tion close to the one desired。 Normally this is not a problem, and when it is, it can be overcome by applying constraints to bound the angle of rotation of a face。 However it is found with the bird example that a little more care is required than with the cartons discussed previously (cf。 also [42])。

To see the difficultly, consider the arrangement of faces shown in Fig。 14。 The point O is taken as the origin of coordinates, with the x-axis along OA, the y-axis as shown, and the z-axis normal to the plane of the figure。

The two main triangles ABR and ACR are joined along line OR and separated along OA。 Each is split into two faces along OP and OQ which lie at angle α to the y-axis shown in the figure。 Folding around these lines is possible and the angle of rotation about each is assumed to be the same angle θ 。 Suppose that face ORBP is

Fig。 15 shows graphs of θ against φ as the latter varies between

zero and 90◦ representing a complete fold。 For small values of α

the initial change in θ is seen to be large。

An instance of this arrangement of faces is present in the net of the bird。 This is in the faces that form the neck。 They lie on either of the cut along diagonal of the square in the top right of the net in Fig。 12。 The large movement corresponding to the initial change in θ is seen in the movement between stages 4 and 5 in Fig。 13。 To ensure that the correct solution is found in the simulation, the appropriate angles (those equivalent to θ ) need to be reset (to around 30◦) before the constraints for the dyad are resolved。 Once the initial jump has been achieved, constraint resolution for subsequent stages can be undertaken starting with the previously found values of the angles。

180 References

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Fig。 15。    Variation of dyad angle θ with main fold angle φ。

[1]  Hine D。 Cartons and cartoning。 Leatherhead: Pira International; 1999。

[2]  Hanlon   JF,   Kelsey   RJ,   Forcinio   HE。   Handbook   of   package  engineering。

Pittsburgh: Technomic Publishing Company; 1998。

[3] Leung SKS, Wong WK, Mok PY。 Multiple-objective genetic optimization of the spatial design for packing and distribution carton boxes。 Computers & Industrial  Engineering 2008;54:899–902。

[4]  van  Roojen  P,  Baardman  J,  Hölscher  J,  Molenaar  K。  Special        packaging。

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[5] Keay M。。 Cartoning machinery moves beyond the box。 Machinery Update 2006 (July/August): 31-7。

[6]  Anon。  Printed  folding  cartons。  Huntingdon:  Mainline  Flatpacks Limited。

http://www。mainlineflatpacks。co。uk/pdfs/cartons。pdf,  accessed   August   2008。 [7] Daniel J, Medland AJ, Mullineux G。 Use of parametric modelling to understand the functional requirements for a reconfigurable packaging system。 In: Proc。 international  conference  on  engineering  design,  ICED,  07  2007;  9  pages on

conference CD。 基于约束的纸箱折叠仿真英文文献和中文翻译(9):http://www.youerw.com/fanyi/lunwen_99512.html