1 Introduction

In view of its high material utilization and flexibility, metal sheet spinning is regarded as an advanced near-net forming technology [1, 2]. For instance, metal sheet spinning is utilized to manufacture axisymmetric thin-walled parts, such as thin-walled ellipsoidal heads [3, 4]. These ellipsoidal alloy heads are widely used in aviation, aerospace, and chemical and nuclear industries, such as heads of rocket fuel tanks and pressure vessels [2, 5, 6]. Normally, the spinning parameters considerably influence the forming quality of thin-walled parts [5, 7]. If these parameters are not suitably employed, however, some forming defects, including wrinkling or vibration [5, 7, 8], wall deviation, and contour deviation, typically appear in the spinning process of thin-walled parts. The design of suitable spinning parameters is therefore particularly important to guarantee the forming quality of alloy parts [9, 10].

In recent years, a number of scholars have studied the metal sheet spinning of different parts by numerical simulations and experiments [1, 2, 11]. Metal sheet spinning can be divided into mandrel spinning and mandrel-free spinning depending on whether there is a specific mandrel in the spinning process [1, 2]. In the mandrel spinning process, Lin et al. [5, 7] obtained the optimal parameters of the staggered spinning for thin-walled Hastelloy C-276 cylindrical part, i.e., the three different roller fillet radii were determined as 3 mm, 4 mm and 6 mm, respectively, and the roller feed ratio was approximately 0.8 mm/r. Xia et al. [6] found that the dynamic recrystallization behavior was enhanced with the increase in the forming temperature and total thinning ratio of wall thickness. Wang and Long [12] found that the wall thickness of the blank decreased in the forward pass and remained unchanged in the backward pass during multi-pass spinning. Zhan et al. [13] reported that the precision of wall thickness in cone spinning could be improved by increasing the feed ratio; however, it reduced the contour precision. El-Khabeery et al. [14] concluded that the dimensional precision of cylindrical aluminum cups was enhanced in conventional spinning as the roller angle and feed ratio were reduced. Xiao et al. [15] revealed that a large clearance between the roller and mandrel as well as a small feed ratio in asymmetric multi-pass spinning could improve wall thickness uniformity. Liu et al. [16] and Hayama et al. [17] discovered that the wall thickness distribution of final workpieces was determined by the first pass in conventional spinning. Xia et al. [18, 19] found that the spinning force was considerably affected by the feed ratio in hollow-part spinning, and it increased with the feed ratio. Kong et al. [20] observed that the feed ratio had a considerable influence on wrinkling, and a low feed ratio could reduce the possibility of its occurrence. Zhang et al. [21] indicated that the flange wrinkling of thin-walled vessel heads easily occurred when the ratio of sheet metal thickness to diameter was extremely small. Xia et al. [22] reported that a large feed ratio could easily induce metal sheet wrinkling in the one-path draw spinning process.

In the mandrel-free spinning, the dependence on the mandrel is eliminated, and the flexibility of conventional spinning is improved. Lin et al. [4] successfully manufactured the thin wall ellipsoidal Al alloy heads with the thickness and diameter of 4 mm and 2.29 m, respectively, by mandrel-free spinning. Zoghi et al. [23] discussed the effects of feed and roller contact start point on strain and residual stress distribution in dome forming of steel tube by spinning at an elevated temperature. Shima et al. [24] developed a die-less shear spinning method, which could be used for sheet metal forming. Rao et al. [25] proposed a multi-constraint spinning method for manufacturing large ellipsoidal parts without wrinkles and cracks and for achieving a wall thickness deviation of less than 10%. Kawai et al. [26, 27] developed a general-purpose mandrel spinning method to successfully manufacture hemispherical and conical shells. Kang et al. [28] discovered that shear spinning was the main deformation mode during the conventional spinning of plates, and wall thickness values conformed with the sine law. Han et al. [29] proposed a method to control the wall thickness distribution in the synchronous die-less shear spinning of oblique cones. Li et al. [30] claimed that a low-degree bending of concave roller paths could reduce contour deviation, and a high-dimensional precision could be obtained with a low-degree bending of convex roller paths. Sekiguchi and Arai [31] discovered that various inclination angles of the flange could cause different wall thickness distributions during oblique shear spinning. Based on a quadratic Bezier curve, Polyblank and Allwood [32] proposed a parametric tool path planning method for metal sheet spinning. Jia et al. [33] found that the edge of the blank flange could easily wrinkle at the 0° area. Liu [34] found that the wrinkling deformation could remarkably increase the strain energy. Sugita and Arai [35] presumed that wrinkling easily occurred if rotational pass sets were employed to form rectangular box shapes.

