Kinematic Geometry of Surface Machining

Author: Stephen P. Radzevich

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KINEMATIC GEOMETRY OF SURFACE MACHINING

© 2008 by Taylor & Francis Group, LLC

KINEMATIC GEOMETRY OF SURFACE MACHINING Stephen P. Radzevich

Boca Raton London New York

CRC Press is an imprint of the Taylor & Francis Group, an informa business

© 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. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the conse‑ quences of their use. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www. copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978‑750‑8400. CCC is a not‑for‑profit organization that provides licenses and registration for a variety of users. 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 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

2007027748

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 Classification 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 Classification 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 Matrices of the Dimension 4 × 4 ..................................... 66 3.2.2 Translations....................................................................................... 67

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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 Conformity 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

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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 Identification 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 Classification 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 Profiling of the Form-Cutting Tools for Sculptured Surface Machining ..................................................................................... 153 5.1.1 Preliminary Remarks .................................................................... 153 5.1.2 On the Concept of Profiling 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

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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 Profiling 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 Profiling of the Form-Cutting Tools for Single-Parametric Kinematic Schemes of Surface Generation ................................ 195 5.3.3 Profiling of the Form-Cutting Tools for Two-Parametric Kinematic Schemes of Surface Generation ................................ 196 5.3.4 Profiling 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 Profiling 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 Defined 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

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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 Modified 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

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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 Sufficient 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 Verification 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 KGR-Mapping of the Surfaces P and T ..... 359 7.3.3.3 Implementation of the Global KGR-Mapping................ 363 7.3.3.4 Selection of an Optimal Cutting Tool for Sculptured Surface Machining...............................364 References ............................................................................................................ 365 8 8.1

Accuracy of Surface Generation ................................................... 367 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

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8.2

Local Approximation of the Contacting Surfaces P and T................... 372 8.2.1 Local Approximation of the Surfaces P and T by Portions of Torus Surfaces....................................................... 373 8.2.2 Local Configuration of the Approximating Torus Surfaces .... 378 8.3 Computation of the Elementary Surface Deviations ............................380 8.3.1 Waviness of the Machined Part Surface .....................................380 8.3.2 Elementary Deviation hss of the Machined Surface .................. 382 8.3.3 An Alternative Approach for the Computation of the Elementary Surface Deviations ........................................ 383 8.4 Total Displacement of the Cutting Tool with Respect to the Part Surface ......................................................................................384 8.4.1 Actual Configuration of the Cutting Tool with Respect to the Part Surface..................................................384 8.4.2 The Closest Distance of Approach between the Surfaces P and T ...................................................................... 390 8.5 Effective Reduction of the Elementary Surface Deviations ................. 396 8.5.1 Method of Gradient ....................................................................... 396 8.5.2 Optimal Feed-Rate and Side-Step Ratio ..................................... 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 ............................................... 411 9.2.2.1 Equation of the Workpiece Surface .............................. 411 9.2.2.2 Mean Chip-Removal Output ........................................ 413 9.2.2.3 Instantaneous Chip-Removal Output ......................... 413 9.2.3 Surface Generation Output............................................................ 417 9.2.4 Limit Parameters of the Cutting Tool Motion ........................... 418 9.2.4.1 Computation of the Limit Feed-Rate Shift.................. 418 9.2.4.2 Computation of the Limit Side-Step Shift................... 420 9.2.5 Maximal Instantaneous Productivity of Surface Generation....................................................................................... 421 9.3 Interpretation of the Surface Generation Output as a Function of Conformity.....................................................................423 References ............................................................................................................ 424 10 Synthesis of Optimal Surface Machining Operations .............. 427 10.1 Synthesis of Optimal Surface Generation: The Local Analysis......... 427 10.1.1 Local Synthesis............................................................................428 10.1.2 Indefiniteness .............................................................................. 432

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10.1.3

A Possibility of Alternative Optimal Configurations of the Cutting Tool...................................................................... 432 10.1.4 Cases of Multiple Points of Contact of the Surfaces P and T..... 434 10.2 Synthesis of Optimal Surface Generation: The Regional Analysis...435 10.3 Synthesis of Optimal Surface Generation: The Global Analysis....... 439 10.3.1 Minimization of Partial Interference of the Neighboring Tool-Paths.................................................. 439 10.3.2 Solution to the Boundary Problem...........................................440 10.3.3 Optimal Location of the Starting Point ...................................442 10.4 Rational Reparameterization of the Part Surface ................................444 10.4.1 Transformation of Parameters ..................................................445 10.4.2 Transformation of Parameters in Connection with the Surface Boundary Contour........................................446 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 ................................................................................... 451 10.5.1 Major Blocks of the DG/K-Based CAD/CAM System............. 451 10.5.2 Representation of the Input Data ............................................. 452 10.5.3 Optimal Workpiece Configuration ..........................................454 10.5.4 Optimal Design of the Form-Cutting Tool .............................454 10.5.5 Optimal Tool-Paths for Sculptured Surface Machining ....... 455 10.5.6 Optimal Location of the Starting Point ................................... 457 References ............................................................................................................ 457 11

Examples 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 Revolution ................................ 476 11.3 Finishing of Involute Gears......................................................................480 References ............................................................................................................ 491 Conclusion................................................................................................ 493 Notation .................................................................................................... 495

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Preface “Gaining time is gaining everything.” John Shebbeare, 1709–1788 This book, based on intensive research I have conducted since the late 1970s, is my attempt to cover in one monograph the modern theory of surface generation with a focus on kinematic geometry of surface machining on a multiaxis numerical control (NC) machine. Although the orientation of this book is toward computer-aided design (CAD) and computer-aided manufacturing (CAM), it is also useful for solving problems that relate to the generation of surfaces on machine tools of conventional design (for example, gear generators, and so forth). Machining of part surfaces can be interpreted as the transformation of a work into the machined part having the desired shape and design parameters. The major characteristics of the machined part surface — its shape and actual design parameters, as well as the properties of the subsurface layer of part material — strongly depend upon the parameters of the surfacegenerating process. In addition to the surface-generating process, there are, of course, many other technical considerations — namely, wear of the cutting tool, stiffness of the machine tool, tool chatter, heat generation, coolant and lubricant supply, and so forth. The analysis in this book is limited to those parameters of the surface-machining process that can be expressed in terms of surface geometry and of kinematics of relative motion of the cutting tool.

Historical Background People have been concerned for centuries with the generation of surfaces. Any machining operation is aimed at the generation of a surface that has appropriate shape and parameters. Enormous practical experience has been accumulated in this area of engineering. Improvements to the surface machining operation are based mostly on generalization of accumulated practical experience. Elements of the theory of surface generation began to appear later. For a long time, scientific developments in the field of surface generation were aimed at solving those problems that are relatively simple in nature. In the late 1970s and early 1980s, the idea of the synthesis of the optimal surface machining operation was, in a manner of speaking, mentioned for the first time. After a decade of gestation, original articles on the subject began to appear. Now, with the passing of a second decade, it is appropriate to attempt

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a consolidated story of some of the many efforts of European and American researchers.

The Importance of the Subject The machining of sculptured part surfaces on a multi-axis NC machine is a widely used process in many industries. The automotive, aerospace, and some other industries are the most advanced in this respect. The ability to quickly introduce new quality products is a decisive factor in capturing market share. For this purpose, the use of multi-axis NC machines is vital. Multi-axis NC machines of modern design are extremely costly. Because of this, machining of sculptured surfaces is costly as well. In order to decrease the cost of machining a sculptured surface on a multi-axis NC machine, the machining time must be as short as possible. Definitely, this is the case where the phrase “Time is money!” applies. Reduction of the machining time is a critical issue when machining sculptured surfaces on multi-axis NC machines. It is also an important consideration when machining surfaces on machine tools of conventional design. Generally speaking, the optimization of surface generation on a multi-axis NC machine results in time savings. Remember, gaining time is gaining everything. Certainly, the subject of this book is of great importance for contemporary industry and engineering.

