KINEMATIC GEOMETRY OF SURFACE MACHINING 2008 by Taylor & Francis Group ppt

KINEMATIC
GEOMETRY
OF SURFACE
MACHINING

© 2008 by Taylor & Francis Group, LLC


KINEMATIC
GEOMETRY
OF SURFACE
MACHINING
Stephen P. Radzevich

Boca Raton London New York

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Taylor & Francis Group, an informa business

© 2008 by Taylor & Francis Group, LLC


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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
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© 2008 by Taylor & Francis Group, LLC

2007027748


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 Classification 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 Classification 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
Matrices of the Dimension 4 ì 4 ..................................... 66
3.2.2 Translations....................................................................................... 67

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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
Conformity 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

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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 Identification 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 Classification 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  Profiling of the Form-Cutting Tools for Sculptured
Surface Machining ..................................................................................... 153
5.1.1 Preliminary Remarks .................................................................... 153
5.1.2 On the Concept of Profiling 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

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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  Profiling 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 Profiling of the Form-Cutting Tools for Single-Parametric

Kinematic Schemes of Surface Generation ................................ 195
5.3.3 Profiling of the Form-Cutting Tools for Two-Parametric
Kinematic Schemes of Surface Generation ................................ 196
5.3.4 Profiling 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 Profiling 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 Defined 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

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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 Modified 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
.

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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 Sufficient 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 Verification 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 KGR-Mapping of the Surfaces P and T ..... 359
7.3.3.3 Implementation of the Global KGR-Mapping............... 363
.
7.3.3.4 Selection of an Optimal Cutting Tool
for Sculptured Surface Machining...............................364
References ............................................................................................................ 365
8
8.1 

Accuracy of Surface Generation ................................................... 367
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

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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 Configuration 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 hss 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 Configuration 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 Efficiency 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 Indefiniteness .............................................................................. 432

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10.1.3

A Possibility of Alternative Optimal Configurations
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 Configuration ..........................................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

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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 generation with a focus on kinematic geometry of surface machining on a multiaxis 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 generators, 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 parameters. 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 surfacegenerating 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
machining operation is aimed at the generation of a surface that has appropriate
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. Elements of the theory of surface generation began to appear later.
For a long time, scientific developments in the field 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 first
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

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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. Definitely, this is
the case where the phrase “Time is money!” applies.
Reduction of the machining time is a critical issue when machining sculptured surfaces on multi-axis NC machines. It is also an important consideration 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 everything. Certainly, the subject of this book is of great importance for contemporary 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 Englishspeaking 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 synthesis of optimal surface generation processes have been published both in
this country and abroad. The rapid intensification 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, filling in as necessary, making comparisons, and elaborating on the implications to give a
well-rounded picture.

In this book, various procedures for handling particular problems constituting the synthesis of optimal surface generation on a multi-axis NC machine

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Preface

xvii

are investigated, compared, and applied. To begin, definitions, 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 multiparametric motion of a rigid body in E3 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 determined in terms of design parameters of the part surface to be machined.
To the best of my knowledge, I was the first to formulate the problem of
synthesizing optimal surface generation, in the early 1980s. In the beginning, the problem was understood mostly intuitively. The first principal
achievements in this field** 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 profiling the form-cutting
tool*** was derived. This solution yields determination of the generating surface 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 specified 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 sculptured surface P. This means that the necessary input information for solving 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 synthesizing optimal surface-generating processes is solvable in nature. On the
premises of these two principal results, dozens of novel methods of part surface 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 indicates 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, filed: October 24, 1983; Radzevich,  S.P.,  A Method of Sculptured Surface Machining on Multi-Axis NC Machine, Patent 1249787, USSR, B23C 3/16, filed:
November 27, 1984.
***Radzevich,  S.P.,  A Method of Design of a Form Cutting Tool for Sculptured Surface Machining on Multi-Axis NC Machine, Patent application  4242296/08  (USSR), filed: March  3, 1987.


© 2008 by Taylor & Francis Group, LLC


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Preface

The principle of Occam’s razor is one of the first 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 formulation. The principle is often expressed in Latin as the lex parsimoniae (law

of succinctness): Entia non sunt multiplicanda praeter necessitatem, which translates 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 hypothetical 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 field is helpful but
not mandatory for solving the problem of synthesizing the optimal machining 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 information 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 first attempt to summarize the obtained results
of the research in the field in 1991. That year, my first books in the field of
surface generation (in Russian) were courageously introduced to the engineering community (Radzevich,  S.P.,  Sculptured Surface Machining on MultiAxis 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 from other sources (www.cse.buffalo.edu/~var2/).
There is a concern that some of today’s mechanical engineers, manufacturing 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


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xix

number of research and application papers and articles. Commonly, isolated
theoretical and practical findings 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 broadbased 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 practitioners 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 theory 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 synthesizing 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 floor environment.
This makes this book not just another book on the subject. For the first 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 surface generation, 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 position 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 benefit from this book: mechanical and manufacturing
engineers involved in continuous process improvement, research workers
who are active or intend to become active in the field, and senior undergraduate
and graduate students of mechanical engineering and manufacturing.

