19 th INTERNATIONAL CONGRESS ON ACOUSTICS MADRID, 2-7 SEPTEMBER 2007 NARROW TUBES ACOUSTIC IMPEDANCE CHARACTERIZATION USING FINITE ELEMENT BASED TOOLS

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1 19 th INTERNATIONAL CONGRESS ON ACOUSTICS MADRID, 2-7 SEPTEMBER 2007 NARROW TUBES ACOUSTIC IMPEDANCE CHARACTERIZATION USING FINITE ELEMENT BASED TOOLS PACS: Ye Brav, Agustín 1 ; Ruiz, Marian 1 ; Recuer, Manuel 1, López, Juan Manuel 1 ; De Arcas, Guillerm 1 ; 1 Instrumentatin and Applied Acustic Research Grup. Universidad Plitécnica de Madrid; Ctra.Valencia Km-7, Madrid, 28031, Spain. abrav@i2a2.upm.es ABSTRACT The lumped parameter apprach has sme built-in limitatins with acustic frequency and physical dimensins and may nt be valid fr higher frequencies where the acustic wavelength is in the range f the narrw tube dimensins. Besides, when sund is prpagated thrugh narrw tubes r in small cavities r ver sharp crner, the lsses becme nticeable because f the thermal and viscus effects appearing. All these effects can nt be mdelled by means f the classical lumped elements mdel apprach s, new mre cmplex and accurate acustic mdels need t be prpsed fr taking int accunt these therm-viscus effects, which are required fr btaining realistic results. Bth kind f lsses can be simulated by means f LMS virtual lab acustic sftware. The finite element mdelling can prvide much mre detailed infrmatin abut the physical phenmena prduced inside narrw tubes, btaining pressure distributin map, s applying definitin z=p/sv, acustic impedance narrw tube is well knwn and trustable. INTRODUCTION Alng the present paper we will illustrate hw the acustic impedance f a narrw tube is mdelled cmparing tw mdels: the traditinal mdel, knwn as lumped parameter mdel and a mdel based n tls f finite elements calculus, in cncrete LMS virtual lab tls have been used. The capture and acustic simulatin f a narrw tube with the dimensinal characteristics f h=3.8mm and r=0.2mm that are indicated in figure 1 will be made. Figure 1.- Narrw tube gemetry and dimensins fr acustic analysis. ACOUSTIC IMPEDANCE IN NARROW TUBES The acustic impedance f a narrw tube is defined in analytical way, as it is knwn, by the Z=P/Sv expressin. Slving the wave equatin fr narrw tube gemetry we btain the acustic impedance which results f cmplex type and depending n data tabs in the Bessel functins [1], as it is shwed in the equatin 1. Z tube iwρh = 2J1( K sr ) 1 K r J ( k r ) s s ( Eq. 1) Where K s is the Stkes wave number, J and J 1 are respectively the rder 0 and 1 Bessel functins, w is the angular frequency,ρ is the air density and h,r are tube dimensins.

