Mathematical Definition of Dimensioning and Tolerancing Principles 7-9Note that it is not always necessary to fully constrain a datum reference frame.. Although form tolerances are conce
Trang 2Mathematical Definition of Dimensioning and Tolerancing Principles 7-9
Note that it is not always necessary to fully constrain a datum reference frame Consider a part thatonly has an orientation tolerance applied to a feature with respect to another datum feature One can seethat it is not necessary or productive to position the datum reference frame in any manner because theorientation of the feature with respect to the datum is not affected by location of the datum nor of thefeature
The rules of datum precedence embodied in Y14.5 can be expressed in terms of degrees of freedom Aprimary datum may arrest one or more of the original six degrees of freedom A secondary datum may
arrest one or more additional available degrees of freedom; that is, a secondary datum may not arrest or
modify any degrees of freedom that the primary datum arrested A tertiary datum may also arrest anyavailable degrees of freedom, though there may be none after the primary and secondary datums havedone their job; in such a case, a tertiary datum is superfluous and can only add confusion
The Y14.5.1 standard contains several tables that capture the finite number of ways that datumreference frames may be constructed using the geometric entities points, lines, and planes Included areconditions between the primary, secondary, and tertiary datums for each case
7.4.5 Form Tolerances
Form tolerances are characterized by the fact that the tolerance zones are not referenced to a datumreference frame Form tolerances do not control the form of a feature with respect to another feature, norwith respect to a coordinate system established by other features Form tolerances are often used to refinethe inherent form control imparted by a size tolerance, but not always Therefore, the mathematical defini-
tions presented in this section reflect the independent application of form tolerances The mathematical
description of the net effect of simultaneously applied multiple tolerance types to a feature is not covered
in this chapter
Although form tolerances are conceptually simple, too many users of geometric dimensioning andtolerancing seem to attribute erroneous characteristics to them, most notably that the orientation and/orlocation of the tolerance zone are related to a part feature As stated in the prior paragraph, form tolerancesare independent of part features or datum reference frames The mathematical definitions that appearbelow describe in vector form the geometric elements of the tolerance zones associated with form toler-ances; these geometric elements are axes, planes, points, and curves in space The description of thesegeometric elements must not be misconstrued to mean that they are specified up front as part of the
application of a form tolerance to a nominal feature; they are not The geometric elements of form
tolerances are dependent only on the characteristics of the toleranced feature itself, and this is tion that cannot be known until the feature actually exists and is measured
informa-7.4.5.1 Circularity
A circularity tolerance controls the form error of a sphere or any other feature that has nominally circular crosssections (there are some exceptions) The cross sections are taken in a plane that is perpendicular to some
spine, which is a term for a curve in space that has continuous first derivative (or tangent) The circularity
tolerance zone for a particular cross-section is an annular area on the cross-section plane, which is centered
on the spine Because circularity is a form tolerance, the tolerance zone is not related to a datum referenceframe, nor is the spine specified as part of the tolerance application Note that the circularity definitiondescribed here is consistent with the ANSI/ASME Y14.5M-1994 definition, but is not entirely consistent withthe 1982 version of the standard See the end of this section for a fuller explanation
The mathematical definition of a circularity tolerance consists of equations that put constraints on a
set of points denoted by Pv
such that these points are in the circularity tolerance zone, and no others
Trang 37-10 Chapter Seven
Consider on Fig 7-4 a point Av
on a spine, and a unit vector T$ which points in the direction of the tangent
to the spine at Av
.The set of points Pv
on the cross-section that passes through Av
is defined by Eq (7.1) as follows
0 ) (
ˆ • P − A =
T v v
(7.1)The zero dot product between the vectors T$ and (Pv−Av) indicates that these vectors are perpendicu-lar to one another Since we know that T$ is perpendicular to the spine at Av, and Pv−Av is a vector thatpoints from Av
to Pv
, then the points Pv
must be on a plane that contains Av
and that is perpendicular to T$.Thus, we have defined all of the points that are on the cross section Next, we need to restrict this set ofpoints to be only those in the circularity tolerance zone
As was stated above, the circularity tolerance zone consists of an annular area, or the area betweentwo concentric circles that are centered on the spine The difference in radius between these circles is thecircularity tolerance t
2
t r A
