كتاب Kinematic Geometry of Surface Machining

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Kinematic Geometry of Surface Machining
Stephen P. Radzevich

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Contents
Preface .
Author .
Acknowledgments
Part I Basics
1 Part Surfaces: Geometry
1.1 Elements of Differential Geometry of Surfaces .
1.2 On the Difference between Classical Differential Geometry
and Engineering Geometry
1.3 On the Classifcation of Surfaces .
1.3.1 Surfaces That Allow Sliding over Themselves
1.3.2 Sculptured Surfaces .
1.3.3 Circular Diagrams .
1.3.4 On Classifcation of Sculptured Surfaces .
References
2 Kinematics of Surface Generation .
2.1 Kinematics of Sculptured Surface Generation .
2.1.1 Establishment of a Local Reference System .
2.1.2 Elementary Relative Motions .
2.2 Generating Motions of the Cutting Tool .
2.3 Motions of Orientation of the Cutting Tool .
2.4 Relative Motions Causing Sliding of a Surface over Itself .
2.5 Feasible Kinematic Schemes of Surface Generation .
2.6 On the Possibility of Replacement of Axodes with Pitch Surfaces.
2.7 Examples of Implementation of the Kinematic Schemes
of Surface Generation
References
3 Applied Coordinate Systems and Linear Transformations
3.1 Applied Coordinate Systems
3.1.1 Coordinate Systems of a Part Being Machined .
3.1.2 Coordinate System of Multi-Axis Numerical Control
(NC) Machine .
3.2 Coordinate System Transformation
3.2.1 Introduction
3.2.1.1 Homogenous Coordinate Vectors
3.2.1.2 Homogenous Coordinate Transformation
Matrices of the Dimension 4 × 4 .
3.2.2 Translations .
3.2.3 Rotation about a Coordinate Axis .
3.2.4 Rotation about an Arbitrary Axis through the Origin .
3.2.5 Eulerian Transformation .
3.2.6 Rotation about an Arbitrary Axis Not through the Origin .
3.2.7 Resultant Coordinate System Transformation
3.2.8 An Example of Nonorthogonal Linear Transformation
3.2.9 Conversion of the Coordinate System Orientation .
3.3 Useful Equations
3.3.1 RPY-Transformation
3.3.2 Rotation Operator
3.3.3 A Combined Linear Transformation
3.4 Chains of Consequent Linear Transformations and a Closed
Loop of Consequent Coordinate System Transformations
3.5 Impact of the Coordinate System Transformations on
Fundamental Forms of the Surface .
References
Part II Fundamentals
4 The Geometry of Contact of Two Smooth, Regular Surfaces
4.1 Local Relative Orientation of a Part Surface and of the Cutting Tool
4.2 The First-Order Analysis: Common Tangent Plane
4.3 The Second-Order Analysis .
4.3.1 Preliminary Remarks: Dupin’s Indicatrix
4.3.2 Surface of Normal Relative Curvature .
4.3.3 Dupin’s Indicatrix of Surface of Relative Curvature
4.3.4 Matrix Representation of Equation of the Dupin’s
Indicatrix of the Surface of Relative Normal Curvature
4.3.5 Surface of Relative Normal Radii of Curvature
4.3.6 Normalized Relative Normal Curvature .
4.3.7 Curvature Indicatrix .
4.3.8 Introduction of the Ir?k(P/T) Characteristic Curve
4.4 Rate of Conformity of Two Smooth, Regular Surfaces
in the First Order of Tangency .
4.4.1 Preliminary Remarks
4.4.2 Indicatrix of Conformity of the Surfaces P and T .
4.4.3 Directions of the Extremum Rate of Conformity
of the Surfaces P and T
4.4.4 Asymptotes of the Indicatrix of Conformity CnfR (P/T) .
4.4.5 Comparison of Capabilities of the Indicatrix of
Conformity CnfR (P/T) and of Dupin’s Indicatrix of the
Surface of Relative Curvature
4.4.6 Important Properties of the Indicatrix
of Conformity CnfR (P/T) .
