Abbes F V Rueda A F del Rincon andM Haddar ldquoAnalysis of planetary gear transmission in non-stationary operationsrdquo Frontiers of Mechanical Engineering vol8 no 1 pp 88ndash94 2013

[20] J Yang and L Zhang ldquoDynamic response and dynamic loadof wind turbine planetary gear transmission system underchanging excitationrdquoAppliedMechanics andMaterials vol 121ndash126 pp 2671ndash2675 2011

[21] Z Feng M Liang and F Chu ldquoRecent advances in time-frequency analysis methods for machinery fault diagnosis areview with application examplesrdquo Mechanical Systems andSignal Processing vol 38 no 1 pp 165ndash205 2013

[22] O Rioul and P Flandrin ldquoTime-scale energy distributions ageneral class extending wavelet transformsrdquo IEEE Transactionson Signal Processing vol 40 no 7 pp 1746ndash1757 1992

[23] F Hlawatsch and G F Boudreaux-Bartels ldquoLinear andquadratic time-frequency signal representationsrdquo IEEE SignalProcessing Magazine vol 9 no 2 pp 21ndash67 1992

[24] L Cohen ldquoTime-frequency distributions a reviewrdquo Proceedingsof the IEEE vol 77 no 7 pp 941ndash981 1989

[25] Z K Peng F L Chu and P W Tse ldquoDetection of the rubbing-caused impacts for rotor-stator fault diagnosis using reassignedscalogramrdquo Mechanical Systems and Signal Processing vol 19no 2 pp 391ndash409 2005

[26] HMa T Yu Q Han Y Zhang BWen and C Xuelian ldquoTimendashfrequency features of two types of coupled rub-impact faults inrotor systemsrdquo Journal of Sound and Vibration vol 321 no 3-5pp 1109ndash1128 2009

[27] X Chen Z Feng and M Liang ldquoFault feature extractionof planetary gearboxes under nonstationary conditions basedon reassigned wavelet scalogramrdquo in Proceedings of the IEEEInternational Instrumentation and Measurement TechnologyConference pp 294ndash299 Pisa Italy May 2015

[28] F Auger and P Flandrin ldquoImproving the readability of time-frequency and time-scale representations by the reassignmentmethodrdquo IEEE Transactions on Signal Processing vol 43 no 5pp 1068ndash1089 1995

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



planetary gearbox vibration signals Feng et al [11] presenteda joint amplitude and frequency demodulation analysis basedon Teager energy operator and ensemble empirical modedecomposition (EEMD) to diagnose planetary gearbox faultBarszcz and Randall [12] employed the spectral kurtosis todetect gear tooth crack in a wind turbine planetary gearboxSun et al [13] customized the multiwavelets based on theredundant symmetric lifting scheme to detect the incipientpitting faults in planetary gearbox Li et al [14] extracted thefault characteristics frequencies of planetary gearbox basedon empirical mode decomposition and adaptive multiscalemorphological gradient filter

Most of above works were based on an assumption ofsignal stationarity that is constant running speed and loadHowever in real-world wind turbine applications volatilewind conditions result in nonstationary vibration signalsFor example the running speed varies with respect to theunpredictable wind power and transient strong wind maylead to a shock Under nonstationary conditions planetarygearbox vibration signals always feature intricate frequencycomponent structure [15 16] manifesting as time-varyingamplitude modulation (AM) and frequency modulation(FM) Due to the nonstationary nature conventional Fouriertransform most likely fails to thoroughly reveal the truefrequency structure To the best of our knowledge theliterature reported onplanetary gearbox fault diagnosis undernonstationary conditions has been very limited [17] Bartel-mus and Zimroz [18] proposed an indicator that reflects thelinear dependence between themeshing frequency amplitudeand the operating condition for condition monitoring ofplanetary gearboxes Chaari et al [19] developed amathemat-ical model to understand the dynamic behavior of planetarygear under variable load condition Yang and Zhang [20]studied the running characteristics of wind turbine planetarygearbox considering time-varying speed and load

Time-frequency analysis (TFA) is capable of revealingtime-frequency structure despite time variation of frequencycomponents and thus is suitable for analyzing nonstation-ary signals [21] Linear time-frequency analysis methodslike short time Fourier transform (STFT) and continuouswavelet transform (CWT) essentially represent signals byweighted sum of a series of bases localized in both time andfrequency domains But their time-frequency resolution isgoverned by Heisenberg uncertainty principle and resolutionin one domain can only be improved at expense of theother resulting in time-frequency smearing [22] Bilineartime-frequency distribution presents the signal energy ontime-frequency plane [23 24] The Wigner-Ville distribution(WVD) is the basis of almost all bilinear TFA methodswith merit of finest time-frequency resolution Howeverit is subjected to inevitable cross-term interferences whenanalyzing multicomponent signals Even for monocompo-nent signals it still suffers from inner interference if the IFcurve is nonlinear Although improved methods smoothedpseudo Wigner-Ville distribution (SPWVD) for examplecan effectively suppress the cross-term interferences they inturn deteriorate time-frequency resolution

