This document provides an overview of ASME Y14.5.1, which establishes a mathematical definition of dimensioning and tolerancing principles. It introduces key concepts such as datum reference frames, vectors, and common tolerance zones like circularity, cylindricity, and flatness. Mathematical definitions are provided for these concepts. It also covers topics like the assumptions behind datum reference frames, interpreting true position tolerances, and profile tolerances for lines and surfaces.

1. ASME Y14.5.1m
By Hassan Habib

2. Mathematical Definition of
Dimensioning and Tolerancing
Principles
Introduction - ASME Y14.5.1
Mathematical Possibilities of
DRFs
Datum Reference Frames
Vectors, Addition, Subtraction,
Scalar Product, Vector Product
Some Mathematics
Circularity, Cylindricity,
Flatness Mathematical
Definition
Common Tolerance Zones
Definition, Use and
Interpretation of Position,
Profile of Line and Surface
Circular and Total Runout
Overview of Location
Lessons Learned
References
Conclusions
Table Of Contents

3. Introduction - ASME Y14.5.1
 Developments of 1980s
 GIDEP Alert 1988
 National Science Foundation
 ASME Meetings (1989~)
 Introduction to the Standard
- Mathematical Definition of
Dimensioning and Tolerancing
Principles
 Important Considerations

4. Introduction - ASME Y14.5.1
Developments of 1980s
 Advent of Computers in Manufacturing Industry
 Decreasing Costs of CMMs
 Integration with PCs
 Invention of Touch Trigger Probes
 Versatile Software Development
 Mismatch Between Different Proprietary Software

5. Introduction - ASME Y14.5.1
GIDEP ALERT, 1988
 Government Industry Data Exchange Program (GIDEP)
 Walker, 1988 tested CMM Software (Form Tolerances)
 Repeatability (Sampling, Strategy, stability, force)
 Flatness, Parallelism, Straightness, Perpendicularity
 Data set = Graphically solvable
 Qty=05 CMMs tested

6. Introduction - ASME Y14.5.1
GIDEP ALERT, 1988
 Results were shocking
 37% worse than actual, 50% better than actual
 Mr. Walker did not hide, he published the results
 Specification crises
 Grant from National Science Foundation
 ASME Board on Research and Development
 Recommended mathematical definition of mechanical tolerances

7. Introduction - ASME Y14.5.1
ASME Meetings, 1989
 ASME sub committee meeting in 1989
 Establishment of Ad hoc ASME Y14.5.1
 15 meetings – 5 years
 Publication of ASME Y14.5.1 in 1994
 First Ever Endeavor in this area
 Reiterated in 1999, 2004

8. Introduction - ASME Y14.5.1
Introduction - Mathematical Definition of Dimensioning and
Tolerancing Principles
 Reiteration of textual tolerance definitions of Y14.5
 Definition of geometric constraints
 Construction of mathematical DRFs
 Easy conversion to programming code

9. Introduction - ASME Y14.5.1
Important Considerations
 Distinction between “measured” and “actual” values
 “Actual” Value is inherently true (Measured perfectly)
 Perfect value can never be obtained
 “Measured” value is the estimated value generated by a
measurement system
 It has Uncertainty associated with it

10. Introduction - ASME Y14.5.1
Important Considerations
 Standard is based on “Actual” Values
 Applies to conceptual design phase
 Compromise between Unique Specification of tolerance &
eventual measurement method

11. Mathematics
 Vectors
 Unit & Position Vectors
 Vector Addition
 Vector Subtraction
 Vector Multiplication - Dot
 Vector Multiplication - Cross

12. Mathematics
Vectors
 Vector is an abstract geometric entity that has length and
magnitude
 In comparison with scalar
 Represented by an arrow on capital letters
𝐴, 𝐵, etc.

