A STUDY OF INPUT MOBILITY FUNCTIONS AT A VIOLIN'S BRIDGE. Jie Pan and Robert Wilkins

Transcription

1 ICSV14 Cairs Ausralia 9-12 July, 27 A STUDY OF INPUT MOBILITY FUNCTIONS AT A VIOLIN'S BRIDGE Jie Pa ad Rober Wilkis School of Mechaical Egieerig, Uiversiy of Weser Ausralia 35 Sirlig Highway, Crawley, WA 9, Ausralia pa@mech.uwa.edu.au Absrac Experimeal compariso of idividual oes from differe violis has revealed some differeces bewee he oal qualiies of violis. These differeces offer clues owards a objecive deermiaio of a violi s acousical qualiy. This paper repors some prelimiary fidigs of he correlaio bewee he oal qualiy ad ipu mobiliy fucios a locaios where he srigs aach of he bridge of wo differe violis. I is demosraed ha he mobiliy fucios are relaed o he violi s bowig sesiiviy ad o he overall soud radiaio o he performer ad liseers. 1. INTRODUCTION The udersadig of violi acousics has log bee a passio ad edeavour of violi players, violi makers ad acousicias. Several impora advaces have bee made i he udersadig of violi s acousics, icludig he self-susaied srig oscillaios due o sickig ad slidig fricioal forces bewee he bow ad srig [1], rockig ad verical vibraio modes of violi bridges [2, 3] ad bi-ri ocave uig of he back ad op free plaes for he bes isrume soud [4]. I parallel wih hose advaces, effor have bee made i ivesigaig a large umber of violis wih high ad moderae qualiies by comparig heir bridge admiace [5] ad shor ime Fourier rasforms of glissadi o each srig [6]. These effors are aimed owards he developme of objecive crieria for he acousical qualiy of violis. I his paper, we cocerae o a sudy of ipu mobiliy fucios a violi bridge. Differe from he overall bridge admiace described i Jasso s measureme [5], we are ieresed i he measureme of he ipu mobiliy fucios a he locaios where each srig aaches o he bridge. Those mobiliy fucios play a impora role i he deermiaio of he violi s bowig sesiiviy o each srig ad soud power radiaed from he violi. This paper aemps o address he followig issues: (1) a violi s acousical qualiy accessed by player ad liseers, (2) wha ca be lear from seady sae oes played? (3) measureme of mobiliy fucios a a violi s bridge, (4) he correlaio of he oal qualiy ad mobiliy 1

2 fucios, (5) he effec of he mobiliy o a violi s bowig sesiiviy, ad (6) he effec of mobiliy o a violi s soud power radiaio. 2. VIOLIN S ACOUSTICAL QUALITY The objecive descripio of a violi s acousical qualiy is based o he imbre of each oe played, which is also called oal qualiy ad is deermied by he relaive ampliude of he soud a he fudameal frequecy heard as pich, ad higher frequecy overoes. The evelope of he soud ad he variaios i imbre wih ime may also play a impora role i deermiig he oal qualiy, bu he discussio of his is beyod he scope of his paper. Whe a music scale or piece is played, he relaive loudess of he oes played coribues o he degree of soud balace of he isrume. I is impora o oe ha he appare qualiy of a violi is ofe perceived differely by he player from ha heard by liseers. As illusraed i Figure 1, he producio of violi soud by a player ivolves a soud feedback mechaism, ad soud received is also depede o he qualiy of he room acousics. By usig his feedback mechaism, a skilful player may be able o maiai he balace of he oe by adjusig he bowig pressure. The feedback mechaism also icludes he player had ad chi respose o he bow/srig ieracio, ad he mechaical vibraio of he isrume body. Tha is why a skilful player may someimes brig a liseer io ears eve wih a sude violi. However, wih a sude violi, he player has o work harder o produce he same effec. Two violis (Figure 2) were seleced for a subjecive compariso es of playig he same piece of music ad music oe by he same player (from WAYO). Alhough, he liseers were o able o ideify overall differeces bewee he wo violis (ideed he deailed oal differece is usually ideified by raied ears or by frequecy aalyser), he player was. Bacher violi was made i 1753 by Marimilia Bacher of Brelau i Germay. I is very ligh, very swee ad quie resposive, alhough o as loud as desirable for a op-ofhe-rage isrume. The sude violi was purchased for he purpose of compariso wih he high qualiy isrumes. No deails exis o whe i was made. I has a umber carved io is eck- a defiie sig of beig facory made. To he player, is soud is quie saisfacory ad resposive. However i is o a all eve, as is easily appare i playig ess.!" Figure 1 Flow diagram of producio of violi soud. 2

