IMPLEMENTATION OF A HYBRID MODEL FOR PREDICTION OF SOUND INSULATION OF FINITE-SIZE MULTILAYERED BUILDING ELEMENTS. Selma Kurra

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1 ICSV14 Cairns Australia 9-1 Jul, 7 IMPLEMENTATION OF A HYBRID MODEL FOR PREDICTION OF SOUND INSULATION OF FINITE-SIZE MULTILAYERED BUILDING ELEMENTS Abstract Selma Kurra Department of Architecture, Acoustics Unit, Bahcesehir Universit manpasa Metebi Soa No: 4 6 Besitas, Istanbul 341 Ture surra@bahcesehir.edu.tr In building acoustics, it is necessar to have a reliable prediction model to obtain the sound transmission loss of laered structures especiall for the building facades eposed to noise. Traditional models are not ver much applicable in most of the cases, since the building elements are multilaered structures comprised of different laer combinations with various phsical characteristics, such as solid homogenous laers, porous or foamed material, elastic damping laers, etc. The previous model that has been developed b integrating the impedance approach for infinite-size laered sstems and the windowing technique for finite-size elements, was computerized as a time efficient program. The predicted results were confirmed b the laborator eperiments as satisfactor for use in insulation practice for acquiring the 1/3 octave band sound transmission losses as well as the rating values; Rw(C; Ctr). This paper describes the model briefl and refers to some implementations in search for the effects of particular laer parameters. 1. INTRODUCTION Structures composed of several laers made of different materials are used for noise control in industr and for the building elements comprising various protective laers and lightweight panels. The multilaered elements revealing a great diversit of phsical and constructional characteristics in buildings with their nature lie isotropic or orthotropic, viscous-damping, fibrous and poro-elastic porous laer, rigid (stiff) (with high stiffness-to-weight ratio lie fibrereinforced plastics) and non-rigid (fleible), plane or corrugated, with rigid or non-rigid laer connections etc., enhance the compleit regarding their acoustical and structural behaviours. Airborne sound transmission through these sstems has been dealt with in the literature however some of the theoretical models are difficult to generalize for practical applications in building elements. The phsical parameters, lie mass, densit, size, thicness, stiffness, elasticit, flow resistivit, porosit, loss factor, Poisson ratio, etc. highl influence the total sound 1

2 ICSV Jul 7 Cairns Australia transmission coefficient of these composite elements, tpes and numbers of connections between laers and size (dimensions) of the panels. Since values of these parameters remain within the definable ranges for the building materials it might be possible to suggest their optimum values to lead to a design criteria. However this stud should be based on an accurate and reliable prediction model that would be applicable to various tpes of constructions within a desired laer composition. Considering the above target, this paper describes a computer model in connection with the former stud of the author et al [1], its validation through the former eperimental stud and implementations in practice.. DESCRIPTION OF THE HYBRID MODEL FOR COMPUTATION OF SOUND TRANSMISSION LOSSES OF MULTILAYERED STRUCTURES The prediction model is based on two approaches: A. The impedance approach for infinite size elements, B. The finite-size effect through the spatial windowing technique..1 Impedance model for infinite size elements The impedance approach was implemented b Au and Brne to calculate the insertion losses of the multilaered elements and the results were eperimentall validated b using the intensit technique []. This approach is capable of applications for the building elements both in the direct field and the diffuse field. The multilaered (lagging) structures consisting of various materials, such as hard, hard-damped with an elastic laer, porous laer and air gap can be taen into account in calculation of sound transmission losses. The boundar conditions of the model of which the two dimensional plane wave model (on - plane) is used, are: Wave number component parallel to the panel surface is the same in all of the laers. Acoustical pressure and the particle