Table Of ContentS e c o n d E d i t i o n
Dudley ’s
H a n d b o o k o f
Practical Gear
Design and
Manufacture
S e c o n d E d i t i o n
Dudley ’s
H a n d b o o k o f
Practical Gear
Design and
Manufacture
Stephen P. Radzevich
Boca Raton London New York
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Contents
Preface..............................................................................................................................................xv
Introduction..................................................................................................................................xix
Acknowledgments.....................................................................................................................xxiii
Author...........................................................................................................................................xxv
1 Gear Design Trends................................................................................................................1
1.1. Small,.Low-Cost.Gears.for.Toys,.Gadgets,.and.Mechanisms..................................4
1.2. Appliance.Gears.............................................................................................................6
1.3. Machine.Tools.................................................................................................................7
1.4. Control.Gears..................................................................................................................8
1.5. Vehicle.Gears................................................................................................................10
1.6. Transportation.Gears...................................................................................................12
1.7. Marine.Gears................................................................................................................13
1.8. Aerospace.Gears...........................................................................................................15
1.9. Industrial.Gearing.......................................................................................................16
1.10. Gears.in.the.Oil.and.Gas.Industry............................................................................18
1.11. Mill.Gears......................................................................................................................18
1.12. External.Spur.Gears.....................................................................................................20
1.13. External.Helical.Gears.................................................................................................23
1.14. Internal.Gears...............................................................................................................24
1.15. Straight.Bevel.Gears.....................................................................................................26
1.16. Zerol.Bevel.Gears.........................................................................................................28
1.17. Spiral.Bevel.Gears........................................................................................................29
1.18. Hypoid.Gears................................................................................................................30
1.19. Face.Gears.....................................................................................................................31
1.20. Crossed-Helical.Gears.(Nonenveloping.Worm.Gears)...........................................32
1.21. Single-Enveloping.Worm.Gears.................................................................................34
1.22. Double-Enveloping.Worm.Gears...............................................................................37
1.23. Spiroid.Gears................................................................................................................39
2 Gear Types and Nomenclature...........................................................................................41
2.1. Types.of.Gears..............................................................................................................41
2.1.1. Classifications..................................................................................................41
2.1.2. Parallel.Axis.Gears.........................................................................................44
2.1.2.1. Spur.Gears........................................................................................44
2.1.2.2. Helical.Gears....................................................................................45
2.1.2.3. Internal.Gears..................................................................................47
2.1.3. Nonparallel,.Coplanar.Gears.(Intersecting.Axes)......................................48
2.1.3.1. Bevel.Gears.......................................................................................48
2.1.3.2. Face.Gears.(On-Center)...................................................................53
2.1.3.3. Conical.Involute.Gearing...............................................................54
2.1.4. Nonparallel,.Noncoplanar.Gears.(Nonintersecting.Axes).......................55
