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

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 Sur

KINEMATIC GEOMETRY

OF SURFACE MACHINING

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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

To my son Andrew

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× 4 66

3.2.2 Translations 67

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

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 AnR(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 F1, F2, and F3 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

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 Tinto 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

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

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 DifferentReference Planes 2686.3.3.5 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 rbetween 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 284References 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

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 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

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 h ssof 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

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

“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-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

Ele-ments 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

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

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 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

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

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

multipa-rametric motion of a rigid body in E3space 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 first to formulate the problem of

synthesizing optimal surface generation, in the early 1980s In the

begin-ning, 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

sur-face of the form-cutting tool as the R-mapping of the sculptured sursur-face 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

sculp-tured surface P This means that the necessary input information for

solv-ing the problem of synthesizsolv-ing the optimal surface-machinsolv-ing 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 (seeChapter 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, filed: October 24, 1983; Radzevich, S.P., A Method of

Sculp-tured 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

Machin-ing on Multi-Axis NC Machine, Patent application 4242296/08 (USSR), filed: March 3, 1987.

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

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 field is helpful but

not mandatory for solving the 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 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

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 from other sources (www.cse.buffalo.edu/~var2/)

There is a concern that some of today’s mechanical engineers,

manufactur-ing engineers, and engineermanufactur-ing students may not be learnmanufactur-ing 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

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

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 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

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 benefit from this book: mechanical and manufacturing

engineers involved in continuous process improvement, research workers

whoareactiveorintendtobecomeactiveinthefield,andseniorundergraduate

and graduate students of mechanical engineering and manufacturing

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 E3space The applied coordinate systems

and linear transformations are briefly 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 classification of surfaces is discussed, and a scientific

classifica-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 definitions and determinations of major applied

coordinate systems are introduced in this chapter The matrix

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

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

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-sufficient 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 first 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 first 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 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

sur-faces 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

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 are classified

Chapter 5 Profiling of the Form-Cutting Tools of Optimal Design

— A novel method of profiling 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

profiling 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 profiling form-cutting tools are discussed

at the end of the chapter

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 cuttgenerat-ing 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 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

con-cludes with the global verification 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-tions 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

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 efficient

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 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

effi-ciency of machining operation are considered Tight connection

of the economical criteria of optimization with geometrical

ana-logues (as established inChapter 4) is illustrated The part surface

generation output is expressed in terms of functions of

confor-mity 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

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

finishing 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

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 finishing

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 multiflow gear trains, (d) design of special-purpose gear cutting and

fin-ishing tools, (e) design and machining (finfin-ishing) 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 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

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

Basics

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 comcom-puter-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, 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

sec-tion may be skipped Then, proceed directly to Secsec-tion 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:

( , )( , )( , )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.

where rP is the position vector of a point of the surface P; U P and V Pare

curvilinear (Gaussian) coordinates of the point of the surface P; X P , Y P , Z P are

Cartesian coordinates of the point of the surface P; U 1.P , U 2.Pare the boundary

values of the closed interval of the U P parameter; and V 1.P , V 2.Pare the

bound-ary values of the closed interval of the V Pparameter

The parameters U P and V P must enter independently, which means that

Y U

Z U X

V

Y V

Z V

P P

P P

P P P

P

P P

P P

has a rank 2 Positions where the rank is 1 or 0 are singular points; when the

rank at all points is 1, then Equation (1.1) represents a curve

The following notation proved the consideration below The first

deriva-tives of rP with respect to Gaussian coordinates U P and V P are designated as

∂rP /∂U P = UP and ∂rP /∂V P = VP, and for the unit tangent vectors uP = UP/ |UP|

and vP = VP/ |VP| correspondingly

Vector uP (as well as vector UP) specifies the direction of the tangent line

to the U P coordinate curve through the given point M on the surface P

Simi-larly, vector vP (as well as VP) specifies the direction of the tangent line to the

V P coordinate curve through that same point M on P.

rP

X P

vP

+V P +U P

Significance of the vectors uP and vP becomes evident from the following

considerations First, tangent vectors uP and vP yield an equation of the

tan-gent plane to the surface P at M:

Tangent plane

t p P M

P P

1 

where rt .P is the position vector of a point of the tangent plane to the surface P

at M, and r P( )M is the position vector of the point M on the surface P.

