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Research Papers

Statistical Tolerance Analysis With Sensitivities Established From Tolerance-Maps and Deviation Spaces

[+] Author and Article Information
Aniket N. Chitale

Design Automation Laboratory,
Department of Mechanical and Aerospace Engineering,
Arizona State University,
Tempe, AZ 85287-6106
e-mail: achitale@asu.edu

J. K. Davidson

Design Automation Laboratory,
Department of Mechanical and Aerospace Engineering,
Arizona State University,
Tempe, AZ 85287-6106
e-mail: j.davidson@asu.edu

Jami J. Shah

Honda Professor of Engineering Design,
Department of Mechanical and Aerospace Engineering,
The Ohio State University,
Columbus, OH 43210
e-mail: shah.493@osu.edu

1Corresponding author.

Contributed by the Computers and Information Division of ASME for publication in the Journal of Computing and Information Science in Engineering. Manuscript received August 7, 2018; final manuscript received January 31, 2019; published online May 16, 2019. Assoc. Editor: Kristina Wärmefjord.

J. Comput. Inf. Sci. Eng 19(4), 041002 (May 16, 2019) (11 pages) Paper No: JCISE-18-1200; doi: 10.1115/1.4042838 History: Received August 07, 2018; Accepted February 01, 2019

Math models aid designers in assessing relationships between tolerances that contribute to variations of a dependent dimension that must be controlled to achieve some design function at a target (functional) feature. The Tolerance-Maps© (T-Maps©) model for representing limits to allowable manufacturing variations is applied to identify the sensitivity of a dependent dimension to each contributing tolerance of the relationship. For each contributing feature and tolerances specified on it, the appropriate T-Map is chosen from a library of T-Maps, each represented in its own respective local reference frame. Each chosen T-Map is then transformed to the coordinate frame at the target feature, and the accumulation T-Map of these is formed with the Minkowski sum. The shape of a functional T-Map/deviation space is circumscribed (fitted) to this accumulation map. Since fitting is accomplished numerically by intersecting geometric shapes, T-Maps/deviation spaces are constructed with linear half-spaces. The sensitivity for each tolerance-and-feature combination is determined by perturbing the tolerance, refitting the functional shape to the modified accumulation map, and forming a ratio of the increment of functional tolerance to the perturbation. Taking tolerance-feature combinations one by one, sensitivities for an entire stack can be built. For certain loop equations, the same sensitivities result by fitting the functional shape to the T-Map/deviation space for each feature, without a Minkowski sum, and forming the overall result as a scalar sum. Sensitivities are used to optimize tolerance assignments by identifying the tolerances that most strongly influence the dependent dimension at the target feature. Form variations are not included in the analysis.

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Figures

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Fig. 1

(a) The tolerance zone for the planar end face of a rectangular bar and (b) its corresponding T-Map [4]. (c) T-Map for the face of a round bar with a size tolerance, and (d) with an additional parallelism tolerance t′′. Adapted from Refs. [2], [3], and [20].

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Fig. 2

(a) Tolerance zone for position (tolerance t) of the axis of a hole or pin. (b)–(e) Four 3D hypersections of the corresponding T-Map: ϕ′ = 0, Δy = 0, ψ′ = 0, and Δx = 0, respectively [20,25]. Every circle has diameter t. Redrawn and adapted from Ref. [25].

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Fig. 3

(a) Cross sections of the T-Map of feature i, of the Minkowski sum M–i from T-Maps for all other features of one tolerance loop and of a circumscribed functional shape. (b) Same as (a) but with a perturbed parallelism tolerance ti + Δti. (c)–(d) Magnified portions of (a) and (b). (e) Magnified perturbation changes.

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Fig. 4

(a) Dimensions for the two similar triangles from Figs. 3(d) and 3(e). (b) A part requiring the Minkowski sum when computing sensitivities.

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Fig. 5

(a) Front and underside views of the two-part assembly, two directions of control, and dependent dimensions Dfy and Dfz. (b) Dimensions and tolerances on the front and underside views for the two parts of the assembly in (a).

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Fig. 6

Coordinate frames for the vertical direction of control

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Fig. 7

T-Maps/deviation spaces for round face 7 in Fig. 5(b) (Sec. 4.1). (a) In its local frame. (b) Transformed to the global frame. (c) Functional shape circumscribing (b).

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Fig. 8

T-Maps/deviation spaces for rectangular face 8 in Fig. 5(b) (Sec. 4.1). (a) In its local frame. (b) Transformed to the global frame. (c) Functional shape circumscribing (b).

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Fig. 9

Coordinate frames for the horizontal direction of control

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Fig. 10

Two views of transformed T-Map/deviation space for rectangular face 8 (Fig. 5(b)) circumscribed by the functional shape (Sec. 4.2)

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Fig. 11

Cross section of T-Maps from which Eqs. (23) and (24) were derived [2]. Characteristic length is d10 (not d10/2) and labels are changed from those in Ref. [2] to be consistent with the labels in Fig. 5(b). Adapted from Ref. [2].

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