A Comparison of Two and Four Microphone Standing Wave Tube Procedures for Estimating the Normal Incidence Absorption Coefficient

Transcription

1 A Comparison of Two an Four Microphone Staning Wave Tube roceures for Estimating the Normal Incience Absorption Coefficient Jason Kunio a Brüel & Kjær Soun an ibration N.A. 37 Bowes R Ste Elgin, IL 623, USA Taewook Yoo b E-A-R TM Aearo Technologies LLC, a 3M company 79 Zionsville R. Inianapolis, IN 46268, USA Kang Hou c J. Stuart Bolton Ray W. Herrick Laboratories School of Mechanical Engineering urue University 4 S. Martin Jischke Drive West Lafayette, IN , USA Jan Enok e Brüel & Kjær Soun an ibration Measurements A/S Skosborgvej 37, DK-285 Nærum, Denmark ABSTRACT A new ASTM stanar has been aopte for characterizing acoustical materials in a tube. This new stanar is ASTM E26-9, Measurement of Normal Incience Soun Transmission of Materials Base on the Transfer Matrix Metho. This test metho escribes the use of a tube, four microphones, an a igital frequency analysis system for the measurement of normal incience transmission loss an other important acoustical properties of materials by etermination of the sample s acoustic transfer matrix. This four microphone test metho is similar to Test Metho E-5-8, which utilizes a two microphone technique to calculate acoustical material properties, in that it also uses a tube with a soun source connecte to one a aress: jason.kunio@bksv.com b aress: taewook.yoo@mmm.com c aress: khou@purue.eu aress: bolton@purue.eu e aress: jenok@bksv.com INTER-NOISE 29 Ottawa, Canaa

2 en an a test sample mounte in the tube. For transmission loss, four microphones, at two locations on each sie of the sample, are mounte so that the iaphragms are flush with the insie surface of the tube perimeter. lane waves propagate in the tube an a broaban signal is use as the noise source. At each frequency the resulting staning wave pattern is ecompose into forwar- an backwar-traveling components by measuring soun pressure simultaneously at the four locations an examining their relative amplitues an phases. The acoustic transfer matrix is calculate from the pressure an particle velocity of the traveling waves evaluate on either sie of the specimen. The transmission loss, as well as several other acoustical properties of the material, incluing the normal incience soun absorption coefficient an the reflection coefficient for a har backe termination, can then be calculate once the transfer matrix elements are known. In this paper the two an four microphone methos for calculating the reflection coefficient an the relate normal incience soun absorption coefficient will be compare with each other an with the results preicte using finite element software.. INTRODUCTION In 24, ASTM began eveloping a new stanar for calculating normal incience properties of acoustical materials. This test metho is similar to ASTM E5 in that it also uses a tube with a soun source connecte to one en an a test sample mounte in the tube. For transmission loss, four microphones, at two locations on each sie of the sample, are mounte so that the iaphragms are flush with the interior surface of the tube. lane waves are generate in the tube using a broaban signal from a noise source. The resulting staning wave pattern is ecompose into forwar- an backwar-traveling components by measuring soun pressure simultaneously at the four locations an examining their relative amplitue an phase. The acoustic transfer matrix, T T2 () T T 2 22 is calculate from a knowlege of the pressure an particle velocity on either sie of the specimen. This transfer matrix relates the soun pressures an normal acoustic particle velocities on the two faces of a sample extening from x to x, an completely escribes the plane wave characteristics of the sample uner test. In aition to normal incience transmission loss, there are many other material properties that can be extracte from the transfer matrix, incluing the reflection coefficient, soun absorption coefficient, characteristic impeance an the wave number of the material 2. Aitionally the 2x2 transfer matrices of various materials can be multiplie to preict the behavior of composite materials With the recent aoption of the ASTM E stanar there are now two stanars that allow the calculation of the normal incience soun absorption coefficient. In theory, these two ifferent measurement techniques an methos of calculating the same physical quantity shoul yiel the same results. Other work has been performe to evaluate the normal incience soun absorption coefficient results that can be calculate from the four microphone technique, but the work thus far has been base on the assumption of an anechoic termination 4. The work presente in this paper compare results measure using both of the ASTM methos, which assumes a har termination behin the sample, an those results are also compare with simulations. INTER-NOISE 29 Ottawa, Ontario Canaa

