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RESEARCH PAPERS

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

[+] Author and Article Information
Gaurav Ameta

Department of Mechanical and Aerospace Engineering, Arizona State University, Tempe, AZgameta@asu.edu

Joseph K. Davidson

Department of Mechanical and Aerospace Engineering, Arizona State University, Tempe, AZj.davidson@asu.edu

Jami J. Shah

Department of Mechanical and Aerospace Engineering, Arizona State University, Tempe, AZ

Preliminary work was done by Chyi-Hwang Lim while he was an undergraduate student at Arizona State University.

J. Mech. Des 129(8), 782-792 (Dec 21, 2006) (11 pages) doi:10.1115/1.2717226 History: Received February 03, 2006; Revised December 21, 2006

## Abstract

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.

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## Figures

Figure 1

A single part of the picture frame with a point-line cluster at each end; in 2-D the target face and face B appear as lines. Dimensions (such as those in an end view) and tolerances (such as on dimension w) that do not enter the analysis are purposely omitted for brevity. For convenience, datum symbols A and B are used for identification; A becomes a datum only in Sec. 4. (a) The part with one basic dimension, three size dimensions, and with two size tolerances; the corresponding tolerance-zones are shown as per the Standard (1). (b) Same as (a) except that the tolerance-zone for t is rectangular for use in constructing the functional map in Sec. 3. (c) Coordinate frames i and j in the individual tolerance-zones for t and T, and the frame k that shows the position of the tolerance-zone t for a general displacement in the tolerance-zone T on angular size.

Figure 2

(a) The four chosen basis-clusters within the tolerance-zone of Fig. 1. (b) The corresponding basis-tetrahedron in which triangles (Sσ)1(Sσ)4(Sσ)2 and (Sσ)1(Sσ)3(Sσ)2 are isosceles and at right angles to one another and length (Sσ)1(Sσ)2=O(Sσ)3=2O(Sσ)4=tcosθt.

Figure 3

The picture frame assembly (shown with an ideal match-up at the target cluster of the assembly). Each part is in the form of Fig. 1 with θ=θt=45deg.

Figure 4

The functional T-Map for the tolerance zones for point S and line σ of the cluster shown in Fig. 2

Figure 5

T-Maps for both tolerance-zones in Fig. 1 when represented in local coordinates. (a) For the target cluster and the target line. (b) For the angular size tolerance on a line of length 2w.

Figure 6

(a) The T-Map (line-segment) in Fig. 5 for tolerance T, now redirected and rescaled to conform with the target cluster in Fig. 5. (b) The composite T-Map for the target point-line cluster on one part which includes variations at all tolerance zones in Fig. 1.

Figure 7

The assembly of two of the parts in Fig. 1

Figure 8

The tolerance-zones for Part 1 and the coordinate frames at the tolerance zones of both the parts in Fig. 7. Frame k shows the position of the target point-line cluster for general displacements in the tolerance-zones on Part 1.

Figure 9

Three Tolerance-Maps for Part 1, all conformable to the target cluster on Part 2. (a) The T-Map for linear size tolerance t applied to the angled face (in 2-D) of Part 1; points α1 and α2 are in the ys-plane. (b) The T-Map for angular size tolerance T of Part 1. (c) The composite T-Map for Part 1 obtained as the Minkowski sum of (a) and (b).

Figure 10

The entire picture-frame showing the tolerance-zones for Parts 1 and 2 and the coordinate frames i and k at the target point-line cluster on Part 4. Frame k is aligned with the position of the target cluster for general displacements in the tolerance-zones on both Parts 1 and 2.

Figure 11

Three T-Maps for Part 1, all conformable to the target cluster on Part 4. (a) The T-Map for linear size tolerance t applied to the angled face (in 2-D) of Part 1; points α1 and α2 are in the ys-plane. (b) The T-Map for angular size tolerance T of Part 1; the projections of the full length of the line-segment onto the q′- and s-axes are wT√2 and wT∕√2, respectively. (c) The composite T-Map for Part 1 obtained as the Minkowski sum of (a) and (b).

Figure 12

Three T-Maps for Part 2, all conformable to the target cluster on Part 4. (a) The T-Map for linear size tolerance t applied to the angled face (in 2-D) of Part 2; points α1 and α2 are in the ys-plane. (b) The T-Map for angular size tolerance T of Part 2. (c) The composite T-Map for Part 2 obtained as the Minkowski sum of (a) and (b).

Figure 13

Solid models of the accumulation T-Map (at right) for the four-part assembly in Fig. 3, and the conformable T-Maps of the individual parts 1, 2, 3, and 4 (l. to r.) from Figs.  111296. Drawn for ℓ∕w=4, tolerances t and T the same on all four parts, and T=t∕2w.

Figure 14

Fit of the accumulation and functional T-Maps. (a) Top view. (b) Front view. (c) Side view. (d) The q′-s sections of the accumulation T-Map and each of the individual part T-Maps from Figs.  691112 located center-to-end. Four vectors show the Minkowski sum for one point on an edge (shown in bold) that is common to the functional and accumulation T-Maps. Drawn for T=t∕2w.

Figure 15

The shape of the picture frame for ℓ=2w

Figure 16

The fill-ratio in terms of size-tolerance T and orientational tolerance t″. Length ℓ=100mm, ℓ∕w=4, and t=0.25mm. The labels along the t″-axis are rounded.

Figure 17

Differential elements for computing the volume of one octant of the functional T-Map. (a) Section in the sy-plane. (b) Isometric view.

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