Comparison of transmission loss prediction using condensed equivalent platemodels
Arasan UTHAYASURIYAN, Fabien CHEVILLOTTE(1) , Luc JAOUEN(1) , Dimitrios CHRONOPOULOS(2) , Emmanuel
GOURDON(3)(1) Matelys-Research Lab, France(2) Faculty of Engineering, University of Nottingham, UK(3) Department of Civil Engineering, ENTPE, France

Abstract
Equivalent plate models are commonly used in order to simulate multilayer system as one single layer. This kindof approach enables for instance to reduce the mesh size and thus the number of degrees of freedom in finite
element models. Usual condensed model rely on thin plate theory and are obviously limited when considering
thick multilayers. In this work we present a comparison of the transmission loss prediction using different
analytical models of solid layers: thin plate, thick plate (first order shear deformation) and 3D modelling (3Dlinear elasticity) . Doing so, analytical limit for the validity of the thin plate theory is derived from wavenumber
analyses. Applications to industry inspired cases are presented.
Keywords: Equivalent plate model, Transmission loss

1INTRODUCTION

In recent years, multi-layered partitions have been used widely for better sound comfort and noise attenuation.
Advanced composite structures are one type of multi-layered systems which are being used increasingly in various industries such as aerospace and aircraft industries. Sandwich composites which exhibit high stiffness withlight weight are widely used in the transportation and construction industries. Since the interactions between
the different layered materials affect the acoustic performance of the multi-layered system, it is not always optimized by including only the best materials. Due to the diversity of materials used in the multi-layered system,
modelling of this system often requires suitable mesh types for the material used and increases the total meshcount in the Finite Element modelling. Additionally, it would lead to high computation time due to these complexities. Therefore, in many engineering cases, it is of high interest to condense the behaviour of a multi-layer
system to a single layer material. This aims at reducing the mesh size of the finite element model which willlead to less computation time. Guyader and Cacciolati[1, 2] had developed an equivalent thin plate model forthe multi-layer plate made of isotropic layers. Ross, Kerwin and Ungar (RKU) [3–6] developed a simplified
dynamic model for a three layer system to find the equivalent material properties. The model is initially developed for beams which was later used for plates as well. For the multi-layer plate consists of general orthotropiclayers, Woodcock[7] developed a vibroacoustic model which was later reformulated by Loredo and Castel[8] foranisotropic layers. Marchetti, Ege, and Leclere[9] recently developed an equivalent plate model for multi-layer
thin plates made of orthotropic layers.
The goal of equivalent plate models is to find a set of parameters for a thin plate model which embeds thebending and shear behaviours of a multilayer. The parameters are computed at each frequency and used witha thin plate model. As the compressional effects are not taken into account in thin or thick plate models, thisrequires a frequency limit to safely apply these models to have only equivalent bending effect. In this direction,
the present paper compares the elastic thin (Love-Kirchoff), thick (Reissner-Mindlin) and solid (linear elasticity)
plate models to find the analytical frequency limit to apply the classical plate theory to find the transmission
loss across the plate.

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Figure 1. Plane wave incidence on the elastic medium

2VIBROACOUSTIC MODEL FOR THE ELASTIC PLATES

2.1 Transfer matrix for elastic solid
Generally, the Transfer Matrix Method (TMM) is used to compute the vibroacoustic indicators of the multilayer system from the transfer matrices of the stratified media. In the present work, this method is adopted tocompute the transmission loss of a single layer of isotropic elastic medium. Consider an infinite layer along
the x and y directions with thickness h along the z direction as shown in the Fig. 1, excited by a plane wave
with incidence angle θ . The layer has z axis origin at the middle surface as a reference. By considering both
longitudinal and shear wave propagation, the associated displacements and stresses can be obtained from theoryof elasticity approach and the transfer matrix ([T ES ]) which relates the state variables (V ES ) between the points
M and M’ is written as follows:
V ES (M) = [T ES ]4×4V ES (M 0 )(1)
The elements of the matrix [T ES ] are given in the book by Allard and Atalla[10] and the state vector (V ES )which describes the acoustic field is given as:
V ES = {u˙x , u˙z , σzz , σxz }T(2)In the above equation, u̇ and σ are the velocities and stresses, respectively. The transmission loss of the elastic
layer can be computed by Transfer Matrix Method (TMM), for example, as detailed by Allard and Atalla[10].
As the acoustic field in the elastic layer is completely described by the longitudinal and shear waves, the TL
solution obtained from [T ES ] is taken as a reference to compare the other vibroacoustic models. Similarly forthe following plate models, by having appropriate transfer matrix, the TL is computed in the same way.
2.2 Transfer matrix for plate theories
If the acoustic field in the elastic layer (Fig. 1) is described by the state vector V P = {σzz , u˙z }T , then the generalform the transfer matrix for the plate theories can be expressed by the following equation:

