Gear Geometry and Applied Theory – SECOND EDITION

Gear Geometry and Applied Theory – SECOND EDITION
Faydor L. Litvin
University of Illinois at Chicago
Alfonso Fuentes
Polytechnic University of Cartagena
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
vvi Contents
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 245Contents 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 365viii 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 482Contents ix
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 Cylindrical Worms 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-Enveloping Worm-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 620x Contents
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 735Contents 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
Index
Addendum, 49, 276, 279, 296, 298, 300, 302, 392
circle, 279, 291, 292, 301, 309
cyclinder, 461, 463
cycloidal gear tooth profile, of, 356, 357, 370
Archimedes screw surface, 82, 93, 94, 96
Archimedes spiral, 27, 96, 272
Area of meshing, 606, 607, 609, 610
Axes of meshing, 128–135, 597, 598
Axial diametral pitch, 618
Axodes, 51, 52, 55, 56, 375, 383, 390, 394,
407, 408, 425, 426, 449, 456, 473, 478,
479, 492, 493, 510, 511, 513, 548, 679,
688
Backlash regulation (planetary gear trains), 697,
711–714, 716
Base circle, cylinder, 268, 273–275, 278, 281,
283, 287–289, 377, 379, 443, 445, 450,
461, 576
Bearing contact, 241, 260, 404, 435, 508, 528, 531,
633, 661, 665
localization, 406, 508, 512, 513
Bevel gear differential, 701, 702
Binormal, 154, 155, 164
Blade installation (for worm generation)
ZA worms, of, 557
ZI worms, of, 578
ZK worms, of, 581
ZN worms, of, 561, 562, 567
Blade profile, 557, 558, 563, 581
Bobilier construction, 352, 353
Boundary conditions (for FEA), 257, 258, 260
Canonical design, 441, 446, 459–461, 467, 473
Carrier, 697
Center distance, 288, 289, 294, 296–298, 302
Centrodes
circular gears, of, 38, 45, 49, 50, 115, 137, 211,
274, 280, 288, 289, 296, 298, 352
noncircular gears, of, 318, 322–334, 337, 338,
341, 343–346, 348, 349, 355, 356, 359,
360
Circular pitch, 390
Clearance, 609
Computer controlled machine (CNC), 746
Phoenix, 747
Star, 748, 749
Cone
generation, of, 31
surface, 31
Contact (of surfaces)
characteristic, 110, 202, 226
line (contact), instantaneous, 97, 110, 111, 128,
202, 226, 242, 249, 262, 404, 405, 408,
427, 440, 441, 444, 445, 450, 456, 508,
538, 656
point (contact), instantaneous, 202, 225, 230,
242, 252, 260, 261, 264, 405, 408, 414,
427, 440, 441, 444, 508, 525, 533, 538,
656
Contact ellipse, 234–240, 247–249
Contact lines, 387–389, 395, 553, 586, 590, 593,
599, 604, 605, 621, 622, 624–626,
721–723, 731
Contact path, 252, 263–265
Contact ratio, 292–294, 398
Coordinate measurement, 782–785
Coordinate transformation, 1, 5
Coordinate transformation, inverse, 4
Coordinates
Cartesian, 78
curvilinear (Gaussian, for surfaces), 78
homogeneous, 1, 2, 14
Cradle, 631, 634
Cradle angle, 635, 653, 692
Crossing angle, 446, 448, 449, 452, 453,
458–464
Crowning
double, 408, 417, 419, 429, 431, 485, 487, 496,
500, 611, 711, 716
longitudinal, 405, 406, 419, 429, 430, 477, 487,
496, 498, 611, 711
profile, 405, 406, 414, 419, 427, 477, 482, 487,
494, 498, 611, 711
795796 Index
Curvature
Gaussian, 189, 190, 193, 194, 529
geodesic, 166, 167, 172–174, 195