Uniqueness of This Publication Literature on the theory of surface generation on a multi-axis NC machine is lacking. A limited number of texts on the topic are available for the Englishspeaking audience. Conventional texts provide an adequate presentation and analysis of a given operation of sculptured surface machining. The problem of surface generation is treated in all recently published books on the topic from the standpoint of analysis, and not of synthesis, of optimal surface generation. In the past 20 years, a wealth of new journal papers relating to the synthesis of optimal surface generation processes have been published both in this country and abroad. The rapid intensification of research in the theory of surface generation for CAD and CAM applications and new needs for advanced technology inspired me to accomplish this work. The present text is an attempt to present a well-balanced and intelligible account of some of the geometric and algebraic procedures, filling in as necessary, making comparisons, and elaborating on the implications to give a well-rounded picture. In this book, various procedures for handling particular problems constituting the synthesis of optimal surface generation on a multi-axis NC machine

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are investigated, compared, and applied. To begin, definitions, concepts, and notations are reviewed and established, and familiar methods of sculptured surface analysis are recapitulated. The fundamental concepts of sculptured surface geometry are introduced, and known results in the theory of multiparametric motion of a rigid body in E3 space are presented. It is postulated in this text that the surface to be machined is the primary element of the surface-generation process. Other elements, for example, the generating surface of the cutting tool and kinematics of their relative motion, are the secondary elements; thus, their optimal parameters must be determined in terms of design parameters of the part surface to be machined. To the best of my knowledge, I was the first to formulate the problem of synthesizing optimal surface generation, in the early 1980s. In the beginning, the problem was understood mostly intuitively. The first principal achievements in this 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 tool as the 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 the entire surface-generation process in terms of design parameters of the sculptured surface P. This means that the necessary input information for solving the problem of synthesizing the optimal surface-machining operation is limited to design parameters of the sculptured part surface. This input information is the minimum feasible. These two important results make evident that the problem of synthesizing optimal surface-generating processes is solvable in nature. On the premises of these two principal results, dozens of novel methods of part surface machining have been developed, and many are successfully used in the industry (see Chapter 11). It is important to stress that the decrease in required input information indicates that the theory is getting closer to the ideal. This concept, which this book strictly adheres to, is widely known as the principle of Occam’s razor. Recall here the old Chinese proverb: The beginning of wisdom is calling things with their right names. ** Radzevich, S.P., A Method of Sculptured Surface Machining on Multi-Axis NC Machine, Patent 1185749, USSR, B23C 3/16, filed: October 24, 1983; Radzevich, S.P., A Method of Sculptured Surface Machining on Multi-Axis NC Machine, Patent 1249787, USSR, B23C 3/16, filed: November 27, 1984. ***Radzevich, S.P., A Method of Design of a Form Cutting Tool for Sculptured Surface Machining on Multi-Axis NC Machine, Patent application 4242296/08 (USSR), filed: March 3, 1987.

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The principle of Occam’s razor is one of the first principles allowing evaluation of how a theory becomes ideal. Minimal feasible input information indicates the strength of a proposed theory. Occam’s razor states that the explanation of any phenomenon should make as few assumptions as possible, eliminating, or “shaving off,” those that make no difference in the observable predictions of the explanatory hypothesis of theory. In short, when given two equally valid explanations for a phenomenon, one should embrace the less-complicated formulation. The principle is often expressed in Latin as the lex parsimoniae (law of succinctness): Entia non sunt multiplicanda praeter necessitatem, which translates to “Entities should not be multiplied beyond necessity.” This is often paraphrased as “All things being equal, the simplest solution tends to be the best one.” In other words, when multiple competing theories are equal in other respects, the principle recommends selecting the theory that introduces the fewest assumptions and postulates the fewest hypothetical entities. It is in this sense that Occam’s razor is usually understood. Following the fundamental principle of Occam’s razor, one can compute optimal values of all the major parameters of sculptured surface machining on a multi-axis NC machine. Previous experience in the field is helpful but not mandatory for solving the problem of synthesizing the optimal machining operation. Important new topics help the reader to solve the challenging problems of synthesizing optimal methods of surface generation. In order to employ the disclosed approach, limited input information is required: For this purpose, only analytical representation of the surface to be generated is necessary. No known theory of surface generation is capable of solving the problems of synthesizing methods of surface generation. Moreover, no known theory is capable of treating the problem on the premises of the geometrical information of the surface being generated alone. The theory of surface generation has been substantionally complemented in this book through recent discoveries made primarily by myself and my colleagues. I have made a first attempt to summarize the obtained results of the research in the field in 1991. That year, my first books in the field of surface generation (in Russian) were courageously introduced to the engineering community (Radzevich, S.P., Sculptured Surface Machining on MultiAxis NC Machine, Kiev, Vishcha Shkola, 1991). Ten years later, a much more comprehensive summary was carried out (Radzevich, S.P., Fundamentals of Surface Generation, Kiev, Rastan, 2001). Both of these monographs are used in Europe, as well as in the United States. They are available from the Library of Congress and from other sources (www.cse.buffalo.edu/~var2/). There is a concern that some of today’s mechanical engineers, manufacturing engineers, and engineering students may not be learning enough about the theory of surface generation. Although containing some vitally important information, books to date do not provide methodological information on the subject which can be helpful in making critical decisions in the process design, design and selection of cutting tools, and implementation of the proper machine tool. The most important information is dispersed throughout a great

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number of research and application papers and articles. Commonly, isolated theoretical and practical findings for a particular surface-generation process are reported instead of methodology, so the question “What would happen if the input parameters are altered?” remains unanswered. Therefore, a broadbased book on the theory of surface generation is needed. The purpose of this book is twofold: To summarize the available information on surface generation with a critical review of previous work, thus helping specialists and practitioners to separate facts from myths. The major problem in the theory of surface generation is the absence of methods by use of which the challenging problem of optimal surface generation can be successfully solved. Other known problems are just 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 design parameters of the part surface to be machined. The distinguishing feature of this book is that the practical ways of synthesizing and optimizing the surface-generation process are considered using just one set of parameters — the design parameters of the part surface to be machined. The desired design parameters of the part surface to be machined are known in a research laboratory as well as in a shop floor environment. This makes this book not just another book on the subject. For the first time, the theory of surface generation is presented as a science that really works. This book is based on the my varied 30 years of experience in research, practical application, and teaching in the theory of surface generation, applied mathematics and mechanics, fundamentals of CAD/CAM, and engineering systems theory. Emphasis is placed on the practical application of the results in everyday practice of part surface machining and cutting-tool design. The application of these recommendations will increase the competitive position of the users through machining economy and productivity. This helps in designing better cutting tools and processes and in enhancing technical expertise and levels of technical services.

Intended Audience Many readers will benefit from this book: mechanical and manufacturing engineers involved in continuous process improvement, research workers who are active or intend to become active in the field, and senior undergraduate and graduate students of mechanical engineering and manufacturing.

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Preface

This book is intended to be used as a reference book as well as a textbook. Chapters that cover geometry of sculptured part surfaces and elementary kinematics of surface generation, and some sections that pertain to design of the form-cutting tools can be used for graduate study; I have used this book for graduate study in my lectures at the National Technical University of Ukraine “Kiev Polytechnic Institute” (Kiev, Ukraine). The design chapters and practical implementation of the proposed theory (Part III) will be of interest for mechanical and manufacturing engineers and for researchers.

The Organization of This Book The book is comprised of three parts entitled “Basics,” “Fundamentals,” and “Application”: Part I: Basics — This section of the book includes analytical description of part surfaces, basics on differential geometry of sculptured surfaces, as well as principal elements of the theory of multiparametric motion of a rigid body in E3 space. The applied coordinate systems and linear transformations are briefly considered. The selected material focuses on the solution to the problem of synthesizing optimal machining of sculptured part surfaces on a multi-axis NC machine. The chapters and their contents are as follows: Chapter 1. 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 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, including generating motion of the cutting tool, motions of orientation, and relative motions that cause sliding of a surface over itself. The chapter concludes with a discussion on all feasible kinematic schemes of surface generation. Several particular issues of kinematics of surface generation are discussed as well. Chapter 3. Applied Coordinate Systems and Linear Transformations — The definitions and determinations of major applied coordinate systems are introduced in this chapter. The matrix

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Preface

approach for the coordinate system transformations is briefly discussed. Here, useful notations and practical equations are provided. Two issues of critical importance are introduced here. The first is chains of consequent linear transformations and a closed loop of consequent coordinate systems transformations. The impact of the coordinate systems transformations on fundamental forms of the surfaces is the second. These tools, rust covered for many readers (the voice of experience), are resharpened in an effort to make the book a self-sufficient unit suited for self-study. Part II: Fundamentals — 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 described conditions of proper part surface generation. A solution to the challenging problem of synthesizing optimal surface machining begins here. The consideration is based on the analytical results presented in the first part of the book. The following chapters are included in this section. Chapter 4. The Geometry of Contact of Two Smooth Regular Surfaces — Local characteristics of contact of two smooth, regular surfaces that make tangency of the first order are considered. The sculptured part surface is one of the contacting surfaces, and the generating surface of the cutting tool is the second. The performed analysis includes local relative orientation of the contacting surfaces and the first- and second-order analyses. The concept of conformity of two smooth, regular surfaces in the first order of tangency is introduced and explained in this chapter. For the purposes of analyses, properties of Plücker’s conoid are implemented. Ultimately, all feasible kinds of contact of the part and of the tool surfaces are classified. Chapter 5. Profiling of the Form-Cutting Tools of Optimal Design — A novel method of profiling the form-cutting tools for sculptured surface machining is disclosed in this chapter. The method is based on the analytical description of the geometry of contact of surfaces that is developed in the previous chapter. Methods of profiling form-cutting tools for machining part surfaces on conventional machine tools are also considered. These methods are based on elements of the theory of enveloping surfaces. Numerous particular issues of profiling form-cutting tools are discussed at the end of the chapter.

