OnlineIntelligentIdentificationofModalParametersforLarge ...downloads.hindawi.com/journals/sv/2020/2040216.pdfnoise in a vibration signal can be reduced in the data ac-quisition stage

Research ArticleOnline Intelligent Identification of Modal Parameters for LargeCable-Stayed Bridges

Peng Wen1 Inamullah Khan 2 He Jie1 Chen Qiaofeng1 and Yang Shiyu1

1Bridge Engineering Department Southwest Jiaotong University Chengdu China2School of Civil Engineering National University of Sciences and Technology Islamabad Pakistan

Correspondence should be addressed to Inamullah Khan inam_bunnyyahoocom

Received 17 August 2019 Accepted 19 December 2019 Published 18 February 2020

Academic Editor Sakdirat Kaewunruen

Copyright copy 2020 Peng Wen et al +is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Realizing online intelligent identification of bridge modal parameters requires not only the adaptive decomposition of structuralresponse signals but also the enforcement of the automatic identification of modal parameters +erefore in this paper the signaldecomposition algorithm-ensemble empirical mode decomposition algorithm (EEMD) is improved to fulfill the above task Firstthe adaptive matching algorithm is introduced to deal with the endpoint effect second the method of classification is used toavoid modal aliasing Finally an index for filtering the effective intrinsic mode function (IMF) components is constructed torealize automatic screening and signal reconstruction of the effective IMF components At the same time the first derivative of thesingular entropy increment is used to automatically determine the order of the system and then the spectral clustering algorithmis combined with the stochastic subspace algorithm to ultimately reach the goal of automatic identification of modal parameters

1 Introduction

A bridge structure health monitoring system is important toguarantee the operational safety of existing bridge structuresand the modal parameter identification [1] of bridgestructures is an important aspect of a bridge health moni-toring system +e accurate identification of these param-eters can reflect the inherent dynamic characteristics of thebridge structure and can aid in the long-term healthmonitoring of the bridge However the existing modalparameter identification algorithms cannot automaticallyidentify parameters and still require human participation todetermine the order of the system [2] and evaluate stabilitydiagrams [3] Additionally the dynamic signals of bridgestructures under environmental excitation generally containnoise and the various sources of the noise are complex +enoise in a vibration signal can be reduced in the data ac-quisition stage by taking measures such as averaging fil-tering and shielding [4] However it is impossible toeliminate the noise altogether In view of this limitation toobtain more accurate bridge modal parameters duringbridge system assessment and improve the damage diagnosis

and health monitoring of structures it is necessary topreprocess the response signals of bridge structures underenvironmental excitation conditions that is the signal mustbe decomposed and reconstructed Based on this approachin this paper the existing ensemble empirical mode de-composition (EEMD) algorithm [5] is improved and then astatistics-based spectral clustering algorithm [6] is combinedwith a stochastic subspace identification method [7] for theonline automatic identification of modal parameters

2 Improvement of the EEMD

+e EEMD algorithm [5] is an important improvement tothe original empirical mode decomposition (EMD) algo-rithm +e specific method involves repeatedly addingrandom white noise of equal amplitude to the original signalto homogenize the specific distribution of the extreme valuepoints in the signal +erefore the influence of the inter-mittent high-frequency component is avoided to some ex-tent In addition to counteract the white noise in theintrinsic mode function (IMF) it is necessary to obtain theensemble mean of the results and retain the IMF component

HindawiShock and VibrationVolume 2020 Article ID 2040216 17 pageshttpsdoiorg10115520202040216

which has physical meaning +e specific implementationsteps are shown in the literature [5]

Based on the core principle of the algorithm the methodstill has the following limitations the processing effect ofendpoints is poor the obtained intrinsic mode function stillincludes mode aliasing and selecting effective IMF com-ponents for signal reconstruction is difficult Consideringthe above three problems this paper proposes a corre-sponding improved algorithm to cope with these problemseffectively

21 Addition of Positive and Negative White Noise In theprocess of EEMD the white noise signal should be added tothe original signal which is then decomposed In thisapproach signal components with different time scales canbe automatically decomposed into the pass band which isdetermined by the white noise signal to overcome themode mixing phenomenon [8] However due to the ad-dition of the white noise signal to the original signal thewhite noise signal will disrupt the original signal which willreduce the signal-to-noise ratio (SNR) of the decomposi-tion results Because the negative effect of white noisecannot be completely eliminated the reconstruction errorof this algorithm is large and the completeness of thedecomposition is poor

Based on these factors in this paper positive andnegative white noise sequences are added to the originalsignal to solve the above problems

+e specific steps in this method are as follows

(1) Addition of randomwhite noise to the original signals(t) including K sets of positive white noise ni(t)

and K sets of negative white noise minus ni(t) +e cal-culation formula is as follows

x(t) s(t) +(minus 1)qa0n

i(t) (1)

where q 1 or 2 (equal to 1 if negative white noise isadded and equal to 2 if positive white noise is added)i 1 2 K K represents the total number oftimes that white noise is added a0 represents thestandard deviation of the amplitude of the addedwhite noise ni(t) represents the white noise that isadded for the ith time s(t) represents the originalsignal and x(t) represents the signal after the whitenoise is added

(2) +e mixed signal with positive and negative pairedwhite noise is an input into the EEMD algorithm andthen decomposed

In practical applications the amplitude of the addedwhite noise cannot be too large or too small if the amplitudeis too large the noise will cover up the effective signal andthe residual noise will be difficult to eliminate However ifthe amplitude is too small the distribution of extreme pointsin the original signal may not be sufficiently disrupted andeliminating the phenomenon of mode aliasing cannot beachieved According to a previous study [8] the amplitude of

the added white noise is usually 01-02 times the standarddeviation of the original signal

22 Adaptive Extreme Point-Matching ContinuationAlgorithm To solve the endpoint effect problem in theEEMD algorithm many scholars have proposed improvedmethods which can be roughly divided into the followingtwo categories

(1) +e extreme point continuation method [9] thebasic principle of this method involves extending theoriginal signal data to a certain number of extremepoints One of the main advantages of this algorithmis that it is simple and easy to implement Howeverthe limitation is that this approach only considers thepartial information of the signal at an endpoint anddoes not consider the signal as a whole

(2) Predictive continuation method [10] in this methodthe original signal is predicted and extended using aneural network [11] ARMA model [12] or supportvector machine method [13] +e effectiveness ofthese algorithms mainly depends on the parametervalues because the approach is not adaptive and thusrequires many computations and has a long oper-ation time

Combining the advantages and disadvantages of theabove twomethods a new continuationmethod is proposedan adaptive extreme point-matching continuation algo-rithm +e main feature of this algorithm is that it candetermine the best-matched extreme points from inside theoriginal signal to match with the extreme points at both endsof the signal +is approach extends the signal consideringthe original trend in a rapid manner and is highly adaptable

+e basic principles of the algorithm are as followsSet the signal equal to x(t) Additionally

mi ni(i 1 2 3 ) are the maximum and minimumpoints respectively and the corresponding times are tmi andtni Taking the extreme left point continuation of x(t) as anexample xtx1 is the starting point at the left and xtx1 minus m1 minus

n1 form an extreme point characteristic wave +en theextreme point waveform that best matches the characteristicwave is determined from the signal In this case xtxi minus mi minus

ni is the matching extreme point wave and the matchingextreme point is extended to the left endpoint of the originalsignal (the right end is extended to the right endpoint)

+en the next specific steps in this method are asfollows

Obtain all xtxi values except the left endpoint xtx1 +ecorresponding time value can then be obtained from thesimilarity relation

txi tm1tmi minus tn1tni

tm1 minus tn1 (2)

where tm1 is assumed to be less than tn1 If the obtained valueof txi is not a sampling point the specific value of xtxi can beobtained by linear interpolation

Calculate the error e between the obtained extremepoints and the extreme points of the feature

2 Shock and Vibration

e(i) mi minus m1

11138681113868111386811138681113868111386811138681113868

m11113868111386811138681113868

1113868111386811138681113868+

ni minus n11113868111386811138681113868

1113868111386811138681113868

n11113868111386811138681113868

1113868111386811138681113868+

xtxi minus xtx11113868111386811138681113868

1113868111386811138681113868

xtx11113868111386811138681113868

1113868111386811138681113868 (3)

In the formula the reason for dividing by the firstmaximum the first minimum and the signal endpoint valueis to standardize the error term

Determine the minimum matching error e(i)min +eextreme point is the matched extreme point at a given timeand the data associated with this matched extreme point areshifted to the left end of the original signal as the contin-uation extremum

According to the steps followed below the right end-point of the original signal is matched and extended and thelast extended signal is set to 1113957x(t)

+en decomposition of the extended signal 1113957x(t) isperformed and the IMF components are obtained accordingto the timing of the original signal In this approach thedecomposition results by considering the improved end-point can be obtained

+e aforementioned continuation algorithm not onlyconsiders the trend and regularity of the signal but also onlyrequires the extension of extreme points +is process is easyto implement and is more self-adaptable as compared toother methods

Currently the direct observation method is generallyused to assess the processing effect of the signal endpointsBased on this method an evaluation index θ is introduced toevaluate the treatment of endpoints +e specific calculationprocess is as follows

First the effective RMS of the original signal and eachIMF component are calculated and used to estimate theenergy of each signal sequence +e specific calculationformula of the RMS is as follows

RMS

1113936ni1 S2(i)

n

1113971

(4)

where S(i) is the signal sequence that is the original signalx(t) of each IMF component and n is the number ofsamples in the signal

According to formula (4) the sum of the effective valuesof each IMF component and the effective value of theoriginal signal is calculated and compared and then theevaluation index θ is obtained

θ 1113936

kf1 RMSf minus RMSx

11138681113868111386811138681113868

11138681113868111386811138681113868

RMSx

(5)

where RMSx is the effective value of the original signalRMSf is the valid value of the fth IMF component and k isthe total number of IMF components including the re-siduals in the decomposed signal

From the definition there is no endpoint effect if θ 0and the larger the value is the greater will be the endpointeffect

23 Mode Aliasing Process When the signals are decom-posed by the EEMD algorithm the phenomenon of modalaliasing may influence the obtained IMFs that is similar

characteristic time scales are distributed among differentIMF components As a result the waveforms of two adjacentIMF components are mixed together thus making it difficultto identify each signal To avoid this phenomenon a clus-tering analysis algorithm [14] is introduced However beforeanalyzing the clustering algorithm it is necessary to un-derstand the basic process of the EEMD algorithm +eflowchart is shown in Figure 1

According to the flowchart the final IMF is obtainedfrom the ensemble average of N numbers of IMFs afterEEMD decomposition+e calculation formula is as follows

IMFj 1N

1113944

N

i1IMFij (6)

where IMFj is the jth IMF obtained from the original signalwhich is decomposed by the EEMD algorithm

+e principle of the EEMD algorithm suggests that theEEMD algorithm only directly takes the average value ofIMFij(i 1 2 N) when calculating IMFj but does notconsider whether the N IMFij(i 1 2 N) values in thesame line belong to the same class At the same time it ispossible for mode aliasing to exist between n minus 1 IMFij(j

1 2 n minus 1) values in the same column Based on thisapproach a clustering analysis method [15] involvingmultivariate data analysis is introduced to solve the aboveproblems and the specific process is as follows

(1) In the modal decomposition of the signal eachdecomposition yields a series of components Toensure that there is no aliasing among these obtainedcomponents clustering analysis can be performed Ifthere is mode aliasing among the obtained com-ponents the results for the relevant signals areeliminated and then white noise is added again +isprocess is repeated until there is no mode aliasing inthe decomposition results

(2) A clustering analysis was performed on N IMFij

values in the same row to select the class of IMF withthe largest number of clusters +en the averagevalue is established as the final IMFj

Introducing the clustering analysis algorithm into theEEMD algorithm not only guarantees that there is no modealiasing among IMF components in each decomposition butalso guarantees that there is no mode aliasing among thefinal IMFs

24 Filtering of Effective IMF Components When the signalis decomposed by the EEMD algorithm multiple intrinsicmode functions (IMFs) can be obtained but the effectiveIMF components cannot be automatically filtered Inpractice it is often necessary to manually participate in thescreening of effective IMF components according to theHilbertndashHuang spectrum of each IMF component therebyreducing the work efficiency and leading to subjectivity inthe selected results due to individual differences

Based on this limitation an algorithm for the automaticscreening of effective IMF components is proposed +is

Shock and Vibration 3

algorithm not only considers the information entropy [16] ofeach IMF component but also merges the energy density andaverage period [17] It is necessary to introduce the infor-mation entropy the energy density and the average periodof the IMF components before the filtering algorithm ispresented

