A Two-Step Marginal-Restraint Mandrel-Free Spinning Method for Accuracy in Forming Large, Special-Shaped Aluminum Alloy Tank Domes

Abstract

The special-shaped tank dome of a launch vehicle is a large, thin-walled, curved structure that is difficult to form using the conventional center-restraint spinning method. This study proposes a two-step marginal-restraint mandrel-free spinning method for forming large domes. The finite element analysis results indicate that a larger roller fillet radius and larger feed ratios lead to a larger upper convex angle and the minimum thickness value for the bottom contour. This study explored the impact of shape parameter variations on the upper convexity and transition rounding angle on forming accuracy. The results show that the convexity of the bottom of the special-shaped domes increases with a larger roller fillet radius and larger feed ratios while the overall height decreases. The forming accuracy is adversely affected by larger transition rounding angles and smaller upper convexities. For the accurate forming of domes, the mutual coupling influence during two-step forming should be considered, and a suitable process and suitable trajectory compensation parameters must be carefully selected. Finally, the study verified that a two-step marginal-restraint mandrel-free spinning method with a 10 mm roller fillet radius, a 2 mm/r feed ratio, and the corresponding trajectory compensation can achieve the precise forming of 2250 mm thin-walled special-shaped domes.

1. Introduction

The structure of a storage tank dome is a key structural component of a launch vehicle, and it has been developed to feature large, integral, thin-walled, and load-bearing optimized configurations [1,2]. In order to increase the compactness of the structure, special-shaped domes with reverse depression, which allows the engine to be connected in a more compact form, have been designed and applied [3,4]. This kind of special-shaped storage tank dome is extremely difficult to manufacture integrally [5,6]. Among the dome processing methods, the integral spinning process offers significant advantages in forming such large, thin-walled dome components [7,8,9,10,11,12,13]. Specifically, compared with the method of welding after piece-by-piece manufacturing, the spinning process reduces the formation of weld seams and provides increased reliability for the storage tank domes, along with a cost-effective cost and forming accuracy. For such special-shaped domes, spinning is an ideal manufacturing method. However, the specific manufacturing techniques and methods are rarely reported. Considering the demands for accurate shape control, it is very necessary to study the spinning technology aiming at the short-process manufacturing of such rotary structures with complex generatrix shapes.

The spinning of domes can be performed using either molded spinning or mandrel-free spinning. As shown in Figure 1, molded spinning involves clamping the workpiece on a core mold with the tail at the top, rotating it together with the machine tool spindle, and pressing the roller onto the material to produce plastic flow, and the curved domes are finally formed with the movement of the roller feed [14,15,16]. When using molded spinning for the processing of the domes of a rocket fuel tank—which is a large, thin-walled dome part with a high diameter–thickness ratio and weak rigidity—equal thickness, uncoordinated deformation always occurs during the core molded spinning. This leads to destabilization phenomena, such as wrinkling, cracking, non-adhesive films, surface rippling, and bonding defects, all of which can seriously affect the processing accuracy of the storage tank. Wrinkling is one of the main forming defects of thin plates, and it is caused by compression instability [17,18,19] during cracking when the material deformation exceeds the tensile limit of the metal [20,21]. Non-adherence to the mold is caused by elastic recovery after unloading of the roller, especially for large-diameter thin-walled domes. Surface ripples are caused by the improper selection of process parameters, which can seriously affect the accuracy and surface quality of the formed parts [22]. Bonding defects are caused by the thermal diffusion welding due to the rapid heating of the mutual contact between the roller and the blank and the roller load [23]. The suppression of these defects is extremely important for guaranteeing forming quality.

According to the restraint position, the process of mandrel-free spinning to create domes involves two main restraint setup methods: center-restraint mandrel-free spinning and marginal-restraint mandrel-free spinning. As shown in Figure 2, center-restraint mandrel-free spinning is a method that adopts pre-forming with a “press drum” and then completes the dome spinning through the combined interaction of a forming roller and a pressure roller on the prefabricated blanks [24]. Compared with the molded spinning method, the mandrel-free spinning replaces the large core mold by forming rollers, and it simplifies the tooling. However, it requires the use of a forming roller and a pressure roller together, which requires high spinning-process-control accuracy and technical experience [25]. In other words, the manufacturing accuracy of the two-step processing is low, and the residual stresses that occur after the end of deformation are hard to reduce [26]. Additionally, domes are formed with large thicknesses of the formed dome parts and low depth-to-diameter ratios. The destabilization and cracking problems are difficult to suppress for both molded spinning and the center-restraint mandrel-free spinning process, especially for the thin-walled blanks with large diameter-to-thickness ratios.

