Geometry of Surfaces E-Book

Stephen P. Radzevich

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 421

Veröffentlichungsjahr: 2013

Beschreibung

Presents an in-depth analysis of geometry of part surfaces and provides the tools for solving complex engineering problems Geometry of Surfaces: A Practical Guide for Mechanical Engineers is a comprehensive guide to applied geometry of surfaces with focus on practical applications in various areas of mechanical engineering. The book is divided into three parts on Part Surfaces, Geometry of Contact of Part Surfaces and Mapping of the Contacting Part Surfaces. Geometry of Surfaces: A Practical Guide for Mechanical Engineers combines differential geometry and gearing theory and presents new developments in the elementary theory of enveloping surfaces. Written by a leading expert of the field, this book also provides the reader with the tools for solving complex engineering problems in the field of mechanical engineering. * Presents an in-depth analysis of geometry of part surfaces * Provides tools for solving complex engineering problems in the field of mechanical engineering * Combines differential geometry and gearing theory * Highlights new developments in the elementary theory of enveloping surfaces Essential reading for researchers and practitioners in mechanical, automotive and aerospace engineering industries; CAD developers; and graduate students in Mechanical Engineering.

Leseprobe

Contents

Cover

Title Page

Copyright

Dedication

About the Author

Preface

Acknowledgments

Glossary

Notation

Introduction

Part I: Part Surfaces

Chapter 1: Geometry of a Part Surface

1.1 On the Analytical Description of Ideal Surfaces

1.2 On the Difference between Classical Differential Geometry and Engineering Geometry of Surfaces

1.3 On the Analytical Description of Part Surfaces

1.4 Boundary Surfaces for an Actual Part Surface

1.5 Natural Representation of a Desired Part Surface

1.6 Elements of Local Geometry of a Desired Part Surface

Chapter 2: On the Possibility of Classification of Part Surfaces

2.1 Sculptured Part Surfaces

2.2 Planar Characteristic Images

2.3 Circular Diagrams at a Surface Point

2.4 One More Useful Characteristic Curve

Part II: Geometry of Contact of Part Surfaces

Chapter 3: Early Works in the Field of Contact Geometry

3.1 Order of Contact

3.2 Contact Geometry of Part Surfaces

3.3 Local Relative Orientation of the Contacting Part Surfaces

3.4 First-Order Analysis: Common Tangent Plane

3.5 Second-Order Analysis

3.6 A Characteristic Curve of Novel Kind

Chapter 4: An Analytical Method Based on Second Fundamental Forms of the Contacting Part Surfaces

Chapter 5: Indicatrix of Conformity of Two Smooth Regular Surfaces in the First Order of Tangency

5.1 Preliminary Remarks

5.2 Indicatrix of Conformity for Two Smooth Regular Part Surfaces in the First Order of Tangency

5.3 Directions of Extremum Degree of Conformity of Two Part Surfaces in Contact

5.4 Asymptotes of the Indicatrix of Conformity CnfR (P1/P2)

5.5 Comparison of Capabilities of Indicatrix of Conformity CnfR(P1/P2) and of Dupin Indicatrix of the Surface of Relative Curvature DUP()

5.6 Important Properties of Indicatrix of Conformity CnfR(P/T) of Two Smooth Regular Part Surfaces

5.7 The Converse Indicatrix of Conformity of Two Regular Part Surfaces in the First Order of Tangency

Chapter 6: Plücker Conoid: More Characteristic Curves

6.1 Plücker Conoid

6.2 On Analytical Description of Local Geometry of a Smooth Regular Part Surface

6.3 Relative Characteristic Curve

Chapter 7: Feasible Kinds of Contact of Two Smooth Regular Part Surfaces in the First Order of Tangency

7.1 On the Possibility of Implementation of the Indicatrix of Conformity for the Purposes of Identification of the Actual Kind of Contact of Two Smooth Regular Part Surfaces

