Modeling and Simulation of Variational Rigid and
Compliant Assembly for Tolerance Analysis
Creators: name: Franciosa, Pasquale
Publisher: name: Università degli Studi di Napoli Federico II
Year: 2009

description: Modern manufacturing processes are strongly affected by part and process variations. Understanding the final shape of the assembly is a crucial task to be achieved during the design stage to reduce cost and production time. Typically, when parts are put together variations propagate part-to-part. This stack-up effect is strictly related to part deviations, assembly sequence, assembly constraints, part flexibility. All these fac-tors combine into a non-linear way. Technical literature provides valid methodologies to analyze tolerance stack-up problems. In this contest, assemblies are usually classified into two main categories: rigid assemblies and compliant assemblies. In the first case, parts are assumed ideal-rigid and the additional variation due to elastic or plastic defor-mation is not accounted. When the flexibility of parts is not negligible, as into manufac-turing processes involving sheet-metal parts, its effect has to be accounted into numeri-cal simulations. This dissertation provides a contribution to the modeling and the simu-lation both of rigid and compliant assemblies. 
With respect to rigid assembly, a general methodology, called SVA-TOL (Sta-tistical Variation Analysis for Tolerancing), is proposed. The methodology is based into two main steps. Tolerance specifications are modeled in the first step following Interna-tional Standard rules. Variational features are so generated. Then, by using these varia-tional features, assembly constraints are modeled as combination of elementary geome-try entities: points, lines and planes. Two assembly solvers are illustrated: the sequential solver and the least squares solver. 
The sequential solver allows to analyze assembly constraints taking into account the assembly sequence. Specific mathematical tools, such as Screw Theory and Graph Theory, are here adopted to automatically calculate the list of degrees of freedom al-lowed for a specific part being assembled. 
The least squares solver, instead, permits to analyze all assembly constraint si-multaneously by best fitting all mating conditions. Case studies have pointed out the field of applicability of proposed assembly solvers.
With respect to compliant assembly, a general frame-work, called SVA-FEA (Statistical Variation Analysis &amp; Finite Element Analysis), is presented. SVA-FEA al-lows to statistically simulate single- and multi-station assembly processes under linear assumptions. A Global Sensitivity Matrix is introduced to link input part or process de-viations and output assembly deviations. In this way, no Monte Carlo simulation is needed. The whole assembly process is based on the Place, Clamp, Fasten and Release (PCFR) cycle. Parts are positioned on fixturing frames, where are clamped and fastened. Then, they are released reaching their final assembly configuration. To numerically cal-culate the sensitivity matrix two FEA runs, performed on the nominal geometry, are re-quired. The first one calculates the fixturing and fastening forces, by applying the method of influence coefficients. These forces are then applied into the second FEA run to simulate the final elastic spring-back. SVA-FEA methodology has been implemented into a friendly MatLAB®’s GUI, allowing to define the whole assembly process, with its variability, and to visualize the final assembly deviations, in terms on mean and standard deviations. Two significant case studies have highlighted how SVA-FEA al-lows to simulate both single- and multi-station assembly processes. Results have been compared with ones coming from commercial CAT software, showing a good numeri-cal correlation. 
SVA-FEA assumes that input statistical deviations are independent among them. This means that no covariance effect is accounted. To overcome this weakness, a new non-linear methodology is also proposed. The non-linear methodology, to do variational analysis of compliant assembly, uses a Monte Carlo approach to statistically generate input free shape geometry. The methodology can be described as follows. For each Monte Carlo step, input geometry is generated according to an automatic morphing mesh procedure. Then, the PCFR cycle is simulated. The geometry is so updated for each phase of the PCFR cycle. To reach more realistic results, contact pairs, defined as surface-to-surface type, may be introduced into the numerical simulation. The morphing mesh approach allows to generated free shape parts starting from few control points, defined on the nominal geometry, and setting the tolerance error value.
Finally, critical remarks are outlined, highlighting future directions of research in the field of tolerance analysis of rigid and compliant assembly processes.