The foregoing review reveals that mandrel spinning requires specific mandrels that limit its application [36]. Meanwhile, the dimensional precision of conventional mandrel-free spinning is not easy to control [37]. Moreover, thin-walled parts are prone to wrinkling during mandrel spinning or mandrel-free spinning and can be more severe in those with large diameter-to-thickness ratios [24]. In this work, a marginal-restraint mandrel-free spinning process that can effectively avoid wrinkling is proposed for the manufacture of thin-walled ellipsoidal heads. In addition, a finite element model (FEM) is established and verified to study the influences of spinning parameters on the wall thickness precision and contour precision of thin-walled ellipsoidal heads. Finally, the optimized spinning parameters are obtained and verified by experiments.

2 Marginal-restraint mandrel-free spinning process

Figure 1 shows the schematic of marginal-restraint mandrel-free spinning process. The process includes two passes. At the beginning of spinning, a blank is restrained on the cylindrical support fixture by the blank-hold ring that can preclude the deformation and wrinkling of metal sheet flange. Moreover, wrinkling, which can be reflected by the stress distribution in spun heads, is avoided in the forming regions. This aspect will be discussed in Section 3.2.2.

Fig. 1
figure 1

Schematic diagrams of the marginal-restraint mandrel-free spinning process

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In the first pass (see Fig. 1a), the blank is rotated around the center axis of the fixture as the roller moves along the target contour. The blank is spun into a conical part with a large deformation as the roller moves along the 0–1–2–3–0 trajectory. The wall thickness can be evaluated as \(t_{1} = t_{0} \, { \cos }\,\alpha\). Here, diameter D0 is the upper limit of the forming range of the blank, and the spinning parameters, F, β, α, N, ρR, are the feed ratio, roller installation angle, forming angle, spindle speed, and roller fillet radius, respectively.

In the second pass (see Fig. 1b), the conical part is gradually expanded into a curved part with a relatively small deformation and wall thickness variation. The wall thickness, t2, can be calculated by the principle of volume invariance. Furthermore, the roller motion direction of the second pass can be divided into forward motion (the roller moves along trajectory 0–1–2–3–0) and backward motion (along trajectory 0–3–2–1–0). The target contour of the curved part is reflected by the angle between passes (i.e., θ, the included angle between the contour tangent of the curved part and conical part profile). Additionally, the other spinning parameters (F, β, N, and ρR) of the second pass are the same as those of the first pass.

3 Development and verification of FEM

3.1 Development of FEM

The metal spinning process is a dynamic forming process with complex deformation mechanisms and contact conditions. In view of this, a three-dimensional (3D) dynamic explicit FEM is established based on the ABAQUS/Explicit code to simulate the marginal-restraint mandrel-free spinning process, as shown in Fig. 2a; X1 and Y1 are the two directions of roller movement. Figures 2b and c show the deformation process of each spinning pass in the simulation. Here, X and Y represent the radial and axial directions, respectively.

Fig. 2
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Schematic diagrams of the FEM and deformation process of the spinning simulation

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In this model, the cylindrical support fixture is simplified into a support ring, which is combined with the edge of the lower surface of the blank by the tie function in ABAQUS software. Meanwhile, the rotating boundary condition is set for the fixture to drive the rotation of the blank. The blank-holder ring is combined with the edge of the upper surface of the blank to restrain the deformation of this edge. The motion of the roller in the X1 and Y1 directions are set to satisfy the feeding requirements of different target profiles in the spinning process, as shown in Fig. 2a. Furthermore, the roller, blank-holder ring, and support ring are all set as discrete rigid bodies to improve computing efficiency. The mass scaling factor of the model is set to 20.

