AN ABSTRACT OF THE THESIS OF
CHOOWONG
TANGKOONSOMBATI for the degree of Master of
Science in Industrial Engineering presented on
August 25, 1994.
Title: Assembly Tolerance Analysis in Geometric Dimensioning
and Tolerancinq
Abstract approved:
Redacted for Privacy
Sheikh
Burhanuddin
Tolerance analysis is a major link between design and
manufacturing.
An assembly or a part should be designed
based on its functions, manufacturing processes,
product quality, and manufacturing cost.
desired
Assembly tolerance
analysis performed at the design stage can reduce potential
manufacturing and assembly problems.
Several commonly used
assembly tolerance analysis models and their limitations are
reviewed in this research.
Also, a new assembly tolerance
analysis model is developed to improve the limitations of the
existing assembly tolerance analysis models.
elucidates the impact of the flatness symbol
The new model
(one of the
Geometric Dimensioning and Tolerancing (GD&T) specification
symbols)
and
reduces
mathematical equations.
design
variables
into
simple
The new model is based on beta
distribution of part dimensions.
In addition, a group of
manufacturing variables, including quality factor, process
tolerance, and mean shift, is integrated in the new assembly
tolerance analysis model.
A computer integrated system has been developed to
handle four support systems for the performance of tolerance
analysis in a single computer application.
systems are:
1)
the CAD drawing system,
2)
These support
the Geometric
Dimensioning and Tolerancing (GD&T) specification system, 3)
the assembly tolerance analysis model, and 4) the tolerance
database operating under the Windows environment.
Data Exchange (DDE)
Dynamic
is applied to exchange the data between
two different window applications, resulting in improvement
of information transfer between the support systems.
In this
way, the user is able to use this integrated system to select
a GD&T specification, determine a critical assembly dimension
and tolerance, and access the tolerance database during the
Examples are presented to
design stage simultaneously.
illustrate
the
analysis system.
application
of
the
integrated
tolerance
Assembly Tolerance Analysis
in
Geometric Dimensioning and Tolerancing
by
Choowong Tangkoonsombati
A THESIS
submitted to
Oregon State University
in partial fulfillment of
the requirements for the
degree of
Master of Science
Completed August 25, 1994
Commencement June 1995
APPROVED:
Redacted for Privacy
Professor of Industrial and Manufacturing Engineering
in charge of major
Redacted for Privacy
Head of Department of Industrial and Manufacturing
Engineering
Redacted for Privacy
Dean of Grad
Schoo
Date thesis is presented
August 25, 1994
Typed by Choowong Tangkoonsombati for Choowona
Tangkoonsombati
TABLE OF CONTENTS
Page
CHAPTER 1 INTRODUCTION
1.1 Problem Statement
1.2 Research Objectives
1.3 Research Methodology
1.4 Thesis Organization
CHAPTER 2 LITERATURE REVIEW
2.1 Existing Tolerance Analysis Models
2.2 Computer Applications and Algorithms for
Tolerance Analysis
CHAPTER 3 GEOMETRIC DIMENSIONING AND TOLERANCING
SPECIFICATIONS IN PART DRAWING
3.1 Geometric Dimensioning and Tolerancing (GD&T)
3.1.1 Definition of Geometric Dimensioning
and Tolerancing
3.1.2 Classification of Geometric
Dimensioning and Tolerancing
Specifications
3.2 AutoCAD System
3.3 Development of GD&T Specification in the
AutoCAD system
3.4 The Applications of GD&T Specifications in
a New AutoCAD Menu
3.5 Tolerance Chain Generation
1
2
6
8
10
12
12
23
28
29
30
31
35
38
40
47
CHAPTER 4 A MODIFIED ASSEMBLY TOLERANCE ANALYSIS MODEL
4.1 Introduction
4.2 Visualization of the Part Dimension
Including Geometric Dimensioning and
Tolerancing
4.3 Development of a Simple Model to Determine
the Nominal Dimension
4.4 The Application of the Beta Distribution
in Part Tolerance
4.5 Development of a New Modified Assembly
Tolerance Model
4.6 An Application Example
52
52
CHAPTER 5 SYSTEM INTEGRATION
5.1 Introduction
5.2 Conceptual Framework
5.3 System Components
5.4 The Integrated System Application Procedure
5.5 An Application Example
89
89
91
56
59
63
71
76
92
103
106
Pane
CHAPTER 6 CONCLUSIONS AND RECOMMENDATIONS
6.1
Conclusions
6.2
Recommendations for Future Research
121
121
124
REFERENCES
127
APPENDICES:
APPENDIX A
APPENDIX B
APPENDIX C
APPENDIX D
User's Guide of Computer Application
Applications of Beta Distribution
in Part Dimension
Normalized Dimension in Beta
Distribution
Determination of Alpha and Beta
Parameters of the Sum Dimension
130
162
168
175
LIST OF FIGURES
Page
Figure
A Part Drawing with Two Different
Conventional Tolerance Specifications
3
102.
A Drawing with GD&T Specifications
5
103.
Research Organization Chart
11
201.
WC Model: Part dimension are stacked at
both extreme sides.
14
RSS Model: Part dimension are normally
distributed with the mean at the nominal
dimension
14
Mean Shift Occurs during the Manufacturing
Process
16
The Resulting Distributions Following the
Assumption of Estimated Mean Shift Model
18
205.
The Variation Shapes of Tolerance Distribution
20
206.
Gear Box Assembly
21
207.
The Mating Features of the Components of the
Gear Box
21
208.
The Tolerance Chain Link of the Gear Box
22
209.
Mechanical Assembly
22
301.
Five Types of GD&T Specifications with Their
Own Group of Symbols
32
302.
A Part Drawing with GD&T Specifications
34
303.
A Part Drawing with the Standard AutoCAD Menu
37
304.
The New AutoCAD Menu Called, "TK menu."
41
305.
A Part Drawing Example with Basic Dimensions
43
306.
A Part Drawing Example with Datum References
44
307.
An Application of TK menu For Creating GD&T
Symbols
46
Tolerance Graphs
48
101.
202.
203.
204.
308.
Page
Figure
309.
An Assembly Drawing Example
49
310.
An Application of TK Menu for Generating a
Simple Tolerance Chain
51
401.
The Variation Shapes of the Beta Distribution
a) Symmetric and b) Asymmetric Shapes
54
402.
Part Drawing Including Both Conventional
Dimension and GD&T Specification
56
403.
Visual Aspects of the Part Surface When the
Flatness Symbol Is Applied
58
404.
Variation of the Flatness Tolerance Zone on the
Part Surface Depending on the Quality
Factor (C,1)
61
The Assigned Variables to the Part Dimension
Specification to Determine the Nominal
Dimension Depending on the Quality Factor(CJ
61
Mean Shift Occurs During the Manufacturing
Process
64
Part Rejection Occurs due to the Mean Shift
When the Natural Part Process Tolerance Is
Equal to One Half of the Flatness Tolerance
Zone
65
408.
The Part Dimension Distribution and Mean Shift
Behavior in the Proposed Model
67
409.
Gradual Skewness Developing in Beta
Distribution as a Increased While 6 is Kept
Constant at 4
70
410.
The Beta Distribution Model of the Sum
Dimension
72
411.
An Assembly Drawing with Conventional
Tolerances and GD&T Symbols (Flatness)
77
412.
Parameter Values of the Sum Dimension in the
Application Example
86
The Sum Dimension in Terms of Symmetrical
Specified Tolerance
87
405.
406.
407.
413.
Figure
Page
A Design Form for Developing a User
Interface in Microsoft Visual Basic
94
502.
The System Components
96
503.
Two User Interfaces for the Modified
Assembly Tolerance Analysis Model
98
501.
The Data Link between the User Interfaces of
the Modified Model Assembly Tolerance Analysis
Model and the Assembly Drawing in the AutoCAD
System
100
505.
Manufacturing Tolerance Database
101
506.
Application of the Pull-Down Menu "Start" to
Open the User Interface of Tolerance Database
102
507.
The Data Link Between the User Interfaces of
the AutoCAD System and the Tolerance Database
104
508.
The Data Link Between the User Interfaces of
the Modified Assembly Tolerance Analysis Model
and the Tolerance Database
105
509.
The Assembly Example with New TK AutoCAD Menu
107
510.
The Function of "Tol+/-" Pull-Down Menu for
Creating Conventional Tolerances in the
Assembly Drawing
108
511.
A Group of GD&T Symbols Slides
110
512.
The Assembly Example with Flatness Symbols
111
513.
Transferring Conventional Tolerance from
AutoCAD System to the User Interface
112
Updating Conventional Tolerance in the
AutoCAD Drawing
113
The Application of the Pull-Down Menu "WC"
for Determining Sum Dimension with Worst Case
Assembly Tolerance Analysis Model
115
The Application of the Pull-Down Menu "RSS"
for Determining Sum Dimension with Root Sum
Squares Assembly Tolerance Analysis Model
116
504.
514.
515.
516.
Page
Figure
517.
518.
519.
User Interfaces of the Modified Model
Including All Supply Variables of the
Assembly Example Following Assigned Values
in Table 403
117
Transferring and Inserting the Assembly
Dimension and Tolerance, "1.999 ±0.0012,"
from the User Interface into the Assembly
Drawing
118
The Application of the Tolerance Database
for Providing a Process Tolerance into the
User Interface of the Modified Model
119
LIST OF TABLES
Page
Table
Computing Error by Summing Up Equal
Rectangular Distributed Dimension
(Confidence Level 99.73%)
55
Normalized Tolerance as a Function of
Confidence Level based on the Normal
Distribution Model
74
403.
Data for the Application Example
79
404.
The Evaluated Variables for Each Part
Dimension
80
401.
402.
LIST OF APPENDICES FIGURES
Figure
Page
APPENDIX A:
A.1.
The Description of TK AutoCAD Menu
133
A.2.
Application of the Pull-Down Menu "Tol+/-"
136
A.3. Application of the Pull-Down Menu "Basic_Dim"
136
A.4. Application of the Pull-Down Menu "GD&T"
138
Two Types of the Sum Dimension; Assembly
Dimension and Gap Dimension
139
Application of the Pull-Down Menu "Sum Dim"
140
A.5.
A.6.
A.7. Application of the Pull-Down Menu "Tol.Chain"
141
A.S. Application of the Pull-Down Menu "DDE"
144
A.9.
The User Interface of
information"
"Part dimension's
A.10. Application of Command Button "Create Table"
A.11.
144
to Initiate Five Rows of Text Box
145
Transferring Data from the AutoCAD Drawing
into the Text Boxes of the User Interface
147
A.12. Application of the Command Button "Analysis"
to Open the User Interface "Assembly Tolerance
Analysis Model"
149
A.13. Application of the Command Button "Exec" to
Determine the Unknown Assembly Dimension and
Tolerance Based on the Modified Model
150
A.14. The User Interface of the Tolerance Database
152
A.15. Application of the Pull-Down Menu "Start" to
A.16.
Open the Tolerance Database Window during
Design Processes
153
Application of the Pull-Down Menu "Design_Tol"
to Update the Selected Conventional Tolerance
"4.0000+0.0300"
154
Page
Figure
Transferring "4.0000 ±0.0300" from the
AutoCAD Drawing into the Text Box of
"Link Data" Frame
155
A.18.
Application of the Command Button "Set"
in Tolerance Database Window
157
A.19.
Application of the Command Button "Update"
158
A.20.
Application of the Command Button "Return"
159
A.21.
Opening the Tolerance Database Window by
Clicking Twice at the Selected Text Box
"Process Tolerance"
160
A.17.
APPENDIX B:
B.1.
The Shape of Symmetrical Beta Distribution
164
LIST OF
APPENDICES TABLES
Page
Table
APPENDIX C:
C.1
Normalized Dimensions (Confidence Level 90.00%)
169
C.2
Normalized Dimensions (Confidence Level 95.00%)
170
C.3
Normalized Dimensions (Confidence Level 99.00%)
171
C.4
Normalized Dimensions (Confidence Level 99.50%)
172
C.5
Normalized Dimensions (Confidence Level 99.73%)
173
C.6
Normalized Dimensions (Confidence Level 99.90%)
174
ASSEMBLY TOLERANCE ANALYSIS IN
GEOMETRIC DIMENSIONING AND TOLERANCING
CHAPTER 1
INTRODUCTION
One of the most difficult research challenges today in
manufacturing is the integration of design and manufacturing
functions for the promotion of cost effective production.
When an engineer designs an assembly, the overall shape and
dimensions
each
of
part
specified.
are
In
is impossible and
manufacturing of exactly similar parts
unnecessary because manufacturing processes
behavior
involve
variability
by
contrast,
their
and material
nature.
Thus,
manufactured features will always deviate from their dimension
For these reasons,
design data.
specified
the
at
design
tolerances are usually
based
stage
on
the
allowable
The assignment of tolerances
variability in part dimensions.
to part dimensions has a critical repercussion on the overall
cost and quality of the assembly.
Both
the
designer
and
the
engineer
in different ways.
recognize tolerance allocations
example, the design engineer is
manufacturing
For
concerned that the tolerance
accumulation will directly affect the assembly tolerance and
function.
The
design
engineer
is
aware
that
tighter
tolerances are preferable to retain the functional behavior of
2
the assembly.
On the other hand, the manufacturing engineer
knows that tight tolerances will increase the cost and the
difficulty of production.
They are aware that improper
tolerance design results in several problems, not only in
manufacturing, but also in assembly functions.
Therefore,
tolerance analysis and design is the major link between design
and manufacturing.
Proper assignment of part tolerances
without violating the assembly design requirements is realized
by development of an assembly tolerance analysis model.
an
assembly
tolerance
analysis
model,
both
Using
design
and
manufacturing engineers may determine where the tolerance
specifications
must
be
made
specifications may be relaxed.
tighter
and
where
the
Finally, manufacturers and
designers can work with the same tolerance specification
system to improve the assembly quality and reduce production
costs.
1.1 Problem Statement
Tolerance assignment to system components is important in
assembly design and production.
An assignment that is too
"loose" may cause a malfunction of the assembly, whereas an
assignment that is too "tight" will definitely raise the cost
of
production.
indispensable.
As
a
result,
tolerance
analysis
is
At present, a number of efforts have been made
3
in the area of tolerance analysis.
Conducting concurrent
tolerance analysis during the design process makes it possible
to discover these problems at an early stage.
Usually,
tolerance analysis of
an assembly requires
statistical analysis to determine the effects of the inherent
variability in size and shape based on some functional or cost
aspect of the design.
tolerance,
For instance, referring to conventional
some statistical distributions,
such as normal
distribution, are assigned to represent the distribution of
actual part dimensions.
Traditionally, tolerance is expressed in a drawing in two
different ways, as shown in Figure 101.
Conventional
Tolerance
3 0000 ±0.2000
Figure 101. A Part Drawing with Two Different
Conventional Tolerance Specifications
The first is shown by the maximum and minimum dimensions.
The second, called the conventional tolerancing (+/- system),
is specified by giving the basic dimension plus the upper and
4
lower deviation.
The maximum and minimum dimensions are
displayed on the left side, whereas the conventional tolerance
is noted on the right side.
Geometric dimensioning and tolerancing (GD&T)
tool
used
specify
to
engineering
design
is a new
and
drawing
requirements [36], supported by the American National Standard
(ANSI
Y14.5M-1982).
The use
GD&T offers
of
tremendous
potential for U.S. companies to improve their productivity and
profitability by providing a clear way to describe
geometry and size of the part.
the
GD&T has been developed
internationally over the past 40 years by committee members
representing a broad spectrum of industrial and academic
disciplines.
By utilizing GD&T, documented in ANSI Y14.5
1982[32], many cost-saving advantages can be gained.
It is an
engineering language which serves as a common communication
medium among designers, manufacturers and inspectors.
Using geometric characteristics and symbols to represent
the actual function or relationship of part features, the
designer and the customer can visualize the functions or
relationship of part features.
As a result, the failure of
products due to unclear definitions can be reduced, thereby
increasing product quality.
Actually, GD&T is composed of
texts, numbers and symbols which are contained in a feature
box,
as
shown
in
Figure
102.
At
present,
drawing
specifications of GD&T are not available in AutoCAD software.
AutoLISP
is
the
computer
language
which
can
connect
a
5
tolerance analysis system to AutoCAD directly. It is possible
to
develop
AutoLISP
an
program
which
allows
GD&T
specifications during the drawing process in AutoCAD.
In
addition, the tolerance information could be stored in a CAD
databases for further manipulation and analysis.
3x00 7071=0.0050
0 0.0005
(f,71.)
0
11251
A
B
rio
75
C
-A-
0.001
_Llo
-c-
_L10.001
_B_
1
A
Figure 102. A Drawing with GD&T Specifications
Mathematical
models
are
used
to
critical dimensions of an assembly.
determine
unknown
Unfortunately,
these
models use only the conventional representation of tolerances.
Also, conventional dimensions have been assumed to follow a
normal distribution, which has been shown by Fortini[29] and
Spotts
[30]
not always to be the case.
Currently, GD&T is
included in 2D drawings but is used only for inspecting the
6
function and the dimension quality of the product.
Tolerance
analysis is not concerned with GD&T. A new assembly tolerance
analysis model is developed in this research to transform GD&T
specifications into conventional dimensions and tolerances,
for integration with original conventional dimensions.
1.2 Research Objectives
There are three main objectives of this research:
1)
creating GD&T specifications in the drawing automatically, 2)
developing an assembly tolerance analysis model,
developing an integrated system.
and
3)
These are briefly discussed
below.
1)
Creating GD&T Specifications in the Drawing
Automatically:
An application of CAD software (AutoCAD) is developed to
automatically insert both conventional tolerance and GD&T in
a 2D drawing during the drawing process. To develop this
application, AutoLISP programming language is chosen.
The
benefit of this language is its ability to access the AutoCAD
system.
In addition, a new screen menu
is
established
to
7
integrate all AutoLISP programs into AutoCAD.
As a result,
new users can easily familiarize themselves within a short
amount of time if they know AutoCAD.
2)
Developing an Assembly Tolerance Analysis Model:
An assembly tolerance analysis model is developed to
estimate the nominal and tolerance size of a critical assembly
dimension. A knowledge of statistics is necessary to use this
model because the distribution of each part dimension in the
assembly has been assumed to be a beta distribution.
GD&T is
considered and included in the assembly tolerance analysis
model,
but
is different from existing assembly tolerance
analysis models.
symbol has
its
There are a lot of symbols in GD&T and each
own definition and
includes only the flatness feature.
form.
This
research
The advantage of this
research model is its flexibility and efficiency because it
involves the evaluation of
assembly tolerance.
the influence of GD&T on the
In addition, the beta distribution is
more flexible than the normal distribution.
It allows the
shape of part dimensions to vary from normal to skewed,
indicated by alpha (0) and beta (le) parameters.
8
3)
Developing an Integrated System:
An integrated system of assembly tolerance analysis and
GD&T specifications is not available.
integrated
system
applications
integrated
of
has
system,
the
developed
been
and
(1)
(2)
user
For this reason, an
to
combine
the
concurrently.
Using
the
can determine
assembly
the
tolerance automatically after finishing an assembly drawing,
and dynamically
insert
the
assembly
tolerance
into
the
assembly drawing.
The
within an
system works
Microsoft Windows.
environment
created by
It combines both applications with a new
linking method, called Dynamic Data Exchange (DDE).
DDE, the data between both applications
(1)
and
(2)
Using
can be
transferred dynamically.
1.3 Research Methodology
Research for the development of tolerance analysis was
based upon
GD&T
1) the development of AutoLISP software to create
specifications
in
AutoCAD
automatically,
2)
the
development of an assembly tolerance analysis model, and 3)
the development of an integrated system, as discussed below.
9
1)
Development of GD&T Specifications in AutoCAD:
The AutoCAD software for Windows is chosen and a new
application is developed to specify GD&T symbols in part
drawings.
Tolerance chain generation is also included to
create a simple tolerance chain matching the assembly drawing.
The development of this application is discussed in Chapter 3.
2)
Development of an Assembly Tolerance Analysis Model:
An assembly tolerance analysis model is developed in a
simple mathematical form.
Design parameters, such as quality
factor, mean shift factor, and nominal dimension, are assigned
to determine the tolerance on critical assembly dimensions.
A knowledge of statistics is required because principle rules
of statistics, such as sum of variance, are used in the model.
