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RESEARCH PAPERS

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

[+] Author and Article Information
Gaurav Ameta

Department of Mechanical and Aerospace Engineering, Arizona State University, Tempe, AZ 85287ameta@asu.edu

Joseph K. Davidson

Department of Mechanical and Aerospace Engineering, Arizona State University, Tempe, AZ 85287j.davidson@asu.edu

Jami J. Shah

Department of Mechanical and Aerospace Engineering, Arizona State University, Tempe, AZ 85287

Plücker coordinates can be augmented with a pitch to form a screw or torsor.

J. Comput. Inf. Sci. Eng 7(4), 347-359 (Jul 13, 2007) (13 pages) doi:10.1115/1.2795308 History: Received January 17, 2007; Revised July 13, 2007

## Abstract

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.

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## Figures

Figure 11

Notional representation of the 5D Tolerance-Map for the engagement of the pin-hole assembly, here drawn for cmin=0. The shaded regions are the hypersurfaces S for the pin (left) and hole (right), placed such that clearance contribution cin′ for the hole is zero. As an example, the numerical LMC and MMC sizes shown correspond to the dimensions and tolerances in Fig. 1.

Figure 1

Two plates, one with a pin (a) at the left and one with a hole (b) at the right. Both are positioned with the tolerance t=0.1mm.

Figure 2

A cylindrical tolerance zone for the axis of a hole determined by circles C and C¯, each with its diameter equal to the positional tolerance t. Basis lines $1⋯$5 are shown, together with four others that complete the symmetry. Lines $1⋯$3 are perpendicular to Datum A.

Figure 3

Four three-dimensional hypersections of the T-Map and its basis 4-simplex for the tolerance-zone on position in Fig. 2; all circles are of diameter t and all squares have diagonals of length t. (a) The central hypersection λ5=0. (b) The hypersection λ3=0. (c) The hypersection λ4=0. (d) The hypersection λ2=0.

Figure 4

5D T-Maps for a cylindrical surface in which the 4D T-Map at each end of the T-Maps is represented notionally with a 2D cross-section from Figs. 3 or 3 (adapted from Ref. 23); increasing size is from left to right. (a) For the modifier M◯ specified, and drawn so that basis point $6 corresponds to the VC size for the hole. (b) For RFS (no modifier M◯); basis-point$6 is now at infinity in the direction of size.

Figure 10

Comparison of the PDF’s for clearance for the hole when MMC is applied and when it is not (RFS). Constructed from the tolerances in Fig. 1: 0⩽c⩽(tin+τin)∕2.

Figure 12

Geometrical representation of computation of the frequency distribution of c′ as the joint frequency distribution of cex and cin. Note that f1 and f2 are identical distributions here only because the same tolerances are used in Figs.  11.

Figure 13

Comparison of two PDF's of clearance for engagement of the pin and hole when MMC is applied and MMC is not applied, i.e., RFS. Constructed from the tolerances in Fig. 1: 0⩽c⩽(tex+τex+tin+τin)∕2.

Figure 5

(a) True position of the hole in Part 2, and Part 1 treated as a go-nogo gage. (b) An instance of the hole showing zero minimum radial clearance. (c) A different instance of the hole, now showing a minimum radial clearance c=c′ and the clearance boundary that identifies all places around the gage-pin where this value of clearance might be located.

Figure 6

Clearance boundary and the corresponding axis zone c′ (bounded by the c′ axis cylinder) of diameter tin−2c′ for a hole in Part 2 of MMC-size. VC is virtual condition size. The infinitesimal axis zone is drawn at an exaggerated scale.

Figure 7

(a) The nested hypersections of the 4D surfaces of the T-Map (here shown notionally with just one of the hypersections in Fig. 3) that correspond to the three frequencies for c′, c′+h and c′+2h. (b) Three corresponding successive instances of axis zones in the tolerance zone (drawn with exaggerated diameters) for increasing values of c′, i.e., c′, c′+h and c′+2h.

Figure 8

Relative frequency distribution (PDF) of clearance when the variations arise only from position of the axis of the hole which is made at the MMC size. Constructed from the tolerances in Fig. 1: 0⩽c⩽tin∕2.

Figure 9

Infinite 5D hyper“pyramidal” surface S (shaded) and its intersection with the 5D T-Maps in Fig. 4 for the hole. Both intersections are shown at a radial clearance c′ for (a) RFS or (b) MMC. For simplicity, the L′Q cross sections are notional representations of the entire 4D T-Map depicted in Fig. 3.

## Errata

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