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Preface Chapter 6. Geometry of Active Part of a Cutting Tool — The generating body of the form-cutting tool is bounded by the generating surface of the cutting tool. Methods of transformation of the generating body of the form-cutting tool into a workable cutting tool are discussed. In addition to two known methods, one novel method for this purpose is proposed. Results of the analytical investigation of the geometry of the active part of cutting tools in both the Tool-in-Hand system as well as the Tool-in-Use system are represented. Numerous practical examples of the computations are also presented. Chapter 7. Conditions of Proper Part Surface Generation — The satisfactory conditions necessary and sufficient for proper part surface machining are proposed and examined. The conditions include the optimal workpiece orientation on the worktable of a multi-axis NC machine and the set of six analytically described conditions of proper part surface generation. The chapter concludes with the global verification of satisfaction of the conditions of proper part surface generation. Chapter 8. Accuracy of Surface Generation — Accuracy is an important issue for the manufacturer of the machined part surfaces. Analytical methods for the analysis and computation of the deviations of the machined part surface from the desired part surface are discussed here. Two principal kinds of deviations of the machined surface from the nominal part surface are distinguished. Methods for the computation of the elementary surface deviations are proposed. The total displacements of the cutting tool with respect to the part surface are analyzed. Effective methods for the reduction of the elementary surface deviations are proposed. Conditions under which the principle of superposition of elementary surface deviations is applicable are established.

Part III: Application — This section illustrates the capabilities of the novel and powerful tool for the development of highly efficient methods of part surface generation. Numerous practical examples of implementation of the theory are disclosed in this part of the monograph. This section of the book is organized as follows: Chapter 9. Selection of the Criterion of Optimization — In order to implement in practice the disclosed Differential Geometry/Kinematics (DG/K)-based method of surface generation, an appropriate criterion of efficiency of part surface machining is necessary. This helps answer the question of what we want to obtain when performing a certain machining operation. Various criteria of efficiency of machining operation are considered. Tight connection of the economical criteria of optimization with geometrical analogues (as established in Chapter 4) is illustrated. The part surface

© 2008 by Taylor & Francis Group, LLC

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xxiii

generation output is expressed in terms of functions of conformity. The last significantly simplifies the synthesizing of optimal operations of part surface machining. Chapter 10. Synthesis of Optimal Surface Machining Operations — The synthesizing of optimal operations of actual part surface machining on both the multi-axis NC machine as well as on a conventional machine tool are explained. For this purpose, three steps of analysis are distinguished: local analysis, regional analysis, and global analysis. A 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 sculptured surfaces on a multi-axis NC machine, novel methods of machining surfaces of revolution, and a novel method of finishing involute gears. The proposed theory of surface generation is oriented on extensive application of a multi-axis NC machine of modern design. In particular cases, implementation of the theory can be useful for machining parts on conventional machine tools. Stephen P. Radzevich Sterling Heights, Michigan

© 2008 by Taylor & Francis Group, LLC

Author Stephen P. Radzevich, Ph.D., is a professor of mechanical engineering and manufacturing engineering. He has received an M.Sc. (1976), a Ph.D. (1982), and a Dr.(Eng)Sc. (1991) in mechanical engineering. Radzevich has extensive industrial experience in gear design and manufacture. He has developed numerous software packages dealing with computer-aided design (CAD) and computer-aided manufacturing (CAM) of precise gear finishing for a variety of industrial sponsors. Dr. Radzevich’s main research interest is kinematic geometry of surface generation with a particular focus on (a) precision gear design, (b) high torque density gear trains, (c) torque share in multiflow gear trains, (d) design of special-purpose gear cutting and finishing tools, (e) design and machining (finishing) of precision gears for lownoise/noiseless transmissions of cars, light trucks, and so forth. He has spent more than 30 years developing software, hardware, and other processes for gear design and optimization. In addition to his work for industry, he trains engineering students at universities and gear engineers in companies. He has authored and coauthored 28 monographs, handbooks, and textbooks; he authored and coauthored more than 250 scientific papers; and he holds more than 150 patents in the field. At the beginning of 2004, he joined EATON Corp. He is a member of several Academies of Sciences around the world.

© 2008 by Taylor & Francis Group, LLC

Acknowledgments I would like to share the credit for any research success with my numerous doctoral students with whom I have tested the proposed ideas and applied them in the industry. The contributions of many friends, colleagues, and students in overwhelming numbers cannot be acknowledged individually, and as much as our benefactors have contributed, even though their kindness and help must go unrecorded.

© 2008 by Taylor & Francis Group, LLC

Part I

Basics

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1 Part Surfaces: Geometry The generation of part surfaces is one of the major purposes of machining operations. An enormous variety of parts are manufactured in various industries. Every part to be machined is bounded with two or more surfaces. Each of the part surfaces is a smooth, regular surface, or it can be composed with a certain number of patches of smooth, regular surfaces that are properly linked to each other. In order to be machined on a numerical control (NC) machine, and for computer-aided design (CAD) and computer-aided manufacturing (CAM) applications, a formal (analytical) representation of a part surface is the required prerequisite. Analytical representation of a part surface (the surface P) is based on analytical representation of surfaces in geometry, specifically, (a) in the differential geometry of surfaces and (b) in the engineering geometry of surfaces. The second is based on the first. For further consideration, it is convenient to briefly consider the principal elements of differential geometry of surfaces that are widely used in this text. If experienced in differential geometry of surfaces, the following section may be skipped. Then, proceed directly to Section 1.2.

1.1

Elements of Differential Geometry of Surfaces

A surface could be uniquely determined by two independent variables. Therefore, we give a part surface P (Figure 1.1), in most cases, by expressing its rectangular coordinates XP, YP, and ZP, as functions of two Gaussian coordinates UP and VP in a certain closed interval: X P (U P , VP )  Y (U , V )  P P P  rP = rP (U P , VP ) =  ; (U1. P ≤ U P ≤ U 2. P ; V1. P ≤ VP ≤ V2. P )  ZP (U P , VP )    1  

(1.1)

The ball of a ball bearing is one of just a few examples of a part surface, which is bounded with the only surface that is the sphere.

© 2008 by Taylor & Francis Group, LLC

Kinematic 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 interval of the UP parameter; and V1.P, V2.P are the boundary values of the closed interval of the VP parameter. The parameters UP and VP must enter independently, which means that the matrix

 ∂ XP  ∂U P M=  ∂ XP  ∂V  P

∂ YP ∂U P ∂ YP ∂VP

∂ ZP  ∂U P   ∂ ZP  ∂VP 

(1.2)

has a rank 2. Positions where the rank is 1 or 0 are singular points; when the rank at all points is 1, then Equation (1.1) represents a curve. The following notation proved the consideration below. The first derivatives of rP with respect to Gaussian coordinates UP and VP are designated as ∂rP/∂UP = UP and ∂rP/∂VP = VP, and for the unit tangent vectors u P = UP/|UP| and vP = VP/|VP| correspondingly. Vector u P (as well as vector UP) specifies the direction of the tangent line to the UP coordinate curve through the given point M on the surface P. Similarly, vector vP (as well as VP) specifies the direction of the tangent line to the VP coordinate curve through that same point M on P.

© 2008 by Taylor & Francis Group, LLC

Part Surfaces: Geometry

Significance of the vectors u P and vP becomes evident from the following considerations. First, tangent vectors u P and vP yield an equation of the tangent plane to the surface P at M:

(

 rt. p − rP( M )   uP Tangent plane ⇒  vP   1

)

 =0  

(1.3)

where rt.P is the position vector of a point of the tangent plane to the surface P at M, and rP( M ) is the position vector of the point M on the surface P. Second, tangent vectors yield an equation of the perpendicular NP, and of the unit normal vector n P to the surface P at M:

N P = U P × VP

nP =

and

NP U × VP = P = uP × v P NP U P × VP

(1.4)

When the order of multipliers in Equation (1.4) is chosen properly, then the unit normal vector n P is pointed outward of the bodily side of the surface P. Unit tangent vectors u P and vP to a surface at a point are of critical importance when solving practical problems in the field of surface generation. Numerous examples, as shown below, prove this statement. Consider two other important issues concerning part surface geometry — both relate to intrinsic geometry in differential vicinity of a surface point. The first issue is the first fundamental form of a surface P. The first fundamental form f1.P of a smooth, regular surface describes the metric properties of the surface P. Usually, it is represented as the quadratic form:

φ1.P ⇒ dsP2 = EP dU P2 + 2 FP dU P dVP + GP dVP2

(1.5)

where sP is the linear element of the surface P (sP is equal to the length of a segment of a certain curve line on the surface P), and EP, FP, GP are fundamental magnitudes of the first order. Equation (1.5) is known from many advanced sources. In the theory of surface generation, another form of analytical representation of the first fundamental form f1.P is proven to be useful:

φ1. P ⇒ dsP2 = [dU P

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dVP

0

EP F P 0] ⋅  0  0

FP GP 0 0

0 0 1 0

0  dU P  0   dVP  ⋅ 0  0     1  0 

(1.6)

Kinematic Geometry of Surface Machining

This kind of analytical representation of the first fundamental form f 1.P is proposed by Radzevich [10]. The practical advantage of Equation (1.6) is that it can easily be incorporated into computer programs using multiple coordinate system transformations, which is vital for CAD/CAM applications. For computation of the fundamental magnitudes of the first order, the following equations can be used:

EP = U P ⋅ U P , FP = U P ⋅ VP , GP = VP ⋅ VP

(1.7)

Fundamental magnitudes EP, FP, and GP of the first order are functions of UP and VP parameters of the surface P. In general form, these relationships can be represented as EP = EP(UP, VP), FP = FP(UP, VP), and GP = GP(UP, VP). Fundamental magnitudes EP and GP are always positive (EP > 0, GP > 0), and the fundamental magnitude FP can equal zero (FP ≥ 0). This results in the first fundamental form always being nonnegative (f1.P ≥ 0). The first fundamental form f1.P yields computation of the following major parameters of geometry of the surface P: (a) length of a curve-line segment on the surface P, (b) square of the surface P portion that is bounded by a closed curve on the surface, and (c) angle between any two directions on the surface P. The first fundamental form represents the length of a curve-line segment, and thus it is always nonnegative — that is, the inequality f1.P ≥ 0 is always observed. The discriminant HP of the first fundamental form f1.P can be computed from the following equation: H P = EP GP − FP2

(1.8)

It is assumed that the discriminant HP is always nonnegative — that is, HP = + EPGP − FP2 . The fundamental form f1.P remains the same while the surface is banding. This is another important feature of the first fundamental form f1.P. The feature can be employed for designing three-dimensional cam for finishing a turbine blade with an abrasive strip as a cutting tool. The second fundamental form of the surface P is another of the two abovementioned important issues. The second fundamental form f 2.P describes the curvature of a smooth, regular surface P. Usually, it is represented as the quadratic form

φ2.P ⇒ − dr P ⋅ dn P = LP dU P2 + 2 M P dU P dVP + N P dVP2

Equation (1.9) is known from many advanced sources.

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(1.9)

Part Surfaces: Geometry

In the theory of surface generation, another analytical representation of the second fundamental form f 2.P is proven useful:

φ2. P ⇒ [dU P

dVP

0

 LP M P 0] ⋅   0   0

MP NP 0 0

0 0 1 0

0  dU P  0   dVP  ⋅ 0  0     1  0 

(1.10)

This analytical representation of the second fundamental form f 2.P is proposed by Radzevich [10]. Similar to Equation (1.6), the practical advantage of Equation (1.10) is that it can be easily incorporated into computer programs using multiple coordinate system transformations, which is vital for CAD/ CAM applications. In Equation (1.10), the parameters LP, MP, NP designate fundamental magnitudes of the second order. Fundamental magnitudes of the second order can be computed from the following equations: ∂U P × U P ⋅ VP ∂U P LP = , EPGP − FP2 ∂U P ∂VP × U P ⋅ VP × U P ⋅ VP ∂VP ∂U P MP = , = EPGP − FP2 EPGP − FP2

∂VP × U P ⋅ VP ∂VP NP = EPGP − FP2

(1.11)

Fundamental magnitudes LP, MP, NP of the second order are also functions of UP and VP parameters of the surface P. These relationships in general form can be represented as LP = LP(UP, VP), MP = MP(UP, VP), and NP = NP(UP, VP). Discriminant TP of the second fundamental form f 2.P can be computed from the following equation: TP = LP N P − M P2

(1.12)

For computation of the principal directions T1.P and T2.P through a given point on the surface P, the fundamental magnitudes of the second order LP, MP, NP, together with the fundamental magnitudes of the first order EP, FP, GP, are used. Principal directions T1.P and T2.P can be computed as roots of the equation

EP dU P + FP dVP LP dU P + M P dVP

FP dU P + GP dVP =0 M P dU P + N P dVP

(1.13)

The first principal plane section C1.P is orthogonal to P at M and passes through the first principal direction T1.P. The second principal plane section

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Kinematic Geometry of Surface Machining

C2.P is orthogonal to P at M and passes through the second principal direction T2.P. In the theory of surface generation, it is often preferred to use not the vectors T1.P and T2.P of the principal directions, but instead to use the unit vectors t1.P and t 2.P of the principal directions. The unit vectors t1.P and t 2.P are computed from equations t1.P = T1.P/|T1.P| and t 2.P = T2.P/|T2.P|, respectively. The first R1.P and the second R 2.P principal radii of curvature of the surface P are measured in the first and in the second principal plane sections C1.P and C2.P, correspondingly. For computation of values of the principal radii of curvature, use the following equation:

RP2 −

EP N P − 2 FP M P + GP LP H RP + P = 0 TP TP

(1.14)

Another two important parameters of local topology of a surface P are (a) mean curvature M P , and intrinsic (Gaussian or full) curvature G P . These parameters can be computed from the following equations:

MP=

k1. P + k 2. P EP N P − 2 FP M P + GP LP = 2 2 ⋅ ( EPGP − FP2 )

GP = k1. P ⋅ k 2. P = The formulae for M P =

k1. P + k 2. P 2

LP N P − M P2 EPGP − FP2

(1.16)

and G P = k1. P ⋅ k 2. P yield a quadratic equation:

k P2 − 2M P k P + GP = 0

(1.15)

(1.17)

with respect to principal curvatures k1.P and k2.P. The expressions

k 1. P = M P + M P 2 − G P

and k 2. P = M P − M P2 − GP

(1.18)

are the solutions to Equation (1.17). Here, k1.P designates the first principal curvature of the surface P, and k2.P designates the second principal curvature of the surface P at that same point. The principal curvatures k1.P and k2.P can be computed from k 1. P = R1−.1P and k2.P = k 2. P = The first principal curvature k1.P always exceeds the second principal curvature k2.P — that is, the inequality k1.P > k2.P is always observed. This brief consideration of elements of surface geometry allow for the introduction of two definitions that are of critical importance for further discussion. Definition 1.1: Sculptured surface P is a smooth, regular surface with major parameters of local topology that differ when in differential vicinity of any two infinitely closed points.

Remember that algebraic values of the radii of principal curvature R1.P and R 2.P relate to each other as R 2.P > R1.P.

© 2008 by Taylor & Francis Group, LLC

Part Surfaces: Geometry

It is instructive to point out here that sculptured surface P does not allow sliding “over itself.” While machining a sculptured surface, the cutting tool rotates about its axis and moves relative to the sculptured surface P. While rotating with a certain angular velocity ωT or while performing relative motion of another kind, the cutting edges of the cutting tool generate a certain surface. We refer to that surface represented by consecutive positions of cutting edges as the generating surface of the cutting tool [11, 13, 14]: Definition 1.2: The generating surface of a cutting tool can be represented as the set of consecutive positions of the cutting edges in their motion relative to the stationary coordinate system, embedded to the cutting tool itself.

In most practical cases, the generating surface T allows sliding over itself. The enveloping surface to consecutive positions of the surface T that performs such a motion is congruent to the surface T. When machining a part, the surface T is conjugate to the sculptured surface P. Bonnet [1] proved that the specification of the first and second fundamental forms determines a unique surface if the Gauss’ characteristic equation and the Codazzi-Mainardi’s relationships of compatibility are satisfied, and those two surfaces that have identical first and second fundamental forms are congruent. Six fundamental magnitudes determine a surface uniquely, except as to position and orientation in space. Specification of a surface in terms of the first and the second fundamental forms is usually called the natural kind of surface parameterization. In general form, it can be represented by a set of two equations:

{

The natural form φ = φ1. P (EP , FP , GP ) of surface ⇒ P = P(φ1. P , φ2. P ) 1. P φ2. P = φ2. P (EP , FP , GP , LP , M P , N P ) P parameterization

(1.19)

Equation (1.19) can be derived from Equation (1.1). Both Equation (1.1) and Equation (1.19) specify that same surface P. In further consideration, the natural parameterization of the surface P plays an important role. Illustrative Example Consider an example of how an analytical representation of a surface in a Cartesian coordinate system can be converted into the natural parameterization of that same surface [13]. A gear tooth surface G is analytically described in a Cartesian coordinate system XgYgZg (Figure 1.2).

Two surfaces with the identical first and second fundamental forms might also be symmetrical. Refer to the literature—Koenderink, J.J., Solid Shape, The MIT Press, Cambridge, MA, 1990, p. 699—on differential geometry of surfaces for details about this specific issue.

© 2008 by Taylor & Francis Group, LLC

10

Kinematic Geometry of Surface Machining

ψb.g

B Zg

Ug

rb.g

H

E

C v*g

Base Cylinder Helix

M

rg A D Yg

ng

ug G

λb.g

F

Vg

Xg

Involute Curve

Figure 1.2 Derivation of the natural form of the gear tooth surface G parameterization. (From Radzevich, S.P., Journal of Mechanical Design, 124, 772–786, 2002. With permission.)

