The book explores the geometric and kinematic design of the various types of gears most commonly used in practical applications, also considering the problems concerning their cutting processes. The cylindrical spur and helical gears are first considered, determining their main geometric quantities in the light of interference and undercut problems, as well as the related kinematic parameters. Particular attention is paid to the profile shift of these types of gears either generated by rack-type cutter or by pinion-rack cutter. Among other things, profile-shifted toothing allows to obtain teeth shapes capable of greater strength and more balanced specific sliding, as well as to reduce the number of teeth below the minimum one to avoid the operating interference or undercut. These very important aspects of geometric-kinematic design of cylindrical spur and helical gears are then generalized and extended to the other examined types of gears most commonly used in practical applications, such as: straight bevel gears; crossed helical gears; worm gears; spiral bevel and hypoid gears. Finally, ordinary gear trains, planetary gear trains and face gear drives are discussed. Includes fully-developed exercises to draw the reader's attention to the problems that are of interest to the designer, as well as to clarify the calculation procedure. Topics are addressed from a theoretical standpoint, but in such a way as not to lose sight of the physical phenomena that characterize the various types of gears which are examined. The analytical and numerical solutions are formulated so as to be of interest not only to academics, but also to designers who deal with actual engineering problems concerning the gears.
Author(s): Vincenzo Vullo
Series: Springer Series In Solid And Structural Mechanics Vol. 10
Edition: 1st Edition
Publisher: Springer
Year: 2020
Language: English
Pages: 880
Tags: Machinery And Machine Elements
Aphorism......Page 6
Preface......Page 8
Contents......Page 18
Symbols, Notations and Units......Page 26
1.1 Introduction......Page 41
1.2 Gear Units and Gears......Page 46
1.3 Efficiency of the Gears......Page 51
1.4 Basic Law of Mating Gear Teeth......Page 54
1.5 Tooth Parts and Some Quantities of the Toothing......Page 61
1.6 Precision and Accuracy Grade of the Gears......Page 70
References......Page 77
2.1 Generation of the Involute and Its Geometry......Page 79
2.2 Parametric Representation of the Involute Curve......Page 84
2.3 Involute Properties and Fundamentals......Page 89
2.4 Characteristic Quantities of the Involute Gears......Page 94
2.5 Gear-Tooth Sizing......Page 104
2.6 Standard Basic Rack Tooth Profile......Page 111
2.7 No-Standard Basic Rack Tooth Profiles......Page 116
References......Page 119
3.1 Minimum Number of Teeth to Avoid Interference......Page 121
3.1.1 Minimum Number of Teeth for Rack-Pinion Pair......Page 122
3.1.3 Minimum Number of Teeth for an Internal Cylindrical Spur Gear......Page 124
3.2 Considerations on the Minimum Number of Teeth......Page 125
3.3 Lengths of the Path and Arc of Contact, and Angles of Contact......Page 129
3.4 Transverse Contact Ratio......Page 133
3.5 Radius of Curvature of Involute Tooth Profiles and Generalized Laws of Gearing......Page 136
3.6 Kinematics of Gearing: Rolling and Sliding Motions of the Teeth Flanks......Page 139
3.7 Relative Sliding and Specific Sliding......Page 148
3.8 Consideration on Wear Damage......Page 151
3.9 Efficiency of Cylindrical Spur Gears......Page 153
3.11.1 Form Cutting Method......Page 166
3.11.2 Generation Cutting Method......Page 170
References......Page 176
4.1 Introduction......Page 179
4.2 Theoretical Interference Between External Spur Gears......Page 181
4.3 Possibility to Realize Gear Pairs with Pinion Having Small Number of Teeth, Through the Generation Process with Rack-Type Cutter......Page 187
4.4 Methods to Avoid Interference in the Cylindrical Spur Gears......Page 191
4.5 Reduction of the Path of Contact Due to Cutting Interference......Page 194