The blank is defined as deformable and meshed with a continuous shell element (SC8R). The number of elements and nodes in the FEM are set as 10 448 and 19 284, respectively. The material of the blank, which is assumed to be homogeneous and isotropic, is a commercial 5052 aluminum alloy, and its properties are obtained by uniaxial tensile test, as listed in Table 1. The elastic capacity of the material is described by Hooke’s law, and its plastic capacity is described by von Mises criterion and Hollomon’s strain-hardening law, \(\sigma\,=\,K\varepsilon^{n}\). The kinematic contact method of finite sliding formulation and Coulomb friction are used to simulate the contact behavior between the roller and blank. The Coulomb friction coefficient is set as 0.02 [12]. A springback analysis model is also established using ABAQUS/Standard after the spinning with ABAQUS/Explicit is simulated.

Table 1 Mechanical properties of 5052 aluminum alloy
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Wall thickness and contour characteristics are the two important geometric dimensions of thin-walled ellipsoidal heads. In order to study the influences of spinning parameters on wall thickness distribution and contour characteristics, orthogonal experimental schemes, L16(45) and L18(2×35), are designed for the first and second passes, as summarized in Tables 2 and 3, respectively. Each spinning parameter is set as having four levels (1–2–3–4) in the first pass and three levels (1–2–3) in the second pass. The level values successively increase.

Table 2 Orthogonal experiment scheme of the first pass spinning
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Table 3 Orthogonal experiment scheme of the second pass spinning
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3.2 Verification of FEM

3.2.1 Energy balance verification

Figure 3 shows the variations in energy ratios (kinetic energy/internal energy and artificial energy/internal energy) in the numerical simulations of two-pass spinning. It is found that in most spinning processes, the ratio of kinetic energy to internal energy is less than 10%, as shown in Fig. 3a. The ratio is initially high because the roller and blank have not been in contact, and there is practically no plastic deformation. Moreover, Fig. 3b shows that the ratio of artificial strain energy to internal energy is less than 5%. This indicates that the influence of inertia resulting from mass scaling (see Section 3.1) is not significant on the simulation results, and the simulation precision is unaffected by the hour glassing problem [38]. The set of parameters of the FEM are therefore reasonable.

Fig. 3
figure 3

Changes of energy ratios in numerical simulations

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3.2.2 Experimental verification

The verification of the FEM based on experimental results is presented in this section. Figure 4a shows the spinning experimental equipment and the spun thin-walled ellipsoidal head. The experimental material has a 330 mm formable diameter and 1mm initial wall thickness. The main spinning parameters (i.e., α, β, N, F, and ρR) in the first pass are 40°, 45°, 360 r/min, 1.4 mm/r, and 8 mm, respectively. In the second pass, the angle between passes (θ) is 8°, and the other spinning parameters (β, N, F, and ρR) are the same as those in the first pass. Additionally, the motion of the roller in the second pass is directed backwards. The spun thin-walled ellipsoidal heads are shown in Figs. 4b and c. It is evident that wrinkling has been avoided as indicated by the stress distributions in the simulated heads, as shown in Figs. 4d and e. After the first and second spinning passes, four stable stress regions are distributed on the forming workpiece surface. Here, the uniformly-distributed von Mises stress in the forming area is not large. Moreover, the red regions shown in Figs. 4d and e are the final positions of roller movements for the first and second passes, respectively. The von Mises stresses in these red regions are not extremely large, and the materials are uniformly distributed.

Fig. 4
figure 4

Spinning experiments and the stress distributions of simulated heads

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To analyze the dimensional precision, only half of the cross section is considered because the workpieces are axisymmetric. The thickness and contour are obtained from a single generatrix. Here, a 3D laser scanning measurement system named FARO is employed to obtain the contour data of the workpiece, whereas the wall thickness data are derived using a dual-ball micrometer. Figures 5a and b compare the simulated and experimental wall thicknesses of the first and second pass spun heads, respectively. It is found that the spinning stage can be divided into three spinning regions, i.e., initial, steady, and final spinning regions, which are similar to the wall thickness distributions of shear spun parts [39]. The steady spinning region where the change in wall thickness is not distinct is mainly discussed here. In the axial direction, the simulated wall thickness distribution is consistent with experimental results. The maximum errors between simulated and experimental results are 4.6% and 5.5% for the first and second passes, respectively. Figures 5c and d compare the simulated and experimental contours of the first and second pass spun heads, respectively. In the radial direction, the maximum errors between simulated and experimental results are 1.83% and 3.06% for the first and second passes, respectively; in the axial direction, these maximum errors for the first and second passes are 2.8% and 3.9%, respectively. These results indicate that the FEM can be effectively utilized to simulate the marginal-restraint mandrel-free spinning.

