Joseph K. Davidson

A New Mathematical Model for Geometric Tolerances as Applied to Round Faces

A new mathematical model for representing geometric tolerances is applied to polygonal faces and is extended to show its sensitivity to the precedence (ordering) of datum reference frames. The model is compatible with the ASME/ISO Standards for geometric tolerances. Central to the new model is a Tolerance-Map® 2 , a hypothetical volume of points that corresponds to all possible locations and variations of a segment of a plane which can arise from tolerances on size, position, form, and orientation. Every Tolerance-Map is a convex set. This model is one part of a bi-level model that we are developing for geometric tolerances. The new model makes stackup relations apparent in an assembly, and these can be used to allocate size and orientational tolerances; the same relations also can be used to identify sensitivities for these tolerances. All stackup relations can be met for 100% interchangeability or for a specified probability. Methods are introduced whereby designers can identify trade-offs and optimize the allocation of tolerances. Examples are presented that illustrate important features of the new model.

A Comparative Study Of Tolerance Analysis Methods

This paper reviews four major methods for tolerance analysis and compares them. The methods discussed are (1) 1D tolerance charts, (2) variational analysis based on Monte Carlo simulation, (3) vector loop (or kinematic) based analysis, and (4) ASU T-Maps© based tolerance analysis. Tolerance charts deal with tolerance analysis in one direction at a time and ignore possible contributions from the other directions. Manual charting is tedious and error-prone, hence attempts have been made for automation. Monte Carlo simulation based tolerance analysis is based on parametric solid modeling; its inherent drawback is that simulation results highly depend on the user-defined modeling scheme, and its inability to obey all Y14.5 rules. The vector loop method uses kinematic joints to model assembly constraints. It is also not fully consistent with Y14.5 standard. ASU T-Maps based tolerance analysis method can model geometric tolerances and their interaction in truly 3-dimensional context. It is completely consistent with Y14.5 standard but its use by designers may be quite challenging. T-Maps based tolerance analysis is still under development. Despite the shortcomings of each of these tolerance analysis methods, each may be used to provide reasonable results under certain circumstances. No guidelines exist for such a purpose. Through a comprehensive comparison of these methods, this paper will develop some guidelines for selecting the best method to use for a given tolerance accumulation problem.Copyright

Navigating the Tolerance Analysis Maze

AbstractThis paper classifies and reviews geometric tolerance analysis methods and software. Two of the most popular methods, 1-D Min/Max Charts and Parametric Simulation, are reviewed in detail. The former is fully consistent with international tolerance standards but limited to decoupled analysis of variations in one direction at a time. It is also hard to automate. The latter can handle variations in all directions but is not fully compatible with the standards. The results are highly dependent on the expertise of the analyst. New methods are emerging to overcome these problems. One such method called T-maps is also reviewed. It maps geometric variations from physical space to 3, 4, or 5 dimensional virtual space and computes tolerance accumulations using Minkowski sums. The method so far has shown to be consistent with tolerance standards and can handle all tolerance classes applicable to planar and cylindrical features.

A mechanism for validating dimensioning and tolerancing schemes in CAD systems

Abstract Dimensioning and tolerancing are inter-related tasks; the validity of the tolerance representation model is dependent on the dimension model. Validation of tolerances is necessary at input and also after any changes of the dimension scheme. This paper presents a computational model for validating the dimensioning scheme and tolerance specifications compatible with dimensioning and tolerancing practice. Validation is implemented through a dimension and tolerance graph. Constrained entities are combined into progressively expanding clusters, which can be used to represent datum reference frames (DRFs), constraint groups of geometric entities, patterns, or the entire part. Validation of all dimensioning and tolerancing relations is performed using the concept of control frames, theoretical dimensions and general rules for forming clusters. The tolerance validation method proposed here supports all tolerance classes defined in the ISO/ANSI/ASME standards, as well as DRFs and special pattern and profile entity relations.

Tolerance-Maps Applied to a Point-Line Cluster of Features

In this paper, groups of individual features, i.e., a point, a line, and a plane, are called clusters and are used to constrain sufficiently the relative location of adjacent parts. A new mathematical model for representing size and geometric tolerances is applied to a point-line cluster of features that is used to align adjacent parts in two-dimensional space. First, tolerance-zones are described for the point-line cluster. A Tolerance-Map® (Patent no. 69638242), a hypothetical volume of points, is then established which is the range of a mapping from all possible locations for the features in the cluster. A picture frame assembly of four parts is used to illustrate the accumulations of manufacturing variations, and the T-Maps® provide stackup relations that can be used to allocate size and orientational tolerances. This model is one part of a bilevel model that we are developing for size and geometric tolerances. At the local level the model deals with the permitted variations in a tolerance zone, while at the global level it interrelates all the frames of reference on a part or assembly.

