Feature Frequency Extraction Based on Principal Component...

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


(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


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


(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


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)


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


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


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


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)


minus1

minus05

0

05

1

Am

plitu

de


(c)

(800786)

002

04

06

08

1

Am

plitu

de

50 1000f (Hz)

(d)



minus1

minus05

0

05

1

Am

plitu

de

(e)

(490688)

50 1000f (Hz)

002

04

06

08

1

Am

plitu

de

(f)


minus05

0

05

Am

plitu

de


(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)


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


D11-1080 rpmx 10

minus5

minus5minus4minus3minus2minus1

012345

X (m

)


(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

)


(c)

D12-1080 rpmPower-line Interference

50Hz

0658105228

Order 1 Order 2

0277101226

Order 5 Order 12

01948 Order 8


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)


(a)

(12247) (2241)

(314) (41396)

D12minus1080 rpme first group

e second group

x 10minus8

005

115

225

Eige

nval

ue o

f PCA

(m)


(b)

Figure 11The eigenvalues distribution of covariancematrixC (a)The eigenvalues distribution of D11 signal (b)The eigenvalues distributionof D12 signal


D11minus1080 rpmx 10

minus5

minus15

minus10

minus05

0051015

X (m

)


(a)


minus5

minus10

minus05

0051015

Y (m

)


(b)

D11minus1080 rpm(1807287)

(360472)

x 10minus5

002040608

1

X (m

)

50 100 150 2000Frequency (Hz)

(c)

(3605105)(1806575)


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]



minus4

minus1

minus05

0

05

1

D11

-X (m

)

1000 2000 3000 40000Sampling numbers i

(a)


minus4

minus1

minus05

0

05

1

D12

-Y (m

)


(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




minus7


0

05

1

15

Eige

nval

ue o

f PCA

(m)

(a)

D12minus2770 rpm(11688) (21687)

e first group

e second group


(31673) (41664)

x 10minus7

0

05

1

15

2

Eige

nval

ue o

f PCA

(m)


(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



minus5

0 1000500Sampling numbers i

minus4

minus2

0

2

4

D11

-X (m

)

(a)


minus5

minus4

minus2

0

2

4

D12

-Y (m

)


(b)

(1375 1185)(46 1009)


minus5

100 200 300 400 5000f (Hz)

0

05

1

15

D11

-X (m

)

(c)

(1375 1218)(46 1164)


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)


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


D11minus2770 rpm

x 10minus5

minus4

minus2

0

2

4

D11

-X (m

)


(a)

D12minus2770 rpm

x 10minus5


minus4

minus2

0

2

4

D12

-Y (m

)

(b)

(1375 1185)(46 1009)

(92 0831)


minus5

00

05

10

15

D11

-X (m

)

100 200 300 400 5000f (Hz)

(c)

(1375 1218)(46 1163)

(92 0898)


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


D11-1080 rpm

x 10minus5

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 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


D11-1080 rpmx 10

minus5


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

)


(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)


(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)


(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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Hindawiwwwhindawicom Volume 2018


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Journal of

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Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

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Journal of



Journal ofEngineeringVolume 2018

SensorsJournal of



RotatingMachinery


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wwwhindawicom Volume 2018

Advances in

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Submit your manuscripts atwwwhindawicom


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)



X =119897

sum119894=1

120572119894y119894 =119897

sum119894=1

120572119894120572T119894 X (8)



x = 119886M+ (9)


=

[[[[[[[[[[

1

1 1

1 1 1

1 1

1

]]]]]]]]]]

M (10)


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


(11)





119909 (119905) =119896

sum119894=1

119860 119894 sin (2120587120596119894119905 + 120593119894) (12)









(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 40 500

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


(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


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)









(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














X = 119886 times[[[[[[



]]]]]]

(13)



X = 1198862119894

times[[[[[[[



]]]]]]]

(14)



X = 1198862119894 times 119890120593119894

times[[[[[[[[



]]]]]]]]


times[[[[[[[[



]]]]]]]](15)



X = 119886 times[[[[[[[[[



]]]]]]]]]

(16)


10038161003816100381610038161003816100381610038161003816100381610038161003816= minussin2 (120596119879119904)

= 0(17)






119897

sum119894=1

C120572119894120572T119894 =119897

sum119894=1

120582119894120572119894120572T119894 (19)


CI119898 = 12058211205721120572T1 + 12058221205722120572T2 (20)


|C|2 = 12058221119898

sum119894=1

12057221198941119898

sum119894=1

12057221198941 + 12059022119898

sum119894=1

12057221198942119898

sum119895=1

12057221198942

+ 21205902112059022119898

sum119894=1

12057211989411205721198942119898

sum119894=1

12057211989411205721198943(21)


|C|2 = 12058221 + 12058222 (22)







minus6minus4minus2

0246

Am

plitu

de


(a)

(200972)

(490693)(800791)

(500468)

0

05

1

Am

plitu

de

50 100 150 2000f (Hz)

(b)



119910 = sin (40120587119905 + 20) + 05sin (100120587119905 + 40)+ 07sin (98120587119905 + 50) + 08sin (160120587119905 + 60)+ 119890 (119899)

(23)





(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











minus2

minus1

0

1

2A

mpl

itude


(a)

(200977)

50 1000f (Hz)

002

04

06

08

1

Am

plitu

de

(b)


minus1

minus05

0

05

1

Am

plitu

de


(c)

(800786)

002

04

06

08

1

Am

plitu

de

50 1000f (Hz)

(d)



minus1

minus05

0

05

1

Am

plitu

de

(e)

(490688)

50 1000f (Hz)

002

04

06

08

1

Am

plitu

de

(f)


minus05

0

05

Am

plitu

de


(g)

(500463)

002

04

06

08

1

Am

plitu

de

50 1000f (Hz)

(h)



A B


Coupling

Sliding Bearing

Rotor

Large Platform

Small Platform

Air spring

BeltSliding Bearing



D13 D14

D12D11













D11-1080 rpmx 10

minus5


012345

X (m

)


(a)


0733404771

Order 1 Order 2

0250502342

Order 6Order 7

D11-1080 rpm

01651 Order 11


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

)


(c)


50Hz

0658105228

Order 1 Order 2

0277101226

Order 5 Order 12

01948 Order 8


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)


x 10minus5

x 10minus5

minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

D12

minusY

(m)



(12751) (22748)e first group


D11minus1080 rpm

x 10minus8

0

1

2

3

4

Eige

nval

ue o

f PCA

(m)


(a)

(12247) (2241)

(314) (41396)


e second group

x 10minus8

005

115

225

Eige

nval

ue o

f PCA

(m)


(b)




minus5

minus15

minus10

minus05

0051015

X (m

)


(a)


minus5

minus10

minus05

0051015

Y (m

)


(b)

D11minus1080 rpm(1807287)

(360472)

x 10minus5

002040608

1

X (m

)

50 100 150 2000Frequency (Hz)

(c)

(3605105)(1806575)


minus5

002040608

1

Y (m

)

50 100 150 2000Frequency (Hz)

(d)


W1

x 10minus5

x 10minus5

minus1

minus05

0

05

1

15

D12

minusY

(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)












minus4

minus1

minus05

0

05

1

D11

-X (m

)