To verify that a measured feature conforms to a circularity tolerance, one must establish that themeasured points meet the restrictions imposed by Eqs (7.1) and (7.2) In geometric terms, one must find aspine that has the circularity tolerance zones that are created according to Eqs (7.1) and (7.2), containingall of the measured points The reader will likely find this definition of circularity foreign, so some explana-tion is in order
As was stated earlier in this section, the details of circularity that are discussed here correspond tothe ANSI/ASME Y14.5M-1994 standard, which contains some changes from the 1982 version The 1982
version of the standard, as written, required that cross sections be taken perpendicular to a straight axis,
and that the circularity tolerance zones be centered on that straight axis, thereby effectively limiting theapplication of circularity to surfaces of revolution In order to expand the applicability of circularitytolerances to other features that have circular cross sections, such as tail pipes and waveguides, the
Figure 7-4 Circularity tolerance zone
Trang 4Mathematical Definition of Dimensioning and Tolerancing Principles 7-11
definition of circularity was modified such that circularity controls form error with respect to a curved
“axis” (a spine) rather than a straight axis The 1994 standard preserves the centering of the circularitytolerance zone on the spine
Unfortunately, the popular interpretation of circularity does not correspond to either the 1982 or the
1994 versions of Y14.5M Rather, a metrology standard (B89.3.1-1972, Measurement of Out of Roundness)seems to have implicitly provided an alternative definition of circularity by virtue of the measurementtechniques that it describes The main difference between the B89 metrology standard and the Y14.5Mtolerance definition standard is that the B89 standard does not require the circularity tolerance zone to becentered on the axis Instead, various fitting criteria are provided for obtaining the “best” center of thetolerance zone for a given cross section Without delving into the details of the B89.3.1-1972 standard,suffice it to say that the four criteria are least squares circle (LSC), minimum radial separation (MRS),maximum inscribed circle (MIC), and minimum circumscribed circle (MCC)
There is a rather serious geometrical ramification to allowing the circularity tolerance zone to “float.”Consider in Fig.7-5 a three-dimensional figure known as an elliptical cylinder which is created by translat-ing or extruding an ellipse perpendicular to the plane in which it lies Obviously, such a figure has elliptical
cross sections, but it also has perfectly circular cross sections if taken perpendicular to a properly titled
Creation of a mathematical definition of circularity revealed the inconsistency between the
Y14.5M-1982 definition of circularity and common measurement practice as described in B89.3.1-1972, and alsorevealed subtle but potentially significant problems with the latter This example illustrates the value thatmathematical definitions have brought to the tolerancing and metrology disciplines
Trang 57-12 Chapter Seven
7.4.5.2 Cylindricity
A cylindricity tolerance controls the form error of cylindrically shaped features The cylindricity tolerancezone consists of a set of points between a pair of coaxial cylinders The axis of the cylinders has no pre-defined orientation or location with respect to the toleranced feature, nor with respect to any datumreference frame Also, the cylinders have no predefined size, although their difference in radii equals the
− , and by the sine of the angle between T$ and Pv−Av The mathematical operations justdescribed are those of the vector cross product Thus, the distance from the axis to a point Pv
is expressedmathematically as T ˆ ( P v A v )
−
× To generate a cylindricity tolerance zone, the points Pv
must be stricted to be between two coaxial cylinders whose radii differ by the cylindricity tolerance t
re-Eq (7.3) constrains the points Pv
such that their distance from the surface of an imaginary cylinder of
radius r is less than half of the cylindricity tolerance.
2 )
(
ˆ P A r t
Trang 6Mathematical Definition of Dimensioning and Tolerancing Principles 7-13
If, when assessing a feature for conformance to a cylindricity tolerance, we can find an axis whosedirection and location in space are defined by T$ and Av , and a radius r such that all of the points of theactual feature consist of a subset of these points Pv
, then the feature meets the cylindricity tolerance
7.4.5.3 Flatness
A flatness tolerance zone controls the form error of a nominally flat feature Quite simply, the tolerancedsurface is required to be contained between two parallel planes that are separated by the flatness toler-ance See Fig 7-7
To express a flatness tolerance mathematically, we define a reference plane by an arbitrary locatingpoint Av
on the plane and a unit direction T$ that points in a direction normal to the plane The quantity
Figure 7-7 Flatness tolerance definition
A
Pv v
− is the vector distance from the reference plane’s locating point to any other point Pv
Of moreinterest though is the component of that distance in the direction normal to the reference plane This isobtained by taking the dot product of Pv Av
− and T$.
