# researchonfaultfeatureextractionmethodbasedon fdm

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Research ArticleResearch on Fault Feature Extraction Method Based onFDM-RobustICA and MOMEDA

Jingzong Yang 1 Xuefeng Li2 and Limei Wu1

1School of Information Baoshan University Baoshan 678000 China2School of Transportation Southeast University Nanjing 211189 China

Correspondence should be addressed to Jingzong Yang yjingzongfoxmailcom

Received 13 April 2020 Revised 22 May 2020 Accepted 28 May 2020 Published 22 June 2020

Academic Editor Stylianos Georgantzinos

Copyright copy 2020 Jingzong Yang 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

Aiming at the difficulty of extracting rolling bearing fault features under strong background noise conditions a method based onthe Fourier decomposition method (FDM) robust independent component analysis (RobustICA) and multipoint optimalminimum entropy deconvolution adjusted (MOMEDA) is proposed Firstly the FDM method is introduced to decompose thesingle-channel bearing fault signal into several Fourier intrinsic band functions (FIBF) Secondly by setting the cross-correlationcoefficient and kurtosis as a new selection criterion it can effectively construct the virtual noise channel and the observation signalchannel which makes RobustICA complete the separation of the useful signal and noise well Finally MOMEDA is introduced toenhance the periodic impact components in the denoised signal and then the filtered signal is analyzed by the Hilbert envelopespectrum to extract the fault characteristic frequency +rough the experimental analysis of the simulated signals and the actualbearing fault signals the results show that the proposed method not only has the ability to suppress noise and accurately extractfault feature information but also has better performance than the traditional method of local mean decomposition (LMD) andintrinsic time-scale decomposition (ITD) highlighting its practicality in the fault diagnosis of rotating machinery

1 Introduction

Bearing is one of the important parts in the mechanicalsystem which has been widely used in metallurgy electricpower aerospace and other fields of national economy Atthe same time due to the high frequency of use and complexworking environment bearings are also the most prone tofailure components Once it breaks down it will not onlyaffect the normal operation of the mechanical system butalso greatly reduce the production efficiency even cause theloss of life and property +erefore it is very important toextract the fault features of bearings and grasp the operationstatus of the equipment in time However in actual oper-ating conditions due to the impact of the componentrsquos ownstructure and working environment the collected signalcontains both useful information about bearing failure anduseless interference noise Meanwhile because the signalpasses through many links in the process of transmissionsome signals will be attenuated to a certain extent Affected

by the aforementioned adverse factors the signal-to-noiseratio of the signals collected by the sensor is very low and thefault characteristics are weak making it difficult to diagnosethe fault +erefore how to effectively extract the faultcharacteristics of bearings is the current research hotspotand difficult point

At present most traditional rolling bearing fault featureextraction methods are developed based on time-frequencyanalysis methods In the field of fault diagnosis there aremany common methods such as short-time Fouriertransform [1 2] WignerndashVille distribution [3] and wavelettransform [4 5] Although the above methods have beenwidely used they are basically based on the idea of integralanalysis Since the set basis functions are fixed the signalscannot be decomposed adaptively +is prevents the noisespectrum from being effectively separated during signalprocessing and may also filter out sudden changes in thesignal To solve the above problem Huang et al [6] proposedthe empirical mode decomposition (EMD) method which

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 6753949 23 pageshttpsdoiorg10115520206753949

can adaptively decompose the signal into several naturalmodal components thereby obtaining the complete time-frequency distribution of the signal It shows some advan-tages when processing stationary and nonstationary signalsbut EMD has problems such as endpoint effects and modalaliasing and lacks strict theoretical proofs [7ndash9] In view ofthe shortcomings of EMD relevant scholars proposedEEMD CEEMD CEEMDAN and other improvedmethods but these methods still have similar problems withEMD [10ndash12] Based on the idea of EMD LMD ITD andother signal processing methods have been proposed oneafter another [13 14] but these methods cannot fully makeup for the shortcomings of EMD In recent years Singh et al[15] proposed a new signal analysis method-Fourier de-composition method (FDM) based on the study of Fouriertransform By using the FDM any complex signal can beadaptively decomposed into a series of Fourier intrinsicband functions (FIBF) +is method changes the defect thatthe Fourier series expansion method can only handle sta-tionary linear signals and can effectively analyze nonsta-tionary nonlinear signals Because the FDM is a completeand orthogonal signal processing method with strict theo-retical basis it has been initially applied in fault diagnosis invarious fields and has shown great advantages Dou and Lin[16] demonstrated the adaptive narrowband filtering char-acteristics of the FDM at low and high frequencies bydetecting white Gaussian noise At the same time aftercomparing the effects of different methods on processinggearbox vibration signals the results show that the FDM canovercome the bottleneck of the EMD in adaptively sepa-rating low-frequency signal components and has superiorperformance Aiming at the problem that the traditionaltime-frequency analysis method is difficult to effectivelyextract the rotor fault characteristics Liu et al [17] intro-duced the FDM to the fault diagnosis of rotor rubbing+rough experimental comparison based on differentmethods the results show that the FDM can achievecomplete signal decomposition and achieve better faultdiagnosis Fu and Gao [18] proposed a fault diagnosismethod for rolling bearings by combining the FDM andsingular value difference spectrum +e experimental sim-ulation and actual engineering tests verify the effectivenessof the proposed method

At the same time as a blind source separation methodindependent component analysis (ICA) provides a new ideafor noise reduction of mechanical equipment fault signals In1994 Comon explained the concept of independent com-ponent analysis (ICA) and pointed out that ICA is an ex-tension of PCA He proposed the target function of theKullbackndashLeibler criterion on the basis of making full use ofthe high-order statistical characteristics of the signal whichbecame themainstream algorithm of blind source separationin the future [19] After that the Finnish scholar Hyvarinenand Oja [20] proposed a fast-independent componentanalysis method based on kurtosis +en the algorithm isimproved using more robust negative entropy as a non-Gaussian measurement criterion and a fast algorithmFastICA based on negative entropy maximization is pro-posed [21] +e algorithm has a very high efficiency so it has

been widely used However in 2004 Zarzoso and Comonet al proposed in [22] that FastICA would fail in the case ofweak or high spatial correlation source signals After thatthey put forward the RobustICA [23 24] and it has de-veloped rapidly in the algorithm of blind source separationAs an algorithm based on kurtosis and optimal step size thisalgorithm has excellent antinoise performance It can sep-arate noise and real signal from the multichannel mixedsignal without prior knowledge of the signal source +ealgorithm is proved to be not only simple but also better thanFastICA In recent years it has been successfully applied tothe noise source separation in the fields of diesel enginegasoline engine and combustion engine [25ndash29] and hasshown high signal separation quality and fast calculationefficiency However in the process of using RobustICA thenumber of sensors must be greater than or equal to thenumber of components separated by them [30] In order tosolve the shortcomings of RobustICA the signal can bedecomposed into many components by time-frequencyanalysis and more virtual observation channels than thesource signal can be generated In this way by combining theadvantages of the above two methods noise reduction of thebearing signal can be achieved However in the process ofuse although the signal-to-noise ratio is improved bysuppressing the noise the useful signal is inevitably sup-pressed Especially under the condition of strong noiseinterference the impact component of the bearing faultsignal after noise reduction is still weak and even the weakfault may be covered by the strong fault which bringsdifficulties for further extraction of fault features

+e essence of fault feature extraction is to extract theperiodic shock components in the signal that is to process thesignal through an optimal filter thereby highlighting the faultshock components +e minimum entropy deconvolution(MED) was first proposed byWiggins in 1978 [31] It uses themaximum kurtosis as the iteration termination condition tosolve the filter which can play a role in highlighting theimpact characteristics in the signal [32 33] Sawalhi et al firstused MED for fault detection of rotating machinery [34]However this iterative method is not only complex but alsonot necessarily the global optimal filter Secondly the methodis only suitable for the single pulse impulse signal +e faultcharacteristics of rotating machinery are mostly periodicpulses In order to overcome the shortcomings of MEDMcDonald et al proposed the maximum correlated kurtosisdeconvolution (MCKD) [35] in 2012 based on the correlationkurtosis To some extent this method can satisfy the need ofdeconvolution with periodic pulse but it is still an iterativeprocess with the maximum correlation kurtosis as the cri-terion +e result is not the optimal solution and it needs theprior knowledge of the fault cycle In view of the limitations ofthe above two methods McDonald and Zhao [36] proposedthe multipoint optimal minimum entropy deconvolutionadjusted (MOMEDA) which has been successfully applied tothe fault diagnosis of the gearbox +is method uses a timetarget vector to define the pulse position obtained by solvingthe deconvolution During the execution of the algorithm theoptimal filter can be obtained without iteration [37] and theperiod of the failure does not need to be determined in

2 Mathematical Problems in Engineering

advance By using this method the periodic impact com-ponents in the signal can be effectively enhanced whichmakes a good foundation for the feature extraction of faultsignals

In summary this paper proposed a fault feature ex-traction method combining FDM-RobustICA andMOMEDA Firstly the FDM is used to decompose thevibration signal of the bearing and obtain several Fourierintrinsic band functions (FIBF) +en by combining thecross-correlation analysis criterion and the kurtosis crite-rion an effective FIBF component is selected as the ob-servation signal and blind source separation is performed+irdly MOMEDA is used to enhance the periodic impactcomponents in the noise-reduced signal Finally the Hilbertenvelope demodulation is used to obtain the fault charac-teristic frequency and the fault characteristic frequency iscompared with the theoretical calculated value for faultdiagnosis

+e structure of this paper is as follows Section 2 introducesthe basic principles of FDM RobustICA and MOMEDASection 3 briefly introduces the implementation details and faultfeature extraction process of themethod proposed in this paperSection 4 uses the proposed method to analyze the simulationsignal Section 5 further verifies the effectiveness of the proposedmethod through the actual bearing fault signal Section 6 givesthe discussion and conclusion

2 Theoretical Background

21 Fourier Decomposition Method Fourier decompositionmethod (FDM) is a new type of the adaptive signal analysismethod based on Fourier transform +is method firstadaptively searches and analyzes the Fourier intrinsic bandfunctions (FIBF) in the entire Fourier domain +en thenonlinear signal is adaptively decomposed into the sum ofseveral physical Fourier intrinsic band functions (FIBF) anda residual component

(1) FIBF has a zero mean value namely 1113938b

ayi(t)dt 0

(2) Any two FIBF are orthogonal to each other namely1113938

b

ayi(t)yj(t)dt 0 ine j

(3) +e analytic function form of FIBF has nonnegativeinstantaneous amplitude and frequency yi(t)

1113954yj(t) ai(t)ejφi(t) namely ai(t)ge 0 φiprime(t)ge 0

forallt isin [a b] and yi(t) isin Cinfin[a b]+erefore FIBF arethe sum of zero-mean sine functions with contin-uous frequency bands

Based on the definition of FIBF the steps of the FDM areas follows

Step 1 make a fast Fourier transformation of the signalx (n) X[k] FFT [x(n)] Step 2 use forward search which scans analytic Fourierintrinsic band functions (AFIBF) from low to highfrequencies

AFIBF 1113944n

kNiminus 1+1X[k]σj2πknN

ai(n)ejφi[n]

X (1)

In order to obtain the minimum number of AFIBF fromlow-frequency to high-frequency scanning for each i 1 2 M start from Niminus 1 + 1 and gradually increase to reachthe maximum Ni where N0 0 and NM (N2 minus 1) Itsatisfies Niminus 1 + 1leNiminus 1 le (N2 minus 1) and also satisfies

ai(t)ge 0 ωi(t) φi[n + 1] minus φi[n minus 1]

2ge 0 forallt (2)

Similarly a reverse search can also be performednamely scanning the AFIBF component from high to lowfrequencies Correspondingly change the upper and lowerlimits of the sum in equation (2) from Ni to Niminus 1 minus 1 wherei 1 2 M N0 N2 and NM 1 +e search shouldstart from Niminus 1 minus 1 and gradually decrease until the smallestNi satisfies 1leNi leNiminus 1 minus 1 and the phase φi[n] is amonotonically increasing function +e instantaneous fre-quency and amplitude can be calculated directly by AFIBFMeanwhile the real part of AFIBF is FIBF

22 Robust Independent Component Analysis Robust in-dependent component analysis (RobustICA) is an inde-pendent component analysis algorithm based on kurtosisand optimal step size that has emerged in recent years on thebasis of the original independent component analysis (ICA)+e algorithm searches for the global optimal step sizethrough the kurtosis comparison function finds the dem-ixing matrix and calculates the approximate value of thesource signal+e frame of independent component analysisis shown in Figure 1 +e purpose of independent com-ponent analysis is to separate independent source signalcomponents from the mixed signal

Let n-dimensional random observation vectors conformto the following model

x AS + N (3)

where S [s1 s2 sm]T and x [x1 x2 xn]T S is the m-dimensional signal vector which means that the number ofmixed signals is m N is the n-dimensional observationvector which means that the number of mixed signals is n Ais an n times m matrix +e basic idea of ICA is to solve theseparation matrix W and make an optimal estimate y of thesource signal S when only the observation signal x is known

y Wx (4)

Compared with the traditional ICA RobustICA does notneed to prewhiten the data which reduces the amount ofcalculation At the same time RobustICA can be used toprocess the sub-Gaussian signal and the super-Gaussiansignal When the Gaussian property of the signal is known itcan extract only the components of interest in the sourcesignal without increasing the complexity of the operationand the estimation error +e linear search method used byRobustICA can ensure that the optimization can be achievedin all the processes of component separation Assuming theoutput signal is y Wx (W is the m times n-dimensionalseparation matrix) the kurtosis equation can be expressed asfollows

Mathematical Problems in Engineering 3

K(W) E |y|41113966 1113967 minus 2E2 |y|21113966 1113967 minus E y21113864 1113865

111386811138681113868111386811138681113868111386811138682

E2 |y|21113966 1113967 (5)

In the equation E middot represents the mathematical ex-pectation RobustICA directly uses the above equation as acomparison function without any simplification and thenoptimizes the kurtosis according to the strategy of the bestlinear optimization step size

μopt argmaxu

|K(W + μg)| (6)

where g is the search direction usually g nablawK(W) whichcan be expressed as follows

nablawK(W) 4

E2 |y|21113966 1113967E |y|

21113966 1113967yx minus E yx1113864 11138651113966 1113967E y

21113966 1113967

minusE |y|41113966 1113967 minus E y21113864 1113865

111386811138681113868111386811138681113868111386811138682

1113872 1113873E yx1113864 1113865

E |y|21113966 1113967

(7)

In the process of iteration the optimization method ofRobustICA can be described as follows

(1) Find the coefficients of the OS polynomial When thekurtosis becomes a comparison function the OSpolynomial can be expressed as follows

P(u) 11139444

k0aku

k (8)

(2) Extract the root a of the OS polynomial uk1113864 11138654k1

(3) Select the root that can make the best optimizationstep μopt argmaxu|K(W + μg)| and take the largestvalue from the direction of iteration

(4) Update μopt based on the value of the updated stepW+ W+ W + μoptg

(5) Normalize W+ W+⟵W+W+(6) Determine whether the iterative operation converges

and meets the termination conditions If it is notsatisfied then return to Step (1)

23 Multipoint Optimal Minimum Entropy DeconvolutionAdjusted If x is the vibration signal y is the shock se-quence Meanwhile h is the system frequency responsefunction e is the noise signal +e symbol lowast is the multi-plication symbol +e collected signal can be expressed as

x hlowasty + e (9)

+e essence of the MOMEDA is to use a noniterativemethod to find an optimal FIR filter Recover the vibrationshock signal from the output signal as much as possiblethrough the deconvolution process and reduce the impact ofnoise +e deconvolution process is as follows

y flowastx 1113944Nminus L

k1fkxk+Lminus 1 (10)

where k 1 2 middot middot middot N minus L According to the characteristics ofthe periodic shock signal multipoint D-norm is introducednamely

MDN(y t) 1

t

tTy

y (11)

MOMEDA maxf

MDN(y t) maxf

tTy

y (12)

In equation (12) t is a constant vector for determiningthe position and weight of the target impact component+eoptimal filter f is obtained by solving the maximum value ofthe multipoint D-norm At this time the deconvolutionprocess also obtains the optimal solution In order to find theextreme value of equation (12) the filter coefficient f

(f1 f2 middot middot middot fL) is first derivedddf

tTy

y1113888 1113889

ddf

t1y1

y+

ddf

t2y2

y+ middot middot middot +

ddf

tNminus LyNminus L

y

(13)

ddf

tkyk

y y

minus 1tkMk minus y

minus 3tkykX0y (14)

Mk xx+Lminus 1 xx+Lminus 2 middot middot middot xk1113858 1113859T (15)

+erefore equation (13) can be written as followsddf

tTy

y1113888 1113889 y

minus 1t1M1 + t2M2 + middot middot middot tNminus LMNminus L( 1113857

minus yminus 3

tT

yX0y

(16)

Let X0 [M1 M1 middot middot middot Mk] in the above equation thenequation (13) is equivalent to the following equation

yminus 1

X0t minus yminus 3

tTyX0y 0 (17)

By sorting out the above equation we can gettTy

yminus 2X0y X0t (18)

Since y XT0 f if (X0X

T0 )minus 1 exists the following

equation can be obtained

Mixedmatrix

(A)

Separationmatrix

(W)

S1 (t)

S2 (t)

Sn (t)

X1 (t)

X2 (t)

Xn (t)

Y1 (t)

Y2 (t)

Yn (t)

n1 (t) n2 (t) nn (t)

Blind source Observed signal Estimated signal

Figure 1 Frame of independent component analysis

4 Mathematical Problems in Engineering

tTy

y21113888 1113889X0f X0X

T01113872 1113873

minus 1X0t (19)

+e special solution of the above equation is a set ofoptimal filters as shown in the following equation

f X0XT01113872 1113873

minus 1X0t (20)

Substituting the above equation into y XT0 f the

original impact signal y can be reconstructed

3 The Process of Fault Feature Extraction

Although the adaptive decomposition of component signalsby the FDM avoids modal overlapping and endpoint effectthe blind selection and rejection of component signals onlyby the FDM may lead to incomplete extraction of faultfeature information RobustICA as a newer method in blindsource separation is not easily interfered by strong noiseHowever the processed bearing vibration signal is often asingle channel which does not meet the conditions of use ofRobustICA To a great extent it restricts the application ofthis kind of algorithm+erefore in order to give full play tothe advantages of FDM and RobustICA in signal processingthis study used a combination of FDM and RobustICA toreduce noise on bearing fault signals When the originalbearing fault signal with noise is decomposed by the FDM anumber of FIBF components from high to low frequencywill be obtained Under actual working conditions the noisesignal is unknown which brings great difficulties to theconstruction of the virtual noise channel If the method ofconstructing the virtual noise channel is inappropriate itwill directly affect the effect of signal noise reductionConsidering the particularity of mechanical signals thisresearch will judge the sensitivity of each component signalto interference noise and fault impact by combining cross-correlation analysis criteria and kurtosis criteria so as tobuild a virtual noise channel

According to the definition of the cross-correlationcriterion the correlation coefficient can be used as an indexto evaluate the degree of correlation between two signals+e larger the correlation coefficient value the higher thecorrelation with the original signal By using this criterionthe degree of correlation between each component signalobtained from the decomposition and the original signal canbe known +e calculation method of the correlationnumber is as follows

r 1113936

ni0 xi minus x( 1113857 yi minus y( 1113857

1113936ni0 xi minus x( 1113857

2yi minus y( 1113857

21113969 (21)

where xi and x are specific values and average values of thesignal x At the same time yi and y are specific values andaverage values of the signal y respectively

Kurtosis can be used tomeasure the peak degree of signalwaveform and it is more sensitive to the impact componentsin the signal +e higher the proportion of impact com-ponents is the higher the kurtosis value will be For bearingsthe kurtosis value is close to the normal distribution under

normal operation and it will increase significantly whenfaults occur +e calculation method is as follows

Ki 1n

1113944

n

i1

xi minus x

σ1113874 1113875

4 (22)

where xi and x are specific values and average values of thesignal x σ is the standard deviation of the signal and n is thenumber of samples

After finding the correlation coefficient and kurtosisvalue of each FIBF component through the above criteriathe virtual channel is further constructed based on this+en RobustICA is used to recover the independentcomponent of the effective signal and the independentcomponent mainly of noise Under the strong noise inter-ference the fault feature of the separated effective signal isstill weak In order to enhance the impact component of thebearing fault signal after noise reduction this research in-troduced the MOMEDA to filter the signal and finallycalculated the Hilbert envelope spectrum of the filteredsignal and extracted the corresponding fault frequency so asto accurately diagnose the fault +e flowchart of the pro-posed method is shown in Figure 2

+e specific steps are as follows

(1) FDM method is used to decompose the vibrationsignals of the rolling bearing under different faultconditions and several FIBF components with highfrequency and low frequency are obtained

(2) According to the cross-correlation and kurtosiscriteria the cross-correlation coefficient and kurtosisvalue of each FIBF component are calculated +enthe signal component is selected that is highly cor-related with the original signal and has a large impactcomponent ratio (the signal components with cross-correlation coefficient greater than 03 and kurtosisvalue greater than 3 are selected in this research) toconstruct an observation signal channel At the sametime the remaining signal components are used toconstruct a virtual noise signal channel

(3) +e observation signal and the virtual noise signalare used as the input matrix of the blind sourceseparation and then the signal is demixed by usingRobustICA to achieve the noise reduction of thevibration signal

(4) +e MOMEDA is used to enhance the periodicimpact component of the noise reduced signal

(5) +e signal obtained in the previous step isdemodulated by Hilbert envelope and then the faultfrequency is extracted for fault diagnosis

4 Simulations and Comparative Analysis

In order to verify the effectiveness of themethod proposed inthis paper the fault signal of the rolling bearing is simulatedand analyzed+e structure of the analog signal is as follows

s(t) y0eminus 2πfnξt sin 2πfn

1 minus ξ21113969

t1113874 1113875 (23)

Mathematical Problems in Engineering 5

where the natural frequency of the bearing fn is 3000Hz thedisplacement constant y0 is 5 the damping coefficient ξ is 1the period of the impact failure is 001 s the samplingfrequency fs is 12 kHz and the number of sampling pointsN is 1000 +rough calculation the fault frequency f0 is100Hz +e time-domain waveform of the original signal s(t) is shown in Figure 3 After the Gaussian white noise witha signal-to-noise ratio of minus 5 dB is added to the originalsignal s (t) the time-domain waveform and frequency-domain waveform of the mixed signal y (t) are shown inFigure 4 It can be seen from Figure 4 that the impactcharacteristics of the bearing have been completely anni-hilated due to the interference of noise and the impactinterval of the bearing cannot be distinguished from thefigure

Next the fault feature extractionmethod proposed in thepaper is used to process and analyze the above analog faultsignals Firstly the FDM is performed on the mixed signal y(t) and Figure 5 shows the results of the decomposition Inorder to highlight the advantages of the method proposed inthis paper the LMD and ITD commonly used in the field offault diagnosis are used to decompose the signal and furtheranalysis is carried out +e decomposition results of theabove two methods can be seen in Figure 6 As shown inFigure 5 the mixed signal y (t) is decomposed into 19Fourier intrinsic band functions (FIBF) from high to lowfrequencies Because FDM overcomes the problems ofmodal aliasing and endpoint effects to a certain extent andhas a strict theoretical basis the components obtained by the

decomposition clearly reflect the local characteristic infor-mation of the original signal

In order to select the signal components that meet theconditions from the decomposition results the correlationcoefficient C (t) and the kurtosis value K (t) of each com-ponent signal are calculated As can be seen from Table 1only the kurtosis values of the y9 y11 and y12 componentsare greater than 3 and their correlation values are greaterthan 03 indicating that the above component signals have ahigh correlation with the original signals and contain moreimpact components +erefore the above three componentsare selected to reconstruct the observation signal channeland the remaining signal components are used to recon-struct the noise signal channel At the same time the cal-culation results of the correlation and kurtosis of the signalcomponents based on the LMD and ITD are shown inTables 2 and 3 respectively It can be seen that the corre-lation and kurtosis values of the PF1 and PF2 components inTable 2 and the PRC2 and PRC3 components in Table 3 meetthe set threshold conditions +erefore the above compo-nent signals are selected to reconstruct the observationsignals and the remaining component signals are used toreconstruct the noise signal channel +en they are sepa-rately demixed by the RobustICA to achieve the separationof the signal and noise +e noise reduction results based onFDM-RobustICA LMD-RobustICA and ITD-RobustICAare shown in Figures 7ndash9 respectively

By comparing the signal and noise separation results inFigures 7ndash9 it can be seen that the impact components in thetarget signal after noise reduction by FDM-RobustICA aremore obvious which can better extract useful informationfrom noise-containing signals and weaken the influence ofstrong background noise However the time-domainwaveform of the target signal based on LMD-RobustICA andITD-RobustICA noise reduction methods is different fromthat of the original signal to some extent Meanwhile be-cause there is no prior information of the fault in the processof using the RobustICA to carry out the demixing operationthe system has uncertainty and the signal amplitude afterdemixing changes to varying degrees but does not affect thenoise reduction analysis

In order to quantitatively analyze the noise reductioneffect based on the above three methods kurtosis valuecorrelation number and noise reduction time are selected asthe evaluation indexes +e larger the kurtosis value is themore fault information in the signal is +e larger thecorrelation number is the more complete the effective

Input signal

FDM decomposition

Calculate the correlationcoefficient and kurtosis of

each FIBF component

Reconstructobservation signal

Reconstruct virtualnoise channel signal

Unmixed by RobustICA

MOMEDA filtering

Select effective FIBFcomponents

Hilbert envelopedemodulation

Extract the fault frequency

Figure 2 +e flowchart of the proposed method

0 001 002 003 004 005 006 007 008 009ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

Figure 3 Original signal

6 Mathematical Problems in Engineering

information of the signal is retained +e shorter the noisereduction time is the higher the operation efficiency of thealgorithm is +e calculation results are shown in Table 4 Itcan be seen from Table 4 that the correlation value andkurtosis value of the noise reduced signal based on FDM-RobustICA are the largest which shows that the noise re-duced signal obtained by the method proposed has a highsimilarity with the original signal At the same time thismethod not only reduces the noise interference but alsoretains the fault characteristic information well Althoughthere is a certain gap in noise reduction time between ITD-RobustICA and FDM-RobustICA it is not significant

In order to further compare the effects of the above threemethods on the separation of the target signal and noise the

signals obtained by the above three methods are separatelydemodulated by Hilbert envelope and then the corre-sponding envelope spectrum is obtained +e results areshown in Figures 10ndash12 It can be seen from Figure 11 thatthe envelope spectrum of the noise signal separated by LMD-RobustICA has components such as one to four doublingfault frequencies and the fault frequency amplitude is rel-atively high It can be seen from Figure 12 that the envelopespectrum of the noise signal obtained by ITD-RobustICAalso contains the components of one two and three dou-bling fault frequencies It shows that after the above twomethods are processed the target signal and noise have notbeen completely separated and some of the fault signals aremissing from the separated target signal It can be seen from

0 001 002 003 004 005 006 007 008 009ndash005

0005

y1

0 001 002 003 004 005 006 007 008 009ndash02

002

y2

0 001 002 003 004 005 006 007 008 009ndash05

005

y3

0 001 002 003 004 005 006 007 008 009ndash05

005

y4

0 001 002 003 004 005 006 007 008 009ndash1

01

y5

0 001 002 003 004 005 006 007 008 009ndash1

01

y6

0 001 002 003 004 005 006 007 008 009ndash1

01

y7

0 001 002 003 004 005 006 007 008 009ndash2

02

y8

0 001 002 003 004 005 006 007 008 009ndash2

02

y9

0 001 002 003 004 005 006 007 008 009ndash1

01

y10

0 001 002 003 004 005 006 007 008 009ndash2

02

y11

0 001 002 003 004 005 006 007 008 009ndash2

02

y12

0 001 002 003 004 005 006 007 008 009ndash1

01

y13

0 001 002 003 004 005 006 007 008 009ndash05

005

y14

0 001 002 003 004 005 006 007 008 009ndash05

005

y15

0 001 002 003 004 005 006 007 008 009ndash05

005

y16

0 001 002 003 004 005 006 007 008 009ndash02

002

y17

0 001 002 003 004 005 006 007 008 009ndash005

0005

y18

0 001 002 003 004 005 006 007 008 009ndash2

02

y19

Time (s)

Time (s)

Figure 5 +e decomposition result by the FDM

0 001 002 003 004 005 006 007 008 009ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

01

02

03

04

Frequency (Hz)

(b)

Figure 4 (a) Time-domain analysis and (b) frequency-domain analysis of the mixed signal with SNR minus 5 dB

Mathematical Problems in Engineering 7

0 001 002 003 004 005 006 007 008 009ndash5

05

PRC1

0 001 002 003 004 005 006 007 008 009ndash2

02

PRC2

0 001 002 003 004 005 006 007 008 009ndash1

01

PRC3

0 001 002 003 004 005 006 007 008 009ndash1

01

PRC4

0 001 002 003 004 005 006 007 008 009ndash05

005

PRC5

0 001 002 003 004 005 006 007 008 009ndash02

002

R

Time (s)

(a)

0 001 002 003 004 005 006 007 008 009ndash5

05

PF1

0 001 002 003 004 005 006 007 008 009ndash2

02

PF2

0 001 002 003 004 005 006 007 008 009ndash2

02

PF3

0 001 002 003 004 005 006 007 008 009ndash1

01

PF4

0 001 002 003 004 005 006 007 008 009ndash500

0500

Time (s)

R

(b)

Figure 6 Signal decomposition results (a) +e decomposition result by ITD (b) +e decomposition result by LMD

Table 1 +e correlation coefficient and kurtosis between the FIBF and the original signal

y1 y2 y3 y4 y5 y6 y7 y8 y9 y10 y11 y12 y13 y14 y15 y16 y17 y18 y19C (t) 0026 0073 0121 0154 0194 0221 0313 0363 0351 0283 0445 0326 0268 0127 0164 0104 0070 0021 0000K (t) 1000 2681 2166 2783 3303 3508 2531 2924 3348 3405 4246 3380 2376 2955 3604 2637 2562 1500 1000

Table 2 +e correlation coefficient and kurtosis between the PF and the original signal

PF1 PF2 PF3 PF4C (t) 0804 0467 0211 0149K (t) 3825 3608 4864 4982

Table 3 +e correlation coefficient and kurtosis between the PRC and the original signal

PRC1 PRC2 PRC3 PRC4 PRC5C (t) 0866 0584 0333 0180 0075K (t) 2929 3768 3121 3692 3312

0 001 002 003 004 005 006 007 008 009ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 001 002 003 004 005 006 007 008 009ndash4

ndash2

0

2

4

Time (s)

(b)

Figure 7 +e waveform of each independent component by the FDM-RobustICA method (a) +e waveform of the target signal (b) +ewaveform of the noise signal

8 Mathematical Problems in Engineering

Figure 10 that the amplitude of the fault characteristicfrequency in the envelope spectrum of the target signal hasbeen increased to a certain extent Meanwhile the faultcharacteristic frequency and its doubling frequency can beobtained In addition there is no fault information in the

envelope spectrum of the noise signal in Figure 10 indi-cating that the fault signal and the noise signal are effectivelyseparated However some irrelevant frequency componentsstill appear in the envelope spectrum of the fault signalwhich have a certain degree of interference to fault diagnosis

Am

plitu

de (m

s2 )

Time (s)0 001 002 003 004 005 006 007 008 009

ndash4

ndash2

0

2

4

(a)

Am

plitu

de (m

s2 )

Time (s)0 001 002 003 004 005 006 007 008 009

ndash4

ndash2

0

2

4

(b)

Figure 8 +e waveform of each independent component by the LMD-RobustICA method (a) +e waveform of the target signal (b) +ewaveform of the noise signal

0 001 002 003 004 005 006 007 008 009

ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 001 002 003 004 005 006 007 008 009ndash4

ndash2

0

2

4

Time (s)

(b)

Figure 9 +e waveform of each independent component by the ITD-RobustICA method (a) +e waveform of the target signal (b) +ewaveform of the noise signal

Table 4 Comparison of noise reduction results of different methods

Evaluation index FDM-RobustICA LMD-RobustICA ITD-RobustICA

Kurtosis and correlation coefficient 39070582

36020374

36300480

Noise reduction time 1378 00889 0103

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

4f 0 5f 03f 0

2f 0f 0

(a)

Am

plitu

de (m

s2 )

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

(b)

Figure 10 +e envelope analysis of each independent component by the FDM-RobustICA method (a) +e envelope analysis of the targetsignal (b) +e envelope analysis of the noise signal

Mathematical Problems in Engineering 9

+erefore it is necessary to further enhance the periodicimpact component of the noise reduced signal throughMOMEDA filtering

Next the experiment uses MOMEDA to enhance thesignal after FDM-RobustICA noise reduction and then thesignal is demodulated by Hilbert envelope Figure 13 showsthe time-domain waveform and envelope spectra of theMOMEDA filter signal It can be seen from Figure 13 thatthe periodic impact component is very obvious after filteringby MOMEDA +ere are more prominent fault features inthe envelope spectrum and there are large peaks at thefrequencies of 1055Hz 1992Hz 3047Hz 3984Hz5039Hz 5977Hz 7031Hz 7969Hz 9023Hz and so on+e above peaks are very close to the simulated faultcharacteristic frequency of 100Hz and the doubling fre-quency from two to ten At the same time the amplitudes ofother noninteger multiples at the characteristic frequency ofthe fault are basically close to 0 indicating that the signalcomponents unrelated to the fault impact have been greatlyweakened +erefore the bearing fault can be accuratelydiagnosed according to the characteristic frequency

5 Application to Roller Bearing Testing

In order to further verify the effectiveness of the methodproposed in this paper the bearing data of CWRU are usedfor analysis [38]+emodel of the rolling bearing at the driveend of the motor is 6205-2RSJEMSKF and the motor speedis 1797 rpm (namely the rotation frequency is 179760Hz 2995Hz)+e sampling frequency of the fault signalis 12 kHz and the length of experimental data is 2048 +especific technical parameters are shown in Table 5

Outer race defect frequency

BPFO Z

21 minus

d

Dcos α1113888 1113889 times fr (24)

Inner race defect frequency

BPFI Z

21 +

d

Dcos α1113888 1113889 times fr (25)

Rolling element frequency

BPFR D

2d1 minus

d

D1113888 1113889

2

cos2α⎛⎝ ⎞⎠ times fr (26)

where Z is the number of rolling elements d is the rollingelement diameter D is the bearing pitch diameter α is thecontact angle and fr is the shaft rotation frequency (Hz)

Based on the bearing parameters shown in Table 5 andequations (24)ndash(26) the fault characteristic frequency ofthe rolling bearing BPFO = 10736Hz BPFI = 16219Hzand BPFR = 1411693Hz is calculated respectively inTable 6 [39]

51 Inner Ring Signal Analysis of CWRU Bearings +e timedomain and the frequency domain of the bearing inner ringfault signal are shown in Figure 14 Although it is disturbedby noise to some extent some periodic pulses can still beseen To test the effectiveness of the proposed method understrong noise interference Gaussian white noise with asignal-to-noise ratio of minus 5 dB is added to the original innerring fault signal in this research +e time domain and thefrequency domain of the generated mixed signal are shown

Am

plitu

de (m

s2 )

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

2f 0 4f 0f 0

(a)

Am

plitu

de (m

s2 )

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

2f 0 4f 03f 0f 0

(b)

Figure 11 +e envelope analysis of each independent component by the LMD-RobustICA method (a) +e envelope analysis of the targetsignal (b) +e envelope analysis of the noise signal

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

f 03f 0 5f 0

4f 02f 0

(a)

Am

plitu

de (m

s2 )

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

3f 02f 0f 0

(b)

Figure 12 +e envelope analysis of each independent component by the ITD-RobustICA method (a) +e envelope analysis of the targetsignal (b) +e envelope analysis of the noise signal

10 Mathematical Problems in Engineering

in Figure 15 Due to the enhancement of noise interference it isimpossible to distinguish the periodic pulse features fromFigure 15 and it is difficult to extract the fault feature frequencydirectly from the time domain and the frequency domain

Next the method proposed in this paper is used toprocess the mixed signal of the inner ring with added noiseAfter the mixed signal is decomposed by the FDM 25 FIBFcomponents are obtained and the decomposition results are

shown in Figure 16 Due to the limited length of the articleonly the first 20 FIBF components are listed in it +en thecross-correlation coefficient and kurtosis value of each signalcomponent are calculated and the results are shown inTable 7 It can be seen from Table 7 that the signal com-ponents of y10 y11 and y12 meet the set threshold conditionsindicating that they have a high degree of correlation withthe original signal and retain a lot of impact features in the

Am

plitu

de (m

s2 )

Am

plitu

de (m

s2 )

0 200 400 600 800 1000 1200ndash500

0

500

1000

Number (n)

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

(a)

0 200 400 600 800 10000

5

10

15

20

Frequency (Hz)

3f 05f 0

6f 0

7f 0 8f 0 9f 0

2f 0

4f 0

Am

plitu

de (m

s2 )

f 0

(b)

Figure 13 +e MOMEDA filter signal and its envelope spectrum (a) Time-domain waveform and envelope spectrum of the MOMEDAfilter signal (b) Part band envelope spectrum of the MOMEDA filter signal

Table 5 +e bearing technical parameters of SKF 6205

Rolling elementnumber (Z) Inner diameter (inches) Outer diameter

(inches)Rolling element

diameter d (inches) Contact angle (α) Pitch circlediameter D (inches)

Speed(rpm)

9 09843 20472 03126 00 15327 1797

Table 6 Fault characteristic frequency (unit Hz)

Inner ring fault Outer ring fault Rolling element fault1621852 1073648 1411693

0 002 004 006 008 01 012 014 016 018ndash2

ndash1

0

1

2

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

005

01

Frequency (Hz)

(b)

Figure 14 (a) Time-domain analysis and (b) frequency-domain analysis of the original inner ring fault signal

Mathematical Problems in Engineering 11

original signal+erefore the above three signal componentsare extracted to construct the target signal channel and theremaining components are reconstructed as the virtual noisechannel +e blind source separation is performed byRobustICA and two sets of independent component time-domain waveforms are obtained +e results are shown inFigure 17

It can be seen from Figure 17 that after the noise re-duction of RobustICA the noise and the target signal areseparated In order to further compare the effect of noisereduction the separated target signal and the mixed signalwith minus 5 dB noise added were analyzed by the Hilbert en-velope spectrum respectively +e results are shown inFigure 18 By analyzing Figure 18 it can be known thatbefore noise reduction the amplitude of the envelope

spectrum of the fault signal is very low due to the influence ofnoise and the interference of irrelevant frequency compo-nents on the fault diagnosis is large Although the faultfrequency of the inner ring can be found as well as 2doubling frequency 5 doubling frequency etc the ampli-tude is low and it is hard to recognize After the noisereduction by RobustICA the envelope spectrum amplitudeof the obtained target signal has been greatly improved andthe fault characteristics are more prominent At the sametime there were large peaks at frequencies of 1641Hz3223Hz 4863Hz 6445Hz 8086Hz 9727Hz and so on+e above peak values are very close to the theoreticalcalculated value of the fault characteristic frequency of thebearing inner ring of 1621852Hz and the 2ndash6 doublingfrequency However there are also many peaks in the

0 002 004 006 008 01 012 014 016 018ndash2

ndash1

0

1

2

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

005

01

Frequency (Hz)

(b)

Figure 15 (a) Time-domain analysis and (b) frequency-domain analysis of the mixed signal with SNR minus 5 dB

0 002 004 006 008 01 012 014 016 018ndash5

05

times10ndash3

y1

0 002 004 006 008 01 012 014 016 018ndash005

0005

y2

0 002 004 006 008 01 012 014 016 018ndash005

0005

y3

0 002 004 006 008 01 012 014 016 018ndash01

001

y4

0 002 004 006 008 01 012 014 016 018ndash02

002

y5

0 002 004 006 008 01 012 014 016 018ndash02

002

y6

0 002 004 006 008 01 012 014 016 018ndash05

005

y7

0 002 004 006 008 01 012 014 016 018ndash05

005

y8

0 002 004 006 008 01 012 014 016 018ndash05

005

y9

0 002 004 006 008 01 012 014 016 018ndash1

01

y10

0 002 004 006 008 01 012 014 016 018ndash1

01

y11

0 002 004 006 008 01 012 014 016 018ndash1

01

y12

0 002 004 006 008 01 012 014 016 018ndash05

005

y13

0 002 004 006 008 01 012 014 016 018ndash05

005

y14

0 002 004 006 008 01 012 014 016 018ndash05

005

y15

0 002 004 006 008 01 012 014 016 018ndash02

002

y16

0 002 004 006 008 01 012 014 016 018ndash05

005

y17

0 002 004 006 008 01 012 014 016 018ndash02

002

y18

0 002 004 006 008 01 012 014 016 018ndash02

002

y19

0 002 004 006 008 01 012 014 016 018ndash01

001

y20

Time (s) Time (s)

Figure 16 FDM decomposition result

12 Mathematical Problems in Engineering

Tabl

e7

+ecorrelationcoeffi

cientandku

rtosisbetweentheFIBF

andtheoriginal

signal

y 1y 2

y 3y 4

y 5y 6

y 7y 8

y 9y 1

0y 1

1y 1

2y 1

3y 1

4y 1

5y 1

6y 1

7y 1

8y 1

9y 2

0y 2

1y 2

2y 2

3y 2

4y 2

5

C(t)

0008

0041

0028

0082

0115

0149

0181

0211

0259

0468

0430

0360

0215

0270

0234

0143

0202

0117

0114

0082

0060

0036

0059

0026

000

K(t)

1000

1958

1928

3032

2501

2708

2375

2470

2847

4302

3221

3183

2793

3417

3383

2424

2881

2620

3039

2866

2444

2743

2073

1500

1000

Mathematical Problems in Engineering 13

envelope spectrum of the signal that have nothing to do withthe characteristic frequency of the fault and these peaks willaffect the results of fault diagnosis +erefore the periodicimpact components in the signal after noise reduction will befurther extracted by MOMEDA filtering and the extractionresults are shown in Figure 19

It can be clearly seen from Figure 19 that the MOMEDAhas effectively extracted the periodic fault shock componentsin the reconstructed signal and a larger peak has appearedMeanwhile the signal components unrelated to the faultshock have been weakened By comparing the envelopespectrum of the signals in Figures 19 and 18 it can be seenthat after MOMEDA filtering the characteristic frequencyof the fault is more obvious Not only the fault frequency ofthe bearing inner ring (1641Hz) can be clearly distinguishedbut also the 2 doubling frequency (3223Hz) 3 doublingfrequency (4863Hz) 4 doubling frequency (6445Hz) 5doubling frequency (8086Hz) and so on are all clearlyvisible and they are very close to the theoretically calculatedfault characteristic frequency and doubling frequency+erefore the fault of bearing inner ring can be accuratelydiagnosed according to fault characteristic frequency

52 Outer Ring Signal Analysis of CWRU Bearings +e timedomain and the frequency domain of the bearing outer ringfault signal are shown in Figure 20 from which the periodicpulse characteristics can be seen After adding Gaussianwhite noise with a signal-to-noise ratio of minus 5 dB to the outerring fault signal the time domain and the frequency domainof the generated mixed signal are shown in Figure 21 Due to

the interference of noise it is difficult to extract the faultfeature frequency

+en the method proposed in this paper is used toprocess the mixed signal of the outer ring with added noiseAfter FDM decomposition 25 FIBF components are ob-tained +e decomposition results are shown in Figure 22Due to the limitation of the length of the article only the first20 FIBF components are listed in the figure By calculatingthe cross-correlation coefficient and kurtosis value of eachsignal component the results are shown in Table 8 Aftercomparing the data results in Table 8 it can be seen that thesignal component of y12 meets the threshold conditions+erefore the above signal component is extracted to re-construct the target signal channel and the remainingcomponents are reconstructed as the virtual noise channel+rough the RobustICA for blind source separation thetime-domain waveforms of two groups of independentcomponents are obtained and the results are shown inFigure 23

After comparing Figures 23 and 21 it can be seen thatmost of the noise is effectively filtered out through blindsource separation processing and the waveform of the targetsignal is highly similar to the original outer ring fault signalwithout the noise added In order to further compare thenoise reduction effect Hilbert envelope spectrum analysis iscarried out for the target signal obtained by separation andthe mixed signal with minus 5 dB noise added +e results areshown in Figure 24 +rough the analysis of Figure 24 it canbe seen that in the envelope spectrum of the outer ring faultsignal without noise reduction only the frequencies of1055Hz and 2168Hz have relatively high peaks which are

0 002 004 006 008 01 012 014 016 018ndash5

0

5

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 002 004 006 008 01 012 014 016 018ndash5

0

5

Time (s)

(b)

Figure 17 +e waveform of each independent component (a) +e waveform of the target signal (b) +e waveform of the noise signal

Am

plitu

de (m

s2 )

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

2f 0

3f 0

5f 0 6f 0

f 0

4f 0

(a)

Am

plitu

de (m

s2 )

2f 0 5f 0

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

f 0

(b)

Figure 18 Envelope spectrum analysis (a) Envelope spectrum analysis of the target signal denoised by FDM-RobustICA (b) Envelopespectrum of the mixed signal

14 Mathematical Problems in Engineering

very close to the theoretical calculation value of 1073648Hzof the bearing outer ring fault characteristic frequency andthe double frequency of the basic fault frequency After noisereduction by RobustICA there are three peaks with higheramplitude in the envelope spectrum of the outer ring faultsignal namely 1055Hz 2168Hz and 3223Hz +e above

three peaks are very close to the fault frequency the 2doubling frequency and the 3 doubling frequency of thebearing outer ring fault feature Although the amplitude ofthe interference spectrum in other places has been effectivelyweakened there are still many peaks in the envelopespectrum that are independent of the fault characteristic

0 200 400 600 800 1000 1200ndash500

0

500

1000

1500A

mpl

itude

(ms

2 )A

mpl

itude

(ms

2 )

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

Number (n)