velocit at the interfaces of the laers are continuous. Laers are uniform having arbitrar thicness. Element is infinite on -z plane. Number of laers (n) is infinite. Both sides of the element is in contact with air. However the model does not tae into account the size of the elements. The major parameters in calculations are; comple characteristic impedance of the laers, comple bending stiffness and comple wave impedances at interfaces. The method is based on the comple wave impedance ratio of both sides of each laer according to the direction of sound intrusion and the ratio of the comple sound pressures. The impedances at input and terminating sides are matched at the interfaces between the laers (Figure 1). The relationship between the transmitted and the input pressures is given below: Z I pi = pt N / m (1) ZT p I, p T represent the sound pressures at source and receiver sides of the wall. The input impedance Z I for each laer is equal to the summation of the terminating impedance Z T and the separating impedance Z s : 3 () ZI = ZT + Zs N. s/ m Z I and Z T values are the comple impedances of the input and transmitted sides of the element. Z s depends on the comple bending stiffness of the laer, ω and which are the wave speed (πf) and wave number (πf/c), ρ s ; the laer surface mass, g/m. As nown, B is related to the thicness, Young s modulus, Poisson ratio and the loss factor of the laer. For the isotropic plates; Z s (separation impedance) and Z I are given below [] : Z s 4 = j( ωρ B / ω) s (3)

3 ICSV Jul 7 Cairns Australia Z I = ZT + j( ωρ s ( B + B + B ) ω, are the comple wave numbers, at and direction (Fig.1) and B is the comple bending stiffness. For impervious solid plates attached with a secondar laer, such as a damping material, the composite bending stiffness and the composite loss factor are calculated b specifing the phsical parameters of this additional laer. For composite structures which include porous material in the cavit, the comple characteristic acoustical impedance of the porous laer as a function of the flow resistivit, is necessar to use and Z I is calculated b inserting the comple wave number ( a ), the comple characteristic acoustical impedance (Z A ) of the bul material and the thicness of the porous laer into the equations. The equations for the porous laer computations are presented in [1-]. Comple acoustical impedances at the receiver side, Z I and sound pressure at the transmitted side, p T are computed b inserting the thicness of the porous laer (h p ). The resultant sound transmission coefficient is calculated as: In diffuse field conditions, the incident sound field is assumed that the plane waves are incident on the element surface from all directions with equal probabilit (-9 o ). Thus, the total sound transmission coefficient is calculated b integrating the results over the range of incident angles of sound waves as given below [3]: τ diffuse θ lim τ ( ω, θ )sinθ cosθ dθ ( ω) = θlim sinθ cosθ dθ τ ( ω, θ ) :Sound transmission inde (coefficient) for each angle of incidence (θ ). The maimum range of the incident angle (θ lim ) depends on the diffuse field conditions. Sound transmission 1 Rdiffuse ( ω ) = 1 log db τ diffuse ( ω) loss and its spectral profile for the diffuse field is calculated b the well nown formula:. Improved model for finite-size structures (Spatial Windowing Technique) The above model was improved for the finite-size elements based on the Spatial Windowing Technique for the finite-size elements that was developed b Villot et al. [4]. The model involves with calculating the vibration velocit field of the infinite structure and then spatiall windowing this field before calculating the radiated field (Figure ). Then the radiated sound power is obtained from the wavenumber spectrum of the velocit field b using the spatial Fourier transform. The radiation efficienc associated with spatial windowing is given below [4] b using the dimensions of the structure, L and L : S σ( p, ψ ) = π a 1 cos(( cosφ cosψ ) L ) 1 cos(( rsinφ sinψ ) L ) π r p p a r [( rcosφ pcosψ ) L ] [( rsinφ psinψ ) L ] a r dφ d p, a, r and φ : the wave number components in the polar coordinate sstem (Figure 3). as: τ ( ϖ, θ ) = P T (1) ( P ) The resultant airborne sound transmission coefficient for the finite element is calculated source r (4) (5) (6) (7) (8) 3

