Acoustic Analysis with Consideration of Damping Effects of Air Viscosity in Sound Pathway

Transcription

1 Aousti Analysis with Consiration of Damping Effts of Air Visosity in Soun Pathway M. Sasajima, M. Watana, T. Yamaguhi, Y. Kurosawa, an Y. Koi Astrat Soun pathways in th nlosurs of small arphons ar vry narrow. In suh narrow pathways, th sp of soun propagation an th phas of soun wavs hang aus of th air visosity. W hav vlop a nw finit lmnt mtho that inlus th ffts of amping u to air visosity for moling th soun pathway. This mtho is vlop as an xtnsion of th xisting finit lmnt mtho for porous soun-asoring matrials. Th numrial alulation rsults using th propos finit lmnt mtho ar valiat against th xisting alulation mthos. Kywors Simulation, FEM, air visosity, amping. I. INTRODUCTION ITH th many avanmnts in th prforman of W omputrs, CAE (Computr Ai Enginring) has n us xtnsivly in rnt yars for aousti analyss. Howvr, th onvntional analysis approah is still ing prominantly us for rlativly larg struturs or larg quipmnt. For xampl, for a strutur with a small volum of a fw ui ntimtrs, suh as an arphon nlosur, vry fw mthos of soun propagation analyss ar availal. In small arphons, nlosurs ar ivi into svral ompartmnts, an th soun pathways onnting ths rooms ar oftn vry narrow. Th visosity of air in ths narrow pathways rsults in amping. Consquntly, th sp of soun propagation rass, an a phas lay ours. Thrfor, to arry out aurat aousti analysis, w n to onsir th ffts of th amping u to air visosity whih ar not onsir in a onvntional aousti analysis. In this stuy, w hav vlop a nw finit lmnt mtho that inlus th ffts of th amping u to air visosity in narrow plas in th soun pathway. This has n vlop as an xtnsion of th aousti finit lmnt mtho propos y Yamaguhi [], [] for a porous soun-asoring M. Sasajima is with th Stratgi Rsarh & Dvlopmnt Division, Fostr ltri Co., Lt., , --9 Tsutsujigaoa, Aishima, Toyo, Japan (phon: -87-; -mail: sasajima@ fostr.o.jp). M. Watana is with th Stratgi Rsarh & Dvlopmnt Division, Fostr ltri Co., Lt., , --9 Tsutsujigaoa, Aishima, Toyo, Japan (phon: -87-; -mail: mtwatana@ fostr.o.jp). T. Yamaguhi is with th Dpartmnt of Mhanial Systm Enginring, Gunma Univrsity, , -5-, Tnjin-ho, Kiryu, Gunma, Japan (-mail: yamagm@gunma-u.a.jp). Y. Kurosawa is with th Dpartmnt of Prision Mhanial Systm Enginring, Tiyo Univrsity, -855, -, Toyosatoai, Utsunomiya, Tohigi, Japan (-mail: yurosawa@mps.tiyo-u.a.jp). Y. Koi is with th Stratgi Rsarh & Dvlopmnt Division, Fostr ltri Co., Lt., , --9 Tsutsujigaoa, Aishima, Toyo, Japan (phon: -87-; -mail: oi@ fostr.o.jp). matrial []. Morovr, w attmpt numrial analysis in th frquny omain with our aousti solvr that uss th propos finit lmnt mtho. For th numrial alulations, w us a tu mol having a irular ross stion. Thn, w ompar th propos finit lmnt mtho with th thortial analysis, an with th gnrally us finit lmnt analysis that os not inlu th ffts of th visosity of th air. II. NUMERICAL PROCEDURES W hav vlop a nw finit lmnt mtho that inorporats th air visosity at small amplitus. Fig. shows th irt Cartsian oorinat systm an a onstant strain lmnt of a thr-imnsional ttrahral. Hr, u x, u y, an u z ar th isplamnts in th x, y, an z irtions at aritrary points in th lmnt. Fig. Dirt Cartsian oorinat systm an a onstant strain lmnt Th strain nrgy U ~ an xprss as follows: ~ x y z U E xyz x y z = + + () whr E is th ul moulus of lastiity of th mium, air. Th tim rivativ of th partil isplamnt is xprss as. u. Thrfor, th inti nrgy T ~ an xprss as follows: ~ T T = ρ{}{}xyz ()