(a)

Am

plitu

de (m

s2 )

0 200 400 600 800 10000

20

40

60

80

100

Frequency (Hz)

1f 0

4f 06f 0

2f 0 3f 0

5f 0

(b)

Figure 19 +e inner ring MOMEDA filter signal and its envelope spectrum (a) Time-domain waveform and the envelope spectrum of theMOMEDA filter signal (b) Part band envelope spectrum of the MOMEDA filter signal

0 002 004 006 008 01 012 014 016 018ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

02

04

Frequency (Hz)

(b)

Figure 20 (a) Time-domain analysis and (b) frequency-domain analysis of the original outer ring fault signal

Am

plitu

de (m

s2 )

0 002 004 006 008 01 012 014 016 018ndash5

0

5

Time (s)

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

01

02

03

04

Frequency (Hz)

(b)

Figure 21 (a) Time-domain analysis and (b) frequency-domain analysis of the mixed signal with SNR minus 5 dB

Mathematical Problems in Engineering 15

frequency which will affect the fault diagnosis results+erefore the periodic impact components in the noisereduced signal will be further extracted through theMOMEDA filtering method and the extraction results areshown in Figure 25

It can be clearly seen from Figure 26 that the periodicfault impact component of the outer ring fault signal hasbeen extracted +e peak value of the fault characteristicfrequency is higher and the amplitude of the signal com-ponent independent of fault impulse is lower From Fig-ure 26 not only the fault frequency (1055Hz) of the bearingouter ring can be clearly distinguished but also the 2 dou-bling frequency (2168Hz) 3 doubling frequency (3223Hz)4 doubling frequency (4277Hz) 5 doubling frequency(5391Hz) 6 doubling frequency (6445Hz) and so on canbe clearly seen which are very close to the fault characteristicfrequency and the doubling frequency calculated theoreti-cally +erefore the fault of the bearing outer ring can beaccurately diagnosed

53 Rolling Element Signal Analysis of CWRU Bearings+e time domain and the frequency domain of the bearingrolling element fault signal are shown in Figure 26 and theperiodic pulse characteristics can be seen from the figureAfter adding Gaussian white noise with a signal-to-noiseratio of minus 5 dB to the rolling element fault signal the timedomain and the frequency domain of the generated mixedsignal are shown in Figure 27 Due to the interference ofnoise it is difficult to extract the fault feature frequency fromthe time domain and the frequency domain

+en the FDM method is used to decompose the mixedsignal of the rolling element with added noise and 25 FIBFcomponents are obtained +e decomposition results areshown in Figure 28 Due to the limitation of the length of thearticle only the first 20 FIBF components are listed inFigure 28 By calculating the cross-correlation coefficientand kurtosis value of each signal component the results areshown in Table 9 By comparing the data in Table 9 it can beseen that only the y9 signal component meets the setthreshold conditions +erefore y9 component is extractedto reconstruct the target signal channel and the remainingcomponents are reconstructed as the virtual noise channel+rough the RobustICA for blind source separation thetime-domain waveforms of two groups of independentcomponents are obtained and the results are shown inFigure 29

After comparing Figures 29 and 27 it can be seen thatthe impact component in the signal has been shown throughblind source separation In order to further compare theeffect of noise reduction Hilbert envelope spectrum analysisis carried out for the separated target signal and the mixedsignal with minus 5 dB noise added +e results are shown inFigure 30 +rough the analysis of Figure 30 it can be seenthat due to the interference of noise the amplitude of lines inthe envelope spectrum of the rolling element fault signal islow so it is difficult to extract the fault characteristic fre-quency After the noise reduction by RobustICA there aremany peaks with high amplitude in the envelope spectrum+e following frequencies can be extracted from the enve-lope spectrum 1406Hz 2813Hz and 5625Hz +e abovefrequencies are very close to the theoretical calculated values

0 002 004 006 008 01 012 014 016 018ndash005

0005

y1

0 002 004 006 008 01 012 014 016 018ndash05

005

y2

0 002 004 006 008 01 012 014 016 018ndash05

005

y3

0 002 004 006 008 01 012 014 016 018ndash05

005

y4

0 002 004 006 008 01 012 014 016 018ndash1

01

y5

0 002 004 006 008 01 012 014 016 018ndash1

01

y6

0 002 004 006 008 01 012 014 016 018ndash05

005

y7

0 002 004 006 008 01 012 014 016 018ndash1

01

y8

0 002 004 006 008 01 012 014 016 018ndash1

01

y9

0 002 004 006 008 01 012 014 016 018ndash2

02

y10

0 002 004 006 008 01 012 014 016 018ndash2

02

y11

0 002 004 006 008 01 012 014 016 018ndash5

05

y12

0 002 004 006 008 01 012 014 016 018ndash2

02

y13

0 002 004 006 008 01 012 014 016 018ndash2

02

y14

0 002 004 006 008 01 012 014 016 018ndash2

02

y15

0 002 004 006 008 01 012 014 016 018ndash1

01

y16

0 002 004 006 008 01 012 014 016 018ndash2

02

y17

0 002 004 006 008 01 012 014 016 018ndash1

01

y18

0 002 004 006 008 01 012 014 016 018ndash1

01

y19

0 002 004 006 008 01 012 014 016 018ndash1

01

y20

Time (s) Time (s)

Figure 22 FDM decomposition result

16 Mathematical Problems in Engineering

Tabl

e8

+ecorrelationcoeffi

cientandku

rtosisbetweentheFIBF

andtheoriginal

signal

y 1y 2

y 3y 4

y 5y 6

y 7y 8

y 9y 1

0y 1

1y 1

2y 1

3y 1

4y 1

5y 1

6y 1

7y 1

8y 1

9Y 2

0y 2

1y 2

2y 2

3y 2

4y 2

5

C(t)

0018

0091

0074

0097

0132

0116

0125

0165

0217

0217

0260

0523

0349

0269

0277

0186

0214

0163

0159

0182

0086

0139

0064

0034

0000

K(t)

1000

2203

2209

2266

3145

4288

2564

4142

2750

2872

2683

4184

2981

2995

3277

3013

3385

2652

3151

2489

2652

4111

2615

2652

1000

Mathematical Problems in Engineering 17

of the rolling element fault characteristic frequency of1411693Hz 2 doubling frequency and 4 doubling fre-quency Although the fault characteristic frequency in theenvelope spectrum has appeared there are still many

unrelated peaks of the fault characteristic frequency and theamplitude is large +is brings a certain degree of inter-ference to the fault diagnosis +erefore the periodic impactcomponents in the signal after noise reduction will be

0 002 004 006 008 01 012 014 016 018ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 002 004 006 008 01 012 014 016 018ndash5

0

5times10ndash3

Time (s)

(b)

Figure 23 +e waveform of each independent component (a) +e waveform of the target signal (b) +e waveform of the noise signal

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 ) f 0 2f 0

3f 0

(a)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

2f 0

f 0

(b)

Figure 24 Envelope spectrum analysis (a) Envelope spectrum analysis of the target signal denoised by FDM-RobustICA (b) Envelopespectrum of the mixed signal

0 200 400 600 800 1000 1200ndash500

0

500

1000

Number (n)

Am

plitu

de (m

s2 )

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

(a)

0 200 400 600 800 10000

20

40

60

80

100

Frequency (Hz)

f 0 3f 0 4f 07f 05f 0

6f 0 8f 0 9f 0

2f 0

Am

plitu

de (m

s2 )

(b)

Figure 25 +e outer ring MOMEDA filter signal and its envelope spectrum (a) Time-domain waveform and the envelope spectrum of theMOMEDA filter signal (b) Part band envelope spectrum of the MOMEDA filter signal

18 Mathematical Problems in Engineering

0 002 004 006 008 01 012 014 016 018ndash05

0

05

Time (s)

Am

plitu

de (m

s2 )

(a)

0 1000 2000 3000 4000 5000 60000

002

004

006

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

Figure 26 (a) Time-domain analysis and (b) frequency-domain analysis of the original rolling element fault signal

0 002 004 006 008 01 012 014 016 018ndash1

0

1

Time (s)

Am

plitu

de (m

s2 )

(a)

0 1000 2000 3000 4000 5000 60000

002

004

006

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

Figure 27 (a) Time-domain analysis and (b) frequency-domain analysis of the mixed signal with SNR minus 5 dB

0 002 004 006 008 01 012 014 016 018ndash2

02

times10ndash3

y1

0 002 004 006 008 01 012 014 016 018ndash005

0005

y2

0 002 004 006 008 01 012 014 016 018ndash01

001

y3

0 002 004 006 008 01 012 014 016 018ndash01

001

y4

0 002 004 006 008 01 012 014 016 018ndash02

002

y5

0 002 004 006 008 01 012 014 016 018ndash02

002

y6

0 002 004 006 008 01 012 014 016 018ndash02

002

y7

0 002 004 006 008 01 012 014 016 018ndash02

002

y8

0 002 004 006 008 01 012 014 016 018ndash05

005

y9

0 002 004 006 008 01 012 014 016 018ndash05

005

y10

0 002 004 006 008 01 012 014 016 018ndash05

005

y11

0 002 004 006 008 01 012 014 016 018ndash05

005

y12

0 002 004 006 008 01 012 014 016 018ndash05

005

y13

0 002 004 006 008 01 012 014 016 018ndash05

005

y14

0 002 004 006 008 01 012 014 016 018ndash02

002

y15

0 002 004 006 008 01 012 014 016 018ndash02

002

y16

0 002 004 006 008 01 012 014 016 018ndash02

002

y7

0 002 004 006 008 01 012 014 016 018ndash02

002

y18

0 002 004 006 008 01 012 014 016 018ndash01

001

y19

0 002 004 006 008 01 012 014 016 018ndash01

001

y20

Time (s) Time (s)

Figure 28 FDM decomposition result

Mathematical Problems in Engineering 19

Tabl

e9

+ecorrelationcoeffi

cientandku

rtosisbetweentheFIBF

andtheoriginal

signal

y 1y 2

y 3y 4

y 5y 6

y 7y 8

y 9y 1

0y 1

1y 1

2y 1

3y 1

4y 1

5y 1

6y 1

7y 1

8y 1

9y 2

0y 2

1y 2

2y 2

3y 2

4y 2

5

C(t)

0004

0054

0126

0116

0120

0171

0196

0173

0306

0296

0490

0379

0219

0254

0206

0181

0209

0154

0113

0107

0051

0088

0017

0025

0000

K(t)

1000

2071

2276

2456

3148

2712

3194

3435

3276

2962

2716

2684

3304

2640

2710

2783

2614

2591

3079

2435

2673

2646

2107

1500

1000

20 Mathematical Problems in Engineering

further extracted by MOMEDA filtering and the extractionresults are shown in Figure 31

It can be clearly seen from Figure 31 that the periodicfault shock component of the rolling element fault signal issuccessfully extracted +e peak value of the characteristic

frequency of the fault is greatly improved and the amplitudeof the signal component unrelated to the impact of the faultis lower From Figure 31 not only the frequency of thebearing rolling element fault (1406Hz) can be clearly dis-tinguished but also the doubling frequency such as the 2

0 002 004 006 008 01 012 014 016 018ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

0 002 004 006 008 01 012 014 016 018ndash04

ndash02

0

02

04

Time (s)

Am

plitu

de (m

s2 )

(b)

Figure 29 +e waveform of each independent component (a) +e waveform of the target signal (b) +e waveform of the noise signal

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

f0 2f0 4f0

(a)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

Figure 30 Envelope spectrum analysis (a) Envelope spectrum analysis of the target signal denoised by FDM-RobustICA (b) Envelopespectrum of the mixed signal

0 200 400 600 800 1000 1200ndash500

0

500

1000

Am

plitu

de (m

s2 )

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

Number (n)

(a)

0 200 400 600 800 10000

20

40

60

80

100

Frequency (Hz)

Am

plitu

de (m

s2 )

f0 2f0 3f0

4f05f 0 6f0 7f0

(b)

Figure 31+e rolling element MOMEDA filter signal and its envelope spectrum (a) Time-domain waveform and the envelope spectrum ofthe MOMEDA filter signal (b) Part band envelope spectrum of the MOMEDA filter signal

Mathematical Problems in Engineering 21

doubling frequency (2813Hz) 3 doubling frequency(4219Hz) 4 doubling frequency (5625Hz) 5 doublingfrequency (7031Hz) and 6 doubling frequency (8436Hz)are clearly visible which are very close to the fault char-acteristic frequency and the multiple frequency theoretically+erefore the fault of the bearing rolling element can beaccurately diagnosed

6 Conclusions

Aiming at the problem of fault feature extraction of vi-bration signals of rolling bearings under the background ofstrong noise this paper proposes a fault feature extractionmethod based on the combination of FDM-RobustICA andMOMEDA After the experimental analysis of the con-structed simulation signals and the actual bearing faultsignals the conclusions reached are as follows

(1) On the basis of cross-correlation analysis and kur-tosis the signal components obtained by the FDMmethod are selected to reconstruct the observationsignal channel and the virtual noise channel whichnot only avoid the mode overlapping and endpointeffect produced by traditional methods but also solvethe problem that RobustICA cannot process thesingle-channel signal and the lack of fault infor-mation caused by blind selection of signalcomponents

(2) By comparing the results of the proposed methodwith the traditional method of LMD-RobustICA andITD-RobustICA the results show that the proposedmethod can more effectively separate fault signalsand noise signals and has achieved relative betterevaluation index value

(3) By using MOMEDA to filter the noise reducedsignal the periodic impact component under thestrong noise background can be effectively enhancedand the irrelevant impact signal can be weakenedBased on the constructed simulation signals and theexperimental analysis of the inner ring outer ringand rolling element of rolling bearings the resultsshow that the proposed method can accurately ex-tract the characteristic frequency of faults in a strongnoise background environment and the amplitude ismore obvious In our future work we will considermore reasonable approaches to optimize the pa-rameters of the proposed method and add otherdenoising techniques

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is work was supported by the Joint Special Fund for BasicResearch of Local Universities in Yunnan Province (no2019FH001-121) and Science Research Fund Project ofBaoshan University (no BYBS201802)

References

[1] P Schniter ldquoShort-time fourier transformrdquo Advanced Topicsin Signal Processing vol 32 pp 289ndash337 2009

[2] D Griffin and J S Jae Lim ldquoSignal estimation from modifiedshort-time fourier transformrdquo IEEE Transactions on Acous-tics Speech and Signal Processing vol 32 no 2 pp 236ndash2431984

[3] A Georgakis L K Stergioulas and G Giakas ldquoWigner fil-tering with smooth roll-off boundary for differentiation ofnoisy non-stationary signalsrdquo Signal Processing vol 82no 10 pp 1411ndash1415 2002

[4] M Holschneider R Kronland-Martinet J Morlet andP Tchamitchian A Real-Time Algorithm for Signal Analysiswith the Help of the Wavelet Transform Springer BerlinGermany 1989

[5] Z Nenadic and J W Burdick ldquoSpike detection using thecontinuous wavelet transformrdquo IEEE Transactions on Bio-medical Engineering vol 52 no 1 pp 74ndash87 2005

[6] N E Huang Z Shen S R Long et al ldquo+e empirical modedecomposition and the Hilbert spectrum for nonlinear andnon-stationary time series analysisrdquo Proceedings of the RoyalSociety of London Series A Mathematical Physical and En-gineering Sciences vol 454 no 1971 pp 903ndash995 1998

[7] V K Wu V Bajaj A Kumar D Sharma and G K SinghldquoAn efficient method for analysis of EMG signals using im-proved empirical mode decompositionrdquo AEU-InternationalJournal of Electronics and Communications vol 72pp 200ndash209 2017

[8] J Cheng D Yu J Tang and Y Yang ldquoApplication of fre-quency family separation method based upon EMD and localHilbert energy spectrum method to gear fault diagnosisrdquoMechanism and Machine Beory vol 43 no 6 pp 712ndash7232008

[9] Z Wu and N E Huang ldquoEnsemble empirical mode de-composition a noise-assisted data analysis methodrdquo Ad-vances in Adaptive Data Analysis vol 1 no 1 pp 1ndash41 2009

[10] H Li T Liu X Wu and Q Chen ldquoApplication of EEMD andimproved frequency band entropy in bearing fault featureextractionrdquo ISA Transactions vol 88 pp 170ndash185 2019

[11] C Wu J Wu and J Yang ldquoResearch on fault diagnosis ofcheck valve based on CEEMD compound screening andimproved SECPSOrdquo Journal of Computers vol 30 pp 128ndash144 2019

[12] K Moshen C Gang P Yusong and Y Li ldquoResearch ofplanetary gear fault diagnosis based on permutation entropyof CEEMDAN and ANFISrdquo Sensors vol 18 no 3 p 7822018

[13] J S Smith and S Jonathan ldquo+e local mean decompositionand its application to EEG perception datardquo Journal of theRoyal Society Interface vol 2 no 5 pp 443ndash454 2005

[14] M G Frei and I Osorio ldquoIntrinsic time-scale decompositiontime-frequency-energy analysis and real-time filtering of non-stationary signalsrdquo Proceedings of the Royal Society AMathematical Physical and Engineering Sciences vol 463no 2078 pp 321ndash342 2007

22 Mathematical Problems in Engineering

[15] P Singh S D Joshi R K Patney and K Saha ldquo+e fourierdecompositionmethod for nonlinear and non-stationary timeseries analysisrdquo Proceedings of the Royal Society A Mathe-matical Physical and Engineering Science vol 473 pp 1ndash122017

[16] C H Dou and J S Lin ldquoExtraction of fault features ofmachinery based on fourier decomposition methodrdquo IEEEAccess vol 7 pp 1ndash11 2019

[17] Y Liu X D Liu and S Liang ldquoAeroengine rotor fault di-agnosis based on fourier decomposition methodrdquo ChinaMechanical Engineering vol 30 pp 2156ndash2163 2019

[18] XW Fu and X Q Gao ldquoRolling bearing fault diagnosis basedon FDM and singular value difference spectrumrdquo ActaMetrologica Sinica vol 39 pp 688ndash692 2018

[19] P Comon ldquoIndependent component analysis a new con-ceptrdquo Signal Processing vol 36 no 3 pp 287ndash314 1994

[20] A Hyvarinen and E Oja ldquoA fast fixed-point algorithm forindependent component analysisrdquo Neural Computationvol 9 no 7 pp 1483ndash1492 1997

[21] A Hyvarinen ldquoFast and robust fixed-point algorithms forindependent component analysisrdquo IEEE Transactions onNeural Networks vol 10 no 3 pp 626ndash634 1999

[22] P Chevalier L Albera P Comon and A Ferreol ldquoCom-parative performance analysis of eight blind source separationmethods on radio communications signalsrdquo in Proceedings ofthe International Joint Conference on Neural Networkspp 25ndash31 Budapest Hungary July 2004

[23] V Zarzoso and P Comon ldquoRobust independent componentanalysis for blind source separation and extraction with ap-plication in electrocardiographyrdquo in Proceedings of theEMBC-2008 30th Annual International Conference of theIEEE Engineering in Medicine and Biology Society pp 3344ndash3347 Vancouver BC Canada August 2008

[24] V Zarzoso and P Comon ldquoRobust independent componentanalysis by iterative maximization of the kurtosis contrastwith algebraic optimal step sizerdquo IEEE Transactions on NeuralNetworks vol 21 pp 248ndash261 2009

[25] J Yao Y Xiang S Qian S Wang and S Wu ldquoNoise sourceidentification of diesel engine based on variational modedecomposition and robust independent component analysisrdquoApplied Acoustics vol 116 pp 184ndash194 2017

[26] F Bi L Li J Zhang and T Ma ldquoSource identification ofgasoline engine noise based on continuous wavelet transformand EEMD-RobustICArdquo Applied Acoustics vol 100 pp 34ndash42 2015

[27] Y Jiachi X Yang Q Sichong andW Shuai ldquoRadiation noiseseparation of internal combustion engine based on Gam-matone-RobustICA methodrdquo Shock and Vibration vol 2017Article ID 7565041 14 pages 2017

[28] Y Jingzong W Xiaodong W Jiande and F Zao ldquoA noisereduction method based on CEEMD-RobustICA and itsapplication in pipeline blockage signalrdquo Journal of AppliedScience and Engineering vol 22 pp 1ndash10 2019

[29] Y Jiachi X Yang Q Sichong and W Shuai ldquoNoise sourceseparation of an internal combustion engine based on asingle-channel algorithmrdquo Shock and Vibration vol 2019Article ID 1283263 19 pages 2019

[30] H Weixin and L Dunwen ldquoMine microseismic signaldenosing based on variational mode decomposition and in-dependent component analysisrdquo Journal of Vibration andShock vol 38 pp 56ndash63 2019

[31] R A Wiggins ldquoMinimum entropy deconvolutionrdquo Geo-exploration vol 16 no 1-2 pp 21ndash35 1978

[32] S Wang and J Xiang ldquoA minimum entropy deconvolution-enhanced convolutional neural networks for fault diagnosis ofaxial piston pumpsrdquo Soft Computing vol 24 no 4pp 2983ndash2997 2020

[33] H Liu and J Xiang ldquoA strategy using variational mode de-composition L-kurtosis and minimum entropy deconvolu-tion to detect mechanical faultsrdquo IEEE Access vol 7pp 70564ndash70573 2019

[34] N Sawalhi R B Randall and H Endo ldquo+e enhancement offault detection and diagnosis in rolling element bearings usingminimum entropy deconvolution combined with spectralkurtosisrdquo Mechanical Systems and Signal Processing vol 21no 6 pp 2616ndash2633 2007

[35] G L McDonald Q Zhao and M J Zuo ldquoMaximum cor-related kurtosis deconvolution and application on gear toothchip fault detectionrdquo Mechanical Systems and Signal Pro-cessing vol 33 pp 237ndash255 2012

[36] G L Mcdonald and Q Zhao ldquoMultipoint optimal minimumentropy deconvolution and convolution fix application tovibration fault detectionrdquo Mechanical Systems and SignalProcessing vol 82 pp 461ndash477 2017

[37] Y Li G Cheng X Chen and Y Pang ldquoResearch on bearingfault diagnosis method based on filter features of MOML-MEDA and LSTMrdquo Entropy vol 21 pp 1ndash15 2019

[38] W A Smith and R B Randall ldquoRolling element bearingdiagnostics using the Case Western Reserve University data abenchmark studyrdquoMechanical Systems and Signal Processingvol 64-65 pp 100ndash131 2015

[39] J Xia L Zhao Y Bai M Yu and Z Wang ldquoFeature ex-traction for rolling element bearing weak fault based onMCKD and VMDrdquo Journal of Vibration and Shock vol 36pp 78ndash83 2017

Mathematical Problems in Engineering 23

can adaptively decompose the signal into several naturalmodal components thereby obtaining the complete time-frequency distribution of the signal It shows some advan-tages when processing stationary and nonstationary signalsbut EMD has problems such as endpoint effects and modalaliasing and lacks strict theoretical proofs [7ndash9] In view ofthe shortcomings of EMD relevant scholars proposedEEMD CEEMD CEEMDAN and other improvedmethods but these methods still have similar problems withEMD [10ndash12] Based on the idea of EMD LMD ITD andother signal processing methods have been proposed oneafter another [13 14] but these methods cannot fully makeup for the shortcomings of EMD In recent years Singh et al[15] proposed a new signal analysis method-Fourier de-composition method (FDM) based on the study of Fouriertransform By using the FDM any complex signal can beadaptively decomposed into a series of Fourier intrinsicband functions (FIBF) +is method changes the defect thatthe Fourier series expansion method can only handle sta-tionary linear signals and can effectively analyze nonsta-tionary nonlinear signals Because the FDM is a completeand orthogonal signal processing method with strict theo-retical basis it has been initially applied in fault diagnosis invarious fields and has shown great advantages Dou and Lin[16] demonstrated the adaptive narrowband filtering char-acteristics of the FDM at low and high frequencies bydetecting white Gaussian noise At the same time aftercomparing the effects of different methods on processinggearbox vibration signals the results show that the FDM canovercome the bottleneck of the EMD in adaptively sepa-rating low-frequency signal components and has superiorperformance Aiming at the problem that the traditionaltime-frequency analysis method is difficult to effectivelyextract the rotor fault characteristics Liu et al [17] intro-duced the FDM to the fault diagnosis of rotor rubbing+rough experimental comparison based on differentmethods the results show that the FDM can achievecomplete signal decomposition and achieve better faultdiagnosis Fu and Gao [18] proposed a fault diagnosismethod for rolling bearings by combining the FDM andsingular value difference spectrum +e experimental sim-ulation and actual engineering tests verify the effectivenessof the proposed method

At the same time as a blind source separation methodindependent component analysis (ICA) provides a new ideafor noise reduction of mechanical equipment fault signals In1994 Comon explained the concept of independent com-ponent analysis (ICA) and pointed out that ICA is an ex-tension of PCA He proposed the target function of theKullbackndashLeibler criterion on the basis of making full use ofthe high-order statistical characteristics of the signal whichbecame themainstream algorithm of blind source separationin the future [19] After that the Finnish scholar Hyvarinenand Oja [20] proposed a fast-independent componentanalysis method based on kurtosis +en the algorithm isimproved using more robust negative entropy as a non-Gaussian measurement criterion and a fast algorithmFastICA based on negative entropy maximization is pro-posed [21] +e algorithm has a very high efficiency so it has

been widely used However in 2004 Zarzoso and Comonet al proposed in [22] that FastICA would fail in the case ofweak or high spatial correlation source signals After thatthey put forward the RobustICA [23 24] and it has de-veloped rapidly in the algorithm of blind source separationAs an algorithm based on kurtosis and optimal step size thisalgorithm has excellent antinoise performance It can sep-arate noise and real signal from the multichannel mixedsignal without prior knowledge of the signal source +ealgorithm is proved to be not only simple but also better thanFastICA In recent years it has been successfully applied tothe noise source separation in the fields of diesel enginegasoline engine and combustion engine [25ndash29] and hasshown high signal separation quality and fast calculationefficiency However in the process of using RobustICA thenumber of sensors must be greater than or equal to thenumber of components separated by them [30] In order tosolve the shortcomings of RobustICA the signal can bedecomposed into many components by time-frequencyanalysis and more virtual observation channels than thesource signal can be generated In this way by combining theadvantages of the above two methods noise reduction of thebearing signal can be achieved However in the process ofuse although the signal-to-noise ratio is improved bysuppressing the noise the useful signal is inevitably sup-pressed Especially under the condition of strong noiseinterference the impact component of the bearing faultsignal after noise reduction is still weak and even the weakfault may be covered by the strong fault which bringsdifficulties for further extraction of fault features

+e essence of fault feature extraction is to extract theperiodic shock components in the signal that is to process thesignal through an optimal filter thereby highlighting the faultshock components +e minimum entropy deconvolution(MED) was first proposed byWiggins in 1978 [31] It uses themaximum kurtosis as the iteration termination condition tosolve the filter which can play a role in highlighting theimpact characteristics in the signal [32 33] Sawalhi et al firstused MED for fault detection of rotating machinery [34]However this iterative method is not only complex but alsonot necessarily the global optimal filter Secondly the methodis only suitable for the single pulse impulse signal +e faultcharacteristics of rotating machinery are mostly periodicpulses In order to overcome the shortcomings of MEDMcDonald et al proposed the maximum correlated kurtosisdeconvolution (MCKD) [35] in 2012 based on the correlationkurtosis To some extent this method can satisfy the need ofdeconvolution with periodic pulse but it is still an iterativeprocess with the maximum correlation kurtosis as the cri-terion +e result is not the optimal solution and it needs theprior knowledge of the fault cycle In view of the limitations ofthe above two methods McDonald and Zhao [36] proposedthe multipoint optimal minimum entropy deconvolutionadjusted (MOMEDA) which has been successfully applied tothe fault diagnosis of the gearbox +is method uses a timetarget vector to define the pulse position obtained by solvingthe deconvolution During the execution of the algorithm theoptimal filter can be obtained without iteration [37] and theperiod of the failure does not need to be determined in

2 Mathematical Problems in Engineering

advance By using this method the periodic impact com-ponents in the signal can be effectively enhanced whichmakes a good foundation for the feature extraction of faultsignals

In summary this paper proposed a fault feature ex-traction method combining FDM-RobustICA andMOMEDA Firstly the FDM is used to decompose thevibration signal of the bearing and obtain several Fourierintrinsic band functions (FIBF) +en by combining thecross-correlation analysis criterion and the kurtosis crite-rion an effective FIBF component is selected as the ob-servation signal and blind source separation is performed+irdly MOMEDA is used to enhance the periodic impactcomponents in the noise-reduced signal Finally the Hilbertenvelope demodulation is used to obtain the fault charac-teristic frequency and the fault characteristic frequency iscompared with the theoretical calculated value for faultdiagnosis

+e structure of this paper is as follows Section 2 introducesthe basic principles of FDM RobustICA and MOMEDASection 3 briefly introduces the implementation details and faultfeature extraction process of themethod proposed in this paperSection 4 uses the proposed method to analyze the simulationsignal Section 5 further verifies the effectiveness of the proposedmethod through the actual bearing fault signal Section 6 givesthe discussion and conclusion

2 Theoretical Background

21 Fourier Decomposition Method Fourier decompositionmethod (FDM) is a new type of the adaptive signal analysismethod based on Fourier transform +is method firstadaptively searches and analyzes the Fourier intrinsic bandfunctions (FIBF) in the entire Fourier domain +en thenonlinear signal is adaptively decomposed into the sum ofseveral physical Fourier intrinsic band functions (FIBF) anda residual component

(1) FIBF has a zero mean value namely 1113938b

ayi(t)dt 0

(2) Any two FIBF are orthogonal to each other namely1113938

b

ayi(t)yj(t)dt 0 ine j

(3) +e analytic function form of FIBF has nonnegativeinstantaneous amplitude and frequency yi(t)

1113954yj(t) ai(t)ejφi(t) namely ai(t)ge 0 φiprime(t)ge 0

forallt isin [a b] and yi(t) isin Cinfin[a b]+erefore FIBF arethe sum of zero-mean sine functions with contin-uous frequency bands

Based on the definition of FIBF the steps of the FDM areas follows

Step 1 make a fast Fourier transformation of the signalx (n) X[k] FFT [x(n)] Step 2 use forward search which scans analytic Fourierintrinsic band functions (AFIBF) from low to highfrequencies

AFIBF 1113944n

kNiminus 1+1X[k]σj2πknN

ai(n)ejφi[n]

X (1)

In order to obtain the minimum number of AFIBF fromlow-frequency to high-frequency scanning for each i 1 2 M start from Niminus 1 + 1 and gradually increase to reachthe maximum Ni where N0 0 and NM (N2 minus 1) Itsatisfies Niminus 1 + 1leNiminus 1 le (N2 minus 1) and also satisfies

ai(t)ge 0 ωi(t) φi[n + 1] minus φi[n minus 1]

2ge 0 forallt (2)

Similarly a reverse search can also be performednamely scanning the AFIBF component from high to lowfrequencies Correspondingly change the upper and lowerlimits of the sum in equation (2) from Ni to Niminus 1 minus 1 wherei 1 2 M N0 N2 and NM 1 +e search shouldstart from Niminus 1 minus 1 and gradually decrease until the smallestNi satisfies 1leNi leNiminus 1 minus 1 and the phase φi[n] is amonotonically increasing function +e instantaneous fre-quency and amplitude can be calculated directly by AFIBFMeanwhile the real part of AFIBF is FIBF

22 Robust Independent Component Analysis Robust in-dependent component analysis (RobustICA) is an inde-pendent component analysis algorithm based on kurtosisand optimal step size that has emerged in recent years on thebasis of the original independent component analysis (ICA)+e algorithm searches for the global optimal step sizethrough the kurtosis comparison function finds the dem-ixing matrix and calculates the approximate value of thesource signal+e frame of independent component analysisis shown in Figure 1 +e purpose of independent com-ponent analysis is to separate independent source signalcomponents from the mixed signal

Let n-dimensional random observation vectors conformto the following model

x AS + N (3)

where S [s1 s2 sm]T and x [x1 x2 xn]T S is the m-dimensional signal vector which means that the number ofmixed signals is m N is the n-dimensional observationvector which means that the number of mixed signals is n Ais an n times m matrix +e basic idea of ICA is to solve theseparation matrix W and make an optimal estimate y of thesource signal S when only the observation signal x is known

y Wx (4)

Compared with the traditional ICA RobustICA does notneed to prewhiten the data which reduces the amount ofcalculation At the same time RobustICA can be used toprocess the sub-Gaussian signal and the super-Gaussiansignal When the Gaussian property of the signal is known itcan extract only the components of interest in the sourcesignal without increasing the complexity of the operationand the estimation error +e linear search method used byRobustICA can ensure that the optimization can be achievedin all the processes of component separation Assuming theoutput signal is y Wx (W is the m times n-dimensionalseparation matrix) the kurtosis equation can be expressed asfollows

Mathematical Problems in Engineering 3

K(W) E |y|41113966 1113967 minus 2E2 |y|21113966 1113967 minus E y21113864 1113865

111386811138681113868111386811138681113868111386811138682

E2 |y|21113966 1113967 (5)

In the equation E middot represents the mathematical ex-pectation RobustICA directly uses the above equation as acomparison function without any simplification and thenoptimizes the kurtosis according to the strategy of the bestlinear optimization step size

μopt argmaxu

|K(W + μg)| (6)

where g is the search direction usually g nablawK(W) whichcan be expressed as follows

nablawK(W) 4

E2 |y|21113966 1113967E |y|

21113966 1113967yx minus E yx1113864 11138651113966 1113967E y

21113966 1113967

minusE |y|41113966 1113967 minus E y21113864 1113865

111386811138681113868111386811138681113868111386811138682

1113872 1113873E yx1113864 1113865

E |y|21113966 1113967

(7)

In the process of iteration the optimization method ofRobustICA can be described as follows

(1) Find the coefficients of the OS polynomial When thekurtosis becomes a comparison function the OSpolynomial can be expressed as follows

P(u) 11139444

k0aku

k (8)

(2) Extract the root a of the OS polynomial uk1113864 11138654k1

(3) Select the root that can make the best optimizationstep μopt argmaxu|K(W + μg)| and take the largestvalue from the direction of iteration

(4) Update μopt based on the value of the updated stepW+ W+ W + μoptg

(5) Normalize W+ W+⟵W+W+(6) Determine whether the iterative operation converges

and meets the termination conditions If it is notsatisfied then return to Step (1)

23 Multipoint Optimal Minimum Entropy DeconvolutionAdjusted If x is the vibration signal y is the shock se-quence Meanwhile h is the system frequency responsefunction e is the noise signal +e symbol lowast is the multi-plication symbol +e collected signal can be expressed as

x hlowasty + e (9)

+e essence of the MOMEDA is to use a noniterativemethod to find an optimal FIR filter Recover the vibrationshock signal from the output signal as much as possiblethrough the deconvolution process and reduce the impact ofnoise +e deconvolution process is as follows

y flowastx 1113944Nminus L

k1fkxk+Lminus 1 (10)

where k 1 2 middot middot middot N minus L According to the characteristics ofthe periodic shock signal multipoint D-norm is introducednamely

MDN(y t) 1

t

tTy

y (11)

MOMEDA maxf

MDN(y t) maxf

tTy

y (12)

In equation (12) t is a constant vector for determiningthe position and weight of the target impact component+eoptimal filter f is obtained by solving the maximum value ofthe multipoint D-norm At this time the deconvolutionprocess also obtains the optimal solution In order to find theextreme value of equation (12) the filter coefficient f

(f1 f2 middot middot middot fL) is first derivedddf

tTy

y1113888 1113889

ddf

t1y1

y+

ddf

t2y2

y+ middot middot middot +

ddf

tNminus LyNminus L

y

(13)

ddf

tkyk

y y

minus 1tkMk minus y

minus 3tkykX0y (14)

Mk xx+Lminus 1 xx+Lminus 2 middot middot middot xk1113858 1113859T (15)

+erefore equation (13) can be written as followsddf

tTy

y1113888 1113889 y

minus 1t1M1 + t2M2 + middot middot middot tNminus LMNminus L( 1113857

minus yminus 3

tT

yX0y

(16)

Let X0 [M1 M1 middot middot middot Mk] in the above equation thenequation (13) is equivalent to the following equation

yminus 1

X0t minus yminus 3

tTyX0y 0 (17)

By sorting out the above equation we can gettTy

yminus 2X0y X0t (18)

Since y XT0 f if (X0X

T0 )minus 1 exists the following

equation can be obtained

Mixedmatrix

(A)

Separationmatrix

(W)

S1 (t)

S2 (t)

Sn (t)

X1 (t)

X2 (t)

Xn (t)

Y1 (t)

Y2 (t)

Yn (t)

n1 (t) n2 (t) nn (t)

Blind source Observed signal Estimated signal

Figure 1 Frame of independent component analysis

4 Mathematical Problems in Engineering

tTy

y21113888 1113889X0f X0X

T01113872 1113873

minus 1X0t (19)

+e special solution of the above equation is a set ofoptimal filters as shown in the following equation

f X0XT01113872 1113873

minus 1X0t (20)

Substituting the above equation into y XT0 f the

original impact signal y can be reconstructed

3 The Process of Fault Feature Extraction

Although the adaptive decomposition of component signalsby the FDM avoids modal overlapping and endpoint effectthe blind selection and rejection of component signals onlyby the FDM may lead to incomplete extraction of faultfeature information RobustICA as a newer method in blindsource separation is not easily interfered by strong noiseHowever the processed bearing vibration signal is often asingle channel which does not meet the conditions of use ofRobustICA To a great extent it restricts the application ofthis kind of algorithm+erefore in order to give full play tothe advantages of FDM and RobustICA in signal processingthis study used a combination of FDM and RobustICA toreduce noise on bearing fault signals When the originalbearing fault signal with noise is decomposed by the FDM anumber of FIBF components from high to low frequencywill be obtained Under actual working conditions the noisesignal is unknown which brings great difficulties to theconstruction of the virtual noise channel If the method ofconstructing the virtual noise channel is inappropriate itwill directly affect the effect of signal noise reductionConsidering the particularity of mechanical signals thisresearch will judge the sensitivity of each component signalto interference noise and fault impact by combining cross-correlation analysis criteria and kurtosis criteria so as tobuild a virtual noise channel

According to the definition of the cross-correlationcriterion the correlation coefficient can be used as an indexto evaluate the degree of correlation between two signals+e larger the correlation coefficient value the higher thecorrelation with the original signal By using this criterionthe degree of correlation between each component signalobtained from the decomposition and the original signal canbe known +e calculation method of the correlationnumber is as follows

r 1113936

ni0 xi minus x( 1113857 yi minus y( 1113857

1113936ni0 xi minus x( 1113857

2yi minus y( 1113857

21113969 (21)

where xi and x are specific values and average values of thesignal x At the same time yi and y are specific values andaverage values of the signal y respectively

Kurtosis can be used tomeasure the peak degree of signalwaveform and it is more sensitive to the impact componentsin the signal +e higher the proportion of impact com-ponents is the higher the kurtosis value will be For bearingsthe kurtosis value is close to the normal distribution under

normal operation and it will increase significantly whenfaults occur +e calculation method is as follows

Ki 1n

1113944

n

i1

xi minus x

σ1113874 1113875

4 (22)

where xi and x are specific values and average values of thesignal x σ is the standard deviation of the signal and n is thenumber of samples

After finding the correlation coefficient and kurtosisvalue of each FIBF component through the above criteriathe virtual channel is further constructed based on this+en RobustICA is used to recover the independentcomponent of the effective signal and the independentcomponent mainly of noise Under the strong noise inter-ference the fault feature of the separated effective signal isstill weak In order to enhance the impact component of thebearing fault signal after noise reduction this research in-troduced the MOMEDA to filter the signal and finallycalculated the Hilbert envelope spectrum of the filteredsignal and extracted the corresponding fault frequency so asto accurately diagnose the fault +e flowchart of the pro-posed method is shown in Figure 2

+e specific steps are as follows

(1) FDM method is used to decompose the vibrationsignals of the rolling bearing under different faultconditions and several FIBF components with highfrequency and low frequency are obtained

(2) According to the cross-correlation and kurtosiscriteria the cross-correlation coefficient and kurtosisvalue of each FIBF component are calculated +enthe signal component is selected that is highly cor-related with the original signal and has a large impactcomponent ratio (the signal components with cross-correlation coefficient greater than 03 and kurtosisvalue greater than 3 are selected in this research) toconstruct an observation signal channel At the sametime the remaining signal components are used toconstruct a virtual noise signal channel

(3) +e observation signal and the virtual noise signalare used as the input matrix of the blind sourceseparation and then the signal is demixed by usingRobustICA to achieve the noise reduction of thevibration signal

(4) +e MOMEDA is used to enhance the periodicimpact component of the noise reduced signal

(5) +e signal obtained in the previous step isdemodulated by Hilbert envelope and then the faultfrequency is extracted for fault diagnosis

4 Simulations and Comparative Analysis

In order to verify the effectiveness of themethod proposed inthis paper the fault signal of the rolling bearing is simulatedand analyzed+e structure of the analog signal is as follows

s(t) y0eminus 2πfnξt sin 2πfn

1 minus ξ21113969

t1113874 1113875 (23)

Mathematical Problems in Engineering 5

where the natural frequency of the bearing fn is 3000Hz thedisplacement constant y0 is 5 the damping coefficient ξ is 1the period of the impact failure is 001 s the samplingfrequency fs is 12 kHz and the number of sampling pointsN is 1000 +rough calculation the fault frequency f0 is100Hz +e time-domain waveform of the original signal s(t) is shown in Figure 3 After the Gaussian white noise witha signal-to-noise ratio of minus 5 dB is added to the originalsignal s (t) the time-domain waveform and frequency-domain waveform of the mixed signal y (t) are shown inFigure 4 It can be seen from Figure 4 that the impactcharacteristics of the bearing have been completely anni-hilated due to the interference of noise and the impactinterval of the bearing cannot be distinguished from thefigure

Next the fault feature extractionmethod proposed in thepaper is used to process and analyze the above analog faultsignals Firstly the FDM is performed on the mixed signal y(t) and Figure 5 shows the results of the decomposition Inorder to highlight the advantages of the method proposed inthis paper the LMD and ITD commonly used in the field offault diagnosis are used to decompose the signal and furtheranalysis is carried out +e decomposition results of theabove two methods can be seen in Figure 6 As shown inFigure 5 the mixed signal y (t) is decomposed into 19Fourier intrinsic band functions (FIBF) from high to lowfrequencies Because FDM overcomes the problems ofmodal aliasing and endpoint effects to a certain extent andhas a strict theoretical basis the components obtained by the

decomposition clearly reflect the local characteristic infor-mation of the original signal

In order to select the signal components that meet theconditions from the decomposition results the correlationcoefficient C (t) and the kurtosis value K (t) of each com-ponent signal are calculated As can be seen from Table 1only the kurtosis values of the y9 y11 and y12 componentsare greater than 3 and their correlation values are greaterthan 03 indicating that the above component signals have ahigh correlation with the original signals and contain moreimpact components +erefore the above three componentsare selected to reconstruct the observation signal channeland the remaining signal components are used to recon-struct the noise signal channel At the same time the cal-culation results of the correlation and kurtosis of the signalcomponents based on the LMD and ITD are shown inTables 2 and 3 respectively It can be seen that the corre-lation and kurtosis values of the PF1 and PF2 components inTable 2 and the PRC2 and PRC3 components in Table 3 meetthe set threshold conditions +erefore the above compo-nent signals are selected to reconstruct the observationsignals and the remaining component signals are used toreconstruct the noise signal channel +en they are sepa-rately demixed by the RobustICA to achieve the separationof the signal and noise +e noise reduction results based onFDM-RobustICA LMD-RobustICA and ITD-RobustICAare shown in Figures 7ndash9 respectively

By comparing the signal and noise separation results inFigures 7ndash9 it can be seen that the impact components in thetarget signal after noise reduction by FDM-RobustICA aremore obvious which can better extract useful informationfrom noise-containing signals and weaken the influence ofstrong background noise However the time-domainwaveform of the target signal based on LMD-RobustICA andITD-RobustICA noise reduction methods is different fromthat of the original signal to some extent Meanwhile be-cause there is no prior information of the fault in the processof using the RobustICA to carry out the demixing operationthe system has uncertainty and the signal amplitude afterdemixing changes to varying degrees but does not affect thenoise reduction analysis

In order to quantitatively analyze the noise reductioneffect based on the above three methods kurtosis valuecorrelation number and noise reduction time are selected asthe evaluation indexes +e larger the kurtosis value is themore fault information in the signal is +e larger thecorrelation number is the more complete the effective

Input signal

FDM decomposition

Calculate the correlationcoefficient and kurtosis of

each FIBF component

Reconstructobservation signal

Reconstruct virtualnoise channel signal

Unmixed by RobustICA

MOMEDA filtering

Select effective FIBFcomponents

Hilbert envelopedemodulation

Extract the fault frequency

Figure 2 +e flowchart of the proposed method

0 001 002 003 004 005 006 007 008 009ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

Figure 3 Original signal

6 Mathematical Problems in Engineering

information of the signal is retained +e shorter the noisereduction time is the higher the operation efficiency of thealgorithm is +e calculation results are shown in Table 4 Itcan be seen from Table 4 that the correlation value andkurtosis value of the noise reduced signal based on FDM-RobustICA are the largest which shows that the noise re-duced signal obtained by the method proposed has a highsimilarity with the original signal At the same time thismethod not only reduces the noise interference but alsoretains the fault characteristic information well Althoughthere is a certain gap in noise reduction time between ITD-RobustICA and FDM-RobustICA it is not significant

In order to further compare the effects of the above threemethods on the separation of the target signal and noise the

signals obtained by the above three methods are separatelydemodulated by Hilbert envelope and then the corre-sponding envelope spectrum is obtained +e results areshown in Figures 10ndash12 It can be seen from Figure 11 thatthe envelope spectrum of the noise signal separated by LMD-RobustICA has components such as one to four doublingfault frequencies and the fault frequency amplitude is rel-atively high It can be seen from Figure 12 that the envelopespectrum of the noise signal obtained by ITD-RobustICAalso contains the components of one two and three dou-bling fault frequencies It shows that after the above twomethods are processed the target signal and noise have notbeen completely separated and some of the fault signals aremissing from the separated target signal It can be seen from

