title fault diagnosis method of gearbox multifeature

Research ArticleTitle Fault Diagnosis Method of Gearbox Multifeature FusionBased on Quadratic Filter and QPSO-KELM

ShuoMeng 1 Jianshe Kang 1 Xupeng Die2 XiaohanWu3 Kuo Chi 4 Zhipeng Dong1

and Weiyi Wu1

1Shijiazhuang Branch Army Engineering University of PLA Shijiazhuang 050000 China231690 Troops Jilin 132000 China3Dingzhou Yangjiazhuang Junior High School Dingzhou 073000 China4China Satellite Maritime Tracking and Control Department Jiangyin 214431 China

Correspondence should be addressed to Shuo Meng 864748906qqcom

Received 15 July 2020 Accepted 4 August 2020 Published 26 September 2020

Academic Editor Xiangyu Meng

Copyright copy 2020 Shuo Meng 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

Effective filtering and noise reduction feature extraction and fault diagnosis and prognostics technology are important toPrognostics and Health Management (PHM) of equipment +erefore a multifeature fusion fault diagnosis method based on thecombination of quadratic filtering and QPSO-KELM algorithm is proposed In the quadratic filtering stable filtering can reducethe impact of noise and fast-kurtogram can filtrate fault frequency bands with rich fault information +en the time-domainfrequency-domain and time-frequency parameters of the secondary filter signal are extracted MSSST was used to analyze thefiltered signal and the time-frequency image was obtained+e time-frequency parameter was extracted from the time-frequencyimage by 2DPCA and all the extracted parameters are taken as the fusion fault feature of the gearbox Finally the fault featureparameters are taken as the training sample and testing sample of QPSO-KELM for training and testing to achieve the purpose offault diagnosis +e experimental results show that the proposed method can effectively filter the noise complete the fault modeidentification of gearbox and improve the fault diagnosis accuracy better than other methods

1 Introduction

As a key component of mechanical transmission systemgearbox failure will lead to equipment shutdown economiclosses and even casualties +e earlier the gearbox fault isfound and repaired in time the less the loss will be causedGearbox is mainly composed of rotating shaft bearing andgear Its common fault types are gear fault and bearing fault+e fault will produce an impact signal but the backgroundnoise interference and complex propagation path make itdifficult to judge the fault state of gearbox timely and ac-curately In order to filter out the noise extract the effectivestate information and better monitor and diagnose the stateof gearbox the related research is being carried outgradually

As the signal is inevitably disturbed by noise in theprocess of acquisition filtering technology as a signal

denoising method plays an important role in the follow-upresearch At present the commonly used methods of fil-tering and denoising can be divided into linear filtering andnonlinear filtering Linear noise reduction algorithm is notsuitable for processing complex vibration signals +ereforethe nonlinear filtering noise reduction method is themainstream noise reduction method in the field of me-chanical equipment fault diagnosis +e commonly usedtechnologies include noise reduction method based onempirical mode decomposition (EMD) [1ndash4] noise reduc-tion method based on wavelet transform [5ndash7] and noisereduction method based on manifold learning [8ndash11] Forexample Boudraa and Cexus [1 2] proposed a noise re-duction method based on EMD threshold Gu et al [7] usedwavelet analysis to denoise data and Song et al [11] andothers used the manifold learning method to denoise Al-though the abovemethods can achieve some effective results

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 7407236 18 pageshttpsdoiorg10115520207407236

the mechanical transmission system is often disturbed byimpact noise in themotion process and the impact noise canbe better filtered by a stable filter

Feature extraction technology is the core of fault diagnosis+e commonly used feature parameters include time-domainfeature [12 13] frequency-domain feature [14] and time-frequency feature [15] Time-domain feature is the featureinformation directly extracted from the time-domain signalwhich can intuitively obtain the signal feature +e frequency-domain feature is extracted from the frequency spectrumobtained by Fourier transform +e time-frequency featurecontains more information +e commonly used time-fre-quency analysismethods include short-time Fourier transform(STFT) [16 17] continuous wavelet transform (CWT)[18 19] and S-transform (ST) [20ndash22] +e S-transform in-herits the advantages of STFT and CWT and develops on thebasis of STFT and CWT However there are still someproblems such as low resolution and energy leakage of thetime-frequency images obtained by using ST technology

Intelligent classification algorithm is widely used in thefield of fault diagnosis and its effect is good Common in-telligent algorithms include neural network [23ndash25] supportvector machine (SVM) [26 27] extreme learning machine(ELM) [28 29] kernel extreme learning machine (KELM)[30] deep learning [31 32] and othermethods Among themthe KELM algorithm has stronger comprehensive advantagesin sample demand network generalization calculation speedand accuracy and is more suitable for solving gearbox faultdiagnosis problems with high calculation speed and accuracyrequirements However the classification ability of KELMnetwork is affected by the value of kernel parameters andpenalty coefficient so it is necessary to optimize the abovestructural parameters +erefore a method using thequantum particle swarm optimization algorithm [33ndash35] tooptimize the structural parameters of KELM (QPSO-KELM)is proposed in this paper +e algorithm based on QPSO canoptimize the structural parameters of KELM with strongglobal search ability so as to improve the learning speed andclassification accuracy of KELM and improve the accuracy ofgearbox fault diagnosis

At present most of the methods are not perfect in termsof diagnosis process and structure and main problems are asfollows

(1) +e effect of signal filtering and noise reduction isnot obvious enough leading to the fault featureextraction which is not obvious enough

(2) +e common time-frequency analysis methods alsohave the problems of energy leakage and low reso-lution which cannot extract effective two-dimen-sional features

(3) +e fault diagnosis accuracy of intelligent classifi-cation algorithm is not enough To a large extent theclassification accuracy can be improved by im-proving the parameters

+erefore filtering noise reduction feature extractionand fault diagnosis technology are the focus of this paper[36]

In order to solve the above problems denoising featureextraction and intelligent fault diagnosis technology arelisted as the research priorities in this paper and the fol-lowing researches are carried out +e related steps involvedare shown in Figure 1

+e working process is summarized as follows

(1) Quadratic filtering collecting the signals of gearboxunder different fault conditions as the signals aresusceptible to noise the stable filter is first used forprimary filtering and then the fast-kurtogram isused to filter frequency bands with rich fault in-formation [37ndash39] to complete quadratic filtering

(2) Feature extraction one-dimensional features andtwo-dimensional features of the filtered signal areextracted One-dimensional features are character-istic parameters in the time domain and frequencydomain +e extraction process of two-dimensionalfeatures is that MSSST is first used for time-fre-quency analysis of the filtered signal and then2DPCA [40 41] is used to extract the two-dimen-sional characteristic parameters of the time-fre-quency image

(3) Fault diagnosis the extracted features are dividedinto training sample and test sample which are usedas the input of the QPSO-KELM model +e testsample is input into the trained QPSO-KELM andthen the result of fault diagnosis can be obtained

2 Basic Theory of the Proposed Method

21 Stable Filter and Fast-Kurtogram Materials andMethods should be described with sufficient details to allowothers to replicate and build on published results Pleasenote that publication of your manuscript implicates that youmust make all materials data computer code and protocolsassociated with the publication available to readers Pleasedisclose at the submission stage any restrictions on theavailability of materials or information New methods andprotocols should be described in detail while well-estab-lished methods can be briefly described and appropriatelycited

Research manuscripts reporting large datasets that aredeposited in a publicly available database should specifywhere the data have been deposited and provide the relevantaccession numbers If the accession numbers have not yetbeen obtained at the time of submission please state thatthey will be provided during review +ey must be providedprior to publication

Interventional studies involving animals or humans andother studies requiring ethical approval must list the au-thority that provided approval and the corresponding ethicalapproval code

Conventional signal processing methods usually filternoise by linear algorithms But when the noise is impact andthe trailing is serious the linear filter cannot deal with iteffectively Stable filter is better for this kind of impact signal

+e standard additive noise model of a signal is asfollows

2 Mathematical Problems in Engineering

xt st + nt t 1 2 3 (1)

where st is the signal of interest and nt is the noise Noisesignals come from a wide range of sources such asequipment operation resonance and environmental noise

+e distribution of such noise can be expressed by sym-metrical stable distribution Stable filter can filter out thenoise to the maximum extent and recover the signal st

For the noise which conforms to the stable distributionthe nonlinear filtering technology based on the same steady-

Stable filter

Fast-kurtogram

MSSST

Time-frequencyimages

Time-frequencyparameters

Fault diagnosis with QPSO-KLEM

First filter

Second filter

Quadraticfiltered signal

Time-domain andfrequency-domain

parameters

Trainingsamples

Testing samples

Is the result satisfactory

Trained QPSO-KLEM

Fault diagnosis

Multifeature

YesNo

Original vibration signal

Denosing

Feature extraction

Fault diagnosis

2 DPCA

Figure 1 +e flow of the proposed method +e structure consists of three parts denoising feature extraction and fault diagnosis

Mathematical Problems in Engineering 3

state distribution model has the best processing effect +esefilters have good robustness and can also effectively deal withthe situation of serious signal tailing distribution Supposeρ(x) minus logf(x) is the negative value of logarithmic den-sity and the cost function is defined as follows

C θ x1 xm( 1113857 1113944m

i1ρ xi minus θ( 1113857 (2)

+e stable filter is defined as the minimum θ value of thecost function

1113954st 1113954θSTABLE xtminus k1 xtminus k1+1 xtminus k2

1113872 1113873

1113954θSTABLE x1 xm( 1113857 argminθ

C θ x1 xm( 1113857

⎧⎪⎨

⎪⎩(3)

When ρ(x) minus logf(x) the minimum value of costfunction is the maximum likelihood estimation of parameterθ When the distribution is the Gaussian distribution ofα 2 ρ(x) minus x22 and the minimum value 1113954θLINEAR can beobviously got When 0lt αlt 2 filtering is nonlinear and theminimum value of equation (3) must be found by numericalmethod

A variety of filtering forms can be obtained by replacingthe cost function equation (2) +e simplest extension is toadd a nonnegative weight ωi

Cweighted θ x1 xm( 1113857 1113944m

i1ρ ωi xi minus θ( 1113857( 1113857 (4)

In the detection problem the known signal s1 sn isextracted from the signal interfered by stable distributednoise which can be expressed as follows

xt θst + nt t 1 2 3 (5)

If there is no known signal in the signal θ 0 if there is aknown mode in the signal θne 0 +e filtering process can berealized by constructing a real filter function

Cmatched θ x1 xm( 1113857 1113944m

i1ρ xi minus θsi( 1113857 (6)

By the minimization equation (6) 1113954θ can be obtained and1113954θ is the estimated value of the extracted known signal

After getting the stable-filtered signal the fast-kurtogram isused to select the frequency band with rich fault information+e basic theory of fast-kurtogram is as follows

Spectral kurtosis is a performance index reflecting theinstantaneous impact strength Its basic theory is to calculatethe kurtosis value of each spectral line so as to find thefrequency band of impact [42]+e calculation equation is asfollows

Kik lang c

ik(n)

11138681113868111386811138681113868

111386811138681113868111386811138684rang

lang cik(n)

11138681113868111386811138681113868

111386811138681113868111386811138682rang2

minus 2 i 0 2kminus 1 k 0 K minus 11113872 1113873

(7)

where cik(n) is the complex envelope of signal x(n) at the

corresponding central frequency fi (i + 2minus 1)2minus kminus 1 andbandwidth (Δf)k 2minus kminus 1

+e spectral kurtosis is proposed by Antoni et al Butthe calculation of kurtosis is cumbersome and not con-ducive to engineering practice +erefore Antoni furtherproposed the concept of fast spectral kurtosis +e fastspectral kurtosis is an inverted pyramid structure +eabscissa represents the frequency the ordinate representsthe number of decomposition layers and the color depthof the image represents the spectral kurtosis under dif-ferent central frequencies and bandwidths According tothe color depth of fast spectral kurtosis the optimalbandwidth and center frequency can be obtained +enthe fault frequency band is extracted by band-pass filterand then the square envelope order spectrum of faultsignal is obtained by square envelope analysis +einverted pyramid structure of the fast-kurtogram is shownin Figure 2

22MSSST-2DPCA For signal x(t) the S-transform of x(t)

can be defined as follows

S(t f) 1113946+infin

minus infinx(τ)g(τ minus t f)e

minus i2πfτdτ (8)

g(t minus τ f) |f|2π

radic eminus f2(tminus τ)22( ) (9)

where the result of S(t f) represents the time spectrum ofthe signal g(t minus τ f) is the Gaussian window function t andf represent time and frequency respectively and τ is a timeparameter that represents the position of the Gaussianwindow on the timeline

According to equation (9) synchronous squeezingS-transformation (SSST) is performed on signal x(t) andthe calculation steps are as follows

+e coefficients of S-transform are Sx(t f) which can beobtained by equation (8)

Calculate the time partial derivative of the S-transformand the equation is as follows

zS(t f)

zt iω middot 1113946

+infin


minus i2πfτdτ

iωS(t f)

(10)

+e instantaneous frequency ωx can be obtained bytransforming equation (10) and the calculated result is

ωx(tω) ztS(t f)

iS(t f) (11)

By compressing the value of the interval[ωx minus (12)Δωωx + (12)Δω] to ωx we can get the syn-chronous compression transformation value SSST +eformula is as follows

SSST(t f) 1113946+infin

minus infinS(t ξ)δ πf minus ωx(t ξ)( 1113857dξ (12)