7.2 Impact of Accuracy of the Computation on the Parameters of the Indicatrices of Conformity CnfR(P1/P2)

7.3 Classification of Possible Kinds of Contact of Two Smooth Regular Part Surfaces

Part III: Mapping of the Contacting Part Surfaces

Chapter 8: R-Mapping of the Interacting Part Surfaces

8.1 Preliminary Remarks

8.2 On the Concept of R-Mapping of the Interacting Part Surfaces

8.3 R-mapping of a Part Surface P1 onto Another Part Surface P2

8.4 Reconstruction of the Mapped Part Surface

8.5 Illustrative Examples of the Calculation of the Design Parameters of the Mapped Part Surface

Chapter 9: Generation of Enveloping Surfaces: General Consideration

9.1 Envelope for Successive Positions of a Moving Planar Curve

9.2 Envelope for Successive Positions of a Moving Surface

9.3 Kinematic Method for Determining Enveloping Surfaces

9.4 Peculiarities of Implementation of the Kinematic Method in Cases of Multi-parametric Relative Motion of the Surfaces

Chapter 10: Generation of Enveloping Surfaces: Special Cases

10.1 Part Surfaces that Allow for Sliding Over Themselves

10.2 Reversibly Enveloping Surfaces: Introductory Remarks

10.3 Generation of Reversibly Enveloping Surfaces

10.4 On the Looseness of Two Olivier Principles

Conclusion

Appendices

Appendix A: Elements of Vector Calculus

A.1 Fundamental Properties of Vectors

A.2 Mathematical Operations over Vectors

Appendix B: Elements of Coordinate System Transformations

B.1 Coordinate System Transformation

B.2 Conversion of the Coordinate System Orientation

B.3 Transformation of Surface Fundamental Forms

Appendix C: Change of Surface Parameters

References

Bibliography

Index

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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 cm Includes 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 2013 516.3024′621–dc23 2012035458

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

ISBN: 978-1-118-52031-4

This book is dedicated to my wife Natasha

About the Author

Dr. Stephen P. Radzevich is a Professor of Mechanical Engineering and a Professor of Manufacturing 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 in gear design and manufacture. He has developed numerous software packages dealing with CAD and CAM of precise gear finishing for a variety of industrial sponsors. His main research interest is in the kinematic geometry of surface generation, particularly focusing on (a) precision gear design, (b) high power density gear trains, (c) torque share in multi-flow gear trains, (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 processes for gear design and optimization. Besides his work for industry, he trains engineering students at universities and gear engineers in companies. He has authored and co-authored over 30 monographs, handbooks and textbooks. Monographs entitled “Generation of Surfaces” (2001), “Kinematic Geometry of Surface Machining” (CRC Press, 2008), “CAD/CAM of Sculptured 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 Practical Gear Design and Manufacture” (CRC Press, 2012) and “Theory of Gearing: Kinematics, Geometry, and Synthesis” (CRC Press, 2012) are among recently published volumes. He has also authored and co-authored over 250 scientific papers, and holds over 200 patents on inventions in the field.

Preface

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

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

Acknowledgments

I would like to share the credit for any research success with my numerous doctoral students, with whom I have tested the proposed ideas and applied them in industry. The many friends, colleagues and students who contributed are overwhelming in number and cannot be acknowledged individually – as much as they have contributed, their kindness and help must go unrecorded.

My thanks also go to those at John Wiley who took over the final stages and will have to manage 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 intermediate (auxiliary) surface when an envelope for successive positions of a moving surface is determined.

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

Cartesian coordinate system   a reference system comprised of three mutually perpendicular straight axes through the common origin. Determination of the location of a point in a Cartesian 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 the designation of the reference system XYZ.

Center-distance   this is the closest distance of approach between the two axes of rotation. In the 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 moving surface that occupies two distinct positions when the distance between the surfaces in these positions is approaching zero. In the limit case, a characteristic line aligns with the line of tangency of the moving surface and with the envelope for successive positions of the moving surface.

Darboux frame   in the differential geometry of surfaces, this is a local moving Cartesian reference system constructed on a surface. The origin of a Darboux frame is at a current point 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 along principal 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 point of 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 flank of one member of a gear pair is to the tooth flank of another member of the gear pair at a point 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.

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

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

Natural kind of surface representation   specification of a surface in terms of the first and second 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.

-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 every point of the mapped surface correspond to normal curvatures at a corresponding point of a given surface.

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

Rotation vector   a vector along an axis of rotation having magnitude equal to the rotation of the axis. The direction of the rotation vector depends upon the direction of the rotation. Commonly, the rotation vector is designated . The magnitude of the rotation vector is commonly denoted . Therefore, the equality is valid. The rotation vector of a gear is designated , the rotation vector of the mating pinion is designated , and the rotation vector of the plane of action is designated .

Sculptured part surface   this is a kind of part surface parameter of local geometry where every two neighboring infinitesimally small patches differ from one another. Free-form surface 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 of the pinion, and operating base pitch, which are equal to one another under various values of the axis misalignment. Gearing of this kind is noiseless and capable of transmitting the highest possible power density.

