Estructura de la substancia Forma y materia, acto y potencia.

The purpose of this work has been to examine the sequence synthesis requirements for the purpose of robust sequence selection. In contrast, prior works have attempted to seek the optimal sequence to minimise variability given particular conditions; which poses a problem if there is no knowledge of the geometric deformation or variation patterns. However, if there is some knowledge of the input variability, then the preferred robust sequence may differ because the analysis domain may change. The preference could further alter since the presented robustness index is a measure of the total variability of the mapping function, which does not take into consideration the shape of the input distribution with respect to the output distribution. Consequently, two curves with slightly different profiles of their mapping curves can result in the same robustness index, but, since the profile of the mapping is different there will be a different output distribution. Although this is not deemed significant for the purpose of analysis in determining a robust sequence, it may be more important to consider if there is knowledge of the input distribution.

6.6

Chapter summary

This chapter has presented a sequence analysis approach that used a broader set of input variables than prior works on sequence analysis. This sequence analysis approach was combined with a method for evaluating performance based on the total variation of mapping curves, to allow for an improved identification of the requirements for a robust joining process synthesis strategy. Unlike previous models used for analysis and synthesis purposes, the beam-based model incorporates a number of physical aspects, specifically: internal stress build-up, stiffness increase and geometric non-linearities. This model was used to simulate prior proposed joining sequence methodologies along with other combinations. A

§6.6 Chapter summary 141

polygon beam assembly was then used to further illustrate these sequence approaches. This analysis has allowed for a comparison of a number of factors relating to sequence selection to address Objective 3 of this thesis and answer the associated research question:

1. What is a robust sequence for minimising output variation given any input dimensional variation pattern?

In the presence of boundary conditions, the preferred sequence should begin at the most rigid area and move towards less rigid areas; a result also observed in Chapter 5.

2. What is the influence of structural design on optimal sequence selection?

The polygon structure has shown a preference for an opposing sequence strategy for robust sequence selection in comparison to a cantilever assembly. In practice, this is more applicable to the more complex freeform geometric surfaces of sheet metal components, because it better replicates the dimensional interactions of the structure.

3. What is the significance of internal stress build-up to the analysis process?

As the degree of restraint increases the importance of considering internal stress-build up becomes greater. Therefore, in the analysis process for determining preferred sequences, it is a factor that needs to be considered primarily when N-2-1 location strategies are followed. However, internal stress build-up does influence the geometric response under all conditions, particularly as the stiffness of the structure decreases.

A joining sequence synthesis

method

As outlined in Section 2.7 of Chapter 2,a method of determining a robust joining sequence is of significant practical importance to industry. The outcomes from Chapters 5 and 6 established the approach that should be taken by such a method; however, currently this relies on visual interpretation of the combined geometric structure and fixture design to determine a sequence that can accommodate variability in the joint gaps and provide a robust output.

To address Objective 4 of this thesis, the establishment of a method for the rapid identification of a robust joining sequence, this chapter presents an algorithm to fulfill this need. The formalised approach of stiffest-to-least-stiff is combined with structural evaluation techniques to determine a robust sequence via an algorithm (Section 7.1). This approach is then applied to the idealised geometric structure presented throughout this work for a simulation-based verification, given the prior gap input variation and fixture boundary conditions (Section 7.2). A discussion of the application and limitations of the developed algorithm is then presented in Section 7.3, followed by the key summary points of this chapter in Section 7.4.

§7.1 A joining process sequencing algorithm 143

7.1

A joining process sequencing algorithm

Although sequence analysis simulations are useful in studying the sequence dependence in sheet metal assembly, they are based on idealisations and assumptions about the underlying mechanics of the process. Furthermore, as the number (n) of joining operations increases, the number of possible sequences increases factorially (n!). As a result, from a practical standpoint, it is nearly impossible with current software and computing technology to simulate all permutations. This computation resource problem is compounded if variation simulations are to be considered. In industry, sequence selection is often based on the experience of the process engineers. Although optimisation procedures have been developed to solve sequence-based problems, they all require exact knowledge about the variation patterns of the incoming components; and then use an iterative procedure. However, it is not always possible to know the incoming component variations, which limits any possible application of these techniques.

