OPTIMIZATION AND CONTROL OF THE BLANKHOLDER FORCE IN SHEET METAL STAMPING WITH APPLICATION TO DEEP DRAWING OF A CYLINDRICAL CUP

The sheet metal forming process plays an important role in the manufacturing industry. The increasing use of high strength steels and the reduction of sheet thickness generate formability difficulties especially for complex parts. Wrinkling, tearing and springback are the principal defects for this process which must be controlled if not avoided. The blankholder force is one of the process parameters, which is able to control effectively the flow of the metal into the die cavity and to improve quality of the workpieces. The present work is based on an original idea which was used by Siegert, Haussermann, Haller and Descamps (1, 2, 3, 4). The goal is to control the movement of the blank under the blankholder by adjusting locally the blankholder force and/or varying it during time in independent zones according to the punch stroke. The objective function is to minimize the external work of the finite element model. An inequality constraint function is formulated to avoid wrinkling: the angle between the die and the blank is limited to a value θmax fixed by the user, as proposed by Gelin and Labergere (5). The value of θmax depends on the strategy of the optimization of the blankholder force (time dependent or not). The optimal control is solved as a sequence of sub-problems using a response surface method based on a moving least squaresapproximation. The maximum value of θmax is then also time-dependent. Three profiles of the angle of inclination, θmax, are analyzed and coupled with the optimal control. This optimization approach is applied to the Numisheet'02 cylindrical cup (6) to optimize both a constant and a time dependent blank holder force.