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Research ArticleFeature Frequency Extraction Based on Principal ComponentAnalysis and Its Application in Axis Orbit
Zhen Li Weiguang Li and Xuezhi Zhao
School of Mechanical and Automotive Engineering South China University of Technology Guangzhou 510640 China
Correspondence should be addressed to Weiguang Li wguangliscuteducn and Xuezhi Zhao mezhaoxzscuteducn
Received 14 March 2018 Accepted 15 May 2018 Published 12 July 2018
Academic Editor Jean-Jacques Sinou
Copyright copy 2018 Zhen Li et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Vibration-based diagnosis has been employed as a powerful tool in maintaining the operating efficiency and safety for largerotating machinery However the extraction of malfunction features is not accurate enough by using traditional vibration signalprocessing techniques owing to their intrinsic shortcomings In this paper the relationship between effective eigenvalues andfrequency components was investigated and a new characteristic frequency separation method based on PCA (CFSM-PCA) wasproposed Certain feature frequency could be purified by reconstructing the specified eigenvalues Furthermore three significantperspectives were studied via the distribution of effective eigenvalues and theoretical derivations were subsequently illustratedMore importantly this proposed scheme could also be used to synthesize axis orbits of larger machines Purified curves were soexplicit and the CFSM-PCA exhibited higher efficiency than harmonic wavelet and wavelet packet
1 Introduction
Principal component analysis (PCA) which can reduce thedimensionality of data set but retainmost of original variables[1ndash3] has been widely used in fields of image processingfault diagnosis pattern recognition neural network datacompression wavelet transform and so on For exampleKirby et al [4] employed PCA algorithm to compress imageand extract main features Moreover the combination ofPCA and Back Propagation (BP) neural network could alsobe applied in reorganization of facial image Xi et al [5]and Malhi et al [6] individually applied PCA approachto reduce the dimension of data and extract the featurevariables Additionally neural network was further used asa classifier to categorize the bearing faults To investigatethe fault diagnosis of impeller in centrifugal compressorPCA was also adopted to decrease the dimensionality ofmultiple time series by Jiangrsquos group [7] Sun et al [8]analyzed the defects of conventional fault diagnosis meth-ods and introduced the data mining technology into faultdiagnosis After that a new scheme used to reduce datafeatures was proposed based on C45 decision tree and PCAalgorithm
Generally when PCA is used to denoise or for data com-pression the number of effective eigenvalues is determinedby the cumulative contribution rate and its deformation [9ndash14] expressed as
119871 119897 = sum119897119894=1 120582119894sum119898119895=1 120582119895 (1)
where 120582119894 and 120582119895 are eigenvalues of covariance matrix respec-tively119898 is the number of eigenvalues of covariance matrices119897 is the number of effective eigenvalues When cumulativecontribution rate119871 119897 is greater than a certain value (80-95)119897 could be decided [2]
Although impressive progress in signal denoising anddimensionality reduction fields has been achieved the stud-ies on extraction or elimination of specific characteristicspectrum (single frequency) via this classical PCA methodhave always been ignored However precise extraction of thefundamental frequency (1X) the second-harmonic (2X) orthe other feature frequencies of raw signal is of significanceto the purifications of axial orbit notch filter [15] speechrecognition [16] fault diagnosis of rolling bearing [17] and soforth Over the past decade many signal processing tools for
HindawiShock and VibrationVolume 2018 Article ID 2530248 17 pageshttpsdoiorg10115520182530248
2 Shock and Vibration
the extraction of certain frequency have been developed suchas wavelet packet transform harmonic wavelet ensembleempirical mode decomposition (EEMD) and sparse decom-position [18] For instance references [19ndash21] adopted mul-tilevel division technique of wavelet packet to select certainfrequency band for the extraction of specific frequency fromwhich axis orbit could be manufactured References [22 23]subdivided random frequency band infinity via harmonicwavelet to extract interesting frequency the refinement ofrotor centerrsquos orbit from one or more interesting frequencybands could be realized subsequently Nevertheless waveletpacket and harmonic packet algorithm are subject to theHeisenberg uncertainty principle and resolutions of timedomain 120590119905 and frequency domain 120590120596 could not be randomlyhigh simultaneously ie 1205902119905 sdot 1205902120596 ge 14 In addition in EEMDmethod signals are adaptively decomposed into several sumsof intrinsic mode functions (IMFs) whose instantaneousfrequencies have physical meanings In practice the IMFis always multicomponent rather than a single componentresulting in unexplainable irregularity in its instantaneousamplitude and blind extraction of the 1X 2X and the othersubharmonics So the EEMD method is not suitable for thedecomposition of signals with multiple components in anarrow band [24] By using singular value decomposition(SVD) strategy reference [25] generated axis orbit by meansof cumulative contribution rate to denoise noisy signalThis method is unable to extract the specified feature fre-quency Therefore other more effective and simple methodswhich suffer free from above disadvantages remain to beexplored
Our group has been committed to studying the faultdiagnosis of large scale equipment [26ndash28] During thecourse of single frequency simulation via PCA an interestingphenomenon was discovered unexpectedly ie a frequencycomponent produces two eigenvalues After intensive studywe found that PCA algorithm could be used to extractthe specified single or multiple feature frequencies from acrude signal Guidelines are summarized as follows (1) Eachcharacteristic frequency in a signal produces only two valideigenvalues (2)Thenumber of effective eigenvalues is relatedto the quantity of raw signal frequencies and has nothing todo with the magnitude of 119891119894 119860 119894 and 120593119894 (3) The sequenceof eigenvalues of covariance matrix in its distribution chartis determined by the amplitude of feature frequency Forthese discoveries a novel frequency separationmethod basedon PCA was proposed through which axis orbits of largerotating machines were readily purified Moreover purifica-tion results are better than that of existing methods such aswavelet packet and harmonic wavelet
Hereafter the paper is organized as follows Section 2briefly introduces the basic principle PCA theory in sig-nal processing and a new method of signal recovery isintroduced In Section 3 the theoretical discovery and thetheoretical verifications of relationship between eigenvaluesand feature frequency are given Section 4 illustrates theapplication on purification of axis orbit of large rotor testbed and compares experimental results with that of harmonicwavelet and wavelet packet proving its high efficiency of theproposed scheme The filtration of single frequency is givenin Section 5 Finally Section 6 draws the conclusions
2 Basic Theories of PCA in Signal Processing
We assume that there are 119898 random vectors (x1 x2 sdot sdot sdot x119898)with each vector x119894 containing 119899 samples (x119894 = (1199091198941 1199091198942 sdot sdot sdot119909119894119899)) An119898 times 119899matrix X with119898 rows and 119899 columns can bedescribed as
X =[[[[[[[
11990911 11990912 sdot sdot sdot 119909111989911990921 11990922 sdot sdot sdot 1199092119899 1199091198981 1199091198982 sdot sdot sdot 119909119898119899
]]]]]]]
(2)
where X = [x1 x2 sdot sdot sdot x119898]T and T denotes the vector trans-pose Supposing that x1 x2 sdot sdot sdot x119898 represent the indicatrix ofcrude variables then 119897 new variables y119894(119894 = 1 2 sdot sdot sdot 119897 isin 119885+) (119897 ⩽119898) could be obtained after principal component analysis ofXdescribed as
y119894 = 1205721198941x1 + 1205721198942x2 + sdot sdot sdot + 120572119894119898x119898 = 120572T119894 X (3)
where y119894 isin R1times119899 According to the definition of PCA [3]120572119894 = (1205721198941 1205721198942 sdot sdot sdot 120572119894119898)T (119894 = 1 2 sdot sdot sdot 119897) and 120572119894 are eigenvectorscorresponding to the ith eigenvalue in descending orderin the covariance matrix of X [2ndash4 6] and (4) should besatisfied
120572T119894 120572119895 = 1 119894 = 119895120572T119894 120572119895 = 0 119894 = 119895
(4)
Covariance matrix of X in (2) is
C =[[[[[[[
cov (x1 x1) cov (x1 x2) sdot sdot sdot cov (x1 x119898)cov (x2 x1) cov (x2 x2) sdot sdot sdot cov (x2 x119898)
cov (x119898 x1) cov (x119898 x2) sdot sdot sdot cov (x119898 x119898)
]]]]]]]
(5)
where cov(x119894 x119895) = 119864[(x119894minus119864(x119894))(x119895minus119864(x119895))T] Based on PCAtheory characteristic equation of covariance matrix C can begiven by
C120572119894 = 120582119894120572119894 (6)
where 120582119894 are eigenvalues of covariance matrix C and 120572119894are eigenvectors corresponding to 120582119894 Given that eigenvaluesranged in descending order that is 1205821 gt 1205822 gt sdot sdot sdot gt 120582119897 datadimensionality reduction can be achieved by using (1) and(3) Then original m variables are converted to new 119897 onesIf signal processing is performed the signal reconstruction isneeded Considering covariance matrix C is a semi-positivesymmetric matrix its eigenvectors are orthogonal to eachother ie sum119898119894=1 120572119894120572T119894 = 119868119898 [1 2] After left multiplication onboth sides by120572119894 of (3) and sum calculation (7) can be given asfollows
119898
sum119894=1
120572119894y119894 =119898
sum119894=1
120572119894120572T119894 X = 119868119898X (7)
Shock and Vibration 3
If the former 119897 principal components are chosen toreconstruct in the light of cumulative contribution rate 119871 119897 anapproximate matrix X can be formulated as
X =119897
sum119894=1
120572119894y119894 =119897
sum119894=1
120572119894120572T119894 X (8)
Compared with original matrix X the reconstructedapproximate matrix X comprises most of information of Xand excludes redundant features such as noise and powerfrequency interference [9 10]
Signal could be recovered from the reconstructed matrixX in terms of matrix composition mode XT is converted to1 row 119898119899 columns vector with the row vectors arranging inhead-to-tail fashion then a new vector is obtained recordedas a a isin 1198771times119898119899 The recovered signal x is obtained via inversetransformation of vector a derived as
x = 119886M+ (9)
where x isin 1198771times119871 L=m+n-1 L is the length of recovered dataM+ = (MTM)minus1MT isin 119877119898119899times119871 and M+ is pseudoinverse ofMM isin 119877119871times119898119899 andM comprises119898 unit matrices where therank of each unit matrix is 119899 Takingm=3 n=3 as an examplein this case L=m+n-1=5 and mn=9 the matrix M can beexpressed as follows
=
[[[[[[[[[[
1
1 1
1 1 1
1 1
1
]]]]]]]]]]
M (10)
The signal x is recovered from the pseudoinverse matrixM+ actually it is to compute the average values of eachelement at counter-diagonal of matrix X which is consistentwith the method reported in [23] SinceM is a sparse matrixespecially when values of 119898 and 119899 are large random-accessmemory and time for calculating of pseudoinverse matrixM+ will increase exponentially Hence signal recovered fromsimple method is extremely desired With this aim in mindwe have developed a new averagingmethod and adopted it torecover signal x from matrix X The expression is shown asfollows
119909 (119896) =
119896
sum119894=1
119909119894119896+1minus119894119896 1 le 119896 lt 119898
119898
sum119894=1
119909119894119896+1minus119894119898 119898 le 119896 lt 119899119898
sum119894=119896minus119899+1
119909119894119896+1minus119894(119898 + 119899 minus 119896) 119899 lt 119896 le 119898 + 119899 minus 1
(11)
where 119909119894119895 are element of the 119894th row and the 119895th column inthe reconstructed approximate matrix X The signal x can berestored facilely according to (11)
3 Internal Law of Effective Eigenvalues andFrequency Components
31The Process of Feature Frequency SeparationMethod Basedon PCA For noise-free signal 119909(119905) the effective eigenvaluesare nonzero ones It can be seen from (7) that signal 119909(119905) has119898 eigenvectors which aremostly generated by noise So for asignal containing certain number of frequency componentswhat is the relationship between the number of eigenvaluesand the number of frequencies
During our research course an important connectionbetween nonzero eigenvalues and the number of frequencycomponents was discovered To illustrate this connectionexplicitly signal with different amplitude 119860 119894 frequency 120596119894and phase 120593119894 was constructed as follows
119909 (119905) =119896
sum119894=1
119860 119894 sin (2120587120596119894119905 + 120593119894) (12)
where 119896 is the number of frequency components4096 data points were collected with sampling frequency
of 1024 Hz and then the Hankel matrix was formed fromsignal 119909(119905) with m rows and n columns Decomposition andreconstruction of signal 119909(119905) were proceeded by employingPCA algorithm in Section 2 [9 13] Effect of the constructedHankel matrix on signal processing was studied by Zhao etal [26] they pointed out that if the number of rows was closeto the number of columns signal processing effect was betterFurthermore when Lwas an evenm=L2 and n=L2+1 whenL was an oddm= (L+1)2 and n= (L+1)2 Hence in our casem = 2048 and n = 2049 were applied The decompositionprocedure consists of the following(1)Given k = 1119860 119894= 1 tri-groups signals were constructed
and their principal component eigenvalue distribution mapsare shown in Figure 1 The number of eigenvalues 119902 rangefrom 1 to 2048 and just the leading 50 eigenvalues are listedin this case(2) Given k = 2 119860 119894= 1 08 tri-groups signals were
reconstructed and each group signal contained two effectivefrequency components with119891119894 as 20Hz and 30Hz 50Hz and60 Hz and 80 Hz and 90 Hz respectively and 120593119894 were 10 and20 40 and 50 and 70 and 80 respectively
When k = 1 each signal contains only one effectivefrequency component These three sets of signals have thesame amplitude119860 119894 but different frequency120596119894 and phase120593119894 Ascan be seen from Figure 1 each signal produces two adjacentnonzero eigenvalues 1205821 and 1205822 Although tri-groups signalsare different nonzero eigenvalues of them are the same toeach other
It can be seen from eigenvalue distribution graphs inFigure 2 that each signal produces two sets of nonzeroeigenvalues each set of eigenvalues contains two adjacenteigenvalues 120582119894 and 120582119894+1 Interestingly the four nonzeroeigenvalues produced from each signal are equal to theircorresponding eigenvalues generated by the other signalseven if the signals are different
Moreover by comparing Figure 2 with Figure 1 the firstset of eigenvalues in both two graphs are exactly sameActually signals in Figure 2 are established by adding an extra
4 Shock and Vibration
(15123) (25118)e first group
(15123) (25118)
(15123) (25118)
e first group
e first group
0
200
400
600
0
200
400
600
10 20 30 40 500Number of eigenvalues q
10 20 30 40 500
10 20 30 40 500
x1(t) = MCH(40t + 10)
x2(t) = MCH(100t + 40)
x3(t) = MCH(160t + 70)
0
200
400
600
Eige
nval
ue o
f PCA
i
Figure 1 Eigenvalues distribution of covariance matrix C with one frequency component
(15123) (25118)
(15123) (25118)
(15123) (25118)
(33278) (43275)
(33278) (43275)
(33278) (43275)
e first group
e second group
e first group
e second group
e first group
e second group
0
200
400
600
10 20 30 40 500
10 20 30 40 500
0
200
400
600
10 20 30 40 500Number of eigenvalues q
x1(t) = MCH(40t + 10) + 08 MCH(60t + 20)
x3(t) = MCH(160t + 70) + 08 MCH(180t + 80)
0
200
400
600
Eige
nval
ue o
f PCA
i
x2(t) = MCH(100t + 40) + 08 MCH(120t + 50)
Figure 2 Eigenvalues distribution of covariance matrix C with twofrequency components
frequency component with amplitude of 08 Thus it couldbe confirmed that the first set of nonzero eigenvalues 1205821 and1205822 in Figure 2 are generated from frequency component withamplitude of 10 while the second set of nonzero eigenvalues
are generated from frequency component with amplitude of08(3)Given 119896= 3119860 119894 = 1 08 and 06 tri-groups signals were
reconstructed and each group signal contained three sets ofeffective frequency components The frequencies of the firstgroup of signal were 20 Hz 30 Hz and 40 Hz respectivelyThe frequencies of the second group of signals were 50 Hz60 Hz and 70 Hz respectively Similarly the frequencies ofthe third group of signals were 80 Hz 90 Hz and 100 Hzrespectively The corresponding phases of these three groupsof signals were taken as 10 20 and 30 40 50 and 60 and 7080 and 90 respectively
When 119896 = 3 119860 119894=1 08 and 06 each group signal pro-duces three effective frequency components The amplitudes119860 119894 of corresponding frequency 119891119894(119894=1 2 3) are same butwith different frequencies 120596119894 and phases 120593119894 As displayed inFigure 3 each group signal produces three sets of nonzeroeigenvalues and each set of eigenvalues contains two adjacenteigenvalues 120582119894 and 120582119894+1 In addition these three sets ofeigenvalues produced by each signal are correspondinglysame
It can be found from Figures 1ndash3 that the magnitudeof eigenvalues of each first group in three graphs is sameAs aforementioned the first set of nonzero eigenvalues1205821 and 1205822 are generated from frequency component withamplitude 119860 119894 of 1 Similarly by comparing Figure 2 withFigure 1 the second set of nonzero eigenvalues 1205823 and 1205824are generated from frequency component with amplitude119860 119894 of 08 And then it is almost certain that the third setof frequency components 1205825 and 1205826 are generated fromfrequency component with amplitude 119860 119894 of 06
The same results can be obtained by continuously increas-ing effective frequency components of signal Therefore an
Shock and Vibration 5
(33278) (43275)
(51844) (61842)
(15123) (25118)
(33278) (43275)
(51844) (61842)
(15123) (25118)
(33278) (43275)
(51844) (61842)
(15123) (25118)e first group
e second group
e third group
e first group
e second group
e third group
e first group
e second group
e third group
x1(t) = MCH(40t + 10) + 08 MCH(60t + 20) + 06 MCH(80t + 30)
x2(t) = MCH(100t + 40) + 08 MCH(120t + 50) + 06 MCH(140t + 60)
x3(t) = MCH(160t + 70) + 08 MCH(180t + 80) + 06 MCH(200t + 90)
0
200
400
600
10 20 30 40 500
0
200
400
600
Eige
nval
ue o
f PCA
i
10 20 30 40 500
0
200
400
600
10 20 30 40 500Number of eigenvalues q
Figure 3 Eigenvalues distribution of covariancematrixCwith threefrequency components
Hankel matrix is derived from certain signal x(t) l=min(mn) with m rows n columns and k effective frequenciesConcerning the fact that the Shannon sampling theorem ismet assuming that lgt2k generic conclusions are summarizedas follows(1) Each frequency component of signal produces two
nonzero eigenvalues with one arranging another closely(2) The number of effective eigenvalues of crude signal
is related to the number of frequency components and has
nothing to do with themagnitude of amplitude119860 119894 frequency120596119894 and phase 120593119894(3) In eigenvalue distribution chart of covariance matrix
C the sequence of nonzero effective eigenvalues is deter-mined by amplitude 119860 119894 of signal The larger amplitude is thelarger eigenvalues will be the more forward rank of the twoeigenvalues produced from corresponding frequencies is
Inspired by the relationship between frequency andeigenvalues a new technique for characteristic frequency sep-aration method based on PCA (CFSM-PCA) was proposedThe concrete steps are listed below(1) For a certain signal 119909(119905) direct component (DC) of
raw signal is filtered out via fast Fourier transform (FFT)firstly and then Hankel matrix X is constructed throughfiltered signal(2) Covariance matrix C of matrix X is obtained with its
eigenvalues 120582119894 arranging in descending order (1205821 1205822 sdot sdot sdot 120582119898)and corresponding eigenvectors are obtained as 12057211205722 sdot sdot sdot120572119898(3) According to the distribution of eigenvalues 120582119894reconstruction is carried out from two eigenvalues and cor-responding eigenvectors of certain frequency For examplefor amplitude perspective if the rank of specific frequencyof raw signal is k a new matrix is received by reconstructingthe eigenvectors corresponding to 2k-1 and 2k eigenval-ues in eigenvalue distribution chart of covariance matrixC(4)Thematrix X can be produced by adding the mean of
original matrix to the new reconstructed matrix(5) The signal x which is the characteristic frequency
component is recovered from matrix X by means of theaveraging method
32Theoretical Deduction In this section deduction processof the three discoveries (Section 31) is provided Supposingthat a signal is expressed as 119909(119905) = 119886 sin(120596119905 + 120593) Samplingtime 119879119904 is used to discretize signal x(t) Hankel matrix withm rows and n columns is derived from signal 119909(119905) exhibitedas
X = 119886 times[[[[[[
sin (0 sdot 120596119879119904 + 120593) sin (1 sdot 120596119879119904 + 120593) sdot sdot sdot sin ((119899 minus 1) sdot 120596119879119904 + 120593)sin (1 sdot 120596119879119904 + 120593) sin (2 sdot 120596119879119904 + 120593) sdot sdot sdot sin (119899 sdot 120596119879119904 + 120593)
sin ((119898 minus 1) sdot 120596119879119904 + 120593) sin (119898 sdot 120596119879119904 + 120593) sdot sdot sdot sin ((119898 + 119899 minus 2) sdot 120596119879119904 + 120593)
]]]]]]
(13)
(1) Each characteristic frequency produces two effectiveeigenvalues
The deduced process of the first conclusion is given belowEquation (13) can be rewritten as (14) based onEulerrsquos Formula
X = 1198862119894
times[[[[[[[
119890(0120596119879119904+120593)119894 minus 119890minus(0120596119879119904+120593)119894 119890(1120596119879119904+120593)119894 minus 119890minus(1120596119879119904+120593)119894 119890(2120596119879119904+120593)119894 minus 119890minus(2120596119879119904+120593)119894 sdot sdot sdot 119890((119899minus1)120596119879119904+120593)119894 minus 119890minus((119899minus1)120596119879119904+120593)119894119890(1120596119879119904+120593)119894 minus 119890minus(1120596119879119904+120593)119894 119890(2120596119879119904+120593)119894 minus 119890minus(2120596119879119904+120593)119894 119890(3120596119879119904+120593)119894 minus 119890minus(3120596119879119904+120593)119894 sdot sdot sdot 119890(119899120596119879119904+120593)119894 minus 119890minus(119899120596119879119904+120593)119894
119890((119898minus1)120596119879119904+120593)119894 minus 119890minus((119898minus1)120596119879119904+120593)119894 119890(119898120596119879119904+120593)119894 minus 119890minus(119898120596119879119904+120593)119894 119890((119898+1)120596119879119904+120593)119894 minus 119890minus((119898+1)120596119879119904+120593)119894 sdot sdot sdot 119890((119898+119899minus2)120596119879119904+120593)119894 minus 119890minus((119898+119899minus2)120596119879119904+120593)119894
]]]]]]]
(14)
6 Shock and Vibration
Equation (14) can be expanded to addition form of twoformulas depicted as
X = 1198862119894 times 119890120593119894
times[[[[[[[[
1198900120596119879119904119894 1198901120596119879119904119894 1198902120596119879119904119894 sdot sdot sdot 119890(119899minus1)1205961198791199041198941198901120596119879119904119894 1198902120596119879119904119894 1198903120596119879119904119894 sdot sdot sdot 119890119899120596119879119904119894
119890(119898minus1)120596119879119904119894 119890119898120596119879119904119894 119890(119898+1)120596119879119904119894 sdot sdot sdot 119890(119898+119899minus2)120596119879119904119894
]]]]]]]]
minus 1198862119894 times 119890minus120593119894
times[[[[[[[[
119890minus0120596119879119904119894 119890minus1120596119879119904119894 119890minus2120596119879119904119894 sdot sdot sdot 119890minus(119899minus1)120596119879119904119894119890minus1120596119879119904119894 119890minus2120596119879119904119894 119890minus3120596119879119904119894 sdot sdot sdot 119890minus119899120596119879119904119894
119890minus(119898minus1)120596119879119904119894 119890minus119898120596119879119904119894 119890minus(119898+1)120596119879119904119894 sdot sdot sdot 119890minus(119898+119899minus2)120596119879119904119894
]]]]]]]](15)
From (15) the rank of both matrices is 1 Based on rankrelationship of two matrices 119877(119860+119861) le 119877(119860) + 119877(119861) Hencerank of matrix X is less than or equal to 2
Additionally when 120593 = 0 (13) is rewritten to (16)According to sum-to-product identities the rank of X is 2when 120596119879119904 = 0 120587 2120587 sdot sdot sdot
X = 119886 times[[[[[[[[[
sin (0 sdot 120596119879119904) sin (1 sdot 120596119879119904) sdot sdot sdot sin ((119899 minus 1) sdot 120596119879119904)sin (1 sdot 120596119879119904) sin (2 sdot 120596119879119904) sdot sdot sdot sin (119899 sdot 120596119879119904)
sin ((119898 minus 1) sdot 120596119879119904) sin (119898 sdot 120596119879119904) sdot sdot sdot sin ((119898 + 119899 minus 2) sdot 120596119879119904)
]]]]]]]]]
(16)
When 120593 = 0 (13) can be deduced into the first-orderprincipal minor of matrix X ie |sin120593| = 0 The principalminor of order 2 of X is shown as10038161003816100381610038161003816100381610038161003816100381610038161003816sin (0 sdot 120596119879119904 + 120593) sin (1 sdot 120596119879119904 + 120593)sin (1 sdot 120596119879119904 + 120593) sin (2 sdot 120596119879119904 + 120593)
10038161003816100381610038161003816100381610038161003816100381610038161003816= minussin2 (120596119879119904)
= 0(17)
where120596119879119904 = 0 120587 2120587 sdot sdot sdot Because the leading principalminorof order 2 ofX in (13) is nonzero values so the rank of matrixX is at least 2
Combining result of (15) 119877(119860 + 119861) le 2 and nonzerovalues of the leading principal submatrix of order 2 hencethe rank of matrix X is 2 Matrix X has two eigenvalues 12058210158401and 12058210158402 Derived from C = XXT119899 (where 119899 is the numberof columns of matrix X) [3] two nonzero eigenvalues ofcovariance matrix C of X are 1205821 and 1205822 respectively(2)The sequence of eigenvalues is determined by ampli-
tudeIn terms of PCA covariance matrix C is described as
C = 119864 [(X minus 119864 (X)) sdot (X minus 119864 (X))T] (18)
Referring to (6) (19) is constructed as follows
119897
sum119894=1
C120572119894120572T119894 =119897
sum119894=1
120582119894120572119894120572T119894 (19)
Two effective eigenvalues are generated from a frequencycomponent that is l=2
CI119898 = 12058211205721120572T1 + 12058221205722120572T2 (20)
Then the energy of matrix C can be deduced as
|C|2 = 12058221119898
sum119894=1
12057221198941119898
sum119894=1
12057221198941 + 12059022119898
sum119894=1
12057221198942119898
sum119895=1
12057221198942
+ 21205902112059022119898
sum119894=1
12057211989411205721198942119898
sum119894=1
12057211989411205721198943(21)
Based upon (4) sum119898119894=1 12057221198941 = 1 sum119898119894=1 12057221198942 = 1 sum119898119894=1 12057211989411205721198942 =0 Then (22) is given by
|C|2 = 12058221 + 12058222 (22)
The constructed Hankel matrix X of signal 119909(119905) is sub-stituted into (18) we can see that the energy of covariancematrix C is proportional to 1198862 The larger 1198862 is the greaterenergy of matrix C is as well as 120582119894 according to (22)Furthermore the larger frequency component amplitude isthe larger corresponding eigenvalue in covariance matrix Ccharacteristic distribution chart is
Based on these conclusions once the amplitude sequenceof a certain frequency in raw signal amplitude spec-trum is determined its corresponding frequency compo-nent can be reconstructed In this way extraction of sin-gle or multiple characteristic frequencies could be real-ized
In view of addition relation as shown in (8) the notchfilter could be achieved through CFSM-PCA The frequencycomponent extracted by this algorithm is subtracted in orig-inal signal ie this frequency component can be eliminatedin raw signal Reader could consult examples in Section 5 formore details
Shock and Vibration 7
Table 1 Amplitude of signal components under the different SNR
SNR SNR1=-1 SNR2=-4 SNR3=-7 SNR4=-10Signal component Raw Process Raw Process Raw Process Raw Processsin(40120587119905 + 20) 09999 09938 10260 10240 10370 10730 1114 111205 sin(100120587119905 + 40) 04703 04826 05317 05482 07353 07533 05177 0523407 sin(98120587119905 + 50) 07299 07152 07116 07235 05110 04968 06800 0662108 sin(160120587119905 + 60) 08340 08324 08264 08142 08389 08338 08379 08923Max difference 00123 00122 00360 00544
minus6minus4minus2
0246
Am
plitu
de
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(a)
(200972)
(490693)(800791)
(500468)
0
05
1
Am
plitu
de
50 100 150 2000f (Hz)
(b)
Figure 4 The raw signal 119910 (a) Time domain (b) Amplitudespectrum
33 Simulation Example To verify the applicability and effec-tiveness of the CFSM-PCA algorithm a simulated sinusoidalsignal 119910 was constructed as
119910 = sin (40120587119905 + 20) + 05sin (100120587119905 + 40)+ 07sin (98120587119905 + 50) + 08sin (160120587119905 + 60)+ 119890 (119899)
(23)
where 119890(119899) is Gaussian white noise with standard deviationof 12 The result is shown in Figure 4(a) and amplitudespectrum (Fourier spectrum) is illustrated in Figure 4(b)after collecting 4096 data points with sampling frequency of1024 Hz
Follow-up work to separate frequencies via the proposedapproach was performed From eigenvalue distribution chartof crude signal in Figure 5 we can see that apart fromdiscernible four sets of eigenvalues most of eigenvaluescaused by noise are nonzero values and distribute at largeeigenvalues region
As demonstrated in Figures 6(a) and 6(b) time domainand frequency spectrum are generated after reconstructingthe first set of eigenvalues 1205821 and 1205822 in Figure 5 Frequencyof the reconstructed signal is 20 Hz with amplitude of 0977which is in correspondence with component sin(40120587t+20) ofy signal
As revealed in Figures 6(c) and 6(d) another time domainand amplitude spectrum are obtained via the reconstruction
(1515) (25144)
(3353) (43527)
(52654) (62643)
(71037) (81036)
e first group
e second group
e third group
e fourth group
0
200
400
600
Eige
nval
ue o
f PCA
i
10 20 30 40 500Number of eigenvalues q
Figure 5The eigenvalues distribution of the covariancematrixC of119910
of the second set of eigenvalues 1205823 and 1205824 Frequencyof this reconstructed signal is 80 Hz with amplitude of0786 which is same as component 08sin(160120587t+60) of rawsignal y Figures 6(e) and 6(f) are brought out through thereconstruction of the third set of eigenvalues 1205825 and 1205826Frequency of reconstructed signal is 49 Hz with amplitude of0688 which is consistent with component 07sin(98120587t+50)of crude signal 119910 Time domain and amplitude spectrum aregained after the reconstruction from the last set of eigenvaluesin Figure 5 exhibited as Figures 6(h) and 6(g) Frequency is50 Hz with amplitude of 0463 which is matched well withcomponent 05sin(100120587t+40) of signal y
Moreover several sets of signals with different signal-to-noise ratio (SNR) were processed by CFSM-PCA Theamplitude difference is summarized in Table 1
It can be seen that when the noise is low the amplitudeof frequency components extracted byCFSM-PCAalgorithmis close to that of the raw signal However the amplitudedifference will increase with the decrease of SNR
From all above-mentioned diagrams (Figure 6) it is obvi-ous that diverse frequency components extracted from rawsignal can be achieved accurately More importantly theseresults are coincident with ideal signal perfectly demon-strating that this CFSM-PCA is a zero phase shift frequencyextraction algorithm Meanwhile it should be noted that twofrequencies could also be perfectly separated via the proposedalgorithm even difference value between the adjacent fre-quency components is only 1 HzThismethod is accurate andseems to be more efficient than other existing filter methodssuch as wavelet filter and finite impulse response (FIR) filterwhich are subject to phase distorting issue
4 Experimental Verification
The fault diagnosis for large rotating machinery has been animportant topic and is attracting more and more attention
8 Shock and Vibration
the primary signalthe recovered signal
minus2
minus1
0
1
2A
mpl
itude
100 200 300 400 5000Sampling numbers i
(a)
(200977)
50 1000f (Hz)
002
04
06
08
1
Am
plitu
de
(b)
the primary signalthe recovered signal
minus1
minus05
0
05
1
Am
plitu
de
100 200 300 400 5000Sampling numbers i
(c)
(800786)
002
04
06
08
1
Am
plitu
de
50 1000f (Hz)
(d)
the primary signalthe recovered signal
100 200 300 400 5000Sampling numbers i
minus1
minus05
0
05
1
Am
plitu
de
(e)
(490688)
50 1000f (Hz)
002
04
06
08
1
Am
plitu
de
(f)
the primary signalthe recovered signal
minus05
0
05
Am
plitu
de
100 200 300 400 5000Sampling numbers i
(g)
(500463)
002
04
06
08
1
Am
plitu
de
50 1000f (Hz)
(h)
Figure 6 The time and frequency domains of reconstructed signal (a) The time domain of the reconstructed signal via the first set ofeigenvalues (b) Amplitude spectrum of Figure 6(a) (c) The time domain of the reconstructed signal using the second set of eigenvalues(d) Amplitude spectrum of Figure 6(c) (e) The time domain of the reconstructed signal using the third set of eigenvalues (f) Amplitudespectrum of Figure 6(e) (g) The time domain of the reconstructed signal using the fourth set of eigenvalues (h) Amplitude spectrum ofFigure 6(g)
Shock and Vibration 9
A B
Servo Motor Unloading device
Coupling
Sliding Bearing
Rotor
Large Platform
Small Platform
Air spring
BeltSliding Bearing
Figure 7 Large rot or vibration test bed
Displacement SensorA-end Face B-end Face
D13 D14
D12D11
Figure 8 Schematic and installation of eddy current displacement sensor
Rotor orbits indicate the symptoms of malfunction andare related to variation of input force or dynamic stiffnessGenerally different curve shapes of rotor orbit in a senserepresent different fault types [19ndash22 29] For example insideldquo8rdquo outside ldquo8rdquo and petal-shaped curves are omens of oil-filmwhirl fault misalignment fault and rubbing fault of rotorsystem respectivelyTherefore extraction of rotor orbit playsa significant role in rotating machinery diagnostics In thissection a simple and efficient CFSM-PCA is applied to purifyrotor orbit precisely
41 Experimental Rigs Figure 7 shows the large rotor vibra-tion test bed system which is constructed by our group Inorder to real-timemonitor rotor working station two KamanKD2306-1S eddy current proximity probes are installedon individual side in orthogonal manners (Figure 8) Theexperimental data were collected through LMS Test Lab
42 The First Group Sample Analysis Experimental datastems from D11 and D12 eddy current displacement sensorin side A 4096 data points were collected with test bed speedof 1080 rmin and sampling frequency of 2048 Hz The timedomain waveforms amplitude spectrums and the order ofamplitudes were obtained after filtering DC component viaFFT (Figure 9)
As demonstrated in Figure 9 raw signal was not onlyaffected by noise but also interfered by power frequency 50Hz and its harmonics frequency The energy of signal mainlyconcentrated on the leading two octaves
Rotor orbit was produced by combining spindle dis-placement signal D11 with signal D12 (Figure 10) X-axispresented raw signal D11 and Y-axis referred to signal D12Manipulating status of test bed is difficult to know from thisunprocessed rotor orbit Next the rotor orbit was attemptedto purify by employing PCA
Generally rotor orbit synthesized by 1X and 2X is cred-itable Orbits included high harmonics incline to becomecomplex and even turn messy Hence extraction of 1Xand 2X of raw signal becomes more important [21 22]Eigenvalue distribution of covariance matrix of D11 and D12signals are displayed in Figure 11 According to amplitudeof each frequency of D11 and D12 in Figure 9 1X and 2Xcorresponded to the first (first and second eigenvalues) andthe second (third and fourth eigenvalues) set of eigenvaluesrespectively The leading four eigenvalues in Figure 11 werereconstructed through CFSM-PCA and results are illustratedin Figure 12 Generated 1X and 2X were legible without effectby noise and disturbance from frequency 50 Hz as well as itsharmonics Meanwhile amplitude of the extracted signal wasclose to that of 1X and 2X in raw signal indicating that thereis no spectral leakage issue
The feature spectrums continued to be separated andeigenvalue distributionmaps of covariance matrix of D11 andD12 are shown in Figure 15
Rotor orbit generated by the purified 1X and 2X is shownin Figure 13The rotor orbit was a caved-in banana-like curvedemonstrating that sideA (near themotor) potentially suffersfrom misalignment fault [18 19] The explicit rotor orbit alsovalidates the high efficiency of this proposed method
43 The Second Group Sample Analysis In this case exper-imental data of D11 and D12 were analyzed when test bedruns steadily for a period of time Similarly 4096 datapoints were measured with test bed speed of 2770 rmin andsampling frequency of 2048Hz Time domainwaveforms andamplitude spectrums are displayed in Figure 14
According to amplitude of the leading trebling frequencyofD11 andD12 in Figure 14 1X corresponded to the second setof eigenvalues (third and fourth eigenvalues) 2X paralleledthe third set of eigenvalues (fifth and sixth eigenvalues) and
10 Shock and Vibration
D11-1080 rpmx 10
minus5
minus5minus4minus3minus2minus1
012345
X (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(a)
Power-line Interference50Hz
0733404771
Order 1 Order 2
0250502342
Order 6Order 7
D11-1080 rpm
01651 Order 11
18Hz36Hz54Hz72Hz90Hz
x 10minus5
1times
2times
3times
4times
5times
1times
2times
3times 4times5times
002040608
1
X (m
)
50 100 150 200 250 300 350 4000Frequency (Hz)
(b)
D12-1080 rpmx 10
minus5
minus6
minus4
minus2
0246
Y (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(c)
D12-1080 rpmPower-line Interference
50Hz
0658105228
Order 1 Order 2
0277101226
Order 5 Order 12
01948 Order 8
18Hz36Hz54Hz72Hz90Hz
x 10minus5
1times
2times
3times
4times
5times
1times
2times
3times
4times5times
0
05
1
Y (m
)
50 100 150 200 250 300 350 4000Frequency (Hz)
(d)
Figure 9The time domain and frequency domain of theD11 andD12 sensors in theA-end face (a)The time domain of D11 (b)The frequencydomain of D11 (c) The time domain of D12 (d) The frequency domain of D12
x 10minus5
x 10minus5
minus5
minus4
minus3
minus2
minus1
0
1
2
3
4
5
D12
minusY
(m)
minus6 minus2minus4 2 4 60D11minusX (m)
Figure 10 A-end face of the D11 and D12 sensors to collect the signal directly synthesized axis orbit
(12751) (22748)e first group
(31243) (41241)e second group
D11minus1080 rpm
x 10minus8
0
1
2
3
4
Eige
nval
ue o
f PCA
(m)
10 20 30 40 500Number of eigenvalues q
(a)
(12247) (2241)
(314) (41396)
D12minus1080 rpme first group
e second group
x 10minus8
005
115
225
Eige
nval
ue o
f PCA
(m)
10 20 30 40 500Number of eigenvalues q
(b)
Figure 11The eigenvalues distribution of covariancematrixC (a)The eigenvalues distribution of D11 signal (b)The eigenvalues distributionof D12 signal
Shock and Vibration 11
D11minus1080 rpmx 10
minus5
minus15
minus10
minus05
0051015
X (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(a)
D12minus1080 rpmx 10
minus5
minus10
minus05
0051015
Y (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(b)
D11minus1080 rpm(1807287)
(360472)
x 10minus5
002040608
1
X (m
)
50 100 150 2000Frequency (Hz)
(c)
(3605105)(1806575)
D12minus1080 rpmx 10
minus5
002040608
1
Y (m
)
50 100 150 2000Frequency (Hz)
(d)
Figure 12 Extracting the time and frequency spectrums of the first two frequencies of D11 and D12 (a)The time domain of D11 (b)The timedomain of D12 (c) The frequency domain of D11 (d) The frequency domain of D12
W1
x 10minus5
x 10minus5
minus1
minus05
0
05
1
15
D12
minusY
(m)
minus1minus15 minus05 05 10 15D11minusX (m)
(a)0
051
152
0
2
t (s)D11minusX (m)
minus2
x 10minus5
x 10minus5
minus1
minus05
0
05
1
15
D12
minusY
(m)
(b)
Figure 13 The synthesis of axis orbit is to extract the first two times fundamental frequencies of the D11 and D12 signals by the CFSM-PCA(a) The axis orbit (b) Relationship between axis orbit and time
3X matched the first set of eigenvalues (first and secondeigenvalues)
As displayed in Figure 15 eigenvalues of 1X and 3Xwere large thus 1X and 3X were utilized to synthesize rotororbit firstly Purification results and orbit of shaft center arerevealed in Figures 16 and 17 respectively The legible orbitshowed a typical plum blossom shape
It can be seen from Figures 16 and 17 that the filteringeffect was also quite obvious The noise and power frequency50 Hz as well as its harmonic interference could be removedsuccessfully Moreover amplitude of refined signal was closeto that of original one and the reconstructed time domainsignal seemed to be steady without any fluctuation
As shown in Figures 14(c) and 14(d) amplitudes of 1X 2Xand 3X in amplitude spectrum of raw signal were largeThus1X 2X and 3X were also extracted to synthesize axis orbit
Time domain frequency domain and orbit curve are shownin Figures 18 and 19 respectively
Axis orbit synthesized by the leading three frequenciesexhibited a quincunx shape which is basically the sameto that synthesized by 1X and 3X frequency componentsindicating that static and dynamic parts friction may exist inA-side
44 Comparison Harmonic Wavelet with Wavelet Packet
(1) Comparison of CFSM-PCA with Harmonic Wavelet Onecommon used method for signal processing harmonicwavelet algorithm proposed by Newland is an improvedwavelet algorithm [29] It can segment entire frequencyband infinitely and avoid the shortcoming of downsamplingmethod of binary wavelet and binary wavelet packet [22 23]
12 Shock and Vibration
D11minus2770 rpmx 10
minus4
minus1
minus05
0
05
1
D11
-X (m
)
1000 2000 3000 40000Sampling numbers i
(a)
D12minus2770 rpmx 10
minus4
minus1
minus05
0
05
1
D12
-Y (m
)
1000 2000 3000 40000Sampling numbers i
(b)
Power-Line interference 50and 150Hz
1X2X
3X 1010D11minus2770 rpm
08441166
1X2X3X
x 10minus5
200 400 600 800 10000f (Hz)
0
05
10
15
D11
-X (m
)
(c)
1X
2X
3XPower-Line interference 50
and 150Hz
D12minus2770 rpm117309021227
1X2X3X
x 10minus5
0
05
10
15
D12
-Y (m
)
200 400 600 800 10000f (Hz)
(d)
Figure 14 The time and frequency domains of the D11 and D12 sensors (a) The time domain of D11 (b) The time domain of D12 (c) Thefrequency domain of D11 (d) The frequency domain of D12
(11489) (21482)e first group
(31237) (41220)e second group
(30348) (40347)e second group
D11minus2770 rpmx 10
minus7
10 20 30 400 50Number of eigenvalues q
0
05
1
15
Eige
nval
ue o
f PCA
(m)
(a)
D12minus2770 rpm(11688) (21687)
e first group
e second group
(30414) (40413)e second group
(31673) (41664)
x 10minus7
0
05
1
15
2
Eige
nval
ue o
f PCA
(m)
10 20 30 40 500Number of eigenvalues q
(b)
Figure 15 The eigenvalues distribution of covariance matrix C (a) The eigenvalues distribution of D11 (b) The eigenvalues distribution ofD12
1X and its frequency doubling signal can be easily extractedto form axis orbit of revolving test bearing
Axis orbit was fabricated by 1X and 2Xof data of example 1(Section 42) through harmonic wavelet algorithm and theresults are demonstrated in Figure 20 The shape of axisorbit was trapped banana-like which is consistent with theresult obtained via CFSM-PCA It should be noted thatorbit in Figure 13 was more obvious than that in Figure 20ie W1ltW2 Meanwhile the amplitude fluctuation of timedomain signal was large (Figures 20(a) and 20(b)) Themajor reason for this phenomenon is that the harmonicwavelet which is a band filter in frequency domain inevitablyextracts the noise near the feature frequency Furthermorethis phenomenon also demonstrates that harmonic waveletis subject to the Heisenberg uncertainty principle ie thephenomenon of the amplitude leakage exists
(2) Comparison of CFSM-PCA with Wavelet Packet Besidesharmonic wavelets wavelet packet is also used to purify axisorbit [19ndash21] Purification results of axis orbit via Daubechieswavelet packet in Figure 21 exhibit that axis orbit was diver-gent seriously This was worse than that of harmonic waveletie W1ltW2ltW3The phenomenon of severe energy leakageoccurred during the extraction of 1X and 2X which cancause divergence of axis orbit Moreover the bandpass filterof wavelet packet also accounts for the divergent orbits
5 A Single Frequency Filtrationwith CFSM-PCA
The application in single frequency filtration is discussedin this section Taking D11 experimental data of example
Shock and Vibration 13
D11minus2770 rpmx 10
minus5
0 1000500Sampling numbers i
minus4
minus2
0
2
4
D11
-X (m
)
(a)
D12minus2770 rpmx 10
minus5
minus4
minus2
0
2
4
D12
-Y (m
)
500 10000Sampling numbers i
(b)
(1375 1185)(46 1009)
D11minus2770 rpmx 10
minus5
100 200 300 400 5000f (Hz)
0
05
1
15
D11
-X (m
)
(c)
(1375 1218)(46 1164)
D12minus2770 rpmx 10
minus5
100 200 300 400 5000f (Hz)
0
05
1
15
D12
-Y (m
)
(d)
Figure 16 Extracting the frequency spectrums of the first two frequencies of D11 and D12 by PCA algorithm (a) The time domain of D11(b) The time domain of D12 (c) The frequency spectrum of D11 (d) The frequency spectrum of D12
x 10minus5
x 10minus5
minus4
minus3
minus2
minus1
0
1
2
3
4
D12
-Y (m
)
210 3 4minus2minus3 minus1minus4
D11-X (m)
(a)0
0204
0608
10
24
t (s)D11-X (m) minus2
minus4
x 10minus5
x 10minus5
minus4
minus2
0
2
4
D12
-Y (m
)
(b)
Figure 17 The synthesis of axis orbit is to extract the first two times fundamental frequencies of the D11 and D12 signals by the CFSM-PCA(a) The axis orbit (b) Relationship between axis orbit and time
1 (Section 42) as a demo we attempt to filter the powerfrequency interference (50 Hz) from D11 by employingCFSM-PCA It can be seen from Figure 9(b) that sequenceof amplitude of 50 Hz in raw signal was 3 so eigenvec-tors corresponding to 5th and 6th eigenvalues in eigen-value distribution map were used to reconstruct Accordingto (8) the filtered signal (Figure 22) without 50 Hz was
obtained by subtracting reconstructed signal from originalone
Results in Figure 22 show that this algorithm filters out50 Hz power frequency interference and has no influenceon the other frequency components of raw signal demon-strating that this method has a potential application in notchfilter
14 Shock and Vibration
D11minus2770 rpm
x 10minus5
minus4
minus2
0
2
4
D11
-X (m
)
200 400 600 800 10000Sampling numbers i
(a)
D12minus2770 rpm
x 10minus5
0 400 600 800 1000200Sampling numbers i
minus4
minus2
0
2
4
D12
-Y (m
)
(b)
(1375 1185)(46 1009)
(92 0831)
D11minus2770 rpmx 10
minus5
00
05
10
15
D11
-X (m
)
100 200 300 400 5000f (Hz)
(c)
(1375 1218)(46 1163)
(92 0898)
D12minus2770 rpmx 10
minus5
00
05
10
15
D12
-Y (m
)
100 200 300 400 5000f (Hz)
(d)
Figure 18 Extracting the time and frequency spectrums of the first three frequencies of D11 and D12 by PCA algorithm (a)The time domainof D11 (b) The time domain of D12 (c) The frequency spectrum of D11 (d) The frequency spectrum of D12
x 10minus5
x 10minus5
minus4
minus3
minus2
minus1
0
1
2
3
4
D12
-Y (m
)
minus3 minus2 minus1minus4 1 2 3 40D11-X (m)
(a)
1008
0604
0200
02
4D11-X (m)
t (s)
minus2
minus4
x 10minus5
x 10minus5
minus4
minus2
0
2
4
D12
-Y (m
)
(b)
Figure 19 The synthesis of axis orbit is to extract the first three times fundamental frequencies of the D11 and D12 by the CFSM-PCA (a)The axis orbit (b) Relationship between axis orbit and time
6 Conclusion
The selection of effective eigenvalues in feature distributionchart of covariance matrix C is crucial in PCA algorithm Inthis paper the relationships between effective eigenvalues andsignal frequency and amplitude are discovered and a series ofconclusions are summarized
First in Hankel matrix mode each valid frequencycomponent of signal only produces two adjacent nonzeroeigenvalues Second the number of valid eigenvalues incharacteristic distribution chart which is independent ofnumerical values of 119891119894 119860 119894 and 120593119894 is related to the number offrequency componentsThird the sequence of signal effectiveeigenvalues in characteristic distribution map is determined
Shock and Vibration 15
D11-1080 rpm
x 10minus5
minus20
minus15
minus05
0
05
10
15
20
D11
-X (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(a)
D12-1080 rpmx 10
minus5
minus15
minus1
minus05
0
05
1
15
D12
-Y (m
)
500 1000 1500 2000 2500 30000 40003500Sampling numbers i
(b)
(16 07335)
(32 04771)
D11-1080 rpm
x 10minus5
10 20 30 40 50 60 70 80 90 1000f (Hz)
0010203040506070809
1
D11
-X (m
)
(c)
(16 06571)
(32 05228)
D12-1080 rpm
x 10minus5
0010203040506070809
1
D12
-Y (m
)
10 20 30 40 50 60 70 80 90 1000f (Hz)
(d)
W2
x 10minus5
x 10minus5
minus15
minus1
minus05
0
05
1
15
Y(D
12) (
m)
minus1 minus05minus15 05 1 15 20X(D11) (m)
(e)
Figure 20The purification effect of harmonic wavelets (a)The time domain of D11 (b)The time domain of D12 (c)The frequency spectrumof D11 (d) The frequency spectrum of D12 (e) The orbit is to extract the first 2 frequencies
by amplitude 119860 119894 of signal frequency component The largermagnitude of amplitude is the greater eigenvalues are andthe higher arrangement will be
Based on aforesaid principle a new CFSM-PCA wasdeveloped and has become an effective tool to extract oreliminate specific characteristic (single frequency) even dif-ference of two frequencies is only 1 Hz Purification of rotororbit using this algorithm shows significant improvement
than those of existing methods such as wavelet packet andharmonic wavelet packet Although the proposed CFSM-PCA is effective some drawbacks are unavoidable Forexample this method suffers from similar issue to otheralgorithms that is the reconstruction error increaseswith thedecrease of SNR In addition the applications of the CFSM-PCA algorithm in purification of speech recognition bearingfault diagnosis and fault diagnosis of power system should
16 Shock and Vibration
D11-1080 rpmx 10
minus5
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
minus20
minus15
minus05
0
05
10
15
20
D11
-X (m
)
(a)
D12-1080 rpmx 10
minus5
minus15
minus1
minus05
0
05
1
15
D12
-Y (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(b)
(16 06557)
(32 04381)
D11-1080 rpmx 10
minus5
10020 30 40 50 60 70 80 90100f (Hz)
0010203040506070809
1
D11
-X (m
)
(c)
(16 06452)(32 04927)
D12-1080 rpm
x 10minus5
0010203040506070809
1
D12
-Y (m
)
9020 30 40 50 60 70 8010 1000f (Hz)
(d)
W3
x 10minus5
x 10minus5
minus15
minus1
minus05
0
05
1
15
Y(D
12) (
m)
minus15 minus05minus1 05 1 15 20X(D11) (m)
(e)
Figure 21 The purification effect of wavelet packet (a) The time domain of D11 (b) The time domain of D12 (c) The frequency spectrum ofD11 (d) The frequency spectrum of D12 (e) The orbit is to extract the first 2 frequencies
be further exploredWe fully believe that the CFSM-PCAwillsparkle in diverse fields
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work is supported by the National High TechnologyResearch andDevelopment Program of China (863 Program)
Shock and Vibration 17
(18 0734)
(36 04634)
(54 02448) (72 02351)(90 01646)
x 10minus5
50 100 150 2000f (Hz)
0
02
04
06
08
1
D11
minusX
(m)
Figure 22 Frequency spectrum of D11 signal after filtering 50Hz
(no 2015AA043005) and the National Natural Science Foun-dation of China (no 51375187)
References
[1] H Abdi and L J Williams ldquoPrincipal component analysisrdquoWiley Interdisciplinary Reviews Computational Statistics vol 2no 4 pp 433ndash459 2010
[2] A Widodo B Yang and T Han ldquoCombination of independentcomponent analysis and support vectormachines for intelligentfaults diagnosis of induction motorsrdquo Expert Systems withApplications vol 32 no 2 pp 299ndash312 2007
[3] S Wold K Esbensen and P Geladi ldquoPrincipal componentanalysisrdquo Chemometrics and Intelligent Laboratory Systems vol2 no 1ndash3 pp 37ndash52 1987
[4] M Kirby and L Sirovich ldquoApplication of the Karhunen-Loeve procedure for the characterization of human facesrdquo IEEETransactions on Pattern Analysis and Machine Intelligence vol12 no 1 pp 103ndash108 1990
[5] J H Xi Y Z Han and R H Su ldquoNew fault diagnosis methodfor rolling bearing based on PCArdquo in Proceedings of the 25thChinese Control and Decision Conference CCDC rsquo13 pp 4123ndash4127 IEEE May 2013
[6] A Malhi and R X Gao ldquoPCA-based feature selection schemefor machine defect classificationrdquo IEEE Transactions on Instru-mentation and Measurement vol 53 no 6 pp 1517ndash1525 2004
[7] S Xu X Jiang J Huang S Yang and X Wang ldquoBayesianwavelet PCA methodology for turbomachinery damage diag-nosis under uncertaintyrdquo Mechanical Systems and Signal Pro-cessing vol 80 pp 1ndash18 2016
[8] W Sun J Chen and J Li ldquoDecision tree and PCA-based faultdiagnosis of rotatingmachineryrdquoMechanical Systems and SignalProcessing vol 21 no 3 pp 1300ndash1317 2007
[9] Z X Li X P Yan C Q Yuan Z Peng and L Li ldquoVirtual pro-totype and experimental research on gear multi-fault diagnosisusing wavelet-autoregressive model and principal componentanalysismethodrdquoMechanical Systems and Signal Processing vol25 no 7 pp 2589ndash2607 2011
[10] K Y Shao M M Cai and G F Zhao ldquoRolling bearing faultdiagnosis based onwavelet energy spectrum PCA andPNNrdquo inProceedings of the 26th Chinese Control andDecisionConferenceCCDC rsquo14 pp 800ndash804 IEEE 2014
[11] T Qiu Q F Zhang and Y J Ding ldquoNonlinear sensor faultdetection and data rebuilding based on principle componentanalysisrdquo Journal of Tsinghua University vol 46 no 5 pp 708ndash711 2006
[12] Y K Gu Z X Yang and F L Zhu ldquoGearbox fault feature fusionbased on principal component analysisrdquo China MechanicalEngineering vol 26 no 11 pp 1532ndash1537 2015
[13] R Dunia S J Qin T F Edgar and T J McAvoy ldquoIdentificationof faulty sensors using principal component analysisrdquo AIChEJournal vol 42 no 10 pp 2797ndash2811 1996
[14] E P de Moura C R Souto A A Silva and M A S IrmaoldquoEvaluation of principal component analysis and neural net-work performance for bearing fault diagnosis from vibrationsignal processed by RS and DF analysesrdquo Mechanical Systemsand Signal Processing vol 25 no 5 pp 1765ndash1772 2011
[15] S Golestan M Monfared F D Freijedo and J M GuerreroldquoDynamics assessment of advanced single-phase PLL struc-turesrdquo IEEE Transactions on Industrial Electronics vol 60 no6 pp 2167ndash2177 2013
[16] A Lima H Zen Y Nankaku C Miyajima K Tokuda andT Kitamura ldquoOn the use of kernel PCA for feature extractionin speech recognitionrdquo IEICE Transaction on Information andSystems vol 12 no 87 pp 2802ndash2811 2004
[17] J D Zheng ldquoImproved hilbert-huang transform and its appli-cations to rolling bearing fault diagnosisrdquo Journal of MechanicalEngineering vol 1 no 51 pp 138ndash145 2015
[18] G He K Ding and H Lin ldquoFault feature extraction of rollingelement bearings using sparse representationrdquo Journal of Soundand Vibration vol 366 pp 514ndash527 2016
[19] J G Yang S B Xia and Y G Liu ldquoWavelet denoising and itsapplication in extraction of shaft centerline orbit featuresrdquoJournal of Harbin Institute of Technology vol 31 no 5 pp 52ndash541999 (Chinese)
[20] J A Duan and X D Zhang ldquoFeature extraction from the orbitof the center of a rotor based on wavelet transformrdquo Journal ofVibration Measurement amp Diagnosis vol 17 no 01 pp 31ndash341997
[21] J Han D X Jiang and W D Ning ldquoApplication of optimalwavelet packets in failure signal extraction of rotor orbitrdquoTurbine Technology vol 43 no 3 pp 133ndash136 2001
[22] W B Zhang X J Zhou andY Lin ldquoRefinement of rotor centerrsquosorbit by a harmonic window methodrdquo Journal of VibrationMeasurement amp Diagnosis vol 30 no 1 pp 87ndash90 2010
