冲压技术英文资料doc.docx

1、冲压技术英文资料doc冲压技术英文资料 Stamping Die Strip Optimization for Paired Parts V.Vamanu, T.j.Nye Mechanical Engineering Department McMaster University Hamilton, Ontario October 30, 2000 Abstract In stamping, operating cost are dominated by raw material costs, which can typically reach 75 of total costs in a s

2、tamping 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

3、and the optimum width for the strip. Examples are given for pairing the same parts together with one rotated 180, and for pairs of different parts nested together. This algorithm is ideally suited for incorporation into die design CAE systems. Keywords Stamping, Die Design, Optimization, Material Ut

4、ilization, 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

5、 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 cru

6、cial to a stamping firms 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 parts, width of the strip, and, in the case of multiple parts b

7、lanked 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

8、 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

9、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 fo

10、r 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 co

11、mpared 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 180, and when two different parts are nested together. In this paper we describe a new algorithm that provides the optimal strip layout for t

12、hese 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

13、into rectangles, then fit the rectangles along the strip2. Variations of this approach have involved fitting blanks into non-overlapping composites of rectangles 3, convex polygons 4,5 and known interlocking shapes6. A fundamental limitation exists with this approach, however, in that the enclosing

14、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

15、 amount, such as 27, the pitch and width of the layout determined and the material utilization calculated. After repeating these steps through a total rotation of 180 due to symmetry, the orientation giving the best utilization is selected. The disadvantage of this method is that, in general, the op

16、timal 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. Meta-heuristic optimization methods have also been applied to the strip layout problem, both simu

17、lated annealing 11, 12 and genetic programming 13. While capable of solving layout problems of great complexity i.e. many different parts nested together, general 2-D nesting of sheets, they are not guaranteed to reach optimal solutions, and may take significant computational effort to converge to a

18、 good solution. Exact optimization algorithms have been developed for fitting a single part on a strip where the strip width is predetermined 14 and where it is determined during the layout process 15. These algorithms are based on a geometric construction in which one shape is grown by another shap

19、e. Similar versions of this construction are found under the names no-fit polygon, obstacle space and Minkowski sum. Fundamentally, they simplify the process of determining relative positions of shapes such that the shapes touch but do not overlap. Through the use of this construction in this paper,

20、 the particular version used is the Minkowski sum, efficient algorithms can be created that find the globally optimal strip layout. For the particular problem of strip layout for pairs of parts, results have been reported using the incremental rotation algorithm 7, 16 and simulated annealing 11, but

21、 so far no exact algorithm has been available. In what follows, the Minkowski sum and its application to strip layout is briefly introduced, and its extension to nesting pairs of parts is described. The Minkowski Sum The shape of blanks to be nested is approximated as a polygon with n vertices, numb

22、ered consecutively in the CCW direction. As the number of vertices increases, curved edges on the blank can be approximated to any desired accuracy. Given two polygons, A and B, the Minkowski sum is defined as the summation of each point in A with each point in B, 1 Intuitively, one can think of thi

23、s process as growing shape A by shape B, or by sliding shape B i.e., B rotated 180 around A and following the trace of some reference point on B. For example, Fig.1 shows an example blank A. If a reference vertex is chosen at 0, 0, and a copy of the blank rotated 180 i.e., A is slid around A, the re

24、ference vertex on A will trace out the path shown as the heavy line in Fig.2. This path is the Minkowski sum . Methods for calculating the Minkowski sum can be found in computational geometry texts such as 17, 18. Sample Part A to be Nested. Minkowski Sum heavy line of sample Part light line. The si

25、gnificance of this is that if the reference vertex on A is on the perimeter of , A and A will touch but not overlap. The two blanks are as close as they can be. Thus, for a layout of a pair of blanks with one rotated 180 relative to the other, defines all feasible relative positions between the pair

26、 of blanks. A corollary of this property is that if the Minkowski sum of a single part is calculated. With its negative, i.e.,. A complete explanation of these properties of the Minkowski sum is given in 15. These observations were the basis for the algorithm for optimally nesting a single part on a

27、 strip. The situation when nesting pairs of parts is more complex, since not only do the optimal orientations of the blanks and the strip width need to be determined, but the optimal relative position of the two blanks needs to be determined as well. To solve this problem, an iterative algorithm is

28、suggested Given Blanks A and B where BA when a blank is paired with itself at 180 1. Select the relative position of B with respect to A. The Minkowski sum defines the set of feasible relative positions Fig.2. 2. Join A and B at this relative position. Call the combined blank C. 3. Nest the combined

29、 blank C on a strip using the Minkowski sum with the algorithm given in 14 or 15. 4. Repeat steps 1-3 to span a full range of potential relative positions of A and B. At each potential position, evaluate if a local optima may be present. If so, numerically optimize the relative positions to maximize

30、 material utilization. Layout Optimization of One Part Paired with Itself The first step in the above procedure is to select a feasible position of blank B relative to A. This position is defined by translation vector t from the origin to a point on, as shown in Fig.3. During the optimization proces

31、s, this translation vector traverses the perimeter of. Relative Part Translation Nodes on, showing Translation Vector t. Initially, a discrete number of nodes are placed on each edge of. The two parts are temporarily joined at a relative position described by each of the translation nodes, then the

32、combined blank is evaluated for optimal orientation and strip width using a single-part layout procedure e.g., as in 14 or 15. In this example, consists of 12 edges, each containing 10 nodes, for a total of 120 translation nodes. The position of each node is found via linear interpolation along each

33、 edge, where is vertex I on the Minkowski sum with a coordinate of ,. Defining a position parameter s such that s 0 at and s 1 at, coordinates of each translation node can be found as 2 3 If m nodes are placed on each edge,the position parameter values for the node, , are found as 4 Calculating the utilization at each of the 1