VibrationandStochasticWaveResponseofaTensionLeg

Platform

WrittenBy:JoshuaHarris,August26,2015

Abstract

Theequationofmotionforasingletendonofatensionlegplatformispresented.Theequationisuncoupled,linearandnon-homogenous.Theforceonthetensionlegplatformismodeledasarandomharmonicload,whichisinterpretedtobethewaveshittingthetensionlegplatformatrandom.Themodelisacompliantstructurethatallowsforsmalldeformationsanddisplacements.Thetensionlegplatformismodeledasarodconnectedtoatorsionalspringwithamassattheend.Thereisviscousdampingthataccountsforthedragthatiscausedbytheseawater.Thefirstmethodofanalysisinvolvesfindingtheinputspectraldensityandmultiplyingitbythetransferfunctiontoreceivetheoutputspectraldensityplot.Thesecondmethodinvolvessolvingtheequationofmotiontofindthemotionofthetensionlegplatformasafunctionoftime.1.Introduction

Atensionlegplatform(TLP)isaverticallymooredfloatingstructurethatisusuallyusedfortheoffshoreproductionofoilandgas,howevertheideahasbeenconsideredforwindturbines.Therearebestsuitedforuseinwaterthatis1000feetindepth.TheusualmethodforinstallingTLPsstartwithfoundationpilesbeingloweredintotheseabedandhammeredintothebottomoftheoceanfloor.Theplatformitselfismooredbytethersortendons,whichareconnectedtothefoundationpiles.Agroupoftethersisknownasatensionleg.Atendonsupportbuoywillnextbeinsertedontopofthetopmosttendons.ThegiantTLPhullisthenboughtinbyboatsandattachedtothetendons.

TherearetypicallytwotypesofTLPstructures:fixedandcompliant.Fixed

structuresarerigidanddonotallowforanymotioninthetethers.Compliantstructures,whicharemorecommonlyused,allowforsmalldeformationsanddisplacementstooccur,thusmakingiteasiertodesignandaccountforthewavesthatarehittingtheTLP.Thedeeperthewaterdepth,thebetteritistousecompliantstructures.

2.FormulationofSpectralDensityPlots ThemostdirectprocessinacquiringthespectraldensityplotofasystemistofindthecorrespondingautocorrelationfunctionandthentaketheFouriertransformofit.Thiswouldgivetheinputspectraldensityplot.Togettheoutput

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spectraldensityplot,whichisofimportancehere,thetransferfunctionmustbefoundandmultipliedbytheinputspectraldensityplot. 2.1PhysicalRepresentationoftheModelandNomenclature

Figure1,showsthephysicalrepresentationofthemodel.ThenomenclatureisfoundinTable1.Someofthesevaluesareborrowedfrom[5],whileotherswerecalculated,inordertocreateamorerealisticmodel

Figure1–SchematicDiagramofTether

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Table1-NomenclatureSymbol Meaning ValueL Length 260mϕ RandomVariable [0,∞)π Pi 3.1415τ Timeconstant Secondst Time [0,∞)secondsk Torsionalstiffness 100kN•mc ViscousDamping 3.036x1010kg•m/sω Frequency [rangeofvalues]rad/sωn NaturalFrequency [rangeofvalues]rad/sωo InitialFrequency(when

n=0)[rangeofvalues]rad/s

m Mass 169,353kgA Amplitude 2000Nθ Angle(ofmotion) RadiansN UpperLimit ∞Boldfacedvaluesareborrowedfrom[5]2.2IdentificationoftheInputForces

Itisnecessarytoderivetheinputforceonthesystem.Theautocorrelationfunctionofthisforcecanthanbefound.Theoceanwavescanbemodeledasastochastic,harmonicforcethatactsonthesystem.Thedifferentnaturalfrequenciesneedtobeaddedtogetherbecausetheycontributetotheoverallforce.Arandomvariablegeneratorcanbeusedtocreatetherandomvariablethatiswithinthesamerangeasthenaturalfrequency.Itdoesnotmatteriftherandomvariableisaddedorsubtracted.

