ON A VIBRATION R E SPO NS E OF A

ON A ^{VIBRATION R} E SPO NS E OF A

PLATE S UB MERG E D I N FLUID

By

©

SUY UNCAO, B.ENG.,M.Sel .

A THESISSUBMITTEDTO THESCHOOLOF GRADUATE STU DIES IN PARTIA LFULFILLMENTOF THE

ItEQUIR EMENTS FOR THEDEGREE OF MASTEROF ENGINEERING

MAY 1994

FACULTYOF ENGINEERINGANDApPLIEDSCIENCE MEMO RI AL UNI VE RSITY OF NEWFOUND LAND

ST.

JOH N'S NEW FOUND LAND CANAD A

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Abstract

This studydealswith the ,Iy namirl"harach'risl;n;tiCplat,'SS1111l1l'''~I..1illwau-r usinganal y ti cal and experimentalml1ll1)(ls.Till'1'111II'Sr'nusids-n..lI10l\""twokiud»lOr boundary conditions,onewit hrl;ullpl~l. rr.'t~·da lllp(..I·rn,,·(('F<:F)s11ppnrts, au"I!I.'r withsimple-frce. simp lc-frcc(SF Sf )support s.TIll'Y 'Iresu b rllC'r g l..1in\\',II,"Til l,lilr"T' ent dept hs.

For the experimental st udy, modnln-sting of 1Ill'1\\'0 plill'·Sist"MTit..l011(,inair andin water. The effectofthede pth ofSI,bl lw r gt 'llc'l!oflire platt·isillVl'SliRillt-,I.Till' simplysupportedconditio nwas sim \llalc dbyusing a notchedI>I,ll cwill." d;uIlIH·,1 end andthe finiteelementso[lwan.'AIJAQ CSisIlSl'(llu.Ic lt'rlll illl'lhl ~.Iill...n.~iulls..f the notc h.The appropriatenessof tiletestedplates asIIMllld~ofthl:(~F( ~F^1111,1SFS)o' plates is verified.The added mass factor.whirhis1ISf":d hIeccounrfo r tl...,I"n"'i~"'"

ill ndural frequencies,and the Inerease ill!nOflal<lalllping ratios fu rt!w\lihratill~

platessubmergedinwaterhave been evaluatedfort1\l~lintlivemoe!C"!.Tile ,·If..<:l..[

theplate bounda rycond it ions011rhevibrationofpliltl~ill waterill alsoiIlV'!>ILiWllI. 1.

Intheanalyticalst udy,a thinplate 1I111lcrg" illg aflexu ralhCllIlill1;vihra Liu lI illi '

bod yofhomogeneous,inec mpre eib lc.'01111invisridf1uie!whose1II0t iol1is irrotntionul, is consid e red.Thegovern ingequatio nfor thesurfacedisplacementof theplall!·lIlli,J sys te misderived.Inthe crforlto solvethevt:!ocitypctcutiel,theg(!IlI:rallill' ~i!.ti~.I·,J fluidfreesu rface bou nd a ryconditionis applied.Till! relationshipbe tweentile luJdl:d mass andthedepthof su bmergenceabOVI!andbelowtheplate isfllitablislulfl.

The analyti cal solut ion is applied to the CFCF and SFS"plates submerge din

willl'l,and llwexperimentalilndanalytical studies on the added mass due to the Jlwst:nn'ofnuldarecompared.The maximum difference betweenthe measured ami thepredic ted ad<..k'<! mass Factors is found to be within 1,916%,

A cknowledgments

Th e author wo uldliketoap pl'l '('i il t"till'sup!'urlnndI'llnlll ril~I'IIl"1I1[ru lllilldh'i,IIl;IL~

inthepro gram ofxlastcr of ElIgill('l,'ring.Partb-ular rhunks Mt'dm-:

ing, for their supervisio n,guidanceatnllillililcial.~ llP IJl)1"1th,'Olll-\hlllll.thisprll.!!,f'" II.

(b)Facilityof EngineeringIlll,l,\ pplil'd Sril'llel'i\11(1till'SelIO'"\If(:r; lOl 'l ill , ' Stud ies fo r theirfinancialsup poriofl-\r1ldllll lt·[('llo\\'shipi1IUIII' <ld li ll l-\;ls.~i sl;ll l lsllil"

(e)~Ir.r\.Burseyfor his I)('lpinpn'pa riugIIll'exp "rlllll'uli,1 d,·\' i.-.'s .

(cl)Xlr.J.Andre ws,~Ir.II.Dy. ~andother staffill the;\li,d,illl' SImpfor tlu-ir techn ica l servl-os andsuggestionsill tlHlllnf,u:luringt.hetankiLndsdllp.

(e) Finally,herFa milymembers,"spl>/'iallyherhllsh'lUdrun]par/' Ut.s ,fertheir deep unders tan ding andencoura gement.

Contents

Abstract Ackn owl edgem en t

Listof Figures List ofTables

List of Symbols 1 Introduction

2 Lit era t u reReview

2.1 PreviousWork011the Fluid-Str uct ureInteraction Problems 2.1.1 Analytica l Approaches..

:U.2 N\,nwricalApproaches 2.1.:1 Experiml.'ntal'\j,proildws .,., SUllImary

3 Modal AnalysisTheory a.1^ThrorctiralBa~i:~lOr^{)IOIlal.\lIalysis.}

jii

vii

ix

10

11

"

"J.:! j·:;l:ller illwnl.al~I()dal.-\Ilillysis

:l.2.l :'>I,'aslltcllll'lliofFtt'lJ1W11" ylh-spons" Fm ICtiol1s

;l.~.2 }Ioda lPnramotorEstiut.uiou

:1.:1AddedMass Faclors :J..l Summa ry

4 Experimen talInvestigati o n .1.1 ExperimentSetup.

1.2 Cnlibrnrion

·1.2.1 Excitation Channel .

·1.2.2 ResponseChannel 4.3 TestedPlates

.1.3.1 ereFPla t e . 4.3.2 SFSF Plate .1..1 Genera l Procedures ofExperiments 1.5 Moda lPara mete r Estimatio n 4.6 Discus sio n.

-

1.6.1 Experime nts inAir.

·\.6.2 ExperimentsinWat er

4.6.3 Fluid Effect V,)rsllsPlilll:HonnduryCond it ions . 1.7 Sum mary

5 Analytic alStud y 5.1 Form ula tionofthe Problem 5.2 Solution oftheProblem

:!ti 2;

:w +1

·17

HI fj::!

fifi

'J_:I nil'A<llbl~tilssofaPlate Subm ergedillFluid

;j,;l. ! Th{~FluidFieldabovetill'Plate(Fd :i.:l.2 TheFluidFieldbelowthe Plate (F1l 1.:1.:1 Added~t assCa1cullll in,lls. 1..1 Discussion oftheResults .

;).,1.[ TheAdded:\tas~fromtllc Fluid Field above theSubmerg.: Plate..

.

; .'1.2 The Added:\la ~sfromth...Flu id Fieldbelowthe Submerged

Plate .

5.'1.;1 Total Added~tas~of thePlates ..

G9 '0 iZ

n

tt

80 80 r).,;Com parisonbetweenExperimentaland Analytic alResults SJ

'>'6 Summa ry 86

6 Conclusion s 87

Refer ences 90

Appendix 95

A Derivat ionof Equations5.20an d5.21 95

List of Figures

3.1 Asystem with noise I!I

:!!l

an 4.1 Exp eriment setup

'1.2 Connccriog rod 4.3 Photograph of thesetup

'l.'! (a)Calibrationsetupfo r theexcitationchannel:(h) Setup usodtoilPllly

loads Lathe force transducer. :1:\

,1.5 Calibrationsetupfortheresponse channel ;W

4.6 (a) Dimensionsof theplates;(b)Glnmpcd -frcc-c1ampc'l]-frl'('(C FCFj plate; (c )S impJe-rrce-si mpl..-ffl'(~( S FS r)plato. :Ii

,1.7 Notchedplatewi;.;;a clampedcnd. :m

4.8 (a) Shell element meshI;(b) Shell clelllclltmesh2;(c)Shelld{~Il1(~lLt

mesh 3. ,1/

4.9 RMSversusnotch dimensions. 'la

·1.10 Mo d e shapes oftheideal SFSFplal(: 'I."i

4.11Mode shapes oftill'simu latedSFSF pla'.t:!withnotch sizeul :\'2 xll.!J:l(j

4.12Thedistrib u t io n ofthedrivingpoint and acquisition points. 1J7

4.13Modeshapes ortil.. CFCF platei'lwa t er. rIO

vil

1.J.1~1(Jdl!.~hilPI!~ofthl!SFSFplateillwater ·jl

·LI,'} Alldl'fllUilSSfactors versus1;-[or

tue

CFCFplate ,jT -l.IfAdd edmass fac tors\'erSHS~for the SFSFplate. .'iT ,1.17TIle increase in modaldamping ratios for theCFCFplatt: 60 ,U S Tilt'increaseinmodal dampingra t ios fortheSFS F plate. fiO

