机械振动_加州大学_University of California_Chapter 2 Free

-Chapter2FreeVibrationNoexternalexcitationObjectives:EquationofmotionNaturalfrequencyDampingPrincipleofconservationofenergy2.1VIBRATIONMODELMass:[kg],[lbs2/in]Masslessspring:linearspringwithkxF,k[N/m],[lb/in]Viscousdamping:xcf,c[N/m/s],[lb/in/s]2.2EQSOFMOTION:NATURALFREQUENCYSimpleUndampedSpring-MassSystem:Assumedtomoveonlyalongtheverticaldirectionisthedeformationofthespringinthestaticequilibriumposition.mgk(2.2.1)xkmgFxmkxxm(2.2.2)Withthestaticequilibriumpositionasreferenceforx,theresultantforceonmissimplythespringforceduetothedisplacementx.mkn2(2.2.3)Eq.(2.2.2)canbewrittenas02xxn.txtxxnnncos0sin0(2.2.6)Thenaturalperiodoftheoscillationkm2(2.2.7)andthenaturalfrequencyismkfn211(2.2.8)Intermsof,gfn21(2.2.9)[Example2.2.1]A41-kgmassissuspendedbyaspringhavingastiffnessof0.1533N/mm.Determineitsnaturalfrequencyincyclespersecond.Determineitsstaticdeflection.[Example2.2.2]DeterminethenaturalfrequencyofthemassMontheendofcantileverbeamofnegligiblemassshowninFig.2.2.2.[Example2.2.3]Anautomobilewheelandtirearesuspendedbyasteelrod0.50cmindiameterand2mlong,asshowninFig.2.2.3.Whenthewheelisgivenanangulardisplacementandreleased,itmakes10oscillationsin30.2s.Determinethepolarmomentofinertiaofthewheelandtire.[Example2.2.4]Figure2.2.4showsauniformbarpivotedaboutpointOwithspringsofequalstiffnesskateachend.ThebarishorizontalintheequilibriumpositionwithspringforcesP1andP2.Determinetheequationofmotionanditsnaturalfrequency.2.3ENERGYMETHODThedifferentialequationmotionbytheprincipleofconservationofenergyThekineticenergyT:StoredinthemassbyvirtueofitsvelocityThepotentialenergyU:Storedintheformofstrainenergyinelasticdeformationorbyaspring,orWorkdoneinaforcefieldsuchasgravityThetotalenergybeingconstant,itsrateofchangeiszero:0UT(2.3.1)0UTdtd(2.3.2)Fornaturalfrequencyofthesystem2211UTUTLetsubscript‘1’bethetimewhenthemassispassingthroughitsstaticequilibrium.01ULetsubscript‘2’bethetimecorrespondingtothemaximumdisplacementofthemass.02T2100UTIfthesystemisundergoingharmonicmotion,T1andU2aremaximumvalues.maxmaxUT[Example2.3.1]DeterminethenaturalfrequencyofthesystemshowninFig.2.3.1.[Example2.3.2]AcylinderofweightwandradiusrrollswithoutslippingonacylindricalsurfaceofradiusR,asshowninFig.2.3.2.Determineitsdifferentialequationofmotionforsmalloscillationsaboutthelowestpoint.Fornoslipping,wehaveRr.PROBLEMS2.6Theratiok/mofaspring-masssystemisgivenas4.0.Ifthemassisdeflected2cmdown,measuredfromitsequilibriumposition,andgivenanupwardvelocityof8cm/s,determineitsamplitudeandmaximumacceleration.2.8Aconnectingrodweighing21.35Noscillates53timesin1minwhensuspendedasshowninFig.P2.9.Determineitsmomentofinertiaaboutitscenterofgravity,whichislocated0.254mfromthepointofsupport.2.4RAYLEIGHMETHOD:EFFECTIVEMASSTheenergymethodcanbeusedformulti-masssystemsorfordistributedmasssystems,providedthemotionofeverypointinthesystemisknown.ThemotionofthevariousmassescanbeexpressedintermsofthemotionxofsomespecificpointandthesystemissimplyoneofasingleDOF.Thekineticenergycanbewrittenas221xmTeff(2.4.1)wheremeffistheeffectivemassoranequivalentlumpedmassatthespecificpoint.Thenaturalfrequency:effnmkIndistributedmasssystemssuchasspringsandbeams,knowledgeofthedistributionofthevibrationamplitudebecomesnecessary.[Example2.4.1]DeterminetheeffectofthemassofthespringonthenaturalfrequencyofthesysteminFig.2.4.1.[Example2.4.2]Asimplysupportedbeamoftotalmassmbhasaconcentratedmassofthesystematmidspan.Determinetheeffectivemassofthesystematmidspanandfinditsfundamentalfrequency.ThedeflectionundertheloadduetoaconcentrationforcePappliedatmidspanisEIPl483.2.5PRINCIPLEOFVIRTUALWORKIfasysteminequilibriumundertheactionofasetofforcesisgivenavirtualdisplacement,thevirtualworkdonebytheforceswillbezero.Avirtualdisplacementrisanimaginaryinfinitesimalvariationofthecoordinategiveninstantaneously.Thevirtualdisplacementmustbecompatiblewiththeconstraintsofthesystem.VirtualworkWistheworkdonebyalltheactiveforcesinavirtualdisplacement.Theinertiaforcesmustbeincludedasactiveforceswhendynamicproblemsareconsidered.[Example2.5.1]Usingthevirtualworkmethod,determinetheequationofmotionfortherigidbeamofmassMloadedasshowninFig.2.5.1[Example2.5.2]Twosimplependulumsareconnectedtogetherwiththebottommassrestrictedtoverticalmotioninafrictionlessguide,asshowninFig.2.5.2.Becauseonlyonecoordinateisnecessary,itrepresentsaninterconnectedsingleDOFsystem.Usingthevirtualworkmethod,determinetheequationofmotionanditsnaturalfrequency.PROBLEMS2.13Ahydrometerfloat,showninFig.P2.13,isusedtomeasurethespecificgravityofliquids.Themassofthefloatis0.0372kg,andthediameterofthecylindricalsectionprotrudingabovethesurfaceis0.0064m.Determinetheperiodofvibrationwhenthefloatisallowedtobobupanddowninafluidofspecificgravity1.20.2.14Asphericalbuoy3ftindiameterisweightedtofloathalfoutorwater,asshowninFig.P2.14.Thecenterofgravityofthebuoyis8in.belowitsgeometriccenter,andtheperiodofoscillationinrollingmotionis1.3s.Determinethemomentofinertiaofth