2.1.4.1. Crossed-Axis.Helical.Gears...........................................................55
2.1.4.2. Cylindrical.Worm.Gearing............................................................57
v
vi Contents
2.1.4.3. Single-Enveloping.Worm.Gearing................................................57
2.1.4.4. Double-Enveloping.Worm.Gearing..............................................59
2.1.4.5. Hypoid.Gears...................................................................................60
2.1.4.6. Spiroid.and.Helicon.Gearing.........................................................61
2.1.4.7. Face.Gears.(Off-Center)..................................................................63
2.1.5. Special.Gear.Types..........................................................................................63
2.1.5.1. Square.or.Rectangular.Gears........................................................63
2.1.5.2. Triangular.Gears.............................................................................64
2.1.5.3. Elliptical.Gears................................................................................64
2.1.5.4. Scroll.Gears......................................................................................65
2.1.5.5. Multiple-Sector.Gears.....................................................................65
2.2. Nomenclature.of.Gears...............................................................................................66
2.2.1. Spur.Gear.Nomenclature.and.Basic.Formulas...........................................68
2.2.2. Helical.Gear.Nomenclature.and.Basic.Formulas.......................................68
2.2.3. Internal.Gear.Nomenclature.and.Formulas...............................................70
2.2.4. Crossed.Helical.Gear.Nomenclature.and.Formulas..................................70
2.2.5. Bevel.Gear.Nomenclature.and.Formulas....................................................73
2.2.5.1. Straight..............................................................................................73
2.2.5.2. Spiral.................................................................................................74
2.2.5.3. Zerol..................................................................................................74
2.2.6. Worm.Gear.Nomenclature.and.Formulas..................................................78
2.2.6.1. Cylindrical.Worm.Gears................................................................80
2.2.6.2. Double-Enveloping.Worm.Gears..................................................81
2.2.7. Face.Gears........................................................................................................81
2.2.8. Spiroid.Gear.Nomenclature.and.Formulas.................................................81
2.2.9. Beveloid.Gears.................................................................................................83
3 Gear Tooth Design................................................................................................................85
3.1. Basic.Requirements.for.Gear.Tooth.Design.............................................................85
3.1.1. Definition.of.Gear.Tooth.Elements...............................................................85
3.1.2. Basic.Considerations.for.Gear.Tooth.Design..............................................88
3.1.2.1. Continuity.of.Action.......................................................................89
3.1.2.2. Conjugate.Action.............................................................................92
3.1.2.3. Pitch.Diameter.................................................................................93
3.1.2.4. Zones.in.Which.Involute.Gear.Teeth.Exist..................................94
3.1.2.5. Pointed.Teeth...................................................................................98
3.1.2.6. Undercut.........................................................................................100
3.1.3. Long-.and.Short-Addendum.Gear.Design................................................101
3.1.3.1. Addendum.Modification.for.Gears.Having.a.Few.Teeth........102
3.1.3.2. Speed-Increasing.Drives..............................................................103
3.1.3.3. Power.Drives.(Optimal.Design)..................................................103
3.1.3.4. Low-Friction.Gearing...................................................................105
3.1.4. Special.Design.Considerations...................................................................107
3.1.4.1. Interchangeability.........................................................................107
3.1.4.2. Tooth.Thickness.............................................................................107
3.1.4.3. Tooth.Profile.Modifications.........................................................109
3.1.4.4. Transverse.of.Profile.Modification..............................................109
3.1.4.5. Allowances.for.Errors.of.Gear.Manufacture.............................109
3.1.4.6. Allowances.for.Deflection.under.Load......................................109
Contents vii
3.1.4.7. Axial.Modifications.......................................................................110
3.1.4.8. Root.Fillets......................................................................................111
