## 1 Introduction  
Power transformer plays an extremely important role in the power system and is the nerve center of the power system, which undertakes the task of voltage and current conversion. Once it breaks down, it will affect a wide range and even lead to the paralysis of a power system [1, 2]. About half of all transformer damages are due to insufficient short-circuit resistance of the transformer. The accumulation of axial deformation of a transformer due to short circuits is the primary cause of transformer short circuit damage accidents [3, 4]. When a transformer is short-circuited, the maximum short-circuit current on the winding is 20-30 times the rated current [5]. Due to the action of the electromagnetic force, the huge electrodynamic force bends and deforms the winding in the axial direction. When the elastic deformation of winding transfers to plastic deformation, the transformer has irreversible structural failure [6, 7]. Axial deformation of transformer winding caused by a short circuit can be attributed to the axial vibration of winding [8, 9]. [PERSON] established a mechanical model of a multi-DOF vibration system with mass particles such as stiffness spring connection in the 18 th century [10], which has further evolved into the \"spring-mass-damper\" model in subsequent studies to describe the axial dynamic characteristics of winding [11, 12]. So far, this model is still the mainstream analysis model for analyzing axial stability. Based on this model, many scholars have studied the axial vibration of winding through simulation and modeling. [PERSON] et al. calculated the short-circuit electrodynamic force of a transformer by using a finite element method and short-circuit electrodynamic force calculation software, and carried out static and dynamic analysis and calculation, respectively [13]; [PERSON] et al. used the finite element method to calculate the short-circuit electrodynamic force acting on each disk of dry-type transformer winding in case of a short circuit by using a finite element method [14]; [PERSON] conducted modal analysis on the vibration signal corresponding to the axial vibration of a transformer under transient operating conditions during power failure by using an operating modal analysis method, and studied and analyzed the vibration characteristics of the transient process and the influence of loose winding pressing force on axial vibration [15]. These literatures studied the mechanical stability changes of winding caused by axial force. However, the influence of the distribution of axial vibration on the transformer cannot be determined. Therefore, some scholars have also conducted research on the distribution model of the axial vibration of the transformer winding. [PERSON] et al. calculated the short-circuit electrodynamic force distribution of transformer winding by using a three-dimensional finite element model [16]. Based on the field-circuit coupling method, [PERSON] et al. used the transient field analysis method to obtain the distribution of leakage magnetic field and electromagnetic force in the winding area and obtained the distribution law of the axial deformation of the transformer in the whole process in the transient structure field [17]; [PERSON]. [PERSON] believed when the transformer is short circuited, the axial displacement of winding presents a \"V\" shape distribution, that is, the axial displacement at the middle of the winding is greater than at the end of the winding [18]. [PERSON] et al. believed that axial vibration acceleration of winding presented an \"M\" shape distribution law, that is, the acceleration amplitude at the height of 1/4 and 3/4 of the winding is greater than at the end and middle of the winding [12; 19]. However, in the process of calculation and solution, the damping parameters are ignored in the above literature, and only the ideal transformer without damping is considered, which is not universal and cannot be applied to a transformer in an actual situation.