0 001 002 003 004 005 006 007 008 009ndash005

0005

y1

0 001 002 003 004 005 006 007 008 009ndash02

002

y2

0 001 002 003 004 005 006 007 008 009ndash05

005

y3

0 001 002 003 004 005 006 007 008 009ndash05

005

y4

0 001 002 003 004 005 006 007 008 009ndash1

01

y5

0 001 002 003 004 005 006 007 008 009ndash1

01

y6

0 001 002 003 004 005 006 007 008 009ndash1

01

y7

0 001 002 003 004 005 006 007 008 009ndash2

02

y8

0 001 002 003 004 005 006 007 008 009ndash2

02

y9

0 001 002 003 004 005 006 007 008 009ndash1

01

y10

0 001 002 003 004 005 006 007 008 009ndash2

02

y11

0 001 002 003 004 005 006 007 008 009ndash2

02

y12

0 001 002 003 004 005 006 007 008 009ndash1

01

y13

0 001 002 003 004 005 006 007 008 009ndash05

005

y14

0 001 002 003 004 005 006 007 008 009ndash05

005

y15

0 001 002 003 004 005 006 007 008 009ndash05

005

y16

0 001 002 003 004 005 006 007 008 009ndash02

002

y17

0 001 002 003 004 005 006 007 008 009ndash005

0005

y18

0 001 002 003 004 005 006 007 008 009ndash2

02

y19

Time (s)

Time (s)

Figure 5 +e decomposition result by the FDM

0 001 002 003 004 005 006 007 008 009ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

01

02

03

04

Frequency (Hz)

(b)

Figure 4 (a) Time-domain analysis and (b) frequency-domain analysis of the mixed signal with SNR minus 5 dB

Mathematical Problems in Engineering 7

0 001 002 003 004 005 006 007 008 009ndash5

05

PRC1

0 001 002 003 004 005 006 007 008 009ndash2

02

PRC2

0 001 002 003 004 005 006 007 008 009ndash1

01

PRC3

0 001 002 003 004 005 006 007 008 009ndash1

01

PRC4

0 001 002 003 004 005 006 007 008 009ndash05

005

PRC5

0 001 002 003 004 005 006 007 008 009ndash02

002

R

Time (s)

(a)

0 001 002 003 004 005 006 007 008 009ndash5

05

PF1

0 001 002 003 004 005 006 007 008 009ndash2

02

PF2

0 001 002 003 004 005 006 007 008 009ndash2

02

PF3

0 001 002 003 004 005 006 007 008 009ndash1

01

PF4

0 001 002 003 004 005 006 007 008 009ndash500

0500

Time (s)

R

(b)

Figure 6 Signal decomposition results (a) +e decomposition result by ITD (b) +e decomposition result by LMD

Table 1 +e correlation coefficient and kurtosis between the FIBF and the original signal

y1 y2 y3 y4 y5 y6 y7 y8 y9 y10 y11 y12 y13 y14 y15 y16 y17 y18 y19C (t) 0026 0073 0121 0154 0194 0221 0313 0363 0351 0283 0445 0326 0268 0127 0164 0104 0070 0021 0000K (t) 1000 2681 2166 2783 3303 3508 2531 2924 3348 3405 4246 3380 2376 2955 3604 2637 2562 1500 1000

Table 2 +e correlation coefficient and kurtosis between the PF and the original signal

PF1 PF2 PF3 PF4C (t) 0804 0467 0211 0149K (t) 3825 3608 4864 4982

Table 3 +e correlation coefficient and kurtosis between the PRC and the original signal

PRC1 PRC2 PRC3 PRC4 PRC5C (t) 0866 0584 0333 0180 0075K (t) 2929 3768 3121 3692 3312

0 001 002 003 004 005 006 007 008 009ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 001 002 003 004 005 006 007 008 009ndash4

ndash2

0

2

4

Time (s)

(b)

Figure 7 +e waveform of each independent component by the FDM-RobustICA method (a) +e waveform of the target signal (b) +ewaveform of the noise signal

8 Mathematical Problems in Engineering

Figure 10 that the amplitude of the fault characteristicfrequency in the envelope spectrum of the target signal hasbeen increased to a certain extent Meanwhile the faultcharacteristic frequency and its doubling frequency can beobtained In addition there is no fault information in the

envelope spectrum of the noise signal in Figure 10 indi-cating that the fault signal and the noise signal are effectivelyseparated However some irrelevant frequency componentsstill appear in the envelope spectrum of the fault signalwhich have a certain degree of interference to fault diagnosis

Am

plitu

de (m

s2 )

Time (s)0 001 002 003 004 005 006 007 008 009

ndash4

ndash2

0

2

4

(a)

Am

plitu

de (m

s2 )

Time (s)0 001 002 003 004 005 006 007 008 009

ndash4

ndash2

0

2

4

(b)

Figure 8 +e waveform of each independent component by the LMD-RobustICA method (a) +e waveform of the target signal (b) +ewaveform of the noise signal

0 001 002 003 004 005 006 007 008 009

ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 001 002 003 004 005 006 007 008 009ndash4

ndash2

0

2

4

Time (s)

(b)

Figure 9 +e waveform of each independent component by the ITD-RobustICA method (a) +e waveform of the target signal (b) +ewaveform of the noise signal

Table 4 Comparison of noise reduction results of different methods

Evaluation index FDM-RobustICA LMD-RobustICA ITD-RobustICA

Kurtosis and correlation coefficient 39070582

36020374

36300480

Noise reduction time 1378 00889 0103

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

4f 0 5f 03f 0

2f 0f 0

(a)

Am

plitu

de (m

s2 )

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

(b)

Figure 10 +e envelope analysis of each independent component by the FDM-RobustICA method (a) +e envelope analysis of the targetsignal (b) +e envelope analysis of the noise signal

Mathematical Problems in Engineering 9

+erefore it is necessary to further enhance the periodicimpact component of the noise reduced signal throughMOMEDA filtering

Next the experiment uses MOMEDA to enhance thesignal after FDM-RobustICA noise reduction and then thesignal is demodulated by Hilbert envelope Figure 13 showsthe time-domain waveform and envelope spectra of theMOMEDA filter signal It can be seen from Figure 13 thatthe periodic impact component is very obvious after filteringby MOMEDA +ere are more prominent fault features inthe envelope spectrum and there are large peaks at thefrequencies of 1055Hz 1992Hz 3047Hz 3984Hz5039Hz 5977Hz 7031Hz 7969Hz 9023Hz and so on+e above peaks are very close to the simulated faultcharacteristic frequency of 100Hz and the doubling fre-quency from two to ten At the same time the amplitudes ofother noninteger multiples at the characteristic frequency ofthe fault are basically close to 0 indicating that the signalcomponents unrelated to the fault impact have been greatlyweakened +erefore the bearing fault can be accuratelydiagnosed according to the characteristic frequency

5 Application to Roller Bearing Testing

In order to further verify the effectiveness of the methodproposed in this paper the bearing data of CWRU are usedfor analysis [38]+emodel of the rolling bearing at the driveend of the motor is 6205-2RSJEMSKF and the motor speedis 1797 rpm (namely the rotation frequency is 179760Hz 2995Hz)+e sampling frequency of the fault signalis 12 kHz and the length of experimental data is 2048 +especific technical parameters are shown in Table 5

Outer race defect frequency

BPFO Z

21 minus

d

Dcos α1113888 1113889 times fr (24)

Inner race defect frequency

BPFI Z

21 +

d

Dcos α1113888 1113889 times fr (25)

Rolling element frequency

BPFR D

2d1 minus

d

D1113888 1113889

2

cos2α⎛⎝ ⎞⎠ times fr (26)

where Z is the number of rolling elements d is the rollingelement diameter D is the bearing pitch diameter α is thecontact angle and fr is the shaft rotation frequency (Hz)

Based on the bearing parameters shown in Table 5 andequations (24)ndash(26) the fault characteristic frequency ofthe rolling bearing BPFO = 10736Hz BPFI = 16219Hzand BPFR = 1411693Hz is calculated respectively inTable 6 [39]

51 Inner Ring Signal Analysis of CWRU Bearings +e timedomain and the frequency domain of the bearing inner ringfault signal are shown in Figure 14 Although it is disturbedby noise to some extent some periodic pulses can still beseen To test the effectiveness of the proposed method understrong noise interference Gaussian white noise with asignal-to-noise ratio of minus 5 dB is added to the original innerring fault signal in this research +e time domain and thefrequency domain of the generated mixed signal are shown

Am

plitu

de (m

s2 )

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

2f 0 4f 0f 0

(a)

Am

plitu

de (m

s2 )

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

2f 0 4f 03f 0f 0

(b)

Figure 11 +e envelope analysis of each independent component by the LMD-RobustICA method (a) +e envelope analysis of the targetsignal (b) +e envelope analysis of the noise signal

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

f 03f 0 5f 0

4f 02f 0

(a)

Am

plitu

de (m

s2 )

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

3f 02f 0f 0

(b)

Figure 12 +e envelope analysis of each independent component by the ITD-RobustICA method (a) +e envelope analysis of the targetsignal (b) +e envelope analysis of the noise signal

10 Mathematical Problems in Engineering

in Figure 15 Due to the enhancement of noise interference it isimpossible to distinguish the periodic pulse features fromFigure 15 and it is difficult to extract the fault feature frequencydirectly from the time domain and the frequency domain

Next the method proposed in this paper is used toprocess the mixed signal of the inner ring with added noiseAfter the mixed signal is decomposed by the FDM 25 FIBFcomponents are obtained and the decomposition results are

shown in Figure 16 Due to the limited length of the articleonly the first 20 FIBF components are listed in it +en thecross-correlation coefficient and kurtosis value of each signalcomponent are calculated and the results are shown inTable 7 It can be seen from Table 7 that the signal com-ponents of y10 y11 and y12 meet the set threshold conditionsindicating that they have a high degree of correlation withthe original signal and retain a lot of impact features in the

Am

plitu

de (m

s2 )

Am

plitu

de (m

s2 )

0 200 400 600 800 1000 1200ndash500

0

500

1000

Number (n)

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

(a)

0 200 400 600 800 10000

5

10

15

20

Frequency (Hz)

3f 05f 0

6f 0

7f 0 8f 0 9f 0

2f 0

4f 0

Am

plitu

de (m

s2 )

f 0

(b)

Figure 13 +e MOMEDA filter signal and its envelope spectrum (a) Time-domain waveform and envelope spectrum of the MOMEDAfilter signal (b) Part band envelope spectrum of the MOMEDA filter signal

Table 5 +e bearing technical parameters of SKF 6205

Rolling elementnumber (Z) Inner diameter (inches) Outer diameter

(inches)Rolling element

diameter d (inches) Contact angle (α) Pitch circlediameter D (inches)

Speed(rpm)

9 09843 20472 03126 00 15327 1797

Table 6 Fault characteristic frequency (unit Hz)

Inner ring fault Outer ring fault Rolling element fault1621852 1073648 1411693

0 002 004 006 008 01 012 014 016 018ndash2

ndash1

0

1

2

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

005

01

Frequency (Hz)

(b)

Figure 14 (a) Time-domain analysis and (b) frequency-domain analysis of the original inner ring fault signal

Mathematical Problems in Engineering 11

original signal+erefore the above three signal componentsare extracted to construct the target signal channel and theremaining components are reconstructed as the virtual noisechannel +e blind source separation is performed byRobustICA and two sets of independent component time-domain waveforms are obtained +e results are shown inFigure 17

It can be seen from Figure 17 that after the noise re-duction of RobustICA the noise and the target signal areseparated In order to further compare the effect of noisereduction the separated target signal and the mixed signalwith minus 5 dB noise added were analyzed by the Hilbert en-velope spectrum respectively +e results are shown inFigure 18 By analyzing Figure 18 it can be known thatbefore noise reduction the amplitude of the envelope

spectrum of the fault signal is very low due to the influence ofnoise and the interference of irrelevant frequency compo-nents on the fault diagnosis is large Although the faultfrequency of the inner ring can be found as well as 2doubling frequency 5 doubling frequency etc the ampli-tude is low and it is hard to recognize After the noisereduction by RobustICA the envelope spectrum amplitudeof the obtained target signal has been greatly improved andthe fault characteristics are more prominent At the sametime there were large peaks at frequencies of 1641Hz3223Hz 4863Hz 6445Hz 8086Hz 9727Hz and so on+e above peak values are very close to the theoreticalcalculated value of the fault characteristic frequency of thebearing inner ring of 1621852Hz and the 2ndash6 doublingfrequency However there are also many peaks in the

0 002 004 006 008 01 012 014 016 018ndash2

ndash1

0

1

2

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

005

01

Frequency (Hz)

(b)

Figure 15 (a) Time-domain analysis and (b) frequency-domain analysis of the mixed signal with SNR minus 5 dB

0 002 004 006 008 01 012 014 016 018ndash5

05

times10ndash3

y1

0 002 004 006 008 01 012 014 016 018ndash005

0005

y2

0 002 004 006 008 01 012 014 016 018ndash005

0005

y3

0 002 004 006 008 01 012 014 016 018ndash01

001

y4

0 002 004 006 008 01 012 014 016 018ndash02

002

y5

0 002 004 006 008 01 012 014 016 018ndash02

002

y6

0 002 004 006 008 01 012 014 016 018ndash05

005

y7

0 002 004 006 008 01 012 014 016 018ndash05

005

y8

0 002 004 006 008 01 012 014 016 018ndash05

005

y9

0 002 004 006 008 01 012 014 016 018ndash1

01

y10

0 002 004 006 008 01 012 014 016 018ndash1

01

y11

0 002 004 006 008 01 012 014 016 018ndash1

01

y12

0 002 004 006 008 01 012 014 016 018ndash05

005

y13

0 002 004 006 008 01 012 014 016 018ndash05

005

y14

0 002 004 006 008 01 012 014 016 018ndash05

005

y15

0 002 004 006 008 01 012 014 016 018ndash02

002

y16

0 002 004 006 008 01 012 014 016 018ndash05

005

y17

0 002 004 006 008 01 012 014 016 018ndash02

002

y18

0 002 004 006 008 01 012 014 016 018ndash02

002

y19

0 002 004 006 008 01 012 014 016 018ndash01

001

y20

Time (s) Time (s)

Figure 16 FDM decomposition result

12 Mathematical Problems in Engineering

Tabl

e7

+ecorrelationcoeffi

cientandku

rtosisbetweentheFIBF

andtheoriginal

signal

y 1y 2

y 3y 4

y 5y 6

y 7y 8

y 9y 1

0y 1

1y 1

2y 1

3y 1

4y 1

5y 1

6y 1

7y 1

8y 1

9y 2

0y 2

1y 2

2y 2

3y 2

4y 2

5

C(t)

0008

0041

0028

0082

0115

0149

0181

0211

0259

0468

0430

0360

0215

0270

0234

0143

0202

0117

0114

0082

0060

0036

0059

0026

000

K(t)

1000

1958

1928

3032

2501

2708

2375

2470

2847

4302

3221

3183

2793

3417

3383

2424

2881

2620

3039

2866

2444

2743

2073

1500

1000

Mathematical Problems in Engineering 13

envelope spectrum of the signal that have nothing to do withthe characteristic frequency of the fault and these peaks willaffect the results of fault diagnosis +erefore the periodicimpact components in the signal after noise reduction will befurther extracted by MOMEDA filtering and the extractionresults are shown in Figure 19

It can be clearly seen from Figure 19 that the MOMEDAhas effectively extracted the periodic fault shock componentsin the reconstructed signal and a larger peak has appearedMeanwhile the signal components unrelated to the faultshock have been weakened By comparing the envelopespectrum of the signals in Figures 19 and 18 it can be seenthat after MOMEDA filtering the characteristic frequencyof the fault is more obvious Not only the fault frequency ofthe bearing inner ring (1641Hz) can be clearly distinguishedbut also the 2 doubling frequency (3223Hz) 3 doublingfrequency (4863Hz) 4 doubling frequency (6445Hz) 5doubling frequency (8086Hz) and so on are all clearlyvisible and they are very close to the theoretically calculatedfault characteristic frequency and doubling frequency+erefore the fault of bearing inner ring can be accuratelydiagnosed according to fault characteristic frequency

52 Outer Ring Signal Analysis of CWRU Bearings +e timedomain and the frequency domain of the bearing outer ringfault signal are shown in Figure 20 from which the periodicpulse characteristics can be seen After adding Gaussianwhite noise with a signal-to-noise ratio of minus 5 dB to the outerring fault signal the time domain and the frequency domainof the generated mixed signal are shown in Figure 21 Due to

the interference of noise it is difficult to extract the faultfeature frequency

+en the method proposed in this paper is used toprocess the mixed signal of the outer ring with added noiseAfter FDM decomposition 25 FIBF components are ob-tained +e decomposition results are shown in Figure 22Due to the limitation of the length of the article only the first20 FIBF components are listed in the figure By calculatingthe cross-correlation coefficient and kurtosis value of eachsignal component the results are shown in Table 8 Aftercomparing the data results in Table 8 it can be seen that thesignal component of y12 meets the threshold conditions+erefore the above signal component is extracted to re-construct the target signal channel and the remainingcomponents are reconstructed as the virtual noise channel+rough the RobustICA for blind source separation thetime-domain waveforms of two groups of independentcomponents are obtained and the results are shown inFigure 23

After comparing Figures 23 and 21 it can be seen thatmost of the noise is effectively filtered out through blindsource separation processing and the waveform of the targetsignal is highly similar to the original outer ring fault signalwithout the noise added In order to further compare thenoise reduction effect Hilbert envelope spectrum analysis iscarried out for the target signal obtained by separation andthe mixed signal with minus 5 dB noise added +e results areshown in Figure 24 +rough the analysis of Figure 24 it canbe seen that in the envelope spectrum of the outer ring faultsignal without noise reduction only the frequencies of1055Hz and 2168Hz have relatively high peaks which are

0 002 004 006 008 01 012 014 016 018ndash5

0

5

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 002 004 006 008 01 012 014 016 018ndash5

0

5

Time (s)

(b)

Figure 17 +e waveform of each independent component (a) +e waveform of the target signal (b) +e waveform of the noise signal

Am

plitu

de (m

s2 )

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

2f 0

3f 0

5f 0 6f 0

f 0

4f 0

(a)

Am

plitu

de (m

s2 )

2f 0 5f 0

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

f 0

(b)

Figure 18 Envelope spectrum analysis (a) Envelope spectrum analysis of the target signal denoised by FDM-RobustICA (b) Envelopespectrum of the mixed signal

14 Mathematical Problems in Engineering

very close to the theoretical calculation value of 1073648Hzof the bearing outer ring fault characteristic frequency andthe double frequency of the basic fault frequency After noisereduction by RobustICA there are three peaks with higheramplitude in the envelope spectrum of the outer ring faultsignal namely 1055Hz 2168Hz and 3223Hz +e above

three peaks are very close to the fault frequency the 2doubling frequency and the 3 doubling frequency of thebearing outer ring fault feature Although the amplitude ofthe interference spectrum in other places has been effectivelyweakened there are still many peaks in the envelopespectrum that are independent of the fault characteristic

0 200 400 600 800 1000 1200ndash500

0

500

1000

1500A

mpl

itude

(ms

2 )A

mpl

itude

(ms

2 )

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

Number (n)

(a)

Am

plitu

de (m

s2 )

0 200 400 600 800 10000

20

40

60

80

100

Frequency (Hz)

1f 0

4f 06f 0

2f 0 3f 0

5f 0

(b)

Figure 19 +e inner ring MOMEDA filter signal and its envelope spectrum (a) Time-domain waveform and the envelope spectrum of theMOMEDA filter signal (b) Part band envelope spectrum of the MOMEDA filter signal

0 002 004 006 008 01 012 014 016 018ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

02

04

Frequency (Hz)

(b)

Figure 20 (a) Time-domain analysis and (b) frequency-domain analysis of the original outer ring fault signal

Am

plitu

de (m

s2 )

0 002 004 006 008 01 012 014 016 018ndash5

0

5

Time (s)

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

01

02

03

04

Frequency (Hz)

(b)

Figure 21 (a) Time-domain analysis and (b) frequency-domain analysis of the mixed signal with SNR minus 5 dB

Mathematical Problems in Engineering 15

frequency which will affect the fault diagnosis results+erefore the periodic impact components in the noisereduced signal will be further extracted through theMOMEDA filtering method and the extraction results areshown in Figure 25

It can be clearly seen from Figure 26 that the periodicfault impact component of the outer ring fault signal hasbeen extracted +e peak value of the fault characteristicfrequency is higher and the amplitude of the signal com-ponent independent of fault impulse is lower From Fig-ure 26 not only the fault frequency (1055Hz) of the bearingouter ring can be clearly distinguished but also the 2 dou-bling frequency (2168Hz) 3 doubling frequency (3223Hz)4 doubling frequency (4277Hz) 5 doubling frequency(5391Hz) 6 doubling frequency (6445Hz) and so on canbe clearly seen which are very close to the fault characteristicfrequency and the doubling frequency calculated theoreti-cally +erefore the fault of the bearing outer ring can beaccurately diagnosed

53 Rolling Element Signal Analysis of CWRU Bearings+e time domain and the frequency domain of the bearingrolling element fault signal are shown in Figure 26 and theperiodic pulse characteristics can be seen from the figureAfter adding Gaussian white noise with a signal-to-noiseratio of minus 5 dB to the rolling element fault signal the timedomain and the frequency domain of the generated mixedsignal are shown in Figure 27 Due to the interference ofnoise it is difficult to extract the fault feature frequency fromthe time domain and the frequency domain

+en the FDM method is used to decompose the mixedsignal of the rolling element with added noise and 25 FIBFcomponents are obtained +e decomposition results areshown in Figure 28 Due to the limitation of the length of thearticle only the first 20 FIBF components are listed inFigure 28 By calculating the cross-correlation coefficientand kurtosis value of each signal component the results areshown in Table 9 By comparing the data in Table 9 it can beseen that only the y9 signal component meets the setthreshold conditions +erefore y9 component is extractedto reconstruct the target signal channel and the remainingcomponents are reconstructed as the virtual noise channel+rough the RobustICA for blind source separation thetime-domain waveforms of two groups of independentcomponents are obtained and the results are shown inFigure 29

After comparing Figures 29 and 27 it can be seen thatthe impact component in the signal has been shown throughblind source separation In order to further compare theeffect of noise reduction Hilbert envelope spectrum analysisis carried out for the separated target signal and the mixedsignal with minus 5 dB noise added +e results are shown inFigure 30 +rough the analysis of Figure 30 it can be seenthat due to the interference of noise the amplitude of lines inthe envelope spectrum of the rolling element fault signal islow so it is difficult to extract the fault characteristic fre-quency After the noise reduction by RobustICA there aremany peaks with high amplitude in the envelope spectrum+e following frequencies can be extracted from the enve-lope spectrum 1406Hz 2813Hz and 5625Hz +e abovefrequencies are very close to the theoretical calculated values

0 002 004 006 008 01 012 014 016 018ndash005

0005

y1

0 002 004 006 008 01 012 014 016 018ndash05

005

y2

0 002 004 006 008 01 012 014 016 018ndash05

005

y3

0 002 004 006 008 01 012 014 016 018ndash05

005

y4

0 002 004 006 008 01 012 014 016 018ndash1

01

y5

0 002 004 006 008 01 012 014 016 018ndash1

01

y6

0 002 004 006 008 01 012 014 016 018ndash05

005

y7

0 002 004 006 008 01 012 014 016 018ndash1

01

y8

0 002 004 006 008 01 012 014 016 018ndash1

01

y9

0 002 004 006 008 01 012 014 016 018ndash2

02

y10

0 002 004 006 008 01 012 014 016 018ndash2

02

y11

0 002 004 006 008 01 012 014 016 018ndash5

05

y12

0 002 004 006 008 01 012 014 016 018ndash2

02

y13

0 002 004 006 008 01 012 014 016 018ndash2

02

y14

0 002 004 006 008 01 012 014 016 018ndash2

02

y15

0 002 004 006 008 01 012 014 016 018ndash1

01

y16

0 002 004 006 008 01 012 014 016 018ndash2

02

y17

0 002 004 006 008 01 012 014 016 018ndash1

01

y18

0 002 004 006 008 01 012 014 016 018ndash1

01

y19

0 002 004 006 008 01 012 014 016 018ndash1

01

y20

Time (s) Time (s)

Figure 22 FDM decomposition result

16 Mathematical Problems in Engineering

Tabl

e8

+ecorrelationcoeffi

cientandku

rtosisbetweentheFIBF

andtheoriginal

signal

y 1y 2

y 3y 4

y 5y 6

y 7y 8

y 9y 1

0y 1

1y 1

2y 1

3y 1

4y 1

5y 1

6y 1

7y 1

8y 1

9Y 2

0y 2

1y 2

2y 2

3y 2

4y 2

5

C(t)

0018

0091

0074

0097

0132

0116

0125

0165

0217

0217

0260

0523

0349

0269

0277

0186

0214

0163

0159

0182

0086

0139

0064

0034

0000

K(t)

1000

2203

2209

2266

3145

4288

2564

4142

2750

2872

2683

4184

2981

2995

3277

3013

3385

2652

3151

2489

2652

4111

2615

2652

1000

Mathematical Problems in Engineering 17

of the rolling element fault characteristic frequency of1411693Hz 2 doubling frequency and 4 doubling fre-quency Although the fault characteristic frequency in theenvelope spectrum has appeared there are still many

unrelated peaks of the fault characteristic frequency and theamplitude is large +is brings a certain degree of inter-ference to the fault diagnosis +erefore the periodic impactcomponents in the signal after noise reduction will be

0 002 004 006 008 01 012 014 016 018ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 002 004 006 008 01 012 014 016 018ndash5

0

5times10ndash3

Time (s)

(b)

Figure 23 +e waveform of each independent component (a) +e waveform of the target signal (b) +e waveform of the noise signal

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 ) f 0 2f 0

3f 0

(a)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

2f 0

f 0

(b)

Figure 24 Envelope spectrum analysis (a) Envelope spectrum analysis of the target signal denoised by FDM-RobustICA (b) Envelopespectrum of the mixed signal

0 200 400 600 800 1000 1200ndash500

0

500

1000

Number (n)

Am

plitu

de (m

s2 )

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

(a)

0 200 400 600 800 10000

20

40

60

80

100

Frequency (Hz)

f 0 3f 0 4f 07f 05f 0

6f 0 8f 0 9f 0

2f 0

Am

plitu

de (m

s2 )

(b)

Figure 25 +e outer ring MOMEDA filter signal and its envelope spectrum (a) Time-domain waveform and the envelope spectrum of theMOMEDA filter signal (b) Part band envelope spectrum of the MOMEDA filter signal

18 Mathematical Problems in Engineering

0 002 004 006 008 01 012 014 016 018ndash05

0

05

Time (s)

Am

plitu

de (m

s2 )

(a)

0 1000 2000 3000 4000 5000 60000

002

004

006

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

Figure 26 (a) Time-domain analysis and (b) frequency-domain analysis of the original rolling element fault signal

0 002 004 006 008 01 012 014 016 018ndash1

0

1

Time (s)

Am

plitu

de (m

s2 )

(a)

0 1000 2000 3000 4000 5000 60000

002

004

006

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

Figure 27 (a) Time-domain analysis and (b) frequency-domain analysis of the mixed signal with SNR minus 5 dB

0 002 004 006 008 01 012 014 016 018ndash2

02

times10ndash3

y1

0 002 004 006 008 01 012 014 016 018ndash005

0005

y2

0 002 004 006 008 01 012 014 016 018ndash01

001

y3

0 002 004 006 008 01 012 014 016 018ndash01

001

y4

0 002 004 006 008 01 012 014 016 018ndash02

002

y5

0 002 004 006 008 01 012 014 016 018ndash02

002

y6

0 002 004 006 008 01 012 014 016 018ndash02

002

y7

0 002 004 006 008 01 012 014 016 018ndash02

002

y8

0 002 004 006 008 01 012 014 016 018ndash05

005

y9

0 002 004 006 008 01 012 014 016 018ndash05

005

y10

0 002 004 006 008 01 012 014 016 018ndash05

005

y11

0 002 004 006 008 01 012 014 016 018ndash05

005

y12

0 002 004 006 008 01 012 014 016 018ndash05

005

y13

0 002 004 006 008 01 012 014 016 018ndash05

005

y14

0 002 004 006 008 01 012 014 016 018ndash02

002

y15

0 002 004 006 008 01 012 014 016 018ndash02

002

y16

0 002 004 006 008 01 012 014 016 018ndash02

002

y7

0 002 004 006 008 01 012 014 016 018ndash02

002

y18

0 002 004 006 008 01 012 014 016 018ndash01

001

y19

0 002 004 006 008 01 012 014 016 018ndash01

001

y20

Time (s) Time (s)

Figure 28 FDM decomposition result

Mathematical Problems in Engineering 19

Tabl

e9

+ecorrelationcoeffi

cientandku

rtosisbetweentheFIBF

andtheoriginal

signal

y 1y 2

y 3y 4

y 5y 6

y 7y 8

y 9y 1

0y 1

1y 1

2y 1

3y 1

4y 1

5y 1

6y 1

7y 1

8y 1

9y 2

0y 2

1y 2

2y 2

3y 2

4y 2

5

C(t)

0004

0054

0126

0116

0120

0171

0196

0173

0306

0296

0490

0379

0219

0254

0206

0181

0209

0154

0113

0107

0051

0088

0017

0025

0000

K(t)

1000

2071

2276

2456

3148

2712

3194

3435

3276

2962

2716

2684

3304

2640

2710

2783

2614

2591

3079

2435

2673

2646

2107

1500

1000

20 Mathematical Problems in Engineering

further extracted by MOMEDA filtering and the extractionresults are shown in Figure 31

It can be clearly seen from Figure 31 that the periodicfault shock component of the rolling element fault signal issuccessfully extracted +e peak value of the characteristic

frequency of the fault is greatly improved and the amplitudeof the signal component unrelated to the impact of the faultis lower From Figure 31 not only the frequency of thebearing rolling element fault (1406Hz) can be clearly dis-tinguished but also the doubling frequency such as the 2

0 002 004 006 008 01 012 014 016 018ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

0 002 004 006 008 01 012 014 016 018ndash04

ndash02

0

02

04

Time (s)

Am

plitu

de (m

s2 )

(b)

Figure 29 +e waveform of each independent component (a) +e waveform of the target signal (b) +e waveform of the noise signal

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

f0 2f0 4f0

(a)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

Figure 30 Envelope spectrum analysis (a) Envelope spectrum analysis of the target signal denoised by FDM-RobustICA (b) Envelopespectrum of the mixed signal

0 200 400 600 800 1000 1200ndash500

0

500

1000

Am

plitu

de (m

s2 )

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

Number (n)

(a)

0 200 400 600 800 10000

20

40

60

80

100

Frequency (Hz)

Am

plitu

de (m

s2 )

f0 2f0 3f0

4f05f 0 6f0 7f0

(b)

Figure 31+e rolling element MOMEDA filter signal and its envelope spectrum (a) Time-domain waveform and the envelope spectrum ofthe MOMEDA filter signal (b) Part band envelope spectrum of the MOMEDA filter signal

Mathematical Problems in Engineering 21

doubling frequency (2813Hz) 3 doubling frequency(4219Hz) 4 doubling frequency (5625Hz) 5 doublingfrequency (7031Hz) and 6 doubling frequency (8436Hz)are clearly visible which are very close to the fault char-acteristic frequency and the multiple frequency theoretically+erefore the fault of the bearing rolling element can beaccurately diagnosed

6 Conclusions

Aiming at the problem of fault feature extraction of vi-bration signals of rolling bearings under the background ofstrong noise this paper proposes a fault feature extractionmethod based on the combination of FDM-RobustICA andMOMEDA After the experimental analysis of the con-structed simulation signals and the actual bearing faultsignals the conclusions reached are as follows

(1) On the basis of cross-correlation analysis and kur-tosis the signal components obtained by the FDMmethod are selected to reconstruct the observationsignal channel and the virtual noise channel whichnot only avoid the mode overlapping and endpointeffect produced by traditional methods but also solvethe problem that RobustICA cannot process thesingle-channel signal and the lack of fault infor-mation caused by blind selection of signalcomponents

(2) By comparing the results of the proposed methodwith the traditional method of LMD-RobustICA andITD-RobustICA the results show that the proposedmethod can more effectively separate fault signalsand noise signals and has achieved relative betterevaluation index value

(3) By using MOMEDA to filter the noise reducedsignal the periodic impact component under thestrong noise background can be effectively enhancedand the irrelevant impact signal can be weakenedBased on the constructed simulation signals and theexperimental analysis of the inner ring outer ringand rolling element of rolling bearings the resultsshow that the proposed method can accurately ex-tract the characteristic frequency of faults in a strongnoise background environment and the amplitude ismore obvious In our future work we will considermore reasonable approaches to optimize the pa-rameters of the proposed method and add otherdenoising techniques

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is work was supported by the Joint Special Fund for BasicResearch of Local Universities in Yunnan Province (no2019FH001-121) and Science Research Fund Project ofBaoshan University (no BYBS201802)

References

[1] P Schniter ldquoShort-time fourier transformrdquo Advanced Topicsin Signal Processing vol 32 pp 289ndash337 2009

[2] D Griffin and J S Jae Lim ldquoSignal estimation from modifiedshort-time fourier transformrdquo IEEE Transactions on Acous-tics Speech and Signal Processing vol 32 no 2 pp 236ndash2431984

[3] A Georgakis L K Stergioulas and G Giakas ldquoWigner fil-tering with smooth roll-off boundary for differentiation ofnoisy non-stationary signalsrdquo Signal Processing vol 82no 10 pp 1411ndash1415 2002

[4] M Holschneider R Kronland-Martinet J Morlet andP Tchamitchian A Real-Time Algorithm for Signal Analysiswith the Help of the Wavelet Transform Springer BerlinGermany 1989

[5] Z Nenadic and J W Burdick ldquoSpike detection using thecontinuous wavelet transformrdquo IEEE Transactions on Bio-medical Engineering vol 52 no 1 pp 74ndash87 2005

[6] N E Huang Z Shen S R Long et al ldquo+e empirical modedecomposition and the Hilbert spectrum for nonlinear andnon-stationary time series analysisrdquo Proceedings of the RoyalSociety of London Series A Mathematical Physical and En-gineering Sciences vol 454 no 1971 pp 903ndash995 1998

[7] V K Wu V Bajaj A Kumar D Sharma and G K SinghldquoAn efficient method for analysis of EMG signals using im-proved empirical mode decompositionrdquo AEU-InternationalJournal of Electronics and Communications vol 72pp 200ndash209 2017

[8] J Cheng D Yu J Tang and Y Yang ldquoApplication of fre-quency family separation method based upon EMD and localHilbert energy spectrum method to gear fault diagnosisrdquoMechanism and Machine Beory vol 43 no 6 pp 712ndash7232008

[9] Z Wu and N E Huang ldquoEnsemble empirical mode de-composition a noise-assisted data analysis methodrdquo Ad-vances in Adaptive Data Analysis vol 1 no 1 pp 1ndash41 2009

[10] H Li T Liu X Wu and Q Chen ldquoApplication of EEMD andimproved frequency band entropy in bearing fault featureextractionrdquo ISA Transactions vol 88 pp 170ndash185 2019

[11] C Wu J Wu and J Yang ldquoResearch on fault diagnosis ofcheck valve based on CEEMD compound screening andimproved SECPSOrdquo Journal of Computers vol 30 pp 128ndash144 2019

[12] K Moshen C Gang P Yusong and Y Li ldquoResearch ofplanetary gear fault diagnosis based on permutation entropyof CEEMDAN and ANFISrdquo Sensors vol 18 no 3 p 7822018

[13] J S Smith and S Jonathan ldquo+e local mean decompositionand its application to EEG perception datardquo Journal of theRoyal Society Interface vol 2 no 5 pp 443ndash454 2005

[14] M G Frei and I Osorio ldquoIntrinsic time-scale decompositiontime-frequency-energy analysis and real-time filtering of non-stationary signalsrdquo Proceedings of the Royal Society AMathematical Physical and Engineering Sciences vol 463no 2078 pp 321ndash342 2007

22 Mathematical Problems in Engineering

[15] P Singh S D Joshi R K Patney and K Saha ldquo+e fourierdecompositionmethod for nonlinear and non-stationary timeseries analysisrdquo Proceedings of the Royal Society A Mathe-matical Physical and Engineering Science vol 473 pp 1ndash122017

[16] C H Dou and J S Lin ldquoExtraction of fault features ofmachinery based on fourier decomposition methodrdquo IEEEAccess vol 7 pp 1ndash11 2019

[17] Y Liu X D Liu and S Liang ldquoAeroengine rotor fault di-agnosis based on fourier decomposition methodrdquo ChinaMechanical Engineering vol 30 pp 2156ndash2163 2019

[18] XW Fu and X Q Gao ldquoRolling bearing fault diagnosis basedon FDM and singular value difference spectrumrdquo ActaMetrologica Sinica vol 39 pp 688ndash692 2018

[19] P Comon ldquoIndependent component analysis a new con-ceptrdquo Signal Processing vol 36 no 3 pp 287ndash314 1994

[20] A Hyvarinen and E Oja ldquoA fast fixed-point algorithm forindependent component analysisrdquo Neural Computationvol 9 no 7 pp 1483ndash1492 1997

[21] A Hyvarinen ldquoFast and robust fixed-point algorithms forindependent component analysisrdquo IEEE Transactions onNeural Networks vol 10 no 3 pp 626ndash634 1999

[22] P Chevalier L Albera P Comon and A Ferreol ldquoCom-parative performance analysis of eight blind source separationmethods on radio communications signalsrdquo in Proceedings ofthe International Joint Conference on Neural Networkspp 25ndash31 Budapest Hungary July 2004

[23] V Zarzoso and P Comon ldquoRobust independent componentanalysis for blind source separation and extraction with ap-plication in electrocardiographyrdquo in Proceedings of theEMBC-2008 30th Annual International Conference of theIEEE Engineering in Medicine and Biology Society pp 3344ndash3347 Vancouver BC Canada August 2008

[24] V Zarzoso and P Comon ldquoRobust independent componentanalysis by iterative maximization of the kurtosis contrastwith algebraic optimal step sizerdquo IEEE Transactions on NeuralNetworks vol 21 pp 248ndash261 2009

[25] J Yao Y Xiang S Qian S Wang and S Wu ldquoNoise sourceidentification of diesel engine based on variational modedecomposition and robust independent component analysisrdquoApplied Acoustics vol 116 pp 184ndash194 2017

[26] F Bi L Li J Zhang and T Ma ldquoSource identification ofgasoline engine noise based on continuous wavelet transformand EEMD-RobustICArdquo Applied Acoustics vol 100 pp 34ndash42 2015

[27] Y Jiachi X Yang Q Sichong andW Shuai ldquoRadiation noiseseparation of internal combustion engine based on Gam-matone-RobustICA methodrdquo Shock and Vibration vol 2017Article ID 7565041 14 pages 2017

[28] Y Jingzong W Xiaodong W Jiande and F Zao ldquoA noisereduction method based on CEEMD-RobustICA and itsapplication in pipeline blockage signalrdquo Journal of AppliedScience and Engineering vol 22 pp 1ndash10 2019

[29] Y Jiachi X Yang Q Sichong and W Shuai ldquoNoise sourceseparation of an internal combustion engine based on asingle-channel algorithmrdquo Shock and Vibration vol 2019Article ID 1283263 19 pages 2019

[30] H Weixin and L Dunwen ldquoMine microseismic signaldenosing based on variational mode decomposition and in-dependent component analysisrdquo Journal of Vibration andShock vol 38 pp 56ndash63 2019

[31] R A Wiggins ldquoMinimum entropy deconvolutionrdquo Geo-exploration vol 16 no 1-2 pp 21ndash35 1978

[32] S Wang and J Xiang ldquoA minimum entropy deconvolution-enhanced convolutional neural networks for fault diagnosis ofaxial piston pumpsrdquo Soft Computing vol 24 no 4pp 2983ndash2997 2020

[33] H Liu and J Xiang ldquoA strategy using variational mode de-composition L-kurtosis and minimum entropy deconvolu-tion to detect mechanical faultsrdquo IEEE Access vol 7pp 70564ndash70573 2019

[34] N Sawalhi R B Randall and H Endo ldquo+e enhancement offault detection and diagnosis in rolling element bearings usingminimum entropy deconvolution combined with spectralkurtosisrdquo Mechanical Systems and Signal Processing vol 21no 6 pp 2616ndash2633 2007

[35] G L McDonald Q Zhao and M J Zuo ldquoMaximum cor-related kurtosis deconvolution and application on gear toothchip fault detectionrdquo Mechanical Systems and Signal Pro-cessing vol 33 pp 237ndash255 2012

[36] G L Mcdonald and Q Zhao ldquoMultipoint optimal minimumentropy deconvolution and convolution fix application tovibration fault detectionrdquo Mechanical Systems and SignalProcessing vol 82 pp 461ndash477 2017

[37] Y Li G Cheng X Chen and Y Pang ldquoResearch on bearingfault diagnosis method based on filter features of MOML-MEDA and LSTMrdquo Entropy vol 21 pp 1ndash15 2019

[38] W A Smith and R B Randall ldquoRolling element bearingdiagnostics using the Case Western Reserve University data abenchmark studyrdquoMechanical Systems and Signal Processingvol 64-65 pp 100ndash131 2015

[39] J Xia L Zhao Y Bai M Yu and Z Wang ldquoFeature ex-traction for rolling element bearing weak fault based onMCKD and VMDrdquo Journal of Vibration and Shock vol 36pp 78ndash83 2017

Mathematical Problems in Engineering 23

advance By using this method the periodic impact com-ponents in the signal can be effectively enhanced whichmakes a good foundation for the feature extraction of faultsignals

In summary this paper proposed a fault feature ex-traction method combining FDM-RobustICA andMOMEDA Firstly the FDM is used to decompose thevibration signal of the bearing and obtain several Fourierintrinsic band functions (FIBF) +en by combining thecross-correlation analysis criterion and the kurtosis crite-rion an effective FIBF component is selected as the ob-servation signal and blind source separation is performed+irdly MOMEDA is used to enhance the periodic impactcomponents in the noise-reduced signal Finally the Hilbertenvelope demodulation is used to obtain the fault charac-teristic frequency and the fault characteristic frequency iscompared with the theoretical calculated value for faultdiagnosis

+e structure of this paper is as follows Section 2 introducesthe basic principles of FDM RobustICA and MOMEDASection 3 briefly introduces the implementation details and faultfeature extraction process of themethod proposed in this paperSection 4 uses the proposed method to analyze the simulationsignal Section 5 further verifies the effectiveness of the proposedmethod through the actual bearing fault signal Section 6 givesthe discussion and conclusion

2 Theoretical Background

21 Fourier Decomposition Method Fourier decompositionmethod (FDM) is a new type of the adaptive signal analysismethod based on Fourier transform +is method firstadaptively searches and analyzes the Fourier intrinsic bandfunctions (FIBF) in the entire Fourier domain +en thenonlinear signal is adaptively decomposed into the sum ofseveral physical Fourier intrinsic band functions (FIBF) anda residual component

(1) FIBF has a zero mean value namely 1113938b

ayi(t)dt 0

(2) Any two FIBF are orthogonal to each other namely1113938

b

ayi(t)yj(t)dt 0 ine j

(3) +e analytic function form of FIBF has nonnegativeinstantaneous amplitude and frequency yi(t)

1113954yj(t) ai(t)ejφi(t) namely ai(t)ge 0 φiprime(t)ge 0

forallt isin [a b] and yi(t) isin Cinfin[a b]+erefore FIBF arethe sum of zero-mean sine functions with contin-uous frequency bands

Based on the definition of FIBF the steps of the FDM areas follows

Step 1 make a fast Fourier transformation of the signalx (n) X[k] FFT [x(n)] Step 2 use forward search which scans analytic Fourierintrinsic band functions (AFIBF) from low to highfrequencies

AFIBF 1113944n

kNiminus 1+1X[k]σj2πknN

ai(n)ejφi[n]

X (1)

In order to obtain the minimum number of AFIBF fromlow-frequency to high-frequency scanning for each i 1 2 M start from Niminus 1 + 1 and gradually increase to reachthe maximum Ni where N0 0 and NM (N2 minus 1) Itsatisfies Niminus 1 + 1leNiminus 1 le (N2 minus 1) and also satisfies

ai(t)ge 0 ωi(t) φi[n + 1] minus φi[n minus 1]

2ge 0 forallt (2)

Similarly a reverse search can also be performednamely scanning the AFIBF component from high to lowfrequencies Correspondingly change the upper and lowerlimits of the sum in equation (2) from Ni to Niminus 1 minus 1 wherei 1 2 M N0 N2 and NM 1 +e search shouldstart from Niminus 1 minus 1 and gradually decrease until the smallestNi satisfies 1leNi leNiminus 1 minus 1 and the phase φi[n] is amonotonically increasing function +e instantaneous fre-quency and amplitude can be calculated directly by AFIBFMeanwhile the real part of AFIBF is FIBF

22 Robust Independent Component Analysis Robust in-dependent component analysis (RobustICA) is an inde-pendent component analysis algorithm based on kurtosisand optimal step size that has emerged in recent years on thebasis of the original independent component analysis (ICA)+e algorithm searches for the global optimal step sizethrough the kurtosis comparison function finds the dem-ixing matrix and calculates the approximate value of thesource signal+e frame of independent component analysisis shown in Figure 1 +e purpose of independent com-ponent analysis is to separate independent source signalcomponents from the mixed signal

Let n-dimensional random observation vectors conformto the following model

x AS + N (3)

where S [s1 s2 sm]T and x [x1 x2 xn]T S is the m-dimensional signal vector which means that the number ofmixed signals is m N is the n-dimensional observationvector which means that the number of mixed signals is n Ais an n times m matrix +e basic idea of ICA is to solve theseparation matrix W and make an optimal estimate y of thesource signal S when only the observation signal x is known

y Wx (4)