In addition, the beta distribution is chosen to represent the
shape of part dimensions.
3)
Development of an Integrated System:
An integrated system is developed to make the application
of tolerance analysis more powerful.
around Microsoft Windows.
The system is structured
The part drawing is created by
AutoCAD for Windows and the assembly tolerance analysis model
10
is operated by Microsoft Visual Basic Version 3.0 for Windows.
Both applications are integrated with DDE and work under the
Windows environment simultaneously.
1.4 Thesis Organization
The organization of the research, as outlined above, is
graphically represented in Figure 103.
11
CHAPTER 1
Introduction
CHAPTER 2
Literature Review
CHAPTER 3
CHAPTER 4
GD&T
Specifications
in Part Drawing
An Assembly
Tolerance
Analysis Model
CHAPTER 5
An Integrated
System
CHAPTER 6
Conclusions and
Recommendations for
Future Work
Figure 103. Research Organization Chart
12
CHAPTER 2
LITERATURE REVIEW
2.1 Existing Tolerance Analysis Models
Many studies have been made in the area of tolerance
Mathematical models are used to determine the
analysis.
assembly, or unknown dimension.
The most commonly used models
are the Worst Case (WC) and the Root Sum Squares (RSS) models,
described below.
Worst Case
Tolasm
:
= toll + to12 + tol3 +...+ tol,
Root Sum Squares
Tolasm
=
:
(Z/3)
+ to122
to132 + ...+ to1,2]
where
Tolasm represents the tolerance of assembly dimension,
tol
and
Z
represents the tolerance of part i (i = 1,2,..n),
represents the confidence level factor.
13
The Worst Case model is not concerned with the distribution of
part dimensions.
If it is assumed that no part exceeds the
tolerance zone, then the parts are distributed on either side
of the nominal dimension as the worst case condition, as shown
in Figure 201.
In contrast,
the Root Sum Squares model
assumes that the distribution of part dimensions is normal
with a mean at the tolerance midpoint.
deviation (0
Moreover, the standard
is usually assumed to be 1/3 of the equal
bilateral tolerance, as shown in Figure 202.
to 3,
When Z is equal
the acceptance rate includes 99.73c+), which means that
there are only 2.7 components per one thousand exceeding
tolerance.
The factor Z can directly influence the acceptance
rate depending on the cost of rejection relative to the cost
of component tolerances.
rejects are expected.
The larger the value of Z, the fewer
Conditionally, the Root Sum Squares
model requires a random procedure so that all part dimensions
are randomly selected and assembled.
In fact, non-random
factors occur during the manufacturing process. This behavior
impacts not only the part dimension distribution, but also the
mean dimension.
The distribution is not normal and the mean
dimension is not located on the midpoint.
To solve this
problem, the Modified Root Sum Squares model was developed.
14
N omin al
Dimension
Figure 201. WC Model: Part dimensions are stacked at
both extreme sides.
Figure 202. RSS Model: Part dimensions are normally
distributed with the mean at the nominal
dimension.
15
Modified Root Sum Squares:
Tol,m
where
= Cf(Z/3)[to112 + to122 + to132 + ...+ to1,2]1/2,
Cf represents the correction factor.
This model improves the accuracy of the Root Sum Squares
model by introducing a correction factor (Cf).
A correction
factor of 1.5, suggested by Bendor[1] and Levy[2], is commonly
used. Gladman[35], however, has recommended the range between
1.4 to 1.8 for the correction factor.
During the manufacturing process,
it is impossible to
keep the mean dimension exactly centered between the tolerance
limits because of the mean shift.
The mean shift results from
effects such as tool wear when the machine produces a large
number of parts.
As a result, the mean shift certainly moves
the mean dimension away from the designed mean (Figure 203).
To incorporate this effect, the process variability should be
less than the tolerance specification. Otherwise, significant
component rejects will occur, resulting in increased costs due
to reworking or scrap waste.
ignored.
GreenWood
and
The mean shift should not be
Chase[3]
have
developed
a
new
mathematical model, called the Estimated Mean Shift Model,
which includes the mean shift as described below.
16
mean shift
Nominal
Dimension
+
Process Tolerance
LSL
Specified Tolerance
USL
Figure 203. Mean Shift Occurs During the Manufacturing
Process (LSL: Lower Specified Limit,
USL: Upper Specified Limit)
17
Estimated Mean Shift Model:
= fltoli + f2to12 + f,to13 +...+ fntoln+ (Z/3) [(1-f1)2to112
+
where
(1-f2)2to122 +
(1-f3)2to132 + ...+ (1-fn)2t01,2] 1/2
fi is the mean shift factor of part i
,
,
0 s fi s 1.
This model combines both Worst Case and Root Sum Squares
together.
The first summation [fItoll
f2t012
f3t013
fntoln] will end up as the Worst Case if all parts have the
same mean shift factor, 1.
On the other hand, if all parts
have the same mean shift factor 0 then the result will be the
same as the Root Sum Squares.
The component variability is
still assumed to be three sigma
(a)
from the mean(not the
midpoint) to the nearest tolerance limit.
increases,
the
standard
deviation
When the mean shift
(a)
must
decrease;
therefore, the shape of part distribution varies between the
Worst Case and the Root Sum Squares depending on the mean
shift factor, as illustrated in Figure 204.
Mansoor[4] presented the use of probability methods and
analyzed the assembly tolerance using different distributions,
such as normal, triangle, sine, moving normal and rectangular
(Figure 205).
A new algorithm by Wong and Ozsoy[5] can automatically
generate the tolerance chain. This method is concerned with
the mating relationship between various components of an
18
UCL
LCL
Ox
mean shift
50X
mean shift
75%T
75x
mean shift
30
1 00%T
<
>
100x
mean shift
Figure 204. The Resulting Distributions Following The
Assumption of Estimated Mean Shift Model
(Greenwood and Chase [3])
19
assembly.
For example, for the assembly shown in Figure 206,
each part can link to other parts with mating links, as shown
in Figure 207, and finally result in the tolerance chain as
shown in Figure 208.
However, the user must do a lot of work
in terms of providing mating relationships if the assembly is
more complicated.
Bjorke[6] has developed the concept of the fundamental
equation based on the relationship between components in an
assembly.
Fundamental equation:
X
=
f (X, )
(2.1)
,
where
XE
is the sum or critical assembly dimension,and
X,
is the individual variable or dimension in the
assembly.
For the assembly shown in Figure 209, the fundamental equation
is:
XE = X, +
X2
X3
X4 -
X5
.
20
NORMAL
Rel. S. D.
= 1
SINE
Rel. S. D.
= 1.31
SEMICIRCLE
TRIANGLE
= 1.22
Rel. S. D.
MOVING NORMAL
Rel. S. D.
= 1_ 4 0
RECTANGLE
= 1.73
Rel. S. D.
= 1.50
Rel. S. D.
= relative stanndard deviation
Rel. S. D.
Figure 205. The Variation Shapes of Tolerance
Distribution (Mansoor [4])
21
Figure 206. Gear Box Assembly (Bjorke [6])
rnC2
Figure 207. The Mating Features of the Components of
the Gear Box (Wong & Ozsoy [5])
22
mfl < > mfl2 < > mfll < > mf5
A
> mfl3 < > mf7
mf3 < > mf16
Figure 208. The Tolerance Chain Link of the Gear Box
(Wong & Ozsoy[5])
X1
-E-
X2
E- X3 -->
----. r-1,---
-.- x5
-->
<- x4
Figure 209. Mechanical Assembly (Bjorke [6])
23
This is simple and linear.
In addition, Bjorke[6] suggested
the beta distribution for manufacturing conditions such as,
small
lot
sizes,
truncated
distributions
skewed
or
distributions. Amore recent work by Turner[7] uses the Monte
Carlo method in solving the tolerance analysis problems in
several simple assemblies.
based,
and
the
Turner's work is geometric-model-
system derives
all
its
necessary
input
geometric relationships from the geometric model.
2.2 Computer Applications and Algorithms for Tolerance
Analysis
Computer Aided Design (CAD)
most efficient tool for designers.
by
Soderberg[8]
can interface
software is currently the
The CATI system developed
the
CAD software with an
external calculation program which determines unknown assembly
analyze
only normal
tolerances.
Unfortunately,
distribution.
Fleming[9] discussed a representation of parts
which includes geometric
it
can
tolerances.
He
introduced the
concept of building networks of tolerance zones and datums for
the analysis
of
parts and assembly tolerances.
In the
network, each arc represents a relationship implied by the
tolerance specification or by contact
condition).
geometrical
(in
the assembled
Porchet and Zhang[10] proposed a way in which
tolerances
and
considered simultaneously.
process
Kapur[11]
inaccuracies
can
be
developed a rational
24
basis to analyze tolerance using quality loss functions.
A
general optimization model is developed to allocate tolerances
to assembly parts.
Different quality loss functions are also
discussed. A new approach to tolerance synthesis
CASCADE-T
(Lu and Wilhelm) [12] uses a complete representation of the
conditional tolerance relations that exist between features of
a part under design.
In CASCADE-T, three errors in tolerance
specification, validity,
consistency, and sufficiency, are
adjusted during design review.
CASCADE-T utilizes several
to deal with emergent
artificial
intelligence techniques
problems.
A detailed account is maintained of functional
requirements,
the
relationship
among
implied
geometric
features, and the checking of computations. Computations that
check for consistency, validity, and sufficiency are done only
for related geometric features.
Karolin and Ahluwalia[13]
have developed a software
called CATC (computer aided tolerance control system).
Their
method of analyzing and controlling tolerances is based on the
tolerance-chart technique developed in the early 1950s.
The
tolerance-chart technique is the simplest method of selecting
the
tolerance
for
the
resulting
dimension.
Machined
dimensions are combined in closed chains of n dimensions, the
resulting one being a combination of the other n-1 independent
working dimensions.
cases;
it
analysis.
This method is limited to simple 1D
is almost impractical for 2D and 3D tolerance
The CATC program is iterative in nature, and uses
25
computer graphics for information display.
Zang's[14] work,
which uses dimensions and tolerancing calculations in CAPP
(computer aided process planning) for precision manufacturing,
is also based on tolerance chain.
16]
Raman and Panchai[15 and
used CAD software in tolerance analysis.
Both works
identify and computerize a procedure for automatic tolerance
assignment in CAD drawing. Martino[17] developed a new method
for
programming
linear
Turner
tolerances.
analysis
and
of
complex
Guilford[18]
geometric
presented
a
representation for the ANSI Y14.5 geometric tolerances in a
solid model, but the representation is concise, using only a
small number of primitives and some generally applicable
concepts.
The work
by Turner and Sodhi[19]
proposed a
method for representing tolerance and assembly information in
a feature-based environment.
are
performed
using
the
positioning capabilities.
Assembly and tolerance analysis
dependency
Irani,
graph
and
relative
Mittal and Lehtihet[20]
introduced a graph theoretic representation of the tolerance
chart and used linear programming to
allocation.
optimize tolerance
The graph representation of tolerance chains is
suitable for representation within, and extraction from, the
CAD database.
Truslove[21] created a concept of data re-use
to link computer aided design and manufacturing (CAD/CAM) by
improving the ability of information flow between design and
manufacturing activities.
However,
the inadequacy of CAD
systems to cater for tolerance information also has some
26
serious implications for the information flows because all
annotational information is removed from a drawing when this
information
passed
is
Shah and Miller[22]
downstream.
describe a product modeling system which combines a geometric
modeler with a form feature modeler and a tolerance modeler.
The
methodology
consists
rules,
of
symbols
and
their
interpretation, definitions, and conditions for inspection.
Using this method,
the user can mark datums and specify
tolerances symbolically by the use of "control frames", so
that new tolerances can be added without major revision.
Unfortunately, their work does not go into enough detail to
determine how each of the Y14.5 tolerances would be expressed
or
what
the
Elmaraghy[23]
measurement
limitations
discuss
data
the
captured
might
be.
geometric
by
a
Moreover,
tolerance
coordinate
Wu
by
and
using
measuring
machine(CMM) in order to detect out-of-tolerance conditions.
This work considers only cylindrical features.
Wozny[25]
Turner and
also provide a mathematical theory of tolerance
based on the M-Space formulation.
The essence of their
approach is the association of instances of in-tolerance parts
with points in a standard vector space of model variations.
Model variables are associated with the boundary features of
the parts.
Although this approach seems to be powerful, the
theory relies on a specific model representation and is quite
difficult to express for general 3-D parts.
developed by Turner and Wozny[26]
GEOTOL software
can solve the tolerance
27
problem involving the use
of
solid modeling technology.
Linear programming and Monte Carlo methods are used in this
software.
It was shown that the worst case tolerance analysis
problem may be formulated as an optimization problem and
solved using linear programming.
Simulation may be applied.
Alternatively, Monte Carlo
The Monte Carlo method allows for
nonlinear design variables and supports statistical as well as
worst case analysis, but is generally more expensive than the
linear programming method.
Srinivasan and Jayaraman[27] have
expressed tolerance constraints in terms of virtual boundaries
between parts, which are shown to accurately describe the
function requirements for certain classes of problems, such as
fit.
In addition, they propose an idea which is the main
objective of this thesis,--to establish a converter.
Using
the converter, one can convert virtual boundary requirements
into condition tolerances.
Chase and Parkinson[281 surveyed
a number of research papers which recommended the analysis of
geometric
tolerances
statistically,
including
geometric
variation in a vector loop model.
Using these features, the
effects
along
on
complex
assemblies
variations can be computed.
with
dimensional
28
CHAPTER 3
GEOMETRIC DIMENSIONING AND
TOLERANCING SPECIFICATIONS IN PART DRAWING
Part drawing
manufacturing
features.
is
the
engineers
language by which design
visualize
and
Drawing a complicated part,
communicate
however,
and
part
is time
consuming. Computer systems are widely used to increase task
potential and reduce drawing time.
Computer aided design
(CAD) software is one such system of design packages used to
enhance
the
technology
capability
of
the
drawing
process.
CAD
has developed greatly during the last decade so
that now a number of CAD packages, such as AutoCAD, and
DesignView, are commonly used in manufacturing and design
engineering.
A part drawing represents the image of part features.
It consists of linear, circular, and curved lines.
specifications are described by numbers, text,
Dimension
or symbols,
and included to represent the size of geometric
features.
Without tolerance specifications, however, the
manufacturer
cannot produce parts of exact design dimensions economically.
Therefore, tolerances must be included in a part drawing.
Tolerance specifications can be displayed in two different
formats: 1)
the minimum and maximum dimensions, and 2) the
29
conventional tolerancing (+/-) system, as was shown in Figure
101. Tolerance specifications produce inadequate information
for designers and manufacturers
images of part features.
to accurately visualize
In addition, significant geometric
characteristics related to assembly functions are needed,
such as flatness, roundness, or straightness.
Even though
parts are produced within tolerance specifications, they may
not fit well during assembly.
specification
drawing
and
design
A new geometric engineering
standard[32]
called,
"geometric dimensioning and tolerancing" or "GD&T," has been
developed
to
improve
geometric
specifications
in
part
drawing.
3.1 Geometric Dimensioning and Tolerancing (GD&T)
Tolerance specifications are used to specify control
limits (maximum and minimum dimensions) of a part.
In the
manufacturing process, a part, if produced within tolerance
specifications,
assembly.
is considered acceptable to be put into
However, an acceptable part may not fit or work
appropriately in the assembly due to some geometric features,
such
as
flatness,
straightness,
or
roundness.
These
specifications are often ignored or not recognized to be
important in a part drawing. Geometric notes, therefore, are
added into the part drawing to describe a geometric feature
30
in a critical section.
Unfortunately, geometric notes are
stated inconsistently.
They require much more time and
space, tend to be scattered on the drawing, and often appear
as footnotes which separate the note from the feature to
which it applies.
Also, notes may require translation if the
drawing is used in another country.
As a result,
the
possibility of misunderstanding geometric notes is apparently
high.
A new geometric specification standard, named geometric
dimensioning and tolerancing (GD&T), has been developed to
improve the ability in geometric specifications in part
drawings.
GD&T specification symbols are used to specify,
not only the geometric characteristics of a part feature, but
also the relationship between part features.
These symbols
are compact, quickly drawn, and can be quickly placed in the
drawing where the control applies.
3.1.1 Definition of Geometric Dimensioning and
Tolerancing (GD&T)
GD&T
specifies
a
geometric
part's
characteristics
(flatness, roundness, straightness, etc.), and identifies the
control value of these features.
31
GD&T is defined in the American National Standards (ANSI
Y14.5-1982) [36] :
Geometric dimensioning and tolerancing is a means of
specifying engineering design and drawing requirements
with respect to actual "function and relationship" of
part features.
Furthermore, it is a technique which,
properly applied, ensures the most economical and
effective production of these features. Thus geometric
dimensioning and tolerancing can be considered both an
engineering design drawing language and a functional
production and inspection technique.
This standard has evolved out of a consolidation of earlier
standards (SAE Automotive Aerospace Drawing Standards Section
A6,
8 September, 1963; MIL-STD-8C, October 1963; USAASI
7
Y14.5-1966; ANSI Y14.5-1973; and ASA Y14.5-1975).
Updated
and expanded practices, however, have been initiated in the
present ANSI Y14.5-1982 standard.
3.1.2 Classification of Geometric Dimensioning and
Tolerancing Specifications
GD&T specifications are classified into five types:
form, profile, orientation, runout and location[10].
Each
type has its own group of symbols, as shown in Figure 301.
First,
the
straightness,
form
type
has
four
symbols:
circularity and cylindricity.
flatness,
Second,
the
profile type has two symbols: profile of line and profile of
surface.
Third,
the orientation type has three symbols:
32
TYFE OF
CHARACTERISTIC
TOLERANCE
FLATNESS
FORM
STRAIGHTNESS
CIRCULARITY
SYMBOL
0
0
CYLINDRICITY
ORIENTATION
PERPENDICULARITY
ANGULARITY
PARALLELISM
LOCATION
RUNOUT
PREFILE
POSITION
CONCENTRICITY
CIRCULAR RUNOUT
TOTAL RUNOUT
PRERILE OF A LINE
PROFILE OF A SURFACE
/
//
0
/
LJ
0
Figure 301. Five Types of GD&T Specifications with
Their Own Group of Symbols
33
perpendicularity, angularity and parallelism.
runout type has
runout.
two symbols:
Fourth, the
circular runout and total
Finally, the location type has two symbols: position
and concentricity.
Each symbol has its own format depending
on its regulation and definition.
parameters than others.
Some symbols require more
For example, the position symbol
requires the datum reference, whereas the flatness does not.
The position tolerance needs the user to specify not only the
material condition, such as maximum material condition(MMC),
but also the datum reference and basic dimensions.
As a
result, there are many variations of different formats within
GD&T that the user could create during the drawing process.
Figure 302 displays some of these various types of GD&T
specifications in a part drawing.
Due to the complexity of part drawings, which include
both conventional tolerancing and GD&T specifications, a CAD
system should be selected to help the design process more
easily and efficiently.
AutoCAD is a well known CAD system
which
several
has
served
architectural
drawing,
mechanical design,
focuses
both
specifications.
interior
design
fields,
design,
and aerospace design.
on adapting
accommodate
follows.
in
the AutoCAD system
conventional
civil
such
as
drawing,
This research
to
tolerancing
efficiently
and
GD&T
A brief explanation of the AutoCAD system
34
J,.
717:1C050
0.0005
(M)
0 001
A
7
-A-
0.75i
A
Figure 302.
A Part Drawing with GD&T Specifications
35
3.2
AutoCAD System
The AutoCAD system design package is a powerful drawing
It is a general purpose computer-aided design/drafting
tool.
application for computer systems.
offers
Using a computer, which
tremendous advantage over hand preparation,
a
drawing
can
be
prepared
and
modified
a
proficiently.
Previously, AutoCAD was available only as a large and costly
system.
Due to sophisticated technology and development, it
now runs in a personal computer (PC) or a network.
With AutoCAD, an engineer can draw and correct drawing
errors easily, producing clean and precise final drawings.
AutoCAD provides a set of entities for constructing a part
drawing.
An entity is a drawing element such as a line,
circle, or text string.
To draw an entity, the user types a
command (for instance, LINE) on the keyboard or selects it
from a menu.
supplying
Then the user responds to prompt on the screen,
parameters
for
the
selected
entity.
These
parameters always include the desired location of the entity
in the drawing.
required.
