﻿ 整体壁板压弯成形的形状控制
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Shape control for press bend forming of integral panels
ZHANG Min, TIAN Xitian, LI Bo
School of Mechanical Engineering, Northwestern Polytechnical University, Xi'an 710072, China
Abstract: To solve the problems of low precision and uncontrollability caused by springback during the press bend forming of integral panels, this paper proposes a shape control method for bent parts. The specimen is designed with the aluminum alloy (7050-T7451) aircraft integral panel as the research object. In addition, the displacement prediction model of the local deformation is established based on the elastoplastic deformation theory and geometric analysis. A finite element simulation prediction model for the overall deformation of press bend forming is then developed, and the simulation results are compared with the test results. Furthermore, on the basis of the prediction model and considering the local-to-global deformation precision, this study constructs the contour curve iterative model based on the iterative compensation mechanism and the step by step approximation method. Compared with the traditional trial and error method, the proposed method can effectively control the shape of the bent parts with higher precision and faster convergence speed.
Keywords: integral panels    press bend forming    shape control    iterative compensation    finite element simulation

 图 1 压弯成形原理 Fig. 1 Press bend forming principle

1 整体壁板压弯成形形状控制方法 1.1 压弯成形整体变形控制原理

 图 2 迭代原理 Fig. 2 Iteration principle

 $\begin{array}{*{20}{c}} {{E^j} = {S^j} - S = \{ \mathit{\boldsymbol{e}}_i^j|\mathit{\boldsymbol{e}}_i^j \in {{\bf{R}}^2}\} }\\ {1 \le i \le m,1 \le j \le m} \end{array}$ （1）

 ${S^{j + 1}} = {S^j} + \alpha {E^j}\;\;\;{\kern 1pt} 1 \le j \le m$ （2）

1.2 压弯成形整体变形控制过程

 $\left\{ {\begin{array}{*{20}{l}} {{E_{{\rm{max}}}} = \mathop {{\rm{max}}}\limits_{0 < i \le m} \{ \Delta {l_i}\} \le \xi }\\ {\Delta e = \sqrt {\frac{1}{m}\sum\limits_{i = 1}^m \Delta l_i^2} \le \zeta } \end{array}} \right.$ （3）

 图 3 压弯成形形状控制过程 Fig. 3 Shape control process of press bend forming
2 整体壁板压弯成形局部-整体变形预测 2.1 整体壁板试件设计

 图 4 整体壁板试件 Fig. 4 Specimen of integral panel
2.2 材料模型

 图 5 7050-T7451铝合金应力-应变曲线 Fig. 5 Stress-strain curves of 7050-T7451 aluminum alloy

 力学性能 数值 密度/(g·cm-3) 2.73 拉伸强度/MPa 502 初始屈服应力σ0/MPa 442 弹性模量E/MPa 66 000 泊松比ν 0.33

 硬化模型 表达式 参数 Swift σ=K(ε0+εp)n ε0=0.006 27K=721.46n=0.11 Ludwik σ=σ0+Kε-pn K=778.8n=0.73 H-S σ=σsat－(σsat－σ0)·exp(－aε-pn) σsat=－3 986.01a=0.17n=0.72

 图 6 实验和不同硬化模型得到的单拉应力-应变曲线 Fig. 6 Uniaxial tensile stress-strain curves obtained from experiments and different hardening models
2.3 整体壁板压弯成形局部变形下压量解析预测

 图 7 下压量与成形半径 Fig. 7 Punch displacement and forming radius

 ${\rm{sin}}\frac{\theta }{2} = \frac{{L/2}}{{{\rho _{\rm{n}}} + t/2 + {R_{{\rm{sb}}}}}}$ （4）

 $\begin{array}{*{20}{l}} {H = ({\rho _{\rm{n}}} + t/2 + {R_{{\rm{sb}}}}){\rm{cos}}\frac{\theta }{2} - {R_{{\rm{sb}}}} = }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} ({\rho _{\rm{n}}} + t/2 + {R_{{\rm{sb}}}})\sqrt {1 - {{\left( {{\rm{sin}}\frac{\theta }{2}} \right)}^2}} - {R_{{\rm{sb}}}}} \end{array}$ （5）

 ${Y_{\rm{p}}} = {\rho _{\rm{n}}} + t/2 - H$ （6）

 $\begin{array}{*{20}{l}} {{Y_{\rm{p}}} = {\rho _{\rm{n}}} + t/2 - [({\rho _{\rm{n}}} + t/2 + {R_{{\rm{sb}}}}) \cdot }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sqrt {1 - {{\left( {\frac{{L/2}}{{{\rho _{\rm{n}}} + t/2 + {R_{{\rm{sb}}}}}}} \right)}^2}} - {R_{{\rm{sb}}}}]} \end{array}$ （7）

 $\frac{1}{{{\rho _{\rm{u}}}}} - \frac{1}{{{\rho _{\rm{n}}}}} = \frac{{M(1 - {\nu ^2})}}{{EI}}$ （8）

 ${\rho _{\rm{n}}} = \frac{{{\rho _{\rm{u}}}EI}}{{EI - {\rho _{\rm{u}}}M(1 - {\nu ^2})}}$ （9）

