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OFFICER-IN-CHARGE CARDEROCK 05

SYSTEMSDEVELOPMENT DEPARTMENT 111

SHIPPERFORMANCE DEPARTMENT15

S RUCTURESDEPARTMENT 1?

S H S PACOUSTICS DEPARTMENT 19

MATERIALSDEPARTMENT 28

OFF ICERIN-CHARGEANNAPOLIS 04

AVIATIONAND SURFACEEFFECTS DEPARTMENT 1 6

COMPUTATIONANDMATHEMATICSDEPARTMENT1{

PROPULSIONANDJAUXILIARYSYSTEMS

OEPARTMrVT21CENTRAL IN S TYUMEN TATi ON DEPARTMENT 29

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II. CONTROLLINGOFFICEMENDADDRESS

1 MONITORINGGENCYNAME 1 AOGRESSffdUtotanlro mControllingOlllca)NavalSeaSystemsCommand Washington,D .C . 20362

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PlaningCrafteakeepingStabilityleasureCraftPorpoisingonmilitaryApplication l f 0 ABBfRAC ContinuenreveriemldoInoceeearyandIdentityaylocknumber)Aheoreticalmethodsderivedorpredictingrimanglean dpeedoeff i c ienttheinceptionofporpoisingofprismaticplaninghulls. Althoughequationsarederivedorth esurge,pitch,andheavedegreesoffreedom,tsseenhatheef fectofsurgessmallatordinaryoperatingtrimangles. Comparisonsoftheoreticalpredictionswithexisting experimentaldataoncoupledpitchan dheaveporpoisingshoweasonablygoodagreementfo rawideangeofspeedoefficients ,oadoefficients ,an ddeadriseangles. Thetheory(Cont inuedonreverseside)DD iA"] 1473 EDITIONOF NOVS SSOBSOLETE

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TABLEOFCONTENTSPageABSTRACT

ADMINISTRATIVEINFORMATIONINTRODUCTIONSTABILITYEQUATIONSDETERMINATIONOFPORPOISINGCONDITIONSCOMPARISONSWITHEXPERIMENTSONPRISMATICHULLSCOMPARISONSWITHEXPERIMENTSONNONPRISMAT1CHULLS1 CONCLUSIONS 3ACKNOWLEDGEMENTS 4APPENDIXA-DERIVATIONOFPORPOISINGSTABILITYEQUATIONS5 APPENDIX-ESTIMATESOFEFFECTOFWINDAGEANDCHINE RADIUSONSTABILITYDERIVATIVES 5REFERENCES 0

L ISTOFF IGURES

1CoordinateSystem 36 2 VariationofLeastStableRealRootan dTrimAnglewithPositionofCenterofGravity 37 3VariationofStabilityRootwithTrimAngleor VariousValuesofSpeed,Coeff i c ient ,DeadriseAngleof0Degree ,LoadVoet f ic ientof 0.72.v...38 4 ComparisonofTheoreticalan dMeasuredPorpoising - Boundariesfo rDeadriseof0Degree 39 -5 ComparisonofTheoreticalan dMeasuredPorpoising BoundariesorDeadriseof10.5Degrees406 ComparisonofTheoreticalan dMeasuredPorpoising Boundariesfo rDeadriseof20.5Degrees 41

in

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L ISTOFTABLESPage

1 omparisonBetweenCalculatedan dMeasuredValuesofX g can dmc,UsingComputedCriticalTrimAnglean dMethodofReference5orMeasuredPorpoising ConditionsofReference 122 rossFlowDragCoeff ic ient 2 1

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NOTATIONaalue satransversep lanehroughboatcenterofgravityBobyleffsunctionofdeadrise;seeEquation54beamofboatCDcrosslowdragcoefficient;seeTable2C fydrodynamicrictioncoefficientCjjjoatiftcoefficient,ondimensionalizedbyhebeam2C^/Cy2Cypeedcoefficient,U/>/ibC^oa dcoefficient,A/pgb3e,. .oordinatemeasuredparallelokeelromransomoftowpointandresultantofwindorce,respectivelyFBSteadystatebuoyancyorceFDynamicpartofhydrodynamicnormalorceonhu l lFDSteadystatepartofFDFp;teadystatehullrictionorceFwsteadystatewinddrag^kn.en)ondimensionalmomentarmabouthecenterofgravityoftoworcen= n dwindorcen=2f(0)eadriseunctionofWagner ;eeEquation53gccelerationofgravity tven^egativeofderivativeoff(kn,n)withrespectorimangleTh(T)eeEquation96)Litchmomentofnertiaabouth eboatcenterofgravity kj,k2oordinatemeasurenormalokeeloftowpointan dresultantwindorce,respectivelykadiusofgyrationofboatwithrespectocenterofgravity LC Gistanceromransomoboatcenterofgravity,measuredparallelokee l1am easLC Glkengthofwettedportionofkee lMydrodynamicpitchmomentelativeocenterofgravity MBSt eadystatepitchmomentdueobuoyancyMpynamicpartofhydrodynamicpitchmomentonhu l lMDSteadystatepartofMD vi ^T ,

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MpSteadystatehu l lrictionpitchmomentMgotalsteadystatepitchmomentactingonhullMTSitchmomentaboutcenterofgravitydueoowingorceMwsitchmomentaboutcenterofgravitydueowindorceMrM^,Mg,etc.artialderivativeofpitchmomentwithrespecttomotionariables,z.0,etc.,respectivelymas sofboatPeeEquation78)QeeEquation(78)soordinatemeasuredalongkeelromoremostmmersedstationofkeel;seeFigure20scleeEquation73)an dF igure20sc2eeEquations(76)an d77)andF igure20TsteadystateowingorcetimeUteadyreferencespeedofboatneetpersecondu .erturbationurgevelocityan daccelerationWoatweightXydrodynamicorcecomponentndirectionofpositivexXDynamicpartofhydrodynamicX-forceXsteadystatepartofX Xy,X^,,Xj,etc.artialderivativeofX-forcewithespectomotionariablesu , ,z,etc.,respectivelyxorizontalcoordinatendirectionofU ZydrodynamicorcecomponentndirectionofpositiveZDynamicpartofhydrodynamicZ-forceZsteadytatepartofZ ZvZ- z,Zg,etc.artialderivativewithrespectomotionariablesz,z,,etc..respectivelyzerticalcoordinate,positivedowneadriseangle;seeFigure20Aoa tweight,W AFDimedependentpartofFDAMDimedependentpartofMDATSimedependentpartoftowingforce

v ii

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AXpim edependentpartofXDAZpim edependentpartofZD{ ,fomponents ,normaloth ekeel ,ofhullvelocityan dacceleration,respect ively6oatpitchangleperturbation,positivebowupXe a nwettedlength-to-beamratioX cengthofwettedchine-to-beamratioX cjondimensionalvalueofscl,scl/ X c2ondimensionalvalueofsc2,sc2/b XondimensionalvalueLCG,LCG/bXalueofX.atinceptionofporpoising X vondimensionalvalueoflk,1 ^ / b X mcalueofXatinceptionofporpoising X yondimensionalvalueofnormaldistanceofcenterofgravityromkeelj uotalsectionaladdedmassM aontributionosectionaladdedmassM sectionaladdedmassa ttransomvinematicviscosityoat-dampingratiopa s sdensityofwaterotabilityroot;se eEquation(22)Tteadytaterimanglemeasuredromkeellinetocalmwaterfreesurfacea teferencespeedU r calueofratinceptionofporpoising 0(X)hree-dimensionaloraspectratiocorrection;seeFigure217olumeofwaL.isplacedtest,utTheprime(' )symbo lsgenerallyusedodenotequantitiesinnondimens iona lorm. Factorsusedfo rnondimensional izingth epreviouslydescribedquantitiesarep ,U ,b. Typicalxamplesaregivenasfollows:

FM-Fgg/d pU2)2)d=sc lM BSM BS/(l/2pU2b3)'=p/l/2pb2M 'M/(l/2pb5)'=ob/Ut'-tU/bviii

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ABSTRACTAheoreticalmethodsderivedorredictingtrimnglendspeedoeffi-

cientthenceptionofporpoisingofprismaticplaninghulls.Althoughequationsar eerivedorthesurge,pitch,andheaveegreesoffreedom,t? seenhatheffectofsurgeissmalltrdinaryoperatingtrimngles.Comparisonsoftheoreticalredictionswithxistingexperimentaldatancoupleditchndheaveorpoisingshoweasonablyoo dagreementorawideangeofspeedoefficients,loadoefficients,ndeadriseangles.Th e theorymayls oesedorestimatingtheaturalrequenciesandamping characteristicsofprismatichullsinhetable,igh-speedlaningrange.ADMINISTRATIVENFORMATION

ThisinvestigationwasauthorizedandundedyheNavaleaSystemsCommand \035)i562-002. (SEA35 )underheGeneralHydrodynamicsResearchProgram,SR-023-0101,WorkUnit

Pernng ,W.Ci.A.ndI .Clauert ."StabilityonheWatofaSeaplanenhePlaningCondit ion,"AeronauticalResearchCounci l ,KVol .423e p1933). Aompleteut ingofreferencessgivenonpages60an d61.Lutowskl ,R.N. ."AComputerProgramfo rVariousPerformanceAspectsofPlaningCraf l ."Thesi ssubmittedtoStevensInstitute ofTechno logy .CastlePoint.Hoboken ,N.J.(1973).Payne.P.R. ."Coup l edPitchan dHeavePorptisingInstabilityinHydrodynamicPlaning,"JournalofHydro.iautict ,Vol .8,No .(Apr1974) .Day ,J.P.an dR.J.llasg,"PlaningBoatPorpoising"Thesi sSubmit tedtoW;bbInstituteofNavalArchitecture.GlenCove .LongIsland,N.Y.(May19S2).