4 ICSV Jul 7 Cairns Australia finite ( θ, ψ ) = τ inf ( θ, ψ )[ σ ( asinθ ψ )cosθ ] τ, τ inf ( θ, ψ ); Sound transmission coefficient for the infinite structure, ψ ; direction angle (from to π) and θ ; incidence angle of plane wave on the surface of the element. The transmission coefficient in a diffuse field is epressed as: τ diffuse/finite = θlim π τ θlim finite π ( θ, ψ )sinθ cosθ dψ dθ sinθ cosθ dψ dθ The above spatial windowing technique has easil been constructed over the eisting impedance model to determine sound transmission loss values of finite-size multilaered structures. Thus transmission inde for the infinite structures, τ inf (θ) is acquired from the impedance model and the resultant diffuse field transmission inde is calculated b using the above equations for the structures whose dimensions are given. (9) (1) 3. DEVELOPMENT OF F MULAY DESCRIPTION OF THE COMPUTERIZED MODEL : F-MULAY Since the model including a comple calculation procedure for both infinite and finite elements involve rather time-consuming processes b using the Matlab [37] due to the comple numbers and iterations, a computer program F-MULAY was developed to increase the computation efficienc and to facilitate the parametric studies. The algorithm on which the model was based consists of discrete functions for input impedances, output pressures and radiation efficienc. It gives the normal, oblique and diffuse sound transmission losses at octave, third octave and narrow bands within the range of 63 Hz - 63 Hz as well as the single-number ratings (i.e., weighted sound reduction inde: R w (C;C tr ). The outputs are presented both in the form of tables and graphs. It is possible to obtain both the infinite and finite size TL values. A database for the phsical characteristics of various building materials has been organized within the model. The program code (programming language) is C++ and MS Visual Basic which is used to build the user interface on the Windows operating sstem. 4. VALIDITATION STUDY The predicted results obtained b emploing this model were checed to search the compatibilit with the original data presented b Au and Brne and a complete match was found when similar material characteristics and the limiting angle of incidence (9 ) were used in the calculations [1]. The model has been verified b the basic double wall theories as of the modal shape in terms of mass-air-mass resonance frequencies (f r ), critical frequencies of laers (f c ), resonance modes as nodes and antinodes (f dm and f dn ) and cross-cavit resonance (f d ) and the critical frequencies. Figure 3 gives the narrow band TLs for a sample structure revealing the complete agreement between the calculated modal behaviour of the infinite panels. The numerical results from the infinite size-version of the model were also compared with the eperimental data obtained through a former stud of the author et al []. The acoustical tests that were performed at the IITRI Riverban Acoustical Laboratories have provided sufficient amount of eperimental data through various multilaered structures. The testing procedure, equipment and laborator facilities were described in detail [1]. The test outputs are compared with the predicted data for both infinite and finite-size elements b considering the aspects to be taen into account in such comparisons with the laborator data. 4

5 ICSV Jul 7 Cairns Australia Total of 8 wall samples were used in the eperimental stud and the two specimen size 11 m and.9 m. The wall samples are: isotrophic panels as single and in multilaered constructions: Steel plate with vinl laer, double panels with identical laers with varing cavit thicnesses and with and without a porous material in the cavit and; various multilaered combinations (gpsum board, steel plate, vinl laer, airgap and glasswool). The phsical properties of the materials used in the calculations corresponding to the test materials are given in Table 1. The comparisons were made both in terms of the third octave bands and R w rating units to provide sufficient evidence regarding the accurac of the model especiall between Hz. Some eamples regarding the comparisons are given below. 