Novikov/Conformal Gearing: Scientific Theory and Practice

Description

Introduction Historical Background Uniqueness of This Publication Intended Audience Organization of This Book Contents Editor and Contributors Chapter 1: Novikov Gearing Is a Kind of Involute Gearing 1.1 Introduction 1.2 Historical Overview 1.3 Principal Design Features of Novikov Gearing 1.3.1 Gear Vector Diagram of a Novikov Gear Pair 1.3.2 Plane of Action in Parallel-Axes Novikov Gearing 1.3.3 A Desirable Line of Contact in Parallel-Axes Gearing 1.3.4 Design Features of Novikov Gearing 1.3.5 Principal Design Parameters of Novikov Gearing 1.4 High-Conformal Gearing 1.4.1 Critical Degree of Conformity in Novikov Gearing 1.4.2 Minimum Required Degree of Conformity at the Point of Contact of Two Interacting Tooth Flanks 1.5 Conformal Gearing with Intermediate Balls 1.6 Conclusion References Bibliography Chapter 2: Meshing Theory for Abnormal Novikov Helical Gears 2.1 Introduction 2.2 Equation and Normal Vector for a Normal Circular-Arc Helical Surface 2.2.1 A Mathematical Model of the Left Convex Tooth Surface of Normal Circular-Arc Gear 1 2.2.2 Mathematical Model of the Left Concave Tooth Surface of Normal Circular-Arc Gear 2 2.3 System of Nonlinear Equations to Determine Instantaneous Contact Point 2.3.1 Equation and Normal Vector of Surface Families in a Static Coordinate System 2.3.2 Derivation and Elimination of Tooth Surface Contact Equations 2.4 Instantaneous Transmission Ratio of a Gear Pair 2.5 Fundamental Quantities of Two Tooth Surfaces 2.6 Moving Frames on Two Tooth Surfaces 2.7 Curvature Parameters of Two Tooth Surfaces 2.8 Relative Curvature Parameters of the Gear Drive 2.9 Relative Principal Curvature and the Relative Principal Direction of the Gear Drive 2.10 Numerical Example Study 2.10.1 Main Technical Parameters 2.10.2 Calculating Method for Instantaneous Contact Point 2.10.3 Numerical Results 2.11 Conclusion References Bibliography Chapter 3: Helical Bevel Novikov Gears 3.1 Introduction 3.2 Mathematical Model of a Gear Mesh 3.2.1 Coordinate Systems 3.2.2 Tooth Surfaces 3.2.3 Modifications 3.3 Tooth Contact Analysis 3.3.1 Meshing Equations 3.3.2 Ease-Off Topography and Transmission Error 3.3.3 Contact Pattern 3.4 Results 3.4.1 Ideal Gear Pair 3.4.2 Effect of Tooth Surface Modifications 3.4.3 Effect of Gear Position Errors 3.4.4 Comparison with a Conventional Helical Bevel Gear Pair 3.5 Conclusion References Chapter 4: Hyperboloidal-Type Normal Circular-Arc Gearing 4.1 Introduction 4.2 Basic Principle of Molding-Surface Conjugation 4.2.1 Molding Surface 4.2.2 Conditional Equation of Molding-Surface Conjugation 4.2.3 Structural Condition of Molding-Surface Conjugation 4.2.4 General Principle of Normal Circular-Arc Gearing 4.3 Geometry of Hyperboloidal-Type Normal Circular-Arc Gears 4.3.1 Determination of Conjugate Directrices 4.3.2 Mathematical Models of Conjugate Tooth Surfaces 4.3.3 Induced Curvatures of Mating Tooth Surfaces 4.4 An Integrated Manufacturing Software System for HNCGing 4.4.1 Functional Framework 4.4.2 Three-Dimensional Modeling 4.4.3 Adaptive Tool Path Programming 4.4.4 Simulation of Meshing and Contact 4.5 Conclusion References Chapter 5: Modern Methods of Estimating and Increasing the Load-Bearing Capacity of Novikov Gearing 5.1 Bending Stresses 5.2 Effective Contact Voltages 5.3 Stiffness of the Teeth References Chapter 6: Some Features of the Contact Strength of Novikov Gearing 6.1 Conclusions References Chapter 7: Tooth Relieving of Worm Hobs for Cutting Novikov