(a)


minus4

minus1

minus05

0

05

1

D12

-Y (m

)


(b)


1X2X


08441166

1X2X3X

x 10minus5

200 400 600 800 10000f (Hz)

0

05

10

15

D11

-X (m

)

(c)

1X

2X


and 150Hz

D12minus2770 rpm117309021227

1X2X3X

x 10minus5

0

05

10

15

D12

-Y (m

)

200 400 600 800 10000f (Hz)

(d)


(11489) (21482)e first group




minus7


0

05

1

15

Eige

nval

ue o

f PCA

(m)

(a)

D12minus2770 rpm(11688) (21687)

e first group

e second group


(31673) (41664)

x 10minus7

0

05

1

15

2

Eige

nval

ue o

f PCA

(m)


(b)









minus5


minus4

minus2

0

2

4

D11

-X (m

)

(a)


minus5

minus4

minus2

0

2

4

D12

-Y (m

)


(b)

(1375 1185)(46 1009)


minus5

100 200 300 400 5000f (Hz)

0

05

1

15

D11

-X (m

)

(c)

(1375 1218)(46 1164)


minus5

100 200 300 400 5000f (Hz)

0

05

1

15

D12

-Y (m

)

(d)


x 10minus5

x 10minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

D12

-Y (m

)


D11-X (m)

(a)0

0204

0608

10

24


minus4

x 10minus5

x 10minus5

minus4

minus2

0

2

4

D12

-Y (m

)

(b)






D11minus2770 rpm

x 10minus5

minus4

minus2

0

2

4

D11

-X (m

)


(a)

D12minus2770 rpm

x 10minus5


minus4

minus2

0

2

4

D12

-Y (m

)

(b)

(1375 1185)(46 1009)

(92 0831)


minus5

00

05

10

15

D11

-X (m

)

100 200 300 400 5000f (Hz)

(c)

(1375 1218)(46 1163)

(92 0898)


minus5

00

05

10

15

D12

-Y (m

)

100 200 300 400 5000f (Hz)

(d)


x 10minus5

x 10minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

D12

-Y (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)


6 Conclusion




D11-1080 rpm

x 10minus5

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

)


(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)


(e)






D11-1080 rpmx 10

minus5


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

)


(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)


(e)



Data Availability




Acknowledgments



(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)



References
































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




X =119897

sum119894=1

120572119894y119894 =119897

sum119894=1

120572119894120572T119894 X (8)



x = 119886M+ (9)


=

[[[[[[[[[[

1

1 1

1 1 1

1 1

1

]]]]]]]]]]

M (10)


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


(11)





119909 (119905) =119896

sum119894=1

119860 119894 sin (2120587120596119894119905 + 120593119894) (12)









(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 40 500

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


(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


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)









(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














X = 119886 times[[[[[[



]]]]]]

(13)



X = 1198862119894

times[[[[[[[



]]]]]]]

(14)



X = 1198862119894 times 119890120593119894

times[[[[[[[[



]]]]]]]]


times[[[[[[[[



]]]]]]]](15)



X = 119886 times[[[[[[[[[



]]]]]]]]]

(16)


10038161003816100381610038161003816100381610038161003816100381610038161003816= minussin2 (120596119879119904)

= 0(17)






119897

sum119894=1

C120572119894120572T119894 =119897

sum119894=1

120582119894120572119894120572T119894 (19)


CI119898 = 12058211205721120572T1 + 12058221205722120572T2 (20)


|C|2 = 12058221119898

sum119894=1

12057221198941119898

sum119894=1

12057221198941 + 12059022119898

sum119894=1

12057221198942119898

sum119895=1

12057221198942

+ 21205902112059022119898

sum119894=1

12057211989411205721198942119898

sum119894=1

12057211989411205721198943(21)


|C|2 = 12058221 + 12058222 (22)







minus6minus4minus2

0246

Am

plitu

de


(a)

(200972)

(490693)(800791)

(500468)

0

05

1

Am

plitu

de

50 100 150 2000f (Hz)

(b)



119910 = sin (40120587119905 + 20) + 05sin (100120587119905 + 40)+ 07sin (98120587119905 + 50) + 08sin (160120587119905 + 60)+ 119890 (119899)

(23)





(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











minus2

minus1

0

1

2A

mpl

itude


(a)

(200977)

50 1000f (Hz)

002

04

06

08

1

Am

plitu

de

(b)


minus1

minus05

0

05

1

Am

plitu

de


(c)

(800786)

002

04

06

08

1

Am

plitu

de

50 1000f (Hz)

(d)



minus1

minus05

0

05

1

Am

plitu

de

(e)

(490688)

50 1000f (Hz)

002

04

06

08

1

Am

plitu

de

(f)


minus05

0

05

Am

plitu

de


(g)

(500463)

002

04

06

08

1

Am

plitu

de

50 1000f (Hz)

(h)



A B


Coupling

Sliding Bearing

Rotor

Large Platform

Small Platform

Air spring

BeltSliding Bearing



D13 D14

D12D11













D11-1080 rpmx 10

minus5


012345

X (m

)


(a)


0733404771

Order 1 Order 2

0250502342

Order 6Order 7

D11-1080 rpm

01651 Order 11


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

)


(c)


50Hz

0658105228

Order 1 Order 2

0277101226

Order 5 Order 12

01948 Order 8


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)


x 10minus5

x 10minus5

minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

D12

minusY

(m)



(12751) (22748)e first group


D11minus1080 rpm

x 10minus8

0

1

2

3

4

Eige

nval

ue o

f PCA

(m)


(a)

(12247) (2241)

(314) (41396)


e second group

x 10minus8

005

115

225

Eige

nval

ue o

f PCA

(m)


(b)




minus5

minus15

minus10

minus05

0051015

X (m

)


(a)


minus5

minus10

minus05

0051015

Y (m

)


(b)

D11minus1080 rpm(1807287)

(360472)

x 10minus5

002040608

1

X (m

)

50 100 150 2000Frequency (Hz)

(c)

(3605105)(1806575)


minus5

002040608

1

Y (m

)

50 100 150 2000Frequency (Hz)

(d)


W1

x 10minus5

x 10minus5

minus1

minus05

0

05

1

15

D12

minusY

(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)












minus4

minus1

minus05

0

05

1

D11

-X (m

)


(a)


minus4

minus1

minus05

0

05

1

D12

-Y (m

)


(b)


1X2X


08441166

1X2X3X

x 10minus5

200 400 600 800 10000f (Hz)

0

05

10

15

D11

-X (m

)

(c)

1X

2X


and 150Hz

D12minus2770 rpm117309021227

1X2X3X

x 10minus5

0

05

10

15

D12

-Y (m

)

200 400 600 800 10000f (Hz)

(d)


(11489) (21482)e first group




minus7


0

05

1

15

Eige

nval

ue o

f PCA

(m)

(a)

D12minus2770 rpm(11688) (21687)

e first group

e second group


(31673) (41664)

x 10minus7

0

05

1

15

2

Eige

nval

ue o

f PCA

(m)


(b)









minus5


minus4

minus2

0

2

4

D11

-X (m

)

(a)


minus5

minus4

minus2

0

2

4

D12

-Y (m

)


