Gears: Volume 1: Geometric And Kinematic Design

Description

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. 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