4.6 The General Problem of the Interference: Theoretical Interference and Fillet Interference......Page 198
4.7 Fillet Profile Generated by a Rack-Type Cutter or Hob......Page 203
4.8 Fillet Profile Generated by a Pinion-Type Cutter......Page 209
4.9 Interference Effects of the Rounded Tip of the Cutter Teeth and Fillet Interference in the Operating Conditions......Page 215
References......Page 219
5.1 Introduction......Page 221
5.2 Theoretical Interference in the Internal Spur Gears......Page 223
5.3 Secondary Interference in the Internal Spur Gears......Page 228
5.4 Possibility to Realize Internal Gear Pairs with Pinion Having a Low Number of Teeth......Page 234
5.5 A Geometric-Analytical Method for Checking of Secondary Interference......Page 235
5.6 Tertiary Interference in the Internal Spur Gears......Page 240
5.7 Fillet Interference Between the Tip of the Pinion, and Root Fillet of the Annulus......Page 243
5.8 Fillet Interference Between the Tip of the Annulus, and Root Fillet of the Pinion......Page 245
5.9 A Design Consideration on the Interference Between Internal Spur Gears......Page 247
5.10 Annulus Fillet Profile Generated by a Pinion-Type Cutter......Page 248
5.11 Undercut or Cutting Interference......Page 253
5.12 Condition to Avoid Rubbing During the Annulus Cutting Process......Page 257
References......Page 260
6.1 Introduction......Page 262
6.2 Fundamentals of Profile Shift......Page 266
6.3.1 External Spur Gear Pairs......Page 273
6.3.2 Internal Spur Gear Pairs......Page 277
6.4 Minimum Profile Shift Coefficient to Avoid Interference......Page 280
6.5 Pointed Teeth and Tooth Thickness......Page 283
6.6.1 External Spur Gear Pairs......Page 286
6.6.2 Internal Spur Gear Pairs......Page 298
6.7 The Sum (or Difference) of the Profile Shift Coefficients: Direct Problem and Inverse Problem......Page 300
6.8.1 Criterion to Avoid the Cutting Interference......Page 304
6.8.2 Criterion to Equalize the Maximum Values of the Specific Sliding and Almen Factors of the Pinion and Wheel......Page 306
6.8.3 Criteria to Balance Different Requirements, and Have Gears Well Compensated......Page 309
6.8.3.1 Method DIN 3992......Page 311
6.8.3.2 Method BSI—BS PD 6457......Page 314
6.8.3.3 Method ISO/TR 4467......Page 315
6.9 Determination of the Profile Shift Coefficient of Internal Gear Pairs......Page 318
6.10 Backlash......Page 320
References......Page 323
7.1 Characteristics of the Pinion-Type Cutter......Page 325
7.3 Characteristic Quantities of the Pinion-Type Cutter Referred to a Pitch Circle Shifted by {\varvec xm}_{0} with Respect to the Nominal Pitch Circle......Page 326
7.4 Characteristic Quantities of a Gear Wheel Having {\varvec z}_{1} Teeth, and Generated with a Profile Shift Coefficient {\varvec x}_{1}......Page 327
7.5.1 Pinion (First Wheel), Having {\varvec z}_{1} Teeth, and Generated with Profile Shift Coefficient {\varvec x}_{1}......Page 329
7.5.2 Wheel (Second Wheel), Having {\varvec z}_{2} Teeth, and Generated with Profile Shift Coefficient {\varvec x}_{2}......Page 330
7.5.3 Meshing Between the Two Wheels with a Backlash-Free Contact......Page 331
7.6 Possibility of Making Profile-Shifted Toothings by Means of a Pinion-Type Cutter......Page 332
7.8 Transverse Contact Ratio......Page 335
References......Page 336
8.1 Introduction......Page 338
8.2 Geometry of Parallel Involute Helical Gears......Page 340
8.3 Main Quantities of a Parallel Helical Gear......Page 348
8.4 Generation of Parallel Cylindrical Helical Gears and Their Sizing......Page 353
8.5 Equivalent Parallel Cylindrical Spur Gear and Virtual Number of Teeth......Page 357
8.6 Parallel Cylindrical Helical Gears with Profile-Shifted Toothing and Variation of Center Distance......Page 362
8.7 Total Length of the Line of Action......Page 367
8.8 Load Analysis of Parallel Cylindrical Helical Gears, and Thrust Characteristics on Shaft and Bearings......Page 370
8.9 Double-Helical Gears......Page 373
8.10 Efficiency of Parallel Helical Gears......Page 375
