Predicting alpha cabin sound absorption in an industrial context

Predicting alpha cabin sound absorption in an industrial context François-Xavier BÉCOT 1 ; Christophe Locqueteau 2 ; Julia Rodenas 1 1 MATELYS - Research Lab, 7 rue des Maraîchers, Bât B, 69120 Vaulx-en-Velin, FRANCE 2 Renault, Le Parc de Gaillon, 27940 Aubevoye, FRANCE ABSTRACT In the automotive industry, sound absorption performance is often assessed in terms of ''alpha cabin'' sound absorption coefficient. This measurement setup was initially proposed to test real trims in conditions which were inspired from those of the ISO 354 standard. This type of measurement is now widely spread in the automotive industry even though it suffers from a number of well known issues : absorption coefficients larger than one, limited frequency range, mounting conditions... These characteristics give rise to a number of stakes from a modeling point of view. Therefore, this paper mainly explores the prediction of the ''alpha cabin'' performance e.g. from surface impedance data obtained using ISO 10534-2 measurement or from fine characterization procedure of each individual layer. Simulated data are compared with measured data obtained on several types of trims, from purely absorbing slabs to impervious surfacings needed to cope with industrial usage constraints. Results show that linear regression may be used to extrapolate some existing results while carefully implemented TMM could be predict with fair accuracy the Alpha cabin results of trims which do not yet exist. Keywords: Alpha cabin, Porous, Absorption I-INCE Classification of Subjects Number(s): 35, 52.3, 73 1. INTRODUCTION The Alpha Cabin quantity is one of the sound absorption data used by material makers and automotive end-users either to prescribe performance levels or to assign levels to be achieved. Therefore it is essential for the two parties to understand the stakes of this quantity and how it could be controlled. Alpha cabin initially originates from trim suppliers as a test of "part" elements as opposed to impedance tube measurements preferred for testing "material" elements. Parts are indeed shaped components, possibly including multiple layers of various nature : impervious, solid, elastic, porous, highly resistive thin layers etc. In addition, most of these elements have varying thickness due to architecture and production constraints and may include curved surfaces used for mechanical and/or mounting issues. The test conditions are inspired from those of the ISO 354 and ASTM C423-02 standards [1,2]. It consists in a reverberant chamber having a volume of about 7 m3 with non parallel walls. Similarly to ISO 354, one or several loudspeakers generate a broad band sound which results in diffuse sound field in a given frequency range. The lower frequency limit may be estimated by the so-called Schroeder frequency which gives a lower frequency limit around 1200 Hz. However generally, the examined frequencies covers a range from several hundreds to several thousands of Hertz. Reverberation times are measured without and with the tested component and the Sabine absorption coefficient is retrieved. The surface area of the material is either measured or known from the design process of the part. Data are finally assessed both in terms of sound absorption area, in square meter, or the dimensionless Alpha cabin quantity which is similar to Sabine absorption coefficient in ISO 354. Several precisions should be made at this point concerning the setup and data accuracy. As the test elements are shaped, namely with varying thickness and eventually with curved surface, there may exist some air gaps between the cabin floor and the rear side of the tested part if the part sides are not sealed. If the parts are only laid down on the cabin floor, additional sound dissipation may be created by these thin layers. In addition, automotive trims have usually a small surface area, less than one 1 fxb@matelys.com 2 christophe.locqueteau@renault.com

