P1:GDZ/SPH P2:GDZ CB672/Litvin-Sample-DVR CB672/Litvin CB672/Litvin-v2.cls April20,2004 10:8 This page intentionally left blank ii P1:GDZ/SPH P2:GDZ CB672/Litvin-Sample-DVR CB672/Litvin CB672/Litvin-v2.cls April20,2004 10:8 GEAR GEOMETRY AND APPLIED THEORY SecondEdition Revisedandexpanded,GearGeometryandAppliedTheory,2ndedition,cov- ersthetheory,design,geometry,andmanufactureofalltypesofgearsandgear drives. Gear Geometry and Applied Theory is an invaluable reference for de- signers, theoreticians, students, and manufacturers. This new edition includes advancesingeartheory,gearmanufacturing,andcomputersimulation.Among thenewtopicsare(1)newgeometryformodifiedspurandhelicalgears,face-gear drives,andcycloidalpumps;(2)newdesignapproachesforone-stageplanetary gear trains and spiral bevel gear drives; (3) an enhanced approach for stress analysisofgeardriveswithFEM;(4)newmethodsofgrindingface-geardrives, generating double-crowned pinions, and generating new types of helical gears; (5)broadapplicationofsimulationofmeshingandTCA;and(6)newtheorieson thesimulationofmeshingformulti-bodysystems,detectionofcaseswhereinthe contactlinesongeneratingsurfacesmayhavetheirownenvelope,anddetection andavoidanceofsingularitiesofgeneratedsurfaces. Faydor L. Litvin is Director of the Gear Research Center and Distinguished ProfessorEmeritusintheDepartmentofMechanicalandIndustrialEngineering, UniversityofIllinoisatChicago.Heholdspatentsfortwenty-fiveinventions,and hewasrecognizedasInventoroftheYearbytheUniversityofIllinoisatChicago in2001. Alfonso Fuentes is Associate Professor of Mechanical Engineering at the PolytechnicUniversityofCartagena. i P1:GDZ/SPH P2:GDZ CB672/Litvin-Sample-DVR CB672/Litvin CB672/Litvin-v2.cls April20,2004 10:8 ii P1:GDZ/SPH P2:GDZ CB672/Litvin-Sample-DVR CB672/Litvin CB672/Litvin-v2.cls April20,2004 10:8 Gear Geometry and Applied Theory SECOND EDITION Faydor L. 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P1:GDZ/SPH P2:GDZ CB672/Litvin-Sample-DVR CB672/Litvin CB672/Litvin-v2.cls April20,2004 10:8 Contents ForewordbyGrazianoCurti pagexii Preface xiv Acknowledgments xv 1 CoordinateTransformation 1 1.1 HomogeneousCoordinates 1 1.2 CoordinateTransformationinMatrixRepresentation 2 1.3 RotationAboutanAxis 6 1.4 RotationalandTranslational4×4Matrices 14 1.5 ExamplesofCoordinateTransformation 15 1.6 ApplicationtoDerivationofCurves 24 1.7 ApplicationtoDerivationofSurfaces 28 2 RelativeVelocity 33 2.1 VectorRepresentation 33 2.2 MatrixRepresentation 39 2.3 ApplicationofSkew-SymmetricMatrices 41 3 Centrodes,Axodes,andOperatingPitchSurfaces 44 3.1 TheConceptofCentrodes 44 3.2 PitchCircle 49 3.3 OperatingPitchCircles 50 3.4 AxodesinRotationBetweenIntersectedAxes 51 3.5 AxodesinRotationBetweenCrossedAxes 52 3.6 OperatingPitchSurfacesforGearswithCrossedAxes 56 4 PlanarCurves 59 4.1 ParametricRepresentation 59 4.2 RepresentationbyImplicitFunction 60 4.3 TangentandNormaltoaPlanarCurve 60 4.4 CurvatureofPlanarCurves 68 5 Surfaces 78 5.1 ParametricRepresentationofSurfaces 78 5.2 CurvilinearCoordinates 78 5.3 TangentPlaneandSurfaceNormal 79 v P1:GDZ/SPH P2:GDZ CB672/Litvin-Sample-DVR CB672/Litvin CB672/Litvin-v2.cls April20,2004 10:8 vi Contents 5.4 RepresentationofaSurfacebyImplicitFunction 82 5.5 ExamplesofSurfaces 82 6 ConjugatedSurfacesandCurves 97 6.1 EnvelopetoaFamilyofSurfaces:NecessaryConditions ofExistence 97 6.2 BasicKinematicRelations 102 6.3 ConditionsofNonundercutting 103 6.4 SufficientConditionsforExistenceofanEnvelope toaFamilyofSurfaces 107 6.5 ContactLines;SurfaceofAction 110 6.6 EnvelopetoFamilyofContactLinesonGenerating Surface(cid:2) 112 1 6.7 FormationofBranchesofEnvelopetoParametric FamiliesofSurfacesandCurves 114 6.8 Wildhaber’sConceptofLimitContactNormal 118 6.9 FilletGeneration 119 6.10 Two-ParameterEnveloping 124 6.11 AxesofMeshing 128 6.12 KnotsofMeshing 134 6.13 Problems 137 7 CurvaturesofSurfacesandCurves 153 7.1 Introduction 153 7.2 SpatialCurvein3D-Space 153 7.3 SurfaceCurves 164 7.4 FirstandSecondFundamentalForms 175 7.5 