GEAR GEOMETRY AND APPLIED THEORY Second Edition docx

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GEAR GEOMETRY AND APPLIED THEORY
Second Edition
Revised and expanded, Gear Geometry and Applied Theory, 2nd edition, cov-
ers the theory, design, geometry, and manufacture of all types of gears and gear
drives. Gear Geometry and Applied Theory is an invaluable reference for de-
signers, theoreticians, students, and manufacturers. This new edition includes
advances in gear theory, gear manufacturing, and computer simulation. Among
the new topics are (1) new geometry for modified spur and helical gears, face-gear
drives, and cycloidal pumps; (2) new design approaches for one-stage planetary
gear trains and spiral bevel gear drives; (3) an enhanced approach for stress
analysis of gear drives with FEM; (4) new methods of grinding face-gear drives,
generating double-crowned pinions, and generating new types of helical gears;
(5) broad application of simulation of meshing and TCA; and (6) new theories on
the simulation of meshing for multi-body systems, detection of cases wherein the
contact lines on generating surfaces may have their own envelope, and detection
and avoidance of singularities of generated surfaces.
Faydor L. Litvin is Director of the Gear Research Center and Distinguished
Professor Emeritus in the Department of Mechanical and Industrial Engineering,
University of Illinois at Chicago. He holds patents for twenty-five inventions, and
he was recognized as Inventor of the Year by the University of Illinois at Chicago
in 2001.
Alfonso Fuentes is Associate Professor of Mechanical Engineering at the
Polytechnic University of Cartagena.
i

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Gear Geometry and Applied Theory
SECOND EDITION
Faydor L. Litvin
University of Illinois at Chicago
Alfonso Fuentes
Polytechnic University of Cartagena
iii
  
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge  , UK
First published in print format
- ----
- ----
© Faydor L. Litvin and Alfonso Fuentes 2004
2004
Information on this title: www.cambrid
g
e.or
g
/9780521815178
This publication is in copyright. Subject to statutory exception and to the provision of
relevant collective licensing agreements, no reproduction of any part may take place
without the written permission of Cambridge University Press.
- ---

- ---
Cambridge University Press has no responsibility for the persistence or accuracy of s
for external or third-party internet websites referred to in this publication, and does not
guarantee that any content on such websites is, or will remain, accurate or appropriate.
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
hardback
eBook (EBL)
eBook (EBL)
hardback
  
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge  , UK
First published in print format
- ----
- ----
© Faydor L. Litvin and Alfonso Fuentes 2004
2004
Information on this title: www.cambrid
g
e.or
g
/9780521815178
This publication is in copyright. Subject to statutory exception and to the provision of
relevant collective licensing agreements, no reproduction of any part may take place
without the written permission of Cambridge University Press.
- ---
- ---
Cambridge University Press has no responsibility for the persistence or accuracy of s

for external or third-party internet websites referred to in this publication, and does not
guarantee that any content on such websites is, or will remain, accurate or appropriate.
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
hardback
eBook (EBL)
eBook (EBL)
hardback
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Contents
Foreword by Graziano Curti page xii
Preface xiv
Acknowledgments xv
1 Coordinate Transformation
1
1.1 Homogeneous Coordinates 1
1.2 Coordinate Transformation in Matrix Representation 2
1.3 Rotation About an Axis 6
1.4 Rotational and Translational 4 ×4 Matrices 14
1.5 Examples of Coordinate Transformation 15
1.6 Application to Derivation of Curves 24
1.7 Application to Derivation of Surfaces 28
2 Relative Velocity 33
2.1 Vector Representation 33
2.2 Matrix Representation 39
2.3 Application of Skew-Symmetric Matrices 41
3 Centrodes, Axodes, and Operating Pitch Surfaces 44
3.1 The Concept of Centrodes 44
3.2 Pitch Circle 49

3.3 Operating Pitch Circles 50
3.4 Axodes in Rotation Between Intersected Axes 51
3.5 Axodes in Rotation Between Crossed Axes 52
3.6 Operating Pitch Surfaces for Gears with Crossed Axes 56
4 Planar Curves 59
4.1 Parametric Representation 59
4.2 Representation by Implicit Function 60
4.3 Tangent and Normal to a Planar Curve 60
4.4 Curvature of Planar Curves 68
5 Surfaces 78
5.1 Parametric Representation of Surfaces 78
5.2 Curvilinear Coordinates 78
5.3 Tangent Plane and Surface Normal 79
v
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5.4 Representation of a Surface by Implicit Function 82
5.5 Examples of Surfaces 82
6 Conjugated Surfaces and Curves 97
6.1 Envelope to a Family of Surfaces: Necessary Conditions
of Existence 97
6.2 Basic Kinematic Relations 102
6.3 Conditions of Nonundercutting 103
6.4 Sufficient Conditions for Existence of an Envelope
to a Family of Surfaces 107
6.5 Contact Lines; Surface of Action 110
6.6 Envelope to Family of Contact Lines on Generating
Surface 
1