INTRODUCTIONPorpoisingisannstabilitynitchndheavexperiencedylaningcraftravelingat

highpeedsonalmwater. Itha see nnownoleadosuchviolentmotionsasoause manyeriousboatingaccidents.Withonstantlyncreasingboatpeeds,hi sphenomenons becomingmorean dmoreofaroblemolaning-boatesigners.

PerhapsheirstattemptatreatingthisroblemnalyticallywismadeyPerring,' ho developedaheoryorporpoisingbasedonow-aspect-ratiowingtheory.Theracticalapplicationofthistheorywa sunsuccessfulsinceheheorywa soversimplified.Sincehen,greatea lofexperimentalworkasee ndoneoncerningporpoisingforwater-basedircraftandlaningboats,andmoreecentlydditionalattempts2,3tdevelopingaheoreticalreat-mentav emetwitharyingdegreesofsuccess. Perhapstheonlysystematicexperimentalinvestigationfo rplaningboatswa sdonebyDa yandHaag onprismaticbodies.hesebodies comprisedawiderangeofdesignparameters.heresultsofthisworkhavebeenwidelyusedas

.in i iimu m il lw-teaWiMBi

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aguideinestimatingtheporpoisinglimitsofplaninghulls. Althoughthisisareasonableempiricalapproach,itdoesnotcontributemuchoagoodtheoreticalunderstandingoftheproblem.Suchannderstandingisrequiredtodetermineheeffectsofvariationsinhullparametersofpracticalboatraswellastoevaluateinnovativeideasforpreventionofporpoising. Furthermore,itsimportantfo rprovidingatoolforestimatingtheeffectsofdesignarametersonnaturaloscillationfrequenciesanddampingcharacterisesoftheoat,sincethesecharacteristicsplayadominantroleindynamicbehaviorinaseaway. Inact,thetheoreticalapproachderivedhereinhasbeensedndevelopmentofalinearizedheoryforpredictingthemotionsofplaningboatsofarbitrarydeadriseangleinwaves.

Althoughtheheoryhasbeendevelopedforprismaticplaninghulls,itappearstobesuitableasaguideforpredictingeffectsofparametervariationonorpoisingofpracticalplaning-boatconfigurations. Furthermore,hemethodsusedlendthemselvestodirectextensionoaheoryfornonprismatichulls.

STABIL ITYEQUATIONSStabilityequationsforthelongitudinalmotionsinsurge,pitch,andheavearederivedin

AppendixA.Tomakecomparisonswiththelargequantityofexistingporpoisingdatafrom towedmodels,rovisionwasmadetoincludetheeffectofthetoworce. Sincethemodels werefowedatconstantspeed,thesurgeequationlayednoroleindeterminationoftheboatmodelstability;onlyhecoupledpitchandheaveequationswereneededoinvestigatetheproblem.However,itisbelievedha theresultsobtainedinthismanneraregenerallyapplicabletoboatswithallhreedegreesoffreedom,since,asshownnAppendixA,themagnitudeofthestabilityderivativesinthesurgeequationreconsiderablysmallerthanhoseinthepitchandheaveequationsformostcasesofinterest.W ethereforeusedhefollowingnondimensional*inearizedstabilityequationsforthedynamicheaveorceanditchmoment equilibrium,respectively.**

^svitaky.[).,"Hydrodyn im i cDesignofPlaningHulls."MarineTechnology(Oct964).''Martin.M. ,"TheoreticalDeterminat ionofMotionofHigh-SpeedPlaningrutnWives"OTNSROCReport76-0069(Apr976) .

'Theprimesymbol ,no:mallyusedtodenoteanondimensionalqual i ty ,isomittedto rconvenience . "Theeffectofth esurgedegreeoffreedomisreadilydeterminedfromth estabilityequat ionsfo rsurge,pitch,an dheaveas derivedinAppendixA.

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(ZJ;-m)'z+Z^ z+Z2z+Zd+ Z Q6+ Zd*01) M j jZ +M ^ z +M zz+ (Mg-ly)+M 0+M 0*02)

Theseequationsdescribehemotionrelativeofixedhorizontalandverticalaxes,0xand0 z, alongandatrightanglestohedirectionofmotion;seeigure.Theorigin0wastakenat theoa tcenterofgravityandroveswithheonstantreferencespeedUoftheboat.ThesymbolsZandrepresenttheverucalforceanddisplacement,respectively,at0,positive inthedowndirection. ThesymbolsM an d0rehepitchmomentandangulardisplacement perturbation,respectively,withrespectoheorigin0,positiveinhesenseofbowup.Thecoefficientsofthevariablesz,z,z,0,9,and0rehestabilityderivatives;e.g.,Z sthe nondimensionalinearizedrateofincreaseofverticalorceZwithondimensionalangularacceleration0. ThestabilityderivativeshaveeenerivedorconstantdeadriseplaninghullsinAppendixAinermsofthegeometricandoperationalcharacteristicsoftheboat.Theywerederivedon theassumptionha thecraftcouldereatedasaslenderbodywithanmpiricalhree-dimensionJorrection. Becauseofthehighroudenumberrangeofoperationandthelow aspectatio,wavemakingandnsteadylifteffectswereassumedegligible.

Thevelocityandccelerationerivatives(AppendixA)areZt-*>(X)cos2rju'ds 3) Zi-2v?(X)*is'cos3T 4)Z f l* ( X )cosrlp (a-s '):ds ' 7)M^VKXJCCK2 Vds'-Xg/VJ 8)M e= -*{\)JM'(a'"s')2d ' 9)M y= -2*(X)cos rh/$'X g2+y/(a'-s ')ds'j10)

where= equilibriumtrimangleX =meanwettedlength-to-beamratio0(X)=three-dimensional correctionfactory . nondimensionalsectionaladdedmass Ht -nondimensionalsectionaladdedmassattransom

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X nondimens iona ldistancefromtransomtocenterofgravityDs' =nondimens iona ldistancefromforemostwettedpointonkeelto an yboatsection;se eFigure20

a' =valueofs'atboatcenterofgravity Theintegrations havebeentakenoverth ewettedlengthofth eboat .xpressionsfo rth esectionaladdedmassdistributionarederivedinAppendixA.

Theso-calledstaticderivativesareobtaineddirectlyromheexpressionsfo rth esteadystateZg'orcean dM . 'moment . Sincebothreunct ionsofXan dT,an dX=X(z,T),we have

leJIi ^l2) 3r 3X or0 or d\ 3T

Th eteadytateZg'orcewhichisth enegativeofth eliftforce(AppendixA)isZc=-ip(X)psinTCOS2+ X 2sinrCOST-XCfsinr/cos15)where=deadriseangle CvU/ \ /gb"=speedcoeffic ient C f =skinrictioncoef f i c ient

Theirstermsth edynamicif tonhehull,th esecondermsth ehydrostaticlift,an dhelastermsth everticalomponentofth eskinrictionorce,assumedoac tparallelohe keeline.

ThexpressionorheteadytatemomentMsaboutheboatenterofgravity (AppendixA)isMs'^XHinrcmr^'d^^^

C f X(X v " 'Ts f(k,,ei)+ F w S f(k2,e2)16)COS ]

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wheregnondimensionaltowingforce=dragFws=nondimensionalaerodynamicdragf(kj,ej)=momentarmoftowingforce

f(k2,e2)=momentarmof aerodynamicforceTheequationforTgEquation(90)ofAppendixA)is

X CfTc'=-Zo'tanT+FW< /17)sOSpCOST waTheirsttermnEquation16)isthehydrodynamicmoment,thesecondtermsthehydro-staticmomentandthethirdtermsthemomentdueoskinfriction.Thelasttwotermsare themomentdu etothetowforceandtheaerodynamicdrag.ThesteadystatevaluesofXandrusedinEquations(3 )through04)aredetermined fromthefollowingequations ofplaningequilibrium.

W'+Zs=0 18 )M s=0 19 )

TheseequationsaresolvedbyaniterationrocessdescribedinAppendixA.ThesolutionoEquations(1 )and(2 )are

z '=Z j +z2e +20) o.Y

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Ineneral,acomplexpairofrootsrepresentsanoscillatorymode ,e .g. ,orth eoo tpaira'-oR' ioj',hez'e sponseis

_.iz'=eRlC ,osaj't'+C2sinOj't')whereC jndC2reea lonstantswhicharedeterminedbyhenitialconditions. Themagnitudeofth eimaginarypartofth eoo tj 'sth enondimensionalnaturalrequencyofth emodalmotion . Indimensionalormhenaturalrequencyan dperiodre

O j=O j 'ad/sec 24)T=ec 25)I

Thef fectofth eea lpartofth eoo toR'maybeillustratedbyomputingth eim eora disturbanceoeitherhalveordoubleitselfinmagni tude . Thus ,ifoRsnegative ,he envelopeofth edisturbancewillbehalvedwhen

e =e =1/2Itollowsthatheim efo rhedisturbancemotionofeachmodeohalveordoubletselfis

*l/2or* 2=0-69/oRec 26)Anotherusefulmeasureofdampingofoscillatorymodesisth edampingratio,whichsdirectlyelatedoth eateofdecayofdisturbanceoscillations. I tsgivenby

,R z -~iT27)I nheicinityofth eesonantencounterfrequencynwaves,hedampingratiosalsoinverselyelatedoheamplificationatioofth eboatesponse . Valuesofbetween0. 6an d1. 0areusuallyonsideredogivewell -dampedmodes . Valueslessthanabout0. 4aregenerally consideiedoproduceunderdampedmodes . Althoughheorgoingmayprovideaough indicationofth everticalplanedynamiccharacteristicsofth eboat,adynamicmotionsanalysisisrequiredoran ydetailedtudy .