[23] S M Li G Z Lu and Q Y Xu ldquoOn obtaining accuraterotor sub-frequency signal with harmonic waveletrdquo Journal ofNorthwestern Polytechincal University vol 19 no 2 pp 220ndash224 2001
[24] F J Wu and L S Qu ldquoAn improved method for restraining theend effect in empiricalmode decomposition and its applicationsto the fault diagnosis of large rotating machineryrdquo Journal ofSound and Vibration vol 314 no 3-5 pp 586ndash602 2008
[25] W G Liu ldquoResearch on method of purification of shaft orbitbased on singular value decompositionrdquo Technical Acousticsvol 5 no 35 pp 30ndash35 2016
[26] X Z Zhao and B Y Ye ldquoThe influence of formation manner ofcomponent on signal processingrdquo Shang Hai Jiao Tong Univer-sity vol 45 no 3 pp 368ndash374 2011
[27] X Z Zhao B Y Ye and T J Chen ldquoPrinciple of singularitydetection based on SVD and its applicationrdquo Journal of Vibra-tion and Shock vol 6 no 27 pp 7ndash10 2008
[28] X Z Zhao B Y Ye and T J Chen ldquoDifference spectrum theoryof singular value and its application to the fault diagnosis ofheadstock of latherdquo Journal of Mechanical Engineering vol 46pp 100ndash108 2010
[29] I A Kougioumtzoglou and P D Spanos ldquoAn identificationapproach for linear and nonlinear time-variant structural sys-tems via harmonic waveletsrdquo Mechanical Systems and SignalProcessing vol 37 no 1-2 pp 338ndash352 2013
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2 Shock and Vibration
the extraction of certain frequency have been developed suchas wavelet packet transform harmonic wavelet ensembleempirical mode decomposition (EEMD) and sparse decom-position [18] For instance references [19ndash21] adopted mul-tilevel division technique of wavelet packet to select certainfrequency band for the extraction of specific frequency fromwhich axis orbit could be manufactured References [22 23]subdivided random frequency band infinity via harmonicwavelet to extract interesting frequency the refinement ofrotor centerrsquos orbit from one or more interesting frequencybands could be realized subsequently Nevertheless waveletpacket and harmonic packet algorithm are subject to theHeisenberg uncertainty principle and resolutions of timedomain 120590119905 and frequency domain 120590120596 could not be randomlyhigh simultaneously ie 1205902119905 sdot 1205902120596 ge 14 In addition in EEMDmethod signals are adaptively decomposed into several sumsof intrinsic mode functions (IMFs) whose instantaneousfrequencies have physical meanings In practice the IMFis always multicomponent rather than a single componentresulting in unexplainable irregularity in its instantaneousamplitude and blind extraction of the 1X 2X and the othersubharmonics So the EEMD method is not suitable for thedecomposition of signals with multiple components in anarrow band [24] By using singular value decomposition(SVD) strategy reference [25] generated axis orbit by meansof cumulative contribution rate to denoise noisy signalThis method is unable to extract the specified feature fre-quency Therefore other more effective and simple methodswhich suffer free from above disadvantages remain to beexplored
Our group has been committed to studying the faultdiagnosis of large scale equipment [26ndash28] During thecourse of single frequency simulation via PCA an interestingphenomenon was discovered unexpectedly ie a frequencycomponent produces two eigenvalues After intensive studywe found that PCA algorithm could be used to extractthe specified single or multiple feature frequencies from acrude signal Guidelines are summarized as follows (1) Eachcharacteristic frequency in a signal produces only two valideigenvalues (2)Thenumber of effective eigenvalues is relatedto the quantity of raw signal frequencies and has nothing todo with the magnitude of 119891119894 119860 119894 and 120593119894 (3) The sequenceof eigenvalues of covariance matrix in its distribution chartis determined by the amplitude of feature frequency Forthese discoveries a novel frequency separationmethod basedon PCA was proposed through which axis orbits of largerotating machines were readily purified Moreover purifica-tion results are better than that of existing methods such aswavelet packet and harmonic wavelet
Hereafter the paper is organized as follows Section 2briefly introduces the basic principle PCA theory in sig-nal processing and a new method of signal recovery isintroduced In Section 3 the theoretical discovery and thetheoretical verifications of relationship between eigenvaluesand feature frequency are given Section 4 illustrates theapplication on purification of axis orbit of large rotor testbed and compares experimental results with that of harmonicwavelet and wavelet packet proving its high efficiency of theproposed scheme The filtration of single frequency is givenin Section 5 Finally Section 6 draws the conclusions
2 Basic Theories of PCA in Signal Processing
We assume that there are 119898 random vectors (x1 x2 sdot sdot sdot x119898)with each vector x119894 containing 119899 samples (x119894 = (1199091198941 1199091198942 sdot sdot sdot119909119894119899)) An119898 times 119899matrix X with119898 rows and 119899 columns can bedescribed as
X =[[[[[[[
11990911 11990912 sdot sdot sdot 119909111989911990921 11990922 sdot sdot sdot 1199092119899 1199091198981 1199091198982 sdot sdot sdot 119909119898119899
]]]]]]]
(2)
where X = [x1 x2 sdot sdot sdot x119898]T and T denotes the vector trans-pose Supposing that x1 x2 sdot sdot sdot x119898 represent the indicatrix ofcrude variables then 119897 new variables y119894(119894 = 1 2 sdot sdot sdot 119897 isin 119885+) (119897 ⩽119898) could be obtained after principal component analysis ofXdescribed as
y119894 = 1205721198941x1 + 1205721198942x2 + sdot sdot sdot + 120572119894119898x119898 = 120572T119894 X (3)
where y119894 isin R1times119899 According to the definition of PCA [3]120572119894 = (1205721198941 1205721198942 sdot sdot sdot 120572119894119898)T (119894 = 1 2 sdot sdot sdot 119897) and 120572119894 are eigenvectorscorresponding to the ith eigenvalue in descending orderin the covariance matrix of X [2ndash4 6] and (4) should besatisfied
120572T119894 120572119895 = 1 119894 = 119895120572T119894 120572119895 = 0 119894 = 119895
(4)
Covariance matrix of X in (2) is
C =[[[[[[[
cov (x1 x1) cov (x1 x2) sdot sdot sdot cov (x1 x119898)cov (x2 x1) cov (x2 x2) sdot sdot sdot cov (x2 x119898)
cov (x119898 x1) cov (x119898 x2) sdot sdot sdot cov (x119898 x119898)
]]]]]]]
(5)
where cov(x119894 x119895) = 119864[(x119894minus119864(x119894))(x119895minus119864(x119895))T] Based on PCAtheory characteristic equation of covariance matrix C can begiven by
C120572119894 = 120582119894120572119894 (6)
where 120582119894 are eigenvalues of covariance matrix C and 120572119894are eigenvectors corresponding to 120582119894 Given that eigenvaluesranged in descending order that is 1205821 gt 1205822 gt sdot sdot sdot gt 120582119897 datadimensionality reduction can be achieved by using (1) and(3) Then original m variables are converted to new 119897 onesIf signal processing is performed the signal reconstruction isneeded Considering covariance matrix C is a semi-positivesymmetric matrix its eigenvectors are orthogonal to eachother ie sum119898119894=1 120572119894120572T119894 = 119868119898 [1 2] After left multiplication onboth sides by120572119894 of (3) and sum calculation (7) can be given asfollows
119898
sum119894=1
120572119894y119894 =119898
sum119894=1
120572119894120572T119894 X = 119868119898X (7)
Shock and Vibration 3
If the former 119897 principal components are chosen toreconstruct in the light of cumulative contribution rate 119871 119897 anapproximate matrix X can be formulated as
X =119897
sum119894=1
120572119894y119894 =119897
sum119894=1
120572119894120572T119894 X (8)
Compared with original matrix X the reconstructedapproximate matrix X comprises most of information of Xand excludes redundant features such as noise and powerfrequency interference [9 10]
Signal could be recovered from the reconstructed matrixX in terms of matrix composition mode XT is converted to1 row 119898119899 columns vector with the row vectors arranging inhead-to-tail fashion then a new vector is obtained recordedas a a isin 1198771times119898119899 The recovered signal x is obtained via inversetransformation of vector a derived as
x = 119886M+ (9)
where x isin 1198771times119871 L=m+n-1 L is the length of recovered dataM+ = (MTM)minus1MT isin 119877119898119899times119871 and M+ is pseudoinverse ofMM isin 119877119871times119898119899 andM comprises119898 unit matrices where therank of each unit matrix is 119899 Takingm=3 n=3 as an examplein this case L=m+n-1=5 and mn=9 the matrix M can beexpressed as follows
=
[[[[[[[[[[
1
1 1
1 1 1
1 1
1
]]]]]]]]]]
M (10)
The signal x is recovered from the pseudoinverse matrixM+ actually it is to compute the average values of eachelement at counter-diagonal of matrix X which is consistentwith the method reported in [23] SinceM is a sparse matrixespecially when values of 119898 and 119899 are large random-accessmemory and time for calculating of pseudoinverse matrixM+ will increase exponentially Hence signal recovered fromsimple method is extremely desired With this aim in mindwe have developed a new averagingmethod and adopted it torecover signal x from matrix X The expression is shown asfollows
119909 (119896) =
119896
sum119894=1
119909119894119896+1minus119894119896 1 le 119896 lt 119898
119898
sum119894=1
119909119894119896+1minus119894119898 119898 le 119896 lt 119899119898
sum119894=119896minus119899+1
119909119894119896+1minus119894(119898 + 119899 minus 119896) 119899 lt 119896 le 119898 + 119899 minus 1
(11)
where 119909119894119895 are element of the 119894th row and the 119895th column inthe reconstructed approximate matrix X The signal x can berestored facilely according to (11)
3 Internal Law of Effective Eigenvalues andFrequency Components
31The Process of Feature Frequency SeparationMethod Basedon PCA For noise-free signal 119909(119905) the effective eigenvaluesare nonzero ones It can be seen from (7) that signal 119909(119905) has119898 eigenvectors which aremostly generated by noise So for asignal containing certain number of frequency componentswhat is the relationship between the number of eigenvaluesand the number of frequencies
During our research course an important connectionbetween nonzero eigenvalues and the number of frequencycomponents was discovered To illustrate this connectionexplicitly signal with different amplitude 119860 119894 frequency 120596119894and phase 120593119894 was constructed as follows
119909 (119905) =119896
sum119894=1
119860 119894 sin (2120587120596119894119905 + 120593119894) (12)
where 119896 is the number of frequency components4096 data points were collected with sampling frequency
of 1024 Hz and then the Hankel matrix was formed fromsignal 119909(119905) with m rows and n columns Decomposition andreconstruction of signal 119909(119905) were proceeded by employingPCA algorithm in Section 2 [9 13] Effect of the constructedHankel matrix on signal processing was studied by Zhao etal [26] they pointed out that if the number of rows was closeto the number of columns signal processing effect was betterFurthermore when Lwas an evenm=L2 and n=L2+1 whenL was an oddm= (L+1)2 and n= (L+1)2 Hence in our casem = 2048 and n = 2049 were applied The decompositionprocedure consists of the following(1)Given k = 1119860 119894= 1 tri-groups signals were constructed
and their principal component eigenvalue distribution mapsare shown in Figure 1 The number of eigenvalues 119902 rangefrom 1 to 2048 and just the leading 50 eigenvalues are listedin this case(2) Given k = 2 119860 119894= 1 08 tri-groups signals were
reconstructed and each group signal contained two effectivefrequency components with119891119894 as 20Hz and 30Hz 50Hz and60 Hz and 80 Hz and 90 Hz respectively and 120593119894 were 10 and20 40 and 50 and 70 and 80 respectively
When k = 1 each signal contains only one effectivefrequency component These three sets of signals have thesame amplitude119860 119894 but different frequency120596119894 and phase120593119894 Ascan be seen from Figure 1 each signal produces two adjacentnonzero eigenvalues 1205821 and 1205822 Although tri-groups signalsare different nonzero eigenvalues of them are the same toeach other
It can be seen from eigenvalue distribution graphs inFigure 2 that each signal produces two sets of nonzeroeigenvalues each set of eigenvalues contains two adjacenteigenvalues 120582119894 and 120582119894+1 Interestingly the four nonzeroeigenvalues produced from each signal are equal to theircorresponding eigenvalues generated by the other signalseven if the signals are different
Moreover by comparing Figure 2 with Figure 1 the firstset of eigenvalues in both two graphs are exactly sameActually signals in Figure 2 are established by adding an extra
4 Shock and Vibration
(15123) (25118)e first group
(15123) (25118)
(15123) (25118)
e first group
e first group
0
200
400
600
0
200
400
600
10 20 30 40 500Number of eigenvalues q
10 20 30 40 500
10 20 30 40 500
x1(t) = MCH(40t + 10)
x2(t) = MCH(100t + 40)
x3(t) = MCH(160t + 70)
0
200
400
600
Eige
nval
ue o
f PCA
i
Figure 1 Eigenvalues distribution of covariance matrix C with one frequency component
(15123) (25118)
(15123) (25118)
(15123) (25118)
(33278) (43275)
(33278) (43275)
(33278) (43275)
e first group
e second group
e first group
e second group
e first group
e second group
0
200
400
600
10 20 30 40 500
10 20 30 40 500
0
200
400
600
10 20 30 40 500Number of eigenvalues q
x1(t) = MCH(40t + 10) + 08 MCH(60t + 20)
x3(t) = MCH(160t + 70) + 08 MCH(180t + 80)
0
200
400
600
Eige
nval
ue o
f PCA
i
x2(t) = MCH(100t + 40) + 08 MCH(120t + 50)
Figure 2 Eigenvalues distribution of covariance matrix C with twofrequency components
frequency component with amplitude of 08 Thus it couldbe confirmed that the first set of nonzero eigenvalues 1205821 and1205822 in Figure 2 are generated from frequency component withamplitude of 10 while the second set of nonzero eigenvalues
are generated from frequency component with amplitude of08(3)Given 119896= 3119860 119894 = 1 08 and 06 tri-groups signals were
reconstructed and each group signal contained three sets ofeffective frequency components The frequencies of the firstgroup of signal were 20 Hz 30 Hz and 40 Hz respectivelyThe frequencies of the second group of signals were 50 Hz60 Hz and 70 Hz respectively Similarly the frequencies ofthe third group of signals were 80 Hz 90 Hz and 100 Hzrespectively The corresponding phases of these three groupsof signals were taken as 10 20 and 30 40 50 and 60 and 7080 and 90 respectively
When 119896 = 3 119860 119894=1 08 and 06 each group signal pro-duces three effective frequency components The amplitudes119860 119894 of corresponding frequency 119891119894(119894=1 2 3) are same butwith different frequencies 120596119894 and phases 120593119894 As displayed inFigure 3 each group signal produces three sets of nonzeroeigenvalues and each set of eigenvalues contains two adjacenteigenvalues 120582119894 and 120582119894+1 In addition these three sets ofeigenvalues produced by each signal are correspondinglysame
It can be found from Figures 1ndash3 that the magnitudeof eigenvalues of each first group in three graphs is sameAs aforementioned the first set of nonzero eigenvalues1205821 and 1205822 are generated from frequency component withamplitude 119860 119894 of 1 Similarly by comparing Figure 2 withFigure 1 the second set of nonzero eigenvalues 1205823 and 1205824are generated from frequency component with amplitude119860 119894 of 08 And then it is almost certain that the third setof frequency components 1205825 and 1205826 are generated fromfrequency component with amplitude 119860 119894 of 06
The same results can be obtained by continuously increas-ing effective frequency components of signal Therefore an
Shock and Vibration 5
(33278) (43275)
(51844) (61842)
(15123) (25118)
(33278) (43275)
(51844) (61842)
(15123) (25118)
(33278) (43275)
(51844) (61842)
(15123) (25118)e first group
e second group
e third group
e first group
e second group
e third group
e first group
e second group
e third group
x1(t) = MCH(40t + 10) + 08 MCH(60t + 20) + 06 MCH(80t + 30)
x2(t) = MCH(100t + 40) + 08 MCH(120t + 50) + 06 MCH(140t + 60)
x3(t) = MCH(160t + 70) + 08 MCH(180t + 80) + 06 MCH(200t + 90)
0
200
400
600
10 20 30 40 500
0
200
400
600
Eige
nval
ue o
f PCA
i
10 20 30 40 500
0
200
400
600
10 20 30 40 500Number of eigenvalues q
Figure 3 Eigenvalues distribution of covariancematrixCwith threefrequency components
Hankel matrix is derived from certain signal x(t) l=min(mn) with m rows n columns and k effective frequenciesConcerning the fact that the Shannon sampling theorem ismet assuming that lgt2k generic conclusions are summarizedas follows(1) Each frequency component of signal produces two
nonzero eigenvalues with one arranging another closely(2) The number of effective eigenvalues of crude signal
is related to the number of frequency components and has
nothing to do with themagnitude of amplitude119860 119894 frequency120596119894 and phase 120593119894(3) In eigenvalue distribution chart of covariance matrix
C the sequence of nonzero effective eigenvalues is deter-mined by amplitude 119860 119894 of signal The larger amplitude is thelarger eigenvalues will be the more forward rank of the twoeigenvalues produced from corresponding frequencies is
Inspired by the relationship between frequency andeigenvalues a new technique for characteristic frequency sep-aration method based on PCA (CFSM-PCA) was proposedThe concrete steps are listed below(1) For a certain signal 119909(119905) direct component (DC) of
raw signal is filtered out via fast Fourier transform (FFT)firstly and then Hankel matrix X is constructed throughfiltered signal(2) Covariance matrix C of matrix X is obtained with its
eigenvalues 120582119894 arranging in descending order (1205821 1205822 sdot sdot sdot 120582119898)and corresponding eigenvectors are obtained as 12057211205722 sdot sdot sdot120572119898(3) According to the distribution of eigenvalues 120582119894reconstruction is carried out from two eigenvalues and cor-responding eigenvectors of certain frequency For examplefor amplitude perspective if the rank of specific frequencyof raw signal is k a new matrix is received by reconstructingthe eigenvectors corresponding to 2k-1 and 2k eigenval-ues in eigenvalue distribution chart of covariance matrixC(4)Thematrix X can be produced by adding the mean of
original matrix to the new reconstructed matrix(5) The signal x which is the characteristic frequency
component is recovered from matrix X by means of theaveraging method
32Theoretical Deduction In this section deduction processof the three discoveries (Section 31) is provided Supposingthat a signal is expressed as 119909(119905) = 119886 sin(120596119905 + 120593) Samplingtime 119879119904 is used to discretize signal x(t) Hankel matrix withm rows and n columns is derived from signal 119909(119905) exhibitedas
X = 119886 times[[[[[[
sin (0 sdot 120596119879119904 + 120593) sin (1 sdot 120596119879119904 + 120593) sdot sdot sdot sin ((119899 minus 1) sdot 120596119879119904 + 120593)sin (1 sdot 120596119879119904 + 120593) sin (2 sdot 120596119879119904 + 120593) sdot sdot sdot sin (119899 sdot 120596119879119904 + 120593)
sin ((119898 minus 1) sdot 120596119879119904 + 120593) sin (119898 sdot 120596119879119904 + 120593) sdot sdot sdot sin ((119898 + 119899 minus 2) sdot 120596119879119904 + 120593)
]]]]]]
(13)
(1) Each characteristic frequency produces two effectiveeigenvalues
The deduced process of the first conclusion is given belowEquation (13) can be rewritten as (14) based onEulerrsquos Formula
X = 1198862119894
times[[[[[[[
119890(0120596119879119904+120593)119894 minus 119890minus(0120596119879119904+120593)119894 119890(1120596119879119904+120593)119894 minus 119890minus(1120596119879119904+120593)119894 119890(2120596119879119904+120593)119894 minus 119890minus(2120596119879119904+120593)119894 sdot sdot sdot 119890((119899minus1)120596119879119904+120593)119894 minus 119890minus((119899minus1)120596119879119904+120593)119894119890(1120596119879119904+120593)119894 minus 119890minus(1120596119879119904+120593)119894 119890(2120596119879119904+120593)119894 minus 119890minus(2120596119879119904+120593)119894 119890(3120596119879119904+120593)119894 minus 119890minus(3120596119879119904+120593)119894 sdot sdot sdot 119890(119899120596119879119904+120593)119894 minus 119890minus(119899120596119879119904+120593)119894
119890((119898minus1)120596119879119904+120593)119894 minus 119890minus((119898minus1)120596119879119904+120593)119894 119890(119898120596119879119904+120593)119894 minus 119890minus(119898120596119879119904+120593)119894 119890((119898+1)120596119879119904+120593)119894 minus 119890minus((119898+1)120596119879119904+120593)119894 sdot sdot sdot 119890((119898+119899minus2)120596119879119904+120593)119894 minus 119890minus((119898+119899minus2)120596119879119904+120593)119894
]]]]]]]
(14)
6 Shock and Vibration
Equation (14) can be expanded to addition form of twoformulas depicted as
X = 1198862119894 times 119890120593119894
times[[[[[[[[
1198900120596119879119904119894 1198901120596119879119904119894 1198902120596119879119904119894 sdot sdot sdot 119890(119899minus1)1205961198791199041198941198901120596119879119904119894 1198902120596119879119904119894 1198903120596119879119904119894 sdot sdot sdot 119890119899120596119879119904119894
119890(119898minus1)120596119879119904119894 119890119898120596119879119904119894 119890(119898+1)120596119879119904119894 sdot sdot sdot 119890(119898+119899minus2)120596119879119904119894
]]]]]]]]
minus 1198862119894 times 119890minus120593119894
times[[[[[[[[
119890minus0120596119879119904119894 119890minus1120596119879119904119894 119890minus2120596119879119904119894 sdot sdot sdot 119890minus(119899minus1)120596119879119904119894119890minus1120596119879119904119894 119890minus2120596119879119904119894 119890minus3120596119879119904119894 sdot sdot sdot 119890minus119899120596119879119904119894
119890minus(119898minus1)120596119879119904119894 119890minus119898120596119879119904119894 119890minus(119898+1)120596119879119904119894 sdot sdot sdot 119890minus(119898+119899minus2)120596119879119904119894
]]]]]]]](15)
From (15) the rank of both matrices is 1 Based on rankrelationship of two matrices 119877(119860+119861) le 119877(119860) + 119877(119861) Hencerank of matrix X is less than or equal to 2
Additionally when 120593 = 0 (13) is rewritten to (16)According to sum-to-product identities the rank of X is 2when 120596119879119904 = 0 120587 2120587 sdot sdot sdot
X = 119886 times[[[[[[[[[
sin (0 sdot 120596119879119904) sin (1 sdot 120596119879119904) sdot sdot sdot sin ((119899 minus 1) sdot 120596119879119904)sin (1 sdot 120596119879119904) sin (2 sdot 120596119879119904) sdot sdot sdot sin (119899 sdot 120596119879119904)
sin ((119898 minus 1) sdot 120596119879119904) sin (119898 sdot 120596119879119904) sdot sdot sdot sin ((119898 + 119899 minus 2) sdot 120596119879119904)
]]]]]]]]]
(16)
When 120593 = 0 (13) can be deduced into the first-orderprincipal minor of matrix X ie |sin120593| = 0 The principalminor of order 2 of X is shown as10038161003816100381610038161003816100381610038161003816100381610038161003816sin (0 sdot 120596119879119904 + 120593) sin (1 sdot 120596119879119904 + 120593)sin (1 sdot 120596119879119904 + 120593) sin (2 sdot 120596119879119904 + 120593)
10038161003816100381610038161003816100381610038161003816100381610038161003816= minussin2 (120596119879119904)
= 0(17)
where120596119879119904 = 0 120587 2120587 sdot sdot sdot Because the leading principalminorof order 2 ofX in (13) is nonzero values so the rank of matrixX is at least 2
Combining result of (15) 119877(119860 + 119861) le 2 and nonzerovalues of the leading principal submatrix of order 2 hencethe rank of matrix X is 2 Matrix X has two eigenvalues 12058210158401and 12058210158402 Derived from C = XXT119899 (where 119899 is the numberof columns of matrix X) [3] two nonzero eigenvalues ofcovariance matrix C of X are 1205821 and 1205822 respectively(2)The sequence of eigenvalues is determined by ampli-
tudeIn terms of PCA covariance matrix C is described as
C = 119864 [(X minus 119864 (X)) sdot (X minus 119864 (X))T] (18)
Referring to (6) (19) is constructed as follows
119897
sum119894=1
C120572119894120572T119894 =119897
sum119894=1
120582119894120572119894120572T119894 (19)
Two effective eigenvalues are generated from a frequencycomponent that is l=2
CI119898 = 12058211205721120572T1 + 12058221205722120572T2 (20)
Then the energy of matrix C can be deduced as
|C|2 = 12058221119898
sum119894=1
12057221198941119898
sum119894=1
12057221198941 + 12059022119898
sum119894=1
12057221198942119898
sum119895=1
12057221198942
+ 21205902112059022119898
sum119894=1
12057211989411205721198942119898
sum119894=1
12057211989411205721198943(21)
Based upon (4) sum119898119894=1 12057221198941 = 1 sum119898119894=1 12057221198942 = 1 sum119898119894=1 12057211989411205721198942 =0 Then (22) is given by
|C|2 = 12058221 + 12058222 (22)
The constructed Hankel matrix X of signal 119909(119905) is sub-stituted into (18) we can see that the energy of covariancematrix C is proportional to 1198862 The larger 1198862 is the greaterenergy of matrix C is as well as 120582119894 according to (22)Furthermore the larger frequency component amplitude isthe larger corresponding eigenvalue in covariance matrix Ccharacteristic distribution chart is
Based on these conclusions once the amplitude sequenceof a certain frequency in raw signal amplitude spec-trum is determined its corresponding frequency compo-nent can be reconstructed In this way extraction of sin-gle or multiple characteristic frequencies could be real-ized
In view of addition relation as shown in (8) the notchfilter could be achieved through CFSM-PCA The frequencycomponent extracted by this algorithm is subtracted in orig-inal signal ie this frequency component can be eliminatedin raw signal Reader could consult examples in Section 5 formore details
Shock and Vibration 7
Table 1 Amplitude of signal components under the different SNR
SNR SNR1=-1 SNR2=-4 SNR3=-7 SNR4=-10Signal component Raw Process Raw Process Raw Process Raw Processsin(40120587119905 + 20) 09999 09938 10260 10240 10370 10730 1114 111205 sin(100120587119905 + 40) 04703 04826 05317 05482 07353 07533 05177 0523407 sin(98120587119905 + 50) 07299 07152 07116 07235 05110 04968 06800 0662108 sin(160120587119905 + 60) 08340 08324 08264 08142 08389 08338 08379 08923Max difference 00123 00122 00360 00544
minus6minus4minus2
0246
Am
plitu
de
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(a)
(200972)
(490693)(800791)
(500468)
0
05
1
Am
plitu
de
50 100 150 2000f (Hz)
(b)
Figure 4 The raw signal 119910 (a) Time domain (b) Amplitudespectrum
33 Simulation Example To verify the applicability and effec-tiveness of the CFSM-PCA algorithm a simulated sinusoidalsignal 119910 was constructed as
119910 = sin (40120587119905 + 20) + 05sin (100120587119905 + 40)+ 07sin (98120587119905 + 50) + 08sin (160120587119905 + 60)+ 119890 (119899)
(23)
where 119890(119899) is Gaussian white noise with standard deviationof 12 The result is shown in Figure 4(a) and amplitudespectrum (Fourier spectrum) is illustrated in Figure 4(b)after collecting 4096 data points with sampling frequency of1024 Hz
Follow-up work to separate frequencies via the proposedapproach was performed From eigenvalue distribution chartof crude signal in Figure 5 we can see that apart fromdiscernible four sets of eigenvalues most of eigenvaluescaused by noise are nonzero values and distribute at largeeigenvalues region
As demonstrated in Figures 6(a) and 6(b) time domainand frequency spectrum are generated after reconstructingthe first set of eigenvalues 1205821 and 1205822 in Figure 5 Frequencyof the reconstructed signal is 20 Hz with amplitude of 0977which is in correspondence with component sin(40120587t+20) ofy signal
As revealed in Figures 6(c) and 6(d) another time domainand amplitude spectrum are obtained via the reconstruction
(1515) (25144)
(3353) (43527)
(52654) (62643)
(71037) (81036)
e first group
e second group
e third group
e fourth group
0
200
400
600
Eige
nval
ue o
f PCA
i
10 20 30 40 500Number of eigenvalues q
Figure 5The eigenvalues distribution of the covariancematrixC of119910
of the second set of eigenvalues 1205823 and 1205824 Frequencyof this reconstructed signal is 80 Hz with amplitude of0786 which is same as component 08sin(160120587t+60) of rawsignal y Figures 6(e) and 6(f) are brought out through thereconstruction of the third set of eigenvalues 1205825 and 1205826Frequency of reconstructed signal is 49 Hz with amplitude of0688 which is consistent with component 07sin(98120587t+50)of crude signal 119910 Time domain and amplitude spectrum aregained after the reconstruction from the last set of eigenvaluesin Figure 5 exhibited as Figures 6(h) and 6(g) Frequency is50 Hz with amplitude of 0463 which is matched well withcomponent 05sin(100120587t+40) of signal y
Moreover several sets of signals with different signal-to-noise ratio (SNR) were processed by CFSM-PCA Theamplitude difference is summarized in Table 1
It can be seen that when the noise is low the amplitudeof frequency components extracted byCFSM-PCAalgorithmis close to that of the raw signal However the amplitudedifference will increase with the decrease of SNR
From all above-mentioned diagrams (Figure 6) it is obvi-ous that diverse frequency components extracted from rawsignal can be achieved accurately More importantly theseresults are coincident with ideal signal perfectly demon-strating that this CFSM-PCA is a zero phase shift frequencyextraction algorithm Meanwhile it should be noted that twofrequencies could also be perfectly separated via the proposedalgorithm even difference value between the adjacent fre-quency components is only 1 HzThismethod is accurate andseems to be more efficient than other existing filter methodssuch as wavelet filter and finite impulse response (FIR) filterwhich are subject to phase distorting issue
4 Experimental Verification
The fault diagnosis for large rotating machinery has been animportant topic and is attracting more and more attention
8 Shock and Vibration
the primary signalthe recovered signal
minus2
minus1
0
1
2A
mpl
itude
100 200 300 400 5000Sampling numbers i
(a)
(200977)
50 1000f (Hz)
002
04
06
08
1
Am
plitu
de
(b)
the primary signalthe recovered signal
minus1
minus05
0
05
1
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plitu
de
100 200 300 400 5000Sampling numbers i
(c)
(800786)
002
04
06
08
1
Am
plitu
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50 1000f (Hz)
(d)
the primary signalthe recovered signal
100 200 300 400 5000Sampling numbers i
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0
05
1
Am
plitu
de
(e)
(490688)
50 1000f (Hz)
002
04
06
08
1
Am
plitu
de
(f)
the primary signalthe recovered signal
minus05
0
05
Am
plitu
de
100 200 300 400 5000Sampling numbers i
(g)
(500463)
002
04
06
08
1
Am
plitu
de
50 1000f (Hz)
(h)
Figure 6 The time and frequency domains of reconstructed signal (a) The time domain of the reconstructed signal via the first set ofeigenvalues (b) Amplitude spectrum of Figure 6(a) (c) The time domain of the reconstructed signal using the second set of eigenvalues(d) Amplitude spectrum of Figure 6(c) (e) The time domain of the reconstructed signal using the third set of eigenvalues (f) Amplitudespectrum of Figure 6(e) (g) The time domain of the reconstructed signal using the fourth set of eigenvalues (h) Amplitude spectrum ofFigure 6(g)
Shock and Vibration 9
A B
Servo Motor Unloading device
Coupling
Sliding Bearing
Rotor
Large Platform
Small Platform
Air spring
BeltSliding Bearing
Figure 7 Large rot or vibration test bed
Displacement SensorA-end Face B-end Face
D13 D14
D12D11
Figure 8 Schematic and installation of eddy current displacement sensor
Rotor orbits indicate the symptoms of malfunction andare related to variation of input force or dynamic stiffnessGenerally different curve shapes of rotor orbit in a senserepresent different fault types [19ndash22 29] For example insideldquo8rdquo outside ldquo8rdquo and petal-shaped curves are omens of oil-filmwhirl fault misalignment fault and rubbing fault of rotorsystem respectivelyTherefore extraction of rotor orbit playsa significant role in rotating machinery diagnostics In thissection a simple and efficient CFSM-PCA is applied to purifyrotor orbit precisely
41 Experimental Rigs Figure 7 shows the large rotor vibra-tion test bed system which is constructed by our group Inorder to real-timemonitor rotor working station two KamanKD2306-1S eddy current proximity probes are installedon individual side in orthogonal manners (Figure 8) Theexperimental data were collected through LMS Test Lab
42 The First Group Sample Analysis Experimental datastems from D11 and D12 eddy current displacement sensorin side A 4096 data points were collected with test bed speedof 1080 rmin and sampling frequency of 2048 Hz The timedomain waveforms amplitude spectrums and the order ofamplitudes were obtained after filtering DC component viaFFT (Figure 9)
As demonstrated in Figure 9 raw signal was not onlyaffected by noise but also interfered by power frequency 50Hz and its harmonics frequency The energy of signal mainlyconcentrated on the leading two octaves
Rotor orbit was produced by combining spindle dis-placement signal D11 with signal D12 (Figure 10) X-axispresented raw signal D11 and Y-axis referred to signal D12Manipulating status of test bed is difficult to know from thisunprocessed rotor orbit Next the rotor orbit was attemptedto purify by employing PCA
Generally rotor orbit synthesized by 1X and 2X is cred-itable Orbits included high harmonics incline to becomecomplex and even turn messy Hence extraction of 1Xand 2X of raw signal becomes more important [21 22]Eigenvalue distribution of covariance matrix of D11 and D12signals are displayed in Figure 11 According to amplitudeof each frequency of D11 and D12 in Figure 9 1X and 2Xcorresponded to the first (first and second eigenvalues) andthe second (third and fourth eigenvalues) set of eigenvaluesrespectively The leading four eigenvalues in Figure 11 werereconstructed through CFSM-PCA and results are illustratedin Figure 12 Generated 1X and 2X were legible without effectby noise and disturbance from frequency 50 Hz as well as itsharmonics Meanwhile amplitude of the extracted signal wasclose to that of 1X and 2X in raw signal indicating that thereis no spectral leakage issue
The feature spectrums continued to be separated andeigenvalue distributionmaps of covariance matrix of D11 andD12 are shown in Figure 15
Rotor orbit generated by the purified 1X and 2X is shownin Figure 13The rotor orbit was a caved-in banana-like curvedemonstrating that sideA (near themotor) potentially suffersfrom misalignment fault [18 19] The explicit rotor orbit alsovalidates the high efficiency of this proposed method
43 The Second Group Sample Analysis In this case exper-imental data of D11 and D12 were analyzed when test bedruns steadily for a period of time Similarly 4096 datapoints were measured with test bed speed of 2770 rmin andsampling frequency of 2048Hz Time domainwaveforms andamplitude spectrums are displayed in Figure 14
According to amplitude of the leading trebling frequencyofD11 andD12 in Figure 14 1X corresponded to the second setof eigenvalues (third and fourth eigenvalues) 2X paralleledthe third set of eigenvalues (fifth and sixth eigenvalues) and
10 Shock and Vibration
D11-1080 rpmx 10
minus5
minus5minus4minus3minus2minus1
012345
X (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(a)
Power-line Interference50Hz
0733404771
Order 1 Order 2
0250502342
Order 6Order 7
D11-1080 rpm
01651 Order 11
18Hz36Hz54Hz72Hz90Hz
x 10minus5
1times
2times
3times
4times
5times
1times
2times
3times 4times5times
002040608
1
X (m
)
50 100 150 200 250 300 350 4000Frequency (Hz)
(b)
D12-1080 rpmx 10
minus5
minus6
minus4
minus2
0246
Y (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(c)
D12-1080 rpmPower-line Interference
50Hz
0658105228
Order 1 Order 2
0277101226
Order 5 Order 12
01948 Order 8
18Hz36Hz54Hz72Hz90Hz
x 10minus5
1times
2times
3times
4times
5times
1times
2times
3times
4times5times
0
05
1
Y (m
)
50 100 150 200 250 300 350 4000Frequency (Hz)
(d)
Figure 9The time domain and frequency domain of theD11 andD12 sensors in theA-end face (a)The time domain of D11 (b)The frequencydomain of D11 (c) The time domain of D12 (d) The frequency domain of D12
x 10minus5
x 10minus5
minus5
minus4
minus3
minus2
minus1
0
1
2
3
4
5
D12
minusY
(m)
minus6 minus2minus4 2 4 60D11minusX (m)
Figure 10 A-end face of the D11 and D12 sensors to collect the signal directly synthesized axis orbit
(12751) (22748)e first group
(31243) (41241)e second group
D11minus1080 rpm
x 10minus8
0
1
2
3
4
Eige
nval
ue o
f PCA
(m)
10 20 30 40 500Number of eigenvalues q
(a)
(12247) (2241)
(314) (41396)
D12minus1080 rpme first group
e second group
x 10minus8
005
115
225
Eige
nval
ue o
f PCA
(m)
10 20 30 40 500Number of eigenvalues q
(b)
Figure 11The eigenvalues distribution of covariancematrixC (a)The eigenvalues distribution of D11 signal (b)The eigenvalues distributionof D12 signal
Shock and Vibration 11
D11minus1080 rpmx 10
minus5
minus15
minus10
minus05
0051015
X (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(a)
D12minus1080 rpmx 10
minus5
minus10
minus05
0051015
Y (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(b)
D11minus1080 rpm(1807287)
(360472)
x 10minus5
002040608
1
X (m
)
50 100 150 2000Frequency (Hz)
(c)
(3605105)(1806575)
D12minus1080 rpmx 10
minus5
002040608
1
Y (m
)
50 100 150 2000Frequency (Hz)
(d)
Figure 12 Extracting the time and frequency spectrums of the first two frequencies of D11 and D12 (a)The time domain of D11 (b)The timedomain of D12 (c) The frequency domain of D11 (d) The frequency domain of D12
W1
x 10minus5
x 10minus5
minus1
minus05
0
05
1
15
D12
minusY
(m)
minus1minus15 minus05 05 10 15D11minusX (m)
(a)0
051
152
0
2
t (s)D11minusX (m)
minus2
x 10minus5
x 10minus5
minus1
minus05
0
05
1
15
D12
minusY
(m)
(b)
Figure 13 The synthesis of axis orbit is to extract the first two times fundamental frequencies of the D11 and D12 signals by the CFSM-PCA(a) The axis orbit (b) Relationship between axis orbit and time
3X matched the first set of eigenvalues (first and secondeigenvalues)
As displayed in Figure 15 eigenvalues of 1X and 3Xwere large thus 1X and 3X were utilized to synthesize rotororbit firstly Purification results and orbit of shaft center arerevealed in Figures 16 and 17 respectively The legible orbitshowed a typical plum blossom shape
It can be seen from Figures 16 and 17 that the filteringeffect was also quite obvious The noise and power frequency50 Hz as well as its harmonic interference could be removedsuccessfully Moreover amplitude of refined signal was closeto that of original one and the reconstructed time domainsignal seemed to be steady without any fluctuation
As shown in Figures 14(c) and 14(d) amplitudes of 1X 2Xand 3X in amplitude spectrum of raw signal were largeThus1X 2X and 3X were also extracted to synthesize axis orbit
Time domain frequency domain and orbit curve are shownin Figures 18 and 19 respectively
Axis orbit synthesized by the leading three frequenciesexhibited a quincunx shape which is basically the sameto that synthesized by 1X and 3X frequency componentsindicating that static and dynamic parts friction may exist inA-side
44 Comparison Harmonic Wavelet with Wavelet Packet
(1) Comparison of CFSM-PCA with Harmonic Wavelet Onecommon used method for signal processing harmonicwavelet algorithm proposed by Newland is an improvedwavelet algorithm [29] It can segment entire frequencyband infinitely and avoid the shortcoming of downsamplingmethod of binary wavelet and binary wavelet packet [22 23]
12 Shock and Vibration
D11minus2770 rpmx 10
minus4
minus1
minus05
0
05
1
D11
-X (m
)
1000 2000 3000 40000Sampling numbers i
(a)
D12minus2770 rpmx 10
minus4
minus1
minus05
0
05
1
D12
-Y (m
)
1000 2000 3000 40000Sampling numbers i
(b)
Power-Line interference 50and 150Hz
1X2X
3X 1010D11minus2770 rpm
08441166
1X2X3X
x 10minus5
200 400 600 800 10000f (Hz)
0
05
10
15
D11
-X (m
)
(c)
1X
2X
3XPower-Line interference 50
and 150Hz
D12minus2770 rpm117309021227
1X2X3X
x 10minus5
0
05
10
15
D12
-Y (m
)
200 400 600 800 10000f (Hz)
(d)
Figure 14 The time and frequency domains of the D11 and D12 sensors (a) The time domain of D11 (b) The time domain of D12 (c) Thefrequency domain of D11 (d) The frequency domain of D12
(11489) (21482)e first group
(31237) (41220)e second group
(30348) (40347)e second group
D11minus2770 rpmx 10
minus7
10 20 30 400 50Number of eigenvalues q
0
05
1
15
Eige
nval
ue o
f PCA
(m)
(a)
D12minus2770 rpm(11688) (21687)
e first group
e second group
(30414) (40413)e second group
(31673) (41664)
x 10minus7
0
05
1
15
2
Eige
nval
ue o
f PCA
(m)
10 20 30 40 500Number of eigenvalues q
(b)
Figure 15 The eigenvalues distribution of covariance matrix C (a) The eigenvalues distribution of D11 (b) The eigenvalues distribution ofD12
1X and its frequency doubling signal can be easily extractedto form axis orbit of revolving test bearing
Axis orbit was fabricated by 1X and 2Xof data of example 1(Section 42) through harmonic wavelet algorithm and theresults are demonstrated in Figure 20 The shape of axisorbit was trapped banana-like which is consistent with theresult obtained via CFSM-PCA It should be noted thatorbit in Figure 13 was more obvious than that in Figure 20ie W1ltW2 Meanwhile the amplitude fluctuation of timedomain signal was large (Figures 20(a) and 20(b)) Themajor reason for this phenomenon is that the harmonicwavelet which is a band filter in frequency domain inevitablyextracts the noise near the feature frequency Furthermorethis phenomenon also demonstrates that harmonic waveletis subject to the Heisenberg uncertainty principle ie thephenomenon of the amplitude leakage exists
(2) Comparison of CFSM-PCA with Wavelet Packet Besidesharmonic wavelets wavelet packet is also used to purify axisorbit [19ndash21] Purification results of axis orbit via Daubechieswavelet packet in Figure 21 exhibit that axis orbit was diver-gent seriously This was worse than that of harmonic waveletie W1ltW2ltW3The phenomenon of severe energy leakageoccurred during the extraction of 1X and 2X which cancause divergence of axis orbit Moreover the bandpass filterof wavelet packet also accounts for the divergent orbits
5 A Single Frequency Filtrationwith CFSM-PCA
The application in single frequency filtration is discussedin this section Taking D11 experimental data of example
Shock and Vibration 13
D11minus2770 rpmx 10
minus5
0 1000500Sampling numbers i
minus4
minus2
0
2
4
D11
-X (m
)
(a)
D12minus2770 rpmx 10
minus5
minus4
minus2
0
2
4
D12
-Y (m
)
500 10000Sampling numbers i
(b)
(1375 1185)(46 1009)
D11minus2770 rpmx 10
minus5
100 200 300 400 5000f (Hz)
0
05
1
15
D11
-X (m
)
(c)
(1375 1218)(46 1164)
D12minus2770 rpmx 10
minus5
100 200 300 400 5000f (Hz)
0
05
1
15
D12
-Y (m
)
(d)
Figure 16 Extracting the frequency spectrums of the first two frequencies of D11 and D12 by PCA algorithm (a) The time domain of D11(b) The time domain of D12 (c) The frequency spectrum of D11 (d) The frequency spectrum of D12
x 10minus5
x 10minus5
minus4
minus3
minus2
minus1
0
1
2
3
4
D12
-Y (m
)
210 3 4minus2minus3 minus1minus4
D11-X (m)
(a)0
0204
0608
10
24
t (s)D11-X (m) minus2
minus4
x 10minus5
x 10minus5
minus4
minus2
0
2
4
D12
-Y (m
)
(b)
Figure 17 The synthesis of axis orbit is to extract the first two times fundamental frequencies of the D11 and D12 signals by the CFSM-PCA(a) The axis orbit (b) Relationship between axis orbit and time
1 (Section 42) as a demo we attempt to filter the powerfrequency interference (50 Hz) from D11 by employingCFSM-PCA It can be seen from Figure 9(b) that sequenceof amplitude of 50 Hz in raw signal was 3 so eigenvec-tors corresponding to 5th and 6th eigenvalues in eigen-value distribution map were used to reconstruct Accordingto (8) the filtered signal (Figure 22) without 50 Hz was
obtained by subtracting reconstructed signal from originalone
Results in Figure 22 show that this algorithm filters out50 Hz power frequency interference and has no influenceon the other frequency components of raw signal demon-strating that this method has a potential application in notchfilter
14 Shock and Vibration
D11minus2770 rpm
x 10minus5
minus4
minus2
0
2
4
D11
-X (m
)
200 400 600 800 10000Sampling numbers i
(a)
D12minus2770 rpm
x 10minus5
0 400 600 800 1000200Sampling numbers i
minus4
minus2
0
2
4
D12
-Y (m
)
(b)
(1375 1185)(46 1009)
(92 0831)
D11minus2770 rpmx 10
minus5
00
05
10
15
D11
-X (m
)
100 200 300 400 5000f (Hz)
(c)
(1375 1218)(46 1163)
(92 0898)
D12minus2770 rpmx 10
minus5
00
05
10
15
D12
-Y (m
)
100 200 300 400 5000f (Hz)
(d)
Figure 18 Extracting the time and frequency spectrums of the first three frequencies of D11 and D12 by PCA algorithm (a)The time domainof D11 (b) The time domain of D12 (c) The frequency spectrum of D11 (d) The frequency spectrum of D12
x 10minus5
x 10minus5
minus4
minus3
minus2
minus1
0
1
2
3
4
D12
-Y (m
)
minus3 minus2 minus1minus4 1 2 3 40D11-X (m)
(a)
1008
0604
0200
02
4D11-X (m)
t (s)
minus2
minus4
x 10minus5
x 10minus5
minus4
minus2
0
2
4
D12
-Y (m
)
(b)
Figure 19 The synthesis of axis orbit is to extract the first three times fundamental frequencies of the D11 and D12 by the CFSM-PCA (a)The axis orbit (b) Relationship between axis orbit and time
6 Conclusion
The selection of effective eigenvalues in feature distributionchart of covariance matrix C is crucial in PCA algorithm Inthis paper the relationships between effective eigenvalues andsignal frequency and amplitude are discovered and a series ofconclusions are summarized
First in Hankel matrix mode each valid frequencycomponent of signal only produces two adjacent nonzeroeigenvalues Second the number of valid eigenvalues incharacteristic distribution chart which is independent ofnumerical values of 119891119894 119860 119894 and 120593119894 is related to the number offrequency componentsThird the sequence of signal effectiveeigenvalues in characteristic distribution map is determined
Shock and Vibration 15
D11-1080 rpm
x 10minus5
minus20
minus15
minus05
0
05
10
15
20
D11
-X (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(a)
D12-1080 rpmx 10
minus5
minus15
minus1
minus05
0
05
1
15
D12
-Y (m
)
500 1000 1500 2000 2500 30000 40003500Sampling numbers i
(b)
(16 07335)
(32 04771)
D11-1080 rpm
x 10minus5
10 20 30 40 50 60 70 80 90 1000f (Hz)
0010203040506070809
1
D11
-X (m
)
(c)
(16 06571)
(32 05228)
D12-1080 rpm
x 10minus5
0010203040506070809
1
D12
-Y (m
)
10 20 30 40 50 60 70 80 90 1000f (Hz)
(d)
W2
x 10minus5
x 10minus5
minus15
minus1
minus05
0
05
1
15
Y(D
12) (
m)
minus1 minus05minus15 05 1 15 20X(D11) (m)
(e)
Figure 20The purification effect of harmonic wavelets (a)The time domain of D11 (b)The time domain of D12 (c)The frequency spectrumof D11 (d) The frequency spectrum of D12 (e) The orbit is to extract the first 2 frequencies
by amplitude 119860 119894 of signal frequency component The largermagnitude of amplitude is the greater eigenvalues are andthe higher arrangement will be
Based on aforesaid principle a new CFSM-PCA wasdeveloped and has become an effective tool to extract oreliminate specific characteristic (single frequency) even dif-ference of two frequencies is only 1 Hz Purification of rotororbit using this algorithm shows significant improvement
than those of existing methods such as wavelet packet andharmonic wavelet packet Although the proposed CFSM-PCA is effective some drawbacks are unavoidable Forexample this method suffers from similar issue to otheralgorithms that is the reconstruction error increaseswith thedecrease of SNR In addition the applications of the CFSM-PCA algorithm in purification of speech recognition bearingfault diagnosis and fault diagnosis of power system should
16 Shock and Vibration
D11-1080 rpmx 10
minus5
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
minus20
minus15
minus05
0
05
10
15
20
D11
-X (m
)
(a)
D12-1080 rpmx 10
minus5
minus15
minus1
minus05
0
05
1
15
D12
-Y (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(b)
(16 06557)
(32 04381)
D11-1080 rpmx 10
minus5
10020 30 40 50 60 70 80 90100f (Hz)
0010203040506070809
1
D11
-X (m
)
(c)
(16 06452)(32 04927)
D12-1080 rpm
x 10minus5
0010203040506070809
1
D12
-Y (m
)
9020 30 40 50 60 70 8010 1000f (Hz)
(d)
W3
x 10minus5
x 10minus5
minus15
minus1
minus05
0
05
1
15
Y(D
12) (
m)
minus15 minus05minus1 05 1 15 20X(D11) (m)
(e)
Figure 21 The purification effect of wavelet packet (a) The time domain of D11 (b) The time domain of D12 (c) The frequency spectrum ofD11 (d) The frequency spectrum of D12 (e) The orbit is to extract the first 2 frequencies
be further exploredWe fully believe that the CFSM-PCAwillsparkle in diverse fields
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work is supported by the National High TechnologyResearch andDevelopment Program of China (863 Program)
Shock and Vibration 17
(18 0734)
(36 04634)
(54 02448) (72 02351)(90 01646)
x 10minus5
50 100 150 2000f (Hz)
0
02
04
06
08
1
D11
minusX
(m)
Figure 22 Frequency spectrum of D11 signal after filtering 50Hz
(no 2015AA043005) and the National Natural Science Foun-dation of China (no 51375187)
References
[1] H Abdi and L J Williams ldquoPrincipal component analysisrdquoWiley Interdisciplinary Reviews Computational Statistics vol 2no 4 pp 433ndash459 2010
[2] A Widodo B Yang and T Han ldquoCombination of independentcomponent analysis and support vectormachines for intelligentfaults diagnosis of induction motorsrdquo Expert Systems withApplications vol 32 no 2 pp 299ndash312 2007
[3] S Wold K Esbensen and P Geladi ldquoPrincipal componentanalysisrdquo Chemometrics and Intelligent Laboratory Systems vol2 no 1ndash3 pp 37ndash52 1987
[4] M Kirby and L Sirovich ldquoApplication of the Karhunen-Loeve procedure for the characterization of human facesrdquo IEEETransactions on Pattern Analysis and Machine Intelligence vol12 no 1 pp 103ndash108 1990
[5] J H Xi Y Z Han and R H Su ldquoNew fault diagnosis methodfor rolling bearing based on PCArdquo in Proceedings of the 25thChinese Control and Decision Conference CCDC rsquo13 pp 4123ndash4127 IEEE May 2013
[6] A Malhi and R X Gao ldquoPCA-based feature selection schemefor machine defect classificationrdquo IEEE Transactions on Instru-mentation and Measurement vol 53 no 6 pp 1517ndash1525 2004
[7] S Xu X Jiang J Huang S Yang and X Wang ldquoBayesianwavelet PCA methodology for turbomachinery damage diag-nosis under uncertaintyrdquo Mechanical Systems and Signal Pro-cessing vol 80 pp 1ndash18 2016
[8] W Sun J Chen and J Li ldquoDecision tree and PCA-based faultdiagnosis of rotatingmachineryrdquoMechanical Systems and SignalProcessing vol 21 no 3 pp 1300ndash1317 2007
[9] Z X Li X P Yan C Q Yuan Z Peng and L Li ldquoVirtual pro-totype and experimental research on gear multi-fault diagnosisusing wavelet-autoregressive model and principal componentanalysismethodrdquoMechanical Systems and Signal Processing vol25 no 7 pp 2589ndash2607 2011
[10] K Y Shao M M Cai and G F Zhao ldquoRolling bearing faultdiagnosis based onwavelet energy spectrum PCA andPNNrdquo inProceedings of the 26th Chinese Control andDecisionConferenceCCDC rsquo14 pp 800ndash804 IEEE 2014
[11] T Qiu Q F Zhang and Y J Ding ldquoNonlinear sensor faultdetection and data rebuilding based on principle componentanalysisrdquo Journal of Tsinghua University vol 46 no 5 pp 708ndash711 2006
[12] Y K Gu Z X Yang and F L Zhu ldquoGearbox fault feature fusionbased on principal component analysisrdquo China MechanicalEngineering vol 26 no 11 pp 1532ndash1537 2015
[13] R Dunia S J Qin T F Edgar and T J McAvoy ldquoIdentificationof faulty sensors using principal component analysisrdquo AIChEJournal vol 42 no 10 pp 2797ndash2811 1996
[14] E P de Moura C R Souto A A Silva and M A S IrmaoldquoEvaluation of principal component analysis and neural net-work performance for bearing fault diagnosis from vibrationsignal processed by RS and DF analysesrdquo Mechanical Systemsand Signal Processing vol 25 no 5 pp 1765ndash1772 2011
[15] S Golestan M Monfared F D Freijedo and J M GuerreroldquoDynamics assessment of advanced single-phase PLL struc-turesrdquo IEEE Transactions on Industrial Electronics vol 60 no6 pp 2167ndash2177 2013
[16] A Lima H Zen Y Nankaku C Miyajima K Tokuda andT Kitamura ldquoOn the use of kernel PCA for feature extractionin speech recognitionrdquo IEICE Transaction on Information andSystems vol 12 no 87 pp 2802ndash2811 2004
[17] J D Zheng ldquoImproved hilbert-huang transform and its appli-cations to rolling bearing fault diagnosisrdquo Journal of MechanicalEngineering vol 1 no 51 pp 138ndash145 2015
[18] G He K Ding and H Lin ldquoFault feature extraction of rollingelement bearings using sparse representationrdquo Journal of Soundand Vibration vol 366 pp 514ndash527 2016
[19] J G Yang S B Xia and Y G Liu ldquoWavelet denoising and itsapplication in extraction of shaft centerline orbit featuresrdquoJournal of Harbin Institute of Technology vol 31 no 5 pp 52ndash541999 (Chinese)
[20] J A Duan and X D Zhang ldquoFeature extraction from the orbitof the center of a rotor based on wavelet transformrdquo Journal ofVibration Measurement amp Diagnosis vol 17 no 01 pp 31ndash341997
[21] J Han D X Jiang and W D Ning ldquoApplication of optimalwavelet packets in failure signal extraction of rotor orbitrdquoTurbine Technology vol 43 no 3 pp 133ndash136 2001
[22] W B Zhang X J Zhou andY Lin ldquoRefinement of rotor centerrsquosorbit by a harmonic window methodrdquo Journal of VibrationMeasurement amp Diagnosis vol 30 no 1 pp 87ndash90 2010
[23] S M Li G Z Lu and Q Y Xu ldquoOn obtaining accuraterotor sub-frequency signal with harmonic waveletrdquo Journal ofNorthwestern Polytechincal University vol 19 no 2 pp 220ndash224 2001
[24] F J Wu and L S Qu ldquoAn improved method for restraining theend effect in empiricalmode decomposition and its applicationsto the fault diagnosis of large rotating machineryrdquo Journal ofSound and Vibration vol 314 no 3-5 pp 586ndash602 2008
[25] W G Liu ldquoResearch on method of purification of shaft orbitbased on singular value decompositionrdquo Technical Acousticsvol 5 no 35 pp 30ndash35 2016
[26] X Z Zhao and B Y Ye ldquoThe influence of formation manner ofcomponent on signal processingrdquo Shang Hai Jiao Tong Univer-sity vol 45 no 3 pp 368ndash374 2011
[27] X Z Zhao B Y Ye and T J Chen ldquoPrinciple of singularitydetection based on SVD and its applicationrdquo Journal of Vibra-tion and Shock vol 6 no 27 pp 7ndash10 2008
[28] X Z Zhao B Y Ye and T J Chen ldquoDifference spectrum theoryof singular value and its application to the fault diagnosis ofheadstock of latherdquo Journal of Mechanical Engineering vol 46pp 100ndash108 2010
[29] I A Kougioumtzoglou and P D Spanos ldquoAn identificationapproach for linear and nonlinear time-variant structural sys-tems via harmonic waveletsrdquo Mechanical Systems and SignalProcessing vol 37 no 1-2 pp 338ndash352 2013
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Shock and Vibration 3
If the former 119897 principal components are chosen toreconstruct in the light of cumulative contribution rate 119871 119897 anapproximate matrix X can be formulated as
X =119897
sum119894=1
120572119894y119894 =119897
sum119894=1
120572119894120572T119894 X (8)
Compared with original matrix X the reconstructedapproximate matrix X comprises most of information of Xand excludes redundant features such as noise and powerfrequency interference [9 10]
Signal could be recovered from the reconstructed matrixX in terms of matrix composition mode XT is converted to1 row 119898119899 columns vector with the row vectors arranging inhead-to-tail fashion then a new vector is obtained recordedas a a isin 1198771times119898119899 The recovered signal x is obtained via inversetransformation of vector a derived as
x = 119886M+ (9)
where x isin 1198771times119871 L=m+n-1 L is the length of recovered dataM+ = (MTM)minus1MT isin 119877119898119899times119871 and M+ is pseudoinverse ofMM isin 119877119871times119898119899 andM comprises119898 unit matrices where therank of each unit matrix is 119899 Takingm=3 n=3 as an examplein this case L=m+n-1=5 and mn=9 the matrix M can beexpressed as follows
