RollingBearingDiagnosisBasedonAdaptiveProbabilisticPCA ...downloads.hindawi.com/journals/sv/2020/8828517.pdfmorphological operator, the construction of morphological operators also

Research ArticleRolling Bearing Diagnosis Based on Adaptive Probabilistic PCAand the Enhanced Morphological Filter

Yuanqing Luo1 Changzheng Chen 1 Siyu Zhao1 Xiangxi Kong1 and Zhong Wang2

1School of Mechanical Engineering Shenyang University of Technology Shenyang 110870 China2School of Mechanical Engineering Liaoning Institute of Science and Technology Benxi 117004 China

Correspondence should be addressed to Changzheng Chen chencz6699sinacom

Received 21 May 2020 Revised 20 July 2020 Accepted 5 August 2020 Published 19 August 2020

Academic Editor Francisco Beltran-Carbajal

Copyright copy 2020 Yuanqing Luo et al-is 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

Early fault diagnosis of rolling element bearing is still a difficult problem Firstly in order to effectively extract the fault impulsesignal of the bearing a new enhanced morphological difference operator (EMDO) is constructed by combining two optimalfeature extraction-type operators Next in the process of processing the test signal in order to reduce the interference problemcaused by strong background noise the probabilistic principal component analysis (PPCA) method is introduced In thetraditional PPCA method two important system parameters (decomposition principal component k and original variable n) areusually set artificially this will greatly reduce the noise reduction performance of PPCA To solve this problem a parameteradaptive PPCA method based on grasshopper optimization algorithm (GOA) is proposed Finally combining the advantages ofthe above algorithms a comprehensive rolling bearing fault diagnosis method named APPCA-EMDF is proposed in this paperExperimental comparison results show that the proposed method can effectively diagnose the vibration signals of rollingelement bearing

1 Introduction

Rolling bearings are widely used in the field of rotatingmachinery It is vulnerable to damage because it operates athigh speed and variable loads -is not only brings diffi-culties to the safety production of the machinery but alsocauses huge economic losses to enterprises When a localfault occurs in the rolling bearing an impulse signal ofperiodic vibration will be generated [1 2] -ese fault signalsare usually subject to background noise and harmonic in-terference caused by other equipment -erefore the earlyweak fault extraction of rolling bearings is very necessary [3]

A number of detection techniques have been developedto address this issue such as empirical mode decomposition(EMD) [4] spectral kurtosis (SK) [5] and variational modedecomposition (VMD) [6]-en traditional time-frequencyanalysis methods tend to denoise while these algorithmsalso have their own shortcomings Different from the abovefault diagnosis analysis methods morphological filter (MF)directly changes the geometry of the original signal through

the structural element (SE) at the same time MF can alsohandle nonlinear and nonsteady signals [7] At first MF wasapplied to image processing by Matheron and Serra [8] andlater it was widely used in mechanical fault diagnosis due toits simple and fast advantages [9ndash14] Two important factorsaffect the performance of MF filter the selection of SE andmorphological operator For the SE a large number of lit-eratures show that the length of SE has a great influence onthe filtering performance of the signal [15ndash17] For themorphological operator the construction of morphologicaloperators also plays an important role

Morphological operators are mainly composed of fourbasic operators opening closing dilation and erosion Atpresent based on these four basic morphological operatorsa variety of new morphological operators are cascaded to-gether and they have been successfully applied to vibrationsignals MG is composed of the difference between the di-lation and erosion operators Raj and Murali [18] Li et al[19] and Hu et al [20] successfully applied the MG operatorto the fault diagnosis of rolling bearings Dong et al [21]

HindawiShock and VibrationVolume 2020 Article ID 8828517 26 pageshttpsdoiorg10115520208828517

introduced the AVG operator with the average of openingand closing operators Hu and Xiang [22] proposed the CMFoperator by using the average of the opening-closing andclosing-opening operators and demonstrated the denoisingperformance of CMF Wang et al [23] and Meng et al [24]used the CMF operator to detect bearing failure DIF iscomposed of the difference between the closing and openingoperators Li et al [25 26] and Zhang et al [27] successfullyapplied the DIF operator to the fault diagnosis of mechanicalsystems

Previous scholars used the morphological operator withsingle structure to deal with the fault of rolling bearings -eresults show that the morphological operator with a singlestructure can no longer meet the requirements when dealingwith signals with complex frequency components -ere-fore some scholars have studied that the new morphologicaloperator constructed by the product can not only enhancethe feature extraction ability but also weaken the interfer-ence of random noise Lv and Yu [13] used basic mor-phological operators to construct some combinationdifference operators and chose two optimal operators fromthem to construct the average combination difference op-erator (ACDIF) -en they used the TEK index factor toselect the optimal scale of the operator and finally applied theproposed method to fault diagnosis of rolling bearing Liet al [15] made a detailed qualitative analysis of somecommon basic morphological operators According to thepurpose of processing the signal these morphological op-erators are divided into two categories noise reduction-typeoperators and feature extraction-type operators and theyconstructed the MGPO operator based on the superiorperformance of the two morphological operators Based onLirsquos research Yan et al [28] derived a new operator namedMHPO and they successfully applied MHPO to the faultdiagnosis of rolling bearings

Unfortunately in the above studies few papers havequantitatively studied the filtering performance of thesemorphological operators In this paper a set of rollingbearing fault simulation signals and four sets of filteringperformance indicators are used to quantitatively analyzethe feature extraction capability of the feature extraction-type operator and rank their filtering performance -en anew enhancement operator named EMDO is constructed

Although EMDO can extract more impulse signals dueto the interference of background noise the filtering per-formance of EMDO will be affected In order to solve thisproblem the probabilistic principal component analysis(PPCA) [29ndash32] is introduced to make up for the deficiencyof EMDO in denoising capability PPCA first establishes anappropriate probability model for each variable and itdecomposes the principal components and noise of thesignal into orthogonal space by increasing the dimensionrealizes the decorrelation between principal componentsand noise and finally achieves the purpose of noise re-duction In the PPCA model two important system pa-rameters the number of decomposed principal componentsk and the original variable n have a great impact on themodelrsquos noise reduction results At present existing papers

have manually selected the system parameters of PPCAwhich greatly affects the noise reduction performance [33]

In order to obtain the optimal parameters of PPCA thispaper chooses grasshopper optimization algorithm (GOA)to optimize the system parameters of PPCA GOA is amathematical optimization algorithm proposed by Saremiet al [34] in 2017 which is used to simulate the behavior ofgrasshoppers in nature Compared with other optimizationmethods GOA has the characteristics of fast convergenceand difficulty in generating local optimal solutions [34]Finally this paper combines the advantages of the above twomethods to propose a new comprehensive fault diagnosismethod for rolling bearing fault diagnosis

-e remainder of this article is organized as followsSection 2 briefly introduces mathematical morphologicaltheory and proposes the EMDO operator Section 3 de-scribes the APPCAmethod for noise reduction as well as theconstruction process of the APPCA-EMDF algorithm Ex-perimental verification of the proposed method is illustratedin Section 4 Conclusions are drawn in Section 5

2 Basic Theory of Mathematical Morphology

21 Morphological Filter Assume that the original signal f(n) is a group of one-dimensional discrete array F (0 1 2 N minus 1) -e structure element g (m) is defined as anotherone-dimensional discrete array G (0 1 2 M minus 1) andsatisfiesNgeM-en the four basic morphological operatorscan be expressed as follows

Dilation

(foplusg)(n) max[f(n minus m) + g(m)] (1)

Erosion

(fΘg)(n) min[f(n + m) minus g(m)] (2)

Opening

(f ∘g)(n) (fΘgoplusg)(n) (3)

Closing

(f bull g)(n) (foplusgΘg)(n) (4)

where oplus Θ and represent the dilation erosionopening and closing operators respectively Besides severalmain morphological filters are introduced as follows

Average of closing and opening (AVG) [21]

AVG(f(n)) (f bull g)(n) +(f ∘g)(n)

2 (5)

Morphological gradient (MG) [18ndash20]

MG(f(n)) (foplusg)(n) minus (fΘg)(n) (6)

Difference filter (DIF) [25ndash27]

DIF(f(n)) (f bull g)(n) minus (f ∘g)(n) (7)

-e opening-closing FOC and closing-opening FCO filtersare defined as

2 Shock and Vibration

FOC(f(n)) (f ∘gbullg)(n) (8)

FCO(f(n)) (f bull g ∘g)(n) (9)

A combination morphological filter with FOC and FCO(CMF) [2324] is defined as

y(n) FCO(f(n)) + FOC(f(n))

2 (10)

FCO and FOC gradient operation (GCOampOC) [35]

Gcoampoc(f(n)) (f bull g ∘g)(n) minus (f ∘g bull g)(n) (11)

Black top-hat (BTH) operation and positive BTH(PBTH) operation [27]

BTH(n) (f bull g)(n) minus f(n) (12)

PBTH f(n) minus (f bull g)(n) (13)

White top-hat (WTH) operation and negative WTH(NWTH) operation [22]

WTH(n) f(n) minus (f ∘g)(n) (14)

NWTH (f ∘g)(n) minus f(n) (15)

In order to verify the filtering performance of themorphological operator the SE needs to be selected Somescholars [17 19 35] have proved that only the length of SEhas a great impact on the filtering results -erefore in orderto improve the calculation efficiency this paper chooses aflat SE with zero height -e relationship between the lengthof SE and the scale is L λ+ 2 and the selection range ofscale is usually from 1 to fsfo where fs and fo representsampling frequency and fault frequency respectively

22 Property of Morphology Operators According to thepurpose of processing fault signals morphological operatorscan generally be divided into two categories noise reduc-tion-type operations and feature extraction-type operations[15 17]-e noise reduction-type operations tend to denoisethe signal while the feature extraction-type operations aremore inclined to extract fault features -e specific classi-fication results of the 15morphological operators introducedin Section 21 are shown in Table 1

In order to extract weak fault signals from strongbackground noise this paper mainly studies the filteringperformance of feature extraction-type operators Li et al[25] introduced the characteristics of some basic morpho-logical operators in detail but they did not quantitativelystudy the fault feature extraction capabilities of morpho-logical operators under different SNRs -erefore this paperquantitatively evaluates the filtering performance of thefeature extraction-type operators In order to analyze theirengineering applicability to rolling bearing faults a set ofbearing fault simulation signal models is established [36] Itis as follows

y(t) y1(t) + y2(t) + y3(t) + δ(t) (16)

-e fault model of the inner race of a bearing is com-posed of four parts the inner race fault signaly1(t) 4eminus 100t sin(2π middot 200 middot t) the random impulse inter-ference signal y2(t) the harmonic interference signaly3(t) 12 sin(2π middot 30 middot t) + 11 cos(2π middot 40 middot t) and thewhite Gaussian noise signal δ(t) Here a local fault isgenerated in the inner race of the bearing -e fault fre-quency of the signal fo 16Hz the number of randomimpulses of the signal is 3 and the resonant frequency is300Hz-e sampling frequency of the signal is 1024Hz andthe number of sampling points is 1024

Feature energy factor (FEF) [17 28] and amplitudes atthe 3th harmonics of the bearing fault frequency (3thA)[3738] have been proved to be an effective index factor forevaluating the feature extraction capability of morphologicaloperators As the statistical measurement index permuta-tion entropy (PE) [25 39] and envelope spectrum sparsity(ESS) [40] represent the randomness and sparsity of signalswhich can effectively reflect the intensity of backgroundnoise in the test signal and they are a set of effective in-dicator factors for measuring signal denoising performance-erefore in this paper four representative evaluationfactors are selected to prove the filtering performance ofthese feature extraction-type operators -e four evaluationindicators are shown in Table 2

For signal y(t) flat SE with a height of zero is selected andthe scale is 5 When the SNR changes from minus 10 dB to 0 dBthe variation of four evaluation index factors with the SNR isshown in Figure 1

For FEF factor the larger the FEF the stronger theoperatorrsquos fault feature extraction ability In formula F1 ofTable 2 M 4 and f 200Hz It can be clearly seen fromFigure 1(a) that the FEF of the 7 operators increases with theincrease of SNR MG received the largest FEF followed byDIF WTH and NWTH obtain the minimum FEF For PEfactor if the signal is more regular then the PE is smaller InFigure 1(b) the PE index of 7 operators does not increasesignificantly with the change of SNR -e PE index of MG isthe global minimum followed by Gco_oc and DIF It showsthat the filtering performance of these three operators isbetter than the other four morphological operators -egreater the ESS the stronger the denoising performance ofthe operator As shown in Figure 1(c) when SNR changesfrom minus 10 dB to minus 6 dB the ESS of Gco_oc is the best WhenSNR changes from minus 6 dB to 0 dB MG has the best ESS -eESS index of DIF is stable between 5 and 6 and its ESS ranksthe third place among all operators BTH PBTH NWTHand WTH operators obtain almost the same ESS indicatingtheir poor filtering performance -e 3thA is a key index toevaluate the feature extraction ability of the operator -e3thA of these operators with SNR is shown in Figure 1(d) InFigure 1(d) when SNR increased from minus 10 dB to 0 dB MGreceived the largest 3thA than other operators while the 3thA of DIF was slightly lower than the MG ranking secondAmong all operators the 3thA index factor of Gco_oc was thelowest

Shock and Vibration 3

Table 2 Evaluation indicator

Number Name DefinitionF1 Feature energy factor FEF (EElowast) (1113936

Mi1 Y2

i (f)1113936 Y(f)2)

F2 Permutation entropy PE (minus 1113936mm1p(π) ln p(π)ln(m))

F3 Envelope spectrum sparsity ESS(Y(f)) (

(1n)1113936nminus 1i0 Y2

i (f)

1113969

(1n)1113936nminus 1i0 |Yi(f)|)

F4 Amplitudes at the 3rd harmonics of the bearing fault frequency 3thA A(3fo)

Note Y (f ) is the Hilbert envelope spectrum of y (t) n is the length of the spectralM is the number of fault frequencies p(π) Y(π)(N minus (m minus 1)τ)N is thelength of the time signal m is the embedding dimension π is a permutation τ is the time lag A represents the amplitude of the signal and fo represents thefault frequency of the signal

MGPBTHBTHWTH

DIFNWTHGco_oc

ndash10 ndash8 ndash6 ndash4 ndash2 0SNR

00

02

04

06

08

FEF

(a)

MGPBTHBTHWTH

DIFNWTHGco_oc


20

20

15

05

PE

(b)

MGPBTHBTHWTH

DIFNWTHGco_oc


2

4

6

8

10

ESS

(c)

MGPBTHBTHWTH

DIFNWTHGco_oc


00

02

04

06

08

3thA

(d)

Figure 1-e results of the evaluation indicator varying with SNR (a) feature energy factor (b) permutation entropy (c) envelope spectrumsparsity (d) 3rd harmonic amplitude

Table 1 Classification of the morphology operators

Noise reduction-type operations Feature extraction-type operationsDilation MG (dilation-erosion)Erosion WTH (signal-opening)Opening BTH (closing-signal)Closing PBTH (signal-closing)FCO (closing opening) DIF (closing-opening)FOC (opening closing) GCOampOC (FCO-FOC)AVG (average of closing and opening) NWTH (opening-signal)CMF (average of FCO and FOC)


It can be seen from the above analysis that the featureextraction ability of BTHWTH PBTH and NWTH is poorwhile the MG operator and DIF operator show superiorfiltering performance

Research through reference [17 19] shows that for thesimulation signal y(t) of equation (16) the optimal scale ofthese operators is 1ndash10 In addition if the scale of the SE istoo large some useful information will be removed In orderto further prove the filtering performance of the featureextraction-type operator at different scales when the SNR ofsignal y(t) is minus 3 the evaluation indicator of the operatorvarying with the scale is shown in Figure 2

In Figure 2(a) when the scale is between 1 and 8 MGreceived the largest FEF than other operators When thescale is between 9 and 10 DIF obtains the best FEE-erefore from FEF factor analysis it is concluded that MGand DIF show excellent feature extraction performance InFigure 2(b) the PE value of the operators decreases andbecomes stable with the increase of scale Compared withother operators MG obtains the smallest PE in the wholescale followed by Gco_oc and DIF ranks third -e PE valuesof PBTH NWTH BTH andWTH are larger indicating thattheir filtering performance is not as good as the MG DIFand Gco_oc operators -e results of ESS index factor of theseoperators changing with scale are shown in Figure 2(c) InFigure 2(c) the change trend of the operatorrsquos ESS with thescale is opposite to PE-e same conclusion can be obtainedfrom the analysis results In Figure 2(d) the variation trendof 3thA index of these operators is similar to FEF MG andDIF still get the largest value at full scale

-rough the comprehensive analysis of the evaluationindexes of the above operators it can be concluded that therank of feature extraction ability of 7 operators is as follows

MGgtDIFgtGco_ocgtBTHPBTHgtNWTHWTH

Based on the above quantitative research on filteringperformance of feature extraction operator it can be con-cluded that MG and DIF have superior fault feature ex-traction ability -erefore a new morphological differentialoperator named EMDO is constructed based on the ex-cellent performance of the two operators

EMDO(n) MG(n) middot DIF(n)

[(foplusg)(n) minus (fΘg)(n)] middot [(f bull g)(n) minus (f ∘g)(n)]

(17)

-e newly proposed EMDO operator has two advan-tages (1) if the fault information is detected by both MG andDIF then EMDO will amplify the fault information (2)because Gaussian noise shows randomness when Gaussiannoise is detected by only one of them EMDO will output asmaller amplitude

23 Comparison of Morphology Operators When SNR ofsignal y(t) is minus 6 dB its time-domain diagram and frequency-domain diagram are shown in Figure 3 respectively InFigure 3(b) the fault frequency fo can no longer be detecteddue to the interference of strong background noise so the

simple envelope analysis can no longer be used to detectearly weak fault of the rolling bearings

In order to further verify the superiority of the proposedoperator in this paper five other morphological operators(MG [19] DIF [25] MGPO [15] MHPO [28] and ACDIF[13]) were selected to compare with the proposed operator infault feature extraction capabilities Since the FEF indexfactor is sensitive to the characteristic information of thesignal it can directly reflect the operatorrsquos ability to extractfault characteristic information -is paper chooses FEF tooptimize the scales of the operators In formula F1 of Table 2M 4 and f 200Hz-e FEF values of abovemorphologicaloperators changing with the scale are shown in Figure 4

It can be seen from Figure 4 that before reaching theoptimal scale the FEF of each operator increases with thescale and when it reaches the optimal scale it decreases withthe increase of the scale -e optimal scale distribution ofthese operators is between 1 and 10 When the optimal scaleis 4 EMDO obtains the highest FEF of 3367 which islarger than other operators-e effectiveness and superiorityof EMDO have been proven -e FEFs of MG and DIF weresignificantly lower than EMDO -is shows that the newlyconstructed operator is enhanced in feature informationextraction capability -e maximum FEFs of MG DIFMGPO MHPO and ACDIF operators are 2337 22612632 2514 and 2336 respectively -e FEF values ofthe above six operators are far greater than the envelopespectrum analysis -e results of processing the signal y(t) inFigure 3(a) by these operators under their optimal scale areshown in Figure 5

It can be seen from Figure 5 that all six operators canextract the fault characteristic frequency fo but 2fo extractedby MG DIF and ACDIF is weak Due to the interference ofbackground noise the frequency components in the range of0ndash200Hz are seriously polluted Although MGPO can ex-tract the fault frequencies fo the amplitude of the faultfrequency is relatively weak and 4fo cannot be detectedMHPO operator can detect 4fo however there are manyinterference frequency components in the range of0ndash200Hz Especially when the background noise furtherincreases MHPO is not as robust as EMDO As shown inFigure 5(f ) the operator proposed in this paper can clearlydetect the fault frequencies fo 2fo 3fo and 4fo Combinedwith the above analysis it is further proved that the EMDO issuperior to other morphological operators in the extractionability of fault features

3 The Method of APPCA-EMDF

It can be seen from the above analysis that the advantages offeature extraction-type operators lie in the ability of faultfeature extraction but their denoising performance is in-sufficient In order to make up the deficiency of featureextraction-type operator in denoising ability this paperpresents an adaptive PPCA denoising method

31 PPCA In the PPCA model [29ndash32] it is assumed thatthe main fault information of the original signal is stored in


the principal component space and the remaining noiseinformation and linear correlation information are saved inthe remaining subspace -e PPCA model for the n-di-mensional original variable X is as follows

X p middot u + E (18)

where X x1 x2 middot middot middot xm1113864 1113865 isin Rntimesm is a matrix (it is generatedby a one-dimensional signal x xr minus xr where xr is the testfailure data and xr is the mean of xr ) of ntimesm n is thedimension of the original variable X and m is the length ofthe signal P p1p2 middot middot middot pm1113864 1113865 isin Rntimesk is a matrix of ntimes kand the limiting condition is klt n where k is the number of

ndash1000 02 04 06 08

Time (s)10

ndash5

0

5

10

Am

plitu

de

(a)

0 40 80 120 160 200Frequency (Hz)

00

02

04

06

Am

plitu

de 1048

(b)

Figure 3 Time-domain diagram and envelope spectrum of simulation signal y(t)

00

02

04

06

FEF

MGPBTHBTHWTH

DIFNWTHGco_oc

2 4 6 8 10Scale

(a)

20

20

15

00

05

PE

MGPBTHBTHWTH

DIFNWTHGco_oc

2 4 6 8 10Scale

(b)

2

0

4

6

8

ESS

MGPBTHBTHWTH

DIFNWTHGco_oc

2 4 6 8 10Scale

(c)

00

02

04

06

08

3thA

MGPBTHBTHWTH

DIFNWTHGco_oc

2 4 6 8 10Scale

(d)

Figure 2-e results of the evaluation indicator varying with scale (a) feature energy factor (b) permutation entropy (c) envelope spectrumsparsity (d) 3rd harmonic amplitude


0 10 20 30Scale

40 50 60 7000

01

02

03

04

FEF

2 4 6 8 10

04

03

02

01

00

Zoom plot

MGMHPODIF

EMDOMGPO

EnvelopeACDIF

Figure 4 FEF factor of different operators with scale

16

3248

64

00

04

08

12

Am

plitu

de

120 16040 80 2000Frequency (Hz)

(a)

16

3248 64

00

04

08

12A

mpl

itude

40 80 120 160 2000Frequency (Hz)

(b)

16

32 48

0

2

4

6

8

10

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

(c)

16

3248 64

0

1

2

3

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

(d)

16

4864

00

04

08

12

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

(e)

16

32 4864

0

4

8

12

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

(f )

Figure 5 -e results of simulation signal y(t) processed by (a) MG (b) DIF (c) MGPO (d) MHPO (e) ACDIF and (f) EMDO


main components u u1u2 middot middot middotum1113864 1113865 isin Rktimesm is the prin-cipal component matrix of ktimesm E is an isotropic Gaussiannoise matrix u and E both satisfy Gaussian distributionusimN(0 I)EsimN(0 σ2I) where I is the identity matrix andσ2 is the variance of the noise variable-ereforeX follows aGaussian distribution N(0PPT + σ2I)

-e probability distribution of u is as follows

p(u) (2π)minus k2

eminus 12σ2( )XTX( ) (19)

-e prior probability distribution of x under the variableu is as follows

p(x | u) (2π)minus n2

eminus 12σ2( )Xminus Pmiddotu2( ) (20)

According to equations (19) and (20) the probabilitydistribution of the original data x is as follows

p(x) 1113946 p(x | u)p(u)dx (2π)minus n2

|C|minus 12

eminus (12)XTCminus 1X( )

(21)

where C PPT + σ2I is the covariance matrix which isdetermined by the parameters P and σ2

According to equations (19)ndash(21) once the values of theparameters P and σ2 are determined the probability model isobtained -e above parameters can be estimated using theexpectation maximization (EM) algorithm [41] and theiterative formula derived is as follows

1113957P SP σ2I + Mminus 1PTSP1113872 1113873minus 1

(22)

1113957σ2 1ntr S minus SPMminus 11113957P

T1113874 1113875 (23)

where S is the covariance matrix of the original data S

(1m) 1113936mi1 xixT

i and tr(middot) is the trace of the matrix -e pa-rameters P and σ2 are obtained by iterative convergence ofequations (22) and (23) Finally a PPCA model is established

When the PPCA model is established the solutionmethod of the reduced-dimensional data is shown below

ui pTi X (24)

From equation (24) each principal component signal(data after dimensionality reduction) is a projection of theoriginal variable data X on the corresponding principalcomponent vector pi Although PPCA is a better noise re-duction algorithm the system parameters of PPCA (thenumber of principal components k and the original variablen) need to be determined by personal experience which willseriously affect its noise reduction performance In PPCAthe parameter n determines the dimension of the con-struction matrix and the parameter k determines thenumber of eigenvalues after dimensionality reduction

