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Control System Design for a Semi-Finished Product Considering Over- and Underbending^{ †}

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

#### 1.1. Freeform Bending State of the Art

#### 1.2. Goals and Assumptions

## 2. Materials and Methods

#### 2.1. Mathematical Model for Curvature in Non-Tangential Bending

^{−1}). It can be inferred that the curvature of the tube undergoes some sort of overshooting upon starting or ending the bending process. One explanation for that behavior is that the circularity of the tube profile can no longer be perfectly maintained in these transition regions, i.e., the tube profile gets distorted and takes, to some extent, an elliptical shape. This profile distortion results in some ripples on the surface of the tube during these transition regions. However, in this work, this dynamic-like behavior is neglected, and only the static behavior was taken into consideration. Therefore, the average of the resulting curvature along the tube is taken further into consideration. The same applies to the other different combinations of y and α. Figure 3 shows a 3D representation of the average curvature against the y and α combinations depicted in Table 1.

**A**in this case, is a matrix composed of the vectors

**y**,

**α**,

**y**,

^{2}**α**that maps between the parameter vector

^{2}**P**with the output vector

**κ**. The vectors

**y**,

**α**, and

**κ**were measured based on the simulation results of the FEM. Since matrix

**A**was a non-square matrix, the pseudo inverse was calculated in order to identify the parameter vector

**P**for each approach (see Equation (8)). Figure 4 shows the identified parameters as well as a comparison between the first approach and the quadratic approach.

^{−3}). Using the quadratic approach, very good compliance between the FEM results and the results of the static model was reached. In order to check whether adding more terms significantly improved the deviation norm, a fifth mixed term was introduced to the quadratic approach. However, this extra term did not significantly improve the error. Therefore, the quadratic approach, Equation (6), was accepted as a mathematical representation of the curvature in conjunction with non-tangential bending.

#### 2.2. Mathematical Model for Residual Stresses in Non-Tangential Bending

## 3. Developing a Closed-Loop Control System

_{act}, as well as the actual curvature κ

_{act}, are to be compared with their respective desired residual stresses σ

_{des}and curvature κ

_{des}. The deviation is then introduced to a PI-Controller that, in turn, produces a correction signal to be augmented on the feedforward signal. The result is then introduced to a mapping block, which, in turn, translates these signals into the respective bending die translation and orientation.

#### 3.1. Deriving the Mathematical Equations for the Mapping Block

_{1,2}in Equation (11), the corresponding bending orientation α can be calculated. In order to determine which value for y and α should be accepted, the boundary conditions should be taken into consideration. Hereby the case where y = 0 and α = 0 is considered. In this case, the curvature and residual stresses are expected to be zero for each. This is also verified using Equations (9) and (10). In order to check whether this condition is not violated, we substitute with σ = 0 MPa and κ = 0 mm

^{−1}in the parameters of the quadratic equation (p and q). In that case, the parameters will be $p=3.15$ and $q=0$; i.e.,${y}_{1,2}=-\left(3.15/2\right)\pm \sqrt{{\left(3.15/2\right)}^{2}-0}$. This shows that the boundary condition $\left(y=0\forall \kappa =0{\mathrm{mm}}^{-1},\sigma =0\mathrm{MPa}\right)$ can only be achieved when:

_{1}= 0 mm and σ = 0 MPa, the bending die orientation α is also 0°. Equations (13) and (14) are implemented inside the mapping block depicted in Figure 7. In the following section, the model earlier introduced is simulated.

#### 3.2. Simulation Results

## 4. Conclusions and Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Tangential bending, Over- and Underbending based on foundations of [2].

**Figure 4.**(

**a**) First approach motivated out of the kinematic model in Equation (5), (

**b**) Quadratic approach, Equation (6).

**Figure 5.**Resulting residual stresses σ out of the combinations depicted in the last row of Table 1.

**Figure 8.**(

**a**) Trajectories for desired curvature (in blue) actual curvature without controller (in red) and with a PI-controller (in yellow). (

**b**) Trajectories for desired residual stresses coinciding with the actual residual stresses without controller (in red). Actual residual stresses with a PI-controller (in yellow).

y (mm) | α (°) | ||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

5 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |

6 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 |

7 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 |

8 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 |

9 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 |

10 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 |

11 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 |

12 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 |

13 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 |

14 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 |

15 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 |

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## Share and Cite

**MDPI and ACS Style**

Ismail, A.; Maier, D.; Stebner, S.; Volk, W.; Münstermann, S.; Lohmann, B.
Control System Design for a Semi-Finished Product Considering Over- and Underbending. *Eng. Proc.* **2022**, *26*, 16.
https://doi.org/10.3390/engproc2022026016

**AMA Style**

Ismail A, Maier D, Stebner S, Volk W, Münstermann S, Lohmann B.
Control System Design for a Semi-Finished Product Considering Over- and Underbending. *Engineering Proceedings*. 2022; 26(1):16.
https://doi.org/10.3390/engproc2022026016

**Chicago/Turabian Style**

Ismail, Ahmed, Daniel Maier, Sophie Stebner, Wolfram Volk, Sebastian Münstermann, and Boris Lohmann.
2022. "Control System Design for a Semi-Finished Product Considering Over- and Underbending" *Engineering Proceedings* 26, no. 1: 16.
https://doi.org/10.3390/engproc2022026016