To address these issues, a marginal-restraint mandrel-free spinning method is proposed [27,28,29,30,31,32,33,34]. As shown in Figure 3, the blank is fixed circumferentially to the cylindrical support fixture with a press plate to achieve the marginal restraint, and it rotates on the spinning machine’s table with the cylindrical support fixture. Compared with molded and mandrel-free spinning with a center restraint, the trajectory of the roller moves from the outer radial center towards the inside, and the final shape is controlled by the roller track and spinning parameters together. A typical set-up for marginal-restraint mandrel-free shearing and expanding forming is shown in Figure 3. This method replaces a mold with marginal-restraint tooling, which can greatly improve the versatility of tooling and save on costs. Thin plates are directly used as the input material for spinning, achieving short-process and low-cost forming.

Single point incremental forming (SPIF) is also a plastic forming method, as shown in Figure 4 [35,36,37]. Although both marginal-restraint mandrel-free spinning and SPIF are circumferentially constrained without a mandrel, and the forming order of the blank is also formed gradually from the outer ring inward, there are still significant differences between the two, mainly as follows: (a) Different trajectories: marginal-restraint mandrel-free spinning blanks do rotate, the roller moves continuously, and the spinning trajectory is a continuous curved bus line. SPIF involves blank-fixed, punch movement-layered forming, and the punch trajectory generally uses a contour trajectory, belonging to a multiple discrete trajectory. The continuous trajectory will cause the workpiece to be continuously and repeatedly crushed, and the springback will be smaller and the forming accuracy will be higher. The characteristics of the two trajectories also lead to a large difference in their forming efficiency and differences in their application sites. Marginal-restraint mandrel-free spinning has obvious advantages for rotary parts, especially for large rotary parts. SPIF has a great advantage in non-rotary complex shapes. (b) Different ways of rotating wheel movement: the marginal-restraint mandrel-free spinning roller belongs to the passive rotating wheel, which is driven by the friction force with the blank; SPIF involves a generally active rotating wheel, and the rotating wheel is actively rotating, so in small SPIF, you can use the industrial robot end clamping rotating wheel forming with high flexibility, but the active rotation of the rotating wheel will lead to the sliding of the rotating wheel and the surface of the workpiece, thus affecting the surface quality of the workpiece. (c) The control parameters are different, as spinning is a simple trajectory with fewer control parameters (feed rate, speed n, etc.), while SPIF involves a discrete trajectory with many control parameters (the entry radius (R_in), exit radius (R_out), forming angle (θ_h), and step-down size (Δ_z), etc.), making it more difficult to achieve accurate forming control. (d) The equipment used is different. Spinning requires special spinning machines, while SPIF can be developed with special equipment or a simple test bench can be built using industrial robots as the main body. However, for large thin-walled rotary parts used in launch vehicles, especially when difficult-to-form materials are used, we believe that marginal-restraint mandrel-free spinning has the advantages of a higher efficiency, simpler equipment, and a higher forming accuracy. The counter SPIF method by Jung [37] is significantly different from the two-step method in this paper: (a) the counter SPIF method by Jung is mainly due to the insufficient forming accuracy of first SPIF, and then reversed for shaping to improve the forming accuracy of SPIF, which is still essentially for forming one feature of the workpiece. Each step in the two-step method is designed to form one feature in the part, such as forming a large contour in the first step and a small contour in the second step, with the aim of forming a multi-feature part. (b) The counter SPIF method by Jung is far from the part constraint end, while the starting forming position of the two-step method is immediately adjacent to the part constraint end. Such an obvious difference also leads to a large difference in the forming accuracy of their starting forming sections.

In this study, special-shaped domes with a diameter of 2250 mm, as shown in Figure 5, were manufactured and studied. The special-shaped dome is a typical large, curved, thin-walled structural member with a complex structure, mainly consisting of outer large and inner small contours. Using the single point incremental forming (SPIF) method, its forming accuracy and forming efficiency need further study. For conventional center-restraint molded spinning, it is difficult to form the inner small contours and there is a great risk of instability. A two-step marginal-restraint mandrel-free spinning forming method for accuracy in forming is proposed to overcome the difficulty of spinning complex-shaped, thin-walled domes. Based on the structural characteristics of the special-shaped domes and the advantages of marginal-restraint mandrel-free spinning, the forming scheme consists of two steps: the first step involves large contour forming, and the small contour surface is formed in the second step using the same set of molds. A finite element model of the forming process is established, verified, and used to discuss the influence of the different process parameters and to study the compensation method. The proposed two-step spinning method provides an improved forming process to achieve the accurate and quality forming of special-shaped domes and eliminates the need for molds, requiring only one set of fixtures for the entire process, which makes it cost-effective and suitable for forming domes with different shapes.

