Crack Subm

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Nearfield Acoustics of a Submerged Cantilever Plate for Fatigue Crack Detection Application

Ry

'DYul anLi, B.Eng., M.Eng.

A ThesisSubmitted tothe Schoolof Graduate Studiesin PartialFulfilmentof the Requireme nt for the Deg reeofMaster of Engineering

April 1996

Facult y ofEngineering and AppliedScie nce Memorial Univers ity of Newfoundland

St. John's Newfoundland Canada

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ABST RAC T

Thisthesispresentsan investigationwhichexaminesexperimentally thechange in modal parameters of a submergedcantilever due tothe existenceof fatiguecracks, through vibratio nalandnearfieldacoustical measurements.Themodalbehaviourofan uncracked cantilever, inairand invarious water submergenceswasalsostudied and co mparedwith nume ricalanalysis.

The modal propertie sof an uncracked cantilever platein air,and inan infinite watermediumwerefirstestimated by solid structure modelling andunde rwater shock analysisusing afinitecleme nt programcalled ABAQUS.Tenmodeswere investigated for the dry cantileverandthree bending modesfor the submergedone.The effectof the depthof submergenceof the cantilever onits modal propertieswas determined experimentally. Itwasobse rved thatamaximum lowerin gin the structure's natural frequencyof 33%occurredinthe firstbendingmode. The dampingratios of the submerged cantilever increasedonetosix timesin accordancewiththe different modes.

When the ratioof immersiondepthto thespan lengthof thecantilever exceeded 0.4.the modal propertiesof the submerged structuretended tobeindependent ofthe depthof the submergence.

In the fatiguelest, acantilever plate wassubjected to acyclic loading with constantmagnitudein air.Vibrationaland near fieldacousticaltestswereconductedat various fatigue cycles both in air and inwater. Modal parametersandnormalized acousticlntenshles (NAIl were monitored as fatigue cyclesaccumulated. Itwas

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iii

fou nd thaithosemodalfrequencies (bending frequenciesinthis study).withwave propagation directions perpendicular to lhe crack propagationdirectionwere scnstnve indicatorsof structuralcracking.Whentheratioof crack areatothetOl:Ucrosssectinnnl areaoftheplate(nonnalizedcrackarea)assumed 33%.thecrackwasdetectable by a maximum natural frequencychangeof 3-4%inboth theairand thewatermedia.Itwas about17%when thenormalizedcrackarea was 57%.1.5to 2timesincreaseindamping ratios and3 to 6.7 dB decreasesin NAImagnitudeswereobservedwhen thenonnalizec.l crackareaexceeded 47%. Fatigueresultsofthe submergedcantilever agreedwellwith thatof thedrycantileverforallmodalfrequenciesandthe lowermodal damping ratios.

Modal parameters extracted from theacoustical measurementsagreedwellwiththose fro m thevibrationalmeasurements madeon thesubmergedcantile ver,andwithmosaof theparametersnotedforthedrycantilever.There werediscrepanciesofmodaldamping ratiosfor the 2nd andthe6rhbendingmodes of the drycantilever.

Tbefrequencyresponse func tions(FRF)andthe nearfieldacousticalpre~sure transferfunctions(PlF)of theuncrackedcantileverinairweresimulated byusing ABAQUScomputer programandcom pared withtheexperimental results.Mea..uredand simulated natural frequencies ofthe dry cantileverplate have good agreements.The maximumdifferencewas2.67%. Goodagreementswere also observedbetween measured andsimulated FRFsforthe first fourbending modesandPTFs forthe3rdand4rh bendingmodes.ThemaximumdiscrepancyofPTfmagnitudeswas found to be 5.9dB furthePTFmagnitudeofthe fIrst bendingmode.

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ACKNOWLEDGMENTS

Theauthor is gratefultoher supervisors.Dr.1.Y.Guigne andDr. A.S.J.

Swarnidus, for their patience. guidanceandencouragement throughout thisprogram.The authoris alsograteful10 Dr.J.Y.Gulgne,totheFaculty ofEngineeringandApplied Science and to theSchoolof Gr aduateStudies for theirfinancialsupportin thisprogram.

Theauthorwouldlike to thankMr.K.Klein.C·CORE.for fatig uingthe specimen and for his helpin theexperiment.Speciallhanksalsogo 10Mr.H.Dye,Mr.J.Andrew and otherstaffinMachine Shopfor theirtechnical servicesandsuggesuone in manufacturing theexperimentalsetup in thedeeptank. Theauthor wouldalsolike 10 thankMr.A. Burseyand otherstaff in Structureand FluidLaboratoriesfortheir technical support intheexperiment. The technical support fromMr.D. Pressand other staff in the Centre for ComputerAided Engineering is also very muchappreciated.

Finally,theauthorwould liketothankher husband,ZhaopengU,her son,Yanling Li,and her parents, for theirlove,understanding,andencouragement.

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5.5.1 Change of Modal Parametersdue to Fatigueand Crack 87 5.5.2 Change ofAcousticIntensityMagnitudeand EnergyFlow

due to Fatigue Crack•.... ,. . 106

5.5.3 Comparison ofModal Propertiesof the Cracked Platein Air

with That in Water .. " 112

S.5A Comparison of FRF and PTF Test Results... . . .. ...115

5.6 Summary ,.116

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6. SIMULATIONOf THE MODALRESPONSES 01'THE CANTILEVER I'l.AT EIN AIRUSING FINIT E ELEMENTANALYSIS 118 6.1 Simulation of FrequencyResponseFunction(FRF)

6.2 Simuluio nof PressureTransferFunction(PTF) 6.3

6.4

Discussion .. .. .. . .... ...•.

Summary .

118 121 126 .. 128

7. CONCLUSION

REFERENCE

APPENDIXA EffectofAirBubbleson FRFS of the Submerged CantileverPlate

APPENDIXB Nonnalized Acoustic IntensitiesObtained from the Experimen ts

APPENDIX C FRFandPTFMagnitudesof theUncracked CantileverPlate

129

136

141

143

147

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List of Figures

Figure 3.t Nyquistcircle 030

Figure 4.1 Thecantileverspecimenunder investigation J8

Figure4.2 A S8RS shellelement ]8

Figure 4.3 Finiteelementmeshofthe uncrackedcantileverwithout the

weld stub

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Figure4.4 Finite elementmeshoftheuncracked cantileverwiththeweld stub

Figure4.5 Figure4.6 Figure4.7

Modeshapes oftheuncrac kedcantileverplaleinair A S4R5shellelement

An USI4 boundaryelement .

39 40 44 44 Figure4.8 Free responseof thesubme rgedcantilever plate. 46 Figure4.9 Free response power spectrumofthesubmergedcant ileverplate 46

Figure 4.10 Experimentalset-upinthe deep tank. 49

Figure4.11 Aphotoofiheexperimental set-up. 5U

Figure 4.12 Measurement system. 51

Figure4.13 Calibra tionset-up. 53

Figure 4.14 Modeshapesofthecantilever inairobtainedfromexperiments 58 Figure4.15 Experime ntalandFEAresul ts of natura!frequency ofuncracked

plate in air. 61

Figure4.1(; Relativechangeofnatural frequen cyof uncrackedcantilever at

various immersiondepths ., 63

Figure 4.17 Increases of dampingraliosofuncracked cantileveratvarious immers ion depths

Figure4.18 Mode shapes of the plate for : "+",air: "0", dII=O.02 and"x", 65

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dII:O.66

Figure4.19 Addedmassesofsubmerge d cantileverasa funct ionof in-vacuo naturalfrequencies

"

.,

Figure5.1 Fatiguctestset-up 73

Figure5.2 l.Dcationandpropagationdirectio n ofthecreek 75

Figure 5.3 (3) Crackprofiles 76

(b) A photo of thecracksurface

Figure 5.4 Acquisition positlo n forFRFandPTFmeasurementsinairand

7.