normal, 166–173, 177, 179, 180, 188, 190, 193,
198
planar curves, of, 68, 72–75, 77
principal (curvatures), 175, 180, 182, 184, 185,
188, 190, 191, 193, 198–200
principal directions, 175, 180–189, 192, 199,
200
radius of, 68, 72, 74
spatial curves, of, 156, 158, 159, 161, 163, 164,
170–173
Curvature matrix, 218, 231, 232
diagonalization, 231–234
Curvature relations
normal curvatures of surfaces, of, 226–230
planar gearing, for, 204–218
principal curvatures, of, 218–225
Cutter mean radius, 637
Cutter point radius, 637, 639, 645, 693
Cutter point width, 637
Cutting ratio, 617, 618, 621
Cycloid
extended, 27
generation, of, 27
ordinary, 27
shortened, 27
Cycloidal gearing, 350
Cycloidal gearing pump, 115, 116
Decomposition of motion, 202, 226
Dedendum, 49, 276, 279, 283, 296, 297, 302,
392
circle, 279, 283, 285, 300, 301
cyclinder, 461, 463
cycloidal gear tooth profile, of, 356, 357, 359,
370
Deviations (of real tooth surface), 787
determination, 762, 763
minimization, 763, 764, 782, 787
Diametral pitch, 49, 50
Direction cosines, 4
Directions (on surface)
principal, 175, 180–189, 192, 199, 200
Disk-shaped milling cutter, 719, 720, 726, 727,
729, 730, 732
Displacement functions (for generation of
non-circular gears)
rack-cutter, by, 345–347
shaper, by, 348, 349
Double-enveloping worm gear drive, 614
modified gearing, 617, 620, 621, 625
unmodified gearing, 617–625
Eccentric gears, 289, 330
Edge contact, 260–266, 404, 408, 439–441, 443,
446, 448, 452, 459, 460
Edge of regression, 80
Efficiency (of a planetary gear train), 709–711
Elastic deformation, 234, 235, 238
Elliptical gears, 318, 319, 326–328
Envelope to family of contact lines, 112, 113, 118,
119
formation of branches of envelope, 114, 115, 117
necessary and sufficient conditions of existence,
112, 113
Envelope to family of surfaces, 97–100, 103,
105–110, 202, 203
necessary conditions of existence, 97, 98
sufficient conditions of existence, 97, 107–110
Envelope to two-parametric family of surfaces, 125,
126, 427, 428
necessary conditions of existence, 126
Epicycloid
extended, 24, 25, 59, 63, 64, 66, 67, 77, 350,
352, 354
generation, of, 24, 351, 352
ordinary, 25, 67, 77, 350, 353
shortened, 25, 350
Equation of meshing, 98–101, 104, 106, 361–363,
366, 372, 383, 392, 583, 584, 586, 588,
593, 595, 596, 599, 603, 604, 620, 621,
641, 648, 650, 652, 653, 657, 695, 737,
740, 742, 754, 757–759, 785
Evolute, 268, 274, 341–343
Fillet, 119–124, 144
face gears, of, 508, 510, 524
internal gears, of, 305–307
Finger-shaped milling cutter, 718–720, 722–725
Finite element analysis, 257–259
Flyblade, 734
Force transmission, 399, 401
Forms (fundamental)
first and second, 175–177, 180
Frenet equations, 71
Frenet trihedron, 69, 70, 73
Gaussian coordinates (surface parameters), 376
Gear ratio, 44, 45, 50–52, 58, 256, 287–289, 318,
322, 323, 327, 330, 407, 418, 549, 550, 552,
555, 604, 617, 697, 698, 701–703, 706, 709
Generating lines (for worm generation), 557,
563–567, 569, 570, 572, 614, 779
Geneva mechanism, 318
Geodesic line, 167, 169, 194–198
Grinding, 404
Grinding wheel, 579–581, 591–594, 597–600, 603,
641, 718, 729
Helical gears, 375
nonstandard, 375
parallel axes, with, 375, 376
profile angles, 385, 386, 402
standard, 375Index 797
Helicoid, 90–95, 376, 395, 547, 597, 603, 611,
731, 773, 774, 781
generalized, 601, 602, 604, 606
generation, of, 28, 90, 718, 721
