Renetal./JZhejiangUnivSciA20089(1):26-3126Dynamicresponseanalysisofamooredcrane-shipwithaflexibleboom*Hui-liREN,Xue-linWANG,Yu-jinHU,Cheng-gangLI(SchoolofMechanicalScienceandEngineering,HuazhongUniversityofScienceandTechnology,Wuhan430074,China)E-mail:；w_ReceivedJune12,2007；revisionacceptedAug.5,2007；publishedonlineDec.14,2007Abstract:Thedynamicresponseofmooredcrane-shipisstudied.Governingequationsforthedynamicresponseofacrane-shipcoupledwiththependulummotionofthepayloadarederivedbasedonLagrangesequations.Theboomismodeledbasedonfiniteelementmethod,whilethepayloadismodeledasaplanarpendulumofpointmass.Thedynamicresponsewasstudiedusingnumericalmethod.Thecalculationresultsshowthatthelarge-amplituderesponsesoccuratwaveperiodsnearthenaturalperiodofthepayload.Loadswingangleissmallerforcrane-shipwithflexibleboom,incomparisonwithrigidboom.Theshipsurgemo-tionshavelargevibrationsforcrane-shipwithflexibleboom,whichwerenotobservedforarigidboom.Theanalysisidentifiesthesignificanceofkeyparametersandrevealshowthesystemdesigncanbeadjustedtoavoidcriticalconditions.Keywords:Dynamicresponse,Mooredcrane-ship,Finiteelementmethod,Rigid-flexiblecouplingdynamicmodeldoi:10.1631/jzus.A071308Documentcode:ACLCnumber:U615.35INTRODUCTIONFloatingcranesplayanimportantroleintheoffshoreprojects.However,theseawaveshakesthecrane-shipandmayexciteitslargemovement.Forheavydutylifting,theoperationsofcrane-shipinwaves,evenastheseaisrelativelyquiet,areoftenrestrictedbytheexcessivemotionsofthecraneload.Insomecases,thedynamicalbehavioroffloatingcranesistobeconsideredascriticalwithrespecttotheamplitudesofthemotionoftheshiportheload.Asmalldisturbanceofthesystemcancausethecolli-sionbetweentheloadandtheshiporotherobjects.Inaddition,theamplitudeofthemotionoftheshiphastostaysmallinordertoachievetherequiredpreciselypositioningandtoavoiddamagestothemooringsystem.Theinvestigationofcrane-shipdynamicshasbeenofinterestinalargenumberofrecentresearches(DongandHan,1993；Masoudetal.,2004；Eller-mann,2005).Arigidmasslesscableandmassivepointloadwereusedtomodelthecraneloadsystem,andtheresultsofthecomputersimulationwereveri-fiedexperimentallyusingathreedegree-of-freedom(DOF)ship-motionsimulationplatform(Henryetal.,2001).Themethodofmultiplescalesisusedtoana-lyzethedynamicsofacable-suspendedload.Theresultsshowthataparametricexcitationattwicethenaturalfrequencyleadstosuddenjumpsinthere-sponseasthecableisunreeled(Chinetal.,2001).Cargopendulationcontrolofanelasticship-mountedcranewasconcernedusingtheMarylandRiggingsystem.Thedynamicsofthecraneisdescribedbyamulti-modelproblemdependingonthecurrentcablelengthandboomluffangle.Avariable-gainobserverandavariable-gaincontrolleraredesigned.Simula-tionandexperimentalresultsshowedthattheex-pressedcontrolstrategyhasasignificanteffectonsuppressingthevibrationsfordifferentoperatingconditionsandpayloadmasses(Al-SweitiandSf-fker,2007).Asthefirstapproachforevaluatingthedynam-JournalofZhejiangUniversitySCIENCEAISSN1673-565X(Print)；ISSN1862-1775(Online)/jzus；E-mail:Correspondingauthor*ProjectsupportedbytheNationalNaturalScienceFoundationofChina(No.50675077)andtheResearchFundfortheDoctoralProgramofHigherEducationofChina(No.20050487047)Renetal./JZhejiangUnivSciA20089(1):26-3127icsofacrane-ship,alineartheoreticalmodelisused.Thelineardifferentialequationofmotionisderivedundertheassumptionofsmallamplitudes,anddoesnotconsideroccurringrestoringforces.Adynamicmodelwasestablishedusingmulti-bodydynamicsmethodsandthedynamicsofthecranewasanalyzed(Chenetal.,2002).Theeffectsofcablereelingandunreelingoncargopendulationswerestudiedwiththeboomcranemodeledasaplanarpendulumandtheshipasarigidbody.Theresultsshownonlinearbe-havior(Kraletal.,1996).Inalmostallofthesestudies,theflexibilityoftheboomisnottakenintoconsideration.Thismaybereasonableforshortcraneboomsandsmallpay-load-to-shipmassratios,butforlongcraneboomsandlargepayload-to-shipmassratios,theinfluenceoftheflexibilityofthecraneboomcannotbeignoredanymore.Totheauthorsbestknowledge,untilrecentlythereisnoreportonthemodelingofsuchsystem.Thestudyaimstopresentarigid-flexiblecouplingdy-namicmodelforthepredictionofthedynamicsofamooredcrane-shipanditspayload.Thepredictionisthefoundationofthedynamicaldesignandanaccu-rateresidualworkinglifeassessment.Itisalsoabasisforthecontrolofthevibrationsofthesystem.MODELDESCRIPTIONTheschematicsystemunderinvestigationisshowninFig.1.Whendimensionsandelasticprop-ertiesofthecrane-shipbodyareconsidered,itissuf-ficienttotakeonlytheboomasflexible.Thefol-lowingassumptionsareusedintheanalysisofthecrane-ship:(1)Themotionisonlyintheverticalplane.Theloadisregardedasapointmass,andcanhavepen-dulummotionintheplane,butwithouttwisting.(2)Theropeistakenasarigidrod.Thisas-sumptionisvalidaslongastheoscillationsoftheloadintheverticaldirectionaresmallandtheroperemainsintension.(3)Thestructuraldampingofthesystemisnottakenintoaccount,becausecranesaretypicallylightlydamped.Toddetal.(1997)reportedthatthedampingofashipmountedcraneisfrom0.1%to0.5%ofthecriticaldamping.AsdepictedinFig.1,acoordinatesystemOXYisfixedtothegroundanddenotedas“Earth-Fixed”,whichistakentobetheinertialframeofreference.TheothersystemO0X0Y0isfixedtoand,hence,moveswiththeship,whichisdenotedas“Body-Fixed”.Jistherotaryinertiaoftheship,mshipandmparethemassesoftheshipandload,respectively.TheelasticdisplacementsoftheboomtippointBaredenotedasuandw.ThecriticalparametersincludingtheboomlengthLb,theluffangleoftheboom,thelengthoftheropeL,thedisplacementinsurgedirec-tionx,thepitchangle,theheavemotionyandtheswingangleoftheloadarealsoindicatedinFig.1.MATHEMATICALMODELExternalforcesactingonthecrane-shipInthemodel,differentexternalforceshavetobeconsidered(Ellermannetal.,2002),thehydrostaticforceTbwwshippm0,(),gAymmgh=+f(1)withthedensityofwaterw,thecross-sectionalareaoftheshipatthestillwaterlevelAwandthemeta-centricheighthm.Themooringlineforces,whichareapproxi-matedbyathirdorderpolynomial3Tm123|,00,cxcxxcx=f(2)wherec1,c2,c3arethelinear,quadratic,andcubiccharacteristiccoefficientsofthemooringsystem.ForcesduetoviscousdragTdwd|/2,0,0cWTxx=f(3)X0Y0O0YOuwPBLLbXFig.1Mooredcrane-shipmodelRenetal./JZhejiangUnivSciA20089(1):26-3128areproportionaltothedensityofwaterw,theem-piricaldragcoefficientcd,thewidthoftheshipWandthedraughtT.Thefrequency-dependentwaveexcitationforces,whichcanbesplitintoaperiodicallychangingpartandtheconstantdriftforces,canbemodeledas:2ridwriri(cos()sin()(cos()sin(),(cos()sin()xxyyAktktApAktktAktkt+=f(4)withthewaveamplitudeA,therealandimaginarypartsofthefrequencydependentcoefficientkrjandkij(j=x,y,),andthecoefficientofthedriftforcepd.TheforcesareabbreviatedbyT123bmdw,.FFF=+Fffff(5)Dynamicequationofcrane-shipThepositionvectorofpayloadPasshowninFig.1canbeexpressedasp(cos()sin)(sin()cos),jjxLuLyLwL=+rij(6)whereiandjareunitvectorsalongtheX-andY-axes,respectively.Itcanbeseenthat,byincludingtheboomtipdisplacementuandw,theeffectoftheelasticdeformationoftheboomisincludedinthepositionequationofthepayload.Itisassumedthattheluffangleoftheboomandthelengthofthepayloadpendulumareconstant.BasedonEq.(6),thevelocityvectorofthepayloadPcanbeobtainedbythetimederivativeofrpaspbb(sin()cos)(cos()sin).xuLLywLL=+Vij(7)Sothekineticenergyofthepayloadcanthenbederivedaspppp222222p22b/2222cos2sin2cos2sin2sin()bTmmxuywLxuywxLyLuLwLLxL=+VVbbb2sin()2sin()cos2cos()2cos()2cos()sin/2.(8)uLLLyLwLLL+Thepotentialenergyofthepayloadppb(sin()cos).UmgyLwL=+(9)Thekineticenergyandthepotentialenergyoftheshipcanberespectivelyexpressedas222shipship()/,TJmxy=+(10)shipship.Umgy=(11)Basedonthefiniteelementdiscretization,thekineticandpotentialenergyofboomcanberespec-tivelyexpressedasTrrrrurwrTbuwrwuww11,22Tumuwmmw=UMMMUUMUMM(12)TrrrrurwrTbuwrwuww11=,22UukuwkkwUKKKUUKUKK(13)whereMandKareglobalmassandstiffnessmatricesoftheboom,UandUaredisplacementandvelocityvectorsoftheflexibleboom；(,)uwand(,)uwarethenodaldisplacementandvelocitiesoftheboomtippointB.UrandrUarevectorsofdisplacementsandvelocitiesfortherestdegreesoffreedomoftheboomstructure.TheLagrangianfunctionofthesystemcanbeexpressedasTrrrrurwr2uruuuwwrwuww22222222shipp22bb11()/2222cos2sin2cos2sin2LTUummuJwmwmxymxuywLxuywxLyLuLwLLxL=+UMMMUMMbsin()2sin(uL+Renetal./JZhejiangUnivSciA20089(1):26-3129bbTrrrrurwruruuuwshipwrwuwwpb)2sin()cos2cos()2cos()2cos()sin/212(sin()cos).(14)LLyLwLLLukkumgywkwmgyLwL+UKKKUKKTheLagrangesequationisd,djjjLLQtqq=(15)whereqjandjqaregeneralcoordinatesandgeneralvelocitiesofthesystem.Qjisgeneralforceofthesystem.SubstitutingEqs.(5)and(14)intoEq.(15)givesrrpp2pb2b2p2bb00(sinsin()cos()cos)sinoscos()sin()muumwwmLLLxLmgyLLLL+=+UUMK000000,(16)2shippppp2bb1()cossinsin()cos(),(17)mmxmumLmLLLF+=2shippppp2shippbb2()sincos()cos()sin(),mmymwmLmLmmgLLF+=(18)pbb22pb3(sin()sin()sin()cossin()sincos()cos()cos()sincos()cos)cos(),JmLLxuLLywLLmgF+=(19)b2bb2bcossincossinsin()coscos()coscos()sinsin()sinsin0.(20)LxyuwLLLLg+=Asinseveralotherinvestigations,theattentionisfocusedonthehorizontalsurgemotion.Inthisspecialcase,equationsofmotionofthesystemcanbere-ducedtorrpp2p2p00(sincos),(2