## 4 Test Results and Discussion
### Characteristic Parameters of the Caisson Motion  
For structural systems subjected to seismic loading, the frequency response function \(H_{fre}(i\omega)\) is employed to identify the characteristic frequency \(f_{c}\), defined as the frequency corresponding to the peak of the amplitude of \(H_{fre}(i\omega)\) of the caisson motion, while the characteristic damping ratio \(\xi_{c}\), defined as the damping ratio at characteristic frequency \(f_{c}\), is calculated using the half-power method [21; 26]. Based on power \((S_{xx})\) and the cross-power \((S_{xy})\) spectral density functions of the input and output acceleration signals within the structural system, calculating \(H_{fre}(i\omega)\):  
\[H_{fre}(i\omega)=\frac{S_{xy}}{S_{xx}} \tag{2}\]  
where \(x\) and \(y\) denote the acceleration signals recorded at the table surface and the top measurement point (A-C-10) of the caisson foundation, respectively. In this section, the influence of the peak acceleration of the shaking table input acceleration on the characteristic parameters of the motion of the caisson foundation is investigated by calculating the \(H_{fre}(i\omega)\) under different loading cases. The \(H_{fre}(i\omega)\) is calculated to obtain the characteristic parameters such as characteristic frequency \(f_{c}\) and characteristic damping ratio \(\xi_{c}\), of the motion of the caisson. These results, along with the displacement amplitude at the top of the caisson, are listed in Table 3. From the results shown in Table 3, it can be seen that: (1) As the peak acceleration of the input wave increases, the \(f_{c}\) of the motion of the caisson foundation decreases, while the displacement amplitude and \(\xi_{c}\) exhibit an increasing trend. (2) The characteristic parameters of the motion of the caisson foundation under the Chichi wave differ significantly from those under the Kobe wave. Under the Chichi wave, the \(f_{c}\) of the motion of the caisson foundation is smaller than that under the Kobe wave, but the displacement amplitude and \(\xi_{c}\) are larger than that under the Kobe wave. The observations mentioned above are due to the fact that for the Chichi wave, the dominant frequency of the table input acceleration is smaller, approximately 12 Hz. Under the Chichi wave, the soil experiences more low-frequency cycles, resulting in larger displacements, leading to increased shear stress and greater plasticity within the soil, causing a notable decrease in the shear modulus and an increase in the damping ratio of the soil. As a consequence, the \(f_{c}\) of the motion of the caisson foundation reduces, while the \(\xi_{c}\) increases. In contrast, for the Kobe wave, the dominant frequency of the table input acceleration is higher, approximately 23.1 Hz, resulting in smaller soil displacements. The soil within the \begin{table}
\begin{tabular}{c c c} \hline
**Test System** & **Ground Motion** & **Peak Acceleration (g)** \\ \hline \multirow{3}{*}{Free-Field} & White noise & 0.1 \\  & Chichi & 0.1, 0.25, 0.5 \\  & Kobe & 0.1 \\ \hline \multirow{3}{*}{Soil–Caisson} & White noise & 0.1 \\  & Chichi & 0.1, 0.25, 0.5, 0.75, 1.0, 1.5 \\ \cline{1-1}  & Kobe & \\ \end{tabular}
\end{table} 
Table 2: Loading cases of the free-field and soil–caisson system model tests. system remains mostly elastic for Kobe 0.1 g\(\sim\)Kobe 0.5 g cases, with displacement amplitudes less than 0.68 mm and characteristic frequencies even higher than the fundamental frequency in the white noise case. Therefore, the \(f_{c}\) and \(\xi_{c}\) of the motion of the caisson foundation do not change significantly with increasing peak acceleration of the table input acceleration. However, for Kobe 0.75 g-Kobe 1.0 g cases, the soil undergoes a certain extent of plasticity, as indicated by displacement amplitudes of 1.36 mm and 4.33 mm, respectively.