=
[[[[[[[[[[
1
1 1
1 1 1
1 1
1
]]]]]]]]]]
M (10)
The signal x is recovered from the pseudoinverse matrixM+ actually it is to compute the average values of eachelement at counter-diagonal of matrix X which is consistentwith the method reported in [23] SinceM is a sparse matrixespecially when values of 119898 and 119899 are large random-accessmemory and time for calculating of pseudoinverse matrixM+ will increase exponentially Hence signal recovered fromsimple method is extremely desired With this aim in mindwe have developed a new averagingmethod and adopted it torecover signal x from matrix X The expression is shown asfollows
119909 (119896) =
119896
sum119894=1
119909119894119896+1minus119894119896 1 le 119896 lt 119898
119898
sum119894=1
119909119894119896+1minus119894119898 119898 le 119896 lt 119899119898
sum119894=119896minus119899+1
119909119894119896+1minus119894(119898 + 119899 minus 119896) 119899 lt 119896 le 119898 + 119899 minus 1
(11)
where 119909119894119895 are element of the 119894th row and the 119895th column inthe reconstructed approximate matrix X The signal x can berestored facilely according to (11)
3 Internal Law of Effective Eigenvalues andFrequency Components
31The Process of Feature Frequency SeparationMethod Basedon PCA For noise-free signal 119909(119905) the effective eigenvaluesare nonzero ones It can be seen from (7) that signal 119909(119905) has119898 eigenvectors which aremostly generated by noise So for asignal containing certain number of frequency componentswhat is the relationship between the number of eigenvaluesand the number of frequencies
During our research course an important connectionbetween nonzero eigenvalues and the number of frequencycomponents was discovered To illustrate this connectionexplicitly signal with different amplitude 119860 119894 frequency 120596119894and phase 120593119894 was constructed as follows
119909 (119905) =119896
sum119894=1
119860 119894 sin (2120587120596119894119905 + 120593119894) (12)
where 119896 is the number of frequency components4096 data points were collected with sampling frequency
of 1024 Hz and then the Hankel matrix was formed fromsignal 119909(119905) with m rows and n columns Decomposition andreconstruction of signal 119909(119905) were proceeded by employingPCA algorithm in Section 2 [9 13] Effect of the constructedHankel matrix on signal processing was studied by Zhao etal [26] they pointed out that if the number of rows was closeto the number of columns signal processing effect was betterFurthermore when Lwas an evenm=L2 and n=L2+1 whenL was an oddm= (L+1)2 and n= (L+1)2 Hence in our casem = 2048 and n = 2049 were applied The decompositionprocedure consists of the following(1)Given k = 1119860 119894= 1 tri-groups signals were constructed
and their principal component eigenvalue distribution mapsare shown in Figure 1 The number of eigenvalues 119902 rangefrom 1 to 2048 and just the leading 50 eigenvalues are listedin this case(2) Given k = 2 119860 119894= 1 08 tri-groups signals were
reconstructed and each group signal contained two effectivefrequency components with119891119894 as 20Hz and 30Hz 50Hz and60 Hz and 80 Hz and 90 Hz respectively and 120593119894 were 10 and20 40 and 50 and 70 and 80 respectively
When k = 1 each signal contains only one effectivefrequency component These three sets of signals have thesame amplitude119860 119894 but different frequency120596119894 and phase120593119894 Ascan be seen from Figure 1 each signal produces two adjacentnonzero eigenvalues 1205821 and 1205822 Although tri-groups signalsare different nonzero eigenvalues of them are the same toeach other
It can be seen from eigenvalue distribution graphs inFigure 2 that each signal produces two sets of nonzeroeigenvalues each set of eigenvalues contains two adjacenteigenvalues 120582119894 and 120582119894+1 Interestingly the four nonzeroeigenvalues produced from each signal are equal to theircorresponding eigenvalues generated by the other signalseven if the signals are different
Moreover by comparing Figure 2 with Figure 1 the firstset of eigenvalues in both two graphs are exactly sameActually signals in Figure 2 are established by adding an extra
4 Shock and Vibration
(15123) (25118)e first group
(15123) (25118)
(15123) (25118)
e first group
e first group
0
200
400
600
0
200
400
600
10 20 30 40 500Number of eigenvalues q
10 20 30 40 500
10 20 30 40 500
x1(t) = MCH(40t + 10)
x2(t) = MCH(100t + 40)
x3(t) = MCH(160t + 70)
0
200
400
600
Eige
nval
ue o
f PCA
i
Figure 1 Eigenvalues distribution of covariance matrix C with one frequency component
(15123) (25118)
(15123) (25118)
(15123) (25118)
(33278) (43275)
(33278) (43275)
(33278) (43275)
e first group
e second group
e first group
e second group
e first group
e second group
0
200
400
600
10 20 30 40 500
10 20 30 40 500
0
200
400
600
10 20 30 40 500Number of eigenvalues q
x1(t) = MCH(40t + 10) + 08 MCH(60t + 20)
x3(t) = MCH(160t + 70) + 08 MCH(180t + 80)
0
200
400
600
Eige
nval
ue o
f PCA
i
x2(t) = MCH(100t + 40) + 08 MCH(120t + 50)
Figure 2 Eigenvalues distribution of covariance matrix C with twofrequency components
frequency component with amplitude of 08 Thus it couldbe confirmed that the first set of nonzero eigenvalues 1205821 and1205822 in Figure 2 are generated from frequency component withamplitude of 10 while the second set of nonzero eigenvalues
are generated from frequency component with amplitude of08(3)Given 119896= 3119860 119894 = 1 08 and 06 tri-groups signals were
reconstructed and each group signal contained three sets ofeffective frequency components The frequencies of the firstgroup of signal were 20 Hz 30 Hz and 40 Hz respectivelyThe frequencies of the second group of signals were 50 Hz60 Hz and 70 Hz respectively Similarly the frequencies ofthe third group of signals were 80 Hz 90 Hz and 100 Hzrespectively The corresponding phases of these three groupsof signals were taken as 10 20 and 30 40 50 and 60 and 7080 and 90 respectively
When 119896 = 3 119860 119894=1 08 and 06 each group signal pro-duces three effective frequency components The amplitudes119860 119894 of corresponding frequency 119891119894(119894=1 2 3) are same butwith different frequencies 120596119894 and phases 120593119894 As displayed inFigure 3 each group signal produces three sets of nonzeroeigenvalues and each set of eigenvalues contains two adjacenteigenvalues 120582119894 and 120582119894+1 In addition these three sets ofeigenvalues produced by each signal are correspondinglysame
It can be found from Figures 1ndash3 that the magnitudeof eigenvalues of each first group in three graphs is sameAs aforementioned the first set of nonzero eigenvalues1205821 and 1205822 are generated from frequency component withamplitude 119860 119894 of 1 Similarly by comparing Figure 2 withFigure 1 the second set of nonzero eigenvalues 1205823 and 1205824are generated from frequency component with amplitude119860 119894 of 08 And then it is almost certain that the third setof frequency components 1205825 and 1205826 are generated fromfrequency component with amplitude 119860 119894 of 06
The same results can be obtained by continuously increas-ing effective frequency components of signal Therefore an
Shock and Vibration 5
(33278) (43275)
(51844) (61842)
(15123) (25118)
(33278) (43275)
(51844) (61842)
(15123) (25118)
(33278) (43275)
(51844) (61842)
(15123) (25118)e first group
e second group
e third group
e first group
e second group
e third group
e first group
e second group
e third group
x1(t) = MCH(40t + 10) + 08 MCH(60t + 20) + 06 MCH(80t + 30)
x2(t) = MCH(100t + 40) + 08 MCH(120t + 50) + 06 MCH(140t + 60)
x3(t) = MCH(160t + 70) + 08 MCH(180t + 80) + 06 MCH(200t + 90)
0
200
400
600
10 20 30 40 500
0
200
400
600
Eige
nval
ue o
f PCA
i
10 20 30 40 500
0
200
400
600
10 20 30 40 500Number of eigenvalues q
Figure 3 Eigenvalues distribution of covariancematrixCwith threefrequency components
Hankel matrix is derived from certain signal x(t) l=min(mn) with m rows n columns and k effective frequenciesConcerning the fact that the Shannon sampling theorem ismet assuming that lgt2k generic conclusions are summarizedas follows(1) Each frequency component of signal produces two
nonzero eigenvalues with one arranging another closely(2) The number of effective eigenvalues of crude signal
is related to the number of frequency components and has
nothing to do with themagnitude of amplitude119860 119894 frequency120596119894 and phase 120593119894(3) In eigenvalue distribution chart of covariance matrix
C the sequence of nonzero effective eigenvalues is deter-mined by amplitude 119860 119894 of signal The larger amplitude is thelarger eigenvalues will be the more forward rank of the twoeigenvalues produced from corresponding frequencies is
Inspired by the relationship between frequency andeigenvalues a new technique for characteristic frequency sep-aration method based on PCA (CFSM-PCA) was proposedThe concrete steps are listed below(1) For a certain signal 119909(119905) direct component (DC) of
raw signal is filtered out via fast Fourier transform (FFT)firstly and then Hankel matrix X is constructed throughfiltered signal(2) Covariance matrix C of matrix X is obtained with its
eigenvalues 120582119894 arranging in descending order (1205821 1205822 sdot sdot sdot 120582119898)and corresponding eigenvectors are obtained as 12057211205722 sdot sdot sdot120572119898(3) According to the distribution of eigenvalues 120582119894reconstruction is carried out from two eigenvalues and cor-responding eigenvectors of certain frequency For examplefor amplitude perspective if the rank of specific frequencyof raw signal is k a new matrix is received by reconstructingthe eigenvectors corresponding to 2k-1 and 2k eigenval-ues in eigenvalue distribution chart of covariance matrixC(4)Thematrix X can be produced by adding the mean of
original matrix to the new reconstructed matrix(5) The signal x which is the characteristic frequency
component is recovered from matrix X by means of theaveraging method
32Theoretical Deduction In this section deduction processof the three discoveries (Section 31) is provided Supposingthat a signal is expressed as 119909(119905) = 119886 sin(120596119905 + 120593) Samplingtime 119879119904 is used to discretize signal x(t) Hankel matrix withm rows and n columns is derived from signal 119909(119905) exhibitedas
X = 119886 times[[[[[[
sin (0 sdot 120596119879119904 + 120593) sin (1 sdot 120596119879119904 + 120593) sdot sdot sdot sin ((119899 minus 1) sdot 120596119879119904 + 120593)sin (1 sdot 120596119879119904 + 120593) sin (2 sdot 120596119879119904 + 120593) sdot sdot sdot sin (119899 sdot 120596119879119904 + 120593)
sin ((119898 minus 1) sdot 120596119879119904 + 120593) sin (119898 sdot 120596119879119904 + 120593) sdot sdot sdot sin ((119898 + 119899 minus 2) sdot 120596119879119904 + 120593)
]]]]]]
(13)
(1) Each characteristic frequency produces two effectiveeigenvalues
The deduced process of the first conclusion is given belowEquation (13) can be rewritten as (14) based onEulerrsquos Formula
X = 1198862119894
times[[[[[[[
119890(0120596119879119904+120593)119894 minus 119890minus(0120596119879119904+120593)119894 119890(1120596119879119904+120593)119894 minus 119890minus(1120596119879119904+120593)119894 119890(2120596119879119904+120593)119894 minus 119890minus(2120596119879119904+120593)119894 sdot sdot sdot 119890((119899minus1)120596119879119904+120593)119894 minus 119890minus((119899minus1)120596119879119904+120593)119894119890(1120596119879119904+120593)119894 minus 119890minus(1120596119879119904+120593)119894 119890(2120596119879119904+120593)119894 minus 119890minus(2120596119879119904+120593)119894 119890(3120596119879119904+120593)119894 minus 119890minus(3120596119879119904+120593)119894 sdot sdot sdot 119890(119899120596119879119904+120593)119894 minus 119890minus(119899120596119879119904+120593)119894
119890((119898minus1)120596119879119904+120593)119894 minus 119890minus((119898minus1)120596119879119904+120593)119894 119890(119898120596119879119904+120593)119894 minus 119890minus(119898120596119879119904+120593)119894 119890((119898+1)120596119879119904+120593)119894 minus 119890minus((119898+1)120596119879119904+120593)119894 sdot sdot sdot 119890((119898+119899minus2)120596119879119904+120593)119894 minus 119890minus((119898+119899minus2)120596119879119904+120593)119894
]]]]]]]
(14)
6 Shock and Vibration
Equation (14) can be expanded to addition form of twoformulas depicted as
X = 1198862119894 times 119890120593119894
times[[[[[[[[
1198900120596119879119904119894 1198901120596119879119904119894 1198902120596119879119904119894 sdot sdot sdot 119890(119899minus1)1205961198791199041198941198901120596119879119904119894 1198902120596119879119904119894 1198903120596119879119904119894 sdot sdot sdot 119890119899120596119879119904119894
119890(119898minus1)120596119879119904119894 119890119898120596119879119904119894 119890(119898+1)120596119879119904119894 sdot sdot sdot 119890(119898+119899minus2)120596119879119904119894
]]]]]]]]
minus 1198862119894 times 119890minus120593119894
times[[[[[[[[
119890minus0120596119879119904119894 119890minus1120596119879119904119894 119890minus2120596119879119904119894 sdot sdot sdot 119890minus(119899minus1)120596119879119904119894119890minus1120596119879119904119894 119890minus2120596119879119904119894 119890minus3120596119879119904119894 sdot sdot sdot 119890minus119899120596119879119904119894
119890minus(119898minus1)120596119879119904119894 119890minus119898120596119879119904119894 119890minus(119898+1)120596119879119904119894 sdot sdot sdot 119890minus(119898+119899minus2)120596119879119904119894
]]]]]]]](15)
From (15) the rank of both matrices is 1 Based on rankrelationship of two matrices 119877(119860+119861) le 119877(119860) + 119877(119861) Hencerank of matrix X is less than or equal to 2
Additionally when 120593 = 0 (13) is rewritten to (16)According to sum-to-product identities the rank of X is 2when 120596119879119904 = 0 120587 2120587 sdot sdot sdot
X = 119886 times[[[[[[[[[
sin (0 sdot 120596119879119904) sin (1 sdot 120596119879119904) sdot sdot sdot sin ((119899 minus 1) sdot 120596119879119904)sin (1 sdot 120596119879119904) sin (2 sdot 120596119879119904) sdot sdot sdot sin (119899 sdot 120596119879119904)
sin ((119898 minus 1) sdot 120596119879119904) sin (119898 sdot 120596119879119904) sdot sdot sdot sin ((119898 + 119899 minus 2) sdot 120596119879119904)
]]]]]]]]]
(16)
When 120593 = 0 (13) can be deduced into the first-orderprincipal minor of matrix X ie |sin120593| = 0 The principalminor of order 2 of X is shown as10038161003816100381610038161003816100381610038161003816100381610038161003816sin (0 sdot 120596119879119904 + 120593) sin (1 sdot 120596119879119904 + 120593)sin (1 sdot 120596119879119904 + 120593) sin (2 sdot 120596119879119904 + 120593)
10038161003816100381610038161003816100381610038161003816100381610038161003816= minussin2 (120596119879119904)
= 0(17)
where120596119879119904 = 0 120587 2120587 sdot sdot sdot Because the leading principalminorof order 2 ofX in (13) is nonzero values so the rank of matrixX is at least 2
Combining result of (15) 119877(119860 + 119861) le 2 and nonzerovalues of the leading principal submatrix of order 2 hencethe rank of matrix X is 2 Matrix X has two eigenvalues 12058210158401and 12058210158402 Derived from C = XXT119899 (where 119899 is the numberof columns of matrix X) [3] two nonzero eigenvalues ofcovariance matrix C of X are 1205821 and 1205822 respectively(2)The sequence of eigenvalues is determined by ampli-
tudeIn terms of PCA covariance matrix C is described as
C = 119864 [(X minus 119864 (X)) sdot (X minus 119864 (X))T] (18)
Referring to (6) (19) is constructed as follows
119897
sum119894=1
C120572119894120572T119894 =119897
sum119894=1
120582119894120572119894120572T119894 (19)
Two effective eigenvalues are generated from a frequencycomponent that is l=2
CI119898 = 12058211205721120572T1 + 12058221205722120572T2 (20)
Then the energy of matrix C can be deduced as
|C|2 = 12058221119898
sum119894=1
12057221198941119898
sum119894=1
12057221198941 + 12059022119898
sum119894=1
12057221198942119898
sum119895=1
12057221198942
+ 21205902112059022119898
sum119894=1
12057211989411205721198942119898
sum119894=1
12057211989411205721198943(21)
Based upon (4) sum119898119894=1 12057221198941 = 1 sum119898119894=1 12057221198942 = 1 sum119898119894=1 12057211989411205721198942 =0 Then (22) is given by
|C|2 = 12058221 + 12058222 (22)
The constructed Hankel matrix X of signal 119909(119905) is sub-stituted into (18) we can see that the energy of covariancematrix C is proportional to 1198862 The larger 1198862 is the greaterenergy of matrix C is as well as 120582119894 according to (22)Furthermore the larger frequency component amplitude isthe larger corresponding eigenvalue in covariance matrix Ccharacteristic distribution chart is
Based on these conclusions once the amplitude sequenceof a certain frequency in raw signal amplitude spec-trum is determined its corresponding frequency compo-nent can be reconstructed In this way extraction of sin-gle or multiple characteristic frequencies could be real-ized
In view of addition relation as shown in (8) the notchfilter could be achieved through CFSM-PCA The frequencycomponent extracted by this algorithm is subtracted in orig-inal signal ie this frequency component can be eliminatedin raw signal Reader could consult examples in Section 5 formore details
Shock and Vibration 7
Table 1 Amplitude of signal components under the different SNR
SNR SNR1=-1 SNR2=-4 SNR3=-7 SNR4=-10Signal component Raw Process Raw Process Raw Process Raw Processsin(40120587119905 + 20) 09999 09938 10260 10240 10370 10730 1114 111205 sin(100120587119905 + 40) 04703 04826 05317 05482 07353 07533 05177 0523407 sin(98120587119905 + 50) 07299 07152 07116 07235 05110 04968 06800 0662108 sin(160120587119905 + 60) 08340 08324 08264 08142 08389 08338 08379 08923Max difference 00123 00122 00360 00544
minus6minus4minus2
0246
Am
plitu
de
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(a)
(200972)
(490693)(800791)
(500468)
0
05
1
Am
plitu
de
50 100 150 2000f (Hz)
(b)
Figure 4 The raw signal 119910 (a) Time domain (b) Amplitudespectrum
33 Simulation Example To verify the applicability and effec-tiveness of the CFSM-PCA algorithm a simulated sinusoidalsignal 119910 was constructed as
119910 = sin (40120587119905 + 20) + 05sin (100120587119905 + 40)+ 07sin (98120587119905 + 50) + 08sin (160120587119905 + 60)+ 119890 (119899)
(23)
where 119890(119899) is Gaussian white noise with standard deviationof 12 The result is shown in Figure 4(a) and amplitudespectrum (Fourier spectrum) is illustrated in Figure 4(b)after collecting 4096 data points with sampling frequency of1024 Hz
Follow-up work to separate frequencies via the proposedapproach was performed From eigenvalue distribution chartof crude signal in Figure 5 we can see that apart fromdiscernible four sets of eigenvalues most of eigenvaluescaused by noise are nonzero values and distribute at largeeigenvalues region
As demonstrated in Figures 6(a) and 6(b) time domainand frequency spectrum are generated after reconstructingthe first set of eigenvalues 1205821 and 1205822 in Figure 5 Frequencyof the reconstructed signal is 20 Hz with amplitude of 0977which is in correspondence with component sin(40120587t+20) ofy signal
As revealed in Figures 6(c) and 6(d) another time domainand amplitude spectrum are obtained via the reconstruction
(1515) (25144)
(3353) (43527)
(52654) (62643)
(71037) (81036)
e first group
e second group
e third group
e fourth group
0
200
400
600
Eige
nval
ue o
f PCA
i
10 20 30 40 500Number of eigenvalues q
Figure 5The eigenvalues distribution of the covariancematrixC of119910
of the second set of eigenvalues 1205823 and 1205824 Frequencyof this reconstructed signal is 80 Hz with amplitude of0786 which is same as component 08sin(160120587t+60) of rawsignal y Figures 6(e) and 6(f) are brought out through thereconstruction of the third set of eigenvalues 1205825 and 1205826Frequency of reconstructed signal is 49 Hz with amplitude of0688 which is consistent with component 07sin(98120587t+50)of crude signal 119910 Time domain and amplitude spectrum aregained after the reconstruction from the last set of eigenvaluesin Figure 5 exhibited as Figures 6(h) and 6(g) Frequency is50 Hz with amplitude of 0463 which is matched well withcomponent 05sin(100120587t+40) of signal y
Moreover several sets of signals with different signal-to-noise ratio (SNR) were processed by CFSM-PCA Theamplitude difference is summarized in Table 1
It can be seen that when the noise is low the amplitudeof frequency components extracted byCFSM-PCAalgorithmis close to that of the raw signal However the amplitudedifference will increase with the decrease of SNR
From all above-mentioned diagrams (Figure 6) it is obvi-ous that diverse frequency components extracted from rawsignal can be achieved accurately More importantly theseresults are coincident with ideal signal perfectly demon-strating that this CFSM-PCA is a zero phase shift frequencyextraction algorithm Meanwhile it should be noted that twofrequencies could also be perfectly separated via the proposedalgorithm even difference value between the adjacent fre-quency components is only 1 HzThismethod is accurate andseems to be more efficient than other existing filter methodssuch as wavelet filter and finite impulse response (FIR) filterwhich are subject to phase distorting issue
4 Experimental Verification
The fault diagnosis for large rotating machinery has been animportant topic and is attracting more and more attention
8 Shock and Vibration
the primary signalthe recovered signal
minus2
minus1
0
1
2A
mpl
itude
100 200 300 400 5000Sampling numbers i
(a)
(200977)
50 1000f (Hz)
002
04
06
08
1
Am
plitu
de
(b)
the primary signalthe recovered signal
minus1
minus05
0
05
1
Am
plitu
de
100 200 300 400 5000Sampling numbers i
(c)
(800786)
002
04
06
08
1
Am
plitu
de
50 1000f (Hz)
(d)
the primary signalthe recovered signal
100 200 300 400 5000Sampling numbers i
minus1
minus05
0
05
1
Am
plitu
de
(e)
(490688)
50 1000f (Hz)
002
04
06
08
1
Am
plitu
de
(f)
the primary signalthe recovered signal
minus05
0
05
Am
plitu
de
100 200 300 400 5000Sampling numbers i
(g)
(500463)
002
04
06
08
1
Am
plitu
de
50 1000f (Hz)
(h)
Figure 6 The time and frequency domains of reconstructed signal (a) The time domain of the reconstructed signal via the first set ofeigenvalues (b) Amplitude spectrum of Figure 6(a) (c) The time domain of the reconstructed signal using the second set of eigenvalues(d) Amplitude spectrum of Figure 6(c) (e) The time domain of the reconstructed signal using the third set of eigenvalues (f) Amplitudespectrum of Figure 6(e) (g) The time domain of the reconstructed signal using the fourth set of eigenvalues (h) Amplitude spectrum ofFigure 6(g)
Shock and Vibration 9
A B
Servo Motor Unloading device
Coupling
Sliding Bearing
Rotor
Large Platform
Small Platform
Air spring
BeltSliding Bearing
Figure 7 Large rot or vibration test bed
Displacement SensorA-end Face B-end Face
D13 D14
D12D11
Figure 8 Schematic and installation of eddy current displacement sensor
Rotor orbits indicate the symptoms of malfunction andare related to variation of input force or dynamic stiffnessGenerally different curve shapes of rotor orbit in a senserepresent different fault types [19ndash22 29] For example insideldquo8rdquo outside ldquo8rdquo and petal-shaped curves are omens of oil-filmwhirl fault misalignment fault and rubbing fault of rotorsystem respectivelyTherefore extraction of rotor orbit playsa significant role in rotating machinery diagnostics In thissection a simple and efficient CFSM-PCA is applied to purifyrotor orbit precisely
41 Experimental Rigs Figure 7 shows the large rotor vibra-tion test bed system which is constructed by our group Inorder to real-timemonitor rotor working station two KamanKD2306-1S eddy current proximity probes are installedon individual side in orthogonal manners (Figure 8) Theexperimental data were collected through LMS Test Lab
42 The First Group Sample Analysis Experimental datastems from D11 and D12 eddy current displacement sensorin side A 4096 data points were collected with test bed speedof 1080 rmin and sampling frequency of 2048 Hz The timedomain waveforms amplitude spectrums and the order ofamplitudes were obtained after filtering DC component viaFFT (Figure 9)
As demonstrated in Figure 9 raw signal was not onlyaffected by noise but also interfered by power frequency 50Hz and its harmonics frequency The energy of signal mainlyconcentrated on the leading two octaves
Rotor orbit was produced by combining spindle dis-placement signal D11 with signal D12 (Figure 10) X-axispresented raw signal D11 and Y-axis referred to signal D12Manipulating status of test bed is difficult to know from thisunprocessed rotor orbit Next the rotor orbit was attemptedto purify by employing PCA
Generally rotor orbit synthesized by 1X and 2X is cred-itable Orbits included high harmonics incline to becomecomplex and even turn messy Hence extraction of 1Xand 2X of raw signal becomes more important [21 22]Eigenvalue distribution of covariance matrix of D11 and D12signals are displayed in Figure 11 According to amplitudeof each frequency of D11 and D12 in Figure 9 1X and 2Xcorresponded to the first (first and second eigenvalues) andthe second (third and fourth eigenvalues) set of eigenvaluesrespectively The leading four eigenvalues in Figure 11 werereconstructed through CFSM-PCA and results are illustratedin Figure 12 Generated 1X and 2X were legible without effectby noise and disturbance from frequency 50 Hz as well as itsharmonics Meanwhile amplitude of the extracted signal wasclose to that of 1X and 2X in raw signal indicating that thereis no spectral leakage issue
The feature spectrums continued to be separated andeigenvalue distributionmaps of covariance matrix of D11 andD12 are shown in Figure 15
Rotor orbit generated by the purified 1X and 2X is shownin Figure 13The rotor orbit was a caved-in banana-like curvedemonstrating that sideA (near themotor) potentially suffersfrom misalignment fault [18 19] The explicit rotor orbit alsovalidates the high efficiency of this proposed method
43 The Second Group Sample Analysis In this case exper-imental data of D11 and D12 were analyzed when test bedruns steadily for a period of time Similarly 4096 datapoints were measured with test bed speed of 2770 rmin andsampling frequency of 2048Hz Time domainwaveforms andamplitude spectrums are displayed in Figure 14
According to amplitude of the leading trebling frequencyofD11 andD12 in Figure 14 1X corresponded to the second setof eigenvalues (third and fourth eigenvalues) 2X paralleledthe third set of eigenvalues (fifth and sixth eigenvalues) and
10 Shock and Vibration
D11-1080 rpmx 10
minus5
minus5minus4minus3minus2minus1
012345
X (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(a)
Power-line Interference50Hz
0733404771
Order 1 Order 2
0250502342
Order 6Order 7
D11-1080 rpm
01651 Order 11
18Hz36Hz54Hz72Hz90Hz
x 10minus5
1times
2times
3times
4times
5times
1times
2times
3times 4times5times
002040608
1
X (m
)
50 100 150 200 250 300 350 4000Frequency (Hz)
(b)
D12-1080 rpmx 10
minus5
minus6
minus4
minus2
0246
Y (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(c)
D12-1080 rpmPower-line Interference
50Hz
0658105228
Order 1 Order 2
0277101226
Order 5 Order 12
01948 Order 8
18Hz36Hz54Hz72Hz90Hz
x 10minus5
1times
2times
3times
4times
5times
1times
2times
3times
4times5times
0
05
1
Y (m
)
50 100 150 200 250 300 350 4000Frequency (Hz)
(d)
Figure 9The time domain and frequency domain of theD11 andD12 sensors in theA-end face (a)The time domain of D11 (b)The frequencydomain of D11 (c) The time domain of D12 (d) The frequency domain of D12
x 10minus5
x 10minus5
minus5
minus4
minus3
minus2
minus1
0
1
2
3
4
5
D12
minusY
(m)
minus6 minus2minus4 2 4 60D11minusX (m)
Figure 10 A-end face of the D11 and D12 sensors to collect the signal directly synthesized axis orbit
(12751) (22748)e first group
(31243) (41241)e second group
D11minus1080 rpm
x 10minus8
0
1
2
3
4
Eige
nval
ue o
f PCA
(m)
10 20 30 40 500Number of eigenvalues q
(a)
(12247) (2241)
(314) (41396)
D12minus1080 rpme first group
e second group
x 10minus8
005
115
225
Eige
nval
ue o
f PCA
(m)
10 20 30 40 500Number of eigenvalues q
(b)
Figure 11The eigenvalues distribution of covariancematrixC (a)The eigenvalues distribution of D11 signal (b)The eigenvalues distributionof D12 signal
Shock and Vibration 11
D11minus1080 rpmx 10
minus5
minus15
minus10
minus05
0051015
X (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(a)
D12minus1080 rpmx 10
minus5
minus10
minus05
0051015
Y (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(b)
D11minus1080 rpm(1807287)
(360472)
x 10minus5
002040608
1
X (m
)
50 100 150 2000Frequency (Hz)
(c)
(3605105)(1806575)
D12minus1080 rpmx 10
minus5
002040608
1
Y (m
)
50 100 150 2000Frequency (Hz)
(d)
Figure 12 Extracting the time and frequency spectrums of the first two frequencies of D11 and D12 (a)The time domain of D11 (b)The timedomain of D12 (c) The frequency domain of D11 (d) The frequency domain of D12
W1
x 10minus5
x 10minus5
minus1
minus05
0
05
1
15
D12
minusY
(m)
minus1minus15 minus05 05 10 15D11minusX (m)
(a)0
051
152
0
2
t (s)D11minusX (m)
minus2
x 10minus5
x 10minus5
minus1
minus05
0
05
1
15
D12
minusY
(m)
(b)
Figure 13 The synthesis of axis orbit is to extract the first two times fundamental frequencies of the D11 and D12 signals by the CFSM-PCA(a) The axis orbit (b) Relationship between axis orbit and time
3X matched the first set of eigenvalues (first and secondeigenvalues)
As displayed in Figure 15 eigenvalues of 1X and 3Xwere large thus 1X and 3X were utilized to synthesize rotororbit firstly Purification results and orbit of shaft center arerevealed in Figures 16 and 17 respectively The legible orbitshowed a typical plum blossom shape
It can be seen from Figures 16 and 17 that the filteringeffect was also quite obvious The noise and power frequency50 Hz as well as its harmonic interference could be removedsuccessfully Moreover amplitude of refined signal was closeto that of original one and the reconstructed time domainsignal seemed to be steady without any fluctuation
As shown in Figures 14(c) and 14(d) amplitudes of 1X 2Xand 3X in amplitude spectrum of raw signal were largeThus1X 2X and 3X were also extracted to synthesize axis orbit
Time domain frequency domain and orbit curve are shownin Figures 18 and 19 respectively
Axis orbit synthesized by the leading three frequenciesexhibited a quincunx shape which is basically the sameto that synthesized by 1X and 3X frequency componentsindicating that static and dynamic parts friction may exist inA-side
44 Comparison Harmonic Wavelet with Wavelet Packet
(1) Comparison of CFSM-PCA with Harmonic Wavelet Onecommon used method for signal processing harmonicwavelet algorithm proposed by Newland is an improvedwavelet algorithm [29] It can segment entire frequencyband infinitely and avoid the shortcoming of downsamplingmethod of binary wavelet and binary wavelet packet [22 23]
12 Shock and Vibration
D11minus2770 rpmx 10
minus4
minus1
minus05
0
05
1
D11
-X (m
)
1000 2000 3000 40000Sampling numbers i
(a)
D12minus2770 rpmx 10
minus4
minus1
minus05
0
05
1
D12
-Y (m
)
1000 2000 3000 40000Sampling numbers i
(b)
Power-Line interference 50and 150Hz
1X2X
3X 1010D11minus2770 rpm
08441166
1X2X3X
x 10minus5
200 400 600 800 10000f (Hz)
0
05
10
15
D11
-X (m
)
(c)
1X
2X
3XPower-Line interference 50
and 150Hz
D12minus2770 rpm117309021227
1X2X3X
x 10minus5
0
05
10
15
D12
-Y (m
)
200 400 600 800 10000f (Hz)
(d)
Figure 14 The time and frequency domains of the D11 and D12 sensors (a) The time domain of D11 (b) The time domain of D12 (c) Thefrequency domain of D11 (d) The frequency domain of D12
(11489) (21482)e first group
(31237) (41220)e second group
(30348) (40347)e second group
D11minus2770 rpmx 10
minus7
10 20 30 400 50Number of eigenvalues q
0
05
1
15
Eige
nval
ue o
f PCA
(m)
(a)
D12minus2770 rpm(11688) (21687)
e first group
e second group
(30414) (40413)e second group
(31673) (41664)
x 10minus7
0
05
1
15
2
Eige
nval
ue o
f PCA
(m)
10 20 30 40 500Number of eigenvalues q
(b)
Figure 15 The eigenvalues distribution of covariance matrix C (a) The eigenvalues distribution of D11 (b) The eigenvalues distribution ofD12
1X and its frequency doubling signal can be easily extractedto form axis orbit of revolving test bearing
Axis orbit was fabricated by 1X and 2Xof data of example 1(Section 42) through harmonic wavelet algorithm and theresults are demonstrated in Figure 20 The shape of axisorbit was trapped banana-like which is consistent with theresult obtained via CFSM-PCA It should be noted thatorbit in Figure 13 was more obvious than that in Figure 20ie W1ltW2 Meanwhile the amplitude fluctuation of timedomain signal was large (Figures 20(a) and 20(b)) Themajor reason for this phenomenon is that the harmonicwavelet which is a band filter in frequency domain inevitablyextracts the noise near the feature frequency Furthermorethis phenomenon also demonstrates that harmonic waveletis subject to the Heisenberg uncertainty principle ie thephenomenon of the amplitude leakage exists
(2) Comparison of CFSM-PCA with Wavelet Packet Besidesharmonic wavelets wavelet packet is also used to purify axisorbit [19ndash21] Purification results of axis orbit via Daubechieswavelet packet in Figure 21 exhibit that axis orbit was diver-gent seriously This was worse than that of harmonic waveletie W1ltW2ltW3The phenomenon of severe energy leakageoccurred during the extraction of 1X and 2X which cancause divergence of axis orbit Moreover the bandpass filterof wavelet packet also accounts for the divergent orbits
5 A Single Frequency Filtrationwith CFSM-PCA
The application in single frequency filtration is discussedin this section Taking D11 experimental data of example
Shock and Vibration 13
D11minus2770 rpmx 10
minus5
0 1000500Sampling numbers i
minus4
minus2
0
2
4
D11
-X (m
)
(a)
D12minus2770 rpmx 10
minus5
minus4
minus2
0
2
4
D12
-Y (m
)
500 10000Sampling numbers i
(b)
(1375 1185)(46 1009)
D11minus2770 rpmx 10
minus5
100 200 300 400 5000f (Hz)
0
05
1
15
D11
-X (m
)
(c)
(1375 1218)(46 1164)
D12minus2770 rpmx 10
minus5
100 200 300 400 5000f (Hz)
0
05
1
15
D12
-Y (m
)
(d)
Figure 16 Extracting the frequency spectrums of the first two frequencies of D11 and D12 by PCA algorithm (a) The time domain of D11(b) The time domain of D12 (c) The frequency spectrum of D11 (d) The frequency spectrum of D12
x 10minus5
x 10minus5
minus4
minus3
minus2
minus1
0
1
2
3
4
D12
-Y (m
)
210 3 4minus2minus3 minus1minus4
D11-X (m)
(a)0
0204
0608
10
24
t (s)D11-X (m) minus2
minus4
x 10minus5
x 10minus5
minus4
minus2
0
2
4
D12
-Y (m
)
(b)
Figure 17 The synthesis of axis orbit is to extract the first two times fundamental frequencies of the D11 and D12 signals by the CFSM-PCA(a) The axis orbit (b) Relationship between axis orbit and time
1 (Section 42) as a demo we attempt to filter the powerfrequency interference (50 Hz) from D11 by employingCFSM-PCA It can be seen from Figure 9(b) that sequenceof amplitude of 50 Hz in raw signal was 3 so eigenvec-tors corresponding to 5th and 6th eigenvalues in eigen-value distribution map were used to reconstruct Accordingto (8) the filtered signal (Figure 22) without 50 Hz was
obtained by subtracting reconstructed signal from originalone
Results in Figure 22 show that this algorithm filters out50 Hz power frequency interference and has no influenceon the other frequency components of raw signal demon-strating that this method has a potential application in notchfilter
14 Shock and Vibration
D11minus2770 rpm
x 10minus5
minus4
minus2
0
2
4
D11
-X (m
)
200 400 600 800 10000Sampling numbers i
(a)
D12minus2770 rpm
x 10minus5
0 400 600 800 1000200Sampling numbers i
minus4
minus2
0
2
4
D12
-Y (m
)
(b)
(1375 1185)(46 1009)
(92 0831)
D11minus2770 rpmx 10
minus5
00
05
10
15
D11
-X (m
)
100 200 300 400 5000f (Hz)
(c)
(1375 1218)(46 1163)
(92 0898)
D12minus2770 rpmx 10
minus5
00
05
10
15
D12
-Y (m
)
100 200 300 400 5000f (Hz)
(d)
Figure 18 Extracting the time and frequency spectrums of the first three frequencies of D11 and D12 by PCA algorithm (a)The time domainof D11 (b) The time domain of D12 (c) The frequency spectrum of D11 (d) The frequency spectrum of D12
x 10minus5
x 10minus5
minus4
minus3
minus2
minus1
0
1
2
3
4
D12
-Y (m
)
minus3 minus2 minus1minus4 1 2 3 40D11-X (m)
(a)
1008
0604
0200
02
4D11-X (m)
t (s)
minus2
minus4
x 10minus5
x 10minus5
minus4
minus2
0
2
4
D12
-Y (m
)
(b)
Figure 19 The synthesis of axis orbit is to extract the first three times fundamental frequencies of the D11 and D12 by the CFSM-PCA (a)The axis orbit (b) Relationship between axis orbit and time
6 Conclusion
The selection of effective eigenvalues in feature distributionchart of covariance matrix C is crucial in PCA algorithm Inthis paper the relationships between effective eigenvalues andsignal frequency and amplitude are discovered and a series ofconclusions are summarized
First in Hankel matrix mode each valid frequencycomponent of signal only produces two adjacent nonzeroeigenvalues Second the number of valid eigenvalues incharacteristic distribution chart which is independent ofnumerical values of 119891119894 119860 119894 and 120593119894 is related to the number offrequency componentsThird the sequence of signal effectiveeigenvalues in characteristic distribution map is determined
Shock and Vibration 15
D11-1080 rpm
x 10minus5
minus20
minus15
minus05
0
05
10
15
20
D11
-X (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(a)
D12-1080 rpmx 10
minus5
minus15
minus1
minus05
0
05
1
15
D12
-Y (m
)
500 1000 1500 2000 2500 30000 40003500Sampling numbers i
(b)
(16 07335)
(32 04771)
D11-1080 rpm
x 10minus5
10 20 30 40 50 60 70 80 90 1000f (Hz)
0010203040506070809
1
D11
-X (m
)
(c)
(16 06571)
(32 05228)
D12-1080 rpm
x 10minus5
0010203040506070809
1
D12
-Y (m
)
10 20 30 40 50 60 70 80 90 1000f (Hz)
(d)
W2
x 10minus5
x 10minus5
minus15
minus1
minus05
0
05
1
15
Y(D
12) (
m)
minus1 minus05minus15 05 1 15 20X(D11) (m)
(e)
Figure 20The purification effect of harmonic wavelets (a)The time domain of D11 (b)The time domain of D12 (c)The frequency spectrumof D11 (d) The frequency spectrum of D12 (e) The orbit is to extract the first 2 frequencies
by amplitude 119860 119894 of signal frequency component The largermagnitude of amplitude is the greater eigenvalues are andthe higher arrangement will be
Based on aforesaid principle a new CFSM-PCA wasdeveloped and has become an effective tool to extract oreliminate specific characteristic (single frequency) even dif-ference of two frequencies is only 1 Hz Purification of rotororbit using this algorithm shows significant improvement
than those of existing methods such as wavelet packet andharmonic wavelet packet Although the proposed CFSM-PCA is effective some drawbacks are unavoidable Forexample this method suffers from similar issue to otheralgorithms that is the reconstruction error increaseswith thedecrease of SNR In addition the applications of the CFSM-PCA algorithm in purification of speech recognition bearingfault diagnosis and fault diagnosis of power system should
16 Shock and Vibration
D11-1080 rpmx 10
minus5
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
minus20
minus15
minus05
0
05
10
15
20
D11
-X (m
)
(a)
D12-1080 rpmx 10
minus5
minus15
minus1
minus05
0
05
1
15
D12
-Y (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(b)
(16 06557)
(32 04381)
D11-1080 rpmx 10
minus5
10020 30 40 50 60 70 80 90100f (Hz)
0010203040506070809
1
D11
-X (m
)
(c)
(16 06452)(32 04927)
D12-1080 rpm
x 10minus5
0010203040506070809
1
D12
-Y (m
)
9020 30 40 50 60 70 8010 1000f (Hz)
(d)
W3
x 10minus5
x 10minus5
minus15
minus1
minus05
0
05
1
15
Y(D
12) (
m)
minus15 minus05minus1 05 1 15 20X(D11) (m)
(e)
Figure 21 The purification effect of wavelet packet (a) The time domain of D11 (b) The time domain of D12 (c) The frequency spectrum ofD11 (d) The frequency spectrum of D12 (e) The orbit is to extract the first 2 frequencies
be further exploredWe fully believe that the CFSM-PCAwillsparkle in diverse fields
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work is supported by the National High TechnologyResearch andDevelopment Program of China (863 Program)
Shock and Vibration 17
(18 0734)
(36 04634)
(54 02448) (72 02351)(90 01646)
x 10minus5
50 100 150 2000f (Hz)
0
02
04
06
08
1
D11
minusX
(m)
Figure 22 Frequency spectrum of D11 signal after filtering 50Hz
(no 2015AA043005) and the National Natural Science Foun-dation of China (no 51375187)
References
[1] H Abdi and L J Williams ldquoPrincipal component analysisrdquoWiley Interdisciplinary Reviews Computational Statistics vol 2no 4 pp 433ndash459 2010
[2] A Widodo B Yang and T Han ldquoCombination of independentcomponent analysis and support vectormachines for intelligentfaults diagnosis of induction motorsrdquo Expert Systems withApplications vol 32 no 2 pp 299ndash312 2007
[3] S Wold K Esbensen and P Geladi ldquoPrincipal componentanalysisrdquo Chemometrics and Intelligent Laboratory Systems vol2 no 1ndash3 pp 37ndash52 1987
[4] M Kirby and L Sirovich ldquoApplication of the Karhunen-Loeve procedure for the characterization of human facesrdquo IEEETransactions on Pattern Analysis and Machine Intelligence vol12 no 1 pp 103ndash108 1990
[5] J H Xi Y Z Han and R H Su ldquoNew fault diagnosis methodfor rolling bearing based on PCArdquo in Proceedings of the 25thChinese Control and Decision Conference CCDC rsquo13 pp 4123ndash4127 IEEE May 2013
[6] A Malhi and R X Gao ldquoPCA-based feature selection schemefor machine defect classificationrdquo IEEE Transactions on Instru-mentation and Measurement vol 53 no 6 pp 1517ndash1525 2004
[7] S Xu X Jiang J Huang S Yang and X Wang ldquoBayesianwavelet PCA methodology for turbomachinery damage diag-nosis under uncertaintyrdquo Mechanical Systems and Signal Pro-cessing vol 80 pp 1ndash18 2016
[8] W Sun J Chen and J Li ldquoDecision tree and PCA-based faultdiagnosis of rotatingmachineryrdquoMechanical Systems and SignalProcessing vol 21 no 3 pp 1300ndash1317 2007
[9] Z X Li X P Yan C Q Yuan Z Peng and L Li ldquoVirtual pro-totype and experimental research on gear multi-fault diagnosisusing wavelet-autoregressive model and principal componentanalysismethodrdquoMechanical Systems and Signal Processing vol25 no 7 pp 2589ndash2607 2011
[10] K Y Shao M M Cai and G F Zhao ldquoRolling bearing faultdiagnosis based onwavelet energy spectrum PCA andPNNrdquo inProceedings of the 26th Chinese Control andDecisionConferenceCCDC rsquo14 pp 800ndash804 IEEE 2014
[11] T Qiu Q F Zhang and Y J Ding ldquoNonlinear sensor faultdetection and data rebuilding based on principle componentanalysisrdquo Journal of Tsinghua University vol 46 no 5 pp 708ndash711 2006
[12] Y K Gu Z X Yang and F L Zhu ldquoGearbox fault feature fusionbased on principal component analysisrdquo China MechanicalEngineering vol 26 no 11 pp 1532ndash1537 2015
[13] R Dunia S J Qin T F Edgar and T J McAvoy ldquoIdentificationof faulty sensors using principal component analysisrdquo AIChEJournal vol 42 no 10 pp 2797ndash2811 1996
[14] E P de Moura C R Souto A A Silva and M A S IrmaoldquoEvaluation of principal component analysis and neural net-work performance for bearing fault diagnosis from vibrationsignal processed by RS and DF analysesrdquo Mechanical Systemsand Signal Processing vol 25 no 5 pp 1765ndash1772 2011
[15] S Golestan M Monfared F D Freijedo and J M GuerreroldquoDynamics assessment of advanced single-phase PLL struc-turesrdquo IEEE Transactions on Industrial Electronics vol 60 no6 pp 2167ndash2177 2013
[16] A Lima H Zen Y Nankaku C Miyajima K Tokuda andT Kitamura ldquoOn the use of kernel PCA for feature extractionin speech recognitionrdquo IEICE Transaction on Information andSystems vol 12 no 87 pp 2802ndash2811 2004
[17] J D Zheng ldquoImproved hilbert-huang transform and its appli-cations to rolling bearing fault diagnosisrdquo Journal of MechanicalEngineering vol 1 no 51 pp 138ndash145 2015
[18] G He K Ding and H Lin ldquoFault feature extraction of rollingelement bearings using sparse representationrdquo Journal of Soundand Vibration vol 366 pp 514ndash527 2016
[19] J G Yang S B Xia and Y G Liu ldquoWavelet denoising and itsapplication in extraction of shaft centerline orbit featuresrdquoJournal of Harbin Institute of Technology vol 31 no 5 pp 52ndash541999 (Chinese)
[20] J A Duan and X D Zhang ldquoFeature extraction from the orbitof the center of a rotor based on wavelet transformrdquo Journal ofVibration Measurement amp Diagnosis vol 17 no 01 pp 31ndash341997
[21] J Han D X Jiang and W D Ning ldquoApplication of optimalwavelet packets in failure signal extraction of rotor orbitrdquoTurbine Technology vol 43 no 3 pp 133ndash136 2001
[22] W B Zhang X J Zhou andY Lin ldquoRefinement of rotor centerrsquosorbit by a harmonic window methodrdquo Journal of VibrationMeasurement amp Diagnosis vol 30 no 1 pp 87ndash90 2010
[23] S M Li G Z Lu and Q Y Xu ldquoOn obtaining accuraterotor sub-frequency signal with harmonic waveletrdquo Journal ofNorthwestern Polytechincal University vol 19 no 2 pp 220ndash224 2001
[24] F J Wu and L S Qu ldquoAn improved method for restraining theend effect in empiricalmode decomposition and its applicationsto the fault diagnosis of large rotating machineryrdquo Journal ofSound and Vibration vol 314 no 3-5 pp 586ndash602 2008
[25] W G Liu ldquoResearch on method of purification of shaft orbitbased on singular value decompositionrdquo Technical Acousticsvol 5 no 35 pp 30ndash35 2016
[26] X Z Zhao and B Y Ye ldquoThe influence of formation manner ofcomponent on signal processingrdquo Shang Hai Jiao Tong Univer-sity vol 45 no 3 pp 368ndash374 2011
[27] X Z Zhao B Y Ye and T J Chen ldquoPrinciple of singularitydetection based on SVD and its applicationrdquo Journal of Vibra-tion and Shock vol 6 no 27 pp 7ndash10 2008
[28] X Z Zhao B Y Ye and T J Chen ldquoDifference spectrum theoryof singular value and its application to the fault diagnosis ofheadstock of latherdquo Journal of Mechanical Engineering vol 46pp 100ndash108 2010
[29] I A Kougioumtzoglou and P D Spanos ldquoAn identificationapproach for linear and nonlinear time-variant structural sys-tems via harmonic waveletsrdquo Mechanical Systems and SignalProcessing vol 37 no 1-2 pp 338ndash352 2013
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4 Shock and Vibration
(15123) (25118)e first group
(15123) (25118)
(15123) (25118)
e first group
e first group
0
200
400
600
0
200
400
600
10 20 30 40 500Number of eigenvalues q
10 20 30 40 500
10 20 30 40 500
x1(t) = MCH(40t + 10)
x2(t) = MCH(100t + 40)
x3(t) = MCH(160t + 70)
0
200
400
600
Eige
nval
ue o
f PCA
i
Figure 1 Eigenvalues distribution of covariance matrix C with one frequency component
(15123) (25118)
(15123) (25118)
(15123) (25118)
(33278) (43275)
(33278) (43275)
(33278) (43275)
e first group
e second group
e first group
e second group
e first group
e second group
0
200
400
600
10 20 30 40 500
10 20 30 40 500
0
200
400
600
10 20 30 40 500Number of eigenvalues q
x1(t) = MCH(40t + 10) + 08 MCH(60t + 20)
x3(t) = MCH(160t + 70) + 08 MCH(180t + 80)
0
200
400
600
Eige
nval
ue o
f PCA
i
x2(t) = MCH(100t + 40) + 08 MCH(120t + 50)
Figure 2 Eigenvalues distribution of covariance matrix C with twofrequency components
frequency component with amplitude of 08 Thus it couldbe confirmed that the first set of nonzero eigenvalues 1205821 and1205822 in Figure 2 are generated from frequency component withamplitude of 10 while the second set of nonzero eigenvalues
are generated from frequency component with amplitude of08(3)Given 119896= 3119860 119894 = 1 08 and 06 tri-groups signals were
reconstructed and each group signal contained three sets ofeffective frequency components The frequencies of the firstgroup of signal were 20 Hz 30 Hz and 40 Hz respectivelyThe frequencies of the second group of signals were 50 Hz60 Hz and 70 Hz respectively Similarly the frequencies ofthe third group of signals were 80 Hz 90 Hz and 100 Hzrespectively The corresponding phases of these three groupsof signals were taken as 10 20 and 30 40 50 and 60 and 7080 and 90 respectively
When 119896 = 3 119860 119894=1 08 and 06 each group signal pro-duces three effective frequency components The amplitudes119860 119894 of corresponding frequency 119891119894(119894=1 2 3) are same butwith different frequencies 120596119894 and phases 120593119894 As displayed inFigure 3 each group signal produces three sets of nonzeroeigenvalues and each set of eigenvalues contains two adjacenteigenvalues 120582119894 and 120582119894+1 In addition these three sets ofeigenvalues produced by each signal are correspondinglysame
It can be found from Figures 1ndash3 that the magnitudeof eigenvalues of each first group in three graphs is sameAs aforementioned the first set of nonzero eigenvalues1205821 and 1205822 are generated from frequency component withamplitude 119860 119894 of 1 Similarly by comparing Figure 2 withFigure 1 the second set of nonzero eigenvalues 1205823 and 1205824are generated from frequency component with amplitude119860 119894 of 08 And then it is almost certain that the third setof frequency components 1205825 and 1205826 are generated fromfrequency component with amplitude 119860 119894 of 06
The same results can be obtained by continuously increas-ing effective frequency components of signal Therefore an
Shock and Vibration 5
(33278) (43275)
(51844) (61842)
(15123) (25118)
(33278) (43275)
(51844) (61842)
(15123) (25118)
(33278) (43275)
(51844) (61842)
(15123) (25118)e first group
e second group
e third group
e first group
e second group
e third group
e first group
e second group
e third group
x1(t) = MCH(40t + 10) + 08 MCH(60t + 20) + 06 MCH(80t + 30)
x2(t) = MCH(100t + 40) + 08 MCH(120t + 50) + 06 MCH(140t + 60)
x3(t) = MCH(160t + 70) + 08 MCH(180t + 80) + 06 MCH(200t + 90)
0
200
400
600
10 20 30 40 500
0
200
400
600
Eige
nval
ue o
f PCA
i
10 20 30 40 500
0
200
400
600
10 20 30 40 500Number of eigenvalues q
Figure 3 Eigenvalues distribution of covariancematrixCwith threefrequency components
Hankel matrix is derived from certain signal x(t) l=min(mn) with m rows n columns and k effective frequenciesConcerning the fact that the Shannon sampling theorem ismet assuming that lgt2k generic conclusions are summarizedas follows(1) Each frequency component of signal produces two
nonzero eigenvalues with one arranging another closely(2) The number of effective eigenvalues of crude signal
is related to the number of frequency components and has
nothing to do with themagnitude of amplitude119860 119894 frequency120596119894 and phase 120593119894(3) In eigenvalue distribution chart of covariance matrix
C the sequence of nonzero effective eigenvalues is deter-mined by amplitude 119860 119894 of signal The larger amplitude is thelarger eigenvalues will be the more forward rank of the twoeigenvalues produced from corresponding frequencies is
Inspired by the relationship between frequency andeigenvalues a new technique for characteristic frequency sep-aration method based on PCA (CFSM-PCA) was proposedThe concrete steps are listed below(1) For a certain signal 119909(119905) direct component (DC) of
raw signal is filtered out via fast Fourier transform (FFT)firstly and then Hankel matrix X is constructed throughfiltered signal(2) Covariance matrix C of matrix X is obtained with its
eigenvalues 120582119894 arranging in descending order (1205821 1205822 sdot sdot sdot 120582119898)and corresponding eigenvectors are obtained as 12057211205722 sdot sdot sdot120572119898(3) According to the distribution of eigenvalues 120582119894reconstruction is carried out from two eigenvalues and cor-responding eigenvectors of certain frequency For examplefor amplitude perspective if the rank of specific frequencyof raw signal is k a new matrix is received by reconstructingthe eigenvectors corresponding to 2k-1 and 2k eigenval-ues in eigenvalue distribution chart of covariance matrixC(4)Thematrix X can be produced by adding the mean of
original matrix to the new reconstructed matrix(5) The signal x which is the characteristic frequency
component is recovered from matrix X by means of theaveraging method
32Theoretical Deduction In this section deduction processof the three discoveries (Section 31) is provided Supposingthat a signal is expressed as 119909(119905) = 119886 sin(120596119905 + 120593) Samplingtime 119879119904 is used to discretize signal x(t) Hankel matrix withm rows and n columns is derived from signal 119909(119905) exhibitedas
X = 119886 times[[[[[[
sin (0 sdot 120596119879119904 + 120593) sin (1 sdot 120596119879119904 + 120593) sdot sdot sdot sin ((119899 minus 1) sdot 120596119879119904 + 120593)sin (1 sdot 120596119879119904 + 120593) sin (2 sdot 120596119879119904 + 120593) sdot sdot sdot sin (119899 sdot 120596119879119904 + 120593)
sin ((119898 minus 1) sdot 120596119879119904 + 120593) sin (119898 sdot 120596119879119904 + 120593) sdot sdot sdot sin ((119898 + 119899 minus 2) sdot 120596119879119904 + 120593)
]]]]]]
(13)
(1) Each characteristic frequency produces two effectiveeigenvalues
The deduced process of the first conclusion is given belowEquation (13) can be rewritten as (14) based onEulerrsquos Formula
X = 1198862119894
times[[[[[[[
119890(0120596119879119904+120593)119894 minus 119890minus(0120596119879119904+120593)119894 119890(1120596119879119904+120593)119894 minus 119890minus(1120596119879119904+120593)119894 119890(2120596119879119904+120593)119894 minus 119890minus(2120596119879119904+120593)119894 sdot sdot sdot 119890((119899minus1)120596119879119904+120593)119894 minus 119890minus((119899minus1)120596119879119904+120593)119894119890(1120596119879119904+120593)119894 minus 119890minus(1120596119879119904+120593)119894 119890(2120596119879119904+120593)119894 minus 119890minus(2120596119879119904+120593)119894 119890(3120596119879119904+120593)119894 minus 119890minus(3120596119879119904+120593)119894 sdot sdot sdot 119890(119899120596119879119904+120593)119894 minus 119890minus(119899120596119879119904+120593)119894
119890((119898minus1)120596119879119904+120593)119894 minus 119890minus((119898minus1)120596119879119904+120593)119894 119890(119898120596119879119904+120593)119894 minus 119890minus(119898120596119879119904+120593)119894 119890((119898+1)120596119879119904+120593)119894 minus 119890minus((119898+1)120596119879119904+120593)119894 sdot sdot sdot 119890((119898+119899minus2)120596119879119904+120593)119894 minus 119890minus((119898+119899minus2)120596119879119904+120593)119894
]]]]]]]
(14)
6 Shock and Vibration
Equation (14) can be expanded to addition form of twoformulas depicted as
X = 1198862119894 times 119890120593119894
times[[[[[[[[
1198900120596119879119904119894 1198901120596119879119904119894 1198902120596119879119904119894 sdot sdot sdot 119890(119899minus1)1205961198791199041198941198901120596119879119904119894 1198902120596119879119904119894 1198903120596119879119904119894 sdot sdot sdot 119890119899120596119879119904119894
119890(119898minus1)120596119879119904119894 119890119898120596119879119904119894 119890(119898+1)120596119879119904119894 sdot sdot sdot 119890(119898+119899minus2)120596119879119904119894
]]]]]]]]
minus 1198862119894 times 119890minus120593119894
times[[[[[[[[
119890minus0120596119879119904119894 119890minus1120596119879119904119894 119890minus2120596119879119904119894 sdot sdot sdot 119890minus(119899minus1)120596119879119904119894119890minus1120596119879119904119894 119890minus2120596119879119904119894 119890minus3120596119879119904119894 sdot sdot sdot 119890minus119899120596119879119904119894
119890minus(119898minus1)120596119879119904119894 119890minus119898120596119879119904119894 119890minus(119898+1)120596119879119904119894 sdot sdot sdot 119890minus(119898+119899minus2)120596119879119904119894
]]]]]]]](15)
From (15) the rank of both matrices is 1 Based on rankrelationship of two matrices 119877(119860+119861) le 119877(119860) + 119877(119861) Hencerank of matrix X is less than or equal to 2
Additionally when 120593 = 0 (13) is rewritten to (16)According to sum-to-product identities the rank of X is 2when 120596119879119904 = 0 120587 2120587 sdot sdot sdot
