HAL Id: hal-03235478
https://hal.archives-ouvertes.fr/hal-03235478
Submitted on 12 Jun 2021
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of
sci-entific research documents, whether they are
pub-lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
Analysis of microporous sheets for transmission loss
applications: an extension to MAA’s equation
Christopher Fuller, Curtis D. Mitchell, Morgan A. Bjerke
To cite this version:
Christopher Fuller, Curtis D. Mitchell, Morgan A. Bjerke. Analysis of microporous sheets for
transmis-sion loss applications: an extentransmis-sion to MAA’s equation. Forum Acusticum, Dec 2020, Lyon, France.
pp.2873-2878, �10.48465/fa.2020.0744�. �hal-03235478�
ANALYSIS OF MICROPOROUS SHEETS FOR
TRANSMISSION LOSS APPLICATIONS: AN EXTENSION TO
MAA’S EQUATION
Christopher. R. Fuller
1Curtis. D. Mitchell
2Morgan A. Bjerke
21Department of Mechanical Engineering, Virginia Tech, USA 2SMD Corp, Manassas, VA, USA
cfuller@vt.edu
ABSTRACT
Recent work has been carried out on the developing and testing of poro-elastic acoustic meta materials(AMM) based upon periodically spaced microporous sheets(MPP) embedded in a poro-elastic matrix. These poro-elastic AMM have been shown in analysis and testing to have both a superior absorption coefficient and increased transmission loss particularly at low frequencies. Analysis of these AMM systems is based upon using a Finite Element code such as COMSOL for the modeling of the dynamics of the poro-elastic matrix in conjunction with transfer impedances derived using Maa’s equations for the dynamics of the MPP sheets. While Maa’s equations are suitable for prediction of absorption of sound by MPP sheet systems they have issues for prediction of the newer application of transmission loss. This paper will discuss these limitations and propose an extension to Maa’s equation for the transfer impedance of MPP, for use in transmission loss analysis. Illustrative numerical examples are presented and the results discussed.
1. INTRODUCTION
Microporous sheets(MPP) have been demonstrated to have high potential as broadband low frequency sound absorbers[1,2]. More recently they have been used by Fuller as the scattering elements of Acoustic Meta Materials(AMM)[3,4,5]. In Fuller’s AMM configuration, the MPP is embedded in an acoustic matrix material such as poro-elastic foam or fiberglass with a periodic spacing. The embedded MPP sheets have the effect of modifying the acoustic impedance and slowing down the acoustic waves of the matrix material. Fuller has analyzed and tested various arrangements of AMM using MPP sheets embedded in acoustic foams. Figure 1 shows and example of a poro-elastic meta material consisting of a Melamine foam matrix supporting three embedded MPP sheets.
Figure 1. Arrangement of poro-elastic AMM[4,5].
In the right hand picture of Figure 1, the MPP sheets have been slid out of the assembled position to show their configuration and axial location.
The measured and calculated absorption coefficient of 4ft by 4ft section of 2 inch thick section of Melamine foam containing three embedded polycarbonate MPP sheets is given in Figure 2. The MPP sheets were of 0.27mm thickness, 0.2mm hole diameter and had 1.7% POA. The measurements were made in a large reverberation chamber. The presence of the MPP sheets can be seen to double the sound absorption coefficient at 200Hz for example, which is very significant. Examining the curves it appears that the MPP sheets have moved them to the left on the frequency axis and this is consistent with the acoustic waves being slowed down while propagating thought the AMM, when compared to the standard matrix material.
Figure 2. Absorption coefficient of AMM versus standard Melamine foam under random incidence [4,5]. The application of the above AMM system to transmission loss applications such as in aircraft trim panels has also been considered. Figure 3 shows the transmission loss of
various AMM configurations measured in a transmission loss test tube. The incident wave is an acoustic plane wave.
Figure 3. Measured normal incidence transmission loss of AMM[4,5].
The MPP sheets were 0.27mm thick, the holes were of 0.2mm diameter and the POA was 0.8%. The addition of the MPP sheets can be seen in Figure 2 to increase the transmission loss of the foam at both high and low frequencies. The dip in the transmission loss is associated with the axial resonance of the AMM material and can be adjusted by using different MPP sheets and their number and spacing. This aspect illustrates an important, desirable characteristic of AMM poro-elastic materials in that their acoustic behavior is designable. In order to analyze the transmission loss of poro-elastic AMM it is desirable to have the transfer impedance of the MPP sheets under acoustic wave transmission. While Maa’s equations have been derived for absorption applications it appears that they need to modified for transmission loss applications and this is the subject of this paper.