Compared with the traditional ICA RobustICA does notneed to prewhiten the data which reduces the amount ofcalculation At the same time RobustICA can be used toprocess the sub-Gaussian signal and the super-Gaussiansignal When the Gaussian property of the signal is known itcan extract only the components of interest in the sourcesignal without increasing the complexity of the operationand the estimation error +e linear search method used byRobustICA can ensure that the optimization can be achievedin all the processes of component separation Assuming theoutput signal is y Wx (W is the m times n-dimensionalseparation matrix) the kurtosis equation can be expressed asfollows

Mathematical Problems in Engineering 3

K(W) E |y|41113966 1113967 minus 2E2 |y|21113966 1113967 minus E y21113864 1113865

111386811138681113868111386811138681113868111386811138682

E2 |y|21113966 1113967 (5)

In the equation E middot represents the mathematical ex-pectation RobustICA directly uses the above equation as acomparison function without any simplification and thenoptimizes the kurtosis according to the strategy of the bestlinear optimization step size

μopt argmaxu

|K(W + μg)| (6)

where g is the search direction usually g nablawK(W) whichcan be expressed as follows

nablawK(W) 4

E2 |y|21113966 1113967E |y|

21113966 1113967yx minus E yx1113864 11138651113966 1113967E y

21113966 1113967

minusE |y|41113966 1113967 minus E y21113864 1113865

111386811138681113868111386811138681113868111386811138682

1113872 1113873E yx1113864 1113865

E |y|21113966 1113967

(7)

In the process of iteration the optimization method ofRobustICA can be described as follows

(1) Find the coefficients of the OS polynomial When thekurtosis becomes a comparison function the OSpolynomial can be expressed as follows

P(u) 11139444

k0aku

k (8)

(2) Extract the root a of the OS polynomial uk1113864 11138654k1

(3) Select the root that can make the best optimizationstep μopt argmaxu|K(W + μg)| and take the largestvalue from the direction of iteration

(4) Update μopt based on the value of the updated stepW+ W+ W + μoptg

(5) Normalize W+ W+⟵W+W+(6) Determine whether the iterative operation converges

and meets the termination conditions If it is notsatisfied then return to Step (1)

23 Multipoint Optimal Minimum Entropy DeconvolutionAdjusted If x is the vibration signal y is the shock se-quence Meanwhile h is the system frequency responsefunction e is the noise signal +e symbol lowast is the multi-plication symbol +e collected signal can be expressed as

x hlowasty + e (9)

+e essence of the MOMEDA is to use a noniterativemethod to find an optimal FIR filter Recover the vibrationshock signal from the output signal as much as possiblethrough the deconvolution process and reduce the impact ofnoise +e deconvolution process is as follows

y flowastx 1113944Nminus L

k1fkxk+Lminus 1 (10)

where k 1 2 middot middot middot N minus L According to the characteristics ofthe periodic shock signal multipoint D-norm is introducednamely

MDN(y t) 1

t

tTy

y (11)

MOMEDA maxf

MDN(y t) maxf

tTy

y (12)

In equation (12) t is a constant vector for determiningthe position and weight of the target impact component+eoptimal filter f is obtained by solving the maximum value ofthe multipoint D-norm At this time the deconvolutionprocess also obtains the optimal solution In order to find theextreme value of equation (12) the filter coefficient f

(f1 f2 middot middot middot fL) is first derivedddf

tTy

y1113888 1113889

ddf

t1y1

y+

ddf

t2y2

y+ middot middot middot +

ddf

tNminus LyNminus L

y

(13)

ddf

tkyk

y y

minus 1tkMk minus y

minus 3tkykX0y (14)

Mk xx+Lminus 1 xx+Lminus 2 middot middot middot xk1113858 1113859T (15)

+erefore equation (13) can be written as followsddf

tTy

y1113888 1113889 y

minus 1t1M1 + t2M2 + middot middot middot tNminus LMNminus L( 1113857

minus yminus 3

tT

yX0y

(16)

Let X0 [M1 M1 middot middot middot Mk] in the above equation thenequation (13) is equivalent to the following equation

yminus 1

X0t minus yminus 3

tTyX0y 0 (17)

By sorting out the above equation we can gettTy

yminus 2X0y X0t (18)

Since y XT0 f if (X0X

T0 )minus 1 exists the following

equation can be obtained

Mixedmatrix

(A)

Separationmatrix

(W)

S1 (t)

S2 (t)

Sn (t)

X1 (t)

X2 (t)

Xn (t)

Y1 (t)

Y2 (t)

Yn (t)

n1 (t) n2 (t) nn (t)

Blind source Observed signal Estimated signal

Figure 1 Frame of independent component analysis

4 Mathematical Problems in Engineering

tTy

y21113888 1113889X0f X0X

T01113872 1113873

minus 1X0t (19)

+e special solution of the above equation is a set ofoptimal filters as shown in the following equation

f X0XT01113872 1113873

minus 1X0t (20)

Substituting the above equation into y XT0 f the

original impact signal y can be reconstructed

3 The Process of Fault Feature Extraction

Although the adaptive decomposition of component signalsby the FDM avoids modal overlapping and endpoint effectthe blind selection and rejection of component signals onlyby the FDM may lead to incomplete extraction of faultfeature information RobustICA as a newer method in blindsource separation is not easily interfered by strong noiseHowever the processed bearing vibration signal is often asingle channel which does not meet the conditions of use ofRobustICA To a great extent it restricts the application ofthis kind of algorithm+erefore in order to give full play tothe advantages of FDM and RobustICA in signal processingthis study used a combination of FDM and RobustICA toreduce noise on bearing fault signals When the originalbearing fault signal with noise is decomposed by the FDM anumber of FIBF components from high to low frequencywill be obtained Under actual working conditions the noisesignal is unknown which brings great difficulties to theconstruction of the virtual noise channel If the method ofconstructing the virtual noise channel is inappropriate itwill directly affect the effect of signal noise reductionConsidering the particularity of mechanical signals thisresearch will judge the sensitivity of each component signalto interference noise and fault impact by combining cross-correlation analysis criteria and kurtosis criteria so as tobuild a virtual noise channel

According to the definition of the cross-correlationcriterion the correlation coefficient can be used as an indexto evaluate the degree of correlation between two signals+e larger the correlation coefficient value the higher thecorrelation with the original signal By using this criterionthe degree of correlation between each component signalobtained from the decomposition and the original signal canbe known +e calculation method of the correlationnumber is as follows

r 1113936

ni0 xi minus x( 1113857 yi minus y( 1113857

1113936ni0 xi minus x( 1113857

2yi minus y( 1113857

21113969 (21)

where xi and x are specific values and average values of thesignal x At the same time yi and y are specific values andaverage values of the signal y respectively

Kurtosis can be used tomeasure the peak degree of signalwaveform and it is more sensitive to the impact componentsin the signal +e higher the proportion of impact com-ponents is the higher the kurtosis value will be For bearingsthe kurtosis value is close to the normal distribution under

normal operation and it will increase significantly whenfaults occur +e calculation method is as follows

Ki 1n

1113944

n

i1

xi minus x

σ1113874 1113875

4 (22)

where xi and x are specific values and average values of thesignal x σ is the standard deviation of the signal and n is thenumber of samples

After finding the correlation coefficient and kurtosisvalue of each FIBF component through the above criteriathe virtual channel is further constructed based on this+en RobustICA is used to recover the independentcomponent of the effective signal and the independentcomponent mainly of noise Under the strong noise inter-ference the fault feature of the separated effective signal isstill weak In order to enhance the impact component of thebearing fault signal after noise reduction this research in-troduced the MOMEDA to filter the signal and finallycalculated the Hilbert envelope spectrum of the filteredsignal and extracted the corresponding fault frequency so asto accurately diagnose the fault +e flowchart of the pro-posed method is shown in Figure 2

+e specific steps are as follows

(1) FDM method is used to decompose the vibrationsignals of the rolling bearing under different faultconditions and several FIBF components with highfrequency and low frequency are obtained

(2) According to the cross-correlation and kurtosiscriteria the cross-correlation coefficient and kurtosisvalue of each FIBF component are calculated +enthe signal component is selected that is highly cor-related with the original signal and has a large impactcomponent ratio (the signal components with cross-correlation coefficient greater than 03 and kurtosisvalue greater than 3 are selected in this research) toconstruct an observation signal channel At the sametime the remaining signal components are used toconstruct a virtual noise signal channel

(3) +e observation signal and the virtual noise signalare used as the input matrix of the blind sourceseparation and then the signal is demixed by usingRobustICA to achieve the noise reduction of thevibration signal

(4) +e MOMEDA is used to enhance the periodicimpact component of the noise reduced signal

(5) +e signal obtained in the previous step isdemodulated by Hilbert envelope and then the faultfrequency is extracted for fault diagnosis

4 Simulations and Comparative Analysis

In order to verify the effectiveness of themethod proposed inthis paper the fault signal of the rolling bearing is simulatedand analyzed+e structure of the analog signal is as follows

s(t) y0eminus 2πfnξt sin 2πfn

1 minus ξ21113969

t1113874 1113875 (23)

Mathematical Problems in Engineering 5

where the natural frequency of the bearing fn is 3000Hz thedisplacement constant y0 is 5 the damping coefficient ξ is 1the period of the impact failure is 001 s the samplingfrequency fs is 12 kHz and the number of sampling pointsN is 1000 +rough calculation the fault frequency f0 is100Hz +e time-domain waveform of the original signal s(t) is shown in Figure 3 After the Gaussian white noise witha signal-to-noise ratio of minus 5 dB is added to the originalsignal s (t) the time-domain waveform and frequency-domain waveform of the mixed signal y (t) are shown inFigure 4 It can be seen from Figure 4 that the impactcharacteristics of the bearing have been completely anni-hilated due to the interference of noise and the impactinterval of the bearing cannot be distinguished from thefigure

Next the fault feature extractionmethod proposed in thepaper is used to process and analyze the above analog faultsignals Firstly the FDM is performed on the mixed signal y(t) and Figure 5 shows the results of the decomposition Inorder to highlight the advantages of the method proposed inthis paper the LMD and ITD commonly used in the field offault diagnosis are used to decompose the signal and furtheranalysis is carried out +e decomposition results of theabove two methods can be seen in Figure 6 As shown inFigure 5 the mixed signal y (t) is decomposed into 19Fourier intrinsic band functions (FIBF) from high to lowfrequencies Because FDM overcomes the problems ofmodal aliasing and endpoint effects to a certain extent andhas a strict theoretical basis the components obtained by the

decomposition clearly reflect the local characteristic infor-mation of the original signal

In order to select the signal components that meet theconditions from the decomposition results the correlationcoefficient C (t) and the kurtosis value K (t) of each com-ponent signal are calculated As can be seen from Table 1only the kurtosis values of the y9 y11 and y12 componentsare greater than 3 and their correlation values are greaterthan 03 indicating that the above component signals have ahigh correlation with the original signals and contain moreimpact components +erefore the above three componentsare selected to reconstruct the observation signal channeland the remaining signal components are used to recon-struct the noise signal channel At the same time the cal-culation results of the correlation and kurtosis of the signalcomponents based on the LMD and ITD are shown inTables 2 and 3 respectively It can be seen that the corre-lation and kurtosis values of the PF1 and PF2 components inTable 2 and the PRC2 and PRC3 components in Table 3 meetthe set threshold conditions +erefore the above compo-nent signals are selected to reconstruct the observationsignals and the remaining component signals are used toreconstruct the noise signal channel +en they are sepa-rately demixed by the RobustICA to achieve the separationof the signal and noise +e noise reduction results based onFDM-RobustICA LMD-RobustICA and ITD-RobustICAare shown in Figures 7ndash9 respectively

By comparing the signal and noise separation results inFigures 7ndash9 it can be seen that the impact components in thetarget signal after noise reduction by FDM-RobustICA aremore obvious which can better extract useful informationfrom noise-containing signals and weaken the influence ofstrong background noise However the time-domainwaveform of the target signal based on LMD-RobustICA andITD-RobustICA noise reduction methods is different fromthat of the original signal to some extent Meanwhile be-cause there is no prior information of the fault in the processof using the RobustICA to carry out the demixing operationthe system has uncertainty and the signal amplitude afterdemixing changes to varying degrees but does not affect thenoise reduction analysis

In order to quantitatively analyze the noise reductioneffect based on the above three methods kurtosis valuecorrelation number and noise reduction time are selected asthe evaluation indexes +e larger the kurtosis value is themore fault information in the signal is +e larger thecorrelation number is the more complete the effective

Input signal

FDM decomposition

Calculate the correlationcoefficient and kurtosis of

each FIBF component

Reconstructobservation signal

Reconstruct virtualnoise channel signal

Unmixed by RobustICA

MOMEDA filtering

Select effective FIBFcomponents

Hilbert envelopedemodulation

Extract the fault frequency

Figure 2 +e flowchart of the proposed method

0 001 002 003 004 005 006 007 008 009ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

Figure 3 Original signal

6 Mathematical Problems in Engineering

information of the signal is retained +e shorter the noisereduction time is the higher the operation efficiency of thealgorithm is +e calculation results are shown in Table 4 Itcan be seen from Table 4 that the correlation value andkurtosis value of the noise reduced signal based on FDM-RobustICA are the largest which shows that the noise re-duced signal obtained by the method proposed has a highsimilarity with the original signal At the same time thismethod not only reduces the noise interference but alsoretains the fault characteristic information well Althoughthere is a certain gap in noise reduction time between ITD-RobustICA and FDM-RobustICA it is not significant

In order to further compare the effects of the above threemethods on the separation of the target signal and noise the

signals obtained by the above three methods are separatelydemodulated by Hilbert envelope and then the corre-sponding envelope spectrum is obtained +e results areshown in Figures 10ndash12 It can be seen from Figure 11 thatthe envelope spectrum of the noise signal separated by LMD-RobustICA has components such as one to four doublingfault frequencies and the fault frequency amplitude is rel-atively high It can be seen from Figure 12 that the envelopespectrum of the noise signal obtained by ITD-RobustICAalso contains the components of one two and three dou-bling fault frequencies It shows that after the above twomethods are processed the target signal and noise have notbeen completely separated and some of the fault signals aremissing from the separated target signal It can be seen from

0 001 002 003 004 005 006 007 008 009ndash005

0005

y1

0 001 002 003 004 005 006 007 008 009ndash02

002

y2

0 001 002 003 004 005 006 007 008 009ndash05

005

y3

0 001 002 003 004 005 006 007 008 009ndash05

005

y4

0 001 002 003 004 005 006 007 008 009ndash1

01

y5

0 001 002 003 004 005 006 007 008 009ndash1

01

y6

0 001 002 003 004 005 006 007 008 009ndash1

01

y7

0 001 002 003 004 005 006 007 008 009ndash2

02

y8

0 001 002 003 004 005 006 007 008 009ndash2

02

y9

0 001 002 003 004 005 006 007 008 009ndash1

01

y10

0 001 002 003 004 005 006 007 008 009ndash2

02

y11

0 001 002 003 004 005 006 007 008 009ndash2

02

y12

0 001 002 003 004 005 006 007 008 009ndash1

01

y13

0 001 002 003 004 005 006 007 008 009ndash05

005

y14

0 001 002 003 004 005 006 007 008 009ndash05

005

y15

0 001 002 003 004 005 006 007 008 009ndash05

005

y16

0 001 002 003 004 005 006 007 008 009ndash02

002

y17

0 001 002 003 004 005 006 007 008 009ndash005

0005

y18

0 001 002 003 004 005 006 007 008 009ndash2

02

y19

Time (s)

Time (s)

Figure 5 +e decomposition result by the FDM

0 001 002 003 004 005 006 007 008 009ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

01

02

03

04

Frequency (Hz)

(b)

Figure 4 (a) Time-domain analysis and (b) frequency-domain analysis of the mixed signal with SNR minus 5 dB

Mathematical Problems in Engineering 7

0 001 002 003 004 005 006 007 008 009ndash5

05

PRC1

0 001 002 003 004 005 006 007 008 009ndash2

02

PRC2

0 001 002 003 004 005 006 007 008 009ndash1

01

PRC3

0 001 002 003 004 005 006 007 008 009ndash1

01

PRC4

0 001 002 003 004 005 006 007 008 009ndash05

005

PRC5

0 001 002 003 004 005 006 007 008 009ndash02

002

R

Time (s)

(a)

0 001 002 003 004 005 006 007 008 009ndash5

05

PF1

0 001 002 003 004 005 006 007 008 009ndash2

02

PF2

0 001 002 003 004 005 006 007 008 009ndash2

02

PF3

0 001 002 003 004 005 006 007 008 009ndash1

01

PF4

0 001 002 003 004 005 006 007 008 009ndash500

0500

Time (s)

R

(b)

Figure 6 Signal decomposition results (a) +e decomposition result by ITD (b) +e decomposition result by LMD

Table 1 +e correlation coefficient and kurtosis between the FIBF and the original signal

y1 y2 y3 y4 y5 y6 y7 y8 y9 y10 y11 y12 y13 y14 y15 y16 y17 y18 y19C (t) 0026 0073 0121 0154 0194 0221 0313 0363 0351 0283 0445 0326 0268 0127 0164 0104 0070 0021 0000K (t) 1000 2681 2166 2783 3303 3508 2531 2924 3348 3405 4246 3380 2376 2955 3604 2637 2562 1500 1000

Table 2 +e correlation coefficient and kurtosis between the PF and the original signal

PF1 PF2 PF3 PF4C (t) 0804 0467 0211 0149K (t) 3825 3608 4864 4982

Table 3 +e correlation coefficient and kurtosis between the PRC and the original signal

PRC1 PRC2 PRC3 PRC4 PRC5C (t) 0866 0584 0333 0180 0075K (t) 2929 3768 3121 3692 3312

0 001 002 003 004 005 006 007 008 009ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 001 002 003 004 005 006 007 008 009ndash4

ndash2

0

2

4

Time (s)

(b)

Figure 7 +e waveform of each independent component by the FDM-RobustICA method (a) +e waveform of the target signal (b) +ewaveform of the noise signal

8 Mathematical Problems in Engineering

Figure 10 that the amplitude of the fault characteristicfrequency in the envelope spectrum of the target signal hasbeen increased to a certain extent Meanwhile the faultcharacteristic frequency and its doubling frequency can beobtained In addition there is no fault information in the

envelope spectrum of the noise signal in Figure 10 indi-cating that the fault signal and the noise signal are effectivelyseparated However some irrelevant frequency componentsstill appear in the envelope spectrum of the fault signalwhich have a certain degree of interference to fault diagnosis

Am

plitu

de (m

s2 )

Time (s)0 001 002 003 004 005 006 007 008 009

ndash4

ndash2

0

2

4

(a)

Am

plitu

de (m

s2 )

Time (s)0 001 002 003 004 005 006 007 008 009

ndash4

ndash2

0

2

4

(b)

Figure 8 +e waveform of each independent component by the LMD-RobustICA method (a) +e waveform of the target signal (b) +ewaveform of the noise signal

0 001 002 003 004 005 006 007 008 009

ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 001 002 003 004 005 006 007 008 009ndash4

ndash2

0

2

4

Time (s)

(b)

Figure 9 +e waveform of each independent component by the ITD-RobustICA method (a) +e waveform of the target signal (b) +ewaveform of the noise signal

Table 4 Comparison of noise reduction results of different methods

Evaluation index FDM-RobustICA LMD-RobustICA ITD-RobustICA

Kurtosis and correlation coefficient 39070582

36020374

36300480

Noise reduction time 1378 00889 0103

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

4f 0 5f 03f 0

2f 0f 0

(a)

Am

plitu

de (m

s2 )

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

(b)

Figure 10 +e envelope analysis of each independent component by the FDM-RobustICA method (a) +e envelope analysis of the targetsignal (b) +e envelope analysis of the noise signal

Mathematical Problems in Engineering 9

+erefore it is necessary to further enhance the periodicimpact component of the noise reduced signal throughMOMEDA filtering

Next the experiment uses MOMEDA to enhance thesignal after FDM-RobustICA noise reduction and then thesignal is demodulated by Hilbert envelope Figure 13 showsthe time-domain waveform and envelope spectra of theMOMEDA filter signal It can be seen from Figure 13 thatthe periodic impact component is very obvious after filteringby MOMEDA +ere are more prominent fault features inthe envelope spectrum and there are large peaks at thefrequencies of 1055Hz 1992Hz 3047Hz 3984Hz5039Hz 5977Hz 7031Hz 7969Hz 9023Hz and so on+e above peaks are very close to the simulated faultcharacteristic frequency of 100Hz and the doubling fre-quency from two to ten At the same time the amplitudes ofother noninteger multiples at the characteristic frequency ofthe fault are basically close to 0 indicating that the signalcomponents unrelated to the fault impact have been greatlyweakened +erefore the bearing fault can be accuratelydiagnosed according to the characteristic frequency

5 Application to Roller Bearing Testing

In order to further verify the effectiveness of the methodproposed in this paper the bearing data of CWRU are usedfor analysis [38]+emodel of the rolling bearing at the driveend of the motor is 6205-2RSJEMSKF and the motor speedis 1797 rpm (namely the rotation frequency is 179760Hz 2995Hz)+e sampling frequency of the fault signalis 12 kHz and the length of experimental data is 2048 +especific technical parameters are shown in Table 5

Outer race defect frequency

BPFO Z

21 minus

d

Dcos α1113888 1113889 times fr (24)

Inner race defect frequency

BPFI Z

21 +

d

Dcos α1113888 1113889 times fr (25)

Rolling element frequency

BPFR D

2d1 minus

d

D1113888 1113889

2

cos2α⎛⎝ ⎞⎠ times fr (26)

where Z is the number of rolling elements d is the rollingelement diameter D is the bearing pitch diameter α is thecontact angle and fr is the shaft rotation frequency (Hz)

Based on the bearing parameters shown in Table 5 andequations (24)ndash(26) the fault characteristic frequency ofthe rolling bearing BPFO = 10736Hz BPFI = 16219Hzand BPFR = 1411693Hz is calculated respectively inTable 6 [39]

51 Inner Ring Signal Analysis of CWRU Bearings +e timedomain and the frequency domain of the bearing inner ringfault signal are shown in Figure 14 Although it is disturbedby noise to some extent some periodic pulses can still beseen To test the effectiveness of the proposed method understrong noise interference Gaussian white noise with asignal-to-noise ratio of minus 5 dB is added to the original innerring fault signal in this research +e time domain and thefrequency domain of the generated mixed signal are shown

Am

plitu

de (m

s2 )

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

2f 0 4f 0f 0

(a)

Am

plitu

de (m

s2 )

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

2f 0 4f 03f 0f 0

(b)

Figure 11 +e envelope analysis of each independent component by the LMD-RobustICA method (a) +e envelope analysis of the targetsignal (b) +e envelope analysis of the noise signal

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

f 03f 0 5f 0

4f 02f 0

(a)

Am

plitu

de (m

s2 )

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

3f 02f 0f 0

(b)

Figure 12 +e envelope analysis of each independent component by the ITD-RobustICA method (a) +e envelope analysis of the targetsignal (b) +e envelope analysis of the noise signal

10 Mathematical Problems in Engineering

in Figure 15 Due to the enhancement of noise interference it isimpossible to distinguish the periodic pulse features fromFigure 15 and it is difficult to extract the fault feature frequencydirectly from the time domain and the frequency domain

Next the method proposed in this paper is used toprocess the mixed signal of the inner ring with added noiseAfter the mixed signal is decomposed by the FDM 25 FIBFcomponents are obtained and the decomposition results are

shown in Figure 16 Due to the limited length of the articleonly the first 20 FIBF components are listed in it +en thecross-correlation coefficient and kurtosis value of each signalcomponent are calculated and the results are shown inTable 7 It can be seen from Table 7 that the signal com-ponents of y10 y11 and y12 meet the set threshold conditionsindicating that they have a high degree of correlation withthe original signal and retain a lot of impact features in the

Am

plitu

de (m

s2 )

Am

plitu

de (m

s2 )

0 200 400 600 800 1000 1200ndash500

0

500

1000

Number (n)

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

(a)

0 200 400 600 800 10000

5

10

15

20

Frequency (Hz)

3f 05f 0

6f 0

7f 0 8f 0 9f 0

2f 0

4f 0

Am

plitu

de (m

s2 )

f 0

(b)

Figure 13 +e MOMEDA filter signal and its envelope spectrum (a) Time-domain waveform and envelope spectrum of the MOMEDAfilter signal (b) Part band envelope spectrum of the MOMEDA filter signal

Table 5 +e bearing technical parameters of SKF 6205

Rolling elementnumber (Z) Inner diameter (inches) Outer diameter

(inches)Rolling element

diameter d (inches) Contact angle (α) Pitch circlediameter D (inches)

Speed(rpm)

9 09843 20472 03126 00 15327 1797

Table 6 Fault characteristic frequency (unit Hz)

Inner ring fault Outer ring fault Rolling element fault1621852 1073648 1411693

0 002 004 006 008 01 012 014 016 018ndash2

ndash1

0

1

2

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

005

01

Frequency (Hz)

(b)

Figure 14 (a) Time-domain analysis and (b) frequency-domain analysis of the original inner ring fault signal

Mathematical Problems in Engineering 11

original signal+erefore the above three signal componentsare extracted to construct the target signal channel and theremaining components are reconstructed as the virtual noisechannel +e blind source separation is performed byRobustICA and two sets of independent component time-domain waveforms are obtained +e results are shown inFigure 17

It can be seen from Figure 17 that after the noise re-duction of RobustICA the noise and the target signal areseparated In order to further compare the effect of noisereduction the separated target signal and the mixed signalwith minus 5 dB noise added were analyzed by the Hilbert en-velope spectrum respectively +e results are shown inFigure 18 By analyzing Figure 18 it can be known thatbefore noise reduction the amplitude of the envelope

spectrum of the fault signal is very low due to the influence ofnoise and the interference of irrelevant frequency compo-nents on the fault diagnosis is large Although the faultfrequency of the inner ring can be found as well as 2doubling frequency 5 doubling frequency etc the ampli-tude is low and it is hard to recognize After the noisereduction by RobustICA the envelope spectrum amplitudeof the obtained target signal has been greatly improved andthe fault characteristics are more prominent At the sametime there were large peaks at frequencies of 1641Hz3223Hz 4863Hz 6445Hz 8086Hz 9727Hz and so on+e above peak values are very close to the theoreticalcalculated value of the fault characteristic frequency of thebearing inner ring of 1621852Hz and the 2ndash6 doublingfrequency However there are also many peaks in the

0 002 004 006 008 01 012 014 016 018ndash2

ndash1

0

1

2

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

005

01

Frequency (Hz)

(b)

Figure 15 (a) Time-domain analysis and (b) frequency-domain analysis of the mixed signal with SNR minus 5 dB

0 002 004 006 008 01 012 014 016 018ndash5

05

times10ndash3

y1

0 002 004 006 008 01 012 014 016 018ndash005

0005

y2

0 002 004 006 008 01 012 014 016 018ndash005

0005

y3

0 002 004 006 008 01 012 014 016 018ndash01

001

y4

0 002 004 006 008 01 012 014 016 018ndash02

002

y5

0 002 004 006 008 01 012 014 016 018ndash02

002

y6

0 002 004 006 008 01 012 014 016 018ndash05

005

y7

0 002 004 006 008 01 012 014 016 018ndash05

005

y8

0 002 004 006 008 01 012 014 016 018ndash05

005

y9

0 002 004 006 008 01 012 014 016 018ndash1

01

y10

0 002 004 006 008 01 012 014 016 018ndash1

01

y11

0 002 004 006 008 01 012 014 016 018ndash1

01

y12

0 002 004 006 008 01 012 014 016 018ndash05

005

y13

0 002 004 006 008 01 012 014 016 018ndash05

005

y14

0 002 004 006 008 01 012 014 016 018ndash05

005

y15

0 002 004 006 008 01 012 014 016 018ndash02

002

y16

0 002 004 006 008 01 012 014 016 018ndash05

005

y17

0 002 004 006 008 01 012 014 016 018ndash02

002

y18

0 002 004 006 008 01 012 014 016 018ndash02

002

y19

0 002 004 006 008 01 012 014 016 018ndash01

001

y20

Time (s) Time (s)

Figure 16 FDM decomposition result

12 Mathematical Problems in Engineering

Tabl

e7

+ecorrelationcoeffi

cientandku

rtosisbetweentheFIBF

andtheoriginal

signal

y 1y 2

y 3y 4

y 5y 6

y 7y 8

y 9y 1

0y 1

1y 1

2y 1

3y 1

4y 1

5y 1

6y 1

7y 1

8y 1

9y 2

0y 2

1y 2

2y 2

3y 2

4y 2

5

C(t)

0008

0041

0028

0082

0115

0149

0181

0211

0259

0468

0430

0360

0215

0270

0234

0143

0202

0117

0114

0082

0060

0036

0059

0026

000

K(t)

1000

1958

1928

3032

2501

2708

2375

2470

2847

4302

3221

3183

2793

3417

3383

2424

2881

2620

3039

2866

2444

2743

2073

1500

1000

Mathematical Problems in Engineering 13

envelope spectrum of the signal that have nothing to do withthe characteristic frequency of the fault and these peaks willaffect the results of fault diagnosis +erefore the periodicimpact components in the signal after noise reduction will befurther extracted by MOMEDA filtering and the extractionresults are shown in Figure 19

It can be clearly seen from Figure 19 that the MOMEDAhas effectively extracted the periodic fault shock componentsin the reconstructed signal and a larger peak has appearedMeanwhile the signal components unrelated to the faultshock have been weakened By comparing the envelopespectrum of the signals in Figures 19 and 18 it can be seenthat after MOMEDA filtering the characteristic frequencyof the fault is more obvious Not only the fault frequency ofthe bearing inner ring (1641Hz) can be clearly distinguishedbut also the 2 doubling frequency (3223Hz) 3 doublingfrequency (4863Hz) 4 doubling frequency (6445Hz) 5doubling frequency (8086Hz) and so on are all clearlyvisible and they are very close to the theoretically calculatedfault characteristic frequency and doubling frequency+erefore the fault of bearing inner ring can be accuratelydiagnosed according to fault characteristic frequency

52 Outer Ring Signal Analysis of CWRU Bearings +e timedomain and the frequency domain of the bearing outer ringfault signal are shown in Figure 20 from which the periodicpulse characteristics can be seen After adding Gaussianwhite noise with a signal-to-noise ratio of minus 5 dB to the outerring fault signal the time domain and the frequency domainof the generated mixed signal are shown in Figure 21 Due to

the interference of noise it is difficult to extract the faultfeature frequency

+en the method proposed in this paper is used toprocess the mixed signal of the outer ring with added noiseAfter FDM decomposition 25 FIBF components are ob-tained +e decomposition results are shown in Figure 22Due to the limitation of the length of the article only the first20 FIBF components are listed in the figure By calculatingthe cross-correlation coefficient and kurtosis value of eachsignal component the results are shown in Table 8 Aftercomparing the data results in Table 8 it can be seen that thesignal component of y12 meets the threshold conditions+erefore the above signal component is extracted to re-construct the target signal channel and the remainingcomponents are reconstructed as the virtual noise channel+rough the RobustICA for blind source separation thetime-domain waveforms of two groups of independentcomponents are obtained and the results are shown inFigure 23

After comparing Figures 23 and 21 it can be seen thatmost of the noise is effectively filtered out through blindsource separation processing and the waveform of the targetsignal is highly similar to the original outer ring fault signalwithout the noise added In order to further compare thenoise reduction effect Hilbert envelope spectrum analysis iscarried out for the target signal obtained by separation andthe mixed signal with minus 5 dB noise added +e results areshown in Figure 24 +rough the analysis of Figure 24 it canbe seen that in the envelope spectrum of the outer ring faultsignal without noise reduction only the frequencies of1055Hz and 2168Hz have relatively high peaks which are

0 002 004 006 008 01 012 014 016 018ndash5

0

5

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 002 004 006 008 01 012 014 016 018ndash5

0

5

Time (s)

(b)

Figure 17 +e waveform of each independent component (a) +e waveform of the target signal (b) +e waveform of the noise signal

Am

plitu

de (m

s2 )

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

2f 0

3f 0

5f 0 6f 0

f 0

4f 0

(a)

Am

plitu

de (m

s2 )

2f 0 5f 0

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

f 0

(b)

Figure 18 Envelope spectrum analysis (a) Envelope spectrum analysis of the target signal denoised by FDM-RobustICA (b) Envelopespectrum of the mixed signal

14 Mathematical Problems in Engineering

very close to the theoretical calculation value of 1073648Hzof the bearing outer ring fault characteristic frequency andthe double frequency of the basic fault frequency After noisereduction by RobustICA there are three peaks with higheramplitude in the envelope spectrum of the outer ring faultsignal namely 1055Hz 2168Hz and 3223Hz +e above

three peaks are very close to the fault frequency the 2doubling frequency and the 3 doubling frequency of thebearing outer ring fault feature Although the amplitude ofthe interference spectrum in other places has been effectivelyweakened there are still many peaks in the envelopespectrum that are independent of the fault characteristic

0 200 400 600 800 1000 1200ndash500

0

500

1000

1500A

mpl

itude

(ms

2 )A

mpl

itude

(ms

2 )

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

Number (n)

(a)

Am

plitu

de (m

s2 )

0 200 400 600 800 10000

20

40

60

80

100

Frequency (Hz)

1f 0

4f 06f 0

2f 0 3f 0

5f 0

(b)

Figure 19 +e inner ring MOMEDA filter signal and its envelope spectrum (a) Time-domain waveform and the envelope spectrum of theMOMEDA filter signal (b) Part band envelope spectrum of the MOMEDA filter signal

0 002 004 006 008 01 012 014 016 018ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

02

04

Frequency (Hz)

(b)

Figure 20 (a) Time-domain analysis and (b) frequency-domain analysis of the original outer ring fault signal

Am

plitu

de (m

s2 )

0 002 004 006 008 01 012 014 016 018ndash5

0

5

Time (s)

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

01

02

03

04

Frequency (Hz)

(b)

Figure 21 (a) Time-domain analysis and (b) frequency-domain analysis of the mixed signal with SNR minus 5 dB

Mathematical Problems in Engineering 15

frequency which will affect the fault diagnosis results+erefore the periodic impact components in the noisereduced signal will be further extracted through theMOMEDA filtering method and the extraction results areshown in Figure 25

It can be clearly seen from Figure 26 that the periodicfault impact component of the outer ring fault signal hasbeen extracted +e peak value of the fault characteristicfrequency is higher and the amplitude of the signal com-ponent independent of fault impulse is lower From Fig-ure 26 not only the fault frequency (1055Hz) of the bearingouter ring can be clearly distinguished but also the 2 dou-bling frequency (2168Hz) 3 doubling frequency (3223Hz)4 doubling frequency (4277Hz) 5 doubling frequency(5391Hz) 6 doubling frequency (6445Hz) and so on canbe clearly seen which are very close to the fault characteristicfrequency and the doubling frequency calculated theoreti-cally +erefore the fault of the bearing outer ring can beaccurately diagnosed

53 Rolling Element Signal Analysis of CWRU Bearings+e time domain and the frequency domain of the bearingrolling element fault signal are shown in Figure 26 and theperiodic pulse characteristics can be seen from the figureAfter adding Gaussian white noise with a signal-to-noiseratio of minus 5 dB to the rolling element fault signal the timedomain and the frequency domain of the generated mixedsignal are shown in Figure 27 Due to the interference ofnoise it is difficult to extract the fault feature frequency fromthe time domain and the frequency domain

+en the FDM method is used to decompose the mixedsignal of the rolling element with added noise and 25 FIBFcomponents are obtained +e decomposition results areshown in Figure 28 Due to the limitation of the length of thearticle only the first 20 FIBF components are listed inFigure 28 By calculating the cross-correlation coefficientand kurtosis value of each signal component the results areshown in Table 9 By comparing the data in Table 9 it can beseen that only the y9 signal component meets the setthreshold conditions +erefore y9 component is extractedto reconstruct the target signal channel and the remainingcomponents are reconstructed as the virtual noise channel+rough the RobustICA for blind source separation thetime-domain waveforms of two groups of independentcomponents are obtained and the results are shown inFigure 29

After comparing Figures 29 and 27 it can be seen thatthe impact component in the signal has been shown throughblind source separation In order to further compare theeffect of noise reduction Hilbert envelope spectrum analysisis carried out for the separated target signal and the mixedsignal with minus 5 dB noise added +e results are shown inFigure 30 +rough the analysis of Figure 30 it can be seenthat due to the interference of noise the amplitude of lines inthe envelope spectrum of the rolling element fault signal islow so it is difficult to extract the fault characteristic fre-quency After the noise reduction by RobustICA there aremany peaks with high amplitude in the envelope spectrum+e following frequencies can be extracted from the enve-lope spectrum 1406Hz 2813Hz and 5625Hz +e abovefrequencies are very close to the theoretical calculated values

0 002 004 006 008 01 012 014 016 018ndash005

0005

y1

0 002 004 006 008 01 012 014 016 018ndash05

005

y2

0 002 004 006 008 01 012 014 016 018ndash05

005

y3

0 002 004 006 008 01 012 014 016 018ndash05

005

y4

0 002 004 006 008 01 012 014 016 018ndash1

01

y5

0 002 004 006 008 01 012 014 016 018ndash1

01

y6

0 002 004 006 008 01 012 014 016 018ndash05

005

y7

0 002 004 006 008 01 012 014 016 018ndash1

01

y8

0 002 004 006 008 01 012 014 016 018ndash1

01

y9

0 002 004 006 008 01 012 014 016 018ndash2

02

y10

0 002 004 006 008 01 012 014 016 018ndash2

02

y11

0 002 004 006 008 01 012 014 016 018ndash5

05

y12

0 002 004 006 008 01 012 014 016 018ndash2

02

y13

0 002 004 006 008 01 012 014 016 018ndash2

02

y14

0 002 004 006 008 01 012 014 016 018ndash2

02

y15

0 002 004 006 008 01 012 014 016 018ndash1

01

y16

0 002 004 006 008 01 012 014 016 018ndash2

02

y17

0 002 004 006 008 01 012 014 016 018ndash1

01

y18

0 002 004 006 008 01 012 014 016 018ndash1

01

y19

0 002 004 006 008 01 012 014 016 018ndash1

01

y20

Time (s) Time (s)

Figure 22 FDM decomposition result

16 Mathematical Problems in Engineering

Tabl

e8

+ecorrelationcoeffi

cientandku

rtosisbetweentheFIBF

andtheoriginal

signal

y 1y 2

y 3y 4

y 5y 6

y 7y 8

y 9y 1

0y 1

1y 1

2y 1

3y 1

4y 1

5y 1

6y 1

7y 1

8y 1

9Y 2

0y 2

1y 2

2y 2

3y 2

4y 2

5

C(t)

0018

0091

0074

0097

0132

0116

0125

0165

0217

0217

0260

0523

0349

0269

0277

0186

0214

0163

0159

0182

0086

0139

0064

0034

0000

K(t)

1000

2203

2209

2266

3145

4288

2564

4142

2750

2872

2683

4184

2981

2995

3277

3013

3385

2652

3151

2489

2652

4111

2615

2652

1000

Mathematical Problems in Engineering 17

of the rolling element fault characteristic frequency of1411693Hz 2 doubling frequency and 4 doubling fre-quency Although the fault characteristic frequency in theenvelope spectrum has appeared there are still many

unrelated peaks of the fault characteristic frequency and theamplitude is large +is brings a certain degree of inter-ference to the fault diagnosis +erefore the periodic impactcomponents in the signal after noise reduction will be

0 002 004 006 008 01 012 014 016 018ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 002 004 006 008 01 012 014 016 018ndash5

0

5times10ndash3

Time (s)

(b)

Figure 23 +e waveform of each independent component (a) +e waveform of the target signal (b) +e waveform of the noise signal

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 ) f 0 2f 0

3f 0

(a)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

2f 0

f 0

(b)

Figure 24 Envelope spectrum analysis (a) Envelope spectrum analysis of the target signal denoised by FDM-RobustICA (b) Envelopespectrum of the mixed signal

0 200 400 600 800 1000 1200ndash500

0

500

1000

Number (n)

Am

plitu

de (m

s2 )

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

(a)

0 200 400 600 800 10000

20

40

60

80

100

Frequency (Hz)

f 0 3f 0 4f 07f 05f 0

6f 0 8f 0 9f 0

2f 0

Am

plitu

de (m

s2 )

(b)

Figure 25 +e outer ring MOMEDA filter signal and its envelope spectrum (a) Time-domain waveform and the envelope spectrum of theMOMEDA filter signal (b) Part band envelope spectrum of the MOMEDA filter signal

18 Mathematical Problems in Engineering

0 002 004 006 008 01 012 014 016 018ndash05

0

05

Time (s)

Am

plitu

de (m

s2 )

(a)

0 1000 2000 3000 4000 5000 60000

002

004

006

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

Figure 26 (a) Time-domain analysis and (b) frequency-domain analysis of the original rolling element fault signal

0 002 004 006 008 01 012 014 016 018ndash1

0

1

Time (s)

Am

plitu

de (m

s2 )

(a)

0 1000 2000 3000 4000 5000 60000

002

004

006

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

Figure 27 (a) Time-domain analysis and (b) frequency-domain analysis of the mixed signal with SNR minus 5 dB

0 002 004 006 008 01 012 014 016 018ndash2

02

times10ndash3

y1

0 002 004 006 008 01 012 014 016 018ndash005

0005

y2

0 002 004 006 008 01 012 014 016 018ndash01

001

y3

0 002 004 006 008 01 012 014 016 018ndash01

001

y4

0 002 004 006 008 01 012 014 016 018ndash02

002

y5

0 002 004 006 008 01 012 014 016 018ndash02

002

y6

0 002 004 006 008 01 012 014 016 018ndash02

002

y7

0 002 004 006 008 01 012 014 016 018ndash02

002

y8

0 002 004 006 008 01 012 014 016 018ndash05

005

y9

0 002 004 006 008 01 012 014 016 018ndash05

005

y10

0 002 004 006 008 01 012 014 016 018ndash05

005

y11

0 002 004 006 008 01 012 014 016 018ndash05

005

y12

0 002 004 006 008 01 012 014 016 018ndash05

005

y13

0 002 004 006 008 01 012 014 016 018ndash05

005

y14

0 002 004 006 008 01 012 014 016 018ndash02

002

y15

0 002 004 006 008 01 012 014 016 018ndash02

002

y16

0 002 004 006 008 01 012 014 016 018ndash02

002

y7

0 002 004 006 008 01 012 014 016 018ndash02

002

y18

0 002 004 006 008 01 012 014 016 018ndash01

001

y19

0 002 004 006 008 01 012 014 016 018ndash01

001

y20

Time (s) Time (s)

Figure 28 FDM decomposition result

Mathematical Problems in Engineering 19

Tabl

e9

+ecorrelationcoeffi

cientandku

rtosisbetweentheFIBF

andtheoriginal

signal

y 1y 2

y 3y 4

y 5y 6

y 7y 8

y 9y 1

0y 1

1y 1

2y 1

3y 1

4y 1

5y 1

6y 1

7y 1

8y 1

9y 2

0y 2

1y 2

2y 2

3y 2

4y 2

5

C(t)

0004

0054

0126

0116

0120

0171

0196

0173

0306

0296

0490

0379

0219

0254

0206

0181

0209

0154

0113

0107

0051

0088

0017

0025

0000

K(t)

1000

2071

2276

2456

3148

2712

3194

3435

3276

2962

2716

2684

3304

2640

2710

2783

2614

2591

3079

2435

2673

2646

2107

1500

1000

20 Mathematical Problems in Engineering

further extracted by MOMEDA filtering and the extractionresults are shown in Figure 31

It can be clearly seen from Figure 31 that the periodicfault shock component of the rolling element fault signal issuccessfully extracted +e peak value of the characteristic

frequency of the fault is greatly improved and the amplitudeof the signal component unrelated to the impact of the faultis lower From Figure 31 not only the frequency of thebearing rolling element fault (1406Hz) can be clearly dis-tinguished but also the doubling frequency such as the 2

0 002 004 006 008 01 012 014 016 018ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

0 002 004 006 008 01 012 014 016 018ndash04

ndash02

0

02

04

Time (s)

Am

plitu

de (m

s2 )

(b)

Figure 29 +e waveform of each independent component (a) +e waveform of the target signal (b) +e waveform of the noise signal

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

f0 2f0 4f0

(a)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

Figure 30 Envelope spectrum analysis (a) Envelope spectrum analysis of the target signal denoised by FDM-RobustICA (b) Envelopespectrum of the mixed signal

0 200 400 600 800 1000 1200ndash500

0

500

1000

Am

plitu

de (m

s2 )

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

Number (n)

(a)

0 200 400 600 800 10000

20

40

60

80

100

Frequency (Hz)

Am

plitu

de (m

s2 )

f0 2f0 3f0

4f05f 0 6f0 7f0

(b)

Figure 31+e rolling element MOMEDA filter signal and its envelope spectrum (a) Time-domain waveform and the envelope spectrum ofthe MOMEDA filter signal (b) Part band envelope spectrum of the MOMEDA filter signal

Mathematical Problems in Engineering 21

doubling frequency (2813Hz) 3 doubling frequency(4219Hz) 4 doubling frequency (5625Hz) 5 doublingfrequency (7031Hz) and 6 doubling frequency (8436Hz)are clearly visible which are very close to the fault char-acteristic frequency and the multiple frequency theoretically+erefore the fault of the bearing rolling element can beaccurately diagnosed

6 Conclusions

Aiming at the problem of fault feature extraction of vi-bration signals of rolling bearings under the background ofstrong noise this paper proposes a fault feature extractionmethod based on the combination of FDM-RobustICA andMOMEDA After the experimental analysis of the con-structed simulation signals and the actual bearing faultsignals the conclusions reached are as follows

(1) On the basis of cross-correlation analysis and kur-tosis the signal components obtained by the FDMmethod are selected to reconstruct the observationsignal channel and the virtual noise channel whichnot only avoid the mode overlapping and endpointeffect produced by traditional methods but also solvethe problem that RobustICA cannot process thesingle-channel signal and the lack of fault infor-mation caused by blind selection of signalcomponents