3.1.4.9. Effective.Outside.Diameter..........................................................113
3.1.4.10. Width.of.Tip.of.Tooth....................................................................113
3.1.4.11. Pointed.Tooth.Diameter...............................................................114
3.1.4.12. Purpose.of.Backlash......................................................................114
3.1.4.13. Backlash:.Recommended.Values.................................................115
3.2. Standard.Systems.of.Gear.Tooth.Proportions.......................................................116
3.2.1. Standard.Systems.for.Spur.Gears...............................................................116
3.2.1.1. Limitations.in.Use.of.Standard.Tables.......................................116
3.2.1.2. Standard.Tooth.Forms.That.Have.Become.Obsolete...............118
3.2.1.3. Brown.and.Sharp.System.............................................................119
3.2.1.4. AGMA.14.5°.Composite.System..................................................119
3.2.1.5. Fellows.20°.Stub.Tooth.System....................................................120
3.2.1.6. AGMA.14.5°.Full-Depth.System..................................................120
3.2.1.7. Cycloidal.Tooth.Profiles...............................................................121
3.2.1.8. Clockwork.and.Timer.Tooth.Profiles.........................................121
3.2.1.9. Specific.Spur.Gear.Calculation.Procedure................................122
3.2.1.10. Explanation.and.Discussion.of.Items.in.Table 3.5....................126
3.2.2. System.for.Helical.Gears..............................................................................128
3.2.2.1. Selection.of.Tooth.Form................................................................130
3.2.2.2. Selection.of.Helix.Angle...............................................................130
3.2.2.3. Face.Width......................................................................................131
3.2.2.4. Specific.Calculation.Procedure.for.Helical.Gears....................132
3.2.3. System.for.Internal.Gears............................................................................133
3.2.3.1. Special.Calculations......................................................................135
3.2.3.2. Specific.Calculation.Procedure.for.Internal.Gears...................136
3.2.4. Standard.Systems.for.Bevel.Gears..............................................................136
3.2.4.1. Discussion.of.20°.Straight.Bevel.Gear.System..........................137
3.2.4.2. Discussion.of.Spiral.Bevel.Gear.System.....................................137
3.2.4.3. Discussion.of.the.Zerol.Bevel.Gear.System...............................138
3.2.4.4. Special.Tooth.Forms......................................................................138
3.2.4.5. Limitations.in.20°.Straight.Bevel.Gear.System.........................139
3.2.4.6. Limitations.in.Spiral.Bevel.Gear.System...................................141
3.2.4.7. Limitations.in.Zerol.Bevel.Gear.System....................................141
3.2.4.8. General.Comments.......................................................................141
3.2.5. Standard.Systems.for.Worm.Gears............................................................143
3.2.5.1. General.Practice.............................................................................143
3.2.5.2. Basic.Tooth.Forms.for.Worm.Gearing........................................144
3.2.5.3. Specific.Calculations.for.Worm.Gears.......................................145
3.2.6. Standard.System.for.Face.Gears.................................................................146
3.2.6.1. Pinion.Design................................................................................148
3.2.6.2. Face.Gear.Design...........................................................................148
3.2.7. System.for.Spiroid.and.Helicon.Gears.......................................................149
3.2.7.1. Spiroid.Gearing.............................................................................151
3.2.7.2. Helicon.Gearing............................................................................154
3.2.7.3. Detailed.Calculations.of.Spiroid.and.Helicon.Tooth.Data......155
3.3. General.Equations.Relating.to.Center.Distance....................................................156
3.3.1. Center.Distance.Equations..........................................................................156
viii Contents
3.3.2. Standard.Center.Distance............................................................................159
3.3.3. Standard.Pitch.Diameters............................................................................160
3.3.4. Operating.Pitch.Diameters..........................................................................160
3.3.5. Operating.Pressure.Angle...........................................................................161
3.3.6. Operating.Center.Distance..........................................................................162
3.3.7. Center.Distance.for.Gears.Operating.on.Nonparallel.
Nonintersecting.Shafts................................................................................162
