19 th INTERNATIONAL CONGRESS ON ACOUSTICS MADRID, 2-7 SEPTEMBER 2007

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1 19 th INTERNATIONAL CONGRESS ON ACOUSTICS MADRID, 2-7 SEPTEMBER 2007 INFLUENCE OF THE PROFILE OF SOME VIBRATING STRUCTURES ON THEIR SOUND RADIATION EFFICIENCY AND DIRECTIVITY PACS: Rj; Rz Alb, Jsus 1 ; Ans, Jg 2 ; Rmis, Jim 3 1 DISAO, Plytchnic Univ. f Vlnci, EPS Gndí 46730, Sin; jslb@fis.uv.s 2 Institut f Acustics, Univ. Austl d Chil, PO Bx 567, Vldivi, Chil; jns@uch.cl 3 DFISTS, Ad. d Cs 99, Univ. d Alicnt, 03080, Alicnt, Sin; jmis@u.s ABSTRACT Bsiclly, this sch ts th ffct f mking slight chngs t th sid viw f sm vibting stuctus n bth, thi sund ditin fficincy nd dictivity. F undtking this, tw numicl mthds cn b usd: 1) Rmbg s bsd numicl intgtin f th sufc intgls invlvd in th blm nd 2) Finit Diffnc Tim Dmin (FDTD) mthd. T find th dictivity, th sund ssu fild is btind t diffnt ints fm th suc using dict numicl intgtin. On th th hnd, th sund ditin fficincy is stimtd fm th intgtin sults f th l t f th mchnicl ditin imdnc, which is ncssy t dtmin th ttl sund w ditd. Th FDTD mthd cn b usd s hl t visuliz th sund fild. Th numicl sults cmd with th ns btind by th mthds td in th littu. In dditin, th mthdlgis lid t cmmcil ludsks. [Wk tilly sutd by FONDECYT N ]. INTRODUCTION In dsigning sund dits f udi lictins, th ximtin f cicul ln istn muntd flush in n infinit bffl is usully cnsidd t fm th clcultins t lw fquncis. This is dn bcus f its simlicity nd th xistnc f clsd fm nd nlyticl xssins f th sund ssu t fild ints in th f-fild, n th xis, nd n th istn fc. Hwv, it is wll-knwn tht, in ctic, mst f dits nt fctly flt nd thy hv ctin fil which is functin f dth. In dditin, th siml clcultin is fmd tkn int ccunt th f-fild ximtin. In this, diffnt tnsvs fils f cicul istns cnsidd, by fming th stil numicl intgtin, in d t stimt thi ffcts n th sund fficincy nd dictivity. THEORETICAL BACKGROUND Th sund ditin in th f-fild ducd by flt cicul istn muntd in n infinit bffl cn sily b btind. This is ttind by ssuming tht th vibtin vlcity n th sufc f th istn is cnstnt nd th sund fild is susitin f lmnty siml sucs f sufc ds, ccding t th gmty shwn in Fig. 1(). Th mthmticl xssin cn b fund in clssicl txtbks [1, 2]. F cicul istn hving givn xisymmtic tnsvs fil, th diffntil sund ssu ducd by siml suc t distnc, vibting with mximum sufc vlcity u nd fquncy f, in fluid mdium f dnsity ρ, cn b wittn s ρ fuds d = j h' jk ( h' ct), (1) wh k is th f-fild wvnumb, c is th sd f sund, nd h is givn by h ' = + σ + z 2( σ sinθ cs Ψ cs Φ + σ sinθ sin Ψ sin Φ + z csθ ). (2)

