## 2 Establishment of Finite Element Model
### Finite Element Analysis of Material Flow Law  
The workpiece is under the tangential, axis and radial force of the roller during spinning, and the stress situation is more complex. However, the thin-walled conical part is mainly affected by axial tension. In this paper, the point tracking method was used to compare the axial displacement before and after the formation of the tracking point. The position change of each point and the material flow law of variable-section thin-walled conical parts could be observed. A set of simulation results were selected for analysis. The simulation parameters were as follows: the initial temperature of the blank was 1000 \({}^{\circ}\)C, the radius of the roller nose was 6 mm, the feed ratio of the roller was 0.4 mm/r, and the mandrel rotation rate was 300 rpm. A total of 21 points were selected at equidistant intervals along the direction of the blank radius, and the position changes of these points prior to and following the formation were analyzed. The point selection is illustrated in Figure 3.  
Figure 1: Structure of target part: (**a**) Part sectional drawing; (**b**) Part partition. Figure 3: The location of material flow law analysis points: (**a**) Before part forming; (**b**) after part forming. Figure 2: The finite element model of spinning. The sampling points P1 to P9 are located in Zone 1, and there is almost no change in the axial position during the hot spinning process. The metal in Zone 2 flows along the circular arc of the mandrel fillet. As depicted in Figure 4, the axial position of sampling point P10 remains unchanged at first, and P10 moves downward in a fluctuating manner as the roller moves along the mandrel. The reason for the fluctuation is that P10 is squeezed upward by the roller when the roller is close to the sampling point. When the roller is distant from the sampling point, P10 is stretched by the outer metal, and causes a downward movement of the axial position. The axial position of sampling point P11 first moves upward and then moves downward in a fluctuating manner, and the moving mechanism is identical to that of P10. The axial position of sampling point P12 decreases rapidly at first and then fluctuates. This fluctuation occurs because the sampling point 12 is in the contact area of Zone 2 and Zone 3. Due to the large thinning rate of the small end of cone and large metal deformation, the axial position of P12 rapidly decreases. After formation, the axial position of sampling points in Zone 2 remains unchanged. The axial displacement distance of sampling points on the outer wall of Zone 3 is depicted in Figure 5. It can be inferred from the figure that Zone 3 is the cone area, which is the main deformation area of variable-section conical spun parts. The axial displacement of sampling points P13 to p15 is large. The axial position of each sampling point in Zone 3 first moves downwards and then remains unchanged. Moreover, the downward movement rate of each sampling point is identical, and the movement rate is equal to the feed rate of the roller. The zigzag shape of the curve in the figure is due to the springback deformation behavior of the spun part in the forming process, leading to the fluctuation of the axial position of each sampling point in Zone 3. Secondly, because the blank rotates along the axial direction, the different contact area of the roller leads to the interaction between the roller extrusion and the metal axial drawing, which also accounts for the serrated fluctuation of the curve. The sampling point p17 is located in Zone 4, but its flow law is similar to that of Zone 5. It can be inferred from the Figure 6 that Zone 5 is the flange area. During the forming process, the change trend of axial movement of each sampling point in this area is almost identical, and the axial displacement rate is almost identical to the feed rate of the roller, indicating that the metal flow law of each sampling point in the flange area is basically the same.