square meter. Therefore, similarly to ISO 354 tests, the Alpha cabin values may exceed unity which is physically not acceptable. It is now admitted that this phenomenon, well known in building acoustics, is due to the incoherence between the predicting theory used to retrieve the sound absorption data and the actual test conditions. The former considers indeed infinitely large samples whereas tested elements are obviously of finite size. The last comment is also valid for large reverberant room measurements. As the sound absorption capacity of the tested elements increases the sound field becomes less diffuse and the underlying hypothesis becomes less valid. In this context, the aim of this paper is to examine the possibilities to predict sound absorption performance as measured in Alpha Cabin test facilities. The calculation of diffuse sound field absorption is firstly examined on a large set of material data by the use of linear regression models. The second one relies on the characterization and the simulation using TMM (Transfer Matrix Method) of diffuse field response of the multi-layer trim. These results are finally discussed with regards to their applicability in an industrial context. 2. DIFFUSE SOUND FIELD PREDICTIONS 2.1 London's prediction models London in 1950 [3] proposed several expressions to compute the diffuse field sound absorption from the absorption coefficient as measured in impedance tube, e.g. according to ISO 10534-2 [4]. Starting from the basic definition, the diffuse field sound absorption is defined as : b = 0 / 2 2 π ( θ ) cosθ sin θ dθ To integrate this relationship, London assumes that the equation for (0) may be verified by an acoustic impedance being real. He call it equivalent impedance, Z e defined by : ( Z 1 0) 1 1 e = Z e + which yields : Z e = 1 + 1 1 (0) 1 (0) Therefore, the previous integral becomes : 2 2 1 1 (0) 2 1 1 (0) 1 1 (0) b = 8 + 2 ln 1 + 1 (0) 1 1 (0) 2 2 where a typographic error has been corrected in the equation (15) of the original paper. This model refers to as the first London's model. Alternatively, the above expression may be significantly simplified using the isotropy of acoustic intensity for perfect diffuse field conditions. In this case, the sound energy in a given direction is proportional to that coming from another direction. If so, the diffuse field absorption may be defined as s = π / 2 0 ( θ ) sin θ dθ which further gives : 1 1 (0) + 2 1 1 (0) s = 4 ln 1 + 1 (0) 1 1 (0) 2 This is referred to as the second model by London. These first two models are used in Section 3 where only sound absorption coefficient data as measured in impedance tube are available. A third and last model is given by equation (6) of London's paper. It expresses the diffuse field sound absorption coefficient from the surface impedance considered as a complex value contrary to the above equations. In this case, 8r r² x² 1 x r P = 1 + tan ln( (1 + r)² + x² ) x² + r² x( x² + r²) 1 + r x² + r²

where x and r are respectively the real and the imaginary part of the surface impedance as measured in impedance tube, hence for plane wave under normal incidence. This model is used only in Section 4 where complex values of surface impedance are available. For the sake of completeness, one should mention that London made other more radical assumptions [3], which are left for interesting readers. 2.2 Data regression models A more experimental approach is based on collecting sound absorption data and building up, or eventually, updating a model based on regression over all collected data. This approach has been proposed for instance in [5]. In this work, the authors use their prior knowledge of the sound dissipation mechanisms by porous materials to introduce additional degrees of freedom with a view to increase the prediction accuracy. At last, the regression model is a function of the sound absorption coefficient as measured in impedance tube (0), a frequency factor Φ(f), as well as the mass density ρ, the thickness Θ and the air flow resistivity σ of the predicted sample : 6 M = 0.945 + 0.245 ln( (0)) + Φ( f ) 0.002ρ + 0.0015Θ + 7.52e σ Note that the air flow resistivity may be measured directly following the corresponding ISO 3 or ASTM standards [6,7]. Typical values range from a few thousands for low density fibrous materials to up to a few millions for highly resistive screens. In total, the influence of the resistivity and of the mass density are of the same order of magnitude, which is two order of magnitude larger than that of the thickness for typical automotive applications. If the information related to e.g. air flow resistivity, is not available, one may reduce the number of degrees of freedom and build up a linear function of the ln( (0) ) in the form of L = A + B ln( (0)) It is obvious that the accuracy of this relationship relies on the number of data points, and the availability of the degrees of freedom. Only this second type of models are tested in Section 3 of the present paper because data of air flow resistivity are not available for the set of data examined there. 2.3 TMM prediction model The previous relationships are adapted to study variations of trim components close to previously studied cases. They are not adapted to design trim components "from scratch", with physical behavior possibly departing largely from those previously measured. In this case, it is necessary to predict the trim response from the properties of the individual components and then to calculate the diffuse field performance using a numerical integration over all angles of incidence. The properties of the individual components may be obtained either by the experimental characterization of their intrinsic properties or by describing their micro-structure, which allows to predict the macroscopic parameters of each individual component. Once the properties of each component are obtained, the sound absorption coefficient of the multi-layer trim in diffuse field condition is computed using the Transfer Matrix Method (TMM) algorithm [8]. If the trim has varying thickness or if the trim consists of patches of different elements, the total trim response may be computed using a weighted average based on the surface area of each patch : i i S i T = S i i where i represents the thickness index and / or the patch index. In this equation, i S i is the total surface area of the considered trim. Even though more cumbersome, this approach allows to change the properties of one component, e.g. the surfacing, without making obsolete the information available for the other components. It is therefore well suited for parametrical studies at early stage of development projects. Moreover, this approach does not require the material to exist and, if it exists, it requires small size samples corresponding to impedance tube diameters. In more details, the experimental characterization for porous materials is described extensively in 3 Interesting readers are referred to paper 98 by Jaouen for a discussion related to the current modification of the ISO 9053 standard.