112
6.7 Formation of Branches of Envelope to Parametric
Families of Surfaces and Curves 114
6.8 Wildhaber’s Concept of Limit Contact Normal 118
6.9 Fillet Generation 119
6.10 Two-Parameter Enveloping 124
6.11 Axes of Meshing 128
6.12 Knots of Meshing 134
6.13 Problems 137
7 Curvatures of Surfaces and Curves 153
7.1 Introduction 153
7.2 Spatial Curve in 3D-Space 153
7.3 Surface Curves 164
7.4 First and Second Fundamental Forms 175
7.5 Principal Directions and Curvatures 180
7.6 Euler’s Equation 188
7.7 Gaussian Curvature; Three Types of Surface
Points 189
7.8 Dupin’s Indicatrix 193
7.9 Geodesic Line; Surface Torsion 194
8 Mating Surfaces: Curvature Relations, Contact Ellipse 202
8.1 Introduction 202
8.2 Basic Equations 203
8.3 Planar Gearing: Relation Between Curvatures 204
8.4 Direct Relations Between Principal Curvatures
of Mating Surfaces 218
8.5 Direct Relations Between Normal Curvatures
of Mating Surfaces 226
8.6 Diagonalization of Curvature Matrix 231
8.7 Contact Ellipse 234

9 Computerized Simulation of Meshing and Contact 241
9.1 Introduction 241
9.2 Predesign of a Parabolic Function of Transmission
Errors 242
9.3 Local Synthesis 245
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9.4 Tooth Contact Analysis 249
9.5 Application of Finite Element Analysis for Design
of Gear Drives 257
9.6 Edge Contact 260
10 Spur Involute Gears 267
10.1 Introduction 267
10.2 Geometry of Involute Curves 268
10.3 Generation of Involute Curves by Tools 273
10.4 Tooth Element Proportions 278
10.5 Meshing of Involute Gear with Rack-Cutter 280
10.6 Relations Between Tooth Thicknesses Measured
on Various Circles 285
10.7 Meshing of External Involute Gears 287
10.8 Contact Ratio 292
10.9 Nonstandard Gears 294
11 Internal Involute Gears 304
11.1 Introduction 304
11.2 Generation of Gear Fillet 305
11.3 Conditions of Nonundercutting 309
11.4 Interference by Assembly 314
12 Noncircular Gears 318
12.1 Introduction 318

12.2 Centrodes of Noncircular Gears 318
12.3 Closed Centrodes 323
12.4 Elliptical and Modified Elliptical Gears 326
12.5 Conditions of Centrode Convexity 329
12.6 Conjugation of an Eccentric Circular Gear with
a Noncircular Gear 330
12.7 Identical Centrodes 331
12.8 Design of Combined Noncircular Gear Mechanism 333
12.9 Generation Based on Application of Noncircular
Master-Gears 335
12.10 Enveloping Method for Generation 336
12.11 Evolute of Tooth Profiles 341
12.12 Pressure Angle 344
Appendix 12.A: Displacement Functions for Generation
by Rack-Cutter 345
Appendix 12.B: Displacement Functions for Generation
by Shaper 348
13 Cycloidal Gearing 350
13.1 Introduction 350
13.2 Generation of Cycloidal Curves 350
13.3 Equations of Cycloidal Curves 354
13.4 Camus’ Theorem and Its Application 355
13.5 External Pin Gearing 359
13.6 Internal Pin Gearing 365
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13.7 Overcentrode Cycloidal Gearing 367
13.8 Root’s Blower 369
14 Involute Helical Gears with Parallel Axes 375

14.1 Introduction 375
14.2 General Considerations 375
14.3 Screw Involute Surface 377
14.4 Meshing of a Helical Gear with a Rack 382
14.5 Meshing of Mating Helical Gears 392
14.6 Conditions of Nonundercutting 396
14.7 Contact Ratio 398
14.8 Force Transmission 399
14.9 Results of Tooth Contact Analysis (TCA) 402
14.10 Nomenclature 403
15 Modified Involute Gears 404
15.1 Introduction 404
15.2 Axodes of Helical Gears and Rack-Cutters 407
15.3 Profile-Crowned Pinion and Gear Tooth Surfaces 411
15.4 Tooth Contact Analysis (TCA) of Profile-Crowned
Pinion and Gear Tooth Surfaces 414
15.5 Longitudinal Crowning of Pinion by a Plunging Disk 419
15.6 Grinding of Double-Crowned Pinion by a Worm 424
15.7 TCA of Gear Drive with Double-Crowned Pinion 430
15.8 Undercutting and Pointing 432
15.9 Stress Analysis 435
16 Involute Helical Gears with Crossed Axes
441
16.1 Introduction 441
16.2 Analysis and Simulation of Meshing of Helical Gears 443
16.3 Simulation of Meshing of Crossed Helical Gears 452
16.4 Generation of Conjugated Tooth Surfaces of Crossed
Helical Gears 455
16.5 Design of Crossed Helical Gears 458
16.6 Stress Analysis 465