Fig. 5
figure 5

Comparisons between the experimental and simulated results

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3.3 Dimensional precision of thin-walled spun heads

In this work, the wall thickness precision is characterized by the average deviation ratio of wall thickness (ADROWT, T) and the standard deviation of wall thickness (SDOWT, t′), as evaluated by Eqs. (1) and (2). Furthermore, the contour precision of the spun part is reflected by the deviation ratio of the forming angle (DROFA, \(\vartriangle \alpha\)) and average radial deviation (ARD, L′), as evaluated by Eqs. (3) and (4), respectively. Here, t0 is the designed wall thickness; t is the simulated wall thickness; \(\bar{t}\) is the average wall thickness; n and n2 are the numbers of extracted wall thickness and contour coordinates, respectively; \(\alpha_{0}\) is the designed forming angle; α is the forming angle of contour coordinate fitting; L and l are the simulated and designed contour coordinates, respectively. In addition, the degree of influence of spinning parameters on the above four indicators is characterized by the range obtained from orthogonal analysis.

$$T = \frac{1}{{t_{0} }}\left( {\frac{1}{n}\sum\limits_{i = 1}^{n} {t_{i} - t_{0} } } \right) \times 100{\text{\%}},$$
(1)
$$t^{'} = \sqrt {\frac{1}{n}\sum\limits_{i = 1}^{n} {\left( {t_{i} - \bar{t}} \right)} } ,$$
(2)
$$\vartriangle \alpha = \frac{{\alpha - \alpha_{0} }}{{\alpha_{0} }} \times 100\%,$$
(3)
$$L^{'} = \frac{1}{{n_{2} }}\sum\limits_{i = 1}^{{n_{2} }} {\left| {L_{i} - l_{i} } \right|}.$$
(4)

4 Simulated results and discussion

4.1 Influences of spinning parameters on dimensional precision in the first pass

4.1.1 Influences of spinning parameters on wall thickness

Figure 6 illustrates the influences of spinning parameters on the ADROWT and SDOWT in the first pass spinning process. As shown in Figs. 6a and b, the influences of spinning parameters α, ρR and F on the ADROWT and SDOWT are relatively evident. For the ADROWT range, the values of α, ρR and F are evaluated as 6.5, 2.6, and 2.1 respectively, and for the SDOWT range, the parameter values are 0.005, 0.002, and 0.001, respectively. The small values of the roller installation angle, \(\beta\), and spindle speed, N, however, indicate that the ADROWT and SDOWT values are not sensitive to these two parameters. The influences of spinning parameters α, ρR, and F, are therefore mainly discussed here. In Figs. 6c and d, it is clearly observed that both the ADROWT and SDOWT values gradually rise when the forming angle or roller fillet radius is increased. This is because the larger forming angle easily induces a large wall thickness reduction ratio, which deteriorates the material flow stability and degrades the thinning precision of wall thickness. Additionally, the contact area between the roller and workpiece increases with increasing fillet radius, which hinders material flow in the axial direction. The suitable forming angle and roller fillet radius are found to be 15° and 3 mm, respectively. Moreover, as the feed ratio is continuously raised, the ADROWT and SDOWT values first decrease and then increase. At a given spindle speed, the contact time between the roller and workpiece generally increases with decreasing feed ratio. This increases the deformation degree, which then leads to an increasingly distinct wall thickness thinning; however, the deformation time is brief. If the feed ratio is extremely large (for example, 2.4 mm/r), then the plastic deformation will be insufficient, and wall thickness thinning will diminish. It is found that the suitable feed ratio is approximately 1.8 mm/r.