.'i.I Art'~tallglllarplatesubmergedin abodyof fluid (H

:i. 2 Clver-susthe nat ural fre quellcy10.. rs

,':l.a AM Flversus

t;-

fortheCFeFplat e 79

.

i 4 AM FIversus

h;-

fortheSFSFplate. i9

.'i.5 AMF1versus~fortheCfeFplate 81

!l.6 ..1:1'1J.;versus~fortheSFSFplate. . Sl

5.7 Comparisonof addedmass factorsfortheCFeFplate in water 8-1 5.8 Comp arisonof addedmass factor sfo r theSFSFplateinwa ter . S·l

vlil

Li st of Table s

,1.1 The readings ofcalibrati onforthcexcitation channel :H 4.2 The sensitivit yfactor oflhe [C'spousechannels. :m -1.3 Natura lIrequcucieefllx]obtainedIlsingtwo meshes \0

,1.'1 NaturalIrequencicsfHa]ofIII{'SFSFplate ¹¹

4.5 Modalfrequenciesandmodaldamping ratios of til!'<'FeFplalt·. .p~

,1.6 Modalfrequen cies andmodaldamping ratios of1Il!'SFSFpl1Ltl~ I!J 4.7 Nat uralfrcquencies(Uz)oftheCFCFplateillair ,~:!

4.8 Natura lfrcqucneieefllx] oftheSFSFplate inair. 7\:\

4.9 Percentage decreas ein naturalfrequencies ofpla l Cllillwate r !iii

5.1 Wavenumbe roftheeFGFandSFSFplates. if)

5.2 Predicted addedmassfactorsfor theCFCFmdSFSFPlatesin water H:!

5.3 Deviation betweenanalyticalandexperimentalstudyOiladdedmass Iactors.

ix

1'!.'j

List of Symbols

x,)',z - Cartesia ncoordinates.

1 - time in seconds.

a,b,h-length,breadthandthicknessoftheplate,m.

t,r,1.1-lengt h,widthofthewater tank. m.

11,, 112 -fluid level abovethe upper surfaceandbelow thelower surfaces ofthe sub- merged plate,respectively.

PI-fluiddens ity,kg /m:l . p'densityoftheplatemateri al,kg/m³• 111==ph-mass ofthe plateperunitarea,kg/m²• E·clastic modulus,Nl m²•

v:Poisson 'sratio.

IJ

=

12[;~l""J

.

flexuralrigidity ofthe plate,Xm.

g - gravityacceleration,m/$1, II•planewavenumber.

k•modalwave number.

ii.' -constants.

W(x ,y,t) .bendingdisplacement ofthe plate,measuredupwa rdfromitsstaticequi- libeium position.

¢/(x,y ,Z',I).velocitypotentia l.

/If.(.r. y , l) ·fluiddynamicpressureon the plate-fluidinterfaceSt.,NJm¹. pu( .r,y,t)-fluiddynamicpressureon theplate-fluidinterfaceSu,NJm¹•

p - dynamic pressure at an}' polmillfluid..\'//II~ . [."- naturalfrequency ofthe platoillair.Hz.

[fluid-naturalfrequency of theplntcsubuwrgcdill lI"i(!.1I,f..

4.1... -natu ra lfrequencyof the plate in air.,·ud/sr c.

WJluid-naturalfrequencyoftheplatesubmerged innllid ,,·(ul./..;t·(:.

4.1 •frequencyofthe wave motionon thefree/Iui(!surface ,1'Il/11-.;'·(·' m"-addedmassper unitarea,kg/m²,

mj•contributionto the added mass fromthe fluid above the[)lllt(l,kg/1tl²•

m;•contribut iontothe addedmass fromthefluidbelowtheplate,/..·y/1II1•

n -numberof degrees-of-freedom.

"'(2 _coher en ce functi on,

s=0

+

jw •Laplace variable.

IAJj•nx nmassmatrix.

Ie] -

nxndamping matrix.

{J(J -n xn stiITnessmatrix.

{z (l)}•nxIvector ofdisplace ments.

{±(l)}•n x 1vector ofvelocit ies.

{x(l)} -nx 1vectorof accelerat ions, {I(t)}.nx 1vectorof forces.

{q(tJ}-n x1vectorof displaceme nts in modal space.

{X(s)}-nXIvectorof displacementsinLa place domain.

{F(s)} - n x1vector offorcesinLaplacedomain.

[H(s )] -transferfunct ionmat rix.

[UI•mode shapematrix,[V]

=

[UI.1-'1.. ,.Un).

xi

1M] .gcneralieedmass matrix,r;rf]

=

[U]TPlI](UJ.

Ie] .

generalizeddamping mat rix,

leI =

^[UjT[CIlU].

W] .

generalizedstiffnessmatrix,

{K1 =

[uj1"[K]{UI·

{P(I)}·nxIvector of forces inmodalspace,{P(I)}

=

[U]T{f (I)}. (A,I·thcr.th residual matrix.

w,-ther-t h complexfrequency, rad/sec.

Pik(j W)•lower residu al.

Qik(jW) •upperresidual.

xii

Chapt er 1 Introduction

Inalargeclassof dynamic problems , thestructure either cont ains or is surrcuuderl by fluid. Examplesinclude safetyanalysis of nuclear reactor s, seism ic analysis

or

largeliquid storagetanks,dynamicsofships,and submarines.Inthese problema, 1I11' structural disp lacementmod ifies the flow field,which inturn affectsthe :;lrucLuf;\1 responses. Gene rally,theseaTCcategorized as fluid-structure interact ionproblems.

Influid-structureinteract ionproblems,naturalfrequencies, dampingra lios al1ll mode shapesofastructure arc different from those inair. The prediction or1IIC$ C changesdue tothepresenceoffluidis important ,asthis makesitpossiblefor;1 des ignerto select appropriatestruct uralparameter s, geometry,dam p ingcoatiugs, etc, to suppressstru ct uralvibration.

Ma ny stud ies on the fluid-struc tur einte ractio nproblemshave been carried0111

byanalytical,numerical and experimentalmethods.Studies performe dbyRay leigh{1877], l.amb{ 192I j,PeakeandThurst on \1954J,Dowell and Voss\1963],Qaisi[1!l881and Hobin- son andPalmer(l99 0Jpresented analyticalsolutio nsforpredict ingthe fluideffcct011 the vibrationofsimplestructuressuchas circ ular orrectangular plate s. Numerical stud iescarriedoutby Mar cusl1978\,Volcyf:ta1.{1979j,Muthu vccra ppan [1978, 1!179,

J!J801and1':v(~r~ tinc[ [99 I Jinvestigatedcomplexstruct uresand/orcomplexboundary conditio ns of structuresallilfluid.Re portsof experimentalstudieson submergedcan- tilcvctI:dplaLe~{Lindholm,1965](Budipriyant o,19931,on perforatedpla t ca[DeSanto, 19811.andonacylindricalstructure[Ra ndall,1985],havebeenpublished .Ingeneral, these studieshaveexposedthemechanismoffluid-st ructure interaction problems and/or presentedmethodologies toestimat e thefluideffecton thevibratio nof sub- merged structures.The fluideffecton a structurecanbe consideredas anaddedmass for lower modes.andasan addedmass and an addeddamping forhigher modes. How- ever, neith eranalyt icalnorexperimenta lstudy which investigatesthe effect ofthe depthofsubmergenceon arectangular plate oriented horizontally has been reported before.Also, thereis110referencein theopen literat urewhichinvestigates exper- iment allytheeffect of boundaryc?nditions onthevibrationofrectangularplates submerged in fluid.

This thesis presents analyt icalandexperimentalstudies on rectangularplates submergedinwater at differentdepths wit h the intent of gainingan insightinto the mentioned subjects. Theplates havetwo differentbounda ryconditions,one has clamped-rree-d amped-free(CFCF)supports andthe other has simple-free-simple- frce(SFSF)supports. The organizatio nof the studyismadeinthe followingmanner:

Litera turereview of previous studies on fluid-structureinte raction problemsis given in Chapter 2.

Thetheoreticalbasisof modal analysisis reviewedin Chapter3.Somepractical considera tionsrelatedto the experimentalmodal analysis,such as the B&K2034 analyzerinvolved indata acquisition,and STAR softwareused inmodalparameter est imati on arealso discussed.

Chapter·1 dealswith theexpcrimcmn l~hlily1111the('FeFamiSF~F pt,'h'~

bothin airand inwater of6 dillcrcm depthsof :<1I1Ilner gl·lICt·,n-spt'Ctin.';.Jo·...\nou-ln-d plate with clampedends issuu cst edtoapproachtheSFSF plate,andfiuih~dl'1I1t'ul software ABAQUSisusedto determine thet1irllClI"ioll>l ufthe notch.H':!lull>lI'll"

disc ussionare presented.