1. The predictions at the third octave bands for the multilaered structures regardless of size revealed divergences from the lab data -as shown in Figure 4- especiall at higher frequencies. This is a tpical case due to the radiation efficienc of finite-size elements.. B taing into account the size of the element and emploing FMULAY, the calculated results for double gpsum walls with air gap, could ield a perfect agreement with the measured results almost all through the frequenc range (Figure 5). When the porous material is inserted into the cavit, the compatibilit is again satisfactor ecept at the critical frequenc where the dips on the calculated TL profile are more pronounced than which could be epected (Fig 6). The two-laered steel panel with an airgap in two widths, 5 and 1 cm, reveals a satisfactor compatibilit to the measured data ecept at ver low (<1 Hz) frequenc (Figure 7). A multilaered wall sample in smaller size is given in Figure 8. The laer composition contains a steel plate damped with a vinl laer, a gpsum board and glasswool of 1cm filled into the cavit. The result of the comparison displas a similar tendenc as described above ecept at the frequencies above 5 Hz. As a result, for ver low frequencies (i.e. at 1 Hz) the range of validit is limited due to the unreliabilit upon the measured results in the laborator. However, the weighted insulation rating for the building elements is not influenced unless special concern eists regarding the lower frequencies. 5. IMPLEMENTATION OF THE MODEL: EFFECT OF POROUS LAYERS As an implementation of the computerized hbrid model for calculation of sound transmission loss (TL) of multilaered elements, the effect of flow resistivit (R 1 ) of porous laer was investigated for various materials which are commonl used in buildings such as light and dense concrete, bric, plwood, aluminium and steel panels. Flow resistivit of rocwool was changed between 5- Ns/m 4. B taing the limiting angle of sound incident as 89, the model was emploed in the narrow frequenc bands with 1 Hz resolution to eamine the resonance modes, as well as in 1/3 octave bands. Figures 9 and 1 give two samples of the predicted narrow band TL s. However it should be noted that the results indicate steeper TL s at high frequencies since the element is accepted as infinite in line with the objective of this stud. As a result, the effect of flow resistivit of the porous material seems to be inversel related to sound transmission losses at ver low frequencies, i.e. the TL decreases while R 1 increases up to N.s/m 4, while at mid and higher frequencies, the identical and non-identical laered walls reveal different behaviours against the variation of the flow resistivit. In general, the modes occurred for the air cavit walls are suppressed when a porous laer is inserted in the cavit. The effect of flow resistivit is more significant for the double walls consisting of identical laers made of materials with lower densit or thicness, whereas it is less significant when the laers are not identical and when at least one laer is relativel thicer and/or denser. 5

6 ICSV Jul 7 Cairns Australia 6. CONCLUSION Below conclusions can be drawn from the validation stud for the hbrid model determining sound transmission losses of finite-size multilaered structures: Based on the narrow band analses performed for the structures consisting of up to 4 laers, the impedance model has been proved to have sufficient accurac in terms of the modal behaviour of the entire structure as well as of the independent laers. The impedance model for infinite-size elements gives satisfactor agreement with the eperimental data for the single laered elements in standard size of 1 m, however for the infinite-size multilaered elements a great discrepanc eists in compliance with the eperimental results. When the windowing technique was combined with the impedance model, the eperimental verification was proved to be more successful according to the results of various comparisons both in the frequenc domain TLs and the insulation rating values. It was revealed that the phsical parameters i.e. limiting angle of incidence, densit, elasticit modulus, loss factor and flow resistivit, pla an important role in the predicted sound transmission losses which are rather sensitive even to minor change in their values. F-MULAY is rather time-efficient program and facilitates the parametric studies because of the Windows interface. Therefore the computerized model can be useful for the problems in building acoustics, as well as in the