## 3 Study on the Identification Method of Damping System Category Based on Interfacial Motion States  
In the dynamical motion of the structure system, the continuous motion states belong to one continuous motion domain, and discontinuous motion states are called discontinuous motion boundaries of the domain. The interface between soil and structure is the motion boundary most likely to appear for the SSI system. The analysis of the continuity at this interface is the crucial point for identifying the SSI damping system. Suppose there exists a material interface where motion states of every position are continuous and smooth, the system is a classical damping system with uniform modes of motion. In other words, motion states at the material interface are all smooth and continuous for the non-classical damping system. The paper takes the horizontal motion as an example. The relationship of motion states at the material interface and the damping system is established by discussing the motion laws of points at the material interface. Suppose there is a classical damping system and the \(i\)th-order transient modal motion of any point \(x\) at the interface can be expressed as Equation (3). \[\left\{ \begin{array}{l}u_{i}(x,t)=\varphi_{i}(x)z_{i}(t)\\ \dot{u}_{i}(x,t)=\varphi_{i}(x)z_{i}(t)\\ \dot{u}_{i}(x,t)=\varphi_{i}(x)z_{i}(t)\end{array} \right. \tag{3}\]  
where \(\varphi_{i}(x)\) represents the \(i\)th-order transient modal shape of the system at point \(x\), \(z_{i}(t)\), \(\dot{z}_{i}(t)\) and \(\ddot{z}_{i}(t)\) represent the displacement, velocity and acceleration of the generalized coordinates at time \(t\). The synthetic motion can be expressed as Equation (4). \[\left\{ \begin{array}{l}u(x,t)=\sum u_{i}(x,t)=\sum\varphi_{i}(x)z_{i}(t)\\ \dot{u}_{i}(x,t)=\sum\ddot{u}_{i}(x,t)=\sum\varphi_{i}(x)\dot{z}_{i}(t)\\ \ddot{u}(x,t)=\sum\ddot{u}_{i}(x,t)=\sum\varphi_{i}(x)\ddot{z}_{i}(t)\end{array} \right. \tag{4}\]  
Taking \(T_{i}(t)\) as the transient period of the \(i\)th-rode modal motion and the corresponding circular frequency is obtained as Equation (5). \[\omega_{i}(t)=\frac{2\pi}{T_{i}(t)} \tag{5}\]  
The \(i\)th-order transient modal motion state of relative motion of two arbitrary points \(x_{1}\) and \(x_{2}\) are shown as Equation (6). \[\left\{ \begin{array}{l}\overline{u}_{i}(x_{1-2},t)=(\varphi_{i}(x_{1})- \varphi_{i}(x_{2}))z_{i}(t)=\overline{\varphi}_{i}^{x_{1-2}}z_{i}(t)\\ \overline{\dot{u}}_{i}(x_{1-2},t)=(\varphi_{i}(x_{1})-\varphi_{i}(x_{2}))\dot {z}_{i}(t)=\overline{\varphi}_{i}^{x_{1-2}}\dot{z}_{i}(t)\\ \overline{\dot{u}}_{i}(x_{1-2},t)=(\varphi_{i}(x_{1})-\varphi_{i}(x_{2}))\ddot {z}_{i}(t)=\overline{\varphi}_{i}^{x_{1-2}}\dot{z}_{i}(t)\end{array} \right. \tag{6}\]  
where \(\overline{\varphi}_{i}^{x_{1-2}}=\varphi_{i}(x_{1})-\varphi_{i}(x_{2})\) is the deviation of modal shape functions of points \(x_{1}\) and \(x_{2}\). The transient frequency of modal relative motion is determined by the motion law of the generalized coordinate because every point of the classical damping system moves with the same frequency \(\omega_{i}(t)\), as shown in Equation (6), which means the transiently circular frequency of the \(i\)th-order modal relative motion is the same as the circular frequency of \(i\)th-order modal motion of the system, as shown in Equation (7). \[\omega_{i}^{\prime}(t)=\omega_{i}(t)=\frac{2\pi}{T_{i}(t)} \tag{7}\]  
The relatively synthetic motion state of two points on the interface is noted as \((\overline{u},\overline{\dot{u}},\overline{\dot{u}})\) and used to analyze the movement rules of the interface. At time \(t\), two sets of relatively synthetic motion states \(A\) and \(B\) at the interface are denoted by \((\overline{u}^{A},\overline{\dot{u}}^{A},\overline{\dot{u}}^{A})\) and \((\overline{u}^{B},\overline{\dot{u}}^{B},\overline{\dot{u}}^{B})\). Relatively synthetic motion law by two orders of modes are discussed first and relatively synthetic motion states can be expressed as Equation (8).