Improvements to algorithms for computing the Minkowski sum of 3-polytopes

A Minkowski sum is a geometric operation that is equivalent either to the vector additions of all points in two operands or to the sweeping of one operand around the profile of the other without changing the relative orientation. Applications of Minkowski sums are found in computer graphics, robotics, spatial planning, and CAD. This paper presents two algorithms for computing Minkowski sum of convex polyhedron in three space (3-polytopes). Both algorithms are improvements on current ones found in the literature. One is based on convex hulls and the other on slope diagrams. The original convex hull based Minkowski algorithm is costly, while the original slope diagram based algorithms require the operation of stereographic projection from 3D to 2D for merging the slope diagrams of the two operands. Implementation of stereographic projection is complicated which increases the computation time and reduces the accuracy of the geometric information that is needed for constructing the resultant solid. This paper reports on improvements that have been made to these two algorithms and their implementation. These improvements include using vector operations to find the interrelations between points, arcs and regions on a unit sphere for the slope diagram algorithm, and addition of a pre-sorting procedure before constructing convex hull for convex hull based Minkowski sum algorithm. With these improvements, the computation time and complexity for both algorithms have been reduced significantly, and the computational accuracy of the slope diagram algorithm has been improved. This paper also compares these two algorithms to each other and to their original counterparts. The potential for extending these algorithms to higher dimensions is briefly discussed.

Geometric tolerances: A new application for line geometry and screws

Abstract A new mathematical model is introduced for the tolerances of cylindrical surfaces. The model is compatible with the ISO/ANSI/ASME standard for geometric tolerances. Central to the new model is a Tolerance-Map®†, a hypothetical volume of points that corresponds to all possible locations and variations of a segment of a line (the axis) that can arise from tolerances on size, location and orientation of the cylindrical surface. Each axis in a tolerance zone will be represented with the six Plücker coordinates. Cylindrical surfaces in a tolerance zone for the same hole can then be treated by attaching a size tolerance to each of the lines, thereby forming a screw. Relationships for the content of line solids for a tolerance zone are developed to correspond to the variations of locations. These are then used to obtain a measure for the increment in cost when a more refined tolerance is specified. This model is one part of a bilevel model that is under development for geometric tolerances.

Tolerance-Maps Applied to the Straightness and Orientation of an Axis

Among the least developed capabilities in well-developed mathemati cal models for geometric tolerances are the representation of toleranc es on form, orientation, and of Rule #1 in the Standards, i.e. the coupling between form and allowable var iations for either size or position of a feature. This paper uses Tolerance-Maps ®1 (T-Maps ®1 ) to describe these aspects of geometric tolerances for the straightness and orientation of an axis within its tolerance-zone on position. A Tolerance-Map is a hypothetical poi nt-space, the size and shape of which reflect all variational possibilities for a targ et feature; for an axis, it is constructed in four-dimensional space. The Tolerance-Map for straight ness is modeled with a geometrically similar, but smaller-sized, four-dimensional s hape to the 4D shape for position; it is a subset within the T-Map for position. Another interna l subset describes the displacement possibilities for the subset T-Maps that limits for m. The T-Map for orientation and position together is formed most reliably by trun cating the T-Map for position alone.

Using Tolerance-Maps to Generate Frequency Distributions of Clearance and Allocate Tolerances for Pin-Hole Assemblies

A new mathematical model for representing the geometric variations of lines is extended to include probabilistic representations of one-dimensional (1D) clearance, which arise from positional variations of the axis of a hole, the size of the hole, and a pin-hole assembly. The model is compatible with the ASME/ ANSI/ISO Standards for geometric tolerances. Central to the new model is a Tolerance-Map (T-Map) (Patent No. 69638242), a hypothetical volume of points that models the 3D variations in location and orientation for a segment of a line (the axis), which can arise from tolerances on size, position, orientation, and form. Here, it is extended to model the increases in yield that occur when maximum material condition (MMC) is specified and when tolerances are assigned statistically rather than on a worst-case basis; the statistical method includes the specification of both size and position tolerances on a feature. The frequency distribution of 1D clearance is decomposed into manufacturing bias, i.e., toward certain regions of a Tolerance-Map, and into a geometric bias that can be computed from the geometry of multidimensional T-Maps. Although the probabilistic representation in this paper is built from geometric bias, and it is presumed that manufacturing bias is uniform, the method is robust enough to include manufacturing bias in the future. Geometric bias alone shows a greater likelihood of small clearances than large clearances between an assembled pin and hole. A comparison is made between the effects of choosing the optional material condition MMC and not choosing it with the tolerances that determine the allowable variations in position.

Comparison of Two Similar Mathematical Models for Tolerance Analysis: T-Map and Deviation Domain

The major part of production cost of a manufacturing product is set during the design stage and especially by the tolerancing choice. Therefore, a lot of work involves trying to simulate the impact of these choices and provide an automatic optimization. For integrating this modeling in computer aided design (cad) software, the tolerancing must be modeled by a mathematical tool. Numerous models have been developed but few of them are really efficient. Two advanced models are “T-map” model developed by Joseph K. Davidson and “deviation domain” developed by Max Giordano. Despite the graphical representation of these two models seems to be similar, they have significant differences in their construction and their resolution method. These similarities and differences highlight the needs of tolerancing modeling tool in each kind of problems, especially in case of assembly with parallel links.