2. The Two-Step Marginal-Restraint Mandrel-Free Spinning Method for Special-Shaped Domes

Based on the structural characteristics of special-shaped domes and the benefits of marginal-restraint mandrel-free spinning, a two-step method is proposed and utilized to achieve the forming of special-shaped domes. The scheme is illustrated in Figure 6, where the initial step involves spin-forming the large contour surface, and the subsequent step involves spin-forming the inner small contour surface. As shown in Figure 6a, the setting consists of a cylindrical support fixture, a blank-hold ring, a roller, and a blank. The blank is screwed onto the blank-hold ring and the cylindrical support fixture, which are mounted on the spinning machine table, and they rotate with it. The roller is mounted on the roller holder and moves following the spinning path to spin the large contour surface. In the second step of forming, as shown in Figure 6b, the semi-finished product of the spun domes is dismantled and then reversed to be reassembled on the cylindrical support fixture, which is still attached to the fixture by the screw connection from the first step. The workpiece is then rotated with the cylindrical support fixture, and the roller moves to the flat bottom of the large contour surface formed in the first step. One or more spinning trajectories are then adopted according to the shape of the small contour surface. The workpiece is finally formed by spinning the special-shaped domes.

The two-step method provides several advantages for the spinning of special-shaped domes. Firstly, it eliminates the need for molds and only requires one set of fixtures throughout the entire process. Secondly, the technique can be used for the formation of domes with diverse shapes but the same opening diameter, enhancing its versatility. Moreover, it is a cost-effective method compared to traditional center-constraint spinning. This technique allows the use of thinner blanks for thin-plate spinning without compromising stability, and it reduces the processing volume after spinning, leading to improved production efficiency.

3. Development and Verification of the FEM Model

3.1. Development of the FEM Model

The FEM model for the forming process for the special-shaped domes was established in ABAQUS/Explicit, as shown in Figure 7. In accordance with the marginal-restraint mandrel-free spinning scheme, the FEM model was mainly composed of the blank, the rollers, and the cylindrical support fixture. The blank was fixed on the cylindrical support fixture and rotated with the table of the spinning machine, and the rollers moved together according to the spinning trajectory.

In the FEM model, a double roller arrangement was used. The roller was defined as the rigid body with the working radius of 10 mm. The diameter of the blank was 2400 mm, and the thickness of the blank was 6 mm. The material was a 2219-O state aluminum alloy with the main components shown in

Table 1. Chemical composition of the 2219 aluminum alloy.

Components	Si	Fe	Cu	Mn	Mg	Ti	Zr	Al
Content	≤0.2	≤0.3	5.8–6.8	0.2–0.4	≤0.02	0.02–0.1	0.1–0.25	The rest

Content in percent by mass (%).

and the mechanical properties shown in

Table 2. 2219-O material performance parameters.

Elastic Modulus E/MPa	Poisson’s Ratio μ	Density ρ/(kg·m−3)	Tensile Strength /MPa	Yield Strength /MPa	Elongation /%
73,100	0.33	2840	179	75.8	18

[31]. To improve the calculation efficiency, the cylindrical support fixture was simplified to a circle and the blank-hold ring was removed since the cylindrical support fixture and blank were bound-constrained. The support fixture was defined as a rigid body as there was almost no deformation during the actual spinning process.

The blank was divided into four regions during meshing: the contact region, with the cylindrical support fixture; the large contour forming region; the small contour forming region; and the middle region. Different mesh densities were used for each region, with varying mesh sizes for efficient computation. In the large contour forming area and small contour forming area, the small mesh size was used, while in the middle region, which was hardly involved in deformation, the mesh size was sparse. The mesh sizes were tested to ensure convergence. The blank is shown with S4R cells [38], and the total number of cells was 26,811.

The forming angle $\alpha ={45}^{°}$ in the large contour area was used and the feed ratio $f=2 \mathrm{m}\mathrm{m}/\mathrm{r}$ was used to realize the first step of the spin-forming of the special-shaped domes [33,34]. Figure 8 illustrates the current state of the FEM (finite element method) simulation process of the domes, and Figure 9 represents the stress diagram of the domes after the roller had completed the spinning trajectory, where it was noticeable that the overall stress was more evenly distributed.

