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Frontiers of Mechanical Engineering

Front. Mech. Eng.    2018, Vol. 13 Issue (3) : 376-384     https://doi.org/10.1007/s11465-018-0500-3
RESEARCH ARTICLE |
Robust optimization of the billet for isothermal local loading transitional region of a Ti-alloy rib-web component based on dual-response surface method
Ke WEI, Xiaoguang FAN(), Mei ZHAN, Miao MENG
State Key Laboratory of Solidification Processing, School of Materials Science and Engineering, Northwestern Polytechnical University, Xi’an 710071, China
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Abstract

Billet optimization can greatly improve the forming quality of the transitional region in the isothermal local loading forming (ILLF) of large-scale Ti-alloy rib-web components. However, the final quality of the transitional region may be deteriorated by uncontrollable factors, such as the manufacturing tolerance of the preforming billet, fluctuation of the stroke length, and friction factor. Thus, a dual-response surface method (RSM)-based robust optimization of the billet was proposed to address the uncontrollable factors in transitional region of the ILLF. Given that the die underfilling and folding defect are two key factors that influence the forming quality of the transitional region, minimizing the mean and standard deviation of the die underfilling rate and avoiding folding defect were defined as the objective function and constraint condition in robust optimization. Then, the cross array design was constructed, a dual-RSM model was established for the mean and standard deviation of the die underfilling rate by considering the size parameters of the billet and uncontrollable factors. Subsequently, an optimum solution was derived to achieve the robust optimization of the billet. A case study on robust optimization was conducted. Good results were attained for improving the die filling and avoiding folding defect, suggesting that the robust optimization of the billet in the transitional region of the ILLF was efficient and reliable.