90 .. ... ... . . . .. 79 Figure5.5 Aphotoof thecantileverplatespecimenandtransd ucers 85 Figure 5.6 Cracklengthas afunctionoffatiguecycles. ... . 88 Figure5.7 Naturalfrequenc yofthe dry can tileveratvario usfatigue

cycles

Figure5.8 Naturalfrequenc y ofthe SUbmergedcant ilever at various fatigue

cycles 93

Figure5.9 Dampingratios of thedrycan tileveraJvarious fatigue cycles. 99 Figure5.10 Dampingratiosofthe submergedcantileveratvarious fatiguecycles101 Figure 5.t t NAIpeakmagnitudesandthe totalacoustic power flowin

zoomedfreque nc y bandsincludinglhesecond bendingmodeofthe

drycantilever 101

Figure5.12 NAIpeakmagnitudesandtheIOlalacoustic: powernowin zoomedfreque nc y bands includingthethi rd bendingmode ofthe

dry cantilever•..• . 108

Figure5.13 NAIpeakmagnitudesandthetot alacoustic powerflowin zoomed frequencybandsincluding thefou rth bendi ngmodeofthe drycantilev er... . . .... .... ... . . . .. .... ... .... 109

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x

Figure 5.1-l- NAIpeakmagnitudes and thetotalacoustic po wer flow in zoomed frequency bands including the second bending mode of the submerged cantilever

Figure5.15 Relative changeof natural frequ encyas a functionofnormalized crackarea

Figure 5.16 Increase of dampingrenoof the cantileveras a functionof fatigue cycles

110

114

114 Figure6.1 ComparisonbetweenFEA andtest results onFRFsof uncrackcdcantilever

plateinair Figure6.2 Air-beam model

.. ..120 ....122 Figure 6.3 Compar ison betweenFEA andlestresultson PTFs ofuncracked cantilever

in air ..125

Figure A.I AFRFof submergedcantilever with airbubbles. 14 1 FigureA.2 A FRF of submerge dcantileve r without air bub bles,. 142

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List of Ta bles

Table 4.1 Naturalfreq uency of the uncrackedcantileverinairfrom FEA

Table4.2 Results of force transducer calibration Table4.3 Results ofcahb rauons ofresponsetransducers

Table4.4 Natural frequencyand damping ratioof theuncrackedcantilever fromtestsin air andin water

47 54 55

59 Table4.5 Relative changeofnaturalfrequenciesof thecantileverplate in

79 64 68 77

80 water.

Table 4.6 Addedmassesofthe submergedcantilever plate Table5.1 Fatigue historyand crack profile

Table5.2 Centre frequencies (f,),frequencyresolutions(Anand averaging numbers (A v g. ) of zoomFRFs.

Table 5.3 Frequencies&dampingratiosof the dry cantileverat specified fatiguecycles fromFRFtests

"fable5A Frequencies&dampingratiosof theSUbmergedcantileverplate at

specified Iauguecyclesfrom FRF tests 81

Table5.5 Frequenciesand damping rsatioof thedrycantileverplateat

variousfatigue cycles fromPTFtests 83

Table5.6 Frequencies&dampingratios of thesubmerged cantileverplateat

various fatiguecyclesfrom PTFtests 84

Table 5.7 Relative decreaseofnaturalfrequencies due10 fatigueand

crack 113

Table6.1 Averag eddamping ratios of thecantileverinairobtained fromthe

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xiii experiments

Table 6.2 FRF and PTF magnitudes of the dry cantilever fromlest and FE,\

simulation

Table 6.3 Natural frequencies of thedry cantilever from testandFEA simulation

Table6.4 ErrorsinsimulatedFRF and PTFmagnitudes

Ill)

127 Table8.1 Normalizedacoustic imensity(NAIl ofthe2ndmodeof the cantileverplate

in air

Table B.2 NAIs of the3rd bendingmode of the cantileverplatein air. 144 TableB.3 NAIs of the4thbending mode of the cantileverplateinair 145 TableB.4 NAls of the2ndbending mode of thecantileverplatein water 146 Table C.I FRF andrTFmagnitudes or the uncrackedcantileverobtained

fromthe experiments . 147

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List of Symbols

a,crack length

A,. -addedmass ofsubmerge dcantilever

Am,bfandA_-addedmasses of thesubmergedstructurefor bending andtorsionalmodes B,~r_complexconstant representingthe influenceofFRFfromother mode tothe

un

mode

C-dampingofa SDOFsystem [c].damping matrix of a MDOFsystem c, • soundspeedofthe fluid

C1•coefficientin fluid boundary conditions

0'111 "fatigue stress

a..·

mean fatigue stress 0...stressin beach marking fatigue 0'••peakmagnit ude of fatigu estress 0....'•diame ter of rth modal circle E·Young'smodulus.N/m^J E -meanfatigue strain [(I ) .excitation force (f(l)l-force vector

F(&),orF(jf,)l- Fourier transformof forcesignal F(s)-transfer functionof force signal f••natura! frequency of thecantilever in air C..•naturalfrequency of thecantileverin water f....-naturalfrequency of the cantileverwith a welded stub fit• natural frequency of the cantilever plateatvarious fatigue cycles Yr.(l)• coherence function

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H~,(andH~2f-frequencyresponsefunction of pressure. or pressuretransferfunction (PTF) H(s) -transfer function

H(7.).or H(it» - frequencyresponse function(FRF)

[H(s)), [HUt>)J•transferfunctionand frequencyresponsefunctio n matrices I,· acousticintensity

1,(7.))-acousticintensityspectrum I.(7.)) -normalized acousticspectrum v • Poisson'sratio

k•stiffness of a SDOFsystem [k)- stiffnessmatrix of a MDOFsystem K.R;..·stiffnessresidual

It, -acousticwavenumberof fluidmedium kl•coefficient influidboundaryconditions 6K -stress intensityfactor

1;..rth modal damping ratio I. L- length of the cantileverplate.mm )., - structuralwavenumber oftherthmode m • mass of a SOOFsystem [m] - massmatrix of aMDOFsystem M'~k•inertiaresidual N-fatiguecycles

lol•angularfrequencyof the fluidcoupled vibratingsystem t>,-therth naturalfrequency of thecantileverplate lol,-samplingfrequencyofthesignalto be processed t>l-angularfrequencyof fatigue load

lollxoI•angularfrequencyof loadInbeachmarkingfatIgue.

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xvi p-acocsuc pressure

P(~)-Fourier transformof acousticpressure r•spacialpctmvector

l!.r •microphooe orhyd rophonespacing Pro- densityof thefluid.kg/nl^l

rb,ft.r

=

^1.2.3•..^•^rth^bendi^ngandtorsional modes of the cantilev er Is,) -rflrmodalvectornormalized with respect to massmatrix S••(~),Srtlot)• autospecrrum ofresponse and force S,,(tl)-cross spectrumof response and force t^ctime •second

u . acoustic velocity

U(~)-Fourier uansform ofaco usticvelocity w -widlb ofthecantileverplate .mm x.y.z•Cartesian coordinales J.(I)-responsesignal Ix(lJl - responsevector 1x,1-rthmodal vector

XIs)•transfer function of respo nsesignals X(lot). or XOlot) •Fouriertransform of response signals YOt.)• mobility functio n

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CHAP TER 1. INTR ODUCTION

Future Canadian oil gasrelated facilitieswillincludea num beroflarg e sect platform s.Thesestructure sresidein anenv ironment that subj ects,them[0cyclicstress reversalsof varyingmagnitudes. inducingfatiguecrackingat thewelded joint swhere stressconcentrations arehigh.Fatiguecrackingis aseriousconcern in thesestructures asitlead sto a co llapse ofjoints.

Modalanalysis. whichisthe processofdetermining thedynam icpropertiesofII

structureeitherbyexperimentalor mathematicalmodelling.orbyaco m bination ofboth, andsubsequently character izingthesepropertiesin terms ofthe stru ctural modesof vibration.has beenstudiedex.tensively bymanyres earchers(GomesandSilva,1990;

SpringerandReznicek.1994;SilvaandGomes.1994:andPerchardandSwamidas,1994) asapromisingno ndestructive dete ctiontechnique (NOT).Detectionoffatigue cracks throughacousticmeasuremen tshas thepotentialadvantageof realizing nondes tructive.

non-cont act monitoringof operatingstructures.

Thefeasibilityto extractaccurate modalparameters from nearfieldacoustic measurementshas been investigatedbyGuigne eral.(1992). Comparisonswere made byKleinet al.(1994)on natural frequenciesobtained fromcontactmodal testingand non - contactnearfieldacoustic modal tes tingfor acantileverbeam in air.Dete ctionoffatigue cracks throughacousticmeasurementsis consideredto be a viableapproachbutwith limitspoorly understoodor docum ented. While currentstud ies on fatiguedetection techniquesconcentrate onthe detec tio nor localizationof fatig uecracks in drystructures.

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studieson subme rgedstr uctures are urgently required.However,onlya fewstudieshave investigatedfa tiguecrac kidentificationofunderwat erstructures(Bud ipriyanto . 1993,and Silvaand Gomes,1992).

Despitethelimitedapplication tofatiguecrackdetectio n,vibrationalandacoustical propertiesofhe avyfluid-structure coupled systems havebeenstudiedextensively through various methods.Experimental studies ontheinfl uenceoffluid0:.structuraldynamic propertieswerecarriedoutbyLindholm et al.(1965),Jezeq ue l(1983)and Bud ipriyanto (1993 ). Numericalanalysis ofthemodal propertiesoffluid- structu re coupledsystems were carriedout by DowellandVoss (19 63), Everstine(1991}, Iitto uneyet at (1992), DaddazioeraI.(1992),and DeRuntz(1989).The influencesof immersio ndepthsand fluid envirenm emon themodaJpropertiesof submergedstructureswere studiedby Lindho lmetaJ.(1965),Muthuvee rappanetaI.(19 7 8,1 919) ,and Cao (1994).