2. EXTENSION TO MAA’S EQUATION Maa derived the original relationship for the transfer impedance of a microporous(MPP) sheets by considering the MPP sheet to be a “lattice of short narrow tubes, separated by distances much larger than their diameter but small compared to the wavelength of impinging sound wave”[1]. Crandall has derived the transfer impedance of a short tube of small diameter as[6],
ܼଵ= ο ௨ = ݆߱ߩݐ 1 െ ଶ ඥି భ൫ඥି൯ బ൫ඥି൯൨ ିଵ (1)
where k= r0 ȡ0ȦȘ 1/2. See Maa’s paper for a list of
symbols[1]. For the MMP sheet which is considered to be an array of small tubes, Maa assumed that the wave motion in each tube was assumed to be in phase and the particle velocity of the same magnitude(valid for normal incidence) and thus the total particle velocity is the direct sum of the particle velocity in each tube. The transfer impedance of the MPP sheet was then derived by Maa to be[1], ݖ= భ ఙఘబ= ݎ + ݆ݔ= ݎ + ݆߱݉ (2) ݎ = ଷଶఎ௧ ఙఘబௗమ݇, ݇= ቂ1 + మ ଷଶቃ ଵ/ଶ + ξଶ ଷଶ݇ ௗ ௧ (2a) ߱݉ = ఠ௧ ఙ݇, ݇= 1 + ቂ1 + మ ଶቃ ିଵ/ଶ + 0.85ௗ ௧ (2b)
and ݇ = ݀ඥ߱ߩ/4ߟ . The important paremeters in the
DERYH HTXDWLRQV IRU WKH GLVFXVVLRQ RI WKLV SDSHU DUH ı which is the area ratio of the holes of the MPP sheet. This is expressed as the ratio of the area of the holes to a unit area of the sheet and is sometimes written as a percentage(POA or percent open area). In addition d is the diameter of the MPP sheet holes. Thus ıFDQDSSURDFK]HUR either by the number of holes approaching zero, the diameter of thHKROHVDSSURDFKLQJ]HURRUERWK:KHQıLV equal to zero the MPP sheet reduces to a continuous sheet with no perforations.
Examining Maa’s assumptions and the resulting equation for the transfer impedance, it is apparent that the matrix of short tubes is essentially embedded in a rigid sheet of infinite impedance. Thus in equation(1), when the area ratio approaches zero(i.e. the microporous hole total area becomes smaller and smaller), the transfer impedance of the MPP sheet approaches infinity or that of a rigid sheet. This behavior is somewhat suitable for the analysis of MPP sheets used in reflection and absorption of incident sound applications where a rigid MPP sheet with ı would give total reflection of sound. However, for transmission loss analysis of arrays of MPP sheets (as in the AMM of Fuller[3,4,5]), this characteristic of Maa’s equation causes problems. Using Maa’s equation, the transmission loss of an array of MPP sheets would approach infinity as the area ratio approaches zero, since the impedance approaches infinity. The correct solution for transmission loss application is that the impedance of the MPP sheet should approach the impedance of the sheet material when the area ratio approaches zero. Here we propose a straightforward modification and an extension of Maa’s equation to represent the transfer impedance of an MMP sheet for use in transmission loss analyses.
Figure 4 shows a schematic arrangement of an MPP sheet considered to be an array of short, small diameter tubes embedded in an MPP sheet. An acoustic plane wave is assumed to impinge on the MPP sheet at normal incidence.
Figure 4. Schematic arrangement of wave transmission through an MPP sheet.
