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Otherthanpurecuriosity,therearemanygoodreasonstoseekhigh-accuracyacous7calmodelsofguitars.Sincetheirinven7on,thedesignofclassicalguitarshasremainedlargelyempiricalwithdifferentluthiersmaintainingtheirowntechniquesandtradi7ons,someeveninconflictwithoneanother.Ifwecouldlinkobjec7ve,measurablephysicalparametersofaguitarwiththemoresubjec7vequalityofitstone,wecouldavoidalotofguessworkintheinstrumentdesign.Fromanenvironmentalperspec7ve,manytonewoodsareonthevergeofex7nc7on,leadingmanyguitarmakerstoexploretheuseofmoresustainableandpredictablesynthe7cmaterials.Inordertoevaluatepoten7almaterials,averyhigh-fidelityacous7calmodeloftheinstrumentisneeded.Image:hHps://en.wikipedia.org/wiki/Classical_guitar#/media/File:Classical_Guitar_two_views2.png

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Inclasswecoveredthefunc7onofanacous7cguitarataverybasiclevel.Thestringsthemselvesdonotradiatemuchatall;rather,theytransferenergythroughthebridgetothebodyoftheguitarwheremostoftheradiatedstoundisproduced.Broadlyspeaking,atlowfrequenciestheguitarbodyandsoundholeactasaHelmholtzoscillator.Thevolumeoftheairintheguitarchangeswhenthesoundboard(topplate)andbackmonopolemodesareinphase,sothesemodesalsocouplewiththeHelmholtzmodeatlowfrequencies.Athigherfrequencies,theradiatedsoundisdominatedbythevibra7onalmodesofthesoundboard,andtheairbodymodesplayamuchsmallerroleinthetotalradiatedsound.Thebackandsidesoftheguitaralsovibrate,buttheydonotcontributesignificantlytotheradiatedsound.Theirmaincontribu7ontotheguitartoneisviacouplingwiththesoundboardandtheairwithintheguitar.Imagereference:hHps://newt.phys.unsw.edu.au/music/guitaracous7cs/construc7on.html

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Withthesefundamentalmodesinmind,wecantrytodescribethephysicsoftheguitarusingalumpedparametermodel,inwhichwesimplifythecomplicatedgeometryofarealguitarintoasmallnumberofcoupledoscillators.Inordertoreproducethefrequencyresponseofaguitaratlowfrequenciesfairlywell,it’sactuallysufficienttouseathree-massmodelinwhichthe“oscillators”aretheHelmholtzresonatoroftheguitarbody,themonopolemo7onofthetopplate,andmonopolemo7onofthebackplate.Allthreeoscillatorscoupletogetherviatheairwithintheguitarbody.Imagereference:Top:Popp,J.E.(2012).Fourmasscoupledoscillatorguitarmodel.TheJournaloftheAcous7calSocietyofAmerica,131(1).BoHom:French,R.(2008).EngineeringtheGuitar:TheoryandPrac7ce.SpringerUS.

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Unfortunately,wequicklyrunupagainstsomemajorlimita7onswiththelumpedparametermodelwhentryingtocapturethetonalqualityofarealinstrument.TheHelmholtzoscillatorapproxima7onisgoodforwavelengthswhicharelargecomparedwiththeeffec7velengthoftheguitarbody,butathigherfrequenciesweneedtoconsidermorecomplicatedmodesoftheairwithintheguitarbody.Thesehighermodesaremuchmoresensi7vetotheactualgeometryoftheguitarbody,andcannotbeapproximatedbyasimpleairpiston.Similarly,thediscretecoupledoscillatormodelofthetopandbackplateisunabletocapturetheeffectofmodeswithinternalnodelines.Themonopolemodesaremuchmoreefficientresonatorsthanthemorecomplicateddipole(andhigher)contribu7onssuchthattheradiatedsoundisdominatedbythemonopolemodes,butthehumanearisverysensi7ve,especiallyinthehigherfrequencyrangeoftheinstrument.Inordertocharacterizethetruetoneoftheinstrument,wecannotaffordtoomitthesehigher-ordercontribu7ons.Imagereference:I.A.PerryandB.E.Richardson,“Radia7onefficiencyandsoundradia7onfieldsofclassicalguitars,”ProceedingsoftheIns0tuteofAcous0cs,vol.35,part1,pp.365–372,2013.