Fortunately reassigned wavelet scalogram has merits ofexcellent time-frequency resolution and good suppression of

cross-terms and thus can clearly exhibit the time-frequencystructure of nonstationary signals However its applicationsinmachinery fault diagnosis have been very limited and onlya few published works focus on this topic Peng et al [25]studied the characteristics of rotor-stator rubbing impactsand applied both scalogram and reassigned scalogram forimpact detection Ma et al [26] summarized the symptomsof rotor-stator rubbing fault based on reassigned scalogram

In this paper we further extend the application of reas-signed wavelet scalogram to planetary gearbox fault diag-nosis under nonstationary conditions [27] and validate itseffectiveness in extracting gear fault features The remainderof this paper is structured as follows In Section 2 thedetailed algorithm of scalogram and reassigned scalogramis introduced In Section 3 a numerical simulated nonsta-tionary signal encompassing a time-varying frequency andperiodical impulses is analyzed to illustrate the performanceof reassigned scalogram In Sections 4 and 5 the effectivenessof reassigned scalogram is further validated by analysis oflab experimental planetary gearbox vibration signals in bothlocal and distributed sun gear fault cases as well as in situwind turbine planetary gearbox vibration signals measuredfrom awind farm Finally conclusions are drawn in Section 6

2 Reassigned Wavelet Scalogram

21 ContinuousWavelet Transform Wavelet transform (WT)has advantage of better time-frequency resolution over shorttime Fourier transform (STFT) Rather than using a constantwindow function for whole time-frequency plane analysis itemploys different window functions for analyzing differentfrequency bands of the signal Therefore WT is popularamong linear TFA methods

For an arbitrary finite-energy signal 119909(119905) isin 1198712(119877) its

CWT is defined as

119882119909(119886 119887 120595) = ⟨119909 (119905) 120595

119886119887(119905)⟩

= 119886minus12

int

infin

minusinfin

119909 (119905) 120595lowast

119886119887(119905) 119889119905

(1)

where asterisk lowast stands for complex conjugate 119886 gt 0 119886 and119887 respectively represent the scale and time offset and 120595

119886119887(119905)

is the family of wavelets generated by dilation and translationfrom the mother wavelet 120595(119905)

120595119886119887

(119905) = 119886minus12

120595(

119905 minus 119887

119886

) (2)

Morlet wavelet is most commonly used in CWT and it isdefined as

120595 (119905) = 120587minus14

(119890minus1198941205960119905

minus 119890minus1205962

02) 119890minus11990522 (3)

The wavelet scalogram is defined as

119878119909(119886 119887 120595) =

1003816100381610038161003816119882119909(119886 119887 120595)

1003816100381610038161003816

2

(4)

22 Reassigned Wavelet Scalogram Although wavelet scalo-gram can reveal the constituent frequency components of


0

Am

plitu

de (V

)

minus06

minus04

minus02

02

04

06

08

1

02 04 06 08 10Time (s)

(a) Waveform

Am

plitu

de (V

)

0

0002

0004

0006

0008

001

200 400 600 800 10000Frequency (Hz)





119909(119886 119887 120595) does




1205960

1198861015840(119887 119886)

=

1205960

119886

+ Im

CWT119909(119886 119887 )CWTlowast

119909(119886 119887 120595)

21205871198861003816100381610038161003816CWT

119909(119886 119887 120595)

1003816100381610038161003816

2

1198871015840(119886 119887)

= 119887 minus Re119886CWT

119909(119886 119887 120595

1015840)CWTlowast

119909(119886 119887 120595)

1003816100381610038161003816CWT

119909(119886 119887 120595)

1003816100381610038161003816

2

(5)

where

1205951015840(119905) = 119905120595 (119905)

(119905) =

119889120595

119889119905

(119905)

(6)


119877119878119909(1198861015840 1198871015840) = int

+infin

minusinfin

int

+infin

minusinfin

(

119886

119886

)

2

119878119909(119886 119887) 120575 [

119887 minus 1198871015840(119886 119887)]

sdot 120575 [119886 minus 1198861015840(119886 119887)] 119889119886 119889119887

(7)



119904 (119905) = 119860 sin(2120587(int800119905

chirp

+ int 50 cos (212058710119905)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

