## 1 Introduction  
The damping system represents the spatial distribution characteristics of the energy dissipation in a dynamic process [1]. Under the assumption of the viscous damping model, the damping system can be divided into two types, namely classical damping systems and non-classical damping systems. The motion law of the classical damping system is consistent. Each part of the system moves coordinated, making the motion-solving process easy and the physical implication of the dynamic results explicit. In contrast, the system is considered a non-classical damping system if there is any mutation of the system's energy consumption, which leads to the motion state of adjacent points being uncoordinated [2]. The motion law is discontinuous in this case, and the solution method of the system is very different from the classical damping system with the complex physical implication of the dynamic results [3; 4; 5]. The type of damping system is a crucial point in the research of vibration control. The reasonable identification of the damping system is the fundamental premise for reasonably choosing dynamic analysis methods and calculating motion states. It is also the key for the research on vibration control. The damping system can forebode the potential damage form or dangerous part of the system, indicating the optimized direction for seismic control and the corresponding design of the system. The lack of in-depth research on the damping system's identification method, especially during the nonlinear dynamic stage, induces no specific standard to identify damping systems. Traditional identification is based on the system's material behaviors. There are two points of view about damping system identification under this simple cognition. Systems composed of different materials are non-classical damping systems [6] because the difference in system damping is the key to the identification. Another point of view is that, systems in nonlinear stages are non-classical damping systems because of their indefinite modal parameters. Soil-structure interaction (SSI) system, comprised of reinforced concrete and foundation soil, is usually chosen as the object for the research of damping system identification due to the different material properties from inharmonic damping characteristics. In addition, foundation soil is easy to get into the nonlinear stage within an increasing kinetic effect. An SSI system is usually treated as a non-classical damping system [7]. However, experimental results of SSI damping system characterization cannot agree with traditional cognition. Shaking table tests of SSI system with soft foundation, designed by Tongji University [8], show that SSI system is classical damping system after some small initial vibrations. However, the nonlinear effect of soft soil foundation is noticeable, and the material properties of soil and structure are quite different. A series of shaking table tests designed by Xi'an Jiaotong University shows that the natural frequency of the SSI system decays gradually in increasing dynamic loads as soon as nonlinear vibration characteristics appear [9; 10]. However, this SSI system can still transform into a unified system with coordinating motion states and show characteristics of a classical damping system even though it is composed of entirely different materials. These results indicate that the traditional identification depending on material characteristics is not completed yet, and nonlinear systems are not necessarily non-classical damping systems. Non-linear processes and non-classical damping systems are quite different. The non-linear process indicates that the system's dynamic characteristics are unsteady, so the stiffness of the system changes with time [11]. However, the characteristics of the damping system show the spatial coordination of the system motion law. Suppose there is one system entering the non-linear process, the stiffness decays with increasing loads, and its modal frequency is not steady anymore.