13. Mathematics
Vectors

14. Mathematics
Vectors

15. Mathematics
Unit & Position Vectors
 Unit Vector is of unit length, describes the direction of a
vector in a coordinate system
 Represented by Hat on Alphabets
𝐴, 𝐵, etc.
 Position Vector is a vector that describes position of a point
in reference coordinate system

16. Mathematics
Vector Addition & Subtraction

17. Mathematics
Vector Multiplication - Dot
 Vectors can be multiplied in two forms
 Dot Product (Scalar Multiplication)
 Cross Product (Vector Multiplication)
 Dot Product yields a scalar quantity.
 Cross Product yields a vector quantity

18. Mathematics
Vector Multiplication - Dot
 Definition:
 𝐴 ● 𝐵 = │ 𝐴 ││ 𝐵 │cos 𝜃
 If 𝐴 II 𝐵 , 𝐴 ● 𝐵 = max, 𝜃=0⁰
 If 𝐴 ⊥ 𝐵, 𝐴 ● 𝐵 = 0, 𝜃=90⁰
 [1,3,5] ● [2,3,4] = (1x2)+(3x3)+(5x4) = 31

19. Mathematics
Vector Multiplication - Dot

20. Mathematics
Vector Multiplication - Cross
 Definition:
 𝐴 x𝐵 = │ 𝐴 ││ 𝐵 │sin 𝜃 𝑛
 Where, 𝑛=Unit Vector ⊥ to the plane containing 𝐴 , 𝐵
 If 𝐴 II 𝐵 , 𝐴 x 𝐵 = 0, 𝜃=0⁰
 If 𝐴 ⊥ 𝐵, 𝐴 x 𝐵 = max, 𝜃=90⁰

21. Mathematics
Vector Multiplication - Cross
 Calculation:
 𝐴 x𝐵 = [a1 𝑖+a2 𝑗+a3 𝑘] x [b1 𝑖+b2 𝑗+b3 𝑘]
 𝐴 x𝐵 = det
𝑖 𝑗 𝑘
a1 a2 a3
b1 b2 b3

22. Datum Reference Frames (DRFs)
 Introduction
 Degrees of Freedom
 Assumptions
 Mathematical Possibilities
 Aggregated Possibilities

23. Datum Reference Frames (DRFs)
Introduction
 Definition:
“A coordinate system that is located and oriented on the
datum features of the part, and from which the location and
orientation of other part features are controlled”

24. Datum Reference Frames (DRFs)
Introduction

25. Datum Reference Frames (DRFs)
Introduction

26. Datum Reference Frames (DRFs)
Assumptions
 Two reasons DRF can yield more than 1 physical datum
 Referenced at MMC and is manufactured b/w MMC & LMC
 Inherent Form Errors
 Therefore, multitude of candidate datum reference frames
 Conclusion, Search for a DRF that yields features within defined
tolerance zones

27. Datum Reference Frames (DRFs)
Assumptions
 Datum is established before a feature is evaluated
 Smoothing of part surface is implied in this standard
 For distinguishing dimension from surface texture, roughness,
material microstructure etc.
 Rule # 1: Size controls the form applies
 Variation of size is based on “spine”

28. Datum Reference Frames (DRFs)
Assumptions
 Spine is a simple non intersecting curve
 0-Dimensional spine is ‘point’
 1-Dimensional spine is a ‘curve’ in space (cylindrical feature)
 2-Dimensional spine is a surface (two Parallel planes)
 ASME Y 14.5M- 1994 establishes a mathematical model of perfect planes,
cylinders, axes, etc. that interact with the infinite point set of imperfectly-
formed features.