Helix (on a surface), 163, 175, 376, 377, 380, 388,
396, 547–550, 553, 555, 569, 573, 574,
578, 680
Helix angle, 375, 390, 554
Herringbone teeth (helical gears), 401
Hobbing, 404
Honing, 441
Hyperboloids of revolution, 55–57, 679
Hypocycloid
extended, 304, 307, 308, 311, 312, 314, 316,
352, 353, 355
ordinary, 353, 354
pseudo, 304, 307, 308
Hypoid gears, 679
face-hobbed, 685, 687, 689
face-milled, 685, 686, 690–692
formate-cut, 687, 690, 691
Instantaneous axis of rotation, 51, 99, 375, 383,
392, 395
Instantaneous center of rotation, 37, 44, 45, 48,
101, 120, 121, 137, 138, 144, 271, 272,
274, 275, 287, 290, 302, 350
Interference, 290
internal gears, of, 314–316
spur gears, of, 291
Internal gears, generation of
axial, 305, 309
axial and step-by-step radial, 305
axial–radial (two-parametric), 305, 309
Involute
conventional, 268–270, 272, 309, 310
extended, 270–272
function, 270
shortened, 270, 271
Involute curve
extended, 25, 59, 76, 95, 572
generation, of, 25, 26
ordinary, 25, 63, 65, 76
shortened, 25
Jacobian, 252, 418, 452
Knots of meshing, 134–137
Lead angle, 552, 553, 734, 768
Limit contact normal (Wildhaber’s concept), 118,
119
Limiting position of rays (for a planar curve), 60, 61
at a regular point, 60
at a singular point, 60
Limiting position of rays (for a surface)
at a regular point, 80
set of rays, 80
Line of action, 33, 126, 129, 252, 255–257, 287,
441, 445–448, 450, 459, 460
cycloidal gears, of, 357, 363, 366, 372
Local synthesis, 245–249, 628, 629, 631, 633, 641,
649, 650, 652, 656
Localized bearing contact, 416
Machine center to back, 642, 644, 692
Machine offset, 692
Machine root angle, 634, 642, 644, 692
Machine-tool settings (of spiral bevel gears), 635,
642–645, 650, 655
Matrix
column, 1, 4
determinant, of, 4
direct, 5
identity, 5, 9
inverse, 4, 5, 15, 17, 19
rotational, 14, 15
row, 1
skew-symmetric, 9, 156, 166
symmetric, 222, 229, 231
translational, 14, 16, 18
transpose, 1
Misalignment, 242, 243, 249, 250, 262, 402,
403
Modified roll, 406, 429, 430, 478, 496, 498,
631–633, 644, 648, 649
Module, 279, 408, 430, 435, 454, 457, 461,
479
Motion
generalized parameter, 39, 98, 100, 106, 109,
110, 115, 122
planar, 44
relative, 47, 102, 103, 129, 161, 202
rotational, 44
transfer, 102, 119, 161, 202
translational, 44
Noise, 404, 406, 416, 417, 419, 475, 477, 478,
485, 487, 627–631, 649
Noncircular gear applications
combined, as, 320
crank-slider linkage, with, 319
Geneva mechanism, 318, 319
instruments, for, 321
liquid meter, for, 320
twisted, as, 321
Noncircular gear generation
enveloping method by rack-cutter, 337–339
enveloping method by shaper, 341
master-gears, by application of, 335, 336
worm-gear master mechanism, by application
of, 336
Nonstandard gears, 280, 284, 294, 295, 300,
302
general system, 295, 298
long–short addendum system, 295798 Index
Normal, unit normal
planar curve, to, 62, 63, 69–71, 74
principal, 154, 155
spatial curve, to, 154
surface, to, 81, 82, 165
Osculating
circle, 68–70
plane, 153–157, 161–164, 168–173,
197
Oval gears, 318, 328
Overcentrode cycloidal gearing, 367–370
Overwire (ball) measurement, 769
Parabola vertex location parameter, 638
Path of contact, 252, 263–265, 514
Phase angle (of planet gears), 707–709
Pin gearing
external, 359–364
internal, 365–367
Pitch
angular, 292
axial, 553
base, 280, 293
circular, 278, 526