3.3.8. Center.Distance.for.Worm.Gearing...........................................................162
3.3.9. Reasons.for.Nonstandard.Center.Distances.............................................163
3.3.10. Nonstandard.Center.Distances...................................................................163
3.4. Elements.of.Center.Distance.....................................................................................165
3.4.1. Effects.of.Tolerances.on.Center.Distance..................................................165
3.4.2. Machine.Elements.that.Require.Consideration.in.Critical.Center.
Distance.Applications..................................................................................168
3.4.3. Control.of.Backlash.......................................................................................171
3.4.4. Effects.of.Temperature.on.Center.Distance..............................................172
3.4.5. Mounting.Distance.......................................................................................174
4 Preliminary Design Considerations................................................................................177
4.1. Stress.Formulas:.Calculation....................................................................................177
4.1.1. Calculated.Stresses.......................................................................................177
4.1.2. Gear.Design.Limits.......................................................................................181
4.1.3. Gear.Strength.Calculations.........................................................................181
4.1.3.1. Worst.Load.....................................................................................184
4.1.3.2. Stress.Concentration.....................................................................185
4.1.3.3. Load.Distribution..........................................................................186
4.1.3.4. Dynamic.Load...............................................................................186
4.1.3.5. Finite.Element................................................................................187
4.1.4. Gear.Surface.Durability.Calculations........................................................188
4.1.4.1. Hertz.Derivations..........................................................................189
4.1.4.2. K-Factor.Derivations.....................................................................192
4.1.4.3. Worst-Load.Position......................................................................193
4.1.4.4. Endurance.Limit............................................................................193
4.1.4.5. Lubrication.Regimens...................................................................194
4.1.5. Gear.Scoring..................................................................................................198
4.1.5.1. Hot.and.Cold.Scoring...................................................................198
4.1.5.2. PVT.Formula..................................................................................199
4.1.5.3. Flash.Temperature........................................................................202
4.1.5.4. Scoring.Criterion...........................................................................203
4.1.6. Thermal.Limits..............................................................................................205
4.1.6.1. Thermal.Limits.at.Regular.Speed...............................................205
4.1.6.2. Thermal.Limits.at.High.Speed....................................................206
4.2. Stress.Formulas:.Special.Considerations................................................................206
4.2.1. Gear.Specifications.......................................................................................207
4.2.2. Size.of.Spur.and.Helical.Gears.by.Q-Factor.Method..............................207
4.2.2.1. Face-Width.Considerations..........................................................210
4.2.2.2. Weight.from.Volume.....................................................................211
4.2.3. Indexes.of.Tooth.Loading............................................................................213
4.2.4. Estimating.Spur.and.Helical.Gear.Size.by.K-Factor................................214
Contents ix
4.2.5. Estimating.Bevel.Gear.Size.........................................................................215
4.2.6. Estimating.Worm.Gear.Size........................................................................220
4.2.7. Estimating.Spiroid.Gear.Size......................................................................223
4.3. Data.Needed.for.Gear.Drawings.............................................................................223
4.3.1. Gear.Dimensional.Data................................................................................223
4.3.2. Gear.Tooth.Tolerances..................................................................................226
4.3.3. Gear.Material.and.Heat.Treatment.Data...................................................234
4.3.4. Enclosed.Gear.Unit.Requirements.............................................................241
5 Design Formulas..................................................................................................................243
5.1. Calculation.of.Gear.Tooth.Data...............................................................................243
5.1.1. Number.of.Pinion.Teeth..............................................................................243
5.1.2. Hunting.Teeth...............................................................................................248
5.1.3. Spur.Gear.Tooth.Proportions......................................................................249
5.1.4. Root.Fillet.Radii.of.Curvature.....................................................................251
5.1.5. Long-Addendum.Pinions............................................................................253
5.1.6. Tooth.Thickness............................................................................................257
5.1.6.1. Backlash..........................................................................................257
5.1.6.2. Tolerances.and.Tooth.Thickness.................................................258
5.1.7. Chordal.Dimensions....................................................................................258
5.1.8. Degrees.Roll.and.Limit.Diameter..............................................................260
5.1.9. Form.Diameter.and.Contact.Ratio.............................................................264
5.1.9.1. Form.Diameter...............................................................................264
5.1.9.2. Contact.Ratio..................................................................................265
5.1.10. Spur.Gear.Dimension.Sheet........................................................................266
5.1.11. Internal.Gear.Dimension.Sheet..................................................................268
5.1.12. Helical.Gear.Tooth.Proportions..................................................................272
5.1.13. Helical.Gear.Dimension.Sheet....................................................................277
5.1.14. Bevel.Gear.Tooth.Proportions.....................................................................279
5.1.15. Straight.Bevel.Gear.Dimension.Sheet........................................................283
5.1.16. Spiral.Bevel.Gear.Dimension.Sheet...........................................................285
5.1.17. Zerol.Bevel.Gear.Dimension.Sheet............................................................288
5.1.18. Hypoid.Gear.Calculations...........................................................................289
5.1.19. Face.Gear.Calculations.................................................................................290
5.1.20. Crossed-Helical.Gear.Proportions.............................................................293
5.1.21. Single-Enveloping.Worm.Gear.Proportions.............................................296
5.1.22. Single-Enveloping.Worm.Gears.................................................................297
5.1.23. Double-Enveloping.Worm.Gears................................................................299
5.2. Gear.Rating.Practice..................................................................................................303