[9, 10]. It is based on the measurement of the intrinsic wave number and wave impedance using impedance tube [11]. It requires the prior knowledge of the static air flow resistivity and the porosity of the material, which are two quantities directly measurable using adapted experimental setups. Finally, it allows to determine analytically, contrary to curve fitting techniques, the parameters of the Johnson-Champoux-Allard-Lafarge (JCAL) model, namely the high frequency limit of the tortuosity, the viscous and thermal characteristic length and the static thermal permeability. Alternatively, so-called micro-macro approaches allows to determine the same set of parameters from the description of the porous material micro-structure, e.g. the pore or the fiber size, the structure morphology, and the value of the porosity [12]. Only the experimental method has been deployed in the present work. This model is used in Section 4 for three different automotive trims with variation composition. 3. VALIDATION ON BLIND TEST DATA In this section, the first two London's models and the data regression models are tested on a large set of measured data collected by Renault. It contains absorption coefficients measured both in impedance tube and in Alpha Cabin test facilities. The data correspond mainly to foam and felt materials. Due to different frequency ranges of validity of the techniques, only the overlapping interval is kept in the study: [400-1600] Hz or [3150-6300] Hz, depending on the origin of the data. Finally, 472 pairs of values are available, which are shown in Figure 1. Figure 1 - Alpha Cabin absorption data Vs Impedance tube absorption data These graph shows that all data seem to be fairly well arranged, with a limited dispersion. 3.1 Test of London's models Next, Figure 2 shows the implementation of the first two models by London, namely b and s. These results show that the predictions given by the two models are close for low levels of absorption. For higher levels of absorption, the two assumptions tend to give slightly different tendencies. For levels lower than approximately 0.5, s tend to predict with more accuracy the measured data, where b gives better prediction for higher levels of absorption. However, they both predict Alpha Cabin values which are lower than unity, which is physical, whereas numerous measured data exceed unity. It should be noted that the full integration b gives simulated data which are larger than s. The reason for that is still unclear to the authors and currently under investigation.

Figure 2 - Test of the London's models for Alpha Cabin prediction 3.2 Test of the regression models Next, the linear regression model is tested. The model is based on all measured data available, namely including those exceeding unity. The obtained regression model is LR = 0.34 ln( (0)) + 1.04 with R² = 0. 94 Results are shown in Figure 3. Figure 3 - Test of the linear regression model including all measured data. These results show that for low levels of absorption, the linear regression model gives absorption coefficient close to those given by London's models. For higher levels of absorption, the inflexion point is observed at a value of absorption around 0.7. In the entire range, it may be concluded that the linear regression model gives results which are in better agreement with measured values of Alpha Cabin absorption than that given by the London's models. To overcome the problem of Alpha cabin values larger than 1, one can first limit the values to the

maximum value physically acceptable, namely 1. A new regression model is built on this set of data : LR 1 = 0.32 ln( (0)) + 1.01 with R² = 0. 95 Alternatively, one can simply omit the points with measured values larger than 1. The regression model becomes in this case : LR 2 = 0.31ln( (0)) + 0.99 with R² = 0. 93 Results are shown in the Figure 4 below. Figure 4 - Test of the linear regression models with measured value limited to 1 (Left) and excluding the values larger than 1 (Right). We observe that excluding values larger than 1 from the set of examined data make the linear regression tend to s values. However, the coefficients are close to each other for the three regression models built here. All standard deviation lie around 0.08 which represent a maximum difference of 5%. It is interesting to note that this corresponds to the uncertainty commonly agreed in Alpha Cabin measurements. The larger determination coefficient is obtained when limiting the unphysical values to 1. Finally, these models are fairly close to the one proposed by Mc Grory (see Section 2.2). The coefficient of the logarithmic function is found to be around 0.3 here against 0.245, which still represents above 20% of variation. Moreover, the constant is found to be here close to 1 whereas Mc Grory would find a slightly lower value, namely 0.945. 4. DETAILED VALIDATION ON THREE PARTICULAR TRIMS In this section, the approach described in Section 2.3 is examined for three type of trims. Again, this approach, yet more demanding, is more adapted for parametrical studies aiming to optimize the performance of the trim. The characterization of each individual component has been carried out by Matelys on samples of diameter 46 mm. Open porosity and static air flow resistivity have been measured on the same samples. Alpha Cabin data have been obtained by Renault. In all three cases, the parts have been laid onto the floor of the cabin without sealing the part perimeter. TMM simulations have been obtained by Matelys using AlphaCell software product version 8.0 [13]. In addition, London's expression P is also computed using the surface impedance data collected during the characterization procedure. For these latter data, the frequency range of computation corresponds to the frequency range of impedance tube measurements, namely [80-4500] Hz. 4.1 Purely porous trim First a purely porous trim is examined. It consists in fiberglass layer with non woven surfacing on the visible side. On the rear side, a polyester screen is present which has been measured to have no