1 −ZP P 0
PP P
0V (M) = [T ]V (M ) =V (M )(3)01Here, ZP is the mechanical impedance of the plate and it is expressed as below based on the theory adopted.
Dkt4
jωms 1 − ω 2 m(s
Zthin
ZP ==(4)Iz ms 2−jDms
4
22
222

kt D − ms ω + G∗ h ω − kt G∗ h ω − kt Iz ωZthick

22
zω

 ω 1+ kt D−I
G∗ hwhere, kt = k0 sin θ with k0 as the wave number in the air, ms is the mass density per unit area, D is the bending
2shstiffness, G∗ = Gξ with G as the shear modulus of the plate, ξ is the shear correction factor, Iz = m12
is themass moment of inertia, and ω is the circular frequency. The mechanical impedances of the thin and thick plates
are obtained from the elastic plate theories of Love-Kirchoff[11] and Reissner-Mindlin[12, 13] respectively.

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2.3 Transfer matrix for equivalent thin plate theories
2.3.1 Guyader model
By applying classical plate theory to the multi-layer viscoelastic plate, Guyader and Cacciolati [1, 2] developed
an analytical model to determine the equivalent complex bending stiffness as a function of the excitation frequency. In this model, each layer is assumed to have Reissner-Mindlin type displacements or in other words
each layer is modeled by considering bending, membrane and shear effects. Displacement and shear stress continuity conditions are applied to obtain the equations of motion of the multi-layer plate which are interestingly
expressed as a function of only the first layer displacement field. Finally, assuming that the transverse displacement is the same for both the equivalent single layer and the multi-layer plate, this model determines theequivalent bending stiffness by using Love-Kirchhoff thin plate theory.
2.3.2 RKU modelRoss, Kerwin and Ungar (RKU) [3–6] developed a vibroacoustic model for a three layered plate by assuming
the second layer (core) to be soft in nature compared to the other two stiff layers (skins), and the core is mainly
stressed in shear. Expressions for the equivalent thin plate properties such as bending stiffness and loss factor
are derived as a function of the shear parameter that describes how well the viscoelastic layer couples flexural
motions of the other two layers. The complete derivation can be found in the articles [3–6] as well as in thebook by Beranek and Ver[14].
2.3.3 Added Stiffness (AS) Model
This model is based on computing the equivalent properties by adding the respective individual layer properties.
The bending stiffness of each layer is obtained with respect to the neutral axis of the multi-layer plate. One
can note that a similar approach is often used for composite materials as described in [15] for beams.
After obtaining the equivalent bending stiffness (Deq ) from the above equivalent thin plate theories, it can besubstituted in Zthin for the TL computation. It must be noted that Deq is a dynamic bending stiffness (i.e,function of the frequency) for the equivalent plate models.

3COMPARISON WITH AS MODEL: A MULTI-LAYER PLATE EXAMPLE

In this section, a multi-layer plate (Plasterboard/Glue/Plasterboard) of practical application is taken to illustrate
the necessity of the frequency limit, for the AS model, to be found. The material properties are listed in theTable 1. The TL curves pertaining to the diffuse field excitation are generated from the AS model (equivalentTable 1. Material properties
Material parameters
Thickness
DensityYoung’s modulus
Loss factor
Poisson’s ratio

Unitsmmkg/m3
GPa-Plasterboard
12.5
700
30.02
0.3

Glue0.5
6000.002
10.3

Concrete
1402300
300.020.3

thin plate model) and compared with linear elasticity solution in the Fig. 2. It can be observed that AS
model starts to differ from the reference solution before the critical frequency (though it predicts the criticalfrequency with good accuracy) of the multi-layer plate. It also shows that the stiffness is predicted with a fairaccuracy while the loss factors is largely underestimated. Furthermore, at higher frequencies, the difference inTL between the reference solution and equivalent plate models is greater than 4 dB. This example the need for275

Figure 2. Transmission loss for the multi-layer plate (Plasterboard/Glue/Plasterboard) under diffuse field excitation

the frequency limit is required to safely use the AS model for the vibroacoustic problems of the multi-layersystem. In this direction, as a primary step, the next section discusses about deriving the analytical frequency
limit for the Love-Kirchoff plate model.