2 ) (
ˆ P A t
Eq (7.4) requires that the points Pv
be within a distance equal to half of the flatness tolerance from thereference plane
In mathematical terms, to determine conformance of a measured feature to a flatness tolerance, wemust find a reference plane from which the distances to the farthest measured point to each side of thereference plane are less than half of the flatness tolerance
Note that Eq (7.4) is not as general as it could be The true requirement for flatness is that the sum of
the normal distances of the most extreme points of the feature to each side of the reference plane be nomore than the flatness tolerance Stated differently, although Eq (7.4) is not incorrect, there is no require-ment that the reference plane equally straddle the most extreme points to either side In fact, manycoordinate measuring machine software algorithms for flatness will calculate a least squares plane throughthe measured data points and assess the distances to the most extreme points to each side of this plane
In general, the least squares plane will not equally straddle the extreme points, but it may serve as anadequate reference plane nevertheless
Trang 77-14 Chapter Seven
7.5 Where Do We Go from Here?
Release of the Y14.5.1 standard in 1994 addressed one of the major recommendations that emanated fromthe NSF Tolerancing Workshop However, the work of the Y14.5.1 subcommittee is not complete TheY14.5.1 standard represents an important first step in increasing the formalism of geometric tolerancing,but many other things must happen before we can claim to have resolved the metrology crisis The goodnews is that things are happening Research efforts related to tolerancing and metrology have acceleratedover the time frame since the GIDEP Alert, and we are moving forward
7.5.1 ASME Standards Committees
Though five years have passed since the release of the Y14.5.1 standard, it is difficult to discern the impactthat it has had on the practitioners of geometric tolerancing However, the impact that it has had on thestandards development scene is easier to measure Advances in standards work are greatly facilitatedwhen standards developers have a minimal dependence on subjective interpretations of the standardizedmaterials Indeed, it is the specific duty and responsibility of standards developers to define their subjectmatter in objectively interpretable terms; otherwise standardization is not achieved The Y14.5.1 standard,and the philosophy that it embodies, provides a means for ensuring a lack of ambiguity in standardizeddefinitions of tolerances
Despite the alphanumeric subcommittee designation (Y14.5.1), which suggests that it sit below theY14.5 subcommittee, the Y14.5.1 subcommittee has the same reporting relationship to the Y14 main com-mittee, as does the Y14.5 subcommittee The new Y14.5.1 effort was truly a parallel effort to that of Y14.5(though certainly not entirely independent) Its value has been sufficiently demonstrated within thesubcommittees to the extent that the leaders of each group are establishing a much closer degree ofcollaboration The result will undoubtedly be better standards, better tools for specifying allowable partvariation, less disagreement between suppliers and customers regarding acceptability of parts, and betterand cheaper products
7.5.2 International Standards Efforts
The impact of the Y14.5.1 standard extends to the international standards scene as well Over the past fewyears, the International Organization for Standardization (ISO) has been engaged in a bold effort tointegrate international standards development across the disciplines from design through inspection As
a participating member body to this effort, the United States has made its share of contributions Amongthese contributions are mathematical definitions of form tolerances These definitions are closely derivedfrom the Y14.5.1 versions, but customized to reflect the particular detailed differences, where they exist,between the Y14.5 definitions and the ISO definitions As other ISO standards are developed or revised,additional mathematical tolerance definitions will be part of the package
7.5.3 CAE Software Developers
Aside from standards developers, computer aided engineering (CAE) software developers should be thekey group of users of mathematical tolerance definitions Recalling the lack of uniformity and correctness
in CMM software as brought to light by the GIDEP Alert, it should not be difficult to see the need forprogrammers of CAE systems (including design, tolerancing, and metrology) to know the detailed aspects
of the tolerance types and code their software accordingly In some cases, this can be achieved by codingthe mathematical expressions from the Y14.5.1 standard directly into their software
We are not yet aware of the actual extent of usage of the mathematical tolerance definitions from theY14.5.1 standard among CAE software developers Where vendors of such software claim compliance to
US dimensioning and tolerancing standards, customers should rightly expect that the vendor owns a
Trang 8Mathematical Definition of Dimensioning and Tolerancing Principles 7-15
copy of the Y14.5.1 standard and has ensured that its algorithms are consistent with its requirements Itmight be reasonable to assume that this is not the case across the board, and it would be a worthyendeavor to determine the extent of any such lack of compliance As of this writing, ten years have passedsince the GIDEP Alert, and perhaps the time is right to see whether the situation has improved withmetrology software
The groundbreaking Y14.5.1 standard was the result of a collective effort by a team of talented and uniqueindividuals with diverse but related backgrounds This author was but one contributor to the effort, and
I would like to sincerely thank the other contributors for their wit, wisdom, and camaraderie; I learned quite
a lot from them through this process Rather than list them here, I refer the reader to page v of the standardfor their names and their sponsoring organizations At the top of that list is Mr Richard Walker whodemonstrated notable dedication and leadership through several years of intense development.Unlike many other countries, standards of these types in the United States are voluntarily specifiedand observed by customers and suppliers rather than mandated by government Moreover, the standardsare developed primarily with private funding by companies that have an interest in the field and havepersonnel with the proper expertise These companies enable committee members to contribute to stan-dards development by providing them with travel expenses for meetings and other tools and resourcesneeded for such work