Sometimes a size or rotation angle is also
After the user supplies this information,
entity is drawn and displayed on the screen.
the
The user can
then enter a new command in order to draw another entity or
perform another AutoCAD function.
Every change the user
makes appears immediately on the screen.
Other AutoCAD
functions let the user modify the drawing in a variety of
36
Changing a parameter of an existing entity, the user
ways.
can then erase, move or copy it to form repeated patterns.
Drawing aids are provided to help the user position entities
accurately.
The simple command format of AutoCAD allows the
user to accomplish all these functions easily.
The user can
enter commands by typing on the keyboard, or by using a
pointing device to select choices from a menu.
The most
updated AutoCAD system is AutoCAD Release 12 [33,34].
There
are a lot of tools to help the user draw an entity more
easily.
A standard AutoCAD menu, as shown in Figure 303, is
available to let the user enter commands by using a pointing
device.
AutoCAD Release 12 provides commands to draw tolerance
specifications.
Using dimension commands (DIMTOL, DIMTP and
DIMTM), two different types of tolerance specifications can
be created in a part drawing:
the maximum and minimum
dimension, and conventional tolerancing (+/-)system.
These
procedures, however, take a lot of time if the values of +
and
tolerance specifications are dissimilar.
A
typical
GD&T
specification
contained
is
in
a
rectangular box which is partitioned into several sections.
These sections consist of symbols, numbers, and letters.
Unfortunately, only one of the GD&T specifications (basic
dimension) is available in the AutoCAD system.
to
basic dimensions,
it
is
possible
to
In addition
introduce GD&T
specifications into a drawing using the standard AutoCAD
37
File
Edit
View
Assist
Draw
Construct
Modify
Settings
Render
Model
Help
16.5262,-0.9303
Command: r
REDRAvV
Command:
Figure 303. A Part Drawing with the Standard AutoCAD
Menu
38
menu, but this is a very time consuming procedure.
Without
enhancing the standard AutoCAD menu, the user may spend a lot
of time creating GD&T specifications from scratch in the
AutoCAD system.
3.3 Development of GD&T Specifications in the AutoCAD system
In the AutoCAD system, a drawing element is represented
Each entity has its own data list, controlling
by an entity.
its appearance on the screen.
For example, the data list of
a line entity consists of an entity name, a coordinate of two
end points, a layer name, and a line type.
In addition,
AutoCAD environment elements, such as the snap mode, screen
size, or gird display, are controlled by a number of system
The response of a system variable is either a
variables.
continuous
or
application.
a
zero
For
or
value,
one
example,
if
the
depending
system
on
its
variable,
"GRIDMODE," controls the appearance of grid display (ON/OFF),
then its response is zero or one.
variable,
In contrast, the system
"GRIDUNIT," controls the distance between grid
points, so its response is a continuous value such as 1.2,
3.4, or 4.8.
To set the AutoCAD environment, the user types
a command on the keyboard or selects choices from a menu with
a pointing device, activating a new value for the chosen
system variable.
39
Tolerance specifications in the AutoCAD system can be
drawn by setting three system variables: DIMTOL, DIMTP, and
DIMTM.
Using the different values for + and
specifications
takes
specifications
within
more
the
time.
tolerance
Therefore,
AutoCAD
system
tolerance
need
to
be
developed.
As previously noted, GD&T specifications in the AutoCAD
system
are
not
currently
available.
Drawing
a
GD&T
specification with a standard AutoCAD menu is time consuming
because it requires a large number of drawing entities, such
as lines, circles, texts, or blocks.
The automatic drawing
of a large number of entities should be combined with the
AutoCAD system.
AutoLISP is AutoCAD's built-in programming language.
Unlike other computer languages, it can directly access the
AutoCAD system.
With AutoLISP, the user can write a command
to draw a drawing entity or set a system variable.
AutoLISP
is defined so well that regular users of AutoCAD software can
start working with it after a minimum of training.
As a
result, AutoLISP is the most suitable tool for AutoCAD users
who are looking for a way to enhance and extend their use of
AutoCAD.
In this research, a group of AutoLISP programs is to be
developed and added to AutoCAD in order to simplify the
addition of tolerances and GD&T specifications on a drawing.
40
3.4 The Application of GD&T Specifications in a New AutoCAD
Menu
The development of new AutoLISP programs along with a
new AutoCAD menu can accommodate both tolerance and GD&T
specifications systematically.
120 AutoLISP programs can be
loaded in the AutoCAD system.
Each program corresponds to
only one typical specification.
A new AutoCAD menu called,
"TK menu," (Figure 304) is applied to organize and run all
Due to the large number of programs,
AutoLISP programs.
specifications
their
are
divided
into
four
groups:
1)
conventional tolerance, 2) basic dimension, 3) datum, and 4)
GD&T symbols.
1)
These groups are briefly discussed below.
Conventional tolerance:
The standard of conventional tolerance has been commonly
used for a long time.
Its format contains two parts.
The
first part is the nominal dimension and the second part is
the
tolerance.
As
illustrated earlier in
Figure
101,
3.000±0.2000 is the conventional tolerance, where 3.000 is
the nominal dimension and ±0.2000 is the tolerance.
This
example illustrates an equal tolerance because the upper
tolerance
0.2000).
(+0.2000)
is
equal
to
the lower tolerance
(-
An unequal tolerance is available but it is not
commonly used in tolerance analysis, therefore, this study
concerns only with equal tolerances.
41
File
Edit
Laver
Tol +j0
AuteCAD UNNAMED
OSNAP Sum Dim.
it 0
gg
Basic Dim.
fis
...
GDT DDE
15 3161 0 0126
Entity
...
Help
ED4Q OR
OfElii
LIM
BE
01151
E3111
MCI
CPA
CIE
Eli
MEI
EEIM
,
L1
Command menu
Loaded menu F \HOME\TANGKOC\GDT mnx
Commend.
KO
MEI
ME
MOO
pa NI
MIN
OM
Figure 304. The New AutoCAD Menu Called, "TK menu."
42
2)
Basic Dimension:
Basic Dimension is a theoretically exact value used to
describe the exact size, profile, orientation, or location of
a feature of the datum target[32]. It is used as the basis
from
which
permissible
variations
are
established
by
tolerances in feature control frames or on other dimensions
or notes. An example is shown in Figure 305.
3)
Datum:
A datum is the point of origin from which the location
or geometric characteristics of
established.
features
of
a part are
It is a theoretically exact point, axis, or
plane derived from the
true geometric counterpart of a
specified datum feature[32].
Datum surfaces and datum
references are actual part surfaces or features used to
establish datums.
inaccuracies.
They include all the surface or feature
Figure 306 illustrates the application of
datum references in a part drawing.
4)
GD&T symbols:
GD&T characteristics are classified into five types:
form, profile, orientation, runout and location)[32], and
43
P 0.75
Figure 305. A Part Drawing Example with Basic
Dimensions
44
-A-
Figure 306. A Part Drawing Example with Datum
References
45
each type has its own symbols, as illustrated in Figure 301.
Each symbol may be of more than one format following the
regulation and the application.
The new TK AutoCAD menu provides a pull-down menu which
consists of a text string.
To run an application, the user
clicks at a pull-down menu by a pointing device, and makes a
selection.
system.
Then a program is loaded and runs in the AutoCAD
For example, Figure 307 shows the application of the
TK menu in which GD&T specification symbols are drawn.
During the time the program is running,
the user should
respond to supply parameters, expressed by text comments on
the command line.
The user prompts a point location, or
types a value from the keyboard.
the information,
After the user supplies all
the GD&T specifications or conventional
tolerances will be drawn in the drawing.
The application of
TK menu is explained in Appendix A.
In addition to drawing an element, AutoLISP can be used
for other applications.
There is increasing interest in
tolerance analysis in the AutoCAD system due to the need to
discover tolerance and assembly problems in the design stage.
For this reason, a tolerance chain generation application is
introduced to be incorporated into the AutoCAD system.
46
File
Edit
Tol+/-
AutoCAD - PIC7
OSNAP Sum Dim.
0
4
1111177;2i..Th
NG
DATUM
3x08.7071&0.0050
+F0 0.0003
t
I
I
el AIBIC1
0
;
e0.751
i2.00-1,
_Lilian i
t
110.001[A 1
-c- 1
=_ _a..
____
_ _ _ _
____
Help
113
IMO
ea
OMR
1210
Del
ERZ
MN!
EIR4
RIM
EIM
ESE
MIN
OMB
11E3
1.o.
zy\
moms.
Basic_Dim. GDT DDE Entity
SYMML:.
10.6301,
Ell
]
I-1
,... ;\
,,s7
Command menu
I Loaded menu FAHOME\TANGKOC\CDT mnx
Command
MD
MIN
PIRO
El
Figure 307. An Application of TK Menu for Creating GD&T
Symbols
47
3.5 Tolerance Chain Generation
A tolerance analysis algorithm dealing with tolerance
chains was developed in 1978 by Bjorke[6], who also defined
"tolerance
chain"
in
Computer-aided
Tolerancinq[6]
as
follows:
A chain is a sequence of elements such that each element
in the sequence has one endpoint in common with its
predecessor in the sequence and its other endpoint in
common with its successor in the sequence. A chain that
does not encounter the same endpoint twice, is called
elementary. A chain that does not use the same element
twice, is called simple.
All other chains are called
interrelated.
The difference between elementary, simple, and interrelated
tolerance chains is shown visually in Figure 308.
to
a
linearized
structure
is
fundamental
associated with
Equation
equation,
the
According
simple
2.1's
chain.
Each
parameter (X) in Equation 2.1 is assigned independently and
its value does not effect the others.
dimension
(XE)
can be simply determined from accumulating
each individual dimension.
If the value of an individual
dimension effects the others,
applied,
resulting
complicated.
Therefore, the sum
in
the
the interrelated chain is
sum dimension becoming more
This research, however, considers only the case
of simple chains.
To show the geometric meaning
of
chain links,
an
assembly in Figure 309 displays each dimension as the vector
in the space span.
An AutoLISP program is written to create
48
TX 2
TX
TX 3
a
Elementary tolerance chain
X5
TX
Simple tolerance chain
TX
TX 2
TX
TX 4
TX5
Interrelated tolerance chain
Figure 308. Tolerance Graphs (Bjorke,[6])
49
1 5 0 00 ± 0 .0 70 0
--- 1.5000 ±0.1000
sun
1.500 ±0.0400-.-
c r-nension
5.5 000 ± .14 00
Figure 309. An Assembly Drawing Example
50
tolerance chains automatically related to the mating link of
First, the unknown or sum dimension
each part dimension.
must be identified.
Then a tolerance chain can be generated,
starting at the left end-point of the unknown dimension and
ending at the right end-point of the unknown dimension like
a closed loop.
unknown
All dimensions in the chain except one (the
dimension)
must
be
specified,
otherwise,
the
tolerance chain cannot be created due to the lack of the
dimension's link.
To avoid a conflict between tolerance
chains and the assembly drawing, the tolerance chains are
created in a different layer labeled, "chain."
The new TK
AutoCAD menu provides a pull-down menu labeled, "Tol.Chain,"
to run the AutoLISP program.
is shown in Figure 310.
An example of this application
51
File
Tol+/-
Edit
_!1/123
0
AutoCAD - PICT II
Basic Dim.
OSNAP Sum Dim.
*
Create
Edit
GAP
ASSEMBLY
GDT
Tat. Chain
DDE
Entity
,T3
Help
EC
magt;i
r-------1
Rani
LIMP
i
BIRO.,I.
z:B:30ii
Tolerance Chain
4x----e---c---f-'4J
101:1
4)
...- 1.5000=0.0700
1 5000/ J000
1500 10 0400
Ira3ME
CZOW
rigli'
EV
i
sun
dimension
EW31531
IESE311
OM
EMI
IIHIM
Y
t..1
<
5500001400
Command Regenerating drawing.
,Command
Ille
BONI
BO
Figure 310. An Application of TK Menu for Generating
a Simple Tolerance Chain
1
i
52
CHAPTER 4
A MODIFIED ASSEMBLY TOLERANCE
ANALYSIS MODEL
4.1 Introduction
The existing assembly tolerance analysis models as
previously discussed, are not useful under real manufacturing
conditions.
For example, using the Worst Case model for
analyzing an assembly with a large number of components will
greatly reduce the component tolerances resulting in higher
cost of production.
Also, the statistical models such as the
Root Sum Squares model, the Modified Root Sum Squares model,
and the Motorola's model [3] are limited by their assumptions
involving statistical part dimension distributions.
models
work
distributed.
only
when
part
dimensions
are
These
normally
In fact, part dimensions do not exactly follow
normal distribution due to non-random factors such as tool
wear and die wear.
process
tolerance
Due to non-random effects, the natural
ends
up
with
a
skewed distribution.
Referring to the central limit theorem, the distribution of
the sum dimension will be approximately normal regardless of
the variation in the shapes of component distributions.
However,
this theorem will not work when the number of
53
components
is
small.
The
existing assembly
tolerance
analysis models are designed to analyze tolerances only in
ideal conditions and need to be modified.
Beta distribution
has been chosen to develop a
modified assembly tolerance analysis model.
The beta
[6)
distribution has the following advantages:
The beta distribution
1.
covers
the
actual
range
of
distribution from normal to rectangular,
2.
the beta distribution has a finite range, and
3.
the beta distribution covers an asymmetrical case.
The
beta
distribution
more
is
flexible
than
normal
distribution for representing part dimensions.
The beta distribution is defined by four parameters: the
lower limit
(a)
((3)
,
(a),
the upper limit
and the beta parameter (p.
parameters
Figure
401.
(b), the alpha parameter
The alpha (a) and the beta
control the distribution shape as shown in
One
should
carefully
evaluate
all
four
parameters for each part dimension in order to understand the
behavior of the sum dimension.
Referring to the central limit theorem, the assembly or
sum dimension distribution is assumed to be approximately
normal,
nevertheless
the
variations
component distributions are assorted.
in
the
shapes
of
In the case of a small
54
a =4, p = 4
CL=7, p = 7
a) Symmetric Beta Distribution
b) Asymmetric Beta Distribution
Figure 401. The Variation Shapes of the Beta
Distribution, a) Symmetric and
b) Asymmetric Shapes
55
number
parts,
of
this
theorem
cannot
be
applied
As displayed in Table 401[6], two different
approximately.
assumptions of the sum dimension distribution, normal and
beta, are compared using the percentage of errors from the
Comparing the data in Table 401,
actual values.
dimension
the
of
normal
distribution
model
the sum
gives
the
percentage of errors approximately nine times greater than
that of the beta distribution model.
Therefore the assembly,
or sum dimension distribution in this study is assumed to be
the beta distribution.
The objective of this chapter is to
develop a new assembly tolerance analysis model based on beta
distribution.
symbol,
In addition, the effectiveness of the flatness
one of
the GD&T specification symbols,
is also
studied and included in the new model, which helps to more
accurately determine sum dimensions and tolerances.
Table 401. Computing Error by Summing Up Equal Rectangular
Distributed Dimensions [6]. (Confidence Level
99.730)
Number of
dimensions
Normal distribution model
Beta distribution model
2
29.7%
3.0%
3
18.5%
2.0%
56
4.2 Visualization of the Part Dimension Including Geometric
Dimensioning and Tolerancing (GD&T)
The existing assembly tolerance analysis models can
accommodate only conventional dimensions (+/- system). A new
tool to specify engineering design and drawing requirements,
Geometric
Dimensioning
and
Tolerancing
(GD&T),
commonly being used in manufacturing design.
is
now
As illustrated
in Figure 402, GD&T is applied in the part drawing.
'1,-2.]:1-1;:i
Figure 402. Part Drawing Including Both Conventional
Dimension and GD&T (Flatness) Specification
Its purpose is to demonstrate not only the actual function
but also the relationship between part features.
two specifications displayed in the drawing.
There are
The first
specification is the conventional dimension (+/-), and the
second
is
the GD&T specification which
is placed in a
57
rectangular block.
The GD&T symbols discussed in Chapter 3
are classified into five types: form, profile, orientation,
runout and location.
Each type has its own group of symbols.
Moreover, each symbol has its own definition and application.
Due to the scope of this research, only the flatness symbol
is studied and included in the modified assembly tolerance
analysis model.
The first specification in Figure 402, the conventional
dimension, is assigned to limit the maximum and minimum size
of the part dimension.
This means that no part is allowed to
exceed the specified tolerance limits.
As a result, the part
in Figure 403 should have dimension values between
3.1.
2.9 and
In addition, the second specification is one of the
GD&T symbols,
flatness,
which includes the value of the
flatness tolerance zone, 0.01, in the rectangular box.
Its
application is to control the surface variation of the part.
Following the rule and the application of
the
flatness
symbol, the visual part feature (flatness) from Figure 402 is
shown in Figure 403.
When the flatness symbol is assigned,
the part dimension is not only limited within the first
specification ( ±0.1), but also limited within the value of
the flatness tolerance zone, 0.01 or ±0.005.
Usually, the
first specification (conventional dimension) implies that the
flatness
is
automatically
tolerance zone.
controlled
For example,
by
the
specified
the first specification in
Figure 402 automatically controls the flatness of the part
58
Conven7,0na
dirnns,on
0,7&T
(i=l0.-tness)
/
0.01
3.0000=0.1000
5.0000± 0.1500
\/
flatness
tolerance
0.01
3.10
3.00
2.90
Figure 403. Visual Aspects of the Part Surface When
the Flatness Symbol Is Applied
59
dimension within a 0.2 flatness tolerance zone.
specification is
requirements of
necessary.
If the first
not sufficient to satisfy the function
a part surface,
the flatness symbol
is
Certainly, the flatness tolerance zone is always
less than the specification tolerance zone.
As a result, the
flatness tolerance zone which is assigned in Figure 402
cannot be greater than 0.2.
When the flatness symbol is assigned to the surface of
the part, there are two visual aspects as shown in Figure
403.
The difference between the two aspects is the slope
between
two parallel planes and the ideal surface.
On the
left visual part, the slope between two parallel planes and
the ideal surface is zero, whereas the slope between two
parallel planes and the ideal surface on the right is not.
In this research, it is assumed that the slope between two
parallel planes and the ideal surface is zero.
As a result,
the left visual part in Figure 403 is considered to verify
the various features of part dimension in the next section.
4.3 Development of a Simple Model to Determine the Nominal
Dimension
The surface of the part dimension in Figure 403
is
enlarged in Figure 404 to illustrate the variation of the
flatness
tolerance
zones,
when
the
flatness
symbol is
60
applied.
Each pair of two parallel planes has its own middle
point which varies within the specified tolerance.
machine quality is more precise,
If the
the middle point of two
parallel planes (flatness tolerance zone) will be located
close to the middle point of specified tolerance.
Otherwise,
the middle point of two parallel planes will deviate from the
middle point of the specified tolerance zone when the machine
quality gradually reduces.
(Cq)
Therefore, the quality factor
is generated to determine the middle point of
flatness
tolerance
zone.
The
center
dimension
the
(Xe)
represents the middle range of the specified tolerance zone
The nominal dimension (X,) represents the middle point
(L).
All variables are shown
of the flatness tolerance zone (T).
in the part drawing in Figure 405.
X,
varies from the
minimum (X8-L+T/2) to the maximum (X,+L-T/2) depending on the
quality factor (C,).
A simple model is developed as shown
below:
Xn
X, + Cq(L-)
-1 Cq
(4.1)
From Equation 4.1, it can be concluded that if the machine is
of high quality, Cy becomes zero, therefore X, is equal to X,.
On the other hand,
the value of
X, varies between the
minimum and the maximum range with the value of Cq,
1, respectively.
-1 and
A manufacturer can estimate the value of Cq
61
nominal
dimension
3.10
0.1
center
dimension 3.00
0.1
'11N"VVV°4'04Y19/1
L
2.90
0.01
flatness
tolerance
one
Figure 404. Variation of the Flatness Tolerance Zone on
the Part Surface Depending on the Quality
Factor (CJ
X c+ L
Xn
A
V
L
X
r
L
1
I
X
- L
L
T
Figure 405. The Assigned Variables to the Part
Dimension Specification to Determine the
Nominal Dimension Depending on the
Quality Factor (C,)
62
by using his experience related to the behavior of
machine.
the
Finally, Equation 4.1 can be developed in terms of
part i. Let:
represent the nominal dimension of part i,
X,,
represent the center dimension of part i,
L,
represent the specified tolerance of part i,
Ti
represent the flatness tolerance zone of
part i, and
C,/,,
represent the quality factor of part i.
