## 2 Mechanical Model
### Structure-Soil-Structure Interaction Model  
The SSSI model used in the present paper is borrowed from [20]. This model is a simplification of the comprehensive model derived in [18; 19]. A notable simplification concerns the dimensions of the foundation platforms. Specifically, in the model proposed in [20], the platforms are assumed to be square-shaped with equal sides of length \(2a\) (see Figure 2). Given the equivalence in the dimensions of the foundation platforms, their masses and polar inertia coincide (\(m_{s}=m_{e}\) and \(I_{s}=I_{e}\)). These parameters are adjusted to incorporate contributions from the underlying soil. To derive such quantities, virtual contributions attributable to the soil (\(m_{vc}\), \(m_{vs}\), \(I_{ve}\), and \(I_{vs}\)) are added to the initial masses (\(m_{fe}=m_{fs}\)) and polar moments of inertia (\(I_{fe}=I_{fs}\)) of the foundation platforms. Specifically, the total mass \(m_{e}\) and the total polar inertia \(I_{e}\) are determined according to Equation (8):  
\[ \begin{split}& m_{e}=m_{fe}+m_{vc},\\ & I_{e}=I_{fe}+I_{vc}.\end{split}  \tag{8}\]  
The initial mass \(m_{fe}\) and polar moment of inertia \(I_{fe}\) only depend on the geometrical and mechanical properties of the foundation platforms, and thus, they are known quantities, defined according to Equation (9):  
\[ \begin{split}& m_{fe}=2a\times 2a\times t\times\rho_{p},\\ & I_{fe}=m_{fe}\times\left[\frac{t^{2}}{12}+\frac{\left(2a \right)^{2}}{12}\right],\end{split}  \tag{9}\]  
where \(t\) and \(\rho_{p}\) are the thickness and mass density of the foundation platform, respectively. The virtual contributions (\(m_{vc}\) and \(I_{vc}\)) are provided as fractional proportions of \(m_{fe}\) and \(I_{fe}\) and are formulated according to Equation (10):  
\[ \begin{split}& mv_{e}=\frac{0.095m_{fe}}{\beta_{e}};\ \ \ \beta_{e}=\frac{(1-v_{max})m_{fe}}{4\rho r_{0}^{3}};\\ & Iv_{e}=\frac{0.095I_{fe}}{\beta_{e}};\ \ \ \beta_{ge}=\frac{3(1-v_{max})m_{fe}}{8\rho r_{0}^{5}}; \end{split}  \tag{10}\]  
where \(r_{0}=2a\Big{/}(3\pi)^{1/4}\) and \(\rho\) is the mass density of the soil. In accordance with [20], the stiffness and damping coefficients of the Kelvin-Voigt devices used to model the interaction between the two foundation platforms and the soil, illustrated in Figure 2, are specified based on Equation (11):  
\[ \begin{split}& k_{e}=\chi\frac{9.2G_{max}a}{2-v_{max}};\ \ k_{\theta e}=\chi\frac{4.0G_{max}a^{3}}{1-v_{max}};\ \ \ c_{e}=\frac{0.163a}{V_{s}}k_{e};\ \ c_{ \theta e}=\frac{0.6a}{V_{s}}k_{\theta e}\\ & k_{s}=\frac{9.2G_{max}a^{2}}{2-v_{max}};\ \ k_{\theta s}=\frac{4.0G_{max}a^{3}}{1-v_{max}};\ \ \ c_{s}=\frac{0.163a}{V_{s}}k_{s};\ \ c_{ \theta s}=\frac{0.6a}{V_{s}}k_{\theta s}.\end{split}  \tag{11}\]  
Moreover, the stiffness and damping of the devices decribing the structure-soil-structure interaction (see Figure 2) are defined according to Equation (12):  
\[ \begin{split}& k_{es}=\Big{[}3.7561\times 10^{-0.18995\left(\frac{t}{ 4}\right)}\Big{]}\frac{G_{max}a}{2-v_{max}};\ \ k_{\theta s}=-\Big{(}0.04234-0.2396\text{ Log}\left(\frac{d}{a}\right)\Big{)}\frac{G_{max}a^{3}}{1-v_{max}}\\ & c_{es}=13.2875\frac{G_{max}a^{2}}{V_{s}(2-v_{max})};\ \ \ c_{ \theta s}=\left(7.3823-6.775\text{ Log}\left(\frac{d}{a}\right)\right)\frac{G_{max}a^{4}}{V_{s}(1-v_{max})}; \end{split}  \tag{12}\]  
where \(G_{max}=V_{s}^{2}\rho\) is the maximum shear modulus of the soil under small strains, \(V_{s}\) is the shear wave velocity, and \(v_{max}\) the Poisson coefficient of the soil under small strains. The parameters referring to the external system, \(k_{e}\) and \(k_{\theta e}\), are scaled by a factor \(\chi\leqslant 1\). Such factor is introduced to model the effect of the application of Geotechnical Seimic Isolation (GSI) [23,24] to the external system. In general, the application of the GSI produces a reduction of the stiffness of the soil.