X = 119886 times[[[[[[[[[
sin (0 sdot 120596119879119904) sin (1 sdot 120596119879119904) sdot sdot sdot sin ((119899 minus 1) sdot 120596119879119904)sin (1 sdot 120596119879119904) sin (2 sdot 120596119879119904) sdot sdot sdot sin (119899 sdot 120596119879119904)
sin ((119898 minus 1) sdot 120596119879119904) sin (119898 sdot 120596119879119904) sdot sdot sdot sin ((119898 + 119899 minus 2) sdot 120596119879119904)
]]]]]]]]]
(16)
When 120593 = 0 (13) can be deduced into the first-orderprincipal minor of matrix X ie |sin120593| = 0 The principalminor of order 2 of X is shown as10038161003816100381610038161003816100381610038161003816100381610038161003816sin (0 sdot 120596119879119904 + 120593) sin (1 sdot 120596119879119904 + 120593)sin (1 sdot 120596119879119904 + 120593) sin (2 sdot 120596119879119904 + 120593)
10038161003816100381610038161003816100381610038161003816100381610038161003816= minussin2 (120596119879119904)
= 0(17)
where120596119879119904 = 0 120587 2120587 sdot sdot sdot Because the leading principalminorof order 2 ofX in (13) is nonzero values so the rank of matrixX is at least 2
Combining result of (15) 119877(119860 + 119861) le 2 and nonzerovalues of the leading principal submatrix of order 2 hencethe rank of matrix X is 2 Matrix X has two eigenvalues 12058210158401and 12058210158402 Derived from C = XXT119899 (where 119899 is the numberof columns of matrix X) [3] two nonzero eigenvalues ofcovariance matrix C of X are 1205821 and 1205822 respectively(2)The sequence of eigenvalues is determined by ampli-
tudeIn terms of PCA covariance matrix C is described as
C = 119864 [(X minus 119864 (X)) sdot (X minus 119864 (X))T] (18)
Referring to (6) (19) is constructed as follows
119897
sum119894=1
C120572119894120572T119894 =119897
sum119894=1
120582119894120572119894120572T119894 (19)
Two effective eigenvalues are generated from a frequencycomponent that is l=2
CI119898 = 12058211205721120572T1 + 12058221205722120572T2 (20)
Then the energy of matrix C can be deduced as
|C|2 = 12058221119898
sum119894=1
12057221198941119898
sum119894=1
12057221198941 + 12059022119898
sum119894=1
12057221198942119898
sum119895=1
12057221198942
+ 21205902112059022119898
sum119894=1
12057211989411205721198942119898
sum119894=1
12057211989411205721198943(21)
Based upon (4) sum119898119894=1 12057221198941 = 1 sum119898119894=1 12057221198942 = 1 sum119898119894=1 12057211989411205721198942 =0 Then (22) is given by
|C|2 = 12058221 + 12058222 (22)
The constructed Hankel matrix X of signal 119909(119905) is sub-stituted into (18) we can see that the energy of covariancematrix C is proportional to 1198862 The larger 1198862 is the greaterenergy of matrix C is as well as 120582119894 according to (22)Furthermore the larger frequency component amplitude isthe larger corresponding eigenvalue in covariance matrix Ccharacteristic distribution chart is
Based on these conclusions once the amplitude sequenceof a certain frequency in raw signal amplitude spec-trum is determined its corresponding frequency compo-nent can be reconstructed In this way extraction of sin-gle or multiple characteristic frequencies could be real-ized
In view of addition relation as shown in (8) the notchfilter could be achieved through CFSM-PCA The frequencycomponent extracted by this algorithm is subtracted in orig-inal signal ie this frequency component can be eliminatedin raw signal Reader could consult examples in Section 5 formore details
Shock and Vibration 7
Table 1 Amplitude of signal components under the different SNR
SNR SNR1=-1 SNR2=-4 SNR3=-7 SNR4=-10Signal component Raw Process Raw Process Raw Process Raw Processsin(40120587119905 + 20) 09999 09938 10260 10240 10370 10730 1114 111205 sin(100120587119905 + 40) 04703 04826 05317 05482 07353 07533 05177 0523407 sin(98120587119905 + 50) 07299 07152 07116 07235 05110 04968 06800 0662108 sin(160120587119905 + 60) 08340 08324 08264 08142 08389 08338 08379 08923Max difference 00123 00122 00360 00544
minus6minus4minus2
0246
Am
plitu
de
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(a)
(200972)
(490693)(800791)
(500468)
0
05
1
Am
plitu
de
50 100 150 2000f (Hz)
(b)
Figure 4 The raw signal 119910 (a) Time domain (b) Amplitudespectrum
33 Simulation Example To verify the applicability and effec-tiveness of the CFSM-PCA algorithm a simulated sinusoidalsignal 119910 was constructed as
119910 = sin (40120587119905 + 20) + 05sin (100120587119905 + 40)+ 07sin (98120587119905 + 50) + 08sin (160120587119905 + 60)+ 119890 (119899)
(23)
where 119890(119899) is Gaussian white noise with standard deviationof 12 The result is shown in Figure 4(a) and amplitudespectrum (Fourier spectrum) is illustrated in Figure 4(b)after collecting 4096 data points with sampling frequency of1024 Hz
Follow-up work to separate frequencies via the proposedapproach was performed From eigenvalue distribution chartof crude signal in Figure 5 we can see that apart fromdiscernible four sets of eigenvalues most of eigenvaluescaused by noise are nonzero values and distribute at largeeigenvalues region
As demonstrated in Figures 6(a) and 6(b) time domainand frequency spectrum are generated after reconstructingthe first set of eigenvalues 1205821 and 1205822 in Figure 5 Frequencyof the reconstructed signal is 20 Hz with amplitude of 0977which is in correspondence with component sin(40120587t+20) ofy signal
As revealed in Figures 6(c) and 6(d) another time domainand amplitude spectrum are obtained via the reconstruction
(1515) (25144)
(3353) (43527)
(52654) (62643)
(71037) (81036)
e first group
e second group
e third group
e fourth group
0
200
400
600
Eige
nval
ue o
f PCA
i
10 20 30 40 500Number of eigenvalues q
Figure 5The eigenvalues distribution of the covariancematrixC of119910
of the second set of eigenvalues 1205823 and 1205824 Frequencyof this reconstructed signal is 80 Hz with amplitude of0786 which is same as component 08sin(160120587t+60) of rawsignal y Figures 6(e) and 6(f) are brought out through thereconstruction of the third set of eigenvalues 1205825 and 1205826Frequency of reconstructed signal is 49 Hz with amplitude of0688 which is consistent with component 07sin(98120587t+50)of crude signal 119910 Time domain and amplitude spectrum aregained after the reconstruction from the last set of eigenvaluesin Figure 5 exhibited as Figures 6(h) and 6(g) Frequency is50 Hz with amplitude of 0463 which is matched well withcomponent 05sin(100120587t+40) of signal y
Moreover several sets of signals with different signal-to-noise ratio (SNR) were processed by CFSM-PCA Theamplitude difference is summarized in Table 1
It can be seen that when the noise is low the amplitudeof frequency components extracted byCFSM-PCAalgorithmis close to that of the raw signal However the amplitudedifference will increase with the decrease of SNR
From all above-mentioned diagrams (Figure 6) it is obvi-ous that diverse frequency components extracted from rawsignal can be achieved accurately More importantly theseresults are coincident with ideal signal perfectly demon-strating that this CFSM-PCA is a zero phase shift frequencyextraction algorithm Meanwhile it should be noted that twofrequencies could also be perfectly separated via the proposedalgorithm even difference value between the adjacent fre-quency components is only 1 HzThismethod is accurate andseems to be more efficient than other existing filter methodssuch as wavelet filter and finite impulse response (FIR) filterwhich are subject to phase distorting issue
4 Experimental Verification
The fault diagnosis for large rotating machinery has been animportant topic and is attracting more and more attention
8 Shock and Vibration
the primary signalthe recovered signal
minus2
minus1
0
1
2A
mpl
itude
100 200 300 400 5000Sampling numbers i
(a)
(200977)
50 1000f (Hz)
002
04
06
08
1
Am
plitu
de
(b)
the primary signalthe recovered signal
minus1
minus05
0
05
1
Am
plitu
de
100 200 300 400 5000Sampling numbers i
(c)
(800786)
002
04
06
08
1
Am
plitu
de
50 1000f (Hz)
(d)
the primary signalthe recovered signal
100 200 300 400 5000Sampling numbers i
minus1
minus05
0
05
1
Am
plitu
de
(e)
(490688)
50 1000f (Hz)
002
04
06
08
1
Am
plitu
de
(f)
the primary signalthe recovered signal
minus05
0
05
Am
plitu
de
100 200 300 400 5000Sampling numbers i
(g)
(500463)
002
04
06
08
1
Am
plitu
de
50 1000f (Hz)
(h)
Figure 6 The time and frequency domains of reconstructed signal (a) The time domain of the reconstructed signal via the first set ofeigenvalues (b) Amplitude spectrum of Figure 6(a) (c) The time domain of the reconstructed signal using the second set of eigenvalues(d) Amplitude spectrum of Figure 6(c) (e) The time domain of the reconstructed signal using the third set of eigenvalues (f) Amplitudespectrum of Figure 6(e) (g) The time domain of the reconstructed signal using the fourth set of eigenvalues (h) Amplitude spectrum ofFigure 6(g)
Shock and Vibration 9
A B
Servo Motor Unloading device
Coupling
Sliding Bearing
Rotor
Large Platform
Small Platform
Air spring
BeltSliding Bearing
Figure 7 Large rot or vibration test bed
Displacement SensorA-end Face B-end Face
D13 D14
D12D11
Figure 8 Schematic and installation of eddy current displacement sensor
Rotor orbits indicate the symptoms of malfunction andare related to variation of input force or dynamic stiffnessGenerally different curve shapes of rotor orbit in a senserepresent different fault types [19ndash22 29] For example insideldquo8rdquo outside ldquo8rdquo and petal-shaped curves are omens of oil-filmwhirl fault misalignment fault and rubbing fault of rotorsystem respectivelyTherefore extraction of rotor orbit playsa significant role in rotating machinery diagnostics In thissection a simple and efficient CFSM-PCA is applied to purifyrotor orbit precisely
41 Experimental Rigs Figure 7 shows the large rotor vibra-tion test bed system which is constructed by our group Inorder to real-timemonitor rotor working station two KamanKD2306-1S eddy current proximity probes are installedon individual side in orthogonal manners (Figure 8) Theexperimental data were collected through LMS Test Lab
42 The First Group Sample Analysis Experimental datastems from D11 and D12 eddy current displacement sensorin side A 4096 data points were collected with test bed speedof 1080 rmin and sampling frequency of 2048 Hz The timedomain waveforms amplitude spectrums and the order ofamplitudes were obtained after filtering DC component viaFFT (Figure 9)
As demonstrated in Figure 9 raw signal was not onlyaffected by noise but also interfered by power frequency 50Hz and its harmonics frequency The energy of signal mainlyconcentrated on the leading two octaves
Rotor orbit was produced by combining spindle dis-placement signal D11 with signal D12 (Figure 10) X-axispresented raw signal D11 and Y-axis referred to signal D12Manipulating status of test bed is difficult to know from thisunprocessed rotor orbit Next the rotor orbit was attemptedto purify by employing PCA
Generally rotor orbit synthesized by 1X and 2X is cred-itable Orbits included high harmonics incline to becomecomplex and even turn messy Hence extraction of 1Xand 2X of raw signal becomes more important [21 22]Eigenvalue distribution of covariance matrix of D11 and D12signals are displayed in Figure 11 According to amplitudeof each frequency of D11 and D12 in Figure 9 1X and 2Xcorresponded to the first (first and second eigenvalues) andthe second (third and fourth eigenvalues) set of eigenvaluesrespectively The leading four eigenvalues in Figure 11 werereconstructed through CFSM-PCA and results are illustratedin Figure 12 Generated 1X and 2X were legible without effectby noise and disturbance from frequency 50 Hz as well as itsharmonics Meanwhile amplitude of the extracted signal wasclose to that of 1X and 2X in raw signal indicating that thereis no spectral leakage issue
The feature spectrums continued to be separated andeigenvalue distributionmaps of covariance matrix of D11 andD12 are shown in Figure 15
Rotor orbit generated by the purified 1X and 2X is shownin Figure 13The rotor orbit was a caved-in banana-like curvedemonstrating that sideA (near themotor) potentially suffersfrom misalignment fault [18 19] The explicit rotor orbit alsovalidates the high efficiency of this proposed method
43 The Second Group Sample Analysis In this case exper-imental data of D11 and D12 were analyzed when test bedruns steadily for a period of time Similarly 4096 datapoints were measured with test bed speed of 2770 rmin andsampling frequency of 2048Hz Time domainwaveforms andamplitude spectrums are displayed in Figure 14
According to amplitude of the leading trebling frequencyofD11 andD12 in Figure 14 1X corresponded to the second setof eigenvalues (third and fourth eigenvalues) 2X paralleledthe third set of eigenvalues (fifth and sixth eigenvalues) and
10 Shock and Vibration
D11-1080 rpmx 10
minus5
minus5minus4minus3minus2minus1
012345
X (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(a)
Power-line Interference50Hz
0733404771
Order 1 Order 2
0250502342
Order 6Order 7
D11-1080 rpm
01651 Order 11
18Hz36Hz54Hz72Hz90Hz
x 10minus5
1times
2times
3times
4times
5times
1times
2times
3times 4times5times
002040608
1
X (m
)
50 100 150 200 250 300 350 4000Frequency (Hz)
(b)
D12-1080 rpmx 10
minus5
minus6
minus4
minus2
0246
Y (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(c)
D12-1080 rpmPower-line Interference
50Hz
0658105228
Order 1 Order 2
0277101226
Order 5 Order 12
01948 Order 8
18Hz36Hz54Hz72Hz90Hz
x 10minus5
1times
2times
3times
4times
5times
1times
2times
3times
4times5times
0
05
1
Y (m
)
50 100 150 200 250 300 350 4000Frequency (Hz)
(d)
Figure 9The time domain and frequency domain of theD11 andD12 sensors in theA-end face (a)The time domain of D11 (b)The frequencydomain of D11 (c) The time domain of D12 (d) The frequency domain of D12
x 10minus5
x 10minus5
minus5
minus4
minus3
minus2
minus1
0
1
2
3
4
5
D12
minusY
(m)
minus6 minus2minus4 2 4 60D11minusX (m)
Figure 10 A-end face of the D11 and D12 sensors to collect the signal directly synthesized axis orbit
(12751) (22748)e first group
(31243) (41241)e second group
D11minus1080 rpm
x 10minus8
0
1
2
3
4
Eige
nval
ue o
f PCA
(m)
10 20 30 40 500Number of eigenvalues q
(a)
(12247) (2241)
(314) (41396)
D12minus1080 rpme first group
e second group
x 10minus8
005
115
225
Eige
nval
ue o
f PCA
(m)
10 20 30 40 500Number of eigenvalues q
(b)
Figure 11The eigenvalues distribution of covariancematrixC (a)The eigenvalues distribution of D11 signal (b)The eigenvalues distributionof D12 signal
Shock and Vibration 11
D11minus1080 rpmx 10
minus5
minus15
minus10
minus05
0051015
X (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(a)
D12minus1080 rpmx 10
minus5
minus10
minus05
0051015
Y (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(b)
D11minus1080 rpm(1807287)
(360472)
x 10minus5
002040608
1
X (m
)
50 100 150 2000Frequency (Hz)
(c)
(3605105)(1806575)
D12minus1080 rpmx 10
minus5
002040608
1
Y (m
)
50 100 150 2000Frequency (Hz)
(d)
Figure 12 Extracting the time and frequency spectrums of the first two frequencies of D11 and D12 (a)The time domain of D11 (b)The timedomain of D12 (c) The frequency domain of D11 (d) The frequency domain of D12
W1
x 10minus5
x 10minus5
minus1
minus05
0
05
1
15
D12
minusY
(m)
minus1minus15 minus05 05 10 15D11minusX (m)
(a)0
051
152
0
2
t (s)D11minusX (m)
minus2
x 10minus5
x 10minus5
minus1
minus05
0
05
1
15
D12
minusY
(m)
(b)
Figure 13 The synthesis of axis orbit is to extract the first two times fundamental frequencies of the D11 and D12 signals by the CFSM-PCA(a) The axis orbit (b) Relationship between axis orbit and time
3X matched the first set of eigenvalues (first and secondeigenvalues)
As displayed in Figure 15 eigenvalues of 1X and 3Xwere large thus 1X and 3X were utilized to synthesize rotororbit firstly Purification results and orbit of shaft center arerevealed in Figures 16 and 17 respectively The legible orbitshowed a typical plum blossom shape
It can be seen from Figures 16 and 17 that the filteringeffect was also quite obvious The noise and power frequency50 Hz as well as its harmonic interference could be removedsuccessfully Moreover amplitude of refined signal was closeto that of original one and the reconstructed time domainsignal seemed to be steady without any fluctuation
As shown in Figures 14(c) and 14(d) amplitudes of 1X 2Xand 3X in amplitude spectrum of raw signal were largeThus1X 2X and 3X were also extracted to synthesize axis orbit
Time domain frequency domain and orbit curve are shownin Figures 18 and 19 respectively
Axis orbit synthesized by the leading three frequenciesexhibited a quincunx shape which is basically the sameto that synthesized by 1X and 3X frequency componentsindicating that static and dynamic parts friction may exist inA-side
44 Comparison Harmonic Wavelet with Wavelet Packet
(1) Comparison of CFSM-PCA with Harmonic Wavelet Onecommon used method for signal processing harmonicwavelet algorithm proposed by Newland is an improvedwavelet algorithm [29] It can segment entire frequencyband infinitely and avoid the shortcoming of downsamplingmethod of binary wavelet and binary wavelet packet [22 23]
12 Shock and Vibration
D11minus2770 rpmx 10
minus4
minus1
minus05
0
05
1
D11
-X (m
)
1000 2000 3000 40000Sampling numbers i
(a)
D12minus2770 rpmx 10
minus4
minus1
minus05
0
05
1
D12
-Y (m
)
1000 2000 3000 40000Sampling numbers i
(b)
Power-Line interference 50and 150Hz
1X2X
3X 1010D11minus2770 rpm
08441166
1X2X3X
x 10minus5
200 400 600 800 10000f (Hz)
0
05
10
15
D11
-X (m
)
(c)
1X
2X
3XPower-Line interference 50
and 150Hz
D12minus2770 rpm117309021227
1X2X3X
x 10minus5
0
05
10
15
D12
-Y (m
)
200 400 600 800 10000f (Hz)
(d)
Figure 14 The time and frequency domains of the D11 and D12 sensors (a) The time domain of D11 (b) The time domain of D12 (c) Thefrequency domain of D11 (d) The frequency domain of D12
(11489) (21482)e first group
(31237) (41220)e second group
(30348) (40347)e second group
D11minus2770 rpmx 10
minus7
10 20 30 400 50Number of eigenvalues q
0
05
1
15
Eige
nval
ue o
f PCA
(m)
(a)
D12minus2770 rpm(11688) (21687)
e first group
e second group
(30414) (40413)e second group
(31673) (41664)
x 10minus7
0
05
1
15
2
Eige
nval
ue o
f PCA
(m)
10 20 30 40 500Number of eigenvalues q
(b)
Figure 15 The eigenvalues distribution of covariance matrix C (a) The eigenvalues distribution of D11 (b) The eigenvalues distribution ofD12
1X and its frequency doubling signal can be easily extractedto form axis orbit of revolving test bearing
Axis orbit was fabricated by 1X and 2Xof data of example 1(Section 42) through harmonic wavelet algorithm and theresults are demonstrated in Figure 20 The shape of axisorbit was trapped banana-like which is consistent with theresult obtained via CFSM-PCA It should be noted thatorbit in Figure 13 was more obvious than that in Figure 20ie W1ltW2 Meanwhile the amplitude fluctuation of timedomain signal was large (Figures 20(a) and 20(b)) Themajor reason for this phenomenon is that the harmonicwavelet which is a band filter in frequency domain inevitablyextracts the noise near the feature frequency Furthermorethis phenomenon also demonstrates that harmonic waveletis subject to the Heisenberg uncertainty principle ie thephenomenon of the amplitude leakage exists
(2) Comparison of CFSM-PCA with Wavelet Packet Besidesharmonic wavelets wavelet packet is also used to purify axisorbit [19ndash21] Purification results of axis orbit via Daubechieswavelet packet in Figure 21 exhibit that axis orbit was diver-gent seriously This was worse than that of harmonic waveletie W1ltW2ltW3The phenomenon of severe energy leakageoccurred during the extraction of 1X and 2X which cancause divergence of axis orbit Moreover the bandpass filterof wavelet packet also accounts for the divergent orbits
5 A Single Frequency Filtrationwith CFSM-PCA
The application in single frequency filtration is discussedin this section Taking D11 experimental data of example
Shock and Vibration 13
D11minus2770 rpmx 10
minus5
0 1000500Sampling numbers i
minus4
minus2
0
2
4
D11
-X (m
)
(a)
D12minus2770 rpmx 10
minus5
minus4
minus2
0
2
4
D12
-Y (m
)
500 10000Sampling numbers i
(b)
(1375 1185)(46 1009)
D11minus2770 rpmx 10
minus5
100 200 300 400 5000f (Hz)
0
05
1
15
D11
-X (m
)
(c)
(1375 1218)(46 1164)
D12minus2770 rpmx 10
minus5
100 200 300 400 5000f (Hz)
0
05
1
15
D12
-Y (m
)
(d)
Figure 16 Extracting the frequency spectrums of the first two frequencies of D11 and D12 by PCA algorithm (a) The time domain of D11(b) The time domain of D12 (c) The frequency spectrum of D11 (d) The frequency spectrum of D12
x 10minus5
x 10minus5
minus4
minus3
minus2
minus1
0
1
2
3
4
D12
-Y (m
)
210 3 4minus2minus3 minus1minus4
D11-X (m)
(a)0
0204
0608
10
24
t (s)D11-X (m) minus2
minus4
x 10minus5
x 10minus5
minus4
minus2
0
2
4
D12
-Y (m
)
(b)
Figure 17 The synthesis of axis orbit is to extract the first two times fundamental frequencies of the D11 and D12 signals by the CFSM-PCA(a) The axis orbit (b) Relationship between axis orbit and time
1 (Section 42) as a demo we attempt to filter the powerfrequency interference (50 Hz) from D11 by employingCFSM-PCA It can be seen from Figure 9(b) that sequenceof amplitude of 50 Hz in raw signal was 3 so eigenvec-tors corresponding to 5th and 6th eigenvalues in eigen-value distribution map were used to reconstruct Accordingto (8) the filtered signal (Figure 22) without 50 Hz was
obtained by subtracting reconstructed signal from originalone
Results in Figure 22 show that this algorithm filters out50 Hz power frequency interference and has no influenceon the other frequency components of raw signal demon-strating that this method has a potential application in notchfilter
14 Shock and Vibration
D11minus2770 rpm
x 10minus5
minus4
minus2
0
2
4
D11
-X (m
)
200 400 600 800 10000Sampling numbers i
(a)
D12minus2770 rpm
x 10minus5
0 400 600 800 1000200Sampling numbers i
minus4
minus2
0
2
4
D12
-Y (m
)
(b)
(1375 1185)(46 1009)
(92 0831)
D11minus2770 rpmx 10
minus5
00
05
10
15
D11
-X (m
)
100 200 300 400 5000f (Hz)
(c)
(1375 1218)(46 1163)
(92 0898)
D12minus2770 rpmx 10
minus5
00
05
10
15
D12
-Y (m
)
100 200 300 400 5000f (Hz)
(d)
Figure 18 Extracting the time and frequency spectrums of the first three frequencies of D11 and D12 by PCA algorithm (a)The time domainof D11 (b) The time domain of D12 (c) The frequency spectrum of D11 (d) The frequency spectrum of D12
x 10minus5
x 10minus5
minus4
minus3
minus2
minus1
0
1
2
3
4
D12
-Y (m
)
minus3 minus2 minus1minus4 1 2 3 40D11-X (m)
(a)
1008
0604
0200
02
4D11-X (m)
t (s)
minus2
minus4
x 10minus5
x 10minus5
minus4
minus2
0
2
4
D12
-Y (m
)
(b)
Figure 19 The synthesis of axis orbit is to extract the first three times fundamental frequencies of the D11 and D12 by the CFSM-PCA (a)The axis orbit (b) Relationship between axis orbit and time
6 Conclusion
The selection of effective eigenvalues in feature distributionchart of covariance matrix C is crucial in PCA algorithm Inthis paper the relationships between effective eigenvalues andsignal frequency and amplitude are discovered and a series ofconclusions are summarized
First in Hankel matrix mode each valid frequencycomponent of signal only produces two adjacent nonzeroeigenvalues Second the number of valid eigenvalues incharacteristic distribution chart which is independent ofnumerical values of 119891119894 119860 119894 and 120593119894 is related to the number offrequency componentsThird the sequence of signal effectiveeigenvalues in characteristic distribution map is determined
Shock and Vibration 15
D11-1080 rpm
x 10minus5
minus20
minus15
minus05
0
05
10
15
20
D11
-X (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(a)
D12-1080 rpmx 10
minus5
minus15
minus1
minus05
0
05
1
15
D12
-Y (m
)
500 1000 1500 2000 2500 30000 40003500Sampling numbers i
(b)
(16 07335)
(32 04771)
D11-1080 rpm
x 10minus5
10 20 30 40 50 60 70 80 90 1000f (Hz)
0010203040506070809
1
D11
-X (m
)
(c)
(16 06571)
(32 05228)
D12-1080 rpm
x 10minus5
0010203040506070809
1
D12
-Y (m
)
10 20 30 40 50 60 70 80 90 1000f (Hz)
(d)
W2
x 10minus5
x 10minus5
minus15
minus1
minus05
0
05
1
15
Y(D
12) (
m)
minus1 minus05minus15 05 1 15 20X(D11) (m)
(e)
Figure 20The purification effect of harmonic wavelets (a)The time domain of D11 (b)The time domain of D12 (c)The frequency spectrumof D11 (d) The frequency spectrum of D12 (e) The orbit is to extract the first 2 frequencies
by amplitude 119860 119894 of signal frequency component The largermagnitude of amplitude is the greater eigenvalues are andthe higher arrangement will be
Based on aforesaid principle a new CFSM-PCA wasdeveloped and has become an effective tool to extract oreliminate specific characteristic (single frequency) even dif-ference of two frequencies is only 1 Hz Purification of rotororbit using this algorithm shows significant improvement
than those of existing methods such as wavelet packet andharmonic wavelet packet Although the proposed CFSM-PCA is effective some drawbacks are unavoidable Forexample this method suffers from similar issue to otheralgorithms that is the reconstruction error increaseswith thedecrease of SNR In addition the applications of the CFSM-PCA algorithm in purification of speech recognition bearingfault diagnosis and fault diagnosis of power system should
16 Shock and Vibration
D11-1080 rpmx 10
minus5
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
minus20
minus15
minus05
0
05
10
15
20
D11
-X (m
)
(a)
D12-1080 rpmx 10
minus5
minus15
minus1
minus05
0
05
1
15
D12
-Y (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(b)
(16 06557)
(32 04381)
D11-1080 rpmx 10
minus5
10020 30 40 50 60 70 80 90100f (Hz)
0010203040506070809
1
D11
-X (m
)
(c)
(16 06452)(32 04927)
D12-1080 rpm
x 10minus5
0010203040506070809
1
D12
-Y (m
)
9020 30 40 50 60 70 8010 1000f (Hz)
(d)
W3
x 10minus5
x 10minus5
minus15
minus1
minus05
0
05
1
15
Y(D
12) (
m)
minus15 minus05minus1 05 1 15 20X(D11) (m)
(e)
Figure 21 The purification effect of wavelet packet (a) The time domain of D11 (b) The time domain of D12 (c) The frequency spectrum ofD11 (d) The frequency spectrum of D12 (e) The orbit is to extract the first 2 frequencies
be further exploredWe fully believe that the CFSM-PCAwillsparkle in diverse fields
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work is supported by the National High TechnologyResearch andDevelopment Program of China (863 Program)
Shock and Vibration 17
(18 0734)
(36 04634)
(54 02448) (72 02351)(90 01646)
x 10minus5
50 100 150 2000f (Hz)
0
02
04
06
08
1
D11
minusX
(m)
Figure 22 Frequency spectrum of D11 signal after filtering 50Hz
(no 2015AA043005) and the National Natural Science Foun-dation of China (no 51375187)
References
[1] H Abdi and L J Williams ldquoPrincipal component analysisrdquoWiley Interdisciplinary Reviews Computational Statistics vol 2no 4 pp 433ndash459 2010
[2] A Widodo B Yang and T Han ldquoCombination of independentcomponent analysis and support vectormachines for intelligentfaults diagnosis of induction motorsrdquo Expert Systems withApplications vol 32 no 2 pp 299ndash312 2007
[3] S Wold K Esbensen and P Geladi ldquoPrincipal componentanalysisrdquo Chemometrics and Intelligent Laboratory Systems vol2 no 1ndash3 pp 37ndash52 1987
[4] M Kirby and L Sirovich ldquoApplication of the Karhunen-Loeve procedure for the characterization of human facesrdquo IEEETransactions on Pattern Analysis and Machine Intelligence vol12 no 1 pp 103ndash108 1990
[5] J H Xi Y Z Han and R H Su ldquoNew fault diagnosis methodfor rolling bearing based on PCArdquo in Proceedings of the 25thChinese Control and Decision Conference CCDC rsquo13 pp 4123ndash4127 IEEE May 2013
[6] A Malhi and R X Gao ldquoPCA-based feature selection schemefor machine defect classificationrdquo IEEE Transactions on Instru-mentation and Measurement vol 53 no 6 pp 1517ndash1525 2004
[7] S Xu X Jiang J Huang S Yang and X Wang ldquoBayesianwavelet PCA methodology for turbomachinery damage diag-nosis under uncertaintyrdquo Mechanical Systems and Signal Pro-cessing vol 80 pp 1ndash18 2016
[8] W Sun J Chen and J Li ldquoDecision tree and PCA-based faultdiagnosis of rotatingmachineryrdquoMechanical Systems and SignalProcessing vol 21 no 3 pp 1300ndash1317 2007
[9] Z X Li X P Yan C Q Yuan Z Peng and L Li ldquoVirtual pro-totype and experimental research on gear multi-fault diagnosisusing wavelet-autoregressive model and principal componentanalysismethodrdquoMechanical Systems and Signal Processing vol25 no 7 pp 2589ndash2607 2011
[10] K Y Shao M M Cai and G F Zhao ldquoRolling bearing faultdiagnosis based onwavelet energy spectrum PCA andPNNrdquo inProceedings of the 26th Chinese Control andDecisionConferenceCCDC rsquo14 pp 800ndash804 IEEE 2014
[11] T Qiu Q F Zhang and Y J Ding ldquoNonlinear sensor faultdetection and data rebuilding based on principle componentanalysisrdquo Journal of Tsinghua University vol 46 no 5 pp 708ndash711 2006
[12] Y K Gu Z X Yang and F L Zhu ldquoGearbox fault feature fusionbased on principal component analysisrdquo China MechanicalEngineering vol 26 no 11 pp 1532ndash1537 2015
[13] R Dunia S J Qin T F Edgar and T J McAvoy ldquoIdentificationof faulty sensors using principal component analysisrdquo AIChEJournal vol 42 no 10 pp 2797ndash2811 1996
[14] E P de Moura C R Souto A A Silva and M A S IrmaoldquoEvaluation of principal component analysis and neural net-work performance for bearing fault diagnosis from vibrationsignal processed by RS and DF analysesrdquo Mechanical Systemsand Signal Processing vol 25 no 5 pp 1765ndash1772 2011
[15] S Golestan M Monfared F D Freijedo and J M GuerreroldquoDynamics assessment of advanced single-phase PLL struc-turesrdquo IEEE Transactions on Industrial Electronics vol 60 no6 pp 2167ndash2177 2013
[16] A Lima H Zen Y Nankaku C Miyajima K Tokuda andT Kitamura ldquoOn the use of kernel PCA for feature extractionin speech recognitionrdquo IEICE Transaction on Information andSystems vol 12 no 87 pp 2802ndash2811 2004
[17] J D Zheng ldquoImproved hilbert-huang transform and its appli-cations to rolling bearing fault diagnosisrdquo Journal of MechanicalEngineering vol 1 no 51 pp 138ndash145 2015
[18] G He K Ding and H Lin ldquoFault feature extraction of rollingelement bearings using sparse representationrdquo Journal of Soundand Vibration vol 366 pp 514ndash527 2016
[19] J G Yang S B Xia and Y G Liu ldquoWavelet denoising and itsapplication in extraction of shaft centerline orbit featuresrdquoJournal of Harbin Institute of Technology vol 31 no 5 pp 52ndash541999 (Chinese)
[20] J A Duan and X D Zhang ldquoFeature extraction from the orbitof the center of a rotor based on wavelet transformrdquo Journal ofVibration Measurement amp Diagnosis vol 17 no 01 pp 31ndash341997
[21] J Han D X Jiang and W D Ning ldquoApplication of optimalwavelet packets in failure signal extraction of rotor orbitrdquoTurbine Technology vol 43 no 3 pp 133ndash136 2001
[22] W B Zhang X J Zhou andY Lin ldquoRefinement of rotor centerrsquosorbit by a harmonic window methodrdquo Journal of VibrationMeasurement amp Diagnosis vol 30 no 1 pp 87ndash90 2010
[23] S M Li G Z Lu and Q Y Xu ldquoOn obtaining accuraterotor sub-frequency signal with harmonic waveletrdquo Journal ofNorthwestern Polytechincal University vol 19 no 2 pp 220ndash224 2001
[24] F J Wu and L S Qu ldquoAn improved method for restraining theend effect in empiricalmode decomposition and its applicationsto the fault diagnosis of large rotating machineryrdquo Journal ofSound and Vibration vol 314 no 3-5 pp 586ndash602 2008
[25] W G Liu ldquoResearch on method of purification of shaft orbitbased on singular value decompositionrdquo Technical Acousticsvol 5 no 35 pp 30ndash35 2016
[26] X Z Zhao and B Y Ye ldquoThe influence of formation manner ofcomponent on signal processingrdquo Shang Hai Jiao Tong Univer-sity vol 45 no 3 pp 368ndash374 2011
[27] X Z Zhao B Y Ye and T J Chen ldquoPrinciple of singularitydetection based on SVD and its applicationrdquo Journal of Vibra-tion and Shock vol 6 no 27 pp 7ndash10 2008
[28] X Z Zhao B Y Ye and T J Chen ldquoDifference spectrum theoryof singular value and its application to the fault diagnosis ofheadstock of latherdquo Journal of Mechanical Engineering vol 46pp 100ndash108 2010
[29] I A Kougioumtzoglou and P D Spanos ldquoAn identificationapproach for linear and nonlinear time-variant structural sys-tems via harmonic waveletsrdquo Mechanical Systems and SignalProcessing vol 37 no 1-2 pp 338ndash352 2013
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Shock and Vibration 5
(33278) (43275)
(51844) (61842)
(15123) (25118)
(33278) (43275)
(51844) (61842)
(15123) (25118)
(33278) (43275)
(51844) (61842)
(15123) (25118)e first group
e second group
e third group
e first group
e second group
e third group
e first group
e second group
e third group
x1(t) = MCH(40t + 10) + 08 MCH(60t + 20) + 06 MCH(80t + 30)
x2(t) = MCH(100t + 40) + 08 MCH(120t + 50) + 06 MCH(140t + 60)
x3(t) = MCH(160t + 70) + 08 MCH(180t + 80) + 06 MCH(200t + 90)
0
200
400
600
10 20 30 40 500
0
200
400
600
Eige
nval
ue o
f PCA
i
10 20 30 40 500
0
200
400
600
10 20 30 40 500Number of eigenvalues q
Figure 3 Eigenvalues distribution of covariancematrixCwith threefrequency components
Hankel matrix is derived from certain signal x(t) l=min(mn) with m rows n columns and k effective frequenciesConcerning the fact that the Shannon sampling theorem ismet assuming that lgt2k generic conclusions are summarizedas follows(1) Each frequency component of signal produces two
nonzero eigenvalues with one arranging another closely(2) The number of effective eigenvalues of crude signal
is related to the number of frequency components and has
nothing to do with themagnitude of amplitude119860 119894 frequency120596119894 and phase 120593119894(3) In eigenvalue distribution chart of covariance matrix
C the sequence of nonzero effective eigenvalues is deter-mined by amplitude 119860 119894 of signal The larger amplitude is thelarger eigenvalues will be the more forward rank of the twoeigenvalues produced from corresponding frequencies is
Inspired by the relationship between frequency andeigenvalues a new technique for characteristic frequency sep-aration method based on PCA (CFSM-PCA) was proposedThe concrete steps are listed below(1) For a certain signal 119909(119905) direct component (DC) of
raw signal is filtered out via fast Fourier transform (FFT)firstly and then Hankel matrix X is constructed throughfiltered signal(2) Covariance matrix C of matrix X is obtained with its
eigenvalues 120582119894 arranging in descending order (1205821 1205822 sdot sdot sdot 120582119898)and corresponding eigenvectors are obtained as 12057211205722 sdot sdot sdot120572119898(3) According to the distribution of eigenvalues 120582119894reconstruction is carried out from two eigenvalues and cor-responding eigenvectors of certain frequency For examplefor amplitude perspective if the rank of specific frequencyof raw signal is k a new matrix is received by reconstructingthe eigenvectors corresponding to 2k-1 and 2k eigenval-ues in eigenvalue distribution chart of covariance matrixC(4)Thematrix X can be produced by adding the mean of
original matrix to the new reconstructed matrix(5) The signal x which is the characteristic frequency
component is recovered from matrix X by means of theaveraging method
32Theoretical Deduction In this section deduction processof the three discoveries (Section 31) is provided Supposingthat a signal is expressed as 119909(119905) = 119886 sin(120596119905 + 120593) Samplingtime 119879119904 is used to discretize signal x(t) Hankel matrix withm rows and n columns is derived from signal 119909(119905) exhibitedas
X = 119886 times[[[[[[
sin (0 sdot 120596119879119904 + 120593) sin (1 sdot 120596119879119904 + 120593) sdot sdot sdot sin ((119899 minus 1) sdot 120596119879119904 + 120593)sin (1 sdot 120596119879119904 + 120593) sin (2 sdot 120596119879119904 + 120593) sdot sdot sdot sin (119899 sdot 120596119879119904 + 120593)
sin ((119898 minus 1) sdot 120596119879119904 + 120593) sin (119898 sdot 120596119879119904 + 120593) sdot sdot sdot sin ((119898 + 119899 minus 2) sdot 120596119879119904 + 120593)
]]]]]]
(13)
(1) Each characteristic frequency produces two effectiveeigenvalues
The deduced process of the first conclusion is given belowEquation (13) can be rewritten as (14) based onEulerrsquos Formula
X = 1198862119894
times[[[[[[[
119890(0120596119879119904+120593)119894 minus 119890minus(0120596119879119904+120593)119894 119890(1120596119879119904+120593)119894 minus 119890minus(1120596119879119904+120593)119894 119890(2120596119879119904+120593)119894 minus 119890minus(2120596119879119904+120593)119894 sdot sdot sdot 119890((119899minus1)120596119879119904+120593)119894 minus 119890minus((119899minus1)120596119879119904+120593)119894119890(1120596119879119904+120593)119894 minus 119890minus(1120596119879119904+120593)119894 119890(2120596119879119904+120593)119894 minus 119890minus(2120596119879119904+120593)119894 119890(3120596119879119904+120593)119894 minus 119890minus(3120596119879119904+120593)119894 sdot sdot sdot 119890(119899120596119879119904+120593)119894 minus 119890minus(119899120596119879119904+120593)119894
119890((119898minus1)120596119879119904+120593)119894 minus 119890minus((119898minus1)120596119879119904+120593)119894 119890(119898120596119879119904+120593)119894 minus 119890minus(119898120596119879119904+120593)119894 119890((119898+1)120596119879119904+120593)119894 minus 119890minus((119898+1)120596119879119904+120593)119894 sdot sdot sdot 119890((119898+119899minus2)120596119879119904+120593)119894 minus 119890minus((119898+119899minus2)120596119879119904+120593)119894
]]]]]]]
(14)
6 Shock and Vibration
Equation (14) can be expanded to addition form of twoformulas depicted as
X = 1198862119894 times 119890120593119894
times[[[[[[[[
1198900120596119879119904119894 1198901120596119879119904119894 1198902120596119879119904119894 sdot sdot sdot 119890(119899minus1)1205961198791199041198941198901120596119879119904119894 1198902120596119879119904119894 1198903120596119879119904119894 sdot sdot sdot 119890119899120596119879119904119894
119890(119898minus1)120596119879119904119894 119890119898120596119879119904119894 119890(119898+1)120596119879119904119894 sdot sdot sdot 119890(119898+119899minus2)120596119879119904119894
]]]]]]]]
minus 1198862119894 times 119890minus120593119894
times[[[[[[[[
119890minus0120596119879119904119894 119890minus1120596119879119904119894 119890minus2120596119879119904119894 sdot sdot sdot 119890minus(119899minus1)120596119879119904119894119890minus1120596119879119904119894 119890minus2120596119879119904119894 119890minus3120596119879119904119894 sdot sdot sdot 119890minus119899120596119879119904119894
119890minus(119898minus1)120596119879119904119894 119890minus119898120596119879119904119894 119890minus(119898+1)120596119879119904119894 sdot sdot sdot 119890minus(119898+119899minus2)120596119879119904119894
]]]]]]]](15)
From (15) the rank of both matrices is 1 Based on rankrelationship of two matrices 119877(119860+119861) le 119877(119860) + 119877(119861) Hencerank of matrix X is less than or equal to 2
Additionally when 120593 = 0 (13) is rewritten to (16)According to sum-to-product identities the rank of X is 2when 120596119879119904 = 0 120587 2120587 sdot sdot sdot
X = 119886 times[[[[[[[[[
sin (0 sdot 120596119879119904) sin (1 sdot 120596119879119904) sdot sdot sdot sin ((119899 minus 1) sdot 120596119879119904)sin (1 sdot 120596119879119904) sin (2 sdot 120596119879119904) sdot sdot sdot sin (119899 sdot 120596119879119904)
sin ((119898 minus 1) sdot 120596119879119904) sin (119898 sdot 120596119879119904) sdot sdot sdot sin ((119898 + 119899 minus 2) sdot 120596119879119904)
]]]]]]]]]
(16)
When 120593 = 0 (13) can be deduced into the first-orderprincipal minor of matrix X ie |sin120593| = 0 The principalminor of order 2 of X is shown as10038161003816100381610038161003816100381610038161003816100381610038161003816sin (0 sdot 120596119879119904 + 120593) sin (1 sdot 120596119879119904 + 120593)sin (1 sdot 120596119879119904 + 120593) sin (2 sdot 120596119879119904 + 120593)
10038161003816100381610038161003816100381610038161003816100381610038161003816= minussin2 (120596119879119904)
= 0(17)
where120596119879119904 = 0 120587 2120587 sdot sdot sdot Because the leading principalminorof order 2 ofX in (13) is nonzero values so the rank of matrixX is at least 2
Combining result of (15) 119877(119860 + 119861) le 2 and nonzerovalues of the leading principal submatrix of order 2 hencethe rank of matrix X is 2 Matrix X has two eigenvalues 12058210158401and 12058210158402 Derived from C = XXT119899 (where 119899 is the numberof columns of matrix X) [3] two nonzero eigenvalues ofcovariance matrix C of X are 1205821 and 1205822 respectively(2)The sequence of eigenvalues is determined by ampli-
tudeIn terms of PCA covariance matrix C is described as
C = 119864 [(X minus 119864 (X)) sdot (X minus 119864 (X))T] (18)
Referring to (6) (19) is constructed as follows
119897
sum119894=1
C120572119894120572T119894 =119897
sum119894=1
120582119894120572119894120572T119894 (19)
Two effective eigenvalues are generated from a frequencycomponent that is l=2
CI119898 = 12058211205721120572T1 + 12058221205722120572T2 (20)
Then the energy of matrix C can be deduced as
|C|2 = 12058221119898
sum119894=1
12057221198941119898
sum119894=1
12057221198941 + 12059022119898
sum119894=1
12057221198942119898
sum119895=1
12057221198942
+ 21205902112059022119898
sum119894=1
12057211989411205721198942119898
sum119894=1
12057211989411205721198943(21)
Based upon (4) sum119898119894=1 12057221198941 = 1 sum119898119894=1 12057221198942 = 1 sum119898119894=1 12057211989411205721198942 =0 Then (22) is given by
|C|2 = 12058221 + 12058222 (22)
The constructed Hankel matrix X of signal 119909(119905) is sub-stituted into (18) we can see that the energy of covariancematrix C is proportional to 1198862 The larger 1198862 is the greaterenergy of matrix C is as well as 120582119894 according to (22)Furthermore the larger frequency component amplitude isthe larger corresponding eigenvalue in covariance matrix Ccharacteristic distribution chart is
Based on these conclusions once the amplitude sequenceof a certain frequency in raw signal amplitude spec-trum is determined its corresponding frequency compo-nent can be reconstructed In this way extraction of sin-gle or multiple characteristic frequencies could be real-ized
In view of addition relation as shown in (8) the notchfilter could be achieved through CFSM-PCA The frequencycomponent extracted by this algorithm is subtracted in orig-inal signal ie this frequency component can be eliminatedin raw signal Reader could consult examples in Section 5 formore details
Shock and Vibration 7
Table 1 Amplitude of signal components under the different SNR
SNR SNR1=-1 SNR2=-4 SNR3=-7 SNR4=-10Signal component Raw Process Raw Process Raw Process Raw Processsin(40120587119905 + 20) 09999 09938 10260 10240 10370 10730 1114 111205 sin(100120587119905 + 40) 04703 04826 05317 05482 07353 07533 05177 0523407 sin(98120587119905 + 50) 07299 07152 07116 07235 05110 04968 06800 0662108 sin(160120587119905 + 60) 08340 08324 08264 08142 08389 08338 08379 08923Max difference 00123 00122 00360 00544
minus6minus4minus2
0246
Am
plitu
de
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(a)
(200972)
(490693)(800791)
(500468)
0
05
1
Am
plitu
de
50 100 150 2000f (Hz)
(b)
Figure 4 The raw signal 119910 (a) Time domain (b) Amplitudespectrum
33 Simulation Example To verify the applicability and effec-tiveness of the CFSM-PCA algorithm a simulated sinusoidalsignal 119910 was constructed as
119910 = sin (40120587119905 + 20) + 05sin (100120587119905 + 40)+ 07sin (98120587119905 + 50) + 08sin (160120587119905 + 60)+ 119890 (119899)
(23)
where 119890(119899) is Gaussian white noise with standard deviationof 12 The result is shown in Figure 4(a) and amplitudespectrum (Fourier spectrum) is illustrated in Figure 4(b)after collecting 4096 data points with sampling frequency of1024 Hz
Follow-up work to separate frequencies via the proposedapproach was performed From eigenvalue distribution chartof crude signal in Figure 5 we can see that apart fromdiscernible four sets of eigenvalues most of eigenvaluescaused by noise are nonzero values and distribute at largeeigenvalues region
As demonstrated in Figures 6(a) and 6(b) time domainand frequency spectrum are generated after reconstructingthe first set of eigenvalues 1205821 and 1205822 in Figure 5 Frequencyof the reconstructed signal is 20 Hz with amplitude of 0977which is in correspondence with component sin(40120587t+20) ofy signal
As revealed in Figures 6(c) and 6(d) another time domainand amplitude spectrum are obtained via the reconstruction
(1515) (25144)
(3353) (43527)
(52654) (62643)
(71037) (81036)
e first group
e second group
e third group
e fourth group
0
200
400
600
Eige
nval
ue o
f PCA
i
10 20 30 40 500Number of eigenvalues q
Figure 5The eigenvalues distribution of the covariancematrixC of119910
of the second set of eigenvalues 1205823 and 1205824 Frequencyof this reconstructed signal is 80 Hz with amplitude of0786 which is same as component 08sin(160120587t+60) of rawsignal y Figures 6(e) and 6(f) are brought out through thereconstruction of the third set of eigenvalues 1205825 and 1205826Frequency of reconstructed signal is 49 Hz with amplitude of0688 which is consistent with component 07sin(98120587t+50)of crude signal 119910 Time domain and amplitude spectrum aregained after the reconstruction from the last set of eigenvaluesin Figure 5 exhibited as Figures 6(h) and 6(g) Frequency is50 Hz with amplitude of 0463 which is matched well withcomponent 05sin(100120587t+40) of signal y
Moreover several sets of signals with different signal-to-noise ratio (SNR) were processed by CFSM-PCA Theamplitude difference is summarized in Table 1
It can be seen that when the noise is low the amplitudeof frequency components extracted byCFSM-PCAalgorithmis close to that of the raw signal However the amplitudedifference will increase with the decrease of SNR
From all above-mentioned diagrams (Figure 6) it is obvi-ous that diverse frequency components extracted from rawsignal can be achieved accurately More importantly theseresults are coincident with ideal signal perfectly demon-strating that this CFSM-PCA is a zero phase shift frequencyextraction algorithm Meanwhile it should be noted that twofrequencies could also be perfectly separated via the proposedalgorithm even difference value between the adjacent fre-quency components is only 1 HzThismethod is accurate andseems to be more efficient than other existing filter methodssuch as wavelet filter and finite impulse response (FIR) filterwhich are subject to phase distorting issue
4 Experimental Verification
The fault diagnosis for large rotating machinery has been animportant topic and is attracting more and more attention
8 Shock and Vibration
the primary signalthe recovered signal
minus2
minus1
0
1
2A
mpl
itude
100 200 300 400 5000Sampling numbers i
(a)
(200977)
50 1000f (Hz)
002
04
06
08
1
Am
plitu
de
(b)
the primary signalthe recovered signal
minus1
minus05
0
05
1
Am
plitu
de
100 200 300 400 5000Sampling numbers i
(c)
(800786)
002
04
06
08
1
Am
plitu
de
50 1000f (Hz)
(d)
the primary signalthe recovered signal
100 200 300 400 5000Sampling numbers i
minus1
minus05
0
05
1
Am
plitu
de
(e)
(490688)
50 1000f (Hz)
002
04
06
08
1
Am
plitu
de
(f)
the primary signalthe recovered signal
minus05
0
05
Am
plitu
de
100 200 300 400 5000Sampling numbers i
(g)
(500463)
002
04
06
08
1
Am
plitu
de
50 1000f (Hz)
(h)
Figure 6 The time and frequency domains of reconstructed signal (a) The time domain of the reconstructed signal via the first set ofeigenvalues (b) Amplitude spectrum of Figure 6(a) (c) The time domain of the reconstructed signal using the second set of eigenvalues(d) Amplitude spectrum of Figure 6(c) (e) The time domain of the reconstructed signal using the third set of eigenvalues (f) Amplitudespectrum of Figure 6(e) (g) The time domain of the reconstructed signal using the fourth set of eigenvalues (h) Amplitude spectrum ofFigure 6(g)
Shock and Vibration 9
A B
Servo Motor Unloading device
Coupling
Sliding Bearing
Rotor
Large Platform
Small Platform
Air spring
BeltSliding Bearing
Figure 7 Large rot or vibration test bed
Displacement SensorA-end Face B-end Face
D13 D14
D12D11
Figure 8 Schematic and installation of eddy current displacement sensor
Rotor orbits indicate the symptoms of malfunction andare related to variation of input force or dynamic stiffnessGenerally different curve shapes of rotor orbit in a senserepresent different fault types [19ndash22 29] For example insideldquo8rdquo outside ldquo8rdquo and petal-shaped curves are omens of oil-filmwhirl fault misalignment fault and rubbing fault of rotorsystem respectivelyTherefore extraction of rotor orbit playsa significant role in rotating machinery diagnostics In thissection a simple and efficient CFSM-PCA is applied to purifyrotor orbit precisely
41 Experimental Rigs Figure 7 shows the large rotor vibra-tion test bed system which is constructed by our group Inorder to real-timemonitor rotor working station two KamanKD2306-1S eddy current proximity probes are installedon individual side in orthogonal manners (Figure 8) Theexperimental data were collected through LMS Test Lab
42 The First Group Sample Analysis Experimental datastems from D11 and D12 eddy current displacement sensorin side A 4096 data points were collected with test bed speedof 1080 rmin and sampling frequency of 2048 Hz The timedomain waveforms amplitude spectrums and the order ofamplitudes were obtained after filtering DC component viaFFT (Figure 9)
As demonstrated in Figure 9 raw signal was not onlyaffected by noise but also interfered by power frequency 50Hz and its harmonics frequency The energy of signal mainlyconcentrated on the leading two octaves
Rotor orbit was produced by combining spindle dis-placement signal D11 with signal D12 (Figure 10) X-axispresented raw signal D11 and Y-axis referred to signal D12Manipulating status of test bed is difficult to know from thisunprocessed rotor orbit Next the rotor orbit was attemptedto purify by employing PCA
Generally rotor orbit synthesized by 1X and 2X is cred-itable Orbits included high harmonics incline to becomecomplex and even turn messy Hence extraction of 1Xand 2X of raw signal becomes more important [21 22]Eigenvalue distribution of covariance matrix of D11 and D12signals are displayed in Figure 11 According to amplitudeof each frequency of D11 and D12 in Figure 9 1X and 2Xcorresponded to the first (first and second eigenvalues) andthe second (third and fourth eigenvalues) set of eigenvaluesrespectively The leading four eigenvalues in Figure 11 werereconstructed through CFSM-PCA and results are illustratedin Figure 12 Generated 1X and 2X were legible without effectby noise and disturbance from frequency 50 Hz as well as itsharmonics Meanwhile amplitude of the extracted signal wasclose to that of 1X and 2X in raw signal indicating that thereis no spectral leakage issue
The feature spectrums continued to be separated andeigenvalue distributionmaps of covariance matrix of D11 andD12 are shown in Figure 15
Rotor orbit generated by the purified 1X and 2X is shownin Figure 13The rotor orbit was a caved-in banana-like curvedemonstrating that sideA (near themotor) potentially suffersfrom misalignment fault [18 19] The explicit rotor orbit alsovalidates the high efficiency of this proposed method
43 The Second Group Sample Analysis In this case exper-imental data of D11 and D12 were analyzed when test bedruns steadily for a period of time Similarly 4096 datapoints were measured with test bed speed of 2770 rmin andsampling frequency of 2048Hz Time domainwaveforms andamplitude spectrums are displayed in Figure 14
According to amplitude of the leading trebling frequencyofD11 andD12 in Figure 14 1X corresponded to the second setof eigenvalues (third and fourth eigenvalues) 2X paralleledthe third set of eigenvalues (fifth and sixth eigenvalues) and
10 Shock and Vibration
D11-1080 rpmx 10
minus5
minus5minus4minus3minus2minus1
012345
X (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(a)
Power-line Interference50Hz
0733404771
Order 1 Order 2
0250502342
Order 6Order 7
D11-1080 rpm
01651 Order 11
18Hz36Hz54Hz72Hz90Hz
x 10minus5
1times
2times
3times
4times
5times
1times
2times
3times 4times5times
002040608
1
X (m
)
50 100 150 200 250 300 350 4000Frequency (Hz)
(b)
D12-1080 rpmx 10
minus5
minus6
minus4
minus2
0246
Y (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(c)
D12-1080 rpmPower-line Interference
50Hz
0658105228
Order 1 Order 2
0277101226
Order 5 Order 12
01948 Order 8
18Hz36Hz54Hz72Hz90Hz
x 10minus5
1times
2times
3times
4times
5times
1times
2times
3times
4times5times
0
05
1
Y (m
)
50 100 150 200 250 300 350 4000Frequency (Hz)
(d)
Figure 9The time domain and frequency domain of theD11 andD12 sensors in theA-end face (a)The time domain of D11 (b)The frequencydomain of D11 (c) The time domain of D12 (d) The frequency domain of D12
x 10minus5
x 10minus5
minus5
minus4
minus3
minus2
minus1
0
1
2
3
4
5
D12
minusY
(m)
minus6 minus2minus4 2 4 60D11minusX (m)
Figure 10 A-end face of the D11 and D12 sensors to collect the signal directly synthesized axis orbit
(12751) (22748)e first group
(31243) (41241)e second group
D11minus1080 rpm
x 10minus8
0
1
2
3
4
Eige
nval
ue o
f PCA
(m)
10 20 30 40 500Number of eigenvalues q
(a)
(12247) (2241)
(314) (41396)
D12minus1080 rpme first group
e second group
x 10minus8
005
115
225
Eige
nval
ue o
f PCA
(m)
10 20 30 40 500Number of eigenvalues q
(b)
Figure 11The eigenvalues distribution of covariancematrixC (a)The eigenvalues distribution of D11 signal (b)The eigenvalues distributionof D12 signal
Shock and Vibration 11
D11minus1080 rpmx 10
minus5
minus15
minus10
minus05
0051015
X (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(a)
D12minus1080 rpmx 10
minus5
minus10
minus05
0051015
Y (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(b)
D11minus1080 rpm(1807287)
(360472)
x 10minus5
002040608
1
X (m
)
50 100 150 2000Frequency (Hz)
(c)
(3605105)(1806575)
D12minus1080 rpmx 10
minus5
002040608
1
Y (m
)
50 100 150 2000Frequency (Hz)
(d)
Figure 12 Extracting the time and frequency spectrums of the first two frequencies of D11 and D12 (a)The time domain of D11 (b)The timedomain of D12 (c) The frequency domain of D11 (d) The frequency domain of D12
W1
x 10minus5
x 10minus5
minus1
minus05
0
05
1
15
D12
minusY
(m)
minus1minus15 minus05 05 10 15D11minusX (m)
(a)0
051
152
0
2
t (s)D11minusX (m)
minus2
x 10minus5
x 10minus5
minus1
minus05
0
05
1
15
D12
minusY
(m)
(b)
Figure 13 The synthesis of axis orbit is to extract the first two times fundamental frequencies of the D11 and D12 signals by the CFSM-PCA(a) The axis orbit (b) Relationship between axis orbit and time
3X matched the first set of eigenvalues (first and secondeigenvalues)
As displayed in Figure 15 eigenvalues of 1X and 3Xwere large thus 1X and 3X were utilized to synthesize rotororbit firstly Purification results and orbit of shaft center arerevealed in Figures 16 and 17 respectively The legible orbitshowed a typical plum blossom shape
It can be seen from Figures 16 and 17 that the filteringeffect was also quite obvious The noise and power frequency50 Hz as well as its harmonic interference could be removedsuccessfully Moreover amplitude of refined signal was closeto that of original one and the reconstructed time domainsignal seemed to be steady without any fluctuation
As shown in Figures 14(c) and 14(d) amplitudes of 1X 2Xand 3X in amplitude spectrum of raw signal were largeThus1X 2X and 3X were also extracted to synthesize axis orbit
Time domain frequency domain and orbit curve are shownin Figures 18 and 19 respectively
Axis orbit synthesized by the leading three frequenciesexhibited a quincunx shape which is basically the sameto that synthesized by 1X and 3X frequency componentsindicating that static and dynamic parts friction may exist inA-side
44 Comparison Harmonic Wavelet with Wavelet Packet
(1) Comparison of CFSM-PCA with Harmonic Wavelet Onecommon used method for signal processing harmonicwavelet algorithm proposed by Newland is an improvedwavelet algorithm [29] It can segment entire frequencyband infinitely and avoid the shortcoming of downsamplingmethod of binary wavelet and binary wavelet packet [22 23]
