Design of ParaMPA: a micro-perforated absorber

Design of ParaMPA: a micro-perforated absorber Onursal Onen and Mehmet Caliskan Department of Mechanical Engineering Middle East Technical University 06531 Ankara, Turkey ABSTRACT Perforated absorbers with sub-millimeter size holes can provide high absorption coefficients. Various types and configurations of micro-perforated absorbers are now available commercially for different applications covering a wide area. This paper presents work to develop an effective single layer micro-perforated absorber from the commercial composite material Parabeam with micro diameter holes drilled on one side. Parabeam is used as a structural material made from glass fiber and epoxy, constitutes an upper, a lower plate and relatively closely spaced, thick fibers connecting the two plates. The paper outlines the analytical model developed for prediction of normal incidence absorption coefficients and finite element solution for 4 selected samples using commercial software MSC.ACTRAN. Results obtained from analytical model and finite element show good agreement. Different absorption characteristics can be obtained by variation of hole diameter, distance between holes and thickness of the intermediate air layer that includes the fibers, as from nature of perforated absorbers. Based on the developed models, a linear optimization is performed to obtain an efficient absorber configuration. Several different and interesting applications can be possible combining structural and absorption properties of the developed micro-perforated absorber. 1 INTRODUCTION Micro perforated absorbers are basically of metal, plastic or wooden panels (or membranes) with very small (micro size) holes drilled on panels and an air gap between the perforated panel and the wall or backing. Their major advantages lie in the flexibility in design, variety of materials, and high sound absorption coefficients with relatively low thicknesses compared with porous and fibrous absorbers. Micro perforated absorbers have been investigated as clean and health-friendly absorbing materials for almost two decades as an alternative to traditional fibrous and porous absorbers. Micro perforated absorbers are resistant to moist oil and dust and they can be built up from any rigid material available in thin plates. They can be painted or a simple surface finish can be applied. In this particular study, it is aimed to develop an effective single layer micro-perforated absorber from the commercial composite material Parabeam with micro diameter holes drilled on one side. Parabeam is special epoxy-cured glass fabric which naturally involves two layers of individual fabric. These layers are connected to each other with fiber groups that are woven inside the individual layers (Figure 1). After a special curing process, the structure turns into a structure with two plates at the bottom and top and an intermediate layer that is composed of thick fibers connecting the two plates through air in-between. Parabeam shows good mechanical properties and is commonly used in marine industry and thin-walled

structures. With its configuration, Parabeam is anticipated to be a perfect candidate for a micro perforated absorber with additional dissipation effect of fibers in the intermediate layer. Prediction of normal incidence absorption coefficients are carried in 6 octave-band frequencies starting from 125 Hz up to 4000 Hz. Analytical modeling is performed for absorber configurations by varying hole diameter and hole spacing. Finite element modeling is performed for comparison with the analytical model and more realistically modeling the whole problem. Commercial acoustic finite element software MSC.ACTRAN is used including acoustical and structural interaction of fibers and air by defining proper boundary conditions and interface conditions. Finite element model is instrumental for prediction of properties of the layer with air and fibers to be also used in the analytical model. An optimization is performed to obtain effective configurations that can be developed from Parabeam using the analytical model. D. Y. Maa [1,2] was firstly proposed developing effective perforated absorbers by reducing hole diameters to sub millimeter size. Starting from Crandall s [3] short tube wave equation, Maa outlined the theory of micro perforated absorbers using electro acoustical analogy. Various efforts were done to improve the theory of micro-perforated absorbers using electro-acoustical analogy [4-8]. Unfortunately, electro acoustical analogy outlined by Maa is limited to absorbers composed only of air cavity and perforated layers. Various approaches are exercised for prediction of absorption properties in the existence of some other materials and structures like fibrous and porous absorbers either with planar or complex geometries, thin films, honeycomb structures [9-17]. Effective predictions were achieved with planar layers of porous absorbing materials used [13,14]. For complex problems, finite element modeling is used with detail physical modeling of the sample of a problem [15] or by application of complex boundary layer conditions to simplify the model [16,17] Figure 1. Side view of Parabeam with eight shaped fibers (left), a sample micro perforated absorber built up from Parabeam (right). 2 THEORY AND ANALYTICAL MODELING 2.1 Acoustic Impedance of micro-perforated layer The acoustic impedance at the top of a micro hole whose length is short compared with the wavelength of the sound wave at the interval of frequency of interest is given as [1] 2 J1( x j) ωρ 0 1 Zh = j t x jj0( x j) (1) where j is the complex constant being equal to 1, ω is the angular frequency, ρ 0 is the density of air under standard conditions, t is the thickness of the hole (also equal to thickness of the perforated layer), J 0 and J 1 are the Bessel s equations of first type and of order zero and one, respectively and perforate constant x which is defined as

