Advances in Gear Theory and Gear Cutting Tool Design

Introduction
Historical Background
Uniqueness of This Publication
Intended Audience
Organization of This Book
I. Accomplishments in the Theory of Gearing
II. Gear Manufacturing Methods
III. Gear Transmissions
IV. In Memoriam of Professor Dmitry T. Babichev
Contents
About the Editors
About the Contributors
Part I: Accomplishments in the Theory of Gearing
Chapter 1: Fundamental Laws of Gearing
1.1 Introduction
1.2 Conditions to Be Fulfilled by Pair of Mating Gears: The Present-Day Practice
1.3 The Law of Contact of Gear and Mating Pinion Tooth Flanks: The First Fundamental Law of Gearing
1.4 Conjugate Action Law: The Second Fundamental Law of Gearing
1.4.1 Equivalent Pulley-and-Belt Transmission
1.4.2 Camus-Euler-Savary Theorem
1.4.3 Euler-Savary Equation
1.5 The Law of Equal Base Pitches of a Gear and a Mating Pinion Tooth Flanks: The Third Fundamental Law of Gearing
1.6 The Correlation of the Results Obtained with the Results Obtained in the Previous Studies
1.7 Concluding Remarks
References
Bibliography
Chapter 2: Gearing Theory Development: Geometry-Kinematic Concepts
2.1 Introduction
2.2 Development of the Shaping Theory Based on New Geometric Images and Concepts
2.2.1 Fundamentals of the Gearing and Shaping Alternative Theory
2.2.2 Multi-parameter Gearings
2.2.3 Development of the Foundations for Gearing Geometry Numerical Modeling
2.3 Development of Methods for Gearing Analysis and Synthesis
2.3.1 Quality Characteristics of Gearing
2.3.2 Higher Kinematic Pairs with a Maximum Load Capacity
2.3.3 Kinematic Methods for Analyzing the Loading Intensity of Cutting Elements in Technological Gearings
2.3.4 Determining the Penetration Speed and Acceleration
2.3.5 Kinematic Profiling Method
2.3.6 Surface Curvature Study
2.4 Conclusion
References
Chapter 3: The Key Mistake in Generation of Conjugate Curves and Surfaces
3.1 Introduction
3.2 Conventional Approach for Generating Conjugate Curves and Surfaces
3.2.1 The Origin and Evolution
3.2.2 Kinematic Method for the Determination of Conjugate Profiles
3.2.3 Differential-Geometric Method for the Determination of Conjugate Profiles
3.3 The Main Reason for Incorrectness of Conventional Method for Generating Conjugate Curves and Surfaces
3.3.1 Basics: Equivalent Pulley-and-Belt Transmission
3.3.2 Equivalent Pulley-and-Belt Transmission
3.3.3 Line of Action and Path of Contact in a Gear Pair
3.3.4 Principal Features of Kinematics of a Gear Pair
3.4 How the Problem Can Be Resolved: Conjugate Action Law (in Parallel-Axes Gearing)
3.5 Examples of Application of the Reported Results
3.5.1 General Comments on Reversibly-Enveloping Surfaces
3.5.2 Illustrative Examples
3.6 The Relationship of the Obtained Results with Previous Research
3.7 Conclusion
References
Bibliography
Chapter 4: Meshing Limit Line of the Archimedes Worm Drive
4.1 Introduction
4.2 Equation and Unit Normal Vector of Archimedes Helicoid
4.2.1 Equation of Archimedes Helicoid
4.2.2 Unit Normal Vector of Archimedes Helicoid
4.3 Meshing Function and Tooth Surface Equation of Archimedes Worm Gear
4.3.1 Equation and Unit Normal Vector of Worm Helicoid Family Formed During Engagement
4.3.2 Meshing Function of Archimedes Worm Drive
4.3.3 Equation of the Worm Gear Tooth Surface
4.4 Meshing Limit Line of Archimedes Worm Drive
4.4.1 Existence of Meshing Limit Line and Its Equations
4.4.2 Computing Method of Meshing Limit Line
4.5 Numerical Case Study
4.5.1 Main Parameters of Worm Pair
4.5.2 Numerical Results and Discussion
4.6 Conclusions
References
Part II: Gear Manufacturing Methods
Chapter 5: Gear Cutting with Disk-Shaped Milling Cutters
5.1 Introduction
5.2 Gear Cutting Processes
5.2.1 Machining with Profile-Dependent Tools
5.2.2 Machining with Profile-Independent Tools
5.2.3 Advantages and Disadvantages of the Gear Cutting Methods
5.3 Strategies for Gear Machining with Disc-Shaped Milling Tools
5.4 Mathematical Model of the Form-Shaping Kinematics
5.4.1 Generalized Form-Shaping Kinematics
5.4.2 Mathematical Model of the Form-Shaping
Simulation of the Form-Shaping Process
5.5 Conclusion
References
Chapter 6: A Novel Design of Cutting Tool for Efficient Finishing of G-Rotors
6.1 Introduction