(b)

(1375 1185)(46 1009)


minus5

100 200 300 400 5000f (Hz)

0

05

1

15

D11

-X (m

)

(c)

(1375 1218)(46 1164)


minus5

100 200 300 400 5000f (Hz)

0

05

1

15

D12

-Y (m

)

(d)


x 10minus5

x 10minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

D12

-Y (m

)


D11-X (m)

(a)0

0204

0608

10

24


minus4

x 10minus5

x 10minus5

minus4

minus2

0

2

4

D12

-Y (m

)

(b)






D11minus2770 rpm

x 10minus5

minus4

minus2

0

2

4

D11

-X (m

)


(a)

D12minus2770 rpm

x 10minus5


minus4

minus2

0

2

4

D12

-Y (m

)

(b)

(1375 1185)(46 1009)

(92 0831)


minus5

00

05

10

15

D11

-X (m

)

100 200 300 400 5000f (Hz)

(c)

(1375 1218)(46 1163)

(92 0898)


minus5

00

05

10

15

D12

-Y (m

)

100 200 300 400 5000f (Hz)

(d)


x 10minus5

x 10minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

D12

-Y (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)


6 Conclusion




D11-1080 rpm

x 10minus5

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

)


(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)


(e)






D11-1080 rpmx 10

minus5


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

)


(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)


(e)



Data Availability




Acknowledgments



(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)



References
































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



(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 40 500

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


(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


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)









(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














X = 119886 times[[[[[[



]]]]]]

(13)



X = 1198862119894

times[[[[[[[



]]]]]]]

(14)



X = 1198862119894 times 119890120593119894

times[[[[[[[[



]]]]]]]]


times[[[[[[[[



]]]]]]]](15)



X = 119886 times[[[[[[[[[



]]]]]]]]]

(16)


10038161003816100381610038161003816100381610038161003816100381610038161003816= minussin2 (120596119879119904)

= 0(17)






119897

sum119894=1

C120572119894120572T119894 =119897

sum119894=1

120582119894120572119894120572T119894 (19)


CI119898 = 12058211205721120572T1 + 12058221205722120572T2 (20)


|C|2 = 12058221119898

sum119894=1

12057221198941119898

sum119894=1

12057221198941 + 12059022119898

sum119894=1

12057221198942119898

sum119895=1

12057221198942

+ 21205902112059022119898

sum119894=1

12057211989411205721198942119898

sum119894=1

12057211989411205721198943(21)


|C|2 = 12058221 + 12058222 (22)







minus6minus4minus2

0246

Am

plitu

de


(a)

(200972)

(490693)(800791)

(500468)

0

05

1

Am

plitu

de

50 100 150 2000f (Hz)

(b)



119910 = sin (40120587119905 + 20) + 05sin (100120587119905 + 40)+ 07sin (98120587119905 + 50) + 08sin (160120587119905 + 60)+ 119890 (119899)

(23)





(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











minus2

minus1

0

1

2A

mpl

itude


(a)

(200977)

50 1000f (Hz)

002

04

06

08

1

Am

plitu

de

(b)


minus1

minus05

0

05

1

Am

plitu

de


(c)

(800786)

002

04

06

08

1

Am

plitu

de

50 1000f (Hz)

(d)



minus1

minus05

0

05

1

Am

plitu

de

(e)

(490688)

50 1000f (Hz)

002

04

06

08

1

Am

plitu

de

(f)


minus05

0

05

Am

plitu

de


(g)

(500463)

002

04

06

08

1

Am

plitu

de

50 1000f (Hz)

(h)



A B


Coupling

Sliding Bearing

Rotor

Large Platform

Small Platform

Air spring

BeltSliding Bearing



D13 D14

D12D11













D11-1080 rpmx 10

minus5


012345

X (m

)


(a)


0733404771

Order 1 Order 2

0250502342

Order 6Order 7

D11-1080 rpm

01651 Order 11


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

)


(c)


50Hz

0658105228

Order 1 Order 2

0277101226

Order 5 Order 12

01948 Order 8


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)


x 10minus5

x 10minus5

minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

D12

minusY

(m)



(12751) (22748)e first group


D11minus1080 rpm

x 10minus8

0

1

2

3

4

Eige

nval

ue o

f PCA

(m)


(a)

(12247) (2241)

(314) (41396)


e second group

x 10minus8

005

115

225

Eige

nval

ue o

f PCA

(m)


(b)




minus5

minus15

minus10

minus05

0051015

X (m

)


(a)


minus5

minus10

minus05

0051015

Y (m

)


(b)

D11minus1080 rpm(1807287)

(360472)

x 10minus5

002040608

1

X (m

)

50 100 150 2000Frequency (Hz)

(c)

(3605105)(1806575)


minus5

002040608

1

Y (m

)

50 100 150 2000Frequency (Hz)

(d)


W1

x 10minus5

x 10minus5

minus1

minus05

0

05

1

15

D12

minusY

(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)












minus4

minus1

minus05

0

05

1

D11

-X (m

)


(a)


minus4

minus1

minus05

0

05

1

D12

-Y (m

)


(b)


1X2X


08441166

1X2X3X

x 10minus5

200 400 600 800 10000f (Hz)

0

05

10

15

D11

-X (m

)

(c)

1X

2X


and 150Hz

D12minus2770 rpm117309021227

1X2X3X

x 10minus5

0

05

10

15

D12

-Y (m

)

200 400 600 800 10000f (Hz)

(d)


(11489) (21482)e first group




minus7


0

05

1

15

Eige

nval

ue o

f PCA

(m)

(a)

D12minus2770 rpm(11688) (21687)

e first group

e second group


(31673) (41664)

x 10minus7

0

05

1

15

2

Eige

nval

ue o

f PCA

(m)


(b)









minus5


minus4

minus2

0

2

4

D11

-X (m

)

(a)


minus5

minus4

minus2

0

2

4

D12

-Y (m

)


(b)

(1375 1185)(46 1009)


minus5

100 200 300 400 5000f (Hz)

0

05

1

15

D11

-X (m

)

(c)

(1375 1218)(46 1164)


minus5

100 200 300 400 5000f (Hz)

0

05

1

15

D12

-Y (m

)

(d)


x 10minus5

x 10minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

D12

-Y (m

)


D11-X (m)

(a)0

0204

0608

10

24


minus4

x 10minus5

x 10minus5

minus4

minus2

0

2

4

D12

-Y (m

)

(b)






D11minus2770 rpm

x 10minus5

minus4

minus2

0

2

4

D11

-X (m

)


(a)

D12minus2770 rpm

x 10minus5


minus4

minus2

0

2

4

D12

-Y (m

)

(b)

(1375 1185)(46 1009)

(92 0831)


minus5

00

05

10

15

D11

-X (m

)

100 200 300 400 5000f (Hz)

(c)

(1375 1218)(46 1163)

(92 0898)


minus5

00

05

10

15

D12

-Y (m

)

100 200 300 400 5000f (Hz)

(d)


x 10minus5

x 10minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

D12

-Y (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)


6 Conclusion




D11-1080 rpm

x 10minus5

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

)


(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)


(e)






D11-1080 rpmx 10

minus5


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

)


(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)