x r ωρ 0 = (2) μ where r is the radius of the hole and μ is the dynamic viscosity of air. Using the impedance of each hole, acoustic impedance of a typical micro perforated layer can be obtained by dividing the corrected impedance of hole by the perforation ratio ε which is defined as the ratio of surface area of the perforations to the total surface area of the layer which is a function of the distance between holes in a square grid a the and radius of a single hole r: Z perf jωρ t 1 ε 0 = 2 J1( x j) x jj0( x j) (3) 2 r ε = π (4) 2 a 2.2 Acoustic Impedance of the Fibrous Layer There are various empirical models for prediction of surface acoustic impedance of fibrous materials. All these models are applicable only to materials with very small fiber diameters void ratios [7]. Such models are not applicable to mid-layer of Parabeam with thick fibers of 0.25 mm x 1 mm of cross section and a total void ratio around 85 % for the whole layer. A finite element model is developed for prediction of acoustic properties of the fibrous part which is to be described in detail in this paper. Acoustic impedance at the top of the layer with its real and imaginary components is given separately in Figure 2 in octave bands as Impedance at the Top of Fibers [N/s] 200-200 -400-600 -800-1000 -1200 Freq (Hz) Figure 2. Real and imaginary parts of acoustic impedance at the top of fibrous layer obtained from finite element modeling

2.3 Determination of Normal Incidence Resulting total acoustic impedance at the top of the micro-perforated layer is simply the sum of acoustic impedance at the top of fibrous layer and acoustic impedance of microperforated layer as Ztotal = Z fibrous + Zperf (5) Using total impedance, reflection coefficient R and normal incidence absorption coefficient α can be found as follows. Ztotal ρ0c R = (6) Z + ρ c total 0 2 α = 1 R (7) 3 FINITE ELEMENT MODELLING In the finite element modeling procedure, a simplified 3D solid model of the absorber and air at the top of the micro perforated layer is developed and meshed in MSC PATRAN with the acoustic elements of the MSC.ACTRAN material library. The model includes a small cross-section of the absorber. The simplified model is built up with mainly keeping ratio of total fiber volume to volume of the whole cavity, which is around 14 %. Care has been exercised to keep the profile and cross-section of modeled fibers as close as to the exact fibers. A simple overview of the model is illustrated in Figure 3. Air cavity inside the Parabeam and above the microperforated layer is modeled with fluid elements and fibers are modeled with elastic solid elements. For modeling the micro perforated layer, a MSC.ACTRAN boundary layer condition is applied rather than directly modeling the holes of micro perforation. This method of modeling about defining such an interface by Mechel s formula as proposed by MSC.ACTRAN [18] avoids time consuming fine meshing both in modeling and solution processes. Also by using the defined interface for application of different micro perforation configurations, necessity of rebuilding of model for each configuration has been evaded. The 4 different samples and an optimized case is analyzed with only changing the required lines of interface definition in the analysis input file. Modal Basis property in Analytic Module of the MSC.ACTRAN is used as excitation model [18]. In Modal Basis, a modal acoustic excitation in terms of duct modes is introduced to the system. The resulting duct modes are identified in terms of longitudinal wave numbers and average modal intensities of the incident and reflected waves are computed. The resulting sound field in the duct is the combination of a known incident sound field and of a reflected sound field. Both the incident and reflected sound fields are defined as linear combinations of duct modes. With the modal basis input to the data file of the finite element model, MSC.ACTRAN solves for the resulting duct modes of the modal basis. The result file includes the average intensity of the reflected wave. The absorption coefficient for the absorber can be calculated by extracting the reflection coefficient from the result file.