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

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

Notation

AP1apex of the base cone of the part surface P1AP2apex of the base cone of the part surface P2Apaapex of the plane of action, PACcenter-distanceC1.P1, C2.P1the first and second principal plane sections of the traveling part surface P1C1.P2, C2.P2the first and second principal plane sections of the generated part surface P2 (the enveloping surface)CnfR(P1/P2)indicatrix of conformity for two smooth regular part surfaces P1 and P2 at a current contact point KCnfk(P1/R2)indicatrix of conformity that is converse to the indicatrix CnfR(P1/P2)a characteristic lineEP1, FP1, GP1fundamental magnitudes of first order of the smooth regular part surface P1EP2, FP2, GP2fundamental magnitudes of first order of the smooth regular part surface P2Gaussian curvature of a part surface P at a point mmean curvature of a surface P at a point mKpoint 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 P2LP1, MP1, NP1fundamental magnitudes of second order of the smooth regular part surface P1LP2, MP2, NP2fundamental magnitudes of second order of the smooth regular part surface P2OP1axis of rotation of the part surface P1OP2axis of rotation of the part surface P2Opaaxis of rotation of the plane of action, PAPAplane of actionPlnaxis of instant rotation of two regular part surfaces P1 and P2 in relation to one anotherthe operator of rolling/sliding (the operator of transition from the plane of action, PA, to the gear, , in crossed-axis gearing)the operator of rolling/sliding (the operator of transition from the plane of action, PA, to the pinion, , in crossed-axis gearing)the operator of rolling over a plane (Y-axis is the axis of rotation, X-axis is the axis of translation)the operator of rolling over a plane (Y-axis is the axis of rotation, Z-axis is the axis of translation)the operator of rolling over a plane (X-axis is the axis of rotation, Y-axis is the axis of translation)the operator of rolling over a plane (X-axis is the axis of rotation, Z-axis is the axis of translation)the operator of rolling over a plane (Z-axis is the axis of rotation, X-axis is the axis of translation)the operator of rolling over a plane (Z-axis is the axis of rotation, Y-axis is the axis of translation)the operator of rolling of two coordinate systemsthe operator of the resultant coordinate system transformation, say from a coordinate system A to a coordinate system Bthe operator of rotation through an angle about the X-axisthe operator of rotation through an angle about the Y-axisthe operator of rotation through an angle about the Z-axisR1.P1, R2.P1the first and second principal radii of the gear tooth flank P1R1.P2, R2.P2the first and second principal radii of the gear tooth flank P2the operator of screw motion about the X-axisthe operator of screw motion about the Y-axisthe 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, VP1curvilinear (Gaussian) coordinates of a point of a smooth regular part surface P1UP2, VP2curvilinear (Gaussian) coordinates of a point of a smooth regular part surface P2UP1, VP1tangent vectors to curvilinear coordinate lines on a smooth regular part surface P1UP2, VP2tangent vectors to curvilinear coordinate lines on a smooth regular part surface P2vector of the resultant motion of the smooth regular part surface P1 in relation to a reference system that the smooth regular part surface P2 will be associated withdmincnfminimal diameter of the indicatrix of conformity CnfR(P1/P2) for two smooth regular part surfaces P1 and P2 at a current contact point Kk1.P1, k2.P1the first and second principal curvatures of the smooth regular part surface P1k1.P2, k2.P2the first and second principal curvatures of the smooth regular part surface P2nPunit normal vector to a smooth regular part surface Ppscscrew parameter (reduced pitch) of instant screw motion of the part surface P1 in relation to the part surface P2rP1position vector of a point of a smooth regular part surface P1rcnfposition vector of a point of the indicatrix of conformity CnfR(P1/P2) for two smooth regular part surfaces P1 and P2 at a current contact point Kt1.P1, t2.P1unit tangent vectors of principal directions on the smooth regular part surface P1t1.P2, t2.P2unit tangent vectors of principal directions on the smooth regular part surface P2UP1, VP1unit tangent vectors to curvilinear coordinate lines on the smooth regular part surface P1UP2, VP2unit tangent vectors to curvilinear coordinate lines on the smooth regular part surface P2xPyPzPlocal Cartesian coordinate system having its origin at a current point of contact of the part surfaces P1 and P2