Gears with Double Lines of Action 7.1 Introduction 7.2 Novikov Gearing: Parameters of the Basic Rack 7.3 Worm Hob Parameters 7.4 Parameters of Radial-Axial Relieving 7.5 Calculation of the Angle φc Installation of the Relieving Support of the Machine 7.6 Choice of the Designed Points to Construct Designed Normals 7.7 Determination of the Angle βw of Inclination of the Grinding Wheel Axis 7.8 Choice of the Angle φ0 for Relieving the Tooth Flanks 7.9 Profiling of Grinding Wheels 7.10 Sensitivity to the Hob Regrinding 7.11 Conclusion References Chapter 8: Mikhail L. Novikov: The Inventor of Novikov Gear System 8.1 Introduction 8.2 Novikov Gear System 8.2.1 A Brief Biographical Sketch of Dr. M.L. Novikov 8.2.2 Kinematics and Geometry of Novikov Gearing 8.3 Wildhaber Gearing 8.3.1 A Brief Biographical Sketch of Dr. E. Wildhaber 8.3.2 Kinematics and Geometry of Wildhaber Gearing 8.4 Fundamental Differences Between Novikov Gearing and Wildhaber Gearing 8.5 Probable Reasons for the Appearance of the Wrong Term ``Wildhaber-Novikov Gearing´´ 8.6 Future Developments in the Realm of Conformal/Novikov Gearing 8.7 Concise Information About the Private Life of M.L. Novikov 8.8 Conclusion References Bibliography Chapter 9: Poor Understanding of the Scientific Theory of Gearing by the Majority of Gear Scientists and Engineers 9.1 Conclusion References Chapter 10: Gear Manufacturing Accuracy Prediction, Control, and Management 10.1 Introduction 10.2 Modeling the Main Regularities of the Processes of Real Shaping of Teeth and Development of the Methods for Analytical Fo... 10.2.1 Development of a System Model of the Process of Real Shaping of Teeth 10.2.1.1 Derivation of the General Equations of the Real Profiles of the Teeth of Machined Gears 10.2.1.2 Establishing the Relationship Between Coordinate Increments and Standardized Accuracy Indicators of Gears 10.2.1.3 Methods for Determining the Numerical Values of the Functions of Reduced Primary Errors 10.2.2 Methods of Analytical Forecasting of Normalized Errors of Gears 10.2.2.1 Calculation-Probability Method 10.2.2.2 Computational-Adaptive Method 10.3 Methods and Systems for Controlling the Accuracy of Gears 10.3.1 A Full Range of Possible Ways to Control the Accuracy of Gears 10.3.2 Analysis of Measurement Errors and the Degree of Risk of Missing a Defective Part in Various Control Systems 10.3.3 Element-by-Element Control Systems 10.3.4 Factor Control Systems 10.3.5 Combined Control Systems 10.4 Technological Methods of Controlling the Accuracy of Machining Gears 10.4.1 The General Characteristics of the Methods of Controlling the Accuracy of Machining Gears 10.4.2 Accuracy Management at the Process Level 10.4.2.1 Possibilities of Accuracy Control Using Two- and Three-Tool Processing Methods 10.4.2.2 Precision Control by Changing the Positions of the Base Surfaces of Gears 10.4.2.3 Influence of Process Input Parameters on the Final Accuracy of Gears 10.4.3 Operational Accuracy Control 10.4.4 Precision Control at the Machining Transition Level 10.4.5 Control of Accuracy at the Level of Passage during Mechanical Processing of Teeth 10.4.5.1 Analysis of Factors that Allow Variation 10.4.5.2 Compensatory Methods of Adaptive Accuracy Control 10.4.6 The Principles of Optimization of the Accuracy of Gears 10.5 Conclusion References Chapter 11: Elliptical Gear Drives 11.1 Introduction 11.2 Geometry: Basic Equations 11.3 Kinematics and Dynamics of Elliptical Gear Drives 11.4 Pressure Angles in Elliptical Gear Drives 11.5 Tooth Load in Elliptical