## 3 Study on the Identification Method of Damping System Category Based on Interfacial Motion States
### Shaking Table Test for SSI Damping System
#### 4.2.3 The Continuity of Motion States of Points at the Material Interface
It can be seen that the relative motion laws of adjacent measuring points at the soil-pile interface are almost the same as motion laws shown in Figure 9 as the vibration condition continues and the vibration intensity increases. The differences between relative displacements and velocities between soil and pile are decreasing, illustrating the coordination of motions between soil and pile is improving and a gradually coordinated trend of motion states is shown at the soil-pile interface. The dynamic response of the measuring point under sinusoidal excitation is generally composed of two parts: the first is the damped free vibration response with the natural frequency, and the second is the harmonic vibration response with excitation frequency as its vibration frequency. The first part of vibration is the transient response with amplitude decaying because of the damping. In contrast, the second part is the response of sinusoidal excitation and the amplitude of the response stays constant, and the frequency is the same with the excitation. Although sinusoidal excitations with a single frequency cannot reflect the whole dynamic characteristics of the system, the consistency of motion states of the system, especially at the contact surface, can be investigated through sinusoidal excitation with different frequencies. Phase-plane curves of \"adjacent\" measuring points at pile cap-soil interface under aforementioned sinusoidal excitations are shown in Figure 12. Furthermore, corresponding acceleration auto-power spectrum density curves of \"adjacent\" measuring points at pile cap-soil interface are shown in Figure 13.  
Figure 9: Relative Phase-Plane Curves of “Adjacent” Measuring Points at Soil-Pile Cap Interface under White Noise: (**a**) Before earthquake, (**b**) After earthquake of small level, (**c**) After earthquake of medium level, (**d**) After earthquake of high level. The sinusoidal excitation causes the working condition of S5-1 shown in Figure 12a with the same frequency as the structure's natural frequency. Motion laws for adjacent points at the pile cap-soil interface are not entirely harmonic vibration responses from the phase plane trajectory of Figure 12a. There are different frequencies than the sinusoidal frequency from acceleration auto-power spectrum density curves of Figure 13a. The response of the measuring point is not entirely the steady-state vibration response at this time, and there is a transient response with damped free vibration through the above analysis. In addition, the amplitude of this transient response decays with time. The working condition of S5-2 shown in Figure 12b is the sinusoidal excitation with the natural frequency of soil foundation. The curve of motion states presents the elliptic phase trajectory initially. The acceleration auto-power spectrum density curves of Figure 13b show that the transient response decays and steady-state vibration response are forming. The working condition of S10 shown in Figure 12c is the sinusoidal excitation with the system's natural frequency. The curve of motion states presents the elliptic phase trajectory, and Figures 12c and 13c show that the transient response disappears and the system is in the steady-state vibration response stage. Figure 10: Relative Phase-Plane Curves of “Adjacent” Measuring Points at Soil-Pile Interface under White Noise: (**a**) Before earthquake, (**b**) After earthquake of small level, (**c**) After earthquake of medium level, (**d**) After earthquake of high level. We can conclude from Figure 12a-c that motion states of adjacent measuring points at pile cap-soil interface are consistent basically on the sinusoidal excitation condition of S5-1 after several working conditions of trim levels of vibration. Namely, motor coordination mechanisms arise and motion states at the interface tend to be consistent after vibrations of several trim levels, whether the transient vibration response exists or the steady-state vibration response form.