An Eigenvalue Based Acoustic Impedance Measurement Technique

A. J. Hull Engineer. Naval Underwater Systems Center, New Lndn, CT 00 C. J. Radcliffe Assciate Prfessr. Department f Mechanical Engineering, Michigan State University, East Lansing, Ml 88 An Eigenvalue Based Acustic Impedance Measurement Technique A methd is develped fr measuring acustic impedance. The methd emplys a nedimensinal tube r duct with excitatin at ne end and an unknwn acustic impedance at the terminatin end. Micrphnes placed in the tube are then emplyed t measure the frequency respnse f the system frm which acustic impedance f the end is calculated. This methd uses fixed instrumentatin and takes advantage f mdern Fast Furier Transfrm analyzers. Cnventinal impedance tube methds have errrs resulting frm mvement f micrphnes t lcate the maxima and minima f the wave pattern in the impedance tube r require phase matched micrphnes with specific micrphne spacing. This technique avids these prblems by calculating the acustic impedance frm measured duct eigenvalues. Labratry tests f the methd are presented t demnstrate its accuracy. Intrductin Measuring the acustic impedance f a bundary is imprtant since the acustic respnse f any acustic system is gverned by the acustic impedance f its bundaries. Accurate mathematical mdels f acustic systems require accurate measurements f acustic impedance. The acustic impedance f bundaries determines the magnitude and frequency f resnant peaks and the spatial distributin f acustic respnse. A variety f acustic impedance measurement techniques have been develped in the past. The first techniques used an impedance tube and a single micrphne (Hall, 99; Beranek, 90; Mrse and Ingard, 98; Dickinsn and Dak, 970; Pierce, 98). They require measurement f maximum and minimum sund pressure levels at an acustic resnance in an impedance tube and their spatial lcatins. These lcatins and magnitudes are then used t calculate the crrespnding impedance (ASTM Standard C 8, 98a). Identifying the lcatin f maximum and minimum sund pressure levels in an impedance tube is nrmally difficult and requires physical changes in micrphne psitin. Tw f the impedance tube measurement methds (Hall, 99; Beranek, 90) use apprximate frmulas fr cmputing impedance which can als lead t impedance measurement errr. A recent acustic impedance measurement technique utilizes a tw micrphne system (Seybert and Rss, 977; Chung and Blaser, 980a, 980b). This technique requires tw similar, phase calibrated, micrphnes at sme lcatin in the tube with a knwn distance between them. The acustic wave respnse is then mathematically decmpsed int its reflected and incident cmpnents using a transfer functin between Cntributed by the Nise Cntrl and Acustics Divisin and presented at the Winter Annual Meeting, Dallas, TX, Nvember 0, 990, f THE AMER ICAN SOCIETY OF MECHANICAL ENGINEERS. Manuscript received at ASME Headquarters, March 990. Paper N. 90WA/NCAl. the acustic pressure at the tw micrphne lcatins. The decmpsitin allws the cmputatin f acustic impedance (ASTM Standard E 00, 98b). ASTM 00 E, althugh better than ASTM C8, requires measurement f the exact distance frm the test sample t the center f the nearest micrphne and the exact spacing f the micrphnes. Bth these physical dimensins can be difficult t measure accurately. The tw micrphne methd wrks best with tw phase matched micrphnes and a surce whse transfer functin has cnstant magnitude arund the frequency f interest. If the micrphnes are nt phase matched, then a crrectin must be included in the cmputatin f acustic impedance. These measurement requirements can lead t errrs when measuring acustic impedance using the tw micrphne technique. This paper develps a methd fr calculating the acustic impedance based n the eigenvalues f a tube with unknwn end impedance. A Fast Furier analyzer is used t measure cmplex frequency respnse frm which the eigenvalues f the system are extracted. Acustic impedance at each resnance is