In general,currentnumeri calandexperimentalstudieshaveinvestigatedvarious metho dologiesand techniquesto examinein detailthe vari ousmec hanismsconnected withafluid-stru cturecou pledvibratingsystem. Itwas observed thatthevibrational prope rt ies of su bmergedstrucmresmaybe independentofthe depthof fluid sub mergence if certaincond itionsarereached. However,studies onthe effectofimmersion depthon struct u ralmodalparameters are restricted to naturalfrequencies. Fewstudieshav e reponedinvesti gationsof nearfield acousticpropertiesof sub mergedstructures . nor have thesepropertie s been use d 10ide ntify thepresence of fatiguecracks.

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1.1 Scopeof TheThesi s

Thisthesisreportsthefindin gsof a comprehensiveexperimentalmvesrigatiun on detectin gfatiguecracks inasubm ergedcantileverplate. YfJeorcticalstudiesinclude estimatio nofmodal frequenciesof theuncrac kedcan tileverplatein airandin water.and simulationof thefrequenc yrespons e functio nsandthe nearfieldacousticpressure transfer function s of the cantilever plateinair.The influenceof immersion dep thson themoda l properties of thesubm erged cantileve r plate wasalso investigated through experime ntation. Thesestudies acepresentedin sevenchapters.

Chapter 2givesa literaturereview oftheprevious studieson topics offatigue crack detectioninsolid structures.acousticmodal testing.andhea vy fluid-s tructure interactionprob lems. A theoreticalbackground tomodalanalysi s andtoacoustic intensity measure menttechn iques. as well assomepracticalconsiderati o nsinco nducting dynamictestinganddata processin g . havebeen brie fly reviewed inChapter3.

Studies on modalproperties of anuncracked cantileve rplate inairand in water are prese ntedinChapter 4,whichincludenumericalestimation ofnatu ralfreq ue nciesof the unc rackedcantilever platein air andinwater.Anexperimental studyon thechange of modal parameters due tothechangeoffluidmediumfromairtowater is also given.

Chapter5highligh tstheresu lts oftheexpe riments onthe ch an geofstructural modalparameters bothin air and in waterdueto thepresence of fatigue cracks.Natural frequenc ies, dampingratio s and normalizedacousticintensities were mon itored atvarious fatigue cyclesand crackdimension s.The resultswer e comp ared and discusse d.

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NumericalsimulationsofzoomFRFsand PTFs of the uncrack edcantileverplate specimeninairare presentedin Chapter6; theseresu lts arecomparedwiththoseobtained from theexperi ments in Chapter 5. Chap ter7prov ides afinalset of conclusions.

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CHAPTER 2. LITERATURE SURVEY

2.1 Previous Studies on Vibrational and Acoustica l Properties

or

Submerged Structures

Studiesonvibrationaland acousticalpropertiesof submergedstruct uresare generaJlycategorizedas acousticfluid-structurecoupled problems. Theacousticfluid propert iesandthe structuralvibrationpropertiesare closely relatedto each other.The combined vibrc-acoustic propertiesare sensitiveindicatorsofthe slateof struc tures.

Thereare two majo rareas inthestudyoffluid-struct ure coupledproblems.One is onthe energy-containedsystem, inwhich the research interest has beenfocused on the dynamic propertiesof submergedstructuressuchasmodalfrequency,modaldamping, modeshapeand harmonic response (Everstine ,199 1;RobinsonandPalmer,1990; and Fu andPrice,1987) . Theother ison(heene rgy-radiating system, which dete rminesthe acoustic radiationandscattering re sponses inducedbyharmonicexcitationof submerged structures(Ettcuneyetal.,1992,Daddazio etal..199 2;Everstineand Henderson,1990 and Marcus.1978).

When examining the dynamic respon sesofsubmergedstructures,studieshave beenrestrictedto responses in thelowfrequencyregime.The fluid actsas an added mass tothe structure,i.e.,thefluid pressureon thewetsurfaceisin phasewith thestructural acceleration. Drymode approxim ation theoryhas been developed forthis case. This

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theory wasfirstintroducedbyDowelland Voss (1963) whentheyinvestigatedthe effect of a cavityonthesupersonicresponseofanaircraftpanel.The freevibrationof a structurein a coupled system wasexpres sedasthe sumof the in-vacuomodes of the struc tu re. Dry mode theory was appliedbyPretlove(1965)toinvestigatethefree vibrationof a rectangular panelina closed rectangularcavity. Prenove setthe expanded functions(dry mode)of the structure tosatisfythe fluidboundary conditionsindividually, thusreconcilingthestructureand thefluidboundary requirements.

Taylorand Duncan(1980)and Robinsonand Palmer(1990) extendedthedry mode theorytoheavy fluid-structure couple d problems.Taylorand Duncandescribedthe dynamicresponsesof a generallysubmergedstructure as the sum of thein-vacu omode shapesofthe structure. Allthehydrodyn amic actionsincluding those inducedby the motionof the structure,wereretainedonthe right handside of the equations of motion, which means theeffect ofthe freesurfac e and thethree-dimensionalfluidflow were retainedin the analysis.Robinson and Palmer deducedthefluid pressure functionor fluid velocitypotential bysolving thewave equationwiththeunsteady Bernoulliequation applie datthefluid-structure interface.

Dry modeapproximationtheoryissimple to perform and is useful to solvelow freque ncy,low amplitude vibrationalrespo nses offluidcoupledstructures. How ever,in- vacuomode shapesarenoteigenfunctions of the coupled system.At higherfrequencies.

thefluid impedance.whichis theratioof fluidpressureto velocity,is mathe matically complex, indieating mass-like anddamping-likeeffects.ntismeans thatthe ad ded mass

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·effectis diminished andthe added damping effectis increased.Realmodalanalysisis nolonger validin chis case. Therefore,when acousticradiation andscatteringofthe fluid coupled structure is themain concern,wei mode theoryshouldbeintroduced.

Wetmodes ofsubmerged structureswereinvestigated byBnouney er al.(1992) asa methodfor derermin⁰meacousticradiation and scatteringresponsesinduced hy harmonicexcitationof a submergedstructure.Wetmodes in the study by Ettouneyetal werenot the complexeigenvectors andeigenvalues of thefreevibration of astructurein fluid,butwerethe complexvectorcomponents of the spatialpart ofthe free harmonic vibration at theforcing frequencyof the fluid-structurecoupledsystem. The surface velocitycouldbeobtainedby muhiplyingthe wet mode withappropriatecoefficients,and thus the far-field pressurescouldbecalculate d.

Everstineand Henderson (1990) investigatedthe combineduse of finiteelement andboundaryelement analysesto deal withfluid-structurecoupled problems.Thesurface fluid pressureand thenormal velocityof an arbitrary three-dimensional elasticstructure were calculatedby coupling thefinite elementmodel with a discrete form of the Helmholtz surfaceintegralequationof theexterior fluid.Farfieldradiatedpressureswere then calculatedfrom the surface solutionusing Helmholtzexterior integralequatio n.

DeRuntz (1989) combinedfiniteelement and boundaryelementanalyses to investigatethe transientresponseof a totally or pantysubmergedsuucture dueto an acousticshock wave of arbitrarypressureprofile and sourcelocation.Theadde~nmess of the structure. was first calculated using b..nmdary elementtreatment of Laplace's

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equationfora flowgeneratedinaninfinite,inviscid,andincompressiblefluidby motions of thestru cture's wetsurface(DcRunizand Geers, 1978).The structuralresponsewas consideredtobelinearelasticandthe mass and stiffnessmatriceswere constructedby standardfinite elementcodes. Doubly asymptoticapproximations wereused to ap proxima teboth thelow-frequencyandhigh -frequency motionsat the fluid-stru cture interface. The fluidandstructure eq uationswere solvedseparatelyandtheresult s were computed forecch timeinc rement. The co m binationof finiteelementandbou ndary element analyseswouldbea suitable techniqu e to deal withthecoupled problems ifthe computationtime couldbereducedbyintrodu c ing advancedcomputingtechniques.

Instead ofinvoking finiteelementanalysis,Jezequel (1983) investigate dthe possibility ofevalu atingthe fluid influence on thestructural respo nse thro ugh experimentallymeasured nat ural freq uency dat a, for both invacuo and incontact with flu id condi tions.The discretemodel which expressesthe dynamic behaviou rofthe fluid structurecoupledsystem was obtaine dthroughanoptimizationprocedure.

