Date post: | 16-Apr-2017 |
Category: |
Documents |
Upload: | joshua-harris |
View: | 41 times |
Download: | 0 times |
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
2
spectraldensityplot,whichisofimportancehere,thetransferfunctionmustbefoundandmultipliedbytheinputspectraldensityplot. 2.1PhysicalRepresentationoftheModelandNomenclature
Figure1,showsthephysicalrepresentationofthemodel.ThenomenclatureisfoundinTable1.Someofthesevaluesareborrowedfrom[5],whileotherswerecalculated,inordertocreateamorerealisticmodel
Figure1–SchematicDiagramofTether
3
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.
4
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)
5
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.
6
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.
7
Figure2-GeneralGraphofEquation10
Theareaunderneaththepowerspectrumisequivalenttothemean-valuesquared,whichalsocanbeexpressedasthevariancesubtractedbythemean.Thus,itcanbeusedtocalculatethevarianceifthemeanisknownorvice-versa.Statistically,thismeansthatthespectraldensityplotisadistributionofthevarianceaccordingtothefrequency.
X(t)issimplytherandomvariableasafunctionoftime.Inthissystem,itisthepositionofthetetherasafunctionoftime.ByusingthenomenclaturefromTable1,aspecificcoefficient,asafunctionofω,inequation10wasfound.Specificfrequencieswerepluggedin,evaluated,andgraphed.Figure3showsthevariousgraphsatvariousfrequencies.Theareaunderneatheachspikebecomesmorevisibleasthefrequenciesbecomehigher.
8
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.
9
(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.
10
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)
11
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
TendonofaTensionLegPlatform."JournalofSoundandVibration293(2005):38-58.Elsevier.Web.26Aug.2015.
2.Benaroya,Haym,andRonAdrezin."ResponseofaTensionLegPlatformto
StochasticWaveForces."ProbabilisticEngineeringMechanics14(1999):3-17.Elsevier.Web.26Aug.2015.
12
3.Benaroya,Haym,andS.M.Han."Non-LinearCoupledTransverseandAxial
VibrationofaCompliantStructurePart1:FormulationandFreeVibration."JournalofSoundandVibration237.5(200):837-73.IdealLibrary.Web.26Aug.2015.
4.Han,SeonMi,andHaymBenaroya."ComparisonofLinearandNon-linear
ResponsesofaCompliantTowertoRandomWaveForces."ChaosSolitonsandFractals14(2001):269-91.Elsevier.Web.26Aug.2015.
5.Mathisen,Jan,OddrunSteinkjer,IngeLotsburg,andOisteinHagen.Guidelinefor
OffshoreStructuralReliabilityAnalysis-ExamplesforTensionLegPlatforms.Tech.no.95-3198.Ed.VigliekHansen.N.p.:n.p.,n.d.JointIndustryProject.DetNorskeVeritas,27Sept.1996.Web.26Aug.2015.
6.Benaroya,Haym,andMarkL.Nagurka.MechanicalVibration:Analysis,
Uncertainties,andControl.3rded.BocaRaton,FL:CRC/Taylor&Francis,2010.Print.