3 James/James 1976 Mathematical Dictionary - 4 th Edition New York, New York: Van Nostrand.
4 Srinivasan V., H.B Voelcker, eds 1993 Proceedings of the 1993 International Forum on Dimensional Tolerancing and Metrology, CRTD-27 New York, New York: The American Society of Mechanical Engineers.
5 The American Society of Mechanical Engineers 1972 ANSI B89.3.1 - Measurement of Out-of-Roundness New
York, New York: The American Society of Mechanical Engineers
6 The American Society of Mechanical Engineers 1994 ASME Y14.5 - Dimensioning and Tolerancing New York,
New York: The American Society of Mechanical Engineers
7 The American Society of Mechanical Engineers 1994 ASME Y14.5.1 - Mathematical Definition of Dimensioning and Tolerancing Principles New York, New York: The American Society of Mechanical Engineers.
8 Tipnis V 1990 Research Needs and Technological Opportunities in Mechanical Tolerancing, CRTD-15 New
York, New York: The American Society of Mechanical Engineers
9 Walker, R.K 1988 CMM Form Tolerance Algorithm Testing, GIDEP Alert, #X1-A-88-01A.
10 Walker R.K., V Srinivasan 1994 Creation and Evolution of the ASME Y14.5.1M Standard Manufacturing Review 7(1): 16-23.
Trang 9He is the Convener of ISO/TC 213/WG 13 on Statistical Tolerancing of Mechanical Parts He holds membership in ASME and SIAM.
8.1 Introduction
Statistical tolerancing is an alternative to worst-case tolerancing In worst-case tolerancing, the designeraims for 100% interchangeability of parts in an assembly In statistical tolerancing, the designer abandonsthis lofty goal and accepts at the outset some small percentage of failures of the assembly
Statistical tolerancing is used to specify a population of parts as opposed to specifying a single part.Statistical tolerances are usually, but not always, specified on parts that are components of an assembly
By specifying part tolerances statistically the designer can take advantage of cancellation of geometricalerrors in the component parts of an assembly — a luxury he does not enjoy in worst-case tolerancing.This results in economic production of parts, which then explains why statistical tolerancing is popular
in industry that relies on mass production
In addition to gain in economy, statistical tolerancing is important for an integrated approach tostatistical quality control It is the first of three major steps - specification, production, and inspection - inany quality control process While national and international standards exist for the use of statisticalmethods in production and inspection, none exists for product specification For example, ASME Y14.5M-
1994 focuses mainly on the worst-case tolerancing By using statistical tolerancing, an integrated tical approach to specification, production, and inspection can be realized
statis-Chapter
8
Trang 108.2 Specification of Statistical Tolerancing
Statistical tolerancing is a language that has syntax (a symbol structure with rules of usage) and semantics(explanation of what the symbol structure means) This section describes the syntax and semantics ofstatistical tolerancing
Statistical tolerancing is specified as an extension to the current geometrical dimensioning and ancing (GD&T) language This extension consists of a statistical tolerance symbol and a statistical toler-ance frame, as described in the next two paragraphs Any geometrical characteristic or condition (such assize, distance, radius, angle, form, location, orientation, or runout, including MMC, LMC, and enveloperequirement) of a feature may be statistically toleranced This is accomplished by assigning an actualvalue to a chosen geometrical characteristic in each part of a population Actual values are defined inASME Y14.5.1M-1994 (See Chapter 7 for details about the Y14.5.1M-1994 standard that provides math-ematical definitions of dimensioning and tolerancing principles.) Some experts think that statisticallytoleranced features should be produced by a manufacturing process that is in a state of statistical controlfor the statistically toleranced geometrical characteristic; this issue is still being debated
toler-The statistical tolerance symbol first appeared in ASME Y14.5M-1994 It consists of the letters STenclosed within a hexagonal frame as shown, for example, in Fig 8-1 For size, distance, radius, and anglecharacteristics the ST symbol is placed after the tolerances specified according to ASME Y14.5M-1994 orISO 129 For geometrical tolerances (such as form, location, orientation, and runout) the ST symbol isplaced after the geometrical tolerance frame specified according to ASME Y14.5M-1994 or ISO 1101 SeeFigs 8-2 and 8-3 for further examples
The statistical tolerance frame is a rectangular frame, which is divided into one or more compartments
