Equivalent Circuit Method Based Double Layer Micro-perforated Panel (MPP) Design to Widen the Sound Absorption Bandwidth Ela Fasllija 1 Bilkent University Department of Interior Architecture and Environmental Design, Bilkent University, Ankara, Turkey Semiha Yilmazer 2 Purdue University Ray W. Herrick Laboratories, Purdue University West Lafayette, Indiana, U.S.A Cengiz Yilmazer 3 CSY R&D and Architecture Engineering, Ankara, Turkey

ABSTRACT

Acoustical solutions that occupy minimum space and simultaneously allow for flexibility in interior spaces are not feasible by using thick absorbers like conventional porous and fibrous materials. Micro-perforated panels(MPPs) are an excellent alternative to those classical materials due to their environmental friendliness, design flexibility on various materials, and easy installation. However, they come with the handicap of having a very narrow absorption frequency bandwidth. This study explores two inhomogeneous microperforated layers arranged in a cascade with different back cavities in order to widen the sound absorption bandwidth. This serial-parallel architected MPP structure is investigated by mathematical models using Maa's model for a single MPP and the Equivalent Circuit Method (ECM). The numerical results were validated through experiments for normal sound incidence in the impedance tube using the transfer function method. Decreasing the back cavity length between the two layers of the structure results in a higher absorption in the high-frequency range. In contrast, combinations of larger cavity lengths and smaller perforation ratios are more effective in the low-frequency spectrum. By tuning the design parameters of those structures, it is possible to achieve wideband absorption by fibreless materials for room acoustic applications.

1 e.fasllija@bilkent.edu.tr

2 syilmaze@purdue.edu

3 censem64@hotmail.com

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1. INTRODUCTION

State of the art in absorbing materials shows that passive absorbers such as fibrous or porous ma- terials are indispensable for high-frequency damping noises. Still, problems are increasingly found at the low frequencies due to room modes 1 . Fibrous and porous materials (mineral wool, fiberglass, acoustic plaster and gypsum, foam, etc.) can absorb the sound energy from waves impinging on them. Their limitations lie in the required large thickness for damping low frequencies and the fra- gility of their open structure, causing severe hygiene problems 2 . Resonant systems such as Micro- Perforated Panels (MPPs) are usually used for treating low-frequency problems encountered in rooms acoustics. The latter are alternative fibreless absorbers that can cope with the highest hygiene demands, but they usually only offer narrowband absorption. Placing microperforated panel absorb- ers at a distance in front of a rigid rear wall launched the development of new acoustic elements that do not require porous/fibrous damping materials. Their acoustic effectiveness can be precisely ad- justed almost independent of the panels' material (as long as it is a non-vibrating material) but solely by choice of its geometric parameters ( d -hole diameter, p -perforation ratio, t -panel thickness, and D -back cavity).

MPPs can be made of transparent or colourful plates or membranes, so they are also in demand by architects for sound quality control in auditoriums. Apart from metals, MPPs have been inte- grated into acrylic glass absorbers 3 and wood-based ones 4 . A series of structures based on MPP net- works in series and parallel have been explored to introduce multiple resonances to broaden the ab- sorption bandwidth. Maa 5 first proposed a Double-layer MPP absorber in 1987, based on which several studies have followed 6-8 . Those studies show an increase in the overall absorption fre- quency band employing multiple-layer structures when the parameters are tuned accordingly. The design of an N-layer MPP depends on 4N parameters generating design complexity. This study probes to investigate the acoustic absorptive performance of a double layer inhomogeneous trans- parent MPP structure having different back cavities and ultimately can be used for architectural ap- plications. 2. THEORETICAL MODEL

This section includes the theoretical model for single-layer MPP, firstly introduced by Maa 9 in 1975. It is followed by introducing the Equivalent Circuit Method (ECM) to calculate the overall surface impedance of inhomogeneous MPPs, and lastly, it presents how to obtain the total impedance of a double layer inhomogeneous MPP having different cavities by using the Transfer Matrix Methods.

2.1. Single Layer MPP

Maa 9 found that if a plate containing holes in the submillimeter range is placed at a distance from a hard reflective surface, absorption occurs because shearing forces damp the vibration of the air in the tiny holes at the neck of the holes. The system's efficiency depends on the mass, the stiffness, and the inherent friction, whereas these parameters can be adjusted following the specific acoustic requirements. The acoustic impedance of an MPP at normal incidence can be calculated as shown in Equation (1).

+ !

+ !

