结构动力学课件-1dyanmicsofstructures-ch1ch

DynamicsofStructuresJunjieWangDept.ofBridgeEngineering2005.01,simpleharmonic;complex;impulsive;(d)long-duration.,FIGURE1-1Characteristicsandsourcesoftypicaldynamicloadings,1.1BACKGROUND,CHAPTER1.OVERVIEWOFSTRUCTURALDYNAMICS,(a)1999年台湾集集地震集鹿大桥破坏状态,TheDamagesofJiluBridge(inTaiwan)inJijiEarthquakeof1999,TheDamagesofKobeBridge(Japan)inKobeEarthquakeof1995,CHAPTER1.OVERVIEWOFSTRUCTURALDYNAMICS,(a)1999年台湾集集地震集鹿大桥破坏状态,SunshineSkywayBridgeTampaBay,Florida(1980),TasmanBridgeDerwentRiver,Hobart,Australia(1975),CHAPTER1.OVERVIEWOFSTRUCTURALDYNAMICS,(a)1999年台湾集集地震集鹿大桥破坏状态,1.2ESSENTIALCHARACTERISTICSOFADYNAMICPROBLEM,timevaryingnatureofthedynamicprobleminertialforces(morefundamentaldistinction),FIGURE1-2Basicdifferencebetweenstaticanddynamicloads:(a)staticloading;(b)dynamicloading.,CHAPTER1.OVERVIEWOFSTRUCTURALDYNAMICS,(a)1999年台湾集集地震集鹿大桥破坏状态,1.3SOLUTIONSTOADYNAMICPROBLEM,ContinuousModels(partialdifferentialequations;generalizeddisplacement,sumofaseries),FIGURE1-3Sine-seriesrepresentationofsimplebeamdeflection.,CHAPTER1.OVERVIEWOFSTRUCTURALDYNAMICS,(a)1999年台湾集集地震集鹿大桥破坏状态,DiscreteModels,FIGURE1-4Lumped-massidealizationofasimplebeam.,CHAPTER1.OVERVIEWOFSTRUCTURALDYNAMICS,(a)1999年台湾集集地震集鹿大桥破坏状态,FEM,Athirdmethodofexpressingthedisplacementsofanygivenstructureintermsofanitenumberofdiscretedisplacementcoordinates,whichcombinescertainfeaturesofboththelumpedmassandthegeneralizedcoordinateprocedures,FIGURE1-5Typicalfinite-elementbeamcoordinates.,CHAPTER1.OVERVIEWOFSTRUCTURALDYNAMICS,1.4FORMULATIONOFTHEEQUATIONSOFMOTION,1.4.1DirectEquilibrationUsingdAlembertsPrinciple,TheequationsofmotionofanydynamicsystemrepresentexpressionsofNewtonssecondlawofmotion,whichstatesthattherateofchangeofmomentumofanymassparticlemisequaltotheforceactingonit.Thisrelationshipcanbeexpressedmathematicallybythedifferentialequation,Formostproblemsinstructuraldynamicsitmaybeassumedthatmassdoesnotvarywithtime,inwhichcaseEq.(13)maybewritten,thesecondtermiscalledtheinertialforceresistingtheaccelerationofthemass.,knownasdAlembertsprinciple,CHAPTER1.OVERVIEWOFSTRUCTURALDYNAMICS,1.4.2PrincipleofVirtualDisplacements,However,ifthestructuralsystemisreasonablycomplexinvolvinganumberofinterconnectedmasspointsorbodiesoffinitesize,thedirectequilibrationofalltheforcesactinginthesystemmaybedifficult.Frequently,thevariousforcesinvolvedmayreadilybeexpressedintermsofthedisplacementdegreesoffreedom,buttheirequilibriumrelationshipsmaybeobscure.Inthiscase,theprincipleofvirtualdisplacementscanbeusedtoformulatetheequationsofmotionasasubstituteforthedirectequilibriumrelationships.,Theprincipleofvirtualdisplacementsmaybeexpressedasfollows.Ifasystemwhichisinequilibriumundertheactionofasetofexternallyappliedforcesissubjectedtoavirtualdisplacement,i.e.,adisplacementpatterncompatiblewiththesystemsconstraints,thetotalworkdonebythesetofforceswillbezero.,1.4.3Hamiltonsprinciple,CHAPTER1.OVERVIEWOFSTRUCTURALDYNAMICS,1.5ORGANIZATIONOFTHETEXT,CHAPTER1