## 2 Axial Vibration Model of Transformer Winding
### Equivalent Model of Axial Vibration of the Transformer
For convenience, Equation (1) is written in matrix form  
\[[m]\big{\{}\overset{x}{x}(t)\big{\}}\,+[c]\big{\{}\overset{x}{x}(t)\big{\}}\,+[k] \big{\{}\overset{x}{x}(t)\big{\}}\,=\{f(t)\} \tag{2}\]  
wherein, \([m]\) is the mass matrix of each disk, \([c]\) is the damping matrix of each disk, \([k]\) is the stiffness matrix of each disk, \([\overset{x}{x}(t)]\) is the vibration acceleration matrix of each disk, \([\overset{x}{x}(t)]\) is the vibration velocity matrix of each disk, \([x(t)]\) is the vibration displacement matrix of each disk, and \(\big{\{}\overset{x}{y}(t)\big{\}}\) is the electrodynamic force matrix received by each disk. When the transformer is short-circuited, only the steady-state part of the short-circuit current is considered,  
\[I=I_{m}\cos(\Omega t+\varphi_{0}) \tag{3}\]  
wherein, \(I_{m}\) is the amplitude of the AC component of the short-circuit current, and \(\varphi_{0}\) is the initial phase. Then the electromagnetic force acting on the winding is  
\[F=\frac{1}{2}pI_{m}^{2}[1+\cos(2\Omega t+\varphi_{0})] \tag{4}\]  
wherein, \(p\) is the electromagnetic force coefficient. It can be seen that the value of the electromagnetic force frequency on the winding is twice the current frequency. The characteristic equation of the system is as follows  
\[\big{|}[k]-{\omega_{n}}^{2}[m]\big{|}=0 \tag{5}\]  
wherein, \(\omega_{n}\) is the natural frequency of vibration mode in \(n\)-th order. Substitute the \(r\)-order natural frequency successively into the eigenvalue problem equation of the system  
\[[k]\big{\{}u\big{\}}={\omega_{n}}^{2}[m]\big{\{}u\big{\}} \tag{6}\]  
wherein, \(\{u\}\) is the modal vector matrix. Manually specify any one of the modal vectors, such as \(\{u_{1}(r)\}=\{1\}\), to solve all other \(r\)-order modal vectors of the system  
\[\{u^{(r)}\}=[{u_{1}}^{(r)},{u_{2}}^{(r)},\cdots,{u_{n}}^{(r)}]^{T} \tag{7}\]  
wherein, \(r=1,\,2,\,3,\,\cdots,\,n\)  
Transform the physical coordinates of the system by using normal modal matrix \([u_{N}]\)  
\[\{q_{N}(t)\}=[u_{N}]^{-1}\{x(t)\} \tag{8}\]  
wherein, \(\{q_{N}(t)\}\) is the displacement matrix converted to normal modal coordinates. Transform the vibration equation of a general multi-DOF system into a normal modal equation  
\[\big{\{}\overset{q}{q}_{N}(t)\big{\}}+[2\zeta_{r}\omega_{nr}]\big{\{}\overset {q}{q}_{N}(t)\big{\}}+[\omega_{nr}]\big{\{}q_{N}(t)\big{\}}=\{N(t)\} \tag{9}\]  
wherein, \(\{\overset{q}{q}_{N}(t)\}\) is the acceleration matrix of each disk under normal modal coordinates, \(\zeta_{r}\) is the \(r\)-order damping ratio, and \(\{N(t)\}\) is the electrodynamic force matrix received by each disk under normal modal coordinates. Therefore, the solution of Equation (9) can be regarded as the solution of n second-order partial differential equations with different coefficients but the same variables,  
\[\left\{ \begin{array}{l}q_{Nr}(t)=Q_{r}\cos(\Omega t-\varphi_{r})\\ \overset{q}{q}_{Nr}(t)=-Q_{r}\Omega\sin(\Omega t-\varphi_{r})\\ \overset{q}{q}_{Nr}(t)=-Q_{r}\Omega^{2}\cos(\Omega t-\varphi_{r})\\ \end{array} \right. \tag{10}\]\[Q_{r}=\frac{N_{r}}{\sqrt{\left(\omega_{nr}{}^{2}-\Omega^{2}\right)^{2}-\left(2\zeta _{r}\omega_{nr}\Omega\right)^{2}}} \tag{11}\]  
\[\varphi_{r}=\arctan\frac{2\zeta_{r}\Omega/\omega_{nr}}{1-\left(\Omega/\omega_{nr }\right)^{2}} \tag{12}\]  
wherein, \(Q_{r}\) is the amplitude of electrodynamic force received by each disk, \(\varphi_{r}\) is the phase angle of electrodynamic force received by each disk, and \(\Omega\) is the frequency of the electric power received by each wire cake. The displacement equation, velocity equation, and acceleration equation can be obtained by transforming it from a modal space coordinate system to a physical coordinate system.