Second, tangent vectors yield an equation of the perpendicular NP, and of

the unit normal vector nP to the surface P at M:

When the order of multipliers in Equation (1.) is chosen properly, then the

unit normal vector nP is pointed outward of the bodily side of the surface P.

Unit tangent vectors uP and vP to a surface at a point are of critical

impor-tance when solving practical problems in the field of surface generation

Numerous examples, as shown below, prove this statement

Consider two other important issues concerning part surface geometry —

both relate to intrinsic geometry in differential vicinity of a surface point

The first issue is the first fundamental form of a surface P The first

funda-mental form f 1.P of a smooth, regular surface describes the metric properties

of the surface P Usually, it is represented as the quadratic form:

φ1.P⇒ds P2 =E dU P 2P+2F dU dV P P P+G dV P P2 (1.)

where s P is the linear element of the surface P (s P is equal to the length of a

segment of a certain curve line on the surface P), and E P , F P , G Pare

funda-mental magnitudes of the first order

Equation (1.) is known from many advanced sources In the theory of

sur-face generation, another form of analytical representation of the first

funda-mental form f 1.P is proven to be useful:

00

(1.6)

This kind of analytical representation of the first fundamental form f 1.P

is proposed by Radzevich [10] The practical advantage of Equation (1.6)

is that it can easily be incorporated into computer programs using

mul-tiple coordinate system transformations, which is vital for CAD/CAM

applications

For computation of the fundamental magnitudes of the first order, the

fol-lowing equations can be used:

E P=U UP⋅ P, F P=U VP⋅ P, G P=V VP⋅ P (1.7)

Fundamental magnitudes E P , F P , and G P of the first order are functions of

U P and V P parameters of the surface P In general form, these relationships

can be represented as E P = E P (U P , V P ), F P = F P (U P , V P ), and G P = G P (U P , V P)

Fundamental magnitudes E P and G P are always positive (E P > 0, G P > 0),

and the fundamental magnitude F P can equal zero (F P ≥ 0) This results in the

first fundamental form always being nonnegative (f 1.P ≥ 0).

The first fundamental form f 1.P yields computation of the following major

parameters of geometry of the surface P: (a) length of a curve-line segment

on the surface P, (b) square of the surface P portion that is bounded by a

closed curve on the surface, and (c) angle between any two directions on the

surface P.

The first fundamental form represents the length of a curve-line

seg-ment, and thus it is always nonnegative — that is, the inequality f 1.P ≥ 0 is

always observed

The discriminant H P of the first fundamental form f 1.P can be computed

from the following equation:

It is assumed that the discriminant H P is always nonnegative — that is, H P=+

E G P P−F P2

The fundamental form f 1.P remains the same while the surface is

band-ing This is another important feature of the first fundamental form f 1.P The

feature can be employed for designing three-dimensional cam for finishing

a turbine blade with an abrasive strip as a cutting tool

The second fundamental form of the surface P is another of the two

above-mentioned important issues The second fundamental form f 2.P describes

the curvature of a smooth, regular surface P Usually, it is represented as the

quadratic form

φ2.P⇒ −drP⋅dnP=L dU P 2P+2M dU dV P P P+N dV P P2 (1.9)

Equation (1.9) is known from many advanced sources

In the theory of surface generation, another analytical representation of

the second fundamental form f 2.P is proven useful:

00

(1.10)

This analytical representation of the second fundamental form f 2.P is

pro-posed by Radzevich [10] Similar to Equation (1.6), the practical advantage of

Equation (1.10) is that it can be easily incorporated into computer programs

using multiple coordinate system transformations, which is vital for CAD/

CAM applications

In Equation (1.10), the parameters L P , M P , N P designate fundamental

mag-nitudes of the second order Fundamental magmag-nitudes of the second order

can be computed from the following equations:

Fundamental magnitudes L P , M P , N P of the second order are also functions

of U P and V P parameters of the surface P These relationships in general form

can be represented as L P = L P (U P , V P ), M P = M P (U P , V P ), and N P = N P (U P , V P)

Discriminant T P of the second fundamental form f 2.P can be computed

from the following equation:

T P= L N P P−M P2 (1.12)