Ananalyticalsolut ionto ancla st icrecta ng ularplate,whichundergoestleXllr;11 ben dingvibration in a. body of bc mogcncous,incomp ressiblenndinvlscld llu i,J whUSt!

mot ionisirro tational,ispresentedinChapter 5.Therelationshi pbetweenrlu-adtl,'d mass and thedcprh orsubmergence is established. Also, thevalid ity ofthe presented solution isassessedbyapplyingittotheCr CFand SFSFplates.

Chap ter 6 makesconclusio nson thestudy,

Chapter 2

Literature Review

2.1 Prev ious Work on the Fluid- St r u ct u r e Inter- action Problems

2.1.1 Analyt icalApproach es

The ana lyticalst udyonfluid-structureinteractionproblemswas init ia ted byRayleigh [1877),He calculat edtheincreaseof inertiafora rigid vibratingplateinaninfini te baffle. Afterwards,Lambl1921J consideredtheproblemfor the first axisymmetric flexural modeofa circularplate clamped atits edge.The plate was placedinan apertureofinfinitelyrigidplane wall in contactwith water. Themethod developed wasbasedon thecalculation ofthekinetic energ yoffluidin terms of thevelocity potential; andtheRayleigh metho d was usedin the calculation ofthe resonance frequencyof theplatein fluid.Lamb'sworkwas ext ended tothe caseofthevibra tio n ofa circularplatebyPeakeand Thurston{ 1954',fortwodifferentboundarycondit ions ,~tits edge,sim plysupportedandclamped.The ir theoret ical result sagreedwith the exper imental data towithin10%.

Kwak and Kim(1991] alsoinvestigat ed thefluideffectonthenaturalfrequen- cicsofcircularplatesinwater.TIleplat e was simply-sup ported,clampedandfree,

respectively,at itsedge and underwent axisyuuuctncvibra tions.TheHankel Trans- formationtechniquewas usedto obtainaqualitativemeasure of thefluil!c1fl·l' I.•h wasfoun d thatthe naturalIrcqucnciosofclampedandsimply supported circular plates weresensitiveto the fluidboundary conditions while the natural rn"luelll:}' of thefreeedgeplate wasnotsensitive to it,

The approachpresentedby Espinosa andGallegc-J uarez[I!IS'Q is for l'v,I!\I;\l- lngthefluid effectonthefrequencies ofcircularplates. Unlikeotheranalyses,lin

assum ptio nwasmade as to the magnitude of the wavelength inrespectto the liJH'ar dimensions ofthe plate. Themet hod was appliedtoa wa ter-loaded circular platt' vibrati ng in its axisymmetricmodes ami confirmed by experimentstudy.

Dowell and Voss[1963]examineda clamp ed rect angular[Jll\t,~illcoutnct with fluidandpresentedametho d toestimatetheeffectof fluidleading011its lla1.llral frequencies. Itwasfound thatthe fluidactedas an aerodynamicspringor as an addedma ss attached to the plate.Prellove{19G5] extended Dowell andVoss'swork considerably. By consideringa simplysuppor ted plate loadedononesideby ufinite fluid-filledcavity,he addresseda problem in reconcilingplat e andfluidboundary conditions . Theplatedisplacements were exp ressed a, aweighted sum ormodeshap'~s of theplatein vacuoandexp anded in termsoforthogonalIunctionswhich;ndivid ually satisfied the fluid boundary condit ions.Thus, fluidandplateboundarycondit ions weresatisfiedsimulta neously. However,thedisplacementexpressionis110 1.illterms ofnorma l modesof the combinedsystem, as themode shapes in vacuumare not eigenfunct ions of thecoupledgoverning equation,

More recently, Qaisi[1!?S8j has studied thesimplysuppo rtedand theclamped plates by similarmethods. Hisstudy exte nded previous workby theapp lication

of matrix methodsto evaluate naturalfrequencies and mode shapes of the platein contact with fluid.

Junger andF<~it[1 9721alsoconside red a simply supportedplate.In theanalysis, the surfacedisplacementwas expanded as a weighted sum ofmodes of the platein vacuo, but fluidside wallboundary conditionswere notincluded. Thenear !h·lt!

and fa.rfield responseswere discussed.Itwasconcluded thattheplate-fluid coupling is smallforlarge plates, inwhichthe onlyeffectofthe fluid is as an added mass.

DaviCII[1971]discussed this couplingin some detail,examining the relationto the damping and added mass effects.

Theanalysis presented by Robinson and Palmer[1990]is of a problem in which theplate andfluid modes arecompatible;the plate modeshapes arenot coupled by thefluid.Unlikethe above analyses,the hydrosta ticpressure exertedby the displaced liquidwasincorpora ted.Free motionin a generalcombined mode was investigated, and constrain tson themode shapes developed.The particularcase ofa.float ingplate with edges constrainedto havezero slopewas then studied.

The above analyticalapproach has advantages forpredictingtheeffect ofthe surrounding fluidonthe vibratingplatesqualitatively,but it isrest ricted to veryspe·

cialcases.In contrast,numericalapproaches suchast~efinite elementmet hod(FEM) andthe boundary elementmethod(BEM)make it possibleto solve complexstructures aswell as complexboundary conditions ofthe structureand fluid.

2.1.2 NumericalApproache s

In the effortto numericallysolvethest ructure-fluid interactionproblems,various approximate methodshave been proposedtodefine thetransientinteractionloading.

They are oftenbased upon the asymptotic behavioursof fluid wave motion, ;'e'.,il l

early time(high frequency) of the interaction thefluidleading tends10beIIdi\11lpilll;

force andatlate timc(lowfrequency)to tends tohe an inerti aforce of ad.lt.->tl lllilSS.

Everstine[1991]investigated a cylindrical shellwith fia.tendclosures sllhulI'rg(,d in fluid by finiteelement met hod and boundary clement method.Thelow frequency vibrationwas consider ed,and fully-coupled added mass rnlltrirt.-'SweTI'calculated.

The FEM wasimplemented usingNASTRt\Nwhile Lhe BEl\'1WIISperformedusing NASHUAandNASTRAN.Bothmethodswere proved capab leof computing nccu rat e submergedresonances,

Another rigorousfinite element methodfor fluidswas developed and discussed by Chowdhury[1972],whichwas used by :'vluth uvecrappa n!1978, 1979,IHRO] to carry out extensive studies on the atructurc-Huid .teraction problems. TilesLudywas carried out fora squarecantileverplate by varying thefollowing factors:(i)depths of immersion;(ii)aspect. ratios; (iii) plate boundary cont.lilions(simp lysupported, clamped andfree at its edges);[iv]plate materials(steel,aluminumandcOPII(:r);

and(v) fiui':! densities. Results presented indicatedthe dependence of the naturnl frequenciesof the subm ergedcantileverplate on depthofwater aboveandbelowtIll!

plate, andon the lateral extentofthe water. As the submergClIdepth increased till' variationin naturalfrequencies became less, and this variation was appreciableonly in thefundam ent alfrequencies.The mode shapesinwatervariedslightlyfrom those!

in air. Relati onshipswereestablishedshowing thedependenceof thenon.dimcnsionul frequency paramet ersonplate materials and fluid densities.

Fu and Price11987] investigated plate-fluidinteractionproblemby usinghydro c- last id ty theory and being ncccmpliahedbythe FEM.Theplat econsideredwasofcan-

tilevcrsupport ,part iallyortotally immersedinlluid.Theinteractions were defined in termsof drymodesha pes, natural frequencies. principalcoordinates,frequency- dependentgeneralizedhydrodynamiccoefficients,etc.

The effectofthe free surfacewas considered.The validityofthe hydroelas- ticitytheoryandthe accuracyof the chosennumericaltechniqueswere assessedby comparing predictionswith the experimentaldataavailable.

Numericalmethods,whetherFEMor BEM,can be applied to solve com- plcx fluid-st ruct ureinteractionproblems.However,both methodsrequire enormous amounts ofcomputationaltimeand effort. Furthermore,it isnot easy to obtain an accuratemeasureorthe fluid effectsbyeither method.Besides,the validityofthe resultsobtained bytheanalyticalmethod orthenumericalmethods can be appro- priatelyassessed onlybyexperimentalstudy.Thus,experimentalmethods have been widely usedin theinvestigat ion oftheaddressed problems.

2.1.3 Expe rimentalApproaches

Jezequel(1983)carriedout astudywhose maingoal was to obtainan experimenta l modelwhichcouldbe used inthe modal synthesismethods.Theaddedmass was identified by usingthe measuredmodesofthestruct ure, bothin air and incon- tact with the fluid.The modelwhich expresses thedynamicbehaviour of thefluid- structuresystemwas obtainedthrough an optimizat ionprocedure.The methodwas confirmed by applyingitto thecase of a platepartiallyimmersedinwater.