+e time-frequency analysis method of MSSST can beobtained by multiple iterations of the SSST and the iterativeequation of MSSST is as follows


SSST[1] SSST

SSST[2](t f) 1113946

+infin

minus infinSSST[1]

(t ξ)δ 2πf minus ωx(t ξ)( 1113857dξ

1113946+infin

minus infinS(t ξ)δ 2πf minus δ 2πf minus ωx(t ξ)( 1113857( 1113857dξ

SSST[3](t f) 1113946

+infin

minus infinSSST[2]


1113946+infin

minus infinS(t ξ)δ 2πf minus δ 2πf minus δ 2πf(((

minus ωx(t ξ)111385711138571113857dξ

⋮

SSST[N](t f) 1113946

+infin

minus infinSSST[Nminus 1]


1113946+infin

minus infinS(t ξ) δ middot middot middot 2πf minus δ 2πf(((

minus δ 2πf minus ωx(t ξ)( 1113857 middot middot middot11138571113857dξ

(13)

where SSST[1] is the SSST and N is the number of iterations(Nge 2) Substituting SSST[kminus 1] into SSST[k] (k 2 3 N)after N minus 1 calculations the final result can be simplified to

MSSST SSST[N](t f)

1113946+infin

minus infinS(t ξ)δ 2πf minus ω[N]

x (t ξ)1113872 1113873dξ(14)

For MSSST it is only required to calculate ω[N]x (t ξ) and

substitute it into the first-order SSST and the calculatedresult is the same as equation (13) +erefore the compu-tational complexity of MSSST is similar to that of the first-order SSST and it is beneficial to improve the calculationspeed +e proposed MSSST is effective to solve energyleakage and low resolution of the spectrum

After getting the time-frequency images 2DPCA isused to extract the feature matrix +e theory of two-dimensional principal component analysis (2DPCA) is asfollows

Suppose X is the n-dimensional column vector A is theimage matrix of mtimes n and Y is the m-dimensional pro-jection vector of A on X after linear transformation +eprocess can be expressed as follows

Y AX (15)

+e trace of covariance matrix Y of Sx is defined as thetotal divergence

J(X) tr Sx( 1113857 (16)

In maximum total divergence criterion when the di-vergence of the projected vector Y is the largest the directionof vector X is the optimal projection direction where theequation of Sx is

Sx E(Y minus EY)(Y minus EY)T

E[AX minus E(AX)][AX minus E(AX)]T

E[(A minus EA)X][(A minus EA)X]T

(17)

+e total divergence of Sx is

tr Sx( 1113857 XT

E(A minus EA)T(A minus EA)1113960 1113961X (18)

+e covariance matrix expression of the image is asfollows

Gt E(A minus EA)T(A minus EA) (19)

According to equation (19) Gt is a nonnegative definitentimes n-dimensional matrix Suppose there are M trainingimages the j-th image is represented as Aj and the meanvalue of all training images is recorded as A then the di-vergence matrix expression is as follows

Gt 1

M1113944

M

j1Aj minus A1113872 1113873

TAj minus A1113872 1113873 (20)

+e normalized expression is as follows

J(X) XTGtX (21)

+e X obtained from the criterion formula of themaximum divergence is called the optimal projection axis+e optimal projection axis X is the eigenvector corre-sponding to the maximum eigenvalue of Gt In general oneoptimal projection axis is not enough +erefore the first dorthogonal unit eigenvectors with the largest eigenvalues areselected as the optimal projection axis which can beexpressed by the following equation

X1 Xd1113864 1113865 argmax J(X)

XTi Xj 0 ine j j 1 d

⎧⎨

⎩ (22)

+e optimal projection vector X1 Xd of 2DPCA isused to extract the two-dimensional features of the imageSuppose sample image A according to the above equationsY1 Yd can be obtained which is called the principalcomponent vector of sample image A

+e matrix B [Y1 Yd] of mtimes d composed ofprincipal component vectors is the feature matrix (featureimage) of sample image A

23 QPSO-KELM KELM algorithm is adding kernelfunction into ELM and the input weights and offsets in ELMare replaced by kernel function mapping which makes theoutput of the ELM more stable and solves the over learningproblems

ELM is a single hidden layer feedforward neural net-work and its hidden layer weight does not need to be ad-justed by feedback regulation +e structure of the ELM isshown in Figure 3

Suppose Z xi ti1113864 1113865N

i1 are N training sample sets wherexi [xi1 xi1 xim]T isin Rm is the input characteristic


parameter and ti [ti1 ti1 tin]T isin Rn is the sample labelthen the ELM training model can be expressed as

1113944

M

i1βig wi middot xi + bi( 1113857 oi i 1 2 N (23)

where wi [w1i w2i wmi] and βi [βi1 βi2 βin]T

represent the input and output weight matrix of the ithhidden layer node respectively oj [oi1 oi2 oin]T

represents the output layer vector of the ELM+e goal of ELM training is to make the error between

the actual output of the training sample and the sample labelclose to 0 and the equation is

1113944N

i1oi minus ti

0βiwibi 1113944

M

i1βig wi middot xi + bi( 1113857 ti

i 1 2 NHβ T

(24)

So βi wi and bi can be used to obtain equation (25) asfollows

1113944

M

i1βig wi middot xi + bi( 1113857 ti i 1 2 N (25)

Equation (23) can also be expressed as a matrix

Hβ T (26)

where H is the output matrix of the hidden layer and theequation of β is as follows

βprime H+T (27)

where H+ represents the inverse matrix of H and βprime is theoptimal quadratic solution to β

According to the structural optimization and ERMcriteria the output weight β of ELM can be determined thatis

minimize LPELM12β

2+ C

12

1113944

N

i1ξi

2

subject to h xi( 1113857β tTi minus ξT

i i 1 N

(28)

where ξi [ξi1 ξim]T represents the network calcula-tion error (12)β2 is the structural error (12) 1113936

Np1 ξp2

is the empirical error and C is the penalty factor+rough the optimization of the solution the optimal

solution of β can be obtained as follows

β HT I

C+ HH

T1113874 1113875

minus 1T (29)

+erefore the output model function of ELM is asfollows

oi hiHT I

C+ HH

T1113874 1113875

minus 1T (30)

By using kernel operation to replace matrix operation Hin ELM the KELM classification model can be established+e kernel matrix is defined as follows

Ω HHT Ωij h xi( 1113857 middot h xj1113872 1113873

K xi xj1113872 1113873(31)

where Ω is the symmetric matrix of N times N and K(x y) isthe kernel function

Equation (30) can be expressed by kernel function asfollows

oi K xi xj1113872 1113873 K xi xj1113872 11138731113960 1113961λ (32)

where λ represents the output weight matrix of KELMDifferent support vector machines can be constructed by

selecting different kernel functions Among the commonlyused kernel functions compared with the polynomial kernelfunction and sigmoid kernel function the radial basis kernelfunction has the advantages of simple parameters and strongadaptability to randomly distributed samples +erefore a

Xi

1

2

m

1

2

L

1

2

n

oj

Inputlayer

Hidden layer

Output layer

Figure 3 ELM model ELM is made up of three parts input layerhidden layer and output layer

k23

k01 k1

1

k02 k1

2 k22 k3

2

k03 k13 k3

3 k43 k5

3 k63 k7

3

fs8 fs2fs4

k0 0

1

2

k

3

Figure 2 Fast-kurtogram +e grid in the diagram represents thefrequency band


SVM based on radial basis kernel function is established inthis paper and its expression is shown as follows

K x xi( 1113857 expminus x minus xi

2

σ2⎛⎝ ⎞⎠ (33)

According to equations (28) and (33) it can be seen thatKELM contains kernel parameter and penalty coefficient Cand it is necessary to find the optimal parameter to make theclassification effect of KELM the best

By quantizing the iterative updating process of particles ofPSO algorithm theQPSO can reduce the algorithm complexityand improve algorithm convergence speed and global searchability +e basic principles of QPSO are as follows

Assume that Ω is the d-dimensional search space andthe population number of particles in the space is b then theposition of the ith particle can be expressed as

Vi vi1 vi2 vi d( 1113857 (34)

Suppose the individual optimal position of the particle ispibest and the global optimal position of the particle is pgbestthe pibest and pgbest are as follows

pibest pi1 pi2 pi d( 1113857

pgbest pg1 pg2 pg d1113872 1113873(35)

+e particle can find and update its individual optimalposition pi and population optimal position pg throughiterative operation and the average optimal position mbestcan be introduced as the population optimal center +enthe particle optimization process can be expressed as

mbest 1

M1113944

M

i1pibest

1

M1113944

M

i1pi1

1M

1113944

M

i1pi2

1M

1113944

M

i1pi d

⎡⎣ ⎤⎦

(36)

pi d φpi d +(1 minus φ)pg d φ isin (0 1) (37)

vi d pi d plusmn α mbest minus vi d

11138681113868111386811138681113868111386811138681113868ln

1u

1113874 1113875 u isin (0 1) (38)

where α is the contraction expansion factor which is dy-namically adjusted in the iterative operation according to thefollowing equation

α α1 minus α2( 1113857 timesN minus t

N+ α2 (39)

where N represents the maximum number of iterations α1and α2 are the initial and final values of α respectivelyusually the two parameters are set as α1 05 and α2 1

According to the basic principle of KELM andQPSO thenetwork training process which used QPSO to optimize thenetwork structure parameters of KELM is as follows

Step 1 Initialize the position of population particlesand set parameters such as particle swarm size iterationstep size and termination conditions

Step 2 Initialize the current position of each particle aspibest define the classification accuracy of KELM asfitness function and calculate the fitness value of eachparticle +e position of the particle with the maximumfitness value was initialized as pgbestStep 3 Update the particle position according toequations (36)ndash(38)Step 4 +e fitness value of each particle is calculatedand the individual optimal position pbest the groupoptimal position pgbest and the group optimal centermbest are updated based on the optimal fitness valueStep 5 Judge whether the termination conditions aremet If so stop the calculation and output the result ifnot return to step 3

According to the above steps the modeling flow chart ofQPSO-KELM can be drawn as shown in Figure 4

3 Experimental Verification and Discussion

To verify whether the proposed method is available threepreset fault tests of the gearbox are conducted +e faultsimulation test rig is shown in Figure 5 +e test rig consistsof four parts the power and control part the bearing faultsimulation part (not used) the gear fault simulation partand the data acquisition part (not shown) +is sectionmainly uses the gearbox fault simulation part +e per-spective and internal structure of the gearbox are shown inFigure 6 +e teeth number in the gearbox from high-speedshaft to low-speed shaft is 41 79 36 and 90 +e fault gear isgear 3 with 36 teeth +e fault types include wear break andmiss teeth +e fault bearing is located in the deep grooveball bearing at the end cover of the gear 3 side of the me-dium-speed shaft (Figure 6(b))+e deep groove ball bearingtype is ER-16k and its fault types include inner race fault andouter race fault

In the test two vibration acceleration sensors are set invertical and horizontal directions respectively (Figure 6(c))+e data acquisition parameters are set as follows samplingfrequency fs 2048 kHz sampling time t 48 s and motorspeed 20 rs +e main dimension parameters of the bearingER-16K are shown in Table 1 According to the speed of themotor and the parameters of the gearbox and the bearingER-16K the relevant frequency can be calculated +ecalculation results are shown in Table 2

fr1 is the rotating frequency of high-speed axis fr2 is therotating frequency of medium-speed axis and fr3 is therotating frequency of low-speed axis fm1 is the meshingfrequency of gear 1 and gear 2 and fm2 is the meshingfrequency of gear 3 and gear 4 fBPFO is the fault characteristicfrequency of the bearing outer race fBPFI is the fault char-acteristic frequency of the bearing inner race and fBSF is thefault characteristic frequency of the bearing ball spin

31 Denoising with Quadratic Filter +e quadratic filteringmethod can effectively filter out the interference noise in thesignal and select the fault frequency band with rich faultinformation Two cases of bearing inner race fault and gear


broken tooth fault are selected as fault cases in the quadraticfiltering part

Firstly the stable filter is used to filter out the noise andthe detailed process is as follows

+e fault signal of bearing inner race and gear brokentooth are filtered with stable filter and the filtering pa-rameters are set as α 15 β 0 and c 1 then the filteredspectrum can be obtained as shown in Figure 7

Comparing with the spectrum before and after filteringthe signal-to-noise ratio (SNR) is improved with a stablefilter +e SNR of bearing inner race fault before filtering isminus 93491 the SNR after filtering is increased to minus 82371 theSNR before filtering is minus 87745 and the SNR after filtering isminus 81016 Comparing with the spectrum before and afterfiltering it can be seen that the trailing part of the signal isreduced which indicates that the stable filter is effective toreduce noise

Secondly the fast-kurtogram technology is used toprocess the filtered signals and the fault frequency bandwith rich fault information can be obtained +e fast-kurtogram diagram and square envelope results of bearinginner race fault and tooth break fault are shown inFigure 8

Figure 9 is the square envelope spectrum of bearing innerring fault and gear broken tooth after quadratic filtering Itcan be seen from Figure 9 that the feature frequency ofbearing inner race fault is obvious and the bearing innerrace fault can be judged according to the fault feature fre-quency In the square envelope spectrum of tooth broken

fault the meshing frequency frequency doubling and thesurrounding side band can be observed so as to judge thegear tooth broken fault

