معى اليوم احبتى فى الله
كتاب Gear Geometry and Applied Theory
Contents
Foreword by Graziano Curti page xii
Preface xiv
Acknowledgments xv
1 Coordinate Transformation 1
1.1 Homogeneous Coordinates 1
1.2 Coordinate Transformation in Matrix Representation 2
1.3 Rotation About an Axis 6
1.4 Rotational and Translational 4 × 4 Matrices 14
1.5 Examples of Coordinate Transformation 15
1.6 Application to Derivation of Curves 24
1.7 Application to Derivation of Surfaces 28
2 Relative Velocity 33
2.1 Vector Representation 33
2.2 Matrix Representation 39
2.3 Application of Skew-Symmetric Matrices 41
3 Centrodes, Axodes, and Operating Pitch Surfaces 44
3.1 The Concept of Centrodes 44
3.2 Pitch Circle 49
3.3 Operating Pitch Circles 50
3.4 Axodes in Rotation Between Intersected Axes 51
3.5 Axodes in Rotation Between Crossed Axes 52
3.6 Operating Pitch Surfaces for Gears with Crossed Axes 56
4 Planar Curves 59
4.1 Parametric Representation 59
4.2 Representation by Implicit Function 60
4.3 Tangent and Normal to a Planar Curve 60
4.4 Curvature of Planar Curves 68
5 Surfaces 78
5.1 Parametric Representation of Surfaces 78
5.2 Curvilinear Coordinates 78
5.3 Tangent Plane and Surface Normal 79
5.4 Representation of a Surface by Implicit Function 82
5.5 Examples of Surfaces 82
6 Conjugated Surfaces and Curves 97
6.1 Envelope to a Family of Surfaces: Necessary Conditions
of Existence 97
6.2 Basic Kinematic Relations 102
6.3 Conditions of Nonundercutting 103
6.4 Sufficient Conditions for Existence of an Envelope
to a Family of Surfaces 107
6.5 Contact Lines; Surface of Action 110
6.6 Envelope to Family of Contact Lines on Generating
Surface 1 112
6.7 Formation of Branches of Envelope to Parametric
Families of Surfaces and Curves 114
6.8 Wildhaber’s Concept of Limit Contact Normal 118
6.9 Fillet Generation 119
6.10 Two-Parameter Enveloping 124
6.11 Axes of Meshing 128
6.12 Knots of Meshing 134
6.13 Problems 137
7 Curvatures of Surfaces and Curves 153
7.1 Introduction 153
7.2 Spatial Curve in 3D-Space 153
7.3 Surface Curves 164
7.4 First and Second Fundamental Forms 175
7.5 Principal Directions and Curvatures 180
7.6 Euler’s Equation 188
7.7 Gaussian Curvature; Three Types of Surface
Points 189
7.8 Dupin’s Indicatrix 193
7.9 Geodesic Line; Surface Torsion 194
8 Mating Surfaces: Curvature Relations, Contact Ellipse 202
8.1 Introduction 202
8.2 Basic Equations 203
8.3 Planar Gearing: Relation Between Curvatures 204
8.4 Direct Relations Between Principal Curvatures
of Mating Surfaces 218
8.5 Direct Relations Between Normal Curvatures
of Mating Surfaces 226
8.6 Diagonalization of Curvature Matrix 231
8.7 Contact Ellipse 234
9 Computerized Simulation of Meshing and Contact 241
9.1 Introduction 241
9.2 Predesign of a Parabolic Function of Transmission
Errors 242
9.3 Local Synthesis 245
Contents vii
9.4 Tooth Contact Analysis 249
9.5 Application of Finite Element Analysis for Design
of Gear Drives 257
9.6 Edge Contact 260
10 Spur Involute Gears 267
10.1 Introduction 267
10.2 Geometry of Involute Curves 268
10.3 Generation of Involute Curves by Tools 273
10.4 Tooth Element Proportions 278
10.5 Meshing of Involute Gear with Rack-Cutter 280
10.6 Relations Between Tooth Thicknesses Measured
on Various Circles 285
10.7 Meshing of External Involute Gears 287
10.8 Contact Ratio 292
10.9 Nonstandard Gears 294
11 Internal Involute Gears 304
11.1 Introduction 304
11.2 Generation of Gear Fillet 305
11.3 Conditions of Nonundercutting 309
11.4 Interference by Assembly 314
12 Noncircular Gears 318
12.1 Introduction 318
12.2 Centrodes of Noncircular Gears 318