then cmputed frm these eigenvalues. The eigenvalue measurement is independent f micrphne psitin a"nd the lcatin f the respnse micrphne in the tube is arbitrary. The cmputatin f the acustic impedance frm the duct eigenvalues is a clsed frm slutin based n the same plane wave assumptins present in previus methds. The nly physical cnstants required are duct length and the speed f sund in the duct. System Mdel The system mdel is f a nedimensinal hardwalled duct excited by a pressure input at ne end and a partially reflective bundary cnditin at the ther end represented by a cmplex bundary impedance. The partially reflective cnditin in the 0/Vl., APRIL 99 Transactins f the ASME Cpyright 99 by ASME Dwnladed 0 Oct 008 t.0... Redistributin subject t ASME license r cpyright; see http://www.asme.rg/terms/terms_use.cfm

c IlKl L g + K Imaginary LL E U+Kj LJ LL B ll+kj LJ [Z",g (^) + f] i [ihltl)] [fam^ I.L Hl+Kj LJ Real mensinal assumptin requires the diameter f the duct t be small cmpared t the wavelength f sund which yields plane wave respnse. The nedimensinal assumptin is usually valid when/ < 0.8(c/tf) where/is the frequency (Hertz) and d is the diameter f the tube (m) (Annual Bk f ASTM Standards, 98a; 98b). The duct end at x = 0 is mdeled as a ttally reflective, pen end. This bundary cnditin is (Set, 97; Hull et al., 990) du (0, 0 = 0. () dx This crrespnds t an pen duct end. Equatin () alng with the right hand side f () mdel the speaker as a pressure surce at x = 0. Althugh speakers are smetimes mdeled as velcity surces, the eigenvalues and eigenvectrs frm the experiment discussed later in the paper crrespnd t the speaker mdeled as a pressure surce. Implicit in () is the assumptin the surce impedance is negligible. If the surce impedance is nt small, it can be incrprated int the mdel (Swansn, 988). The acustic pressure f the system is related t the spatial gradient f the particle displacement by (Set, 97) LL Hl+Kj LJ, du P(x, t)=~p(?^x (x, t). () Fig. System eigenvalues A, fr cnstant K duct allws sme energy t be dissipated ut the end while the rest is reflected back int the system. The terminatin end impedance is a rati between the pressure and the particle velcity at x = L and is expressed as (Set, 97; Pierce, 98; Spiekermann and Radcliffe, 988) du dx 0 (L,t) du dt (L,t) () where K = cmplex acustic impedance f the terminatin end (dimensinless), u(l, t) = particle displacement at x = L (m), c = wave speed in the duct (m/s), t = time(s), x = spatial lcatin (m) and L = length f the duct (m). Implicit in () is the acustic analgy with electrical systems in which vlume velcity is analgus t current and duct pressure is analgus t vltage. The reciprcal acustic mbility analgy is als smetimes used; and if applied t this system, the parameter Km (I) wuld be the acustic admittance. Acustic impedance K=Q + 0i crrespnds t an ideal fully reflective terminatin and #=+0/ crrespnds t ideal fully absrptive terminatin. In general, K is a cmplex value which des nt match either f these ideal cnditins. The real part f K (acustic resistance) is assciated with nncnservative pwer dissipatin at the end while the imaginary part (acustic reactance) is assciated with cnservative inertial and/r cmpliant characteristics f the end. The linear secnd rder wave equatin mdeling particle displacement in a hardwalled, nedimensinal duct is (Set, 97; Dak, 97) d u(x,t),d u(x,t) di c dx d_ dx Hx)PeW where u(x,t) = particle displacement (m), p = density f the medium (kg/m ), P e (t) = pressure excitatin at pint x = 0 (N/m ), and 8(x) = the Dirac delta functin. The wave equatin assumes an adiabatic system, n mean flw in the duct, unifrm duct crss sectin and negligible air viscsity effects. The hardwall assumptin yields a system with dissipatin nly at the terminatin end; the lsses at the duct walls due t heat transfer, viscsity, and vibratin are negligible. The nedi () The abve fur equatins represent a mathematical mdel f a lng, thin duct with a speaker at ne