A statistical tolerance analysis approach for over-constrained mechanism based on optimization and Monte Carlo simulation

Abstract

Tolerancing decisions can profoundly impact the quality and cost of the mechanism. To evaluate the impact of tolerance on mechanism quality, designers need to simulate the influences of tolerances with respect to the functional requirements. This paper proposes a mathematical formulation of tolerance analysis which integrates the notion of quantifier: ''For all acceptable deviations (deviations which are inside tolerances), there exists a gap configuration such as the assembly requirements and the behavior constraints are verified'' & ''For all acceptable deviations (deviations which are inside tolerances), and for all admissible gap configurations, the assembly and functional requirements and the behavior constraints are verified''. The quantifiers provide a univocal expression of the condition corresponding to a geometrical product requirement. This opens a wide area for research in tolerance analysis. To solve the mechanical problem, an approach based on optimization is proposed. Monte Carlo simulation is implemented for the statistical analysis. The proposed approach is tested on an over-constrained mechanism.

Tolerance analysis — Form defects modeling and simulation by modal decomposition and optimization

“…These mathematical representations of tolerances allowed computing accumulation of the tolerances by Minkowski sums or intersections and the operations depended on the kinematic chains. Based on the definition of the probability of defects due to the manufacturing imperfections by different techniques which aim to provide suitable quality level of designed mechanical systems at lower costs, some tolerance analysis models have also been proposed which are based on the statistical analysis method [4,5,46,47].…”

“…The approach is developed for a required parametrization of surfaces to deal with defect probability computation in an assembly simulation procedure. The assembly simulation is moreover, currently based on constrained optimization problems and defect probability assessment [4,47,58]. To add value to the existing research, the proposed MMD integrates form defects into simulation.…”

“…The assembly requirement was given by Qureshi et al [4]: ''For all admissible deviations, there exists a gap configuration such as the assembly requirements and the behavior constraints are respected''. The technique to compute the requirement verification is to minimize one constraint with respect to all geometrical constraints.…”

“…Moreover, tolerancing activities, when simulating geometric deviations, are generally based on the hypothesis that parts are ideals and surfaces have no form defects [4][5][6][7]. The consideration of parts' form defects is integral to the tolerancing activities and hence, make it significant to highlight its impacts on the assembly probability estimation in the product quality and cost assessment [8,9].…”

“…By applying the statistical analysis (e.g., based on Monte-Carlo simulation), the tolerances can ensure acceptability of a certain large number of assemblies [3,4]. The worst-case methods, on the other hand, generate a functional assembly from the combination of dimensions and tolerances of individual components.…”