REAFFIRMED 2004

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FOR CURRENT COMMITTEE PERSONNEL PLEASE E-MAIL [email protected]

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AN AMERICAN NATIONAL STANDARD ENGINEERING DRAWINGAND RELATED DOCUMENTATION PRACTICES

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Mathematical Definition of Dimensioning and Tolerancing Principles

ASME Y14.5.1M-1994

The American Society of Mechanical Engineers 345 East 47th Street, New York, N.Y. 10017 Copyright ASME International Provided by IHS under license with ASME

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Date of Issuance: January 31, 1995

This Standard will be revised when the Society approves the issuance of a new edition. There will be no addenda or written interpretations of the requirements of this Standard issued to this Edition.

ASME is the registered trademark of The American Society of Mechanical Engineers. This code or standard was developed under procedures accredited as meeting the criteria for American National Standards. The Consensus Committee that approved the code or standard was balanced to assure that individualsfrom competent and concerned interests have had an opportunity to participate. The proposed code or standard was made available for public review and comment which provides an opportunity for additional public input from industry,academia, regulatory agencies, and the public-at-large. ASME does not "approve," "rate," or "endorse" any item, construction, proprietary device, or activity. ASME does not take any position with respect to the validity of any patent rights asserted in connection with any items mentionedin this document, and does not undertake to insure anyone utilizing a standard against liabilityfor infringement of any applicable Letters Patent, nor assume any such liability. Users of a code or standard are expressly advised that determination of the validity of any such patent rights, andthe risk of infringement of such rights, is entirely their own responsibility. Participation by federal agency representative(s1 or person(s) affiliated with industry is not to be interpreted as government or industry endorsementof this code or standard. ASME accepts responsibilityfor only those interpretationsissued in accordance with governing ASMEproceduresandpolicies whichprecludetheissuanceofinterpretationsbyindividual volunteers.

No part of this document may be reproducedin any form, in an electronic retrieval system or otherwise, without the prior written permission of the publisher. Copyright 0 1995 by THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS All Rights Resewed Printed in U.S.A --`,``,,`,`,,,,,`,``,`,,```,``,-`-`,,`,,`,`,,`---

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(This Foreword is not a part of ASME Y14.5.1M-1994.)

The Y14 Committee created the Y14.5.1 Subcommittee in response to a need identified during a National Science Foundation (NSF) workshop. The International Workshop on Mechanical Tolerancing was held in Orlando, Florida, in late 1988. The workshop report strongly identifiedaneed for amathematical definition for the current tolerancingstandards. Tom Charlton coined the phrase “mathematization of tolerances.” The phrase meant to add mathematical rigor to the Y 14.5M standard.The response is the present standard, ASME Y 14.5.1M1994. This new standard creates explicit definitions for use in such areas as Computer Aided Design (CAD) and Computer Aided Manufacturing (CAM). The Committee has met three times a year since their first meeting af January of1989in Long Boat Key, Florida.Initial discussionscovered scope of the document, boundary definitions, size, and datums. The Committee identified four major divisions of a tolerance: 1) the mathematical definition of the tolerance zone; 2) the mathematical definition of conformance to the tolerance; 3) the mathematical definition of the actual value; 4) the mathematical definition of the measured value. The Subcommittee later decided that the measured value was beyond the scope of this Standard. When this Standard defines part conformance, it consists of the infinite set of points that make up all the surfaces of the part, and it is addressing imperfect form semantics. This Standard does not fully address the issue of boundary, that is where one surface stops and the other surface starts. The Subcommittee hopes to define this in the next edition of this Standard. The definition of size took up many days of discussions and interaction with the Y14.5 Subcommittee. It always came back to the statement of a micrometer-type two point crosssectional measurement. The difficulty comes from the method of defining the cross-section. Consider a figure such as an imperfectly formed cylinder. When considering the infinite set of points that make up the surface, what is the intent behind a two point measurement? Most of the reasons appear to be for strength. Yet, a two pointcross-sectional definition doesn’t define strength on, for instance, a three-lobed part. These and other considerations led to the existing definition. The pictorial definition, presented in Section 2, is the smallest of the largest elastic perfect spheres thatcan be passed through the part without breaking the surface. ws Standard does not address measurement, yet often a two point cross-sectional measurement is adequate for form, fit, and function. The subject of datums also led to many hours of work by the Subcommittee. The current definitions, presented in Section 4, were theresult of evaluating a number of approaches against four criteria: 1) conformance to Y14.5M; 2) whether a unique datum is defined; 3) whether the definition is mathematically unambiguous; and 4) whether the definition conveys design for reasons intent. A fifth criterion, whetherthedefinitionwasmeasurable,wasnotused discussed above. The end result of this work was based feedback on from the Y 14.5M Subcommittee when Y14.5.1 presentedits analysis, and involved change a in its thinking aboutdatums. The initial view of a datum was as something established before a part feature is evaluated. The current definitions involve a different view that a datum existsfor the sake ofthe features related to it. The result was a consolidation of the issues involved with “wobbling” datums andthe issues involvedwith datum features of size at MMC or LMC. These apparently ...

Ill

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dissimilar issues are unified mathematicallyin the concepts of “candidate datum” and “candidate datum reference frame.” A special thanks to the Y14 Main Committee and Y14.5 Subcommittee members in their support and encouragement in the development of this Standard. Also of note are the participation and contributions of Professor Ari Requicha of University of Rensselaer Polytechnic Institute, and Professor of Southern California, Professor Josh Turner Herb Voelcker of Cornell University. Suggestions for improvement of this Standard are welcome. They should be sent to the American Society of Mechanical Engineers, Att: Secretary, Y14 Main Committee, 345 East 47th Street, New York, NY 10017. This Standard was approved as an American National Standard on November 14, 1994.

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ASME STANDARDS COMMllTEE Y14 Engineering Drawing and Related Documentation Practices (The following is the roster of the Committee at the time of approval of this Standard.)

OFFICERS P. E. McKim, Chair F. Bakos, Jr., Vice-Chair C. J. Gomez, Secretary

COMMIITEE PERSONNEL A. R. Anderson, Trikon Corp. F. Bakos, Jr., Eastman Kodak Co. T. D. Benoit, Alternate, Pratt b Whitney CEB D. E. Bowerman, Copeland Corp. J. V. Burleigh, The Boeing Co. L. Burns

R. A. Chaddedon, Southwest Consultants F. A. Christiana, ASEA Brown Boveri, Combustion Engineering Systems M. E. Curtis, Jr., Rexnord Corp. R. W. DeBolt, Motorola, Inc., Government and Space Technology Group H. L. Dubocq L. W. Foster, L. W. Foster Associates, Inc. C. J. Gomez, The American Society of Mechanical Engineers D. Hagler, E-Systems, Inc., Garland Div. E. L. Kardas, Pratt b Whitney CEB C. G. Lance, Santa Cruz Technology Center P. E. McKim, Caterpillar, Inc. C. D. Merkley, IBM Corp. E. Niemiec, Westinghouse Electric Corp. R. J. Poliui D. L. Ragon, Deere b Company, John Deere Dubuque Works R. L. Tennis, Caterpillar, Inc. R. P. Tremblay, US. Department of the Army, ARDEC R. K. Walker, Westinghouse Marine Division G. H. Whitmire, TEC/TREND K. E. Wiegandt, Sandia National Laboratory P. Wreede, E-Systems, Inc.

SUBCOMMllTEE 5.1 MATHEMATICALDEFINITIONOF DIMENSIONING AND TOLERANCING PRINCIPLES R. K. Walker, Chair, Westinghouse Marine Division T. H. Hopp, Vice-Chair, National Institute of Standards and Technology M. A. Nasson, Vice-Chair, The Charles Stark Draper Laboratory, Inc. A. M. Nickles, Secretary, The American Society of Mechanical Engineers V

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M. E. Algeo, National Institute of Standards and Technology R. E. Coombes, Caterpillar Inc. L. W. Foster, Lowell W. Foster Associates Inc. M. T. Gale, Giddings b Lewis Measurement Systems J. D. Guilford, Rensselaer Design Research Center R. J. Hocken, University of North Carolina R. K. Hook, Metcon J. Hurt, SDRC D. P. Karl, Ford Motor Co. C. G. Lance, Santa Cruz Technology Center J. D. Meadows, Institute for Engineering and Design A. G. Neumann, Technical Consultants, Inc. R. W. Nickey, Naval Warfare Assessment Center F. G. Parsons, Federal Products Co. K. L. Sheehan, Brown b Sharpe V. Srinivasan, IBM, Research Division B. R. Taylor, Renishaw PLC W. B. Taylor, Westinghouse Electric Corp. S. Thompson, Lawrence Livermore National Laboratory T. Woo, National Science Foundation

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CONTENTS iii

1 ScopeandDefinitions .................................................. 1.1 General ........................................................... 1.2 References......................................................... 1.3 Mathematical Notation ............................................. 1.4 Definitions ........................................................ 1.5 Summary ofConventionalDesignations ............................. 1.6 Format ............................................................

1 1 1 1 2 5 5

2

GeneralTolerancingandRelatedPrinciples ........................... 2.1 Feature Boundary .................................................. 2.2 DimensionOrigin .................................................. 2.3 Limits of Size .....................................................

7 7 7 7

3

Symbology .............................................................

11

4

DatumReferencing ..................................................... 4.1 General ........................................................... 4.2 Concepts .......................................................... 4.3 Establishing Datums ............................................... 4.4 Establishing Datum ReferenceFrames ............................... 4.5 Datum Reference Frames for Composite Tolerances .................. 4.6 Multiple Patterns of Features ....................................... 4.7 Tabulation ofDatum Systems.......................................

13 13 13 13 16 17 17 18

5

TolerancesofLocation .................................................. 5.1 General ........................................................... 5.2 Positional Tolerancing.............................................. 5.3 Projected Tolerance Zone ........................................... 5.4 Conical Tolerance Zone ............................................ 5.5 Bidirectional Positional Tolerancing ................................. 5.6 Position Tolerancing atMMC for Boundaries of Elongated Holes ..... 5.7 Concentricity andSymmetry ........................................

21

Tolerances of Form, Profile, Orientation, and Runout ................. 6.1 General ........................................................... 6.2 Formand Orientation Control....................................... 6.3 Specifying Formand Orientation Tolerances ......................... 6.4 Form Tolerances ................................................... 6.5 Profile Control..................................................... 6.6 Orientation Tolerances ............................................. 6.7 Runout Tolerance .................................................. 6.8 Free State Variation ................................................

35 35 35 35 35 39 40 45 48

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Foreword .................................................................... Standards Committee Roster ..................................................

6

V

21

22 24 25 27 32 32

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Appendix

AConsolidation of Parallelism.Perpendicularity.andAngularity ....... 49 A1 General ........................................................... 49 A2 Planar Orientation.................................................. 49 A3 Cylindrical Orientation ............................................. 57 A4 Linear Orientation.................................................. 66 Index .......................................................................

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79

ASME Y14.5.1M-1994

ENGINEERINGDRAWINGANDRELATEDDOCUMENTATION

PRACTICES

MATHEMATICAL DEFINITION OF DIMENSIONING AND TOLERANCING PRINCIPLES 1 SCOPE AND DEFINITIONS

This Standard presents a mathematical definition of geometrical dimensioning and tolerancing consistentwiththeprinciplesandpracticesofASME Y14.5M-1994, enabling determination of actual values. While the general format of this Standard parallels thatof ASMEY 14.5M- 1994, the latter document should be consulted for practices relating to dimensioning and tolerancing for use on engineering drawings and in related documentation. Textualreferencesareincludedthroughout this StandardwhicharedirectquotationsfromASME Y14.5M-1994. All such quotations are identified by italicized type. Any direct references to other documents are identified by an immediate citation. The definitions established in this Standard apply to product specifications in any representation, includingdrawings,electronicexchangeformats,or data bases. When reference is made in this Standard to a part drawing, it applies to any form of product specification.

1.1.4 Referenceto Gaging. This Standard is not intended as a gaging standard. Any reference to gaging is included for explanatory purposes only. 1.2References

When the following American National Standards referred to inthis Standard are superseded by a revision approved by the American National Standards Institute, the revision shall apply. ANSI B46.1-1985, Surface Texture Dimensioning and ASME Y 14.5M-1994, Tolerancing 1.3 Mathematical Notation

This Subsection describes the mathematical notation used throughout this Standard, including symbology (typographic conventions) and algebraic notation. 1.3.1 Symbology.All mathematical equations in this Standard are relationships between real numbers,

three-dimensional vectors, coordinate systems associated with datum reference frames, and sets of these quantities. The symbol conventions shown in Table 1.3 are used for these quantities. These symbols may be subscripted to distinguish between distinct quantities. Such subscripts do not 1.1.2 Figures. Thefiguresin this Standardare intended only as illustrations to aid the user in under-change the nature of the designated quantity. Technically, thereis a difference between a vector standing the principles and methods described in the this Stantext. In some instances figures show added detail for and a vector with position. Generally in dard, vectors do not have location. In particular, diemphasis; in other instances figures are incomplete rection vectors, which are often defined for specific by intent. Any numerical values of dimensions and points on curves or surfaces, are functions of position tolerances are illustrative only. on the geometry, but are not located at those points. (Anotherconventionalview is thatallvectorsare 1.1.3 Notes. Notesshownincapitallettersare located at the origin.) Throughoutthis Standard, pointended to appear on finished drawings. Notes in sitionvectorsareusedtodenotepointsinspace. lower case letters are explanatory only and are not While thereis a technical difference between a vector intended to appear on drawings. 1.1.1 Units. TheInternationalSystem ofUnits (SI) is featured in the Standard because SI units are expected to supersede United States (U.S.) customary units specified on engineering drawings.