The equation of the screw involute surface G is represented in matrix form:

rb. g cos Vg + U g cos τ b. g sin Vg   r sin V − U sin τ sin V  b. g g g b. g g  r g (U g , Vg ) =   rb. g tan τ b. g − U g sin τ b. g    1  

(1.20)

This equation yields the computation of two tangent vectors Ug(Ug,Vg) and Vg(Ug,Vg) that are correspondingly equal:  cos τ b. g sin Vg  − cos τ cos V  b. g g Ug =    − sin τ b. g   1  

− rb. g sin Vg + U g cos τ b. g cos Vg   r cos V + U cos τ sin V  b. g g g b. g g  Vg =    rb. g tan τ b. g   1  

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(1.21)

11

Part Surfaces: Geometry

Substituting the computed vectors Ug and Vg into Equation (1.7), one can come up with formulae for computation of the fundamental magnitudes of the first order:

Eg = 1, Fg = −

rb. g cos τ b. g

and Gg =

U g2 cos τ b. g + rb2. g cos 2 τ b. g

(1.22)

These equations can be substituted directly to Equation (1.5) for the first fundamental form:

φ1. g ⇒ dU g2 − 2

rb. g cos τ b. g

dU g dVg +

U g2 cos τ b. g + rb2. g cos 2 τ b. g

dVg2

(1.23)

The computed values of the fundamental magnitudes Eg, Fg, and Gg can be substituted to Equation (1.6) for f1.g. In this way, matrix representation of the first fundamental form f1.g can be computed. The interested reader may wish to complete this formulae transformation on his or her own. The discriminant Hg of the first fundamental form of the surface G can be computed from the formula Hg = Ugcosf b.g. In order to derive an equation for the second fundamental form f 2.g of the gear-tooth surface G, the second derivatives of rg(Ug, Vg) with respect to Ug and Vg parameters are necessary. The above derived equations for the vectors Ug and Vg yield the following computation:

cos τ b. g cos Vg  ∂U g ∂Vg  cos τ b. g sin Vg  = ≡ and  0 ∂Vg ∂U g    1    − rb. g cos Vg − U g cos τ b. g sin Vg  ∂Vg − rb. g sin Vg + U g cos τ b. g cos Vg  =  0 ∂Vg    1  

0  ∂U g 0  = , ∂U P 0    1 

(1.24)

Further, substitute these derivatives (see Equation 1.24 and Equation 1.8 into Equation 1.11). After the necessary formulae transformations are complete, then Equation (1.11) casts into the set of formulae for computation of the second fundamental magnitudes of the surface G is as follows:

Lg = 0 M g = 0 and N g = − U g sin τ b. g cos τ b. g

(1.25)

After substituting Equation (1.25) into Equation (1.9), an equation for the computation of the second fundamental form of the surface G can be obtained:

φ2. g ⇒ − dr g ⋅ d N g = −U g sin τ b. g cos τ b. g dVg2

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(1.26)

12

Kinematic Geometry of Surface Machining

Similar to Equation (1.23), the computed values of the fundamental magnitudes Lg, Mg, and Ng can be substituted into Equation (1.10) for f 2.g. In this way, matrix representation of the second fundamental form f 2.g can be computed. The interested reader may wish to complete this formulae transformation on his or her own. Discriminant Tg of the second fundamental form f 2.g of the surface G is equal to Tg = Lg M g − N g2 = 0 . The derived set of six equations for computation of the fundamental magnitudes represents the natural parameterization of the surface P: Eg = 1 Fg = − Gg =

Lg = 0 rb. g

Mg = 0

cos τ b. g

U g2 cos τ b. g + rb2. g

N g = − U g sin τ b. g cos τ b. g

cos τ b. g 2

All major elements of geometry of the gear-tooth surface can be computed based on the fundamental magnitudes of the first f1.g and of the second f 2.g fundamental forms. Location and orientation of the surface G are the two parameters that remain indefinite. Once a surface is represented in natural form — that is, it is expressed in terms of six fundamental magnitudes of the first and of the second order — then further computation of parameters of the surface P becomes much easier. In order to demonstrate significant simplification of the computation of parameters of the surface P, several useful equations are presented below as examples. Examples 1. Length of a curve segment UP = UP(t), VP = VP(t), t0 ≤ t ≤ t1 is given by t

s=

∫ t0

2

2

dU P dVP  dU P   dV  E + 2F + G  P  dt  dt   dt  dt dt

(1.27)

2. Value of the angle q between two given directions through a certain point M on the surface P can be computed from one of the equations: cos θ =

© 2008 by Taylor & Francis Group, LLC

FP , sin θ = EP GP

HP H , tan θ = P FP EP GP

(1.28)

13

Part Surfaces: Geometry 3. For computation of square S P of a surface patch S, which is bounded by a closed line on the surface P, the following equation can be used:

SP =

∫∫

EPGP − FP2 dU P dVP

(1.29)

S

4. Value of radius of curvature R P of the surface P in normal plane section through M at a given direction can be computed from the following equation: RP =

φ1 .p φ2 .p

(1.30)

5. Euler’s equation for the computation of R P is kθ . P = k1. P cos 2 θ + k 2. P sin 2 θ

(1.31)

This is also a good illustration of the above statement. (Here q is the angle that the normal plane section CP makes with the first principal plane section C1.P. In other words, θ = ∠(t P , t 1. P ); here t P designates the unit tangent vector within the normal plane section CP.) Shape-index and the curve of the surface are two other useful properties that are also drawn from the principal curvatures. The shape-index, S P , is a generalized measure of concavity and convexity. It can be defined [4] by

SP = −

k 1. P + k 2. P 2 arctan k 1. P − k 2. P π

(1.32)

The shape-index varies from −1 to +1. It describes the local shape at a surface point independent of the scale of the surface. A shape-index value of +1 corresponds to a concave local portion of the surface P for which the principal directions are unidentified; thus, normal radii of curvature in all directions are identical. A shape-index of 0 corresponds to a saddle-like local portion of the surface P with principal curvatures of equal magnitude but opposite sign. The curvedness RP, is another measure derived from the principal curvatures [4]:

© 2008 by Taylor & Francis Group, LLC

RP =

k12. P + k 22. P 2

(1.33)

14

Kinematic Geometry of Surface Machining

The curvedness describes the scale of the surface P independent of its shape. These quantities S P and RP are the primary differential properties of the surface. Note that they are properties of the surface itself and do not depend upon its parameterization except for a possible change of sign. In order to get a profound understanding of differential geometry of surfaces, the interested reader may wish to go to advanced monographs in the field. Systematic discussion of the topic is available from many sources. The author would like to turn the reader’s attention to the monographs by doCarmo [2], Eisenhart [3], Struik [16], and others.

1.2

On the Difference between Classical Differential Geometry and Engineering Geometry

Classical differential geometry is developed mostly for the purpose of investigation of smooth, regular surfaces. Engineering geometry also deals with the surfaces. What is the difference between these two geometries? The difference between classical differential geometry and between engineering geometry is mostly due to how the surfaces are interpreted. Only phantom surfaces are studied in classical differential geometry. Surfaces of this kind do not exist in reality. They can be imagined as a thin film of an appropriate shape and with zero thickness. Such film can be accessed from both of the surface sides. This causes the following indefiniteness. As an example, consider a surface having positive Gaussian curvature GP at a surface point ( GP > 0 ). Classical differential geometry gives no answer to the question of whether the surface P is convex ( M P > 0 ) or concave ( M P < 0 ) at this point. In classical differential geometry, the answer to this question can be given only by convention. A similar observation is made when Gaussian curvature GP at a certain surface point is negative ( G P < 0 ). Surfaces in classical differential geometry strictly follow the equation they are specified by. No deviation of the surface shape from what is predetermined by the equation is allowed. More examples can be found in the following chapters of this book. In turn, surfaces that are treated in engineering geometry bound a part (or machine element). This part can be called a real object (Figure 1.3). The real object is the bearer of the surface shape. Surfaces that bound real objects are accessible from only one side (Figure 1.4). We refer to this side of the surface as the open side of a surface. The opposite side of the surface P is not accessible. Because of this, we refer to the opposite side of the surface P as the closed side of a surface. The positively directed normal unit vector +nP is pointed outward from the part body — that is, from its bodily side to the void side. The negative normal unit vector −nP is pointed opposite to +nP. The existence of open and closed sides of a surface P eliminates the

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15

Part Surfaces: Geometry P

P

Figure 1.3 Examples of surfaces that mechanical engineers are dealing with. (From Radzevich, S.P., Computer-Aided Design, 37 (7), 767–778, 2005. With permission.)

problem of identifying whether the surface is convex or concave. No convention in this concern is required. Another principal difference is due to the nature of the real object. No real object can be machined or manufactured precisely without deviations of its actual shape from the desired shape of the real object. Smaller or bigger deviations of shape of the real object from the desired shape are unavoidable in nature. We do not go into detail here about this concern. Because of the deviations, the actual part surface Pact deviates from its nominal surface P (Figure 1.5). However, the deviations do not exceed a reasonable range. Otherwise, the real object will become useless. In practice, the

Open Side of the Surface +nP

P

M −nP

Closed Side of the Surface

ZP XP YP

Figure 1.4 Open and closed sides of surface P. (From Radzevich, S.P., Computer-Aided Design, 34 (10), 727– 740, 2002. With permission.)

