Gear Geometry and Applied Theory

Cover......Page 2
Half-title......Page 4
Title......Page 5
Copyright......Page 6
Contents......Page 7
Foreword......Page 14
Preface......Page 16
Acknowledgments......Page 17
1.1 HOMOGENEOUS COORDINATES......Page 19
1.2 COORDINATE TRANSFORMATION IN MATRIX REPRESENTATION......Page 20
Two Main Problems......Page 24
Employment of Additional Coordinate Systems......Page 30
1.4 ROTATIONAL AND TRANSLATIONAL 4 × 4 MATRICES......Page 32
1.5 EXAMPLES OF COORDINATE TRANSFORMATION......Page 33
Generation of Epicycloid......Page 42
Generation of Involute Curves......Page 43
Generation of a Cycloid......Page 45
1.7 APPLICATION TO DERIVATION OF SURFACES......Page 46
2.1 VECTOR REPRESENTATION......Page 51
2.2 MATRIX REPRESENTATION......Page 57
2.3 APPLICATION OF SKEW-SYMMETRIC MATRICES......Page 59
3.1 THE CONCEPT OF CENTRODES......Page 62
3.2 PITCH CIRCLE......Page 67
3.3 OPERATING PITCH CIRCLES......Page 68
3.4 AXODES IN ROTATION BETWEEN INTERSECTED AXES......Page 69
3.5 AXODES IN ROTATION BETWEEN CROSSED AXES......Page 70
3.6 OPERATING PITCH SURFACES FOR GEARS WITH CROSSED AXES......Page 74
4.1 PARAMETRIC REPRESENTATION......Page 77
4.3 TANGENT AND NORMAL TO A PLANAR CURVE......Page 78
Introduction......Page 86
Frenet Trihedron......Page 87
Curvature of curve represented by vector function r(s)......Page 89
Curvature of parametric curve represented by vector function r(θ)......Page 90
Modification of Eq. (4.4.31)......Page 92
Curvature of curves represented by explicit or implicit functions......Page 93
5.2 CURVILINEAR COORDINATES......Page 96
5.3 TANGENT PLANE AND SURFACE NORMAL......Page 97
5.5 EXAMPLES OF SURFACES......Page 100
Surface of Revolution......Page 101
Spherical Surface......Page 103
Cone Surface......Page 106
General Equations of a Helicoid......Page 108
Helicoid with Ruled Surface......Page 109
Relationship Between Helicoid Coordinates and the Surface Normal Projections......Page 112
Cross Section of a Helicoid......Page 113
Introduction......Page 115
Engineering Approach......Page 116
Particular Cases......Page 117
Planar Gearing......Page 118
6.2 BASIC KINEMATIC RELATIONS......Page 120
6.3 CONDITIONS OF NONUNDERCUTTING......Page 121
Classical Approach......Page 125
Engineering Approach (proposed by Litvin [1968, 1989])......Page 127
6.5 CONTACT LINES; SURFACE OF ACTION......Page 128
6.6 ENVELOPE TO FAMILY OF CONTACT LINES ON GENERATING SURFACE  1......Page 130
6.7 FORMATION OF BRANCHES OF ENVELOPE TO PARAMETRIC FAMILIES OF SURFACES AND CURVES......Page 132
Branches of Tooth Profiles in a Cycloidal Pump......Page 133
6.8 WILDHABER’S CONCEPT OF LIMIT CONTACT NORMAL......Page 136
Planar Gearing......Page 137
Spatial Gearing......Page 139
Introduction......Page 142
Equation of Meshing......Page 143
Conditions of Nonundercutting......Page 144
Basic Concept......Page 146
Worm-Gear Drive with Cylindrical\rWorm......Page 148
Generation of\rWorm by Peripheral Tool......Page 150
6.12 KNOTS OF MESHING......Page 152
6.13 PROBLEMS......Page 155
Space Curve Trihedron......Page 171
Frenet–Serret Equations......Page 174
Determination of  and … for a Curve Represented by r(s)\r......Page 176
Determination of κo and τ for a Curve Represented by Vector Function r(phi)......Page 177
Determination of κo......Page 179
Structure of Spatial Curve at the Curve Point......Page 180