29. Datum Reference Frames (DRFs)
Assumptions
 Part is fixed in space DRFs are established in relation to the part
 In contrast, ASME Y14.5 assumes that DRF is fixed and part is moved into
the DRF
 Does not apply to screw threads, gears, splines, or mathematically defined
surfaces (Sculptured Surfaces)

30. Datum Reference Frames (DRFs)
Mathematical Possibilities
 Conventions

31. Datum Reference Frames (DRFs)
Mathematical Possibilities
 Point as Primary Datum

32. Datum Reference Frames (DRFs)
Mathematical Possibilities
 Line as Primary Datum

33. Datum Reference Frames (DRFs)
Mathematical Possibilities
 Plane as Primary Datum

34. Datum Reference Frames (DRFs)
Mathematical Possibilities
 Aggregated Possibilities

35. Common Tolerance Zones
 Overview of Form Tolerances
 Circularity
 Cylindricity
 Flatness

36. Common Tolerance Zones
Overview of Form Tolerances
 Form tolerances refine the inherent form control imparted by a size tolerance
 They are not referenced from a datum reference frame
 They are not specified on a nominal feature
 Form tolerances are dependent on the on the characteristics of the tolerance
feature itself

37. Common Tolerance Zones
Circularity
 Circularity controls the form error of a sphere or
any other feature that has nominally circular
cross sections
 Cross sections exist on a spine
 Spine is a curve in space with continuous slope
(1st Derivative)
 Tolerance zone is on annular area on the cross
section plane, centered on spine

38. Common Tolerance Zones
Circularity
 Definition puts constraint on points denoted by 𝑃
 Point 𝐴 is on spine
 𝑇 is a unit vector (Tangent to the spine at 𝐴)
 Points are defined by:
 𝑇 • (𝑃- 𝐴) = 0
 𝑇 is ⊥ to 𝐴
 (𝑃- 𝐴) points from 𝐴 to 𝑃
 To restrict these points in tolerance zone t
 ||𝑃- 𝐴| - r | ≤
𝑡
2

39. Common Tolerance Zones
Cylindricity
 Cylindricity tolerance controls the form error of
cylindrically shaped features.
 Consists of a set of points existing in a pair of
coaxial cylinders
 Axis of the cylinder does not have any defined
orientation

40. Common Tolerance Zones
Cylindricity
 Definition puts constraint on points denoted by 𝑃
 Point 𝐴 is position vector for axis
 𝑇 is a unit vector (Defines cylindricity axis at 𝐴)
 Points are defined by:
 || 𝑇 x (𝑃- 𝐴) | - r | ≤
𝑡
2

41. Common Tolerance Zones
Flatness
 Flatness tolerance zone controls the form error
of a nominally flat feature
 Surface to be constrained by two parallel
planes

42. Common Tolerance Zones
Flatness
 Definition puts constraint on points denoted by 𝑃
 Point 𝐴 is an arbitrary locating point
 𝑇 is a unit vector (Defines normal to plane)
 Points are defined by:
 | 𝑇 • (𝑃- 𝐴) | ≤
𝑡
2

43. Overview of Location
 Assumptions, Definition, & Interpretation of
True Position
 Profile of Line & Surface
 Circular and Total Runout

44. Overview of Location
Assumptions, Definition, & Interpretation of TP
 Assumptions:
 Surface Interpretation
 Surface of the actual feature
 Resolved Geometry Interpretation
 Size and resolved geometry (Center Point, Axis,
or center plane) of applicable (Mating or Minimum
Material) actual envelope

45. Overview of Location
 Consider the hole with 0 TP at MMC
 The MMC VC has a Dia equal to MMC Dia of Hole
 Assume that manufactured hole is within limits of size
(LOS) (Does not violate LOS)
 It would be acceptable as per surface interpretation
 As per resolved geometry interpretation it would be
rejected (Hole is further away from TP than allowed
by combined effects of TP (zero) and bonus tolerance
resulting from actual mating size of hole
Assumptions, Definition, & Interpretation of TP

46. Overview of Location
 Conversely, consider the opposite
 Shaft is controlled by TP = t at MMC
 Radius of shaft = rAM and MMC radius is rMMC
 Radius of tolerance zone = rMMC-rAM+t/2
 Height of shaft = h
 Axis of actual shaft is tilted to extreme
 As per resolved geometry interpretation, the part is
acceptable
 Points 𝑃 lie outside the tolerance as per surface
interpretation
Assumptions, Definition, & Interpretation of TP