optimization studies. REFERENCES [1] S. Kurra and D., Arditi, Determination of sound transmission loss of multilaered elements, Part 1: Predicted and measured results, International Journal on Acoustics: Acta Acustica, S.H.Verlag, 87 n.5 (1) [] A.C.K. Au and. K.P. Brne, On the insertion losses produced b acoustic lagging structures which incorporate fleurall orthotropic impervious barriers, Acustica 7, (199) [3] L.Berane, I.L. Ver, Noise and Vibration Control Engineering, Principles and Applications, John Wile and Sons Inc. 199 [4] M. Villot, C.Guigou and L. Gagliardini, Predicting the acoustical radiation of finite size multilaered structures b appling spatial windowing on infinite structures, Journal of Sound and Vibration, 45 (3), (1) [5] Z. Maeawa and P. Lord, Environmental and Architectural Acoustics, Chapter 5, E&FN Spon, 1994 RECEIVER SIDE PT(1) laer# θ transmitted sound interface # ZT(1) 1 r φ PT() PI() PT(3) PI(3) PT(4) PI(4) 1 3 ZT() ZI() ZT(3) ZI(3) ZT(4) ZI(4) 3 4 ZT :Termination wave impedance ZI : Input wave impedance PT :Termination pressure PI : Input pressure L a L p ψ θ a PT(N+1) PI(N+1) SOURCE SIDE N incident sound θ ZT(N+1) ZI(N+1) N+1 Figure 1. Calculation process of the impedance model for multi-laered elements (Arrows indicate the order of computations) Figure. Wave parameters in windowing technique [4]. a) Wave number components contributing to radiated power in solid angle. b) Parameters relative to incident wave on a finite element. 6

7 ICSV Jul 7 Cairns Australia 8 Comparison of calculated and measured results for double gpsum wall with 1cm glasswool 8 Transmission Loss Values for Different Limiting Angles TL(dB) 6 4 frd1=91. Hz 6881 Hz 5761 Hz 344 Hz 174 Hz fc=77 TL,dB R w (C ;C tr) 47(-6;-14) 43(-8;-17) 41 (-7;-16) measured (RD 1:large size:11.7m) calculated (-78) R=3 calculated (-9) R=3 calculated (-78) R=6 calculated (-9) R= Narrow bands,hz Third octave bands,hz Figure 3. Predicted narrow band TL for a multilaered structure b using the infinite model MULAY and the calculated modes according to the basic theories. (Double gpsum board with 1 cm airgap) Figure 4. Comparison of calculated (as infinite) and the measured results for double gpsum board with 1cm glasswool ( R: flow resistivit, angles of incidence are shown in the legend) [1] 6 Doublegpsum board with 1cm air 1. m X.43m 7 Double gpsumboard with 1cm glasswool.74m X 4.6 m Measured RD9 finite size infinite size Measured RD1 Finite size Infinite Figure 5. Comparison between the measured and calculated data for double gpsum board with 1cm air in small size: 1. mx.43m Figure 6. Comparison between the measured and calculated data for double gpsum board with 1 cm glasswool in large size:.74 m X4.6m 7 6 Steel plate+1cm air+steel plate 1.1 m X.43 m Gpsum board, 1 cm air, steel plate with vinl 1.1m X.43m Measured RD8 Finite size Infinite size Infinite size 9 Finite size Measured RD Figure 7. Comparison between the measured and calculated data for double steel plates with 1 cm air in small size: 1.1 m X.43 m Figure 8. Comparison between the measured and calculated data for a multilaered structure in small size: 1.1 m X.43 m. 7

8 Analsis of the low frequenc variation of the flow resistivit effect for wall in Fig.b N.s/m narrow bands, Hz air ICSV Jul 7 Cairns Australia 9 effect of flow resistivit of the porous material in the cavit of the double wall ( double gpsum board with 1cm gapwidth) 8 7 N.s/m Low frequenc detail of the plot 3 5 N.s/m4 frd3=344 Hz fc=574 Hz 1 N.s/m4 (air) frm(air)=frd1=96 Hz db/octave frd4=5761 Hz Figure 9. Variation of the narrow band sound transmission losses of double gpsum wall with.1m cavit (infinite size). 15 Light concrete bloc of 15 cm with 1 cm porous laer and gpsum board as a second laer N.s/m4 9 Figure 1. Variation of narrow band sound transmission losses of the wall tpe in Fig.4a with.1m gapwidth (infinite size). TL,dB frm(po)=5.3 Hz frm(air)=fd1=7 Hz 5dB/octave 16dB fc(concrete)=15 Hz N.s/m4 fd fd3 fc(gp)= 547 Hz fd4 with air 5 N.s/m4 1 N.s/m4 N.s/m4 Table1. Phsical characteristics of the materials used in the sample walls during the eperiment Material Densit (av.),g/m 3 Thicness m Elasticit Loss factor Poisson modulus,n/m ratio Flow resistivit of porous laer, N.s/m 4 Gpsum board e9-3 e _ Steel plate e9-31 e _ ( ) Vinl plate e9-9e9.5.6 _ Ma:.6 Glass-wool _ Air _ Light concrete e