12 Shock and Vibration
D11minus2770 rpmx 10
minus4
minus1
minus05
0
05
1
D11
-X (m
)
1000 2000 3000 40000Sampling numbers i
(a)
D12minus2770 rpmx 10
minus4
minus1
minus05
0
05
1
D12
-Y (m
)
1000 2000 3000 40000Sampling numbers i
(b)
Power-Line interference 50and 150Hz
1X2X
3X 1010D11minus2770 rpm
08441166
1X2X3X
x 10minus5
200 400 600 800 10000f (Hz)
0
05
10
15
D11
-X (m
)
(c)
1X
2X
3XPower-Line interference 50
and 150Hz
D12minus2770 rpm117309021227
1X2X3X
x 10minus5
0
05
10
15
D12
-Y (m
)
200 400 600 800 10000f (Hz)
(d)
Figure 14 The time and frequency domains of the D11 and D12 sensors (a) The time domain of D11 (b) The time domain of D12 (c) Thefrequency domain of D11 (d) The frequency domain of D12
(11489) (21482)e first group
(31237) (41220)e second group
(30348) (40347)e second group
D11minus2770 rpmx 10
minus7
10 20 30 400 50Number of eigenvalues q
0
05
1
15
Eige
nval
ue o
f PCA
(m)
(a)
D12minus2770 rpm(11688) (21687)
e first group
e second group
(30414) (40413)e second group
(31673) (41664)
x 10minus7
0
05
1
15
2
Eige
nval
ue o
f PCA
(m)
10 20 30 40 500Number of eigenvalues q
(b)
Figure 15 The eigenvalues distribution of covariance matrix C (a) The eigenvalues distribution of D11 (b) The eigenvalues distribution ofD12
1X and its frequency doubling signal can be easily extractedto form axis orbit of revolving test bearing
Axis orbit was fabricated by 1X and 2Xof data of example 1(Section 42) through harmonic wavelet algorithm and theresults are demonstrated in Figure 20 The shape of axisorbit was trapped banana-like which is consistent with theresult obtained via CFSM-PCA It should be noted thatorbit in Figure 13 was more obvious than that in Figure 20ie W1ltW2 Meanwhile the amplitude fluctuation of timedomain signal was large (Figures 20(a) and 20(b)) Themajor reason for this phenomenon is that the harmonicwavelet which is a band filter in frequency domain inevitablyextracts the noise near the feature frequency Furthermorethis phenomenon also demonstrates that harmonic waveletis subject to the Heisenberg uncertainty principle ie thephenomenon of the amplitude leakage exists
(2) Comparison of CFSM-PCA with Wavelet Packet Besidesharmonic wavelets wavelet packet is also used to purify axisorbit [19ndash21] Purification results of axis orbit via Daubechieswavelet packet in Figure 21 exhibit that axis orbit was diver-gent seriously This was worse than that of harmonic waveletie W1ltW2ltW3The phenomenon of severe energy leakageoccurred during the extraction of 1X and 2X which cancause divergence of axis orbit Moreover the bandpass filterof wavelet packet also accounts for the divergent orbits
5 A Single Frequency Filtrationwith CFSM-PCA
The application in single frequency filtration is discussedin this section Taking D11 experimental data of example
Shock and Vibration 13
D11minus2770 rpmx 10
minus5
0 1000500Sampling numbers i
minus4
minus2
0
2
4
D11
-X (m
)
(a)
D12minus2770 rpmx 10
minus5
minus4
minus2
0
2
4
D12
-Y (m
)
500 10000Sampling numbers i
(b)
(1375 1185)(46 1009)
D11minus2770 rpmx 10
minus5
100 200 300 400 5000f (Hz)
0
05
1
15
D11
-X (m
)
(c)
(1375 1218)(46 1164)
D12minus2770 rpmx 10
minus5
100 200 300 400 5000f (Hz)
0
05
1
15
D12
-Y (m
)
(d)
Figure 16 Extracting the frequency spectrums of the first two frequencies of D11 and D12 by PCA algorithm (a) The time domain of D11(b) The time domain of D12 (c) The frequency spectrum of D11 (d) The frequency spectrum of D12
x 10minus5
x 10minus5
minus4
minus3
minus2
minus1
0
1
2
3
4
D12
-Y (m
)
210 3 4minus2minus3 minus1minus4
D11-X (m)
(a)0
0204
0608
10
24
t (s)D11-X (m) minus2
minus4
x 10minus5
x 10minus5
minus4
minus2
0
2
4
D12
-Y (m
)
(b)
Figure 17 The synthesis of axis orbit is to extract the first two times fundamental frequencies of the D11 and D12 signals by the CFSM-PCA(a) The axis orbit (b) Relationship between axis orbit and time
1 (Section 42) as a demo we attempt to filter the powerfrequency interference (50 Hz) from D11 by employingCFSM-PCA It can be seen from Figure 9(b) that sequenceof amplitude of 50 Hz in raw signal was 3 so eigenvec-tors corresponding to 5th and 6th eigenvalues in eigen-value distribution map were used to reconstruct Accordingto (8) the filtered signal (Figure 22) without 50 Hz was
obtained by subtracting reconstructed signal from originalone
Results in Figure 22 show that this algorithm filters out50 Hz power frequency interference and has no influenceon the other frequency components of raw signal demon-strating that this method has a potential application in notchfilter
14 Shock and Vibration
D11minus2770 rpm
x 10minus5
minus4
minus2
0
2
4
D11
-X (m
)
200 400 600 800 10000Sampling numbers i
(a)
D12minus2770 rpm
x 10minus5
0 400 600 800 1000200Sampling numbers i
minus4
minus2
0
2
4
D12
-Y (m
)
(b)
(1375 1185)(46 1009)
(92 0831)
D11minus2770 rpmx 10
minus5
00
05
10
15
D11
-X (m
)
100 200 300 400 5000f (Hz)
(c)
(1375 1218)(46 1163)
(92 0898)
D12minus2770 rpmx 10
minus5
00
05
10
15
D12
-Y (m
)
100 200 300 400 5000f (Hz)
(d)
Figure 18 Extracting the time and frequency spectrums of the first three frequencies of D11 and D12 by PCA algorithm (a)The time domainof D11 (b) The time domain of D12 (c) The frequency spectrum of D11 (d) The frequency spectrum of D12
x 10minus5
x 10minus5
minus4
minus3
minus2
minus1
0
1
2
3
4
D12
-Y (m
)
minus3 minus2 minus1minus4 1 2 3 40D11-X (m)
(a)
1008
0604
0200
02
4D11-X (m)
t (s)
minus2
minus4
x 10minus5
x 10minus5
minus4
minus2
0
2
4
D12
-Y (m
)
(b)
Figure 19 The synthesis of axis orbit is to extract the first three times fundamental frequencies of the D11 and D12 by the CFSM-PCA (a)The axis orbit (b) Relationship between axis orbit and time
6 Conclusion
The selection of effective eigenvalues in feature distributionchart of covariance matrix C is crucial in PCA algorithm Inthis paper the relationships between effective eigenvalues andsignal frequency and amplitude are discovered and a series ofconclusions are summarized
First in Hankel matrix mode each valid frequencycomponent of signal only produces two adjacent nonzeroeigenvalues Second the number of valid eigenvalues incharacteristic distribution chart which is independent ofnumerical values of 119891119894 119860 119894 and 120593119894 is related to the number offrequency componentsThird the sequence of signal effectiveeigenvalues in characteristic distribution map is determined
Shock and Vibration 15
D11-1080 rpm
x 10minus5
minus20
minus15
minus05
0
05
10
15
20
D11
-X (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(a)
D12-1080 rpmx 10
minus5
minus15
minus1
minus05
0
05
1
15
D12
-Y (m
)
500 1000 1500 2000 2500 30000 40003500Sampling numbers i
(b)
(16 07335)
(32 04771)
D11-1080 rpm
x 10minus5
10 20 30 40 50 60 70 80 90 1000f (Hz)
0010203040506070809
1
D11
-X (m
)
(c)
(16 06571)
(32 05228)
D12-1080 rpm
x 10minus5
0010203040506070809
1
D12
-Y (m
)
10 20 30 40 50 60 70 80 90 1000f (Hz)
(d)
W2
x 10minus5
x 10minus5
minus15
minus1
minus05
0
05
1
15
Y(D
12) (
m)
minus1 minus05minus15 05 1 15 20X(D11) (m)
(e)
Figure 20The purification effect of harmonic wavelets (a)The time domain of D11 (b)The time domain of D12 (c)The frequency spectrumof D11 (d) The frequency spectrum of D12 (e) The orbit is to extract the first 2 frequencies
by amplitude 119860 119894 of signal frequency component The largermagnitude of amplitude is the greater eigenvalues are andthe higher arrangement will be
Based on aforesaid principle a new CFSM-PCA wasdeveloped and has become an effective tool to extract oreliminate specific characteristic (single frequency) even dif-ference of two frequencies is only 1 Hz Purification of rotororbit using this algorithm shows significant improvement
than those of existing methods such as wavelet packet andharmonic wavelet packet Although the proposed CFSM-PCA is effective some drawbacks are unavoidable Forexample this method suffers from similar issue to otheralgorithms that is the reconstruction error increaseswith thedecrease of SNR In addition the applications of the CFSM-PCA algorithm in purification of speech recognition bearingfault diagnosis and fault diagnosis of power system should
16 Shock and Vibration
D11-1080 rpmx 10
minus5
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
minus20
minus15
minus05
0
05
10
15
20
D11
-X (m
)
(a)
D12-1080 rpmx 10
minus5
minus15
minus1
minus05
0
05
1
15
D12
-Y (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(b)
(16 06557)
(32 04381)
D11-1080 rpmx 10
minus5
10020 30 40 50 60 70 80 90100f (Hz)
0010203040506070809
1
D11
-X (m
)
(c)
(16 06452)(32 04927)
D12-1080 rpm
x 10minus5
0010203040506070809
1
D12
-Y (m
)
9020 30 40 50 60 70 8010 1000f (Hz)
(d)
W3
x 10minus5
x 10minus5
minus15
minus1
minus05
0
05
1
15
Y(D
12) (
m)
minus15 minus05minus1 05 1 15 20X(D11) (m)
(e)
Figure 21 The purification effect of wavelet packet (a) The time domain of D11 (b) The time domain of D12 (c) The frequency spectrum ofD11 (d) The frequency spectrum of D12 (e) The orbit is to extract the first 2 frequencies
be further exploredWe fully believe that the CFSM-PCAwillsparkle in diverse fields
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work is supported by the National High TechnologyResearch andDevelopment Program of China (863 Program)
Shock and Vibration 17
(18 0734)
(36 04634)
(54 02448) (72 02351)(90 01646)
x 10minus5
50 100 150 2000f (Hz)
0
02
04
06
08
1
D11
minusX
(m)
Figure 22 Frequency spectrum of D11 signal after filtering 50Hz
(no 2015AA043005) and the National Natural Science Foun-dation of China (no 51375187)
References
[1] H Abdi and L J Williams ldquoPrincipal component analysisrdquoWiley Interdisciplinary Reviews Computational Statistics vol 2no 4 pp 433ndash459 2010
[2] A Widodo B Yang and T Han ldquoCombination of independentcomponent analysis and support vectormachines for intelligentfaults diagnosis of induction motorsrdquo Expert Systems withApplications vol 32 no 2 pp 299ndash312 2007
[3] S Wold K Esbensen and P Geladi ldquoPrincipal componentanalysisrdquo Chemometrics and Intelligent Laboratory Systems vol2 no 1ndash3 pp 37ndash52 1987
[4] M Kirby and L Sirovich ldquoApplication of the Karhunen-Loeve procedure for the characterization of human facesrdquo IEEETransactions on Pattern Analysis and Machine Intelligence vol12 no 1 pp 103ndash108 1990
[5] J H Xi Y Z Han and R H Su ldquoNew fault diagnosis methodfor rolling bearing based on PCArdquo in Proceedings of the 25thChinese Control and Decision Conference CCDC rsquo13 pp 4123ndash4127 IEEE May 2013
[6] A Malhi and R X Gao ldquoPCA-based feature selection schemefor machine defect classificationrdquo IEEE Transactions on Instru-mentation and Measurement vol 53 no 6 pp 1517ndash1525 2004
[7] S Xu X Jiang J Huang S Yang and X Wang ldquoBayesianwavelet PCA methodology for turbomachinery damage diag-nosis under uncertaintyrdquo Mechanical Systems and Signal Pro-cessing vol 80 pp 1ndash18 2016
[8] W Sun J Chen and J Li ldquoDecision tree and PCA-based faultdiagnosis of rotatingmachineryrdquoMechanical Systems and SignalProcessing vol 21 no 3 pp 1300ndash1317 2007
[9] Z X Li X P Yan C Q Yuan Z Peng and L Li ldquoVirtual pro-totype and experimental research on gear multi-fault diagnosisusing wavelet-autoregressive model and principal componentanalysismethodrdquoMechanical Systems and Signal Processing vol25 no 7 pp 2589ndash2607 2011
[10] K Y Shao M M Cai and G F Zhao ldquoRolling bearing faultdiagnosis based onwavelet energy spectrum PCA andPNNrdquo inProceedings of the 26th Chinese Control andDecisionConferenceCCDC rsquo14 pp 800ndash804 IEEE 2014
[11] T Qiu Q F Zhang and Y J Ding ldquoNonlinear sensor faultdetection and data rebuilding based on principle componentanalysisrdquo Journal of Tsinghua University vol 46 no 5 pp 708ndash711 2006
[12] Y K Gu Z X Yang and F L Zhu ldquoGearbox fault feature fusionbased on principal component analysisrdquo China MechanicalEngineering vol 26 no 11 pp 1532ndash1537 2015
[13] R Dunia S J Qin T F Edgar and T J McAvoy ldquoIdentificationof faulty sensors using principal component analysisrdquo AIChEJournal vol 42 no 10 pp 2797ndash2811 1996
[14] E P de Moura C R Souto A A Silva and M A S IrmaoldquoEvaluation of principal component analysis and neural net-work performance for bearing fault diagnosis from vibrationsignal processed by RS and DF analysesrdquo Mechanical Systemsand Signal Processing vol 25 no 5 pp 1765ndash1772 2011
[15] S Golestan M Monfared F D Freijedo and J M GuerreroldquoDynamics assessment of advanced single-phase PLL struc-turesrdquo IEEE Transactions on Industrial Electronics vol 60 no6 pp 2167ndash2177 2013
[16] A Lima H Zen Y Nankaku C Miyajima K Tokuda andT Kitamura ldquoOn the use of kernel PCA for feature extractionin speech recognitionrdquo IEICE Transaction on Information andSystems vol 12 no 87 pp 2802ndash2811 2004
[17] J D Zheng ldquoImproved hilbert-huang transform and its appli-cations to rolling bearing fault diagnosisrdquo Journal of MechanicalEngineering vol 1 no 51 pp 138ndash145 2015
[18] G He K Ding and H Lin ldquoFault feature extraction of rollingelement bearings using sparse representationrdquo Journal of Soundand Vibration vol 366 pp 514ndash527 2016
[19] J G Yang S B Xia and Y G Liu ldquoWavelet denoising and itsapplication in extraction of shaft centerline orbit featuresrdquoJournal of Harbin Institute of Technology vol 31 no 5 pp 52ndash541999 (Chinese)
[20] J A Duan and X D Zhang ldquoFeature extraction from the orbitof the center of a rotor based on wavelet transformrdquo Journal ofVibration Measurement amp Diagnosis vol 17 no 01 pp 31ndash341997
[21] J Han D X Jiang and W D Ning ldquoApplication of optimalwavelet packets in failure signal extraction of rotor orbitrdquoTurbine Technology vol 43 no 3 pp 133ndash136 2001
[22] W B Zhang X J Zhou andY Lin ldquoRefinement of rotor centerrsquosorbit by a harmonic window methodrdquo Journal of VibrationMeasurement amp Diagnosis vol 30 no 1 pp 87ndash90 2010
[23] S M Li G Z Lu and Q Y Xu ldquoOn obtaining accuraterotor sub-frequency signal with harmonic waveletrdquo Journal ofNorthwestern Polytechincal University vol 19 no 2 pp 220ndash224 2001
[24] F J Wu and L S Qu ldquoAn improved method for restraining theend effect in empiricalmode decomposition and its applicationsto the fault diagnosis of large rotating machineryrdquo Journal ofSound and Vibration vol 314 no 3-5 pp 586ndash602 2008
[25] W G Liu ldquoResearch on method of purification of shaft orbitbased on singular value decompositionrdquo Technical Acousticsvol 5 no 35 pp 30ndash35 2016
[26] X Z Zhao and B Y Ye ldquoThe influence of formation manner ofcomponent on signal processingrdquo Shang Hai Jiao Tong Univer-sity vol 45 no 3 pp 368ndash374 2011
[27] X Z Zhao B Y Ye and T J Chen ldquoPrinciple of singularitydetection based on SVD and its applicationrdquo Journal of Vibra-tion and Shock vol 6 no 27 pp 7ndash10 2008
[28] X Z Zhao B Y Ye and T J Chen ldquoDifference spectrum theoryof singular value and its application to the fault diagnosis ofheadstock of latherdquo Journal of Mechanical Engineering vol 46pp 100ndash108 2010
[29] I A Kougioumtzoglou and P D Spanos ldquoAn identificationapproach for linear and nonlinear time-variant structural sys-tems via harmonic waveletsrdquo Mechanical Systems and SignalProcessing vol 37 no 1-2 pp 338ndash352 2013
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6 Shock and Vibration
Equation (14) can be expanded to addition form of twoformulas depicted as
X = 1198862119894 times 119890120593119894
times[[[[[[[[
1198900120596119879119904119894 1198901120596119879119904119894 1198902120596119879119904119894 sdot sdot sdot 119890(119899minus1)1205961198791199041198941198901120596119879119904119894 1198902120596119879119904119894 1198903120596119879119904119894 sdot sdot sdot 119890119899120596119879119904119894
119890(119898minus1)120596119879119904119894 119890119898120596119879119904119894 119890(119898+1)120596119879119904119894 sdot sdot sdot 119890(119898+119899minus2)120596119879119904119894
]]]]]]]]
minus 1198862119894 times 119890minus120593119894
times[[[[[[[[
119890minus0120596119879119904119894 119890minus1120596119879119904119894 119890minus2120596119879119904119894 sdot sdot sdot 119890minus(119899minus1)120596119879119904119894119890minus1120596119879119904119894 119890minus2120596119879119904119894 119890minus3120596119879119904119894 sdot sdot sdot 119890minus119899120596119879119904119894
119890minus(119898minus1)120596119879119904119894 119890minus119898120596119879119904119894 119890minus(119898+1)120596119879119904119894 sdot sdot sdot 119890minus(119898+119899minus2)120596119879119904119894
]]]]]]]](15)
From (15) the rank of both matrices is 1 Based on rankrelationship of two matrices 119877(119860+119861) le 119877(119860) + 119877(119861) Hencerank of matrix X is less than or equal to 2
Additionally when 120593 = 0 (13) is rewritten to (16)According to sum-to-product identities the rank of X is 2when 120596119879119904 = 0 120587 2120587 sdot sdot sdot
X = 119886 times[[[[[[[[[
sin (0 sdot 120596119879119904) sin (1 sdot 120596119879119904) sdot sdot sdot sin ((119899 minus 1) sdot 120596119879119904)sin (1 sdot 120596119879119904) sin (2 sdot 120596119879119904) sdot sdot sdot sin (119899 sdot 120596119879119904)
sin ((119898 minus 1) sdot 120596119879119904) sin (119898 sdot 120596119879119904) sdot sdot sdot sin ((119898 + 119899 minus 2) sdot 120596119879119904)
]]]]]]]]]
(16)
When 120593 = 0 (13) can be deduced into the first-orderprincipal minor of matrix X ie |sin120593| = 0 The principalminor of order 2 of X is shown as10038161003816100381610038161003816100381610038161003816100381610038161003816sin (0 sdot 120596119879119904 + 120593) sin (1 sdot 120596119879119904 + 120593)sin (1 sdot 120596119879119904 + 120593) sin (2 sdot 120596119879119904 + 120593)
10038161003816100381610038161003816100381610038161003816100381610038161003816= minussin2 (120596119879119904)
= 0(17)
where120596119879119904 = 0 120587 2120587 sdot sdot sdot Because the leading principalminorof order 2 ofX in (13) is nonzero values so the rank of matrixX is at least 2
Combining result of (15) 119877(119860 + 119861) le 2 and nonzerovalues of the leading principal submatrix of order 2 hencethe rank of matrix X is 2 Matrix X has two eigenvalues 12058210158401and 12058210158402 Derived from C = XXT119899 (where 119899 is the numberof columns of matrix X) [3] two nonzero eigenvalues ofcovariance matrix C of X are 1205821 and 1205822 respectively(2)The sequence of eigenvalues is determined by ampli-
tudeIn terms of PCA covariance matrix C is described as
C = 119864 [(X minus 119864 (X)) sdot (X minus 119864 (X))T] (18)
Referring to (6) (19) is constructed as follows
119897
sum119894=1
C120572119894120572T119894 =119897
sum119894=1
120582119894120572119894120572T119894 (19)
Two effective eigenvalues are generated from a frequencycomponent that is l=2
CI119898 = 12058211205721120572T1 + 12058221205722120572T2 (20)
Then the energy of matrix C can be deduced as
|C|2 = 12058221119898
sum119894=1
12057221198941119898
sum119894=1
12057221198941 + 12059022119898
sum119894=1
12057221198942119898
sum119895=1
12057221198942
+ 21205902112059022119898
sum119894=1
12057211989411205721198942119898
sum119894=1
12057211989411205721198943(21)
Based upon (4) sum119898119894=1 12057221198941 = 1 sum119898119894=1 12057221198942 = 1 sum119898119894=1 12057211989411205721198942 =0 Then (22) is given by
|C|2 = 12058221 + 12058222 (22)
The constructed Hankel matrix X of signal 119909(119905) is sub-stituted into (18) we can see that the energy of covariancematrix C is proportional to 1198862 The larger 1198862 is the greaterenergy of matrix C is as well as 120582119894 according to (22)Furthermore the larger frequency component amplitude isthe larger corresponding eigenvalue in covariance matrix Ccharacteristic distribution chart is
Based on these conclusions once the amplitude sequenceof a certain frequency in raw signal amplitude spec-trum is determined its corresponding frequency compo-nent can be reconstructed In this way extraction of sin-gle or multiple characteristic frequencies could be real-ized
In view of addition relation as shown in (8) the notchfilter could be achieved through CFSM-PCA The frequencycomponent extracted by this algorithm is subtracted in orig-inal signal ie this frequency component can be eliminatedin raw signal Reader could consult examples in Section 5 formore details
Shock and Vibration 7
Table 1 Amplitude of signal components under the different SNR
SNR SNR1=-1 SNR2=-4 SNR3=-7 SNR4=-10Signal component Raw Process Raw Process Raw Process Raw Processsin(40120587119905 + 20) 09999 09938 10260 10240 10370 10730 1114 111205 sin(100120587119905 + 40) 04703 04826 05317 05482 07353 07533 05177 0523407 sin(98120587119905 + 50) 07299 07152 07116 07235 05110 04968 06800 0662108 sin(160120587119905 + 60) 08340 08324 08264 08142 08389 08338 08379 08923Max difference 00123 00122 00360 00544
minus6minus4minus2
0246
Am
plitu
de
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(a)
(200972)
(490693)(800791)
(500468)
0
05
1
Am
plitu
de
50 100 150 2000f (Hz)
(b)
Figure 4 The raw signal 119910 (a) Time domain (b) Amplitudespectrum
33 Simulation Example To verify the applicability and effec-tiveness of the CFSM-PCA algorithm a simulated sinusoidalsignal 119910 was constructed as
119910 = sin (40120587119905 + 20) + 05sin (100120587119905 + 40)+ 07sin (98120587119905 + 50) + 08sin (160120587119905 + 60)+ 119890 (119899)
(23)
where 119890(119899) is Gaussian white noise with standard deviationof 12 The result is shown in Figure 4(a) and amplitudespectrum (Fourier spectrum) is illustrated in Figure 4(b)after collecting 4096 data points with sampling frequency of1024 Hz
Follow-up work to separate frequencies via the proposedapproach was performed From eigenvalue distribution chartof crude signal in Figure 5 we can see that apart fromdiscernible four sets of eigenvalues most of eigenvaluescaused by noise are nonzero values and distribute at largeeigenvalues region
As demonstrated in Figures 6(a) and 6(b) time domainand frequency spectrum are generated after reconstructingthe first set of eigenvalues 1205821 and 1205822 in Figure 5 Frequencyof the reconstructed signal is 20 Hz with amplitude of 0977which is in correspondence with component sin(40120587t+20) ofy signal
As revealed in Figures 6(c) and 6(d) another time domainand amplitude spectrum are obtained via the reconstruction
(1515) (25144)
(3353) (43527)
(52654) (62643)
(71037) (81036)
e first group
e second group
e third group
e fourth group
0
200
400
600
Eige
nval
ue o
f PCA
i
10 20 30 40 500Number of eigenvalues q
Figure 5The eigenvalues distribution of the covariancematrixC of119910
of the second set of eigenvalues 1205823 and 1205824 Frequencyof this reconstructed signal is 80 Hz with amplitude of0786 which is same as component 08sin(160120587t+60) of rawsignal y Figures 6(e) and 6(f) are brought out through thereconstruction of the third set of eigenvalues 1205825 and 1205826Frequency of reconstructed signal is 49 Hz with amplitude of0688 which is consistent with component 07sin(98120587t+50)of crude signal 119910 Time domain and amplitude spectrum aregained after the reconstruction from the last set of eigenvaluesin Figure 5 exhibited as Figures 6(h) and 6(g) Frequency is50 Hz with amplitude of 0463 which is matched well withcomponent 05sin(100120587t+40) of signal y
Moreover several sets of signals with different signal-to-noise ratio (SNR) were processed by CFSM-PCA Theamplitude difference is summarized in Table 1
It can be seen that when the noise is low the amplitudeof frequency components extracted byCFSM-PCAalgorithmis close to that of the raw signal However the amplitudedifference will increase with the decrease of SNR
From all above-mentioned diagrams (Figure 6) it is obvi-ous that diverse frequency components extracted from rawsignal can be achieved accurately More importantly theseresults are coincident with ideal signal perfectly demon-strating that this CFSM-PCA is a zero phase shift frequencyextraction algorithm Meanwhile it should be noted that twofrequencies could also be perfectly separated via the proposedalgorithm even difference value between the adjacent fre-quency components is only 1 HzThismethod is accurate andseems to be more efficient than other existing filter methodssuch as wavelet filter and finite impulse response (FIR) filterwhich are subject to phase distorting issue
4 Experimental Verification
The fault diagnosis for large rotating machinery has been animportant topic and is attracting more and more attention
8 Shock and Vibration
the primary signalthe recovered signal
minus2
minus1
0
1
2A
mpl
itude
100 200 300 400 5000Sampling numbers i
(a)
(200977)
50 1000f (Hz)
002
04
06
08
1
Am
plitu
de
(b)
the primary signalthe recovered signal
minus1
minus05
0
05
1
Am
plitu
de
100 200 300 400 5000Sampling numbers i
(c)
(800786)
002
04
06
08
1
Am
plitu
de
50 1000f (Hz)
(d)
the primary signalthe recovered signal
100 200 300 400 5000Sampling numbers i
minus1
minus05
0
05
1
Am
plitu
de
(e)
(490688)
50 1000f (Hz)
002
04
06
08
1
Am
plitu
de
(f)
the primary signalthe recovered signal
minus05
0
05
Am
plitu
de
100 200 300 400 5000Sampling numbers i
(g)
(500463)
002
04
06
08
1
Am
plitu
de
50 1000f (Hz)
(h)
Figure 6 The time and frequency domains of reconstructed signal (a) The time domain of the reconstructed signal via the first set ofeigenvalues (b) Amplitude spectrum of Figure 6(a) (c) The time domain of the reconstructed signal using the second set of eigenvalues(d) Amplitude spectrum of Figure 6(c) (e) The time domain of the reconstructed signal using the third set of eigenvalues (f) Amplitudespectrum of Figure 6(e) (g) The time domain of the reconstructed signal using the fourth set of eigenvalues (h) Amplitude spectrum ofFigure 6(g)
Shock and Vibration 9
A B
Servo Motor Unloading device
Coupling
Sliding Bearing
Rotor
Large Platform
Small Platform
Air spring
BeltSliding Bearing
Figure 7 Large rot or vibration test bed
Displacement SensorA-end Face B-end Face
D13 D14
D12D11
Figure 8 Schematic and installation of eddy current displacement sensor
Rotor orbits indicate the symptoms of malfunction andare related to variation of input force or dynamic stiffnessGenerally different curve shapes of rotor orbit in a senserepresent different fault types [19ndash22 29] For example insideldquo8rdquo outside ldquo8rdquo and petal-shaped curves are omens of oil-filmwhirl fault misalignment fault and rubbing fault of rotorsystem respectivelyTherefore extraction of rotor orbit playsa significant role in rotating machinery diagnostics In thissection a simple and efficient CFSM-PCA is applied to purifyrotor orbit precisely
41 Experimental Rigs Figure 7 shows the large rotor vibra-tion test bed system which is constructed by our group Inorder to real-timemonitor rotor working station two KamanKD2306-1S eddy current proximity probes are installedon individual side in orthogonal manners (Figure 8) Theexperimental data were collected through LMS Test Lab
42 The First Group Sample Analysis Experimental datastems from D11 and D12 eddy current displacement sensorin side A 4096 data points were collected with test bed speedof 1080 rmin and sampling frequency of 2048 Hz The timedomain waveforms amplitude spectrums and the order ofamplitudes were obtained after filtering DC component viaFFT (Figure 9)
As demonstrated in Figure 9 raw signal was not onlyaffected by noise but also interfered by power frequency 50Hz and its harmonics frequency The energy of signal mainlyconcentrated on the leading two octaves
Rotor orbit was produced by combining spindle dis-placement signal D11 with signal D12 (Figure 10) X-axispresented raw signal D11 and Y-axis referred to signal D12Manipulating status of test bed is difficult to know from thisunprocessed rotor orbit Next the rotor orbit was attemptedto purify by employing PCA
Generally rotor orbit synthesized by 1X and 2X is cred-itable Orbits included high harmonics incline to becomecomplex and even turn messy Hence extraction of 1Xand 2X of raw signal becomes more important [21 22]Eigenvalue distribution of covariance matrix of D11 and D12signals are displayed in Figure 11 According to amplitudeof each frequency of D11 and D12 in Figure 9 1X and 2Xcorresponded to the first (first and second eigenvalues) andthe second (third and fourth eigenvalues) set of eigenvaluesrespectively The leading four eigenvalues in Figure 11 werereconstructed through CFSM-PCA and results are illustratedin Figure 12 Generated 1X and 2X were legible without effectby noise and disturbance from frequency 50 Hz as well as itsharmonics Meanwhile amplitude of the extracted signal wasclose to that of 1X and 2X in raw signal indicating that thereis no spectral leakage issue
The feature spectrums continued to be separated andeigenvalue distributionmaps of covariance matrix of D11 andD12 are shown in Figure 15
Rotor orbit generated by the purified 1X and 2X is shownin Figure 13The rotor orbit was a caved-in banana-like curvedemonstrating that sideA (near themotor) potentially suffersfrom misalignment fault [18 19] The explicit rotor orbit alsovalidates the high efficiency of this proposed method
43 The Second Group Sample Analysis In this case exper-imental data of D11 and D12 were analyzed when test bedruns steadily for a period of time Similarly 4096 datapoints were measured with test bed speed of 2770 rmin andsampling frequency of 2048Hz Time domainwaveforms andamplitude spectrums are displayed in Figure 14
According to amplitude of the leading trebling frequencyofD11 andD12 in Figure 14 1X corresponded to the second setof eigenvalues (third and fourth eigenvalues) 2X paralleledthe third set of eigenvalues (fifth and sixth eigenvalues) and
10 Shock and Vibration
D11-1080 rpmx 10
minus5
minus5minus4minus3minus2minus1
012345
X (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(a)
Power-line Interference50Hz
0733404771
Order 1 Order 2
0250502342
Order 6Order 7
D11-1080 rpm
01651 Order 11
18Hz36Hz54Hz72Hz90Hz
x 10minus5
1times
2times
3times
4times
5times
1times
2times
3times 4times5times
002040608
1
X (m
)
50 100 150 200 250 300 350 4000Frequency (Hz)
(b)
D12-1080 rpmx 10
minus5
minus6
minus4
minus2
0246
Y (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(c)
D12-1080 rpmPower-line Interference
50Hz
0658105228
Order 1 Order 2
0277101226
Order 5 Order 12
01948 Order 8
18Hz36Hz54Hz72Hz90Hz
x 10minus5
1times
2times
3times
4times
5times
1times
2times
3times
4times5times
0
05
1
Y (m
)
50 100 150 200 250 300 350 4000Frequency (Hz)
(d)
Figure 9The time domain and frequency domain of theD11 andD12 sensors in theA-end face (a)The time domain of D11 (b)The frequencydomain of D11 (c) The time domain of D12 (d) The frequency domain of D12
x 10minus5
x 10minus5
minus5
minus4
minus3
minus2
minus1
0
1
2
3
4
5
D12
minusY
(m)
minus6 minus2minus4 2 4 60D11minusX (m)
Figure 10 A-end face of the D11 and D12 sensors to collect the signal directly synthesized axis orbit
(12751) (22748)e first group
(31243) (41241)e second group
D11minus1080 rpm
x 10minus8
0
1
2
3
4
Eige
nval
ue o
f PCA
(m)
10 20 30 40 500Number of eigenvalues q
(a)
(12247) (2241)
(314) (41396)
D12minus1080 rpme first group
e second group
x 10minus8
005
115
225
Eige
nval
ue o
f PCA
(m)
10 20 30 40 500Number of eigenvalues q
(b)
Figure 11The eigenvalues distribution of covariancematrixC (a)The eigenvalues distribution of D11 signal (b)The eigenvalues distributionof D12 signal
Shock and Vibration 11
D11minus1080 rpmx 10
minus5
minus15
minus10
minus05
0051015
X (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(a)
D12minus1080 rpmx 10
minus5
minus10
minus05
0051015
Y (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(b)
D11minus1080 rpm(1807287)
(360472)
x 10minus5
002040608
1
X (m
)
50 100 150 2000Frequency (Hz)
(c)
(3605105)(1806575)
D12minus1080 rpmx 10
minus5
002040608
1
Y (m
)
50 100 150 2000Frequency (Hz)
(d)
Figure 12 Extracting the time and frequency spectrums of the first two frequencies of D11 and D12 (a)The time domain of D11 (b)The timedomain of D12 (c) The frequency domain of D11 (d) The frequency domain of D12
W1
x 10minus5
x 10minus5
minus1
minus05
0
05
1
15
D12
minusY
(m)
minus1minus15 minus05 05 10 15D11minusX (m)
(a)0
051
152
0
2
t (s)D11minusX (m)
minus2
x 10minus5
x 10minus5
minus1
minus05
0
05
1
15
D12
minusY
(m)
(b)
Figure 13 The synthesis of axis orbit is to extract the first two times fundamental frequencies of the D11 and D12 signals by the CFSM-PCA(a) The axis orbit (b) Relationship between axis orbit and time
3X matched the first set of eigenvalues (first and secondeigenvalues)
As displayed in Figure 15 eigenvalues of 1X and 3Xwere large thus 1X and 3X were utilized to synthesize rotororbit firstly Purification results and orbit of shaft center arerevealed in Figures 16 and 17 respectively The legible orbitshowed a typical plum blossom shape
It can be seen from Figures 16 and 17 that the filteringeffect was also quite obvious The noise and power frequency50 Hz as well as its harmonic interference could be removedsuccessfully Moreover amplitude of refined signal was closeto that of original one and the reconstructed time domainsignal seemed to be steady without any fluctuation
As shown in Figures 14(c) and 14(d) amplitudes of 1X 2Xand 3X in amplitude spectrum of raw signal were largeThus1X 2X and 3X were also extracted to synthesize axis orbit
Time domain frequency domain and orbit curve are shownin Figures 18 and 19 respectively
Axis orbit synthesized by the leading three frequenciesexhibited a quincunx shape which is basically the sameto that synthesized by 1X and 3X frequency componentsindicating that static and dynamic parts friction may exist inA-side
44 Comparison Harmonic Wavelet with Wavelet Packet
(1) Comparison of CFSM-PCA with Harmonic Wavelet Onecommon used method for signal processing harmonicwavelet algorithm proposed by Newland is an improvedwavelet algorithm [29] It can segment entire frequencyband infinitely and avoid the shortcoming of downsamplingmethod of binary wavelet and binary wavelet packet [22 23]
12 Shock and Vibration
D11minus2770 rpmx 10
minus4
minus1
minus05
0
05
1
D11
-X (m
)
1000 2000 3000 40000Sampling numbers i
(a)
D12minus2770 rpmx 10
minus4
minus1
minus05
0
05
1
D12
-Y (m
)
1000 2000 3000 40000Sampling numbers i
(b)
Power-Line interference 50and 150Hz
1X2X
3X 1010D11minus2770 rpm
08441166
1X2X3X
x 10minus5
200 400 600 800 10000f (Hz)
0
05
10
15
D11
-X (m
)
(c)
1X
2X
3XPower-Line interference 50
and 150Hz
D12minus2770 rpm117309021227
1X2X3X
x 10minus5
0
05
10
15
D12
-Y (m
)
200 400 600 800 10000f (Hz)
(d)
Figure 14 The time and frequency domains of the D11 and D12 sensors (a) The time domain of D11 (b) The time domain of D12 (c) Thefrequency domain of D11 (d) The frequency domain of D12
(11489) (21482)e first group
(31237) (41220)e second group
(30348) (40347)e second group
D11minus2770 rpmx 10
minus7
10 20 30 400 50Number of eigenvalues q
0
05
1
15
Eige
nval
ue o
f PCA
(m)
(a)
D12minus2770 rpm(11688) (21687)
e first group
e second group
(30414) (40413)e second group
(31673) (41664)
x 10minus7
0
05
1
15
2
Eige
nval
ue o
f PCA
(m)
10 20 30 40 500Number of eigenvalues q
(b)
Figure 15 The eigenvalues distribution of covariance matrix C (a) The eigenvalues distribution of D11 (b) The eigenvalues distribution ofD12
1X and its frequency doubling signal can be easily extractedto form axis orbit of revolving test bearing
Axis orbit was fabricated by 1X and 2Xof data of example 1(Section 42) through harmonic wavelet algorithm and theresults are demonstrated in Figure 20 The shape of axisorbit was trapped banana-like which is consistent with theresult obtained via CFSM-PCA It should be noted thatorbit in Figure 13 was more obvious than that in Figure 20ie W1ltW2 Meanwhile the amplitude fluctuation of timedomain signal was large (Figures 20(a) and 20(b)) Themajor reason for this phenomenon is that the harmonicwavelet which is a band filter in frequency domain inevitablyextracts the noise near the feature frequency Furthermorethis phenomenon also demonstrates that harmonic waveletis subject to the Heisenberg uncertainty principle ie thephenomenon of the amplitude leakage exists
(2) Comparison of CFSM-PCA with Wavelet Packet Besidesharmonic wavelets wavelet packet is also used to purify axisorbit [19ndash21] Purification results of axis orbit via Daubechieswavelet packet in Figure 21 exhibit that axis orbit was diver-gent seriously This was worse than that of harmonic waveletie W1ltW2ltW3The phenomenon of severe energy leakageoccurred during the extraction of 1X and 2X which cancause divergence of axis orbit Moreover the bandpass filterof wavelet packet also accounts for the divergent orbits
5 A Single Frequency Filtrationwith CFSM-PCA
The application in single frequency filtration is discussedin this section Taking D11 experimental data of example
Shock and Vibration 13
D11minus2770 rpmx 10
minus5
0 1000500Sampling numbers i
minus4
minus2
0
2
4
D11
-X (m
)
(a)
D12minus2770 rpmx 10
minus5
minus4
minus2
0
2
4
D12
-Y (m
)
500 10000Sampling numbers i
(b)
(1375 1185)(46 1009)
D11minus2770 rpmx 10
minus5
100 200 300 400 5000f (Hz)
0
05
1
15
D11
-X (m
)
(c)
(1375 1218)(46 1164)
D12minus2770 rpmx 10
minus5
100 200 300 400 5000f (Hz)
0
05
1
15
D12
-Y (m
)
(d)
Figure 16 Extracting the frequency spectrums of the first two frequencies of D11 and D12 by PCA algorithm (a) The time domain of D11(b) The time domain of D12 (c) The frequency spectrum of D11 (d) The frequency spectrum of D12
x 10minus5
x 10minus5
minus4
minus3
minus2
minus1
0
1
2
3
4
D12
-Y (m
)
210 3 4minus2minus3 minus1minus4
D11-X (m)
(a)0
0204
0608
10
24
t (s)D11-X (m) minus2
minus4
x 10minus5
x 10minus5
minus4
minus2
0
2
4
D12
-Y (m
)
(b)
Figure 17 The synthesis of axis orbit is to extract the first two times fundamental frequencies of the D11 and D12 signals by the CFSM-PCA(a) The axis orbit (b) Relationship between axis orbit and time
1 (Section 42) as a demo we attempt to filter the powerfrequency interference (50 Hz) from D11 by employingCFSM-PCA It can be seen from Figure 9(b) that sequenceof amplitude of 50 Hz in raw signal was 3 so eigenvec-tors corresponding to 5th and 6th eigenvalues in eigen-value distribution map were used to reconstruct Accordingto (8) the filtered signal (Figure 22) without 50 Hz was
obtained by subtracting reconstructed signal from originalone
Results in Figure 22 show that this algorithm filters out50 Hz power frequency interference and has no influenceon the other frequency components of raw signal demon-strating that this method has a potential application in notchfilter
14 Shock and Vibration
D11minus2770 rpm
x 10minus5
minus4
minus2
0
2
4
D11
-X (m
)
200 400 600 800 10000Sampling numbers i
(a)
D12minus2770 rpm
x 10minus5
0 400 600 800 1000200Sampling numbers i
minus4
minus2
0
2
4
D12
-Y (m
)
(b)
(1375 1185)(46 1009)
(92 0831)
D11minus2770 rpmx 10
minus5
00
05
10
15
D11
-X (m
)
100 200 300 400 5000f (Hz)
(c)
(1375 1218)(46 1163)
(92 0898)
D12minus2770 rpmx 10
minus5
00
05
10
15
D12
-Y (m
)
100 200 300 400 5000f (Hz)
(d)
Figure 18 Extracting the time and frequency spectrums of the first three frequencies of D11 and D12 by PCA algorithm (a)The time domainof D11 (b) The time domain of D12 (c) The frequency spectrum of D11 (d) The frequency spectrum of D12
x 10minus5
x 10minus5
minus4
minus3
minus2
minus1
0
1
2
3
4
D12
-Y (m
)
minus3 minus2 minus1minus4 1 2 3 40D11-X (m)
(a)
1008
0604
0200
02
4D11-X (m)
t (s)
minus2
minus4
x 10minus5
x 10minus5
minus4
minus2
0
2
4
D12
-Y (m
)
(b)
Figure 19 The synthesis of axis orbit is to extract the first three times fundamental frequencies of the D11 and D12 by the CFSM-PCA (a)The axis orbit (b) Relationship between axis orbit and time
6 Conclusion
The selection of effective eigenvalues in feature distributionchart of covariance matrix C is crucial in PCA algorithm Inthis paper the relationships between effective eigenvalues andsignal frequency and amplitude are discovered and a series ofconclusions are summarized
First in Hankel matrix mode each valid frequencycomponent of signal only produces two adjacent nonzeroeigenvalues Second the number of valid eigenvalues incharacteristic distribution chart which is independent ofnumerical values of 119891119894 119860 119894 and 120593119894 is related to the number offrequency componentsThird the sequence of signal effectiveeigenvalues in characteristic distribution map is determined
Shock and Vibration 15
D11-1080 rpm
x 10minus5
minus20
minus15
minus05
0
05
10
15
20
D11
-X (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(a)
D12-1080 rpmx 10
minus5
minus15
minus1
minus05
0
05
1
15
D12
-Y (m
)
500 1000 1500 2000 2500 30000 40003500Sampling numbers i
(b)
(16 07335)
(32 04771)
D11-1080 rpm
x 10minus5
10 20 30 40 50 60 70 80 90 1000f (Hz)
0010203040506070809
1
D11
-X (m
)
(c)
(16 06571)
(32 05228)
D12-1080 rpm
x 10minus5
0010203040506070809
1
D12
-Y (m
)
10 20 30 40 50 60 70 80 90 1000f (Hz)
(d)
W2
x 10minus5
x 10minus5
minus15
minus1
minus05
0
05
1
15
Y(D
12) (
m)
minus1 minus05minus15 05 1 15 20X(D11) (m)
(e)
Figure 20The purification effect of harmonic wavelets (a)The time domain of D11 (b)The time domain of D12 (c)The frequency spectrumof D11 (d) The frequency spectrum of D12 (e) The orbit is to extract the first 2 frequencies
by amplitude 119860 119894 of signal frequency component The largermagnitude of amplitude is the greater eigenvalues are andthe higher arrangement will be
Based on aforesaid principle a new CFSM-PCA wasdeveloped and has become an effective tool to extract oreliminate specific characteristic (single frequency) even dif-ference of two frequencies is only 1 Hz Purification of rotororbit using this algorithm shows significant improvement
than those of existing methods such as wavelet packet andharmonic wavelet packet Although the proposed CFSM-PCA is effective some drawbacks are unavoidable Forexample this method suffers from similar issue to otheralgorithms that is the reconstruction error increaseswith thedecrease of SNR In addition the applications of the CFSM-PCA algorithm in purification of speech recognition bearingfault diagnosis and fault diagnosis of power system should
16 Shock and Vibration
D11-1080 rpmx 10
minus5
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
minus20
minus15
minus05
0
05
10
15
20
D11
-X (m
)
(a)
D12-1080 rpmx 10
minus5
minus15
minus1
minus05
0
05
1
15
D12
-Y (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(b)
(16 06557)
(32 04381)
D11-1080 rpmx 10
minus5
10020 30 40 50 60 70 80 90100f (Hz)
0010203040506070809
1
D11
-X (m
)
(c)
(16 06452)(32 04927)
D12-1080 rpm
x 10minus5
0010203040506070809
1
D12
-Y (m
)
9020 30 40 50 60 70 8010 1000f (Hz)
(d)
W3
x 10minus5
x 10minus5
minus15
minus1
minus05
0
05
1
15
Y(D
12) (
m)
minus15 minus05minus1 05 1 15 20X(D11) (m)
(e)
Figure 21 The purification effect of wavelet packet (a) The time domain of D11 (b) The time domain of D12 (c) The frequency spectrum ofD11 (d) The frequency spectrum of D12 (e) The orbit is to extract the first 2 frequencies
be further exploredWe fully believe that the CFSM-PCAwillsparkle in diverse fields
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work is supported by the National High TechnologyResearch andDevelopment Program of China (863 Program)
Shock and Vibration 17
(18 0734)
(36 04634)
(54 02448) (72 02351)(90 01646)
x 10minus5
50 100 150 2000f (Hz)
0
02
04
06
08
1
D11
minusX
(m)
Figure 22 Frequency spectrum of D11 signal after filtering 50Hz
(no 2015AA043005) and the National Natural Science Foun-dation of China (no 51375187)
References
[1] H Abdi and L J Williams ldquoPrincipal component analysisrdquoWiley Interdisciplinary Reviews Computational Statistics vol 2no 4 pp 433ndash459 2010
[2] A Widodo B Yang and T Han ldquoCombination of independentcomponent analysis and support vectormachines for intelligentfaults diagnosis of induction motorsrdquo Expert Systems withApplications vol 32 no 2 pp 299ndash312 2007
[3] S Wold K Esbensen and P Geladi ldquoPrincipal componentanalysisrdquo Chemometrics and Intelligent Laboratory Systems vol2 no 1ndash3 pp 37ndash52 1987
[4] M Kirby and L Sirovich ldquoApplication of the Karhunen-Loeve procedure for the characterization of human facesrdquo IEEETransactions on Pattern Analysis and Machine Intelligence vol12 no 1 pp 103ndash108 1990
[5] J H Xi Y Z Han and R H Su ldquoNew fault diagnosis methodfor rolling bearing based on PCArdquo in Proceedings of the 25thChinese Control and Decision Conference CCDC rsquo13 pp 4123ndash4127 IEEE May 2013
[6] A Malhi and R X Gao ldquoPCA-based feature selection schemefor machine defect classificationrdquo IEEE Transactions on Instru-mentation and Measurement vol 53 no 6 pp 1517ndash1525 2004
[7] S Xu X Jiang J Huang S Yang and X Wang ldquoBayesianwavelet PCA methodology for turbomachinery damage diag-nosis under uncertaintyrdquo Mechanical Systems and Signal Pro-cessing vol 80 pp 1ndash18 2016
[8] W Sun J Chen and J Li ldquoDecision tree and PCA-based faultdiagnosis of rotatingmachineryrdquoMechanical Systems and SignalProcessing vol 21 no 3 pp 1300ndash1317 2007
[9] Z X Li X P Yan C Q Yuan Z Peng and L Li ldquoVirtual pro-totype and experimental research on gear multi-fault diagnosisusing wavelet-autoregressive model and principal componentanalysismethodrdquoMechanical Systems and Signal Processing vol25 no 7 pp 2589ndash2607 2011
[10] K Y Shao M M Cai and G F Zhao ldquoRolling bearing faultdiagnosis based onwavelet energy spectrum PCA andPNNrdquo inProceedings of the 26th Chinese Control andDecisionConferenceCCDC rsquo14 pp 800ndash804 IEEE 2014
[11] T Qiu Q F Zhang and Y J Ding ldquoNonlinear sensor faultdetection and data rebuilding based on principle componentanalysisrdquo Journal of Tsinghua University vol 46 no 5 pp 708ndash711 2006
[12] Y K Gu Z X Yang and F L Zhu ldquoGearbox fault feature fusionbased on principal component analysisrdquo China MechanicalEngineering vol 26 no 11 pp 1532ndash1537 2015
[13] R Dunia S J Qin T F Edgar and T J McAvoy ldquoIdentificationof faulty sensors using principal component analysisrdquo AIChEJournal vol 42 no 10 pp 2797ndash2811 1996
[14] E P de Moura C R Souto A A Silva and M A S IrmaoldquoEvaluation of principal component analysis and neural net-work performance for bearing fault diagnosis from vibrationsignal processed by RS and DF analysesrdquo Mechanical Systemsand Signal Processing vol 25 no 5 pp 1765ndash1772 2011
[15] S Golestan M Monfared F D Freijedo and J M GuerreroldquoDynamics assessment of advanced single-phase PLL struc-turesrdquo IEEE Transactions on Industrial Electronics vol 60 no6 pp 2167ndash2177 2013
[16] A Lima H Zen Y Nankaku C Miyajima K Tokuda andT Kitamura ldquoOn the use of kernel PCA for feature extractionin speech recognitionrdquo IEICE Transaction on Information andSystems vol 12 no 87 pp 2802ndash2811 2004
[17] J D Zheng ldquoImproved hilbert-huang transform and its appli-cations to rolling bearing fault diagnosisrdquo Journal of MechanicalEngineering vol 1 no 51 pp 138ndash145 2015
[18] G He K Ding and H Lin ldquoFault feature extraction of rollingelement bearings using sparse representationrdquo Journal of Soundand Vibration vol 366 pp 514ndash527 2016
[19] J G Yang S B Xia and Y G Liu ldquoWavelet denoising and itsapplication in extraction of shaft centerline orbit featuresrdquoJournal of Harbin Institute of Technology vol 31 no 5 pp 52ndash541999 (Chinese)
[20] J A Duan and X D Zhang ldquoFeature extraction from the orbitof the center of a rotor based on wavelet transformrdquo Journal ofVibration Measurement amp Diagnosis vol 17 no 01 pp 31ndash341997
[21] J Han D X Jiang and W D Ning ldquoApplication of optimalwavelet packets in failure signal extraction of rotor orbitrdquoTurbine Technology vol 43 no 3 pp 133ndash136 2001
[22] W B Zhang X J Zhou andY Lin ldquoRefinement of rotor centerrsquosorbit by a harmonic window methodrdquo Journal of VibrationMeasurement amp Diagnosis vol 30 no 1 pp 87ndash90 2010
[23] S M Li G Z Lu and Q Y Xu ldquoOn obtaining accuraterotor sub-frequency signal with harmonic waveletrdquo Journal ofNorthwestern Polytechincal University vol 19 no 2 pp 220ndash224 2001
[24] F J Wu and L S Qu ldquoAn improved method for restraining theend effect in empiricalmode decomposition and its applicationsto the fault diagnosis of large rotating machineryrdquo Journal ofSound and Vibration vol 314 no 3-5 pp 586ndash602 2008
[25] W G Liu ldquoResearch on method of purification of shaft orbitbased on singular value decompositionrdquo Technical Acousticsvol 5 no 35 pp 30ndash35 2016
[26] X Z Zhao and B Y Ye ldquoThe influence of formation manner ofcomponent on signal processingrdquo Shang Hai Jiao Tong Univer-sity vol 45 no 3 pp 368ndash374 2011
[27] X Z Zhao B Y Ye and T J Chen ldquoPrinciple of singularitydetection based on SVD and its applicationrdquo Journal of Vibra-tion and Shock vol 6 no 27 pp 7ndash10 2008
[28] X Z Zhao B Y Ye and T J Chen ldquoDifference spectrum theoryof singular value and its application to the fault diagnosis ofheadstock of latherdquo Journal of Mechanical Engineering vol 46pp 100ndash108 2010
[29] I A Kougioumtzoglou and P D Spanos ldquoAn identificationapproach for linear and nonlinear time-variant structural sys-tems via harmonic waveletsrdquo Mechanical Systems and SignalProcessing vol 37 no 1-2 pp 338ndash352 2013
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Shock and Vibration 7
Table 1 Amplitude of signal components under the different SNR
SNR SNR1=-1 SNR2=-4 SNR3=-7 SNR4=-10Signal component Raw Process Raw Process Raw Process Raw Processsin(40120587119905 + 20) 09999 09938 10260 10240 10370 10730 1114 111205 sin(100120587119905 + 40) 04703 04826 05317 05482 07353 07533 05177 0523407 sin(98120587119905 + 50) 07299 07152 07116 07235 05110 04968 06800 0662108 sin(160120587119905 + 60) 08340 08324 08264 08142 08389 08338 08379 08923Max difference 00123 00122 00360 00544
minus6minus4minus2
0246
Am
plitu
de
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(a)
(200972)
(490693)(800791)
(500468)
0
05
1
Am
plitu
de
50 100 150 2000f (Hz)
(b)
Figure 4 The raw signal 119910 (a) Time domain (b) Amplitudespectrum
33 Simulation Example To verify the applicability and effec-tiveness of the CFSM-PCA algorithm a simulated sinusoidalsignal 119910 was constructed as
119910 = sin (40120587119905 + 20) + 05sin (100120587119905 + 40)+ 07sin (98120587119905 + 50) + 08sin (160120587119905 + 60)+ 119890 (119899)
(23)
where 119890(119899) is Gaussian white noise with standard deviationof 12 The result is shown in Figure 4(a) and amplitudespectrum (Fourier spectrum) is illustrated in Figure 4(b)after collecting 4096 data points with sampling frequency of1024 Hz
Follow-up work to separate frequencies via the proposedapproach was performed From eigenvalue distribution chartof crude signal in Figure 5 we can see that apart fromdiscernible four sets of eigenvalues most of eigenvaluescaused by noise are nonzero values and distribute at largeeigenvalues region
As demonstrated in Figures 6(a) and 6(b) time domainand frequency spectrum are generated after reconstructingthe first set of eigenvalues 1205821 and 1205822 in Figure 5 Frequencyof the reconstructed signal is 20 Hz with amplitude of 0977which is in correspondence with component sin(40120587t+20) ofy signal
As revealed in Figures 6(c) and 6(d) another time domainand amplitude spectrum are obtained via the reconstruction
(1515) (25144)
(3353) (43527)
(52654) (62643)
(71037) (81036)
e first group
e second group
e third group
e fourth group
0
200
400
600
Eige
nval
ue o
f PCA
i
10 20 30 40 500Number of eigenvalues q
Figure 5The eigenvalues distribution of the covariancematrixC of119910
of the second set of eigenvalues 1205823 and 1205824 Frequencyof this reconstructed signal is 80 Hz with amplitude of0786 which is same as component 08sin(160120587t+60) of rawsignal y Figures 6(e) and 6(f) are brought out through thereconstruction of the third set of eigenvalues 1205825 and 1205826Frequency of reconstructed signal is 49 Hz with amplitude of0688 which is consistent with component 07sin(98120587t+50)of crude signal 119910 Time domain and amplitude spectrum aregained after the reconstruction from the last set of eigenvaluesin Figure 5 exhibited as Figures 6(h) and 6(g) Frequency is50 Hz with amplitude of 0463 which is matched well withcomponent 05sin(100120587t+40) of signal y
Moreover several sets of signals with different signal-to-noise ratio (SNR) were processed by CFSM-PCA Theamplitude difference is summarized in Table 1
It can be seen that when the noise is low the amplitudeof frequency components extracted byCFSM-PCAalgorithmis close to that of the raw signal However the amplitudedifference will increase with the decrease of SNR
From all above-mentioned diagrams (Figure 6) it is obvi-ous that diverse frequency components extracted from rawsignal can be achieved accurately More importantly theseresults are coincident with ideal signal perfectly demon-strating that this CFSM-PCA is a zero phase shift frequencyextraction algorithm Meanwhile it should be noted that twofrequencies could also be perfectly separated via the proposedalgorithm even difference value between the adjacent fre-quency components is only 1 HzThismethod is accurate andseems to be more efficient than other existing filter methodssuch as wavelet filter and finite impulse response (FIR) filterwhich are subject to phase distorting issue
4 Experimental Verification
The fault diagnosis for large rotating machinery has been animportant topic and is attracting more and more attention
8 Shock and Vibration
the primary signalthe recovered signal
minus2
minus1
0
1
2A
mpl
itude
100 200 300 400 5000Sampling numbers i
(a)
(200977)
50 1000f (Hz)
002
04
06
08
1
Am
plitu
de
(b)
the primary signalthe recovered signal
minus1
minus05
0
05
1
Am
plitu
de
100 200 300 400 5000Sampling numbers i
(c)
(800786)
002
04
06
08
1
Am
plitu
de
50 1000f (Hz)
(d)
the primary signalthe recovered signal
100 200 300 400 5000Sampling numbers i
minus1
minus05
0
05
1
Am
plitu
de
(e)
(490688)
50 1000f (Hz)
002
04
06
08
1
Am
plitu
de
(f)
the primary signalthe recovered signal
minus05
0
05
Am
plitu
de
100 200 300 400 5000Sampling numbers i
(g)
(500463)
002
04
06
08
1
Am
plitu
de
50 1000f (Hz)
(h)
Figure 6 The time and frequency domains of reconstructed signal (a) The time domain of the reconstructed signal via the first set ofeigenvalues (b) Amplitude spectrum of Figure 6(a) (c) The time domain of the reconstructed signal using the second set of eigenvalues(d) Amplitude spectrum of Figure 6(c) (e) The time domain of the reconstructed signal using the third set of eigenvalues (f) Amplitudespectrum of Figure 6(e) (g) The time domain of the reconstructed signal using the fourth set of eigenvalues (h) Amplitude spectrum ofFigure 6(g)
Shock and Vibration 9
A B
Servo Motor Unloading device
Coupling
Sliding Bearing
Rotor
Large Platform
Small Platform
Air spring
BeltSliding Bearing
Figure 7 Large rot or vibration test bed
Displacement SensorA-end Face B-end Face
D13 D14
D12D11
Figure 8 Schematic and installation of eddy current displacement sensor
Rotor orbits indicate the symptoms of malfunction andare related to variation of input force or dynamic stiffnessGenerally different curve shapes of rotor orbit in a senserepresent different fault types [19ndash22 29] For example insideldquo8rdquo outside ldquo8rdquo and petal-shaped curves are omens of oil-filmwhirl fault misalignment fault and rubbing fault of rotorsystem respectivelyTherefore extraction of rotor orbit playsa significant role in rotating machinery diagnostics In thissection a simple and efficient CFSM-PCA is applied to purifyrotor orbit precisely
41 Experimental Rigs Figure 7 shows the large rotor vibra-tion test bed system which is constructed by our group Inorder to real-timemonitor rotor working station two KamanKD2306-1S eddy current proximity probes are installedon individual side in orthogonal manners (Figure 8) Theexperimental data were collected through LMS Test Lab
42 The First Group Sample Analysis Experimental datastems from D11 and D12 eddy current displacement sensorin side A 4096 data points were collected with test bed speedof 1080 rmin and sampling frequency of 2048 Hz The timedomain waveforms amplitude spectrums and the order ofamplitudes were obtained after filtering DC component viaFFT (Figure 9)
