Geometry of SurfaceS

a Practical Guidefor mechanical enGineerS

Stephen p. Radzevich

GEOMETRY OFSURFACES

GEOMETRY OFSURFACESA PRACTICAL GUIDE FORMECHANICAL ENGINEERS

Stephen P. RadzevichPrincipal Gear Engineer, USA

A John Wiley & Sons, Ltd., Publication

This edition first published 2013C© 2013 John Wiley & Sons, Ltd

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Library of Congress Cataloging-in-Publication Data

Radzevich, S. P. (Stephen P.)Geometry of surfaces : a practical guide for mechanical engineers / by Stephen P. Radzevich.

pages cmIncludes bibliographical references and index.ISBN 978-1-118-52031-4 (hardback : alk. paper) – ISBN 978-1-118-52243-1 (mobi) –

ISBN 978-1-118-52270-7 (ebook) – ISBN 978-1-118-52271-4 (epub) – ISBN 978-1-118-52272-1 (ebook/epdf)1. Mechanical engineering–Mathematics. 2. Surfaces (Technology)–Mathematical models. 3. Geometry,Differential. I. Title.

TA418.7.R33 2013516.3024′621–dc23

2012035458

A catalogue record for this book is available from the British Library.

ISBN: 978-1-118-52031-4

Typeset in 10/12pt Times by Aptara Inc., New Delhi, India

This book is dedicated to my wife Natasha

Contents

About the Author xiii

Preface xv

Acknowledgments xvii

Glossary xix

Notation xxi

Introduction xxv

Part I PART SURFACES 1

1 Geometry of a Part Surface 31.1 On the Analytical Description of Ideal Surfaces 31.2 On the Difference between Classical Differential Geometry and Engineering

Geometry of Surfaces 61.3 On the Analytical Description of Part Surfaces 71.4 Boundary Surfaces for an Actual Part Surface 91.5 Natural Representation of a Desired Part Surface 11

1.5.1 First fundamental form of a desired part surface 121.5.2 Second fundamental form of a desired part surface 141.5.3 Illustrative example 16

1.6 Elements of Local Geometry of a Desired Part Surface 191.6.1 Unit tangent vectors 191.6.2 Tangent plane 201.6.3 Unit normal vector 201.6.4 Unit vectors of principal directions on a part surface 211.6.5 Principal curvatures of a part surface 211.6.6 Other parameters of curvature of a part surface 23

2 On the Possibility of Classification of Part Surfaces 272.1 Sculptured Part Surfaces 27

2.1.1 Local patches of ideal part surfaces 272.1.2 Local patches of real part surfaces 29

viii Contents

2.2 Planar Characteristic Images 332.2.1 Dupin indicatrix 332.2.2 Curvature indicatrix 382.2.3 Circular chart for local patches of smooth regular part surfaces

based on curvature indicatrix 412.3 Circular Diagrams at a Surface Point 42

2.3.1 Circular diagrams 422.3.2 Circular chart for local patches of smooth regular part surfaces

based on circular diagrams 522.4 One More Useful Characteristic Curve 53

Part II GEOMETRY OF CONTACT OF PART SURFACES 55

3 Early Works in the Field of Contact Geometry 573.1 Order of Contact 573.2 Contact Geometry of Part Surfaces 593.3 Local Relative Orientation of the Contacting Part Surfaces 593.4 First-Order Analysis: Common Tangent Plane 643.5 Second-Order Analysis 65

3.5.1 Comments on analytical description of the local geometry ofcontacting surfaces loaded by a normal force: Hertzproportional assumption 65