Randall[1985] carriedoutdynamicstudieson a cylindricalstructure inairand submergedinwater.Modal analysistechniques were used to identifytheno::.tural frequencies and mode shapes underbothconditions.

De Santo[198l]accomplished experimentalstudy on perforatedplates vibrat.ing in wate-r.The water effect was expressed in terms of addedmassandhydrcdynmuic damping. Dimensionlessformulas which gave accuratevalues ror the addedIll;L'\S

and lower bounds (orthehydrodynamic damping force in the linearand ucnllncnr damping ranges were presented.

Lindholm et ai.[1965j studied extensively cantilever plates ill airand inwaterby experiments.The plates were vertically-placedor tilted,withdifferent aspect ratios, chordrat ios andthickness.The resultswere compared with thc.oretical prcdlctlons based on simplebeam theory orthin-platetheoryandthechorclwisehydrodynamic strip theory. An empiricalcorrection(actorwasintrod uced to achievegoodtheo ret ical and experimentalcorrelation.The fluid free surfaceandpartial submergencedfeds were alsoinvestigate d. It was concluded that:(i)nat ural frequencies or theplate decreased andnode lines of mode shapesshifted whenit wassubmerged inIluld;

(ii)the added mass factorchangedwith thesubmergeddepthorthe plate,hut lhe significantchangeoccurredonly when thesubmerged depth was less tha.n about one halfspan lengthofthe plate .

Theexperimenta lstudiescarriedoutby Budipriyantol1993]examined uncracked andcrackedcantileveredplates in airand submerged in water.Itwasfound lhatthe natu ralfrequencies reduced by39highas 26.8% andthe damping increasedby 5 times in-air value whenthewater level was justaboutthe midd leoftheplatethickness. For full submergencewherethewater level was about 230mmabove theupper surface or the plate, a maximum natur al frequency reduction of 40%and dampingincreaseof about 6timesin-air valuewere observed.

2.2 Summary

Previous studieswhichhavebeen reviewedabove haveinvest iga ted thedyna mic char- acteristicsofstruct ur esin fluid. Mos t of these stu dieshaveexamined thechange in natura lfrequenciesofthe structuresdue to thepresenceof the fluid. Some have studiedthechangesindamping ratios andmode shapes.Themechanism of thefluid- structure interaction problemshas been exposed. Thestudypresented iu this thesis is focusedonthestudyofthe effectof the depth of submergenceon the vibration of recta ngular plates oriented horizontally.The effect of plateboundary conditionson the vibrat ion of platessubmerged in fluidis also invcsdgated. Both analyticaland experimental methodsareused.

10

Chapter 3

Modal Analysis Theory

Modal ana lysisisthe processof characterizingvibrationproperties andIJI'h:wiulir of a linearsystembymoda lparameters. This process is ofte nreferredto as llKJlJlI1 modelling.Modal modelling focusesOilthree keypropertiesofavibralillg>lys1.elll:

naturalfrequencies, dampingratios,IIIIQmodeshapes .

Inthe subsequent sections,theoretical basis

o r

modalanalysiswill hercvit~w, sl;

then experimental modal analysiswillhe addressed.Inthediscussion of tlw cxpurl- mental method of modal analysis,considerab leconsiderationswillbe given toS()IIlC~

practical problemsrelated to the B&J< 2034analyzerandtheSTArtsortwilr(~flc'·

velcped by StructuralMeas rrcrncnt Systems, which arc usedin theanalysisoftill!

experimenta llyobtai nedfreque ncyresponsefunctionsana themodalparalilderl~sti·

mation.ALso,the calculatio nof addedmassfactorsfrom thenaturalfrequencies of;1 vibratingstructure in airand in fluidi~discussed.

3.1 The ore tical B a si s of Modal A nalys is

Inorderto applymodalanalysis to a.system,this system is assumedto satisfy the followingconditionslA Jlemaug,1987J:

II

(i)Thesystem is linear.This impliesthat the response ofthe system,due to any combinationof forceswhich applied simultaneously,isequal tothe sumof the individualresponse to each of the forcesactingalone.Underthis assumpt ion, the systembehaviou rcan becharacterized bya controlledexperiment,ir. which forces applied tothesystem haveaformconvenient for measurementandpa- rameterestima tion ratherthan being similartothe forces actuall yapplied to thesystemin its normalenvironment.

(ii) The system is time-invariant .Thismeans thatthesystem parameters such as theequivalent mass,thestiffness andthe damping ratioareconstants instead of functionsintime.

(iii )Thesystemisconsideredto followBerti-Maxwell'srecip rocal theorem.The theorem states thatthedeformat ion atpointjdueto a forceappliedat point kis equal to thedeformat ionat pointkdue to aforceat pointj.Underthis assumption, itis required tomeasureonly a columnor a rowof the system frequency responsefunctions.

Considerthe governingequationof vibrationfora multipledegree-of-freedom systemwithviscousdamping

[MHi(t ))

+

[cHi(t))

+

[KHz(t ))

=

(f(t)) where

{MI:II Xnmass matrix;

(el:11x ndam ping matrix;

[Al n xnstiffnessmatrix;

12

(3.1)

{x(I )} :11XIvecto rofaccelerations:

{r( l)}:nxIvecto r ofvelocities:

{.r(l)} :nxIvectorofdisplaceme nts:

U(t )}:n xIvector offc eees,

Application of Laplacetra nsformto Equat iona.! yields

I,'IMJ

+

,[CI+IKJI{X(,)}={F{,))+(lMI

+

[CIIIX(O))

+

[M H.~(O)} (:1.'1

wheres

=

6+ jwis theLaplacevariable:(S(O)}and{X (O)}arctheinitial diNpllll'l"

ment andveloci tyvectorsattimet ::0:(.\'(.~)}and{V(.~)}arc thediNplan'll1t?llt and force vectorsintheLaplacedomain.Ifthe initialconditions arcZI?rO,I~tlililliull 3.2becomes

["1M)+,rC)+[KIJ{X(.•))={F{.•)}

Let[B(s))

=

^,,2(MI

+

.lI[G]

+

(KJ.then Equation:1.3canberewrit tenas (~"'I Thisisanequ ivalen~representationof Equation:U intheLaplacedomain. By definingIH(s))

=

(B(.lI))- I.Equation3.4becomes

where[H(,,)jiscalled thetra nsferfunctionmatrix.LettheLaplacevarlablcawjw, thenonehas

[If{'

)1 =

[J3{'

It ' =

,dj[l3{jw)J )W JW dct[B(jl<ll ]

where[B(jwl]=1-(I\-lI w²

+

jw[Cj+[1\']];adj[B(jIlJ)jistheadjoi nt matrix of(lJ{j w)/

anddet{B(jw )listhedet erminantof[LJ(jw)j.

Sincebot h the adjointmatrix ofB[(jw )]andthe determinantof[B (j w)/ are polynomials injw,the eleme ntof[I/(jw) ]isa rationalfractionin jw.Therefore, itis possibleto represent any clementof thefrequency responsefunction mat rix[ilU.,.;) ] in a partia lfractionCorm:

(lI(jw)]~ :tl~

+ .

[A; I

.J

r=I}W -W, JW-w, (3.7)

wncrcIA, ]isthe r-th residualmat rix whichreflectsthe corresponding mo de shape;

W ris the r-thcomplex frequency,whoseimaginarypart givesthedamped natural frequencyand whoserealpart givesthe dampingcoefficient . A·designates the corresponding complexconjugate .

For the homogeneous solution;

[B(jw)]{ X (j w ))={OJ

The characteristicpolynomialequa tionis ob tained from

do/[B(jw) j~0

(3.8)

(3.9)

The rootsWiofthe characteristicEquat ion3.9 are called eigenvalues or complex Ircqucucics.Substi tutinganeigenvalue into the equationof motion3.8,solvingfor XUw) and normalizing the valuesofXUw)to unity yield the eigenvectorU,corre- spondingto the eigenvalueWj.

Itcanbe shown that modalvectors are orthogonalwithrespectto one another iftheyareweighted with respectto themassmat rix[M] andthe stiffnessmatrix JK].Itcan alsoheshown that modalvectorsare orthogonalto one anotherifthey areweightedwithrespectto thedampingmatrix[C]whenproportionaldampingis assumed.

14

Equation 3.1can hewritten in themodal span'. Thetrnnsforma riou Irom physical space to modalspaceis given by

{x(l)}

=

^{[UJ[q(l ))}

where

[£II:modalmat rix,[ll ] "'"(U\,U2, •• ,U"I;

{q(t)} : displacementvectorin moda lspace.