241 2e Information Entropy of IMF Components A seriesof components with different bandwidths can be obtained bythe mode decomposition of the response signal Eachcomponent contains different frequencies of the signal fromhigh to low and these frequencies and bandwidths willchange as the signal changes In addition studies of in-formation entropy have indicated that the more orderly thesignal components contained in the IMF are and the betterthe aggregation of the time-frequency distribution is thesmaller the resulting value of information entropy is Incontrast the more disorderly the signal is and the worse theaggregation of the time-frequency distribution is the greaterthe value of the resulting information entropy is

+e following steps detail how to calculate the infor-mation entropy of each IMF component

(a) It is assumed that f(t) is a set of components of theobtained IMF after the decomposition of signal x(t)+e maximum value of f(t) is fmax and the min-imum value is fmin

(b) Ai is set equal toNwithin the interval [fmin fmax] andthe interval [fmin A1] (A1 A2] [ANminus 1 fmax) is

the discrete range B isin B1 B2 BN1113864 1113865 of the char-acteristic quantityWhen the value of a sample with aqualifying attribute falls into the interval (Ai Ai+1]this sample has the corresponding discrete attributevalue Bi

(c) When the total number of sampling points for thediscrete signals is n the number of sample points off(t) that fall in the ith interval is mi +us theprobability P(Bi) min of f(t) falling within the ith

interval can be calculated according to the corre-sponding statistics +erefore according to the thirddefinition [16] of information entropy the infor-mation entropy of the IMF can be calculated asfollows

H minus 1113944N

i1p xi( 1113857logP xi( 1113857 (7)

242 Energy Density and Average Period According toreference [17] for a signal with a white noise sequencethe product of the energy density and the average periodof each component is a constant when EMD is used todecompose the signal +erefore an algorithm for fil-tering components is proposed +e specific process is asfollows

First the mode decomposition of the response signal isperformed and the energy density (E) and average period(T) corresponding to each IMF component are calculatedNext the product of these two components is calculated and

Original signal

EMD

White noise 1 White noise iWhite noise 2 White noise N

X1 (t)

hellip

(IMF1(nndash1)

(IMF11

hellip

IMFi1

hellip

IMFN1)N

hellip

IMFN(nndash1))N

+

+

+

+

IMF1

hellip

Reconstructed signal

(r1n

+

+

+

+

+ rNn)N+

X2 (t) Xi (t) XN (t)

(IMF12 IMF22

IMF21

IMF2(nndash1) IMFi(nndash1)

ri(nndash1)r2n

IMF2

IMFnndash1

rn

IMFi2 IMFN2)N

Figure 1 Flowchart of the EEMD algorithm


is defined as a mathematical expression of the energy co-efficient (ET) as follows

Ej 1N

1113944

N

i1fj(i)1113872 1113873

2Tj

2N

Oj

(8)

where N is the length of the original signal fj is the am-plitude of the jth IMF component and Oj is the number ofexisting extreme points for the jth IMF component

+e energy coefficient (ET) of each IMF component isused to calculate the final effective coefficient RP of each IMFcomponent +e calculation formula is as follows

RPj ETj minus (1j minus 1)1113936

jminus 1i1 ETi

(1j minus 1)1113936jminus 1i1 ETi

1113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868 (9)

When the effective coefficient RPj of the jth IMFcomponent is greater than or equal to 1 the energy coef-ficient corresponding to the jth IMF component exponen-tially increases compared with the average value of theprevious j minus 1th IMF component In this case the product ofthe energy density of the previous j minus 1th IMF componentand the average period can be considered a constant that isthe previous j minus 1th IMF component can be regarded asnoise and deleted +erefore what is left is the effectivecomponent

243 Comprehensive Evaluation Algorithm If the effectiveIMF is only selected according to the information entropy ofeach IMF component there may be errors in the selectionresults +erefore a comprehensive evaluation algorithm[18] is proposed by combining the information entropy (H)and the effectiveness coefficient (RP) +is algorithm is atype of comprehensive bid evaluation method based onfuzzy mathematics and mainly transforms a qualitativeevaluation into a quantitative evaluation according to themembership theory of fuzzy mathematics A new index(effectiveness degree) can be constructed using the infor-mation entropy and the effectiveness coefficient to quantifythe effectiveness degree Y between each IMF component andthe original signal

+e specific formula is as follows

Yi wH 1 minusHi minus Hmin

Hmax minus Hmin1113888 1113889 + wRP

RPi minus RPmin

RPmax minus RPmin1113888 1113889

(10)

where wH is the weight of the information entropy H andwRP is the weight of the effectiveness coefficient RP

In the literature [19] a processing method for deter-mining the weight value is given Considering the primaryand secondary relations between information entropy andthe effectiveness coefficient it is assumed that both weightcoefficients in formula (10) are 05

+e above analysis shows that the closer Yi is to 1 thehigher the degree of association is between the ith compo-nent and the original signal In combination with reference[20] this paper assumes that when Yge 08 the IMF com-ponent is considered effective Finally summing all effective

IMF components the final reconstruction signal can beobtained

+e flowchart of the improved EEMD algorithm is givenin Figure 2

3 Improvement of the Stochastic SubspaceIdentification Algorithm

Currently the stochastic subspace identification (SSI) [21]algorithm is a commonly used structural modal parameteridentification algorithm +is method is mainly used inlinear systems and is a type of time domain identificationmethod as compared with other modal parameter identi-fication methods +is algorithm does not require a pa-rameterized system model but adopts basic matrix analysismethods such as QR decomposition and SVD decomposi-tion However this algorithm still has the following twolimitations

(1) +e real order of the system is difficult to determineand there is no unified method or theory to deter-mine the real order of the system

(2) Currently the selection of extreme points in thestability diagram often requires manual participationin the selection However in practice because dif-ferent points are manually selected the final modalparameter identification results can differ Based onthe above two problems a new system order de-termination algorithm and a real modal filteringalgorithm are proposed

31 Automatic Determination of the SystemOrder +e mostcommonly used order determination method is the stabilitydiagram but it still has two limitations

(1) Modal distortion is mainly caused by external factorsthat do not belong to the system itself or the mode ofthe system itself cannot be identified due to thisdistortion

(2) +e large number of calculations is the main reasonwhy stability diagram theory is based on structuralparameters such as the frequency mode shape anddamping as the basis for determining the stabilitypoint +erefore it is necessary to calculate a largenumber of modal parameters for a structure of givenorder Based on this requirement singular entropytheory is introduced to solve the problem of deter-mining order difficulty +e algorithm does not needto use the modal parameters of the structure todetermine the stability point which can greatly re-duce the calculation workload

A detailed description of how to use the singular entropyto determine the system order is given as follows

By analyzing the decomposition principle of the singularvalue of the signal [22] a real matrix Q with dimensions ofm times n can be decomposed into a matrix R with dimensionsof m times l a diagonal matrix Λ with dimensions of l times l and a


matrix S with dimensions of n times l +e following relation-ships exist among these matrices

Q R middot Λ middot ST

Λ diag λ1 λ2 λl( 1113857(11)

An analysis of the principal diagonal elements λi

(singular values of the matrix Q) in matrix Λ indicates thatthe number of the principal diagonal elements is closelyrelated to the complexity of the frequency componentscontained in the signal +e more elements there are themore complex the original signal components are Whenthe original signal is disturbed by noise the principaldiagonal elements are likely to be nonzero which meansthat the frequency component of the original signal isrelatively simple According to this characteristic thevibration signal information can be objectively reflected

by the matrix Λ +e definition of singular entropy is asfollows

Ek 1113944k

i1ΔEi kle l

ΔEi minusλi

1113936lk1 λk

⎛⎝ ⎞⎠lnλi

1113936lk1 λk

⎛⎝ ⎞⎠

(12)

where k is the order of the singular entropy and ΔEi is theincrement of the singular entropy of the ith order

As the order continuously increases the rate of increasein the singular entropy gradually decreases and finallystabilizes +is characteristic does not change with noiselevel of the signal Based on this relation the change insingular entropy can be regarded as a criterion of the systemorder that is when the change in the singular entropy

Informationentropy

Energydensity

Effective IMFcomponents

Fuzzy comprehensiveevaluation method

Averageperiod

Reconstructedsignals

Positive and negative white noise

Original signal

Identification of modal aliasing phenomenon by clustering algorithm

(IMF11

(IMF1(nndash1)

+

+

+ IMF2(nndash1) IMFi(nndash1)

IMFN1)N

IMFN(nndash1))N

+

+

+

IMF components

(r1n + r2n ri(nndash1) rNn)N+

IMF1

IMFnndash1

rn

Empirical mode decomposition

Clusteringalgorithm

+Residual quantity

Adaptive extreme point-matchingcontinuation algorithm

IMF21 IMFi1

hellip hellip hellip helliphellip Clusteringalgorithm

Clusteringalgorithm

Clusteringalgorithm

Figure 2 Flowchart of improved EEMD


stabilizes the corresponding order is the real order of thesystem +erefore in practice a derivative of the change insingular entropy is often taken When the derivative valueapproaches 0 the change in singular entropy tends to sta-bilize and the corresponding order can be regarded as thereal order of the system

32 Automatic Identification ofModal Parameters +ere aretwo typical difficulties in modal parameter identificationusing the DATS-SSI algorithm system order determinationand physical mode selection +e traditional SSI method isbased on the stability diagram and manual steps which notonly reduces the working efficiency but also leads to sub-jective differences in the identification results due to indi-vidual differences

Based on this this paper makes in-depth research on therelevant principles of a stabilized diagram It is found thatthe modal parameters of a bridge structure will not changegreatly in a short time +at is the real modals will exist inboth the stabilized diagram at this moment and the stabilizeddiagram at the next moment Only the false modals willchange which is mainly because the response signals col-lected by the sensors at different times will be affected bydifferent degrees of noise but it will still contain thestructural information of the bridge structure

According to this property the real modes of structurescan be selected from the multiple stability diagrams In thiscase the structures that appear in the multiple diagrams areused However in practice it is time consuming and laborintensive to identify the real modes manually and individ-ually and the selection of real modes is influenced bysubjectivity due to the differences among individuals+erefore the hierarchical clustering method in statistics isintroduced to perform the automatic selection of real modesand the automatic identification of modal parameters

Detailed descriptions of the implementation steps of themodal parameter automatic identification algorithm basedon improved EEMD and the hierarchical clustering analysisare given below

321 Adaptive Decomposition and the Reconstruction ofStructural Signals +e improved EEMD algorithm proposedin this paper is used for the adaptive decomposition and re-construction of the structural response signals that are col-lected by the sensor Assuming that the sensor has collected theresponse data for a bridge structure for a total of N days N

groups of reconstructed signals can be obtained in this period

322 Determination of the System Order +e reconstructedsignals are decomposed by singular values and the real orderof the bridge structure which is represented by letter n isdetermined using the algorithm in Section 21

323 Order Range Determination Based on the StabilityDiagram +e order range of the stability diagram is [1 2n]that is the calculation starts from the first order of thesystem and the maximum calculation order is 2n

324 Modal Parameter Identification +emodal parametersof N groups of reconstructed signals are identified by DATA-SSI and N groups of identification results are obtained Inaddition the results of each group include the correspondingfrequency value damping ratio and mode coefficient

325 Establishment of the Distance Matrix It is assumedthat N groups of recognition results belong to a distinctcategory that is N subsets can be established and arerepresented by X1 X2 XN In addition the results foreach group include the corresponding frequency valuedamping ratio and mode coefficient Starting from the firstsubset the distance (similarity) between the ith subset andthe i + 1th subset is calculated in turn and a distance matrixD(i) with dimensions of n times n is obtained where n is the realorder of the system

To establish the distance matrix it is necessary to definethe statistical magnitude that can reflect the distance be-tween modes Because the stability diagram is composed ofnatural frequencies damping ratios and mode shapes thesethree parameters are used as clustering factors to define thedistance dij between mode i and mode j +e calculationformula is as follows

dij wf

df

fi minus fj

11138681113868111386811138681113868

11138681113868111386811138681113868

max fi fj1113872 1113873+

wξ

dξ

ξi minus ξj

11138681113868111386811138681113868

11138681113868111386811138681113868

max ξi ξj1113872 1113873+

wψ

dψ(1 minus MAC(i j))

(13)where df dξ and dψ represent the frequency damping ra-tio and modal shape tolerances respectively According tothe literature [19] the tolerance values are determined asdf 001 dξ 005 and dψ 002 where wf wξ andwψrepresent the frequency damping ratio and mode shapeweights in the calculation of the modal distance +e sum ofthe three components is 1 and in this paper the values ofthese three variables are as follows wf 04

wξ 03 andwψ 03 +e corresponding frequencyweight value is larger than that of the other two variablesbecause the focus of this paper is identifying the exactfrequency value therefore the corresponding frequencyvalue should be larger than the other two values iedamping ratio and mode shape