3.2. Verification of the FEM Model

Energy balance verification: The mass scaling factor was adopted in this study to speed up the calculation process. During the finite element numerical simulation of the forming process, it is generally recommended that the ratio of the kinetic energy to the internal energy of the blank does not exceed 5–10% [39]. As shown in Figure 10a, the ratio of the kinetic energy to the internal energy of the blank remained at less than 10% for nearly the whole spinning process. Therefore, the mass scaling factor used in this study could be considered to be within a reasonable range. As shown in Figure 10b, the ratio of the artificial energy to the internal energy was less than 5% most of the time, indicating that the hourglass phenomenon had a limited influence on the finite element calculation results, and thus, the established FEM model could be considered reliable. This study used two FEM models: one was a scaled model and the other was a 1:1 proportion model. Both models were validated. The research results showed that the various laws and trends of the scaled model were similar to those of the 1:1 proportion model. In order to improve the analysis efficiency, the subsequent parameter discussion uses the scaled model.

Experimental verification: As shown in Figure 11, the mandrel-free spinning experimental platform was built and used to carry out the spinning verification experiment. The first-step spinning product was obtained, as shown in Figure 12. A 3D/Scan 3D scanner and PX-7 ultrasonic thickness measuring instrument, as shown in Figure 13, were used to extract the contour and thickness data of the experimental part. Figure 14 shows the comparison between the measured and simulated results. As can be seen from this figure, the predicted results were in good agreement with the experiment except for the deviation at the two bends and the middle part of the bottom. The deviation at the two bends was due to the grid size limitation in the finite element calculation. At the bottom, the mesh was sparse in the middle part to improve the calculation efficiency, and so the deformation at the bottom of the workpiece was not fully expressed.

According to the measured data, the shape of the first-step product was an upward convex structure, which is consistent with the simulation. The maximum error between the simulation results and the experimental results was 1.29%. Figure 15 shows the error of thickness between the experimental part and the simulated results. As can be seen from the figure, the thickness of the domes of the experimental piece rapidly thinned from the initial blank thickness to the minimum thickness, and then the thickness gradually increased until it reached the blank thickness at the flat bottom. The thickness change trend corresponded exactly to the contour shape of the domes as the thickness at the thinnest point was 3.53 mm and the maximum thinning rate was 40%. The deviations in the thicknesses of the experimental and simulated parts also mainly appeared at the two bending points while the rest of the parts met well. The maximum error of thickness between the simulation results and the experimental results was 3.11%. In summary, the FEM model had good accuracy and could be used to optimize the forming parameters.

4. Results and Discussion

According to the two-step forming method for the special-shaped dome, the second step of small-contour forming is based on the large contour that was formed in the first step of spinning. Therefore, in addition to the influence of the process parameters on the forming accuracy, the influence of the first-step forming accuracy on the second step should be further considered. The large contour forming accuracy mainly refers to the transition rounding angle and the upper convexity. In order to improve the efficiency, this study employed finite element calculations of scaled parts (blank diameter, 586 mm; roller diameter, 150 mm; blank rotation speed, 15 rpm) to investigate the influence of the process parameters, such as the roller fillet radius and feed ratio, on the forming accuracy of the large contour. Furthermore, when forming the small contour, this study considered the changes in the roller fillet radius, feed ratio, and shape parameters such as the transition rounding angle and the upper convexity. Based on the finite element analysis results, this study provided suitable process parameters for engineering experimentation to verify the feasibility of the spinning solution.

4.1. First-Step Large Contour Forming Study

In the pre-stage of the marginal-restraint mandrel-free spinning, various process parameters were investigated for their influence on the forming accuracy of the main working surface of the workpiece. Specifically, the large contour forming shape of the first step, including the bottom bulge and the size of the transition rounding angle radius resulting from the roller’s different fillet radii, may affect the shape of the blank in the second spinning step and, ultimately, impact the forming accuracy of the workpiece. Therefore, it was crucial to explore the effects of the fillet radius and feeding rates of the roller on the forming shape of the first step before the second spinning step, particularly, the transition rounding angle radius and the impact on the outsole’s shape.

