Research Article Acoustical Source Tracing Using ...downloads.hindawi.com/journals/sv/2015/571206.pdf · Acoustical Source Tracing Using Independent Component Analysis and Correlation

Research ArticleAcoustical Source Tracing Using Independent ComponentAnalysis and Correlation Analysis

Wei Cheng Zhousuo Zhang Jie Zhang and Jiantao Lu

State Key Laboratory for Manufacturing System Engineering Xirsquoan Jiaotong University Xirsquoan 710049 China

Correspondence should be addressed to Wei Cheng chengwmailxjtueducn

Received 4 January 2015 Accepted 25 February 2015

Academic Editor Yumin He

Copyright copy 2015 Wei Cheng et al This 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

Acoustical signals from mechanical systems reveal the operating conditions of mechanical components and thus benefit formachinery condition monitoring and fault diagnosis However the acoustical signals directly measured by the sensors in essentialare the mixed signals of all the sources and normally it is very difficult to be used for source identification or operating featureextraction Therefore this paper studies the acoustical source tracing problem using independent component analysis (ICA) andidentifies the sources using correlation analysis the measured acoustical signals are separated into independent componentsby independent component analysis method and thus all the independent information of all the sources is obtained theseindependent components are identified based on the prior information of the sources and correlation analysis Therefore all thesource information contained in the measured acoustical signals can be independently separated and traced which can providemore purer source information for condition monitoring and fault diagnosis

1 Introduction

Vibration and acoustical signals caused by the collision andfriction of mechanical components normally can reduce theoperational precision and even shorten the service life of themachineryHowever the vibration and acoustical signals pro-vide important information of the operating conditions andthusmachinery conditionmonitoring and fault diagnosis canbe carried out based on a nondestructive measurement Asthemeasured acoustical signals are normallymixed signals ofall the sources and noises they have great significance to tracethe source information from the measured signals and thusprovide pure and independent information of mechanicalcomponents for an effective condition monitoring or faultdiagnosis

Generally the acoustical signals directly measured bythe sensors are complicated and rough information of themechanical systems because they are mixed signals of allthe sources and the transmission effects of the transmissionpaths Recently many researchers have devoted their effortsto transmission effects of vibration and acoustical signals Xinet al [1] investigated the vibroacoustic performance of a rect-angular double-panel partition with enclosed air cavity and

simply mounted on an infinite acoustic rigid baffle HuangandNutt [2] presented an analytical study of sound transmis-sion through unbounded panels of functionally gradedmate-rials Bravo et al [3] analyzed the sound absorption and trans-mission properties ofmicroperforated panels backed by an aircavity and a thin plate Qian et al [4] described an approachon the prediction of sound transmission loss for a finite sand-wich panel with honeycomb core Posson et al [5] developedan analytical model for the sound transmission through anannular stator row in a configuration without mean flow Leeand Kim [6] investigated the effects of structure on soundabsorption and transmission loss of a composite sheet Chan-dra et al [7] analyzed the vibroacoustic and transmissionloss characteristics of functionally graded material platesAll these articles studied the transmission characteristics ofstructures and can help to carry out a passive vibration andsound monitoring and control However normally it is achallenging task and costs plenty of time to build a precisemodel of acoustical transmission for complicatedmechanicalsystems and thus the engineering applications are limited

To reveal the operating conditions of the mechanicalcomponents signal processing methods are developed toextract the features hidden in the noisy response signals and

Hindawi Publishing CorporationShock and VibrationVolume 2015 Article ID 571206 8 pageshttpdxdoiorg1011552015571206

2 Shock and Vibration

thus benefit for system analysis condition monitoring orfault diagnosis To clearly reveal the operating conditions ofeach mechanical component blind source separation (BSS)and independent component analysis (ICA) are developedto separate the mixed signals into uncorrelated componentswithout the knowledge of the sources and their mixingmodes Jutten and Herault [8] solved blind source separationproblem with an adaptive algorithm based on neuromimeticarchitecture Belouchrani et al [9] proposed a techniqueto blindly separate the sources using high-order statisticsComon [10] reexamined the concept of independent compo-nent analysis (ICA) and proposed an efficient ICA algorithmHyvarinen [11 12] proposed a very famous ICA algorithmcalled fast-fixed algorithm for independent component anal-ysis which is very effective and efficient for linear systemsSince then independent component analysis has attracteda worldwide attention and has been widely used in manyengineering applications such as image processing [13]EEGMEG analysis [14] communications [15] radar [16]and modal parameter estimation [17] In recent years inde-pendent component analysis has been applied to mechanicalsignal processing and some practical algorithms have beenconstructed Antoni [18] addressed the issues of blind separa-tion of vibration components Cheng et al studied the sourcenumber estimation [19] source separation [20] and sourcecontribution evaluation [21] methods for mechanical systemsbased on an effective source separation Henriquez et al [22]provided a review of automatic fault diagnosis using indepen-dent component analysis and vibration signals In this paperindependent component analysis is applied to acousticalsource tracing (acoustical source separation and identifi-cation) and the separation performances of independentcomponent analysis for acoustical signals are quantitativelyevaluated by a numerical case study and an experimentalstudy on a mechanical system with shell structures

The remainder of this paper is organized as follows InSection 2 basic theory and key principals of independentcomponent analysis are introduced In Section 3 the sepa-ration performance of the independent component analysisalgorithm is tested by a numerical case study In Section 4the independent component analysis algorithm is applied toseparate the acoustical signals and then acoustical sourcetracing is carried out based on the source separation andcorrelation analysis In Section 5 the conclusions of this studyare summarized

2 Theory of Independent Component Analysis

21 PrincipalTheory of Blind Source Separation Blind sourceseparation (BSS) including independent component analysis(ICA) is a powerful tool to separate a set of source signalsS(119905) = [s

1(119905) s2(119905) s

119899(119905)]119879 from the mixed signals X(119905)=

[x1(119905) x2(119905) x

119898(119905)]119879 without any information of the sour-

ces and their mixing approaches [8 23 24] which can bedescribed as follows

x119894(119905) =

119899

sum119895=1

119886119894119895s119895(119905) 119894 = 1 2 119898 119895 = 1 2 119899

X = AS Matrix form

(1)

where119898 is the number of the mixed signals 119899 is the numberof the sources x

119894is the 119894th mixed signal collected at location

119894 s119895is the 119895th source signal 119886

119894119895is a mixing coefficient and

A = 119886119894119895 is a mixing matrix

The key process of BSS or ICA is to find a separatingmatrixW= [120596

11205962 120596

119899]119879 and recover the estimated sour-

ces Y(119905) = [y1(119905) y2(119905) y

119899(119905)]119879 by

Y = WX = W (AS) (2)