Tne influenceofimmersiondepth onthe natural frequencyof subme rgedstructures werestudiedbyLindholmetaI.(1965), Muthuveerap panetaI. (1978),Fu and Price (1987),Budipriyanto (1993)andCao (1994). Lindho lmetal.carried out ex te nsive experimentalstudieson thevibrational propertiesofsubmergedplates. Differentplate geometriesandfluid immersiondepths wereinvestigated.Thenatural frequenciesofthe SUbmerged plates were foun d 10 decrease asthe immersiondepthincreased.However, sig nificant changes occurred only whenthesu b mergeddepthwas lessthan halfthe span

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lengthofthe plates. Fuand Price(1987) disc ussed thedynamiccharacteristics ofa part iallyortotally immersedvibrating cantilever plate.using abydroelasuctheory.The generalizedhydrodynamiccoefficients werefoundtoreducerapidly as theplate was placedfurther fromthe freesurface.whichis in agreeme ntwith Lindholm's study.

Muthuveerap panera1.(1978) showed that the firstnon-dimensional frequencywas alwayschangingunti l a certain water depth wasreached. The highernon-dimensional frequencies tendedtobeconstantwhen the immersiondepth ratio,which istheratio of immersion depth to thespanlength of the plate.was2.75.Theinfluence of thewater depth under the plateandthe watervolumein the horizontal direction ofthecantilever plate was also investigated byMuthuveerappanetaI.(1979). It wasfound that the vari ationof naturalfrequenciesof asubmerged structurewasnegligiblein a fairlybroad fluidmedium.Thisanalysiswasconducted by usingfiniteelementanalysisunderthe assumptionthattheacoustic pressurewaszero at the free surface andthe pressure grad ient,aswell, was zero at theinterface betwe e nthefluidand the rigidside walls .

Budipriyan to(1993)studied the changeof naturalfrequenciesof a deep submerged canti lever plate.A 40% reductionofnaturalfrequencywas observedwhentheimmersion depthratio was 0.76.Cao (1994)investigated theeffect of immersiondepthsonthe added mass factorsof a SFSF (S-simplysupported,F-Iree) ; and a CFCF (C-clamped.F·

free ) plate.The maximum immersiondepth ratioconsidered was 0.5.

In summary.the vibretionalandacousticalpropertiesof submerged structures have been examined;yet the acousticbehaviours of he avyfluid-structure coupledsystem s as

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10

measured from experimentsremainpoorlyunderstoo d. Experimentaland numerical investigations ofthe effectofimmersiondepth s onthe modal parametersof submerged structures have been restricte dtonaturalfrequencies only.

2.2 PreviousStudyon Fatigu e Cra ck Detect ion

Fatigue crac ksin struct uralelementsare usuallymonito redbynon-destructive techniques (NOT)suchas visual inspection, magnetic field, thermal contours,X-ray, strain gauge,laserinterferom euy , acousticemissionandmodal analysis (Mannan etaJ..

1994).Studiesonacoustic-relatedNOTmethods arebriefly reviewedinthefollo wing sections.

2.2.1 Detection ofPropa gat ing Cr acks by Acoustic Emission

Detections ofpropagatingcracks, withinsolidstructures,bysoundemission techniqueswereinvestigated by Chishko(1990,1992)andKrylov(19l.l3). Krylov developed ageneralized approach,which analyzedtheradiationofsound bycracksof arbitraryconfigurationsbased onHuygens'principle. Hispaper showed that it was possible to determine themovementof the edgesof a crackunder the action of applied externalstresses,and capture theradiationsfromit. The resultsindicatedthai the subsequentstages oftheintermediategrowthofthecrackwereaccompaniedbya sharpeningofthe directionofthesound wavesgeneratedby it.

Chishko (1992) modellediheacoustic emission from an axisynunetricalcrack

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II

growing in an unboundedhomogeneous.isotropicelasticmedium.Theextensionofthe crack was interpretedasaprocess where thefluxes of prismaticdislocation loo ps within thevolume of themediumwereannihilatedat thesurfaces of thecracks.ucccmpamed bytransitional radiation of sound. Based on this assumption,the space-time characteristicsof thegrowingcracksand theiracousticemissionpropertieswere determinedbythedynamics of thedislocation fluxesin the volumeof themedium.which in tum, dependedon several factors not directlyrelatedtofracturekine tics.Tilevolume mobility ofthe dislocations indirectlyinfluencedtheprocess of the crackgrowth.

2.2.2 Detection of Fatigue Cra cks byModalAnalysis

Theoretical andexperimental modal analyses havebeencarriedout by many researchers suchas Gomesand Silva(1990),Springerand Reznicek (1994),Silvaand Gomes (1994)andPerchardandSwamidas (1994), aseffectivetechniques to detect fatigue cracks through changes of modalparameters. The detectionandlocalization of cracksin a thick cantilever beam were investigated by Silva andGomes (994).

Significantreductionsofnaturalfrequencies were observedonthecrackedbeamhoth from numerical andexperimentalstudies.Changes in naturalfrequencies were used to predictandlocatethepositionanddepthofan"unknown"crack.A theoretical model was setupfor a cantilever beamwitha concentratedload atthe free end.Thecracked beam wasmodelleda.stwo shorterbeams connected through a torsionspring, which simulated the stif fnessof theoriginal beamat the sectionwhere the crackexisted. Based

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12

on the knowledgeofthe ratiosof the naturalfrequenciesof thecracked/ unt rackedbeam , a computer program determinedthelocation and depthof a crack in the structural element.The computationtimeofthis programwouldbequitelengthyfor complex structures.Sometimesthese computationswouldbedifficult and evenimpossible.since the use of a relativelysmallnumberofmodal parametersyieldsarank-deficient setof equatio ns.

Investigationsintothe changesofmodal parametersofa slenderplatewithnotch typecrac ks werecarriedQUIbyPerchardand Swamidas(1994).Results showe d that the presenceofa crackcaused reductions ofthenaturalfrequenciesfor a number of modes, when themode shapesexhibitedhighdegreesof curvature inthevicinityofthe crack.

Thisphenomenon.namely, "Theposition ofthe crackObvio uslyaffccts themodes differently",wasalsoobserved by ether researchers.forexam ple, Springerand Reznicek (199.1.),Silva andGomes (1994).and Meneghetti and Maggiore (1994).

Perchard andSwamidas (1994)and Chance eral.(1994)investigatedthe change ofmode shapes and curvaturesof structuresduetothe existenceof cracks. II was assumed thatat thecrack location.therewouldbeacharacteri stic change in themode shapethatdid notoccur in theuncracked model.Thereforethemodal displacementor the curvature, whi ch was obtainedby differ entiationofthemode shapes.could be used as sensitive indicators offatiguecracks.evenforthemodes thatexhibitedvery little frequency shifts.Curvaturesobtained from the experimental data inthestudy byChance eraI.werefound tobelimitedin highlightingthe presence of cracks. Chenand

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IJ

Swamidas(199 4)studiedthe changeof mod alparamet ersof a platewitha smallgrow ing surfacecrack.Different methods were comparedfortheirsensitivitiestilindicatethe presence ofasur facecra ck.Localstrai nfreque ncy respon se fu nctionwas foundto he a sensitive method for detecting surfacecracks.

Springer andReznicek(1994) investigatedanappropriate modelthat couldheused to representthephysical propertiesof a crack. Four linearspringstiffnessconstants, namely. oneaxial spri ngstiffnessconstant. two bend ingspringsti ffnessconslants, and onetorsional springstiffness constant,wereusedtomodelthe crackedsectionof a L- sectionbeam.Numericalresultsshowed a much greaterreduction innatural frequencies than that observedfromexperiments, indicatingthat some couplingeffects in till'actual beamwere not properlyrepresentedin thenumericalmodel.

Thesignal processingpart ofafatigue crackidentification pr\Jgramis sometimes calleda neuralsystem. Aneuralsystem mimicsthe pattern recognitioncapabilityof a humanbrain. Basedonmeasured changes ofstructural modalparameters caused hy fatigue cracks.theneuralsystem locatesandquantifies the changesofmass.stiffnessand dampingina structure.Aprerequ isiteforthe usc:ofa neural network is that it'must be trained . Mannan etaI.(1994) trainedtheneural systembystructuraldynamic modification(SDM) procedure.whichsetsup therelation between themodified structure andthemodificationinits mass.stiffnessand dampingmatrices. Chanceetal.(1994) showed thatmode shapes andcurva tures calculatedfromfinite element approaches cou ld alsobeused 10train the neuralsystem. In general.changesofmodal parameters. of

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,4

cracked/uncrackedstructures,obtained from experimentaltests,serve as abasicindicator of physical fatiguecracks instructural elements.However, studiesondetection ofthe presenceoffatigue cracks insubmergedstructuresarequite limi ted.Themechanismin whicha crack influences themodal propertiesofdryand submergedstructuresrequires furtherstudy .