+e above results show that the fault information ofgearbox can be reflected to a great extent through the signalanalysis after quadratic filtering

32 Feature Parameters of Time Domain and FrequencyDomain Time-domain feature parameters after quadraticfiltering are as follows mean square value (P1) root meansquare (P2) variance (P3) standard deviation (P4) energy(P5) root square amplitude (P6) mean square amplitude(P7) mean square amplitude (P8) kurtosis (P9) skewness(P10) waveform index (P11) peak index (P12) pulse index(P13) margin index (P14) and clearance coefficient (P15)+ese characteristic parameters are extracted from theoriginal signal Frequency-domain characteristic parametersinclude mean frequency (MF P16) frequency center (FCP17) root mean square frequency (RMSF P18) and stan-dard deviation frequency (STDF P19) In order to facilitatecalculation the extracted feature parameters are normalizedand the normalized parameters are expressed by NP asshown in Table 3

33 Feature Images Extraction with MSSST-2DPCATime-frequency analysis is carried out on the filtered innerrace fault signal of bearing and tooth break fault signal andthe time-frequency analysis results of STFT CWT ST andMSSST are compared +e time-frequency analysis resultsare shown in Figures 10 and 11

From the time-frequency analysis images it can be seenthat the order of time-frequency resolution and energyaggregation is MSSSTgt STgtCWTgt STFT both in bearinginner race fault or gear broken tooth fault

After getting the time-frequency images 2DPCA is usedto extract the six cases of normal tooth miss tooth breaktooth wear and bearing inner race fault and outer race fault+e parameter settings are as follows the original image sizeis 258 lowast 330 and the image size after 2DPCA feature ex-traction is 72 lowast 99 By using the MSSSTmethod 960 imagesare generated for each case and a total of 5760 images can beobtained +en the 2DPCA algorithm is used to extract thefeature matrix of the image Comparing STFT and MSSSTand the results are as follows

+e images in Figure 12 and 13 are the time-frequencyfeatures of the filtered fault signal Comparing the results of2DPCA-STFT and 2DPCA-MSSST it can be seen that thefeature extracted by 2DPCA-MSSST contains more abun-dant fault information

34 Feature Fusion In the previous paper 19 feature pa-rameters are extracted by calculation and the feature matrixcorresponding to the time-frequency image is obtained byusing MSSST-2DPCA technology +e fusion features are

Initialize particle swarm size evolutionary algebra and particle position (C σ)

Determine particle fitness function(the test classification accuracy of KELM)

Calculate the fitness function value of each particle

Update individual optimal location group optimal location and group optimal center

Output the optimal parameter combination of KELM(C σ)

Whether the terminationconditions are satisfied

No

Yes

Figure 4 Flow chart of QPSO-KELM QPSO is used to searchglobal optimal parameters of KELM


obtained by fusing the feature parameters and the featurematrix +e specific steps are as follows

(1) Since the size of the feature image is 72 lowast 99 in orderto ensure the same length of the feature vector 99segments of the same length signal are interceptedfrom the collected signals 19 feature parameters areextracted respectively and the feature parametermatrix of 19 lowast 99 can be obtained

(2) For the two sensors (Figure 6(c)) the matrix offeature parameters is extracted according to themethod in step 1 and the eigenvector of the featurematrix is calculated

(3) Calculate the feature vectors of feature image(4) By combining the feature vectors corresponding to

the two feature parameters and the eigenvectors

corresponding to the feature images fusion featurematrix can be obtained

+e corresponding image is shown in Figure 14

35 Fault Diagnosis Based on QPSO-KELM +e time-do-main frequency-domain and time-frequency feature param-eters obtained above are summarized then 960 groups offeatures can be obtained and the features are randomly dividedinto training samples and testing samples according to the ratioof 3 1 the training samples number is 720 and the testingsamples number is 240 +e 720 training samples are used totrain the QPSO-KELM After training the 240 testing samplesare input into the QPSO-KELM and then the test results areobtained as shown in Figure 15 +ere is no error point inFigure 15 so the classification accuracy is 100

Power and control part Bearing part Gearbox part

Motorcontroller

Speedindicator

Wheels

Bearing 1 Bearing 2

ShaftGearbox Brake

Brake controller

Motor

Figure 5 +e mechanical fault simulation test bench

Gear 1

Inputshaft

Output shaft

Gear 2

Gear 3

Gear 4

(a)

Gear 141 teeth

Gear 279 teeth

Gear 490 teeth

Gear 336 teeth

High-speedshaft

Low-speedshaft

Faultbearing

Medium-speedshaft

(b)

Sensor 2

Sensor 1

(c)

Figure 6 +e structure of gearbox (a) perspective view (b) the inner structure (c) the outer structure

Table 1 Main dimensions of bearing ER-16K

Number of rolling elements Nb Ball diameter d (inch) Pitch diameter D (inch) Contact angle θ (deg)9 03125 1516 0

Table 2 Relevant frequencies of the gearbox

Rotating frequency (rmiddotsminus 1) Meshing frequency (Hz) Fault feature frequency of bearing (Hz)fr1 fr2 fr3 fm1 fm2 fBPFO fBPFI fBSF20 10 4 820 37167 36588 54965 23787


4

35

3

25

2

15

1

05

00 2000 4000 6000 8000 10000

Frequency (HZ)

Am

plitu

de (m

s2 )

SNR = ndash93491

times10ndash3

(a)

4

35

3

25

2

15

1

05

00 2000 4000 6000 8000 10000

Frequency (HZ)

Am

plitu

de (m

s2 )

SNR = ndash82371

times10ndash3

(b)

3

25

2

15

1

05

00 2000 4000 6000 8000 10000

Frequency (HZ)

Am

plitu

de (m

s2 )

SNR = ndash87445

times10ndash3

(c)

3

25

2

15

1

05

00 2000 4000 6000 8000 10000

Frequency (HZ)

Am

plitu

de (m

s2 )

SNR = ndash81016

times10ndash3

(d)

Figure 7 +e spectrum of fault signal before and after filtering (a) inner race fault-without filter (b) inner race fault-stable filter (c) toothbreak-without filter (d) tooth break-stable filter

02

025

015

01

005

00 2000 4000 6000 8000 10000

Frequency (HZ)

01

162

263

364

465

566

Leve

l k

fb-kurt1 ndash Kmax = 03level 3 Bw = 1280Hz fc = 3200Hz

(a)

2

15

1

05

00 2000 4000 6000 8000 10000

Frequency (HZ)

01

162

263

364

465

566

Leve

l k


(b)

Figure 8 +e fast-kurtogram of fault signal (a) inner race fault (b) tooth break fault


Frequency (HZ)

Am

plitu

de (m

s2 )

8

6

4

2

0

f BPM

2fBP

M

3fBP

M

4fBP

M

5fBP

M

times10ndash10

0 200100 400300 600500 800700

(a)

times10ndash10

Frequency (HZ)

Am

plitu

de (m

s2 )

3

2

1

0

f m2 2f

m2

3fm

2

4fm

2

0 500 1000 1500 2000

(b)

Figure 9 +e square envelope spectrum of fault signal (a) inner race fault (b) tooth break fault

6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)002 004 006 008 01 012 014 016 018

(a)

5000

4500

4000

3000

3500

2500

2000

1000

1500

500

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(b)

Figure 10 Continued

Table 3 Normalized fault characteristic parameters

Parameters Normal Tooth miss Tooth break Tooth wear Bearing inner race Bearing outer raceNP1 minus 01722 minus 02070 minus 05214 minus 04619 02612 minus 00560NP2 02163 00154 minus 02039 05636 minus 00097 minus 02371NP3 03545 minus 00780 01330 minus 00686 minus 06341 minus 00873NP4 minus 03568 minus 07051 minus 07467 03346 minus 05576 minus 00312NP5 minus 02626 minus 03410 00753 04721 04040 minus 01450NP6 00315 06561 minus 03575 minus 05123 03392 03301NP7 02004 01586 01160 01654 minus 01836 minus 01569NP8 minus 00779 02327 00570 minus 04184 minus 00005 minus 05677NP9 minus 00356 minus 00050 minus 00466 minus 00359 minus 00488 minus 01385NP10 02045 02679 02693 minus 03857 01252 02183NP11 minus 03601 00658 00311 00880 minus 01048 minus 02282NP12 minus 01763 minus 01525 minus 02617 minus 04034 minus 02318 01652NP13 minus 05131 minus 00078 03056 minus 03098 minus 00876 minus 06876NP14 minus 04827 03851 minus 01396 04718 00795 minus 03908NP15 minus 01136 01951 minus 03124 minus 05820 minus 01669 01302NP16 minus 02533 minus 00725 04445 03444 minus 00275 minus 03765NP17 minus 03963 minus 03155 minus 02906 00142 minus 02101 minus 02109NP18 minus 06192 minus 00573 01311 00116 01430 01435NP19 minus 02871 02230 minus 00180 05400 01824 01822


10000

9000

8000

7000

6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)002 004 06 008 01 012 014 016 018

(a)

9000

8000

7000

6000

5000

4000

3000

2000

1000

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(b)

10000

9000

8000

7000

6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(c)

9000

10000

8000

7000

6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(d)

Figure 11 Time-frequency analysis of tooth break fault (a) STFT (b) CWT (c) ST (d) MSSST

6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(c)

6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(d)

Figure 10 Time-frequency analysis of inner race fault (a) STFT (b) CWT (c) ST (d) MSSST


4 Comparison and Discussion

In order to verify the effectiveness of the proposed methodELM and PSO-KELM algorithms are used to compare withthe QPSO-KELM algorithm Meanwhile in order to verify

the noise filtering effect of quadratic filter the fault classi-fication accuracy before filtering is also analyzed based onthe above three methods which is compared with the faultdiagnosis effect after filtering +e experimental results areshown in Table 4 and Figure 16

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(a)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(b)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(c)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(d)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(e)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(f )

Figure 12 Time-frequency feature based on 2DPCA-STFT (a) normal (b) miss tooth fault (c) break tooth fault (d) wear tooth fault (e)bearing inner race fault (f ) bearing outer race fault


10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(a)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(b)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(c)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(d)

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

10

(e)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(f )

Figure 13 Time-frequency feature based on 2DPCA-MSSST (a) normal (b) miss tooth fault (c) break tooth fault (d) wear tooth fault (e)bearing inner race fault (f ) bearing outer race fault


In Figure 16 and Table 4 it can be seen that among allthe classification methods QPSO-KELM has the best clas-sification effect and the highest fault diagnosis accuracy Bycomparing the fault diagnosis accuracy before and after

filtering it can be concluded that no matter which method isused the fault diagnosis accuracy after filtering has beenimproved +erefore the effectiveness of the proposedmethod is proved

Sensor 1

Sensor 2

Sensor 1

Vibrationsignal

Vibrationsignal

Vibrationsignal

Matrix of featureparameters (19lowast99)


Matrix of featureimages (72lowast99)

Featurevectors

Featurevectors

Featurevectors

Featurevectors

Featurevectors

Fusion featurematrix

Figure 14 Fusion feature flow Feature vectors are extracted from parameter matrix and image matrix respectively for feature fusion

2

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se

t

NormalRollerInner

OuterInnerOuter

Sample number

Figure 15 QPSO-KELM classification results with quadratic filter +e horizontal axis is the number of samples and the vertical axis is theoutput of QPSO-KELM

Table 4 Comparison of fault diagnosis results

ELM PSO-KELM QPSO-KELMWithout filter Filter Without filter Filter Without filter () Filter ()

Training accuracy 5334 7567 7267 9817 7983 100Testing accuracy 5334 7567 6667 9875 7083 100


0 50 100 150 200 250Sample number

15

1

05

0

ndash05

ndash1

ndash15

ndash2

KELM

out

put f

or te

st se

t

NormalRollerInnerOuter

InnerOuterError point

(a)

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se

tNormalRollerInnerOuter


Sample number

(b)

1

05

0

ndash05

ndash1

ndash150 50 100 150 200 250

Sample number

KELM

out

put f

or te

st se

t



(c)

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se

t

Sample number



(d)

Figure 16 Continued


5 Conclusion

In this paper a fault diagnosis method based on gearboxmultifeature fusion quadratic filter and QPSO-KELM algo-rithm is proposed +rough the analysis of the experimentalresults the following conclusions can be drawn after stablefiltering of the collected signal the noise interference can beeffectively filtered out and the SNR can be improved Com-bined with the method of fast-kurtogram the fault frequencyband with rich fault characteristic signals can be screened outwhich is also conducive to the improvement of fault diagnosisaccuracy By extracting the time-domain frequency-domainand time-frequency characteristic parameters more faultcharacteristic information can be extracted which is beneficialto improve the accuracy of fault diagnosis +eMSSSTmethodis proposed to improve the time-frequency analysis methodwhich can effectively solve the problems of energy leakage andlow resolution+e KELM algorithm optimized by QPSO to alarge extent improves the learning ability and classificationaccuracy of the KELM algorithm Compared with othermethods the classification effect is better

Although the method proposed in this paper is betterthan the traditional fault diagnosis method it still needs tocontinue to explore in some aspects looking for some newfault characteristic parameters to make it more conducive tomonitor the current state of the equipment In this paperonly some single fault modes of the gearbox are analyzedbut in the actual situation the compound fault of gearbox ismore common So the next step of the study is to extract

more sensitive and effective fault feature parameters and tryto analyze more complex gearbox failure