For computation of the principal directions T1.P and T2.P through a given point

on the surface P, the fundamental magnitudes of the second order L P , M P , N P,

together with the fundamental magnitudes of the first order E P , F P , G P, are used

Principal directions T1.P and T2.P can be computed as roots of the equation

The first principal plane section C 1.P is orthogonal to P at M and passes

through the first principal direction T1.P The second principal plane section

C 2.P is orthogonal to P at M and passes through the second principal

direc-tion T2.P

In the theory of surface generation, it is often preferred to use not the

vec-tors T1.P and T2.P of the principal directions, but instead to use the unit vectors

t1.P and t2.P of the principal directions The unit vectors t1.P and t2.P are

com-puted from equations t1.P = T1.P/|T1.P| and t2.P = T2.P/|T2.P|, respectively

The first R 1.P and the second R 2.P principal radii of curvature of the surface

P are measured in the first and in the second principal plane sections C 1.P

and C 2.P, correspondingly For computation of values of the principal radii of

curvature, use the following equation:

H T

P P

Another two important parameters of local topology of a surface P are (a) mean

curvature MP, and intrinsic (Gaussian or full) curvature GP These

param-eters can be computed from the following equations:

The formulae for MP= k1P+k2P

2 and GP=k1 P⋅k2 P yield a quadratic equation:

k P2−2MP P k +GP=0 (1.17)

with respect to principal curvatures k 1.P and k 2.P The expressions

k1 P=MP+ MP2−GP and k2 P=MP− MP2−GP (1.18)

are the solutions to Equation (1.17)

Here, k 1.P designates the first principal curvature of the surface P, and k 2.P

des-ignates the second principal curvature of the surface P at that same point The

principal curvatures k 1.P and k 2.P can be computed from k1.P=R1−.P and k 2.P= k2.P=

The first principal curvature k 1.P always exceeds the second principal curvature

k 2.P — that is, the inequality k 1.P> k 2.P is always observed

This brief consideration of elements of surface geometry allow for the

intro-duction of two definitions that are of critical importance for further discussion

Definition 1.1: Sculptured surface P is a smooth, regular surface with

major parameters of local topology that differ when in differential

vicin-ity of any two infinitely closed points.

 Remember that algebraic values of the radii of principal curvature R 1.P and R 2.P relate to each

other as R 2.P > R 1.P.

It is instructive to point out here that sculptured surface P does not allow

slid-ing “over itself.”

While machining a sculptured surface, the cutting tool rotates about its axis

and moves relative to the sculptured surface P While rotating with a certain

angular velocity ωT or while performing relative motion of another kind, the

cutting edges of the cutting tool generate a certain surface We refer to that

face represented by consecutive positions of cutting edges as the generating

sur-face of the cutting tool [11, 1, 1]:

Definition 1.2: The generating surface of a cutting tool can be represented

as the set of consecutive positions of the cutting edges in their motion

rela-tive to the stationary coordinate system, embedded to the cutting tool itself.

In most practical cases, the generating surface T allows sliding over itself The

enveloping surface to consecutive positions of the surface T that performs

such a motion is congruent to the surface T When machining a part, the

surface T is conjugate to the sculptured surface P.

Bonnet [1] proved that the specification of the first and second

fundamen-tal forms determines a unique surface if the Gauss’ characteristic equation

and the Codazzi-Mainardi’s relationships of compatibility are satisfied, and

those two surfaces that have identical first and second fundamental forms

are congruent. Six fundamental magnitudes determine a surface uniquely,

except as to position and orientation in space

Specification of a surface in terms of the first and the second fundamental

forms is usually called the natural kind of surface parameterization In

gen-eral form, it can be represented by a set of two equations:

The natural form

Equation (1.19) specify that same surface P In further consideration, the

nat-ural parameterization of the surface P plays an important role.

Illustrative Example

Consider an example of how an analytical representation of a surface in a

Cartesian coordinate system can be converted into the natural

parameteriza-tion of that same surface [1]

A gear tooth surface G is analytically described in a Cartesian coordinate

system X g Y g Z g (Figure 1.2)

 Two surfaces with the identical first and second fundamental forms might also be

symmetri-cal Refer to the literature— Koenderink, J.J., Solid Shape, The MIT Press, Cambridge, MA, 1990,

p 699—on differential geometry of surfaces for details about this specific issue.