Fig. 6
figure 6

Influences of spinning parameters on wall thicknes in the first pass spinning process

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4.1.2 Influences of spinning parameters on contour precision

Figure 7 displays the influences of spinning parameters on the contour precision in the first pass spinning process. As shown in Fig. 7a, the influences of spinning parameters α, β, and F on the DROFA are relatively evident. For the DROFA range, the values of α, β and F are evaluated as 1.04, 0.77 and 1.53 respectively. The relatively small values of roller fillet radius, ρR, and spindle speed, N, however, indicate that the DROFA value is not sensitive to these two parameters; accordingly, the influences of spinning parameters α, β, and F, are mainly discussed in this section. Figure 7b shows that the DROFA value first decreases and thereafter increases with the rise in the forming angle or feed ratio. This is because both shear and bending deformations occur in mandrel-free spinning [14]. A small forming angle increases the bending radius of workpieces that easily induces a large springback. The reduction in wall thickness is evident at a larger forming angle, which reduces material flowability and forming precision. A low feed ratio benefits the diameter expansion of workpieces, whereas a large feed ratio induces a large springback. The suitable forming angle and feed ratio are found to be 25° and 1.2 mm/r, respectively. In addition, the DROFA value first slowly decreases and then rapidly drops with increasing roller installation angle. This is because a large roller installation angle improves the material flow in the axial direction, making the shear deformation sufficient and reducing the springback. The suitable roller installation angle is accordingly found to be approximately 65°.

Fig. 7
figure 7

Influences of spinning parameters on the contour precision in the first pass spinning process

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4.2 Influences of spinning parameters on dimensional precision in the second pass

4.2.1 Influences of roller motion direction on dimensional precision

Figure 8 displays the influences of roller motion direction on wall thickness and contour precision in the second pass spinning process. In Figs. 8a and b, it is clearly observed that the ADROWT value decreases, whereas the SDOWT value increases as the roller’s motion direction is changed from backward to forward. This is because forward spinning promotes axial material flow and causes the contact region between the roller and workpiece to gradually decrease. This interpretation may be reflected by the changes in the spinning force, as shown in Fig. 8d. The contact area reduction in the forward spinning process decreases the spinning force, whereas the spinning force of backward spinning is relatively stable because of the small change in the contact area [18, 19]. Thereafter, the workpieces cannot be sufficiently deformed in the middle and later stages of forward spinning, causing poor wall thickness uniformity and thinning degree. In view of the small difference in the theoretical wall thickness between the first and second passes, however, the low thinning degree of wall thickness reduces the ADROWT value. In addition, because of the insufficient workpiece deformation, the ARD value in the forward spinning process is also larger than that in the backward spinning process, as shown in Fig. 8c. As a result, the contour precision in the forward spinning process deteriorates. Accordingly, the suitable roller motion direction is backwards.

Fig. 8
figure 8

Influences of roller motion direction on dimensional precision in the second pass pinning process

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4.2.2 Influences of major spinning parameters on wall thickness

Figure 9 illustrates the influences of spinning parameters on wall thickness in the second pass spinning process. As shown in Figs. 9a and b, the influences of spinning parameters θ, β and F on the ADROWT and SDOWT values are also relatively considerable. For the ADROWT range, the values of θ, β and F are evaluated as 0.81, 0.27, and 0.33, respectively, and for the SDOWT range, these are 0.0055, 0.001 and 0.0014, respectively. The influences of the roller fillet radius, ρR, and spindle speed, N, on the ADROWT and SDOWT values are not considerable. In this section, therefore, the influences of θ, β and F, are mainly discussed. Figures 9c and d show that the ADROWT and SDOWT values sharply rise with the increase in the angle between passes or the decrease in the feed ratio. This trend is related to the degree of material deformation, as presented in Section 4.1.1. The suitable angle between passes and feed ratio are found to be 6° and 2.4 mm/r, respectively. In addition, the ADROWT and SDOWT values first increase and then slightly decrease as the roller installation angle is continuously increased. This is because a small roller installation angle can increase the workpiece radial strain, which is conducive for reducing wall thickness. A large roller installation angle induces a larger spinning force axial component, which enhances the wall thickness thinning effect. The appropriate roller installation angle is accordingly found to be approximately 20°.