In order to test the noise reduction performance aftersignal processing Deng et al [40] proposed a new and ef-fective signal noise reduction index SIESS by combining ESSand PE Because the kurtosis Kr is suitable for early faultdiagnosis of rolling bearings this paper proposed a new

dimensionless comprehensive evaluation index KSP to de-tect the noise reduction performance of the signal

KSP Kr times ESS

PE (25)

where the expressions of ESS and PE are shown in Table 2-e larger the KSP the better noise reduction performanceof the signal -erefore the maxKSP can indicate the op-timal noise reduction result of PPCA For the signal y(t) inFigure 3(a) the distribution of KSP under different systemparameters (n k) is shown in Figure 6

From Figure 6 the distribution of KSP is relativelydiscrete and random with the change of n and k Whenn 19 and k 8 the maximum value of KSP is 1498 -enoise reduction effect of PPCA is affected by the unrea-sonable setting of parameters n and k -erefore in order toachieve the best noise reduction effect of PPCA this articlechooses GOA to adaptively optimize the system parametersof PPCA -e GOA [34] is described as follows

32 GOA GOA is a mathematical optimization algorithmthat simulates the behavior of grasshopper swarms in nature-ere are two distinct main features of the grasshopperswarm behavior (1) during the larval stage the grasshoppersmove slowly and the number of steps is small the grass-hopper behavior in adulthood shows a long distance ofmovement and agility (2) the process by which grasshopperswarms look for food sources can be divided into two trendsexploration and exploitation During exploration the searchswarms are encouraged to move suddenly and in exploi-tation they are encouraged to move locally Grasshopperswarms naturally accomplish both functions and seek goals-erefore the mathematical model used to simulate thebehavior of the grasshopper swarm is as follows

Xi Si + Gi + Ai (26)

where Xi is the position of the ith grasshopper Si representssocial interaction Gi defines the gravity force on the ithgrasshopper and Ai represents wind advection

Si 1113944N

j1jneis dij1113872 11138731113954dij 1113944

N

j1jne i

s xi minus xj

11138681113868111386811138681113868

111386811138681113868111386811138681113874 1113875xi minus xj

dij

(27)

s(r) feminus rl

minus eminus r

(28)

where dij |xi minus xj| indicates the distance between the ithand jth grasshopper (the interval of dij is usually 1 to 4 in theGOA) 1113954dij (xi minus xj)dij shows a unit vector from the ithgrasshopper to the jth grasshopper s(bull) is used to representthe function of social forces between grasshoppers f rep-resents the intensity of attraction l indicates attractive lengthscale and f and l are two important parameters of the GOA(they are described in [34]) Usually f and l are equal to 15and 05

Gi minus g1113954eg (29)


where g represents the gravitational constant and 1113954eg is aunity vector towards the center of Earth

Ai u1113954ew (30)

where u indicates a constant drift and 1113954ew represents a unityvector of the wind direction

Substituting S G and A into equation (26)

Xi 1113944N

j1jne i

s xj minus xi

11138681113868111386811138681113868

111386811138681113868111386811138681113874 1113875xj minus xi

dij

minus g1113954eg + u1113954ew (31)

where the expression of s(bull) is as shown in equation (28) andN is the number of grasshoppers

However the above mathematical model cannot bedirectly used to solve the optimization problem because thegrasshoppers reach the comfort zone too fast and the swarmscannot converge in the designated area -erefore thefollowing modified equation is used to solve the aboveproblem [34]

Xdi c 1113944

N

j1jne i

cubd minus lbd

2s x

dj minus x

di

11138681113868111386811138681113868

111386811138681113868111386811138681113874 1113875xj minus xi

dij

⎛⎝ ⎞⎠ + 1113954Td

(32)

where ubd is the upper boundary ofD-dimensional space lbdis the lower boundary of D-dimensional space 1113954Td indicatesthe optimal target found so far in the Dth dimension and crepresents a decreasing coefficient its equation is as follows

c cmax minus lcmax minus cmin

L (33)

where cmax represents the maximum value of the decreasingcoefficient c cmin represents the minimum value of thedecreasing coefficient c l indicates the current number ofiterations and L indicates the total number of iterations Inthis paper cmin and cmax are taken as 000001 and 1respectively

-e mathematical model of the GOA is described above-e specific optimization process is as follows Firstly ini-tialize the swarm and randomly generate the swarms Xiinitialize system parameters (eg cmax cmin l and L)

calculate the fitness of each search agent and let T be equalto the optimal fitness Secondly using equation (32) toupdate the position of the current target search agent inorder to avoid the optimization model falling into the localoptimal solution the decreasing coefficient c is updated eachtime by using equation (33) and the distance betweengrasshoppers is normalized in each iteration Finally whenthe iteration condition is satisfied the global optimal so-lution is obtained A more detailed introduction to the GOAis given in [34]

33APPCA-EMDF -eproposed parameter adaptive PPCAmethod takes the maxKSP value as the objective function andoptimizes the parameters through the GOA -e specificobjective function is shown in the following equation

fitness maxc(nk)

KSPi1113864 1113865

stn 1 2 20

k 1 2 3 20

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(34)

where fitness represents the objective function of the opti-mization KSPi (i 1 2 c) is the KSP index after PPCAdecomposition and reconstruction of the signal and c (n k)represents the optimized system parameters of PPCA -isarticle proposes an early weak fault detection method forrolling bearings based on a combination of adaptive PPCAand enhanced morphological differential filtering

-e specific flowchart is shown in Figure 7-e specific working steps are as follows

Step 1 input the vibration signal y(t) set the optimi-zation range of PPCA parameters (n k) and determinethe initial parameters of GOA (the maximum iterationsteps L and the number of population N) [34]Step 2 in the process of each iteration the signal y(t) isdecomposed and the fitness of the signal is calculated aftereach decomposition -e maximum fitness is retainedStep 3 determine the termination condition of theprogram If lge L return yes otherwise return no thenl l+ 1 and continue iteration until the condition issatisfied [34]Step 4 save the optimal system parameters (n k) andobtain the maximum objective function value PPCAapplies optimal system parameters to the signal fornoise reductionStep 5 the morphological analysis scale range is de-termined and the SE is determinedStep 6 calculate the FEF value at each scale and de-termine the optimal scale λ for EMDFStep 7 EMDF uses the optimal scale λ to perform themorphological calculationStep 8 envelope analysis finally realizes fault diagnosis

4 Simulation

41 Case 1eTraditional Bearing FaultModel -emethodproposed in this paper is used to process the simulation

KSP

4

6

8

10

12

14

20

15

10

5

n

10 15 205k

Figure 6 KSP obtained after noise reduction by PPCA


signal y(t) in Figure 3(a) -e range of optimization pa-rameters (n k) of APPCA is selected according to equation(34) -e parameters of GOA are selected as followsmaximum number of iterations L 10 and search agentsN 20 -e GOA convergence curve of the maxKSP valueobtained after each iteration is shown in Figure 8(a) InFigure 8(a) when the number of iterations reaches 3 thecurve starts to converge -e optimal n k and maxKSPvalues obtained by GOA are 16 8 and 14981 respectivelyPPCA processes the signal y(t) with the optimal n and k -enoise reduction result is shown in Figure 8(b) ComparingFigure 8(b) with Figure 3(a) it can be clearly found that thebackground noise is effectively suppressed Next the cal-culated full-scale FEF value of the denoising signal is shownin Figure 9(a) In Figure 9(a) it can be seen that the optimalscale obtained by EMDF is 4 -e signal of Figure 8(b) isprocessed with EMDF and the time-domain and frequency-domain diagrams of the calculation results are shown inFigures 9(b) and 9(c) respectively Comparing the calculatedresults with Figures 5(a)ndash5(f) it can be found that theAPPCA-EMDF method can clearly detect the bearing faultcharacteristic frequency of 16Hz and its harmonic

frequencies of 32Hz 48Hz and 64Hz At the same time thebackground noise is effectively suppressed and the featureextraction capability of the fault is greatly enhanced

42 Case 2 Complex Fault Model of Bearings In order tofurther verify the anti-interference ability and weak faultextraction ability of the algorithm proposed in this paper anew set of rolling bearing simulation fault model is estab-lished whose expression is as follows

yprime(t) y1(t) + y2(t) + y3(t) + y4(t) + δ(t)

y1(t) 1113944I

i1Aisi t minus iTa minus τi( 1113857

y2(t) 1113944N

n1Bnsn t minus nTb( 1113857

y3(t) 1113944M

m1Cm sin 2πfm + φm( 1113857

y4(t) 1113944P

p1Cp cos 2πfp + φp1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(35)

Set PPCA parameter range and initialize GOA

Input vibration signal y(t)

Denoise and reconstructsignals with PPCA

Calculate fitness aer eachreconstructed signal

l ge L

l = l + 1No

Save optimal parameters

Use the optimal parameters(nk) to denoise the signal to

obtain the signal y(t)rsquo

Yes

Step 1

e analytical scale range andthe structural element (SE) are determined

e optimal scale λ of EMDF isobtained through FEF optimization

Feature extraction-type operators

FEF PE ESS 3thA

Evaluation indicator

e filtering performance of thefeature extraction-type operators is

ranked

EMDO operator is constructed

Step 2

e signal y(t)rsquo isprocessed

Spectrum analysis and fault diagnosis

Step 3

Figure 7 Flowchart of the proposed method


-e fault model of the bearing is composed of fiveparts the fault signal y1(t) the random pulse interferencesignal y2(t) the harmonic interference signals y3(t) andy4(t) and the white Gaussian noise signal δ(t) In themodel A represents the amplitude of the fault signal Trepresents the time interval between the two shock signalsand τ represents a random variable that is usually used tosimulate random sliding of the roller bearing B is avariable representing the amplitude of the random impulsesignal C represents the amplitude of the harmonic in-terference signal f represents the frequency of the har-monic signal and φ represents the phase angle s(t)

represents the impulse response function of the me-chanical system and can be described as follows

s(t) Meminus at sin 2πfnt + φ( 1113857 (36)

whereM represents the amplitude a is the decay factor fn isthe resonance frequency and φ is the phase angle

A simulated signal is established based on equation (35)Here a local fault is generated in the inner race of thebearing -e fault frequency of the signal fo 16Hz theamplitude A 4 the decay factor a is set to 100 the naturalfrequency fn 200Hz and the phase is zero -e number of

The best scale

00

02

04

06

FEF

10 20 30 40 50 60 700Scale λ

(a)

times106

0

1

2

3

4

Am

plitu

de

02 04 06 08 1000Time (s)

(b)

times104

16

32

4864

0

4

8

12

16

Am

plitu

de

40 80 120 160 1200Frequency (Hz)

(c)

Figure 9 Case1 (a) correlation curve between FEF and scale λ (b) time-domain spectrum processed by the EMDF method (c) frequency-domain spectrum processed by the EMDF method

12

13

14

15

16Be

st sc

ore

4 6 8 102Iteration

(a)

times102

ndash10

ndash5

0

5

10

Am

plitu

de

02 04 06 08 1000Time (s)

(b)

Figure 8 Case1 (a) fitness convergence curve for PPCA optimization (b) time-domain diagram after noise reduction by APPCA


random impulses of the signal is 10 and the resonant fre-quency is 300Hz In harmonic interference the harmonicfrequencies of the signals y3(t) are 30Hz and 40Hz with anamplitude of 12 and 11 respectively -e harmonic fre-quencies of the signals y4(t) are 30Hz and 50Hz with anamplitude of 1 and 12 respectively -e SNR of whiteGaussian noise is minus 10 dB -e sampling frequency of thesignal is 2048Hz and the number of sampling points is 2048-e components of the simulated signal are shown inFigure 10 -e obtained simulated signal yprime(t) and its en-velope spectrum are shown in Figure 11 It can be found inFigure 11(b) that due to the serious interference of harmonicfrequency and background noise only 10Hz 20Hz and50Hz interference frequencies can be identified making itdifficult to detect the bearing fault frequency

-e method proposed in this paper is used to process thesimulation signal yprime(t) -e obtained GOA optimizationcurve and the time-domain diagram after noise reductionare shown in Figures 12(a) and 12(b) respectively InFigure 12(a) when iterating to 4 the objective function valueconverges the maximum KSP value obtained is 12945 andthe optimized PPCA parameters n 18 and k 7 Nextfeature extraction is carried out for the denoised signal -eprocessed results are shown in Figures 13(a)ndash13(c) InFigure 13(c) the fault characteristic frequencies 16Hz32Hz and 48Hz still can be clearly detected-is shows thatthe algorithm proposed in this paper has strong anti-in-terference ability and weak fault feature extraction ability

For comparison the three algorithms (ACDIF [13]VMD [6] and SK [5]) were used to process the simulationsignal in Figure 11(a) and the processing results are shownin Figures 14ndash16 In Figure 14(c) ACDIF can only detect aweak fault frequency 16Hz and the harmonic interferencefrequency 10Hz is clearly present In Figure 15(c) althoughVMD can identify the fault frequency of 16Hz the fre-quency components of 32Hz and 48Hz have weak ampli-tude and they are seriously disturbed It can be seen inFigure 16(c) that the SK method cannot effectively detect thefault characteristics of the bearing if the background noiseand other frequency interference components are serious

5 Experimental Validations

51 Case 1 Bearing Fault Diagnosis of Wind TurbineWind energy is used as a clean energy source and thenumber of wind turbines in the world is gradually in-creasing Due to the long-term service of wind turbines inextreme environments the transmission system of windturbines is easily damaged Once the wind turbine isdamaged it will bring huge economic losses to the windturbine operator -erefore fault monitoring of wind tur-bines is particularly important

By monitoring the vibration of multiple wind turbines inTuoshan Wind Farm the abnormal vibration amplitude ofthe front bearing of No 16 wind turbine can be found so thefault monitoring is carried out -e structural model of thewind turbine and its sensor arrangement are shown inFigure 17(a) -e test system monitors 8 positions in thecabin (test points 1ndash8 in Figure 17(a)) During the test

abnormal vibration signal of measuring point 2 was foundand the field measured photos are shown in Figure 17(b)-e tested generator front-end bearing model is SKFNU1030 -e specific structural geometric parameters of thebearing are shown in Table 3 -e rated power of the windturbine is 1500 kW the blade speed is stable at 223 rpm themotor speed is 1740 rpm and the rotation frequency of thegenerator fr 29Hz -e sampling frequency of the testsystem is 16384Hz and the total length of the test signal is81920 -e fault frequencies of the inner ring outer ringcage and rolling element of the rolling bearing calculatedfrom Table 4 are fi 3847Hz fo 3159Hz fc 1311Hzand fB 971Hz

-e time-domain diagram and the FFT diagram of thecollected generator front-end vibration signals are shown inFigures 18(a) and 18(b) respectively In Figure 18(a) im-pulse signals with different energy amplitudes can be seenbut due to the interference of background noise the periodicvibration impulse signals of the bearings are not very ob-vious so further processing and analysis are needed InFigure 18(b) the vibration energy of the test signal is mainlyconcentrated in the middle frequency band In the lowfrequency band the weak outer ring failure frequency of therolling bearing can only be identified at 3156Hz

-erefore the method proposed in this paper is used toprocess the experimental signal to detect the bearing fault atthe front of the generator Firstly the experimental signalwas denoised using PPCA and the GOA was used toadaptively optimize the system parameters (n k) of PPCA-e obtained objective function convergence curve is shownin Figure 19(a) In Figure 19(a) when iterating to 3 theobjective function converges the maxKSP obtained is16684 and the optimal system parameters of PPCA aren 20 and k 8 -e optimal parameters are used to denoisethe experimental signals and the results are shown inFigure 19(b) Compared with Figure 18(a) it can be foundthat the signal denoised by APPCA presents relatively cleanimpulse vibration It can be seen in Figure 19(c) that there isan impulse vibration signal with a period interval of 32mswhich is fully consistent with the characteristic frequency ofthe outer ring fault of the rolling bearing of 3156Hz Finallyin order to further enhance the fault information the EMDFis used to process the denoised signal In formula F1 ofTable 2 M 3 and f 1000Hz As can be seen fromFigure 20(a) when the scale is 3 EMDF obtains the largestFEF factor -e time-domain and frequency-domain resultsof the denoised signals processed using the optimal scale areshown in Figures 20(b) and 20(c) respectively InFigure 20(c) in addition to the rotation frequency of 29Hzthe fault frequency of bearing outer ring of 3158Hz and itsharmonic frequencies of 630Hz and 9469Hz can be clearlydetected -e analysis results show that the method pro-posed in this paper can both detect the bearing faults in thefront end of the generator and enhance the faultinformation

For comparison the three algorithms (ACDIF VMDand SK) were still used to process the experimental signalin Figure 18(a) and the processing results are shown inFigures 21ndash23 In the ACDIF method the optimal scale


ndash4

0

4

Am

plitu

de

02 04 06 08 1000Time (s)

(a)

ndash6

ndash3

0

3

6

Am

plitu

de

02 04 06 08 1000Time (s)

(b)

ndash3

0

3

Am

plitu

de

02 04 06 08 1000Time (s)

(c)

ndash3

0

3

Am

plitu

de

02 04 06 08 1000Time (s)

(d)

Figure 10 (a) Impulsive signal y1(t) (b) random impulsive signal y2(t) (c) harmonic signal y3(t) and (d) harmonic signal y4(t)

ndash8

ndash4

0

4

8

Am

plitu

de

02 04 06 08 1000Time (s)

(a)

00

02

04

06

08

10

Am

plitu

de

20 40 60 80 100 120 140 160 180 2000Frequency (Hz)

10Hz

20Hz

50Hz

(b)

Figure 11 (a) -e simulation signal yprime(t) and (b) envelope spectrum for simulation signal y(t)

10

11

12

13

14

Best

scor

e

4 6 8 102Iteration

(a)

ndash2

0

2

Am

plitu

de

02 04 06 08 1000Time (s)

times103

(b)



05

10152025303540

TEK

10 20 30 40 50 600Scale λ

The best scale

(a)

5

10

15

20

25

Am

plitu

de

02 04 06 08 1000Time (s)

(b)

0

1

2

3

Am

plitu

de

10

16

40 80 120 160 2000Frequency (Hz)

(c)

Figure 14 Analysis results of simulation signals by the ACDIF method (a) correlation curve between TEK and scale λ (b) time-domainspectrum (c) envelope spectrum

10 20 30 40 50 60 700Scale λ

00

01

02

03

04

05

FEF

e best scale

(a)

0

2

4

6

8

Am

plitu

de

02 04 06 08 1000Time (s)

times106

(b)

00

05

10

15

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

16

32

48

times106

(c)

Figure 13 Case2 (a) correlation curve between FEF and scale λ (b) time-domain spectrum processed by the EMDFmethod (c) frequency-domain spectrum processed by the EMDF method


00

05

10

15

20A

mpl

itude

200 400 600 800 10000Frequency (Hz)

Selected

u1u2

u3u4

(a)

ndash5

0

5

Am

plitu

de

02 04 06 08 1000Time (s)

(b)

00

02

04

06

08

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

16

3248

(c)

Figure 15 Analysis results of simulation signals by the VMD method with k 4 and α 2000 (a) frequency spectrum of the fourdecomposed modes (b) time-domain spectrum of mode u2 (c) envelope spectrum of mode u2

200 400 600 800 10000Frequency (Hz)

3

26

2

16

1

0

Leve

l k

0

01

02

03

04

05

(a)

0

1

2

3

4

Am

plitu

de

0604 08 100200Time (s)

(b)

Figure 16 Continued


000

005

010

015

020

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

16

50

(c)

Figure 16 Analysis results of simulation signals by the SK method (a) fast kurtogram (b) envelope signal (c) envelope spectrum

12

3 45 6

78

Back bearing of generator

Front bearing of generator

Gearbox Main sha

Blade

Monitoring system

(a)

(b)

Figure 17 (a) Diagram of experimental device and distribution of measuring points (b) Data collection on field wind turbine generator


Table 3 Structural parameters of rolling bearing

Structural parameters Parameter valuesBearing type NU 1030Inside diameter 150mmPitch diameter 1875mmRoller diameter 18mmOutside diameter 225mm-e number of roller 24Contact angle 0deg

Table 4 Mathematical equations of bearing fault frequencies

Fault frequency FormulaInner race fault frequency fi fr(N2)(1 + (RdPd)cosφ)

Outer race fault frequency fo fr(N2)(1 minus (RdPd)cosφ)

Cage fault frequency fC (fr2)(1 minus (RdPd)cosφ)

Ball fault frequency fB fr(Pd2Rd)(1 minus (R2dP

2d)cos2φ)

Note N is the number of roller Rd is the roller diameter Pd is the pitch diameter and φ is the contact angle

ndash50

ndash25

0

25

50

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(a)

0

1

2

3

Am

plitu

de (m

s2 )

1000 1000 3000 4000 5000 6000 7000 80000Frequency (Hz)

02

01

00200 250 300 350 400

3156Hz

(b)

Figure 18 -e experimental signal of the wind turbine generator (a) time-domain spectrum (b) FFT spectrum

4 6 8 102Iteration

05

10

15

20

Best

scor

e

(a)

ndash15

00

15

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

times107

(b)

Figure 19 Continued


selection of the operator is shown in Figure 21(a) and theprocessed frequency-domain results are shown inFigure 21(c) In Figure 21(c) the rotation frequency of29 Hz can be detected but the amplitude of the faultfrequency fo of the bearing outer ring is relatively weakand the effect of fault feature extraction is not obvious Inthe VMD method the penalty factor α is 2000 andthe decomposition mode k is 5 According to the max-imum weight kurtosis index the most sensitive modeselected is u5 and the selection result is shown in

Figure 22(a) -e final processing result is shown inFigure 22(c) -e fault frequency fo and its harmonicfrequency 2fo of the bearing outer ring can be detectedbut the amplitude of the fault characteristic frequency isrelatively weak and the fault feature extraction capabilityis not strong In Figure 23(c) from the results obtained byprocessing the experimental signals with the SK methodthe fault frequency fo is severely contaminated due tointerference from the background noise and can hardlybe detected

5 10 15 20 25 30 35 40 45 500Scale λ

000

002

004

006

008

010

FEF

e best scale

(a)

00

05

10

15

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

times 1016

(b)

00

05

10

15

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

times 1010

29Hz (fr)3158Hz (fo)

630Hz (2lowastfo)

9469Hz (3lowastfo)

(c)

Figure 20 Processing results of wind turbine generator denoising signal by the EMDFmethod (a) correlation curve between FEF and scaleλ (b) time-domain spectrum (c) envelope spectrum

ndash08

00

08

Am

plitu

de (m

s2 )

0655 0660 0665 0670 0675 06800650Time (s)

32ms

times107

(c)

Figure 19 -e experimental signal denoising results of wind turbine generator (a) fitness convergence curve for PPCA optimization (b)time-domain diagram after noise reduction by APPCA (c) a partial enlarged view of the denoising signal


e best scale

10

15

20

25

30TE

K

1510 20 2550Scale λ

(a)

0

20

40

60

80

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

0

2

4

6

8

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

29Hz

3158Hz

(c)

Figure 21 Analysis results of experimental signals by the ACDIF method (a) correlation curve between TEK and scale λ (b) time-domainspectrum (c) envelope spectrum

00

05

10

15

20

Am

plitu

de (m

s2 )

2000 4000 6000 8000 100000Frequency (Hz)

Selected

u1u2u3

u4u5

(a)

ndash30

ndash15

0

15

30

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

00

05

10

15

20

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

29Hz

3158Hz

630Hz

(c)

Figure 22 Analysis results of experimental signals by the VMD method with k 5 and α 2000 (a) frequency spectrum of the fivedecomposed modes (b) time-domain spectrum of mode u5 (c) envelope spectrum of mode u5


52 Case 2 Early Fault Diagnosis of Bearings In order toverify the early fault feature extraction ability of the pro-posed algorithm the bearing life cycle data collected by theUniversity of Cincinnatirsquos Intelligent Maintenance System(IMS) [42] were applied Bearing test benches sensors andtheir schematic diagrams are shown in Figures 24(a) and24(b) respectively During the experiment a total of threesets of experimental data were tested In this paper thesecond set of experimental data is selected for processing-is group of data consists of 984 files Each group of data iscollected every 10 minutes -e specific geometric pa-rameters of the tested bearing are shown in Table 5 Duringthe experiment the motor speed was 2000 rpm and thesampling frequency was 20 kHz According to Table 5 thefault frequency of bearing outer ring (fBPFO) is 2364 Hz-e root mean square (RMS) value of the full life cycle ofthe bearing is shown in Figure 25(a) As can be seen inFigure 25(a) the damage form of the bearing is divided intothree stages early middle and late stages corresponding tothe A B and C points in the Figure 25(a) -eir corre-sponding time-domain diagrams are shown inFigures 25(b)ndash25(d) respectively It can be seen fromFigure 25(b) that the impulse amplitude of bearing earlyfault is relatively weak

In this paper the time-domain signal in Figure 25(b) isprocessed Its FFT spectrum and envelope spectrum areshown in Figures 26(a) and 26(b) respectively In

