机械振动加州大学UniversityofCaliforniaChapterFreeVibration

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1、-Chapter2FreeVibration·Noexternalexcitation·Objectives:-Equationofmotion-Naturalfrequency·Damping·Principleofconservationofenergy2.1VIBRATIONMODEL·Mass:[kg],[lb×s2/in]·Masslessspring:linearspringwith,k[N/m],[lb/in]·Viscousdamping:,c[N/m/s],[lb/in/s]2.2EQSOFMOTION:NATURALFREQUE

2、NCY·SimpleUndampedSpring-MassSystem:-Assumedtomoveonlyalongtheverticaldirection-Disthedeformationofthespringinthestaticequilibriumposition.-(2.2.1)33/33--(2.2.2)-Withthestaticequilibriumpositionasreferenceforx,theresultantforceonmissimplythespringforceduetothedisplacementx.-(2

3、.2.3)-Eq.(2.2.2)canbewrittenas.-(2.2.6)-Thenaturalperiodoftheoscillation(2.2.7)andthenaturalfrequencyis(2.2.8)-IntermsofD,(2.2.9)33/33[Example2.2.1]A-kgmassissuspendedbyaspringhavingastiffnessof0.1533N/mm.Determineitsnaturalfrequencyincyclespersecond.Determineitsstaticdeflecti

4、on.33/33[Example2.2.2]DeterminethenaturalfrequencyofthemassMontheendofcantileverbeamofnegligiblemassshowninFig.2.2.2.33/33[Example2.2.3]Anautomobilewheelandtirearesuspendedbyasteelrod0.50cmindiameterand2mlong,asshowninFig.2.2.3.Whenthewheelisgivenanangulardisplacementandreleas

5、ed,itmakes10oscillationsin30.2s.Determinethepolarmomentofinertiaofthewheelandtire.33/33[Example2.2.4]Figure2.2.4showsauniformbarpivotedaboutpointOwithspringsofequalstiffnesskateachend.ThebarishorizontalintheequilibriumpositionwithspringforcesP1andP2.Determinetheequationofmotio

6、nanditsnaturalfrequency.33/332.3ENERGYMETHOD·Thedifferentialequationmotionbytheprincipleofconservationofenergy·ThekineticenergyT:-Storedinthemassbyvirtueofitsvelocity·ThepotentialenergyU:-Storedintheformofstrainenergyinelasticdeformationorbyaspring,or-Workdoneinaforcefieldsuch

7、asgravity·Thetotalenergybeingconstant,itsrateofchangeiszero:-(2.3.1)-(2.3.2)·Fornaturalfrequencyofthesystem--Letsubscript‘1’bethetimewhenthemassispassingthroughitsstaticequilibrium.Þ-Letsubscript‘2’bethetimecorrespondingtothemaximumdisplacementofthemass.Þ--Ifthesystemisundergo

8、ingharmonicmotion,T1andU2aremaximumvalues.Þ[Example2.3.1]Dete