Noncircular Gears: Design and Generation

by Faydor L. Litvin , Alfonso Fuentes-Aznar , Ignacio Gonzalez-Perez , Kenichi Hayasaka

• ISBN13: 9780521761703
• ISBN10: 0521761700
• Format: Hardcover
• Copyright: 2009-09-14
• Publisher: Cambridge University Press

Summary

Noncircular Gears: Design and Generation represents the extension of the modern theory of gearing applied to the design and manufacture of the main types of noncircular gears: conventional and modified elliptical gears, eccentric gears, oval gears, gears with lobes, and twisted gears. This book is enhanced by updated theoretical description of the methods of generation of noncircular gears by enveloping methods similar to those applied to the generation of circular gears. Noncircular Gears: Design and Generation also offers new developments directed to extend the application of noncircular gears for output speed variation and generation of functions. Numerous numerical examples show the application of the developed theory. This book aims to extend the application of noncircular gear drives in mechanisms and industry.

Author Biography

Faydor L. Litvin has been a Professor at the University of Illinois at Chicago for the past 30 years, after 30 years as a Professor and Department Head of Leningrad Polytechnic University and Leningrad Institute of Precise Mechanics and Optics. Dr. Litvin is the author of more than 300 publications (including 10 monographs) as well as the inventor and co-inventor of 25 inventions. Among his many honors, Dr. Litvin was made Doctor Honoris Causa of Miskolc University, Hungary, in 1999. He was named Inventor of the Year 2001 by the University of Illinois at Chicago and has been awarded 12 NASA Tech Brief awards; the 2001 Thomas Bernard Hall Prize (Institution of Mechanical Engineers, UK); and the 2004 Thomas A. Edison Award (ASME). He was elected a Fellow of ASME and is an American Gear Manufacturers Association (AGMA) member. He has supervised 84 Ph.D. students. In addition to his deep interest in teaching, Dr. Litvin has conducted seminal research on the theory of mechanisms and the theory and design of gears. Alfonso Fuentes-Aznar is a Professor of Mechanical Engineering at the Polytechnic University of Cartagena (UPCT). He has more than 15 years of teaching experience in machine element design and is the author of more than 65 publications. He was a Visiting Scholar and Research Scientist at the Gear Research Center of the University of Illinois at Chicago from 1999 to 2001. Among his many honors, he was awarded the 2001 Thomas Bernard Hall Prize (Institution of Mechanical Engineers, UK) and the NASA Tech Brief Award No. 17596-I for the development of a new technology - New Geometry of Face Worms Gear Drives with Conical and Cylindrical Worms. He is a member of the editorial boards of Mechanism and Machine Theory, the Open Mechanical Engineering Journal, and Recent Patents on Mechanical Engineering. Dr. Fuentes-Aznar is a member of AGMA and ASME. This is his second book. Ignacio Gonzalez-Perez earned his Ph.D. from the Polytechnic University of Cartagena. He was a Visiting Scholar at the Gear Research Center of the University of Illinois at Chicago, from 2001 to 2003 and a Visiting Scientist at the Gear Research and Development Department of Yamaha Motor Co., Ltd., in Japan, in 2005. Dr. Gonzalez-Perez is currently an Associate Professor in the Department of Mechanical Engineering of the Polytechnic University of Cartagena. He is a member of AGMA and ASME. He received an AGMA student paper award in 2002 and a UPCT research award for his thesis in 2006. Kenichi Hayasaka graduated from Tohoku University, Japan, in 1977 and earned his master's degree in 1983 from the University of Illinois at Chicago. He is the author of 16 research publications and inventor and co-inventor of 23 inventions. Mr. Hayasaka has been a researcher and engineer at Yamaha Motor Co., Ltd., since 1977. He is involved in the research and design of gears and transmissions applied in motorcycles and marine propulsion systems, and in the development of enhanced computerized design systems for optimizing low noise and durability of spiral bevel gears, helical gears, and spur gears.

