Gear Cutting Tools – Fundamentals of Design and Computation
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Stephen P. Radzevich
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Gear Cutting Tools – Fundamentals of Design and Computation
Stephen P. Radzevich
Contents
Preface . xix
Acknowledgments xxi
Introduction .xxiii
Syntax . xxxi
Section I Basics
1. Gears: Geometry of Tooth Flanks .3
1.1 Basic Types of Gears .3
1.2. Analytical Description of Gear Tooth Flanks .6
1.2 1 Tooth Flank of an Involute Spur Gear .9
1.2 2. Tooth Flank of an Involute Helical Gear . 10
1.2 3 Tooth Flank of a Bevel Gear 14
1.2 4 Tooth Flank of a Helical Bevel Gear 15
1.3 Gear Tooth for Surfaces That Allow Sliding . 16
2. Principal Kinematics of a Gear Machining Process 19
2 1 Relative Motions in Gear Machining . 19
2 1.1 Elementary Relative Motions of the Work Gear and the Gear
Cutting Tool .2.0
2 1.2. Feasible Relative Motions of the Work Gear and the Gear
Cutting Tool . 2.1
2 2 Rolling of the Conjugate Surfaces 2.3
3. Kinematics of Continuously Indexing Methods of Gear Machining
Processes .2.5
3.1 Vector Representation of the Gear Machining Mesh 2.5
3.2. Kinematic Relationships for the Gear Machining Mesh 32.
3.3 Configuration of the Vectors of Relative Motions 37
3.3.1 Principal Features of Configuration of the Rotation Vectors . 37
3.3.2. Classification of Gear Machining Meshes 39
3.4 Kinematics of Gear Machining Processes .42.
4. Elements of Coordinate Systems Transformations .43
4.1 Coordinate System Transformation .43
4.1.1 Introduction .43
4.1.2. Translations .44
4.1.3 Rotation about Coordinate Axis .46
4.1.4 Resultant Coordinate System Transformation .47
4.1.5 Screw Motion about a Coordinate Axis 48
4.1.6 Rolling Motion of a Coordinate System 50
4.1.7 Rolling of Two Coordinate Systems . 52.viii Contents
4.2. Conversion of the Coordinate System Orientation 54
4.3 Direct Transformation of Surfaces Fundamental Forms 55
Section II Form Gear Cutting Tools
5. Gear Broaching Tools 59
5.1 Kinematics of the Gear Broaching Process .59
5.2. Generating Surface of a Gear Broach .60
5.3 Cutting Edges of the Gear Broaching Tools 61
5.3.1 Rake Surface of Finishing Teeth of a Gear Broach . 61
5.3.2. Clearance Surface of Gear Broach Teeth .64
5.4 Chip Removal Diagrams .65
5.5 Sharpening of Gear Broaches .66
5.6 A Concept of Precision Gear Broaching Tool for Machining Involute
Gears . 70
5.7 Application of Gear Broaching Tools .72.
5.7.1 Broaching Internal Gears 73
5.7.2. Broaching External Gears 73
5.8 Shear-Speed Cutting 74
5.8.1 Principle of Shear-Speed Cutting of Gears . 74
5.8.2. Profiling of Form Tools for Shear-Speed Cutting of Gears . 76
5.8.3 Application of Shear-Speed Cutting 79
5.9 Rotary Broaches: Slater Tools 80
5.10 Broaching Bevel Gear Teeth 81
5.10.1 Principle of the Revacycle Process of Cutting of Gear Teeth .82.
5.10.2. Revacycle Cutting Tools .83
5.10.3 Profiling of a Cutter for Machining Bevel Gears Using the
Revacycle Process .85
5.10.4 Application of the Revacycle Process of Cutting of Gear Teeth .90
6. End-Type Gear Milling Cutters . 91
6.1 Kinematics of Gear Cutting with End-Type Milling Cutter . 91
6.2. Generating Surface of the End-Type Gear Milling Cutter 92.
6.2 1 Equation for the Generating Surface of an End-Type Milling
Cutter for Machining Spur Involute Gears .92.
6.2 2. Equation for the Generating Surface of an End-Type Milling
Cutter for Machining Helical Involute Gears .96
6.2 3 Elements of Intrinsic Geometry of the Generating Surface of
End-Type Milling Cutters 100
6.3 Cutting Edges of the End-Type Gear Milling Cutter . 101
6.3.1 Rake Surface of the Milling Cutter for Machining of
Involute Gears . 101
6.3.2. Clearance Surface of the Milling Cutter for Machining of
Involute Gears . 105
6.3.3 Cutting Edge Geometry of the End-Type Milling Cutter . 108
6.4 Accuracy of Machining of Gear Tooth Flanks with End-Type Milling
Cutters 115Contents ix
6.4.1 Cusps on Tooth Flanks of Spur Gear . 115
6.4.2. Cusps on the Tooth Flanks of a Helical Gear . 117
6.5 Application of Gear Milling Cutters 119
7. Disk-Type Gear Milling Cutters 12.3
7.1 Kinematics of Gear Cutting with Disk-Type Milling Cutter 12.3
7.2. Generating Surface of the Disk-Type Gear Milling Cutter . 12.4
7.2 1 Equation for the Generating Surface of Disk-Type Milling Cutters
for Machining Spur Involute Gears .12.5
7.2 2. Equation for the Generating Surface of the Disk-Type Milling
Cutter for Machining Helical Involute Gears . 12.7
7.2 3 Elements of the Intrinsic Geometry of the Generating Surface of
Disk-Type Milling Cutters . 130
7.3 Cutting Edges of the Disk-Type Gear Milling Cutter 132.
7.3.1 Rake Surface of the Milling Cutter for Machining Involute Gears . 132.
7.3.2. Clearance Surface of the Milling Cutter for Machining
Involute Gears . 134
7.4 Profiling of the Disk-Type Gear Milling Cutters 136
7.4.1 Use of the Descriptive Geometry–Based Method of Profiling . 136
7.4.2. Analytical Profiling of Disk-Type Gear Milling Cutters . 138
7.5 Cutting Edge Geometry of the Disk-Type Milling Cutter 143
7.6 Disk-Type Milling Cutters for Roughing of Gears . 147
7.7 Accuracy of Gear Tooth Flanks Machined with Disk-Type Milling Cutters . 152.
7.8 Application of Disk-Type Gear Milling Cutters . 154
8. Nontraditional Methods of Gear Machining with Form Cutting Tools . 163
8.1 Plurality of Single Parametric Motions . 163
8.2. Implementation of the Single Parametric Motions for Designing of Form
Gear Cutting Tool . 166
8.2 1 End-Type Gear Milling Cutter 166
8.2 2. Disk-Type Gear Milling Cutter . 167
8.2 3 Face Gear Milling Cutter . 168
8.2 4 Internal Round Broach for Cutting Spur and Helical Gears 169
8.2 5 Internal Round Broach for Machining Straight Bevel Gears . 170
8.3 Diversity of Form Tools for Machining a Given Gear . 172.
8.3.1 Machining of an Involute Worm on a Lathe . 172.
8.3.2. Milling of an Involute Worm 175
8.3.3 Thread Whirling . 177
8.3.4 Grinding of an Involute Worm . 178
8.4 Classification of Form Gear Tools 181
Section III Cutting Tools for Gear Generating: Parallel-Axis
Gear Machining Mesh
9. Rack Cutters for Planing of Gears 187
9.1 Generating Surface of a Rack Cutter 187
9.2. On the Variety of Feasible Tooth Profiles of Rack Cutters 191x Contents
9.3 Cutting Edges of the Rack Cutter . 193
9.3.1 Rake Surface of a Rack Cutter . 194
9.3.2. Clearance Surface of a Rack Cutter 195
9.4 Profiling of Rack Cutters . 196
9.4.1 Profiling of Rack Cutters Using DG-Based Methods 197
9.4.2. Analytical Profiling of Rack Cutters 198
9.5 Cutting Edge Geometry of the Rack Cutter 199
9.5.1 Computation of the Cutting Edge Geometry for Lateral Cutting
Edges 2.00
9.5.2. Possible Improvements in the Geometry of Lateral Cutting
Edges 2.03
9.6 Chip Thickness Cut by Cutting Edges of the Rack Cutter Tooth 2.07
9.7 Accuracy of the Machined Gear . 2.12.
9.7.1 Satisfaction of the Fifth Condition of Proper PSG . 2.12.
9.7.2. Satisfaction of the Sixth Condition of Proper PSG . 2.15
9.8 Application of Rack Cutters 2.18
9.9 Potential Methods of Gear Cutting and Designs of Rack-Type Gear
Cutting Tools .2.2.0
10. Gear Shaper Cutters I: External Gear Machining Mesh 2.2.3
10.1 Kinematics of Gear Shaping Operation .2.2.3
10.2. Generating Surface of a Gear Shaper Cutter .2.2.5
10.3 Cutting Edges of the Shaper Cutter .2.2.9
10.3.1 Rake Surface of a Shaper Cutter .2.2.9
10.3.2. Clearance Surface of a Shaper Cutter Tooth . 2.32.
10.4 Profiling of Gear Shaper Cutters 2.33
10.5 Critical Distance to the Nominal Cross Section of the Gear Shaper Cutter 2.36
10.6 Cutting Edge Geometry of a Gear Shaper Cutter Tooth . 2.39
10.6.1 Angle of Inclination of the Lateral Cutting Edge . 2.40
10.6.2. Rake Angle of the Lateral Cutting Edge . 2.41
10.6.3 Clearance Angle of the Lateral Cutting Edge . 2.43
10.6.4 Improvement in the Geometry of Lateral Cutting Edges . 2.45
10.7 Desired Corrections to the Gear Shaper Cutter Tooth Profile . 2.48
10.8 Thickness of Chip Cut by Gear Shaper Cutter Tooth 2.51
10.9 Accuracy of Gears Cut with the Gear Shaper Cutter 2.55
10.9.1 Satisfaction of the Fifth Condition of Proper PSG .2.55
10.9.2. Satisfaction of the Sixth Condition of Proper PSG .2.58
10.10 Application of Gear Shaper Cutters .2.60
10.10.1 Design of Shaper Cutters . 2.61
10.10.2. Special Features of the Shaper Cutter Tooth Profile 2.63
10.10.3 Shaper Cutters for Machining of Helical and Herringbone Gears .2.64
10.10.4 Special Designs of Gear Shaper Cutters 2.65
10.10.5 Typical Gear Shaping Operations 2.73
10.10.6 Grinding of Shaper Cutters . 2.74
11. Gear Shaper Cutters II: Internal Gear Machining Mesh . 2.81
11.1 Kinematics of Shaping Operation of an Internal Gear . 2.81
11.2. Design of Shaper Cutters .2.83
11.2 1 Generating Surface of Gear Shaper Cutters 2.83Contents xi
11.2 2. Profiling of Gear Shaper Cutters 2.83
11.2 3 Cutting Edge Geometry of Gear Shaper Cutters .2.85
11.3 Thickness of Chip Cut by the Gear Shaper Cutter Tooth .2.86
11.4 Accuracy of Shaped Internal Gears .2.90
11.5 Enveloping Gear Shaper Cutters 2.93
11.6 Application of Gear Shaper Cutters . 2.93
Section IV Cutting Tools for Gear Generating:
Intersecting-Axis Gear Machining Mesh
12. Gear Shapers with a Tilted Axis of Rotation 301
12 1 Kinematics of Gear Shaper Operation with the Shaper Cutters Having a
Tilted Axis of Rotation . 301
12 2. Determination of the Generating Surface of a Gear Shaper Cutter Having
a Tilted Axis of Rotation 304
12 3 Illustration of Capabilities of the External Intersecting-Axis
Gear Machining Mesh . 311
12 3.1 Shaping of Conical Involute Gears . 311
12 3.2. Shaping of Face Gears 311
13. Gear Cutting Tools for Machining Bevel Gears 315
13.1 Principal Elements of the Kinematics of Bevel Gear Generation 315
13.2. Geometry of Interacting Tooth Surfaces . 317
13.2 1 Principal Elements of the Geometry of the Involute Straight Bevel
Gear Tooth Flank 318
13.2 2. Generating Surface of the Gear Cutting Tool . 319
13.2 3 Geometry of Tooth Flanks of the Generated Gear . 32.3
13.3 Peculiarities of Generation of Straight Bevel Gears with Offset Teeth . 32.5
13.3.1 Generating Surface of the Gear Cutting Tool . 32.6
13.3.2. Generating Surface of the Gear Cutting Tool . 32.6
13.4 Generation of Straight Bevel Gear Teeth . 32.8
13.4.1 Generation of the Plane T
a by Straight Motion of the Cutting Edge . 32.8
13.4.2. Machining of Straight Bevel Gears 32.9
13.4.3 Gear Cutting Tools for Machining Straight Bevel Gears 330
13.5 Peculiarities of Straight Bevel Gear Cutting .333
13.6 Milling of Straight Bevel Gears 334
13.6.1 Peculiarities of the Gear Machining Operation .334
13.6.2. Design of Milling Cutters 335
13.6.3 Specific Features of the Shape of Finished Bevel Gear Flanks 336
13.7 Machining of Bevel Gears with Curved Teeth . 337
13.7.1 Peculiarities of the Gear Machining Operation .338
13.7.2. Design of Cutters 340
14. Gear Shaper Cutters Having a Tilted Axis of Rotation: Internal Gear
Machining Mesh .343
14.1 Principal Kinematics of Internal Gear Machining Mesh 343
14.2. Peculiarities of the Gear Cutting Tool Design 344xii Contents
14.2 1 Shaping of Internal Gear .344
14.2 2. Shaping a Spur Gear with Enveloping Shaper Cutter .346
14.2 3 Shaping of External Recessed Tooth Forms with Enveloping
Shaper Cutter .347
Section V Cutting Tools for Gear Generating: Spatial Gear
Machining Mesh
Section V-A Design of Gear Cutting Tools: External Gear
Machining Mesh
15. Generating Surface of the Gear Cutting Tool .353
15.1 Kinematics of External Spatial Gear Machining Mesh .353
15.2. Auxiliary Generating Surface of the Gear Cutting Tool . 357
15.3 Examples of Possible Types of Auxiliary Generating Surfaces of Gear
Cutting Tools .363
15.4 Generation of Generating Surface of a Gear Cutting Tool 363
15.4.1 Design Parameters of the Generating Surface of the Gear
Cutting Tool .365
15.4.2. Equation of the Generating Surface of the Gear Cutting Tool . 371
15.4.3 Setting Angle of the Gear Cutting Tool . 374
15.4.4 Complementary Equations 376
15.5 Use of the DG-Based Methods for Determining the Design Parameters of
the Generating Surfaces of the Gear Cutting Tools . 378
15.5.1 Base Helix Angle ψ b.c of the Generating Surface of the Gear
Cutting Tool . 378
15.5.2. Base Diameter db.c of the Generating Surface of the Gear Cutting
Tool 380
15.6 Possible Types of Generating Surfaces of Gear Cutting Tools . 381
15.6.1 Generating Surface of the Gear Cutting Tool with a Zero Profile
Angle 381
15.6.2. Conical Generating Surface of the Gear Cutting Tool .383
15.6.3 Generating Surface of a Gear Cutting Tool with an Asymmetric
Tooth Profile 389
15.6.4 Generating Surfaces of the Gear Cutting Tools Featuring TorusShaped Pitch Surfaces 390
15.7 Constraints on the Design Parameters of the Generating Surface of a Gear
Cutting Tool . 392.
16. Hobs for Machining Gears . 395
16.1 Transformation of the Generating Surface into a Workable Gear Cutting
Tool 395
16.2. Geometry and Generation of Rake Surface of a Gear Hob 399
16.2 1 Geometry of the Rake Surface 399
16.2 2. Generation of the Rake Surface 403
16.2 2 1 Generation of a Rake Surface in the Form of a Plane .403
16.2 2 2. Generation of a Screw Rake Surface .405Contents xiii
16.2 2 3 Peculiarities of Generation of a Screw Rake Surface
of a Multistart Hob 407
16.2 2 4 Methods for Generation of an Intermittent Rake
Surface of the Special-Purpose Gear Hob 409
16.3 Geometry and Generation of Clearance Surfaces of Gear Hobs . 411
16.3.1 Equation of the Desired Clearance Surface of the Hob Tooth . 411
16.3.2. Generation of the Clearance Surface of the Hob Tooth . 415
16.3.2 1 Cutting of the Relieved Clearance Surfaces of the
Hob Teeth 415
16.3.2 2. Grinding of the Relieved Clearance Surfaces of the
Hob Teeth 42.3
16.4 Accuracy of Hobs for Machining of Involute Gears 433
16.4.1 Preliminary Remarks .433
16.4.2. Accuracy of an Involute Gear Hob as a Function of Its Design
Parameters 435
16.4.2 1 Analytical Description of the Desired Lateral
Cutting Edge .436
16.4.2 2. Analytical Description of the Actual Lateral
Cutting Edge .436
16.4.2 3 Machining Surface of an Involute Hob .437
16.4.2 4 Deviation of the Actual Machining Surface from
the Desired Generating Surface of an Involute Hob .438
16.4.3 Impact of Pitch Diameter on the Accuracy of a Gear Hob .445
16.4.3.1 Peculiarities of the Relative Motion of the Work Gear
and the Hob 446
16.4.3.2. Principal Design Parameters of an Involute Hob 449
16.4.3.3 Elements of Kinematic Geometry of an Involute Hob .454
16.5 Design of Gear Hobs 462.
16.5.1 Design Parameters of a Gear Hob 462.
16.5.2. Tooth Profile of the Gear Hob .465
16.5.3 Precision Involute Hobs with Straight Lateral Cutting Edges .468
16.5.3.1 Principal Design Parameters of the Precision
Involute Hob . 470
16.5.3.2. A Method for Resharpening the Precision Involute Hob 477
16.5.3.3 An Involute Hob for Machining Gear with a
Modified Tooth Profile 481
16.5.4 Examples of Nonstandard Designs of Involute Hobs .487
16.5.4.1 Cylindrical Hobs of Nonstandard Design .487
16.5.4.2. Conical Gear Hobs . 493
16.5.4.3 Toroidal Gear Hobs 497
16.6 The Cutting Edge Geometry of a Gear Hob Tooth 498
16.6.1 The Penetration Curve and the Machining Zone in a Gear
Hobbing Operation .500
16.6.1.1 Parameters of the G/H
pc Penetration Curve . 501
16.6.1.2. Partitioning of the Machining Zone .503
16.6.2. The Cutting Edge Geometry of a Hob Tooth in the Tool-in-Use
Reference System 505
16.6.2 1 The Tool-in-Use Reference System in a Gear Hobbing
Operation 505xiv Contents
16.6.2 2. Geometrical Parameters of the Hob Cutting Edge in the
Tool-in-Use Reference System 508
16.6.2 3 The Possibility of Improving a Hob Design on the
Premise of the Results of Investigating the Cutting Edge
Geometry 514
16.7 Constraints on the Parameters of Modification of the Hob Tooth Profile 515
16.7.1 The Applied Reference Systems . 516
16.7.2. Kinematics of the Elementary Gear Drive 517
16.7.3 Computation of the Maximum Allowed Value of the Modification
of the Tooth Profile of an Involute Hob . 518
16.7.4 Normalized Deviation Δ
m of the Tooth Profile of the Hobbed Gear 52.2.
16.7.5 Peculiarities of Involute Hobs with Reduced Addendum 52.4
16.7.6 Illustrative Examples of the Computation 52.7
16.8 Application of Hobs for Machining Gears 52.8
16.8.1 Peculiarities of a Gear Hobbing Operation . 52.8
16.8.2. Cycles of Gear Hobbing Operations 534
16.8.3 Minimum Hob Travel Distance 536
16.8.3.1 Hobbing Time as a Function of the Hob Total Travel
Distance .536
16.8.3.2. Impact of the Hob’s Idle Distance on the Minimal Neck
Width of the Hobbed Cluster Gear . 537
16.8.3.3 Selection of a Proper Value of the Setting Angle of the
Hob .538
16.8.3.4 Computation of the Shortest Allowed Hob Idle
Distance .540
16.8.3.5 Impact of Tolerance onto the Shortest Possible Hob Idle
Distance .545
16.8.3.6 Computation of the Shortest Allowable Approach
Distance of the Hob . 552.
