## 3 Study on the Identification Method of Damping System Category Based on Interfacial Motion States
Based on the rotating vector method, the displacement amplitudes of relatively synthetic motions of groups \(A\) and \(B\), synthesized by the first two modes, can be presented by projections of the relatively synthetic displacement vectors \(\overset{\rightarrow}{\overline{u}}^{A}\) and \(\overset{\rightarrow}{\overline{u}}^{B}\) along the \(u\) axis, respectively, as shown in Figure 1. Besides, the first and second modal relative motion vectors of groups \(A\) and \(B\) are collinear, respectively, due to the equality of \(i\)-th order of circular modal frequencies of groups \(A\) and \(B\). As per the trigonometric relationship of relatively synthetic motion, the amplitudes of the relatively synthetic motion of group \(A\) and group \(B\) can be obtained as Equations (14) and (15), respectively. \[\left|\overline{u}^{A}\right|=\sqrt{\left(\overline{\varphi}_{1}^{A}\alpha_{1 }\right)^{2}+\left(\overline{\varphi}_{2}^{A}\alpha_{2}\right)^{2}+2\alpha_{1 }\alpha_{2}\overline{\varphi}_{1}^{A}\overline{\varphi}_{2}^{A}\cos(\omega_{2 }-\omega_{1})t} \tag{14}\]  
\[\left|\overline{u}^{B}\right|=\sqrt{\left(\overline{\varphi}_{1}^{B}\alpha_{1 }\right)^{2}+\left(\overline{\varphi}_{2}^{B}\alpha_{2}\right)^{2}+2\alpha_{1 }\alpha_{2}\overline{\varphi}_{1}^{B}\overline{\varphi}_{2}^{B}\cos(\omega_{2 }-\omega_{1})t} \tag{15}\]  
It can be seen that the magnitude of this amplitude varies periodically with time. The transient circular frequency of the relatively synthetic motion of groups \(A\) and \(B\) are shown as Equations (16) and (17), respectively. \[\overline{\omega}^{A}=\frac{\omega_{1}+\omega_{2}}{2}+\frac{\left(\overline {\varphi}_{1}^{A}\alpha_{1}\right)^{2}-\left(\overline{\varphi}_{2}^{A} \alpha_{2}\right)^{2}}{\left(\overline{\varphi}_{1}^{A}\alpha_{1}\right)^{2}+ \left(\overline{\varphi}_{2}^{A}\alpha_{2}\right)^{2}+2\alpha_{1}\alpha_{2} \overline{\varphi}_{1}^{A}\overline{\varphi}_{2}^{A}\cos(\omega_{2}-\omega_{1 })t}\frac{\omega_{2}-\omega_{1}}{2} \tag{16}\]  
\[\overline{\omega}^{B}=\frac{\omega_{1}+\omega_{2}}{2}+\frac{\left(\overline {\varphi}_{1}^{B}\alpha_{1}\right)^{2}-\left(\overline{\varphi}_{2}^{B} \alpha_{2}\right)^{2}}{\left(\overline{\varphi}_{1}^{B}\alpha_{1}\right)^{2}+ \left(\overline{\varphi}_{2}^{B}\alpha_{2}\right)^{2}+2\alpha_{1}\alpha_{2} \overline{\varphi}_{1}^{B}\overline{\varphi}_{2}^{B}\cos(\omega_{2}-\omega_{1 })t}\frac{\omega_{2}-\omega_{1}}{2} \tag{17}\]  
If displacement vectors of relatively synthetic motion \(\overset{\rightarrow}{\overline{u}}^{A}\) and \(\overset{\rightarrow}{\overline{u}}^{B}\) are collinear in the diagram of the synthesis motion law, as shown in Figure 2, angles between the displacement vector of relatively synthetic motion \(\overset{\rightarrow}{\overline{u}}^{i}\) and corresponding deviation of initial position coordinates \(\left|\overline{x}^{i}\right|\) of groups \(A\) and \(B\) are equal, namely, proportions between the amplitude  
Figure 1: The diagram of the synthesis motion law. of relatively synthetic motion and its own initial position coordinates \(\left|\overline{x}^{i}\right|\) of groups \(A\) and \(B\) are equal, shown as Equation (18);  
\[\frac{\left|\overline{u}^{A}\right|}{\overline{x}^{A}}=\frac{\left|\overline{u} ^{B}\right|}{\left|\overline{x}^{B}\right|} \tag{18}\]  
Furthermore, the relationship of modal motion vectors shown as Equation (19) can be satisfied as soon as the relatively synthetic motion \(\overset{\rightarrow}{\overline{u}}^{A}\) and \(\overset{\rightarrow}{\overline{u}}^{B}\) are collinear. \[\frac{\overline{\varphi}_{1}^{A}\alpha_{1}}{\overline{\varphi}_{1}^{B}\alpha_ {1}}=\frac{\overline{\varphi}_{2}^{A}\alpha_{2}}{\overline{\varphi}_{2}^{B} \alpha_{2}} \tag{19}\]  
Suppose  
\[k=\frac{\left|\overline{u}^{A}\right|}{\left|\overline{u}^{B}\right|} \tag{20}\]  
Then  
\[\frac{\overline{\varphi}_{1}^{A}\alpha_{1}}{\overline{\varphi}_{1}^{B}\alpha_ {1}}=\frac{\overline{\varphi}_{2}^{A}\alpha_{2}}{\overline{\varphi}_{2}^{B} \alpha_{2}}=k \tag{21}\]  
Namely,  
\[\left\{