Boundary elements for noise barriers above an impedance ground

Transcription

1 Boudary elemet for oie arrier aove a imedace groud afael PICOYA; Marti OCHMANN Beut Hoccule für Tecik Berli, ermay ABTACT Te iertio lo (IL) of a oie arrier tadig o a imedace groud i ivetigated uig te Boudary Elemet Metod (BEM). Te aortio of te groud i caracterized y a fiite comlex imedace. Te BEM formulatio icororate a tailored ree fuctio tat atifie te imedace oudary coditio o te groud, terefore te groud i automatically take ito accout ad oly te oie arrier mut e dicretized. Te tailored ree fuctio ca e decried a a ueroitio of ource wit comlex oitio exreed a a lie itegral over te imagiary axi. uc formulatio i very uitale for a BEM imlemetatio ice te lie itegral exit for all kid of imedace ad coverge fater ta oter formula. I te reet aer, te ifluece of te tye ad magitude of te groud imedace o te ieldig roertie of te arrier i tudied i detail. Te oie arrier itelf i coidered rigid. Keyword: BEM, alface, iulatio I-INCE Claificatio of uject Numer(): 75.5, 51., 51.4, INTODUCTION oadway oie i a imortat cotriutio to te total oie exitig i moder citie. Te oie roduced y car a maily tree ource: te motor, te road-tyre iteractio ad te erturatio i te airflow. Oe way to reduce te oie i to act directly o te oud ource. egardig te road-tyre oie, everal tye of avemet ave ee develoed to cage te imact force etwee road ad tyre urface. uc material ca rig a reductio i te oie level tat varie from 3 db(a) u to 8 db(a). Ufortuately, due to te cot, tee tye of avemet are oly eig italled ofte we te roadeedtoereairedoryte cotructioofewroad. A differet aroac to reduce te oie affectig te eole i to uild olid otacle to lock te oud wave, i.e. oie arrier. Due to diffractio, te oud wave caot e comletely locked ut te overall oie level ca e reduced. Te amout of level reductio deed o te dimeio of te arrier. For a very log arrier, it eigt will e te determiat dimeio. If a aorig groud i reet, te diffracted wave eid te arrier will uffer a additioal atteuatio ad te effective IL of te arrier may icreae. I te reet work, te IL of a oie arrier coiderig a imedace groud will e calculated.. NUMEICAL MODEL For ti tudy we coider ti rigid arrier tadig o a aorig groud. I order to imlify te umerical model, te arrier will e aumed ifiitely ti, i.e. to avezeroticke.oterwie, te ize of te elemet ould e very mall i order to avoid arrow ga rolem. Te dimeio of te arrier are fixed, wit legt L=4m ad eigt H= m. Te aortio of te groud i caracterized y a comlex imedace Z.Itatcae,te oudary coditio o te groud read Z jk 0 (1) ρc Te oud ource i a oit ource laced at a fixed oitio, 0.8 m away from te arrier ad 0.4 m aove te groud. Te oud arrier i laced at te YZ lae (x=0). Te total oud reure a tree comoet: te icidet reure ( ifiite groud ( ic ref. ic ), it reflectio o te ref ) ad te cattered reure due to te reece of te ody ( ), 4357

2 Figure 1 Model of te oud arrier.1 BEM aroac Te direct BEM aroac alied to te ti arrier (Fig. 1a) i aed o te itegral equatio [1] c g g g d y g d a a, () were g i ay ree fuctio atifyig te Helmoltz equatio, a i te ormal vector o te ic ref urface of te arrier oitig toward te acoutic domai ad. For oit i te acoutic domai, c = 1, for oit o te urface, c = 0.5 if te oudarie are moot ad c =0 oterwie. By takig te limit 0,ot ad collae to te middle urface ad a, [] (Fig. 1). Terefore, a g g g g, a a ad te itegral equatio ecome c g, a, g a, (3) μ σg d g d, (4) wit μ ad σ. Eq. (4) i referred a a idirect BEM aroac. I te aece of te catterer, 0 ad,eceot ad imultaeouly atify te oudary coditio. Terefore, te ecod itegral i (4) ca e writte a j k Z jkg ρc g d, (5) If te ree fuctio i coe o tat it alo atifie te oudary coditio o te groud, Z g g g jkg 0, te itegral (5) vaie. ice we coider a rigid arrier, ρc σ 0 ad te itegral exreio for te oud reure i writte a c g μ d, (6) Eq. (6) i ot very ueful ice o te urface, c /,wit. Hece tere i oe 4358

