## 4 Test Results and Discussion
### The Effect of Soil-Caisson Kinematic Interaction
#### 4.3.2 Effect on the Rotation Component  
Under seismic loading, the horizontal displacement response of the soil at various embedment depths in the free field exhibits differences in amplitudes and phases, referred to as the \"embedment depth effect\" [31]. These disparities lead to an asynchronization of the soil motion at different embedment depths, i.e., \"pseudo-rotation\". The stiffness difference between the caisson foundation and the surrounding soil results in the kinematic interaction, modifying the rotation of the caisson foundation distinct from the free-field \"pseudo-rotation\". This section explores the impact of the kinematic interaction on the rotation component of the caisson foundation. It presents the rotation component of the caisson foundation and its comparison with the free-field \"pseudo-rotation\", as depicted in Figure 9. The rotation histories of the caisson foundation are calculated from the responses of the vertical displacement meters U1-U6 (as shown in Figure 4b) positioned at the top of the caisson, as well as the horizontal distances between the displacement meters. Figure 9a illustrates the comparison between the rotation amplitude of the caisson foundation and the amplitude of the free-field \"pseudo-rotation\" under different Chichi cases. The results clearly show that the caisson foundation exhibits a distinct rotational response under seismic loading, especially when the peak acceleration of the table input acceleration is larger than 0.8 g. However, for the same table input acceleration, the rotation amplitude of the caisson foundation is smaller than that of the free-field \"pseudo-rotation\", which indicates that the kinematic interaction suppresses the rotation component of the caisson foundation compared to the free-field \"pseudo-rotation\". For a deeper understanding of how the kinematic interaction affects the rotational component of the caisson foundation, the relationship between the rotational response of the caisson foundation and the \"pseudo-rotation\" response of the free-field is investigated in the frequency domain by using the transfer function of Equation (5) and the coherence function of Equation (6), and the results of the transfer function \(\left|H_{cp}(\omega)\right|\) under Chichi and Kobe waves are presented in Figure 9b,c, where the results of \(\left|H_{cp}(\omega)\right|\) with a high coherency are also demonstrated with dots at the same time. \[\left|H_{cp}(\omega)\right|=\sqrt{\frac{S_{cr}}{S_{pr}}} \tag{5}\]  
Figure 9: The impact of the kinematic interaction on the rotational component of the caisson foundation under different loading cases: (**a**) Comparison between caisson rotation amplitude and free-field “pseudo-rotation” rotation amplitude; (**b**) Rotational transfer function under the Chichi wave; (**c**) Rotational transfer function under the Kobe wave. \[\gamma^{2}(\omega)=\frac{\left|S_{cp}(\omega)\right|^{2}}{S_{cr}(\omega)S_{pr}( \omega)} \tag{6}\]  
where \(S_{cr}\) and \(S_{pr}\) are the power spectral density functions of the caisson foundation rotation response (\(\theta_{cr}\)) and the free-field \"pseudo-rotation\" response (\(\theta_{pr}\)), respectively, and \(S_{cp}\) is the cross-power spectral density function. From the results shown in Figure 9b,c, it can be seen that: (1) In the Chichi 0.1 g and Kobe 0.1 g cases, the rotation angle of the caisson is less than 0.1\({}^{\circ}\), and the effect of noise on the signal is more pronounced. As the peak acceleration of the table input wave increases gradually, the angle of rotation of the caisson increases, and the impact of noise on the signal decreases gradually. (2) The rotational transfer function starts at zero and gradually increases with increasing frequency. (3) Similar to the transfer function \(\left|H_{gf}(\omega)\right|\), the transfer function \(\left|H_{cp}(\omega)\right|\) also displays frequency-dependent behavior, consistently staying below unity at all frequencies.