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Whatisthefrequencyrangeweshouldconsiderwhencalcula7ngthesoundresponseofaguitar?Wewouldliketocaptureanyresonanceswhichalistenertendstoassociatewithaqualityinstrument.RecalltheISOcontoursofequalloudness(showninfigure),whichdefinethesoundpressurelevelrequiredtoachieveaspecificloudnessasafunc7onoffrequency.Humanhearingispar7cularlysensi7vearound3kHz,soweshouldnotbesurprisedtofindimportanttonalfeaturesinthisfrequencyrange.Ina2012paper,Czajkowskadirectlymeasuredfrequencyresponsecurvesforawiderangeoflow-tomid-rangeclassicalguitarsinanaHempttocorrelatespecificmodalfeatureswithhigherquality.Inthiscase,thepriceoftheguitarwasusedasaproxyfortonalquality,sinceusuallypricetrendsaresimilartosubjec7vequalityassessments.Theyfoundseveralhigh-frequencyfeaturestobecorrelatedwithquality,suchasFRFvarianceinthe1-2kHzbandandstrongerstructuralresonancesat4-5kHz.Imagereference:hHps://commons.wikimedia.org/wiki/File:Lindos1.svg.Equal-loudnesscontourscreatedbylindoslandusingOpenOfficeDrawbyreferencetoISO226:2003standard.1December2005,22:07(UTC)

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Inordertoachieveatrulyhigh-fidelitymodeloftheguitar,weneedtosolvesomenon-trivialdifferen7alequa7onswhiletakingintoaccountthecompletegeometryoftheinstrument.Thisisacommonprobleminbasicallyallareasofphysicsandengineering,andthemostcommonsolu7onistodividethesystemintomanysmallfiniteelements.Thephysicsofa7nyhomogeneousbrick-shapedelement,forexample,arefarsimplerthanthephysicsofawholeguitar.Finiteelementmodelsaresophis7catedmathema7caltoolsforcombininglargenumbersofsmallfiniteelementsintoalmostanydesiredgeometryandsolvingtherelevantdifferen7alequa7onsusingnumericalmethods.Thesedays,manyhigh-performanceFEsolversarereadilyavailableascommercialsolwarepackages,andevenasopen-sourceprojects(seetheFEniCSproject).Imagereference:Elejabarrieta,M.J.,Ezcurra,A.,andSantamaria,C.(2002b).Coupledmodesoftheresonanceboxoftheguitar.TheJournaloftheAcous7calSocietyofAmerica,111(5):2283–2292

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Evenwiththeuseofasophis7catedsolverfromaready-madesolwarepackage,itiss7lluptoustocarefullydefinethearrangement,proper7es,andbehaviorofourfiniteelements.Finiteelementscancomeinallshapesandsizes,andarenotrestrictedtoLego-shapedbricks.Forexample,onemightmodelthetopandbackplatesusingsecond-orderbrickelementsinordertocapturethedetailsofhigher-ordervibra7onalmodes,butuseplanarelementsforthesidesoftheguitarsincethoseexperiencerela7velyliHlemo7on.Inordertoachieveahigh-accuracysimula7onoftheradiatedsound,weneedtostartwithaveryhigh-accuracyCADmodeloftheinstrumentitself.Ofpar7cularinterestarethevariousbracesacrossthetopandbackplates.Thesebracess7ffenandaddweighttotheplates,drama7callyalteringtheacous7cbehavior.Imagereferences:Lel:J.Dean.Introduc7ontothefiniteelementmethod(fem)lecture2firstandsecondorderone.[Online].Available:hHp://www.ccg.msm.cam.ac.uk/images/FEMORLecture2.pdfRight:hHps://cadguitarplans.com/product/cad-antonio-de-torres-style-classical-guitar-plan-CADGuitarPlans.RickeyBrunet.P.O.Box5082,Cleveland,TN37320