The equation of the screw involute surface G is represented in matrix form:

This equation yields the computation of two tangent vectors Ug (U g ,V g ) and

Vg (U g ,V g ) that are correspondingly equal:

.

τττ1

.

ττ1

(1.21)

Ug rb.g

G

ng

Xg H

D

C M

Figure   1.2

Derivation of the natural form of the gear tooth surface G parameterization (From Radzevich,

S.P., Journal of Mechanical Design, 12, 772–786, 2002 With permission.)

Substituting the computed vectors Ug and Vg into Equation (1.7), one can

come up with formulae for computation of the fundamental magnitudes of

the first order:

τ

τand

cos

2

The computed values of the fundamental magnitudes E g , F g, and Gg can be

substituted to Equation (1.6) for f 1.g In this way, matrix representation of the

first fundamental form f 1.g can be computed The interested reader may wish

to complete this formulae transformation on his or her own

The discriminant H g of the first fundamental form of the surface G can be

computed from the formula H g = U g cosf b.g

In order to derive an equation for the second fundamental form f 2.g of the

gear-tooth surface G, the second derivatives of r g (U g , Vg ) with respect to U g

and V g parameters are necessary The above derived equations for the vectors

Ug and Vg yield the following computation:

0001

U

V V

cos coscos sin

.

ττ01



and

Further, substitute these derivatives (see Equation 1.2 and Equation 1.8

into Equation 1.11) After the necessary formulae transformations are

com-plete, then Equation (1.11) casts into the set of formulae for computation of

the second fundamental magnitudes of the surface G is as follows:

L g=0 M g=0 and N g= −U gsinτb g. cosτb g. (1.2)

After substituting Equation (1.2) into Equation (1.9), an equation for the

computation of the second fundamental form of the surface G can be obtained:

φ2 g⇒ −drg⋅dNg = −U gsinτb g. cosτb g. dV g2 (1.26)

Similar to Equation (1.2), the computed values of the fundamental

mag-nitudes L g , M g , and N g can be substituted into Equation (1.10) for f 2.g In this

way, matrix representation of the second fundamental form f 2.g can be

com-puted The interested reader may wish to complete this formulae

transfor-mation on his or her own

Discriminant T g of the second fundamental form f 2.g of the surface G is

equal to T g= L M g g−N g2 =0

The derived set of six equations for computation of the fundamental

mag-nitudes represents the natural parameterization of the surface P:

F g r b g

b g

= −

.

τ τ

N g= −U gsin τb g cos τb g.

All major elements of geometry of the gear-tooth surface can be computed

based on the fundamental magnitudes of the first f 1.g and of the second f 2.g

fundamental forms Location and orientation of the surface G are the two

parameters that remain indefinite

Once a surface is represented in natural form — that is, it is expressed in

terms of six fundamental magnitudes of the first and of the second order —

then further computation of parameters of the surface P becomes much

eas-ier In order to demonstrate significant simplification of the computation of

parameters of the surface P, several useful equations are presented below as

dV

dV dt

2 Value of the angle q between two given directions through a certain

point M on the surface P can be computed from one of the equations:

cosθ= F , sinθ= , tanθ=

E G

H

E G

H F P

P P

P

P P

P P

(1.28)

 For computation of square SP of a surface patch S, which is bounded

by a closed line on the surface P, the following equation can be used:

SP=∫∫ E G P P−F dU dV P2 P P

S

(1.29)

 Value of radius of curvature R P of the surface P in normal plane

section through M at a given direction can be computed from the

following equation:

p

P= φφ1 2

 Euler’s equation for the computation of R P is

kθ P=k1 Pcos2θ+k2 Psin2θ (1.1)This is also a good illustration of the above statement (Here q is the angle

that the normal plane section C P makes with the first principal plane section

C 1.P In other words, θ = ∠( ,t tP 1 P); here tP designates the unit tangent vector

within the normal plane section C P.)

