## 3 Study on the Identification Method of Damping System Category Based on Interfacial Motion States
### Evaluation of SSI Damping System Based on Interface Motion State  
The pile-soil interface is the crucial point for the classification of the SSI damping system. The coordination degree of adjacent points at the interaction surface can reflect the adjusting ability of motor coordination of the interface. It can be the direct criterion for the type of damping system. From the test data analysis in Section 4.2, we can determine that motion states at the interface of the SSI system are uncoordinated on the S5 working condition, and the system is a non-classical damping system; in contrast, motion states at the interface are coordinated on the working condition of S10 and the system is classical damping system. Then, the identification method of the damping system category mentioned in Section 2 of this paper is used for damping system identification on conditions of S5 and S10, respectively. Moreover, the validity of this identification method can be verified compared with the conclusion of Section 4.2. Figure 12: Phase-Plane Curves of “Adjacent” Measuring Points at Pile Cap-Soil Interface under Sinusoidal Excitation: (**a**) S5-1, (**b**) S5-2, (**c**) S10. Figure 14: Phase-Plane Curves of “Adjacent” Measuring Points at Soil-Pile Interface under Sine Excitation: (**a**) S5-1, (**b**) S10. Figure 13: Acceleration Auto-power Spectrum Density curves of “Adjacent” Measuring Points at Pile Cap-Soil Interface: (**a**) S5-1, (**b**) S5-2, (**c**) S10. Sidewalls of piles are the primary interaction interfaces of the SSI system. There are two measuring points of soil and structure, respectively, namely points AX34 and AX3 at the structure interface and points AX5 and AX23 for that of soil. The position coordinates of AX34 and AX5, and coordinates of AX3 and AX23, are the same, respectively. The relative position coordinates are the same, shown as Equation (30). \[\left|\overline{x}^{\text{pile}}\right|=\left|\overline{x}^{\text{soil}}\right| \tag{30}\]  
Displacements of relatively synthetic motions of two groups are the same because of the restraint between pile and soil, shown as Equation (31):  
\[\left|\overline{x}^{\text{pile}}\right|=\left|\overline{x}^{\text{soil}}\right| \tag{31}\]  
Then  
\[\frac{\left|\overline{u}^{\text{pile}}\right|}{\left|\overline{x}^{\text{pile} }\right|}=\frac{\left|\overline{u}^{\text{soil}}\right|}{\left|\overline{x}^{ \text{soil}}\right|} \tag{32}\]  
Therefore, displacement vectors of relatively synthetic motions of two groups are collinear in the diagram of the synthesis motion law, and the corresponding synthetic motion law satisfies Equation (18). Relative motion states at the soil-structure interaction interface are investigated under S5 and S10 working conditions. The relative displacement, velocity and acceleration of two measuring points of the pile are calculated through Equation (33). \[\left\{ \begin{array}{l}u^{(1)}\bigg{|}_{x=l_{1}}=u_{\text{AX34}}-u_{\text{ AX3}}\\ \left.\dot{u}^{(1)}\right|_{x=l_{1}}=\dot{u}_{\text{AX34}}-\dot{u}_{\text{ AX3}}\\ \left.\ddot{u}^{(1)}\right|_{x=l_{1}}=\ddot{u}_{\text{AX34}}-\ddot{u}_{\text{ AX3}}\end{array} \right. \tag{33}\]  
Figure 15: Acceleration Auto-power Spectrum Density curves of “Adjacent” Measuring Points at Soil-Pile Interface: (**a**) S5-1, (**b**) S10. The relative displacement, velocity and acceleration of two measuring points of soil are calculated through Equation (34). \[\left\{ \begin{array}{l}u^{(2)}\big{|}_{x=l_{1}}=u_{\text{AX5}}-u_{\text{AX23}}\\ \dot{u}^{(2)}\big{|}_{x=l_{1}}=\dot{u}_{\text{AX5}}-\dot{u}_{\text{AX23}}\\ \ddot{u}^{(2)}\big{|}_{x=l_{1}}=\ddot{u}_{\text{AX5}}-\ddot{u}_{\text{AX23}}\\ \end{array} \right. \tag{34}\]  
Amplitude-frequency curves of relative acceleration of soil and pile are shown in Figure 16, respectively. Furthermore, the working condition of Figure 16a is S5, and Figure 16b is S10.