Greek symbols

, the first and second fundamental forms of the smooth regular part surface P1, the first and second fundamental forms of the smooth regular part surface P2normal pressure angleμangle of the part surfaces' P1 and P2 local relative orientationrotation vector of the regular part surface P1rotation vector of the part surface P2vector of instant rotation of the part surfaces P1 and P2 in relation to one another

Subscripts

cnfconformitymaxmaximumminminimumnnormaloptoptimal

Introduction

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

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

The geometry of part surfaces is discussed in Chapter 1. The discussion begins with an analytical description of ideal surfaces. Here, the ideal surface is interpreted as a zero-thickness film. The difference between classical differential geometry and engineering geometry of surfaces 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 that while it remains unknown, a real part surface is located between two boundary surfaces. The said boundary surfaces are represented by two ideal surfaces, of upper tolerance and lower tolerance. The specification of surfaces ends with a discussion of the natural representation of a desired part surface. This consideration involves the first and second fundamental forms of a smooth regular part surface. For an analytically specified surface, the elements of its local geometry are outlined. This consideration includes but is not limited to an analytical representation of the unit tangent vectors, the tangent plane, the unit normal vector, the unit vectors of principal directions on a part surface, etc. Ultimately, the parameters of part surface curvature are discussed. Mostly, the equations for principal surface curvatures along with normal curvatures at a surface point are considered. In addition to the mean curvature, the Gaussian curvature, absolute curvature, shape operator and curvedness of a surface at a point are considered. The classification of local part surface patches is proposed in this section of the book. The classification is followed by a circular chart comprising all possible kinds of local 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 global sense is feasible in nature, local part surface patches can be classified. For an investigation of the geometry of local part surface patches, planar characteristic images are employed. In this analysis the Dupin indicatrix, curvature indicatrix and circular diagrams at a part surface point are covered in detail. Based on the results of the analysis, two more circular charts are developed. One of them employs the part surface curvature indicatrices, while the other is based on the properties of circular diagrams at a current part surface point. This section of the book ends with a brief consideration of one more useful characteristic curve, which can be helpful for analytical description of the geometry of a part surface locally.

In Part II the geometry of contact of two smooth regular part surfaces is considered. This part of the book comprises four chapters.

In Chapter 3 the discussion begins with a review of earlier works in the field of contact geometry of surfaces. This includes the order of contact of two surfaces, the local relative orientation of the surfaces at a point of their contact, and the first- and second-order analysis. The first-order analysis is limited just to the common tangent plane. The second-order analysis begins with the author's comments on the analytical description of the local geometry of contacting surfaces loaded by a normal force: Hertz's proportional assumption. Then, the surface of relative normal curvature is considered. The Dupin indicatrix and curvature indicatrix of the surface of relative normal curvature are discussed. This analysis is followed by a discussion of the surface of relative normal radii of curvature, normalized relative normal curvature along with a characteristic curve of novel kind.

This section of the book is followed by Chapter 4, in which an analytical method based on second fundamental forms of the contacting part surfaces is discussed. It is shown here that the resultant deviation of one of the contacting surfaces from the other contacting surface expressed in terms of the second fundamental forms of the contacting surfaces could be the best possible criterion for the analytical description of the contact geometry of two smooth regular surfaces. Such a criterion is legitimate, but computationally impractical. Thus, other analytical methods need to be developed for this purpose.

In Chapter 5 a novel kind of characteristic curve for the purpose of analytical description of contact geometry of two smooth regular part surfaces in the first order of tangency is discussed in detail. The discussion begins with preliminary remarks, followed by the introduction and derivation of an equation of the indicatrix of conformity CnfR(P1/P2) of two part surfaces. Then, the directions of extremum degree of conformity of two part surfaces in contact are specified and described analytically. This analysis is followed by the determination and derivation of corresponding equations of asymptotes of the indicatrix of conformity CnfR(P1/P2). The capabilities of the indicatrix of conformity CnfR(P1/P2) of two smooth regular part surfaces P1 and P2 in the first order of tangency are compared with the corresponding capabilities of the Dupin indicatrix of the surface of relative curvature . Important properties of the indicatrix of conformity of two smooth regular part surfaces are outlined. Ultimately, the converse indicatrix of conformity CnfcnvR(P1/P2) of two regular part surfaces in the first order of tangency is introduced and discussed briefly as an alternative to the regular indicatrix of conformity CnfR(P1/P2).