© 2008 by Taylor & Francis Group, LLC

16

Kinematic Geometry of Surface Machining nP

P act

P+ P

δ+(UP ,VP)

M+ M

P−

M–

r+P r–P

Zp XP

δ–(UP ,VP)

rP

YP Figure 1.5 An example of actual part surface Pact.

selection of appropriate tolerances on shape and dimensions of the actual surface Pact easily solve this particular problem. Similar to measuring deviations, the tolerances are measured in the direc+ tion of the unit normal vector n P to the surface P. Positive tolerance d is measuring along the positive direction of n P, and negative tolerance d is measuring along the negative direction of n P. In a particular case, one of the + tolerances, either d or d can be equal to zero. + Often, the value of tolerances d and d are constant within the entire patch of the surface P. However, in special cases, for example when machining a sculptured surface on a multi-axis numerical control machine, the actual + value of the tolerances d and d can be set as functions of coordinates of current point M on P. This results in the tolerances being represented in terms + + of UP and VP parameters of the surface P, say in the form d = d (UP, VP) and d = d (UP, VP). + The endpoint of the vector d ∙ n P at a current surface point M produces + point M . Similarly, the endpoint of the vector d ∙ n P produces the corresponding point M . The surface P+ of the upper tolerance is represented by loci of the points + + M (i.e., by loci of endpoints of the vector d ∙ n P). This yields an analytical representation of the surface of upper tolerance in the form

© 2008 by Taylor & Francis Group, LLC

rP+ (U P , VP ) = rP + δ + ⋅ n P

(1.34)

17

Part Surfaces: Geometry +

Usually, the surface P of the upper tolerance is located above the nominal surface P. Similarly, the surface P - of lower tolerance is represented by loci of the points M− (i.e., by the loci of endpoints of the vector d ∙ n P). This also yields an analytical representation of the surface of lower tolerance in the form rP− (U P , VP ) = rP + δ − ⋅ n P

(1.35)

-

Commonly, the surface P of lower tolerance is located beneath the nominal surface P. The actual part surface Pact cannot be represented analytically. Moreover, the above-considered parameters of local topology of the surface P cannot be computed for the surface Pact. However, because the tolerances + d and d are small compared to the normal radii of curvature of the nominal surfaces P, it is assumed below that the surface Pact possesses the same geometrical properties as the surface P does, and that the difference in corresponding geometrical parameters of the surfaces Pact and P is negligibly small. In further consideration, this yields replacement of the actual surface Pact with the nominal surface P, which is much more convenient for performing computations. The consideration above illustrates the second principal difference between classical differential geometry and the engineering geometry of surfaces. Because of the differences, engineering geometry often presents problems that were not envisioned in classical (pure) differential geometry.

1.3

On the Classification of Surfaces

The number of different surfaces that bound real objects is infinitely large. A systematic consideration of surfaces for the purposes of surface generation is of critical importance. 1.3.1 Surfaces That Allow Sliding over Themselves In industry, a small number of surfaces with relatively simple geometry are in wide use. Surfaces of this kind allow for sliding over themselves. The property of a surface that allows sliding over itself means that for a certain

Actually, surface Pact is unknown — any surface located within the surfaces of upper tolerance + − P and lower tolerance P satisfies the requirements of the part blueprint; thus, every such surface can be considered an actual surface Pact. An equation of the surface Pact cannot be represented in the form P act = P act (U P , VP ), because the actual value of deviation δ act at the current surface point is not known. CMM data yields only an approximation for δ act as well as the corresponding approximation for Pact.

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18

Kinematic Geometry of Surface Machining

surface P there exists a corresponding motion of a special kind. When performing this motion, the enveloping surface to the consecutive position of the moving surface P is congruent to the surface P itself. The motion of the mentioned kind can be monoparametric, biparametric, or triparametric. The screw surface of constant pitch (px = Const) is the most general kind of surface that allows sliding over itself. While performing the screw motion of that same pitch px, the surface P is sliding over itself, similar to the “boltand-nut” pair. When the pitch of a screw surface reduces to zero (px = 0), then the screw surface degenerates to the surface of revolution. Every surface of revolution is sliding over itself when rotating. When the pitch of a screw surface rises to an infinitely large value, then the screw surface degenerates into a general cylinder. Surfaces of that kind allow straight motion along straight generating lines of the surface. The considered kinds of surface motion are (a) screw motion of constant pitch (px = Const), (b) rotation, and (c) straight motion, correspondingly. All of these motions are monoparametric. Surfaces like that of a circular cylinder allow rotation as well as straight motion along the axis of the cylinder. In this case, the surface motion is biparametric (rotation and translation can be performed independently). A sphere allows for rotations about three axes independently. A plane surface allows straight motion in two different directions as well as a rotation about an axis that is orthogonal to the plane. The surface motion in the last two cases (for a sphere and for plane) is triparametric. Ultimately, one can summarize that surfaces allowing sliding over themselves are limited to screw surfaces of constant pitch, cylinders of general kind, surfaces of revolution, circular cylinders, spheres, and planes. It is proven [12–15] that there are no other kinds of surfaces that allow for sliding over themselves. Surfaces that allow sliding over themselves proved to be very convenient in manufacturing as well as in industrial applications. Most of the surfaces being machined in various industries are surfaces of this nature. 1.3.2 Sculptured Surfaces Many products are designed with aesthetic sculptured surfaces to enhance their aesthetic appeal, an important factor in customer satisfaction, especially for automotive and consumer-electronics products. In other cases, products have sculptured surfaces to meet functional requirements. Examples of functional surfaces can be easily found in aero-, gas- and hydrodynamic applications (turbine blades), optical (lamp reflector) and medical (parts of anatomical reproduction) applications, manufacturing surfaces (molding die, die face), and so forth. Functional surfaces interact with the environment or with other surfaces. Due to this, functional surfaces can also be called dynamic surfaces.

© 2008 by Taylor & Francis Group, LLC

19

Part Surfaces: Geometry

A functional surface does not possess the property to slide over itself. This causes significant complexity in the machining of sculptured surfaces. The application of a multi-axis NC machine is the only way to efficiently machine sculptured surfaces. At every instant of surface machining on a multi-axis NC machine, the sculptured surface being machined and the generating surface of the cutting tool make point contact. In order to develop an advanced technology of sculptured surface generation, a comprehensive understanding of the local topology of a sculptured surface is highly desired. 1.3.3 Circular Diagrams For the purpose of precisely describing the local topology of a surface P, circular diagrams can be implemented. Circular diagrams are a powerful tool for analysis and in-depth understanding of the topology of a sculptured surface. A circular diagram reflects the principal properties of a sculptured surface in differential vicinity of a surface point. Euler’s equation for normal surface curvature, kθ . P = k1. P cos 2 θ + k 2. P sin 2 θ

(1.36)

together with Germain’s equation (or Bertrand’s equation in other interpretations),

τ θ . P = ( k 2. P − k1. P )sin θ cos θ

(1.37)

are the foundation of circular diagrams of a sculptured surface. Here in the last equation, t q.P designates torsion of a surface in the direction specified by the value of angle q. Figure 1.6 illustrates an example of a circular diagram constructed for a convex local patch of the elliptic kind. It is important to point out here that due to the algebraic value of the first principal curvature k1.P always exceeding the algebraic value of the second principal curvature k2.P, the circular diagram point with coordinates (0, k1.P) is always located at the far right relative to the circular diagram point with coordinates (0, k2.P).

Initially proposed by C.O. Mohr (1835–1918) for the purposes of solving problems in the field of strength of materials, circular diagrams later gained wider application. The origination of application of circular diagrams for the purposes of differential geometry of surfaces can be traced back to the publications by Miron [7] and Vaisman [17]. Lowe [5,6] applied circular diagrams in studying surface geometry with special reference to twist, as well as in developing plate theory. A profound analysis of properties of circular diagrams can be found in publications by Nutbourn [8] and Nutbourn and Martin [9]. The application of circular diagrams in the field of sculptured surface machining on a multi-axis NC machine is known from the monograph by Radzevich [13].

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20

Kinematic Geometry of Surface Machining

τP

k1.P

k2.P

kP τP(θ)

2θ kP(θ)

Figure 1.6 An example of a circular diagram constructed for a convex elliptic patch of a sculptured surface.