Data Availability

+e experiment data are obtained through the laboratorybench of our unit which is not open to the public

Conflicts of Interest

+e authors declare that there are no conflicts of interest

References

[1] A-O Boudraa and J-C Cexus ldquoEMD-based signal filteringrdquoIEEE Transactions on Instrumentation and Measurementvol 56 no 6 pp 2196ndash2202 2007

[2] A O Boudraa and J C Cexus ldquoDenoising via empirical modedecompositionrdquo International Symposium on Communica-tions Control and Signal Processing vol 6 2006

[3] R J Hao X J An and Y L Shi ldquoSingle-channel blind sourceseparation based on EMD and CICA and its application togearbox multi-fault diagnosisrdquo Journal of Vibration andShock vol 38 no 8 pp 230ndash236 2019

[4] Y Shen Z Zhao and S Yang ldquoImproved EMD based de-noising methodrdquo Journal of Vibration and Shock vol 28no 12 pp 35ndash37 2009

[5] M Y Wang ldquo+e fault monitor of planetary gear grindingbased on wavelet and Hilbert transformrdquoMachine Design andManufacturing Engineering vol 48 no 5 pp 105ndash108 2019

15

2

1

05

0

ndash05

ndash1

ndash15

ndash2

ndash250 50 100 150 200 250

Sample number

KELM

out

put f

or te

st se

t



(e)

2

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se

tNormalRollerInner

OuterInnerOuter

Sample number

(f)

Figure 16 Comparison of fault diagnosis results (a) ELM classification results without filter (b) ELM classification results with quadraticfiltering (c) PSO-KELM classification results without filter (d) PSO-KELM classification results with quadratic filter (e) QPSO-ELMclassification results without filter (f ) QPSO-KELM classification results with quadratic filter


[6] Z L Wang H L Ren and J G Ning ldquoAcoustic emissionsource location based on wavelet transform de-noisingrdquoJournal of Vibration and Shock vol 37 no 4 pp 226ndash2322018

[7] H Gu J Zhao and X Zhang ldquoHybrid methodology ofdegradation feature extraction for bearing prognosticsrdquoEksploatacja I Niezawodnosc-Maintenance and Reliabilityvol 15 no 2 pp 195ndash201 2013

[8] H S Seung and D D Lee ldquoCOGNITION the manifold waysof perceptionrdquo Science vol 290 no 5500 pp 2268-22692000

[9] S T Roweis and L Saul ldquoNonlinear dimensionality reductionby locally linear embeddingrdquo Science vol 290 no 5500pp 2323ndash2326 2000

[10] J B Tenenbaum V Silva and J Langford ldquoA global geo-metric framework for nonlinear dimensionality reductionrdquoScience vol 290 no 5500 pp 2319ndash2323 2000

[11] T Song B P Tang and F Li ldquoFault diagnosis method forrotating machinery based on manifold learning and K-nearestneighbor classifierrdquo Journal of Vibration and Shock vol 32no 5 pp 149ndash153 2013

[12] J F Wang and F C Li ldquoSignal processing methods in faultdiagnosis of machinery-analyses in time domanrdquo Noise ampVibration Control vol 4 no 2 pp 198ndash202 2013

[13] C Li and M Liang ldquoContinuous-scale mathematical mor-phology-based optimal scale band demodulation of impulsivefeature for bearing defect diagnosisrdquo Journal of Sound andVibration vol 331 no 26 pp 5864ndash5879 2012

[14] J F Wang and F C Li ldquoSignal processing methods in faultdiagnosis of machinery-analyses in frequency domanrdquo Noiseamp Vibration Control vol 2 no 1 pp 173ndash180 2013

[15] J F Wang and F C Li ldquoReview of signal processing methods infault diagnosis for machinery using time-frequency analysisrdquoNoise and Vibration Control vol 3 no 2 pp 202ndash206 2013

[16] L H Wang X P Zhao J X Wu et al ldquoMotor fault diagnosisbased on short-time fourier transform and convolutionalneural networkrdquo Chinese Journal of Mechanical Engineeringvol 30 no 6 pp 272ndash277 2017

[17] M T Zhu X J Wu G F Zheng et al ldquoLoad signal editionmethod based on the short-time fourier transform to dura-bility test of vehicle componentrdquo Journal of MechanicalEngineering vol 55 no 4 pp 143ndash151 2019

[18] J D Jia C Z Wu X Y Jia et al ldquoA time-frequency analysismethod suitable for engine vibration signalsrdquo AutomotiveEngineering vol 39 no 1 pp 97ndash101 2017

[19] J D Zheng H Y Pan S B Yang et al ldquoGeneralized vari-ational mode decomposition and its applications to gearboxfault diagnosis under variable conditionsrdquo Journal of Vi-bration Engineering vol 30 no 3 pp pp502ndash509 2017

[20] Q Wang J Gao N Liu and X Jiang ldquoHigh-resolutionseismic time-frequency analysis using the synchrosqueezinggeneralized S-transformrdquo IEEE Geoscience and RemoteSensing Letters vol 15 no 3 pp 374ndash378 2018

[21] Z Huang and J Zhang ldquoSynchrosqueezing S-transformrdquoScientia Sinica Informationis vol 46 no 5 pp 643ndash650 2016

[22] D-H Pham and S Meignen ldquoHigh-order synchrosqueezingtransform for multicomponent signals analysis-with an ap-plication to gravitational-wave signalrdquo IEEE Transactions onSignal Processing vol 65 no 12 pp 3168ndash3178 2017

[23] J Li B Li and M Q Wang ldquoExpert system of gearrsquos faultdiagnosis based on neural networkrdquo Journal of MechanicalTransmission vol 31 no 5 pp 81ndash83 2007

[24] J L Qu L Yu T Yuan et al ldquoAdaptive fault diagnosis al-gorithm for rolling bearings based on one-dimensional

convolutional neural networkrdquo Chinese Journal of ScientificInstrument vol 39 no 7 pp 134ndash143 2018

[25] Y K Wang ldquoA diagnosis method for electrical submersiblepump based on RSGWPT-MSE and PNNrdquo Mechanical En-gineering vol 1 pp 167ndash170 2018

[26] H L Tian H R Li and B H Xu ldquoFault prediction forhydraulic pump based on EEMD and SVMrdquo China Me-chanical Engineering vol 24 no 20 pp 926ndash931 2013

[27] Y Yu R Bai and P Yang ldquoApplication of support vectormachine based on genetic algorithm optimization on acousticemission detection of gear faultrdquo Journal of MechanicalTransmission vol 42 no 1 pp 163ndash166 2018

[28] G B Huang H Zhou X Ding and R Zhang ldquoExtremelearning machine for regression and multiclass classificationrdquoIEEE Transactions on Systems Man and Cybernetics Part B(Cybernetics) vol 42 no 2 pp 513ndash529 2012

[29] F Gao H Li and B Xu ldquoApplications of extreme learningmachine optimized by ICPSO in fault diagnosisrdquo ChinaMechanical Engineering vol 24 no 20 pp 2753ndash2757 2013

[30] A Iosifidis A Tefas and I Pitas ldquoOn the kernel extremelearning machine classifierrdquo Pattern Recognition Lettersvol 54 no 1 pp 11ndash17 2015

[31] P M Shi K Liang and N Zhao ldquoAn SJ Intelligent fault di-agnosis for gears based on deep learning feature extraction andparticle swarm optimization SVM state identificationrdquo ChinaMechanical Engineering vol 28 no 9 pp 1056ndash1061 2017

[32] J T Wen C H Yan J D Sun et al ldquoBearing fault diagnosismethod based on compression acquisition and deep learningrdquoChinese Journal of Scientific Instrument vol 39 no 1pp 171ndash179 2018

[33] J Xu Y Wang and Z Ji ldquoFault diagnosis method of rollingbearing based on WKELM optimized by whale optimizationalgorithmrdquo Journal of System Simulation vol 29 no 9pp 2189ndash2197 2017

[34] H B Duan D B Wu and X F Yu ldquoAnt colony algorithmsurvey and prospectrdquo Engineering Science vol 9 no 2pp 98ndash102 2007

[35] H SWu F M Zhang and L SWu ldquoNew swarm intelligencealgorithm-wolf pack algorithmrdquo Systems Engineering andElectronics vol 35 no 11 pp 2430ndash2438 2013

[36] K L Tsui N Chen Q Zhou Y Hai and W WangldquoPrognostics and Health management a review on datadriven approachesrdquo Mathematical Problems in Engineeringvol 2015 no 1 17 pages Article ID 793161 2015

[37] C Jiang S L Liu and R H Jiang ldquoIMF of EEMD selectionbased on fast spectral kurtosisrdquo Journal of Vibration Mea-surement amp Diagnosis vol 35 no 170 pp 1173ndash1178 2015

[38] R Xueping L Pan and W Chaoge ldquoRolling bearing earlyfault diagnosis based on VMD and FSKrdquo Bearing vol 11no 12 pp 39ndash43 2017

[39] Z Liu L Tang and L Cao ldquoFault diagnosis for rollingbearings based on kalman filter and spectral kurtosisrdquoBearing vol 12 no 8 pp 39ndash43 2014

[40] W H Li L Lin and W P Shan ldquoFault diagnosis for rollingbearings based on GST and bidirectional 2DPCArdquo Journal ofVibration Measurement amp Diagnosis vol 16 no 3pp 499ndash506 2015

[41] S Zhang Y Cai and W Mu ldquoCoding feature extraction anddiagnosis of ICengine vibration time-frequency imagesrdquoJournal of Mechanical Science and Technology vol 37 no 3pp 391ndash395 2018

[42] P Lian J W Yang and D C Yao ldquoFault diagnosis method ofgear based on adaptive demodulated resonancerdquo Journal ofMechanical Transmission vol 42 no 3 pp 169ndash174 2018


the mechanical transmission system is often disturbed byimpact noise in themotion process and the impact noise canbe better filtered by a stable filter

Feature extraction technology is the core of fault diagnosis+e commonly used feature parameters include time-domainfeature [12 13] frequency-domain feature [14] and time-frequency feature [15] Time-domain feature is the featureinformation directly extracted from the time-domain signalwhich can intuitively obtain the signal feature +e frequency-domain feature is extracted from the frequency spectrumobtained by Fourier transform +e time-frequency featurecontains more information +e commonly used time-fre-quency analysismethods include short-time Fourier transform(STFT) [16 17] continuous wavelet transform (CWT)[18 19] and S-transform (ST) [20ndash22] +e S-transform in-herits the advantages of STFT and CWT and develops on thebasis of STFT and CWT However there are still someproblems such as low resolution and energy leakage of thetime-frequency images obtained by using ST technology

Intelligent classification algorithm is widely used in thefield of fault diagnosis and its effect is good Common in-telligent algorithms include neural network [23ndash25] supportvector machine (SVM) [26 27] extreme learning machine(ELM) [28 29] kernel extreme learning machine (KELM)[30] deep learning [31 32] and othermethods Among themthe KELM algorithm has stronger comprehensive advantagesin sample demand network generalization calculation speedand accuracy and is more suitable for solving gearbox faultdiagnosis problems with high calculation speed and accuracyrequirements However the classification ability of KELMnetwork is affected by the value of kernel parameters andpenalty coefficient so it is necessary to optimize the abovestructural parameters +erefore a method using thequantum particle swarm optimization algorithm [33ndash35] tooptimize the structural parameters of KELM (QPSO-KELM)is proposed in this paper +e algorithm based on QPSO canoptimize the structural parameters of KELM with strongglobal search ability so as to improve the learning speed andclassification accuracy of KELM and improve the accuracy ofgearbox fault diagnosis

At present most of the methods are not perfect in termsof diagnosis process and structure and main problems are asfollows

(1) +e effect of signal filtering and noise reduction isnot obvious enough leading to the fault featureextraction which is not obvious enough

(2) +e common time-frequency analysis methods alsohave the problems of energy leakage and low reso-lution which cannot extract effective two-dimen-sional features

(3) +e fault diagnosis accuracy of intelligent classifi-cation algorithm is not enough To a large extent theclassification accuracy can be improved by im-proving the parameters

+erefore filtering noise reduction feature extractionand fault diagnosis technology are the focus of this paper[36]

In order to solve the above problems denoising featureextraction and intelligent fault diagnosis technology arelisted as the research priorities in this paper and the fol-lowing researches are carried out +e related steps involvedare shown in Figure 1

+e working process is summarized as follows

(1) Quadratic filtering collecting the signals of gearboxunder different fault conditions as the signals aresusceptible to noise the stable filter is first used forprimary filtering and then the fast-kurtogram isused to filter frequency bands with rich fault in-formation [37ndash39] to complete quadratic filtering

(2) Feature extraction one-dimensional features andtwo-dimensional features of the filtered signal areextracted One-dimensional features are character-istic parameters in the time domain and frequencydomain +e extraction process of two-dimensionalfeatures is that MSSST is first used for time-fre-quency analysis of the filtered signal and then2DPCA [40 41] is used to extract the two-dimen-sional characteristic parameters of the time-fre-quency image

(3) Fault diagnosis the extracted features are dividedinto training sample and test sample which are usedas the input of the QPSO-KELM model +e testsample is input into the trained QPSO-KELM andthen the result of fault diagnosis can be obtained