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1.1General

MATHEMATICAL DEFINITION OF DIMENSIONING AND TOLERANCINGPRINCIPLES

ASME Y14.5.1M-1994

I

Quantity

Symbol Plain-face, italic, lower-case English or lower-case Greek letters ( t ,r , 8, etc.)

Real Numbers

Bold-face, italic English letters with an arrow diacritical mark (7,etc.)

Vectors Unit Vectors

Bold-face, italic English letters with a carat diacritical mark @, etc.)

Functions (real or vector-valued) Datum Reference Frames (coordinate systems) Sets

A real number or vector symbol (depending on the kind of value of the function) followed etc.] by the parameters of the function in parentheses [r

(3,

Plain-face, upper case Greek letter

(r,etc.)

Plain-face, italic, upper-case English letters (S, F , etc.)

and a point in space, the equivalence used in Standard should not cause confusion.

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I

TABLE 1.3.1 SYMBOLOGY

this

The magnitudeof the cross productis equal in value two vectors times to the product of the lengths of the the sine of the angle between them. For a given feature, the notagon r($, r ) will de1.3.2 Algebraic Notation. A vector can be exP to true position (see panded into scalar components (with the components note the distance from a point Subsection 1.4) in datum reference frame r. When 0, j', distinguished by subscripts, if necessary). Let thedatumreference f r q e is understoodfromthe and k be the unit vectors along the x, y, andz axes, context, the notation r(P) will be used. Figure 1-1 qspectively, of a coordinate system. Then a vector shows a case of a true-position axis. If the axis is V can be uniquely expanded as: represented by a point Po on the axis and a direction (a unit vector), then r($) can be evaluated by either of the following formulas: +

-+

The vector can be written V = (u,b,c).The magnitude (length) of vector ? is denoted by I? I and can be evaluated by: Iv'l =

d

m

r(P)=

-

- [(P' - z0). l0]2

r$) = I ($-$o)

-3

+ - +

s0t2

or

A unit vector9 is any vector with magnitude equal to one. The scalar pr2duct (dot product; innzr product) of two vectors V , = (?, b,, c , ) and V , = (u,, b,, c,) is denoted by V , V,. The scalar product is a real number given by: VI * V2= ala2+ b,b,

41.'-

x 8 1 i .

The first equationis a version of the Pythagorean Theorem. The second equationis based on the properties of the cross product. 1.4DEFINITIONS

The following terms are defined as their use applies to this Standard. ASME Y14.5M-1994 should be consulted for definitions applying to dimensioning and tolerancing.

+ clc2

and is equal in value to the product of the lengthsof the two vectors times the cosine of the angle between 1.4.1 Actualmating surface. A surface of perthem. The vector product (cross product; outer prod+ fect form which corresponds to an actual part feature. uct) of two vectorsand ?, is denoted by V , x -+ + For a cylindrical or spherical feature, the actual matV , . The cross product is a vector V, = (a,, b,, c,) ing surface is the actual mating envelope. For a plawith components given by: nar feature, it is defined by the procedures defining a primary datum plane. a3 = b, c2 - b2c, b3 = a2 ~1 - ~2 1.4.2 Actual value. A uniquenumericalvalue c3 = a, b, - a2b, representingageometriccharacteristicassociated 2 Copyright ASME International Provided by IHS under license with ASME

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MATHEMATICALDEFINITIONOF DIMENSIONINGANDTOLERANCINGPRINCIPLES

ASME Y14.5.1M-1994

A

f

N

1.4.10 Direction vector.A unit vector. Conventionally, directions are associated with various geometries as follows. The direction vector of a straight line (or pairof parallel lines)is parallel to the line@). The direction vectorof a plane (or a pairof parallel planes) is normal to the plane. The direction vector of a cylinderis the direction vectorof the cylinderaxis.

FIG. 1-1

THEDISTANCEFROMAPOINT

1.4.1 1 Element, circular surface. Given an axis and a surfaceof revolution about thataxis, a circular surface element is a set of points determined by the intersection of the given surface with one of the following, as applicable: (a) a plane perpendicular to the given axis (b) a cone with axis coincident with the given axis (c) a cylinderwithaxiscoincidentwiththe given axis

TO ALINE

1.4.12 Element, surface line. The intersection between an actual surface and a cutting plane.

with a workpiece. Example characteristics are flatness, perpendicularity, position, size, etc. Later sections of this Standard give rulesfor the determination of actual values for specific characteristics. 1.4.3Candidatedatum. from a datum feature.

1.4.13 Envelope, actual mating. A surface, or pair of parallel surfaces, of perfect form, which cor-

respond to an actual part feature of size, as follows: (a) ForanExternal Feature. Asimilarperfect featurecounterpart of smallestsizewhichcanbe circumscribed about the feature so that it just contacts the surface. (b) For an InternalFeature. A similar perfect featurecounterpart of largestsizewhichcanbeinscribed within the featureso that it just contacts the surface. Figure 1-2 illustrates the definition of both actual mating envelope and actual minimum material envelope (see definition below) for both internal and external features. In certain cases (e.g., a secondary datum cylinder) the orientation or positionof an actual mating envelope may be restricted in one or more directions. (See Fig. 1-3.)

Adatumestablished

1.4.4Candidate datum referenceframe. A datumreferenceframeestablishedfromcandidate datums. 1.4.5 Candidate datum reference frame set. The set of all candidate datum reference frames established from a set of referenced datums. 1.4.6 Candidatedatum set. The set of all candidate datums that can be established from a datum feature.

1.4.7 Conformance. Applied to a part feature, that condition in which the feature does not violate the constraints defined by the tolerance. For tolerances that reference datums, if the feature does not 1.4.14Envelope,actual minimum material. violate the constraints definedby the tolerance for at A surface, or pair of parallel surfaces,of perfect form least one datum reference frame in the candidate da- which correspond to an actual part feature of size, tum reference frame set, then the part feature is in as follows: conformance to the tolerance. (a) ForanExternal Feature. Asimilarperfect feature counterpart of largest size which can be in1.4.8 Cutting plane. A plane used to establisha scribed within the feature so that it just contacts the planar curve in a feature. The curve is the intersecsurface. tion of the cutting plane with the feature. 3 Copyright ASME International Provided by IHS under license with ASME

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1.4.9 Cutting vector. A unit vector on the actual mating surface which, together with the normal to the actual mating surface, defines the direction of the cutting plane. Cutting vectors will be designated &.

MATHEMATICAL DEFINITION OF DIMENSIONING AND TOLERANCING PRINCIPLES

ASME Y14.5.1M-1994 Actual mating envelope

---ti

-i

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Actual minimum material envelope INTERNAL FEATURE

Actual minimum material envelope

-:

i

t

Actual mating envelope

11 i

True position axir

Actual mating envelope

I

Actual mating envelope at basic orientation

EXTERNAL FEATURE AS DRAWN

FIG. 1-2 ILLUSTRATION OF ACTUAL MATING ENVELOPE AND ACTUAL MINIMUM MATERIAL ENVELOPE

FIG. 1-3 ACTUAL MATING ENVELOPE AND ACTUAL MATING ENVELOPE AT BASIC ORIENTATION

(b) For an Internal Feature.A similar perfect feature counterpart of smallest size which can be circumscribed about the feature so that it just contacts the surface. Figure 1-2 illustrates the definition of both actual matingenvelope(seedefinitionbelow)andactual minimum material envelope for both internal and external features. In certain cases the orientation or position of the actual minimum material envelope may be restricted in one or more directions. 1.4.15 Mating surfacenormal. Foragiven point on a part feature, the direction vector of a line passing through the point and normal to the actual mating surface at the point of intersection of the line with the actual mating surface. (See Fig. 1-4.)

AS PRODUCED

Actual matir surface

I

I

k Mating surface normal

FIG. 1-4

#

ILLUSTRATION OF MATING SURFACE NORMAL

1.4.16 Perfect form.A geometric form that corresponds to the nominal (design) geometry except for possible variations in size, position, or orientation. 1.4.17 Resolved geometry. A collectiveterm for the center point of a sphere, theofaxis a cylinder, orthecenterplane of awidth.Conceptually,the resolved geometry of a feature of size is a correspondingfeatureofsizehavingperfectformand zero size. 1.4.18 Set of support. For a planar feature, a plane that contacts the feature at one or more points, such that the feature is not on both sides of the plane. The conceptof set of support is illustrated in Fig. 1-5.

----- Not a set of support

1.4.19 Size, actual mating. The dimension of the actual mating envelope. (The dimension may be a radius or diameter for a spherical or cylindricd,\

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MATHEMATICALDEFINITION OF DIMENSIONINGANDTOLERANCINGPRINCIPLES

ASME Y14.5.1M-1994

envelope, or width or half-width for a parallel-plane envelope, depending on the context. The radius of a cylindrical or spherical actual mating envelope will be designated TAM')

1.4.24 True position. The theoretically exact position of a feature in a particular datum reference frame.Insomecontexts,theterm"trueposition" refers to the resolved geometryby which the feature is located.

1.4.20 Size, actualminimum material. The dimension of the actual minimum material envelope. (The dimension maybe a radius or diameter for a 1.5 SUMMARY OF CONVENTIONAL DESIGNATIONS spherical or cylindrical envelope, or width or halfwidth for a parallel-plane envelope, depending on the Throughout this Standard, conventional designa: context. The radius of a cylindrical or spherical acThis Subsection tions are used for various quantities. tual minimum material envelope will be designated summarizes these conventions. rA")

tP= direction vector of a cutting plane

1.4.21Size, true positionmating. Thesize, optimized over the candidate datum reference frame set, R, of the actual mating envelope constrained to belocatedandorientedattrueposition.The tru~ position mating size will be de2ignated rTp. If r(P, r) is the distance from a pointP to the true position in datum reference framer (an element of the candidate datum reference frame set), and if F is the feature, then the true position mating size is given by:

[

b1= direction vector of the primary datum 8,

plane

10,= direction vector of the surface normal

P = position vector r(Z, r) = the distance of a point P to true position -)

in datum reference

r(P,r)external features

r a PEF

For more information, see Subsection4.2. 1.4.22 Size,true position minimum material. The size, optimized over the candidate datum reference frame set, R, of the actual minimum material envelope constrained to be located and oriented at true position. The true position mi$mum material size will be designa5d TTpMW If r(P,r) is the distance from a point P to the true position in datum reference framer (an element of the candidate datum reference frame set), andif F is the feature, then the true position minimum material sizeis given by:

~ T P M M=

I

r a P€F

frmz r

r ( P ) = the distance of a point P to true position, in the case that the datum reference frame is understood from context r,, = actual mating size (radius) rAMM = actual minimum material size (radius) rTp= true position mating size (radius) rTpMM = true position minimum material size (radius) 9 = direction vector of a tolerance zone t = an unspecified tolerance value to = a specific tolerance given on a drawing or part specification r = a candidate datum reference frame 1.6FORMAT

min % a x r(P,r) internal features r a PEF max y n

plane

8, = direction vector of the tertiary datum

max $n .(Pi r) internal features ra PEF min qax

plane

= direction vector of the secondary datum

The format used in this Standard for explanation of geometric characteristics is as follows:

r(P,r) external features

Definition - narrativeandmathematical scription of the tolerance zone

For more information, see Subsection 4.2.

Conformance - mathematical definition of the conformance

1.4.23Spine. A point,simple(nonself-intersecting) curve, or simple surface. Spines are used in the definitions of size and circularity.

Actual value - mathematical definition of actual value 5

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de-

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rTP =

= cutting vector

MATHEMATICALDEFINITION OF DIMENSIONINGANDTOLERANCINGPRINCIPLES

ASME Y14.5.1M-1994

2GENERALTOLERANCING

AND RELATEDPRINCIPLES

provided for primary datum feahres. See Section 4, Datum Referencing.

2.1 FEATUREBOUNDARY

Tolerances are applied to features of a part. Generally, features are well-defined .only in drawings and computer models. This Section establishes the rules foridentifyingfeaturesonactualparts.Twosteps are involved: establishing the surface points ofthe part and establishing the feature boundary.