326 Clustering of Similar Items

(a) +e distance matrix D(k) composed of the kth groupand the k + 1th group can be obtained via Section325

(b) +e clustering of the same modes can be performedby determining the numerical values in the distancematrix +at is when d

(k)ij le 1 the ith mode of the kth

group and the jth modal of the k + 1th group haveconsistent frequency damping ratio and modalshape values In other words these two modes areconsidered to be of the same type +erefore Xk(i)

and Xk+1(j) should be clustered together(c) All the modes of the same type in groups k and k + 1

should be clustered into one category and the


category should be used to obtain a new subset Xk+1with the remainder of the modes

(d) According to the same principle the subsets Xk+1and Xk+2 obtained in Section 323 are clustered anda new subset Xk+2 is constructed +en this subset isclustered with subset Xk+3 and the process con-tinues until all N subsets are clustered and the finalsubset XN is obtained

(e) +e number of modal clustering elements of eachorder in the new subset XN is counted andaccording to reference [19] if the number of clus-tering elements is greater than 06n (where n is thereal order of the system) the mode is considered tobe real and a corresponding stability diagram isconstructed

+e basic flowchart of the automatic identification ofmodal parameters for bridge structures based on the im-proved EEMD and hierarchical clustering analysis is shownin Figure 3

4 Verification of the Simulated Signal

Both the improved EEMD algorithm and the EEMD algo-rithm are used to decompose the analog signals and theresults obtained by the two methods are compared +eanalog signal is composed of sinusoidal signals of 1Hz 3Hzand 7Hz superimposed with stochastic noise (the noise levelis approximately 10)+e analog signal can be expressed asfollows

s(t) 7 sin(14πt) + 3 sin(6πt) + sin(2πt) + 10rand

(14)

+e time of the analog signal is 10 seconds and every001 second represents one test point +ere are a total of1000 test points +e time domain diagram of the super-imposed signal and noise is shown in Figure 4

In the following section the improved EEMD algorithmand the EEMD algorithm are used for the modal decom-position of the above analog signals +e specific analysis isas follows

41Comparisonof theEndpointEffect +e results of the IMFcomponents corresponding to the two decomposition al-gorithms are shown in Figures 5 and 6 and the followingconclusions can be obtained by comparing IMFs 3ndash5 inFigure 5 and IMFs 2ndash4 in Figure 6

(1) Comparing the endpoint waveform shape ofIMF3 (which is calculated by IEEMD algorithm)to IMF2 (which is calculated by EEMD algo-rithm) IFM4 (IEEMD) to IFM3 (EEMD) andIFM5 (IEEMD) to IFM4 (EEMD) it can be de-duced that the improved EEMD algorithm canimprove the endpoint effects of the decomposi-tion results

(2) Using the endpoint effect evaluation index in Section22 for quantitative determination it can be

concluded that the evaluation index correspondingto the EEMD algorithm is 07358 and that of theadaptive extreme matching continuation method is00084

Based on the above results the adaptive extreme point-matching algorithm proposed in this paper can improve theendpoint determination in the EEMD algorithm

42 Mode Aliasing To verify whether mode aliasing existsamong the obtained IMF components the clustering resultscorresponding to the IMF components of the two decom-position algorithms are presented and compared as shownin Figures 7 and 8 +e following conclusions can be drawnfrom the comparison of the results

(1) Figure 7 shows that each IMF component is a sep-arate category that is there is no similar information


Order determination of singularentropy increment derivative method

Determination of orderscale of stabilized diagram

DATA-SSI

Frequency Damping ratio Mode

Distance matrix

Hierarchical clusteringanalysis

Response signal ofstructure

Improved EEMD

Automatic identification ofmodal parameters

Figure 3 Flowchart of the automatic identification of modalparameters


among the IMF components which indicates thatthere is no mode aliasing among these IMFcomponents

(2) Figure 8 shows that IMF1 and IMF2 are classified inthe same category and IMF5 and IMF6 are classifiedin the same category which indicates that there issome similar information between these compo-nents ie there is mode aliasing

+e above findings indicate that in the process of signaldecomposition the clustering analysis algorithm can beintroduced to avoid mode aliasing

43 Filtering of Effective IMF Components A new compre-hensive evaluation algorithm is proposed to solve theproblem with the EEMD algorithm Specifically the EEMDalgorithm cannot filter the effective IMF components +einformation entropy energy density and average period ofeach IMF component are used to construct a screeningindex which can be used to screen the effective IMFcomponents +e specific analyses are shown in Tables 1 and2 By comparing the data in the two tables the followingconclusions can be drawn

(1) According to Table 1 the corresponding coefficientsof effectiveness for IMF3 IMF4 and IMF5 are greater

Time (s)

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

ndash505

IMF 1

ndash100

10

IMF 2

ndash505

IMF 3

ndash202

IMF 4

ndash050

05

IMF 5

ndash050

05

IMF 6

ndash020

02

IMF 7

ndash0010

001

IMF 8

ndash050

05

IMF 9

Figure 6 Results of decomposition (EEMD)

1 2 6 7 4 5 3 84

5

6

7

8

9

10

11

12

13

IMFs

Dist

ance

Figure 7 Results of clustering (improved EEMD)

0 2 4 6 8

0 2 4 6 8

0 2 4 6 8

0 2 4 6 8

0 2 4 6 8

10

10

10

10

10

ndash200

20

All

signa

ls

Analog signal

ndash100

10

7Hz

ndash505

3Hz

ndash202

1Hz

ndash101

Noi

se

Time (s)

Figure 4 Signal and noise

Time (s)

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

ndash101

IMF 1

ndash050

05

IMF 2

ndash100

10

IMF 3

ndash505

IMF 4

ndash202

IMF 5

ndash020

02

IMF 6

ndash050

05

IMF 7

ndash020

02

IMF 8

Figure 5 Results of decomposition (improved EEMD)


than 08 thus all these components are effective IMFcomponents that can be used to reconstruct thesignals

(2) According to Table 2 the corresponding coefficientsof effectiveness for IMF2 IMF3 and IMF4 are greaterthan 08 therefore all these components are effectiveIMF components that can be used to reconstruct thesignals

(3) An analysis of the filtered IMF components based onthe decomposition diagrams shown in Figures 5 and6 indicates that the comprehensive evaluation al-gorithm proposed in this paper can screen out thecorresponding signals of 7Hz 3Hz and 1Hz fromthe superimposed signals +us this algorithm canautomatically filter the effective IMF components

However to further verify that the improved EEMDalgorithm can better handle endpoint effects the instanta-neous frequencies corresponding to the effective IMFcomponents in the results of these two decomposition al-gorithms are given as shown in Figures 9 and 10 Acomparison of the two figures suggests that both decom-position algorithms can highlight the frequency components

of analog signals Based on the change in the instantaneousfrequency at the endpoint the improved EEMD algorithmproposed in this paper can better handle endpoint effectscompared to the traditional EEMD method

Table 1 Table of effectiveness degree coefficients (improved EEMD)

Indicators IMF1 IMF2 IMF3 IMF4 IMF5 IMF6 IMF7 IMF8Information entropy 266 232 031 032 034 099 097 057Effectiveness coefficient 012 045 100 100 086 001 022 002Effectiveness degree coefficient 000 031 100 100 093 036 047 025

Table 2 Table of effectiveness degree coefficients (EEMD)

Indicators IMF1 IMF2 IMF3 IMF4 IMF5 IMF6 IMF7 IMF8 IMF9Information entropy 273 033 043 070 179 169 135 149 102Effectiveness coefficient 023 100 078 086 021 012 008 003 001Effectiveness degree coefficient 011 100 087 085 030 027 032 027 036

5 6 1 2 7 3 4 9 86

7

8

9

10

11

12

13

14

IMFs

Dist

ance

Figure 8 Results of clustering (EEMD)

Freq

uenc

y (H

z)

Number of points0 100 200 300 400 500 600 700 800 900 1000

0

5

10

15

20

25

30

35

40

45

50

ndash20

ndash18

ndash16

ndash14

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

0

Figure 9 HilbertndashHuang plot (improved EEMD)


5 Automatic Identification of ModalParameters for Actual Bridge Structures

In this paper the modal parameters for Sutong Bridge areautomatically identified +e specific steps are as follows

(1) First the structural response signals collected bysensors are analyzed with an exploratory dataanalysis method and the structural signals satisfyinga normal distribution are screened and removed

(2) Second the improved EEMD algorithm is used todecompose and reconstruct the acceleration re-sponse signal

(3) +ird the reconstructed signal is decomposed basedon a singular value and the system order is deter-mined according to the algorithm proposed inSection 21

(4) Fourth the DATA-SSI algorithm is used to identifythe modal parameters of the reconstructed responsesignal and various group identification results areobtained

(5) Fifth the automatic identification of modal pa-rameters is performed using the algorithm intro-duced in Section 22

(6) Finally the automatic identification results arecompared with the real results to verify the feasibilityof the proposed algorithm

51 Project Profile +e Sutong Bridge shown in Figure 11 isa large cable-stayed bridge +e main span is a 108 m cable-stayed bridge with twin towers and a cable plane +e bridgeis located on the Yangtze River+e total length of the bridgestructure is 2088m and the specific span arrangement is(2 times 100 + 300 + 1088 + 300 + 2 times 100) m

Because the bridge is a large cable-stayed bridge acomplete health monitoring system was designed duringconstruction to constantly monitor the operation status ofthe bridge and evaluate its performance +ere are manytypes of sensors on the bridge including temperature

sensors cable force sensors and acceleration sensors with asampling frequency of 20Hz In this paper the verticalresponse signals measured by 14 acceleration sensors on themain bridge are selected for modal parameter identification+e layout of the acceleration sensors is shown in Figure 12

52 Filtering of the Structural Response Signal +e healthmonitoring system of the bridge structure can continuouslycollect data collection and large datasets are available If theresponse signals for an entire year need to be analyzed thenthe workload will be very large In view of this limitation thispaper uses a quarterly approach and only one representativemonth in each quarter is selected for investigation andparameter change analysis +e selected months are March(spring) July (summer) October (autumn) and December(winter) +e histogram method of exploratory data analysisis used to analyze the acceleration response signals in theabove four months Figure 13 shows the correspondinghistograms of the response signals collected by 14 sensors inJuly According to the figure the structural response signalscollected by the second sensor do not satisfy a normaldistribution and only the response signals collected by theother 13 sensors can be used for modal decomposition andparameter identification

53 Order-Based Implementation of the Incremental SingularEntropyMethod +e improved EEMD algorithm is used todecompose and reconstruct the acceleration response signalscollected by the sensors +en the singular values of thereconstructed signals are decomposed and the corre-sponding singular entropy is calculated to obtain the sin-gular entropy variations as the order increases as shown inFigure 14 To determine whether the singular entropy sta-bilizes the first-order derivative of the change in the singularentropy is obtained as shown in Figure 15

Figures 14 and 15 show that the change in the singularentropy stabilizes and the first-order derivative approacheszero when the system order is 150 that is the real order ofthe bridge structure is considered to be 150

Number of points0 100 200 300 400 500 600 700 800 900 1000

Freq

uenc

y (H

z)

0

5

10

15

20

25

30

35

40

45

50

ndash20

ndash18

ndash16

ndash14

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

0

Figure 10 HilbertndashHuang plot (EEMD)


54 Automatic Identification of Modal Parameters To verifythat the improved EEMD algorithm proposed in this papercan decompose and reconstruct the signals of a real bridgebetter than the EEMD algorithm the corresponding ac-celeration response signals on July 1 2018 were selected foranalysis and the above two decomposition algorithms wereused to decompose and reconstruct the response signalsseparately +en the obtained reconstructed signals wereused as inputs to the DATA-SSI algorithm leading to twostability diagrams as shown in Figures 16 and 17 In thisfigure the red color represents the stable frequency bluecolor represents the stable damping ratio and green colorrepresents the stable mode shape

A comparison of the two figures suggests that the signalsthat are processed by the improved EEMD algorithm resultin a stability diagram with few false modes +e stability axisis much clearer especially between 3Hz and 5Hz+ereforecompared with the EEMD algorithm the improved EEMDalgorithm can extract more accurate structural information

However it is difficult to select the real modes of thebridge structure according to Figures 16 and 17 becausethere are many false modes in the figures and there is no

criterion for distinguishing the true modes from the falsemodes +e corresponding final stability diagram for eachmonth can be obtained as follows

