Geometry of Surfaces A Practical Guide for Mechanical Engineers

by Radzevich, Stephen P.

• ISBN13: 9781118520314
• ISBN10: 1118520319
• Edition: 1st
• Format: Hardcover
• Copyright: 2013-03-04
• Publisher: Wiley

Summary

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.

Author Biography

Stephen P. Radzevich, Principal Gear Engineer, USA

Table of Contents

About the Author

Foreword/Preface

Introduction

Acknowledgements

I PART SURFACES

1 Geometry of a Part Surface

1.1. On Analytical Description of Ideal Surfaces

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

1.3. On Analytical Description of Part Surfaces

1.4. Boundary Surfaces for an Actual Part Surface

1.5. Natural Representation of a Desired Part Surface

1.5.1. First fundamental form of a desired part surface

1.5.2. Second fundamental form of a desired part surface

1.5.3. Illustrative example

1.6. Elements of Local Geometry of a Desired Part Surface

1.6.1. Unit tangent vectors

1.6.2. Tangent plane

1.6.3. Unit normal vector

1.6.4. Unit vectors of principal directions on a part surface

1.6.5. Principal curvatures of a part surface

1.6.6. Other parameters of curvature of a part surface

1.6.6.1. Mean curvature of a surface

1.6.6.2. Gaussian curvature of a surface

1.6.6.3. Absolute curvature of a surface

1.6.6.4. Shape operator of a surface

1.6.6.5. Curvedness of a surface

2 On a Possibility of Classification of Part Surfaces

2.1. Sculptured Part Surfaces

2.1.1. Local patches of ideal part surfaces

2.1.2. Local patches of real part surfaces

2.2. Planar Characteristic Images

2.2.1. Dupin indicatrix

2.2.2. Curvature indicatrix

2.2.3. Circular chart for local patches of smooth regular part surfaces based on curvature indicatrix

2.3. Circular Diagrams at a Surface Point

2.3.1. Circular diagrams

2.3.2. Circular chart for local patches of smooth regular part surfaces based on circular diagrams

2.4. One More Useful Characteristic Curve

Part II: Geometry of Contact of Part Surfaces

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. The First Order Analysis: Common Tangent Plane

3.5. The Second Order Analysis

3.5.1. Comments on analytical description of the local geometry of contacting surfaces loaded by a normal force: Hertz proportional assumption

3.5.2. Surface of normal relative curvature

3.5.3. Dupin indicatrix of the surface of relative normal curvature

3.5.4. Matrix representation of equation of the Dupin indicatrix of the surface of relative normal curvature

3.5.5. Surface of relative normal radii of curvature

3.5.6. Normalized relative normal curvature

3.5.7. Curvature indicatrix of the surface of relative normal curvature

3.6. A characteristic Curve of Novel Kind

4 An Analytical Method based on Second Fundamental Forms of the Contacting Part Surfaces

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 (P/T)

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

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

6 Plücker Conoid: More Characteristic Curves

6.1. Plücker Conoid

6.1.1. Basics

6.1.2. Analytical representation

6.1.3. Local properties

6.1.4. Auxiliary formulae

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

6.2.1. Preliminary remarks

6.2.2. The Plücker conoid

6.2.3. Plücker curvature indicatrix

6.2.4. AnR (P1) -indicatrix of a part surface

6.3. Relative Characteristic Curve

6.3.1. On a possibility of implementation of two Plücker conoids

6.3.2. AnR(P1/P2)-relative indicatrix of two contacting part surfaces P1 and P2

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

7.1. On a Possibility of Implementation of the Indicatrix of Conformity for the Purposes of Identification of 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

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

Part III Mapping of the Contacting Part Surfaces

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

9 Generating of Enveloping Surfaces: General Consideration

9.1. Envelope to Successive Positions of a Moving Planar Curve

9.2. Envelope to Successive Positions of a Moving Surface

9.2.1. Envelope to a one-parametric family of surfaces

9.2.2. Envelope to a two-parametric family of surfaces

9.3 Kinematic Method for the Determining of Enveloping Surfaces

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

10 Generation of Enveloping Surfaces: Special Cases

10.1. Part Surfaces those Allow for Sliding Over Themselves

10.2. Introductory Remarks: Reversibly-Enveloping Surfaces

10.3. Generation of Reversibly-Enveloping Surfaces

10.3.1. Kinematics of crossed-axis gearing

10.3.2. Base cones in crossed-axis gear pairs

10.3.3. Tooth flanks of geometrically accurate (ideal) crossed-axis gear pairs

10.3.4. Tooth flank of a crossed-axis gear

10.4. On Looseness of Two Olivier Principles

10.4.1. An example of implementation of the first Olivier principle for generation of enveloping surfaces in a degenerate case

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

10.4.3. Concluding remarks

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 Systems Transformations

B.1. Coordinate System Transformation

B.1.1. Introduction

B.1.2. Translations

B.1.3. Rotation about coordinate axis

B.1.4. Resultant coordinate system transformation

B.1.5. Screw motion about a coordinate axis

B.1.6. Rolling motion of a coordinate system

B.1.7. Rolling of two coordinate systems

B.2. Conversion of the Coordinate System Orientation

B.3. Transformation of Surfaces Fundamental Forms

Appendix C: Change of Surface Parameters

Notation

Glossary

References

Bibliography

Index