4FREQUENCY LIMIT FOR THE LOVE-KIRCHOFF PLATE MODEL

The behaviour of the propagating waves in the framework of the thick plate model is studied in this section
followed by the discussion on the analytical frequency limit for the thin plate model. The dispersion equationof thick plate can be obtained by letting Zthick = 0.
Iz ms 2422 Dms
kt D − ms ω +ω − kt ∗ ω 2 − kt2 Iz ω 2 = 0(5)G∗ hG hThe propagating wave number solution of the above dispersion relation is written as:vsuu m ω2 I ω2ms ω 2 Iz ω 2 2
ms ω 2t sz
kt = ±+++−2G∗ h2DD2G∗ h2D
By defining kb2 = ωms ω 2
2D , ks = G∗ hp ms(6)2and kr2 = IzDω , Eq.(6) is rewritten in a compact form as:s q
1 2
kt = ±
ks + kr2 + 4kb4 + (ks2 − kr2 )22(7)It can be observed from Eq.(6) that at low frequencies kb >> ks , kr , leading to the pure bending wave propaga-276

tion. In this case, we haves rms(at low frequencies)
kt ≈ kb = ωDAt higher frequencies, ks >> kb , leading to pure shear wave propagation.
rms
kt ≈ ks = ω(at high frequencies)G∗ h(8)(9)With this regard, kb and ks are referred as bending and shear wave numbers respectively in the further discussions. Based on the percentage error between the wave number pertaining to the bending wave (Eq.(8)) andthe full propagating wave (Eq.(6)), the limiting frequency for the use of thin plate model for the vibroacoustic
problems can be found. By defining C = kb /ks , it can be seen from the Eq.(10a) that, the ratio between kt andkb (thus the error,ε) is the function of only the Poisson’s ratio, shear correction factor and C.
vsu


1u
2 − ξ (1 − ν) 2kt
t 2 + ξ (1 − ν)=+ 16 +(10a)
kb
2C2C2

ε = 1−1100%
kt /kb(10b)The value for C can be chosen such that ε is below the accepted error percentage and the frequency limit for
the thin isotropic plate model can be found as given by Eq.(11).
sξ
12E 1 − νωlimit = 2(11)
2C h
ρ 1+ν
It can be observed that the influence of material properties on the frequency limit is small when the plate
thickness is large and vice versa. To illustrate this frequency limit, the transmission loss (TL) curves for theplasterboard and concrete (Table 1) are computed from the transfer matrices of thin plate, thick plate, Guyader
model and compared with linear elasticity solution for an oblique plane wave incidence with θ = 60o . TheTL curves obtained for plasterboard (Fig. 3) show that the thick plate and Guyader models estimate better withelasticity solution. It is to be noted that the frequency limit is computed from the Eq.(11) by having C = 4,
which corresponds to the error percentage (ε) of 1.94%. The frequency limit for the thin plate model in thiscase is found to be before the coincidence frequency. But in the case of concrete material, Fig. 4 shows thatthe frequency limit spotted after the coincidence frequency, which is interesting to note.

277

Figure 3. Transmission loss for the plasterboard under plane wave excitation with θ = 60oFigure 4. Transmission loss for the concrete under plane wave excitation with θ = 60o278

5COMPARISON OF PLATE MODELS FOR THE MULTI-LAYER PLATE EXAMPLE

In this section, the same multi-layer plate (discussed in section 3) is taken for the comparison of equivalentthin plate models with linear elasticity solution. From Fig. 5, it can be observed that, since Guyader andRKU models include the shear behaviour of the multi-layer plate in their formulation, they predict the criticalfrequency of the plate with good approximation though the equivalent properties corresponds to thin plate. But
it is also seen that they exhibit poor correspondence with elasticity solution at higher frequencies, because thesemodels do not account for compressional effect in their theories. Also, it must be noted that, as the parametersof equivalent plate models are fitted with a thin plate model, they can not be used with other models as thickplate or full elasticity.

Figure 5. Transmission loss for the multi-layer plate (Plasterboard/Glue/Plasterboard) under diffuse field excitation

6CONCLUSION

A multi-layer plate of practical application is taken to illustrate the need to find the frequency limit for theAdded Stiffness (AS) model, as it is applicable only for thin multi-layer plates. From the wave propagation
analysis of the thick plate model, based on the Reissner-Mindlin plate hypothesis, the analytical expression of
the frequency limit for the thin plate model is derived. From the comparison plots, the deviations of transmission loss curves obtained from different models are observed after this frequency limit. It is also observed thatthe thick plate (Reissner-Mindlin), Guyader and RKU models provide good approximation with the linear elasticity solution by including shear effect of the plate but do not exhibit good approximation for the compressional
effect of the plate.

ACKNOWLEDGEMENTS
The authors would like to gratefully acknowledge Marie-Skłodowska Curie Actions (MSCA) Project 765472‘N2N: No2Noise’ for the financial support.

279

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