Substitut ionofEquat ion3.10into Equation3.\gives

(:1.111)

IMIIUJ[q(l ))

+

ICIIUJ[;(I))

+

IKIIUJ[q(') }=!fl')} (:1.11) Pre-multiplyingby[Ufgives

[UfIMIIUJ[q(I)}

+

IUf\CIIUJ[;(I)}

+

1U)"'\J,IIUJ[q( l))=1U1'·{fII)} (:1.12) When the dampingofthe system ispropor tional, applicat ionof thcorthogonnluy propertiesof modalvectorsyieldsthegeneralizeddiagonalmass,stiffnessunddamp- ing matrices:

Rewrit ingEquat ion3.12as

IM\

=

[U\T[MIIU\

le\=

[U)"'[C\IU\

{K]

=

[/J)"'[J,IIUI

(:/.1:1) (:t.I1) (:I.J,'i)

(MJ[q(l )}

+

/GJ[q(I)}

+

(KJ{q(l))={P(I)) (:t.I6) where{P(t )}

=

[UjT{j(t His the forcevectorinmodal space.Itis noticedthat each equati onuncoup lesfrom theotherandrepresentsallindividua lmodalrespons eof thesystem[Ewins, 19841.

15

3.2 Experimental M odal Anal ysis

Experimentalmo da lanalysisisthe processof experimen tallydetermining the modal par ameters ofIt.linea r,lime-inva riant system.Equation3.7 isthe generalmatrix for m that is usedin modal analysis . Continuoussystems have aninfinitenumberof degree-of-freedom:however,in general,onlya finitenumber of modes areneeded to describe the dyna micbehaviour ofasystem.In thefrequency rangeofinterest,the modalparameterscan beestimated to be consiste ntwith Equa tion3.7.In the lower andhigher frequency ranges,resid ualtermscan beincluded tohandle modesin these ranges.ln thiscase,Equation 3.7becomes

whereP;k(j W) islower residu al,andQ;k(jw)upperresidual. In manycasesthe lower residual iscalled theresidual inertia which ref lectstheiner tiaofthelowermodesand isan inversefunction ofthe frequency squar ed;andtheupperresidualis calledthe residualflexibilitywhichreflects the flexibility of the uppermodesand isconsta nt withfrequency,

In exper imentalmodal analysis, frequencyresponsefunctions areused as input s so thatmodal parameterscould be estimated. Since it isassumed thattheBetti- Maxwell theoremcan beapplied to the system,modalparameters canheestimat ed by measuringeithera column orarow ofthesystem frequencyresponse functions .Thus, the frequen cyresponsefunctionplays an impor tan t role intheexperimentalmodal anal ysis. Thefrequencyrespo nse funct ions arefirst lydeterminedby experiments andthen usedtoestima tethe natur alfrequencies,modaldampingratio,and mode sha pes of thesystem . Thus theexperimental modalanalysis comprises mainlyoftwo

16

phases, the measurement of frequency response[unctionsand the modnlparnnwtur estimat ion.

3.2.1 Measure mentofFrequency Resp on seFunctio ns

Before discussingthe measurementof frequencyresponse{unctions, an appropriat e' excitationsignalshouldbe chosen.

Excitat ionswhichare widelyused to drivethe testedetrucurrc illorderlu measurefrequency responsefunctions includeslow-sinesweep,Iast-slncSWIWp.illl- pact(impulse),steprelaxation,and random.Thefast-sine sweep was used to _Irivl' the platesinthis experimentalstudy,The advantagesof\Ising the last-sinesweep are: therelativelyshort measurementtime, the possibilityof reducingleakageerror andthe highsignaltonoise ratio[Ewins.1981J,

The fast-sweep sineis a periodic deterministicsignal.Itis formulate dhySW(~ r)' ing a sine wavesignalup and down withina frequency band of lnterestIluringasiuglo sampleperiod . The suitability of theSWL~[Jre tecan becheckedbytrlnlMillerror.

Measureme ntis madetwice by once sweeping upand the second time sweeping down throughthe frequencyrange.Ifthe samecurveresultsinthetwo cases,then the sweep rateisappropriate.

Inthis experimenta lstudy,theB&K 203'1 analyzer is chosen to take themen- surement of frequency responsefunctions.Itis aFast,flexible,a.ndfully self-contai ned two-channelFa st Fourie r Transformanalysis system.The resolut ion linesam 801.

Noisewillbe unavoidebtvinvolved ill the measurement. This noisecanresult fromthree sources:(i) noncoherentnoiseresultingfromstrayelectricalsignal~or unmeasuredexcitationsources;(ii)signalprocessingnoiseproduced during analysis

17

due to theuseofdiscretefouri erTran sform toconvert time into frequencydomain orviceversa;and(iii)nonlinearnoise due tonon-linear behavioursofthestudied syste mneeded -to bcconsideredin the signal processing methods. These noisas arc eliminated by utilizinga signalaveraging procedure and sufficient lengthofdata.with proper windowsand frequency resolution.

The B&K 2034analyzerprovides five errorreduction techn iques to minimize errorsinthemeasurementoffrequency responsefunctio ns: (i)choiceofthe appropri- ate frequency response function estimator;(ii)use of signalaveragingmethods;(iii) choiceofproperfrequencyresolution,and(iv)use of!I.suitable weighti ng function or window. Appropriateconsiderationshould begiventoeach oftheseinordertc ensurethat'correct'measurement ismade.

FrequencyRespons e Function Estimator

Based ontheassumednoiseinputintothesystem,the frequency respon sefunction estimatingprocedurecan be grouped intothreedifferentmethods, i.e.,H"H,and H...HIassumesthatnoiseexist s intheoutputand theinput isfree ofnoise.Hence

[H,J{X)={V) -{,) (3.18)

H'lassumes thatthe noise ispresentintheinputandabsent inthe output,

(H,J{{X)-{OJ={V) (3.19)

H..assumes the noise to existbothinthe inputand output signals,consequently (/1,1{{Xl-W)~{V) -{. I

18

(3.20)

(,

I ^{x J} 1 _>_''1

x '- l-' l~11-'"

(, - - -0 - -6 '/3-~O--v'

Figure 3.1:A systemwith noise

The B&K2034 analyzerutilizesthe spectralapproachfor estimatingfrequency response functionsgivenby[B&1<2031 analyzermanual,W871

(:1.21)

where

G~;<isthe inputauto-spectrum and equalto[1'..1XiX;";

Gn is the outpu tauto-spectrumand cquallo Ei=lYiY;";

Gs:~=GlIt:is the cross-spec tru m,and equal10 Ei=lXiY,-=t;'"lY;Xi.

In anactualmeasurementsituation, the r.oiso occurs hot h inthe inputandoutput as shownin Figure 3,1inwhichapostrophe denotesthetrue measuredinputaile!

output.Therefo re,HIandH,arefurther definedae

1/1= 1/(1+,.,)

"

where

II is thetrue frequencyresponse functionofthesystem;

'I::=g~isthe relative amount of noise at the input,{::= {I

+{,;

(2::=.~is therell.l.jvc &mountof noisealtheout put ,'I=='II

+

'12' ftcanbe seenthatHIis the lower boundandH2is the upper boundofH.

Thelevelofconfidenceinfrequencyresponse functionmeasurementCIInbe estimatedbythecoherence functionwhichisdefinedas

,..2::=~::=GrwGwr GuGn ^GzrG~~

,..2isrealvalued and varies from 0 to 1.Zerovalu eof tho coherencefunction means thattheresponseisgeneratedbynoiseorasourceotherthanmeasuredinput .When thecoherencefunction is one,however,the measurementis perfect,i.e.,the output is caused bythemeasuredinput.

LeakageAnd Wind owing

Leakageisa direct consequenceofthe truncation whichoccurs during samplinga finitelengthof timehistory coupled withthe assumptionofperiodicity.Onepractical solutiontotheleabgeproblem involvesthe useofweighting funct ions or windows.

Windowinginvolvestheimpositionof a prescribed profile on the timesignalpriorto performingthe FourierTransfo:m.Byapplyingaweightingfunctionor awindow, error in frequencyresponsefunct ionscanbereduced.

Windows available in the analyzerare Rectan gular,Transient,Exponential, lIanning,FlatTopand KaiserBessel[B&K 2034analyzermanual,19871. Aile- mang[19871suggeststheuseofHanningwindowforstationarysignals. Inthisst udy, Hanningwindowis usedinthesignal ana lysis.

20

Av er aging

Signal averagingis the processinvolvin g severn! individualtime records. or Sillllpll'li, beforea result, which canbe usedwithconfidence,isobtnincd. Error(a:j,~l!l,lby leakageandspur iousrandom noise can be reducedhyavcfilging sothattheaccuracy andstatisticalreliabilityoffrequencyresponse function sarc improved.

Severaltime averagingmethods,suchasLinear,Exponential.•ndPeak Aver- aging, areavailablein the analyzer.These can beperformedwithor without1.1ll' overla ppingof thelime record. The overla pprocessinintended 10enhancetill!

measureddata by including consecutive historydatabefore thepreviousdataare completed.Inthe analyzer, thodegree or overla pcouldbe50%,75% ora maximum of85%.Thenumberofaveraging required aredeterm ined bytwomajor co usldem- tions,theaccur acy andstat ist icalreliabilitydesired,an dthenoise levelinsigu'lls.III theexper imentofthisstudy,Linear Averaging meth od with75%oVI~rlapwasUS"I( andthe numberof averagingwas20.