The simple model to determine the nominal dimension of part
i is:
XThi
q Si
Ti
+
-1SC
After the nominal dimension of part
flatness tolerance zone
i
,
is specified,
allowed
Generally,
to
exceed
the
automatically limits the upper
(Ti)
and lower value of the actual part dimension.
not
(4.2)
X,,±
T1/2
(instead
Now the part is
of
X,,,±
L1).
the natural part tolerance is produced by the
natural process tolerance
machining process.
(±tj
depending on the type of
Each type of machine gives a specific
range of natural process tolerance.
For example,
a part
processed by a broaching machine results in a natural process
tolerance less than that processed by a drilling machine.
If
63
the natural part tolerance is less than or equal to one half
of
flatness
tolerance
zone
( ±T: /2),
the
part
will
be
acceptable provided the nominal dimension remains centered.
Unfortunately, it is impossible to keep the nominal dimension
centered
during
the
production
occurrence of the mean shift.
period
because
of
the
The mean shift usually develops
in part manufacturing due to uncontrollable factors such as
tool wear or die wear when the machine produces a large number
of parts (Figure 406).
If the natural process tolerance (±t1)
is assigned to be equal to one half of the flatness tolerance
zone
(±T1/2)
,
as shown in Figure 407,
cause parts rejection.
the mean shift will
To allow mean shift to occur,
the
natural process tolerance is always less than one half of the
flatness tolerance zone, as displayed in Figure 407.
4.4 The Application of the Beta Distribution in Part Tolerance
An assembly consists of several components.
In the case
of a small number of parts, the central limit theorem is too
imprecise to assume that the assembly sum dimension tolerance
is normally distributed.
As discussed in 4.1,
the beta
distribution in this research is applied not only to each part
dimension, but also to the sum, or assembly dimension [6].
Before evaluating the assembly dimension and tolerance, the
parameters assigned to each part dimension are considered.
64
mean shift
:
Ir
natural
process
tolerance
X n,
N
Ti/2
Ti/2
flatness tolerance zone
Figure 406. Mean Shift Occurs During the Manufacturing
Process
65
ti = TiA
mean
shift
Fart
Rejection
Figure 407. Part Rejection Occurs Due to the Mean Shift
When the Natural Part Process Tolerance Is
Equal to One Half of the Flatness Tolerance
Zone (MX:
the nominal dimension after
including mean shift effect)
66
Each part dimension i is assigned as a group of variables
as follows:
1.
the specified tolerance (±L,) is always greater than one
half of the flatness tolerance zone (T1/2),
2.
the natural process tolerance (+t,)
is always less than
one half of the flatness tolerance zone (T1/2),
3.
the
part
dimensions
are
assumed
to
have
beta
distribution; the distribution shape is allowed to be
skewed depending on the alpha(a) and beta (f3) parameters
(Figure 401), and
4.
the mean shift generally exists (Figure 408).
Actually, the actual part tolerance is directly produced
by the natural process tolerance, and the middle point of the
natural process tolerance (MX) is located in the middle of
the natural process tolerance zone.
In addition,
MX,
is
ideally located at the middle point of the flatness tolerance
zone or the nominal dimension (X,1).
However, the occurrence
of the mean shift will gradually change the position of MX,
from X,, as shown in Figure 408,
so that the maximum mean
shift for part i is limited to (T,/2
the natural process tolerance
(MX,)
t ).
can be mathematically
determined using the mean shift factor (f1):
Si
(Ti/2)
The midpoint of
67
mean shift (Si)
-t
LSL
2
> <.<
+t-
MXi
+TV2
USL
Figure 408. The Part Dimension Distribution and Mean
Shift Behavior in the Proposed Model
68
where
Si
the mean shift for part i, and
is
varies from 0 to 1 for part i.
where
T1/2
CP,
Cp
ti
Cp,, is the process capability index, and
C.
C,
and
1
Ci
is the inverse process capability index.
is a factor indicating how much process tolerance
covers
half
one
of
the
flatness
tolerance
zone
(t1)
(T1/2)
.
Therefore, the midpoint of the natural process tolerance (MX,)
after accounting
for the mean shift
factor
is
given in
Equation 4.3:
Xn,
+
fi (1-C1)
(4.3)
Ti
The shape of part dimension i's distribution is indicated by
two
parameters:
parameter
(R)
the
alpha
parameter
(xi)
as shown in Figure 401.
and
the
Because of
beta
the
variation of shapes available in the beta distribution, the
distribution of
the natural process
tolerance should be
simulated in terms of one specified shape value which means
that the values of (m1) and
(3;) must be carefully chosen for
compatibility with the true part dimension's distribution.
69
In manufacturing,
if
the part dimension is normally
distributed, the natural process tolerance is equal to 3 times
the standard deviation (30) and the acceptance rate is 99.73%.
When the symmetrical shape is simulated in terms of a and 13
parameters, the
a and
shown in Figure 401,
parameters are always the same.
13
As
there are various values of a and p
parameters resulting in a symmetrical shape.
The a and p
parameters are simulated with the values 4 and 4 because these
values make the shape look closely like normal distribution
with 3
times the standard deviation equal to the natural
process tolerances as proven in Appendix B.
However,
the
acceptable rate is 100% (instead of 99.73% in normal case)
because the range in beta distribution is
result,
finite.
As a
each part must be inspected before its assigned
parameters: minimum dimension, maximum dimension, alpha and
beta parameters, are evaluated.
If all rejected parts were
removed before they are sent to the assembly line, the result
will be 100% acceptable assemblies.
The
beta
distribution.
distribution
can
represent
When the shape is asymmetric,
parameters will be different.
asymmetrical
the a and p
If the a parameter is greater
than the /3 parameter, the shape will skew to the right side.
On the other hand, if the p parameter is greater than the a
parameter, the shape will skew to the left side, as shown in
Figure 401.
As shown in Figure 409,
symmetrical, the a and
13
when the shape is
parameters are defined with the same
70
Next, when the shape begins to skew to the right
value (4,4).
side, the
a parameter is assumed to increase, whereas the
p parameter remains constant at 4.
Before
determining
the
sum
all
dimension,
eight
parameters of part dimension i must be specified: the center
dimension (X,,),
the specified tolerance
(Li),
the flatness
tolerance zone (Ti, the quality factor (CO, the mean shift
factor (f,), the natural process tolerance (ti) , and the cri and
pi parameters.
In addition, the nominal dimension (X,i) and
the midpoint of the natural process tolerance
(MXi)
determined by Equations 4.2 and 4.3, respectively.
all
ten parameters
in
can be
Finally,
each part dimension are used
to
determine the behavior of the sum dimension which is discussed
in the following section.
CC = 4
s
U=6
a= 7
a =s
p = 4
p= 4
P=4
p = 4
3 =4
Figure 409. Gradual Skewness Developing in Beta
Distribution as a Increases While p is Kept
Constant at 4
71
4.5 Development of a New Modified Assembly Tolerance Model
Four parameters of the sum dimension are calculated as
follows:
1. The middle range of the sum dimension (MXm)
2. The range of the sum dimension (RAXE)
3. The alpha parameter of the sum dimension (040
4. The beta parameter of the sum dimension (PO
Using the four parameters above, the mean and the variance of
the sum dimension are determined and two critical values of
the sum dimension, the tolerance zone and the middle point in
the tolerance zone, are calculated.
The beta distribution model of the sum dimension shown in
Figure 410, displays the tolerance of the sum dimension (TXE)
represented by the hatched area, and the middle point in the
tolerance zone (MXE).
Both TXE and MXE are easily calculated
by using normalized dimensions:
TXE
TWE VARA XE
and
MA XE
MWE VVARA XE
72
ELX1
A
f
(x-)
....
X
1=1.6XE
M XE
R
M
X
MXE
Figure 410. The Beta Distribution Model of the Sum
Dimension
73
where
TWE
is the normalized tolerance,
MWE
is the normalized middle point movement,
VARAXE
is the variance of the sum dimension, and
is the middle point movement.
MAXE
The middle point in the tolerance zone (MXE)
MXE
where
MXER
+
MAXE,
is the middle range of the sum dimension.
MXER
According to the normal distribution,
tolerance
is given by
(TWE)
the normalized
is dependent only on the confidence level
(Table 402), and the normalized middle point movement (MWE) is
always zero due to the symmetrical shape[6].
Thus, the middle
point of the tolerance zone (MXE) is equal to the middle range
of the sum dimension as follows:
MXE
=
MXE,
+
MAXE
when
MWE = 0, MAXE= 0
[6],
therefore
MXE
=
MXE for symmetrical beta distribution.
74
Table 402. Normalized Tolerance as a Function of
Confidence Level based on the Normal
Distribution Model [6]
Confidence
level
Normalized
tolerance
in 96-
TWE
100
00
99.99
7.78
99.9
6.58
99.73
6.00
99.5
5.60
99.0
5.15
95.0
3.92
90.0
3.29
In the beta distribution model, four parameters -the mean
of the sum dimension, the variance of the sum dimension, the
range of the sum dimension and the asymmetry of the sum
dimension- are required to evaluate the critical values of the
assembly dimension: assembly tolerance zone
(TXE)
middle point in the assembly tolerance zone (MXE).
and the
All four
parameters can be calculated using four specified parameters
in each part dimension: the lower limit (a), the upper limit
(b),
the alpha parameter
(o),
and the beta parameter
(p).
Moreover, TXE and MXE can be estimated by using normalized
dimensions depending on the confidence level.
In normalized
beta distributions, the mean is 0 and the variance is 1.
The
75
normalized range of the sum dimension and the normalized
asymmetry are tabulated resulting in the normalized assembly
tolerance (TWE) and the normalized middle point movement (MAW.
The normalized range of the sum dimension (RWE)
is given by
Bjorke [6] as:
RWE
where
RAIXE
is
RA XE
VVARAXE
the range of the sum dimension and the
asymmetry (FWE) is measured by:
FWz
where
EAXm
EAXm
RAXE
is the difference between the mean and the middle
range of the sum dimension.
The normalized tolerance(TWE) and middle point movement
(MWE) are tabulated as functions of RWE and FWm for different
confidence levels.
Appendix C
[6]
displays the normalized
dimensions with six different confidence levels (Tables C.1,
C.2, C.3, C.4, C.5, and C.6).
Appendix describes a procedure for determining alpha (a)
and beta
((3)
parameters of the sum dimension. Using the sum
mean (Equation d.3) and the sum variance (Equation d.4), as
76
shown in Appendix D,
the alpha parameter of sum dimension
(%), the beta parameter (PO of sum dimension, the variation
of sum dimension (VAR6XE), the mean of sum dimension [E(XE)),
the range of sum dimension (RAAXE), and the middle range of sum
dimension (MXm) are calculated.
Using normalized dimensions,
the normalized range (RWE) and the asymmetry (FWE) is computed
resulting in the normalized tolerance (TWE) and the normalized
middle point movement (MWE) depending on the confidence level.
Finally, the tolerance zone
(TXE)
of sum dimension and the
middle point in tolerance zone (MXE) are determined.
However,
the sum dimension in the part drawing is a
symmetrical tolerance specification so that the tolerance zone
(TXE) of the sum dimension and the midpoint of the tolerance
zone (MXE) can be easily transformed into symmetrical form as
MXE ± TXE/2, consistent with Figure 410.
4.6 An Application Example
The assembly shown
horizontal
part
in Figure
dimensions.
The
411
consists
of
center dimension,
five
the
specified tolerance and the flatness tolerance for each part
is shown.
All five part dimensions are assembled in linear
relationship.
Moreover,
nine
parameters
for
each part
component: the center dimension (X,,,), the specified tolerance
(Li),
the
flatness
tolerance
(T ),
the
natural
process
77
L
0.00=,
0.005
003
I
1.0000-1-0.0300
2.0000±0.01
:.0000 =0.0400
0 p00±0.0500
j
00C12.±0.010C
2 0000±0 .02
0
Figure 411. An Assembly Drawing with Conventional
Tolerances and GD&T Symbols (Flatness)
78
tolerance
factor
(t,),
the quality factor
(Cq,,),
the mean shift
the inverse process capability index (C,), the
(f,),
alpha parameter (x), and the beta parameter (5.) are displayed
in Table 403.
The nominal dimension for each part (X,i)
is
determined following Equation 4.2 and shown in the last row of
Table 403.
Next, the middle range of part dimension (MX,) and
the modified variables for each part: q,
t,2/qi, ti2/c1,2,
(following Appendix D) are shown in Table 404.
and t /(q,+1)
Four parameters of the sum dimension are evaluated as
follows:
1. The middle range of the sum dimension (MX2R)
MXIIR
=
MX2- (MX1+MX3+MX4+MX5)
8.011568
(1.99638 + 1.024384
+ 0.982422 + 2.00988)
=
1.998465
2. The range of the sum dimension (RAXm)
E ti
5
RA XE
2
=
2*(0.0004 + 0.0012 + 0.0004
+ 0.0003 + 0.0006)
=
0.0058
79
Table 403. Data for The Application Example
Component
Dimension
X
X2
X;
X3
4
,
Center
Dimension
2.00
1.00
8.00
(X,1)
Specified
Tolerance (Li)
+
0.01
+
--1-
Flatness
Tolerance (Ti)
Natural
Process
Tolerance (t,)
Quality
Factor(C,,,)
0.005
+
.0004
.0012
-0.7
Inverse
Process
Capability
0.78
0.16
.0003
0.6
0.6
+
-r
-r
.0006
-0.7
0.84
0.46
0.003
+
+
+
+
0.007
.0004
0.4
0.02
-1-
+
+
1-
Mean shift
factor (f.
0.006
+
+
0.03
-r
--1-
0.004
+
-I-
0.04
0.03
2.00
1.00
0.5
-r
0.31
+
-r
.1333
0.71
.0857
0.4
index (C1)
Alpha
parameter (al)
+
8
+
7
+
Beta
+
10
4
9
+
+
+
+
parameter (V
4
4
4
4
4
Nominal
Dimension
1.9947
8.0012
1.0222
0.9814
2.0092
(X, j
80
Table 404. The Evaluated Variables for Each Part
Dimension.
Part
number
X,
X2
X3
X4
1.996388
8.011568
1.024384
0.982442
r
t,
.0004
r
r
qi
3.33E-5
1.09E-4
t.f
qi
1.33E-8
r
F
I-
h
r
F
F
.
14
.
4.29E-5
3.75E-5
1.23E-8
1.31E-7
r
8
3.08E-5
.
.0006
.0003
13
r
2.00988
r
r
r
11
12
ti
.0004
.0012
r
r
X
F
2.57E-8
1.13E-8
F
.
r
.
2
1.11E-9
1.19E-8
9.47E-10
1.41E-9
1.84E-9
cri
r
tl
(q1+1)
1.23E-8
r
r
1.2E-7
1.14E-8
r
1E-8
-
2.4E-8
81
3. The alpha parameter of the sum dimension (c)
Before calculating the alpha parameter (%), the modified
variables (Appendix D): A, B, C, J, K ,and M are determined as
follows:
=
5
t.
=
i
3.33x10-5 + 1.09x10-4 + 3.08x10-5
+ 3.75x10-5 + 4.29x10-5
=
0.000254
5
A
I=1
=
tZ
-1
1.33x10-9 + 1.31x10-7 + 1.23x10-9
+ 1.13x10-9 + 2.57x10-9
=
B
1.94x10-'
5
t.
2
= 1.11x10-9 + 1.19x10-9 + 9.47x10-'°
+ 1.41x10-9 + 1.84x10-9
= 1.72x10-9
82
5
ti
gi+1
= 1.23x10-8 + 1.2x10-7 + 1.14x10-8
+ 10-8 + 2.4x10-8
= 1.78x10-7
5
E
ti
2=1
= 0.0004 + 0.0012 + 0.0004 + 0.0003 + 0.0006
= 0.0029
5A-4B-5C
= 5x1.94x10-"
4x1.72x10-8
5x1.78x10-7
= 1.01x10-8
Finally, the alpha parameter (00 is evaluated as follows:
(M H43) 2J-(M-4J)K
az
KM
=
30.18324
4. The beta parameter of the sum dimension (PO
4J aE
M-4J
PE
=
16.23288
83
After calculating all four parameters of the sum dimension,
the mean and the variance of the sum dimension are determined
as follows:
1. The mean of the sum dimension E(Xz)
E(Xz)
E
MXE R
al+PE
1.998465 +
1
RAX
30.
30.18
30.18+16.23
11
2
0.0058
1.999337
2. The variance of the sum dimension (VARAX2)
VARAX2
(RAxz)2azpz
(aE+121E+1) (aE+PE)2
0.00582x30.18x16.23
(30.18 +16.23 +1) (30.18 +16.23)2
1.61x10-7
84
The normalized asymmetry (FWE)and the normalized range of the
sum (RWE) are determined as follows:
EAXE
RAXE
Fr 4rm
MXER
E(XE)
RA 21
1.999337-1.998765
0.0058
0.150275
RAXE
RWE
XE
0.0058
V1.61x10
14.43947
The normalized middle point (MMFE) and the normalized tolerance
(TWE) are chosen from Table C.5 (Appendix C) as a
function of
normalized asymmetry (FWE) and the normalized range of the sum
dimension
(RWE).
With
a
normalized middle point movement
normalized
tolerance
(TWE)
confidence
99.73%
is
tolerance of the sum dimension
(MWE)
5.81.
(TXE)
level,
the
and
the
Consequently,
the
is
1.24
and the middle point
movement (MAXE) are both evaluated as follows:
85
MA XE
MWE 1,1 VARA XE
1.24q1.61x10-7
0.000498
TXE
TWE V VARA .A1
5.81 x -V1.61x10-7
0.002334
The middle point in the tolerance zone (MXE) is determined and
the sum dimension distribution is displayed in Figure 412.
MJCE
=
MXER +MA21
=
1.998465 + 0.000498
1.998963
Finally,
all
sum dimension parameters
assembly drawing (Figure 413).
half
of
the
tolerance
zone
are
shown in
the
The assembly tolerance is one
(TXE)
so
that
the
assembly
tolerance is equal to 0.001167.
To show the improvement of the new model, the tolerances
of
the
sum dimension using
the WC and RSS
determined and compared with each other.
models
are
86
EA X E = 0.000872
E(Xz) = 1.999337
f (Xi)
a.
R
E
= 16.23
RLX
MX
VARA Xz = 1.61 X 107
= 30.18
R = 1.998465
=
0.0058
TX
= 0.002334
MAX
0.000498
M XE = 1.998963
Figure 412. Parameter Values of the Sum Dimension in
the Application Example
87
0.007
/ /10.006
0.005
//
0.003
;I
C 00-0.03000
11
Figure 413.
The Sum Dimension in Terms of Symmetrical
Specified Tolerance
88
The Worst Case Model:
5
To1asn,
= 0.0004 + 0.0012 + 0.0004 + 0.0003 + 0.0006
=
0.0029
The RSS Model with 99.7396 confidence level:
2=1
ti
[0.00042 + 0.00122 + 0.00042
+ 0.00032 + 0.00062] 1"
0.001487
Comparing the assembly tolerances from the WC model (0.0029)
and the RSS model (0.001487), the assembly tolerance of the
modified model (0.001167) is the least of all three.
The determination of critical assembly dimensions and
tolerances from the modified assembly tolerance analysis model
is tedious because this model involves a large number of
variables
and
calculations.
demanding
calculation procedures,
a
computerized
(discussed in the next chapter)
model.
To
facilitate
integrated system
is incorporated into the
89
CHAPTER 5
SYSTEM INTEGRATION
5.1
Introduction
Designers and manufacturers treat tolerance analysis in
different ways.
In the design stage, tolerance analysis is
used to control dimension limits for each part without
violating the assembly dimension limits.
On the other hand,
tolerance analysis in the manufacturing process is applied to
handle
the
production
indicated
quality,
by
process
capabilities and mean shift behavior.
Integration of tolerance analysis from both stages is a
great challenge in the field of Integrated Manufacturing
Engineering.
However, an integrated system must deal with
several support systems.
For example, designers might employ
assembly tolerance analysis models to analyze a critical
assembly tolerance.
In contrast,
to produce sufficient
information for tolerance analysis
process,
measure
manufacturing
process
tolerance
capabilities
and
in the manufacturing
databases
control
are
mean
used
to
shifts.