As shown in Figure 16, the variations in process parameters resulted in different degrees of upward convexity at the bottom of the workpiece in the spinning process. To investigate the impact of this phenomenon on the second step of the forming accuracy, this study simplified the bottom bulge as a cone, as shown in Figure 17. The cone was characterized by the middle flat bottom diameter d1, the maximum convex height $h_t$, and the upward convex angle ${\theta }_p=\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}(h_t/(d_1/2\left)\right)$. This simplified model was employed to evaluate the bottom bulge phenomenon and study the influence of the bottom bulge on the second step of the forming accuracy.

4.1.1. Influence of the Roller Fillet Radius on the Forming Accuracy

To study the effect of the roller fillet radius (${\rho }_R$) on the contour accuracy and thickness uniformity of the workpiece, this study adopted a one-curve spinning trajectory with ${\rho }_R$ values of 10 mm, 20 mm, and 30 mm, as well as a feed ratio ($f$) of 2 mm/r and the forming angle $\alpha ={45}^{°}$.

The cross-section contour under a different roller fillet radius is shown in Figure 18. While the roller fillet radius ${\rho }_R$ was 10 mm, the maximum convexity of the bottom $h_t$ was 3.92 mm and the upper convexity angle was ${\theta }_p={2.72}^{°}$, and when the roller fillet radius ${\rho }_R$ was 30 mm, the maximum convexity of the bottom $h_t$ was 6.07 mm and the upper convex angle was ${\theta }_p={4.21}^{°}$. The different roller fillet radii induced the different contact areas and statements during spinning, which led to the significantly different sizes of the transition corners and the different shapes of the bottoms of the workpiece after final forming. It could be concluded that the larger the transition rounding angle, the larger the upper convexity of the bottom and the larger the transition rounding angle.

The thickness of the workpiece decreased rapidly from the blank to the lowest thickness, and then it increased quickly and then gradually, and, finally, it returned to the thickness of the blank, as shown in Figure 19. This thickness variation followed the sub-circular spinning track. As the ${\rho }_R$ value increased, the thickness uniformity of the workpiece improved and the minimum thickness increased. This was due to the edge of the workpiece being fixed on the cylinder support. With an increase in the radius of the rotary roller, the load on the workpiece increased and part of the shear deformation turned into a bending deformation, leading to an increase in the minimum thickness value. Figure 20 illustrates the workpiece’s condition obtained from the spinning experiment using the different roller fillet radius values, with the transition rounding angle circled in red. The size of the transition rounding angle varied and increased with the increasing roller fillet radii. A larger contact area between the roller and the workpiece was identified, and it led to a larger transition rounding angle.

4.1.2. Influence of Feed Ratio on Forming Accuracy

The feed ratio was a crucial parameter that significantly affected the workpiece forming. In this study, the roller fillet radius ${\rho }_R$ was fixed at 10 mm and the spinning trajectory was a secondary curve with a forming angle of $\alpha ={45}^{°}$, similar to the previous experiment. Three different feed ratios, namely, 1 mm/r, 2 mm/r, and 4 mm/r, were used to investigate their impacts on the contour accuracies and thickness uniformities of the workpieces.

Figure 21 demonstrates that increasing the feed ratio led to a more prominent bottom bulge effect on the workpiece contour, with a maximum bulge height of 3.52 mm and an upper bulge angle ${\theta }_p$ of 2.44° at a feed ratio of 1 mm/r. When the feed ratio was increased to 4 mm/r, the maximum bulge height and upper convex angle were 4.94 mm and 3.43°, respectively. Figure 22 shows that the thickness of the workpiece followed a similar variation rule as the radius of the roller corner, and the uniformity improved as the feed ratio increased, resulting in a larger minimum thickness value. However, a higher feed ratio also led to an increased spinning load, which may have negatively affected the surface quality of the workpiece.

4.2. Second-Step Small Contour Forming Study

In the previous section, an analysis was conducted on the effects of different roller fillet radii and feed ratios on the forming accuracies of the large contour surfaces. The results revealed that the bulge at the bottom of the large contour surface became more pronounced as the roller fillet radii and feed ratios increased, while the overall thickness uniformities and minimum thickness values of the surfaces improved. Additionally, an increase in the roller fillet radius resulted in a larger transition rounding angle of the large contour surface. The large contour surface served as the blank for the small contour forming in the second step, and its forming quality inevitably affected the forming of the small contour surface. Therefore, this study aimed to investigate the impacts of the roller fillet radius, feed ratio, transition rounding angle size, and large base convexity degree on the small contour surface forming.

