## 4 Vibration-Suppression-System Layout and Individuation of the Appropriate MRD  
The adoption of dissipative bracings that integrate magnetorheological fluid dampers (MRDs) was considered as a simple and effective solution to mitigate the seismic response of the reference structure. Figure 19: Seismic response, interstory speed at the 5 th floor (\(\Delta v_{\text{y5}}=v_{\text{y5}}-v_{\text{y4}}\)). \begin{table}
\begin{tabular}{c c c c c c} \hline \multicolumn{2}{c}{**Story (j) \(\rightarrow\)**} & **1** & **2** & **3** & **4** & **5** \\ \hline \(\mathbf{a_{yj,min}}\) & **(m/s\({}^{2}\))** & \(-1.8519\) & \(-3.3017\) & \(-1.8979\) & \(-1.8335\) & \(\mathbf{-3.9358}\) \\ \hline \(\mathbf{a_{yj,max}}\) & **(m/s\({}^{2}\))** & \(2.1315\) & \(2.9111\) & \(1.9864\) & \(1.9148\) & \(\mathbf{3.8772}\) \\ \hline \(\Delta\mathbf{a_{yj,min}}\) & **(m/s\({}^{2}\))** & \(-1.8519\) & \(-1.865\) & \(-2.6038\) & \(\mathbf{-3.1147}\) & \(-3.1007\) \\ \hline \(\Delta\mathbf{a_{yj,max}}\) & **(m/s\({}^{2}\))** & \(2.1315\) & \(1.6767\) & \(2.0622\) & \(\mathbf{3.0768}\) & \(2.5508\) \\ \hline \(\Delta\mathbf{V_{min}}\) & **(m/s)** & \(-0.0661\) & \(-0.0484\) & \(-0.0491\) & \(\mathbf{-0.105}\) & \(-0.0947\) \\ \hline \(\Delta\mathbf{V_{max}}\) & **(m/s)** & \(0.0564\) & \(0.0375\) & \(0.0695\) & \(\mathbf{0.1005}\) & \(0.0724\) \\ \hline \end{tabular}
\end{table} 
Table 3: Seismic response of the reference structure; peak values of relevant parameters (maximum and minimum values in bold). The bracings were positioned according to the layout in Figure 20, which is in compliance with three practical design requirements:  
* To use the lowest number of bracings per floor;
* To preserve the symmetry of the structure with respect to the Y-Z plane (thus preventing relevant changes in its seismic response after the installation of the bracings);
* To dampen the vibrations of the structure along the direction of excitation (\(Y\)-axis). The selection of the best candidate among the MRD models that are available on the market resulted from the qualitative analysis of the acceleration-response spectrum to the seismic signal (Figure 21). Figure 20: FEM of the reference structure equipped with the vibration-suppression system. The spectrum is a graphical representation of the transient acceleration input, in terms of how a single-degree-of-freedom (SDOF) system (such as a mass on a spring and a damper) would respond to that input. The horizontal axis shows the natural frequency of a hypothetical SDOF. The vertical axis shows the peak acceleration that this SDOF would undergo as a consequence of the input signal and in correspondence with different damping levels [29]. From the curves in Figure 21, it was found that, to mitigate the acceleration response by 50%, it would have been necessary to reach at least a value of 0.04 for the modal damping of all the modes participating in the response. A generalized damping matrix characterized by 4% modal damping for all modes was then assumed for the structure and was converted into physical coordinates. The result was found to be consistent with the expression of the damping matrix that was obtained by assembling the contributions of the linear viscous dampers installed on each bracing, with each one characterized by a damping coefficient in the order of \(10^{5}\)Ns/m. The MRD-1005-3 from Lord Co.(*) (Cary, NC, USA) proved to ensure equivalent damping right in the order of \(10^{5}\)Ns/m, and it was therefore selected as a suitable device for the specific application (Figure 22; see datasheet in Appendix B). The equivalent physical damping is intended here as the damping (Ceq) that is exhibited by a linear viscous damper that dissipates the same energy as the MRD during the seismic event. The evaluation of this parameter can be carried out by means of the following approach. The damping force (\(D\)) produced by an MRD is a nonlinear function (\(f\)) of the piston speed (\(\dot{\delta}\): \(D=f\left(\dot{\delta}\right)\)).