.OVERVIEWOFSTRUCTURALDYNAMICS,PARTISINGLEDEGREEOFFREEDOMSYSTEMS,CHAPTER2.ANALYSISOFFREEVIBRATION,21COMPONENTSOFTHEBASICDYNAMICSYSTEM,Theessentialphysicalpropertiesofanylinearlyelasticstructuralormechanicalsystemsubjectedtoanexternalsourceofexcitationordynamicloadingareitsmass,elasticproperties(exibilityorstiffness),andenergylossmechanismordamping.InthesimplestmodelofaSDOFsystem,eachofthesepropertiesisassumedtobeconcentratedinasinglephysicalelement.AsketchofsuchasystemisshowninFig.21a.,FIGURE2-1IdealizedSDOFsystem:(a)basiccomponents;(b)forcesinequilibrium.,CHAPTER2.ANALYSISOFFREEVIBRATION,22EQUATIONOFMOTIONOFTHEBASICDYNAMICSYSTEM,TheequationofmotionforthesimplesystemismosteasilyformulatedbydirectlyexpressingtheequilibriumofallforcesactingonthemassusingdAlembertsprinciple.,Theequationofmotionismerelyanexpressionoftheequilibriumoftheseforcesasgivenby,InaccordancewithdAlembertsprinciple,theinertialforceistheproductofthemassandacceleration,Assumingaviscousdampingmechanism,thedampingforceistheproductofthedampingconstantcandthevelocity,Finally,theelasticforceistheproductofthespringstiffnessandthedisplacement,CHAPTER2.ANALYSISOFFREEVIBRATION,23INFLUENCEOFGRAVITATIONALFORCES,FIGURE2-2InfluenceofgravityonSDOFequilibrium.,CHAPTER2.ANALYSISOFFREEVIBRATION,ifthetotaldisplacementv(t)isexpressedasthesumofthestaticdisplacementcausedbytheweightWplustheadditionaldynamicdisplacementasshowninFig.22c,i.e.,thenthespringforceisgivenby,thenwehave,andnotingthatleadsto,notingthatdoesnotvarywithtime,itisevidentthatand,thenwehave,Itdemonstratesthattheequationofmotionexpressedwithreferencetothestaticequilibriumpositionofthedynamicsystemisnotaffectedbygravityforces.Forthisreason,displacementsinallfuturediscussionswillbereferencedfromthestaticequilibriumposition,CHAPTER2.ANALYSISOFFREEVIBRATION,24ANALYSISOFUNDAMPEDFREEVIBRATIONS,Ithasbeenshownintheprecedingsectionsthattheequationofmotionofasimplespringmasssystemwithdampingcanbeexpressedas,Thesolutionoftheaboveequationwillbeobtainedbyconsideringfirstthehomogeneousformwiththerightsidesetequaltozero,i.e.,Followthethetheoryofdifferentialequationswithconstantcoefficients,thesolutionoftheaboveequationcanbeobtainedstepbystep,Egin-equation,Ifthedampingiszero,i.e.,c=0,thenonehas,Frequencyoffreevibrationofundampedsystemmeasuredinradians/second,CHAPTER2.ANALYSISOFFREEVIBRATION,Theinitialconditionsare,Thenonehas,v(t)canberewritten,inwhich,Thesolutionofv(t)presentsasimpleharmonicmotion.,CHAPTER2.ANALYSISOFFREEVIBRATION,FIGURE2-3Undampedfree-vibrationresponse,Angularvelocity,orCircularfrequency,Cyclicfrequency,usuallyreferredtoasthefrequencyofmotion(cycles/sec),Theperiodofmotion(measuredinseconds),Itsreciprocal,CHAPTER2.ANALYSISOFFREEVIBRATION,26ANALYSISOFDAMPEDFREEVIBRATIONS,Themotionequationis,Egin-equation,Critically-Dampedsystems,Iftheradicaltermintheaboveequationissetequaltozero,itisevidentthat,Define,CriticalDamping,CHAPTER2.ANALYSISOFFREEVIBRATION,Usingtheinitialconditions,Notethatthisfreeresponseofacriticallydampedsystemdoesnotincludeoscillationaboutthezerodeectionposition;insteaditsimplyreturnstozeroasymptoticallyinaccordancewiththeexponentialterm.However,asinglezerodisplacementcrossingwouldoccurifthesignsoftheinitialvelocityanddisplacementweredifferentfromeachother.Averyusefuldenitionofthecriticallydampedconditiondescribedaboveisthatitrepresentsthesmallestamountofdampingforwhichnooscillationoccursinthefreevibrationresponse.,FIGURE2-4Free-vibrationresponsewithcriticaldamping.,CHAPTER2.ANALYSISOFFREEVIBRATION,Undercritically-Dampedsystems,Ifdampingislessthancritical,thatis,ifccc,itisapparentthatthequantityundertheradicalsigninEgin-equationisnegative.Toevaluatethefreevibrationresponseinthiscase,itisconvenienttoexpressdampingintermsofadampingratiowhichistheratioofthegivendampingtothecriticalvalue,Thenwehave,DampingRatio,DampedFrequency,CHAPTER2.ANALYSISOFFREEVIBRATION,Alternatively,thisresponsecanbewrittenintheform,Inwhich,Notethatforlowdampingvalueswhicharetypicalofmostpracticalstructures,20%,thefrequencyratioisnearlyequaltounity.TherelationbetweendampingratioandfrequencyratiomaybedepictedgraphicallyasacircleofunitradiusasshowninthefollowingFigure.,FIGURE2-5Relationshipbetweenfrequencyratioanddampingratio.,CHAPTER2.ANALYSISOFFREEVIBRATION,Itisofinteresttonotethattheunderdampedsystemoscillatesabouttheneutralposition,withaconstantcircularfrequency.,FIGURE2-6Free-vibrationresponseofundercritically-dampedsystem.,CHAPTER2.ANALYSISOFFREEVIBRATION,Thetruedampingcharacteristicsoftypicalstructuralsystemsareverycomplexanddifficulttodefine.However,itiscommonpracticetoexpressthedampingofsuchrealsystemsintermsofequivalentviscousdampingratioswhichshowsimilardecayratesunderfreevibrationconditions.,Consideranytwosuccessivepositivepeaks,oneobtainsthesocalledlogarithmicdecrementofdamping,Forlowvaluesofdampingcanbeapproximatedby,SufficientaccuracyisobtainedbyretainingonlythersttwotermsintheTaylorsseriesexpansionontherighthandside,inwhichcase,CHAPTER2.ANALYSISOFFREEVIBRATION,FIGURE2-7Damping-ratiocorrectionfactor,Thisgraphpermitsonetocorrectthedampingratioobtainedbytheapproximatemethod.,Forlightlydampedsystems,greateraccuracyinevaluatingthedampingratiocanbeobtainedbyconsideringresponsepeakswhichareseveralcyclesapart,saymcycles;then,whichcanbesimpliedforlowdampingtoanapproximaterelation,Whendampedfreevibrationsareobservedexperimentally,aconvenientmethodforestimatingthedampingratioistocountthenumberofcyclesrequiredtogivea50percentreductioninamplitude.,CHAPTER2.ANALYSISOFFREEVIBRATION,FIGURE2-8Dampingratiovs.numberofcyclesrequiredtoreducepeakamplitudeby50percent.,ExampleE21.Aonestorybuildingisidealizedasarigidgirdersupportedbyweightlesscolumns,asshowninFig.E21.Inordertoevaluatethedynamicpropertiesofthisstructure,afreevibrationtestismade,inwhichtheroofsystem(rigidgirder)isdisplacedlaterallybyahydraulicjackandthensuddenlyreleased.Duringthejackingoperation,itisobservedthataforceof9,072kgisrequiredtodisplace