3 2. GENERAL THEORY FOR ASTM E26 The general theory presente here focuses on the four microphone technique use for measuring normal incience transmission loss an absorption coefficient accoring to the ASTM E26 stanar 3,5. The ASTM E5 stanar was originally publishe over a ecae ago an has been thoroughly ocumente.,6,7 A. Normal Incience Soun Transmission Loss accoring to the ASTM E26 In Figure, the four-microphone tube setup is shown. A sample with a thickness of is locate between microphone 2 an microphone 3. The soun pressure at each microphone can be escribe as Mic Mic 2 Mic 3 Mic 4 B A D C Speaker x 3 x Figure : Four-microphone tube. jkx jkx jωt jkx2 jkx2 jωt ( Ae + Be ) e 2 ( Ae + Be ) e jkx3 jkx3 jωt jkx4 jkx4 jωt 3 ( Ce + De ) e 4 ( Ce + De ) e (2) where k is a complex wave number an A, B, C an D are the amplitues of the four plane waves as shown in Figure. By aing an subtracting those equations, the coefficients, A, B, C an D can be calculate. The soun pressures an particle velocities at the front an back surfaces of the sample can be efine in terms of a transfer matrix for the two-loa metho: i.e., T T2 a b a b T T. (3) ( ) ( ) ( ) ( ) 2 22 To solve the 2 x 2 transfer matrix problem, it is necessary to perform tests using two ifferent terminations. Here, nearly-anechoic an open terminations were use to solve this matrix equation. Superscripts a an b are use to enote results obtaine using the ifferent terminations. The transfer matrix can then be ientifie by using the expression T T 2 2 T T22 x + The latter equation shows that by multiplying by the inverse of the right-most matrix in equation (3), the transfer matrix elements can be calculate base on the soun pressure an particle velocities at the front an back surface of the sample. In aition, by using equation (2) an the relationship between soun pressure an particle velocity for plane waves, the soun pressures an particle velocities at the front an back of the sample can be escribe as jk x + R Te jk R Te x x ρc ρ c x 2 x 4. (5) +. (4) INTER-NOISE 29 Ottawa, Ontario Canaa

4 In this case, it is assume that the incient wave is of unit amplitue an that an anechoic termination was applie in the ownstream section. Here, RB/A an TC/A, an they are the reflection an transmission coefficients, respectively. By applying equations (5) to equation (), the anechoic reflection an transmission coefficients can be written as functions of the transfer matrix elements: i.e., jk 2e Ta T + ( T2 / ρc) + ρct2+ T22 (6) R T+ ( T2 / ρc) ρct2 T22 a T + ( T / ρ c) + ρ ct +. T 2 The normal incience soun transmission loss can then be calculate as STL n a log. (7) T B. Normal Incience Absorption Coefficient with Har Termination accoring to ASTM E26 In the normal incience absorption coefficient test efine using the ASTM E5-8 stanar the rear surface of the sample is place ajacent to the har termination at the en of the tube. In the transfer matrix metho, it is possible to simulate that conition by enforcing a zero particle velocity conition at : i.e.,. After substituting the last expression into equation (), the normal incience reflection coefficient for the har termination case can be calculate as T ρct2 R h. (8) T + ρ ct 2 The normal incience absorption coefficient with a har termination can then be calculate as h 2 Rh α. (9) 3. SIMULATION Calculations mae using a finite element moel were use to verify the above theory. Finite element moels were built to simulate the two ifferent measurement configurations shown in Figs. 3 an 4 (in Section 4, below). A two-imensional soli mesh was built in atran with the same imensions as the circular tubes use in Section 4, an it was then importe into COMET/SAFE. The soun pressures at the two or four microphone locations were calculate at the corresponing mesh noes. Then base on either ASTM E5 or ASTM 26, the acoustical property of the sample material can be calculate exactly as in the experimental calculation. For the four-microphone tube simulation, the one-loa approach was implemente for the sake of simplicity. A limp, poroelastic moel was use to represent the porous material: for this type of material ege constraint conitions are not important. Here, white an yellow fibrous materials were moele as poroelastic materials in the FEM simulation. The material properties of the white an yellow materials use in this simulation are liste in Table. INTER-NOISE 29 Ottawa, Ontario Canaa