2 Basic Theory of the Proposed Method

21 Stable Filter and Fast-Kurtogram Materials andMethods should be described with sufficient details to allowothers to replicate and build on published results Pleasenote that publication of your manuscript implicates that youmust make all materials data computer code and protocolsassociated with the publication available to readers Pleasedisclose at the submission stage any restrictions on theavailability of materials or information New methods andprotocols should be described in detail while well-estab-lished methods can be briefly described and appropriatelycited

Research manuscripts reporting large datasets that aredeposited in a publicly available database should specifywhere the data have been deposited and provide the relevantaccession numbers If the accession numbers have not yetbeen obtained at the time of submission please state thatthey will be provided during review +ey must be providedprior to publication

Interventional studies involving animals or humans andother studies requiring ethical approval must list the au-thority that provided approval and the corresponding ethicalapproval code

Conventional signal processing methods usually filternoise by linear algorithms But when the noise is impact andthe trailing is serious the linear filter cannot deal with iteffectively Stable filter is better for this kind of impact signal

+e standard additive noise model of a signal is asfollows


xt st + nt t 1 2 3 (1)




Stable filter

Fast-kurtogram

MSSST




First filter

Second filter



parameters

Trainingsamples

Testing samples


Trained QPSO-KLEM

Fault diagnosis

Multifeature

YesNo


Denosing

Feature extraction

Fault diagnosis

2 DPCA




C θ x1 xm( 1113857 1113944m

i1ρ xi minus θ( 1113857 (2)



1113872 1113873


C θ x1 xm( 1113857

⎧⎪⎨

⎪⎩(3)




i1ρ ωi xi minus θ( 1113857( 1113857 (4)


xt θst + nt t 1 2 3 (5)


Cmatched θ x1 xm( 1113857 1113944m





Kik lang c

ik(n)

11138681113868111386811138681113868

111386811138681113868111386811138684rang

lang cik(n)

11138681113868111386811138681113868

111386811138681113868111386811138682rang2


(7)















zS(t f)


+infin


minus i2πfτdτ

iωS(t f)

(10)


ωx(tω) ztS(t f)

iS(t f) (11)






SSST[1] SSST

SSST[2](t f) 1113946

+infin

minus infinSSST[1]


1113946+infin


SSST[3](t f) 1113946

+infin

minus infinSSST[2]


1113946+infin


minus ωx(t ξ)111385711138571113857dξ

⋮

SSST[N](t f) 1113946

+infin



1113946+infin



(13)


MSSST SSST[N](t f)

1113946+infin


x (t ξ)1113872 1113873dξ(14)





Y AX (15)


J(X) tr Sx( 1113857 (16)





(17)


tr Sx( 1113857 XT





Gt 1

M1113944

M

j1Aj minus A1113872 1113873

TAj minus A1113872 1113873 (20)


J(X) XTGtX (21)


X1 Xd1113864 1113865 argmax J(X)


⎧⎨

⎩ (22)









1113944

M






1113944N

i1oi minus ti

0βiwibi 1113944

M


i 1 2 NHβ T

(24)


1113944

M



Hβ T (26)


βprime H+T (27)



minimize LPELM12β

2+ C

12

1113944

N

i1ξi

2


i i 1 N

(28)


Np1 ξp2



β HT I

C+ HH

T1113874 1113875

minus 1T (29)


oi hiHT I

C+ HH

T1113874 1113875

minus 1T (30)



K xi xj1113872 1113873(31)



oi K xi xj1113872 1113873 K xi xj1113872 11138731113960 1113961λ (32)



Xi

1

2

m

1

2

L

1

2

n

oj

Inputlayer

Hidden layer

Output layer


k23

k01 k1

1

k02 k1

2 k22 k3

2

k03 k13 k3

3 k43 k5

3 k63 k7

3

fs8 fs2fs4

k0 0

1

2

k

3





2

σ2⎛⎝ ⎞⎠ (33)




Vi vi1 vi2 vi d( 1113857 (34)



pgbest pg1 pg2 pg d1113872 1113873(35)


mbest 1

M1113944

M

i1pibest

1

M1113944

M

i1pi1

1M

1113944

M

i1pi2

1M

1113944

M

i1pi d

⎡⎣ ⎤⎦

(36)



11138681113868111386811138681113868111386811138681113868ln

1u

1113874 1113875 u isin (0 1) (38)



N+ α2 (39)
































No

Yes












Motorcontroller

Speedindicator

Wheels

Bearing 1 Bearing 2

ShaftGearbox Brake

Brake controller

Motor


Gear 1

Inputshaft

Output shaft

Gear 2

Gear 3

Gear 4

(a)

Gear 141 teeth

Gear 279 teeth

Gear 490 teeth

Gear 336 teeth

High-speedshaft

Low-speedshaft

Faultbearing

Medium-speedshaft

(b)

Sensor 2

Sensor 1

(c)







4

35

3

25

2

15

1

05

00 2000 4000 6000 8000 10000

Frequency (HZ)

Am

plitu

de (m

s2 )

SNR = ndash93491

times10ndash3

(a)

4

35

3

25

2

15

1

05

00 2000 4000 6000 8000 10000

Frequency (HZ)

Am

plitu

de (m

s2 )

SNR = ndash82371

times10ndash3

(b)

3

25

2

15

1

05

00 2000 4000 6000 8000 10000

Frequency (HZ)

Am

plitu

de (m

s2 )

SNR = ndash87445

times10ndash3

(c)

3

25

2

15

1

05

00 2000 4000 6000 8000 10000

Frequency (HZ)

Am

plitu

de (m

s2 )

SNR = ndash81016

times10ndash3

(d)


02

025

015

01

005

00 2000 4000 6000 8000 10000

Frequency (HZ)

01

162

263

364

465

566

Leve

l k


(a)

2

15

1

05

00 2000 4000 6000 8000 10000

Frequency (HZ)

01

162

263

364

465

566

Leve

l k


(b)



Frequency (HZ)

Am

plitu

de (m

s2 )

8

6

4

2

0

f BPM

2fBP

M

3fBP

M

4fBP

M

5fBP

M

times10ndash10

0 200100 400300 600500 800700

(a)

times10ndash10

Frequency (HZ)

Am

plitu

de (m

s2 )

3

2

1

0

f m2 2f

m2

3fm

2

4fm

2

0 500 1000 1500 2000

(b)


6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)002 004 006 008 01 012 014 016 018

(a)

5000

4500

4000

3000

3500

2500

2000

1000

1500

500

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(b)

Figure 10 Continued




10000

9000

8000

7000

6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)002 004 06 008 01 012 014 016 018

(a)

9000

8000

7000

6000

5000

4000

3000

2000

1000

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(b)

10000

9000

8000

7000

6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(c)

9000

10000

8000

7000

6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(d)


6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(c)

6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(d)






10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(a)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(b)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(c)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(d)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(e)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(f )



10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(a)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(b)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(c)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(d)

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

10

(e)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(f )





Sensor 1

Sensor 2

Sensor 1

Vibrationsignal

Vibrationsignal

Vibrationsignal




Featurevectors

Featurevectors

Featurevectors

Featurevectors

Featurevectors



2

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se

t

NormalRollerInner

OuterInnerOuter

Sample number






0 50 100 150 200 250Sample number

15

1

05

0

ndash05

ndash1

ndash15

ndash2

KELM

out

put f

or te

st se

t



(a)

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se



Sample number

(b)

1

05

0

ndash05

ndash1

ndash150 50 100 150 200 250

Sample number

KELM

out

put f

or te

st se

t



(c)

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se

t

Sample number



(d)

Figure 16 Continued


5 Conclusion




Data Availability




References






15

2

1

05

0

ndash05

ndash1

ndash15

ndash2

ndash250 50 100 150 200 250

Sample number

KELM

out

put f

or te

st se

t



(e)

2

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se

tNormalRollerInner

OuterInnerOuter

Sample number

(f)










































xt st + nt t 1 2 3 (1)




Stable filter

Fast-kurtogram

MSSST




First filter

Second filter



parameters

Trainingsamples

Testing samples


Trained QPSO-KLEM

Fault diagnosis

Multifeature

YesNo


Denosing

Feature extraction

Fault diagnosis

2 DPCA




C θ x1 xm( 1113857 1113944m

i1ρ xi minus θ( 1113857 (2)



1113872 1113873


C θ x1 xm( 1113857

⎧⎪⎨

⎪⎩(3)




i1ρ ωi xi minus θ( 1113857( 1113857 (4)


xt θst + nt t 1 2 3 (5)


Cmatched θ x1 xm( 1113857 1113944m





Kik lang c

ik(n)

11138681113868111386811138681113868

111386811138681113868111386811138684rang

lang cik(n)

11138681113868111386811138681113868

111386811138681113868111386811138682rang2


(7)















zS(t f)


+infin


minus i2πfτdτ

iωS(t f)

(10)


ωx(tω) ztS(t f)

iS(t f) (11)






SSST[1] SSST

SSST[2](t f) 1113946

+infin

minus infinSSST[1]


1113946+infin


SSST[3](t f) 1113946

+infin

minus infinSSST[2]


1113946+infin


minus ωx(t ξ)111385711138571113857dξ

⋮

SSST[N](t f) 1113946

+infin



1113946+infin



(13)


MSSST SSST[N](t f)

1113946+infin


x (t ξ)1113872 1113873dξ(14)





Y AX (15)


J(X) tr Sx( 1113857 (16)





(17)


tr Sx( 1113857 XT





Gt 1

M1113944

M

j1Aj minus A1113872 1113873

TAj minus A1113872 1113873 (20)


J(X) XTGtX (21)


X1 Xd1113864 1113865 argmax J(X)


⎧⎨

⎩ (22)









1113944

M






1113944N

i1oi minus ti

0βiwibi 1113944

M


i 1 2 NHβ T

(24)


1113944

M



Hβ T (26)


βprime H+T (27)



minimize LPELM12β

2+ C

12

1113944

N

i1ξi

2


i i 1 N

(28)


Np1 ξp2



β HT I

C+ HH

T1113874 1113875

minus 1T (29)


oi hiHT I

C+ HH

T1113874 1113875

minus 1T (30)



K xi xj1113872 1113873(31)



oi K xi xj1113872 1113873 K xi xj1113872 11138731113960 1113961λ (32)



Xi

1

2

m

1

2

L

1

2

n

oj

Inputlayer

Hidden layer

Output layer


k23

k01 k1

1

k02 k1

2 k22 k3

2

k03 k13 k3

3 k43 k5

3 k63 k7

3

fs8 fs2fs4

k0 0

1

2

k

3





2

σ2⎛⎝ ⎞⎠ (33)




Vi vi1 vi2 vi d( 1113857 (34)



pgbest pg1 pg2 pg d1113872 1113873(35)


mbest 1

M1113944

M

i1pibest

1

M1113944

M

i1pi1

1M

1113944

M

i1pi2

1M

1113944

M

i1pi d

⎡⎣ ⎤⎦

(36)



11138681113868111386811138681113868111386811138681113868ln

1u

1113874 1113875 u isin (0 1) (38)



N+ α2 (39)
































No

Yes












Motorcontroller

Speedindicator

Wheels

Bearing 1 Bearing 2

ShaftGearbox Brake

Brake controller

Motor


Gear 1

Inputshaft

Output shaft

Gear 2

Gear 3

Gear 4

(a)

Gear 141 teeth

Gear 279 teeth

Gear 490 teeth

Gear 336 teeth

High-speedshaft

Low-speedshaft

Faultbearing

Medium-speedshaft

(b)

Sensor 2

Sensor 1

(c)







4

35

3

25

2

15

1

05

00 2000 4000 6000 8000 10000

Frequency (HZ)

Am

plitu

de (m

s2 )

SNR = ndash93491

times10ndash3

(a)

4

35

3

25

2

15

1

05

00 2000 4000 6000 8000 10000

Frequency (HZ)

Am

plitu

de (m

s2 )

SNR = ndash82371

times10ndash3

(b)

3

25

2

15

1

05

00 2000 4000 6000 8000 10000

Frequency (HZ)

Am

plitu

de (m

s2 )

SNR = ndash87445

times10ndash3

(c)

3

25

2

15

1

05

00 2000 4000 6000 8000 10000

Frequency (HZ)

Am

plitu

de (m

s2 )

SNR = ndash81016

times10ndash3

(d)


02

025

015

01

005

00 2000 4000 6000 8000 10000

Frequency (HZ)

01

162

263

364

465

566

Leve

l k


(a)

2

15

1

05

00 2000 4000 6000 8000 10000

Frequency (HZ)

01

162

263

364

465

566

Leve

l k


(b)



Frequency (HZ)

Am

plitu

de (m

s2 )

8

6

4

2

0

f BPM

2fBP

M

3fBP

M

4fBP

M

5fBP

M

times10ndash10

0 200100 400300 600500 800700

(a)

times10ndash10

Frequency (HZ)

Am

plitu

de (m

s2 )

3

2

1

0

f m2 2f

m2

3fm

2

4fm

2

0 500 1000 1500 2000

(b)


6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)002 004 006 008 01 012 014 016 018

(a)

5000

4500

4000

3000

3500

2500

2000

1000

1500

500

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(b)

Figure 10 Continued




10000

9000

8000

7000

6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)002 004 06 008 01 012 014 016 018

(a)

9000

8000

7000

6000

5000

4000

3000

2000

1000

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(b)

10000

9000

8000

7000

6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(c)

9000

10000

8000

7000

6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(d)


6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(c)

6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(d)






10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(a)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(b)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(c)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(d)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(e)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(f )



10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(a)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(b)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(c)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(d)