FM

))

+ 120575 (119905)

impulse+ 119899 (119905)

(8)





0

200

400

600

800

1000Fr

eque

ncy

(Hz)

02 04 06 08 10Time (s)


0

200

400

600

800

1000

Freq

uenc

y (H

z)

02 04 06 08 10Time (s)


0

200

400

600

800

1000

Freq

uenc

y (H

z)

02 04 06 08 10Time (s)
















1 3 4 5 6 72






119889(119905) =


1198981(119905) = 2222Hz 119891(119903)

1199041(119905) = 133Hz and

1198911199041(119905) = 444Hz for stage 1 and 119891

1198982(119905) = 486Hz 119891(119903)

1199042(119905) =





119898(119905) (10027) 119891

119889(119905) (175216) 119891

119889(119905)

Sun rotating 119891(119903)119904(119905) (29) 119891

119889(119905) (127) 119891

119889(119905)


119889(119905) (7864) 119891

119889(119905)

Sun fault 119891119904(119905) (2027) 119891

119889(119905) (1751512) 119891

119889(119905)

Planet fault 119891119901(119905) (554) 119891

119889(119905) (1757776) 119891

119889(119905)

Ring fault 119891119903(119905) (427) 119891

119889(119905) (1755400) 119891

119889(119905)



1198981(119905) minus 119891

1199041(119905)


1198981(119905) minus 4119891

(119903)





1198981(119905) minus 119891



(119903)




1199041(119905) and


1198981(119905) + 3119891






0 1 2 3 4 5Time (s)

minus8

minus6

minus4

minus2

0

Am

plitu

de (m

s2)

2

4

6

8

(a) Waveform

50 100 150 200 250 300 350 4000Frequency (Hz)

0

001

002

003

Am

plitu

de (m

s2)

004

005

006


45

50

55

60

65

Spee

d (H

z)

1 2 3 4 50Time (s)

(c) Motor speed





1 2 3 4 50Time (s)

0

100

200

300

400Fr

eque

ncy

(Hz)



1 2 3 4 50Time (s)

0

20

40

60

80

Freq

uenc

y (H

z)



Am

plitu

de (m

s2)

minus8

minus6

minus4

minus2

0

2

4

6

8

1 2 3 4 50Time (s)

(a) Waveform

Am

plitu

de (m

s2)

50 100 150 200 250 300 350 4000Frequency (Hz)

0

001

002

003

004

005

006


Spee

d (H

z)

45

50

55

60

65

41 2 3 50Time (s)

(c) Motor speed





1198982minus3[119891(119903)

1199042+






0

100

200

300

400

Freq

uenc

y (H

z)

1 2 3 4 50Time (s)

fm1


fm1 + 3fs1

fm1 + (74)fs1


minus1

minus05

0

05

1

Am

plitu

de (m

s2)

1 2 3 4 50Time (s)

(a) Waveform

0

005

01

015

02

025

03

035

Am

plitu

de (m

s2)

10 20 30 40 50 60 70 800Frequency (Hz)


0 1 2 3 4 5Time (s)

Spee

d (H

z)

45

50

55

60

65

(c) Motor speed








1 2 3 4 50Time (s)

0

20

40

60

80

Freq

uenc

y (H

z)










52 Signal Analysis








minus10

minus5

0

5

10

Am

plitu

de (m

s2)

5 10 15 20 25 300Time (s)

(a) Waveform

0

0002

0004

0006

0008

001

0012

0014

0016

0018

Am

plitu

de (m

s2)

10 20 30 40 50 60 700Frequency (Hz)



0 5 10 15 20 25 30

Time (s)

0

10

20

30

40

50

60

70

Freq

uenc

y (H

z)


Am

plitu

de (m

s2)

minus40

minus30

minus20

minus10

0

10

20

30

40

5 10 15 20 25 300Time (s)

(a) Waveform

0 10 20 30 40 50 60 70Frequency (Hz)

Am

plitu

de (m

s2)

0

001

002

003

004

005




0 5 10 15 20 25 30

A B C D E F G

Freq

uenc

y (H

z)

0

10

20

30

40

50

60

70

Time (s)


Freq

uenc

y (H

z)

B C

a b c d e f

1 2 3 4 5 6 7 8 90 10

Time (s)

0

10

20

30

40

50

60

70



6 Conclusions




Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

Am

plitu

de (V

)

minus06

minus04

minus02

02

04

06

08

1

02 04 06 08 10Time (s)

(a) Waveform

Am

plitu

de (V

)

0

0002

0004

0006

0008

001

200 400 600 800 10000Frequency (Hz)





119909(119886 119887 120595) does




1205960

1198861015840(119887 119886)

