## 3 Methodology  
Dynamic System Coupling Rocking and Horizontal Motions: For the dynamic system shown in Figure 1, the equations of motion include two translational and two rotational components:  
\[m\left[\ddot{u}_{g}+\ddot{u}_{0}+\ddot{u}+(H+h)\ddot{v}_{0}\right]+m_{0}\left( \ddot{u}_{g}+\ddot{u}_{0}+e\ddot{v}_{0}\right)+Ku_{0}+Ci_{0}=0 \tag{10}\]  
\[m\left[\ddot{u}_{g}+\ddot{u}_{0}+\ddot{u}+(H+h)\ddot{v}_{0}\right]+ku+ci\dot{u}=0 \tag{11}\]  
\[m\left[\ddot{u}_{g}+\ddot{u}_{0}+\ddot{u}+(H+h)\ddot{v}_{0}\right](H+h)+I_{e} \ddot{v}_{0}+c_{v}\dot{v}_{0}+k_{v}v_{0}-\frac{k_{v}}{3}(v_{0}-v_{1})-Ce\dot{u }_{0}-Keu_{0}=0 \tag{12}\]  
\[-c_{v}\dot{v}_{1}-\frac{k_{v}}{3}(v_{1}-v_{0})=0 \tag{13}\]  
where \(I_{e}\) is the mass moment of inertia of the system, which comprises the structure and foundation block. It is calculated relative to the rotation axis and considers the mass of the superstructure (\(m\)) and the eccentricity (\(e\)). The corresponding relation is  
\[I_{e}=I+m_{0}e^{2}+m\left(h+H\right)^{2} \tag{14}\]  
Linear SSI analyses offer significant advantages in many engineering contexts, though they have limitations. They are valuable because they provide an efficient means of assessing structural responses, particularly under lower-level seismic events where nonlinear effects may not be prominent. Linear analysis can serve as a preliminary tool, allowing for simpler, approximate solutions that facilitate early-stage design. It also serves as a supporting framework for interpreting complex nonlinear results and can yield insights into systems with strong nonlinearities, as the results of nonlinear analyses sometimes mirror those of a linear model. However, linear analysis alone cannot fully capture the behavior of systems under high-intensity seismic events or other extreme loads, where soil and structural nonlinearities play a crucial role. Nonlinear effects are essential for accurately representing complex interactions, such as sliding, soil-structure separation, and base isolation behavior during severe earthquakes. For maximum credible and other high-seismic events, nonlinear modeling becomes necessary to accurately reflect the behavior of the soil and structure. Future SSI analyses may incorporate nonlinearity by using advanced soil models, integrating material and geometric nonlinearities in the structure, and employing coupled nonlinear numerical techniques such as finite element or finite difference methods. Although the developed dynamic model is approximate, it provides sufficient accuracy for practical purposes. Given the inherent uncertainties discussed, a deviation of \(\pm 20\%\) from the rigorous solution for a single set of parameters is typically acceptable, as supported by the dynamic stiffness coefficients presented in [24]. Furthermore, for transient loads such as earthquakes, these deviations are distributed over the frequency range, reducing the impact of the maximum error at any single frequency. While simplified physical models may lose some precision compared to boundary element or finite element methods, they offer substantial practical advantages, as highlighted in this work. To replace each derivative in the differential equations given in Equations (10)-(13) with an appropriate difference quotient approximation, the following central-difference formulas are used:  
\[\dot{U}(x)=\left(\frac{dU}{dx}\right)=\frac{1}{2\Delta x}\{U(x+\Delta x)-U(x- \Delta x)\} \tag{15}\]  
\[\ddot{U}(x)=\left(\frac{d^{2}U}{dx^{2}}\right)=\frac{1}{(\Delta x)^{2}}\{U(x+ \Delta x)-2U(x)+U(x-\Delta x)\} \tag{16}\]  
Here, \(\Delta x\) represents a constant step size, where \(x_{j}-x_{j-1}=\Delta x\), and the derivatives of the \(U(x)\) function are single-valued, finite, and continuous. The finite difference (FD) method uses discrete approximations for the derivatives in the differential equation with these approximations derived from Taylor series expansions. There are three types of approximations: central, forward, and backward.