2 Th vlus f ch mt in Eq. (2) givn in Fig. 1(b). H, is th vlu f th vitin in th -xis f givn fil. Intgtin f Eq. (1) sults in th sund ssu t int P. Clly, Eq. (1) hs t b numiclly intgtd. Ψ ds h P θ P σ θ Ψ h h' z Y 1 Pistón ln cicul X Φ σ () (b) Figu 1.-Gmtis usd t dscib th blm: () Cicul ln istn; (b) Cicul istn hving givn tnsvs fil. In th f-fild h, nd f smll fil dth vlus Thus, th ttl sund ssu cn b ximtd by h ' + σ sinθ cs Ψ. (3) ρ fu (, θ, t) j ρ fu = j σ = jk ( ct) jk ( ct) σ = 0 σ jkz ( σ ) 4J1( ksinθ ) ksinθ dσ σ = σ = 0 Ψ= 2π jkσ sinθ cs Ψ Ψ= 0 σ jkz ( σ ) dσ, dψ = (4) wh J 1 is th Bssl functin f d 1. Slutin f Eq. (4) dnds n th givn fil. Mchnicl ditin imdnc Th mchnicl ditin imdnc is ltd t th fcs tht th fluid xts n th vibting sufc. Ths fc in hs with th sufc vlcity tht ccunts f ditin lsss nd fc tinl t th sufc ccltin tht ccunts f n incs f th sufc mss. Mthmticlly, f tim-hmnic vibtin f th sufc, th mchnicl ditin imdnc is givn by [2] 1 = ds'. (5) jkct u Thf, by intgting Eq. (1) nd thn intgting Eq. (5), th vlu f cn b btind. APPLICATION TO A DYNAMIC LOUDSPEAKER Th l t f th ditin imdnc is vy imtnt t dict th fficincy f vibting systms, such s dynmic ludsk [3]. In fct, it cn b shwn tht th fficincy η f such dvic, whn muntd flush in bffl (sund w ditd/ttl w sulid by th lctic gnt), is givn s [4] (%) = 100 ( R g + R E ) MT 2 S ' 2 ( Bl) R ( Bl) (2R + R ) η, (6)

3 wh is MT (kg/s) is th ttl mchnicl imdnc, B (T) is th stdy i-g flux dnsity, l (m) is th lngth f wi n th vic-cil winding, Rg (Ω) is th gnt sistnc, R E (Ω) is th sistnc f th vic cil, nd R (kg/s) is th mchnicl sistnc f th dihgm. Th ttl mchnicl imdnc cn b wittn s wh (kg/s) snts th u mchnicl imdnc: 1 = R + j ωm ωc = +, (7) MT, (8) wh ω (d/s) is th cicul fquncy, M (kg) is th mss f th dihgm nd th vic cil, C (m/n) is th mchnicl cmlinc f th dihgm, nd (kg/s) is th mchnicl ditin imdnc [s Eq. (5)]: = 2 R + j2x = 2R + 2 jωm, (9) wh R (kg/s) is th mchnicl ditin sistnc, X (kg/s) is th mchnicl ditin ctnc, nd M (kg) is th mchnicl ditin mss. In gnl, th fficincy is clcultd t 1 khz. It is bsvd tht bth incsing R nd ducing M nhncs th fficincy. NUMERICAL EXAMPLES Mchnicl ditin imdnc Fiv tnsvs fils csnding t tyicl ludsk dihgm w cnsidd s xmls t ly th mthd. Th fils sntd in Tbl I. Tbl I.- Pfils cnsidd f th numicl xmls Pfil Gmty Equtin lin x = d d qudtic x = d 2 d xnntil z d( x = ) 1 squ t = d x d lin + cicul 2 2 x = d( x ) d x < x 3

4 Th Rmbg s mthd ws usd t fm th numicl intgtins nd th mthd ws imlmntd int Mtlb cmut cd. Th vibting sufcs w disctizd in dil nd ngul dictins, hving ttl f lmnts. This chic is bcus th Rmbg intgtin mthd uss w-f-tw nds f th functin. Dtils n th intgtin schm cn b fund in th littu [5, 6]. Figu 2 snts th cmutd sults f nmlizd mchnicl ditin imdnc f th fiv tnsvs fils nd th ln cicul istn s functin f k. Th numicl sults hv bn nmlizd by th duct f th sufc S nd ρ c. A dth d=0.25 hs bn cnsidd f ll th numicl xmls. Figu 2.- Rl (R ) nd imginy (X ) ts f th nmlizd mchnicl ditin imdnc ( / Sρ c) f diffnt tnsvs fils clcultd by numicl intgtin. Th sults cmd with th clssicl sults f cicul ln istn. It is bsvd in Fig. 2 tht th fil cn duc significnt chngs in th l nd imginy ts f th mchnicl ditin imdnc. Th mximum vlu f mchnicl ditin sistnc is chivd by th qudtic fil f k=2.5, ximtly, whil th minimum vlu is bsvd t th sm fquncy f th lin fil. Thf, it shuld b xctd tht th dictivity ttns nd vlus f fficincy cn b quit diffnt whn cmd t th sults btind by using th ln istn mdl. As xctd, f ll css, th nmlizd sistnc tnds hmniclly t unity t high fquncis. Figu 2 (ight) shws tht, f fquncis bv k=3, ngtiv sults f ctnc btind. It is xctd tht, f vy high fquncis, th vlu f ctnc must g t z. This cn b du t th numb f intgtin sts nd msh usd t numiclly vlut th intgls. Btt sults shuld b btind by incsing th disct msh ints ducing th intgtin sts. Hwv, this cn hv hug cst in cmuttinl tim. Dictivity ttns Sinc sund ssu ditd by th vibting stuctu cn b btind t fixd distnc fm th bffl, it is ssibl t cmut th sund ssu n ln ndicul t th vibting stuctu. Thus, it is ssibl t lt th dictinl chctistics f ch fil. F cmisn, Figs. 3 nd 4 shw th numicl sults f dictivity ttns f ludsk f dimt 8 hving diffnt tnsvs fils, t fquncis 1kHz nd 4kHz, sctivly. Th sund ssu fild is lttd t fixd distnc f 1 m vy 2.5 O. It cn b nticd th ffct f th fils n th dictivity ttns. Th svl diffncs in th sund fild ditd f nn ln fil whn cmd with th n ducd by ln istn. 4