8.11.1 Form Cutting Methods......Page 378
8.11.2 Generation Cutting Methods......Page 379
8.12 Short Notes on Cutting Methods of Double-Helical Gears......Page 381
8.12.2 Generation Cutting Methods......Page 382
References......Page 385
9.1 Introduction......Page 387
9.2 Geometry and Characteristics of the Bevel Gears......Page 390
9.3 Main Equations of Straight Bevel Gears......Page 395
9.4 Spherical Involute Toothing and Octoidal Toothing, and Their Implications on the Cutting Process......Page 397
9.5 Main Conical Surfaces and Equivalent Cylindrical Gear......Page 405
9.6 Minimum Number of Teeth to Avoid Interference......Page 410
9.7 Reference Profile Modifications......Page 416
9.8 Straight Bevel Gears with Profile-Shifted Toothing and Variation of Shaft Angle......Page 419
9.9 Straight Bevel Gears with Profile-Shifted Toothing and Variation of Shaft Angle, as Result of a Transverse Tooth Thickness Modification......Page 429
9.10 Load Analysis for Straight Bevel Gears and Thrust Characteristics on Shaft and Bearings......Page 434
9.11 Efficiency of Straight Bevel Gears......Page 437
9.12.1 Form Cutting Methods......Page 443
9.12.2 Generation Cutting Methods......Page 444
9.13 Construction and Assembly Solutions for Bevel Gears......Page 448
References......Page 450
10.1 Fundamentals of General Rigid Kinematic Pairs......Page 453
10.2 Hyperboloid Pitch Surfaces......Page 457
10.3 Generation of a Crossed Helical Gear Pair......Page 463
10.4 Fundamental Kinematic Properties......Page 467
10.5 Determination of Other Characteristic Quantities of the Crossed Helical Gear Pairs......Page 472
10.6 Path of Contact, Face Width, and Contact Ratio......Page 477
10.7 Longitudinal Sliding and Sliding Velocity......Page 480
10.8 Load Analysis for Crossed Helical Gears and Thrust Characteristics on Shafts and Bearings......Page 485
10.9 Efficiency of Crossed Helical Gears......Page 487
10.10 Profile-Shifted Toothing for Crossed Helical Gears......Page 497
References......Page 499
11.1 Introduction......Page 501
11.2.1 Type A Worm, with Straight-Sided Axial Profile......Page 505
11.2.2 Type I Worm, with Involute Helicoid Flanks, and Generation Straight Line in a Plane Tangent to the Base Cylinder......Page 507
11.2.4 Type K Worm, with Convex Thread Profiles in Axial Plane, and Helicoid Generated by Biconical Grinding Wheel or Milling Cutter......Page 509
11.2.5 Type C Worm, with Concave Axial Profile Formed by Machining with a Concave Circular Profile Disk-Type Cutter or Grinding Wheel......Page 512
11.2.6 Worm Wheel Cutting Process......Page 514
11.3 Coordinate Systems and Main Geometric Quantities of Worm and Worm Wheel......Page 515
11.4 Gear Ratio and Interdependences Between Worm and Worm Wheel Quantities......Page 525
11.5 Elements of Differential Geometry of Surfaces......Page 529
11.6 Parametric Equations of a Helicoid......Page 533
11.7 Relative Velocity and Coordinate Transformation......Page 540
11.8 Worm and Worm Wheel Meshing, and Lines of Contact......Page 544
11.9 Surface of Contact: General Concepts and Determination by Analytical Methods......Page 551
11.10 Surface of Contact: Determination by Graphic-Analytical Methods......Page 562
11.10.1 Schiebel’s Method for Archimedean Spiral Worms......Page 564
11.10.2 Ingrisch’s Method for Involute Worm......Page 570
11.11 Outside Surface of the Worm Wheel and Related Interference Problems......Page 575
11.12 Load Analysis of Worm Gears, Thrust Characteristics on Shafts and Bearings, and Efficiency......Page 579
11.13 Worm and Worm Wheel Sizing: Further Considerations......Page 587
11.14 Double-Enveloping Worm Gear Pairs......Page 590
11.15 Standard and Non-standard Worm Drives, and Special Worm Drives......Page 594
References......Page 601
12.1 Introduction......Page 605
12.2 Considerations on the Spiral Angle......Page 611
12.3 Geometry and Cutting Process of the Main Types of Spiral Bevel and Hypoid Gears......Page 617
12.3.1 Gleason Spiral Bevel Gears......Page 618