16.8.3.7 Designing a Hob Featuring a Prescribed Value of the
Setting Angle 555
17. Gear Shaving Cutters . 559
17.1 Transforming the Generating Surface into a Workable Gear Shaving
Cutter 559
17.1.1 Generating Surface of a Shaving Cutter 559
17.1.2. Rake Surface of the Cutting Teeth of a Shaving Cutter .560
17.1.3 Clearance Surface of the Cutting Teeth of a Shaving Cutter 562.
17.1.4 Inclination Angle of the Cutting Edges of a Shaving Cutter 562.
17.2. Design of the Gear Shaving Cutters 566
17.2 1 Design Parameters of a Shaving Cutter . 567
17.2 2. Serrations on the Tooth Flanks of a Shaving Cutter 568
17.2 3 Resharpening of a Shaving Cutter . 571
17.3 Axial Method of the Gear Shaving Process 576
17.3.1 Kinematics of the Axial Method of the Gear Shaving Process 576
17.3.2. Cutting Speed in the Axial Method of Rotary Shaving of
the Gear 578
17.3.2 1 Impact of the Crossed-Axis Angle 578
17.3.2 2. Impact of the Traverse Motion . 579Contents xv
17.3.2 3 Impact of Profile Sliding .580
17.3.2 4 A Resultant Formula for Cutting Speed in Axial Gear
Shaving 587
17.4 Diagonal Method of the Gear Shaving Process . 587
17.4.1 Kinematics of the Diagonal Method of the Gear Shaving
Process 588
17.4.2. Traverse Angle in Diagonal Method of the Rotary Shaving of a
Gear . 589
17.4.3 Cutting Speed in the Diagonal Method of the Rotary Shaving of a
Gear .590
17.4.4 Optimization of the Kinematics in the Diagonal Method of the
Rotary Shaving of a Gear 592.
17.4.4.1 The Concept of the Optimization 592.
17.4.4.2. Local Topology of the Contacting Tooth Flanks . 595
17.4.4.3 Applied Coordinate Systems 596
17.4.4.4 Geometry of Contact of the Tooth Flanks G and T 598
17.4.4.5 Optimal Design Parameters of a Shaving Cutter and
Optimal Parameters of the Kinematics of the Rotary
Shaving Operation .602.
17.5 Tangential Method of the Gear Shaving Process .603
17.5.1 Kinematics of the Tangential Method of the Gear Shaving
Process 603
17.5.2. Cutting Speed in the Tangential Method of the Rotary Shaving of
a Gear 604
17.5.3 Tangential Shaving of Shoulder Gear: Descriptive
Geometry–Based Approach 605
17.5.3.1 Maximum Allowed Outer Diameter of a Shaving
Cutter .606
17.5.3.2. Minimum Required Overlap of the Work Gear and the
Shaving Cutter 608
17.5.3.3 Minimum Required Face Width of a Shaving Cutter . 612.
17.5.4 Tangential Shaving of Shoulder Gear: Analytical Approach . 612.
17.5.4.1 Optimal Design Parameters of a Shaving Cutter 612.
17.5.4.2. Influence of the Overlap of a Shaving Cutter over the
Work Gear onto the Accuracy of the Finished Tooth
Flanks 615
17.6 Plunge Method of the Gear Shaving Process . 619
17.6.1 Kinematics of the Plunge Method of the Gear Shaving
Process 619
17.6.2. Cutting Speed in the Plunge Method of the Rotary Shaving of a
Gear . 62.0
17.6.3 Plunge Gear Shaving Process . 62.0
17.6.4 Plunge Shaving of Topologically Modified Gears . 62.1
17.6.4.1 Geometry of a Topologically Modified Gear Tooth
Flank 62.1
17.6.4.2. Geometry of the Desired Topologically Modified Tooth
Flank of a Shaving Cutter . 62.4
17.6.4.3 Grinding a Topologically Modified Tooth Flank of the
Shaving Cutter 62.5xvi Contents
17.6.5 Satisfaction of Conditions of Proper Part Surface Generation
When Designing a Shaving Cutter for Plunge Shaving of
Gears . 62.8
17.6.5.1 Circular Mapping of Tooth Flanks of a Work Gear and
the Shaving Cutter . 62.9
17.6.5.2. Shaving Cutter of a Special Design for Plunge Shaving
of Precision Gears 631
17.7 Advances in the Design of the Shaving Cutter 633
17.7.1 Elements of the Geometry of the Cutting Edges 633
17.7.2. Utilization of Features of the Generating Surface of a Shaving
Cutter 636
17.8 Peculiarities of the Gear Shaving Process .638
17.8.1 Shaving Cutter Selection .639
17.8.2. Requirements for Preshaved Work Gear .639
17.8.3 Manufacturing Aspects of Gear Shaving Operation 640
17.8.4 Modification of Tooth Form and Shape .641
17.8.5 Shaving of Worm Gear 641
18. Examples of Implementation of the Classification of the Gear
Machining Meshes .643
18.1 A Hob for Tangential Gear Hobbing .643
18.2. A Hob for Plunge Gear Hobbing 644
18.3 Hobbing of a Face Gear 645
18.4 A Worm-Type Gear Cutting Tool with a Continuous Helix-Spiral Cutting
Edge 646
18.5 Cutting Tools for Scudding Gears 649
18.5.1 Essentials of the Gear Scudding Process 649
18.5.2. A Design Concept of a Precision Cutting Tool for the Gear
Scudding Process 649
18.5.3 Applications of the Gear Scudding Process 651
18.6 A Shaper Cutter with a Tilted Axis of Rotation for Shaping Cylindrical
Gears . 651
18.6.1 The Kinematics of Shaping a Helical Gear with the Straight-Tooth
Shaper Cutter . 651
18.6.2. Principal Elements of Design of the Gear Cutting Tool 652.
18.6.3 A Possible Application for the Gear Shaper Cutter with a Tilted
Axis of Rotation 652.
18.7 A Gear Cutting Tool for Machining a Worm in the Continuously
Indexing Method 653
18.8 Rack Shaving Cutters .654
18.8.1 Rack-Type Shaving Cutter .655
18.8.2. Kinematics of the Rack Shaving Process .655
18.9 A Tool for Gear Reinforcement by Surface Plastic Deformation .657
18.10 Conical Hob for the Palloid Method of Gear Cutting .658
18.10.1 Preamble 659
18.10.2. Design of the Conical Hob 659
18.10.3 Kinematics of the Palloid Gear Hobbing Process 660
18.10.4 Peculiarities of Design of a Conical Hob for Machining a Work
Gear with Crowned Teeth 662.Contents xvii
Section V-B Design of Gear Cutting Tools: Quasi-Planar Gear
Machining Mesh
19. Gear Cutting Tools for Machining of Bevel Gears 665
19.1 Design of a Gear Cutting Tool for the Plunge Method of Machining of
Bevel Gears 665
19.1.1 Kinematics .665
19.1.2. Possible Designs of Tools for Machining Bevel Gears 667
19.2. Face Hob for Cutting Bevel Gear 668
19.3 More Possibilities for Designing Gear Cutting Tools Based on
Quasi-Planar Gear Machining Meshes . 669
Section V-C Design of Gear Cutting Tools: Internal Gear Machining Mesh
20. Gear Cutting Tools with an Enveloping Generating Surface . 673
2.0.1 Gear Cutting Tools with a Cylindrical Generating Surface . 673
2.0.1.1 Generating Surface of an Internal Cylindrical Gear Cutting Tool 673
2.0.1.2. Solution to the Inverse Problem of Part Surface Generation 674
2.0.1.3 Examples of Gear Cutting Tools with an Enveloping Cylindrical
Generating Surface . 676
2.0.2. Gear Cutting Tools with a Conical Generating Surface 679
2.0.2 1 Generating Surface of an Enveloping Conical Gear Cutting Tool . 679
2.0.2 2. Examples of Gear Cutting Tools with an Enveloping Conical
Generating Surface .680
2.0.3 Gear Cutting Tools with a Toroidal Generating Surface . 681
2.0.3.1 Generating Surface of an Enveloping Toroidal Gear Cutting Tool . 681
2.0.3.2. Examples of Gear Cutting Tools with an Enveloping Toroidal
Generating Surface .684
21. Gear Cutting Tools for Machining Internal Gears 687
2.1.1 Principal Design Parameters of a Gear Cutting Tool for Machining an
Internal Gear .687
2.1.1.1 Geometry of an Internal Gear .687
2.1.1.2. Kinematics of Machining an Internal Gear 687
2.1.1.3 Determination of the Generating Surface of a Gear Cutting Tool
for Machining an Internal Gear .689
2.1.2. Examples of Gear Cutting Tools for Machining an Internal Gear 689
Conclusion . 693
Appendix A: Engineering Formulae for the Specification of Gear Tooth . 695
Appendix B: Conditions of Proper Part Surface Generation 699
Appendix C: Change of Surface Parameters 703
Appendix D: Cutting Edge Geometry: Definition of the Major Parameters . 705
Notation . 719
References .72.3
Index .733xix
A
Addendum, 310, 695t, 697t
Analytical profiling, of rack cutter, 198–199
Angle of inclination, 2.40–2.41
Angular pitches, 2.68
Archimedean screw surface, 62.–63
Archimedean worm, machining, 173
Asymmetric tooth profile, 389–90
Auxiliary generating surface, 357–363
characteristic line E
g, 362.
coordinate systems, 357–359
coordinate systems transformations, 359
pitch diameter, 361–362.
profile angle, 362.
rack surface, 357–358
relative motion of work gear, 358
tooth flank surface, 359–360
types of, 363
Auxiliary parameter, 475t
Auxiliary rack, 371–372., 608–610
Axial gear shaving, 576–587. See also shaving
cutters
coordinate systems transformations, 585
cutting speed, 578–587
equivalent base diameters, 586–587
formula for cutting speed, 587
impact of crossed-axis angle, 578–579
impact of profile sliding, 580–587
impact of traverse motion, 579–580
kinematics of, 576–578
line of action, 584
workgear to shaving cutter meshing, 580,
581f, 583f
Axial pitch, 475t, 696t
B
Backlash per pair, 697t
Barreled spur gear, 17
Base diameter
equation, 695t, 697t
of generating surface, 380–381
Base diameter of shaper cutter, 310
Base helix angle, 378–379, 696t
Base pitch, 696t
Bevel gear machining, 315–342.
generating surface of gear cutting tool, 32.5–32.8
geometry of interacting tooth surfaces, 318–32.5
generating surface of gear cutting tool,
319–32.3
involute straight bevel gear tooth flank,
318–319
octoidal profile, 319
tooth flanks of generated gear, 32.3–32.5
kinematics of, 315–317
axodes, 316
coordinate systems, 317
pitch angle, 316
rotation vectors, 315–316
straight bevel gear teeth, 32.8–333
gear cutting tools, 330–333
machining of straight bevel gears, 32.9–330
plane Ta by straight motion of cutting
edge, 32.8–32.9
straight bevel gears with offset teeth, 32.5–32.8
Bevel gears, 14–16
face hob for, 668, 669f
helical, 15–16
internal round broach, 170–172.
spiral, 337–342.
straight, 14–15
teeth, broaching, 81–90
tooth flank, 14–15
Beveloids, 347
Biparametric motion, 16
Broaching, 73–74
bevel gear teeth, 81–90
external gears, 73–74
internal gears, 73
push-up pot, 74
Built-up hobs, 463–464
Burnishing button, 73
Butt-end hob, 496, 496f
C
Carbide gear shaper cutter, 2.70–2.73
Cartesian coordinate system, 10f, 2.0–2.1
conversion of orientation, 54
of gear shaper cutters, 2.2.5–2.2.6
screw motion, 48–49
Center distance, 2.6
zero value, 40
Index734 Index
Chamfers, 2.63–2.64, 677–678
Characteristic E, 115
Chip per tooth, 59, 65–66
Chip removal diagrams, 65–66
Chip-flow rake angle, 717–718
Chip-flow reference plane, 716–717
Circular mapping, 62.9–631
Circular pitch, 695t, 697t
Circular tooth thickness, 697t
Clearance angle, 710–711
definition of, 708–709, 710–711
disk-type gear milling cutters, 144–146
of hob tooth cutting edge, 510, 515–516
of lateral cutting edge, 2.43–2.48
normal, 710–711
of rack cutter, 2.02.
of rotary breaches, 80
Clearance surface, 64–65
of disk-type gear milling cutter, 134–136
of gear hobs, 411–433
clearance angle, 413, 416–42.3
cutting edges, 411
cutting of relieved clearance surface,
415–42.3
equation of desired clearance surface,
411–415
grinding of relieved clearance surface,
42.3–433
helix angle, 414
operator of linear transformation, 415
position vector, 413
rake plane, 413
reduced pitch, 413
tooth relieving operation, 415
grinding of, 2.74–2.75
of rack cutter, 195–196
of shaper cutter tooth, 2.32.–2.33
Climb hopping, 530
Cluster gears, 5, 6f
Coarse-pitch gears, 12.0
Combined rake surface, 2.66
Cone drive, 5
Conical gear cutting tools, 679–681
Conical generating surface, 383–389
Conical hobs, 493–497
butt-end hob, 496, 496f
design of, 661–662.
face hob, 496, 496f
palloid gear hobbing with, 658–662.
kinematics of, 660–661
machining with crowned teeth, 662.
work gear to cutting tool penetration curve,
493, 494f
Conical involute gears, shaping of, 311, 312.f
Conical rake surface, 62.–63
Conjugate surfaces, rolling of, 2.3–2.4
Contact ellipse, 594
Continuously indexing methods, 2.5–42.
classification of gear machining meshes,
39–42.
configuration of rotation vectors, 37–39
kinematic relationships for gear machining
mesh, 32.–37
kinematics of gear machining, 42.
vector representation of gear machining
mesh, 2.5–32.
Conventional hopping, 530
Coordinate axis, rotation about, 46–47
Coordinate systems transformations, 43–56
conversion of coordinate system orientation,
54
direct transformation, 45
direct transformation of surfaces
fundamental forms, 55–56
homogenous coordinate transformation
matrices, 44
homogenous coordinate vectors, 43–44
opposite transformation, 54
orientation-preserving transformation, 45
orientation-reversing transformation, 54
overview, 43
resultant, 47–48
rolling motion of coordinate system, 50–51
rolling of two coordinate systems, 52.–54
rotation about coordinate axis, 46–47
screw motions about coordinate axis, 48–49
translations, 44–45
Coordinate transformation matrices,
homogenous, 43–44
Coordinate vectors, homogenous, 43–44
Critical configuration, 38
Crossed-axis angle, 2.7, 12.3
impact of, 578–579
zero value, 40
Crowned spur gear, 17
Cut per tooth, 65–66
Cutting edge geometry, 705–718
chip flow rake angle, 717–718
chip-flow reference plane, 716–717
clearance angle, 708–709
correlation measured within major and
normal section planes, 713–715
cutting edge, 705
of disk-type gear milling cutters, 143–146
clearance angle, 144–146
pressure angle, 145–146Index 735
profile angle, 144–145
rake angle, 143, 146
of external gear machining meshes, 2.39–2.48
gear hobs, 498–515
of gear shaper cutters
external gear machining mesh, 2.39–2.48
internal gear machining mesh, 2.85–2.86
hobs for machining gears, 498–515
of internal gear machining meshes, 2.85–2.86
in major section plane, 707–708
mandatory relationship, 711–712.
minor cutting edge approach angle, 715–716
normal clearance angle, 710–711
normal rake angle, 709–710
in normal section plane, 709
in plane of cut, 706
of rack cutters, 199–2.07
clearance angle, 2.02.
computation, 2.00–2.02.
inclination angle, 2.00
for lateral cutting edges, 2.00–2.02.
modification of clearance surface,
2.04–2.07
modification of rake surface, 2.03–2.04
roundness of cutting edge, 712.
vector of resultant motion, 705
Cutting edge vector, 705
Cutting edges, 705
definition of, 705
end-type gear milling cutters, 101–114
helix angle, 113
rake surface in form of plane, 109–111
rake surface in form of screw surface,
111–114
of gear broach, 61–65
roundness of, 712.
Cutting motion, 2.08
Cutting tools, 1–2.
cylindrical, 673–679
direct problem of design, 2.
elementary relative motions, 2.0–2.1
with enveloping generating surface, 673–686
feasible relative motions, 2.1–2.3
for machining bevel gears, 315–342.
for machining internal gears, 687–691
Shear-Speed cutting, 74–80
Cylinders, rolling motion over, 53
Cylindrical gear cutting tool, 673–679
examples of, 676–679
gear tooth with chamfers, 677–678
gear tooth with lengthwise modification,
678f
generating surface of, 673–674
part surface generation, 674–676
vector diagram, 677f
Cylindrical hobs, 487–93
for finishing hardened gears, 490–493
number of hob threads, 489
for smooth roughing of coarse pitch gears,
487–489
unfolded cross section, 488f, 490f
D
Deep counterbore-type shaper cutters, 2.61,
2.84–2.85
Descriptive geometry-based methods, 136–138
generating surfaces, 378–381
for rack cutter, 197, 198f
tangential shaving, 605–612.
Diagonal shaving, 587–602 See also shaving
cutters
coordinate systems, 596–598
cutting speed, 590–592.
diagonal-underpass, 590
geometry of contact of tooth flanks, 598–602.
kinematics of, 588–589
local topology of contacting tooth flanks,
595–596
multistroke, 589
optimal kinetic parameters, 602.
optimization of kinematics, 592.–595
traverse angle, 589–590
Diagonal-underpass shaving, 590
Differential serrations, 605
Direct problem of gear-cutting tool design, 2.
Direct transformation, 45
of surface fundamental forms, 55–56
Disk-type gear milling cutters, 12.3–161. See also
end-type gear milling cutters
15-gear, 155
accuracy of tooth flanks machined with,
152.–154
application of, 154–161
circular saws, 159–160
cutting edge geometry of, 143–146
clearance angle, 144–146
pressure angle, 145–146
profile angle, 144–145
rake angle, 143, 146
cutting edges of, 132.–136
clearance surface, 132.f, 134–136
rake surface, 132.–134, 132.f
eight-gear, 155
generating surface of, 12.5–12.7
elements of intrinsic geometry, 130–132.736 Index
Disk-type gear milling cutters (continued)
equation for machining spur involute
gears, 12.5–12.7
for machining helical involute gears,
12.7–130
inclined tooth profile, 155, 156f
indexing, 161
kinematics of gear cutting, 12.3–12.4
crossed-axis angle, 12.3
feed motion, 12.3
pitch helix angle, 12.3
rotation vector, 12.3–12.4
screw feed motion, 12.4
multiple-tooth, 161
overview, 12.3
profiling of, 136–143
analytical, 138–143
descriptive geometry-based methods,
136–138
for roughing of gears, 147–152.
assembled cutter, 151f
clearance angle, 147–148
design parameters, 147f
even-numbered teeth, 150
odd-numbered teeth, 150
progressive cutting diagram, 149
rake angle, 147
top-loaded cutting diagram, 150
wavy cutting edges, 151
single parametric motions in design of, 167–168
span measurement, 158f
Disk-type gear shaper cutter, 2.61
Double-enveloping worm gear drive, 5
Double-helical gears, 3, 6f
Dupin indicatrix, 596
E
End-type gear milling cutters, 91–12.2 See also
disk-type gear milling cutters
application of, 12.0–12.2.
cutting edges, 101–114
clearance surface, 105–108
geometry of, 108–114
helix angle, 113
pressure angle, 114
rake surface in form of plane, 109–111
rake surface in form of screw surface,
111–114
generating surface of, 92.–101
elements of intrinsic geometry, 100–101
equation for machining helical involute
gears, 96–100
equation for machining spur involute
gears, 92.–96
secondary, 92.
for helical gears, 12.0–12.1
indexing mechanism, 12.2.
kinematics of gear machining, 91, 92.f
machining gear tooth flanks with, 115–119
characteristic E, 115
curvature of helical gear tooth flank, 118
cusps on tooth flanks of helical gear,
117–119
cusps on tooth flanks of spur gear, 115–117
maximal deviation, 117
waviness of tooth flank, 116
overview, 91
single parametric motions in design of,
166–167
for spur gears, 12.0
Engineering approach, 8
Enveloping generating surface, 673–686
conical, 679–681
cylindrical, 673–679
toroidal, 681–686
Enveloping shaper cutters, 2.93, 2.95f
shaping external recessed tooth forms with,
347–349
shaping spur gears with, 346–347
Epicyclical motion, 619
Equivalent base diameters, 586–587
External gear machining meshes, 41f, 2.2.3–2.79.
