## 4 Results  
In this paper, the distribution law of winding axial deformation in the process of transformer short circuit is studied; a multi-DOF winding \"spring-mass-damping\" model considering damping is established; and the variation laws of axial displacement and acceleration when the transformer with different damping ratios is short circuited at different frequencies is analyzed, which provide a new idea for the design of winding to a certain extent, further improving the short-circuit-bearing capacity of a transformer. The main conclusions are as follows:  
1. In this paper, an axial vibration model of the transformer winding, considering the system damping parameters, is built. Compared with the previous V-type and M-type mode shapes, the mode shape structure calculated in this paper is more detailed and refined. At the same time, the displacement and acceleration distributions of the model at different frequencies are illustrated by sweeping the frequency.
2. Under short-circuit conditions, the acceleration and displacement of the winding are distributed in a \"VMV\" type, and the displacement and acceleration reach peak values at 1/8 of the upper end and 1/8 of the lower end at the same time, and the displacement and acceleration at both ends of the winding are approaching to zero; in  
Figure 9: Acceleration of each disk under different damping ratios.  
addition, the numerical distribution of displacement and acceleration in the middle of the winding is relatively flat. With the increase of the damping ratio, the vibration amplitude shows a decaying trend.
3. According to the calculation model, suggestions for the operation and maintenance of the transformer can be put forward. The positions of the sensors for measuring the vibration acceleration can be located at the two ends, the middle, one quarter, and one-eighth of the winding. In actual operation, attention should be paid to the vibration of the upper and lower ends of the winding and the deformation of the winding. When designing the structure of the transformer winding, it is especially necessary to improve the anti-deformation and stretching ability near both ends of the winding. In future research, we can further expand the following aspects based on the research in this paper: (1) Comparative analysis with the actual transformer short-circuit test to verify whether the theory of axial vibration of transformer windings is insufficient. (2) Further analyze the influence of different short-circuit conditions or transformer overload on the axial vibration. (3) Considering the response of transformer winding radial vibration under short-circuit conditions, comprehensive axial vibration transformer winding vibration characteristics reflect the transformer winding vibration characteristics more comprehensively.  
Conceptualization, S.G. and L.S.; methodology, S.G., L.S. and Y.T.; validation, L.S. and H.L.; formal analysis, S.G.; investigation, Y.T.; resources, S.G., L.S. and Y.T.; data curation, H.L.; writing--original draft preparation, S.G.; writing--review and editing, Y.T.; visualization, H.L.; supervision, L.S.; project administration, S.G. All authors have read and agreed to the published version of the manuscript. This research was funded by Science and Technology Project of State Grid Hebei Electric Power, Fund No. Kj2019-060 and the Natural Science Foundation of Hebei Province, Fund No. E2021521004.  
Not applicable.  
The authors declare no conflict of interest.