Table Of ContentGunter Kirschling
Quality Assurance
and Tolerance
With 278 Figures
Springer-Verlag
Berlin Heidelberg NewY ork
London Paris Tokyo
Hong Kong Barcelona Budapest
Professor Dr.-lng, Gunter Kirscbling
Universitiit Kassel. FRG
Translatedfrom thefirsl German Edition 'Toleranzen und
QUalifiitssicherung" (Springer-Verlag) by
Sue Greener. Springfield. Five Ash Down. Uckfield.
Easl Sussex TN12 3AH. England.
ISBN 978-3-642-48691-3 ISBN 978-3-642-48689-0 (eBook)
DOl 10,1007/978-3-642-48689-0
l.ibrary of Cong~ Cataloging_in-Pub!;"ation Data
Kirschling, Glinlu. [Qualiliilssicherung und Toleranzen. English] QualilY assurance and
lolerances I Giinler Kirschling. p. cm. Translalion ot: Qua1iliilssicherung und TQleranzen.
lndudes bibliographiQIl refel"(:n<;U and index,
rSBN-13: 978-3-642-48691-3
I. Qualily oonl.ol- Statistical methl)ds.
I. Title. TSIS6.K49213 1991 658S62'OI519S- dc20.
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10 Springer-Verlag llerlin Heidelberg 1991
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Foreword of the First German Edition
It has been well known for decades that it is possible to calculate the tolerances
of single dimensions within chains using statistical criteria. However, in practice,
this is rarely done.
Measurement tolerances which are part of a series are almost exclusively
calculated using conventional arithmetic. This is generally preferred, because
the statistical method for calculating tolerances is complicated and difficult to
comprehend.
This book therefore offers a comprehensive presentation of tolerance
problems and their solution by statistical methods, successfully presenting all
calculated solutions in clear figures or graphical form. These figures consist of
with additional material on tolerance zones and information on the distribution
assumptions of position, shape and width with further information on respective
proportional non-conformities.
Trials with students and engineers in industry have shown that the use of
these figures, as an aid to exact calculated solutions, make the approach to this
complicated matter considerably easier.
This book is particularly appropriate for those working on the one hand
in the field of development and construction and on the other hand in production
and quality control, especially in mechanical engineering and related fields.
Readers in the quality assurance field who are familiar with the underlying
principles of statistics with quantitative characteristics will have no difficulty
in understanding the contents of this book. For readers who have either no
knowledge or an insufficient knowledge of statistics we have provided Chaps. 2
to 5. These set out the theoretical principles of statistics necessary for an
understanding of the tolerance analyses and process analyses in following
chapter and are illustrated with practical examples.
Tolerance problems involve not only the accurate calculation of tolerances
with respect to safeguarding the function of the dimensional chains, but also
their realisation in production. This theme is dealt with extensively in
Chaps. 13 to 15 and should be of particular interest to those who worK in
Production areas and in Quality Assurance.
Modern machining processes are distinguished by their high efficiency levels
with the result that these processes show significant trends with machine tool
wear. Because of that the familiar technical expertise on process control has to
be extended in many ways using quality control charts see Chap. 15.
VI Foreword of the First German Edition
It is strongly recommended that the reader keeps constantly to hand a
scientific calculator with xis-automatic facility, so as to be able to calculate
mathematically the solutions given in examples and exercises. For this reason
all calculated parameters, characteristics and estimate rates are given to six or
more decimal places, a "precision" which is not necessary or practicable either
in theory or in practice but which allows readers to check the accuracy of their
calculations.
Since the literature offers little on the subjects of "process capability process
control and process correction" and what is offered is partly contradictory, in
many respects odd definitions and solutions may be found. I am conscious
therefore that some figures and examples may contain obscurities or deficiencies;
I am always grateful for advice on these.
I would like to express sincere thanks to Prof. Dr.-Ing. P. Th. Wilrich, Freie
Universitat Berlin, for his critical examination of the manuscript and for his
suggestions for improvement.
I would also like to thank Springer-Verlag for their interest in the production
of this book.