2.3 LIMITS OF SIZE

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A feature of size is one cylindrical or spherical surface, or a setof two opposed elements or opposed parallel surfaces, associated with a size dimension. 2.1.1 Establishing the Surface Points. A cerA feature of size may be either an internal feature or tain amount of smoothing of the part surface is iman external feature. This Section establishes definiplied in this Standard. This smoothing is necessary tionsforthesizelimits,conformance,andactual to distinguish dimensional from roughness, surface value of size for such features. Subject to Rule of #1 texture,materialmicrostructure,andevensmallerASME Y 14.5M-1994, size limits also control form scale variations of the part. This Standard incorpovariation. The method by which Rule #1 is applied rates by reference the smoothing functions defined in is discussed below in para. 2.3.2, Envelope Principle. ANSI B46.1-1985, Surface Texture. These functions For the definitionof form controls, refer to Sections define how the variations of physical part surfaces 5 and 6. must be smoothed to distinguish dimension and form 2.3.1 Variation of Size variation from smaller-scale variations. When refer(a)Definition. A size tolerance zoneis the volume ence is made in this Standard to points on a feature, between two half-space boundaries, to be described itreferstopointsonthesurfacethatresultafter below. The tolerance zone does not have a unique smoothing. form. Each half-space boundary is formed by sweep2.1.2 Establishing Feature Boundaries. It is ing a ball of appropriate radius along an acceptable possible to subdivide the boundary of a part in many spine, as discussed below. The radiiof the balls are ways. No rules for establishing a unique subdivision determined by the size limits: one ball radius is the are provided. Any subdivision of an actual part surleast-material condition limit (rye) and one is the face into features (subject to some mild limitations maximum-material condition limt (rMMc). such as having one-to-one correspondence with the A 0-dimensional spine is a point, and applies to nominal part features, and preserving adjacency rela- spherical features.A 1-dimensional spineis a simple tionships) which favors conformance to all applica(non self-intersecting) curve in space, and applies to ble tolerances is acceptable. While tolerance require- cylindrical features. A 2-dimensional spineis a simmentsmaybesimultaneousorindependent,the ple (non self-intersecting) surface, and applies to parsubdivision of a part surface into features cannot varyallel-plane features. These three types of spines can from one tolerance to another. bemorerigorouslydefined,respectively, as connected regular (in the relative topology) subsets of d-manifolds, for d = 0, 1,and2. A d-dimensional spine will be denoted as 9.Also, a (solid) ball of 2.2 DIMENSION ORIGIN radius r will be denoted as B,. When a dimension origin is specified for the disA solid G(Sd,B,) is obtained by sweeping the ball tance between two features, the feature from which B, so that its center lies in 9.G(sO,B,) is a single the dimension originates defines an origin plane for ball bounded by a sphere. If S' is a line segment, defining the tolerance zone. In such cases, the origin G(S',B,) is a solid bounded by a cylindrical surface plane shall be established using the same rules as are and two spherical end caps. If S2 is a planar patch, 7

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When perfect form at MMC not required ~