end and a partially reflective terminatin end. Separatin f Variables The eigenvalues f the mdel are fund by applying separatin f variables t () and () and the hmgeneus versin f (). Separatin f variables assumes each term f the series slutin is a prduct f a functin in the spatial dmain multiplied by a functin in the time dmain: v(x,t)=x(x)t(t). () Substituting () int the hmgeneus versin f () prduces tw independent rdinary differential equatins, each with cmplex valued separatin cnstant A, namely and d X(x) dx cfit(t) di A X{x)=0 A c T(t)=0. The separatin cnstant A = 0 is a special case where X(x) = T(t) = t satisfy () and (). Althugh A = 0 is a separatin cnstant f the system, it des nt cntribute t the pressure field in the duct, therefre it is ignred fr further cmputatinal purpses (Hull et al., 990; MacCluer et al., 990). The spatial rdinary differential equatin () is slved fr A T± 0 using the bundary cnditin () yielding () (7) X(x)=e Ax +e~ Ax. (8) The time dependent rdinary differential equatin yields the fllwing general slutin T{t)=Ae Acl + Be Act. (9) Applying bundary cnditin () t (8) and (9) yields B = 0 and the separatin cnstant, /A niri A " = l l0 HT+^J 7' n = 0 >^± (0) The system eigenvalues X are equal t the separatin cnstant multiplied by the wave speed c (A = ca ). An eigenvalue plt is shwn in Fig.. These eigenvalues are each functins f Jurnal f Vibratin and Acustics APRIL 99, Vl. / Dwnladed 0 Oct 008 t.0... Redistributin subject t ASME license r cpyright; see http://www.asme.rg/terms/terms_use.cfm

Table Measured duct eigenvalues fr the fam end Excitatin Speaker \ Fast Furier Transfrm Input Reference Micrphne PCO.t) H((0): P(X,(ll) P(0,O>) Frequency Respnse Respnse Measurement Micrphne P(x.t) Curve Fit Impedance Calculatin Acustic Impedance t be Measured Eigenvalue (n) Table Re(X n ).9.0.7.7.. ReC\,) 0.0 0. 0.09 0. 0. 0.7 Im(X n ) (Hertz) 7.89. 7.7.8 9.7 0. ImOn) (Hertz) 0.0 0.0 0. 0.9 0. 0.9 Calculated acustic impedance fr the fam end HPA Structural Dynamics Analyzer Fig. Labratry cnfiguratin acustic impedance, K. The inverse functin will allw impedance, K, t be cmputed frm measured eigenvalues. Acustic Impedance Cmputatin The acustic impedance K f the end can be determined at each duct resnance frm the eigenvalue at that resnance. This cmputatin assumes the eigenvalues f the system are knwn. Measuring these duct system eigenvalues is discussed in the next sectin. Directly slving fr K in terms f X is very difficult, therefre an intermediate variable fi is intrduced t simplify the acustic impedance cmputatin. The variable / is related t the «th eigenvalue X by Re(\)+ilm(\ n ) = lg e [Re(p ) + Hm{P )] nwci () where Re( )dentes the real part, Im( ) dentes the imaginary part, and the subscript "«" dentes the nth term. Equatin () is nw brken int tw parts, ne equating the real cefficients and the ther equating the imaginary cefficients. The cmplex lgarithm n the right hand side is rewritten as lg e [Re(l n )+iim(l n )]=lg e \l n \ +i argu ) () where l( l is the magnitude f / and arg({ ) is the argument f /, i.e., the arctangent f [Vwi(/ )/7?e(/ )]. The intermediate variable ( is nw slved fr in terms f the real and imaginary parts f the eigenvalues. The real part f / is exp / ZJ?e(A )V Re(P ) where d = Im(k ) + tan (a) The sign f Re(l ) in (a) is determined by sgn[re(0 )] = + if 0 <IAI<0. if 0. < IAI<0.0 () where A (f) If the value f A is less than 0. r greater than 0., the eigenvalue index n is incrrect and crrespnds t an eigenvalue ther than the nth ne. The value, n, must then be changed t prduce a A between 0. and 0. which will crrespnd t the crrect eigenvalue index. Once Re(P ) is fund, Im(P ) is fund by the equatin Re(K n ) 0.0 0.7 0.98 0.00 0. 