2 Acustic impedance first apprach: the lumped parameter mdel. Traditinally electracustic analgy has been used fr making a similarity f the narrw tube acustic impedance by means f a resistance and an inductance. This analgy cmes frm linear aprximatin n Bessel's functins shwn in equatin 1. Of such way, the narrw tube acustic impedance wuld be the same ne that it wuld have an electric circuit cmpsed by a resistance cnnected in series with an inductance, RL netwrk, that cnstitutes the denminated mdel LPM (Lumped Parameter Mdel). Fr the dimensins shwn in figure 1, an apprach fr the acustic impedance will be the indicated ne in figure 2, where ne acustic hm is equivalent t ne pasm-3. Figure 2.- Lumped parameter mdel f a narrw tube. The frequency respnse fr acustic impedance f LPM is in figure 3. It is pssible t bserve the split effect f acustic impedance real and imaginary part in the mentined analgy, acustic resistance and acustic mass respectively. 200 Z a c s u t i c ( d B ) Hz 30Hz 100Hz 300Hz 1.0KHz 3.0KHz 10KHz 20*LOG10(100000*R((V(VIN:+)/ -I(VIN)))) 20*LOG10(100000*IMG((V(VIN:+)/ -I(VIN)))) Frequency Figure 3.- Acustic impedance dependant f frequency fr a Narrw tube accrding t lumped mdel. Acustic impedance finite element apprach: BEM (Bundary Element Methd) mdel. The finite elements mdel type BEM (Bundary Element Methd) f the narrw tube, shwed in figure 1, is frmed by a three-dimensinal structure that is mdelled by elements f type TR3, linear triangle [6], elements f surface in CATIA V5 Advanced meshing methds [7]. The number f elements in the mdel will depend n the gemetry f the shape t analyze. The mdel results are shwn in figure 4. As it can be bserved in figure 4, narrw tube BEM mdel has 632 ndes x 1260 elements TR3 type and the quality figures f the finite element mdel allw making an acustic analysis until a frequency f 39.5 KHz. The acustic analysis f the narrw tube has been made giving t the prperties f the air that flws in its interir the values f 1.2 Kg/m3 fr air density and 340 m/s fr the speed f sund. 2

3 Figure 4.- Features f BEM Finite elements mdel fr a narrw tube. Thermal and viscsity lsses in narrw tubes. The lsses in the prpagatin f acustic waves thrugh very narrw elements cause that the acustic energy is transfrmed int ther frms f energy and that is mdelled, in its mre generic frm, by the Navier Stkes equatins. These effects are widely dcumented in the bibligraphy [3], [4], [5]. Tw mechanisms f lsses exist: Viscsity lss; it is the main mechanism f lsses and it is prduced by frictin phenmena within the tube. Thermal lss; It is the secnd mechanism f lsses and nrmally it is nt cnsidered. It is caused by heat cnductin between the fluid and the wall. Bth types f lsses can be simulated in LMS virtual Lab finite element calculus mdule fr the range f frequencies included between 20Hz and 10kHz. This suppses a remarkable imprvement in relatin t the lumped parameter mdel where these viscsity and thermal lsses are nt cnsidered in. BEM setup with lsses. First f all, we will capture the narrw tube gemetry with dimensins f r=0.2mm and h=3.8mm. Next, the narrw tube meshing is built and finally the cmplex density and speed f sund parameters, that allw us t simulate the lsses, will be intrduced. - Gemetry f the element; the capture f a narrw tube with the indicated dimensins is made as shwn in figure 1, using CATIA V5 tl. 3

4 - Viscsity lsses are mdelled by means f cmplex density f air [2] and can be shaped by the expressins indicated in equatin J1( K ) = 1 sr ρ ρ ( Eq. 2) K sr J ( k sr ) - Thermal lsses will be mdelled by cmplex speed f sund thrugh the bulk mdulus [2] indicated in equatin 3. K = 2( γ 1) J ( B r ) P 1 + Bk r J ( Bk r ) 1 1 k γ ( Eq. 3) Where K s is the Stkes wave number, J and J 1 are respectively the rder 0 and 1 Bessel functins, γ is the air adiabatic cnstant, P is the standard air pressure, B k is the thermal diffusivity f the air, ρ is the nrmal density f the air and h,r are tube dimensins. Results f BEM analysis with lsses. Resistance and acustic mass fr narrw tube f figure 1, after type BEM finite elements analysis is finished, is shwed in figure 5. Narrw tube acustic impedance frm BEM finite element mdel Acustic Impedance (db) Acustic_resistance(dB) Acustic_mass(dB) 130 Figure 5. Acustic impedance f a narrw tube btained frm BEM finite element mdel. As it can be bserved there are clear differences between the traditinal mdel, as it appears in figure 3, and the BEM mdel with lsses shwn in figure 5, mainly with respect t resistive value. These differences are shwn in detail in figures 6, 7 and 8, as much by the real part f the acustic impedance R, acustic resistance, like fr the imaginary part f the same ne, L, acustic mass. Narrw tube acustic impedance cmparisn definitin vs lumped vs BEM mdels Acustic Impedance (db) Acustic resistance_definitin Acustic resistance_lumped Acustic resistance_bem 155 Figure 6.- Cmparisn in BEM mdel versus lumped mdel versus theretical mdel fr acustic resistance f a narrw tube. 4