Some of the acoustic energy will transmit through the tubes and some will transmit through the actual MPP sheet material surrounding the holes. We consider uholesand usheet
to be the spatially averaged particle velocity over the holes and sheet area. The hole and sheet dynamics can be considered in parallel thus the total impedance of the MPP can be simply modelled as two impedances in parallel or,
ଵ ௭ೢ=
ఙ௨ೞ ା(ଵି ఙ)௨ೞ
௱ (3)
where uholesis the particle velocity through the holes and
usheetis the particle velocity transmitted through the sheet
at the surface of the sheet. Manipulating equation(3) we get, ଵ ௭ೢ= ଵ ௭ೌೌ + (ଵିఙ) ௭ೞ (4) and ଵ ௭ೢ= ௭ೌೌ(ଵିఙ) ା ௭ೞ ௭ೌೌ ௭ೞ (5)
where Zmaa is the impedance value for the MPP sheet
originally derived by Maa in equation(2). Note Maa’s LPSHGDQFHLQFOXGHVWKHDUHDUDWLRQıRIWKHKROHVFinally we derive the expression for the total MPP sheet impedance including sound transmission through the sheet material surrounding the holes as,
ݖ௪ =
௭ೌೌ ௭ೞ/(ଵିఙ)
௭ೌೌା ௭ೞ/(ଵିఙ) (6)
where all the impedances are specific. It can be seen from equation (6) that, as expected, the impedances of the holes(the impedance originally derived by Maa for the MPP sheet[1]) and the impedance of the sound transmission through the sheet material surrounding the holes sum as if in parallel. ,IıDSSURDFKHV]HUR=newwill
approach Zsheet, the impedance of the sheet material which
is physically consistent. Note ZmaaDSSURDFKHVLQILQLW\DVı
approaches zero (see equations (2), (2a) and (2b)). From equations (2), (2a), 2(b) and (6) we can see that the modified impedance will approach the impedance of the sheet material as the area ratio of the MPP sheet approaches zero. On the other hand, Maa’s impedance, equations (2), (2a) and (2b) will approach infinity as generally required for reflection control using an MPP sheet. Note that the corrected transfer impedance of the MPP sheet, due to transmission through the sheet material, will also affect the MPP sheet absorption coefficient and this needs to be investigated in the future.
If we consider a very thin sheet of MPP material made of polycarbonate or metal embedded in air, then as shown in Kinsler et al[6], for bulk wave transmission, the impedance of the sheet will be very close to the impedance of air. Thus, since we are considering very thin MPP sheets at low frequencies, we can approximate the sheet material impedance in equation (3) as the impedance of the surrounding acoustic medium. In other words, there will
be complete transmission of the sound through the sheet. Thus, since we are considering very thin MPP sheets at low frequencies we can represent the sheet material impedance in equation (3) as the impedance of the surrounding acoustic medium. An improvement of this approach might be to calculate the transfer impedance of the sheet material by considering it a thin, distributed elastic structure rather than as a bulk material as modeled here.
3. NUMERICAL EXAMPLES
In order to illustrate the use of this extension of Maa’s equation to the analysis of acoustic transmission through MPP sheets, we consider the system of Figure 5.
Figure 5. Schematic arrangement of system used to analyze sound transmission through an array of MPP
sheets.
In the system of Figure 5, we analyze the acoustic response and transmission loss of three MPP sheets located in an air filled standing wave tube with an infinite termination at the right hand end.
A plane wave is considered incident on the MPP sheets. The analysis is carried out using the COMSOL Multi-Physics Finite Element package. The MPP sheets are considered to made from polycarbonate with a thickness of 0.5 mm and hole diameter of 0.2 mm. These values, in conjunction with the POA, are used to calculate the MPP sheet transfer impedance using either Maa’s equation or the modified impedance equation (6) and the values are then used in the COMSOL model discussed below. Various Percent Open Area(POA) ranging from 0.2% to 1.5% MPP sheets are analyzed. POA is area ratio ı expressed as a percentage. The transfer impedance of the MPP sheets were calculated using both Maa’s impedance of equation(2) and the corrected impedance of equation(6). These two MPP sheet impedances were used in the COMSOL model to calculate the transmission loss of the array of MPP sheets in Figure 5 and the results compared. Figure 6 shows the COMSOL model of a wave tube geometry with a one-inch total air space matrix and three equally-spaced MPP sheets used in the transmission loss study. Boundary conditions include perfectly absorbing inlet and exit ports and sound hard boundaries on the inner walls of the duct. The MPP sheets are
represented as a transfer impedance surface in the finite element analysis of the COMSOL model. The noise source incident on the MPP sheet arrangement is considered to be a plane acoustic wave with 1 Pa amplitude. Transmission loss as a function of frequency is computed utilizing the original Maa’s impedance equation(see equation (2)) model and the modified impedance equation(see equation(6)) for POA's 0.2, 0.5, 1.0, and 1.5 as shown in Figures 7-10 below. Sound Pressure Level contour plots showing the dB distribution down the wave tube is provided in Figures 11 and 12 for the original and modified analytical impedance models, respectively.
Figure 6. COMSOL model used for AMM acoustic analysis.
The results of Figure 7 show that, as expected, when the POA(aQG ı DUH VPDOO WKH 033 WUDQVIHU LPSHGDQFH provided by Maa’s equation is very high. Consequently, the COMSOL predicted transmission loss of the MPP sheet arrangement is large and of the order of 40dB. In contrast, using the corrected MPP sheet transfer impedance gives a much lower, more physically reasonable, transmission loss of around 10dB. This result can be compared to the experimental result of Figure 2 and can be seen to be much closer to the measured value for a similar system (note the results Figure 2 are for AMM sheets embedded in Melamine foam, not air as in the case of this paper). Similar values of transmission loss of MPP sheets in air were measured by Slagle and Fuller[8].