PrincipalDirectionsandCurvatures 180 7.6 Euler’sEquation 188 7.7 GaussianCurvature;ThreeTypesofSurface Points 189 7.8 Dupin’sIndicatrix 193 7.9 GeodesicLine;SurfaceTorsion 194 8 MatingSurfaces:CurvatureRelations,ContactEllipse 202 8.1 Introduction 202 8.2 BasicEquations 203 8.3 PlanarGearing:RelationBetweenCurvatures 204 8.4 DirectRelationsBetweenPrincipalCurvatures ofMatingSurfaces 218 8.5 DirectRelationsBetweenNormalCurvatures ofMatingSurfaces 226 8.6 DiagonalizationofCurvatureMatrix 231 8.7 ContactEllipse 234 9 ComputerizedSimulationofMeshing andContact 241 9.1 Introduction 241 9.2 PredesignofaParabolicFunctionofTransmission Errors 242 9.3 LocalSynthesis 245 P1:GDZ/SPH P2:GDZ CB672/Litvin-Sample-DVR CB672/Litvin CB672/Litvin-v2.cls April20,2004 10:8 Contents vii 9.4 ToothContactAnalysis 249 9.5 ApplicationofFiniteElementAnalysisfor Design ofGearDrives 257 9.6 EdgeContact 260 10 SpurInvoluteGears 267 10.1 Introduction 267 10.2 GeometryofInvoluteCurves 268 10.3 GenerationofInvoluteCurvesbyTools 273 10.4 ToothElementProportions 278 10.5 MeshingofInvoluteGearwithRack-Cutter 280 10.6 RelationsBetweenToothThicknessesMeasured onVariousCircles 285 10.7 MeshingofExternalInvoluteGears 287 10.8 ContactRatio 292 10.9 NonstandardGears 294 11 InternalInvoluteGears 304 11.1 Introduction 304 11.2 GenerationofGearFillet 305 11.3 ConditionsofNonundercutting 309 11.4 InterferencebyAssembly 314 12 NoncircularGears 318 12.1 Introduction 318 12.2 CentrodesofNoncircularGears 318 12.3 ClosedCentrodes 323 12.4 EllipticalandModifiedEllipticalGears 326 12.5 ConditionsofCentrodeConvexity 329 12.6 ConjugationofanEccentricCircularGearwith aNoncircularGear 330 12.7 IdenticalCentrodes 331 12.8 DesignofCombinedNoncircularGearMechanism 333 12.9 GenerationBasedonApplicationofNoncircular Master-Gears 335 12.10 EnvelopingMethodforGeneration 336 12.11 EvoluteofToothProfiles 341 12.12 PressureAngle 344 Appendix12.A:DisplacementFunctionsforGeneration byRack-Cutter 345 Appendix12.B:DisplacementFunctionsforGeneration byShaper 348 13 CycloidalGearing 350 13.1 Introduction 350 13.2 GenerationofCycloidalCurves 350 13.3 EquationsofCycloidalCurves 354 13.4 Camus’TheoremandItsApplication 355 13.5 ExternalPinGearing 359 13.6 InternalPinGearing 365 P1:GDZ/SPH P2:GDZ CB672/Litvin-Sample-DVR CB672/Litvin CB672/Litvin-v2.cls April20,2004 10:8 viii Contents 13.7 OvercentrodeCycloidalGearing 367 13.8 Root’sBlower 369 14 InvoluteHelicalGearswithParallelAxes 375 14.1 Introduction 375 14.2 GeneralConsiderations 375 14.3 ScrewInvoluteSurface 377 14.4 MeshingofaHelicalGearwithaRack 382 14.5 MeshingofMatingHelicalGears 392 14.6 ConditionsofNonundercutting 396 14.7 ContactRatio 398 14.8 ForceTransmission 399 14.9 ResultsofToothContactAnalysis(TCA) 402 14.10 Nomenclature 403 15 ModifiedInvoluteGears 404 15.1 Introduction 404 15.2 AxodesofHelicalGearsandRack-Cutters 407 15.3 Profile-CrownedPinionandGearToothSurfaces 411 15.4 ToothContactAnalysis(TCA)ofProfile-Crowned PinionandGearToothSurfaces 414 15.5 LongitudinalCrowningofPinionbyaPlungingDisk 419 15.6 GrindingofDouble-CrownedPinionbyaWorm 424 15.7 TCAofGearDrivewithDouble-CrownedPinion 430 15.8 UndercuttingandPointing 432 15.9 StressAnalysis 435 16 InvoluteHelicalGearswithCrossedAxes 441 16.1 Introduction 441 16.2 AnalysisandSimulationofMeshingofHelicalGears 443 16.3 SimulationofMeshingofCrossedHelicalGears 452 16.4 GenerationofConjugatedToothSurfacesofCrossed HelicalGears 455 16.5 DesignofCrossedHelicalGears 458 16.6 StressAnalysis 465 Appendix16.A:DerivationofShortestCenterDistancefor CanonicalDesign 467 Appendix16.B:DerivationofEquationofCanonicalDesign f(γ ,α ,λ ,λ )=0 472 o on b1 b2 Appendix16.C:RelationsBetweenParametersα andα 473 pt pn Appendix16.D:DerivationofEquation(16.5.5) 473 Appendix16.E:DerivationofAdditionalRelationsBetween α andα 474 ot1 ot2 17 NewVersionofNovikov–WildhaberHelicalGears 475 17.1 Introduction 475 17.2 AxodesofHelicalGearsandRack-Cutter 478 17.3 ParabolicRack-Cutters 479 17.4 Profile-CrownedPinionandGearToothSurfaces 482