5 Table : Sample properties for material use in simulation Material White Fiber Yellow Fiber Thickness [m].2.25 orosity Flow resistivity [Rayls/m] Tortuosity.. iscous characteristic Length [m] 2.24E E-5 Thermal characteristic Length [m].42e-4.2e-4 Bulk ensity [kg/m3] The simulation results for both the two-microphone an the four-microphone methos are presente in figure 2 for the white an yellow materials. As expecte, both proceures give exactly the same absorption curves in the simulation for a limp material that can be moele as an effective flui with complex properties. Thus it may be conclue that for ieal, limp materials, e.g., lightweight fibrous material, the two approaches to the calculation of absorption coefficient give the same results. Figure 2: Comparison of results for white (left) an yellow (right) fibrous materials. 4. EXERIMENTAL SETU With the objective of comparing the normal incience soun absorption results that are calculate using equation (23) from ASTM E5 with those calculate using equation (28) from ASTM E26 3, two separate test setups were require: they are picture in figures 3 an 4. Test Sample Microphones Louspeaker Figure 3: ASTM E5 2 microphone setup. Test Sample Holer Figure 4: ASTM E26 4 microphone setup. Termination Section INTER-NOISE 29 Ottawa, Ontario Canaa

6 The measurements for the ASTM E5 an ASTM E26 tests were mae using the mm iameter tube that is part of the Brüel & Kjær Transmission Loss Tube Kit Type 426-T. Type 487 microphones were use with a spacing of 5 mm in both test setups. The usable frequency range epens on the iameter of the tube an the spacing between the microphone positions. A louspeaker at one en of the tube was use to generate a broaban ranom signal over the frequency range to 6 Hz, as escribe in section of both stanars, with 2 Hz resolution an a BT prouct of as prescribe in section 9.2. The frequency response functions between the reference microphone locate closest to the louspeaker an the other measurement positions were measure simultaneously by using a Brüel & Kjær Type 356-B- 3 ata acquisition front en an the ulse FFT & CB Analysis Type 77 software in conjunction with the Type 7758 Material Test application running on a personal computer. Even though the yellow an white samples were symmetric front-to-back, an so their properties coul be measure using the one-loa metho, all of the measurements were in fact mae using the two-loa metho as escribe in section of ASTM E26. Yellow fibrous material White fibrous material Gray foam Figure 5: ictures of sample materials teste. 5. EXERIMENTAL RESULTS The experimental results for both the two-microphone ASTM E5 an the four-microphone ASTM E26 are presente in figures 3 an 4 for the white an yellow fibrous materials, an for an aitional material here enote as gray foam. The ASTM E5 test was performe accoring to the stanar. For the testing performe accoring to the ASTM E26, no microphone switching was use. This methoology was use since accoring to reference 8 it was foun that a non-switching-microphone metho in the four-microphone cases generally gives better performance than o switching-microphone methos especially for ifficult materials. One test was performe with a single sample of both white an yellow fibrous materials an three tests were performe with three samples of poroelastic urethane gray foam. INTER-NOISE 29 Ottawa, Ontario Canaa

7 A. Easy Material to Characterize Figure 6: Narrow ban result comparisons for white (left) an yellow (right) fibrous materials. While the results were calculate using a narrow ban FFT, the tabulate results below are presente at the /3 r octave center frequencies. To generate the ata in Table 2, the absorption coefficient at the FFT line that was closest to the one-thir octave center frequency was use. Table 2: /3 r octave soun absorption comparison White Fiber Yellow Fiber ASTM E26 ASTM E5 ASTM E26 ASTM E5 Frequency (Hz) 4 Mic Metho 2 Mic Metho 4 Mic Metho 2 Mic Metho For the ASTM E5 test, the within- an between-laboratory precision expresse in terms of the within-laboratory, 95 percent Repeatability Interval, I(r), an the between-laboratory, 95 percent, Reproucibility Interval, I(R), are liste in Table 3. These statistics are base on the results of a roun-robin test program involving ten laboratories. The Repeatability Interval, I(r) is use for testing conucte in the same laboratory on the same material, in which case the absolute value of the ifference in two test results will be expecte to excee I(r) only about 5 percent of the time. The Reproucibility Interval, I(R) is use for testing conucte in ifferent laboratories on the same material, in which case the absolute value of the ifference in two test results will be expecte to excee I(R) only about 5 % of the time. Table 3: Repeatability Interval, I(r) an Reproucibility Interval, I(R) from the ASTM E5 stanar ariable Statistic α I(r) I(R) INTER-NOISE 29 Ottawa, Ontario Canaa