FIG. 2-1

~

FIG. 2-2

~~~

CONFORMANCETO LIMITS OF SIZE

SYMBOLSUSEDINTHEDEFINITIONOFSIZE

G(S2,I?,.)is a solid bounded by two planar patches andsome canal surfaces. (Canalsurfacesareobtainedbysweepingspheres,orballs, so that their centers lie on a curve in space.) Figure 2-1 shows a one-dimensional spine and its associated solid for a ball of radius r . S' and S2 need not be portions of lines or planes, respectively. If necessary, S' or S2 can be extended to infinity, or closed upon itself,so that the resulting solid G is a half-space. The spine, along with the balls, also defines the symmetricaxis transformation of such solids. (b) Confomnce. A feature of size, F , conforms to the limitsof size rMc and rMMcif there exist two spines, S, corresponding to rMc and S, corresponding to rMMc,and two associated solids, G, = G(Sl, BrMc) and G, = G(S,, ByMMC),that satisfy three conditions described below. If F is an external feature, then let H,= GI and H, = G,. If F is an internal feature, then let HI be the complement of G, and H, be the complementof G, . (See Fig.2-2.) F conforms to its limits of size if: HI c H, FcH,-H, F surrounds (encloses) the boundary of HI(if F is an external feature) or the boundary of H, (if F is an internal feature). The purposeof the third conditionis illustrated by Fig. 2-3. The figure shows a configuration where an external feature F satisfies the first two conditions of conformance but does not surround the boundary of H,. (c) Actual value. Two actual values are defined. The actual external (to the material) size of an external (respectively, internal) featureis the smallest (re-

FIG. 2-3

VIOLATION OFTHESURROUNDCONDITION FOR AN EXTERNALFEATURE

spectively, largest) size of the ball to which the feature conforms. The actual internal size is the largest (respectively, smallest) size of the ball to which the feature conforms. The size may be expressed as a radius or diameter, as appropriate to the application. 2.3.2 Variation of Size Under the Envelope Principle (a)Definition. If Rule #1 applies, the M M C limit of size establishes a boundary of perfect form (envelope) at MMC. The tolerance zone for a feature of size under the envelope principle is as described in the previous section, with the further constraint that

8

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MATHEMATICAL DEFINITION OF DIMENSIONING AND TOLERANCINGPRINCIPLES

ASME Y14.5.1M-1994

MATHEMATICAL DEFINITION OF DIMENSIONING AND TOLERANCINGPRINCIPLES

ASME Y14.5.1M-1994

feature conforms. The actual internal size is the largest(respectively,smallest)diameter of theballto which the feature conforms. The boundary associated with S, for the actual external sizeis the actual mating envelope. Note that under the envelope principle, the actual external sizeis identical to the actual mating size; however, the actual internal size is not the actual minimum material size because the boundary associated with S, is not of perfect form.

S,,, is of perfect form (a straight line for cylindrical

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features or a plane for parallel plane features). (b) Conformance. A feature conforms to the envelope principle if it conforms to a size tolerance zone , a perfect-form spine. with S ( c ) Actual value. Two actual values are defined. The actual external (to the material) size of an external (respectively, internal) featureis the smallest (respectively, largest) diameterof the ball to which the

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MATHEMATICALDEFINITION OF DIMENSIONINGANDTOLERANCINGPRINCIPLES

3 SYMBOLOGY Thereare noconcepts in Section 3 of ASME Y 14.5M-1994that require mathematical definition.

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ASME Y14.5.1M-1994

MATHEMATICALDEFINITION OF DIMENSIONINGANDTOLERANCINGPRINCIPLES

4

ASME Y14.5.1M-1994

DATUM REFERENCING

adatumfeatureiscalleda candidate datum set. Since a datum feature may generate more than one datum, multiple datum reference frames may exist for a single feature control frame. (Or, from the viewpoint of ASME Y14.5M-1994, the part may move in the datum reference frame.) The set of datum reference frames established from one or more datum features is called acandidate datum referenceframe set. For tolerances that reference datums, if the feature does not violate the constraints defined by the tolerance for at least one datum reference frame in the candidate datum reference frame set, then the part feature is in conformance to the tolerance. There is a candidate actual value associated with each candidate datum reference frame in the candidate datum reference frame set. The actual value associated with the tolerance is the minimum candidate actual value.

4.1 GENERAL

This Section contains mathematical methods for establishing datums from datum features and for establishing datum reference frames. Datum reference framesarecoordinatesystemsusedtolocateand orient part features. Constructing a datum reference frame is a two step process. The first step is to develop datum(s) from one or more datum features on the part. The second step is to determine the position and orientation of the datum reference frame from the datums. These steps are described in Subsections 4.3and4.4,respectively.Subsection4.2discusses basic concepts of datum referencing. 4.2CONCEPTS

The concept of datums established within ASME

Y 14.5M-1994 refers to “processing equipment’’ such

as “machine tables and surface plates” that are used 4.3ESTABLISHING DATUMS to simulate datums for applications such as measureThe methodof establishing datums depends on the ment. This present Standard establishes mathematical type of datum feature (flat surface, cylinder, width, concepts for datums and datum reference frames. It considers all points on each feature, and deals with or sphere), the datum precedence (primary, secondary, or tertiary) in the feature control frame, and (for imperfect-form features. The attempt here is to refine the conceptsof ASME Y 14.5M- 1994 by establishing datum features of size) the material condition of the a mathematical model of perfect planes, cylinders, datum reference. axes, etc. that interact with the infinite point set of 4.3.1 List of Datum Feature Types. The folimperfectly-formedfeatures. This Standarddefines lowing classificationof datum featuresis used inthis the datums that are simulated by processing Section: equipment. Datum features not subject to size variation: In this Standard, the part is assumed to be fixed Planes in space and the datums and datum reference frames Datum featuressubjecttosizevariation are established in relation to the part.This approach (Cylinders, Widths, Spheres): can be contrasted to that of ASME Y14.5M-1994, Features referenced atRFS where the datums and datum reference frames are Features referenced at MMC assumed to be fixed in space and the part is moved Features referenced at LMC into the datum reference frame. The two approaches This Standard does not specify how to establish daare different, but the end results are identical. tums for screw threads, gears, splines, or mathematiMultiple valid datums may be established from a cally defined surfaces (such as sculptured surfaces). datum feature. This may happen, for example, if the 4.3.2Planar Datum Features. Thecandidate feature “rocks” orif a datum feature of size is referdatum set for a nominally flat datum feature is deenced at MMC and manufactured away from MMC fined in a procedural manner. This empirical definisize. The set of datums that can be established from 13 --`,``,,`,`,,,,,`,``,`,,```,``,-`-`,,`,,`,`,,`---

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MATHEMATICALDEFINITION OF DIMENSIONING AND TOLERANCING PRINCIPLES

ASME Y14.5.1M-1994

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Datum feature

contact points

\

P -candidate datum plane Contact points - points where the datum feature contactsP L -any line on planeP L’ - line segment corresponding to the projection of the datum feature Lon Regions 1 , 2 - regions at the end ofL’, each with lengthxL‘ (default x=%) FIG. 4-1

CONSTRUCTION FOR TESTINGWHETHERAPLANE

IS A VALID DATUM PLANE

endpoints of L’. Unless otherwise specified on the drawing, the value of x shall be 113. If all of the orthogonal projections of the points in C are within P is rejected as a either single region, then plane valid datum plane. (3) Do this for all lines inP. (Note that parallel If no line rejectsP , lines.wil1 yield identical results.) then P is a candidate datum for the datum feature. Theprocedure is illustratedinFig. 4-1, which shows one line direction L . The line segment L‘ is bounded by the projection of the datum feature onto L . The particular line direction illustrated in the figure does not reject P as a valid datum plane since the projections ofthe contact points are not all in region 1 or all in region 2 of L’. Note that only the direction of L in the plane P is important. P is a candidate datum for the datum feature if it is not rejected by any line direction in P. (b) Secondary Planar Datum Features. The candidate datum set for a secondary planar datum feature is determined by one of the following:

tion specifies a set of datums which are reasonable from a functional standpoint. If the datum feature is perfectlyflat,thecandidatedatumsetconsists of only one datum; otherwise it may consist of more than one datum. This is equivalent to “rocking” the datum featureon a perfect surface plate. The definition below limits the amount that the datum feature can ‘‘rock‘‘ in a manner that is roughly proportional to the form variation of the datum feature. (a) Primary Planar Datum Features. The candidate datum set for a nominally flat primary datum feature is defined by the following procedure: ( I ) Consider a plane P which is an external set of support for the datum feature. LetC be the set of contact points of the datum feature and P. (2) Consider an arbitrary line L in P . Orthogonally project each point on the boundary of the datum feature ontoL , giving the line segmentL’. Consider regions of L’ that are within some fraction x of the L’ is n , endpoints of L’. That is, if the length of consider regions of L’ within a distance xn of the 14

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MATHEMATICALDEFINITION OF DIMENSIONINGANDTOLERANCINGPRINCIPLES

ASME Y14.5.1M-1994

( I ) If the primary datumis a point, use the procedure for a primary planar datum feature to establish the secondary datum. (2) If the primary datum is an axis nominally normal to the secondary datum, then,for each candidate datum in the candidate datum set for the primary datum feature, the candidate datum set for the secondarydatumfeatureincludestheuniqueplane which is basically oriented relative to that primary datum and which forms a set of support for the sec2 x 0 2.0-2.1 ondary datum feature. @I00.2@IAIBIC] (3) If neither (1) nor (2) apply, use the procedure for a primary planar datum feature modified in thefollowingways:Givenaprimarydatumfrom the primary candidate datum set, each plane P being T Datum reference frame considered as a secondary datum must be basically oriented (not located) to the primary datum. Also, each lineL in P being considered mustbe perpendicular to the direction vector of theprimarydatum. (Only one line in P must be considered.) (c) Tertiary Planar Datum Features. If the first two datums leave a rotational degree of freedom (see is Subsection 4.4), thenthecandidatedatumset formed by the procedure for a primary planar datum LCenter plane must bf feature modified such that each plane P being conorientated parallel to sidered mustbe basically oriented (but not necessardatum axis B ily basicallylocated) relative to the datums of higher precedence, and oneline L is to be considered, which FIG. 4-2 TERTIARY DATUM IS BASICALLY must be perpendicular to the axis established by the ORIENTEDONLY higher precedence datums.If the fust two datums do not leave a rotational degree of freedom, the candidate datum set consists of the plane which is basidatums. Datum plane C in the figure must be estabcally oriented (but not necessarily basically located) lished from the datum feature under the constraint relative to thedatums of higherprecedenceand that the plane is parallel to datumB (and hence perwhich forms a set of support for the datum feature. pendicular to datumA). Datum planeC does not need to contain datum axis B . 4.3.3 Datum Features Subject to Size Vari(c) Sphere (both internal and external). The can,ation, RFS. didatedatumsetforasphereistheset of center ( a ) Cylinder (both internaland external). The points of all actual mating envelopes of the datum candidate datum set for a cylinder is the set of axes feature. of all actual mating envelopes of the datum feature. 4.3.4 Datum Features Subject to Size VariFor secondary or tertiary datum features, the actual MMC. ation, mating envelopes are constrainedto be basically ori(a) Cylinder [External] {Znternal}. The candidate ented (not located) to the higher precedence datums. datumset for acylinder is theset of axes of all (b) Width (both internal and external). The candicylinders of MMC virtualconditionsizethat[endate datum set for a width is the set of all center close] {are enclosed within} the datum feature. For planes of all actual mating envelopes of the datum secondaryortertiarydatumfeatures,thecylinders feature. For secondaryor tertiary datum features, the are constrained to be basically oriented and, as appliactual mating envelopes are constrained to be basicable,basicallylocatedtothehigherprecedence cally oriented (not located) to the higher precedence datums. datums. (b) Width [External] {Znternal}. Thecandidate Figure 4-2 shows an example of a tertiary datum at RFS oriented and not located to higher precedence datum set for a width is the set of center planes of

4

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MATHEMATICAL DEFINITION OF DIMENSIONINGANDTOLERANCINGPRINCIPLES

ASMEY14.5.1M-1994

all pairs of parallel planes separated by the MMC virtualconditionsizethat[enclose] {are enclosed within} the datum feature. For secondary or tertiary datumfeatures,theparallelplanesareconstrained to be basically oriented and, as applicable, basically located to the higher precedence datums. (c) Sphere [External] {Znternal}. Thecandidate datum set for a sphere is the set of center points of all spheres of MMC virtual condition size that [enclose] {are enclosed within} the datum feature. For secondary or tertiary datum features, the spheres are constrained as applicable to be basically located to the higher precedence datums.

section establishes rules for constructing candidate datum reference frames and candidate datum reference frame sets. The construction of a particular candidate datum reference frame proceeds as follows. A primary datum is selected from the candidate datum set associated with the primary datum feature. If a secondary datumiscalledout,thechoiceofprimarydatum establishes, by the rules of the previous subsection, a candidate datum set for the secondary datum feature. A secondary datum is chosen from this latter set. Similarly, if a tertiary datum is called out, the choice of primary and secondary datums establishes a candidate datum set for the tertiary datum feature. A tertiary datum is chosen from this last set. The choice of particular datums from the candidate datum sets determines a particular candidate datum reference frame, in a way described below. The set of candidatedatumreferenceframesobtainedby all possible choices of datums is the candidate datum reference frame set.

4.3.5 Datum Features Subject to Size Variation, LMC. (a) Cylinder [External] {Internal}. The candidate datumset for acylinderistheset of axes of all cylinders of LMC virtual condition size that [are enclosed within] {enclose} the datum feature. For secondary or tertiary datum features, the cylinders are constrained to be basically oriented and, as applicable,basicallylocatedtothehigherprecedence datums. (b) Width [External] {Internal}. Thecandidate datum set for a width is the set of center planes of allpairs of parallelplanesseparated bytheLMC virtual condition size that [are enclosed within]{enclose} the datum feature. For secondary or tertiary datumfeatures,theparallelplanesareconstrained to be basically oriented and, as applicable, basically located to the higher precedence datums. (c) Sphere [External] {Internal}. Thecandidate datum set for a sphere is the set of center points of all spheres ofLMC virtual condition size that [are enclosedwithin]{enclose}thedatumfeature..For secondary or tertiary datum features, the spheres are constrained as applicable to be basically located to the higher precedence datums.

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4.4.1 Degrees of Freedom. The tables in Subsection 4.7tabulate invariants for all possible datum referenceframes.Thesetablesshowthepossible combinations of datum geometries that can be used to establish datum reference frames. A datum reference frame may not fully constrain the coordinate system for locating and orienting tolerance zones. In other words, a datum reference frame may leave certain translational or rotational transformations free. An invariant in a datum reference frame is a quantity (distance or angle) that does not change under free transformations allowed by that frame. The location of a completely defined datum reference frame is restricted in three translational directions (x, y, andz) and in three rotational orientations (u, v, and w) where u is rotation around the x axis, v is rotationaroundthe y axisand w is rotation around the z axis. Tables 4-2through 4-4show the various combinations of datums and, for each case, 4.4 ESTABLISHING DATUM REFERENCE the free transformations (degrees of freedom) and the FRAMES invariants. Dependingon how a designated featureis The previous section establishes the rules for asso- toleranced from a datum reference frame, a partially constrained datum reference framemay be sufficient ciating candidate datum sets with individual datum to evaluate the designated feature. For example, if features. (Whlle a candidate datum set is associated the datum reference frame is defined by only a plane with an individual datum feature, datum precedence is used in the definition.) The candidate datums from (case 3.1 in Table 4-4),which leaves one rotational these sets are used to construct candidate datum ref- and two translational degrees of freedom, then it is sufficientlydefinedtoevaluatetheparallelismor erenceframes. The collection ofdatumreference perpendicularity of a designated feature with respect frames that can be constructed in this way is called to the plane. the candidate datum reference frame set. This Sub16 Copyright ASME International Provided by IHS under license with ASME