Table of Contents

Foreword	p. xi
Preface	p. xiii
Acknowledgments	p. xv
Introduction to Theory of Gearing, Design, and Generation of Noncircular Gears	p. 1
Historical Comments	p. 1
Toward Design and Application of Noncircular Gears	p. 6
Examples of Previous Designs	p. 6
Examples of New Designs	p. 11
Developments Related with Theory of Gearing	p. 14
Centrodes of Noncircular Gears	p. 18
Introduction	p. 18
Centrode as the Trajectory of the Instantaneous Center of Rotation	p. 20
Concept of Polar Curve	p. 21
Derivation of Centrodes	p. 21
Tangent to Polar Curve	p. 22
Conditions for Design of Centrodes as Closed Form Curves	p. 24
Observation of Closed Centrodes for Function Generation	p. 27
Basic and Alternative Equations of Curvature of Polar Curve	p. 27
Conditions of Centrode Convexity	p. 30
Evolutes and Involutes	p. 31
Introduction and Terminology	p. 31
Determination of Evolutes	p. 33
Local Representation of a Noncircular Gear	p. 36
Pressure Angle	p. 37
Elliptical Gears and Gear Drives	p. 40
Introduction	p. 40
Basic Concepts	p. 40
Ellipse Parameters	p. 40
Polar Equation of an Ellipse	p. 41
External Elliptical Gear Drives	p. 43
Basic Equations	p. 43
Conventional Elliptical Gear Drives	p. 45
Centrodes and Transmission Function	p. 45
Influence of Ellipse Parameters and Design Recommendations	p. 47
Gear Drive with Elliptical Pinion and Conjugated Gear	p. 52
External Gear Drive with Modified Elliptical Gears	p. 53
Modification of the Ellipse	p. 53
Derivation of Modified Centrode ?1	p. 55
Derivative Functions m 21 (I)(?1) and m 21 (II) (?1)	p. 55
Relation between Rotations of Gears 1 and 2	p. 55
Derivation of Centrode ?2	p. 56
External Gear Drive with Oval Centrodes	p. 57
Equation of Oval Centrode	p. 57
Derivative Function m21(?1)	p. 58
Relation between Rotations of Gears 1 and 2	p. 58
Transmission Function ?2(?1)	p. 60
Design of Noncircular Gears with Lobes	p. 60
Design of Gear Drives with Different Number of Lobes for Pinion and Gear	p. 63
Transmission Function of Elliptical Gears as Curve of Second Order	p. 65
Functional of Identical Centrodes	p. 66
Generation of Planar and Helical Elliptical Gears	p. 71
Introduction	p. 71
Generation of Elliptical Gears by Rack Cutter	p. 71
Generation of Elliptical Gears by Hob	p. 79
Generation of Elliptical Gears by Shaper	p. 86
Examples of Design of Planar and Helical Elliptical Gears	p. 90
Planar Elliptical Gears	p. 90
Helical Elliptical Gears	p. 92
Design of Gear Drives Formed by Eccentric Circular Gear and Conjugated Noncircular Gear	p. 94
Introduction	p. 94
Centrodes of Eccentric Gear Drive	p. 94
Equations of Mating Centrodes	p. 94
Curvature of Centrode a2 and Applications	p. 96
Generation of the Noncircular Gear by Shaper and Hob	p. 101
Generation of Noncircular Gear by a Noneccentric Shaper	p. 101
Generation of the Noncircular Gear by a Hob	p. 105
Generation of the Eccentric Gear Providing Localized Contact	p. 112
Design of Internal Noncircular Gears	p. 115
Introduction	p. 115
Derivation of Centrodes	p. 115
Preliminary Considerations of Kinematics of Internal Gear Drive	p. 115
Basic Equations of Centrodes	p. 116
Design of Centrodes ?1 and ?2 as Closed-Form Curves	p. 118
Examples of Design of Internal Noncircular Gear Drives	p. 118
Gear Drive with Elliptical Pinion	p. 118
Gear Drive with Modified Elliptical Pinion	p. 120
Gear Drive with Oval Pinion	p. 121
Gear Drive with Eccentric Pinion	p. 123
Generation of Planar Internal Noncircular Gears by Shaper	p. 126
Conditions of Nonundercutting of Planar Internal Noncircular Gears	p. 132
Approach A	p. 133
Approach B	p. 134
Numerical Example	p. 136
Application for Design of Planetary Gear Train with Noncircular and Circular Gears	p. 138
Introduction	p. 138
Structure and Basic Kinematic Concept of Planetary Train	p. 138
Planetary Gear Train with Elliptical Gears	p. 139
Planetary Gear Train with Noncircular and Circular Gears	p. 141
Transformation of Rotation into Translation with Variation of Gear Ratio	p. 143
Introduction	p. 143
Determination of Centrodes of Noncircular Gear and Rack	p. 144
Application of Mechanism Formed by a Noncircular Gear and Rack	p. 144
Tandem Design of Mechanisms for Function Generation and Output Speed Variation	p. 147
Introduction	p. 147
General Aspects of Generation of Functions	p. 152
Generation of Function with Varied Sign of Derivative	p. 153
Introduction to Design of Multigear Drive with Noncircular Gears	p. 156
Basic Functionals	p. 156
Interpretations of Lagrange's Theorem	p. 160
Illustration of Application of Lagrange's Theorem for Functional ?(a) = g2(gi(a))	p. 161
Previous Solutions for ?(a) = f(f(a))	p. 161
Computational Procedure for Functional (10.4.6)	p. 163
Design of Multigear Drive	p. 166
Basic Equations	p. 166
Design of Centrodes	p. 168
Design of Planar Linkages Coupled with Noncircular Gears	p. 171
Tandem Design of Double-Crank Mechanism Coupled with Two Pairs of Noncircuiar Gears	p. 171
Tandem Design of Slider-Crank Mechanism Coupled with Modified Elliptical Gears	p. 174
Preliminary Information	p. 174
Basic Ideas of Modification of Elliptical Centrodes	p. 174
Analytical Determination of Modified Elliptical Centrodes	p. 177
Numerical Examples	p. 179
Tandem Design of Scotch-Yoke Mechanism Coupled with Noncircular Gears	p. 180
Tandem Design of Mechanism Formed by Two Pairs of Noncircular Gears and Racks	p. 183
Generation of Function	p. 186
Additional Numerical Problems	p. 188
References	p. 201
Index	p. 203
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