Zoom

Beca useor the constrai ntsimposedby thelimited num berof discr ete pointsavailable dueto theblock size,themaximumfrequencyrangeto be coveredand/ orthelengt hor time sampleavailab leornecessaryto providegooddata, thelimi tationsof i1lilllcquit t.c frequencyresolu tionarise. Thecommon solutio n is to 'zoom'ill thefrequencyrange of interestand to concentr ate allthespect ra linesintoa narrowbandbetw eenI,ni"

and/mu'

Inaddition , increaseofthe frequencyresolutio nminim izesthe leakageerror.

Zoommea ns the redu ctionin Irequencyspanof measurement whichautoma t ically

21

requiresa longer timehistoryrecord.When thefrequencyresponsefunctionpeak is narrower than the frequency resolution,erroroccurs.In thiscasethefrequency respofl5efunctionwill·Ieak" orbewiderthanthe regionofinterest.Resonantand anti-resonantpeaksofthe frequencyresponse function are susceptibletothis typeof error.Intheexperimentcarried outinthis stu dy,zoomof50Hz frequencyspan is usedtoobtainmore accuratemeasurement.

3.2.2 ModalPar amet erEstimation

Modalparameter est imationisthe estimationof frequency,damping, and mode shape fromthemeasureddatawhich maybe(i) inarela tivelyrawformin terms of force andresponse datainthetimeorfrequency domain or(ii)in aprocessed formsuch as frequencyresponseor impulse-response functions. Modalparameterestimation carried out inthisstudyis baseduponthemeasured databeingthefrequencyre- sponse Iunct icns.Thecomputersoftware package usedto performtheestimationis STAR(Stru ctu ral Testing, Analysis and Reporting),version 4.00:developedby Struc- tural MeasurementSystems. Besides its capabilityto identify themodal propert iesof mechanical st ructures,STARprovides functions suchasmeasurementdisplay,real- time animatedmodeshape displayand datahandling.

When astruct ure is excitedusing a broadband input force,manymodes of vibra tion are excitedsimultaneously. Since the structureisassumedto belinear, frequencyresponse functionsarereally madeup ofthesumsof the resonancecurves for each vibrationmode. Inotherwords, at any given frequency,thefrequency responsefunction ismadeup of thesumof motionsof allthe modesofvibration whichhave beenexcited.However,the contributionof eachparticular modetothe

22

overall mot ion isgenerally greatestin thevicinityof itsresonancepeaks.

The amountof the contribut iondue to adjacentmodal resonance curves1.0 theoverallfrequencyresponse functionvalueat a mode'snatural frequencyis called modalcoupling. The degreeofmodal coupling ina frequencyresponsefunction is governedby themodaldampingratio and thefrequencyseparati onofthemodes.

Accordingto the mode-coupling degreeof a system, twomodalparametercsli llla- tiontechniques,(i)single aegree-of-freedom(SDOF)approximation s and(ii)multiple degree-of-freedolO(MDOF )approximations, can beused .Thesetwo techniques'HI!

available inSTARISTAR manual,1990\.

SDOF Techniques

In caseswhere modal couplingislight , the frequencyresponsefunction data ill the vicinity of each modal resonancepeakcanbe treatedas ifthey aretheresponses of a SDOFsystem or a single mode ofvibration.Inot herwords, inthe measurement offrequencyresponse functionsofa system withlightmedalcoupling,it isassumed that the contr ibutio n of the tails ofadjacent modesnear each modal resonancepeak is negligible. Algebraically, thismeans that themagnit ude ofthefrequencyresponse functionis effectively cont rolled by oneof the termsintheseries, that beingthe one relatingto the modewhose resonance is beingobserved.Inthiscase, SDOFcurve fittingmethods canbe used toidentifythemoda lparameters.

STAR provides severalSDOFtechniques,suchlUICoincident Peak,Quadralurc Peak,Complex Peakand Polynomial.WhenusingaSDOF technique, modesin the testdlt a should be relat ivelyuncoupled.A peakon an frequencyresponsefunction plot shouldoccur inrelati veisolat ionfromotherpeaksandshould be relatively

23

unconta minated byresidual contrib utions fromhigherand lowermodes. One key disadva ntage isthatthey tend to be less accurate thanthe moresophisticatedMDOF techniques. Ifaccuracyis nota top priority,SDortechniquescanprovide results with minimum effortandcomp uterresources.

MDOF Techniques

wheremodal couplingis heavy,an SOOFmodal parameter estimationmetho d may yield parame ters withlargeerrors.In cases of heavymodalcoupling,the parameters of all themodesmust beestimated simultaneously using a MDQFmethod. STAR provides users wit h a MOOF estimationtechniquenamed the RationalFraction Least Squares( RFLS)po lynomialmethod. This method fits a polynomialfunction in a rationalfractionform,toa frequencyresponsefunction usinga least squareerror technique.

TheRFtS methodallowsusers tohave better chance of accuratelydetermining howmany modesareactu ally inthe specified frequ encyrange.This isaccomplished byover-specifying themax imumnumber of modes.There mayberepeat ed modes, or modesveryclose in frequency, thatappearas one mode.By over-specifyingthe max- imum numberof expecte dmodes, onehas better chanceof accura tely determining mode size inthe specifiedfrequencyrange.The true(act ual) modes and compu- tational(artificial) modes ofthestructure are identifiedaccordingto the "Stability Diagram" window[STARman~al,^{19921.As stat}ed earlier,MDOFestimati on tech- niquea offer improvedaccuracyincases wherethe frequencyresponse functio n data contain heavymoda lcoupling and/ or thestructure haslocalmodes.However, they aremore demanding orcomputerresources.

24

3.3 Added Mass Factors

Whena structure is subme rgedinHuid.neturnlIrcqucnck-s,ll'tTe,)II'-awldilill p itl ~ ratiosincrease.Itis moreappropr iate to lisethe uon-dimcnsicnal\'allle.iHldl'!lnlil.~S faclo r( AMF),to measure thechangeinnat uralfreque nciesduettlIIll'lIuid- ln;lI lill ~ effect . The addedmassfactor to liedetermine disbasedOiltheoluaim-d lIIt1tLd parameters.The necessaryparam etersarcnaturalfrCI!U('IU:iL'lIofI.I's1.I,<:1st ructun-s ill vacuumandin fluid.Forst ruc t ur es havingIlrelatively higher Ilt'Uslt.y, lht·lIill uri,1 frequencyinairissufficient lydosetotheI,hiltin vacuum. ThisallolVs\.lll~mw oftlu- in- airnaturalfrequencyfor added massca lc ula tions .

Forcontin uoussyst ems such asplmcs ,Ihesyst eTll Shaveill]infinite1I1t1ll! J('ruf degrees-of-freedom.However,ingeneral,onlyafinitenumberofllllllll':'areIlf 'l 'l lt ~ltfl describethe dynam icbeha viou rofa.system.As stat edearlier,modalparal1lf'1.f~r~uf thes e modescanheestimatedfrom the measuredfrcq uf'lt/:yresponsefuu diunsin tilt' frequencyrange of interest.Abo, becauseoftheorthogonalityofmass, st ilfllf'Ss111111 dampingmatriceswith respecttothe modesha pematrix,thedynamicchar;\(:te~risl.il·s of thesystemcanberepresentedas,illmodalspace ,

M;ii;+G;"q;+K,qi

=

0 i

=

I, ..,/I

whereMi,CiandKiarethemodalmuss,modalda m pingand1I1001alslilrrll~ss/IflIw l-thmode,respect ively.Theundampednaturalfrequency;s

When a structu revibratesin fluid. itissuhjecttofluiddfcd,Asstated/~arfi/~r,the fluideffectcan beaccountedfor II}'anadd!:11Illassand addeddiUllpiligratln.Frum

2.')

this pointofview,Equation3.24 is valid foravibrating plate both in air andin fluid.

A15o,because physical stiffnessof the plate remainsunchanged whetherit is in air or ill fluid,modal stiffness1?;remainsthe same if mode shapes areassumed unchanged duetothe presenceof fluid.Inthis case, onehas

(3.25)

whereMidi.andMi/h,"'darethe modalmasses ofthei-thmode of the structurein air andinwater.respectively.IfMtis used to designate the added mass duetothe fluid effect on thest ructure, thenMi/ l"i'/'can be written as

We define an added mass factor as

then from Equation3.25, onehas

p.26)

3.4 Summar y

IIIthischap ter,the theoreticalbasis of modalanalysisis reviewed,and experimental modal analysistech niqueshe-..e beendiscussed.Some practic alproblems related to the measurementoffrequency response functionsand modal parameterestimation have beenconsideredalso.In addition,thecalculation of added mass factors from the naturalfrequencies of avibrat ing structure in air and in fluidis discussed.