Fig. 9
figure 9

Influences of spinning parameters on wall thickness in the second pass spinning process

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4.2.3 Influences of major spinning parameters on contour precision

Figure 10 displays the influences of spinning process parameters on the contour precision in the second pass. As shown in Fig. 10a, the influences of spinning parameters θ, β and ρR on the ARD value are relatively considerable; for the ARD range, the parameter values are 0.42, 0.2 and 0.15, respectively. It can be found, however, that the ARD value is not sensitive to the spindle speed, N, and feed ratio, F, whose values are relatively small. Accordingly, the influences of θ, β and ρR are mainly discussed in this section. In Fig. 10b, it is clearly observed that the ARD value first slightly decreases and then increases with the rise in the roller installation angle or roller fillet radius. This is because the small roller installation angle can increase the workpiece radial strain, which can improve the diameter expansion of the workpiece. A large roller installation angle enhances the axial material flow and easily causes a larger depth deviation; thereafter, the contour precision deteriorates. Moreover, a small roller fillet radius can reduce the contact area between the roller and workpiece, thus increasing the material axial flow and aggravating the depth deviation of the workpiece. The material axial flow is reduced when a larger roller fillet radius is used, resulting in the insufficient deformation of the material. The suitable roller installation angle and roller fillet radius are accordingly found to be 40° and 6 mm, respectively. The ARD value also gradually increases with the increase in the angle between passes. This is because a large angle between passes increases the wall thickness thinning degree, which deteriorates material flowability and forming precision. The suitable angle between passes is found to be approximately 6°.

Fig. 10
figure 10

Influences of spinning parameters on contour precision in the second pass spinning process

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4.3 Verification of optimized spinning parameters

In the orthogonal experimental scheme, which involves the comprehensive consideration of wall thickness precision and contour precision, the optimized values of spinning parameters α, β, F and ρR for the first pass spinning process are approximately 25°, 65°, 1.8 mm/r, and 3 mm, respectively. For the second pass, the optimized values of α, β, F and ρR are approximately 6°, 20°, 2.4 mm/r, and 6 mm, respectively. Moreover, the selected roller motion direction for the second pass spinning process is backwards. The spinning experiments on thin-walled ellipsoidal heads are performed to validate the reliability of optimized spinning parameters. Here, the spindle speed, N, is maintained at 300 r/min.

Figure 11 shows the spun thin-walled ellipsoidal heads formed by using unsuitable spinning parameters, such as a large forming angle (more than 55°) or feed ratio (more than 2.8 mm/r); some cracks and large ripples are evident. On the contrary, it can be observed that wrinkling is effectively avoided, and if the optimized spinning parameters are used, the surface quality of spun heads is ideal, as shown in Figs. 12a and b. Figure 12c compares the experimental and designed wall thickness distributions. It can be seen that the experimental wall thickness in each pass approximates the designed thickness. It is evident that the ADROWT values in both passes are less than 0.5%. The SDOWT values in the first and second spinning passes are 0.0014 mm and 0.0017 mm, respectively. In Fig. 12d, it is clearly observed that the experimental contour characteristics considerably coincide with the designed values. The DROFA value in the first pass spinning is less than 0.5%, and the ARD value in the second pass spinning is 0.805 3 mm. The optimized spinning parameters can therefore be effectively employed to fabricate thin-walled ellipsoidal heads.

Fig. 11
figure 11

Spun thin-walled ellipsoidal heads based on the unsuitable spinning parameters

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Fig. 12
figure 12

Experimental results based on the optimized spinning parameters

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5 Conclusions

In this work, a marginal-restraint mandrel-free spinning process for the fabrication of thin-walled ellipsoidal heads without wrinkles is proposed. Moreover, an FEM is established to study the influences of spinning parameters on the wall thickness and contour characteristics of spun parts. Some crucial conclusions are summarized as follows.

  1. (i)

    In the first pass spinning process, the wall thickness precision decreases with the increase in the forming angle or roller fillet radius, whereas it first increases and then decreases with the increase in feed ratio. The contour precision first increases and then decreases with the increase in the forming angle or feed ratio. The high roller installation angle can enhance the contour precision.

  2. (ii)

    In the second pass spinning process, the wall thickness precision decreases with the increase in the angle between passes or the decrease in feed ratio, whereas it first decreases and then increases with the increase in the roller installation angle. The contour precision first increases and then decreases with the increase in the roller installation angle or roller fillet radius. The increase in the angle between passes can decrease the contour precision.

  3. (iii)

    For the first pass spinning, the optimized forming angle, roller installation angle, feed ratio, and roller fillet radius are approximately 25°, 65°, 1.8 mm/r, and 3 mm, respectively. For the second pass spinning, the suitable roller motion direction is backwards, and the optimized angle between passes, roller installation angle, roller fillet radius, and feed ratio are approximately 6°, 20°, 6 mm, and 2.4 mm/r, respectively.