As demonstrated in Figure 9 raw signal was not onlyaffected by noise but also interfered by power frequency 50Hz and its harmonics frequency The energy of signal mainlyconcentrated on the leading two octaves
Rotor orbit was produced by combining spindle dis-placement signal D11 with signal D12 (Figure 10) X-axispresented raw signal D11 and Y-axis referred to signal D12Manipulating status of test bed is difficult to know from thisunprocessed rotor orbit Next the rotor orbit was attemptedto purify by employing PCA
Generally rotor orbit synthesized by 1X and 2X is cred-itable Orbits included high harmonics incline to becomecomplex and even turn messy Hence extraction of 1Xand 2X of raw signal becomes more important [21 22]Eigenvalue distribution of covariance matrix of D11 and D12signals are displayed in Figure 11 According to amplitudeof each frequency of D11 and D12 in Figure 9 1X and 2Xcorresponded to the first (first and second eigenvalues) andthe second (third and fourth eigenvalues) set of eigenvaluesrespectively The leading four eigenvalues in Figure 11 werereconstructed through CFSM-PCA and results are illustratedin Figure 12 Generated 1X and 2X were legible without effectby noise and disturbance from frequency 50 Hz as well as itsharmonics Meanwhile amplitude of the extracted signal wasclose to that of 1X and 2X in raw signal indicating that thereis no spectral leakage issue
The feature spectrums continued to be separated andeigenvalue distributionmaps of covariance matrix of D11 andD12 are shown in Figure 15
Rotor orbit generated by the purified 1X and 2X is shownin Figure 13The rotor orbit was a caved-in banana-like curvedemonstrating that sideA (near themotor) potentially suffersfrom misalignment fault [18 19] The explicit rotor orbit alsovalidates the high efficiency of this proposed method
43 The Second Group Sample Analysis In this case exper-imental data of D11 and D12 were analyzed when test bedruns steadily for a period of time Similarly 4096 datapoints were measured with test bed speed of 2770 rmin andsampling frequency of 2048Hz Time domainwaveforms andamplitude spectrums are displayed in Figure 14
According to amplitude of the leading trebling frequencyofD11 andD12 in Figure 14 1X corresponded to the second setof eigenvalues (third and fourth eigenvalues) 2X paralleledthe third set of eigenvalues (fifth and sixth eigenvalues) and
10 Shock and Vibration
D11-1080 rpmx 10
minus5
minus5minus4minus3minus2minus1
012345
X (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(a)
Power-line Interference50Hz
0733404771
Order 1 Order 2
0250502342
Order 6Order 7
D11-1080 rpm
01651 Order 11
18Hz36Hz54Hz72Hz90Hz
x 10minus5
1times
2times
3times
4times
5times
1times
2times
3times 4times5times
002040608
1
X (m
)
50 100 150 200 250 300 350 4000Frequency (Hz)
(b)
D12-1080 rpmx 10
minus5
minus6
minus4
minus2
0246
Y (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(c)
D12-1080 rpmPower-line Interference
50Hz
0658105228
Order 1 Order 2
0277101226
Order 5 Order 12
01948 Order 8
18Hz36Hz54Hz72Hz90Hz
x 10minus5
1times
2times
3times
4times
5times
1times
2times
3times
4times5times
0
05
1
Y (m
)
50 100 150 200 250 300 350 4000Frequency (Hz)
(d)
Figure 9The time domain and frequency domain of theD11 andD12 sensors in theA-end face (a)The time domain of D11 (b)The frequencydomain of D11 (c) The time domain of D12 (d) The frequency domain of D12
x 10minus5
x 10minus5
minus5
minus4
minus3
minus2
minus1
0
1
2
3
4
5
D12
minusY
(m)
minus6 minus2minus4 2 4 60D11minusX (m)
Figure 10 A-end face of the D11 and D12 sensors to collect the signal directly synthesized axis orbit
(12751) (22748)e first group
(31243) (41241)e second group
D11minus1080 rpm
x 10minus8
0
1
2
3
4
Eige
nval
ue o
f PCA
(m)
10 20 30 40 500Number of eigenvalues q
(a)
(12247) (2241)
(314) (41396)
D12minus1080 rpme first group
e second group
x 10minus8
005
115
225
Eige
nval
ue o
f PCA
(m)
10 20 30 40 500Number of eigenvalues q
(b)
Figure 11The eigenvalues distribution of covariancematrixC (a)The eigenvalues distribution of D11 signal (b)The eigenvalues distributionof D12 signal
Shock and Vibration 11
D11minus1080 rpmx 10
minus5
minus15
minus10
minus05
0051015
X (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(a)
D12minus1080 rpmx 10
minus5
minus10
minus05
0051015
Y (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(b)
D11minus1080 rpm(1807287)
(360472)
x 10minus5
002040608
1
X (m
)
50 100 150 2000Frequency (Hz)
(c)
(3605105)(1806575)
D12minus1080 rpmx 10
minus5
002040608
1
Y (m
)
50 100 150 2000Frequency (Hz)
(d)
Figure 12 Extracting the time and frequency spectrums of the first two frequencies of D11 and D12 (a)The time domain of D11 (b)The timedomain of D12 (c) The frequency domain of D11 (d) The frequency domain of D12
W1
x 10minus5
x 10minus5
minus1
minus05
0
05
1
15
D12
minusY
(m)
minus1minus15 minus05 05 10 15D11minusX (m)
(a)0
051
152
0
2
t (s)D11minusX (m)
minus2
x 10minus5
x 10minus5
minus1
minus05
0
05
1
15
D12
minusY
(m)
(b)
Figure 13 The synthesis of axis orbit is to extract the first two times fundamental frequencies of the D11 and D12 signals by the CFSM-PCA(a) The axis orbit (b) Relationship between axis orbit and time
3X matched the first set of eigenvalues (first and secondeigenvalues)
As displayed in Figure 15 eigenvalues of 1X and 3Xwere large thus 1X and 3X were utilized to synthesize rotororbit firstly Purification results and orbit of shaft center arerevealed in Figures 16 and 17 respectively The legible orbitshowed a typical plum blossom shape
It can be seen from Figures 16 and 17 that the filteringeffect was also quite obvious The noise and power frequency50 Hz as well as its harmonic interference could be removedsuccessfully Moreover amplitude of refined signal was closeto that of original one and the reconstructed time domainsignal seemed to be steady without any fluctuation
As shown in Figures 14(c) and 14(d) amplitudes of 1X 2Xand 3X in amplitude spectrum of raw signal were largeThus1X 2X and 3X were also extracted to synthesize axis orbit
Time domain frequency domain and orbit curve are shownin Figures 18 and 19 respectively
Axis orbit synthesized by the leading three frequenciesexhibited a quincunx shape which is basically the sameto that synthesized by 1X and 3X frequency componentsindicating that static and dynamic parts friction may exist inA-side
44 Comparison Harmonic Wavelet with Wavelet Packet
(1) Comparison of CFSM-PCA with Harmonic Wavelet Onecommon used method for signal processing harmonicwavelet algorithm proposed by Newland is an improvedwavelet algorithm [29] It can segment entire frequencyband infinitely and avoid the shortcoming of downsamplingmethod of binary wavelet and binary wavelet packet [22 23]
12 Shock and Vibration
D11minus2770 rpmx 10
minus4
minus1
minus05
0
05
1
D11
-X (m
)
1000 2000 3000 40000Sampling numbers i
(a)
D12minus2770 rpmx 10
minus4
minus1
minus05
0
05
1
D12
-Y (m
)
1000 2000 3000 40000Sampling numbers i
(b)
Power-Line interference 50and 150Hz
1X2X
3X 1010D11minus2770 rpm
08441166
1X2X3X
x 10minus5
200 400 600 800 10000f (Hz)
0
05
10
15
D11
-X (m
)
(c)
1X
2X
3XPower-Line interference 50
and 150Hz
D12minus2770 rpm117309021227
1X2X3X
x 10minus5
0
05
10
15
D12
-Y (m
)
200 400 600 800 10000f (Hz)
(d)
Figure 14 The time and frequency domains of the D11 and D12 sensors (a) The time domain of D11 (b) The time domain of D12 (c) Thefrequency domain of D11 (d) The frequency domain of D12
(11489) (21482)e first group
(31237) (41220)e second group
(30348) (40347)e second group
D11minus2770 rpmx 10
minus7
10 20 30 400 50Number of eigenvalues q
0
05
1
15
Eige
nval
ue o
f PCA
(m)
(a)
D12minus2770 rpm(11688) (21687)
e first group
e second group
(30414) (40413)e second group
(31673) (41664)
x 10minus7
0
05
1
15
2
Eige
nval
ue o
f PCA
(m)
10 20 30 40 500Number of eigenvalues q
(b)
Figure 15 The eigenvalues distribution of covariance matrix C (a) The eigenvalues distribution of D11 (b) The eigenvalues distribution ofD12
1X and its frequency doubling signal can be easily extractedto form axis orbit of revolving test bearing
Axis orbit was fabricated by 1X and 2Xof data of example 1(Section 42) through harmonic wavelet algorithm and theresults are demonstrated in Figure 20 The shape of axisorbit was trapped banana-like which is consistent with theresult obtained via CFSM-PCA It should be noted thatorbit in Figure 13 was more obvious than that in Figure 20ie W1ltW2 Meanwhile the amplitude fluctuation of timedomain signal was large (Figures 20(a) and 20(b)) Themajor reason for this phenomenon is that the harmonicwavelet which is a band filter in frequency domain inevitablyextracts the noise near the feature frequency Furthermorethis phenomenon also demonstrates that harmonic waveletis subject to the Heisenberg uncertainty principle ie thephenomenon of the amplitude leakage exists
(2) Comparison of CFSM-PCA with Wavelet Packet Besidesharmonic wavelets wavelet packet is also used to purify axisorbit [19ndash21] Purification results of axis orbit via Daubechieswavelet packet in Figure 21 exhibit that axis orbit was diver-gent seriously This was worse than that of harmonic waveletie W1ltW2ltW3The phenomenon of severe energy leakageoccurred during the extraction of 1X and 2X which cancause divergence of axis orbit Moreover the bandpass filterof wavelet packet also accounts for the divergent orbits
5 A Single Frequency Filtrationwith CFSM-PCA
The application in single frequency filtration is discussedin this section Taking D11 experimental data of example
Shock and Vibration 13
D11minus2770 rpmx 10
minus5
0 1000500Sampling numbers i
minus4
minus2
0
2
4
D11
-X (m
)
(a)
D12minus2770 rpmx 10
minus5
minus4
minus2
0
2
4
D12
-Y (m
)
500 10000Sampling numbers i
(b)
(1375 1185)(46 1009)
D11minus2770 rpmx 10
minus5
100 200 300 400 5000f (Hz)
0
05
1
15
D11
-X (m
)
(c)
(1375 1218)(46 1164)
D12minus2770 rpmx 10
minus5
100 200 300 400 5000f (Hz)
0
05
1
15
D12
-Y (m
)
(d)
Figure 16 Extracting the frequency spectrums of the first two frequencies of D11 and D12 by PCA algorithm (a) The time domain of D11(b) The time domain of D12 (c) The frequency spectrum of D11 (d) The frequency spectrum of D12
x 10minus5
x 10minus5
minus4
minus3
minus2
minus1
0
1
2
3
4
D12
-Y (m
)
210 3 4minus2minus3 minus1minus4
D11-X (m)
(a)0
0204
0608
10
24
t (s)D11-X (m) minus2
minus4
x 10minus5
x 10minus5
minus4
minus2
0
2
4
D12
-Y (m
)
(b)
Figure 17 The synthesis of axis orbit is to extract the first two times fundamental frequencies of the D11 and D12 signals by the CFSM-PCA(a) The axis orbit (b) Relationship between axis orbit and time
1 (Section 42) as a demo we attempt to filter the powerfrequency interference (50 Hz) from D11 by employingCFSM-PCA It can be seen from Figure 9(b) that sequenceof amplitude of 50 Hz in raw signal was 3 so eigenvec-tors corresponding to 5th and 6th eigenvalues in eigen-value distribution map were used to reconstruct Accordingto (8) the filtered signal (Figure 22) without 50 Hz was
obtained by subtracting reconstructed signal from originalone
Results in Figure 22 show that this algorithm filters out50 Hz power frequency interference and has no influenceon the other frequency components of raw signal demon-strating that this method has a potential application in notchfilter
14 Shock and Vibration
D11minus2770 rpm
x 10minus5
minus4
minus2
0
2
4
D11
-X (m
)
200 400 600 800 10000Sampling numbers i
(a)
D12minus2770 rpm
x 10minus5
0 400 600 800 1000200Sampling numbers i
minus4
minus2
0
2
4
D12
-Y (m
)
(b)
(1375 1185)(46 1009)
(92 0831)
D11minus2770 rpmx 10
minus5
00
05
10
15
D11
-X (m
)
100 200 300 400 5000f (Hz)
(c)
(1375 1218)(46 1163)
(92 0898)
D12minus2770 rpmx 10
minus5
00
05
10
15
D12
-Y (m
)
100 200 300 400 5000f (Hz)
(d)
Figure 18 Extracting the time and frequency spectrums of the first three frequencies of D11 and D12 by PCA algorithm (a)The time domainof D11 (b) The time domain of D12 (c) The frequency spectrum of D11 (d) The frequency spectrum of D12
x 10minus5
x 10minus5
minus4
minus3
minus2
minus1
0
1
2
3
4
D12
-Y (m
)
minus3 minus2 minus1minus4 1 2 3 40D11-X (m)
(a)
1008
0604
0200
02
4D11-X (m)
t (s)
minus2
minus4
x 10minus5
x 10minus5
minus4
minus2
0
2
4
D12
-Y (m
)
(b)
Figure 19 The synthesis of axis orbit is to extract the first three times fundamental frequencies of the D11 and D12 by the CFSM-PCA (a)The axis orbit (b) Relationship between axis orbit and time
6 Conclusion
The selection of effective eigenvalues in feature distributionchart of covariance matrix C is crucial in PCA algorithm Inthis paper the relationships between effective eigenvalues andsignal frequency and amplitude are discovered and a series ofconclusions are summarized
First in Hankel matrix mode each valid frequencycomponent of signal only produces two adjacent nonzeroeigenvalues Second the number of valid eigenvalues incharacteristic distribution chart which is independent ofnumerical values of 119891119894 119860 119894 and 120593119894 is related to the number offrequency componentsThird the sequence of signal effectiveeigenvalues in characteristic distribution map is determined
Shock and Vibration 15
D11-1080 rpm
x 10minus5
minus20
minus15
minus05
0
05
10
15
20
D11
-X (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(a)
D12-1080 rpmx 10
minus5
minus15
minus1
minus05
0
05
1
15
D12
-Y (m
)
500 1000 1500 2000 2500 30000 40003500Sampling numbers i
(b)
(16 07335)
(32 04771)
D11-1080 rpm
x 10minus5
10 20 30 40 50 60 70 80 90 1000f (Hz)
0010203040506070809
1
D11
-X (m
)
(c)
(16 06571)
(32 05228)
D12-1080 rpm
x 10minus5
0010203040506070809
1
D12
-Y (m
)
10 20 30 40 50 60 70 80 90 1000f (Hz)
(d)
W2
x 10minus5
x 10minus5
minus15
minus1
minus05
0
05
1
15
Y(D
12) (
m)
minus1 minus05minus15 05 1 15 20X(D11) (m)
(e)
Figure 20The purification effect of harmonic wavelets (a)The time domain of D11 (b)The time domain of D12 (c)The frequency spectrumof D11 (d) The frequency spectrum of D12 (e) The orbit is to extract the first 2 frequencies
by amplitude 119860 119894 of signal frequency component The largermagnitude of amplitude is the greater eigenvalues are andthe higher arrangement will be
Based on aforesaid principle a new CFSM-PCA wasdeveloped and has become an effective tool to extract oreliminate specific characteristic (single frequency) even dif-ference of two frequencies is only 1 Hz Purification of rotororbit using this algorithm shows significant improvement
than those of existing methods such as wavelet packet andharmonic wavelet packet Although the proposed CFSM-PCA is effective some drawbacks are unavoidable Forexample this method suffers from similar issue to otheralgorithms that is the reconstruction error increaseswith thedecrease of SNR In addition the applications of the CFSM-PCA algorithm in purification of speech recognition bearingfault diagnosis and fault diagnosis of power system should
16 Shock and Vibration
D11-1080 rpmx 10
minus5
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
minus20
minus15
minus05
0
05
10
15
20
D11
-X (m
)
(a)
D12-1080 rpmx 10
minus5
minus15
minus1
minus05
0
05
1
15
D12
-Y (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(b)
(16 06557)
(32 04381)
D11-1080 rpmx 10
minus5
10020 30 40 50 60 70 80 90100f (Hz)
0010203040506070809
1
D11
-X (m
)
(c)
(16 06452)(32 04927)
D12-1080 rpm
x 10minus5
0010203040506070809
1
D12
-Y (m
)
9020 30 40 50 60 70 8010 1000f (Hz)
(d)
W3
x 10minus5
x 10minus5
minus15
minus1
minus05
0
05
1
15
Y(D
12) (
m)
minus15 minus05minus1 05 1 15 20X(D11) (m)
(e)
Figure 21 The purification effect of wavelet packet (a) The time domain of D11 (b) The time domain of D12 (c) The frequency spectrum ofD11 (d) The frequency spectrum of D12 (e) The orbit is to extract the first 2 frequencies
be further exploredWe fully believe that the CFSM-PCAwillsparkle in diverse fields
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work is supported by the National High TechnologyResearch andDevelopment Program of China (863 Program)
Shock and Vibration 17
(18 0734)
(36 04634)
(54 02448) (72 02351)(90 01646)
x 10minus5
50 100 150 2000f (Hz)
0
02
04
06
08
1
D11
minusX
(m)
Figure 22 Frequency spectrum of D11 signal after filtering 50Hz
(no 2015AA043005) and the National Natural Science Foun-dation of China (no 51375187)
References
[1] H Abdi and L J Williams ldquoPrincipal component analysisrdquoWiley Interdisciplinary Reviews Computational Statistics vol 2no 4 pp 433ndash459 2010
[2] A Widodo B Yang and T Han ldquoCombination of independentcomponent analysis and support vectormachines for intelligentfaults diagnosis of induction motorsrdquo Expert Systems withApplications vol 32 no 2 pp 299ndash312 2007
[3] S Wold K Esbensen and P Geladi ldquoPrincipal componentanalysisrdquo Chemometrics and Intelligent Laboratory Systems vol2 no 1ndash3 pp 37ndash52 1987
[4] M Kirby and L Sirovich ldquoApplication of the Karhunen-Loeve procedure for the characterization of human facesrdquo IEEETransactions on Pattern Analysis and Machine Intelligence vol12 no 1 pp 103ndash108 1990
[5] J H Xi Y Z Han and R H Su ldquoNew fault diagnosis methodfor rolling bearing based on PCArdquo in Proceedings of the 25thChinese Control and Decision Conference CCDC rsquo13 pp 4123ndash4127 IEEE May 2013
[6] A Malhi and R X Gao ldquoPCA-based feature selection schemefor machine defect classificationrdquo IEEE Transactions on Instru-mentation and Measurement vol 53 no 6 pp 1517ndash1525 2004
[7] S Xu X Jiang J Huang S Yang and X Wang ldquoBayesianwavelet PCA methodology for turbomachinery damage diag-nosis under uncertaintyrdquo Mechanical Systems and Signal Pro-cessing vol 80 pp 1ndash18 2016
[8] W Sun J Chen and J Li ldquoDecision tree and PCA-based faultdiagnosis of rotatingmachineryrdquoMechanical Systems and SignalProcessing vol 21 no 3 pp 1300ndash1317 2007
[9] Z X Li X P Yan C Q Yuan Z Peng and L Li ldquoVirtual pro-totype and experimental research on gear multi-fault diagnosisusing wavelet-autoregressive model and principal componentanalysismethodrdquoMechanical Systems and Signal Processing vol25 no 7 pp 2589ndash2607 2011
[10] K Y Shao M M Cai and G F Zhao ldquoRolling bearing faultdiagnosis based onwavelet energy spectrum PCA andPNNrdquo inProceedings of the 26th Chinese Control andDecisionConferenceCCDC rsquo14 pp 800ndash804 IEEE 2014
[11] T Qiu Q F Zhang and Y J Ding ldquoNonlinear sensor faultdetection and data rebuilding based on principle componentanalysisrdquo Journal of Tsinghua University vol 46 no 5 pp 708ndash711 2006
[12] Y K Gu Z X Yang and F L Zhu ldquoGearbox fault feature fusionbased on principal component analysisrdquo China MechanicalEngineering vol 26 no 11 pp 1532ndash1537 2015
[13] R Dunia S J Qin T F Edgar and T J McAvoy ldquoIdentificationof faulty sensors using principal component analysisrdquo AIChEJournal vol 42 no 10 pp 2797ndash2811 1996
[14] E P de Moura C R Souto A A Silva and M A S IrmaoldquoEvaluation of principal component analysis and neural net-work performance for bearing fault diagnosis from vibrationsignal processed by RS and DF analysesrdquo Mechanical Systemsand Signal Processing vol 25 no 5 pp 1765ndash1772 2011
[15] S Golestan M Monfared F D Freijedo and J M GuerreroldquoDynamics assessment of advanced single-phase PLL struc-turesrdquo IEEE Transactions on Industrial Electronics vol 60 no6 pp 2167ndash2177 2013
[16] A Lima H Zen Y Nankaku C Miyajima K Tokuda andT Kitamura ldquoOn the use of kernel PCA for feature extractionin speech recognitionrdquo IEICE Transaction on Information andSystems vol 12 no 87 pp 2802ndash2811 2004
[17] J D Zheng ldquoImproved hilbert-huang transform and its appli-cations to rolling bearing fault diagnosisrdquo Journal of MechanicalEngineering vol 1 no 51 pp 138ndash145 2015
[18] G He K Ding and H Lin ldquoFault feature extraction of rollingelement bearings using sparse representationrdquo Journal of Soundand Vibration vol 366 pp 514ndash527 2016
[19] J G Yang S B Xia and Y G Liu ldquoWavelet denoising and itsapplication in extraction of shaft centerline orbit featuresrdquoJournal of Harbin Institute of Technology vol 31 no 5 pp 52ndash541999 (Chinese)
[20] J A Duan and X D Zhang ldquoFeature extraction from the orbitof the center of a rotor based on wavelet transformrdquo Journal ofVibration Measurement amp Diagnosis vol 17 no 01 pp 31ndash341997
[21] J Han D X Jiang and W D Ning ldquoApplication of optimalwavelet packets in failure signal extraction of rotor orbitrdquoTurbine Technology vol 43 no 3 pp 133ndash136 2001
[22] W B Zhang X J Zhou andY Lin ldquoRefinement of rotor centerrsquosorbit by a harmonic window methodrdquo Journal of VibrationMeasurement amp Diagnosis vol 30 no 1 pp 87ndash90 2010
[23] S M Li G Z Lu and Q Y Xu ldquoOn obtaining accuraterotor sub-frequency signal with harmonic waveletrdquo Journal ofNorthwestern Polytechincal University vol 19 no 2 pp 220ndash224 2001
[24] F J Wu and L S Qu ldquoAn improved method for restraining theend effect in empiricalmode decomposition and its applicationsto the fault diagnosis of large rotating machineryrdquo Journal ofSound and Vibration vol 314 no 3-5 pp 586ndash602 2008
[25] W G Liu ldquoResearch on method of purification of shaft orbitbased on singular value decompositionrdquo Technical Acousticsvol 5 no 35 pp 30ndash35 2016
[26] X Z Zhao and B Y Ye ldquoThe influence of formation manner ofcomponent on signal processingrdquo Shang Hai Jiao Tong Univer-sity vol 45 no 3 pp 368ndash374 2011
[27] X Z Zhao B Y Ye and T J Chen ldquoPrinciple of singularitydetection based on SVD and its applicationrdquo Journal of Vibra-tion and Shock vol 6 no 27 pp 7ndash10 2008
[28] X Z Zhao B Y Ye and T J Chen ldquoDifference spectrum theoryof singular value and its application to the fault diagnosis ofheadstock of latherdquo Journal of Mechanical Engineering vol 46pp 100ndash108 2010
[29] I A Kougioumtzoglou and P D Spanos ldquoAn identificationapproach for linear and nonlinear time-variant structural sys-tems via harmonic waveletsrdquo Mechanical Systems and SignalProcessing vol 37 no 1-2 pp 338ndash352 2013
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8 Shock and Vibration
the primary signalthe recovered signal
minus2
minus1
0
1
2A
mpl
itude
100 200 300 400 5000Sampling numbers i
(a)
(200977)
50 1000f (Hz)
002
04
06
08
1
Am
plitu
de
(b)
the primary signalthe recovered signal
minus1
minus05
0
05
1
Am
plitu
de
100 200 300 400 5000Sampling numbers i
(c)
(800786)
002
04
06
08
1
Am
plitu
de
50 1000f (Hz)
(d)
the primary signalthe recovered signal
100 200 300 400 5000Sampling numbers i
minus1
minus05
0
05
1
Am
plitu
de
(e)
(490688)
50 1000f (Hz)
002
04
06
08
1
Am
plitu
de
(f)
the primary signalthe recovered signal
minus05
0
05
Am
plitu
de
100 200 300 400 5000Sampling numbers i
(g)
(500463)
002
04
06
08
1
Am
plitu
de
50 1000f (Hz)
(h)
Figure 6 The time and frequency domains of reconstructed signal (a) The time domain of the reconstructed signal via the first set ofeigenvalues (b) Amplitude spectrum of Figure 6(a) (c) The time domain of the reconstructed signal using the second set of eigenvalues(d) Amplitude spectrum of Figure 6(c) (e) The time domain of the reconstructed signal using the third set of eigenvalues (f) Amplitudespectrum of Figure 6(e) (g) The time domain of the reconstructed signal using the fourth set of eigenvalues (h) Amplitude spectrum ofFigure 6(g)
Shock and Vibration 9
A B
Servo Motor Unloading device
Coupling
Sliding Bearing
Rotor
Large Platform
Small Platform
Air spring
BeltSliding Bearing
Figure 7 Large rot or vibration test bed
Displacement SensorA-end Face B-end Face
D13 D14
D12D11
Figure 8 Schematic and installation of eddy current displacement sensor
Rotor orbits indicate the symptoms of malfunction andare related to variation of input force or dynamic stiffnessGenerally different curve shapes of rotor orbit in a senserepresent different fault types [19ndash22 29] For example insideldquo8rdquo outside ldquo8rdquo and petal-shaped curves are omens of oil-filmwhirl fault misalignment fault and rubbing fault of rotorsystem respectivelyTherefore extraction of rotor orbit playsa significant role in rotating machinery diagnostics In thissection a simple and efficient CFSM-PCA is applied to purifyrotor orbit precisely
41 Experimental Rigs Figure 7 shows the large rotor vibra-tion test bed system which is constructed by our group Inorder to real-timemonitor rotor working station two KamanKD2306-1S eddy current proximity probes are installedon individual side in orthogonal manners (Figure 8) Theexperimental data were collected through LMS Test Lab
42 The First Group Sample Analysis Experimental datastems from D11 and D12 eddy current displacement sensorin side A 4096 data points were collected with test bed speedof 1080 rmin and sampling frequency of 2048 Hz The timedomain waveforms amplitude spectrums and the order ofamplitudes were obtained after filtering DC component viaFFT (Figure 9)
As demonstrated in Figure 9 raw signal was not onlyaffected by noise but also interfered by power frequency 50Hz and its harmonics frequency The energy of signal mainlyconcentrated on the leading two octaves
Rotor orbit was produced by combining spindle dis-placement signal D11 with signal D12 (Figure 10) X-axispresented raw signal D11 and Y-axis referred to signal D12Manipulating status of test bed is difficult to know from thisunprocessed rotor orbit Next the rotor orbit was attemptedto purify by employing PCA
Generally rotor orbit synthesized by 1X and 2X is cred-itable Orbits included high harmonics incline to becomecomplex and even turn messy Hence extraction of 1Xand 2X of raw signal becomes more important [21 22]Eigenvalue distribution of covariance matrix of D11 and D12signals are displayed in Figure 11 According to amplitudeof each frequency of D11 and D12 in Figure 9 1X and 2Xcorresponded to the first (first and second eigenvalues) andthe second (third and fourth eigenvalues) set of eigenvaluesrespectively The leading four eigenvalues in Figure 11 werereconstructed through CFSM-PCA and results are illustratedin Figure 12 Generated 1X and 2X were legible without effectby noise and disturbance from frequency 50 Hz as well as itsharmonics Meanwhile amplitude of the extracted signal wasclose to that of 1X and 2X in raw signal indicating that thereis no spectral leakage issue
The feature spectrums continued to be separated andeigenvalue distributionmaps of covariance matrix of D11 andD12 are shown in Figure 15
Rotor orbit generated by the purified 1X and 2X is shownin Figure 13The rotor orbit was a caved-in banana-like curvedemonstrating that sideA (near themotor) potentially suffersfrom misalignment fault [18 19] The explicit rotor orbit alsovalidates the high efficiency of this proposed method
43 The Second Group Sample Analysis In this case exper-imental data of D11 and D12 were analyzed when test bedruns steadily for a period of time Similarly 4096 datapoints were measured with test bed speed of 2770 rmin andsampling frequency of 2048Hz Time domainwaveforms andamplitude spectrums are displayed in Figure 14
According to amplitude of the leading trebling frequencyofD11 andD12 in Figure 14 1X corresponded to the second setof eigenvalues (third and fourth eigenvalues) 2X paralleledthe third set of eigenvalues (fifth and sixth eigenvalues) and
10 Shock and Vibration
D11-1080 rpmx 10
minus5
minus5minus4minus3minus2minus1
012345
X (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(a)
Power-line Interference50Hz
0733404771
Order 1 Order 2
0250502342
Order 6Order 7
D11-1080 rpm
01651 Order 11
18Hz36Hz54Hz72Hz90Hz
x 10minus5
1times
2times
3times
4times
5times
1times
2times
3times 4times5times
002040608
1
X (m
)
50 100 150 200 250 300 350 4000Frequency (Hz)
(b)
D12-1080 rpmx 10
minus5
minus6
minus4
minus2
0246
Y (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(c)
D12-1080 rpmPower-line Interference
50Hz
0658105228
Order 1 Order 2
0277101226
Order 5 Order 12
01948 Order 8
18Hz36Hz54Hz72Hz90Hz
x 10minus5
1times
2times
3times
4times
5times
1times
2times
3times
4times5times
0
05
1
Y (m
)
50 100 150 200 250 300 350 4000Frequency (Hz)
(d)
Figure 9The time domain and frequency domain of theD11 andD12 sensors in theA-end face (a)The time domain of D11 (b)The frequencydomain of D11 (c) The time domain of D12 (d) The frequency domain of D12
x 10minus5
x 10minus5
minus5
minus4
minus3
minus2
minus1
0
1
2
3
4
5
D12
minusY
(m)
minus6 minus2minus4 2 4 60D11minusX (m)
Figure 10 A-end face of the D11 and D12 sensors to collect the signal directly synthesized axis orbit
(12751) (22748)e first group
(31243) (41241)e second group
D11minus1080 rpm
x 10minus8
0
1
2
3
4
Eige
nval
ue o
f PCA
(m)
10 20 30 40 500Number of eigenvalues q
(a)
(12247) (2241)
(314) (41396)
D12minus1080 rpme first group
e second group
x 10minus8
005
115
225
Eige
nval
ue o
f PCA
(m)
10 20 30 40 500Number of eigenvalues q
(b)
Figure 11The eigenvalues distribution of covariancematrixC (a)The eigenvalues distribution of D11 signal (b)The eigenvalues distributionof D12 signal
Shock and Vibration 11
D11minus1080 rpmx 10
minus5
minus15
minus10
minus05
0051015
X (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(a)
D12minus1080 rpmx 10
minus5
minus10
minus05
0051015
Y (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(b)
D11minus1080 rpm(1807287)
(360472)
x 10minus5
002040608
1
X (m
)
50 100 150 2000Frequency (Hz)
(c)
(3605105)(1806575)
D12minus1080 rpmx 10
minus5
002040608
1
Y (m
)
50 100 150 2000Frequency (Hz)
(d)
Figure 12 Extracting the time and frequency spectrums of the first two frequencies of D11 and D12 (a)The time domain of D11 (b)The timedomain of D12 (c) The frequency domain of D11 (d) The frequency domain of D12
W1
x 10minus5
x 10minus5
minus1
minus05
0
05
1
15
D12
minusY
(m)
minus1minus15 minus05 05 10 15D11minusX (m)
(a)0
051
152
0
2
t (s)D11minusX (m)
minus2
x 10minus5
x 10minus5
minus1
minus05
0
05
1
15
D12
minusY
(m)
(b)
Figure 13 The synthesis of axis orbit is to extract the first two times fundamental frequencies of the D11 and D12 signals by the CFSM-PCA(a) The axis orbit (b) Relationship between axis orbit and time
3X matched the first set of eigenvalues (first and secondeigenvalues)
As displayed in Figure 15 eigenvalues of 1X and 3Xwere large thus 1X and 3X were utilized to synthesize rotororbit firstly Purification results and orbit of shaft center arerevealed in Figures 16 and 17 respectively The legible orbitshowed a typical plum blossom shape
It can be seen from Figures 16 and 17 that the filteringeffect was also quite obvious The noise and power frequency50 Hz as well as its harmonic interference could be removedsuccessfully Moreover amplitude of refined signal was closeto that of original one and the reconstructed time domainsignal seemed to be steady without any fluctuation
As shown in Figures 14(c) and 14(d) amplitudes of 1X 2Xand 3X in amplitude spectrum of raw signal were largeThus1X 2X and 3X were also extracted to synthesize axis orbit
Time domain frequency domain and orbit curve are shownin Figures 18 and 19 respectively
Axis orbit synthesized by the leading three frequenciesexhibited a quincunx shape which is basically the sameto that synthesized by 1X and 3X frequency componentsindicating that static and dynamic parts friction may exist inA-side
44 Comparison Harmonic Wavelet with Wavelet Packet
(1) Comparison of CFSM-PCA with Harmonic Wavelet Onecommon used method for signal processing harmonicwavelet algorithm proposed by Newland is an improvedwavelet algorithm [29] It can segment entire frequencyband infinitely and avoid the shortcoming of downsamplingmethod of binary wavelet and binary wavelet packet [22 23]
12 Shock and Vibration
D11minus2770 rpmx 10
minus4
minus1
minus05
0
05
1
D11
-X (m
)
1000 2000 3000 40000Sampling numbers i
(a)
D12minus2770 rpmx 10
minus4
minus1
minus05
0
05
1
D12
-Y (m
)
1000 2000 3000 40000Sampling numbers i
(b)
Power-Line interference 50and 150Hz
1X2X
3X 1010D11minus2770 rpm
08441166
1X2X3X
x 10minus5
200 400 600 800 10000f (Hz)
0
05
10
15
D11
-X (m
)
(c)
1X
2X
3XPower-Line interference 50
and 150Hz
D12minus2770 rpm117309021227
1X2X3X
x 10minus5
0
05
10
15
D12
-Y (m
)
200 400 600 800 10000f (Hz)
(d)
Figure 14 The time and frequency domains of the D11 and D12 sensors (a) The time domain of D11 (b) The time domain of D12 (c) Thefrequency domain of D11 (d) The frequency domain of D12
(11489) (21482)e first group
(31237) (41220)e second group
(30348) (40347)e second group
D11minus2770 rpmx 10
minus7
10 20 30 400 50Number of eigenvalues q
0
05
1
15
Eige
nval
ue o
f PCA
(m)
(a)
D12minus2770 rpm(11688) (21687)
e first group
e second group
(30414) (40413)e second group
(31673) (41664)
x 10minus7
0
05
1
15
2
Eige
nval
ue o
f PCA
(m)
10 20 30 40 500Number of eigenvalues q
(b)
Figure 15 The eigenvalues distribution of covariance matrix C (a) The eigenvalues distribution of D11 (b) The eigenvalues distribution ofD12
1X and its frequency doubling signal can be easily extractedto form axis orbit of revolving test bearing
Axis orbit was fabricated by 1X and 2Xof data of example 1(Section 42) through harmonic wavelet algorithm and theresults are demonstrated in Figure 20 The shape of axisorbit was trapped banana-like which is consistent with theresult obtained via CFSM-PCA It should be noted thatorbit in Figure 13 was more obvious than that in Figure 20ie W1ltW2 Meanwhile the amplitude fluctuation of timedomain signal was large (Figures 20(a) and 20(b)) Themajor reason for this phenomenon is that the harmonicwavelet which is a band filter in frequency domain inevitablyextracts the noise near the feature frequency Furthermorethis phenomenon also demonstrates that harmonic waveletis subject to the Heisenberg uncertainty principle ie thephenomenon of the amplitude leakage exists
(2) Comparison of CFSM-PCA with Wavelet Packet Besidesharmonic wavelets wavelet packet is also used to purify axisorbit [19ndash21] Purification results of axis orbit via Daubechieswavelet packet in Figure 21 exhibit that axis orbit was diver-gent seriously This was worse than that of harmonic waveletie W1ltW2ltW3The phenomenon of severe energy leakageoccurred during the extraction of 1X and 2X which cancause divergence of axis orbit Moreover the bandpass filterof wavelet packet also accounts for the divergent orbits
5 A Single Frequency Filtrationwith CFSM-PCA
The application in single frequency filtration is discussedin this section Taking D11 experimental data of example
Shock and Vibration 13
D11minus2770 rpmx 10
minus5
0 1000500Sampling numbers i
minus4
minus2
0
2
4
D11
-X (m
)
(a)
D12minus2770 rpmx 10
minus5
minus4
minus2
0
2
4
D12
-Y (m
)
500 10000Sampling numbers i
(b)
(1375 1185)(46 1009)
D11minus2770 rpmx 10
minus5
100 200 300 400 5000f (Hz)
0
05
1
15
D11
-X (m
)
(c)
(1375 1218)(46 1164)
D12minus2770 rpmx 10
minus5
100 200 300 400 5000f (Hz)
0
05
1
15
D12
-Y (m
)
(d)
Figure 16 Extracting the frequency spectrums of the first two frequencies of D11 and D12 by PCA algorithm (a) The time domain of D11(b) The time domain of D12 (c) The frequency spectrum of D11 (d) The frequency spectrum of D12
x 10minus5
x 10minus5
minus4
minus3
minus2
minus1
0
1
2
3
4
D12
-Y (m
)
210 3 4minus2minus3 minus1minus4
D11-X (m)
(a)0
0204
0608
10
24
t (s)D11-X (m) minus2
minus4
x 10minus5
x 10minus5
minus4
minus2
0
2
4
D12
-Y (m
)
(b)
Figure 17 The synthesis of axis orbit is to extract the first two times fundamental frequencies of the D11 and D12 signals by the CFSM-PCA(a) The axis orbit (b) Relationship between axis orbit and time
1 (Section 42) as a demo we attempt to filter the powerfrequency interference (50 Hz) from D11 by employingCFSM-PCA It can be seen from Figure 9(b) that sequenceof amplitude of 50 Hz in raw signal was 3 so eigenvec-tors corresponding to 5th and 6th eigenvalues in eigen-value distribution map were used to reconstruct Accordingto (8) the filtered signal (Figure 22) without 50 Hz was
obtained by subtracting reconstructed signal from originalone
Results in Figure 22 show that this algorithm filters out50 Hz power frequency interference and has no influenceon the other frequency components of raw signal demon-strating that this method has a potential application in notchfilter
14 Shock and Vibration
D11minus2770 rpm
x 10minus5
minus4
minus2
0
2
4
D11
-X (m
)
200 400 600 800 10000Sampling numbers i
(a)
D12minus2770 rpm
x 10minus5
0 400 600 800 1000200Sampling numbers i
minus4
minus2
0
2
4
D12
-Y (m
)
(b)
(1375 1185)(46 1009)
(92 0831)
D11minus2770 rpmx 10
minus5
00
05
10
15
D11
-X (m
)
100 200 300 400 5000f (Hz)
(c)
(1375 1218)(46 1163)
(92 0898)
D12minus2770 rpmx 10
minus5
00
05
10
15
D12
-Y (m
)
100 200 300 400 5000f (Hz)
(d)
Figure 18 Extracting the time and frequency spectrums of the first three frequencies of D11 and D12 by PCA algorithm (a)The time domainof D11 (b) The time domain of D12 (c) The frequency spectrum of D11 (d) The frequency spectrum of D12
x 10minus5
x 10minus5
minus4
minus3
minus2
minus1
0
1
2
3
4
D12
-Y (m
)
minus3 minus2 minus1minus4 1 2 3 40D11-X (m)
(a)
1008
0604
0200
02
4D11-X (m)
t (s)
minus2
minus4
x 10minus5
x 10minus5
minus4
minus2
0
2
4
D12
-Y (m
)
(b)
Figure 19 The synthesis of axis orbit is to extract the first three times fundamental frequencies of the D11 and D12 by the CFSM-PCA (a)The axis orbit (b) Relationship between axis orbit and time
6 Conclusion
The selection of effective eigenvalues in feature distributionchart of covariance matrix C is crucial in PCA algorithm Inthis paper the relationships between effective eigenvalues andsignal frequency and amplitude are discovered and a series ofconclusions are summarized
First in Hankel matrix mode each valid frequencycomponent of signal only produces two adjacent nonzeroeigenvalues Second the number of valid eigenvalues incharacteristic distribution chart which is independent ofnumerical values of 119891119894 119860 119894 and 120593119894 is related to the number offrequency componentsThird the sequence of signal effectiveeigenvalues in characteristic distribution map is determined
Shock and Vibration 15
D11-1080 rpm
x 10minus5
minus20
minus15
minus05
0
05
10
15
20
D11
-X (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(a)
D12-1080 rpmx 10
minus5
minus15
minus1
minus05
0
05
1
15
D12
-Y (m
)
500 1000 1500 2000 2500 30000 40003500Sampling numbers i
(b)
(16 07335)
(32 04771)
D11-1080 rpm
x 10minus5
10 20 30 40 50 60 70 80 90 1000f (Hz)
0010203040506070809
1
D11
-X (m
)
(c)
(16 06571)
(32 05228)
D12-1080 rpm
x 10minus5
0010203040506070809
1
D12
-Y (m
)
10 20 30 40 50 60 70 80 90 1000f (Hz)
(d)
W2
x 10minus5
x 10minus5
minus15
minus1
minus05
0
05
1
15
Y(D
12) (
m)
minus1 minus05minus15 05 1 15 20X(D11) (m)
(e)
Figure 20The purification effect of harmonic wavelets (a)The time domain of D11 (b)The time domain of D12 (c)The frequency spectrumof D11 (d) The frequency spectrum of D12 (e) The orbit is to extract the first 2 frequencies
by amplitude 119860 119894 of signal frequency component The largermagnitude of amplitude is the greater eigenvalues are andthe higher arrangement will be
Based on aforesaid principle a new CFSM-PCA wasdeveloped and has become an effective tool to extract oreliminate specific characteristic (single frequency) even dif-ference of two frequencies is only 1 Hz Purification of rotororbit using this algorithm shows significant improvement
than those of existing methods such as wavelet packet andharmonic wavelet packet Although the proposed CFSM-PCA is effective some drawbacks are unavoidable Forexample this method suffers from similar issue to otheralgorithms that is the reconstruction error increaseswith thedecrease of SNR In addition the applications of the CFSM-PCA algorithm in purification of speech recognition bearingfault diagnosis and fault diagnosis of power system should
16 Shock and Vibration
D11-1080 rpmx 10
minus5
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
minus20
minus15
minus05
0
05
10
15
20
D11
-X (m
)
(a)
D12-1080 rpmx 10
minus5
minus15
minus1
minus05
0
05
1
15
D12
-Y (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(b)
(16 06557)
(32 04381)
D11-1080 rpmx 10
minus5
10020 30 40 50 60 70 80 90100f (Hz)
0010203040506070809
1
D11
-X (m
)
(c)
(16 06452)(32 04927)
D12-1080 rpm
x 10minus5
0010203040506070809
1
D12
-Y (m
)
9020 30 40 50 60 70 8010 1000f (Hz)
(d)
W3
x 10minus5
x 10minus5
minus15
minus1
minus05
0
05
1
15
Y(D
12) (
m)
minus15 minus05minus1 05 1 15 20X(D11) (m)
(e)
Figure 21 The purification effect of wavelet packet (a) The time domain of D11 (b) The time domain of D12 (c) The frequency spectrum ofD11 (d) The frequency spectrum of D12 (e) The orbit is to extract the first 2 frequencies
be further exploredWe fully believe that the CFSM-PCAwillsparkle in diverse fields
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work is supported by the National High TechnologyResearch andDevelopment Program of China (863 Program)
Shock and Vibration 17
(18 0734)
(36 04634)
(54 02448) (72 02351)(90 01646)
x 10minus5
50 100 150 2000f (Hz)
0
02
04
06
08
1
D11
minusX
(m)
Figure 22 Frequency spectrum of D11 signal after filtering 50Hz
(no 2015AA043005) and the National Natural Science Foun-dation of China (no 51375187)
References
[1] H Abdi and L J Williams ldquoPrincipal component analysisrdquoWiley Interdisciplinary Reviews Computational Statistics vol 2no 4 pp 433ndash459 2010
[2] A Widodo B Yang and T Han ldquoCombination of independentcomponent analysis and support vectormachines for intelligentfaults diagnosis of induction motorsrdquo Expert Systems withApplications vol 32 no 2 pp 299ndash312 2007
[3] S Wold K Esbensen and P Geladi ldquoPrincipal componentanalysisrdquo Chemometrics and Intelligent Laboratory Systems vol2 no 1ndash3 pp 37ndash52 1987
[4] M Kirby and L Sirovich ldquoApplication of the Karhunen-Loeve procedure for the characterization of human facesrdquo IEEETransactions on Pattern Analysis and Machine Intelligence vol12 no 1 pp 103ndash108 1990
[5] J H Xi Y Z Han and R H Su ldquoNew fault diagnosis methodfor rolling bearing based on PCArdquo in Proceedings of the 25thChinese Control and Decision Conference CCDC rsquo13 pp 4123ndash4127 IEEE May 2013
[6] A Malhi and R X Gao ldquoPCA-based feature selection schemefor machine defect classificationrdquo IEEE Transactions on Instru-mentation and Measurement vol 53 no 6 pp 1517ndash1525 2004
[7] S Xu X Jiang J Huang S Yang and X Wang ldquoBayesianwavelet PCA methodology for turbomachinery damage diag-nosis under uncertaintyrdquo Mechanical Systems and Signal Pro-cessing vol 80 pp 1ndash18 2016
[8] W Sun J Chen and J Li ldquoDecision tree and PCA-based faultdiagnosis of rotatingmachineryrdquoMechanical Systems and SignalProcessing vol 21 no 3 pp 1300ndash1317 2007
[9] Z X Li X P Yan C Q Yuan Z Peng and L Li ldquoVirtual pro-totype and experimental research on gear multi-fault diagnosisusing wavelet-autoregressive model and principal componentanalysismethodrdquoMechanical Systems and Signal Processing vol25 no 7 pp 2589ndash2607 2011
[10] K Y Shao M M Cai and G F Zhao ldquoRolling bearing faultdiagnosis based onwavelet energy spectrum PCA andPNNrdquo inProceedings of the 26th Chinese Control andDecisionConferenceCCDC rsquo14 pp 800ndash804 IEEE 2014
[11] T Qiu Q F Zhang and Y J Ding ldquoNonlinear sensor faultdetection and data rebuilding based on principle componentanalysisrdquo Journal of Tsinghua University vol 46 no 5 pp 708ndash711 2006
[12] Y K Gu Z X Yang and F L Zhu ldquoGearbox fault feature fusionbased on principal component analysisrdquo China MechanicalEngineering vol 26 no 11 pp 1532ndash1537 2015
[13] R Dunia S J Qin T F Edgar and T J McAvoy ldquoIdentificationof faulty sensors using principal component analysisrdquo AIChEJournal vol 42 no 10 pp 2797ndash2811 1996
[14] E P de Moura C R Souto A A Silva and M A S IrmaoldquoEvaluation of principal component analysis and neural net-work performance for bearing fault diagnosis from vibrationsignal processed by RS and DF analysesrdquo Mechanical Systemsand Signal Processing vol 25 no 5 pp 1765ndash1772 2011
[15] S Golestan M Monfared F D Freijedo and J M GuerreroldquoDynamics assessment of advanced single-phase PLL struc-turesrdquo IEEE Transactions on Industrial Electronics vol 60 no6 pp 2167ndash2177 2013
[16] A Lima H Zen Y Nankaku C Miyajima K Tokuda andT Kitamura ldquoOn the use of kernel PCA for feature extractionin speech recognitionrdquo IEICE Transaction on Information andSystems vol 12 no 87 pp 2802ndash2811 2004
[17] J D Zheng ldquoImproved hilbert-huang transform and its appli-cations to rolling bearing fault diagnosisrdquo Journal of MechanicalEngineering vol 1 no 51 pp 138ndash145 2015
[18] G He K Ding and H Lin ldquoFault feature extraction of rollingelement bearings using sparse representationrdquo Journal of Soundand Vibration vol 366 pp 514ndash527 2016
[19] J G Yang S B Xia and Y G Liu ldquoWavelet denoising and itsapplication in extraction of shaft centerline orbit featuresrdquoJournal of Harbin Institute of Technology vol 31 no 5 pp 52ndash541999 (Chinese)
[20] J A Duan and X D Zhang ldquoFeature extraction from the orbitof the center of a rotor based on wavelet transformrdquo Journal ofVibration Measurement amp Diagnosis vol 17 no 01 pp 31ndash341997
[21] J Han D X Jiang and W D Ning ldquoApplication of optimalwavelet packets in failure signal extraction of rotor orbitrdquoTurbine Technology vol 43 no 3 pp 133ndash136 2001
[22] W B Zhang X J Zhou andY Lin ldquoRefinement of rotor centerrsquosorbit by a harmonic window methodrdquo Journal of VibrationMeasurement amp Diagnosis vol 30 no 1 pp 87ndash90 2010
[23] S M Li G Z Lu and Q Y Xu ldquoOn obtaining accuraterotor sub-frequency signal with harmonic waveletrdquo Journal ofNorthwestern Polytechincal University vol 19 no 2 pp 220ndash224 2001
[24] F J Wu and L S Qu ldquoAn improved method for restraining theend effect in empiricalmode decomposition and its applicationsto the fault diagnosis of large rotating machineryrdquo Journal ofSound and Vibration vol 314 no 3-5 pp 586ndash602 2008
[25] W G Liu ldquoResearch on method of purification of shaft orbitbased on singular value decompositionrdquo Technical Acousticsvol 5 no 35 pp 30ndash35 2016
[26] X Z Zhao and B Y Ye ldquoThe influence of formation manner ofcomponent on signal processingrdquo Shang Hai Jiao Tong Univer-sity vol 45 no 3 pp 368ndash374 2011
[27] X Z Zhao B Y Ye and T J Chen ldquoPrinciple of singularitydetection based on SVD and its applicationrdquo Journal of Vibra-tion and Shock vol 6 no 27 pp 7ndash10 2008
[28] X Z Zhao B Y Ye and T J Chen ldquoDifference spectrum theoryof singular value and its application to the fault diagnosis ofheadstock of latherdquo Journal of Mechanical Engineering vol 46pp 100ndash108 2010
[29] I A Kougioumtzoglou and P D Spanos ldquoAn identificationapproach for linear and nonlinear time-variant structural sys-tems via harmonic waveletsrdquo Mechanical Systems and SignalProcessing vol 37 no 1-2 pp 338ndash352 2013
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Shock and Vibration 9
A B
Servo Motor Unloading device
Coupling
Sliding Bearing
Rotor
Large Platform
Small Platform
Air spring
BeltSliding Bearing
Figure 7 Large rot or vibration test bed
Displacement SensorA-end Face B-end Face
D13 D14
D12D11
Figure 8 Schematic and installation of eddy current displacement sensor
Rotor orbits indicate the symptoms of malfunction andare related to variation of input force or dynamic stiffnessGenerally different curve shapes of rotor orbit in a senserepresent different fault types [19ndash22 29] For example insideldquo8rdquo outside ldquo8rdquo and petal-shaped curves are omens of oil-filmwhirl fault misalignment fault and rubbing fault of rotorsystem respectivelyTherefore extraction of rotor orbit playsa significant role in rotating machinery diagnostics In thissection a simple and efficient CFSM-PCA is applied to purifyrotor orbit precisely
41 Experimental Rigs Figure 7 shows the large rotor vibra-tion test bed system which is constructed by our group Inorder to real-timemonitor rotor working station two KamanKD2306-1S eddy current proximity probes are installedon individual side in orthogonal manners (Figure 8) Theexperimental data were collected through LMS Test Lab
42 The First Group Sample Analysis Experimental datastems from D11 and D12 eddy current displacement sensorin side A 4096 data points were collected with test bed speedof 1080 rmin and sampling frequency of 2048 Hz The timedomain waveforms amplitude spectrums and the order ofamplitudes were obtained after filtering DC component viaFFT (Figure 9)
As demonstrated in Figure 9 raw signal was not onlyaffected by noise but also interfered by power frequency 50Hz and its harmonics frequency The energy of signal mainlyconcentrated on the leading two octaves
Rotor orbit was produced by combining spindle dis-placement signal D11 with signal D12 (Figure 10) X-axispresented raw signal D11 and Y-axis referred to signal D12Manipulating status of test bed is difficult to know from thisunprocessed rotor orbit Next the rotor orbit was attemptedto purify by employing PCA
Generally rotor orbit synthesized by 1X and 2X is cred-itable Orbits included high harmonics incline to becomecomplex and even turn messy Hence extraction of 1Xand 2X of raw signal becomes more important [21 22]Eigenvalue distribution of covariance matrix of D11 and D12signals are displayed in Figure 11 According to amplitudeof each frequency of D11 and D12 in Figure 9 1X and 2Xcorresponded to the first (first and second eigenvalues) andthe second (third and fourth eigenvalues) set of eigenvaluesrespectively The leading four eigenvalues in Figure 11 werereconstructed through CFSM-PCA and results are illustratedin Figure 12 Generated 1X and 2X were legible without effectby noise and disturbance from frequency 50 Hz as well as itsharmonics Meanwhile amplitude of the extracted signal wasclose to that of 1X and 2X in raw signal indicating that thereis no spectral leakage issue
The feature spectrums continued to be separated andeigenvalue distributionmaps of covariance matrix of D11 andD12 are shown in Figure 15
Rotor orbit generated by the purified 1X and 2X is shownin Figure 13The rotor orbit was a caved-in banana-like curvedemonstrating that sideA (near themotor) potentially suffersfrom misalignment fault [18 19] The explicit rotor orbit alsovalidates the high efficiency of this proposed method
43 The Second Group Sample Analysis In this case exper-imental data of D11 and D12 were analyzed when test bedruns steadily for a period of time Similarly 4096 datapoints were measured with test bed speed of 2770 rmin andsampling frequency of 2048Hz Time domainwaveforms andamplitude spectrums are displayed in Figure 14
According to amplitude of the leading trebling frequencyofD11 andD12 in Figure 14 1X corresponded to the second setof eigenvalues (third and fourth eigenvalues) 2X paralleledthe third set of eigenvalues (fifth and sixth eigenvalues) and
10 Shock and Vibration
D11-1080 rpmx 10
minus5
minus5minus4minus3minus2minus1
012345
X (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(a)
Power-line Interference50Hz
0733404771
Order 1 Order 2
0250502342
Order 6Order 7
D11-1080 rpm
01651 Order 11
18Hz36Hz54Hz72Hz90Hz
x 10minus5
1times
2times
3times
4times
5times
1times
2times
3times 4times5times
002040608
1
X (m
)
50 100 150 200 250 300 350 4000Frequency (Hz)
(b)
D12-1080 rpmx 10
minus5
minus6
minus4
minus2
0246
Y (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(c)
D12-1080 rpmPower-line Interference
50Hz
0658105228
Order 1 Order 2
0277101226
Order 5 Order 12
01948 Order 8
18Hz36Hz54Hz72Hz90Hz
x 10minus5
1times
2times
3times
4times
5times
1times
2times
3times
4times5times
0
05
1
Y (m
)
50 100 150 200 250 300 350 4000Frequency (Hz)
(d)
Figure 9The time domain and frequency domain of theD11 andD12 sensors in theA-end face (a)The time domain of D11 (b)The frequencydomain of D11 (c) The time domain of D12 (d) The frequency domain of D12
x 10minus5
x 10minus5
minus5
minus4
minus3
minus2
minus1
0
1
2
3
4
5
D12
minusY
(m)
minus6 minus2minus4 2 4 60D11minusX (m)
Figure 10 A-end face of the D11 and D12 sensors to collect the signal directly synthesized axis orbit
(12751) (22748)e first group
(31243) (41241)e second group
D11minus1080 rpm
x 10minus8
0
1
2
3
4
Eige
nval
ue o
f PCA
(m)
10 20 30 40 500Number of eigenvalues q
(a)
(12247) (2241)
(314) (41396)
D12minus1080 rpme first group
e second group
x 10minus8
005
115
225
Eige
nval
ue o
f PCA
(m)
10 20 30 40 500Number of eigenvalues q
(b)
Figure 11The eigenvalues distribution of covariancematrixC (a)The eigenvalues distribution of D11 signal (b)The eigenvalues distributionof D12 signal
Shock and Vibration 11
D11minus1080 rpmx 10
minus5
minus15
minus10
minus05
0051015
X (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(a)
D12minus1080 rpmx 10
minus5
minus10
minus05
0051015
Y (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(b)
D11minus1080 rpm(1807287)
(360472)
x 10minus5
002040608
1
X (m
)
50 100 150 2000Frequency (Hz)
(c)
(3605105)(1806575)
D12minus1080 rpmx 10
minus5
002040608
1
Y (m
)
50 100 150 2000Frequency (Hz)
(d)
Figure 12 Extracting the time and frequency spectrums of the first two frequencies of D11 and D12 (a)The time domain of D11 (b)The timedomain of D12 (c) The frequency domain of D11 (d) The frequency domain of D12
W1
x 10minus5
x 10minus5
minus1
minus05
0
05
1
15
D12
minusY
(m)
minus1minus15 minus05 05 10 15D11minusX (m)
(a)0
051
152
0
2
t (s)D11minusX (m)
minus2
x 10minus5
x 10minus5
minus1
minus05
0
05
1
15
D12
minusY
(m)
(b)
Figure 13 The synthesis of axis orbit is to extract the first two times fundamental frequencies of the D11 and D12 signals by the CFSM-PCA(a) The axis orbit (b) Relationship between axis orbit and time
3X matched the first set of eigenvalues (first and secondeigenvalues)
As displayed in Figure 15 eigenvalues of 1X and 3Xwere large thus 1X and 3X were utilized to synthesize rotororbit firstly Purification results and orbit of shaft center arerevealed in Figures 16 and 17 respectively The legible orbitshowed a typical plum blossom shape
It can be seen from Figures 16 and 17 that the filteringeffect was also quite obvious The noise and power frequency50 Hz as well as its harmonic interference could be removedsuccessfully Moreover amplitude of refined signal was closeto that of original one and the reconstructed time domainsignal seemed to be steady without any fluctuation
As shown in Figures 14(c) and 14(d) amplitudes of 1X 2Xand 3X in amplitude spectrum of raw signal were largeThus1X 2X and 3X were also extracted to synthesize axis orbit
Time domain frequency domain and orbit curve are shownin Figures 18 and 19 respectively
Axis orbit synthesized by the leading three frequenciesexhibited a quincunx shape which is basically the sameto that synthesized by 1X and 3X frequency componentsindicating that static and dynamic parts friction may exist inA-side
44 Comparison Harmonic Wavelet with Wavelet Packet
(1) Comparison of CFSM-PCA with Harmonic Wavelet Onecommon used method for signal processing harmonicwavelet algorithm proposed by Newland is an improvedwavelet algorithm [29] It can segment entire frequencyband infinitely and avoid the shortcoming of downsamplingmethod of binary wavelet and binary wavelet packet [22 23]
12 Shock and Vibration
D11minus2770 rpmx 10
minus4
minus1
minus05
0
05
1
D11
-X (m
)
1000 2000 3000 40000Sampling numbers i
(a)
D12minus2770 rpmx 10
minus4
minus1
minus05
0
05
1
D12
-Y (m
)
1000 2000 3000 40000Sampling numbers i
(b)
Power-Line interference 50and 150Hz
1X2X
3X 1010D11minus2770 rpm
08441166
1X2X3X
x 10minus5
200 400 600 800 10000f (Hz)
0
05
10
15
D11
-X (m
)
(c)
1X
2X
3XPower-Line interference 50
and 150Hz
D12minus2770 rpm117309021227
1X2X3X
x 10minus5
0
05
10
15
D12
-Y (m
)
200 400 600 800 10000f (Hz)
(d)
Figure 14 The time and frequency domains of the D11 and D12 sensors (a) The time domain of D11 (b) The time domain of D12 (c) Thefrequency domain of D11 (d) The frequency domain of D12
(11489) (21482)e first group
(31237) (41220)e second group
(30348) (40347)e second group
D11minus2770 rpmx 10
minus7
10 20 30 400 50Number of eigenvalues q
0
05
1
15
Eige
nval
ue o
f PCA
(m)
(a)
D12minus2770 rpm(11688) (21687)
e first group
e second group
(30414) (40413)e second group
(31673) (41664)
x 10minus7
0
05
1
15
2
Eige
nval
ue o
f PCA
(m)
10 20 30 40 500Number of eigenvalues q
(b)
Figure 15 The eigenvalues distribution of covariance matrix C (a) The eigenvalues distribution of D11 (b) The eigenvalues distribution ofD12