(e)



Data Availability




Acknowledgments



(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)



References
































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



(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














X = 119886 times[[[[[[



]]]]]]

(13)



X = 1198862119894

times[[[[[[[



]]]]]]]

(14)



X = 1198862119894 times 119890120593119894

times[[[[[[[[



]]]]]]]]


times[[[[[[[[



]]]]]]]](15)



X = 119886 times[[[[[[[[[



]]]]]]]]]

(16)


10038161003816100381610038161003816100381610038161003816100381610038161003816= minussin2 (120596119879119904)

= 0(17)






119897

sum119894=1

C120572119894120572T119894 =119897

sum119894=1

120582119894120572119894120572T119894 (19)


CI119898 = 12058211205721120572T1 + 12058221205722120572T2 (20)


|C|2 = 12058221119898

sum119894=1

12057221198941119898

sum119894=1

12057221198941 + 12059022119898

sum119894=1

12057221198942119898

sum119895=1

12057221198942

+ 21205902112059022119898

sum119894=1

12057211989411205721198942119898

sum119894=1

12057211989411205721198943(21)


|C|2 = 12058221 + 12058222 (22)







minus6minus4minus2

0246

Am

plitu

de


(a)

(200972)

(490693)(800791)

(500468)

0

05

1

Am

plitu

de

50 100 150 2000f (Hz)

(b)



119910 = sin (40120587119905 + 20) + 05sin (100120587119905 + 40)+ 07sin (98120587119905 + 50) + 08sin (160120587119905 + 60)+ 119890 (119899)

(23)





(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











minus2

minus1

0

1

2A

mpl

itude


(a)

(200977)

50 1000f (Hz)

002

04

06

08

1

Am

plitu

de

(b)


minus1

minus05

0

05

1

Am

plitu

de


(c)

(800786)

002

04

06

08

1

Am

plitu

de

50 1000f (Hz)

(d)



minus1

minus05

0

05

1

Am

plitu

de

(e)

(490688)

50 1000f (Hz)

002

04

06

08

1

Am

plitu

de

(f)


minus05

0

05

Am

plitu

de


(g)

(500463)

002

04

06

08

1

Am

plitu

de

50 1000f (Hz)

(h)



A B


Coupling

Sliding Bearing

Rotor

Large Platform

Small Platform

Air spring

BeltSliding Bearing



D13 D14

D12D11













D11-1080 rpmx 10

minus5


012345

X (m

)


(a)


0733404771

Order 1 Order 2

0250502342

Order 6Order 7

D11-1080 rpm

01651 Order 11


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

)


(c)


50Hz

0658105228

Order 1 Order 2

0277101226

Order 5 Order 12

01948 Order 8


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)


x 10minus5

x 10minus5

minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

D12

minusY

(m)



(12751) (22748)e first group


D11minus1080 rpm

x 10minus8

0

1

2

3

4

Eige

nval

ue o

f PCA

(m)


(a)

(12247) (2241)

(314) (41396)


e second group

x 10minus8

005

115

225

Eige

nval

ue o

f PCA

(m)


(b)




minus5

minus15

minus10

minus05

0051015

X (m

)


(a)


minus5

minus10

minus05

0051015

Y (m

)


(b)

D11minus1080 rpm(1807287)

(360472)

x 10minus5

002040608

1

X (m

)

50 100 150 2000Frequency (Hz)

(c)

(3605105)(1806575)


minus5

002040608

1

Y (m

)

50 100 150 2000Frequency (Hz)

(d)


W1

x 10minus5

x 10minus5

minus1

minus05

0

05

1

15

D12

minusY

(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)












minus4

minus1

minus05

0

05

1

D11

-X (m

)


(a)


minus4

minus1

minus05

0

05

1

D12

-Y (m

)


(b)


1X2X


08441166

1X2X3X

x 10minus5

200 400 600 800 10000f (Hz)

0

05

10

15

D11

-X (m

)

(c)

1X

2X


and 150Hz

D12minus2770 rpm117309021227

1X2X3X

x 10minus5

0

05

10

15

D12

-Y (m

)

200 400 600 800 10000f (Hz)

(d)


(11489) (21482)e first group




minus7


0

05

1

15

Eige

nval

ue o

f PCA

(m)

(a)

D12minus2770 rpm(11688) (21687)

e first group

e second group


(31673) (41664)

x 10minus7

0

05

1

15

2

Eige

nval

ue o

f PCA

(m)


(b)









minus5


minus4

minus2

0

2

4

D11

-X (m

)

(a)


minus5

minus4

minus2

0

2

4

D12

-Y (m

)


(b)

(1375 1185)(46 1009)


minus5

100 200 300 400 5000f (Hz)

0

05

1

15

D11

-X (m

)

(c)

(1375 1218)(46 1164)


minus5

100 200 300 400 5000f (Hz)

0

05

1

15

D12

-Y (m

)

(d)


x 10minus5

x 10minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

D12

-Y (m

)


D11-X (m)

(a)0

0204

0608

10

24


minus4

x 10minus5

x 10minus5

minus4

minus2

0

2

4

D12

-Y (m

)

(b)






D11minus2770 rpm

x 10minus5

minus4

minus2

0

2

4

D11

-X (m

)


(a)

D12minus2770 rpm

x 10minus5


minus4

minus2

0

2

4

D12

-Y (m

)

(b)

(1375 1185)(46 1009)

(92 0831)


minus5

00

05

10

15

D11

-X (m

)

100 200 300 400 5000f (Hz)

(c)

(1375 1218)(46 1163)

(92 0898)


minus5

00

05

10

15

D12

-Y (m

)

100 200 300 400 5000f (Hz)

(d)


x 10minus5

x 10minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

D12

-Y (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)


6 Conclusion




D11-1080 rpm

x 10minus5

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

)


(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)


(e)






D11-1080 rpmx 10

minus5


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

)


(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)


(e)



Data Availability




Acknowledgments



(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)



References
































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




X = 1198862119894 times 119890120593119894

times[[[[[[[[



]]]]]]]]


times[[[[[[[[



]]]]]]]](15)



X = 119886 times[[[[[[[[[



]]]]]]]]]

(16)


10038161003816100381610038161003816100381610038161003816100381610038161003816= minussin2 (120596119879119904)

= 0(17)






119897

sum119894=1

C120572119894120572T119894 =119897

sum119894=1

120582119894120572119894120572T119894 (19)


CI119898 = 12058211205721120572T1 + 12058221205722120572T2 (20)


|C|2 = 12058221119898

sum119894=1

12057221198941119898

sum119894=1

12057221198941 + 12059022119898

sum119894=1

12057221198942119898

sum119895=1

12057221198942

+ 21205902112059022119898

sum119894=1

12057211989411205721198942119898

sum119894=1

12057211989411205721198943(21)


|C|2 = 12058221 + 12058222 (22)







minus6minus4minus2

0246

Am

plitu

de


(a)

(200972)

(490693)(800791)

(500468)

0

05

1

Am

plitu

de

50 100 150 2000f (Hz)

(b)



119910 = sin (40120587119905 + 20) + 05sin (100120587119905 + 40)+ 07sin (98120587119905 + 50) + 08sin (160120587119905 + 60)+ 119890 (119899)

