Springback Analysis and Die Optimization Design of Ellipsoid Rings with Large Diameter-thickness Ratio during Nosing Processes

doi:10.3969/j.issn.1004-132X.2026.06.022

Abstract

Abstract:

To improve the nosing forming accuracy of the ellipsoid rings with large diameter-thickness ratio， the springback deformation and the die optimization design were studied. The springback distribution of the nosed rings was analyzed by the finite element model. The influences of forming parameters on the springback of the nosed rings were investigated by the orthogonal design. Based on the displacement adjustment method， the springback compensation of the nosed rings was studied. The nosing die with reinforced ribs was proposed based on the equal-thickness bowl structure. The sensitive structure parameters were obtained by Plackett-Burman Design screening method. The response surface model was established by Box-Behnken Design response surface method， and the optimized structural parameters were obtained by the automatic optimization algorithm. The results show that the springback of the nosed rings at the mouth region is obvious， and the springback in the middle and bottom regions is small. The maximum springback of the nosed rings with the diameter-thickness ratio of 127.66 is 0.92 mm. The degree of influences of forming parameters on the springback of ellipsoid rings was ranked as follows： nosing die ellipticity，rings diameter-thickness ratio，friction coefficient，nosing speed. The maximum springback of the nosed rings is reduced from 0.92 mm to 0.05 mm after two springback compensations. The peak stress of the nosing die is reduced by about 27.5% by adjusting the structure layout and optimizing the structure parameters， resulting in effectively reducing the stress concentration. Finally， the verification of the nosing process was conducted for the ring part with large diameter-thickness ratio. The nosed ellipsoid rings exhibite a smooth surface without wrinkling， and the errors of the mouth diameter are much less than 5%. The maximum deviations between different contours are less than 0.25 mm， which clearly demonstrates the feasibility of the design method of the nosing dies.

Key words: ellipsoid ring with large diameter-thickness ratio, nosing process, springback analysis, die optimization, response surface method

摘要：

为提高大径厚比椭球环件收口成形精度，对收口件回弹变形及模具优化进行了研究。利用有限元模型分析了收口件回弹分布规律，通过正交试验法分析了成形参数对收口件回弹的影响，采用位移调整法进行了回弹补偿。提出基于等厚碗状结构的带加强肋收口模具设计方案，利用Plackett-Burman Design筛选试验设计方法筛选出敏感结构参数；通过Box-Behnken Design响应面方法建立响应面模型，并结合自动寻优算法获得优化后模具结构参数。结果表明：椭球收口件口部区域回弹明显，中部及底部区域回弹较小；径厚比为127.66时收口件最大回弹量为0.92 mm。成形参数对收口环件回弹的影响显著程度大小排序为：模具长短轴比，环件径厚比，摩擦因数，冲压速度。采用位移调整法进行2次回弹补偿，最大回弹量从0.92 mm降低至0.05 mm。通过结构模式调整与响应面优化后收口模具峰值应力减小约27.5%，有效减少了模具应力集中。最后，针对外径638 mm、厚度5 mm、高度143 mm的环件进行了收口成形验证，其收口件表面光滑，无起皱等缺陷，且口部直径误差远小于5%，成形及模拟轮廓与理论轮廓的最大偏差小于0.25 mm，说明了该收口成形模具设计及优化方法的可行性。

关键词: 大径厚比椭球环件, 收口成形, 回弹分析, 模具优化, 响应面法

CLC Number:

TG381

OUYANG Yawen, ZHAO Yixi, LI Xiaokai, HU Lan. Springback Analysis and Die Optimization Design of Ellipsoid Rings with Large Diameter-thickness Ratio during Nosing Processes[J]. China Mechanical Engineering, 2026, 37(6): 1497-1507.

欧阳亚文, 赵亦希, 李晓凯, 胡蓝. 大径厚比椭球收口环件回弹分析与模具优化研究[J]. 中国机械工程, 2026, 37(6): 1497-1507.