Shape-index and the curve of the surface are two other useful properties

that are also drawn from the principal curvatures

The shape-index, SP, is a generalized measure of concavity and convexity

The shape-index varies from −1 to +1 It describes the local shape at a

surface point independent of the scale of the surface A shape-index value

of +1 corresponds to a concave local portion of the surface P for which the

principal directions are unidentified; thus, normal radii of curvature in all

directions are identical A shape-index of 0 corresponds to a saddle-like local

portion of the surface P with principal curvatures of equal magnitude but

The curvedness describes the scale of the surface P independent of its

shape

These quantities SP and R P are the primary differential properties of the

surface Note that they are properties of the surface itself and do not depend

upon its parameterization except for a possible change of sign

In order to get a profound understanding of differential geometry of

sur-faces, the interested reader may wish to go to advanced monographs in the

field Systematic discussion of the topic is available from many sources The

author would like to turn the reader’s attention to the monographs by doCarmo

[2], Eisenhart [], Struik [16], and others

1.2  On the Difference between Classical Differential 

Geometry and Engineering Geometry

Classical differential geometry is developed mostly for the purpose of

inves-tigation of smooth, regular surfaces Engineering geometry also deals with

the surfaces What is the difference between these two geometries?

The difference between classical differential geometry and between

engi-neering geometry is mostly due to how the surfaces are interpreted Only

phantom surfaces are studied in classical differential geometry Surfaces of

this kind do not exist in reality They can be imagined as a thin film of an

appropriate shape and with zero thickness Such film can be accessed from

both of the surface sides This causes the following indefiniteness

As an example, consider a surface having positive Gaussian curvature GP

at a surface point ( GP>0 ) Classical differential geometry gives no answer to

the question of whether the surface P is convex (M P>0 ) or concave (MP<0 )

at this point In classical differential geometry, the answer to this question can

be given only by convention A similar observation is made when Gaussian

curvature GP at a certain surface point is negative ( GP<0 )

Surfaces in classical differential geometry strictly follow the equation they

are specified by No deviation of the surface shape from what is predetermined

by the equation is allowed More examples can be found in the following

chap-ters of this book

In turn, surfaces that are treated in engineering geometry bound a part (or

machine element) This part can be called a real object (Figure 1.) The real

object is the bearer of the surface shape

Surfaces that bound real objects are accessible from only one side (Figure 1.)

We refer to this side of the surface as the open side of a surface The opposite side

of the surface P is not accessible Because of this, we refer to the opposite side

of the surface P as the closed side of a surface The positively directed normal unit

vector +nP is pointed outward from the part body — that is, from its bodily

side to the void side The negative normal unit vector −nP is pointed opposite

to +nP The existence of open and closed sides of a surface P eliminates the

problem of identifying whether the surface is convex or concave No

conven-tion in this concern is required

Another principal difference is due to the nature of the real object No

real object can be machined or manufactured precisely without deviations

of its actual shape from the desired shape of the real object Smaller or bigger

deviations of shape of the real object from the desired shape are unavoidable

in nature We do not go into detail here about this concern

Because of the deviations, the actual part surface P act deviates from its

nominal surface P (Figure 1.) However, the deviations do not exceed a

rea-sonable range Otherwise, the real object will become useless In practice, the

Figure   1.3

Examples of surfaces that mechanical engineers are dealing with (From Radzevich, S.P.,

Computer-Aided Design, 7 (7), 767–778, 200 With permission.)

Open Side of the Surface

Closed Side of the Surface

selection of appropriate tolerances on shape and dimensions of the actual

surface P act easily solve this particular problem

Similar to measuring deviations, the tolerances are measured in the

direc-tion of the unit normal vector nP to the surface P Positive tolerance d+ is

measuring along the positive direction of  nP, and negative tolerance d− is

measuring along the negative direction of nP In a particular case, one of the

tolerances, either d+ or d− can be equal to zero

Often, the value of tolerances d+ and d− are constant within the entire patch

of the surface P However, in special cases, for example when machining a

sculptured surface on a multi-axis numerical control machine, the actual

value of the tolerances d+ and d− can be set as functions of coordinates of

cur-rent point M on P This results in the tolerances being represented in terms

of U P and V P parameters of the surface P, say in the form d+= d+(U P , V P) and

d−= d−(U P , V P)

The endpoint of the vector d+∙ nP at a current surface point M produces

point M+ Similarly, the endpoint of the vector d−∙ nP produces the

corre-sponding point M−

The surface P+ of the upper tolerance is represented by loci of the points

M+ (i.e., by loci of endpoints of the vector d+∙ nP) This yields an analytical

representation of the surface of upper tolerance in the form

r–

Figure   1.5

An example of actual part surface P act.