(23)





(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











minus2

minus1

0

1

2A

mpl

itude


(a)

(200977)

50 1000f (Hz)

002

04

06

08

1

Am

plitu

de

(b)


minus1

minus05

0

05

1

Am

plitu

de


(c)

(800786)

002

04

06

08

1

Am

plitu

de

50 1000f (Hz)

(d)



minus1

minus05

0

05

1

Am

plitu

de

(e)

(490688)

50 1000f (Hz)

002

04

06

08

1

Am

plitu

de

(f)


minus05

0

05

Am

plitu

de


(g)

(500463)

002

04

06

08

1

Am

plitu

de

50 1000f (Hz)

(h)



A B


Coupling

Sliding Bearing

Rotor

Large Platform

Small Platform

Air spring

BeltSliding Bearing



D13 D14

D12D11













D11-1080 rpmx 10

minus5


012345

X (m

)


(a)


0733404771

Order 1 Order 2

0250502342

Order 6Order 7

D11-1080 rpm

01651 Order 11


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

)


(c)


50Hz

0658105228

Order 1 Order 2

0277101226

Order 5 Order 12

01948 Order 8


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)


x 10minus5

x 10minus5

minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

D12

minusY

(m)



(12751) (22748)e first group


D11minus1080 rpm

x 10minus8

0

1

2

3

4

Eige

nval

ue o

f PCA

(m)


(a)

(12247) (2241)

(314) (41396)


e second group

x 10minus8

005

115

225

Eige

nval

ue o

f PCA

(m)


(b)




minus5

minus15

minus10

minus05

0051015

X (m

)


(a)


minus5

minus10

minus05

0051015

Y (m

)


(b)

D11minus1080 rpm(1807287)

(360472)

x 10minus5

002040608

1

X (m

)

50 100 150 2000Frequency (Hz)

(c)

(3605105)(1806575)


minus5

002040608

1

Y (m

)

50 100 150 2000Frequency (Hz)

(d)


W1

x 10minus5

x 10minus5

minus1

minus05

0

05

1

15

D12

minusY

(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)












minus4

minus1

minus05

0

05

1

D11

-X (m

)


(a)


minus4

minus1

minus05

0

05

1

D12

-Y (m

)


(b)


1X2X


08441166

1X2X3X

x 10minus5

200 400 600 800 10000f (Hz)

0

05

10

15

D11

-X (m

)

(c)

1X

2X


and 150Hz

D12minus2770 rpm117309021227

1X2X3X

x 10minus5

0

05

10

15

D12

-Y (m

)

200 400 600 800 10000f (Hz)

(d)


(11489) (21482)e first group




minus7


0

05

1

15

Eige

nval

ue o

f PCA

(m)

(a)

D12minus2770 rpm(11688) (21687)

e first group

e second group


(31673) (41664)

x 10minus7

0

05

1

15

2

Eige

nval

ue o

f PCA

(m)


(b)









minus5


minus4

minus2

0

2

4

D11

-X (m

)

(a)


minus5

minus4

minus2

0

2

4

D12

-Y (m

)


(b)

(1375 1185)(46 1009)


minus5

100 200 300 400 5000f (Hz)

0

05

1

15

D11

-X (m

)

(c)

(1375 1218)(46 1164)


minus5

100 200 300 400 5000f (Hz)

0

05

1

15

D12

-Y (m

)

(d)


x 10minus5

x 10minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

D12

-Y (m

)


D11-X (m)

(a)0

0204

0608

10

24


minus4

x 10minus5

x 10minus5

minus4

minus2

0

2

4

D12

-Y (m

)

(b)






D11minus2770 rpm

x 10minus5

minus4

minus2

0

2

4

D11

-X (m

)


(a)

D12minus2770 rpm

x 10minus5


minus4

minus2

0

2

4

D12

-Y (m

)

(b)

(1375 1185)(46 1009)

(92 0831)


minus5

00

05

10

15

D11

-X (m

)

100 200 300 400 5000f (Hz)

(c)

(1375 1218)(46 1163)

(92 0898)


minus5

00

05

10

15

D12

-Y (m

)

100 200 300 400 5000f (Hz)

(d)


x 10minus5

x 10minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

D12

-Y (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)


6 Conclusion




D11-1080 rpm

x 10minus5

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

)


(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)


(e)






D11-1080 rpmx 10

minus5


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

)


(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)


(e)



Data Availability




Acknowledgments



(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)



References
































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





minus6minus4minus2

0246

Am

plitu

de


(a)

(200972)

(490693)(800791)

(500468)

0

05

1

Am

plitu

de

50 100 150 2000f (Hz)

(b)



119910 = sin (40120587119905 + 20) + 05sin (100120587119905 + 40)+ 07sin (98120587119905 + 50) + 08sin (160120587119905 + 60)+ 119890 (119899)

(23)





(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











minus2

minus1

0

1

2A

mpl

itude


(a)

(200977)

50 1000f (Hz)

002

04

06

08

1

Am

plitu

de

(b)


minus1

minus05

0

05

1

Am

plitu

de


(c)

(800786)

002

04

06

08

1

Am

plitu

de

50 1000f (Hz)

(d)



minus1

minus05

0

05

1

Am

plitu

de

(e)

(490688)

50 1000f (Hz)

002

04

06

08

1

Am

plitu

de

(f)


minus05

0

05

Am

plitu

de


(g)

(500463)

002

04

06

08

1

Am

plitu

de

50 1000f (Hz)

(h)



A B


Coupling

Sliding Bearing

Rotor

Large Platform

Small Platform

Air spring

BeltSliding Bearing



D13 D14

D12D11













D11-1080 rpmx 10

minus5


012345

X (m

)


(a)


0733404771

Order 1 Order 2

0250502342

Order 6Order 7

D11-1080 rpm

01651 Order 11


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

)


(c)


50Hz

0658105228

Order 1 Order 2

0277101226

Order 5 Order 12

01948 Order 8


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)


x 10minus5

x 10minus5

minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

D12

minusY

(m)



(12751) (22748)e first group


D11minus1080 rpm

x 10minus8

0

1

2

3

4

Eige

nval

ue o

f PCA

(m)


(a)

(12247) (2241)

(314) (41396)


e second group

x 10minus8

005

115

225

Eige

nval

ue o

f PCA

(m)


(b)




minus5

minus15

minus10

minus05

0051015

X (m

)


(a)


minus5

minus10

minus05

0051015

Y (m

)


(b)

D11minus1080 rpm(1807287)

(360472)

x 10minus5

002040608

1

X (m

)

50 100 150 2000Frequency (Hz)

(c)

(3605105)(1806575)


minus5

002040608

1

Y (m

)

50 100 150 2000Frequency (Hz)

(d)


W1

x 10minus5

x 10minus5

minus1

minus05

0

05

1

15

D12

minusY

(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)












minus4

minus1

minus05

0

05

1

D11

-X (m

)


(a)


minus4

minus1

minus05

0

05

1

D12

-Y (m

)


(b)


1X2X


08441166

1X2X3X

x 10minus5

200 400 600 800 10000f (Hz)

0

05

10

15

D11

-X (m

)

(c)

