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Research Papers

Statistical Tolerance Allocation for Tab-Slot Assemblies Utilizing Tolerance-Maps

[+] Author and Article Information
Gaurav Ameta

School of Mechanical and Materials Engineering, Washington State University, Pullman, WA 99163gameta@wsu.edu

Joseph K. Davidson

School of Mechanical, Aerospace, Chemical and Materials Engineering, Arizona State University, Tempe, AZ 85827-6106j.davidson@asu.edu

Jami J. Shah

School of Mechanical, Aerospace, Chemical and Materials Engineering, Arizona State University, Tempe, AZ 85827-6106

J. Comput. Inf. Sci. Eng 10(1), 011005 (Feb 16, 2010) (13 pages) doi:10.1115/1.3249576 History: Received July 17, 2008; Revised July 30, 2009; Published February 16, 2010; Online February 16, 2010

A new mathematical model for representing the geometric variations of tabs/slots is extended to include probabilistic representations of 1D clearance. The 1D clearance can be determined from multidimensional variations of the medial-plane for a slot or a tab, and from variations of both medial-planes in a tab-slot assembly. The model is compatible with the ASME/ANSI/ISO Standards for geometric tolerances. Central to the new model is a Tolerance-Map (Patent No. 6963824) (T-Map), a hypothetical volume of points that models the range of 3D variations in location and orientation for a segment of a plane (the medial-plane), which can arise from tolerances on size, position, orientation, and form. Here it is extended to model the increases in yield that occur when the optional maximum material condition (MMC) is specified and when tolerances are assigned statistically rather than on a worst-case basis. 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 tab and slot. A comparison is made between the effects of specifying the optional MMC and not specifying it with the tolerance that determines the allowable variations in position of a tab, a slot, or of both in a tab-slot assembly. Statistical tolerance assignment for the tab-slot assembly is computed based on initial worst-case tolerances and for (a) constant size of tab and slot at maximum material condition, and (b) constant virtual-condition size.

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Copyright © 2010 by American Society of Mechanical Engineers
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Figures

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Figure 1

(a) Dimensions ℓ, dx, and dy, tolerance t, and the tolerance-zone for a rectangular target face. dy>dx. (b) The Tolerance-Map for the tolerance-zone shown in (a); σ1σ2=Oσ3=t and Oσ4′=δt.

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Figure 2

The size and position tolerances, related by the modifier M◯, on two parts that share the common datum A□ and are to be engaged with a tab and a slot. (a) The part with the tab. (d) The part with the slot. (b) and (c) The exaggerated ranges of positions for the lower and upper surfaces of the tab at the MMC and at the LMC, respectively. (e) and (f) The corresponding ranges for the slot.

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Figure 3

Four-dimensional T-Maps for the tab in Fig. 2. (a) Drawn for the modifier M◯ specified and drawn with an exaggerated scale in the τ-direction. Basis-point σ5 corresponds to the VC-size for the tab. (b) Drawn when the modifier M◯ is missing, i.e., for RFS; basis-point σ5 is now at infinity in the direction of τ.

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Figure 4

The adjoined four-dimensional T-Maps for tabs and slots that are to be engaged in an assembly and are made to the specifications in Fig. 2. The common VC-size is 10.0 mm at vertex σ5.

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Figure 5

The slot on Part 2 engaging the tab on Part 1 that is regarded as a go-nogo gauge, i.e., with the tab at its true position and made at the VC-size (10.0 mm) of the slot. (a) The two parts. (b) An instance of the slot showing zero minimum clearance with the tab. (c) A different instance of the slot, now showing a value of clearance c=c′ and the clearance boundary that identifies all places around the gauge-tab where this value of c could be located.

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Figure 6

A slot made to give a clearance c′ with the tab-gauge. The clearance boundary and the corresponding clearance-zone c′ of height tin−2c′ for a slot cut in Part 2 (Fig. 2). The VC-size (Figs.  45) is the 10.0 mm size of the gauge-tab used to measure the position of the slot.

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Figure 7

Relative frequency distribution (PDF) of clearance when the variations arise only from position of the medial-plane of the slot. Constructed from the tolerances in Fig. 2: 0≤c≤tin/2.

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Figure 8

Infinite 4D pyramidal hypersurface S (shaded) and its intersection with the 4D T-Maps in Fig. 3 for the slot. Both intersections are shown at a clearance c′ for (a) RFS or (b) MMC. For simplicity, the 3D cross sections of both S and the T-Map at each feature size are represented notionally by 2D cross sections σ1σ4′σ2σ8′ (Fig. 1).

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Figure 9

Comparison of the PDFs for clearance for the slot in Fig. 2 when MMC is applied and when it is not (RFS). Constructed from the tolerances in Fig. 2: 0≤c≤(tin+τin)/2.

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Figure 10

A notional representation of the adjoined 4D Tolerance-Maps in Fig. 4. The shaded regions are the hypersurfaces S for the tab (left) and slot (right), placed such that clearance contribution c′in for the slot is zero. The 3D bases of the hyperpyramids have been shown notionally in two dimensions so as to make the intent of the shadings more apparent.

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Figure 11

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.  22.

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Figure 12

Comparison of PDFs of clearance for engagement of tabs and slots when MMC is applied and MMC is not applied, i.e., RFS. Constructed from the tolerances in Fig. 2; 0≤c≤(tex+τex+tin+τin)/2.

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Figure 13

Comparison of PDFs of one-dimensional clearance for engagement of gauge-tab and slot and gauge pin-and-hole. The variations arise only from position-variations (position tolerance 0.5 mm) of the medial-plane (for the slot) and axis (for the hole). The corresponding tab and the pin are considered as gauges.

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Figure 14

Comparison of PDFs of one-dimensional clearance for engagement of tab and slot and pin-and-hole. The variations arise from position-variations (position tolerance 0.5 mm) and size-variation (0.5 mm) of the slot and the hole. The tab and the pin are considered as gauges of VC-sizes.

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