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1、Chapter1Lagrangiandynamicsofmechanicalsystems1.1IntroductionThisbookconsidersthemodellingofelectromechanicalsystemsinanunifiedwaybasedonHamilton’sprinciple.ThischapterstartswithareviewoftheLagrangiandynamicsofmechanicalsystems;nextchapterproceedswiththeLagrangiandynamicsofelectricaln
2、etworksandtheremainingchaptersaddressawideclassofelectromechanicalsystems,includingpiezoelectricstructures.TheLagrangiandynamics(oranalyticaldynamics)hasbeenmotivatedbythesub-stitutionofscalarquantities(energyandwork)tovectorquantities(force,momentum,torque,angularmomentum)inclassic
3、alvectordynamics.Generalizedcoordinatesaresubstitutedtophysicalcoordinates,whichallowsaformulationindependentoftheref-erenceframe.Thesystemsareconsideredglobally,ratherthaneverycomponentinde-pendently,withtheadvantageofeliminatingautomaticallytheinteractionforces(con-straints)betwee
4、nthevariouselementarypartsofthesystem.Thechoiceofgeneralizedcoordinatesisnotunique.Thederivationofthevariationalformoftheequationsofdynamicsfromitsvectorcounterpart(Newton’slaws)isdonethroughtheprincipleofvirtualwork,extendedtodynamicsthankstod’Alembert’sprinciple,leadingeventuallyt
5、oHamilton’sprincipleandtheLagrange’sequationsfordiscretesystems.Hamilton’sprincipleisanalternativetoNewton’slawsanditcanbearguedthat,assuch,itisafundamentallawofphysicswhichcannotbederived.Webelieve,however,thatitsformmaynotbeimmediatelycomprehensibletotheunexperiencedreaderandthati
6、tsderivationforasystemofparticleswillhelpitsacceptanceasanalternativeformulationofthedynamicequilibrium.Hamilton’sprincipleisinfactmoregeneralthanNewton’slaws,becauseitcanbegeneralizedtodistributedsystems(governedbypartialdifferentialequations)and,asweshallseelater,toelectromechanica
7、lsystems.Itisalsothestartingpointfortheformulationofmanynumericalmethodsindynamics,includingthefiniteelementmethod.12Mechatronics1.2KineticstatefunctionsConsideraparticletravellinginthedirectionxwithalinearmomentump.AccordingtoNewton’slaw,theforceactingontheparticleequalstherateofcha
8、ngeofthemomentum:dp