22 Measures of Independence To effectively separate themixed signals into independent components some quanti-tative measures of the independence for a random variableshould be developed A well-known theory of independenceis ldquonon-Gaussian is independentrdquo and a classical measure ofnon-Gaussianity is kurtosis or the fourth-order cumulantThe kurtosis of y (zero-mean and unit-variance) is classicallydefined by

kurt (y) = 119864 y4 minus 3 (119864 y2)2

(3)

The main reason of kurtosis as a widely used measure ofindependence is its computational and theoretical simplicityComputationally kurtosis can be estimated simply by usingthe fourth moment of the sample dataTheoretically kurtosishas good performances for linear transform

kurt (y1+ y2) = kurt (y

1) + kurt (y

2)

kurt (120572y) = 1205724kurt (y)

(4)

where 120572 is a scalarAnother and very important measure of non-Gaussianity

is given by negentropy which is based on the informationtheoretical quantity of entropy The more ldquorandomrdquo thevariable is the larger its entropy will be Entropy is normallydefined as

119867(y) = minusint119891 (y) log119891 (y) 119889y (5)

where 119891(y) is the density of a random vector yNegentropy is used to obtain an effective measure of non-

Gaussianity and can be defined as follows

119869 (y) = 119867 (ygauss) minus 119867 (y) (6)

where ygauss is a Gaussian variable of the same covariancematrix as y

The advantage of using negentropy as an effective mea-sure of non-Gaussianity is that it is well justified by sta-tistical theory However the estimation of negentropy isdifficult and therefore some approximations have to be

Shock and Vibration 3

used to simplify the calculation The classical method ofapproximating negentropy is using higher-order moments

119869 (y) asymp 1

12119864 y3

2

+1

48kurt (y)2 (7)

23 Efficient ICA Algorithm

231 Centering Themost basic and necessary process beforean effective source separation is to center x

x = x minusmean (x) (8)

232 Whitening One popular method for whitening is touse the eigenvalue decomposition (EVD) of the covariancematrix 119864xx119879 = EDE119879 where E is the orthogonal matrixof eigenvectors of 119864xx119879 and D is the diagonal matrix of itseigenvaluesD = diag(119889

1 119889119899)

x = EDminus12E119879x (9)

whereDminus12 = diag(119889minus121

119889minus12119899

)

233 Fixed Point Iteration Scheme for One Component

(1) Choose an initial weight vector w

(2) Let w+ = 119864x119892(w119879x) minus 1198641198921015840(w119879x)w where 119892is the derivatives of the nonquadratic function andnormally has

1198921(u) = tanh (119886u) 119892

2(u) = u exp(minusu

2

2) (10)

where 1 le 119886 le 2 is a suitable constant

(3) Let w = w+w+

(4) If not converged go back to (2)

234 Fixed Point Iteration Scheme for Several Components

(1) Let w119901+1

= w119901+1

minus sum119901

119895=1w119879119901+1

w119895w119895

(2) Let w119901+1

= w119901+1

radicw119879119901+1

w119901+1

Comparing the separating vector w for one componentthe separating matrixW = (w

1 w

119899)119879 for several compo-

nents can be obtained as follows

W =W

radic1003817100381710038171003817WW1198791003817100381710038171003817

W =3

2W minus

1

2WW119879W

(11)

24 Source Identification For mechanical systems someinformation of the sources normally can be obtained bytheory study or instructionsTherefore waveform correlationcan be used to identify the source information by a correla-tion analysis between the sources and the separated signalsFor discrete signals s and y waveform correlation coefficient120588119882is defined as

120588119882=

max119877sy (119896)

radicmax119877ss (119896) timesmax119877yy (119896) 0 le 119896 le 119879 (12)

where 119877sy(119896) = sum119879minus119896minus1

119894=0s(119894 + 119896)y(119894) 119877ss(119896) = sum

119879minus119896minus1

119894=0s(119894 +

119896)s(119894) 119877yy(119896) = sum119879minus119896minus1

119894=0y(119894 + 119896)y(119894) forall119896 ge 0 and 119879 is the data

length

3 Numerical Case Study

In this section typical vibration and acoustical signals ofmechanical systems are artificially generated to test the sep-aration performance of the ICA algorithm based on negen-tropy Since acoustical signals transmit from the sources tothe measured points through air the mixing mode of all theacoustical source signals tends to be a linear superpositionFurthermore the generating mechanism of different sourcesis far from each other Therefore all the sources are consid-ered to be linearly mixed and independent of each other

The given source signals are typical vibration and acous-tical signals of mechanical systems signal s

1(119905) is a sinusoidal

wave that simulates a vibration signal of rotational equip-ment signal s

2(119905) is a periodic wave of oscillating attenuation

that simulates mechanical shocks signal s3(119905) is a periodic

wave that simulates amplitude modulation signal s4(119905) is

a periodic wave that simulates frequency modulation andsignal s

5(119905) is a white noise that simulates noises produced

by environment and structural transmission The generatingfunctions of sources are as follows

S (119905) =[[[[[

[

s1(119905)

s2(119905)

s3(119905)

s4(119905)

s5(119905)

]]]]]

]

=

[[[[[[[[

[

sin (15120587119905)

sum sin (1000 (119905 minus 01119899 minus 002)) exp (minus150 (119905 minus 01119899 minus 002)

2) 119906 (119905 minus 01119899 minus 002)

sin (10120587119905) cos (150120587119905)sin (6120587 cos (6120587119905))

119899 (119905)

]]]]]]]]

]

(13)

where 119906(119905) is a step function


01

Ag

Ag

Ag

05

01

0 02 04 06 08 1

04

t (s)

0 02 04 06 08 1t (s)

0 02 04 06 08 1t (s)

0 02 04 06 08 1t (s)

0 02 04 06 08 1t (s)

minus4minus4

minus1

Ag

Ag

01

minus1

minus05

minus1

s 1s 2

s 3s 4

s 5

Figure 1 Waveforms of the source signals

Generally the measured acoustical signals are mixedsignals of all the sources with noises Furthermore as theacoustical signals transmit from the sources to themeasuringpoints through air linear superposition is applied to arti-ficially produce the mixed signals The mixing matrix A israndomly generated as the following matrix

A =

[[[[[[[[[[[

[

023 minus058 073 minus081 095

028 minus089 049 minus075 097

056 015 minus072 058 099

minus038 minus052 045 061 099

minus048 038 084 minus051 098

088 minus061 054 minus020 095

]]]]]]]]]]]

]

(14)

The waveforms of the source signals and the mixedsignals are shown in Figures 1 and 2 Obviously all thesource signals have typical waveform features However it isvery difficult to directly identify the features of the mixedsignals as the complicated waveforms Therefore generallysignal processing methods are required to help to extractthe desired features clearly In this numerical case study theICA algorithm based on negentropy is applied to separatethe mixed signals into 5 independent components and thenthese separated components are identified by the correlationanalysis