1X

2X


and 150Hz

D12minus2770 rpm117309021227

1X2X3X

x 10minus5

0

05

10

15

D12

-Y (m

)

200 400 600 800 10000f (Hz)

(d)


(11489) (21482)e first group




minus7


0

05

1

15

Eige

nval

ue o

f PCA

(m)

(a)

D12minus2770 rpm(11688) (21687)

e first group

e second group


(31673) (41664)

x 10minus7

0

05

1

15

2

Eige

nval

ue o

f PCA

(m)


(b)









minus5


minus4

minus2

0

2

4

D11

-X (m

)

(a)


minus5

minus4

minus2

0

2

4

D12

-Y (m

)


(b)

(1375 1185)(46 1009)


minus5

100 200 300 400 5000f (Hz)

0

05

1

15

D11

-X (m

)

(c)

(1375 1218)(46 1164)


minus5

100 200 300 400 5000f (Hz)

0

05

1

15

D12

-Y (m

)

(d)


x 10minus5

x 10minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

D12

-Y (m

)


D11-X (m)

(a)0

0204

0608

10

24


minus4

x 10minus5

x 10minus5

minus4

minus2

0

2

4

D12

-Y (m

)

(b)






D11minus2770 rpm

x 10minus5

minus4

minus2

0

2

4

D11

-X (m

)


(a)

D12minus2770 rpm

x 10minus5


minus4

minus2

0

2

4

D12

-Y (m

)

(b)

(1375 1185)(46 1009)

(92 0831)


minus5

00

05

10

15

D11

-X (m

)

100 200 300 400 5000f (Hz)

(c)

(1375 1218)(46 1163)

(92 0898)


minus5

00

05

10

15

D12

-Y (m

)

100 200 300 400 5000f (Hz)

(d)


x 10minus5

x 10minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

D12

-Y (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)


6 Conclusion




D11-1080 rpm

x 10minus5

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

)


(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)


(e)






D11-1080 rpmx 10

minus5


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

)


(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)


(e)



Data Availability




Acknowledgments



(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)



References
































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




minus2

minus1

0

1

2A

mpl

itude


(a)

(200977)

50 1000f (Hz)

002

04

06

08

1

Am

plitu

de

(b)


minus1

minus05

0

05

1

Am

plitu

de


(c)

(800786)

002

04

06

08

1

Am

plitu

de

50 1000f (Hz)

(d)



minus1

minus05

0

05

1

Am

plitu

de

(e)

(490688)

50 1000f (Hz)

002

04

06

08

1

Am

plitu

de

(f)


minus05

0

05

Am

plitu

de


(g)

(500463)

002

04

06

08

1

Am

plitu

de

50 1000f (Hz)

(h)



A B


Coupling

Sliding Bearing

Rotor

Large Platform

Small Platform

Air spring

BeltSliding Bearing



D13 D14

D12D11













D11-1080 rpmx 10

minus5


012345

X (m

)


(a)


0733404771

Order 1 Order 2

0250502342

Order 6Order 7

D11-1080 rpm

01651 Order 11


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

)


(c)


50Hz

0658105228

Order 1 Order 2

0277101226

Order 5 Order 12

01948 Order 8


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)


x 10minus5

x 10minus5

minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

D12

minusY

(m)



(12751) (22748)e first group


D11minus1080 rpm

x 10minus8

0

1

2

3

4

Eige

nval

ue o

f PCA

(m)


(a)

(12247) (2241)

(314) (41396)


e second group

x 10minus8

005

115

225

Eige

nval

ue o

f PCA

(m)


(b)




minus5

minus15

minus10

minus05

0051015

X (m

)


(a)


minus5

minus10

minus05

0051015

Y (m

)


(b)

D11minus1080 rpm(1807287)

(360472)

x 10minus5

002040608

1

X (m

)

50 100 150 2000Frequency (Hz)

(c)

(3605105)(1806575)


minus5

002040608

1

Y (m

)

50 100 150 2000Frequency (Hz)

(d)


W1

x 10minus5

x 10minus5

minus1

minus05

0

05

1

15

D12

minusY

(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)












minus4

minus1

minus05

0

05

1

D11

-X (m

)


(a)


minus4

minus1

minus05

0

05

1

D12

-Y (m

)


(b)


1X2X


08441166

1X2X3X

x 10minus5

200 400 600 800 10000f (Hz)

0

05

10

15

D11

-X (m

)

(c)

1X

2X


and 150Hz

D12minus2770 rpm117309021227

1X2X3X

x 10minus5

0

05

10

15

D12

-Y (m

)

200 400 600 800 10000f (Hz)

(d)


(11489) (21482)e first group




minus7


0

05

1

15

Eige

nval

ue o

f PCA

(m)

(a)

D12minus2770 rpm(11688) (21687)

e first group

e second group


(31673) (41664)

x 10minus7

0

05

1

15

2

Eige

nval

ue o

f PCA

(m)


(b)









minus5


minus4

minus2

0

2

4

D11

-X (m

)

(a)


minus5

minus4

minus2

0

2

4

D12

-Y (m

)


(b)

(1375 1185)(46 1009)


minus5

100 200 300 400 5000f (Hz)

0

05

1

15

D11

-X (m

)

(c)

(1375 1218)(46 1164)


minus5

100 200 300 400 5000f (Hz)

0

05

1

15

D12

-Y (m

)

(d)


x 10minus5

x 10minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

D12

-Y (m

)


D11-X (m)

(a)0

0204

0608

10

24


minus4

x 10minus5

x 10minus5

minus4

minus2

0

2

4

D12

-Y (m

)

(b)






D11minus2770 rpm

x 10minus5

minus4

minus2

0

2

4

D11

-X (m

)


(a)

D12minus2770 rpm

x 10minus5


minus4

minus2

0

2

4

D12

-Y (m

)

(b)

(1375 1185)(46 1009)

(92 0831)


minus5

00

05

10

15

D11

-X (m

)

100 200 300 400 5000f (Hz)

(c)

(1375 1218)(46 1163)

(92 0898)


minus5

00

05

10

15

D12

-Y (m

)

100 200 300 400 5000f (Hz)

(d)


x 10minus5

x 10minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

D12

-Y (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)


6 Conclusion




D11-1080 rpm

x 10minus5

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

)


(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)


(e)






D11-1080 rpmx 10

minus5


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

)


(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)


(e)



Data Availability




Acknowledgments



(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)



References
































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



A B


Coupling

Sliding Bearing

Rotor

Large Platform

Small Platform

Air spring

BeltSliding Bearing



D13 D14

D12D11













D11-1080 rpmx 10

minus5


012345

X (m

)


(a)


0733404771

Order 1 Order 2

0250502342

Order 6Order 7

D11-1080 rpm

01651 Order 11


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

)


(c)


50Hz

0658105228

Order 1 Order 2

0277101226

Order 5 Order 12

01948 Order 8


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)


x 10minus5

x 10minus5

minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

D12

minusY

(m)



(12751) (22748)e first group


D11minus1080 rpm

x 10minus8

0

1

2

3

4

Eige

nval

ue o

f PCA

(m)


(a)

(12247) (2241)

(314) (41396)


e second group

x 10minus8

005

115

225

Eige

nval

ue o

f PCA

(m)