(1) In units of days the improved EEMD algorithm wasused to decompose and reconstruct the accelerationresponse signals each day and the reconstructedsignals were retained +ere were 31 groups in total

(2) +e DATA-SSI algorithm was used to identify themodal parameters of each group of reconstructedsignals and the results were retained including thefrequency values damping ratios and modal shapes

(3) According to the theory introduced in Section 22 31groups of recognition results were analyzed based onhierarchical clustering analysis which can auto-matically select the real modes and be used toconstruct the final stability diagram

According to the above steps the automatic identifica-tion of modal parameters can be performed +e corre-sponding stability diagram of the bridge structure in July isgiven as shown in Figure 18 +e proposed automaticidentification algorithm for modal parameters which is

Figure 11 Photography of Sutong Bridge

100 100 300

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Accelerationsensor

AccelerationsensorUpstrean Downstream

10882 10882 300 100 100

Figure 12 Acceleration sensor arrangement


based on the improved EEMD and hierarchical clusteringanalysis can eliminate the false modes and preserve the realmodes To further verify that the identified parameter valuesare similar to the real parameter values of the bridgestructure the top five frequency values obtained from theidentification are compared with the calculated values basedon the theoretical vertical frequency which is given inreference [23]

Table 3 shows the comparison between the results obtainedby improved EEMD and the real values in reference [23] andTable 4 shows the comparison between the results obtained byEEMD and the real values in reference [23] Analyzing the datain two tables we know that the results from improved EEMD isbetter than results from EEMD and the improved EEMD canbe used for the automatic identification of modal parametersfor the actual bridge structure and the frequency values based

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 3

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 7

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 4

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 8

ndash002 0 0020

50

100

150Fr

eque

ncy

Amplitude range

Sensor 1

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 5

19 2 210

500

1000Sensor 2

Freq

uenc

y

Amplitude rangetimes10ndash3

ndash002 0 002 0040

50

100

150Sensor 6

Freq

uenc

y

Amplitude range

ndash002 0 0020

50

100

150

Freq

uenc

yAmplitude range

Sensor 11

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 12

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 9

ndash002 0 0020

50

100

150Sensor 10

Freq

uenc

y

Amplitude range

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 13

ndash002 0 0020

50

100

150Sensor 14

Freq

uenc

y

Amplitude range

Figure 13 Data histograms

0 50 100 150 200 250 3000

005

01

015

02

025

03

035

Incr

emen

t of s

ingu

lar e

ntro

py

System orders

Figure 14 +e trend of the change in singular entropy

0 50 100 150 200 250 300

0

005

01

015

Incr

emen

t of s

ingu

lar e

ntro

py-fi

rst d

eriv

ativ

e

System orders

Figure 15 +e slope trend of the change in singular entropy


0 1 2 3 4 5 6 7 8 9 100

50

100

150

200

250

300

Syste

m o

rder

sFrequency (Hz)

Figure 16 Stability diagram of the improved EEMD method

0 1 2 3 4 5 6 7 8 9 10Frequency (Hz)

0

50

100

150

200

250

300

Syste

m o

rder

s

Figure 17 Stability diagram of EEMD

0

50

100

150

05 1 15 2 25 3 35 4 45 5

018508023776

046347055536

117211859

2485331657

3635643286

50873

Syet

em o

rder

s

Frequency (Hz)

070871

Figure 18 Stability diagram of hierarchical clustering (July)

Table 3 Comparison between improved EEMD results and real values (Hz)

Model order True value [23] March July October December Average Tolerance Percentage difference ()First 01856 01845 01851 01845 01849 01847 00009 046Second 02191 02217 02378 02200 02378 02293 00102 322+ird 04531 04624 04635 04634 04644 04634 00103 228Fourth 05746 05542 05554 05546 05389 05508 00238 415Fifth 06928 06911 07087 07062 06858 06979 00051 074


on identification are very similar to the values calculated basedon theory with an error of less than 5

MIDAS software was used for modelling of bridge andthe top three modes were obtained as shown in Figure 19and Figure 20 is a modal shape diagram of the cable-stayedbridge for the top three orders identified by the proposedalgorithm Comparing the two results we can know that thisdiagram is very similar to the actual modal diagram with asimilarity of approximately 95 which further verifies thefeasibility of the proposed algorithm

55 Quarterly Frequency Changes in the Longitudinal Di-rection of the Bridge To study the specific frequency valuetrends of the top five orders in the longitudinal direction ofthe bridge structure in each quarter the correspondingfrequency values in March July October and Decemberwere identified In total 124 days in these four months werestudied and the frequency value trends of each order overtime are shown in Figure 21+e frequency values of the firstand second orders are relatively stable in 2018 for the bridgestructure however the values of the other three orders

1 2 3 4 5 6 7 8 9 10 1101020ndash1

0

1

(a)

2 4 6 8 100

1020ndash1

01

(b)

2 4 6 8 100

1020ndash1

01

(c)

Figure 20 Modal shape diagrams of the top three orders (algorithm of this paper) (a) +e first-order mode shape (b) +e second-ordermode shape (c) +e third-order mode shape

Table 4 Comparison between EEMD results and real values (Hz)

Model order True value [23] March July October December Average Tolerance Percentage difference ()First 01856 01914 01791 01736 01815 01711 00063 337Second 02191 02017 02164 02684 02711 02064 -00137 minus 625+ird 04531 04208 04311 04217 04319 03754 00369 815Fourth 05746 05099 05054 05158 04904 05398 00624 1085Fifth 06928 07602 07370 07415 07064 07677 minus 00498 minus 718

(a)

(b)

(c)

Figure 19 Modal shape diagrams of the top three orders (MIDAS model) (a) +e first-order mode shape (b) +e second-order modeshape (c) +e third-order mode shape


fluctuate with time but within a small range Because themodal parameters of the bridge structure can reflect theinherent dynamic characteristics of the bridge structure andthese characteristics did not obviously change in 2018 thebridge structure was in a healthy and stable state

6 Conclusion

+e following conclusions were obtained by applying theproposed algorithm to the analog signals and measured datafor an actual bridge

(1) +e proposed adaptive extreme point-matchingcontinuation algorithm can improve the endpointeffect problems in the EEMD algorithm

(2) When clustering analysis is applied in the process ofmodal decomposition the mode aliasing among theobtained IMFs can be avoided

(3) +e effectiveness degree coefficient a new indexconstructed based on the information entropy en-ergy density and average period can be used toautomatically select the effective IMF components

(4) By taking the derivative of the singular entropychange the real order of a system can be automat-ically determined when identifying the modal pa-rameters of an actual bridge structure

(5) By introducing the hierarchical clustering analysisalgorithm the real modes in the stability diagram canbe automatically selected and the modal parameterscan be automatically identified

(6) +e processing results based on test data for actualbridges indicate that the proposed automatic iden-tification algorithm of modal parameters for bridgestructures based on improved EEMD and hierar-chical clustering analysis can effectively perform theadaptive decomposition and reconstruction ofstructural response signals and the automatic de-termination of the system order and modalparameters

Data Availability

+e data used to support the findings of this paper are freelyavailable It can be made available on request if and whenrequired

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is work was funded by the National Basic ResearchProgram of China (no2013CB036302)

References

[1] F Magalhatildees A Cunha and E Caetano ldquoOnline automaticidentification of the modal parameters of a long span archbridgerdquo Mechanical Systems and Signal Processing vol 23no 2 pp 316ndash329 2009

[2] R J Allemang D L Brown and A W Phillips ldquoSurvey ofmodal techniques applicable to autonomoussemi-autono-mous parameter identificationrdquo in Proceedings of the Inter-national Conference on Noise and Vibration Engineering(ISMArsquo10) p 42 Katholieke Universiteit Leuven LeuvenBelgium September 2010

[3] R J Allemang AW Phillips and D L Brown ldquoAutonomousmodal parameter estimation statisical considerationsrdquo inProceedings of the 29th IMAC Conference on Structural Dy-namics (IMACrsquo11) pp 385ndash401 Leuven Belgium February2011

[4] E Ntotsios C Papadimitriou P Panetsos G KaraiskosK Perros and P C Perdikaris ldquoBridge health monitoringsystem based on vibration measurementsrdquo Bulletin ofEarthquake Engineering vol 7 no 2 pp 469ndash483 2008

[5] N E Huang M-L C Wu S R Long et al ldquoA confidencelimit for the empirical mode decomposition and Hilbertspectral analysisrdquo Proceedings of the Royal Society of LondonSeries A Mathematical Physical and Engineering Sciencesvol 459 no 2037 pp 2317ndash2345 2003

[6] M E Torres M A Colominas G Schlotthauer andP Flandrin ldquoA complete ensemble empirical mode decom-position with adaptive noiserdquo in Proceedings of the 36th IEEEInternational Conference on Acoustics Speech and SignalProcessing pp 4144ndash4147 Prague Czech Republic May 2011

[7] B Moaveni and E Asgarieh ldquoDeterministic-stochastic sub-space identification method for identification of nonlinearstructures as time-varying linear systemsrdquo Mechanical Sys-tems and Signal Processing vol 31 no 8 pp 40ndash55 2012

[8] L Mingliang W Keqi S Laijun and Z Jianju ldquoApplyingempirical mode decomposition (EMD) and entropy to di-agnose circuit breaker faultsrdquo Optik vol 126 no 20pp 2338ndash2342 2015

[9] L Han C Li and H Liu ldquoFeature extraction method ofrolling bearing fault signal based on EEMD and cloud modelcharacteristic entropyrdquo Entropy vol 17 no 12pp 6683ndash6697 2015

[10] T Wang X Wu T Liu and Z M Xiao ldquoGearbox faultdetection and diagnosis based on EEMD De-noising andpower spectrumrdquo in Proceedings of the IEEE InternationalConference on Information and Automation (ICIArsquo15)pp 1528ndash1531 IEEE Lijiang China August 2015

March July October December01

02

03

04

05

06

07

08

Freq

uenc

y (H

z)

Third order

First orderSecond order

Fourth orderFifth order

Figure 21 Frequency value trends for the top five orders


[11] J Doucette Advances on design and analysis of mesh-re-storable networks PhD thesis University of AlbertaEdmonton Canada 2004

[12] M I S Bezerra Y Iano and M H Tarumoto ldquoEvaluatingsome Yule-Walker methods with the maximum-likelihoodestimator for the spectral ARMA modelrdquo Tendencias emMatematica Aplicada e Computacional vol 9 no 2pp 175ndash184 2008

[13] H P Graf E Cosatto L Bottou I Dourdanovic andV Vapnik ldquoParallel support vector machines the cascadesvmrdquo in Advances in Neural Information Processing SystemsL K Saul Y Weiss and L Bottou Eds vol 17 pp 521ndash528MIT Press Cambridge MA USA 2005

[14] M A Sanchez O Castillo and J R Castro ldquoGeneralizedtype-2 fuzzy systems for controlling a mobile robot and aperformance comparison with interval type-2 and type-1fuzzy systemsrdquo Expert Systems with Applications vol 42no 14 pp 5904ndash5914 2015

[15] S V Guttula A Allam and R S Gumpeny ldquoAnalyzingmicroarray data of Alzheimerrsquos using cluster analysis toidentify the biomarker genesrdquo International Journal of Alz-heimerrsquos Disease vol 2012 Article ID 649456 5 pages 2012

[16] Y Y Chen and Y M Chen ldquoAttribute reduction algorithmbased on information entropy and ant colony optimizationrdquoJournal of Chinese Computer System vol 36 no 3 pp 586ndash590 2015

[17] X L Wang L Y Lu and W P Tai ldquoResearch of a newalgorithm of words similarity based on information entropyrdquoComputer Technology and Development vol 25 no 9pp 119ndash122 2015

[18] J-F Chen H-N Hsieh and Q H Do ldquoEvaluating teachingperformance based on fuzzy AHP and comprehensive eval-uation approachrdquo Applied Soft Computing vol 28 pp 100ndash108 2015

[19] Z Song H Zhu G Jia and C He ldquoComprehensive evalu-ation on self-ignition risks of coal stockpiles using fuzzy AHPapproachesrdquo Journal of Loss Prevention in the Process In-dustries vol 32 no 1 pp 78ndash94 2014

[20] Y-N HanW Zhou and X-Q Zhang ldquoFuzzy comprehensiveevaluation of the adaptability of an expressway systemrdquoJournal of Highway and Transportation Research and Devel-opment (English Edition) vol 8 no 4 pp 97ndash103 2014

[21] G Zhang B Tang and G Tang ldquoAn improved stochasticsubspace identification for operational modal analysisrdquoMeasurement vol 45 no 5 pp 1246ndash1256 2012