With the ICA algorithm 6 mixed signals are separatedinto 5 independent components and the waveforms of theseindependent components are shown in Figure 3 ComparingFigure 3 with Figure 1 obviously the waveform features ofeach source signal are well separated Waveform correlationanalysis is used to quantitatively evaluate the separationperformances of the ICA algorithm and identify the sources

0 02 04 06 08 1t (s)

0 02 04 06 08 1t (s)

0 02 04 06 08 1t (s)

0 02 04 06 08 1t (s)

0 02 04 06 08 1t (s)

0 02 04 06 08 1t (s)

Ag 0

4

minus4

x 6Ag 0

4

minus4

x 5Ag 0

4

minus4

x 4Ag 0

4

minus4

x 3Ag 0

4

minus4

x 2Ag 0

4

minus4

x 1

Figure 2 Waveforms of the mixed signals

and the waveform correlation coefficients 120588119882

are shown inthe following matrix

Ω119882=

[[[[[[[[

[

099 001 004 000 004

001 099 003 017 012

003 003 099 001 009

000 015 001 099 003

005 012 009 004 099

]]]]]]]]

]

(15)

Each correlation coefficient 120588119894119895inΩ119882= 120588119894119895 indicates the

similarity between the separated component 119894 and the source119895 From the correlation matrix Ω

119882= 120588119894119895 the elements

in the diagonal show the correlation coefficients betweenthe separated components and related source signals thecorrelation coefficient of s

1(s2 s3 s4 and s

5) and related

separated component 1 (2 3 4 and 5) is 099 which indicatesthat all the source information of s

1(s2 s3 s4 and s

5) has

been completely separated from themixed signals by the ICAalgorithm The high correlation coefficients indicate that theICA algorithm can effectively separate the mixed signals intoindependent components for the given sources and linearlymixed mode However the correlation coefficients betweenthe independent components and the unrelated sources areall less than 017 which indicates that all the separatedcomponents have good independent properties Thereforethe ICA algorithm based on negentropy is applied to extractthe source information from the measured acoustical signalsfrom real mechanical systems and the separated information


0 02 04 06 08 1t (s)

0 02 04 06 08 1t (s)

0 02 04 06 08 1t (s)

0 02 04 06 08 1t (s)

0 02 04 06 08 1t (s)

Ag 0

minus4

Ag 0

2

minus2

Ag 0

2

minus2

Ag 0

5

minus5

Ag 1

minus1

y 4y 3

y 2y 1

y 5

Figure 3 Separated components by DSS

is used to validate the effectiveness of ICAalgorithmand tracethe source information hidden in the measured signals

4 Experimental Study

41 Introductions of the Test Bed A test bed of a mechan-ical system with shell structures is constructed to test theseparation performances of the ICA algorithm based onnegentropy which is composed of four components an endcover a shell two clapboards and supports Rubber airsprings are used to support the whole test bed and reducethe effects of the ground vibrations and environmental noisesThree acoustical sources are designed in the test bed twoloudspeakers controlled by the signal generators and onemotor controlled by the frequency converter The structureof the test bed is shown in Figure 4

Six sound pressure sensors are used to measure theacoustical signals and they are located in six directions ofthe test bed with a distance of 500 millimeters HBM Gen2idata acquisition system is applied to collect the acousticaldata from these six sensors The framework of the measuringsystem is shown in Figure 5 and the test parameters areshown in Table 1

42 Acoustical Signals of the Test Bed The acoustical signalsmeasured by the six sound pressure sensors are the mixedsignals of all the sources as they are working together Inthe experimental study the acoustical signals from sensors1 4 and 6 are used as the mixed signals so as to satisfythe assumptions of ICA the number of the mixed signalsshould be no less than the number of the source signalsFurthermore the directions of these sensors represent adiversified mixing mode of all the sources

(a) (b) (c) (d) (e) (f) (g) (h) (i)

(a) End cover(b) Loudspeaker I(c) Left clapboard(d) Loudspeaker II(e) Shell

(f) Motor(g) Right clapboard(h) Rubber springs(i) Supports

Figure 4 The structure of the test bed

The test bedHBM Gen2i

data acquisition

Sound pressures6

s5s4

s3

s2

s1

500mm

500mm

system

sensor

Figure 5 The framework of the measuring system

Table 1 The test parameters of the measuring system

Parameters Values and unitsSound pressure sensors 6HBM Gen2i data acquisition system 1Test bed with shell structure 1Sampling frequency 10240HzSampling length 10 secondsRotational speed of motor 1800 rmin (20Hz)Frequency of Loudspeaker I 1750HzFrequencies of Loudspeaker II 500 1700 and 2000Hz

The waveforms and spectrums of the mixed signals areshown in Figures 6 and 7 respectively From waveforms inFigure 6 it is difficult to identify the waveform features of thesource signals except some periodic waves which indicatesthat the waveforms of the mixed signals are complicated andnormally signal processing method is required to extract thedesired features From spectrums in Figure 7 some majorcomponents of 20 1100 1700 1750 and 2000Hz are clearwhich represent the characteristic features of the sourcesas the experimental settings Generally the independentinformation of the source signals cannot be directly identifiedfrom the measured mixed signals as each mixed signalcontains all the major components of the source signals


0 0005 001 0015 002 0025

0 0005 001 0015 002 0025

0 0005 001 0015 002 0025

t (s)

t (s)

t (s)

Ag 0

0204

minus02

x 3Ag 0

0204

minus02minus04

x 2Ag 0

0204

minus02minus04

x 1


0 500 1000 1500 2000 25000

002004006

0 500 1000 1500 2000 25000

002004006

0 500 1000 1500 2000 25000

002004

f (Hz)

f (Hz)

f (Hz)

2000

20

20

20

1100

11001750

17001100

1750

Agx 1

Agx 2

Agx 3

Figure 7 Spectrums of the mixed signals

43 Acoustical Source Separation The ICA algorithm basedon negentropy is applied to separate the mixed signals intoindependent components and 3 independent componentsare extracted from the given mixed signals The waveformsand spectrums of the independent components are shownin Figures 8 and 9 respectively Figure 8 clearly shows thatthe waveform of the separated component y

1has typically

periodic and amplitude-modulation features and the basiccomponents are sine waves which are normally caused by theeccentric vibration of mechanical systems The spectrums inFigure 9 also show that the separated component y

1hasmajor

components of 20 and 2000Hz The waveform of the sepa-rated component y

2has typical features of sine waves and its

spectrum also clearly shows themajor component of 1750HzThe waveform of the separated component y

3has an obvious

feature of sine waves with a periodic amplitude-modulation

0 0005 001 0015 002 0025

0 0005 001 0015 002 0025

0 0005 001 0015 002 0025

t (s)

t (s)

t (s)

Ag 0

2

minus2

y 1Ag 0

1

minus1

y 2Ag 0

12

minus1

y 3

Figure 8 Waveforms of separated components

0 500 1000 1500 2000 25000

010203

0 500 1000 1500 2000 2500

0 500 1000 1500 2000 2500

110020

1750

20005001700

f (Hz)

f (Hz)

f (Hz)