=

1205960

119886

+ Im

CWT119909(119886 119887 )CWTlowast

119909(119886 119887 120595)

21205871198861003816100381610038161003816CWT

119909(119886 119887 120595)

1003816100381610038161003816

2

1198871015840(119886 119887)

= 119887 minus Re119886CWT

119909(119886 119887 120595

1015840)CWTlowast

119909(119886 119887 120595)

1003816100381610038161003816CWT

119909(119886 119887 120595)

1003816100381610038161003816

2

(5)

where

1205951015840(119905) = 119905120595 (119905)

(119905) =

119889120595

119889119905

(119905)

(6)


119877119878119909(1198861015840 1198871015840) = int

+infin

minusinfin

int

+infin

minusinfin

(

119886

119886

)

2

119878119909(119886 119887) 120575 [

119887 minus 1198871015840(119886 119887)]

sdot 120575 [119886 minus 1198861015840(119886 119887)] 119889119886 119889119887

(7)



119904 (119905) = 119860 sin(2120587(int800119905

chirp

+ int 50 cos (212058710119905)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

FM

))

+ 120575 (119905)

impulse+ 119899 (119905)

(8)





0

200

400

600

800

1000Fr

eque

ncy

(Hz)

02 04 06 08 10Time (s)


0

200

400

600

800

1000

Freq

uenc

y (H

z)

02 04 06 08 10Time (s)


0

200

400

600

800

1000

Freq

uenc

y (H

z)

02 04 06 08 10Time (s)
















1 3 4 5 6 72






119889(119905) =


1198981(119905) = 2222Hz 119891(119903)

1199041(119905) = 133Hz and

1198911199041(119905) = 444Hz for stage 1 and 119891

1198982(119905) = 486Hz 119891(119903)

1199042(119905) =





119898(119905) (10027) 119891

119889(119905) (175216) 119891

119889(119905)

Sun rotating 119891(119903)119904(119905) (29) 119891

119889(119905) (127) 119891

119889(119905)


119889(119905) (7864) 119891

119889(119905)

Sun fault 119891119904(119905) (2027) 119891

119889(119905) (1751512) 119891

119889(119905)

Planet fault 119891119901(119905) (554) 119891

119889(119905) (1757776) 119891

119889(119905)

Ring fault 119891119903(119905) (427) 119891

119889(119905) (1755400) 119891

119889(119905)



1198981(119905) minus 119891

1199041(119905)


1198981(119905) minus 4119891

(119903)





1198981(119905) minus 119891



(119903)




1199041(119905) and


1198981(119905) + 3119891






0 1 2 3 4 5Time (s)

minus8

minus6

minus4

minus2

0

Am

plitu

de (m

s2)

2

4

6

8

(a) Waveform

50 100 150 200 250 300 350 4000Frequency (Hz)

0

001

002

003

Am

plitu

de (m

s2)

004

005

006


45

50

55

60

65

Spee

d (H

z)

1 2 3 4 50Time (s)

(c) Motor speed





1 2 3 4 50Time (s)

0

100

200

300

400Fr

eque

ncy

(Hz)



1 2 3 4 50Time (s)

0

20

40

60

80

Freq

uenc

y (H

z)



Am

plitu

de (m

s2)

minus8

minus6

minus4

minus2

0

2

4

6

8

1 2 3 4 50Time (s)

(a) Waveform

Am

plitu

de (m

s2)

50 100 150 200 250 300 350 4000Frequency (Hz)

0

001

002

003

004

005

006


Spee

d (H

z)

45

50

55

60

65

41 2 3 50Time (s)

(c) Motor speed





1198982minus3[119891(119903)

1199042+






0

100

200

300

400

Freq

uenc

y (H

z)

1 2 3 4 50Time (s)

fm1


fm1 + 3fs1

fm1 + (74)fs1


minus1

minus05

0

05

1

Am

plitu

de (m

s2)

1 2 3 4 50Time (s)

(a) Waveform

0

005

01

015

02

025

03

035

Am

plitu

de (m

s2)

10 20 30 40 50 60 70 800Frequency (Hz)


0 1 2 3 4 5Time (s)

Spee

d (H

z)

45

50

55

60

65

(c) Motor speed








1 2 3 4 50Time (s)

0

20

40

60

80

Freq

uenc

y (H

z)










52 Signal Analysis








minus10

minus5

0

5

10

Am

plitu

de (m

s2)

5 10 15 20 25 300Time (s)

(a) Waveform

0

0002

0004

0006

0008

001

0012

0014

0016

0018

Am

plitu

de (m

s2)

10 20 30 40 50 60 700Frequency (Hz)