2.3 Nearfield Acoustic Measurementsfor Fat igueCrackDetection Application

Traditionalmodal analysis wascarried our through contactmeasurementsby accelerometersorstraingauges,which is inconvenientand sometimesimpossibleinthe monitoring ofcomplicatedoperating structures.Acoustic modaltestingand acoustic

~nsitivi lYanalysis have beeninvestigatedby Guigneetal.(1987 (a),(b),1989,1992) and Okubo and Masuda (1990)aseffectivenondestructive andnon-contactdetectorsof fatiguecracks. Aconceptualideaof detecting fatigue cracks throughacoustic measurementwasintroducedby Gutgneetal. (1987a).Analysisofthedeformation of a paniallycrackedweldedT-platewascarriedout usingthis methodologyby Guigneet ul.(I987b).The feasibility10extractmodalinformation from nearfieJd acoustic profiles wasillustrated by Guign6etal.(1992).Vibro-acousticstudiesof a cantileverplate were carriedoutinair.using the finite element programABAQ US.The system wasmodelled using beamelementsand acousticelements.The structurewasexcitedatafrequency aroundoneof its natural frequencies.Acoustic pressure variation wasinvestigatedina

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15

plane perpendicular10the surface ofthe plate along its length. The acoustic pressure variationat the platesurface was observed to be in the shape of the specificbending mode ofthe plate.The feasibilityof extracting modal data fromexperi men tally ohtained acoustic pressuremeasurementsand acoustic intensitymeasurementswere alsoillustrated byGuign!!etaI.(1992).The relationbetween the mode shape and the verticalcomponent of acoustic intensity was given on a theoreticalbasis and proved experimentally.

Monitoringchange of structural modal parameters,dueto the presenceof fatigue cracks, through vibrational and nearfield acousticalmeasurementswas studied by Klein et al.(1994).FRF(frequencyResponse function)and PTF (fressureIransfcrfunction) measurements from accelerometers and microphones. were conducted on a cantilever beam in air at differentfatiguecyclesunlit the plate failed.The reductionof natural frequency was observed from both FRF and PTF measurements.

DynamicAcoustic IntensityScanning (DAIS) (Guigne etaJ.•1989). orNearlield Acoustical Holography(NAH)(Hallmanetal.•1994) werealso used to detect acoustic sources or sinks in light or heavy medium. Guigne etaJ.(1989) studied a DAIS technique for its potentialcapability of detecting fatigue cracks of submerged structures.

A plate was put into a cylindrical watertank.Acousticintensitiesatdifferentlocations near the plate surface were measuredby using a pair of hydrophones. Measuremems from repeated tests showed good agreement.indicatingthat it is possible to detect fatigue cracks through a DAIS type oftechnique. Hallman etaI.(1994) used the NAH technique to localesources in a full enclosure with hard, rectangular walls. An aluminum plate was

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16

excited mechan ically by a pointforce.andthe velocityof theplate surface and the acousticimensity ncartheplate surface werereconstructed usingtheNAHtechnique.

Experimentalresults showedthat the acousticene rgy entered and left thevibratingsystem through the driven pointandthe leaks (loosemountings ofthe plate)atlow frequencies.

Athigher frequencies,the effect of the leaksbecamedominant. Theseleaks were detectedas sourcesorsinksofacousticenergyon theplatesurface.Theresultsshowed promisein detectingstructuralcracks(leaks) by thesinkorsourceeffectof thenormal acousticintensitymeasuredor reconstructed.

2.4 Summary

Insummary,thevibration and acoustic properties ofsubmergedstructureshave beenstudied.Studieson the influenceof immersiondepths onthenatural frequency of submerged structureshavebeenexplored.Reports ontheeffectofimmersiondepthson damping ratiosofsubmergedstructuresarehowever limited. Detection of structural fatiguecracks throughvibro-acoustics is considered feasible in experimentsand advantageous in practice. Currentlyreported studiesfocused mainly on fatigue crack detection of plate structures in air,Hence the studies presentedinthis thesis focuson the detection offatiguecracksin submergedstructures throughvibrational and nearfield acousticalmeasurements. The influences ofimmersion depths on bothnatural frequencies and dampingratios ofa submergedcantileverplatewereinvestigatedwithregards tothe first ten structural modes. Anumerical studyconsideredthevibrationaland nearfield

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17

acoustical properties ofthe uncrackedcantilever plateinairmedium.

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CHAPTER 3. MODALANALYSISTHEORY

Structuralmodesare representedbytheir naturalfrequencies.dampingratiosand modeshapes inamodalanalysis.Thebasicaspects of amodalanalysis include three parts: I) mathematicalmodelling,using differentassumptions about thevibrating systems suchassingle:degree ormulti-degreesystems, undampedordamped systems etc., 2) modaltesring,using varioustestingtechniquesandsignalprocessingmethodsto obtain frequency response functionsof thestructure; and3)modal parameter extraction thatserves as a bridge10linkthemathematical model and themodaltesting data.Simple or complicatedsoftwareisinvolve d inthe idemific alion process,wh.ichextrac res modal parametersofthestructure.A brief reviewis givenin thefollowingsectionsonvarious aspects ofmodalanalysis.withanemphasisonthoseaspectsused inthis study.

3.1 Metheme tlcal Model

Experimentalmodal analysisis socloselyrelatedtovibrationtheory that mathematicalmodelling becomes abasic part oftheprocess.Thesystem modelledis assumedtobelinear.time-invariant.and satisfiesBent-M axwell's reciprocitytheory.

3. 1.1SingleDegrH- or-Freedom (SDOF) Model

The simplestmathematicalmodel of a dynamicsystemis aslngle-degree-or- freedom model;the practicalsystemismodelled asaspring-mass system withhysteric

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19 orvisco us damping.The governing equationis:

mX(t )+cx(t )..kxLt.]"f( t) (3. n wherem,c. and k arethe mass,dampingand stiffnessofthesystem.TakingLaplace transform ofEquation(3.1)and assuming theinitial displacementand velocityof the system tobezero.the transfer functionof thevibrating systemcanbeobtainedas:

H(s)_s · !i!l.._ _

s·_

F(s) msz"'cs+k ^(3.2)

wheres:::(J+jrois the Laplacevariable;andF(s) and X(s)are lhe Laplacetransforms oftheinput force and the outputresponse. H(s) representsdifferentresponsequantities inmeasurementswhen a isassigneddifferentvalues. Itisadisplaceme nttransfer function orreceprance when a=0,IIvelocity transfer functionor mobilitywhenfl

=

^I,

and an accelerationtransferfunctionor inenancewhena:::2. When(hesystemis excitedby a sinusoidalforce, thetransfer function H(s) iscalledfrequencyresponse functionor FRF,and for the case of receptanceit is:

(3.3)

whereFUm) andX(jW) are the fouriertransformsoftheforce and displaceme ntsignals. Itisclearthai HUw)is complexinnaturedueto theexistenceof dampingin the

For a systemwithviscous damping,the Nyquistplot ofthe mobilityfunctio n

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20 yOW}=jw •HOw) represents a modalcircle.Its equationis:

[Re(Y)-1 j2+[ Im(Y)]2=(1)2

1C 1C ^(3.4)

Equation(3.4)fo rmsthefoundation ofvario us curve-fittingmethodsused toextract modalparametersfromexperimentally measureddata.

TheSOOF modelstated above isusefulbecauseit is aprincipalanalysis that can beadopted tothe analysis of a multi-degree-cf-freedcm (MDOF)system.Itisapplicable to the analysis ofaMOOFsystem whenthe vibrationmodesof thesystem are relatively separated betweeneach other.

3.1.2 Multi Degree.o r.F reedom(MDOF) Model A MOOF modelcanbeexpressed as:

(mJ{l<(tl}.(c ] lx(t)}.(k ]lx (til-I'(til ^(3.5)

Followingsimilar proceduresdescribed for theSOaFsystem, thedisplacementtransfer function of aMDOFsystem canbeexpressedas:

(H(s»)"( B(s ) j -l. wh er e [B( s ) )-s 2[m]+s [ c]+{k]

The frequencyresponsefunctionmatrix(HUm)]is obtainedbyreplacingswithjoo in

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Equation(3.6) and is givenby:

[H\jlo))J =[B(j(,))

J -l ·be(t(~(~l~) 1

^13.71

wherelAOw)]isthe adjoin!matrixof {B(jw» .and Det[B(jto)Jisthe determinantof (B(jw)].Thenaturalfrequenciesw,(r- 1.2.3•...) ofthesystemcanbeobtainedbyselling the determinantof [B(jto»tozero:

CombiningEquations(3.7)and(3.8).thefollowing condition is deduced: [BU"'.))IAU"'.)]=[O]

(3.H)

(3. 9) Matrix[AGro.)] in Equation(3.9) containsinformation about theeigenvectors of the system.Toillustratethis. considerthehomogeneous form of Equation(3.6):

lBU"'.)) [' .1= 101

1'=1,2•...•0 (3.10) where(XGw)l has been replacedby (x.lto indicatethe satisfactionof thehomogeneous condition;to,andIx.)are therlh naturalfrequencyandeigenvector.respectively.