In Chapter 6 more characteristic curves are derived on the premise of the Plücker conoid constructed at a point of a smooth regular part surface. Initially, the main properties of the surface of the Plücker conoid are briefly outlined. This includes, but is not limited to, the basics, analytical representation and local properties along with auxiliary formulae. This analysis is followed by an analytical description of the local geometry of a smooth regular part surface. Ultimately, expressions for two more characteristic curves are derived. These newly introduced characteristic curves are referred to as the Plücker curvature indicatrix and -indicatrix of a part surface. The analysis performed makes possible the derivation of equations for two more planar characteristic curves for analytical description of the contact geometry of two smooth regular part surfaces P1 and P2 at a point of their contact. One of the newly derived characteristic curves is referred to as the -relative indicatrix of the first kind of two contacting part surfaces P1 and P2. Another is a curve inverse to the characteristic curve . This second characteristic curve is referred to as the -relative indicatrix of the second kind. The main properties of both the characteristic curves are discussed briefly.

The feasible kinds of contact of two smooth regular part surfaces in the first order of tangency are discussed in Chapter 7. This analysis begins with an investigation of the possibility of implementing the indicatrix of conformity for the purpose of identification of the actual kind of contact of two smooth regular part surfaces. Then, the impact of accuracy of the computation of the parameters of the indicatrix of conformity CnfR(P1/P2) of two part surfaces is investigated. Ultimately, a classification of all possible kinds of contact of two smooth regular part surfaces in the first order of tangency is developed.

Various kinds of mapping of one part surface onto another part surface are discussed in Part III. The discussion in this part of the book begins with a novel kind of surface mapping, the so-called -mapping of the interacting part surfaces.

In Chapter 8 a novel method of surface mapping, namely -mapping of the interacting part surface, is disclosed. The preliminary remarks on the developed approach are followed by a detailed consideration of the concept underlying the -mapping of the interacting part surfaces. Then, the principal features of -mapping of a part surface P1 onto another part surface P2 are disclosed. Because -mapping of a surface returns an equation of the mapped surface in a natural representation, namely in terms of the fundamental magnitudes of the first and second order, the derived equation of the mapped surface must be reconstructed and represented in a convenient reference system. This issue receives comprehensive coverage in this chapter. The chapter ends with a consideration of two examples of implementation of the discussed method of part surface mapping.

A general consideration of the generation of enveloping surfaces is discussed in Chapter 9. The consideration begins with the analysis of generation of an envelope for successive positions of a moving planar curve. Then, the discussion is extended to the generation of the enveloping surface for successive positions of a moving smooth regular part surface. Enveloping surfaces for one-parametric, as well as two-parametric, families of surfaces are covered in this section. Further, the kinematic method for generation of enveloping surfaces is introduced. This method was developed in the 1940s by V.A. Shishkov. Implementation of the kinematic method for generation of one-parametric enveloping surfaces is discussed. Then, the approach is extended to multi-parametric motion of a smooth regular part surface.

In Chapter 10 special cases of generation of enveloping surfaces are disclosed. For this purpose a concept of reversibly enveloping surfaces is introduced. For the generation of reversibly enveloping surfaces, a novel method is proposed. This method is illustrated by an example of the generation of reversibly enveloping surfaces in the case of tooth flanks for geometrically accurate (ideal) crossed-axis gear pairs. The performed analysis makes possible a conclusion that two Olivier principles of generation of enveloping surfaces

in the general case are not valid, andin a degenerate case are useless.

Ultimately, there is no sense in applying Olivier principles for the purpose of generation of reversibly enveloping smooth regular part surfaces.

Part surfaces that allow for sliding over themselves are considered as a particular degenerate case of enveloping surfaces.

The appendices contain reference material that is useful in practical applications. The elements of vector algebra are briefly outlined in Appendix A. In Appendix B, the elements of coordinate system transformation are represented. This section of the book also includes direct transformation of the surface fundamental forms. The latter makes it possible to avoid calculation of the first and second derivatives of the part surface equation after the equation is represented in a new reference system. Formulae for changing surface parameters are represented in Appendix C.

A book of this size is likely to contain omissions and errors. If you have any constructive suggestions, please communicate them to the author via e-mail: [email protected].

Part I

Part Surfaces

The design, production and implementation of parts for products are common practice for most mechanical and manufacturing engineers. Any part can be understood as a solid bounded by a certain number of surfaces. Two kinds of bounding surfaces are recognized in this text: they can be either working surfaces of a part, or not working surfaces of the part. The consideration below is focused mostly on the geometry of working part surfaces.

