CRC Press Kinematic Geometry of Surface Machining potx

KINEMATIC GEOMETRY OF SURFACE MACHINING © 2008 by Taylor & Francis Group, LLC KINEMATIC GEOMETRY OF SURFACE MACHINING Stephen P. Radzevich CRC Press is an imprint of the Taylor & Francis Group, an informa business Boca Raton London New York © 2008 by Taylor & Francis Group, LLC CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487‑2742 © 2008 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 Printed in the United States of America on acid‑free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number‑13: 978‑1‑4200‑6340‑0 (Hardcover) This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. 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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 identification and explanation without intent to infringe. Library of Congress Cataloging‑in‑Publication Data Radzevich, S. P. (Stephen Pavlovich) Kinematic geometry of surface machining / Stephen P. Radzevich. p. cm. Includes bibliographical references and index. ISBN 978‑1‑4200‑6340‑0 (alk. paper) 1. Machinery, Kinematics of. I. Title. TJ175.R345 2008 671.3’5‑‑dc22 2007027748 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com © 2008 by Taylor & Francis Group, LLC Dedication To my son Andrew © 2008 by Taylor & Francis Group, LLC Contents Preface xv Author xxv Acknowledgments xxvii Part I Basics 1 Part Surfaces: Geometry 3 1.1 Elements of Differential Geometry of Surfaces 3 1.2 On the Difference between Classical Differential Geometry and Engineering Geometry 14 1.3 On the Classication of Surfaces 17 1.3.1 Surfaces That Allow Sliding over Themselves 17 1.3.2 Sculptured Surfaces 18 1.3.3 Circular Diagrams 19 1.3.4 On Classication of Sculptured Surfaces 24 References 25 2 Kinematics of Surface Generation 27 2.1 Kinematics of Sculptured Surface Generation 29 2.1.1 Establishment of a Local Reference System 30 2.1.2 Elementary Relative Motions 33 2.2 Generating Motions of the Cutting Tool 34 2.3 Motions of Orientation of the Cutting Tool 39 2.4 Relative Motions Causing Sliding of a Surface over Itself 42 2.5 Feasible Kinematic Schemes of Surface Generation 45 2.6 On the Possibility of Replacement of Axodes with Pitch Surfaces 51 2.7 Examples of Implementation of the Kinematic Schemes of Surface Generation 53 References 59 3 Applied Coordinate Systems and Linear Transformations 63 3.1 Applied Coordinate Systems 63 3.1.1 Coordinate Systems of a Part Being Machined 63 3.1.2 Coordinate System of Multi-Axis Numerical Control (NC) Machine 64 3.2 Coordinate System Transformation 65 3.2.1 Introduction 66 3.2.1.1 Homogenous Coordinate Vectors 66 3.2.1.2 Homogenous Coordinate Transformation Matr ices of the Dimension 4 × 4 66 3.2.2 Translations 67 © 2008 by Taylor & Francis Group, LLC viii Contents 3.2.3 Rotation about a Coordinate Axis 69 3.2.4 Rotation about an Arbitrary Axis through the Origin 70 3.2.5 Eulerian Transformation 71 3.2.6 Rotation about an Arbitrary Axis Not through the Origin 71 3.2.7 Resultant Coordinate System Transformation 72 3.2.8 An Example of Nonorthogonal Linear Transformation 74 3.2.9 Conversion of the Coordinate System Orientation 74 3.3 Useful Equations 75 3.3.1 RPY-Transformation 76 3.3.2 Rotation Operator 76 3.3.3 A Combined Linear Transformation 76 3.4 Chains of Consequent Linear Transformations and a Closed Loop of Consequent Coordinate System Transformations 77 3.5 Impact of the Coordinate System Transformations on Fundamental Forms of the Surface 83 References 85 Part II Fundamentals 4 The Geometry of Contact of Two Smooth, Regular Surfaces 89 4.1 Local Relative Orientation of a Part Surface and of the Cutting Tool 90 4.2 The First-Order Analysis: Common Tangent Plane 94 4.3 The Second-Order Analysis 94 4.3.1 Preliminary Remarks: Dupin’s Indicatrix 95 4.3.2 Surface of Normal Relative Curvature 97 4.3.3 Dupin’s Indicatrix of Surface of Relative Curvature 101 4.3.4 Matrix Representation of Equation of the Dupin’s Indicatrix of the Surface of Relative Normal Curvature 102 4.3.5 Surface of Relative Normal Radii of Curvature 102 4.3.6 Normalized Relative Normal Curvature 103 4.3.7 Curvature Indicatrix 103 4.3.8 Introduction of the Ir  k (P/T) Characteristic Curve 106 4.4 Rate of Conformity of Two Smooth, Regular Surfaces in the First Order of Tangency 107 4.4.1 Preliminary Remarks 108 4.4.2 Indicatrix of Conformity of the Surfaces