Noncircular Gears: Design and Generation Illustrated Edition by Faydor L. Litvin, ISBN-13: 978-0521761703
[PDF eBook eTextbook]
- Publisher: Cambridge University Press; Illustrated edition (September 14, 2009)
- Language: English
- 214 pages
- ISBN-10: 0521761700
- ISBN-13: 978-0521761703
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.
Table of Contents:
Foreword page xi
Preface xiii
Acknowledgments xv
1 Introduction to Theory of Gearing, Design, and Generation of
Noncircular Gears . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Historical Comments 1
1.2 Toward Design and Application of Noncircular Gears 6
1.2.1 Examples of Previous Designs 6
1.2.2 Examples of New Designs 11
1.3 Developments Related with Theory of Gearing 14
2 Centrodes of Noncircular Gears . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.1 Introduction 18
2.2 Centrode as the Trajectory of the Instantaneous Center of
Rotation 20
2.3 Concept of Polar Curve 21
2.4 Derivation of Centrodes 21
2.5 Tangent to Polar Curve 22
2.6 Conditions for Design of Centrodes as Closed Form Curves 24
2.7 Observation of Closed Centrodes for Function Generation 27
2.8 Basic and Alternative Equations of Curvature of Polar Curve 27
2.9 Conditions of Centrode Convexity 30
3 Evolutes and Involutes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.1 Introduction and Terminology 31
3.2 Determination of Evolutes 33
3.3 Local Representation of a Noncircular Gear 36
3.4 Pressure Angle 37
4 Elliptical Gears and Gear Drives . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.1 Introduction 40
4.2 Basic Concepts 40
4.2.1 Ellipse Parameters 40
4.2.2 Polar Equation of an Ellipse 41
4.3 External Elliptical Gear Drives 43
4.3.1 Basic Equations 43
4.3.2 Conventional Elliptical Gear Drives 45
4.3.2.1 Centrodes and Transmission Function 45
4.3.2.2 Influence of Ellipse Parameters and Design
Recommendations 47
4.3.3 Gear Drive with Elliptical Pinion and Conjugated Gear 52
4.3.4 External Gear Drive with Modified Elliptical Gears 53
4.3.4.1 Modification of the Ellipse 53
4.3.4.2 Derivation of Modified Centrode σ1 55
4.3.4.3 Derivative Functions m(I )
21 (φ1) and m(I I)
21 (φ1) 55
4.3.4.4 Relation between Rotations of Gears 1 and 2 55
4.3.4.5 Derivation of Centrode σ2 56
4.3.5 External Gear Drive with Oval Centrodes 57
4.3.5.1 Equation of Oval Centrode 57
4.3.5.2 Derivative Function m21(φ1) 58
4.3.5.3 Relation between Rotations of Gears 1 and 2 58
4.3.5.4 Transmission Function φ2(φ1) 60
4.3.6 Design of Noncircular Gears with Lobes 60
4.3.6.1 Design of Gear Drives with Different Number of
Lobes for Pinion and Gear 63
4.4 Transmission Function of Elliptical Gears as Curve of Second
Order 65
4.5 Functional of Identical Centrodes 66
5 Generation of Planar and Helical Elliptical Gears . . . . . . . . . . . . . . . 71
5.1 Introduction 71
5.2 Generation of Elliptical Gears by Rack Cutter 71
5.3 Generation of Elliptical Gears by Hob 79
5.4 Generation of Elliptical Gears by Shaper 86
5.5 Examples of Design of Planar and Helical Elliptical Gears 90
5.5.1 Planar Elliptical Gears 90
5.5.2 Helical Elliptical Gears 92
6 Design of Gear Drives Formed by Eccentric Circular Gear and
Conjugated Noncircular Gear . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6.1 Introduction 94
6.2 Centrodes of Eccentric Gear Drive 94
6.2.1 Equations of Mating Centrodes 94
6.2.2 Curvature of Centrode σ2 and Applications 96
6.3 Generation of the Noncircular Gear by Shaper and Hob 101
6.3.1 Generation of Noncircular Gear by a Noneccentric Shaper 101
6.3.2 Generation of the Noncircular Gear by a Hob 105
6.4 Generation of the Eccentric Gear Providing Localized Contact 112
7 Design of Internal Noncircular Gears . . . . . . . . . . . . . . . . . . . . . . 115
7.1 Introduction 115
7.2 Derivation of Centrodes 115
7.2.1 Preliminary Considerations of Kinematics of Internal Gear
Drive 115
7.2.2 Basic Equations of Centrodes 116
7.2.3 Design of Centrodes σ1 and σ2 as Closed-Form Curves 118
7.3 Examples of Design of Internal Noncircular Gear Drives 118
7.3.1 Gear Drive with Elliptical Pinion 118
7.3.2 Gear Drive with Modified Elliptical Pinion 120
7.3.3 Gear Drive with Oval Pinion 121
7.3.4 Gear Drive with Eccentric Pinion 123
7.4 Generation of Planar Internal Noncircular Gears by Shaper 126
7.5 Conditions of Nonundercutting of Planar Internal Noncircular
Gears 132
7.5.1 Approach A 133
7.5.2 Approach B 134
7.5.3 Numerical Example 136
8 Application for Design of Planetary Gear Train with Noncircular
and Circular Gears . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
8.1 Introduction 138
8.2 Structure and Basic Kinematic Concept of Planetary Train 138
8.3 Planetary Gear Train with Elliptical Gears 139
8.4 Planetary Gear Train with Noncircular and Circular Gears 141
9 Transformation of Rotation into Translation with Variation of
Gear Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
9.1 Introduction 143
9.2 Determination of Centrodes of Noncircular Gear and Rack 144
9.3 Application of Mechanism Formed by a Noncircular Gear and
Rack 144
10 Tandem Design of Mechanisms for Function Generation and Output
Speed Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
10.1 Introduction 147
10.2 General Aspects of Generation of Functions 152
10.3 Generation of Function with Varied Sign of Derivative 153
10.4 Introduction to Design of Multigear Drive with Noncircular
Gears 156
10.4.1 Basic Functionals 156
10.4.2 Interpretations of Lagrange’s Theorem 160
10.4.3 Illustration of Application of Lagrange’s Theorem for
Functional ψ(α) = g2(g1(α)) 161
10.4.3.1 Previous Solutions for ψ(α) = f ( f (α)) 161
10.4.3.2 Computational Procedure for Functional (10.4.6) 163
10.5 Design of Multigear Drive 166
10.5.1 Basic Equations 166
10.5.2 Design of Centrodes 168
10.6 Design of Planar Linkages Coupled with Noncircular Gears 171
10.6.1 Tandem Design of Double-Crank Mechanism Coupled
with Two Pairs of Noncircular Gears 171
10.6.2 Tandem Design of Slider-Crank Mechanism Coupled with
Modified Elliptical Gears 174
10.6.2.1 Preliminary Information 174
10.6.2.2 Basic Ideas of Modification of Elliptical Centrodes 174
10.6.2.3 Analytical Determination of Modified Elliptical
Centrodes 177
10.6.2.4 Numerical Examples 179
10.6.3 Tandem Design of Scotch-Yoke Mechanism Coupled with
Noncircular Gears 180
10.6.4 Tandem Design of Mechanism Formed by Two Pairs of
Noncircular Gears and Racks 183
10.6.4.1 Generation of Function 186
11 Additional Numerical Problems . . . . . . . . . . . . . . . . . . . . . . . . . .188
References 201
Index 203
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