## 2 Study on Damping System Type Based on Motion State  
For a multiple-degree-of-freedom system, motion states can be expressed by a second-order differential equation as below [4]. \[\mathbf{M\ddot{u}}(x,t)+\mathbf{\ddot{f}}(\mathbf{u}(x,t),\mathbf{\dot{u}}(x,t))+ \mathbf{Ku}(x,t)=\mathbf{P}(x,t) \tag{1}\]  
where \(x\) is the axis of the physical position of the system, \(t\) is time, \(\ddot{\mathbf{u}}(x,t)\), \(\dot{\mathbf{u}}(x,t)\) and \(\mathbf{u}(x,t)\) are acceleration, velocity and displacement with respect to coordinate \(x\) and time \(t\), \(\mathbf{M}\) and \(\mathbf{K}\) are the generalized mass and stiffness of the system, \(\mathbf{P}(x,t)\) is the external excitation subjected to the system, \(\mathbf{\ddot{f}}(\mathbf{u}(x,t),\mathbf{\dot{u}}(x,t))\) is the restoring force of the system related to the velocity and displacement. Under the assumption of the viscous damping model, dynamical equations of the classical damping system can be decoupled by linear or nonlinear modes. Motion states can be expressed by the linear combination of shape functions and generalized coordinates [12], as shown in Equation (2). \[\left\{ \begin{array}{l}u(x,t)=\sum\limits_{i=1}^{n}\varphi_{i}(x)z_{i}(t)\\ \dot{u}(x,t)=\sum\limits_{i=1}^{n}\varphi_{i}(x)\dot{z}_{i}(t)\\ \ddot{u}(x,t)=\sum\limits_{i=1}^{n}\varphi_{i}(x)\dot{z}_{i}(t)\end{array} \right. \tag{2}\]  
where \(\varphi_{i}(x)\) is the \(i\)th-order shape function of the system with respect to position \(x\), \(z_{i}(t)\) is the \(i\)th-order generalized coordinate of the system with respect to time \(t\). It is often considered that the appearance of nonlinear process means the emergence of non-classical damping system characteristics in traditional analyses. However, the system's nonlinearity indicates the unsteady change of the dynamic characteristics. On the contrary, the non-classical damping system indicates the incoordination of motion states of the system in space. [PERSON] proposed the nonlinear normal modes theory [13; 14; 15]. If the system keeps moving coordinately, every point of the system can reach the equilibrium position and maximal position at the same time. Motion states of the system accord with a specific spatial law and obey the nonlinear normal mode. Nonlinear normal mode is the transient mode with a time-dependent amplitude influenced by the development of the nonlinearity. In summary, the existence of modes is the necessary and sufficient condition for a classical damping system. The \(i\)th-order shape function \(\varphi_{i}(x)\) and the corresponding modal frequency are constant in linear conditions. On the contrary, the motion law of the system is time-dependent. The shape function \(\varphi(x,t)\) changes with time t, and the corresponding modal frequency is instantaneous. The characteristics of the damping system are easy to observe under linear conditions but hard to confirm under nonlinear conditions, so a comprehensive identification via motion states is necessary for the type of damping system. If the motion state of the system can be expressed by the linear combination of shape functions and generalized coordinates, the shape functions \(\varphi_{i}(x)\) of every order modal are continuous, and the acceleration \(\tilde{z}_{i}(t)\) of the reference point exists, which can be deduced as follows [16; 17]. \[\left\{ \begin{array}{l}\varphi_{\mathrm{i}}(x)\mathrm{is\ continuous}\\ \tilde{z}_{\mathrm{i}}(t)\mathrm{exist}\end{array} \right.\Leftrightarrow\left\{ \begin{array}{l}\varphi_{\mathrm{i}}(x)\mathrm{is\ continuous}\\ \tilde{z}_{\mathrm{i}}(t)\mathrm{is\ derivable}\end{array} \right.\Leftrightarrow \left\{ \begin{array}{l}\varphi_{\mathrm{i}}(x)\mathrm{is\ continuous}\\ \tilde{z}_{\mathrm{i}}(t)\mathrm{is\ smooth\ and\ continuous}\end{array} \right.\]  
Suppose the \(i\)th-order shape function \(\varphi_{i}(x)\) is continuous, and the velocity of the reference point \(\tilde{z}_{i}(t)\) is smooth and continuous. The real-valued models of the system exist, and the system is a classical damping system.