Equation of Osculating Plane......Page 181
Surface Curve Trihedron......Page 182
Determination of Derivatives ts , ds , ns......Page 183
Velocity and Acceleration......Page 185
Curvature of Spatial Curve......Page 188
Geodesic Curvature......Page 190
First Fundamental Form......Page 193
Interpretation of Fundamental Forms......Page 194
Determination of Normal Curvature......Page 195
Approach 1......Page 198
Rodrigues’ Formula......Page 200
Approach 2......Page 201
Particular Case 2......Page 203
7.6 EULER’S EQUATION......Page 206
7.7 GAUSSIAN CURVATURE; THREE TYPES OF SURFACE POINTS......Page 207
Hyperbolic Point......Page 211
Geodesic Line......Page 212
Surface Torsion as the Curve Torsion of a Geodesic Line......Page 214
Relations Between Surface Torsion and Surface Principal Curvatures......Page 216
Relation Between Surface Normal Curvatures and Torsions in Directions of t(1) and t(2) (Fig. 7.9.5)......Page 219
8.1 INTRODUCTION......Page 220
8.2 BASIC EQUATIONS......Page 221
8.3 PLANAR GEARING: RELATION BETWEEN CURVATURES......Page 222
Transformation of Translation into Rotation and Rotation into Translation......Page 226
Auxiliary Equations......Page 236
Basic System of Linear Equations......Page 238
Case 1......Page 240
Case 3......Page 243
Case 1......Page 244
Derivation of First Two Equations of System (8.5.4)......Page 245
Derivation of Third Equation of System (8.5.4)......Page 246
Case 2......Page 247
Particular Case......Page 248
8.6 DIAGONALIZATION OF CURVATURE MATRIX......Page 249
Basic Equation of Elastic Deformations......Page 252
Determination of Contact Ellipse......Page 254
9.1 INTRODUCTION......Page 259
9.2 PREDESIGN OF A PARABOLIC FUNCTION OF TRANSMISSION ERRORS......Page 260
Effect of Application of a Predesigned Parabolic Function of Transmission Errors......Page 261
Determination of Derivative m\'21......Page 262
9.3 LOCAL SYNTHESIS......Page 263
Relation Between Directions of Paths of Contact......Page 264
Relations Between the Magnitude of the Major Axis of the Contact Ellipse, Its Orientation, and Principal Curvatures and Directions of Contacting Surfaces......Page 265
Conditions of Continuous Tangency......Page 267
Analysis of Meshing......Page 269
9.5 APPLICATION OF FINITE ELEMENT ANALYSIS FOR DESIGN OF GEAR DRIVES......Page 275
9.6 EDGE CONTACT......Page 278
Edge Contact of Gear Tooth Surfaces That Are Initially in Line Contact......Page 280
Edge Contact of Gear Tooth Surfaces That Are Initially in Point Contact......Page 282
10.1 INTRODUCTION......Page 285
Involute Curve Used for Spur Gears......Page 286
Extended and Shortened Involute Curves......Page 288
Generation by a Rack-Cutter......Page 291
Design Parameters of Rack-Cutter......Page 293
Generation by Hob......Page 294
10.4 TOOTH ELEMENT PROPORTIONS......Page 296
Conditions of Nonundercutting......Page 298
Change of Gear Tooth Thickness and Dedendum Height......Page 301
10.6 RELATIONS BETWEEN TOOTH THICKNESSES MEASURED ON VARIOUS CIRCLES......Page 303
Line of Action......Page 305
Change of Center Distance......Page 306
Involute Profiles as Equidistant Curves......Page 307
Interference......Page 308
10.8 CONTACT RATIO......Page 310
10.9 NONSTANDARD GEARS......Page 312
Long–Short Addendum System......Page 313
Long–Short Addendum System: Computational Procedure......Page 314
General Nonstandard Gear System: Computational Procedure......Page 316