47. Overview of Location
 For the purposes of this standard all tolerances of location are
considered to apply to pattern of features (PLTZF)
 Definition
 A positional tolerance can be explained in terms of a zone
within which the resolved geometry (center point, axis or
center plane) of a Feature of Size is permitted to vary
from TP
 Notation r(𝑃) denotes the distance of points 𝑃 to the TP
Assumptions, Definition, & Interpretation of TP

48. Overview of Location
 In terms of Surface of a Feature
 Definition
 For a Pattern of Feature of Size, a TP specifies that the
surface of each actual feature must not violate the
boundary of a corresponding TP zone
 Each TP is volume defined by all points 𝑃 that satisfy:
 b = radius or half width
Assumptions, Definition, & Interpretation of TP

49. Overview of Location
 In terms of Surface of a Feature
Assumptions, Definition, & Interpretation of TP

50. Overview of Location
 In terms of Surface of a Feature
 Conformance
Assumptions, Definition, & Interpretation of TP

51. Overview of Location
 In terms of Resolved Geometry of Feature
 Definition
 For features within a pattern, a position tolerance specifies
that the resolved geometry (center point, axis, or center
plane, as applicable) of each actual mating envelope (for
features at MMC or RFS) or actual minimum material
envelope (for features at LMC) must lie within a
corresponding positional tolerance zone
 Each TP is volume defined by all points 𝑃 that satisfy r(𝑃) ≤ b
Assumptions, Definition, & Interpretation of TP

52. Overview of Location
 In terms of Resolved Geometry of Feature
 Definition
Assumptions, Definition, & Interpretation of TP

53. Overview of Location
 In terms of Resolved Geometry of Feature
 Conformance
Assumptions, Definition, & Interpretation of TP

54. Overview of Location
 Conical Tolerance Zones
 Bi-Directional Tolerance Zone
 Polar Bi-Directional Tolerance Zone
Assumptions, Definition, & Interpretation of TP

55. Overview of Location
 A profile is the outline of an object in a given plane (2D
figure). Profiles are formed by projecting a 3D figure onto a
plane or taking cross sections through the figure. The
elements of a profile are straight lines, arcs, and other curved
lines. With profile tolerancing, the true profile may be defined
by basic radii, basic angular dimensions, basic coordinate
dimensions, basic size dimensions, un-dimensioned
drawings, or formulas
Profile of Line & Surface

56. Overview of Location
 Definition:
 A profile tolerance zone is an area (profile of a line) or a
volume (profile of a surface) generated by offsetting each
point on the nominal surface in a direction normal to the
nominal surface at that point.
Profile of Line & Surface

57. Overview of Location
 For a given point 𝑃N on a nominal surface there is a unit
vector 𝑁, normal to the nominal surface either into or out of
material.
 A profile tolerance t consists of sum of two intermediate
tolerances t+ and t-. +ve and –ve disposition of tolerance in
surface normal 𝑁 at 𝑃N
Profile of Line & Surface

58. Overview of Location
 Conformance:
 Surface conforms to profile tolerance t0 if all points 𝑃S of the surface
conform to either of intermediate tolerances t+ or t- disposed about
some corresponding point 𝑃N on nominal surface
 𝑃S conforms to t+ if 𝑃S is between 𝑃N and 𝑃N + 𝑁t+
 𝑃S conforms to t- if 𝑃S is between 𝑃N and 𝑃N − 𝑁t+
 2 values are necessarily calculated: 1 for surface variations in positive
direction and 1 for negative direction. Actual value is the smallest
intermediate tolerance to which the surface conforms
Profile of Line & Surface

59. Overview of Location
 Runout is a composite tolerance used to control the functional
relationship of one or more features of a part to a datum axis. The
types of features controlled by runout tolerances include those
surfaces constructed around a datum axis1 and those
constructed at right angles to a datum axis2
 The mathematical definition of 1 and 2 are different
Runout