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MATHEMATICAL DEFINITION OF DIMENSIONINGANDTOLERANCINGPRINCIPLES

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4.4.2 Datum Precedence. The datum precedence (primary, secondary, and tertiary) in the feature control frame determines which datums arrest or constrain each ofthe degrees of freedom. The primary datum arrests three or more of the original six degrees of freedom. The secondary datum, if specified, arrests additional degrees of freedom that were not previously arrested by the primary datum. In some cases (for example, two orthogonal axes), two datums are sufficient to fully constrain the coordinate system. In such cases, a third datum cannot be meaningfully applied. Otherwise, the tertiary datum, if specified, arrests the balance of the degrees of freedom that were not previously arrested by the primary and secondary datums.

TABLE 4-1 SYMBOLS FOR DATUM TABLES Svmbol

A

B C PT Ax PL {LI {LI

...}

... : ... }

1

Description

primary datum secondary datum tertiary datum point axis plane line through ... line through

... such that ... is true

#

not coincidental with

C

contained within

a

not contained within

I1 I

parallel with

A

perpendicular to logical AND

..

logical OR (one or the other, or

7

logical NOT

n x, Y. u. v, w

In composite tolerancing, the feature-relating tolerances (lower tiers of the feature control frame) control only the orientation of the pattern. The candidate datum reference frame set for such a tolerance is the closure under translation of the candidate datum reference frame set established by the procedures given above.

intersection position in a Cartesian coordinate system rotation about x, y, z axis, respectively; yaw, pitch, roll, respectively angle relative to datum axis z sphericalradius:

-

4.5 DATUM REFERENCE FRAMES FOR COMPOSITE TOLERENCES

entry no

x

+ y +z

4.6 MULTIPLE PAlTERNS OF FEATURES

Where two or more patterns of features arerelated to commondatum features referenced in the same

(e.g., not applicable, none)

TABLE 4-2 POINT ASPRIMARY DATUM

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TABLE 4-3 LINE AS PRIMARY DATUM

2.16 2.17

Ax

2.18

Ax Ax Ax

2.19 2.20

Ax

PL PL PL PL

-

PT Ax Ax PL PL PL

-

order of precedence and at the same material condi-

tion, as applicable, they are considered a composite

pattern with the geometric tolerances applied simultaneously. If such interrelationship is not required, a notation such as SEP REQT is placed adjacent to each applicable feature control frame. When tolerances apply simultaneously, conformance and actual value of all features in a composite pattern must be evaluated with respect to a common datum reference frame selected from the candidate datum reference frame set. When a tolerance applies to each pattern

all all all all all

(A // B) A 1 (A // C) (A IB) A (A # C)

(A // B) A 1 (A // C) (A IB) A 7 (A IC)

-

ancing is used, the lower entries are always considered to be separate requirements for each pattern. 4.7 TABULATION OF DATUM SYSTEMS

This Subsection presents tables of datum systems. The first table presents the symbols usedin the rest of the Section. The next three tables, organized by the geometry of the primary datum, present detailed information about each type of datum system. They as a separate requirement, conformance and actual list all valid combination of datum geometries, the value for each pattern can be evaluated using a differfree transformations remainingin the coordinate system associated with the datum reference frame, the ent datum reference frame taken from the candidate datum reference frame set. All features within each invariant quantities under the free transformations, pattern, however, must be evaluated with respect to and the conditions under which the system is a common datum reference frame. If composite toler- valid. The following example is from Table 4-3:

dam

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MATHEMATICALDEFINITION OF DIMENSIONINGANDTOLERANCINGPRINCIPLES

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EXAMPLE FROM TABLE 4-2 LINE AS PRIMARY DATUM Datums Case

A

0

2.8

Ax

Ax

C

-

Free xfrms z

Validity Conditions x, y, u, v. w

(A # B) A (A I/ B)

Invariants .

TABLE 4-5 GENERIC INVARIANT CASES Index

Cases r

1.1

p z, Yz

2.1 1.2, 1.7, 1.11, 2.3, 2.14, 3.2, 3.7

1

2 3 4

Case Number(s) Invariant

z.

Y2’

Pz

3.1

5

Yz x, Y. u, v, w or y, z, u. v. w

2.8, 2.13, 3.8, 3.15

6

all

all others

2,

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TABLE 4-4 PLANE AS PRIMARY DATUM

MATHEMATICALDEFINITION OF DIMENSIONING AND TOLERANCINGPRINCIPLES

ASME Y14.5.1M-1994

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This shows case 2.8, a datum system consisting of athrough datum pointA. In this case, for the purpose primary datum axis, a secondary dam axis, and no ofestablishingthedatumreferenceframe,datum tertiary datum. The validity conditions indicate that axis B shouldbetranslated so astopassthrough this case applies only if the two axes (A and B) are datum point A. is For cases 2.3 through 2.6 in Table 4-3, if datum parallel, but not equal. The only free transformation translation along the axis. z As a result, the invariants B is called out atRFS then datum pointB as derived include x and y coordinates, and all angle relationshipsfrom the actual datum feature will in general not lie between features and the datum reference frame. ondatumaxis A. In this case,forthepurpose of establishingthedatumreferenceframe,datum Coordinate system labels are somewhat arbitrary. B The following conventions apply.If the primary dashould be projected onto datum axis A. tum is a point,it establishes the origin.If the primary Table 4-5 summarizes the contents of Tables 4-2 datum is a line, it establishes the z coordinate axis. through 4-4 according to the free transformations and If it is a plane,it establishes the x-y coordinate plane invariants. It can be seen that, subject to renaming of directions in the coordinate systems, there are only (and hence the direction of the z axis). Secondary and tertiary datums establish additional elements of six distinct casesof datum systems. Table 4-5 crossthe coordinate system. references the distinct cases to the cases in the previFor cases 1.7 through 1.10 in Table 4-2, if datum ous tables. B is called out at RFS then datum axis B as derived The symbols used in Tables 4-2 through 4-5 are from the actual datum feature will in general not pass presented in Table4- 1.

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ASME Y14.5.1M-1994

5 TOLERANCES OF LOCATION 5.1 GENERAL This Section establishes the principles of tolerances of location; included are position, concentricity, andsymmetryused to control the following relationships: (a) center distance between ffeatures of size, such as] holes, slots, bosses, and tabs; ( b ) location of features (such as in (a) above) as a group with respect to datum reference frames; (c) coaxiality of features; (d) concentricity or symmetry of features - center distances of correspondingly-located feature elements equally disposed about a datum axis or plane.

4 surface

\I

Position erroraxis interpretation

f Trut:position

Axis of actual

FIG. 5-1 ILLUSTRATIONOFDIFFERENCEBETWEEN 5.1.1 Material Condition Basis. Positional tolSURFACE AND SIZEINTERPRETATIONS OF POSITION erances are applied on an MMC, RFS, or LMC basis. TOLERANCING FOR ACYLINDRICALHOLE A position tolerance may be explained either in terms of the surface of the actual feature or in terms of size and the resolved geometry (center point, axis, or true position than allowed by the combined effects of center plane) of the applicable (mating or minimum the position tolerance (zero) and the bonus tolerance material) actual envelope. These two interpretations resultingfromtheactualmatingsize of thehole. will be called the sulface interpretation and the re(Thevirtualconditionboundaryextendsbeyond solved geometry interpretation, respectively.(See theactualmatingenvelope.)Theholewouldnot Subsection 5.2 and the following for the precise defi-beacceptedaccordingtotheresolvedgeometry nitions of positional tolerancing interpretations.) For interpretation. MMC and LMC callouts, these explanations are not Figure 5-2 shows the converse situation. Assume equivalent. They differ in part because the resolved that the shaft shown in the figure is controlled by a geometry interpretation relies on an assumption that position tolerance t at MMC. Assume also that the the feature is of perfect form and in part because shaft was manufactured with perfect form. (This asthe derivation of the surface interpretation assumes sumption is not necessary, but simplifies the examperfect orientation. ple.) If the radius of the shaft is rAMand the MMC Two examples will illustrate the issues. Consider radius is rMMc,the radius of the tolerance zone for theillustrationinFig.5-1.Assumethatthehole the axis is rMMc- rAM+ t/2. If the height of the shaft shown in the figure is controlled by a zero position is tilted to an is h , and the axis of the actual shaft toleranceatMMC.TheMMCvirtualcondition extreme orientation within the tolerance$one, a simboundary has a diameter equal to the MMC diameter ple geometric analysis shows that point P lies outside of the hole. The actual hole was manufactured with the virtual condition boundary by a distance poor form, but assume that it is within the limits of size. (See para. 2.3.1.) The hole does not violate the rAM[++[(r+2(rMMc - rAM))/hl2 - 11. virtual condition boundary, and, as explained below, would be accepted according to the surface interpre- Thus,afeaturemaybeacceptableaccordingtoa resolved geometry interpretation but fail according tation.Usingtheresolvedgeometryinterpretation, to the surface interpretation. however, the hole is apparently further away from .

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boundary matin

envelope

ASME Y14.5.1M-1994

Virtual condition envelope

-

MATHEMATICALDEFINITIONOF DIMENSIONINGANDTOLERANCINGPRINCIPLES

\

TABLE 5-1 DEFINITIONOFPOSITION TOLERANCE ZONE, SURFACE INTERPRETATION

True position axis

Shaft as produced

Material Condition Basis

Axis tolerance zone

Feature

4

Internal

r($) c b

envelope axis

FIG. 5-2 THESHAFTSATISFIESTHERESOLVED GEOMETRYINTERPRETATIONBUTVIOLATESTHE VIRTUALCONDITIONBOUNDARY

Throughout most of this Section, both a surface interpretation and a resolved geometry interpretation are supplied.In a few cases (e.g., projected tolerance zones)onlyasurfaceinterpretationisprovided. Wheneverthetwointerpretationsdonotproduce equivalentresults,thesurfaceinterpretationshall take precedence.

axis, or center plane)of a featureof size is permitted to vary from true (theoretically exact) position. Basic dimensions establish the true position from specified datumfeaturesandbetweeninterrelatedfeatures. (Note: some position tolerance specifications can be explained in terms of a surface boundary.) Throughout this Subsection, whenever the true p,osition is understood from context, the n p t i o n r(P ) will denote the distanceJrom a point P to the true position. For spheres, r(P) is the distanse to the true position centerqoint. For cylinders, r(P) is the distance betweenP andge true position axis. For pyalle1 plane features, r(P ) is the distance between Pand thetruepositioncenterplane.Thesedefinitions shouldalsobeunderstoodtobefor a particular choice of datum reference frame from the candidate datum reference frame set. Throughout this Section, allsphericalandcylindricalsizesareinterms of radius unless otherwise specified. All tolerance values are assumed to be diameters for spheres and cylinders, and full widths for parallel planes, in accord with common practice.

5.1.2 Patternsof Features. For the purposes of this Standard all tolerances of location are considered to apply to patternsof features, where a pattern may consist of only a single feature. The control of the location of the pattern as a group is called the patrem-locating tolerance zone framework (PLl'ZF). When the pattern consists of two or more features, there is the possibility, through the use of composite tolerancing, to control the relative location of features within the pattern. This is done by specifying 5.2.1 In Terms of the Surface of a Feature. a secondary location tolerance, called the feature(a) Definition. For a pattern of features of size, a position tolerance specifies that the surface of each. relating tolerance zone framework (FRlZF),in conactual feature must not violate the boundary junction with the PLTZF. There may be more than of a one FRTZF for a pattern. All features within a single corresponding position tolerance zone. Each boundpattern are controlled simultaneously. That all is,feaary is a sphere, cylinder, or pairof parallel planes of tures must be evaluated with respect to a single dasize equal to the collective effect of the limitsof size, tum reference frame from the candidate datum refer- materialconditionbasis,andapplicablepositional ence frame set for the control. tolerance. Each boundary is located and oriented by the basic dimensions of the pattern. Each positio% tolerance zone is a volume defined by all points P 5.2 POSITIONAL TOLERANCING that satisfy the appropriate equation from Table5-1, where b is a position tolerance zone size parameter This Subsection presents a general explanationof (radius or half-width). positional tolerancing for features of size. A positional tolerance can be explained in terms of a zone Figure 5-3 illustrates the tolerance zone for a cylindrical hole at MMC or RFS, or a shaft at LMC. withinwhichtheresolvedgeometry(centerpoint, 22 --`,``,,`,`,,,,,`,``,`,,```,``,-`-`,,`,,`,`,,`---

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r T o l e r a n c e zone

Tolerance zone

I-/ri6

True position center plane FIG. 5-4

tion zone as defined above withb determined by the appropriate value from Table 5-2. The surface must conform to the applicable size limits. In the case ofan internal feature (spherical hollow, hole, or slot), there is a further condition that the feature must surround the tolerance zone. For MMC or LMC Taterial condition basis, the boundary defined by r(P ) = b ,with b as given here, is called the virtual condition. (c) Actual value. The actual value of position deviation is the smallest valueof to to which the feature conforms.

FIG. 5-3 TOLERANCE ZONE AND CONFORMANCE: HOLES AT MMC OR RFS; SHAFTS AT LMC --`,``,,`,`,,,,,`,``,`,,```,``,-`-`,,`,,`,`,,`---

TABLE 5-2 SIZE OF POSITION TOLERANCE ZONE, SURFACE INTERPRETATION Material Condition Basis

b

I

External]

MMC

rMMc

I

RFS

1

TOLERANCEZONEANDCONFORMANCE:TABS AT MMC OR RFS; SLOTSATLMC

LMC

+

The tolerance zone is a cylindrical volume.’Figure 5-4 illustrates the tolerance zone for a tab at MMC is two or RFS, or a slot at LMC. The tolerance zone disjoint half-spaces bounded by parallel planes. (b) Conformance. A feature conforms to a position tolerance t,, at a specified material condition basis if all points of the feature lie outside some posi-

5.2.2 In Terms of the Resolved Geometry of a Feature. (a) 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 featuresatLMC)mustliewithinacorresponding positional tolerance zone. Each zone is bounded by a sphere, cylinder, or pair of parallel planes of size equal to the total allowable tolerance for the corres‘For LMC and MMC controls the actual value of deviation can be negative. A negative actualvaluecan be interpreted as the unused portion of the bonus tolerance resulting from the departure of the feature from the applicable limit of size.

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TABLE 5-3 SIZE OF POSITION TOLERANCE ZONE, RESOLVED GEOMETRY INTERPRETATION

11

/-Tolerance

I

zone (radius=b)

Material Condition Basis

Feature

Internal

it0

MMC

2 + ( r M - r"C)

R r

LMC

5 + (rLMc - r M M ) Actual mating envelope (hole MMC, RFS)

ponding feature. Each zone is located and oriented by the basic dimensions of the pattern. A position tolerance zone is a spherical, cylindrjcal, or parallelplane volum: defined by all points P that satisfy the equation r(P ) 5 b, where b is the radius or halfwidth of the tolerance zone. Figure 5-5 illustratesthedefinitionforholesat MMC and RFS and for shafts at LMC. The figure shows the positionof a point on the axis of the actual envelope thatis outside the tolerance zone.A similar figure for holes at LMC or shafts at MMC or RFS would show the actual envelope surrounding the featuresurface.Thefeatureaxisextendsforthe full length of the feature. (b) Confomnce. A feature conforms to a position tolerance to at a specified material condition basis if all pointsof the resolved geometry of the applicableenvelope(asdeterminedbythematerial condition basis)lie within some position zone as definedabovewith b determinedbytheappropriate formulafromTable 5-3. Furthermore,thesurface must conform to the applicable size limits. (c) Actual value. The position deviation of a feature is the diameter of the smallest tolerance zone (smallest valueof b) which contains the center point or all points on the axis or center plane (within the extent of the feature) of the applicable actual envelope of the feature.