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

10

(e)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(f )





Sensor 1

Sensor 2

Sensor 1

Vibrationsignal

Vibrationsignal

Vibrationsignal




Featurevectors

Featurevectors

Featurevectors

Featurevectors

Featurevectors



2

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se

t

NormalRollerInner

OuterInnerOuter

Sample number






0 50 100 150 200 250Sample number

15

1

05

0

ndash05

ndash1

ndash15

ndash2

KELM

out

put f

or te

st se

t



(a)

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se



Sample number

(b)

1

05

0

ndash05

ndash1

ndash150 50 100 150 200 250

Sample number

KELM

out

put f

or te

st se

t



(c)

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se

t

Sample number



(d)

Figure 16 Continued


5 Conclusion




Data Availability




References






15

2

1

05

0

ndash05

ndash1

ndash15

ndash2

ndash250 50 100 150 200 250

Sample number

KELM

out

put f

or te

st se

t



(e)

2

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se

tNormalRollerInner

OuterInnerOuter

Sample number

(f)











































C θ x1 xm( 1113857 1113944m

i1ρ xi minus θ( 1113857 (2)



1113872 1113873


C θ x1 xm( 1113857

⎧⎪⎨

⎪⎩(3)




i1ρ ωi xi minus θ( 1113857( 1113857 (4)


xt θst + nt t 1 2 3 (5)


Cmatched θ x1 xm( 1113857 1113944m





Kik lang c

ik(n)

11138681113868111386811138681113868

111386811138681113868111386811138684rang

lang cik(n)

11138681113868111386811138681113868

111386811138681113868111386811138682rang2


(7)















zS(t f)


+infin


minus i2πfτdτ

iωS(t f)

(10)


ωx(tω) ztS(t f)

iS(t f) (11)






SSST[1] SSST

SSST[2](t f) 1113946

+infin

minus infinSSST[1]


1113946+infin


SSST[3](t f) 1113946

+infin

minus infinSSST[2]


1113946+infin


minus ωx(t ξ)111385711138571113857dξ

⋮

SSST[N](t f) 1113946

+infin



1113946+infin



(13)


MSSST SSST[N](t f)

1113946+infin


x (t ξ)1113872 1113873dξ(14)





Y AX (15)


J(X) tr Sx( 1113857 (16)





(17)


tr Sx( 1113857 XT





Gt 1

M1113944

M

j1Aj minus A1113872 1113873

TAj minus A1113872 1113873 (20)


J(X) XTGtX (21)


X1 Xd1113864 1113865 argmax J(X)


⎧⎨

⎩ (22)









1113944

M






1113944N

i1oi minus ti

0βiwibi 1113944

M


i 1 2 NHβ T

(24)


1113944

M



Hβ T (26)


βprime H+T (27)



minimize LPELM12β

2+ C

12

1113944

N

i1ξi

2


i i 1 N

(28)


Np1 ξp2



β HT I

C+ HH

T1113874 1113875

minus 1T (29)


oi hiHT I

C+ HH

T1113874 1113875

minus 1T (30)



K xi xj1113872 1113873(31)



oi K xi xj1113872 1113873 K xi xj1113872 11138731113960 1113961λ (32)



Xi

1

2

m

1

2

L

1

2

n

oj

Inputlayer

Hidden layer

Output layer


k23

k01 k1

1

k02 k1

2 k22 k3

2

k03 k13 k3

3 k43 k5

3 k63 k7

3

fs8 fs2fs4

k0 0

1

2

k

3





2

σ2⎛⎝ ⎞⎠ (33)




Vi vi1 vi2 vi d( 1113857 (34)



pgbest pg1 pg2 pg d1113872 1113873(35)


mbest 1

M1113944

M

i1pibest

1

M1113944

M

i1pi1

1M

1113944

M

i1pi2

1M

1113944

M

i1pi d

⎡⎣ ⎤⎦

(36)



11138681113868111386811138681113868111386811138681113868ln

1u

1113874 1113875 u isin (0 1) (38)



N+ α2 (39)
































No

Yes












Motorcontroller

Speedindicator

Wheels

Bearing 1 Bearing 2

ShaftGearbox Brake

Brake controller

Motor


Gear 1

Inputshaft

Output shaft

Gear 2

Gear 3

Gear 4

(a)

Gear 141 teeth

Gear 279 teeth

Gear 490 teeth

Gear 336 teeth

High-speedshaft

Low-speedshaft

Faultbearing

Medium-speedshaft

(b)

Sensor 2

Sensor 1

(c)







4

35

3

25

2

15

1

05

00 2000 4000 6000 8000 10000

Frequency (HZ)

Am

plitu

de (m

s2 )

SNR = ndash93491

times10ndash3

(a)

4

35

3

25

2

15

1

05

00 2000 4000 6000 8000 10000

Frequency (HZ)

Am

plitu

de (m

s2 )

SNR = ndash82371

times10ndash3

(b)

3

25

2

15

1

05

00 2000 4000 6000 8000 10000

Frequency (HZ)

Am

plitu

de (m

s2 )

SNR = ndash87445

times10ndash3

(c)

3

25

2

15

1

05

00 2000 4000 6000 8000 10000

Frequency (HZ)

Am

plitu

de (m

s2 )

SNR = ndash81016

times10ndash3

(d)


02

025

015

01

005

00 2000 4000 6000 8000 10000

Frequency (HZ)

01

162

263

364

465

566

Leve

l k


(a)

2

15

1

05

00 2000 4000 6000 8000 10000

Frequency (HZ)

01

162

263

364

465

566

Leve

l k


(b)



Frequency (HZ)

Am

plitu

de (m

s2 )

8

6

4

2

0

f BPM

2fBP

M

3fBP

M

4fBP

M

5fBP

M

times10ndash10

0 200100 400300 600500 800700

(a)

times10ndash10

Frequency (HZ)

Am

plitu

de (m

s2 )

3

2

1

0

f m2 2f

m2

3fm

2

4fm

2

0 500 1000 1500 2000

(b)


6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)002 004 006 008 01 012 014 016 018

(a)

5000

4500

4000

3000

3500

2500

2000

1000

1500

500

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(b)

Figure 10 Continued




10000

9000

8000

7000

6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)002 004 06 008 01 012 014 016 018

(a)

9000

8000

7000

6000

5000

4000

3000

2000

1000

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(b)

10000

9000

8000

7000

6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(c)

9000

10000

8000

7000

6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(d)


6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(c)

6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(d)






10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(a)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(b)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(c)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(d)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(e)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(f )



10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(a)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(b)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(c)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(d)

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

10

(e)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(f )





Sensor 1

Sensor 2

Sensor 1

Vibrationsignal

Vibrationsignal

Vibrationsignal




Featurevectors

Featurevectors

Featurevectors

Featurevectors

Featurevectors



2

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se

t

NormalRollerInner

OuterInnerOuter

Sample number






0 50 100 150 200 250Sample number

15

1

05

0

ndash05

ndash1

ndash15

ndash2

KELM

out

put f

or te

st se

t



(a)

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se



Sample number

(b)

1

05

0

ndash05

ndash1

ndash150 50 100 150 200 250

Sample number

KELM

out

put f

or te

st se

t



(c)

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se

t

Sample number



(d)

Figure 16 Continued


5 Conclusion




Data Availability




References






15

2

1

05

0

ndash05

ndash1

ndash15

ndash2

ndash250 50 100 150 200 250

Sample number

KELM

out

put f

or te

st se

t



(e)

2

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se

tNormalRollerInner

OuterInnerOuter

Sample number

(f)










































SSST[1] SSST

SSST[2](t f) 1113946

+infin

minus infinSSST[1]


1113946+infin


SSST[3](t f) 1113946

+infin

minus infinSSST[2]


1113946+infin


minus ωx(t ξ)111385711138571113857dξ

⋮

SSST[N](t f) 1113946

+infin



1113946+infin



(13)


MSSST SSST[N](t f)

1113946+infin


x (t ξ)1113872 1113873dξ(14)





Y AX (15)


J(X) tr Sx( 1113857 (16)





(17)


tr Sx( 1113857 XT





Gt 1

M1113944

M

j1Aj minus A1113872 1113873

TAj minus A1113872 1113873 (20)


J(X) XTGtX (21)


X1 Xd1113864 1113865 argmax J(X)


⎧⎨

⎩ (22)









1113944

M






1113944N

i1oi minus ti

0βiwibi 1113944

M


i 1 2 NHβ T

(24)


1113944

M



Hβ T (26)


βprime H+T (27)



minimize LPELM12β

2+ C

12

1113944

N

i1ξi

2


i i 1 N

(28)


Np1 ξp2



β HT I

C+ HH

T1113874 1113875

minus 1T (29)


oi hiHT I

C+ HH

T1113874 1113875

minus 1T (30)



K xi xj1113872 1113873(31)



oi K xi xj1113872 1113873 K xi xj1113872 11138731113960 1113961λ (32)



Xi

1

2

m

1

2

L

1

2

n

oj

Inputlayer

Hidden layer

Output layer


k23

k01 k1

1

k02 k1

2 k22 k3

2

k03 k13 k3

3 k43 k5

3 k63 k7

3

fs8 fs2fs4

k0 0

1

2

k

3





2

σ2⎛⎝ ⎞⎠ (33)




Vi vi1 vi2 vi d( 1113857 (34)



pgbest pg1 pg2 pg d1113872 1113873(35)


mbest 1

M1113944

M

i1pibest

1

M1113944

M

i1pi1

1M

1113944

M

i1pi2

1M

1113944

M

i1pi d

⎡⎣ ⎤⎦

(36)



11138681113868111386811138681113868111386811138681113868ln

1u

1113874 1113875 u isin (0 1) (38)



N+ α2 (39)
































No

Yes












Motorcontroller

Speedindicator

Wheels

Bearing 1 Bearing 2

ShaftGearbox Brake

Brake controller

Motor


Gear 1

Inputshaft

Output shaft

Gear 2

Gear 3

Gear 4

(a)

Gear 141 teeth

Gear 279 teeth

Gear 490 teeth

Gear 336 teeth

High-speedshaft

Low-speedshaft

Faultbearing

Medium-speedshaft

(b)

Sensor 2

Sensor 1

(c)







4

35

3

25

2

15

1

05

00 2000 4000 6000 8000 10000

Frequency (HZ)

Am

plitu

de (m

s2 )

SNR = ndash93491

times10ndash3

(a)

4

35

3

25

2

15

1

05

00 2000 4000 6000 8000 10000

Frequency (HZ)

Am

plitu

de (m

s2 )

SNR = ndash82371

times10ndash3

(b)

3

25

2

15

1

05

00 2000 4000 6000 8000 10000

Frequency (HZ)

Am

plitu

de (m

s2 )

SNR = ndash87445

times10ndash3

(c)

3

25

2

15

1

05

00 2000 4000 6000 8000 10000

Frequency (HZ)

Am

plitu

de (m

s2 )

SNR = ndash81016

times10ndash3

(d)


02

025

015

01

005

00 2000 4000 6000 8000 10000

Frequency (HZ)

01

162

263

364

465

566

Leve

l k


(a)

2

15

1

05

00 2000 4000 6000 8000 10000

Frequency (HZ)

01

162

263

364

465

566

Leve

l k


(b)



Frequency (HZ)

Am

plitu

de (m

s2 )

8

6

4

2

0

f BPM

2fBP

M

3fBP

M

4fBP

M

5fBP

M

times10ndash10

0 200100 400300 600500 800700

(a)

times10ndash10

Frequency (HZ)

Am

plitu

de (m

s2 )

3

2

1

0

f m2 2f

m2

3fm

2

4fm

2

0 500 1000 1500 2000

(b)


6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)002 004 006 008 01 012 014 016 018

(a)

5000

4500

4000

3000

3500

2500

2000

1000

1500

500

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(b)

Figure 10 Continued




10000

9000

8000

7000

6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)002 004 06 008 01 012 014 016 018

(a)

9000

8000

7000

6000

5000

4000

3000

2000

1000

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(b)

10000

9000

8000

7000

6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(c)

9000

10000

8000

7000

6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(d)


6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(c)

6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(d)






10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(a)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(b)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(c)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(d)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(e)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(f )



10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(a)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(b)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(c)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(d)

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

10

(e)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(f )





Sensor 1

Sensor 2

Sensor 1

Vibrationsignal

Vibrationsignal

Vibrationsignal




Featurevectors

Featurevectors

Featurevectors

Featurevectors

Featurevectors



2

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se

t

NormalRollerInner

OuterInnerOuter

Sample number






0 50 100 150 200 250Sample number

15

1

05

0

ndash05

ndash1

ndash15

ndash2

KELM

out

put f

or te

st se

t



(a)

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se



Sample number

(b)

1

05

0

ndash05

ndash1

ndash150 50 100 150 200 250

Sample number

KELM

out

put f

or te

st se

t



(c)

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se

t

Sample number



(d)

Figure 16 Continued


5 Conclusion




Data Availability




References






15

2

1

05

0

ndash05

ndash1

ndash15

ndash2

ndash250 50 100 150 200 250

Sample number

KELM

out

put f

or te

st se

t



(e)

2

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se

tNormalRollerInner

OuterInnerOuter

Sample number

(f)











































1113944

M






1113944N

i1oi minus ti

0βiwibi 1113944

M


i 1 2 NHβ T

(24)


1113944

M



Hβ T (26)