Appendix 16.A: Derivation of Shortest Center Distance for
Canonical Design 467
Appendix 16.B: Derivation of Equation of Canonical Design
f (γ
o
,α
on
,λ
b1
,λ
b2
) = 0 472
Appendix 16.C: Relations Between Parameters α
pt
and α
pn
473
Appendix 16.D: Derivation of Equation (16.5.5) 473
Appendix 16.E: Derivation of Additional Relations Between
α
ot1
and α
ot2
474
17 New Version of Novikov–Wildhaber Helical Gears 475
17.1 Introduction 475
17.2 Axodes of Helical Gears and Rack-Cutter 478
17.3 Parabolic Rack-Cutters 479
17.4 Profile-Crowned Pinion and Gear Tooth Surfaces 482
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17.5 Tooth Contact Analysis (TCA) of Gear Drive with
Profile-Crowned Pinion 485
17.6 Longitudinal Crowning of Pinion by a Plunging Disk 487
17.7 Generation of Double-Crowned Pinion by a Worm 491
17.8 TCA of a Gear Drive with a Double-Crowned Pinion 497
17.9 Undercutting and Pointing 500
17.10 Stress Analysis 502
18 Face-Gear Drives 508
18.1 Introduction 508
18.2 Axodes, Pitch Surfaces, and Pitch Point 510
18.3 Face-Gear Generation 512
18.4 Localization of Bearing Contact 512
18.5 Equations of Face-Gear Tooth Surface 515
18.6 Conditions of Nonundercutting of Face-Gear Tooth
Surface (Generated by Involute Shaper) 519
18.7 Pointing of Face-Gear Teeth Generated by Involute
Shaper 522
18.8 Fillet Surface 524
18.9 Geometry of Parabolic Rack-Cutters 525
18.10 Second Version of Geometry: Derivation of Tooth
Surfaces of Shaper and Pinion 527
18.11 Second Version of Geometry: Derivation of Face-Gear
Tooth Surface 529
18.12 Design Recommendations 529
18.13 Tooth Contact Analysis (TCA) 531
18.14 Application of Generating Worm 535
18.15 Stress Analysis 541
19 Worm-Gear Drives with Cylindrical Worms 547

19.1 Introduction 547
19.2 Pitch Surfaces and Gear Ratio 548
19.3 Design Parameters and Their Relations 552
19.4 Generation and Geometry of ZA Worms 557
19.5 Generation and Geometry of ZN Worms 561
19.6 Generation and Geometry of ZI (Involute) Worms 574
19.7 Geometry and Generation of K Worms 581
19.8 Geometry and Generation of F-I Worms (Version I) 590
19.9 Geometry and Generation of F-II Worms (Version II) 597
19.10 Generalized Helicoid Equations 601
19.11 Equation of Meshing of Worm and Worm-Gear
Surfaces 603
19.12 Area of Meshing 606
19.13 Prospects of New Developments 609
20 Double-Enveloping Worm-Gear Drives 614
20.1 Introduction 614
20.2 Generation of Worm and Worm-Gear Surfaces 614
20.3 Worm Surface Equations 618
20.4 Equation of Meshing 620
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20.5 Contact Lines 622
20.6 Worm-Gear Surface Equations 622
21 Spiral Bevel Gears 627
21.1 Introduction 627
21.2 Basic Ideas of the Developed Approach 628
21.3 Derivation of Gear Tooth Surfaces 633
21.4 Derivation of Pinion Tooth Surface 644
21.5 Local Synthesis and Determination of Pinion

