# Numerical Analysis of Vibroacoustic Response of Timber Floor Panels Damped with Porous Materials
## 4 Dynamic Analyses and Parametric Study
### Effect of Porous Material  
In this section, the effects of the porous material on the vibroacoustic response of the system are presented. Porous materials with open cells consist of an elastic or rigid frame which is surrounded by air. The porosity, \(\phi\), is the ratio of the air volume, inside the sample, to the total volume, occupied by the porous material. Table 4 describes the main properties of the porous material that were used in this study [33; 34]. Figure 6 presents and compares the averages of 1/3-octave-band sound-pressure levels for the different loadings under two boundary conditions, with and without the presence of the porous material, within the frequency range of 25 to 250 Hz. The sound pressure has been calculated for all panels at the same exact point, at the center of the floor, i.e., the point of applied load. The results are shown for the following cases: rigid-connected panel under load walking (Figure 6a), hinge-connected panel under load walking (Figure 6b), rigid-connected panel under washing machine load (Figure 6c), and hinge-connected panel under washing machine load (Figure 6d). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & **Value** & **Description** \\ \hline \(E\) & 1.5 MPa & [PERSON]’s modulus \\ \(\rho_{f}\) & 1141 Kg/m\({}^{3}\) & Density \\ \(\sigma\) & 5200 Pas/m\({}^{3}\) & Airflow resistivity \\ \(\Lambda\) & 37 \(\upmu\)m & Viscous characteristic length \\ \(\Lambda^{\prime}\) & 100 \(\upmu\)m & Thermal characteristic length \\ \(\phi\) & 0.617 & Porosity \\ \(\tau\) & 2.749 & Tortuosity \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Properties of porous material. Figure 6: Comparison of a sound-pressure level in the 1/3 octave band at the middle point of the floor for (**a**) rigid connected panel submitted to load walking, (**b**) hinge connected panel submitted to load walking, (**c**) rigid connected panel submitted to washing machine load, and (**d**) hinge connected panel submitted to washing machine load. It can be observed that regardless of the type of excitation and the boundary conditions applied to the floor, the incorporation of a layer of porous material leads to a significant reduction in the sound level by approximately 9 to 11 dB. On average, this represents an impressive reduction rate of about 18%. This remarkable reduction using the porous material can be attributed to its effective sound-absorbing properties. In fact, this material acts as an efficient barrier, absorbing and dissipating a substantial portion of sound energy within its complex microstructure. As sound waves encounter the porous layer, they undergo viscous and thermal losses within the material, thereby diminishing their intensity. This phenomenon remains consistent across the entire frequency range studied, resulting in a consistent transmission loss and lower sound-pressure levels for all tested cases. Moreover, the porous material not only contributes to sound attenuation but also plays a crucial role in vibration isolation and control. By effectively dampening structural vibrations induced by different types of excitation, such as load walking or washing machine operation, the porous layer further enhances the overall noise reduction achieved in the environment. Figure 7a,c and Figure 8a,c present a comprehensive comparison, in narrow band representation, of the radiated sound pressure, while Figure 7b,d and Figure 8b,d show the vertical displacement at the middle point of the panel for two loading scenarios: the load walking and the washing machine loading, respectively. These figures reveal prominent max-amplitude peaks at specific frequencies, corresponding to the critical eigenfrequencies of the panel. These identified max-amplitude peaks represent the panel's vibroacoustic response, indicating the frequencies where it exhibits its most significant vibrational and acoustic behavior. Both displacement and radiated acoustic pressure reach their peak at these specific eigenfrequencies, resulting in amplified vibrations and increased radiated-noise levels.