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Research Papers

Tolerance Analysis and Allocation for Design of a Self-Aligning Coupling Assembly Using Tolerance-Maps

[+] Author and Article Information
Gagandeep Singh

Salesforce.com, Inc.,
The Landmark @ One Market,
Suite 300,
San Francisco, CA, 94105

Gaurav Ameta

School of Mechanical and Materials Engineering,
Washington State University,
Pullman, WA 99164
e-mail: gameta@wsu.edu

Joseph K. Davidson

e-mail: j.davidson@asu.edu

Jami J. Shah

School of Mechanical, Aerospace, Chemical, and Materials Engineering,
Arizona State University,
Tempe, AZ 85287-6106

Contributed by the Design Automation Committee of ASME for publication in the Journal of Mechanical Design. Manuscript received September 14, 2010; final manuscript received November 26, 2012; published online February 7, 2013. Assoc. Editor: Karthik Ramani.

J. Mech. Des 135(3), 031005 (Feb 07, 2013) (14 pages) Paper No: MD-10-1355; doi: 10.1115/1.4023279 History: Received September 14, 2010; Revised November 26, 2012

A self-aligning coupling is used as a vehicle to show that the Tolerance-Map (T-Map) mathematical model for geometric tolerances can distinguish between related and unrelated actual mating envelopes as described in the ASME/ISO standards. The coupling example illustrates how T-Maps (Patent No. 6963824) may be used for tolerance assignment during design of assemblies that contain non-congruent features in contact. Both worst-case and statistical measures are obtained for the variation in alignment of the axes of the two engaged parts of the coupling in terms of the tolerances. The statistical study is limited to contributions from the geometry of toleranced features and their tolerance-zones. Although contributions from characteristics of manufacturing machinery are presumed to be uniform, the method described in the paper is robust enough to include different types of manufacturing bias in the future. An important result is that any misalignment in the coupling depends only on tolerances, not on any dimension of the coupling.

Copyright © 2013 by ASME
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References