Figure 26(b) only the weak fault frequency fFBPO can beidentified and the bearing fault information is seriouslydisturbed by background noise

-e method proposed in this paper is used to process theexperimental data First the GOA optimization curve andthe noise reduction results are shown in Figures 27(a) and27(b) respectively In Figure 27(a) when iterating to 4 theobjective function value converges the maximum KSP valueobtained is 17059 and the optimized PPCA parametersn 15 and k 6 By comparing Figure 27(b) withFigure 25(b) it can be found that significant bearing faultimpulse signals can be detected after noise reduction pro-cessing by PPCA Next feature extraction is performed onthe denoised signal -e processed results are shown inFigures 28(a) to 28(c) In Figure 28(c) the fault characteristicfrequency (fBPFO) of the bearing outer ring can be clearlydetected and its corresponding high-order characteristicfrequency (2fBPFO 3fBPFO and 4fBPFO) can also be detected-e fault feature extraction effect is obvious

ACDIF VMD and SK were used to process the sameexperimental signals for comparison -e experimental re-sults obtained are shown in Figure 29ndashFigure 31 respec-tively In Figure 29(b) although the fault frequency (fBPFO2fBPFO and 3fBPFO) of the bearing outer ring can be detectedthe fault characteristics are more seriously polluted by noiseand there are many interference components It can be seenfrom Figures 30(b) and 31(b) that the fault feature extraction

2000 4000 6000 80000Frequency (Hz)

546

436

326

216

10

Leve

l k

0

02

04

06

08

1

12

(a)

000

005

010

015

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

0

2

4

6

8

Am

plitu

de (m

s2 )

100 200 300 400 500 600 7000Frequency (Hz)

times10ndash5

29Hz

3158Hz

(c)

Figure 23 Analysis results of experimental signals by the SK method (a) fast kurtogram (b) envelope signal (c) envelope spectrum


Accelerometers

Thermocouples

(a)

Motor

Bearing 1

Bearing 2 Bearing 4

Bearing 3

Accelerometers

Radial loadThermocouples

Motor

Bearing 1

Bearing 2 Bearing 4

Bearing 3

AccelerometersThermocouples

(b)

Figures 24 (a) -e whole life cycle test bench for rolling element bearings (b) Diagram of experimental device


Structural parameters Parameter valuesBearing type ZA-2115Inside diameter 4921mmPitch diameter 7150mmRoller diameter 841mmAxial load 267 kN-e number of roller 16Contact angle 1517deg

00

02

04

06

08

RMS

20 40 60 80 100 120 140 1600Time (hours)

02

01

0086 88 90 92 94 96

Zoom plot

Test ATest C

Test B

Stage 1

Stage 3

Stage 2

(a)

01 02 03 0400Time (s)

ndash06ndash03

0003

A06

Am

plitu

de

(b)

Figure 25 Continued


01 02 03 0400Time (s)

ndash10ndash05

000510 B

Am

plitu

de

(c)

01 02 03 0400Time (s)

ndash4ndash2

0

C24

Am

plitu

de

(d)

Figure 25-e test signal for experimental rolling bearing (a) RMS for the whole life cycle (b) early-stage vibration waveform of measuringpoint A (c) mid-stage vibration waveform of measuring point B (d) late-stage vibration waveform of measuring point C

14

15

16

17

18

Best

scor

e

8 104 62Iteration

(a)

01 02 03 0400Time (s)

ndash8

ndash4

0

4

8

Am

plitu

de (m

s2 )

(b)

Figure 27 -e experimental signal denoising results (a) fitness convergence curve for PPCA optimization (b) time-domain diagram afternoise reduction by APPCA

000

001

002

003

Am

plitu

de (m

s2 )

2000 4000 6000 8000 100000Frequency (Hz)

(a)

000

001

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

fFBPO

(b)

Figure 26 -e experimental signal (a) FFT spectrum (b) envelope spectrum

000

003

006

009

012

FEF

20 40 60 800Scale λ

The best scale

(a)

01 02 03 0400Time (s)

0

40

80

120

160

Am

plitu

de (m

s2 )

(b)

Figure 28 Continued


capabilities of the VMD method and the SK method are stillrelatively weak

-rough the comparison of the above experiments andengineering applications it can be clearly found that themethod proposed in this paper can effectively extract thefault characteristic information of rolling bearings -ismethod not only has good noise reduction performance butalso can play a role in enhancing the early weak faults andhas good filtering performance

6 Discussion

-e previous analysis results can qualitatively prove that themethod proposed is effective In order to further comparethe superiority of the APPCA-EMDF in this paper with theexisting decomposition methods (ACDIF VMD and SK)some quantitative indicators are applied such as kurtosisand FEF indicators -e larger the kurtosis and FEF indexthe stronger the feature extraction ability of the signal At the

0

4

8

12

16

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

fBPFO

2fBPFO3fBPFO

4fBPFO

(c)

Figure 28 Processing results of denoising signal by the EMDF method (a) correlation curve between FEF and scale λ (b) time-domainspectrum (c) envelope spectrum

01 02 03 0400Time (s)

00

02

04

06

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

000

001

002

003

Am

plitu

de (m

s2 )

fBPFO

2fBPFO

3fBPFO

(b)

Figure 29 Analysis results of experimental signals by the ACDIF method (a) time-domain spectrum (b) envelope spectrum

01 02 03 0400Time (s)

ndash01

00

01

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

000

001

002

Am

plitu

de (m

s2 )

fFBPO

2fFBPO

(b)

Figure 30 Analysis results of experimental signals by the VMDmethod with k 5 and α 2000 (a) time-domain spectrum of mode u4 (b)envelope spectrum of mode u4


same time the calculation efficiency of the above methods isalso compared -e test was performed on a computerconfigured with Inter (R) Core (TM) i7-6700HQ CPU26Hz and RAM 8GB-e comparison results are shown inTables 6ndash8 respectively

From the comparison results of Table 6ndashTable 8 it can beseen that although the SK method has the highest com-putational efficiency its fault feature extraction ability is theweakest -e APPCA-EMDF method in this paper obtainsthe largest kurtosis value and FEF value which shows that itsfault feature extraction ability is better than other decom-position methods (ACDIF VMD and SK) However themethod proposed in this article is less computationally ef-ficient -e reason for this phenomenon is that GOA is usedto optimize PPCA system parameters and a lot of calcu-lation costs are caused in the process of iterative optimi-zation -erefore we need to further improve thecomputational efficiency of the algorithm in the future

In future work we can try to use the multiscale mor-phological analysis method because morphological opera-tors can contain more abundant fault informationcomponents at full scale We can also try to combine thealgorithm proposed in this paper with the intelligent rec-ognition algorithm to achieve accurate classification ofbearing faults Finally we can try to find out how the noiseelimination performance of PPCA is under the condition ofother interference signals How to improve the calculation

efficiency of the algorithm proposed in this paper is also aproblem to be studied in the future

7 Conclusions

In this paper a new rolling element bearing fault methodnamed APPCA-EMDF is proposed By quantitatively ana-lyzing the fault extraction capability of the feature extrac-tion-type operators a morphological operator namedEMDO was proposed Compared with MG DIF MGPOMHPO and ACDIF operators the fault extraction ability ofEMDO is superior and it has an enhanced effect on faultcharacteristics Next in order to solve the interferenceproblem caused by strong background noise a parameteradaptive PPCA method based on the GOA method isproposed APPCA can make up for the deficiency of EMDFin denoising capability Simulation and engineering appli-cation results show that the method proposed in this paper iseffective in detecting early fault of rolling element bearingsCompared with ACDIF VMD and SK methods the pro-posed method has certain advantages

Nomenclature

PPCA Probabilistic principal component analysisEMDF Enhanced morphological difference filterEMDO Enhanced morphological difference operatorGOA Grasshopper optimization algorithmMF Morphological filterSE Structural elementMG Morphology gradientAVG Average of opening and closing operatorsCMF Average of the opening-closing and closing-

opening operatorsACDIF Average combination difference morphological

filterTEK Teager energy kurtosis

01 02 03 0400Time (s)

000

002

004

006

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

00

05

10

15

Am

plitu

de (m

s2 ) fBPFO

(b)

Figure 31 Analysis results of experimental signals by the SK method (a) envelope signal (b) envelope spectrum

Table 6-e comparison results of APPCA-EMDF ACDIF VMDand SK methods in simulation case 2

Method APPCA-EMDF ACDIF VMD SKCPU time (s) 4002 1681 3516 395Kurtosis 128540 49885 50105 45118FEF 03635 01026 02375 00074

Table 7-e comparison results of APPCA-EMDF ACDIF VMDand SK methods in experimental case 1





MGPO Morphology gradient product operationMHPO Morphology-hat product operationBTH Black top-hatPBTH Positive black top-hatWTH White top-hatNWTH Negative white top-hatGCOampOC Closing-opening and opening-closing gradient

operationFEF Feature energy factorPE Permutation entropyESS Envelope spectrum sparsity3thA Amplitudes at the 3rd harmonics of the bearing

fault frequencyVMD Variational mode decompositionSK Spectral kurtosis

Data Availability

-e experimental data were acquired from the PrognosticsCenter of Excellence (PCoE) through the prognostic datarepository donated by the Center of Intelligent MaintenanceSystems (IMS) University of Cincinnati (J Lee H Qiu GYu J Lin RT Services Bearing data set in UO Cincinnati(Ed) IMS University of Cincinnati NASA Ames Prog-nostics Data Repository 2007)

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is study was supported by the National Natural ScienceFoundation of China under grant nos 51675350 and51705337 Natural Science Foundation of Liaoning Provinceunder grant nos 2019MS245 and 20180551036 and the PhDStart-Up Foundation of Liaoning Province under grant no20170520111

References

[1] H Aijun L Jianfeng S Shangfei and X Ling ldquoA novelapproach of impulsive signal extraction for early fault de-tection of rolling element bearingrdquo Shock and Vibrationvol 2017 Article ID 9375491 11 pages 2017

[2] S Dong L Chen B Tang X Xu Z Gao and J Liu ldquoRotatingmachine fault diagnosis based on optimal morphological filterand local tangent space alignmentrdquo Shock and Vibrationvol 2015 Article ID 893504 9 pages 2015

[3] J Chen J Pan Z Li Y Zi and X Chen ldquoGenerator bearingfault diagnosis for wind turbine via empirical wavelettransform using measured vibration signalsrdquo RenewableEnergy vol 89 pp 80ndash92 2016

[4] J Dybała and R Zimroz ldquoRolling bearing diagnosing methodbased on Empirical Mode Decomposition of machine vi-bration signalrdquo Applied Acoustics vol 77 pp 195ndash203 2014

[5] J Antoni ldquoFast computation of the kurtogram for the de-tection of transient faultsrdquo Mechanical Systems and SignalProcessing vol 21 no 1 pp 108ndash124 2007

[6] K Dragomiretskiy and D Zosso ldquoVariational mode de-compositionrdquo IEEE Transactions on Signal Processing vol 62no 3 pp 531ndash544 2014

[7] J Serra ldquoMorphological filtering an overviewrdquo Signal Pro-cessing vol 38 no 1 pp 3ndash11 1994

[8] J Serra ldquoImage analysis and mathematical morphologyrdquoAcademic Press Cambridge MA USA 1982

[9] C Shen Q He F Kong and P W Tse ldquoA fast and adaptivevarying-scale morphological analysis method for rolling el-ement bearing fault diagnosisrdquo Proceedings of the Institutionof Mechanical Engineers Part C Journal of Mechanical En-gineering Science vol 227 no 6 pp 1362ndash1370 2013

[10] Y Li X Liang andM J Zuo ldquoA new strategy of using a time-varying structure element for mathematical morphologicalfilteringrdquo Measurement vol 106 pp 53ndash65 2017

[11] Q Chen Z Chen W Sun G Yang A Palazoglu and Z RenldquoA new structuring element for multi-scale morphologyanalysis and its application in rolling element bearing faultdiagnosisrdquo Journal of Vibration and Control vol 21 no 4pp 765ndash789 2013

[12] B Li P-L Zhang S-S Mi R-X Hu and D-S Liu ldquoAnadaptive morphological gradient lifting wavelet for detectingbearing defectsrdquo Mechanical Systems and Signal Processingvol 29 pp 415ndash427 2012

[13] J Lv and J Yu ldquoAverage combination difference morpho-logical filters for fault feature extraction of bearingrdquo Me-chanical Systems and Signal Processing vol 100 pp 827ndash8452018

[14] Z Dong X Sun F Xu and W Liu ldquoA low-rank and sparsedecomposition-based method of improving the accuracy ofsub-pixel grayscale centroid extraction for spot imagesrdquo IEEESensors Journal vol 20 no 11 pp 5845ndash5854 2020

[15] Y Li M J Zuo Y Chen and K Feng ldquoAn enhancedmorphology gradient product filter for bearing fault detec-tionrdquo Mechanical Systems and Signal Processing vol 109pp 166ndash184 2018

[16] Y Li X Liang and M J Zuo ldquoDiagonal slice spectrumassisted optimal scale morphological filter for rolling elementbearing fault diagnosisrdquo Mechanical Systems and SignalProcessing vol 85 pp 146ndash161 2017

[17] X Yan M Jia W Zhang and L Zhu ldquoFault diagnosis ofrolling element bearing using a new optimal scale morphologyanalysis methodrdquo ISA Transactions vol 73 pp 165ndash180 2018

[18] A S Raj and N Murali ldquoEarly classification of bearing faultsusing morphological operators and fuzzy inferencerdquo IEEETransactions on Industrial Electronics vol 60 no 2pp 567ndash574 2013

[19] B Li P-L Zhang Z-J Wang S-S Mi and D-S Liu ldquoAweighted multi-scale morphological gradient filter for rollingelement bearing fault detectionrdquo ISA Transactions vol 50no 4 pp 599ndash608 2011

[20] Z Hu C Wang J Zhu X Liu and F Kong ldquoBearing faultdiagnosis based on an improved morphological filterrdquoMeasurement vol 80 pp 163ndash178 2016

[21] Y Dong M Liao X Zhang and F Wang ldquoFaults diagnosis ofrolling element bearings based on modified morphologicalmethodrdquo Mechanical Systems and Signal Processing vol 25no 4 pp 1276ndash1286 2011

[22] A Hu and L Xiang ldquoSelection principle of mathematicalmorphological operators in vibration signal processingrdquoJournal of Vibration and Control vol 22 no 14 pp 3157ndash3168 2014

[23] J Wang G Xu Q Zhang and L Liang ldquoApplication ofimproved morphological filter to the extraction of impulsive


attenuation signalsrdquo Mechanical Systems and Signal Pro-cessing vol 23 no 1 pp 236ndash245 2009

[24] L Meng J Xiang Y Wang Y Jiang and H Gao ldquoA hybridfault diagnosis method using morphological filter-translationinvariant wavelet and improved ensemble empirical modedecompositionrdquo Mechanical Systems and Signal Processingvol 50-51 pp 101ndash115 2015

[25] Y Li G Li Y Yang X Liang and M Xu ldquoA fault diagnosisscheme for planetary gearboxes using adaptive multi-scalemorphology filter and modified hierarchical permutationentropyrdquo Mechanical Systems and Signal Processing vol 105pp 319ndash337 2018

[26] B Li P-L Zhang Z-J Wang S-S Mi and Y-T ZhangldquoGear fault detection using multi-scale morphological filtersrdquoMeasurement vol 44 no 10 pp 2078ndash2089 2011

[27] L Zhang J Xu J Yang D Yang and D Wang ldquoMultiscalemorphology analysis and its application to fault diagnosisrdquoMechanical Systems and Signal Processing vol 22 no 3pp 597ndash610 2008

[28] X Yan Y Liu and M Jia ldquoResearch on an enhanced scalemorphological-hat product filtering in incipient fault detec-tion of rolling element bearingsrdquo Measurement vol 147p 106856 2019

[29] A Bellas C Bouveyron M Cottrell and J Lacaille ldquoModel-based clustering of high-dimensional data streams with onlinemixture of probabilistic PCArdquo Advances in Data Analysis andClassification vol 7 no 3 pp 281ndash300 2013

[30] X Yuan L Ye L Bao Z Ge and Z Song ldquoNonlinear featureextraction for soft sensor modeling based on weightedprobabilistic PCArdquo Chemometrics and Intelligent LaboratorySystems vol 147 pp 167ndash175 2015

[31] D Kim and I-B Lee ldquoProcess monitoring based on proba-bilistic PCArdquo Chemometrics and Intelligent Laboratory Sys-tems vol 67 no 2 pp 109ndash123 2003

[32] P Zuccolotto ldquoPrincipal component analysis with intervalimputed missing valuesrdquo AStA Advances in StatisticalAnalysis vol 96 no 1 pp 1ndash23 2012

[33] B Jiang J Xiang and Y Wang ldquoRolling bearing fault di-agnosis approach using probabilistic principal componentanalysis denoising and cyclic bispectrumrdquo Journal of Vibra-tion and Control vol 22 no 10 pp 2420ndash2433 2014

[34] S Saremi S Mirjalili and A Lewis ldquoGrasshopper optimi-sation algorithm theory and applicationrdquo Advances in En-gineering Software vol 105 pp 30ndash47 2017

[35] Y Li M J Zuo J Lin and J Liu ldquoFault detection method forrailway wheel flat using an adaptive multiscale morphologicalfilterrdquo Mechanical Systems and Signal Processing vol 84pp 642ndash658 2017

[36] Y Miao M Zhao and J Lin ldquoIdentification of mechanicalcompound-fault based on the improved parameter-adaptivevariational mode decompositionrdquo ISA Transactions vol 84pp 82ndash95 2019

[37] Y Li X Liang J Lin Y Chen and J Liu ldquoTrain axle bearingfault detection using a feature selection scheme based multi-scale morphological filterrdquo Mechanical Systems and SignalProcessing vol 101 pp 435ndash448 2018

[38] X Zhao M J Zuo Z Liu and M R Hoseini ldquoDiagnosis ofartificially created surface damage levels of planet gear teethusing ordinal rankingrdquo Measurement vol 46 no 1pp 132ndash144 2013

[39] J Zheng Z Dong H Pan Q Ni T Liu and J ZhangldquoComposite multi-scale weighted permutation entropy andextreme learning machine based intelligent fault diagnosis forrolling bearingrdquo Measurement vol 143 pp 69ndash80 2019

[40] F Deng S Yang G Tang R Hao and M Zhang ldquoSelfadaptive multi-scale morphology AVG-Hat filter and itsapplication to fault feature extraction for wheel bearingrdquoMeasurement Science and Technology vol 28 no 4 Article ID045011 2017

[41] M I Jordan and R A Jacobs ldquoHierarchical mixtures ofexperts and the EM algorithmrdquo in Proceedings of the 1993International Conference on Neural Networks IEEE NagoyaJapan October 1993

[42] J Lee H Qiu G Yu J Lin and R T Services ldquoBearing datasetrdquoldquoBearing data setrdquo Edited by U O Cincinnati Ed IMSUniversity of Cincinnati NASA Ames Prognostics DataRepository Cincinnati OH USA 2007


introduced the AVG operator with the average of openingand closing operators Hu and Xiang [22] proposed the CMFoperator by using the average of the opening-closing andclosing-opening operators and demonstrated the denoisingperformance of CMF Wang et al [23] and Meng et al [24]used the CMF operator to detect bearing failure DIF iscomposed of the difference between the closing and openingoperators Li et al [25 26] and Zhang et al [27] successfullyapplied the DIF operator to the fault diagnosis of mechanicalsystems

Previous scholars used the morphological operator withsingle structure to deal with the fault of rolling bearings -eresults show that the morphological operator with a singlestructure can no longer meet the requirements when dealingwith signals with complex frequency components -ere-fore some scholars have studied that the new morphologicaloperator constructed by the product can not only enhancethe feature extraction ability but also weaken the interfer-ence of random noise Lv and Yu [13] used basic mor-phological operators to construct some combinationdifference operators and chose two optimal operators fromthem to construct the average combination difference op-erator (ACDIF) -en they used the TEK index factor toselect the optimal scale of the operator and finally applied theproposed method to fault diagnosis of rolling bearing Liet al [15] made a detailed qualitative analysis of somecommon basic morphological operators According to thepurpose of processing the signal these morphological op-erators are divided into two categories noise reduction-typeoperators and feature extraction-type operators and theyconstructed the MGPO operator based on the superiorperformance of the two morphological operators Based onLirsquos research Yan et al [28] derived a new operator namedMHPO and they successfully applied MHPO to the faultdiagnosis of rolling bearings

Unfortunately in the above studies few papers havequantitatively studied the filtering performance of thesemorphological operators In this paper a set of rollingbearing fault simulation signals and four sets of filteringperformance indicators are used to quantitatively analyzethe feature extraction capability of the feature extraction-type operator and rank their filtering performance -en anew enhancement operator named EMDO is constructed

Although EMDO can extract more impulse signals dueto the interference of background noise the filtering per-formance of EMDO will be affected In order to solve thisproblem the probabilistic principal component analysis(PPCA) [29ndash32] is introduced to make up for the deficiencyof EMDO in denoising capability PPCA first establishes anappropriate probability model for each variable and itdecomposes the principal components and noise of thesignal into orthogonal space by increasing the dimensionrealizes the decorrelation between principal componentsand noise and finally achieves the purpose of noise re-duction In the PPCA model two important system pa-rameters the number of decomposed principal componentsk and the original variable n have a great impact on themodelrsquos noise reduction results At present existing papers

have manually selected the system parameters of PPCAwhich greatly affects the noise reduction performance [33]

In order to obtain the optimal parameters of PPCA thispaper chooses grasshopper optimization algorithm (GOA)to optimize the system parameters of PPCA GOA is amathematical optimization algorithm proposed by Saremiet al [34] in 2017 which is used to simulate the behavior ofgrasshoppers in nature Compared with other optimizationmethods GOA has the characteristics of fast convergenceand difficulty in generating local optimal solutions [34]Finally this paper combines the advantages of the above twomethods to propose a new comprehensive fault diagnosismethod for rolling bearing fault diagnosis

-e remainder of this article is organized as followsSection 2 briefly introduces mathematical morphologicaltheory and proposes the EMDO operator Section 3 de-scribes the APPCAmethod for noise reduction as well as theconstruction process of the APPCA-EMDF algorithm Ex-perimental verification of the proposed method is illustratedin Section 4 Conclusions are drawn in Section 5

2 Basic Theory of Mathematical Morphology

21 Morphological Filter Assume that the original signal f(n) is a group of one-dimensional discrete array F (0 1 2 N minus 1) -e structure element g (m) is defined as anotherone-dimensional discrete array G (0 1 2 M minus 1) andsatisfiesNgeM-en the four basic morphological operatorscan be expressed as follows

Dilation

(foplusg)(n) max[f(n minus m) + g(m)] (1)

Erosion

(fΘg)(n) min[f(n + m) minus g(m)] (2)

Opening

(f ∘g)(n) (fΘgoplusg)(n) (3)

Closing

(f bull g)(n) (foplusgΘg)(n) (4)

where oplus Θ and represent the dilation erosionopening and closing operators respectively Besides severalmain morphological filters are introduced as follows

Average of closing and opening (AVG) [21]

AVG(f(n)) (f bull g)(n) +(f ∘g)(n)

2 (5)

Morphological gradient (MG) [18ndash20]

MG(f(n)) (foplusg)(n) minus (fΘg)(n) (6)

Difference filter (DIF) [25ndash27]

DIF(f(n)) (f bull g)(n) minus (f ∘g)(n) (7)

-e opening-closing FOC and closing-opening FCO filtersare defined as






2 (10)












y(t) y1(t) + y2(t) + y3(t) + δ(t) (16)








Mi1 Y2

i (f)1113936 Y(f)2)



(1n)1113936nminus 1i0 Y2

i (f)

1113969

(1n)1113936nminus 1i0 |Yi(f)|)



MGPBTHBTHWTH

DIFNWTHGco_oc


00

02

04

06

08

FEF

(a)

MGPBTHBTHWTH

DIFNWTHGco_oc


20

20

15

05

PE

(b)

MGPBTHBTHWTH

DIFNWTHGco_oc


2

4

6

8

10

ESS

(c)

MGPBTHBTHWTH

DIFNWTHGco_oc


00

02

04

06

08

3thA

(d)













(17)














ndash1000 02 04 06 08

Time (s)10

ndash5

0

5

10

Am

plitu

de

(a)

0 40 80 120 160 200Frequency (Hz)

00

02

04

06

Am

plitu

de 1048

(b)


00

02

04

06

FEF

MGPBTHBTHWTH

DIFNWTHGco_oc

2 4 6 8 10Scale

(a)

20

20

15

00

05

PE

MGPBTHBTHWTH

DIFNWTHGco_oc

2 4 6 8 10Scale

(b)

2

0

4

6

8

ESS

MGPBTHBTHWTH

DIFNWTHGco_oc

2 4 6 8 10Scale

(c)

