## 4 Results
### Properties of TMD1 and TMD2  
As discussed in Section 3, the TMD design is based on the main vibration modes of the building. Since the installation of the TMD on the building's rooftop requires the steel frame structure described in Section 3, the vibration modes to consider are those of the OB with the steel structure on the top (OBS). Table 4 shows the periods and the participating mass ratios of the most significant modes of the OBS. From Table 4, it can be seen that the first two modes are translational in the \(X\) and \(Y\) directions, respectively, and the third is rotational. These three modes have similar periods, ranging between 0.35 s and 0.41 s, and very similar participating mass ratios in the main direction, varying between 62.2 % and 68.3%. The remaining modes have lower periods, ranging between 0.08 s and 0.09 s. The modes of the OBS are practically identical to those of the OB (Table 3). The design method requires that the values of \(a_{max}\) and \(d_{max}\) be fixed. In order to maintain the structural members in the elastic field and to avoid damage to masonry infill \begin{table}
\begin{tabular}{c c c c c c c c} \hline \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 & 0.41 s & 66.3\% & 0.2\% & 1.0\% & 66.3\% & 0.2\% & 1.0\% \\
2 & 0.40 s & 0.2\% & 68.3\% & 0.0\% & 66.5\% & 68.5\% & 1.0\% \\
3 & 0.35 s & 1.0\% & 0.0\% & 66.2\% & 67.5\% & 68.6\% & 67.3\% \\
4 & 0.09 s & 0.0\% & 18.4\% & 0.0\% & 67.5\% & 87.0\% & 67.3\% \\
5 & 0.09 s & 19.5\% & 0.0\% & 0.1\% & 87.1\% & 87.0\% & 67.5\% \\
6 & 0.08 s & 0.1\% & 0.0\% & 19.2\% & 87.2\% & 87.0\% & 86.8\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Periods and mass participating ratios of the most significant modes of OBS. Figure 11: Bilinear force–velocity relationship of the ‘Damper–Bilinear’ element. panels under the seismic actions at the SLV, the values \(a_{max}=3.26\) m/s\({}^{2}\) and \(d_{max}=0.011\) m were taken. These values lead to a mass ratio equal to 0.51. As the effective mass of mode 2, \(M_{2}\) equal to 1183 \(t\), an RC slab thickness of \(z_{TMD}\) equal to 0.90 m is derived. If the TMD design was made taking different values of \(a_{max}\) and \(d_{max}\) in the \(X\) and \(Y\) directions, the values of \(a_{max}\) and \(d_{max}\) to take in the \(X\) direction would be 3.50 m/s\({}^{2}\) and 0.012 m, respectively, which would lead to \(\mu=0.38\). Consequently, with \(M_{1}=1148\) \(t\), a thickness of the RC slab equal to 0.65 m would be obtained. From the values of \(k\) and \(\xi\), the number and characteristics of LRIs, for TMD1, and those of LDRIs and VLDs, for TMD2, can be determined. Then, the strength verifications of structural members are made, and the maximum values of the flexural demand over capacity ratio \(\frac{M_{Ld}}{M_{Rd}},_{max}\) equal to 1.45 for TMD1 and equal to 1.94 for TMD2 are found. However, the values obtained for \(\mu\) and \(z_{TMD}\) are considered to be too large. Therefore, the performance of smaller TMDs is investigated. The trend of \(\frac{M_{Ld}}{M_{Rd}},_{max}\) obtained for TMD1 and TMD2 as \(\mu\) varies between 0 (OB) and 0.51 is shown in Figure 12.  
In this figure, the mean values of the flexural demand over capacity ratios obtained for the two proposed solutions are also shown. From Figure 12, it can be seen that, in both solutions, \(\frac{M_{Ld}}{M_{Rd}},_{mean}\) decreases monotonically as \(\mu\) increases, passing from 1.3 for the OB to 0.53 for TMD1 and 0.52 for TMD2, but the decrease is negligible from \(\mu=0.11\). Moreover, from Figure 12, it can be observed that the designed TMDs do not lead to the minimum values of \(\frac{M_{Ld}}{M_{Rd}},_{max}\) as the minimum values are achieved with lower values of \(\mu\), and in particular with \(\mu=0.38\) for TMD1 and \(\mu=0.28\) for TMD2. Therefore, a greater mass of the TMD does not mean better performance.