See also internal gear machining
meshes
accuracy of gears cut with, 2.55–2.60
fifth condition of proper post surface
generation, 2.55–2.58
gear-to-shaper cutter meshing diagram,
2.57
application of, 12.0–12.2., 2.60–2.79
critical distance to nominal cross section,
2.36–2.39
circular tooth thickness, 2.39
length of top cutting edge, 2.37–2.38
maximal length of distance, 2.37
cutting edge geometry of, 2.39–2.48
angle of inclination of lateral cutting
edge, 2.40–2.41
clearance angle of lateral cutting edge,
2.43–2.48
improvement in geometry, 2.45–2.48, 2.65–2.69
rake angle of lateral cutting edge, 2.41–2.43
cutting edges of, 2.2.9–2.33
clearance surface of tooth, 2.32.–2.33
rake surface, 2.2.9–2.32.Index 737
definition of, 39
design of shaper cutters, 2.61–2.63
deep counterbore-type, 2.61
disk-type, 2.61
for machining helical and herringbone
gears, 2.64–2.65
tooth profiles, 2.63–2.64
desired corrections to tooth profile, 2.48–2.50
generating surface of, 2.2.5–2.2.9
Cartesian coordinate system, 2.2.5–2.2.6
for machining involute shapes, 2.2.8
pitch helix angle, 2.2.8
position vector, 2.2.5–2.2.6
profile angle, 2.2.7
Shishkov’s equation of contact, 2.2.7–2.2.8
geometry of clearance surface, 2.46–2.48
grinding of, 2.74–2.79
clearance surfaces, 2.74–2.75
rake surfaces, 2.75–2.79
helical gears for, 12.0–12.1
indexing mechanism, 12.2.
kinematics of gear shaping, 2.2.3–2.2.5
principal elements, 2.2.4f
reciprocation, 2.2.3
screw motion, 2.2.3–2.2.4
vector diagram, 2.2.4f
lateral cutting edges
angle of inclination, 2.40–2.41
clearance angle, 2.43–2.48
rake angle, 2.41–2.43
overview, 2.2.3–2.79
profiling of, 2.33–2.36
angular displacement of tooth profile, 2.36
base diameter, 2.35
circular tooth thickness, 2.34–2.35
meshing of auxiliary rack surface, 2.33–2.34
profile shift correction coefficient, 2.35
tooth addendum, 2.35
tooth dedendum, 2.35
tooth thickness, 2.35–2.36
rake surface of, 2.2.9–2.32.
geometry, 2.30–2.31
helical, geometry of, 2.31–2.32.
improvement in geometry, 2.45–2.46
special designs of, 2.65–2.73
angular pitches, 2.68
combined rake surface, 2.66
improved cutting edge geometry, 2.65–2.69
for machining hardened gears, 2.70–2.72.
precision gear, 2.69–2.70
two gears, 2.67–2.69
spur gears for, 12.0
thickness of chip cut by, 2.51–2.55
feed rate motion, 2.51
operator of rolling, 2.53–2.54
position vector, 2.53–2.54
roll angle, 2.52.
speed of instant rotation, 2.51
tooth ratio, 2.52.
tooth profiles
chamfer, 2.63–2.64
full topping, 2.64
modified pressure angle, 2.64
protuberance, 2.64
root fillet, 2.63
semitopping, 2.63–2.64
tip relief, 2.63
typical gear shaping operations, 2.73–2.74
combination operations, 2.73–2.74
cutting two gears with different
diametral pitch, 2.74
cutting two gears with similar diametral
pitch, 2.74
cutting two gears with up and down
shaping, 2.74
length of stroke, 2.73, 2.74f
spiral infeed, 2.74
External gears, broaching, 73–74
External recessed tooth forms, shaping of, 347–349
External spatial gear machining, kinematics of,
353–357
F
Face angle radius, 62.
Face gear milling cutters, 168–169
Face gears, 6, 311–314
Face hob, 496, 496f, 668, 669f
Feed motion, 2.8, 12.3, 2.08–2.09
Fiat spiral bevel-gear generation, 668
Finish rack cutter, 2.18–2.19
Finishing teeth, 59
of gear broach, 61–64
Archimedean screw surface, 62.–63
clearance surface, 64–65
conical, 62.–63
face angle radius, 62.
gullet, 62.
pitch, 62.
tooth depth, 62.
Form gear cutting tools, 163–181
analytical profiling of, 77–79
classification of, 181
disk-type gear milling cutters, 12.3–161
end-type gear milling cutters, 91–12.2.
gear broaching tools, 59–90738 Index
Form gear cutting tools (continued)
grinding involute worm, 178–181
machining involute worm on lathe, 172.–175
milling involute worm, 175–177
single parametric motions in design, 166–172.
disk-type gear milling cutter, 167–168
end-type gear milling cutter, 166–172.
face gear milling cutter, 168–169
internal round broach for cutting spur
and helical gears, 169–170
internal round broach for machining
bevel gears, 170–172.
single parametric motions, plurality of, 163–166
coordinate system, 163–164
position vector, 163–164
rotations, 165
translations, 164–165
thread whirling, 177–178
Form milling cutter, 12.0
Full topping, 2.64
Fundamental forms, direct transformation of,
55–56
G
Gear broaching, 72.–73
definition of, 72.
kinematics of, 59, 60f
pull broaches, 73
push broaches, 73
side-shaving section of, 73
Gear broaching tools, 59–90
application of, 72.–74
broaching external gears, 73–74
broaching internal gears, 73
bevel gear teeth broaching, 81–90
burnishing button, 73
chip per tooth, 59, 65–66
chip removal diagrams, 65–66
cut per tooth, 65–66
cutting edges, 61–65
finishing teeth, 59
generating surface of, 60–61
for machining involute gears, 70–72.
cross section of auxiliary rack, 72.
shapes and configurations of cutting
edges, 70–71
straight lateral cutting edges, 71–72.
pot broaching, 74
rake angle, 68
rake surface of finishing teeth, 61–64
Archimedean screw surface, 62.–63
clearance surface, 64–65
conical, 62.–63
face angle radius, 62.
gullet, 62.
pitch, 62.
tooth depth, 62.
resharpening of, 69f
Revacycle process, 81–90
cutting tools, 83–84
kinematics of, 82.f
principle of, 82.–83
profiling of cutter for bevel gear
machining, 85–90
tooth space generation, 83f
rotary broaches, 80–81
sharpening of, 66–70
generating of rake surface, 68
generating surface of grinding wheel,
66–67
helical gullet, 68
maximal feasible outer diameter of
grinding wheel, 67
schematic diagram, 69f
support at center, 67
Shear-Speed cutting, 74–80
application, 79–80
principle of, 74
profiling of form tools, 76–79
slater tools, 80–81
Gear cutting tools, 1–2.
cylindrical, 673–679
examples of, 676–679
gear tooth with chamfers, 677–678
gear tooth with lengthwise modification,
678f
generating surface of, 673–674
part surface generation, 674–676
vector diagram, 677f
direct problem of design, 2.
elementary relative motions, 2.0–2.1
with enveloping generating surface, 673–686
conical generating surface, 679–681
cylindrical generating surface, 673–679
toroidal generating surface, 681–686
feasible relative motions, 2.1–2.3
generating body of, 395
inverse problem of design, 2.
for machining internal gears, 687–691
examples of, 689–691
generating surface of, 689
geometry of internal gear, 687
kinematics of machining internal gear,
689–690
Shear-Speed cutting, 74–80Index 739
Gear cutting tools, for machining bevel gears,
315–342.
curved teeth, 337–342.
clearance angle, 342.
design of cutters, 340–342.
diagrammatic arrangement of, 339f
gear machining operation, 338–340
profile angle, 341
rake angle, 342.
spiral angles, 337
face hob for, 668, 669f
generating surface of, 32.5–32.8
geometry of interacting tooth surfaces,
318–32.5
generating surface of gear cutting tool,
319–32.3
involute straight bevel gear tooth flank,
318–319
octoidal profile, 319
tooth flanks of generated gear, 32.3–32.5
kinematics of bevel gear generation, 315–317
axodes, 316
coordinate systems, 317
pitch angle, 316
rotation vectors, 315–316
overview, 315–42.
plunge method, 665–668
face hob, 668, 669f
kinematics, 665–667
possible designs, 667–668
quasi-planar gear machining meshes, 669–670
straight bevel gear cutting, 333–334
straight bevel gear milling, 334–337
design of milling cutters, 335–336
disk-type milling cutters, 336
lateral cutting edges, 2.5–2.6
shape of finished flanks, 336–337
straight bevel gear teeth, 32.8–333
gear cutting tools, 330–333
machining of straight bevel gears, 32.9–330
plane Ta by straight motion of cutting
edge, 32.8–32.9
straight bevel gears with offset teeth, 32.5–32.8
Gear hobs, 395–559
accuracy for machining involute gears,
433–462.
actual machining surface vs. desired
generating surface, 438–445
analytical description of actual lateral
cutting edge, 436–437
analytical description of desired lateral
cutting edge, 436
deviations of rack surface, 434f
as function of design parameters, 435–445
impact of lead angle of screw rake surface
on tooth profile deviation, 440–443
impact of pitch diameter, 445–462.
impact of rake angle on tooth profile
deviation, 438–440
kinematic geometry of involute hob,
454–462.
machining surface, 437–438
maximum deviation of hob tooth profile,
438
pitch helix angle, 447
principal design parameters of, 449–454
relative motions of work gear and hob,
446–449
clearance surface of, 411–433
clearance angle, 413, 416–42.3
cutting edges, 411
cutting of relieved clearance surface,
415–42.3
equation of desired clearance surface,
411–415
grinding of relieved clearance surface,
42.3–433
helix angle, 414
operator of linear transformation, 415
position vector, 413
rake plane, 413
reduced pitch, 413
tooth relieving operation, 415
conical hobs, 493–497
butt-end hob, 496, 496f
face hob, 496, 496f
work gear to cutting tool penetration
curve, 493, 495f
cutting edge geometry of gear hob tooth,
498–515
clearance angle, 510, 513–514
cutting edge roundness, 511
improvement of hob design, 514–515
inclination angle, 510–512.
machining zone, 500, 503–505
penetration curve, 500–503
rake angle, 509–510, 512.–513
resultant motion, 506–508
tool-in-use reference system, 505–515
unit normal vector to surface of cut,
505–506
unit tangent vector, 508
cylindrical hobs of nonstandard design,
487–493
for finishing hardened gears, 490–493
number of hob threads, 489740 Index
Gear hobs (continued)
for smooth roughing of coarse pitch
gears, 487–489
unfolded cross section, 488f, 490f
definition of, 395
design of, 462.–498
design parameters, 462.–465
tooth profiles, 465–468
for face gear, 645–646
hobbing operations, 52.8–534
climb hopping, 530
conventional hopping, 530
cycles, 534–536
hob total travel distance, 536–537
hobbing time, 536–537
idle distance and neck width of cluster
gear, 538–540
plunge method, 644–645
prescribed value of setting angle, 555–558
setting angle, 538–540
shortest allowable approach distance,
552.–554
shortest allowed idle distance, 540–545
tangential method, 643, 644f
tolerance and shortest possible idle
distance, 545–552.
kinematic geometry of involute hob
change to normal profile angle, 454–457
correction to configuration of rake
surface, 458–560
value of center-distance in gear hobbing
operation, 457–458
multistart, 407–408
precision involute hubs with lateral cutting
edges, 468–487
analytical approach to configuration of
rake plane, 471–475
computation, 475–477
design parameters, 470–477
DG-based approach to configuration of
rake plane, 470–471
modified tooth profile, 481–487
resharpening, 477–481
rake surface of, 399–410
auxiliary axis of projections, 407
auxiliary reference systems, 401
Cartesian coordinate system, 399–400
cone angle, 407
in form of plane, 403–404
generation of, 403–410
geometry, 399–403
intermittent, 409–410
of multistart hob, 407–408
plane of projections, 407
position vectors, 401
rake angle, 403
screw rake surface, 405–407
straight-generating lines, 405–406
straight-line generator, 400
zero rake angle, 402.
tool-in-use reference system, 505–515
clearance angle, 510, 513–514
inclination angle, 510–512.
rake angle, 509–510, 512.–513
tooth profiles, 515–52.8
actual value of deviation, 52.7–52.8
allowed interval for profile angle, 52.7–52.8
applied reference systems, 516–517
elementary gear drive, 517–518
form diameter of gear, 517–518
limit diameter of gear, 517
maximum allowed value of modification,
516–52.2.
modification of, 517
normalized deviation, 52.2.–52.4
reduced addendum, 52.4–52.7
toroidal hobs, 497–498
transformation of generating surface into
gear tool, 395–399
feed motion, 395–396
grinding worm, 395, 396f
screw motion, 396–398
Gear honing, 685–686
Gear machining, 19–2.4
continuously indexing methods, 2.5–42.
classification of gear machining meshes,
39–42.
configuration of rotation vectors, 37–39
kinematic relationships for gear
machining mesh, 32.–37
kinematics of gear machining, 42.
vector representation of gear machining
mesh, 2.5–32.
relative motions, 19–2.3
elementary, 19–2.3
feasible, 2.1–2.3
rolling of conjugate surfaces, 2.3–2.4
Gear machining mesh, 39–42.
applications of, 643–662.
conical hob for palloid gear cutting,
658–662.
cutting tool for machining worm in
continuously indexing method, 653–654
cutting tools for scudding gears, 649–651
gear reinforcement by surface plastic
deformation, 657–658Index 741
hob for face gear, 645–646
hob for plunge gear hobbing, 644–645
hob for tangential gear hobbing, 643, 644f
rack shaving cutters, 654–657
shaper cutter with tilted axis of rotation,
651–653
worm-type cutting tool with continuous
helix-spiral cutting edge, 646–649
center distance, 40
crossed-axis angle, 40
definition of, 2.5
external, 39–40, 41f
internal, 40, 41f
kinematics of, 42.
kinematics relationships, 32.–37
magnitude of rotation vectors, 40
planar, 39–40, 41f
vector diagram, 2.9, 30f
vector representation of, 2.5–32.
Gear milling cutters, disk-type, 12.3–161
15-gear, 155
accuracy of tooth flanks machined with,
152.–154
application of, 154–161
circular saws, 159–160
cutting edge geometry of, 143–146
clearance angle, 144–146
pressure angle, 145–146
profile angle, 144–145
rake angle, 143, 146
cutting edges of, 132.–136
clearance surface, 132.f, 134–136
rake surface, 132.–134, 132.f
eight-gear, 155
generating surface of, 12.5–12.7
elements of intrinsic geometry, 130–132.
equation for machining spur involute
gears, 12.5–12.7
for machining helical involute gears, 12.7–130
inclined tooth profile, 155, 156f
indexing, 161
kinematics of gear cutting, 12.3–12.4
crossed-axis angle, 12.3
feed motion, 12.3
pitch helix angle, 12.3
rotation vector, 12.3–12.4
screw feed motion, 12.4
multiple-tooth, 161
overview, 12.3
profiling of, 136–143
analytical, 138–143
descriptive geometry-based methods,
136–138
for roughing of gears, 147–152.
assembled cutter, 151f
clearance angle, 147–148
design parameters, 147f
even-numbered teeth, 150
odd-numbered teeth, 150
progressive cutting diagram, 149
rake angle, 147
top-loaded cutting diagram, 150
wavy cutting edges, 151
single parametric motions in design of,
167–168
span measurement, 158f
Gear milling cutters, end-type, 91–12.2.
application of, 12.0–12.2.
cutting edges, 101–114
clearance surface, 105–108
geometry of, 108–114
helix angle, 113
pressure angle, 114
rake surface in form of plane, 109–111
rake surface in form of screw surface,
111–114
generating surface of, 92.–101
elements of intrinsic geometry, 100–101
equation for machining helical involute
gears, 96–100
equation for machining spur involute
gears, 92.–96
secondary, 92.
for helical gears, 12.0–12.1
indexing mechanism, 12.2.
kinematics of gear machining, 91, 92.f
machining gear tooth flanks with, 115–119
characteristic E, 115
curvature of helical gear tooth flank, 118
cusps on tooth flanks of helical gear,
117–119
cusps on tooth flanks of spur gear, 115–117
maximal deviation, 117
waviness of tooth flank, 116
overview, 91
single parametric motions in design of,
166–167
for spur gears, 12.0
Gear modification matrix, 62.2.
Gear scudding, 649–651
applications of, 651
cutting tools for, 649–651
design concept for tools, 649–651
Gear shaper cutters, external gear machining
mesh, 2.2.3–2.79
accuracy of gears cut with, 2.55–2.60742 Index
Gear shaper cutters (continued)
fifth condition of proper post surface
generation, 2.55–2.58
gear-to-shaper cutter meshing diagram, 2.57
sixth condition of proper post surface
generation, 2.58–2.60
application of, 2.60–2.79
critical distance to nominal cross section,
2.36–2.39
circular tooth thickness, 2.39
length of top cutting edge, 2.37–2.38
maximal length of distance, 2.37
cutting edge geometry of, 2.39–2.48
angle of inclination of lateral cutting
edge, 2.40–2.41
clearance angle of lateral cutting edge,
2.43–2.48
improvement in geometry, 2.45–2.48,
2.65–2.69
rake angle of lateral cutting edge, 2.41–2.43
cutting edges of, 2.2.9–2.33
clearance surface of tooth, 2.32.–2.33
rake surface, 2.2.9–2.32.
design of shaper cutters, 2.61–2.63
deep counterbore-type, 2.61
disk-type, 2.61
for machining helical and herringbone
gears, 2.64–2.65
tooth profiles, 2.63–2.64
desired corrections to tooth profile, 2.48–2.50
generating surface of, 2.2.5–2.2.9
Cartesian coordinate system, 2.2.5–2.2.6
for machining involute shapes, 2.2.8
pitch helix angle, 2.2.8
position vector, 2.2.5–2.2.6
profile angle, 2.2.7
Shishkov’s equation of contact, 2.2.7–2.2.8
grinding of, 2.74–2.79
clearance surfaces, 2.74–2.75
rake surfaces, 2.75–2.79
improvement in geometry of clearance
surface, 2.46–2.48
kinematics of gear shaping, 2.2.3–2.2.5
principal elements, 2.2.4f
reciprocation, 2.2.3
screw motion, 2.2.3–2.2.4
vector diagram, 2.2.4f
lateral cutting edges
angle of inclination, 2.40–2.41
clearance angle, 2.43–2.48
rake angle, 2.41–2.43
overview, 2.2.3–2.79
profiling of, 2.33–2.36
angular displacement of tooth profile, 2.36
base diameter, 2.35
circular tooth thickness, 2.34–2.35
meshing of auxiliary rack surface, 2.33–2.34
profile shift correction coefficient, 2.35
tooth addendum, 2.35
tooth dedendum, 2.35
tooth thickness, 2.35–2.36
rake surface of, 2.2.9–2.32.
geometry, 2.30–2.31
helical, geometry of, 2.31–2.32.
improvement in geometry, 2.45–2.46
special designs of, 2.65–2.73
angular pitches, 2.68
combined rake surface, 2.66
improved cutting edge geometry, 2.65–2.69
for machining hardened gears, 2.70–2.72.
precision gear, 2.69–2.70
two gears, 2.67–2.69
thickness of chip cut by, 2.51–2.55
feed rate motion, 2.51
operator of rolling, 2.53–2.54
position vector, 2.53–2.54
roll angle, 2.52.
speed of instant rotation, 2.51
tooth ratio, 2.52.
tooth profiles, 2.63–2.64
chamfer, 2.63–2.64
full topping, 2.64
modified pressure angle, 2.64
protuberance, 2.64
root fillet, 2.63
semitopping, 2.63–2.64
tip relief, 2.63
typical gear shaping operations, 2.73–2.74
combination operations, 2.74
cutting two gears with different
diametral pitch, 2.74
cutting two gears with similar diametral
pitch, 2.74
cutting two gears with up and down
shaping, 2.74
length of stroke, 2.73, 2.74f
spiral infeed, 2.74
Gear shaper cutters, internal gear machining
mesh, 2.81–2.95
accuracy of shaped internal gears, 2.90–2.92.
interference of internal work gear and
shaper cutter teeth, 2.91–2.92.
tooth number, 2.92.
transverse generating pressure angle, 2.91
application of, 2.93–2.95
cutting edge geometry of, 2.85–2.86Index 743
design of shaper cutters, 2.83
enveloping shaper cutters, 2.93
generating surface of, 2.83
kinematics of shaping operation, 2.81–2.83
rotation vector, 2.81–2.82.
screw motion, 2.83
materials used in, 2.93
profiling of shape cutters
deep counterbore-type, 2.84–2.85
shank-type, 2.84–2.85
profiling of shaper cutters, 2.83–2.85
clearance surface, 2.83–2.84
generating surface, 2.83–2.84
rake surface, 2.83–2.84
thickness of chip cut by, 2.86–2.90
feed motion, 2.86
motion of cut, 2.86
operator of rolling, 2.88–2.89
Gear shaper cutters, with tilted axis of rotation,
301–314
capabilities of external intersecting-axis gear
machining mesh, 311–314
shaping conical involute gears, 311, 312.f
shaping of face gears, 311–314
generating surface of, 304–311
addendum, 310
base diameter of shaper cutter, 310
coordinate systems transformations, 306–307
coordinates of points within tooth profile,
307–308
dedendum, 310
left-hand-oriented coordinate system,
304–305
internal gear machining mesh, 343–349
axodes, 345
kinematics of internal gear machining
mesh, 343
motion of cut, 344
operating pitch surfaces, 345
rake surface, 345–346
rotation vectors, 345
shaping of external recessed tooth forms,
347–349
shaping of internal gear, 344–346
shaping of spur gear, 346–347
kinematics of shaping operation, 301–304
rake surface, 303–304
reference systems, 302.
rotation vector, 301–302.