Usually, the surface P+ of the upper tolerance is located above the nominal

surface P.

Similarly, the surface P − of lower tolerance is represented by loci of the

points M− (i.e., by the loci of endpoints of the vector d−∙ nP) This also yields

an analytical representation of the surface of lower tolerance in the form

rP−(U V P, )P = +rP δ−⋅nP (1.)

Commonly, the surface P− of lower tolerance is located beneath the nominal

surface P.

The actual part surface P act cannot be represented analytically.

More-over, the above-considered parameters of local topology of the surface P

cannot be computed for the surface P act However, because the tolerances

d+ and d− are small compared to the normal radii of curvature of the

nomi-nal surfaces P, it is assumed below that the surface Pact possesses the same

geometrical properties as the surface P does, and that the difference in

corresponding geometrical parameters of the surfaces P act and P is

negli-gibly small In further consideration, this yields replacement of the actual

surface P act with the nominal surface P, which is much more convenient for

performing computations

The consideration above illustrates the second principal difference between

classical differential geometry and the engineering geometry of surfaces

Because of the differences, engineering geometry often presents problems

that were not envisioned in classical (pure) differential geometry

1.3  On the Classification of Surfaces

The number of different surfaces that bound real objects is infinitely large A

systematic consideration of surfaces for the purposes of surface generation is

of critical importance

1.3.1  Surfaces That Allow Sliding over Themselves

In industry, a small number of surfaces with relatively simple geometry are

in wide use Surfaces of this kind allow for sliding over themselves The

property of a surface that allows sliding over itself means that for a certain

 Actually, surface P act is unknown — any surface located within the surfaces of upper tolerance

P+ and lower tolerance P− satisfies the requirements of the part blueprint; thus, every such

surface can be considered an actual surface P act An equation of the surface P act cannot be

rep-resented in the form P act P U V act

surface P there exists a corresponding motion of a special kind When

per-forming this motion, the enveloping surface to the consecutive position of

the moving surface P is congruent to the surface P itself The motion of the

mentioned kind can be monoparametric, biparametric, or triparametric

The screw surface of constant pitch (px = Const) is the most general kind

of surface that allows sliding over itself While performing the screw motion

of that same pitch px, the surface P is sliding over itself, similar to the

“bolt-and-nut” pair

When the pitch of a screw surface reduces to zero (px = 0), then the screw

surface degenerates to the surface of revolution Every surface of revolution

is sliding over itself when rotating

When the pitch of a screw surface rises to an infinitely large value, then the

screw surface degenerates into a general cylinder Surfaces of that kind allow

straight motion along straight generating lines of the surface

The considered kinds of surface motion are (a) screw motion of constant

pitch (px = Const), (b) rotation, and (c) straight motion, correspondingly All of

these motions are monoparametric

Surfaces like that of a circular cylinder allow rotation as well as straight

motion along the axis of the cylinder In this case, the surface motion is

biparametric (rotation and translation can be performed independently)

A sphere allows for rotations about three axes independently A plane

sur-face allows straight motion in two different directions as well as a rotation

about an axis that is orthogonal to the plane The surface motion in the last

two cases (for a sphere and for plane) is triparametric

Ultimately, one can summarize that surfaces allowing sliding over

them-selves are limited to screw surfaces of constant pitch, cylinders of general

kind, surfaces of revolution, circular cylinders, spheres, and planes It is

proven [12–1] that there are no other kinds of surfaces that allow for sliding

over themselves

Surfaces that allow sliding over themselves proved to be very convenient

in manufacturing as well as in industrial applications Most of the surfaces

being machined in various industries are surfaces of this nature

1.3.2  Sculptured Surfaces

Many products are designed with aesthetic sculptured surfaces to enhance

their aesthetic appeal, an important factor in customer satisfaction, especially

for automotive and consumer-electronics products In other cases,

prod-ucts have sculptured surfaces to meet functional requirements Examples

of functional surfaces can be easily found in aero-, gas- and hydrodynamic

applications (turbine blades), optical (lamp reflector) and medical (parts of

anatomical reproduction) applications, manufacturing surfaces (molding

die, die face), and so forth Functional surfaces interact with the environment

or with other surfaces Due to this, functional surfaces can also be called

dynamic surfaces