4.2.1. Influence of the Roller Fillet Radius on the Forming Accuracy

The size of the roller fillet radius has a large influence on the forming of the contoured surface. For a large contour surface with an upper convex angle of ${\theta }_p=0^{°}$ at the bottom and a transition rounding angle of $R_b=10 \mathrm{m}\mathrm{m}$, the roller fillet radii ${\rho }_R$ were taken as 6 mm, 10 mm, 20 mm, and 40 mm, respectively, and the feed ratio $f=2 \mathrm{m}\mathrm{m}/\mathrm{s}$ and a forming angle of 25° for the circular arc trajectory were used. The effects of the different roller fillet radii on the contour accuracies and thickness uniformities of the workpieces were investigated.

As shown in Figure 23, the impact of varying the roller fillet radius on the forming accuracy of the workpiece, with a specific focus on the transition corners and the bottom of the small contour surface, was studied. The results indicated that the larger the roller fillet radius, the greater the offset between the top of the small contour surface and the bottom of the large contour surface, resulting in more pronounced sinking of the small contour surface. This sinking phenomenon was primarily due to the larger contact area between the rotary roller and the workpiece, and it resulted in a larger load on the workpiece. Given the limited load-bearing capacity of the large contour surface, the increased load caused greater deformation downwards. The maximum contour deflection at the transition corner increased from 1.81 mm to 3.01 mm as the roller fillet radius increased from 6 mm to 40 mm, while the maximum bulge height at the bottom of the small contour surface rose from 0.28 mm to 3.09 mm and the upper convex angle ${\theta }_p$ increased from 0.49° to 5.43°.

As shown in Figure 24, the thickness variation law of the large contour surface was the same as that in the first step of forming, and the thickness deviation began to appear at the transition round corner. This indicated that the larger the roller fillet radius, the more powerful the thinning at the transition round corner. The thickness of the small contour part changed as the thickness decreased and then increased, and the inflection point was at the end of the spinning of the small contour surface, indicating that the larger the roller fillet radius, the smaller the minimum thickness of the small contour surface.

4.2.2. Influence of Feed Ratio on Forming Accuracy

To investigate the effects of different feed ratios on the contour accuracies and thickness uniformities of the workpieces, a large contour surface with the bottom convexity ${\theta }_p=0^{°}$ and the transition rounding angle $R_b=10 \mathrm{m}\mathrm{m}$ was formed using a roller fillet radius of ${\rho }_R=10 \mathrm{m}\mathrm{m}$ and a forming angle of 25° arc trajectory at feed ratios of 1 mm/r, 2 mm/r, and 4 mm/r. The small contour surface was still offset at the transition corners, but the increase in the feed ratio had little effect on the contour offset of the workpiece. The convexity of the unspun area at the bottom of the small contour increased as the feed ratio increased. The minimum thickness of the small contour part decreased as the feed ratio increased, while the relative uniformity of the workpiece improved. However, the higher feed ratio required greater spin pressure and resulted in a poorer surface quality of the workpiece.

As shown in Figure 25, the small contour surface was still offset at the transition corners. Different feed ratios had little effect on the contour offset of the workpiece because, although the increase in the feed ratio brought the increase in the load, the load increased while the spin roller was gradually moving away from the transition corners so that more material was available to bear the increased load. The larger the feed ratio was, the more serious the convexity of the unspun area at the bottom of the small contour was. When the feed ratio $f$ was 1 mm/r, the maximum convexity was 0.33 mm and the upper convexity angle ${\theta }_p$ was 0.58°, and when the feed ratio $f$ was 4 mm/r, the maximum convexity was 0.88 mm and the upper convexity angle ${\theta }_p$ was 1.55°. As shown in Figure 26, the small contour forming did not affect the thickness value of the large contour part. The larger the feed ratio, the larger the minimum thickness and the better the relative uniformity of the workpiece. However, the larger the feed ratio, the larger the spin pressure required in the spinning process, which may result in a poor surface quality.

4.2.3. Influence of the Transition Rounding Angle on the Forming Accuracy

From the above, it was known that different roller fillet radii resulted in different transition corner radii of the large contour surfaces. The effects of different transition rounding angles in the first step on the contour accuracies and thickness uniformities of the workpieces were investigated.