Gear shaving cutters, 559–642.
advances in design of, 633–638
constant inclination angle within cutting
edges, 635–636
for finishing modified work gear tooth
flank, 637–638
near optimal angle of inclination of
cutting edges, 633–635
precision cutter with straight cutting
edges, 636–637
axial method of shaving process, 576–587
coordinate systems transformations, 585
cutting speed, 578–587
equivalent base diameters, 586–587
formula for cutting speed, 587
impact of crossed-axis angle, 578–579
impact of profile sliding, 580–587
impact of traverse motion, 579–580
kinematics of, 576–578
line of action, 584
workgear to shaving cutter meshing, 580,
581f, 583f
clearance surface of cutting teeth, 562.
design of shaper cutters, 566–576
design parameters, 567, 575t
resharpening, 571–576
serrations on tooth flanks, 568–571
diagonal method of shaving process,
587–602.
coordinate systems, 596–598
cutting speed, 590–592.
diagonal-underpass, 590
geometry of contact of tooth flanks,
598–602.
kinematics of, 588–589
local topology of contacting tooth flanks,
595–596
multistroke, 589
optimal kinetic parameters, 602.
optimization of kinematics, 592.–595
traverse angle, 589–590
generating surface of, 559–560
inclination angle of cutting edges, 564–568
manufacturing aspects of shaving operation,
640–641
modification of tooth form and shape, 641
plunge method of shaving processing,
619–633
circular mapping of tooth flanks, 62.9–631
cutting speed, 619
design of shaving cutters, 631–633
epicyclical motion in, 619
kinematics of, 619–62.0
part surface generation, 62.8–633
topologically modified gears, 62.1–62.8
rake surface of cutting teeth, 560–561
requirements for preshaved work gear, 639–640744 Index
Gear shaving cutters (continued)
selection of, 639
shaving of worm gear, 641–642.
tangential method of shaving process,
603–618
analytical approach, 612.–618
cutting speed, 604–605
descriptive geometry-based methods,
605–612.
kinematics of, 603–604
serrations, 605
tooth flanks, 595–596
curvature of tooth surface, 596
Dupin indicatrix of tooth surface, 596
equation of tooth flank of, 595
first fundamental form of tooth surface,
595
indicatrix of conformity, 599
local orientation, 598
maximum rate of conformity, 600
optimal design parameters, 600
principal directions, 598
second fundamental form of tooth
surface, 595
Gear tooth
engineering formula for, 695–697
for surfaces that allow sliding, 16–18
Gear tooth flank surface, natural
parameterization of, 13
Gears, 3–18
definition of, 3
tooth flanks, 6–16
types of, 3–6
cluster, 5, 6f
double-helical, 3, 6o
face, 6
helical, 3
helical rack, 3
herringbone, 3
spur, 3, 4f
straight bevel, 5–6
Generating body, 395
Generating surfaces, 353–394
auxiliary, 357–363
characteristic line E
g, 362.
coordinate systems, 357–359
coordinate systems transformations, 359
pitch diameter, 361–362.
profile angle, 362.
rack surface, 357–358
relative motion of work gear, 358
tooth flank surface, 359–360
types of, 363
base diameter, 380–381
base helix angle, 378–379, 696t
of conical gear cutting tools, 679–680
of cylindrical gear cutting tools, 673–674
definition of, 363–378
descriptive geometry-based methods, 378–381
design parameters, 363–371
base diameter of generating surface, 376–378
characteristic line, 367
complementary equations, 376–378
coordinate systems, 365
envelope to successive positions of plane
with screw motion, 365–368
helix angle, 377
principal elements of geometry, 368–370
setting angle, 374–376
straight rack, 365
disk-type gear milling cutters, 12.4–132.
for machining helical involute gears,
12.7–130
for machining spur involute gears, 12.5–12.7
equation, 371–374
auxiliary rack, 371–372.
screw motion, 372.–373
unit normal vector, 374
of gear broach, 60–61
of gear shaper cutters, 2.83
for machining involute shapes, 2.2.8
pitch helix angle, 2.2.8
profile angle, 2.2.7
Shishkov’s equation of contact, 2.2.7–2.2.8
of grinding wheel, 66–67
kinematics of external spatial gear
machining mesh, 353–357
axode of gear cutting tool, 357
center distance, 353
crossed-axis angle, 353
hyperboloid, 356
opposite-directed vector, 356
pure rolling of axodes, 354–355
rotation of cutting tool, 353
rotation of work gear, 353
rotation vectors, 353–354
vector of linear velocity of sliding of
axodes, 355
straight bevel gears with offset teeth, 32.5–32.8
of toroidal gear cutting tools, 681–686
types of, 364f, 381–392.
asymmetric tooth profile, 389–390
conical, 383–389
torus-shaped pitch surfaces, 390–392.
zero profile angle, 381–383
Gleason method, 668Index 745
Grinding, relief, 42.3–433
of assembled gear hobs, 42.5–433
position for machining, 42.5, 431f
of solid gear hobs, 42.3–42.5
technological worm, 42.8
working position, 42.5, 431f
Grinding wheel, 66–67
generating surface of, 66–67
maximal feasible outer diameter, 67
Gullet, 62.
H
Hardened gears, machining, 2.70–2.72.
Helical bevel gear, 15–16
Helical gears, 3
disk-type gear milling cutters, 12.7–130
end-type gear milling cutter, 96–100
gear milling cutter for, 12.0–12.1
internal round broach for, 169–170
involute, 10–14
shaper cutters for, 2.64–2.65
tooth flanks, 10–14, 15–16
cusps, 117–119
deviation from desired shape, 118–119
normal curvature of, 118
Helical gullet, 68
Helical rack gears, 3
Helical shaper cutters, rake surface of, 2.31–2.32.
Helix angle, 113
equation, 696t
of involute hobs, 476
Herringbone gears, 3, 2.64–2.65
High speed steels (HSS), 2.61, 2.93
Hob base diameter, 475t
Hobs for machining gears, 395–559
accuracy for machining involute gears,
433–462.
actual machining surface vs. desired
generating surface, 438–445
analytical description of actual lateral
cutting edge, 436–437
analytical description of desired lateral
cutting edge, 436
deviations of rack surface, 434f
as function of design parameters, 435–445
impact of lead angle of screw rake surface
on tooth profile deviation, 440–443
impact of pitch diameter, 445–462.
impact of rake angle on tooth profile
deviation, 438–440
kinematic geometry of involute hob,
454–462.
machining surface, 437–438
maximum deviation of hob tooth profile,
438
pitch helix angle, 447
principal design parameters of, 449–454
relative motions of work gear and hob,
446–449
clearance surface of, 411–433
clearance angle, 413, 416–42.3
cutting edges, 411
cutting of relieved clearance surface,
415–42.3
equation of desired clearance surface,
411–415
grinding of relieved clearance surface,
42.3–433
helix angle, 414
operator of linear transformation, 415
position vector, 413
rake plane, 413
reduced pitch, 413
tooth relieving operation, 415
conical hobs, 493–7
butt-end hob, 496, 496f
face hob, 496, 496f
work gear to cutting tool penetration
curve, 493, 494f
cutting edge geometry of gear hob tooth,
498–515
clearance angle, 510, 513–514
cutting edge roundness, 511
improvement of hob design, 514–515
inclination angle, 510–512.
machining zone, 500, 503–505
penetration curve, 500–503
rake angle, 509–510, 512.–513
resultant motion, 506–508
tool-in-use reference system, 505–515
unit normal vector to surface of cut, 505–506
unit tangent vector, 508
cylindrical hobs of nonstandard design,
487–493
for finishing hardened gears, 490–493
number of hob threads, 489
for smooth roughing of coarse pitch
gears, 487–489
unfolded cross section, 488f, 490f
definition of, 395
design of, 462.–498
design parameters, 462.–465
tooth profiles, 465–468
for face gear, 645–646
hobbing operations, 52.8–534746 Index
Hobs for machining gears (continued)
climb hopping, 530
conventional hopping, 530
cycles, 534–536
hob total travel distance, 536–537
hobbing time, 536–537
idle distance and neck width of cluster
gear, 537–538
prescribed value of setting angle, 555–558
setting angle, 538–540
shortest allowable approach distance,
552.–554
shortest allowed idle distance, 540–545
tolerance and shortest possible idle
distance, 545–552.
kinematic geometry of involute hob
change to normal profile angle, 454–457
correction to configuration of rake
surface, 458–560
value of center-distance in gear hobbing
operation, 457–458
multistart, 407–408
for plunge gear hobbing, 644–645
precision involute hubs with lateral cutting
edges, 468–487
analytical approach to configuration of
rake plane, 471–475
computation, 475–477
design parameters, 470–477
DG-based approach to configuration of
rake plane, 470–471
modified tooth profile, 481–487
resharpening, 477–481
rake surface of, 399–410
auxiliary axis of projections, 407
auxiliary reference systems, 401
Cartesian coordinate system, 399–400
cone angle, 407
in form of plane, 403–404
generation of, 403–410
geometry, 399–403
intermittent, 409–410
of multistart hob, 407–408
plane of projections, 407
position vectors, 401
rake angle, 403
screw rake surface, 405–407
straight-generating lines, 405–406
straight-line generator, 400
zero rake angle, 402.
for tangential gear hobbing, 643, 644f
tool-in-use reference system, 505–515
clearance angle, 510, 513–514
inclination angle, 510–512.
rake angle, 509–510, 512.–513
tooth profiles, 515–52.8
actual value of deviation, 52.7–52.8
allowed interval for profile angle, 52.7–52.8
applied reference systems, 516–517
elementary gear drive, 517–518
form diameter of gear, 517–518
limit diameter of gear, 517
maximum allowed value of modification,
516–52.2.
modification of, 517
normalized deviation, 52.2.–52.4
reduced addendum, 52.4–52.7
toroidal hobs, 497–498
transformation of generating surface into
gear tool, 395–399
feed motion, 395–396
grinding worm, 395, 396f
screw motion, 396–398
Homogenous coordinate transformation
matrices, 43–44
Homogenous coordinate vectors, 43–44
I
Inclination angle, 2.00, 511–513
Indexing, continuous, 2.5–42.
classification of gear machining meshes,
39–42.
configuration of rotation vectors, 37–39
kinematic relationships for gear machining
mesh, 32.–37
kinematics of gear machining, 42.
vector representation of gear machining
mesh, 2.5–32.
Indicatrix of conformity, 599
Integral-shank hobs, 463
Intermittent rake surface, 409–410
Internal gear machining meshes, 41f, 2.81–2.95. See
also external gear machining meshes
accuracy of shaped internal gears, 2.90–2.92.
interference of internal work gear and
shaper cutter teeth, 2.91–2.92.
tooth number, 2.92.
transverse generating pressure angle, 2.91
application of, 2.93–2.95
cutting edge geometry of, 2.85–2.86
definition of, 40
design of shaper cutters, 2.83
enveloping shaper cutters, 2.93
gear cutting tools for machining internal
gears, 687–691Index 747
gear cutting tools with enveloping
generating surface, 673–686
conical generating surface, 679–681
cylindrical generating surface, 673–679
toroidal generating surface, 681–686
generating surface of, 2.83
kinematics of shaping operation, 2.81–2.83
rotation vector, 2.81–2.82.
screw motion, 2.83
materials used in, 2.93
profiling of shaper cutters, 2.83–2.85
clearance surface, 2.83–2.84
deep counterbore-type, 2.84–2.85
generating surface, 2.83–2.84
rake surface, 2.83–2.84
shank-type, 2.84–2.85
thickness of chip cut by, 2.86–2.90
feed motion, 2.86
motion of cut, 2.86
operator of rolling, 2.88–2.89
Internal gears, 5f, 687–691
broaching, 73
examples of gear cutting tool, 689–691
generating surface of gear cutting tool, 689
geometry of, 687
hob for cutting, 690f
kinematics of machining internal gear, 687–688
shaping of, 344–346
Internal round broach
for bevel gears, 170–172.
for helical gears, 169–170
for spur gears, 169–170
Intersecting-axis gear machining mesh
gear cutting tools, for machining bevel
gears, 315–342.
gear shapers with tilted axis of rotation,
301–304, 343–349
Inverse problem of gear-cutting tool design, 2.
Involute curve, 10f
Involute function of pressure angle, 696t
Involute gears
curvature of tooth surface, 596
Dupin indicatrix of tooth surface, 596
equation of tooth flank of, 595
first fundamental form of tooth surface, 595
machining, 70–72.
cross section of auxiliary rack, 72.
gear broaching tools for, 70–72.
gear milling cutter for, 12.1
helical gears, 96–100
shapes and configurations of cutting
edges, 70–71
spur gears, 92.–100
straight lateral cutting edges, 71–72.
second fundamental form of tooth surface,
595
tooth flanks, 9–14
helical gears, 10–14
spur gears, 9–10
Involute helical gears, tooth flank, 10–14
Involute hobs, 433–462., 468–487
actual machining surface vs. desired
generating surface, 438–445
analytical approach to configuration of rake
plane, 471–475
analytical description of actual lateral
cutting edge, 436–437
analytical description of desired lateral
cutting edge, 436
change to normal profile angle, 454–457
computation, 475–477
conical hobs, 493–497
butt-end hob, 496, 496f
face hob, 496, 496f
work gear to cutting tool penetration
curve, 493, 494f
correction to configuration of rake surface,
458–560
cylindrical hobs of nonstandard design,
487–493
for finishing hardened gears, 490–493
number of hob threads, 489
for smooth roughing of coarse pitch
gears, 487–489
unfolded cross section, 488f, 490f
design parameters, 470–477
deviations of rack surface, 434f
DG-based approach to configuration of rake
plane, 470–471
as function of design parameters, 435–445
impact of lead angle of screw rake surface
on tooth profile deviation, 440–443
impact of normal profile angle, 478f
impact of number of starts, 478f
impact of pitch diameter, 445–462.
impact of rake angle on tooth profile
deviation, 438–440
impact of setting angle, 479f
machining surface, 437–438
maximum deviation of hob tooth profile, 438
modified tooth profile, 481–487
pitch helix angle, 447
principal design parameters of, 449–454, 460t
relative motions of work gear and hob, 446–449
tooth profiles, 515–52.8
actual value of deviation, 52.7–52.8748 Index
Involute hobs (continued)
allowed interval for profile angle, 52.7–52.8
applied reference systems, 516–517
elementary gear drive, 517–518
form diameter of gear, 517–518
limit diameter of gear, 517
maximum allowed value of modification,
516–52.2.
modification of, 517
normalized deviation, 52.2.–52.4
reduced addendum, 52.4–52.7
toroidal hobs, 497–498
value of center-distance in gear hobbing
operation, 457–458
Involute worm, 171–181
grinding, 178–181
machining, 172.–175
milling, 175–177
J
Jacobian matrix, 703
K
Kinematics of gear cutting, 12.3–12.4
crossed-axis angle, 12.3
feed motion, 12.3
gear milling cutters, disk-type, 12.3–12.4
pitch helix angle, 12.3
rotation vector, 12.3–12.4
Kinematics of gear machining, 19–2.4
bevel gears, 315–317
continuously indexing methods, 42.
elementary relative motions, 2.0–2.1
external spatial gears, 353–357
feasible relative motions, 2.1–2.3
gear broaching, 60f
gear machining mesh, 42.
gear milling cutters, end-type, 91, 92.f
internal gears, 687–688
invertibility of, 2.8
rolling of conjugate surfaces, 2.3–2.4
Kinematics of gear shaping
external gear machining mesh, 2.2.3–2.2.5
gear shapers with tilted axis of rotation,
301–304
internal gear machining mesh, 2.81–2.83
Kinematics of gear shaving
axial method, 576–578
diagonal method, 588–589
plunge method, 619–62.0
tangential method, 603–604
L
Lateral cutting edges
analytical descriptions of, 436–437
angle of inclination, 2.40–2.41
clearance angle of, 2.43–2.48
computation of geometry for, 2.00–2.02.
of involute hobs, 436–437
rake angle of, 2.41–2.43
Lathe, machining worm on, 172.–175
Lead, 695t
Line of action, 584
M
M-2. high-speed steel, 80
Machining meshes, 2.5
applications of, 643–662.
conical hob for palloid gear cutting,
658–662.
cutting tool for machining worm in
continuously indexing method, 653–654
cutting tools for scudding gears, 649–651
gear reinforcement by surface plastic
deformation, 657–658
hob for face gear, 645–646
hob for plunge gear hobbing, 644–645
hob for tangential gear hobbing, 643, 644f
rack shaving cutters, 656–659
shaper cutter with tilted axis of rotation,
651–653
worm-type cutting tool with continuous
helix-spiral cutting edge, 646–649
center distance, 40
crossed-axis angle, 40
external, 39–40, 41f
internal, 40, 41f
kinematics of, 42.
kinematics relationships, 32.–37
magnitude of rotation vectors, 40
planar, 39–40, 41f
spatial gear, 353–357
gear shaving cutters, 559–642.
generating surface of gear cutting tool,
353–394
hobs for machining gears, 395–559
kinematics of, 353–357
vector diagram, 2.9, 30f
vector representation of, 2.5–32.
Machining surface, of involute hobs, 437–438
Machining zone, 500, 503–505
Magnitude of rotation vectors, 40
Major section plane, 706–708Index 749
Mensnier’s formula, 117, 152.
Merchant’s formula, 511
Metric-to-pitch system conversion, 697t
Minor cutting edge approach angle, 715–716
Modified pressure angle, 2.64
Multistart hobs, 407–408
Multistroke shaving, 589
N
Natural parameterization, 13
NC grinder, 62.5–62.7
Nitriding surface treatment, 2.93
Normal circular pitch, 695t
Normal diametral pitch, 696t
Normal pitch, 475t
Normal pressure angle, 696t
Normal rake angle, 709–710
Normal section plane, 709
Normalized velocity, 34
O
Octoidal profile, 319
Oerlikon method, 668
Offset teeth, 32.5–32.8
Operating pitch diameter, 696t
Operating pressure angle, 696t
Operators of reflections, 54
Operators of rolling motion, 50–51
Operators of rotation, 46–47
Operators of screw motion, 48–49
Operators of translation, 44–45
Opposite transformation, 54
Orientation-preserving transformation, 45
Orientation-reversing transformation, 54
Outside diameter, 695t, 697t
P
Palloid gear hobbing, 658–662.
design of conical hob, 659–660
kinematics of, 660–661
machining with crowned teeth, 662.
overview, 659
Parallel-axis gear machining mesh, 183–185
gear shaper cutters, 2.2.3–2.95
external gear machining mesh, 2.2.3–2.79
internal gear machining mesh, 2.81–2.95
rack cutter, 187–2.2.2.