2 whr ρ is th fftiv nsity of th lmnt. T rprsnts a transpos. Th visosity nrgy D ~ of a visous flui an xprss as follows: ~ T D = {}{}xyz T Γ () whr { T} is th strss vtor attriutal to visosity. Th rlationship twn th partil vloity an th strss an xprss as follows: { T} μ μ μ x y z μ μ μ τ xx x y z τ yy μ μ μ x τ zz x y z = y τ xy μ μ u z τ yz y x τ zx μ μ z y μ μ z x... whr ux, uy, an uz ar th partil vloitis in th x, y, an z irtions at aritrary points in th lmnt, an μ is th offiint of visosity of th mium. In th aov quation, {} Γ is th strain vtor. Th rlationship twn th partil vloity an th strain an xprss y th onstant strain lmnt of a thr-imnsional ttrahral, as shown in Fig.. {} Γ γ xx γ yy γ zz = γ xy 6V γ yz γ zx (5) () x y z x y z x y z x y z V is th volum of th lmnt an - ar onstants. Ths onstants an xprss as follows: = ~ ε { yl ( zn zm ) + ym( zl zn ) + yn( zm zl )} = ~ ε { zl ( xn xm ) + zm( xl xn ) + zn( xm xl )} = ~ ε { xl ( yn ym ) + xm( yl yn ) + xn( ym yl )} (6) ~ ( =,) ε = ( =,) Fig. Rlationship twn partil vloity an strain whr susripts, l, m, an n rprsnt th irular rotation of,,, an. Nxt, w onsir th formulation of th motion quation of an lmnt, for th aousti analysis mol that onsirs visous amping. Th potntial nrgy V ~ an xprss as follows: T T { u} { P} Γ + {}{ u F} V ~ xyz (7) = Γ whr { P } is th surfa for vtor. { F } is th oy for vtor. An Γ rprsnts th intgral of th lmnt Γ ounary. Th total nrgy E ~ an riv y using th following xprssion: ~ ~ ~ ~ ~ E = U + D T V (8) W an otain th following isrtiz quation of an lmnt y using Lagrang s quations: whr t ~ ~ ~ ~ ~ T T U V D + + = i i i i i (9) u i is th i-th omponnt of th noal isplamnt u an i is th i-th omponnt of th noal partil vtor { } vloity vtor { }. W an otain th following isrtiz quation of an lmnt y sustituting () (7) in (9): [ M ]{ u } + [ K ]{ u } + jω[ C ]{ u } = { f } ω () u in this quation aus a prioi M, [ K ], f ar th lmnt mass matrix, lmnt stiffnss W us { u } = jω{ } motion having angular frquny ω is assum. [ ] [ C ] an { } matrix, lmnt visosity matrix, an noal for vtor, rsptivly.