3 equatio for te two ukow μ ad ormal directio :. A ueful itegral equatio i otaied y derivig (6) i te g 0 μ d, (7) were ad are te ormal vector of ource ad receiver oit at te urface reectively.. Tailored ree fuctio g Te ree fuctio aove a ifiite lae wit a urface imedace ree fuctio for te rigid lae lu a correctio term C (γ ). g g C(γ ), g 1 4 Z ca e writte a te jk1 jk e e, (8) 4π π were x y 1, x y, 0 γ jk / Z ad Z0 Z / ρc (eefig.a). Figure eal ad comlex ource oitio Ocma [3] reeted a olutio aed o te comlex image metod. Te correctio term i foud to e C( γ ) jγ e 4π 0 ( η) jk jγη e dη, (9) ( η) were x yc ad yc y jβ,wit β ( 0,0, η ). Eq. (9) ca e ee a a um of te cotriutio of mooole wit comlex ource oitio. Te real art of te ource oitio i give y te image ource oitio elow te ifiite groud ad te image art i give y te real vector β. Te itegrad i (9) a a weak igularity i te cae tat ource ad receiver lie o te lae. Ti igularity i itegrale if ource ad receiver do ot coicide..3 Numerical imulatio I ractical cae, te erformace of oie arrier i ot evaluated at all frequecie. Ofte, te acoutic iertio lo of te arrier i evaluated at te 500 Hz ave ad ad it i take a a rereetative value. I te reet tudy, te imulatio are erformed at te 500 Hz ave ad. Te field oit are laced eredicular to te arrier at a ditace from te arrier varyig from 0. m to 15. m. I order to determie te averaged IL value, te calculatio are made at 11 frequecie witi te ave ad. Te averaged IL i te ratio of te total reure witout te arrier ( ) to te total reure wit te w arrier ( ) 4359

4 IL 1 10 lg, w i1 i (10) Figure 3 imulatio arameter Te geometrical arameter of te imulatio are ow i Fig. 3. Te ifluece of te tye of imedace of te groud o te iertio lo i ivetigated. Tree tye of imedace are coidered: a) aorig imedace, wic a a ure real value; ) ma-like imedace, wic a a ure imagiary ad oitive value ad c) rig-like imedace, wic a alo a ure imagiary ut egative value. For eac tye of imedace, eigt value of Z0 are aumed: 0.1, 0.5, 0.5, 1,.5, 5, 10 ad 5. Te itegral i (9) ad it firt ad ecod derivative are evaluated umerically uig a gloal adative quadrature imlemeted i te fuctio itegral from Matla. Due to te ocillatig term j( k γη ) e, te time of calculatio deed o te value of γ ad o te field oitio. From te tree tye of imedace, te rig-like tye a te iget calculatio time. Ti calculatio time i ow i Fig. 4 for te cetral frequecy f=500 Hz. Figure 4 Calculatio time for differet value of Z0 of te rig-like imedace.4 eult Te curve of te averaged IL for te differet value of imedace are lotted togeter wit te two limitig cae, amely, oft groud ( Z 0 ) ad rigid groud ( Z ). Fig. 5, 6 ad 7 ow te reult for a aorig imedace, ma-like imedace ad rig-like imedace reectively. 4360

5 Figure 5 IL of a arrier o aorig groud It ca e oerved, tat a oft groud roduce a iger IL ta a rigid groud i two zoe 0<x<3.1 ad 6.7<x<8. wile te rigid groud roduce a iger IL at 3.1<x<6.7 ad x>8.. Far from te arrier, te oft groud a a mall effect or eve a egative oe. For te aorig or ma-like tye imedace te traitio etwee te oft groud ad te rigid groud i relatively moot. Te limitig cae cotitute te lower limit of te IL ractically at all field oit. Figure 6 IL of a arrier o a ma-like groud 4361

6 Figure 7 IL of a arrier o a rig-like groud Oe ca fid toug ome value of te imedace ta rovide a iger IL ta te limitig cae, for examle: Z. 0 5 for x>9 ad Z 5 0 j for x>10. Te imrovemet i igger for te aorig cae ta for te ma-like cae. For te rig-like tye imedace, te traitio etwee te oft groud ad te rigid groud i alo moot ut te deviatio from te value of te limitig cae ca e very large, for examle y Z0. 5 j. Te limitig cae cotitute te uer limit of te IL at all field oit wile te value rovided y te rig-like tye groud ca e muc lower ta te limitig cae at everal field oit. 3. CONCLUION Te reet tudy a ow tat te reece of a groud wit a fiite imedace ca affect te IL of oie arrier. It a ee demotrated tat a imrovemet of everal db i te IL of a rigid arrier ca e otaied. Accordig to te reult, te major imrovemet i otaied y a aorig imedace followed y te ma-like imedace. Te rig-like imedace o te cotrary caot rovide ad advatage comared to te rigid ad oft cae. ACKNOWLEDEMENT Ti work wa uorted y te erma eearc Foudatio (DF) witi te roject Teory ad alicatio of acoutic multiole radiator wit comlex igularitie. EFEENCE 1. eyert A.F., oearko B. adiatio ad catterig of acoutic wave from odie of aritrary ae i a tree-dimeioal alf ace. Traactio of te AME 1988;110: Ce Z, Hoftetter, Mag H. A alerki-tye BE-formulatio for acoutic radiatio ad catterig of tructure wit aritrary ae. I: Marurg, Nolte B, editor. Comutatioal acoutic of oie roagatio i fluid - fiite ad oudary elemet metod. Berli - Heidelerg, ermay: riger-verlag; Ocma M. Te comlex equivalet ource metod for oud roagatio over a imedace lae. J Acout oc Am. 004;116(6):