5.2.1. General.Considerations.in.Rating.Calculations.......................................303
5.2.1.1. Calculation.Procedure..................................................................305
5.2.1.2. Grades.of.Material.Quality..........................................................306
5.2.1.3. Reliability.of.Gears........................................................................307
5.2.2. General.Formulas.for.Tooth.Bending.Strength.and.Tooth.Surface.
Durability.......................................................................................................307
5.2.2.1. Strength.Formula..........................................................................308
5.2.2.2. Durability.Formula.......................................................................308
5.2.2.3. Rating.Curves.of.Stress.versus.Cycles.......................................313
5.2.2.4. Rating.Bevel.Gears........................................................................315
x Contents
5.2.3. Geometry.Factors.for.Strength...................................................................316
5.2.3.1. Lack.of.Load.Sharing....................................................................317
5.2.3.2. Helical.Gears.with.Narrow.Face.Width.....................................317
5.2.3.3. Geometry.Factors.for.Strength.for.Some.Standard.Designs.....318
5.2.4. Overall.De-Rating.Factor.for.Strength.......................................................319
5.2.4.1. Application.Factor.K ....................................................................322
a
5.2.4.2. Load.Distribution.Factor.K ........................................................322
m
5.2.4.3. Effect.of.Helix.Error.and.Shaft.Misalignment..........................324
5.2.4.4. Aspect.Ratio.Effects......................................................................327
5.2.4.5. Load.Distribution.Factor.K .for.Bevel.Gears............................328
m
5.2.4.6. Size.Factor.K..................................................................................330
s
5.2.4.7. Dynamic.Load.Factors.K .and.C ................................................330
v v
5.2.5. Geometry.Factors.for.Durability................................................................333
5.2.6. Overall.De-Rating.Factor.for.Surface.Durability.....................................335
5.2.6.1. Size.Factor.C..................................................................................336
s
5.2.6.2. Complementary.Considerations.................................................338
5.2.7. Load.Rating.of.Worm.Gearing...................................................................339
5.2.7.1. Crossed-Helical.Gear.Durability................................................339
5.2.7.2. Cylindrical.Worm.Gear.Durability............................................341
5.2.7.3. Double-Enveloping.Worm.Gear.Durability..............................346
5.2.7.4. Comparison.of.Double-Enveloping.and.Cylindrical.
Worm.Gear.Rating.Procedures...................................................349
5.2.8. Design.Formulas.for.Scoring......................................................................350
5.2.8.1. Hot.Scoring....................................................................................350
5.2.8.2. Cold.Scoring...................................................................................355
5.2.8.3. Design.Practice.to.Handle.Scoring.............................................360
5.2.9. Trade.Standards.for.Rating.Gears..............................................................361
5.2.10. Vehicle.Gear.Rating.Practice.......................................................................363
5.2.11. Marine.Gear.Rating.Practice.......................................................................365
5.2.12. Oil.and.Gas.Industry.Gear.Rating.............................................................367
5.2.13. Aerospace.Gear.Rating.Practice.................................................................368
6 Gear Materials......................................................................................................................371
6.1. Steels.for.Gears...........................................................................................................371
6.1.1. Mechanical.Properties.................................................................................371
6.1.2. Heat-Treating.Techniques............................................................................375
6.1.3. Heat-Treating.Data........................................................................................379
6.1.4. Hardness.Tests..............................................................................................381
6.2. Localized.Hardening.of.Gear.Teeth........................................................................383
6.2.1. Carburizing...................................................................................................383
6.2.2. Nitriding.........................................................................................................389
6.2.2.1. Features.of.Nitriding.Process......................................................389
6.2.2.2. Nitride.Case.Depth.......................................................................392
6.2.3. Induction.Hardening.of.Steel......................................................................393
6.2.3.1. Induction.Hardening.by.Scanning.............................................395
6.2.3.2. Load-Carrying.Capacity.of.Induction-Hardened.Gear.Teeth......398
6.2.4. Flame.Hardening.of.Steel............................................................................400
6.2.5. Combined.Heat.Treatments........................................................................401
6.2.6. Metallurgical.Quality.of.Steel.Gears..........................................................402
