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Bell clapper impact dynamics and the voicing of a cari Bell clapper impact dynamics and the voicing of a cari

Bell clapper impact dynamics and the voicing of a cari - PDF document

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Bell clapper impact dynamics and the voicing of a cari - PPT Presentation

H Fletcher a Research School of Physical Sciences and Engineering Australian National University Canberra 0200 Australia W T McGee 61 Calder Crescent Holder 2611 Australia A Z Tarnopolsky School of Aerospace and Mechanical Engineering University Col ID: 67398

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BellclapperimpactdynamicsandthevoicingofacarillonN.H.FletcherResearchSchoolofPhysicalSciencesandEngineering,AustralianNationalUniversity,Canberra0200,W.T.McGee61CalderCrescent,Holder2611,AustraliaA.Z.TarnopolskySchoolofAerospaceandMechanicalEngineering,UniversityCollege,TheUniversityofNewSouthWales,AustralianDefenceForceAcademy,Canberra2600,AustraliaReceived31August2001;revised22November2001;accepted13December2001 Electronicmail:neville.¯etcher@anu.edu.auJ.Acoust.Soc.Am.(3),March20020001-4966/2002/111(3)/1437/8/$19.002002AcousticalSocietyofAmerica instrumenthasreceived,theimpactsiteonthebellwallhasbeenalmost¯attenedoveranellipticalareasomewhatlargerthanthe¯atdevelopedontheclapper,thisexaggerationbe-ingduetosmallvariationsinthepositionoftheimpactsitebecauseofmechanicalslackinthemechanism.Themechanismitselfconsistsofapear-shapedclapperofcastiron,mountedonahingedrod.Theoriginalmecha-nismwasmodi®edsomeyearsago,andtheweightofeachclapperisnowbalancedbyaspiralspring.Theclapperisdrawnintoimpactwiththebellbya¯exiblesteelcabletypicallyabout30cmlongthatconnectsittotherods,le-vers,andcablesleadingupfromtheclavier.Itisnotourpurposeheretosurveywhatisknownoftheacousticsofbellsorofcarillondesign.Theinterestedreaderwill®ndappropriatematerialinabookbyFletcherandandinacollectionofpaperseditedbyRossing,includedamongwhichisanextensivemonographbyBigelow.II.MEASUREMENTSToexaminetheeffectsofre-voicingtheclappers,threebellswereselectedforstudy:Nos.9(E),29(C),and47),effectivelyspanningthecompassoftheinstrument.Theresultsforthemidrangebell(C)willbereportedinsomedetailandthencomparableresultsfortheotherbellsBell29isabout43cmindiameterand36cminheighttoitsshoulderandiscastfrombronze.Itsmassisabout59kg.Thesphericalpartofitspear-shapedcast-ironclapperhasadiameterofabout95mm.Theestimatedmassoftheclap-perisabout5kg.Theimpactdamage¯atontheclapperwasapproximatelyelliptical,withdimensions17mm14mm,whilethatontheinsideofthebellwasslightlylargerat2014mm.Thegroovesonthebellsurfacehadbeenplas-ticallydeformedsothatthesurfacewasessentially¯at,thoughtracesofthegroovepatterndidremain.There-voicingitselfwascarriedoutbyhandandeye,usingapoweredanglegrinderandthena®lefor®nishing.Theresultantclappersurfacewasofapproximatelyuniformsphericalcurvaturebuthadsome®lemarksacrossitofap-proximatedepth0.1mm.Sincetheplasticwashersusedinthebellmountisolateditelectricallyfromtheclapper,itwaspossibletoexaminethecontacttimesimplybyusingtheimpactasaswitchinasimpleelectricalcircuitconsistingofa9Vcellanda1000resistorinseriesanddisplayingthevoltageacrosstheresis-toronastorageoscilloscope.WhenasimpleimpactwasachievedÐapointtobeconsideredagainlaterÐitsdurationwasabout0.6ms.Thesameexperimentalarrangementwasusedtomeasurecontacttimeafterre-voicing,thetypicaldurationthenbeingabout1.0ms,thoughbecomingaslongas1.3msforparticularlygentlestrikes.Aninterestingfeatureofthisexperiment,whichwilllaterbeseentobesigni®cant,isthat,whileimpactswidelyseparatedintimesothatthebellhasceasedvibratingwerequitecleanandreproducible,impactsinratherquicksucces-sionwithonlyafewsecondsseparationdisplayedjitterintheformofrepeatedverybriefcontacts.Thisjittercanbeattributedtotheeffectsofresidualvibrationofthebell.Theradiatedsoundsignalwasrecordedusingamicro-phoneandtaperecorderlocatedinthebellchamberabout4mfromthebell.Themicrophonepositionwasthesameforrecordingsbeforeandafterre-voicingsothatcomparisonwaspossibleirrespectiveofresonancesorothereffectswithinthebellchamber.Recordedsignalsfromanumberofbellstrokesbeforeandafterre-voicingwerelatersubjecttofrequencyanalysisusingafastFouriertransformtechnique,thesamplebeingselectedtocommence40msaftertheini-tiatingimpulse.Atypicalpairofspectraforbell29isshowninFig.1.Thespectralenvelopecanbeadequatelydescribedintermsofthetwostraightlinesshown,andthefrequencyattheintersectionoftheselinesgivesameasureofthe``bright-ness''ofthesound.Thisfrequencyisabout4kHzbeforere-voicingandabout2kHzafterwards.Thedistributionof TABLEI.Experimentalresults.¯atdiam)120950200181.53.5900Hz600Hz29(C)435995150.61.24kHz1.8kHz47(F207.75670.60.84kHz4kHzUncertainbecauseofwindnoiseandotherproblems. FIG.1.Soundspectrumofbell29beforere-voicing;soundspectrumofthesamebellafterre-voicing.1438J.Acoust.Soc.Am.,Vol.111,No.3,March2002Fletcheretal.:Bellclapperdynamics modeswithinthespectralenvelopeisdeterminedbythede-signandtuningofthebellsandneednotbeconsideredherethoughitis,ofcourse,theprimaryconcernofthede-signerandbellfounder.Similaranalyseswerecarriedoutfortheothertwobells,andthecompleteresultsaresummarizedinTableI.Theclapper¯atdiameterquotedisthegeometricmeanofthetwodiametersofthecontactellipse.Itcanbeseenthatthebe-haviorscalesquiteconsistentlyforallthreebells.III.THEORYTwoaspectsoftheimpactbehaviorrequiretheoreticalinvestigation.The®rstistheimpactitself,includingcontacttime,behaviorofthecontactforce,andthewayinwhichthesequantitiesdependuponthesizeofthedamage¯atanduponparameterssuchasimpactspeed.Thesecondisthevibrationalexcitationandsoundoutputtobeexpectedfromthistypeofimpact.Thewholesubjectoftheimpactofonebodyuponan-otherisofgreatpracticalimportance,andhasbeenthesub-jectofmuchtheoreticalandexperimentalinvestigation,dat-ingbacktotheclassicworkofHertzinthenineteenthcentury.AnexcellentexpositionwithextensivereferenceshasbeengivenbyGoldsmith.Arigoroustheoryofimpactinthepresentcaseneces-sarilyinvolvesthecomplexelasticdeformationofthebellandclapperduringtheimpactandalsothevibrationandbodilymotionofthebellduringthattime.Suchananalysiswouldbeextremelycomplicatedandisinanycaseunneces-saryforthepresentproblem.Theanalysistobepresentedherethereforeincludesonlyenoughdetailtoestablishtheeffectsofthemajorphysicalparametersandtogivearea-sonableapproximationtothenumericalvaluesinvolved.A.HertzianimpacttheoryMostreasonablysimpletreatmentsofimpactarebaseduponHertz'sassumptionthattheimpacttakesplacesuf®-cientlyslowlythatastatictreatmentoftheelasticdistortionsisadequate.Histheorycoversthecaseofimpactofspheresonspheresandofspheresonplaneobjects,butcouldrea-sonablybeextendedtothemorecomplexgeometryofaspherewithacontact¯at.FollowingthetreatmentinChapter4ofGoldsmith,we®ndthat,forasphereofmassimpactingonaverymassiveplateataspeed,theHertztheorygivesacontacttime ~1!whered1512m12 pE1;d2512m22 arethePoisson'sratiosandtheYoung'smoduliofthesphereandplate,respectively.Duringtheim-pact,theforcefollowsacurvelike ,whichapplyonlyforanundamagedclap-per,serveasacheckfortheextendedbutlessaccuratetheorytobedevelopedinSec.IIIB.Thesequasistaticequationscanapplyonlyinthelimitofveryslowimpactswhichallowanyelasticvibrationsgener-atedtodissipatecompletelyinanegligiblefractionoftheimpacttime.Formorerapidimpactsitisnecessarytocon-sidertheeffectsofwavegenerationandre¯ectionfromtheboundariesoftheimpactingobjects.Thisisdif®cultenoughinthecaseofasmallsphereimpactingonaclampedrodandbecomesimpossiblycomplicatedformorecomplexge-ometries.Forveryrapidimpacts,ofthetypeexempli®edbyimpactofasmallsphereonalargeobjectsuchasanex-tendedplate,thesituationissomewhatsimpli®ed,sincethedeformationofthespherecanbetakentobequasistatic,andthegenerationofelasticwavesintheplatecanbeapproxi-matedbyasimplewaveimpedance,providedthereisnottimeduringtheimpactforanyre¯ectionsfromtheplateboundariestoreturntotheimpactsite.Thelatterassumptionisareasonableapproximationforthecaseofclapperimpactonabell,andwillbethebasisofthetheorydevelopedinSec.IIIB.B.AnextendedimpacttheoryItisnecessarynowtoformulateasimpletheoryfromwhichpredictionscanbemadeoftheimpactbehaviorofaspherewithaninitial¯atuponthesurfaceofabell.ThegeometryassumedduringtheimpactisasshowninFig.2.Thecoordinateofthecenteroftheclapper,assumedspheri-calforsimplicity,relativetoa®xedplaneis,andthatofthesurfaceofthebell,assumedplane,is,thecoordinateofthissurfacebeforetheimpactbeing.Duringtheim- FIG.2.Assumedgeometryofthebellandclapperduringimpact.Thebellisdisplacedfromitsinitialpositionbecauseofelasticwavegeneration,whiletheclapperdeformsthesurfaceelasticallyandisitselfdeformed,thediameterofthecontact¯atincreasingfromitsinitialvalueJ.Acoust.Soc.Am.,Vol.111,No.3,March2002Fletcheretal.:Bellclapperdynamics pact,theclapperpenetratesthebellsurfacebyadistanceandthedepthofthe¯atbelowtheextensionofthesphericalsurfaceincreasesfrom.Atthesametimethediameterofthecontact¯atincreasesfromitsinitialmagni-toanewtime-dependentmagnitude.Itisassumedthatalldistortionsareelasticratherthanplastic,atleastdur-ingthecourseofasingleimpactevent.The®rstapproximationtobemadeistoassumethatthebellislargeenoughandtheimpactshortenoughthattherearenosigni®cantre¯ectionsduringthecontacttime.Sincethebellhasaxialsymmetry,thisassumptionisequivalenttotheprovisothatthereisnotenoughtimeduringtheimpactforawavetotravelroundthecircumferenceofthebellandreturntotheimpactpointbeforetheclapperrebounds,amatterthatwillbediscussedlater.Thelarge-scalebehaviorofthebellduringimpactcannowbeapproximatedby istheforceimposedbytheclapperimpact,isthecoordinateofthebellsurfaceasshowninFig.2,andisthecharacteristicimpedancefor¯exuralwavesonthebell.Thetotalmassofthebellissuf®cientlymuchlargerthanthatoftheclapperthatitisreasonabletoignoreitsbodilydisplace-ment.Exactevaluationofisdif®cult,becauseofthecom-plexgeometryofthebell,butanapproximationcanbemadebyassumingittobehavelikeanin®nite¯atplate.Ananaly-sishasbeengivenbySkudrzyk,whoshowsthat,forthecaseofaplateorironorsoftsteel,thecharacteristicimped-anceiskgsisthethicknessoftheplateinmillimeters.Becausetheelasticparametersofbellbronzeandcastironareverysimilar,weexpectthisresulttobeagoodapproximationforthebellmaterial.Forothermaterials,isproportionalto,whereistheYoung'smodulusandthematerialdensity.Since,however,theclapperstrikesthebellclosetoafreeedge,theeffectiveimpedanceisexpectedtobeabout0.5.Thethicknessofthebellwallisnotuniform,butisintherange20±30mmforbell29.Adoptingthisapproximationasaguide,therefore,weexpectthatisintherange2±5kgsforthisbell.Insomeofthecal-culationstofollow,avalueof3kgswillbeas-ReferringnowtoFig.2,ifisthediameteroftheimpactdamage¯at,thenthiscanberelatedtotheparameter