0.8 0.7 0.08 Im(K n ) 0.0 0.07 0.0 0.0 0.0 0.0 0.0 0.0 /Ld \ Im{p )=Retf H )tan (^j () where i?e(/ ) is given in (). The term ( K) / ( + K) is nw equated t the intermediate variable /? using (0) and () as Re(P n )+iimtf n ) = \Re{K )iim{k n) \+Re(K n ) + iim(k ) () where Re (K ) is the real part f K and Im (K n ) is the imaginary part f K fr the nth. eigenvalue. Breaking () int tw equatins, and slving fr K as a functin f / yields the acustic impedance as Re(K H ) = Im(K n ): l[j?e(0 [/m(g )] [Re(p n )+l] +[Im(p )] /wi(js B) [Re{p n )+\?+[Im(p n )] () (7) Acustic impedance measurement K n represents the acustic impedance at the «th resnant frequency. Experiment The viability f the abve acustic impedance methd was investigated thrugh labratry tests. The test used a 0.07 m ( in) circular PVC schedule 0 duct that was.9 m (9. ft) lng driven by a 0. m (0 in) diameter speaker (Realistic 0B). The impedance f a piece f 0 mm thick packing fam inserted in the terminatin end was tested. The packing fam will be shwn t have acustic impedance which is nearly cnstant with frequency (Table ), allwing fr frequency respnse cmparisn t knwn thery. Speaker input pressure was measured in the exit plane f the input speaker with a Bruel and Kjaer Type half inch micrphne (input reference micrphne) attached t a HewlettPackard A digital signal analyzer. The respnse f the tube was measured at varius lcatins with anther Bruel and Kjaer Type half inch micrphne (respnse measurement micrphne) at /Vl., APRIL 99 Transactins f the ASME Dwnladed 0 Oct 008 t.0... Redistributin subject t ASME license r cpyright; see http://www.asme.rg/terms/terms_use.cfm

Table Measured duct eigenvalues fr the capped end Re(X n ) Im^) (Hertz) 7 0.7 0.87.00.79.9 0.9.08 9. 88. 7. 0.0.. 8.9 Table Calculated acustic impedance fr the capped end Re(K n ) Im(K n ) 7.7.0.0 8. 8.07..8.88 0. 9.0 7..8 7.7 8.8 Fig. 00 00 Frequency (Hertz) Frequency respnse f duct with fam end at x = 0.79 m tached t the signal analyzer (Fig. ). Bth micrphnes were calibrated using a Bruel and Kjaer Type 0 Sund Level Calibratr. The impedance measurement technique develped here des nt require phase matched micrphnes nr des it require cmpensatin fr phase mismatched micrphnes. Phase mismatch in the micrphnes is neglected since the measurements are made at a duct resnant frequency, i.e. the measurements are made when the system phase angles are changing rapidly thrugh 80 degrees. Micrphnes perating under 00 Hertz rarely have phase errr greater than degrees (Bruel and Kjaer, 98). Steady state eigenvalue measurements are amplitude dminated. The distance between the micrphnes is nt critical, since the duct eigenvalues are independent f measurement lcatin. This is unlike previus methds (Seybert and Rss, 977; Chung and Blaser, 980a, 980b) where micrphne spacing is a required parameter in the analysis and phase matched micrphnes (r a cmpensatin functin) are necessary because wave prpagatin acrss the micrphnes is detected. Errrs in the methd develped here are nly a functin f errrs assciated with measuring the eigenvalues f the duct, the duct length, and the speed f sund. The cmputatin f acustic impedance frm duct eigenvalues is a clsed frm slutin. The methd uses the input micrphne as an amplitude reference and the excitatin speaker des nt require a flat respnse arund the frequency f interest, because the respnse is nrmalized by the pressure input reference when the Fast Furier Transfrm is cmputed. The A Structural Dynamics Analyzer used here is capable f prviding a number f real time analyses including determining the transfer functin (frequency respnse) f a system and calculating the crrespnding eigenvalues. The A Structural Dynamics Analyzer des this by curve fitting a single mde vibratin mdel (tw first rder states) t the experimental data using the fllwing equatin (HewlettPackard, 979) J/(ft>) = Re(A n) + iim(a ) i Re(\ ) ilm(\ n ) Re(A )iim(a ) iire(\ ) +ilm(\ ) + B,w + B, (8) where H(a) = the transfer functin, A = the system residue, and.biandb = cmpensatin cnstants fr verlapping mdes. Included in the single mde vibratin mdel is cmpensatin fr ther mdes which may be verlapping at that particular frequency. During the curve fitting prcess, the real