8 If we compare the calculate results for the normal incience soun absorption coefficients from the two methos with the Repeatability an Reproucibility prescribe in the ASTM E5 it can be seen that the correlation between the two methos is equal to or less than the Repeatability in all of the octave bans within the frequency range that was vali for the mm tube. Thus, it is possible to conclue that the two methos give the same results for lightweight, relatively limp fibrous materials. B. Difficult Material to Characterize Figure 7: Narrow ban result comparison of results for gray foam material sample. Figure 8: Narrow ban result comparison of results for gray foam material sample 2 (left) an 3 (right) As an example of a ifficult material, a flexible poroelastic urethane foam (gray foam) was use. Here, three samples were teste an the results are ivie into two conitions: goo an poor agreement as shown in figures 7 an 8. That is, ifferent results were foun epening on the sample even though all samples were nominally the same. In figure 7, sample shows a reasonably goo agreement between the two calculations. However, in figure 8, poorer agreement is shown for samples 2 an 3. In particular, the calculate absorptions from the four microphone test in figure 8 show large fluctuations near 4 Hz an 7 Hz. These effects come from shearing resonances in the material an they epen on the sample-tube bounary conition 9, i.e., how the sample is cut an how it fits into the tube. For the two microphone test, the sample is hel against a har termination; thus, the sample s movement is restricte by that har surface. However, in the four microphone test, the sample is hel only aroun its circumference not at its back surface. Thus, the sample can move more freely in the four microphone test. As a result, ifferent absorptions are foun in the two-microphone an fourmicrophone tests. Generally foam is a stiffer material than fibrous material: that increase INTER-NOISE 29 Ottawa, Ontario Canaa

9 stiffness can increase the shearing resonance effect an so increases the iscrepancy between the two methos. Even though the same metho was applie when putting the samples insie the tube, the results show ifferences in agreement level from sample to sample. Due to the relatively higher stiffness of the foam, more variation tens to occur in the circumferential bounary conitions, which cause a larger variability in the soun absorption coefficient results. 6. CONCLUSIONS In this work we were successful in correlating the experimental absorption results of the twoan four-microphone methos escribe in the ASTM E5 an E26 stanars, for porous materials that can be moele as fluis. Aitionally, FEA simulation showe excellent agreement between the two- an four-microphone methos. Measurement results showe very goo agreement for fibrous materials an reasonably goo agreement for the poroelastic gray foam material samples. The reason for the ifference between the materials is the ifferent bounary conitions applie to the sample insie the tube. The agreement between the two- an four-microphone methos was better for fibrous materials than for the poroelastic urethane foam material, since fibrous samples usually have a lower stiffness. Fibrous materials act more like fluis, whereas poroelastic urethane foams act more as a combination of flui an soli. The test result for the foam is more sensitive to the bounary conition ue to its increase stiffness that causes a shear resonance. As a result, the ege constraint effect was noticeably visible in the four-microphone test 9,. However, in the foam test, reasonable agreement was still foun. Future work inclues preiction of ranom incience absorption base on the normal incience four microphone measurement REFERENCES Stanar test metho for impeance an absorption of acoustical materials using a tube, two microphones an a igital frequency analysis system, ASTM E-5-8 (ASTM International, West Conshohocken, A, USA, 28). Oliviero Olivieri, J. S. Bolton an Taewook Yoo 26 Measurement of transmission loss of materials using a staning wave tube, INTERNOISE 26 Stanar Test Metho for Measurement of Normal Incience Soun Transmission of Acoustical Materials Base on the Transfer Matrix Metho, E 26-9 (International, West Conshohocken, A, USA, 29). Leping Feng 28 Comparison of ifferent methos for surface impeance an transmission loss measurements in uct, INTERNOISE 28 B. H. Song an J. S. Bolton, A transfer matrix approach for estimating the characteristic impeance an wave numbers of limp an rigi porous materials, J. Acoust. Soc. Am., 7 (3), 3-52 (2). Chung, J. Y., an Blaser, D. A., Transfer Function Metho of Measuring In-Duct Acoustic roperties I. Theory an II. Experiment, J. Acoust. Soc. Am., 68 (3), pp , 98. Oliviero Olivieri, J. S. Bolton an Taewook Yoo Transmission Loss Measurements in a Staning Wave Tube, Bruel an Kjaer 27 Technical Review B 59 ISSN J. S. Bolton an Kang Hou, 28 Calibration methos for four-microphone staning wave tubes, NOISE-CON 28 B.H. Song an J.S. Bolton, Effect of circumferential ege constraint on the acoustical properties of glass fiber materials, J. Acoust. Soc. Am., (2) B.H. Song an J.S. Bolton Investigation of the vibrational moes of ege-constraine fibrous samples place in a staning wave tube, J. Acoust. Soc. Am. 3, (April 23) Taewook Yoo, J. Stuart Bolton, Jonathan H. Alexaner, 25 reiction of Ranom Incience Transmission Loss base on Normal Incience Four-Microphone Measurements, INTERNOISE 25 INTER-NOISE 29 Ottawa, Ontario Canaa