00

02

04

06

08

3thA

MGPBTHBTHWTH

DIFNWTHGco_oc

2 4 6 8 10Scale

(d)



0 10 20 30Scale

40 50 60 7000

01

02

03

04

FEF

2 4 6 8 10

04

03

02

01

00

Zoom plot

MGMHPODIF

EMDOMGPO

EnvelopeACDIF


16

3248

64

00

04

08

12

Am

plitu

de

120 16040 80 2000Frequency (Hz)

(a)

16

3248 64

00

04

08

12A

mpl

itude

40 80 120 160 2000Frequency (Hz)

(b)

16

32 48

0

2

4

6

8

10

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

(c)

16

3248 64

0

1

2

3

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

(d)

16

4864

00

04

08

12

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

(e)

16

32 4864

0

4

8

12

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

(f )





p(u) (2π)minus k2







|C|minus 12


(21)




(22)


T1113874 1113875 (23)


(1m) 1113936mi1 xixT



ui pTi X (24)




KSP Kr times ESS

PE (25)






Si 1113944N

j1jneis dij1113872 11138731113954dij 1113944

N

j1jne i

s xi minus xj

11138681113868111386811138681113868

111386811138681113868111386811138681113874 1113875xi minus xj

dij

(27)

s(r) feminus rl

minus eminus r

(28)





Ai u1113954ew (30)



Xi 1113944N

j1jne i

s xj minus xi

11138681113868111386811138681113868

111386811138681113868111386811138681113874 1113875xj minus xi

dij

minus g1113954eg + u1113954ew (31)



Xdi c 1113944

N

j1jne i

cubd minus lbd

2s x

dj minus x

di

11138681113868111386811138681113868

111386811138681113868111386811138681113874 1113875xj minus xi

dij

⎛⎝ ⎞⎠ + 1113954Td

(32)



L (33)





fitness maxc(nk)

KSPi1113864 1113865

stn 1 2 20

k 1 2 3 20

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(34)




4 Simulation


KSP

4

6

8

10

12

14

20

15

10

5

n

10 15 205k







y1(t) 1113944I


y2(t) 1113944N


y3(t) 1113944M


y4(t) 1113944P

p1Cp cos 2πfp + φp1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(35)





l ge L

l = l + 1No




Yes

Step 1




FEF PE ESS 3thA



ranked


Step 2



Step 3








The best scale

00

02

04

06

FEF

10 20 30 40 50 60 700Scale λ

(a)

times106

0

1

2

3

4

Am

plitu

de

02 04 06 08 1000Time (s)

(b)

times104

16

32

4864

0

4

8

12

16

Am

plitu

de

40 80 120 160 1200Frequency (Hz)

(c)


12

13

14

15

16Be

st sc

ore

4 6 8 102Iteration

(a)

times102

ndash10

ndash5

0

5

10

Am

plitu

de

02 04 06 08 1000Time (s)

(b)














ndash4

0

4

Am

plitu

de

02 04 06 08 1000Time (s)

(a)

ndash6

ndash3

0

3

6

Am

plitu

de

02 04 06 08 1000Time (s)

(b)

ndash3

0

3

Am

plitu

de

02 04 06 08 1000Time (s)

(c)

ndash3

0

3

Am

plitu

de

02 04 06 08 1000Time (s)

(d)


ndash8

ndash4

0

4

8

Am

plitu

de

02 04 06 08 1000Time (s)

(a)

00

02

04

06

08

10

Am

plitu

de

20 40 60 80 100 120 140 160 180 2000Frequency (Hz)

10Hz

20Hz

50Hz

(b)


10

11

12

13

14

Best

scor

e

4 6 8 102Iteration

(a)

ndash2

0

2

Am

plitu

de

02 04 06 08 1000Time (s)

times103

(b)



05

10152025303540

TEK

10 20 30 40 50 600Scale λ

The best scale

(a)

5

10

15

20

25

Am

plitu

de

02 04 06 08 1000Time (s)

(b)

0

1

2

3

Am

plitu

de

10

16

40 80 120 160 2000Frequency (Hz)

(c)


10 20 30 40 50 60 700Scale λ

00

01

02

03

04

05

FEF

e best scale

(a)

0

2

4

6

8

Am

plitu

de

02 04 06 08 1000Time (s)

times106

(b)

00

05

10

15

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

16

32

48

times106

(c)



00

05

10

15

20A

mpl

itude

200 400 600 800 10000Frequency (Hz)

Selected

u1u2

u3u4

(a)

ndash5

0

5

Am

plitu

de

02 04 06 08 1000Time (s)

(b)

00

02

04

06

08

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

16

3248

(c)


200 400 600 800 10000Frequency (Hz)

3

26

2

16

1

0

Leve

l k

0

01

02

03

04

05

(a)

0

1

2

3

4

Am

plitu

de

0604 08 100200Time (s)

(b)

Figure 16 Continued


000

005

010

015

020

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

16

50

(c)


12

3 45 6

78



Gearbox Main sha

Blade

Monitoring system

(a)

(b)










2d)cos2φ)


ndash50

ndash25

0

25

50

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(a)

0

1

2

3

Am

plitu

de (m

s2 )

1000 1000 3000 4000 5000 6000 7000 80000Frequency (Hz)

02

01

00200 250 300 350 400

3156Hz

(b)


4 6 8 102Iteration

05

10

15

20

Best

scor

e

(a)

ndash15

00

15

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

times107

(b)

Figure 19 Continued




5 10 15 20 25 30 35 40 45 500Scale λ

000

002

004

006

008

010

FEF

e best scale

(a)

00

05

10

15

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

times 1016

(b)

00

05

10

15

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

times 1010


630Hz (2lowastfo)

9469Hz (3lowastfo)

(c)


ndash08

00

08

Am

plitu

de (m

s2 )

0655 0660 0665 0670 0675 06800650Time (s)

32ms

times107

(c)



e best scale

10

15

20

25

30TE

K

1510 20 2550Scale λ

(a)

0

20

40

60

80

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

0

2

4

6

8

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

29Hz

3158Hz

(c)


00

05

10

15

20

Am

plitu

de (m

s2 )

2000 4000 6000 8000 100000Frequency (Hz)

Selected

u1u2u3

u4u5

(a)

ndash30

ndash15

0

15

30

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

00

05

10

15

20

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

29Hz

3158Hz

630Hz

(c)








2000 4000 6000 80000Frequency (Hz)

546

436

326

216

10

Leve

l k

0

02

04

06

08

1

12

(a)

000

005

010

015

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

0

2

4

6

8

Am

plitu

de (m

s2 )

100 200 300 400 500 600 7000Frequency (Hz)

times10ndash5

29Hz

3158Hz

(c)



Accelerometers

Thermocouples

(a)

Motor

Bearing 1

Bearing 2 Bearing 4

Bearing 3

Accelerometers


Motor

Bearing 1

Bearing 2 Bearing 4

Bearing 3


(b)




00

02

04

06

08

RMS

20 40 60 80 100 120 140 1600Time (hours)

02

01

0086 88 90 92 94 96

Zoom plot

Test ATest C

Test B

Stage 1

Stage 3

Stage 2

(a)

01 02 03 0400Time (s)

ndash06ndash03

0003

A06

Am

plitu

de

(b)

Figure 25 Continued


01 02 03 0400Time (s)

ndash10ndash05

000510 B

Am

plitu

de

(c)

01 02 03 0400Time (s)

ndash4ndash2

0

C24

Am

plitu

de

(d)


14

15

16

17

18

Best

scor

e

8 104 62Iteration

(a)

01 02 03 0400Time (s)

ndash8

ndash4

0

4

8

Am

plitu

de (m

s2 )

(b)


000

001

002

003

Am

plitu

de (m

s2 )

2000 4000 6000 8000 100000Frequency (Hz)

(a)

000

001

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

fFBPO

(b)


000

003

006

009

012

FEF

20 40 60 800Scale λ

The best scale

(a)

01 02 03 0400Time (s)

0

40

80

120

160

Am

plitu

de (m

s2 )

(b)

Figure 28 Continued




6 Discussion


0

4

8

12

16

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

fBPFO

2fBPFO3fBPFO

4fBPFO

(c)


01 02 03 0400Time (s)

00

02

04

06

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

000

001

002

003

Am

plitu

de (m

s2 )

fBPFO

2fBPFO

3fBPFO

(b)


01 02 03 0400Time (s)

ndash01

00

01

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

000

001

002

Am

plitu

de (m

s2 )

fFBPO

2fFBPO

(b)







7 Conclusions


Nomenclature




01 02 03 0400Time (s)

000

002

004

006

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

00

05

10

15

Am

plitu

de (m

s2 ) fBPFO

(b)












Data Availability




Acknowledgments


References


















































2 (10)












y(t) y1(t) + y2(t) + y3(t) + δ(t) (16)








Mi1 Y2

i (f)1113936 Y(f)2)



(1n)1113936nminus 1i0 Y2

i (f)

1113969

(1n)1113936nminus 1i0 |Yi(f)|)



MGPBTHBTHWTH

DIFNWTHGco_oc


00

02

04

06

08

FEF

(a)

MGPBTHBTHWTH

DIFNWTHGco_oc


20

20

15

05

PE

(b)

MGPBTHBTHWTH

DIFNWTHGco_oc


2

4

6

8

10

ESS

(c)

MGPBTHBTHWTH

DIFNWTHGco_oc


00

02

04

06

08

3thA

(d)













(17)














ndash1000 02 04 06 08

Time (s)10

ndash5

0

5

10

Am

plitu

de

(a)

0 40 80 120 160 200Frequency (Hz)

00

02

04

06

Am

plitu

de 1048

(b)


00

02

04

06

FEF

MGPBTHBTHWTH

DIFNWTHGco_oc

2 4 6 8 10Scale

(a)

20

20

15

00

05

PE

MGPBTHBTHWTH

DIFNWTHGco_oc

2 4 6 8 10Scale

(b)

2

0

4

6

8

ESS

MGPBTHBTHWTH

DIFNWTHGco_oc

2 4 6 8 10Scale

(c)

00

02

04

06

08

3thA

MGPBTHBTHWTH

DIFNWTHGco_oc

2 4 6 8 10Scale

(d)



0 10 20 30Scale

40 50 60 7000

01

02

03

04

FEF

2 4 6 8 10

04

03

02

01

00

Zoom plot

MGMHPODIF

EMDOMGPO

EnvelopeACDIF


16

3248

64

00

04

08

12

Am

plitu

de

120 16040 80 2000Frequency (Hz)

(a)

16

3248 64

00

04

08

12A

mpl

itude

40 80 120 160 2000Frequency (Hz)

(b)

16

32 48

0

2

4

6

8

10

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

(c)

16

3248 64

0

1

2

3

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

(d)

16

4864

00

04

08

12

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

(e)

16

32 4864

0

4

8

12

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

(f )





p(u) (2π)minus k2







|C|minus 12


(21)




(22)


T1113874 1113875 (23)


(1m) 1113936mi1 xixT



ui pTi X (24)




KSP Kr times ESS

PE (25)






Si 1113944N

j1jneis dij1113872 11138731113954dij 1113944

N

j1jne i

s xi minus xj

11138681113868111386811138681113868

111386811138681113868111386811138681113874 1113875xi minus xj

dij

(27)

s(r) feminus rl

minus eminus r

(28)





Ai u1113954ew (30)



Xi 1113944N

j1jne i

s xj minus xi

11138681113868111386811138681113868

111386811138681113868111386811138681113874 1113875xj minus xi

dij

minus g1113954eg + u1113954ew (31)



Xdi c 1113944

N

j1jne i

cubd minus lbd

2s x

dj minus x

di

11138681113868111386811138681113868

111386811138681113868111386811138681113874 1113875xj minus xi

dij

⎛⎝ ⎞⎠ + 1113954Td

(32)



L (33)





fitness maxc(nk)

KSPi1113864 1113865

stn 1 2 20

k 1 2 3 20

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(34)




4 Simulation


KSP

4

6

8

10

12

14

20

15

10

5

n

10 15 205k







y1(t) 1113944I


y2(t) 1113944N


y3(t) 1113944M


y4(t) 1113944P

p1Cp cos 2πfp + φp1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(35)





l ge L

l = l + 1No




Yes

Step 1




FEF PE ESS 3thA



ranked


Step 2



Step 3








The best scale

00

02

04

06

FEF

10 20 30 40 50 60 700Scale λ

(a)

times106

0

1

2

3

4

Am

plitu

de

02 04 06 08 1000Time (s)

(b)

times104

16

32

4864

0

4

8

12

16

Am

plitu

de

40 80 120 160 1200Frequency (Hz)

(c)


12

13

14

15

16Be

st sc

ore

4 6 8 102Iteration

(a)

times102

ndash10

ndash5

0

5

10

Am

plitu

de

02 04 06 08 1000Time (s)

(b)














ndash4

0

4

Am

plitu

de

02 04 06 08 1000Time (s)

(a)

ndash6

ndash3

0

3

6

Am

plitu

de

02 04 06 08 1000Time (s)

(b)

ndash3

0

3

Am

plitu

de

02 04 06 08 1000Time (s)

(c)

ndash3

0

3

Am

plitu

de

02 04 06 08 1000Time (s)

(d)


ndash8

ndash4

0

4

8

Am

plitu

de

02 04 06 08 1000Time (s)

(a)

00

02

04

06

08

10

Am

plitu

de

20 40 60 80 100 120 140 160 180 2000Frequency (Hz)

10Hz

20Hz

50Hz

(b)


10

11

12

13

14

Best

scor

e

4 6 8 102Iteration

(a)

ndash2

0

2

Am

plitu

de

02 04 06 08 1000Time (s)

times103

(b)



05

10152025303540

TEK

10 20 30 40 50 600Scale λ

The best scale

(a)

5

10

15

20

25

Am

plitu

de

02 04 06 08 1000Time (s)

(b)

0

1

2

3

Am

plitu

de

10

16

40 80 120 160 2000Frequency (Hz)

(c)


10 20 30 40 50 60 700Scale λ

00

01

02

03

04

05

FEF

e best scale

(a)

0

2

4

6

8

Am

plitu

de

02 04 06 08 1000Time (s)

times106

(b)

00

05

10

15

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

16

32

48

times106

(c)



00

05

10

15

20A

mpl

itude

200 400 600 800 10000Frequency (Hz)

Selected

u1u2

u3u4

(a)

ndash5

0

5

Am

plitu

de

02 04 06 08 1000Time (s)

(b)

00

02

04

06

08

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

16

3248

(c)


200 400 600 800 10000Frequency (Hz)

3

26

2

16

1

0

Leve

l k

0

01

02

03

04

05

(a)

0

1

2

3

4

Am

plitu

de

0604 08 100200Time (s)

(b)

Figure 16 Continued


000

005

010

015

020

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

16

50

(c)


12

3 45 6

78



Gearbox Main sha

Blade

Monitoring system

(a)

(b)










2d)cos2φ)


ndash50

ndash25

0

25

50

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(a)

0

1

2

3

Am

plitu

de (m

s2 )

1000 1000 3000 4000 5000 6000 7000 80000Frequency (Hz)

02

01

00200 250 300 350 400

3156Hz

(b)


4 6 8 102Iteration

05

10

15

20

Best

scor

e

(a)

ndash15

00

15

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

times107

(b)

Figure 19 Continued




5 10 15 20 25 30 35 40 45 500Scale λ

000

002

004

006

008

010

FEF

e best scale

(a)

00

05

10

15

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

times 1016

(b)

00

05

10

15

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

times 1010


630Hz (2lowastfo)

9469Hz (3lowastfo)

(c)


ndash08

00

08

Am

plitu

de (m

s2 )

0655 0660 0665 0670 0675 06800650Time (s)

32ms

times107

(c)



e best scale

10

15

20

25

30TE

K

1510 20 2550Scale λ

(a)

0

20

40

60

80

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

0

2

4

6

8

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

29Hz

3158Hz

(c)


00

05

10

15

20

Am

plitu

de (m

s2 )

2000 4000 6000 8000 100000Frequency (Hz)

Selected

u1u2u3

u4u5

(a)

ndash30

ndash15

0

15

30

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

00

05

10

15

20

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

29Hz

3158Hz

630Hz

(c)








2000 4000 6000 80000Frequency (Hz)

546

436

326

216

10

Leve

l k

0

02

04

06

08

1

12

(a)

000

005

010

015

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

0

2

4

6

8

Am

plitu

de (m

s2 )

100 200 300 400 500 600 7000Frequency (Hz)

times10ndash5

29Hz

3158Hz

(c)



Accelerometers

Thermocouples

(a)

Motor

Bearing 1

Bearing 2 Bearing 4

Bearing 3

Accelerometers


Motor

Bearing 1

Bearing 2 Bearing 4

Bearing 3


(b)




00

02

04

06

08

RMS

20 40 60 80 100 120 140 1600Time (hours)

02

01

0086 88 90 92 94 96

Zoom plot

Test ATest C

Test B

Stage 1

Stage 3

Stage 2

(a)

01 02 03 0400Time (s)

ndash06ndash03

0003

A06

Am

plitu

de

(b)

Figure 25 Continued


01 02 03 0400Time (s)

ndash10ndash05

000510 B

Am

plitu

de

(c)

01 02 03 0400Time (s)

ndash4ndash2

0

C24

Am

plitu

de

(d)


14

15

16

17

18

Best

scor

e

8 104 62Iteration

(a)

01 02 03 0400Time (s)

ndash8

ndash4

0

4

8

Am

plitu

de (m

s2 )

(b)


000

001

002

003

Am

plitu

de (m

s2 )

2000 4000 6000 8000 100000Frequency (Hz)

(a)

000

001

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

fFBPO

(b)


000

003

006

009

012

FEF

20 40 60 800Scale λ

The best scale

(a)

01 02 03 0400Time (s)

0

40

80

120

160

Am

plitu

de (m

s2 )

(b)

Figure 28 Continued




6 Discussion


0

4

8

12

16

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

fBPFO

2fBPFO3fBPFO

4fBPFO

(c)


01 02 03 0400Time (s)

00

02

04

06

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

000

001

002

003

Am

plitu

de (m

s2 )

fBPFO

2fBPFO

3fBPFO

(b)


01 02 03 0400Time (s)

ndash01

00

01

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

000

001

002

Am

plitu

de (m

s2 )

fFBPO

2fFBPO

(b)







7 Conclusions


Nomenclature




01 02 03 0400Time (s)

000

002

004

006

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

00

05

10

15

Am

plitu

de (m

s2 ) fBPFO

(b)












Data Availability




Acknowledgments


References
















































Mi1 Y2

i (f)1113936 Y(f)2)



(1n)1113936nminus 1i0 Y2

i (f)

1113969

(1n)1113936nminus 1i0 |Yi(f)|)



MGPBTHBTHWTH

DIFNWTHGco_oc


00

02

04

06

08

FEF

(a)

MGPBTHBTHWTH

DIFNWTHGco_oc


20

20

15

05

PE

(b)

MGPBTHBTHWTH

DIFNWTHGco_oc


2

4

6

8

10

ESS

(c)

MGPBTHBTHWTH

DIFNWTHGco_oc


00

02

04

06

08

3thA

(d)













(17)














ndash1000 02 04 06 08

Time (s)10

ndash5

0

5

10

Am

plitu

de

(a)

0 40 80 120 160 200Frequency (Hz)

00

02

04

06

Am

plitu

de 1048

(b)


00

02

04

06

FEF

MGPBTHBTHWTH

DIFNWTHGco_oc

2 4 6 8 10Scale

(a)

20

20

15

00

05

PE

MGPBTHBTHWTH

DIFNWTHGco_oc

2 4 6 8 10Scale

(b)

2

0

4

6

8

ESS

MGPBTHBTHWTH

DIFNWTHGco_oc

2 4 6 8 10Scale

(c)

00

02

04

06

08

3thA

MGPBTHBTHWTH

DIFNWTHGco_oc

2 4 6 8 10Scale

(d)



0 10 20 30Scale

40 50 60 7000

01

02

03

04

FEF

2 4 6 8 10

04

03

02

01

00

Zoom plot

MGMHPODIF

EMDOMGPO

EnvelopeACDIF


16

3248

64

00

04

08

12

Am

plitu

de

120 16040 80 2000Frequency (Hz)

(a)

16

3248 64

00

04

08

12A

mpl

itude

40 80 120 160 2000Frequency (Hz)

(b)

16

32 48

0

2

4

6

8

10

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

(c)

16

3248 64

0

1

2

3

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

(d)

16

4864

00

04

08

12

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

(e)

16

32 4864

0

4

8

12

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

(f )





p(u) (2π)minus k2







|C|minus 12


(21)




(22)


T1113874 1113875 (23)


(1m) 1113936mi1 xixT



ui pTi X (24)




KSP Kr times ESS

PE (25)






Si 1113944N

j1jneis dij1113872 11138731113954dij 1113944

N

j1jne i

s xi minus xj

11138681113868111386811138681113868

111386811138681113868111386811138681113874 1113875xi minus xj

dij

(27)

s(r) feminus rl

minus eminus r

(28)





Ai u1113954ew (30)



Xi 1113944N

j1jne i

s xj minus xi

11138681113868111386811138681113868

111386811138681113868111386811138681113874 1113875xj minus xi

dij

minus g1113954eg + u1113954ew (31)



Xdi c 1113944

N

j1jne i

cubd minus lbd

2s x

dj minus x

di

11138681113868111386811138681113868

111386811138681113868111386811138681113874 1113875xj minus xi

dij

⎛⎝ ⎞⎠ + 1113954Td

(32)



L (33)





fitness maxc(nk)

KSPi1113864 1113865

stn 1 2 20

k 1 2 3 20

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(34)




4 Simulation


KSP

4

6

8

10

12

14

20

15

10

5

n

10 15 205k







y1(t) 1113944I


y2(t) 1113944N


y3(t) 1113944M


y4(t) 1113944P

p1Cp cos 2πfp + φp1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(35)





l ge L

l = l + 1No




Yes

Step 1




FEF PE ESS 3thA



ranked


Step 2



Step 3








The best scale

00

02

04

06

FEF

10 20 30 40 50 60 700Scale λ

(a)

times106

0

1

2

3

4

Am

plitu

de

02 04 06 08 1000Time (s)

(b)

times104

16

32

4864

0

4

8

12

16

Am

plitu

de

40 80 120 160 1200Frequency (Hz)

(c)


12

13

14

15

16Be

st sc

ore

4 6 8 102Iteration

(a)

times102

ndash10

ndash5

0

5

10

Am

plitu

de

02 04 06 08 1000Time (s)

(b)














ndash4

0

4

Am

plitu

de

02 04 06 08 1000Time (s)

(a)

ndash6

ndash3

0

3

6

Am

plitu

de

02 04 06 08 1000Time (s)

(b)

ndash3

0

3

Am

plitu

de

02 04 06 08 1000Time (s)

(c)

ndash3

0

3

Am

plitu

de

02 04 06 08 1000Time (s)

(d)


ndash8

ndash4

0

4

8

Am

plitu

de

02 04 06 08 1000Time (s)

(a)

00

02

04

06

08

10

Am

plitu

de

20 40 60 80 100 120 140 160 180 2000Frequency (Hz)

10Hz

20Hz

50Hz

(b)


10

11

12

13

14

Best

scor

e

4 6 8 102Iteration

(a)

ndash2

0

2

Am

plitu

de

02 04 06 08 1000Time (s)

times103

(b)



05

10152025303540

TEK

10 20 30 40 50 600Scale λ

The best scale

(a)

5

10

15

20

25

Am

plitu

de

02 04 06 08 1000Time (s)

(b)

0

1

2

3

Am

plitu

de

10

16

40 80 120 160 2000Frequency (Hz)

(c)


10 20 30 40 50 60 700Scale λ

00

01

02

03

04

05

FEF

e best scale

(a)

0

2

4

6

8

Am

plitu

de

02 04 06 08 1000Time (s)

times106

(b)

00

05

10

15

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

16

32

48

times106

(c)



00

05

10

15

20A

mpl

itude

200 400 600 800 10000Frequency (Hz)

Selected

u1u2

u3u4

(a)

ndash5

0

5

Am

plitu

de

02 04 06 08 1000Time (s)

(b)

00

02

04

06

08

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

16

3248

(c)


200 400 600 800 10000Frequency (Hz)

3

26

2

16

1

0

Leve

l k

0

01

02

03

04

05

(a)

0

1

2

3

4

Am

plitu

de

0604 08 100200Time (s)

(b)

Figure 16 Continued


000

005

010

015

020

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

16

50

(c)


12

3 45 6

78



Gearbox Main sha

Blade

Monitoring system

(a)

(b)










2d)cos2φ)


ndash50

ndash25

0

25

50

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(a)

0

1

2

3

Am

plitu

de (m

s2 )

1000 1000 3000 4000 5000 6000 7000 80000Frequency (Hz)

02

01

00200 250 300 350 400

3156Hz

(b)


4 6 8 102Iteration

05

10

15

20

Best

scor

e

(a)

ndash15

00

15

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

times107

(b)