The roller fillet radius ${\rho }_R=10 \mathrm{m}\mathrm{m}$, the feed ratio $f=2 \mathrm{m}\mathrm{m}/\mathrm{s}$, and a 25° circular arc spinning trajectory with a spin-forming angle were selected for the second step. As shown in Figure 27, the small contour surface was offset at the transition rounding angle, and the larger the radius of the transition rounding angle, the more serious the offset. The reason for this phenomenon was that the starting position of the spin roller was the same, and when the radius of the transition rounding angle increased, the height of the large contour surface within the transition rounding angle decreased, and so the starting deformation position was easily lower when the spin roller was pressed down, which was equivalent to making the small contour. The larger the offset between the small contour surface and the bottom of the large contour surface, the larger the offset. When the radius of the transition circle was 10 mm, the maximum contour offset at the transition circle was 2.29 mm, and when the radius of the transition circle was 30 mm, the maximum contour offset at the transition circle was 4.44 mm. The smaller the radius of the transition circle, the more serious the convexity of the unspun area at the bottom of the small contour. When the radius of the transition circle was 10 mm, the maximum convexity was 1.43 mm and the upper convexity angle ${\theta }_p$ was 2.52°, and when the transition rounding angle radius was 30 mm, the maximum projection height was 0.73 mm and the upper convex angle ${\theta }_p$ was 1.59°. As shown in Figure 28, the effect of the size of the transition rounding angle radius on the thickness value of the workpiece was mainly in the vicinity of the transition rounding angle, and the larger the radius of the excessive fillet, the larger the thickness value. However, the effect was very small, and the overall thickness deviation value was within 0.1 mm, and so it could also be considered that the size of the transition rounding angle radius has almost no effect on the thickness of the workpiece.

4.2.4. Influence of the Upper Convexity on the Forming Accuracy

From the above, it was known that spinning with different process parameters would yield preformed large contour domes with different upward convexities at their bottoms. Therefore, in this study, the roller fillet radius ${\rho }_R=10 \mathrm{m}\mathrm{m}$, the feed ratio $f=2 \mathrm{m}\mathrm{m}/\mathrm{s}$, and a 25° circular arc spinning trajectory with a spin-forming angle were selected. The effects of the different amounts of large-base up-convexities on the contour accuracies and thickness uniformities of the workpieces were investigated.

As shown in Figure 29, the small contour was offset at the transition corner, and the smaller the upper convex angle ${\theta }_p$ was, the more serious the offset was. The reason for this phenomenon was that the spin roller’s starting position was the same, and when the upper convex angle ${\theta }_p$ increased, the press-down of the spin roller on the blank would be reduced, which means the spin pressure would be reduced, thus reducing the small contour offset at the transition corner. When the upper convex angle ${\theta }_p$ was 0°, the maximum contour offset at the transition round corner was 2.29 mm, and when the upper convex angle ${\theta }_p$ was 6°, the maximum contour offset at the transition round corner was 1.81 mm. The smaller the upper convex angle ${\theta }_p$ was, the more serious the convexity of the unspun area at the bottom of the small contour was. When the upper convex angle ${\theta }_p$ was 0°, the maximum convexity was 1.43 mm and the upper convex angle ${\theta }_p$ was 2.52°, and when the upper convex angle ${\theta }_p$ was 6°, the maximum bump height was −0.22 mm and the upper convex angle ${\theta }_p$ was −0.37°. As shown in Figure 30, the effect of the size of the upper convexity on the thickness value of the workpiece was mainly at the bottom, and the larger the upper convexity angle was, the larger the thickness value was. However, the effect was very small and the overall thickness deviation value was within 0.1 mm, and so it could also be considered that the size of the upper convexity of the large bottom had almost no effect on the thickness of the workpiece.

4.3. 2250 mm Special-Shaped Dome Spin-Forming Study

From the previous analysis results, the forming parameters were selected as follows. The blank rotation speed was 15 rpm and the roller diameter was 400 mm; in the first step of the large contour forming, the roller fillet radius was 10 mm and the feed ratio was 2 mm/r. For the small contour forming, the roller fillet radius was 10 mm and the feed ratio was 2 mm/r. Considering the sink deflection, a trajectory compensation method was employed in the first step of the large contour forming to make the special-shaped domes meet the accuracy requirements.