Part surface generation, 62.8–633
conditions for, 699–701
cylindrical gear cutting tool, 674–676
plunge shaving, 62.8–633
Penetration curve, 498–515
Pitch angle, 475t
Pitch diameter
conversion to metric system, 697t
equation, 695t
of involute hobs, 475t
Pitch helix angle
in disk-type milling cutters, 12.3–12.4
in gear broaching, 59
of gear shaper cutters, 2.2.8
Pitch line, 2.9
Pitch point, 33
Pitch-to-metric system conversion, 697t
Planar gear machining mesh, 39–40, 41f
Plane of cut, 705–706
Plastic deformation, gear reinforcement by,
657–658
Plunge shaving, 619–33. See also shaving cutters
circular mapping of tooth flanks, 62.9–631
cutting speed, 619
design of shaving cutters, 631–633
epicyclical motion in, 619
kinematics of, 619–62.0
part surface generation, 62.8–633
topologically modified gears, 62.1–62.8
auxiliary rack modification matrix, 62.7
gear modification matrix, 62.2.
generating surface of form grinding
wheel, 62.7–62.9
geometry of, 62.1–62.4
grinding of tooth flanks, 62.5–62.8
NC grinder, 62.5–62.7
tooth flanks of shaving cutter, 62.4–62.5
PM4 powder metal, 80
Position for machining, relief grinding, 42.5,
431f
Position vector
of gear shaper cutters, 2.2.5–2.2.6
involute tooth profile, 9
tooth flank of helical bevel gear, 16
tooth flank of involute spur gear, 9
tooth flank of work gear, 2.2.6–2.2.7
Pot broaching, 74
Precision gear shaper cutters, 2.69–2.70
carbide, 2.70–2.73
Pregrind rack cutter, 2.18–2.19
Pressure angle
disk-type gear milling cutters, 145–146
end-type gear milling cutters, 114
involute function of, 696t
of involute hobs, 475t
normal, 696t750 Index
Pressure angle (continued)
transverse, 696t
Profile angle
disk-type gear milling cutters, 144–145
of gear shaper cutters, 2.2.7
Profile sliding, 580–587
Progressive cutting diagram, 149
Proper post surface generation
accuracy of gears and, 2.55–2.60
fifth condition of, 2.12.–2.15, 2.55–2.58
sixth condition of, 2.15–2.18, 2.55–2.58
Protuberance, 2.64
Pull broaches, 73
Pull-up broaching, 74
Push broaches, 73
Push-up pot broaching, 74
Q
Quasi-planar gear machining meshes, 351, 669–670
R
Rack cutters, 187–2.2.2.
accuracy of machined gear, 2.12.–2.18
fifth condition of proper post surface
generation, 2.12.–2.15
gear-to-rack cutter meshing diagram,
2.14–2.15
sixth condition of proper post surface
generation, 2.15–2.18
application of, 2.18–2.19
chip thickness cut by cutting edges of,
2.07–2.12.
clearance angle, 2.02.
computation, 2.00–2.02.
cutting edge geometry of, 199–2.07
cutting motion, 2.08
direction of reciprocation, 2.07–2.08
feed motion, 2.08–2.09
inclination angle, 2.00
for lateral cutting edges, 2.00–2.02.
modification of clearance surface, 2.04–2.07
modification of rake surface, 2.03–2.04
operator of rolling, 2.11
position vector, 2.11
translational motion, 2.08
cutting edges of, 193–196
clearance surface, 195–196
rake surface, 194–195
feasible tooth profiles, 191–193
helix angle, 191
radius of pitch cylinder, 191
tooth profile angle, 191
tooth thickness of generating surface, 192.
finish, 2.18–2.19
generating surface of, 187–191
equation for, 187–190
indexing time, 2.19
methods and designs, 2.2.0–2.2.2.
pregrind, 2.18–2.19
profiling of, 196–199
analytical, 198–199
descriptive geometry-based, 197, 198f
rough, 2.18–2.19
tooth profile angle, 198–199
Rack shaving cutters, 654–657
Rake angle, 709–710
chip-flow, 717–718
definition of, 709–710
disk-type gear milling cutters, 143, 146
of gear broach, 68
of hob tooth cutting edge, 509–510, 512.–513
of lateral cutting edge, 2.41–2.43
normal, 709–710
Rake surface
of gear hobs, 399–410
auxiliary axis of projections, 407
auxiliary reference systems, 401
Cartesian coordinate system, 399–400
cone angle, 407
in form of plane, 403–404
generation of, 403–410
geometry, 399–403
intermittent, 409–410
of multistart hob, 407–408
plane of projections, 407
position vectors, 401
rake angle, 403
screw rake surface, 405–407
straight-generating lines, 405–406
straight-line generator, 400
zero rake angle, 402.
of milling cutter for machining involute
gears, 132.–134
modification of, 2.03–2.04
of rack cutter, 194–195
of shaper cutter, 2.2.9–2.32.
combined rake surface, 2.66
geometry, 2.30–2.31
grinding, 2.75–2.79
helical, geometry of, 2.31–2.32.
Ratio of translation V
cut, 59
Reciprocation, 2.07–12., 2.2.3
Reference plane of chip flow, 716–717
Reinforcement, 657–658Index 751
Relief grinding, 42.3–433
of assembled gear hobs, 42.5–433
position for machining, 42.5, 431f
of solid gear hobs, 42.3–42.5
technological worm, 42.8
working position, 42.5, 431f
Resharpening
precision involute hobs, 477–481
shaving cutters, 571–576
Resultant coordinate systems transformations,
47–48
Revacycle process, 81–90. See also broaching
cutting tools, 83–84
kinematics of, 82.f
principle of, 82.–83
profiling of bevel gear cutter, 85–90
application, 88–90
finishing teeth, 85–88
reference systems, 85f
shape of roughing teeth, 88–90
tooth space generation, 83f
Ring-type pot broaching tool, 74
Rolling motion, 50–54
of coordinate system, 50–51
over cylinder, 53
operators of, 50–51
over plane, 50
of two coordinate systems, 52.–54
Root fillet, 2.63
Rotary broaches, 80–81
clearance angle, 80
form size, 81
shank axis, 80
tool holder, 81
Rotation
frequencies, 33
inversion of, 2.9f
Rotation vector, 2.6–2.9
configuration of, 37–39
critical configuration of, 38
of disk-type milling cutter, 12.3–12.4
in gear machining mesh, 32.–33
magnitude of, 36, 40–42.
superimposition of translation vector and, 36
Rotations, 19
Rough rack cutter, 2.18–2.19
Round rack, 319
S
Scientific approach, 8
Screw surface, Archimedean, 62.–63
Screw feed motion, 12.4
Screw involute surface, 10–13, 10f, 13t
Screw motion, 2.6
about coordinate axis, 48–49
gear shaper cutters, 2.2.3–2.2.4
operators of, 48–49
parameter, 36
Screw rake surface, 405–407
Scudding, 651–653
applications of, 651
cutting tools for, 649–651
design concept for tools, 649–651
Secondary generating surface T2, 92.
Semitopping, 2.63–2.64
Serrations, 605
Setting angle, 374–376, 475t
Shank axis, 80
Shank-type shaper cutters, 2.84–2.85, 2.94f
Shaper cutters
cutting edge geometry of, 2.85–2.86
deep counterbore-type, 2.84–2.85
enveloping, 2.93, 2.95f, 346–348
interference of internal work gear and
shaper cutter teeth, 2.91–2.92.
profiling of, 2.83–2.85
clearance surface, 2.83–2.84
generating surface, 2.83–2.84
rake surface, 2.83–2.84
shank-type, 2.84–2.85, 2.94f
with tilted axis of rotation, 651–653
application, 652.–653
kinematics of shaping helical gear, 651–652.
principal design elements, 652.
tooth number, 2.92.
transverse generating pressure angle, 2.91
Sharpening of gear broaches, 66–70
generating of rake surface, 68
generating surface of grinding wheel, 66–67
helical gullet, 68
maximal feasible outer diameter of grinding
wheel, 67
schematic diagram, 69f
support at center, 67
Shave grinding, 685
Shaving cutters, 559–642.
advances in design of, 635–640
constant inclination angle within cutting
edges, 635–636
for finishing modified work gear tooth
flank, 637–638
near optimal angle of inclination of
cutting edges, 633–635
precision cutter with straight cutting
edges, 636–637752 Index
Shaving cutters (continued)
axial method, 576–587
coordinate systems transformations, 585
cutting speed, 578–587
equivalent base diameters, 586–587
formula for cutting speed, 587
impact of crossed-axis angle, 578–579
impact of profile sliding, 580–587
impact of traverse motion, 579–580
kinematics of, 576–578
line of action, 584
workgear to shaving cutter meshing, 580,
581f, 583f
clearance surface of cutting teeth, 562.
design of shaper cutters, 566–576
design parameters, 567, 575t
resharpening, 571–576
serrations on tooth flanks, 568–571
diagonal method, 587–602.
coordinate systems, 596–598
cutting speed, 590–592.
diagonal-underpass, 590
geometry of contact of tooth flanks,
598–602.
kinematics of, 588–589
local topology of contacting tooth flanks,
595–596
multistroke, 589
optimal kinetic parameters, 602.
optimization of kinematics, 592.–595
traverse angle, 589–590
generating surface of, 559–560
inclination angle of cutting edges, 562.–566
manufacturing aspects of shaving operation,
640–641
modification of tooth form and shape, 641
plunge method, 619–633
circular mapping of tooth flanks, 62.9–631
cutting speed, 619
design of shaving cutters, 631–633
epicyclical motion in, 619
kinematics of, 619–62.0
part surface generation, 62.8–633
topologically modified gears, 62.1–62.8
rack-type, 654–657
rake surface of cutting teeth, 560–561
requirements for preshaved work gear,
639–640
selection of, 639
shaving of worm gear, 641–642.
tangential method of shaving process,
603–618
analytical approach, 612.–618
cutting speed, 604–605
descriptive geometry-based methods,
605–612.
kinematics of, 603–604
serrations, 605
tooth flanks, 595–596
curvature of tooth surface, 596
Dupin indicatrix of tooth surface, 596
equation of tooth flank of, 595
first fundamental form of tooth surface,
595
indicatrix of conformity, 599
local orientation, 598
maximum rate of conformity, 600
optimal design parameters, 600
principal directions, 598
second fundamental form of tooth
surface, 595
Shear-Speed cutting, 74–80
kinematics of, 75f
principle of, 74
profiling of form tools for, 76–79
Shell-type gear hobs, 462.f, 463–464
Shishkov’s equation of contact, 2.2.7–2.2.8
Shoulder gears, tangential shaving of, 605–618
analytical approach, 612.–618
closed vector polygon, 614–615
operators of linear transformations, 613–614
optimal design parameters, 612.–615
overlap of shaving cutter over work gear,
615–618
reference systems, 612.–613
descriptive geometry-based method, 605–612.
maximum allowed diameter of cutter,
606–608
minimum permissible width of auxiliary
generating rack, 612.–614
minimum permissible width of work
gear auxiliary rack, 608–610
minimum required face width, 612.
minimum required overlap of gear and
cutter, 608–612.
Single parametric motion, 16
Skiving hobs, 463–464
Slater tools, 80–81
Sliding motion over surfaces, 2.3–2.4
Sliding vectors, 2.6
Spatial gear machining mesh, 353–357. See also
machining meshes
gear shaving cutters, 559–642.
generating surface of gear cutting tool,
353–394
hobs for machining gears, 395–559Index 753
kinematics of, 353–357
axode of gear cutting tool, 357
center distance, 353
crossed-axis angle, 353
hyperboloid, 356
opposite-directed vector, 356
pure rolling of axodes, 354–355
rotation of cutting tool, 353
rotation of work gear, 353
rotation vectors, 353–354
vector of linear velocity of sliding of
axodes, 355
Special roughing hobs, 463
Spiral bevel gears, 337–342.
clearance angle, 342.
design of cutters, 340–342.
diagrammatic arrangement of, 339f
gear machining operation, 338–340
profile angle, 341
rake angle, 342.
spiral angles, 337
Spiral gullet, 68
Spiral infeed, 2.74
Spur gears, 3, 4f
barreled, 17
crowned, 17
cusps on tooth flanks, 115–117
disk-type gear milling cutter, 12.5–12.7
gear milling cutter for, 12.0
internal round broach for, 169–170
involute, 9–10
machining, 92.–100
modified, 17
shaping of, 346–347
tooth flank, 9–10
Spur rack, 3
Stabler’s equation, 718
Staggered pattern serration, 605
Standard circular tooth thickness, 697t
Standard normal circular thickness, 696t
Standard outside diameter, 697t
Start of active profile (SAP), 9
Stick-type pot broaching tool, 74
Straight bevel gears, 5–6
cutting, 333–334
gear cutting tools, 330–333
design parameters, 330–331
tool geometry, 331–333
tool height, 330
machining of, 32.9–330
milling, 334–337
design of milling cutters, 335–336
disk-type milling cutters, 336
lateral cutting edges, 2.5–2.6
shape of finished flanks, 336–337
with offset teeth, 32.5–32.8
teeth, 32.8–33
gear cutting tools, 330–333
machining of straight bevel gears, 32.9–330
plane Ta by straight motion of cutting
edge, 32.8–32.9
tooth flank, 14–15
Straight feed motion Fc, 12.3
Straight rack, 365
Surface of tolerance, 594
Surface parameters, 703–704
Surface plastic deformation, gear reinforcement
by, 657–658
Surfaces, fundamental forms, 55–56
Surfaces that allow sliding, gear tooth, 16–18
T
Tangential shaving, 603–618. See also shaving
cutters
analytical approach, 612.–618
cutting speed, 604–605
descriptive geometry-based methods, 605–612.
kinematics of, 603–604
serrations, 605
Technological worm, 42.8
Thickness of chip cut, 2.86–2.90
feed motion, 2.86
motion of cut, 2.86
operator of rolling, 2.88–2.89
Thread whirling, 177–178
Tip relief, 2.63
Titanium nitride coating, 2.61, 2.94
Tool-in-use reference system, 505–515
Tooth depth, 62.
Tooth flanks, 6–16
of bevel gear, 14–15
circular mapping of, 62.9–631
of helical gear, 117–119
cusps, 117–119
deviation from desired shape, 118–119
normal curvature, 118
of involute helical gear, 10–14
of involute spur gears, 9–10
serrations, 568–571
of shaving cutters, 568–571, 595–596
curvature of tooth surface, 596
Dupin indicatrix of tooth surface, 596
equation of tooth flank of, 595
first fundamental form of tooth surface,
595754 Index
Tooth flanks (continued)
indicatrix of conformity, 599
local orientation, 598
maximum rate of conformity, 600
optimal design parameters, 600
principal directions, 598
second fundamental form of tooth
surface, 595
specification, engineering approach, 8
specification, scientific approach, 8
of spur gear, 115–117
waviness of, 116
Tooth profiles
asymmetric, 389–390
gear hobs, 465–468, 515–52.8
gear shaper cutters, 2.63–2.64
chamfer, 2.63–2.64
full topping, 2.64
modified pressure angle, 2.64
protuberance, 2.64
root fillet, 2.63
semitopping, 2.63–2.64
tip relief, 2.63
involute hobs, 515–52.8
rack cutters, 191–193
Tooth relieving operation, clearance surface of,
415
Top-loaded cutting diagram, 150
Topologically modified gears, 62.1–62.8
auxiliary rack modification matrix, 62.7
gear modification matrix, 62.2.
generating surface of form grinding wheel,
62.9–30
geometry of, 62.1–62.4
grinding of tooth flanks, 62.5–62.8
NC grinder, 62.5–62.7
tooth flanks of shaving cutter, 62.4–62.5
Toroidal gear cutting tools, 681–686
Toroidal gear hobs, 497–498
Torus-shaped pitch surfaces, 390–392.
Transformation matrices, 44
Transformations, coordinate-system, 43–56
conversion of coordinate system orientation,
54
direct transformation, 45
direct transformation of surfaces
fundamental forms, 55–56
homogenous coordinate transformation
matrices, 44
homogenous coordinate vectors, 43–44
opposite transformation, 54
orientation-preserving transformation, 45
orientation-reversing transformation, 54
overview, 43
resultant, 47–48
rolling motion of coordinate system, 50–51
rolling of two coordinate systems, 52.–54
rotation about coordinate axis, 46–47
screw motions about coordinate axis, 48–49
translations, 44–45
Translation vector, 2.6
magnitude of speed, 36
superimposition of rotation vector and, 36
Translational motion, 2.08
Translations, 44–45
Transverse circular pitch, 696t
Transverse pressure angle, 696t
Traverse angle, 589–590
Traverse motion, 579–580
Triparametric motion, 16
U
Unit normal vector, 374
V
Vector of instant rotation, magnitude of, 34–36
W
Work gear, 2.0–2.3
auxiliary rack, 608–610
elementary relative motions, 2.0–2.1
feasible relative motions, 2.1–2.3
Work gear to cutting tool penetration curve,
493, 494f
Work gear to generation surface mesh, 2.5
Work gear to shaving cutter meshing, 580, 581f,
583f
Working position, relief grinding, 42.5, 431f
Worm, 5, 7f
grinding, 178–181
machining in continuously indexing
method, 653–654
machining on lathe, 172.–175
milling, 175–177
technological
Worm gear, 7f
Z
Zero profile angle, 381–383734 Index
Chamfers, 2.63–2.64, 677–678
Characteristic E, 115
Chip per tooth, 59, 65–66
Chip removal diagrams, 65–66
Chip-flow rake angle, 717–718
Chip-flow reference plane, 716–717
Circular mapping, 62.9–631
Circular pitch, 695t, 697t
Circular tooth thickness, 697t
Clearance angle, 710–711
definition of, 708–709, 710–711
disk-type gear milling cutters, 144–146
of hob tooth cutting edge, 510, 515–516
of lateral cutting edge, 2.43–2.48
normal, 710–711
of rack cutter, 2.02.
of rotary breaches, 80
Clearance surface, 64–65
of disk-type gear milling cutter, 134–136
of gear hobs, 411–433
clearance angle, 413, 416–42.3
cutting edges, 411
cutting of relieved clearance surface,
415–42.3
equation of desired clearance surface,
411–415
grinding of relieved clearance surface,
42.3–433
helix angle, 414
operator of linear transformation, 415
position vector, 413
rake plane, 413
reduced pitch, 413
tooth relieving operation, 415
grinding of, 2.74–2.75
of rack cutter, 195–196
of shaper cutter tooth, 2.32.–2.33
Climb hopping, 530
Cluster gears, 5, 6f
Coarse-pitch gears, 12.0
Combined rake surface, 2.66
Cone drive, 5
Conical gear cutting tools, 679–681
Conical generating surface, 383–389
Conical hobs, 493–497
butt-end hob, 496, 496f
design of, 661–662.
face hob, 496, 496f
palloid gear hobbing with, 658–662.
kinematics of, 660–661
machining with crowned teeth, 662.
work gear to cutting tool penetration curve,
493, 494f
Conical involute gears, shaping of, 311, 312.f
Conical rake surface, 62.–63
Conjugate surfaces, rolling of, 2.3–2.4
Contact ellipse, 594
Continuously indexing methods, 2.5–42.
classification of gear machining meshes,
39–42.
configuration of rotation vectors, 37–39
kinematic relationships for gear machining
mesh, 32.–37
kinematics of gear machining, 42.
vector representation of gear machining
mesh, 2.5–32.
Conventional hopping, 530
Coordinate axis, rotation about, 46–47
Coordinate systems transformations, 43–56
conversion of coordinate system orientation,
54
direct transformation, 45
direct transformation of surfaces
fundamental forms, 55–56
homogenous coordinate transformation
matrices, 44
homogenous coordinate vectors, 43–44
opposite transformation, 54
orientation-preserving transformation, 45
orientation-reversing transformation, 54
overview, 43
resultant, 47–48
rolling motion of coordinate system, 50–51
rolling of two coordinate systems, 52.–54
rotation about coordinate axis, 46–47
screw motions about coordinate axis, 48–49
translations, 44–45
Coordinate transformation matrices,
homogenous, 43–44
Coordinate vectors, homogenous, 43–44
Critical configuration, 38
Crossed-axis angle, 2.7, 12.3
impact of, 578–579
zero value, 40
Crowned spur gear, 17
Cut per tooth, 65–66
Cutting edge geometry, 705–718
chip flow rake angle, 717–718
chip-flow reference plane, 716–717
clearance angle, 708–709
correlation measured within major and
normal section planes, 713–715
cutting edge, 705
of disk-type gear milling cutters, 143–146
clearance angle, 144–146
pressure angle, 145–146Index 735
profile angle, 144–145
rake angle, 143, 146
of external gear machining meshes, 2.39–2.48
gear hobs, 498–515
of gear shaper cutters
external gear machining mesh, 2.39–2.48
internal gear machining mesh, 2.85–2.86
hobs for machining gears, 498–515
of internal gear machining meshes, 2.85–2.86
in major section plane, 707–708
mandatory relationship, 711–712.
minor cutting edge approach angle, 715–716
normal clearance angle, 710–711
normal rake angle, 709–710
in normal section plane, 709
in plane of cut, 706
of rack cutters, 199–2.07
clearance angle, 2.02.
computation, 2.00–2.02.
inclination angle, 2.00
for lateral cutting edges, 2.00–2.02.
modification of clearance surface,
2.04–2.07
modification of rake surface, 2.03–2.04
roundness of cutting edge, 712.
vector of resultant motion, 705
Cutting edge vector, 705
Cutting edges, 705
definition of, 705
end-type gear milling cutters, 101–114
helix angle, 113
rake surface in form of plane, 109–111
rake surface in form of screw surface,
111–114
of gear broach, 61–65
roundness of, 712.