Gear Drives 11.6 Calculation of the Tooth Strength of Elliptical Gears Using the Colloquial Calculations of Involute Spur Gears 11.7 Example of the Design of an Elliptical Gear Drive 11.8 Conclusion References Appendices Appendix A: Elements of Vector Calculus A.1 Fundamental Properties of Vectors A.1.1 Addition A.1.2 Equality A.1.3 Negation A.1.4 Subtraction A.1.5 Scalar Multiplication A.2 Mathematical Operations over Vectors A.2.1 Components of Vectors A.2.2 Scalar Product (or Dot Product) of Two Vectors A.2.3 Vector Product (or Cross Product) of Two Vectors A.2.4 Triple Scalar Product of Three Vectors A.2.5 Triple Vector Product of Three Vectors A.2.6 Lagrange Equation for Vectors A.3 Similarity and Difference Between Vectors and Matrices Appendix B: Elements of the Differential Geometry of Surfaces B.1 Specification of a Gear Tooth Flank B.2 Tangent Vectors and Tangent Plane; Unit Normal Vector B.3 Local Frame B.4 Fundamental Forms of a Surface B.5 Principal Directions on a Gear Tooth Flank B.6 Curvatures at a Point of a Part Surface B.7 Illustrative Example B.8 Few More Useful Equations Appendix C: Contact Geometry of the Tooth Flanks of a Gear and a Mating Pinion C.1 Local Relative Orientation at a Point of Contact of the Tooth Flanks of a Gear and a Mating Pinion C.2 The Second-Order Analysis: Planar Characteristic Images C.2.1 Preliminary Remarks: Dupin Indicatrix C.2.2 Matrix Representation of the Equation of a Dupin Indicatrix at a Point of a Gear Tooth Flank C.3 Degree of Conformity at a Point of Contact of the Tooth Flanks of a Gear and a Mating Pinion (in the First Order of Tangen... C.3.1 Preliminary Remarks C.3.2 The Indicatrix of Conformity at a Point of Contact of the Tooth Flanks of a Gear and a Mating Pinion C.3.3 Directions of the Extremum Degree of Conformity at a Point of Contact of the Tooth Flanks of a Gear and a Mating Pinion C.3.4 Important Properties of the Indicatrix of Conformity at a Point of Contact of the Tooth Flanks of a Gear and a Mating P... C.3.5 Converse Indicatrix of Conformity at a Point of Contact of the Tooth Flanks of a Gear and a Mating Pinion Appendix D: Applied Coordinate Systems and Linear Transformations D.1 Coordinate System Transformation D.1.1 Homogeneous Coordinate Vectors D.1.2 Homogeneous Coordinate Transformation Matrices of the Dimension 4 x 4 D.1.3 Translations D.1.4 Rotation About a Coordinate Axis D.1.5 Rotation About an Arbitrary Axis Through the Origin Conventional Approach Eulerian Transformation D.1.6 Rotation About an Arbitrary Axis Not Through the Origin D.1.7 Resultant Coordinate System Transformation D.2 Complex Coordinate System Transformation D.2.1 Linear Transformation Describing a Screw Motion About a Coordinate Axis D.2.2 Linear Transformation Describing the Rolling Motion of a Coordinate System D.2.3 Linear Transformation Describing the Rolling of Two Coordinate Systems D.2.4 Coupled Linear Transformation D.2.5 An Example of Non-orthogonal Linear Transformation D.2.6 Conversion of a Coordinate System Hand D.3 Useful Equations D.3.1 RPY Transformation D.3.2 Operator of Rotation About an Axis in Space D.3.3 Combined Linear Transformation D.4 Chains of Consequent Linear Transformations and a Closed Loop of Consequent Coordinate System Transformations D.5 Impact of the Coordinate System Transformations on the Fundamental Forms of the Surface Appendix E: Closest Distance of Approach Between the Tooth Flanks of a Gear and a Mating Pinion Appendix F: Selected Bibliography on Novikov/Conformal Gearing Index