diametral, 274, 278, 279, 526
normal, 553
transverse, 553
Pitch circle, 49, 274–276, 278, 379
operating, 50
Pitch cones, 51, 511, 514, 679–681
design, 685–690
operating, 679, 680
Pitch cylinder, 390, 404, 420, 424, 425, 428, 449,
459–461, 488, 491, 495, 496
operating, 375, 418, 449, 459
Pitch diameter
hob, of, 611
worm, of, 552, 553
Pitch line, 511, 512, 514
Pitch plane, 681–684, 686–689
Pitch point, 510, 511, 513, 514, 517, 518, 522,
681–684, 686
Pitch surfaces, 510, 511, 548
operating, 548
ordinary, 548
Planar curve
parametric representation, 59, 63, 72
regular, 59, 60
representation by implicit function, 60
simple, 59, 60
Plane
normal to surface, 154, 164, 169–172, 178, 180,
197, 198
osculating, 153–157, 161–164, 168–173, 197
rectifying, 154
surface parameters, of, 78
Plane of action, 389, 396, 443–445
Plunging (of a tool), 406, 419, 429–432, 438
Plunging motion, 422, 423, 487, 489, 490
Pointing
face gears, of, 522–524
modified helical gears, of, 434, 435
Novikov-Wildhaber helical gears, of, 501
spur gears, of, 283
Points of planar curve
regression, 60, 61, 66, 67
singular, 60, 61, 67
Points of surface
elliptic, 190, 193, 194, 529
hyperbolic, 190–195, 529
parabolic, 191, 192, 194, 529
rectification, 153, 155
Poisson’s ratio, 440, 506, 542, 665
Predesigned parabolic function of transmission
errors, 242–245, 477, 478, 482, 498,
500
Pressure angle, 288, 291, 298, 344, 409, 415, 418,
430, 434, 523
Profile angles, of worm
axial, 555, 556, 586
normal, 555, 556, 581
transverse, 555, 556, 576, 581
Rack, rack-cutter
circular arc profile, with, 145, 146
external pin gearing, for, 361, 364
internal pin gearing, for, 367
parabolic profile, with, 408, 410, 477, 479, 481,
482, 487, 502, 515, 525, 526
straight line profile, with, 105, 119, 273–276,
280–283, 408, 411, 525, 526
watch gearing, for, 359
Radial distance, 635
Radial setting, 692
Ratio of roll, 644, 648
Root’s blower, 369, 371
Rotation
between crossed axes, 33, 52, 441
between intersected axes, 51, 52
between parallel axes, 35, 44, 441
Screw involute surface, 405, 443–445, 456
Screw motion, 28, 54, 414, 415, 421, 423, 428,
443, 444, 456, 488–490, 495, 496
instantaneous axis, of, 55
parameter, 28, 32, 55, 90, 94
Screw rotors
compressors, of, 351
pump, of, 351
Screw surface
generation, of, 32
Settings (or rack-cutter)
conventional, 280–282
limiting, 280, 282
non-conventional, 280, 282, 283, 295Index 799
Shaper, 508, 512, 513, 527
edged top, with, 541–545
rounded top, with, 524, 541
Shaping, 404
Shaving, 441
Singularities (see undercutting), 103
Sliding base, 642, 644, 692
Spiral bevel gears, 627
face-milled generated, 627, 650, 666
formate-cut, 627, 633–635, 641–644, 650,
656
Spur involute gears, 267, 268, 273
generation by a hob, 276, 277
generation by a rack-cutter, 273–276
generation by a shaper, 278
Standard center distance, 50
Stress analysis
crossed helical gears, of, 465–467
face gears, of, 541–546
modified helical gears, of, 435–440
Novikov-Wildhaber helical gears, of,
502–507
spiral bevel gears, of, 670–676
Surface
normal, 6
point, 6
revolution, of, 29
spherical, 30
Surface of action, 112, 134, 389, 392, 395,
606
Surface points
pseudosingular, 86, 88
regular, 80
singular, 80, 81
Surface representation
by implicit function, 82
in parametric form, 78
Surface types
cone, of, 82, 83, 88, 183, 185, 186
helicoid, 90–95