Figures/Tables 30

Fig.1 Schematic diagram of the ellipsoid nosed ring with large diameter-thickness ratio

Fig.2 Uniaxial tensile stress-strain curve of Al 2219-T351

Fig.3 Finite element model of the nosing process for the ring part with large diameter-thickness ratio

Fig.4 Simulation result of the ellipsoid ring with large diameter-thickness ratio during the nosing process

Fig.5 Springback results of the ellipsoid nosed ring with large diameter-thickness ratio

Tab.1 Factors and levels in orthogonal test

水平	因素
	A	B	C	D
	模具长短轴比	摩擦因数	收口速度/ （mm∙s $- 1$ ）	环件径厚比
1	1.5	0.05	25	95.76
2	1.6	0.10	50	109.42
3	1.7	0.15	100	127.66
4	1.8	0.20	200	153.20

Tab.1 Factors and levels in orthogonal test

水平	因素
	A	B	C	D
	模具长短轴比	摩擦因数	收口速度/ （mm∙s $- 1$ ）	环件径厚比
1	1.5	0.05	25	95.76
2	1.6	0.10	50	109.42
3	1.7	0.15	100	127.66
4	1.8	0.20	200	153.20

Tab.2 Orthogonal test results

试验序号	因素				最大回弹变形量/mm
	A	B	C	D
	模具长短轴比	摩擦因数	收口速度/（mm∙s $- 1$ ）	环件径厚比
1	1.5	0.05	25	153.20	0.9040
2	1.5	0.1	50	127.66	0.9002
3	1.5	0.15	100	109.42	0.9066
4	1.5	0.2	200	95.76	0.9154
5	1.6	0.05	50	109.42	0.9155
6	1.6	0.1	25	95.76	0.9250
7	1.6	0.15	200	153.20	0.9186
8	1.6	0.2	100	127.66	0.9181
9	1.7	0.05	100	95.76	0.9369
10	1.7	0.1	200	109.42	0.9310
11	1.7	0.15	25	127.66	0.9300
12	1.7	0.2	50	153.20	0.9368
13	1.8	0.05	200	127.66	0.9374
14	1.8	0.1	100	153.20	0.9431
15	1.8	0.15	50	95.76	0.9549
16	1.8	0.2	25	109.42	0.9515

Tab.2 Orthogonal test results

试验序号	因素				最大回弹变形量/mm
	A	B	C	D
	模具长短轴比	摩擦因数	收口速度/（mm∙s $- 1$ ）	环件径厚比
1	1.5	0.05	25	153.20	0.9040
2	1.5	0.1	50	127.66	0.9002
3	1.5	0.15	100	109.42	0.9066
4	1.5	0.2	200	95.76	0.9154
5	1.6	0.05	50	109.42	0.9155
6	1.6	0.1	25	95.76	0.9250
7	1.6	0.15	200	153.20	0.9186
8	1.6	0.2	100	127.66	0.9181
9	1.7	0.05	100	95.76	0.9369
10	1.7	0.1	200	109.42	0.9310
11	1.7	0.15	25	127.66	0.9300
12	1.7	0.2	50	153.20	0.9368
13	1.8	0.05	200	127.66	0.9374
14	1.8	0.1	100	153.20	0.9431
15	1.8	0.15	50	95.76	0.9549
16	1.8	0.2	25	109.42	0.9515

Tab.3 Analysis of springback indicators results mm

参数	因素
	A	B	C	D
	模具长短轴比	摩擦因数	收口速度/（mm∙s $- 1$ ）	环件径厚比
K₁	3.6262	3.6937	3.7105	3.7024
K₂	3.6771	3.6993	3.7074	3.6857
K₃	3.7347	3.7101	3.7047	3.7046
K₄	3.7870	3.7218	3.7024	3.7322
k₁	0.9066	0.9234	0.9276	0.9256
k₂	0.9193	0.9248	0.9269	0.9214
k₃	0.9337	0.9275	0.9262	0.9262
k₄	0.9468	0.9305	0.9256	0.9331
R	0.0402	0.0070	0.0020	0.0116
显著性顺序	A（模具长短轴比）>D（环件径厚比）>B（摩擦因数）>C（收口速度）

Tab.3 Analysis of springback indicators results mm

参数	因素
	A	B	C	D
	模具长短轴比	摩擦因数	收口速度/（mm∙s $- 1$ ）	环件径厚比
K₁	3.6262	3.6937	3.7105	3.7024
K₂	3.6771	3.6993	3.7074	3.6857
K₃	3.7347	3.7101	3.7047	3.7046
K₄	3.7870	3.7218	3.7024	3.7322
k₁	0.9066	0.9234	0.9276	0.9256
k₂	0.9193	0.9248	0.9269	0.9214
k₃	0.9337	0.9275	0.9262	0.9262
k₄	0.9468	0.9305	0.9256	0.9331
R	0.0402	0.0070	0.0020	0.0116
显著性顺序	A（模具长短轴比）>D（环件径厚比）>B（摩擦因数）>C（收口速度）