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DETERMINATIONOFPORPOIS INGCONDITIONS Perhapstheonlysystematicexperimentalinvestigationofporpoisingwascarriedou tin

1952byDa yandHaag.4Measurementbyotherinvestigators7,8,9avegenerallybeenincidentaltoabroaderprogramrimarilyconcernedwithesistance.TheexperimentsofDayandHaagwerecarriedoutwith3.8-inch-beam,prismaticwoodmodelstowedbyalightlinefromapointslightlyforwardofan dabovethecenterofgravity.Deadriseanglesof0,C.6,and0.5degreeswereinvestigated.Theloadcoefficientswere0.36,0.48,and0.72.Foreachspeedtheboatcenterof gravitywasgraduallymovedaftuntiltheboatporpoised.hetrimatwhichthisoccurreddefinedthecriticaltrimangleandrovidedapointnthetrimangleversusspeedtabilityoundary.Valuesofthemassan dmomentofinertiaforpracticallyall themodelsatporpoisingwereprovided.Thus,itwaspossibletomakeatheoreticalcalculationforeaches tcondition.mallallowancesforaerodynamiceffectsandhinecornerradius weremade.ThesearedescribedinAppendix.

Calculationsofeachoftheou rstabilityrootsforeachtestconditionwereobtainedrom Equation(22). Inhevicinityofporpoising,theleaststablerootwascomplexineachcase,indicatingthatheresponsetoadisturbancewouldalwaysbeoscillatory. Figuresaypicalplotofthevariationoftherealar toftheleaststableroota 'withhenondimensional longitudinalistanceX Rofthecenterofgravityromhetransomorthreeofthees tspeeds with 10.6degreesandC^=0.48. Itisseenthatthestabilityrootsbecomenegative(stable)orvaluesofX lessthanbout0.20andgreaterthanabout0.85orthecasesshown.0W enoteha tnhestableregion,correspondingtothesmallX range,heequilibriumrim anglesaremuchhigherthaninhestableregionofthehighX range.Also,themagnitudeof thestabilityootinhemallX rangeisusuallyquitesmall,sothatthedampingoftheboatoscillationsma ysuallybeexpectedobeoor.Takinganexampleromheigure,weind fromEquation(26)thatthetimeforadisturbanceodamptohalfamplitudeatCy .67 wouldemorethan5econdsforaboatwitha5-fcoteam,andR1'=-0.01.This wouldbeintolerableinhepresenceofevenverysmalldisturbances.Ontheotherhand,itis seenromthesameigureha tmovingthecenterofgravityorwardoftheuppervalueat whichheootecomesnegativeresultsinincreasinglyegativevaluesofthestabilityrootand.Iridsma,G. ,"A SyitemtticStudyofRough-WaterPerformanceofPlaningBoats,"DavidsonLaboratory,StevensInstituteofTechnology,Hoboken ,N.J. .Report27 5(Nov1969).

aC l ement .F.P,an dD .Blount,"Resistance Tests o lSystematicSeriesofPlaningHullForms ,"Transact ions SocietyofNavalArchi tectsandMarineEngineers ,Vol .7 1,pp .491-S611963).9Dav idon ,K.S.M.an dA.Suarez,"Tests ofTwen tyRelatedModelsofV-BottomMotorBoats-EMB Series50,"DavidTaylor ModelBasinReportR-47(1949).

mmmmmmmmm

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therefore,considerablyigherdampingforthismode. Forexample,forthesamespeedan d\=.5,wefindoR1'=0.3,andthetimeforadisturbancetodamptohalfamplitudeisoftheorderofonlyne-halfsecond.Sincethecalculatedvalueofo..'atthisconditionsabout0.5,thedampingratioromEquation(27)sfoundtobeapproximately0.50;whereas,forthe previousconditionitwa sonlyabout0.02.

AnalternativeplotoFigureshowingthevariationofthecalculatedR1' withrim angle,foracompleterangeoftestspeeds,with0=0degreesandC =0.72,sshowninFigure3.Thisplotstypicalofthoseusedtodeterminethecriticaltrimangleasafunctionofthespeedcoefficient.Wheretw otrimanglesatzerorootcrossingareshown,thesmallertrimanglesareaken.Theporpoisingboundariesthusdeterminedarediscussedinthenextsection.

COMPARISONSWITHEXPER IMENTS ONPRISMATICHULLSFigure4showsthevariation,withhespeedcoefficientCy,ofthecriticaltrimangleof

aprismaticplaningconfiguration,withadeadriseangleofzerodegrees,orthreevaluesofloadcoefficientTheointsintheigureepresenttheexperimentaldata.4Thecurveswereobtainedfromthepresenttheory. Itisseenthatthetheorygreesreasonablywellwithhe experimentsforC =0.36and0.72.FortheintermediateloadcoefficientC^=0.48,thetheorygivesintermediatevaluesofcriticaltrimangle;whereas,thedataforthisconditionare closeothoseobtainedatC =0.36.

Figureisasimilarsetofgraphsfortheprismatichullwithadeadriseangleof10.6degrees.Here,theagreementbetweenheheoryndtheexperimentalresultsoverthewholespeedrangeisquitegoodforC^=0.48and0.72. However,orC =0.36.thetheoryends tounderpredicthecriticalrimangleaboutegreeatthehigherspeeds.Partofthis discrepancysdu eoheactha thechineecamenwettedattheransom,andthetheorydoesnotaccountforthis.*Thediscrepancyendstobegreatestatthehigh-stspeedsandfor thelowestloadingconditions.Theregionwherethewholechineisou tofthewaterisindi-catedapproximatelyyheashedportionoftheurve.

Figure6sasimilarcomparisonforahullwitheadriseangleof20.5egrees. Herehe theoreticalcurvetendstofallslightlylowerthanthedata,atthelowerspeeds,andunderpre-dictsthecriticalrimangleyaboutonedegreeathehigherspeeds. Inthiscasehechinebecameunwettedsoonerthanfortheboatwitha0.6degreedeadriseangle.Theapproxi-materegionofoccurrenceisindicatedbythedashedportionofthecurves.Anattemptwas Thetheorysbe ingmodi f i edtoinc ludetheseeffects

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madeoetermineheffectfmodifyinghe agnitudefheowerimitfntegrationinpartoftheaddedmassntegralsdiscussednAppendixA.ThishadheeffectoffurtherimprovinghepredictionsshownbyhebrokenurvesnFigure.Thislightalterationoiheheoryhadpracticallynoeffectonhepredictedporpoisingboundariesorhehullshaving0-and0.6-degreedeadrisengles.

D ayndHaagoundhatbyplottinghemeasuredrimnglegainst(CJJ,/2)''2 r/C /Cy,heeparateurvesoreachoadcoefficientollapsedntonarrowband.Thisresultwaslsooundromheheoreticalcalculations.Figures hrough9howcomparisonsbetweenheexperimentaldataandhecalculationsplottednhismanner. Exceptsnotedearlier,tseenhatheoverallgreementbetweenheheoryndhedataseasonablygoodforal lofhedeadriseconditionsnvestigated.

ForachriticalrimngleC,heorrespondingcriticalpositionoftheboatcenterofgravity mustatisfyheteadytatequilibriumequations.ThushedegreeofagreementbetweenhemeasuredandpredictedmagnitudeofXs ,npartateast,measureoftheaccuracyoftheteadytateequations. Figure0howscomparisonwithheoryofthemeasuredvariationofX withpeedcoefficientoreachoftheconditionsnvestigatedby5 *DayndHaag.tsseenhathebestoverallagreementwasobtainedorhehullwithzerodeadrisengleorhedeadrisenglesof0.6nd20.5degrees,healculationswereabout10o5ercentowerhanhemeasurements.

ThisesulteemsoeconsistentwithheendencyoftheteadytateEquations(18)and19)ounderestimateheteadyrimngle,husequiringamoreftpositionofthecenterofgravityoobtain ivenrimngle.ComparisonwithheteadytaterimdatanRelerenceofaimilarormulationecentlyroposedby rown exhibitsheameend-encyoapproximatelyheamedegree. However,hempiricalormulationresentedbySavitsky5oesnothowhisendencyndives,onheverage,betteragreementwithmeasuredteadytaterimngles. Aswilleeenater,heuseofthisormulationgenerallygivesargervaluesofX for ivenvalueofr.*

AnothernterestingeatureofFigure0shatheeffectofloadoefficientppearsobemall.Thissmainlydueoheacthathemorehighlyoadedonfigurationsofthe D aynd aagmodelshadmalleradiiofgyration. Aswilleeenater,heheoryhowsthathesewoeffectsregenerallycompensatorynheireffectonheriticalcenterofgravityosition. Itslsoworthmentioninghatheheoreticalvaluesoftheocationofthe

Brown,P.W.,"Anhx perimen l Ian dTheoreticalStudyofPlaningSurfacewithTrimlaps."David ionLaboratory,StevensInstituteofTechnology .Hoboken .N.J . ,Report1463(Apr1971) . Thereasonsfo rthisdi screpancyarebein:investigated.