Comparing Equations (3.9)with (3.10).itis clear thai anyonecolumn ofmatrixlAGro,)!

isproportional tothe eigenvector Ix,Ifor the same orderofnaturalfrequency.

Thetransferfunction inEquation(3.7) canfurtherbeexpressed as alinear sum ofthe modalresponsesofthe structure:

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22

Equation(3.lll or(3.12) isthe mathematicalmodel fora linearMDOFsystemwith generalviscous damping,wherew,.istt'.etthcomplexfrequency, and Is.)isthe rrh eigenvector

1x,1

nonnaJized withrespect(0the massmatrix(m).Itcanbeshownthat matrix(A(j<o)JinEquation(3.7)isasyrrunetric matrix.Inpracticalmeasurements. only oneroworone columnofmatrix {A(jfll)) needs 10 bemeasuredto detennine the eigenvectors compk tcly.

The physicalmeaningof[H(jm})canfunherbestated as:

(3.13)

whereFtUoo)is the Fouriertransfonnof the excitation force acting at pointkof a structure; andX;(jCil)istheFourier transformoftheresponsemeasuredat pointiofthe structure.Thereciprocityof the systemensures thatHoIl(jCl) ",HtJ(jOO).

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23

Ifa system tsproponionaJlydamped.itwill S3lis( y:[li.llm)"1[c]=Icllml..[k].

Asimple decouplcdformof EqUlllion(3. 12) canbeobtainn1as:

(3.14)

wherero,is therthundampedfrequency.~istherthdamping ratio ofthesystem. and ls.lis therthrealmode vector nonnalized to therlhprincipalmass.Equation(3.4)and (3.1 1) through(3.14) are the basic mathematical modelsthatarewide ly used in practical modal parameterextractionprocedures.

3.2 ModalTesting

1beobjectiveinmodaltests is 10 obtains\.lUctural responsefunctions, whic hare usedas inp utsinthemodal parameterexuaet ionprocess..Variouskinds ofexcneuonand dataprocessingmethodshave been used 10 obtai n accuratemeasurementdata. andthey arebrie fly reviewedinthefollowin gsub-sections.

3.2.1ChoiceofEXc:lta tlo nSig na l

Different measurement techniques are closelyrelatedtodifferent excitationsignals

suchas fast-sweepsineandpseudo-random signals.Ina fast-sweepsineexcitation.the systemis excitedbyasine signal with asweeping frequencyand thelinear responseof thestructureis measured. Itisimponanttomaintaina propersweeprate 10eliminate the

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24

dlstorncn effects of FRFsin these tests. Ina pseudo-randomexcitation. a"periodic random"signal, which consistsofamixtureofmagn itudes andphases of various frequency compone ntsisusedtoexcit ethe struc ture, andtheresponsesof all freq uency co mpone ntsaremeasu redat the same lime.A "pseudo- random"signal maybeadjusted tosatisfy certainrequirement s suchas equalenergy at eachfrequenc y orlandexact periodicityin theanaly zerbandwidth,toeffectively reducetheleak ageand aliasing errors usuallyencounteredin discrete signal analyses.

3.2.2MeasurementorTra n s rerFunetlo ns Irem PowerSpedralAnalysis The accelerationor aco usticpressure response of astructureexcitedbya random signalf(t)can be writtenas a randomprocessX(I), whe re((I)and x(t)are assumed10be ergod icforastationary randomexcitation.The propertiesof thes erandom processes can be represent edby their correl ations Ru(t)or R.n<t)in thetimedomain and theirpower spectralden sities 5 ..(00) or 5n<oo)in thefrequencydomain.Thefreque ncy response function ofthesyste m H(CI) is related 10 the spectraldensityof theinput and output signals by:

(3.1S)

The aboveequationdoesno tconta ininformationaboutthephaseof the transferfunction.

In practical analyzers such as aB&K2032dual channel analyzer.aFRF isdefinedby thecross spe ctral densityof the two channels.IfchannelArepresentstheforceinput

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25

(I)andchannelB represents theresponsextu,the FRFcanthenbewrittenas :

".(101)..S",,(lol}

s:;TWr

13.16)

(3.171

where SU<OJ)isthe one-sided, real valuedensemble average ofthe squaredmagnitudesof the one-sided instantaneouscomplexspectrumas shownin Equation(3.18).

(J.IS ) The crossspectrum betweenthe forteand responsesignalsisthecomplex ensemble averageof thecomplexproductofthe complexconjugatedone-sided instantaneous spectrum F'(ro)of force and the one-sidedinstantaneous spectrumX(W)ofresponse:

(3.t 9)

Aone-sidedspectrum isdouble the value of thetwo-sided insta-.aneousspectrumat the positivereal frequencyaxisand zeroal the negativerealaxis.

In practice,thechoiceof HjorHz in Equations(3.16) and (3.17)dependsonthe influenceof noise.Ifthe inputsignalismoreprone to noise.forexample.in thecase of resonantmeasurements,Sn<w)will giveerroneous results, andHI willbetheappropriate choice.Iftheoutputis moreprone to noise, asinthe case of anti-resonantor acoustic

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26

measurements.S..(£Il)willbeerroneousandHIwillbethe properchoice.The presence of noiseis verifiedbythe coherence function.whichis definedas:

(3.20)

where 0<}'r/( O)<:I.

3.2.3DlgilalSignal Proc es sing Techniqu es

Duetothetruncation and discretization ofmeasured signals,errors are introduced inthe estimationsofFRFs, which includealiasing andleakage. Mostanalyzers have built inmethods to provide forpropercompensation.A brief review abouttheseproblems and compensationmethods are giveninthefollowing sub-sections.

3.2.3.1Aliasing

Aliasing isthe resultof a low sampling rate inanAIDconversion procedure.The frequencycomponentthaiis higherthantheNyquist frequency willbereflec!cdback into thecorresponding lower frequencyrange,causingspuriousresuas.To overcomealiasing, ananti-aliasing low passfilter was automaticallyinstalled intheB&K 2032 analyzer.

However. a rejection range or (0.5- 1.0)(ro,l2) is stillrecomme ndedbyEwins (1984), whereto,is the sampling fre quency inthe measurements.

3.2.3.2 Leakage

leakage is caused by the use or afinite lengthoftime history insignal analysis andby the assumption thatthe datil. ate periodicin thaifinitelength. As a resultthe

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27

spectral magnitudeisreduced inthevicinityof the truefrequency vulueandspreadover the adjacentfrequencies.To overco meleakage,weightingfunc tions,orwindows. arc usedtomodify thetime domainsignals.Differenttype sof windowsare availablein practicalanalyzers,forexam ple.Cosine Taper,Hanning,Resse l,Flat topand Force windows.

3.2.3.3 Zoom

In structureswithligh t damping,resonant peaksare vel}' sharp.Zoomingof the frequency spectrum within anarrowfrequency band isneededtoobtainaccurate estimationsof modal dampingratios. Inorderto eliminate theenergy outside the interestedfrequencyrange,theexcitation should alsobebanded ifzoo misapplied.

Whenmeasurement quantities arenot transfer functions,recallingofthe weighteddata isneededtocompensatetheattenuatio nofthesignalbythe application ofthewindow.

3.2.3.4 Avera geand OverlapAverage

Averagingoftheprocessed data is requiredwhen random ordynam icsignals are processedto extractcharacteristic parameters. The number of averagesrequired is governedby the statisticalreliabilityand theremoval of spuriousrandomnoisefrom the measured signals. Overlapaverageisanotherfeatureofmodemanalyzers.Overlap effectivelyreducesthe data length needed inthe measurement. Italsoproducesa smootherspectrum than wouldbe obtained if each data sample is used only once.

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28 3.3 Modal Par a meter Extraction

Foragenerally damped MDOFsystem.therelationbetweena FRFobtained from experimentationand that obtainedfromamathematicalmodelis:

wherethesummation from01to"2is themodal responsein the interested frequency band;and thequantitiesM.R;~and K.R,k aretheinertia and the stiffness residualsofthe response.whichreflect theinfluences of the modesbelowandabove theinterested frequencyband and are usually treated as constants.In thevici nityofa natural mode, the lotalfrequencyresponse ofa structure is assumedtobemainly contributedby(he modal responseofthaI mode. According 10 the treatment of the influencefromO~t modes,modal parameter extractionmethodscanbedividedas SDOFmethods andMDOF methods.