26

Chapter 4

Experimental Investigation

Experimentalst udiesarean important toolilltheinvestiga tion of the behaviouruf vib ratingstruct uressubmerged in fluid. Experimentalstudies makeitpossibletu gain adirectinsightintoproblems whose aualytical scluticn is difficu ltto obtaln.

Inthis chapterwedescribe experime ntsto !I1udy the vibration res ponse of twn flat plates which arehorizontallyorientedandwithdifferentbounda ryconditions.

Experiments on the platesin airand submergedin water arereport ed.Oneof the plateshasdamped·Cree-clamped·Cree(CFCF)supports while the other ha." siltlflk..

free-simple-free(SFSF)supports. The effectof the depth ofsubmers ion011the vi- bration res po nse has been-audicd by varing the depth [rom0lo327.5 mmillfive steps.

Inthe subseq uentsections,experimenta lsetupused inthestudy followr.dI,y meth odofcalibrati onare introduced. Detailedstudyon modelling ofthesirnple sup ports involved intheboundary condition softheSFSF plateusingfiniteelement software ABAQUS ispresented . Descriptionofthe modaltcsting a onCFC F and SF SF platesbothinairaru]inthesubmersionofsixdifferent water levelsisgiven.

Resul ts and discussion arej.resented.

27

4 .1 Ex p eriment Setup

To measurethe vibrationof eachplate,15 PCB 330Aacceleromete rs are used and mountedat acquisition pointson the plate.Theseaccelerometers arc selectednot only because they are light,each weighting 3 grams together withit s mounting socket,and as small as4Jllx 21.5mm,butalso because they have high sensitivity of 200=F 40 mY/g.

The experiment setup usedin the experimentis shownin Figure4.1.Fast sine sweep excitation was selectedso that the best resultsfor coherence and frequency response functionswouldbe obtained. Asweepsine signalgenerated by the func- tion generatorwas amplifiedby a power amplifier theninputtothevibr at ion exciter 1.0excitethe tested platethrougha connectingrod shown in Figure4.2. A force transducer was attached betweenthe connectingrod and the tested plateto measure theexcitation force whichthen was amplifiedby the dual mode amplifier.There- sponse signalof the vibrating plate was measured by accelerometers, then amplified by thedifferential amplifier. Finally, both force andresponse signalswere inputto the oscilloscopeto monitor and the analyzerto perform Fa.st Fourier Transform so thatfrequency response functions and coherence functions were obtained. These data werethen exportedand storedin a PC computerforanalysis usingSTAR.

The connecti ng rodbet weenthe vibrationexciterand the force transduceris comprisedof three parts,as shownin Figure4.2. Part 1 has two funct ions.One isto connectthe vibrationexcitertothe long steel rod part 2;anothe r is to adapt preloadby adjusting the couplinglength with part 2.Part 3 is designed to protect instrumentsfrom overloading;the middlepart whose diameteris

¢h

~is made of mild steels and will be yieldedwhenthe exciting force becomeslarge.

28

o

^{KiSTL ER}912-2010 FORCE TRANSDUCE R

0

PCB3JOA ACCELEROMET E R Figure 4.1:Experi mentsetup

29

,. ,

-- _. -- _ . -

--- -

-

-~

.

---

--

---

--

-

--

--.,

,

B&:K4809 VtBRAT~ONt:;XClTOR

:~::' . . . .' . . . .. J

rcecsraoeouc

. ^~

^"---

I~,

^-__;

^{'--- -i}

^{, ..1-}

Figure402:Connecting rod

30

Tocarryout the measuremen tsin water ,a 1300x550xSOO mill t.anKwasbuil~.

Thewallofthe water lank wumadeof transpare ntplasticsandconstructed onIitu.-o\

frameswhichwereusedtostrengt henthe wall.Thelevelof the free water~lIrracr.

withrespectto the plLtoewu controlledby thedept hof waterin the tan k.The tank was fixed to arigid platformso thatit. did not vibratewhentheplate was excited. Whenthe experimentin waterwascarri ed out,theaccelerometersanrlall connectiolls of wires submergedinwater werewater -tightusing vaselinc.Aphotogra ph ofthe setup is shownin Figure4.3.

4.2 Calibration

The calibrat ionofthe testingsystem is carried outin two ste ps. The first step involv('ll thecalibrat ionof the excitat ionchannelwhichincludes theforce transducerandlhcl dual mode amplifier.The secondstep involves the calibratte>llof theresponsechannel whichinclud es t.heaccelerometers,the differentialamplifierandthescanner.

4.2.1 Exci t a tionCha n nel

The setup forthecalibrationisshownin Figure 4.4.Thestandardweightwas applied onthe forcetransd ucer. Thesignalprod ucedbythetra nsducer was amplifiedbythe dual modeamplifier whichwas properlyset so that.theexpectedchannelsensitiv ity of 2VlIb wasobtained.Themult.imete r wasusedtomeas ure theout put.

In orderto apply theweighton the force tra nsd ucer, two piecesof 98)C71)C(0 mm aluminu mplateswereused. The forcetransduc erwas attachedto the platc:c by screws, andtheeffect fromtheweight. oftheplat eandscrews waa avoidedby groundin g the amplifierbeforethe sta ndardweightWa.1applied.The calibrationWAll

31

:t !

'0'

~~STANll AlUl lVm nl1T S

~ . rOIlC~inANS\)\1CEII

os.-

ALUMINU~'I·L"'Tr.

'b'

Figure 4.4: (a)Calibratio nsetupfortheexcit ati on cha nnel;(1)) Setup used toapply loadsto theforce transd ucer

repeate dfivetimesand the readi ngs were shownin Table4.1. Thesensitivity «quut ion wasestimated using the Least Square Met hod .As expected,the function oftill:input and theoutp ut islinear in the calibrationrange, whichis expressed ljy

!J

=

0.021428

+

2.118572z Where x is aload inpounds and yis a readinginvoltages.

4.2.2 Response Channel

Thesetupforthe responsecalibrationis shownin Figure 4.S. Tileaccelerometer 10 be calibrated was attachedto thesocket ofthe calibrato r, whichoutPlll.~acons tant level of 19(R MS) at afreq uency of 80.l25 Hz.The output

o r

the acceleromete rwas amplifiedby thedifferen t ialpoweramplifierandsen t. to theanalyzer tocalculal.nthe auto spectr um.Thesam plingtimewas Bs,averaging num berwas20andHannin g

33

Table 4.1:The readings of calibrationforthe excitation channel Readi"ngt7T --

to,drlb) 1 2 3 4 5

0.5 1.050

uo

1.08 l.08 1../15 1.0 2.15 2.14 2.16 2.09 2.05 1.5 3.05 3.03 3.02 2.98 3.06 2.0 4.10 4.12 3.99 4.15 4.03 2.5 5.15 5.10 5.08 5.10 5.06 3.0 6.13 6.03 6.04 5.98 6.02 3.5 7.10 6.95 7.00 7.10 7.04 4.0 8.12 8.09 8.14 7.96 13.02 4.5 9.05 9.07 9.01 9.00 9.12 5.0 10.10 10.12 to.10 10.05 10.00

window was used. The calibration foreach cha nnelwas done one byone, and the scaled sensitivity facto r ofeach channel was then estimated a.ndshownin Tab le4.2.

4.3 Te sted Plates

The sample len gt h of tested platesis measured655 x 201.65x 9.36 rom,which were cutfrom AISIC-1020cold-rolledsteelplate stock. In order to exclude the effect ofmat erialdiffere nce between plates,only one pla te is used forthetwodifferent boundarycondi tions.One is CFCF supported,anotheris SFSFsupported,whichare shown inFigur e4.6.Only thefirst five modes areof interest.

4.3.1 CFCF Plat e

Inthe experime nt,the platewas damped horizon tallybetween two300 x 300x 27.5 rom steel blocks over a plate length of300mm ateach chordwiseend. The lower

34

Table 4.2:Thesensitivityfactoror theresponsechannels

Sq

Channel No. AccelerometersevieeNo, RMS(V) Sco.lcdfador

1 20502 1.1533 {.3873

2 HJ912 1.2570 1.2729

3 20093 1.1832 1.352:1

4 199H 1.6000 1.0000

5 20397 1.1662 1.:.1720

6 19593 1.2369 1.2!J:JG

1 20082 1.4849 1.0775

8 20505 1.3360 1.197fi

9 20403 1.2570 1.2729

10 20236 1.5652 1.0222

11 19579 1.1492 LID"!

12 19612 1.3620 1.1747

13 19906 1.34M 1.1892

14 19907 1.4306 1.1184

15 19568 1.2884 1.2WJ

TRM is the Root ofMean uarc.

Figure 4.5: Calibrationsetup for the response channel

blockwas attachedto a heavy Ibeam support,whichwaswelded to the bottom of the tank.The support height is 275mm.