acoustic influence. The maximum thickness is measured to be 15 mm. The minimal thickness is 2 mm on the perimeter as a consequence of both production and mechanical constraints. A picture of the part is shown in the Figure 5 below. Figure 5 - Purely porous trim: rear side (Left), visible side (Right). Three different thicknesses have been considered. Characterization has been carried out on the thicker material region to increase accuracy. In addition, a compression procedure inspired by works in [14] and adapted from Matelys return of experience, has been applied to obtain simulated data for the other two thicknesses. It should be underlined that effects of the compression have been accounted for the sets of acoustic parameters (JCAL model) as well as elastic parameters (elastic isotropic assumption). The surface area of each thickness has been measured manually. From a measurement point of view, 4 parts were tested to obtain the Alpha Cabin data. Due to the absorbing character of both sides of the part, it was considered that both surfaces contribute to the sound dissipation in the cabin. Comparisons between measured data, simulations using London's P expression and TMM are shown in Figure 6. Figure 6 - Purely porous trim : Alpha Cabin prediction using London and TMM simulations. These results show that TMM simulations correspond well to measured data up to 2000 Hz. At higher frequencies, predicted levels largely overestimates the measured ones. In this frequency range, because of the large absorption level of the tested element, it may be argued that the sound field is no longer diffuse, as already pointed out in [15]. This tends to overestimates the sound absorption levels compared to the actually measured ones. Levels predicted by London's equation are fairly close to those given by TMM. However, the frequency behavior is not correctly.

4.2 Trim with impervious surfacing The second part to be tested presents a portion of its surface which is covered by an aluminum sheet to satisfy thermal constraints. Otherwise and below this sheet, the trim is composed of non woven surfacing and a so-called shoddy (see Figure 7 below). From a measurement point of view, 5 parts have been tested simultaneously to achieve a sufficient measurement dynamic. From a modeling point of view, the impervious layer has been represented as a septum with a surface mass corresponding to the one actually measured. The surface of this region has been estimated manually to be 70% of the total trim surface. Moreover, only one thickness value and two patch regions were considered. It also was considered that the two faces of the part contribute to the sound dissipation in the cabin. Comparison between measured and simulated data are shown in Figure 8 below. Figure 7 - Trim with impervious surfacing: rear side (Left), visible side (Right). Figure 8 - Trim with impervious surfacing: Alpha Cabin prediction using London and TMM simulations. Levels predicted by TMM approach provide a good correspondence in the low frequency range. Around 1000 Hz, a mass-spring resonance effect is visible which is not observed in the measured data. In this frequency range, simulations largely overestimates the measured levels. At higher frequencies, levels are in better correspondence again. London's predictions do not correspond, nor in levels nor in frequency behavior.

4.3 Trim with plastic rigid element The last part to be examined in this paper contains a plastic component, which is totally impervious (see Figure 9). The presence of this plastic element is due to mechanical constraints. The sound absorbing material is made of a polyester fiber layer of constant thickness covered by a non woven surfacing. Three parts were tested simultaneously to obtain sufficient measurement accuracy. From a modeling point of view, the plastic element has not been actually modeled. However it was considered in this case that only the visible side contribute to the sound dissipation in the cabin. Results of measurement and simulations are shown in Figure 10. Figure 9 - Trim with plastic rigid element: rear side (Left), visible side (Right). Figure 10 - Trim with plastic rigid element: Alpha Cabin prediction using London and TMM simulations. In this case, TMM simulations and London's model yield similar levels of absorption, both in levels and frequency tendency. These levels however are underestimated compared to measured levels. One hypothesis is that the lacking level of dissipation originates from the dissipation which may be created by the small gap on the rear side of the part, between the plastic element and the cabin floor. To check this, Alpha Cabin measurements have been carried out with only the plastic element. This dissipation has been added to the levels simulated by the TMM approach. Results are shown in the same figure. Levels are now in good correspondence with the measured ones up to 3000 Hz. Above levels are overestimated for a reason which is still to be determined.

