Acoustical Porous Material Recipes

In the recent years, structures with multiple layers (for example, fuselage in aircraft, windshield in automotives, glass windows in buildings etc.) are commonly used to enhance sound comfort and noise reduction. Transport and building industries use sandwich constructions which provide higher stiffness and damping while having lighter weight. These solutions are often consist three layers where a soft layer is embedded between two hard skins. While the skins provide the structural stiffness, the soft core helps to increase the energy dissipation.

The main advantage of condensed models is that it reduces the multilayer structure into a single layer structure, mimicking the same natural behaviour of the actual multilayer structure. By achieving this, it would greatly help us to save or

In this page, the following two condensed models are discussed:

$\bullet$ **Sigmoid model** - applicable to three-layer structure of isotropic layers:

(see Arasan U., Marchetti F., Chevillotte F., Jaouen L., Chronopoulos D. and Gourdon E., 2021, "A simple equivalent plate model for dynamic bending stiffness of three-layer sandwich panels with shearing core", Journal of Sound and Vibration, 500, p. 116025. published version or preprint version)

$\bullet$ **New condensed model** - applicable to symmetric multilayers:

(see Marchetti F., Arasan U., Chevillotte F. and Ege K., 2021, "On the condensation of thick symmetric multilayer panels including dilatational motion", Journal of Sound and Vibration, 502, p. 116078. published version or preprint version)

For more information, you're also invited to look at the recording of Arasan's PhD defense:

Therefore, a simple model is developed to predict the vibroacoustic behaviour a three-layer system which facilitates the implementation process compared to other models. The new model is called as the

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These four characteristic behaviours of the three-layer structure is discussed well in this journal article. Once these characteristic parameters are calculated, the equivalent property ($D_\text{eq}$) as function of the frequency ($f$) is calculated as shown in Figure 3 below.

Although this model can be applied to symmetric multilayer structure with any number of layers, for the sake of explanation,

Two impedances are needed to fully describe the motions of the structure:

$\bullet$ $D_\text{eq}$ - can be computed from any existing equivalent plate model (RKU, Guyader, Sigmoid etc.)

$\bullet$ $\tilde{\rho}_\text{A}$ - average (or mean) density of the multilayer

$\bullet$ $\tilde{\rho}_\text{S}$ - it is computed by assuming the three-layer structure mass-spring-mass system \[ \tilde{\rho}_\text{S}=2\frac{\mathcal{M}}{h}\left(1-\frac{2}{\omega^2\mathcal{C}\mathcal{M}}\right), \] where $\mathcal{M}$ is the mass per unit area of the skin and $\mathcal{C}$ is the compliance of the core layer.

The following interactive application shows the equivalent quantities computed by the Sigmoid model for three-layer symmetric sandwich structure. Properties of these layers could changed below to see their effects
on the computed quantities such as $D_\text{eq},\eta_\text{eq},k_\text{eq}$ and transmission loss (TL). The TL is also computed from the new condensed model to compare its accuracy with the Sigmoid model and the
Transfer Matrix Method (TMM).

**Observations:**

* Plot:* $D_\text{eq}$$\eta_\text{eq}$$k_\text{eq}$TL

**Sigmoid model parameters:**

$D_\mathrm{low}=$ N m; $D_\mathrm{high}=$ N m

$f_T=$ Hz; $R=$

$\bullet$ $D_\text{low}$ is mainly controlled by the skin properties and core thickness

$\bullet$ Young's modulus of the core $(E_c)$ influences the transition frequnecy $(f_T)$ strongly than other properties

$\bullet$ If the core is too soft (or has low value of Young's modulus $(E_c)$), the new condensed model shows good correspondance with the TMM (in TL computations),
whereas the Sigmoid model shows good correspondance only at mass-law region

$\bullet$ If $E_c$ is increased, then the sigmoid model shows good correspondance with the new condensed model as well as with the TMM. This is because
the anti-symmetric motions (such as bending, shear) controlls the motion of the structure as the core becomes harder, for which the Sigmoid model is sufficent

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Arasan Uthayasuryian & Luc Jaouen (@ljaouen), ISSN 2606-4138.

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