11.1 INTRODUCTION......Page 322
11.2 GENERATION OF GEAR FILLET......Page 323
Pseudohypocycloid......Page 325
Envelope to Family of Extended Hypocycloids......Page 326
Internal Gear Involute Profile......Page 327
Nonundercutting by Axial Generation......Page 329
Two-Parameter Generation......Page 330
Axial Assembly......Page 332
Radial Assembly......Page 333
Nomenclature......Page 334
12.2 CENTRODES OF NONCIRCULAR GEARS......Page 336
12.3 CLOSED CENTRODES......Page 341
Modification of Elliptical Centrode......Page 344
12.5 CONDITIONS OF CENTRODE CONVEXITY......Page 347
12.6 CONJUGATION OF AN ECCENTRIC CIRCULAR GEAR WITH A NONCIRCULAR GEAR......Page 348
12.7 IDENTICAL CENTRODES......Page 349
12.8 DESIGN OF COMBINED NONCIRCULAR GEAR MECHANISM......Page 351
12.9 GENERATION BASED ON APPLICATION OF NONCIRCULAR MASTER-GEARS......Page 353
12.10 ENVELOPING METHOD FOR GENERATION......Page 354
Generation by a Rack-Cutter: Relations Between Motions......Page 356
12.11 EVOLUTE OF TOOTH PROFILES......Page 359
12.12 PRESSURE ANGLE......Page 362
Gear Centrode......Page 363
Equations of Centrode Tangency......Page 364
Computational Procedure......Page 365
APPENDIX 12.B: DISPLACEMENT FUNCTIONS FOR GENERATION BY SHAPER......Page 366
13.2 GENERATION OF CYCLOIDAL CURVES......Page 368
Extended Epicycloid......Page 372
13.4 CAMUS’ THEOREM AND ITS APPLICATION......Page 373
Tooth Addendum–Dedendum Profiles......Page 374
Watch Gearing......Page 376
13.5 EXTERNAL PIN GEARING......Page 377
Conjugation of Tooth Profiles......Page 378
Equation of Meshing......Page 379
Gear 2 Tooth Profile......Page 381
Rack-Cutter Tooth Profile......Page 382
Applied Coordinate Systems......Page 383
Line of Action......Page 384
Basic Idea......Page 385
Relation Between Gear Tooth Numbers......Page 386
13.8 ROOT’S BLOWER......Page 387
Conjugation of Profiles......Page 388
Applied Coordinate Systems......Page 389
Equation of Meshing; Line of Action......Page 390
Equations of Dedendum Curve Σ2 of Rotor 2......Page 392
14.2 GENERAL CONSIDERATIONS......Page 393
14.3 SCREW INVOLUTE SURFACE......Page 395
14.4 MESHING OF A HELICAL GEAR WITH A RACK......Page 400
Rack Surface Σr......Page 401
Interpretation of Σr......Page 402
Sections of Σr......Page 403
Lines of Contact on Σr......Page 405
Lines of Contact L1r on Surface Σ1......Page 406
Surface of Action......Page 407
Relations Between Design Parameters......Page 408
Equation of Meshing......Page 410
Derivation of Surface Σ2......Page 411
Surface of Action......Page 413
14.6 CONDITIONS OF NONUNDERCUTTING......Page 414
14.7 CONTACT RATIO......Page 416
14.8 FORCE TRANSMISSION......Page 417
14.9 RESULTS OF TOOTH CONTACT ANALYSIS (TCA)......Page 420
14.10 NOMENCLATURE......Page 421
15.1 INTRODUCTION......Page 422
15.2 AXODES OF HELICAL GEARS AND RACK-CUTTERS......Page 425
Pinion Parabolic Rack-Cutter......Page 426
Gear Rack-Cutter......Page 428
Generation of Σσ......Page 429
Necessary and Sufficient Conditions of Existence of an Envelope to a Parametric Family of Surfaces......Page 430
Representation of Envelope Σσ in Two-Parameter Form......Page 431
Meshing of Profile-Crowned Helicoids: Conceptual Considerations......Page 432
Algorithm of Analytical Simulation......Page 434
Application of a Plunging Disk......Page 437
Worm Installment......Page 442
Determination of Worm Thread Surface Σw......Page 443