istheradiusoftheclapperball,ormorepreciselyitsradiusofcurvature,andtheapproximationisgoodprovided.Similarly,theparameter,whichistheextentofthecompression,isrelatedtothediameterofthecontact¯atduringtheimpactandtotheinitialgeometricparameter ,asisthecaseinpractice.Calculationoftheelasticstrainintheclapperisverycomplicated,butitisadequateforourpresentpurposestoassumethatitisdis-tributedoveravolumeofdiameteranddepth,whereisrathergreaterthanunity.Thetotalforceontheim-pactsurfacethenapproximatelysatis®es dh5K1 dSpd2 4DE1 12m125K1d aretheYoung'smodulusandPoisson'sratio,respectively,oftheclappermaterialandisde®nedbyEq..Theuncertaintyimplicitinthisresultisencapsu-latedinthefactor.ItsmagnitudewillbederivedlaterbycomparisonwiththemoreaccurateHertztheoryforacasewheretheyoverlap.SubstitutingEq.intoEq.integratinggives Theelasticdeformationofthebellsurfacepresentsasimilarlydif®cultproblembut,tothesamedegreeofap-proximation,andbythesameargumentsusedtotreattheclapper,wecanwrite isanotherconstantoforderunity.Itistobeex-pectedthat.FromEqs. K2d1~h11h0!S12K1d2 Thereisalsoafurthergeometricalconnectionbetweenthecoordinates,namelyNotethat,inalltheseequations,,andhavebeende®nedsoastobealwayspositive,andmustalsobepositivesincethereisnoadhesionatthecontact.Finally,themotionofthecenterofmassoftheclapper 5F wherethemassoftheclapperisrathergreaterthanthesphericalmassbecauseofitspearshapeandattachedmecha-nism.Equationsnowde®nethemotionofthewholesystem.C.NumericalsolutionForanumericalsolution,Eqs.arewritteninthe®rst-orderform 1440J.Acoust.Soc.Am.,Vol.111,No.3,March2002Fletcheretal.:Bellclapperdynamics dt5F m,~16! 52 withboundaryconditions0,and0,whereisanarbitraryconstantandisde®nedbythediameteroftheinitial¯atbyEq..Itisnecessarytowritetheforceexclusivelyintermsofthecoordinates,andtodothismustbeeliminatedfromEqs.andtheresultingequationsolvednumericallyfor.Thethreeequationsthenserveasthebasisforthecomputation.Inthecarillonunderstudy,thebellsaremadeofcastbronzeforwhichPaand0.36,whiletheclappersarecastironwithThesevaluesaresuf®cientlysimilarthatitisrea-sonabletoassumeequality,sothat.Inaddition,sinceexactvaluesoftheconstantsinEqs.areunknown,itmakessensetoassumethatThesesimplifyingassumptionsthenleadtotheresultsothat,fromEq. Theseassumptionsleadtoagreatsimpli®cationinthesub-sequentcalculations.Theonemajorformaluncertaintyintheextendedtheoryisencapsulatedinthevalueoftheelasticparameter,whenallthatisknownisthatitisoforderunity.Theaccompany-inguncertaintycan,however,beresolvedbycomparingthecomputedresultforcontacttimeinthecase0andwiththeknownaccurateresultsoftheHertziantreat-mentforthiscaseasgivenbyEq..Thiscomparisoncanbemadebyassumingthatinthesimplemodelandtoobtainagreement,andleadstotheresult5.Thisvalueisthenusedinsubsequentevaluations.Thetheorycannowbeusedtoinvestigatetheeffectofimpact¯atdiameteruponimpacttime.Becauseofuncer-taintyintheeffectivevalueofforthebell,andtoillustratetheimportanceofthisparameter,arangeofvaluesisstudied.Onceagain,thecalculationsallrefertobell29.Figure3showsthecalculatedbehavioroftheforcebe-tweentheclapperandthebellforanimpactvelocityof1andthreeassumedvaluesoftheimpedanceparameter.Forthecasetheforcecurveissymmetricalaboutitsmidpoint,inagreementwiththeHertzianresult,whilefora®nitevalueof,asinreality,theforcecurvehasalongertail.Thisasymmetryincreasesastheimpedancelowervalues.Itisencouragingtonotethatthesesamefea-turesareseenintheshapesofcurvescalculatedfortheim-pactofasphereonaplateofin®nitesize,asshowninFig.71ofGoldsmith.Becauseoftheasymmetryofthecurvesitishelpfultode®netheimpacttimetobethetimefrominitialcontacttothemaximumintheforcecurve.ForaHertzianimpact()clearly/2,whileislessthanhalfthecontacttimefor®nitevaluesofFigure4showsthecalculatedimpacttime,asde-rivedfromthecurvesofFig.3,asafunctionofimpact¯at FIG.3.Calculatedtimeevolutionofforcebetweentheclapperandthebellforanimpactvelocityof1msandthreedifferentimpedancevalues.Thediameteroftheimpactdamage¯atisshowninmillimetersasaparameter.Inthecasekgs,theclapperremainsincontactwiththebellsurface. FIG.4.Calculatedimpacttimeforasphericalclapperofmass5kgonabellsurfacewitheffectiveimpedance,asafunctionofthediameterofthedamage¯atontheclappersurface.J.Acoust.Soc.Am.,Vol.111,No.3,March2002Fletcheretal.:Bellclapperdynamics diameterforthreevaluesoftheimpedance,oneofwhichisthein®niteimpedanceimplicitlyassumedintheHertzmodel.Thebehaviorforkgsisnotverydifferentfromthatfor,thoughimpacttimesarealittleshorterandtotalcontacttimesalittlelonger.Forvaluessmallerthanabout5kgs,however,andspeci®callyforthevalue3kgsconsideredappropriateforthisbell,theclapperactuallycomestorestagainstthebellsurfaceaftertheimpact.Theimportanceofthispredictionwillbeexam-inedinSec.IIID.Finally,Fig.5showstheeffectofclapperimpactveloc-ityonimpacttimeforarangeofvaluesofimpact¯atdiameter.Asalreadydiscussed,itisassumedthatkgsforthisbell,sothattheclapperremainsincontactwiththebellafterimpactandthecontacttimein®nite.Fortheundamagedclapper,thereisasigni®cantshorteningofimpacttimeastheimpactvelocityisincreased,butoncea¯atofmoderatediameterdevelopsindependentofimpactvelocity.Thisfeatureisanotherrea-sonfortheclappervoicingprogram.D.OtherbellsAsmentionedearlier,similarmeasurementsandre-voicingprocedureswerecarriedoutontwootherbells,onemuchlargerandonemuchsmallerthanbell29.Themea-surementsonallthreebellsareshowninTableI.HerethetotalcontacttimefromelectricalmeasurementsandthespectralturnoverfrequencyasshowninFig.6.There-sultsforthelargerbell,No.9,arebroadlyconsistentwiththoseanalyzedindetail,bell29.Thoseforbell47arelessdecisive,sincethe¯atwassmallerandthespectralrecord-ingswereobscuredbywindnoiseandotherproblems.Itwouldbeofinteresttoundertakeacompletestudyofspec-tralbalanceacrossthewholecarillon,butthiswasnotat-IV.CONTACTTIMEDISCREPANCYThereisoneaspectinwhichthetheoryappearstobeinsubstantialdisagreementwithexperiment,andthatistheac-tualmagnitudeofthecontacttimeduringimpact.Theex-perimentalcontacttimeforbell29rangesfrom0.6msforthedamagedclapperto1.0±1.2mswhenithasbeenre-voiced.Ifthebellimpedanceis10kgsorhigher,sothattheclapperreboundsaftercontact,thenthecalculatedcontacttimesareabout0.15and0.3ms,respectivelyÐaboutone-quarterofthemeasuredtimes.Ifthebellimpedanceissubstantiallylower,asanalysisindicates,thenthecalculatedcontacttimesbecomein®nite.Whileitmightbepossibleto®ndaprecisevalueofgivingapproximateagreementwithexperimentforcontacttimes,thisbalancewouldbeunreal-isticallysensitive.Onemightbetemptedtoquestionthereliabilityoftheelectricalcontactmeasurements.Otherinvestigators,how-ever,haveusedalmostidenticaltechniquestomeasurecon-tacttimesandhaveobtainedresultsthatagreeingeneralmagnitudewithourown.Joneslongagomademeasure-mentsonalargebellinplaceinachimeandfoundacontacttimeofabout0.8ms.MorerecentlyGruetal.measuredcontacttimesintherange0.6±1.5msinalabora-torysetup,inapproximateagreementwithourresults,andalsofoundadecreaseincontacttimewithincreasingimpactvelocity.Anexplanationofthisdiscrepancyisactuallysimple,andissuggestedbytheobservedinstabilityofthecontactmeasurementswhenthebellisalreadyvibrating.Supposethattheimpedancethatthebellpresentstotheclapperislessthanabout5kgs,asisappropriatefromtheplateanalogydiscussedpreviously.Thenafterimpacttheclapperwillremainincontactwiththebellsurfaceuntilitis FIG.5.Calculatedvariationoftheimpacttimeasafunctionofimpactdamage¯atdiameterforarangeofimpactspeeds,showninmeterspersecondasaparameter.Itisassumedthatkgs,atwhichim-pedancevaluetheclappercomestorestagainstthebellafterimpact. FIG.6.Fouriertransformoftheimpactforce)foraveryhighbell,givingaHertzianimpactasinFig.3alowbellim-pedancegivinganasymmetricimpactandin®nitecontacttimeasinFig..Thetimevariationoftheforceisindicatedintheinsetineachcase.1442J.Acoust.Soc.Am.,Vol.111,No.3,March2002Fletcheretal.:Bellclapperdynamics dislodgedbysomemechanicalimpulsefromthebell.FromFig.3,theclapperimpacthasasharprisetimebyaslowdecay,andthuslaunchesatransversepulseofthisshapeinbothdirectionsaroundthesound-bowofthebell.Thesepulsesreturntotheclapperpositionaftermakingacompletecircuitofthebell,duringwhichtheyaredistortedbydispersioneffects,anddislodgetheclapperfromcontactwiththesurfaceastheyreturntotheimpactsite.Theclapperthenmovesawayfromthebellsurfaceandisfurtheraccel-eratedbythereturnspringoftheaction.Iftheclapperhasanimpact¯at,thentheimpacttimeisshorter,andthegen-eratedpulsewave-packetishigherinfrequency.Thephasevelocityof¯exuralwaves,however,isproportionaltothesquarerootofthefrequency,andthegroupvelocityistwicethephasevelocity,sothatthishigher-frequencypulsetravelsaroundthebellmorequicklyanddislodgestheclapperafterasmallerdelay.Thesestatementsaboutphaseandgroupvelocitiesfol-lowfromtheequationforbendingwavepropagationonaplate,whichis isthenormaldisplacementandistheplatebendingstiffnessdividedbyitsmassperunitarea.Ifthewavehasthe),thenEq.showsthatand,sincethephasevelocityis,thisgives.Thegroupvelocity,however,isgivenbywhichisjust2Tocon®rmthisexplanationoflongcontacttimes,itisnecessarytointroducequantitativeconsiderations.FromtheanalysisofSkudrzyk,thephasevelocityof¯exuralwavesoffrequencyonanin®niteplateis istheplatemassperunitareaandisthecharac-teristicimpedancegivenbyEq..Forbell29,thewallthicknessisabout20mmsothatkgs.At2kHz,whichisthefrequencyofmaximumexcitationasjudgedfromthesoundspectrumforthevoicedclapper,Eq.givesaphasevelocityofabout540msandagroupvelocityof1080ms.Thisgivesapulsetransittimeofabout1.2ms,infairlygoodagreementwiththeexperimentalcontacttimeof1.0±1.2msforthevoicedclapper.Fortheunvoicedclapper,theexcitationmaximumisatabout4kHz,givingapulsereturntimeofabout0.8ms,whichiscompa-rablewith,thoughalittlelongerthan,theexperimentalvalueof0.6ms.Itispossible,however,toestimatethesepropagationvelocitiesquiteindependentlyofthevalidityofthein®nite-plateimpedanceapproximation.Inthefundamental``hum''mode,thebellvibrateswithtwonodaldiameters,andthemodecanbeconsideredtobeasuperpositionoftwocoun-terpropagatingwaves,eachwithtwowavelengthsaroundthecircumferenceofthesound-bow.Sincethebellcircumfer-enceisabout140cmatthestrikeposition,thisgivesaphasevelocityofabout350msatthisfrequency,andhenceagroupvelocityofabout1400msat2kHz,inmoderatelygoodagreementwiththeabove-givenestimate.Thecontacttimespredictedfromthisestimateareabout0.9msforthevoicedclapperand0.6msfortheunvoicedclapper,againinfairlygoodagreementwithexperiment.V.SOUNDGENERATIONSoundgenerationbythebellisacombinationoftheexcitationeventandsubsequentsoundradiation.Sincethebellitselfisnotmodi®edduringthevoicingprocess,itsmodefrequencies,impedances,andradiationef®cienciesre-mainunaltered,anditisnecessaryonlytoconsidertheex-citationevent.Theabove-developedtheorypredictsanim-pactforce)withatimeevolutionthatdependsupontheofthebellwallasshowninFig.3.ThespectralshapeofthisimpacteventisgivenbyitsFouriertransform,whichiswrittenmostconveniently,towithinaconstantfac-tor,astdttdtTheshapeofthisfunction,plottedinthesamewayastheexperimentalresultsofFig.1,isshowninFig.6fortwodifferentcases:onewithlarge,givinganearlyHertzianimpact,andonewithsmall,givingalingeringcontact.Inbothcasestheenvelopeofthespectrumconsistsoftwostraightlinesintersectingatafrequencycloseto0.2/istheimpacttimeaspreviouslyde®nedandindi-catedintheinsets.Whenallowanceismadeforaradiationef®ciencythatincreasessmoothlywithincreasingfrequency,thereisclosequalitativesimilaritybetweenthespectralen-velopeofFig.6andthatofthebellsoundinFig.1,forbothunvoicedandvoicedclappers.Comparisonwithexperiment.Inmakingcomparisonsbetweentheoreticalpredictionsandexperimentalmeasure-ments,itmustbeborneinmindthatthiswasa®eldexperi-mentratherthanalaboratoryexperiment.Thismeantthatseveralimportantparametershadtobeestimatedratherthanmeasuredinacontrolledway.Withthisproviso,thepredic-tionsofthetheoryareinacceptablygoodagreementwithexperimentalresults.Theturnoverfrequenciespredictedforthebell29onthebasisofthetheorysetoutaboveandtheimpacttimesgiveninFig.4,assumingthatkgs,areabout8kHzfortheunvoicedclapperand2kHzforthevoicedclap-per,comparedwiththemeasuredvaluesof4and1.8kHz,respectively.Whilefarfromexact,thisagreementshouldbeconsideredassatisfactoryina®eldexperimentratherthanalaboratoryexperiment,particularlysincethebellimpedance,theclappermass,andtheimpactvelocityhaveallbeenestimatedratherthanmeasured.Alloftheseparametersaffecttheimpacttimesigni®cantly,asindi-catedinFigs.4and5.Choiceofaratherlargervalueforwouldimprovetheagreement.Themeasuredbehavioroftheotherbellsinvestigatedscalesinjustabouttheexpectedmanner,asindicatedinTableI,thoughtheexperimentaldataforbell47isinconclu-J.Acoust.Soc.Am.,Vol.111,No.3,March2002Fletcheretal.:Bellclapperdynamics sive.Forbell9,thelinearsizescalingisbyafactor2.8,sothat,byEq.,thescaledvalueofshouldbeabout2.4kgs.Thescalingoftheclapperdiameterindicatesaofabout46kg.Insertingthesevaluesinthetheory,togetherwiththeimpact¯atdiameterfromTableI,givesvaluesforthedamagedandundamagedclapperof0.1and0.23ms,respectively.Theresultingpredictedturnoverfre-quenciesare2000and870Hz,respectively,comparedwiththemeasuredvaluesof900and600Hz.Thisagreementissimilartothatforbell29andwouldagainbeimprovedbychoiceofalargervalueforItisnotappropriateinsucha®eldexperimenttoseektore®netheagreementbetweentheoryandmeasurementbyvaryingtheparameters.Bearinginmindthemanyapproxi-mationsinvolved,theagreementisfairlysatisfactory.VI.CONCLUSIONSThisstudyofthedynamicsofbellclapperimpacthaselucidatedmanyfeaturesoftheproblemthatwerepresum-ablysolvedbythedesignersandbuildersofhistoricbellsandcarillonsonthebasisofexperiencefoundedupontrialanderror.Itissomewhatsurprisingthatthebellclapperac-tuallycomestorestagainstthebellsurfaceuntilitispushedawaybyareturningvibrationpulse.Onre¯ection,however,suchadesignresultsinthetransferofmaximumenergyfromtheclappertothebell,andthustheachievementoftheloud-estpossiblesoundforagivenimpactvelocity.Theparameterthatprincipallydeterminesthisbehavior,assumingthebelldesigntobe®xed,istheclappermassÐtoolightaclapperwillbounce,aswellashavingratherlittleenergytotransfer.Bellfoundershavelongrecognizedthatthesoundofacarillonchangesratherrapidlyoverthe®rstyearorsoofitsuse,andthenbecomesmorestable.Thischangeisassociatedwiththedevelopmentofimpact¯atsontheclappers,andinitiallyproceedsquiterapidlyuntiltheclappers,andtheinsidesurfaceofthebell,becomemoreresistanttoplasticdeformationthroughworkhardening.Avoicingprocessthatrestorestheshapeoftheclappersbringsthesoundclosertoitsoriginalstate,andthework-hardenedclappersarenowmoreresistanttoplasticdeformation.Theworkhardeningitself,whichhasalmostnoeffectontheelasticmoduli,hasnodirectin¯uenceonimpactdynamicsorsound.Generallynothingisdoneaboutthedeformationofthebellsurface,thoughsomecarillonsdoallowa180Érotationofthebellstotheotherequivalentimpactposition,aroutinethatultimatelyhalvestherateofbelldeformation.Itiscertainthatacarillonsoundsdifferenttothemusi-calearandtothecarillonneurafterithasbeenre-voiced.Itisinterestingtodeterminewhatarethequalitiesofare-voicedcarillonthatmakeitanimprovementonitsstatebeforetheTheeffectsofre-voicingaretwo:theimpacttimeislengthenedsothattheturnoverfrequencyisreduced,givingamore``mellow''sound;andtheimpacttimebecomesmoredependentupontheimpactvelocity,sothat``soft''noteshaverelativelylittleharmonicdevelopmentand``loud''notesarebothlouderand``brighter,''asinmostothermusicalinstruments.Theseconsiderationsarethosethatprincipallymotivatethecarilloncuratortoundertaketheop-eration.Inthis,acarillonisratherdifferentfromanisolatedbellorfromapealofbells,sincethemusicitiscalledupontoproduceinvolvesharmonies,ratherthansimplymelodicExperiencewithanalysisofthere-voicingofthispar-ticularcarillonshowsthatdetailsoftheclapperimpactareimportant.Theanalysisalsopointsuptheimportanceofmatchingeachclappertoitsbellinordertogiveawell-balancedtonequalityandloudnessacrossthecompass.Itisthesethings,aswellasthedesignofthebellsthemselves,thatdistinguishareally®necarillon.X.BoutillonandB.David,``Assessingtuninganddampingofhistoricalcarillonbellsandtheirchangesthroughrestoration,''Appl.Acoust.N.H.FletcherandT.D.Rossing,ThePhysicsofMusicalInstruments,2ndSpringer,NewYork,1998,Chap.21.AcousticsofBells,editedbyT.D.RossingVanNostrandReinhold,NewYork,1984A.L.Bigelow,TheAcousticallyBalancedCarillonSchoolofEngineer-ing,PrincetonUniversity,Princeton,NJ,1961LargelyreprintedinRef.W.Goldsmith,Arnold,London,1960E.Skudrzyk,SimpleandComplexVibratorySystemsPennsylvaniaStateUniversityPress,UniversityPark,PA,1968,Chap.9.APhysicist'sDeskReference,2nded.,editedbyH.L.AndersoncanInstituteofPhysics,NewYork,1989,p.37.A.T.Jones,``Thestrikenoteofbells,''J.Acoust.Soc.Am.,373±381ReprintedinRef.3M.Grutzmacher,W.Kallenbach,andE.Nellessen,``AkustischeUntersu-chungenanKirchenglocken,''Acustica,34±45intranslationas``Acousticalinvestigationsonchurchbells''inRef.3.1444J.Acoust.Soc.Am.,Vol.111,No.3,March2002Fletcheretal.:Bellclapperdynamics