## 6 Conclusions  
This paper studied the reduction in the seismic response of a multi-story frame structure through the utilization of an external vibrating mass coupled with an inerter device and their interaction with the soil. The system under study comprised two identical foundation platforms. The frame structure intended for protection was considered placed on the first platform. The second platform was considered as support for a vibrating mass connected to the platform via a visco-elastic device. The studied system also included an inerter device having one terminal linked the vibrating mass and the other directly to the ground. Notably, the two platforms were not connected through any device. Both the horizontal displacements and the rotations of the platforms were considered as Lagrangian parameters, along with the horizontal displacement of the vibrating mass. Given that the multi-degree-of-freedom frame structure was represented by a 2-degree-of-freedom equivalent system, two additional Lagrangian parameters were required to fully characterize the complete system. Consequently, a seven-degrees-of-freedom mechanical model was employed to model the interacting system. The interactions between the frame structure, the soil, and the protection system were modeled using linear visco-elastic devices. The damping and stiffness of these devices were determined by utilizing established soil-structure and structure-soil-structure interaction models. An extensive parametric analysis was conducted to investigate the performance of the system. The obtained results were organized into performance maps, which present contour plots of performance indexes specifically defined for this study. These indexes capture the ratios between the displacements and drifts of the frame structure when interacting with the soil through the vibrating mass, compared to those of the standalone frame structure. The study focused on examining the influence of parameters related to the vibrating mass and the inerter, such as the stiffness of their connection with the foundation platform and the virtual mass of the inerter device. It was observed that the system's response is significantly influenced by these parameters. Specifically, within the parameter space defined by these two factors, a region characterized by relative minimum values of both performance indexes was consistently identified. This region could be used to define preliminary optimal design parameters for the system under study. Further results of the parametric analyses can be summarized as follows:  
* The efficacy of the interaction with the external protection system (i.e., the vibrating mass and the inerter device) is enhanced by reducing the separation distance between the two foundation platforms. * The effectiveness of the interaction with the external protection is greater in soils with low shear wave velocity (\(V_{s}\)). Figure 10: Absolute acceleration maps for Frame 2 under several earthquakes (\(\alpha=0.2\), \(Vs=300\), \(\chi=0.2\)). * The performance of the interaction with the external protection system is heightened by reducing the stiffness of the soil beneath the foundation of the external protection system, such as through the implementation of a Geotechnical Seismic Isolation strategy. In conclusion, optimal attenuation of the seismic response in the frame structure is achieved when the entire external protection system, encompassing the foundation platform, undergoes nearly counter-phase movement with the frame structure platform. This synchronization results in a diminished motion intensity at the base of the frame structure, thereby ensuring superior seismic performance of the protected structure. Conceptualization, A.D.E. and A.C.; Methodology, A.D.E. and A.C.; Formal analysis, A.D.E. and A.C.; Investigation, A.D.E. and A.C.; Writing--original draft, A.C.; Writing--review & editing, A.D.E. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Not applicable.