Profile Crowning of Pinion......Page 445
Double Crowning of Pinion......Page 447
15.7 TCA OF GEAR DRIVE WITH DOUBLE-CROWNED PINION......Page 448
Undercutting......Page 450
Pointing......Page 452
15.9 STRESS ANALYSIS......Page 453
Numerical Example......Page 457
16.1 INTRODUCTION......Page 459
Conceptual Considerations......Page 461
Analytical Determination of Line of Action of Crossed Helical Gears......Page 464
Case 1: Error γ of the Crossing Angle (Shaft Angle)......Page 466
Case 2: Error E of the Center Distance......Page 469
16.3 SIMULATION OF MESHING OF CROSSED HELICAL GEARS......Page 470
Numerical Example:\rWorm-Gear Drive......Page 472
Generation of a Helical Gear......Page 473
Generation of Conjugated Crossed Helical Gears......Page 474
Direction of Helices......Page 476
Canonical Design......Page 477
Numerical Example 1: Design of Standard Gears......Page 478
Numerical Example 2: Approach 1 for Design of Nonstandard Crossed Helical Gears......Page 480
Numerical Example 3: Approach 2 for Design of Nonstandard Crossed Helical Gears......Page 481
Numerical Example......Page 483
APPENDIX 16.A: DERIVATION OF SHORTEST CENTER DISTANCE FOR CANONICAL DESIGN......Page 485
APPENDIX 16.B: DERIVATION OF EQUATION OF CANONICAL
DESIGN f (γo, αon, λb1, λb2) = 0......Page 490
APPENDIX 16.D: DERIVATION OF EQUATION (16.5.5)......Page 491
APPENDIX 16.E: DERIVATION OF ADDITIONAL RELATIONS
BETWEEN αot1 AND αot2......Page 492
17.1 INTRODUCTION......Page 493
17.2 AXODES OF HELICAL GEARS AND RACK-CUTTER......Page 496
Mismatched Parabolic Rack-Cutters......Page 497
Pinion Parabolic Rack-Cutter......Page 499
Generation of Σσ......Page 500
Generation of Gear Tooth Surface Σ2......Page 501
Representation of Envelope Σσ in Two-Parameter Form......Page 502
17.5 TOOTH CONTACT ANALYSIS (TCA) OF GEAR DRIVE WITH PROFILE-CROWNED PINION......Page 503
Application of a Disk-Shaped Tool......Page 505
Worm Installment......Page 509
Determination ofWorm Thread Surface Σw......Page 510
Profile Crowning of Pinion......Page 512
Double Crowning of Pinion......Page 514
17.8 TCA OF A GEAR DRIVE WITH A DOUBLE-CROWNED PINION......Page 515
Undercutting......Page 518
Pointing......Page 519
17.10 STRESS ANALYSIS......Page 520
Development of Finite Element Models......Page 521
Numerical Example......Page 523
18.1 INTRODUCTION......Page 526
Axodes......Page 528
Pitch Surfaces......Page 529
18.4 LOCALIZATION OF BEARING CONTACT......Page 530
Shaper Tooth Surfaces......Page 533
Face-Gear Tooth Surface Σ2......Page 535
18.6 CONDITIONS OF NONUNDERCUTTING OF FACE-GEAR TOOTH SURFACE (GENERATED BY INVOLUTE SHAPER)......Page 537
Directions for Computations......Page 539
18.7 POINTING OF FACE-GEAR TEETH GENERATED BY INVOLUTE SHAPER......Page 540
18.8 FILLET SURFACE......Page 542
Basic Concept......Page 543
Reference and Parabolic Rack-Cutters......Page 544
Shaper Tooth Surface......Page 545
Pinion Tooth Surface......Page 546
18.12 DESIGN RECOMMENDATIONS......Page 547
Applied Coordinate Systems......Page 549
Computational Procedure......Page 551
Results of Investigation......Page 552
Concept of Generating\rWorm......Page 553
Crossing Angle Between Axes of Shaper and\rWorm......Page 554
Determination of Worm Surface Σw......Page 555
Generation of Surface Σ2 by Worm Surface Σw......Page 556
Dressing of the\rWorm......Page 557
