## 5.1 Longitudinal bending deformation analysis

Figure 8 shows the longitudinal curvature of spherical surfaces formed by different central-section diameters of the rollers with three maximum reduction conditions of 0.06, 0.08 and 0.10mm. The central cross-section diameters of the rollers are 40, 60, 80, 100 and 120 mm, respectively. When the central cross-section diameter of the rollers is small (less than 80 mm), the longitudinal curvature of the formed spherical surface increases with the increase of the central cross-section diameter; when the central cross-section diameter of the rollers is larger (greater than 80 mm), the longitudinal curvature of the formed spherical surface decreases with the increase of the central cross-section diameter.

It is analyzed that when the central cross-section diameter of the rollers is less than 80 mm, the variation in the dimension of the deformation zone will dominate the longitudinal bending deformation of the formed surface. At this time, the diameter of the rollers is relatively small, which will not inhibit the longitudinal bending of the sheet during rolling, so the longitudinal curvature of the formed surface increases with the increase of the central cross-section diameters. However, when the central cross-section diameter of the rollers is greater than 80 mm, the larger roller diameter will have greater mechanical interference on the longitudinal bending of the sheet metal, thereby inhibiting the longitudinal bending of the sheet (see Fig. 2b), resulting in the longitudinal curvature of the formed curved surface decreases with the increase of the central cross-section diameters.

When the maximum rolling reduction is 0.06, 0.08 and 0.10 mm, the longitudinal curvature range of the spherical surfaces formed with different central-section diameters are 0.124×10− 3, 0.375×10− 3 and 0.676×10− 3 mm− 1, respectively, which shows that the influence of the central cross-section diameter of the rollers on the longitudinal bending deformation of the formed spherical surface increases with the increase of the rolling reduction. That is to say, the greater the rolling reduction (the greater the longitudinal curvature of the target surface), the central cross-section diameter of the rollers has a greater influence on the longitudinal bending deformation of the formed spherical surface (see Fig. 8).

Figure 9 shows the difference between the longitudinal center extension and the longitudinal edge extension of the formed spherical surfaces with different center-section diameters of the rollers. When the central cross-section diameter of the rollers is relatively small (less than 80 mm), the difference in the longitudinal extension of the formed spherical surface increases with the increase of the central cross-section diameter. When the central cross-section diameter of the rollers is relatively large (greater than 80 mm), the difference in the longitudinal extension of the formed spherical surface decreases with the increase of the central cross-section diameter. And the greater the rolling reduction, the greater the influence of the roll diameter on the difference in longitudinal extension of the formed spherical surface (see Fig. 9). Since the longitudinal bending deformation of the formed curved surface is mainly determined by the difference in the longitudinal extension between the center and the edge, the influence of the roller diameter on the longitudinal extension difference of the formed surface is consistent with its influence on the longitudinal curvature of the formed surface.

## 5.2 Stability analysis of forming process

Figures 10, 11 and 12 are the equivalent plastic strain distributions of formed spherical surfaces with the maximum rolling reduction of 0.06, 0.08 and 0.10 mm, respectively; each figure contains the formed spherical surfaces with five different central cross-section diameters of the arc-shaped rollers (40, 60, 80, 100 and 120mm). When the central cross-section diameter is 40 mm, the equivalent plastic strain fluctuates significantly along the rolling direction. As the central cross-section diameter of the rollers increases, the equivalent plastic strain of the formed spherical surfaces along the rolling direction becomes smoother. This indicates that when the central cross-section diameter of the rollers is small, the stability of the rolling process is insufficient, and the formed surface cannot obtain uniform longitudinal bending deformation.

In order to characterize the stability of the rolling process, select the sheet metal tail node (see Fig. 13), and output its vertical displacement component U2 during the forming process. When using rollers with different central cross-sectional diameters to roll spherical surfaces, the displacement component of the tail node in the vertical direction during the sheet metal forming process is shown in Fig. 14. The first 0.1 second of the forming process is the time for convex roller to press down. After the 0.1 second, the rollers rotate synchronously, and the forming speed is 300 mm/s.

The rolling of doubly-curved surfaces belongs to the line forming method, and the formed metal plate is usually a thin plate, and the rolling reduction is usually very small. Therefore, during the rolling process, the length of the deformation zone where the sheet metal is squeezed in the roll gap along the rolling direction is small. Due to the weight of the sheet metal, mechanical vibration and other factors, there is a slight swing at the end of the sheet during rolling. It can be found from Fig. 14 that the larger the central cross-section diameter of the rollers, the smaller the swing amplitude of the sheet tail during rolling, and the more stable the forming process. When the central cross-section diameter of the rollers is 40 mm, the swing amplitude of the sheet tail increases with the increase of the rolling reduction; when the central cross-section diameter of the rollers is 60 mm, the swing amplitude of the sheet tail is not affected by the rolling reduction; when the central cross-section diameter of the rollers is 80, 100 and 120 mm, the swing amplitude of the sheet tail decreases with the increase of the rolling reduction. Obviously, the stability of the rolling process increases with the increase of the central cross-section diameter of the rollers.

In order to characterize the shape accuracy of the formed spherical surface, the longitudinal curvature deviation is defined

(1)

In Eq. (1), *E**Li* is the longitudinal curvature deviation of the formed spherical surface; *ρ**Li* is the fitted value of the local longitudinal curvature of the formed spherical surface; *ρ**L* is the fitted value of the overall longitudinal curvature of the formed spherical surface.

Figure 15 shows the longitudinal curvature deviation of the spherical surfaces formed by rollers with different central cross-sectional diameters. The longitudinal curvature deviation of the formed spherical surface is consistent with the vertical displacement of the sheet tail (compare Fig. 14 and Fig. 15). The greater the vertical displacement of the sheet tail, the greater the longitudinal curvature deviation. When the central cross-section diameter of the rollers is 40 mm, the longitudinal curvature deviation of the formed spherical surface is the largest. The longitudinal curvature deviation of the stable forming area (except the head and tail transition area) of the formed spherical surface decreases with the increase of the central cross-section diameter. And it can be found that when the central cross-section diameter is 80, 100 and 120 mm, their difference in the longitudinal curvature deviation is very small. This shows that when the central cross-section diameter of the rollers increases to a certain extent, the rolling process will reach a relatively stable state. In this state, the longitudinal curvature deviation of the stable forming area is small and the shape accuracy is good for the formed spherical surfaces.