## 4 Results
### Vibration Modes of the Building after the Interventions  
The modal information of the building equipped with TMD1 is shown in Table 8.  
From this table, it can be seen that the modes can be subdivided into three groups, each constituted by three modes: the first two translational in the \(X\) and \(Y\) directions, respectively, and the third rotational. The first group, composed of modes 1 to 3, is characterized by higher periods than those of OBS (Table 4), ranging from 0.62 s (mode 1) to 0.47 s (mode 3). The participating mass ratios of these modes range between 36.7% and 48.7%. The second group of modes, composed of modes 4, 5 and 6, is characterized by periods that are slightly lower than those of modes 1, 2 and 3 of the OBS and mass participating ratios that are about half, ranging between 26.2% (mode 6) and 38.3% (mode 4). The periods and mass participating ratios of the last group of modes (modes 7-9) are similar to those of modes 4-6 of the OBS. These modes are fundamentally those of the OBS, since the TMD does not participate in them.  
Similarly to what was performed for the building equipped with TMD1, the modal properties of the building equipped with TMD2 are shown in Table 9.  
Figure 17: Layout of the base isolation system.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Mode \(i\)** & \(T_{i}\) & \(\alpha_{i,X}\) & \(\alpha_{i,Y}\) & \(\alpha_{i,Rz}\) & \(\Sigma\,\alpha_{i,X}\) & \(\Sigma\,\alpha_{i,Y}\) & \(\Sigma\,\alpha_{i,Rz}\) \\ \hline
1 & 0.62 s & 0.0\% & 36.9\% & 0.0\% & 0.0\% & 36.9\% & 0.0\% \\
2 & 0.61 s & 36.7\% & 0.0\% & 0.0\% & 36.8\% & 36.9\% & 0.0\% \\
3 & 0.47 s & 0.0\% & 0.0\% & 48.7\% & 36.8\% & 36.9\% & 48.7\% \\
4 & 0.31 s & 0.1\% & 38.3\% & 0.0\% & 36.9\% & 75.2\% & 48.7\% \\
5 & 0.31 s & 37.8\% & 0.1\% & 0.0\% & 74.7\% & 75.3\% & 48.7\% \\
6 & 0.24 s & 0.1\% & 0.0\% & 26.2\% & 74.7\% & 75.3\% & 75.0\% \\
7 & 0.08 s & 0.0\% & 14.1\% & 0.0\% & 74.7\% & 89.4\% & 75.0\% \\
8 & 0.08 s & 14.9\% & 0.0\% & 0.1\% & 89.6\% & 89.4\% & 75.0\% \\
9 & 0.07 s & 0.1\% & 0.0\% & 14.2\% & 89.7\% & 89.4\% & 89.2\% \\ \hline \end{tabular}
\end{table}
Table 8: Periods and mass participating ratios of the most significant modes of the building equipped with TMD1.  
From the comparison with Table 8, it can be observed that the effect of TMD2 on the vibration modes of the building is very similar to that of TMD1, since the periods and participating mass ratios of translational modes are almost identical. A slight difference emerges in rotational modes 3 and 6, whose participating mass ratios are, respectively, higher and lower than those of TMD1.  
The modal information of the original building with the BIS is shown in Table 10.  
It can be seen that the dynamic behavior of the building is fully described by only three modes, which altogether provide participating mass ratios of 100% for each DOF of the building. The periods of the translational modes, equal to 2.87 s for modes 1 and 2, fall in the last branch of the acceleration site spectrum in Figure 5.  
From the comparison of the modal analysis results of the OB (Table 4) with those of the building equipped with TMD1 (Table 8), TMD2 (Table 9) and BIS (Table 10), it can be noticed that all three interventions produce significant modifications of the building modes.