0 5 10 15 20 25 30

Time (s)

0

10

20

30

40

50

60

70

Freq

uenc

y (H

z)


Am

plitu

de (m

s2)

minus40

minus30

minus20

minus10

0

10

20

30

40

5 10 15 20 25 300Time (s)

(a) Waveform

0 10 20 30 40 50 60 70Frequency (Hz)

Am

plitu

de (m

s2)

0

001

002

003

004

005




0 5 10 15 20 25 30

A B C D E F G

Freq

uenc

y (H

z)

0

10

20

30

40

50

60

70

Time (s)


Freq

uenc

y (H

z)

B C

a b c d e f

1 2 3 4 5 6 7 8 90 10

Time (s)

0

10

20

30

40

50

60

70



6 Conclusions




Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

200

400

600

800

1000Fr

eque

ncy

(Hz)

02 04 06 08 10Time (s)


0

200

400

600

800

1000

Freq

uenc

y (H

z)

02 04 06 08 10Time (s)


0

200

400

600

800

1000

Freq

uenc

y (H

z)

02 04 06 08 10Time (s)
















1 3 4 5 6 72






119889(119905) =


1198981(119905) = 2222Hz 119891(119903)

1199041(119905) = 133Hz and

1198911199041(119905) = 444Hz for stage 1 and 119891

1198982(119905) = 486Hz 119891(119903)

1199042(119905) =





119898(119905) (10027) 119891

119889(119905) (175216) 119891

119889(119905)

Sun rotating 119891(119903)119904(119905) (29) 119891

119889(119905) (127) 119891

119889(119905)


119889(119905) (7864) 119891

119889(119905)

Sun fault 119891119904(119905) (2027) 119891

119889(119905) (1751512) 119891

119889(119905)

Planet fault 119891119901(119905) (554) 119891

119889(119905) (1757776) 119891

119889(119905)

Ring fault 119891119903(119905) (427) 119891

119889(119905) (1755400) 119891

119889(119905)



1198981(119905) minus 119891

1199041(119905)


1198981(119905) minus 4119891

(119903)





1198981(119905) minus 119891



(119903)




1199041(119905) and


1198981(119905) + 3119891






0 1 2 3 4 5Time (s)

minus8

minus6

minus4

minus2

0

Am

plitu

de (m

s2)

2

4

6

8

(a) Waveform

50 100 150 200 250 300 350 4000Frequency (Hz)

0

001

002

003

Am

plitu

de (m

s2)

004

005

006


45

50

55

60

65

Spee

d (H

z)

1 2 3 4 50Time (s)

(c) Motor speed





1 2 3 4 50Time (s)

0

100

200

300

400Fr

eque

ncy

(Hz)



1 2 3 4 50Time (s)

0

20

40

60

80

Freq

uenc

y (H

z)



Am

plitu

de (m

s2)

minus8

minus6

minus4

minus2

0

2

4

6

8

1 2 3 4 50Time (s)

(a) Waveform

Am

plitu

de (m

s2)

50 100 150 200 250 300 350 4000Frequency (Hz)

0

001

002

003

004

005

006


Spee

d (H

z)

45

50

55

60

65

41 2 3 50Time (s)

(c) Motor speed





1198982minus3[119891(119903)

1199042+






0

100

200

300

400

Freq

uenc

y (H

z)

1 2 3 4 50Time (s)

fm1


fm1 + 3fs1

fm1 + (74)fs1


minus1

minus05

0

05

1

Am

plitu

de (m

s2)

1 2 3 4 50Time (s)

(a) Waveform

0

005

01

015

02

025

03

035

Am

plitu

de (m

s2)

10 20 30 40 50 60 70 800Frequency (Hz)


0 1 2 3 4 5Time (s)

Spee

d (H

z)

45

50

55

60

65

(c) Motor speed








1 2 3 4 50Time (s)

0

20

40

60

80

Freq

uenc

y (H

z)










52 Signal Analysis








minus10

minus5

0

5

10

Am

plitu

de (m

s2)

5 10 15 20 25 300Time (s)

(a) Waveform

0

0002

0004

0006

0008

001

0012

0014

0016

0018

Am

plitu

de (m

s2)

10 20 30 40 50 60 700Frequency (Hz)



0 5 10 15 20 25 30

Time (s)

0

10

20

30

40

50

60

70

Freq

uenc

y (H

z)


Am

plitu

de (m

s2)

minus40

minus30

minus20

minus10

0

10

20

30

40

5 10 15 20 25 300Time (s)

(a) Waveform

0 10 20 30 40 50 60 70Frequency (Hz)

Am

plitu

de (m

s2)