[22] A C Altunisik A Bayraktar and B Sevim ldquoOperationalmodal analysis of a scaled bridge model using EFDD and SSImethodsrdquo Indian Journal of Engineering and Materials Sci-ences vol 19 no 5 pp 320ndash330 2012

[23] I Khan D Shan and Q Li ldquoModal parameter identificationof cable stayed bridge based on exploratory data analysisrdquoArchives of Civil Engineering vol 61 no 2 pp 3ndash22 2015


which has physical meaning +e specific implementationsteps are shown in the literature [5]

Based on the core principle of the algorithm the methodstill has the following limitations the processing effect ofendpoints is poor the obtained intrinsic mode function stillincludes mode aliasing and selecting effective IMF com-ponents for signal reconstruction is difficult Consideringthe above three problems this paper proposes a corre-sponding improved algorithm to cope with these problemseffectively

21 Addition of Positive and Negative White Noise In theprocess of EEMD the white noise signal should be added tothe original signal which is then decomposed In thisapproach signal components with different time scales canbe automatically decomposed into the pass band which isdetermined by the white noise signal to overcome themode mixing phenomenon [8] However due to the ad-dition of the white noise signal to the original signal thewhite noise signal will disrupt the original signal which willreduce the signal-to-noise ratio (SNR) of the decomposi-tion results Because the negative effect of white noisecannot be completely eliminated the reconstruction errorof this algorithm is large and the completeness of thedecomposition is poor

Based on these factors in this paper positive andnegative white noise sequences are added to the originalsignal to solve the above problems

+e specific steps in this method are as follows

(1) Addition of randomwhite noise to the original signals(t) including K sets of positive white noise ni(t)

and K sets of negative white noise minus ni(t) +e cal-culation formula is as follows

x(t) s(t) +(minus 1)qa0n

i(t) (1)

where q 1 or 2 (equal to 1 if negative white noise isadded and equal to 2 if positive white noise is added)i 1 2 K K represents the total number oftimes that white noise is added a0 represents thestandard deviation of the amplitude of the addedwhite noise ni(t) represents the white noise that isadded for the ith time s(t) represents the originalsignal and x(t) represents the signal after the whitenoise is added

(2) +e mixed signal with positive and negative pairedwhite noise is an input into the EEMD algorithm andthen decomposed

In practical applications the amplitude of the addedwhite noise cannot be too large or too small if the amplitudeis too large the noise will cover up the effective signal andthe residual noise will be difficult to eliminate However ifthe amplitude is too small the distribution of extreme pointsin the original signal may not be sufficiently disrupted andeliminating the phenomenon of mode aliasing cannot beachieved According to a previous study [8] the amplitude of

the added white noise is usually 01-02 times the standarddeviation of the original signal

22 Adaptive Extreme Point-Matching ContinuationAlgorithm To solve the endpoint effect problem in theEEMD algorithm many scholars have proposed improvedmethods which can be roughly divided into the followingtwo categories

(1) +e extreme point continuation method [9] thebasic principle of this method involves extending theoriginal signal data to a certain number of extremepoints One of the main advantages of this algorithmis that it is simple and easy to implement Howeverthe limitation is that this approach only considers thepartial information of the signal at an endpoint anddoes not consider the signal as a whole

(2) Predictive continuation method [10] in this methodthe original signal is predicted and extended using aneural network [11] ARMA model [12] or supportvector machine method [13] +e effectiveness ofthese algorithms mainly depends on the parametervalues because the approach is not adaptive and thusrequires many computations and has a long oper-ation time

Combining the advantages and disadvantages of theabove twomethods a new continuationmethod is proposedan adaptive extreme point-matching continuation algo-rithm +e main feature of this algorithm is that it candetermine the best-matched extreme points from inside theoriginal signal to match with the extreme points at both endsof the signal +is approach extends the signal consideringthe original trend in a rapid manner and is highly adaptable

+e basic principles of the algorithm are as followsSet the signal equal to x(t) Additionally

mi ni(i 1 2 3 ) are the maximum and minimumpoints respectively and the corresponding times are tmi andtni Taking the extreme left point continuation of x(t) as anexample xtx1 is the starting point at the left and xtx1 minus m1 minus

n1 form an extreme point characteristic wave +en theextreme point waveform that best matches the characteristicwave is determined from the signal In this case xtxi minus mi minus

ni is the matching extreme point wave and the matchingextreme point is extended to the left endpoint of the originalsignal (the right end is extended to the right endpoint)

+en the next specific steps in this method are asfollows

Obtain all xtxi values except the left endpoint xtx1 +ecorresponding time value can then be obtained from thesimilarity relation

txi tm1tmi minus tn1tni

tm1 minus tn1 (2)

where tm1 is assumed to be less than tn1 If the obtained valueof txi is not a sampling point the specific value of xtxi can beobtained by linear interpolation

Calculate the error e between the obtained extremepoints and the extreme points of the feature


e(i) mi minus m1

11138681113868111386811138681113868111386811138681113868

m11113868111386811138681113868

1113868111386811138681113868+

ni minus n11113868111386811138681113868

1113868111386811138681113868

n11113868111386811138681113868

1113868111386811138681113868+

xtxi minus xtx11113868111386811138681113868

1113868111386811138681113868

xtx11113868111386811138681113868

1113868111386811138681113868 (3)








RMS

1113936ni1 S2(i)

n

1113971

(4)



θ 1113936

kf1 RMSf minus RMSx

11138681113868111386811138681113868

11138681113868111386811138681113868

RMSx

(5)






IMFj 1N

1113944

N

i1IMFij (6)



















H minus 1113944N

i1p xi( 1113857logP xi( 1113857 (7)



Original signal

EMD


X1 (t)

hellip

(IMF1(nndash1)

(IMF11

hellip

IMFi1

hellip

IMFN1)N

hellip

IMFN(nndash1))N

+

+

+

+

IMF1

hellip


(r1n

+

+

+

+

+ rNn)N+


(IMF12 IMF22

IMF21


ri(nndash1)r2n

IMF2

IMFnndash1

rn

IMFi2 IMFN2)N




Ej 1N

1113944

N

i1fj(i)1113872 1113873

2Tj

2N

Oj

(8)




jminus 1i1 ETi


1113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868 (9)






RPi minus RPmin


(10)


















Λ diag λ1 λ2 λl( 1113857(11)




Ek 1113944k

i1ΔEi kle l

ΔEi minusλi

1113936lk1 λk

⎛⎝ ⎞⎠lnλi

1113936lk1 λk

⎛⎝ ⎞⎠

(12)



Informationentropy

Energydensity



Averageperiod



Original signal


(IMF11

(IMF1(nndash1)

+

+


IMFN1)N

IMFN(nndash1))N

+

+

+

IMF components


IMF1

IMFnndash1

rn


Clusteringalgorithm

+Residual quantity


IMF21 IMFi1


Clusteringalgorithm

Clusteringalgorithm















dij wf

df

fi minus fj

11138681113868111386811138681113868

11138681113868111386811138681113868

max fi fj1113872 1113873+

wξ

dξ

ξi minus ξj

11138681113868111386811138681113868

11138681113868111386811138681113868

max ξi ξj1113872 1113873+

wψ



















(14)













DATA-SSI


Distance matrix



Improved EEMD









Time (s)

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

ndash505

IMF 1

ndash100

10

IMF 2

ndash505

IMF 3

ndash202

IMF 4

ndash050

05

IMF 5

ndash050

05

IMF 6

ndash020

02

IMF 7

ndash0010

001

IMF 8

ndash050

05

IMF 9


1 2 6 7 4 5 3 84

5

6

7

8

9

10

11

12

13

IMFs

Dist

ance


0 2 4 6 8

0 2 4 6 8

0 2 4 6 8

0 2 4 6 8

0 2 4 6 8

10

10

10

10

10

ndash200

20

All

signa

ls

Analog signal

ndash100

10

7Hz

ndash505

3Hz

ndash202

1Hz

ndash101

Noi

se

Time (s)


Time (s)

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

ndash101

IMF 1

ndash050

05

IMF 2

ndash100

10

IMF 3

ndash505

IMF 4

ndash202

IMF 5

ndash020

02

IMF 6

ndash050

05

IMF 7

ndash020

02

IMF 8












5 6 1 2 7 3 4 9 86

7

8

9

10

11

12

13

14

IMFs

Dist

ance


Freq

uenc

y (H

z)

Number of points0 100 200 300 400 500 600 700 800 900 1000

0

5

10

15

20

25

30

35

40

45

50

ndash20

ndash18

ndash16

ndash14

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

0

















Number of points0 100 200 300 400 500 600 700 800 900 1000

Freq

uenc

y (H

z)

0

5

10

15

20

25

30

35

40

45

50

ndash20

ndash18

ndash16

ndash14

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

0












100 100 300

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Accelerationsensor


10882 10882 300 100 100





ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 3

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 7

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 4

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 8

ndash002 0 0020

50

100

150Fr

eque

ncy

Amplitude range

Sensor 1

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 5

19 2 210

500

1000Sensor 2

Freq

uenc

y


ndash002 0 002 0040

50

100

150Sensor 6

Freq

uenc

y

Amplitude range

ndash002 0 0020

50

100

150

Freq

uenc

yAmplitude range

Sensor 11

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 12

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 9

ndash002 0 0020

50

100

150Sensor 10

Freq

uenc

y

Amplitude range

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 13

ndash002 0 0020

50

100

150Sensor 14

Freq

uenc

y

Amplitude range


0 50 100 150 200 250 3000

005

01

015

02

025

03

035

Incr

emen

t of s

ingu

lar e

ntro

py

System orders


0 50 100 150 200 250 300

0

005

01

015

Incr

emen

t of s

ingu

lar e

ntro

py-fi

rst d

eriv

ativ

e

System orders



0 1 2 3 4 5 6 7 8 9 100

50

100

150

200

250

300

Syste

m o

rder

sFrequency (Hz)


0 1 2 3 4 5 6 7 8 9 10Frequency (Hz)

0

50

100

150

200

250

300

Syste

m o

rder

s


0

50

100

150

05 1 15 2 25 3 35 4 45 5

018508023776

046347055536

117211859

2485331657

3635643286

50873

Syet

em o

rder

s

Frequency (Hz)

070871








1 2 3 4 5 6 7 8 9 10 1101020ndash1

0

1

(a)

2 4 6 8 100

1020ndash1

01

(b)

2 4 6 8 100

1020ndash1

01

(c)




(a)

(b)

(c)




6 Conclusion








Data Availability




Acknowledgments


References












02

03

04

05

06

07

08

Freq

uenc

y (H

z)

Third order



















e(i) mi minus m1

11138681113868111386811138681113868111386811138681113868

m11113868111386811138681113868

1113868111386811138681113868+

ni minus n11113868111386811138681113868

1113868111386811138681113868

n11113868111386811138681113868

1113868111386811138681113868+

xtxi minus xtx11113868111386811138681113868

1113868111386811138681113868

xtx11113868111386811138681113868

1113868111386811138681113868 (3)








RMS

1113936ni1 S2(i)

n

1113971

(4)



θ 1113936

kf1 RMSf minus RMSx

11138681113868111386811138681113868

11138681113868111386811138681113868

RMSx

(5)






IMFj 1N

1113944

N

i1IMFij (6)



















H minus 1113944N

i1p xi( 1113857logP xi( 1113857 (7)



Original signal

EMD


X1 (t)

hellip

(IMF1(nndash1)

(IMF11

hellip

IMFi1

hellip

IMFN1)N

hellip

IMFN(nndash1))N

+

+

+

+

IMF1

hellip


(r1n

+

+

+

+

+ rNn)N+


(IMF12 IMF22

IMF21


ri(nndash1)r2n

IMF2

IMFnndash1

rn

IMFi2 IMFN2)N




Ej 1N

1113944

N

i1fj(i)1113872 1113873

2Tj

2N

Oj

(8)




jminus 1i1 ETi


1113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868 (9)






RPi minus RPmin


(10)


















Λ diag λ1 λ2 λl( 1113857(11)




Ek 1113944k

i1ΔEi kle l

ΔEi minusλi

1113936lk1 λk

⎛⎝ ⎞⎠lnλi

1113936lk1 λk

⎛⎝ ⎞⎠

(12)



Informationentropy

Energydensity



Averageperiod



Original signal


(IMF11

(IMF1(nndash1)

+

+


IMFN1)N

IMFN(nndash1))N

+

+

+

IMF components


IMF1

IMFnndash1

rn


Clusteringalgorithm

+Residual quantity


IMF21 IMFi1


Clusteringalgorithm

Clusteringalgorithm















dij wf

df

fi minus fj

11138681113868111386811138681113868

11138681113868111386811138681113868

max fi fj1113872 1113873+

wξ

dξ

ξi minus ξj

11138681113868111386811138681113868

11138681113868111386811138681113868

max ξi ξj1113872 1113873+

wψ



















(14)













