哈工大课件机械系统动力学DynamicsofMechanicalSyst

机械系统动力学机电工程学院机械设计系ChenZhaobo(陈照波)Tel:[phone]E-mail:[e-mail]机械楼一楼1020室参考书1.闻邦椿等《机械振动理论及应用》高等教育出版社2.胡海岩《机械振动基础》北京航空航天大学出版社3.师汉民《机械振动系统》华中科技大学出版社4.W.T.Thomson《振动理论及应用》清华大学出版社………考核办法—累加式1.大作业110%2.大作业210%3.平时表现10%4.期终考试70%PrefaceWhatisdynamics？Dynamicsfocusesonunderstandingwhyobjectsmovethewaytheydo.Dynamics=Kinematics+KineticsKinematicsisthestudyofthemotionofpointmassesorrigidbodies.Kineticsisthestudyoftheforceswhichcauseandaffectmotion.Whatismechanicalsystem?StructureMechanismMachineDevelopmentstagesofdynamicsofmechanicalsystems:StaticanalysisKineto-staticanalysisDynamicanalysisElasto-dynamicanalysisWhytostudydynamicsofmechanicalsystem?HigherspeedHigherprecisionMoreflexibleMorecomplicatedResonanceWhenaforcingfrequencyisequaltoanaturalfrequencyPrerequisites:Themostimportantprerequisiteisordinarydifferentialequations.Youshouldbepreparedtoreviewundergraduatedifferentialequationsifnecessary.COURSEGOALS:1.Tobecomeproficientatmodelingvibratingmechanicalsystems.2.ToperformdynamicanalysissuchasfreeandforcedresponseofSDOFandMDOFsystems.3.Tounderstandconceptsofmodalanalysis.4.Tounderstandconceptsinpassiveandactivevibrationcontrolsystems.Chapter1SingledegreeoffreedomsystemsObjectivesRecognizeaSDOFsystemBeabletosolvethefreevibrationequationofaSDOFsystemwithandwithoutdampingUnderstandtheeffectofdampingonthesystemvibrationApplynumericaltoolstoobtainthetimeresponseofSDOFsystemSingledegreeoffreedomsystems•Whenonevariablecandescribethemotionofastructureorasystemofbodies,thenwemaycallthesystema1-Dsystemorasingledegreeoffreedom(SDOF)system.LkmxmstUnloadedSpringBodyinequilibrium(atrest)BodyinmotionAtequilibrium,kst=mgUndampedSpring-MassSystemk(st+x)xmg=mx..FreeBodyDiagramEquationsofmotionxmxFstk-mg:Sumforces:Rearrangetoyieldthefamiliarequationofmotion:PhysicalmodelMathematicalmodelRestoringforceDampingelementSpringDampingforceUndampedFreeVibrationDifferentialequation00(0),(0)xxxv()txtaeSolvingODEProposedsolution:IntoODEyougetthecharacteristicequation:20ttkaeaemGiving:Theproposedsolutionbecomes:-12()kkjtjtmmxtaeaejkm2kmForsimplicity,let’sdefine:nkmGiving:-12()nnjtjtxtaeaeLet’smanipulatethesolutionRecallcos()sin()sin()sin()cos()cos()sin()jejManipulatingthesolution-12()nnjtjtxtaeaeSolutionwehave:Rewriting:121212()(cossin)+(cossin)()cos(-)sinnnnnnnxtatjtatjtaatjaatGiving:12()cossinnnxtAtAtFurthermanipulation2212AAASolutionwehave:Let:Giving:12()cossinnnxtAtAt1A2AA21cos/sin/AAAA()sincoscossinnnxtAtAt()sin()nxtAtDifferentformsofthesolution12()cossinnnxtAtAt()sin()nxtAt-12()nnjtjtxtaeaeNote!Naturalfrequency()sin()nxtAtInthepreviouslyobtainedsolution:ThefrequencyofvibrationisnItdependsonlyonthecharacteristicsofthevibrationsystem.Thatiswhyitiscalledthenaturalfrequencyofvibration.nkmNaturalfrequencynaturalfrequencyfromstaticdeflection.nstgnaturalfrequencyfromenergymethod.Recall:InitialConditionsAmplitude&phasefromICs002210000sin(0)sincos(0)cosSolvingyieldAtannnnnnnxAAvAAvxxvUndampedfreeresponseAddingDampingDampingDampingissomeformoffriction!Insolids,frictionbetweenmoleculesresultindampingInfluids,viscosityistheformofdampingthatismostobservedInthiscourse,wewillusetheviscousdampingmodel;i.e.dampingproportionaltovelocitySpring-mass-dampersystems00()()()0(0)(0)mxtcxtkxtxxxvFromNewton’sLaw:Solution(datesto1743byEuler)2nn()2()()0xtxtxtDividetheequationofmotionbymWherethedampingRatioisgivenby:(dimensionless)2cmkLet()txtae2220tttnnaeaeaeWhichisnowanalgebraicequationin&substituteintoequationofmotion21,2nn1Herethediscriminant,,determinesthenatureoftheroots.2111)Rootsarerepeated&equal.Threepossibilities:Calledcriticallydamped1210200122()Usingtheinitialconditions:nncrnttnccmkmxtaeateaxavxNooscillationoccurs.CriticalDamping22,1112200122002),()()(1)21(1)21nnntttnnnnxteaeaevxavxa212nn221calledover-dampingtwodistinctrealroots:1whereOver-Damping,),212nn31calledunderdampedmotion-MostCommontwocomplexrootsasconjugatepairs:j1MostInterestingCase!2211122220002000()()sin()1()tannnnnjtjtttddnnddnxteaeaeAetvxAxxvx1Under-dampingDampednaturalfrequencyUnder-DampedfreeresponseUnder-DampedfreeresponseLogarithmicdecrementLogarithmicdecrementisnaturallogarithmofratioofanytwosuccesiveamplitudesandisequaltofollowingformi2i1iij2ln11lnj2(when1)AAAADampingestimatesHomework#11.Theamplitudeofvibrationofanundampedsystemismeasuredtobe1mm.thephaseshiftismeasuredtobe2radandthefrequency5rad/sec.Calculatetheinitialconditions.2.Usingtheequation:evaluatetheconstantA1andA2intermsoftheinitialconditions.12()cossinxtAtAt3.Anautomobileismodeledas1000kgmasssupportedbyastiffnessk=400000N/m.Whenitoscillates,themaximumdeflectionis10cm.whenloadedwiththepassengers,themassbecomes1300kg.calculatethechangeinthefrequency,vel