Previously developed sequence synthesis algorithms have been based on simplified models. In Chapter 6, a more detailed model of the factors contributing to sequence dependence was developed and used to illustrate the sequencing influence on different geometric forms. General sequencing guidelines were shown to be applicable for robust sequence selection.

This section derives a computational algorithm, based on structural mechanics, that will provide a comparatively faster solution to determine a joining sequence in accordance with these sequencing guidelines. At a minimum, all that is required for this technique is the Computer Aided Design (CAD) geometry, join locations, and the fixture strategy.

While no single joining sequence can provide an ideal solution for minimising all types of dimensional variation patterns, the purpose of this work is to quickly identify a robust solution; for example, by applying the stiffest-to-least-stiff strategy. To achieve this, the geometric structure must be evaluated and a decision process

performed to determine the location of the next join after any prior joining operation is made. This follows from the conclusions of Chapter 5, which concluded that the current state of the assembly must be considered when selecting the next join, including any existing joins and the fixture design.

The algorithm must follow the logic of the sequencing guidelines, as presented in the following procedural manner:

1. Locate component according to referencing strategy 2. Apply clamps to hold component

3. Forn join iterations

(a) For each remaining join from the pool of candidate joins:

i. Evaluate the local stiffness given all prior joins and the fixture strategy (b) Select join at the stiffest location

(c) Apply join to structure

To ensure the computational expense is not too large, the number of complete iterations of the algorithm can be determined and compared to the maximum possible number of calculations. The total number of iterations for this algorithm is equal to the nth triangular number, or n(n₂+1) evaluations, this is significantly less than the possible n! computational combinations. While optimisation procedures could be applied here to minimise the number of iterations, the primary focus is on an implementation of the guidelines algorithm, therefore optimisation procedures were not considered.

To evaluate the stiffness at each successive join a number of approaches could be used. The most obvious option is by using static implicit force-displacement FEA. In this case, by applying a local force at the join location and measuring the local deflection of the structure. The local deflection could be compared against all other alternative local deflections in the evaluation procedure until all candidate joins

§7.1 A joining process sequencing algorithm 145

had been evaluated and the stiffest location could be selected. Liu and Hu (1995b) described a similar approach but in an attempt to find the weakest-to-strongest join locations.

Alternatively, structural stiffness can be evaluated via natural frequency extraction of the stiffness matrix. Structural natural frequency extraction is a common tool for stiffness estimations and is used in industry for fixture and tooling design. A significant benefit of this approach is a reduction in computational expense and required analysis of the results. One example of natural frequency extraction and its use in the automotive Body In White (BIW) structure evaluation field is by Bhatti et al. (2011). In Bhatti et al.’s (2011) work, structural stiffness was maximised based on join positions of the assembly utilising a number of methods, including utilising the natural frequency shift as an evaluation method.

To order the welds from the most to least rigid area, the influence of each successive join is evaluated by analysing the frequency shift that results in the addition of that join from the prior state of the assembly. The algorithm presented, combined with this decision making process, seeks to minimise the change in natural frequency of the structure between successive joins. There are numerous possible structural natural frequencies against which this could be evaluated; however, the first compliant natural frequency is utilised in this work since it represents the most compliant direction of the component.

7.1.1 Summary

The proposed method uses the global structural stiffness at any particular joined configuration to evaluate the next possible join in the sequence. The basis of this method relates to the frequency shift that occurs as the joining operations are performed. When each join connection is made, this associated stiffness change dictates the level of spring-back local to the joining tool upon release. This allows the areas of highest stiffness to lowest stiffness to be evaluated numerically for any

complex form.