Research Papers

Effects of Size, Orientation, and Form Tolerances on the Frequency Distributions of Clearance Between Two Planar Faces

[+] Author and Article Information
Gaurav Ameta

School of Mechanical and Materials Engineering, Washington State University, Pullman, WA 99163-2960gameta@wsu.edu

Joseph K. Davidson, Jami J. Shah

School of Mechanical, Aerospace, Chemical, and Materials Engineering, Arizona State University, Tempe, AZ 85287-6106j.davidson@asu.edu

J. Comput. Inf. Sci. Eng 11(1), 011002 (Mar 30, 2011) (10 pages) doi:10.1115/1.3503881 History: Received November 23, 2009; Revised August 04, 2010; Published March 30, 2011; Online March 30, 2011

A new mathematical model for representing the geometric variations of a planar surface is extended to include probabilistic representations for a 1D dimension of interest, which can be determined from multidimensional variations of the planar surface on a part. 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. 6963824), a hypothetical volume of points that models the 3D variations in location and orientation of a feature, which can arise from tolerances on size, position, orientation, and form. The 3D variations of a planar surface are decomposed into manufacturing bias, i.e., toward certain regions of a Tolerance-Map, and into geometric bias that can be computed from the geometry of T-Maps. The geometric bias arises from the shape of the feature, the tolerance-zone, and the control used on the mating envelope. Influence of manufacturing bias on the frequency distribution of 1D dimension of interest is demonstrated with two examples: the multidimensional truncated Gaussian distribution and the uniform distribution. In this paper, form and orientation variations are incorporated as subsets in order to model the coupling between size and form variations, as permitted by the ASME Standard when the amounts of these variations differ. Two distributions for flatness, i.e., the uniform distribution and the Gaussian distribution that has been truncated symmetrically to six standard deviations, are used as examples to illustrate the influence of form on the dimension of interest. The influence of orientation (parallelism and perpendicularity) refinement on the frequency distribution for the dimension of interest is demonstrated. Although rectangular faces are utilized in this paper to illustrate the method, the same techniques may be applied to any convex plane-segment that serves as a target face.

Copyright © 2011 by American Society of Mechanical Engineers
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Figure 1

The basis-tetrahedron with its basis points

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Figure 2

The tolerance-zone on size for the end of a rectangular bar and a coordinate frame centered within it

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Figure 3

The T-Map for the size tolerance shown in Fig. 2. Distance σ1σ2=t.

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Figure 4

(a) The T-Map for size on the length of a round bar. Distance σ1σ2=t. (b) The trade-off between the array of subsets for form and their companion locations within the T-Map of (a). This figure was prepared by Mr. S. Ramaswami while he was at the Design Automation Laboratory at ASU.

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Figure 5

Tolerance-Maps for (a) circular and (b) rectangular planar faces with both size and parallelism tolerances specified

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Figure 6

Tolerance-Maps of a rectangular plane with both size and perpendicularity tolerances specified

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Figure 7

T-Map for the planar rectangular end of the bar in Fig. 2 to which orientation tolerances for both parallelism and perpendicularity have been applied. δ=dy/dx.

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Figure 8

(a) Cross section of an assembly with clearance c that is bounded by congruent rectangular faces on both parts. Nominal dimensions are shown. (b) Possible tolerances on part 2 of the assembly shown in (a).

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Figure 9

(a) The yz-section of the tolerance-zone shown in Fig. 2. (b) The q′s-section of the T-Map shown in Fig. 3. Clearance c′ measures the distance between planes σ2 and σ′2.

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Figure 10

The Tolerance-Map for the target face showing the shaded pyramidal area S that represents the frequency of occurrence for clearance value c′

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Figure 11

Frequency distribution of clearance (plot of Eq. 4) constructed from the three-dimensional variations of the target plane

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Figure 12

The array of subsets for form (upper dipyramids) and their companion possibilities for location (lower dipyramids) within the T-Map of Fig. 2. The Pw-curve shows the linear trade-off between the two arrays, and the truncated PDFs represent illustrative distributions for form variation: Gaussian over the size zone t and uniform up to the limit t′.

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Figure 13

Comparison of the relative frequency distributions (plots of PDFs) of clearance corresponding to the three-dimensional variations of the target plane of part 2 in Fig. 8. The distribution of form subsets is assumed to be uniform and Gaussian while the range of variation of the form subset varies from 0 to t′. Size tolerance t2=t=1.0 mm (Fig. 8).

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Figure 14

The Tolerance-Map when size and parallelism tolerances are applied to the target face. The shaded pyramidal area represents the frequency of occurrence for clearance value c′=σ′2σ2.

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Figure 15

Comparison of relative frequency distributions (PDFs) of clearance for different parallelism tolerances (in mm) applied to part 2 in Fig. 8. In all cases, t2=1 mm.

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Figure 16

The Tolerance-Map for size, parallelism, and perpendicularity tolerances applied to the target face of part 2 in Fig. 8. The shaded pyramidal area represents the frequency of occurrence for one clearance value c′.

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Figure 17

Comparison of relative frequency distributions (PDFs) of clearance for different values of parallelism and perpendicularity tolerances applied to part 2 in Fig. 8

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Figure 18

(a) Notional representation of manufacturing bias using the trivariate Gaussian distribution of weights within the T-Map. (b) Illustration of an infinitesimal area on the surface S and its corresponding weight from the trivariate Gaussian distribution, both to be used for computation of the frequency for a particular clearance.

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Figure 19

Comparison of the relative frequency distributions of clearance for two distributions of manufacturing bias: uniform and Gaussian. Only the size tolerance for part 2 in Fig. 8 is considered.



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