Therefore, integration of tolerance analysis is extremely
tedious.
It requires not only demanding
calculations, but
90
also a user interface to display tolerance information.
facilitate the application of
the
To
integrated system,
a
computerized integrated system, as discussed in this chapter,
is incorporated to handle four support systems; a drawing
system, a GD&T specifications system, an assembly tolerance
analysis model, and a manufacturing tolerance database.
Four
main objectives of the integrated system are:
1)
Drawing a part or an assembly in a CAD system,
2)
Drawing GD&T specifications in the part or assembly
drawing,
3)
Determining the critical assembly tolerance of
the
assembly drawing using assembly tolerance analysis
models, and
4)
Dynamic support of the tolerance database during the
design stage.
This
integrated
system
concurrently
objectives in a user-friendly interface.
applies
all
four
The advantage of
this system is that the user can perform advanced tolerance
analysis and design at an early stage, resulting in cost
reduction and production quality improvement.
91
5.2
Conceptual Framework
As discussed in Chapter 2, a number of computer programs
have been developed in tolerance analysis.
Most of them are
designed to solve one specific problem and run under a single
For example, Karolin and Ahluwalia[13] developed
overhead.
an
independent
computer aided
tolerance
control
(CATC)
program based solely upon the tolerance-chart technique.
Martino[17]
used a simple linear programming approach to
generate design functions based on the use of solid models
and variational geometry.
The difficulty in using existing
tolerance analysis programs is the lack of flexibility to
accommodate different problem aspects.
To extend their
abilities, they require several different support systems,
resulting in larger and more complex programs.
the productivity of the user remains low.
As a result,
The integration of
different support systems in a single computer application
has been greatly needed.
During the
last
several
years,
the
development
of
hardware and operating systems, such as Microsoft Windows 3.1
and IBM OS/2.x,
powerful.
(PCs)
more
Specifically, Microsoft Windows 3.1 provides an
advanced feature
unified
has made personal computers
system.
the ability of mutitasking in a single
Using
applications can be run
this
feature,
various
simultaneously under
software
the Windows
92
environment.
In addition, a new linking method, the Dynamic
Data Exchange (DDE), is available to transfer data between
Under
resulting in an integrated system.
two applications,
Windows 3.1, the applications that support DDE can establish
links between several other applications and exchange data or
pass remote commands dynamically.
Thus, DDE is considered a
key feature in developing a computerized integrated system.
5.3
System Components
To approach the objectives of the integrated system in
this
research,
four support systems
all
assembly
specifications,
tolerance
GD&T
(drawing,
analysis
model,
and
manufacturing tolerance database) have been implemented under
Windows.
Each support system requires a suitable software
package to operate as an individual user interface.
Most
commercial
software packages,
such
as
spread
sheets, databases, or CAD are established for serving the
Windows system.
Some of these are Excel for Windows, Paradox
for Windows, and AutoCAD for Windows, respectively.
them support DDE.
Most of
In this research, two software packages
are included in the integrated system: AutoCAD for Windows
and Microsoft Visual Basic Version 3.0 for Windows.
93
AutoCAD for Windows is chosen to handle two support
systems; the drawing and GD&T specifications support systems.
The drawing capability of AutoCAD software is well known
among design and manufacturing engineers.
A part or an
assembly can easily be drawn by using the AutoCAD standard
menu.
In addition, a group of AutoLISP programs has been
developed to create the ability of GD&T specifications in the
AutoCAD system, as stated in Chapter 3.
The AutoCAD software
also supports DDE, so that the data in a part drawing can be
transferred to other applications, or vise versa.
However,
the AutoCAD system is not able to exchange data with the
other support systems, such as the tolerance database system.
A user interface design software package is greatly needed to
handle this.
Microsoft Visual Basic Version 3.0 for Windows
has been used for this purpose because it can easily handle
implementations such as databases or calculations.
To build
a visual basic application, the user can create the user
interface by "drawing"
controls,
such as text boxes and
command buttons on a form, illustrated in Figure 501.
Next,
all properties of form and controls are freely set to such
values as caption, color, or size.
The source code written
in Basic enables the application to function.
Under the
Windows environment, the user can open the designed form in
a manner similar to that of a user interface.
The capability
of the designed form is freely activated depending on the
user assignment.
As a result, the user can apply his work
94
File
edit View Bun
Microsoft Visual Basic idesigoi
J2ebug
.ptions
Window Help
Textl TextBox
iti
14601
285
H ideS election
T rue
4
H dpContextID 0
Index
Lel t
6000
Link! tem
Linkki ode
LinkTimeout
0
-
None
50
LinkT opic
MaxLength
MousePointer
0
Multiine
False
0
-
Del ault
Name
T eldl
PasswordChar
ScrollBars
0 - None
T abl nclex
0
T abS top
True
Tag
Te<r:::-:.:-.::.:::.:.:.:.:.:1't1-.,,...,.:
Top
600
Figure 501. A Design Form for Developing a User
Interface in Microsoft Visual Basic
.
95
visually with this software.
For these reasons, Microsoft
Visual Basic Version 3.0 for Windows was chosen to process
and display the tolerance database support system.
the CAD drawing
There are three system components:
program, the Assembly tolerance analysis program, and the
database
tolerance
system.
program,
operated
in
integrated
the
DDE links all three system components in the Windows
environment which is graphically summarized in Figure 502.
The
system
components
are
described
in
the
following
paragraphs:
1) CAD Drawing Program
CAD drawing program is
incorporated by AutoCAD for
Windows, handling two support systems; the drawing and the
GD&T specifications support system.
The standard AutoCAD
menu is applied to help the user to develop a part or an
assembly drawing.
In addition,
a new AutoCAD menu using
AutoLISP programming has been developed to
specifications
for
drawing.
The
create GD&T
inclusion
of
GD&T
specifications has enhanced the capabilities of standard
AutoCAD.
The AutoCAD screen can be set like a user interface
in the Windows,
so that the user can apply other user
interfaces concurrently without violating the AutoCAD system.
Additionally, the data from the drawing can be transferred to
other user interfaces dynamically through DDE.
96
Microsoft
Windows 3.1
DDE
CAD
Drawing
Tolerance
Analysis
Model
DDE
DDE
Tolerance
Database
Figure 502. The System Components
97
2) Assembly Tolerance Analysis Program
Determining
a
assembly
critical
tolerance
is
an
essential application to provide maximum and minimum limits
Assembly tolerance analysis
of the assembly dimension.
models have been developed to calculate a critical assembly
tolerance
by
accumulating
each
part
The
dimension.
integrated system includes only three assembly tolerance
analysis
models;
the
Worst
Case(WC),
the
Root
Sum
Squares(RSS), and the new model as described in Chapter 4.
Due
to their simplicity of calculations,
assembly tolerance analysis models
operated in the AutoCAD system.
the first two
(WC and RSS)
can be
In contrast, the new model,
as stated in Chapter 4, employs a number of input variables,
such as flatness tolerance zone, process tolerance, quality
factor, mean shift factor, and the alpha and beta parameters.
Therefore, the determination of a critical assembly tolerance
with the new model is complex.
handle this application.
AutoCAD system itself can not
Two user interfaces, as shown in
Figure 503, have been developed using Microsoft Visual Basic
Version 3.0 to handle the requirements of the new model.
Using the assembly tolerance analysis program,
the part
dimensions in the drawing are dynamically extracted and a
critical assembly tolerance can be determined and inserted in
the drawing automatically.
Though the critical assembly
tolerance with the modified model is determined in a user
98
Part dimension 's information
If doh
Shift
hhoh5fult
rather
SU.*
paitiotthhtt
atiddloPtalt:
:41mmortim4nagyie!
Itfletenes
AartmablfktuamOm:.
Figure 503. Two User Interfaces for the Modified
Assembly Tolerance Analysis Model
99
interface
(outside the AutoCAD system),
its value can be
transferred and activated in the AutoCAD system by DDE, as
shown in Figure 504.
3) Tolerance Database Program
During the design stage, the designer needs some general
tolerance information to specify manufacturable tolerances
and
violate
not
the
assembly
dimension
The
limits.
integration of a tolerance database in the tolerance analysis
system is greatly needed.
The database
of conventional tolerance selected for
this study consists of two parameters: the range of part
dimension and the machining process.
varies depending on both parameters.
The tolerance value
The tolerance database
used in this research is shown in Figure 505 [37].
range
of
part
dimension
and
machining
In each
process,
the
conventional tolerance is actually limited in a specified
range.
0.999
For example, the range of dimension between 0.600 to
inches,
processed by a broaching machine has the
conventional tolerance between ±0.00025 to ±0.001 inches.
The
user
interface
of
the
tolerance
database
was
developed in Microsoft Visual Basic Version 3.0 for Windows.
The form of this application shown in Figure 506 consists of
text boxes, scroll bars, and command buttons.
The objective
100
Elle
EtIN
Tel
OSNAP
Ao1n0.141 FICIt
Seen Ohm. Beele_DIm.
/.
GUT
MMW
'CM
-..-4,40.444...,404 %, kbmo,AbA, - ,
c181
107e6re MARI
15 3563 0 We
5014ie
gwelmweamwsx-:$;
47
K
ass
**.
Command
17414
Pat
cemert Second font
kssvnttv aftzow.c Meow:4,4*dgF
Stilt
144,4 Sidle.
facto
Stliow
orsoster
CO11.711110
11M
Idelmor*Pa:
LT"?
Figure 504. The Data Link between the User Interfaces
of the New Assembly Tolerance Analysis
Model and the Assembly Drawing in the
AutoCAD System
101
RANGES OF SEES
FROM
TOLERANCES
TO
0. 000
0. 599
0. 600
1. 000
0. 999
1. 499
2. 799
4. 499
0.
00015
0. 0002
0. 00015 0. 00025
0. 0003
0. 0005
0.
0.
0006
0. 0009
0009
_t_
0. 0012
0. 002
0. 001
0. 0015
0. 0025
0.
0. 0012
0. 002
0. 003
0. 005
0. 0025
0. 003
0. 005
004
0. 006
0. 0002
0. 0003
0004
0. 0005
0. 00025
0.
0004
0. 0006
0. 001
0. 0015
0. 004
0.
0. 0003
0. 0005
0. 0009
0. 0012
0. 002
0.003
0. 005
0. 008
0. 012
4. 500
7. 799 0. 0004
7. 800 13. 599 0. 0005
13. 600 20. 999 0. 0006
0. 0006
0. 001
0. 0015
0. 0025
0. 004
0. 006
0. 010
0. 015
1. 500
2. 900
0.
006
0.
002
0. 010
0.
0009
0. 0012
0. 002
0. 003
0. 005
0.008
0. 012
0. 020
0.
001
0. 0015
0. 0025
0. 004
0.
006
0. 010
0. 015
0. 025
LAPPING & HONING
GRINDING, DIAMOND
TURNING. BORING
BROACHING
REAMING
TURNING, BORING. SLOTTING,
PLANING, & SHAPING
MILLING
DRILLING
Figure 505. Manufacturing Tolerance Database [37]
102
Eile
Edit
To1+1-
OSNAP
Sinn Dim.
Basic _Dim.
nicuaravw
M000.0100
3 0003± 00200
SDI
8.3943. 8.9874
Design_Tol
Analysis
1
re..!traZINI
Laad_DOE
l
Range 21114 tale react, veldt,
4 zace0000a
ter upper limit 0.3000
-ter lower krot
^mmond
Nominal rare
Talesance (.1-)
Figure 506. Application of the Pull-Down Menu "Start"
to Open the User Interface of Tolerance
Database
103
of this user interface is to handle the requirements of the
tolerance database support system.
The tolerance database
program can link both CAD drawing and assembly tolerance
analysis programs.
For example, the user can visualize the
tolerance database and also automatically update a part
dimension in CAD drawing program, as shown in Figure 507.
With the application of
the modified assembly tolerance
analysis model, the tolerance databases can also be activated
to introduce process tolerance as dynamic information with a
user interface, as displayed in Figure 508.
5.4
The Integrated System Application Procedure
The application of the integrated system is controlled
by the new TK AutoCAD menu, as developed and described in
Chapter 3.
The TK menu is used to load
AutoLISP programs
and operate all three system components: the CAD drawing,
assembly tolerance analysis and tolerance database programs.
To run a program, a number of pull-down menus are provided to
be clicked upon by a pointing device.
The functions of pull-
down menus are explained in Appendix A.
When a program is
running, a series of text strings emerges in the command line
(of the AutoCAD system) to let the user follow the program
procedures.
The details of the program procedures, including
examples, are described in Appendix A.
104
File
Edit
Tol+/-
OSNAP Sum Dim.
Basic_Dim.
GDT DDE
Entity
Help
Range and tolerwe -value
1.0000±0.0100
3.0000=0.0200
Z800 - 4.499
I._41
Machine Pmeess
.0001
Broaching
Reaming
Turning. Boring. Slotting. Planning & Shaping
Milling
Drillin.
Lsixt Pala
Commend. Regenerating drawing
Command
4 0000
Nominal value
Figure 507. The Data Link Between the User Interfaces
of the AutoCAD System and the Tolerance
Database
105
fiiiiiiiiiHAiiiiiiiii : Tliiiiiiiiiiiiiiiii&mmin
.,
Number
Part
er
part
5 El
/1:reate f able:
Update
Artailpv.4
,MMSNgNg
Specified
relevance
Center
Dimension
Set Table
Flatness
Tolerance
Tolesance
Quality
Facto;
PIC/COSI
.-..NV:xtM
Nominal
dimension.. ...
1
+2.0000
0.0100
0.005
0.0004
0
±1
10
2
-8.0000
0.0300
0.004
0
0
±1
0
3
+1.0000
0.0400
0.006
10
0
If.
0
4
+1.0000
0.0300
0.007
0
0
121
0
5
+2.0000
0.0200
0.003
0
0
14
0
1
k*,:excXig,Z,
.....<:..e.vfor;,,,,;:::tAV.,MV::::42tv.c,
C4*(0:41illitt4.40*
2
E
4M
Grinding. Diamond Turning. Boring
Broaching
Reaming
Turning, Boring, Slotting, Planning & Shaping
Milling
Drilling
Figure 508. The Data Link Between the User Interfaces of the Modified Assembly Tolerance
Analysis Model and the Tolerance Database
106
5.5
An Application Example
To present the application of the integrated system, an
assembly drawing and the TK menu are provided in the AutoCAD
system (Figure 509).
in
the
computer
All AutoLISP programs should be loaded
following
explained in Appendix A.
the
installation
procedures
The application procedures are:
1) Inserting a conventional tolerance in the drawing
To insert a conventional tolerance, a pull-down menu
"Tol+/-" is designed to provide various types of dimension
features such as a horizontal line, a vertical line, or a
The program procedures require two end points
curved line.
of a selected line and the location of the conventional
tolerance.
In addition, the value of conventional tolerance
is freely set by typing in the keyboard.
Figure 510 displays
an assembly drawing with conventional tolerances.
2) Inserting a GD&T specification in the drawing
GD&T specification is divided into three categories: 1)
basic dimension, 2) datum, and 3) GD&T symbols.
down menu "Basic Dim",
insert
"Datum", and "symbol" are used to
a basic dimension,
respectively.
The pull-
a
datum,
and a GD&T symbol,
Due to the various types of GD&T symbols, a
107
File
4tittICAD- P1C11
Edit
14Y.gri
ToIfil13
OSNAP Sum Dim.
1±
0 Eirj
Basic Dim.
GDT DDE
la 6154,1 6756
Entity
Help
CUM
0 EllnE;531111
CILO
MI
10111
ff
Ilain
0111
A Ng
DIN
cam,
EEG
mei
EBB!
illOil
EVE
Y\
Select objects:
jCommand '_redraw
Command
ME
Cr
Elll
,e
,..,
r
Figure 509. The Assembly Example with New TK AutoCAD
Menu
108
Alf faCAD- RC] I
File
Edit
Taft OSNAP Sum Dim.
Basic Dim.
EMU
1 EDIT
______
CREATE
CYLINDER
LINEAR
RADIAL
ERASE
...
GDT DDE
Entity
1 3663. 9 8755
HN1
VER
....
Help
ELM ell
20
UM
Ilia
101E1
DP
A
Aligned
Rotate
1;1
MBA
1- 1 0000 , 0 0300
r ec000io moo
1.0
--i)00= 0.0500
i
h- 1 000ozo osoo
20001001W
1-*- 2300320 CFO
1
i
'-1/' ,
L;
000000020
I211
El Lli
EUN3
E963
3 ga
Ella
gi
BONI
A
Command
'REDRAW
Command
Es
Figure 510. The Function of "Tol+/-" Pull-Down Menu for
Creating Conventional Tolerances in the
Assembly Drawing
109
group of symbol slides has been designed, as shown in Figure
511.
To specify a GD&T symbol, the user selects a choice
from the slides and supplies the required parameters during
the procedures.
In Figure 512, only the flatness symbol is
inserted in the drawing.
3) Adjusting a conventional tolerance with a tolerance
database
A user interface of the tolerance database has been
designed,
as
shown
in
Figure
open
To
506.
the
user
interface, the user selects "DDE," "START," and "Design_tol,"
respectively.
To run the program, the user selects "DDE" and
"Design_Tol."
tolerance
For
example,
"8.0000 ±0.0300"
the
is
interface, as shown in Figure 513.
selected
transferred
conventional
to
user
the
Its value can be adjusted
following the tolerance database, and then updated in the
drawing
dynamically.
conventional
tolerance
As
shown
in
"8.0000 ±0.0300"
Figure
is
514,
changed
the
to
"8.0000 ±0.0100."
4) Determining a critical assembly tolerance
Three assembly tolerance analysis models are included in
the integrated system: 1)
the Worst Case model (WC), 2) the
Root Sum Squares model (RSS), and 3) the new model.
The WC
110
File
Edit
Layer
Tol+/-
Aititte*D- UNNAMED
Basic_Dim.
OSNAP Sum Dim.
31 2253 61.5
10
In
GDI
DDE
DATUM
Entity
Help
IMED
Select SYMBOL
FORM
FLATNESS
STRAIGHTNESS
CIRCULARITY
CYLINDRICITY
**ORIENTATION**
I4
Command. R
Command
PERPENDICULARTIY
ANGULARITY
PARALLELISM
FLATNESS
STRAIGHTNESS
PE wonccuuerry
ANGULARITY
PARALLELISM
CIRO-LAP RU UT
TOTAL RUMOR.,
CONGO...MI icrr
r
CIRCULARITY
CY LINDR IC IT Y
7/
POSITION
***LOCATION***
POSITION
CONCENTRICITY
PIRTLE OF L SIPPla
****RUNOUT****
CIRCULAR RUNOUT
Pf
.11
OK
1
Figure 511. A Group of GD&T Symbol Slides
Cancel
111
File
Tol+/-
Edit
Layer
OSNAP Sum_Dim.
0
0
Basic_Dim.
,RI
GDT DDE
14.8502,0.7414
.7 10.007
=10.605
1
0
.
'Command:
Command: _OPEN
Command:
ElEl
MEI
II
MI
I
il
I
I-1.0000±0.0400
Ella
111
00±00500
i
1
2.001110±0M00
G.0000 4.0300
.
E3ITGOIN
ICIER
I
I2.000 0±0.02 DO
LA
:::::ti:?*112011
1.0000±0.0300
--2,0000±0.01
1.0
Help
IMO
.= I 0.006
I
=10.003
Entity
=
OUR
4
ga
raillill
IN
I
BO
jo.004
I
OE
KIWI
hC el
Figure 512. The Assembly Example with Flatness Symbols
112
:
File
Edit
Tolft-
°SNAP
SornDiro.
Basic
G131
B
DOE
4 747,10 00O2
Entity
telp
6EP
CITNN
=taco,
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o.c.00s
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I
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2.00C SO.
Uri
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501
Housirmi Ofmensise
±..1
11 T----1+3
10 0000
0
isleirgrice
32)45(0
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Brushing
10.03000
Dt77d
So
Rearng
2...7C 0 0
Turning. Prong. Slotting. Planning & Shaping
DC
Wile&
ap4.sre
[Nang
I
80000
detallnk
ink
I .1- Dela
1
I
N canine/ vnue
0030
Tolnonce WA
Figure 513. Transferring Conventional Tolerance from
AutoCAD System to the User Interface
113
File
Edit
Tol+/0
Folk/CgDS.P 14:11
OSNAP Sum Dim
#0
Dim.