All part surfaces are reproduced on a solid. Appropriate manufacturing methods are used for these purposes. Therefore, part surfaces are often referred to as engineering surfaces, in contrast to those surfaces which cannot be reproduced on a solid, and which can exist only virtually [30, 33, 34, 36, 45].

Interaction with the environment is the main purpose of all working part surfaces. Therefore, working part surfaces are also referred to as dynamic surfaces. Air, gases, fluids, solids and powders are good examples of the environments which part surfaces commonly interact with. Moreover, part surfaces may interact with light and other electromagnetic fields, with sound waves, etc. Favorable parameters of part surface geometry are usually outputs of a solution to complex problems in aerodynamics, hydrodynamics, contact interaction of solids with other solids, or solids with powders, etc.

In order to be able to design and produce products with favorable performance, the design and manufacture of part surfaces having favorable geometry is of critical importance. An appropriate analytical description of part surfaces is the first step to better understanding of what we need to design and how a desired part surface can be reproduced on a solid or, in other words, how a desired part surface can be manufactured.

1

Geometry of a Part Surface

The number of different kinds of part surfaces approaches infinity. Planes, surfaces of revolution, cylinders of general type (including, but not limited to, cylinders of revolution) and screw surfaces of constant axial pitch can all be found in the design of parts produced in industry. Examples of part surfaces are illustrated in Fig. 1.1. This figure shows part surfaces featuring simple geometry. Most surfaces of such types allow for sliding over themselves [33].

Figure 1.1 Examples of smooth regular part surfaces: a plane (1); an outer cylinder of revolution (2); an inner cylinder of revolution (3); a cone of revolution (4); a torus (5). Reproduced with permission from Industrial Model, Inc.

Part surfaces of complex geometry are widely used in practice as well. The working surface of an impeller blade is a perfect example of a part surface having complex geometry. Part surfaces of this kind are commonly referred to as sculptured part surfaces or free-form part surfaces. An example of a sculptured part surface is depicted in Fig. 1.2.

Figure 1.2 Working surface of impeller is an example of a smooth regular sculptured part surface. Reproduced from Somani Engineering.

Sculptured part surfaces do not allow for sliding over themselves. Moreover, the parameters of local geometry of a sculptured part surface at any two infinitesimally close points within the surface patch differ from each other.

More examples of part surfaces of complex geometry can be found in various industries, in the field of design and in the production of gear cutting tools in particular [35].

1.1 On the Analytical Description of Ideal Surfaces

A smooth regular surface could be specified uniquely by two independent variables. Therefore, we give a surface P (Fig. 1.3), in most cases, by expressing its rectangular coordinates XP, YP and ZP as functions of two Gaussian coordinates, UP and VP, in a certain closed interval:

(1.1)

Here we define:

rP– position vector of a point of the surface PUP and VP– curvilinear (Gaussian) coordinates of the point of the surface PXP, YP, ZP– Cartesian coordinates of the point of the surface PU1.P, U2.P– boundary values of the closed interval of the UP-parameterV1.P, V2.P– boundary values of the closed interval of the VP-parameter

Figure 1.3 Analytical description of an ideal part surface P (adapted from [33]).

The parameters UP and VP must enter independently, which means that the matrix

(1.2)

has rank 2. Positions where the rank is 1 or 0 are singular points; when the rank at all points is 1, then Eq. (1.1) represents a curve.

The following notations will be convenient in the consideration below.

The first derivatives of rP with respect to the Gaussian coordinates UP and VP are designated and , and for the unit tangent vectors and correspondingly.

The unit tangent vector uP (as well as the tangent vector uP) specifies a direction of the tangent line to the UP-coordinate curve through the given point m on the surface P. Similarly, the unit tangent vector vP (as well as the corresponding tangent vector vP) specifies a direction of the tangent line to the VP-coordinate curve through that same point m on the surface P.

The significance of the unit tangent vectors uP and vP becomes evident from the considerations immediately following.

First, the unit tangent vectors uP and vP allow for an equation of the tangent plane to the surface P at m:

(1.3)

Here we define:

rt.P– position vector of a point of the tangent plane to the surface P at mr(m)P– position vector of the point m on the surface P

Second, the unit tangent vectors uP and vP allow for an equation of the perpendicular, NP, and of the unit normal vector, NP, to the surface P at m:

(1.4)

(1.5)

When the order of multipliers in Eq. (1.4) [as well as in Eq. (1.5)] is chosen properly, then the unit normal vector NP is pointed outward from the body side bounded by the surface P. (It should be pointed out here that the unit tangent vectors uP and vP, as well as the unit normal vector NP, are dimensionless parameters of the geometry of the surface P. This feature of the unit vectors uP, vP and NP is convenient when performing practical calculations.)