(2) By comparing the results of the proposed methodwith the traditional method of LMD-RobustICA andITD-RobustICA the results show that the proposedmethod can more effectively separate fault signalsand noise signals and has achieved relative betterevaluation index value

(3) By using MOMEDA to filter the noise reducedsignal the periodic impact component under thestrong noise background can be effectively enhancedand the irrelevant impact signal can be weakenedBased on the constructed simulation signals and theexperimental analysis of the inner ring outer ringand rolling element of rolling bearings the resultsshow that the proposed method can accurately ex-tract the characteristic frequency of faults in a strongnoise background environment and the amplitude ismore obvious In our future work we will considermore reasonable approaches to optimize the pa-rameters of the proposed method and add otherdenoising techniques

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is work was supported by the Joint Special Fund for BasicResearch of Local Universities in Yunnan Province (no2019FH001-121) and Science Research Fund Project ofBaoshan University (no BYBS201802)

References

[1] P Schniter ldquoShort-time fourier transformrdquo Advanced Topicsin Signal Processing vol 32 pp 289ndash337 2009

[2] D Griffin and J S Jae Lim ldquoSignal estimation from modifiedshort-time fourier transformrdquo IEEE Transactions on Acous-tics Speech and Signal Processing vol 32 no 2 pp 236ndash2431984

[3] A Georgakis L K Stergioulas and G Giakas ldquoWigner fil-tering with smooth roll-off boundary for differentiation ofnoisy non-stationary signalsrdquo Signal Processing vol 82no 10 pp 1411ndash1415 2002

[4] M Holschneider R Kronland-Martinet J Morlet andP Tchamitchian A Real-Time Algorithm for Signal Analysiswith the Help of the Wavelet Transform Springer BerlinGermany 1989

[5] Z Nenadic and J W Burdick ldquoSpike detection using thecontinuous wavelet transformrdquo IEEE Transactions on Bio-medical Engineering vol 52 no 1 pp 74ndash87 2005

[6] N E Huang Z Shen S R Long et al ldquo+e empirical modedecomposition and the Hilbert spectrum for nonlinear andnon-stationary time series analysisrdquo Proceedings of the RoyalSociety of London Series A Mathematical Physical and En-gineering Sciences vol 454 no 1971 pp 903ndash995 1998

[7] V K Wu V Bajaj A Kumar D Sharma and G K SinghldquoAn efficient method for analysis of EMG signals using im-proved empirical mode decompositionrdquo AEU-InternationalJournal of Electronics and Communications vol 72pp 200ndash209 2017

[8] J Cheng D Yu J Tang and Y Yang ldquoApplication of fre-quency family separation method based upon EMD and localHilbert energy spectrum method to gear fault diagnosisrdquoMechanism and Machine Beory vol 43 no 6 pp 712ndash7232008

[9] Z Wu and N E Huang ldquoEnsemble empirical mode de-composition a noise-assisted data analysis methodrdquo Ad-vances in Adaptive Data Analysis vol 1 no 1 pp 1ndash41 2009

[10] H Li T Liu X Wu and Q Chen ldquoApplication of EEMD andimproved frequency band entropy in bearing fault featureextractionrdquo ISA Transactions vol 88 pp 170ndash185 2019

[11] C Wu J Wu and J Yang ldquoResearch on fault diagnosis ofcheck valve based on CEEMD compound screening andimproved SECPSOrdquo Journal of Computers vol 30 pp 128ndash144 2019

[12] K Moshen C Gang P Yusong and Y Li ldquoResearch ofplanetary gear fault diagnosis based on permutation entropyof CEEMDAN and ANFISrdquo Sensors vol 18 no 3 p 7822018

[13] J S Smith and S Jonathan ldquo+e local mean decompositionand its application to EEG perception datardquo Journal of theRoyal Society Interface vol 2 no 5 pp 443ndash454 2005

[14] M G Frei and I Osorio ldquoIntrinsic time-scale decompositiontime-frequency-energy analysis and real-time filtering of non-stationary signalsrdquo Proceedings of the Royal Society AMathematical Physical and Engineering Sciences vol 463no 2078 pp 321ndash342 2007

22 Mathematical Problems in Engineering

[15] P Singh S D Joshi R K Patney and K Saha ldquo+e fourierdecompositionmethod for nonlinear and non-stationary timeseries analysisrdquo Proceedings of the Royal Society A Mathe-matical Physical and Engineering Science vol 473 pp 1ndash122017

[16] C H Dou and J S Lin ldquoExtraction of fault features ofmachinery based on fourier decomposition methodrdquo IEEEAccess vol 7 pp 1ndash11 2019

[17] Y Liu X D Liu and S Liang ldquoAeroengine rotor fault di-agnosis based on fourier decomposition methodrdquo ChinaMechanical Engineering vol 30 pp 2156ndash2163 2019

[18] XW Fu and X Q Gao ldquoRolling bearing fault diagnosis basedon FDM and singular value difference spectrumrdquo ActaMetrologica Sinica vol 39 pp 688ndash692 2018

[19] P Comon ldquoIndependent component analysis a new con-ceptrdquo Signal Processing vol 36 no 3 pp 287ndash314 1994

[20] A Hyvarinen and E Oja ldquoA fast fixed-point algorithm forindependent component analysisrdquo Neural Computationvol 9 no 7 pp 1483ndash1492 1997

[21] A Hyvarinen ldquoFast and robust fixed-point algorithms forindependent component analysisrdquo IEEE Transactions onNeural Networks vol 10 no 3 pp 626ndash634 1999

[22] P Chevalier L Albera P Comon and A Ferreol ldquoCom-parative performance analysis of eight blind source separationmethods on radio communications signalsrdquo in Proceedings ofthe International Joint Conference on Neural Networkspp 25ndash31 Budapest Hungary July 2004

[23] V Zarzoso and P Comon ldquoRobust independent componentanalysis for blind source separation and extraction with ap-plication in electrocardiographyrdquo in Proceedings of theEMBC-2008 30th Annual International Conference of theIEEE Engineering in Medicine and Biology Society pp 3344ndash3347 Vancouver BC Canada August 2008

[24] V Zarzoso and P Comon ldquoRobust independent componentanalysis by iterative maximization of the kurtosis contrastwith algebraic optimal step sizerdquo IEEE Transactions on NeuralNetworks vol 21 pp 248ndash261 2009

[25] J Yao Y Xiang S Qian S Wang and S Wu ldquoNoise sourceidentification of diesel engine based on variational modedecomposition and robust independent component analysisrdquoApplied Acoustics vol 116 pp 184ndash194 2017

[26] F Bi L Li J Zhang and T Ma ldquoSource identification ofgasoline engine noise based on continuous wavelet transformand EEMD-RobustICArdquo Applied Acoustics vol 100 pp 34ndash42 2015

[27] Y Jiachi X Yang Q Sichong andW Shuai ldquoRadiation noiseseparation of internal combustion engine based on Gam-matone-RobustICA methodrdquo Shock and Vibration vol 2017Article ID 7565041 14 pages 2017

[28] Y Jingzong W Xiaodong W Jiande and F Zao ldquoA noisereduction method based on CEEMD-RobustICA and itsapplication in pipeline blockage signalrdquo Journal of AppliedScience and Engineering vol 22 pp 1ndash10 2019

[29] Y Jiachi X Yang Q Sichong and W Shuai ldquoNoise sourceseparation of an internal combustion engine based on asingle-channel algorithmrdquo Shock and Vibration vol 2019Article ID 1283263 19 pages 2019

[30] H Weixin and L Dunwen ldquoMine microseismic signaldenosing based on variational mode decomposition and in-dependent component analysisrdquo Journal of Vibration andShock vol 38 pp 56ndash63 2019

[31] R A Wiggins ldquoMinimum entropy deconvolutionrdquo Geo-exploration vol 16 no 1-2 pp 21ndash35 1978

[32] S Wang and J Xiang ldquoA minimum entropy deconvolution-enhanced convolutional neural networks for fault diagnosis ofaxial piston pumpsrdquo Soft Computing vol 24 no 4pp 2983ndash2997 2020

[33] H Liu and J Xiang ldquoA strategy using variational mode de-composition L-kurtosis and minimum entropy deconvolu-tion to detect mechanical faultsrdquo IEEE Access vol 7pp 70564ndash70573 2019

[34] N Sawalhi R B Randall and H Endo ldquo+e enhancement offault detection and diagnosis in rolling element bearings usingminimum entropy deconvolution combined with spectralkurtosisrdquo Mechanical Systems and Signal Processing vol 21no 6 pp 2616ndash2633 2007

[35] G L McDonald Q Zhao and M J Zuo ldquoMaximum cor-related kurtosis deconvolution and application on gear toothchip fault detectionrdquo Mechanical Systems and Signal Pro-cessing vol 33 pp 237ndash255 2012

[36] G L Mcdonald and Q Zhao ldquoMultipoint optimal minimumentropy deconvolution and convolution fix application tovibration fault detectionrdquo Mechanical Systems and SignalProcessing vol 82 pp 461ndash477 2017

[37] Y Li G Cheng X Chen and Y Pang ldquoResearch on bearingfault diagnosis method based on filter features of MOML-MEDA and LSTMrdquo Entropy vol 21 pp 1ndash15 2019

[38] W A Smith and R B Randall ldquoRolling element bearingdiagnostics using the Case Western Reserve University data abenchmark studyrdquoMechanical Systems and Signal Processingvol 64-65 pp 100ndash131 2015

[39] J Xia L Zhao Y Bai M Yu and Z Wang ldquoFeature ex-traction for rolling element bearing weak fault based onMCKD and VMDrdquo Journal of Vibration and Shock vol 36pp 78ndash83 2017

Mathematical Problems in Engineering 23

K(W) E |y|41113966 1113967 minus 2E2 |y|21113966 1113967 minus E y21113864 1113865

111386811138681113868111386811138681113868111386811138682

E2 |y|21113966 1113967 (5)

μopt argmaxu

|K(W + μg)| (6)

where g is the search direction usually g nablawK(W) whichcan be expressed as follows

nablawK(W) 4

E2 |y|21113966 1113967E |y|

21113966 1113967yx minus E yx1113864 11138651113966 1113967E y

21113966 1113967

minusE |y|41113966 1113967 minus E y21113864 1113865

111386811138681113868111386811138681113868111386811138682

1113872 1113873E yx1113864 1113865

E |y|21113966 1113967

(7)

In the process of iteration the optimization method ofRobustICA can be described as follows

P(u) 11139444

k0aku

k (8)

(2) Extract the root a of the OS polynomial uk1113864 11138654k1

(4) Update μopt based on the value of the updated stepW+ W+ W + μoptg

(5) Normalize W+ W+⟵W+W+(6) Determine whether the iterative operation converges

and meets the termination conditions If it is notsatisfied then return to Step (1)

x hlowasty + e (9)

y flowastx 1113944Nminus L

k1fkxk+Lminus 1 (10)

MDN(y t) 1

t

tTy

y (11)

MOMEDA maxf

MDN(y t) maxf

tTy

y (12)

(f1 f2 middot middot middot fL) is first derivedddf

tTy

y1113888 1113889

ddf

t1y1

y+

ddf

t2y2

y+ middot middot middot +

ddf

tNminus LyNminus L

y

(13)

ddf

tkyk

y y

minus 1tkMk minus y

minus 3tkykX0y (14)

Mk xx+Lminus 1 xx+Lminus 2 middot middot middot xk1113858 1113859T (15)

+erefore equation (13) can be written as followsddf

tTy

y1113888 1113889 y

minus 1t1M1 + t2M2 + middot middot middot tNminus LMNminus L( 1113857

minus yminus 3

tT

yX0y

(16)

yminus 1

X0t minus yminus 3

tTyX0y 0 (17)

By sorting out the above equation we can gettTy

yminus 2X0y X0t (18)

Since y XT0 f if (X0X

T0 )minus 1 exists the following

equation can be obtained

Mixedmatrix

(A)

Separationmatrix

(W)

S1 (t)

S2 (t)

Sn (t)

X1 (t)

X2 (t)

Xn (t)

Y1 (t)

Y2 (t)

Yn (t)

n1 (t) n2 (t) nn (t)

Blind source Observed signal Estimated signal

Figure 1 Frame of independent component analysis

4 Mathematical Problems in Engineering

tTy

y21113888 1113889X0f X0X

T01113872 1113873

minus 1X0t (19)

f X0XT01113872 1113873

minus 1X0t (20)

Substituting the above equation into y XT0 f the

original impact signal y can be reconstructed

3 The Process of Fault Feature Extraction

r 1113936

ni0 xi minus x( 1113857 yi minus y( 1113857

1113936ni0 xi minus x( 1113857

2yi minus y( 1113857

21113969 (21)

Ki 1n

1113944

n

i1

xi minus x

σ1113874 1113875

4 (22)

+e specific steps are as follows

(4) +e MOMEDA is used to enhance the periodicimpact component of the noise reduced signal

4 Simulations and Comparative Analysis

s(t) y0eminus 2πfnξt sin 2πfn

1 minus ξ21113969

t1113874 1113875 (23)

Mathematical Problems in Engineering 5

decomposition clearly reflect the local characteristic infor-mation of the original signal

Input signal

FDM decomposition

Calculate the correlationcoefficient and kurtosis of

each FIBF component

Reconstructobservation signal

Reconstruct virtualnoise channel signal

Unmixed by RobustICA

MOMEDA filtering

Select effective FIBFcomponents

Hilbert envelopedemodulation

Extract the fault frequency

Figure 2 +e flowchart of the proposed method

0 001 002 003 004 005 006 007 008 009ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

Figure 3 Original signal

6 Mathematical Problems in Engineering

0 001 002 003 004 005 006 007 008 009ndash005

0005

y1

0 001 002 003 004 005 006 007 008 009ndash02

002

y2

0 001 002 003 004 005 006 007 008 009ndash05

005

y3

0 001 002 003 004 005 006 007 008 009ndash05

005

y4

0 001 002 003 004 005 006 007 008 009ndash1

01

y5

0 001 002 003 004 005 006 007 008 009ndash1

01

y6

0 001 002 003 004 005 006 007 008 009ndash1

01

y7

0 001 002 003 004 005 006 007 008 009ndash2

02

y8

0 001 002 003 004 005 006 007 008 009ndash2

02

y9

0 001 002 003 004 005 006 007 008 009ndash1

01

y10

0 001 002 003 004 005 006 007 008 009ndash2

02

y11

0 001 002 003 004 005 006 007 008 009ndash2

02

y12

0 001 002 003 004 005 006 007 008 009ndash1

01

y13

0 001 002 003 004 005 006 007 008 009ndash05

005

y14

0 001 002 003 004 005 006 007 008 009ndash05

005

y15

0 001 002 003 004 005 006 007 008 009ndash05

005

y16

0 001 002 003 004 005 006 007 008 009ndash02

002

y17

0 001 002 003 004 005 006 007 008 009ndash005

0005

y18

0 001 002 003 004 005 006 007 008 009ndash2

02

y19

Time (s)

Time (s)

Figure 5 +e decomposition result by the FDM

0 001 002 003 004 005 006 007 008 009ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

01

02

03

04

Frequency (Hz)

(b)

Mathematical Problems in Engineering 7

0 001 002 003 004 005 006 007 008 009ndash5

05

PRC1

0 001 002 003 004 005 006 007 008 009ndash2

02

PRC2

0 001 002 003 004 005 006 007 008 009ndash1

01

PRC3

0 001 002 003 004 005 006 007 008 009ndash1

01

PRC4

0 001 002 003 004 005 006 007 008 009ndash05

005

PRC5

0 001 002 003 004 005 006 007 008 009ndash02

002

R

Time (s)

(a)

0 001 002 003 004 005 006 007 008 009ndash5

05

PF1

0 001 002 003 004 005 006 007 008 009ndash2

02

PF2

0 001 002 003 004 005 006 007 008 009ndash2

02

PF3

0 001 002 003 004 005 006 007 008 009ndash1

01

PF4

0 001 002 003 004 005 006 007 008 009ndash500

0500

Time (s)

R

(b)

Table 1 +e correlation coefficient and kurtosis between the FIBF and the original signal

Table 2 +e correlation coefficient and kurtosis between the PF and the original signal

PF1 PF2 PF3 PF4C (t) 0804 0467 0211 0149K (t) 3825 3608 4864 4982

Table 3 +e correlation coefficient and kurtosis between the PRC and the original signal

PRC1 PRC2 PRC3 PRC4 PRC5C (t) 0866 0584 0333 0180 0075K (t) 2929 3768 3121 3692 3312

0 001 002 003 004 005 006 007 008 009ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 001 002 003 004 005 006 007 008 009ndash4

ndash2

0

2

4

Time (s)

(b)

8 Mathematical Problems in Engineering

Am

plitu

de (m

s2 )

Time (s)0 001 002 003 004 005 006 007 008 009

ndash4

ndash2

0

2

4

(a)

Am

plitu

de (m

s2 )

Time (s)0 001 002 003 004 005 006 007 008 009

ndash4

ndash2

0

2

4

(b)

0 001 002 003 004 005 006 007 008 009

ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 001 002 003 004 005 006 007 008 009ndash4

ndash2

0

2

4

Time (s)

(b)

Table 4 Comparison of noise reduction results of different methods

Evaluation index FDM-RobustICA LMD-RobustICA ITD-RobustICA

Kurtosis and correlation coefficient 39070582

36020374

36300480

Noise reduction time 1378 00889 0103

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

4f 0 5f 03f 0

2f 0f 0

(a)

Am

plitu

de (m

s2 )

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

(b)

Mathematical Problems in Engineering 9

5 Application to Roller Bearing Testing

Outer race defect frequency

BPFO Z

21 minus

d

Dcos α1113888 1113889 times fr (24)

Inner race defect frequency

BPFI Z

21 +

d

Dcos α1113888 1113889 times fr (25)

Rolling element frequency

BPFR D

2d1 minus

d

D1113888 1113889

2

cos2α⎛⎝ ⎞⎠ times fr (26)

Am

plitu

de (m

s2 )

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

2f 0 4f 0f 0

(a)

Am

plitu

de (m

s2 )

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

2f 0 4f 03f 0f 0

(b)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

f 03f 0 5f 0

4f 02f 0

(a)

Am

plitu

de (m

s2 )

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

3f 02f 0f 0

(b)

10 Mathematical Problems in Engineering

Am

plitu

de (m

s2 )

Am

plitu

de (m

s2 )

0 200 400 600 800 1000 1200ndash500

0

500

1000

Number (n)

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

(a)

0 200 400 600 800 10000

5

10

15

20

Frequency (Hz)

3f 05f 0

6f 0

7f 0 8f 0 9f 0

2f 0

4f 0

Am

plitu

de (m

s2 )

f 0

(b)

Table 5 +e bearing technical parameters of SKF 6205

Rolling elementnumber (Z) Inner diameter (inches) Outer diameter

(inches)Rolling element

diameter d (inches) Contact angle (α) Pitch circlediameter D (inches)

Speed(rpm)

9 09843 20472 03126 00 15327 1797

Table 6 Fault characteristic frequency (unit Hz)

Inner ring fault Outer ring fault Rolling element fault1621852 1073648 1411693

0 002 004 006 008 01 012 014 016 018ndash2

ndash1

0

1

2

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

005

01

Frequency (Hz)

(b)

Mathematical Problems in Engineering 11

0 002 004 006 008 01 012 014 016 018ndash2

ndash1

0

1

2

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

005

01

Frequency (Hz)

(b)

0 002 004 006 008 01 012 014 016 018ndash5

05

times10ndash3

y1

0 002 004 006 008 01 012 014 016 018ndash005

0005

y2

0 002 004 006 008 01 012 014 016 018ndash005

0005

y3

0 002 004 006 008 01 012 014 016 018ndash01

001

y4

0 002 004 006 008 01 012 014 016 018ndash02

002

y5

0 002 004 006 008 01 012 014 016 018ndash02

002

y6

0 002 004 006 008 01 012 014 016 018ndash05

005

y7

0 002 004 006 008 01 012 014 016 018ndash05

005

y8

0 002 004 006 008 01 012 014 016 018ndash05

005

y9

0 002 004 006 008 01 012 014 016 018ndash1

01

y10

0 002 004 006 008 01 012 014 016 018ndash1

01

y11

0 002 004 006 008 01 012 014 016 018ndash1

01

y12

0 002 004 006 008 01 012 014 016 018ndash05

005

y13

0 002 004 006 008 01 012 014 016 018ndash05

005

y14

0 002 004 006 008 01 012 014 016 018ndash05

005

y15

0 002 004 006 008 01 012 014 016 018ndash02

002

y16

0 002 004 006 008 01 012 014 016 018ndash05

005

y17

0 002 004 006 008 01 012 014 016 018ndash02

002

y18

0 002 004 006 008 01 012 014 016 018ndash02

002

y19

0 002 004 006 008 01 012 014 016 018ndash01

001

y20

Time (s) Time (s)

Figure 16 FDM decomposition result

12 Mathematical Problems in Engineering

Tabl

e7

+ecorrelationcoeffi

cientandku

rtosisbetweentheFIBF

andtheoriginal

signal

y 1y 2

y 3y 4

y 5y 6

y 7y 8

y 9y 1

0y 1

1y 1

2y 1

3y 1

4y 1

5y 1

6y 1

7y 1

8y 1

9y 2

0y 2

1y 2

2y 2

3y 2

4y 2

5

C(t)

0008

0041

0028

0082

0115

0149

0181

0211

0259

0468

0430

0360

0215

0270

0234

0143

0202

0117

0114

0082

0060

0036

0059

0026

000

K(t)

1000

1958

1928

3032

2501

2708

2375

2470

2847

4302

3221

3183

2793

3417

3383

2424

2881

2620

3039

2866

2444

2743

2073

1500

1000

Mathematical Problems in Engineering 13

the interference of noise it is difficult to extract the faultfeature frequency

0 002 004 006 008 01 012 014 016 018ndash5

0

5

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 002 004 006 008 01 012 014 016 018ndash5

0

5

Time (s)

(b)

Am

plitu

de (m

s2 )

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

2f 0

3f 0

5f 0 6f 0

f 0

4f 0

(a)

Am

plitu

de (m

s2 )

2f 0 5f 0

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

f 0

(b)

14 Mathematical Problems in Engineering

0 200 400 600 800 1000 1200ndash500

0

500

1000

1500A

mpl

itude

(ms

2 )A

mpl

itude

(ms

2 )

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

Number (n)

(a)

Am

plitu

de (m

s2 )

0 200 400 600 800 10000

20

40

60

80

100

Frequency (Hz)

1f 0

4f 06f 0

2f 0 3f 0

5f 0

(b)

0 002 004 006 008 01 012 014 016 018ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

02

04

Frequency (Hz)

(b)

Am

plitu

de (m

s2 )

0 002 004 006 008 01 012 014 016 018ndash5

0

5

Time (s)

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

01

02

03

04

Frequency (Hz)

(b)

Mathematical Problems in Engineering 15

0 002 004 006 008 01 012 014 016 018ndash005

0005

y1

0 002 004 006 008 01 012 014 016 018ndash05

005

y2

0 002 004 006 008 01 012 014 016 018ndash05

005

y3

0 002 004 006 008 01 012 014 016 018ndash05

005

y4

0 002 004 006 008 01 012 014 016 018ndash1

01

y5

0 002 004 006 008 01 012 014 016 018ndash1

01

y6

0 002 004 006 008 01 012 014 016 018ndash05

005

y7

0 002 004 006 008 01 012 014 016 018ndash1

01

y8

0 002 004 006 008 01 012 014 016 018ndash1

01

y9

0 002 004 006 008 01 012 014 016 018ndash2

02

y10

0 002 004 006 008 01 012 014 016 018ndash2

02

y11

0 002 004 006 008 01 012 014 016 018ndash5

05

y12

0 002 004 006 008 01 012 014 016 018ndash2

02

y13

0 002 004 006 008 01 012 014 016 018ndash2

02

y14

0 002 004 006 008 01 012 014 016 018ndash2

02

y15

0 002 004 006 008 01 012 014 016 018ndash1

01

y16

0 002 004 006 008 01 012 014 016 018ndash2

02

y17

0 002 004 006 008 01 012 014 016 018ndash1

01

y18

0 002 004 006 008 01 012 014 016 018ndash1

01

y19

0 002 004 006 008 01 012 014 016 018ndash1

01

y20

Time (s) Time (s)

Figure 22 FDM decomposition result

16 Mathematical Problems in Engineering

Tabl

e8

+ecorrelationcoeffi

cientandku

rtosisbetweentheFIBF

andtheoriginal

signal

y 1y 2

y 3y 4

y 5y 6

y 7y 8

y 9y 1

0y 1

1y 1

2y 1

3y 1

4y 1

5y 1

6y 1

7y 1

8y 1

9Y 2

0y 2

1y 2

2y 2

3y 2

4y 2

5

C(t)

0018

0091

0074

0097

0132

0116

0125

0165

0217

0217

0260

0523

0349

0269

0277

0186

0214

0163

0159

0182

0086

0139

0064

0034

0000

K(t)

1000

2203

2209

2266

3145

4288

2564

4142

2750

2872

2683

4184

2981

2995

3277

3013

3385

2652

3151

2489

2652

4111

2615

2652

1000

Mathematical Problems in Engineering 17

0 002 004 006 008 01 012 014 016 018ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 002 004 006 008 01 012 014 016 018ndash5

0

5times10ndash3

Time (s)

(b)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 ) f 0 2f 0

3f 0

(a)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

2f 0

f 0

(b)

0 200 400 600 800 1000 1200ndash500

0

500

1000

Number (n)

Am

plitu

de (m

s2 )

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

(a)

0 200 400 600 800 10000

20

40

60

80

100

Frequency (Hz)

f 0 3f 0 4f 07f 05f 0

6f 0 8f 0 9f 0

2f 0

Am

plitu

de (m

s2 )

(b)

18 Mathematical Problems in Engineering

0 002 004 006 008 01 012 014 016 018ndash05

0

05

Time (s)

Am

plitu

de (m

s2 )

(a)

0 1000 2000 3000 4000 5000 60000

002

004

006

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

0 002 004 006 008 01 012 014 016 018ndash1

0

1

Time (s)

Am

plitu

de (m

s2 )

(a)

0 1000 2000 3000 4000 5000 60000

002

004

006

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

0 002 004 006 008 01 012 014 016 018ndash2

02

times10ndash3

y1

0 002 004 006 008 01 012 014 016 018ndash005

0005

y2

0 002 004 006 008 01 012 014 016 018ndash01

001

y3

0 002 004 006 008 01 012 014 016 018ndash01

001

y4

0 002 004 006 008 01 012 014 016 018ndash02

002

y5

0 002 004 006 008 01 012 014 016 018ndash02

002

y6

0 002 004 006 008 01 012 014 016 018ndash02

002

y7

0 002 004 006 008 01 012 014 016 018ndash02

002

y8

0 002 004 006 008 01 012 014 016 018ndash05

005

y9

0 002 004 006 008 01 012 014 016 018ndash05

005

y10

0 002 004 006 008 01 012 014 016 018ndash05

005

y11

0 002 004 006 008 01 012 014 016 018ndash05

005

y12

0 002 004 006 008 01 012 014 016 018ndash05

005

y13

0 002 004 006 008 01 012 014 016 018ndash05

005

y14

0 002 004 006 008 01 012 014 016 018ndash02

002

y15

0 002 004 006 008 01 012 014 016 018ndash02

002

y16

0 002 004 006 008 01 012 014 016 018ndash02

002

y7

0 002 004 006 008 01 012 014 016 018ndash02

002

y18

0 002 004 006 008 01 012 014 016 018ndash01

001

y19

0 002 004 006 008 01 012 014 016 018ndash01

001

y20

Time (s) Time (s)

Figure 28 FDM decomposition result

Mathematical Problems in Engineering 19

Tabl

e9

+ecorrelationcoeffi

cientandku

rtosisbetweentheFIBF

andtheoriginal

signal

y 1y 2

y 3y 4

y 5y 6

y 7y 8

y 9y 1

0y 1

1y 1

2y 1

3y 1

4y 1

5y 1

6y 1

7y 1

8y 1

9y 2

0y 2

1y 2

2y 2

3y 2

4y 2

5

C(t)

0004

0054

0126

0116

0120

0171

0196

0173

0306

0296

0490

0379

0219

0254

0206

0181

0209

0154

0113

0107

0051

0088

0017

0025

0000

K(t)

1000

2071

2276

2456

3148

2712

3194

3435

3276

2962

2716

2684

3304

2640

2710

2783

2614

2591

3079

2435

2673

2646

2107

1500

1000

20 Mathematical Problems in Engineering

further extracted by MOMEDA filtering and the extractionresults are shown in Figure 31

0 002 004 006 008 01 012 014 016 018ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

0 002 004 006 008 01 012 014 016 018ndash04

ndash02

0

02

04

Time (s)

Am

plitu

de (m

s2 )

(b)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

f0 2f0 4f0

(a)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

0 200 400 600 800 1000 1200ndash500

0

500

1000

Am

plitu

de (m

s2 )

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

Number (n)

(a)

0 200 400 600 800 10000

20

40

60

80

100

Frequency (Hz)

Am

plitu

de (m

s2 )

f0 2f0 3f0

4f05f 0 6f0 7f0

(b)

Mathematical Problems in Engineering 21

6 Conclusions

Data Availability

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

References

22 Mathematical Problems in Engineering

Mathematical Problems in Engineering 23

tTy

y21113888 1113889X0f X0X

T01113872 1113873

minus 1X0t (19)

f X0XT01113872 1113873

minus 1X0t (20)

Substituting the above equation into y XT0 f the

original impact signal y can be reconstructed

3 The Process of Fault Feature Extraction

r 1113936

ni0 xi minus x( 1113857 yi minus y( 1113857

1113936ni0 xi minus x( 1113857

2yi minus y( 1113857

21113969 (21)

Ki 1n

1113944

n

i1

xi minus x

σ1113874 1113875

4 (22)

+e specific steps are as follows

(4) +e MOMEDA is used to enhance the periodicimpact component of the noise reduced signal

4 Simulations and Comparative Analysis

s(t) y0eminus 2πfnξt sin 2πfn

1 minus ξ21113969

t1113874 1113875 (23)

Mathematical Problems in Engineering 5

decomposition clearly reflect the local characteristic infor-mation of the original signal

Input signal

FDM decomposition

Calculate the correlationcoefficient and kurtosis of

each FIBF component

Reconstructobservation signal

Reconstruct virtualnoise channel signal

Unmixed by RobustICA

MOMEDA filtering

Select effective FIBFcomponents

Hilbert envelopedemodulation

Extract the fault frequency

Figure 2 +e flowchart of the proposed method

0 001 002 003 004 005 006 007 008 009ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

Figure 3 Original signal

6 Mathematical Problems in Engineering

0 001 002 003 004 005 006 007 008 009ndash005

0005

y1

0 001 002 003 004 005 006 007 008 009ndash02

002

y2

0 001 002 003 004 005 006 007 008 009ndash05

005

y3

0 001 002 003 004 005 006 007 008 009ndash05

005

y4

0 001 002 003 004 005 006 007 008 009ndash1

01

y5

0 001 002 003 004 005 006 007 008 009ndash1

01

y6

0 001 002 003 004 005 006 007 008 009ndash1

01

y7

0 001 002 003 004 005 006 007 008 009ndash2

02

y8

0 001 002 003 004 005 006 007 008 009ndash2

02

y9

0 001 002 003 004 005 006 007 008 009ndash1

01

y10

0 001 002 003 004 005 006 007 008 009ndash2

02

y11

0 001 002 003 004 005 006 007 008 009ndash2

02

y12

0 001 002 003 004 005 006 007 008 009ndash1

01

y13

0 001 002 003 004 005 006 007 008 009ndash05

005

y14

0 001 002 003 004 005 006 007 008 009ndash05

005

y15

0 001 002 003 004 005 006 007 008 009ndash05

005

y16

0 001 002 003 004 005 006 007 008 009ndash02

002

y17

0 001 002 003 004 005 006 007 008 009ndash005

0005

y18

0 001 002 003 004 005 006 007 008 009ndash2

02

y19

Time (s)

Time (s)

Figure 5 +e decomposition result by the FDM

0 001 002 003 004 005 006 007 008 009ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

01

02

03

04

Frequency (Hz)

(b)

Mathematical Problems in Engineering 7

0 001 002 003 004 005 006 007 008 009ndash5

05

PRC1

0 001 002 003 004 005 006 007 008 009ndash2

02

PRC2

0 001 002 003 004 005 006 007 008 009ndash1

01

PRC3

0 001 002 003 004 005 006 007 008 009ndash1

01

PRC4

0 001 002 003 004 005 006 007 008 009ndash05

005

PRC5

0 001 002 003 004 005 006 007 008 009ndash02

002

R

Time (s)

(a)

0 001 002 003 004 005 006 007 008 009ndash5

05

PF1

0 001 002 003 004 005 006 007 008 009ndash2

02

PF2

0 001 002 003 004 005 006 007 008 009ndash2

02

PF3

0 001 002 003 004 005 006 007 008 009ndash1

01

PF4

0 001 002 003 004 005 006 007 008 009ndash500

0500

Time (s)

R

(b)

Table 1 +e correlation coefficient and kurtosis between the FIBF and the original signal

Table 2 +e correlation coefficient and kurtosis between the PF and the original signal

PF1 PF2 PF3 PF4C (t) 0804 0467 0211 0149K (t) 3825 3608 4864 4982

Table 3 +e correlation coefficient and kurtosis between the PRC and the original signal

PRC1 PRC2 PRC3 PRC4 PRC5C (t) 0866 0584 0333 0180 0075K (t) 2929 3768 3121 3692 3312

0 001 002 003 004 005 006 007 008 009ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 001 002 003 004 005 006 007 008 009ndash4

ndash2

0

2

4

Time (s)

(b)

8 Mathematical Problems in Engineering

Am

plitu

de (m

s2 )

Time (s)0 001 002 003 004 005 006 007 008 009

ndash4

ndash2

0

2

4

(a)

Am

plitu

de (m

s2 )

Time (s)0 001 002 003 004 005 006 007 008 009

ndash4

ndash2

0

2

4

(b)

0 001 002 003 004 005 006 007 008 009

ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 001 002 003 004 005 006 007 008 009ndash4

ndash2

0

2

4

Time (s)

(b)

Table 4 Comparison of noise reduction results of different methods

Evaluation index FDM-RobustICA LMD-RobustICA ITD-RobustICA

Kurtosis and correlation coefficient 39070582

36020374

36300480

Noise reduction time 1378 00889 0103

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

4f 0 5f 03f 0

2f 0f 0

(a)

Am

plitu

de (m

s2 )

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

(b)

Mathematical Problems in Engineering 9

5 Application to Roller Bearing Testing

Outer race defect frequency

BPFO Z

21 minus

d

Dcos α1113888 1113889 times fr (24)

Inner race defect frequency

BPFI Z

21 +

d

Dcos α1113888 1113889 times fr (25)

Rolling element frequency

BPFR D

2d1 minus

d

D1113888 1113889

2

cos2α⎛⎝ ⎞⎠ times fr (26)

Am

plitu

de (m

s2 )

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

2f 0 4f 0f 0

(a)

Am

plitu

de (m

s2 )

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

2f 0 4f 03f 0f 0

(b)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

f 03f 0 5f 0

4f 02f 0

(a)

Am

plitu

de (m

s2 )

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

3f 02f 0f 0

(b)

10 Mathematical Problems in Engineering

Am

plitu

de (m

s2 )

Am

plitu

de (m

s2 )

0 200 400 600 800 1000 1200ndash500

0

500

1000

Number (n)

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

(a)

0 200 400 600 800 10000

5

10

15

20

Frequency (Hz)

3f 05f 0

6f 0

7f 0 8f 0 9f 0

2f 0

4f 0

Am

plitu

de (m

s2 )

f 0

(b)

Table 5 +e bearing technical parameters of SKF 6205

Rolling elementnumber (Z) Inner diameter (inches) Outer diameter

(inches)Rolling element

diameter d (inches) Contact angle (α) Pitch circlediameter D (inches)

Speed(rpm)

9 09843 20472 03126 00 15327 1797

Table 6 Fault characteristic frequency (unit Hz)

Inner ring fault Outer ring fault Rolling element fault1621852 1073648 1411693

0 002 004 006 008 01 012 014 016 018ndash2

ndash1

0

1

2

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

005

01

Frequency (Hz)

(b)

Mathematical Problems in Engineering 11

0 002 004 006 008 01 012 014 016 018ndash2

ndash1

0

1

2

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

005

01

Frequency (Hz)

(b)

0 002 004 006 008 01 012 014 016 018ndash5

05

times10ndash3

y1

0 002 004 006 008 01 012 014 016 018ndash005

0005

y2

0 002 004 006 008 01 012 014 016 018ndash005

0005

y3

0 002 004 006 008 01 012 014 016 018ndash01

001

y4

0 002 004 006 008 01 012 014 016 018ndash02

002

y5

0 002 004 006 008 01 012 014 016 018ndash02

002

y6

0 002 004 006 008 01 012 014 016 018ndash05

005

y7

0 002 004 006 008 01 012 014 016 018ndash05

005

y8

0 002 004 006 008 01 012 014 016 018ndash05

005

y9

0 002 004 006 008 01 012 014 016 018ndash1

01

y10

0 002 004 006 008 01 012 014 016 018ndash1

01

y11

0 002 004 006 008 01 012 014 016 018ndash1

01

y12

0 002 004 006 008 01 012 014 016 018ndash05

005

y13

0 002 004 006 008 01 012 014 016 018ndash05

005

y14

0 002 004 006 008 01 012 014 016 018ndash05

005

y15

0 002 004 006 008 01 012 014 016 018ndash02

002

y16

0 002 004 006 008 01 012 014 016 018ndash05

005

y17

0 002 004 006 008 01 012 014 016 018ndash02

002

y18

0 002 004 006 008 01 012 014 016 018ndash02

002

y19

0 002 004 006 008 01 012 014 016 018ndash01

001

y20

Time (s) Time (s)

Figure 16 FDM decomposition result

12 Mathematical Problems in Engineering

Tabl

e7

+ecorrelationcoeffi

cientandku

rtosisbetweentheFIBF

andtheoriginal

signal

y 1y 2

y 3y 4

y 5y 6

y 7y 8

y 9y 1

0y 1

1y 1

2y 1

3y 1

4y 1

5y 1

6y 1

7y 1

8y 1

9y 2

0y 2

1y 2

2y 2

3y 2

4y 2

5

C(t)

0008

0041

0028

0082

0115

0149

0181

0211

0259

0468

0430

0360

0215

0270

0234

0143

0202

0117

0114

0082

0060

0036

0059

0026

000

K(t)

1000

1958

1928

3032

2501

2708

2375

2470

2847

4302

3221

3183

2793

3417

3383

2424

2881

2620

3039

2866

2444

2743

2073

1500

1000

Mathematical Problems in Engineering 13

the interference of noise it is difficult to extract the faultfeature frequency

0 002 004 006 008 01 012 014 016 018ndash5

0

5

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 002 004 006 008 01 012 014 016 018ndash5

0

5

Time (s)

(b)

Am

plitu

de (m

s2 )

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

2f 0

3f 0

5f 0 6f 0

f 0

4f 0

(a)

Am

plitu

de (m

s2 )

2f 0 5f 0

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

f 0

(b)

14 Mathematical Problems in Engineering

0 200 400 600 800 1000 1200ndash500

0

500

1000

1500A

mpl

itude

(ms

2 )A

mpl

itude

(ms

2 )

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

Number (n)

(a)

Am

plitu

de (m

s2 )

0 200 400 600 800 10000

20

40

60

80

100

Frequency (Hz)

1f 0

4f 06f 0

2f 0 3f 0

5f 0

(b)

0 002 004 006 008 01 012 014 016 018ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

02

04

Frequency (Hz)

(b)

Am

plitu

de (m

s2 )

0 002 004 006 008 01 012 014 016 018ndash5

0

5

Time (s)

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

01

02

03

04

Frequency (Hz)

(b)

Mathematical Problems in Engineering 15

0 002 004 006 008 01 012 014 016 018ndash005

0005

y1

0 002 004 006 008 01 012 014 016 018ndash05

005

y2

0 002 004 006 008 01 012 014 016 018ndash05

005

y3

0 002 004 006 008 01 012 014 016 018ndash05

005

y4

0 002 004 006 008 01 012 014 016 018ndash1

01

y5

0 002 004 006 008 01 012 014 016 018ndash1

01

y6

0 002 004 006 008 01 012 014 016 018ndash05

005

y7

0 002 004 006 008 01 012 014 016 018ndash1

01

y8

0 002 004 006 008 01 012 014 016 018ndash1

01

y9

0 002 004 006 008 01 012 014 016 018ndash2

02

y10

0 002 004 006 008 01 012 014 016 018ndash2

02

y11

0 002 004 006 008 01 012 014 016 018ndash5

05

y12

0 002 004 006 008 01 012 014 016 018ndash2

02

y13

0 002 004 006 008 01 012 014 016 018ndash2

02

y14

0 002 004 006 008 01 012 014 016 018ndash2

02

y15

0 002 004 006 008 01 012 014 016 018ndash1

01

y16

0 002 004 006 008 01 012 014 016 018ndash2

02

y17

0 002 004 006 008 01 012 014 016 018ndash1

01

y18

0 002 004 006 008 01 012 014 016 018ndash1

01

y19

0 002 004 006 008 01 012 014 016 018ndash1

01

y20

Time (s) Time (s)

Figure 22 FDM decomposition result

16 Mathematical Problems in Engineering

Tabl

e8

+ecorrelationcoeffi

cientandku

rtosisbetweentheFIBF

andtheoriginal

signal

y 1y 2

y 3y 4

y 5y 6

y 7y 8

y 9y 1

0y 1

1y 1

2y 1

3y 1

4y 1

5y 1

6y 1

7y 1

8y 1

9Y 2

0y 2

1y 2

2y 2

3y 2

4y 2

5

C(t)

0018

0091

0074

0097

0132

0116

0125

0165

0217

0217

0260

0523

0349

0269

0277

0186

0214

0163

0159

0182

0086

0139

0064

0034

0000

K(t)

1000

2203

2209

2266

3145

4288

2564

4142

2750

2872

2683

4184

2981

2995

3277

3013

3385

2652

3151

2489

2652

4111

2615

2652

1000

Mathematical Problems in Engineering 17

0 002 004 006 008 01 012 014 016 018ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 002 004 006 008 01 012 014 016 018ndash5

0

5times10ndash3

Time (s)

(b)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 ) f 0 2f 0

3f 0

(a)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

2f 0

f 0

(b)

0 200 400 600 800 1000 1200ndash500

0

500

1000

Number (n)

Am

plitu

de (m

s2 )

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

(a)

0 200 400 600 800 10000

20

40

60

80

100

Frequency (Hz)

f 0 3f 0 4f 07f 05f 0

6f 0 8f 0 9f 0

2f 0

Am

plitu

de (m

s2 )

(b)

18 Mathematical Problems in Engineering

0 002 004 006 008 01 012 014 016 018ndash05

0

05

Time (s)

Am

plitu

de (m

s2 )

(a)

0 1000 2000 3000 4000 5000 60000

002

004

006

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

0 002 004 006 008 01 012 014 016 018ndash1

0

1

Time (s)

Am

plitu

de (m

s2 )

(a)

0 1000 2000 3000 4000 5000 60000

002

004

006

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

0 002 004 006 008 01 012 014 016 018ndash2

02

times10ndash3

y1

0 002 004 006 008 01 012 014 016 018ndash005

0005

y2

0 002 004 006 008 01 012 014 016 018ndash01

001

y3

0 002 004 006 008 01 012 014 016 018ndash01

001

y4

0 002 004 006 008 01 012 014 016 018ndash02

002

y5

0 002 004 006 008 01 012 014 016 018ndash02

002

y6

0 002 004 006 008 01 012 014 016 018ndash02

002

y7

0 002 004 006 008 01 012 014 016 018ndash02

002

y8

0 002 004 006 008 01 012 014 016 018ndash05

005

y9

0 002 004 006 008 01 012 014 016 018ndash05

005

y10

0 002 004 006 008 01 012 014 016 018ndash05

005

y11

0 002 004 006 008 01 012 014 016 018ndash05

005

y12

0 002 004 006 008 01 012 014 016 018ndash05

005

y13

0 002 004 006 008 01 012 014 016 018ndash05

005

y14

0 002 004 006 008 01 012 014 016 018ndash02

002

y15

0 002 004 006 008 01 012 014 016 018ndash02

002

y16

0 002 004 006 008 01 012 014 016 018ndash02

002

y7

0 002 004 006 008 01 012 014 016 018ndash02

002

y18

0 002 004 006 008 01 012 014 016 018ndash01

001

y19

0 002 004 006 008 01 012 014 016 018ndash01

001

y20

Time (s) Time (s)

Figure 28 FDM decomposition result

Mathematical Problems in Engineering 19

Tabl

e9

+ecorrelationcoeffi

cientandku

rtosisbetweentheFIBF

andtheoriginal

signal

y 1y 2

y 3y 4

y 5y 6

y 7y 8

y 9y 1

0y 1

1y 1

2y 1

3y 1

4y 1

5y 1

6y 1

7y 1

8y 1

9y 2

0y 2

1y 2

2y 2

3y 2

4y 2

5

C(t)

0004

0054

0126

0116

0120

0171

0196

0173

0306

0296

0490

0379

0219

0254

0206

0181

0209

0154

0113

0107

0051

0088

0017

0025

0000

K(t)

1000

2071

2276

2456

3148

2712

3194

3435

3276

2962

2716

2684

3304

2640

2710

2783

2614

2591

3079

2435

2673

2646

2107

1500

1000

20 Mathematical Problems in Engineering

further extracted by MOMEDA filtering and the extractionresults are shown in Figure 31

0 002 004 006 008 01 012 014 016 018ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

0 002 004 006 008 01 012 014 016 018ndash04

ndash02

0

02

04

Time (s)

Am

plitu

de (m

s2 )

