Multi-station models

The previous section discussed finite element methods for single station assembly simu- lation, which importantly captures the exibility of assembly components. Sheet metal assembly is an iterative process involving many sub-assembly stations between initial stampings and the final BIW, so multi-station assembly analysis forms an important piece of the virtual assembly toolset. This section discusses some key approaches for multi-station assembly variation modelling.

There are many well established methods for the analysis of variation or tolerance propagation in mechanical assemblies based primarily around vector-loop equations (or assembly functions). Root-sum-of-squares (RSS) approaches use a first order approxi- mation to derive a linear propagation model and are popular due to their ease of use (Gao et al., 1995). The inability of RSS approaches to deal with non-linear input and output distributions has led to the proposal of higher order taylor series approximations for more accurate depictions of tolerance propagation (Shapiro and Gross, 1981). A second order method proposed by Glancy and Chase (1999) combined finite difference approximations of partial derivatives, the Method of System Moments for estimating

20 CHAPTER 2. BACKGROUND

nonlinear system outputs (Cox, 1979), and nonlinear distribution fits, to allow for the approximation of nonlinear assembly tolerances. Another perhaps more straightfor- ward method of variation or tolerance propagation analysis in complicated nonlinear assemblies is the Monte Carlo method, which involves running thousands of random inputs to the assembly function to estimate final distributions. A drawback of this method is obvious time constraints, although there are advancements that have been made to reduce computational effort of such approaches (Huang et al., 2004; Wu et al., 2006).

A more recent and increasingly popular form of modelling variation propagation in multi-station assemblies has been referred to as the state space modelling approach. It uses established control theory methods to describe the positioning of each component throughout each stage of the assembly process, and is often combined with Monte Carlo simulation for variation or tolerance analysis. A state space modelling approach was presented by Jin and Shi (1999) to determine the variation propagation throughout all steps in multi-stage assembly. Part orientations were transformed according to par- ticular manufacturing stage conditions. Tooling error, such as locating pin deviations, part error, and re-orientation are the key sources describing the positioning of parts throughout the stages of the model. State space modelling of assembly variation prop- agation has become popularly known as the \Stream of variation theory" (Hu, 1997). Xiong et al. (2002) present a statistical variation analysis model for multi-station as- sembly that is implemented in IDEAS. As with the state space modelling approach, this is achieved by a series of transformations of assembly components at each assembly stage. These approaches have provided valuable tools for the investigation of variation propagation and tolerance allocation in multi-station models, however, they have only focused on rigid component assemblies which limits the predictive accuracy for sheet metal assembly.

Camelio et al. (2003) presented a method that combines state space modelling techniques for multi-station variation propagation analysis with mechanistic variation analysis. Linearized mechanistic variation simulation analysis (Liu and Hu, 1997a) was incorporated into the discrete step model at the assembly stage. At each assembly step, parts are located, clamped, welded and released according to the mechanistic variation model, and this is cycled for each step. Part, fixture and welding gun variation are considered in this final model. The limitations of this model include: model complexity as the number of assembly stations increase, there is no consideration of non-linear behaviour (such as deformation induced part-to-part and part-to-tool interactions), and it is discontinuous for variables such as fixture positioning. This means that a new model must be developed for alternative welding sequences. However, the combination of multi-station assembly and exible component modelling provides a key advancement in virtual process investigation.

2.2. VIRTUAL ASSEMBLY 21 While multi-station modelling for rigid component assemblies is well developed, the application of multi-station modelling to sheet metal assembly presents additional chal- lenges through the need to capture the exibility of components. The combination of linearized finite element simulation with multi-station models is practically achievable, however, linear mechanistic simulation approaches are not as accurate as non-linear contact models. Advancements in the field of sheet metal assembly could involve devel- oping computationally reasonable methods for capturing nonlinear contact behavior in exible assemblies. For example, the linearised contact assembly approach proposed by Soderberg (2006) could be integrated into the multi-station approach. For the purposes of this thesis, virtual multi-station modelling is not considered due to the adoption of computationally intensive nonlinear mechanistic simulation models, and to make the experimental set-up more manageable. Moreover, a natural extension to the research presented later in this thesis is to extend the virtual analysis of multi-station mod- els using the shape characterisation tools developed for stampings and single station assemblies.

As with the mentioned finite element models, multi-station simulation approaches for sheet metal assembly provide valuable tools for the prediction of variation propaga- tion and can be used to identify more optimal assembly approaches. However, in order to make best use of these tools, a well defined measure of quality must exist that can be targeted in quality control efforts. This thesis proposes a range of methods that can assist in providing improved quality objectives for multi-station analysis.