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Top1. Introduction
Tolerance analysis has a considerable importance in concurrent engineering and it represents the best way to solve problem in order to ensure higher quality and lower costs. The need to assign dimensional and geometric tolerances to assembly components is due to the standardization of the production and to the correct working of the assembly. The appropriate allocation of tolerances among the different parts of an assembly can result in lower costs per assembly and higher probability of fit, reducing the number of rejects or the amount of rework required on components. The dimensions and the tolerances of the assembly components combine, according to the assembly sequences, and generate the tolerance stack-up functions. Solving a tolerance stack-up function means to determine the nominal value and the tolerance range of a product function by combining the nominal values and the tolerance ranges assigned to the assembly components.
The first step towards translation of the functional requirements and geometric relations in form of quantifiable mathematical expression is represented by the tolerance models (Morse et al., 2018). In the last decade many research efforts have been carried out to explore the mathematical models for geometric deviation representation: variational geometry approach, skin model shape, modal representation, and others. Among the most commonly used are variational geometry approaches. In these approaches, the form defects are neglected. These approaches are based on the parameterization of deviations from theoretic geometry. The real geometry of parts is considered by a variation of nominal dimension or it is bounded by a variation (position and orientation) of the nominal geometry (Gupta, & Turner, 1993). The orientation and position deviations of each surface could be represented by TTRS (Clément, Rivière, Serré, & Valade, 1998), kinematic formulation (Sacks, & Joskowicz, 1997), small displacement torsor (SDT) (Bourdet, Mathieu, Lartigue, & Ballu, 1996), matrix representation (Hillyard, & Braid, 1978), or vectorial tolerancing (Wirtz, 1991). Thirteen different constraints for dimensioning and tolerancing were proposed by Clemént et al. (1998). The geometric or dimensioning tolerances are represented by deviation domain (Joskowicz, Sacks, & Srinivasan, 1997), Tolerance-Map (Mujezinovic, Davidson, & Shah, 2004) or specification hull (Dantan, Mathieu, Ballu, & Martin, 2005). The TTRS model, the Deviation Domain model and the T-map model were analysed and compared by Ameta et al. (2011). The methods are quite similar in their aim, using different mathematical approaches. No model is fully complete when it comes to representing the tolerancing standard but provide a good support for tolerance analysis. Specifically, modelling of form errors is not possible.
Methods to model part variation by superposing different variation modes had been proposed by Chase et al. (1995, 1996, 2004). Manufacturing signature for tolerance analysis was addressed in (Corrado, & Polini, 2017a; Corrado, & Polini, 2017b; Corrado, Polini, & Moroni, 2017). The skin model is a comprehensive framework that includes position, orientation and form defects (Schleich, Anwer, Mathieu, & Wartzack, 2014). The modal representation method of geometrical deviation decomposition has extensively been studied. A discrete-cosine transformation (DCT) based on decomposition method for form defects modelling was proposed in (Huang, & Ceglarek, 2002). The Discrete Modal Decomposition (DMD) considering modal shapes of a discretized feature was developed by (Samper, Petit, & Giordano, 2004).