60. Overview of Location
 Evaluation
 Total Runout on tapered or contoured surfaces require establishment
of actual mating normal. Nominal Diameters, lengths, radii, and
angles establish cross sectional desired contour having perfect form
and orientation. It:
 may be translated axially and/or radially
 May not be tilted/scaled with respect to a datum axis
 When a tolerance band is equally disposed about this contour and
then revolved around datum axis, a volumetric tolerance zone is
generated.
Runout

61. Overview of Location
 Circular Runout
 Surfaces constructed at right angles to a datum axis
 The tolerance zone for each circular element on a surface
constructed at right angles to a datum axis is generated by revolving
a line segment about the datum axis.
Runout

62. Overview of Location
 For a surface point 𝑃S, a circular runout tolerance is a set of points
𝑃 satisfying ∶
 | 𝐷1 𝑥(𝑃- 𝐴) | = r and
 | 𝐷1 • (𝑃- 𝐵) | ≤
𝑡
2
Runout

63. Overview of Location
 Circular Runout
 Surfaces constructed around a Datum Axis
 The tolerance zone for each circular element on a surface
constructed around a datum axis is generated by revolving a line
segment about the datum axis.
Runout

64. Overview of Location
 For a surface point 𝑃S, a datum axis [ 𝐴 , 𝐷1 ] and a given mating
surface, a circular runout tolerance is a set of points 𝑃 satisfying ∶

𝐷1•(𝑃− 𝐵)
|𝑃− 𝐵|
= 𝐷1 • 𝑁 and
 | |𝑃− 𝐵|-d| ≤
𝑡
2
 𝑁 • (𝑃s− 𝐵) > 0
Runout

65. Overview of Location
Runout

66. Overview of Location
 Total Runout
 Surfaces constructed at right angles to a datum axis
 A total runout tolerance for a surface constructed at right angles to a
datum axis specifies that all points of the surface must lie in a zone
bounded by two parallel planes perpendicular to the datum axis and
separated by the specified tolerance
Runout

67. Overview of Location
 For a surface constructed at right angles to a datum axis, total runout
zone is a volume consisting of point 𝑃 satisfying ∶
 | 𝐷1 • (𝑃- 𝐵) | ≤
𝑡
2
 𝐷1 = direction vector for datum axis
 𝐵 = Position vector locating midplane of tolerance zone
 t = size of tolerance zone
Runout

68. Overview of Location
 Total Runout
 Surfaces constructed around a datum axis
 A total runout tolerance zone for a ,surface constructed around a
datum axis is a volume of revolution generated by revolving an area
about the datum axis.
Runout

69. Overview of Location
 For a surface point 𝑃S, a datum axis [ 𝐴 , 𝐷1 ], Let 𝐵 be a point on
datum axis locating one end of desired contour and r is distance from
datum axis to desired contour. Then for given 𝐵 and r, C(𝐵 ,r) denotes
the desired contour. For each C(𝐵 ,r) runout zone is a set of points
𝑃 satisfying ∶
 |𝑃− 𝑃′| ≤
𝑡
2
 𝑃′ = projection of 𝑃 onto surface generated by rotating C(𝐵 ,r) about
datum axis
 t = size of tolerance zone
Runout

70. Conclusions
 Lessons Learned
 References

71. Conclusions
 This standard is a mathematical translation of ASME Y14.5
 It is a guideline for software developers and users to understand how the
calculations are made
 The information helps in better comprehending the design intent in later
stages of product development, specially manufacturing and inspection
 The understanding helps in better decisions regarding “Acceptable” and
“Conforming” parts.
 It takes into account the theoretical calculations used to establish definitions
GD&T as defined in Y14.5
 It does not take into account measureability
Lessons Learned

72. Conclusions
 Basic Theme is a courtesy of SlideModel
 Dimensioning and tolerancing handbook by Paul J. Drake
 https://betterexplained.com/articles/cross-product/
 https://www.grc.nasa.gov/WWW/K-12/airplane/vectpart.html
 https://en.wikipedia.org/wiki/Multiplication_of_vectors
References