3.5.2 Surface of normal relative curvature 683.5.3 Dupin indicatrix of the surface of relative normal curvature 713.5.4 Matrix representation of equation of the Dupin indicatrix

of the surface of relative normal curvature 723.5.5 Surface of relative normal radii of curvature 733.5.6 Normalized relative normal curvature 733.5.7 Curvature indicatrix of the surface of relative normal

curvature 743.6 A Characteristic Curve Irk(R ) of Novel Kind 75

4 An Analytical Method Based on Second Fundamental Forms of theContacting Part Surfaces 79

5 Indicatrix of Conformity of Two Smooth Regular Surfaces in theFirst Order of Tangency 83

5.1 Preliminary Remarks 835.2 Indicatrix of Conformity for Two Smooth Regular Part Surfaces in the First

Order of Tangency 875.3 Directions of Extremum Degree of Conformity of Two Part Surfaces

in Contact 945.4 Asymptotes of the Indicatrix of Conformity CnfR (P1/P2) 97

Contents ix

5.5 Comparison of Capabilities of Indicatrix of Conformity Cnf R(P1/P2) and ofDupin Indicatrix of the Surface of Relative Curvature Dup (R ) 98

5.6 Important Properties of Indicatrix of Conformity Cnf R(P/T ) of Two SmoothRegular Part Surfaces 99

5.7 The Converse Indicatrix of Conformity of Two Regular Part Surfaces in theFirst Order of Tangency 99

6 Plucker Conoid: More Characteristic Curves 1016.1 Plucker Conoid 101

6.1.1 Basics 1016.1.2 Analytical representation 1026.1.3 Local properties 1036.1.4 Auxiliary formulae 104

6.2 On Analytical Description of Local Geometry of a Smooth RegularPart Surface 1056.2.1 Preliminary remarks 1056.2.2 The Plucker conoid 1066.2.3 Plucker curvature indicatrix 1086.2.4 AnR(P1)-indicatrix of a part surface 109

6.3 Relative Characteristic Curve 1126.3.1 On a possibility of implementation of two Plucker conoids 1126.3.2 AnR(P1/P2)-relative indicatrix of two contacting part surfaces

P1 and P2 113

7 Feasible Kinds of Contact of Two Smooth Regular Part Surfaces in theFirst Order of Tangency 117

7.1 On the Possibility of Implementation of the Indicatrix of Conformity for thePurposes of Identification of the Actual Kind of Contact of Two SmoothRegular Part Surfaces 117

7.2 Impact of Accuracy of the Computation on the Parameters of the Indicatricesof Conformity Cnf R(P1/P2) 121

7.3 Classification of Possible Kinds of Contact of Two Smooth RegularPart Surfaces 122

Part III MAPPING OF THE CONTACTING PART SURFACES 131

8 R-Mapping of the Interacting Part Surfaces 1338.1 Preliminary Remarks 1338.2 On the Concept of R-Mapping of the Interacting Part Surfaces 1348.3 R-mapping of a Part Surface P1 onto Another Part Surface P2 1368.4 Reconstruction of the Mapped Part Surface 1408.5 Illustrative Examples of the Calculation of the Design Parameters of the

Mapped Part Surface 141

x Contents

9 Generation of Enveloping Surfaces: General Consideration 1459.1 Envelope for Successive Positions of a Moving Planar Curve 1459.2 Envelope for Successive Positions of a Moving Surface 149

9.2.1 Envelope for a one-parametric family of surfaces 1499.2.2 Envelope for a two-parametric family of surfaces 152

9.3 Kinematic Method for Determining Enveloping Surfaces 1549.4 Peculiarities of Implementation of the Kinematic Method in Cases of

Multi-parametric Relative Motion of the Surfaces 164

10 Generation of Enveloping Surfaces: Special Cases 16710.1 Part Surfaces that Allow for Sliding Over Themselves 16710.2 Reversibly Enveloping Surfaces: Introductory Remarks 16910.3 Generation of Reversibly Enveloping Surfaces 180

10.3.1 Kinematics of crossed-axis gearing 18010.3.2 Base cones in crossed-axis gear pairs 18210.3.3 Tooth flanks of geometrically accurate (ideal)

crossed-axis gear pairs 18610.3.4 Tooth flank of a crossed-axis gear 192

10.4 On the Looseness of Two Olivier Principles 19710.4.1 An example of implementation of the first Olivier principle for

generation of enveloping surfaces in a degenerate case 19810.4.2 An example of implementation of the second Olivier principle for

generation of enveloping surfaces in a degenerate case 19910.4.3 Concluding remarks 200

Conclusion 203

APPENDICES 205

Appendix A: Elements of Vector Calculus 207A.1 Fundamental Properties of Vectors 207A.2 Mathematical Operations over Vectors 207