0

001

002

003

004

005




0 5 10 15 20 25 30

A B C D E F G

Freq

uenc

y (H

z)

0

10

20

30

40

50

60

70

Time (s)


Freq

uenc

y (H

z)

B C

a b c d e f

1 2 3 4 5 6 7 8 90 10

Time (s)

0

10

20

30

40

50

60

70



6 Conclusions




Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













1 3 4 5 6 72






119889(119905) =


1198981(119905) = 2222Hz 119891(119903)

1199041(119905) = 133Hz and

1198911199041(119905) = 444Hz for stage 1 and 119891

1198982(119905) = 486Hz 119891(119903)

1199042(119905) =





119898(119905) (10027) 119891

119889(119905) (175216) 119891

119889(119905)

Sun rotating 119891(119903)119904(119905) (29) 119891

119889(119905) (127) 119891

119889(119905)


119889(119905) (7864) 119891

119889(119905)

Sun fault 119891119904(119905) (2027) 119891

119889(119905) (1751512) 119891

119889(119905)

Planet fault 119891119901(119905) (554) 119891

119889(119905) (1757776) 119891

119889(119905)

Ring fault 119891119903(119905) (427) 119891

119889(119905) (1755400) 119891

119889(119905)



1198981(119905) minus 119891

1199041(119905)


1198981(119905) minus 4119891

(119903)





1198981(119905) minus 119891



(119903)




1199041(119905) and


1198981(119905) + 3119891






0 1 2 3 4 5Time (s)

minus8

minus6

minus4

minus2

0

Am

plitu

de (m

s2)

2

4

6

8

(a) Waveform

50 100 150 200 250 300 350 4000Frequency (Hz)

0

001

002

003

Am

plitu

de (m

s2)

004

005

006


45

50

55

60

65

Spee

d (H

z)

1 2 3 4 50Time (s)

(c) Motor speed





1 2 3 4 50Time (s)

0

100

200

300

400Fr

eque

ncy

(Hz)



1 2 3 4 50Time (s)

0

20

40

60

80

Freq

uenc

y (H

z)



Am

plitu

de (m

s2)

minus8

minus6

minus4

minus2

0

2

4

6

8

1 2 3 4 50Time (s)

(a) Waveform

Am

plitu

de (m

s2)

50 100 150 200 250 300 350 4000Frequency (Hz)

0

001

002

003

004

005

006


Spee

d (H

z)

45

50

55

60

65

41 2 3 50Time (s)

(c) Motor speed





1198982minus3[119891(119903)

1199042+






0

100

200

300

400

Freq

uenc

y (H

z)

1 2 3 4 50Time (s)

fm1


fm1 + 3fs1

fm1 + (74)fs1


minus1

minus05

0

05

1

Am

plitu

de (m

s2)

1 2 3 4 50Time (s)

(a) Waveform

0

005

01

015

02

025

03

035

Am

plitu

de (m

s2)

10 20 30 40 50 60 70 800Frequency (Hz)


0 1 2 3 4 5Time (s)

Spee

d (H

z)

45

50

55

60

65

(c) Motor speed








1 2 3 4 50Time (s)

0

20

40

60

80

Freq

uenc

y (H

z)










52 Signal Analysis








minus10

minus5

0

5

10

Am

plitu

de (m

s2)

5 10 15 20 25 300Time (s)

(a) Waveform

0

0002

0004

0006

0008

001

0012

0014

0016

0018

Am

plitu

de (m

s2)

10 20 30 40 50 60 700Frequency (Hz)



0 5 10 15 20 25 30

Time (s)

0

10

20

30

40

50

60

70

Freq

uenc

y (H

z)


Am

plitu

de (m

s2)

minus40

minus30

minus20

minus10

0

10

20

30

40

5 10 15 20 25 300Time (s)

(a) Waveform

0 10 20 30 40 50 60 70Frequency (Hz)

Am

plitu

de (m

s2)

0

001

002

003

004

005




0 5 10 15 20 25 30

A B C D E F G

Freq

uenc

y (H

z)

0

10

20

30

40

50

60

70

Time (s)


Freq

uenc

y (H

z)

B C

a b c d e f

1 2 3 4 5 6 7 8 90 10

Time (s)

0

10

20

30

40

50

60

70



6 Conclusions




Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












0 1 2 3 4 5Time (s)

minus8

minus6

minus4

minus2

0

Am

plitu

de (m

s2)

2

4

6

8

(a) Waveform

50 100 150 200 250 300 350 4000Frequency (Hz)

0

001

002

003

Am

plitu

de (m

s2)

004

005

006


45

50

55

60

65

Spee

d (H

z)

1 2 3 4 50Time (s)