The application of circular diagrams enables one to easily identify the kind of local surface patch in the differential vicinity of a point on a sculptured surface. Circular diagrams of convex ( M P > 0, M P > 0) and concave ( M P < 0, G P > 0 ) elliptic local patches (Figure 1.7a) are depicted as the circles that are remote at a certain distance from the tP axis. Radii of the circles are equal to half a sum of the surface principal curvatures k1.P and k2.P. For umbilic local patches, principal directions are not identified. Thus, normal curvatures in all directions are of the same value. This results in that for umbilic local patches, circular diagrams shrink into points (Figure 1.7b). Coordinates of the points are (kP > 0, 0) for convex ( M P > 0, G P > 0 ) local surface patch, and (kP < 0, 0) for concave ( M P < 0, G P > 0 ) surface patch, respectively. Circular diagrams of convex ( M P > 0, G P = 0 ), and of concave ( M P < 0, GP = 0 ) parabolic local patches (Figure 1.7c) are passing through the origin of the coordinate system kPt P. It is recommended that quasi-convex ( M P > 0 ), and quasi-concave ( M P < 0 ) local surface patches be distinguished when they are of hyperbolic type—that is, for saddle-like local surface patches ( GP < 0 ). Circular diagrams of saddlelike local surface patches intersect the t P axis (Figure 1.7d). A particular case of saddle-like local surface patches is distinguished for the hyperbolic surface local patch ( GP < 0 ) of zero mean curvature ( M P = 0 ). Surface local patches of this kind are referred to as minimal surface local patches (Figure 1.7e).

We refer to a sculptured surface area in differential vicinity of a surface point as the local surface patch, and not as a surface point. The name of local surface patches corresponds to the name of surface points. However, from the standpoint of surface machining, local surface patches instead of surface points is preferred.

© 2008 by Taylor & Francis Group, LLC

21

Part Surfaces: Geometry MP < 0

GP > 0

k2.P

MP > 0

τP

k1.P

GP > 0 k1.P

k2.P

MP < 0

GP > 0

MP > 0

τP

GP > 0

kP < 0

kP > 0

kP Concave Elliptic

Concave Elliptic

kP Concave Umbilic

(a) MP < 0

GP = 0

k2.P

(b) MP > 0

τP

Convex Umbilic

k2.P

GP = 0

k2.P

k1.P

k1.P

GP < 0

MP < 0

MP > 0

τP k2.P

kP

GP < 0 k1.P

k1.P

kP

Concave Parabolic Convex Parabolic Saddle-Like (pseudoconcave) Saddle-Like (pseudoconvex) (c) MP = 0

(d)

GP < 0 τ P

MP = 0

GP = 0

τP

k1.P

k2.P

kP = 0

kP Saddle-Like (minimal) (e)

kP

Planar (f)

Figure 1.7 Circular diagrams of smooth, regular local patches of a sculptured surface.

Finally, the circular diagram of the planar surface local patch ( M P = 0, GP = 0 ) is degenerated into the point that coincides with the origin of the coordinate system kPt P (Figure 1.7f). All points of plane can be considered as parabolic umbilics. As follows from Figure 1.7, the circular diagram clearly illustrates the major local properties of sculptured surface geometry. Principal curvatures, normal curvatures, and surface torsion can be easily seen from the diagram. Moreover, actual values of mean M P and Gaussian GP curvatures can also be gained from the circular diagram. Figure 1.8 illustrates examples of how the mean M P and the Gaussian GP curvatures can be constructed. The examples are given for convex and concave elliptic (Figure 1.7a) local surface patches, as well as for quasi-convex, and quasi-concave saddle-like (Figure 1.7a) local surface patches. The above consideration yields the conclusion that the circular diagram is a simple characteristic image that provides the researcher with comprehensive

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22

Kinematic Geometry of Surface Machining

τP

+GP

τP

+GP

k2.P

k1.P k1.P

k2.P

−MP

+MP (a)

k2.P

kP

−GP

−GP

k2.P −MP

k1.P

k1.P kP +MP

(b)

Figure 1.8 Geometric interpretation of mean, and of full (Gaussian) curvature of a sculptured surface.

information on the local topology of the surface. This information includes (a) principal curvatures k1.P and k2.P, (b) normal curvature kP in a given direction on the surface, (c) extremum values of the surface torsion τ Pmin and τ Pmax, (d) the surface torsion t P in a given direction on the surface, (e) mean curvature M P , and (f) Gaussian curvature G P . No other characteristic image of that simple nature (as the circular diagram is) provides the researcher with such comprehensive information on local topology of a sculptured surface. Circular diagrams are used for solving problems in surface generation. One such problem relates to the classification of surfaces. The classification of sculptured surfaces is necessary for the purpose of developing efficient technology of surface machining on a multi-axis NC machine. We will take a brief look at surface classification from this point of view. Sculptured surfaces are geometrical objects of complex nature. It is shown [4] that no scientific classification of sculptured surfaces can be developed. It is recommended [4] that classification of local surface patches be considered instead of classification of sculptured surfaces. The recommendation is based on the consideration of point contact that a sculptured surface makes with the generating surface of a cutting tool when machining. Figure 1.9 is insightful for understanding the relationship between local surface patches of different kinds. The shift of a circular diagram along the kP axis in V − + or in V + − direction reflects a corresponding change of shape and kind of local surface patches. On the premises of these changes, a classification of local patches of sculptured surfaces can be worked out. Depicted in Figure 1.9a, the initial circular diagram represents a convex elliptic local patch with certain values of the surface principal curvatures. Due to a shift in the V − + direction, the shape of the convex elliptic surface patch changes; however, for some extent, it remains of the elliptic kind. When the circular diagram passes through the origin of the coordinate system kP t P, this leads to a dramatic change in the shape of the local surface patch. Instead of being of the elliptic kind, it transforms to a local surface patch of the parabolic kind.

© 2008 by Taylor & Francis Group, LLC

23

Part Surfaces: Geometry τP

kP +

V− (a) τP

kP V+– (b) Figure 1.9 Various locations of a circular diagram correspond to local surface patches of different kinds.

A further shift of the circular diagram in the V − + direction results in the consequent change of shape of the local surface patch to a quasi-convex saddlelike local surface patch, a minimum saddle-like local surface patch, and so on up to a concave elliptic local surface patch of certain values of principal curvatures. Similar changes in shape and in kind of local surface patch are observed when the circular diagram shifts in the direction of V + − , which is opposite to the direction of V − + . In order to understand the relationship between the local surface patches of different kinds, it is convenient to also consider shift of a circular diagram together with change of ratio between principal curvatures (k1.P/k2.P) of the surface. Following this, one can come up with the idea of circular distribution of circular diagrams. An example of the circular distribution of the

The author would like to credit the idea of circular disposition of local surface patches of different kinds to J. Koenderink. To the best of the author’s knowledge, Koenderink is the first who used circular disposition of images of local surface patches for the purpose of illustrating the relationship between local surface patches of different geometries. Reading the monograph by Koenderink [4] inspired the author to apply the circular disposition of circular diagrams of local surface patches to the needs of kinematical geometry of surface machining.

© 2008 by Taylor & Francis Group, LLC

24

Kinematic Geometry of Surface Machining τP �P > 0 �P < 0 τP

�P = 0 �P < 0 C

C

τP

kP

C

kP

�P < 0 �P < 0 kP

�P > 0 �P = 0 τP

τP

C

C

kP

τP kP �P

�P > 0

�P > 0

τP

�P = 0 �P < 0

C

kP > 0 C kP

C

kP

�P > 0 �P = 0

C

kP

τP �P = 0 C kP = 0 kP

�P = 0 �P < 0

kP < 0 C

�P = 0 �P < 0 τP

�P > 0 �P < 0

τP kP

C kP �P = 0

C

τP

�P = 0 �P < 0

C

τP

kP

kP

kP �P > 0

�P > 0

�P < 0 C

�P < 0

kP

C

kP

τP kP

�P < 0 �P = 0

τP C

kP

τP

�P

τP C

C

kP

�P = 0

kP

�P > 0

τP

τP

�P < 0 �P = 0 τP

kP �P < 0 �P < 0

Figure 1.10 Smooth, regular local patches of a sculptured surface: relationship between local patches of various kinds.

circular diagrams of all possible local patches of a smooth, regular sculptured surface is shown in Figure 1.10. 1.3.4 On Classification of Sculptured Surfaces Figure 1.10 is helpful for understanding the local topology of the sculptured surface being machined. It also yields classification of local patches of smooth, regular surface P (Figure 1.11). The classification includes ten kinds of local surface patches and is an accomplished one. Based on the analysis of sculptured surface geometry, as well as on the classification of local surface patches (Figure 1.11), a profound scientific classification of all feasible kinds of local surface patches is developed by Radzevich [10]. It is proven that the total number of feasible kinds of local surface patches is limited. Hence, local surface patches of every kind can be

© 2008 by Taylor & Francis Group, LLC

25

Part Surfaces: Geometry

Local Surface Patches �P = 0 �P < 0

�P = 0

�P > 0

�P < 0

�P = 0

�P > 0

Planar Kind: Parabolical Umbilic

Convex Parabolic

Quasi-Concave Saddle-Like

Minimal: Saddle-Like

Quasi-Convex Saddle-Like

kP

Convex Umbilic

kP �P

Concave Umbilic

Concave Elliptic

�P

�P > 0

Concave Parabolic

�P < 0

�P > 0

Convex Elliptic

�P < 0

Figure 1.11 Ten kinds of local patches of smooth, regular sculptured surface.

investigated separately. No surface patches would be left out of the consideration. This means that the problem of synthesis of optimal machining of sculptured surface is solvable. See Reference [13] for details on the developed classification of local surface patches and its use in surface generation.