Cutting motion, 2.08
Cutting tools, 1–2.
cylindrical, 673–679
direct problem of design, 2.
elementary relative motions, 2.0–2.1
with enveloping generating surface, 673–686
feasible relative motions, 2.1–2.3
for machining bevel gears, 315–342.
for machining internal gears, 687–691
Shear-Speed cutting, 74–80
Cylinders, rolling motion over, 53
Cylindrical gear cutting tool, 673–679
examples of, 676–679
gear tooth with chamfers, 677–678
gear tooth with lengthwise modification,
678f
generating surface of, 673–674
part surface generation, 674–676
vector diagram, 677f
Cylindrical hobs, 487–93
for finishing hardened gears, 490–493
number of hob threads, 489
for smooth roughing of coarse pitch gears,
487–489
unfolded cross section, 488f, 490f
D
Deep counterbore-type shaper cutters, 2.61,
2.84–2.85
Descriptive geometry-based methods, 136–138
generating surfaces, 378–381
for rack cutter, 197, 198f
tangential shaving, 605–612.
Diagonal shaving, 587–602 See also shaving
cutters
coordinate systems, 596–598
cutting speed, 590–592.
diagonal-underpass, 590
geometry of contact of tooth flanks, 598–602.
kinematics of, 588–589
local topology of contacting tooth flanks,
595–596
multistroke, 589
optimal kinetic parameters, 602.
optimization of kinematics, 592.–595
traverse angle, 589–590
Diagonal-underpass shaving, 590
Differential serrations, 605
Direct problem of gear-cutting tool design, 2.
Direct transformation, 45
of surface fundamental forms, 55–56
Disk-type gear milling cutters, 12.3–161. See also
end-type gear milling cutters
15-gear, 155
accuracy of tooth flanks machined with,
152.–154
application of, 154–161
circular saws, 159–160
cutting edge geometry of, 143–146
clearance angle, 144–146
pressure angle, 145–146
profile angle, 144–145
rake angle, 143, 146
cutting edges of, 132.–136
clearance surface, 132.f, 134–136
rake surface, 132.–134, 132.f
eight-gear, 155
generating surface of, 12.5–12.7
elements of intrinsic geometry, 130–132.736 Index
Disk-type gear milling cutters (continued)
equation for machining spur involute
gears, 12.5–12.7
for machining helical involute gears,
12.7–130
inclined tooth profile, 155, 156f
indexing, 161
kinematics of gear cutting, 12.3–12.4
crossed-axis angle, 12.3
feed motion, 12.3
pitch helix angle, 12.3
rotation vector, 12.3–12.4
screw feed motion, 12.4
multiple-tooth, 161
overview, 12.3
profiling of, 136–143
analytical, 138–143
descriptive geometry-based methods,
136–138
for roughing of gears, 147–152.
assembled cutter, 151f
clearance angle, 147–148
design parameters, 147f
even-numbered teeth, 150
odd-numbered teeth, 150
progressive cutting diagram, 149
rake angle, 147
top-loaded cutting diagram, 150
wavy cutting edges, 151
single parametric motions in design of, 167–168
span measurement, 158f
Disk-type gear shaper cutter, 2.61
Double-enveloping worm gear drive, 5
Double-helical gears, 3, 6f
Dupin indicatrix, 596
E
End-type gear milling cutters, 91–12.2 See also
disk-type gear milling cutters
application of, 12.0–12.2.
cutting edges, 101–114
clearance surface, 105–108
geometry of, 108–114
helix angle, 113
pressure angle, 114
rake surface in form of plane, 109–111
rake surface in form of screw surface,
111–114
generating surface of, 92.–101
elements of intrinsic geometry, 100–101
equation for machining helical involute
gears, 96–100
equation for machining spur involute
gears, 92.–96
secondary, 92.
for helical gears, 12.0–12.1
indexing mechanism, 12.2.
kinematics of gear machining, 91, 92.f
machining gear tooth flanks with, 115–119
characteristic E, 115
curvature of helical gear tooth flank, 118
cusps on tooth flanks of helical gear,
117–119
cusps on tooth flanks of spur gear, 115–117
maximal deviation, 117
waviness of tooth flank, 116
overview, 91
single parametric motions in design of,
166–167
for spur gears, 12.0
Engineering approach, 8
Enveloping generating surface, 673–686
conical, 679–681
cylindrical, 673–679
toroidal, 681–686
Enveloping shaper cutters, 2.93, 2.95f
shaping external recessed tooth forms with,
347–349
shaping spur gears with, 346–347
Epicyclical motion, 619
Equivalent base diameters, 586–587
External gear machining meshes, 41f, 2.2.3–2.79.
See also internal gear machining
meshes
accuracy of gears cut with, 2.55–2.60
fifth condition of proper post surface
generation, 2.55–2.58
gear-to-shaper cutter meshing diagram,
2.57
application of, 12.0–12.2., 2.60–2.79
critical distance to nominal cross section,
2.36–2.39
circular tooth thickness, 2.39
length of top cutting edge, 2.37–2.38
maximal length of distance, 2.37
cutting edge geometry of, 2.39–2.48
angle of inclination of lateral cutting
edge, 2.40–2.41
clearance angle of lateral cutting edge,
2.43–2.48
improvement in geometry, 2.45–2.48, 2.65–2.69
rake angle of lateral cutting edge, 2.41–2.43
cutting edges of, 2.2.9–2.33
clearance surface of tooth, 2.32.–2.33
rake surface, 2.2.9–2.32.Index 737
definition of, 39
design of shaper cutters, 2.61–2.63
deep counterbore-type, 2.61
disk-type, 2.61
for machining helical and herringbone
gears, 2.64–2.65
tooth profiles, 2.63–2.64
desired corrections to tooth profile, 2.48–2.50
generating surface of, 2.2.5–2.2.9
Cartesian coordinate system, 2.2.5–2.2.6
for machining involute shapes, 2.2.8
pitch helix angle, 2.2.8
position vector, 2.2.5–2.2.6
profile angle, 2.2.7
Shishkov’s equation of contact, 2.2.7–2.2.8
geometry of clearance surface, 2.46–2.48
grinding of, 2.74–2.79
clearance surfaces, 2.74–2.75
rake surfaces, 2.75–2.79
helical gears for, 12.0–12.1
indexing mechanism, 12.2.
kinematics of gear shaping, 2.2.3–2.2.5
principal elements, 2.2.4f
reciprocation, 2.2.3
screw motion, 2.2.3–2.2.4
vector diagram, 2.2.4f
lateral cutting edges
angle of inclination, 2.40–2.41
clearance angle, 2.43–2.48
rake angle, 2.41–2.43
overview, 2.2.3–2.79
profiling of, 2.33–2.36
angular displacement of tooth profile, 2.36
base diameter, 2.35
circular tooth thickness, 2.34–2.35
meshing of auxiliary rack surface, 2.33–2.34
profile shift correction coefficient, 2.35
tooth addendum, 2.35
tooth dedendum, 2.35
tooth thickness, 2.35–2.36
rake surface of, 2.2.9–2.32.
geometry, 2.30–2.31
helical, geometry of, 2.31–2.32.
improvement in geometry, 2.45–2.46
special designs of, 2.65–2.73
angular pitches, 2.68
combined rake surface, 2.66
improved cutting edge geometry, 2.65–2.69
for machining hardened gears, 2.70–2.72.
precision gear, 2.69–2.70
two gears, 2.67–2.69
spur gears for, 12.0
thickness of chip cut by, 2.51–2.55
feed rate motion, 2.51
operator of rolling, 2.53–2.54
position vector, 2.53–2.54
roll angle, 2.52.
speed of instant rotation, 2.51
tooth ratio, 2.52.
tooth profiles
chamfer, 2.63–2.64
full topping, 2.64
modified pressure angle, 2.64
protuberance, 2.64
root fillet, 2.63
semitopping, 2.63–2.64
tip relief, 2.63
typical gear shaping operations, 2.73–2.74
combination operations, 2.73–2.74
cutting two gears with different
diametral pitch, 2.74
cutting two gears with similar diametral
pitch, 2.74
cutting two gears with up and down
shaping, 2.74
length of stroke, 2.73, 2.74f
spiral infeed, 2.74
External gears, broaching, 73–74
External recessed tooth forms, shaping of, 347–349
External spatial gear machining, kinematics of,
353–357
F
Face angle radius, 62.
Face gear milling cutters, 168–169
Face gears, 6, 311–314
Face hob, 496, 496f, 668, 669f
Feed motion, 2.8, 12.3, 2.08–2.09
Fiat spiral bevel-gear generation, 668
Finish rack cutter, 2.18–2.19
Finishing teeth, 59
of gear broach, 61–64
Archimedean screw surface, 62.–63
clearance surface, 64–65
conical, 62.–63
face angle radius, 62.
gullet, 62.
pitch, 62.
tooth depth, 62.
Form gear cutting tools, 163–181
analytical profiling of, 77–79
classification of, 181
disk-type gear milling cutters, 12.3–161
end-type gear milling cutters, 91–12.2.
gear broaching tools, 59–90738 Index
Form gear cutting tools (continued)
grinding involute worm, 178–181
machining involute worm on lathe, 172.–175
milling involute worm, 175–177
single parametric motions in design, 166–172.
disk-type gear milling cutter, 167–168
end-type gear milling cutter, 166–172.
face gear milling cutter, 168–169
internal round broach for cutting spur
and helical gears, 169–170
internal round broach for machining
bevel gears, 170–172.
single parametric motions, plurality of, 163–166
coordinate system, 163–164
position vector, 163–164
rotations, 165
translations, 164–165
thread whirling, 177–178
Form milling cutter, 12.0
Full topping, 2.64
Fundamental forms, direct transformation of,
55–56
G
Gear broaching, 72.–73
definition of, 72.
kinematics of, 59, 60f
pull broaches, 73
push broaches, 73
side-shaving section of, 73
Gear broaching tools, 59–90
application of, 72.–74
broaching external gears, 73–74
broaching internal gears, 73
bevel gear teeth broaching, 81–90
burnishing button, 73
chip per tooth, 59, 65–66
chip removal diagrams, 65–66
cut per tooth, 65–66
cutting edges, 61–65
finishing teeth, 59
generating surface of, 60–61
for machining involute gears, 70–72.
cross section of auxiliary rack, 72.
shapes and configurations of cutting
edges, 70–71
straight lateral cutting edges, 71–72.
pot broaching, 74
rake angle, 68
rake surface of finishing teeth, 61–64
Archimedean screw surface, 62.–63
clearance surface, 64–65
conical, 62.–63
face angle radius, 62.
gullet, 62.
pitch, 62.
tooth depth, 62.
resharpening of, 69f
Revacycle process, 81–90
cutting tools, 83–84
kinematics of, 82.f
principle of, 82.–83
profiling of cutter for bevel gear
machining, 85–90
tooth space generation, 83f
rotary broaches, 80–81
sharpening of, 66–70
generating of rake surface, 68
generating surface of grinding wheel,
66–67
helical gullet, 68
maximal feasible outer diameter of
grinding wheel, 67
schematic diagram, 69f
support at center, 67
Shear-Speed cutting, 74–80
application, 79–80
principle of, 74
profiling of form tools, 76–79
slater tools, 80–81
Gear cutting tools, 1–2.
cylindrical, 673–679
examples of, 676–679
gear tooth with chamfers, 677–678
gear tooth with lengthwise modification,
678f
generating surface of, 673–674
part surface generation, 674–676
vector diagram, 677f
direct problem of design, 2.
elementary relative motions, 2.0–2.1
with enveloping generating surface, 673–686
conical generating surface, 679–681
cylindrical generating surface, 673–679
toroidal generating surface, 681–686
feasible relative motions, 2.1–2.3
generating body of, 395
inverse problem of design, 2.
for machining internal gears, 687–691
examples of, 689–691
generating surface of, 689
geometry of internal gear, 687
kinematics of machining internal gear,
689–690
Shear-Speed cutting, 74–80Index 739
Gear cutting tools, for machining bevel gears,
315–342.
curved teeth, 337–342.
clearance angle, 342.
design of cutters, 340–342.
diagrammatic arrangement of, 339f
gear machining operation, 338–340
profile angle, 341
rake angle, 342.
spiral angles, 337
face hob for, 668, 669f
generating surface of, 32.5–32.8
geometry of interacting tooth surfaces,
318–32.5
generating surface of gear cutting tool,
319–32.3
involute straight bevel gear tooth flank,
318–319
octoidal profile, 319
tooth flanks of generated gear, 32.3–32.5
kinematics of bevel gear generation, 315–317
axodes, 316
coordinate systems, 317
pitch angle, 316
rotation vectors, 315–316
overview, 315–42.
plunge method, 665–668
face hob, 668, 669f
kinematics, 665–667
possible designs, 667–668
quasi-planar gear machining meshes, 669–670
straight bevel gear cutting, 333–334
straight bevel gear milling, 334–337
design of milling cutters, 335–336
disk-type milling cutters, 336
lateral cutting edges, 2.5–2.6
shape of finished flanks, 336–337
straight bevel gear teeth, 32.8–333
gear cutting tools, 330–333
machining of straight bevel gears, 32.9–330
plane Ta by straight motion of cutting
edge, 32.8–32.9
straight bevel gears with offset teeth, 32.5–32.8
Gear hobs, 395–559
accuracy for machining involute gears,
433–462.
actual machining surface vs. desired
generating surface, 438–445
analytical description of actual lateral
cutting edge, 436–437
analytical description of desired lateral
cutting edge, 436
deviations of rack surface, 434f
as function of design parameters, 435–445
impact of lead angle of screw rake surface
on tooth profile deviation, 440–443
impact of pitch diameter, 445–462.
impact of rake angle on tooth profile
deviation, 438–440
kinematic geometry of involute hob,
454–462.
machining surface, 437–438
maximum deviation of hob tooth profile,
438
pitch helix angle, 447
principal design parameters of, 449–454
relative motions of work gear and hob,
446–449
clearance surface of, 411–433
clearance angle, 413, 416–42.3
cutting edges, 411
cutting of relieved clearance surface,
415–42.3
equation of desired clearance surface,
411–415
grinding of relieved clearance surface,
42.3–433
helix angle, 414
operator of linear transformation, 415
position vector, 413
rake plane, 413
reduced pitch, 413
tooth relieving operation, 415
conical hobs, 493–497
butt-end hob, 496, 496f
face hob, 496, 496f
work gear to cutting tool penetration
curve, 493, 495f
cutting edge geometry of gear hob tooth,
498–515
clearance angle, 510, 513–514
cutting edge roundness, 511
improvement of hob design, 514–515
inclination angle, 510–512.
machining zone, 500, 503–505
penetration curve, 500–503
rake angle, 509–510, 512.–513
resultant motion, 506–508
tool-in-use reference system, 505–515
unit normal vector to surface of cut,
505–506
unit tangent vector, 508
cylindrical hobs of nonstandard design,
487–493
for finishing hardened gears, 490–493
number of hob threads, 489740 Index
Gear hobs (continued)
for smooth roughing of coarse pitch
gears, 487–489
unfolded cross section, 488f, 490f
definition of, 395
design of, 462.–498
design parameters, 462.–465
tooth profiles, 465–468
for face gear, 645–646
hobbing operations, 52.8–534
climb hopping, 530
conventional hopping, 530
cycles, 534–536
hob total travel distance, 536–537
hobbing time, 536–537
idle distance and neck width of cluster
gear, 538–540
plunge method, 644–645
prescribed value of setting angle, 555–558
setting angle, 538–540
shortest allowable approach distance,
552.–554
shortest allowed idle distance, 540–545
tangential method, 643, 644f
tolerance and shortest possible idle
distance, 545–552.
kinematic geometry of involute hob
change to normal profile angle, 454–457
correction to configuration of rake
surface, 458–560
value of center-distance in gear hobbing
operation, 457–458
multistart, 407–408
precision involute hubs with lateral cutting
edges, 468–487
analytical approach to configuration of
rake plane, 471–475
computation, 475–477
design parameters, 470–477
DG-based approach to configuration of
rake plane, 470–471
modified tooth profile, 481–487
resharpening, 477–481
rake surface of, 399–410
auxiliary axis of projections, 407
auxiliary reference systems, 401
Cartesian coordinate system, 399–400
cone angle, 407
in form of plane, 403–404
generation of, 403–410
geometry, 399–403
intermittent, 409–410
of multistart hob, 407–408
plane of projections, 407
position vectors, 401
rake angle, 403
screw rake surface, 405–407
straight-generating lines, 405–406
straight-line generator, 400
zero rake angle, 402.
tool-in-use reference system, 505–515
clearance angle, 510, 513–514
inclination angle, 510–512.
rake angle, 509–510, 512.–513
tooth profiles, 515–52.8
actual value of deviation, 52.7–52.8
allowed interval for profile angle, 52.7–52.8
applied reference systems, 516–517
elementary gear drive, 517–518
form diameter of gear, 517–518
limit diameter of gear, 517
maximum allowed value of modification,
516–52.2.
modification of, 517
normalized deviation, 52.2.–52.4
reduced addendum, 52.4–52.7
toroidal hobs, 497–498
transformation of generating surface into
gear tool, 395–399
feed motion, 395–396
grinding worm, 395, 396f
screw motion, 396–398
Gear honing, 685–686
Gear machining, 19–2.4
continuously indexing methods, 2.5–42.
classification of gear machining meshes,
39–42.
configuration of rotation vectors, 37–39
kinematic relationships for gear
machining mesh, 32.–37
kinematics of gear machining, 42.
vector representation of gear machining
mesh, 2.5–32.
relative motions, 19–2.3
elementary, 19–2.3
feasible, 2.1–2.3
rolling of conjugate surfaces, 2.3–2.4
Gear machining mesh, 39–42.
applications of, 643–662.
conical hob for palloid gear cutting,
658–662.
cutting tool for machining worm in
continuously indexing method, 653–654
cutting tools for scudding gears, 649–651
gear reinforcement by surface plastic
deformation, 657–658Index 741
hob for face gear, 645–646
hob for plunge gear hobbing, 644–645
hob for tangential gear hobbing, 643, 644f
rack shaving cutters, 654–657
shaper cutter with tilted axis of rotation,
651–653
worm-type cutting tool with continuous
helix-spiral cutting edge, 646–649
center distance, 40
crossed-axis angle, 40
definition of, 2.5
external, 39–40, 41f
internal, 40, 41f
kinematics of, 42.
kinematics relationships, 32.–37
magnitude of rotation vectors, 40
planar, 39–40, 41f
vector diagram, 2.9, 30f
vector representation of, 2.5–32.
Gear milling cutters, disk-type, 12.3–161
15-gear, 155
accuracy of tooth flanks machined with,
152.–154
application of, 154–161
circular saws, 159–160
cutting edge geometry of, 143–146
clearance angle, 144–146
pressure angle, 145–146
profile angle, 144–145
rake angle, 143, 146
cutting edges of, 132.–136
clearance surface, 132.f, 134–136
rake surface, 132.–134, 132.f
eight-gear, 155
generating surface of, 12.5–12.7
elements of intrinsic geometry, 130–132.
equation for machining spur involute
gears, 12.5–12.7
for machining helical involute gears, 12.7–130
inclined tooth profile, 155, 156f
indexing, 161
kinematics of gear cutting, 12.3–12.4
crossed-axis angle, 12.3
feed motion, 12.3
pitch helix angle, 12.3
rotation vector, 12.3–12.4
screw feed motion, 12.4
multiple-tooth, 161
overview, 12.3
profiling of, 136–143
analytical, 138–143
descriptive geometry-based methods,
136–138
for roughing of gears, 147–152.
assembled cutter, 151f
clearance angle, 147–148
design parameters, 147f
even-numbered teeth, 150
odd-numbered teeth, 150
progressive cutting diagram, 149
rake angle, 147
top-loaded cutting diagram, 150
wavy cutting edges, 151
single parametric motions in design of,
167–168
span measurement, 158f
Gear milling cutters, end-type, 91–12.2.
application of, 12.0–12.2.
cutting edges, 101–114
clearance surface, 105–108
geometry of, 108–114
helix angle, 113
pressure angle, 114
rake surface in form of plane, 109–111
rake surface in form of screw surface,
111–114
generating surface of, 92.–101
elements of intrinsic geometry, 100–101
equation for machining helical involute
gears, 96–100
equation for machining spur involute
gears, 92.–96
secondary, 92.
for helical gears, 12.0–12.1
indexing mechanism, 12.2.
kinematics of gear machining, 91, 92.f
machining gear tooth flanks with, 115–119
characteristic E, 115
curvature of helical gear tooth flank, 118
cusps on tooth flanks of helical gear,
117–119
cusps on tooth flanks of spur gear, 115–117
maximal deviation, 117
waviness of tooth flank, 116
overview, 91
single parametric motions in design of,
166–167
for spur gears, 12.0
Gear modification matrix, 62.2.