American National Standard ASME Y14.5M, 2009, Dimensioning and Tolerancing, The American Society of Mechanical Engineers, New York.
Davidson, J. K., Mujezinović, A., and Shah, J. J., 2002, “A New Mathematical Model for Geometric Tolerances as Applied to Round Faces,” ASME J. Mech. Des., 124, pp. 609–622. [CrossRef]
Mujezinović, A., Davidson, J. K., and Shah, J. J., 2001, “A New Mathematical Model for Geometric Tolerances as Applied to Polygonal Faces,” ASME J. Mech. Des., 126, pp. 504–518. [CrossRef]
Davidson, J. K., and Shah, J. J., 2002, “Geometric Tolerances: A New Application for Line Geometry and Screws,” IMechE J. Mech. Eng. Sci., Part C, 216(C1), pp. 95–104. [CrossRef]
Bhide, S., Davidson, J. K., and Shah, J. J., “A New Mathematical Model for Geometric Tolerances as Applied to Axes,” Proceedings of ASME Design Technical Conferences, Chicago, IL, #DETC2003/DAC-48736 (CDROM).
Ameta, G., Davidson, J. K., and Shah, J. J., 2007, “Tolerance-Maps Applied to a Point-Line Cluster of Features,” ASME J. Mech. Des., 29, pp. 782–792. [CrossRef]
Giordano, M., Kataya, B., and Samper, S., 2001, “Tolerance Analysis and Synthesis by Means of Clearance and Deviation Spaces,” Geometric Product Specification and Verification, Proceedings of the 7th CIRP International Seminar on Computer-Aided Tolerancing, P.Bourdet and L.Mathieu, eds., Ecole Norm. Supérieure, Cachan, France, April 24–25, Kluwer, Dordrecht, pp. 345–354.
Giordano, M., Pairel, E., and Samper, S., 1999, “Mathematical Representation of Tolerance Zones,” Global Consistency of Tolerances, Proceedings of the 6th CIRP International Seminar on Computer-Aided Tolerancing, F.vanHouten and H.Kals, eds., University of Twente, Enschede, Netherlands, Mar. 22–24, Kluwer, pp. 177–186.
Roy, U., and Li, B., 1999, “Representation and Interpretation of Geometric Tolerances for Polyhedral Objects–I: Form Tolerance,” Comput.-Aided Des., 30, pp. 151–161. [CrossRef]
Roy., U., and Li., B., 1999, “Representation and Interpretation of Geometric Tolerances for Polyhedral Objects–II: Size, Orientation, and Position Tolerances,” Comput.-Aided Des., 31, pp. 273–285. [CrossRef]
Pasupathy, T. M. K., Morse, E. P., and Wilhelm, R. G., 2003, “A Survey of Mathematical Methods for the Construction of Geometric Tolerance Zones,” J. Comput. Inf. Sci. Eng., 3, pp. 64–75. [CrossRef]
Hong, Y. S., and Chang, T. C., 2002, “A Comprehensive Review of Tolerancing Research,” Int. J. Prod. Res., 40(11), pp. 2425–2459. [CrossRef]
Mathieu, L., and Villeneuve, F., 2010, Geometric Toleracing of Products, Wiley, New York, (Adapted and updated from Tolerancement geometrique des produits, 2007, Hermes Science/Lavoisier).
Whitney, D. E., Gilbert, O. L., and Jastrzebski, M., 1994, “Representation of Geometric Variations Using Matrix Transforms for Statistical Tolerance Analysis in Assemblies,” Res. Eng. Des., 6, pp. 191–210. [CrossRef]
LeeS., and Yi, C., 1998, “Statistical Representation and Computation of Tolerance and Clearance for Assemblability Evaluation,” Robotica, 16, pp. 251–264. [CrossRef]
Lehtihet, E. A., and Gunasena, U. N., 1988, “Models for the Position and Size Tolerance of a Single Hole,” Manufacturing Metrology, ASME PED-29, Presented at the Winter Annual Meeting of the ASME, Chicago, Illinois, November, pp. 49–63.
Teissandier, D., Couétard, Y., and Gérard, A., 1999, “A Computer Aided Tolerancing Model: Proportioned Assembly Clearance Volume,” Comput.-Aided Des., 31, pp. 805–817. [CrossRef]
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Giordano, M., Petit, J., and Samper, S., 2003, “Minimum Clearance for Tolerance Analysis of a Vacuum Pump,” CD-ROM, 8th CIRP Seminar on Computer Aided Tolerancing, Charlotte, NC.
Ameta, G., Davidson, J. K., and Shah, J. J., 2004, “The Effects of Different Specifications on the Tolerance-Maps for an Angled Face,” Proceedings of American Society of Mechanical Engineering—Design Engineering Technical Conference, Salt Lake City, Utah, Sept. 28–Oct. 2, Paper No. DAC-57199.
Banchoff, T. F., 1996, Beyond the Third Dimension: Geometry, Computer Graphics, and Higher Dimensions, Scientific American Library, New York.
Ameta, G., Davidson, J. K., and Shah, J. J., 2010, “Statistical Tolerance Allocation for Tab-Slot Assemblies Using Tolerance-Maps,” J. Comput. Inf. Sci. Eng., 10(1), p. 011005. [CrossRef]
Uicker, J. J., Pennock, G. R., and Shigley, J. E., 2003, Theory of Machines and Mechanisms, 3rd ed., Oxford, New York.
Ameta, G., Davidson, J. K., and Shah, J. J., 2007, “Using Tolerance-Maps to Generate Frequency Distributions of Clearance and Allocate Tolerances for Pin-Hole Assemblies,” ASME J. Comput. Inf. Sci. Eng., 7(4), pp. 347–359. [CrossRef]
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Singh, G., Ameta, G., Davidson, J. K., and Shah, J. J., 2009, “Worst-Case Tolerance Analysis of a Self-Aligning Coupling Assembly using Tolerance-Maps,” Proc., 11th CIRP Int'l Conference on Computer-Aided Tolerancing, ed. F. Villeneuve and M. Giordano, March 26–27, Annecy, France. (CDROM).

Figures

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Fig. 1

(a) Top and front views of a plate with a pin of length ℓ in it. The position tolerance for the pin is t = 0.1 mm. (b) Tolerance zone for the position variation of the axis of the pin specified in Fig. 1(a). (c) The 3D hypersection P = 0 (one of four of this shape, see Ref. [5]) of the 4D T-Map that represents the range of the position variation of an axis. The only edges are the two circles shown; both have diameter t.

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Fig. 2

(a) A slot with size and position tolerances and with MMC specified. (b) Tolerance zone for the position variation of the medial plane. (c) The allowable ranges in location of the upper and lower surfaces of the slot when the size of the slot is at MMC- or LMC-size, respectively.

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Fig. 3

The dipyramidal Tolerance-Map for the rectangular tolerance-zone in Fig. 2(b). Coordinates p′ and q′ measure small rotations about the y- and x-axes, respectively.

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Fig. 4

The 4D truncated hyperpyramidal Tolerance-Map for all the variations of the slot in Fig. 2(a). 3D dipyramids form the large and small bases.

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Fig. 5

A simplified form of the 5D truncated hyper “pyramidal” Tolerance-Map for the variations of the pin in Fig. 1(a). For simplicity, the LQ-sections at MMC and LMC are notional representations of the 4D T-Map described in Fig. 1(c).

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Fig. 6

The self-aligning coupling in which three equally spaced pins on Part 2 engage three slots on Part 1 to coaxially align the short cylinder B on Part 2 with the big cylinder D on Part 1. The axis of the big cylinder (Datum D) is the datum for the assembly and the axis of short cylinder (Datum B) on Part 2 is the target feature for the assembly.