DATA-SSI


Distance matrix



Improved EEMD









Time (s)

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

ndash505

IMF 1

ndash100

10

IMF 2

ndash505

IMF 3

ndash202

IMF 4

ndash050

05

IMF 5

ndash050

05

IMF 6

ndash020

02

IMF 7

ndash0010

001

IMF 8

ndash050

05

IMF 9


1 2 6 7 4 5 3 84

5

6

7

8

9

10

11

12

13

IMFs

Dist

ance


0 2 4 6 8

0 2 4 6 8

0 2 4 6 8

0 2 4 6 8

0 2 4 6 8

10

10

10

10

10

ndash200

20

All

signa

ls

Analog signal

ndash100

10

7Hz

ndash505

3Hz

ndash202

1Hz

ndash101

Noi

se

Time (s)


Time (s)

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

ndash101

IMF 1

ndash050

05

IMF 2

ndash100

10

IMF 3

ndash505

IMF 4

ndash202

IMF 5

ndash020

02

IMF 6

ndash050

05

IMF 7

ndash020

02

IMF 8












5 6 1 2 7 3 4 9 86

7

8

9

10

11

12

13

14

IMFs

Dist

ance


Freq

uenc

y (H

z)

Number of points0 100 200 300 400 500 600 700 800 900 1000

0

5

10

15

20

25

30

35

40

45

50

ndash20

ndash18

ndash16

ndash14

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

0

















Number of points0 100 200 300 400 500 600 700 800 900 1000

Freq

uenc

y (H

z)

0

5

10

15

20

25

30

35

40

45

50

ndash20

ndash18

ndash16

ndash14

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

0












100 100 300

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Accelerationsensor


10882 10882 300 100 100





ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 3

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 7

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 4

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 8

ndash002 0 0020

50

100

150Fr

eque

ncy

Amplitude range

Sensor 1

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 5

19 2 210

500

1000Sensor 2

Freq

uenc

y


ndash002 0 002 0040

50

100

150Sensor 6

Freq

uenc

y

Amplitude range

ndash002 0 0020

50

100

150

Freq

uenc

yAmplitude range

Sensor 11

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 12

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 9

ndash002 0 0020

50

100

150Sensor 10

Freq

uenc

y

Amplitude range

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 13

ndash002 0 0020

50

100

150Sensor 14

Freq

uenc

y

Amplitude range


0 50 100 150 200 250 3000

005

01

015

02

025

03

035

Incr

emen

t of s

ingu

lar e

ntro

py

System orders


0 50 100 150 200 250 300

0

005

01

015

Incr

emen

t of s

ingu

lar e

ntro

py-fi

rst d

eriv

ativ

e

System orders



0 1 2 3 4 5 6 7 8 9 100

50

100

150

200

250

300

Syste

m o

rder

sFrequency (Hz)


0 1 2 3 4 5 6 7 8 9 10Frequency (Hz)

0

50

100

150

200

250

300

Syste

m o

rder

s


0

50

100

150

05 1 15 2 25 3 35 4 45 5

018508023776

046347055536

117211859

2485331657

3635643286

50873

Syet

em o

rder

s

Frequency (Hz)

070871








1 2 3 4 5 6 7 8 9 10 1101020ndash1

0

1

(a)

2 4 6 8 100

1020ndash1

01

(b)

2 4 6 8 100

1020ndash1

01

(c)




(a)

(b)

(c)




6 Conclusion








Data Availability




Acknowledgments


References












02

03

04

05

06

07

08

Freq

uenc

y (H

z)

Third order



























H minus 1113944N

i1p xi( 1113857logP xi( 1113857 (7)



Original signal

EMD


X1 (t)

hellip

(IMF1(nndash1)

(IMF11

hellip

IMFi1

hellip

IMFN1)N

hellip

IMFN(nndash1))N

+

+

+

+

IMF1

hellip


(r1n

+

+

+

+

+ rNn)N+


(IMF12 IMF22

IMF21


ri(nndash1)r2n

IMF2

IMFnndash1

rn

IMFi2 IMFN2)N




Ej 1N

1113944

N

i1fj(i)1113872 1113873

2Tj

2N

Oj

(8)




jminus 1i1 ETi


1113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868 (9)






RPi minus RPmin


(10)


















Λ diag λ1 λ2 λl( 1113857(11)




Ek 1113944k

i1ΔEi kle l

ΔEi minusλi

1113936lk1 λk

⎛⎝ ⎞⎠lnλi

1113936lk1 λk

⎛⎝ ⎞⎠

(12)



Informationentropy

Energydensity



Averageperiod



Original signal


(IMF11

(IMF1(nndash1)

+

+


IMFN1)N

IMFN(nndash1))N

+

+

+

IMF components


IMF1

IMFnndash1

rn


Clusteringalgorithm

+Residual quantity


IMF21 IMFi1


Clusteringalgorithm

Clusteringalgorithm















dij wf

df

fi minus fj

11138681113868111386811138681113868

11138681113868111386811138681113868

max fi fj1113872 1113873+

wξ

dξ

ξi minus ξj

11138681113868111386811138681113868

11138681113868111386811138681113868

max ξi ξj1113872 1113873+

wψ



















(14)













DATA-SSI


Distance matrix



Improved EEMD









Time (s)

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

ndash505

IMF 1

ndash100

10

IMF 2

ndash505

IMF 3

ndash202

IMF 4

ndash050

05

IMF 5

ndash050

05

IMF 6

ndash020

02

IMF 7

ndash0010

001

IMF 8

ndash050

05

IMF 9


1 2 6 7 4 5 3 84

5

6

7

8

9

10

11

12

13

IMFs

Dist

ance


0 2 4 6 8

0 2 4 6 8

0 2 4 6 8

0 2 4 6 8

0 2 4 6 8

10

10

10

10

10

ndash200

20

All

signa

ls

Analog signal

ndash100

10

7Hz

ndash505

3Hz

ndash202

1Hz

ndash101

Noi

se

Time (s)


Time (s)

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

ndash101

IMF 1

ndash050

05

IMF 2

ndash100

10

IMF 3

ndash505

IMF 4

ndash202

IMF 5

ndash020

02

IMF 6

ndash050

05

IMF 7

ndash020

02

IMF 8












5 6 1 2 7 3 4 9 86

7

8

9

10

11

12

13

14

IMFs

Dist

ance


Freq

uenc

y (H

z)

Number of points0 100 200 300 400 500 600 700 800 900 1000

0

5

10

15

20

25

30

35

40

45

50

ndash20

ndash18

ndash16

ndash14

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

0

















Number of points0 100 200 300 400 500 600 700 800 900 1000

Freq

uenc

y (H

z)

0

5

10

15

20

25

30

35

40

45

50

ndash20

ndash18

ndash16

ndash14

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

0












100 100 300

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Accelerationsensor


10882 10882 300 100 100





ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 3

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 7

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 4

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 8

ndash002 0 0020

50

100

150Fr

eque

ncy

Amplitude range

Sensor 1

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 5

19 2 210

500

1000Sensor 2

Freq

uenc

y


ndash002 0 002 0040

50

100

150Sensor 6

Freq

uenc

y

Amplitude range

ndash002 0 0020

50

100

150

Freq

uenc

yAmplitude range

Sensor 11

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 12

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 9

ndash002 0 0020

50

100

150Sensor 10

Freq

uenc

y

Amplitude range

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 13

ndash002 0 0020

50

100

150Sensor 14

Freq

uenc

y

Amplitude range


0 50 100 150 200 250 3000

005

01

015

02

025

03

035

Incr

emen

t of s

ingu

lar e

ntro

py

System orders


0 50 100 150 200 250 300

0

005

01

015

Incr

emen

t of s

ingu

lar e

ntro

py-fi

rst d

eriv

ativ

e

System orders



0 1 2 3 4 5 6 7 8 9 100

50

100

150

200

250

300

Syste

m o

rder

sFrequency (Hz)


0 1 2 3 4 5 6 7 8 9 10Frequency (Hz)

0

50

100

150

200

250

300

Syste

m o

rder

s


0

50

100

150

05 1 15 2 25 3 35 4 45 5

018508023776

046347055536

117211859

2485331657

3635643286

50873

Syet

em o

rder

s

Frequency (Hz)

070871








1 2 3 4 5 6 7 8 9 10 1101020ndash1

0

1

(a)

2 4 6 8 100

1020ndash1

01

(b)

2 4 6 8 100

1020ndash1

01

(c)




(a)

(b)

(c)




6 Conclusion








Data Availability




Acknowledgments


References












02

03

04

05

06

07

08

Freq

uenc

y (H

z)

Third order




















Ej 1N

1113944

N

i1fj(i)1113872 1113873

2Tj

2N

Oj

(8)




jminus 1i1 ETi


1113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868 (9)






RPi minus RPmin


(10)


















Λ diag λ1 λ2 λl( 1113857(11)




Ek 1113944k

i1ΔEi kle l

ΔEi minusλi

1113936lk1 λk

⎛⎝ ⎞⎠lnλi

1113936lk1 λk

⎛⎝ ⎞⎠

(12)



Informationentropy

Energydensity



Averageperiod



Original signal


(IMF11

(IMF1(nndash1)

+

+


IMFN1)N

IMFN(nndash1))N

+

+

+

IMF components


IMF1

IMFnndash1

rn


Clusteringalgorithm

+Residual quantity


IMF21 IMFi1


Clusteringalgorithm

Clusteringalgorithm















dij wf

df

fi minus fj

11138681113868111386811138681113868

11138681113868111386811138681113868

max fi fj1113872 1113873+

wξ

dξ

ξi minus ξj

11138681113868111386811138681113868

11138681113868111386811138681113868

max ξi ξj1113872 1113873+

wψ



















(14)













DATA-SSI


Distance matrix



Improved EEMD









Time (s)

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

ndash505

IMF 1

ndash100

10

IMF 2

ndash505

IMF 3

ndash202

IMF 4

ndash050

05

IMF 5

ndash050

05

IMF 6

ndash020

02

IMF 7

ndash0010

001

IMF 8

ndash050

05

IMF 9


1 2 6 7 4 5 3 84

5

6

7

8

9

10

11

12

13

IMFs

Dist

ance


0 2 4 6 8

0 2 4 6 8

0 2 4 6 8

0 2 4 6 8

0 2 4 6 8

10

10

10

10

10

ndash200

20

All

signa

ls

Analog signal

ndash100

10

7Hz

ndash505

3Hz

ndash202

1Hz

ndash101

Noi

se

Time (s)


Time (s)

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

ndash101

IMF 1

ndash050

05

IMF 2

ndash100

10

IMF 3

ndash505

IMF 4

ndash202

IMF 5

ndash020

02

IMF 6

ndash050

05

IMF 7

ndash020

02

IMF 8












5 6 1 2 7 3 4 9 86

7

8

9

10

11

12

13

14

IMFs

Dist

ance


Freq

uenc

y (H

z)

Number of points0 100 200 300 400 500 600 700 800 900 1000

0

5

10

15

20

25

30

35

40

45

50

ndash20

ndash18

ndash16

ndash14

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

0

















Number of points0 100 200 300 400 500 600 700 800 900 1000

Freq

uenc

y (H

z)

0

5

10

15

20

25

30

35

40

45

50

ndash20

ndash18

ndash16

ndash14

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

0












100 100 300

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Accelerationsensor


10882 10882 300 100 100





ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 3

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 7

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 4

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 8

ndash002 0 0020

50

100

150Fr

eque

ncy

Amplitude range

Sensor 1

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 5

19 2 210

500

1000Sensor 2

Freq

uenc

y


ndash002 0 002 0040

50

100

150Sensor 6

Freq

uenc

y

Amplitude range

ndash002 0 0020

50

100

150

Freq

uenc

yAmplitude range

Sensor 11

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 12

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 9

ndash002 0 0020

50

100

150Sensor 10

Freq

uenc

y

Amplitude range

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 13

ndash002 0 0020

50

100

150Sensor 14

Freq

uenc

y

Amplitude range


0 50 100 150 200 250 3000

005

01

015

02

025

03

035

Incr

emen

t of s

ingu

lar e

ntro

py

System orders


0 50 100 150 200 250 300

0

005

01

015

Incr

emen

t of s

ingu

lar e

ntro

py-fi

rst d

eriv

ativ

e

System orders



0 1 2 3 4 5 6 7 8 9 100

50

100

150

200

250

300

Syste

m o

rder

sFrequency (Hz)


0 1 2 3 4 5 6 7 8 9 10Frequency (Hz)