(c) Motor speed





1 2 3 4 50Time (s)

0

100

200

300

400Fr

eque

ncy

(Hz)



1 2 3 4 50Time (s)

0

20

40

60

80

Freq

uenc

y (H

z)



Am

plitu

de (m

s2)

minus8

minus6

minus4

minus2

0

2

4

6

8

1 2 3 4 50Time (s)

(a) Waveform

Am

plitu

de (m

s2)

50 100 150 200 250 300 350 4000Frequency (Hz)

0

001

002

003

004

005

006


Spee

d (H

z)

45

50

55

60

65

41 2 3 50Time (s)

(c) Motor speed





1198982minus3[119891(119903)

1199042+






0

100

200

300

400

Freq

uenc

y (H

z)

1 2 3 4 50Time (s)

fm1


fm1 + 3fs1

fm1 + (74)fs1


minus1

minus05

0

05

1

Am

plitu

de (m

s2)

1 2 3 4 50Time (s)

(a) Waveform

0

005

01

015

02

025

03

035

Am

plitu

de (m

s2)

10 20 30 40 50 60 70 800Frequency (Hz)


0 1 2 3 4 5Time (s)

Spee

d (H

z)

45

50

55

60

65

(c) Motor speed








1 2 3 4 50Time (s)

0

20

40

60

80

Freq

uenc

y (H

z)










52 Signal Analysis








minus10

minus5

0

5

10

Am

plitu

de (m

s2)

5 10 15 20 25 300Time (s)

(a) Waveform

0

0002

0004

0006

0008

001

0012

0014

0016

0018

Am

plitu

de (m

s2)

10 20 30 40 50 60 700Frequency (Hz)



0 5 10 15 20 25 30

Time (s)

0

10

20

30

40

50

60

70

Freq

uenc

y (H

z)


Am

plitu

de (m

s2)

minus40

minus30

minus20

minus10

0

10

20

30

40

5 10 15 20 25 300Time (s)

(a) Waveform

0 10 20 30 40 50 60 70Frequency (Hz)

Am

plitu

de (m

s2)

0

001

002

003

004

005




0 5 10 15 20 25 30

A B C D E F G

Freq

uenc

y (H

z)

0

10

20

30

40

50

60

70

Time (s)


Freq

uenc

y (H

z)

B C

a b c d e f

1 2 3 4 5 6 7 8 90 10

Time (s)

0

10

20

30

40

50

60

70



6 Conclusions




Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1 2 3 4 50Time (s)

0

100

200

300

400Fr

eque

ncy

(Hz)



1 2 3 4 50Time (s)

0

20

40

60

80

Freq

uenc

y (H

z)



Am

plitu

de (m

s2)

minus8

minus6

minus4

minus2

0

2

4

6

8

1 2 3 4 50Time (s)

(a) Waveform

Am

plitu

de (m

s2)

50 100 150 200 250 300 350 4000Frequency (Hz)

0

001

002

003

004

005

006


Spee

d (H

z)

45

50

55

60

65

41 2 3 50Time (s)

(c) Motor speed





1198982minus3[119891(119903)

1199042+






0

100

200

300

400

Freq

uenc

y (H

z)

1 2 3 4 50Time (s)

fm1


fm1 + 3fs1

fm1 + (74)fs1


minus1

minus05

0

05

1

Am

plitu

de (m

s2)

1 2 3 4 50Time (s)

(a) Waveform

0

005

01

015

02

025

03

035

Am

plitu

de (m

s2)

10 20 30 40 50 60 70 800Frequency (Hz)


0 1 2 3 4 5Time (s)

Spee

d (H

z)

45

50

55

60

65

(c) Motor speed








1 2 3 4 50Time (s)

0

20

40

60

80

Freq

uenc

y (H

z)










52 Signal Analysis








minus10

minus5

0

5

10

Am

plitu

de (m

s2)

5 10 15 20 25 300Time (s)

(a) Waveform

0

0002

0004

0006

0008

001

0012

0014

0016

0018

Am

plitu

de (m

s2)

10 20 30 40 50 60 700Frequency (Hz)



0 5 10 15 20 25 30

Time (s)

0

10

20

30

40

50

60

70

Freq

uenc

y (H

z)


Am

plitu

de (m

s2)

minus40

minus30

minus20

minus10

0

10

20

30

40

5 10 15 20 25 300Time (s)

(a) Waveform

0 10 20 30 40 50 60 70Frequency (Hz)

Am

plitu

de (m

s2)

0

001

002

003

004

005




0 5 10 15 20 25 30

A B C D E F G

Freq

uenc

y (H

z)

0

10

20

30

40

50

60

70

Time (s)


