## 2 Mechanical Model
### Model of the Frame Structure Connected to Vibrating Mass and an Inerter Device  
The SSSI problem is investigated using a low-dimensional mechanical model designed to capture the fundamental dynamics of the actual structure. This low-dimensional model, termed the archetype model (see Figure 2a), comprises two primary components: (i) the external system acting as a protection device and (ii) the system to be protected against earthquakes. The external system consists of a foundation platform with a mass \(m_{e}\), upon which is situated the vibrating mass \(M_{r}\) interacting with the inerter device having a virtual mass \(M_{I}\) (referred to as inertance). Withing the external system, the parameters \(k_{M}\) and \(c_{M}\) define the linear elastic connection between the vibrating mass \(M_{r}\) and the foundation platform. The system to be seismically protected consists of a foundation platform with a mass \(m_{s}\) supporting the frame structure. The multi-degree-of-freedom (M-DOF) structure is represented by a dynamically equivalent 2-degree-of-freedom (2-DOF) shear-type model with masses \(m_{1}\) and \(m_{2}\). The definition of such equivalent 2-DOF model exploits the dynamic equivalence proposed in [21; 22]. This dynamic equivalence provides the mass \(m_{i}\), stiffness \(k_{i}\), and damping \(c_{i}\) (\(i=\{1,2\}\)) associated with each of the two degrees of freedom. More specifically, the 2-DOF equivalent system is constructed such that the degree of freedom associated with \(m_{1}\) models the displacement of the first story of the real M-DOF frame structure, whereas the second degree of freedom associated with \(m_{2}\) models the displacement of the final story of the real structure. The connection between the two platforms and the soil is represented by Kelvin-Voigt devices characterized by linear and rotational stiffness coefficients \(k_{j}\) and \(k_{\theta j}\), along with the corresponding damping coefficients \(c_{j}\) and \(c_{\theta j}\), where \(j=\{e,s\}\). Subscript \(j=e\) denotes parameters pertaining to the external platform, whereas those with subscript \(j=s\) pertain to the platform supporting the frame structure. Additionally, the interaction between the external system and the frame structure occurring through the soil is modeled through an additional Kelvin-Voigt device characterized by the four coefficients \(k_{es}\), \(k_{\theta es}\), \(c_{es}\), and \(c_{\theta es}\). The two coefficients with subscript \(\theta\) refer to rotational components of the Kelvin-Voigt devices, essential for capturing the rotational dynamics of the two platforms. However, for the sake of simplicity, these rotational components are omitted from the graphical representation in Figure 2a. Overall, the archetype system is described by seven Lagrangian parameters, whose positive directions are shown in Figure 2b. Specifically, \(u_{j}\) and \(\theta_{j}\), where \(j=\{e,s\}\), describe the displaced configuration of the two foundation platforms, whereas \(u_{M}\), \(u_{1}\), and \(u_{2}\) are the relative displacements of the vibrating mass \(M_{r}\) and the 2-DOF equivalent frame system with respect to the corresponding foundation platforms. The equations of motion are derived through a Lagrangian approach. The displacement components of the masses defining the system, expressed as functions of the Lagrangian parameters according to Equation (1):  
\[ \begin{split} x_{e}(t)&=u_{e}(t)+u_{g}(t),\\ x_{M_{r}}(t)&=u_{e}(t)+u_{M}(t)-\theta_{e}(t)h_{e} +u_{g}(t),\\ x_{s}(t)&=u_{s}(t)+u_{g}(t),\\ x_{1}(t)&=u_{s}(t)+u_{1}(t)-\theta_{s}(t)h_{1}+u _{g}(t),\\ x_{2}(t)&=u_{s}(t)+u_{2}(t)-\theta_{s}(t)h_{2}+u _{g}(t).\end{split}  \tag{1}\]  
Figure 1: Interaction through the soil between vibrating mass with inerter and frame structure. Such displacements are needed to write the kinetic energy possessed by the system according to Equation (2):  
\[