## 5 Seismic Performance of the SSSI System
### Spectral Analysis  
This section explores the effectiveness of SSSI system in mitigating the seismic response of frame structures. Given the linearity of the mechanical system, conducting a spectral analysis becomes a logical step. Specifically, analyzing the vibration periods and modal shapes across distinct locations within the parameter space \(\eta-\gamma\) can yield critical insights into the system's functionality. Figure 8 presents the vibration periods and modal shapes of both the standalone and SSSI systems. The spectral analysis of the latter is conducted for two configurations identified by points \(A\) and \(B\) on the performance maps related to \(\alpha_{1}\) and \(\alpha_{2}\) that have been obtained for the San Jose earthquake, in Figure 7. Specifically, Figure 8a illustrates the first two vibration periods and associated modal shapes of the standalone structure. It should be emphasized that the standalone system is defined by four Lagrangian parameters (illustrated in the right section of the system in Figure 2b), thereby admitting four modes. Figure 8b,c display the initial four vibration periods and associated modal shapes of the SSSI system, which possesses a total of seven modes. The analysis deliberately concentrates on the initial modes of both systems, based on the assumption that the primary dynamic effects are predominantly influenced by these modes. At Point \(A\), both \(\alpha_{1}\) and \(\alpha_{2}\) exhibit values slightly less than unity. This point is situated in a region of the performance maps, where the effectiveness of the external protection system in reducing the response of the frame structure is comparatively lower, as indicated by the darker shade of gray. Conversely, Point \(B\) is located along the locus (represented by a dotted thick curve) where relative minima of the performance indices are observed. Consequently, for the selected value of \(\eta=50\), Point \(B\) corresponds to the optimal performance of the SSSI system, while Point \(A\) demonstrates the least effective performance. Independently from the chosen combination of parameters, in both \(A\) and \(B\), the SSSI system generally exhibits higher periods for the first modes compared to the standalone system. This effect can be attributed to the substantial mass of the external protection system, which can be theoretically increased as desired through the incorporation of an inerter device. Consequently, the first modes of the SSSI system are shifted towards spectral regions where seismic activity exhibits reduced power content. The analysis of results for Point \(A\) (Figure 8b) reveals that the two foundation platforms consistently move in phase across all four modes. Furthermore, in the first three modes, the entire translating mass of the external protection system also moves in phase with the foundation platform of the frame structure. It is reasonable to hypothesize that the dynamic contributions from these first three modes enhance the motion intensity at the base of the frame structure. Conversely, in the fourth mode, the vibrating mass moves in counter-phase relative to the foundation platform, which may lead to an improvement in the response. It is also observed that the periods of Mode 2 and Mode 4 in the SSSI system are close to the 5-story transition, which is consistent with the observation of the 5-story transition. Figure 7: Performance \(\alpha_{1}\) and \(\alpha_{2}\) maps of the 5-story frame structure under several earthquakes (\(\alpha=0.2\), \(V_{s}=300\) m/s, \(\chi=0.2\)). the periods of the first two modes of the standalone structure. The modest enhancement in the response of the SSSI system compared to the standalone system, as indicated by performance index values slightly less than unity, is primarily attributable to Mode 4. In this mode, the vibrating mass moves in counter-phase relative to both foundation platforms, thereby mitigating the dynamic effects transferred to the base of the frame structure.