12.3.3 Oerlikon-Spiromatic Spiral Bevel Gears......Page 621
12.3.4 Klingelnberg-Ziclo-Palloid Spiral Bevel Gears......Page 623
12.3.5 Klingelnberg-Palloid Spiral Bevel Gears......Page 624
12.3.8 Branderberger Spiral Bevel Gears......Page 626
12.4 Generation Process of the Tooth Active Flank of Spiral Bevel Gears......Page 627
12.5 Spiral Bevel Gears: Main Quantities and Equivalent Cylindrical Gears......Page 634
12.6 Load Analysis for Spiral Bevel Gears and Thrust Characteristics on Shafts and Bearings......Page 639
12.7 Hypoid Gears: Basic Concepts......Page 643
12.8.1 First Analytical Method......Page 652
12.8.2 Second Analytical Method......Page 658
12.9 Main Characteristics of the Hypoid Gears, and Some Indications of Design Choices......Page 669
12.10 Load Analysis for Hypoid Gears......Page 674
12.11 Bevel and Hypoid Gear Geometry: Unified Discussion......Page 678
12.11.1 First Step of Calculation......Page 686
12.11.2 Second Step of Calculation......Page 695
12.12.1 Determination of the Pitch Cone Parameters......Page 701
12.12.1.2 Method 1......Page 702
12.12.1.3 Method 2......Page 706
12.12.1.4 Method 3......Page 710
12.12.2.1 Determination of the Basic Data......Page 712
12.12.2.2 Determination of the Tooth Depth at Calculation Point......Page 714
12.12.2.3 Determination of the Root Angles and Face Angles......Page 715
12.12.2.4 Determination of the Pinion Face Width......Page 716
12.12.2.5 Determination of the Inner and Outer Spiral Angles......Page 719
12.12.2.7 Determination of the Tooth Thickness......Page 722
12.12.3 Undercut Check......Page 724
References......Page 727
13.1 Introduction......Page 731
13.2 Ordinary Gear Trains......Page 732
13.3 Epicyclic or Planetary Gear Trains: Definitions and Generalities......Page 738
13.4.1 Algebraic Method and Willis Formula......Page 744
13.4.2 Torque Balance on Single Members......Page 748
13.4.3 Analysis of Tangential Velocity Vectors......Page 749
13.5 Transmission Ratios Achievable with Planetary Gear Trains......Page 750
13.6 Some Problems Related to Simple Planetary Gear Train......Page 753
13.7 Main Characteristics of Some Planetary Gear Trains......Page 755
13.7.1 Planetary Gear Train Type 1 (Example 1)......Page 756
13.7.2 Planetary Gear Train Type 2 (Example 2)......Page 758
13.7.3 Planetary Gear Train Type 3 (Example 3)......Page 759
13.7.4 Planetary Gear Train Type 4 (Example 4)......Page 762
13.7.5 Planetary Gear Train Type 5 (Example 5)......Page 763
13.7.6 Planetary Gear Train Type 6 (Example 6)......Page 766
13.8.1 Summarizing Planetary Gear Trains......Page 768
13.8.2 Differential Planetary Gear Trains......Page 771
13.9 Multi-stage Planetary Gear Trains......Page 776
13.10 Braking Torque on Members of a Planetary Gear System Held at Rest......Page 783
13.11 Parallel Mixed Power Trains......Page 786
13.12 Efficiency of a Planetary Gear Train......Page 791
13.13 Efficiency of Planetary Gear Trains for Large Transmission Ratios......Page 803
References......Page 806
14.1 Introduction and General News......Page 809
14.2 About the Face Gear Cutting Process......Page 814
14.3 Geometric Elements of a Face Gear Pair, and Its Generation......Page 816
14.4 Basic Topics of Face Gear Manufacturing and Bearing Contact......Page 820
14.5 Tooth Flank Surfaces of Face Gears, and Their Equations......Page 823
14.6 Conditions of Non-undercut of Face Gear Tooth Surface......Page 829
14.7 Conditions to Avoid Face Gear Pointed Teeth, and Fillet Surface......Page 833
14.8 Meshing and Contact Condition Between Tooth Surfaces of a Face Gear Pair......Page 835
14.9.1 General Considerations......Page 840
14.9.2 Determination of the Shaft Angle......Page 842
14.9.3 Determination of igma_{w}-Surface......Page 843
14.9.4 Generation of Surface igma_{2} by the Worm Surface igma_{w}......Page 846
14.9.5 Singularities of the Worm Thread Surface......Page 848
14.9.6 Dressing of the Worm......Page 850
References......Page 852
Index of Standards......Page 855
Name of Index......Page 857
Subject Index......Page 863