GDT DDE
Entity help
1747 13 J052
L7 C.037
I
It L-7 10X06
I
tkoarnal 0 ~Asam
"--71n.aca
1.000014=0
o nnn
n 5991.1
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.1
FR nnon
Toiemme
0.0 100
mrn,°F,nammommummu
na
Gnnchnq, DIenvond T utr.n,
O
achang
Re.wrng
Turing. Booing. Slotting. Planntno L S1apevj
CO): t:,0109
MINN,
data link
fT,raon
Figure 514. Updating Conventional Tolerance in the
AutoCAD Drawing
114
and RSS models are operated into the AutoCAD system.
user chooses the pull-down menu,
The
"Sum_Dim," and selects
The assembly dimension and tolerance
either "WC" or "RSS."
will be determined following the selected models (WC or RSS)
and inserted in the drawing, as shown in Figures 515 and 516.
Due
calculations
demanding
to
of
the
new
determination is applied in a user interface.
model,
its
All data from
the part drawing are transferred to the user interface.
In
addition, all supply variables can be freely set by typing in
All data in the user interface, as
the desired text box.
shown in Figure 517, follows the data in Table 403.
Finally,
the assembly dimension and tolerances "1.999±0.0012" are
determined and inserted in the drawing, as shown in Figure
518.
the
Moreover, the value of text box "process tolerance" in
user
interface
of
the
new model
can
be
assigned
automatically by connecting with a user interface of a
tolerance database.
To open the tolerance database window,
the user clicks twice at the selected text box "Process
tolerance."
In the tolerance database window,
the user
adjusts the value of process tolerance, and then clicks the
command button on "Close."
The adjusted process tolerance
will be transferred to the user interface of the modified
model
and
the
tolerance
database
automatically, as shown in Figure 519.
will
be
closed
115
File
Edit
Layer
Tol+/0
OSNAP
a
AutoCAD - PICfI
SurnDim.
Basic Dim.
GAP
ASSEMBLY
GDT
Create
Edit
Tol. Chain
Entity
DDE
WC
Help
RSS
sum dimension
2.0000 =0.1200
amoo=am
1 300010 0300
I-1000010044o
];oo:to )30
1
2.000010 0100
F..000 CO.C.n0
e wool°
07300
r
0
90
Command nil
Command '_redraw
Command.
Figure 515. The Application of Pull-Down Menu "WC"
for Determining Sum Dimension with Worst
Case Assembly Tolerance Analysis Model
116
AittoCAE2 PIC1
Edit
File
1
La sr
Tol+/-
OSNAP
0
I
Sufft_Dinf. > Basic Dim.
GDT DDE
Cteite ,- HIM=
i....GAP
ASSEMBLY
Edit
Tol. Chain
Entity
--.:: FISS.
Help
la
01:MUMII
LIM
Wil
sum dimension
-2 0000/0 0624
^.- E.0000 3 0.01
0
111
13.2
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h I 0000=0 0400
Lowo=onn
I
2.00CCt l 3100
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iY
I
Ile
EZ
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EEO
'-ii 1131
1E1
11E3
'
8.000030.0300
LI
Enter upper limit 0 06244
Enter lower limit:
Command
1111
II
111
IS
Figure 516. The Application of Pull-Down Menu "RSS"
for Determining Sum Dimension with Root
Sum Squares Assembly Tolerance Analysis
Model
117
Number of past
el Update
W.
Center
Part
Specified
Tolerance
Dimension
Flatness
Tolerant=
Process
Tolerance
... BC22:31
Quality
Nominal
dimension
-0.7 1±1
1.99475
Factor
1+2.0000
0.M N
0.005
1.0004
-8.0000
a 000
0.004
.0012
+1.0000
0.0400
0.006
+1.0000
0.0300
0.007
.0003
.98145
+2_0000
0.0200
0_003
.0006
2.00925
;
-8.0112
0.4
1.0222
Nominal
Dimension
Exec
1.99475
. Middle point
11.9985
-8.0112
1.0222
.9M 45
1.0058
011111111111:1
2.00925
Alpha parameter
30.18
16.23
1.999
Expect Value :
Standard
deriation
1.9993
.0004
Normalized Range :
Asymmetry Parameter
14.4395
.1503
.0012
Link
j Fitrtl.n
Figure 517. User Interface of the Modified Model
Including All Supply Variables of the
Assembly Example Following Assigned Values
in Table 403
118
File Edit
Delp
AtitOCATP- MIT
Telq- OSNAP Sum Dim.
Basic Dim.
an.
GDT
et part
0 s. < I5.351171737
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Cuarr.edelon- 11.9133
Stated
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AnAMa, PA aac
114.4395
1.1503
Figure 518. Transferring and Inserting the Assembly
Dimension and Tolerance, "1.999+0.0012,"
from the User Interface into the Assembly
Drawing
1
I
I
I
119
.
1.-.."...-.^2;.;";:=:-.:^5'
:444-41:44:42;4:"Nr6'5
5
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R earning
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Milling
Drillin
,,,06,v,,,,wosm>wwwww.3
Clow
Figure 519. The Application of the Tolerance Database
for Providing a Process Tolerance into the
User Interface of the New Model
120
Integration of
specifications,
four support systems
assembly
(drawing,
tolerance analysis models,
GD&T
and
tolerance databases) has been developed in a single computer
application.
Windows 3.1 software is applied to address and
operate the three system components (CAD drawing, assembly
tolerance analysis, and tolerance database programs).
Using
DDE, all three programs can communicate and transfer their
data, making the integrated system more powerful and able to
solve tolerance analysis problems in a large scale software
package.
All source files are developed and stored in
diskettes 1, 2, and 3.
in Appendix A.
The program installation is explained
It is important to note that adequate RAM and
hard disk capacity is required, since the working files have
substantial memory requirements to access system integration
functions.
121
CHAPTER 6
CONCLUSIONS AND RECOMMENDATIONS
6.1 Conclusions
In any manufacturing process,
the manufactured part
dimensions will always deviate from their nominal design data
due to random and/or systematic effects.
Tolerance data is
an important part of design information, as expressed in a
part or an assembly drawing.
During the design stage,
however, the specification of tolerance constraints can have
major consequences for product quality and cost.
tolerances
result
in
better
product
quality,
increase manufacturing costs considerably.
Tighter
but
they
In addition, the
tolerance constraints on part dimensions control the assembly
dimension and tolerance which should not exceed the relevant
design constraints.
Critical assembly dimensions and tolerances are often
evaluated by using existing assembly tolerance analysis
models, such as the Worst Case(WC) and Root Sum Squares(RSS)
models.
These models can handle only conventional tolerance.
Rather than conventional tolerance,
a
new standard,
"geometric dimensioning and tolerancing,"
or,
"GD&T,"
the
is
defined by ANSI standard Y14.5-1932 in order to give an
122
unambiguous interpretation for each geometric feature.
It
consists of rules, symbols, and their representations.
To
improve the applicability of the existing assembly tolerance
analysis models, a new assembly tolerance analysis model was
developed to include the effects of the flatness symbol (one
In addition, the part
of the GD&T specifications symbols).
dimensions are assumed to have beta distributions, making
their
shapes
distribution
However,
vary
normal
from
skewed.
to
the calculation of the modified model is highly
complicated because it involves a large number of variables.
These variables are divided into two groups.
The first group
of variables must follow the data in the part drawing, such
as the center dimension, specification tolerance and flatness
tolerance
zone.
other
The
information provided
by
the
group
user
is
or
dependent
extracted
upon
from
a
database.
For example, the process tolerance variable is
extracted
from
the
tolerance
manufacturing
database.
Therefore, tolerance analysis in the design stage requires
different support systems.
A computer integrated system has been developed to
reduce the application complexity.
The operation of the
system deals with four support systems: 1) a drawing system,
2) a GD&T specifications
analysis model, and 4)
system, 3) an assembly
a manufacturing
tolerance
tolerance database
123
system.
To take advantage of the multitasking feature,
integrated system runs under the Windows environment.
the
Using
the Dynamic Data Exchange(DDE), all three system components
the CAD drawing program, the assembly tolerance analysis
program, and the tolerance database program, can transfer
information dynamically.
The CAD drawing program allows the
allocation of conventional tolerances and GD&T specifications
in
the AutoCAD system.
This application uses computer
programs written in AutoLISP.
Using the assembly tolerance
analysis program, the user can access several commonly used
assembly tolerance analysis models as well as the modified
model developed in this thesis.
The manufacturing tolerance
database was prepared such that it could be applied like a
user
interface
to
provide dynamic
support
information to the integrated system.
generation is also included in the
of
tolerance
A tolerance chain
system to initiate a
closed loop of dimension links around a critical assembly
dimension.
All application programs are controlled by a new
AutoCAD menu named
"TK menu"
using
a
pointing
device.
Finally, the application of the integrated system improves
the capability of tolerance analysis concurrently with CAD
modeling
database.
while
making
use
of
a
manufacturing
process
124
6.2 Recommendations for Future Research
only covers
research not
This
a
tolerance analysis
methodology, but also improves the ability of AutoCAD computer
application.
However,
four additional research areas have
been identified for further refinement of the methodology and
for the extensions of its applications as follows:
1).
Model
Analysis
Tolerance
the
of
Application
Associated with Interrelated Tolerance Chains
Only the simple tolerance chains are dealt with in this
Each individual part dimension is independent of the
thesis.
others
whereas those of the interrelated
in an assembly,
tolerance chains are not.
In fact, some assemblies result in
an interrelated tolerance chain if each part
directly
affected by
the
others.
dimension is
Certainly,
tolerance
analysis becomes more complicated and needs to be explored.
2).
and
Alpha
Appropriate
of
Determination
Parameters for Individual Part Dimensions
Beta
Although the exact distribution of part dimensions are
not
known,
the
distribution of
research is assumed to be
part
the beta.
dimensions
in
this
The part distribution
125
varies depending on the beta
and alpha parameters.
If the
estimated alpha and beta parameters are close to the exact
distribution of the part dimension,
the tolerance analysis
will be more precise and meritorious.
Therefore, some studies
for finding appropriate values of alpha and beta parameters
These studies are laborious because
need to be initiated.
they are associated with a great deal of data collection and
analysis.
3).
Determination of Appropriate Natural Parameters for
Individual Part Dimensions
The modified model includes significant natural factors
such
as
the
quality
factor
and
the
mean
shift
factor.
However, the expected values of these factors still need to be
explored because their values are independent of the actual
manufacturing
recommended
process.
for
Some
future
works
experiment
dealing
with
designs
the
actual
conditions of manufacturing.
4).
Application of the Tolerance Analysis Model with
Other GD&T Symbols
The modified model developed in this thesis includes only
the
flatness
symbol.
The
rules
flatness symbol are discussed and
and applications
applied
in
of
the
the modified
126
model.
Other GD&T symbols, such as straightness, circularity
and cylindricity, need to be considered and included in the
tolerance analysis model.
127
REFERENCES
it Relates to
1.
Bender,A.,"Statistical Tolerancing as
2.
Levy,S J.,"Applied Geometric Tolerancing",TAD Products
Corp,MA,Appendix C,1974.
3.
4.
Quality Control and the Designer", Society of Automotive
Engineers, SAE paper No 680490,1968.
Greenwood,W H.,Chase, K W.,"A New Tolerance Analysis
Method for Designers and Manufacturers",Journal of
Enaineerinu for Industrv,Vol 109,pp 112-116,1987.
to
Probability
Application of
Mansoor,E M.,"The
Tolerances used in Engineering Designs", Proc. Instn
pp29-39,1963.
Mech Enunr,Vo1178 Ptl Not
,
5.
6.
7.
Wang, N and Ozsoy, T.,"Representation of Assemblies for
Automatic Tolerance chain Generation", Enuineerina with
Computers, Vo16,No2,pp121-126,1990.
Bjorke,O.,Computer-Aided Tolerancina,Tapir Publishers,
Trandheim, Norway,1978.
Wozny,M J.,Hoh,D D. and Turner,J U.,"Tolerance analysis
in a solid modeling environment", Proceedings of the
1987 ASME, Computers in Engineering Conference, New
York, NY. pp9-13,1987.
R.,"CATI: A computer Aided Tolerancing
interface",Advances in Design Automation,Vo12,ASME1992
pp157-164.
8.
Soderberg,
9.
Fleming,A.,"Geometric Relationships between Toleranced
Features",ArtificialIntelliuence,Vo137,pp403-412,1988.
10.
11.
12.
Porchet,M.,Zhang,G.,"Incorporating
Geometrical
Tolerances and Processing Inaccuracy into Dimensioning
and Tolerancing", Advances in Design Automation, Vo12,
ASME 1992, pp151-156.
Kapur,C K.,"Methodology for tolerance
functionn,Computers
loss
quality
Enaineerina, Vo119, pp254-257, 1990.
design
using
Industrial
Lu,S.,Wilhelm,G R.,"Automating Tolerance Synthesis: A
Manufacturing
of
Journal
and
Tools",
Framework
System, Vo110, No4,1990.
128
13.
Ahluwalia, R S.,Karolin,A V.,"CATC
Tolerance Control System", Journal
Systems, Vo13, pp153-160, 1986.
14.
Zhang,Y Z.,Zhang, J K L."Operational Dimensional and
Tolerances Calculation in CAPP System for Precision
A Computer Aided
of Manufacturing
Manufacturing", Annals of CIRP 38(1),1989.
15.
Raman,S.,Panchai,K.,"Computer-Aided Tolerances Assignment",Computer Industrial Enuineerinu,Vo121,p66-71,1991.
16.
Raman,S.,Pachchai,K.,"Computer-Aided
Tolerance
assignment
procedure (CATAP)
for
design
dimensioning",International
Journal
Research.,Vo130,No3,pp599-610,1992.
17.
of
Production
Martino,P.,"Analysis of Complex tolerances by Linear
Programming", Computers in Engineering, Volt, pp351-360,
1992.
18.
Turner,J and Guilford,
J.
"Representing Geometric
Tolerance in Solid Models", Computer in Engineering,
Vol 1, pp319-329,1992.
19.
Turner, J and Sadhi, R., "Representing Tolerance and
Assembly
Information
in
a
Feature-based
Design
Environment",Advances in Design Automation, Volt, pp101-
108,1991.
S
A.,
Mittal,
R 0.,
and Lehtiher, E A.,
"Tolerance Chart Optimization", International Journal of
Productions Research, Vol 27,No9,1531-1552,1989.
20.
Irani,
21.
Truslove, K C E., "The Implications on Tolerancing for
Computer-aided
Mechanical
Design",
Computer-Aided
Engineering Journal, pp79-85,1988.
D., Shah, J J., "A Structure for Integrating
Geometric Tolerance with Form Features and Geometric
Models",
Proceeding
of
1989
ASME
Computer
in
22. Miller,
Enuineerina.
D., Shah J J.,
A Structure for Supporting
Geometric Tolerances in Product Definition System for
CIM", Manufacturing Review, Vo13, Not, pp23-31,1990.
23.
Miller,
24.
Elmaraghy, W H. and Wu X., "Determination of Actual
Geometric Deviations Using Coordinate Measuring Machine
Data", Manufacturing Review, Vo13,Nol,pp32- 39,1990.
"
129
25.
26.
27.
Turner, J U and Wozny, M J., "The M-Space Theory of
Tolerances", Advances in Design Automation, Volt, Not
pp217-225,1990.
Turner, J U. and Wozny, M J., " Tolerances in ComputerAided Geometric Design", Visual Computer, Vol3, pp214226,1987.
Sririvasan, V.,Jayaranam, R., "Issues in Conditional
Tolerances for CAD Systems",IEEE Conferences Robotics
and Automation, 1985.
28.
Chase K W, and Parkinson, A R., "A Survey of Research in
the Application of Tolerance Analysis to the Design of
Mechanical Assemblies", Research in Enaineerina Design,
Vo13,pp23-37,1991.
29.
Interchangeable
for
"Dimensioning
E.T.,
Fortini,
Manufacturing", Industrial Press, pp 48, 1967.
30.
Spotts, M.F., "Dimensioning Stacked Assemblies ", Machine
Design, April 20, pp60,1978.
31.
Roy,
U., Pollard, M.D., Mantooth, K.,and Liu, C.R.,
"Tolerance Representation Scheme in Solid Model: Part
1&2", ASME Advances Design Automation, Vol 19-1, ppl-
18,1989.
Addison-Wesley,
Reading,
32.
Foster, L.W, GEO-METRIC
Massachusetts, 1986.
33.
AUTODESK, Inc., AutoCAD Release 12: Reference Manual,
2,
1992.
AutoLISP
Release
12:
Programmer's
34.
Inc.,
AUTODESK,
Reference, 1992.
35.
Gladman,
36.
American Society of Mechanical Engineers, New York, N.Y.
Dimensioning and Tolerancina, 1983 ANSIY14.5M-1982.
37.
Trucks,
C.D.,
"Applying
Technology", Trans. Inst.
Vol.ME5, No. 2, p82, 1980.
Probability in Tolerance
Eng. Austral., Mech. Encr.,
H.E., Designing for Economical Production.,
Society of Manufacturing Engineers, Dearbon, Michigan,
1987
APPENDICES
130
APPENDIX A
USER'S GUIDE OF COMPUTER APPLICATION
Setting Up
All source codes are stored in diskettes 1,
2, and 3.
Before you install all source files in a hard disk, make sure
that the computer meets the requirements as follows:
Any IBM-compatible make with an 80386 processor or higher
A hard disk (drive C)
A 3 1/2" floppy drive (drive A)
A super VGA or compatible display
One megabyte of memory in addition to run AutoCAD program
A mouse
Microsoft Windows 3.1 or later
AutoCad for Windows Release 12 or later
After checking the system requirements, the user should
open the File Manager in Windows and follow the steps below:
Select drive C and create directory named JOB
Insert Disk 1 in drive A
Copy all files from drive A to C:\JOB\
131
Insert Disk 2 in drive A
Copy all files from drive A to C:\JOB\
Select drive C and create directory named PLAY
Insert Disk 3 in drive A
Copy all files form dive A to C:\PLAY\
After processing, all research files are kept in a specific
directory (C:\JOB\ and C:\PLAY\).
Getting Started
This user's guide assumes that the user knows how to use
AutoCad for Windows and the basic techniques to work with
Windows.
First, the AutoCad system must be initiated and an
assembly drawing must be completed by using the standard
AutoCAD menu or drawing
program,
commands.
To
run an AutoLISP
the completed assembly drawing must be saved in
directory C:\JOB\ because all AutoLISP programs
research are saved in directory C:\JOB\.
in this
A new AutoCAD menu
named TK menu has been developed to group and initiate all
application programs
in the integrated system.
AutoCAD menu must be called and loaded as follows:
Type "menu" in the command line
Select C:\JOB\TK.mnu
The new
132
The functions of the TK menu are briefly displayed in Figure
A.1.
The following sections describe the application of the
TK menu to serve as a complementary tool for the application
of the integrated system.
The Application of the Integrated System
The application of the integrated system is divided into
five sections.
These sections are described as follows:
1. Creating a Conventional Tolerance
2. Creating a GD&T specification
3. Creating a Simple Tolerance Chain
4. Determining a Critical Assembly Tolerance
5. Dynamic Support with a Tolerance Databases
All sections are controlled by an individual group of pulldown menus.
The following paragraphs describe the functions
of the pull-down menu for each section.
133
File
Edit
Tol +j-
AutoCAD NCI I
OSNAP Sum_Dim. Basic_Dim.
GDT DDE
Entity
10.9012A1667
II
New..
Open..
Save..
Save As..
Plot
Help
an
f Dt EdIE
New Project
Open Project
Save Project
Print Project
Exit
Undo
Redo
Undo Step
Redo Step
Copy Image
Copy Vectors
Copy Embed
Copy Link
Copy Picture
Text Window
EDIT
CREATE
CYLINDER
ERASE
Change an existing conventional tolerance
Creat a new conventional tolerance
Creat a new cylindrical tolerance
Delete an existing conventional tolerance
Center
Endpoint
Insert
Intersection
Midpoint
Nearest
Perpendicular
Quadrant
Tangent
Object snap
Nod
Figure A.1. The Describtion of TK AutoCAD Menu
134
Sum Dim.
1:113G
Basic Dim.
GDT DDE
Entity
10.9012,4.1667
Sum Dim.