1.2 On the Difference between Classical Differential Geometry and Engineering Geometry of Surfaces

Classical differential geometry has been developed mostly for the purpose of investigation of smooth regular surfaces. Engineering geometry also deals with smooth regular surfaces. What is the difference between these two geometries?

The difference between classical differential geometry and engineering geometry of surfaces is due mostly to how surfaces are interpreted.

Only phantom surfaces are investigated in classical differential geometry. Surfaces of this kind do not exist physically. They can be understood as a zero-thickness film of appropriate shape. Such a film can be accessed from both sides of the surface. This causes the following indefiniteness.

As an example, consider a surface, at a certain point m, with Gaussian curvature of the surface having positive value (). Classical differential geometry gives no answer to the question of whether the surface P is convex or concave in the vicinity of the point m. In the first case (when the surface P is convex), the mean curvature of the surface P at the point m is of positive value, , while in the second case (when the surface P is concave), the mean curvature of the surface P at the point m is of negative value, .

A similar situation is observed when the Gaussian curvature at a certain surface point is of negative value ().

In classical differential geometry, the answer to the question of whether a surface is convex or concave in the vicinity of a certain point m can be given only by convention.

In turn, surfaces that are treated in engineering geometry bound a solid – a machine part (or machine element). This part can be called a real object (Figs 1.1 and 1.2). The real object is the bearer of the surface shape.

Surfaces that bound real objects are accessible only from one side, as illustrated schematically in Fig. 1.4. We refer to this side of the surface as the open side of a part 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 part surface.

Figure 1.4 Open and closed sides of a part surface P (adapted from [33]).

The positively directed unit normal vector +nP is pointed outward from the part body, i.e. it is pointed from the body side to the void side. The negative unit normal vector −nP is pointed oppositely to +nP.

The existence of the open and closed sides of a part surface P eliminates the problem of identifying whether a surface is convex or concave. No convention is required in this respect.

The description of a smooth regular surface in differential geometry of surfaces and in engineering geometry provides more differences between surfaces treated in these two different branches of geometry.

1.3 On the Analytical Description of Part Surfaces

Another principal difference in this respect is due to the nature of the real object. We should point out here again that a real object is the bearer of a surface shape. No real object can be machined/manufactured precisely without deviations of its actual shape from the desired shape of the real object. Smaller or larger deviations in shape of the real object from its desired shape are inevitable in nature. We won't go into detail here on the nature of the deviations. We should simply realize that such deviations always exist.

As an example, let's consider how the surface of a round cylinder is specified in differential geometry of surfaces and compare it with that in engineering geometry.

In differential geometry of surfaces, the coordinates of the current point m of the surface of a cylinder of revolution can be specified by the position vector rm of the point m [Fig. 1.5(a)]. In the case under consideration, the position vector rm of a point within the surface of a cylinder of radius r, and having the Z-axis as its axis of rotation, can be expressed in matrix form as

(1.6)

Here, the surface curvilinear coordinates are denoted by and Zm, accordingly. They are equivalents of the curvilinear coordinates UP and VP in Eq. (1.1).

Figure 1.5 Specification of (a) an ideal and (b) a real part surface.

Mechanical engineers have no other option than to treat a desired (nominal) part surface P, which is given by the part blueprint, and which is specified by the tolerance for the surface P accuracy.

As manufacturing errors are inevitable, the current surface point mact actually deviates from its desired location m. The position vector ractm of a current point mact of the actual part surface deviates from rm for an ideal surface point m. Without loss of generality, the surface deviations in the direction of the Z-axis are ignored. Instead, the surface deviations in the directions of the X- and Y-axes are considered.

The deviation of a point mact from the corresponding surface point m that is measured perpendicular to the desired part surface P is designated as [Fig. 1.5(b)]. Formally, the position vector ractm of a current point mact of the actual part surface can be described analytically in matrix form as

(1.7)

where the deviation is understood as a signed value. It is positive for points mact located outside the surface [see Eq. (1.6)] and negative for points mact located inside the surface [see Eq. (1.6)].

Unfortunately, the actual value of the deviation is never known. Thus, Eq. (1.7) cannot be used for the purpose of analytical description of real part surfaces.