(b)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

f0 2f0 4f0

(a)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

0 200 400 600 800 1000 1200ndash500

0

500

1000

Am

plitu

de (m

s2 )

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

Number (n)

(a)

0 200 400 600 800 10000

20

40

60

80

100

Frequency (Hz)

Am

plitu

de (m

s2 )

f0 2f0 3f0

4f05f 0 6f0 7f0

(b)

Mathematical Problems in Engineering 21

6 Conclusions

Data Availability

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

References

22 Mathematical Problems in Engineering

Mathematical Problems in Engineering 23

decomposition clearly reflect the local characteristic infor-mation of the original signal

Input signal

FDM decomposition

Calculate the correlationcoefficient and kurtosis of

each FIBF component

Reconstructobservation signal

Reconstruct virtualnoise channel signal

Unmixed by RobustICA

MOMEDA filtering

Select effective FIBFcomponents

Hilbert envelopedemodulation

Extract the fault frequency

Figure 2 +e flowchart of the proposed method

0 001 002 003 004 005 006 007 008 009ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

Figure 3 Original signal

6 Mathematical Problems in Engineering

0 001 002 003 004 005 006 007 008 009ndash005

0005

y1

0 001 002 003 004 005 006 007 008 009ndash02

002

y2

0 001 002 003 004 005 006 007 008 009ndash05

005

y3

0 001 002 003 004 005 006 007 008 009ndash05

005

y4

0 001 002 003 004 005 006 007 008 009ndash1

01

y5

0 001 002 003 004 005 006 007 008 009ndash1

01

y6

0 001 002 003 004 005 006 007 008 009ndash1

01

y7

0 001 002 003 004 005 006 007 008 009ndash2

02

y8

0 001 002 003 004 005 006 007 008 009ndash2

02

y9

0 001 002 003 004 005 006 007 008 009ndash1

01

y10

0 001 002 003 004 005 006 007 008 009ndash2

02

y11

0 001 002 003 004 005 006 007 008 009ndash2

02

y12

0 001 002 003 004 005 006 007 008 009ndash1

01

y13

0 001 002 003 004 005 006 007 008 009ndash05

005

y14

0 001 002 003 004 005 006 007 008 009ndash05

005

y15

0 001 002 003 004 005 006 007 008 009ndash05

005

y16

0 001 002 003 004 005 006 007 008 009ndash02

002

y17

0 001 002 003 004 005 006 007 008 009ndash005

0005

y18

0 001 002 003 004 005 006 007 008 009ndash2

02

y19

Time (s)

Time (s)

Figure 5 +e decomposition result by the FDM

0 001 002 003 004 005 006 007 008 009ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

01

02

03

04

Frequency (Hz)

(b)

Mathematical Problems in Engineering 7

0 001 002 003 004 005 006 007 008 009ndash5

05

PRC1

0 001 002 003 004 005 006 007 008 009ndash2

02

PRC2

0 001 002 003 004 005 006 007 008 009ndash1

01

PRC3

0 001 002 003 004 005 006 007 008 009ndash1

01

PRC4

0 001 002 003 004 005 006 007 008 009ndash05

005

PRC5

0 001 002 003 004 005 006 007 008 009ndash02

002

R

Time (s)

(a)

0 001 002 003 004 005 006 007 008 009ndash5

05

PF1

0 001 002 003 004 005 006 007 008 009ndash2

02

PF2

0 001 002 003 004 005 006 007 008 009ndash2

02

PF3

0 001 002 003 004 005 006 007 008 009ndash1

01

PF4

0 001 002 003 004 005 006 007 008 009ndash500

0500

Time (s)

R

(b)

Table 1 +e correlation coefficient and kurtosis between the FIBF and the original signal

Table 2 +e correlation coefficient and kurtosis between the PF and the original signal

PF1 PF2 PF3 PF4C (t) 0804 0467 0211 0149K (t) 3825 3608 4864 4982

Table 3 +e correlation coefficient and kurtosis between the PRC and the original signal

PRC1 PRC2 PRC3 PRC4 PRC5C (t) 0866 0584 0333 0180 0075K (t) 2929 3768 3121 3692 3312

0 001 002 003 004 005 006 007 008 009ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 001 002 003 004 005 006 007 008 009ndash4

ndash2

0

2

4

Time (s)

(b)

8 Mathematical Problems in Engineering

Am

plitu

de (m

s2 )

Time (s)0 001 002 003 004 005 006 007 008 009

ndash4

ndash2

0

2

4

(a)

Am

plitu

de (m

s2 )

Time (s)0 001 002 003 004 005 006 007 008 009

ndash4

ndash2

0

2

4

(b)

0 001 002 003 004 005 006 007 008 009

ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 001 002 003 004 005 006 007 008 009ndash4

ndash2

0

2

4

Time (s)

(b)

Table 4 Comparison of noise reduction results of different methods

Evaluation index FDM-RobustICA LMD-RobustICA ITD-RobustICA

Kurtosis and correlation coefficient 39070582

36020374

36300480

Noise reduction time 1378 00889 0103

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

4f 0 5f 03f 0

2f 0f 0

(a)

Am

plitu

de (m

s2 )

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

(b)

Mathematical Problems in Engineering 9

5 Application to Roller Bearing Testing

Outer race defect frequency

BPFO Z

21 minus

d

Dcos α1113888 1113889 times fr (24)

Inner race defect frequency

BPFI Z

21 +

d

Dcos α1113888 1113889 times fr (25)

Rolling element frequency

BPFR D

2d1 minus

d

D1113888 1113889

2

cos2α⎛⎝ ⎞⎠ times fr (26)

Am

plitu

de (m

s2 )

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

2f 0 4f 0f 0

(a)

Am

plitu

de (m

s2 )

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

2f 0 4f 03f 0f 0

(b)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

f 03f 0 5f 0

4f 02f 0

(a)

Am

plitu

de (m

s2 )

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

3f 02f 0f 0

(b)

10 Mathematical Problems in Engineering

Am

plitu

de (m

s2 )

Am

plitu

de (m

s2 )

0 200 400 600 800 1000 1200ndash500

0

500

1000

Number (n)

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

(a)

0 200 400 600 800 10000

5

10

15

20

Frequency (Hz)

3f 05f 0

6f 0

7f 0 8f 0 9f 0

2f 0

4f 0

Am

plitu

de (m

s2 )

f 0

(b)

Table 5 +e bearing technical parameters of SKF 6205

Rolling elementnumber (Z) Inner diameter (inches) Outer diameter

(inches)Rolling element

diameter d (inches) Contact angle (α) Pitch circlediameter D (inches)

Speed(rpm)

9 09843 20472 03126 00 15327 1797

Table 6 Fault characteristic frequency (unit Hz)

Inner ring fault Outer ring fault Rolling element fault1621852 1073648 1411693

0 002 004 006 008 01 012 014 016 018ndash2

ndash1

0

1

2

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

005

01

Frequency (Hz)

(b)

Mathematical Problems in Engineering 11

0 002 004 006 008 01 012 014 016 018ndash2

ndash1

0

1

2

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

005

01

Frequency (Hz)

(b)

0 002 004 006 008 01 012 014 016 018ndash5

05

times10ndash3

y1

0 002 004 006 008 01 012 014 016 018ndash005

0005

y2

0 002 004 006 008 01 012 014 016 018ndash005

0005

y3

0 002 004 006 008 01 012 014 016 018ndash01

001

y4

0 002 004 006 008 01 012 014 016 018ndash02

002

y5

0 002 004 006 008 01 012 014 016 018ndash02

002

y6

0 002 004 006 008 01 012 014 016 018ndash05

005

y7

0 002 004 006 008 01 012 014 016 018ndash05

005

y8

0 002 004 006 008 01 012 014 016 018ndash05

005

y9

0 002 004 006 008 01 012 014 016 018ndash1

01

y10

0 002 004 006 008 01 012 014 016 018ndash1

01

y11

0 002 004 006 008 01 012 014 016 018ndash1

01

y12

0 002 004 006 008 01 012 014 016 018ndash05

005

y13

0 002 004 006 008 01 012 014 016 018ndash05

005

y14

0 002 004 006 008 01 012 014 016 018ndash05

005

y15

0 002 004 006 008 01 012 014 016 018ndash02

002

y16

0 002 004 006 008 01 012 014 016 018ndash05

005

y17

0 002 004 006 008 01 012 014 016 018ndash02

002

y18

0 002 004 006 008 01 012 014 016 018ndash02

002

y19

0 002 004 006 008 01 012 014 016 018ndash01

001

y20

Time (s) Time (s)

Figure 16 FDM decomposition result

12 Mathematical Problems in Engineering

Tabl

e7

+ecorrelationcoeffi

cientandku

rtosisbetweentheFIBF

andtheoriginal

signal

y 1y 2

y 3y 4

y 5y 6

y 7y 8

y 9y 1

0y 1

1y 1

2y 1

3y 1

4y 1

5y 1

6y 1

7y 1

8y 1

9y 2

0y 2

1y 2

2y 2

3y 2

4y 2

5

C(t)

0008

0041

0028

0082

0115

0149

0181

0211

0259

0468

0430

0360

0215

0270

0234

0143

0202

0117

0114

0082

0060

0036

0059

0026

000

K(t)

1000

1958

1928

3032

2501

2708

2375

2470

2847

4302

3221

3183

2793

3417

3383

2424

2881

2620

3039

2866

2444

2743

2073

1500

1000

Mathematical Problems in Engineering 13

the interference of noise it is difficult to extract the faultfeature frequency

0 002 004 006 008 01 012 014 016 018ndash5

0

5

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 002 004 006 008 01 012 014 016 018ndash5

0

5

Time (s)

(b)

Am

plitu

de (m

s2 )

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

2f 0

3f 0

5f 0 6f 0

f 0

4f 0

(a)

Am

plitu

de (m

s2 )

2f 0 5f 0

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

f 0

(b)

14 Mathematical Problems in Engineering

0 200 400 600 800 1000 1200ndash500

0

500

1000

1500A

mpl

itude

(ms

2 )A

mpl

itude

(ms

2 )

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

Number (n)

(a)

Am

plitu

de (m

s2 )

0 200 400 600 800 10000

20

40

60

80

100

Frequency (Hz)

1f 0

4f 06f 0

2f 0 3f 0

5f 0

(b)

0 002 004 006 008 01 012 014 016 018ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

02

04

Frequency (Hz)

(b)

Am

plitu

de (m

s2 )

0 002 004 006 008 01 012 014 016 018ndash5

0

5

Time (s)

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

01

02

03

04

Frequency (Hz)

(b)

Mathematical Problems in Engineering 15

0 002 004 006 008 01 012 014 016 018ndash005

0005

y1

0 002 004 006 008 01 012 014 016 018ndash05

005

y2

0 002 004 006 008 01 012 014 016 018ndash05

005

y3

0 002 004 006 008 01 012 014 016 018ndash05

005

y4

0 002 004 006 008 01 012 014 016 018ndash1

01

y5

0 002 004 006 008 01 012 014 016 018ndash1

01

y6

0 002 004 006 008 01 012 014 016 018ndash05

005

y7

0 002 004 006 008 01 012 014 016 018ndash1

01

y8

0 002 004 006 008 01 012 014 016 018ndash1

01

y9

0 002 004 006 008 01 012 014 016 018ndash2

02

y10

0 002 004 006 008 01 012 014 016 018ndash2

02

y11

0 002 004 006 008 01 012 014 016 018ndash5

05

y12

0 002 004 006 008 01 012 014 016 018ndash2

02

y13

0 002 004 006 008 01 012 014 016 018ndash2

02

y14

0 002 004 006 008 01 012 014 016 018ndash2

02

y15

0 002 004 006 008 01 012 014 016 018ndash1

01

y16

0 002 004 006 008 01 012 014 016 018ndash2

02

y17

0 002 004 006 008 01 012 014 016 018ndash1

01

y18

0 002 004 006 008 01 012 014 016 018ndash1

01

y19

0 002 004 006 008 01 012 014 016 018ndash1

01

y20

Time (s) Time (s)

Figure 22 FDM decomposition result

16 Mathematical Problems in Engineering

Tabl

e8

+ecorrelationcoeffi

cientandku

rtosisbetweentheFIBF

andtheoriginal

signal

y 1y 2

y 3y 4

y 5y 6

y 7y 8

y 9y 1

0y 1

1y 1

2y 1

3y 1

4y 1

5y 1

6y 1

7y 1

8y 1

9Y 2

0y 2

1y 2

2y 2

3y 2

4y 2

5

C(t)

0018

0091

0074

0097

0132

0116

0125

0165

0217

0217

0260

0523

0349

0269

0277

0186

0214

0163

0159

0182

0086

0139

0064

0034

0000

K(t)

1000

2203

2209

2266

3145

4288

2564

4142

2750

2872

2683

4184

2981

2995

3277

3013

3385

2652

3151

2489

2652

4111

2615

2652

1000

Mathematical Problems in Engineering 17

0 002 004 006 008 01 012 014 016 018ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 002 004 006 008 01 012 014 016 018ndash5

0

5times10ndash3

Time (s)

(b)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 ) f 0 2f 0

3f 0

(a)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

2f 0

f 0

(b)

0 200 400 600 800 1000 1200ndash500

0

500

1000

Number (n)

Am

plitu

de (m

s2 )

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

(a)

0 200 400 600 800 10000

20

40

60

80

100

Frequency (Hz)

f 0 3f 0 4f 07f 05f 0

6f 0 8f 0 9f 0

2f 0

Am

plitu

de (m

s2 )

(b)

18 Mathematical Problems in Engineering

0 002 004 006 008 01 012 014 016 018ndash05

0

05

Time (s)

Am

plitu

de (m

s2 )

(a)

0 1000 2000 3000 4000 5000 60000

002

004

006

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

0 002 004 006 008 01 012 014 016 018ndash1

0

1

Time (s)

Am

plitu

de (m

s2 )

(a)

0 1000 2000 3000 4000 5000 60000

002

004

006

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

0 002 004 006 008 01 012 014 016 018ndash2

02

times10ndash3

y1

0 002 004 006 008 01 012 014 016 018ndash005

0005

y2

0 002 004 006 008 01 012 014 016 018ndash01

001

y3

0 002 004 006 008 01 012 014 016 018ndash01

001

y4

0 002 004 006 008 01 012 014 016 018ndash02

002

y5

0 002 004 006 008 01 012 014 016 018ndash02

002

y6

0 002 004 006 008 01 012 014 016 018ndash02

002

y7

0 002 004 006 008 01 012 014 016 018ndash02

002

y8

0 002 004 006 008 01 012 014 016 018ndash05

005

y9

0 002 004 006 008 01 012 014 016 018ndash05

005

y10

0 002 004 006 008 01 012 014 016 018ndash05

005

y11

0 002 004 006 008 01 012 014 016 018ndash05

005

y12

0 002 004 006 008 01 012 014 016 018ndash05

005

y13

0 002 004 006 008 01 012 014 016 018ndash05

005

y14

0 002 004 006 008 01 012 014 016 018ndash02

002

y15

0 002 004 006 008 01 012 014 016 018ndash02

002

y16

0 002 004 006 008 01 012 014 016 018ndash02

002

y7

0 002 004 006 008 01 012 014 016 018ndash02

002

y18

0 002 004 006 008 01 012 014 016 018ndash01

001

y19

0 002 004 006 008 01 012 014 016 018ndash01

001

y20

Time (s) Time (s)

Figure 28 FDM decomposition result

Mathematical Problems in Engineering 19

Tabl

e9

+ecorrelationcoeffi

cientandku

rtosisbetweentheFIBF

andtheoriginal

signal

y 1y 2

y 3y 4

y 5y 6

y 7y 8

y 9y 1

0y 1

1y 1

2y 1

3y 1

4y 1

5y 1

6y 1

7y 1

8y 1

9y 2

0y 2

1y 2

2y 2

3y 2

4y 2

5

C(t)

0004

0054

0126

0116

0120

0171

0196

0173

0306

0296

0490

0379

0219

0254

0206

0181

0209

0154

0113

0107

0051

0088

0017

0025

0000

K(t)

1000

2071

2276

2456

3148

2712

3194

3435

3276

2962

2716

2684

3304

2640

2710

2783

2614

2591

3079

2435

2673

2646

2107

1500

1000

20 Mathematical Problems in Engineering

further extracted by MOMEDA filtering and the extractionresults are shown in Figure 31

0 002 004 006 008 01 012 014 016 018ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

0 002 004 006 008 01 012 014 016 018ndash04

ndash02

0

02

04

Time (s)

Am

plitu

de (m

s2 )

(b)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

f0 2f0 4f0

(a)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

0 200 400 600 800 1000 1200ndash500

0

500

1000

Am

plitu

de (m

s2 )

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

Number (n)

(a)

0 200 400 600 800 10000

20

40

60

80

100

Frequency (Hz)

Am

plitu

de (m

s2 )

f0 2f0 3f0

4f05f 0 6f0 7f0

(b)

Mathematical Problems in Engineering 21

6 Conclusions

Data Availability

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

References

22 Mathematical Problems in Engineering

Mathematical Problems in Engineering 23

0 001 002 003 004 005 006 007 008 009ndash005

0005

y1

0 001 002 003 004 005 006 007 008 009ndash02

002

y2

0 001 002 003 004 005 006 007 008 009ndash05

005

y3

0 001 002 003 004 005 006 007 008 009ndash05

005

y4

0 001 002 003 004 005 006 007 008 009ndash1

01

y5

0 001 002 003 004 005 006 007 008 009ndash1

01

y6

0 001 002 003 004 005 006 007 008 009ndash1

01

y7

0 001 002 003 004 005 006 007 008 009ndash2

02

y8

0 001 002 003 004 005 006 007 008 009ndash2

02

y9

0 001 002 003 004 005 006 007 008 009ndash1

01

y10

0 001 002 003 004 005 006 007 008 009ndash2

02

y11

0 001 002 003 004 005 006 007 008 009ndash2

02

y12

0 001 002 003 004 005 006 007 008 009ndash1

01

y13

0 001 002 003 004 005 006 007 008 009ndash05

005

y14

0 001 002 003 004 005 006 007 008 009ndash05

005

y15

0 001 002 003 004 005 006 007 008 009ndash05

005

y16

0 001 002 003 004 005 006 007 008 009ndash02

002

y17

0 001 002 003 004 005 006 007 008 009ndash005

0005

y18

0 001 002 003 004 005 006 007 008 009ndash2

02

y19

Time (s)

Time (s)

Figure 5 +e decomposition result by the FDM

0 001 002 003 004 005 006 007 008 009ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

01

02

03

04

Frequency (Hz)

(b)

Mathematical Problems in Engineering 7

0 001 002 003 004 005 006 007 008 009ndash5

05

PRC1

0 001 002 003 004 005 006 007 008 009ndash2

02

PRC2

0 001 002 003 004 005 006 007 008 009ndash1

01

PRC3

0 001 002 003 004 005 006 007 008 009ndash1

01

PRC4

0 001 002 003 004 005 006 007 008 009ndash05

005

PRC5

0 001 002 003 004 005 006 007 008 009ndash02

002

R

Time (s)

(a)

0 001 002 003 004 005 006 007 008 009ndash5

05

PF1

0 001 002 003 004 005 006 007 008 009ndash2

02

PF2

0 001 002 003 004 005 006 007 008 009ndash2

02

PF3

0 001 002 003 004 005 006 007 008 009ndash1

01

PF4

0 001 002 003 004 005 006 007 008 009ndash500

0500

Time (s)

R

(b)

Table 1 +e correlation coefficient and kurtosis between the FIBF and the original signal

Table 2 +e correlation coefficient and kurtosis between the PF and the original signal

PF1 PF2 PF3 PF4C (t) 0804 0467 0211 0149K (t) 3825 3608 4864 4982

Table 3 +e correlation coefficient and kurtosis between the PRC and the original signal

PRC1 PRC2 PRC3 PRC4 PRC5C (t) 0866 0584 0333 0180 0075K (t) 2929 3768 3121 3692 3312

0 001 002 003 004 005 006 007 008 009ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 001 002 003 004 005 006 007 008 009ndash4

ndash2

0

2

4

Time (s)

(b)

8 Mathematical Problems in Engineering

Am

plitu

de (m

s2 )

Time (s)0 001 002 003 004 005 006 007 008 009

ndash4

ndash2

0

2

4

(a)

Am

plitu

de (m

s2 )

Time (s)0 001 002 003 004 005 006 007 008 009

ndash4

ndash2

0

2

4

(b)

0 001 002 003 004 005 006 007 008 009

ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 001 002 003 004 005 006 007 008 009ndash4

ndash2

0

2

4

Time (s)

(b)

Table 4 Comparison of noise reduction results of different methods

Evaluation index FDM-RobustICA LMD-RobustICA ITD-RobustICA

Kurtosis and correlation coefficient 39070582

36020374

36300480

Noise reduction time 1378 00889 0103

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

4f 0 5f 03f 0

2f 0f 0

(a)

Am

plitu

de (m

s2 )

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

(b)

Mathematical Problems in Engineering 9

5 Application to Roller Bearing Testing

Outer race defect frequency

BPFO Z

21 minus

d

Dcos α1113888 1113889 times fr (24)

Inner race defect frequency

BPFI Z

21 +

d

Dcos α1113888 1113889 times fr (25)

Rolling element frequency

BPFR D

2d1 minus

d

D1113888 1113889

2

cos2α⎛⎝ ⎞⎠ times fr (26)

Am

plitu

de (m

s2 )

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

2f 0 4f 0f 0

(a)

Am

plitu

de (m

s2 )

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

2f 0 4f 03f 0f 0

(b)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

f 03f 0 5f 0

4f 02f 0

(a)

Am

plitu

de (m

s2 )

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

3f 02f 0f 0

(b)

10 Mathematical Problems in Engineering

Am

plitu

de (m

s2 )

Am

plitu

de (m

s2 )

0 200 400 600 800 1000 1200ndash500

0

500

1000

Number (n)

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

(a)

0 200 400 600 800 10000

5

10

15

20

Frequency (Hz)

3f 05f 0

6f 0

7f 0 8f 0 9f 0

2f 0

4f 0

Am

plitu

de (m

s2 )

f 0

(b)

Table 5 +e bearing technical parameters of SKF 6205

Rolling elementnumber (Z) Inner diameter (inches) Outer diameter

(inches)Rolling element

diameter d (inches) Contact angle (α) Pitch circlediameter D (inches)

Speed(rpm)

9 09843 20472 03126 00 15327 1797

Table 6 Fault characteristic frequency (unit Hz)

Inner ring fault Outer ring fault Rolling element fault1621852 1073648 1411693

0 002 004 006 008 01 012 014 016 018ndash2

ndash1

0

1

2

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

005

01

Frequency (Hz)

(b)

Mathematical Problems in Engineering 11

0 002 004 006 008 01 012 014 016 018ndash2

ndash1

0

1

2

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

005

01

Frequency (Hz)

(b)

0 002 004 006 008 01 012 014 016 018ndash5

05

times10ndash3

y1

0 002 004 006 008 01 012 014 016 018ndash005

0005

y2

0 002 004 006 008 01 012 014 016 018ndash005

0005

y3

0 002 004 006 008 01 012 014 016 018ndash01

001

y4

0 002 004 006 008 01 012 014 016 018ndash02

002

y5

0 002 004 006 008 01 012 014 016 018ndash02

002

y6

0 002 004 006 008 01 012 014 016 018ndash05

005

y7

0 002 004 006 008 01 012 014 016 018ndash05

005

y8

0 002 004 006 008 01 012 014 016 018ndash05

005

y9

0 002 004 006 008 01 012 014 016 018ndash1

01

y10

0 002 004 006 008 01 012 014 016 018ndash1

01

y11

0 002 004 006 008 01 012 014 016 018ndash1

01

y12

0 002 004 006 008 01 012 014 016 018ndash05

005

y13

0 002 004 006 008 01 012 014 016 018ndash05

005

y14

0 002 004 006 008 01 012 014 016 018ndash05

005

y15

0 002 004 006 008 01 012 014 016 018ndash02

002

y16

0 002 004 006 008 01 012 014 016 018ndash05

005

y17

0 002 004 006 008 01 012 014 016 018ndash02

002

y18

0 002 004 006 008 01 012 014 016 018ndash02

002

y19

0 002 004 006 008 01 012 014 016 018ndash01

001

y20

Time (s) Time (s)

Figure 16 FDM decomposition result

12 Mathematical Problems in Engineering

Tabl

e7

+ecorrelationcoeffi

cientandku

rtosisbetweentheFIBF

andtheoriginal

signal

y 1y 2

y 3y 4

y 5y 6

y 7y 8

y 9y 1

0y 1

1y 1

2y 1

3y 1

4y 1

5y 1

6y 1

7y 1

8y 1

9y 2

0y 2

1y 2

2y 2

3y 2

4y 2

5

C(t)

0008

0041

0028

0082

0115

0149

0181

0211

0259

0468

0430

0360

0215

0270

0234

0143

0202

0117

0114

0082

0060

0036

0059

0026

000

K(t)

1000

1958

1928

3032

2501

2708

2375

2470

2847

4302

3221

3183

2793

3417

3383

2424

2881

2620

3039

2866

2444

2743

2073

1500

1000

Mathematical Problems in Engineering 13

the interference of noise it is difficult to extract the faultfeature frequency

0 002 004 006 008 01 012 014 016 018ndash5

0

5

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 002 004 006 008 01 012 014 016 018ndash5

0

5

Time (s)

(b)

Am

plitu

de (m

s2 )

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

2f 0

3f 0

5f 0 6f 0

f 0

4f 0

(a)

Am

plitu

de (m

s2 )

2f 0 5f 0

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

f 0

(b)

14 Mathematical Problems in Engineering

0 200 400 600 800 1000 1200ndash500

0

500

1000

1500A

mpl

itude

(ms

2 )A

mpl

itude

(ms

2 )

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

Number (n)

(a)

Am

plitu

de (m

s2 )

0 200 400 600 800 10000

20

40

60

80

100

Frequency (Hz)

1f 0

4f 06f 0

2f 0 3f 0

5f 0

(b)

0 002 004 006 008 01 012 014 016 018ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

02

04

Frequency (Hz)

(b)

Am

plitu

de (m

s2 )

0 002 004 006 008 01 012 014 016 018ndash5

0

5

Time (s)

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

01

02

03

04

Frequency (Hz)

(b)

Mathematical Problems in Engineering 15

0 002 004 006 008 01 012 014 016 018ndash005

0005

y1

0 002 004 006 008 01 012 014 016 018ndash05

005

y2

0 002 004 006 008 01 012 014 016 018ndash05

005

y3

0 002 004 006 008 01 012 014 016 018ndash05

005

y4

0 002 004 006 008 01 012 014 016 018ndash1

01

y5

0 002 004 006 008 01 012 014 016 018ndash1

01

y6

0 002 004 006 008 01 012 014 016 018ndash05

005

y7

0 002 004 006 008 01 012 014 016 018ndash1

01

y8

0 002 004 006 008 01 012 014 016 018ndash1

01

y9

0 002 004 006 008 01 012 014 016 018ndash2

02

y10

0 002 004 006 008 01 012 014 016 018ndash2

02

y11

0 002 004 006 008 01 012 014 016 018ndash5

05

y12

0 002 004 006 008 01 012 014 016 018ndash2

02

y13

0 002 004 006 008 01 012 014 016 018ndash2

02

y14

0 002 004 006 008 01 012 014 016 018ndash2

02

y15

0 002 004 006 008 01 012 014 016 018ndash1

01

y16

0 002 004 006 008 01 012 014 016 018ndash2

02

y17

0 002 004 006 008 01 012 014 016 018ndash1

01

y18

0 002 004 006 008 01 012 014 016 018ndash1

01

y19

0 002 004 006 008 01 012 014 016 018ndash1

01

y20

Time (s) Time (s)

Figure 22 FDM decomposition result

16 Mathematical Problems in Engineering

Tabl

e8

+ecorrelationcoeffi

cientandku

rtosisbetweentheFIBF

andtheoriginal

signal

y 1y 2

y 3y 4

y 5y 6

y 7y 8

y 9y 1

0y 1

1y 1

2y 1

3y 1

4y 1

5y 1

6y 1

7y 1

8y 1

9Y 2

0y 2

1y 2

2y 2

3y 2

4y 2

5

C(t)

0018

0091

0074

0097

0132

0116

0125

0165

0217

0217

0260

0523

0349

0269

0277

0186

0214

0163

0159

0182

0086

0139

0064

0034

0000

K(t)

1000

2203

2209

2266

3145

4288

2564

4142

2750

2872

2683

4184

2981

2995

3277

3013

3385

2652

3151

2489

2652

4111

2615

2652

1000

Mathematical Problems in Engineering 17

0 002 004 006 008 01 012 014 016 018ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 002 004 006 008 01 012 014 016 018ndash5

0

5times10ndash3

Time (s)

(b)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 ) f 0 2f 0

3f 0

(a)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

2f 0

f 0

(b)

0 200 400 600 800 1000 1200ndash500

0

500

1000

Number (n)

Am

plitu

de (m

s2 )

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

(a)

0 200 400 600 800 10000

20

40

60

80

100

Frequency (Hz)

f 0 3f 0 4f 07f 05f 0

6f 0 8f 0 9f 0

2f 0

Am

plitu

de (m

s2 )

(b)

18 Mathematical Problems in Engineering

0 002 004 006 008 01 012 014 016 018ndash05

0

05

Time (s)

Am

plitu

de (m

s2 )

(a)

0 1000 2000 3000 4000 5000 60000

002

004

006

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

0 002 004 006 008 01 012 014 016 018ndash1

0

1

Time (s)

Am

plitu

de (m

s2 )

(a)

0 1000 2000 3000 4000 5000 60000

002

004

006

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

0 002 004 006 008 01 012 014 016 018ndash2

02

times10ndash3

y1

0 002 004 006 008 01 012 014 016 018ndash005

0005

y2

0 002 004 006 008 01 012 014 016 018ndash01

001

y3

0 002 004 006 008 01 012 014 016 018ndash01

001

y4

0 002 004 006 008 01 012 014 016 018ndash02

002

y5

0 002 004 006 008 01 012 014 016 018ndash02

002

y6

0 002 004 006 008 01 012 014 016 018ndash02

002

y7

0 002 004 006 008 01 012 014 016 018ndash02

002

y8

0 002 004 006 008 01 012 014 016 018ndash05

005

y9

0 002 004 006 008 01 012 014 016 018ndash05

005

y10

0 002 004 006 008 01 012 014 016 018ndash05

005

y11

0 002 004 006 008 01 012 014 016 018ndash05

005

y12

0 002 004 006 008 01 012 014 016 018ndash05

005

y13

0 002 004 006 008 01 012 014 016 018ndash05

005

y14

0 002 004 006 008 01 012 014 016 018ndash02

002

y15

0 002 004 006 008 01 012 014 016 018ndash02

002

y16

0 002 004 006 008 01 012 014 016 018ndash02

002

y7

0 002 004 006 008 01 012 014 016 018ndash02

002

y18

0 002 004 006 008 01 012 014 016 018ndash01

001

y19

0 002 004 006 008 01 012 014 016 018ndash01

001

y20

Time (s) Time (s)

Figure 28 FDM decomposition result

Mathematical Problems in Engineering 19

Tabl

e9

+ecorrelationcoeffi

cientandku

rtosisbetweentheFIBF

andtheoriginal

signal

y 1y 2

y 3y 4

y 5y 6

y 7y 8

y 9y 1

0y 1

1y 1

2y 1

3y 1

4y 1

5y 1

6y 1

7y 1

8y 1

9y 2

0y 2

1y 2

2y 2

3y 2

4y 2

5

C(t)

0004

0054

0126

0116

0120

0171

0196

0173

0306

0296

0490

0379

0219

0254

0206

0181

0209

0154

0113

0107

0051

0088

0017

0025

0000

K(t)

1000

2071

2276

2456

3148

2712

3194

3435

3276

2962

2716

2684

3304

2640

2710

2783

2614

2591

3079

2435

2673

2646

2107

1500

1000

20 Mathematical Problems in Engineering

further extracted by MOMEDA filtering and the extractionresults are shown in Figure 31

0 002 004 006 008 01 012 014 016 018ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

0 002 004 006 008 01 012 014 016 018ndash04

ndash02

0

02

04

Time (s)

Am

plitu

de (m

s2 )

(b)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

f0 2f0 4f0

(a)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

0 200 400 600 800 1000 1200ndash500

0

500

1000

Am

plitu

de (m

s2 )

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

Number (n)

(a)

0 200 400 600 800 10000

20

40

60

80

100

Frequency (Hz)

Am

plitu

de (m

s2 )

f0 2f0 3f0

4f05f 0 6f0 7f0

(b)

Mathematical Problems in Engineering 21

6 Conclusions

Data Availability

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

References

22 Mathematical Problems in Engineering

Mathematical Problems in Engineering 23

0 001 002 003 004 005 006 007 008 009ndash5

05

PRC1

0 001 002 003 004 005 006 007 008 009ndash2

02

PRC2

0 001 002 003 004 005 006 007 008 009ndash1

01

PRC3

0 001 002 003 004 005 006 007 008 009ndash1

01

PRC4

0 001 002 003 004 005 006 007 008 009ndash05

005

PRC5

0 001 002 003 004 005 006 007 008 009ndash02

002

R

Time (s)

(a)

0 001 002 003 004 005 006 007 008 009ndash5

05

PF1

0 001 002 003 004 005 006 007 008 009ndash2

02

PF2

0 001 002 003 004 005 006 007 008 009ndash2

02

PF3

0 001 002 003 004 005 006 007 008 009ndash1

01

PF4

0 001 002 003 004 005 006 007 008 009ndash500

0500

Time (s)

R

(b)

Table 1 +e correlation coefficient and kurtosis between the FIBF and the original signal

Table 2 +e correlation coefficient and kurtosis between the PF and the original signal

PF1 PF2 PF3 PF4C (t) 0804 0467 0211 0149K (t) 3825 3608 4864 4982

Table 3 +e correlation coefficient and kurtosis between the PRC and the original signal

PRC1 PRC2 PRC3 PRC4 PRC5C (t) 0866 0584 0333 0180 0075K (t) 2929 3768 3121 3692 3312

0 001 002 003 004 005 006 007 008 009ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 001 002 003 004 005 006 007 008 009ndash4

ndash2

0

2

4

Time (s)

(b)

8 Mathematical Problems in Engineering

Am

plitu

de (m

s2 )

Time (s)0 001 002 003 004 005 006 007 008 009

ndash4

ndash2

0

2

4

(a)

Am

plitu

de (m

s2 )

Time (s)0 001 002 003 004 005 006 007 008 009

ndash4

ndash2

0

2

4

(b)

0 001 002 003 004 005 006 007 008 009

ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 001 002 003 004 005 006 007 008 009ndash4

ndash2

0

2

4

Time (s)

(b)

Table 4 Comparison of noise reduction results of different methods

Evaluation index FDM-RobustICA LMD-RobustICA ITD-RobustICA

Kurtosis and correlation coefficient 39070582

36020374

36300480

Noise reduction time 1378 00889 0103

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

4f 0 5f 03f 0

2f 0f 0

(a)

Am

plitu

de (m

s2 )

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

(b)

Mathematical Problems in Engineering 9

5 Application to Roller Bearing Testing

Outer race defect frequency

BPFO Z

21 minus

d

Dcos α1113888 1113889 times fr (24)

Inner race defect frequency

BPFI Z

21 +

d

Dcos α1113888 1113889 times fr (25)

Rolling element frequency

BPFR D

2d1 minus

d

D1113888 1113889

2

cos2α⎛⎝ ⎞⎠ times fr (26)

Am

plitu

de (m

s2 )

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

2f 0 4f 0f 0

(a)

Am

plitu

de (m

s2 )

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

2f 0 4f 03f 0f 0

(b)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

f 03f 0 5f 0

4f 02f 0

(a)

Am

plitu

de (m

s2 )

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

3f 02f 0f 0

(b)

10 Mathematical Problems in Engineering

Am

plitu

de (m

s2 )

Am

plitu

de (m

s2 )

0 200 400 600 800 1000 1200ndash500

0

500

1000

Number (n)

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

(a)

0 200 400 600 800 10000

5

10

15

20

Frequency (Hz)

3f 05f 0

6f 0

7f 0 8f 0 9f 0

2f 0

4f 0

Am

plitu

de (m

s2 )

f 0

(b)

Table 5 +e bearing technical parameters of SKF 6205

Rolling elementnumber (Z) Inner diameter (inches) Outer diameter

(inches)Rolling element

diameter d (inches) Contact angle (α) Pitch circlediameter D (inches)

Speed(rpm)

9 09843 20472 03126 00 15327 1797

Table 6 Fault characteristic frequency (unit Hz)

Inner ring fault Outer ring fault Rolling element fault1621852 1073648 1411693

0 002 004 006 008 01 012 014 016 018ndash2

ndash1

0

1

2

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

005

01

Frequency (Hz)

(b)

Mathematical Problems in Engineering 11

0 002 004 006 008 01 012 014 016 018ndash2

ndash1

0

1

2

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

005

01

Frequency (Hz)

(b)

0 002 004 006 008 01 012 014 016 018ndash5

05

times10ndash3

y1

0 002 004 006 008 01 012 014 016 018ndash005

0005

y2

0 002 004 006 008 01 012 014 016 018ndash005

0005

y3

0 002 004 006 008 01 012 014 016 018ndash01

001

y4

0 002 004 006 008 01 012 014 016 018ndash02

002

y5

0 002 004 006 008 01 012 014 016 018ndash02

002

y6

0 002 004 006 008 01 012 014 016 018ndash05

005

y7

0 002 004 006 008 01 012 014 016 018ndash05

005

y8

0 002 004 006 008 01 012 014 016 018ndash05

005

y9

0 002 004 006 008 01 012 014 016 018ndash1

01

y10

0 002 004 006 008 01 012 014 016 018ndash1

01

y11

0 002 004 006 008 01 012 014 016 018ndash1

01

y12

0 002 004 006 008 01 012 014 016 018ndash05

005

y13

0 002 004 006 008 01 012 014 016 018ndash05

005

y14

0 002 004 006 008 01 012 014 016 018ndash05

005

y15

0 002 004 006 008 01 012 014 016 018ndash02

002

y16

0 002 004 006 008 01 012 014 016 018ndash05

005

y17

0 002 004 006 008 01 012 014 016 018ndash02

002

y18

0 002 004 006 008 01 012 014 016 018ndash02

002

y19

0 002 004 006 008 01 012 014 016 018ndash01

001

y20

Time (s) Time (s)

Figure 16 FDM decomposition result

12 Mathematical Problems in Engineering

Tabl

e7

+ecorrelationcoeffi

cientandku

rtosisbetweentheFIBF

andtheoriginal

signal

y 1y 2

y 3y 4

y 5y 6

y 7y 8

y 9y 1

0y 1

1y 1

2y 1

3y 1

4y 1

5y 1

6y 1

7y 1

8y 1

9y 2

0y 2

1y 2

2y 2

3y 2

4y 2

5

C(t)

0008

0041

0028

0082

0115

0149

0181

0211

0259

0468

0430

0360

0215

0270

0234

0143

0202

0117

0114

0082

0060

0036

0059

0026

000

K(t)

1000

1958

1928

3032

2501

2708

2375

2470

2847

4302

3221

3183

2793

3417

3383

2424

2881

2620

3039

2866

2444

2743

2073

1500

1000

Mathematical Problems in Engineering 13

the interference of noise it is difficult to extract the faultfeature frequency

0 002 004 006 008 01 012 014 016 018ndash5

0

5

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 002 004 006 008 01 012 014 016 018ndash5

0

5

Time (s)

(b)

Am

plitu

de (m

s2 )

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

2f 0

3f 0

5f 0 6f 0

f 0

4f 0

(a)

Am

plitu

de (m

s2 )

2f 0 5f 0

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

f 0

(b)

14 Mathematical Problems in Engineering

0 200 400 600 800 1000 1200ndash500

0

500

1000

1500A

mpl

itude

(ms

2 )A

mpl

itude

(ms

2 )

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

Number (n)

(a)

Am

plitu

de (m

s2 )

0 200 400 600 800 10000

20

40

60

80

100

Frequency (Hz)

1f 0

4f 06f 0

2f 0 3f 0

5f 0

(b)

0 002 004 006 008 01 012 014 016 018ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

02

04

Frequency (Hz)

(b)

Am

plitu

de (m

s2 )

0 002 004 006 008 01 012 014 016 018ndash5

0

5

Time (s)

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

01

02

03

04

Frequency (Hz)

(b)

Mathematical Problems in Engineering 15

0 002 004 006 008 01 012 014 016 018ndash005

0005

y1

0 002 004 006 008 01 012 014 016 018ndash05

005

y2

0 002 004 006 008 01 012 014 016 018ndash05

005

y3

0 002 004 006 008 01 012 014 016 018ndash05

005

y4

0 002 004 006 008 01 012 014 016 018ndash1

01

y5

0 002 004 006 008 01 012 014 016 018ndash1

01

y6

0 002 004 006 008 01 012 014 016 018ndash05

005

y7

0 002 004 006 008 01 012 014 016 018ndash1

01

y8

0 002 004 006 008 01 012 014 016 018ndash1

01

y9

0 002 004 006 008 01 012 014 016 018ndash2

02

y10

0 002 004 006 008 01 012 014 016 018ndash2

02

y11

0 002 004 006 008 01 012 014 016 018ndash5

05

y12

0 002 004 006 008 01 012 014 016 018ndash2

02

y13

0 002 004 006 008 01 012 014 016 018ndash2

02

y14

0 002 004 006 008 01 012 014 016 018ndash2

02

y15

0 002 004 006 008 01 012 014 016 018ndash1

01

y16

0 002 004 006 008 01 012 014 016 018ndash2

02

y17

0 002 004 006 008 01 012 014 016 018ndash1

01

y18

0 002 004 006 008 01 012 014 016 018ndash1

01

y19

0 002 004 006 008 01 012 014 016 018ndash1

01

y20

Time (s) Time (s)

Figure 22 FDM decomposition result

16 Mathematical Problems in Engineering

Tabl

e8

+ecorrelationcoeffi

cientandku

rtosisbetweentheFIBF

andtheoriginal

signal

y 1y 2

y 3y 4

y 5y 6

y 7y 8

y 9y 1

0y 1

1y 1

2y 1

3y 1

4y 1

5y 1

6y 1

7y 1

8y 1

9Y 2

0y 2

1y 2

2y 2

3y 2

4y 2

5

C(t)

0018

0091

0074

0097

0132

0116

0125

0165

0217

0217

0260

0523

0349

0269

0277

0186

0214

0163

0159

0182

0086

0139

0064

0034

0000

K(t)

1000

2203

2209

2266

3145

4288

2564

4142

2750

2872

2683

4184

2981

2995

3277

3013

3385

2652

3151

2489

2652

4111

2615

2652

1000

Mathematical Problems in Engineering 17

0 002 004 006 008 01 012 014 016 018ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 002 004 006 008 01 012 014 016 018ndash5

0

5times10ndash3

Time (s)

(b)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 ) f 0 2f 0

3f 0

(a)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

2f 0

f 0

(b)

0 200 400 600 800 1000 1200ndash500

0

500

1000

Number (n)

Am

plitu

de (m

s2 )

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

(a)

0 200 400 600 800 10000

20

40

60

80

100

Frequency (Hz)

f 0 3f 0 4f 07f 05f 0

6f 0 8f 0 9f 0

2f 0

Am

plitu

de (m

s2 )

(b)

18 Mathematical Problems in Engineering

0 002 004 006 008 01 012 014 016 018ndash05

0

05

Time (s)

Am

plitu

de (m

s2 )

(a)

0 1000 2000 3000 4000 5000 60000

002

004

006

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

0 002 004 006 008 01 012 014 016 018ndash1

0

1

Time (s)

Am

plitu

de (m

s2 )

(a)

0 1000 2000 3000 4000 5000 60000

002

004

006

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

0 002 004 006 008 01 012 014 016 018ndash2

02

times10ndash3

y1

0 002 004 006 008 01 012 014 016 018ndash005

0005

y2

0 002 004 006 008 01 012 014 016 018ndash01

001

y3

0 002 004 006 008 01 012 014 016 018ndash01

001

y4

0 002 004 006 008 01 012 014 016 018ndash02

002

y5

0 002 004 006 008 01 012 014 016 018ndash02

002

y6

0 002 004 006 008 01 012 014 016 018ndash02

002

y7

0 002 004 006 008 01 012 014 016 018ndash02

002

y8

0 002 004 006 008 01 012 014 016 018ndash05

005

y9

0 002 004 006 008 01 012 014 016 018ndash05

005

y10

0 002 004 006 008 01 012 014 016 018ndash05

005

y11

0 002 004 006 008 01 012 014 016 018ndash05

005

y12

0 002 004 006 008 01 012 014 016 018ndash05

005

y13

0 002 004 006 008 01 012 014 016 018ndash05

005

y14

0 002 004 006 008 01 012 014 016 018ndash02

002

y15

0 002 004 006 008 01 012 014 016 018ndash02

002

y16

0 002 004 006 008 01 012 014 016 018ndash02

002

y7

0 002 004 006 008 01 012 014 016 018ndash02

002

y18

0 002 004 006 008 01 012 014 016 018ndash01

001

y19

0 002 004 006 008 01 012 014 016 018ndash01

001

y20

Time (s) Time (s)

Figure 28 FDM decomposition result

Mathematical Problems in Engineering 19

Tabl

e9

+ecorrelationcoeffi

cientandku

rtosisbetweentheFIBF

andtheoriginal

signal

y 1y 2

y 3y 4

y 5y 6

y 7y 8

y 9y 1

0y 1

1y 1

2y 1

3y 1

4y 1

5y 1

6y 1

7y 1

8y 1

9y 2

0y 2

1y 2

2y 2

3y 2

4y 2

5

C(t)

0004

0054

0126

0116

0120

0171

0196

0173

0306

0296

0490

0379

0219

0254

0206

0181

0209

0154

0113

0107

0051

0088

0017

0025

0000

K(t)