It is placed after the ST symbol as shown in Figs 8-1, 8-2, and 8-3 Statistical tolerance requirements can
be indicated in the ST frame in one of the three ways defined in sections 8.2.1, 8.2.2, and 8.2.3
8.2.1 Using Process Capability Indices
Three sets of process capability indices are defined as follows
Trang 11Statistical Tolerancing 8-3
The process capability indices are nondimensional parameters involving the mean and the standarddeviation of the population The nondimensionality is achieved using the upper and lower specificationlimits Cp is a measure of the spread of the population about the average Cc is a measure of the location
of the average of the population from the target value Cpk is a measure of both the location and the spread
For the example illustrated in Fig 8-1, the population of actual values for the specified size shouldhave its Cp value at or above 1.5, Cpk value at or above 1.0, and Cc value at or below 0.5 For theindicated parallelism, the population of out-of-parallelism values (that is, the actual values for parallelism)should have its Cpu value at or above 1.0, and its Ccu value at or below 0.3
Limits on the process capability indices also imply limits on the mean and the standard deviation ofthe population of actual values through the formulas shown at the beginning of this section Such limits
on µ and σ can be visualized as zones in the µ−σ plane, as described in section 8.3.1 To derive the limits
on µ and σ , values of L, U, and τ should be obtained from the specification For the example illustrated in Fig 8-1, consider the size first From the size specification, the lower specification limit L = 9.95, the upper specification limit U = 10.05, and the target value τ = 10.00 because it is the midpoint of the allowable size variation Next consider the specified parallelism, from which it can be inferred that L = 0.00, U = 0.01, and
τ = 0.00 because zero is the intended target value.
Using Cpl, Cpu, or Cpk in the ST tolerance frame implies only that these values should be within thelimits indicated Caution must be exercised in any further interpretation, such as the fraction of population
lying outside the L and/or U limits, because it requires further assumption about the type of distribution,
such as normality, of the population Note that such additional assumptions are not part of the tion, and their invocation, if any, should be separately justified
specifica-Figure 8-1 Statistical tolerancing using process capability indices
Trang 128-4 Chapter Eight
Process capability indices are used quite extensively in industrial production, both in the US andabroad, to quantify manufacturing process capability and process potential Their use in product specifi-cation may seem to be in conflict with the time-honored “process independence” principle of the ASMEY14.5 This apparent conflict is false; the process capability indices do not dictate what manufacturingprocess should be used — they place demand only on some statistical characteristics of whatever pro-cess that is chosen
Issues raised in the last two paragraphs have led to some rethinking of the use of the phrase “processcapability indices” in statistical tolerancing We will come back to this point in section 8.5, after theintroduction of a powerful concept called population parameter zones in section 8.3.1
8.2.2 Using RMS Deviation Index
RMS (root-mean-square) deviation index is defined as Cpm = U−L
+ −
6 σ2 (µ τ)2 A numerical lower limit forCpm is indicated as shown in Fig 8-2 using the ≥ symbol The requirement here is that the mean andstandard deviation of the population of actual values should be such that the Cpm index is within thespecified limit
For the example illustrated in Fig 8-2, the population of actual values for the size should have aCpm value that is greater than or equal to 2.0 For the specified parallelism, the population of out-of-parallelism values (that is, the actual values for parallelism) should have a Cpm value that is greater than
or equal to 1.0
Cpm is called the RMS deviation index because σ2 + ( µ τ − )2 is the square root of the mean ofthe square of the deviation of actual values from the target value τ Limiting Cpm also limits the mean andthe standard deviation, and this can be visualized as a zone in the µ−σ plane Section 8.3.1 describes suchzones To derive the limits on µ and σ, values for L, U, and τ should be obtained from the specification of
Fig 8-2 as explained in section 8.2.1
Cpm is closely related to Taguchi’s quadratic cost function, which states that the total cost to society
of producing a part whose actual value deviates from a specified target value increases quadratically withthe deviation Specifying an upper limit for Cpm is equivalent to specifying an upper limit to the average
Figure 8-2 Statistical tolerancing using
RMS deviation index