## 1 Introduction
In alignment with this approach, the present paper shares a similar objective and employs a low-dimensional model to represent structure-soil-structure interaction. Specifically, it investigates reducing seismic responses in frame structures by utilizing soil interaction with a vibrating mass equipped with an inerter device. In this paper, a mechanical type of inerter device, designed to exert a force that is directly proportional to the relative acceleration between two connection points, or terminals, is considered. A common design approach involves using a rack-pinion-flywheel assembly [15; 16]. It is pertinent to clarify that this paper does not aim to refine existing soil-structure interaction (SSI) and structure-soil-structure interaction (SSSI) models. Instead, these models serve as indispensable tools for modeling the interaction between structures and the soil, as well as between adjacent structures. Specifically, the models employed in this study characterize the soil utilizing linear viscoelastic Kelvin-Voigt devices, with their damping and stiffness properties derived from specific relationships. For further elucidation on the utilized SSI and SSSI models, readers are directed to Section 2.2. Compared to [13; 14], the original contributions primarily pertain to the following aspects:  
* The rotational motion of a portion of soil is accounted for, leading to a dual level of interaction between structures and the soil. * The structure to be protected is a frame structure and is modeled using a 2-DOF dynamically equivalent system. Employing a 2-DOF model, rather than a simpler 1-DOF system, enables the capture of additional information on the behavior of the structure. * The parametric analysis extends to parameters defining the soil properties beneath both the protected structure and the external system (comprising the vibrating mass and inerter device). This comprehensive approach allows for the consideration of soil heterogeneity across different locations. * The visualization of parametric results through performance maps, constructed by varying the characteristics of the vibrating mass and inerter, facilitates the identification of optimal parameter ranges where the system exhibits superior effectiveness. The system under investigation comprises two foundation platforms. One of these platforms supports the vibrating mass, which is connected to the platform via a viscoelastic device and to an inerter device [17]; consequently, the two terminals of the inerter are linked with the vibrating mass and directly with the ground. Positioned atop the second platform is the frame structure intended for protection. Both the horizontal displacements and rotations of the platforms, along with the horizontal displacement of the vibrating mass, are considered Lagrangian parameters. Conversely, the multi-story frame structure to be protected is modeled using a 2-degree-of-freedom equivalent system, employing two Lagrangian parameters. Hence, to describe the interaction system, a seven-degree-of-freedom mechanical system is employed. Two distinct series of parametric analyses are undertaken. The initial series involves the systematic variation of the stiffness of the device linking the vibrating mass to the foundation platform and the virtual mass provided by the inerter device. The results are presented in performance maps, which illustrate the performance metrics across the parameter space. These metrics are the ratios between two specific displacements of the structure interacting with the vibrating mass and the corresponding displacements of the standalone structure, i.e., the structure without the seismic protection system. The two displacements are that of the first-story drift and the total drift of the structure. The second series of parametric analyses explores the influence on the dynamics of the structure of parameters characterizing the soil and certain geometrical characteristics of the foundation platforms.