4.3.2 SFSF Plat e

Itis more complicated to obtainthe SFSF plateinpract ice than the CFCF plate.

Thussome detaileddiscussions are called for.

Surveyof Different Approache sto A SimpleSuppor t

"Hinged" support anJ "edge" supportare widelyusedto accomplish asimplesup- port .However,ahinge always involvessome friction. Thusthemomentata "hinged"

supportwill notbe equalto zero. The friction moment may bea constantor propor- tionalto the reactionexerted bythe support .For anedge support,then,sincethe supportis off-cente r,rotationwillresult ina tangentialfriction force whichproduces

36

U)<Irrs: n,,"

-~}" _I

(a)

I

(b)

(e)

Figure 4.6:(a)Dimensionsof the plates;(b)Clamped·frcc·d amped·frt.,'c(CFCF)platl,';

(c)Simple-free-simple-free(SFSF)plate

37

botharesisting moment and anaxialload.

Theeffect ofIrlction moment s atsimplesupportsis simila rtotllat of decreasing thedimensions oraplate ,by an amountrangingfromdosetozerofor'Trlctl onless"

hinges to a maximumof about onehalfthethickness.For support sat thetwoends, theamountis one timesthethickness[Donnell,1976J.

Beca useoftheshortcomingsInvolved in thesetwoapproaches, a"not ched"

plate wit ha dampedendis suggestedtoapproachthe conditio nofsimplesup ports.

Thus,aplatehavingnotc hesisclamped at theimmediateoutwardends,as shownin Figure 4.7.canbe used to simula tetheSFSF plate.Obviously,it is the dimensions ofthenotches that govern the accuracyof theapproach.

Dime nsion softheNotch

Thefinite elementmethod isa powerfultooltopredictthe dyna mic behaviour of systcme.Inthis study,thegeneralpurpose finiteelement program ABAQUS,version 5.1,developed byHiobit t,Karls son and Sorensen Inc., wasused todetermine the optimal dimensions ofthenotch.

S8R5shell elementwa s chosento discretize the plate inthis study.Itis a 8-node doublycur ved shellelement;eachnodehas5degrecs-of-Ireedornlthreedisplacements endtwoin-surfacerotations ).It hasa 3 x 3middle surface integ rat ionfor mass,body forcesandsurfacepressurecalcula tionand a 2 x 2 reducedintegrationfor consti tu tive calculati onand output .5 integrat ion points arechosen throughthe shellthickness.

Theyarelocatedfromthebotto m to thetop surfaceof the shell, withequaldista nces betweentwoadjacentpoint s[ABAQ US manua l,1992}.

In order to selectanappropriatemesh fortheplate,te ntative calculatio nsarc

38

~ r-r-r

--- -

--

- - - - /:_- _{-~~ ~~~~}

--- --

,

" ' ,".' ^,".'

" '''''",' ""

(l ^{B . • "- ---~ . --} ---1---- - - --(

·0 · .""., ":

:{EF - ---:0 : --- ---)

Figure 4.7:Notchedplatewitha clampedend

:19

Tablt· 4.3:Naturalfrequencicll(Hz)obtainedusing twomeshes

-

MorleNo .

r:--

Me.51!/

^r-l

Mesh l?

I 51.33 .j1.33

2 204.01 203.98 3 222.58 221.22

<\ ·159.47 459.39

5 -178.55 476.40

performedto a notchedplate with notchesmeasured 3 x 0.6552 mm.Twodifferent meshesare used;mesh 1has 175clements,butthe size ofelement changes abruptly;

mesh 2 has ,113clementsand thesize ofelementchanges gradually,as showninFigure ,1.8.The firstfivenat ura l frequencies ofthesimu lat edSFSF plate arccalculated and listedinTable4.3. The result sobtai nedwith two meshes corr elate quitewell betweenthemselves, suggest ing that meshI isreasona ble andreliabl e to perform the calculatio n ofthe notched plate.

1'0 investigatethe effect ofthe notchdimensionson boundary condition, three notchesarc investigated .The not cheshave the width of3 mm,4 mm and 5mm whichare approxima tely

!h, fh

^and

til,

resp ectively.The depthof thenot ches is measured0.07h,0.08h, ...,and0.13h,respectively. The naturalfrequenciesofthe first five modes forthe notchedplateMecalculat edusingABAQUS. Also,the first fivenatural frequencies of the idealSFSFplateareestimatedbyABAQUSin which mesh ;1 is used.These frequenciesarelisted inTable 4.4 forcomparison.

Inorder to judge which notchis the best ,the root of meansQuare(RMS)ofthe .\0

(a)

(b)

(0)

Figure4.8:(a) Shell elementmesh 1 (b)Shell element mesh2 (c) Shell elementmesh3

'1

Table4.'1: Nat ura l (rcquenciesfHe] of theSFSF plate

- - - _ ._ - -- --- -

/l,/orlcNo. I 2 3 1 5

IdealSFSF ,51.015 205.660 228.600 465.510 491.600

10m 51.332 204.0tO 222.650 459.520 478.650 0.08 5U'1I 204.570 223.700 460.630 '180 .920 0.09 52 .25t 205.t80 224.480 461.590 482.540 t3mm O.tO 52.871 205,880 225.090 462.520 483.8 10 Simulfllefl 0.11 53.606 206.690 225.610 463.500 484 .870 0. 12 54.460 207.630 226.110 464.570 485.820 0.13 55.435 208.710 226.580 465.750 486 .730

0,07 50.932 202.600 218.670 455,150 469.840 0.08 51.256 203.230 220.500 457.000 473. 950 0.09 51.652 203.820 221.780 458.350 476. 760 amm 0.10 52.l 3 t 204.440 222.730459.470 478.810

e .u

52. 700 205.110 223.490 460.490 480.4 00 0.12 53.363 205.870 224.130 461. 490 481.710 0.13 5<1.124 206.720 224.690 462,540 482.840

0.07 50.596 200.980 213.550 449.350 458.3tO 0.08 50.877 201.810 216.520 452.470 ,165.080

SFSF 0.09 51.210 202.490 218.510 454.560 469.590

5mm O.tO 51.606 203.120 219.950 456.110 472.780 0.11 52 .074 203.760 221.050 457,380 475.160 0.12 52 .619 204.430 221.930458.500 477.030 0.13 53.245 205.170 222.660 459.570 478.570

tnotehwidth; coefficientor the notch thickness .

"

, ' - ~ ~ J 0.07

- 3mm - - 4mm - -~m m

Figure4.9:RMSversusnotchdimensions

non-d imensiondized deviation(NDD) ofeach natu ralfrequencyforillsimulatedSFSF platefromtheidealoneis estimated andshown inFigure4.9,inwhich

NDD. ="-f.

f.

whereI;and Ii' atethe naturalfrequencyofthe i-th mode foran idealSFSfplalc and the simulatedSFSFplate, respect ively.

FromFigure4.9,itisfoundthat.the minimumRMS 1.595%isobtainedwhen the notchhasillwidthof3mm anditdepthof0.7488mrn.However,consideringthe

43

notch strength, the handling convenienceduring the lest, thedifficu lty inmanufactur- ing andthe sizesof the milling tools available, the notchis measured3.2)( 0.936 mm, Thenaturalfrequenciesof the first five modes for thenotched plate are esti matedin the same way, and the RMS is1.885%.

Naturalfrequencies are one aspect of concern;anotherone isthe concurrence in mode shapesbetween theideal and simulated SFSF plates.Figures 4.10and4.11 showthe mode ehepcs ofthe two plates. Itisfoundthat mode shapesof the first livemode for the twoplatesare in good agreementwith each other. The consensus of naturalfrequencies and mode shap es ind icates thatthesimulated SFSFplate is a good approachto theidealSFSF plate.

4.4 General Procedures of Experiments

The experimentwas performedfirstlyto theCFCF plate thenthe SFSF plateboth in air and in water. For the experrnents inwater, thedepthof submersionabove the plateshivaried from0 to327.5 mm infive steps while thewate rlevelbelowthe plales remained275mm.The drivingpointwas locatedat327.50 mmfromeach end, 30 mm offcenter;theresponses of the plateswere collectedonlyatacquisition points of the

i

platearea because ofthe symmetry of theplates,as shownin Figure 4.12.The positionsof these points were determined,basedon thepredicted dynamic behaviour ofthe platesinthe firstfivemodes,so thatallof themodal responses could be obtained.

Each testwas performed intwo steps.Firstly, sinusoida lsweep excitationwith normalfrequencyspan200Hzwas donetoroughlylocate theresona nt frequency for thefirstfivemodes. Secondly, sinusoidalsweepexcitat ionwithfrequencyspan 50Hz

44

51.02"1

778.60HI

2OS.60111

-165.51"1

191.601h

Figure'I.~O:~Io"..)hitpcso£the idealSFSrplate

'5