In practice, the permissible deviations of surfaces in engineering geometry are limited to a certain tolerance band. An example of a tolerance band is shown schematically in Fig. 1.5(b). The positive deviation must not exceed the upper limit , and the negative deviation must not be greater than the lower limit . That is, in order to meet the requirements specified by the blueprint, the deviation must be within the tolerance band

(1.8)

The total width of the tolerance band is equal to . In this expression for the deviation , both limits and are signed values. They can be either of positive value, or of negative value, as well as equal to zero.

Under such a scenario not only does the desired part surface meet the requirements specified by the part blueprint, but any and all actual part surfaces located within the tolerance band meet the requirements given by the blueprint. In other words, if a surface is specified by a tolerance band , and a surface is specified by a tolerance band , then an actual part surface is always located between the surfaces and . And, of course, the actual part surface always differs from the desired part surface . However, the deviation of the surface from the surface is always the tolerance band .

An intermediate summarization is as follows: we know everything about ideal surfaces, which do not exist in reality, and we know nothing about real surfaces, which exist physically (or, at least, our knowledge about real surfaces is very limited).

In addition, the entire endless surface of the cylinder of revolution is not considered in engineering geometry. Only a portion of this surface is of importance in practice. Therefore, in the axial direction, the length of the cylinder is limited to an interval , where H is a pre-specified length of the cylinder of revolution.

With that said, we can now proceed with a more general consideration of the analytical representation of surfaces in engineering geometry.

1.4 Boundary Surfaces for an Actual Part Surface

Owing to the deviations, an actual part surface Pact deviates from its nominal (desired) surface (Fig. 1.6). However, the deviations are within pre-specified tolerance bands. Otherwise, the real object could become useless. In practice, this particular problem is easily solved by selecting appropriate tolerance bands for the shape and dimensions of the actual surface Pact.

Figure 1.6 Analytical description of an actual part surface Pact located between the boundary surfaces and (adapted from [33]).

Similar to measuring deviations, the tolerances are also measured in the direction of the unit normal vector NP to the desired (nominal) part surface P. Positive tolerance is measured along the positive direction of the vector NP, while negative tolerance is measured along the negative direction of the vector NP. In a particular case, one of the tolerances, either or , can be zero.

Often, the values of the tolerance bands and are constant within the entire patch of the surface P. However, in special cases, for example when machining a sculptured part surface on a multi-axis NC machine, the actual values of the tolerances and can be set as functions of the coordinates of the current point m on the surface P. This results in the tolerances being represented in terms of UP- and VP-parameters of the surface P, say in the form and .

The endpoint of the vector at a current surface point m produces the point m+. Similarly, the endpoint of the vector produces the corresponding point m−.

The surface P+ of upper tolerance is represented by the loci of the points m+ (i.e. by the loci of the endpoints of the vector ). This makes it possible to have an analytical representation of the surface P+ of upper tolerance in the form

(1.9)

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

Similarly, the surface P− of lower tolerance is represented by the loci of the points m− (i.e. by the loci of the endpoints of the vector ). This also makes it possible to have an analytical representation of the surface P− of lower tolerance in the form

(1.10)

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

The surfaces P+ and P− are the boundary surfaces. The actual part surface Pact is located between the surfaces P+ and P−, as illustrated schematically in Fig. 1.6.

The actual part surface Pact cannot be represented analytically. Actually, the surface Pact is unknown – any surface that is located between the surfaces of upper tolerance P+ and lower tolerance P− meets the requirements of the part blueprint, and thus every such surface can be considered as an actual surface Pact. The equation of the surface Pact cannot be represented in the form , because the actual value of the deviation at the current surface point is not known. CMM data yields only an approximation for as well as the corresponding approximation for Pact. Moreover, the parameters of the local topology of the surface P considered above cannot be calculated for the surface Pact. However, owing to the tolerances and being small enough to compare the normal radii of curvature of the nominal surfaces P, it is assumed below that the surface Pact possesses the same geometrical properties as the surface P, and that the difference between the corresponding geometrical parameters of the surfaces Pact and P is negligibly small. In further consideration, this allows for a replacement of the actual surface Pact with the nominal surface P, which is much more convenient for performing calculations.

The consideration in this section illustrates the second principal difference between classical differential geometry and engineering geometry of surfaces.

Because of these differences, engineering geometry of surfaces often presents problems that were not envisioned in classical (pure) differential geometry of surfaces.

1.5 Natural Representation of a Desired Part Surface

The specification of a surface in terms of the first and second fundamental forms is commonly called the natural kind of surface representation. In general form, it can be represented by a set of two equations

(1.11)