1000

2071

2276

2456

3148

2712

3194

3435

3276

2962

2716

2684

3304

2640

2710

2783

2614

2591

3079

2435

2673

2646

2107

1500

1000

20 Mathematical Problems in Engineering

further extracted by MOMEDA filtering and the extractionresults are shown in Figure 31

0 002 004 006 008 01 012 014 016 018ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

0 002 004 006 008 01 012 014 016 018ndash04

ndash02

0

02

04

Time (s)

Am

plitu

de (m

s2 )

(b)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

f0 2f0 4f0

(a)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

0 200 400 600 800 1000 1200ndash500

0

500

1000

Am

plitu

de (m

s2 )

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

Number (n)

(a)

0 200 400 600 800 10000

20

40

60

80

100

Frequency (Hz)

Am

plitu

de (m

s2 )

f0 2f0 3f0

4f05f 0 6f0 7f0

(b)

Mathematical Problems in Engineering 21

6 Conclusions

Data Availability

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

References

22 Mathematical Problems in Engineering

Mathematical Problems in Engineering 23

Am

plitu

de (m

s2 )

Time (s)0 001 002 003 004 005 006 007 008 009

ndash4

ndash2

0

2

4

(a)

Am

plitu

de (m

s2 )

Time (s)0 001 002 003 004 005 006 007 008 009

ndash4

ndash2

0

2

4

(b)

0 001 002 003 004 005 006 007 008 009

ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 001 002 003 004 005 006 007 008 009ndash4

ndash2

0

2

4

Time (s)

(b)

Table 4 Comparison of noise reduction results of different methods

Evaluation index FDM-RobustICA LMD-RobustICA ITD-RobustICA

Kurtosis and correlation coefficient 39070582

36020374

36300480

Noise reduction time 1378 00889 0103

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

4f 0 5f 03f 0

2f 0f 0

(a)

Am

plitu

de (m

s2 )

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

(b)

Mathematical Problems in Engineering 9

5 Application to Roller Bearing Testing

Outer race defect frequency

BPFO Z

21 minus

d

Dcos α1113888 1113889 times fr (24)

Inner race defect frequency

BPFI Z

21 +

d

Dcos α1113888 1113889 times fr (25)

Rolling element frequency

BPFR D

2d1 minus

d

D1113888 1113889

2

cos2α⎛⎝ ⎞⎠ times fr (26)

Am

plitu

de (m

s2 )

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

2f 0 4f 0f 0

(a)

Am

plitu

de (m

s2 )

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

2f 0 4f 03f 0f 0

(b)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

f 03f 0 5f 0

4f 02f 0

(a)

Am

plitu

de (m

s2 )

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

3f 02f 0f 0

(b)

10 Mathematical Problems in Engineering

Am

plitu

de (m

s2 )

Am

plitu

de (m

s2 )

0 200 400 600 800 1000 1200ndash500

0

500

1000

Number (n)

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

(a)

0 200 400 600 800 10000

5

10

15

20

Frequency (Hz)

3f 05f 0

6f 0

7f 0 8f 0 9f 0

2f 0

4f 0

Am

plitu

de (m

s2 )

f 0

(b)

Table 5 +e bearing technical parameters of SKF 6205

Rolling elementnumber (Z) Inner diameter (inches) Outer diameter

(inches)Rolling element

diameter d (inches) Contact angle (α) Pitch circlediameter D (inches)

Speed(rpm)

9 09843 20472 03126 00 15327 1797

Table 6 Fault characteristic frequency (unit Hz)

Inner ring fault Outer ring fault Rolling element fault1621852 1073648 1411693

0 002 004 006 008 01 012 014 016 018ndash2

ndash1

0

1

2

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

005

01

Frequency (Hz)

(b)

Mathematical Problems in Engineering 11

0 002 004 006 008 01 012 014 016 018ndash2

ndash1

0

1

2

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

005

01

Frequency (Hz)

(b)

0 002 004 006 008 01 012 014 016 018ndash5

05

times10ndash3

y1

0 002 004 006 008 01 012 014 016 018ndash005

0005

y2

0 002 004 006 008 01 012 014 016 018ndash005

0005

y3

0 002 004 006 008 01 012 014 016 018ndash01

001

y4

0 002 004 006 008 01 012 014 016 018ndash02

002

y5

0 002 004 006 008 01 012 014 016 018ndash02

002

y6

0 002 004 006 008 01 012 014 016 018ndash05

005

y7

0 002 004 006 008 01 012 014 016 018ndash05

005

y8

0 002 004 006 008 01 012 014 016 018ndash05

005

y9

0 002 004 006 008 01 012 014 016 018ndash1

01

y10

0 002 004 006 008 01 012 014 016 018ndash1

01

y11

0 002 004 006 008 01 012 014 016 018ndash1

01

y12

0 002 004 006 008 01 012 014 016 018ndash05

005

y13

0 002 004 006 008 01 012 014 016 018ndash05

005

y14

0 002 004 006 008 01 012 014 016 018ndash05

005

y15

0 002 004 006 008 01 012 014 016 018ndash02

002

y16

0 002 004 006 008 01 012 014 016 018ndash05

005

y17

0 002 004 006 008 01 012 014 016 018ndash02

002

y18

0 002 004 006 008 01 012 014 016 018ndash02

002

y19

0 002 004 006 008 01 012 014 016 018ndash01

001

y20

Time (s) Time (s)

Figure 16 FDM decomposition result

12 Mathematical Problems in Engineering

Tabl

e7

+ecorrelationcoeffi

cientandku

rtosisbetweentheFIBF

andtheoriginal

signal

y 1y 2

y 3y 4

y 5y 6

y 7y 8

y 9y 1

0y 1

1y 1

2y 1

3y 1

4y 1

5y 1

6y 1

7y 1

8y 1

9y 2

0y 2

1y 2

2y 2

3y 2

4y 2

5

C(t)

0008

0041

0028

0082

0115

0149

0181

0211

0259

0468

0430

0360

0215

0270

0234

0143

0202

0117

0114

0082

0060

0036

0059

0026

000

K(t)

1000

1958

1928

3032

2501

2708

2375

2470

2847

4302

3221

3183

2793

3417

3383

2424

2881

2620

3039

2866

2444

2743

2073

1500

1000

Mathematical Problems in Engineering 13

the interference of noise it is difficult to extract the faultfeature frequency

0 002 004 006 008 01 012 014 016 018ndash5

0

5

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 002 004 006 008 01 012 014 016 018ndash5

0

5

Time (s)

(b)

Am

plitu

de (m

s2 )

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

2f 0

3f 0

5f 0 6f 0

f 0

4f 0

(a)

Am

plitu

de (m

s2 )

2f 0 5f 0

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

f 0

(b)

14 Mathematical Problems in Engineering

0 200 400 600 800 1000 1200ndash500

0

500

1000

1500A

mpl

itude

(ms

2 )A

mpl

itude

(ms

2 )

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

Number (n)

(a)

Am

plitu

de (m

s2 )

0 200 400 600 800 10000

20

40

60

80

100

Frequency (Hz)

1f 0

4f 06f 0

2f 0 3f 0

5f 0

(b)

0 002 004 006 008 01 012 014 016 018ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

02

04

Frequency (Hz)

(b)

Am

plitu

de (m

s2 )

0 002 004 006 008 01 012 014 016 018ndash5

0

5

Time (s)

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

01

02

03

04

Frequency (Hz)

(b)

Mathematical Problems in Engineering 15

0 002 004 006 008 01 012 014 016 018ndash005

0005

y1

0 002 004 006 008 01 012 014 016 018ndash05

005

y2

0 002 004 006 008 01 012 014 016 018ndash05

005

y3

0 002 004 006 008 01 012 014 016 018ndash05

005

y4

0 002 004 006 008 01 012 014 016 018ndash1

01

y5

0 002 004 006 008 01 012 014 016 018ndash1

01

y6

0 002 004 006 008 01 012 014 016 018ndash05

005

y7

0 002 004 006 008 01 012 014 016 018ndash1

01

y8

0 002 004 006 008 01 012 014 016 018ndash1

01

y9

0 002 004 006 008 01 012 014 016 018ndash2

02

y10

0 002 004 006 008 01 012 014 016 018ndash2

02

y11

0 002 004 006 008 01 012 014 016 018ndash5

05

y12

0 002 004 006 008 01 012 014 016 018ndash2

02

y13

0 002 004 006 008 01 012 014 016 018ndash2

02

y14

0 002 004 006 008 01 012 014 016 018ndash2

02

y15

0 002 004 006 008 01 012 014 016 018ndash1

01

y16

0 002 004 006 008 01 012 014 016 018ndash2

02

y17

0 002 004 006 008 01 012 014 016 018ndash1

01

y18

0 002 004 006 008 01 012 014 016 018ndash1

01

y19

0 002 004 006 008 01 012 014 016 018ndash1

01

y20

Time (s) Time (s)

Figure 22 FDM decomposition result

16 Mathematical Problems in Engineering

Tabl

e8

+ecorrelationcoeffi

cientandku

rtosisbetweentheFIBF

andtheoriginal

signal

y 1y 2

y 3y 4

y 5y 6

y 7y 8

y 9y 1

0y 1

1y 1

2y 1

3y 1

4y 1

5y 1

6y 1

7y 1

8y 1

9Y 2

0y 2

1y 2

2y 2

3y 2

4y 2

5

C(t)

0018

0091

0074

0097

0132

0116

0125

0165

0217

0217

0260

0523

0349

0269

0277

0186

0214

0163

0159

0182

0086

0139

0064

0034

0000

K(t)

1000

2203

2209

2266

3145

4288

2564

4142

2750

2872

2683

4184

2981

2995

3277

3013

3385

2652

3151

2489

2652

4111

2615

2652

1000

Mathematical Problems in Engineering 17

0 002 004 006 008 01 012 014 016 018ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 002 004 006 008 01 012 014 016 018ndash5

0

5times10ndash3

Time (s)

(b)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 ) f 0 2f 0

3f 0

(a)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

2f 0

f 0

(b)

0 200 400 600 800 1000 1200ndash500

0

500

1000

Number (n)

Am

plitu

de (m

s2 )

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

(a)

0 200 400 600 800 10000

20

40

60

80

100

Frequency (Hz)

f 0 3f 0 4f 07f 05f 0

6f 0 8f 0 9f 0

2f 0

Am

plitu

de (m

s2 )

(b)

18 Mathematical Problems in Engineering

0 002 004 006 008 01 012 014 016 018ndash05

0

05

Time (s)

Am

plitu

de (m

s2 )

(a)

0 1000 2000 3000 4000 5000 60000

002

004

006

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

0 002 004 006 008 01 012 014 016 018ndash1

0

1

Time (s)

Am

plitu

de (m

s2 )

(a)

0 1000 2000 3000 4000 5000 60000

002

004

006

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

0 002 004 006 008 01 012 014 016 018ndash2

02

times10ndash3

y1

0 002 004 006 008 01 012 014 016 018ndash005

0005

y2

0 002 004 006 008 01 012 014 016 018ndash01

001

y3

0 002 004 006 008 01 012 014 016 018ndash01

001

y4

0 002 004 006 008 01 012 014 016 018ndash02

002

y5

0 002 004 006 008 01 012 014 016 018ndash02

002

y6

0 002 004 006 008 01 012 014 016 018ndash02

002

y7

0 002 004 006 008 01 012 014 016 018ndash02

002

y8

0 002 004 006 008 01 012 014 016 018ndash05

005

y9

0 002 004 006 008 01 012 014 016 018ndash05

005

y10

0 002 004 006 008 01 012 014 016 018ndash05

005

y11

0 002 004 006 008 01 012 014 016 018ndash05

005

y12

0 002 004 006 008 01 012 014 016 018ndash05

005

y13

0 002 004 006 008 01 012 014 016 018ndash05

005

y14

0 002 004 006 008 01 012 014 016 018ndash02

002

y15

0 002 004 006 008 01 012 014 016 018ndash02

002

y16

0 002 004 006 008 01 012 014 016 018ndash02

002

y7

0 002 004 006 008 01 012 014 016 018ndash02

002

y18

0 002 004 006 008 01 012 014 016 018ndash01

001

y19

0 002 004 006 008 01 012 014 016 018ndash01

001

y20

Time (s) Time (s)

Figure 28 FDM decomposition result

Mathematical Problems in Engineering 19

Tabl

e9

+ecorrelationcoeffi

cientandku

rtosisbetweentheFIBF

andtheoriginal

signal

y 1y 2

y 3y 4

y 5y 6

y 7y 8

y 9y 1

0y 1

1y 1

2y 1

3y 1

4y 1

5y 1

6y 1

7y 1

8y 1

9y 2

0y 2

1y 2

2y 2

3y 2

4y 2

5

C(t)

0004

0054

0126

0116

0120

0171

0196

0173

0306

0296

0490

0379

0219

0254

0206

0181

0209

0154

0113

0107

0051

0088

0017

0025

0000

K(t)

1000

2071

2276

2456

3148

2712

3194

3435

3276

2962

2716

2684

3304

2640

2710

2783

2614

2591

3079

2435

2673

2646

2107

1500

1000

20 Mathematical Problems in Engineering

further extracted by MOMEDA filtering and the extractionresults are shown in Figure 31

0 002 004 006 008 01 012 014 016 018ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

0 002 004 006 008 01 012 014 016 018ndash04

ndash02

0

02

04

Time (s)

Am

plitu

de (m

s2 )

(b)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

f0 2f0 4f0

(a)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

0 200 400 600 800 1000 1200ndash500

0

500

1000

Am

plitu

de (m

s2 )

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

Number (n)

(a)

0 200 400 600 800 10000

20

40

60

80

100

Frequency (Hz)

Am

plitu

de (m

s2 )

f0 2f0 3f0

4f05f 0 6f0 7f0

(b)

Mathematical Problems in Engineering 21

6 Conclusions

Data Availability

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

References

22 Mathematical Problems in Engineering

Mathematical Problems in Engineering 23

5 Application to Roller Bearing Testing

Outer race defect frequency

BPFO Z

21 minus

d

Dcos α1113888 1113889 times fr (24)

Inner race defect frequency

BPFI Z

21 +

d

Dcos α1113888 1113889 times fr (25)

Rolling element frequency

BPFR D

2d1 minus

d

D1113888 1113889

2

cos2α⎛⎝ ⎞⎠ times fr (26)

Am

plitu

de (m

s2 )

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

2f 0 4f 0f 0

(a)

Am

plitu

de (m

s2 )

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

2f 0 4f 03f 0f 0

(b)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

f 03f 0 5f 0

4f 02f 0

(a)

Am

plitu

de (m

s2 )

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

3f 02f 0f 0

(b)

10 Mathematical Problems in Engineering

Am

plitu

de (m

s2 )

Am

plitu

de (m

s2 )

0 200 400 600 800 1000 1200ndash500

0

500

1000

Number (n)

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

(a)

0 200 400 600 800 10000

5

10

15

20

Frequency (Hz)

3f 05f 0

6f 0

7f 0 8f 0 9f 0

2f 0

4f 0

Am

plitu

de (m

s2 )

f 0

(b)

Table 5 +e bearing technical parameters of SKF 6205

Rolling elementnumber (Z) Inner diameter (inches) Outer diameter

(inches)Rolling element

diameter d (inches) Contact angle (α) Pitch circlediameter D (inches)

Speed(rpm)

9 09843 20472 03126 00 15327 1797

Table 6 Fault characteristic frequency (unit Hz)

Inner ring fault Outer ring fault Rolling element fault1621852 1073648 1411693

0 002 004 006 008 01 012 014 016 018ndash2

ndash1

0

1

2

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

005

01

Frequency (Hz)

(b)

Mathematical Problems in Engineering 11

0 002 004 006 008 01 012 014 016 018ndash2

ndash1

0

1

2

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

005

01

Frequency (Hz)

(b)

0 002 004 006 008 01 012 014 016 018ndash5

05

times10ndash3

y1

0 002 004 006 008 01 012 014 016 018ndash005

0005

y2

0 002 004 006 008 01 012 014 016 018ndash005

0005

y3

0 002 004 006 008 01 012 014 016 018ndash01

001

y4

0 002 004 006 008 01 012 014 016 018ndash02

002

y5

0 002 004 006 008 01 012 014 016 018ndash02

002

y6

0 002 004 006 008 01 012 014 016 018ndash05

005

y7

0 002 004 006 008 01 012 014 016 018ndash05

005

y8

0 002 004 006 008 01 012 014 016 018ndash05

005

y9

0 002 004 006 008 01 012 014 016 018ndash1

01

y10

0 002 004 006 008 01 012 014 016 018ndash1

01

y11

0 002 004 006 008 01 012 014 016 018ndash1

01

y12

0 002 004 006 008 01 012 014 016 018ndash05

005

y13

0 002 004 006 008 01 012 014 016 018ndash05

005

y14

0 002 004 006 008 01 012 014 016 018ndash05

005

y15

0 002 004 006 008 01 012 014 016 018ndash02

002

y16

0 002 004 006 008 01 012 014 016 018ndash05

005

y17

0 002 004 006 008 01 012 014 016 018ndash02

002

y18

0 002 004 006 008 01 012 014 016 018ndash02

002

y19

0 002 004 006 008 01 012 014 016 018ndash01

001

y20

Time (s) Time (s)

Figure 16 FDM decomposition result

12 Mathematical Problems in Engineering

Tabl

e7

+ecorrelationcoeffi

cientandku

rtosisbetweentheFIBF

andtheoriginal

signal

y 1y 2

y 3y 4

y 5y 6

y 7y 8

y 9y 1

0y 1

1y 1

2y 1

3y 1

4y 1

5y 1

6y 1

7y 1

8y 1

9y 2

0y 2

1y 2

2y 2

3y 2

4y 2

5

C(t)

0008

0041

0028

0082

0115

0149

0181

0211

0259

0468

0430

0360

0215

0270

0234

0143

0202

0117

0114

0082

0060

0036

0059

0026

000

K(t)

1000

1958

1928

3032

2501

2708

2375

2470

2847

4302

3221

3183

2793

3417

3383

2424

2881

2620

3039

2866

2444

2743

2073

1500

1000

Mathematical Problems in Engineering 13

the interference of noise it is difficult to extract the faultfeature frequency

0 002 004 006 008 01 012 014 016 018ndash5

0

5

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 002 004 006 008 01 012 014 016 018ndash5

0

5

Time (s)

(b)

Am

plitu

de (m

s2 )

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

2f 0

3f 0

5f 0 6f 0

f 0

4f 0

(a)

Am

plitu

de (m

s2 )

2f 0 5f 0

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

f 0

(b)

14 Mathematical Problems in Engineering

0 200 400 600 800 1000 1200ndash500

0

500

1000

1500A

mpl

itude

(ms

2 )A

mpl

itude

(ms

2 )

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

Number (n)

(a)

Am

plitu

de (m

s2 )

0 200 400 600 800 10000

20

40

60

80

100

Frequency (Hz)

1f 0

4f 06f 0

2f 0 3f 0

5f 0

(b)

0 002 004 006 008 01 012 014 016 018ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

02

04

Frequency (Hz)

(b)

Am

plitu

de (m

s2 )

0 002 004 006 008 01 012 014 016 018ndash5

0

5

Time (s)

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

01

02

03

04

Frequency (Hz)

(b)

Mathematical Problems in Engineering 15

0 002 004 006 008 01 012 014 016 018ndash005

0005

y1

0 002 004 006 008 01 012 014 016 018ndash05

005

y2

0 002 004 006 008 01 012 014 016 018ndash05

005

y3

0 002 004 006 008 01 012 014 016 018ndash05

005

y4

0 002 004 006 008 01 012 014 016 018ndash1

01

y5

0 002 004 006 008 01 012 014 016 018ndash1

01

y6

0 002 004 006 008 01 012 014 016 018ndash05

005

y7

0 002 004 006 008 01 012 014 016 018ndash1

01

y8

0 002 004 006 008 01 012 014 016 018ndash1

01

y9

0 002 004 006 008 01 012 014 016 018ndash2

02

y10

0 002 004 006 008 01 012 014 016 018ndash2

02

y11

0 002 004 006 008 01 012 014 016 018ndash5

05

y12

0 002 004 006 008 01 012 014 016 018ndash2

02

y13

0 002 004 006 008 01 012 014 016 018ndash2

02

y14

0 002 004 006 008 01 012 014 016 018ndash2

02

y15

0 002 004 006 008 01 012 014 016 018ndash1

01

y16

0 002 004 006 008 01 012 014 016 018ndash2

02

y17

0 002 004 006 008 01 012 014 016 018ndash1

01

y18

0 002 004 006 008 01 012 014 016 018ndash1

01

y19

0 002 004 006 008 01 012 014 016 018ndash1

01

y20

Time (s) Time (s)

Figure 22 FDM decomposition result

16 Mathematical Problems in Engineering

Tabl

e8

+ecorrelationcoeffi

cientandku

rtosisbetweentheFIBF

andtheoriginal

signal

y 1y 2

y 3y 4

y 5y 6

y 7y 8

y 9y 1

0y 1

1y 1

2y 1

3y 1

4y 1

5y 1

6y 1

7y 1

8y 1

9Y 2

0y 2

1y 2

2y 2

3y 2

4y 2

5

C(t)

0018

0091

0074

0097

0132

0116

0125

0165

0217

0217

0260

0523

0349

0269

0277

0186

0214

0163

0159

0182

0086

0139

0064

0034

0000

K(t)

1000

2203

2209

2266

3145

4288

2564

4142

2750

2872

2683

4184

2981

2995

3277

3013

3385

2652

3151

2489

2652

4111

2615

2652

1000

Mathematical Problems in Engineering 17

0 002 004 006 008 01 012 014 016 018ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 002 004 006 008 01 012 014 016 018ndash5

0

5times10ndash3

Time (s)

(b)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 ) f 0 2f 0

3f 0

(a)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

2f 0

f 0

(b)

0 200 400 600 800 1000 1200ndash500

0

500

1000

Number (n)

Am

plitu

de (m

s2 )

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

(a)

0 200 400 600 800 10000

20

40

60

80

100

Frequency (Hz)

f 0 3f 0 4f 07f 05f 0

6f 0 8f 0 9f 0

2f 0

Am

plitu

de (m

s2 )

(b)

18 Mathematical Problems in Engineering

0 002 004 006 008 01 012 014 016 018ndash05

0

05

Time (s)

Am

plitu

de (m

s2 )

(a)

0 1000 2000 3000 4000 5000 60000

002

004

006

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

0 002 004 006 008 01 012 014 016 018ndash1

0

1

Time (s)

Am

plitu

de (m

s2 )

(a)

0 1000 2000 3000 4000 5000 60000

002

004

006

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

0 002 004 006 008 01 012 014 016 018ndash2

02

times10ndash3

y1

0 002 004 006 008 01 012 014 016 018ndash005

0005

y2

0 002 004 006 008 01 012 014 016 018ndash01

001

y3

0 002 004 006 008 01 012 014 016 018ndash01

001

y4

0 002 004 006 008 01 012 014 016 018ndash02

002

y5

0 002 004 006 008 01 012 014 016 018ndash02

002

y6

0 002 004 006 008 01 012 014 016 018ndash02

002

y7

0 002 004 006 008 01 012 014 016 018ndash02

002

y8

0 002 004 006 008 01 012 014 016 018ndash05

005

y9

0 002 004 006 008 01 012 014 016 018ndash05

005

y10

0 002 004 006 008 01 012 014 016 018ndash05

005

y11

0 002 004 006 008 01 012 014 016 018ndash05

005

y12

0 002 004 006 008 01 012 014 016 018ndash05

005

y13

0 002 004 006 008 01 012 014 016 018ndash05

005

y14

0 002 004 006 008 01 012 014 016 018ndash02

002

y15

0 002 004 006 008 01 012 014 016 018ndash02

002

y16

0 002 004 006 008 01 012 014 016 018ndash02

002

y7

0 002 004 006 008 01 012 014 016 018ndash02

002

y18

0 002 004 006 008 01 012 014 016 018ndash01

001

y19

0 002 004 006 008 01 012 014 016 018ndash01

001

y20

Time (s) Time (s)

Figure 28 FDM decomposition result

Mathematical Problems in Engineering 19

Tabl

e9

+ecorrelationcoeffi

cientandku

rtosisbetweentheFIBF

andtheoriginal

signal

y 1y 2

y 3y 4

y 5y 6

y 7y 8

y 9y 1

0y 1

1y 1

2y 1

3y 1

4y 1

5y 1

6y 1

7y 1

8y 1

9y 2

0y 2

1y 2

2y 2

3y 2

4y 2

5

C(t)

0004

0054

0126

0116

0120

0171

0196

0173

0306

0296

0490

0379

0219

0254

0206

0181

0209

0154

0113

0107

0051

0088

0017

0025

0000

K(t)

1000

2071

2276

2456

3148

2712

3194

3435

3276

2962

2716

2684

3304

2640

2710

2783

2614

2591

3079

2435

2673

2646

2107

1500

1000

20 Mathematical Problems in Engineering

further extracted by MOMEDA filtering and the extractionresults are shown in Figure 31

0 002 004 006 008 01 012 014 016 018ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

0 002 004 006 008 01 012 014 016 018ndash04

ndash02

0

02

04

Time (s)

Am

plitu

de (m

s2 )

(b)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

f0 2f0 4f0

(a)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

0 200 400 600 800 1000 1200ndash500

0

500

1000

Am

plitu

de (m

s2 )

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

Number (n)

(a)

0 200 400 600 800 10000

20

40

60

80

100

Frequency (Hz)

Am

plitu

de (m

s2 )

f0 2f0 3f0

4f05f 0 6f0 7f0

(b)

Mathematical Problems in Engineering 21

6 Conclusions

Data Availability

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

References

22 Mathematical Problems in Engineering

Mathematical Problems in Engineering 23

Am

plitu

de (m

s2 )

Am

plitu

de (m

s2 )

0 200 400 600 800 1000 1200ndash500

0

500

1000

Number (n)

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

(a)

0 200 400 600 800 10000

5

10

15

20

Frequency (Hz)

3f 05f 0

6f 0

7f 0 8f 0 9f 0

2f 0

4f 0

Am

plitu

de (m

s2 )

f 0

(b)

Table 5 +e bearing technical parameters of SKF 6205

Rolling elementnumber (Z) Inner diameter (inches) Outer diameter

(inches)Rolling element

diameter d (inches) Contact angle (α) Pitch circlediameter D (inches)

Speed(rpm)

9 09843 20472 03126 00 15327 1797

Table 6 Fault characteristic frequency (unit Hz)

Inner ring fault Outer ring fault Rolling element fault1621852 1073648 1411693

0 002 004 006 008 01 012 014 016 018ndash2

ndash1

0

1

2

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

005

01

Frequency (Hz)

(b)

Mathematical Problems in Engineering 11

0 002 004 006 008 01 012 014 016 018ndash2

ndash1

0

1

2

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

005

01

Frequency (Hz)

(b)

0 002 004 006 008 01 012 014 016 018ndash5

05

times10ndash3

y1

0 002 004 006 008 01 012 014 016 018ndash005

0005

y2

0 002 004 006 008 01 012 014 016 018ndash005

0005

y3

0 002 004 006 008 01 012 014 016 018ndash01

001

y4

0 002 004 006 008 01 012 014 016 018ndash02

002

y5

0 002 004 006 008 01 012 014 016 018ndash02

002

y6

0 002 004 006 008 01 012 014 016 018ndash05

005

y7

0 002 004 006 008 01 012 014 016 018ndash05

005

y8

0 002 004 006 008 01 012 014 016 018ndash05

005

y9

0 002 004 006 008 01 012 014 016 018ndash1

01

y10

0 002 004 006 008 01 012 014 016 018ndash1

01

y11

0 002 004 006 008 01 012 014 016 018ndash1

01

y12

0 002 004 006 008 01 012 014 016 018ndash05

005

y13

0 002 004 006 008 01 012 014 016 018ndash05

005

y14

0 002 004 006 008 01 012 014 016 018ndash05

005

y15

0 002 004 006 008 01 012 014 016 018ndash02

002

y16

0 002 004 006 008 01 012 014 016 018ndash05

005

y17

0 002 004 006 008 01 012 014 016 018ndash02

002

y18

0 002 004 006 008 01 012 014 016 018ndash02

002

y19

0 002 004 006 008 01 012 014 016 018ndash01

001

y20

Time (s) Time (s)

Figure 16 FDM decomposition result

12 Mathematical Problems in Engineering

Tabl

e7

+ecorrelationcoeffi

cientandku

rtosisbetweentheFIBF

andtheoriginal

signal

y 1y 2

y 3y 4

y 5y 6

y 7y 8

y 9y 1

0y 1

1y 1

2y 1

3y 1

4y 1

5y 1

6y 1

7y 1

8y 1

9y 2

0y 2

1y 2

2y 2

3y 2

4y 2

5

C(t)

0008

0041

0028

0082

0115

0149

0181

0211

0259

0468

0430

0360

0215

0270

0234

0143

0202

0117

0114

0082

0060

0036

0059

0026

000

K(t)

1000

1958

1928

3032

2501

2708

2375

2470

2847

4302

3221

3183

2793

3417

3383

2424

2881

2620

3039

2866

2444

2743

2073

1500

1000

Mathematical Problems in Engineering 13

the interference of noise it is difficult to extract the faultfeature frequency

0 002 004 006 008 01 012 014 016 018ndash5

0

5

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 002 004 006 008 01 012 014 016 018ndash5

0

5

Time (s)

(b)

Am

plitu

de (m

s2 )

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

2f 0

3f 0

5f 0 6f 0

f 0

4f 0

(a)

Am

plitu

de (m

s2 )

2f 0 5f 0

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

f 0

(b)

14 Mathematical Problems in Engineering

0 200 400 600 800 1000 1200ndash500

0

500

1000

1500A

mpl

itude

(ms

2 )A

mpl

itude

(ms

2 )

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

Number (n)

(a)

Am

plitu

de (m

s2 )

0 200 400 600 800 10000

20

40

60

80

100

Frequency (Hz)

1f 0

4f 06f 0

2f 0 3f 0

5f 0

(b)

0 002 004 006 008 01 012 014 016 018ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

02

04

Frequency (Hz)

(b)

Am

plitu

de (m

s2 )

0 002 004 006 008 01 012 014 016 018ndash5

0

5

Time (s)

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

01

02

03

04

Frequency (Hz)

(b)

Mathematical Problems in Engineering 15

0 002 004 006 008 01 012 014 016 018ndash005

0005

y1

0 002 004 006 008 01 012 014 016 018ndash05

005

y2

0 002 004 006 008 01 012 014 016 018ndash05

005

y3

0 002 004 006 008 01 012 014 016 018ndash05

005

y4

0 002 004 006 008 01 012 014 016 018ndash1

01

y5

0 002 004 006 008 01 012 014 016 018ndash1

01

y6

0 002 004 006 008 01 012 014 016 018ndash05

005

y7

0 002 004 006 008 01 012 014 016 018ndash1

01

y8

0 002 004 006 008 01 012 014 016 018ndash1

01

y9

0 002 004 006 008 01 012 014 016 018ndash2

02

y10

0 002 004 006 008 01 012 014 016 018ndash2

02

y11

0 002 004 006 008 01 012 014 016 018ndash5

05

y12

0 002 004 006 008 01 012 014 016 018ndash2

02

y13

0 002 004 006 008 01 012 014 016 018ndash2

02

y14

0 002 004 006 008 01 012 014 016 018ndash2

02

y15

0 002 004 006 008 01 012 014 016 018ndash1

01

y16

0 002 004 006 008 01 012 014 016 018ndash2

02

y17

0 002 004 006 008 01 012 014 016 018ndash1

01

y18

0 002 004 006 008 01 012 014 016 018ndash1

01

y19

0 002 004 006 008 01 012 014 016 018ndash1

01

y20

Time (s) Time (s)

Figure 22 FDM decomposition result

16 Mathematical Problems in Engineering

Tabl

e8

+ecorrelationcoeffi

cientandku

rtosisbetweentheFIBF

andtheoriginal

signal

y 1y 2

y 3y 4

y 5y 6

y 7y 8

y 9y 1

0y 1

1y 1

2y 1

3y 1

4y 1

5y 1

6y 1

7y 1

8y 1

9Y 2

0y 2

1y 2

2y 2

3y 2

4y 2

5

C(t)

0018

0091

0074

0097

0132

0116

0125

0165

0217

0217

0260

0523

0349

0269

0277

0186

0214

0163

0159

0182

0086

0139

0064

0034

0000

K(t)

1000

2203

2209

2266

3145

4288

2564

4142

2750

2872

2683

4184

2981

2995

3277

3013

3385

2652

3151

2489

2652

4111

2615

2652

1000

Mathematical Problems in Engineering 17

0 002 004 006 008 01 012 014 016 018ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 002 004 006 008 01 012 014 016 018ndash5

0

5times10ndash3

Time (s)

(b)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 ) f 0 2f 0

3f 0

(a)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

2f 0

f 0

(b)

0 200 400 600 800 1000 1200ndash500

0

500

1000

Number (n)

Am

plitu

de (m

s2 )

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

(a)

0 200 400 600 800 10000

20

40

60

80

100

Frequency (Hz)

f 0 3f 0 4f 07f 05f 0

6f 0 8f 0 9f 0

2f 0

Am

plitu

de (m

s2 )

(b)

18 Mathematical Problems in Engineering

0 002 004 006 008 01 012 014 016 018ndash05

0

05

Time (s)

Am

plitu

de (m

s2 )

(a)

0 1000 2000 3000 4000 5000 60000

002

004

006

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

0 002 004 006 008 01 012 014 016 018ndash1

0

1

Time (s)

Am

plitu

de (m

s2 )

(a)

0 1000 2000 3000 4000 5000 60000

002

004

006

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

0 002 004 006 008 01 012 014 016 018ndash2

02

times10ndash3

y1

0 002 004 006 008 01 012 014 016 018ndash005

0005

y2

0 002 004 006 008 01 012 014 016 018ndash01

001

y3

0 002 004 006 008 01 012 014 016 018ndash01

001

y4

0 002 004 006 008 01 012 014 016 018ndash02

002

y5

0 002 004 006 008 01 012 014 016 018ndash02

002

y6

0 002 004 006 008 01 012 014 016 018ndash02

002

y7

0 002 004 006 008 01 012 014 016 018ndash02

002

y8

0 002 004 006 008 01 012 014 016 018ndash05

005

y9

0 002 004 006 008 01 012 014 016 018ndash05

005

y10

0 002 004 006 008 01 012 014 016 018ndash05

005

y11

0 002 004 006 008 01 012 014 016 018ndash05

005

y12

0 002 004 006 008 01 012 014 016 018ndash05

005

y13

0 002 004 006 008 01 012 014 016 018ndash05

005

y14

0 002 004 006 008 01 012 014 016 018ndash02

002

y15

0 002 004 006 008 01 012 014 016 018ndash02

002

y16

0 002 004 006 008 01 012 014 016 018ndash02

002

y7

0 002 004 006 008 01 012 014 016 018ndash02

002

y18

0 002 004 006 008 01 012 014 016 018ndash01

001

y19

0 002 004 006 008 01 012 014 016 018ndash01

001

y20

Time (s) Time (s)

Figure 28 FDM decomposition result

Mathematical Problems in Engineering 19

Tabl

e9

+ecorrelationcoeffi

cientandku

rtosisbetweentheFIBF

andtheoriginal

signal

y 1y 2

y 3y 4

y 5y 6

y 7y 8

y 9y 1

0y 1

1y 1

2y 1

3y 1

4y 1

5y 1

6y 1

7y 1

8y 1

9y 2

0y 2

1y 2

2y 2

3y 2

4y 2

5

C(t)

0004

0054

0126

0116

0120

0171

0196

0173

0306

0296

0490

0379

0219

0254

0206

0181

0209

0154

0113

0107

0051

0088

0017

0025

0000

K(t)

1000

2071

2276

2456

3148

2712

3194

3435

3276

2962

2716

2684

3304

2640

2710

2783

2614

2591

3079

2435

2673

2646

2107

1500

1000

20 Mathematical Problems in Engineering

further extracted by MOMEDA filtering and the extractionresults are shown in Figure 31

0 002 004 006 008 01 012 014 016 018ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

0 002 004 006 008 01 012 014 016 018ndash04

ndash02

0

02

04

Time (s)

Am

plitu

de (m

s2 )

(b)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

f0 2f0 4f0

(a)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

0 200 400 600 800 1000 1200ndash500

0

500

1000

Am

plitu

de (m

s2 )

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

Number (n)

(a)

0 200 400 600 800 10000

20

40

60

80

100

Frequency (Hz)

Am

plitu

de (m

s2 )

f0 2f0 3f0

4f05f 0 6f0 7f0

(b)

Mathematical Problems in Engineering 21

6 Conclusions

Data Availability

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

References

22 Mathematical Problems in Engineering

Mathematical Problems in Engineering 23

0 002 004 006 008 01 012 014 016 018ndash2

ndash1

0

1

2

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

005

01

Frequency (Hz)

(b)

0 002 004 006 008 01 012 014 016 018ndash5

05

times10ndash3

y1

0 002 004 006 008 01 012 014 016 018ndash005

0005

y2

0 002 004 006 008 01 012 014 016 018ndash005

0005

y3

0 002 004 006 008 01 012 014 016 018ndash01

001

y4

0 002 004 006 008 01 012 014 016 018ndash02

002

y5

0 002 004 006 008 01 012 014 016 018ndash02

002

y6

0 002 004 006 008 01 012 014 016 018ndash05

005

y7

0 002 004 006 008 01 012 014 016 018ndash05

005

y8

0 002 004 006 008 01 012 014 016 018ndash05

005

y9

0 002 004 006 008 01 012 014 016 018ndash1

01

y10

0 002 004 006 008 01 012 014 016 018ndash1

01

y11

0 002 004 006 008 01 012 014 016 018ndash1

01

y12

0 002 004 006 008 01 012 014 016 018ndash05

005

y13

0 002 004 006 008 01 012 014 016 018ndash05

005

y14

0 002 004 006 008 01 012 014 016 018ndash05

005

y15

0 002 004 006 008 01 012 014 016 018ndash02

002

y16

0 002 004 006 008 01 012 014 016 018ndash05

005

y17

0 002 004 006 008 01 012 014 016 018ndash02

002

y18

0 002 004 006 008 01 012 014 016 018ndash02

002

y19

0 002 004 006 008 01 012 014 016 018ndash01

001

y20

Time (s) Time (s)

Figure 16 FDM decomposition result

12 Mathematical Problems in Engineering

Tabl

e7

+ecorrelationcoeffi

cientandku

rtosisbetweentheFIBF

andtheoriginal

signal

y 1y 2

y 3y 4

y 5y 6

y 7y 8

y 9y 1

0y 1

1y 1

2y 1

3y 1

4y 1

5y 1

6y 1

7y 1

8y 1

9y 2

0y 2

1y 2

2y 2

3y 2

4y 2

5

C(t)

0008

0041

0028

0082

0115

0149

0181

0211

0259

0468

0430

0360

0215

0270

0234

0143

0202

0117

0114

0082

0060

0036

0059

0026

000

K(t)

1000

1958

1928

3032

2501

2708

2375

2470

2847

4302

3221

3183

2793

3417

3383

2424

2881

2620

3039

2866

2444

2743

2073

1500

1000

Mathematical Problems in Engineering 13

the interference of noise it is difficult to extract the faultfeature frequency

0 002 004 006 008 01 012 014 016 018ndash5

0

5

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 002 004 006 008 01 012 014 016 018ndash5

0

5

Time (s)

(b)

Am

plitu

de (m

s2 )

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

2f 0

3f 0

5f 0 6f 0

f 0

4f 0

(a)

Am

plitu

de (m

s2 )

2f 0 5f 0

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

f 0

(b)

14 Mathematical Problems in Engineering

0 200 400 600 800 1000 1200ndash500

0

500

1000

1500A

mpl

itude

(ms

2 )A

mpl

itude

(ms

2 )

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

Number (n)

(a)

Am

plitu

de (m

s2 )

0 200 400 600 800 10000

20

40

60

80

100

Frequency (Hz)

1f 0

4f 06f 0

2f 0 3f 0

5f 0

(b)

0 002 004 006 008 01 012 014 016 018ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

02

04

Frequency (Hz)

(b)

Am

plitu

de (m

s2 )

0 002 004 006 008 01 012 014 016 018ndash5

0

5

Time (s)

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

01

02

03

04

Frequency (Hz)

(b)

Mathematical Problems in Engineering 15

0 002 004 006 008 01 012 014 016 018ndash005

0005

y1

0 002 004 006 008 01 012 014 016 018ndash05

005

y2

0 002 004 006 008 01 012 014 016 018ndash05

005

y3

0 002 004 006 008 01 012 014 016 018ndash05

005

y4

0 002 004 006 008 01 012 014 016 018ndash1

01

y5

0 002 004 006 008 01 012 014 016 018ndash1

01

y6

0 002 004 006 008 01 012 014 016 018ndash05

005

y7

0 002 004 006 008 01 012 014 016 018ndash1

01

y8

0 002 004 006 008 01 012 014 016 018ndash1

01

y9

0 002 004 006 008 01 012 014 016 018ndash2

02

y10

0 002 004 006 008 01 012 014 016 018ndash2

02

y11

0 002 004 006 008 01 012 014 016 018ndash5

05

y12

0 002 004 006 008 01 012 014 016 018ndash2

02

y13

0 002 004 006 008 01 012 014 016 018ndash2

02

y14

0 002 004 006 008 01 012 014 016 018ndash2

02

y15

0 002 004 006 008 01 012 014 016 018ndash1

01

y16

0 002 004 006 008 01 012 014 016 018ndash2

02

y17

0 002 004 006 008 01 012 014 016 018ndash1

01

y18

0 002 004 006 008 01 012 014 016 018ndash1

01

y19

0 002 004 006 008 01 012 014 016 018ndash1

01

y20

Time (s) Time (s)

Figure 22 FDM decomposition result

16 Mathematical Problems in Engineering

Tabl

e8

+ecorrelationcoeffi

cientandku

rtosisbetweentheFIBF

andtheoriginal

signal

y 1y 2

y 3y 4

y 5y 6

y 7y 8

y 9y 1

0y 1

1y 1

2y 1

3y 1

4y 1

5y 1

6y 1

7y 1

8y 1

9Y 2

0y 2

1y 2

2y 2

3y 2

4y 2

5

C(t)

0018

0091

0074

0097

0132

0116

0125

0165

0217

0217

0260

0523

0349

0269

0277

0186

0214

0163

0159

0182

0086

0139

0064

0034

0000

K(t)

1000

2203

2209

2266

3145

4288

2564

4142

2750

2872

2683

4184

2981

2995

3277

3013

3385

2652

3151

2489

2652

4111

2615

2652

1000

Mathematical Problems in Engineering 17

0 002 004 006 008 01 012 014 016 018ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 002 004 006 008 01 012 014 016 018ndash5

0

5times10ndash3

Time (s)

(b)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 ) f 0 2f 0

3f 0

(a)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

2f 0

f 0

(b)

0 200 400 600 800 1000 1200ndash500

0

500

1000

Number (n)

Am

plitu

de (m

s2 )

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

(a)

0 200 400 600 800 10000

20

40

60

80

100

Frequency (Hz)

f 0 3f 0 4f 07f 05f 0

6f 0 8f 0 9f 0

2f 0

Am

plitu

de (m

s2 )

(b)

18 Mathematical Problems in Engineering

0 002 004 006 008 01 012 014 016 018ndash05

0

05

Time (s)

Am

plitu

de (m

s2 )

(a)

0 1000 2000 3000 4000 5000 60000

002

004

006

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

0 002 004 006 008 01 012 014 016 018ndash1

0

1

Time (s)

Am

plitu

de (m

s2 )

(a)

0 1000 2000 3000 4000 5000 60000

002

004

006

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

0 002 004 006 008 01 012 014 016 018ndash2

02

times10ndash3

y1

0 002 004 006 008 01 012 014 016 018ndash005

0005

y2

0 002 004 006 008 01 012 014 016 018ndash01

001

y3

0 002 004 006 008 01 012 014 016 018ndash01

001

y4

0 002 004 006 008 01 012 014 016 018ndash02

002

y5

0 002 004 006 008 01 012 014 016 018ndash02

002

y6

0 002 004 006 008 01 012 014 016 018ndash02

002

y7

0 002 004 006 008 01 012 014 016 018ndash02

002

y8

0 002 004 006 008 01 012 014 016 018ndash05

005

y9

0 002 004 006 008 01 012 014 016 018ndash05

005

y10

0 002 004 006 008 01 012 014 016 018ndash05

005

y11

0 002 004 006 008 01 012 014 016 018ndash05

005

y12

0 002 004 006 008 01 012 014 016 018ndash05

005

y13

0 002 004 006 008 01 012 014 016 018ndash05

005

y14

0 002 004 006 008 01 012 014 016 018ndash02

002

y15

0 002 004 006 008 01 012 014 016 018ndash02

002

y16

0 002 004 006 008 01 012 014 016 018ndash02

002

y7

0 002 004 006 008 01 012 014 016 018ndash02

002

y18

0 002 004 006 008 01 012 014 016 018ndash01

001

y19

0 002 004 006 008 01 012 014 016 018ndash01

001

y20

Time (s) Time (s)

Figure 28 FDM decomposition result

Mathematical Problems in Engineering 19

Tabl

e9

+ecorrelationcoeffi

cientandku

rtosisbetweentheFIBF

andtheoriginal

signal

y 1y 2

y 3y 4

y 5y 6

y 7y 8

y 9y 1

0y 1

1y 1

2y 1

3y 1

4y 1

5y 1

6y 1

7y 1

8y 1

9y 2

0y 2

1y 2

2y 2

3y 2

4y 2

5

C(t)

0004

0054

0126

0116

0120

0171

0196

0173

0306

0296

0490

0379

0219

0254

0206

0181

0209

0154

0113

0107

0051

0088

0017

0025

0000

K(t)

1000

2071

2276

2456

3148

2712

3194

3435

3276

2962

2716

2684

3304

2640

2710

2783

2614

2591

3079

2435

2673

2646

2107

1500

1000

20 Mathematical Problems in Engineering

further extracted by MOMEDA filtering and the extractionresults are shown in Figure 31

0 002 004 006 008 01 012 014 016 018ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

