# Seismic Benefits from Coupling Frame Structures with a Hysteretic Mass Damper Inerter
## 5 Seismic Analysis
### Spectral Gain Curve of a Single Seismic Event  
The first set of analyses aims to obtain the spectral gain curves using a single seismic record. These curves provide the values of one gain coefficient, either \(\alpha_{1}\) or \(\alpha_{2}\), versus \(T_{ext}\). All the spectra presented in this section are for the 4-story frame structure with mechanical characteristics reported in the second lines of Tables 1 and 2. Within this set of analyses, the first investigation is performed by varying the parameter \(\psi\), which characterizes the post-elastic behavior of the hysteretic connection. Figure 8 shows the gain spectra of \(\alpha_{1}\) (left graphs) and \(\alpha_{2}\) (right graphs) for the Kobe (first line of the graphs) and Newhall (second line of the graphs) earthquake records. The spectra are obtained for fixed values of \(\
u=10\), \(\beta_{y}=1.5\), and \(\gamma=0.1\). It is important to note that, by fixing \(\gamma\) and \(\
u\) (and thus \(m\) and \(m_{l}\)), a variation of the parameter \(T_{ext}\) corresponds to a variation of the elastic stiffness \(k_{c}\) only. Each graph contains two spectra, one obtained for \(\psi=1\) (thin line) and the other for \(\psi=0.1\) (thick line). The value of \(\psi=1\) refers to a connection device that has a linear elastic behavior with a stiffness \(k_{c}\) (see Equation (2)). Each graph includes a horizontal dashed line passing through the unitary value of the relevant gain coefficient. According to the definition of the gain coefficients in Equation (11), if the spectrum lies below the unitary line, then the HMDI is effective in reducing the displacements of the frame structure. On the other hand, if the spectrum is above the unitary line, the displacements of the coupled structure are larger than those of the uncoupled structure, and the coupling with the HMDI does not provide any benefits to the structure. It is worth noting that the presence of an external oscillating mass improves the seismic behavior of the structure, both in the case of elastic (\(\psi=1\)) and hysteretic (\(\psi=0.1\)) connections since the gain spectra are below the unitary horizontal line in almost all the investigated range of \(T_{ext}\). The case with \(\psi=1\) corresponds to a classical Tuned Mass Damper Inerter (TMDI) [26; 27; 28; 31]. The comparison between the spectra obtained for \(\psi=1\) and \(\psi=0.1\) reveals that the hysteretic behavior of the connection (\(\psi=0.1\)) results in reduced values of the gain coefficients in almost all the range of \(T_{ext}\) (i.e., the HMDI is effective in reducing the displacements of the frame structure) and an increase in the range of values of \(T_{ext}\) where \(\alpha_{1}\) decreases compared to the case of a linear elastic connection (\(\psi=1\)). It is noteworthy that in all cases, the spectra have points of absolute minimum where the reduction of displacements is maximized. As shown before, the minimum point of the \(\alpha_{1}\) spectrum is located close to \(\hat{T}_{ext}\). Conversely, the value \(\hat{T}_{ext}\) does not represent a point of particular interest for the \(\alpha_{2}\) spectra, but it is expected that the absolute minimum of \(\alpha_{2}\) occurs for values of \(T_{ext}\) less than \(\hat{T}_{ext}\). Figure 7: Spectra of the selected set of seven earthquakes, (dashed, solid thick, and solid thin lines refer to the average spectrum, the design spectrum, and the record spectra, respectively). A further analysis investigates the role of the parameter \(\beta_{y}\), which defines the threshold between the elastic and plastic behavior of the hysteretic connection. Figure 9 shows the gain spectra of \(\alpha_{1}\) (left graphs) and \(\alpha_{2}\) (right graphs), referring to the Kobe (first line of graphs) and Newhall (second line of graphs) earthquake records. The spectra are obtained for fixed values of \(\
u=10\), \(\psi=0.1\), and \(\gamma=0.1\). Each graph contains four spectra related to different values of \(\beta_{y}\), ranging from 0.2 to 3.0, as indicated in the graphs.