Actual minimum material envelope (shaft LMC)

' 2-True ' position axis FIG.5-5TOLERANCEZONEANDCONFORMANCE: HOLES AT MMC OR RFS;SHAFTS AT LMC

TABLE 5 4 DEFINITION OF VERIFYING VOLUME FOR PROJECTED TOLERANCE ZONE Material Condition Basis

Feature

Internal

f(64 w

f(F)> w

itself a boundary of perfectform extending outward from the feature by the specified projection length. Second, the surface of the feature does not violate the verifying volume. A projected position tolerance zone is a CylincJical or parallel-plane volumz defined byall points P that satisfy the equation r(P ) I b , where b is the radius 5.3 PROJECTED TOLERANCE ZONE or half-width of the tolerance zone.A verifying vol(a) Definition. For a cylindrical or parallel-plane ume is a cylind$cal or'parallel-plane volume defined feature, a projected tolerance specifies that a volume,by all points P that satisfy the appropriate equation called a verifying volume, with a boundary of perfect from Table 5-4, where w is a size parameter for the form,calledaverifyingboundary,canbedefined verifying volume. such that the following two conditions hold. First, Figure 5-6 illustrates a typical case. The projected tolerancezone is positionedandoriented by the the axis or center planeof the verifying boundaryis contained within a projected position tolerance zone, choice of datum reference frame. A plane perpendic24

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b

MATHEMATICALDEFINITION OF DIMENSIONINGANDTOLERANCINGPRINCIPLES

ASME Y14.5.1M-1994

TABLE 5-5 DEFINITION OFCONICAL TOLERANCE ZONE, SURFACE INTERPRETATION

, r Projected tolerance zone (1to

true positionaxis)

Material Condition Basis

’ 1,

p\\\smsw

,/

Verifying volume

Feature

L V e r i f y i n g envelope

\True

position axis

FIG.5-6PROJECTEDTOLERANCEZONE

FOR A HOLE

5.4 CONICAL TOLERANCE ZONE

--`,``,,`,`,,,,,`,``,`,,```,``,-`-`,,`,,`,`,,`---

A conical position tolerance zone is specified by different position tolerance values at each end of a cylindrical feature. A conical tolerance can be interpreted either in terms of the surface of thefeature or ular to the true position axis is located to contact the in terms of the =is of the feature. part surface that defines the end of the cylindrical 5.4.1 In Terms of the Surface of the Feature. feature. The height of the zone is the specified pro(a) Definition. Forapattern of cylindricalfeajection length and starts at the point where the true tures, a position control tighter at one end of the position axis intersects the contacting plane. The verfeatures than the other specifies that the surface of ifying volume is shown for a hole at MMC or RFS. each actual feature must not violate a corresponding (A similar picture would apply for a shaft at LMC. perfect form conical boundary. This boundary is a For a shaft at MMC or RFS, or a hole at LMC, the frustum of height and diameters equal to the collecverifying envelope would surround the feature.) tive effects of the limits of size, material condition (b) Conformance. A feature conforms to a posibasis, andapplicablepositionaltolerancesateach tiontolerance to, projectedadistance h , andata end of the feature. The boundary is located and orispecified material condition basis, if there exists at ented by the basic dimensions of the feature. A posileast one verifying volume for which the following tion toleratye zone is a conical volume defined by conditions hold. All points of the feature lie outside all points P that satisfy the appropriate equation from w deterthe verifying volume as defined above with 5-5, where+ b($) is the radiJus of the tolerance Table mined from the material basis as follows: for MMC, zone at height P . The radius b(P ) is related to the w = rM4c;for RFS, w = and for+LMC, w = rMc. position tolerance zone size parametersrl and ‘2 by: The veqfying envelope sabsfies 41‘)I t0/2 for all points P ontheresolvedgeometrystartingatthe intersection of the resolved geometry with the contacting plane and ending at the intersection of the resolved geometry with a second plane parallel to where the contacting plane and separated fromit by a distance h . Note that forRFS features, this definition can also be considered the resolved geometry interpretation. No resolved geometry interpretation is provided for 5 the position of P‘ gong the axis tetween $1 and MMC or LMC tolerances. P2, scaled so that y(P1) = 0 and y(P2) = 1. (c) Actual value. The position deviation of a feaFigure5-7illustratesthedefinitionforholesat ture is the size of the smallest projected tolerance MMC and RFS. (Shafts would have a similar piczone such that the resolved geometry of the actual ture.) A similar figure for holes at LMC or shafts at mating envelope lies within the tolerance zone for MMC or RFS would show the envelope surrounding the full projection height.

A ‘ M;.

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TABLE 5-6 SIZES OF CONICAL TOLERANCE ZONE, SURFACE INTERPRETATION Material Condition Basis

Feature

-

. I

Y

7

I

Nominal feature extent position position true True ( axis) 1 axis to

AT SURF C

FIG. 5-7 SURFACEINTERPRETATIONOFCONICAL TOLERANCEZONEFORHOLES AT MMC OR RFS I

the feature surfaze. The+tolerance zone axis extends between points PIand P2,which are the intersection of the true position axis with two planes, one at each end of thebasicfeature,atnominaldistanceand nominally located and oriented relative to the datum reference frame. (b) Confomance. A cylindrical feature conforms to position tolerancestl and t2 at a specified material condition basis if all points of the feature lie outside

I, 1

/

ATSURFD

some position zone as defined above, with ri(i = 1, 1) determined according to Table 5-6. The surface

must also conform to the applicable size limits. In the case of a hole, there is a further conditi hat the hole must surround the tolerance zone. (c)Actual va2ue. No definition for actual valueof position deviation is provided for the surface interpretation. Refer below to the axis interpretation for a definition of actual value.

8

UUI\I

u

-

,

I

SURF D

5.4.2 In Terms of the Axis of the Feature. FIG. 5-8 AXIS INTERPRETATIONOFCONICAL (a) Definition. For the axes of cylindrical features TOLERANCE ZONE FOR HOLES AT MMC OR RFS within a pattern, a position tolerance tighter at one end specifies that the axes of the actual mating envelopes (for features atM M C or RFS) or of the actual minimum material envelopes (for features at LMC) facesboundingthefeature.Thepositiontolerance mustliewithincorrespondingpositionaltolerance zones. Each of these zones is bounded by a frustum zone is a conicalvoluqe defiydby all pin@ P that of height and diameters equal to the collective effects satisfy the equation r(P ) I b(P ), where b(P ) is the radius os the tolerance zone at the height along the of the limits of size, material condition basis, and axis of P . (See the surface interpretation, para. 5.4.1, applicable positional tolerances at each end of the for details.) feature. The axis of the frustum is located and oriFigure 5-8 illustrates the axis definition for hole2. entedbythebasicdimensionsofthefeature.The frustum is located along the axis by the nominal sur- The tolerance zone axis extends between points P, 26 --`,``,,`,`,,,,,`,``,`,,```,``,-`-`,,`,,`,`,,`---

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TABLE 5-7 SIZES OF CONICAL TOLERANCE TABLE ZONE, AXIS INTERPRETATION POSITION TOLERANCE ZONE,

5-8 DEFINITION OF BIDIRECTIONAL

SURFACE INTERPRETATION

Material Condition Basis MMC

'i

Feature Type

Internal External

f

t.

+ (r,

t.

+ (rMMc - r,)

f

- rMMc)

RFS

LMC

ti

t + (rLMc - rmM)

ti

t.

Material Condition Basis

t.

Feature Type

+ (rMM - rMc)

Internal

r($ < b

r($ > b

External

GENERALNOTE: In this kble, r($ is the iistance from $to the resolved geometryof the tolerance zone boundary. Thetolerance zone boundary is a cylinder for a hole at MMC or RFS and for shafts at LMC; it is a pair of parallel planes for shafts at MMC or RFS and for holes at LMC.

s2,

--`,``,,`,`,,,,,`,``,`,,```,``,-`-`,,`,,`,`,,`---

and which are the intersection of the true position axis with the nominal surfaces bounding the feature. (b) Confomnce. A cylindrical feature conforms to position tolerancestl and t2 at a specified material coordinatesystem.Rectangularbidirectionaltolercondition basisif all points on the axis of the applicaof either the surface ble envelope (as determined by the material condition ancing can be explained in terms or the axis of the feature. An axis interpretation only basis) lie withinsomepositionzoneasdefined is provided for polar bidirectional tolerancing. above, with rl (i = 1, 2) determined by the appropriate formula from Table5-7. Furthermore, the sur5.5.1 In Terms of the Surface of the Feature. face must conform to the applicable size limits. This Section establishes the surface interpretationof (c) Actual value. A cylindrical feature controlled bidirectional positional tolerancing when applied in by a conical tolerance zone has two actual values for a rectangular coordinate system. position deviation, one at each end of the feature. (a) Definition. For a pattern of cylindrical feaThe actual value at each end is the smallest diameter tures, each bidirectional positional tolerance specifies circle that contains the axisof the actual mating enthat each surface must not violate a tolerance boundvelope at that end. Each circle is in the plane perpenary. For holes at MMC or RFS and shafts at LMC, dicular to the true position axis at the end point of each tolerance boundary is a cylinder of diameter the feature axis, and is centered on the true position equal to the collective effects of the limits of size, axis. In the case that the actual mating envelope can material condition basis, and applicable position tolrock, it may be possible to decrease the actual value erance. Each boundary is located and oriented, by the of position deviation at one end at the expense of the basic dimensionsof the pattern and by the applicable deviationattheotherend. No rule is definedfor direction of tolerance control, such that the axis of selecting among possible pairsof actual values. each boundary lies in the plane containing the true position axis of the corresponding feature and normal to the direction in which the tolerance applies. The 5.5 BIDERECTIONAL POSITIONAL orientation and position of the boundary axis within TOLERANCING this plane is unconstrained. A bidirectional positional tolerance zone for a cyFor holes at LMC and shafts at MMC or RFS, lindrical featureis specified by different position tol- each tolerance boundary is a pair of parallel planes erance valuesin different directions perpendicular to separated by a distance equal to the collective effects of the limits of size, material condition basis, and the basic feature axis. Bidirectional positional tolerapplicablepositiontolerance.Thecenterplane of ancing results in two distinct tolerance zones for locating each cylindrical feature. Each tolerance zone each boundary is that plane containing the axis of is consideredseparatelyinthefollowing. As with the corresponding feature and normal to the direction other tolerances, however, rules for simultaneous or in which the tolerance applies. separaterequirementsapplytothecomponents of A positional+ tolerance zone is a volume defined a bidirectional positional tolerance. (See Subsection by all points P that satisfy the appropriate equation from Table 5-8, where b is a position tolerance zone 4.6.) Bidirectional positional tolerancing may be applied in either a rectangular or a polar (cylindrical) size parameter (radius or half-width). 21 Copyright ASME International Provided by IHS under license with ASME

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(b) Confomnce. A cylindrical feature conforms to at a specified to a bidirectional positional tolerance material condition basis if all points of the feature lie outside some position tolerance as defined above with b determined by the appropriate value fromTable 5-9. Figure 5-9 shows an example of bidirectional tolerancing of a hole at MMC. Each callout creates its owncylindricalpositiontolerancezone.Thezone corresponding to the0.4 mm tolerance, shown in the bottom left,is free tobe positioned and oriented only in the plane indicated by the vertical dashed line. Similarly, the zone corresponding to the 0.2 mm tolerance, shown in the bottom right, can only move and tilt left-to-right in the view shown. Each of these planes of motion are determined by the basic dimensions from the indicated datums. A similar exampleis illustrated for shafts at MMC in Fig. 5-10. In this case, each callout creates a toler-

\@

TABLE 5-9 SIZE OF BIDIRECTIONAL POSITION TOLERANCE ZONE, SURFACE INTERPRETATION

b

1 FIG. 5-9

I

I

RFS

surface

L 0.2 tolerance zone

BIDIRECTIONALHOLETOLERANCES AT MMC. THE AXIS OFEACHTOLERANCEBOUNDARY TO LIE IN THEINDICATEDPLANCE

IS CONSTRAINED

28 --`,``,,`,`,,,,,`,``,`,,```,``,-`-`,,`,,`,`,,`---

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LMC

Uponstrained direction for 0.2 tolerance zone

/ L0.4 tolerance zone

MMC

4

16.2 16.0

I Unconstrained direction for 0.4 tolerance zone

I

I

Material Condition Basis

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--`,``,,`,`,,,,,`,``,`,,```,``,-`-`,,`,,`,`,,`---

FIG. 5-10

ILLUSTRATION OF BIDIRECTIONALPOSITIONTOLERANCINGOFASHAFTAT MMC. THETOLERANCEZONE FOR EACHCALLOUT IS BOUNDEDBYAPAIR OF PARALLELPLANES

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usually be controlled by more $an one directional tolerance;there is adistinctactualvalueforeach tolerance callout.)

TABLE 5-10 SIZE OF BIDIRECTIONAL POSITION TOLERANCE ZONE, AXIS INTERPRETATION

--`,``,,`,`,,,,,`,``,`,,```,``,-`-`,,`,,`,`,,`---

5.5.3 Polar Bidirectional Tolerancing in Terms of the Axis of the Feature. This Section establishes the axis (resolved geometry) interpretation of bidirectional positional tolerancing when applied in a cylindrical coordinate system. (While the term “polar” is used in ASME Y14.5M-1994, and used herein for consistency, a cylindrical coordinate system is actuallybeingused.Thetolerancesare specified in the plane normal to the axis of the cylindrical coordinate system.) (a) Definition. Foraxes of cylindricalfeatures ance zone bounded by parallel planes. The zone corwithin a pattern, polar bidirectional position tolerresponding to the 0.4 mm tolerance is bounded by ances specify that the axes of the actual mating envethe vertical planes separated by 16.6 mm. The zone lopes (for features at MMC or RFS) or of the minicorresponding to the 0.2 mm tolerance is bounded mum material envelopes (for features at LMC) must by the horizontal planes separated by 16.4 mm. lie within corresponding positional tolerance zones. (c)Actual value. No definition for actual valueof Each zone is bounded radially by two concentric cybidirectional position deviation is provided in terms lindrical arcs and tangentially by two planes symmetof the surface of the feature. Refer below to the axis rically disposed about the true position of the feature interpretation for a definition of actual value. and oriented at the basic polar angle of the feature. 5.5.2 In Terms of the Axis of the Feature. The plane separation and the difference in cylindrical This Section establishes theaxis (resolved geometry) arc radii are each equal to the total allowable tolerinterpretation of bidirectional positional tolerancing ance for the corresponding feature, including any efwhen applied in a rectangular coordinate system. fects of feature size. Each zone is located and ori(a) Definition. Foraxes of cylindricalfeatures A polar ented by the basic dimensions of the pattern. withinapattern,bidirectionalpositiontolerances bidirectional position tolerancezonejs a (cylindrical specify that the axis of each actual mating envelope shell) volume defined by all pointsP that satisfy the (for features at MMC or RFS) or minimum material two equations: envelope (for features at LMC) must lie within two corresponding positional tolerance zones. Each zone is boundedbytwoparallelplanesseparatedbya IP,‘ - Po I < b, distanceequaltothetotalallowabletolerancefor the corresponding feature, including any effects of and feature size. Each zone is located and oriented by the basic dimensions of the pattern. A bidirectional I (Z-2). l c r t l l b, position tolerance zone is a (slab) voluqe defined by all points P that satisfy the equationr(P ) I b , where where b is half the thickness of the tolerance zone. --t A = apointonthetrueposition axis of the (b) Conformance. A cylindrical feature conforms feature to a position toleranceto at a specified material conpp = the distance of P‘ from the axis of the polar dition basis if all points on the axis of the applicable (cylindrical) coordinate system envelope (as determined by the material condition basis) lie within some position zone as defined above po = the distance of the true position axis from the axis of the polar (cylindrical) coordinate with b determined by the appropriate formula from A system Table 5-10. Furthermore, the surface must conform Nt = the direction vector of the plane containing to the applicable size limits. the axis of the polar (cylindrical) coordinate (c) Actual value. The position deviation of a feasystemandthetrueposition axis of the ture is the thickness of the smallest tolerance zone feature towhichtheaxisconforms.(Note:afeaturewill 30

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t

-b-

P-

P

t

Axis of polar (cylindrical) coordinate system FIG. 5-11

DEFINITION OF THE TOLERANCE ZONE FORPOLAR

BIDIRECTIONAL TOLERANCING

