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Renetal./JZhejiangUnivSciA200891263126DynamicresponseanalysisofamooredcraneshipwithaflexibleboomHuiliREN†,XuelinWANG†‡,YujinHU,ChenggangLISchoolofMechanicalScienceandEngineering,HuazhongUniversityofScienceandTechnology,Wuhan430074,China†Emailrenhustsina.comw_stevensina.comReceivedJune12,2007revisionacceptedAug.5,2007publishedonlineDec.14,2007AbstractThedynamicresponseofmooredcraneshipisstudied.GoverningequationsforthedynamicresponseofacraneshipcoupledwiththependulummotionofthepayloadarederivedbasedonLagrangesequations.Theboomismodeledbasedonfiniteelementmethod,whilethepayloadismodeledasaplanarpendulumofpointmass.Thedynamicresponsewasstudiedusingnumericalmethod.Thecalculationresultsshowthatthelargeamplituderesponsesoccuratwaveperiodsnearthenaturalperiodofthepayload.Loadswingangleissmallerforcraneshipwithflexibleboom,incomparisonwithrigidboom.Theshipsurgemotionshavelargevibrationsforcraneshipwithflexibleboom,whichwerenotobservedforarigidboom.Theanalysisidentifiesthesignificanceofkeyparametersandrevealshowthesystemdesigncanbeadjustedtoavoidcriticalconditions.KeywordsDynamicresponse,Mooredcraneship,Finiteelementmethod,Rigidflexiblecouplingdynamicmodeldoi10.1631/jzus.A071308DocumentcodeACLCnumberU615.35INTRODUCTIONFloatingcranesplayanimportantroleintheoffshoreprojects.However,theseawaveshakesthecraneshipandmayexciteitslargemovement.Forheavydutylifting,theoperationsofcraneshipinwaves,evenastheseaisrelativelyquiet,areoftenrestrictedbytheexcessivemotionsofthecraneload.Insomecases,thedynamicalbehavioroffloatingcranesistobeconsideredascriticalwithrespecttotheamplitudesofthemotionoftheshiportheload.Asmalldisturbanceofthesystemcancausethecollisionbetweentheloadandtheshiporotherobjects.Inaddition,theamplitudeofthemotionoftheshiphastostaysmallinordertoachievetherequiredpreciselypositioningandtoavoiddamagestothemooringsystem.TheinvestigationofcraneshipdynamicshasbeenofinterestinalargenumberofrecentresearchesDongandHan,1993Masoudetal.,2004Ellermann,2005.Arigidmasslesscableandmassivepointloadwereusedtomodelthecraneloadsystem,andtheresultsofthecomputersimulationwereverifiedexperimentallyusingathreedegreeoffreedomDOFshipmotionsimulationplatformHenryetal.,2001.Themethodofmultiplescalesisusedtoanalyzethedynamicsofacablesuspendedload.TheresultsshowthataparametricexcitationattwicethenaturalfrequencyleadstosuddenjumpsintheresponseasthecableisunreeledChinetal.,2001.CargopendulationcontrolofanelasticshipmountedcranewasconcernedusingtheMarylandRiggingsystem.Thedynamicsofthecraneisdescribedbyamultimodelproblemdependingonthecurrentcablelengthandboomluffangle.Avariablegainobserverandavariablegaincontrolleraredesigned.SimulationandexperimentalresultsshowedthattheexpressedcontrolstrategyhasasignificanteffectonsuppressingthevibrationsfordifferentoperatingconditionsandpayloadmassesAlSweitiandSöffker,2007.AsthefirstapproachforevaluatingthedynamJournalofZhejiangUniversitySCIENCEAISSN1673565XPrintISSN18621775Onlinewww.zju.edu.cn/jzuswww.springerlink.comEmailjzuszju.edu.cn‡CorrespondingauthorProjectsupportedbytheNationalNaturalScienceFoundationofChinaNo.50675077andtheResearchFundfortheDoctoralProgramofHigherEducationofChinaNo.20050487047Renetal./JZhejiangUnivSciA200891263127icsofacraneship,alineartheoreticalmodelisused.Thelineardifferentialequationofmotionisderivedundertheassumptionofsmallamplitudes,anddoesnotconsideroccurringrestoringforces.AdynamicmodelwasestablishedusingmultibodydynamicsmethodsandthedynamicsofthecranewasanalyzedChenetal.,2002.Theeffectsofcablereelingandunreelingoncargopendulationswerestudiedwiththeboomcranemodeledasaplanarpendulumandtheshipasarigidbody.TheresultsshownonlinearbehaviorKraletal.,1996.Inalmostallofthesestudies,theflexibilityoftheboomisnottakenintoconsideration.Thismaybereasonableforshortcraneboomsandsmallpayloadtoshipmassratios,butforlongcraneboomsandlargepayloadtoshipmassratios,theinfluenceoftheflexibilityofthecraneboomcannotbeignoredanymore.Totheauthorsbestknowledge,untilrecentlythereisnoreportonthemodelingofsuchsystem.Thestudyaimstopresentarigidflexiblecouplingdynamicmodelforthepredictionofthedynamicsofamooredcraneshipanditspayload.Thepredictionisthefoundationofthedynamicaldesignandanaccurateresidualworkinglifeassessment.Itisalsoabasisforthecontrolofthevibrationsofthesystem.MODELDESCRIPTIONTheschematicsystemunderinvestigationisshowninFig.1.Whendimensionsandelasticpropertiesofthecraneshipbodyareconsidered,itissufficienttotakeonlytheboomasflexible.Thefollowingassumptionsareusedintheanalysisofthecraneship1Themotionisonlyintheverticalplane.Theloadisregardedasapointmass,andcanhavependulummotionintheplane,butwithouttwisting.2Theropeistakenasarigidrod.Thisassumptionisvalidaslongastheoscillationsoftheloadintheverticaldirectionaresmallandtheroperemainsintension.3Thestructuraldampingofthesystemisnottakenintoaccount,becausecranesaretypicallylightlydamped.Toddetal.1997reportedthatthedampingofashipmountedcraneisfrom0.1to0.5ofthecriticaldamping.AsdepictedinFig.1,acoordinatesystemOXYisfixedtothegroundanddenotedasEarthFixed,whichistakentobetheinertialframeofreference.TheothersystemO0X0Y0isfixedtoand,hence,moveswiththeship,whichisdenotedasBodyFixed.Jistherotaryinertiaoftheship,mshipandmparethemassesoftheshipandload,respectively.TheelasticdisplacementsoftheboomtippointBaredenotedasuandw.ThecriticalparametersincludingtheboomlengthLb,theluffangleoftheboomβ,thelengthoftheropeL,thedisplacementinsurgedirectionx,thepitchangleθ,theheavemotionyandtheswingangleoftheloadαarealsoindicatedinFig.1.MATHEMATICALMODELExternalforcesactingonthecraneshipInthemodel,differentexternalforceshavetobeconsideredEllermannetal.,2002,thehydrostaticforceTbwwshippm0,,,gAymmghρθ−−f1withthedensityofwaterρw,thecrosssectionalareaoftheshipatthestillwaterlevelAwandthemetacentricheighthm.Themooringlineforces,whichareapproximatedbyathirdorderpolynomial3Tm123||,00,cxcxxcx−−−f2wherec1,c2,c3arethelinear,quadratic,andcubiccharacteristiccoefficientsofthemooringsystem.ForcesduetoviscousdragTdwd||/2,0,0cWTxxρ−f3X0Y0αθO0YOβuwPBLLbXFig.1MooredcraneshipmodelRenetal./JZhejiangUnivSciA200891263128areproportionaltothedensityofwaterρw,theempiricaldragcoefficientcd,thewidthoftheshipWandthedraughtT.Thefrequencydependentwaveexcitationforces,whichcanbesplitintoaperiodicallychangingpartandtheconstantdriftforces,canbemodeledas2ridwriricossincossin,cossinxxyyAktktApAktktAktktθθωωωω⎡⎤−⎢⎥−−⎣⎦f4withthewaveamplitudeA,therealandimaginarypartsofthefrequencydependentcoefficientkrjandkijjx,y,θ,andthecoefficientofthedriftforcepd.TheforcesareabbreviatedbyT123bmdw,,.FFFFffff5DynamicequationofcraneshipThepositionvectorofpayloadPasshowninFig.1canbeexpressedaspcossinsincos,jjxLuLyLwLβθαβθα⋅−⋅rij6whereiandjareunitvectorsalongtheXandYaxes,respectively.Itcanbeseenthat,byincludingtheboomtipdisplacementuandw,theeffectoftheelasticdeformationoftheboomisincludedinthepositionequationofthepayload.Itisassumedthattheluffangleoftheboomandthelengthofthepayloadpendulumareconstant.BasedonEq.6,thevelocityvectorofthepayloadPcanbeobtainedbythetimederivativeofrpaspbbsincoscossin.xuLLywLLθβθααθβθαα−⋅⋅Vij7Sothekineticenergyofthepayloadcanthenbederivedaspppp222222p22b/2222cos2sin2cos2sin2sinbTmmxuywLxuywxLyLuLwLLxLαααααααααθθβθ⋅−VVbbb2sin2sincos2cos2cos2cossin/2.8uLLLyLwLLLθβθθαβθαθβθθβθθαβθα−−Thepotentialenergyofthepayloadppbsincos.UmgyLwLβθα−9Thekineticenergyandthepotentialenergyoftheshipcanberespectivelyexpressedas222shipship/,TJmxyθ10shipship.Umgy11Basedonthefiniteelementdiscretization,thekineticandpotentialenergyofboomcanberespectivelyexpressedasTrrrrurwrTbuwrwuww11,22Tumuwmmw⎧⎫⎧⎫⎡⎤⎪⎪⎪⎪⎢⎥⎨⎬⎨⎬⎪⎣⎦⎩⎭⎩⎭UMMMUUMUMM12TrrrrurwrTbuwrwuww11,22Uukuwkkw⎡⎤⎧⎫⎧⎫⎪⎪⎪⎪⎢⎥⎨⎬⎨⎬⎪⎩⎭⎩⎭⎣⎦UKKKUUKUKK13whereMandKareglobalmassandstiffnessmatricesoftheboom,UandUaredisplacementandvelocityvectorsoftheflexibleboom,uwand,uwarethenodaldisplacementandvelocitiesoftheboomtippointB.UrandrUarevectorsofdisplacementsandvelocitiesfortherestdegreesoffreedomoftheboomstructure.TheLagrangianfunctionofthesystemcanbeexpressedasTrrrrurwr2uruuuwwrwuww22222222shipp22bb11/2222cos2sin2cos2sin2LTUummuJwmwmxymxuywLxuywxLyLuLwLLxLθαααααααααθθ⎧⎫⎧⎫⎡⎤⎪⎪⎪⎪⎢⎥−⎨⎬⎨⎬⎪⎪⎪⎪⎣⎦⎩⎭⎩⎭−UMMMUMMbsin2sinuLβθθβ−Renetal./JZhejiangUnivSciA200891263129bbTrrrrurwruruuuwshipwrwuwwpb2sincos2cos2cos2cossin/212sincos.14LLyLwLLLukkumgywkwmgyLwLθθαβθαθβθθβθθαβθαβθα−⎡⎤⎧⎫⎧⎫⎪⎪⎪⎪⎢⎥−−⎨⎬⎨⎬⎪⎪⎪⎪⎩⎭⎩⎭⎣⎦−−UKKKUKKTheLagrangesequationisd,djjjLLQtqq⎛⎞∂∂−⎜⎟∂∂⎝⎠15whereqjandjqaregeneralcoordinatesandgeneralvelocitiesofthesystem.Qjisgeneralforceofthesystem.SubstitutingEqs.5and14intoEq.15givesrrpp2pb2b2p2bb00sinsincoscossinoscossinmuumwwmLLLxLmgyLLLLααθβθθβθααααααθβθθβθ⎛⎞⎡⎤⎡⎤⎡⎤⎜⎟⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦⎣⎦⎝⎠⎡⎤⎢⎥−−−−−−⎢⎢⎣⎦UUMK000000,16⎥⎥2shippppp2bb1cossinsincos,17mmxmumLmLLLFααααθβθθβθ−−−2shippppp2shippbb2sincoscossin,mmymwmLmLmmgLLFααααθβθθβθ−18pbb22pb3sinsinsincossinsincoscoscossincoscoscos,JmLLxuLLywLLmgFθθβθβθαβθααβθαβθβθαβθααβθαβθ−−−19b2bb2bcossincossinsincoscoscoscossinsinsinsin0.20LxyuwLLLLgαααααθβααθβααθβααθβααα−−−Asinseveralotherinvestigations,theattentionisfocusedonthehorizontalsurgemotion.Inthisspecialcase,equationsofmotionofthesystemcanbereducedtorrpp2p2p00sincos,21sincosmuumwwmLLxmgLLαααααααα⎛⎞⎡⎤⎡⎤⎡⎤⎜⎟⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦⎣⎦⎝⎠⎡⎤−−⎢⎥−−−⎣⎦UUMK0000002shippppp1cossin,mmxmumLmLFαααα−22coscossinsin0.Lxuwgααααα23Iftheboomstructureisassumedtoberigid,thedeformationoftheboomvanishesinEqs.2123.Inthiscase,theequationsofmotionofthesystemcanbereducedto2shipppp1cossin,mmxmLmLFαααα−24cossin0.Lxgααα25SIMULATIONRESULTSANDDISCUSSIONThedynamicresponseofthecraneshipisinvestigatedintimedomainbasedonaNewmarkmethodandaniterativeapproachJuetal.,2006.Thetypeoffiniteelementusedinmodelingthecraneboomisspaceframeelement.Thebasicparametervaluesoftheshipusedfortheanalysisaregivenasfollowsc121200N/m,c29440N/m2,c313.82N/m3,kr5540N/m,ki426000N/m,cd0.2,pd15800Nm2,mp200000kg,mship1920000kg,B25m,Lb60m,T1.69m,ρ1000kg/m3,β60°,A1.2m.InfluenceofwaveexcitationfrequencyConsidertheeffectofdifferentwaveexcitationfrequenciesontheloadswingangles.Thecablelengthis30m,whichcorrespondstoaloadnaturalfrequencyof0.626Hz.Thewavefrequenciesare0.313Hz,0.626Hz,and0.8138Hz.ThedynamicresponsesofloadswinganglesαareshowninFig.2.