## 5 Seismic Performance of the SSSI System
### Spectral Analysis
Results obtained for the system designated as Point \(B\) (Figure 8c) indicate that the external protection system and the foundation platform of the frame structure move in phase only in the first two modes. However, these modes exhibit periods longer than the initial pe  
Figure 8: Periods and modal shapes of systems labeled in point \(A\) and \(B\) in Figure 7. (**a**) Standalone structure. (**b**) Point \(A\) (\(\eta=50\), \(\gamma=37.5\), \(\alpha=0.2\), \(V_{s}=300\) m/s, \(\chi=0.2\)). (**c**) Point \(B\) (\(\eta=50\), \(\gamma=137.5\), \(\alpha=0.2\), \(V_{s}=300\) m/s, \(\chi=0.2\)). riod of the standalone structure, generally reducing their dynamic significance. Conversely, Modes 3 and 4, which have periods closely matching those of the standalone structure, display either total or partial movement of the external protection system in counter-phase relative to the foundation platform of the frame structure. Specifically, in Mode 3, the entire external protection system moves in counter-phase relative to the foundation platform of the frame structure, whereas in Mode 4, only the vibrating mass moves in counter-phase relative to both foundation platforms. Therefore, since Modes 3 and 4 of the SSSI system effectively replace the corresponding modes of the standalone structure due to the proximity of their periods, the benefit of the SSSI system is greater at Point \(B\) compared to Point \(A\). This is because both the third and fourth modes contribute to a reduction in the dynamic effects transmitted to the base of the frame structure. A further confirmation of the previous assertions can be obtained by analyzing the time histories of the SSSI system. Figure 9 shows the time histories of some referring displacement of systems characterized by parameters \(\eta-\gamma\) of Point \(A\), Point \(B\), and Point \(C\) on the San Jose performance maps of Figure 7. Specifically, in Figure 9a, each graph shows the \(u_{e}\) and \(u_{M}\) components, whereas in Figure 9b, each graph shows the \(u_{e}\) and \(u_{s}\) components. Each row of Figure 9 shows graphs whose time histories refer to the three different points \(A\), \(B\), and \(C\). Further validation of the previous assertions is provided by the analysis of the time histories of the SSSI system. Figure 9 displays the time histories of specific displacements associated with systems characterized by parameters \(\eta-\gamma\) for Points \(A\), \(B\), and \(C\) on the San Jose performance maps of Figure 7. Specifically, Figure 9a presents the time histories of the \(u_{e}\) and \(u_{M}\) displacements, while Figure 9b displays the time histories of \(u_{e}\) and \(u_{s}\). Each row of Figure 9 corresponds to the time histories associated with one of the three distinct points \(A\), \(B\), and \(C\). Starting from Point \(A\), the comparison of the time histories for the displacements \(u_{e}\) and \(u_{M}\) indicates that the displacement of the vibrating mass \(u_{M}\) is significantly smaller than that of the foundation platform of the external protection system \(u_{e}\). Additionally, these two displacements are substantially in phase with each other. The comparison between displacements \(u_{e}\) and \(u_{s}\) also shows that these two components are in phase, corroborating the results from the spectral analysis which indicated that in almost all modal shapes at Point \(A\), the components of the two foundation platforms are in phase. Consequently, the benefits of the external protection system in mitigating seismic effects on the frame structure are minimal, as reflected by the proximity of \(\alpha_{1}\) and \(\alpha_{2}\) to unity. Results pertaining to Point \(B\) continue to demonstrate the relatively small magnitude of \(u_{M}\) compared to \(u_{e}\), with these two displacements remaining in phase with each other. Conversely, the motion of the two foundation platforms, described by \(u_{e}\) and \(u_{s}\), is significantly out of phase.