5 In figure 6, the theretical value f equatin 1 and the value btained in BEM simulatin are ttally cincident until 6kHz. The theretical mdel des nt cnsider the thermal lsses and that is why it separates slightly frm finite elements mdelling. Als, the lumped mdel mves away enugh f the expected behaviur, nt being able t mdel bth the thermal and viscsity lsses. In figure 7, a clear difference between lumped mdel and BEM analysis can be bserved mainly frm 400Hz frequency. The lumped mdel des nt cnsider neither viscsity nr thermal lsses and appreciate errr is incurred. Lumped mdel is valid nly fr lw frequencies. Acustic Impedance (db)) 10 5 Acustic Resistance difference between lumped and BEM mdels Acustic resistance_bemvslmp 0 Figure 7.- Detailed cmparisn between BEM and lumped mdels fr acustic resistance f a narrw tube. In the case f the acustic mass, the difference is less accusing. It is in between ±1dB fr all the frequency range as it is shwn in figure 8. 1,00 Acustic mass difference between lumped and BEM mdels Acustic Impedance (db) 0,50 0,00-0,50-1,00 Acustic mass_bemvslmp Figure 8.- Detailed cmparisn between BEM and lumped mdels fr acustic mass f a narrw tube. The acustic pressures map distributin fr several frequencies can be bserved in figure 9, which prvides additinal infrmatin f the physical phenmena that takes place inside f the narrw tube. This acustic pressure map is nt pssible t anticipate with the lumped mdel. CONCLUSIONS As it has being indicated previusly, a cmplete mdel f narrw tube finite elements fr acustic impedance determinatin has been develped. That mdel cnsiders the thermal and viscsity effects taking place in the interir f a narrw tube. After cmparing the results f the simulatin with the traditinal mdel, it is stated that the develped mdel is mre realistic, mainly, fr frequencies upper t 400Hz, where the traditinal mdel presents clear limitatins inherent t the incapacity f the lumped mdel t simulate any kind f lsses. In additin, it is pssible in the BEM finite elements mdel t btain a three-dimensinal distributin f sund pressures, thrughut all the tube, that can help t understand the physical phenmena taking place in its interir which fully surpasses the lumped parameter mdel capacities. 5

6 20Hz 100Hz 400Hz 1000Hz 1600Hz 3150Hz 6300Hz 10000HZ Figure 9.- Acustic pressures distributin in BEM finiete element mdel fr a narrw tube. References:[1] K. K. Ahuja and R. J. Gaeta, Jr. Active Cntrl f Liner Impedance by Varying Perfrate Orifice Gemetry Gergia Institute f Technlgy, Atlanta, Gergia. NASA STI Prgram. [2] C. Zwicker and C.V. Ksten, Sund absrving material, Elsevier publising (1949). [3] UNO INGARD, Acustic nn linearity f an rifice, Acustics Labratry, Massachusetts Institute f Technlgy, Cambridge, Massachusetts (1967),The jurnal f the acustic Sciety f America. [4] L.J.SILVIAN, Acustic impedance f small rifice, Bell Telephne labratries (1935), The jurnal f the acustic Sciety f America. [5] V. Cutanda, Numerical transducer mdelling, Ph.D. thesis, Department f Acustic Technlgy, Orsted, DTU, Technical University f Denmark,2002. [6] Finite Element Reference guide fr CATIA V5 Release 16, 2005.DASSAULT SYSTEMES. [7] CATIA V5 Manuals f DASSAULT SYSTEMES. 6

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