0

50

100

150

200

250

300

Syste

m o

rder

s


0

50

100

150

05 1 15 2 25 3 35 4 45 5

018508023776

046347055536

117211859

2485331657

3635643286

50873

Syet

em o

rder

s

Frequency (Hz)

070871








1 2 3 4 5 6 7 8 9 10 1101020ndash1

0

1

(a)

2 4 6 8 100

1020ndash1

01

(b)

2 4 6 8 100

1020ndash1

01

(c)




(a)

(b)

(c)




6 Conclusion








Data Availability




Acknowledgments


References












02

03

04

05

06

07

08

Freq

uenc

y (H

z)

Third order





















Λ diag λ1 λ2 λl( 1113857(11)




Ek 1113944k

i1ΔEi kle l

ΔEi minusλi

1113936lk1 λk

⎛⎝ ⎞⎠lnλi

1113936lk1 λk

⎛⎝ ⎞⎠

(12)



Informationentropy

Energydensity



Averageperiod



Original signal


(IMF11

(IMF1(nndash1)

+

+


IMFN1)N

IMFN(nndash1))N

+

+

+

IMF components


IMF1

IMFnndash1

rn


Clusteringalgorithm

+Residual quantity


IMF21 IMFi1


Clusteringalgorithm

Clusteringalgorithm















dij wf

df

fi minus fj

11138681113868111386811138681113868

11138681113868111386811138681113868

max fi fj1113872 1113873+

wξ

dξ

ξi minus ξj

11138681113868111386811138681113868

11138681113868111386811138681113868

max ξi ξj1113872 1113873+

wψ



















(14)













DATA-SSI


Distance matrix



Improved EEMD









Time (s)

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

ndash505

IMF 1

ndash100

10

IMF 2

ndash505

IMF 3

ndash202

IMF 4

ndash050

05

IMF 5

ndash050

05

IMF 6

ndash020

02

IMF 7

ndash0010

001

IMF 8

ndash050

05

IMF 9


1 2 6 7 4 5 3 84

5

6

7

8

9

10

11

12

13

IMFs

Dist

ance


0 2 4 6 8

0 2 4 6 8

0 2 4 6 8

0 2 4 6 8

0 2 4 6 8

10

10

10

10

10

ndash200

20

All

signa

ls

Analog signal

ndash100

10

7Hz

ndash505

3Hz

ndash202

1Hz

ndash101

Noi

se

Time (s)


Time (s)

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

ndash101

IMF 1

ndash050

05

IMF 2

ndash100

10

IMF 3

ndash505

IMF 4

ndash202

IMF 5

ndash020

02

IMF 6

ndash050

05

IMF 7

ndash020

02

IMF 8












5 6 1 2 7 3 4 9 86

7

8

9

10

11

12

13

14

IMFs

Dist

ance


Freq

uenc

y (H

z)

Number of points0 100 200 300 400 500 600 700 800 900 1000

0

5

10

15

20

25

30

35

40

45

50

ndash20

ndash18

ndash16

ndash14

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

0

















Number of points0 100 200 300 400 500 600 700 800 900 1000

Freq

uenc

y (H

z)

0

5

10

15

20

25

30

35

40

45

50

ndash20

ndash18

ndash16

ndash14

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

0












100 100 300

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Accelerationsensor


10882 10882 300 100 100





ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 3

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 7

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 4

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 8

ndash002 0 0020

50

100

150Fr

eque

ncy

Amplitude range

Sensor 1

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 5

19 2 210

500

1000Sensor 2

Freq

uenc

y


ndash002 0 002 0040

50

100

150Sensor 6

Freq

uenc

y

Amplitude range

ndash002 0 0020

50

100

150

Freq

uenc

yAmplitude range

Sensor 11

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 12

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 9

ndash002 0 0020

50

100

150Sensor 10

Freq

uenc

y

Amplitude range

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 13

ndash002 0 0020

50

100

150Sensor 14

Freq

uenc

y

Amplitude range


0 50 100 150 200 250 3000

005

01

015

02

025

03

035

Incr

emen

t of s

ingu

lar e

ntro

py

System orders


0 50 100 150 200 250 300

0

005

01

015

Incr

emen

t of s

ingu

lar e

ntro

py-fi

rst d

eriv

ativ

e

System orders



0 1 2 3 4 5 6 7 8 9 100

50

100

150

200

250

300

Syste

m o

rder

sFrequency (Hz)


0 1 2 3 4 5 6 7 8 9 10Frequency (Hz)

0

50

100

150

200

250

300

Syste

m o

rder

s


0

50

100

150

05 1 15 2 25 3 35 4 45 5

018508023776

046347055536

117211859

2485331657

3635643286

50873

Syet

em o

rder

s

Frequency (Hz)

070871








1 2 3 4 5 6 7 8 9 10 1101020ndash1

0

1

(a)

2 4 6 8 100

1020ndash1

01

(b)

2 4 6 8 100

1020ndash1

01

(c)




(a)

(b)

(c)




6 Conclusion








Data Availability




Acknowledgments


References












02

03

04

05

06

07

08

Freq

uenc

y (H

z)

Third order































dij wf

df

fi minus fj

11138681113868111386811138681113868

11138681113868111386811138681113868

max fi fj1113872 1113873+

wξ

dξ

ξi minus ξj

11138681113868111386811138681113868

11138681113868111386811138681113868

max ξi ξj1113872 1113873+

wψ



















(14)













DATA-SSI


Distance matrix



Improved EEMD









Time (s)

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

ndash505

IMF 1

ndash100

10

IMF 2

ndash505

IMF 3

ndash202

IMF 4

ndash050

05

IMF 5

ndash050

05

IMF 6

ndash020

02

IMF 7

ndash0010

001

IMF 8

ndash050

05

IMF 9


1 2 6 7 4 5 3 84

5

6

7

8

9

10

11

12

13

IMFs

Dist

ance


0 2 4 6 8

0 2 4 6 8

0 2 4 6 8

0 2 4 6 8

0 2 4 6 8

10

10

10

10

10

ndash200

20

All

signa

ls

Analog signal

ndash100

10

7Hz

ndash505

3Hz

ndash202

1Hz

ndash101

Noi

se

Time (s)


Time (s)

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

ndash101

IMF 1

ndash050

05

IMF 2

ndash100

10

IMF 3

ndash505

IMF 4

ndash202

IMF 5

ndash020

02

IMF 6

ndash050

05

IMF 7

ndash020

02

IMF 8












5 6 1 2 7 3 4 9 86

7

8

9

10

11

12

13

14

IMFs

Dist

ance


Freq

uenc

y (H

z)

Number of points0 100 200 300 400 500 600 700 800 900 1000

0

5

10

15

20

25

30

35

40

45

50

ndash20

ndash18

ndash16

ndash14

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

0

















Number of points0 100 200 300 400 500 600 700 800 900 1000

Freq

uenc

y (H

z)

0

5

10

15

20

25

30

35

40

45

50

ndash20

ndash18

ndash16

ndash14

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

0












100 100 300

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Accelerationsensor


10882 10882 300 100 100





ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 3

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 7

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 4

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 8

ndash002 0 0020

50

100

150Fr

eque

ncy

Amplitude range

Sensor 1

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 5

19 2 210

500

1000Sensor 2

Freq

uenc

y


ndash002 0 002 0040

50

100

150Sensor 6

Freq

uenc

y

Amplitude range

ndash002 0 0020

50

100

150

Freq

uenc

yAmplitude range

Sensor 11

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 12

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 9

ndash002 0 0020

50

100

150Sensor 10

Freq

uenc

y

Amplitude range

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 13

ndash002 0 0020

50

100

150Sensor 14

Freq

uenc

y

Amplitude range


0 50 100 150 200 250 3000

005

01

015

02

025

03

035

Incr

emen

t of s

ingu

lar e

ntro

py

System orders


0 50 100 150 200 250 300

0

005

01

015

Incr

emen

t of s

ingu

lar e

ntro

py-fi

rst d

eriv

ativ

e

System orders



0 1 2 3 4 5 6 7 8 9 100

50

100

150

200

250

300

Syste

m o

rder

sFrequency (Hz)


0 1 2 3 4 5 6 7 8 9 10Frequency (Hz)

0

50

100

150

200

250

300

Syste

m o

rder

s


0

50

100

150

05 1 15 2 25 3 35 4 45 5

018508023776

046347055536

117211859

2485331657

3635643286

50873

Syet

em o

rder

s

Frequency (Hz)

070871








1 2 3 4 5 6 7 8 9 10 1101020ndash1

0

1

(a)

2 4 6 8 100

1020ndash1

01

(b)

2 4 6 8 100

1020ndash1

01

(c)




(a)

(b)

(c)




6 Conclusion








Data Availability




Acknowledgments


References












02

03

04

05

06

07

08

Freq

uenc

y (H

z)

Third order


























(14)













DATA-SSI


Distance matrix



Improved EEMD









Time (s)

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

ndash505

IMF 1

ndash100

10

IMF 2

ndash505

IMF 3

ndash202

IMF 4

ndash050

05

IMF 5

ndash050

05

IMF 6

ndash020

02

IMF 7

ndash0010

001

IMF 8

ndash050

05

IMF 9


1 2 6 7 4 5 3 84

5

6

7

8

9

10

11

12

13

IMFs

Dist

ance


0 2 4 6 8

0 2 4 6 8

0 2 4 6 8

0 2 4 6 8

0 2 4 6 8

10

10

10

10

10

ndash200

20

All

signa

ls

Analog signal

ndash100

10

7Hz

ndash505

3Hz

ndash202

1Hz

ndash101

Noi

se

Time (s)


Time (s)

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

ndash101

IMF 1

ndash050

05

IMF 2

ndash100

10

IMF 3

ndash505

IMF 4

ndash202

IMF 5

ndash020

02

IMF 6

ndash050

05

IMF 7

ndash020

02

IMF 8












5 6 1 2 7 3 4 9 86

7

8

9

10

11

12

13

14

IMFs

Dist

ance


Freq

uenc

y (H

z)

Number of points0 100 200 300 400 500 600 700 800 900 1000

0

5

10

15

20

25

30

35

40

45

50

ndash20

ndash18

ndash16

ndash14

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

0

















Number of points0 100 200 300 400 500 600 700 800 900 1000

Freq

uenc

y (H

z)

0

5

10

15

20

25

30

35

40

45

50

ndash20

ndash18

ndash16

ndash14

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

0












100 100 300

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Accelerationsensor


10882 10882 300 100 100





ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 3

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 7

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 4

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 8

ndash002 0 0020

50

100

150Fr

eque

ncy

Amplitude range

Sensor 1

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 5

19 2 210

500

1000Sensor 2

Freq

uenc

y


ndash002 0 002 0040

50

100

150Sensor 6

Freq

uenc

y

Amplitude range

ndash002 0 0020

50

100

150

Freq

uenc

yAmplitude range

Sensor 11

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 12

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 9

ndash002 0 0020

50

100

150Sensor 10

Freq

uenc

y

Amplitude range

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 13

ndash002 0 0020

50

100

150Sensor 14

Freq

uenc

y

Amplitude range


0 50 100 150 200 250 3000

005

01

015

02

025

03

035

Incr

emen

t of s

ingu

lar e

ntro

py

System orders


0 50 100 150 200 250 300

0

005

01

015

Incr

emen

t of s

ingu

lar e

ntro

py-fi

rst d

eriv

ativ

e

System orders



0 1 2 3 4 5 6 7 8 9 100

50

100

150

200

250

300

Syste

m o

rder

sFrequency (Hz)


0 1 2 3 4 5 6 7 8 9 10Frequency (Hz)

0

50

100

150

200

250

300

Syste

m o

rder

s


0

50

100

150

05 1 15 2 25 3 35 4 45 5

018508023776

046347055536

117211859

2485331657

3635643286

50873

Syet

em o

rder

s

Frequency (Hz)

070871








1 2 3 4 5 6 7 8 9 10 1101020ndash1

0

1

(a)

2 4 6 8 100

1020ndash1

01

(b)

2 4 6 8 100

1020ndash1

01

(c)




(a)

(b)

(c)




6 Conclusion








Data Availability




Acknowledgments


References












02

03

04

05

06

07

08

Freq

uenc

y (H

z)

Third order
























Time (s)

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

ndash505

IMF 1

ndash100

10

IMF 2

ndash505

IMF 3

ndash202

IMF 4

ndash050

05

IMF 5

ndash050

05

IMF 6

ndash020

02

IMF 7

ndash0010

001

IMF 8

ndash050

05

IMF 9


1 2 6 7 4 5 3 84

5

6

7

8

9

10

11

12

13

IMFs

Dist

ance


0 2 4 6 8

0 2 4 6 8

0 2 4 6 8

0 2 4 6 8

0 2 4 6 8

10

10

10

10

10

ndash200

20

All

signa

ls

Analog signal

ndash100

10

7Hz

ndash505

3Hz

ndash202

1Hz

ndash101

Noi

se

Time (s)


