2008 Design and modeling of a magnetic shock absor

JOURNALOFSOUNDANDVIBRATIONJournalofSoundandVibration315(2008)875–889DesignandmodelingofamagneticshockabsorberbasedoneddycurrentdampingeffectBabakEbrahimia,MirBehradKhameseea,,M.FaridGolnaraghibaDepartmentofMechanicalandMechatronicsEngineering,UniversityofWaterloo,Waterloo,Ontario,CanadaN2L3G1bMechatronicSystemsEngineering,SimonFraserUniversity,Surrey,BritishColumbia,CanadaV3T0A3Received21March2007;receivedinrevisedform17January2008;accepted7February2008HandlingEditor:S.BoltonAbstractThispaperdescribesthedesign,modeling,andanalysisofanovelmagneticspring–damper.Thiscost-effective,self-poweredmagneticspring–damperutilizestwopermanentmagnetsandaconductivealuminumplatetogeneratebothspringandvariabledampingeffects.Eddycurrentsaregeneratedinthealuminumplateduetoitsrelativemotionwithrespecttothemagnets.Theseeddycurrentsproducearepulsiveforcethatisproportionaltothevelocityoftheconductorsuchthatthemovingmagnetandconductoractasaviscousdamper.Thestructureoftheproposedpassivemagneticspring–damperissimple,anddoesnotrequireanexternalpowersupplyoranyotherelectronicdevice.Anaccurate,analyticalmodelofthesystemisobtainedbyusingtheelectromagnetictheorytoestimatetheelectromagneticforces,inducedinthesystem.Thenewlydevelopedmodelcanbeusedtodesignhigh-performancedampersforvariousapplications.Tooptimizethedesign,simulationsareconductedandthedesignparametersareevaluated.Afteramagneticspring–damperprototypeisfabricated,experimentsareconductedtoverifytheaccuracyofthetheoreticalmodel.Theeddycurrentmodelexhibitsa7.5%RMSerrorforthedampingratioestimation,andadampingratioashighas40Ns/misachievedbythefabricatedprototype.r2008ElsevierLtd.Allrightsreserved.1.IntroductionEddycurrentsaregeneratedinaconductorinatime-varyingmagneticfield.Theyareinducedeitherbythemovementoftheconductorinthestaticfieldorbychangingthestrengthofthemagneticfield,initiatingmotionalandtransformerelectromotiveforces(emfs),respectively.Sincethegeneratededdycurrentscreatearepulsiveforcethatisproportionaltothevelocityoftheconductor,themovingmagnetandconductorbehavelikeaviscousdamper.Gravesetal.[1]havederivedamathematicalrepresentationforeddycurrentdampers,basedonthemotionalandtransformeremf,andhavedevelopedananalyticalapproachtocomparetheefficiencyofthedampersintermsofthesetwosources.Formorethantwodecades,theapplicationofeddycurrentsfordampingpurposeshasbeeninvestigated,includingmagneticbrakingsystems[2–4],vibrationARTICLEINPRESS:10.1016/j.jsv.2008.02.022Correspondingauthor.Tel.:+1519888456735095.E-mailaddress:[e-mail](M.B.Khamesee).controlofrotarymachinery[5],structuralvibrationsuppression[6–8],andvibrationisolationenhancementinlevitationsystems[9,10].Sodanoetal.[6]haveanalyzedthesuppressionofcantileverbeamvibrations,whereapermanentmagnetisfixedsothatitisperpendiculartothebeammotion,andaconductingsheetisattachedtothebeamtip.InRef.[8],atheoreticalmodelofaneddycurrentdamperhasbeenmodifiedandfurtherdevelopedbyapplyinganimagemethodtosatisfytheboundaryconditionofthezeroeddycurrentdensityattheconductingplate’sboundaries.Anadditionalpermanentmagnetisincludedinthedevicewiththelike-polesfixedinthesamedirection,intensifyingtheradialmagneticfluxdensity,enhancingthedampingeffect.Teshimaetal.[9]havestudiedtheeffectofaneddycurrentdamperonthevibrationisolationpropertiesofsuperconductinglevitation,anddemonstratedthatintheverticaldirection,thedampingisimprovedbyapproximately100timesbyusingeddycurrentdampers.Elbukenetal.[10]haveinvestigatedtheeddycurrentdampingforhighprecisionmagneticlevitation.TheauthorsinRef.[10]havesuggestedaneddycurrentdampertosuppressthevibrationofthelevitatedobject.Karnopp[11]hasdevisedanewelectromechanicaldamperforvehicleapplications,consistingofcopperwireandpermanentmagnets.Hehasdemonstratedthatforoscillationfrequencies,expectedinroadvehiclesuspensions,electrodynamicvariableshockabsorbersarefeasible.SchmidandVaraga[12]havedesignedandanalyzedavibrationattenuationsystemfortheconstructionofhigh-resolutionnanotechnologystructuressuchasthescanningtunnellingmicroscopebyusingeddycurrentdampers.Foreddycurrentdampersandcouplersunderdynamicconditions,Tonoli[13]haspresentedaphysical,dynamicmodel.Theapplicationofthemagneticspringeffect[14,15],andeddycurrenteffect[6–8]invibrationsuppressionstudieshavebeenreported,but,tothebestoftheauthors’knowledge,theapplicationofeddycurrentswithamagneticspringeffecthasnotbeenaddressedinpriorpublications.Magneticspringdampersareappealinginvariousvibrationisolationsystemssuchasprecisionmachinery,structurevibrationisolation,andvehiclesuspensionsystems.Inthisarticle,theeddycurrentdampingeffectisexploitedforthedevelopmentofaspring–damper,andamodelofthemagneticandeddycurrentforcesforthedesignedmagneticspring–damperisproposed.Fig.1isaschematicoftheproposedsystem,whichconsistsofaring-shapedconductingaluminumplateandtwocylindricalpermanentmagnets,whoselike-polesareincloseproximity.Thelowerpermanentmagnetandthealuminumplatearestationary,whiletheuppermagnet