Profile Tolerances Modeling: A Unified Framework for Representing Geometric Variations for Line Profiles
Authors: Chen Jiao , Li Yuan , Yu Jianfeng , Tang Wenbin 
Published: March 2016

Tolerance modeling is the most basic issue in Computer Aided Tolerancing (CAT). It will negatively influence the performance of subsequent activities such as tolerance analysis to a great extent if the resultant model cannot accurately represent variations in tolerance zone. According to ASME Y14.5M Standard [1], there is a class of profile tolerances for lines and surfaces which should also be interpreted correctly. Aim at this class of tolerances, the paper proposes a unified framework called DOFAS for representing them which composed of three parts: a basic DOF (Degrees of Freedom) model for interpreting geometric variations for profiles, an assessment method for filtering out and rejecting those profiles cannot be accurately represented and a split algorithm for splitting rejected profiles into sub profiles to make their variations interpretable. The scope of discussion in this paper is restricted to the line profiles; we will focus on the surface profiles in forthcoming papers.
From the DOF model, two types of errors result from the rotations of the features are identified and formulized. One type of the errors is the result of the misalignment between profile boundary and tolerance zone boundary (noted as type 1); and if the feature itself exceeds the range of tolerance zone the other type of errors will form (noted as type 2). Specifically, it is required that the boundary points of the line profile should align with the corresponding boundary lines of the tolerance zone and an arbitrary point of the line profile should lie within the tolerance zone when line profile rotates in the tolerance zone.
To make DOF model as accurate as possible, an assessment method and a split algorithm are developed to evaluate and eliminate these two type errors. It is clear that not all the line features carry the two type errors; as such the assessment method is used as a filter for checking and reserving such features that are consistent with the error conditions. In general, feature with simple geometry is error-free and selected by the filter whereas feature with complex geometry is rejected. According to the two type errors, two sub-procedures of the assessment process are introduced. The first one mathematically is a scheme of solving the maximum deviation of rotation trajectories of profile boundary, so as to neglect the type 1 error if it approaches to zero. The other one is to solve the maximum deviation of trajectories of all points of the feature: type 2 error can be ignored when the retrieved maximum deviation is not greater than prescribed threshold, so that the feature will always stay within the tolerance zone.
For such features rejected by the filter which are inconsistent with the error conditions, the split algorithm, which is spread into the three cases of occurrence of type 1 error, occurrence of type 2 error and concurrence of two type errors, is developed to ease their errors. By utilizing and analyzing the geometric and kinematic properties of the feature, the split point is recognized and obtained accordingly. Two sub-features are retrieved from the split point and then substituted into the DOFAS framework recursively until all split features can be represented in desired resolution. The split algorithm is efficient and self-adapting lies in the fact that the rules applied can ensure high convergence rate and expected results.
Finally, the implementation with two examples indicates that the DOFAS framework is capable of representing profile tolerances with enhanced accuracy thus supports the feasibility of the proposed approach.