(1)

Althoughthecosinefunctionisusedhere,thesinefunctioncanbeusedaswell.Theupperlimitcanbeadjustedbasedonhowmanyfrequencyvaluesneedtobeevaluated.Itshouldbenotedthatthenaturalfrequencyisdefinedas:

(2)

WhereTisafixedtimeperiod,butniffromtherangeof0,1,2,..N.

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2.3DerivationofInputandOutputSpectralDensitiesAmajorassumptionthatismadeisthattheentireprocessisstationary.Inaphysicalsense,itistobeassumedthatthefunctionandtheprocesswillentersteadystateafteralongperiodoftime.Therearenosuddenimpulseforcesactingonthesystem.Statisticallythismeansthatthejointprobabilitydensityfunctionatadistinctsetoftimeswillequalthejointprobabilitydensityfunctionandanentirelydifferentsetoftimes.Thus,thisslightlychangesthedefinitionoftheautocorrelationfunction:

(3)

ThisistheautocorrelationfunctionofastationaryrandomprocesswhereF(t)istheinputforceasafunctionoftime.Whentheforceispluggedintoequation3,

(4)

Thetwoexpectationsarefirstfoundseparatelyandthenmultipliedtogether.Afterfurtherevaluationandsimplificationandfactoredoutcoefficients,thefinalresultisanautocorrelationfunctionthatisafunctionofthetimeconstant.

(5)

Thedetailedprocessoffindingtheseexpectationsandsimplifyingtermsisfoundintheappendix.TheinputspectraldensityisfoundbytakingtheFouriertransformoftheautocorrelationfunction.ByusingatableofFouriertransforms,theInputspectraldensityisequalto:

(6)

δrepresentstheDiracdeltafunctionevaluatedatthevaluewithintheparenthesis.Thenextstepistofindthetransferfunction,H(iω),andmultiplythetransferfunctionbytheinputspectraldensitytogettheoutputspectraldensity.

(7)

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Equation7isthefundamentalresultforlinear,stationarysystems.ThemostreliablewaytofindthetransferfunctionistotaketheFouriertransformoftheequationofmotion.Becausethereareimaginarynumbersinthetransferfunction,themagnitudeofitistaken,beforebeingmultipliedbytheinputspectraldensity,becausetheoutputspectraldensityplotmustberealandonlyafunctionofthefrequency.Thus,itisnecessarytofindtheequationofmotionforthesystem.TheequationofmotionisfoundbyapplyingNewton’ssecondlawtothesystemandthenequatingittotheforce(1)previouslydescribed.AsnoticedinFigure1,thereisamass,viscousdamper,andtorsionalstiffnessthatareassociatedwiththesystem.Moreover,thetether’smotionismodeledasanangularmotionthatrotatesinradians.Theequationofmotionisrepresentedbelow,withasingleover-dotrepresentingthefirstderivativeandtwoover-dotsrepresentingasecondderivative.

(8)

AgainF(t)isdenotedbyequation1.Thissameequationofmotionwillbeusedinsolvingthepositionasafunctionoftime.OnceF(t)ispluggedintotheequation,theFouriertransformcanbetakentogetthetransferfunction.However,inordertomakeitsimplertotaketheFouriertransform,asimplesubstitutiontoconvertequation8asafunctionofθtoafunctionofx.

Thiscanbeappliedtoanyrotationalsystem.Becausethetetherisfixedaboutatorsionalspring,itcanbeusedhereaswell.ItistobenotedthathererisequivalenttoL.TheFouriertransformoftherightsideoftheequationofmotiongivestheinputfunctionF(ω)andthetransformoftheleftsideoftheequationofmotiongivestheoutputfunctionX(iω).Dividingthelatterbytheformeristhetransferfunction.

TheFouriertransferistakenandthensubstitutedintotheformaldefinitionofthetransferfunction.

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Asaforementionedfromequation7,themagnitudeofH(iω)mustbetaken.

(9)

Alloftheunknownsinequation7arefound.Afterfactoringtheappropriateconstants,theoutputspectraldensityisobtained.Notethatωcanequalωn.