Figure 19 Continued




5 10 15 20 25 30 35 40 45 500Scale λ

000

002

004

006

008

010

FEF

e best scale

(a)

00

05

10

15

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

times 1016

(b)

00

05

10

15

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

times 1010


630Hz (2lowastfo)

9469Hz (3lowastfo)

(c)


ndash08

00

08

Am

plitu

de (m

s2 )

0655 0660 0665 0670 0675 06800650Time (s)

32ms

times107

(c)



e best scale

10

15

20

25

30TE

K

1510 20 2550Scale λ

(a)

0

20

40

60

80

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

0

2

4

6

8

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

29Hz

3158Hz

(c)


00

05

10

15

20

Am

plitu

de (m

s2 )

2000 4000 6000 8000 100000Frequency (Hz)

Selected

u1u2u3

u4u5

(a)

ndash30

ndash15

0

15

30

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

00

05

10

15

20

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

29Hz

3158Hz

630Hz

(c)








2000 4000 6000 80000Frequency (Hz)

546

436

326

216

10

Leve

l k

0

02

04

06

08

1

12

(a)

000

005

010

015

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

0

2

4

6

8

Am

plitu

de (m

s2 )

100 200 300 400 500 600 7000Frequency (Hz)

times10ndash5

29Hz

3158Hz

(c)



Accelerometers

Thermocouples

(a)

Motor

Bearing 1

Bearing 2 Bearing 4

Bearing 3

Accelerometers


Motor

Bearing 1

Bearing 2 Bearing 4

Bearing 3


(b)




00

02

04

06

08

RMS

20 40 60 80 100 120 140 1600Time (hours)

02

01

0086 88 90 92 94 96

Zoom plot

Test ATest C

Test B

Stage 1

Stage 3

Stage 2

(a)

01 02 03 0400Time (s)

ndash06ndash03

0003

A06

Am

plitu

de

(b)

Figure 25 Continued


01 02 03 0400Time (s)

ndash10ndash05

000510 B

Am

plitu

de

(c)

01 02 03 0400Time (s)

ndash4ndash2

0

C24

Am

plitu

de

(d)


14

15

16

17

18

Best

scor

e

8 104 62Iteration

(a)

01 02 03 0400Time (s)

ndash8

ndash4

0

4

8

Am

plitu

de (m

s2 )

(b)


000

001

002

003

Am

plitu

de (m

s2 )

2000 4000 6000 8000 100000Frequency (Hz)

(a)

000

001

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

fFBPO

(b)


000

003

006

009

012

FEF

20 40 60 800Scale λ

The best scale

(a)

01 02 03 0400Time (s)

0

40

80

120

160

Am

plitu

de (m

s2 )

(b)

Figure 28 Continued




6 Discussion


0

4

8

12

16

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

fBPFO

2fBPFO3fBPFO

4fBPFO

(c)


01 02 03 0400Time (s)

00

02

04

06

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

000

001

002

003

Am

plitu

de (m

s2 )

fBPFO

2fBPFO

3fBPFO

(b)


01 02 03 0400Time (s)

ndash01

00

01

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

000

001

002

Am

plitu

de (m

s2 )

fFBPO

2fFBPO

(b)







7 Conclusions


Nomenclature




01 02 03 0400Time (s)

000

002

004

006

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

00

05

10

15

Am

plitu

de (m

s2 ) fBPFO

(b)












Data Availability




Acknowledgments


References






















































(17)














ndash1000 02 04 06 08

Time (s)10

ndash5

0

5

10

Am

plitu

de

(a)

0 40 80 120 160 200Frequency (Hz)

00

02

04

06

Am

plitu

de 1048

(b)


00

02

04

06

FEF

MGPBTHBTHWTH

DIFNWTHGco_oc

2 4 6 8 10Scale

(a)

20

20

15

00

05

PE

MGPBTHBTHWTH

DIFNWTHGco_oc

2 4 6 8 10Scale

(b)

2

0

4

6

8

ESS

MGPBTHBTHWTH

DIFNWTHGco_oc

2 4 6 8 10Scale

(c)

00

02

04

06

08

3thA

MGPBTHBTHWTH

DIFNWTHGco_oc

2 4 6 8 10Scale

(d)



0 10 20 30Scale

40 50 60 7000

01

02

03

04

FEF

2 4 6 8 10

04

03

02

01

00

Zoom plot

MGMHPODIF

EMDOMGPO

EnvelopeACDIF


16

3248

64

00

04

08

12

Am

plitu

de

120 16040 80 2000Frequency (Hz)

(a)

16

3248 64

00

04

08

12A

mpl

itude

40 80 120 160 2000Frequency (Hz)

(b)

16

32 48

0

2

4

6

8

10

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

(c)

16

3248 64

0

1

2

3

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

(d)

16

4864

00

04

08

12

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

(e)

16

32 4864

0

4

8

12

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

(f )





p(u) (2π)minus k2







|C|minus 12


(21)




(22)


T1113874 1113875 (23)


(1m) 1113936mi1 xixT



ui pTi X (24)




KSP Kr times ESS

PE (25)






Si 1113944N

j1jneis dij1113872 11138731113954dij 1113944

N

j1jne i

s xi minus xj

11138681113868111386811138681113868

111386811138681113868111386811138681113874 1113875xi minus xj

dij

(27)

s(r) feminus rl

minus eminus r

(28)





Ai u1113954ew (30)



Xi 1113944N

j1jne i

s xj minus xi

11138681113868111386811138681113868

111386811138681113868111386811138681113874 1113875xj minus xi

dij

minus g1113954eg + u1113954ew (31)



Xdi c 1113944

N

j1jne i

cubd minus lbd

2s x

dj minus x

di

11138681113868111386811138681113868

111386811138681113868111386811138681113874 1113875xj minus xi

dij

⎛⎝ ⎞⎠ + 1113954Td

(32)



L (33)





fitness maxc(nk)

KSPi1113864 1113865

stn 1 2 20

k 1 2 3 20

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(34)




4 Simulation


KSP

4

6

8

10

12

14

20

15

10

5

n

10 15 205k







y1(t) 1113944I


y2(t) 1113944N


y3(t) 1113944M


y4(t) 1113944P

p1Cp cos 2πfp + φp1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(35)





l ge L

l = l + 1No




Yes

Step 1




FEF PE ESS 3thA



ranked


Step 2



Step 3








The best scale

00

02

04

06

FEF

10 20 30 40 50 60 700Scale λ

(a)

times106

0

1

2

3

4

Am

plitu

de

02 04 06 08 1000Time (s)

(b)

times104

16

32

4864

0

4

8

12

16

Am

plitu

de

40 80 120 160 1200Frequency (Hz)

(c)


12

13

14

15

16Be

st sc

ore

4 6 8 102Iteration

(a)

times102

ndash10

ndash5

0

5

10

Am

plitu

de

02 04 06 08 1000Time (s)

(b)














ndash4

0

4

Am

plitu

de

02 04 06 08 1000Time (s)

(a)

ndash6

ndash3

0

3

6

Am

plitu

de

02 04 06 08 1000Time (s)

(b)

ndash3

0

3

Am

plitu

de

02 04 06 08 1000Time (s)

(c)

ndash3

0

3

Am

plitu

de

02 04 06 08 1000Time (s)

(d)


ndash8

ndash4

0

4

8

Am

plitu

de

02 04 06 08 1000Time (s)

(a)

00

02

04

06

08

10

Am

plitu

de

20 40 60 80 100 120 140 160 180 2000Frequency (Hz)

10Hz

20Hz

50Hz

(b)


10

11

12

13

14

Best

scor

e

4 6 8 102Iteration

(a)

ndash2

0

2

Am

plitu

de

02 04 06 08 1000Time (s)

times103

(b)



05

10152025303540

TEK

10 20 30 40 50 600Scale λ

The best scale

(a)

5

10

15

20

25

Am

plitu

de

02 04 06 08 1000Time (s)

(b)

0

1

2

3

Am

plitu

de

10

16

40 80 120 160 2000Frequency (Hz)

(c)


10 20 30 40 50 60 700Scale λ

00

01

02

03

04

05

FEF

e best scale

(a)

0

2

4

6

8

Am

plitu

de

02 04 06 08 1000Time (s)

times106

(b)

00

05

10

15

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

16

32

48

times106

(c)



00

05

10

15

20A

mpl

itude

200 400 600 800 10000Frequency (Hz)

Selected

u1u2

u3u4

(a)

ndash5

0

5

Am

plitu

de

02 04 06 08 1000Time (s)

(b)

00

02

04

06

08

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

16

3248

(c)


200 400 600 800 10000Frequency (Hz)

3

26

2

16

1

0

Leve

l k

0

01

02

03

04

05

(a)

0

1

2

3

4

Am

plitu

de

0604 08 100200Time (s)

(b)

Figure 16 Continued


000

005

010

015

020

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

16

50

(c)


12

3 45 6

78



Gearbox Main sha

Blade

Monitoring system

(a)

(b)










2d)cos2φ)


ndash50

ndash25

0

25

50

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(a)

0

1

2

3

Am

plitu

de (m

s2 )

1000 1000 3000 4000 5000 6000 7000 80000Frequency (Hz)

02

01

00200 250 300 350 400

3156Hz

(b)


4 6 8 102Iteration

05

10

15

20

Best

scor

e

(a)

ndash15

00

15

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

times107

(b)

Figure 19 Continued




5 10 15 20 25 30 35 40 45 500Scale λ

000

002

004

006

008

010

FEF

e best scale

(a)

00

05

10

15

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

times 1016

(b)

00

05

10

15

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

times 1010


630Hz (2lowastfo)

9469Hz (3lowastfo)

(c)


ndash08

00

08

Am

plitu

de (m

s2 )

0655 0660 0665 0670 0675 06800650Time (s)

32ms

times107

(c)



e best scale

10

15

20

25

30TE

K

1510 20 2550Scale λ

(a)

0

20

40

60

80

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

0

2

4

6

8

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

29Hz

3158Hz

(c)


00

05

10

15

20

Am

plitu

de (m

s2 )

2000 4000 6000 8000 100000Frequency (Hz)

Selected

u1u2u3

u4u5

(a)

ndash30

ndash15

0

15

30

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

00

05

10

15

20

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

29Hz

3158Hz

630Hz

(c)








2000 4000 6000 80000Frequency (Hz)

546

436

326

216

10

Leve

l k

0

02

04

06

08

1

12

(a)

000

005

010

015

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

0

2

4

6

8

Am

plitu

de (m

s2 )

100 200 300 400 500 600 7000Frequency (Hz)

times10ndash5

29Hz

3158Hz

(c)



Accelerometers

Thermocouples

(a)

Motor

Bearing 1

Bearing 2 Bearing 4

Bearing 3

Accelerometers


Motor

Bearing 1

Bearing 2 Bearing 4

Bearing 3


(b)




00

02

04

06

08

RMS

20 40 60 80 100 120 140 1600Time (hours)

02

01

0086 88 90 92 94 96

Zoom plot

Test ATest C

Test B

Stage 1

Stage 3

Stage 2

(a)

01 02 03 0400Time (s)

ndash06ndash03

0003

A06

Am

plitu

de

(b)

Figure 25 Continued


01 02 03 0400Time (s)

ndash10ndash05

000510 B

Am

plitu

de

(c)

01 02 03 0400Time (s)

ndash4ndash2

0

C24

Am

plitu

de

(d)


14

15

16

17

18

Best

scor

e

8 104 62Iteration

(a)

01 02 03 0400Time (s)

ndash8

ndash4

0

4

8

Am

plitu

de (m

s2 )

(b)


000

001

002

003

Am

plitu

de (m

s2 )

2000 4000 6000 8000 100000Frequency (Hz)

(a)

000

001

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

fFBPO

(b)


000

003

006

009

012

FEF

20 40 60 800Scale λ

The best scale

(a)

01 02 03 0400Time (s)

0

40

80

120

160

Am

plitu

de (m

s2 )

(b)

Figure 28 Continued




6 Discussion


0

4

8

12

16

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

fBPFO

2fBPFO3fBPFO

4fBPFO

(c)


01 02 03 0400Time (s)

00

02

04

06

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

000

001

002

003

Am

plitu

de (m

s2 )

fBPFO

2fBPFO

3fBPFO

(b)


01 02 03 0400Time (s)

ndash01

00

01

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

000

001

002

Am

plitu

de (m

s2 )

fFBPO

2fFBPO

(b)







7 Conclusions


Nomenclature




01 02 03 0400Time (s)

000

002

004

006

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

00

05

10

15

Am

plitu

de (m

s2 ) fBPFO

(b)












Data Availability




Acknowledgments


References

















































ndash1000 02 04 06 08

Time (s)10

ndash5

0

5

10

Am

plitu

de

(a)

0 40 80 120 160 200Frequency (Hz)

00

02

04

06

Am

plitu

de 1048

(b)


00

02

04

06

FEF

MGPBTHBTHWTH

DIFNWTHGco_oc

2 4 6 8 10Scale

(a)

20

20

15

00

05

PE

MGPBTHBTHWTH

DIFNWTHGco_oc

2 4 6 8 10Scale

(b)

2

0

4

6

8

ESS

MGPBTHBTHWTH

DIFNWTHGco_oc

2 4 6 8 10Scale

(c)

00

02

04

06

08

3thA

MGPBTHBTHWTH

DIFNWTHGco_oc

2 4 6 8 10Scale

(d)



0 10 20 30Scale

40 50 60 7000

01

02

03

04

FEF

2 4 6 8 10

04

03

02

01

00

Zoom plot

MGMHPODIF

EMDOMGPO

EnvelopeACDIF


16

3248

64

00

04

08

12

Am

plitu

de

120 16040 80 2000Frequency (Hz)

(a)

16

3248 64

00

04

08

12A

mpl

itude

40 80 120 160 2000Frequency (Hz)

(b)

16

32 48

0

2

4

6

8

10

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

(c)

16

3248 64

0

1

2

3

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

(d)

16

4864

00

04

08

12

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

(e)

16

32 4864

0

4

8

12

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

(f )





p(u) (2π)minus k2







|C|minus 12


(21)




(22)


T1113874 1113875 (23)


(1m) 1113936mi1 xixT



ui pTi X (24)




KSP Kr times ESS

PE (25)






Si 1113944N

j1jneis dij1113872 11138731113954dij 1113944

N

j1jne i

s xi minus xj

11138681113868111386811138681113868

111386811138681113868111386811138681113874 1113875xi minus xj

dij

(27)

s(r) feminus rl

minus eminus r

(28)





Ai u1113954ew (30)



Xi 1113944N

j1jne i

s xj minus xi

11138681113868111386811138681113868

111386811138681113868111386811138681113874 1113875xj minus xi

dij

minus g1113954eg + u1113954ew (31)



Xdi c 1113944

N

j1jne i

cubd minus lbd

2s x

dj minus x

di

11138681113868111386811138681113868

111386811138681113868111386811138681113874 1113875xj minus xi

dij

⎛⎝ ⎞⎠ + 1113954Td

(32)



L (33)





fitness maxc(nk)

KSPi1113864 1113865

stn 1 2 20

k 1 2 3 20

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(34)




4 Simulation


KSP

4

6

8

10

12

14

20

15

10

5

n

10 15 205k







y1(t) 1113944I


y2(t) 1113944N


y3(t) 1113944M


y4(t) 1113944P

p1Cp cos 2πfp + φp1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(35)





l ge L

l = l + 1No




Yes

Step 1




FEF PE ESS 3thA



ranked


Step 2



Step 3








The best scale

00

02

04

06

FEF

10 20 30 40 50 60 700Scale λ

(a)

times106

0

1

2

3

4

Am

plitu

de

02 04 06 08 1000Time (s)

(b)

times104

16

32

4864

0

4

8

12

16

Am

plitu

de

40 80 120 160 1200Frequency (Hz)

(c)


12

13

14

15

16Be

st sc

ore

4 6 8 102Iteration

(a)

times102

ndash10

ndash5

0

5

10

Am

plitu

de

02 04 06 08 1000Time (s)

(b)














ndash4

0

4

Am

plitu

de

02 04 06 08 1000Time (s)

(a)

ndash6

ndash3

0

3

6

Am

plitu

de

02 04 06 08 1000Time (s)

(b)

ndash3

0

3

Am

plitu

de

02 04 06 08 1000Time (s)

(c)

ndash3

0

3

Am

plitu

de

02 04 06 08 1000Time (s)

(d)


ndash8

ndash4

0

4

8

Am

plitu

de

02 04 06 08 1000Time (s)

(a)

00

02

04

06

08

10

Am

plitu

de

20 40 60 80 100 120 140 160 180 2000Frequency (Hz)

10Hz

20Hz

50Hz

(b)


10

11

12

13

14

Best

scor

e

4 6 8 102Iteration

(a)

ndash2

0

2

Am

plitu

de

02 04 06 08 1000Time (s)

times103

(b)



05

10152025303540

TEK

10 20 30 40 50 600Scale λ

The best scale

(a)

5

10

15

20

25

Am

plitu

de

02 04 06 08 1000Time (s)

(b)

0

1

2

3

Am

plitu

de

10

16

40 80 120 160 2000Frequency (Hz)

(c)


10 20 30 40 50 60 700Scale λ

00

01

02

03

04

05

FEF

e best scale

(a)

0

2

4

6

8

Am

plitu

de

02 04 06 08 1000Time (s)

times106

(b)

00

05

10

15

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

16

32

48

times106

(c)



00

05

10

15

20A

mpl

itude

200 400 600 800 10000Frequency (Hz)

Selected

u1u2

u3u4

(a)

ndash5

0

5

Am

plitu

de

02 04 06 08 1000Time (s)

(b)

00

02

04

06

08

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

16

3248

(c)


200 400 600 800 10000Frequency (Hz)

3

26

2

16

1

0

Leve

l k

0

01

02

03

04

05

(a)

0

1

2

3

4

Am

plitu

de

0604 08 100200Time (s)

(b)

Figure 16 Continued


000

005

010

015

020

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

16

50

(c)


12

3 45 6

78



Gearbox Main sha

Blade

Monitoring system

(a)

(b)










2d)cos2φ)


ndash50

ndash25

0

25

50

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(a)

0

1

2

3

Am

plitu

de (m

s2 )

1000 1000 3000 4000 5000 6000 7000 80000Frequency (Hz)

02

01

00200 250 300 350 400

3156Hz

(b)


4 6 8 102Iteration

05

10

15

20

Best

scor

e

(a)

ndash15

00

15

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

times107

(b)

Figure 19 Continued




5 10 15 20 25 30 35 40 45 500Scale λ

000

002

004

006

008

010

FEF

e best scale

(a)

00

05

10

15

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

times 1016

(b)

00

05

10

15

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

times 1010


630Hz (2lowastfo)

9469Hz (3lowastfo)

(c)


ndash08

00

08

Am

plitu

de (m

s2 )

0655 0660 0665 0670 0675 06800650Time (s)

32ms

times107

(c)



e best scale

10

15

20

25

30TE

K

1510 20 2550Scale λ

(a)

0

20

40

60

80

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

0

2

4

6

8

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

29Hz

3158Hz

(c)


00

05

10

15

20

Am

plitu

de (m

s2 )

2000 4000 6000 8000 100000Frequency (Hz)

Selected

u1u2u3

u4u5

(a)

ndash30

ndash15

0

15

30

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

00

05

10

15

20

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

29Hz

3158Hz

630Hz

(c)








2000 4000 6000 80000Frequency (Hz)

546

436

326

216

10

Leve

l k

0

02

04

06

08

1

12

(a)

000

005

010

015

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

0

2

4

6

8

Am

plitu

de (m

s2 )

100 200 300 400 500 600 7000Frequency (Hz)

times10ndash5

29Hz

3158Hz

(c)



Accelerometers

Thermocouples

(a)

Motor

Bearing 1

Bearing 2 Bearing 4

Bearing 3

Accelerometers


Motor

Bearing 1

Bearing 2 Bearing 4

Bearing 3


(b)




00

02

04

06

08

RMS

20 40 60 80 100 120 140 1600Time (hours)

02

01

0086 88 90 92 94 96

Zoom plot

Test ATest C

Test B

Stage 1

Stage 3

Stage 2

(a)

01 02 03 0400Time (s)

ndash06ndash03

0003

A06

Am

plitu

de

(b)

Figure 25 Continued


01 02 03 0400Time (s)

ndash10ndash05

000510 B

Am

plitu

de

(c)

01 02 03 0400Time (s)

ndash4ndash2

0

C24

Am

plitu

de

(d)


14

15

16

17

18

Best

scor

e

8 104 62Iteration

(a)

01 02 03 0400Time (s)

ndash8

ndash4

0

4

8

Am

plitu

de (m

s2 )

(b)


000

001

002

003

Am

plitu

de (m

s2 )

2000 4000 6000 8000 100000Frequency (Hz)

(a)

000

001

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

fFBPO

(b)


000

003

006

009

012

FEF

20 40 60 800Scale λ

The best scale

(a)

01 02 03 0400Time (s)

0

40

80

120

160

Am

plitu

de (m

s2 )

(b)

Figure 28 Continued




6 Discussion


0

4

8

12

16

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

fBPFO

2fBPFO3fBPFO

4fBPFO

(c)


01 02 03 0400Time (s)

00

02

04

06

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

000

001

002

003

Am

plitu

de (m

s2 )

fBPFO

2fBPFO

3fBPFO

(b)


01 02 03 0400Time (s)

ndash01

00

01

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

000

001

002

Am

plitu

de (m

s2 )

fFBPO

2fFBPO

(b)







7 Conclusions


Nomenclature




01 02 03 0400Time (s)

000

002

004

006

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

00

05

10

15

Am

plitu

de (m

s2 ) fBPFO

(b)












Data Availability




Acknowledgments


References














































0 10 20 30Scale

40 50 60 7000

01

02

03

04

FEF

2 4 6 8 10

04

03

02

01

00

Zoom plot

MGMHPODIF

EMDOMGPO

EnvelopeACDIF


16

3248

64

00

04

08

12

Am

plitu

de

120 16040 80 2000Frequency (Hz)

(a)

16

3248 64

00

04

08

12A

mpl

itude

40 80 120 160 2000Frequency (Hz)

(b)

16

32 48

0

2

4

6

8

10

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

(c)

16

3248 64

0

1

2

3

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

(d)

16

4864

00

04

08

12

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

(e)

16

32 4864

0

4

8

12

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

(f )





p(u) (2π)minus k2







|C|minus 12


(21)




(22)


T1113874 1113875 (23)


(1m) 1113936mi1 xixT



ui pTi X (24)




KSP Kr times ESS

PE (25)






Si 1113944N

j1jneis dij1113872 11138731113954dij 1113944

N

j1jne i

s xi minus xj

11138681113868111386811138681113868

111386811138681113868111386811138681113874 1113875xi minus xj

dij

(27)

s(r) feminus rl

minus eminus r

(28)





Ai u1113954ew (30)



Xi 1113944N

j1jne i

s xj minus xi

11138681113868111386811138681113868

111386811138681113868111386811138681113874 1113875xj minus xi

dij

minus g1113954eg + u1113954ew (31)



Xdi c 1113944

N

j1jne i

cubd minus lbd

2s x

dj minus x

di

11138681113868111386811138681113868

111386811138681113868111386811138681113874 1113875xj minus xi

dij

⎛⎝ ⎞⎠ + 1113954Td

(32)



L (33)





fitness maxc(nk)

KSPi1113864 1113865

stn 1 2 20

k 1 2 3 20

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(34)




4 Simulation


KSP

4

6

8

10

12

14

20

15

10

5

n

10 15 205k







y1(t) 1113944I


y2(t) 1113944N


y3(t) 1113944M


y4(t) 1113944P

p1Cp cos 2πfp + φp1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(35)





l ge L

l = l + 1No




Yes

Step 1




FEF PE ESS 3thA



ranked


Step 2



Step 3








The best scale

00

02

04

06

FEF

10 20 30 40 50 60 700Scale λ

(a)

times106

0

1

2

3

4

Am

plitu

de

02 04 06 08 1000Time (s)

(b)

times104

16

32

4864

0

4

8

12

16

Am

plitu

de

40 80 120 160 1200Frequency (Hz)

(c)


12

13

14

15

16Be

st sc

ore

4 6 8 102Iteration

(a)

times102

ndash10

ndash5

0

5

10

Am

plitu

de

02 04 06 08 1000Time (s)

(b)














ndash4

0

4

Am

plitu

de

02 04 06 08 1000Time (s)

(a)

ndash6

ndash3

0

3

6

Am

plitu

de

02 04 06 08 1000Time (s)

(b)

ndash3

0

3

Am

plitu

de

02 04 06 08 1000Time (s)

(c)

ndash3

0

3

Am

plitu

de

02 04 06 08 1000Time (s)

(d)


ndash8

ndash4

0

4

8

Am

plitu

de

02 04 06 08 1000Time (s)

(a)