5 Figu 3.- Dictivity ttns f ludsk (=4 ) hving diffnt tnsvs fil clcultd t 1kHz. Figu 4.- Dictivity ttns f ludsk (=4 ) hving diffnt tnsvs fil clcultd t 4kHz. 5

6 Hwv, it is bsvd tht f 1kHz th bmwidth f th min lbs mins cnstnt nd it is lmst indndnt f th fil. Sm chngs n th sid lbs cn b bsvd. F 4kHz, th fils shw lg bmwidth whn cmd with th ln istn. Thn, it is suggstd tht mdifictin f th fil f ludsk culd ld t imtnt chngs in th dictivity ttn. This fct cn b usd t dsign btt ditin dvics. Efficincy Fm th sults f mchnicl ditin imdnc btind by numicl intgtin f Eq. (5), it is ssibl t vlut th fficincy f dynmic ludsk s givn in Eq. (6), tking int ccunt its tnsvs fil. As n xml, Tbl II shws th sults f fficincy η t 1kHz f cmmcil 8 ludsk mdl 8M70 md by Bym in Sin. Efficincy hs bn clcultd using th imdnc f ln istn, th mthd sntd by Pnz [7], nd th numicl mthd sntd bv. In dditin, th sults cmd with th msud fficincy in lbty. Th fficincy sttd by th mnufctu ws 1.1. Accding t th hysicl dimnsins nd sh f th ludsk dihgm, th numicl sults w btind using cubic fil hving 3 cm-dth. Tbl II.- Efficincy t 1kHz btind by fu diffnt mthds f cmmcil ludsk. Ludsk Efficincy η by Pistn by Pnz Numicl Eximntl 8M It cn b bsvd tht th fficincy clcultd by th snt mthd gs quit wll with th msud sult. Th fficincy clcultd by using th ln istn mdl undstimts th fficincy by 12.5%. Th fficincy clcultd by th mthd td by Pnz vstimts th vlu f fficincy by 14.2%. CONCLUSIONS A numicl mthd t cnsid th ffct ducd by tnsvs fil f vibting sufc muntd flush in bffl hs bn sntd. Th mthd uss numicl intgtin t find th sund ssu fild nd th mchnicl ditin imdnc. Th sults cn b lid t th cticl cs f dynmic ludsk in d t stimt its dictivity ttns nd fficincy. Th numicl sults shw tht mdifictin f th tnsvs fil f ludsk cn duc imtnt vitin in th sund fild ditd. Th sult f fficincy clcultd by th mthd sntd h mtchs fctly with th sult btind by ximntl mthd. Of cus, this mthdlgy lis whn w cn ssum tht th ludsk dihgm bhvs s mmbn [8]. Futh studis shuld b fmd in d t tst th mthdlgy with th cmmcil ludsks. Rfncs: [1] P.M. Ms: Vibtin nd sund, Acusticl Scity f Amic, futh inting, [2] L.E. Kinsl, A.R. Fy, A.B. Cns, J.V. Snds: Fundmntls f custics, Jhn Wily & Sns, Thid Editin, Nw Yk, [3] M. Cllms: High fmnc ludsk, Jhn Wily & Sns, Nw Yk, [4] L. L. Bnk: Acustics, Acusticl Scity f Amic, Nw Yk, [5] W.H. Pss, S.A. Tuklsky, W.T. Vttling, B.P. Flnny: Numicl cis in C, Scnd Editin, Cmbidg Univsity Pss, Cmbidg, [6] G. Lindfild, J. Pnny: Numicl mthds using Mtlb, Ellis Hwd Limitd, Lndn, [7] J. Pnz: Rditin imdnc f cns t high fquncis, 112th AES Cnvntin, Munich, Gmny, My [8] A.L. Gldstin, S.N.Y. Ggs: Numicl mdlling nd msumnt f th vibcustic chctistics f ludsks, Intnis 97,