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hydrodynamiccenterofpressurean denterofgravitywerewithinpercentofeachotherin nearlyal lases.sonemightexpect,hiswa shecasehroughouthetableoperatingrangtaswell.Figure1salo toftheheoreticalnondimensionalmeanwettedengthoftheboatatporpoisingagainstpeedoefficientorheariousconditionsinvestigated.Unfortunately,DayndHaagdidnotmeasurehewettedengthoftheirmodels ,oodirectomparisonwithhedataispossible. Averagealuesofobservedwettedengthsofthespraysheetthe chine,whichidnotarymuchwithpeed,rehownoraoughomparison.Ason emightexpectheprayheetengthelativechemeanwettedlengthecomessmallerwithncreas-ingdeadriseangle.

Foraypicalonfiguration,igures2nd3ho wheeparateffectofnondimensiona;loadingC^ndadiusofgyrationc/bonhemagnitudeoftheriticalrimngleCs determinedro mheheory. ItsseenhatQecomessmallerwithbothncreasingradiusofgyrationndoa doefficient. However,heeffectsnotlarge -especiallyorsmallvaluesof^Lb* '*w' eeca"edthathehighertheloadingthesmallerwa siheradiusofgyrationorthemodelsinvestigatedyDayndHaag.Thiseffectalsocontributed,osomeextent,ocollapseoftheirdataordifferentloadonditions.Aimilareffect,houghmorepronounced,ma yehownoholdwithespectoheriticalondimensiou? alueoftheongitudinalcenteiofgravityositionndmeanwettedength. Figures4hrough6showhathealueoi andmcncreaseswithbothncreasingC^ndncreasingky/b. HereheseeffectsaccountalmostntirelyortheendencyoftheX versusCvurves(Figure0)an dhe XmcersusCvurves(Figure1)tocollapseintoanarrowandorthevariousloadings shown.

Although,healculatedvaluesofX tendoeow .igures4nd5evertheless illustratesomenterestingtrends. ItsseenhathevalueofX ha sanncreasingtendencytoreachalimitwithincreasingdeadriseangleanddecreasingloadcoefficient. ls othevalueofX becomessmallerwithdecreasingnondimensionalradiusofgyrationan dincreasingdeadrise angle.tthusappearsthathighdeadriseangle,lowloading,an dasmallradius ofgyrationpermitamoreaftlocationofthecriticalcenterofgravity.urthermore,itappearsthatthetendencyforthecurvetoturndownatthehigherdeadriseanglesmayexplaintheobservationbyStolz and othersthat"deadrisesurfaceswhichar eno tdeepintotheporpoisingrangeoftenregainstabilityatigherspeeds."

RelativelyecentlyFridsma'eportedatherlimitedmountofexperimentaldataon theorpoisingofprismaticullswitheadriseaiiglesof0,0,an d30degreesandwith 'Theoryan dexper imenthow,however ,thatnaturalfrequencyofoscil latorymotiondecreaaessignificantlywithncreasing momen tofinertia.

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variousloadoefficients.Experimentsforthe0degreeeadriseas ewerecarriedoutwith modelsoflength-to-beamatioof4,,an dsothatdifferentvaluesoflc,/bwereobtainedfo rthesameC^nom ecases. Figures(17a)nd(17b)showomparisonofmeasuredvaluesofcriticalrimanglemadeyr idsmawithhosecomputedro mheheory.The theoreticalurvesareasicallyrossplotsofcurvessuchsthosenigure2orvariousload coefficients.Theyrovideaonvenientmethodofindicatingtheeffectofvariationsinky/b,separatelyro mCndC^onhemagnitudeofT C.tmayeseenro mheigurehathe theoryredictsT_uitewellnmostasesbuthatherearensufficientdataoverifyhe cvariationwithc,/bpredictedbyheheory.

ThealculatedvaluesofX or ,ndXmrnheseaseswereabout8and5percentlower, respectively,thanhemeasuredvalues.Th eormerismorehantwicethedifferencecomputedforheDayndHaagexperiments. Asnotedearlier,hi swasel ttobeuemainlyohe facthatheteadytatequationstendounderpredictteadytaterimangle.ValuesofX werehe necalculatedoreachoftheheoreticalvaluesofrcro mheollowingsteady statequations,adaptedromReference.

X=0.75Xi-*X28)F5.21Cv2/X 2)+.39 * vwhereXmcsdefinedbyheollowingequationsCrc'-'(0.0120xJ/20.0055\J2ICyhCLbCLo -0065Cj>*

an dX _stheondimensionalistanceofthecenterofpressureorwardoftheransom.Th e assumptionhatheenterofpressurendcenterofgravityreoincidentwasfoundro m theheoryobevalidowithinewpercentnearlyllases. Theus eoftheaboveequationsresultedinbettergreementwithheexperimentalvalueswhenheheoreticalvalues ofT(weresubstituted.Table howsacomparisonofX thusobtainedwithhemeasuredvalues. Alsoshownrecomparisonswithhemeasuredmeanwettedlengths. Itsseenhatthealculatedaluesarewithinaewpercentofthemeasurementsinnearlyllases.

COMPARISONS WITHXPERIMENTS ONNONPRISMATIC HULLSAlthoughheheorynddatapresentedhu sfa rareorprismatichullorms,he yho w

trendswhichrecloseothoseobservedonavarietyofnonprismaticmodels.Twoprinciplesetsofdata8,9reavailableorcomparison.Th eDTMBSeries62modelsofClementnd Blount8reur emonohedranhullswitharansom-to-beamatioetween0.64an d0.80an d constantfteadrisengleof2.5egrees.ThoseofDavidsonndSuare/ havelowhine

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TABLE-COMPARISONBETWEENCALCULATEDANDMEASURED VALUESOFX cANDXmc,USINGCOMPUTEDCRITICALTRIM ANGLEANDMETHODOFREFERENCE 5FORMEASURED PORPOISINGCONDITIONSOFREFERENCE7

Deadrise 0deg Load Coefficient Speed Coefficient cv

X opCalculated X .Measured XmcCalculadXmMeasured mc

10

20

30

0.912 0.6080.3040.3040.6080.6080.3040.3040.6090.912

3.833.003.33 2.00 3.89 2.73 2.662.98 2.73 3.85

0.860.900.930.94 1.07 1.031.00 1.02 1.021.07

0.94 0.991.06 1.001.170.97 1.12 1.000.951.00

linendwarpedbottomwithtransom-to-beamatioof0.88ndameaneadriseanglebetween4.0and.3degrees. Inviewoftheapproximatenatureofthecomparisonitwas felthatexistingcalculationsorthe0.6-degreedeadriseonfigurationswouldgiveasuffi-cientlyoo drepresentationoftherendsmeasurednhereviouslymentioneddata.

Th eorpoisingboundaryescribedyClementandBlountwa spresentedsalo tofC,b/X versusthevolumetricFroudeumberFywhere

Cu, CA/CV2Fv U/ CV/(CA)1/6

Itsseenhatheriticalrimngleoe sno tppearanywhere,an dheorpoisingboundaryisexpressedmainlynermsofthecriticalositionoftheenterofgravity.Thismethodofplottingwa sanapparentattemptocollapsehemeasuredvaluesofX forthewideangeofloadoefficientsnvestigated. Figure8howsaomparisonoftheheoreticalalculatis withheSeries62data.Theheoreticalcurvesar econstructedfromhecalculatedcurvesofFigure0for0=0.6degreesan dCA=0.36.0.48,an d0.72.TheC^valuesar enhe approximateangeofmostoftheata.Th evaluesofCLb/X gcivennReferencewerenondimensionalizedwithespectoheea mtheenterofgravity,whilehoseshownnFigure8haveee nasedonhemeanofthemaximumea mndheea mtheransom inordertoprovideamorerealisticomparisonwithheonstantea mcase. Itsseenhattheheoreticalurveollowshesamerendasthedatautsabout0to5percentigher.

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theoryoncludemoregeneral-typehullonfigurations. CONCLUSIONS

A theoreticalmethodhasbeenderivedforpredictingtheonditionsleadingtoporpoising inhesurge,itch,andheavedegreesoffreedomofprismatichullswitharbitrarydeadriseangle.Comparisonsofthetheorywithheporpoisingboundariesmeasuredonowedmodels

Agnelli.I.C.,"Kvaluilionofth eTrimof iPlaningBoattInceptionofPor^'iising."presentedatSpringMeetingcfSocietyofNivalArchitectsandMarineFnginecrs,LakeHm-'uVista ,I'll.(Apr1973).

1 3

ThissmainlyreflectionofthedifferencebetweenredictedandmeasuredvaluesofX _ showninigure0. Itma ybeseenhathepointsfordifferentloadingconditionswillall onseparateinesandha thetrendwithloadingforeachmodelisclosetothatgivenbythetheory;e.g.,compareConditions3,4,andwithhetrendsshowninigure5.

Itseasyoshewthatthemeasuredeffectofradiusofgyrationxhibitsthesametrends asthathownyheheory;seeFigure4.Althoughheradiusofgyration/bofthe modelswasno tmeasured,itsreasonableoassumethatitwasincreasinglylargerforthe modelswithargerlength-to-mean-beamratioLp/Bp^. Bycomparingthetrendsofthedatafordifferentmodelswithpproximatelyhesameoadoefficient(Conditionswith ndwith)weseeha thevalueofA followsthetrendswithkbpredictedbyhetheory.

GementandBlountfoundthattheslopeofastraightlinethroughthedatapointswas about-2.5.Thisimpliesthat,foragivenvalueofCA

-sinceCy, C /Cy Fromigure0itsseenha thepreviouslyescribedquationwouldfitheheoryndexperimentaldatafortheprismauchullsquitewell. Althoughheequationmayeagoodapproximationorthedatashown,itsseenromheheoreticalurvesof Figures4and5orkbconstantha ttsnotsafeoassumeha thisstruenallcases.