6.1.1 Properties and Design of G-Rotor Pair
6.1.2 Determining the Existant Conditions of the Curves Outlining the Working Profiles in G-Rotor Pair
6.1.3 Synthesis of Worm Tool Profiles for Highly Efficient Satellite Machining Technology
6.1.4 A New Tool for High-Performance Finishing of Spur Wheels
6.2 Conclusion
References
Chapter 7: Interactive Control of the Teeth Gear Shaping in the Cutting Tools Design
7.1 Introduction
7.2 Bezier Curves
7.3 Generating Surface
7.4 Gear Geometry
7.5 Parameters of Teeth Surfaces of Cut Gears
7.6 Interactive Control of the Shaping Process for Kinematic Shape Generating Schemes of the 2nd Class
7.7 Geometric and Kinematic Parameters of Gears and Their Visual Analysis
7.7.1 Relative Sliding Velocity of Working Srfaces
7.7.2 The Total Velocity of Contact Points Movement in the Direction Perpendicular to the Contact Line
7.7.3 Specific Sliding Coefficients
7.7.4 The Angle Between the Relative Velocity and the Direction of the Contact Line
7.7.5 The Reduced Curvature of the Teeth Surfaces in the Direction Perpendicular to the Contact Line
7.8 Conclusion
References
Chapter 8: Sinusoidal Gears and Alternative Method of Tooth Generation
8.1 Introduction: Advantages and Disadvantages of Traditional Involute Gears
8.2 Comparative Investigation of Involute and Sinusoidal Spur Gears by the Operating Parameters
8.2.1 Friction Forces in Gear Mating
8.2.2 Stress and Strain Simulation
8.3 Generating Radial-Circular Method (RCM) for Sinusoidal Gear Cutting - Tool Design
8.4 Hobbing and Radial Circular Gear Cutting Processes Simulation
8.4.1 3D Chips Modeling
8.4.2 The Cutting Force and Spatial Loading in the Gear Cutting Processes Modeling
8.5 Formed Teeth Surfaces Roughness
8.6 The Sinusoidal Gears and RC Method Limitations
8.7 The RC Method Improvement Capabilities
8.8 Conclusions
References
Chapter 9: Design of Technological Systems for Gear Finishing
9.1 Introduction
9.2 Principles of Technological Systems Design
9.3 Synthesized Technological Systems
9.3.1 Systems with a Rigid Kinematic Connection
9.3.2 Systems with Free Rolling
9.3.3 Technological Systems of Combined Machining
9.3.4 Multi-Tool Setup Systems
9.3.5 Selective Tracking Systems
9.3.6 Shaping Systems for Gear Tools
9.4 Conclusion
References
Part III: Gear Transmissions
Chapter 10: Calculation of Gear Trains in Transmission Systems of Vehicles
10.1 Introduction
10.2 Belarusian Scientific School on Gears and Transmissions
10.2.1 Creation of the Theory of Transport and Traction Machines Based on Dynamic Models of Their Interacting Units
10.2.2 Statistical Regularities of the Load Mode of Vehicle Transmissions
10.2.3 Development of Probabilistic Calculations of Machine Parts
10.3 Modern Stage: Lifetime Mechanics of Machines
10.3.1 The Features of Lifetime Mechanics of Machines
10.3.2 LMM and Digitalization
10.3.3 Architectonics of the Informational Model of a Gear Train in Vehicle Transmission System
10.4 Main Components of the Gear Train Information Model
10.4.1 Synthesis of Kinematic Diagrams
10.4.2 Evaluation of Kinematic Diagrams
10.4.3 Kinematic and Quasi-Static Calculations
10.4.4 Dynamics
10.4.5 Strength as Lifetime Under Defined (Given) Operation Conditions
10.4.6 Reliability Calculation: General Approach
10.4.7 Reliability Calculation: Special Approach
10.4.8 Diagnostics
10.4.9 Lifetime Expense and PHM
10.5 Conclusion
References
Chapter 11: Multispeed Planetary-Layshaft Transmissions with Multipower Flow
11.1 Introduction
11.2 Representation of Transmission Mechanisms by Generalized Diagrams
11.2.1 Planetary Transmissions
11.2.2 Layshaft Transmissions
11.3 Structural Diagrams Classification and Operating Modes of Planetary-Layshaft Transmissions with Multipower Flows
11.4 Estimation of the Total Amount of Speeds in Planetary-Layshaft Transmissions with Multipower Flow
11.5 Synthesis Methodology of Planetary-Layshaft Transmission with Multipower Flow
11.5.1 Conditions and Restrictions Adopted in the Synthesis of Transmissions Kinematic Diagrams
11.5.2 Parametric Synthesis Technique
11.6 Synthesis of Transmissions with Two Power Flows
11.6.1 Synthesis of an Eight-Speed ID Transmission
11.6.2 Synthesis of a 12-Speed DO Transmission