00

02

04

06

08

10

Am

plitu

de

20 40 60 80 100 120 140 160 180 2000Frequency (Hz)

10Hz

20Hz

50Hz

(b)


10

11

12

13

14

Best

scor

e

4 6 8 102Iteration

(a)

ndash2

0

2

Am

plitu

de

02 04 06 08 1000Time (s)

times103

(b)



05

10152025303540

TEK

10 20 30 40 50 600Scale λ

The best scale

(a)

5

10

15

20

25

Am

plitu

de

02 04 06 08 1000Time (s)

(b)

0

1

2

3

Am

plitu

de

10

16

40 80 120 160 2000Frequency (Hz)

(c)


10 20 30 40 50 60 700Scale λ

00

01

02

03

04

05

FEF

e best scale

(a)

0

2

4

6

8

Am

plitu

de

02 04 06 08 1000Time (s)

times106

(b)

00

05

10

15

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

16

32

48

times106

(c)



00

05

10

15

20A

mpl

itude

200 400 600 800 10000Frequency (Hz)

Selected

u1u2

u3u4

(a)

ndash5

0

5

Am

plitu

de

02 04 06 08 1000Time (s)

(b)

00

02

04

06

08

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

16

3248

(c)


200 400 600 800 10000Frequency (Hz)

3

26

2

16

1

0

Leve

l k

0

01

02

03

04

05

(a)

0

1

2

3

4

Am

plitu

de

0604 08 100200Time (s)

(b)

Figure 16 Continued


000

005

010

015

020

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

16

50

(c)


12

3 45 6

78



Gearbox Main sha

Blade

Monitoring system

(a)

(b)










2d)cos2φ)


ndash50

ndash25

0

25

50

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(a)

0

1

2

3

Am

plitu

de (m

s2 )

1000 1000 3000 4000 5000 6000 7000 80000Frequency (Hz)

02

01

00200 250 300 350 400

3156Hz

(b)


4 6 8 102Iteration

05

10

15

20

Best

scor

e

(a)

ndash15

00

15

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

times107

(b)

Figure 19 Continued




5 10 15 20 25 30 35 40 45 500Scale λ

000

002

004

006

008

010

FEF

e best scale

(a)

00

05

10

15

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

times 1016

(b)

00

05

10

15

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

times 1010


630Hz (2lowastfo)

9469Hz (3lowastfo)

(c)


ndash08

00

08

Am

plitu

de (m

s2 )

0655 0660 0665 0670 0675 06800650Time (s)

32ms

times107

(c)



e best scale

10

15

20

25

30TE

K

1510 20 2550Scale λ

(a)

0

20

40

60

80

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

0

2

4

6

8

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

29Hz

3158Hz

(c)


00

05

10

15

20

Am

plitu

de (m

s2 )

2000 4000 6000 8000 100000Frequency (Hz)

Selected

u1u2u3

u4u5

(a)

ndash30

ndash15

0

15

30

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

00

05

10

15

20

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

29Hz

3158Hz

630Hz

(c)








2000 4000 6000 80000Frequency (Hz)

546

436

326

216

10

Leve

l k

0

02

04

06

08

1

12

(a)

000

005

010

015

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

0

2

4

6

8

Am

plitu

de (m

s2 )

100 200 300 400 500 600 7000Frequency (Hz)

times10ndash5

29Hz

3158Hz

(c)



Accelerometers

Thermocouples

(a)

Motor

Bearing 1

Bearing 2 Bearing 4

Bearing 3

Accelerometers


Motor

Bearing 1

Bearing 2 Bearing 4

Bearing 3


(b)




00

02

04

06

08

RMS

20 40 60 80 100 120 140 1600Time (hours)

02

01

0086 88 90 92 94 96

Zoom plot

Test ATest C

Test B

Stage 1

Stage 3

Stage 2

(a)

01 02 03 0400Time (s)

ndash06ndash03

0003

A06

Am

plitu

de

(b)

Figure 25 Continued


01 02 03 0400Time (s)

ndash10ndash05

000510 B

Am

plitu

de

(c)

01 02 03 0400Time (s)

ndash4ndash2

0

C24

Am

plitu

de

(d)


14

15

16

17

18

Best

scor

e

8 104 62Iteration

(a)

01 02 03 0400Time (s)

ndash8

ndash4

0

4

8

Am

plitu

de (m

s2 )

(b)


000

001

002

003

Am

plitu

de (m

s2 )

2000 4000 6000 8000 100000Frequency (Hz)

(a)

000

001

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

fFBPO

(b)


000

003

006

009

012

FEF

20 40 60 800Scale λ

The best scale

(a)

01 02 03 0400Time (s)

0

40

80

120

160

Am

plitu

de (m

s2 )

(b)

Figure 28 Continued




6 Discussion


0

4

8

12

16

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

fBPFO

2fBPFO3fBPFO

4fBPFO

(c)


01 02 03 0400Time (s)

00

02

04

06

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

000

001

002

003

Am

plitu

de (m

s2 )

fBPFO

2fBPFO

3fBPFO

(b)


01 02 03 0400Time (s)

ndash01

00

01

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

000

001

002

Am

plitu

de (m

s2 )

fFBPO

2fFBPO

(b)







7 Conclusions


Nomenclature




01 02 03 0400Time (s)

000

002

004

006

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

00

05

10

15

Am

plitu

de (m

s2 ) fBPFO

(b)












Data Availability




Acknowledgments


References
















































p(u) (2π)minus k2







|C|minus 12


(21)




(22)


T1113874 1113875 (23)


(1m) 1113936mi1 xixT



ui pTi X (24)




KSP Kr times ESS

PE (25)






Si 1113944N

j1jneis dij1113872 11138731113954dij 1113944

N

j1jne i

s xi minus xj

11138681113868111386811138681113868

111386811138681113868111386811138681113874 1113875xi minus xj

dij

(27)

s(r) feminus rl

minus eminus r

(28)





Ai u1113954ew (30)



Xi 1113944N

j1jne i

s xj minus xi

11138681113868111386811138681113868

111386811138681113868111386811138681113874 1113875xj minus xi

dij

minus g1113954eg + u1113954ew (31)



Xdi c 1113944

N

j1jne i

cubd minus lbd

2s x

dj minus x

di

11138681113868111386811138681113868

111386811138681113868111386811138681113874 1113875xj minus xi

dij

⎛⎝ ⎞⎠ + 1113954Td

(32)



L (33)





fitness maxc(nk)

KSPi1113864 1113865

stn 1 2 20

k 1 2 3 20

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(34)




4 Simulation


KSP

4

6

8

10

12

14

20

15

10

5

n

10 15 205k







y1(t) 1113944I


y2(t) 1113944N


y3(t) 1113944M


y4(t) 1113944P

p1Cp cos 2πfp + φp1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(35)





l ge L

l = l + 1No




Yes

Step 1




FEF PE ESS 3thA



ranked


Step 2



Step 3








The best scale

00

02

04

06

FEF

10 20 30 40 50 60 700Scale λ

(a)

times106

0

1

2

3

4

Am

plitu

de

02 04 06 08 1000Time (s)

(b)

times104

16

32

4864

0

4

8

12

16

Am

plitu

de

40 80 120 160 1200Frequency (Hz)

(c)


12

13

14

15

16Be

st sc

ore

4 6 8 102Iteration

(a)

times102

ndash10

ndash5

0

5

10

Am

plitu

de

02 04 06 08 1000Time (s)

(b)














ndash4

0

4

Am

plitu

de

02 04 06 08 1000Time (s)

(a)

ndash6

ndash3

0

3

6

Am

plitu

de

02 04 06 08 1000Time (s)

(b)

ndash3

0

3

Am

plitu

de

02 04 06 08 1000Time (s)

(c)

ndash3

0

3

Am

plitu

de

02 04 06 08 1000Time (s)

(d)


ndash8

ndash4

0

4

8

Am

plitu

de

02 04 06 08 1000Time (s)

(a)

00

02

04

06

08

10

Am

plitu

de

20 40 60 80 100 120 140 160 180 2000Frequency (Hz)

10Hz

20Hz

50Hz

(b)


10

11

12

13

14

Best

scor

e

4 6 8 102Iteration

(a)

ndash2

0

2

Am

plitu

de

02 04 06 08 1000Time (s)

times103

(b)



05

10152025303540

TEK

10 20 30 40 50 600Scale λ

The best scale

(a)

5

10

15

20

25

Am

plitu

de

02 04 06 08 1000Time (s)

(b)

0

1

2

3

Am

plitu

de

10

16

40 80 120 160 2000Frequency (Hz)

(c)


10 20 30 40 50 60 700Scale λ

00

01

02

03

04

05

FEF

e best scale

(a)

0

2

4

6

8

Am

plitu

de

02 04 06 08 1000Time (s)

times106

(b)

00

05

10

15

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

16

32

48

times106

(c)



00

05

10

15

20A

mpl

itude

200 400 600 800 10000Frequency (Hz)

Selected

u1u2

u3u4

(a)

ndash5

0

5

Am

plitu

de

02 04 06 08 1000Time (s)

(b)

00

02

04

06

08

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

16

3248

(c)


200 400 600 800 10000Frequency (Hz)

3

26

2

16

1

0

Leve

l k

0

01

02

03

04

05

(a)

0

1

2

3

4

Am

plitu

de

0604 08 100200Time (s)

(b)

Figure 16 Continued


000

005

010

015

020

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

16

50

(c)


12

3 45 6

78



Gearbox Main sha

Blade

Monitoring system

(a)

(b)










2d)cos2φ)


ndash50

ndash25

0

25

50

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(a)

0

1

2

3

Am

plitu

de (m

s2 )

1000 1000 3000 4000 5000 6000 7000 80000Frequency (Hz)

02

01

00200 250 300 350 400

3156Hz

(b)


4 6 8 102Iteration

05

10

15

20

Best

scor

e

(a)

ndash15

00

15

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

times107

(b)

Figure 19 Continued




5 10 15 20 25 30 35 40 45 500Scale λ

000

002

004

006

008

010

FEF

e best scale

(a)

00

05

10

15

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

times 1016

(b)

00

05

10

15

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

times 1010


630Hz (2lowastfo)

9469Hz (3lowastfo)

(c)


ndash08

00

08

Am

plitu

de (m

s2 )

0655 0660 0665 0670 0675 06800650Time (s)

32ms

times107

(c)



e best scale

10

15

20

25

30TE

K

1510 20 2550Scale λ

(a)

0

20

40

60

80

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

0

2

4

6

8

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

29Hz

3158Hz

(c)


00

05

10

15

20

Am

plitu

de (m

s2 )

2000 4000 6000 8000 100000Frequency (Hz)

Selected

u1u2u3

u4u5

(a)

ndash30

ndash15

0

15

30

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

00

05

10

15

20

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

29Hz

3158Hz

630Hz

(c)








2000 4000 6000 80000Frequency (Hz)

546

436

326

216

10

Leve

l k

0

02

04

06

08

1

12

(a)

000

005

010

015

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

0

2

4

6

8

Am

plitu

de (m

s2 )

100 200 300 400 500 600 7000Frequency (Hz)

times10ndash5

29Hz

3158Hz

(c)



Accelerometers

Thermocouples

(a)

Motor

Bearing 1

Bearing 2 Bearing 4

Bearing 3

Accelerometers


Motor

Bearing 1

Bearing 2 Bearing 4

Bearing 3


(b)




00

02

04

06

08

RMS

20 40 60 80 100 120 140 1600Time (hours)

02

01

0086 88 90 92 94 96

Zoom plot

Test ATest C

Test B

Stage 1

Stage 3

Stage 2

(a)

01 02 03 0400Time (s)

ndash06ndash03

0003

A06

Am

plitu

de

(b)

Figure 25 Continued


01 02 03 0400Time (s)

ndash10ndash05

000510 B

Am

plitu

de

(c)

01 02 03 0400Time (s)

ndash4ndash2

0

C24

Am

plitu

de

(d)


14

15

16

17

18

Best

scor

e

8 104 62Iteration

(a)

01 02 03 0400Time (s)

ndash8

ndash4

0

4

8

Am

plitu

de (m

s2 )

(b)


000

001

002

003

Am

plitu

de (m

s2 )

2000 4000 6000 8000 100000Frequency (Hz)

(a)

000

001

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

fFBPO

(b)


000

003

006

009

012

FEF

20 40 60 800Scale λ

The best scale

(a)

01 02 03 0400Time (s)

0

40

80

120

160

Am

plitu

de (m

s2 )

(b)

Figure 28 Continued




6 Discussion


0

4

8

12

16

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

fBPFO

2fBPFO3fBPFO

4fBPFO

(c)


01 02 03 0400Time (s)

00

02

04

06

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

000

001

002

003

Am

plitu

de (m

s2 )

fBPFO

2fBPFO

3fBPFO

(b)


01 02 03 0400Time (s)

ndash01

00

01

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

000

001

002

Am

plitu

de (m

s2 )

fFBPO

2fFBPO

(b)







7 Conclusions


Nomenclature




01 02 03 0400Time (s)

000

002

004

006

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

00

05

10

15

Am

plitu

de (m

s2 ) fBPFO

(b)












Data Availability




Acknowledgments


References















































Ai u1113954ew (30)



Xi 1113944N

j1jne i

s xj minus xi

11138681113868111386811138681113868

111386811138681113868111386811138681113874 1113875xj minus xi

dij

minus g1113954eg + u1113954ew (31)



Xdi c 1113944

N

j1jne i

cubd minus lbd

2s x

dj minus x

di

11138681113868111386811138681113868

111386811138681113868111386811138681113874 1113875xj minus xi

dij

⎛⎝ ⎞⎠ + 1113954Td

(32)



L (33)





fitness maxc(nk)

KSPi1113864 1113865

stn 1 2 20

k 1 2 3 20

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(34)




4 Simulation


KSP

4

6

8

10

12

14

20

15

10

5

n

10 15 205k







y1(t) 1113944I


y2(t) 1113944N


y3(t) 1113944M


y4(t) 1113944P

p1Cp cos 2πfp + φp1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(35)





l ge L

l = l + 1No




Yes

Step 1




FEF PE ESS 3thA



ranked


Step 2



Step 3








The best scale

00

02

04

06

FEF

10 20 30 40 50 60 700Scale λ

(a)

times106

0

1

2

3

4

Am

plitu

de

02 04 06 08 1000Time (s)

(b)

times104

16

32

4864

0

4

8

12

16

Am

plitu

de

40 80 120 160 1200Frequency (Hz)

(c)


12

13

14

15

16Be

st sc

ore

4 6 8 102Iteration

(a)

times102

ndash10

ndash5

0

5

10

Am

plitu

de

02 04 06 08 1000Time (s)

(b)














ndash4

0

4

Am

plitu

de

02 04 06 08 1000Time (s)

(a)

ndash6

ndash3

0

3

6

Am

plitu

de

02 04 06 08 1000Time (s)

(b)

ndash3

0

3

Am

plitu

de

02 04 06 08 1000Time (s)

(c)

ndash3

0

3

Am

plitu

de

02 04 06 08 1000Time (s)

(d)


ndash8

ndash4

0

4

8

Am

plitu

de

02 04 06 08 1000Time (s)

(a)

00

02

04

06

08

10

Am

plitu

de

20 40 60 80 100 120 140 160 180 2000Frequency (Hz)

10Hz

20Hz

50Hz

(b)


10

11

12

13

14

Best

scor

e

4 6 8 102Iteration

(a)

ndash2

0

2

Am

plitu

de

02 04 06 08 1000Time (s)

times103

(b)



05

10152025303540

TEK

10 20 30 40 50 600Scale λ

The best scale

(a)

5

10

15

20

25

Am

plitu

de

02 04 06 08 1000Time (s)

(b)

0

1

2

3

Am

plitu

de

10

16

40 80 120 160 2000Frequency (Hz)

(c)


10 20 30 40 50 60 700Scale λ

00

01

02

03

04

05

FEF

e best scale

(a)

0

2

4

6

8

Am

plitu

de

02 04 06 08 1000Time (s)

times106

(b)

00

05

10

15

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

16

32

48

times106

(c)



00

05

10

15

20A

mpl

itude

200 400 600 800 10000Frequency (Hz)

Selected

u1u2

u3u4

(a)

ndash5

0

5

Am

plitu

de

02 04 06 08 1000Time (s)

(b)

00

02

04

06

08

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

16

3248

(c)


200 400 600 800 10000Frequency (Hz)

3

26

2

16

1

0

Leve

l k

0

01

02

03

04

05

(a)

0

1

2

3

4

Am

plitu

de

0604 08 100200Time (s)

(b)

Figure 16 Continued


000

005

010

015

020

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

16

50

(c)


12

3 45 6

78



Gearbox Main sha

Blade

Monitoring system

(a)

(b)










2d)cos2φ)


ndash50

ndash25

0

25

50

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(a)

0

1

2

3

Am

plitu

de (m

s2 )

1000 1000 3000 4000 5000 6000 7000 80000Frequency (Hz)

02

01

00200 250 300 350 400

3156Hz

(b)


4 6 8 102Iteration

05

10

15

20

Best

scor

e

(a)

ndash15

00

15

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

times107

(b)

Figure 19 Continued




5 10 15 20 25 30 35 40 45 500Scale λ

000

002

004

006

008

010

FEF

e best scale

(a)

00

05

10

15

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

times 1016

(b)

00

05

10

15

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

times 1010


630Hz (2lowastfo)

9469Hz (3lowastfo)

(c)


ndash08

00

08

Am

plitu

de (m

s2 )

0655 0660 0665 0670 0675 06800650Time (s)

32ms

times107

(c)



e best scale

10

15

20

25

30TE

K

1510 20 2550Scale λ

(a)

0

20

40

60

80

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

0

2

4

6

8

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

29Hz

3158Hz

(c)


00

05

10

15

20

Am

plitu

de (m

s2 )

2000 4000 6000 8000 100000Frequency (Hz)

Selected

u1u2u3

u4u5

(a)

ndash30

ndash15

0

15

30

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

00

05

10

15

20

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

29Hz

3158Hz

630Hz

(c)








2000 4000 6000 80000Frequency (Hz)

546

436

326

216

10

Leve

l k

0

02

04

06

08

1

12

(a)

000

005

010

015

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

0

2

4

6

8

Am

plitu

de (m

s2 )

100 200 300 400 500 600 7000Frequency (Hz)

times10ndash5

29Hz

3158Hz

(c)



Accelerometers

Thermocouples

(a)

Motor

Bearing 1

Bearing 2 Bearing 4

Bearing 3

Accelerometers


Motor

Bearing 1

Bearing 2 Bearing 4

Bearing 3


(b)




00

02

04

06

08

RMS

20 40 60 80 100 120 140 1600Time (hours)

02

01

0086 88 90 92 94 96

Zoom plot

Test ATest C

Test B

Stage 1

Stage 3

Stage 2

(a)

01 02 03 0400Time (s)

ndash06ndash03

0003

A06

Am

plitu

de

(b)

Figure 25 Continued


01 02 03 0400Time (s)

ndash10ndash05

000510 B

Am

plitu

de

(c)

01 02 03 0400Time (s)

ndash4ndash2

0

C24

Am

plitu

de

(d)


14

15

16

17

18

Best

scor

e

8 104 62Iteration

(a)

01 02 03 0400Time (s)

ndash8

ndash4

0

4

8

Am

plitu

de (m

s2 )

(b)


000

001

002

003

Am

plitu

de (m

s2 )

2000 4000 6000 8000 100000Frequency (Hz)

(a)

000

001

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

fFBPO

(b)


000

003

006

009

012

FEF

20 40 60 800Scale λ

The best scale

(a)

01 02 03 0400Time (s)

0

40

80

120

160

Am

plitu

de (m

s2 )

(b)

Figure 28 Continued




6 Discussion


0

4

8

12

16

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

fBPFO

2fBPFO3fBPFO

4fBPFO

(c)


01 02 03 0400Time (s)

00

02

04

06

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

000

001

002

003

Am

plitu

de (m

s2 )

fBPFO

2fBPFO

3fBPFO

(b)


01 02 03 0400Time (s)

ndash01

00

01

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

000

001

002

Am

plitu

de (m

s2 )

fFBPO

2fFBPO

(b)







7 Conclusions


Nomenclature




01 02 03 0400Time (s)

000

002

004

006

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

00

05

10

15

Am

plitu

de (m

s2 ) fBPFO

(b)












Data Availability




Acknowledgments


References


















































y1(t) 1113944I


y2(t) 1113944N


y3(t) 1113944M


y4(t) 1113944P

p1Cp cos 2πfp + φp1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(35)





l ge L

l = l + 1No




Yes

Step 1




FEF PE ESS 3thA



ranked


Step 2



Step 3








The best scale

00

02

04

06

FEF

10 20 30 40 50 60 700Scale λ

(a)

times106

0

1

2

3

4

Am

plitu

de

02 04 06 08 1000Time (s)

(b)

times104

16

32

4864

0

4

8

12

16

Am

plitu

de

40 80 120 160 1200Frequency (Hz)

(c)


12

13

14

15

16Be

st sc

ore

4 6 8 102Iteration

(a)

times102

ndash10

ndash5

0

5

10

Am

plitu

de

02 04 06 08 1000Time (s)

(b)














ndash4

0

4

Am

plitu

de

02 04 06 08 1000Time (s)

(a)

ndash6

ndash3

0

3

6

Am

plitu

de

02 04 06 08 1000Time (s)

(b)

ndash3

0

3

Am

plitu

de

02 04 06 08 1000Time (s)

(c)

ndash3

0

3

Am

plitu

de

02 04 06 08 1000Time (s)

(d)


ndash8

ndash4

0

4

8

Am

plitu

de

02 04 06 08 1000Time (s)

(a)

00

02

04

06

08

10

Am

plitu

de

20 40 60 80 100 120 140 160 180 2000Frequency (Hz)

10Hz

20Hz

50Hz

(b)


10

11

12

13

14

Best

scor

e

4 6 8 102Iteration

(a)

ndash2

0

2

Am

plitu

de

02 04 06 08 1000Time (s)

times103

(b)



05

10152025303540

TEK

10 20 30 40 50 600Scale λ

The best scale

(a)

5

10

15

20

25

Am

plitu

de

02 04 06 08 1000Time (s)

(b)

0

1

2

3

Am

plitu

de

10

16

40 80 120 160 2000Frequency (Hz)

(c)


10 20 30 40 50 60 700Scale λ

00

01

02

03

04

05

FEF

e best scale

(a)

0

2

4

6

8

Am

plitu

de

02 04 06 08 1000Time (s)

times106

(b)

00

05

10

15

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

16

32

48

times106

(c)



00

05

10

15

20A

mpl

itude

200 400 600 800 10000Frequency (Hz)

Selected

u1u2

u3u4

(a)

ndash5

0

5

Am

plitu

de

02 04 06 08 1000Time (s)

(b)

00

02

04

06

08

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

16

3248

(c)


200 400 600 800 10000Frequency (Hz)

3

26

2

16

1

0

Leve

l k

0

01

02

03

04

05

(a)

0

1

2

3

4

Am

plitu

de

0604 08 100200Time (s)

(b)

Figure 16 Continued


000

005

010

015

020

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

16

50

(c)


12

3 45 6

78



Gearbox Main sha

Blade

Monitoring system

(a)

(b)










2d)cos2φ)


ndash50

ndash25

0

25

50

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(a)

0

1

2

3

Am

plitu

de (m

s2 )

1000 1000 3000 4000 5000 6000 7000 80000Frequency (Hz)

02

01

00200 250 300 350 400

3156Hz

(b)


4 6 8 102Iteration

05

10

15

20

Best

scor

e

(a)

ndash15

00

15

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

times107

(b)

Figure 19 Continued




5 10 15 20 25 30 35 40 45 500Scale λ

000

002

004

006

008

010

FEF

e best scale

(a)

00

05

10

15

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

times 1016

(b)

00

05

10

15

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

times 1010


630Hz (2lowastfo)

9469Hz (3lowastfo)

(c)


ndash08

00

08

Am

plitu

de (m

s2 )

0655 0660 0665 0670 0675 06800650Time (s)

32ms

times107

(c)



e best scale

10

15

20

25

30TE

K

1510 20 2550Scale λ

(a)

0

20

40

60

80

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

0

2

4

6

8

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

29Hz

3158Hz

(c)


00

05

10

15

20

Am

plitu

de (m

s2 )

2000 4000 6000 8000 100000Frequency (Hz)

Selected

u1u2u3

u4u5

(a)

ndash30

ndash15

0

15

30

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

00

05

10

15

20

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

29Hz

3158Hz

630Hz

(c)








2000 4000 6000 80000Frequency (Hz)

546

436

326

216

10

Leve

l k

0

02

04

06

08

1

12

(a)

000

005

010

015

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

0

2

4

6

8

Am

plitu

de (m

s2 )

100 200 300 400 500 600 7000Frequency (Hz)

times10ndash5

29Hz

3158Hz

(c)



Accelerometers

Thermocouples

(a)