3 III. EXPRESSIONS FOR DAMPING IN POROUS MATERIALS In th motion quation (), w us th following mol having a omplx fftiv nsity, an a omplx ul moulus of lastiity in th lmnt stiffnss matrix[ K ], an th lmnt mass matrix[ M ] [] []. ρ ρ = + E E = E R + je ρ R jρi I () () whr j nots th imaginary omponnt. Not that ρ R an ρ I ar th ral an imaginary parts of ρ, rsptivly. Hr ρ R is th nsity of air insi th porous soun-asoring matrial an ρ I is rlat to th flow rsistan of air insi th soun-asoring matrial. W onsir th rsistan to th air flow at a high frquny y using th omplx nsity. Th varials E R an E I ar th ral an imaginary parts of E, rsptivly an E I is rlat to th hystrsis twn th soun prssur an th volum strain. Th issipat nrgy u to E I is onvrt into hat nrgy orrsponing to th nlos ara of th hystrsis urv otain in on yl; this is nown as th attnuation fft. As a rsult, th quations of th soun fil in th lmnt that inlus amping ar formulat using omplx linar simultanous quations. Fig. Thr-imnsional tu for finit lmnt mtho ω whr [ ([ M ] + j[ M ] ){ u } + ([ K ] + j[ K ] ){ u } + jω[ C ]{ u } = { f } R ] R I R () I K ar th ral an imaginary parts of ar th ral an M, rsptivly. All noal partil K an [ ] I [ K ], rsptivly an [ M ] R an [ M ] I imaginary parts of [ ] isplamnts an alulat y solving quation () for th partil isplamnt. Furthrmor, th strain an th soun prssur of ah lmnt an alulat from th noal partil isplamnts. IV. CALCULATION A. Damping Analysis y th Thr Dimnsional Finit Elmnt Mtho To vrify our mtho, w arri out an aousti amping analysis for tus using thr-imnsional finit lmnts mol. Whn w ma this mol, w us HyprMsh v. (Altair Enginring In.) at mshing. As shown in Fig., this mol is / soli mol symmtrial aout x-z plan an x-y plan. An th air insi th tu is mol using thr-imnsional ttrahral lmnts having four nos. Th numr of ivi lmnts was in th axial irtion, an in th raial irtion. Both ns of th tu wr los. Th raius of mol was.5 mm. Th lngth of th tu was 6.6 mm. Fig. Th istriution of partil isplamnts ontour an th isosurfa viw W slt th fftiv nsity ρ R =. g/m, offiint of visosity μ =.8 5 N s/m, ral part of th omplx volum lastiity E R =. 5 Pa, an soun propagation sp = m/s for air. As th ounary onitions, th partil isplamnts of all nos on th outsi in ontat with th tu wr fix, xpt for th plan of symmtry. Fig. shows th ontours of th alulat partil isplamnts an th isosurfa viw of th mol tu for th propos finit lmnt mtho, nar th rsonan onitions (, Hz an, Hz). As an sn, th magnitu of th isplamnt of th partils hangs signifiantly nar th outsi of th tu. Howvr, th isplamnt is flat los to th ntr of th tu.

4 B. Damping Analysis y Thortial Analysis W hav arri out thortial analysis of th rsonant rspons of th tu for omparison an vrifiation of th propos finit lmnt mtho. W onsir a straight ut with irular ross stion. In this as, th frquny rspons of th prssur an xprss y th following gnral xprssion, jωt os x ( l) P = jρv () sin l whr ρ is nsity of th air, is th sp of soun, lis th lngth of th tu, x is a position of rfrn point, is ω/. v is an xitation vloity, an tis tim. Fig. 5 Thr-imnsional tu mol an vloity V (y) apillary flow thory [6]. R j tanh j μ = tanh j j (8) whr is th iamtr of th irular ross stion tu. An is xprss as follows. ρ ω = (9) μ = κ γp = ρ ρ () whr κ is th ul moulus, p is atmosphri prssur an γis th spifi hat at onstant volum. For th as of th.5 mm raius tu, th omplx nsity an omplx soun vloity ar shown in Figs. 6 an 7 rsptivly. In this quation, w introu th omplx soun sp an th omplx fftiv nsity ρ to inlu th attnuation u to th visosity of th air. W rpla th sp of soun an th nsity with th omplx soun sp an th omplx fftiv nsity as shown low. ρ ρ (5) (6) Using th two sustitutions aov in quitation () an using th notation of Craggs an Hilrant [5], th fftiv nsity an writtn as Fig. 6 Dnsity trn for th thr-imnsional tu mol ρ ρ + R jω (7) whr ρ is mass nsity, an R is flow rsistan. Th ral part of th omplx fftiv nsity ρ is th nsity of th air rlat to th inrtial for. Imaginary part is a trm that rprsnts th visous rsistan or rsistan to flow. In aition, th flow rsistan pr unit ara R in th high frquny rgion an xprss as follows, using th Fig. 7 Vloity trn for th thr-imnsional tu mol

5 Fig. 8 Prssur vrsus frquny rspons for a thr-imnsional tu mol (Diamtr is.5 mm for th mol tu) (a) Propos FEM () Thortial mtho Fig.9 Efft of iamtrs ompar twn Propos FEM an Thortial mtho at,hz (a) Propos FEM () Thortial mtho Fig. Efft of iamtrs ompar twn Propos FEM an Thortial mtho at,hz 5