4.4.7 The Converse Indicatrix of Conformity of the Surfaces
P and T in the First Order of Tangency
4.5 Plücker’s Conoid: More Characteristic Curves .
4.5.1 Plücker’s Conoid .
4.5.1.1 Basics .
4.5.1.2 Analytical Representation
4.5.1.3 Local Properties .
4.5.1.4 Auxiliary Formulas .
4.5.2 Analytical Description of Local Topology of the
Smooth, Regular Surface P
4.5.2.1 Preliminary Remarks
4.5.2.2 Plücker’s Conoid
4.5.2.3 Plücker’s Curvature Indicatrix
4.5.2.4 AnR (P)-Indicatrix of the Surface P
4.5.3 Relative Characteristic Curves
4.5.3.1 On a Possibility of Implementation of
Two of Plücker’s Conoids
4.5.3.2 AnR(P/T)-Relative Indicatrix of the Surfaces
P and T
4.6 Feasible Kinds of Contact of the Surfaces P and T .
4.6.1 On a Possibility of Implementation of the Indicatrix of
Conformity for Identifcation of Kind of Contact of the
Surfaces P and T
4.6.2 Impact of Accuracy of the Computations on the Desired
Parameters of the Indicatrices of Conformity CnfR(P/T)
4.6.3 Classifcation of Kinds of Contact of the Surfaces P and T .
References .
5 Profiling of the Form-Cutting Tools of the Optimal Design .
5.1 Profling of the Form-Cutting Tools for Sculptured
Surface Machining
5.1.1 Preliminary Remarks .
5.1.2 On the Concept of Profling the Optimal
Form-Cutting Tool
5.1.3 R-Mapping of the Part Surface P on the Generating
Surface T of the Form-Cutting Tool .
5.1.4 Reconstruction of the Generating Surface T of the
Form-Cutting Tool from the Precomputed Natural
Parameterization .
5.1.5 A Method for the Determination of the Rate
of Conformity Functions F 1, F 2, and F 3 .
5.1.6 An Algorithm for the Computation of the Design
Parameters of the Form-Cutting Tool
5.1.7 Illustrative Examples of the Computation of the
Design Parameters of the Form-Cutting Tool .
5.2 Generation of Enveloping Surfaces
5.2.1 Elements of Theory of Envelopes .
5.2.1.1 Envelope to a Planar Curve
5.2.1.2 Envelope to a One-Parametric Family of Surfaces
5.2.1.3 Envelope to a Two-Parametric Family of Surfaces .
5.2.2 Kinematical Method for the Determining
of Enveloping Surfaces .
5.3 Profling of the Form-Cutting Tools for Machining Parts
on Conventional Machine Tools .
5.3.1 Two Fundamental Principles by Theodore Olivier
5.3.2 Profling of the Form-Cutting Tools for Single-Parametric
Kinematic Schemes of Surface Generation .
5.3.3 Profling of the Form-Cutting Tools for Two-Parametric
Kinematic Schemes of Surface Generation .
5.3.4 Profling of the Form-Cutting Tools for Multiparametric
Kinematic Schemes of Surface Generation .
5.4 Characteristic Line E of the Part Surface P and of the Generating
Surface T of the Cutting Tool .
5.5 Selection of the Form-Cutting Tools of Rational Design .
5.6 The Form-Cutting Tools Having a Continuously
Changeable Generating Surface
5.7 Incorrect Problems in Profling the Form-Cutting Tools
5.8 Intermediate Conclusion
References .
6 The Geometry of the Active Part of a Cutting Tool
6.1 Transformation of the Body Bounded by the Generating Surface
T into the Cutting Tool .
6.1.1 The First Method for the Transformation of the
Generating Body of the Cutting Tool into the
Workable Edge Cutting Tool .
6.1.2 The Second Method for the Transformation of the
Generating Body of the Cutting Tool into the
Workable Edge Cutting Tool .
6.1.3 The Third Method for the Transformation of the
Generating Body of the Cutting Tool into the
Workable Edge Cutting Tool .
6.2 Geometry of the Active Part of Cutting Tools in the
Tool-in-Hand System
6.2.1 Tool-in-Hand Reference System .
6.2.2 Major Reference Planes: Geometry of the Active Part of a
Cutting Tool Defned in a Series of Reference Planes .
