## 4 Seismic Excitation
### Comparison of Results from Various Earthquake Records  
This section explores the comparison of results obtained with the three previously presented earthquake records (Kobe, Newhall, and Pacoima) for the two frame structures detailed in Tables 1 and 2. The analysis was carried out under a constant \(M\) (\(\eta=0.1\)) and \(r=0.7\). In Figure 6, gain maps for both coefficients related to Frame 2 are depicted; each row represents results from a different earthquake. As expected, the values of the gain coefficient significantly depended on the earthquake records. The size of the advantage regions, the minimum values of \(\alpha_{1}\) and \(\alpha_{2}\), and the position of the relative minimum points all varied according to the used record. Overall, the best seismic behavior (i.e., the smallest displacements) of the coupled system was observed under the Pacoima earthquake. In this case, the advantage regions covered the entire parameter plane, and the protection system reduced the displacements of the structure (both \(u_{1}\) and \(\Delta u\)) by up to 50% of those of the uncoupled structure. Figure 7 presents the analysis of the time histories of the coupled systems characterized by the values of \(\beta_{c}\) and \(\rho_{e}\) at points \(A\) and \(B\) marked in Figure 6. Such analysis is intended to provide a deeper understanding of the protection system's behavior in reducing the displacements of the frame structure. These points were carefully chosen to be in proximity to the relative minimum points of both the \(\alpha_{1}\) and \(\alpha_{2}\) maps in correspondence with the same values of \(\beta_{c}\). The graph on the left corresponds to point \(A\), while that on the right pertain to point \(B\). Both graphs depict the time histories of the displacement of the connection story  
Figure 4: Gain maps of the coupled frame structure excited by the Kobe earthquake for different values of the restitution coefficient \(r\) (Frame 1 and \(\eta=0.1\)). (dotted line) and drift of the superstructure \(\Delta u\) (dashed line) together with the relative motion of the vibro-impacting mass \(u_{M}-u_{1}\) (solid line). The horizontal dashed lines correspond to the clearance \(d\) of the VI-NES, thereby representing the upper threshold of the displacement of the vibro-impacting mass at which an impact occurs. Since the two graphs were derived for the same value of \(\beta_{e}\), the value of \(d\) was also consistent between the two cases. In correspondence with the initial impacts of the vibro-impacting mass, its motion was almost in a counterphase with the displacements of the frame structure. Therefore, the protection system functions similarly to a tuned mass damper. After the initial impacts, the vibro-impacting mass remained slightly out of phase with the motion of the frame structure, thus limiting the displacements of the frame structure and providing energy dissipation through the impacts. Finally, Figure 8 displays the \(\alpha_{1}\) and \(\alpha_{2}\) gain maps of Frame 1. The maps in each row correspond to a different earthquake. Also for Frame 1, the gain maps strongly depend on the characteristics of the earthquake record, thereby affecting the extension of the advantage regions, the minimum values of the gain coefficients, and their positions. Unlike the results presented for Frame 2, in this case, the smallest displacements were obtained under the Newhall earthquake record. Although the results in Figure 6 and Figure 8 consider the same protection system (\(\eta=0.1\) and \(r=0.7\)), the gain coefficients for Frame 1 were slightly higher than those for Frame 2. This difference could be attributed to the varying height between Frame 1, which consists of four stories, and Frame 2, a three-story frame (see Table 1). One might reasonably expect that the effectiveness of the protection system, which was consistently connected to the first story of the frame, would decrease with an increase in the number of stories.