## 2 Axial Vibration Model of Transformer Winding
### Equivalent Model of Axial Vibration of the Transformer  
The lumped parameters of a linear vibration system are composed of mass, damping, and spring stiffness, which is usually called the mass-damping-spring model. The disk is regarded as a lumped mass; the insulating cushion block, gasket, and pressing plate between windings are regarded as elastic elements. The iron core and iron yoke are regarded as rigid bodies. The \"mass-spring-damping\" model is shown in Figure 1[20]. As is shown in Figure 1, the mass block \(m\) in the figure corresponds to the mass of a single-layer disk, and spring stiffness \(k_{i}\) and damping \(c_{i}\) correspond to the equivalent stiffness and damping of an insulating paper and insulating cushion block between two layers of the disk. According to the equivalent model and stress analysis of each disk, the dynamic equations of each disk can be written as follows:  
\[\left\{ \begin{array}{l}m_{1}\ddot{x}_{1}+c_{t}\dot{x}_{1}-c_{1}(\dot{x}_{2}- \dot{x}_{1})+k_{t}x_{1}-k_{1}(x_{2}-x_{1})=f_{1}(t)\\ m_{2}\ddot{x}_{2}+c_{1}(\dot{x}_{2}-\dot{x}_{1})-c_{2}(\dot{x}_{3}-\dot{x}_{2})+ k_{1}(x_{2}-x_{1})-k_{2}(x_{3}-x_{2})=f_{2}(t)\\ \vdots\\ m_{i}\ddot{x}_{i}+c_{i-1}(\dot{x}_{i}-\dot{x}_{i-1})-c_{i}(\dot{x}_{i+1}-\dot{ x}_{i})+k_{i-1}(x_{i}-x_{i-1})-k_{i}(x_{i+1}-x_{i})=f_{i}(t)\\ \vdots\\ m_{n}\ddot{x}_{n}+c_{n-1}(\dot{x}_{n}-\dot{x}_{i-1})-c_{b}\dot{x}_{n}+k_{n-1}( x_{n}-x_{n-1})-k_{b}x_{n}=f_{n}(t)\end{array} \right. \tag{1}\]  
wherein, \(m_{n}\) is the mass of the \(n\)-th layer of disk; \(k_{t}\) and \(k_{b}\) are the equivalent stiffness coefficients between the upper and lower pressing plates, and between the first disk and the last disk; \(c_{t}\) and \(c_{b}\) are the equivalent damping coefficients between the upper and lower pressing plates and between the first disk and the last disk; \(k_{n}\) is the equivalent stiffness of the \(n\)-th disk; \(c_{n}\) is the equivalent stiffness of the \(n\)-th disk; \(c_{n}\) is the equivalent stiffness of the \(n\)-th disk; and \(f_{n}(t)\) is the electrodynamic force received by the \(n\)-th disk. As shown in Table 1, the element definitions required for the solution of the axial multi-DOF vibration of the transformer winding. \begin{table}
\begin{tabular}{c c} \hline \hline
**Elements** & **Definition** \\ \hline \(m_{n}\) & the quality of the \(n\)-th layer of disk \\ \(k_{t}\) and \(k_{b}\) & equivalent stiffness between clamping rings and winding end \\ \(k_{n}\) & equivalent stiffness between the \(n\)-th disk and the (\(n+1\))-th disk \\ \(c_{t}\) and \(c_{b}\) & equivalent viscous damping factors between clamping rings and winding end \\ \(c_{n}\) & equivalent viscous damping factors between the \(n\)-th disk and the (\(n+1\))-th disk \\ \(x_{n}(t)\) & displacement of \(n\)-th disk \\ \(\dot{x}_{n}(t)\) & the speed of the \(n\)-th disk \\ \(\ddot{x}_{n}(t)\) & acceleration of the \(n\)-th disk \\ \(\dot{q}_{n}(t)\) & displacement of \(n\)-th disk in canonical modal coordinates \\ \(\dot{q}_{n}(t)\) & the speed of \(n\)-th disk in canonical modal coordinates \\ \(\ddot{q}_{n}(t)\) & acceleration of \(n\)-th disk in canonical modal coordinates \\ \(\ddot{\zeta}_{r}\) & \(r\)-th order damping ratio \\ \(\dot{\Omega}_{r}\) & the magnitude of the electric power received by each disk \\ \(\omega_{nr}\) & the r-order component of the natural frequency of the \(n\)-th mode shape \\ \(\Omega\) & the frequency of the electric power received by each disk \\ \(\varphi_{r}\) & phase angle of electric power received by each disk \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Definition of elements in the multi-DOF model. Figure 1: Winding axial mass-damping-spring model.