P and T 110 4.4.3 Directions of the Extremum Rate of Conformity of the Surfaces P and T 117 4.4.4 Asymptotes of the Indicatrix of Conformity Cnf R (P/T) 120 4.4.5 Comparison of Capabilities of the Indicatrix of Con formity Cnf R (P/T) and of Dupin’s Indicatrix of the Surface of Relative Curvature 121 4.4.6 Important Properties of the Indicatrix of Conformity Cnf R (P/T) 122 4.4.7 The Converse Indicatrix of Conformity of the Surfaces P and T in the First Order of Tangency 122 © 2008 by Taylor & Francis Group, LLC Contents ix 4.5 Plücker’s Conoid: More Characteristic Curves 124 4.5.1 Plücker’s Conoid 124 4.5.1.1 Basics 124 4.5.1.2 Analytical Representation 124 4.5.1.3 Local Properties 126 4.5.1.4 Auxiliary Formulas 127 4.5.2 Analytical Description of Local Topology of the Smooth, Regular Surface P 127 4.5.2.1 Preliminary Remarks 128 4.5.2.2 Plücker’s Conoid 128 4.5.2.3 Plücker’s Curvature Indicatrix 131 4.5.2.4 An R (P)-Indicatrix of the Surface P 132 4.5.3 Relative Characteristic Curves 134 4.5.3.1 On a Possibility of Implementation of Two of Plücker’s Conoids 134 4.5.3.2 An R (P/T)-Relative Indicatrix of the Surfaces P and T 135 4.6 Feasible Kinds of Contact of the Surfaces P and T 138 4.6.1 On a Possibility of Implementation of the Indicatrix of Conformity for Identication of Kind of Contact of the Surfaces P and T 138 4.6.2 Impact of Accuracy of the Computations on the Desired Parameters of the Indicatrices of Conformity Cnf R (P/T) 142 4.6.3 Classication of Kinds of Contact of the Surfaces P and T 143 References 151 5 Profiling of the Form-Cutting Tools of the Optimal Design 153 5.1 Proling of the Form-Cutting Tools for Sculptured Surface Machining 153 5.1.1 Preliminary Remarks 153 5.1.2 On the Concept of Proling the Optimal Form-Cutting Tool 156 5.1.3 R-Mapping of the Part Surface P on the Generating Surface T of the Form-Cutting Tool 160 5.1.4 Reconstruction of the Generating Surface T of the Form-Cutting Tool from the Precomputed Natural Parameterization 164 5.1.5 A Method for the Determination of the Rate of Conformity Functions F 1 , F 2 , and F 3 165 5.1.6 An Algorithm for the Computation of the Design Parameters of the Form-Cutting Tool 173 5.1.7 Illustrative Examples of the Computation of the Design Parameters of the Form-Cutting Tool 175 5.2 Generation of Enveloping Surfaces 177 5.2.1 Elements of Theory of Envelopes 178 © 2008 by Taylor & Francis Group, LLC x Contents 5.2.1.1 Envelope to a Planar Curve 178 5.2.1.2 Envelope to a One-Parametric Family of Surfaces 182 5.2.1.3 Envelope to a Two-Parametric Family of Surfaces 184 5.2.2 Kinematical Method for the Determining of Enveloping Surfaces 186 5.3 Proling of the Form-Cutting Tools for Machining Parts on Conventional Machine Tools 193 5.3.1 Two Fundamental Principles by Theodore Olivier 194 5.3.2 Proling of the Form-Cutting Tools for Single-Parametric Kinematic Schemes of Surface Generation 195 5.3.3 Proling of the Form-Cutting Tools for Two-Parametric Kinematic Schemes of Surface Generation 196 5.3.4 Proling of the Form-Cutting Tools for Multiparametric Kinematic Schemes of Surface Generation 200 5.4 Characteristic Line E of the Part Surface P and of the Generating Surface T of the Cutting Tool 201 5.5 Selection of the Form-Cutting Tools of Rational Design 203 5.6 The Form-Cutting Tools Having a Continuously Changeable Generating Surface 210 5.7 Incorrect Problems in Proling the Form-Cutting Tools 210 5.8 Intermediate Conclusion 214 References 215 6 The Geometry of the Active Part of a Cutting Tool 217 6.1 Transformation of the Body Bounded by the Generating Surface T into the Cutting Tool 218 6.1.1 The First Method for the Transformation of the Generating Body of the Cutting Tool into the Workable Edge Cutting Tool 219 6.1.2 The Second Method for the Transformation of the Generating Body of the Cutting Tool into the Workable Edge Cutting Tool 222 6.1.3 The Third Method for the Transformation of the Generating Body of the Cutting Tool into the Workable Edge Cutting Tool 225 6.2 Geometry of the Active Part of Cutting Tools in the Tool-in-Hand System 234 6.2.1 Tool-in-Hand Reference System 235 6.2.2 Major Reference Planes: Geometry of the Active Part of a Cutting Tool Dened in a Series of Reference Planes 237 6.2.3 Major Geometric Parameters of the Cutting Edge of a Cutting Tool 240 6.2.3.1 Main Reference Plane 240 6.2.3.2 Assumed Reference