involute screw, 82, 83, 93, 95, 185, 547, 570,
573, 575, 576, 579, 588, 712, 724
regular, 82, 97, 107, 108, 110
revolution, of, 83–85, 185, 195, 196
ruled, 82, 83, 91–93, 191, 192, 195, 196
screw, 82, 149, 150
simple, 78, 82
spherical, 85–88, 171, 172, 182, 195,
196
Swivel angle, 692
Tangent plane
“half” tangent plane, 80
surface regular point, at, 79–82, 153, 176,
179
Tangent, unit tangent
“half” tangent (to planar curve), 60, 61, 66,
67
planar curve regular point, at, 60, 61, 63, 69, 70,
77
spatial curve, to, 153–155, 161, 164, 165, 167,
168, 170–172, 176, 180
Theorem
Bonnet, 167, 196
Camus, 355, 356
Clariaut, 195
Dupin’s (for indicatrix), 193–195
Euler, 188, 189
Euler–Rodrigues, 8, 9, 14
Frenet, 69, 70, 73
Frenet-Serret, 156, 159
Gauss, 175, 189
Implicit function system existence, 106, 108, 251,
413, 418, 452, 484, 518
Lewis, 101
Meusnier, 171, 172
Rodrigues, 8, 182
Wildhaber, 118, 119
Zalgaller, 60, 80, 107, 110
Thickness of tooth, 279, 283, 285–287, 294, 295,
298, 299, 380, 392
Tilt angle, 692
Tooth contact analysis (TCA), 249–256, 531
Tooth thickness, 455, 457, 461–463, 473
Torsion
spatial curve, of, 156–159, 162–164
surface, of, 166, 167, 194, 196–198, 227
Transfer of meshing, 289, 293
Transition point, 242
Transmission errors, function of, 404, 408, 415,
417–419, 430, 431, 498, 500, 533, 548,
606, 611, 612, 661, 665, 714, 716
integrated one, 716, 717
linear one, 242–244, 264, 404, 406, 416, 419,
431, 475, 487, 500, 548, 611, 629,
712
parabolic one, 496, 498, 500, 535, 611, 613,
628–631, 633, 667, 711, 712
predesigned parabolic one, 242–245, 406, 416,
429–431, 526, 535
Transmission function, 242–244
Trihedron
planar curve, of, 69, 70
spatial curve, of, 153–155, 158, 162, 163
surface spatial curve, 164–166, 172
Undercutting, 103–107, 109, 110, 118, 126, 127
face gears, of, 519–522, 531, 539
helical gears, of, 396, 397
internal involute gears, of, 304, 307, 309, 311,
312
modified helical gears, of, 432–434
Novikov-Wildhaber helical gears, of, 500, 501
spur involute gears, of, 280, 281, 285
Unitless coefficient of face gears, 529
Unitless stress parameter, 545, 546800 Index
Vector
free, 6
free, components of, 6
moment, 33, 54
sliding, 6, 33, 38, 54
unit, 1, 3, 4, 6, 8–11
Velocity
relative, 33, 34, 39, 102, 128, 584, 603, 652,
657, 658, 660, 679, 683, 709, 755, 757, 785
relative angular, 304
sliding, 34–36, 38, 41, 42, 99, 102, 106, 203,
213, 701
sliding, matrix representation, 39
transfer, 102
Vector
free, 6
free, components of, 6
moment, 33, 54
sliding, 6, 33, 38, 54
unit, 1, 3, 4, 6, 8–11
Velocity
relative, 33, 34, 39, 102, 128, 584, 603, 652,
657, 658, 660, 679, 683, 709, 755, 757, 785
relative angular, 304
sliding, 34–36, 38, 41, 42, 99, 102, 106, 203,
213, 701
sliding, matrix representation, 39
transfer, 102
Velocity ratio, 644, 648
Vibration, 404, 406, 416, 417, 419, 475, 485, 487,
606, 611, 629, 649
Watch gearing, 358
Width (of space), 279, 283, 295, 297–299, 377,
379, 392, 455, 473
Willis’ equation, 698
Worm surface types
ZA (Archimedes) worm, 547, 557, 561, 573,
604, 605, 609, 610, 738
ZF (Flender) F-I worm, 590, 591, 594–597,
599, 743
ZF (Flender) F-II worm, 597–600, 744
ZI (Involute) worm, 547, 574–579
ZK (Klingelnberg) worm, 547, 581,
740
ZN (Convolute) worm, 547, 561–573
Young’s Modulus, 440, 506, 542,
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