Figure 3. A general view of finite element model with whole model (a), a closer view to absorber (b) and fibers modeled inside the cavity (c). The finite element model is composed of 52198 3D elements including 336 HEX6 and 51862 TET8 elements and fibers and also 1606 2D QUAD elements of interfaces and modal excitation corresponding to a sample size of 6 x 9.6 mm and thickness of 18 mm. Average length of the elements is around 0.5 mm. A model very similar to model described above is built for determination of the surface acoustic impedance at the top of the fibrous layer. Fibers and air between the fibers and air at the top is modeled in a similar manner without the interface definition of micro perforated layer. Field points are defined just at the nodes of air elements which are placed at elevation of top of the fibers. Pressure and particle velocity information are obtained separately at each node corresponding to these field points. Such information is later averaged to obtain the surface acoustic impedance at the top of the fibers by division of complex averaged pressure amplitude to complex averaged particle velocity. 4 RESULTS For verification of the models, 4 samples of different micro-perforation configurations are used. Apart from unchanged parameters: thickness of the fibrous layer tf = 18mm, thickness of the micro-perforated layer tp = 1mm, the two parameters: the distance between holes in the square grid a and radius of holes r are varied. The sample configurations are given in Table 1 below. Table 1: Sample Configurations Sample # a (mm) r (mm) 1 16 0.5 2 8 0.5 3 16 0.3 4 8 0.3

Predictions for normal incidence absorption coefficients by analytical and finite element models are carried and illustrated in the Figures 4-7 for each sample. 1,00 0,90 0,80 0,70 0,60 0,50 0,40 0,30 0,20 0,10 Figure 4. Predicted normal incident absorption coefficients for Sample 1. analytical model (blue-solid), finite element model (dashed-pink). 1,00 0,90 0,80 0,70 0,60 0,50 0,40 0,30 0,20 0,10 Figure 5. Predicted normal incident absorption coefficients for Sample 2, analytical model (blue-solid), finite element model (dashed-pink).

1,00 0,90 0,80 0,70 0,60 0,50 0,40 0,30 0,20 0,10 Figure 6. Predicted normal incident absorption coefficients for Sample 3, analytical model (blue-solid), finite element model (dashed-pink). 1,00 0,90 0,80 0,70 0,60 0,50 0,40 0,30 0,20 0,10 Figure 7. Predicted normal incident absorption coefficients for Sample 4, analytical model (blue-solid), finite element model (dashed-pink). Through inspection of the Figures 4-7, predictions of absorption coefficients from analytical and finite element models show fairly good agreement, thus analytical model can be applied for design and optimization of this particular type of absorber. The analytical model is employed for a simple optimization procedure with two control parameters by assignment of a linear objective function by Solver add-in of MS EXCEL. In this optimization procedure, weightings are applied to absorption coefficients in octave-bands in the desired manner to be able to optimize absorption characteristics. The weighted individual absorption coefficients are summed up to set up the objective function. By this simple optimization technique, an optimum configuration with distance a between each hole being 13 mm apart and radius r of a single hole being 0,4 mm is obtained as follows:

1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00 Figure 8. Predicted normal incident absorption coefficients for the optimized sample. a = 13mm, r = 0, 4mm, analytical model (blue-solid), finite element model (dashed-pink). 5 CONCLUSIONS From the results obtained, it can be concluded that the analytical and finite element predictions show good agreement except the peak values of results obtained from the configurations with hole diameter of 0.6 mm. Apart from that, general characteristics of absorption performances of the micro perforated Parabeam are fairly good. The acoustic impedance values at the top of the layer with fibers (Figure 1) show that the effect of fibers are not felt in real part but in imaginary part of the impedance. From this, it can be concluded the air cavity itself still dominates the absorption characteristics with existence of fibers being a modification to the impedance. The absorption characteristics are still driven by combined effect of the micro perforations and the air cavity of behind the perforated plate. Even with the presence of fibers and the optimization performed, it seems it is difficult to obtain a wideband absorber with relatively high thickness of the perforated face. For effective low frequency absorption, a higher value of acoustic resistance is needed and this Parabeam configuration lacks sufficient acoustic resistance in this frequency range. Effective absorbers can be possibly obtained by lowering the thickness of the perforated face and hole radius. However, these treatments will require further processing of the material. Still Parabeam offers an alternative solution for special areas due to its low weight and moisture resistance. 6 ACKNOWLEDGEMENTS Aydın Kuntay of BIAS Inc. of Turkey, Jonathan Jacqmot of FFT Technologies Pty. Ltd. and Jaap Jan Kleef of Parabeam 3D Glass Fabrics are gratefully acknowledged for their help and assistance. 7 REFERENCES [1] D.Y. Maa, Microperforated-panel wideband absorbers, Noise Control Engineering Journal, Vol. 29, No. 3, pp. 77 84, 1987 [2] D.Y. Maa, Potential of microperforated panel absorber, Journal of Acoustical Society of America, Vol. 104, No. 5, pp. 2861-2866, 1998 [3] C. Zwikker, C. W. Kosten, Sound Absorbing Materials, Elsevier Publishing Company, Inc., New York, 1949

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