Machine-Tool Settings 649
21.6 Relationships Between Principal Curvatures and
Directions of Mating Surfaces 656
21.7 Simulation of Meshing and Contact 661
21.8 Application of Finite Element Analysis for the Design
of Spiral Bevel Gear Drives 665
21.9 Example of Design and Optimization of a Spiral Bevel
Gear Drive 666
21.10 Compensation of the Shift of the Bearing Contact 676
22 Hypoid Gear Drives 679
22.1 Introduction 679
22.2 Axodes and Operating Pitch Cones 679
22.3 Tangency of Hypoid Pitch Cones 680
22.4 Auxiliary Equations 682
22.5 Design of Hypoid Pitch Cones 685
22.6 Generation of Face-Milled Hypoid Gear Drives 690
23 Planetary Gear Trains 697
23.1 Introduction 697
23.2 Gear Ratio 697
23.3 Conditions of Assembly 703
23.4 Phase Angle of Planet Gears 707
23.5 Efficiency of a Planetary Gear Train 709
23.6 Modifications of Gear Tooth Geometry 711
23.7 Tooth Contact Analysis (TCA) 712
23.8 Illustration of the Effect of Regulation of Backlash 716
24 Generation of Helicoids 718
24.1 Introduction 718
24.2 Generation by Finger-Shaped Tool: Tool Surface is
Given 718
24.3 Generation by Finger-Shaped Tool: Workpiece Surface

is Given 723
24.4 Generation by Disk-Shaped Tool: Tool Surface is Given 726
24.5 Generation by Disk-Shaped Tool: Workpiece Surface is
Given 730
25 Design of Flyblades 734
25.1 Introduction 734
25.2 Two-Parameter Form Representation of Worm Surfaces 735
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25.3 Three-Parameter Form Representation of Worm
Surfaces 737
25.4 Working Equations 738
26 Generation of Surfaces by CNC Machines 746
26.1 Introduction 746
26.2 Execution of Motions of CNC Machines 747
26.3 Generation of Hypoid Pinion 750
26.4 Generation of a Surface with Optimal Approximation 752
27 Overwire (Ball) Measurement 769
27.1 Introduction 769
27.2 Problem Description 769
27.3 Measurement of Involute Worms, Involute Helical
Gears, and Spur Gears 773
27.4 Measurement of Asymmetric Archimedes Screw 779
28 Minimization of Deviations of Gear Real Tooth Surfaces 782
28.1 Introduction 782
28.2 Overview of Measurement and Modeling Method 783
28.3 Equations of Theoretical Tooth Surface 
t
784

28.4 Coordinate Systems Used for Coordinate
Measurements 785
28.5 Grid and Reference Point 786
28.6 Deviations of the Real Surface 787
28.7 Minimization of Deviations 787
References
789
Index 795
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Foreword
The main topics of the book are the theory of gearing, computerized design, generation,
simulation of meshing, and stress analysis of gear drives. The first edition of the book
is already considered the leading reference in the field by the engineering community,
but this edition complements the first with new chapters and thoughtful revision of the
previous version, which will make it very useful for the design and manufacture of gear
drives.
New ideas of gear design presented in the book include:
(1) Development of gear drives with improved bearing contact, reduced sensitivity
to misalignment, and reduced transmission errors and vibration. These goals are
achieved by (i) simultaneous application of local synthesis of gear drives and com-
puterized simulation of meshing and contact and (ii) application of a predesigned
parabolic function of transmission errors that is able to absorb linear functions of
transmission errors caused by misalignments.
(2) Development of enhanced finite element analysis of stresses with the following
features: (i) the contacting model of teeth is developed automatically, on the basis
of analytical representation of equations of tooth surfaces; (ii) the formation of
bearing contact is investigated for several pairs of teeth in order to detect and
avoid areas of severe contact stresses.
(3) Improved conditions of load distribution in planetary gear trains by modification

of the applied geometry and regulation of installment of planet gears on the carrier.
New approaches are presented for gear manufacture that enable (i) grinding of face-
gear drives by application of a grinding worm of a special shape and (ii) design and
manufacture of new types of helical gears with double-crowned pinions for obtaining
localization of bearing contact and reduction of transmission errors.
The developed theory of gearing presented in the book will make the authors the
experts in this area. The book includes the solution to the following important complex
problems:
(i) development of new approaches for determination of an envelope to the family of
surfaces including the formation of the envelope by two branches;
(ii) avoidance of singularities of tooth surfaces and undercutting in the process of
generation; and
xii
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Foreword xiii
(iii) simplification of the contacting problem by a new approach for the determination
of principal curvatures and directions of an envelope.
The developed ideas have been applied to the design of gear drives, including a new
version of Wildhaber–Novikov helical gear drives, spiral bevel gears, and worm-gear
drives. Computerized simulation of meshing and contact and testing of prototypes of
gear drives have confirmed the effectiveness of the ideas presented in the book. Three
patents for new manufacturing approaches have been obtained by Professor Faydor L.
Litvin and representatives of gear companies.
The main ideas in the book have been developed by the authors and their associates at
the Gear Research Center of the University of Illinois at Chicago. They have also been the
subject of a great number of international publications of permanent interest. Thanks to
the wonderful leadership of Professor Faydor L. Litvin, who is universally well known
in the field of gears, this Center has involved representatives of various universities in
the United States, Italy, Spain, and Japan in gear research. The publication of this book

will certainly enhance the education and training of engineers in the area of gear theory
and design of gear transmissions.
Prof. Eng. Graziano Curti
Politecnico di Torino, Italy
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Preface
The contents of the second edition of the book have been thoroughly revised and sub-
stantially augmented in comparison with the first edition of 1994.
New topics in the second edition include the following new developments:
(1) A new geometry of modified spur gears, helical gears with parallel and crossed
axes, a new version of Novikov–Wildhaber helical gears, a new geometry of face-
gear drives, geometry of cycloidal pumps, a new approach for design of one-stage
planetary gear trains with improved conditions of load distribution, and a new
approach for design of spiral bevel gear drives with a reduced level of noise and
vibration and improved bearing contact.
(2) Development of an enhanced approach for stress analysis of gear drives by applica-
tion of the finite element method. The advantage of the developed approach is the
analytical design of the contacting model based on the analytical representation of
the gear tooth surfaces.
(3) Development of a new method of grinding of face-gear drives, new methods of
generation of double-crowned pinions for localization of the bearing contact and
reduction of transmission errors, and application of modified roll for reduction of
transmission errors.
(4) Broad application of simulation of meshing and tooth contact analysis (TCA) for
determination of the influence of errors of alignment on transmission errors and
shift of the bearing contact. This approach has been applied for almost all types of
gear drives discussed in the book.
(5) The authors have contributed to the development of the modern theory of gearing.
In particular, they have developed in this new edition of the book (i) formation of an

envelope by two branches, (ii) an extension of simulation of meshing for multi-body
systems, (iii) detection of cases wherein the contact lines on the generating surface
may have their own envelope, and (iv) detection and avoidance of singularities of
generated surfaces (for avoidance of undercutting during the process of generation).
The authors are grateful to the companies and institutions that have supported their
research and to the members of the Gear Research Center of the University of Illinois
at Chicago who tested their ideas as co-authors of joint papers (see Acknowledgments).
xiv
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Acknowledgments
The authors express their deep gratitude to the institutions and companies that have
supported their research and to colleagues and co-authors of accomplished research
projects. The following list of names only partially covers those to whom the authors
are obliged for their valuable help and inspiration:
(1) Dr. John J. Coy, Assistant Director, NASA Airspace System Program, NASA
Ames Research Center; formerly Manager, Mechanical Components Branch, NASA
Glenn Research Center
(2) Dr. Robert Bill, Director; James Zakrajsak, Transmission Chief; Dr. Robert F. Hand-
schuh, Senior Researcher; NASA John Glenn Research Center and Army Research
Laboratory
(3) Dr. Gary L. Anderson, U.S. Army Research Office
(4) James S. Gleason, Chairman, The Gleason Corporation; Gary J. Kimmet, Vice-
President, Worldwide Sales & Marketing, The Gleason Corporation; Ralph E.
Harper, Secretary and Treasurer, The Gleason Corporation; John V. Thomas,
Director of Gear Technology, The Gleason Works
(5) Ryuichi Yamashita, Vice-President; Kenichi Hayasaka, Manager, Gear Research
and Development Group; Yamaha Motor Co., Japan
(6) Terrel W. Hansen, Manager; Robert J. King, former Manager; Gregory F. Heath,
Project Engineer; The Boeing Company – McDonnell Douglas Helicopter Systems

(7) Dr. Robert B. Mullins, Director of Engineering; Ron Woods, Technical Resource
Specialist; Bell Helicopter Textron
(8) Edward Karedes, Chief – Transmissions; Bruce Hansen, Manager, Research and
Development; Sikorsky Aircraft Corporation
(9) Daniel V. Sagady P.E., Vice-President, Engineering & Product Development;
Theresa M. Barrett, Executive Engineer; Dr. Mauro De Donno, Area Manager –
Gears (Guanajuato Gear & Axle); Dr. Jui S. Chen, Senior Engineer Gear Design;
American Axle & Manufacturing
(10) Tom M. Sep, Senior Technical Fellow; Visteon Corporation
xv
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xvi Acknowledgments
(11) Matt Hawkins, Gear Specialist; Rolls-Royce Corporation (in Indiana)
(12) Daniele Vecchiato, Gear Research Center for UIC
(13) Ignacio Gonz
´
alez-P
´
erez, Gear Research Center for UIC
And the following scholars formerly associated with the Gear Research Center of UIC:
(14) Dr. C.B. Patrick Tsay
(15) Dr. Wei-Jiung Tsung
(16) Dr. Sergei A. Lagutin
(17) Dr. Wei-Shing Chaing
(18) Dr. Ningxin Chen
(19) Dr. Andy Feng
(20) Dr. Yi Zhang
(21) Dr. Chinping Kuan
(22) Dr. Yyh-Chiang Wang

(23) Dr. Jian Lu
(24) Dr. Hong-Tao Lee
(25) Dr. Chun-Liang Hsiao
(26) Dr. Vadim Kin
(27) Dr. Inwan Seol
(28) Dr. David Kim
(29) Dr. Shawn Zhao
(30) Dr. Anngwo Wang
(31) Giuseppe Argentieri
(32) Alberto Demenego
(33) Dr. Kazumasa Kawasaki
(34) Dr. Qi Fan
(35) Claudio Zanzi
(36) Matteo Pontiggia
(37) Alessandro Nava
(38) Luca Carnevali
(39) Alessandro Piscopo
(40) Paolo Ruzziconi
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1 Coordinate Transformation
1.1 HOMOGENEOUS COORDINATES
A position vector in a three-dimensional space (Fig. 1.1.1) may be represented (i) in
vector form as
r
m
= O
m
M = x
m

i
m
+ y
m
j
m
+ z
m
k
m
(1.1.1)
where (i
m
, j
m
, k
m
) are the unit vectors of coordinate axes, and (ii) by the column matrix
r
m
=


x
m
y
m
z
m



. (1.1.2)
The subscript “m” indicates that the position vector is represented in coordinate system
S
m
(x
m
, y
m
, z
m
). To save space while designating a vector, we will also represent the
position vector by the row matrix,
r
m
=
[
x
m
y
m
z
m
]
T
. (1.1.3)
The superscript “T” means that r
T
m
is a transpose matrix with respect to r

m
.
A point – the end of the position vector – is determined in Cartesian coordinates with
three numbers: x, y, z. Generally, coordinate transformation in matrix operations
needs mixed matrix operations where both multiplication and addition of matrices
must be used. However, only multiplication of matrices is needed if position vectors are
represented with homogeneous coordinates. Application of such coordinates for
coordinate transformation in theory of mechanisms has been proposed by Denavit &
Hartenberg [1955] and by Litvin [1955]. Homogeneous coordinates of a point in a three-
dimensional space are determined by four numbers (x
∗
, y
∗
, z
∗
, t
∗
) which are not equal
to zero simultaneously and of which only three are independent. Assuming that t
∗
= 0,
ordinary coordinates and homogeneous coordinates may be related as follows:
x =
x
∗
t
∗
y =
y
∗

t
∗
z =
z
∗
t
∗
. (1.1.4)
1
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2 Coordinate Transformation
Figure 1.1.1: Position vector in Cartesian coordi-
nate system.
With t
∗
= 1, a point may be specified by homogeneous coordinates such as (x, y, z, 1),
and a position vector may be represented by
r
m
=




x
m
y
m
z

m
1




or
r
m
=
[
x
m
y
m
z
m
1
]
T
.
1.2 COORDINATE TRANSFORMATION IN MATRIX REPRESENTATION
Consider two coordinate systems S
m
(x
m
, y
m
, z
m

) and S
n
(x
n
, y
n
, z
n
) (Fig. 1.2.1). Point
M is represented in coordinate system S
m
by the position vector
r
m
=
[
x
m
y
m
z
m
1
]
T
. (1.2.1)
The same point M can be determined in coordinate system S
n
by the position vector
r

n
=
[
x
n
y
n
z
n
1
]
T
(1.2.2)
with the matrix equation
r
n
= M
nm
r
m
. (1.2.3)
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1.2 Coordinate Transformation in Matrix Representation 3
Figure 1.2.1: Derivation of coordinate transforma-
tion.
Matrix M
nm
is represented by
M

nm
=






a
11
a
12
a
13
a
14
a
21
a
22
a
23
a
24
a
31
a
32
a
33

a
34
0001






=






(i
n
· i
m
)(i
n
· j
m
)(i
n
· k
m
)(O
n

O
m
· i
n
)
(j
n
· i
m
)(j
n
· j
m
)(j
n
· k
m
)(O
n
O
m
· j
n
)
(k
n
· i
m
)(k
n

· j
m
)(k
n
· k
m
)(O
n
O
m
· k
n
)
00 0 1






=






cos(

x

n
, x
m
) cos(

x
n
, y
m
) cos(

x
n
, z
m
) x
(O
m
)
n
cos(

y
n
, x
m
) cos(

y
n

, y
m
) cos(

y
n
, z
m
) y
(O
m
)
n
cos(

z
n
, x
m
) cos(

z
n
, y
m
) cos(

z
n
, z

m
) z
(O
m
)
n
0001






. (1.2.4)
Here, (i
n
, j
n
, k
n
) are the unit vectors of the axes of the “new” coordinate system;
(i
m
, j
m
, k
m
) are the unit vectors of the axes of the “old” coordinate system; O
n
and

O
m
are the origins of the “new” and “old” coordinate systems; subscript “nm” in the
designation M
nm
indicates that the coordinate transformation is performed from S
m
to
P1: JYT
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4 Coordinate Transformation
S
n
. The determination of elements a
lk
(k = 1, 2, 3; l = 1, 2, 3) of matrix M
nm
is based
on the following rules:
(i) Elements of the 3 × 3 submatrix
L
nm
=


a
11
a
12
a

13
a
21
a
22
a
23
a
31
a
32
a
33


(1.2.5)
represent the direction cosines of the “old” unit vectors (i
m
, j
m
, k
m
) in the “new”
coordinate systems S
n
. For instance, a
21
= cos(

y

n
, x
m
), a
32
= cos(

z
n
, y
m
), and so
on. The subscripts of elements a
kl
in matrix (1.2.5) indicate the number l of the
“old” coordinate axis and the number k of the “new” coordinate axis. Axes x, y, z
are given numbers 1, 2, and 3, respectively.
(ii) Elements a
14
, a
24
, and a
34
represent the “new” coordinates x
(O
m
)
n
, y
(O

m
)
n
, z
(O
m
)
n
of
the “old” origin O
m
.
Recall that nine elements of matrix L
nm
are related by six equations that express the
following:
(1) Elements of each row (or column) are direction cosines of a unit vector. Thus,
a
2
11
+ a
2
12
+ a
2
13
= 1, a
2
11
+ a

2
21
+ a
2
31
= 1, ···. (1.2.6)
(2) Due to orthogonality of unit vectors of coordinate axes, we have
[
a
11
a
12
a
13
][
a
21
a
22
a
23
]
T
= 0
[
a
11
a
21
a

31
][
a
12
a
22
a
32
]
T
= 0. (1.2.7)
An element of matrix L
nm
can be represented by a respective determinant of the second
order [Strang, 1988]. For instance,
a
11
=




a
22
a
23
a
32
a
33





, a
23
= (−1)




a
11
a
12
a
31
a
32




. (1.2.8)
To determine the new coordinates (x
n
, y
n
, z
n

, 1) of point M, we have to use the rule
of multiplication of a square matrix (4 ×4) and a column matrix (4 × 1). (The number
of rows in the column matrix is equal to the number of columns in matrix M
nm
.)
Equation (1.2.3) yields
x
n
= a
11
x
m
+ a
12
y
m
+ a
13
z
m
+ a
14
y
n
= a
21
x
m
+ a
22

y
m
+ a
23
z
m
+ a
24
z
n
= a
31
x
m
+ a
32
y
m
+ a
33
z
m
+ a
34
.
(1.2.9)
The purpose of the inverse coordinate transformation is to determine the coordinates
(x
m
, y

m
, z
m
), taking as given coordinates (x
n
, y
n
, z
n
). The inverse coordinate transfor-
mation is represented by
r
m
= M
mn
r
n
. (1.2.10)
The inverse matrix M
mn
indeed exists if the determinant of matrix M
nm
differs from
zero.
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1.2 Coordinate Transformation in Matrix Representation 5
There is a simple rule that allows the elements of the inverse matrix to be determined
in terms of elements of the direct matrix. Consider that matrix M
nm

is given by
M
nm
=





a
11
a
12
a
13
a
14
a
21
a
22
a
23
a
24
a
31
a
32
a

33
a
34
0001





. (1.2.11)
It is necessary to determine the elements of matrix M
mn
represented by
M
mn
=




b
11
b
12
b
13
b
14
b
21

b
22
b
23
b
24
b
31
b
32
b
33
b
34
0001




. (1.2.12)
Here,
M
mn
= M
−1
nm
, M
mn
M
nm

= I
where I is the identity matrix.
The submatrix L
mn
of the order (3 × 3) is determined as follows:
L
mn
=


b
11
b
12
b
13
b
21
b
22
b
23
b
31
b
32
b
33



=


a
11
a
21
a
31
a
12
a
22
a
32
a
13
a
23
a
33


= L
T
nm
. (1.2.13)
The remaining elements (b
14
, b

24
, and b
34
) are determined with the following equations:
b
14
=−(a
11
a
14
+ a
21
a
24
+ a
31
a
34
) ⇒−