3 ICSV July 27 Cairs Ausralia Bacher (1753) Sude ($5) Figure 2. Two violis used for he es. 3. QUALITY OF STEADY STATE NOTES The deailed oe-o-oe qualiy comparisos were coduced bewee he wo violis. Figure 3 shows he power specral desiy of he ope G srigs, which were produced by he same player ad measured a he same locaio wih respec o he violis i a aechoic room. The player kep he same bowig speed ad pressure as much as possible whe bowig differe violis. I is assumed ha he relaive ampliudes of he higher frequecies overoes wih respec o ha a he fudameal frequecy do o vary sigificaly wih he bowig pressure of he player. This assumpio implies ha he feedback mechaism of violi playig may affec he overall loudess of a oe, bu o he relaive ampliudes, which is used o describe he oal qualiy. Sude G Soud Pressure (db) Soud Pressure (db) Bacher G Figure 3. PSD of G3 (196 Hz) of Bacher ad sude violis. Lile circles i he figures are he ideified peaks i he arrow frequecy specrum. The circle wih larges value i a major peak is used o represe he peak value of he fudameals ad overoes. The compariso of he relaive ampliudes a he higher frequecy overoes of he wo violis shows sigifica level differece a a hree frequecy regios as show i Figure 4. 3

4 Those differeces occur a he hird harmoics (588 Hz) by 5 db, a frequecy rage ceered a 2kHz by more ha 15 db ad aroud 33Hz by more ha 1dB. For all hese hree cases, he Bacher violi has higher ampliudes. 75 Bacher G3/Sude G3 Soud Pressure (db) Figure 4. PSD of G3 of Bacher ad sude violis Similar compariso resuls were rue for he A3 oes (22Hz). As show i Figure 5, level differeces of he overoes bewee he wo violis occur a he same hree frequecy regios as ideified i Figure 4. A exra regio of ampliude differece is also observed a frequecies ear 1Hz. Soud Pressure (db) Bacher A3/Sude A Figure 5. PSD of A3 of Bacher ad sude violis A quesio was raised a his poi ha if such level differeces a overoes are relaed o he 4

5 characerisics of he violis, which moivaed he measureme of he mobiliy fucios a he violi bridges. 4. MEASUREMENT OF MOBILITY FUNCTIONS AT VIOLIN S BRIDGE The experime is desiged o measure he mobiliy fucios a he bridge locaios where he srigs are aached. The domia mobiliy fucios a each locaio are he ageial mobiliy: v M ( ω ) =, (1) F ad ormal mobiliy v M ( ω ) =, (2) F where F ad F respecively are he ageial ad ormal forces (as show i Figure 6 for he G srig locaio o he bridge), v ad v are he correspodig ageial ad ormal velociies due o he forces. F F Figure 6. Tageial ad ormal forces a G srig locaio of he bridge. The forces ad velociies a he bridge are geeraed by a B&K mii-shaker ad measured by a B&K impedace head. Figure 7 shows he experimeal se-up ad a coecor which jois he impedace head ad he srig o he bridge. (a) (b) (c) Figure 7. Mobiliy measureme experimeal se-up ad a coecor bewee impedace head ad he srig o he bridge. (a) coecor for ormal mobiliy, (b) mii-shaker, impedace head ad coecor, (c) coecor for ageial mobiliy. 5

6 The mobiliy fucios of a he G srig locaio of he bridge of he wo violis are show i Figure 7. I he mobiliy measureme, he vibraio of he srigs is damped. The differece bewee he mobiliy fucios of he violis is also ideified a he same frequecy regios where he differece of he overoe ampliudes is sigifica. The oly excepio is ha he differeces of he mobiliy fucios ear 2Hz are small, while ha of he overoe ampliudes is more ha 1 db. Trageial Mobiliy of G Srig Bacher Sude Normal Mobiliy of G Srig Bacher Sude -3-3 M (db) - -5? M (db) - -5? Figure 8. Measured ageial ad ormal mobiliy fucios of a he G srig locaio of he wo violis. The properies of he overoe ampliude disribuio seem o be explaiable usig he mobiliy fucios. Figures 8 ad 9 sugges: (1) The 5dB differece a 588 Hz is maily due o ha he hird harmoics of he ope G srig of he Bacher violi overlaps wih he peak frequecy of he ageial mobiliy, while ha of he sude violi does o. (2) The 1dB differece ear 3Hz correlaes well wih he drop of he ageial mobiliy (15 db) of he sude violi a ha frequecy. (3) The differece ear 1Hz whe A3 is played may due o he Bacher srig beig coupled well wih he violi body hrough he ormal mobiliy (which is early 1 db higher ha ha of he ageial mobiliy) a his figerig posiio, while he sude srig drives he violi oly ageial. -15 Bacher G Normal Tageial Seady Sae G3-15 Sude G Normal Tageial Seady Sae G SSN & M (db) SSN & M (db) Figure 9. Measured ageial ad ormal mobiliy fucios of a he G srig locaio of he wo violis. 6