ThedataofDavidsonndSuarez9remostonvenientlyomparedinheormfa porpoisingboundaryresentedyAgnelli.1 Thissshowninigure9asaplotofmeasured valuesofthecriticaltrimangleT Qagainst2/(Cljbf ) .ncludedalsoarehedataointsforthe Series62ulls.ThesewereecentlybtainedbyheauthorfromMr.Blountandareot quitethesameasthoseshownnhelotfReference1. Shownorcomparisonwithhe trendspredictedyheoryrestraightinesdrawnhroughointsobtainedromhealculatedcurvesofFiguresnd0forhecaseof0=0.6degreesandCA 0.36.0.48,and.72.Althoughmanyfheointsdonotallnhelines,tsclearthatherendsarepredicted.Moreetailedcomparisonswithheindividualmodelsmustwaiturtherextensionofthe

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withreedomonlyinitchandheaveshowedreasonablygoodagreement.Sincenoporpoisingdatawithallhreedegreesoffreedomareavailable,itwasno tpossibletocheckthetheoryor thiscase.However,fromacomparisonoftherelativemagnitudesofthecoefficientsinthe surgeequationtappearsthatthiseffectissmall. Inanycasethismayeadilybeinvestigated inmoredetailwiththethreedegreeoffreedomstabilityequationsinAppendixA.

Thestabilityootsobtainedromhecharacteristicequationrovideestimatesofthe dynamicbehaviour,suchsoscillationaturalfrequenciesanddampingcharacteristicsoftheboatnhestableregion.

Althoughheheorywasdevelopedorprismatichullforms,itappearstobesuitableas aguidenestimatingtheorpoisinglimitsanddynamiccharacteristicsofmoreconventional- typeplaninghulls,aswellastheeffectsofvariationsinseveraloftheparameters. Itselt thatynextensionoftheanalyticalmethodsusedinhepresentanalysisevencloseragree-mentwithataonrismatichullsandconventionalboatscouldbeachieved,andtooluitableforinvestigatingtheeffectsofdetaileddesignmodificationscouldbeobtained.

ACKNOWLEDGMENTSIwishoexpressmyeepgratitudetoMr.JacquesB .Hadler,underwhosesupervision

theresentworkwasstarted,andoMr.GrantR.Hagen,HeadShipDynamicsDivision,or theircontinuingencouragementandnterestnhepresenteffort. MyppreciationsalsoextendedtoDr.William.Cummins,HeadShipPerformanceDepartmentandtoMr.VincentJ.Monacellafortheirvaluableassistanceandsupport. Furthermore,specialhanksaredueto M s .NadineHubble,whodevelopedhecomputerprogramandcarriedou tllofthecalcula-tionswithhercharacteristicskillndefficiency.

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APPENDIXADERIVAT IONOFPORPOIS INGSTABIL ITYQUATIONS

Itsas sumedhathelaningcraftasarismatichullofconstanteadrise,ismovingatconstantpeedaralleloaal mwatersurface,ndsreeoperformmallperturbationmotionsinitch,heave,ndurgeabouttssteadyquilibriumattitude.Sinceheheorys concernedmainlywithheigh-speed,ow-aspectatiocondition,itsassumedhatheraftma yereatedsaslenderbodywithnempiricalhree-dimensionalorrection,ndunsteadyeffectsar esmall.Theindofanalysistobesedwasfirstsedn924byMunk' ndaterbyJones inonnectionwithheanalysisofairshipsandlenderwings,respectively.Morerecentlyhi smethodha see neneralizedyBryson1 fo rcompletelyubmergedlender- finnedmissiles. Itasalsobeenppliedoheroblemofpureranslationalmpactofsea-planesncalmwatersurfaceyMayo1 ndothers.16,

FORCESDUETOPERTURBATIONS INVELOCITYANDACCELERATIONTh elowve rtheul lsassumedooccurinransverselaneswhichreixednpace

andorientedormaloheeel;seeigure0. Themomentumofeachayerofwaterrans-verseoheee ls/ifds,where(ishewo-dimensionalddedmassoftheectionoftheul latoint,nteractingwithheectionofth elowlaneoflengths ,ndsthecomponentoftheelocityoftheodyormaloheeelthatoint.Theoordinateismeasured fromheoremostmmersedtationlongtheeel. Theormalorcenheectionsoftheul lstheim eateofchangeofthemomentumoftheayerofwaterdsat.

'"Muni. .M.M.."TheAerodynamicForce*on AirshipHull."NationalAdvisoryCommitteefo rAeronauticsReport184(1924).Jones ,R.T. ,"PropertiesofLow-Aspect-RatioWingsatSpeedsBelowan dAboveth eSpeed sofSound,"NationalAdvisory Commit t e efo rAeronauticsRepor '53(1946).Bryion,A.E . ,Jr.,"StabilityDerivativ. raSlenderMissilewithApplicationtoaWing-BodyVerticalTrailConf iguration ,"JournalofAeronauticalSciences .Vol . No .S,pp .29'/308(I9Sj).Mayo,W.L.,"Aiul>s i an dModif icationofTheoryfo rImpactofSeaplaneson Water,"NationalAdvisoryCommitteefo rAeronauticsReport810(1945).Milwitzky,t"AGeneralizedTheoreticalan dExper imentalInvestigationofth eMotionsan dHydrodynamicLoadsExper iencedhyV-Bottom SeaplanesDuringStep-LandingImpacts,"NationalAdvisoryCommitteeorAeronautics TN1516(1948).Schni tzer .E .,"TheoryndProcedurefo rDeterminingLoadsan dMotionsinChine-ImmersedHydrodynamicImpactsofPrismaticBodies,"NationalAdvisoryCommitteefo rAeronauticsReport152(1053).

1 5

n* nmn fmm mmmmmmmmummmmmmmmmmmmmm*- .- -w

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dFD= inds 29) Bothnandfwillineneralbefunctionsofthelongitudinalpositionoordinateandime.Theimederivativeistherefore

dtx dt whereisthesteadystatespeed.ThenormalhydrodynamicforceovertheentirehullisobtainedbyintegratingEquatior. (29)alongthewettedlengthofthehullkndmultiplyingbyacorrectionactor0(X)to accountforthethree-dimensionalityoftheflow.

rckF r >*(*)/ -r:(Mr)ds 31 )D V . T ,whereX sthemeanwettedlengthividedyhebeam. Aplotof< p ( \ )obtainedempirically byabst1 i sshownonigure1.Theintegralmayeexpressedasthesumfavelocitytermandanaccelerationerm.

Theongitudinalandheaveerturbationelocitiesandaccelerationsarc,espectively,denotedby,,z,z .Thepitchangleerturbationsare0,0. Fromiguresnd0we obtainheollowingrelationships:

=-cosT 33)dx =~sinT 34 )3x

3tr-=us inr+zcosT0(a-s)35 )otf0 36)

whererstheequilibriumtrimngleoftheboat,andaisthevalueofsathetranverselow-planethroughtheboatcenterofgravity.romheseequationsandEquation(30)wehaveto theirstordernheperturbations18Pibit,W.,"LandingImpactofSeapbutei ,"NationalAdvisoryCommi t t eeT o tAeronautic TM624(1931).

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f=Us in inr+zosT0(a-s )f=U0os+us in zcosr-0(a-s )

dp_dfi_sdndt 9f 3s

(37)(38)(39)

OnubstitutingtheseequationsintoEquation32),droppingthecondorderperturba-tionterms,andntegrating,weobtain

(40)D=FDS+AFDwhere

FDS=i f i C K )p2inTcosrA Fn=(2 Fn./U)(u+zco tr+ 0C/sinT) D D S I'

+

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2MDS AMD-u cot)(X)2UcosTMsg2 P(a-s)ds

+ip(X)(uin osT) p.(a-s)dsV .I L(\)0I ju(a-srdso (46)

TheirstermMpgsheteadytatehydrodynamicmomentabouthecenterofgravityoftheoat.heemainingermsAMQreheinearizedcontributionsofheurge,pitch,ndheaveperturbationsnvelocityndacceleration.

VELOCITYANDACCELERATIONSTABILITYDERIVATIVESTheverticalndhorizontalomponentsoftheorcetabilityderivativeswithespecto

thevelocityndccelerationerturbationsrehecoefficientsoftheperturbationermsn theverticalndhorizontalcomponentsoftheperturbationorceAFDnEquation42).W ewriteheseequationsinnondimensionalorm*bydividinghroughby/2pU Thustheverticalomponentoftheperturbationorceequationecomeswithpositivedown

AZD 2FDS'osr(u'z'cosT6 Xsin)* 5 < X ) (uincos :cos2r)fVds'+9fV(a's')ds'

where FDS'p(\)* ss inco sT

6 0b/U tc,

(47)

(48)

'Nondin*ns ionalquant irsKepresentedbyaprimetymbol .

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LikewisethehorizontalcomponentwithXpositiveforwardbecomesAXD' AZD tanT (49)

Theorrespondingnondimensionalstabilityderivativesorthemomentequationareobtained inananalogousmannerfromthenondimensionalformofEquation(46)as

AMD'=2MDS'(u'+z 'cotT)

where

-*(X)2cosTS'g2+/V(a '-s ')ds '8+ (X) (u 's inr+i cosT)I fi d -s ')ds '

-*>(X)0'/(a'-s')2ds ' rXjjM DS'= (X)/ n'(a'-s')2d '

(50)

(51)

Typicalotationforthenondimensionalstabilityderivativesareshownasfollows z '-2FD SgcotT K=zu 'tn'M ->p(\)J V(a'-s')2ds '

ZJ--

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SECTIONALADDEDMASSDISTRIBUTIONThenextstepstofindanexpressionorthedistributionftheboatsectionaladded

massasafunctionofs .A generalheoreticalexpressionorthisquantityisno tavailable,evenforprismatichullswithconstantdeadrise. However,relativelysimpleapproximateexpressionsforaddedmasshavebeensedsuccessfullyinthepastintheanalysisofhydrody- namicimpactofprismaticodies. 1 Forthesectionsofthatportionofthebodywithhe ;hinesabovethewatersurface,thesectionalddedmasswasestimatedbyhpfollowingequation

,-pm 52> where

f(0)=-l 53 )=deadriseangleinadians

ThisexpressionsbasedonheworkofWagner.'9Thequantityf f(0)stheradiusofthe semicircularcylinderrepresentingtheaddedmassofthesection. Forthesectionsofthehullwithhehinesubmerged,theollowingexpressionwasused.

M=-~f(0)tan0)2B|b(f-rc)54 )wheresaunctionftheangleofdeadrise,andsthebeateam. TheirstermsthecontributionoftheV-shapedottomaloneathenstantofchineimm.rsion.ThisisobtainedfromEquation(52)byuttingf=fc /2an . Thesecondermsanestimateofthe effectfhineepthssuggestedySchnitzer1 andsasedonheheoryofBobylefP^ fornfinitemmersion. TheBobylefffunctionsshownnigure2.

ThexpressionsorngivenyEquations(52)and54 )willesednEquations(40)and(44)todetermineheormalorceQndmomentM Q. However,itsfirstecessaryodefineherangeofkeelengthverwhichachcontributionopsvalid. Itsclearthatno singlelocation,suchssuggestedyEquations(52)and(54),existsatwhichheeffectof chinemmersiontarts,sincehelowsmuchmorecomplexhanheseequationssuggest.Thissespeciallyruenheicinityofchineimmersion. However,aracticalsolutionothisproblemhasbeenmadeossibleymakingtheheoryorthesteadyar tfthenormal forceandmomentonsistentwithhelargeamountfexistingsteadytatedata.19 Wagner,H. ,"ThePhenomenaofImpactan dPlaningonWater."NationalAdvisoryCommitteeforAeronautic Translation1366,ZAMMBd2.Heft4.pp.93 -15(Aug1932).Lamb.H."Hydrodynamics,"SixthEdition.CambridgeUniversityPress.England(1932).

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Byssumingthathelaninghullormalorceatighpeedswa smadepoftheum oflow-aspect-ratiowingliftndarosslowdragerm,Shuford21rrivedtheasicor m ofanexpressionwhichewa sableoiterywellodataobtainedymanyinvestigators.

Thesedatacoveraangeoftrimanglesbetweenand30degrees,wettedlengthsfrom onetoseveneams,anddeadriseanglesbetv/een0and0degrees.Hisexpressionorthenormalorcesgiveny

rADS=>>S[^A)2 sinrcosT1-sin

where +CD sin^oszos

Cpc rosslowdragcoefficientS=lanformareaofthewettedportionofthehullorX b2A-aspectratioofS,i.e.

A=b2/S=/X

(55)

(56)Th ealuesofCD andhedependencyondeadriseanglewereobtainedyaitohedata.Th erossflowdragcoefficientswereoundohavethealuesshowninTable.

TABLE2-CROSSFLOWDRAGCOEFFICIENT 21vD. c1.33SectionhapeV-bottom,onstanteadriseV-bottom,horizontalhinelare.330.01470 V-bottom.verticalhinestrips1.60+0.0147Wewillowdefineheariouscontributionsu aohesectionalddedmassalongthehullyheollowingequations,whichreomewhatlessrestrictiveha nEquations(52)nd (54).

Bln*a2fU 2an2r

BbtanTs-sc2)

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wherewehavemadethesubstitutionf=stan 60)Thesectionalddedmassn tanyectionssimplyhes umofth econtributionsathatsection.

Therincipledifferencero mEquations(52)and(54)ishatsclndsc2av enotee nassumedobenownnadvance. Equations(57)and(58)areheontributionsfromhebottomofthehullohehine.Thequantityclsthealueofsathepointwherehechineseffectivelymmersed. Equation(59)isthecontributioncorrespondingtoheseconjtermnEquation(54),andc2shealueof atwhichhisbeginstogrow.

FromEquations(5 7hrough9)wereadilyindhat(61)M ,yf(0)sclan)2+Bbanr(fifcsc2)

r^k air / 2 \ o ^ksc2'J Mds=i~(f(0)sclanr)2 -yc]j+-|Bbtanr62)oSubstitutingEquation61 )ntoEquation(41)gives FDS~-^()n(({)sclan)2inosr+ (X)Bbsin2T(Kk-2)] 63)

ThisequationecomesidenticalwithheormulationofShufordnEquation55)rovided* ,x ' =rn vh* >

y(f()sc,anr)2 -1s in0)65)*p{\)B=CDcos2cosj3 66)

B ksc2 b 67)Thehree-dimensionalorrectionactorofEquation64)ha sarendimilartoheesultobtainedyabstsshownyigure1. AlsohevariationofBwithdeadriseanglesseentoeproportionaloosinigure2.

IfweubstituteEquations(64)hrough(67)ntoEquation(63)andondimensionalize, weobtain

FDS'r-r = s inTosr(1s in ) +CD Xs in2cos2cos68)

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LikewisefweubstituteEquations(61)hrough67)intoEquation45)anddividey1/2pU2b3weobtainheondimensionaldynamicmomentaboutheenterofgravity

MDS'=fTxsnrcosT(1"sn)(X k3xc i-\)+CDc in2os 2os(--X g) 69)

whered cl/b\l=c2/bX g -g/bX k =Ck/b

DETERMINATIONOFXcl,Xc2,kBecauseofwaveis eonmpact,heeffectivedepthoftheV-bottomsgreaterthanhe

depthelativeothealmwaterfreesurface.Wagner19oundhateforehineimmersion, theffectivedepthwasgreaterbyaactorofir/2.Thereforeheeffectiveradiuscofthe semicircularcylinderofwaterrepresentingM(S)stakenas

(70)irV tanta nCombining with Equation(57)gives

M(s) it '[(-ifftan22 fors

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Anexpressionforthemeanwettedlength-to-beamratioXhasbeenobtainedbyBrown1fromextensivephotographicobservationsas

X=0.5X kX c)+0.03 74)whereX c,theratioofthewettedlengthofchinetobeam,is

(75)c=X k-(0.57+0.001) (tan0/(2tanr)-0.0060)providedX c>.ThelastterminEquation(74)isanallowancefo rstagnationlinecurvature.Wethenfindfromthepreviousequationsthat

X c20.5(0.57+0.001)(tan0/(2tanT )-0.0060)-0.0376) sinceaccordingtoEquation(67) Xk X+Xc2 77)

Equations(73),(76),an d(77)completelydefinetherangesofthesectionaladdedmass distributionsintermsof0 ,X ,andr.othXandTreobtainedfromthesteadystateequili-briumconditionstobediscussedinalatersection.

ItisnotedthatthevalueofXc2sdefinedbyEquations(59)and(76)issmallerthanX j,X cwhichorrespondstothelowerlimitofintegrationatwhichf=fcntherepresenta-tionofthesecondtermoftheaddedmassinEquation(54). Althoughitappearsmorereasonabletous e -X corthelowerlimitofintegration,thebestfittothedata10equiresthatweseheX 0fEquation(76). Inthenumericalanalysis,theeffectofusingX vX.i nlaceofXwasfoundtobeinsignificantexceptforthe0-degree-deadrisecase.CalculationsofrsingX k-X creshownbyhebrokencurvesinFigure6whereitisseentoproduceasmallmprovementinthefittothedata.ADDEDMASSFUNCTIONS

Theaddedmassfunctionsusednevaluatingthestabilityerivativesma ynowbe expressedintermsofthehullgeometryandtheintegrationlimitsX cl,X c2,and\.

Thenondimensionalsectionaladdedmassatthestern,includingthethree-dimensionaleffectsisreadilybtainedfromEquations(61)and(64)through(67)as *(X)M$'2(PQ) 78)

where

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Q = r X sinrcosrco sTh enondimensionaladdedmassi nheaveis,romEquation(62)

where

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mayeestimatedbymeansofexistingtechniques,22,23ndwillnotbeconsideredfurtherhere.However,theeffectofatowingforcean dmomentwillbeincludedinordertofacilitatecomparisonwithowedmodeldata.Th eontributionsdueoperturbationsinpitchan dheavedisplacementmayeadilye obtainedfromheompleteexpressionsforthesteadytateorceandmoment.hedditionaltermsrequiredoompletethesexpressionsareescribedasfollows. BUOYANCYFORCEANDMOMENT

Wewillconsideronlyhecasesfo rspeedcoefficientCygreaterthan0.5,wherehewater breaksclearoftheransom,thusfullyventilatingthebacksideoftheboattotheatmosphere.Forthiscasethehydrostaticorcema yes sumedtoac tnormaltothekeel.Theollowing expressionwa sfoundtoitthedatareasonablywell.10'24

FgS'KX 2inrjQyj (82)whereKsanempiricalcorrectionfactorwhichaccountsforventilationeffectsonthestaticpressure.Ontheasisofpreliminaryanalysisofplaningboattestdata,avalueofKof0.7 wa stentativelyuggestedbyHsu. However,ro mecentextensiveexperimentswith 0-degreeeadriserismaticplaninghull,Brown'0obtainedhebestagreementwiththedataby putting

K=0.624 83)andassumingthattactsaton ehirdofthemeanwettedlengthfromhestern.incehe presentnalysisdealswithrisr.iatichullsEquation(83)willbeusedinheollowing.he momentbouthecenterofgravityisclearly

MB S FWih\) (84)SKINRICTION

Theontributiondu etoskinfrictionsas sumedoacttangentialohebottomandmid-wa yetweenheee indchinelines. Itsgivennermsofthemeanwettedareaby Savitsky5

2324

"Uadler,J.B. ,"ThePredictionofPowerPerformanceon PlaningCraft ."Trantact ioniSocietyof NavalArchitectan dMarineIng inee ii.Vol.74.pp .563 6 0 (1966).Rtbner.U.S. ,"PropellerinYaw."NationalAdvitoryCommittee fo rAeronaut i c iReport820(194V).Htu.C .C . ,"Onth eMotionscfHighSpeedPlaningCraft ,"HydronautktRepor tMl VIMt*1967) .

