# Effect of Multiple Reclosing Time Intervals on Axial Vibration of Winding
## 2 Dynamic Model of Winding Axial Vibration
### Solving the Axial Vibration Model  
The transformer winding axial vibration system is a linear vibration system. A linear vibration system consists of mass, damping, and spring stiffness. In the system, the disks are regarded as a lumped mass, the insulating pad between the disks and the pressure plate is regarded as an elastic element, and the oil-filled area between the disks is regarded as damping. Therefore, the transformer winding can be abstracted as a \"mass-damping-spring\" model and the solution method of the response of the transformer winding axial vibration is an analytical method. The model is presented in Figure 1[18]. In Figure 1, the mass of the individual disk is \(m_{i}\), \(i=1\), \(2\),..., \(n\); \(n\) is the number of winding turns; the equivalent stiffness of the insulating pad between \(i-1\)-th disk and \(i\)-th disk is \(k_{i}\); and the equivalent viscous damping of the insulating oil between the lines is \(c_{i}\)[19]. According to the \"mass-damping-spring\" model of the actual transformer and force analysis, the vibration of each disk can be expressed by a second-order differential equation system, which is as follows:  
\[\left\{ \begin{array}{l}m_{t}\ddot{x}_{t}=k_{t}(x_{1}-x_{t})+c_{t}\big{(} \dot{x}_{1}-\dot{x}_{t}\big{)}+k_{t}(x_{b}-x_{t})+c_{c}\big{(}\dot{x}_{b}-\dot{ x}_{t}\big{)}-F_{c}-m_{t}g\\ m_{1}\ddot{x}_{1}=k_{t}(x_{t}-x_{1})+c_{t}\big{(}\dot{x}_{t}-\dot{x}_{1}\big{)} +k_{1}(x_{2}-x_{1})+c_{1}\big{(}\dot{x}_{2}-\dot{x}_{1}\big{)}-F_{c}-m_{1}g+f_ {1}\\ m_{2}\ddot{x}_{2}=k_{1}(x_{1}-x_{2})+c_{1}\big{(}\dot{x}_{1}-\dot{x}_{2}\big{)} +k_{2}(x_{3}-x_{2})+c_{2}\big{(}\dot{x}_{3}-\dot{x}_{2}\big{)}-F_{c}-m_{2}g+f_ {2}\\ \hskip 28.452756 pt\cdots\cdots\\ m_{n}\ddot{x}_{n}=k_{n-1}(x_{n-1}-x_{n})+c_{n-1}\big{(}\dot{x}_{n-1}-\dot{x}_{n} \big{)}+k_{b}(x_{b}-x_{n})+c_{b}\big{(}\dot{x}_{b}-\dot{x}_{n}\big{)}-F_{c}-m_ {n}g+f_{n}\\ m_{b}\ddot{x}_{b}=k_{b}(x_{n}-x_{b})+c_{b}\big{(}\dot{x}_{n}-\dot{x}_{b}\big{)} +k_{c}(x_{t}-x_{b})+c_{c}\big{(}\dot{x}_{t}-\dot{x}_{b}\big{)}-k_{d}x_{b}-c_{ d}\dot{x}_{b}-F_{c}-m_{b}g\\ \end{array} \right. \tag{1}\]  
where \(m_{n}\) is the mass of the \(n\)-th disk; \(n\) is the order of the disk from top to bottom; \(m_{t}\) is the mass of top platen, \(m_{b}\) is the mass of the bottom platen; \(k_{t}\) and \(c_{t}\) are the equivalent stiffness of the insulating pad and damping of oil between the top platen and the first disk; \(k_{b}\) and \(c_{b}\) are the equivalent stiffness of the insulating pad and damping of oil between the bottom platen and the last disk; \(k_{c}\) is the equivalent stiffness of the winding brace and external compression structure; \(c_{c}\) is the corresponding equivalent damping coefficient; \(k_{d}\) and \(c_{d}\) are the equivalent stiffness of the insulating pad and damping of oil between the bottom platen and the transformer base; \(x_{i}\) is the vibration displacement of the \(i\)-th disk; and \(f_{i}(t)\) is the electric force applied to the \(i\)-th disk. Compared with the finite element simulation model, the calculation speed of this model is faster under the same mesh. In contrast to the conventional mass-damping-spring model, the equivalent stiffness and damping of the fixed parts of the winding braces and top and bottom platers, as well as the equivalent stiffness and damping of the bottom telos of the winding and the transformer base, are considered in this model. Additionally, the influence of each part on the axial vibration of the winding is considered more comprehensively.