6 By alulating th frquny rspons of () using th valus of ths paramtrs, th thortial solution that inlus th visosity is otain. C. Vrifiation an Comparison of th Propos Mtho W hav analyz frquny rsponss of th propos finit lmnt mtho (Propos FEM), an ompar it with th aov sri thortial mtho (Thortial mtho) that inlus th visosity, an with th onvntional aousti finit lmnt mtho (Convntional FEM) that os not inlu th attnuation. Fig. 8 shows th omparison of th analysis rsults for a mol tu of raius.5 mm. Th onition of xitation was th onstant isplamnt xitation. From Fig. 8, w trmin th fft of amping on th alulat rsults y using th propos FEM an th thortial mtho ass. Th onvntional FEM os not show attnuation for th rsonan pas. In aition, w analyz th rsonant rsponss with iffrnt tu iamtrs using oth th propos FEM an th thortial mtho ass. Fig. 9 shows th fft of iamtrs of irular tu mols on th rspons, for th propos FEM an th thortial mtho ass at aroun,hz. An Fig. shows th fft of iamtrs of irular tu mols on th rspons at aroun,hz. Th iamtr of th irular tus wr.5 mm,.8 mm, an. mm. Th onition of xitation was th onstant isplamnt xitation. As an sn from Fig.9 an Fig., whn th tu iamtr is narrow, th rsonan pa ras aus flow rsistan inrass. This trn is th sam for oth mthos. A omparison of th rsults of th propos mtho with that of th thortial mtho shows that th propos mtho shows slightly largr attnuation. W thin this is oming u to th influn of th msh siz an orr nar th ounary layr. As a rsult of th first orr lmnts us in this analysis, th msh siz was somwhat largr nar ounary layr that ha larg hang of isplamnt. Damping for Automotiv Doul Walls with a Porous Matrial, Journal of Soun an Viration, Vol. 5, pp. 6-5, 9. [] M. Sasajima, T. Yamaguhi an A. Hara, Aousti Analysis Using Finit Elmnt Mtho Consiring Effts of Damping Caus y Air Visosity in Auio Equipmnt, Appli Mhanis an Matrials, Vol. 6, pp. 8-86,. [] H. Utsuno, T. Tanaa, Y. Morisawa an T. Yoshimura, Prition of Normal Soun Asorption Coffiint for Multi-Layr Soun Asoring Matrials y Using th Bounary Elmnt Mtho, Transations of Japan Soity of Mhanial Enginrs, Vol. 56-5C, pp. 8-5, 99. [5] A. Craggs an J.G.Hilrant, Efftiv nsitis an rsistivitis for aousti propagation in narrow tus, Journal of Soun an Viration, Vol.9, pp-, 98. [6] M. A. Biot, Thory of Propagation of Elasti Wavs in a Flui-Saturat Porous Soli. Ⅱ. Highr Frquny Rang, Journal of th Aoustial Soity of Amria, Vol.8, pp79-9, 956. V. CONCLUSION W hav vlop a nw aousti finit lmnt mtho that onsirs th ffts of amping y th visosity of air. W ompar alulation rsults of soun prssur vrsus frquny haratristis using th propos mtho with that of th thortial mtho, an th onvntional aousti finit lmnt mtho without visosity of air. Th omparison shows that th gnral shaps of th haratristis ar vry los. For futur, w ar planning to xtn this rsarh furthr, to fully unrstan th amping ffts of air on soun wavs. Thry, w hop to stalish a thnology that allows onsiration arly in th sign stag, an to provi xllnt soun solutions. REFERENCES [] T. Yamaguhi, J. Tsugawa, H. Enomoto an Y. Kurosawa, Layout of Soun Asoring Matrials in D Rooms Using Damping Contriutions with Eignvtors as Wight Coffiints, Journal of Systm Dsign an Dynamis, Vol. -, pp ,. [] T. Yamaguhi, Y. Kurosawa an H. Enomoto, Damp Viration Analysis Using Finit Elmnt Mtho with Approximat Moal 6