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FFS '=X Cf/cos0 85)whereth efrictionoeffic ientCfisgivenby

0.2427q"lo g10hmasaunctionofReynoldsnumber . Themomentaboutth ecenterofgravitysclearlyMR-F FS(\-S) 86)

TOWINGANDAERODYNAMICFORCESTermssimilartoth eaboveEquation(86)maybewrittenorth eowingan daerodynamic

forcean dmomentontributions. Theseareassumedhereoac tnain eparalleloheteady partofth etraightlinemotion.* Themoment saboutheenterofgravity,du etoth eow forceTg 'an dwindorceF^',respective ly,are

MT S *-Tjf0c,,e,) 87)

where

steadytateorcean dmomentequations. Theforceequationsresolvedn toavertical(lift)componentan dahorizontal(drag)component . Theif tequationisreadilyobtainedbyetting th eboatweightW equalohesumofth everticalomponentofth evariousorcecontribu-tionsde f inedearlier. Thusinnondimensionalormwehave

W--Zj(Fjjg'4 -FK)cosr-FFSsinr89)ITmii foodMunipiionforthemodelexperimentducuiardkter.

2 7

MWSFWS(k2-e2} 88)f(kn,en)=(X^X v)cosT+(XCTX g)sinr\\ \\=nondimensionaloordinatesofth eowpointwithespectto

keelathestern;se eFigure23Nc2* e2 oordinatesofth eesultantwindforceNperpendiculardistanceromhekeeloth eenterofgravity

STEADYSTATEEQUILIBRIUMThesteadytaterimangleran dmeanwettedlengthXbarereadilydeterminedfromhe

rfrni

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whereZ 'sth enondimensionalhydrodynamiclift,positivedown .imilarlyth enond imen-sionalowforceFTS 'sse tequalohedragorhorizontalcomponen tofth eorcecontribu-tions. V=-Xs=(FDS+FM)sin+FFS 'osrFws'

=-Zsan+FFS'/cosr+Fws'=Ds90)whereDs isth enondimens ionaldragofth eboat. Themomentequationsobtainedby summingth eomponents

Ms=MDS'+MBS'+MFS'+MTS'+Mws 091)whereheomponentmomentsaregivenwithespecttoth eenterofgravity.

Ifwesubsti tuteEquations(68),(82),an d(85)intoEquation(89)weobtaininallyor th esteadytateif tequation W'=-ZP'=5rTsincos2(1-sin)+Cn.X sin2cos3os1+A 2 ,t

0.624 2^+X^inTcosT-XCfsinT/COSp (92)Ifwemultiplyhisequationby+XweobtainheollowingformofEquation(92)asacubicequationinX .

whereDX3+(C+D+E)X2+(B+C-E-W)X-W 0

B-^ s inTcos21-sin)CC JJ in2cos 3cos

(93)

D 0.624sinrcosr/CvE=-CfsinT/COS0 94 )

Subst itut ingEquations(69),(77),(84),(86),(87),(88),an d(90)intoEquation(91)yields

, ir sin2r(l-s in XMS=4X0.624

(z s

1*M(X+h(r)-X g) CDc(sin2r)2co tj i-X,) -XCrJf(k,,e,)+Fwsf(k2,e2)=0an -FWo'cosTcosp wa (95)

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whereh(r)=X c2--cl (Q6)

Equations(93)and(95)mustbesolvedorX andTbyaniterativerocedureinwhichsuccessivevaluesofrreassumed. Equation(93)issolvedorXoreachassumedvalueof T. SuccessivepairsofXandThu sdeterminedaresubstitutedintoEquation(95)untilts magnitudebecomesequalozerowithinsomeprespecifiedamount. Inheresentnalysis thiswastakenas0.002/Cv2.

STATICSTABIL ITYDER IVATIVESAsnotedeafertheorceandmomentstabilityerivativeswithretnectoheheaveand

pitchvelocityandaccelerationerturbationsarereadilyobtainedromEquations(47)and(50).Thereonlyemainstheas kofobtainingthederivativeswithrespectoheheaveandpitch angledisplacementerturbations.ThesearereadilyobtainedwithheaidoftheexpressionsforZsndMg'inEquations(92)an d(95). Since,oragivendeadriseandspeed,heyare functionsonlyfX(z\T)ndwehaveforthederivativeswithespectotheheaveandpitcherturbations

a zd z

a x 'a _5 73 Z30

*3 03M30

3X3Xs3X3Ms3X

T

3XsdrM

3 z

3Xa? 3X?

hbJ* 3 X3 XT3 Xs3 X3 M

3XT

S3X

( 9 7 )

( 9 8 )

( 9 9 )

( 1 0 0 )

( 1 0 1 )

( 1 0 1 )dr 3X 3rwhereallotationsarewithrespectohecenterofgravity;z '= z/bandz equalsperturbationinheavedisplacement,ositivedown. FromEquations (90),(92),and(95)wer:adilyind

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3X

3T

where

3X 3X 3T

3X

. /cosr\2-Gs in -- -Hs iniCOSJ-2JinosT+Cfs inT/COS-G-co s3 s in2COST) -HX(2s inTos4-3os 2ts in3)- X-osTC j -XosT/COS3ZS'FFS' 3ZS':-rr-an -rr-rr-an3XX cosr 3\

Zo ' Fccan 3ZS's ta n3TOC-3Z S'S ^ FS_tanT+y-+ 77o"os T Gs inosT K m ) 2 ( X +h(r)-X g)+h]+Hin2cos2X-X g)+JinX(X-2X g)

3Ms* 3r

1

ta n cos2r(X+h(r)-X g)(0.1570.00025)

&&-0 (-- afe)'*-T*l['+HXXg)sin2rcos2r+JX 2Xgjc/3ZS's CfXtnA+ ( , anT+ snr^/fk, ,e i- stanr-Tf-Fws) -

g(kn.f)=X kn-X v)sinr-(Xen-g)COST G=ir(lsin0)/2H-CDccos?J 624/Cv2

30

(103)

(104)(105)

(106)

(107)

e,)-Fws'g(k2.e)108)(109)

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AlsowithheaidofEquations(76)an d( 7 7 )an dFigure20itseasyoshowhat 3z ' sir.T g=_(X 1 +X v)+( 0 . 5 7 0 . 0 0 1f )JSLLr \anr v/sin2f (110)(111)

STABIL ITYEQUATIONSTheotalnondimens ionalforcean dmomentequationsmaynowbewrittenas

(D 3z ' 30' u '= T s '+X s'+A X D ' +0+|f+A T sm'z'=W'+ZsAZD '+ p-z * 80

whereV* . ' *+ f

m'-=nondimens ionalmassofth eboatpb3

Iy1 ' =nondimens ionalpitchmomentofinertiaabouthe 2penterofgravity ATS'- ATcIpU^b2 =perturbationnoworce

FromhetaticequilibriumEquations(89)hrough(91)weequirehat V+*S=0w+V=0Ms =0

(112)

(113)

(114)

(115)Ifth ebodysfreeosurgehenATS'szero. Ifih ebodysrestrainednurgeu'an d'areero,an dhepitchan dheavemotionsareuncoupledromsurge.

Assumingthathebodysre etosurge,hestabilityquationsatheequilibriumlying conditionsareobtainedSyputt ingATS qualozeroan dsubstitutingEquations(47),(49),(50),an d(115)intoEquations(112)through(114). Thisleadstoth eollowingequat ions

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wheretheprimeshavebeenomittedorconvenience.(X y-m) +\u+X ^z+X zz+X zz+XQ6XQ6X00Z ^ +Zuu+(Z~m)z+Zzz+Zzz+ZQ Z 6 Zd=0M+MuuMz+Mzz+Mzz+(M g-Iy)0MgM90

whereheorcederivativesareZ=- f i C K )s inTcosrJn ds'Zu=-*>(X)M$'inrcos2Z-z-- < p ( X )cos2/ ds ' Zi-2

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Thentegrationsarerom0toX k.heevaluationoftheaddedmasstermsareeadilyobtainedwiththeaidofEquations(78)through(81).hestaticderivativesaregivenbyEquations(97)through(108).TheexpressionsforFDS andD< j aregivenyEquations(68)and(69).

ItsseenromEquation(119)thatorsmallrimanglesthederivativesoftheX-equation areconsiderablysmallerthanthoseoftheZ-equation.Undertheseconditionsitisreasonable toexpectha thenfluenceofthesurgedegreeoffreedomonheorpoisingstabilitywouldbesmallenoughothatheX-equationcouldbeomitted. Inanyase,iftheboatisbeingtowedtonstantspeed,then,aspreviouslynoted,valuesofuandinEquations(117)and(118)arezero.Thiseffectivelyncouplesthesurgedegreeoffreedom,andhestabilitymaybedeterminedromhepitchandheaveequationsalone. Inheresentinvestigation,allof thedataavailableforcheckingtheheorywerebtainedbyusingplaningboatmodelstowedatonstantspeed. Forthesereasonsnumericalalculationshavebeenmadeusingonlythe pitchandheavestabilityquations.