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Almostasimportantasthegeometryofthemodelarethemechanicalproper7esoftheelements.Thesoundboardsofclassicalguitarsarebook-matchedquarter-sawnlogs,suchthatYoung’smodulusdiffersgreatlyintwoorthogonaldirec7onsacrosstheplate.Thesematerialproper7esmaketheirwayintotheelas7citymatrixshown.Simula7onofcompositematerialswithmul7pleindividuallayersintroducesfurthercomplica7on.Olen,inordertoevaluatetheaccuracyoftheFEmodel,thenormalmodesofaverysimplegeometry(suchasasquareplate)willbecalculatedandexperimentallyverified.Imagereferences:Top-lelandtop-right:Abaeian,N.(2017).FiniteElementDesignandManufacturingofaNylon-StringGuitarSoundboardfromSandwich-StructuredComposites.PhDthesis,SchulichSchoolofMusic,McGillUniversity.BoHom:Ono,Teruaki,andIsomura,Daisuke."Acous7cCharacteris7csofCarbonFiber-reinforcedSynthe7cWoodforMusicalInstrumentSoundboards."Acous7calScienceandTechnology25.6(2004):475-77.Web.

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Boundarycondi7onsarethefinalingredienttothefiniteelementmodel.Wecouldcalculatethenormalmodesoftheguitarbodyintheabsenceofanyair,butwhatwearereallyinterestedinistheactualsoundpressurelevelthatresultsfarfromtheguitar,wherealistenerwouldbe.Afluidmodeloftheairaroundtheguitaraswellaswithintheguitarbodywillgiveusjustthat.Inordertoavoidunwantedinterac7onwiththesimula7onboundary,thefluidvolumemustbemany7mesgreaterthanthevolumeoftheguitaritself.Intheexampleshowninthefigurehere,theexternalvolumeneededtobeabout30007mesthatoftheguitartostabilizeboundaryinterac7ons.Imagereferences:Elejabarrieta,M.,Santamaria,C.,andEzcurra,A.(2002).Aircavitymodesintheresonanceboxoftheguitar:theeffectofthesoundhole.JournalofSoundandVibra7on-JSOUNDVIB,252:584–590.

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Puungeverythingtogether,wecanuseagoodfiniteelementmodeltoproducedetailedpicturesofthenormalmodesandfrequencyresponseoftheinstrument.Inthefigureshownhere,onthelelwecanseenormalmodesofthesoundboardsimulatedbyafiniteelementmodel.Ontherightaretheanalogousmodesmeasuredintherealinstrument,showingafairlygoodagreement.Imagereferences:Top-lelandtop-right:Abaeian,N.(2017).FiniteElementDesignandManufacturingofaNylon-StringGuitarSoundboardfromSandwich-StructuredComposites.PhDthesis,SchulichSchoolofMusic,McGillUniversity.

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Inresearchingthistopic,Iwasdelightedtocomeacrossanumberofthesisprojectswhichrevolvedaroundthedesignandconstruc7onofanacous7cguitarfromnon-standardmaterialsusingfiniteelementmodelstosimulatetheacous7calresponseoftheinstrument.Theguitarshownontherightandbelowisanexampleofonesuchinstrument,whichapparentlysoundedquitegoodincomparisonwiththeYamahaFG401whichwasusedasthemodel.Theguitarsatthetoplelareexamplesofcommerciallyavailableguitarsconstructedfromsynthe7cmaterials,someofwhichareconsideredtop-endconcertinstruments.ThemanufacturerOva7oninpar7cularhasbeenusingsynthe7cmaterialsforguitardesignforover50yearsnow.Itisencouragingtoseesignificantprogressbeingmadetowardssustainable,reliableguitardesign.Imagereferences:Top-lel:Abaeian,N.(2017).FiniteElementDesignandManufacturingofaNylon-StringGuitarSoundboardfromSandwich-StructuredComposites.PhDthesis,SchulichSchoolofMusic,McGillUniversity.BoHom-lelandright:Probert,S.M.(2007).Design,ManufactureandAnalysisofaCarbonFiberEpoxyCompositeAcous7cGuitar.PhDthesis,UniversityofWashington,MechanicalEngineering.

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