Agy 1

0010203

Agy 2

0

02

04

Agy 3

Figure 9 Spectrums of separated components

and its spectrum contains three major components of 5001700 and 2000Hz

Comparing the spectrums of the separated componentswith the parameters of the experimental settings it canbe speculated that the separated component y

1represents

the typical feature of the source 1 from a motor whilethe separated components y

2and y

3represent the typical

features of the sources 2 and 3 from Loudspeakers I and IIrespectively However this is just based on the parameters ofthe experimental settings and it is still not a convincing andreliable source tracing method

44 Acoustical Source Tracing andValidation To intelligentlytrace the sources and validate the effectiveness of the ICAalgorithm in real mechanical systems acoustical sourcesignals are measured independently by the closest sensors inthe condition that only one source is working with the given


0 0005 001 0015 002 0025

0 0005 001 0015 002 0025

0 0005 001 0015 002 0025

t (s)

t (s)

t (s)

00204

minus02

Ags 1

0

0

01

minus01

Ags 2

01

minus01

Ags 3


0 500 1000 1500 2000 25000

005

01

0 500 1000 1500 2000 25000

002004

0 500 1000 1500 2000 25000

001002

1100

1750

1700 2000500

20Ags 1

Ags 2

Ags 3

f (Hz)

f (Hz)

f (Hz)

Figure 11 Spectrums of the source signals

parametersThe independent source waveforms from sensors1 4 and 6 in the condition that only the motor LoudspeakerI or Loudspeaker II is working with the given experimentalsettings are shown in Figure 10 and their spectrums areshown in Figure 11

Comparing the waveforms and spectrums of the sourcesignals with those of the independent components thewaveforms of the independent components are very similarto those of the related source signals the independentcomponent y

1and the source signal s

1have a typical feature

of sine waves with an uncertainty amplitude modulationwhich can be caused by the eccentric vibration of the motorand the uncertainty amplitude modulation can be causedby the rubbing the independent component y

2and the

source signal s2have the same major component of 1750Hz

which agrees with the experimental settings the independentcomponent y


3have three samemajor

components of 500 1700 and 2000Hz as the experimentalsettings Therefore all the major components of the sourcesignals can be effectively traced by the ICA algorithm basedon negentropy

To quantitatively and intelligently trace the sources thewaveform correlation analysis is used to evaluate the simi-larity between independent components and the sources Allthe independent components are made a correlation analysiswith all the sources and the correlation coefficients are listedin the correlation matrixΩ

119882

Ω119882=[[

[

090 007 006

005 076 008

010 016 085

]]

]

(16)

The correlation matrix Ω119882

shows that the correlationcoefficients between the independent components and therelated sources are 090 076 and 085 which indicatehigh correlation coefficients and high similarity between theindependent components and the related sources (Liu et al[25] obtained waveform correlation coefficients of 077 plusmn

003 for ECG signals with noises and Farina et al [26]obtained correlation coefficients of 070 plusmn 009 for nonsta-tionary surface myoelectric signals) while the correlatedcoefficients between the independent components and theunrelated sources are less than 016 which indicates thatall the independent components and all the sources havegood independence property Therefore a threshold 120574 canbe set as 120574 isin (016 076) (in practice 120574 isin (050 070)) tointelligently identify and trace the acoustical sources and thehigh correlation coefficients between the independent com-ponents and the related sources validate the effectiveness ofthe ICA algorithm based on negentropy in source separationand acoustical source tracing

5 Conclusions

This paper presents the fundamental theory of blind sourceseparation and key principals of independent componentanalysis and validates the effectiveness of the ICA algorithmbased on negentropy according to a numerical case studyand an experimental study on a mechanical system with shellstructures The experimental study indicates that the acous-tical sources can be effectively separated and intelligentlytraced

In the numerical case study five typical acoustical sourcesignals of mechanical systems are effectively separated fromsix linearly mixed signals and the correlation coefficientsbetween the independent components and the related sourcesignals are all more than 099 which indicates a completeand highly effective source separation of the ICA algorithmfor the given mixed signals while in the experimental studyon a mechanical system with shell structures the correlationcoefficients between the independent components and therelated source signals are all more than 076 which alsoreveals an effective acoustical source separation If artificiallygiving a threshold 120574 isin (050 070) for the correlation coef-ficients all the acoustical sources can be intelligently identi-fied and traced


This work can provide pure source information for ma-chinery condition monitoring and fault diagnosis and theindependent source information can also be of benefit fornoise identification reduction and control

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the projects of National NatureScience Foundation of China (no 51305329 51035007) theChina Postdoctoral Science Foundation (nos 2013M532032and 2014T70911) the Doctoral Foundation of EducationMinistry of China (no 20130201120040) and the ShaanxiPostdoctoral Scientific Research Project

References

[1] F X Xin T J Lu and C Q Chen ldquoSound transmission throughsimply supported finite double-panel partitions with enclosedair cavityrdquo Journal of Vibration andAcoustics vol 132 no 1 2010

[2] C Z Huang and S Nutt ldquoAn analytical study of soundtransmission through unbounded panels of functionally gradedmaterialsrdquo Journal of Sound and Vibration vol 330 no 6 pp1153ndash1165 2011

[3] T Bravo C Maury and C Pinhede ldquoSound absorption andtransmission through flexible micro-perforated panels backedby an air layer and a thin platerdquo Journal of the Acoustical Societyof America vol 131 no 5 pp 3853ndash3863 2012

[4] Z C Qian D Q Chang B L Liu and K Liu ldquoPrediction ofsound transmission loss for finite sandwich panels based ona test procedure on beam elementsrdquo Journal of Vibration andAcoustics Transactions of the ASME vol 135 no 6 2013

[5] H Posson H Beriot and S Moreau ldquoOn the use of an ana-lytical cascade response function to predict sound transmissionthrough an annular cascaderdquo Journal of Sound and Vibrationvol 332 no 15 pp 3706ndash3739 2013

[6] B C Lee and S R Kim ldquoEffect of structure on sound absorptionand sound transmission loss of composite sheetrdquo AdvancedComposite Materials vol 23 no 4 pp 319ndash325 2014

[7] N Chandra S Raja and K V N Gopal ldquoVibro-acousticresponse and sound transmission loss analysis of functionallygraded platesrdquo Journal of Sound and Vibration vol 333 no 22pp 5786ndash5802 2014

[8] C Jutten and J Herault ldquoBlind separation of sources part I anadaptive algorithm based on neuromimetic architecturerdquo SignalProcessing vol 24 no 1 pp 1ndash10 1991

[9] A Belouchrani K Abed-Meraim J-F Cardoso and EMoulines ldquoA blind source separation technique using second-order statisticsrdquo IEEE Transactions on Signal Processing vol 45no 2 pp 434ndash444 1997