© 2008 by Taylor & Francis Group, LLC


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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 
and practical implementation of the proposed theory (Part III) will be of

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 surfaces, as well as principal elements of the theory of multiparametric
motion of a rigid body in E3 space. The applied coordinate systems
and linear transformations are briefly considered. The selected material 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 classification of surfaces is discussed, and a scientific classification of local patches of sculptured surfaces is proposed.
Chapter  2.  Kinematics of Surface Generation — The generalized 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 relative  motions of the cutting tool are discovered, including generating 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 kinematics of surface generation are discussed as well.
Chapter 3. Applied Coordinate Systems and Linear Transformations — The definitions and determinations of major applied
coordinate systems are introduced in this chapter. The matrix

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Preface

approach for the coordinate system transformations is briefly
discussed. Here, useful notations and practical equations are
provided. Two issues of critical importance are introduced here.
The first is chains of consequent linear transformations and a
closed loop of consequent coordinate systems transformations.
The impact of the coordinate systems transformations on fundamental 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-sufficient unit suited for
self-study.
Part II: Fundamentals — Fundamentals of the theory of surface generation 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 first order of tangency; a novel
kind of mapping of one surface onto another surface; a novel analytical 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 first part of the book. The following chapters
are included in this section.
Chapter 4. The Geometry of Contact of Two Smooth Regular Surfaces — Local characteristics of contact of two smooth, regular
surfaces that make tangency of the first 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 surfaces and the first- and second-order analyses. The concept of
conformity of two smooth, regular surfaces in the first order of
tangency is introduced and explained in this chapter. For the purposes of analyses, properties of Plücker’s conoid are implemented.
Ultimately, all feasible kinds of contact of the part and of the tool

surfaces are classified.
Chapter  5.  Profiling of the Form-Cutting Tools of Optimal Design
— A novel method of profiling the form-cutting tools for sculptured 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
profiling form-cutting tools for machining part surfaces on conventional machine tools are also considered. These methods are
based on elements of the theory of enveloping surfaces. Numerous particular issues of profiling form-cutting tools are discussed
at the end of the chapter.

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Preface
Chapter  6.  Geometry of Active Part of a Cutting Tool — The generating body of the form-cutting tool is bounded by the generating 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 computations are also presented.
Chapter  7.  Conditions of Proper Part Surface Generation — The
satisfactory conditions necessary and sufficient for proper part
surface machining are 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 concludes with the global verification of satisfaction of the conditions of proper part surface generation.
Chapter 8. Accuracy of Surface Generation — Accuracy is an important issue for the manufacturer of the machined part surfaces.
Analytical methods for the analysis and computation of the deviations of the machined part surface from the desired part surface are

discussed here. Two principal kinds of deviations of the machined
surface from the nominal part surface are distinguished. Methods
for the computation of the elementary surface deviations are proposed. 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 efficient
methods of part surface generation. Numerous practical examples of
implementation of the theory are disclosed in this part of the monograph. 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/Kinematics (DG/K)-based method of surface generation, an appropriate criterion of efficiency 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 efficiency of machining operation are considered. Tight connection
of the economical criteria of optimization with geometrical analogues (as established in Chapter  4) is illustrated. The part surface

© 2008 by Taylor & Francis Group, LLC


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xxiii

generation output is expressed in terms of functions of conformity. The last significantly simplifies 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 surface 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 developed on the premises of implementation of the proposed DG/Kbased 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
finishing involute gears.
The proposed theory of surface generation is oriented on extensive application of a multi-axis NC machine of modern design. In particular cases,
implementation of the theory can be useful for machining parts on conventional 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 extensive industrial experience in gear design and manufacture. He has developed numerous software packages dealing with computer-aided design
(CAD) and computer-aided manufacturing (CAM) of precise gear finishing
for a variety of industrial sponsors. Dr. Radzevich’s main research interest 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 multiflow gear trains, (d)  design of special-purpose gear cutting and finishing tools, (e)  design and machining (finishing) of precision gears for lownoise/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 scientific papers; and he holds more
than 150 patents in the field. 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 kindness and help must go unrecorded.

© 2008 by Taylor & Francis Group, LLC


Part I

Basics

© 2008 by Taylor & Francis Group, LLC


1
Part Surfaces: Geometry
The generation of part surfaces is one of the major purposes of machining operations. An enormous variety of parts are manufactured in various
industries. Every part to be machined is bounded with two or more surfaces. 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 computer-aided design (CAD) and computer-aided manufacturing (CAM) applications, 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, specifically, (a) in
the differential geometry of surfaces and (b) in the engineering geometry of
surfaces. The second is based on the first.
For further consideration, it is convenient to briefly 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 section 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 XP, YP, and ZP, as functions of two Gaussian coordinates UP and VP in a certain closed interval:
X P (U P , VP )
 Y (U , V ) 
P
P
P 
rP = rP (U P , VP ) = 
; (U1. P ≤ U P ≤ U 2. P ; V1. P ≤ VP ≤ V2. P )
 ZP (U P , VP ) 


1





(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