Statistical Tolerance Analysis—A Survey on Awareness, Use and Need in German Industry

The importance of geometric deviations of components for the aesthetic and functional quality of products has been undisputed for decades. So, it is not surprising that not only have numerous researchers devoted themselves to this field, but also commercial software tools for the analysis and optimization of tolerance specifications (currently already fully integrated in 3D-CAD systems) have been available for around 30 years. However, it is even more surprising that the well-founded specification of tolerances and their analysis using a so-called statistical tolerance analysis are only established in a few companies. There is thus a contradiction between the proclaimed relevance of tolerances and their actual consideration in everyday business life. Thus, the question of the significance of geometric deviations and tolerances as well as the use of statistical tolerance analysis arises. Therefore, a survey among 102 German companies was carried out. The results are presented and discussed in this paper.

Statistical Tolerance Analysis of 3D-Printed Non-Assembly Mechanisms in Motion Using Empirical Predictive Models

Fused Deposition Modelling (FDM) enables the fabrication of entire non-assembly mechanisms within a single process step, making previously required assembly steps dispensable. Besides the advantages of FDM, the manufacturing of these mechanisms implies some shortcomings such as comparatively large joint clearances and geometric deviations depending on machine-specific process parameters. The current state-of-the-art concerning statistical tolerance analysis lacks in providing suitable methods for the consideration of these shortcomings, especially for 3D-printed mechanisms. Therefore, this contribution presents a novel methodology for ensuring the functionality of fully functional non-assembly mechanisms in motion by means of a statistical tolerance analysis considering geometric deviations and joint clearance. The process parameters and hence the geometric deviations are considered in terms of empirical predictive models using machine learning (ML) algorithms, which are implemented in the tolerance analysis for an early estimation of tolerances and resulting joint clearances. Missing information concerning the motion behaviour of the clearance affected joints are derived by a multi-body-simulation (MBS). The exemplarily application of the methodology to a planar 8-bar mechanism shows its applicability and benefits. The presented methodology allows evaluation of the design and the chosen process parameters of 3D-printed non-assembly mechanisms through a process-oriented tolerance analysis to fully exploit the potential of Additive Manufacturing (AM) in this field along with its ambition: ‘Print first time right’.

Statistical tolerance analysis applied on overconstrained mechanisms with form deviations

One method for modeling geometric variations in hyperstatic (i.e. overconstrained) systems is to use sets of constraints. Different models have been developed in this way, e.g. domains, T-maps, and polytopes. In general, if the intersection of the contact constraints between two parts potentially in contact is nonempty, the parts can be assembled without interference, and their relative positions determined. In this study, the polytope method is used with a statistical approach to define the behavior of an assembly. In the first part, geometric variations including form deviations of individual parts are defined. The relations between these variations resulting from the architecture of a mechanism are then defined. In the second part, contact constraints are introduced and the general method to conform the constraints into double description polytopes is presented. The general process to simulate the compliance of the mechanism with respect to functional conditions is described. A failure rate is obtained for a simulated population of manufactured parts using the Monte Carlo method. In the third part, an application to a flange is described, an example from an industrial case study. We show how to take advantage of double description of polytopes when simulating the assembly and the misalignment of the two parts that make up the flange. Finally, we present our conclusions and prospects for future studies.

Statistical Tolerance Analysis With Sensitivities Established From Tolerance-Maps and Deviation Spaces

Math models aid designers in assessing relationships between tolerances that contribute to variations of a dependent dimension that must be controlled to achieve some design function at a target (functional) feature. The Tolerance-Maps© (T-Maps©) model for representing limits to allowable manufacturing variations is applied to identify the sensitivity of a dependent dimension to each contributing tolerance of the relationship. For each contributing feature and tolerances specified on it, the appropriate T-Map is chosen from a library of T-Maps, each represented in its own respective local reference frame. Each chosen T-Map is then transformed to the coordinate frame at the target feature, and the accumulation T-Map of these is formed with the Minkowski sum. The shape of a functional T-Map/deviation space is circumscribed (fitted) to this accumulation map. Since fitting is accomplished numerically by intersecting geometric shapes, T-Maps/deviation spaces are constructed with linear half-spaces. The sensitivity for each tolerance-and-feature combination is determined by perturbing the tolerance, refitting the functional shape to the modified accumulation map, and forming a ratio of the increment of functional tolerance to the perturbation. Taking tolerance-feature combinations one by one, sensitivities for an entire stack can be built. For certain loop equations, the same sensitivities result by fitting the functional shape to the T-Map/deviation space for each feature, without a Minkowski sum, and forming the overall result as a scalar sum. Sensitivities are used to optimize tolerance assignments by identifying the tolerances that most strongly influence the dependent dimension at the target feature. Form variations are not included in the analysis.

Statistical Tolerance Analysis of Over-Constrained Mechanical Assemblies With Form Defects Considering Contact Types

Tolerance analysis aims toward the verification impact of the individual tolerances on the assembly and functional requirements of a mechanism. The manufactured products have several types of contact and are inherent in imperfections, which often causes the failure of the assembly and its functioning. Tolerances are, therefore, allocated to each part of the mechanism in purpose to obtain an optimal quality of the final product. Three main issues are generally defined to realize the tolerance analysis of a mechanical assembly: the geometrical deviations modeling, the geometrical behavior modeling, and the tolerance analysis techniques. In this paper, a method is proposed to realize the tolerance analysis of an over-constrained mechanical assembly with form defects by considering the contacts nature (fixed, sliding, and floating contacts) in its geometrical behavior modeling. Different optimization methods are used to study the different contact types. The overall statistical tolerance analysis of the over-constrained mechanical assembly is carried out by determining the assembly and the functionality probabilities based on optimization techniques combined with a Monte Carlo simulation (MCS). An application to an over-constrained mechanical assembly is given at the end.

Tolerance Analysis of Over-Constrained Assembly Considering Gravity Influence: Constraints of Multiple Planar Hole-Pin-Hole Pairs

Over-constrained assembly of rigid parts is widely adopted in aircraft assembly to yield higher stiffness and accuracy of assembly. Unfortunately, the quantitative tolerance analysis of over-constrained assembly is challenging, subject to the coupling effect of geometrical and physical factors. Especially, gravity will affect the geometrical gaps in mechanical joints between different parts, and thus influence the deviations of assembled product. In the existing studies, the influence of gravity is not considered in the tolerance analysis of over-constrained assembly. This paper proposes a novel tolerance analysis method for over-constrained assembly of rigid parts, considering the gravity influence. This method is applied to a typical over-constrained assembly with constraints of multiple planar hole-pin-hole pairs. This type of constraints is non-linear, which makes the tolerance analysis more challenging. Firstly, the deviation propagation analysis of an over-constrained assembly is conducted. The feasibility of assembly is predicted, and for a feasible assembly, the assembly deviations are determined with the principle of minimum potential energy. Then, the statistical tolerance analysis is performed. The probabilities of assembly feasibility and quality feasibility are computed, and the distribution of assembly deviations is estimated. Two case studies are presented to show the applicability of the proposed method.