Synthesis-of-mechanical-networks-the-inerter

1648IEEETRANSACTIONSONAUTOMATICCONTROL,VOL.47,NO.10,OCTOBER2002SynthesisofMechanicalNetworks:TheInerterMalcolmC.Smith,Fellow,IEEEAbstract—Thispaperisconcernedwiththeproblemofsynthesisof(passive)mechanicalone-portnetworks.Oneofthemaincon-tributionsofthispaperistheintroductionofadevice,whichwillbecalledtheinerter,whichisthetruenetworkdualofthespring.Thiscontrastswiththemasselementwhich,bydefinition,alwayshasoneterminalconnectedtoground.Theinerterallowselectricalcircuitstobetranslatedovertomechanicalonesinacompletelyanalogousway.Theinerterneednothavelargemass.Thisallowsanyarbitrarypositive-realimpedancetobesynthesizedmechan-icallyusingphysicalcomponentswhichmaybeassumedtohavesmallmasscomparedtootherstructurestowhichtheymaybeat-tached.Thepossibleapplicationoftheinerterisconsideredtoavibrationabsorptionproblem,asuspensionstrutdesign,andasasimulatedmass.IndexTerms—Brunesynthesis,Darlingtonsynthesis,elec-trical–mechanicalanalogies,mechanicalnetworks,networksynthesis,passivity,suspensionsystems,vibrationabsorption.I.INTRODUCTIONTHEREisastandardanalogybetweenmechanicalandelectricalnetworksinwhichforce(respectively,velocity)correspondstocurrent(respectively,voltage)andafixedpointinaninertialframeofreferencecorrespondstoelectricalground[9],[26].Inthisanalogy,thespring(respectively,damper,mass)correspondstotheinductor(respectively,resistor,capacitor).Itiswellknownthatthecorrespondenceisperfectinthecaseofthespringanddamper,butthereisarestrictioninthecaseofthemass.Thisrestrictionisduetothefactthattheforce–velocityrelationshipsatisfiedbythemass,namelyNewton’sSecondLaw,relatestheaccelerationofthemassrelativetoafixedpointintheinertialframe.Effectivelythismeansthatone“terminal”ofthemassisthegroundandtheother“terminal”isthepositionofthecenterofmassitself[26,p.111],[15,pp.10–15].Clearly,intheelectricalcontext,itisnotrequiredthatoneterminalofthecapacitorisgrounded.Thismeansthatanelectricalcircuitmaynothaveadirectspring–mass-dampermechanicalanalog.Thereisafurtherdrawbackwiththemasselementastheanalogofthecapacitorinthecontextofsynthesisofmechan-icalimpedances.Namely,itmaybeimportanttoassumethatthemechanicaldeviceassociatedwiththe“black-boximpedance”tobedesignedhasnegligiblemasscomparedtoothermassesinthesystem(cf.,asuspensionstrutforavehiclecomparedtothesprungandunsprungmasses).Clearlythispresentsaproblemif(possibly)largemassesmayberequiredforitsrealization.Itappearsthattheaforementionedtwodifficultieshavepre-ventedelectricalcircuitsynthesisfrombeingfullyexploitedforManuscriptreceivedNovember1,2001;revisedApril9,2002.Recom-mendedbyAssociateEditorK.Gu.ThisworkwassupportedinpartbytheEPSRC.TheauthoriswiththeDepartmentofEngineering,UniversityofCambridge,CambridgeCB21PZ,U.K.(e-mail:[e-mail]).DigitalObjectIdentifier10.1109/TAC.2002.803532.thesynthesisofmechanicalnetworks.Itseemsinterestingtoaskifthesedrawbacksareessentialones?Itisthepurposeofthispapertoshowthattheyarenot.Thiswillbeachievedbyintroducingamechanicalcircuitelement,whichwillbecalledtheinerter,whichisagenuinetwo-terminaldeviceequivalenttotheelectricalcapacitor.Thedeviceiscapableofsimplere-alization,andmaybeconsideredtohavenegligiblemassandsufficientlineartravel,formodelingpurposes,asiscommonlyassumedforspringsanddampers.Theinerterallowsclassicalresultsfromelectricalcircuitsynthesistobecarriedoverexactlytomechanicalsystems.Threeapplicationsoftheinerterideawillbepresented.Thefirstisavibrationabsorptionproblemwhoseclassicalsolutionisatunedspring–massattachedtothemainbody.Itwillbeshownthattheinerteroffersanalternativeapproachwhichdoesnotrequireadditionalelementstobemountedonthemainbody.Thesecondapplicationisasuspensionstrutdesign.Traditionalstrutsemployspringsanddampersonly,whichgreatlyrestrictstheavailablemechanicaladmittances.Inparticular,theirphasecharacteristicisalwayslagging.Byconsideringageneralclassofthirdorderadmittancesitwillbeshownthattheuseofin-ertersoffersapossibilitytoreduceoscillationinstifflysprungsuspensionsystems.TheproceduresofBruneandDarlingtonwillbeemployedtoobtainnetworkrealizationsoftheseadmit-tances.Thethirdapplicationistheuseoftheinertertosimulateamasselement.Theapproachusedforthemechanicaldesignproblemsinthispaperowesadebttothemethodsofmoderncontrol.Firstly,theproblemsareviewedasaninterconnectionbetweenagivenpartofthesystem(analogoustotheplant)andaparttobedesigned(analogoustothecontroller).Secondly,theparttobedesignedisadynamicalelementwhoseadmissibilityisdefinedasbroadlyaspossible—passiveinthepresentcase(stabilizingforfeed-backcontrol).Theadvantageofthisviewpointisthatsynthesismethodscomeintoplay,andthatnewsolutionsemergewhichwouldotherwisebemissed.II.MECHANICALNETWORKSA.ClassicalNetworkAnalogiesHistorically,thefirstanalogytobeusedbetweenelectricalandmechanicalsystemswastheforce–voltageanalogy,asisreadilyseenintheearlyuseofthetermelectromotiveforce.Thealternativeforce–currentanalogyisusuallyattributedtoFire-stone[9],thoughitappearstohavebeenindependentlydiscov-eredin[12],[7].Firestonealsointroducedtheideasofthroughandacrossvariableswhichprovidea