Plane 241 6.2.3.3 Tool Cutting Edge Plane 242 6.2.3.4 Tool Back Plane 242 © 2008 by Taylor & Francis Group, LLC Contents xi 6.2.3.5 Orthogonal Plane 242 6.2.3.6 Cutting Edge Normal Plane 242 6.2.4 Analytical Representation of the Geometric Parameters of the Cutting Edge of a Cutting Tool 243 6.2.5 Correspondence between Geometric Parameters of the Active Part of Cutting Tools That Are Measured in Different Reference Planes 244 6.2.6 Diagrams of Variation of the Geometry of the Active Part of a Cutting Tool 253 6.3 Geometry of the Active Part of Cutting Tools in the Tool-in-Use System 255 6.3.1 The Resultant Speed of Relative Motion in the Cutting of Materials 257 6.3.2 Tool-in-Use Reference System 258 6.3.3 Reference Planes 261 6.3.3.1 The Plane of Cut Is Tangential to the Surface of Cut at the Point of Interest M 261 6.3.3.2 The Normal Reference Plane 263 6.3.3.3 The Major Section Plane 266 6.3.3.4 Correspondence between the Geometric Parameters Measured in Different Reference Planes 268 6.3.3.5 The Main Reference Plane 269 6.3.3.6 The Reference Plane of Chip Flow 272 6.3.4 A Descriptive-Geometry-Based Method for the Determination of the Chip-Flow Rake Angle 276 6.4 On Capabilities of the Analysis of Geometry of the Active Part of Cutting Tools 277 6.4.1 Elements of Geometry of Active Part of a Skiving Hob 277 6.4.2 Elements of Geometry of the Active Part of a Cutting Tool for Machining Modied Gear Teeth 279 6.4.3 Elements of Geometry of the Active Part of a Precision Involute Hob 281 6.4.3.1 An Auxiliary Parameter R 281 6.4.3.2 The Angle f r between the Lateral Cutting Edges of the Hob Tooth 282 6.4.3.3 The Angle x of Intersection of the Rake Surface and of the Hob Axis of Rotation 284 References 285 7 Conditions of Proper Part Surface Generation 287 7.1 Optimal Workpiece Orientation on the Worktable of a Multi-Axis Numerical Control (NC) Machine 287 7.1.1 Analysis of a Given Workpiece Orientation 288 7.1.2 Gaussian Maps of a Sculptured Surface P and of the Generating Surface T of the Cutting Tool 290 © 2008 by Taylor & Francis Group, LLC xii Contents 7.1.3 The Area-Weighted Mean Normal to a Sculptured Surface P 293 7.1.4 Optimal Workpiece Orientation 295 7.1.5 Expanded Gaussian Map of the Generating Surface of the Cutting Tool 297 7.1.6 Important Peculiarities of Gaussian Maps of the Surfaces P and T 299 7.1.7 Spherical Indicatrix of Machinability of a Sculptured Surface 302 7.2 Necessary and Sufcient Conditions of Proper Part Surface Generation 309 7.2.1 The First Condition of Proper Part Surface Generation 309 7.2.2 The Second Condition of Proper Part Surface Generation 313 7.2.3 The Third Condition of Proper Part Surface Generation 314 7.2.4 The Fourth Condition of Proper Part Surface Generation 323 7.2.5 The Fifth Condition of Proper Part Surface Generation 324 7.2.6 The Sixth Condition of Proper Part Surface Generation 329 7.3 Global Verication of Satisfaction of the Conditions of Proper Part Surface Generation 330 7.3.1 Implementation of the Focal Surfaces 330 7.3.1.1 Focal Surfaces 331 7.3.1.2 Cutting Tool (CT)-Dependent Characteristic Surfaces 336 7.3.1.3 Boundary Curves of the CT-Dependent Characteristic Surfaces 338 7.3.1.4 Cases of Local-Extremal Tangency of the Surfaces P and T 341 7.3.2 Implementation of R-Surfaces 343 7.3.2.1 Local Consideration 343 7.3.2.2 Global Interpretation of the Results of the Local Analysis 346 7.3.2.3 Characteristic Surfaces of the Second Kind 355 7.3.3 Selection of the Form-Cutting Tool of Optimal Design 357 7.3.3.1 Local K LR -Mapping of the Surfaces P and T 357 7.3.3.2 The Global K GR -Mapping of the Surfaces P and T 359 7.3.3.3 Implementation of the Global K GR -Mapping 363 7.3.3.4 Selection of an Optimal Cutting Tool for Sculptured Surface Machining 364 References 365 8 Accuracy of Surface Generation 367 8.1 Two Principal Kinds of Deviations of the Machined Surface from the Nominal Part Surface 368 8.1.1 Principal Deviations of the First Kind 368 8.1.2 Principal Deviations of the Second Kind 369 8.1.3 The Resultant Deviation of the Machined Part Surface 370 © 2008 by Taylor & Francis Group, LLC [...]