## 1 Introduction  
Power transformers are essential electrical equipment in the electrical grid that are responsible for critical functions such as energy transmission, flexible allocation, and voltage transformation within the entire power system [1, 2]. Consequently, the operational status of transformers directly affects the safety and stability of both local and overall power systems. During short-circuit incidents, the short-circuit forces acting on transformers can cause insulation and structural component damage, resulting in significant winding deformation and wire breakage [3]. Copper, the most common material used for making transformer windings, is a typical elastoplastic material, while cushion blocks possess a stiffness that gradually hardens [4]. The ability of transformers to withstand short circuits is almost unaffected by minor deformations in windings. However, the cumulative effect of subtle plastic deformations caused by multiple short-circuit impacts can lead to irreversible changes in the mechanical characteristics of windings, resulting in a reduced capacity to withstand short circuits [5; 6; 7]. Under the axial short-circuit electromagnetic force, windings generate axial vibrations, leading to resonance phenomena, collisions, separation between cushion blocks and coil disks, and this ultimately causes winding looseness, distortion, and even collapse [8; 9]. Therefore, the study of improving transformer windings, cushion block performance, or structures to enhance their short-circuit withstand capability is of significant importance in engineering. Numerous scholars have conducted research on the short-circuit strength of transformers. References [10; 11; 12] established a mass-spring model that includes cushion blocks, coil disks, pressure plates, and iron cores. Since coil disks experience non-uniform forces on the radial surface, the model divided a coil disk into two mass units and connected them with axial bending springs. Reference [13] researched how increasing the radial width of windings, the number of bracing ribs, and the size of winding subwires can enhance winding strength based on a specific transformer short-circuit test and calculation results. Reference [14] developed a coupled three-dimensional model considering electric, magnetic, and structural fields to investigate the influence of factors such as the insulation paper wrapped around the wire, the tightness of insulation paper tubes, and the Young's modulus on winding strength. Reference [15] studied the distribution of axial forces in windings during transformer short circuits through simulation. Reference [16] proposed a transformer winding deformation online detection method based on VFTO and validated its effectiveness. Reference [17] established a nonlinear vibration model considering axial vibration of windings that deduced and calculated the winding compression force, cushion block and pressure plate compression force, and the axial bending stress of the wires. Reference [18] investigated the impact of cushion blocks on the axial winding strength and of bracing ribs on the radial winding strength, with specific consideration of factors such as the number and width of cushion blocks. References [19; 20] indicated that, over the transformer's lifespan, cushion blocks were immersed in dielectric fluid and exposed to high temperatures, thermal aging, and chemical reactions, resulting in the degradation of their dielectric and mechanical properties and weakening the transformer short-circuit resistance. In references [21; 22], a two-dimensional simulation was established and the leakage flux and stress distribution of high- and low-voltage winding were analyzed; short-circuit tests were carried out and it was pointed out that the electromagnetic force generated by the fault current might lead to instability in and the collapse of the winding. An improved radial buckling analysis method was proposed to study the radial stability of the inner winding.