## 3 Results and Discussion
### Shaker Excitation Tests  
The results obtained from the analyses of the shaker excitation measurements for the first ten vibration modes are summarized in Table 4, where subscripts _PN_, _PY_, and \(i\) refer to panel _PN_, panel _PY_, and mode ranking, respectively. Such results are computed as an average of the values obtained from the different modal identification processes, by also varying the excitation profile/level and the computational algorithm (in the time and frequency domains, respectively), while keeping fixed the frequency range (5-200 Hz). Only the results characterized by satisfactory values of the parameters MPC, MPD, and MP have been included in the average. The mode shapes are described by means of two parameters, namely \(m\) and \(n\), according to a conventional notation frequently adopted for the vibrations of rectangular plates [34]. In particular, in the case of a rectangular plate made of a homogeneous material with all the sides clamped, \(m\) and \(n\) refer to the number of half-sine waves along the direction of the long and short sides, respectively. The comparison between the natural frequencies and the damping ratios of the _PN_ and _PY_ panels is also reported, in terms of percentage variations \(\Delta f_{i}\) and \(\Delta\zeta_{i}\), respectively, normalized to the panel PN values; moreover, factors of damping increment (i.e., the ratios between the damping ratios) \(\rho_{\gamma N,i}=\zeta_{PY,i}/\zeta_{PN,i}\) are provided as well. The result presentation is limited to the first ten modes, below 200 Hz, since, for higher ranking modes--above all for panel _PY_--the mode shape estimation proves not extremely reliable (with fuzzy values for MPC, MPD, and MP parameters) and a robust association with the _PN_ panel ones is not straightforward, thus making the direct comparison uncertain. Figure 4: Sum of frequency response functions (FRFs), limited to 5–1000 Hz, estimated in two tests performed on panel _PN_. The comparison between the FE model (FEM) and the EMA results for the \(PN\) panel vibration modes #1, #6, and #9 (chosen as examples) is reported in Figure 5.  
In particular, it can be observed that the limited resolution of the modal geometry (\(Pij\), \(C1\), and \(in1\) nodes) causes a very similar representation for the mode shapes #1 and #9. This confirm the usefulness  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multicolumn{2}{c}{**Mode**} & \multicolumn{2}{c}{**Mode**} & \multicolumn{2}{c}{**Natural**} & \multicolumn{2}{c}{**Damping**} & \multicolumn{2}{c}{**Comparison**} \\ \multicolumn{2}{c}{**Ranking**} & \multicolumn{2}{c}{**Shape**} & \multicolumn{2}{c}{**Frequency**} & \multicolumn{2}{c}{**Ratio**} & \multicolumn{2}{c}{**} & \multicolumn{2}{c}{**Comparison**} \\ \hline \multirow{2}{*}{\#\({}_{PN,i}\)} & \multirow{2}{*}{\#\({}_{PY,i}\)} & \multirow{2}{*}{**(_Xi_)**} & \(f_{PN,i}\) & \(f_{PY,i}\) & \(\zeta_{PN,i}\) & \(\zeta_{PY,i}\) & \(\Delta f_{i}\)  
of the FEM results for properly identifying the mode shapes estimated through the EMA, thus, partially solving the shape aliasing problem. The natural frequency variations between the two panels remain rather limited for all the vibration modes, the maximum difference being less than 10%. The mode ranking remains the same except for modes #3 and #4. The trend of \(\Delta f_{i}\) is not monotonic: in particular \(\Delta f_{i}>0\) for modes with \(n\geq m\) (apart from \(n=1\)), due to the stiffening effect of the additional thickness of the SMACWRAP(r) layer on the mode shapes featuring more waves along the short side with respect to the long one (e.g., mode #9 in Figure 5e). On the contrary, the slight decrement in other natural frequencies is due to the additional mass effect. The damping ratios characterizing the \(PY\) panel are significantly higher than in the \(PN\) panel, for all the vibration modes, with percentage increment ranging from 123% to 303%, depending--apparently inconsistently--on the single modes.