Appendix B: Elements of Coordinate System Transformations 211B.1 Coordinate System Transformation 211

B.1.1 Introduction 211B.1.2 Translations 213B.1.3 Rotation about a coordinate axis 214B.1.4 Resultant coordinate system transformation 215B.1.5 Screw motion about a coordinate axis 216B.1.6 Rolling motion of a coordinate system 218B.1.7 Rolling of two coordinate systems 220

B.2 Conversion of the Coordinate System Orientation 222B.3 Transformation of Surface Fundamental Forms 223

Contents xi

Appendix C: Change of Surface Parameters 225

References 227

Bibliography 229

Index 233

About the Author

Dr. Stephen P. Radzevich is a Professor of Mechanical Engineering and a Professor ofManufacturing Engineering. He received the M.Sc. (1976), Ph.D. (1982) and Dr.(Eng.)Sc.(1991) – all in mechanical engineering. Dr. Radzevich has extensive industrial experience ingear design and manufacture. He has developed numerous software packages dealing withCAD and CAM of precise gear finishing for a variety of industrial sponsors. His main researchinterest is in the kinematic geometry of surface generation, particularly focusing on (a) pre-cision gear design, (b) high power density gear trains, (c) torque share in multi-flow geartrains, (d) design of special purpose gear cutting/finishing tools, (e) design and machining(finishing) of precision gears for low-noise/noiseless transmissions for cars, light trucks, etc.Dr. Radzevich has spent about 40 years developing software, hardware and other processesfor gear design and optimization. Besides his work for industry, he trains engineering stu-dents at universities and gear engineers in companies. He has authored and co-authored over30 monographs, handbooks and textbooks. Monographs entitled “Generation of Surfaces”(2001), “Kinematic Geometry of Surface Machining” (CRC Press, 2008), “CAD/CAM of Sculp-tured Surfaces on Multi-Axis NC Machine: The DG/K-Based Approach” (M&C Publishers,2008), “Gear Cutting Tools: Fundamentals of Design and Computation” (CRC Press, 2010),“Precision Gear Shaving” (Nova Science Publishers, 2010), “Dudley’s Handbook of PracticalGear Design and Manufacture” (CRC Press, 2012) and “Theory of Gearing: Kinematics,Geometry, and Synthesis” (CRC Press, 2012) are among recently published volumes. Hehas also authored and co-authored over 250 scientific papers, and holds over 200 patents oninventions in the field.

Preface

This book is about the geometry of part surfaces, their generation and interaction with oneanother. Written by a mechanical engineer, this book is not on the differential geometry ofsurfaces. Instead, this book is devoted to the application of methods developed in the differentialgeometry of surfaces, for the purpose of solving problems in mechanical engineering.

A paradox exists in our present understanding of geometry of surfaces: we know everythingabout ideal surfaces, which do not exist in reality, and we know almost nothing about realsurfaces, which exist physically. Therefore, one of the main goals of this book is to adjustour knowledge of ideal surfaces for the purpose of better understanding the geometry of realsurfaces. In other words: to bridge a gap between ideal and real surfaces. One of the significantadvantages of the book is that it has been written not by a mathematician, but by a mechanicalengineer for mechanical engineers.

Acknowledgments

I would like to share the credit for any research success with my numerous doctoral stu-dents, with whom I have tested the proposed ideas and applied them in industry. The manyfriends, colleagues and students who contributed are overwhelming in number and cannot beacknowledged individually – as much as they have contributed, their kindness and help mustgo unrecorded.

My thanks also go to those at John Wiley who took over the final stages and will have tomanage the marketing and sale of the fruit of my efforts.

Glossary

We list, alphabetically, the most commonly used terms in engineering geometry of surfaces.In addition, most of the newly introduced terms are listed below as well.

Auxiliary generating surface R a smooth regular surface that is used as an intermedi-ate (auxiliary) surface when an envelope for successive positions of a moving surface isdetermined.

CA-gearing crossed-axis gearing, or a gearing having axes of rotation of the gear and of thepinion that are skewed in relation to one another.