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APPEND IXBESTIMATES OFEFFECTOFWINDAGEANDCHINEADIUSONSTABIL ITYDER IVATIVES

AhighmetalrameworkwasattachedohemodelsofDayan dHaag.4 Itspurposewas tosupportscribeorrecordingth emotionafterth eboatstartedoporpoise . Thisintroduced asmallaerodynamicdragan dmomentonhemodel . Theollowingestimatedvaluesfo rthisdragwereusednheomputat ions

Fws=O.O32X e2=.3\k2=2.6

Shuford21oundhatasmalladiusonhechinewillaccountoraeductionnliftof5o0percent-correspondingtoa1/64-an d1/16-inchadius,respectively ,on4-inch-beam,f lat-bottom,planingsurface. Onhebasisofchineradiusmeasurementsobtainedromatypicalwoodmodel ,twasestimatedhathemodelsusedbyD ayan dHaaghadaboutapercentos sindynamicift. Theheoreticalalculationswerehereforemodifiedoak ehis intoaccount. Thisamountedoreducingth emagnitudeofPan d0inEquations(78)through (81)an dGan dHnEquations(103)hrough108)bypercent .

35 HtSCEDINPi

**""' ' * IHI I

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M,0,0.0

Z,Z,2,2Figure-CoordinateSystem

36

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0.31

0.2

0. 1a .>

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82n* O

VUJUio r >O 5 2uJC M U M" o t> o osU J

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14

12

10c/jLLUe r oLU QLU-IUZ

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THEORYS E E.4)OAD MEASUREMENTS

0=0.5

REGIMEOFPORPO I S ING

REG IMEOFSTABLE PLANING13

S P E E DCOEFFICIENT,CvFigure6-Compar i sonofTheoret ica la ndMeasuredPorpois ingBoundar iesfo readriseo f20.5egrees

4

im

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16

14-

12-C O U iU JU J OUI1 8 0.48

CA=0.36rv

/kA

yDREG IMEIFSTABLEPLANING

- So A 0AEASUREMENTS0 A

4-

0. 1 0.2 0.3 0 .4LOADSPEEDACTOR,VcT/2Figure-VariationofCriticalTrimAnglewithLoadSpeedFactorfo rVariousLoadings.DeadriseAngleof0Degrees

42

*M*

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16

14

12C OUlL U cUlD 10KUJ-ICz

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16

14

12V ) UJ UJ o coUJQut>

U J-IZ

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1 .5

O 1 .0

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2.0E

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Figure2-VariationofCriticalTrimAnglewithLoadSpeed Factoran dRadiusofGyration-BeamRatio

47

12

1 0

Vu iU Jtr 8(3U IQUJ_JZ 6

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0. 1 0.2.3LOADS P E EDACTOR,^CLb/2 0. 4Figure3-VariationofCriticalTrimAnglewithLoadSpeedFactoran dLoadCoef f i c i ent

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1.5

0. 51. 5

2.

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1 .5

2 1 .0

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2.4

2.0

1 .6

: 1 .2

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Figure7-ComparisonofTheoreticalValuesofCriticalTrim AngleswithDavidsonLaboratoryMeasurements12.0

10.0-

o mQ

L U-Ioz

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12.0,v/p0.286

10.0

COUJL UD C C 3 LUQ

ut.LL I-Jo z

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0.40

0.30

0.20

It* V 0.100 .08h0 .060 .050 .040 .03-0 .02-

I iCONDITION

I PROJECTEDCHINEENGTHCA MODEL MEANCHINEBEAM

1 2 0A 0.38 4665 )> 2.360.24 4665 3 f c > 0.35 4666

0 v 4 D 0.50 4666 3.69_ _ 5 o 0.80 4666 /6 V 0.86 466/1 5.007 t > 1.33 4668 6.72N \

HULLS ,=10.6.3 6 UNSTABLE\A .0.48

2.5\-0.72 \ mm 1 A\ - STABLE\ > [ C O

I I I1567890VOLUMETRICFROUDEUMBER ,U>/gV1/FigureH-Compar i sonofApprox imateEst imatesofPorpois ingBoundarieso fReference8withTMBSeries62ModelData

54

mtrnti

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r

o00

o3 fto~O -Iu

_o

I _o00 otoI

oin ooe n?v oto oin o c s i o

* '310NVwiaiivouiao

s .za50 S QZcoS :/ ;^=0Sao IS 'S0 1I ~

8 >rtOC *S.2'7" -I* a*e MeeS IIa1U g

a cM

S5

JU

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30e '( 7 5>.

c /5 ^ =i .'J E

8g3S OII

56mmwmwiftw

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{\

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i jJ < > z /U J /o c /tu /"- /U J /

//c a . ;/C O //O //

/7/ -////i ff

oCO

00d

o

s8

-3

-8oCM

01 SOc

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FwsWINDORCETsOWINGORCE

Figure ^-CoordinatesofTowPointandCenterofWindForce

59

--

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REFERENCES1 .erring,W.G.A.an dH.dauert,"StabilityntheWaterofaSeaplaneinthePlaningCondition,"AeronauticalRssearchCouncil,TRVol.42(Sep933).2.utowski,R.N.,"AComputerProgramorVariousPerformanceAspectsofPlaning Craft,"ThesissubmittedtoStevensInstituteofTechnology,CastlePoint,Hoboken,N.J.1973).3.ayne,.R.,CoupleditchndHeavePorpoisingInstabilityinHydrodynamicPlaning,"JournalofHydronautics,Vol.,No .(Apr974).4.ay ,.P .ndR.J.Haag,PlaningBoatPorpoising"ThesisSubmittedoWebbInstituteofNavalArchitecture,GlenCove,LongIsland,N.Y.May952).5.avitsky,D.,"HydrodynamicDesignofPlaningHulls,"MarineTechnology,(Oct964).6.artin,M.,"TheoreticalDeterminationofMotionofHigh-SpeedPlaningCraftin

Waves."DTNSRDCReport6-0069(Apr976).7.r idsma,G.,ASystematicStudyofRough-WaterPerformanceofPlaningBoats,"DavidsonLaboratory,StevensnstituteofTechnology,Hoboken,N.J.,Report27 5Nov969).8.lement,E.P.ndD.Blount,ResistanceTestsofSystematicSeriesofPlaningHullForms."TransactionsSocietyofNavalArchitectsandMarineEngineers,Vol.1,p.491-561 (1963).9.avidson,K.S.M.ndA.Suarez,"TestsofTwentyRelatedModelsofV-Bottorr.MotorBoats-EM BSeries50,"DavidTaylorModelBasinReportRA1(1949).

10.rown,P.W.,"AnExperimentalan dTheoreticalStudyofPlaningSurfaceswithTrim Flaps."DavidsonLaboratory.StevensnstituteofTechnology,Hoboken,N.J.,Report463 (Apr971).

11.gnelli,J.C.,EvaluationoftheTrimofaPlaningBoattnceptionofPorpoising," presentedtSpringMeetingofSocietyofNavalArchitectsandMarineEngineers,LnkeBuenaVista.ia .Apr973).12.unk,M.,"TheAerodynamicForcesonAirshipHulls"NationalAdvisoryCommitteeforAeronauticsReport84(1924).13.ones,R.T.,"PropertiesofLow-Aspect-RatioWingsatSpeedsBelowndAbovethe SpeedofSound,"NationalAdvisoryCommitteeorAeronauticsReport351946).14.ryson,A.E.,Jr.,"StabilityDerivativesforaSlenderMissilewithApplicationtoa Wing-BodyVerticalTailConfiguration,"JournalofAeronauticalSciences,Vol.0,No .,pp .297-308(1953).15.ayo,W.L..Analysisan dModificationofTheoryorImpactofSeaplanesonWater,"NationalAdvisoryCommitteeorAeronauticsReport10(1945).

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16.ilwitzky,B.,"AGeneralizedTheoret icalan dExperimentalnvestigationofth eMotionsan dHydrodynamicLoadsExper iencedbyV-Bot tomSeaplanesDuringStep-LandingImpacts,"NationalAdvisoryCommit teeorAeronauticsTN1516(1948).17 .chnitzer,E .,"Theoryan dProcedureforDetermin ingLoadsan dMotionsinChine -ImmersedHydrodynamicImpactsofPrismaticBodies ,"NationalAdvisoryCommit teeforAeronauticsReport15 2(1953).18.abst,W.,"LandingImpactofSeaplanes,"NationalAdvisoryCommit teeorAeronauticsTM624(1931).19.agner,H. ,"ThePhenomenaofImpactan dPlaningonWater,"NationalAdvisory Commit teeorAeronauticsTranslation1366,ZAMMBd12,Heft4,pp.93-215Aug1932).20.amb,H. ,"Hydrodynamics,"SixthEdit ion ,CambridgeUniversityPress,England

(1932).21 .huford ,C.L. ,Jr.,"ATheoret icalan dExperimentalStudyofPlaningSurfacesIncludingEffect sofCrossSectionan dPlanForm,"NationalAdvisoryCommit teeor AeionauticsReport13551957).22.adler,J.B.,"ThePredictiono i* PowerPerformanceonPlaningCraft,"TransactionsSocietyofNavalArchitectsan dMarineEngineers,Vol .4,pp .563-610(1966).23 .ibner,H.S.,"PropellersinYaw,"NationalAdvisoryCommitteeorAeronautics Report820(1949).24.su ,C.C . ,"OnheMotionsofHighSpeedPlaningCraft,"HydronauticsReport603-1May967).

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