[10] P Comon ldquoIndependent component analysis a new conceptrdquoSignal Processing vol 36 no 3 pp 287ndash314 1994

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

[12] A Hyvarinen and E Oja ldquoIndependent component analysisalgorithms and applicationsrdquo Neural Networks vol 13 no 4-5pp 411ndash430 2000

[13] D-M Tsai and S-C Lai ldquoIndependent component analysis-based background subtraction for indoor surveillancerdquo IEEETransactions on Image Processing vol 18 no 1 pp 158ndash167 2009

[14] A Hyvarinen P Ramkumar L Parkkonen and R Hari ldquoInde-pendent component analysis of short-time Fourier transformsfor spontaneous EEGMEG analysisrdquo NeuroImage vol 49 no1 pp 257ndash271 2010

[15] J Gao X Zhu and A K Nandi ldquoIndependent componentanalysis for multiple-input multiple-output wireless communi-cation systemsrdquo Signal Processing vol 91 no 4 pp 607ndash6232011

[16] J Ji ldquoRobust approach to independent component analysis forSAR image analysisrdquo IET Image Processing vol 6 no 3 pp 284ndash291 2012

[17] R Rodrıguez-Rocha F J Rivero-Angeles and E Gomez-Ramırez ldquoApplication of the baseline stiffness method fordamage detection of an RC building without baseline modalparameters utilizing the independent component analysis formodal extractionrdquo International Journal of Structural Stabilityand Dynamics vol 13 no 3 Article ID 1250074 2013

[18] J Antoni ldquoBlind separation of vibration components principlesand demonstrationsrdquoMechanical Systems and Signal Processingvol 19 no 6 pp 1166ndash1180 2005

[19] W Cheng S Lee Z S Zhang and Z J He ldquoIndependentcomponent analysis based source number estimation and itscomparison for mechanical systemsrdquo Journal of Sound andVibration vol 331 no 23 pp 5153ndash5167 2012

[20] W Cheng Z J He and Z S Zhang ldquoA comprehensive studyof vibration signals for a thin shell structure using enhancedindependent component analysis and experimental validationrdquoJournal of Vibration and Acoustics vol 136 no 4 Article ID041011 2014

[21] W Cheng Z S Zhang S Lee and Z J He ldquoSource contribu-tion evaluation of mechanical vibration signals via enhancedindependent component analysisrdquo Transactions of the ASMEJournal of Manufacturing Science and Engineering vol 134 no2 Article ID 021014 2012

[22] P Henriquez J B Alonso M A Ferrer and C M TraviesoldquoReview of automatic fault diagnosis systems using audio andvibration signalsrdquo IEEE Transactions on Systems Man andCybernetics Systems vol 44 no 5 pp 642ndash652 2014

[23] P ComonC Jutten and JHerault ldquoBlind separation of sourcesPart II problems statementrdquo Signal Processing vol 24 no 1 pp11ndash20 1991

[24] E Sorouchyari ldquoBlind separation of sources Part III Stabilityanalysisrdquo Signal Processing vol 24 no 1 pp 21ndash29 1991

[25] H Liu C Q Chang K D K Luk and Y Hu ldquoComparison ofblind source separation methods in fast somatosensory-evokedpotential detectionrdquo Journal of Clinical Neurophysiology vol 28no 2 pp 170ndash177 2011

[26] D Farina C Fevotte C Doncarli and R Merletti ldquoBlindseparation of linear instantaneous mixtures of nonstationarysurface myoelectric signalsrdquo IEEE Transactions on BiomedicalEngineering vol 51 no 9 pp 1555ndash1567 2004

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



thus benefit for system analysis condition monitoring orfault diagnosis To clearly reveal the operating conditions ofeach mechanical component blind source separation (BSS)and independent component analysis (ICA) are developedto separate the mixed signals into uncorrelated componentswithout the knowledge of the sources and their mixingmodes Jutten and Herault [8] solved blind source separationproblem with an adaptive algorithm based on neuromimeticarchitecture Belouchrani et al [9] proposed a techniqueto blindly separate the sources using high-order statisticsComon [10] reexamined the concept of independent compo-nent analysis (ICA) and proposed an efficient ICA algorithmHyvarinen [11 12] proposed a very famous ICA algorithmcalled fast-fixed algorithm for independent component anal-ysis which is very effective and efficient for linear systemsSince then independent component analysis has attracteda worldwide attention and has been widely used in manyengineering applications such as image processing [13]EEGMEG analysis [14] communications [15] radar [16]and modal parameter estimation [17] In recent years inde-pendent component analysis has been applied to mechanicalsignal processing and some practical algorithms have beenconstructed Antoni [18] addressed the issues of blind separa-tion of vibration components Cheng et al studied the sourcenumber estimation [19] source separation [20] and sourcecontribution evaluation [21] methods for mechanical systemsbased on an effective source separation Henriquez et al [22]provided a review of automatic fault diagnosis using indepen-dent component analysis and vibration signals In this paperindependent component analysis is applied to acousticalsource tracing (acoustical source separation and identifi-cation) and the separation performances of independentcomponent analysis for acoustical signals are quantitativelyevaluated by a numerical case study and an experimentalstudy on a mechanical system with shell structures

The remainder of this paper is organized as follows InSection 2 basic theory and key principals of independentcomponent analysis are introduced In Section 3 the sepa-ration performance of the independent component analysisalgorithm is tested by a numerical case study In Section 4the independent component analysis algorithm is applied toseparate the acoustical signals and then acoustical sourcetracing is carried out based on the source separation andcorrelation analysis In Section 5 the conclusions of this studyare summarized

2 Theory of Independent Component Analysis

21 PrincipalTheory of Blind Source Separation Blind sourceseparation (BSS) including independent component analysis(ICA) is a powerful tool to separate a set of source signalsS(119905) = [s

1(119905) s2(119905) s

119899(119905)]119879 from the mixed signals X(119905)=

[x1(119905) x2(119905) x

119898(119905)]119879 without any information of the sour-

ces and their mixing approaches [8 23 24] which can bedescribed as follows

x119894(119905) =

119899

sum119895=1

119886119894119895s119895(119905) 119894 = 1 2 119898 119895 = 1 2 119899

X = AS Matrix form

(1)

where119898 is the number of the mixed signals 119899 is the numberof the sources x

119894is the 119894th mixed signal collected at location

119894 s119895is the 119895th source signal 119886

119894119895is a mixing coefficient and

A = 119886119894119895 is a mixing matrix

The key process of BSS or ICA is to find a separatingmatrixW= [120596

11205962 120596

119899]119879 and recover the estimated sour-

ces Y(119905) = [y1(119905) y2(119905) y

119899(119905)]119879 by

Y = WX = W (AS) (2)