Melsungen, 1988 Gunter Kirschling
Contents
Symbols x
1 Introduction
2 Concepts 3
3 Probability Calculation 8
3.1 Definition of Probability 8
3.2 Addition Formula (OR Formula) 9
3.3 Multiplication Formula (AND Formula) 9
4 Probability Distributions of Quantitative Characteristics 11
4.1 General 11
4.2 Normal Distribution 12
4.3 Abnormal Probability Distributions 19
4.4 Random Spread for Mean Values 25
5 Mathematical and Graphical Evaluation of Dimensional Series 28
5.1 General 28
5.2 Mathematical Evaluation Without Classification 29
5.3 Graphical Evaluation Without Classification 30
5.4 Mathematical Evaluation of Classified Dimensional Series 31
5.5 Graphical Evaluation of Classified Dimensional Series 34
5.6 Other Statistical Characteristics . . . . 36
5.7 The Significance of Statistical Characteristics 40
6 Mixed Distributions 43
6.1 Types of Mixed Distribution 43
6.2 Mixed Distribution Type 1 43
6.3 Mixed Distribution Type 2 48
6.4 TschebyschefT's Extreme Case 67
VIII Contents
7 Distribution Folds 71
7.1 Description of Fold Methods 71
7.2 Examples of Folding Methods 73
8 Relationship Between Tolerances and Production Distributions 97
8.1 Direct and Indirect Function Characteristics 97
8.2 Establishment and Observance Tolerances 99
8.3 Tolerances and Costs . . . . . 101
8.4 Errors and Waste 102
8.5 Arithmetical Tolerance Calculation 103
8.6 Examples for Arithmetically Calculated Linear Dimensional
Chains . . . . . . . . . . . 106
8.7 The Sorting of Non-Conforming Parts 115
9 Tolerance Calculation Using Square Law . 132
10 Statistical Calculation of Tolerances of Individual Values
with Rectangular Distributions. . . . . . . . . 142
10.1 General 142
10.2 Derivation of Reduction and Expansion Factors
with Equal Individual Tolerances ..... 142
10.3 Calculation of Dimensional Chains with Unequal Individual
Tolerances and Rectangular Individual Distributions 148
10.4 Advantages of the Assumption of the Presence
of Rectangular Distributions . . . . . . 152
11 Statistical Tolerance Calculation of Individual Values
with Trapezoid Distribution or with Triangular Distribution 156
11.1 Method of Calculation 156
11.2 Simple Formula to Check Fixed Linear Dimensional Chains 172
12 The Tolerance Model 176
13 Assessment of Completed Batches and Production Batches 182
13.1 General 182
13.2 Random Sample Dimension Tests ..... . 184
13.3 Selection of AQL Values for Random Sample Tests
of Dimensions (Variable Testing) 190
13.4 Quality Control Charts for Dimensions 216
14 Process Analysis and Process Capability 229
14.1 Process Analysis 229
14.2 Process Capability 250
Contents IX
15 Process Control 257
15.1 General Aspects ............ 257
15.2 Process Control by Mean Value Quality Control Charts 259
15.3 Process Control by Original Value QCC's 280
15.4 Process Correction 292
16 Bibliography 318
17 Appendix 321
Tables 321
Subject Index 331
Symbols
d anticipated value of R/(J
n
EG cut off point
N range of basic totals or of clearances or of a probability distribution
as a result of a simulated discrete distribution using integral x-values
Nominal size
N extent of the j-th class in a probability distribution
j
n random sample test range
extent of the j-th class in a random sample test
nj
p margin of error (exceeding limit value) in basic totals or in a probability
distribution
p margin of error in the random sample test
G(u) area value (probability total) from - 00 to u
P a probability of acceptability
Q(u) area value from u to + 00
u standardised, normally distributed random sizes (/1 = 0; (J = 1)
x characteristic
Xi i-th observed value of X
middle value of the j-th class
Xj
g(x) probability of value x
G(x) probability total up to and including x.
Other symbols in Table 2.1 and in the text.
1 Introduction
Employees working in areas of development and design as well as those in
production and quality control are constantly confronted by the fact that it is
impossible to manufacture technical products in accordance with exact theore
tical dimensions. In fact -depending on the variability of technical resources
some deviations from the theoretical dimensions are unavoidable, and these
deviation may be either assignable (systematic) or unassignable (random).
This is especially true for technical products which are manufactured in
large batches or mass produced. At the same time, these products must be
interchangeable, i.e. they must fulfill their given function despite unspecified
mating in subsequent assembly or fabrication processes.
The designer must take account of unavoidable deviations by specifying
direct or indirect limit values for all dimensions, between which the actual
dimensions of the component must lie. In other words:
Each dimension has tolerances.
The selection oftolerances presents no problem to the designer if close tolerances
i.e. binomial dimensional chains are involved. For decades there has been the
ISO tolerance system for tolerances on fit, and this has been updated and
expanded by many national and international standards. In this system, many
concepts and their interrelationships are specified. Also specified are the values
for tolerances (tolerance groups) in relation to nominal value ranges (tolerance
progressions). Tolerances are also listed by zone.
It is also straightforward for the designer to specify tolerances when
dimensions do not interact with other dimensions (indirect working dimensions
more exact definitions later). These dimensions are governed by general tole
rances which are set as standard.
There is a problem, however, in selecting tolerances according to the value
and position of the tolerance range when there are three or more components
in the dimensional chain. In these cases, there are no standards and the designer
has to rely on his own judgement, often without knowing the decision criteria.
Consequently many designers apply the principle:
"Tolerances should be as wide as possible and as narrow as necessary"
with the rider:
"if in doubt, better too narrow"
and there usually is some doubt.