5. DISCUSSION This work presents several approaches to predict the sound absorption as measured in Alpha Cabin. Examined approaches include analytical approaches based on the sound absorption coefficient measured in impedance tube and data regression models obtained on a large set of previously measured data. In addition, direct TMM simulations obtained from the characterized set of intrinsic parameters are compared to measured data obtained on three different types of trims. Results show that the analytical approaches yield predictions in close correspondence to measured data. However, these approaches are only adapted to study small variations of designs around a previously measured configuration. TMM simulations seem to give tendencies which correspond well in terms of both physical behavior and of frequency behavior. However, these simulations are largely sensitive to (i) the actual mounting condition of the parts in the Alpha Cabin and (ii) the surface area of the different regions of the tested part. Work are currently in progress to implement the linear regression for the parts tested in the last section of this paper and to obtain lacking information. These results will be presented during the conference. ACKNOWLEDGEMENTS This work has been carried out in the framework of the French research project EcOBEx which is funded by BPI France, Région Rhône-Alpes Auvergne and Grand Lyon La Métropole. All EcOBEx partners are warmly thanked for their support in this work : suppliers RJP, MECAPLAST, and Isover, and research laboratories CrittM2A, UTC, and engineering companies ESI and MicrodB. In particular, the authors thank MECAPLAST for the provision of the additional measured data in Section 4.3 Finally, Christophe Locqueteau would like to thank Olivier Caldier from the Direction of Material Engineering of Renault for the provision of test data. REFERENCES [1] NF EN ISO 354, Acoustics - Measurement of sound absorption in a reverberation room, International Standard Organisation, pp. 9 (2003). [2] ASTM C423-02, Standard Test Method for Sound Absorption and Sound Absorption Coefficients by the Reverberation Room Method, American Society for Testing and Materials, pp. 11 (2002) [3] London, A., The determination of reverberant sound absorption coefficients from acoustic impedance measurements, J. Acoust. Soc. Am., 1950: 22 (2), pp. 263-269. [4] NF EN ISO 10534-2, Acoustics - Determination of sound absorption coefficient and impedance in impedance tubes - Part 2: Transfer-function method, International Standard Organisation, pp. 27 (1998) [5] McGrory, M. and Castro Cirac, D. and Gaussen, O. and Cabrera, D., Sound absorption coefficient measurement: Re-examining the relationship between impedance tube and reverberant room methods, In Proceedings of Acoustics 12 - Fremantle, Fremantle, Australia (2012) [6] ISO 9053, Acoustics - Materials for acoustical application - Determination of airflow resistance, International Standard Organisation, pp. 10 (1991) [7] ASTM C522-03, Standard Test Method for Airflow Resistance of Acoustical Materials, American Society for Testing and Materials, Vol. pp. 11 (2016) [8] Brouard, B. and Lafarge, D. and Allard, J. F., A general method of modelling sound propagation in layered media, J. Sound Vib., 1995: 183 (1), pp. 129-142 [9] Panneton, R. and Olny, X., Acoustical determination of the parameters governing viscous dissipation in porous media, J. Acoust. Soc. Am., 2006:119 (4), pp. 2027-2040 [10] Olny, X. and Panneton, R., Acoustical determination of the parameters governing thermal dissipation in porous media, J. Acoust. Soc. Am., 2008:123 (2), pp. 814-824 [11] Iwase, T. and Izumi, Y. and Kawabata, R., A new measuring method for sound propagation constant by using sound tube without any air spaces back of a test material, In Proceedings of inter.noise 98, Christchurch, New Zealand (1998) [12] Perrot, C. and Chevillotte, F. and Hoang, M. T. and Bonnet, G. and Bécot, F.-X. and Gautron, L. and Duval, A., Microstructure, transport, and acoustic properties of open-cell foam samples: Experiments and three-dimensional numerical simulations, J. Appl Phy, 2012:111 (1), pp. 014911 [13] http://alphacell.matelys.com/, last visit May, 17th of 2016 [14] Castagnède, B. and Aknine, A. and Brouard, B. and Tarnow, V., Effects of compression on the sound absorption of fibrous materials, Appl. Acoust., 2000:61 pp. 173-182 [15] Duval, A. and Rondeau, J.-F. and Bischoff, L. and Dejaeger, L. and Morgenstern, C. and De Bree,

H.-E., In situ impedance and absorption coefficient measurements compared to poro-elastic simulation in free, diffuse or semi-statistical fields using microflown p-u probes, In Proceedings of DAGA 2006, Braunschweig, Germany (2006).