0 002 004 006 008 01 012 014 016 018ndash04

ndash02

0

02

04

Time (s)

Am

plitu

de (m

s2 )

(b)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

f0 2f0 4f0

(a)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

0 200 400 600 800 1000 1200ndash500

0

500

1000

Am

plitu

de (m

s2 )

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

Number (n)

(a)

0 200 400 600 800 10000

20

40

60

80

100

Frequency (Hz)

Am

plitu

de (m

s2 )

f0 2f0 3f0

4f05f 0 6f0 7f0

(b)

Mathematical Problems in Engineering 21

6 Conclusions

Data Availability

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

References

22 Mathematical Problems in Engineering

Mathematical Problems in Engineering 23

Tabl

e7

+ecorrelationcoeffi

cientandku

rtosisbetweentheFIBF

andtheoriginal

signal

y 1y 2

y 3y 4

y 5y 6

y 7y 8

y 9y 1

0y 1

1y 1

2y 1

3y 1

4y 1

5y 1

6y 1

7y 1

8y 1

9y 2

0y 2

1y 2

2y 2

3y 2

4y 2

5

C(t)

0008

0041

0028

0082

0115

0149

0181

0211

0259

0468

0430

0360

0215

0270

0234

0143

0202

0117

0114

0082

0060

0036

0059

0026

000

K(t)

1000

1958

1928

3032

2501

2708

2375

2470

2847

4302

3221

3183

2793

3417

3383

2424

2881

2620

3039

2866

2444

2743

2073

1500

1000

Mathematical Problems in Engineering 13

the interference of noise it is difficult to extract the faultfeature frequency

0 002 004 006 008 01 012 014 016 018ndash5

0

5

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 002 004 006 008 01 012 014 016 018ndash5

0

5

Time (s)

(b)

Am

plitu

de (m

s2 )

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

2f 0

3f 0

5f 0 6f 0

f 0

4f 0

(a)

Am

plitu

de (m

s2 )

2f 0 5f 0

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

f 0

(b)

14 Mathematical Problems in Engineering

0 200 400 600 800 1000 1200ndash500

0

500

1000

1500A

mpl

itude

(ms

2 )A

mpl

itude

(ms

2 )

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

Number (n)

(a)

Am

plitu

de (m

s2 )

0 200 400 600 800 10000

20

40

60

80

100

Frequency (Hz)

1f 0

4f 06f 0

2f 0 3f 0

5f 0

(b)

0 002 004 006 008 01 012 014 016 018ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

02

04

Frequency (Hz)

(b)

Am

plitu

de (m

s2 )

0 002 004 006 008 01 012 014 016 018ndash5

0

5

Time (s)

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

01

02

03

04

Frequency (Hz)

(b)

Mathematical Problems in Engineering 15

0 002 004 006 008 01 012 014 016 018ndash005

0005

y1

0 002 004 006 008 01 012 014 016 018ndash05

005

y2

0 002 004 006 008 01 012 014 016 018ndash05

005

y3

0 002 004 006 008 01 012 014 016 018ndash05

005

y4

0 002 004 006 008 01 012 014 016 018ndash1

01

y5

0 002 004 006 008 01 012 014 016 018ndash1

01

y6

0 002 004 006 008 01 012 014 016 018ndash05

005

y7

0 002 004 006 008 01 012 014 016 018ndash1

01

y8

0 002 004 006 008 01 012 014 016 018ndash1

01

y9

0 002 004 006 008 01 012 014 016 018ndash2

02

y10

0 002 004 006 008 01 012 014 016 018ndash2

02

y11

0 002 004 006 008 01 012 014 016 018ndash5

05

y12

0 002 004 006 008 01 012 014 016 018ndash2

02

y13

0 002 004 006 008 01 012 014 016 018ndash2

02

y14

0 002 004 006 008 01 012 014 016 018ndash2

02

y15

0 002 004 006 008 01 012 014 016 018ndash1

01

y16

0 002 004 006 008 01 012 014 016 018ndash2

02

y17

0 002 004 006 008 01 012 014 016 018ndash1

01

y18

0 002 004 006 008 01 012 014 016 018ndash1

01

y19

0 002 004 006 008 01 012 014 016 018ndash1

01

y20

Time (s) Time (s)

Figure 22 FDM decomposition result

16 Mathematical Problems in Engineering

Tabl

e8

+ecorrelationcoeffi

cientandku

rtosisbetweentheFIBF

andtheoriginal

signal

y 1y 2

y 3y 4

y 5y 6

y 7y 8

y 9y 1

0y 1

1y 1

2y 1

3y 1

4y 1

5y 1

6y 1

7y 1

8y 1

9Y 2

0y 2

1y 2

2y 2

3y 2

4y 2

5

C(t)

0018

0091

0074

0097

0132

0116

0125

0165

0217

0217

0260

0523

0349

0269

0277

0186

0214

0163

0159

0182

0086

0139

0064

0034

0000

K(t)

1000

2203

2209

2266

3145

4288

2564

4142

2750

2872

2683

4184

2981

2995

3277

3013

3385

2652

3151

2489

2652

4111

2615

2652

1000

Mathematical Problems in Engineering 17

0 002 004 006 008 01 012 014 016 018ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 002 004 006 008 01 012 014 016 018ndash5

0

5times10ndash3

Time (s)

(b)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 ) f 0 2f 0

3f 0

(a)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

2f 0

f 0

(b)

0 200 400 600 800 1000 1200ndash500

0

500

1000

Number (n)

Am

plitu

de (m

s2 )

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

(a)

0 200 400 600 800 10000

20

40

60

80

100

Frequency (Hz)

f 0 3f 0 4f 07f 05f 0

6f 0 8f 0 9f 0

2f 0

Am

plitu

de (m

s2 )

(b)

18 Mathematical Problems in Engineering

0 002 004 006 008 01 012 014 016 018ndash05

0

05

Time (s)

Am

plitu

de (m

s2 )

(a)

0 1000 2000 3000 4000 5000 60000

002

004

006

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

0 002 004 006 008 01 012 014 016 018ndash1

0

1

Time (s)

Am

plitu

de (m

s2 )

(a)

0 1000 2000 3000 4000 5000 60000

002

004

006

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

0 002 004 006 008 01 012 014 016 018ndash2

02

times10ndash3

y1

0 002 004 006 008 01 012 014 016 018ndash005

0005

y2

0 002 004 006 008 01 012 014 016 018ndash01

001

y3

0 002 004 006 008 01 012 014 016 018ndash01

001

y4

0 002 004 006 008 01 012 014 016 018ndash02

002

y5

0 002 004 006 008 01 012 014 016 018ndash02

002

y6

0 002 004 006 008 01 012 014 016 018ndash02

002

y7

0 002 004 006 008 01 012 014 016 018ndash02

002

y8

0 002 004 006 008 01 012 014 016 018ndash05

005

y9

0 002 004 006 008 01 012 014 016 018ndash05

005

y10

0 002 004 006 008 01 012 014 016 018ndash05

005

y11

0 002 004 006 008 01 012 014 016 018ndash05

005

y12

0 002 004 006 008 01 012 014 016 018ndash05

005

y13

0 002 004 006 008 01 012 014 016 018ndash05

005

y14

0 002 004 006 008 01 012 014 016 018ndash02

002

y15

0 002 004 006 008 01 012 014 016 018ndash02

002

y16

0 002 004 006 008 01 012 014 016 018ndash02

002

y7

0 002 004 006 008 01 012 014 016 018ndash02

002

y18

0 002 004 006 008 01 012 014 016 018ndash01

001

y19

0 002 004 006 008 01 012 014 016 018ndash01

001

y20

Time (s) Time (s)

Figure 28 FDM decomposition result

Mathematical Problems in Engineering 19

Tabl

e9

+ecorrelationcoeffi

cientandku

rtosisbetweentheFIBF

andtheoriginal

signal

y 1y 2

y 3y 4

y 5y 6

y 7y 8

y 9y 1

0y 1

1y 1

2y 1

3y 1

4y 1

5y 1

6y 1

7y 1

8y 1

9y 2

0y 2

1y 2

2y 2

3y 2

4y 2

5

C(t)

0004

0054

0126

0116

0120

0171

0196

0173

0306

0296

0490

0379

0219

0254

0206

0181

0209

0154

0113

0107

0051

0088

0017

0025

0000

K(t)

1000

2071

2276

2456

3148

2712

3194

3435

3276

2962

2716

2684

3304

2640

2710

2783

2614

2591

3079

2435

2673

2646

2107

1500

1000

20 Mathematical Problems in Engineering

further extracted by MOMEDA filtering and the extractionresults are shown in Figure 31

0 002 004 006 008 01 012 014 016 018ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

0 002 004 006 008 01 012 014 016 018ndash04

ndash02

0

02

04

Time (s)

Am

plitu

de (m

s2 )

(b)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

f0 2f0 4f0

(a)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

0 200 400 600 800 1000 1200ndash500

0

500

1000

Am

plitu

de (m

s2 )

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

Number (n)

(a)

0 200 400 600 800 10000

20

40

60

80

100

Frequency (Hz)

Am

plitu

de (m

s2 )

f0 2f0 3f0

4f05f 0 6f0 7f0

(b)

Mathematical Problems in Engineering 21

6 Conclusions

Data Availability

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

References

22 Mathematical Problems in Engineering

Mathematical Problems in Engineering 23

the interference of noise it is difficult to extract the faultfeature frequency

0 002 004 006 008 01 012 014 016 018ndash5

0

5

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 002 004 006 008 01 012 014 016 018ndash5

0

5

Time (s)

(b)

Am

plitu

de (m

s2 )

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

2f 0

3f 0

5f 0 6f 0

f 0

4f 0

(a)

Am

plitu

de (m

s2 )

2f 0 5f 0

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

f 0

(b)

14 Mathematical Problems in Engineering

0 200 400 600 800 1000 1200ndash500

0

500

1000

1500A

mpl

itude

(ms

2 )A

mpl

itude

(ms

2 )

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

Number (n)

(a)

Am

plitu

de (m

s2 )

0 200 400 600 800 10000

20

40

60

80

100

Frequency (Hz)

1f 0

4f 06f 0

2f 0 3f 0

5f 0

(b)

0 002 004 006 008 01 012 014 016 018ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

02

04

Frequency (Hz)

(b)

Am

plitu

de (m

s2 )

0 002 004 006 008 01 012 014 016 018ndash5

0

5

Time (s)

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

01

02

03

04

Frequency (Hz)

(b)

Mathematical Problems in Engineering 15

0 002 004 006 008 01 012 014 016 018ndash005

0005

y1

0 002 004 006 008 01 012 014 016 018ndash05

005

y2

0 002 004 006 008 01 012 014 016 018ndash05

005

y3

0 002 004 006 008 01 012 014 016 018ndash05

005

y4

0 002 004 006 008 01 012 014 016 018ndash1

01

y5

0 002 004 006 008 01 012 014 016 018ndash1

01

y6

0 002 004 006 008 01 012 014 016 018ndash05

005

y7

0 002 004 006 008 01 012 014 016 018ndash1

01

y8

0 002 004 006 008 01 012 014 016 018ndash1

01

y9

0 002 004 006 008 01 012 014 016 018ndash2

02

y10

0 002 004 006 008 01 012 014 016 018ndash2

02

y11

0 002 004 006 008 01 012 014 016 018ndash5

05

y12

0 002 004 006 008 01 012 014 016 018ndash2

02

y13

0 002 004 006 008 01 012 014 016 018ndash2

02

y14

0 002 004 006 008 01 012 014 016 018ndash2

02

y15

0 002 004 006 008 01 012 014 016 018ndash1

01

y16

0 002 004 006 008 01 012 014 016 018ndash2

02

y17

0 002 004 006 008 01 012 014 016 018ndash1

01

y18

0 002 004 006 008 01 012 014 016 018ndash1

01

y19

0 002 004 006 008 01 012 014 016 018ndash1

01

y20

Time (s) Time (s)

Figure 22 FDM decomposition result

16 Mathematical Problems in Engineering

Tabl

e8

+ecorrelationcoeffi

cientandku

rtosisbetweentheFIBF

andtheoriginal

signal

y 1y 2

y 3y 4

y 5y 6

y 7y 8

y 9y 1

0y 1

1y 1

2y 1

3y 1

4y 1

5y 1

6y 1

7y 1

8y 1

9Y 2

0y 2

1y 2

2y 2

3y 2

4y 2

5

C(t)

0018

0091

0074

0097

0132

0116

0125

0165

0217

0217

0260

0523

0349

0269

0277

0186

0214

0163

0159

0182

0086

0139

0064

0034

0000

K(t)

1000

2203

2209

2266

3145

4288

2564

4142

2750

2872

2683

4184

2981

2995

3277

3013

3385

2652

3151

2489

2652

4111

2615

2652

1000

Mathematical Problems in Engineering 17

0 002 004 006 008 01 012 014 016 018ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 002 004 006 008 01 012 014 016 018ndash5

0

5times10ndash3

Time (s)

(b)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 ) f 0 2f 0

3f 0

(a)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

2f 0

f 0

(b)

0 200 400 600 800 1000 1200ndash500

0

500

1000

Number (n)

Am

plitu

de (m

s2 )

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

(a)

0 200 400 600 800 10000

20

40

60

80

100

Frequency (Hz)

f 0 3f 0 4f 07f 05f 0

6f 0 8f 0 9f 0

2f 0

Am

plitu

de (m

s2 )

(b)

18 Mathematical Problems in Engineering

0 002 004 006 008 01 012 014 016 018ndash05

0

05

Time (s)

Am

plitu

de (m

s2 )

(a)

0 1000 2000 3000 4000 5000 60000

002

004

006

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

0 002 004 006 008 01 012 014 016 018ndash1

0

1

Time (s)

Am

plitu

de (m

s2 )

(a)

0 1000 2000 3000 4000 5000 60000

002

004

006

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

0 002 004 006 008 01 012 014 016 018ndash2

02

times10ndash3

y1

0 002 004 006 008 01 012 014 016 018ndash005

0005

y2

0 002 004 006 008 01 012 014 016 018ndash01

001

y3

0 002 004 006 008 01 012 014 016 018ndash01

001

y4

0 002 004 006 008 01 012 014 016 018ndash02

002

y5

0 002 004 006 008 01 012 014 016 018ndash02

002

y6

0 002 004 006 008 01 012 014 016 018ndash02

002

y7

0 002 004 006 008 01 012 014 016 018ndash02

002

y8

0 002 004 006 008 01 012 014 016 018ndash05

005

y9

0 002 004 006 008 01 012 014 016 018ndash05

005

y10

0 002 004 006 008 01 012 014 016 018ndash05

005

y11

0 002 004 006 008 01 012 014 016 018ndash05

005

y12

0 002 004 006 008 01 012 014 016 018ndash05

005

y13

0 002 004 006 008 01 012 014 016 018ndash05

005

y14

0 002 004 006 008 01 012 014 016 018ndash02

002

y15

0 002 004 006 008 01 012 014 016 018ndash02

002

y16

0 002 004 006 008 01 012 014 016 018ndash02

002

y7

0 002 004 006 008 01 012 014 016 018ndash02

002

y18

0 002 004 006 008 01 012 014 016 018ndash01

001

y19

0 002 004 006 008 01 012 014 016 018ndash01

001

y20

Time (s) Time (s)

Figure 28 FDM decomposition result

Mathematical Problems in Engineering 19

Tabl

e9

+ecorrelationcoeffi

cientandku

rtosisbetweentheFIBF

andtheoriginal

signal

y 1y 2

y 3y 4

y 5y 6

y 7y 8

y 9y 1

0y 1

1y 1

2y 1

3y 1

4y 1

5y 1

6y 1

7y 1

8y 1

9y 2

0y 2

1y 2

2y 2

3y 2

4y 2

5

C(t)

0004

0054

0126

0116

0120

0171

0196

0173

0306

0296

0490

0379

0219

0254

0206

0181

0209

0154

0113

0107

0051

0088

0017

0025

0000

K(t)

1000

2071

2276

2456

3148

2712

3194

3435

3276

2962

2716

2684

3304

2640

2710

2783

2614

2591

3079

2435

2673

2646

2107

1500

1000

20 Mathematical Problems in Engineering

further extracted by MOMEDA filtering and the extractionresults are shown in Figure 31

0 002 004 006 008 01 012 014 016 018ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

0 002 004 006 008 01 012 014 016 018ndash04

ndash02

0

02

04

Time (s)

Am

plitu

de (m

s2 )

(b)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

f0 2f0 4f0

(a)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

0 200 400 600 800 1000 1200ndash500

0

500

1000

Am

plitu

de (m

s2 )

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

Number (n)

(a)

0 200 400 600 800 10000

20

40

60

80

100

Frequency (Hz)

Am

plitu

de (m

s2 )

f0 2f0 3f0

4f05f 0 6f0 7f0

(b)

Mathematical Problems in Engineering 21

6 Conclusions

Data Availability

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

References

22 Mathematical Problems in Engineering

Mathematical Problems in Engineering 23

0 200 400 600 800 1000 1200ndash500

0

500

1000

1500A

mpl

itude

(ms

2 )A

mpl

itude

(ms

2 )

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

Number (n)

(a)

Am

plitu

de (m

s2 )

0 200 400 600 800 10000

20

40

60

80

100

Frequency (Hz)

1f 0

4f 06f 0

2f 0 3f 0

5f 0

(b)

0 002 004 006 008 01 012 014 016 018ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

02

04

Frequency (Hz)

(b)

Am

plitu

de (m

s2 )

0 002 004 006 008 01 012 014 016 018ndash5

0

5

Time (s)

(a)

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

01

02

03

04

Frequency (Hz)

(b)

Mathematical Problems in Engineering 15

0 002 004 006 008 01 012 014 016 018ndash005

0005

y1

0 002 004 006 008 01 012 014 016 018ndash05

005

y2

0 002 004 006 008 01 012 014 016 018ndash05

005

y3

0 002 004 006 008 01 012 014 016 018ndash05

005

y4

0 002 004 006 008 01 012 014 016 018ndash1

01

y5

0 002 004 006 008 01 012 014 016 018ndash1

01

y6

0 002 004 006 008 01 012 014 016 018ndash05

005

y7

0 002 004 006 008 01 012 014 016 018ndash1

01

y8

0 002 004 006 008 01 012 014 016 018ndash1

01

y9

0 002 004 006 008 01 012 014 016 018ndash2

02

y10

0 002 004 006 008 01 012 014 016 018ndash2

02

y11

0 002 004 006 008 01 012 014 016 018ndash5

05

y12

0 002 004 006 008 01 012 014 016 018ndash2

02

y13

0 002 004 006 008 01 012 014 016 018ndash2

02

y14

0 002 004 006 008 01 012 014 016 018ndash2

02

y15

0 002 004 006 008 01 012 014 016 018ndash1

01

y16

0 002 004 006 008 01 012 014 016 018ndash2

02

y17

0 002 004 006 008 01 012 014 016 018ndash1

01

y18

0 002 004 006 008 01 012 014 016 018ndash1

01

y19

0 002 004 006 008 01 012 014 016 018ndash1

01

y20

Time (s) Time (s)

Figure 22 FDM decomposition result

16 Mathematical Problems in Engineering

Tabl

e8

+ecorrelationcoeffi

cientandku

rtosisbetweentheFIBF

andtheoriginal

signal

y 1y 2

y 3y 4

y 5y 6

y 7y 8

y 9y 1

0y 1

1y 1

2y 1

3y 1

4y 1

5y 1

6y 1

7y 1

8y 1

9Y 2

0y 2

1y 2

2y 2

3y 2

4y 2

5

C(t)

0018

0091

0074

0097

0132

0116

0125

0165

0217

0217

0260

0523

0349

0269

0277

0186

0214

0163

0159

0182

0086

0139

0064

0034

0000

K(t)

1000

2203

2209

2266

3145

4288

2564

4142

2750

2872

2683

4184

2981

2995

3277

3013

3385

2652

3151

2489

2652

4111

2615

2652

1000

Mathematical Problems in Engineering 17

0 002 004 006 008 01 012 014 016 018ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 002 004 006 008 01 012 014 016 018ndash5

0

5times10ndash3

Time (s)

(b)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 ) f 0 2f 0

3f 0

(a)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

2f 0

f 0

(b)

0 200 400 600 800 1000 1200ndash500

0

500

1000

Number (n)

Am

plitu

de (m

s2 )

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

(a)

0 200 400 600 800 10000

20

40

60

80

100

Frequency (Hz)

f 0 3f 0 4f 07f 05f 0

6f 0 8f 0 9f 0

2f 0

Am

plitu

de (m

s2 )

(b)

18 Mathematical Problems in Engineering

0 002 004 006 008 01 012 014 016 018ndash05

0

05

Time (s)

Am

plitu

de (m

s2 )

(a)

0 1000 2000 3000 4000 5000 60000

002

004

006

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

0 002 004 006 008 01 012 014 016 018ndash1

0

1

Time (s)

Am

plitu

de (m

s2 )

(a)

0 1000 2000 3000 4000 5000 60000

002

004

006

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

0 002 004 006 008 01 012 014 016 018ndash2

02

times10ndash3

y1

0 002 004 006 008 01 012 014 016 018ndash005

0005

y2

0 002 004 006 008 01 012 014 016 018ndash01

001

y3

0 002 004 006 008 01 012 014 016 018ndash01

001

y4

0 002 004 006 008 01 012 014 016 018ndash02

002

y5

0 002 004 006 008 01 012 014 016 018ndash02

002

y6

0 002 004 006 008 01 012 014 016 018ndash02

002

y7

0 002 004 006 008 01 012 014 016 018ndash02

002

y8

0 002 004 006 008 01 012 014 016 018ndash05

005

y9

0 002 004 006 008 01 012 014 016 018ndash05

005

y10

0 002 004 006 008 01 012 014 016 018ndash05

005

y11

0 002 004 006 008 01 012 014 016 018ndash05

005

y12

0 002 004 006 008 01 012 014 016 018ndash05

005

y13

0 002 004 006 008 01 012 014 016 018ndash05

005

y14

0 002 004 006 008 01 012 014 016 018ndash02

002

y15

0 002 004 006 008 01 012 014 016 018ndash02

002

y16

0 002 004 006 008 01 012 014 016 018ndash02

002

y7

0 002 004 006 008 01 012 014 016 018ndash02

002

y18

0 002 004 006 008 01 012 014 016 018ndash01

001

y19

0 002 004 006 008 01 012 014 016 018ndash01

001

y20

Time (s) Time (s)

Figure 28 FDM decomposition result

Mathematical Problems in Engineering 19

Tabl

e9

+ecorrelationcoeffi

cientandku

rtosisbetweentheFIBF

andtheoriginal

signal

y 1y 2

y 3y 4

y 5y 6

y 7y 8

y 9y 1

0y 1

1y 1

2y 1

3y 1

4y 1

5y 1

6y 1

7y 1

8y 1

9y 2

0y 2

1y 2

2y 2

3y 2

4y 2

5

C(t)

0004

0054

0126

0116

0120

0171

0196

0173

0306

0296

0490

0379

0219

0254

0206

0181

0209

0154

0113

0107

0051

0088

0017

0025

0000

K(t)

1000

2071

2276

2456

3148

2712

3194

3435

3276

2962

2716

2684

3304

2640

2710

2783

2614

2591

3079

2435

2673

2646

2107

1500

1000

20 Mathematical Problems in Engineering

further extracted by MOMEDA filtering and the extractionresults are shown in Figure 31

0 002 004 006 008 01 012 014 016 018ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

0 002 004 006 008 01 012 014 016 018ndash04

ndash02

0

02

04

Time (s)

Am

plitu

de (m

s2 )

(b)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

f0 2f0 4f0

(a)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

0 200 400 600 800 1000 1200ndash500

0

500

1000

Am

plitu

de (m

s2 )

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

Number (n)

(a)

0 200 400 600 800 10000

20

40

60

80

100

Frequency (Hz)

Am

plitu

de (m

s2 )

f0 2f0 3f0

4f05f 0 6f0 7f0

(b)

Mathematical Problems in Engineering 21

6 Conclusions

Data Availability

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

References

22 Mathematical Problems in Engineering

Mathematical Problems in Engineering 23

0 002 004 006 008 01 012 014 016 018ndash005

0005

y1

0 002 004 006 008 01 012 014 016 018ndash05

005

y2

0 002 004 006 008 01 012 014 016 018ndash05

005

y3

0 002 004 006 008 01 012 014 016 018ndash05

005

y4

0 002 004 006 008 01 012 014 016 018ndash1

01

y5

0 002 004 006 008 01 012 014 016 018ndash1

01

y6

0 002 004 006 008 01 012 014 016 018ndash05

005

y7

0 002 004 006 008 01 012 014 016 018ndash1

01

y8

0 002 004 006 008 01 012 014 016 018ndash1

01

y9

0 002 004 006 008 01 012 014 016 018ndash2

02

y10

0 002 004 006 008 01 012 014 016 018ndash2

02

y11

0 002 004 006 008 01 012 014 016 018ndash5

05

y12

0 002 004 006 008 01 012 014 016 018ndash2

02

y13

0 002 004 006 008 01 012 014 016 018ndash2

02

y14

0 002 004 006 008 01 012 014 016 018ndash2

02

y15

0 002 004 006 008 01 012 014 016 018ndash1

01

y16

0 002 004 006 008 01 012 014 016 018ndash2

02

y17

0 002 004 006 008 01 012 014 016 018ndash1

01

y18

0 002 004 006 008 01 012 014 016 018ndash1

01

y19

0 002 004 006 008 01 012 014 016 018ndash1

01

y20

Time (s) Time (s)

Figure 22 FDM decomposition result

16 Mathematical Problems in Engineering

Tabl

e8

+ecorrelationcoeffi

cientandku

rtosisbetweentheFIBF

andtheoriginal

signal

y 1y 2

y 3y 4

y 5y 6

y 7y 8

y 9y 1

0y 1

1y 1

2y 1

3y 1

4y 1

5y 1

6y 1

7y 1

8y 1

9Y 2

0y 2

1y 2

2y 2

3y 2

4y 2

5

C(t)

0018

0091

0074

0097

0132

0116

0125

0165

0217

0217

0260

0523

0349

0269

0277

0186

0214

0163

0159

0182

0086

0139

0064

0034

0000

K(t)

1000

2203

2209

2266

3145

4288

2564

4142

2750

2872

2683

4184

2981

2995

3277

3013

3385

2652

3151

2489

2652

4111

2615

2652

1000

Mathematical Problems in Engineering 17

0 002 004 006 008 01 012 014 016 018ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 002 004 006 008 01 012 014 016 018ndash5

0

5times10ndash3

Time (s)

(b)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 ) f 0 2f 0

3f 0

(a)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

2f 0

f 0

(b)

0 200 400 600 800 1000 1200ndash500

0

500

1000

Number (n)

Am

plitu

de (m

s2 )

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

(a)

0 200 400 600 800 10000

20

40

60

80

100

Frequency (Hz)

f 0 3f 0 4f 07f 05f 0

6f 0 8f 0 9f 0

2f 0

Am

plitu

de (m

s2 )

(b)

18 Mathematical Problems in Engineering

0 002 004 006 008 01 012 014 016 018ndash05

0

05

Time (s)

Am

plitu

de (m

s2 )

(a)

0 1000 2000 3000 4000 5000 60000

002

004

006

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

0 002 004 006 008 01 012 014 016 018ndash1

0

1

Time (s)

Am

plitu

de (m

s2 )

(a)

0 1000 2000 3000 4000 5000 60000

002

004

006

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

0 002 004 006 008 01 012 014 016 018ndash2

02

times10ndash3

y1

0 002 004 006 008 01 012 014 016 018ndash005

0005

y2

0 002 004 006 008 01 012 014 016 018ndash01

001

y3

0 002 004 006 008 01 012 014 016 018ndash01

001

y4

0 002 004 006 008 01 012 014 016 018ndash02

002

y5

0 002 004 006 008 01 012 014 016 018ndash02

002

y6

0 002 004 006 008 01 012 014 016 018ndash02

002

y7

0 002 004 006 008 01 012 014 016 018ndash02

002

y8

0 002 004 006 008 01 012 014 016 018ndash05

005

y9

0 002 004 006 008 01 012 014 016 018ndash05

005

y10

0 002 004 006 008 01 012 014 016 018ndash05

005

y11

0 002 004 006 008 01 012 014 016 018ndash05

005

y12

0 002 004 006 008 01 012 014 016 018ndash05

005

y13

0 002 004 006 008 01 012 014 016 018ndash05

005

y14

0 002 004 006 008 01 012 014 016 018ndash02

002

y15

0 002 004 006 008 01 012 014 016 018ndash02

002

y16

0 002 004 006 008 01 012 014 016 018ndash02

002

y7

0 002 004 006 008 01 012 014 016 018ndash02

002

y18

0 002 004 006 008 01 012 014 016 018ndash01

001

y19

0 002 004 006 008 01 012 014 016 018ndash01

001

y20

Time (s) Time (s)

Figure 28 FDM decomposition result

Mathematical Problems in Engineering 19

Tabl

e9

+ecorrelationcoeffi

cientandku

rtosisbetweentheFIBF

andtheoriginal

signal

y 1y 2

y 3y 4

y 5y 6

y 7y 8

y 9y 1

0y 1

1y 1

2y 1

3y 1

4y 1

5y 1

6y 1

7y 1

8y 1

9y 2

0y 2

1y 2

2y 2

3y 2

4y 2

5

C(t)

0004

0054

0126

0116

0120

0171

0196

0173

0306

0296

0490

0379

0219

0254

0206

0181

0209

0154

0113

0107

0051

0088

0017

0025

0000

K(t)

1000

2071

2276

2456

3148

2712

3194

3435

3276

2962

2716

2684

3304

2640

2710

2783

2614

2591

3079

2435

2673

2646

2107

1500

1000

20 Mathematical Problems in Engineering

further extracted by MOMEDA filtering and the extractionresults are shown in Figure 31

0 002 004 006 008 01 012 014 016 018ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

0 002 004 006 008 01 012 014 016 018ndash04

ndash02

0

02

04

Time (s)

Am

plitu

de (m

s2 )

(b)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

f0 2f0 4f0

(a)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

0 200 400 600 800 1000 1200ndash500

0

500

1000

Am

plitu

de (m

s2 )

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

Number (n)

(a)

0 200 400 600 800 10000

20

40

60

80

100

Frequency (Hz)

Am

plitu

de (m

s2 )

f0 2f0 3f0

4f05f 0 6f0 7f0

(b)

Mathematical Problems in Engineering 21

6 Conclusions

Data Availability

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

References

22 Mathematical Problems in Engineering

Mathematical Problems in Engineering 23

Tabl

e8

+ecorrelationcoeffi

cientandku

rtosisbetweentheFIBF

andtheoriginal

signal

y 1y 2

y 3y 4

y 5y 6

y 7y 8

y 9y 1

0y 1

1y 1

2y 1

3y 1

4y 1

5y 1

6y 1

7y 1

8y 1

9Y 2

0y 2

1y 2

2y 2

3y 2

4y 2

5

C(t)

0018

0091

0074

0097

0132

0116

0125

0165

0217

0217

0260

0523

0349

0269

0277

0186

0214

0163

0159

0182

0086

0139

0064

0034

0000

K(t)

1000

2203

2209

2266

3145

4288

2564

4142

2750

2872

2683

4184

2981

2995

3277

3013

3385

2652

3151

2489

2652

4111

2615

2652

1000

Mathematical Problems in Engineering 17

0 002 004 006 008 01 012 014 016 018ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 002 004 006 008 01 012 014 016 018ndash5

0

5times10ndash3

Time (s)

(b)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 ) f 0 2f 0

3f 0

(a)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

2f 0

f 0

(b)

0 200 400 600 800 1000 1200ndash500

0

500

1000

Number (n)

Am

plitu

de (m

s2 )

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

(a)

0 200 400 600 800 10000

20

40

60

80

100

Frequency (Hz)

f 0 3f 0 4f 07f 05f 0

6f 0 8f 0 9f 0

2f 0

Am

plitu

de (m

s2 )

(b)

18 Mathematical Problems in Engineering

0 002 004 006 008 01 012 014 016 018ndash05

0

05

Time (s)

Am

plitu

de (m

s2 )

(a)

0 1000 2000 3000 4000 5000 60000

002

004

006

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

0 002 004 006 008 01 012 014 016 018ndash1

0

1

Time (s)

Am

plitu

de (m

s2 )

(a)

0 1000 2000 3000 4000 5000 60000

002

004

006

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

0 002 004 006 008 01 012 014 016 018ndash2

02

times10ndash3

y1

0 002 004 006 008 01 012 014 016 018ndash005

0005

y2

0 002 004 006 008 01 012 014 016 018ndash01

001

y3

0 002 004 006 008 01 012 014 016 018ndash01

001

y4

0 002 004 006 008 01 012 014 016 018ndash02

002

y5

0 002 004 006 008 01 012 014 016 018ndash02

002

y6

0 002 004 006 008 01 012 014 016 018ndash02

002

y7

0 002 004 006 008 01 012 014 016 018ndash02

002

y8

0 002 004 006 008 01 012 014 016 018ndash05

005

y9

0 002 004 006 008 01 012 014 016 018ndash05

005

y10

0 002 004 006 008 01 012 014 016 018ndash05

005

y11

0 002 004 006 008 01 012 014 016 018ndash05

005

y12

0 002 004 006 008 01 012 014 016 018ndash05

005

y13

0 002 004 006 008 01 012 014 016 018ndash05

005

y14

0 002 004 006 008 01 012 014 016 018ndash02

002

y15

0 002 004 006 008 01 012 014 016 018ndash02

002

y16

0 002 004 006 008 01 012 014 016 018ndash02

002

y7

0 002 004 006 008 01 012 014 016 018ndash02

002

y18

0 002 004 006 008 01 012 014 016 018ndash01

001

y19

0 002 004 006 008 01 012 014 016 018ndash01

001

y20

Time (s) Time (s)

Figure 28 FDM decomposition result

Mathematical Problems in Engineering 19

Tabl

e9

+ecorrelationcoeffi

cientandku

rtosisbetweentheFIBF

andtheoriginal

signal

y 1y 2

y 3y 4

y 5y 6

y 7y 8

y 9y 1

0y 1

1y 1

2y 1

3y 1

4y 1

5y 1

6y 1

7y 1

8y 1

9y 2

0y 2

1y 2

2y 2

3y 2

4y 2

5

C(t)

0004

0054

0126

0116

0120

0171

0196

0173

0306

0296

0490

0379

0219

0254

0206

0181

0209

0154

0113

0107

0051

0088

0017

0025

0000

K(t)

1000

2071

2276

2456

3148

2712

3194

3435

3276

2962

2716

2684

3304

2640

2710

2783

2614

2591

3079

2435

2673

2646

2107

1500

1000

20 Mathematical Problems in Engineering

further extracted by MOMEDA filtering and the extractionresults are shown in Figure 31

0 002 004 006 008 01 012 014 016 018ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

0 002 004 006 008 01 012 014 016 018ndash04

ndash02

0

02

04

Time (s)

Am

plitu

de (m

s2 )

(b)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

f0 2f0 4f0

(a)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

0 200 400 600 800 1000 1200ndash500

0

500

1000

Am

plitu

de (m

s2 )

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

Number (n)

(a)

0 200 400 600 800 10000

20

40

60

80

100

Frequency (Hz)

Am

plitu

de (m

s2 )

f0 2f0 3f0

4f05f 0 6f0 7f0

(b)

Mathematical Problems in Engineering 21

6 Conclusions

Data Availability

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

References

22 Mathematical Problems in Engineering

Mathematical Problems in Engineering 23

0 002 004 006 008 01 012 014 016 018ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

Am

plitu

de (m

s2 )

0 002 004 006 008 01 012 014 016 018ndash5

0

5times10ndash3

Time (s)

(b)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 ) f 0 2f 0

3f 0

(a)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

2f 0

f 0

(b)

0 200 400 600 800 1000 1200ndash500

0

500

1000

Number (n)

Am

plitu

de (m

s2 )

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

(a)

0 200 400 600 800 10000

20

40

60

80

100

Frequency (Hz)

f 0 3f 0 4f 07f 05f 0

6f 0 8f 0 9f 0

2f 0

Am

plitu

de (m

s2 )

(b)

18 Mathematical Problems in Engineering

0 002 004 006 008 01 012 014 016 018ndash05

0

05

Time (s)

Am

plitu

de (m

s2 )

(a)

0 1000 2000 3000 4000 5000 60000

002

004

006

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

0 002 004 006 008 01 012 014 016 018ndash1

0

1

Time (s)

Am

plitu

de (m

s2 )

(a)

0 1000 2000 3000 4000 5000 60000

002

004

006

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

0 002 004 006 008 01 012 014 016 018ndash2

02

times10ndash3

y1

0 002 004 006 008 01 012 014 016 018ndash005

0005

y2

0 002 004 006 008 01 012 014 016 018ndash01

001

y3

0 002 004 006 008 01 012 014 016 018ndash01

001

y4

0 002 004 006 008 01 012 014 016 018ndash02

002

y5

0 002 004 006 008 01 012 014 016 018ndash02

002

y6

0 002 004 006 008 01 012 014 016 018ndash02

002

y7

0 002 004 006 008 01 012 014 016 018ndash02

002

y8

0 002 004 006 008 01 012 014 016 018ndash05

005

y9

0 002 004 006 008 01 012 014 016 018ndash05

005

y10

0 002 004 006 008 01 012 014 016 018ndash05

005

y11

0 002 004 006 008 01 012 014 016 018ndash05

005

y12

0 002 004 006 008 01 012 014 016 018ndash05

005

y13

0 002 004 006 008 01 012 014 016 018ndash05

005

y14

0 002 004 006 008 01 012 014 016 018ndash02

002

y15

0 002 004 006 008 01 012 014 016 018ndash02

002

y16

0 002 004 006 008 01 012 014 016 018ndash02

002

y7

0 002 004 006 008 01 012 014 016 018ndash02

002

y18

0 002 004 006 008 01 012 014 016 018ndash01

001

y19

0 002 004 006 008 01 012 014 016 018ndash01

001

y20

Time (s) Time (s)

Figure 28 FDM decomposition result

Mathematical Problems in Engineering 19

Tabl

e9

+ecorrelationcoeffi

cientandku

rtosisbetweentheFIBF

andtheoriginal

signal

y 1y 2

y 3y 4

y 5y 6

y 7y 8

y 9y 1

0y 1

1y 1

2y 1

3y 1

4y 1

5y 1

6y 1

7y 1

8y 1

9y 2

0y 2

1y 2

2y 2

3y 2

4y 2

5

C(t)

0004

0054

0126

0116

0120

0171

0196

0173

0306

0296

0490

0379

0219

0254

0206

0181

0209

0154

0113

0107

0051

0088

0017

0025

0000

K(t)

1000

2071

2276

2456

3148

2712

3194

3435

3276

2962

2716

2684

3304

2640

2710

2783

2614

2591

3079

2435

2673

2646

2107

1500

1000

20 Mathematical Problems in Engineering

further extracted by MOMEDA filtering and the extractionresults are shown in Figure 31

0 002 004 006 008 01 012 014 016 018ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

0 002 004 006 008 01 012 014 016 018ndash04

ndash02

0

02

04

Time (s)

Am

plitu

de (m

s2 )

(b)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

f0 2f0 4f0

(a)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

0 200 400 600 800 1000 1200ndash500

0

500

1000

Am

plitu

de (m

s2 )

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

Number (n)

(a)

0 200 400 600 800 10000

20

40

60

80

100

Frequency (Hz)

Am

plitu

de (m

s2 )

f0 2f0 3f0

4f05f 0 6f0 7f0

(b)

Mathematical Problems in Engineering 21

6 Conclusions

Data Availability

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

References

22 Mathematical Problems in Engineering

Mathematical Problems in Engineering 23

0 002 004 006 008 01 012 014 016 018ndash05

0

05

Time (s)

Am

plitu

de (m

s2 )

(a)

0 1000 2000 3000 4000 5000 60000

002

004

006

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

0 002 004 006 008 01 012 014 016 018ndash1

0

1

Time (s)

Am

plitu

de (m

s2 )

(a)

0 1000 2000 3000 4000 5000 60000

002

004

006

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

0 002 004 006 008 01 012 014 016 018ndash2

02

times10ndash3

y1

0 002 004 006 008 01 012 014 016 018ndash005

0005

y2

0 002 004 006 008 01 012 014 016 018ndash01

001

y3

0 002 004 006 008 01 012 014 016 018ndash01

001

y4

0 002 004 006 008 01 012 014 016 018ndash02

002

y5

0 002 004 006 008 01 012 014 016 018ndash02

002

y6

0 002 004 006 008 01 012 014 016 018ndash02

002

y7

0 002 004 006 008 01 012 014 016 018ndash02

002

y8

0 002 004 006 008 01 012 014 016 018ndash05

005

y9

0 002 004 006 008 01 012 014 016 018ndash05

005

y10

0 002 004 006 008 01 012 014 016 018ndash05

005

y11

0 002 004 006 008 01 012 014 016 018ndash05

005

y12

0 002 004 006 008 01 012 014 016 018ndash05

005

y13

0 002 004 006 008 01 012 014 016 018ndash05

005

y14

0 002 004 006 008 01 012 014 016 018ndash02

002

y15

0 002 004 006 008 01 012 014 016 018ndash02

002

y16

0 002 004 006 008 01 012 014 016 018ndash02

002

y7

0 002 004 006 008 01 012 014 016 018ndash02

002

y18

0 002 004 006 008 01 012 014 016 018ndash01

001

y19

0 002 004 006 008 01 012 014 016 018ndash01

001

y20

Time (s) Time (s)

Figure 28 FDM decomposition result

Mathematical Problems in Engineering 19

Tabl

e9

+ecorrelationcoeffi

cientandku

rtosisbetweentheFIBF

andtheoriginal

signal

y 1y 2

y 3y 4

y 5y 6

y 7y 8

y 9y 1

0y 1

1y 1

2y 1

3y 1

4y 1

5y 1

6y 1

7y 1

8y 1

9y 2

0y 2

1y 2

2y 2

3y 2

4y 2

5

C(t)

0004

0054

0126

0116

0120

0171

0196

0173

0306

0296

0490

0379

0219

0254

0206

0181

0209

0154

0113

0107

0051

0088

0017

0025

0000

K(t)

1000

2071

2276

2456

3148

2712

3194

3435

3276

2962

2716

2684

3304

2640

2710

2783

2614

2591

3079

2435

2673

2646

2107

1500

1000

20 Mathematical Problems in Engineering

further extracted by MOMEDA filtering and the extractionresults are shown in Figure 31

0 002 004 006 008 01 012 014 016 018ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

0 002 004 006 008 01 012 014 016 018ndash04

ndash02

0

02

04

Time (s)

Am

plitu

de (m

s2 )

(b)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

f0 2f0 4f0

(a)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

0 200 400 600 800 1000 1200ndash500

0

500

1000

Am

plitu

de (m

s2 )

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

Number (n)

(a)

0 200 400 600 800 10000

20

40

60

80

100

Frequency (Hz)

Am

plitu

de (m

s2 )

f0 2f0 3f0

4f05f 0 6f0 7f0

(b)

Mathematical Problems in Engineering 21

6 Conclusions

Data Availability

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

References

22 Mathematical Problems in Engineering

Mathematical Problems in Engineering 23

Tabl

e9

+ecorrelationcoeffi

cientandku

rtosisbetweentheFIBF

andtheoriginal

signal

y 1y 2

y 3y 4

y 5y 6

y 7y 8

y 9y 1

0y 1

1y 1

2y 1

3y 1

4y 1

5y 1

6y 1

7y 1

8y 1

9y 2

0y 2

1y 2

2y 2

3y 2

4y 2

5

C(t)

0004

0054

0126

0116

0120

0171

0196

0173

0306

0296

0490

0379

0219

0254

0206

0181

0209

0154

0113

0107

0051

0088

0017

0025

0000

K(t)

1000

2071

2276

2456

3148

2712

3194

3435

3276

2962

2716

2684

3304

2640

2710

2783

2614

2591

3079

2435

2673

2646

2107

1500

1000

20 Mathematical Problems in Engineering

further extracted by MOMEDA filtering and the extractionresults are shown in Figure 31

0 002 004 006 008 01 012 014 016 018ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

0 002 004 006 008 01 012 014 016 018ndash04

ndash02

0

02

04

Time (s)

Am

plitu

de (m

s2 )

(b)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

f0 2f0 4f0

(a)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

0 200 400 600 800 1000 1200ndash500

0

500

1000

Am

plitu

de (m

s2 )

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

Number (n)

(a)

0 200 400 600 800 10000

20

40

60

80

100

Frequency (Hz)

Am

plitu

de (m

s2 )

f0 2f0 3f0

4f05f 0 6f0 7f0

(b)

Mathematical Problems in Engineering 21

6 Conclusions

Data Availability

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

References

22 Mathematical Problems in Engineering

Mathematical Problems in Engineering 23

further extracted by MOMEDA filtering and the extractionresults are shown in Figure 31

0 002 004 006 008 01 012 014 016 018ndash4

ndash2

0

2

4

Time (s)

Am

plitu

de (m

s2 )

(a)

0 002 004 006 008 01 012 014 016 018ndash04

ndash02

0

02

04

Time (s)

Am

plitu

de (m

s2 )

(b)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

f0 2f0 4f0

(a)

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

Frequency (Hz)

Am

plitu

de (m

s2 )

(b)

0 200 400 600 800 1000 1200ndash500

0

500

1000

Am

plitu

de (m

s2 )

Am

plitu

de (m

s2 )

0 1000 2000 3000 4000 5000 60000

50

100

Frequency (Hz)

Number (n)

(a)

0 200 400 600 800 10000

20

40

60

80

100

Frequency (Hz)

Am

plitu

de (m

s2 )

f0 2f0 3f0

4f05f 0 6f0 7f0

(b)

Mathematical Problems in Engineering 21

6 Conclusions

Data Availability

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

References

22 Mathematical Problems in Engineering

Mathematical Problems in Engineering 23

6 Conclusions

Data Availability

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

References

22 Mathematical Problems in Engineering

Mathematical Problems in Engineering 23

Mathematical Problems in Engineering 23