12.3 Closed Centrodes 323
12.4 Elliptical and Modified Elliptical Gears 326
12.5 Conditions of Centrode Convexity 329
12.6 Conjugation of an Eccentric Circular Gear with
a Noncircular Gear 330
12.7 Identical Centrodes 331
12.8 Design of Combined Noncircular Gear Mechanism 333
12.9 Generation Based on Application of Noncircular
Master-Gears 335
12.10 Enveloping Method for Generation 336
12.11 Evolute of Tooth Profiles 341
12.12 Pressure Angle 344
Appendix 12.A: Displacement Functions for Generation
by Rack-Cutter 345
Appendix 12.B: Displacement Functions for Generation
by Shaper 348
13 Cycloidal Gearing 350
13.1 Introduction 350
13.2 Generation of Cycloidal Curves 350
13.3 Equations of Cycloidal Curves 354
13.4 Camus’ Theorem and Its Application 355
13.5 External Pin Gearing 359
13.6 Internal Pin Gearing 365
viii Contents
13.7 Overcentrode Cycloidal Gearing 367
13.8 Root’s Blower 369
14 Involute Helical Gears with Parallel Axes 375
14.1 Introduction 375
14.2 General Considerations 375
14.3 Screw Involute Surface 377
14.4 Meshing of a Helical Gear with a Rack 382
14.5 Meshing of Mating Helical Gears 392
14.6 Conditions of Nonundercutting 396
14.7 Contact Ratio 398
14.8 Force Transmission 399
14.9 Results of Tooth Contact Analysis (TCA) 402
14.10 Nomenclature 403
15 Modified Involute Gears 404
15.1 Introduction 404
15.2 Axodes of Helical Gears and Rack-Cutters 407
15.3 Profile-Crowned Pinion and Gear Tooth Surfaces 411
15.4 Tooth Contact Analysis (TCA) of Profile-Crowned
Pinion and Gear Tooth Surfaces 414
15.5 Longitudinal Crowning of Pinion by a Plunging Disk 419
15.6 Grinding of Double-Crowned Pinion by a Worm 424
15.7 TCA of Gear Drive with Double-Crowned Pinion 430
15.8 Undercutting and Pointing 432
15.9 Stress Analysis 435
16 Involute Helical Gears with Crossed Axes 441
16.1 Introduction 441
16.2 Analysis and Simulation of Meshing of Helical Gears 443
16.3 Simulation of Meshing of Crossed Helical Gears 452
16.4 Generation of Conjugated Tooth Surfaces of Crossed
Helical Gears 455
16.5 Design of Crossed Helical Gears 458
16.6 Stress Analysis 465
Appendix 16.A: Derivation of Shortest Center Distance for
Canonical Design 467
Appendix 16.B: Derivation of Equation of Canonical Design
f (γo, αon, λb1, λb2) = 0 472
Appendix 16.C: Relations Between Parameters αpt and αpn 473
Appendix 16.D: Derivation of Equation (16.5.5) 473
Appendix 16.E: Derivation of Additional Relations Between
αot1 and αot2 474
17 New Version of Novikov–Wildhaber Helical Gears 475
17.1 Introduction 475
17.2 Axodes of Helical Gears and Rack-Cutter 478
17.3 Parabolic Rack-Cutters 479
17.4 Profile-Crowned Pinion and Gear Tooth Surfaces 482
17.5 Tooth Contact Analysis (TCA) of Gear Drive with
Profile-Crowned Pinion 485
17.6 Longitudinal Crowning of Pinion by a Plunging Disk 487
17.7 Generation of Double-Crowned Pinion by a Worm 491
17.8 TCA of a Gear Drive with a Double-Crowned Pinion 497
17.9 Undercutting and Pointing 500
17.10 Stress Analysis 502
18 Face-Gear Drives 508
18.1 Introduction 508
18.2 Axodes, Pitch Surfaces, and Pitch Point 510
18.3 Face-Gear Generation 512
18.4 Localization of Bearing Contact 512
18.5 Equations of Face-Gear Tooth Surface 515
18.6 Conditions of Nonundercutting of Face-Gear Tooth
Surface (Generated by Involute Shaper) 519
18.7 Pointing of Face-Gear Teeth Generated by Involute
Shaper 522
18.8 Fillet Surface 524
18.9 Geometry of Parabolic Rack-Cutters 525
18.10 Second Version of Geometry: Derivation of Tooth
Surfaces of Shaper and Pinion 527
18.11 Second Version of Geometry: Derivation of Face-Gear
Tooth Surface 529