 $\begin{array}{*{20}{l}} {{Y_{\rm{p}}} = \frac{{{\rho _{\rm{u}}}EI}}{{EI - {\rho _{\rm{u}}}M(1 - {\nu ^2})}} + \frac{t}{2} - }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{1}{2} \cdot \sqrt {4{{\left( {\frac{{{\rho _{\rm{u}}}EI}}{{EI - {\rho _{\rm{u}}}M(1 - {\nu ^2})}} + \frac{t}{2} + {R_{{\rm{sb}}}}} \right)}^2} - {L^2} + {R_{{\rm{sb}}}}} } \end{array}$ （10）

 $M = \omega \int_{{\rho _{\rm{i}}}}^{{\rho _{\rm{o}}}} {{\sigma _\theta }} |\rho - {\rho _{\rm{n}}}|{\rm{d}}\rho$ （11）

1) 弹性区域(ρnye < ρ < ρn+ye)：

 $\begin{array}{*{20}{l}} {{\sigma _\theta } = \frac{E}{{1 - {\nu ^2}}} \cdot \frac{{\rho - {\rho _{\rm{n}}}}}{{{\rho _{\rm{n}}}}} + \frac{{\nu E}}{{(1 - {\nu ^2})(1 - 2\nu )}} \cdot \frac{{{y_{\rm{e}}}}}{{{\rho _{\rm{n}}}}} - }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{\nu }{{1 - 2\nu }} \cdot \frac{2}{{\sqrt 3 }}K{{\left( {{\varepsilon _0} + \frac{2}{{\sqrt 3 }}{\rm{ln}}\frac{{{\rho _{\rm{n}}} + {y_{\rm{e}}}}}{{{\rho _{\rm{n}}}}}} \right)}^n}} \end{array}$ （12）

2) 弯板内侧塑性区域(ρiρρnye)：

 $\begin{array}{*{20}{l}} {{\sigma _\theta } = \frac{2}{{\sqrt 3 }}K{{\left( {{\varepsilon _0} - \frac{2}{{\sqrt 3 }}{\rm{ln}}\frac{\rho }{{{\rho _{\rm{n}}}}}} \right)}^n}\left( {{\rm{ln}}\frac{{{\rho _{\rm{n}}}}}{\rho } - 1} \right) - }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{2}{{\sqrt 3 }}K{{\left( {{\varepsilon _0} - \frac{2}{{\sqrt 3 }}{\rm{ln}}\frac{{{\rho _{\rm{i}}}}}{{{\rho _{\rm{n}}}}}} \right)}^n}{\rm{ln}}\frac{{{\rho _{\rm{n}}}}}{{{\rho _{\rm{i}}}}}} \end{array}$ （13）

3) 弯板外侧塑性区域(ρn+yeρρo)：

 $\begin{array}{*{20}{l}} {{\sigma _\theta } = \frac{2}{{\sqrt 3 }}K{{\left( {{\varepsilon _0} + \frac{2}{{\sqrt 3 }}{\rm{ln}}\frac{\rho }{{{\rho _{\rm{n}}}}}} \right)}^n}\left( {1 + {\rm{ln}}\frac{\rho }{{{\rho _{\rm{n}}}}}} \right) - }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{2}{{\sqrt 3 }}K{{\left( {{\varepsilon _0} + \frac{2}{{\sqrt 3 }}{\rm{ln}}\frac{{{\rho _{\rm{o}}}}}{{{\rho _{\rm{n}}}}}} \right)}^n}{\rm{ln}}\frac{{{\rho _{\rm{o}}}}}{{{\rho _{\rm{n}}}}}} \end{array}$ （14）

 图 8 局部变形下压量计算流程 Fig. 8 Flowchart of punch displacement calculation of local deformation
2.4 整体壁板压弯成形整体变形有限元仿真预测

 图 9 有限元仿真模型 Fig. 9 Finite element simulation model

 图 10 压弯成形实验 Fig. 10 Bending forming experiments

 图 11 有限元模拟与压弯实验对比 Fig. 11 Comparison between finite element simulation and bending experiment
3 结果分析

 图 12 目标形状 Fig. 12 Target shape

 图 13 基于本文方法的成形曲线迭代过程 Fig. 13 Iterative process of contour curves based on the proposed method

 图 14 基于试错法的成形曲线迭代过程 Fig. 14 Iterative process of contour curves based on trial and error method
 图 15 整体变形误差随迭代的变化 Fig. 15 Evolution of global deformation error with iterations
4 结论

1) 采用理论解析和有限元模拟相结合的方法，对整体壁板压弯成形局部-整体变形进行预测；利用迭代补偿机制与逐步逼近思想，提出了整体壁板压弯成形的形状控制方法。

2) 以设计的机翼整体壁板实验样件为例，对所提方法进行了验证。经过一次迭代后，变形件整体变形误差由0.298 mm降低至0.004 7 mm，压弯成形形状整体变形精度提高了98.4%。

3) 通过与传统的试错法进行对比，本文所提方法能够以更高的精度、更快的收敛速度有效控制压弯件的成形形状。

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http://dx.doi.org/10.7527/S1000-6893.2020.23620

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#### 文章信息

ZHANG Min, TIAN Xitian, LI Bo

Shape control for press bend forming of integral panels

Acta Aeronautica et Astronautica Sinica, 2020, 41(7): 623620.
http://dx.doi.org/10.7527/S1000-6893.2020.23620