Fig.6 Springback compensation process based on the displacement adjustment method

Fig.7 Contour comparison of the ellipsoid nosed ring with large diameter-thickness ratio after two springback compensations

Fig.8 Stress distribution of cylindrical nosing die during the nosing forming process

Fig.9 Nosing die structure and the layout of reinforced rib

Fig.10 Equivalent stress distribution of nosing die under different layout types

Fig.11 Schematic diagram of structural parameters of the nosing die with reinforced ribs

Tab.4 PBD test parameter level and coding

参数

顶端

厚度h/mm

模具

壁厚

t/mm

圆角

半径

R₁/mm

圆角

半径

R₂/mm

倾斜角α/（°）

底端

宽度

L₂/mm

Tab.5 PBD test combination and peak stress response

组合	顶端厚度 h/mm	模具壁厚 t/mm	圆角半径 R₁/mm	圆角半径 R₂/mm	倾斜角 α/（°）	底端宽度 L₂/mm	应力峰值σ_ND/MPa
1	5.50	9	10.5	3	4	15.0	195.7
2	8.25	6	10.5	3	4	22.5	232.5
3	5.50	6	7.0	2	4	15.0	266.9
4	5.50	9	10.5	2	6	22.5	199.4
5	8.25	6	7.0	2	6	15.0	253.8
6	8.25	6	10.5	3	6	15.0	232.0
7	5.50	6	10.5	2	6	22.5	265.3
8	8.25	9	7.0	2	4	22.5	208.2
9	8.25	9	7.0	3	6	22.5	218.1
10	8.25	9	10.5	2	4	15.0	186.1
11	5.50	9	7.0	3	6	15.0	224.4
12	5.50	6	7.0	3	4	22.5	294.7

Fig.12 Sorting chart of die structural parameters

Tab.6 Regression analysis results of nosing die parameters

因素	相关参数
因素	平方和/ $103$	自由度	均方根/ $103$	F 值	P 值	显著性
本文模型	11.625	6	1.937	23.71	0.0016	显著
A	1.115	1	1.115	13.65	0.0141	显著
B	8.179	1	8.179	100.07	0.0002	显著
C	2.004	1	2.004	24.53	0.0043	显著
D	2.611	1	2.611	0.3194	0.5964
E	0.006	1	0.006	0.0808	0.7877
F	0.293	1	0.293	3.59	0.1168

Tab.6 Regression analysis results of nosing die parameters

因素	相关参数
因素	平方和/ $103$	自由度	均方根/ $103$	F 值	P 值	显著性
本文模型	11.625	6	1.937	23.71	0.0016	显著
A	1.115	1	1.115	13.65	0.0141	显著
B	8.179	1	8.179	100.07	0.0002	显著
C	2.004	1	2.004	24.53	0.0043	显著
D	2.611	1	2.611	0.3194	0.5964
E	0.006	1	0.006	0.0808	0.7877
F	0.293	1	0.293	3.59	0.1168

Tab.7 BBD parameter level of response surface test

参数	参数水平
参数	低	中	高
模具壁厚t/mm	6.0	8.0	10.0
圆角半径R₁/mm	6.5	8.0	9.5
顶端厚度h/mm	6.0	7.5	9.0

Tab.8 BBD test combination and peak stress response

组合	模具壁厚 t/mm	圆角半径R₁/mm	顶端厚度h/mm	峰值应力 σ_ND/MPa
1	6	8.0	6.0	259.4
2	8	8.0	7.5	204.6
3	8	8.0	7.5	204.2
4	8	6.5	6.0	233.9
5	10	8.0	9.0	170.8
6	6	9.5	7.5	237.8
7	10	6.5	7.5	204.5
8	8	8.0	7.5	204.6
9	6	8.0	9.0	237.7
10	6	6.5	7.5	260.5
11	8	9.5	6.0	225.1
12	8	6.5	9.0	212.3
13	10	9.5	7.5	179.5
14	8	9.5	9.0	191.5
15	10	8.0	6.0	208.2

Fig.13 Comparison between predicted value and test value

Tab.9 Regression analysis of response surface model

响应目标	R²	R $a d j 2$	P 值	显著性
峰值应力	0.9953	0.9868	<0.0001	显著

Tab.9 Regression analysis of response surface model

响应目标	R²	R $a d j 2$	P 值	显著性
峰值应力	0.9953	0.9868	<0.0001	显著

Fig.14 Response surface diagram and contour map of t and R1 to the die peak stress

