Case 2: Polygon beam assembly - Beam assembly joining sequence results

6.3 Beam assembly joining sequence results

6.3.2 Case 2: Polygon beam assembly

A circular structure simulation was established as an example of a closed geometric form for the sequencing problem. This polygon representation considers two hexagonal geometries. The first geometry was at the ideal location, and the second hexagon was deformed by three different deformation patterns, and the variation

therein. These were then overlayed to the same location strategy: node one constrained in both translation ofx andy, representing a pin location; and node six constrained in the y direction, representing a slot constraint. The joining sequence brought the components to the nominal join locations perpendicular to the tangent of the encompassing circle. This geometry is represented in Figure 6.12.

Figure 6.12: Two-dimensional, polygon-based beam assembly model with an illustrative deformation applied.

It is important to note that there was no free end. The free-end of the structure was thus considered to be the least rigid area - the place furthest away from the fixed location controls on the bottom edge of the component. This structural design contrasts the cantilevered beam assembly presented in Section 6.3.1. The outgoing deformation measure used to evaluate the performance of the sequence was the Hausdorff distance of the located assembly, considering both polygons, to its nominal geometry - which is the position of the un-deformed perfect polygon. Figure 6.13 illustrated the three variation patterns used for the polygon geometry.

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(a)

(b)

(c)

Figure 6.13: Variation patterns for the polygon geometry (a) Pattern one, (b) Pattern two, and (c) Pattern three.

robustness across the domain specified. The robustness index for each particular sequence, across all variation patterns was then summed. Although this approach will bias the results towards the input variation pattern that is most sensitive to sequence selection, this is considered acceptable because the robustness index for each sequence and variation patterns all closely follow the same trend, and it is the overall robustness that is important for a production system.

In all cases the poorest performing sequences began furthest from the rigid location point and then progressed towards the most rigid. A summary of the best, worst and common strategy results is presented in Table 6.4.

Table 6.4: Robustness measure for selected sequences based on given input variations for the polygon geometry.

Sequence classification Variation pattern one Variation pattern two Variation pattern three Best 1-2-4-3 (7.56) 4-3-2-1 (0.48) 4-3-2-1 (7.52) 1-4-2-3 (7.57) 1-2-3-4 (7.53) 4-3-1-2 (7.69) 4-1-3-2 (7.68) Worst 2-1-3-4 (16.23) 2-1-4-3 (9.18) 3-4-1-2 (13.28) 2-3-1-4 (16.06) 2-4-1-3 (9.16) 3-1-4-2 (13.28) Weak-to-Strong 3-2-4-1 (14.34) 3-2-4-1 (2.03) 3-2-4-1 (11.07) Fixed-to-free 1-4-2-3 (7.57) 1-4-2-3 (2.14) 1-4-2-3 (8.55)

The sequences that were the most robust to the first input variation pattern began close to the locator supports and worked outwards. There were a number of variants of the fixed-to-free strategy that all performed comparably, and represent a linear mapping curve with a low constant gradient (Figure 6.14 (a-c)(ii)). The preferred sequence for the second and third variation pattern proceeded around the polygon loop. For each variation pattern the curves illustrated here are the fixed-to-free (1-

§6.3 Beam assembly joining sequence results 123

4-2-3), weak-to-strong (3-2-4-1), the overall most robust, the overall least robust, and the respective best and worst against the particular variation pattern.

a(i)

a(ii)

Figure 6.14: Mapping and gradient curves for each input variation pattern illustrating the selected sequences. (a) Pattern one (i) Mapping (ii) Gradient, (b) Pattern two (i) Mapping (ii) Gradient, and (c) Pattern three (i) Mapping (ii) Gradient.

b(i)

b(ii)

§6.3 Beam assembly joining sequence results 125

c(i)

c(ii)

The fixed-to-free strategy outperformed the weak-to-strong strategy by a significant margin in the cases of the polygon geometry, where stress build-up was dominant in determining the final geometry. A visual representation of the polygon geometries both with and without internal stress build-up for each of the variation patterns is shown in Figure 6.15. When stress build-up is not considered, for this geometry and variation combination, an overestimation of the final geometric deflection occurs.

a(i)

a(ii)

Figure 6.15: Shape deformations for the weak-to-strong and fixed-to-free methodologies modelled with and without internal stress build-up. (a) Pattern one (i) Sequence 1-4-2-3 (ii) Sequence 3-2-4-1, (b) Pattern two (i) Sequence 1-4-2-3 (ii) Sequence 3-2-4-1, and (c) Pattern three (i) Sequence 1-4-2-3 (ii) Sequence 3-2-4-1.

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b(i)

b(ii)

c(i)

c(ii)

§6.3 Beam assembly joining sequence results 129

A general relationship exists between an increase in the mean stress in the structure and an increase in the robustness measure of total variation. This is illustrated in Figure 6.16, where the robustness measure for each sequence and variation pattern combination is plotted against the mean stress in the structure.

Figure 6.16: Illustration of the increasing structural stress against the increase in total variation of the polygon structure.

In document Robust dimensional variation control of compliant assemblies through the application of sheet metal joining process sequencing (Page 145-155)