Gear scudding, 649–651
applications of, 651
cutting tools for, 649–651
design concept for tools, 649–651
Gear shaper cutters, external gear machining
mesh, 2.2.3–2.79
accuracy of gears cut with, 2.55–2.60742 Index
Gear shaper cutters (continued)
fifth condition of proper post surface
generation, 2.55–2.58
gear-to-shaper cutter meshing diagram, 2.57
sixth condition of proper post surface
generation, 2.58–2.60
application of, 2.60–2.79
critical distance to nominal cross section,
2.36–2.39
circular tooth thickness, 2.39
length of top cutting edge, 2.37–2.38
maximal length of distance, 2.37
cutting edge geometry of, 2.39–2.48
angle of inclination of lateral cutting
edge, 2.40–2.41
clearance angle of lateral cutting edge,
2.43–2.48
improvement in geometry, 2.45–2.48,
2.65–2.69
rake angle of lateral cutting edge, 2.41–2.43
cutting edges of, 2.2.9–2.33
clearance surface of tooth, 2.32.–2.33
rake surface, 2.2.9–2.32.
design of shaper cutters, 2.61–2.63
deep counterbore-type, 2.61
disk-type, 2.61
for machining helical and herringbone
gears, 2.64–2.65
tooth profiles, 2.63–2.64
desired corrections to tooth profile, 2.48–2.50
generating surface of, 2.2.5–2.2.9
Cartesian coordinate system, 2.2.5–2.2.6
for machining involute shapes, 2.2.8
pitch helix angle, 2.2.8
position vector, 2.2.5–2.2.6
profile angle, 2.2.7
Shishkov’s equation of contact, 2.2.7–2.2.8
grinding of, 2.74–2.79
clearance surfaces, 2.74–2.75
rake surfaces, 2.75–2.79
improvement in geometry of clearance
surface, 2.46–2.48
kinematics of gear shaping, 2.2.3–2.2.5
principal elements, 2.2.4f
reciprocation, 2.2.3
screw motion, 2.2.3–2.2.4
vector diagram, 2.2.4f
lateral cutting edges
angle of inclination, 2.40–2.41
clearance angle, 2.43–2.48
rake angle, 2.41–2.43
overview, 2.2.3–2.79
profiling of, 2.33–2.36
angular displacement of tooth profile, 2.36
base diameter, 2.35
circular tooth thickness, 2.34–2.35
meshing of auxiliary rack surface, 2.33–2.34
profile shift correction coefficient, 2.35
tooth addendum, 2.35
tooth dedendum, 2.35
tooth thickness, 2.35–2.36
rake surface of, 2.2.9–2.32.
geometry, 2.30–2.31
helical, geometry of, 2.31–2.32.
improvement in geometry, 2.45–2.46
special designs of, 2.65–2.73
angular pitches, 2.68
combined rake surface, 2.66
improved cutting edge geometry, 2.65–2.69
for machining hardened gears, 2.70–2.72.
precision gear, 2.69–2.70
two gears, 2.67–2.69
thickness of chip cut by, 2.51–2.55
feed rate motion, 2.51
operator of rolling, 2.53–2.54
position vector, 2.53–2.54
roll angle, 2.52.
speed of instant rotation, 2.51
tooth ratio, 2.52.
tooth profiles, 2.63–2.64
chamfer, 2.63–2.64
full topping, 2.64
modified pressure angle, 2.64
protuberance, 2.64
root fillet, 2.63
semitopping, 2.63–2.64
tip relief, 2.63
typical gear shaping operations, 2.73–2.74
combination operations, 2.74
cutting two gears with different
diametral pitch, 2.74
cutting two gears with similar diametral
pitch, 2.74
cutting two gears with up and down
shaping, 2.74
length of stroke, 2.73, 2.74f
spiral infeed, 2.74
Gear shaper cutters, internal gear machining
mesh, 2.81–2.95
accuracy of shaped internal gears, 2.90–2.92.
interference of internal work gear and
shaper cutter teeth, 2.91–2.92.
tooth number, 2.92.
transverse generating pressure angle, 2.91
application of, 2.93–2.95
cutting edge geometry of, 2.85–2.86Index 743
design of shaper cutters, 2.83
enveloping shaper cutters, 2.93
generating surface of, 2.83
kinematics of shaping operation, 2.81–2.83
rotation vector, 2.81–2.82.
screw motion, 2.83
materials used in, 2.93
profiling of shape cutters
deep counterbore-type, 2.84–2.85
shank-type, 2.84–2.85
profiling of shaper cutters, 2.83–2.85
clearance surface, 2.83–2.84
generating surface, 2.83–2.84
rake surface, 2.83–2.84
thickness of chip cut by, 2.86–2.90
feed motion, 2.86
motion of cut, 2.86
operator of rolling, 2.88–2.89
Gear shaper cutters, with tilted axis of rotation,
301–314
capabilities of external intersecting-axis gear
machining mesh, 311–314
shaping conical involute gears, 311, 312.f
shaping of face gears, 311–314
generating surface of, 304–311
addendum, 310
base diameter of shaper cutter, 310
coordinate systems transformations, 306–307
coordinates of points within tooth profile,
307–308
dedendum, 310
left-hand-oriented coordinate system,
304–305
internal gear machining mesh, 343–349
axodes, 345
kinematics of internal gear machining
mesh, 343
motion of cut, 344
operating pitch surfaces, 345
rake surface, 345–346
rotation vectors, 345
shaping of external recessed tooth forms,
347–349
shaping of internal gear, 344–346
shaping of spur gear, 346–347
kinematics of shaping operation, 301–304
rake surface, 303–304
reference systems, 302.
rotation vector, 301–302.
Gear shaving cutters, 559–642.
advances in design of, 633–638
constant inclination angle within cutting
edges, 635–636
for finishing modified work gear tooth
flank, 637–638
near optimal angle of inclination of
cutting edges, 633–635
precision cutter with straight cutting
edges, 636–637
axial method of shaving process, 576–587
coordinate systems transformations, 585
cutting speed, 578–587
equivalent base diameters, 586–587
formula for cutting speed, 587
impact of crossed-axis angle, 578–579
impact of profile sliding, 580–587
impact of traverse motion, 579–580
kinematics of, 576–578
line of action, 584
workgear to shaving cutter meshing, 580,
581f, 583f
clearance surface of cutting teeth, 562.
design of shaper cutters, 566–576
design parameters, 567, 575t
resharpening, 571–576
serrations on tooth flanks, 568–571
diagonal method of shaving process,
587–602.
coordinate systems, 596–598
cutting speed, 590–592.
diagonal-underpass, 590
geometry of contact of tooth flanks,
598–602.
kinematics of, 588–589
local topology of contacting tooth flanks,
595–596
multistroke, 589
optimal kinetic parameters, 602.
optimization of kinematics, 592.–595
traverse angle, 589–590
generating surface of, 559–560
inclination angle of cutting edges, 564–568
manufacturing aspects of shaving operation,
640–641
modification of tooth form and shape, 641
plunge method of shaving processing,
619–633
circular mapping of tooth flanks, 62.9–631
cutting speed, 619
design of shaving cutters, 631–633
epicyclical motion in, 619
kinematics of, 619–62.0
part surface generation, 62.8–633
topologically modified gears, 62.1–62.8
rake surface of cutting teeth, 560–561
requirements for preshaved work gear, 639–640744 Index
Gear shaving cutters (continued)
selection of, 639
shaving of worm gear, 641–642.
tangential method of shaving process,
603–618
analytical approach, 612.–618
cutting speed, 604–605
descriptive geometry-based methods,
605–612.
kinematics of, 603–604
serrations, 605
tooth flanks, 595–596
curvature of tooth surface, 596
Dupin indicatrix of tooth surface, 596
equation of tooth flank of, 595
first fundamental form of tooth surface,
595
indicatrix of conformity, 599
local orientation, 598
maximum rate of conformity, 600
optimal design parameters, 600
principal directions, 598
second fundamental form of tooth
surface, 595
Gear tooth
engineering formula for, 695–697
for surfaces that allow sliding, 16–18
Gear tooth flank surface, natural
parameterization of, 13
Gears, 3–18
definition of, 3
tooth flanks, 6–16
types of, 3–6
cluster, 5, 6f
double-helical, 3, 6o
face, 6
helical, 3
helical rack, 3
herringbone, 3
spur, 3, 4f
straight bevel, 5–6
Generating body, 395
Generating surfaces, 353–394
auxiliary, 357–363
characteristic line E
g, 362.
coordinate systems, 357–359
coordinate systems transformations, 359
pitch diameter, 361–362.
profile angle, 362.
rack surface, 357–358
relative motion of work gear, 358
tooth flank surface, 359–360
types of, 363
base diameter, 380–381
base helix angle, 378–379, 696t
of conical gear cutting tools, 679–680
of cylindrical gear cutting tools, 673–674
definition of, 363–378
descriptive geometry-based methods, 378–381
design parameters, 363–371
base diameter of generating surface, 376–378
characteristic line, 367
complementary equations, 376–378
coordinate systems, 365
envelope to successive positions of plane
with screw motion, 365–368
helix angle, 377
principal elements of geometry, 368–370
setting angle, 374–376
straight rack, 365
disk-type gear milling cutters, 12.4–132.
for machining helical involute gears,
12.7–130
for machining spur involute gears, 12.5–12.7
equation, 371–374
auxiliary rack, 371–372.
screw motion, 372.–373
unit normal vector, 374
of gear broach, 60–61
of gear shaper cutters, 2.83
for machining involute shapes, 2.2.8
pitch helix angle, 2.2.8
profile angle, 2.2.7
Shishkov’s equation of contact, 2.2.7–2.2.8
of grinding wheel, 66–67
kinematics of external spatial gear
machining mesh, 353–357
axode of gear cutting tool, 357
center distance, 353
crossed-axis angle, 353
hyperboloid, 356
opposite-directed vector, 356
pure rolling of axodes, 354–355
rotation of cutting tool, 353
rotation of work gear, 353
rotation vectors, 353–354
vector of linear velocity of sliding of
axodes, 355
straight bevel gears with offset teeth, 32.5–32.8
of toroidal gear cutting tools, 681–686
types of, 364f, 381–392.
asymmetric tooth profile, 389–390
conical, 383–389
torus-shaped pitch surfaces, 390–392.
zero profile angle, 381–383
Gleason method, 668Index 745
Grinding, relief, 42.3–433
of assembled gear hobs, 42.5–433
position for machining, 42.5, 431f
of solid gear hobs, 42.3–42.5
technological worm, 42.8
working position, 42.5, 431f
Grinding wheel, 66–67
generating surface of, 66–67
maximal feasible outer diameter, 67
Gullet, 62.
H
Hardened gears, machining, 2.70–2.72.
Helical bevel gear, 15–16
Helical gears, 3
disk-type gear milling cutters, 12.7–130
end-type gear milling cutter, 96–100
gear milling cutter for, 12.0–12.1
internal round broach for, 169–170
involute, 10–14
shaper cutters for, 2.64–2.65
tooth flanks, 10–14, 15–16
cusps, 117–119
deviation from desired shape, 118–119
normal curvature of, 118
Helical gullet, 68
Helical rack gears, 3
Helical shaper cutters, rake surface of, 2.31–2.32.
Helix angle, 113
equation, 696t
of involute hobs, 476
Herringbone gears, 3, 2.64–2.65
High speed steels (HSS), 2.61, 2.93
Hob base diameter, 475t
Hobs for machining gears, 395–559
accuracy for machining involute gears,
433–462.
actual machining surface vs. desired
generating surface, 438–445
analytical description of actual lateral
cutting edge, 436–437
analytical description of desired lateral
cutting edge, 436
deviations of rack surface, 434f
as function of design parameters, 435–445
impact of lead angle of screw rake surface
on tooth profile deviation, 440–443
impact of pitch diameter, 445–462.
impact of rake angle on tooth profile
deviation, 438–440
kinematic geometry of involute hob,
454–462.
machining surface, 437–438
maximum deviation of hob tooth profile,
438
pitch helix angle, 447
principal design parameters of, 449–454
relative motions of work gear and hob,
446–449
clearance surface of, 411–433
clearance angle, 413, 416–42.3
cutting edges, 411
cutting of relieved clearance surface,
415–42.3
equation of desired clearance surface,
411–415
grinding of relieved clearance surface,
42.3–433
helix angle, 414
operator of linear transformation, 415
position vector, 413
rake plane, 413
reduced pitch, 413
tooth relieving operation, 415
conical hobs, 493–7
butt-end hob, 496, 496f
face hob, 496, 496f
work gear to cutting tool penetration
curve, 493, 494f
cutting edge geometry of gear hob tooth,
498–515
clearance angle, 510, 513–514
cutting edge roundness, 511
improvement of hob design, 514–515
inclination angle, 510–512.
machining zone, 500, 503–505
penetration curve, 500–503
rake angle, 509–510, 512.–513
resultant motion, 506–508
tool-in-use reference system, 505–515
unit normal vector to surface of cut, 505–506
unit tangent vector, 508
cylindrical hobs of nonstandard design,
487–493
for finishing hardened gears, 490–493
number of hob threads, 489
for smooth roughing of coarse pitch
gears, 487–489
unfolded cross section, 488f, 490f
definition of, 395
design of, 462.–498
design parameters, 462.–465
tooth profiles, 465–468
for face gear, 645–646
hobbing operations, 52.8–534746 Index
Hobs for machining gears (continued)
climb hopping, 530
conventional hopping, 530
cycles, 534–536
hob total travel distance, 536–537
hobbing time, 536–537
idle distance and neck width of cluster
gear, 537–538
prescribed value of setting angle, 555–558
setting angle, 538–540
shortest allowable approach distance,
552.–554
shortest allowed idle distance, 540–545
tolerance and shortest possible idle
distance, 545–552.
kinematic geometry of involute hob
change to normal profile angle, 454–457
correction to configuration of rake
surface, 458–560
value of center-distance in gear hobbing
operation, 457–458
multistart, 407–408
for plunge gear hobbing, 644–645
precision involute hubs with lateral cutting
edges, 468–487
analytical approach to configuration of
rake plane, 471–475
computation, 475–477
design parameters, 470–477
DG-based approach to configuration of
rake plane, 470–471
modified tooth profile, 481–487
resharpening, 477–481
rake surface of, 399–410
auxiliary axis of projections, 407
auxiliary reference systems, 401
Cartesian coordinate system, 399–400
cone angle, 407
in form of plane, 403–404
generation of, 403–410
geometry, 399–403
intermittent, 409–410
of multistart hob, 407–408
plane of projections, 407
position vectors, 401
rake angle, 403
screw rake surface, 405–407
straight-generating lines, 405–406
straight-line generator, 400
zero rake angle, 402.
for tangential gear hobbing, 643, 644f
tool-in-use reference system, 505–515
clearance angle, 510, 513–514
inclination angle, 510–512.
rake angle, 509–510, 512.–513
tooth profiles, 515–52.8
actual value of deviation, 52.7–52.8
allowed interval for profile angle, 52.7–52.8
applied reference systems, 516–517
elementary gear drive, 517–518
form diameter of gear, 517–518
limit diameter of gear, 517
maximum allowed value of modification,
516–52.2.
modification of, 517
normalized deviation, 52.2.–52.4
reduced addendum, 52.4–52.7
toroidal hobs, 497–498
transformation of generating surface into
gear tool, 395–399
feed motion, 395–396
grinding worm, 395, 396f
screw motion, 396–398
Homogenous coordinate transformation
matrices, 43–44
Homogenous coordinate vectors, 43–44
I
Inclination angle, 2.00, 511–513
Indexing, continuous, 2.5–42.
classification of gear machining meshes,
39–42.
configuration of rotation vectors, 37–39
kinematic relationships for gear machining
mesh, 32.–37
kinematics of gear machining, 42.
vector representation of gear machining
mesh, 2.5–32.
Indicatrix of conformity, 599
Integral-shank hobs, 463
Intermittent rake surface, 409–410
Internal gear machining meshes, 41f, 2.81–2.95. See
also external gear machining meshes
accuracy of shaped internal gears, 2.90–2.92.
interference of internal work gear and
shaper cutter teeth, 2.91–2.92.
tooth number, 2.92.
transverse generating pressure angle, 2.91
application of, 2.93–2.95
cutting edge geometry of, 2.85–2.86
definition of, 40
design of shaper cutters, 2.83
enveloping shaper cutters, 2.93
gear cutting tools for machining internal
gears, 687–691Index 747
gear cutting tools with enveloping
generating surface, 673–686
conical generating surface, 679–681
cylindrical generating surface, 673–679
toroidal generating surface, 681–686
generating surface of, 2.83
kinematics of shaping operation, 2.81–2.83
rotation vector, 2.81–2.82.
screw motion, 2.83
materials used in, 2.93
profiling of shaper cutters, 2.83–2.85
clearance surface, 2.83–2.84
deep counterbore-type, 2.84–2.85
generating surface, 2.83–2.84
rake surface, 2.83–2.84
shank-type, 2.84–2.85
thickness of chip cut by, 2.86–2.90
feed motion, 2.86
motion of cut, 2.86
operator of rolling, 2.88–2.89
Internal gears, 5f, 687–691
broaching, 73
examples of gear cutting tool, 689–691
generating surface of gear cutting tool, 689
geometry of, 687
hob for cutting, 690f
kinematics of machining internal gear, 687–688
shaping of, 344–346
Internal round broach
for bevel gears, 170–172.
for helical gears, 169–170
for spur gears, 169–170
Intersecting-axis gear machining mesh
gear cutting tools, for machining bevel
gears, 315–342.
gear shapers with tilted axis of rotation,
301–304, 343–349
Inverse problem of gear-cutting tool design, 2.
Involute curve, 10f
Involute function of pressure angle, 696t
Involute gears
curvature of tooth surface, 596
Dupin indicatrix of tooth surface, 596
equation of tooth flank of, 595
first fundamental form of tooth surface, 595
machining, 70–72.
cross section of auxiliary rack, 72.
gear broaching tools for, 70–72.
gear milling cutter for, 12.1
helical gears, 96–100
shapes and configurations of cutting
edges, 70–71
spur gears, 92.–100
straight lateral cutting edges, 71–72.
second fundamental form of tooth surface,
595
tooth flanks, 9–14
helical gears, 10–14
spur gears, 9–10
Involute helical gears, tooth flank, 10–14
Involute hobs, 433–462., 468–487
actual machining surface vs. desired
generating surface, 438–445
analytical approach to configuration of rake
plane, 471–475
analytical description of actual lateral
cutting edge, 436–437
analytical description of desired lateral
cutting edge, 436
change to normal profile angle, 454–457
computation, 475–477
conical hobs, 493–497
butt-end hob, 496, 496f
face hob, 496, 496f
work gear to cutting tool penetration
curve, 493, 494f
correction to configuration of rake surface,
458–560
cylindrical hobs of nonstandard design,
487–493
for finishing hardened gears, 490–493
number of hob threads, 489
for smooth roughing of coarse pitch
gears, 487–489
unfolded cross section, 488f, 490f
design parameters, 470–477
deviations of rack surface, 434f
DG-based approach to configuration of rake
plane, 470–471
as function of design parameters, 435–445
impact of lead angle of screw rake surface
on tooth profile deviation, 440–443
impact of normal profile angle, 478f
impact of number of starts, 478f
impact of pitch diameter, 445–462.
impact of rake angle on tooth profile
deviation, 438–440
impact of setting angle, 479f
machining surface, 437–438
maximum deviation of hob tooth profile, 438
modified tooth profile, 481–487
pitch helix angle, 447
principal design parameters of, 449–454, 460t
relative motions of work gear and hob, 446–449
tooth profiles, 515–52.8
actual value of deviation, 52.7–52.8748 Index
Involute hobs (continued)
allowed interval for profile angle, 52.7–52.8
applied reference systems, 516–517
elementary gear drive, 517–518
form diameter of gear, 517–518
limit diameter of gear, 517
maximum allowed value of modification,
516–52.2.
modification of, 517
normalized deviation, 52.2.–52.4
reduced addendum, 52.4–52.7
toroidal hobs, 497–498
value of center-distance in gear hobbing
operation, 457–458
Involute worm, 171–181
grinding, 178–181
machining, 172.–175
milling, 175–177
J
Jacobian matrix, 703
K
Kinematics of gear cutting, 12.3–12.4
crossed-axis angle, 12.3
feed motion, 12.3
gear milling cutters, disk-type, 12.3–12.4
pitch helix angle, 12.3
rotation vector, 12.3–12.4
Kinematics of gear machining, 19–2.4
bevel gears, 315–317
continuously indexing methods, 42.
elementary relative motions, 2.0–2.1
external spatial gears, 353–357
feasible relative motions, 2.1–2.3
gear broaching, 60f
gear machining mesh, 42.
gear milling cutters, end-type, 91, 92.f
internal gears, 687–688
invertibility of, 2.8
rolling of conjugate surfaces, 2.3–2.4
Kinematics of gear shaping
external gear machining mesh, 2.2.3–2.2.5
gear shapers with tilted axis of rotation,
301–304
internal gear machining mesh, 2.81–2.83
Kinematics of gear shaving
axial method, 576–578
diagonal method, 588–589
plunge method, 619–62.0
tangential method, 603–604
L
Lateral cutting edges
analytical descriptions of, 436–437
angle of inclination, 2.40–2.41
clearance angle of, 2.43–2.48
computation of geometry for, 2.00–2.02.
of involute hobs, 436–437
rake angle of, 2.41–2.43
Lathe, machining worm on, 172.–175
Lead, 695t
Line of action, 584
M
M-2. high-speed steel, 80
Machining meshes, 2.5
applications of, 643–662.
conical hob for palloid gear cutting,
658–662.
cutting tool for machining worm in
continuously indexing method, 653–654
cutting tools for scudding gears, 649–651
gear reinforcement by surface plastic
deformation, 657–658
hob for face gear, 645–646
hob for plunge gear hobbing, 644–645
hob for tangential gear hobbing, 643, 644f
rack shaving cutters, 656–659
shaper cutter with tilted axis of rotation,
651–653
worm-type cutting tool with continuous
helix-spiral cutting edge, 646–649
center distance, 40
crossed-axis angle, 40
external, 39–40, 41f
internal, 40, 41f
kinematics of, 42.
kinematics relationships, 32.–37
magnitude of rotation vectors, 40
planar, 39–40, 41f
spatial gear, 353–357
gear shaving cutters, 559–642.
generating surface of gear cutting tool,
353–394
hobs for machining gears, 395–559
kinematics of, 353–357
vector diagram, 2.9, 30f
vector representation of, 2.5–32.