Motor

Bearing 1

Bearing 2 Bearing 4

Bearing 3

Accelerometers


Motor

Bearing 1

Bearing 2 Bearing 4

Bearing 3


(b)




00

02

04

06

08

RMS

20 40 60 80 100 120 140 1600Time (hours)

02

01

0086 88 90 92 94 96

Zoom plot

Test ATest C

Test B

Stage 1

Stage 3

Stage 2

(a)

01 02 03 0400Time (s)

ndash06ndash03

0003

A06

Am

plitu

de

(b)

Figure 25 Continued


01 02 03 0400Time (s)

ndash10ndash05

000510 B

Am

plitu

de

(c)

01 02 03 0400Time (s)

ndash4ndash2

0

C24

Am

plitu

de

(d)


14

15

16

17

18

Best

scor

e

8 104 62Iteration

(a)

01 02 03 0400Time (s)

ndash8

ndash4

0

4

8

Am

plitu

de (m

s2 )

(b)


000

001

002

003

Am

plitu

de (m

s2 )

2000 4000 6000 8000 100000Frequency (Hz)

(a)

000

001

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

fFBPO

(b)


000

003

006

009

012

FEF

20 40 60 800Scale λ

The best scale

(a)

01 02 03 0400Time (s)

0

40

80

120

160

Am

plitu

de (m

s2 )

(b)

Figure 28 Continued




6 Discussion


0

4

8

12

16

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

fBPFO

2fBPFO3fBPFO

4fBPFO

(c)


01 02 03 0400Time (s)

00

02

04

06

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

000

001

002

003

Am

plitu

de (m

s2 )

fBPFO

2fBPFO

3fBPFO

(b)


01 02 03 0400Time (s)

ndash01

00

01

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

000

001

002

Am

plitu

de (m

s2 )

fFBPO

2fFBPO

(b)







7 Conclusions


Nomenclature




01 02 03 0400Time (s)

000

002

004

006

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

00

05

10

15

Am

plitu

de (m

s2 ) fBPFO

(b)












Data Availability




Acknowledgments


References



















































The best scale

00

02

04

06

FEF

10 20 30 40 50 60 700Scale λ

(a)

times106

0

1

2

3

4

Am

plitu

de

02 04 06 08 1000Time (s)

(b)

times104

16

32

4864

0

4

8

12

16

Am

plitu

de

40 80 120 160 1200Frequency (Hz)

(c)


12

13

14

15

16Be

st sc

ore

4 6 8 102Iteration

(a)

times102

ndash10

ndash5

0

5

10

Am

plitu

de

02 04 06 08 1000Time (s)

(b)














ndash4

0

4

Am

plitu

de

02 04 06 08 1000Time (s)

(a)

ndash6

ndash3

0

3

6

Am

plitu

de

02 04 06 08 1000Time (s)

(b)

ndash3

0

3

Am

plitu

de

02 04 06 08 1000Time (s)

(c)

ndash3

0

3

Am

plitu

de

02 04 06 08 1000Time (s)

(d)


ndash8

ndash4

0

4

8

Am

plitu

de

02 04 06 08 1000Time (s)

(a)

00

02

04

06

08

10

Am

plitu

de

20 40 60 80 100 120 140 160 180 2000Frequency (Hz)

10Hz

20Hz

50Hz

(b)


10

11

12

13

14

Best

scor

e

4 6 8 102Iteration

(a)

ndash2

0

2

Am

plitu

de

02 04 06 08 1000Time (s)

times103

(b)



05

10152025303540

TEK

10 20 30 40 50 600Scale λ

The best scale

(a)

5

10

15

20

25

Am

plitu

de

02 04 06 08 1000Time (s)

(b)

0

1

2

3

Am

plitu

de

10

16

40 80 120 160 2000Frequency (Hz)

(c)


10 20 30 40 50 60 700Scale λ

00

01

02

03

04

05

FEF

e best scale

(a)

0

2

4

6

8

Am

plitu

de

02 04 06 08 1000Time (s)

times106

(b)

00

05

10

15

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

16

32

48

times106

(c)



00

05

10

15

20A

mpl

itude

200 400 600 800 10000Frequency (Hz)

Selected

u1u2

u3u4

(a)

ndash5

0

5

Am

plitu

de

02 04 06 08 1000Time (s)

(b)

00

02

04

06

08

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

16

3248

(c)


200 400 600 800 10000Frequency (Hz)

3

26

2

16

1

0

Leve

l k

0

01

02

03

04

05

(a)

0

1

2

3

4

Am

plitu

de

0604 08 100200Time (s)

(b)

Figure 16 Continued


000

005

010

015

020

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

16

50

(c)


12

3 45 6

78



Gearbox Main sha

Blade

Monitoring system

(a)

(b)










2d)cos2φ)


ndash50

ndash25

0

25

50

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(a)

0

1

2

3

Am

plitu

de (m

s2 )

1000 1000 3000 4000 5000 6000 7000 80000Frequency (Hz)

02

01

00200 250 300 350 400

3156Hz

(b)


4 6 8 102Iteration

05

10

15

20

Best

scor

e

(a)

ndash15

00

15

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

times107

(b)

Figure 19 Continued




5 10 15 20 25 30 35 40 45 500Scale λ

000

002

004

006

008

010

FEF

e best scale

(a)

00

05

10

15

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

times 1016

(b)

00

05

10

15

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

times 1010


630Hz (2lowastfo)

9469Hz (3lowastfo)

(c)


ndash08

00

08

Am

plitu

de (m

s2 )

0655 0660 0665 0670 0675 06800650Time (s)

32ms

times107

(c)



e best scale

10

15

20

25

30TE

K

1510 20 2550Scale λ

(a)

0

20

40

60

80

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

0

2

4

6

8

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

29Hz

3158Hz

(c)


00

05

10

15

20

Am

plitu

de (m

s2 )

2000 4000 6000 8000 100000Frequency (Hz)

Selected

u1u2u3

u4u5

(a)

ndash30

ndash15

0

15

30

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

00

05

10

15

20

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

29Hz

3158Hz

630Hz

(c)








2000 4000 6000 80000Frequency (Hz)

546

436

326

216

10

Leve

l k

0

02

04

06

08

1

12

(a)

000

005

010

015

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

0

2

4

6

8

Am

plitu

de (m

s2 )

100 200 300 400 500 600 7000Frequency (Hz)

times10ndash5

29Hz

3158Hz

(c)



Accelerometers

Thermocouples

(a)

Motor

Bearing 1

Bearing 2 Bearing 4

Bearing 3

Accelerometers


Motor

Bearing 1

Bearing 2 Bearing 4

Bearing 3


(b)




00

02

04

06

08

RMS

20 40 60 80 100 120 140 1600Time (hours)

02

01

0086 88 90 92 94 96

Zoom plot

Test ATest C

Test B

Stage 1

Stage 3

Stage 2

(a)

01 02 03 0400Time (s)

ndash06ndash03

0003

A06

Am

plitu

de

(b)

Figure 25 Continued


01 02 03 0400Time (s)

ndash10ndash05

000510 B

Am

plitu

de

(c)

01 02 03 0400Time (s)

ndash4ndash2

0

C24

Am

plitu

de

(d)


14

15

16

17

18

Best

scor

e

8 104 62Iteration

(a)

01 02 03 0400Time (s)

ndash8

ndash4

0

4

8

Am

plitu

de (m

s2 )

(b)


000

001

002

003

Am

plitu

de (m

s2 )

2000 4000 6000 8000 100000Frequency (Hz)

(a)

000

001

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

fFBPO

(b)


000

003

006

009

012

FEF

20 40 60 800Scale λ

The best scale

(a)

01 02 03 0400Time (s)

0

40

80

120

160

Am

plitu

de (m

s2 )

(b)

Figure 28 Continued




6 Discussion


0

4

8

12

16

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

fBPFO

2fBPFO3fBPFO

4fBPFO

(c)


01 02 03 0400Time (s)

00

02

04

06

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

000

001

002

003

Am

plitu

de (m

s2 )

fBPFO

2fBPFO

3fBPFO

(b)


01 02 03 0400Time (s)

ndash01

00

01

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

000

001

002

Am

plitu

de (m

s2 )

fFBPO

2fFBPO

(b)







7 Conclusions


Nomenclature




01 02 03 0400Time (s)

000

002

004

006

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

00

05

10

15

Am

plitu

de (m

s2 ) fBPFO

(b)












Data Availability




Acknowledgments


References

























































ndash4

0

4

Am

plitu

de

02 04 06 08 1000Time (s)

(a)

ndash6

ndash3

0

3

6

Am

plitu

de

02 04 06 08 1000Time (s)

(b)

ndash3

0

3

Am

plitu

de

02 04 06 08 1000Time (s)

(c)

ndash3

0

3

Am

plitu

de

02 04 06 08 1000Time (s)

(d)


ndash8

ndash4

0

4

8

Am

plitu

de

02 04 06 08 1000Time (s)

(a)

00

02

04

06

08

10

Am

plitu

de

20 40 60 80 100 120 140 160 180 2000Frequency (Hz)

10Hz

20Hz

50Hz

(b)


10

11

12

13

14

Best

scor

e

4 6 8 102Iteration

(a)

ndash2

0

2

Am

plitu

de

02 04 06 08 1000Time (s)

times103

(b)



05

10152025303540

TEK

10 20 30 40 50 600Scale λ

The best scale

(a)

5

10

15

20

25

Am

plitu

de

02 04 06 08 1000Time (s)

(b)

0

1

2

3

Am

plitu

de

10

16

40 80 120 160 2000Frequency (Hz)

(c)


10 20 30 40 50 60 700Scale λ

00

01

02

03

04

05

FEF

e best scale

(a)

0

2

4

6

8

Am

plitu

de

02 04 06 08 1000Time (s)

times106

(b)

00

05

10

15

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

16

32

48

times106

(c)



00

05

10

15

20A

mpl

itude

200 400 600 800 10000Frequency (Hz)

Selected

u1u2

u3u4

(a)

ndash5

0

5

Am

plitu

de

02 04 06 08 1000Time (s)

(b)

00

02

04

06

08

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

16

3248

(c)


200 400 600 800 10000Frequency (Hz)

3

26

2

16

1

0

Leve

l k

0

01

02

03

04

05

(a)

0

1

2

3

4

Am

plitu

de

0604 08 100200Time (s)

(b)

Figure 16 Continued


000

005

010

015

020

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

16

50

(c)


12

3 45 6

78



Gearbox Main sha

Blade

Monitoring system

(a)

(b)










2d)cos2φ)


ndash50

ndash25

0

25

50

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(a)

0

1

2

3

Am

plitu

de (m

s2 )

1000 1000 3000 4000 5000 6000 7000 80000Frequency (Hz)

02

01

00200 250 300 350 400

3156Hz

(b)


4 6 8 102Iteration

05

10

15

20

Best

scor

e

(a)

ndash15

00

15

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

times107

(b)

Figure 19 Continued




5 10 15 20 25 30 35 40 45 500Scale λ

000

002

004

006

008

010

FEF

e best scale

(a)

00

05

10

15

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

times 1016

(b)

00

05

10

15

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

times 1010


630Hz (2lowastfo)

9469Hz (3lowastfo)

(c)


ndash08

00

08

Am

plitu

de (m

s2 )

0655 0660 0665 0670 0675 06800650Time (s)

32ms

times107

(c)



e best scale

10

15

20

25

30TE

K

1510 20 2550Scale λ

(a)

0

20

40

60

80

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

0

2

4

6

8

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

29Hz

3158Hz

(c)


00

05

10

15

20

Am

plitu

de (m

s2 )

2000 4000 6000 8000 100000Frequency (Hz)

Selected

u1u2u3

u4u5

(a)

ndash30

ndash15

0

15

30

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

00

05

10

15

20

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

29Hz

3158Hz

630Hz

(c)








2000 4000 6000 80000Frequency (Hz)

546

436

326

216

10

Leve

l k

0

02

04

06

08

1

12

(a)

000

005

010

015

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

0

2

4

6

8

Am

plitu

de (m

s2 )

100 200 300 400 500 600 7000Frequency (Hz)

times10ndash5

29Hz

3158Hz

(c)



Accelerometers

Thermocouples

(a)

Motor

Bearing 1

Bearing 2 Bearing 4

Bearing 3

Accelerometers


Motor

Bearing 1

Bearing 2 Bearing 4

Bearing 3


(b)




00

02

04

06

08

RMS

20 40 60 80 100 120 140 1600Time (hours)

02

01

0086 88 90 92 94 96

Zoom plot

Test ATest C

Test B

Stage 1

Stage 3

Stage 2

(a)

01 02 03 0400Time (s)

ndash06ndash03

0003

A06

Am

plitu

de

(b)

Figure 25 Continued


01 02 03 0400Time (s)

ndash10ndash05

000510 B

Am

plitu

de

(c)

01 02 03 0400Time (s)

ndash4ndash2

0

C24

Am

plitu

de

(d)


14

15

16

17

18

Best

scor

e

8 104 62Iteration

(a)

01 02 03 0400Time (s)

ndash8

ndash4

0

4

8

Am

plitu

de (m

s2 )

(b)


000

001

002

003

Am

plitu

de (m

s2 )

2000 4000 6000 8000 100000Frequency (Hz)

(a)

000

001

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

fFBPO

(b)


000

003

006

009

012

FEF

20 40 60 800Scale λ

The best scale

(a)

01 02 03 0400Time (s)

0

40

80

120

160

Am

plitu

de (m

s2 )

(b)

Figure 28 Continued




6 Discussion


0

4

8

12

16

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

fBPFO

2fBPFO3fBPFO

4fBPFO

(c)


01 02 03 0400Time (s)

00

02

04

06

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

000

001

002

003

Am

plitu

de (m

s2 )

fBPFO

2fBPFO

3fBPFO

(b)


01 02 03 0400Time (s)

ndash01

00

01

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

000

001

002

Am

plitu

de (m

s2 )

fFBPO

2fFBPO

(b)







7 Conclusions


Nomenclature




01 02 03 0400Time (s)

000

002

004

006

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

00

05

10

15

Am

plitu

de (m

s2 ) fBPFO

(b)












Data Availability




Acknowledgments


References














































ndash4

0

4

Am

plitu

de

02 04 06 08 1000Time (s)

(a)

ndash6

ndash3

0

3

6

Am

plitu

de

02 04 06 08 1000Time (s)

(b)

ndash3

0

3

Am

plitu

de

02 04 06 08 1000Time (s)

(c)

ndash3

0

3

Am

plitu

de

02 04 06 08 1000Time (s)

(d)


ndash8

ndash4

0

4

8

Am

plitu

de

02 04 06 08 1000Time (s)

(a)

00

02

04

06

08

10

Am

plitu

de

20 40 60 80 100 120 140 160 180 2000Frequency (Hz)

10Hz

20Hz

50Hz

(b)


10

11

12

13

14

Best

scor

e

4 6 8 102Iteration

(a)

ndash2

0

2

Am

plitu

de

02 04 06 08 1000Time (s)

times103

(b)



05

10152025303540

TEK

10 20 30 40 50 600Scale λ

The best scale

(a)

5

10

15

20

25

Am

plitu

de

02 04 06 08 1000Time (s)

(b)

0

1

2

3

Am

plitu

de

10

16

40 80 120 160 2000Frequency (Hz)

(c)


10 20 30 40 50 60 700Scale λ

00

01

02

03

04

05

FEF

e best scale

(a)

0

2

4

6

8

Am

plitu

de

02 04 06 08 1000Time (s)

times106

(b)

00

05

10

15

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

16

32

48

times106

(c)



00

05

10

15

20A

mpl

itude

200 400 600 800 10000Frequency (Hz)

Selected

u1u2

u3u4

(a)

ndash5

0

5

Am

plitu

de

02 04 06 08 1000Time (s)

(b)

00

02

04

06

08

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

16

3248

(c)


200 400 600 800 10000Frequency (Hz)

3

26

2

16

1

0

Leve

l k

0

01

02

03

04

05

(a)

0

1

2

3

4

Am

plitu

de

0604 08 100200Time (s)

(b)

Figure 16 Continued


000

005

010

015

020

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

16

50

(c)


12

3 45 6

78



Gearbox Main sha

Blade

Monitoring system

(a)

(b)










2d)cos2φ)


ndash50

ndash25

0

25

50

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(a)

0

1

2

3

Am

plitu

de (m

s2 )

1000 1000 3000 4000 5000 6000 7000 80000Frequency (Hz)

02

01

00200 250 300 350 400

3156Hz

(b)


4 6 8 102Iteration

05

10

15

20

Best

scor

e

(a)

ndash15

00

15

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

times107

(b)

Figure 19 Continued




5 10 15 20 25 30 35 40 45 500Scale λ

000

002

004

006

008

010

FEF

e best scale

(a)

00

05

10

15

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

times 1016

(b)

00

05

10

15

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

times 1010


630Hz (2lowastfo)

9469Hz (3lowastfo)

(c)


ndash08

00

08

Am

plitu

de (m

s2 )

0655 0660 0665 0670 0675 06800650Time (s)

32ms

times107

(c)



e best scale

10

15

20

25

30TE

K

1510 20 2550Scale λ

(a)

0

20

40

60

80

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

0

2

4

6

8

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

29Hz

3158Hz

(c)


00

05

10

15

20

Am

plitu

de (m

s2 )

2000 4000 6000 8000 100000Frequency (Hz)

Selected

u1u2u3

u4u5

(a)

ndash30

ndash15

0

15

30

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

00

05

10

15

20

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

29Hz

3158Hz

630Hz

(c)








2000 4000 6000 80000Frequency (Hz)

546

436

326

216

10

Leve

l k

0

02

04

06

08

1

12

(a)

000

005

010

015

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

0

2

4

6

8

Am

plitu

de (m

s2 )

100 200 300 400 500 600 7000Frequency (Hz)

times10ndash5

29Hz

3158Hz

(c)



Accelerometers

Thermocouples

(a)

Motor

Bearing 1

Bearing 2 Bearing 4

Bearing 3

Accelerometers


Motor

Bearing 1

Bearing 2 Bearing 4

Bearing 3


(b)




00

02

04

06

08

RMS

20 40 60 80 100 120 140 1600Time (hours)

02

01

0086 88 90 92 94 96

Zoom plot

Test ATest C

Test B

Stage 1

Stage 3

Stage 2

(a)

01 02 03 0400Time (s)

ndash06ndash03

0003

A06

Am

plitu

de

(b)

Figure 25 Continued


01 02 03 0400Time (s)

ndash10ndash05

000510 B

Am

plitu

de

(c)

01 02 03 0400Time (s)

ndash4ndash2

0

C24

Am

plitu

de

(d)


14

15

16

17

18

Best

scor

e

8 104 62Iteration

(a)

01 02 03 0400Time (s)

ndash8

ndash4

0

4

8

Am

plitu

de (m

s2 )

(b)


000

001

002

003

Am

plitu

de (m

s2 )

2000 4000 6000 8000 100000Frequency (Hz)

(a)

000

001

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

fFBPO

(b)


000

003

006

009

012

FEF

20 40 60 800Scale λ

The best scale

(a)

01 02 03 0400Time (s)

0

40

80

120

160

Am

plitu

de (m

s2 )

(b)

Figure 28 Continued




6 Discussion


0

4

8

12

16

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

fBPFO

2fBPFO3fBPFO

4fBPFO

(c)


01 02 03 0400Time (s)

00

02

04

06

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

000

001

002

003

Am

plitu

de (m

s2 )

fBPFO

2fBPFO

3fBPFO

(b)


01 02 03 0400Time (s)

ndash01

00

01

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

000

001

002

Am

plitu

de (m

s2 )

fFBPO

2fFBPO

(b)







7 Conclusions


Nomenclature




01 02 03 0400Time (s)

000

002

004

006

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

00

05

10

15

Am

plitu

de (m

s2 ) fBPFO

(b)












Data Availability




Acknowledgments


References














































05

10152025303540

TEK

10 20 30 40 50 600Scale λ

The best scale

(a)

5

10

15

20

25

Am

plitu

de

02 04 06 08 1000Time (s)

(b)

0

1

2

3

Am

plitu

de

10

16

40 80 120 160 2000Frequency (Hz)

(c)


10 20 30 40 50 60 700Scale λ

00

01

02

03

04

05

FEF

e best scale

(a)

0

2

4

6

8

Am

plitu

de

02 04 06 08 1000Time (s)

times106

(b)

00

05

10

15

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

16

32

48

times106

(c)



00

05

10

15

20A

mpl

itude

200 400 600 800 10000Frequency (Hz)

Selected

u1u2

u3u4

(a)

ndash5

0

5

Am

plitu

de

02 04 06 08 1000Time (s)

(b)

00

02

04

06

08

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

16

3248

(c)


200 400 600 800 10000Frequency (Hz)

3

26

2

16

1

0

Leve

l k

0

01

02

03

04

05

(a)

0

1

2

3

4

Am

plitu

de

0604 08 100200Time (s)

(b)

Figure 16 Continued


000

005

010

015

020

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

16

50

(c)


12

3 45 6

78



Gearbox Main sha

Blade

Monitoring system

(a)

(b)










2d)cos2φ)


ndash50

ndash25

0

25

50

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(a)

0

1

2

3

Am

plitu

de (m

s2 )

1000 1000 3000 4000 5000 6000 7000 80000Frequency (Hz)

02

01

00200 250 300 350 400

3156Hz

(b)


4 6 8 102Iteration

05

10

15

20

Best

scor

e

(a)

ndash15

00

15

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

times107

(b)

Figure 19 Continued




5 10 15 20 25 30 35 40 45 500Scale λ

000

002

004

006

008

010

FEF

e best scale

(a)

00

05

10

15

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

times 1016

(b)

00

05

10

15

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

times 1010


630Hz (2lowastfo)

9469Hz (3lowastfo)

(c)


ndash08

00

08

Am

plitu

de (m

s2 )

0655 0660 0665 0670 0675 06800650Time (s)

32ms

times107

(c)



e best scale

10

15

20

25

30TE

K

1510 20 2550Scale λ

(a)

0

20

40

60

80

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

0

2

4

6

8

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

29Hz

3158Hz

(c)


00

05

10

15

20

Am

plitu

de (m

s2 )

2000 4000 6000 8000 100000Frequency (Hz)

Selected

u1u2u3

u4u5

(a)

ndash30

ndash15

0

15

30

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

00

05

10

15

20

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

29Hz

3158Hz

630Hz

(c)








2000 4000 6000 80000Frequency (Hz)

546

436

326

216

10

Leve

l k

0

02

04

06

08

1

12

(a)

000

005

010

015

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

0

2

4

6

8

Am

plitu

de (m

s2 )

100 200 300 400 500 600 7000Frequency (Hz)

times10ndash5

29Hz

3158Hz

(c)



Accelerometers

Thermocouples

(a)

Motor

Bearing 1

Bearing 2 Bearing 4

Bearing 3

Accelerometers


Motor

Bearing 1

Bearing 2 Bearing 4

Bearing 3


(b)




00

02

04

06

08

RMS

20 40 60 80 100 120 140 1600Time (hours)

02

01

0086 88 90 92 94 96

Zoom plot

Test ATest C

Test B

Stage 1

Stage 3

Stage 2

(a)

01 02 03 0400Time (s)

ndash06ndash03

0003

A06

Am

plitu

de

(b)

Figure 25 Continued


01 02 03 0400Time (s)

ndash10ndash05

000510 B

Am

plitu

de

(c)

01 02 03 0400Time (s)

ndash4ndash2

0

C24

Am

plitu

de

(d)


14

15

16

17

18

Best

scor

e

8 104 62Iteration

(a)

01 02 03 0400Time (s)

ndash8

ndash4

0

4

8

Am

plitu

de (m

s2 )

(b)


000

001

002

003

Am

plitu

de (m

s2 )

2000 4000 6000 8000 100000Frequency (Hz)

(a)

000

001

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

fFBPO

(b)


000

003

006

009

012

FEF

20 40 60 800Scale λ

The best scale

(a)

01 02 03 0400Time (s)

0

40

80

120

160

Am

plitu

de (m

s2 )

(b)

Figure 28 Continued




6 Discussion


0

4

8

12

16

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

fBPFO

2fBPFO3fBPFO

4fBPFO

(c)


01 02 03 0400Time (s)

00

02

04

06

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

000

001

002

003

Am

plitu

de (m

s2 )

fBPFO

2fBPFO

3fBPFO

(b)


01 02 03 0400Time (s)

ndash01

00

01

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

000

001

002

Am

plitu

de (m

s2 )

fFBPO

2fFBPO

(b)







7 Conclusions


Nomenclature




01 02 03 0400Time (s)

000

002

004

006

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

00

05

10

15

Am

plitu

de (m

s2 ) fBPFO

(b)












Data Availability




Acknowledgments


References














































00

05

10

15

20A

mpl

itude

200 400 600 800 10000Frequency (Hz)

Selected

u1u2

u3u4

(a)

ndash5

0

5

Am

plitu

de

02 04 06 08 1000Time (s)