Numerical Example......Page 559
19.1 INTRODUCTION......Page 565
19.2 PITCH SURFACES AND GEAR RATIO......Page 566
Worm Pitch Diameter, Lead Angle, and Axial Pitch......Page 570
Relation Between\rWorm and Worm-Gear Pitches......Page 571
Radius of\rWorm-Gear Operating Pitch Cylinder......Page 572
Relations Between Profile Angles in Axial, Normal, and Transverse Sections......Page 573
19.4 GENERATION AND GEOMETRY OF ZA WORMS......Page 575
Generation......Page 579
Representation of Generating Lines in Coordinate Systems Sa......Page 582
Note 1: Determination of Expressions for cos δ and sin δ......Page 583
Determination of ρ......Page 584
Equations of Surfaces of\rWorm Thread......Page 585
Kinematic Interpretation of Surface Generation......Page 587
Particular Cases......Page 591
Surface Equations......Page 592
Methods for Generation......Page 596
Generation......Page 599
Worm Surface Equations......Page 600
Particular Case......Page 606
19.8 GEOMETRY AND GENERATION OF F-I WORMS (VERSION I)......Page 608
Installment of the Grinding Wheel for F-I......Page 609
Equations of Generating Surface Σc......Page 610
Lines of Contact on\rWorm Surface......Page 611
Method for Grinding......Page 615
Equation of Meshing......Page 617
19.10 GENERALIZED HELICOID EQUATIONS......Page 619
19.11 EQUATION OF MESHING OF WORM AND WORM-GEAR SURFACES......Page 621
19.12 AREA OF MESHING......Page 624
Introductory Remarks......Page 627
Application of Oversized Hob......Page 629
Worm Generation......Page 632
Worm-Gear Generation......Page 633
Worm-Gear Surface......Page 634
Unmodified and Modified Gearing......Page 635
20.3 WORM SURFACE EQUATIONS......Page 636
Unmodified Gearing......Page 638
Modified Gearing......Page 639
20.6 WORM-GEAR SURFACE EQUATIONS......Page 640
Unmodified Gearing......Page 641
Modified Gearing......Page 643
21.1 INTRODUCTION......Page 645
21.2 BASIC IDEAS OF THE DEVELOPED APPROACH......Page 646
Introduction......Page 651
Applied Coordinate Systems......Page 652
Head-Cutter Surfaces......Page 654
Equations of the Generated Gear Tooth Surface......Page 657
Equations of the Formate-Cut Gear Tooth Surface......Page 659
Head-Cutter Surfaces......Page 662
Families of Pinion Tooth Surfaces......Page 665
Equation of Meshing......Page 666
21.5 LOCAL SYNTHESIS AND DETERMINATION OF PINION MACHINE-TOOL SETTINGS......Page 667
Local Synthesis of Face-Milled Generated Spiral Bevel Gear Drives......Page 668
21.6 RELATIONSHIPS BETWEEN PRINCIPAL CURVATURES AND DIRECTIONS OF MATING SURFACES......Page 674
Meshing of Surfaces Σg and Σ2......Page 675
Meshing of Surfaces Σ2 and Σ1......Page 676
Procedure of Determination of kf , kh, and σ(12)......Page 677
Meshing of Surfaces Σ1 and Σp......Page 678
Applied Coordinate Systems......Page 679
Simulation Algorithm......Page 680
21.8 APPLICATION OF FINITE ELEMENT ANALYSIS FOR THE DESIGN OF SPIRAL BEVEL GEAR DRIVES......Page 683
21.9 EXAMPLE OF DESIGN AND OPTIMIZATION OF A SPIRAL BEVEL GEAR DRIVE......Page 684
21.10 COMPENSATION OF THE SHIFT OF THE BEARING CONTACT......Page 694
22.2 AXODES AND OPERATING PITCH CONES......Page 697
22.3 TANGENCY OF HYPOID PITCH CONES......Page 698
22.4 AUXILIARY EQUATIONS......Page 700
Tooth Longitudinal Shapes......Page 701
Sliding Velocity at the Pitch Point......Page 702
Derivation of Equations (22.5.1) and (22.5.2)......Page 703