and imaginary parts f the eigenvalues are calculated. It is beynd the scpe f this paper t describe this prcess; hwever, there exist additinal alternative methds t extract mdal parameters frm the transfer functin f a system (HewlettPackard, 979; Structural Dynamics Research Crpratin, 98). Eigenvalue extractin is a cmmn functin f cmmercial Fast Furier analyzers. The first part f the experiment measured the transfer functin f the duct with the fam end impedance. The A Structural Dynamics Analyzer des this by sending a randm nise signal t the speaker and then cmputing the rati f the Fast Furier Transfrms f the input and respnse signals. Once the transfer functin was knwn, the eigenvalues f the duct were fund using the curve fitting prcess discussed in (8). Frm the eigenvalues, the acustic impedance f the fam was determined using equatins () (7). The mean and standard deviatin f the measured eigenvalues f the system with the fam end impedance are shwn in Table. These values are derived frm five independent sets f measurements at x = 0.79 m (.0 ft) thrugh x =. m (.7 ft) at 0.7 m (0. ft) increments. Each individual eigenvalue was measured frm a transfer functin cmpsed f 0 averaged Fast Furier transfrms. The calculated acustic impedance f the fam is shwn in Table. In this case, the real part f the acustic impedance dminates the respnse. Figure shws the measured frequency respnse at x= 0.79 m (.0 ft) cmpared t the theretical frequency respnse fr K = 0.7 + 0.0/ at the same pint (Spiekermann and Jurnal f Vibratin and Acustics APRIL 99, Vl. / Dwnladed 0 Oct 008 t.0... Redistributin subject t ASME license r cpyright; see http://www.asme.rg/terms/terms_use.cfm

index n fr large impedances. Frm the abve measurements, a value f cefficient, A =. was calculated fr the ne f the eigenvalues. Because the measurements were nly accurate t tw significant figures, the cefficient was runded t A = 0.. Fr mst materials the reflectivity is nt large enugh fr this t be a cncern. Cnclusins Calculatin f the acustic impedance f a duct end frm experimentally btained impedance tube eigenvalues is develped here. These eigenvalues are easily determined frm a measured tube transfer functin by cmmercially available Fast Furier analyzers. This methd has the advantage f statinary micrphne psitining at any lcatin in the impedance tube. The cmputatinal step frm eigenvalue t acustic impedance is a clsed frm slutin. Errrs in measured impedance can arise nly frm errrs in measured system eigenvalues, duct length, and the speed f sund. Experimental results shw that duct respnse can be accurately predicted frm measured impedances and demnstrate the methd is bth accurate and insensitive t measurement errrs. Future wrk will quantify and cmpare that accuracy and errr sensitivity with previus standard methds. 00 00 Frequency (Hertz) Fig. Frequency respnse f duct with clsed end at x = 0.79 m Radcliffe, 988; Hull et al., 990). The value f P/P 0 is the rati f the respnse t the input and the measured respnses are marked by X's while the theretical respnse is dented by a slid line. The impedance, K, used in the theretical respnse is the average f the six individual acustic impedance measurements taken at different lcatins alng the duct. Figure demnstrates that the theretical mdel using the measured acustic impedance f a material can accurately predict duct respnse. There is a high degree f accuracy in bth the magnitude and the phase angles. The acustic impedances f the abve experiment were calculated by increasing the magnitudes f bth the real and imaginary parts f the measured eigenvalues by ne, tw, and three standard deviatins frm their mean values. After these changes, the magnitude f the calculated impedance K nly changed by an average f.7 percent,. percent, and.0 percent, respectively. This shws the high stability f the measurement technique, its resistance t errr prpagatin, and the accuracy f acustic impedances determined using it. The experiment was repeated