22 Measures of Independence To effectively separate themixed signals into independent components some quanti-tative measures of the independence for a random variableshould be developed A well-known theory of independenceis ldquonon-Gaussian is independentrdquo and a classical measure ofnon-Gaussianity is kurtosis or the fourth-order cumulantThe kurtosis of y (zero-mean and unit-variance) is classicallydefined by

kurt (y) = 119864 y4 minus 3 (119864 y2)2

(3)

The main reason of kurtosis as a widely used measure ofindependence is its computational and theoretical simplicityComputationally kurtosis can be estimated simply by usingthe fourth moment of the sample dataTheoretically kurtosishas good performances for linear transform

kurt (y1+ y2) = kurt (y

1) + kurt (y

2)

kurt (120572y) = 1205724kurt (y)

(4)

where 120572 is a scalarAnother and very important measure of non-Gaussianity

is given by negentropy which is based on the informationtheoretical quantity of entropy The more ldquorandomrdquo thevariable is the larger its entropy will be Entropy is normallydefined as

119867(y) = minusint119891 (y) log119891 (y) 119889y (5)

where 119891(y) is the density of a random vector yNegentropy is used to obtain an effective measure of non-

Gaussianity and can be defined as follows

119869 (y) = 119867 (ygauss) minus 119867 (y) (6)

where ygauss is a Gaussian variable of the same covariancematrix as y

The advantage of using negentropy as an effective mea-sure of non-Gaussianity is that it is well justified by sta-tistical theory However the estimation of negentropy isdifficult and therefore some approximations have to be



119869 (y) asymp 1

12119864 y3

2

+1

48kurt (y)2 (7)





1 119889119899)



119889minus12119899

)




1198921(u) = tanh (119886u) 119892

2(u) = u exp(minusu

2

2) (10)


(3) Let w = w+w+



(1) Let w119901+1

= w119901+1

minus sum119901

119895=1w119879119901+1

w119895w119895

(2) Let w119901+1

= w119901+1

radicw119879119901+1

w119901+1


1 w



W =W

radic1003817100381710038171003817WW1198791003817100381710038171003817

W =3

2W minus

1

2WW119879W

(11)


120588119882=

max119877sy (119896)



119894=0s(119894 + 119896)y(119894) 119877ss(119896) = sum


119894=0s(119894 +



length












S (119905) =[[[[[

[

s1(119905)

s2(119905)

s3(119905)

s4(119905)

s5(119905)

]]]]]

]

=

[[[[[[[[

[

sin (15120587119905)


2) 119906 (119905 minus 01119899 minus 002)

sin (10120587119905) cos (150120587119905)sin (6120587 cos (6120587119905))

119899 (119905)

]]]]]]]]

]

(13)



01

Ag

Ag

Ag

05

01

0 02 04 06 08 1

04

t (s)

0 02 04 06 08 1t (s)

0 02 04 06 08 1t (s)

0 02 04 06 08 1t (s)

0 02 04 06 08 1t (s)

minus4minus4

minus1

Ag

Ag

01

minus1

minus05

minus1

s 1s 2

s 3s 4

s 5



A =

[[[[[[[[[[[

[

023 minus058 073 minus081 095

028 minus089 049 minus075 097

056 015 minus072 058 099

minus038 minus052 045 061 099

minus048 038 084 minus051 098

088 minus061 054 minus020 095

]]]]]]]]]]]

]

(14)



0 02 04 06 08 1t (s)

0 02 04 06 08 1t (s)

0 02 04 06 08 1t (s)

0 02 04 06 08 1t (s)

0 02 04 06 08 1t (s)

0 02 04 06 08 1t (s)

Ag 0

4

minus4

x 6Ag 0

4

minus4

x 5Ag 0

4

minus4

x 4Ag 0

4

minus4

x 3Ag 0

4

minus4

x 2Ag 0

4

minus4

x 1




Ω119882=

[[[[[[[[

[

099 001 004 000 004

001 099 003 017 012

003 003 099 001 009

000 015 001 099 003

005 012 009 004 099

]]]]]]]]

]

(15)



119882= 120588119894119895 the elements


1(s2 s3 s4 and s

5) and related


1(s2 s3 s4 and s

5) has



0 02 04 06 08 1t (s)

0 02 04 06 08 1t (s)

0 02 04 06 08 1t (s)

0 02 04 06 08 1t (s)

0 02 04 06 08 1t (s)

Ag 0

minus4

Ag 0

2

minus2

Ag 0

2

minus2

Ag 0

5

minus5

Ag 1

minus1

y 4y 3

y 2y 1

y 5







(a) (b) (c) (d) (e) (f) (g) (h) (i)





data acquisition

Sound pressures6

s5s4

s3

s2

s1

500mm

500mm

system

sensor






0 0005 001 0015 002 0025

0 0005 001 0015 002 0025

0 0005 001 0015 002 0025

t (s)

t (s)

t (s)

Ag 0

0204

minus02

x 3Ag 0

0204

minus02minus04

x 2Ag 0

0204

minus02minus04

x 1


0 500 1000 1500 2000 25000

002004006

0 500 1000 1500 2000 25000

002004006

0 500 1000 1500 2000 25000

002004

f (Hz)

f (Hz)

f (Hz)

2000

20

20

20

1100

11001750

17001100

1750

Agx 1

Agx 2

Agx 3



1has typically


1hasmajor




3has an obvious


0 0005 001 0015 002 0025

0 0005 001 0015 002 0025

0 0005 001 0015 002 0025

t (s)

t (s)

t (s)

Ag 0

2

minus2

y 1Ag 0

1

minus1

y 2Ag 0

12

minus1

y 3


0 500 1000 1500 2000 25000

010203

0 500 1000 1500 2000 2500

0 500 1000 1500 2000 2500

110020

1750

20005001700

f (Hz)

f (Hz)

f (Hz)

Agy 1

0010203

Agy 2

0

02

04

Agy 3




1represents


2and y





0 0005 001 0015 002 0025

0 0005 001 0015 002 0025

0 0005 001 0015 002 0025

t (s)

t (s)

t (s)

00204

minus02

Ags 1

0

0

01

minus01

Ags 2

01

minus01

Ags 3


0 500 1000 1500 2000 25000

005

01

0 500 1000 1500 2000 25000

002004

0 500 1000 1500 2000 25000

001002

1100

1750

1700 2000500

20Ags 1

Ags 2

Ags 3

f (Hz)

f (Hz)

f (Hz)







2and the







119882

Ω119882=[[

[

090 007 006

005 076 008

010 016 085

]]

]

(16)




5 Conclusions







Acknowledgments


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119869 (y) asymp 1

12119864 y3

2

+1

48kurt (y)2 (7)





1 119889119899)



119889minus12119899

)




1198921(u) = tanh (119886u) 119892

2(u) = u exp(minusu

2

2) (10)