Figure 7. Transmission loss for 0.2 POA, MPP arrangement.
We now consider the comparison of increasing values of POA shown in Figures 8, 9 and 10 for POA’s of 0.5, 1.0 and 1.5 respectively. The results show that as the POA is increased, the predicted transmission loss using a transfer impedance from Maa’s equation reduces as expected and the difference between the calculated transmission losses using Maa’s equation and the modified impedance equation reduce. This is also expected because the transmission of sound through the sheet material becomes less dominant as the POA is increased.
What is surprising is that the transmission loss of the MPP system calculated using the modified impedance does not change significantly as the POA is increased nor with frequency. It can be seen to reduce by a few dB’s when the POA is increased from 0.2 to 1.5. This behavior is presently under investigation and may differ when the AMM matrix material is a different medium, such as Melamine foam etc.
Figure 8. Transmission loss for 0.5 POA, MPP arrangement.
Figure 9. Transmission loss for 1.0 POA, MPP arrangement.
Figure 10. Transmission loss for 1.5 POA, MPP arrangement.
Finally, Figure 11-14 show the COMSOL predicted SPL distribution throughout the duct and the MMP systems using the same color bar scale for dB. Results were calculated using Maa’s impedance and the modified impedance for comparison for a frequency of 500Hz. The results show that the Maa impedance leads to much stronger attenuation of sound through the MPP sheet particularly when the POA is small. Interestingly, when the POA is larger there is more attenuation of sound distributed through the MPP sheet system suggesting that the periodic arrangement dynamics come more into play when the POA is larger. There is little evidence of an upstream standing wave as in the inlet duct before the MPP sheet as it is only 2.5 inches length in the COMSOL model.
Figure 11. Sound Pressure Level (dB) for 0.2 POA, using original Maa impedance model.
Figure 12. Sound Pressure Level (dB) for 0.2 POA, using modified impedance model.
Figure 13. Sound Pressure Level (dB) for 1.5 POA, using original Maa impedance model.
Figure 14. Sound Pressure Level (dB) for 1.5 POA, using modified impedance model.
4. CONCLUSIONS
An extension to Maa’s relationship for predicting the transfer impedance of MPP sheets is proposed in this paper. The extension takes account of the transmission of sound through the MMP sheet material as well as through the micro holes. When used as the basis to calculate the transmission loss through a periodic arrangement of MPP sheets(an acoustic meta material), the modified impedance predicts more realistic values of transmission loss than is obtained when by Maa’s equation. The modified impedance should provide the basis for more accurate analysis of MPP sheet acoustic behavior under sound transmission.
5. ACKNOWLEDGEMENTS
The first author gratefully acknowledges the financial support of this work by NASA Langley Research Center. Helpful discussions with Dr. Noah Schiller of SAB, NASA Langley Research Center on the subject of this paper, and assistance with the graphics by his VT graduate student, Melissa Polen, are also gratefully acknowledged.
6. REFERENCES
[1] D.Y. Maa, “Potential of Microperforated Panel Absorber”, Journal of the Acoustical Society of
America, Vol. 104(5), pp. 2861-2866, 1998.
[2] D. Herrin, J. Lui and A. Seybert, “Properties and Applications of Microperforated Panels”, Sound and
Vibration, Vol.45(7), pp.6-9, 2011.
[3] C.R. Fuller and T.D. Saux, “Sound Absorption Using Poro-elastic Acoustic Meta Materials”, Proceeding of
Inter Noise 2012, New York, NY, 2012.
[4] C.R. Fuller and G.P. Mathur,” Poro-elastic Acoustic Meta Materials”, Proceedings of Inter Noise 2017, Hong Kong, 2017.
[5] C.R. Fuller, ”Poro-elastic Acoustic Meta Materials with Improved Sound Absorption”, Proceedings of
22nd International Conference on Composite
Materials(ICCM22)”, Melbourne, Australia, 2019.
[6] I.B. Crandall, “Theory of Vibration System and Sound”, Van Nostrand, New York, 1926.
[7] L.H. Kinsler, A.R. Frey, A.B. Coppens and J.V. Sanders, “Fundamentals of Acoustics”, John Wiley
and Sons, 4th Edition, New York, 2000.
[8] A.C. Slagle and C.R. Fuller, “Low Frequency Noise Reduction Using Poro-Elastic Acoustic Meta Materials”, Proceeding of the 21st AIAA/CEAS