() #1 + 1 $3 $ +

#$%&

√ $

*

-&

*

'()* ! #$1 +

& ( (1)

𝑍 !"" =

#$ +

#$ 𝑘

& ( +

$ * + 0.85

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When 𝑘= 𝑑 2 * $ 𝜔𝜌 𝜂 * is the perforation constant, w =2 p f is the angular frequency, 𝜂 is viscosity,

r = air density, d,t,p being the hole diameter, panel thickness, and perforation ratio, respectively. When the MPP is placed at a distance from the wall, the resulting surface impedance Z tot adds the impedance of the panel with the impedance of the air cavity having a depth of D as explained in Equations (2) and (3).

𝑍 &.& = 𝑍 !"" + 𝑍 /01 (2) 𝑍 /01 = −𝑗cot

-∗3

) (3) 2.2. Inhomogeneous MPP

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According to the electric equivalent circuit model, when n number of MPP is connected in paral- lel, the total impedance of the panel is calculated as shown in Equation (4).

:9

𝑍 &.& = :∑ ∅ " 5 #$$"

6 789 <

(4)

When ∅ is the ratio each panel has on the overall surface panel. A schematic diagram of a simple example of two MPPs linked in parallel having different cavities is shown below (Figure 1) with its equivalent circuit.

=] =

Figure 1. A schematic diagram of two MPPs with N cavities and

its respective equivalent circuit

2.3. Double-layer inhomogeneous MPP

A wider band absorption can be achieved by cascading multiple MPP layers into a composite. The resulting total impedance of this structure can be derived by using the transfer matrix method. This method calculates the surface impedance of single and multiple layers of absorbers. Transfer parameters show the relation between initial pressure and velocity, the change they undergo in a transmission region, and the outgoing pressure and particle velocity. Below in Equations (5) and (6) are presented the transfer matrices for an MPP and a single cavity. Equation (7) shows how to cal- culate the total impedance of the arrangement by transfer matrices.

𝑇 !"" = ? 1 𝑍 !"" 0 1 @ (5)

Te inal

cos 𝑘𝐷 −𝑖𝑍 0 sin 𝑘𝐷 1 𝑍 0 * sin 𝑘𝐷 cos 𝑘𝐷 B (6)

𝑇 /01 = A

𝑇 &.&0; = 𝑇 !"" ∗𝑇 /01 ∗… (7)

Where 𝑘= = w /c and 𝑍 < is the characteristic impedance of air. Below in Figure 2, a double layer inhomogeneous MPP is presented with its respective equivalent circuit.

5 0

5 5 0

5

0

0

SF

='

='

Figure 2. A sc hema tic diagram of double-la yer Inhomogeneous MPP and

its respective equivalent circuit

In order to find the normal incidence absorption coefficient of the double-layer structure, the following equations (Eq.8 and Eq.9 can be used).

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𝛼= 1 − |𝑅| $ (8)

" % " " =

5&.&:5<

𝑅 =

5&.&=5< (9)

In the context of this study, a double layer inhomogeneous MPP structure having four different MPPs in each layer, a 2cm cavity between two layers that are partitioned with rigid walls, is being investigated. Moreover, the second layer is backed by different length cavities facing different MPPs. Table 1 explains the parameters belonging to each layer, and Figure 3 shows one section of the proposed structure with its respective equivalent circuit. Table 1:Parameters of the double laye r i nhomogeneous MPP absorber

MPP layer 1 MPP Layer 2

D (mm)

T (mm)

p %

D1 (mm)

D (mm)

T (mm)

P %

D2 (mm)

MPP1-a 0.7 6 2,8 20 MPP2-a 2 6 6,4 40

MPP1-b 2 6 6,4 20 MPP2-b 1 6 4,9 80

MPP1-c 1 6 4,9 20 MPP2-c 1 6 1,6 60

MPP1-d 1 6 1,6 20 MPP3-d 0.7 6 2,8 20

=] =

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Figure 3. A schematic diagram of the proposed double-layer Inhomogeneous MPP and

its respective equivalent circuit 3. MATERIALS AND METHODS

3.1. Digital fabrication

The MPP circular prototypes seen in Figure 4 were firstly designed in AutoCAD2021 and sent to the fabrication tools such as CNC and laser cutter. CNC stands for Computerized Numerical Control and is a computerized manufacturing process in which pre-programmed software and code control the movement of production equipment. Prototypes with diameters of nearly 100mm and 30mm were manufactured to normal incidence absorption data according to TS EN ISO 10354-2 10 .