18.12 Design Recommendations 529
18.13 Tooth Contact Analysis (TCA) 531
18.14 Application of Generating Worm 535
18.15 Stress Analysis 541
19 Worm-Gear Drives with CylindricalWorms 547
19.1 Introduction 547
19.2 Pitch Surfaces and Gear Ratio 548
19.3 Design Parameters and Their Relations 552
19.4 Generation and Geometry of ZA Worms 557
19.5 Generation and Geometry of ZN Worms 561
19.6 Generation and Geometry of ZI (Involute) Worms 574
19.7 Geometry and Generation of K Worms 581
19.8 Geometry and Generation of F-I Worms (Version I) 590
19.9 Geometry and Generation of F-II Worms (Version II) 597
19.10 Generalized Helicoid Equations 601
19.11 Equation of Meshing of Worm and Worm-Gear
Surfaces 603
19.12 Area of Meshing 606
19.13 Prospects of New Developments 609
20 Double-EnvelopingWorm-Gear Drives 614
20.1 Introduction 614
20.2 Generation of Worm and Worm-Gear Surfaces 614
20.3 Worm Surface Equations 618
20.4 Equation of Meshing 620
20.5 Contact Lines 622
20.6 Worm-Gear Surface Equations 622
21 Spiral Bevel Gears 627
21.1 Introduction 627
21.2 Basic Ideas of the Developed Approach 628
21.3 Derivation of Gear Tooth Surfaces 633
21.4 Derivation of Pinion Tooth Surface 644
21.5 Local Synthesis and Determination of Pinion
Machine-Tool Settings 649
21.6 Relationships Between Principal Curvatures and
Directions of Mating Surfaces 656
21.7 Simulation of Meshing and Contact 661
21.8 Application of Finite Element Analysis for the Design
of Spiral Bevel Gear Drives 665
21.9 Example of Design and Optimization of a Spiral Bevel
Gear Drive 666
21.10 Compensation of the Shift of the Bearing Contact 676
22 Hypoid Gear Drives 679
22.1 Introduction 679
22.2 Axodes and Operating Pitch Cones 679
22.3 Tangency of Hypoid Pitch Cones 680
22.4 Auxiliary Equations 682
22.5 Design of Hypoid Pitch Cones 685
22.6 Generation of Face-Milled Hypoid Gear Drives 690
23 Planetary Gear Trains 697
23.1 Introduction 697
23.2 Gear Ratio 697
23.3 Conditions of Assembly 703
23.4 Phase Angle of Planet Gears 707
23.5 Efficiency of a Planetary Gear Train 709
23.6 Modifications of Gear Tooth Geometry 711
23.7 Tooth Contact Analysis (TCA) 712
23.8 Illustration of the Effect of Regulation of Backlash 716
24 Generation of Helicoids 718
24.1 Introduction 718
24.2 Generation by Finger-Shaped Tool: Tool Surface is
Given 718
24.3 Generation by Finger-Shaped Tool: Workpiece Surface
is Given 723
24.4 Generation by Disk-Shaped Tool: Tool Surface is Given 726
24.5 Generation by Disk-Shaped Tool: Workpiece Surface is
Given 730
25 Design of Flyblades 734
25.1 Introduction 734
25.2 Two-Parameter Form Representation of Worm Surfaces 735
Contents xi
25.3 Three-Parameter Form Representation of Worm
Surfaces 737
25.4 Working Equations 738
26 Generation of Surfaces by CNC Machines 746
26.1 Introduction 746
26.2 Execution of Motions of CNC Machines 747
26.3 Generation of Hypoid Pinion 750
26.4 Generation of a Surface with Optimal Approximation 752
27 Overwire (Ball) Measurement 769
27.1 Introduction 769
27.2 Problem Description 769
27.3 Measurement of Involute Worms, Involute Helical
Gears, and Spur Gears 773
27.4 Measurement of Asymmetric Archimedes Screw 779
28 Minimization of Deviations of Gear Real Tooth Surfaces 782
28.1 Introduction 782
28.2 Overview of Measurement and Modeling Method 783
28.3 Equations of Theoretical Tooth Surface t 784
28.4 Coordinate Systems Used for Coordinate
Measurements 785
28.5 Grid and Reference Point 786
28.6 Deviations of the Real Surface 787
28.7 Minimization of Deviations 787
References 789
Index 795
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