7 (4) The overoe level differece i he regio of 2Hz idicaes ha he srig eergy rasmissio io he Bacher violi is hrough he ormal mobiliy, while ha io he sude violi sill relies o he ageial mobiliy. 5. EFFECT OF THE MOBILITY ON VIOLIN S BOWING SENSITIVITY The relevace of he bridge mobiliy fucios o he bowig sesiiviy ca be illusraed by a simple esioed srig model as show i Figure 1, where he effec of he eire violi o he mobiliy a he bowig locaio is represeed by he bridge mobiliy fucio (oly ageial mobiliy is used here for illusraio). Bowig Exciaio F d M Bridge Mobiliy x d x b x = x = L Figure 1. Simple srig ad bridge mobiliy model for sudyig bowig sesiiviy. The aalysis of he srig rasverse vibraio show ha a he h overoe of he srig, he mobiliy a he bowig posiio x d (bowig sesiiviy) is relaed o he bridge mobiliy a x b by: π xd V ( x, ) si d ω M d ( ω) = L M ( ω). (3) Fd ( ω) π xb si L I he becomes clear ha he bowig sesiiviy of a violi srig is direcly proporioal o he bridge mobiliy whe x d is close o x b. If he fudameal frequecy or ay of he overoe frequecy overlaps wih he oe of he peak frequecies of he mobiliy fucio, he a large vibraio a he bridge locaio ca be excied. O he oher had, if he frequecy of srig oscillaio is a he valley of he mobiliy, he a large amou of bowig pressure is required o geerae he same bridge vibraio. I seems desirable o have all he frequecies associae wih impora oes of a violi coicidig wih hose frequecies a which he mobiliy fucios have similarly high values (a or ear he heir peaks). This will resul i a relaively eve or uiform bowig sesiiviy. 6. EFFECT OF MOBILITY ON VIOLIN S SOUND POWER RADIATION The mobiliy fucios a he bridge allow he evaluaio of he mechaical power rasmied io he violi a each of he oscillaio frequecy of he bowed srig: P( ω) = Re{ M ( ω)} F ( ω) + Re{ M ( ω)} F ( ω ) (4) 2 2 where F ad F are he ageial ad ormal forces a he srig aachme posiio of he bridge. This rasmied mechaical power is he supply of he kieic eergy i violi srucure, which is dissipaed by mechaical ad radiaio dampig mechaisms. Thus he 7

8 eergy-balace equaio likig his ipu power o he spaial averaged velociy V ( ω ) of he violi srucure ca be expressed as: 1 2 P( ω) = η m V ( ω ) (5) 2 where m is he violi mass, ad η = η + η (6) m r is he loss facor cosisig of mechaical ( η m ) ad radiaio ( η r ) loss facors. η m is a simple fucio of frequecy. η r is deermied by he radiaio efficiecies of all violi body modes ad couplig of hese modes wih he soud field wihi he violi caviy a ω. Alhough he deermiaio of he loss facors requires furher experimeal ad compuaioal effors, he spaial averaged body velociy of he sude violi measured by a laser scaig vibromeer already shows a promisig correlaio bewee he ipu mobiliy ad he body velociy (see Figure 11). I becomes clear ha he characerisics of he mobiliy fucios are exhibied i he averaged velociy ad herefore he radiaed soud power. However, he relaive ampliudes of he power ad soud pressure a each of he oscillaig frequecies of a bowig srig will be adjused byη m, η r, radiaio direciviy ad characerisics of he room, which makes he violi research more challegig. Compariso Bewee Frequecy Respose, Mobiliy ad Soud Pressure; A srig, Sude Violi Laser Acceleraio Measureme Mobiliy Measureme Soud Pressure 2 Magiude (db) - 7. CONCLUSIONS Figure 11. Spaial averaged velociy, mobiliy ad soud pressure (iside violi body) of he sude violi. ACKNOWLEDGEMENTS The experimeal work of Claire Garcia-Webb, Chrisia Rogerso ad Michael O'Halo o he UWA s violi projec is graefully ackowledged. REFERENCES [1] L. Cremer. The Physics of he Violi, MIT Press, [2] W. Reiicke, The bridge properies of srig isrumes, Disseraio, Isiue for Techical Acousics, Techical Uiversiy of Berli, [3] H. A. Muller, The fucio of he violi bridge, Cagu Acousical Sociey Newsleer 31, 19-22, [4] C. M. Huchis, The acousics of violi plaes, Scieific America, , [5] E. V. Jasso, Admiace measureme of 25 high qualiy violis, ACUSTICA aca acousica, Vol. 83, , [6] O. E. Rodgers, Toal ess of prize wiig violis a he 24 VSA compeiio, Cagu Acousical Sociey Newsleer, summer, 75-94, 25. 8