(3) Let w = w+w+



(1) Let w119901+1

= w119901+1

minus sum119901

119895=1w119879119901+1

w119895w119895

(2) Let w119901+1

= w119901+1

radicw119879119901+1

w119901+1


1 w



W =W

radic1003817100381710038171003817WW1198791003817100381710038171003817

W =3

2W minus

1

2WW119879W

(11)


120588119882=

max119877sy (119896)



119894=0s(119894 + 119896)y(119894) 119877ss(119896) = sum


119894=0s(119894 +



length












S (119905) =[[[[[

[

s1(119905)

s2(119905)

s3(119905)

s4(119905)

s5(119905)

]]]]]

]

=

[[[[[[[[

[

sin (15120587119905)


2) 119906 (119905 minus 01119899 minus 002)

sin (10120587119905) cos (150120587119905)sin (6120587 cos (6120587119905))

119899 (119905)

]]]]]]]]

]

(13)



01

Ag

Ag

Ag

05

01

0 02 04 06 08 1

04

t (s)

0 02 04 06 08 1t (s)

0 02 04 06 08 1t (s)

0 02 04 06 08 1t (s)

0 02 04 06 08 1t (s)

minus4minus4

minus1

Ag

Ag

01

minus1

minus05

minus1

s 1s 2

s 3s 4

s 5



A =

[[[[[[[[[[[

[

023 minus058 073 minus081 095

028 minus089 049 minus075 097

056 015 minus072 058 099

minus038 minus052 045 061 099

minus048 038 084 minus051 098

088 minus061 054 minus020 095

]]]]]]]]]]]

]

(14)



0 02 04 06 08 1t (s)

0 02 04 06 08 1t (s)

0 02 04 06 08 1t (s)

0 02 04 06 08 1t (s)

0 02 04 06 08 1t (s)

0 02 04 06 08 1t (s)

Ag 0

4

minus4

x 6Ag 0

4

minus4

x 5Ag 0

4

minus4

x 4Ag 0

4

minus4

x 3Ag 0

4

minus4

x 2Ag 0

4

minus4

x 1




Ω119882=

[[[[[[[[

[

099 001 004 000 004

001 099 003 017 012

003 003 099 001 009

000 015 001 099 003

005 012 009 004 099

]]]]]]]]

]

(15)



119882= 120588119894119895 the elements


1(s2 s3 s4 and s

5) and related


1(s2 s3 s4 and s

5) has



0 02 04 06 08 1t (s)

0 02 04 06 08 1t (s)

0 02 04 06 08 1t (s)

0 02 04 06 08 1t (s)

0 02 04 06 08 1t (s)

Ag 0

minus4

Ag 0

2

minus2

Ag 0

2

minus2

Ag 0

5

minus5

Ag 1

minus1

y 4y 3

y 2y 1

y 5







(a) (b) (c) (d) (e) (f) (g) (h) (i)





data acquisition

Sound pressures6

s5s4

s3

s2

s1

500mm

500mm

system

sensor






0 0005 001 0015 002 0025

0 0005 001 0015 002 0025

0 0005 001 0015 002 0025

t (s)

t (s)

t (s)

Ag 0

0204

minus02

x 3Ag 0

0204

minus02minus04

x 2Ag 0

0204

minus02minus04

x 1


0 500 1000 1500 2000 25000

002004006

0 500 1000 1500 2000 25000

002004006

0 500 1000 1500 2000 25000

002004

f (Hz)

f (Hz)

f (Hz)

2000

20

20

20

1100

11001750

17001100

1750

Agx 1

Agx 2

Agx 3



1has typically


1hasmajor




3has an obvious


0 0005 001 0015 002 0025

0 0005 001 0015 002 0025

0 0005 001 0015 002 0025

t (s)

t (s)

t (s)

Ag 0

2

minus2

y 1Ag 0

1

minus1

y 2Ag 0

12

minus1

y 3


0 500 1000 1500 2000 25000

010203

0 500 1000 1500 2000 2500

0 500 1000 1500 2000 2500

110020

1750

20005001700

f (Hz)

f (Hz)

f (Hz)

Agy 1

0010203

Agy 2

0

02

04

Agy 3




1represents


2and y





0 0005 001 0015 002 0025

0 0005 001 0015 002 0025

0 0005 001 0015 002 0025

t (s)

t (s)

t (s)

00204

minus02

Ags 1

0

0

01

minus01

Ags 2

01

minus01

Ags 3


0 500 1000 1500 2000 25000

005

01

0 500 1000 1500 2000 25000

002004

0 500 1000 1500 2000 25000

001002

1100

1750

1700 2000500

20Ags 1

Ags 2

Ags 3

f (Hz)

f (Hz)

f (Hz)







2and the







119882

Ω119882=[[

[

090 007 006

005 076 008

010 016 085

]]

]

(16)




5 Conclusions







Acknowledgments


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










01

Ag

Ag

Ag

05

01

0 02 04 06 08 1

04

t (s)

0 02 04 06 08 1t (s)

0 02 04 06 08 1t (s)

0 02 04 06 08 1t (s)

0 02 04 06 08 1t (s)

minus4minus4

minus1

Ag

Ag

01

minus1

minus05

minus1

s 1s 2

s 3s 4

s 5



A =

[[[[[[[[[[[

[

023 minus058 073 minus081 095

028 minus089 049 minus075 097

056 015 minus072 058 099

minus038 minus052 045 061 099

minus048 038 084 minus051 098

088 minus061 054 minus020 095

]]]]]]]]]]]

]

(14)



0 02 04 06 08 1t (s)

0 02 04 06 08 1t (s)

0 02 04 06 08 1t (s)

0 02 04 06 08 1t (s)

0 02 04 06 08 1t (s)

0 02 04 06 08 1t (s)

Ag 0

4

minus4

x 6Ag 0

4

minus4

x 5Ag 0

4

minus4

x 4Ag 0

4

minus4

x 3Ag 0

4

minus4

x 2Ag 0

4

minus4

x 1




Ω119882=

[[[[[[[[

[

099 001 004 000 004

001 099 003 017 012

003 003 099 001 009

000 015 001 099 003

005 012 009 004 099

]]]]]]]]

]

(15)



119882= 120588119894119895 the elements


1(s2 s3 s4 and s

5) and related


1(s2 s3 s4 and s

5) has



0 02 04 06 08 1t (s)

0 02 04 06 08 1t (s)

0 02 04 06 08 1t (s)

0 02 04 06 08 1t (s)

0 02 04 06 08 1t (s)

Ag 0

minus4

Ag 0

2

minus2

Ag 0

2

minus2

Ag 0

5

minus5

Ag 1

minus1

y 4y 3

y 2y 1

y 5







(a) (b) (c) (d) (e) (f) (g) (h) (i)





data acquisition

Sound pressures6

s5s4

s3

s2

s1

500mm

500mm

system

sensor






0 0005 001 0015 002 0025

0 0005 001 0015 002 0025

0 0005 001 0015 002 0025

t (s)

t (s)

t (s)