1X and its frequency doubling signal can be easily extractedto form axis orbit of revolving test bearing
Axis orbit was fabricated by 1X and 2Xof data of example 1(Section 42) through harmonic wavelet algorithm and theresults are demonstrated in Figure 20 The shape of axisorbit was trapped banana-like which is consistent with theresult obtained via CFSM-PCA It should be noted thatorbit in Figure 13 was more obvious than that in Figure 20ie W1ltW2 Meanwhile the amplitude fluctuation of timedomain signal was large (Figures 20(a) and 20(b)) Themajor reason for this phenomenon is that the harmonicwavelet which is a band filter in frequency domain inevitablyextracts the noise near the feature frequency Furthermorethis phenomenon also demonstrates that harmonic waveletis subject to the Heisenberg uncertainty principle ie thephenomenon of the amplitude leakage exists
(2) Comparison of CFSM-PCA with Wavelet Packet Besidesharmonic wavelets wavelet packet is also used to purify axisorbit [19ndash21] Purification results of axis orbit via Daubechieswavelet packet in Figure 21 exhibit that axis orbit was diver-gent seriously This was worse than that of harmonic waveletie W1ltW2ltW3The phenomenon of severe energy leakageoccurred during the extraction of 1X and 2X which cancause divergence of axis orbit Moreover the bandpass filterof wavelet packet also accounts for the divergent orbits
5 A Single Frequency Filtrationwith CFSM-PCA
The application in single frequency filtration is discussedin this section Taking D11 experimental data of example
Shock and Vibration 13
D11minus2770 rpmx 10
minus5
0 1000500Sampling numbers i
minus4
minus2
0
2
4
D11
-X (m
)
(a)
D12minus2770 rpmx 10
minus5
minus4
minus2
0
2
4
D12
-Y (m
)
500 10000Sampling numbers i
(b)
(1375 1185)(46 1009)
D11minus2770 rpmx 10
minus5
100 200 300 400 5000f (Hz)
0
05
1
15
D11
-X (m
)
(c)
(1375 1218)(46 1164)
D12minus2770 rpmx 10
minus5
100 200 300 400 5000f (Hz)
0
05
1
15
D12
-Y (m
)
(d)
Figure 16 Extracting the frequency spectrums of the first two frequencies of D11 and D12 by PCA algorithm (a) The time domain of D11(b) The time domain of D12 (c) The frequency spectrum of D11 (d) The frequency spectrum of D12
x 10minus5
x 10minus5
minus4
minus3
minus2
minus1
0
1
2
3
4
D12
-Y (m
)
210 3 4minus2minus3 minus1minus4
D11-X (m)
(a)0
0204
0608
10
24
t (s)D11-X (m) minus2
minus4
x 10minus5
x 10minus5
minus4
minus2
0
2
4
D12
-Y (m
)
(b)
Figure 17 The synthesis of axis orbit is to extract the first two times fundamental frequencies of the D11 and D12 signals by the CFSM-PCA(a) The axis orbit (b) Relationship between axis orbit and time
1 (Section 42) as a demo we attempt to filter the powerfrequency interference (50 Hz) from D11 by employingCFSM-PCA It can be seen from Figure 9(b) that sequenceof amplitude of 50 Hz in raw signal was 3 so eigenvec-tors corresponding to 5th and 6th eigenvalues in eigen-value distribution map were used to reconstruct Accordingto (8) the filtered signal (Figure 22) without 50 Hz was
obtained by subtracting reconstructed signal from originalone
Results in Figure 22 show that this algorithm filters out50 Hz power frequency interference and has no influenceon the other frequency components of raw signal demon-strating that this method has a potential application in notchfilter
14 Shock and Vibration
D11minus2770 rpm
x 10minus5
minus4
minus2
0
2
4
D11
-X (m
)
200 400 600 800 10000Sampling numbers i
(a)
D12minus2770 rpm
x 10minus5
0 400 600 800 1000200Sampling numbers i
minus4
minus2
0
2
4
D12
-Y (m
)
(b)
(1375 1185)(46 1009)
(92 0831)
D11minus2770 rpmx 10
minus5
00
05
10
15
D11
-X (m
)
100 200 300 400 5000f (Hz)
(c)
(1375 1218)(46 1163)
(92 0898)
D12minus2770 rpmx 10
minus5
00
05
10
15
D12
-Y (m
)
100 200 300 400 5000f (Hz)
(d)
Figure 18 Extracting the time and frequency spectrums of the first three frequencies of D11 and D12 by PCA algorithm (a)The time domainof D11 (b) The time domain of D12 (c) The frequency spectrum of D11 (d) The frequency spectrum of D12
x 10minus5
x 10minus5
minus4
minus3
minus2
minus1
0
1
2
3
4
D12
-Y (m
)
minus3 minus2 minus1minus4 1 2 3 40D11-X (m)
(a)
1008
0604
0200
02
4D11-X (m)
t (s)
minus2
minus4
x 10minus5
x 10minus5
minus4
minus2
0
2
4
D12
-Y (m
)
(b)
Figure 19 The synthesis of axis orbit is to extract the first three times fundamental frequencies of the D11 and D12 by the CFSM-PCA (a)The axis orbit (b) Relationship between axis orbit and time
6 Conclusion
The selection of effective eigenvalues in feature distributionchart of covariance matrix C is crucial in PCA algorithm Inthis paper the relationships between effective eigenvalues andsignal frequency and amplitude are discovered and a series ofconclusions are summarized
First in Hankel matrix mode each valid frequencycomponent of signal only produces two adjacent nonzeroeigenvalues Second the number of valid eigenvalues incharacteristic distribution chart which is independent ofnumerical values of 119891119894 119860 119894 and 120593119894 is related to the number offrequency componentsThird the sequence of signal effectiveeigenvalues in characteristic distribution map is determined
Shock and Vibration 15
D11-1080 rpm
x 10minus5
minus20
minus15
minus05
0
05
10
15
20
D11
-X (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(a)
D12-1080 rpmx 10
minus5
minus15
minus1
minus05
0
05
1
15
D12
-Y (m
)
500 1000 1500 2000 2500 30000 40003500Sampling numbers i
(b)
(16 07335)
(32 04771)
D11-1080 rpm
x 10minus5
10 20 30 40 50 60 70 80 90 1000f (Hz)
0010203040506070809
1
D11
-X (m
)
(c)
(16 06571)
(32 05228)
D12-1080 rpm
x 10minus5
0010203040506070809
1
D12
-Y (m
)
10 20 30 40 50 60 70 80 90 1000f (Hz)
(d)
W2
x 10minus5
x 10minus5
minus15
minus1
minus05
0
05
1
15
Y(D
12) (
m)
minus1 minus05minus15 05 1 15 20X(D11) (m)
(e)
Figure 20The purification effect of harmonic wavelets (a)The time domain of D11 (b)The time domain of D12 (c)The frequency spectrumof D11 (d) The frequency spectrum of D12 (e) The orbit is to extract the first 2 frequencies
by amplitude 119860 119894 of signal frequency component The largermagnitude of amplitude is the greater eigenvalues are andthe higher arrangement will be
Based on aforesaid principle a new CFSM-PCA wasdeveloped and has become an effective tool to extract oreliminate specific characteristic (single frequency) even dif-ference of two frequencies is only 1 Hz Purification of rotororbit using this algorithm shows significant improvement
than those of existing methods such as wavelet packet andharmonic wavelet packet Although the proposed CFSM-PCA is effective some drawbacks are unavoidable Forexample this method suffers from similar issue to otheralgorithms that is the reconstruction error increaseswith thedecrease of SNR In addition the applications of the CFSM-PCA algorithm in purification of speech recognition bearingfault diagnosis and fault diagnosis of power system should
16 Shock and Vibration
D11-1080 rpmx 10
minus5
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
minus20
minus15
minus05
0
05
10
15
20
D11
-X (m
)
(a)
D12-1080 rpmx 10
minus5
minus15
minus1
minus05
0
05
1
15
D12
-Y (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(b)
(16 06557)
(32 04381)
D11-1080 rpmx 10
minus5
10020 30 40 50 60 70 80 90100f (Hz)
0010203040506070809
1
D11
-X (m
)
(c)
(16 06452)(32 04927)
D12-1080 rpm
x 10minus5
0010203040506070809
1
D12
-Y (m
)
9020 30 40 50 60 70 8010 1000f (Hz)
(d)
W3
x 10minus5
x 10minus5
minus15
minus1
minus05
0
05
1
15
Y(D
12) (
m)
minus15 minus05minus1 05 1 15 20X(D11) (m)
(e)
Figure 21 The purification effect of wavelet packet (a) The time domain of D11 (b) The time domain of D12 (c) The frequency spectrum ofD11 (d) The frequency spectrum of D12 (e) The orbit is to extract the first 2 frequencies
be further exploredWe fully believe that the CFSM-PCAwillsparkle in diverse fields
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work is supported by the National High TechnologyResearch andDevelopment Program of China (863 Program)
Shock and Vibration 17
(18 0734)
(36 04634)
(54 02448) (72 02351)(90 01646)
x 10minus5
50 100 150 2000f (Hz)
0
02
04
06
08
1
D11
minusX
(m)
Figure 22 Frequency spectrum of D11 signal after filtering 50Hz
(no 2015AA043005) and the National Natural Science Foun-dation of China (no 51375187)
References
[1] H Abdi and L J Williams ldquoPrincipal component analysisrdquoWiley Interdisciplinary Reviews Computational Statistics vol 2no 4 pp 433ndash459 2010
[2] A Widodo B Yang and T Han ldquoCombination of independentcomponent analysis and support vectormachines for intelligentfaults diagnosis of induction motorsrdquo Expert Systems withApplications vol 32 no 2 pp 299ndash312 2007
[3] S Wold K Esbensen and P Geladi ldquoPrincipal componentanalysisrdquo Chemometrics and Intelligent Laboratory Systems vol2 no 1ndash3 pp 37ndash52 1987
[4] M Kirby and L Sirovich ldquoApplication of the Karhunen-Loeve procedure for the characterization of human facesrdquo IEEETransactions on Pattern Analysis and Machine Intelligence vol12 no 1 pp 103ndash108 1990
[5] J H Xi Y Z Han and R H Su ldquoNew fault diagnosis methodfor rolling bearing based on PCArdquo in Proceedings of the 25thChinese Control and Decision Conference CCDC rsquo13 pp 4123ndash4127 IEEE May 2013
[6] A Malhi and R X Gao ldquoPCA-based feature selection schemefor machine defect classificationrdquo IEEE Transactions on Instru-mentation and Measurement vol 53 no 6 pp 1517ndash1525 2004
[7] S Xu X Jiang J Huang S Yang and X Wang ldquoBayesianwavelet PCA methodology for turbomachinery damage diag-nosis under uncertaintyrdquo Mechanical Systems and Signal Pro-cessing vol 80 pp 1ndash18 2016
[8] W Sun J Chen and J Li ldquoDecision tree and PCA-based faultdiagnosis of rotatingmachineryrdquoMechanical Systems and SignalProcessing vol 21 no 3 pp 1300ndash1317 2007
[9] Z X Li X P Yan C Q Yuan Z Peng and L Li ldquoVirtual pro-totype and experimental research on gear multi-fault diagnosisusing wavelet-autoregressive model and principal componentanalysismethodrdquoMechanical Systems and Signal Processing vol25 no 7 pp 2589ndash2607 2011
[10] K Y Shao M M Cai and G F Zhao ldquoRolling bearing faultdiagnosis based onwavelet energy spectrum PCA andPNNrdquo inProceedings of the 26th Chinese Control andDecisionConferenceCCDC rsquo14 pp 800ndash804 IEEE 2014
[11] T Qiu Q F Zhang and Y J Ding ldquoNonlinear sensor faultdetection and data rebuilding based on principle componentanalysisrdquo Journal of Tsinghua University vol 46 no 5 pp 708ndash711 2006
[12] Y K Gu Z X Yang and F L Zhu ldquoGearbox fault feature fusionbased on principal component analysisrdquo China MechanicalEngineering vol 26 no 11 pp 1532ndash1537 2015
[13] R Dunia S J Qin T F Edgar and T J McAvoy ldquoIdentificationof faulty sensors using principal component analysisrdquo AIChEJournal vol 42 no 10 pp 2797ndash2811 1996
[14] E P de Moura C R Souto A A Silva and M A S IrmaoldquoEvaluation of principal component analysis and neural net-work performance for bearing fault diagnosis from vibrationsignal processed by RS and DF analysesrdquo Mechanical Systemsand Signal Processing vol 25 no 5 pp 1765ndash1772 2011
[15] S Golestan M Monfared F D Freijedo and J M GuerreroldquoDynamics assessment of advanced single-phase PLL struc-turesrdquo IEEE Transactions on Industrial Electronics vol 60 no6 pp 2167ndash2177 2013
[16] A Lima H Zen Y Nankaku C Miyajima K Tokuda andT Kitamura ldquoOn the use of kernel PCA for feature extractionin speech recognitionrdquo IEICE Transaction on Information andSystems vol 12 no 87 pp 2802ndash2811 2004
[17] J D Zheng ldquoImproved hilbert-huang transform and its appli-cations to rolling bearing fault diagnosisrdquo Journal of MechanicalEngineering vol 1 no 51 pp 138ndash145 2015
[18] G He K Ding and H Lin ldquoFault feature extraction of rollingelement bearings using sparse representationrdquo Journal of Soundand Vibration vol 366 pp 514ndash527 2016
[19] J G Yang S B Xia and Y G Liu ldquoWavelet denoising and itsapplication in extraction of shaft centerline orbit featuresrdquoJournal of Harbin Institute of Technology vol 31 no 5 pp 52ndash541999 (Chinese)
[20] J A Duan and X D Zhang ldquoFeature extraction from the orbitof the center of a rotor based on wavelet transformrdquo Journal ofVibration Measurement amp Diagnosis vol 17 no 01 pp 31ndash341997
[21] J Han D X Jiang and W D Ning ldquoApplication of optimalwavelet packets in failure signal extraction of rotor orbitrdquoTurbine Technology vol 43 no 3 pp 133ndash136 2001
[22] W B Zhang X J Zhou andY Lin ldquoRefinement of rotor centerrsquosorbit by a harmonic window methodrdquo Journal of VibrationMeasurement amp Diagnosis vol 30 no 1 pp 87ndash90 2010
[23] S M Li G Z Lu and Q Y Xu ldquoOn obtaining accuraterotor sub-frequency signal with harmonic waveletrdquo Journal ofNorthwestern Polytechincal University vol 19 no 2 pp 220ndash224 2001
[24] F J Wu and L S Qu ldquoAn improved method for restraining theend effect in empiricalmode decomposition and its applicationsto the fault diagnosis of large rotating machineryrdquo Journal ofSound and Vibration vol 314 no 3-5 pp 586ndash602 2008
[25] W G Liu ldquoResearch on method of purification of shaft orbitbased on singular value decompositionrdquo Technical Acousticsvol 5 no 35 pp 30ndash35 2016
[26] X Z Zhao and B Y Ye ldquoThe influence of formation manner ofcomponent on signal processingrdquo Shang Hai Jiao Tong Univer-sity vol 45 no 3 pp 368ndash374 2011
[27] X Z Zhao B Y Ye and T J Chen ldquoPrinciple of singularitydetection based on SVD and its applicationrdquo Journal of Vibra-tion and Shock vol 6 no 27 pp 7ndash10 2008
[28] X Z Zhao B Y Ye and T J Chen ldquoDifference spectrum theoryof singular value and its application to the fault diagnosis ofheadstock of latherdquo Journal of Mechanical Engineering vol 46pp 100ndash108 2010
[29] I A Kougioumtzoglou and P D Spanos ldquoAn identificationapproach for linear and nonlinear time-variant structural sys-tems via harmonic waveletsrdquo Mechanical Systems and SignalProcessing vol 37 no 1-2 pp 338ndash352 2013
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10 Shock and Vibration
D11-1080 rpmx 10
minus5
minus5minus4minus3minus2minus1
012345
X (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(a)
Power-line Interference50Hz
0733404771
Order 1 Order 2
0250502342
Order 6Order 7
D11-1080 rpm
01651 Order 11
18Hz36Hz54Hz72Hz90Hz
x 10minus5
1times
2times
3times
4times
5times
1times
2times
3times 4times5times
002040608
1
X (m
)
50 100 150 200 250 300 350 4000Frequency (Hz)
(b)
D12-1080 rpmx 10
minus5
minus6
minus4
minus2
0246
Y (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(c)
D12-1080 rpmPower-line Interference
50Hz
0658105228
Order 1 Order 2
0277101226
Order 5 Order 12
01948 Order 8
18Hz36Hz54Hz72Hz90Hz
x 10minus5
1times
2times
3times
4times
5times
1times
2times
3times
4times5times
0
05
1
Y (m
)
50 100 150 200 250 300 350 4000Frequency (Hz)
(d)
Figure 9The time domain and frequency domain of theD11 andD12 sensors in theA-end face (a)The time domain of D11 (b)The frequencydomain of D11 (c) The time domain of D12 (d) The frequency domain of D12
x 10minus5
x 10minus5
minus5
minus4
minus3
minus2
minus1
0
1
2
3
4
5
D12
minusY
(m)
minus6 minus2minus4 2 4 60D11minusX (m)
Figure 10 A-end face of the D11 and D12 sensors to collect the signal directly synthesized axis orbit
(12751) (22748)e first group
(31243) (41241)e second group
D11minus1080 rpm
x 10minus8
0
1
2
3
4
Eige
nval
ue o
f PCA
(m)
10 20 30 40 500Number of eigenvalues q
(a)
(12247) (2241)
(314) (41396)
D12minus1080 rpme first group
e second group
x 10minus8
005
115
225
Eige
nval
ue o
f PCA
(m)
10 20 30 40 500Number of eigenvalues q
(b)
Figure 11The eigenvalues distribution of covariancematrixC (a)The eigenvalues distribution of D11 signal (b)The eigenvalues distributionof D12 signal
Shock and Vibration 11
D11minus1080 rpmx 10
minus5
minus15
minus10
minus05
0051015
X (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(a)
D12minus1080 rpmx 10
minus5
minus10
minus05
0051015
Y (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(b)
D11minus1080 rpm(1807287)
(360472)
x 10minus5
002040608
1
X (m
)
50 100 150 2000Frequency (Hz)
(c)
(3605105)(1806575)
D12minus1080 rpmx 10
minus5
002040608
1
Y (m
)
50 100 150 2000Frequency (Hz)
(d)
Figure 12 Extracting the time and frequency spectrums of the first two frequencies of D11 and D12 (a)The time domain of D11 (b)The timedomain of D12 (c) The frequency domain of D11 (d) The frequency domain of D12
W1
x 10minus5
x 10minus5
minus1
minus05
0
05
1
15
D12
minusY
(m)
minus1minus15 minus05 05 10 15D11minusX (m)
(a)0
051
152
0
2
t (s)D11minusX (m)
minus2
x 10minus5
x 10minus5
minus1
minus05
0
05
1
15
D12
minusY
(m)
(b)
Figure 13 The synthesis of axis orbit is to extract the first two times fundamental frequencies of the D11 and D12 signals by the CFSM-PCA(a) The axis orbit (b) Relationship between axis orbit and time
3X matched the first set of eigenvalues (first and secondeigenvalues)
As displayed in Figure 15 eigenvalues of 1X and 3Xwere large thus 1X and 3X were utilized to synthesize rotororbit firstly Purification results and orbit of shaft center arerevealed in Figures 16 and 17 respectively The legible orbitshowed a typical plum blossom shape
It can be seen from Figures 16 and 17 that the filteringeffect was also quite obvious The noise and power frequency50 Hz as well as its harmonic interference could be removedsuccessfully Moreover amplitude of refined signal was closeto that of original one and the reconstructed time domainsignal seemed to be steady without any fluctuation
As shown in Figures 14(c) and 14(d) amplitudes of 1X 2Xand 3X in amplitude spectrum of raw signal were largeThus1X 2X and 3X were also extracted to synthesize axis orbit
Time domain frequency domain and orbit curve are shownin Figures 18 and 19 respectively
Axis orbit synthesized by the leading three frequenciesexhibited a quincunx shape which is basically the sameto that synthesized by 1X and 3X frequency componentsindicating that static and dynamic parts friction may exist inA-side
44 Comparison Harmonic Wavelet with Wavelet Packet
(1) Comparison of CFSM-PCA with Harmonic Wavelet Onecommon used method for signal processing harmonicwavelet algorithm proposed by Newland is an improvedwavelet algorithm [29] It can segment entire frequencyband infinitely and avoid the shortcoming of downsamplingmethod of binary wavelet and binary wavelet packet [22 23]
12 Shock and Vibration
D11minus2770 rpmx 10
minus4
minus1
minus05
0
05
1
D11
-X (m
)
1000 2000 3000 40000Sampling numbers i
(a)
D12minus2770 rpmx 10
minus4
minus1
minus05
0
05
1
D12
-Y (m
)
1000 2000 3000 40000Sampling numbers i
(b)
Power-Line interference 50and 150Hz
1X2X
3X 1010D11minus2770 rpm
08441166
1X2X3X
x 10minus5
200 400 600 800 10000f (Hz)
0
05
10
15
D11
-X (m
)
(c)
1X
2X
3XPower-Line interference 50
and 150Hz
D12minus2770 rpm117309021227
1X2X3X
x 10minus5
0
05
10
15
D12
-Y (m
)
200 400 600 800 10000f (Hz)
(d)
Figure 14 The time and frequency domains of the D11 and D12 sensors (a) The time domain of D11 (b) The time domain of D12 (c) Thefrequency domain of D11 (d) The frequency domain of D12
(11489) (21482)e first group
(31237) (41220)e second group
(30348) (40347)e second group
D11minus2770 rpmx 10
minus7
10 20 30 400 50Number of eigenvalues q
0
05
1
15
Eige
nval
ue o
f PCA
(m)
(a)
D12minus2770 rpm(11688) (21687)
e first group
e second group
(30414) (40413)e second group
(31673) (41664)
x 10minus7
0
05
1
15
2
Eige
nval
ue o
f PCA
(m)
10 20 30 40 500Number of eigenvalues q
(b)
Figure 15 The eigenvalues distribution of covariance matrix C (a) The eigenvalues distribution of D11 (b) The eigenvalues distribution ofD12
1X and its frequency doubling signal can be easily extractedto form axis orbit of revolving test bearing
Axis orbit was fabricated by 1X and 2Xof data of example 1(Section 42) through harmonic wavelet algorithm and theresults are demonstrated in Figure 20 The shape of axisorbit was trapped banana-like which is consistent with theresult obtained via CFSM-PCA It should be noted thatorbit in Figure 13 was more obvious than that in Figure 20ie W1ltW2 Meanwhile the amplitude fluctuation of timedomain signal was large (Figures 20(a) and 20(b)) Themajor reason for this phenomenon is that the harmonicwavelet which is a band filter in frequency domain inevitablyextracts the noise near the feature frequency Furthermorethis phenomenon also demonstrates that harmonic waveletis subject to the Heisenberg uncertainty principle ie thephenomenon of the amplitude leakage exists
(2) Comparison of CFSM-PCA with Wavelet Packet Besidesharmonic wavelets wavelet packet is also used to purify axisorbit [19ndash21] Purification results of axis orbit via Daubechieswavelet packet in Figure 21 exhibit that axis orbit was diver-gent seriously This was worse than that of harmonic waveletie W1ltW2ltW3The phenomenon of severe energy leakageoccurred during the extraction of 1X and 2X which cancause divergence of axis orbit Moreover the bandpass filterof wavelet packet also accounts for the divergent orbits
5 A Single Frequency Filtrationwith CFSM-PCA
The application in single frequency filtration is discussedin this section Taking D11 experimental data of example
Shock and Vibration 13
D11minus2770 rpmx 10
minus5
0 1000500Sampling numbers i
minus4
minus2
0
2
4
D11
-X (m
)
(a)
D12minus2770 rpmx 10
minus5
minus4
minus2
0
2
4
D12
-Y (m
)
500 10000Sampling numbers i
(b)
(1375 1185)(46 1009)
D11minus2770 rpmx 10
minus5
100 200 300 400 5000f (Hz)
0
05
1
15
D11
-X (m
)
(c)
(1375 1218)(46 1164)
D12minus2770 rpmx 10
minus5
100 200 300 400 5000f (Hz)
0
05
1
15
D12
-Y (m
)
(d)
Figure 16 Extracting the frequency spectrums of the first two frequencies of D11 and D12 by PCA algorithm (a) The time domain of D11(b) The time domain of D12 (c) The frequency spectrum of D11 (d) The frequency spectrum of D12
x 10minus5
x 10minus5
minus4
minus3
minus2
minus1
0
1
2
3
4
D12
-Y (m
)
210 3 4minus2minus3 minus1minus4
D11-X (m)
(a)0
0204
0608
10
24
t (s)D11-X (m) minus2
minus4
x 10minus5
x 10minus5
minus4
minus2
0
2
4
D12
-Y (m
)
(b)
Figure 17 The synthesis of axis orbit is to extract the first two times fundamental frequencies of the D11 and D12 signals by the CFSM-PCA(a) The axis orbit (b) Relationship between axis orbit and time
1 (Section 42) as a demo we attempt to filter the powerfrequency interference (50 Hz) from D11 by employingCFSM-PCA It can be seen from Figure 9(b) that sequenceof amplitude of 50 Hz in raw signal was 3 so eigenvec-tors corresponding to 5th and 6th eigenvalues in eigen-value distribution map were used to reconstruct Accordingto (8) the filtered signal (Figure 22) without 50 Hz was
obtained by subtracting reconstructed signal from originalone
Results in Figure 22 show that this algorithm filters out50 Hz power frequency interference and has no influenceon the other frequency components of raw signal demon-strating that this method has a potential application in notchfilter
14 Shock and Vibration
D11minus2770 rpm
x 10minus5
minus4
minus2
0
2
4
D11
-X (m
)
200 400 600 800 10000Sampling numbers i
(a)
D12minus2770 rpm
x 10minus5
0 400 600 800 1000200Sampling numbers i
minus4
minus2
0
2
4
D12
-Y (m
)
(b)
(1375 1185)(46 1009)
(92 0831)
D11minus2770 rpmx 10
minus5
00
05
10
15
D11
-X (m
)
100 200 300 400 5000f (Hz)
(c)
(1375 1218)(46 1163)
(92 0898)
D12minus2770 rpmx 10
minus5
00
05
10
15
D12
-Y (m
)
100 200 300 400 5000f (Hz)
(d)
Figure 18 Extracting the time and frequency spectrums of the first three frequencies of D11 and D12 by PCA algorithm (a)The time domainof D11 (b) The time domain of D12 (c) The frequency spectrum of D11 (d) The frequency spectrum of D12
x 10minus5
x 10minus5
minus4
minus3
minus2
minus1
0
1
2
3
4
D12
-Y (m
)
minus3 minus2 minus1minus4 1 2 3 40D11-X (m)
(a)
1008
0604
0200
02
4D11-X (m)
t (s)
minus2
minus4
x 10minus5
x 10minus5
minus4
minus2
0
2
4
D12
-Y (m
)
(b)
Figure 19 The synthesis of axis orbit is to extract the first three times fundamental frequencies of the D11 and D12 by the CFSM-PCA (a)The axis orbit (b) Relationship between axis orbit and time
6 Conclusion
The selection of effective eigenvalues in feature distributionchart of covariance matrix C is crucial in PCA algorithm Inthis paper the relationships between effective eigenvalues andsignal frequency and amplitude are discovered and a series ofconclusions are summarized
First in Hankel matrix mode each valid frequencycomponent of signal only produces two adjacent nonzeroeigenvalues Second the number of valid eigenvalues incharacteristic distribution chart which is independent ofnumerical values of 119891119894 119860 119894 and 120593119894 is related to the number offrequency componentsThird the sequence of signal effectiveeigenvalues in characteristic distribution map is determined
Shock and Vibration 15
D11-1080 rpm
x 10minus5
minus20
minus15
minus05
0
05
10
15
20
D11
-X (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(a)
D12-1080 rpmx 10
minus5
minus15
minus1
minus05
0
05
1
15
D12
-Y (m
)
500 1000 1500 2000 2500 30000 40003500Sampling numbers i
(b)
(16 07335)
(32 04771)
D11-1080 rpm
x 10minus5
10 20 30 40 50 60 70 80 90 1000f (Hz)
0010203040506070809
1
D11
-X (m
)
(c)
(16 06571)
(32 05228)
D12-1080 rpm
x 10minus5
0010203040506070809
1
D12
-Y (m
)
10 20 30 40 50 60 70 80 90 1000f (Hz)
(d)
W2
x 10minus5
x 10minus5
minus15
minus1
minus05
0
05
1
15
Y(D
12) (
m)
minus1 minus05minus15 05 1 15 20X(D11) (m)
(e)
Figure 20The purification effect of harmonic wavelets (a)The time domain of D11 (b)The time domain of D12 (c)The frequency spectrumof D11 (d) The frequency spectrum of D12 (e) The orbit is to extract the first 2 frequencies
by amplitude 119860 119894 of signal frequency component The largermagnitude of amplitude is the greater eigenvalues are andthe higher arrangement will be
Based on aforesaid principle a new CFSM-PCA wasdeveloped and has become an effective tool to extract oreliminate specific characteristic (single frequency) even dif-ference of two frequencies is only 1 Hz Purification of rotororbit using this algorithm shows significant improvement
than those of existing methods such as wavelet packet andharmonic wavelet packet Although the proposed CFSM-PCA is effective some drawbacks are unavoidable Forexample this method suffers from similar issue to otheralgorithms that is the reconstruction error increaseswith thedecrease of SNR In addition the applications of the CFSM-PCA algorithm in purification of speech recognition bearingfault diagnosis and fault diagnosis of power system should
16 Shock and Vibration
D11-1080 rpmx 10
minus5
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
minus20
minus15
minus05
0
05
10
15
20
D11
-X (m
)
(a)
D12-1080 rpmx 10
minus5
minus15
minus1
minus05
0
05
1
15
D12
-Y (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(b)
(16 06557)
(32 04381)
D11-1080 rpmx 10
minus5
10020 30 40 50 60 70 80 90100f (Hz)
0010203040506070809
1
D11
-X (m
)
(c)
(16 06452)(32 04927)
D12-1080 rpm
x 10minus5
0010203040506070809
1
D12
-Y (m
)
9020 30 40 50 60 70 8010 1000f (Hz)
(d)
W3
x 10minus5
x 10minus5
minus15
minus1
minus05
0
05
1
15
Y(D
12) (
m)
minus15 minus05minus1 05 1 15 20X(D11) (m)
(e)
Figure 21 The purification effect of wavelet packet (a) The time domain of D11 (b) The time domain of D12 (c) The frequency spectrum ofD11 (d) The frequency spectrum of D12 (e) The orbit is to extract the first 2 frequencies
be further exploredWe fully believe that the CFSM-PCAwillsparkle in diverse fields
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work is supported by the National High TechnologyResearch andDevelopment Program of China (863 Program)
Shock and Vibration 17
(18 0734)
(36 04634)
(54 02448) (72 02351)(90 01646)
x 10minus5
50 100 150 2000f (Hz)
0
02
04
06
08
1
D11
minusX
(m)
Figure 22 Frequency spectrum of D11 signal after filtering 50Hz
(no 2015AA043005) and the National Natural Science Foun-dation of China (no 51375187)
References
[1] H Abdi and L J Williams ldquoPrincipal component analysisrdquoWiley Interdisciplinary Reviews Computational Statistics vol 2no 4 pp 433ndash459 2010
[2] A Widodo B Yang and T Han ldquoCombination of independentcomponent analysis and support vectormachines for intelligentfaults diagnosis of induction motorsrdquo Expert Systems withApplications vol 32 no 2 pp 299ndash312 2007
[3] S Wold K Esbensen and P Geladi ldquoPrincipal componentanalysisrdquo Chemometrics and Intelligent Laboratory Systems vol2 no 1ndash3 pp 37ndash52 1987
[4] M Kirby and L Sirovich ldquoApplication of the Karhunen-Loeve procedure for the characterization of human facesrdquo IEEETransactions on Pattern Analysis and Machine Intelligence vol12 no 1 pp 103ndash108 1990
[5] J H Xi Y Z Han and R H Su ldquoNew fault diagnosis methodfor rolling bearing based on PCArdquo in Proceedings of the 25thChinese Control and Decision Conference CCDC rsquo13 pp 4123ndash4127 IEEE May 2013
[6] A Malhi and R X Gao ldquoPCA-based feature selection schemefor machine defect classificationrdquo IEEE Transactions on Instru-mentation and Measurement vol 53 no 6 pp 1517ndash1525 2004
[7] S Xu X Jiang J Huang S Yang and X Wang ldquoBayesianwavelet PCA methodology for turbomachinery damage diag-nosis under uncertaintyrdquo Mechanical Systems and Signal Pro-cessing vol 80 pp 1ndash18 2016
[8] W Sun J Chen and J Li ldquoDecision tree and PCA-based faultdiagnosis of rotatingmachineryrdquoMechanical Systems and SignalProcessing vol 21 no 3 pp 1300ndash1317 2007
[9] Z X Li X P Yan C Q Yuan Z Peng and L Li ldquoVirtual pro-totype and experimental research on gear multi-fault diagnosisusing wavelet-autoregressive model and principal componentanalysismethodrdquoMechanical Systems and Signal Processing vol25 no 7 pp 2589ndash2607 2011
[10] K Y Shao M M Cai and G F Zhao ldquoRolling bearing faultdiagnosis based onwavelet energy spectrum PCA andPNNrdquo inProceedings of the 26th Chinese Control andDecisionConferenceCCDC rsquo14 pp 800ndash804 IEEE 2014
[11] T Qiu Q F Zhang and Y J Ding ldquoNonlinear sensor faultdetection and data rebuilding based on principle componentanalysisrdquo Journal of Tsinghua University vol 46 no 5 pp 708ndash711 2006
[12] Y K Gu Z X Yang and F L Zhu ldquoGearbox fault feature fusionbased on principal component analysisrdquo China MechanicalEngineering vol 26 no 11 pp 1532ndash1537 2015
[13] R Dunia S J Qin T F Edgar and T J McAvoy ldquoIdentificationof faulty sensors using principal component analysisrdquo AIChEJournal vol 42 no 10 pp 2797ndash2811 1996
[14] E P de Moura C R Souto A A Silva and M A S IrmaoldquoEvaluation of principal component analysis and neural net-work performance for bearing fault diagnosis from vibrationsignal processed by RS and DF analysesrdquo Mechanical Systemsand Signal Processing vol 25 no 5 pp 1765ndash1772 2011
[15] S Golestan M Monfared F D Freijedo and J M GuerreroldquoDynamics assessment of advanced single-phase PLL struc-turesrdquo IEEE Transactions on Industrial Electronics vol 60 no6 pp 2167ndash2177 2013
[16] A Lima H Zen Y Nankaku C Miyajima K Tokuda andT Kitamura ldquoOn the use of kernel PCA for feature extractionin speech recognitionrdquo IEICE Transaction on Information andSystems vol 12 no 87 pp 2802ndash2811 2004
[17] J D Zheng ldquoImproved hilbert-huang transform and its appli-cations to rolling bearing fault diagnosisrdquo Journal of MechanicalEngineering vol 1 no 51 pp 138ndash145 2015
[18] G He K Ding and H Lin ldquoFault feature extraction of rollingelement bearings using sparse representationrdquo Journal of Soundand Vibration vol 366 pp 514ndash527 2016
[19] J G Yang S B Xia and Y G Liu ldquoWavelet denoising and itsapplication in extraction of shaft centerline orbit featuresrdquoJournal of Harbin Institute of Technology vol 31 no 5 pp 52ndash541999 (Chinese)
[20] J A Duan and X D Zhang ldquoFeature extraction from the orbitof the center of a rotor based on wavelet transformrdquo Journal ofVibration Measurement amp Diagnosis vol 17 no 01 pp 31ndash341997
[21] J Han D X Jiang and W D Ning ldquoApplication of optimalwavelet packets in failure signal extraction of rotor orbitrdquoTurbine Technology vol 43 no 3 pp 133ndash136 2001
[22] W B Zhang X J Zhou andY Lin ldquoRefinement of rotor centerrsquosorbit by a harmonic window methodrdquo Journal of VibrationMeasurement amp Diagnosis vol 30 no 1 pp 87ndash90 2010
[23] S M Li G Z Lu and Q Y Xu ldquoOn obtaining accuraterotor sub-frequency signal with harmonic waveletrdquo Journal ofNorthwestern Polytechincal University vol 19 no 2 pp 220ndash224 2001
[24] F J Wu and L S Qu ldquoAn improved method for restraining theend effect in empiricalmode decomposition and its applicationsto the fault diagnosis of large rotating machineryrdquo Journal ofSound and Vibration vol 314 no 3-5 pp 586ndash602 2008
[25] W G Liu ldquoResearch on method of purification of shaft orbitbased on singular value decompositionrdquo Technical Acousticsvol 5 no 35 pp 30ndash35 2016
[26] X Z Zhao and B Y Ye ldquoThe influence of formation manner ofcomponent on signal processingrdquo Shang Hai Jiao Tong Univer-sity vol 45 no 3 pp 368ndash374 2011
[27] X Z Zhao B Y Ye and T J Chen ldquoPrinciple of singularitydetection based on SVD and its applicationrdquo Journal of Vibra-tion and Shock vol 6 no 27 pp 7ndash10 2008
[28] X Z Zhao B Y Ye and T J Chen ldquoDifference spectrum theoryof singular value and its application to the fault diagnosis ofheadstock of latherdquo Journal of Mechanical Engineering vol 46pp 100ndash108 2010
[29] I A Kougioumtzoglou and P D Spanos ldquoAn identificationapproach for linear and nonlinear time-variant structural sys-tems via harmonic waveletsrdquo Mechanical Systems and SignalProcessing vol 37 no 1-2 pp 338ndash352 2013
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
Shock and Vibration 11
D11minus1080 rpmx 10
minus5
minus15
minus10
minus05
0051015
X (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(a)
D12minus1080 rpmx 10
minus5
minus10
minus05
0051015
Y (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(b)
D11minus1080 rpm(1807287)
(360472)
x 10minus5
002040608
1
X (m
)
50 100 150 2000Frequency (Hz)
(c)
(3605105)(1806575)
D12minus1080 rpmx 10
minus5
002040608
1
Y (m
)
50 100 150 2000Frequency (Hz)
(d)
Figure 12 Extracting the time and frequency spectrums of the first two frequencies of D11 and D12 (a)The time domain of D11 (b)The timedomain of D12 (c) The frequency domain of D11 (d) The frequency domain of D12
W1
x 10minus5
x 10minus5
minus1
minus05
0
05
1
15
D12
minusY
(m)
minus1minus15 minus05 05 10 15D11minusX (m)
(a)0
051
152
0
2
t (s)D11minusX (m)
minus2
x 10minus5
x 10minus5
minus1
minus05
0
05
1
15
D12
minusY
(m)
(b)
Figure 13 The synthesis of axis orbit is to extract the first two times fundamental frequencies of the D11 and D12 signals by the CFSM-PCA(a) The axis orbit (b) Relationship between axis orbit and time
3X matched the first set of eigenvalues (first and secondeigenvalues)
As displayed in Figure 15 eigenvalues of 1X and 3Xwere large thus 1X and 3X were utilized to synthesize rotororbit firstly Purification results and orbit of shaft center arerevealed in Figures 16 and 17 respectively The legible orbitshowed a typical plum blossom shape
It can be seen from Figures 16 and 17 that the filteringeffect was also quite obvious The noise and power frequency50 Hz as well as its harmonic interference could be removedsuccessfully Moreover amplitude of refined signal was closeto that of original one and the reconstructed time domainsignal seemed to be steady without any fluctuation
As shown in Figures 14(c) and 14(d) amplitudes of 1X 2Xand 3X in amplitude spectrum of raw signal were largeThus1X 2X and 3X were also extracted to synthesize axis orbit
Time domain frequency domain and orbit curve are shownin Figures 18 and 19 respectively
Axis orbit synthesized by the leading three frequenciesexhibited a quincunx shape which is basically the sameto that synthesized by 1X and 3X frequency componentsindicating that static and dynamic parts friction may exist inA-side
44 Comparison Harmonic Wavelet with Wavelet Packet
(1) Comparison of CFSM-PCA with Harmonic Wavelet Onecommon used method for signal processing harmonicwavelet algorithm proposed by Newland is an improvedwavelet algorithm [29] It can segment entire frequencyband infinitely and avoid the shortcoming of downsamplingmethod of binary wavelet and binary wavelet packet [22 23]
12 Shock and Vibration
D11minus2770 rpmx 10
minus4
minus1
minus05
0
05
1
D11
-X (m
)
1000 2000 3000 40000Sampling numbers i
(a)
D12minus2770 rpmx 10
minus4
minus1
minus05
0
05
1
D12
-Y (m
)
1000 2000 3000 40000Sampling numbers i
(b)
Power-Line interference 50and 150Hz
1X2X
3X 1010D11minus2770 rpm
08441166
1X2X3X
x 10minus5
200 400 600 800 10000f (Hz)
0
05
10
15
D11
-X (m
)
(c)
1X
2X
3XPower-Line interference 50
and 150Hz
D12minus2770 rpm117309021227
1X2X3X
x 10minus5
0
05
10
15
D12
-Y (m
)
200 400 600 800 10000f (Hz)
(d)
Figure 14 The time and frequency domains of the D11 and D12 sensors (a) The time domain of D11 (b) The time domain of D12 (c) Thefrequency domain of D11 (d) The frequency domain of D12
(11489) (21482)e first group
(31237) (41220)e second group
(30348) (40347)e second group
D11minus2770 rpmx 10
minus7
10 20 30 400 50Number of eigenvalues q
0
05
1
15
Eige
nval
ue o
f PCA
(m)
(a)
D12minus2770 rpm(11688) (21687)
e first group
e second group
(30414) (40413)e second group
(31673) (41664)
x 10minus7
0
05
1
15
2
Eige
nval
ue o
f PCA
(m)
10 20 30 40 500Number of eigenvalues q
(b)
Figure 15 The eigenvalues distribution of covariance matrix C (a) The eigenvalues distribution of D11 (b) The eigenvalues distribution ofD12
1X and its frequency doubling signal can be easily extractedto form axis orbit of revolving test bearing
Axis orbit was fabricated by 1X and 2Xof data of example 1(Section 42) through harmonic wavelet algorithm and theresults are demonstrated in Figure 20 The shape of axisorbit was trapped banana-like which is consistent with theresult obtained via CFSM-PCA It should be noted thatorbit in Figure 13 was more obvious than that in Figure 20ie W1ltW2 Meanwhile the amplitude fluctuation of timedomain signal was large (Figures 20(a) and 20(b)) Themajor reason for this phenomenon is that the harmonicwavelet which is a band filter in frequency domain inevitablyextracts the noise near the feature frequency Furthermorethis phenomenon also demonstrates that harmonic waveletis subject to the Heisenberg uncertainty principle ie thephenomenon of the amplitude leakage exists
(2) Comparison of CFSM-PCA with Wavelet Packet Besidesharmonic wavelets wavelet packet is also used to purify axisorbit [19ndash21] Purification results of axis orbit via Daubechieswavelet packet in Figure 21 exhibit that axis orbit was diver-gent seriously This was worse than that of harmonic waveletie W1ltW2ltW3The phenomenon of severe energy leakageoccurred during the extraction of 1X and 2X which cancause divergence of axis orbit Moreover the bandpass filterof wavelet packet also accounts for the divergent orbits
5 A Single Frequency Filtrationwith CFSM-PCA
The application in single frequency filtration is discussedin this section Taking D11 experimental data of example
Shock and Vibration 13
D11minus2770 rpmx 10
minus5
0 1000500Sampling numbers i
minus4
minus2
0
2
4
D11
-X (m
)
(a)
D12minus2770 rpmx 10
minus5
minus4
minus2
0
2
4
D12
-Y (m
)
500 10000Sampling numbers i
(b)
(1375 1185)(46 1009)
D11minus2770 rpmx 10
minus5
100 200 300 400 5000f (Hz)
0
05
1
15
D11
-X (m
)
(c)
(1375 1218)(46 1164)
D12minus2770 rpmx 10
minus5
100 200 300 400 5000f (Hz)
0
05
1
15
D12
-Y (m
)
(d)
Figure 16 Extracting the frequency spectrums of the first two frequencies of D11 and D12 by PCA algorithm (a) The time domain of D11(b) The time domain of D12 (c) The frequency spectrum of D11 (d) The frequency spectrum of D12
x 10minus5
x 10minus5
minus4
minus3
minus2
minus1
0
1
2
3
4
D12
-Y (m
)
210 3 4minus2minus3 minus1minus4
D11-X (m)
(a)0
0204
0608
10
24
t (s)D11-X (m) minus2
minus4
x 10minus5
x 10minus5
minus4
minus2
0
2
4
D12
-Y (m
)
(b)
Figure 17 The synthesis of axis orbit is to extract the first two times fundamental frequencies of the D11 and D12 signals by the CFSM-PCA(a) The axis orbit (b) Relationship between axis orbit and time
1 (Section 42) as a demo we attempt to filter the powerfrequency interference (50 Hz) from D11 by employingCFSM-PCA It can be seen from Figure 9(b) that sequenceof amplitude of 50 Hz in raw signal was 3 so eigenvec-tors corresponding to 5th and 6th eigenvalues in eigen-value distribution map were used to reconstruct Accordingto (8) the filtered signal (Figure 22) without 50 Hz was
obtained by subtracting reconstructed signal from originalone
Results in Figure 22 show that this algorithm filters out50 Hz power frequency interference and has no influenceon the other frequency components of raw signal demon-strating that this method has a potential application in notchfilter
14 Shock and Vibration
D11minus2770 rpm
x 10minus5
minus4
minus2
0
2
4
D11
-X (m
)
200 400 600 800 10000Sampling numbers i
(a)
D12minus2770 rpm
x 10minus5
0 400 600 800 1000200Sampling numbers i
minus4
minus2
0
2
4
D12
-Y (m
)
(b)
(1375 1185)(46 1009)
(92 0831)
D11minus2770 rpmx 10
minus5
00
05
10
15
D11
-X (m
)
100 200 300 400 5000f (Hz)
(c)
(1375 1218)(46 1163)
(92 0898)
D12minus2770 rpmx 10
minus5
00
05
10
15
D12
-Y (m
)
100 200 300 400 5000f (Hz)
(d)
Figure 18 Extracting the time and frequency spectrums of the first three frequencies of D11 and D12 by PCA algorithm (a)The time domainof D11 (b) The time domain of D12 (c) The frequency spectrum of D11 (d) The frequency spectrum of D12
x 10minus5
x 10minus5
minus4
minus3
minus2
minus1
0
1
2
3
4
D12
-Y (m
)
minus3 minus2 minus1minus4 1 2 3 40D11-X (m)
(a)
1008
0604
0200
02
4D11-X (m)
t (s)
minus2
minus4
x 10minus5
x 10minus5
minus4
minus2
0
2
4
D12
-Y (m
)
(b)
Figure 19 The synthesis of axis orbit is to extract the first three times fundamental frequencies of the D11 and D12 by the CFSM-PCA (a)The axis orbit (b) Relationship between axis orbit and time
6 Conclusion
The selection of effective eigenvalues in feature distributionchart of covariance matrix C is crucial in PCA algorithm Inthis paper the relationships between effective eigenvalues andsignal frequency and amplitude are discovered and a series ofconclusions are summarized
First in Hankel matrix mode each valid frequencycomponent of signal only produces two adjacent nonzeroeigenvalues Second the number of valid eigenvalues incharacteristic distribution chart which is independent ofnumerical values of 119891119894 119860 119894 and 120593119894 is related to the number offrequency componentsThird the sequence of signal effectiveeigenvalues in characteristic distribution map is determined
Shock and Vibration 15
D11-1080 rpm
x 10minus5
minus20
minus15
minus05
0
05
10
15
20
D11
-X (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(a)
D12-1080 rpmx 10
minus5
minus15
minus1
minus05
0
05
1
15
D12
-Y (m
)
500 1000 1500 2000 2500 30000 40003500Sampling numbers i
(b)
(16 07335)
(32 04771)
D11-1080 rpm
x 10minus5
10 20 30 40 50 60 70 80 90 1000f (Hz)
0010203040506070809
1
D11
-X (m
)
(c)
(16 06571)
(32 05228)
D12-1080 rpm
x 10minus5
0010203040506070809
1
D12
-Y (m
)
10 20 30 40 50 60 70 80 90 1000f (Hz)
(d)
W2
x 10minus5
x 10minus5
minus15
minus1
minus05
0
05
1
15
Y(D
12) (
m)
minus1 minus05minus15 05 1 15 20X(D11) (m)
(e)
Figure 20The purification effect of harmonic wavelets (a)The time domain of D11 (b)The time domain of D12 (c)The frequency spectrumof D11 (d) The frequency spectrum of D12 (e) The orbit is to extract the first 2 frequencies
by amplitude 119860 119894 of signal frequency component The largermagnitude of amplitude is the greater eigenvalues are andthe higher arrangement will be
Based on aforesaid principle a new CFSM-PCA wasdeveloped and has become an effective tool to extract oreliminate specific characteristic (single frequency) even dif-ference of two frequencies is only 1 Hz Purification of rotororbit using this algorithm shows significant improvement
than those of existing methods such as wavelet packet andharmonic wavelet packet Although the proposed CFSM-PCA is effective some drawbacks are unavoidable Forexample this method suffers from similar issue to otheralgorithms that is the reconstruction error increaseswith thedecrease of SNR In addition the applications of the CFSM-PCA algorithm in purification of speech recognition bearingfault diagnosis and fault diagnosis of power system should
16 Shock and Vibration
D11-1080 rpmx 10
minus5
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
minus20
minus15
minus05
0
05
10
15
20
D11
-X (m
)
(a)
D12-1080 rpmx 10
minus5
minus15
minus1
minus05
0
05
1
15
D12
-Y (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(b)
(16 06557)
(32 04381)
D11-1080 rpmx 10
minus5
10020 30 40 50 60 70 80 90100f (Hz)
0010203040506070809
1
D11
-X (m
)
(c)
(16 06452)(32 04927)
D12-1080 rpm
x 10minus5
0010203040506070809
1
D12
-Y (m
)
9020 30 40 50 60 70 8010 1000f (Hz)
(d)
W3
x 10minus5
x 10minus5
minus15
minus1
minus05
0
05
1
15
Y(D
12) (
m)
minus15 minus05minus1 05 1 15 20X(D11) (m)
(e)
Figure 21 The purification effect of wavelet packet (a) The time domain of D11 (b) The time domain of D12 (c) The frequency spectrum ofD11 (d) The frequency spectrum of D12 (e) The orbit is to extract the first 2 frequencies
be further exploredWe fully believe that the CFSM-PCAwillsparkle in diverse fields
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work is supported by the National High TechnologyResearch andDevelopment Program of China (863 Program)
Shock and Vibration 17
(18 0734)
(36 04634)
(54 02448) (72 02351)(90 01646)
x 10minus5
50 100 150 2000f (Hz)
0
02
04
06
08
1
D11
minusX
(m)
Figure 22 Frequency spectrum of D11 signal after filtering 50Hz
(no 2015AA043005) and the National Natural Science Foun-dation of China (no 51375187)
References
[1] H Abdi and L J Williams ldquoPrincipal component analysisrdquoWiley Interdisciplinary Reviews Computational Statistics vol 2no 4 pp 433ndash459 2010
[2] A Widodo B Yang and T Han ldquoCombination of independentcomponent analysis and support vectormachines for intelligentfaults diagnosis of induction motorsrdquo Expert Systems withApplications vol 32 no 2 pp 299ndash312 2007
[3] S Wold K Esbensen and P Geladi ldquoPrincipal componentanalysisrdquo Chemometrics and Intelligent Laboratory Systems vol2 no 1ndash3 pp 37ndash52 1987
[4] M Kirby and L Sirovich ldquoApplication of the Karhunen-Loeve procedure for the characterization of human facesrdquo IEEETransactions on Pattern Analysis and Machine Intelligence vol12 no 1 pp 103ndash108 1990
[5] J H Xi Y Z Han and R H Su ldquoNew fault diagnosis methodfor rolling bearing based on PCArdquo in Proceedings of the 25thChinese Control and Decision Conference CCDC rsquo13 pp 4123ndash4127 IEEE May 2013
[6] A Malhi and R X Gao ldquoPCA-based feature selection schemefor machine defect classificationrdquo IEEE Transactions on Instru-mentation and Measurement vol 53 no 6 pp 1517ndash1525 2004
[7] S Xu X Jiang J Huang S Yang and X Wang ldquoBayesianwavelet PCA methodology for turbomachinery damage diag-nosis under uncertaintyrdquo Mechanical Systems and Signal Pro-cessing vol 80 pp 1ndash18 2016
[8] W Sun J Chen and J Li ldquoDecision tree and PCA-based faultdiagnosis of rotatingmachineryrdquoMechanical Systems and SignalProcessing vol 21 no 3 pp 1300ndash1317 2007
[9] Z X Li X P Yan C Q Yuan Z Peng and L Li ldquoVirtual pro-totype and experimental research on gear multi-fault diagnosisusing wavelet-autoregressive model and principal componentanalysismethodrdquoMechanical Systems and Signal Processing vol25 no 7 pp 2589ndash2607 2011
[10] K Y Shao M M Cai and G F Zhao ldquoRolling bearing faultdiagnosis based onwavelet energy spectrum PCA andPNNrdquo inProceedings of the 26th Chinese Control andDecisionConferenceCCDC rsquo14 pp 800ndash804 IEEE 2014
[11] T Qiu Q F Zhang and Y J Ding ldquoNonlinear sensor faultdetection and data rebuilding based on principle componentanalysisrdquo Journal of Tsinghua University vol 46 no 5 pp 708ndash711 2006
[12] Y K Gu Z X Yang and F L Zhu ldquoGearbox fault feature fusionbased on principal component analysisrdquo China MechanicalEngineering vol 26 no 11 pp 1532ndash1537 2015
[13] R Dunia S J Qin T F Edgar and T J McAvoy ldquoIdentificationof faulty sensors using principal component analysisrdquo AIChEJournal vol 42 no 10 pp 2797ndash2811 1996
[14] E P de Moura C R Souto A A Silva and M A S IrmaoldquoEvaluation of principal component analysis and neural net-work performance for bearing fault diagnosis from vibrationsignal processed by RS and DF analysesrdquo Mechanical Systemsand Signal Processing vol 25 no 5 pp 1765ndash1772 2011
[15] S Golestan M Monfared F D Freijedo and J M GuerreroldquoDynamics assessment of advanced single-phase PLL struc-turesrdquo IEEE Transactions on Industrial Electronics vol 60 no6 pp 2167ndash2177 2013
[16] A Lima H Zen Y Nankaku C Miyajima K Tokuda andT Kitamura ldquoOn the use of kernel PCA for feature extractionin speech recognitionrdquo IEICE Transaction on Information andSystems vol 12 no 87 pp 2802ndash2811 2004
[17] J D Zheng ldquoImproved hilbert-huang transform and its appli-cations to rolling bearing fault diagnosisrdquo Journal of MechanicalEngineering vol 1 no 51 pp 138ndash145 2015
[18] G He K Ding and H Lin ldquoFault feature extraction of rollingelement bearings using sparse representationrdquo Journal of Soundand Vibration vol 366 pp 514ndash527 2016
[19] J G Yang S B Xia and Y G Liu ldquoWavelet denoising and itsapplication in extraction of shaft centerline orbit featuresrdquoJournal of Harbin Institute of Technology vol 31 no 5 pp 52ndash541999 (Chinese)
[20] J A Duan and X D Zhang ldquoFeature extraction from the orbitof the center of a rotor based on wavelet transformrdquo Journal ofVibration Measurement amp Diagnosis vol 17 no 01 pp 31ndash341997
[21] J Han D X Jiang and W D Ning ldquoApplication of optimalwavelet packets in failure signal extraction of rotor orbitrdquoTurbine Technology vol 43 no 3 pp 133ndash136 2001
[22] W B Zhang X J Zhou andY Lin ldquoRefinement of rotor centerrsquosorbit by a harmonic window methodrdquo Journal of VibrationMeasurement amp Diagnosis vol 30 no 1 pp 87ndash90 2010
[23] S M Li G Z Lu and Q Y Xu ldquoOn obtaining accuraterotor sub-frequency signal with harmonic waveletrdquo Journal ofNorthwestern Polytechincal University vol 19 no 2 pp 220ndash224 2001
[24] F J Wu and L S Qu ldquoAn improved method for restraining theend effect in empiricalmode decomposition and its applicationsto the fault diagnosis of large rotating machineryrdquo Journal ofSound and Vibration vol 314 no 3-5 pp 586ndash602 2008
[25] W G Liu ldquoResearch on method of purification of shaft orbitbased on singular value decompositionrdquo Technical Acousticsvol 5 no 35 pp 30ndash35 2016
[26] X Z Zhao and B Y Ye ldquoThe influence of formation manner ofcomponent on signal processingrdquo Shang Hai Jiao Tong Univer-sity vol 45 no 3 pp 368ndash374 2011
[27] X Z Zhao B Y Ye and T J Chen ldquoPrinciple of singularitydetection based on SVD and its applicationrdquo Journal of Vibra-tion and Shock vol 6 no 27 pp 7ndash10 2008
[28] X Z Zhao B Y Ye and T J Chen ldquoDifference spectrum theoryof singular value and its application to the fault diagnosis ofheadstock of latherdquo Journal of Mechanical Engineering vol 46pp 100ndash108 2010
[29] I A Kougioumtzoglou and P D Spanos ldquoAn identificationapproach for linear and nonlinear time-variant structural sys-tems via harmonic waveletsrdquo Mechanical Systems and SignalProcessing vol 37 no 1-2 pp 338ndash352 2013
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
12 Shock and Vibration
D11minus2770 rpmx 10
minus4
minus1
minus05
0
05
1
D11
-X (m
)
1000 2000 3000 40000Sampling numbers i
(a)
D12minus2770 rpmx 10
minus4
minus1
minus05
0
05
1
D12
-Y (m
)
1000 2000 3000 40000Sampling numbers i
(b)
Power-Line interference 50and 150Hz
1X2X
3X 1010D11minus2770 rpm
08441166
1X2X3X
x 10minus5
200 400 600 800 10000f (Hz)
0
05
10
15
D11
-X (m
)
(c)
1X
2X
3XPower-Line interference 50
and 150Hz
D12minus2770 rpm117309021227
1X2X3X
x 10minus5
0
05
10
15
D12
-Y (m
)
200 400 600 800 10000f (Hz)
(d)
Figure 14 The time and frequency domains of the D11 and D12 sensors (a) The time domain of D11 (b) The time domain of D12 (c) Thefrequency domain of D11 (d) The frequency domain of D12
(11489) (21482)e first group
(31237) (41220)e second group
(30348) (40347)e second group
D11minus2770 rpmx 10
minus7
10 20 30 400 50Number of eigenvalues q
0
05
1
15
Eige
nval
ue o
f PCA
(m)
(a)
D12minus2770 rpm(11688) (21687)
e first group
e second group
(30414) (40413)e second group
(31673) (41664)
x 10minus7
0
05
1
15
2
Eige
nval
ue o
f PCA
(m)
10 20 30 40 500Number of eigenvalues q
(b)
Figure 15 The eigenvalues distribution of covariance matrix C (a) The eigenvalues distribution of D11 (b) The eigenvalues distribution ofD12
1X and its frequency doubling signal can be easily extractedto form axis orbit of revolving test bearing
Axis orbit was fabricated by 1X and 2Xof data of example 1(Section 42) through harmonic wavelet algorithm and theresults are demonstrated in Figure 20 The shape of axisorbit was trapped banana-like which is consistent with theresult obtained via CFSM-PCA It should be noted thatorbit in Figure 13 was more obvious than that in Figure 20ie W1ltW2 Meanwhile the amplitude fluctuation of timedomain signal was large (Figures 20(a) and 20(b)) Themajor reason for this phenomenon is that the harmonicwavelet which is a band filter in frequency domain inevitablyextracts the noise near the feature frequency Furthermorethis phenomenon also demonstrates that harmonic waveletis subject to the Heisenberg uncertainty principle ie thephenomenon of the amplitude leakage exists
(2) Comparison of CFSM-PCA with Wavelet Packet Besidesharmonic wavelets wavelet packet is also used to purify axisorbit [19ndash21] Purification results of axis orbit via Daubechieswavelet packet in Figure 21 exhibit that axis orbit was diver-gent seriously This was worse than that of harmonic waveletie W1ltW2ltW3The phenomenon of severe energy leakageoccurred during the extraction of 1X and 2X which cancause divergence of axis orbit Moreover the bandpass filterof wavelet packet also accounts for the divergent orbits
5 A Single Frequency Filtrationwith CFSM-PCA
The application in single frequency filtration is discussedin this section Taking D11 experimental data of example
Shock and Vibration 13
D11minus2770 rpmx 10
minus5
0 1000500Sampling numbers i
minus4
minus2
0
2
4
D11
-X (m
)
(a)
D12minus2770 rpmx 10
minus5
minus4
minus2
0
2
4
D12
-Y (m
)
500 10000Sampling numbers i
(b)
(1375 1185)(46 1009)
D11minus2770 rpmx 10
minus5
100 200 300 400 5000f (Hz)
0
05
1
15
D11
-X (m
)
(c)
(1375 1218)(46 1164)
D12minus2770 rpmx 10
minus5
100 200 300 400 5000f (Hz)
0
05
1
15
D12
-Y (m
)
(d)
Figure 16 Extracting the frequency spectrums of the first two frequencies of D11 and D12 by PCA algorithm (a) The time domain of D11(b) The time domain of D12 (c) The frequency spectrum of D11 (d) The frequency spectrum of D12
x 10minus5
x 10minus5
minus4
minus3
minus2
minus1
0
1
2
3
4
D12
-Y (m
)
210 3 4minus2minus3 minus1minus4
D11-X (m)
(a)0
0204
0608
10
24
t (s)D11-X (m) minus2
minus4
x 10minus5
x 10minus5
minus4
minus2
0
2
4
D12
-Y (m
)
(b)
Figure 17 The synthesis of axis orbit is to extract the first two times fundamental frequencies of the D11 and D12 signals by the CFSM-PCA(a) The axis orbit (b) Relationship between axis orbit and time
1 (Section 42) as a demo we attempt to filter the powerfrequency interference (50 Hz) from D11 by employingCFSM-PCA It can be seen from Figure 9(b) that sequenceof amplitude of 50 Hz in raw signal was 3 so eigenvec-tors corresponding to 5th and 6th eigenvalues in eigen-value distribution map were used to reconstruct Accordingto (8) the filtered signal (Figure 22) without 50 Hz was
obtained by subtracting reconstructed signal from originalone
Results in Figure 22 show that this algorithm filters out50 Hz power frequency interference and has no influenceon the other frequency components of raw signal demon-strating that this method has a potential application in notchfilter
14 Shock and Vibration
D11minus2770 rpm
x 10minus5
minus4
minus2
0
2
4
D11
-X (m
)
200 400 600 800 10000Sampling numbers i
(a)
D12minus2770 rpm
x 10minus5
0 400 600 800 1000200Sampling numbers i
minus4
minus2
0
2
4
D12
-Y (m
)
(b)
(1375 1185)(46 1009)