Figure 4. The prototypes fabricated for the scope of this study and the back cavity structure 3.2. Acoustic measurements with preliminary results

The proposed structure's normally incident sound absorption coefficient was measured using the transfer function method by an impedance tube. Each layer was attached to the partitions with a double-sided table, and the structure was fit as a whole inside the impedance tube. Since it did not have any other enveloping structure, the walls of the impedance tube formed the specific back cavities for each MPP. Figure 5 shows the measurement setup used to which data was used to obtain the total absorption coefficients. Using the microphones, frequency-dependent measurements were taken, and a frequency-sound absorption coefficient graphic was formed (See Figure 6). To test the repeatability of the experimental measurements and reproducibility of the materials, each measurement was repeated with three samples in order to perform reliability analysis.

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Figure 5. Impedance tube setup of Turk Standartlar Ensitutusu (TSE), Tuzla, Istanbul

The preliminary results show a good agreement between the mathematical model developed us- ing the ECM and the measurements obtained by experimental data. Compared to traditional MPPs, this serial-parallel proposed structure of MPPs has wider absorption bandwidth, especially in the low frequencies (250 - 600 Hz), due to the inhomogeneous MPPs, which welcome the incident wave and the back cavities (D2) after the second layer. Another resonance peak is present in the high-frequency range at 1600 Hz, which happens due to the small cavity (D1=20mm) between the two inhomogeneous layers. Also, when designing double layer structures, it is crucial to design waveguides of the incident sound, which significantly affect the overall absorption.

Figure 6. Measured results and theoretical prediction model by ECM of the double-layer

inhomogeneous MPPs

a a a a Frequency (H2)

4. CONCLUSIONS

This study investigates the possibility of designing transparent, double-layer inhomogeneous MPPs that can be used instead of conventional absorbing materials in architectural applications to attenuate low frequencies. The results show that mathematical models using ECM provide rela- tively accurate results in line with experimental data. Combining serial-parallel MPPs increases the resonance peaks of the overall structure. However, the design of such arrangements may be rela- tively complex but is quite desirable in spaces where the thickness of the panels and usage of fi- brous/porous materials is a critical constraint. Those structures' parameters can be tuned and opti- mized according to the frequencies required to be absorbed. 5. ACKNOWLEDGEMENTS

The authors would like to thank the Ortac Research group at UNAM, Bilkent University, and Sertek Interactive Exhibition for the help provided while producing the prototypes.

6. REFERENCES

1. Fuchs, H. Applied Acoustics: Concepts, absorbers, and silencers for acoustical comfort and noise control: Alternative solutions - Innovative tools - Practical examples . Springer-Verlag Berlin Heidelberg, 2013. 2. Qiu, X. Principles of Sound Absorbers. Acoustic Textiles. Textile Science and Clothing Technology . Springer, Singapore, 2016. 3. Sakagami, K., & Okuzono, T. Some considerations on the use of space sound absorbers with next-generation materials reflecting COVID situations in Japan: additional sound absorption for post-pandemic challenges in indoor acoustic environments. UCL Open Environment , 1 , 1–10 (2020). 4. Sheng, S., & Mo, F. Study on wooden micro-perforated panel and its application. Proceedings of 20th International Congress on Acoustics ICA 2010, 23-27 August 2010, Sydney, Australia 5. Maa, D. Potential of microperforated panel absorber. Journal of the Acoustical Society of America 104( November 1997) , 2861–2866. (1988). 6. Wang, S., & Li, F. A broadband sound absorber of hybrid-arranged perforated panels with perforated partitions. Applied Acoustics , 188, 108547, (2022). 7. Mosa, A. I., Putra, A., Ramlan, R., & Esraa, A. A. (2020). Wideband sound absorption of a double-layer microperforated panel with inhomogeneous perforation. Applied Acoustics , 161 , 107167 (2020). 8. Lu, C. H., Chen, W., Zhu, Y. W., Du, S. Z., & Liu, Z. E. Comparison analysis and optimization of composite micro-perforated absorbers in sound absorption bandwidth. Acoustics Australia , 46(3), 305-315, (2018). 9. Maa, D. Theory and design of microperforated panel sound-absorbing constructions. Scient. Sin. , 18(1) , 55–71, (1975). 10. ISO Standard 10534–2 Acoustics – Determination of Sound Absorption Coefficient and Impedance in Impedance Tubes – Transfer Function Method International Organization for Standardization (1998)

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