Ag 0

0204

minus02

x 3Ag 0

0204

minus02minus04

x 2Ag 0

0204

minus02minus04

x 1


0 500 1000 1500 2000 25000

002004006

0 500 1000 1500 2000 25000

002004006

0 500 1000 1500 2000 25000

002004

f (Hz)

f (Hz)

f (Hz)

2000

20

20

20

1100

11001750

17001100

1750

Agx 1

Agx 2

Agx 3



1has typically


1hasmajor




3has an obvious


0 0005 001 0015 002 0025

0 0005 001 0015 002 0025

0 0005 001 0015 002 0025

t (s)

t (s)

t (s)

Ag 0

2

minus2

y 1Ag 0

1

minus1

y 2Ag 0

12

minus1

y 3


0 500 1000 1500 2000 25000

010203

0 500 1000 1500 2000 2500

0 500 1000 1500 2000 2500

110020

1750

20005001700

f (Hz)

f (Hz)

f (Hz)

Agy 1

0010203

Agy 2

0

02

04

Agy 3




1represents


2and y





0 0005 001 0015 002 0025

0 0005 001 0015 002 0025

0 0005 001 0015 002 0025

t (s)

t (s)

t (s)

00204

minus02

Ags 1

0

0

01

minus01

Ags 2

01

minus01

Ags 3


0 500 1000 1500 2000 25000

005

01

0 500 1000 1500 2000 25000

002004

0 500 1000 1500 2000 25000

001002

1100

1750

1700 2000500

20Ags 1

Ags 2

Ags 3

f (Hz)

f (Hz)

f (Hz)







2and the







119882

Ω119882=[[

[

090 007 006

005 076 008

010 016 085

]]

]

(16)




5 Conclusions







Acknowledgments


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 02 04 06 08 1t (s)

0 02 04 06 08 1t (s)

0 02 04 06 08 1t (s)

0 02 04 06 08 1t (s)

0 02 04 06 08 1t (s)

Ag 0

minus4

Ag 0

2

minus2

Ag 0

2

minus2

Ag 0

5

minus5

Ag 1

minus1

y 4y 3

y 2y 1

y 5







(a) (b) (c) (d) (e) (f) (g) (h) (i)





data acquisition

Sound pressures6

s5s4

s3

s2

s1

500mm

500mm

system

sensor






0 0005 001 0015 002 0025

0 0005 001 0015 002 0025

0 0005 001 0015 002 0025

t (s)

t (s)

t (s)

Ag 0

0204

minus02

x 3Ag 0

0204

minus02minus04

x 2Ag 0

0204

minus02minus04

x 1


0 500 1000 1500 2000 25000

002004006

0 500 1000 1500 2000 25000

002004006

0 500 1000 1500 2000 25000

002004

f (Hz)

f (Hz)

f (Hz)

2000

20

20

20

1100

11001750

17001100

1750

Agx 1

Agx 2

Agx 3



1has typically


1hasmajor




3has an obvious


0 0005 001 0015 002 0025

0 0005 001 0015 002 0025

0 0005 001 0015 002 0025

t (s)

t (s)

t (s)

Ag 0

2

minus2

y 1Ag 0

1

minus1

y 2Ag 0

12

minus1

y 3


0 500 1000 1500 2000 25000

010203

0 500 1000 1500 2000 2500

0 500 1000 1500 2000 2500

110020

1750

20005001700

f (Hz)

f (Hz)

f (Hz)

Agy 1

0010203

Agy 2

0

02

04

Agy 3




1represents


2and y





0 0005 001 0015 002 0025

0 0005 001 0015 002 0025

0 0005 001 0015 002 0025

t (s)

t (s)

t (s)

00204

minus02

Ags 1

0

0

01

minus01

Ags 2

01

minus01

Ags 3


0 500 1000 1500 2000 25000

005

01

0 500 1000 1500 2000 25000

002004

0 500 1000 1500 2000 25000

001002

1100

1750

1700 2000500

20Ags 1

Ags 2

Ags 3

f (Hz)

f (Hz)

f (Hz)







2and the







119882

Ω119882=[[

[

090 007 006

005 076 008

010 016 085

]]

]

(16)




5 Conclusions







Acknowledgments


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 0005 001 0015 002 0025

0 0005 001 0015 002 0025

0 0005 001 0015 002 0025

t (s)

t (s)

t (s)

Ag 0

0204

minus02

x 3Ag 0

0204

minus02minus04

x 2Ag 0

0204

minus02minus04

x 1


0 500 1000 1500 2000 25000

002004006

0 500 1000 1500 2000 25000

002004006

0 500 1000 1500 2000 25000

002004

f (Hz)

f (Hz)

f (Hz)

2000

20

20

20

1100

11001750

17001100

1750

Agx 1

Agx 2

Agx 3



1has typically


1hasmajor




3has an obvious


0 0005 001 0015 002 0025

0 0005 001 0015 002 0025

0 0005 001 0015 002 0025

t (s)

t (s)

t (s)

Ag 0

2

minus2

y 1Ag 0

1

minus1

y 2Ag 0

12

minus1

y 3


0 500 1000 1500 2000 25000

010203

0 500 1000 1500 2000 2500

0 500 1000 1500 2000 2500

110020

1750

20005001700

f (Hz)

f (Hz)

f (Hz)

Agy 1

0010203

Agy 2

0

02

04

Agy 3




1represents


2and y





0 0005 001 0015 002 0025

0 0005 001 0015 002 0025

0 0005 001 0015 002 0025

t (s)

t (s)

t (s)

00204

minus02

Ags 1

0

0

01

minus01

Ags 2

01

minus01

Ags 3


0 500 1000 1500 2000 25000

005

01

0 500 1000 1500 2000 25000

002004

0 500 1000 1500 2000 25000

001002

1100

1750

1700 2000500

20Ags 1

Ags 2

Ags 3

f (Hz)

f (Hz)

f (Hz)







2and the







119882

Ω119882=[[

[

090 007 006

005 076 008

010 016 085

]]

]

(16)




5 Conclusions







Acknowledgments


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 0005 001 0015 002 0025

0 0005 001 0015 002 0025

0 0005 001 0015 002 0025

t (s)

t (s)

t (s)

00204

minus02

Ags 1

0

0

01

minus01

Ags 2

01

minus01

Ags 3


0 500 1000 1500 2000 25000

005

01

0 500 1000 1500 2000 25000

002004

0 500 1000 1500 2000 25000

001002

1100

1750

1700 2000500

20Ags 1

Ags 2

Ags 3

f (Hz)

f (Hz)

f (Hz)







2and the







119882

Ω119882=[[

[

090 007 006

005 076 008

010 016 085

]]

]

(16)




5 Conclusions







Acknowledgments


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













Acknowledgments


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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