(92 0831)
D11minus2770 rpmx 10
minus5
00
05
10
15
D11
-X (m
)
100 200 300 400 5000f (Hz)
(c)
(1375 1218)(46 1163)
(92 0898)
D12minus2770 rpmx 10
minus5
00
05
10
15
D12
-Y (m
)
100 200 300 400 5000f (Hz)
(d)
Figure 18 Extracting the time and frequency spectrums of the first three frequencies of D11 and D12 by PCA algorithm (a)The time domainof D11 (b) The time domain of D12 (c) The frequency spectrum of D11 (d) The frequency spectrum of D12
x 10minus5
x 10minus5
minus4
minus3
minus2
minus1
0
1
2
3
4
D12
-Y (m
)
minus3 minus2 minus1minus4 1 2 3 40D11-X (m)
(a)
1008
0604
0200
02
4D11-X (m)
t (s)
minus2
minus4
x 10minus5
x 10minus5
minus4
minus2
0
2
4
D12
-Y (m
)
(b)
Figure 19 The synthesis of axis orbit is to extract the first three times fundamental frequencies of the D11 and D12 by the CFSM-PCA (a)The axis orbit (b) Relationship between axis orbit and time
6 Conclusion
The selection of effective eigenvalues in feature distributionchart of covariance matrix C is crucial in PCA algorithm Inthis paper the relationships between effective eigenvalues andsignal frequency and amplitude are discovered and a series ofconclusions are summarized
First in Hankel matrix mode each valid frequencycomponent of signal only produces two adjacent nonzeroeigenvalues Second the number of valid eigenvalues incharacteristic distribution chart which is independent ofnumerical values of 119891119894 119860 119894 and 120593119894 is related to the number offrequency componentsThird the sequence of signal effectiveeigenvalues in characteristic distribution map is determined
Shock and Vibration 15
D11-1080 rpm
x 10minus5
minus20
minus15
minus05
0
05
10
15
20
D11
-X (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(a)
D12-1080 rpmx 10
minus5
minus15
minus1
minus05
0
05
1
15
D12
-Y (m
)
500 1000 1500 2000 2500 30000 40003500Sampling numbers i
(b)
(16 07335)
(32 04771)
D11-1080 rpm
x 10minus5
10 20 30 40 50 60 70 80 90 1000f (Hz)
0010203040506070809
1
D11
-X (m
)
(c)
(16 06571)
(32 05228)
D12-1080 rpm
x 10minus5
0010203040506070809
1
D12
-Y (m
)
10 20 30 40 50 60 70 80 90 1000f (Hz)
(d)
W2
x 10minus5
x 10minus5
minus15
minus1
minus05
0
05
1
15
Y(D
12) (
m)
minus1 minus05minus15 05 1 15 20X(D11) (m)
(e)
Figure 20The purification effect of harmonic wavelets (a)The time domain of D11 (b)The time domain of D12 (c)The frequency spectrumof D11 (d) The frequency spectrum of D12 (e) The orbit is to extract the first 2 frequencies
by amplitude 119860 119894 of signal frequency component The largermagnitude of amplitude is the greater eigenvalues are andthe higher arrangement will be
Based on aforesaid principle a new CFSM-PCA wasdeveloped and has become an effective tool to extract oreliminate specific characteristic (single frequency) even dif-ference of two frequencies is only 1 Hz Purification of rotororbit using this algorithm shows significant improvement
than those of existing methods such as wavelet packet andharmonic wavelet packet Although the proposed CFSM-PCA is effective some drawbacks are unavoidable Forexample this method suffers from similar issue to otheralgorithms that is the reconstruction error increaseswith thedecrease of SNR In addition the applications of the CFSM-PCA algorithm in purification of speech recognition bearingfault diagnosis and fault diagnosis of power system should
16 Shock and Vibration
D11-1080 rpmx 10
minus5
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
minus20
minus15
minus05
0
05
10
15
20
D11
-X (m
)
(a)
D12-1080 rpmx 10
minus5
minus15
minus1
minus05
0
05
1
15
D12
-Y (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(b)
(16 06557)
(32 04381)
D11-1080 rpmx 10
minus5
10020 30 40 50 60 70 80 90100f (Hz)
0010203040506070809
1
D11
-X (m
)
(c)
(16 06452)(32 04927)
D12-1080 rpm
x 10minus5
0010203040506070809
1
D12
-Y (m
)
9020 30 40 50 60 70 8010 1000f (Hz)
(d)
W3
x 10minus5
x 10minus5
minus15
minus1
minus05
0
05
1
15
Y(D
12) (
m)
minus15 minus05minus1 05 1 15 20X(D11) (m)
(e)
Figure 21 The purification effect of wavelet packet (a) The time domain of D11 (b) The time domain of D12 (c) The frequency spectrum ofD11 (d) The frequency spectrum of D12 (e) The orbit is to extract the first 2 frequencies
be further exploredWe fully believe that the CFSM-PCAwillsparkle in diverse fields
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work is supported by the National High TechnologyResearch andDevelopment Program of China (863 Program)
Shock and Vibration 17
(18 0734)
(36 04634)
(54 02448) (72 02351)(90 01646)
x 10minus5
50 100 150 2000f (Hz)
0
02
04
06
08
1
D11
minusX
(m)
Figure 22 Frequency spectrum of D11 signal after filtering 50Hz
(no 2015AA043005) and the National Natural Science Foun-dation of China (no 51375187)
References
[1] H Abdi and L J Williams ldquoPrincipal component analysisrdquoWiley Interdisciplinary Reviews Computational Statistics vol 2no 4 pp 433ndash459 2010
[2] A Widodo B Yang and T Han ldquoCombination of independentcomponent analysis and support vectormachines for intelligentfaults diagnosis of induction motorsrdquo Expert Systems withApplications vol 32 no 2 pp 299ndash312 2007
[3] S Wold K Esbensen and P Geladi ldquoPrincipal componentanalysisrdquo Chemometrics and Intelligent Laboratory Systems vol2 no 1ndash3 pp 37ndash52 1987
[4] M Kirby and L Sirovich ldquoApplication of the Karhunen-Loeve procedure for the characterization of human facesrdquo IEEETransactions on Pattern Analysis and Machine Intelligence vol12 no 1 pp 103ndash108 1990
[5] J H Xi Y Z Han and R H Su ldquoNew fault diagnosis methodfor rolling bearing based on PCArdquo in Proceedings of the 25thChinese Control and Decision Conference CCDC rsquo13 pp 4123ndash4127 IEEE May 2013
[6] A Malhi and R X Gao ldquoPCA-based feature selection schemefor machine defect classificationrdquo IEEE Transactions on Instru-mentation and Measurement vol 53 no 6 pp 1517ndash1525 2004
[7] S Xu X Jiang J Huang S Yang and X Wang ldquoBayesianwavelet PCA methodology for turbomachinery damage diag-nosis under uncertaintyrdquo Mechanical Systems and Signal Pro-cessing vol 80 pp 1ndash18 2016
[8] W Sun J Chen and J Li ldquoDecision tree and PCA-based faultdiagnosis of rotatingmachineryrdquoMechanical Systems and SignalProcessing vol 21 no 3 pp 1300ndash1317 2007
[9] Z X Li X P Yan C Q Yuan Z Peng and L Li ldquoVirtual pro-totype and experimental research on gear multi-fault diagnosisusing wavelet-autoregressive model and principal componentanalysismethodrdquoMechanical Systems and Signal Processing vol25 no 7 pp 2589ndash2607 2011
[10] K Y Shao M M Cai and G F Zhao ldquoRolling bearing faultdiagnosis based onwavelet energy spectrum PCA andPNNrdquo inProceedings of the 26th Chinese Control andDecisionConferenceCCDC rsquo14 pp 800ndash804 IEEE 2014
[11] T Qiu Q F Zhang and Y J Ding ldquoNonlinear sensor faultdetection and data rebuilding based on principle componentanalysisrdquo Journal of Tsinghua University vol 46 no 5 pp 708ndash711 2006
[12] Y K Gu Z X Yang and F L Zhu ldquoGearbox fault feature fusionbased on principal component analysisrdquo China MechanicalEngineering vol 26 no 11 pp 1532ndash1537 2015
[13] R Dunia S J Qin T F Edgar and T J McAvoy ldquoIdentificationof faulty sensors using principal component analysisrdquo AIChEJournal vol 42 no 10 pp 2797ndash2811 1996
[14] E P de Moura C R Souto A A Silva and M A S IrmaoldquoEvaluation of principal component analysis and neural net-work performance for bearing fault diagnosis from vibrationsignal processed by RS and DF analysesrdquo Mechanical Systemsand Signal Processing vol 25 no 5 pp 1765ndash1772 2011
[15] S Golestan M Monfared F D Freijedo and J M GuerreroldquoDynamics assessment of advanced single-phase PLL struc-turesrdquo IEEE Transactions on Industrial Electronics vol 60 no6 pp 2167ndash2177 2013
[16] A Lima H Zen Y Nankaku C Miyajima K Tokuda andT Kitamura ldquoOn the use of kernel PCA for feature extractionin speech recognitionrdquo IEICE Transaction on Information andSystems vol 12 no 87 pp 2802ndash2811 2004
[17] J D Zheng ldquoImproved hilbert-huang transform and its appli-cations to rolling bearing fault diagnosisrdquo Journal of MechanicalEngineering vol 1 no 51 pp 138ndash145 2015
[18] G He K Ding and H Lin ldquoFault feature extraction of rollingelement bearings using sparse representationrdquo Journal of Soundand Vibration vol 366 pp 514ndash527 2016
[19] J G Yang S B Xia and Y G Liu ldquoWavelet denoising and itsapplication in extraction of shaft centerline orbit featuresrdquoJournal of Harbin Institute of Technology vol 31 no 5 pp 52ndash541999 (Chinese)
[20] J A Duan and X D Zhang ldquoFeature extraction from the orbitof the center of a rotor based on wavelet transformrdquo Journal ofVibration Measurement amp Diagnosis vol 17 no 01 pp 31ndash341997
[21] J Han D X Jiang and W D Ning ldquoApplication of optimalwavelet packets in failure signal extraction of rotor orbitrdquoTurbine Technology vol 43 no 3 pp 133ndash136 2001
[22] W B Zhang X J Zhou andY Lin ldquoRefinement of rotor centerrsquosorbit by a harmonic window methodrdquo Journal of VibrationMeasurement amp Diagnosis vol 30 no 1 pp 87ndash90 2010
[23] S M Li G Z Lu and Q Y Xu ldquoOn obtaining accuraterotor sub-frequency signal with harmonic waveletrdquo Journal ofNorthwestern Polytechincal University vol 19 no 2 pp 220ndash224 2001
[24] F J Wu and L S Qu ldquoAn improved method for restraining theend effect in empiricalmode decomposition and its applicationsto the fault diagnosis of large rotating machineryrdquo Journal ofSound and Vibration vol 314 no 3-5 pp 586ndash602 2008
[25] W G Liu ldquoResearch on method of purification of shaft orbitbased on singular value decompositionrdquo Technical Acousticsvol 5 no 35 pp 30ndash35 2016
[26] X Z Zhao and B Y Ye ldquoThe influence of formation manner ofcomponent on signal processingrdquo Shang Hai Jiao Tong Univer-sity vol 45 no 3 pp 368ndash374 2011
[27] X Z Zhao B Y Ye and T J Chen ldquoPrinciple of singularitydetection based on SVD and its applicationrdquo Journal of Vibra-tion and Shock vol 6 no 27 pp 7ndash10 2008
[28] X Z Zhao B Y Ye and T J Chen ldquoDifference spectrum theoryof singular value and its application to the fault diagnosis ofheadstock of latherdquo Journal of Mechanical Engineering vol 46pp 100ndash108 2010
[29] I A Kougioumtzoglou and P D Spanos ldquoAn identificationapproach for linear and nonlinear time-variant structural sys-tems via harmonic waveletsrdquo Mechanical Systems and SignalProcessing vol 37 no 1-2 pp 338ndash352 2013
International Journal of
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Active and Passive Electronic Components
VLSI Design
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Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
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Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
Shock and Vibration 13
D11minus2770 rpmx 10
minus5
0 1000500Sampling numbers i
minus4
minus2
0
2
4
D11
-X (m
)
(a)
D12minus2770 rpmx 10
minus5
minus4
minus2
0
2
4
D12
-Y (m
)
500 10000Sampling numbers i
(b)
(1375 1185)(46 1009)
D11minus2770 rpmx 10
minus5
100 200 300 400 5000f (Hz)
0
05
1
15
D11
-X (m
)
(c)
(1375 1218)(46 1164)
D12minus2770 rpmx 10
minus5
100 200 300 400 5000f (Hz)
0
05
1
15
D12
-Y (m
)
(d)
Figure 16 Extracting the frequency spectrums of the first two frequencies of D11 and D12 by PCA algorithm (a) The time domain of D11(b) The time domain of D12 (c) The frequency spectrum of D11 (d) The frequency spectrum of D12
x 10minus5
x 10minus5
minus4
minus3
minus2
minus1
0
1
2
3
4
D12
-Y (m
)
210 3 4minus2minus3 minus1minus4
D11-X (m)
(a)0
0204
0608
10
24
t (s)D11-X (m) minus2
minus4
x 10minus5
x 10minus5
minus4
minus2
0
2
4
D12
-Y (m
)
(b)
Figure 17 The synthesis of axis orbit is to extract the first two times fundamental frequencies of the D11 and D12 signals by the CFSM-PCA(a) The axis orbit (b) Relationship between axis orbit and time
1 (Section 42) as a demo we attempt to filter the powerfrequency interference (50 Hz) from D11 by employingCFSM-PCA It can be seen from Figure 9(b) that sequenceof amplitude of 50 Hz in raw signal was 3 so eigenvec-tors corresponding to 5th and 6th eigenvalues in eigen-value distribution map were used to reconstruct Accordingto (8) the filtered signal (Figure 22) without 50 Hz was
obtained by subtracting reconstructed signal from originalone
Results in Figure 22 show that this algorithm filters out50 Hz power frequency interference and has no influenceon the other frequency components of raw signal demon-strating that this method has a potential application in notchfilter
14 Shock and Vibration
D11minus2770 rpm
x 10minus5
minus4
minus2
0
2
4
D11
-X (m
)
200 400 600 800 10000Sampling numbers i
(a)
D12minus2770 rpm
x 10minus5
0 400 600 800 1000200Sampling numbers i
minus4
minus2
0
2
4
D12
-Y (m
)
(b)
(1375 1185)(46 1009)
(92 0831)
D11minus2770 rpmx 10
minus5
00
05
10
15
D11
-X (m
)
100 200 300 400 5000f (Hz)
(c)
(1375 1218)(46 1163)
(92 0898)
D12minus2770 rpmx 10
minus5
00
05
10
15
D12
-Y (m
)
100 200 300 400 5000f (Hz)
(d)
Figure 18 Extracting the time and frequency spectrums of the first three frequencies of D11 and D12 by PCA algorithm (a)The time domainof D11 (b) The time domain of D12 (c) The frequency spectrum of D11 (d) The frequency spectrum of D12
x 10minus5
x 10minus5
minus4
minus3
minus2
minus1
0
1
2
3
4
D12
-Y (m
)
minus3 minus2 minus1minus4 1 2 3 40D11-X (m)
(a)
1008
0604
0200
02
4D11-X (m)
t (s)
minus2
minus4
x 10minus5
x 10minus5
minus4
minus2
0
2
4
D12
-Y (m
)
(b)
Figure 19 The synthesis of axis orbit is to extract the first three times fundamental frequencies of the D11 and D12 by the CFSM-PCA (a)The axis orbit (b) Relationship between axis orbit and time
6 Conclusion
The selection of effective eigenvalues in feature distributionchart of covariance matrix C is crucial in PCA algorithm Inthis paper the relationships between effective eigenvalues andsignal frequency and amplitude are discovered and a series ofconclusions are summarized
First in Hankel matrix mode each valid frequencycomponent of signal only produces two adjacent nonzeroeigenvalues Second the number of valid eigenvalues incharacteristic distribution chart which is independent ofnumerical values of 119891119894 119860 119894 and 120593119894 is related to the number offrequency componentsThird the sequence of signal effectiveeigenvalues in characteristic distribution map is determined
Shock and Vibration 15
D11-1080 rpm
x 10minus5
minus20
minus15
minus05
0
05
10
15
20
D11
-X (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(a)
D12-1080 rpmx 10
minus5
minus15
minus1
minus05
0
05
1
15
D12
-Y (m
)
500 1000 1500 2000 2500 30000 40003500Sampling numbers i
(b)
(16 07335)
(32 04771)
D11-1080 rpm
x 10minus5
10 20 30 40 50 60 70 80 90 1000f (Hz)
0010203040506070809
1
D11
-X (m
)
(c)
(16 06571)
(32 05228)
D12-1080 rpm
x 10minus5
0010203040506070809
1
D12
-Y (m
)
10 20 30 40 50 60 70 80 90 1000f (Hz)
(d)
W2
x 10minus5
x 10minus5
minus15
minus1
minus05
0
05
1
15
Y(D
12) (
m)
minus1 minus05minus15 05 1 15 20X(D11) (m)
(e)
Figure 20The purification effect of harmonic wavelets (a)The time domain of D11 (b)The time domain of D12 (c)The frequency spectrumof D11 (d) The frequency spectrum of D12 (e) The orbit is to extract the first 2 frequencies
by amplitude 119860 119894 of signal frequency component The largermagnitude of amplitude is the greater eigenvalues are andthe higher arrangement will be
Based on aforesaid principle a new CFSM-PCA wasdeveloped and has become an effective tool to extract oreliminate specific characteristic (single frequency) even dif-ference of two frequencies is only 1 Hz Purification of rotororbit using this algorithm shows significant improvement
than those of existing methods such as wavelet packet andharmonic wavelet packet Although the proposed CFSM-PCA is effective some drawbacks are unavoidable Forexample this method suffers from similar issue to otheralgorithms that is the reconstruction error increaseswith thedecrease of SNR In addition the applications of the CFSM-PCA algorithm in purification of speech recognition bearingfault diagnosis and fault diagnosis of power system should
16 Shock and Vibration
D11-1080 rpmx 10
minus5
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
minus20
minus15
minus05
0
05
10
15
20
D11
-X (m
)
(a)
D12-1080 rpmx 10
minus5
minus15
minus1
minus05
0
05
1
15
D12
-Y (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(b)
(16 06557)
(32 04381)
D11-1080 rpmx 10
minus5
10020 30 40 50 60 70 80 90100f (Hz)
0010203040506070809
1
D11
-X (m
)
(c)
(16 06452)(32 04927)
D12-1080 rpm
x 10minus5
0010203040506070809
1
D12
-Y (m
)
9020 30 40 50 60 70 8010 1000f (Hz)
(d)
W3
x 10minus5
x 10minus5
minus15
minus1
minus05
0
05
1
15
Y(D
12) (
m)
minus15 minus05minus1 05 1 15 20X(D11) (m)
(e)
Figure 21 The purification effect of wavelet packet (a) The time domain of D11 (b) The time domain of D12 (c) The frequency spectrum ofD11 (d) The frequency spectrum of D12 (e) The orbit is to extract the first 2 frequencies
be further exploredWe fully believe that the CFSM-PCAwillsparkle in diverse fields
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work is supported by the National High TechnologyResearch andDevelopment Program of China (863 Program)
Shock and Vibration 17
(18 0734)
(36 04634)
(54 02448) (72 02351)(90 01646)
x 10minus5
50 100 150 2000f (Hz)
0
02
04
06
08
1
D11
minusX
(m)
Figure 22 Frequency spectrum of D11 signal after filtering 50Hz
(no 2015AA043005) and the National Natural Science Foun-dation of China (no 51375187)
References
[1] H Abdi and L J Williams ldquoPrincipal component analysisrdquoWiley Interdisciplinary Reviews Computational Statistics vol 2no 4 pp 433ndash459 2010
[2] A Widodo B Yang and T Han ldquoCombination of independentcomponent analysis and support vectormachines for intelligentfaults diagnosis of induction motorsrdquo Expert Systems withApplications vol 32 no 2 pp 299ndash312 2007
[3] S Wold K Esbensen and P Geladi ldquoPrincipal componentanalysisrdquo Chemometrics and Intelligent Laboratory Systems vol2 no 1ndash3 pp 37ndash52 1987
[4] M Kirby and L Sirovich ldquoApplication of the Karhunen-Loeve procedure for the characterization of human facesrdquo IEEETransactions on Pattern Analysis and Machine Intelligence vol12 no 1 pp 103ndash108 1990
[5] J H Xi Y Z Han and R H Su ldquoNew fault diagnosis methodfor rolling bearing based on PCArdquo in Proceedings of the 25thChinese Control and Decision Conference CCDC rsquo13 pp 4123ndash4127 IEEE May 2013
[6] A Malhi and R X Gao ldquoPCA-based feature selection schemefor machine defect classificationrdquo IEEE Transactions on Instru-mentation and Measurement vol 53 no 6 pp 1517ndash1525 2004
[7] S Xu X Jiang J Huang S Yang and X Wang ldquoBayesianwavelet PCA methodology for turbomachinery damage diag-nosis under uncertaintyrdquo Mechanical Systems and Signal Pro-cessing vol 80 pp 1ndash18 2016
[8] W Sun J Chen and J Li ldquoDecision tree and PCA-based faultdiagnosis of rotatingmachineryrdquoMechanical Systems and SignalProcessing vol 21 no 3 pp 1300ndash1317 2007
[9] Z X Li X P Yan C Q Yuan Z Peng and L Li ldquoVirtual pro-totype and experimental research on gear multi-fault diagnosisusing wavelet-autoregressive model and principal componentanalysismethodrdquoMechanical Systems and Signal Processing vol25 no 7 pp 2589ndash2607 2011
[10] K Y Shao M M Cai and G F Zhao ldquoRolling bearing faultdiagnosis based onwavelet energy spectrum PCA andPNNrdquo inProceedings of the 26th Chinese Control andDecisionConferenceCCDC rsquo14 pp 800ndash804 IEEE 2014
[11] T Qiu Q F Zhang and Y J Ding ldquoNonlinear sensor faultdetection and data rebuilding based on principle componentanalysisrdquo Journal of Tsinghua University vol 46 no 5 pp 708ndash711 2006
[12] Y K Gu Z X Yang and F L Zhu ldquoGearbox fault feature fusionbased on principal component analysisrdquo China MechanicalEngineering vol 26 no 11 pp 1532ndash1537 2015
[13] R Dunia S J Qin T F Edgar and T J McAvoy ldquoIdentificationof faulty sensors using principal component analysisrdquo AIChEJournal vol 42 no 10 pp 2797ndash2811 1996
[14] E P de Moura C R Souto A A Silva and M A S IrmaoldquoEvaluation of principal component analysis and neural net-work performance for bearing fault diagnosis from vibrationsignal processed by RS and DF analysesrdquo Mechanical Systemsand Signal Processing vol 25 no 5 pp 1765ndash1772 2011
[15] S Golestan M Monfared F D Freijedo and J M GuerreroldquoDynamics assessment of advanced single-phase PLL struc-turesrdquo IEEE Transactions on Industrial Electronics vol 60 no6 pp 2167ndash2177 2013
[16] A Lima H Zen Y Nankaku C Miyajima K Tokuda andT Kitamura ldquoOn the use of kernel PCA for feature extractionin speech recognitionrdquo IEICE Transaction on Information andSystems vol 12 no 87 pp 2802ndash2811 2004
[17] J D Zheng ldquoImproved hilbert-huang transform and its appli-cations to rolling bearing fault diagnosisrdquo Journal of MechanicalEngineering vol 1 no 51 pp 138ndash145 2015
[18] G He K Ding and H Lin ldquoFault feature extraction of rollingelement bearings using sparse representationrdquo Journal of Soundand Vibration vol 366 pp 514ndash527 2016
[19] J G Yang S B Xia and Y G Liu ldquoWavelet denoising and itsapplication in extraction of shaft centerline orbit featuresrdquoJournal of Harbin Institute of Technology vol 31 no 5 pp 52ndash541999 (Chinese)
[20] J A Duan and X D Zhang ldquoFeature extraction from the orbitof the center of a rotor based on wavelet transformrdquo Journal ofVibration Measurement amp Diagnosis vol 17 no 01 pp 31ndash341997
[21] J Han D X Jiang and W D Ning ldquoApplication of optimalwavelet packets in failure signal extraction of rotor orbitrdquoTurbine Technology vol 43 no 3 pp 133ndash136 2001
[22] W B Zhang X J Zhou andY Lin ldquoRefinement of rotor centerrsquosorbit by a harmonic window methodrdquo Journal of VibrationMeasurement amp Diagnosis vol 30 no 1 pp 87ndash90 2010
[23] S M Li G Z Lu and Q Y Xu ldquoOn obtaining accuraterotor sub-frequency signal with harmonic waveletrdquo Journal ofNorthwestern Polytechincal University vol 19 no 2 pp 220ndash224 2001
[24] F J Wu and L S Qu ldquoAn improved method for restraining theend effect in empiricalmode decomposition and its applicationsto the fault diagnosis of large rotating machineryrdquo Journal ofSound and Vibration vol 314 no 3-5 pp 586ndash602 2008
[25] W G Liu ldquoResearch on method of purification of shaft orbitbased on singular value decompositionrdquo Technical Acousticsvol 5 no 35 pp 30ndash35 2016
[26] X Z Zhao and B Y Ye ldquoThe influence of formation manner ofcomponent on signal processingrdquo Shang Hai Jiao Tong Univer-sity vol 45 no 3 pp 368ndash374 2011
[27] X Z Zhao B Y Ye and T J Chen ldquoPrinciple of singularitydetection based on SVD and its applicationrdquo Journal of Vibra-tion and Shock vol 6 no 27 pp 7ndash10 2008
[28] X Z Zhao B Y Ye and T J Chen ldquoDifference spectrum theoryof singular value and its application to the fault diagnosis ofheadstock of latherdquo Journal of Mechanical Engineering vol 46pp 100ndash108 2010
[29] I A Kougioumtzoglou and P D Spanos ldquoAn identificationapproach for linear and nonlinear time-variant structural sys-tems via harmonic waveletsrdquo Mechanical Systems and SignalProcessing vol 37 no 1-2 pp 338ndash352 2013
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
14 Shock and Vibration
D11minus2770 rpm
x 10minus5
minus4
minus2
0
2
4
D11
-X (m
)
200 400 600 800 10000Sampling numbers i
(a)
D12minus2770 rpm
x 10minus5
0 400 600 800 1000200Sampling numbers i
minus4
minus2
0
2
4
D12
-Y (m
)
(b)
(1375 1185)(46 1009)
(92 0831)
D11minus2770 rpmx 10
minus5
00
05
10
15
D11
-X (m
)
100 200 300 400 5000f (Hz)
(c)
(1375 1218)(46 1163)
(92 0898)
D12minus2770 rpmx 10
minus5
00
05
10
15
D12
-Y (m
)
100 200 300 400 5000f (Hz)
(d)
Figure 18 Extracting the time and frequency spectrums of the first three frequencies of D11 and D12 by PCA algorithm (a)The time domainof D11 (b) The time domain of D12 (c) The frequency spectrum of D11 (d) The frequency spectrum of D12
x 10minus5
x 10minus5
minus4
minus3
minus2
minus1
0
1
2
3
4
D12
-Y (m
)
minus3 minus2 minus1minus4 1 2 3 40D11-X (m)
(a)
1008
0604
0200
02
4D11-X (m)
t (s)
minus2
minus4
x 10minus5
x 10minus5
minus4
minus2
0
2
4
D12
-Y (m
)
(b)
Figure 19 The synthesis of axis orbit is to extract the first three times fundamental frequencies of the D11 and D12 by the CFSM-PCA (a)The axis orbit (b) Relationship between axis orbit and time
6 Conclusion
The selection of effective eigenvalues in feature distributionchart of covariance matrix C is crucial in PCA algorithm Inthis paper the relationships between effective eigenvalues andsignal frequency and amplitude are discovered and a series ofconclusions are summarized
First in Hankel matrix mode each valid frequencycomponent of signal only produces two adjacent nonzeroeigenvalues Second the number of valid eigenvalues incharacteristic distribution chart which is independent ofnumerical values of 119891119894 119860 119894 and 120593119894 is related to the number offrequency componentsThird the sequence of signal effectiveeigenvalues in characteristic distribution map is determined
Shock and Vibration 15
D11-1080 rpm
x 10minus5
minus20
minus15
minus05
0
05
10
15
20
D11
-X (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(a)
D12-1080 rpmx 10
minus5
minus15
minus1
minus05
0
05
1
15
D12
-Y (m
)
500 1000 1500 2000 2500 30000 40003500Sampling numbers i
(b)
(16 07335)
(32 04771)
D11-1080 rpm
x 10minus5
10 20 30 40 50 60 70 80 90 1000f (Hz)
0010203040506070809
1
D11
-X (m
)
(c)
(16 06571)
(32 05228)
D12-1080 rpm
x 10minus5
0010203040506070809
1
D12
-Y (m
)
10 20 30 40 50 60 70 80 90 1000f (Hz)
(d)
W2
x 10minus5
x 10minus5
minus15
minus1
minus05
0
05
1
15
Y(D
12) (
m)
minus1 minus05minus15 05 1 15 20X(D11) (m)
(e)
Figure 20The purification effect of harmonic wavelets (a)The time domain of D11 (b)The time domain of D12 (c)The frequency spectrumof D11 (d) The frequency spectrum of D12 (e) The orbit is to extract the first 2 frequencies
by amplitude 119860 119894 of signal frequency component The largermagnitude of amplitude is the greater eigenvalues are andthe higher arrangement will be
Based on aforesaid principle a new CFSM-PCA wasdeveloped and has become an effective tool to extract oreliminate specific characteristic (single frequency) even dif-ference of two frequencies is only 1 Hz Purification of rotororbit using this algorithm shows significant improvement
than those of existing methods such as wavelet packet andharmonic wavelet packet Although the proposed CFSM-PCA is effective some drawbacks are unavoidable Forexample this method suffers from similar issue to otheralgorithms that is the reconstruction error increaseswith thedecrease of SNR In addition the applications of the CFSM-PCA algorithm in purification of speech recognition bearingfault diagnosis and fault diagnosis of power system should
16 Shock and Vibration
D11-1080 rpmx 10
minus5
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
minus20
minus15
minus05
0
05
10
15
20
D11
-X (m
)
(a)
D12-1080 rpmx 10
minus5
minus15
minus1
minus05
0
05
1
15
D12
-Y (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(b)
(16 06557)
(32 04381)
D11-1080 rpmx 10
minus5
10020 30 40 50 60 70 80 90100f (Hz)
0010203040506070809
1
D11
-X (m
)
(c)
(16 06452)(32 04927)
D12-1080 rpm
x 10minus5
0010203040506070809
1
D12
-Y (m
)
9020 30 40 50 60 70 8010 1000f (Hz)
(d)
W3
x 10minus5
x 10minus5
minus15
minus1
minus05
0
05
1
15
Y(D
12) (
m)
minus15 minus05minus1 05 1 15 20X(D11) (m)
(e)
Figure 21 The purification effect of wavelet packet (a) The time domain of D11 (b) The time domain of D12 (c) The frequency spectrum ofD11 (d) The frequency spectrum of D12 (e) The orbit is to extract the first 2 frequencies
be further exploredWe fully believe that the CFSM-PCAwillsparkle in diverse fields
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work is supported by the National High TechnologyResearch andDevelopment Program of China (863 Program)
Shock and Vibration 17
(18 0734)
(36 04634)
(54 02448) (72 02351)(90 01646)
x 10minus5
50 100 150 2000f (Hz)
0
02
04
06
08
1
D11
minusX
(m)
Figure 22 Frequency spectrum of D11 signal after filtering 50Hz
(no 2015AA043005) and the National Natural Science Foun-dation of China (no 51375187)
References
[1] H Abdi and L J Williams ldquoPrincipal component analysisrdquoWiley Interdisciplinary Reviews Computational Statistics vol 2no 4 pp 433ndash459 2010
[2] A Widodo B Yang and T Han ldquoCombination of independentcomponent analysis and support vectormachines for intelligentfaults diagnosis of induction motorsrdquo Expert Systems withApplications vol 32 no 2 pp 299ndash312 2007
[3] S Wold K Esbensen and P Geladi ldquoPrincipal componentanalysisrdquo Chemometrics and Intelligent Laboratory Systems vol2 no 1ndash3 pp 37ndash52 1987
[4] M Kirby and L Sirovich ldquoApplication of the Karhunen-Loeve procedure for the characterization of human facesrdquo IEEETransactions on Pattern Analysis and Machine Intelligence vol12 no 1 pp 103ndash108 1990
[5] J H Xi Y Z Han and R H Su ldquoNew fault diagnosis methodfor rolling bearing based on PCArdquo in Proceedings of the 25thChinese Control and Decision Conference CCDC rsquo13 pp 4123ndash4127 IEEE May 2013
[6] A Malhi and R X Gao ldquoPCA-based feature selection schemefor machine defect classificationrdquo IEEE Transactions on Instru-mentation and Measurement vol 53 no 6 pp 1517ndash1525 2004
[7] S Xu X Jiang J Huang S Yang and X Wang ldquoBayesianwavelet PCA methodology for turbomachinery damage diag-nosis under uncertaintyrdquo Mechanical Systems and Signal Pro-cessing vol 80 pp 1ndash18 2016
[8] W Sun J Chen and J Li ldquoDecision tree and PCA-based faultdiagnosis of rotatingmachineryrdquoMechanical Systems and SignalProcessing vol 21 no 3 pp 1300ndash1317 2007
[9] Z X Li X P Yan C Q Yuan Z Peng and L Li ldquoVirtual pro-totype and experimental research on gear multi-fault diagnosisusing wavelet-autoregressive model and principal componentanalysismethodrdquoMechanical Systems and Signal Processing vol25 no 7 pp 2589ndash2607 2011
[10] K Y Shao M M Cai and G F Zhao ldquoRolling bearing faultdiagnosis based onwavelet energy spectrum PCA andPNNrdquo inProceedings of the 26th Chinese Control andDecisionConferenceCCDC rsquo14 pp 800ndash804 IEEE 2014
[11] T Qiu Q F Zhang and Y J Ding ldquoNonlinear sensor faultdetection and data rebuilding based on principle componentanalysisrdquo Journal of Tsinghua University vol 46 no 5 pp 708ndash711 2006
[12] Y K Gu Z X Yang and F L Zhu ldquoGearbox fault feature fusionbased on principal component analysisrdquo China MechanicalEngineering vol 26 no 11 pp 1532ndash1537 2015
[13] R Dunia S J Qin T F Edgar and T J McAvoy ldquoIdentificationof faulty sensors using principal component analysisrdquo AIChEJournal vol 42 no 10 pp 2797ndash2811 1996
[14] E P de Moura C R Souto A A Silva and M A S IrmaoldquoEvaluation of principal component analysis and neural net-work performance for bearing fault diagnosis from vibrationsignal processed by RS and DF analysesrdquo Mechanical Systemsand Signal Processing vol 25 no 5 pp 1765ndash1772 2011
[15] S Golestan M Monfared F D Freijedo and J M GuerreroldquoDynamics assessment of advanced single-phase PLL struc-turesrdquo IEEE Transactions on Industrial Electronics vol 60 no6 pp 2167ndash2177 2013
[16] A Lima H Zen Y Nankaku C Miyajima K Tokuda andT Kitamura ldquoOn the use of kernel PCA for feature extractionin speech recognitionrdquo IEICE Transaction on Information andSystems vol 12 no 87 pp 2802ndash2811 2004
[17] J D Zheng ldquoImproved hilbert-huang transform and its appli-cations to rolling bearing fault diagnosisrdquo Journal of MechanicalEngineering vol 1 no 51 pp 138ndash145 2015
[18] G He K Ding and H Lin ldquoFault feature extraction of rollingelement bearings using sparse representationrdquo Journal of Soundand Vibration vol 366 pp 514ndash527 2016
[19] J G Yang S B Xia and Y G Liu ldquoWavelet denoising and itsapplication in extraction of shaft centerline orbit featuresrdquoJournal of Harbin Institute of Technology vol 31 no 5 pp 52ndash541999 (Chinese)
[20] J A Duan and X D Zhang ldquoFeature extraction from the orbitof the center of a rotor based on wavelet transformrdquo Journal ofVibration Measurement amp Diagnosis vol 17 no 01 pp 31ndash341997
[21] J Han D X Jiang and W D Ning ldquoApplication of optimalwavelet packets in failure signal extraction of rotor orbitrdquoTurbine Technology vol 43 no 3 pp 133ndash136 2001
[22] W B Zhang X J Zhou andY Lin ldquoRefinement of rotor centerrsquosorbit by a harmonic window methodrdquo Journal of VibrationMeasurement amp Diagnosis vol 30 no 1 pp 87ndash90 2010
[23] S M Li G Z Lu and Q Y Xu ldquoOn obtaining accuraterotor sub-frequency signal with harmonic waveletrdquo Journal ofNorthwestern Polytechincal University vol 19 no 2 pp 220ndash224 2001
[24] F J Wu and L S Qu ldquoAn improved method for restraining theend effect in empiricalmode decomposition and its applicationsto the fault diagnosis of large rotating machineryrdquo Journal ofSound and Vibration vol 314 no 3-5 pp 586ndash602 2008
[25] W G Liu ldquoResearch on method of purification of shaft orbitbased on singular value decompositionrdquo Technical Acousticsvol 5 no 35 pp 30ndash35 2016
[26] X Z Zhao and B Y Ye ldquoThe influence of formation manner ofcomponent on signal processingrdquo Shang Hai Jiao Tong Univer-sity vol 45 no 3 pp 368ndash374 2011
[27] X Z Zhao B Y Ye and T J Chen ldquoPrinciple of singularitydetection based on SVD and its applicationrdquo Journal of Vibra-tion and Shock vol 6 no 27 pp 7ndash10 2008
[28] X Z Zhao B Y Ye and T J Chen ldquoDifference spectrum theoryof singular value and its application to the fault diagnosis ofheadstock of latherdquo Journal of Mechanical Engineering vol 46pp 100ndash108 2010
[29] I A Kougioumtzoglou and P D Spanos ldquoAn identificationapproach for linear and nonlinear time-variant structural sys-tems via harmonic waveletsrdquo Mechanical Systems and SignalProcessing vol 37 no 1-2 pp 338ndash352 2013
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
Shock and Vibration 15
D11-1080 rpm
x 10minus5
minus20
minus15
minus05
0
05
10
15
20
D11
-X (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(a)
D12-1080 rpmx 10
minus5
minus15
minus1
minus05
0
05
1
15
D12
-Y (m
)
500 1000 1500 2000 2500 30000 40003500Sampling numbers i
(b)
(16 07335)
(32 04771)
D11-1080 rpm
x 10minus5
10 20 30 40 50 60 70 80 90 1000f (Hz)
0010203040506070809
1
D11
-X (m
)
(c)
(16 06571)
(32 05228)
D12-1080 rpm
x 10minus5
0010203040506070809
1
D12
-Y (m
)
10 20 30 40 50 60 70 80 90 1000f (Hz)
(d)
W2
x 10minus5
x 10minus5
minus15
minus1
minus05
0
05
1
15
Y(D
12) (
m)
minus1 minus05minus15 05 1 15 20X(D11) (m)
(e)
Figure 20The purification effect of harmonic wavelets (a)The time domain of D11 (b)The time domain of D12 (c)The frequency spectrumof D11 (d) The frequency spectrum of D12 (e) The orbit is to extract the first 2 frequencies
by amplitude 119860 119894 of signal frequency component The largermagnitude of amplitude is the greater eigenvalues are andthe higher arrangement will be
Based on aforesaid principle a new CFSM-PCA wasdeveloped and has become an effective tool to extract oreliminate specific characteristic (single frequency) even dif-ference of two frequencies is only 1 Hz Purification of rotororbit using this algorithm shows significant improvement
than those of existing methods such as wavelet packet andharmonic wavelet packet Although the proposed CFSM-PCA is effective some drawbacks are unavoidable Forexample this method suffers from similar issue to otheralgorithms that is the reconstruction error increaseswith thedecrease of SNR In addition the applications of the CFSM-PCA algorithm in purification of speech recognition bearingfault diagnosis and fault diagnosis of power system should
16 Shock and Vibration
D11-1080 rpmx 10
minus5
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
minus20
minus15
minus05
0
05
10
15
20
D11
-X (m
)
(a)
D12-1080 rpmx 10
minus5
minus15
minus1
minus05
0
05
1
15
D12
-Y (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(b)
(16 06557)
(32 04381)
D11-1080 rpmx 10
minus5
10020 30 40 50 60 70 80 90100f (Hz)
0010203040506070809
1
D11
-X (m
)
(c)
(16 06452)(32 04927)
D12-1080 rpm
x 10minus5
0010203040506070809
1
D12
-Y (m
)
9020 30 40 50 60 70 8010 1000f (Hz)
(d)
W3
x 10minus5
x 10minus5
minus15
minus1
minus05
0
05
1
15
Y(D
12) (
m)
minus15 minus05minus1 05 1 15 20X(D11) (m)
(e)
Figure 21 The purification effect of wavelet packet (a) The time domain of D11 (b) The time domain of D12 (c) The frequency spectrum ofD11 (d) The frequency spectrum of D12 (e) The orbit is to extract the first 2 frequencies
be further exploredWe fully believe that the CFSM-PCAwillsparkle in diverse fields
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work is supported by the National High TechnologyResearch andDevelopment Program of China (863 Program)
Shock and Vibration 17
(18 0734)
(36 04634)
(54 02448) (72 02351)(90 01646)
x 10minus5
50 100 150 2000f (Hz)
0
02
04
06
08
1
D11
minusX
(m)
Figure 22 Frequency spectrum of D11 signal after filtering 50Hz
(no 2015AA043005) and the National Natural Science Foun-dation of China (no 51375187)
References
[1] H Abdi and L J Williams ldquoPrincipal component analysisrdquoWiley Interdisciplinary Reviews Computational Statistics vol 2no 4 pp 433ndash459 2010
[2] A Widodo B Yang and T Han ldquoCombination of independentcomponent analysis and support vectormachines for intelligentfaults diagnosis of induction motorsrdquo Expert Systems withApplications vol 32 no 2 pp 299ndash312 2007
[3] S Wold K Esbensen and P Geladi ldquoPrincipal componentanalysisrdquo Chemometrics and Intelligent Laboratory Systems vol2 no 1ndash3 pp 37ndash52 1987
[4] M Kirby and L Sirovich ldquoApplication of the Karhunen-Loeve procedure for the characterization of human facesrdquo IEEETransactions on Pattern Analysis and Machine Intelligence vol12 no 1 pp 103ndash108 1990
[5] J H Xi Y Z Han and R H Su ldquoNew fault diagnosis methodfor rolling bearing based on PCArdquo in Proceedings of the 25thChinese Control and Decision Conference CCDC rsquo13 pp 4123ndash4127 IEEE May 2013
[6] A Malhi and R X Gao ldquoPCA-based feature selection schemefor machine defect classificationrdquo IEEE Transactions on Instru-mentation and Measurement vol 53 no 6 pp 1517ndash1525 2004
[7] S Xu X Jiang J Huang S Yang and X Wang ldquoBayesianwavelet PCA methodology for turbomachinery damage diag-nosis under uncertaintyrdquo Mechanical Systems and Signal Pro-cessing vol 80 pp 1ndash18 2016
[8] W Sun J Chen and J Li ldquoDecision tree and PCA-based faultdiagnosis of rotatingmachineryrdquoMechanical Systems and SignalProcessing vol 21 no 3 pp 1300ndash1317 2007
[9] Z X Li X P Yan C Q Yuan Z Peng and L Li ldquoVirtual pro-totype and experimental research on gear multi-fault diagnosisusing wavelet-autoregressive model and principal componentanalysismethodrdquoMechanical Systems and Signal Processing vol25 no 7 pp 2589ndash2607 2011
[10] K Y Shao M M Cai and G F Zhao ldquoRolling bearing faultdiagnosis based onwavelet energy spectrum PCA andPNNrdquo inProceedings of the 26th Chinese Control andDecisionConferenceCCDC rsquo14 pp 800ndash804 IEEE 2014
[11] T Qiu Q F Zhang and Y J Ding ldquoNonlinear sensor faultdetection and data rebuilding based on principle componentanalysisrdquo Journal of Tsinghua University vol 46 no 5 pp 708ndash711 2006
[12] Y K Gu Z X Yang and F L Zhu ldquoGearbox fault feature fusionbased on principal component analysisrdquo China MechanicalEngineering vol 26 no 11 pp 1532ndash1537 2015
[13] R Dunia S J Qin T F Edgar and T J McAvoy ldquoIdentificationof faulty sensors using principal component analysisrdquo AIChEJournal vol 42 no 10 pp 2797ndash2811 1996
[14] E P de Moura C R Souto A A Silva and M A S IrmaoldquoEvaluation of principal component analysis and neural net-work performance for bearing fault diagnosis from vibrationsignal processed by RS and DF analysesrdquo Mechanical Systemsand Signal Processing vol 25 no 5 pp 1765ndash1772 2011
[15] S Golestan M Monfared F D Freijedo and J M GuerreroldquoDynamics assessment of advanced single-phase PLL struc-turesrdquo IEEE Transactions on Industrial Electronics vol 60 no6 pp 2167ndash2177 2013
[16] A Lima H Zen Y Nankaku C Miyajima K Tokuda andT Kitamura ldquoOn the use of kernel PCA for feature extractionin speech recognitionrdquo IEICE Transaction on Information andSystems vol 12 no 87 pp 2802ndash2811 2004
[17] J D Zheng ldquoImproved hilbert-huang transform and its appli-cations to rolling bearing fault diagnosisrdquo Journal of MechanicalEngineering vol 1 no 51 pp 138ndash145 2015
[18] G He K Ding and H Lin ldquoFault feature extraction of rollingelement bearings using sparse representationrdquo Journal of Soundand Vibration vol 366 pp 514ndash527 2016
[19] J G Yang S B Xia and Y G Liu ldquoWavelet denoising and itsapplication in extraction of shaft centerline orbit featuresrdquoJournal of Harbin Institute of Technology vol 31 no 5 pp 52ndash541999 (Chinese)
[20] J A Duan and X D Zhang ldquoFeature extraction from the orbitof the center of a rotor based on wavelet transformrdquo Journal ofVibration Measurement amp Diagnosis vol 17 no 01 pp 31ndash341997
[21] J Han D X Jiang and W D Ning ldquoApplication of optimalwavelet packets in failure signal extraction of rotor orbitrdquoTurbine Technology vol 43 no 3 pp 133ndash136 2001
[22] W B Zhang X J Zhou andY Lin ldquoRefinement of rotor centerrsquosorbit by a harmonic window methodrdquo Journal of VibrationMeasurement amp Diagnosis vol 30 no 1 pp 87ndash90 2010
[23] S M Li G Z Lu and Q Y Xu ldquoOn obtaining accuraterotor sub-frequency signal with harmonic waveletrdquo Journal ofNorthwestern Polytechincal University vol 19 no 2 pp 220ndash224 2001
[24] F J Wu and L S Qu ldquoAn improved method for restraining theend effect in empiricalmode decomposition and its applicationsto the fault diagnosis of large rotating machineryrdquo Journal ofSound and Vibration vol 314 no 3-5 pp 586ndash602 2008
[25] W G Liu ldquoResearch on method of purification of shaft orbitbased on singular value decompositionrdquo Technical Acousticsvol 5 no 35 pp 30ndash35 2016
[26] X Z Zhao and B Y Ye ldquoThe influence of formation manner ofcomponent on signal processingrdquo Shang Hai Jiao Tong Univer-sity vol 45 no 3 pp 368ndash374 2011
[27] X Z Zhao B Y Ye and T J Chen ldquoPrinciple of singularitydetection based on SVD and its applicationrdquo Journal of Vibra-tion and Shock vol 6 no 27 pp 7ndash10 2008
[28] X Z Zhao B Y Ye and T J Chen ldquoDifference spectrum theoryof singular value and its application to the fault diagnosis ofheadstock of latherdquo Journal of Mechanical Engineering vol 46pp 100ndash108 2010
[29] I A Kougioumtzoglou and P D Spanos ldquoAn identificationapproach for linear and nonlinear time-variant structural sys-tems via harmonic waveletsrdquo Mechanical Systems and SignalProcessing vol 37 no 1-2 pp 338ndash352 2013
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
16 Shock and Vibration
D11-1080 rpmx 10
minus5
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
minus20
minus15
minus05
0
05
10
15
20
D11
-X (m
)
(a)
D12-1080 rpmx 10
minus5
minus15
minus1
minus05
0
05
1
15
D12
-Y (m
)
500 1000 1500 2000 2500 3000 3500 40000Sampling numbers i
(b)
(16 06557)
(32 04381)
D11-1080 rpmx 10
minus5
10020 30 40 50 60 70 80 90100f (Hz)
0010203040506070809
1
D11
-X (m
)
(c)
(16 06452)(32 04927)
D12-1080 rpm
x 10minus5
0010203040506070809
1
D12
-Y (m
)
9020 30 40 50 60 70 8010 1000f (Hz)
(d)
W3
x 10minus5
x 10minus5
minus15
minus1
minus05
0
05
1
15
Y(D
12) (
m)
minus15 minus05minus1 05 1 15 20X(D11) (m)
(e)
Figure 21 The purification effect of wavelet packet (a) The time domain of D11 (b) The time domain of D12 (c) The frequency spectrum ofD11 (d) The frequency spectrum of D12 (e) The orbit is to extract the first 2 frequencies
be further exploredWe fully believe that the CFSM-PCAwillsparkle in diverse fields
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work is supported by the National High TechnologyResearch andDevelopment Program of China (863 Program)
Shock and Vibration 17
(18 0734)
(36 04634)
(54 02448) (72 02351)(90 01646)
x 10minus5
50 100 150 2000f (Hz)
0
02
04
06
08
1
D11
minusX
(m)
Figure 22 Frequency spectrum of D11 signal after filtering 50Hz
(no 2015AA043005) and the National Natural Science Foun-dation of China (no 51375187)
References
[1] H Abdi and L J Williams ldquoPrincipal component analysisrdquoWiley Interdisciplinary Reviews Computational Statistics vol 2no 4 pp 433ndash459 2010
[2] A Widodo B Yang and T Han ldquoCombination of independentcomponent analysis and support vectormachines for intelligentfaults diagnosis of induction motorsrdquo Expert Systems withApplications vol 32 no 2 pp 299ndash312 2007
[3] S Wold K Esbensen and P Geladi ldquoPrincipal componentanalysisrdquo Chemometrics and Intelligent Laboratory Systems vol2 no 1ndash3 pp 37ndash52 1987
[4] M Kirby and L Sirovich ldquoApplication of the Karhunen-Loeve procedure for the characterization of human facesrdquo IEEETransactions on Pattern Analysis and Machine Intelligence vol12 no 1 pp 103ndash108 1990
[5] J H Xi Y Z Han and R H Su ldquoNew fault diagnosis methodfor rolling bearing based on PCArdquo in Proceedings of the 25thChinese Control and Decision Conference CCDC rsquo13 pp 4123ndash4127 IEEE May 2013
[6] A Malhi and R X Gao ldquoPCA-based feature selection schemefor machine defect classificationrdquo IEEE Transactions on Instru-mentation and Measurement vol 53 no 6 pp 1517ndash1525 2004
[7] S Xu X Jiang J Huang S Yang and X Wang ldquoBayesianwavelet PCA methodology for turbomachinery damage diag-nosis under uncertaintyrdquo Mechanical Systems and Signal Pro-cessing vol 80 pp 1ndash18 2016
[8] W Sun J Chen and J Li ldquoDecision tree and PCA-based faultdiagnosis of rotatingmachineryrdquoMechanical Systems and SignalProcessing vol 21 no 3 pp 1300ndash1317 2007
[9] Z X Li X P Yan C Q Yuan Z Peng and L Li ldquoVirtual pro-totype and experimental research on gear multi-fault diagnosisusing wavelet-autoregressive model and principal componentanalysismethodrdquoMechanical Systems and Signal Processing vol25 no 7 pp 2589ndash2607 2011
[10] K Y Shao M M Cai and G F Zhao ldquoRolling bearing faultdiagnosis based onwavelet energy spectrum PCA andPNNrdquo inProceedings of the 26th Chinese Control andDecisionConferenceCCDC rsquo14 pp 800ndash804 IEEE 2014
[11] T Qiu Q F Zhang and Y J Ding ldquoNonlinear sensor faultdetection and data rebuilding based on principle componentanalysisrdquo Journal of Tsinghua University vol 46 no 5 pp 708ndash711 2006
[12] Y K Gu Z X Yang and F L Zhu ldquoGearbox fault feature fusionbased on principal component analysisrdquo China MechanicalEngineering vol 26 no 11 pp 1532ndash1537 2015
[13] R Dunia S J Qin T F Edgar and T J McAvoy ldquoIdentificationof faulty sensors using principal component analysisrdquo AIChEJournal vol 42 no 10 pp 2797ndash2811 1996
[14] E P de Moura C R Souto A A Silva and M A S IrmaoldquoEvaluation of principal component analysis and neural net-work performance for bearing fault diagnosis from vibrationsignal processed by RS and DF analysesrdquo Mechanical Systemsand Signal Processing vol 25 no 5 pp 1765ndash1772 2011
[15] S Golestan M Monfared F D Freijedo and J M GuerreroldquoDynamics assessment of advanced single-phase PLL struc-turesrdquo IEEE Transactions on Industrial Electronics vol 60 no6 pp 2167ndash2177 2013
[16] A Lima H Zen Y Nankaku C Miyajima K Tokuda andT Kitamura ldquoOn the use of kernel PCA for feature extractionin speech recognitionrdquo IEICE Transaction on Information andSystems vol 12 no 87 pp 2802ndash2811 2004
[17] J D Zheng ldquoImproved hilbert-huang transform and its appli-cations to rolling bearing fault diagnosisrdquo Journal of MechanicalEngineering vol 1 no 51 pp 138ndash145 2015
[18] G He K Ding and H Lin ldquoFault feature extraction of rollingelement bearings using sparse representationrdquo Journal of Soundand Vibration vol 366 pp 514ndash527 2016
[19] J G Yang S B Xia and Y G Liu ldquoWavelet denoising and itsapplication in extraction of shaft centerline orbit featuresrdquoJournal of Harbin Institute of Technology vol 31 no 5 pp 52ndash541999 (Chinese)
[20] J A Duan and X D Zhang ldquoFeature extraction from the orbitof the center of a rotor based on wavelet transformrdquo Journal ofVibration Measurement amp Diagnosis vol 17 no 01 pp 31ndash341997
[21] J Han D X Jiang and W D Ning ldquoApplication of optimalwavelet packets in failure signal extraction of rotor orbitrdquoTurbine Technology vol 43 no 3 pp 133ndash136 2001
[22] W B Zhang X J Zhou andY Lin ldquoRefinement of rotor centerrsquosorbit by a harmonic window methodrdquo Journal of VibrationMeasurement amp Diagnosis vol 30 no 1 pp 87ndash90 2010
[23] S M Li G Z Lu and Q Y Xu ldquoOn obtaining accuraterotor sub-frequency signal with harmonic waveletrdquo Journal ofNorthwestern Polytechincal University vol 19 no 2 pp 220ndash224 2001
[24] F J Wu and L S Qu ldquoAn improved method for restraining theend effect in empiricalmode decomposition and its applicationsto the fault diagnosis of large rotating machineryrdquo Journal ofSound and Vibration vol 314 no 3-5 pp 586ndash602 2008
[25] W G Liu ldquoResearch on method of purification of shaft orbitbased on singular value decompositionrdquo Technical Acousticsvol 5 no 35 pp 30ndash35 2016
[26] X Z Zhao and B Y Ye ldquoThe influence of formation manner ofcomponent on signal processingrdquo Shang Hai Jiao Tong Univer-sity vol 45 no 3 pp 368ndash374 2011
[27] X Z Zhao B Y Ye and T J Chen ldquoPrinciple of singularitydetection based on SVD and its applicationrdquo Journal of Vibra-tion and Shock vol 6 no 27 pp 7ndash10 2008
[28] X Z Zhao B Y Ye and T J Chen ldquoDifference spectrum theoryof singular value and its application to the fault diagnosis ofheadstock of latherdquo Journal of Mechanical Engineering vol 46pp 100ndash108 2010
[29] I A Kougioumtzoglou and P D Spanos ldquoAn identificationapproach for linear and nonlinear time-variant structural sys-tems via harmonic waveletsrdquo Mechanical Systems and SignalProcessing vol 37 no 1-2 pp 338ndash352 2013
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
Shock and Vibration 17
(18 0734)
(36 04634)
(54 02448) (72 02351)(90 01646)
x 10minus5
50 100 150 2000f (Hz)
0
02
04
06
08
1
D11
minusX
(m)
Figure 22 Frequency spectrum of D11 signal after filtering 50Hz
(no 2015AA043005) and the National Natural Science Foun-dation of China (no 51375187)
References
[1] H Abdi and L J Williams ldquoPrincipal component analysisrdquoWiley Interdisciplinary Reviews Computational Statistics vol 2no 4 pp 433ndash459 2010
[2] A Widodo B Yang and T Han ldquoCombination of independentcomponent analysis and support vectormachines for intelligentfaults diagnosis of induction motorsrdquo Expert Systems withApplications vol 32 no 2 pp 299ndash312 2007
[3] S Wold K Esbensen and P Geladi ldquoPrincipal componentanalysisrdquo Chemometrics and Intelligent Laboratory Systems vol2 no 1ndash3 pp 37ndash52 1987
[4] M Kirby and L Sirovich ldquoApplication of the Karhunen-Loeve procedure for the characterization of human facesrdquo IEEETransactions on Pattern Analysis and Machine Intelligence vol12 no 1 pp 103ndash108 1990
[5] J H Xi Y Z Han and R H Su ldquoNew fault diagnosis methodfor rolling bearing based on PCArdquo in Proceedings of the 25thChinese Control and Decision Conference CCDC rsquo13 pp 4123ndash4127 IEEE May 2013
[6] A Malhi and R X Gao ldquoPCA-based feature selection schemefor machine defect classificationrdquo IEEE Transactions on Instru-mentation and Measurement vol 53 no 6 pp 1517ndash1525 2004
[7] S Xu X Jiang J Huang S Yang and X Wang ldquoBayesianwavelet PCA methodology for turbomachinery damage diag-nosis under uncertaintyrdquo Mechanical Systems and Signal Pro-cessing vol 80 pp 1ndash18 2016
[8] W Sun J Chen and J Li ldquoDecision tree and PCA-based faultdiagnosis of rotatingmachineryrdquoMechanical Systems and SignalProcessing vol 21 no 3 pp 1300ndash1317 2007
[9] Z X Li X P Yan C Q Yuan Z Peng and L Li ldquoVirtual pro-totype and experimental research on gear multi-fault diagnosisusing wavelet-autoregressive model and principal componentanalysismethodrdquoMechanical Systems and Signal Processing vol25 no 7 pp 2589ndash2607 2011
[10] K Y Shao M M Cai and G F Zhao ldquoRolling bearing faultdiagnosis based onwavelet energy spectrum PCA andPNNrdquo inProceedings of the 26th Chinese Control andDecisionConferenceCCDC rsquo14 pp 800ndash804 IEEE 2014
[11] T Qiu Q F Zhang and Y J Ding ldquoNonlinear sensor faultdetection and data rebuilding based on principle componentanalysisrdquo Journal of Tsinghua University vol 46 no 5 pp 708ndash711 2006
[12] Y K Gu Z X Yang and F L Zhu ldquoGearbox fault feature fusionbased on principal component analysisrdquo China MechanicalEngineering vol 26 no 11 pp 1532ndash1537 2015
[13] R Dunia S J Qin T F Edgar and T J McAvoy ldquoIdentificationof faulty sensors using principal component analysisrdquo AIChEJournal vol 42 no 10 pp 2797ndash2811 1996
[14] E P de Moura C R Souto A A Silva and M A S IrmaoldquoEvaluation of principal component analysis and neural net-work performance for bearing fault diagnosis from vibrationsignal processed by RS and DF analysesrdquo Mechanical Systemsand Signal Processing vol 25 no 5 pp 1765ndash1772 2011
[15] S Golestan M Monfared F D Freijedo and J M GuerreroldquoDynamics assessment of advanced single-phase PLL struc-turesrdquo IEEE Transactions on Industrial Electronics vol 60 no6 pp 2167ndash2177 2013
[16] A Lima H Zen Y Nankaku C Miyajima K Tokuda andT Kitamura ldquoOn the use of kernel PCA for feature extractionin speech recognitionrdquo IEICE Transaction on Information andSystems vol 12 no 87 pp 2802ndash2811 2004
[17] J D Zheng ldquoImproved hilbert-huang transform and its appli-cations to rolling bearing fault diagnosisrdquo Journal of MechanicalEngineering vol 1 no 51 pp 138ndash145 2015
[18] G He K Ding and H Lin ldquoFault feature extraction of rollingelement bearings using sparse representationrdquo Journal of Soundand Vibration vol 366 pp 514ndash527 2016
[19] J G Yang S B Xia and Y G Liu ldquoWavelet denoising and itsapplication in extraction of shaft centerline orbit featuresrdquoJournal of Harbin Institute of Technology vol 31 no 5 pp 52ndash541999 (Chinese)
[20] J A Duan and X D Zhang ldquoFeature extraction from the orbitof the center of a rotor based on wavelet transformrdquo Journal ofVibration Measurement amp Diagnosis vol 17 no 01 pp 31ndash341997
[21] J Han D X Jiang and W D Ning ldquoApplication of optimalwavelet packets in failure signal extraction of rotor orbitrdquoTurbine Technology vol 43 no 3 pp 133ndash136 2001
[22] W B Zhang X J Zhou andY Lin ldquoRefinement of rotor centerrsquosorbit by a harmonic window methodrdquo Journal of VibrationMeasurement amp Diagnosis vol 30 no 1 pp 87ndash90 2010
[23] S M Li G Z Lu and Q Y Xu ldquoOn obtaining accuraterotor sub-frequency signal with harmonic waveletrdquo Journal ofNorthwestern Polytechincal University vol 19 no 2 pp 220ndash224 2001
[24] F J Wu and L S Qu ldquoAn improved method for restraining theend effect in empiricalmode decomposition and its applicationsto the fault diagnosis of large rotating machineryrdquo Journal ofSound and Vibration vol 314 no 3-5 pp 586ndash602 2008
[25] W G Liu ldquoResearch on method of purification of shaft orbitbased on singular value decompositionrdquo Technical Acousticsvol 5 no 35 pp 30ndash35 2016
[26] X Z Zhao and B Y Ye ldquoThe influence of formation manner ofcomponent on signal processingrdquo Shang Hai Jiao Tong Univer-sity vol 45 no 3 pp 368ndash374 2011
[27] X Z Zhao B Y Ye and T J Chen ldquoPrinciple of singularitydetection based on SVD and its applicationrdquo Journal of Vibra-tion and Shock vol 6 no 27 pp 7ndash10 2008
[28] X Z Zhao B Y Ye and T J Chen ldquoDifference spectrum theoryof singular value and its application to the fault diagnosis ofheadstock of latherdquo Journal of Mechanical Engineering vol 46pp 100ndash108 2010
[29] I A Kougioumtzoglou and P D Spanos ldquoAn identificationapproach for linear and nonlinear time-variant structural sys-tems via harmonic waveletsrdquo Mechanical Systems and SignalProcessing vol 37 no 1-2 pp 338ndash352 2013
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom