CRC Press Kinematic Geometry of Surface Machining potx

KINEMATIC
GEOMETRY
OF SURFACE
MACHINING

© 2008 by Taylor & Francis Group, LLC
KINEMATIC
GEOMETRY
OF SURFACE
MACHINING
Stephen P. Radzevich
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Library of Congress Cataloging‑in‑Publication Data
Radzevich, S. P. (Stephen Pavlovich)
Kinematic geometry of surface machining / Stephen P. Radzevich.
p. cm.
Includes bibliographical references and index.
ISBN 978‑1‑4200‑6340‑0 (alk. paper)
1. Machinery, Kinematics of. I. Title.
TJ175.R345 2008
671.3’5‑‑dc22 2007027748
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© 2008 by Taylor & Francis Group, LLC
Dedication
To my son Andrew

© 2008 by Taylor & Francis Group, LLC

Contents
Preface xv
Author xxv
Acknowledgments xxvii
Part I Basics
1 Part Surfaces: Geometry 3
1.1 Elements of Differential Geometry of Surfaces 3
1.2 On the Difference between Classical Differential Geometry
and Engineering Geometry 14
1.3 On the Classication of Surfaces 17
1.3.1 Surfaces That Allow Sliding over Themselves 17
1.3.2 Sculptured Surfaces 18
1.3.3 Circular Diagrams 19
1.3.4 On Classication of Sculptured Surfaces 24
References 25
2 Kinematics of Surface Generation 27
2.1
Kinematics of Sculptured Surface Generation 29
2.1.1 Establishment of a Local Reference System 30
2.1.2 Elementary Relative Motions 33
2.2 Generating Motions of the Cutting Tool 34
2.3 Motions of Orientation of the Cutting Tool 39
2.4 Relative Motions Causing Sliding of a Surface over Itself 42
2.5 Feasible Kinematic Schemes of Surface Generation 45
2.6 On the Possibility of Replacement of Axodes with Pitch Surfaces 51
2.7 Examples of Implementation of the Kinematic Schemes
of Surface Generation 53
References 59
3 Applied Coordinate Systems and Linear Transformations 63
3.1 Applied Coordinate Systems 63

3.1.1 Coordinate Systems of a Part Being Machined 63
3.1.2 Coordinate System of Multi-Axis Numerical Control
(NC) Machine 64
3.2 Coordinate System Transformation 65
3.2.1 Introduction 66
3.2.1.1 Homogenous Coordinate Vectors 66
3.2.1.2 Homogenous Coordinate Transformation
Matr
ices of the Dimension 4 × 4 66
3.2.2 Translations 67

© 2008 by Taylor & Francis Group, LLC
viii Contents
3.2.3 Rotation about a Coordinate Axis 69
3.2.4 Rotation about an Arbitrary Axis through the Origin 70
3.2.5 Eulerian Transformation 71
3.2.6 Rotation about an Arbitrary Axis Not through the Origin 71
3.2.7 Resultant Coordinate System Transformation 72
3.2.8 An Example of Nonorthogonal Linear Transformation 74
3.2.9 Conversion of the Coordinate System Orientation 74
3.3 Useful Equations 75
3.3.1 RPY-Transformation 76
3.3.2 Rotation Operator 76
3.3.3 A Combined Linear Transformation 76
3.4 Chains of Consequent Linear Transformations and a Closed
Loop of Consequent Coordinate System Transformations 77
3.5 Impact of the Coordinate System Transformations on
Fundamental Forms of the Surface 83
References 85
Part II Fundamentals

4 The Geometry of Contact of Two Smooth, Regular Surfaces 89
4.1 Local Relative Orientation of a Part Surface and of the Cutting Tool 90
4.2 The First-Order Analysis: Common Tangent Plane 94
4.3 The Second-Order Analysis 94
4.3.1 Preliminary Remarks: Dupin’s Indicatrix 95
4.3.2 Surface of Normal Relative Curvature 97
4.3.3 Dupin’s Indicatrix of Surface of Relative Curvature 101
4.3.4 Matrix Representation of Equation of the Dupin’s
Indicatrix of the Surface of Relative Normal Curvature 102
4.3.5 Surface of Relative Normal Radii of Curvature 102
4.3.6 Normalized Relative Normal Curvature 103
4.3.7 Curvature Indicatrix 103
4.3.8 Introduction of the Ir 
k
(P/T) Characteristic Curve 106
4.4 Rate of Conformity of Two Smooth, Regular Surfaces
in the First Order of Tangency 107
4.4.1 Preliminary Remarks 108
4.4.2 Indicatrix of Conformity of the Surfaces P and T 110
4.4.3 Directions of the Extremum Rate of Conformity
of the Surfaces P and T 117
4.4.4 Asymptotes of the Indicatrix of Conformity Cnf
R
(P/T) 120
4.4.5 Comparison of Capabilities of the Indicatrix of
Con
formity Cnf
R
(P/T) and of Dupin’s Indicatrix of the
Surface of Relative Curvature 121

4.4.6 Important Properties of the Indicatrix
of Conformity Cnf
R
(P/T) 122
4.4.7 The Converse Indicatrix of Conformity of the Surfaces
P and T in the First Order of Tangency 122

© 2008 by Taylor & Francis Group, LLC
Contents ix
4.5 Plücker’s Conoid: More Characteristic Curves 124
4.5.1 Plücker’s Conoid 124
4.5.1.1 Basics 124
4.5.1.2 Analytical Representation 124
4.5.1.3 Local Properties 126
4.5.1.4 Auxiliary Formulas 127
4.5.2 Analytical Description of Local Topology of the
Smooth, Regular Surface P 127
4.5.2.1 Preliminary Remarks 128
4.5.2.2 Plücker’s Conoid 128
4.5.2.3 Plücker’s Curvature Indicatrix 131
4.5.2.4 An
R
(P)-Indicatrix of the Surface P 132
4.5.3 Relative Characteristic Curves 134
4.5.3.1 On a Possibility of Implementation of
Two of Plücker’s Conoids 134
4.5.3.2 An
R
(P/T)-Relative Indicatrix of the Surfaces
P and T 135

4.6 Feasible Kinds of Contact of the Surfaces P and T 138
4.6.1 On a Possibility of Implementation of the Indicatrix of
Conformity for Identication of Kind of Contact of the
Surfaces P and T 138
4.6.2 Impact of Accuracy of the Computations on the Desired
Parameters of the Indicatrices
of
Conformity Cnf
R
(P/T) 142
4.6.3 Classication of Kinds of Contact of the Surfaces P and T 143
References 151
5 Profiling of the Form-Cutting Tools of the Optimal Design 153
5.1 Proling of the Form-Cutting Tools for Sculptured
Surface Machining 153
5.1.1 Preliminary Remarks 153
5.1.2 On the Concept of Proling the Optimal
Form-Cutting Tool 156
5.1.3 R-Mapping of the Part Surface P on the Generating
Surface T of the Form-Cutting Tool 160
5.1.4 Reconstruction of the Generating Surface T of the
Form-Cutting Tool from the Precomputed Natural
Parameterization 164
5.1.5 A Method for the Determination of the Rate
of Conformity Functions F
1
, F
2
, and F
3

165
5.1.6 An Algorithm for the Computation of the Design
Parameters of the Form-Cutting Tool 173
5.1.7 Illustrative Examples of the Computation of the
Design Parameters of the Form-Cutting Tool 175
5.2 Generation of Enveloping Surfaces 177
5.2.1 Elements of Theory of Envelopes 178

© 2008 by Taylor & Francis Group, LLC
x Contents
5.2.1.1 Envelope to a Planar Curve 178
5.2.1.2 Envelope to a One-Parametric Family of Surfaces 182
5.2.1.3 Envelope to a Two-Parametric Family of Surfaces 184
5.2.2 Kinematical Method for the Determining
of Enveloping Surfaces 186
5.3 Proling of the Form-Cutting Tools for Machining Parts
on Conventional Machine Tools 193
5.3.1 Two Fundamental Principles by Theodore Olivier 194
5.3.2 Proling of the Form-Cutting Tools for Single-Parametric
Kinematic Schemes of Surface Generation 195
5.3.3 Proling of the Form-Cutting Tools for Two-Parametric
Kinematic Schemes of Surface Generation 196
5.3.4 Proling of the Form-Cutting Tools for Multiparametric
Kinematic Schemes of Surface Generation 200
5.4 Characteristic Line E of the Part Surface P and of the Generating
Surface T of the Cutting Tool 201
5.5 Selection of the Form-Cutting Tools of Rational Design 203
5.6 The Form-Cutting Tools Having a Continuously
Changeable Generating Surface 210
5.7 Incorrect Problems in Proling the Form-Cutting Tools 210

5.8 Intermediate Conclusion 214
References 215
6 The Geometry of the Active Part of a Cutting Tool 217
6.1 Transformation of the Body Bounded by the Generating Surface
T into the Cutting Tool 218
6.1.1 The First Method for the Transformation of the
Generating Body of the Cutting Tool into the
Workable Edge Cutting Tool 219
6.1.2 The Second Method for the Transformation of the
Generating Body of the Cutting Tool into the
Workable Edge Cutting Tool 222
6.1.3 The Third Method for the Transformation of the
Generating Body of the Cutting Tool into the
Workable Edge Cutting Tool 225
6.2 Geometry of the Active Part of Cutting Tools in the
Tool-in-Hand System 234
6.2.1 Tool-in-Hand Reference System 235
6.2.2 Major Reference Planes: Geometry of the Active Part of a
Cutting Tool Dened in a Series of Reference Planes 237
6.2.3 Major Geometric Parameters of the Cutting Edge
of a Cutting Tool 240
6.2.3.1 Main Reference Plane 240
6.2.3.2 Assumed Reference Plane 241
6.2.3.3 Tool Cutting Edge Plane 242
6.2.3.4 Tool Back Plane 242

© 2008 by Taylor & Francis Group, LLC
Contents xi
6.2.3.5 Orthogonal Plane 242
6.2.3.6 Cutting Edge Normal Plane 242

6.2.4 Analytical Representation of the Geometric Parameters
of the Cutting Edge of a Cutting Tool 243
6.2.5 Correspondence between Geometric Parameters of
the Active Part of Cutting Tools That Are Measured in
Different Reference Planes 244
6.2.6 Diagrams of Variation of the Geometry of the Active
Part of a Cutting Tool 253
6.3 Geometry of the Active Part of Cutting Tools in the
Tool-in-Use System 255
6.3.1 The Resultant Speed of Relative Motion in the Cutting
of Materials 257
6.3.2 Tool-in-Use Reference System 258
6.3.3 Reference Planes 261
6.3.3.1 The Plane of Cut Is Tangential to the Surface
of Cut at the Point of Interest M 261
6.3.3.2 The Normal Reference Plane 263
6.3.3.3 The Major Section Plane 266
6.3.3.4 Correspondence between the Geometric
Parameters Measured in Different
Reference Planes 268
6.3.3.5 The Main Reference Plane 269
6.3.3.6 The Reference Plane of Chip Flow 272
6.3.4 A Descriptive-Geometry-Based Method for the
Determination of the Chip-Flow Rake Angle 276
6.4 On Capabilities of the Analysis of Geometry of the Active
Part of Cutting Tools 277
6.4.1 Elements of Geometry of Active Part of a Skiving Hob 277
6.4.2 Elements of Geometry of the Active Part of a Cutting Tool
for Machining Modied Gear Teeth 279
6.4.3 Elements of Geometry of the Active Part of a

Precision Involute Hob 281
6.4.3.1 An Auxiliary Parameter R 281
6.4.3.2 The Angle f
r
between the Lateral Cutting Edges
of the Hob Tooth 282
6.4.3.3 The Angle x of Intersection of the Rake Surface
and of the Hob Axis of Rotation 284
References 285
7 Conditions of Proper Part Surface Generation 287
7.1 Optimal Workpiece Orientation on the Worktable
of a Multi-Axis Numerical Control (NC) Machine 287
7.1.1 Analysis of a Given Workpiece Orientation 288
7.1.2 Gaussian Maps of a Sculptured Surface P and of the
Generating Surface T of the Cutting Tool 290

© 2008 by Taylor & Francis Group, LLC
xii Contents
7.1.3 The Area-Weighted Mean Normal to a
Sculptured Surface P 293
7.1.4 Optimal Workpiece Orientation 295
7.1.5 Expanded Gaussian Map of the Generating Surface
of the Cutting Tool 297
7.1.6 Important Peculiarities of Gaussian Maps
of the Surfaces P and T 299
7.1.7 Spherical Indicatrix of Machinability
of a Sculptured Surface 302
7.2 Necessary and Sufcient Conditions of Proper
Part Surface Generation 309
7.2.1 The First Condition of Proper Part Surface Generation 309

7.2.2 The Second Condition of Proper Part Surface Generation 313
7.2.3 The Third Condition of Proper Part Surface Generation 314
7.2.4 The Fourth Condition of Proper Part Surface Generation 323
7.2.5 The Fifth Condition of Proper Part Surface Generation 324
7.2.6 The Sixth Condition of Proper Part Surface Generation 329
7.3 Global Verication of Satisfaction of the Conditions
of Proper Part Surface Generation 330
7.3.1 Implementation of the Focal Surfaces 330
7.3.1.1 Focal Surfaces 331
7.3.1.2 Cutting Tool (CT)-Dependent Characteristic
Surfaces 336
7.3.1.3 Boundary Curves of the CT-Dependent
Characteristic Surfaces 338
7.3.1.4 Cases of Local-Extremal Tangency of the Surfaces
P and T 341
7.3.2 Implementation of R-Surfaces 343
7.3.2.1 Local Consideration 343
7.3.2.2 Global Interpretation of the Results
of the Local Analysis 346
7.3.2.3 Characteristic Surfaces of the Second Kind 355
7.3.3 Selection of the Form-Cutting Tool of Optimal Design 357
7.3.3.1 Local K
LR
-Mapping of the Surfaces P and T 357
7.3.3.2 The Global K
GR
-Mapping of the Surfaces P and T 359
7.3.3.3 Implementation of the Global K
GR
-Mapping 363

7.3.3.4 Selection of an Optimal Cutting Tool
for Sculptured Surface Machining 364
References 365
8 Accuracy of Surface Generation 367
8.1 Two Principal Kinds of Deviations of the Machined Surface
from the Nominal Part Surface 368
8.1.1 Principal Deviations of the First Kind 368
8.1.2 Principal Deviations of the Second Kind 369
8.1.3 The Resultant Deviation of the Machined Part Surface 370

© 2008 by Taylor & Francis Group, LLC
Contents xiii
8.2 Local Approximation of the Contacting Surfaces P and T 372
8.2.1 Local Approximation of the Surfaces P and T
by Portions of Torus Surfaces 373
8.2.2 Local Conguration of the Approximating Torus Surfaces 378
8.3 Computation of the Elementary Surface Deviations 380
8.3.1 Waviness of the Machined Part Surface 380
8.3.2 Elementary Deviation h
ss
of the Machined Surface 382
8.3.3 An Alternative Approach for the Computation
of the Elementary Surface Deviations 383
8.4 Total Displacement of the Cutting Tool with Respect
to the Part Surface 384
8.4.1 Actual Conguration of the Cutting Tool
with Respect to the Part Surface 384
8.4.2 The Closest Distance of Approach between
the Surfaces P and T 390
8.5 Effective Reduction of the Elementary Surface Deviations 396

8.5.1 Method of Gradient 396
8.5.2 Optimal Feed-Rate and Side-Step Ratio 397
8.6 Principle of Superposition of Elementary Surface Deviations 399
References 403
Part III Application
9 Selection of the Criterion of Optimization 407
9.1 Criteria of the Efciency of Part Surface Machining 408
9.2 Productivity of Surface Machining 409
9.2.1 Major Parameters of Surface Machining Operation 409
9.2.2 Productivity of Material Removal 411
9.2.2.1 Equation of the Workpiece Surface 411
9.2.2.2 Mean Chip-Removal Output 413
9.2.2.3 Instantaneous Chip-Removal Output 413
9.2.3 Surface Generation Output 417
9.2.4 Limit Parameters of the Cutting Tool Motion 418
9.2.4.1 Computation of the Limit Feed-Rate Shift 418
9.2.4.2 Computation of the Limit Side-Step Shift 420
9.2.5 Maximal Instantaneous Productivity of Surface
Generation 421
9.3 Interpretation of the Surface Generation Output
as a Function of Conformity 423
References 424
10 Synthesis of Optimal Surface Machining Operations 427
10.1 Synthesis of Optimal Surface Generation: The Local Analysis 427
10.1.1 Local Synthesis 428
10.1.2 Indeniteness 432

© 2008 by Taylor & Francis Group, LLC
xiv Contents
10.1.3 A Possibility of Alternative Optimal Congurations

of the Cutting Tool 432
10.1.4 Cases of Multiple Points of Contact of the Surfaces P and T 434
10.2 Synthesis of Optimal Surface Generation: The Regional Analysis 435
10.3 Synthesis of Optimal Surface Generation: The Global Analysis 439
10.3.1 Minimization of Partial Interference
of the Neighboring Tool-Paths 439
10.3.2 Solution to the Boundary Problem 440
10.3.3 Optimal Location of the Starting Point 442
10.4 Rational Reparameterization of the Part Surface 444
10.4.1 Transformation of Parameters 445
10.4.2 Transformation of Parameters in Connection
with the Surface Boundary Contour 446
10.5 On a Possibility of the Differential Geometry/Kinematics
(DG/K)-Based Computer-Aided Design/Computer-Aided
Manufacturing (CAD/CAM) System for Optimal Sculptured
Surface Machining 451
10.5.1 Major Blocks of the DG/K-Based CAD/CAM System 451
10.5.2 Representation of the Input Data 452
10.5.3 Optimal Workpiece Conguration 454
10.5.4 Optimal Design of the Form-Cutting Tool 454
10.5.5 Optimal Tool-Paths for Sculptured Surface Machining 455
10.5.6 Optimal Location of the Starting Point 457
References 457
11 Examples of Implementation of the Differential Geometry/
Kinematics (DG/K)-Based Method of Surface Generation 459
11.1 Machining of Sculptured Surfaces on a Multi-Axis Numerical
Control (NC) Machine 459
11.2 Machining of Surfaces of Revolution 469
11.2.1 Turning Operations 469
11.2.2 Milling Operations 474

11.2.3 Machining of Cylinder Surfaces 475
11.2.4 Reinforcement of Surfaces of Revolution 476
11.3 Finishing of Involute Gears 480
References 491
Conclusion 493
Notation 495

© 2008 by Taylor & Francis Group, LLC
Preface
“GAINING TIME IS GAINING EVERYTHING.”
John Shebbeare, 1709–1788
This book, based on intensive research I have conducted since the late 1970s,
is my attempt to cover in one monograph the modern theory of surface gen-
eration with a focus on kinematic geometry of surface machining on a multi-
axis numerical control (NC) machine. Although the orientation of this book
is toward computer-aided design (CAD) and computer-aided manufacturing
(CAM), it is also useful for solving problems that relate to the generation of
surfaces on machine tools of conventional design (for example, gear genera-
tors, and so forth).
Machining of part surfaces can be interpreted as the transformation of a
work into the machined part having the desired shape and design param-
eters. The major characteristics of the machined part surface — its shape
and actual design parameters, as well as the properties of the subsurface
layer of part material — strongly depend upon the parameters of the surface-
generating process. In addition to the surface-generating process, there are,
of course, many other technical considerations — namely, wear of the cutting
tool, stiffness of the machine tool, tool chatter, heat generation, coolant and
lubricant supply, and so forth. The analysis in this book is limited to those
parameters of the surface-machining process that can be expressed in terms
of surface geometry and of kinematics of relative motion of the cutting tool.

Historical Background
People have been concerned for centuries with the generation of surfaces. Any
machiningoperation isaimedat thegeneration ofasurfacethathasappropriate
shape and parameters. Enormous practical experience has been accumulated
in this area of engineering. Improvements to the surface machining operation
are based mostly on generalization of accumulated practical experience. Ele-
ments of the theory of surface generation began to appear later.
For a long time, scientic developments in the eld of surface generation
were aimed at solving those problems that are relatively simple in nature. In
the late 1970s and early 1980s, the idea of the synthesis of the optimal surface
machining operation was, in a manner of speaking, mentioned for the rst
time. After a decade of gestation, original articles on the subject began to
appear. Now, with the passing of a second decade, it is appropriate to attempt

© 2008 by Taylor & Francis Group, LLC
xvi Preface
a consolidated story of some of the many efforts of European and American
researchers.
The Importance of the Subject
The machining of sculptured part surfaces on a multi-axis NC machine is
a widely used process in many industries. The automotive, aerospace, and
some other industries are the most advanced in this respect.
The ability to quickly introduce new quality products is a decisive factor in
capturing market share. For this purpose, the use of multi-axis NC machines
is vital. Multi-axis NC machines of modern design are extremely costly.
Because of this, machining of sculptured surfaces is costly as well. In order
to decrease the cost of machining a sculptured surface on a multi-axis NC
machine, the machining time must be as short as possible. Denitely, this is
the case where the phrase “Time is money!” applies.
Reduction of the machining time is a critical issue when machining sculp-

tured surfaces on multi-axis NC machines. It is also an important consider-
ation when machining surfaces on machine tools of conventional design.
Generally speaking, the optimization of surface generation on a multi-axis
NC machine results in time savings. Remember, gaining time is gaining every-
thing. Certainly, the subject of this book is of great importance for contempo-
rary industry and engineering.
Uniqueness of This Publication
Literature on the theory of surface generation on a multi-axis NC machine is
lacking. A limited number of texts on the topic are available for the English-
speaking audience. Conventional texts provide an adequate presentation and
analysis of a given operation of sculptured surface machining. The problem of
surface generation is treated in all recently published books on the topic from
the standpoint of analysis, and not of synthesis, of optimal surface generation.
In the past 20 years, a wealth of new journal papers relating to the syn-
thesis of optimal surface generation processes have been published both in
this country and abroad. The rapid intensication of research in the theory
of surface generation for CAD and CAM applications and new needs for
advanced technology inspired me to accomplish this work.
The present text is an attempt to present a well-balanced and intelligible
account of some of the geometric and algebraic procedures, lling in as nec-
essary, making comparisons, and elaborating on the implications to give a
well-rounded picture.
In this book, various procedures for handling particular problems constitut-
ing the synthesis of optimal surface generation on a multi-axis NC machine

© 2008 by Taylor & Francis Group, LLC
Preface xvii
are investigated, compared, and applied. To begin, denitions, concepts, and
notations are reviewed and established, and familiar methods of sculptured
surface analysis are recapitulated.* The fundamental concepts of sculptured

surface geometry are introduced, and known results in the theory of multipa-
rametric motion of a rigid body in E
3
space are presented.
It is postulated in this text that the surface to be machined is the primary
element of the surface-generation process. Other elements, for example, the
generating surface of the cutting tool and kinematics of their relative motion,
are the secondary elements; thus, their optimal parameters must be deter-
mined in terms of design parameters of the part surface to be machined.
To the best of my knowledge, I was the rst to formulate the problem of
synthesizing optimal surface generation, in the early 1980s. In the begin-
ning, the problem was understood mostly intuitively. The rst principal
achievements in this eld** allowed expression of the optimal parameters of
kinematics of the sculptured surface machining on a multi-axis NC machine
in terms of geometry of the part surface and of the generating surface of the
form-cutting tool.
A bit later, a principal solution to the problem of proling the form-cutting
tool*** was derived. This solution
yi
elds determination of the generating sur-
face of the form-cutting tool as the R-mapping of the sculptured surface to
be machined. Therefore, optimal parameters of the generating surface of the
form-cutting tool can be expressed in terms of design parameters of the part
surface to be machined. Taking into account that the optimal parameters
of kinematics of surface machining are already specied in terms of the
surfaces P and T, the last solution allows an analytical representation of the
entire surface-generation process in terms of design parameters of the sculp-
tured surface P. This means that the necessary input information for solv-
ing the problem of synthesizing the optimal surface-machining operation
is limited to design parameters of the sculptured part surface. This input

information is the minimum feasible.
These two important results make evident that the problem of synthe-
sizing optimal surface-generating processes is solvable in nature. On the
premises of these two principal results, dozens of novel methods of part sur-
face machining have been developed, and many are successfully used in the
industry (see Chapter 11).
It is important to stress that the decrease in required input information indi-
cates that the theory is getting closer to the ideal. This concept, which this
book strictly adheres
to
, is widely known as the principle of Occam’s razor.
*
Recall here the old Chinese proverb: The beginning of wisdom is calling things with their
right names.
**
Radzevich, S.P., A Method of Sculptured Surface Machining on Multi-Axis NC Machine,
Patent 1185749, USSR, B23C 3/16, led: October 24, 1983; Radzevich, S.P., A Method of Sculp-
tured Surface Machining on Multi-Axis NC Machine, Patent 1249787, USSR, B23C 3/16, led:
November 27, 1984.
***
Radzevich, S.P., A Method of Design of a Form Cutting Tool for Sculptured Surface Machin-
ing on Multi-Axis NC Machine, Patent application 4242296/08 (USSR), led: March 3, 1987.

© 2008 by Taylor & Francis Group, LLC
xviii Preface
The principle of Occam’s razor is one of the rst principles allowing evaluation
of how a theory becomes ideal. Minimal feasible input information indicates
the strength of a proposed theory. Occam’s razor states that the explanation of
any phenomenon should make as few assumptions as possible, eliminating,
or “shaving off,” those that make no difference in the observable predictions of

the explanatory hypothesis of theory. In short, when given two equally valid
explanations for a phenomenon, one should embrace the less-complicated for-
mulation. The principle is often expressed in Latin as the lex parsimoniae (law
of succinctness): Entia non sunt multiplicanda praeter necessitatem, which trans-
lates to “Entities should not be multiplied beyond necessity.”
This is often paraphrased as “All things being equal, the simplest solution
tends to be the best one.” In other words, when multiple competing theories
are equal in other respects, the principle recommends selecting the theory
that introduces the fewest assumptions and postulates the fewest hypotheti-
cal entities. It is in this sense that Occam’s razor is usually understood.
Following the fundamental principle of Occam’s razor, one can compute
optimal values of all the major parameters of sculptured surface machining
on a multi-axis NC machine. Previous experience in the eld is helpful but
not mandatory for solving
th
e problem of synthesizing the optimal machin-
ing operation.
Important new topics help the reader to solve the challenging problems of
synthesizing optimal methods of surface generation. In order to employ the
disclosed approach, limited input information is required: For this purpose,
only analytical representation of the surface to be generated is necessary.
No known theory of surface generation is capable of solving the problems of
synthesizing methods of surface generation. Moreover, no known theory is
capable of treating the problem on the premises of the geometrical informa-
tion of the surface being generated alone.
The theory of surface generation has been substantionally complemented
in
this book through recent discoveries made primarily by myself and my
colleagues. I have made a rst attempt to summarize the obtained results
of the research in the eld in 1991. That year, my rst books in the eld of

surface generation (in Russian) were courageously introduced to the engi-
neering community (Radzevich, S.P., Sculptured Surface Machining on Multi-
Axis NC Machine, Kiev, Vishcha Shkola, 1991). Ten years later, a much more
comprehensive summary was carried out (Radzevich, S.P., Fundamentals of
Surface Generation, Kiev, Rastan, 2001). Both of these monographs are used in
Europe, as well as in the United States. They are available from the Library of
Congress and
fro
m other sources (www.cse.buffalo.edu/~var2/).
There is a concern that some of today’s mechanical engineers, manufactur-
ing engineers, and engineering students may not be learning enough about
the theory of surface generation. Although containing some vitally important
information, books to date do not provide methodological information on
the subject which can be helpful in making critical decisions in the process
design, design and selection of cutting tools, and implementation of the proper
machine tool. The most important information is dispersed throughout a great

© 2008 by Taylor & Francis Group, LLC
Preface xix
number of research and application papers and articles. Commonly, isolated
theoretical and practical ndings for a particular surface-generation process
are reported instead of methodology, so the question “What would happen if
the input parameters are altered?” remains unanswered. Therefore, a broad-
based book on the theory of surface generation is needed.
The purpose of this book is twofold:
To summarize the available information on surface generation with a
critical review of previous work, thus helping specialists and prac-
titioners to separate facts from myths. The major problem in the
theory of surface generation is the absence of methods by use of
which the challenging problem of optimal surface generation can be

successfully solved. Other known problems are just consequences of
the absence of the said methods of surface generation.
To present, explain, and exemplify a novel principal concept in the the-
ory of surface generation, namely that the part surface is the primary
element of the part surface-machining operation. The rest of the
elements are the secondary elements of the part surface-machining
operation; thus, all of them can be expressed in terms of the desired
design parameters of the part surface to be machined.
The distinguishing feature of this book is that the practical ways of synthe-
sizing and optimizing the surface-generation process are considered using
just one set of parameters — the design parameters of the part surface to be
machined. The desired design parameters of the part surface to be machined
are known in a research laboratory as well as in a shop oor environment.
This makes this book not just another book on the subject. For the rst time,
the theory of surface generation is presented as a science that really works.
This book is based on the my varied 30 years of experience in research,
practical application, and teaching in the theory of surfacegeneration,applied
mathematics and mechanics, fundamentals of CAD/CAM, and engineering
systems theory. Emphasis is placed on the practical application of the results
in everyday practice of part surface machining and cutting-tool design. The
application of these recommendations will increase the competitive posi-
tion of the users through machining economy and productivity. This helps
in designing better cutting tools and processes and in enhancing technical
expertise and levels of technical services.
Intended Audience
Many readers will benet from this book: mechanical and manufacturing
engineers involved in continuous process improvement, research workers
whoareactiveorintendtobecomeactiveintheeld,andseniorundergraduate
and graduate students of mechanical engineering and manufacturing.

© 2008 by Taylor & Francis Group, LLC
xx Preface
This book is intended to be used as a reference book as well as a textbook.
Chapters that cover geometry of sculptured part surfaces and elementary
kinematics of surface generation, and some sections that pertain to design
of the form-cutting tools can be used for graduate study; I have used this
book for graduate study in my lectures at the National Technical University
of Ukraine “Kiev Polytechnic Institute” (Kiev, Ukraine). The design chapters
interest for mechanical and manufacturing engineers and for researchers.
The Organization of This Book
The book is comprised of three parts entitled “Basics,” “Fundamentals,” and
“Application”:
Part I: Basics — This section of the book includes analytical description
of part surfaces, basics on differential geometry of sculptured sur-
faces, as well as principal elements of the theory of multiparametric
motion of a rigid body in E
3
space. The applied coordinate systems
and linear transformations are briey considered. The selected mate-
rial focuses on the solution to the problem of synthesizing optimal
machining of sculptured part surfaces on a multi-axis NC machine.
The chapters and their contents are as follows:
Chapter 1. Part Surfaces: Geometry — The basics of differential
geometry of sculptured part surfaces are explained. The focus
here is on the difference between classical differential geometry
and engineering geometry of surfaces. Numerous examples of the
computation of major surface elements are provided. A feasibility
of classication of surfaces is discussed, and a scientic classica-
tion of local patches of sculptured surfaces is proposed.
Chapter 2. Kinematics of Surface Generation — The general-

ized analysis of kinematics of sculptured surface generation
is presented. Here, a generalized kinematics of instant relative
motion of the cutting tool relative to the work is proposed. For
the purposes of the profound investigation, novel kinds of rela-
tive motions of the cutting tool are discovered, including gen-
erating motion of the cutting tool, motions of orientation, and
relative motions that cause sliding of a surface over itself. The
chapter concludes with a discussion on all feasible kinematic
schemes of surface generation. Several particular issues of kine-
matics of surface generation are discussed as well.
Chapter 3. Applied Coordinate Systems and Linear Transforma-
tions — The denitions and determinations of major applied
coordinate systems are introduced in this chapter. The matrix

© 2008 by Taylor & Francis Group, LLC
and practical implementation of the proposed theory (Part III) will be of
Preface xxi
approach for the coordinate system transformations is briey
discussed. Here, useful notations and practical equations are
provided. Two issues of critical importance are introduced here.
The rst is chains of consequent linear transformations and a
closed loop of consequent coordinate systems transformations.
The impact of the coordinate systems transformations on funda-
mental forms of the surfaces is the second.
These tools, rust covered for many readers (the voice of experience), are
resharpened in an effort to make the book a self-sufcient unit suited for
self-study.
Part II: Fundamentals — Fundamentals of the theory of surface genera-
tion are the core of the book. This part of the book includes a novel
powerful method of analytical description of the geometry of contact

of two smooth, regular surfaces in the rst order of tangency; a novel
kind of mapping of one surface onto another surface; a novel analyti-
cal method of investigation of the cutting-tool geometry; and a set of
analytically described conditions of proper part surface generation. A
solution to the challenging problem of synthesizing optimal surface
machining begins here. The consideration is based on the analytical
results presented in the rst part of the book. The following chapters
are included in this section.
Chapter 4. The Geometry of Contact of Two Smooth Regular Sur-
faces — Local characteristics of contact of two smooth, regular
surfaces that make tangency of the rst order are considered. The
sculptured part surface is one of the contacting surfaces, and the
generating surface of the cutting tool is the second. The performed
analysis includes local relative orientation of the contacting sur-
faces and the rst- and second-order analyses. The concept of
conformity of two smooth, regular surfaces in the rst order of
tangency is introduced and explained in this chapter. For the pur-
poses of analyses, properties of Plücker’s conoid are implemented.
Ultimately, all feasible kinds of contact of the part and of the tool
surfaces
ar
e classied.
Chapter 5. Proling of the Form-Cutting Tools of Optimal Design
— A novel method of proling the form-cutting tools for sculp-
tured surface machining is disclosed in this chapter. The method
is based on the analytical description of the geometry of contact
of surfaces that is developed in the previous chapter. Methods of
proling form-cutting tools for machining part surfaces on con-
ventional machine tools are also considered. These methods are
based on elements of the theory of enveloping surfaces. Numer-

ous particular issues of proling form-cutting tools are discussed
at the end of the chapter.

© 2008 by Taylor & Francis Group, LLC
xxii Preface
Chapter 6. Geometry of Active Part of a Cutting Tool — The gen-
erating body of the form-cutting tool is bounded by the generat-
ing surface of the cutting tool. Methods of transformation of the
generating body of the form-cutting tool into a workable cutting
tool are discussed. In addition to two known methods, one novel
method for this purpose is proposed. Results of the analytical
investigation of the geometry of the active part of cutting tools in
both the Tool-in-Hand system as well as the Tool-in-Use system
are represented. Numerous practical examples of the computa-
tions are also presented.
Chapter 7. Conditions of Proper Part Surface Generation — The
satisfactory conditions necessary and sufcient for proper part
surface machining
ar
e proposed and examined. The conditions
include the optimal workpiece orientation on the worktable of a
multi-axis NC machine and the set of six analytically described
conditions of proper part surface generation. The chapter con-
cludes with the global verication of satisfaction of the condi-
tions of proper part surface generation.
Chapter 8. Accuracy of Surface Generation — Accuracy is an impor-
tant issue for the manufacturer of the machined part surfaces.
Analytical methods for the analysis and computation of the devia-
tionsofthe machinedpart surfacefromthedesiredpartsurface are
discussed here. Two principal kinds of deviations of the machined

surface from the nominal part surface
ar
e distinguished. Methods
for the computation of the elementary surface deviations are pro-
posed. The total displacements of the cutting tool with respect to
the part surface are analyzed. Effective methods for the reduction
of the elementary surface deviations are proposed. Conditions
under which the principle of superposition of elementary surface
deviations is applicable are established.
Part III: Application — This section illustrates the capabilities of the
novel and powerful tool for the development of highly efcient
methods of part surface generation. Numerous practical examples of
implementation of the theory are disclosed in this part of the mono-
graph. This section of the book is organized as follows:
Chapter 9. Selection of the Criterion of Optimization — In order to
implement in practice the disclosed Differential Geometry/Kine-
matics (DG/K)-based method of surface generation, an appropri-
ate criterion of efciency of part surface machining is necessary.
This helps answer the question of what we want to obtain when
performing a certain machining operation. Various criteria of ef-
ciency of machining operation are considered. Tight connection
of the economical criteria of optimization with geometrical ana-
logues (as established in Chapter 4) is illustrated. The part surface

© 2008 by Taylor & Francis Group, LLC
Preface xxiii
generation output is expressed in terms of functions of confor-
mity. The last signicantly simplies the synthesizing of optimal
operations of part surface machining.
Chapter 10. Synthesis of Optimal Surface Machining Operations

— The synthesizing of optimal operations of actual part sur-
face machining on both the multi-axis NC machine as well as
on a conventional machine tool are explained. For this purpose,
three steps of analysis are distinguished: local analysis, regional
analysis, and global analysis. A possibility of the development of
the DG/K-based CAD/CAM system for the optimal sculptured
surface machining is shown.
Chapter 11. Examples of Implementation of the DG/K-Based
Method of Surface Generation — This chapter demonstrates
numerous novel methods of surface machining — those devel-
oped on the premises of implementation of the proposed DG/K-
based method surface generation. Addressed are novel methods of
machining sculptured surfaces on a multi-axis NC machine, novel
methods of machining surfaces of revolution, and a novel method of
nishing involute gears.
The proposed theory of surface generation is oriented on extensive appli-
cation of a multi-axis NC machine of modern design. In particular cases,
implementation of the theory can be useful for machining parts on conven-
tional machine tools.
Stephen P. Radzevich
Sterling Heights, Michigan

© 2008 by Taylor & Francis Group, LLC
Author
Stephen P. Radzevich, Ph.D., is a professor of mechanical engineering and
manufacturing engineering. He has received an M.Sc. (1976), a Ph.D. (1982),
and a Dr.(Eng)Sc. (1991) in mechanical engineering. Radzevich has exten-
sive industrial experience in gear design and manufacture. He has devel-
oped numerous software packages dealing with computer-aided design
(CAD) and computer-aided manufacturing (CAM) of precise gear nishing

for a variety of industrial sponsors. Dr. Radzevich’s main research inter-
est is kinematic geometry of surface generation with a particular focus on
(a) precision gear design, (b) high torque density gear trains, (c) torque share
in multiow gear trains, (d) design of special-purpose gear cutting and n-
ishing tools, (e) design and machining (nishing) of precision gears for low-
noise/noiseless transmissions of cars, light trucks, and so forth. He has spent
more than 30 years developing software, hardware, and other processes for
gear design and optimization. In addition to his work for industry, he trains
engineering students at universities and gear engineers in companies. He
has authored and coauthored 28 monographs, handbooks, and textbooks; he
authored and coauthored more than 250 scientic papers; and he holds more
than 150 patents in the eld. At the beginning of 2004, he joined EATON
Corp. He is a member of several Academies of Sciences around the world.

© 2008 by Taylor & Francis Group, LLC
Acknowledgments
I would like to share the credit for any research success with my numerous
doctoral students with whom I have tested the proposed ideas and applied
them in the industry. The contributions of many friends, colleagues, and
students in overwhelming numbers cannot be acknowledged individually,
and as much as our benefactors have contributed, even though their kind-
ness and help must go unrecorded.

© 2008 by Taylor & Francis Group, LLC
Part I
Basics

© 2008 by Taylor & Francis Group, LLC
3
1

Part Surfaces: Geometry
The generation of part surfaces is one of the major purposes of machin-
ing operations. An enormous variety of parts are manufactured in various
industries. Every part to be machined is bounded with two or more sur-
faces.* Each of the part surfaces is a smooth, regular surface, or it can be
composed with a certain number of patches of smooth, regular surfaces that
are properly linked to each other.
In order to be machined on a numerical control (NC) machine, and for com-
puter-aided design (CAD) and computer-aided manufacturing (CAM) appli-
cations, a formal (analytical) representation of a part surface is the required
prerequisite. Analytical representation of a part surface (the surface P) is
based on analytical representation of surfaces in geometry, specically, (a) in
the differential geometry of surfaces and (b) in the engineering geometry of
surfaces. The second is based on the rst.
For further consideration, it is convenient to briey consider the principal
elements of differential geometry of surfaces that are widely used in this
text. If experienced in differential geometry of surfaces, the following sec-
tion may be skipped. Then, proceed directly to Section 1.2.
1.1 Elements of Differential Geometry of Surfaces
A surface could be uniquely determined by two independent variables.
Therefore, we give a part surface P (Figure 1.1), in most cases, by expressing
its rectangular coordinates X
P
, Y
P
, and Z
P
, as functions of two Gaussian coor-
dinates U
P

and V
P
in a certain closed interval:
r r
P P P P
P P P
P P P
P P P
U V
X U V
Y U V
Z U V
= =


( , )
( , )
( , )
( , )
1











≤ ≤ ≤ ≤; ( ; )
. . . .
U U U V V
P P P P P P1 2 1 2
V


(1.1)
*
The ball of a ball bearing is one of just a few examples of a part surface, which is bounded
with the only surface that is the sphere.

© 2008 by Taylor & Francis Group, LLC