# Robust Design of Deep Drawing Process through In-Line Feedback Control of the Draw-In
## 3 Results
### Feedback Control of the Draw-in by Regulating the BHF  
The control methodology described in Section 2.2 was implemented by varying the friction coefficient in a range around the 0.11 value. Keeping the optimal blank holder force obtained for the 0.11 friction coefficient (650 kN), a friction reduction on the blank-tool interfaces shows surface deflections in the central region of the part (S\({}_{\text{D}}\) region) due to the greater blank draw-in, as confirmed by observing the draw-in in sensors A, B, and C. With this assumption, the control system intervenes by increasing the force on the blank holder until it reaches the limit allowed by the press machine (1000 kN). The results obtained show that this limit condition is reached for a friction coefficient equal to 0.07. Conversely, an increase in the friction coefficient causes a decrease in the draw-in in each sensor and an increase in the split risk in the T\({}_{\text{A}}\) and T\({}_{\text{B}}\) regions. With a friction coefficient equal to 0.15, the thinning in these critical regions reaches 30%. Figure 19 summarizes the results obtained for a BHF equal to 650 kN and with the investigated friction condition limits (0.07, 0.15), comparing these results with those obtained for the optimized solution (u0.11-650 kN). In the case of friction coefficients lower than 0.11, the maximum draw-in deviation occurs in sensor C, and the control system applies the feedback procedures on this sensor. For friction coefficients greater than 0.11, the control system applies the feedback procedures on sensor A because it is precisely in this sensor that the maximum draw-in deviations are recorded. Regarding the boundary friction condition investigated (u = 0.07 and u = 0.15), Figure 20 shows draw-in curves simulated in the controlled sensor and the blank holder force imposed to reduce draw-in deviations. In particular, the draw-in curves show the draw-in measured with a BHF equal to 650 kN (dashed curve), the draw-in taken as reference for deviation computation (continuous-line curve), and the draw-in measured following the blank holder force variations imposed by the control system (marked curve). These results have been obtained choosing a maximum draw-in deviation equal to 10% (\(e_{accepted}=0.1\)). Furthermore, an adjustment time of the blank holder force of at least double compared to the response time of the hydraulic actuation (equal to 100 ms) was chosen. Figure 18: Draw-in over the time at point (**a**) A, (**b**) B, and (**c**) C for friction coefficient equal to 0.05, 0.11, and 0.15. Assuming a friction coefficient equal to 0.07 (Figure 20a), the first sensor to have a draw-in deviation greater than \(e_{accepted}\) is sensor C. To minimize the error between the draw-in measured in the optimal condition (\(\mu\)0.11-650 kN) and that obtained for a friction coefficient equal to 0.07 (\(\mu\)0.07-650 kN), the BHF was adjusted by increasing the force up to the 1000 kN limit value and then maintaining it constant (BHFadj). It is observed that, thanks to the BHF adjustment phase, the draw-in at C (\(\mu\)0.07-BHFadj) approaches the optimal one, although the error obtained at the last time step is greater than 10% because the maximum admissible BHF was reached. The drawn component at the end of the process meets the quality  
Figure 19: Formed parts’ curvature and thinning values simulated with a BHF equal to 650 kN and different friction coefficients. Figure 20: Draw-in curves and BHF adjusted in the controlled sensor with a friction coefficient equal to (**a**) 0.07 and (**b**) 0.15.  
requirements. Specifically, the curvature in region S\({}_{\rm D}\) is about 8.5 \(\times\) 10\({}^{-3}\) 1/mm, and the thinning percentage in regions T\({}_{\rm A}\) and T\({}_{\rm B}\) is equal to approximately 22%. The drawn part after BHF adjustment is like that obtained under the \(\upmu\)0.11-650 kN condition (Figure 19). Assuming a friction coefficient equal to 0.15 (Figure 20), the first sensor to have a draw-in deviation greater than \(e_{accepted}\) is sensor A.