(b)




minus5

minus15

minus10

minus05

0051015

X (m

)


(a)


minus5

minus10

minus05

0051015

Y (m

)


(b)

D11minus1080 rpm(1807287)

(360472)

x 10minus5

002040608

1

X (m

)

50 100 150 2000Frequency (Hz)

(c)

(3605105)(1806575)


minus5

002040608

1

Y (m

)

50 100 150 2000Frequency (Hz)

(d)


W1

x 10minus5

x 10minus5

minus1

minus05

0

05

1

15

D12

minusY

(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)












minus4

minus1

minus05

0

05

1

D11

-X (m

)


(a)


minus4

minus1

minus05

0

05

1

D12

-Y (m

)


(b)


1X2X


08441166

1X2X3X

x 10minus5

200 400 600 800 10000f (Hz)

0

05

10

15

D11

-X (m

)

(c)

1X

2X


and 150Hz

D12minus2770 rpm117309021227

1X2X3X

x 10minus5

0

05

10

15

D12

-Y (m

)

200 400 600 800 10000f (Hz)

(d)


(11489) (21482)e first group




minus7


0

05

1

15

Eige

nval

ue o

f PCA

(m)

(a)

D12minus2770 rpm(11688) (21687)

e first group

e second group


(31673) (41664)

x 10minus7

0

05

1

15

2

Eige

nval

ue o

f PCA

(m)


(b)









minus5


minus4

minus2

0

2

4

D11

-X (m

)

(a)


minus5

minus4

minus2

0

2

4

D12

-Y (m

)


(b)

(1375 1185)(46 1009)


minus5

100 200 300 400 5000f (Hz)

0

05

1

15

D11

-X (m

)

(c)

(1375 1218)(46 1164)


minus5

100 200 300 400 5000f (Hz)

0

05

1

15

D12

-Y (m

)

(d)


x 10minus5

x 10minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

D12

-Y (m

)


D11-X (m)

(a)0

0204

0608

10

24


minus4

x 10minus5

x 10minus5

minus4

minus2

0

2

4

D12

-Y (m

)

(b)






D11minus2770 rpm

x 10minus5

minus4

minus2

0

2

4

D11

-X (m

)


(a)

D12minus2770 rpm

x 10minus5


minus4

minus2

0

2

4

D12

-Y (m

)

(b)

(1375 1185)(46 1009)

(92 0831)


minus5

00

05

10

15

D11

-X (m

)

100 200 300 400 5000f (Hz)

(c)

(1375 1218)(46 1163)

(92 0898)


minus5

00

05

10

15

D12

-Y (m

)

100 200 300 400 5000f (Hz)

(d)


x 10minus5

x 10minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

D12

-Y (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)


6 Conclusion




D11-1080 rpm

x 10minus5

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

)


(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)


(e)






D11-1080 rpmx 10

minus5


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

)


(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)


(e)



Data Availability




Acknowledgments



(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)



References
































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



D11-1080 rpmx 10

minus5


012345

X (m

)


(a)


0733404771

Order 1 Order 2

0250502342

Order 6Order 7

D11-1080 rpm

01651 Order 11


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

)


(c)


50Hz

0658105228

Order 1 Order 2

0277101226

Order 5 Order 12

01948 Order 8


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)


x 10minus5

x 10minus5

minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

D12

minusY

(m)



(12751) (22748)e first group


D11minus1080 rpm

x 10minus8

0

1

2

3

4

Eige

nval

ue o

f PCA

(m)


(a)

(12247) (2241)

(314) (41396)


e second group

x 10minus8

005

115

225

Eige

nval

ue o

f PCA

(m)


(b)




minus5

minus15

minus10

minus05

0051015

X (m

)


(a)


minus5

minus10

minus05

0051015

Y (m

)


(b)

D11minus1080 rpm(1807287)

(360472)

x 10minus5

002040608

1

X (m

)

50 100 150 2000Frequency (Hz)

(c)

(3605105)(1806575)


minus5

002040608

1

Y (m

)

50 100 150 2000Frequency (Hz)

(d)


W1

x 10minus5

x 10minus5

minus1

minus05

0

05

1

15

D12

minusY

(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)












minus4

minus1

minus05

0

05

1

D11

-X (m

)


(a)


minus4

minus1

minus05

0

05

1

D12

-Y (m

)


(b)


1X2X


08441166

1X2X3X

x 10minus5

200 400 600 800 10000f (Hz)

0

05

10

15

D11

-X (m

)

(c)

1X

2X


and 150Hz

D12minus2770 rpm117309021227

1X2X3X

x 10minus5

0

05

10

15

D12

-Y (m

)

200 400 600 800 10000f (Hz)

(d)


(11489) (21482)e first group




minus7


0

05

1

15

Eige

nval

ue o

f PCA

(m)

(a)

D12minus2770 rpm(11688) (21687)

e first group

e second group


(31673) (41664)

x 10minus7

0

05

1

15

2

Eige

nval

ue o

f PCA

(m)


(b)









minus5


minus4

minus2

0

2

4

D11

-X (m

)

(a)


minus5

minus4

minus2

0

2

4

D12

-Y (m

)


(b)

(1375 1185)(46 1009)


minus5

100 200 300 400 5000f (Hz)

0

05

1

15

D11

-X (m

)

(c)

(1375 1218)(46 1164)


minus5

100 200 300 400 5000f (Hz)

0

05

1

15

D12

-Y (m

)

(d)


x 10minus5

x 10minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

D12

-Y (m

)


D11-X (m)

(a)0

0204

0608

10

24


minus4

x 10minus5

x 10minus5

minus4

minus2

0

2

4

D12

-Y (m

)

(b)






D11minus2770 rpm

x 10minus5

minus4

minus2

0

2

4

D11

-X (m

)


(a)

D12minus2770 rpm

x 10minus5


minus4

minus2

0

2

4

D12

-Y (m

)

(b)

(1375 1185)(46 1009)

(92 0831)


minus5

00

05

10

15

D11

-X (m

)

100 200 300 400 5000f (Hz)

(c)

(1375 1218)(46 1163)

(92 0898)


minus5

00

05

10

15

D12

-Y (m

)

100 200 300 400 5000f (Hz)

(d)


x 10minus5

x 10minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

D12

-Y (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)


6 Conclusion




D11-1080 rpm

x 10minus5

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

)


(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)


(e)






D11-1080 rpmx 10

minus5


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

)


(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)


(e)



Data Availability




Acknowledgments



(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)



References
































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




minus5

minus15

minus10

minus05

0051015

X (m

)


(a)


minus5

minus10

minus05

0051015

Y (m

)


(b)

D11minus1080 rpm(1807287)

(360472)

x 10minus5

002040608

1

X (m

)

50 100 150 2000Frequency (Hz)

(c)

(3605105)(1806575)


minus5

002040608

1

Y (m

)

50 100 150 2000Frequency (Hz)

(d)


W1

x 10minus5

x 10minus5

minus1

minus05

0

05

1

15

D12

minusY

(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)












minus4

minus1

minus05

0

05

1

D11

-X (m

)


(a)


minus4

minus1

minus05

0

05

1

D12

-Y (m

)


(b)


