# Seismic Benefits from Coupling Frame Structures with a Hysteretic Mass Damper Inerter
## 5 Seismic Analysis
### Spectral Gain Maps of a Single Earthquake Record  
This section presents the results of an analysis performed to obtain the spectral gain maps of the coupled structure under a single earthquake record. Such maps are contour plots of one gain coefficient, either \(\alpha_{1}\) or \(\alpha_{2}\), in the parameter plane \(v\)-\(T_{ext}\). As in the previous analyses, all other parameters characterizing the coupled system are fixed. In these gain maps, the light gray regions are those where the gain coefficient is smaller than unity; thus, these are regions where the HMDI reduces \(u_{1}\) or \(u_{2}-u_{1}\), compared with those of the uncoupled structure. On the contrary, the dark gray regions are those where the gain  
Figure 10: Spectral gain curve of the four-story frame structure under Kobe and Newhall earthquakes: parameters \(m\) and \(m_{1}\) variable (thick line: \(m_{I}=10(m_{1}+m_{2})\), \(m=0.1m_{1}\); thin line: \(m_{I}=0.1m_{1}\), \(m=10(m_{1}+m_{2})\)); (\(\beta_{y}=1.5\); \(\psi=0.1\)). Figure 9: Spectral gain curve of the four-story frame structure under Kobe and Newhall earthquakes: parameter \(\beta_{y}\) variable (\(\
u=10\); \(\psi=0.1\); \(\gamma=0.1\)). coefficient is higher than unity. It is worth noting that the sections of the gain maps parallel to the \(T_{ext}\) axis are the spectral gain curves described in the previous section. Figure 11 shows the gain maps of \(\alpha_{1}\) (left column) and \(\alpha_{2}\) (right column) refers to the four-story frame structure (see Tables 1 and 2), which is excited by the three earthquake records, i.e., Kobe (first row), Newhall (middle row), and Parkfield (third row). As can be observed, the advantage regions where the gain coefficients are smaller than unity (light gray regions) cover almost all of the parameter plane \(\
u\)-\(T_{ext}\). Each map in Figure 11 shows a thick dashed line representing the value of \(\hat{T}_{ext}\). From the gain maps of the Kobe earthquake (first row), it can be observed that the lowest values of \(\alpha_{1}\) can be found for \(T_{ext}=\hat{T}_{ext}\). The same does not occur for \(\alpha_{2}\); the lowest values of \(\alpha_{2}\) can be found for \(T_{ext}<\hat{T}_{ext}\), as anticipated in the previous sections. The other two earthquake records lead to similar results. Specifically, when the Newhall and Parkfield earthquake records are considered, the lowest values of \(\alpha_{1}\) are not in correspondence with \(\hat{T}_{ext}\) but can be obtained for values of \(T_{ext}\) that are slightly lower than \(\hat{T}_{ext}\). Moreover, for these two additional earthquake records, the lowest values of \(\alpha_{2}\) are found for \(T_{ext}<\hat{T}_{ext}\). The contour levels of the gain maps show that the HMDI significantly reduces the displacement of the story connected to the HMDI and/or the drift of the upper stories. Since the regions where \(\alpha_{1}\) and \(\alpha_{2}\) take the lowest values do not coincide, it is not possible to obtain design parameters that simultaneously minimize \(u_{1}\) and \(u_{2}-u_{1}\). However, it is always possible to choose design parameters where both \(\alpha_{1}\) and \(\alpha_{2}\) are smaller than unity. It is important to note that, in the maps, the position of the lowest value of \(\alpha_{1}\) with respect to \(\hat{T}_{ext}\) depends on the seismic record and, in particular, on its frequency spectral content. For instance, considering the Kobe earthquake record, the lowest values of \(\alpha_{1}\) are obtained for values of \(T_{ext}\) that are significantly close to \(\hat{T}_{ext}\), whereas using the Newhall and Parkfield earthquake records the lowest values of \(\alpha_{1}\) are obtained for values lower than \(T_{ext}\). Figure 12 shows time histories used to investigate the interaction between the frame structure and HMDI further. The graphs are arranged in four rows, each referring to one of the points labeled from \(A\) to \(D\) in the gain maps obtained with the Kobe and Newhall earthquake records (Figure 11).