3.3.1 SDOF meth ods

Thesimplest SDOF method isthePeak-Pickingmethod,which assumesthata FRF isonlycontributedbythe nh modal responsein thevicinlryoftherlhmode.

Therefore.Equation (3.14) becomes:

(3.21)

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29

The modal parametersof therthmodecan beextrac ted as follows:

I. The frequency thaI correspond s tothepeakmagnitudeofthe FRFis[he estimated naturalfrequency<0,;

2. The damping ratio~risrela tedtothefrequencies00.and~as:

whereOJ.and0\are the half- powerpoints of the FRF magnitud espectru m.

3. Therthmodeshape is obta ined from Equation (3.21)as:

(3.23)

The signofthe modalvector canbeobtainedby comparingtherelative phasediffe rence of Hik(ro,)wheniandklakedifferentvalues. ThePeak-Pickingmethodisarough estimation becauseitdoesnotconsid er the influence ofother modesand usesdatar~l1'l a small number ofmeasureddatapoints.

ThemostwidelyusedSOOF methodin practiceistheSDOFcurve-fittingmethod.

Inthevicinityof therthmode.the influe ncefrom othermodesis assumed tobea co mplex constantBI/:" Equation (3.21) becomes:

(3.24)

ThefirstpartofEquation(3.24)satisfies Equation(3.4)andis amod alcircleinthe

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30

complexdomain. Modal parameterscan then be extractedin the followingsteps:

The frequency thatcorrespondsto the maximumrate of sweepofthe arcar the successive measuredpoints nearresonanceis therthnatural frequency. Thedamping ratio canbeobtainedbythe followingequation:

(3.25)

where!Il.and

ro..

are two angularfrequenci es oneithersidesof angularfrequency fil" and a,and 9. are anglesshownin Fig ure3.1.In practice~fis obtainedbychoosingdiffere nt setsof00,and0\andby averag ingtheresults obtained from Equati on(3.25 ).

1m loci Re lee) _._-t--- - -

Figure3.1 Nyquisteleele

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31

3. Therthmod e shape isrelated10 the diameter( Olio' )oftheNyquis tcircle by:

13.16)

3.3 .2 MDOFMethods

Ifmodal frequencies ofastruct ureare veryclosely coupled,MDOF methodsere required.Thesimplest MDOF methodistheextensionof theSOOFcurve-filling met hod. Amoredet ailedinfluenceof modalresponsefromothermodesis cons idered inthe vicinity of therthmode. Equation(3.14)isexpressed as:

(3.21)

Thesecondtenn in Eqt'aticn(3.21)rep resents theinfluence of theother modes ontherlk modal response.

The SOOFand MDOFmethod smentionedabovearcused in real modalanalysis.

whichassumesproportional structuraldamping.Ifthestructuraldamping is of general cas e,complex modalanalysishastobecarriedOUI.Naturalfrequenciesandeigenvectors ofthe structurewill becomplexin nature.

Tim edomainmethods.such as ComplexExponentialMethod and AbrahamTime DomainMethod.havebeenusedto dealwithcomplexmodal analysis. Adetailed discussion ofthesemethodscanbefoundin Ewins' book (1984) or Perchard and

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32

Swamidas'soverview(1990) .These methodsmaygivespurious modesin theestimations, andthe computationtime wouldbe quite le ngthy (orhandling practicalproblems.

3.4 Acousti calMeasurementto ExtractModalParamete rs

Acousticintensity is theamountof acousticene rgypassingthroughaunit areaper second. Itsunit is joulepersquarecentimetrepersecond. Themeasure ment ofacoustic intensityinvolves measuring thetime-avera gedsoundintensity:

(3.28) whererrepresen tstheme asureme nt position,panduaretheacoustic pressureand velocity respective ly. The accuracy of inte ns itymeas urementis limitedby theaccurate measurementof paniclevelocityu.

CrossspectrummethodwasproposedbyChugandPope (1978) andFahy(1977) to measuretheacousticinten sity.Twopressu re signals. whichare measuredfrom closely spacedmicrophon es, areusedtoestimatethe acousticintensity I,andintensityspectrum l,(OJ)atthe midpo intbythefollowingequa tions:

(3,29)

(3,30) whereSIl,P,(W)isthe one-sidedcross spectrum betweenpressuresignalPIandPl '.toris the

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3J

micro phonespaci ng,andPristhe densityofthefluidmedium.A simple verificationof Eq uation (3.30)isgiven as follows.

The acousticintensity slatedinEquation (3.28)is the crosscorrelationR",,( t)of press ureandparticlevelocity whent eq uals to zero.R,..(O)can be expressedas:

",d

Ir (w)=Re[S",, (w)1

(J.3D

(3.32)

If theacoustic pressureP.andthevelocityII,ofa spatialpointrisapproximated bytwo signalsPIand PI,whic h ate closelypositio ned nearpointr.as:

the n the correspo nding Fouriertransforms are:

Pr(W)"[P1( CaJ) ;P 2«(IllI

Ur(CaJ).-)

CaJ:

f

Ar[P2(foI)-P1(w))

(3 .34 )

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)4

Assuming ,P,(w );: P,.+jPI,and PilJ) =PJt+jPlI,theacoustic inlensity spectrum canbeobtained from Equation(3.321as:

(3.3.5)

According toEquation (3.19),the crossspec trum5'1,, (11))ofpressuresignalsPIand PI can beobtainedas:

Comparing Equations(3.35) and (3.36), itisCoundthat:

(3.37)

To reduceerrors related10themeasurement,the twomicrophonepositionshave tobe very closelymatchedcompared withlite acousticwave lengthoftheinterested freque ncy, Le••~AC«1.whereIt,EWCtistheacousticwavenumberandc,isthesound speedintheFluidmedium.This condition isrelatively easiertobesatisfied for intensity measurementin watermedium,becausethespeedofsound inwateris much fasterthan thatinair.In fact,thehydrophonespac ing bastobeextended10 reduce theerror of phase-mismatchbetweenthetwo byd!'o?honemeasurementchannels.

Thephase mismatch erroriscausedbythephase mismalc:bbetweenthetwo

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35

microphone s orhydrophones.amplifiers. and filtersinthetwomeasurement cha nnels.

A circuit-s wi tchingtechniquecanbeinvok edto eliminatethis error. Thecorrected formulato estimatetheacoustic intensity spe c trumis:

(3.38)

whereHI andHIarethefrequencyresponsefunctionsof the first and second microphone channels.respectively.

Thephasemismatcherrorisdirectlycausedbythe use oftwo differentsetsof microphonechanne ls. Petterson (1979)impr o ved themethodbyusingonlyoneset of microphonesystem . Onemicrophone movesfrom point to point.with phase referencing to thedrivin g system .A separatesensorsuchas an acce lerometeroranothermicrophone canbeusedas aphase reference.The soundintensityspecuum is thencalculatedas:

(3.39)

whereS..is the auto-spectrumofthe reference signal.

S".

isthecross spectrumbetween thepressuresignal at firstposition andthereference signal.andSP:zIistheresultwhen the pressuresignalfromthemoved position isused.Bysetting upareference signal, this methodremovesthestrict requirement of phase-match betweenthe twomicrophone cha

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36

nnels.The measu rementaccuracyis actuallycontrolledbythe movingdistance of the microphone.

In order 10 setup commonreference forcomparisonbetweendifferen t tests, Guigaeetal,(199 2 ) furtherderived thenormalizedacou sticinte nsityspectrum,whichis:

If thestruc turedriving force C(t)is used asthereference signal.Equation(3.40)becomes:

(3.41)

whereH9r^f^(w)andHlIlf (00)arethe pressuretransferfunctions(PTf)measured attwo points separatedbydistanceJl.f,and they canbeexpressedas:

wherep.(ro) and P1(ro}arethe Fouriertransformof PIand PIand F(ro)isthe Fourier transformofthedriving force(t).Equation(3 .41)isusefulin acousticmodalanalysis wherethe drivingforcevaries withdifferentresonantmodes.Theconstructionof mode shape. howe ver,requires the drivingforcetobeconstant throug houtthetests.

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CHAPTER 4. NUMERICAL ANALYSIS OF AN UNC RACKE D CANTILEVER PLATE USING FINITE ELEMENT ANALYSIS(F EAl

4.1 Estimationof ModalParameters by FEA

Finiteelementanalysis(FEA) was carriedout toestimatethenatural Frequencies andmode shapesof an uncrackedcantileverplatespecime n in airand in water,usinga softwarecalledABAQUS.developed byHibbiu,Karlsson&Sorensen Inc.(ABAQUS manual,1994 ).The firsttenmodesweredetermined forthe cantileverplateinair.