Fig.15 Response surface diagram and contour map of t and h to the die peak stress

Fig.16 Response surface diagram and contour map of R1 and h to the die peak stress

Tab.10 Structural parameters of the nosing die after the response surface optimization

结构参数	模具壁厚t	圆角半径R₁	顶端厚度h
数值	9.9	9.4	8.8

Fig.17 Stress distribution of nosing die under initial structure and optimized structure

Fig.18 The nosing forming experiment device for the ring parts with large diameter-to-thickness ratio

Fig.19 Forming results and measurement diagram of the ellipsoid nosed rings with large diameter-thickness ratio

Fig.20 Comparison results of the contour curves for the ellipsoid nosed rings with large diameter-thickness ratio

References 21

[1]	张卫红，唐长红. 航空航天装备的轻量化：挑战与未来［J］. 航空学报， 2024， 45（5）：529965.
	ZHANG Weihong， TANG Changhong. Lightweighting of Aerospace and Aeronautical Equipment： Challenges and perspectives［J］. Acta Aeronautica of Astronautica Sinica， 2024， 45（5）：529965.
[2]	李雨，王建刚，杨菊鹏，等. 火箭贮箱结构材料应用及发展现状［J］. 宇航材料工艺， 2024， 54（2）：29-42.
	LI Yu， WANG Jiangang， YANG Jupeng， et al. Application and Development Status of Rocket Tank Structural Materials［J］. Aerospace Materials & Technology， 2024， 54（2）：29-42.
[3]	刘观日，吴迪，姚重阳，等. 航天运载器结构先进材料及工艺技术应用与发展展望［J］. 宇航材料工艺， 2021， 51（4）：1-9.
	LIU Guanri， WU Di， YAO Chongyang， et al. Application and Development of Advanced Materials and Processing Technology in Aerospace Vehicle Structure［J］. Aerospace Materials & Technology， 2021， 51（4）：1-9.
[4]	何巍，牟宇，朱海洋，等. 下一代主力运载火箭发展思考［J］. 宇航总体技术， 2023， 7（2）：1-12.
	HE Wei， MOU Yu， ZHU Haiyang， et al. Reflections on the Development of Next Generation Main Launch Vehicle［J］.Astronautical Systems Engineering Technology， 2023， 7（2）：1-12.
[5]	张志明，吴歆，符韵，等. 热收口成形对模具寿命影响的研究［J］. 锻压技术， 2007（1）：59-62.
	ZHANG Zhiming， WU Xin， Fu Yun， et al. Research on the Die Life under the Influence of Hot Shell Nosing［J］. Forging & Stamping Technology， 2007（1）：59-62.
[6]	陈露，陈一进，丁福建，等. 基于Deform-3D的某弹壳收口模具结构优化设计［J］. 自动化应用， 2021，（12）：143-146.
	CHEN Lu， CHEN Yijin， DING Fujian， et al. Structure Optimization Design of a Shell Closing Die Based on Deform-3D［J］. Automation Application， 2021，（12）： 143-146.
[7]	吴昂，吴莹，李国俊，等. 基于响应曲面法的大型锥形件缩口成形工艺设计及多几何参数优化［J］. 机械工程学报， 2019， 55（24）：83-92.
	WU Ang， WU Ying， LI Guojun， et al. Process Design and Multi-geometric Parameter Optimization of Large Cones Based on Response Surface Methodology［J］. Journal of Mechanical Engineering， 2019， 55（24）：83-92.
[8]	李碧聪，李国俊，张治民，等. 基于响应面法的大型锥筒件内凸缘缩口成形工艺参数优化［J］. 精密成形工程， 2023， 15（4）：197-204.
	LI Bicong， LI Guojun， ZHANG Zhimin， et al. Optimization of Necking Forming Process Parameters for Inner Flange of Large Conical Cylinder Part Based on Response Surface Method［J］. Journal of Netshape Forming Engineering， 2023， 15（4）：197-204.
[9]	TAKASHI K， YASUHIKO K， MAKOTO M， et al. Tool Design for Suppressing Fracture at the Tip of Tube after Nosing with Die［C］∥Ⅷ International Conference on Computational Plasticity. Barcelona， 2005：1-4.