(10)

Thecomplete,detailedmathematicalsolutionforeachstepisfoundintheappendix.2.4PlotsofPowerSpectraBydefinitionoftheDiracdeltafunction,itiszeroeverywhereexceptwhenthetermevaluatedinsidetheparenthesisisequivalenttozero.Also,itsareamustequalone.Sointhiscase,theDiracdeltafunctionisonlyzerowheneitherωn=-ω0forthefirsttermofthesummationorwhenωn=ω0forthesecondtermofthesummation.AsFigure2shows,thefollowingresultwillbespikesattheaforementionedvaluesofω.Thespectraldensityplotcanbeusedtomeasuretheamountofenergyinastochasticprocess.Higheramplitudesingraphicalresultsindicatedthatthereismoreenergyatthoseparticularfrequencies.Therefore,thephysicalsignificanceofthesegraphicalresultsliesinthefactthattheenergyofthesystemishighlyconcentratedattwodifferentfrequencies.Thecoefficientinfrontofthesummationofequation10iswhatdeterminestheamountofareaunderthespike.Again,theareaundertheDiracdeltafunctionmustequalone.Thus,thecoefficientcanincreaseordecreasetheamountofareaunderneaththespike.

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Figure2-GeneralGraphofEquation10

Theareaunderneaththepowerspectrumisequivalenttothemean-valuesquared,whichalsocanbeexpressedasthevariancesubtractedbythemean.Thus,itcanbeusedtocalculatethevarianceifthemeanisknownorvice-versa.Statistically,thismeansthatthespectraldensityplotisadistributionofthevarianceaccordingtothefrequency.

X(t)issimplytherandomvariableasafunctionoftime.Inthissystem,itisthepositionofthetetherasafunctionoftime.ByusingthenomenclaturefromTable1,aspecificcoefficient,asafunctionofω,inequation10wasfound.Specificfrequencieswerepluggedin,evaluated,andgraphed.Figure3showsthevariousgraphsatvariousfrequencies.Theareaunderneatheachspikebecomesmorevisibleasthefrequenciesbecomehigher.

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a.Thefrequencyωequals0.Thevalueof

266.87ismultipliedbyδ(0).

b.Thefrequencyωequals10rad/s.Thevalueof2.9x10-11ismultipliedbyδ(0).

c.Frequencyofωequals100rad/s.Thevalueof2.9x10-13ismultipliedbyδ(0).

Figure3–SpectralDensityPlotsatDifferentFrequencies

3.FormulationofthePositionFunctionandItsMean-ValueSquared

Theprocessbehindfindingthepositionasafunctionoftimerequiressolvingtheequationofmotion.Thispositionfunctioncanbeusedasa“randomvariable”andbeusedtofindthemean-valueandthemean-valuesquared.Equation8showstheequationofmotionintermsofθ,butbyusingtherelationshipofbetweenθandx,theequationofmotioncanbeafunctionofx.However,thiswillbedoneaftertheequationissolvedintermsofθfirst.3.1SolvingtheDifferentialEquation Aspreviouslymentioned,equation8isasecond-order,linear,uncoupled,nonhomogeneousdifferentialequation.Thesolutionisthereforethehomogeneoussolutionaddedtothenonhomogeneoussolution.Tofindthehomogeneoussolutiontotheequation,thecharacteristicequationissolvedanditssolutionisusedasexponentsofe,multipliedbyt.

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(11)Thecoefficientsc1andc2,arefoundbytheinitialconditionsthatcanbeuniquelyformulatedbasedonthespecificconditionssurroundingthesystem.Itisnottobeconfusedwiththecintheexponent,whichistheviscousdampingcausedbythewater.Tofindthenonhomogeneoussolution,themethodofundeterminedcoefficientscanbeused,especiallysincetheforceisaharmonicfunctionofcosine.

(12)

Thenextstepistoaddequations11and12togethertogetthegeneralsolution.Itisstillafunctionofθ.

(13)

Thefinalstepistorelateθtox.Aspreviouslymentioned,risequaltoL.Therefore,Lisinthefinalsolutioninplaceofr.

