## 1 Introduction  
The seismic protection of new or existing structures stands as a cornerstone within the field of Civil Engineering. Numerous possibilities exists for the preliminary assessment of the effectiveness of new protection techniques. When a proposed technique aims to influence the behaviour of an entire structure rather than focusing on local interventions, one viable approach involves evaluating its effectiveness using a simplified archetype mechanical model. This initial step yields insights that warrant more exhaustive analyses employing refined models, despite their higher computational costs. The present paper seamlessly fits into this narrative, employing an archetypal model to explore the feasibility of implementing a cutting-edge seismic protection system for frame structures. The use of low-dimensional models to explore methods for mitigating seismic responses in structures has been extensive. Within scholarly literature, two-degrees-of-freedom (2-DOF) models have been often employed to study the conceptual aspects related to seismic protection methodologies. For instance, comprehensive studies on Base Isolation (BI) have been conducted in [1], while investigations into Tuned Mass Damper (TMD)systems have been detailed in [2]. These simplified models have also facilitated the analysis of various modifications and combinations of these techniques. Notably, [PERSON] [3] and [PERSON] et al. [4] delved into the reduction of base displacement in base-isolated systems through the integration of TMDs. Recently, the same objective has been pursued in [5; 6] using TMDIs, i.e., combinations of TMDs and inerter devices. The effectiveness of such devices in controlling seismic responses has also been investigated using low-dimensional mechanical models, as demonstrated in [7; 8]. More complex low-dimensional mechanical models have been recently employed to investigate the seismic protection of frame structures by coupling them with external systems. Noteworthy, a Hysteretic Mass Damper Inerter (HMDI) [9] and an external elastic structure equipped with an inerter [10] have been used as coupled external systems to enhance the seismic response of frame structures. The prevailing trend observed in numerous papers, including those aforementioned, is the direct attachment of external systems to the target structure for protection. However, recent studies have investigated the prospect of enhancing structural response by considering the interaction mediated through the soil between an external system and the structure under consideration. In essence, this entails a scenario where the structure and the external system are not directly coupled but interact indirectly through the soil medium. The topic of structure-soil-structure interaction (SSSI) provides the tools to model interactions among different structures through the soil medium. The use of vibrating masses or barriers, enhanced with inerter devices, interacting through the soil with a target structure for sesimic protection, requires the application of SSSI models found in the scientific literature. In [11], a comprehensive review of the state of the art in recent papers addressing the SSSI problem is presented. In [12; 13; 14], researchers investigated the use of vibrating masses or barriers equipped with inerter devices, to improve the seismic response of structures, and the interaction between the vibrating mass and the structure was modeled using existing SSSI models. The analysis method applied in this paper is similar to that in [13; 14], where a low-dimensional model was used to describe the structure-soil-structure interaction between the vibrating mass and the structure to be protected, both represented by one-dimensional systems, with only the soil's horizontal dynamics considered. These studies pioneered the concept of mitigating seismic responses in structures by leveraging soil interaction with an external system.