Derivation of Equation (22.5.3) Case 1: Hypoid gear drive with face-milled teeth is considered......Page 704
Case 2: Hypoid gear drive with face-hobbed teeth......Page 705
Computational Procedure for Determination of γ1, γ2, and β2......Page 707
Gear Generation......Page 708
Pinion Generation......Page 710
Pinion Tool Surface Equations......Page 711
Equation of Meshing......Page 713
Pinion Tooth Surface......Page 714
Planetary Mechanisms of Figs. 23.2.1 (a) and (b)......Page 715
Planetary Mechanism of Fig. 23.2.2......Page 717
Planetary Mechanism of Fig. 23.2.3......Page 718
Bevel Gear Differential of Fig. 23.2.5......Page 719
Observation of Assigned Backlash Between Planet Gears [Litvin et al., 2002e]......Page 721
Relation Between Tooth Numbers of Planetary Train of Fig. 23.2.4......Page 722
Determination of m1(k), m3(k), δ1(k), and δ3(k), (k = 1, . . . , n)......Page 724
23.4 PHASE ANGLE OF PLANET GEARS......Page 725
23.5 EFFICIENCY OF A PLANETARY GEAR TRAIN......Page 727
Modification of Geometry of Planet Gears......Page 729
Conventional Gear Drive......Page 730
Function of Transmission Errors of Sub-Gear Drives......Page 732
23.8 ILLUSTRATION OF THE EFFECT OF REGULATION OF BACKLASH......Page 734
24.2 GENERATION BY FINGER-SHAPED TOOL: TOOL SURFACE IS GIVEN......Page 736
Equation of Meshing......Page 738
Derivation of Generated Surface Σp......Page 739
24.3 GENERATION BY FINGER-SHAPED TOOL: WORKPIECE SURFACE IS GIVEN......Page 741
Equation of Meshing......Page 744
Generated Surface......Page 746
Equation of Meshing......Page 748
Determination of the Tool Profile......Page 749
25.1 INTRODUCTION......Page 752
25.2 TWO-PARAMETER FORM REPRESENTATION OF WORM SURFACES......Page 753
25.3 THREE-PARAMETER FORM REPRESENTATION OF WORM SURFACES......Page 755
ZN (Convolute)\rWorm......Page 756
ZI (Involute)\rWorm......Page 757
ZK (Klingelnberg)\rWorm......Page 758
F-I (Flender Version I)\rWorm......Page 759
F-II (Flender Version II)\rWorm......Page 762
26.1 INTRODUCTION......Page 764
Coordinate Systems Applied for “Phoenix” CNC machine......Page 765
Basic Principle of Execution of Motions......Page 766
Derivation of L(G) and (OtOp)…......Page 768
Execution of Motions of CNC Machine......Page 769
Introduction......Page 770
Equation of Meshing Between Σt and Σg......Page 772
Determination of Generated Surface Σg......Page 777
Optimal Approximation of Generated Surface Σg to Ideal Surface Σp......Page 778
Curvatures of Ground Surface Σg......Page 782
Numerical Example: Grinding of an Archimedes\rWorm Surface......Page 784
27.2 PROBLEM DESCRIPTION......Page 787
Procedure of Computation......Page 789
Basic Equations......Page 791
Representation of the Unit Vectors of Wire Axes and the Shortest Distance Between the Axes of Two Wires......Page 792
Determination of the Overwire Measurement M......Page 794
Numerical Example: Measurement of Involute Helical Gear......Page 796
27.4 MEASUREMENT OF ASYMMETRIC ARCHIMEDES SCREW......Page 797
Numerical Example: Measurement of Asymmetric Screw......Page 799
28.1 INTRODUCTION......Page 800
28.2 OVERVIEW OF MEASUREMENT AND MODELLING METHOD......Page 801
28.3 EQUATIONS OF THEORETICAL TOOTH SURFACE…......Page 802
28.4 COORDINATE SYSTEMS USED FOR COORDINATE MEASUREMENTS......Page 803
28.5 GRID AND REFERENCE POINT......Page 804
28.7 MINIMIZATION OF DEVIATIONS......Page 805
References......Page 807
Index......Page 813