Abstract In stamping, operating cost are dominated by raw material costs, which can typically reach 75% of total costs in a stamping facility. In this paper, a new algorithm is described that determines stamping strip layouts for pairs of parts such that the layout optimizes material utilization efficiency. This algorithm predicts the jointly-optimal blank orientation on the strip, relative positions of the paired blanks and the optimum width for the strip. Examples are given for pairing the same parts together with one rotated 180º, and for pairs of different parts nested together. This algorithm is ideally suited for incorporation into die design CAE systems.33102
Keywords: Stamping, Die Design, Optimization, Material Utilization, Minkowski Sum, Design Tools
Introduction
    In stamping, sheet metal parts of various levels of complexity are produced rapidly, often in very high volumes, using hard tooling. The production process operates efficiently, and material costs can typically represent 75% of total operating costs in a stamping facility [1]. Not all of this material is used in the parts, however, due to the need to trim scrap material from around irregularly-shaped parts. The amount of scrap produced is directly related to the efficiency of the stamping strip layout. Clearly, using optimal strip layouts is crucial to a stamping firm’s competitiveness.

    The degree of this trim loss is determined at the tooling design stage when the strip layout is created. As a part or parts are laid out on the strip, the designer chooses the orientation of the part(s), width of the strip, and, in the case of multiple parts blanked together, their relative positions. Ideally, the material utilization is maximized. The value of even tiny improvements in material utilization can be great; for example, in a stamping operation running at 200 strokes per minute, a savings of just 10 grams of material per part will accumulate into a savings of more than a tonne of raw material per eight-hour shift. The material utilization is set during the tooling design stage, and remains fixed for the (usually long) life of the tool. Thus, there is significant value in determining the optimal strip layout before tooling is built.

    This task is complicated, however, since changing each variable in the layout can change both the pitch (distance along the strip between adjacent parts) and strip width simultaneously. Evaluating layout efficiency manually is extremely challenging, and while exact optimal algorithms have been described for the layout of a single part on a strip, so far only approximate algorithms have been available for the layout of pairs of parts together. Nesting solutions for pairs of parts is an important problem since it is empirically known that nesting pairs of parts can often improve material utilization compared to nesting each part on a separate strip. This paper addresses the common cases in which a given part is nested with a second copy of itself rotated at 180º, and when two different parts are nested together. In this paper we describe  a new algorithm that provides the optimal strip layout for these two cases.

Previous Work
    Originally, strip layout problems were solved manually, for example, by cutting blanks from cardboard and manipulating them to obtain a good layout. The introduction of computers into the design process led to algorithmic approaches. Perhaps the first was to fit blanks into rectangles, then fit the rectangles along the strip[2]. Variations of this approach have involved fitting blanks into non-overlapping composites of rectangles [3], convex polygons [4,5] and known interlocking shapes[6]. A fundamental limitation exists with this approach, however, in that the enclosing shape adds material to the blank that cannot be removed later during the layout process. This added material may prevent optimal layouts from being found.

    A popular approach to performing strip layout is the incremental rotation algorithm [6-10, 16]. In it, the blank, or blanks, are rotated by a fixed amount, such as  2º[7], the pitch and width of the layout determined and the material utilization calculated. After repeating these steps through a total rotation of 180º (due to symmetry), the orientation giving the best utilization is selected. The disadvantage of this method is that, in general, the optimal blank orientation will fall between the rotation increments, and will not be found. Although small, this inefficiency per part can accumulate into significant material losses in volume production. 冲压模具英文参考文献和中文翻译:http://www.751com.cn/fanyi/lunwen_29993.html