Keywords isothermal local loading forming      rib-web component      transitional region      robust optimization      dual response surface method     
Corresponding Authors: Xiaoguang FAN   
Just Accepted Date: 15 January 2018   Online First Date: 08 March 2018    Issue Date: 11 June 2018
 Cite this article:   
Ke WEI,Xiaoguang FAN,Mei ZHAN, et al. Robust optimization of the billet for isothermal local loading transitional region of a Ti-alloy rib-web component based on dual-response surface method[J]. Front. Mech. Eng., 2018, 13(3): 376-384.
 URL:  
http://journal.hep.com.cn/fme/EN/10.1007/s11465-018-0500-3
http://journal.hep.com.cn/fme/EN/Y2018/V13/I3/376
Fig.1  Illustration of the local loading forming
Fig.2  Folding defect and die underfilling of the transitional region in the ILLF of the LTRC [9]
Fig.3  Illustration of the UTB and size parameters
Control factors Uncontrolled factors
Outer array Inner array
Z1 1 1 3
Z2 1 2 3
Z3 1 2 2
Z4 1 2 1
x1 x2 x3 x4
1 1 2 3 f1,1 f1,2 f1,j
1 2 3 2 f2,1 f2,2 f2,j
1 3 1 1 f3,1 f3,2 f3,j
1 2 2 2 fi,1 fi,2 fi,j
Tab.1  Cross array with control and uncontrolled factors
Fig.4  Flowchart of the robust optimization of the billet in the transitional region during the ILLF of LTRC
Fig.5  Eigenstructure of the transitional region
Feature Parameter Value
Rib 1 Width, W1/mm 15
Height, H1/mm 44
Rib 2 Width, W2/mm 13
Height, H2/mm 43
Rib 3 Width, W3/mm 12
Height, H3/mm 45
Rib 4 Width, W4/mm 12
Height, H4/mm 44
Rib 5 Width, W5/mm 16
Height, H5/mm 45
Distance to the left side of Rib 1 D01/mm 35
Distance between Ribs 1 and 2 D12/mm 60
Distance between Ribs 2 and 3 D23/mm 75
Distance to the right side of Rib 3 D30/mm 30
Thickness of Web 1 Tweb1/mm 13
Thickness of Web 2 Tweb2/mm 13
Thickness of Web 3 Tweb3/mm 14
Thickness of Web 4 Tweb4/mm 14
Thickness of Web 5 Tweb5/mm 12
Thickness of Web 6 Tweb6/mm 12
All fillet radii R/mm 5
All drafts γ/(° ) 2
Width of the eigenstructure Weigen/mm 100
Volume of the eigenstructure Veigen/mm3 5.63×105
Tab.2  Geometric parameters of the eigenstructure
Fig.6  FE model in the transitional region of ILLF [9]. (a) First-loading step; (b) second-loading step
Factor type Variables Minimum values Maximum values
Control factors a: H1/He 0.75 1.15
b: H3/He 0.75 1.15
c: lleft/Lleft 0.60 1.00
d: lright/Lright 0.60 1.00
Uncontrolled factors MFerror −0.1 mm 0.1 mm
m 0.3 0.5
Lerror 13.9 mm 14 mm
Tab.3  Ranges of the control and uncontrolled factors
BBD NO. Control factors Uncontrolled factors Responses
a b c d UD 1 2 3 4 5 fiμ fiσ
MFerror –0.10 –0.05 0.00 0.05 0.10
m 0.35 0.45 0.30 0.40 0.50
Lerror 13.975 13.950 13.925 13.900 14.000
1 1.15 0.95 0.8 0.6 1.29 1.32 1.49 1.30 1.58 1.40 0.13
2 0.95 1.15 0.8 0.6 4.30 5.24 4.00 5.06 5.32 4.78 0.59
3 0.75 0.95 0.6 0.8 4.88 4.93 4.07 4.82 5.07 4.75 0.39
4 1.15 1.15 0.8 0.8 1.84 2.58 1.30 2.60 2.34 2.13 0.56
5 0.95 0.95 1.0 1.0 1.23 1.39 1.12 1.12 1.62 1.29 0.21
6 0.95 0.95 0.8 0.8 1.43 1.48 1.27 1.30 1.65 1.43 0.15
7 1.15 0.95 1.0 0.8 1.22 1.13 1.29 1.04 0.92 1.12 0.15
8 0.95 0.95 0.6 0.6 1.73 1.35 1.55 1.52 1.59 1.55 0.14
9 0.95 0.95 0.6 1.0 1.51 1.28 1.52 1.54 1.62 1.49 0.13
10 0.75 0.95 1.0 0.8 2.20 2.58 2.08 3.25 3.79 2.78 0.73
11 0.95 1.15 0.6 0.8 3.95 4.82 3.46 4.29 4.77 4.26 0.57
12 0.95 0.75 0.6 0.8 1.35 1.80 1.35 1.42 1.24 1.43 0.21
13 0.75 0.95 0.8 0.6 2.91 3.46 2.69 3.69 3.85 3.32 0.50
14 0.75 0.95 0.8 1.0 3.48 3.96 3.23 4.23 4.41 3.86 0.50
15 0.95 0.95 0.8 0.8 1.59 1.25 1.37 1.30 1.63 1.43 0.17
16 0.95 0.75 0.8 1.0 1.35 1.49 1.10 1.53 1.05 1.30 0.22
17 0.95 0.95 0.8 0.8 1.56 1.60 1.32 1.32 1.59 1.48 0.15
18 1.15 0.95 0.8 1.0 1.23 1.26 1.35 1.34 1.52 1.34 0.11
19 1.15 0.75 0.8 0.8 2.76 2.81 2.63 2.59 2.79 2.72 0.10
20 0.95 0.75 1.0 0.8 1.21 1.14 1.03 1.35 0.57 1.06 0.30
21 0.95 0.75 0.8 0.6 1.35 1.54 1.24 1.46 1.01 1.32 0.21
22 0.75 0.75 0.8 0.8 1.96 1.97 0.97 2.81 2.42 2.03 0.69
23 0.95 1.15 1.0 0.8 4.14 5.11 3.15 4.93 4.66 4.40 0.79
24 0.95 0.95 0.8 0.8 1.45 1.21 1.42 1.36 1.61 1.41 0.15
25 0.95 0.95 0.8 0.8 1.46 1.23 1.40 1.32 1.63 1.41 0.15
26 0.95 1.15 0.8 1.0 3.92 4.60 3.34 4.42 4.63 4.18 0.55
27 1.15 0.95 0.6 0.8 1.67 1.73 1.76 1.89 1.91 1.79 0.11
28 0.75 1.15 0.8 0.8 7.05 8.20 6.42 7.59 7.99 7.45 0.73
29 0.95 0.95 1.0 0.6 1.03 1.20 1.21 1.37 0.87 1.14 0.19
Tab.4  Cross array design
Optimization type a b c d fiμ fiσ
Robust optimization 1.12 0.93 0.65 1 1.23 0.09
Deterministic optimization 1.13 0.96 1.00 1 1.26 0.15
Tab.5  Comparison between robust optimization and deterministic optimization
Fig.7  Contour graphs demonstrating the effects of Factors a and b on Mt. (a) Factor c=0.65, Factor d=1; (b) Factor c=1, Factor d=1
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