Help
131:1=531111
Determine a gap dimension
GAP
ASSEMBLY
Determine an assembly dimension
essm
LINEAR
ANGULAR
ERASE
Creat a linear basic dimension
Create an angular basic dimension
Delete a basic dimension
SYMBOL
Create a GD&T symbol
Create a datum reference
DATUM
Load_DDE
Start
Design_Tol
Analysis
SELECT
BLOCK
LAYER
Load DDE operation
Open an user interface
Update conventional tolerances from database
Determine the sum dimension using
the modified assembly tolerance
analysis model
List data of a selected object
List data of a selected box
Chage the layer
Figure A.1. The Describtion of TK AutoCAD Menu
(continued)
135
Creating a Conventional Tolerance
In the TK menu, a group of AutoLisp programs for crating
a conventional tolerance is initiated by clicking the pull-
There are different types of
down menu named "Tol+/-".
dimension lines associated with conventional tolerances such
as a vertical line, a horizontal line, or a circular line.
A group of pull-down menus is designed to serve all types of
dimension lines and their description is shown in Figure A.2.
clicking
After
at
a
choice
menu,
the AutoLISP program
associated with the selected choice will be initiated and run
The user follows the comments at the command
immediately.
line, and finally, a conventional tolerance will be inserted,
related
to
the
Only
dimension.
selected
symmetrical
conventional tolerances are included in the system.
Creating a GD&T Specifications
In the TK menu, GD&T specifications are divided into
three
categories:
references.
down menu
depending
basic dimensions,
GD&T
symbols,
datum
Figure A.3 displays the application of the pull"Basic Dim"
on
the
which can draw a basic dimension
type
vertical, or circle).
of
dimension
line
(horizontal,
In addition, the pull-down menu "GDT"
can establish both GD&T symbols and datum references as
136
-Change a linear conventional tolerance
-Change a radial conventional tolerance
LINEAR
RADIAL
HOR
VER
Aligned
Rotate
DIAMETER
RADIUS
Create
Create
Create
-- Create
a horizontal dimension
a vertical dimension
an actual dimension
a rotated dimension
Create a diameter dimension
Create a radius dimension
Create a cylindrial dimension in vertical line
Figure A.2. Application of the Pull-Down Menu "Tol+/-"
Basic Dim.
LINEAR
ANGULAR
ERASE
HOR
VER
Aligned
Create a basic dimension in horizontal line
Create a basic dimension in vertical line
Create an actual basic dimension
Create an angular basic dimension
Delete a basic dimension
Figure A.3. Application of the Pull-Down Menu
"Basic Dim"
137
To crate a GD&T symbol, the user
described in Figure A.4.
clicks the pull-down menu
"symbol",
and then the figure
(Figure A.4).
frames of GD&T symbols will be visualized
Next, the user selects the choice and actives following the
comments in the command line. Finally, A datum reference or
a GD&T symbol will be inserted in the drawing.
Creating a Simple Tolerance Chain
A simple tolerance chain can be generated in different
types of the sum dimension.
The sum dimension in the TK menu
is divided into two features: assembly and gap dimensions.
An
assembly
dimension
is
considered
dimension from all part dimensions.
dimension is the
part sections.
as
an
accumulated
In contrast,
a gap
unknown sum dimension located between two
Both types of sum dimensions are displayed in
Figure A.5.
The pull-down "Sum Dim" is divided into two sections as
shown in Figure A.6.
The user chooses the pull-down menu
named Tol_chain, and then specifies two end points of the
unknown sum dimension.
A simple tolerance chain will be
created following the successive link between each part
dimension, as displayed in Figure A.7.
Moreover, the simple
tolerance chain will be located in a new layer named "chain".
However,
all part dimensions except one
(the unknown sum
138
SYMBOL
Selm:IS'YMI3OL
DATUM
**"* FORM "*"
FLATNESS
STRAIGHTNESS
CIRCULARITY
_
/17
(:)
FLATNESS
STRAIGHTNESS
CIRCULARITY
PEW0111111:ULARITY
ANGULARITY
PARALLELISM
CYLI.NORICITY
CYLINDRICITY
*
" ORIENTATION **
PERPENDICULARTIY
ANGULARITY
PARALLELISM
POSITIEN
(Th
LJ
COVCCMIRWITY
TOTA, RUNOUT
.
PRORLZ CI
*
*** LOCATION'**
POSITION
CONCENTRICITY
*
"*" RUNOUT
""
CIRCULAR RUNOUT
[tNeviour;
1
PLANAR FEATURE
FEATURE of SIZE
mcFat V
L___
1-±
OK
Cancel
Create a planar datum reference
Create a feature of size datum
Figure A.4. Application of the Pull-Down Menu
"GDT"
OW
139
1.0000/ 0.0040 --w-
2.0000/ 0.0080
3.0000/0.01.00
ow
assembly
dimension
a-
2.0000/ 0.0030
1--Loo
0/0.0040
gap
dimension
4.0000/0.1000
Figure A.5. Two Types of the Sum Dimension; Assembly
Dimension and Gap Dimension
140
1
Determine a gap dimension
with the Worst Case model
Determine a gap dimension
with the Root Sum Squares model
Determine an existing gap dimension
with the Worst Case model
Determine an existing gap dimension
with the Root Sum Squares model
GAP
ASSEMBLY
Create a simple tolerance chain
associated with a gap dimension
Determine an assembly dimension
with the Worst Case model
Determine an assembly dimension
with the Root Sum Squares model
Determine an existing assembly dimension
with the Worst Case model
Determine an existing assembly dimension
with the Root Sum Squares model
Create a simple tolerance chain
associated with an assembly dimension
Figure A.6. Application of the Pull-Down Menu
"Sum Dim"
141
File
Edit
l'.Layer
Tol+/-
OSNAP
Anta-CAD PICIO
Som_Dim. Basic Dim.
GAP
0
ASSEMBLY
GDT DDE
Entity
Help
Create
Edit
Tolerance Chain
O."
1.500020.0700
1.50001 J000
1 500
1004M+
INF
sum
idnension
71
5.500010_1400
Command: Regenerating drawing
,Command:
Figure A.7. Application of the Pull-Down Menu
"Tol.Chain"
Q
142
Otherwise, the program cannot
dimension) must be specified.
find
the
link
between
two
successive
part
dimensions
resulting in an unsuccessful application.
Determining a Critical Assembly Tolerance
Assembly tolerance analysis models are used to determine
a critical unknown sum dimension of an assembly.
Only three
assembly tolerance analysis models are available in
integrated system.
and Root
the
The first two models are the Worst Case
Sum Squares
models.
Both assembly
tolerance
analysis models can operated within AutoCad system due to
their simple calculations.
Choosing the pull-down menu as
shown in Figure A.6, the sum dimension from a selected model,
either the Worst Case or the Root Sum Squares, is determined
and
inserted
in
the
part drawing.
However,
the
third
assembly tolerance analysis model or the modified assembly
tolerance analysis model, described in Chapter 4,
complicated and associated with a large number of
variables.
Therefore,
is too
input
the modified model should be used
outside the AutoCAD system.
Microsoft Visual Basic was
chosen to serve in the execution of this third model as a
user
interface under Windows
environment.
In
addition,
143
Dynamic Data Exchange (DDE) links
AutoCAD
system and
the
user
the
data
interface
of
between both
modified
the
assembly tolerance analysis model.
To
determine
the
sum dimension using
the modified
assembly tolerance analysis model, the pull-down menu "DDE"
is used and its application is displayed in Figure A.B.
The
assembly drawing in Figure 501 of Chapter 5 is selected to
illustrate the application of the user interface.
To examine
this illustration, the user should follow these steps:
1.
Make sure that the assembly drawing and the TK menu are
already initiated in the AutoCAD system
2.
Click the toolbar "DDE" and select the pull-down menu
"Start"
3.
Select "MODEL", then the user interface of the modified
assembly tolerance analysis model will be loaded and
opened in the form of a window named "Part dimension's
information" as shown in Figure A.9.
4.
Click "Create Table" command button, and rows of table
boxes will be created corresponding to the number in
the
combobox
assembly
"Number
of
part".
Referring
to
the
drawing in Figure 501, five part dimensions
are involved so that five rows of text boxes will be
created as shown in
Figure A.10.
144
Load_DDE
Start
Design_Tol
Open the tolerance
database window
Design_Tol
Analysis
Analysis
Open the window of
the modified model
Load the AutoLISP program
to update the selected
conventional tolerance
Assembly_Dim
Determine the assembly
dimension and tolerance
using the modified model
Gap_Dim
Figure A.8. Application of the Pull-Down Menu "DDE"
Pad ditfielnlett cs.ilddmaddly
V i 1 I a; 6."2 M.3 Vt1 k ,..' IffiaM41
Rumba, of part
[Pert
2
t
KreateTatgel
Set iofile
,
..
4
1
Center
Specified
Flattuus
Ptuateut
Quality
Noirmud
Distenleun
Tole-ram=
Tolerance
Tolerance
Facto;
dimension
i
Figure A.9. The User Interface of "Part dimension's
information"
145
Edit
File
MSS°
Att loCAD MIT
0 ton
Tali+
OSNAP Sum_Dim.
Basic_Dim.
..
GOT DDE
z:7-10.0o3
0111:1
gli
ig
..- L00001-0.0300
0
El -
1--1.0000±0o400
.. 0.05G0
1,0
MIK4
Eliffit
I
OKA
2.00C10±0.0100
k-2.000020.
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Y
9.000010.0200
IA
IRE
I
,
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CIO
113
Eh"
I
<
,
M:..AW
Pluinbef of
Past
f.1
,
Pat F71
c,,,,, Tab;c
2
1
I
Process
Flatness
1
I
I
1
I
3
4
5
I
I
I
1
I
Quebtp
Faclot
Tolerance
1
I
I
I
I
I
I
::m;5:,,:::
Update
Toletence
Toietanee
121151
..:91112
'---Ismmmi
Set rabte
Speciect
Center
Dimemtion
, fm,
i imiiiiiituoikr:::
...-
.-
1
-.-.-2,0000±0.01
I
=10.6 1
I
Help
Enrimmoil
UN
UM
ME
0E1
14174710 0052
=10.007
,t0.005
Entity
0
r4
0
It
0
1±
0
it
IIIN
Noesinet
&mention
1
1
1
I
Figure A.10. Application of Command Button "Create
Table" to Initiate Five Rows of Text Box
146
5.
Click the toolbar "DDE" and select the pull-down menu
"Load DDE". This process establishes a link between both
windows.
application
6.
Click the pull-down menu "Analysis" and select either
"Assembly_Dim" or "Gap_Dim" depending on the types of
sum
the
Following
dimension.
the
drawing
example,
"Gap_Dim" is selected.
7.
An Autolisp program will be loaded and run.
8.
After following the comments in the command line, all
part dimensions' data from the part drawing such as:
center dimension
flatness
and
(X_
specification dimension
),
tolerance
zone
(T)
are
copied
(L ),
and
transferred to the text boxes as shown in Figure A.11.
In addition, the plus and minus signs (+/-) in front of
the center dimension indicate the direction of the part
dimension in a closed dimension loop, associated with
the simple tolerance chain (Figure 310).
9.
In
"Part
dimension's
information"
window,
should assigned the natural process tolerance
the quality factor (CO.
the user
(t.)
and
In the text boxes of the
natural process tolerance, the user can directly type
the value in the text box.
10.
Click "Update" command button, all nominal dimensions
(X
.)
are calculated and appear in the
Moreover,
if
the
assigned values
computer generates an error message.
text boxes.
are violated
the
For example, the
147
Edit
File
Tol+/-
Sum Dim.
OSNAP
Basic_Dim.
GDT DDE
Entity
Help
1411747.10 0052
12Ea
.10.007
=10.005
T
RIM
z_-/--i 0.006
I
0
2.0000±0.01
IIIN
i
00.003
0
I
-1.0030±
0
OD
allgi
I
1121112
ni
0
Eili
1.-I000±0.0400
1.0000±0.0500
MPH
ESIE1
I
2.000010.0100
®
i
2.00.00 =0.02 DO
Z\
Y
1:2153
0.0300
EIGE
6E1E1
1
ESN
MCI
ME
5.0000±0.0300
IA
10.004
1.-----4Y
1--'1.75.7:-,7ZM;57=
...,.
%i:::::i%:::4:::*:::6:::::::'4:::::<.:".::5::::*;;6:::::.
114
r
u m bat of Pad
Part
5
13
Center
Dimemion
L_
CHIS
f-...:00004000
#* .
:,
Specified
I otetance
+2.0000
2
-8.0000
0.0300
3
+1.0000
10.0400
4
+1_0000
0.0300
5
+2.0000
0.0200
I
Nominal
Tolerance
Process
Tolerance
Quality
Factor
0.005
0
0
j±:
0.004
0
0
±
0
0.006
0
0
±
0
0.007
0
0
0
0
Flatness
0.0100
1
'. :g?.; '...
:,fc
Ma::::::,::M:,...t.,,,,:*:M.:,::.''":;:::,...: ,
Update
Set Tattle
t.....watf, r..14,..
"..
______
0.003
t
+:
dimension
0
o
Figure A.11. Transferring Data from the AutoCAD
Drawing into the Text Boxes of the User
Interface
148
natural process tolerance is
greater than one half of
the flatness tolerance zone or the specified tolerance
is less than one half of the flatness tolerance zone.
The nominal dimensions are not determined and a message
error related to the error value will be displayed.
11.
Click "Analysis" command button, the "Assembly Tolerance
Analysis Model" window will appear (Figure A.12). The
user adjusts the mean shift factor by moving the scroll
bar or typing in the text box directly.
Furthermore,
the alpha and beta parameters are needed to be set.
12.
Click
"Exec"
command
all
button,
dimension
sum
parameters are determined as displayed in Figure A.13.
13.
Click
"Link"
and then click
dimension and tolerance
are
"Return",
transferred
the assembly
to AutoCad
system.
14.
Go back to AutoCad system and
follow the comments in
the command line to specify the location of the sum
dimension.
Finally,
the estimated sum dimension and
tolerance are inserted in the drawing.
Dynamic Support with a Tolerance Database
The user interface of a tolerance databases
[37]
is
developed to support the integrated system in two system
processes; the drawing process, and the
assembly tolerance
149
.6#4::****4-Ak
Sum cismenzion
Middle point
iwAZR
SuniRange:
Esstect Value :
Standard
deviation
Asymmetry Parameter
Figure A.12. Application of the Command Button
"Analysis" to Open the User Interface
"Assembly Tolerance Analysis Model"
150
Number of Dad
1.99475
.78
-8.0112
.46
3
1.0222
1-.84
4
.98145
.31
5
2.00925
n11111:1
.71
7.741711]
9.4
MM-±.
NicklispoitA:
11.9985
Sum Range ;
Alpha pa/aloof= *.
30.18
1, Bela paronort ea
116.23
Middle Point
11.999
Tofef once:
Expect .Value
Standasd
deviation
Aspoeutetry Pewoatotof
Figure A.13. Application of the Command Button "Exec"
to Determine the Unknown Assembly
Dimension and Tolerance Based on the
Modified Model
151
analysis process.
two
different
Two user interfaces are created to use in
processes.
The
application
of
the
user
interfaces in the two processes are described as follows:
1) Drawing Process
During the drawing process,
a tolerance database is
needed to help the designer assign a realistic manufacturing
tolerance.
A.14)
A user interface of tolerance database (Figure
is developed to link the AutoCAD system.
Its main
application is to pull a dimension from the drawing and
update dynamically.
To apply this user interface, the user
should follow the steps below:
1.
Make sure that a drawing and the TK menu is already
initiated in the AutoCAD system
2.
Click the pull-down menu "DDE".
3.
Click "START" and then click "Design_Tol" to open the
user interface of tolerance database (Figure A.15)
4.
Click "Load DDE" to establish a link program
5.
Click the pull-down menu "Design_Tol" (Figure A.16), an
AutoLISP will be loaded and run.
6.
Select a dimension in the drawing, then the nominal and
tolerance values of the selected dimension will transfer
and appear in the
appropriate text boxes, in the Link
152
Range and tolerance value
Nominal Dimension
Range of size
2.800 - 4.499
Machine Process
1014,
Grinding. Diamond Turning_ Boring
Broaching
Reaming
Turning. Boring. Slotting. Planning & Shaping
Milling
Drillin
Set
Update
Link
link Data
Return
Nominal value
Tolerance (+1-)
Figure A.14. The User Interface of the Tolerace Database
153
1.000(00.0100
0 3:00
awer .pper
-nter over imrt
'-orrmftrld
Figure A.15. Application of the Pull-Down Menu "Start"
to Open the Tolerance Database Window
during Drawing Processes
154
file
agrrori.Ta"
Edit
Tol
0 SNAP Sum Dim.
Basic Dim.
10.1133 2 1062
GOT
Load_DDE
Start
Design Tat
Analysis
- *-L0000 =0.0100
3.003310.0200
Range ol *am
4.000010.0300
2E03
4.491 ;!.?;
Hi
ti
Naminat Dimension
Totetance
Machine Process
GrinCling, Dietect
Broaching
Reaming
Tienitg, Boring
"Cancer
Set
it
Tuning, Boling. Slotting, Planning i Shaping
DrJing
1
it
L tl
ate
'Corr meed:
Lint Data
R elan
Nominal value
Tolerance (.1-)
Figure A.16. Application of the Pull-Down Menu
"Design_Tol" to Update the Selected
Conventional Tolerance "4.0000 ±0.0300"
155
Tot./
Edit
OSNAP Sum Dim.
Layer 0
Ililma
Basic Dim.
10 1189.9.1062
GDT
Entity
Help
Load_DDE
Start
rzigtr_To;
Analysis
L0000x0.0103
3.0000±0.0200
rinalriaf CO1101112111001
Range of we
4300020.0300
4.499 1.tt
2.1300
C3
11111111112
Tolerance-
Machine Protersg
CIMI11111
Grinding, Diamond Turning. Boring
Broaching
Reaming
Can:el"
Turning. Boring. Slotting. Planning & Shaping
Milling
Drilling
command
Lin* 0.46r
4
Irmo
Nominal value
0 0300
Return
Toleiance (1-1
Figure A.17. Transferring "4.0000 ±0.0300" from the
AutoCAD Drawing into the Text Box of
"Link Data" Frame
1
156
Data frame.
As shown in Figure A.17,
the dimension
4.000 ±0.0300 is selected and its values appear in the
text boxes.
7.
Click "set" command button in the user interface, the
nominal and tolerance values from the Link Data frame
will transfer to the text boxes named Nominal Dimension
and Tolerance in the upper right corner of the window,
as displayed in Figure A.18.
8.
Select the types of machines in Machine Process list box
9.
Move the scroll bar on the right side of the Machine
Process label and the tolerance value in the text box
will
change
from the minimum to
the maximum value
following the tolerance database in Figure 504.
In this
example, 4.000 ±0.0300 is changed to 4.000 ±0.0001.
10.
Click "Update" command button,
the updated tolerance
will appear in the Link Data frame (Figure A.19).
11.
Click "Link" command button, making a linking object
between two application windows.
12.
Click "Return" command button, the selected dimension
will be changed from 4.000 ±0.0300 to 4.000 ±0.0001, as
displayed in Figure A.20.
157
File
Edit
To 141-
itistcoM - NW'S
OSNAP Sum Dim. Basic Dim.
GDT
ICIE10111111IMEMI
Entity
Load_DDE
Start
Help
Design ref
Analysis
e-3.0003.10.0200
Bang* of tine
4.70010.03CD
12 WU
NateinaF Ditsonsten
4 (IWO
4 4T.1 _l&rj
T Ditdh012*
Machine Pro-cats
I*j
Gunding, Diamond T L11 nong Boring
8loachng
Reaming
Cancer
Turning. Bating. Slotting. Planning i Shaping
II [ling
Doting
10.0300
Set
Update
ommand
Link D.ile
Return
Nominal value
Tolerance [4-)
Figure A.18. Application of the Command Button "Set"
in Tolerance Database Window
158
File
.344oCAO PI:03
TOW- OSNAP Sum Dim.
Edit
MiGE
0
1 ±.1
OEM
Basic Dim.
GDT
DDE
Entity
Help
L7,2E:gm rtam
;tootle aodleteraher
1.0000t0D100
Range of tiztt
.4-- 10000* 0 02 LO
2800
4499 'Ltji
Wiirrina/ Dimension
1LI
1.1
ITiTinn
I
LJ
0001
4 000C,-0.3000
Tolmence
Machine Proces:
Grinding. Diamond Turning. Boring
Broachong
Reaming
Tuining, Bating, Slotting, Planning & Shaping
illing
Set
titsctstit j
Ink Data
Erter upper limit 1j01.10
14.0000
.0001
Command-
Nominal value
Tolerance (.1-)
Erter lower limit
Figure A.19. Application of the Command Button
"Update"
Return
I
159
File
Edit
To1-14
OSNAP Sum Dim.
11[0 I SIPI
Basic Dim.
GDT DDE
Entity
Help
I 13 2435,2 4440
1. 000030 0 107
itominall Dimension
flange et wrzia
-- 3 0 00030 .02 0 0
1 2.800
4.499
1
14.0000
,t,1
4.00 0 0304001
Tolerance
Machine Maces:
Data Link
Broaching
Reaming
Tuining, Boring, Slotting, Planning i Shaping
Milling
Drilling
Lit Data
Nominal value
Tolerance (1
Figure A.20. Application of the Command Button
"Return"
160
rg'707<s:CVA'SZVOigkiW744,4FRRNPIAMVM&:W?;,;;;z
.....,,,t.,`,$,":4::::45.0Mi:0W,5A.
.
Update
olerance
.,,z.z*,.:,r,wa=44*(00-4,,,:,
U/
ti
Factor
Nominal
dimension
.
Grinding, Diamond Turning, Boring
Broaching
R earning
Turning, Boring, Slotting, Planning & Shaping
Milling
Drilling
Figure A.21. Opening the Tolerance Database Window by
Clicking Twice at the Selected Text Box
"Process Tolerance"
161
2) Assembly Tolerance Analysis Process
The modified assembly tolerance analysis model runs in
the user interface because it employs a large number of input
variables.
must
be
First, the window "Part dimension's information"
initiated
following
by
the
steps
in
section
"Determining a Critical Assembly Tolerance", as previously
described in this Appendix.
Assigning the input variable,
the process tolerance (t,), is required tolerance information,
so that a user interface of tolerance database is created to
support this requirement.
To open the user interface, the
user clicks twice at the text box of the process tolerance,
as
shown
in
Figure A.21.
In
the user
interface,
the
tolerance value will change depending on part dimension range
and the type of the machining process.
Click "Close" command
button, the user interface will close and the tolerance value
will
automatically be
inserted
selected process tolerance.
in
the
text
box of
the
162
APPENDIX B
APPLICATION OF BETA DISTRIBUTION
IN PART DIMENSION
Generalized Beta Distribution
The probability density function of a generalized beta
distribution in the interval [a,b]
f (x,
is:
x-a\a-111_ x-a\13-1
1
a, p, a, b)
(b-a)B(a,
13)
\b-a)
b-a)
where
a>
0,
B(a, 13)
13
> 0
and
a sx s b
(the gamma function) is defined by the equation:
r(a)r(p)
B(a,(3)
r(a+(3)
and
r(a+1)
=
ar(a)
163
where
E(X)
is
the expected value of X,
and VAR(X)
is
the
variance of X,
E(X)
a
+
[(b-a) x
co-P]
(b-a) 2a13
VAR (X)
(a+P+1) (a+(3) 2
Determination of Alpha and Beta Parameters with a Symmetrical
Shape in Beta Distribution
Let
a
be the lower limit of interval range [a,b]
let
b
be the upper limit of interval range (a,b)
let
a
be the standard deviation
VAR(X)
Since
(b-a) 2aP
(a+13+1) (a+0)2
a2
VAR (X)
Therefore
a
((b-a) 2aP
(a+(3 +1) (a+p)2
)
(0.1)
164
middle
range
Figure B.1. The Shape of Symmetrical Beta Distribution
In Figure B.1, the process tolerance (t)
is one half of the
interval range [a,b].
2xt
t
=
(b -a)
(b -a)
2
(b.2)
In manufacturing, when a part dimension is assumed to be
normally distributed, the process tolerance is often assigned
to be equal to 3 times the standard deviation with acceptance
165
rate of 99.73%.
To simulate the symmetrical shape in beta
distribution, the alpha and beta parameters are specified by
using the same rule.
Unlike the normal distribution, the beta
distribution has a finite range and each part dimension must
For
be within upper and lower limits (interval range la,b1).
this reason, all parts are assumed to be inspected before they
are sent to the assembly line.
Since
30
((b-a)2aP
3
t
(a+P+1) (a+P) 2
(b.3)
When the shape is symmetric, the alpha and beta parameters are
equal so
a
(b.4)
=
Substituting (b.4) in (b.3):
t
3
(
(b-a) 2 a2
(2a+1) (2a) 2
)
(b. 5)
166
From (b.2) and (b.5), the alpha and beta parameters can be
computed as follows:
.
(b-a)
2
3,\
j (b-a) \
2
(2a+1) (2a) 2
(b -a) 2 a2
32
)
(b-a)2
(2a+1) (2a) 2
(b-a) 2a2
(2a+1) (2a) 2
9
.
4
1
1
4 (2a+1)
36
36
.
4 (2a+1)
9
=
2a+1
a
From (b.4)
((b-a) 2a2
=
4
167
Therefore, the alpha and beta parameters, which indicate the
symmetrical shape in beta distribution, are assigned the same
value (4,4).
168
APPENDIX C
NORMALIZED DIMENSION IN BETA DISTRIBUTION
Tolerances and middle point movements for normalized
beta distributions
The following tables[6] give the tolerances and middle point
movements for normalized beta distributions at six different
confidence levels.
The parameters in the tables are as follows:
The abscissa FW
the asymmetry of a normalized dimension
The ordinate RW
range of normalized dimension
Upper number in pair TW
tolerance of a normalized dimension
Lower number in pair MW
normalized dimension
middle point movement
of
a
169
Table C.1 Normalized Dimensions (Confidence Level
90.0n)[6]
-.4
_.
5
-.2
:
:
1
1
7
I
r
I
I
1
r
1
r
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1
1
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9
r
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1
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1
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3.10
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1
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:
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3
1
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1
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1
i
3.19
-'."4
3.21
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3.22
-2.44
3.24
-2.77
3.26
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3.26
1
r
i
:
7
1
1
r
1
1
1
r
i
1
r
i
1
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1
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1
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1
1
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1
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3.28
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3.29
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r
1
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3.22
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3.24
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1
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1
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1
1
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1
1
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6
2.93
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3.27
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3.29
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3.,..?
1
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1
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1
1
1
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t
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3.29
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1
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1
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3.29
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9
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1
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1
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3.29
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4.96
3.22
5.96
1
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1
1
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1
1
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1
:
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3.65
3.41
1
,.'7
.
-
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:
1
3.26
3.73
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t
3.25
3.09
1
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r
1
1
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3.10
4.35
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3.12
1
1
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4.8_
3.14
1
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r
3.7
4.35
3.7
,
1
1
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1
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3.22
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i
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3._
5.'3
r
1
1
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3.23
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1
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8.93
3.18
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3.28
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5.65
1
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3.28
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I
:
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t
1
r
i
1
r
5
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7.33
3.22
7.4
3.24
9.79
1
I
11.83
170
Table C.2 Normalized Dimensions (Confidence Level
95.0M
-.2
-.3
-.4
-.1
I
_
i
1
"..=',..
1
:
r
1
r
_
3.7:
1
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,
3.74
1
1
-.
3.7
i
1
:
r
r
1
9
i
1
r
1
10
-1.92
3.79
-2.24
3.92
-2.63
i
1
r
1
r
1
1
r
1
1
-1.36
3.85
-1.59
3.86
1
1
r
-1.91
12
3.71
I
1.c.,5
1
-3."7?
r
1
13
-4.54
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.-1.--
-5.41
1
r
I
1
r
-3.62
I
I
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I
1
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.._..-
-4.26
:
r
1
1
1
r
1
1
r
-6.73
,
..i,..,]
:
r
2=,
L86
3.88
r
i
1
r
-11.72
i
1
I
..
I
I
r
1
i
i
:
r
1
r
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1
1
1
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1
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i
:
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3.9:
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1
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1
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1
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,
5.28
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3.95
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r
.00
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r
i
:
r
3.87
-.81
1
3.99
-.92
,
3.99
-1.03
3.?
-1.13
390
-1.24
3.?)
-7.3,3
3.91
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1
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1
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I
-1.-17
5.-.
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3.'1
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3.97
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3.86
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3.87
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3.89
1
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3.30
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3.78
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3.92
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3.91
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3.91
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3.91
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1
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1
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3.90
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3.88
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3.91
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3.9
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-2.97
1
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1
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1.34
3.90
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3.90
1.55
3.91
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13.711
r
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r
3.9?
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1
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3.,3
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3.83
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r
13.7.1
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:.-.....
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1
3.63
1
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1
1
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1
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1
3.83
1.36
3.85
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3.86
1.81
3.87
2.03
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2.24
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1
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1
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1
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3.71
1.55
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1
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1
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3.76
1.92
3.79
2.28
3.82
2.63
3.83
3.96
3.95
3.29
I
:
r
:
3.62
..
i
1
r
-
i
1
r
-
i
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1
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3.71
1
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4.9
1
3.73
4.54
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3.7
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2.46
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;
1
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1
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:
r
r
IIIIII._I
I,-I---I--,I,I-,--I---I,--I..
r
r
-:.
1.89
1
1
3
-2.03
.32
1
III!!
1
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I'
--''.96
3.69
1
T
13.731
3.79
3..3,
1
r
.70
i
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1
:
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3.78
-.45
r
r
r
i
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i
1
r
r
r
6
1
3.291
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3.91
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3.91
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3.9'
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1
1
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:
r
1
1
1
1
r
1
1
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1
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3.89
2.67
1
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...90
t
r
1
r
I
I
3.09
3.9;
1.76
3.91
1.86
3.91
1.96
3.91
2.47
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2.97
I
1
r
i
:
r
1
1
1
r
i
1
r
1
:
:
3.87
3.94
I
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1
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.1..:
3.5'
3.91
1.70
r
1
:
r
1
4.26
3.99
4.57
1.88
I
I
1
r
I
:
r
3.91
3.91
1
3.91
4.93
1
3.91
5.94
r
r
I
I
r
r
3.?,
r
1
1
r
i
1
r
I
1
3.99
5.28
3.89
5.51
3.89
5.89
3.90
7.36
3.91
8.83
I
1
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3.-7
5.43
I
:
r
.7-....7
I D
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r
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1
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I
r
1
1
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1
1
1
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1
1
1
r
1
:
3.8')
7.15
3.83
7.57
3.86
9.56
3.89
11.7
171
Table C.3 Normalized Dimensions (Confidence Level
99.00%) [6]
2
-.
rr
1
1
1
4.24
1
1
,
1
1
r
i
r
,
1
1
r
4.61
-1.16
4.72
1
1
r
10
1 1
1
r
1
11
1
1
r
4.93
-3.52
1-
!.94
-4.70
,
4.?6
1
1
r
i
1
r
i
i
4.90
-1.94
4.87
-2.31
4.91
-2.67
4.95
-3.02
4.99
-3.36
5.77
4.':r'
-4.94
4.?9
1
r
1
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r
1
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1
1
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1
1
1
r
1
1
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1
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1
1
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;
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r
r
1
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9
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1
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1
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25
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5.07
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:
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1
1
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1
1
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5..,
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1
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1
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1
1
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1
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5.1.:,'
-7.
5.12
-8.77
4.5:
-.63
4.69
-.9:
4.79
-1.16
4.6
-1.4
4.92
-1.64
4.96
-1.89
4.99
-2.1:
5.72
-2.32
5.74
-2.54
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-2.7
1
1
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1
1
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i
1
1
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1
1
r
i
1
r
1 i
1
r
1 1
1
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1
1
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4.30
-.35
4.57
-.34
4.73
-.49
4.93
-.60
4.90
-.73
4.95
-.84
4.99
-.96
5.01
-1.07
5.0
I
1
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I
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1
1
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1
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Table C.4 Normalized Dimensions (Confidence Level
99.50%) [6]
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11.33
173
Level
Table C.5 Normalized Dimensions (Confidence
99.73-U[6]
.0
-.4
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1
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1
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5.96
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5.1?
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5.37
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5.50
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5.83
5.96
5.84
5.42
5.85
6.87
5.90
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174
Table C.6 Normalized Dimensions (Confidence Level
99.9M[6]
1
4.56
1
4.7d
1
r
r
r
1
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4.72
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4.76
1
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4.65
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5.46
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5.67
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5.83
1
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5.96
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1
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175
APPENDIX D
DETERMINATION OF ALPHA AND BETA PARAMETERS
OF THE SUM DIMENSION
When the beta distribution is assumed for each part
dimension, four parameters: the alpha parameter, the beta
parameter, the process tolerance, and the middle range of
process tolerance must be assigned.
let
be the nominal dimension of part i
be the flatness tolerance zone of part i
be the process tolerance of part i
be the mean shift factor of part i
MX:
C
the middle range of part i
be the inverse process capability index
of part i
Referring to Figure 408, the nominal dimension
is effected by the mean shift factor (f,).
of part i (MX) is determined as follows:
MXi
+ fi(1-Ci) Ti
(X.,_)
The middle range
176
where
ti
ci
Ti/ 2
The mean and the variance of the beta distribution can be
determined as follows:
Let
be the minimum dimension of part i
be the maximum dimension of part i
E(Xj
VAR(X.)
be the expected value of part i
be the variance of part i
The middle range of part i is equal to the middle point of
interval a and b, and the process tolerance is equal to one
half of the range of interval [a,b]:
/irfi
ti
ai+bi
2
2
177
The expected value and the variance of part i are modified in
terms of part dimension as follows:
From Appendix B,
[(bi-ai) x
al
E ( Xi)
ai+bi
b1 -ai
2
2
+
ai
[(b2-ai)
x
since
ai+bi
M-X1
and
2
(bi-ai)
ti
2
Therefore
MX
E (
b.-ai
2
+
E (Xi)
=
MXi
+
2
+
-a )
ai
ai+Pi
2
211
x
al
aPi
"
2 ti
(d.1)
178
The varince:
VAR(Xi)
VAR(Xi)
4
ti aiNi
(CCa+Pi+l) (a2+132) 2
(bi-a1) 2 oci Ri
(ai÷Pi+1) (cci+P) 2
(d.2)
To investigate the behavior of the assembly sum (critical)
dimension, four parameters are determined.
1. The range of the sum dimension.
2. The middle range of the sum dimension.
3. The alpha parameter of the sum dimension.
4. The beta parameter of the sum dimension.
1. The range of the sum dimension
The range of the sum dimension is equal to the sum of
the range dimensions, or process tolerance limits of each
part component.
179
let
be the range of the sum dimension:
RAXE
E ti
n
Rd XE
where
2
1=1
n is the number of part dimensions.
2. The middle range of the sum dimension
The middle range of the sum dimension is equal to the
sum of the middle range of part dimension.
Let
be the middle range of the sum dimension.
MX11,
ItIXE R
1=1
since
MXi
=
Xn,1 + fi(1-Ci) Ti
then
E xn
i=i
and 4.
dimension
3.
fi(1 -Ci)Ti
E
i=i
The alpha and the beta parameters of the sum
If Xl and X, are random variables with joint pdf f(xl,x,)
then
E(X
+ Xi)
= E(X,)
+ E(X,)
180
If a, ,
a, ,
a, .
.
.
a, are constants and X X, .
.
.
.
X, are j ointly
distributed random variables, then
1k
k
E E aiXi)
=
\i =1
If
X:
E aiE(Xi)
1=1
(d.3)
j oint pdf f (xl ,
and X, are random variables with
then:
VAR (Xi
+
X2)
= VAR (X- )
+ VAR (X2) + 2 Coy(
VAR (X,
+ X2 )
= VAR (X: )
+ VAR (X, )
X2 )
and
whenever
X: and
Xi are independent.
It can also be verified that if
variables and
VAR(E
al,
X,,..,X
are random
a.; are constants, then
a2,
E a ITAR (Xi)
a iX
+ 2ZEajaiCov(xi, Xj)
Ri
and if X1,..,X, are independent, then
VAR
(E aiXi)
=
E calTAR (Xi)
(d.4)
Assuming that each part dimension is independent and the
assembly is linear, the
mean and the
variance of the sum
181
dimension can be determined by using equations
(d.3)
and
First, the alpha and the beta parameters of the sum
(d.4).
dimension are computed as follows:
let
ocE
be the alpha parameter of the sum dimension
be the beta parameter of the sum dimension
E(XE)
VARAIUm
be the mean of the sum dimension
be the variance of the sum dimension
From Appendix B, the mean and the variance can be found as
follows:
az
E(XE)
111XER
az+13z
11
RAX
(d. 5)
(RAXE)2azPz
VARA Xz
(CCE+13E+1) (CCE+PE) 2
(d.6)
Substituting equations (d.1) and (d.5) in equation (d.3):
E(XX)
=
EE(Xi)
1=1
(d.7)
182
Substituting equations (d.2) and (d.6) in equation
VARA XE
(d. 8)
E VAR (Xi)
.2=-1
where
is the number of part dimensions
n
Finally,
unknown
two
the
variables,
mathematically found (equations d.1, d.3,
and
ac
PE,
are
d.5 and d.7) as
follows:
RAXz
MXZ R
E (mxi
2=1
+
ccE
az4-13z
n
R
E NIX
1-1
2
RAX
and
[
ai+Pi
a,
E mxi
RA Xi
[ai+Pi
E
2=1.
2 ti
112 ti)
2
-12t.
2
2
183
then,
azi-Pz
az
RE
RAX
11
2
E 2 ti- E
i=1
i =1
a
azE R 1=3.
and
cc
as
ti
1=1 aiPi
ai
ci
2 ti
ti
2=-1
(d.9)
ti
i=1 -i+Pi
As discussed in Appendix B,
4)
2112 ti
may be kept constant (equal to
1
is assumed to be an integer not less than 4.
Equation d.9 results in:
E
az+13z
-
=-
(d.10)
+ 4 t=
i=2.
where
is an integer and not less than 4.
From equations (d.2),
(RAXE) 2 °COE
(ccE+pE+1)
(d.6), and (d.8):
(d.4),
(aE413E)2
4 taii3i
=
1=1
(ai+pi4-1) (cci+pi) 2
184
Since
then
= 4
(E 2 ti)
(ccE÷13E+1)
2
ccE
PE
n
16 1:
(az+PE) 2
(d.11)
t.fai
(ai+5) (ai+4)
2
To solve both (d.10) and (d.11), the following procedure is
followed:
let
n
2=1
L,
or
E ti
n
A
2
qi
n
t )2
2=3.
n
4_ 2
Li
qi+1
185
q.,
J, and M are substituted in equation d.10
CC
«.+P.
i
i
Et
ai+ 4
q
az
i
aEl-PE
az
CC i
-4
) t.-4)
n
n
c(E-13z
az
=
PE
Substituting q ,
i
4J
M
az+PE
az M
t
qi
i=1
M
ti
(M
4J) (az+PE)
4J
PP-4J
az
(d.12)
A, B, C, and M in equation (d.11):
n
E tij2 az Pz
2
i=1
(az+PE+1) (az+13E)2
n
ti ai
i=1
(ai+5) (ai+4) 2
16
186
m2azpz
(az+pz+i)
(c,E+p)2
m2aBpm
(ccE+PE÷3-)
4
4
(aE÷PE÷1) (aE+PE) 2
i=1
m2a0m
(C4Z+PZ+1) (CCE+13z)
2
5
n
5 tl
qi
t
(g-i+i) (qi) 2
i=1
(«E+13E) 2
m2ccEpE
(qi-4)
4
(qi+i)
2
4
t
2
1.
5
(q1
4 [5A-4B-5C]
2
Using equation (d.12), a and p parameters are solved:
4J
mt LixE
M-4J az
m44LT
\ 2
az)
4 [5A-4B-5C]
t2
187
(M-4J) 2J
az/v/A-M-4J
=
5A-4B-5C
let
K
=
4B
5A
5C
then
(M -4J) 2J
=
K az M+ KM- 4 JK
Finally,
ocE
(M-4J) 2J- (M-4(3) K
(d.13)
KM
and
PE
(M -4J) J2 -JK )
KM
(d.14)