6.2.3 Major Geometric Parameters of the Cutting Edge
of a Cutting Tool
6.2.3.1 Main Reference Plane
6.2.3.2 Assumed Reference Plane
6.2.3.3 Tool Cutting Edge Plane .
6.2.3.4 Tool Back Plane
6.2.3.5 Orthogonal Plane
6.2.3.6 Cutting Edge Normal Plane
6.2.4 Analytical Representation of the Geometric Parameters
of the Cutting Edge of a Cutting Tool .
6.2.5 Correspondence between Geometric Parameters of
the Active Part of Cutting Tools That Are Measured in
Different Reference Planes .
6.2.6 Diagrams of Variation of the Geometry of the Active
Part of a Cutting Tool
6.3 Geometry of the Active Part of Cutting Tools in the
Tool-in-Use System .
6.3.1 The Resultant Speed of Relative Motion in the Cutting
of Materials .
6.3.2 Tool-in-Use Reference System
6.3.3 Reference Planes
6.3.3.1 The Plane of Cut Is Tangential to the Surface
of Cut at the Point of Interest M
6.3.3.2 The Normal Reference Plane
6.3.3.3 The Major Section Plane
6.3.3.4 Correspondence between the Geometric
Parameters Measured in Different
Reference Planes .
6.3.3.5 The Main Reference Plane .
6.3.3.6 The Reference Plane of Chip Flow .
6.3.4 A Descriptive-Geometry-Based Method for the
Determination of the Chip-Flow Rake Angle .
6.4 On Capabilities of the Analysis of Geometry of the Active
Part of Cutting Tools
6.4.1 Elements of Geometry of Active Part of a Skiving Hob .
6.4.2 Elements of Geometry of the Active Part of a Cutting Tool
for Machining Modifed Gear Teeth .
6.4.3 Elements of Geometry of the Active Part of a
Precision Involute Hob
6.4.3.1 An Auxiliary Parameter R .
6.4.3.2 The Angle fr between the Lateral Cutting Edges
of the Hob Tooth .
6.4.3.3 The Angle x of Intersection of the Rake Surface
and of the Hob Axis of Rotation .
References
7 Conditions of Proper Part Surface Generation
7.1 Optimal Workpiece Orientation on the Worktable
of a Multi-Axis Numerical Control (NC) Machine
7.1.1 Analysis of a Given Workpiece Orientation
7.1.2 Gaussian Maps of a Sculptured Surface P and of the
Generating Surface T of the Cutting Tool
7.1.3 The Area-Weighted Mean Normal to a
Sculptured Surface P
7.1.4 Optimal Workpiece Orientation .
7.1.5 Expanded Gaussian Map of the Generating Surface
of the Cutting Tool
7.1.6 Important Peculiarities of Gaussian Maps
of the Surfaces P and T .
7.1.7 Spherical Indicatrix of Machinability
of a Sculptured Surface
7.2 Necessary and Suffcient Conditions of Proper
Part Surface Generation
7.2.1 The First Condition of Proper Part Surface Generation
7.2.2 The Second Condition of Proper Part Surface Generation .
7.2.3 The Third Condition of Proper Part Surface Generation
7.2.4 The Fourth Condition of Proper Part Surface Generation
7.2.5 The Fifth Condition of Proper Part Surface Generation .
7.2.6 The Sixth Condition of Proper Part Surface Generation .
7.3 Global Verifcation of Satisfaction of the Conditions
of Proper Part Surface Generation
7.3.1 Implementation of the Focal Surfaces
7.3.1.1 Focal Surfaces .
7.3.1.2 Cutting Tool (CT)-Dependent Characteristic
Surfaces .
7.3.1.3 Boundary Curves of the CT-Dependent
Characteristic Surfaces
7.3.1.4 Cases of Local-Extremal Tangency of the Surfaces
P and T
7.3.2 Implementation of R-Surfaces .
7.3.2.1 Local Consideration
7.3.2.2 Global Interpretation of the Results
of the Local Analysis .
7.3.2.3 Characteristic Surfaces of the Second Kind .
7.3.3 Selection of the Form-Cutting Tool of Optimal Design
7.3.3.1 Local KLR-Mapping of the Surfaces P and T
7.3.3.2 The Global KGR-Mapping of the Surfaces P and T
7.3.3.3 Implementation of the Global KGR-Mapping .
7.3.3.4 Selection of an Optimal Cutting Tool
for Sculptured Surface Machining
References .