fr a capped end. The results are shwn in Tables and. An ideal clsed end wuld have an impedance f infinity; hwever, the real material used here has sme absrptin. The large impedances shwn in Table indicate this trend and the variatin f impedance with frequency in this case. Figure shws the measured frequency respnse cmpared t a theretical frequency respnse calculated using the measured impedances. The theretical frequency respnse was prduced by assembling a state space mdel which used the measured acustic impedances at each eigenvalue (Hull et al., 990). As in Fig., there is a high degree f accuracy in bth the magnitude and phase angles. It is imprtant t mnitr the value f A when testing extremely reflective ends. It is pssible fr eigenvalue cmputatin errrs t yield a A greater than 0.0 with the crrect References The American Sciety fr Testing and Materials, 98a, "Standard Test Methd fr Impedance and Absrptin f Acustical Materials by the Impedance Tube Methd," Annual Bk f ASTM Standards, Designatin: C 8 8, Vl. 00, pp.. The American Sciety fr Testing and Materials, 98b, "Standard Test Methd fr Impedance and Absrptin f Acustical Materials Using a Tube, Tw Micrphnes, and a Digital Frequency Analysis System," Annual Bk f ASTM Standards, Designatin: E 008a, Vl. 00, pp. 9097. Beranek, L. L., 90, "Acustic Impedance f Cmmercial Materials and the Perfrmance f Rectangular Rms with One Treated Surface," Jurnal f the Acustical Sciety f America, Vl., pp.. Bruel and Kjaer, 98, "Cndenser Micrphnes and Micrphne Preamplifiers," Bruel and Kjaer Instruments Inc., Naerum, Denmark. Chung, J. Y., and Blaser, D. A., 980, "Transfer Functin Methd f Measuring InDuct Acustic Prperties. I. Thery," Jurnal f the Acustical Sciety f America, Vl. 8, N., pp. 9079. Chung, J. Y., and Blaser, D. A., 980, "Transfer Functin Methd f Measuring InDuct Acustic Prperties. II. Experiment," Jurnal f the Acustical Sciety f America, Vl. 8, N., pp. 99. Dickinsn, P. J., and Dak, P. E., 970, "Measurements f the Nrmal Acustic Impedance f Grund Surfaces," Jurnal f Sund and Vibratin, Vl., N., pp. 09. Dak, P. E., 97, "Excitatin, Transmissin and Radiatin f Sund Frm Surce Distributins in HardWalled Ducts f Finite Length (I): The Effects f Duct CrssSectin Gemetry and Surce Distributin SpaceTime Pattern," Jurnal f Sund and Vibratin, Vl., N., pp. 7. Hall, W. M., 99, "An Acustic Transmissin Line fr Impedance Measurement," Jurnal f the Acustical Sciety f America, Vl., pp. 0. HewlettPackard, 979, "User's Guide, A Structural Dynamics Analyzer," HewlettPackard Cmpany, Santa Clara, Califrnia. Hull, A. J., Radcliffe, C. J., Miklavic, M., andmaccluer, C. R., 990, "State Space Representatin f the NnselfAdjint Acustic Duct System," ASME Jurnal f Vibratin and Acustics, in press. MacCluer, C. R., Radcliffe, C. J., and Hull, A. J., 990, "Diagnalizing Acustic Mdels," SI AM Jurnal f Applied Mathematics, in press. Mrse, P. M., and Ingard, K. U., 98, "Theretical Acustics," McGraw Hill Bk Cmpany, New Yrk, pp. 97. Pierce, A. D., 98, "Acustics: An Intrductin t Its Physical Principles and Applicatins," McGrawHill Bk Cmpany, New Yrk, pp. and p.. Set, William W., 97, Thery and Prblems f Acustics, McGrawHill Bk Cmpany, New Yrk. Seybert, A. F., and Rss, D. F., 977, "Experimental Determinatin f Acustic Prperties Using a TwMicrphne RandmExcitatin Technique," Jurnal f the Acustical Sciety f America, Vl., pp. 70. Spiekermann, C. E., and Radcliffe, C. J., 988, "Decmpsing Onedimensinal Acustic Respnse int Prpagating and Standing Wave Cmpnents," Jurnal f the Acustical Sciety f America, Vl. 8, N., pp. 8. Structural Dynamics Research Crpratin, 98, "User Manual fr MODAL ANALYSIS 8.0," Structural Dynamics Research Crpratin, Milfrd, Ohi. Swansn.D. C, 988, "The Rle f Impedance Cupling in Achieving Glbal Active Attenuatin f Nise," ASME Winter Annual Meeting, Chicag, Paper number 88WA/NCA. /Vl., APRIL 99 Transactins f the ASME Dwnladed 0 Oct 008 t.0... 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