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JOURNALOFSOUNDANDVIBRATIONJournalofSoundandVibration3082007548–562amongtheenergystateaftercollisionbetweenabarandabeam,therestitutionstateandthesoundgeneratedwhenatestofaweightandreportedintermittenttypecollisionwithinashorttime2.WiththeadvancementofARTICLEINPRESSwww.elsevier.com/locate/jsvi0022460X/seefrontmatterr2007ElsevierLtd.Allrightsreserved.doi10.1016/j.jsv.2007.05.008C3Correspondingauthor.Emailaddressnarakeyaki.cc.utokai.ac.jpT.Narabayashi.beamandabarcollide.Therefore,itisimportanttoanalyticallystudytheeffectofkeyparametersonvariousenergystates.Thus,asourprincipalresult,thecomplicatedmotionofasysteminvolvinganintermittentcollisionphenomenonnamely,anelasticbarandanelasticbeamsupportedatbothends,wasclarifiedtheoreticallyandexperimentally.r2007ElsevierLtd.Allrightsreserved.1.IntroductionWhenabarcollidesperpendicularlywithabeam,longitudinalvibrationisgeneratedinthebar,whilelateralvibrationisgeneratedinthebeam.Boththebarandbeamhaveadefiniteelasticpropertyhowever,collisionphenomenabetweenabarandabeamareverycomplicated,sincetheirelasticpropertiesinfluenceeachother.Regardingcollision,particularlylateralcollision,ofabeamhavingatypicalsimplestructure,Timoshenkoreportedanalyticalresultsforafallingsteelballmanyyearsago1.MajimaandNishidacarriedoutadropEffectsofkeyparametersonenergydistributionandkineticcharacteristicsincollisionofbarandbeamT.Narabayashia,C3,KazuoShibaikeb,A.Ishizakaa,K.OzakicaDepartmentofPrecisionEngineering,SchoolofEngineering,TokaiUniversity,1117,Kitakaname,Hiratsukasi,Kanagawa,JapanbTOSHIBATECGotanda2172,Shinagawaku,Tokyo,JapancDepartmentofPrimemoverEngineering,SchoolofEngineering,TokaiUniversity,1117,Kitakaname,Hiratsukasi,Kanagawa,JapanAccepted3May2007ThepeerreviewofthisarticlewasorganisedbytheGuestEditorAvailableonline10July2007AbstractElasticcollisionproblemsarethefoundationsofallcollisionproblems,andabarandabeamcanrepresentatypicalandsimplestructure.However,theelasticcollisionofabarandabeamisacomplicatedphenomenon.Thepurposeofthisresearchistoclarifytherelationshipbetweentheprincipalparameterandthekineticcharacteristicsthatinfluencethiscomplicatedelasticcollisionphenomenon,andtoexplaintheseonthebasisofanimpactforcewaveform,theapparentcoefficientofrestitution,andenergydistribution.Weemployedananalysismethodinwhichtheimpactforceassumesanexponentialfunctionforthebasicequationoftheonedimensionalelasticvibrationcorrespondingtothelongitudinalvibrationofthebarandthelateralvibrationofthebeam,andinwhichtheforcesatisfiestheconditionofcontinuityofthedisplacement.Intheexperiment,assumingflatsurfacecontact,wedirectlydetectedcollisionphenomenonelectricallyusingPZTpastedontotheendsurfaceofthecollisionbar.Onthebasisofthisstudy,thekeyparameterintheabovementionedcollisionphenomenonisclarified,andanenergydistributionsituationandtheinfluenceofeachparametercannowbeconsideredforarbitrarycombinationsofthebarandbeam.Furthermore,itisassumedthatthereisacorrelationmeasurementinstruments,thenumberofreportsonmeasurementmethodsusingstraingaugesandpiezoelectricelementsPZTshasbeenincreasing3–5.However,detaileddiscussionhasnotbeencarriedoutonintermittenttypecollisionbetweenabarandabeam.AttheATEM99andICEM12InternationalConference,theauthorsdemonstratedandreportedtheeffectivenessof1theanalyticalmethodbasedonaonedimensionalfundamentalequationoncollisionphenomenabetweenabarandabeamtosatisfytheconditionofcontinuityand2anexperimentalmethodofobservingthecollisionphenomena6andrestitutioncharacteristics7intermsofelectricalcharacteristics.Inthisanalyticalmethod,rotatoryinertiaandsheardeformationofthebeamaredisregardedforsimplicity.Inaddition,assumingimpactforcetobeanexponentialfunctionoftime8,9,theequationsaresolvedsuchthattheconditionofcontinuitybetweenthebarandthebeamaresatisfied,inordertoclarifythecomplicatedmotionphenomenaduringcollision.Inthispaper,abothendssupportedbeam,whichiseasilyrealizedasasupportingcondition,isused,andPZTisARTICLEINPRESST.Narabayashietal./JournalofSoundandVibration3082007548–5625492.2.EquationofmotionandelasticdisplacementofbarincollisionImpactforcePtisappliedtotheendofthebarandthebarissubjectedtoelasticdeformation.Assumingtheelasticdeformationofthebarendtobeu1t,theequationofmotionisq2u1ðtÞqt2¼c21q2u1ðtÞqx2.2l2−l1,00,l2/20,−l2/20,00,lpyx2.1.EquationofmotionforcenterofgravityofbarAssumingimpactforcePtisappliedtotheendofthebarintheleftwarddirectionduetocollision,thecenterofgravityofthebarUtisobtainedbyintegratingundertheinitialconditionofU¼0anddU/dt¼Vatt¼0r1l1A1d2UðtÞdt2¼C0PðtÞ.1pastedontotheendsurfaceofthecollidingbarandthecollisionphenomenaareobservedintermsofelectricalcharacteristics.Acomparativestudywasalsocarriedoutonthekeyparametersintheanalysisandexperiment,whilechangingcollisionposition.Sincetheexperimentalandtheoreticalresultsgenerallywereingoodagreement,weconcludedthatbothourtheoryandexperimentalmethodareeffectiveevenundertheconditionthatcollisionpositionchanges.Furthermore,thechangesofinthekineticcharacteristicsofthebarandbeamarediscussedfromtheviewpointofchangesincollisionposition,restitutioncharacteristicsandenergydistribution.2.SettingproblemandoutlineoffundamentalequationsforanalysisAcaseinwhichanelasticbarlengthl1,crosssectionalareaA1,densityr1,longitudinalelasticcoefficientE1,velocityoflongitudinalwavec1collidedperpendicularlywithanelasticbeaml2,A2,r2,E2andc2atthecenterofabeamoratapositionlpfromthecenterinthesupportedenddirection,ataninitialvelocityofV,isconsideredFig.1.Thetimeatwhichthebarcollideswiththebeamisassumedtobetheoriginoftimet.Fig.1.Barandbeam.ARTICLEINPRESST.Narabayashietal./JournalofSoundandVibration3082007548–562550Theboundaryconditionsareexpressedasatx¼0,E1A1qu1ðtÞqx¼C0PðtÞð0ptpTÞ0ðTotÞ3atx¼C0l1,qu1ðtÞqx¼0.4Theinitialconditionisgivenbyu1ðtÞ¼0qu1ðtÞqt¼0.5Theelasticdeformationofthebarendcanbedeterminedbyexpandingtheequationtoaninfiniteseriesusingthenaturalangularfrequencyandtheeigenfunctionofabothendsfreebar.Thenaturalangularfrequencyandeigenfunctionofabothendsfreebararen1n¼npc1l1Un¼ffiffiffiffiffiffiffiffiffi2l1pcosðnpxl1Þ.6Theelasticdeformationofthebarendisgivenbyu1ðtÞ¼C02r1A1l1X1n¼11n1nZt0PðxÞsinn1nðtC0xÞdx7consideringthedisplacementatx¼0.here,nisthemodenumber.2.3.EquationofmotionanddeflectiondisplacementofbeamattimeofcollisionTheequationofmotionofthebeamisexpressedbyq2u2ðtÞqt2þE2I2r2A2q4u2ðtÞqy4¼0,8whereI2isthemomentofinertiaofareaandu2tthedeflectiondisplacementofthebeam.ThedeflectiondisplacementofthebeamcanbecalculatedbyexpandingtheequationtoaninfiniteseriesusingeigenfunctionWvyforbothendssupportedbeamu2ðtÞ¼1r2A2X1n¼11n2nWnðlrÞ2Zt0PðzÞsinn2nðtC0zÞdz.9Here,assumingthefirstmodenaturalangularfrequencyofthebothendssupportedbeamisn21,n2n¼n2n21holds.Underthiscondition,theeigenfunctionisexpressedasWnðlpÞ¼ffiffiffiffi2l2rsinnplpl2þ12C18C1910andthedeflectiondisplacementofthebeamatthetimeofcollisionisobtained.Here,lp/l2isthecollisionpositionratio,i.e.,theratioofthedistancebetweenthebeamcenterandthecollisionpositiontothelengthbetweenthesupportedends.2.4.ConditionofcontinuityandapparentcoefficientofrestitutionAcaseinwhichabarcollidedwithabothendssupportedbeamatapositionlpfromthecenterofthebeamisconsidered.ThedisplacementofthebarendisthesumofthedisplacementofthecenterofgravityUtobtainedbyintegratingEq.1andtheelasticdisplacementofthebarendu1t,andisgivenbyUðtÞþu1ðtÞ.11Next,thedisplacementofthebeamatthecollisionposition,u2t,whentheimpactforcePtisappliedtothecollisionposition,isassumedtobeequaltoEq.11,sincethedisplacementofthebarendandthatofthebeamatthecollisionpositionareequalduringcollision,andholdsUðtÞþu1ðtÞ¼u2ðtÞ.12AssumingthedisplacementofthecenterofgravityofthebarbeforecollisiontobeU0andthataddedaftercollisiontobeU1,theaboveequationisexpressedEq.13ARTICLEINPRESST.Narabayashietal./JournalofSoundandVibration3082007548–562551U0¼Vt¼u2ðtÞC0ðU1þu1Þ.13BecausethelefthandsideofEq.13changeslinearlywithtime,therighthandsideshouldalsochangesimilarly.TheimpactforcePtisapproximatedusingthefollowingequationsuchthattherighthandsideofEq.13changesaslinearlyaspossiblewithtime,asshowninFig.2b,andbothP1andm1areadjustedsothatthecontinuityconditionofdisplacementissatisfiedPðtÞ¼P1eC0m1t.14OnceEq.14isdetermined,boththeelasticdeformationgeneratedinthebarandthedeflectiondisplacementgeneratedatthecollisionpositionofthebeamcanbeobtainedasinfiniteseriesusingtheeigenfunctionsofthebarandbeam.Here,theequationsofdisplacementofthebarandbeamareobtainedasafunctionoft,inwhichthemassratioandnaturalfrequencyratioofthebarandbeamareusedasparametersandm1asanunknown.Byconsideringthecollisionpositionratiocontainedintheeigenfunctionofthebeam,itbecomespossibletocalculatethemotionofthebarandthebeam.Basedontheequationforthedistancebetweenthebarandthebeam¼u2ðtÞC0fUðtÞþu1ðtÞg40,itisfoundthatthebarendseparatesfromthebeam.Collisionsceaseatthispoint,andthebarandbeamwillnotcollideagainunlessthedistancebetweenthetwobecomes0beforeapproximately3/4oftheperiodofthebeam.Inthiscase,thebarandthebeamcollideonlyonce.Incontrast,whentheycollidentimes,theycomeintocontactagainattimeTnatwhichthedistancebetweenthebarandthebeamis0orless.AssuminganimpactforceexpressedbyPðtÞ¼PneC0mnðtC0TnÞ15isappliedtothebar,Pnandmnaredeterminedsuchthattheconditionofcontinuityissatisfied,andthedisplacementiscalculatedforthenthorlatercollisions.Bythisprocedure,variousimpactforcepatternsareobtainedundervariouscombinationsofthethreeparameters,i.e.,1thenaturalfrequencyratioofthebartothebeamn11/n21,2themassratioofthebartothebeamm1/m21and2refertothebarandbeam,respectively,and3thecollisionpositionratiolp/l2wherelpthedistancebetweenthebeamcenterandthecollisionpositionandl2thelengthbetweenthesupportedends.Whenthebarandthebeamcollidentimes,b1n11tdRightsideofeq.13PtP1e−afii98391tTimeP1ImpactForce0T0Leftsideofeq.1302πFig.2.Approximationofimpactforcebyadjustingm1/n11andcontinuityconditionofdisplacement8.
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