## 3 Results and Discussion
### Damping Model of the Specimens  
The vibration modes identified through the analyses of the shaker excitation measurements in the frequency range 5-1000 Hz are 39 and 21 for the \(PN\) and \(PY\) panels, respectively. The damping ratio experimental estimates are plotted over the corresponding natural frequencies in Figure 6. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & \(\zeta_{PN}\) & \(\zeta_{PY}\) & \(\rho_{YN}\) \\ \hline Mean value, \(\mu\) & 0.55 & 1.61 & 3.04 \\ Standard deviation, \(\sigma\) & 0.11 & 0.20 & 0.59 \\ Relative standard deviation, \(\sigma^{\star}\) & 19.2\% & 12.1\% & 19.3\% \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Statistical parameter values of the EMA damping ratios and corresponding fraction. Figure 6: Modal parameters with proportional damping interpolating curves. Firstly, data fitting by means of a proportional damping model is evaluated. With this popular model, also known as Rayleigh model [35; 36], solving the structural dynamics/multibody dynamics problem is remarkably simpler. Indeed, it is largely adopted (e.g., for FE modelling and analysis), even if it may lead to inaccurate results [37]. Under the hypothesis of proportional damping, the following relation holds:  
\[2\zeta_{i}\omega_{i}=\frac{c_{i}}{m_{i}}=\frac{\alpha m_{i}+\beta k_{i}}{m_{i}}= \alpha+\beta\omega_{i}^{2} \tag{1}\]  
where \(\alpha\) and \(\beta\) are the proportional damping coefficients, \(\omega_{i}=2\pi f_{i}\), and \(m_{i}\), \(k_{i}\), \(c_{i}\), are the modal mass, modal stiffness, and modal damping of the \(i\)-th mode, respectively. After determining the modal parameters \(f_{i}\) and \(\zeta_{i}\) for \(N\) vibration modes, the coefficients \(\alpha\) and \(\beta\) can be estimated through least-squares approximation, by using the following expression:  
\[\left\{ \begin{array}{c}\alpha\\ \beta\end{array} \right\}=\left(\mathbf{A}^{T}\mathbf{A}\right)^{-1}\mathbf{A} ^{T}\mathbf{b} \tag{2}\]  
where  
\[\mathbf{A}=\left[ \begin{array}{ccc}1&\omega_{1}^{2}\\ \vdots&\vdots\\ \vdots&\vdots\\ 1&\omega_{N}^{2}\end{array} \right]\! ;\;\mathbf{b}=2\!\left\{ \begin{array}[] {c}\zeta_{1}\omega_{1}\\ \vdots\\ \vdots\\ \zeta_{N}\omega_{N}\end{array} \right\}\!. \tag{3}\]  
The proportional damping curves computed for the \(PN\) and \(PY\) panels are plotted in Figure 6. From the comparison with the corresponding experimental data, such models do not appear suitable for the tested specimens, since the damping ratios of modes at the lowest natural frequencies are not correctly fitted. The EMA results show that the \(PY\) panel exhibits a quite evident growth of the damping ratios as the natural frequencies increase. A slight increment in \(\zeta_{i}\) apparently characterizes the \(PN\) panel as well. Linear regression models, described by the expression  
\[\zeta_{i}=\lambda_{0}+\lambda_{1}f_{i}, \tag{4}\]  
are, thus, computed for possibly describing the observed trends of the damping ratios as linear functions of the natural frequencies. The experimental values of the damping ratios and the corresponding regression lines are plotted in Figure 7 for both the panels. The estimates of the intercept (\(\lambda_{0}\)) and of the slope (\(\lambda_{1}\)) of the regression lines are reported in Table 6, with the corresponding \(p\)-values. It is worth recalling that the \(p\)-value measures the probability that the estimated parameter has a correlation with the measured data, i.e., whether it is relevant for the regression model (a \(p\)-value of 0.05 or lower being the level of significance commonly accepted to reject the null hypothesis). For each specimen, the coefficient of determination, \(R^{2}\) (also known as \(R\)-_squared_), and the \(p\)-value of the regression model as a whole are also shown. The former, ranging from 0 to 1, indicates which fraction of the variability exhibited by the dependent variable is explained by the regression model.