b, = the tolerance zone size parameter for the cy- TABLE 5-11 SIZE OF POLAR BIDIRECTIONAL lindrical boundaries of the tolerance zone, POSITION TOLERANCE ZONE, equal in value to half the difference in radii AXIS INTERPRETATION of the boundaries Material Condition Basis b, = the tolerance zone size parameter for the planar boundariesof the zone, equal in value to LMC MMC I RFS b, or b, half of the distance between the boundaries tInternal The relationship between these quantitiesis illus2 Feature trated in Fig. 5-11. Type (b) Conformance. A cylindrical feature conforms External to a polar, bidirectional position tolerance with radial component f, and tangential component f,, each applied at a specified material condition basis, if all points on the axis of the applicable envelope(asdetermined by the material condition basis) lie within (c) Actual value. As withrectangular,bidirectional positional tolerancing, some position zone as defined above with b, and b, two actual valuesof position deviation are defined. The actual value of posidetermined by the appropriate formula from Table tiondeviationineithertheradialortangential 5-11, witht = t, and f = t,, respectively. Furthermore, direction is the thickness of the smallest tolerance thesurfacemustconformtotheapplicablesize zone to which the applicable axis conforms. limits. 31

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fined above, with W = WMMc - t,/2 and L = LMMC - tL/2,where WMMc is the MMC width of the elongated hole andLMMC is the MMC length of the elongated hole. Furthermore, the hole must surround the tolerance zoneandmustconformtothelimitsofsize. An elongated hole conforms to the limits of size if there exist two right,elongated-holecylinders(unconstrained in location or orientation), such that the following conditions hold. One cylinder, withW and L equal t o the MMC limits of size, is -surrounded by W and L the hole surface. The other cylinder, with equal to the LMC limits of size, surrounds the hole surface. (c)Actual value. No actualvalue of positiondeviation for elongated holes is defined.

r T r u e position axis

I

FIG. 5-12 TOLERANCEZONEANDCONFORMANCE; ELONGATED HOLE AT MMC. THE TOLERANCE ZONE RIGHT A CYLINDER SHOWN IN CROSS SECTION

I

IS

5.7 CONCENTRICITY AND SYMMETRY

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I

This Section provides definitions of concentricity andsymmetrytolerancesthatcontrolconcentricity and symmetry of features. Concentricity and symmetry controls are similar concepts and are treated toAn elongated holeis an internal feature consisting gether in this Section. Concentricityis that condition of two parallel, opposed, planar faces terminated by cylindrical end caps, tangent to the planar faces, with where the median points (centroids) of all diametriaxes inside the hole. For purposes of positional toler-cally opposed elements of a figure of revolution (or correspondingly located elementsof two or more raancing, an elongated hole is considered a feature of dially disposed features) are congruent with a datum size, characterized bytwo size parameters, its length axis orcenterpoint.Symmetryisthatcondition and width. Positional tolerancing can be applied to elongated holes on an MMC basis. Such tolerancing where one or more features is equally disposed about a datum plane. A symmetry toleranceis used for the is always considered to be bidirectional in nature, mathematical conceptof symmetry about a plane and even if a single tolerance value is applied. Only a a concentricity toleranceis used for the mathematical surface interpretation is provided. conceptofsymmetryaboutapointorsymmetry (a) Definition. For a pattern of elongated holes, a about an axis. Concentricity and symmetry controls position tolerance at MMC specifies that the surface are applied to features on anRFS basis only. Datum of each actual hole must not violate the boundary of references must also be RFS. a corresponding tolerance zone. Each boundary is a (a) Definition. A concentricity or symmetry tolerright cylinder with an elongated cross section of perance specifies that the centroid of corresponding fect form as shown in Fig. 5-12. Each boundary is point elements on the surfaces of the actual features located and oriented by the basic dimensions of the must lie in some symmetry tolerance zone. The zone pattern. Each position tolerance zone is the volume is bounded by a sphere, cylinder, or pair of parallel interior to the corresponding boundary (the shaded planes of size equal to the total allowable tolerance area in Fig.5-12). The boundary size is characterized for the features. The zone is located and oriented by by two size parameters, L and W , representing, respectively, the half-length and half-width of the zone.the basic dimensionsof the feature(s). The zoneis a spherical, cylindrical, or parallel-plane volume+de(b) Confomnce. A positiontolerance for an fined by all points P that satisfy the equationr(P ) I elongated hole specifies two values: t,, controlling position deviation in the direction of the hole width, b , where b is the radius or half-width of the tolerance zone. and tL, controllingpositiondeviationalongthe Corresponding point elements are obtained by inlength of the hole. An elongated hole conforms to tersecting a patternof symmetry rays with the actual position tolerances t , and tL if all points of the hole feature.Therays ofsymmetryaredeterminedacsurface lie outside a position tolerance zone as de5.6 POSITION TOLERANCING AT MMC FOR BOUNDARIES OF ELONGATED HOLES

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TABLE 5-12 SYMMETRY PATTERNSFOR OBTAINING CORRESPONDING FEATURE ELEMENTS Symmetry Type

symmetry axis

Tolerance Type Patterns of Symmetry Rays

Point Axis

Concentricity Concentricity

Plane

Symmetry

Rays from the datum point Rays from, and perpendicular to, the datum axis Rays from, and perpendicular to, the datum plane

centroid

Three-fold symmetry Six-fold symmetry cording to Table 5-12. If the feature is symmetric about a plane, a two-fold symmetry pattern is always FIG. 5-13 RAYSAREARRANGED IN THELOWEST used.Forpointandaxissymmetry,thesymmetry ORDER OF SYMMETRYABOUT AN AXIS OR POINT pattern is constructed using the lowest order of symmetry of the basic feature. One consequence of this is that surfaces of revolution use two-fold patterns of symmetry rays about the axis or center of symmetry. The feature elements are located at the intersec- pattern of rays if the centroid of corresponding points of intersection of the rays with the feature all lie tionofthesymmetryraysandtheactual feature b= within a tolerance zone as defined above with surface. t0/2. A feature conforms to a concentricity or symThis principle is illustrated in Fig.5-13. A feature metry tolerancetoif it conforms to symmetry patterns that has basic three-fold symmetry about a point or of (as shown in the figure) an axis results in a three-fold raysatallpossibleorientations(forsymmetry symmetry for the symmetry rays. If the symmetry of point), orientations and positions (symmetry axis), or positions (symmetry plane). the feature is six-fold, however, the symmetry rays (c) Actual value. The actual valueof concentricity are arranged in a two-fold pattern. or symmetry deviationis the smallest tolerance value (b) Conformance. For a concentricity or symmetry tolerance of to, a feature conforms to a symmetry to which the feature will conform.

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ORIENTATION, AND RUNOUT

zone is limited to the actual feature surface. For a feature axis, tangent plane, or center plane the extent is defined by projecting the actual surface points onto the axis, tangent plane, or center plane.

6.1 GENERAL

This Section establishes the principles and methods for mathematical evaluation of ASME Y14.5M1994 dimensioning and tolerancing which controls form, profile, orientation, and runout of various geometrical shapes and free state variations.

6.4 FORMTOLERANCES Form tolerances are applicableto single (individw1)features or elements of single features; therefore, form tolerances are not relatedto datums. The following subparagraphs cover the particulars of the form tolerances:straightness, flatness, circularity, and cylindricity.

6.2 FORM AND ORIENTATION CONTROL Form tolerances control straightness, flatness, circularity, and cylindricity. Orientation tolerances control angularity, parallelism, and perpendicularity. A profile tolerance may control form,orientation, size, and location depending on how it is applied.

6.4.1 Straightness. Straightnessis a condition where an element of a sulface, or an axis,is a straight line.A straightness tolerance specifiesa tolerance zone within which the considered element or derived median line must lie. A straightness tolerance is applied in the view where the elements to be controlled are represented by a straight line. 6.4.1.1 Straightness of a Derived Median Line (a)Definition. A straightness tolerance for the derived median line of a feature specifies that the derived median line must lie within some cylindrical zone whose diameter is the specified tolerance. A straightness zone for a derived median @e is a cylindrical volume consisting of all points P satisfying the condition:

6.3 SPECIFYING FORM AND ORIENTATION TOLERANCES

Form and orientation tolerances critical to function and interchangeability are spec$ed where the tolerances of size and location do not provide suflcient control.A tolerance of form or orientation may be spec$ed where no tolerance of size is given, for example, the controlofflatness afer assembly of the a parts. A form ororientationtolerancespec$es zonewithinwhichtheconsidered feature, its line elements, its axis, or its center plane must be contained. Where the tolerance value represents the diameter of a cylindrical zone, it is preceded by the diametersymbol.Inallothercases,thetolerance t If'x(P-A)ISvaluerepresentsthetotallineardistancebetween 2 two geometric boundaries and no symbol is required. Whiletheshapeofthetolerancezone is wellwhere defined (a cylinder, a zone bounded by two parallel f = the direction vector of the straightness axis planes, or a zone bounded by two parallel lines), the = apositionvectorlocatingthestraightness extent of the tolerance zone (e.g., the length of the axis cylinder)mustalsobeconsidered.Therearetwo t = thediameter of thestraightnesstolerance cases to be considered: zone (a) The extent of the tolerance zone is restricted tocontrolalimitedareaorlength ofthesurface (b) Conformance. A feature conforms to a straightness tolerance to if all points of the derived shown by a chain line drawn parallel to the surface median line lie within some straightness zone as deprofile dimensioned for length and location. fined above with t = to. That is, there exist f and (b) In all other cases, the extent of the tolerance - - - f

A'

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6 TOLERANCES OF FORM, PROFILE,

MATHEMATICALDEFINITION OF DIMENSIONINGANDTOLERANCINGPRINCIPLES

ASME Y14.5.1M-1994

A'

A surface conforms to the straightness tolerance to if it conforms simultaneously for all toleranced surface line elements corresponding to some actual mating surface. (c) Actual value. The actual value of straightness for a surface is the smallest straightness tolerance to whichthesurfacewillconform.

such that with t = to, all points of the derived median line are within the straightness zone. (c) Actual value. The actual value of straightness for the derived median line of a feature is the smallest straightness tolerance to which the derived median line will conform. 6.4.1.2 Straightness of Surface Line Elements (a) Definition. A straightnesstoleranceforthe

6.4.2 Flatness. Flatnessisthecondition of a sugace having all elements in one plane. A flatness tolerance specijies a tolerance Zone defined by two parallel planes which the su,$ace must lie. (a) Definition. A flatness tolerance specifies that allpointsofthesurfacemustlieinsomezone bounded by two parallelplaneswhichareseparated by the specified tolerance.

of a feature specifies that each line must lie in a 'One bounded by lines .which are separated by the specified and which are in the cutting plane defining the line element. A straightness 'One for a surface line is ~ area JI between parallel lines consisting of all points P satisfying the condition: line

A flatnesszone is avolumeconsisting - of all points P' satisfying the condition:

IP.(?-2)lI5t

and where

f = the direction vectorof the parallel planes de-

ep (2- Zs)= 0

+ finingtheflatnesszone A = a position vector locating the mid-plane of the flatness zone t = the size of the flatness zone (the separation of the parallel planes)

*

ep4=0

where

P = the direction vector of the center fine of the

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(b) Conformance. A feature conformsto a flatness tolerance to if all pointsof the featurelie within some flatness zone as defined above, with t = to. That is, there exist f and A such that with t = to, all points of the feature are within the flatness zone. (c) Actual value. The actual value of flatness for a surface is the smallest flatness tolerance to which the surface will conform.

straightness zone A = a position vector locating the center line of the straightness zone t = the sizeof the straightness zone (the separation between the parallel lines) = the normal to the cutting plane defined as being parallel to the cross product of the desired cuttin3 vector and the mating surface normal at Ps Ss= a point on the surface, contained by the cutting plane +

eP

Figure 6-1 illustrates a straightness tolerance zone for surface line elements of a cylindrical feature. Figure 6-2illustrates a straightness tolerance zone for surface line elements of a planar feature. (b) Conformance. A surfacelineelementconforms to the straightness tolerance to for a cutting of thesurfacelineelementlie planeifallpoints within some straightness zone ascJefined above with t = to. That is, there existf and A such that with t = to, all points of the surface line element are within the straightness zone.

6.4.3 Circularity (Roundness). Circularity is a condition of a surface where: (a) for a feature other than a sphere, all points of the su$ace intersected by any plane perpendicular to an axis are equidistantfrom that axis; (b) for a sphere, all points of the surface intersected by anyplane passing througha commn center are equidistant from that center. A circularity tolerance specifies a tolerance zone bounded by two concentric circles within which each circular elementof the sugace must lie, and applies independently at any plane described in (a) and (b) above. (a) Definition. A circularitytolerancespecifies that all points of each circular element of the surface

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h

STRAIGHTNESS ZONE

FIG. 6-1

EVALUATIONOFSTRAIGHTNESSOFACYLINDRICALSURFACE

must lie in some zone bounded by two concentric circles whose radii differ by the specified tolerance. Circular elements are obtained by taking cross-sections perpendicular to some spine. For a sphere, the spine is 0-dimensional (a point), and for a cylinder or conethespineis1-dimensional(asimple,non self-intersecting, tangent-continuous curve). The concentriccirclesdefiningthecircularityzoneare centeredon,andinaplaneperpendicularto,the spine. A circularity zone at a given crospection is an annular area consistingof all points P satisfying the conditions:

where

f' = for a cylinder or cone,g unit vector that is tangent to the spine atA . For a sphere, f' is

a unit vector+that points radially in all directions fromA A = apositionvectorlocatingapointonthe spine r = a radial distance (which may vary between circular elements) from the spine to the center of the circularity zone( r > 0 for all circular elements) t = the size of the circularity zone -9

Figure 6-3 illustrates a circularity zone for a circular element of a cylindrical or conical feature. (b) Conformance. A cylindrical or conical feature conforms to a circularity tolerance to if there e5ists a 1-dimensional spine such that at each point A of the spine the circular-element perpendicular to the tangentvector f' at A conformstothecircularity

rp.(p'-i)=O

and

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ZONE

FIG. 6-2

EVALUATION OF STRAIGHTNESSOF

toleran+ce to. That is, for each circular element there exist A and r such that with t = to all points of the circular element are within the circularity zone. A spherical feature conforms toa circularity tolerance to if there exists a point (a 0-dimensional spine) such that each circular element in each cutting plane containing the point conforms to the circularity tolerance to. That is, for e y h circular element there exist 9, r , and a common A such that witht = tn all points of thecircularelementarewithinthecircularity zone.

A PLANARSURFACE

(c) Actual value. The actual value of circularity for a feature is the smallest circularity tolerance to which the feature will conform. 6.4.4 Cylindricity. Cylindricity is a condition of a suflace of revolution in which all points of the suflace are equidistantfrom a common axis. A cylindricity tolerance specifies a tolerance zone bounded by two concentric cylinders within which thesuqace must lie. Inthe case of cylindricity, unlike that of circularity, the tolerance applies simultaneously to

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MATHEMATICALDEFINITION OF DIMENSIONING AND TOLERANCINGPRINCIPLES \

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/

\

ASME Y14.5.1M-1994

6.5 PROFILE CONTROL

CIRCULARITY

/ZONE

.,

--/%LOCUS OF TOLERANCE ZONE CENTERS FIG. 6-3 ILLUSTRATIONOFCIRCULARITYTOLERANCE ZONEFOR ACYLINDRICAL OR CONICALFEATURE

both circular and longitudinal elements of the surface (the entire surface). Note: The cylindricity tolerance is a composite control of form which includes circularity, straightness, and taper of a cylindrical feature.

(a) Definition. A cylindricitytolerancespecifies

that all points of the surface must lie in some zone

bounded by two coaxial cylinders whose radii differ by the specified tolerance. A cylindricity zoneis a volumebetyeen two coaxial cylinders consisting of all points P satisfying the condition:

