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A set of critical distances is shown as below:
YPij is the width of the jth point of the ith element (in this case
(XPcrij XÁijk )2 + (YPcrij YÁijk )2 (1)
critical points are considered according to Y-axis), « is a set
where is the Ymax is the maximum element of « ; Ymin is the minimum element of « ; max is the length of the circumscrib-
ing rectangle; S is the area of the circumscribing rectangle; Af
is the area of the main shape; W is the area of scrap.
The area of scrap is shown in Fig. 2A. It is noticeable that the length of the circumscribing rectangle equals the maximum
Fig. 4 - Variation of scrap area according to different rotation angles (time taken 0.01 s).
element of which is equal to the progress feed. Also, the line, the first derivative of both elements in the tangent point
width of this rectangle is equal to the minimum possible width of strip. Obviously the strip will be wider than this because of scrap allowance and other technical considerations.
If the shape is rotated repeatedly and the above calculation is performed for each rotation angle, an array will be produced which contains the amount of scrap for each rotation angle. Finally, the angle of rotation which results in minimum scrap can be found in this array.
Fig. 3 illustrates the shape in Fig. 2 at different angles of rotation. Fig. 4 shows the scrap area variation at different angles of rotation for the same shape. Critical points on con- vex and concave arcs are another matter and will be briefly described in Section 4.1.2.
are equal. In more general terms it can be said that for any
shape consisting of lines and arcs, apart from all vertices, all other points with the same Y coordinates whose tangents are equal can be regarded as critical points. For instance, points A and B in Fig. 5 are critical points. In the figure an arc is
4.1.2. Critical points on arcs
As mentioned in the previous section, to find critical points
on arcs other factors need to be considered. The main point to note is that when an arc is tangent to another arc or a results in better utilization of the die and a better accuracy of parts.
Since the designer (Software) does not know beforehand which piloting system is the best one, there is no other option than to determine all kinds of possible piloting systems and select the best one according to minimum scrap strategy.
In the piloting module, the possibility of direct piloting is first determined. Also indirect piloting is determined for each part. If indirect pilots do not enlarge the strip width they will be considered as major pilots. Otherwise direct pilots are taken to be major pilots. Piloting module involves three subroutines and each one of them has two branches. In the following sec- tion the piloting module is described.
Fig. 8 - Single direct pilot and critical distance.
5.1. Direct piloting
semi-direct pilots. Figs. 7 and 8 show the results of software
Initially the software recognizes all of the entire holes. The
shape is rotated by the rotation angle which results in min- imum scrap. If there is a circular hole with an acceptable
for direct pilots for two typical components. Critical distances are also shown in the figures.
diameter around the central line, which is in feed direction, it 5.2. Semi-direct pilots
will be considered as a single direct pilot. It is noticeable that
when a single pilot is used far from the central line this leads to undesirable forces and moments which cause difficulties in strip feed. If there are two circular holes with acceptable diam- eters and acceptable distances from the central line, they will be considered as direct pilots. These holes should be at a suffi- cient distance from the edges of the shape to avoid distortion in the delicate and weak parts of strip. Also the smaller the diameter of the pilot the lower its mechanical strength. The software determines a suitable diameter and distance from the edges in keeping with the technical considerations of the die design, material, strip thickness and accuracy (Progressive Dies, 1994). They can also be changed by the operator. Time complexity order of direct piloting algorithm is exponential O(n2) where n is the number of suitable holes in the compo- nent. When all possible direct pilots have been considered, the best single or double pilot system according to techni- cal considerations will be saved to compare with indirect and