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Fig. 7

The left sides of one engaged MMC pin and MMC slot in Fig. 6, as viewed from the center of rotation, with the theoretical medial line for the pin incident with the theoretical medial plane for the slot. (a) With variations p′  =  s  =  L′  =  Q  =  0, the related and unrelated envelopes coincide, so c  =  u  =  (tin + tex)/2. (b) With variations p′  =  –s  =  tin/4, L′  =  Q  =  tex/4, the related and unrelated envelopes are distinct, c  =  tin/2, and u  =  tin.

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Fig. 8

The 3D T-Map for a single engaged pin and slot for cmin = 0. The size of a pin or slot progresses continuously from left to right. The variable ΔF represents the width of a slot less the diameter of its engaged pin; for the tolerances in Fig. 6, it ranges from 0.8 mm to 1.8 mm. The virtual condition size is 7.2 mm.

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Fig. 9

Relationships among the LMC sizes and tolerances for one engaged pin and slot in Fig. 6 for u = c = 0 (corresponds to the envelopes in Fig. 7(a)). The dashed circle shows the pin contacting the other side of the slot when the two parts and their theoretical medial features have been displaced fully.

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Fig. 10

Variations in position of the theoretical center P of the pattern of pins on Part 2 which arise from variations of position and size only for Pin 1 and Slot 1

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Fig. 11

Geometry to determine the variations in position of the theoretical center of pattern of pins on Part 2 which arise from variations in position of one pin-slot pair and the clearance between them. The exaggerated displacement LM, shown for point G, is 2ΔFmax, the same as the range of displacement shown in Fig. 10.

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Fig. 12

Application of the Minkowski Sum to yield the hexagonal T-Map for the end point of the axis of the short cylinder B on Part 2 relative to the axis for Datum D on Part 1

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Fig. 13

The functional T-Map of diameter tf circumscribing the accumulation T-Map, both for the target point on cylinder B of Part 2. Also shown is the accumulation T-Map (dashed lines) when the coupling is used as a unidirectional drive.

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Fig. 14

Unrelated envelopes for MMC sizes of the pin and slot, as viewed from the center of rotation. Dimensions in mm are for the tolerances specified in Fig. 6. (a) The pin and slot shown centered (points $1 and σ0 in Fig. 15) on the theoretical medial line and plane, and with the pin shifted leftward by the radial clearance (tin +  tex)/2; the related and unrelated envelopes coincide. (b) The pin made at its leftward positional limit and the slot made at its rightward limit (points $8 and σ1 in Fig. 15); shift of the datum features is zero. The related and unrelated envelopes coincide. (c) The pin and slot each shown centered and made with its maximum CW tilt, and shown (long dashed lines) with the pin shifted leftward by the amount tin. The short dashed lines are for a CCW tilted slot; no shift is possible.

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Fig. 15

A portion of Fig. 8, showing the subset of basis-points used in its constructing the two T-Maps for the medial plane and axis. The x− and +-locations correspond to the unrelated actual mating envelopes for the MMC sizes in Fig. 7(b) and the last row of Table 1, respectively. The dashed lines represent Eqs. (12) and (13).

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Fig. 16

The T-Maps in Fig. 8 augmented with notional representations for surfaces 4Sex and 3Sin (shaded) that are used in generating the frequency distribution for unrelated clearance u.

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Fig. 17

The Tolerance-Map for the medial plane of the slot showing the shaded area 2Sin formed as the intersection of the V-shaped trough with the dipyramidal T-Map in Fig. 3. Area 2Sin represents the frequency of occurrence for clearance value uin. The ray-pair for one size of slot is represented by σ1σ4′ and σ1σ8′. Labeling is for the MMC size.

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Fig. 18

(a) Wire frame view with labels for the shaded area obtained as one central section (P = 0) of the 3D solid formed as the intersection of the inverted right-angled 3D hypertrough with the 4D T-Map for the axis of a pin. (b) and (c) Wire frame and solid views from a computer-aided design (CAD) CAD model.

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Fig. 19

Relative frequency distributions of the unrelated clearances u′ for the pin and slot. Clearance values are for the tolerances specified in Fig. 6.

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Fig. 20

Relative frequency distribution of total unrelated clearance u′ between one engaged pin-slot pair.

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Fig. 21

(a) Three relative frequency distributions for unrelated clearance along the three directions at right-angles to the engaged pin-slot pairs for the unidirectional drive in Fig. 6. (b) The superimposed circular functional T-Map, dashed hexagonal accumulation T-Map (Fig. 13), and projected contours of the resultant bi-variate relative frequency distribution. Cylinder 1Smis shows the locus of all misalignments of the same amount (radius m′i) between Datum axes B and D.

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Fig. 22

The bi-variate relative frequency distribution arranged over the plane of the hexagonal accumulation T-Map

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Fig. 23

Relative frequency distribution of misalignment of the axes in the coupling when it is used as a mechanical drive

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