[10]	KWAN C T. An Analysis of the Eccentric Nosing Process of Metal Tubes［J］. Journal of Materials Processing Technology， 2003， 140（1/2/3）： 530-534.
[11]	WANG Xiaoliang， CUI Yunxian， YUAN Tiebing. Nosing Process of 2A12 Aluminum Alloy Spacing Shims for Non-disengagement Fastening Devices for Spacecrafts［J］. Metals， 2022， 12（8）： 1348.
[12]	闫华军，邢博，张双杰，等. 基于Dynaform的前防撞梁回弹分析及模具补偿研究［J］. 塑性工程学报， 2023， 30（8）：35-41．
	YAN Huajun， XING Bo， ZHANG Shuangjie， et al. Study on Springback Analysis and Die Compensation of Front Anti-collision Beam Based on Dynaform［J］. Journal of Plasticity Engineering， 2023， 30（8）：35-41．
[13]	刘飞，范昇，廖宏谊，等. 基于Dynaform的整体式波形片成形回弹研究及其模具参数优化［J］. 锻压技术， 2018， 43（9）：29-36.
	LIU Fei， FAN Sheng， LIAO Hongyi， et al. Study on Springback and Optimization on Die Structure Parameters for Integral Waveform Piece Based on Dynaform［J］. Forging & Stamping Technology， 2018， 43（9）：29-36.
[14]	丁华锋，潘俊杰，晏洋，等. DP780高强钢U形弯曲回弹影响因素及优化［J］. 精密成形工程， 2022， 14（10）： 49-55.
	DING Huafeng， PAN Junjie， YAN yang， et al. Influence Factors and Optimization of U-shaped Bending Springback of DP780 High-strength Steel［J］. Journal of Netshape Forming Engineering， 2022， 14（10）： 49-55.
[15]	王涌纲，何仕荣，顾猛，等. 金属双极板冲压成形数值分析及回弹补偿［J］. 塑性工程学报， 2024， 31（2）：43-50.
	WANG Yonggang， HE Shirong， GU Meng， et al. Numerical Analysis and Springback Compensation of Metal Bipolar Plate Stamping Forming［J］. Journal of Plasticity Engineering， 2024， 31（2）：43-50.
[16]	熊文韬，谢三山，黄兆飞，等. 基于神经网络遗传算法函数寻优与回弹补偿技术的某型汽车前幅拉延成形优化［J］. 塑性工程学报， 2020， 27（6）：38-45.
	XIONG Wentao， XIE Sanshan， HUANG Zhaofei， et al. Optimization of Drawing Forming for Front Panel of an Automobile Based on Neural Network Genetic Algorithm Function Optimization and SCP Technology［J］. Journal of Plasticity Engineering， 2020， 27（6）：38-45.
[17]	张凌云，弥晓东，黄杰，等. 一种橡皮囊成形弯边零件的回弹补偿方法［J］. 锻压技术， 2021， 46（8）：49-54.
	ZHANG Lingyun， MI Xiaodong， HUANG Jie， et al. A Springback Compensation Method for Rubber Bladder Forming of Flanged Parts［J］. Forging & Stamping Technology， 2021， 46（8）：49-54.
[18]	韩飞.金属板材数控渐进成形回弹机理与补偿方法研究［D］.武汉：华中科技大学，2008
	HAN Fei. Research on Springback Mechanism and Compensation Method for NC Incremental Sheet Metal Forming［D］.Wuhan： Huazhong University of Science and Technology， 2008.
[19]	曾路平.汽车外板件的回弹及其补偿研究［D］.长沙：湖南大学，2021
	ZENG Luping. Research on Springback and Compensation of Automobile Outer Panels［D］. Changsha： Hunan University， 2021.
[20]	宋家祺，王彤. 便携式呼吸装置高速离心风机的设计与研究［J］. 机械设计与研究， 2023， 39（3）：227-231.
	SONG Jiaqi， WANG Tong. Development and Experimental Research of Small High-speed Centrifugal Fan Based on Portable Respiratory Device［J］. Machine Design and Research， 2023， 39（3）：227-231.
[21]	王良文，刘飞飞，汪曙光，等. 大腔体六面顶金刚石压机铰链梁锻造模具结构多参数优化［J］. 塑性工程学报， 2024， 31（5）：94-105.
	WANG Liangwen， LIU Feifei， WANG Shuguang， et al. Multi-parameter Optimization of Forging Die Structure for Hinge Beam of Large Cavity Hexahedral Top Diamond Press［J］. Journal of Plasticity Engineering， 2024， 31（5）：94-105.