8 Accuracy of Surface Generation .
8.1 Two Principal Kinds of Deviations of the Machined Surface
from the Nominal Part Surface .
8.1.1 Principal Deviations of the First Kind .
8.1.2 Principal Deviations of the Second Kind
8.1.3 The Resultant Deviation of the Machined Part Surface
8.2 Local Approximation of the Contacting Surfaces P and T .
8.2.1 Local Approximation of the Surfaces P and T
by Portions of Torus Surfaces
8.2.2 Local Confguration of the Approximating Torus Surfaces .
8.3 Computation of the Elementary Surface Deviations .
8.3.1 Waviness of the Machined Part Surface
8.3.2 Elementary Deviation hss of the Machined Surface .
8.3.3 An Alternative Approach for the Computation
of the Elementary Surface Deviations .
8.4 Total Displacement of the Cutting Tool with Respect
to the Part Surface .
8.4.1 Actual Confguration of the Cutting Tool
with Respect to the Part Surface .
8.4.2 The Closest Distance of Approach between
the Surfaces P and T .
8.5 Effective Reduction of the Elementary Surface Deviations
8.5.1 Method of Gradient
8.5.2 Optimal Feed-Rate and Side-Step Ratio
8.6 Principle of Superposition of Elementary Surface Deviations .
References .
Part III Application
9 Selection of the Criterion of Optimization .
9.1 Criteria of the Effciency of Part Surface Machining
9.2 Productivity of Surface Machining
9.2.1 Major Parameters of Surface Machining Operation
9.2.2 Productivity of Material Removal
9.2.2.1 Equation of the Workpiece Surface .
9.2.2.2 Mean Chip-Removal Output .
9.2.2.3 Instantaneous Chip-Removal Output
9.2.3 Surface Generation Output .
9.2.4 Limit Parameters of the Cutting Tool Motion
9.2.4.1 Computation of the Limit Feed-Rate Shift .
9.2.4.2 Computation of the Limit Side-Step Shift
9.2.5 Maximal Instantaneous Productivity of Surface
Generation
9.3 Interpretation of the Surface Generation Output
as a Function of Conformity
References .
10 Synthesis of Optimal Surface Machining Operations
10.1 Synthesis of Optimal Surface Generation: The Local Analysis
10.1.1 Local Synthesis .
10.1.2 Indefniteness .
10.1.3 A Possibility of Alternative Optimal Confgurations
of the Cutting Tool 4
10.1.4 Cases of Multiple Points of Contact of the Surfaces P and T . 4
10.2 Synthesis of Optimal Surface Generation: The Regional Analysis .4
10.3 Synthesis of Optimal Surface Generation: The Global Analysis .4
10.3.1 Minimization of Partial Interference
of the Neighboring Tool-Paths 4
10.3.2 Solution to the Boundary Problem .4
10.3.3 Optimal Location of the Starting Point .4
10.4 Rational Reparameterization of the Part Surface 4
10.4.1 Transformation of Parameters 4
10.4.2 Transformation of Parameters in Connection
with the Surface Boundary Contour 4
10.5 On a Possibility of the Differential Geometry/Kinematics
(DG/K)-Based Computer-Aided Design/Computer-Aided
Manufacturing (CAD/CAM) System for Optimal Sculptured
Surface Machining . 4
10.5.1 Major Blocks of the DG/K-Based CAD/CAM System . 4
10.5.2 Representation of the Input Data . 4
10.5.3 Optimal Workpiece Confguration 4
10.5.4 Optimal Design of the Form-Cutting Tool .4
10.5.5 Optimal Tool-Paths for Sculptured Surface Machining .4
10.5.6 Optimal Location of the Starting Point . 4
References 4
11 Examples of Implementation of the Differential Geometry/
Kinematics (DG/K)-Based Method of Surface Generation 4
11.1 Machining of Sculptured Surfaces on a Multi-Axis Numerical
Control (NC) Machine .4
11.2 Machining of Surfaces of Revolution 4
11.2.1 Turning Operations 4
11.2.2 Milling Operations . 4
11.2.3 Machining of Cylinder Surfaces .4
11.2.4 Reinforcement of Surfaces of Revolution 4
11.3 Finishing of Involute Gears 4
References 4
Conclusion 4
Notation 4

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