Based on the simulated parameters for the finite element simulation of the 2250 mm contour domes, this paper presents Figure 8 and Figure 9, which illustrate the simulation process and results of the first step of the large contour forming. Figure 31 presents the initial stress diagram of the second step of the small contour simulation, where it was observed that a ring constraint existed between the outer ring of the workpiece and the cylindrical support fixture. Additionally, when the roller began spinning, a cantilevered force structure transferred the stress to the outer ring of the workpiece, leading to a significant stress concentration around the contact with the cylindrical support fixture. Moreover, a high stress area was identified between the roller and the workpiece due to the spinning contact. Figure 32 shows the stress diagram of the special-shaped domes after the completion of the second step of the small contour forming simulation. The results indicated that the force’s arm length affected the load on the outer ring of the workpiece. Furthermore, it was observed that a smaller transition rounding angle was achieved for the entire contour domes by using a smaller roller fillet radius in the first step of the forming, and then a smaller roller fillet radius was further used in the second step of the small contour forming, with a suitable spinning trajectory. In contrast, a larger transition rounding angle was obtained by spinning with a large roller fillet radius in the first step, with continuous spinning using a large roller fillet radius in the second small contour forming step, as shown in Figure 33. Subsequently, a special-shaped dome with a large transition rounding angle radius was obtained, as presented in Figure 34.

4.4. Experimental Verification

The spinning experiment was performed according to the process parameters simulated in the previous section. Figure 35 shows the preparation state of the small profile spinning, and the large profile perimeter is fixed on the barrel and support using the press plate. Figure 36 shows the 2250 mm profiled head. The contour and wall thickness data of the experimental part were extracted with a 3D scanner and a wall-thickness measuring instrument, and they were compared with the simulation results.

As can be seen from Figure 37, except for a certain deviation at the bottom of the small contour, the simulation contour and the experimental contour were in good agreement. The deviation of the bottom was because the spinning wheel did not exit immediately when the spinning track of the small contour was finished during the experiment due to limitations in the equipment’s movement trajectory, and the material there was fully deformed and a ring groove was formed, which aggravated the convexity of the bottom. The maximum radial deviation of the bottom was 1.4 mm and the maximum height deviation was 2.1 mm, both of which were within the allowable error range. As shown in Figure 37, a local downward deviation occurred at the transition rounding angle. This was because when the small contour started to spin, the spin wheel drove the large contour at the transition round corner downward with a sink offset. This phenomenon existed in both the simulation and experimental processes shown in Figure 38.

From Figure 39, it can be seen that the thickness of the experimental domes rapidly thinned from the initial blank thickness to the minimum thickness, and then the thickness gradually increased until reaching the transition corners of the large and small contours. The thickness value continued to slowly decrease until reaching the stable state and then rapidly rose to the blank thickness. The change trend in the thickness corresponded exactly to the contour shape of the domes, and the thickness at the thinnest point was 3.53 mm, with a maximum thinning rate of 40%, which was greater than the theoretical requirement of 3 mm. Then, the theoretical thickness of the domes could be realized by a small amount of the processing mode.

Therefore, it could be concluded that this two-step marginal-restraint mandrel-free spinning method could realize special-shaped aluminum alloy tank dome spinning and forming based on a thin-walled blank and the selection of suitable spinning process parameters.

5. Conclusions

In this paper, we addressed the challenge of spinning and forming complex, large, thin-walled, special-shaped domes by proposing a two-step forming scheme based on our previous work on marginal-restraint mandrel-free spinning. An FEM model with high simulation accuracy was established and used to study the influence of the different process parameters on the first step of large contour forming and to introduce the shape parameters that would characterize the contour shape. The impacts of various process parameters and shape parameters on the forming accuracy of special-shaped domes were investigated. Based on the influence laws obtained, the simulation process parameters of the 2250 mm special-shaped domes were selected and further verified experimentally. The main conclusions are as follows:

(1)

For large contour forming, increasing the roller fillet radius leads to more serious convexity of the bottom, a larger transition rounding angle, and a larger minimum thickness, and increasing the feed ratio leads to more serious convexity of the bottom of the large contour, a larger minimum thickness, and a poorer surface quality.

(2)

For small contour forming, increasing the roller fillet radius leads to a larger sinking offset of the transition circle and a larger bottom upper convexity, and increasing the feed ratio leads to a larger bottom upper convexity. A larger transition circle results in a more serious sinking offset of the transition circle and a smaller bottom upper convexity. A larger bottom upper convexity of the large contour leads to a smaller sinking offset of the transition circle, a smaller bottom convexity, and even a flat or negative angle bottom.

(3)

The two-step forming results affect each other, and suitable process parameters and trajectory compensation can achieve the spinning forming of special-shaped domes with different shapes. The two-step solution of marginal-restraint mandrel-free spinning, with a 10 mm roller fillet radius, a 2 mm/r feed ratio, and trajectory compensation, can achieve the accurate forming of 2250 mm thin-walled, special-shaped domes.