A profile is theoutline of an object in a given plane (two-dimensionalfigure). Profiles are formed by projecting a three-dimensional f i g ure onto plane a or taking cross sections through thefigure. The elements of a profile are straight lines, arcs, and other curved lines. With profile tolerancing, the true proby basic radii, basic angular file maybedefined dimensions, basic coordinate dimensions, basic size dimensions, undimensioned drawings, or formulas. (a) 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 surat that point. For unilateral profile tolerances the face is offset totally in one direction or the otherby an amount equal to the profile tolerance. For bilateral profile tolerances the surface is offset in both directions by a combined amount equal to the profile tolerance. The offsets in each direction may, or may not, be disposed equgly. For a given pointPN on the nominal surface there is aunitvector 20 normaltothenominalsurface whosepositivedirection is arbitrary;itmaypoint either into or outof the material. A profile tolerance t consists of the sum of two intermediate tolerances t+ and t- . The intermediate tolerancest+ and t- represent the amount of tolerance to be disposed in the positive and negative dir-ctions of the surface normal fi, respectively, at PN.For unilateral profile tolerances either t+ or t- equals zero. t+ and t- are always non-negative numbers. The contribution of the nominal surface point towardsthetotaltolerancezoneisalinesegment normal to the nominal surfa3e and bounded by points at distances t+ and t- from P N .The profile tolerance zone is theunionof line segments obtained from each of the points on the nominal surface. (b) Conformance. A surJace conforms to a profile tolerance to if all points Ps of the surface conform to either of the intermediate tolerances t + p r t- disposedaboutsome correyonding point PN onthe nominal surface.A point Psconfoqs to th%intermediate tolefance t+ if it is between P? and PN + fit+. A point Ps confops to t h intermediate ~ tolerancetif it is between PN and PN - f i t - . Mathe5atically, this is the condition that there exists some PN on the %omin$ surface and someu , -t- Iu I t + , for which PS=pN+fiu. (c) Actual value. For both unilateral and bilateral profile tolerances two actual values are necessarily calculated: one for surface variations in the positive

sN

where

$ = the direction vector of the cylindricity axis A = apositionvectorlocatingthecylindricity axis r = the radial distance from the cylindricity axis to the center of the tolerance zone t = the size of the cylindricity zone (b) Conformance. A feature conforms to a cylindricitytolerance to if all points ofthefeature lie within some cylindricity zone+as defined above with t = to. That is, there existf,A , and r such that with t = to, all points of the feature are within the cylindricity zone. (c) Actual value. The actual value of cylindricity for a surface is the smallest cylindricity tolerance to which it will conform. 39

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direction and one for the negative direction. For each lines. Each of these cases is defined separately bedirection, the actual value of profile is the smallest low. If the tolerance valueis preceded by the diameintermediatetolerancetowhichthesurfaceconter symbol then the tolerance zone is a cylindrical forms. Note that no single actual value may be calcu-volume; if the notation EACH ELEMENT or EACH lated for comparison to the tolerance value in the RADIAL ELEMENTappearsthenthetolerance feature control frame, except in the case of unilateral zone is an area between parallel lines; in all other profile tolerances. cases the tolerance zone is a volume between parallel planes by default. 6.6 ORIENTATIONTOLERANCES

Angularity, parallelism, perpendicularity, and in some instancesprofile are orientation tolerancesapplicable to related features. These tolerances control the orientation of features to one another. In specifying orientation tolerancesto control angularity, parallelism, perpendicularity, and in some cases profile, the consideredfeature is related to one or more datum features. Relation to more than one datum feature is specified to stabilize the tolerance zone in more than one direction. Tolerance zones are total in valuerequiringan axis, or all elementsof the considered sulface to fall within this zone. Where it is a requirement to control only individual line elementsof a surface, a qualifying notation, such as EACH ELEMENT or EACH RALMAL ELEMEm, is added to the drawing. This permits control of individual elementsof the surface independently in relation to the datum and does not limit the total surface to an encompassing zone. Where it is desired to control a feature sulface established by the contacting points of that sulface, the tangentplane symbol is added in the feature control frame afer the stated tolerance. Angularity is the condition of a surface or center plane or axis at a specijiedangle (other than 90 deg.) from a datum plane or axis. Parallelism is the condition of a s u ~ a c eor center plane, equidistant at all points from a datum plane or an axis, equidistant along its length from one or more datum planes or a datum axis. Perpendicularity is the conditionof a surface, center plane, or axis at a right angle to a datum plane or axis. Mathematically, the equations describing angularity, parallelism, and perpendicularity are identical for a given orientation zone type when generalized in terms of the angle@) between the tolerance zone and the related datum@). Accordingly, the generic term orientation is used in place of angularity, parallelism, and perpendicularity in the definitions. See Appendix A. An orientation zone is bounded by a pair of parallel planes, a cylindrical surface, or a pairof parallel

6.6.1 Planar Orientation Zone (a) Definition. An orientation tolerance which is not preceded by the diameter symbol and which does not include the notation EACH ELEMENT or EACH RADIAL ELEMENTspecifiesthatthetoleranced surface, center plane, tangent plane, oraxis must lie in a zone bounded by two parallel planes separated by the specified tolerance and basically oriented to the primary datum and,if specified, to the secondary datum as well. A planar ogentation zone is a volume consisting of all points P satisfying the condition:

where

P = the direction vector

A'

of the planar orientation zone = a position vector locating the mid-plane of the planar orientation zone

t = the size of the planar orientation zone (the

separation of the parallel planes)

The planar orientation zone is oriented such that, if b1is the direction vector of the primary datum, then: IP.b,I=

I

I cos 0 I for a primary datum axis I sin 0 I for a primary datum plane

where 0 is the basic angle between the primary datum and the direction vector of the planar orientation zone. If a secondary datum is specified, the orientation zone is further restricted to be oriented relative to the direction vector,d2,of the secondary datum by: IP.b,l=

I cos a I for a secondary datum axis

I sin a I for a secondary datum plane

where f' is the normalized projection of P onto a plane normal to8,,and a is the basic angle between the secondary datum and f'. f' is given by:

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/ /

FIG. 6-4

PLANARORIENTATIONZONE

p'=

ASME Y14.5.1M-1994

\

I I I

WITH PRIMARYANDSECONDARY

P - (P 8,>8, I P - (f B,>B,I

DATUM PLANESSPECIFIED

preceded by the diameter symbol specifies that the toleranced axis must lie in a zone bounded by a cylinder with a diameter equal to the specified tolerance and whose axis is basically oriented to the primary datum and, if specified, to the secondary datum as well. A cylindrical orieption zone is a volume consisting of all points P satisfying the condition:

*

*

Figure 6-4 shows the relationshipof the tolerance zone direction vector to the primary and secondary datums. Figure6-5 illustrates the projection of f onto the primary datum plane to form P . (b) Conformance. A surface, center plane, tangent plane, or axis S conforms to an orientation tolerance to if all points of S lie within some planar orientation zone as tefined above with t = to. That is, there exist f and A such that with t = to, all points of S are within the planar orientation zone. Note that if the orientation tolerance refers to both a primary datum and a secondary datum, then f is fully determined. (c) Actual value. The actual value of orientation for S is the smallest orientation tolerance to whichS will conform.

where

'i = the direction vectorof the a x i s of the cylindrical orientation zone

A' = a position vector locating thea x i s of the cylindrical orientation zone

t = the diameter of the cylindrical orientation

zone

6.6.2 Cylindrical Orientation Zone (a) Definition. An orientation tolerance which is

The axis of the cylindrical orientation zoneis ori41

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FIG. 6-5

PROJECTION OF TOLERANCE VECTOR ONTO PRIMARY DATUM PLANE

ented such that, if 8,is the direction vector of the primary datum, then:

IP.8,k

I

I cos 0 I for a primary datum axis I sin 0 I for a primary datum plane

where 0 is the basic angle between the primary datum and the direction vector of the axis of the cylindrical orientation zone. If a secondary datum is specified, the orientation zone is further restricted to be oriented relative to the direction vector,h2,of the secondary datum by:

IP -8,I=

I

I cos OL I for a secondary datum axis I sin OL I for a secondary datum plane

where f" is the normalized projection of f onto a plane normal to8,,and OL is the basic angle between the secondary datum and f: f' is given by:

Figure 6-6 illustrates a cylindrical orientation tolerance zone. (b) Confomnce. An axis S conforms to an orientation tolerance to if all points of S lie within some cylindrical orientation zoneas defined above witht = to. That is, there exists f and A' such that with t = to, all points of S are within the orientation zone. Note that if the orientation tolerance refers to both a primarydatumandasecondarydatum,then f is fully determined. (c) Actual value. The actual value of orientation for S is the smallest orientation tolerance to whichS will conform. 6.6.3 Linear Orientation Zone (a) Definition. An orientation tolerance which includes the notation EACH ELEMENTor EACH RA-

42 --`,``,,`,`,,,,,`,``,`,,```,``,-`-`,,`,,`,`,,`---

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ASME Y14.5.1M-1994

3 .

CYL

FIG. 6-6

ORIENTATIONZONEBOUNDED

BYACYLINDER

WITH RESPECT TO APRIMARY DATUM PLANE

where

DIAL ELEMENT specifies that each line element of

'f = the direction vector of the center lineof the

the toleranced surface must lie in a zone bounded by two parallel lines which are (1) in the cutting plane defining the line element,(2) separated by the specified tolerance, and (3) are basically oriented to the primarydatumand,ifspecified,tothesecondary datum as well. --t For a surface point Ps,a line5 orientation zone is an area consisting ofall points P in J cutting plane OJ direction vector that contains pS.The points P satisfy the conditions:

+ linearorientationzone A = a position vector locating the center line of the linear orientation zone --t Ps = a point on S = the normal to the cutting plane and basically oriented to the datum reference frame t = the size of the linear orientation zone (the separation between the parallel lines) The cutting planeis oriented to the primary datum by the constraint:

e,

tp

ep 8, = 0 *

and

If a secondary datum is specified, the cutting plane is further restricted to be oriented to the direction vector of the secondary datum, fi2, by the constraint: 43

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MATHEMATICALDEFINITION OF DIMENSIONINGANDTOLERANCINGPRINCIPLES

ASME Y14.5.1M-1994

--`,``,,`,`,,,,,`,``,`,,```,``,-`-`,,`,,`,`,,`---

Z f+X

/

Z

ORIENTATION ZONE FIG. 6-7

ORIENTATIONZONEBOUNDEDBYPARALLELLINES

eP. b2I = I cos a Ifor secondary datum axis I ep.b, I = I sin a I for a secondary datum plane I

Thedirectionvectorofthecenterlineofthelinear orientation zone,f', is constrained to liein the cutting planeby:

a

The position vector i, which locates the center orientation linear line the locates the also zone, of cutting plane through the following constraint:

&,.9=0 The center line of the linear orientation zone is oriented such that, if b1is the direction vectorof the primary datum, then:

&f(+A')=O

If a primary or secondary datum axis is specified, and the toleranced feature in its nominal conditionis rotationally symmetric about that datum axis, then the cutting planes are further restricted to contain the datum axis as follows:

IP*b,l=

I

I cos 0 I for a primary datum axis

I sin 0 I for a primary datum plane

where 0 is the basic angle between the primary daand turn direction thevector oflinear orientathe tion zone. 6-7 illustrates anorientationzonebounded by parallel lines on a cutting plane for a contoured

ep.(&li)=O

to

y+x

where B' is apositionvectorthatlocatesthedatumFigure axis. Otherwise, the cutting planes are required to be parallel 44

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MATHEMATICAL DEFINITION OF DIMENSIONING AND TOLERANCING PRINCIPLES

ASME Yl4.5.1M-1994

(b) Conformunce. A surface, center plane, or tangent plane S conforms to an orientation tolerance to for a cutting plane if all pointsof the intersection lie within some linear orientation zone of s with as deJfined above with t = to. That is, there exist f' and A such that witht = to, all points of S are within the linear orientation zone. A surface S conforms to the orientation tolerance to if it conforms simultaneously for all surface points and cutting planes Note that if the orientation tolerance refers to both a primary datum and a secondary datum, then f' is fully deterinined. (c) Actual value. The actual value of orientation S for S is the smallest orientation tolerance to which will conform.

eP

eP

eP.

6.7.1CircularRunout

6.7.1.1 Surfaces Constructed at Right Angles to a Datum Axis (a) Definition. The tolerance zone for each circular element on a surface constructed at right angles to a datum axisis generated by revolving a line segment about the datum axis. The line segmentis parallel to the datum axis and is of length to, where to is the specified tolerance. The resulting tolerance zone is the surface of a cylindzr of height to. For a surface pointPs,a circular runout tolerance zone is the-surface ofa cylinder consistingof the set of points P satisfying the conditions: +

+

lblx(P-A)I=r

and

l b , . ( P ' - ~ )2l d

6.7RUNOUTTOLERANCE

Runout is a composite tolerance used to control where the functional relationship of one or more features of a part to a datum axis. The types of features conr = the radial distance from to the axis trolled by runout tolerances include those sulfaces 8,= the direction vector of the datum axis constructed around a datum axis and those conA+ = a position vector locating the datumaxis structed at right angles to a datum axis. B = a position vector locating the center of the Surfaces constructed around a datum axis are those tolerance zone surfaces that are either parallel to the datum axis or t = the size of the tolerance zone (the height of are at some angle other than 90 deg. to the datum the cylindrical surface) axis. The mathematical definitionof runout is necessarily separated into two definitions: one for surfaces (b) Confomynce. The circular element through a surface pointPs conforms to the circular runout tolconstructed around the datum axis, and one forsurerance to if all points of the element lie within some faces constructed at right angles to the datum axis. circular runout tolerance zo2e as defined above with A feature may consist of surfaces constructed both t = to. That is, there existsB such that with t = to all around and at right angles to the datum axis. Separate points of the surface element are within the circular immathematicaldefinitionsdescribethecontrols runout zone. posedby a single runout tolerance on the distinct A surface conforms to the circular runout tolerance surfaces that comprise such a feature. Circular and if all circular surface elements conform. 6.7.1 and 6.7.2 totalrunoutarehandledinparas. (c)Actual value. The actual valueof circular runrespectively. out for a surface constructed at right angles to daa Evaluation of runout (especially total runout) on tum axis is the smallest circular runout tolerance to tapered or contoured surfaces requires establishment which it will conform. of actualmatingnormals.Nominaldiameters,and 6.7.1.2SurfacesConstructed Around aDa(as applicable) lengths, radii, and angles establish a tum Axis cross-sectional desired contour having perfect form (a) Definition. The tolerance zone for each circuand orientation. The desired contour may be translated axially and/or radially, but may not be tilted or lar element on a surface constructed around a datum axis is generated by revolving a line segment about scaled with respect to the datum axis. Whena tolerthe datum axis. The line segment is normal to the ance band is equally disposed aboutthis contour and desired surface and is of length to, where to is the then revolved around the datum axis, a volumetric specified tolerance. Depending on the orientationof tolerance zone is generated.

S,,

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the resulting tolerance zone will be either a flat annu- 6.7.2 Total Runout lar area, or the surface+of a truncated co$e. Surfaces Constructed at Right Angles to a DaFor a surface pointPs,a datum axis [ A ,hl], and tum Axis a given mating surface, a circular runout tolerance (a)Definition. A total runout tolerance for a surzone for a surface constructed aro5nd a datum axis axis speciface constructed at right angles to a datum consists of theset of points P satisfyingthe fies that all points of the surface must lie in a zone conditions: bounded by two parallel planes perpendicular to the datum axis and separated by the specified tolerance. For a surface constructed at right angles to a datum axis, a total runout tolerance zone is a volume consisting of the points P satisfying: and +

+

IIP-BI-dl