Freq

uenc

y (H

z)

B C

a b c d e f

1 2 3 4 5 6 7 8 90 10

Time (s)

0

10

20

30

40

50

60

70



6 Conclusions




Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

100

200

300

400

Freq

uenc

y (H

z)

1 2 3 4 50Time (s)

fm1


fm1 + 3fs1

fm1 + (74)fs1


minus1

minus05

0

05

1

Am

plitu

de (m

s2)

1 2 3 4 50Time (s)

(a) Waveform

0

005

01

015

02

025

03

035

Am

plitu

de (m

s2)

10 20 30 40 50 60 70 800Frequency (Hz)


0 1 2 3 4 5Time (s)

Spee

d (H

z)

45

50

55

60

65

(c) Motor speed








1 2 3 4 50Time (s)

0

20

40

60

80

Freq

uenc

y (H

z)










52 Signal Analysis








minus10

minus5

0

5

10

Am

plitu

de (m

s2)

5 10 15 20 25 300Time (s)

(a) Waveform

0

0002

0004

0006

0008

001

0012

0014

0016

0018

Am

plitu

de (m

s2)

10 20 30 40 50 60 700Frequency (Hz)



0 5 10 15 20 25 30

Time (s)

0

10

20

30

40

50

60

70

Freq

uenc

y (H

z)


Am

plitu

de (m

s2)

minus40

minus30

minus20

minus10

0

10

20

30

40

5 10 15 20 25 300Time (s)

(a) Waveform

0 10 20 30 40 50 60 70Frequency (Hz)

Am

plitu

de (m

s2)

0

001

002

003

004

005




0 5 10 15 20 25 30

A B C D E F G

Freq

uenc

y (H

z)

0

10

20

30

40

50

60

70

Time (s)


Freq

uenc

y (H

z)

B C

a b c d e f

1 2 3 4 5 6 7 8 90 10

Time (s)

0

10

20

30

40

50

60

70



6 Conclusions




Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











1 2 3 4 50Time (s)

0

20

40

60

80

Freq

uenc

y (H

z)










52 Signal Analysis








minus10

minus5

0

5

10

Am

plitu

de (m

s2)

5 10 15 20 25 300Time (s)

(a) Waveform

0

0002

0004

0006

0008

001

0012

0014

0016

0018

Am

plitu

de (m

s2)

10 20 30 40 50 60 700Frequency (Hz)



0 5 10 15 20 25 30

Time (s)

0

10

20

30

40

50

60

70

Freq

uenc

y (H

z)


Am

plitu

de (m

s2)

minus40

minus30

minus20

minus10

0

10

20

30

40

5 10 15 20 25 300Time (s)

(a) Waveform

0 10 20 30 40 50 60 70Frequency (Hz)

Am

plitu

de (m

s2)

0

001

002

003

004

005




0 5 10 15 20 25 30

A B C D E F G

Freq

uenc

y (H

z)

0

10

20

30

40

50

60

70

Time (s)


Freq

uenc

y (H

z)

B C

a b c d e f

1 2 3 4 5 6 7 8 90 10

Time (s)

0

10

20

30

40

50

60

70



6 Conclusions




Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












minus10

minus5

0

5

10

Am

plitu

de (m

s2)

5 10 15 20 25 300Time (s)

(a) Waveform

0

0002

0004

0006

0008

001

0012

0014

0016

0018

Am

plitu

de (m

s2)

10 20 30 40 50 60 700Frequency (Hz)



0 5 10 15 20 25 30

Time (s)

0

10

20

30

40

50

60

70

Freq

uenc

y (H

z)


Am

plitu

de (m

s2)

minus40

minus30

minus20

minus10

0

10

20

30

40

5 10 15 20 25 300Time (s)

(a) Waveform

0 10 20 30 40 50 60 70Frequency (Hz)

Am

plitu

de (m

s2)

0

001

002

003

004

005




0 5 10 15 20 25 30

A B C D E F G

Freq

uenc

y (H

z)

0

10

20

30

40

50

60

70

Time (s)


Freq

uenc

y (H

z)

B C

a b c d e f

1 2 3 4 5 6 7 8 90 10

Time (s)

0

10

20

30

40

50

60

70



6 Conclusions




Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 5 10 15 20 25 30

A B C D E F G

Freq

uenc

y (H

z)

0

10

20

30

40

50

60

70

Time (s)


Freq

uenc

y (H

z)

B C

a b c d e f

1 2 3 4 5 6 7 8 90 10

Time (s)

0

10

20

30

40

50

60

70



6 Conclusions




Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Research Article Application of Reassigned Wavelet...

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Transcript of Research Article Application of Reassigned Wavelet...