(14)

Thecomplete,detailedmathematicalsolutionforeachstepisfoundintheappendix.

3.2FindingtheMean-ValueSquared Thereisarelativelysimplywaytofindthemean-valuedsquaredofacontinuousrandomvariablethatinvolvesonlycalculus.

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Inthiscase,n=2.However,aproblemarisesbecausethereisnofunctionf(x)anditcannotbederivedorfoundwithease.Therefore,theassumptionofergodicityhastobemade.Formally,astationaryrandomprocessisergodicifthetimeaverageofaneventatasingletimeperiodisequaltotheensembleaverage.Inotherwords,theaverageisconstant.Withergodicity,anaverageoveralongperiodoftimecanbetaken,insteadofnumerousaveragesatmanydifferenttimeperiods.Also,anergodicprocessisalwaysstationary.SinceTLPsaredesignedtolastoverlongperiodsoftime,theassumptionofergodicitycanbemade.Itissafetosaythattheaverageoveralongperiodoftimeisthesameasoveraverylongperiodoftime.Forexample,theaverageoverasix-monthperiodwillnotbetoodifferentfromanaverageoveratwo-yearperiod.Mathematically,asthetimeapproachesinfinity,theexponentialswillapproachzero;itistheexponentialpartthatwouldprovidethemostchangeanddiscrepancyinaverages.Withthisnewergodicassumption,thedefinitionofthemean-valuesquaredchanges,anditdoesnotrequireafunctionofx.

(15)

Thecoefficientsthatprecedethesineandcosinetermsarenowconstant,andthroughouttheintegrationprocess,canbefactoredoutinfrontoftheintegral.Foreaseofcalculation,thecoefficientsarerenamedasAandB.

Asseen,thecoefficientsofAandBarenotfunctionsofT,sotheyareconstantalways.Thesesamecoefficientsareusedandrepresentedinthefinalanswer.

(16)

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Asaforementioned,themean-valuedsquaredcanbecomparedtothespectraldensityplotandcanbeusedtofindandevaluatethevariance.Thecomplete,detailedmathematicalsolutionforeachstepisfoundintheappendix.4.SummaryandConclusions AmodelofatendoninaTensionLegPlatform(TLP)ismodeledasarotatingbeamaboutatorsionalspring.Ithasviscousdamping,torsionalstiffness,andarandomharmonicload.Theprocessisassumedtobeergodic.TakingtheFouriertransformoftheautocorrelationfunctionprovidedthefunctionofthepowerspectrum,specificallytheinputspectraldensity.AfterusingaFourierTransformfortheequationofmotiontogetthetransferfunctionofthesystem,theoutputspectraldensityisfound.Thesecondmethodofanalysisinvolvedsolvingtheequationofmotion.Theresultwasusedtofindthemean-valuesquaredofthesystem,whichisquiteusefulinfindingotherstatisticallyproperties.ThemanyresultsstemmingfromsuchanalysescanbeusedforthedesignprocessofTLPs.Thespectraldensityplotscanshowwherealloftheenergyisconcentrated.Whentesting,suchfrequenciescanbefocusedon.Ifthemotionofatetherneedstobelimitedorevenmademoremovable,valuesofviscousdampingandtorsionalspringconstantscanbeusedinthepositionfunction,toseewhichparameteraffectsthemotionofthetendonthemost.Differentfrequencies,lengths,andmassescanbeeasilysubstitutedtoseehowthesystemreactstochangesintheseparameters.Byusingthevarianceandmean-valuesquared,thechangesinthepositioncanbeevaluated,analyzed,andaccountedforduringthedesignprocess.Adesignercanrecognizeiftheaveragepositionwouldleadtofailureorinstability.Althoughtheforcesduetotheoceanwavesarestochastic,theycanstillbepredictedusingprobabilisticmodels.5.Appendix Theappendixprovidesallofthedetailedcalculationsdoneinthisworkintheiroriginalform.Itincludesthekeyassumptionsthatweremadeinordertoapplycertainformulasandcarryoutthemathematics.References1.Benaroya,Haym,andMangalaM.Gadagi."DynamicResponseofanAxiallyLoaded

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