Cartesian coordinate system a reference system comprised of three mutually perpendicularstraight axes through the common origin. Determination of the location of a point in aCartesian coordinate system is based on the distances along the coordinate axes. Commonly,the axes are labeled X , Y and Z . Often, either a subscript or a superscript is added to thedesignation of the reference system XYZ.

Center-distance this is the closest distance of approach between the two axes of rotation. Inthe particular case of hypoid gearing, the center-distance is often referred to as the offset.

Characteristic line this is a limit configuration of the line of intersection of a movingsurface that occupies two distinct positions when the distance between the surfaces in thesepositions is approaching zero. In the limit case, a characteristic line aligns with the lineof tangency of the moving surface and with the envelope for successive positions of themoving surface.

Darboux frame in the differential geometry of surfaces, this is a local moving Cartesianreference system constructed on a surface. The origin of a Darboux frame is at a currentpoint of interest on the surface. The axes of the Darboux frame are along three unit vectors,namely along the unit normal vector to the surface, and two unit tangent vectors alongprincipal directions of the gear tooth flank. The Darboux frame is analogous to the Frenet–Serret frame applied to surface geometry. A Darboux frame exists at any non-umbilic pointof a surface. It is named after the French mathematician Jean Gaston Darboux.

Degree of conformity this is a qualitative parameter to evaluate how close the tooth flankof one member of a gear pair is to the tooth flank of another member of the gear pair at apoint of their contact (or at a point within the line of contact of the teeth flanks).

Dynamic surface this is a part surface that is interacting with the environment.Engineering surface this is a part surface that can be reproduced on a solid using for these

purposes any production method.

xx Glossary

Free-form surface this is a kind of part surface (see: sculptured part surface for moredetails).

Indicatrix of conformity this is a planar centro-symmetrical characteristic curve of fourthorder that is used for the purpose of analytical description of geometry of contact of the geartooth flank and of the pinion tooth flank. In particular cases, the indicatrix of conformityalso possesses the property of mirror symmetry.

Natural kind of surface representation specification of a surface in terms of the first andsecond fundamental forms, commonly referred to as a natural kind of surface representation.

Part surface this is one of numerous surfaces that bound a solid.Point of contact this is any point at which two tooth profiles touch each other.R-gearing a kind of crossed-axis gearing that features line contact between tooth flanks of

the gear and pinion.R-mapping of the interacting part surfaces a kind of mapping of one smooth regular part

surface onto another smooth regular part surface, under which normal curvatures at everypoint of the mapped surface correspond to normal curvatures at a corresponding point of agiven surface.

Reversibly enveloping surfaces (or just Re-surfaces) a pair of smooth regular part surfacesthat are enveloping one another regardless of which one of the surfaces is traveling and whichone of them is enveloping.

Rotation vector a vector along an axis of rotation having magnitude equal to the rotationof the axis. The direction of the rotation vector depends upon the direction of the rotation.Commonly, the rotation vector is designated ω. The magnitude of the rotation vector iscommonly denoted ω. Therefore, the equality ω = |ω| is valid. The rotation vector of a gearis designated ωg , the rotation vector of the mating pinion is designated ωp, and the rotationvector of the plane of action is designated ωpa .

Sculptured part surface this is a kind of part surface parameter of local geometry whereevery two neighboring infinitesimally small patches differ from one another. Free-formsurface is another terminology that is used for part surfaces of this particular kind.

Spr-gearing a kind of crossed-axis gearing that features base pitch of the gear, base pitch ofthe pinion, and operating base pitch, which are equal to one another under various valuesof the axis misalignment. Gearing of this kind is noiseless and capable of transmitting thehighest possible power density.

Surface that allows for sliding over itself a smooth regular surface for which there existsa motion that results in the envelope for successive positions of the moving surface beingcongruent to the surface itself.

Vector of instant rotation a vector along the axis of instant rotation, either of the pinionin relation to the gear or of the gear in relation to the pinion. The direction of the vectorof instant rotation depends upon the direction of rotation of the gear and of the pinion.Commonly, the rotation vector is designated ωpl .

