Concept and model of eddy current damper for vibra

JOURNALOFSOUNDANDVIBRATIONJournalofSoundandVibration288(2005)1177–1196ConceptandmodelofeddycurrentdamperforvibrationsuppressionofabeamHenryA.Sodanoa,
,Jae-SungBaeb,DanielJ.Inmana,W.KeithBelvincaCenterforIntelligentMaterialsSystemsandStructures,VirginiaPolytechnicInstituteandStateUniversity,Blacksburg,VA24061-0261,USAbSchoolofAerospaceandMechanicalEngineering,HankukAviationUniversity,200-1,Hwajeon-dong,Deogyang-gu,Goyang-city,Geonggi-do,412-791,KoreacStructuralDynamicsBranch,NASALangleyResearchCenter,Hampton,VA23681-0001,USAReceived12April2004;receivedinrevisedform12January2005;accepted27January2005Availableonline29March2005AbstractElectromagneticforcesaregeneratedbythemovementofaconductorthroughastationarymagneticfieldoratimevaryingmagneticfieldthroughastationaryconductorandcanbeusedtosuppressthevibrationsofaflexiblestructure.Inthepresentstudy,anewelectromagneticdampingmechanismisintroduced.Thismechanismisdifferentfrompreviouslydevelopedelectromagneticbrakingsystemsandeddycurrentdampersbecausethesysteminvestigatedinthesubsequentmanuscriptusestheradialmagneticfluxtogeneratetheelectromagneticdampingforceratherthanthefluxperpendiculartothemagnet’sfaceasdoneinotherstudies.Oneimportantadvantageoftheproposedmechanismisthatitissimpleandeasytoapply.Additionally,asinglemagnetcanbeusedtodampthetransversevibrationsthatarepresentinmanystructures.Furthermore,itdoesnotrequireanyelectronicdevicesorexternalpowersupplies,thereforefunctioningasanon-contactingpassivedamper.Atheoreticalmodelofthesystemisderivedusingelectromagnetictheoryenablingustoestimatetheelectromagneticdampingforceinducedonthestructure.Theproposededdycurrentdamperwasconstructedandexperimentswereperformedtoverifytheprecisionofthetheoreticalmodel.Itisfoundthattheproposededdycurrentdampingmechanismcouldincreasethedampingratiobyupto150timesandprovidesufficientdampingforcetoquicklysuppressthebeam’svibration.r2005ElsevierLtd.Allrightsreserved.ARTICLEINPRESS:10.1016/j.jsv.2005.01.016
Correspondingauthor.Fax:[phone].E-mailaddress:[e-mail](H.A.Sodano).1.IntroductionWhenanon-magneticconductivemetalisplacedinamagneticfield,eddycurrentsaregenerated.Theseeddycurrentscirculateinsuchawaythattheyinducetheirownmagneticfieldwithoppositepolarityoftheappliedfieldcausingaresistiveforce.However,duetotheelectricalresistanceofthemetal,theinducedcurrentswillbedissipatedintoheatattherateofI2Randtheforcewilldisappear.Inthecaseofadynamicsystemtheconductivemetaliscontinuouslymovinginthemagneticfieldandexperiencesacontinuouschangeinfluxthatinducesanelectromotiveforce(emf)allowingtheinducedcurrentstoregenerateandinturnproducearepulsiveforcethatisproportionaltothevelocityoftheconductivemetal.Thisprocesscausestheeddycurrentstofunctionlikeaviscousdamperanddissipateenergycausingthevibrationstodieoutfaster.Theuseofeddycurrentsfordampingofdynamicsystemshasbeenknownfordecadesanditsapplicationtomagneticbrakingsystems[1–4]andlateralvibrationcontrolofrotatingmachinery[5,6]hasbeenthoroughlyinvestigated.Whilethetheoryandapplicationsofrotarymagneticbrakingsystemshavebeenwelldocumented,therearemanymoreapplicationsofeddycurrentdampers.Karnopp[7]introducedtheideathatalinearelectrodynamicmotorconsistingofcoilsofcopperwireandpermanentmagnetscouldbeusedasanelectromechanicaldamperforvehiclesuspensionsystems.Itwasshownthatthisactuatorcouldbemuchsmallerandlighterthanconventionaldamperswhilestillprovidingeffectivedampinginthefrequencyrangetypicallyencounteredbyroadvehiclesuspensionsystems;howeveritwasunabletoeffectivelyisolatethevehiclefromshockexcitation.SchmidandVarga[8]studiedavibration-reducingsystemwitheddycurrentdampers(ECDs)forhighresolutionandnanotechnologydevicessuchasascanningtunnelingmicroscope(STM).Teshimaetal.[9]investigatedtheeffectsofaneddycurrentdamperonthevibrationalcharacteristicsofsuperconductinglevitationandshowedthatthedampingofverticalvibrationsARTICLEINPRESSNomenclaturebradiusofthecircularmagnetBmagneticfluxdensityBrresidualmagneticfluxceeddycurrentdampingforceCdampingmatrixCbdampingmatrixofbeamdthicknessofconductorEmodulusofelasticityFdampingforceJeddycurrentdensityKstiffnessmatrixLlengthofthemagnetlggaplengthbetweenmagnetandcon-ductorMmassmatrixM0magnetizationm0permeabilityoffreespacefassumedmodeshapedrdensityQenon-conservativeforcesrtemporalcoordinatercequivalentradiusoftheconductorSstrainsconductivityttimeTkineticenergyudisplacementUpotentialenergyvvelocityofconductorinzdirectionvvelocityofconductorVvolumeoinaturalfrequencyH.A.Sodanoetal./JournalofSoundandVibration288(2005)1177–11961178wasabout100timesimprovedbyeddycurrentdampers.TheconceptofusingaviscoelasticmaterialtodissipateenergyfromastructurewasmodifiedtoincorporatedmagnetsbyOhetal.[10].Thestudysandwichedaviscoelasticmaterialbetweenmagneticstripsthatwereconfiguredtoattracteachotherinonecaseandtorepelintheother.Itwasdeterminedthatthepassivemagne-ticcomposite(PMC)treatmentsfunctionbestwhenthemagnetsweresettoattracteachother.However,thismethod