βprime H+T (27)



minimize LPELM12β

2+ C

12

1113944

N

i1ξi

2


i i 1 N

(28)


Np1 ξp2



β HT I

C+ HH

T1113874 1113875

minus 1T (29)


oi hiHT I

C+ HH

T1113874 1113875

minus 1T (30)



K xi xj1113872 1113873(31)



oi K xi xj1113872 1113873 K xi xj1113872 11138731113960 1113961λ (32)



Xi

1

2

m

1

2

L

1

2

n

oj

Inputlayer

Hidden layer

Output layer


k23

k01 k1

1

k02 k1

2 k22 k3

2

k03 k13 k3

3 k43 k5

3 k63 k7

3

fs8 fs2fs4

k0 0

1

2

k

3





2

σ2⎛⎝ ⎞⎠ (33)




Vi vi1 vi2 vi d( 1113857 (34)



pgbest pg1 pg2 pg d1113872 1113873(35)


mbest 1

M1113944

M

i1pibest

1

M1113944

M

i1pi1

1M

1113944

M

i1pi2

1M

1113944

M

i1pi d

⎡⎣ ⎤⎦

(36)



11138681113868111386811138681113868111386811138681113868ln

1u

1113874 1113875 u isin (0 1) (38)



N+ α2 (39)
































No

Yes












Motorcontroller

Speedindicator

Wheels

Bearing 1 Bearing 2

ShaftGearbox Brake

Brake controller

Motor


Gear 1

Inputshaft

Output shaft

Gear 2

Gear 3

Gear 4

(a)

Gear 141 teeth

Gear 279 teeth

Gear 490 teeth

Gear 336 teeth

High-speedshaft

Low-speedshaft

Faultbearing

Medium-speedshaft

(b)

Sensor 2

Sensor 1

(c)







4

35

3

25

2

15

1

05

00 2000 4000 6000 8000 10000

Frequency (HZ)

Am

plitu

de (m

s2 )

SNR = ndash93491

times10ndash3

(a)

4

35

3

25

2

15

1

05

00 2000 4000 6000 8000 10000

Frequency (HZ)

Am

plitu

de (m

s2 )

SNR = ndash82371

times10ndash3

(b)

3

25

2

15

1

05

00 2000 4000 6000 8000 10000

Frequency (HZ)

Am

plitu

de (m

s2 )

SNR = ndash87445

times10ndash3

(c)

3

25

2

15

1

05

00 2000 4000 6000 8000 10000

Frequency (HZ)

Am

plitu

de (m

s2 )

SNR = ndash81016

times10ndash3

(d)


02

025

015

01

005

00 2000 4000 6000 8000 10000

Frequency (HZ)

01

162

263

364

465

566

Leve

l k


(a)

2

15

1

05

00 2000 4000 6000 8000 10000

Frequency (HZ)

01

162

263

364

465

566

Leve

l k


(b)



Frequency (HZ)

Am

plitu

de (m

s2 )

8

6

4

2

0

f BPM

2fBP

M

3fBP

M

4fBP

M

5fBP

M

times10ndash10

0 200100 400300 600500 800700

(a)

times10ndash10

Frequency (HZ)

Am

plitu

de (m

s2 )

3

2

1

0

f m2 2f

m2

3fm

2

4fm

2

0 500 1000 1500 2000

(b)


6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)002 004 006 008 01 012 014 016 018

(a)

5000

4500

4000

3000

3500

2500

2000

1000

1500

500

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(b)

Figure 10 Continued




10000

9000

8000

7000

6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)002 004 06 008 01 012 014 016 018

(a)

9000

8000

7000

6000

5000

4000

3000

2000

1000

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(b)

10000

9000

8000

7000

6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(c)

9000

10000

8000

7000

6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(d)


6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(c)

6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(d)






10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(a)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(b)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(c)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(d)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(e)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(f )



10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(a)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(b)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(c)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(d)

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

10

(e)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(f )





Sensor 1

Sensor 2

Sensor 1

Vibrationsignal

Vibrationsignal

Vibrationsignal




Featurevectors

Featurevectors

Featurevectors

Featurevectors

Featurevectors



2

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se

t

NormalRollerInner

OuterInnerOuter

Sample number






0 50 100 150 200 250Sample number

15

1

05

0

ndash05

ndash1

ndash15

ndash2

KELM

out

put f

or te

st se

t



(a)

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se



Sample number

(b)

1

05

0

ndash05

ndash1

ndash150 50 100 150 200 250

Sample number

KELM

out

put f

or te

st se

t



(c)

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se

t

Sample number



(d)

Figure 16 Continued


5 Conclusion




Data Availability




References






15

2

1

05

0

ndash05

ndash1

ndash15

ndash2

ndash250 50 100 150 200 250

Sample number

KELM

out

put f

or te

st se

t



(e)

2

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se

tNormalRollerInner

OuterInnerOuter

Sample number

(f)












































2

σ2⎛⎝ ⎞⎠ (33)




Vi vi1 vi2 vi d( 1113857 (34)



pgbest pg1 pg2 pg d1113872 1113873(35)


mbest 1

M1113944

M

i1pibest

1

M1113944

M

i1pi1

1M

1113944

M

i1pi2

1M

1113944

M

i1pi d

⎡⎣ ⎤⎦

(36)



11138681113868111386811138681113868111386811138681113868ln

1u

1113874 1113875 u isin (0 1) (38)



N+ α2 (39)
































No

Yes












Motorcontroller

Speedindicator

Wheels

Bearing 1 Bearing 2

ShaftGearbox Brake

Brake controller

Motor


Gear 1

Inputshaft

Output shaft

Gear 2

Gear 3

Gear 4

(a)

Gear 141 teeth

Gear 279 teeth

Gear 490 teeth

Gear 336 teeth

High-speedshaft

Low-speedshaft

Faultbearing

Medium-speedshaft

(b)

Sensor 2

Sensor 1

(c)







4

35

3

25

2

15

1

05

00 2000 4000 6000 8000 10000

Frequency (HZ)

Am

plitu

de (m

s2 )

SNR = ndash93491

times10ndash3

(a)

4

35

3

25

2

15

1

05

00 2000 4000 6000 8000 10000

Frequency (HZ)

Am

plitu

de (m

s2 )

SNR = ndash82371

times10ndash3

(b)

3

25

2

15

1

05

00 2000 4000 6000 8000 10000

Frequency (HZ)

Am

plitu

de (m

s2 )

SNR = ndash87445

times10ndash3

(c)

3

25

2

15

1

05

00 2000 4000 6000 8000 10000

Frequency (HZ)

Am

plitu

de (m

s2 )

SNR = ndash81016

times10ndash3

(d)


02

025

015

01

005

00 2000 4000 6000 8000 10000

Frequency (HZ)

01

162

263

364

465

566

Leve

l k


(a)

2

15

1

05

00 2000 4000 6000 8000 10000

Frequency (HZ)

01

162

263

364

465

566

Leve

l k


(b)



Frequency (HZ)

Am

plitu

de (m

s2 )

8

6

4

2

0

f BPM

2fBP

M

3fBP

M

4fBP

M

5fBP

M

times10ndash10

0 200100 400300 600500 800700

(a)

times10ndash10

Frequency (HZ)

Am

plitu

de (m

s2 )

3

2

1

0

f m2 2f

m2

3fm

2

4fm

2

0 500 1000 1500 2000

(b)


6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)002 004 006 008 01 012 014 016 018

(a)

5000

4500

4000

3000

3500

2500

2000

1000

1500

500

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(b)

Figure 10 Continued




10000

9000

8000

7000

6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)002 004 06 008 01 012 014 016 018

(a)

9000

8000

7000

6000

5000

4000

3000

2000

1000

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(b)

10000

9000

8000

7000

6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(c)

9000

10000

8000

7000

6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(d)


6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(c)

6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(d)






10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(a)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(b)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(c)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(d)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(e)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(f )



10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(a)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(b)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(c)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(d)

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

10

(e)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(f )





Sensor 1

Sensor 2

Sensor 1

Vibrationsignal

Vibrationsignal

Vibrationsignal




Featurevectors

Featurevectors

Featurevectors

Featurevectors

Featurevectors



2

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se

t

NormalRollerInner

OuterInnerOuter

Sample number






0 50 100 150 200 250Sample number

15

1

05

0

ndash05

ndash1

ndash15

ndash2

KELM

out

put f

or te

st se

t



(a)

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se



Sample number

(b)

1

05

0

ndash05

ndash1

ndash150 50 100 150 200 250

Sample number

KELM

out

put f

or te

st se

t



(c)

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se

t

Sample number



(d)

Figure 16 Continued


5 Conclusion




Data Availability




References






15

2

1

05

0

ndash05

ndash1

ndash15

ndash2

ndash250 50 100 150 200 250

Sample number

KELM

out

put f

or te

st se

t



(e)

2

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se

tNormalRollerInner

OuterInnerOuter

Sample number

(f)






























































No

Yes












Motorcontroller

Speedindicator

Wheels

Bearing 1 Bearing 2

ShaftGearbox Brake

Brake controller

Motor


Gear 1

Inputshaft

Output shaft

Gear 2

Gear 3

Gear 4

(a)

Gear 141 teeth

Gear 279 teeth

Gear 490 teeth

Gear 336 teeth

High-speedshaft

Low-speedshaft

Faultbearing

Medium-speedshaft

(b)

Sensor 2

Sensor 1

(c)







4

35

3

25

2

15

1

05

00 2000 4000 6000 8000 10000

Frequency (HZ)

Am

plitu

de (m

s2 )

SNR = ndash93491

times10ndash3

(a)

4

35

3

25

2

15

1

05

00 2000 4000 6000 8000 10000

Frequency (HZ)

Am

plitu

de (m

s2 )

SNR = ndash82371

times10ndash3

(b)

3

25

2

15

1

05

00 2000 4000 6000 8000 10000

Frequency (HZ)

Am

plitu

de (m

s2 )

SNR = ndash87445

times10ndash3

(c)

3

25

2

15

1

05

00 2000 4000 6000 8000 10000

Frequency (HZ)

Am

plitu

de (m

s2 )

SNR = ndash81016

times10ndash3

(d)


02

025

015

01

005

00 2000 4000 6000 8000 10000

Frequency (HZ)

01

162

263

364

465

566

Leve

l k


(a)

2

15

1

05

00 2000 4000 6000 8000 10000

Frequency (HZ)

01

162

263

364

465

566

Leve

l k


(b)



Frequency (HZ)

Am

plitu

de (m

s2 )

8

6

4

2

0

f BPM

2fBP

M

3fBP

M

4fBP

M

5fBP

M

times10ndash10

0 200100 400300 600500 800700

(a)

times10ndash10

Frequency (HZ)

Am

plitu

de (m

s2 )

3

2

1

0

f m2 2f

m2

3fm

2

4fm

2

0 500 1000 1500 2000

(b)


6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)002 004 006 008 01 012 014 016 018

(a)

5000

4500

4000

3000

3500

2500

2000

1000

1500

500

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(b)

Figure 10 Continued




10000

9000

8000

7000

6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)002 004 06 008 01 012 014 016 018

(a)

9000

8000

7000

6000

5000

4000

3000

2000

1000

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(b)

10000

9000

8000

7000

6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(c)

9000

10000

8000

7000

6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(d)


6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(c)

6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(d)






10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(a)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(b)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(c)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(d)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(e)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(f )



10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(a)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(b)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(c)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(d)

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

10

(e)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(f )





Sensor 1

Sensor 2

Sensor 1

Vibrationsignal

Vibrationsignal

Vibrationsignal




Featurevectors

Featurevectors

Featurevectors

Featurevectors

Featurevectors



2

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se

t

NormalRollerInner

OuterInnerOuter

Sample number






0 50 100 150 200 250Sample number

15

1

05

0

ndash05

ndash1

ndash15

ndash2

KELM

out

put f

or te

st se

t



(a)

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se



Sample number

(b)

1

05

0

ndash05

ndash1

ndash150 50 100 150 200 250

Sample number

KELM

out

put f

or te

st se

t



(c)

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se

t

Sample number



(d)

Figure 16 Continued


5 Conclusion




Data Availability




References






15

2

1

05

0

ndash05

ndash1

ndash15

ndash2

ndash250 50 100 150 200 250

Sample number

KELM

out

put f

or te

st se

t



(e)

2

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se

tNormalRollerInner

OuterInnerOuter

Sample number

(f)



















































Motorcontroller

Speedindicator

Wheels

Bearing 1 Bearing 2

ShaftGearbox Brake

Brake controller

Motor


Gear 1

Inputshaft

Output shaft

Gear 2

Gear 3

Gear 4

(a)

Gear 141 teeth

Gear 279 teeth

Gear 490 teeth

Gear 336 teeth

High-speedshaft

Low-speedshaft

Faultbearing

Medium-speedshaft

(b)

Sensor 2

Sensor 1

(c)







4

35

3

25

2

15

1

05

00 2000 4000 6000 8000 10000

Frequency (HZ)

Am

plitu

de (m

s2 )

SNR = ndash93491

times10ndash3

(a)

4

35

3

25

2

15

1

05

00 2000 4000 6000 8000 10000

Frequency (HZ)

Am

plitu

de (m

s2 )

SNR = ndash82371

times10ndash3

(b)

3

25

2

15

1

05

00 2000 4000 6000 8000 10000

Frequency (HZ)

Am

plitu

de (m

s2 )