Notation

AP1 apex of the base cone of the part surface P1

AP2 apex of the base cone of the part surface P2

Apa apex of the plane of action, PAC center-distanceC 1.P1, C 2.P1 the first and second principal plane sections of the traveling part surface P1

C 1.P2, C 2.P2 the first and second principal plane sections of the generated part surfaceP2 (the enveloping surface)

Cnf R(P1/P2) indicatrix of conformity for two smooth regular part surfaces P1 and P2 ata current contact point K

Cnf k(P1/R 2) indicatrix of conformity that is converse to the indicatrix Cnf R(P1/P2)E a characteristic lineEP1, FP1, G P1 fundamental magnitudes of first order of the smooth regular part surface

P1

EP2, FP2, G P2 fundamental magnitudes of first order of the smooth regular part surfaceP2

GP Gaussian curvature of a part surface P at a point mMP mean curvature of a surface P at a point mK point of contact of two smooth regular part surfaces P1 and P2 (or a point

within a line of contact of the part surfaces P1 and P2)Lc (or LC) line of contact between two regular part surfaces P1 and P2

L P1, MP1, NP1 fundamental magnitudes of second order of the smooth regular part surfaceP1

L P2, MP2, NP2 fundamental magnitudes of second order of the smooth regular part surfaceP2

OP1 axis of rotation of the part surface P1

OP2 axis of rotation of the part surface P2

Opa axis of rotation of the plane of action, PAPA plane of actionPln axis of instant rotation of two regular part surfaces P1 and P2 in relation to

one anotherRc (PA �→ G ) the operator of rolling/sliding (the operator of transition from the plane of

action, PA, to the gear, G , in crossed-axis gearing)Rc (PA �→ P ) the operator of rolling/sliding (the operator of transition from the plane of

action, PA, to the pinion, P , in crossed-axis gearing)

xxii Notation

Rlx (ϕy, Y ) the operator of rolling over a plane (Y -axis is the axis of rotation, X -axis isthe axis of translation)

Rlz(ϕy, Y ) the operator of rolling over a plane (Y -axis is the axis of rotation, Z -axis isthe axis of translation)

Rly(ϕx , X ) the operator of rolling over a plane (X -axis is the axis of rotation, Y -axis isthe axis of translation)

Rlz(ϕx , X ) the operator of rolling over a plane (X -axis is the axis of rotation, Z -axis isthe axis of translation)

Rlx (ϕz, Z ) the operator of rolling over a plane (Z -axis is the axis of rotation, X -axis isthe axis of translation)

Rly(ϕz, Z ) the operator of rolling over a plane (Z -axis is the axis of rotation, Y -axis isthe axis of translation)

Rru(ϕ, Z ) the operator of rolling of two coordinate systemsRs (A �→ B) the operator of the resultant coordinate system transformation, say from a

coordinate system A to a coordinate system BRt (ϕx , X ) the operator of rotation through an angle ϕx about the X -axisRt (ϕy, Y ) the operator of rotation through an angle ϕy about the Y -axisRt (ϕz, Z ) the operator of rotation through an angle ϕz about the Z -axisR 1.P1, R 2.P1 the first and second principal radii of the gear tooth flank P1

R 1.P2, R 2.P2 the first and second principal radii of the gear tooth flank P2

Scx (ϕx , px ) the operator of screw motion about the X -axisScy(ϕy, py) the operator of screw motion about the Y -axisScz(ϕz, pz) the operator of screw motion about the Z -axisTr (ax , X ) the operator of translation at a distance ax along the X -axisTr (ay, Y ) the operator of translation at a distance ay along the Y -axisTr (az, Z ) the operator of translation at a distance az along the Z -axisUP1, VP1 curvilinear (Gaussian) coordinates of a point of a smooth regular part

surface P1

UP2, VP2 curvilinear (Gaussian) coordinates of a point of a smooth regular partsurface P2

UP1, VP1 tangent vectors to curvilinear coordinate lines on a smooth regular partsurface P1

UP2, VP2 tangent vectors to curvilinear coordinate lines on a smooth regular partsurface P2

