Table Of Content

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 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 CRC Press is an imprint of the Taylor & Francis Group, an informa business CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2012 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 2011922 International Standard Book Number-13: 978-1-4398-6602-3 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material repro- duced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copy- right.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identifica- tion and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com 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

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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 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 CRC Press is an imprint of the Taylor & Francis Group, an informa business CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2012 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 2011922 International Standard Book Number-13: 978-1-4398-6602-3 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material repro- duced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copy- right.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identifica- tion and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com 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

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