Machining surface, of involute hobs, 437–438
Machining zone, 500, 503–505
Magnitude of rotation vectors, 40
Major section plane, 706–708Index 749
Mensnier’s formula, 117, 152.
Merchant’s formula, 511
Metric-to-pitch system conversion, 697t
Minor cutting edge approach angle, 715–716
Modified pressure angle, 2.64
Multistart hobs, 407–408
Multistroke shaving, 589
N
Natural parameterization, 13
NC grinder, 62.5–62.7
Nitriding surface treatment, 2.93
Normal circular pitch, 695t
Normal diametral pitch, 696t
Normal pitch, 475t
Normal pressure angle, 696t
Normal rake angle, 709–710
Normal section plane, 709
Normalized velocity, 34
O
Octoidal profile, 319
Oerlikon method, 668
Offset teeth, 32.5–32.8
Operating pitch diameter, 696t
Operating pressure angle, 696t
Operators of reflections, 54
Operators of rolling motion, 50–51
Operators of rotation, 46–47
Operators of screw motion, 48–49
Operators of translation, 44–45
Opposite transformation, 54
Orientation-preserving transformation, 45
Orientation-reversing transformation, 54
Outside diameter, 695t, 697t
P
Palloid gear hobbing, 658–662.
design of conical hob, 659–660
kinematics of, 660–661
machining with crowned teeth, 662.
overview, 659
Parallel-axis gear machining mesh, 183–185
gear shaper cutters, 2.2.3–2.95
external gear machining mesh, 2.2.3–2.79
internal gear machining mesh, 2.81–2.95
rack cutter, 187–2.2.2.
Part surface generation, 62.8–633
conditions for, 699–701
cylindrical gear cutting tool, 674–676
plunge shaving, 62.8–633
Penetration curve, 498–515
Pitch angle, 475t
Pitch diameter
conversion to metric system, 697t
equation, 695t
of involute hobs, 475t
Pitch helix angle
in disk-type milling cutters, 12.3–12.4
in gear broaching, 59
of gear shaper cutters, 2.2.8
Pitch line, 2.9
Pitch point, 33
Pitch-to-metric system conversion, 697t
Planar gear machining mesh, 39–40, 41f
Plane of cut, 705–706
Plastic deformation, gear reinforcement by,
657–658
Plunge shaving, 619–33. See also shaving cutters
circular mapping of tooth flanks, 62.9–631
cutting speed, 619
design of shaving cutters, 631–633
epicyclical motion in, 619
kinematics of, 619–62.0
part surface generation, 62.8–633
topologically modified gears, 62.1–62.8
auxiliary rack modification matrix, 62.7
gear modification matrix, 62.2.
generating surface of form grinding
wheel, 62.7–62.9
geometry of, 62.1–62.4
grinding of tooth flanks, 62.5–62.8
NC grinder, 62.5–62.7
tooth flanks of shaving cutter, 62.4–62.5
PM4 powder metal, 80
Position for machining, relief grinding, 42.5,
431f
Position vector
of gear shaper cutters, 2.2.5–2.2.6
involute tooth profile, 9
tooth flank of helical bevel gear, 16
tooth flank of involute spur gear, 9
tooth flank of work gear, 2.2.6–2.2.7
Pot broaching, 74
Precision gear shaper cutters, 2.69–2.70
carbide, 2.70–2.73
Pregrind rack cutter, 2.18–2.19
Pressure angle
disk-type gear milling cutters, 145–146
end-type gear milling cutters, 114
involute function of, 696t
of involute hobs, 475t
normal, 696t750 Index
Pressure angle (continued)
transverse, 696t
Profile angle
disk-type gear milling cutters, 144–145
of gear shaper cutters, 2.2.7
Profile sliding, 580–587
Progressive cutting diagram, 149
Proper post surface generation
accuracy of gears and, 2.55–2.60
fifth condition of, 2.12.–2.15, 2.55–2.58
sixth condition of, 2.15–2.18, 2.55–2.58
Protuberance, 2.64
Pull broaches, 73
Pull-up broaching, 74
Push broaches, 73
Push-up pot broaching, 74
Q
Quasi-planar gear machining meshes, 351, 669–670
R
Rack cutters, 187–2.2.2.
accuracy of machined gear, 2.12.–2.18
fifth condition of proper post surface
generation, 2.12.–2.15
gear-to-rack cutter meshing diagram,
2.14–2.15
sixth condition of proper post surface
generation, 2.15–2.18
application of, 2.18–2.19
chip thickness cut by cutting edges of,
2.07–2.12.
clearance angle, 2.02.
computation, 2.00–2.02.
cutting edge geometry of, 199–2.07
cutting motion, 2.08
direction of reciprocation, 2.07–2.08
feed motion, 2.08–2.09
inclination angle, 2.00
for lateral cutting edges, 2.00–2.02.
modification of clearance surface, 2.04–2.07
modification of rake surface, 2.03–2.04
operator of rolling, 2.11
position vector, 2.11
translational motion, 2.08
cutting edges of, 193–196
clearance surface, 195–196
rake surface, 194–195
feasible tooth profiles, 191–193
helix angle, 191
radius of pitch cylinder, 191
tooth profile angle, 191
tooth thickness of generating surface, 192.
finish, 2.18–2.19
generating surface of, 187–191
equation for, 187–190
indexing time, 2.19
methods and designs, 2.2.0–2.2.2.
pregrind, 2.18–2.19
profiling of, 196–199
analytical, 198–199
descriptive geometry-based, 197, 198f
rough, 2.18–2.19
tooth profile angle, 198–199
Rack shaving cutters, 654–657
Rake angle, 709–710
chip-flow, 717–718
definition of, 709–710
disk-type gear milling cutters, 143, 146
of gear broach, 68
of hob tooth cutting edge, 509–510, 512.–513
of lateral cutting edge, 2.41–2.43
normal, 709–710
Rake surface
of gear hobs, 399–410
auxiliary axis of projections, 407
auxiliary reference systems, 401
Cartesian coordinate system, 399–400
cone angle, 407
in form of plane, 403–404
generation of, 403–410
geometry, 399–403
intermittent, 409–410
of multistart hob, 407–408
plane of projections, 407
position vectors, 401
rake angle, 403
screw rake surface, 405–407
straight-generating lines, 405–406
straight-line generator, 400
zero rake angle, 402.
of milling cutter for machining involute
gears, 132.–134
modification of, 2.03–2.04
of rack cutter, 194–195
of shaper cutter, 2.2.9–2.32.
combined rake surface, 2.66
geometry, 2.30–2.31
grinding, 2.75–2.79
helical, geometry of, 2.31–2.32.
Ratio of translation V
cut, 59
Reciprocation, 2.07–12., 2.2.3
Reference plane of chip flow, 716–717
Reinforcement, 657–658Index 751
Relief grinding, 42.3–433
of assembled gear hobs, 42.5–433
position for machining, 42.5, 431f
of solid gear hobs, 42.3–42.5
technological worm, 42.8
working position, 42.5, 431f
Resharpening
precision involute hobs, 477–481
shaving cutters, 571–576
Resultant coordinate systems transformations,
47–48
Revacycle process, 81–90. See also broaching
cutting tools, 83–84
kinematics of, 82.f
principle of, 82.–83
profiling of bevel gear cutter, 85–90
application, 88–90
finishing teeth, 85–88
reference systems, 85f
shape of roughing teeth, 88–90
tooth space generation, 83f
Ring-type pot broaching tool, 74
Rolling motion, 50–54
of coordinate system, 50–51
over cylinder, 53
operators of, 50–51
over plane, 50
of two coordinate systems, 52.–54
Root fillet, 2.63
Rotary broaches, 80–81
clearance angle, 80
form size, 81
shank axis, 80
tool holder, 81
Rotation
frequencies, 33
inversion of, 2.9f
Rotation vector, 2.6–2.9
configuration of, 37–39
critical configuration of, 38
of disk-type milling cutter, 12.3–12.4
in gear machining mesh, 32.–33
magnitude of, 36, 40–42.
superimposition of translation vector and, 36
Rotations, 19
Rough rack cutter, 2.18–2.19
Round rack, 319
S
Scientific approach, 8
Screw surface, Archimedean, 62.–63
Screw feed motion, 12.4
Screw involute surface, 10–13, 10f, 13t
Screw motion, 2.6
about coordinate axis, 48–49
gear shaper cutters, 2.2.3–2.2.4
operators of, 48–49
parameter, 36
Screw rake surface, 405–407
Scudding, 651–653
applications of, 651
cutting tools for, 649–651
design concept for tools, 649–651
Secondary generating surface T2, 92.
Semitopping, 2.63–2.64
Serrations, 605
Setting angle, 374–376, 475t
Shank axis, 80
Shank-type shaper cutters, 2.84–2.85, 2.94f
Shaper cutters
cutting edge geometry of, 2.85–2.86
deep counterbore-type, 2.84–2.85
enveloping, 2.93, 2.95f, 346–348
interference of internal work gear and
shaper cutter teeth, 2.91–2.92.
profiling of, 2.83–2.85
clearance surface, 2.83–2.84
generating surface, 2.83–2.84
rake surface, 2.83–2.84
shank-type, 2.84–2.85, 2.94f
with tilted axis of rotation, 651–653
application, 652.–653
kinematics of shaping helical gear, 651–652.
principal design elements, 652.
tooth number, 2.92.
transverse generating pressure angle, 2.91
Sharpening of gear broaches, 66–70
generating of rake surface, 68
generating surface of grinding wheel, 66–67
helical gullet, 68
maximal feasible outer diameter of grinding
wheel, 67
schematic diagram, 69f
support at center, 67
Shave grinding, 685
Shaving cutters, 559–642.
advances in design of, 635–640
constant inclination angle within cutting
edges, 635–636
for finishing modified work gear tooth
flank, 637–638
near optimal angle of inclination of
cutting edges, 633–635
precision cutter with straight cutting
edges, 636–637752 Index
Shaving cutters (continued)
axial method, 576–587
coordinate systems transformations, 585
cutting speed, 578–587
equivalent base diameters, 586–587
formula for cutting speed, 587
impact of crossed-axis angle, 578–579
impact of profile sliding, 580–587
impact of traverse motion, 579–580
kinematics of, 576–578
line of action, 584
workgear to shaving cutter meshing, 580,
581f, 583f
clearance surface of cutting teeth, 562.
design of shaper cutters, 566–576
design parameters, 567, 575t
resharpening, 571–576
serrations on tooth flanks, 568–571
diagonal method, 587–602.
coordinate systems, 596–598
cutting speed, 590–592.
diagonal-underpass, 590
geometry of contact of tooth flanks,
598–602.
kinematics of, 588–589
local topology of contacting tooth flanks,
595–596
multistroke, 589
optimal kinetic parameters, 602.
optimization of kinematics, 592.–595
traverse angle, 589–590
generating surface of, 559–560
inclination angle of cutting edges, 562.–566
manufacturing aspects of shaving operation,
640–641
modification of tooth form and shape, 641
plunge method, 619–633
circular mapping of tooth flanks, 62.9–631
cutting speed, 619
design of shaving cutters, 631–633
epicyclical motion in, 619
kinematics of, 619–62.0
part surface generation, 62.8–633
topologically modified gears, 62.1–62.8
rack-type, 654–657
rake surface of cutting teeth, 560–561
requirements for preshaved work gear,
639–640
selection of, 639
shaving of worm gear, 641–642.
tangential method of shaving process,
603–618
analytical approach, 612.–618
cutting speed, 604–605
descriptive geometry-based methods,
605–612.
kinematics of, 603–604
serrations, 605
tooth flanks, 595–596
curvature of tooth surface, 596
Dupin indicatrix of tooth surface, 596
equation of tooth flank of, 595
first fundamental form of tooth surface,
595
indicatrix of conformity, 599
local orientation, 598
maximum rate of conformity, 600
optimal design parameters, 600
principal directions, 598
second fundamental form of tooth
surface, 595
Shear-Speed cutting, 74–80
kinematics of, 75f
principle of, 74
profiling of form tools for, 76–79
Shell-type gear hobs, 462.f, 463–464
Shishkov’s equation of contact, 2.2.7–2.2.8
Shoulder gears, tangential shaving of, 605–618
analytical approach, 612.–618
closed vector polygon, 614–615
operators of linear transformations, 613–614
optimal design parameters, 612.–615
overlap of shaving cutter over work gear,
615–618
reference systems, 612.–613
descriptive geometry-based method, 605–612.
maximum allowed diameter of cutter,
606–608
minimum permissible width of auxiliary
generating rack, 612.–614
minimum permissible width of work
gear auxiliary rack, 608–610
minimum required face width, 612.
minimum required overlap of gear and
cutter, 608–612.
Single parametric motion, 16
Skiving hobs, 463–464
Slater tools, 80–81
Sliding motion over surfaces, 2.3–2.4
Sliding vectors, 2.6
Spatial gear machining mesh, 353–357. See also
machining meshes
gear shaving cutters, 559–642.
generating surface of gear cutting tool,
353–394
hobs for machining gears, 395–559Index 753
kinematics of, 353–357
axode of gear cutting tool, 357
center distance, 353
crossed-axis angle, 353
hyperboloid, 356
opposite-directed vector, 356
pure rolling of axodes, 354–355
rotation of cutting tool, 353
rotation of work gear, 353
rotation vectors, 353–354
vector of linear velocity of sliding of
axodes, 355
Special roughing hobs, 463
Spiral bevel gears, 337–342.
clearance angle, 342.
design of cutters, 340–342.
diagrammatic arrangement of, 339f
gear machining operation, 338–340
profile angle, 341
rake angle, 342.
spiral angles, 337
Spiral gullet, 68
Spiral infeed, 2.74
Spur gears, 3, 4f
barreled, 17
crowned, 17
cusps on tooth flanks, 115–117
disk-type gear milling cutter, 12.5–12.7
gear milling cutter for, 12.0
internal round broach for, 169–170
involute, 9–10
machining, 92.–100
modified, 17
shaping of, 346–347
tooth flank, 9–10
Spur rack, 3
Stabler’s equation, 718
Staggered pattern serration, 605
Standard circular tooth thickness, 697t
Standard normal circular thickness, 696t
Standard outside diameter, 697t
Start of active profile (SAP), 9
Stick-type pot broaching tool, 74
Straight bevel gears, 5–6
cutting, 333–334
gear cutting tools, 330–333
design parameters, 330–331
tool geometry, 331–333
tool height, 330
machining of, 32.9–330
milling, 334–337
design of milling cutters, 335–336
disk-type milling cutters, 336
lateral cutting edges, 2.5–2.6
shape of finished flanks, 336–337
with offset teeth, 32.5–32.8
teeth, 32.8–33
gear cutting tools, 330–333
machining of straight bevel gears, 32.9–330
plane Ta by straight motion of cutting
edge, 32.8–32.9
tooth flank, 14–15
Straight feed motion Fc, 12.3
Straight rack, 365
Surface of tolerance, 594
Surface parameters, 703–704
Surface plastic deformation, gear reinforcement
by, 657–658
Surfaces, fundamental forms, 55–56
Surfaces that allow sliding, gear tooth, 16–18
T
Tangential shaving, 603–618. See also shaving
cutters
analytical approach, 612.–618
cutting speed, 604–605
descriptive geometry-based methods, 605–612.
kinematics of, 603–604
serrations, 605
Technological worm, 42.8
Thickness of chip cut, 2.86–2.90
feed motion, 2.86
motion of cut, 2.86
operator of rolling, 2.88–2.89
Thread whirling, 177–178
Tip relief, 2.63
Titanium nitride coating, 2.61, 2.94
Tool-in-use reference system, 505–515
Tooth depth, 62.
Tooth flanks, 6–16
of bevel gear, 14–15
circular mapping of, 62.9–631
of helical gear, 117–119
cusps, 117–119
deviation from desired shape, 118–119
normal curvature, 118
of involute helical gear, 10–14
of involute spur gears, 9–10
serrations, 568–571
of shaving cutters, 568–571, 595–596
curvature of tooth surface, 596
Dupin indicatrix of tooth surface, 596
equation of tooth flank of, 595
first fundamental form of tooth surface,
595754 Index
Tooth flanks (continued)
indicatrix of conformity, 599
local orientation, 598
maximum rate of conformity, 600
optimal design parameters, 600
principal directions, 598
second fundamental form of tooth
surface, 595
specification, engineering approach, 8
specification, scientific approach, 8
of spur gear, 115–117
waviness of, 116
Tooth profiles
asymmetric, 389–390
gear hobs, 465–468, 515–52.8
gear shaper cutters, 2.63–2.64
chamfer, 2.63–2.64
full topping, 2.64
modified pressure angle, 2.64
protuberance, 2.64
root fillet, 2.63
semitopping, 2.63–2.64
tip relief, 2.63
involute hobs, 515–52.8
rack cutters, 191–193
Tooth relieving operation, clearance surface of,
415
Top-loaded cutting diagram, 150
Topologically modified gears, 62.1–62.8
auxiliary rack modification matrix, 62.7
gear modification matrix, 62.2.
generating surface of form grinding wheel,
62.9–30
geometry of, 62.1–62.4
grinding of tooth flanks, 62.5–62.8
NC grinder, 62.5–62.7
tooth flanks of shaving cutter, 62.4–62.5
Toroidal gear cutting tools, 681–686
Toroidal gear hobs, 497–498
Torus-shaped pitch surfaces, 390–392.
Transformation matrices, 44
Transformations, coordinate-system, 43–56
conversion of coordinate system orientation,
54
direct transformation, 45
direct transformation of surfaces
fundamental forms, 55–56
homogenous coordinate transformation
matrices, 44
homogenous coordinate vectors, 43–44
opposite transformation, 54
orientation-preserving transformation, 45
orientation-reversing transformation, 54
overview, 43
resultant, 47–48
rolling motion of coordinate system, 50–51
rolling of two coordinate systems, 52.–54
rotation about coordinate axis, 46–47
screw motions about coordinate axis, 48–49
translations, 44–45
Translation vector, 2.6
magnitude of speed, 36
superimposition of rotation vector and, 36
Translational motion, 2.08
Translations, 44–45
Transverse circular pitch, 696t
Transverse pressure angle, 696t
Traverse angle, 589–590
Traverse motion, 579–580
Triparametric motion, 16
U
Unit normal vector, 374
V
Vector of instant rotation, magnitude of, 34–36
W
Work gear, 2.0–2.3
auxiliary rack, 608–610
elementary relative motions, 2.0–2.1
feasible relative motions, 2.1–2.3
Work gear to cutting tool penetration curve,
493, 494f
Work gear to generation surface mesh, 2.5
Work gear to shaving cutter meshing, 580, 581f,
583f
Working position, relief grinding, 42.5, 431f
Worm, 5, 7f
grinding, 178–181
machining in continuously indexing
method, 653–654
machining on lathe, 172.–175
milling, 175–177
technological
Worm gear, 7f
Z
Zero profile angle, 381–383
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