Time (s)

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

ndash101

IMF 1

ndash050

05

IMF 2

ndash100

10

IMF 3

ndash505

IMF 4

ndash202

IMF 5

ndash020

02

IMF 6

ndash050

05

IMF 7

ndash020

02

IMF 8












5 6 1 2 7 3 4 9 86

7

8

9

10

11

12

13

14

IMFs

Dist

ance


Freq

uenc

y (H

z)

Number of points0 100 200 300 400 500 600 700 800 900 1000

0

5

10

15

20

25

30

35

40

45

50

ndash20

ndash18

ndash16

ndash14

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

0

















Number of points0 100 200 300 400 500 600 700 800 900 1000

Freq

uenc

y (H

z)

0

5

10

15

20

25

30

35

40

45

50

ndash20

ndash18

ndash16

ndash14

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

0












100 100 300

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Accelerationsensor


10882 10882 300 100 100





ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 3

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 7

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 4

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 8

ndash002 0 0020

50

100

150Fr

eque

ncy

Amplitude range

Sensor 1

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 5

19 2 210

500

1000Sensor 2

Freq

uenc

y


ndash002 0 002 0040

50

100

150Sensor 6

Freq

uenc

y

Amplitude range

ndash002 0 0020

50

100

150

Freq

uenc

yAmplitude range

Sensor 11

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 12

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 9

ndash002 0 0020

50

100

150Sensor 10

Freq

uenc

y

Amplitude range

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 13

ndash002 0 0020

50

100

150Sensor 14

Freq

uenc

y

Amplitude range


0 50 100 150 200 250 3000

005

01

015

02

025

03

035

Incr

emen

t of s

ingu

lar e

ntro

py

System orders


0 50 100 150 200 250 300

0

005

01

015

Incr

emen

t of s

ingu

lar e

ntro

py-fi

rst d

eriv

ativ

e

System orders



0 1 2 3 4 5 6 7 8 9 100

50

100

150

200

250

300

Syste

m o

rder

sFrequency (Hz)


0 1 2 3 4 5 6 7 8 9 10Frequency (Hz)

0

50

100

150

200

250

300

Syste

m o

rder

s


0

50

100

150

05 1 15 2 25 3 35 4 45 5

018508023776

046347055536

117211859

2485331657

3635643286

50873

Syet

em o

rder

s

Frequency (Hz)

070871








1 2 3 4 5 6 7 8 9 10 1101020ndash1

0

1

(a)

2 4 6 8 100

1020ndash1

01

(b)

2 4 6 8 100

1020ndash1

01

(c)




(a)

(b)

(c)




6 Conclusion








Data Availability




Acknowledgments


References












02

03

04

05

06

07

08

Freq

uenc

y (H

z)

Third order




























5 6 1 2 7 3 4 9 86

7

8

9

10

11

12

13

14

IMFs

Dist

ance


Freq

uenc

y (H

z)

Number of points0 100 200 300 400 500 600 700 800 900 1000

0

5

10

15

20

25

30

35

40

45

50

ndash20

ndash18

ndash16

ndash14

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

0

















Number of points0 100 200 300 400 500 600 700 800 900 1000

Freq

uenc

y (H

z)

0

5

10

15

20

25

30

35

40

45

50

ndash20

ndash18

ndash16

ndash14

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

0












100 100 300

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Accelerationsensor


10882 10882 300 100 100





ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 3

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 7

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 4

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 8

ndash002 0 0020

50

100

150Fr

eque

ncy

Amplitude range

Sensor 1

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 5

19 2 210

500

1000Sensor 2

Freq

uenc

y


ndash002 0 002 0040

50

100

150Sensor 6

Freq

uenc

y

Amplitude range

ndash002 0 0020

50

100

150

Freq

uenc

yAmplitude range

Sensor 11

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 12

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 9

ndash002 0 0020

50

100

150Sensor 10

Freq

uenc

y

Amplitude range

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 13

ndash002 0 0020

50

100

150Sensor 14

Freq

uenc

y

Amplitude range


0 50 100 150 200 250 3000

005

01

015

02

025

03

035

Incr

emen

t of s

ingu

lar e

ntro

py

System orders


0 50 100 150 200 250 300

0

005

01

015

Incr

emen

t of s

ingu

lar e

ntro

py-fi

rst d

eriv

ativ

e

System orders



0 1 2 3 4 5 6 7 8 9 100

50

100

150

200

250

300

Syste

m o

rder

sFrequency (Hz)


0 1 2 3 4 5 6 7 8 9 10Frequency (Hz)

0

50

100

150

200

250

300

Syste

m o

rder

s


0

50

100

150

05 1 15 2 25 3 35 4 45 5

018508023776

046347055536

117211859

2485331657

3635643286

50873

Syet

em o

rder

s

Frequency (Hz)

070871








1 2 3 4 5 6 7 8 9 10 1101020ndash1

0

1

(a)

2 4 6 8 100

1020ndash1

01

(b)

2 4 6 8 100

1020ndash1

01

(c)




(a)

(b)

(c)




6 Conclusion








Data Availability




Acknowledgments


References












02

03

04

05

06

07

08

Freq

uenc

y (H

z)

Third order

































Number of points0 100 200 300 400 500 600 700 800 900 1000

Freq

uenc

y (H

z)

0

5

10

15

20

25

30

35

40

45

50

ndash20

ndash18

ndash16

ndash14

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

0












100 100 300

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Accelerationsensor


10882 10882 300 100 100





ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 3

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 7

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 4

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 8

ndash002 0 0020

50

100

150Fr

eque

ncy

Amplitude range

Sensor 1

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 5

19 2 210

500

1000Sensor 2

Freq

uenc

y


ndash002 0 002 0040

50

100

150Sensor 6

Freq

uenc

y

Amplitude range

ndash002 0 0020

50

100

150

Freq

uenc

yAmplitude range

Sensor 11

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 12

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 9

ndash002 0 0020

50

100

150Sensor 10

Freq

uenc

y

Amplitude range

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 13

ndash002 0 0020

50

100

150Sensor 14

Freq

uenc

y

Amplitude range


0 50 100 150 200 250 3000

005

01

015

02

025

03

035

Incr

emen

t of s

ingu

lar e

ntro

py

System orders


0 50 100 150 200 250 300

0

005

01

015

Incr

emen

t of s

ingu

lar e

ntro

py-fi

rst d

eriv

ativ

e

System orders



0 1 2 3 4 5 6 7 8 9 100

50

100

150

200

250

300

Syste

m o

rder

sFrequency (Hz)


0 1 2 3 4 5 6 7 8 9 10Frequency (Hz)

0

50

100

150

200

250

300

Syste

m o

rder

s


0

50

100

150

05 1 15 2 25 3 35 4 45 5

018508023776

046347055536

117211859

2485331657

3635643286

50873

Syet

em o

rder

s

Frequency (Hz)

070871








1 2 3 4 5 6 7 8 9 10 1101020ndash1

0

1

(a)

2 4 6 8 100

1020ndash1

01

(b)

2 4 6 8 100

1020ndash1

01

(c)




(a)

(b)

(c)




6 Conclusion








Data Availability




Acknowledgments


References












02

03

04

05

06

07

08

Freq

uenc

y (H

z)

Third order




























100 100 300

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Accelerationsensor


10882 10882 300 100 100





ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 3

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 7

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 4

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 8

ndash002 0 0020

50

100

150Fr

eque

ncy

Amplitude range

Sensor 1

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 5

19 2 210

500

1000Sensor 2

Freq

uenc

y


ndash002 0 002 0040

50

100

150Sensor 6

Freq

uenc

y

Amplitude range

ndash002 0 0020

50

100

150

Freq

uenc

yAmplitude range

Sensor 11

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 12

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 9

ndash002 0 0020

50

100

150Sensor 10

Freq

uenc

y

Amplitude range

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 13

ndash002 0 0020

50

100

150Sensor 14

Freq

uenc

y

Amplitude range


0 50 100 150 200 250 3000

005

01

015

02

025

03

035

Incr

emen

t of s

ingu

lar e

ntro

py

System orders


0 50 100 150 200 250 300

0

005

01

015

Incr

emen

t of s

ingu

lar e

ntro

py-fi

rst d

eriv

ativ

e

System orders



0 1 2 3 4 5 6 7 8 9 100

50

100

150

200

250

300

Syste

m o

rder

sFrequency (Hz)


0 1 2 3 4 5 6 7 8 9 10Frequency (Hz)

0

50

100

150

200

250

300

Syste

m o

rder

s


0

50

100

150

05 1 15 2 25 3 35 4 45 5

018508023776

046347055536

117211859

2485331657

3635643286

50873

Syet

em o

rder

s

Frequency (Hz)

070871








1 2 3 4 5 6 7 8 9 10 1101020ndash1

0

1

(a)

2 4 6 8 100

1020ndash1

01

(b)

2 4 6 8 100

1020ndash1

01

(c)




(a)

(b)

(c)




6 Conclusion








Data Availability




Acknowledgments


References












02

03

04

05

06

07

08

Freq

uenc

y (H

z)

Third order





















ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 3

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 7

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 4

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 8

ndash002 0 0020

50

100

150Fr

eque

ncy

Amplitude range

Sensor 1

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 5

19 2 210

500

1000Sensor 2

Freq

uenc

y


ndash002 0 002 0040

50

100

150Sensor 6

Freq

uenc

y

Amplitude range

ndash002 0 0020

50

100

150

Freq

uenc

yAmplitude range

Sensor 11

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 12

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 9

ndash002 0 0020

50

100

150Sensor 10

Freq

uenc

y

Amplitude range

ndash002 0 0020

50

100

150

Freq

uenc

y

Amplitude range

Sensor 13

ndash002 0 0020

50

100

150Sensor 14

Freq

uenc

y

Amplitude range


0 50 100 150 200 250 3000

005

01

015

02

025

03

035

Incr

emen

t of s

ingu

lar e

ntro

py

System orders


0 50 100 150 200 250 300

0

005

01

015

Incr

emen

t of s

ingu

lar e

ntro

py-fi

rst d

eriv

ativ

e

System orders



0 1 2 3 4 5 6 7 8 9 100

50

100

150

200

250

300

Syste

m o

rder

sFrequency (Hz)


0 1 2 3 4 5 6 7 8 9 10Frequency (Hz)

0

50

100

150

200

250

300

Syste

m o

rder

s


0

50

100

150

05 1 15 2 25 3 35 4 45 5

018508023776

046347055536

117211859

2485331657

3635643286

50873

Syet

em o

rder

s

Frequency (Hz)

070871








1 2 3 4 5 6 7 8 9 10 1101020ndash1

0

1

(a)

2 4 6 8 100

1020ndash1

01

(b)

2 4 6 8 100

1020ndash1

01

(c)




(a)

(b)

(c)




6 Conclusion








Data Availability




Acknowledgments


References












02

03

04

05

06

07

08

Freq

uenc

y (H

z)

Third order



















0 1 2 3 4 5 6 7 8 9 100

50

100

150

200

250

300

Syste

m o

rder

sFrequency (Hz)


0 1 2 3 4 5 6 7 8 9 10Frequency (Hz)

0

50

100

150

200

250

300

Syste

m o

rder

s


0

50

100

150

05 1 15 2 25 3 35 4 45 5

018508023776

046347055536

117211859

2485331657

3635643286

50873

Syet

em o

rder

s

Frequency (Hz)

070871








1 2 3 4 5 6 7 8 9 10 1101020ndash1

0

1

(a)

2 4 6 8 100

1020ndash1

01

(b)

2 4 6 8 100

1020ndash1

01

(c)




(a)

(b)

(c)




6 Conclusion








Data Availability




Acknowledgments


References












02

03

04

05

06

07

08

Freq

uenc

y (H

z)

Third order






















1 2 3 4 5 6 7 8 9 10 1101020ndash1

0

1

(a)

2 4 6 8 100

1020ndash1

01

(b)

2 4 6 8 100

1020ndash1

01

(c)




(a)

(b)

(c)




6 Conclusion








Data Availability




Acknowledgments


References












02

03

04

05

06

07

08

Freq

uenc

y (H

z)

Third order




















6 Conclusion








Data Availability




Acknowledgments


References












02

03

04

05

06

07

08

Freq

uenc

y (H

z)

Third order

































Documents

OnlineIntelligentIdentificationofModalParametersforLarge ...downloads.hindawi.com/journals/sv/2020/2040216.pdfnoise in a vibration signal can be reduced in the data ac-quisition stage