Abstract
AbstractIn the frame of a statistical tolerance analysis of complex assemblies, for example an aircraft wing, the capability to predict accurately and fast specified, very small quantiles of the distribution of the assembly key characteristic becomes crucial. The problem is significantly magnified, when the tolerance synthesis problem is considered in which several tolerance analyses are performed and thus, a reliability analysis problem is nested inside an optimisation one in a fully probabilistic approach. The need to reduce the computational time and accurately estimate the specified probabilities is critical. Therefore, herein, a systematic study on several state of the art simulation methods is performed whilst they are critically evaluated with respect to their efficiency to deal with tolerance analysis problems. It is demonstrated that tolerance analysis problems are characterised by high dimensionality, high non-linearity of the state functions, disconnected failure domains, implicit state functions and small probability estimations. Therefore, the successful implementation of reliability methods becomes a formidable task. Herein, advanced simulation methods are combined with in-house developed assembly models based on the Homogeneous Transformation Matrix method as well as off-the-self Computer Aided Tolerance tools. The main outcome of the work is that by using an appropriate reliability method, computational time can be reduced whilst the probability of defected products can be accurately predicted. Furthermore, the connection of advanced mathematical toolboxes with off-the-self 3D tolerance tools into a process integration framework introduces benefits to successfully deal with the tolerance allocation problem in the future using dedicated and powerful computational tools.
Highlights
High-value mechanical assemblies e.g. an aircraft wing needs a strict dimensional management procedure in place in order to control and manage variation stemming from the various manufacturing processes to fabricate the parts as well as to assemble them and form the final product
To summarise the characteristics of the statistical tolerance analysis problem, they are distinguished by the estimation of small probability values, involving a moderate to high number of random variables when considering complex assemblies in which tens of tolerances will be included in the tolerance chain of the assembly key characteristics (AKC), exhibiting highly non-linear limit state functions as well as disconnected failure domains
The normalisation was performed with respect to the expectation of the probability estimator evaluated by Crude Monte Carlo (CMC) for 1E+06 iterations
Summary
High-value mechanical assemblies e.g. an aircraft wing needs a strict dimensional management procedure in place in order to control and manage variation stemming from the various manufacturing processes to fabricate the parts as well as to assemble them and form the final product. The main core of any dimensional management methodology is the ability to perform tolerance analysis and synthesis at the early design stage and to predict the variance or the entire distribution of specified assembly key characteristics (AKC) as well as to optimise and allocate design tolerances for the assembly features in the various parts by minimising manufacturing cost For the latter case, several studies have been performed, e.g. in [1], in which the optimisation problem in most of the cases was formulated using mainly one objective function, the manufacturing cost, and constraint functions based on the worst case error or on the root sum square variance of the AKC. It is already common that crude Monte Carlo method can become very time consuming when estimation of small probabilities is involved This probably explains the fact that tolerance synthesis is still treated in commercial CAT tools by using linearized assembly models and linear optimisation techniques e.g. in [3]. This is the first step to establish a process integration and design optimization framework in order to deal with the more complex problem, the tolerance allocation one
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