(b)

00

02

04

06

08

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

16

3248

(c)


200 400 600 800 10000Frequency (Hz)

3

26

2

16

1

0

Leve

l k

0

01

02

03

04

05

(a)

0

1

2

3

4

Am

plitu

de

0604 08 100200Time (s)

(b)

Figure 16 Continued


000

005

010

015

020

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

16

50

(c)


12

3 45 6

78



Gearbox Main sha

Blade

Monitoring system

(a)

(b)










2d)cos2φ)


ndash50

ndash25

0

25

50

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(a)

0

1

2

3

Am

plitu

de (m

s2 )

1000 1000 3000 4000 5000 6000 7000 80000Frequency (Hz)

02

01

00200 250 300 350 400

3156Hz

(b)


4 6 8 102Iteration

05

10

15

20

Best

scor

e

(a)

ndash15

00

15

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

times107

(b)

Figure 19 Continued




5 10 15 20 25 30 35 40 45 500Scale λ

000

002

004

006

008

010

FEF

e best scale

(a)

00

05

10

15

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

times 1016

(b)

00

05

10

15

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

times 1010


630Hz (2lowastfo)

9469Hz (3lowastfo)

(c)


ndash08

00

08

Am

plitu

de (m

s2 )

0655 0660 0665 0670 0675 06800650Time (s)

32ms

times107

(c)



e best scale

10

15

20

25

30TE

K

1510 20 2550Scale λ

(a)

0

20

40

60

80

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

0

2

4

6

8

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

29Hz

3158Hz

(c)


00

05

10

15

20

Am

plitu

de (m

s2 )

2000 4000 6000 8000 100000Frequency (Hz)

Selected

u1u2u3

u4u5

(a)

ndash30

ndash15

0

15

30

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

00

05

10

15

20

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

29Hz

3158Hz

630Hz

(c)








2000 4000 6000 80000Frequency (Hz)

546

436

326

216

10

Leve

l k

0

02

04

06

08

1

12

(a)

000

005

010

015

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

0

2

4

6

8

Am

plitu

de (m

s2 )

100 200 300 400 500 600 7000Frequency (Hz)

times10ndash5

29Hz

3158Hz

(c)



Accelerometers

Thermocouples

(a)

Motor

Bearing 1

Bearing 2 Bearing 4

Bearing 3

Accelerometers


Motor

Bearing 1

Bearing 2 Bearing 4

Bearing 3


(b)




00

02

04

06

08

RMS

20 40 60 80 100 120 140 1600Time (hours)

02

01

0086 88 90 92 94 96

Zoom plot

Test ATest C

Test B

Stage 1

Stage 3

Stage 2

(a)

01 02 03 0400Time (s)

ndash06ndash03

0003

A06

Am

plitu

de

(b)

Figure 25 Continued


01 02 03 0400Time (s)

ndash10ndash05

000510 B

Am

plitu

de

(c)

01 02 03 0400Time (s)

ndash4ndash2

0

C24

Am

plitu

de

(d)


14

15

16

17

18

Best

scor

e

8 104 62Iteration

(a)

01 02 03 0400Time (s)

ndash8

ndash4

0

4

8

Am

plitu

de (m

s2 )

(b)


000

001

002

003

Am

plitu

de (m

s2 )

2000 4000 6000 8000 100000Frequency (Hz)

(a)

000

001

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

fFBPO

(b)


000

003

006

009

012

FEF

20 40 60 800Scale λ

The best scale

(a)

01 02 03 0400Time (s)

0

40

80

120

160

Am

plitu

de (m

s2 )

(b)

Figure 28 Continued




6 Discussion


0

4

8

12

16

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

fBPFO

2fBPFO3fBPFO

4fBPFO

(c)


01 02 03 0400Time (s)

00

02

04

06

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

000

001

002

003

Am

plitu

de (m

s2 )

fBPFO

2fBPFO

3fBPFO

(b)


01 02 03 0400Time (s)

ndash01

00

01

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

000

001

002

Am

plitu

de (m

s2 )

fFBPO

2fFBPO

(b)







7 Conclusions


Nomenclature




01 02 03 0400Time (s)

000

002

004

006

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

00

05

10

15

Am

plitu

de (m

s2 ) fBPFO

(b)












Data Availability




Acknowledgments


References














































000

005

010

015

020

Am

plitu

de

40 80 120 160 2000Frequency (Hz)

16

50

(c)


12

3 45 6

78



Gearbox Main sha

Blade

Monitoring system

(a)

(b)










2d)cos2φ)


ndash50

ndash25

0

25

50

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(a)

0

1

2

3

Am

plitu

de (m

s2 )

1000 1000 3000 4000 5000 6000 7000 80000Frequency (Hz)

02

01

00200 250 300 350 400

3156Hz

(b)


4 6 8 102Iteration

05

10

15

20

Best

scor

e

(a)

ndash15

00

15

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

times107

(b)

Figure 19 Continued




5 10 15 20 25 30 35 40 45 500Scale λ

000

002

004

006

008

010

FEF

e best scale

(a)

00

05

10

15

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

times 1016

(b)

00

05

10

15

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

times 1010


630Hz (2lowastfo)

9469Hz (3lowastfo)

(c)


ndash08

00

08

Am

plitu

de (m

s2 )

0655 0660 0665 0670 0675 06800650Time (s)

32ms

times107

(c)



e best scale

10

15

20

25

30TE

K

1510 20 2550Scale λ

(a)

0

20

40

60

80

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

0

2

4

6

8

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

29Hz

3158Hz

(c)


00

05

10

15

20

Am

plitu

de (m

s2 )

2000 4000 6000 8000 100000Frequency (Hz)

Selected

u1u2u3

u4u5

(a)

ndash30

ndash15

0

15

30

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

00

05

10

15

20

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

29Hz

3158Hz

630Hz

(c)








2000 4000 6000 80000Frequency (Hz)

546

436

326

216

10

Leve

l k

0

02

04

06

08

1

12

(a)

000

005

010

015

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

0

2

4

6

8

Am

plitu

de (m

s2 )

100 200 300 400 500 600 7000Frequency (Hz)

times10ndash5

29Hz

3158Hz

(c)



Accelerometers

Thermocouples

(a)

Motor

Bearing 1

Bearing 2 Bearing 4

Bearing 3

Accelerometers


Motor

Bearing 1

Bearing 2 Bearing 4

Bearing 3


(b)




00

02

04

06

08

RMS

20 40 60 80 100 120 140 1600Time (hours)

02

01

0086 88 90 92 94 96

Zoom plot

Test ATest C

Test B

Stage 1

Stage 3

Stage 2

(a)

01 02 03 0400Time (s)

ndash06ndash03

0003

A06

Am

plitu

de

(b)

Figure 25 Continued


01 02 03 0400Time (s)

ndash10ndash05

000510 B

Am

plitu

de

(c)

01 02 03 0400Time (s)

ndash4ndash2

0

C24

Am

plitu

de

(d)


14

15

16

17

18

Best

scor

e

8 104 62Iteration

(a)

01 02 03 0400Time (s)

ndash8

ndash4

0

4

8

Am

plitu

de (m

s2 )

(b)


000

001

002

003

Am

plitu

de (m

s2 )

2000 4000 6000 8000 100000Frequency (Hz)

(a)

000

001

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

fFBPO

(b)


000

003

006

009

012

FEF

20 40 60 800Scale λ

The best scale

(a)

01 02 03 0400Time (s)

0

40

80

120

160

Am

plitu

de (m

s2 )

(b)

Figure 28 Continued




6 Discussion


0

4

8

12

16

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

fBPFO

2fBPFO3fBPFO

4fBPFO

(c)


01 02 03 0400Time (s)

00

02

04

06

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

000

001

002

003

Am

plitu

de (m

s2 )

fBPFO

2fBPFO

3fBPFO

(b)


01 02 03 0400Time (s)

ndash01

00

01

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

000

001

002

Am

plitu

de (m

s2 )

fFBPO

2fFBPO

(b)







7 Conclusions


Nomenclature




01 02 03 0400Time (s)

000

002

004

006

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

00

05

10

15

Am

plitu

de (m

s2 ) fBPFO

(b)












Data Availability




Acknowledgments


References





















































2d)cos2φ)


ndash50

ndash25

0

25

50

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(a)

0

1

2

3

Am

plitu

de (m

s2 )

1000 1000 3000 4000 5000 6000 7000 80000Frequency (Hz)

02

01

00200 250 300 350 400

3156Hz

(b)


4 6 8 102Iteration

05

10

15

20

Best

scor

e

(a)

ndash15

00

15

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

times107

(b)

Figure 19 Continued




5 10 15 20 25 30 35 40 45 500Scale λ

000

002

004

006

008

010

FEF

e best scale

(a)

00

05

10

15

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

times 1016

(b)

00

05

10

15

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

times 1010


630Hz (2lowastfo)

9469Hz (3lowastfo)

(c)


ndash08

00

08

Am

plitu

de (m

s2 )

0655 0660 0665 0670 0675 06800650Time (s)

32ms

times107

(c)



e best scale

10

15

20

25

30TE

K

1510 20 2550Scale λ

(a)

0

20

40

60

80

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

0

2

4

6

8

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

29Hz

3158Hz

(c)


00

05

10

15

20

Am

plitu

de (m

s2 )

2000 4000 6000 8000 100000Frequency (Hz)

Selected

u1u2u3

u4u5

(a)

ndash30

ndash15

0

15

30

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

00

05

10

15

20

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

29Hz

3158Hz

630Hz

(c)








2000 4000 6000 80000Frequency (Hz)

546

436

326

216

10

Leve

l k

0

02

04

06

08

1

12

(a)

000

005

010

015

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

0

2

4

6

8

Am

plitu

de (m

s2 )

100 200 300 400 500 600 7000Frequency (Hz)

times10ndash5

29Hz

3158Hz

(c)



Accelerometers

Thermocouples

(a)

Motor

Bearing 1

Bearing 2 Bearing 4

Bearing 3

Accelerometers


Motor

Bearing 1

Bearing 2 Bearing 4

Bearing 3


(b)




00

02

04

06

08

RMS

20 40 60 80 100 120 140 1600Time (hours)

02

01

0086 88 90 92 94 96

Zoom plot

Test ATest C

Test B

Stage 1

Stage 3

Stage 2

(a)

01 02 03 0400Time (s)

ndash06ndash03

0003

A06

Am

plitu

de

(b)

Figure 25 Continued


01 02 03 0400Time (s)

ndash10ndash05

000510 B

Am

plitu

de

(c)

01 02 03 0400Time (s)

ndash4ndash2

0

C24

Am

plitu

de

(d)


14

15

16

17

18

Best

scor

e

8 104 62Iteration

(a)

01 02 03 0400Time (s)

ndash8

ndash4

0

4

8

Am

plitu

de (m

s2 )

(b)


000

001

002

003

Am

plitu

de (m

s2 )

2000 4000 6000 8000 100000Frequency (Hz)

(a)

000

001

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

fFBPO

(b)


000

003

006

009

012

FEF

20 40 60 800Scale λ

The best scale

(a)

01 02 03 0400Time (s)

0

40

80

120

160

Am

plitu

de (m

s2 )

(b)

Figure 28 Continued




6 Discussion


0

4

8

12

16

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

fBPFO

2fBPFO3fBPFO

4fBPFO

(c)


01 02 03 0400Time (s)

00

02

04

06

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

000

001

002

003

Am

plitu

de (m

s2 )

fBPFO

2fBPFO

3fBPFO

(b)


01 02 03 0400Time (s)

ndash01

00

01

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

000

001

002

Am

plitu

de (m

s2 )

fFBPO

2fFBPO

(b)







7 Conclusions


Nomenclature




01 02 03 0400Time (s)

000

002

004

006

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

00

05

10

15

Am

plitu

de (m

s2 ) fBPFO

(b)












Data Availability




Acknowledgments


References
















































5 10 15 20 25 30 35 40 45 500Scale λ

000

002

004

006

008

010

FEF

e best scale

(a)

00

05

10

15

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

times 1016

(b)

00

05

10

15

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

times 1010


630Hz (2lowastfo)

9469Hz (3lowastfo)

(c)


ndash08

00

08

Am

plitu

de (m

s2 )

0655 0660 0665 0670 0675 06800650Time (s)

32ms

times107

(c)



e best scale

10

15

20

25

30TE

K

1510 20 2550Scale λ

(a)

0

20

40

60

80

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

0

2

4

6

8

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

29Hz

3158Hz

(c)


00

05

10

15

20

Am

plitu

de (m

s2 )

2000 4000 6000 8000 100000Frequency (Hz)

Selected

u1u2u3

u4u5

(a)

ndash30

ndash15

0

15

30

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

00

05

10

15

20

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

29Hz

3158Hz

630Hz

(c)








2000 4000 6000 80000Frequency (Hz)

546

436

326

216

10

Leve

l k

0

02

04

06

08

1

12

(a)

000

005

010

015

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

0

2

4

6

8

Am

plitu

de (m

s2 )

100 200 300 400 500 600 7000Frequency (Hz)

times10ndash5

29Hz

3158Hz

(c)



Accelerometers

Thermocouples

(a)

Motor

Bearing 1

Bearing 2 Bearing 4

Bearing 3

Accelerometers


Motor

Bearing 1

Bearing 2 Bearing 4

Bearing 3


(b)




00

02

04

06

08

RMS

20 40 60 80 100 120 140 1600Time (hours)

02

01

0086 88 90 92 94 96

Zoom plot

Test ATest C

Test B

Stage 1

Stage 3

Stage 2

(a)

01 02 03 0400Time (s)

ndash06ndash03

0003

A06

Am

plitu

de

(b)

Figure 25 Continued


01 02 03 0400Time (s)

ndash10ndash05

000510 B

Am

plitu

de

(c)

01 02 03 0400Time (s)

ndash4ndash2

0

C24

Am

plitu

de

(d)


14

15

16

17

18

Best

scor

e

8 104 62Iteration

(a)

01 02 03 0400Time (s)

ndash8

ndash4

0

4

8

Am

plitu

de (m

s2 )

(b)


000

001

002

003

Am

plitu

de (m

s2 )

2000 4000 6000 8000 100000Frequency (Hz)

(a)

000

001

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

fFBPO

(b)


000

003

006

009

012

FEF

20 40 60 800Scale λ

The best scale

(a)

01 02 03 0400Time (s)

0

40

80

120

160

Am

plitu

de (m

s2 )

(b)

Figure 28 Continued




6 Discussion


0

4

8

12

16

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

fBPFO

2fBPFO3fBPFO

4fBPFO

(c)


01 02 03 0400Time (s)

00

02

04

06

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

000

001

002

003

Am

plitu

de (m

s2 )

fBPFO

2fBPFO

3fBPFO

(b)


01 02 03 0400Time (s)

ndash01

00

01

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

000

001

002

Am

plitu

de (m

s2 )

fFBPO

2fFBPO

(b)







7 Conclusions


Nomenclature




01 02 03 0400Time (s)

000

002

004

006

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

00

05

10

15

Am

plitu

de (m

s2 ) fBPFO

(b)












Data Availability




Acknowledgments


References














































e best scale

10

15

20

25

30TE

K

1510 20 2550Scale λ

(a)

0

20

40

60

80

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

0

2

4

6

8

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

29Hz

3158Hz

(c)


00

05

10

15

20

Am

plitu

de (m

s2 )

2000 4000 6000 8000 100000Frequency (Hz)

Selected

u1u2u3

u4u5

(a)

ndash30

ndash15

0

15

30

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

00

05

10

15

20

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

29Hz

3158Hz

630Hz

(c)








2000 4000 6000 80000Frequency (Hz)

546

436

326

216

10

Leve

l k

0

02

04

06

08

1

12

(a)

000

005

010

015

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

0

2

4

6

8

Am

plitu

de (m

s2 )

100 200 300 400 500 600 7000Frequency (Hz)

times10ndash5

29Hz

3158Hz

(c)



Accelerometers

Thermocouples

(a)

Motor

Bearing 1

Bearing 2 Bearing 4

Bearing 3

Accelerometers


Motor

Bearing 1

Bearing 2 Bearing 4

Bearing 3


(b)




00

02

04

06

08

RMS

20 40 60 80 100 120 140 1600Time (hours)

02

01

0086 88 90 92 94 96

Zoom plot

Test ATest C

Test B

Stage 1

Stage 3

Stage 2

(a)

01 02 03 0400Time (s)

ndash06ndash03

0003

A06

Am

plitu

de

(b)

Figure 25 Continued


01 02 03 0400Time (s)

ndash10ndash05

000510 B

Am

plitu

de

(c)

01 02 03 0400Time (s)

ndash4ndash2

0

C24

Am

plitu

de

(d)


14

15

16

17

18

Best

scor

e

8 104 62Iteration

(a)

01 02 03 0400Time (s)

ndash8

ndash4

0

4

8

Am

plitu

de (m

s2 )

(b)


000

001

002

003

Am

plitu

de (m

s2 )

2000 4000 6000 8000 100000Frequency (Hz)

(a)

000

001

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

fFBPO

(b)


000

003

006

009

012

FEF

20 40 60 800Scale λ

The best scale

(a)

01 02 03 0400Time (s)

0

40

80

120

160

Am

plitu

de (m

s2 )

(b)

Figure 28 Continued




6 Discussion


0

4

8

12

16

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

fBPFO

2fBPFO3fBPFO

4fBPFO

(c)


01 02 03 0400Time (s)

00

02

04

06

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

000

001

002

003

Am

plitu

de (m

s2 )

fBPFO

2fBPFO

3fBPFO

(b)


01 02 03 0400Time (s)

ndash01

00

01

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

000

001

002

Am

plitu

de (m

s2 )

fFBPO

2fFBPO

(b)







7 Conclusions


Nomenclature




01 02 03 0400Time (s)

000

002

004

006

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

00

05

10

15

Am

plitu

de (m

s2 ) fBPFO

(b)












Data Availability




Acknowledgments


References



















































2000 4000 6000 80000Frequency (Hz)

546

436

326

216

10

Leve

l k

0

02

04

06

08

1

12

(a)

000

005

010

015

Am

plitu

de (m

s2 )

02 04 06 08 1000Time (s)

(b)

0

2

4

6

8

Am

plitu

de (m

s2 )

100 200 300 400 500 600 7000Frequency (Hz)

times10ndash5

29Hz

3158Hz

(c)



Accelerometers

Thermocouples

(a)

Motor

Bearing 1

Bearing 2 Bearing 4

Bearing 3

Accelerometers


Motor

Bearing 1

Bearing 2 Bearing 4

Bearing 3


(b)




00

02

04

06

08

RMS

20 40 60 80 100 120 140 1600Time (hours)

02

01

0086 88 90 92 94 96

Zoom plot

Test ATest C

Test B

Stage 1

Stage 3

Stage 2

(a)

01 02 03 0400Time (s)

ndash06ndash03

0003

A06

Am

plitu

de

(b)

Figure 25 Continued


01 02 03 0400Time (s)

ndash10ndash05

000510 B

Am

plitu

de

(c)

01 02 03 0400Time (s)

ndash4ndash2

0

C24

Am

plitu

de

(d)


14

15

16

17

18

Best

scor

e

8 104 62Iteration

(a)

01 02 03 0400Time (s)

ndash8

ndash4

0

4

8

Am

plitu

de (m

s2 )

(b)


000

001

002

003

Am

plitu

de (m

s2 )

2000 4000 6000 8000 100000Frequency (Hz)

(a)

000

001

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

fFBPO

(b)


000

003

006

009

012

FEF

20 40 60 800Scale λ

The best scale

(a)

01 02 03 0400Time (s)

0

40

80

120

160

Am

plitu

de (m

s2 )

(b)

Figure 28 Continued




6 Discussion


0

4

8

12

16

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

fBPFO

2fBPFO3fBPFO

4fBPFO

(c)


01 02 03 0400Time (s)

00

02

04

06

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

000

001

002

003

Am

plitu

de (m

s2 )

fBPFO

2fBPFO

3fBPFO

(b)


01 02 03 0400Time (s)

ndash01

00

01

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

000

001

002

Am

plitu

de (m

s2 )

fFBPO

2fFBPO

(b)







7 Conclusions


Nomenclature




01 02 03 0400Time (s)

000

002

004

006

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

00

05

10

15

Am

plitu

de (m

s2 ) fBPFO

(b)












Data Availability




Acknowledgments


References














































Accelerometers

Thermocouples

(a)

Motor

Bearing 1

Bearing 2 Bearing 4

Bearing 3

Accelerometers


Motor

Bearing 1

Bearing 2 Bearing 4

Bearing 3


(b)




00

02

04

06

08

RMS

20 40 60 80 100 120 140 1600Time (hours)

02

01

0086 88 90 92 94 96

Zoom plot

Test ATest C

Test B

Stage 1

Stage 3

Stage 2

(a)

01 02 03 0400Time (s)

ndash06ndash03

0003

A06

Am

plitu

de

(b)

Figure 25 Continued


01 02 03 0400Time (s)

ndash10ndash05

000510 B

Am

plitu

de

(c)

01 02 03 0400Time (s)

ndash4ndash2

0

C24

Am

plitu

de

(d)


14

15

16

17

18

Best

scor

e

8 104 62Iteration

(a)

01 02 03 0400Time (s)

ndash8

ndash4

0

4

8

Am

plitu

de (m

s2 )

(b)


000

001

002

003

Am

plitu

de (m

s2 )

2000 4000 6000 8000 100000Frequency (Hz)

(a)

000

001

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

fFBPO

(b)


000

003

006

009

012

FEF

20 40 60 800Scale λ

The best scale

(a)

01 02 03 0400Time (s)

0

40

80

120

160

Am

plitu

de (m

s2 )

(b)

Figure 28 Continued




6 Discussion


0

4

8

12

16

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

fBPFO

2fBPFO3fBPFO

4fBPFO

(c)


01 02 03 0400Time (s)

00

02

04

06

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

000

001

002

003

Am

plitu

de (m

s2 )

fBPFO

2fBPFO

3fBPFO

(b)


01 02 03 0400Time (s)

ndash01

00

01

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

000

001

002

Am

plitu

de (m

s2 )

fFBPO

2fFBPO

(b)







7 Conclusions


Nomenclature




01 02 03 0400Time (s)

000

002

004

006

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

00

05

10

15

Am

plitu

de (m

s2 ) fBPFO

(b)












Data Availability




Acknowledgments


References














































01 02 03 0400Time (s)

ndash10ndash05

000510 B

Am

plitu

de

(c)

01 02 03 0400Time (s)

ndash4ndash2

0

C24

Am

plitu

de

(d)


14

15

16

17

18

Best

scor

e

8 104 62Iteration

(a)

01 02 03 0400Time (s)

ndash8

ndash4

0

4

8

Am

plitu

de (m

s2 )

(b)


000

001

002

003

Am

plitu

de (m

s2 )

2000 4000 6000 8000 100000Frequency (Hz)

(a)

000

001

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

fFBPO

(b)


000

003

006

009

012

FEF

20 40 60 800Scale λ

The best scale

(a)

01 02 03 0400Time (s)

0

40

80

120

160

Am

plitu

de (m

s2 )

(b)

Figure 28 Continued




6 Discussion


0

4

8

12

16

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

fBPFO

2fBPFO3fBPFO

4fBPFO

(c)


01 02 03 0400Time (s)

00

02

04

06

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

000

001

002

003

Am

plitu

de (m

s2 )

fBPFO

2fBPFO

3fBPFO

(b)


01 02 03 0400Time (s)

ndash01

00

01

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

000

001

002

Am

plitu

de (m

s2 )

fFBPO

2fFBPO

(b)







7 Conclusions


Nomenclature




01 02 03 0400Time (s)

000

002

004

006

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

00

05

10

15

Am

plitu

de (m

s2 ) fBPFO

(b)












Data Availability




Acknowledgments


References
















































6 Discussion


0

4

8

12

16

Am

plitu

de (m

s2 )

200 400 600 800 10000Frequency (Hz)

fBPFO

2fBPFO3fBPFO

4fBPFO

(c)


01 02 03 0400Time (s)

00

02

04

06

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

000

001

002

003

Am

plitu

de (m

s2 )

fBPFO

2fBPFO

3fBPFO

(b)


01 02 03 0400Time (s)

ndash01

00

01

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

000

001

002

Am

plitu

de (m

s2 )

fFBPO

2fFBPO

(b)







7 Conclusions


Nomenclature




01 02 03 0400Time (s)

000

002

004

006

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

00

05

10

15

Am

plitu

de (m

s2 ) fBPFO

(b)












Data Availability




Acknowledgments


References


















































7 Conclusions


Nomenclature




01 02 03 0400Time (s)

000

002

004

006

Am

plitu

de (m

s2 )

(a)

200 400 600 800 10000Frequency (Hz)

00

05

10

15

Am

plitu

de (m

s2 ) fBPFO

(b)












Data Availability




Acknowledgments


References

















































Data Availability




Acknowledgments


References



































































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