1X2X


08441166

1X2X3X

x 10minus5

200 400 600 800 10000f (Hz)

0

05

10

15

D11

-X (m

)

(c)

1X

2X


and 150Hz

D12minus2770 rpm117309021227

1X2X3X

x 10minus5

0

05

10

15

D12

-Y (m

)

200 400 600 800 10000f (Hz)

(d)


(11489) (21482)e first group




minus7


0

05

1

15

Eige

nval

ue o

f PCA

(m)

(a)

D12minus2770 rpm(11688) (21687)

e first group

e second group


(31673) (41664)

x 10minus7

0

05

1

15

2

Eige

nval

ue o

f PCA

(m)


(b)









minus5


minus4

minus2

0

2

4

D11

-X (m

)

(a)


minus5

minus4

minus2

0

2

4

D12

-Y (m

)


(b)

(1375 1185)(46 1009)


minus5

100 200 300 400 5000f (Hz)

0

05

1

15

D11

-X (m

)

(c)

(1375 1218)(46 1164)


minus5

100 200 300 400 5000f (Hz)

0

05

1

15

D12

-Y (m

)

(d)


x 10minus5

x 10minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

D12

-Y (m

)


D11-X (m)

(a)0

0204

0608

10

24


minus4

x 10minus5

x 10minus5

minus4

minus2

0

2

4

D12

-Y (m

)

(b)






D11minus2770 rpm

x 10minus5

minus4

minus2

0

2

4

D11

-X (m

)


(a)

D12minus2770 rpm

x 10minus5


minus4

minus2

0

2

4

D12

-Y (m

)

(b)

(1375 1185)(46 1009)

(92 0831)


minus5

00

05

10

15

D11

-X (m

)

100 200 300 400 5000f (Hz)

(c)

(1375 1218)(46 1163)

(92 0898)


minus5

00

05

10

15

D12

-Y (m

)

100 200 300 400 5000f (Hz)

(d)


x 10minus5

x 10minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

D12

-Y (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)


6 Conclusion




D11-1080 rpm

x 10minus5

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

)


(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)


(e)






D11-1080 rpmx 10

minus5


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

)


(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)


(e)



Data Availability




Acknowledgments



(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)



References
































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




minus4

minus1

minus05

0

05

1

D11

-X (m

)


(a)


minus4

minus1

minus05

0

05

1

D12

-Y (m

)


(b)


1X2X


08441166

1X2X3X

x 10minus5

200 400 600 800 10000f (Hz)

0

05

10

15

D11

-X (m

)

(c)

1X

2X


and 150Hz

D12minus2770 rpm117309021227

1X2X3X

x 10minus5

0

05

10

15

D12

-Y (m

)

200 400 600 800 10000f (Hz)

(d)


(11489) (21482)e first group




minus7


0

05

1

15

Eige

nval

ue o

f PCA

(m)

(a)

D12minus2770 rpm(11688) (21687)

e first group

e second group


(31673) (41664)

x 10minus7

0

05

1

15

2

Eige

nval

ue o

f PCA

(m)


(b)









minus5


minus4

minus2

0

2

4

D11

-X (m

)

(a)


minus5

minus4

minus2

0

2

4

D12

-Y (m

)


(b)

(1375 1185)(46 1009)


minus5

100 200 300 400 5000f (Hz)

0

05

1

15

D11

-X (m

)

(c)

(1375 1218)(46 1164)


minus5

100 200 300 400 5000f (Hz)

0

05

1

15

D12

-Y (m

)

(d)


x 10minus5

x 10minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

D12

-Y (m

)


D11-X (m)

(a)0

0204

0608

10

24


minus4

x 10minus5

x 10minus5

minus4

minus2

0

2

4

D12

-Y (m

)

(b)






D11minus2770 rpm

x 10minus5

minus4

minus2

0

2

4

D11

-X (m

)


(a)

D12minus2770 rpm

x 10minus5


minus4

minus2

0

2

4

D12

-Y (m

)

(b)

(1375 1185)(46 1009)

(92 0831)


minus5

00

05

10

15

D11

-X (m

)

100 200 300 400 5000f (Hz)

(c)

(1375 1218)(46 1163)

(92 0898)


minus5

00

05

10

15

D12

-Y (m

)

100 200 300 400 5000f (Hz)

(d)


x 10minus5

x 10minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

D12

-Y (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)


6 Conclusion




D11-1080 rpm

x 10minus5

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

)


(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)


(e)






D11-1080 rpmx 10

minus5


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

)


(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)


(e)



Data Availability




Acknowledgments



(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)



References
































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




minus5


minus4

minus2

0

2

4

D11

-X (m

)

(a)


minus5

minus4

minus2

0

2

4

D12

-Y (m

)


(b)

(1375 1185)(46 1009)


minus5

100 200 300 400 5000f (Hz)

0

05

1

15

D11

-X (m

)

(c)

(1375 1218)(46 1164)


minus5

100 200 300 400 5000f (Hz)

0

05

1

15

D12

-Y (m

)

(d)


x 10minus5

x 10minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

D12

-Y (m

)


D11-X (m)

(a)0

0204

0608

10

24


minus4

x 10minus5

x 10minus5

minus4

minus2

0

2

4

D12

-Y (m

)

(b)






D11minus2770 rpm

x 10minus5

minus4

minus2

0

2

4

D11

-X (m

)


(a)

D12minus2770 rpm

x 10minus5


minus4

minus2

0

2

4

D12

-Y (m

)

(b)

(1375 1185)(46 1009)

(92 0831)


minus5

00

05

10

15

D11

-X (m

)

100 200 300 400 5000f (Hz)

(c)

(1375 1218)(46 1163)

(92 0898)


minus5

00

05

10

15

D12

-Y (m

)

100 200 300 400 5000f (Hz)

(d)


x 10minus5

x 10minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

D12

-Y (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)


6 Conclusion




D11-1080 rpm

x 10minus5

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

)


(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)


(e)






D11-1080 rpmx 10

minus5


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

)


(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)


(e)



Data Availability




Acknowledgments



(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)



References
































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



D11minus2770 rpm

x 10minus5

minus4

minus2

0

2

4

D11

-X (m

)


(a)

D12minus2770 rpm

x 10minus5


minus4

minus2

0

2

4

D12

-Y (m

)

(b)

(1375 1185)(46 1009)

(92 0831)


minus5

00

05

10

15

D11

-X (m

)

100 200 300 400 5000f (Hz)

(c)

(1375 1218)(46 1163)

(92 0898)


minus5

00

05

10

15

D12

-Y (m

)

100 200 300 400 5000f (Hz)

(d)


x 10minus5

x 10minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

D12

-Y (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)


6 Conclusion




D11-1080 rpm

x 10minus5

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

)


(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)


(e)






D11-1080 rpmx 10

minus5


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

)


(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)


(e)



Data Availability




Acknowledgments



(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)



References
































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



D11-1080 rpm

x 10minus5

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

)


(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)


(e)






D11-1080 rpmx 10

minus5


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

)


(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)


(e)



Data Availability




Acknowledgments



(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)



References
































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



D11-1080 rpmx 10

minus5


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

)


(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)


(e)



Data Availability




Acknowledgments



(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)



References
































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



(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)



References
































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


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