V� vector of the resultant motion of the smooth regular part surface P1 inrelation to a reference system that the smooth regular part surface P2 willbe associated with

dmincnf minimal diameter of the indicatrix of conformity Cnf R(P1/P2) for two

smooth regular part surfaces P1 and P2 at a current contact point Kk 1.P1, k 2.P1 the first and second principal curvatures of the smooth regular part surface

P1

k 1.P2, k 2.P2 the first and second principal curvatures of the smooth regular part surfaceP2

nP unit normal vector to a smooth regular part surface Ppsc screw parameter (reduced pitch) of instant screw motion of the part surface

P1 in relation to the part surface P2

Notation xxiii

rP1 position vector of a point of a smooth regular part surface P1

rcnf position vector of a point of the indicatrix of conformity Cnf R(P1/P2) fortwo smooth regular part surfaces P1 and P2 at a current contact point K

t 1.P1, t 2.P1 unit tangent vectors of principal directions on the smooth regular partsurface P1

t 1.P2, t 2.P2 unit tangent vectors of principal directions on the smooth regular partsurface P2

uP1, vP1 unit tangent vectors to curvilinear coordinate lines on the smooth regularpart surface P1

uP2, vP2 unit tangent vectors to curvilinear coordinate lines on the smooth regularpart surface P2

xP yP zP local Cartesian coordinate system having its origin at a current point ofcontact of the part surfaces P1 and P2

Greek symbols

� 1.P1, � 2.P1 the first and second fundamental forms of the smooth regular part surface P1

� 1.P2, � 2.P2 the first and second fundamental forms of the smooth regular part surface P2

φn.ω normal pressure angleμ angle of the part surfaces’ P1 and P2 local relative orientationωP1 rotation vector of the regular part surface P1

ωP2 rotation vector of the part surface P2

ωpl vector of instant rotation of the part surfaces P1 and P2 in relation to oneanother

Subscripts

cnf conformitymax maximummin minimumn normalopt optimal

Introduction

The performance of parts depends largely on the geometry of the interacting surfaces. Anin-depth investigation of the geometry of smooth regular part surfaces is undertaken in thisbook. An analytical description of the surfaces, and the methods of their generation, alongwith an analytical approach for description of the geometry of contact of the interacting partsurfaces, is covered. The book comprises three parts, and appendices.

The specification of part surfaces in terms of the corresponding nominal smooth regularsurface is considered in Part I of the book.

The geometry of part surfaces is discussed in Chapter 1. The discussion begins with ananalytical description of ideal surfaces. Here, the ideal surface is interpreted as a zero-thicknessfilm. The difference between classical differential geometry and engineering geometry ofsurfaces is analyzed. This analysis is followed by an analytical description of real part surfaces,based largely on an analytical description of the corresponding ideal surface. It is shown thatwhile it remains unknown, a real part surface is located between two boundary surfaces. Thesaid boundary surfaces are represented by two ideal surfaces, of upper tolerance and lowertolerance. The specification of surfaces ends with a discussion of the natural representationof a desired part surface. This consideration involves the first and second fundamental formsof a smooth regular part surface. For an analytically specified surface, the elements of itslocal geometry are outlined. This consideration includes but is not limited to an analyticalrepresentation of the unit tangent vectors, the tangent plane, the unit normal vector, the unitvectors of principal directions on a part surface, etc. Ultimately, the parameters of part surfacecurvature are discussed. Mostly, the equations for principal surface curvatures along withnormal curvatures at a surface point are considered. In addition to the mean curvature, theGaussian curvature, absolute curvature, shape operator and curvedness of a surface at a pointare considered. The classification of local part surface patches is proposed in this section ofthe book. The classification is followed by a circular chart comprising all possible kinds oflocal part surface patches.

Chapter 2 is devoted to the analysis of a possibility of classification of part surfaces.Regardless of the fact that no scientific classification of smooth regular surfaces in a globalsense is feasible in nature, local part surface patches can be classified. For an investigationof the geometry of local part surface patches, planar characteristic images are employed. Inthis analysis the Dupin indicatrix, curvature indicatrix and circular diagrams at a part surfacepoint are covered in detail. Based on the results of the analysis, two more circular charts aredeveloped. One of them employs the part surface curvature indicatrices, while the other isbased on the properties of circular diagrams at a current part surface point. This section of the