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.
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
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.
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
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”
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)
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
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
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