Natural frequenciesof the submergedcantileve rplatewereestimated hy usingthe underwater shockanalysis(USA) program incorpo ratedin ABAQUS.OnlytheIirstthree bendingmodes werecomputed.

4.1.1 The Ca ntileve r PlaleSpecimen

Figure4.1 showsthe geometryof a cantileve r plate specimen.A stubwaswelded nearits clampingend 10 simulatea"Tejoim".whichisanacceptedstructuralunit for tests. Theplate was made of mildsteelwitha massdensity ofQs=7860kglm'.a Young'smodulu sofE=2.01X10"N/m'and a Poisson'sratioofv

=

0.3.The physical propertiesofthe watermedium.used inthenume rical simulation are al atemperatur e of Tr=20"C.amassde nsity ofQr=998 kglm'anda soundspeedof cr=1450mls.

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38

t ===. ^Z :l:'= "=3!I=I~ F'= _I -C" =;" =· ==,,=======::J

'C!C!4D

y

, . "

,..,

fJ'!

~~

l,/1f"o.ll'Igo.ng&>

V ·.. . - i

~ ill!...

Figure4.)Thecantileve r platespeefmenunderInnstigation

(}

1

Figure4.2: A S8R5shellele mene

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J9

F1ellrc4.3Finitedementmesh or1Il1Cr.ck ed anlllner..lthoultheweldJhlh

f1aurt4,4 Finite dement ID"hor untr ackedantllntrwiththe "tid stub

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40

Figure4.5Modeshapes

o r

^theunc:rac:ked c:antlleverplate Inair

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41

Figure 4.5(cont.) Mode sha pes

o r

theuncrackedcantileverplate Inair

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4 '

Figure4.5(cont. ) Modelib. pes of the uDuuktd cant lln erplateIn aIr

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43

Figure4.5 (cont.) Modesha pes of the uncrackedcantileverplateInair

4.1.1 Modal Parameters of tbe Cantilever Plate In Air

S8RSsh elleleme nt,shown inFigure 4.2,wasused10modeltheplate.Thisis a 8 noded doubly-curvedshellelement withfiveactive degrees of freedom per node.

namelythreedisplacements and two in-plane rutauons.Thecantilever plate wasfirst modelledwithoutconside ring the weld stub;200ele ments wereusedasshownin Figure 4.3. Natural frequenciesand modeshapes of thecantileverplatein airareshown in Table4.1andFigure4.5respective lyforthe firstten modes.In asecondcase,the weld stub was modelledbyanadditional20 S8RSshell elements as shown inFigure4.4.

Table4.1 showsthe natural frequenciesof thecantileverplatewiththe weldstub. The mode shapes are almostthe same as that shownin Figure4.5.

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44 4.1.3 Underwater Shock Analysis(USA)

When astructure is submerged in weer.its dynamicproperties are influenced by the coupling motionof the fluid.Theinfluence isusuallycategorizedinto twoaspects, vts.,Incadded massandthe addeddamping ofthestructurecaused by thedeformation ofthefluid surroundingit. Asa result,themodalparameters ofthe coupledfluid- struct uresys tem aredifferentfrom thaiofthe drystructur e. Unde rwatershoc k analysis (USA)program,available in ABAQUS. was used to investigate the changes innatural frequencyand damping due to deepsubmergenceof thestructure.A smoothcantilever

.:'. 'D _~2 ¹ ² 3

Figure4.6 A S4RS shell element Figure 4.7An USI4boundaryelement

platewas usedin theUSA modelling.S4R5shellelements, which are four noded shell elements withfiveactive degrees of freedom per nodeas shownin Figure4.6,wereused to model halfthecantileverplate.Symmetric boundaryconditionswere applied along thecentral lineof the cantilever plate.USl4elements. whichare four noded boundary

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45

elementsas shown in Figure 4.7.were usedto modelthe fluidmedium.USI" elements coupledthe structuralvibrationwiththatof an infinitefluid mediu mhy:Idoubly asymptoticapprox imation(OAA). "Doubly asymptotic approximation s are differential equationsforthe simplified analysis ofthetransie nt interactio nbetweenaflexible struc tureand a surrounding infi nite medium" (Geers,1978). The first order approximationOAAlwas giveninclassical matrixnotationby DeRuntz(1989) as:

(4.1)

where{M,]is thesymmetric fluidmass matrix,(p.1 is the scattered-wavepressure vector, p,andc_fare mass density and soundvelocity of the fluid, respectively,(Ar)is the (diago nal) malrix offluid areas.and{J1.1is the vectorof scattered-wave fluid velocities normal tothe structure'ssurface,whichis coupled10thestructure's response by:

(4.2)

where[0]1islhe transformationmatrix Ihat convensthe structuralvelocitiesI

x

Iinto normal surfacevelocitiesat the centroid of eachfluidelementandI

u ll

is the fluid incident velocity.

Anacousticimpulseexcitationwasappliedtothecentre ofthecantileverplate's freeend.This simulated anexplosionoccurring inthez-axisfar from the cantilever.The

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46

,Xl0·'

:[0.2

" 0 '6-0.2

0.03 0.04 Mme(l)

Figure 4.8Freeresponse

or

the submergedeantllenrplate

300 <00

IrequencyIHZ)

Figure4.9 Free response power spedrum

or

^tbe

submerged canlllenr plate

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47

distancebetween theexplosion source(or charge)10thelocationofthecharge stand

orr .

i.e..thecentre ofthefree endofthecantilever,\113$given3.\999910approxtmatetbe source wave as a plane wave.Thepressure profileat the charge stand off wasa step functionwith amagnitude of1.0.Thechargewas releasedat 0.00075second and the free responseofthesubmergedcantileverplait wascalculated at16384 timeincrements withafixed time interval of 0.000025second.The freevibration displacementat the centre point ofthe cantilever free endis shown inFigure4.8.FastFouriertransform (FFT)of thefree displacementresponsewas obtainedbyusingcomputerprogram MATLAB.Thepowerspectrum of the free responsewasthencalculatedand is shown in Figure 4.9.Thepeakfrequencies in Figure4.9 are alsotabulated in Table4.1.The choiceoflimeintervalandtime incre ments inthe analysisisa compromi se of data qualityand computatio n time.While smalltime intervalscould catch highfrequency components.inthefreq uencydomain , theyalso increasethecomputation time and decrease the frequencyresolution.TbefrequencyresclcncninFigure4.9is2.44Hz.

Table4.1 Nalura l rrequ ent}'or Ibeuncra cked cantllenrrromrEAesllmatlon Mode

1. 2.

Il 3. 21

4.

^3. ^4t

s. ..

lC. (Hz) 25.841613 165.1 451.8 :516.2885.2926.1 1424 14:551708 Air IC.,{Hz)25.87 161.4 165.44:51.8 :517.0 884.4 927.1 1424 1452 1708 Water l£wCHz) 17.08112.3 336.8

If..If.,: natural frequencies of the uncrackedcantileverin airwitooucand with the weldstub 2f.: naturalfrequencyoftbeunoothcantileverin^Wiler^.

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48

4.2 Experi mental Study on Modal Prop ert ies

o r

the Uncrac ked CantileverPlatein water

4.2.1 Set-upandMu sur r mrn tSystem

Theexperimentwa.sconductedin a12xI2,;13.5 itdeep lank.AnJ-beam of height 1673mm wasfixedonto a IIOOx650x300 rom concreteblocktosimulate arigid base.

Thecantileverplatespecimenshown in Figure4.1wasclampedto thetop oftheI-beam witha coverplateandfour bolts.each tightenedwitha 250 It-lbtorque.The coverplate wasalsoclamped tothel-Beam alonewithanotherfour bolts.each tightenedwith aISO ft-lbtorque.

Thecantileverplatewasexcitedbyapointforce acting at the centreof itsfree end.AKistlerMOD912force transducer wasusedtoacquiretheexcita tionsignal. A B&K4809exciterwasanachedtoan aluminumlube.whichin tum,

w as

clamped through analuminum clamping systemtoasteelframethatsat onthelopofthedeeptank.A hollow stainlesspushrodand a steel stinger connectedthevibrationexciter andtheforce transducer.Theoverallset-up isshowninFigures4.10 and 4.11.Theexciter-hanging system was designed10provide sufficient dynamic stiffnessin thevertical direction.The clamping framealsoallowed theexciter tobeadjusted ill three displacementdirections and threerotationaldirections.Thisallowedliteprecisealignmentadj ustmentsbetween thecentral line of theexciterand that oftheforce transducer. The pushrodwas designed inviewof varyingthe water depthabovethecantileverplatespecimen.

Randomandpseudo-random excitations wereusedintheacceleration andacoustic