... Fundamentals of the theory of surface generation are the core of the book This part of the book includes a novel powerful method of analytical description of the geometry of contact of two smooth, regular surfaces in the first order of tangency; a novel kind of mapping of one surface onto another surface; a novel analytical method of investigation of the cutting-tool geometry; and a set of analytically... field** allowed expression of the optimal parameters of kinematics of the sculptured surface machining on a multi-axis NC machine in terms of geometry of the part surface and of the generating surface of the form-cutting tool A bit later, a principal solution to the problem of profiling the form-cutting tool*** was derived This solution yields determination of the generating surface of the form-cutting... Part Surfaces: Geometry — The basics of differential geometry of sculptured part surfaces are explained The focus here is on the difference between classical differential geometry and engineering geometry of surfaces Numerous examples of the computation of major surface elements are provided A feasibility of classification of surfaces is discussed, and a scientific classification of local patches of. .. Implementation of the Differential Geometry/ Kinematics (DG/K)-Based Method of Surface Generation 459 11.1  Machining of Sculptured Surfaces on a Multi-Axis Numerical Control (NC) Machine 459 11.2  Machining of Surfaces of Revolution 469 11.2.1 Turning Operations 469 11.2.2 Milling Operations 474 11.2.3 Machining of Cylinder Surfaces 475 11.2.4 Reinforcement of Surfaces of. .. R-mapping of the sculptured surface to be machined Therefore, optimal parameters of the generating surface of the form-cutting tool can be expressed in terms of design parameters of the part surface to be machined Taking into account that the optimal parameters of kinematics of surface machining are already specified in terms of the surfaces P and T, the last solution allows an analytical representation of. .. consequences of the absence of the said methods of surface generation To present, explain, and exemplify a novel principal concept in the theory of surface generation, namely that the part surface is the primary element of the part surface- machining operation The rest of the elements are the secondary elements of the part surface- machining operation; thus, all of them can be expressed in terms of the desired... 397 8.6  Principle of Superposition of Elementary Surface Deviations 399 References 403 Part III  Application 9   Selection of the Criterion of Optimization 407 9.1  Criteria of the Efficiency of Part Surface Machining 408 9.2  Productivity of Surface Machining 409 9.2.1 Major Parameters of Surface Machining Operation .409 9.2.2 Productivity of Material Removal ... possibility of the development of the DG/K-based CAD/CAM system for the optimal sculptured surface machining is shown Chapter 11 Examples of Implementation of the DG/K-Based Method of Surface Generation — This chapter demonstrates numerous novel methods of surface machining — those developed on the premises of implementation of the proposed DG/Kbased method surface generation Addressed are novel methods of machining. .. Geometry of Surface Machining nP UP – curve vP P VP – curve M ZP +UP uP rP XP +VP YP Figure 1.1 Principal parameters of local topology of a surface P where rP is the position vector of a point of the surface P; UP and VP are curvilinear (Gaussian) coordinates of the point of the surface P; XP, YP, ZP are Cartesian coordinates of the point of the surface P; U1.P, U2.P are the boundary values of the closed... local patches of sculptured surfaces is proposed Chapter  2.  Kinematics of Surface Generation — The generalized analysis of kinematics of sculptured surface generation is presented Here, a generalized kinematics of instant relative motion of the cutting tool relative to the work is proposed For the purposes of the profound investigation, novel kinds of relative  motions of the cutting tool are discovered, . allowed expression of the optimal parameters of kinematics of the sculptured surface machining on a multi-axis NC machine in terms of geometry of the part surface and of the generating surface of the form-cutting. KINEMATIC GEOMETRY OF SURFACE MACHINING © 2008 by Taylor & Francis Group, LLC KINEMATIC GEOMETRY OF SURFACE MACHINING Stephen P. Radzevich CRC Press is an imprint of the Taylor. Elements of Geometry of Active Part of a Skiving Hob 277 6.4.2 Elements of Geometry of the Active Part of a Cutting Tool for Machining Modied Gear Teeth 279 6.4.3 Elements of Geometry of the Active