: a
11
: a
12
a
13

: a
14
:
: a
21
: a
22
a
23
: a
24
:
: a
31
: a
32
a
33
: a
34
:
:0:00:1:





b
24
=−(a

12
a
14
+ a
22
a
24
+ a
32
a
34
) ⇒−





a
11
: a
12
: a
13
: a
14
:
a
21
: a
22

: a
23
: a
24
:
a
31
: a
32
: a
33
: a
34
:
0:0:0:1:





b
34
=−(a
13
a
14
+ a
23
a
24

+ a
33
a
34
) ⇒−




a
11
a
12
: a
13
::a
14
:
a
21
a
22
: a
23
::a
24
:
a
31
a

32
: a
33
::a
34
:
0 0 :0: :1:




. (1.2.14)
The columns to be multiplied are marked.
To perform successive coordinate transformation, we need only to follow the product
rule of matrix algebra. For instance, the matrix equation
r
p
= M
p( p−1)
M
(p−1)( p−2)
···M
32
M
21
r
1
(1.2.15)
represents successive coordinate transformation from S
1

to S
2
, from S
2
to S
3
, ,from
S
p−1
to S
p
.
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6 Coordinate Transformation
To perform transformation of components of free vectors, we need only to apply
3 × 3 submatrices L, which may be obtained by eliminating the last row and the last
column of the corresponding matrix M. This results from the fact that the free-vector
components (projections on coordinate axes) do not depend on the location of the origin
of the coordinate system.
The transformation of vector components of a free vector A from system S
m
to S
n
is
represented by the matrix equation
A
n
= L
nm

A
m
(1.2.16)
where
A
n
=



A
xn
A
yn
A
zn



, L
nm
=



a
11
a
12
a

13
a
21
a
22
a
23
a
31
a
32
a
33



, A
m
=



A
xm
A
ym
A
zm




. (1.2.17)
A normal to the gear tooth surface is a sliding vector because it may be translated along
its line of action. However, we may transform the surface normal as a free vector if the
surface point where the surface normal is considered will be transferred simultaneously.
1.3 ROTATION ABOUT AN AXIS
Two Main Problems
We consider a general case in which the rotation is performed about an axis that does
not coincide with any axis of the employed coordinate system. We designate the unit
vector of the axis of rotation by c (Fig. 1.3.1) and assume that the rotation about c may
be performed either counterclockwise or clockwise.
Henceforth we consider two coordinate systems: (i) the fixed one, S
a
; and (ii) the
movable one, S
b
. There are two typical problems related to rotation about c. The first
one can be formulated as follows.
Consider that a position vector is rigidly connected to the movable body. The initial
position of the position vector is designated by
OA = ρ (Fig. 1.3.1). After rotation
through an angle φ about c, vector ρ will take a new position designated by
OA
∗
= ρ
∗
.
Both vectors, ρ and ρ
∗
(Fig. 1.3.1), are considered to be in the same coordinate system,

say S
a
. Our goal is to develop an equation that relates components of vectors ρ
a
and ρ
∗
a
.
(The subscript “a” indicates that the two vectors are represented in the same coordinate
system S
a
.) Matrix equation
ρ
∗
a
= L
a
ρ
a
(1.3.1)
describes the relation between the components of vectors ρ and ρ
∗
that are represented
in the same coordinate system S
a
.
The other problem concerns representation of the same position vector in different
coordinate systems. Our goal is to derive matrix L
ba
in matrix equation

ρ
b
= L
ba
ρ
a
. (1.3.2)
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1.3 Rotation About an Axis 7
Figure 1.3.1: Rigid body rotation.
The designations ρ
a
and ρ
b
indicate that the same position vector ρ is represented
in coordinate systems S
a
and S
b
, respectively. Although the same position vector is
considered, the components of ρ in coordinate systems S
a
and S
b
are different and we
designate them by
ρ
a
= a

1
i
a
+ a
2
j
a
+ a
3
k
a
(1.3.3)
and
ρ
b
= b
1
i
b
+ b
2
j
b
+ b
3
k
b
. (1.3.4)
Matrix L
ba

is an operator that transforms the components [a
1
a
2
a
3
]
T
into
[b
1
b
2
b
3
]
T
. It will be shown below that operators L
a
and L
ba
are related.
Problem 1. Relations between components of vectors ρ
a
and ρ
∗
a
.
Recall that ρ
a

and ρ
∗
a
are two position vectors that are represented in the same coordinate
system S
a
. Vector ρ represents the initial position of the position vector, before rotation,
and ρ
∗
represents the position vector after rotation about c. The following derivations
are based on the assumption that rotation about c is performed counterclockwise. The
procedure of derivations (see also Suh & Radcliffe, 1978, Shabana, 1989, and others)
is as follows.
Step 1: We represent ρ
∗
a
by the equation (Fig. 1.3.1)
ρ
∗
a
= OM + MN + NA
∗
(1.3.5)