11.7 Synthesis of Transmissions with Three Power Flows by DD and IDD Structures
11.7.1 Synthesis of a 14-Speed DD Transmission with A1 and B1 Node Points
11.7.2 Synthesis of an 11-Speed DDO Transmission
11.8 Synthesis of Transmissions with Three Power Flows and Single Transition Shifts between Neighbor Speeds
11.8.1 Synthesis of a IDD Transmission with Single Transition Shifts
11.8.2 Synthesis of an IDDO Transmission with Single Transition Shifts
11.9 Conclusion
References
Chapter 12: Multiparameter Gears and Gear-Type Variators
12.1 Introduction
12.1.1 General Classification of Multiparameter Gears and Gear-Type Variators
12.1.2 Structure and Design of Multiparameter Gears
12.1.3 Structure and Design of Gear Variators
12.1.4 Gear (Gear-Lever) Variators with One Pair of Meshing Wheels
12.1.5 Gear Variators with Multiple Pairs of Engaging Wheels
Kinematically Accurate Variator
Approximate Variators: Impulse Type
12.1.6 Features of Contact and Selection of Parameters of Teeth of Multiparameter Gears
12.1.7 Multiparameter Gears and Variators Cogwheel Tooth Shaping
12.2 Conclusion
References
Chapter 13: Generalizing Structural Unified Model of the Synthesis of Links of Flat-Toothed Gearing Systems
13.1 Actuality. Formulation of the Issue and the Purpose of the Study
13.1.1 Actuality
13.1.2 Formulation of the Issue
13.1.3 Purpose of the Study
13.2 Literature Survey and Objectives of the Study
13.2.1 Literature Survey
13.2.2 Main Tasks of Studies
13.3 The Body of Research
13.3.1 Development of a Generalized Structural Diagram of Theoretical and Technological Shaping
13.4 Conclusions and Program of Further Research
References
Chapter 14: Evolution, State of the Art, and Trends to Improve Gear Tooth Strength
14.1 Introduction
14.2 Evolution of Gear Meshing: Development of the Gear Theory and Technology
14.3 Raw Materials for the Production of Gears
14.4 The Main Processes of Gear Machining
14.5 Application of Surface Engineering Provisions to Gears
14.6 Processes for Gear Tooth Flank Hardening
14.7 Conclusions
References
Part IV: In Memoriam of Professor Dmitry T. Babichev
Chapter 15: Carefully, Scrupulously, Responsibly: In Memoriam of Professor Dmitry T. Babichev
15.1 Introduction
15.2 Global Frustrated Project: Gear Train: From ``Birth´´ to ``Death´´
15.3 A Suggestion was Entertained
15.4 IFToMM Workshop on History of MMS
15.5 Meetings with BDT in Izhevsk
15.6 International Conference ``KOD-2018´´
15.7 The 15th IFToMM World Congress
15.8 Conclusions
References
Bibliography
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.2.2. Vector Product (or Cross Product) of Two Vectors
A.2.3. Triple Scalar Product of Three Vectors
A.2.4. Triple Vector Product of Three Vectors
A.2.5. Lagrange Equation for Vectors
A.3. On the Similarity and Difference Between Vectors and Matrices
Appendix B: Elements of 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 a Gear and a Mating Pinion Tooth Flanks
C.1. Local Relative Orientation at a Point of Contact of a Gear and a Mating Pinion Tooth Flanks
C.2. The Second-Order Analysis: Planar Characteristic Images
C.2.1. Preliminary Remarks
C.2.2. Matrix Representation of Equation of Dupin indicatrix at a Point of a Gear Tooth Flank
C.3. Degree of Conformity at a Point of Contact of a Gear and a Mating Pinion Tooth Flanks (in the First Order of Tangency)
C.3.1. Preliminary Remarks
C.3.2. Indicatrix of Conformity at a Point of Contact of a Gear and a Mating Pinion Tooth Flanks
C.3.3. Directions of Extremum Degree of Conformity at a Point of Contact of a Gear and a Mating Pinion Tooth Flanks
C.3.4. Important Properties of Indicatrix of Conformity at Point of Contact of a Gear and a Mating Pinion Tooth Flanks
C.3.5. Converse Indicatrix of Conformity at Point of Contact of a Gear and a Mating Pinion Tooth Flanks
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
D.1.6. Rotation About an Arbitrary Axis Not Through the Origin
D.1.6. 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 Rolling Motion of a Coordinate System
D.2.3. Linear Transformation Describing 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 Fundamental Forms of the Surface
Appendix E: Closest Distance of Approach Between a Gear and a Mating Pinion Tooth Flanks
Index