SNR = ndash87445

times10ndash3

(c)

3

25

2

15

1

05

00 2000 4000 6000 8000 10000

Frequency (HZ)

Am

plitu

de (m

s2 )

SNR = ndash81016

times10ndash3

(d)


02

025

015

01

005

00 2000 4000 6000 8000 10000

Frequency (HZ)

01

162

263

364

465

566

Leve

l k


(a)

2

15

1

05

00 2000 4000 6000 8000 10000

Frequency (HZ)

01

162

263

364

465

566

Leve

l k


(b)



Frequency (HZ)

Am

plitu

de (m

s2 )

8

6

4

2

0

f BPM

2fBP

M

3fBP

M

4fBP

M

5fBP

M

times10ndash10

0 200100 400300 600500 800700

(a)

times10ndash10

Frequency (HZ)

Am

plitu

de (m

s2 )

3

2

1

0

f m2 2f

m2

3fm

2

4fm

2

0 500 1000 1500 2000

(b)


6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)002 004 006 008 01 012 014 016 018

(a)

5000

4500

4000

3000

3500

2500

2000

1000

1500

500

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(b)

Figure 10 Continued




10000

9000

8000

7000

6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)002 004 06 008 01 012 014 016 018

(a)

9000

8000

7000

6000

5000

4000

3000

2000

1000

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(b)

10000

9000

8000

7000

6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(c)

9000

10000

8000

7000

6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(d)


6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(c)

6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(d)






10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(a)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(b)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(c)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(d)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(e)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(f )



10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(a)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(b)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(c)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(d)

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

10

(e)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(f )





Sensor 1

Sensor 2

Sensor 1

Vibrationsignal

Vibrationsignal

Vibrationsignal




Featurevectors

Featurevectors

Featurevectors

Featurevectors

Featurevectors



2

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se

t

NormalRollerInner

OuterInnerOuter

Sample number






0 50 100 150 200 250Sample number

15

1

05

0

ndash05

ndash1

ndash15

ndash2

KELM

out

put f

or te

st se

t



(a)

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se



Sample number

(b)

1

05

0

ndash05

ndash1

ndash150 50 100 150 200 250

Sample number

KELM

out

put f

or te

st se

t



(c)

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se

t

Sample number



(d)

Figure 16 Continued


5 Conclusion




Data Availability




References






15

2

1

05

0

ndash05

ndash1

ndash15

ndash2

ndash250 50 100 150 200 250

Sample number

KELM

out

put f

or te

st se

t



(e)

2

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se

tNormalRollerInner

OuterInnerOuter

Sample number

(f)










































4

35

3

25

2

15

1

05

00 2000 4000 6000 8000 10000

Frequency (HZ)

Am

plitu

de (m

s2 )

SNR = ndash93491

times10ndash3

(a)

4

35

3

25

2

15

1

05

00 2000 4000 6000 8000 10000

Frequency (HZ)

Am

plitu

de (m

s2 )

SNR = ndash82371

times10ndash3

(b)

3

25

2

15

1

05

00 2000 4000 6000 8000 10000

Frequency (HZ)

Am

plitu

de (m

s2 )

SNR = ndash87445

times10ndash3

(c)

3

25

2

15

1

05

00 2000 4000 6000 8000 10000

Frequency (HZ)

Am

plitu

de (m

s2 )

SNR = ndash81016

times10ndash3

(d)


02

025

015

01

005

00 2000 4000 6000 8000 10000

Frequency (HZ)

01

162

263

364

465

566

Leve

l k


(a)

2

15

1

05

00 2000 4000 6000 8000 10000

Frequency (HZ)

01

162

263

364

465

566

Leve

l k


(b)



Frequency (HZ)

Am

plitu

de (m

s2 )

8

6

4

2

0

f BPM

2fBP

M

3fBP

M

4fBP

M

5fBP

M

times10ndash10

0 200100 400300 600500 800700

(a)

times10ndash10

Frequency (HZ)

Am

plitu

de (m

s2 )

3

2

1

0

f m2 2f

m2

3fm

2

4fm

2

0 500 1000 1500 2000

(b)


6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)002 004 006 008 01 012 014 016 018

(a)

5000

4500

4000

3000

3500

2500

2000

1000

1500

500

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(b)

Figure 10 Continued




10000

9000

8000

7000

6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)002 004 06 008 01 012 014 016 018

(a)

9000

8000

7000

6000

5000

4000

3000

2000

1000

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(b)

10000

9000

8000

7000

6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(c)

9000

10000

8000

7000

6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(d)


6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(c)

6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(d)






10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(a)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(b)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(c)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(d)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(e)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(f )



10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(a)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(b)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(c)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(d)

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

10

(e)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(f )





Sensor 1

Sensor 2

Sensor 1

Vibrationsignal

Vibrationsignal

Vibrationsignal




Featurevectors

Featurevectors

Featurevectors

Featurevectors

Featurevectors



2

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se

t

NormalRollerInner

OuterInnerOuter

Sample number






0 50 100 150 200 250Sample number

15

1

05

0

ndash05

ndash1

ndash15

ndash2

KELM

out

put f

or te

st se

t



(a)

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se



Sample number

(b)

1

05

0

ndash05

ndash1

ndash150 50 100 150 200 250

Sample number

KELM

out

put f

or te

st se

t



(c)

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se

t

Sample number



(d)

Figure 16 Continued


5 Conclusion




Data Availability




References






15

2

1

05

0

ndash05

ndash1

ndash15

ndash2

ndash250 50 100 150 200 250

Sample number

KELM

out

put f

or te

st se

t



(e)

2

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se

tNormalRollerInner

OuterInnerOuter

Sample number

(f)










































Frequency (HZ)

Am

plitu

de (m

s2 )

8

6

4

2

0

f BPM

2fBP

M

3fBP

M

4fBP

M

5fBP

M

times10ndash10

0 200100 400300 600500 800700

(a)

times10ndash10

Frequency (HZ)

Am

plitu

de (m

s2 )

3

2

1

0

f m2 2f

m2

3fm

2

4fm

2

0 500 1000 1500 2000

(b)


6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)002 004 006 008 01 012 014 016 018

(a)

5000

4500

4000

3000

3500

2500

2000

1000

1500

500

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(b)

Figure 10 Continued




10000

9000

8000

7000

6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)002 004 06 008 01 012 014 016 018

(a)

9000

8000

7000

6000

5000

4000

3000

2000

1000

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(b)

10000

9000

8000

7000

6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(c)

9000

10000

8000

7000

6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(d)


6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(c)

6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(d)






10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(a)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(b)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(c)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(d)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(e)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(f )



10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(a)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(b)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(c)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(d)

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

10

(e)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(f )





Sensor 1

Sensor 2

Sensor 1

Vibrationsignal

Vibrationsignal

Vibrationsignal




Featurevectors

Featurevectors

Featurevectors

Featurevectors

Featurevectors



2

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se

t

NormalRollerInner

OuterInnerOuter

Sample number






0 50 100 150 200 250Sample number

15

1

05

0

ndash05

ndash1

ndash15

ndash2

KELM

out

put f

or te

st se

t



(a)

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se



Sample number

(b)

1

05

0

ndash05

ndash1

ndash150 50 100 150 200 250

Sample number

KELM

out

put f

or te

st se

t



(c)

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se

t

Sample number



(d)

Figure 16 Continued


5 Conclusion




Data Availability




References






15

2

1

05

0

ndash05

ndash1

ndash15

ndash2

ndash250 50 100 150 200 250

Sample number

KELM

out

put f

or te

st se

t



(e)

2

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se

tNormalRollerInner

OuterInnerOuter

Sample number

(f)










































10000

9000

8000

7000

6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)002 004 06 008 01 012 014 016 018

(a)

9000

8000

7000

6000

5000

4000

3000

2000

1000

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(b)

10000

9000

8000

7000

6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(c)

9000

10000

8000

7000

6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(d)


6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(c)

6000

5000

4000

3000

2000

1000

0

Freq

uenc

y (H

z)

Seconds (s)0 005 01 015 02

(d)






10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(a)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(b)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(c)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(d)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(e)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(f )



10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(a)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(b)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(c)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(d)

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

10

(e)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(f )





Sensor 1

Sensor 2

Sensor 1

Vibrationsignal

Vibrationsignal

Vibrationsignal




Featurevectors

Featurevectors

Featurevectors

Featurevectors

Featurevectors



2

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se

t

NormalRollerInner

OuterInnerOuter

Sample number






0 50 100 150 200 250Sample number

15

1

05

0

ndash05

ndash1

ndash15

ndash2

KELM

out

put f

or te

st se

t



(a)

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se



Sample number

(b)

1

05

0

ndash05

ndash1

ndash150 50 100 150 200 250

Sample number

KELM

out

put f

or te

st se

t



(c)

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se

t

Sample number



(d)

Figure 16 Continued


5 Conclusion




Data Availability




References






15

2

1

05

0

ndash05

ndash1

ndash15

ndash2

ndash250 50 100 150 200 250

Sample number

KELM

out

put f

or te

st se

t



(e)

2

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se

tNormalRollerInner

OuterInnerOuter

Sample number

(f)













































10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(a)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(b)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(c)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(d)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(e)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(f )



10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(a)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(b)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(c)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(d)

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

10

(e)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(f )





Sensor 1

Sensor 2

Sensor 1

Vibrationsignal

Vibrationsignal

Vibrationsignal




Featurevectors

Featurevectors

Featurevectors

Featurevectors

Featurevectors



2

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se

t

NormalRollerInner

OuterInnerOuter

Sample number






0 50 100 150 200 250Sample number

15

1

05

0

ndash05

ndash1

ndash15

ndash2

KELM

out

put f

or te

st se

t



(a)

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se



Sample number

(b)

1

05

0

ndash05

ndash1

ndash150 50 100 150 200 250

Sample number

KELM

out

put f

or te

st se

t



(c)

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se

t

Sample number



(d)

Figure 16 Continued


5 Conclusion




Data Availability




References






15

2

1

05

0

ndash05

ndash1

ndash15

ndash2

ndash250 50 100 150 200 250

Sample number

KELM

out

put f

or te

st se

t



(e)

2

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se

tNormalRollerInner

OuterInnerOuter

Sample number

(f)










































10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(a)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(b)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(c)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(d)

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

10

(e)

10

20

30

40

50

60

70

10 20 30 40 50 60 70 80 90

(f )





Sensor 1

Sensor 2

Sensor 1

Vibrationsignal

Vibrationsignal

Vibrationsignal




Featurevectors

Featurevectors

Featurevectors

Featurevectors

Featurevectors



2

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se

t

NormalRollerInner

OuterInnerOuter

Sample number






0 50 100 150 200 250Sample number

15

1

05

0

ndash05

ndash1

ndash15

ndash2

KELM

out

put f

or te

st se

t



(a)

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se



Sample number

(b)

1

05

0

ndash05

ndash1

ndash150 50 100 150 200 250

Sample number

KELM

out

put f

or te

st se

t



(c)

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se

t

Sample number



(d)

Figure 16 Continued


5 Conclusion




Data Availability




References






15

2

1

05

0

ndash05

ndash1

ndash15

ndash2

ndash250 50 100 150 200 250

Sample number

KELM

out

put f

or te

st se

t



(e)

2

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se

tNormalRollerInner

OuterInnerOuter

Sample number

(f)












































Sensor 1

Sensor 2

Sensor 1

Vibrationsignal

Vibrationsignal

Vibrationsignal




Featurevectors

Featurevectors

Featurevectors

Featurevectors

Featurevectors



2

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se

t

NormalRollerInner

OuterInnerOuter

Sample number






0 50 100 150 200 250Sample number

15

1

05

0

ndash05

ndash1

ndash15

ndash2

KELM

out

put f

or te

st se

t



(a)

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se



Sample number

(b)

1

05

0

ndash05

ndash1

ndash150 50 100 150 200 250

Sample number

KELM

out

put f

or te

st se

t



(c)

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se

t

Sample number



(d)

Figure 16 Continued


5 Conclusion




Data Availability




References






15

2

1

05

0

ndash05

ndash1

ndash15

ndash2

ndash250 50 100 150 200 250

Sample number

KELM

out

put f

or te

st se

t



(e)

2

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se

tNormalRollerInner

OuterInnerOuter

Sample number

(f)










































0 50 100 150 200 250Sample number

15

1

05

0

ndash05

ndash1

ndash15

ndash2

KELM

out

put f

or te

st se

t



(a)

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se



Sample number

(b)

1

05

0

ndash05

ndash1

ndash150 50 100 150 200 250

Sample number

KELM

out

put f

or te

st se

t



(c)

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se

t

Sample number



(d)

Figure 16 Continued


5 Conclusion




Data Availability




References






15

2

1

05

0

ndash05

ndash1

ndash15

ndash2

ndash250 50 100 150 200 250

Sample number

KELM

out

put f

or te

st se

t



(e)

2

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se

tNormalRollerInner

OuterInnerOuter

Sample number

(f)










































5 Conclusion




Data Availability




References






15

2

1

05

0

ndash05

ndash1

ndash15

ndash2

ndash250 50 100 150 200 250

Sample number

KELM

out

put f

or te

st se

t



(e)

2

15

1

05

0

ndash05

ndash1

ndash15

ndash20 50 100 150 200 250

KELM

out

put f

or te

st se

tNormalRollerInner

OuterInnerOuter

Sample number

(f)










































title fault diagnosis method of gearbox multifeature

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