References [1] Bonnet, P.O., Journ. Ec. Polytech., xiii, 31, 1867. [2] doCarmo, M.P., Differential Geometry of Curves and Surfaces, Prentice Hall, Englewood Cliffs, NJ, 1976. [3] Eisenhart, L.P., A Treatise on the Differential Geometry of Curves and Surfaces, Dover Publications, London, 1909; New York, reprint 1960. [4] Koenderink, J.J., Solid Shape, MIT Press, Cambridge, MA, 1990. [5] Lowe, P.G., A Note on Surface Geometry with Special Reference to Twist, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 87, 1980, pp. 481–487. [6] Lowe, P.G., Basic Principles of Plate Theory, Surrey University Press, Glasgow, 1982. [7] Miron, R., Observatii a Supra Unor Formule din Geometria Varietatilor Neonolonome E2 , Bulletinul Institutuini Politechnic din Iasi, IPI Press, Iasi, 1958. [8] Nutbourn, A.W., A Circle Diagram for Local Differential Geometry. In J. Gregory (Ed.), Mathematics of Surfaces, Conference Proceedings, Institute of Mathematics and Its Application, 1984, Oxford University Press, Oxford, 1986. [9] Nutbourn, A.W., and Martin, R.R., Differential Geometry Applied to Curve and Surface Design, Volume 1: Foundations, Ellis Horwood, Chichester, 1988. [10] Radzevich, S.P., Classification of Surfaces, UkrNIINTI, Kiev, No. 1440-Yk88, 1988. [11] Radzevich, S.P., Conditions of Proper Sculptured Surface Machining, Computer-Aided Design, 34, 10, 1 September, 727–740, 2002. [12] Radzevich, S.P., Differential-Geometric Method of Surface Generation, DrSci thesis, Tula Polytechnic Institute, Tula, Russia, 1991.

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Kinematic Geometry of Surface Machining

[13] Radzevich, S.P., Fundamentals of Surface Generation, Kiev, Rastan, 2001. Copy of the monograph is available from the Library of Congress, call number: MLCM 2006/04297. [14] Radzevich, S.P., Sculptured Surface Machining on Multi-Axis NC Machine, Vishcha Schola, Kiev, 1991. Copy of the monograph is available from the Library of Congress, call number: TJ1189 .R26 1991. [15] Radzevich, S.P., and Petrenko, T. Yu., Part and Tool Surfaces That Allow Sliding over Themselves, Mechanika ta Mashinobuduvann’a, Kharkiv, Ukraine, 1, 231–240, 1999. [16] Struik, D.J., Lectures on Classical Differential Geometry, 2nd ed., Addison-Wesley, Reading, MA, 1961. [17] Vaisman, I., Unele Observatii Privind Suprafetele si Varietatile Neonolonome din S Euclidian, Mathematica, Academia R.P.R., Filila Iasi, Studii si Cercetari Stiintifice, 10 (1), 195.

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2 Kinematics of Surface Generation A motion of the cutting tool relative to the work is necessary for machining a given part surface on the machine tool. Relative motions of the work and of the cutting tool are necessary for generation of a part surface being machined. No machining operation can be performed without a certain relative motion of the work and of the cutting tool. The work and the cutting tool relative motion are observed for all methods of machining of surfaces on a machine tool. The work and the cutting tool relative motions of different natures are observed. The following relative motions, among others, can be recognized: • A setup motion of the work and of the cutting tool. Motions of that kind are required for proper positioning of the cutting tool relative to the work being machined in its initial position prior to the start of the machining operation. • Cutting motions necessary for the removal of stock out of the workpiece. • Feed-rate motions required for continuation of the process of stock removal. • Surface-generating motions. Motions of this kind are required for the purpose of generating the entire surface being machined. • Orientation motions of the cutting tool. In addition to those listed above, relative motions of other kinds can also be distinguished. Below, the generation motions, feed-rate motions, and orientation motions of the cutting tool are investigated. Cutting tools are used for machining various surfaces shapes and geometries. When machining, the cutting wedge cuts off a stock from the work. The shape and parameters of the machined part surface can be considered as a result of interaction of the work and of the cutting edge of a certain shape, which is performing a certain motion relative to the work. This means that the kinematics of the machining operation directly affects the shape, accuracy, and surface quality of the machined part surface. The required kinematics of a machining operation determines the corresponding kinematical structure of a conventional machine tool. It also establishes exact requirements for an appropriate code for controlling a multi-axis numerical control (NC) machine. In order to develop the most efficient machining operation of a given part surface, it is necessary to determine the optimal parameters of the relative 27 © 2008 by Taylor & Francis Group, LLC

28

Kinematic Geometry of Surface Machining

motion of the work and of the cutting tool at every instant of machining. An appropriate criterion of optimization is of critical importance. Relative motion of the work and of the cutting tool can be represented as a superposition of a certain number of elementary motions that are performed by the machine tool of a certain design. The mentioned elementary motions are the translations along and the rotations about various axes that are differently oriented relative to each other. Ultimately, a combination of a certain number of translations and rotations can result in a complex relative motion of the work and of the cutting tool. Elementary motions generated by cams, copiers, and computer codes can be incorporated in the resultant motion of the work and of the cutting tool as well. All the elementary motions are timed (synchronized) with one another in a proper manner. Consider a simple example of drilling a hole on a drilling press. In order to drill the hole, it is necessary to rotate a twist drill and feed it into the work. A combination of the rotation and of the translation of the twist drill results in the hole being machined in the work. Machining of that same hole can be performed under another scenario. The twist drill is rotating as in the previous machining operation. However, not the twist drill, but the worktable with the work is feeding into the rotating twist drill. The combination of the rotation of the twist drill and of the translation of the worktable, which is carrying the work, results in the hole being machined in the work. Drilling of that same hole can be performed on a lathe. Under such a scenario, the work is rotating about its axis and the twist drill is feeding into the rotating work. This also results in the hole being machined in the work. Feasible combinations of the translation and of the rotation of the work and of the twist drill are not limited to the three scenarios considered above. However, it is important to point out here that in all cases of drilling of the hole, the relative motion of the work and that of the twist drill remain the same. It is represented with a screw relative motion regardless of the absolute motions performed by the twist drill, and of the absolute motions performed by the work itself. In the example considered, the translation and the rotation represent elementary absolute motions. These motions are considered as motions that are performing relative to the frame (body) of the machine tool. The example reveals that even in cases of machining a relatively simple surface (the surface of the hole), the number of feasible combinations of elementary absolute motions in an actual machining operation can be large. In the case of machining a sculptured surface on a multi-axis NC machine, the number of feasible combinations of elementary motions increases dramatically. Moreover, parameters of the elementary motions vary in time. The Differential Geometry/Kinematics (DG/K ) approach to part surface generation is based on the consideration of the relative motion of the work and of the cutting tool regardless of the kind of elementary absolute motions that are performing the work and the cutting tool. Use of the concept of relative motion yields computation of the optimal parameters of the desired relative motion of the work and of the cutting tool. After computation, the optimal relative motion

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Kinematics of Surface Generation

29

of the work and of the cutter can be further decomposed into elementary absolute motions that the machine tool of certain design is able to perform. The machining of a sculptured surface on a multi-axis NC machine represents the most general case of kinematics of surface machining. NC makes any desired relative motion of the work and of the cutting tool feasible. Therefore, it is natural to begin investigation of the kinematics of surface generation from consideration of the most general case of surface machining — that is, from the case of machining a sculptured surface on a multi-axis NC machine.

2.1

Kinematics of Sculptured Surface Generation

For machining of sculptured surfaces on a multi-axis NC machine, cutting tools of relatively simple design are often used. When machining the sculptured surface, the cutting tool is performing complex motion relative to the work. This relative motion can be interpreted as a summa of in numerous instant translations as well as numerous instant rotations about the instant axes. Due to the geometry of the sculptured part surface, the relative translations and the relative rotations are instant in nature. Because of this, not a kinematic scheme of surface generation but a principal instant kinematics of sculptured surface generation has to be investigated instead. Kinematics of sculptured surface generation features instant values of the relative motions. Accelerations and decelerations, both linear and angular, are inherent to the instant kinematics of sculptured surface generation. In order to investigate the principal instant kinematics of surface generation, it is necessary to specify the sculptured surface P to be machined. It is assumed below that the equation of the surface P is represented either in vector form rP = rP(UP,VP), or in matrix form. It is also assumed that the generating surface T of a cutting tool is given. This means that the equation of the surface T is represented in vector form r T = r T(U T,V T), or in matrix form. The analysis of kinematics of the multiparametric sculptured surface generation is based on the presumption that any desired relative motion of the work and of the cutting tool can be performed on the NC machine. Without loss of generality, the analysis of kinematics of sculptured surface generation can be substantially simplified if the principle of inversion is used. In order to implement this fundamental principle of mechanics for investigation of surface generation, an additional motion is applied to both the work and the cutting tool. The speed of the additional motion is equal to the speed of the motion that the work is performing in a certain machining operation. The direction of the additional motion is reversed to the motion that the work is performing in the machining operation. When summarized with the initial motions, then the work gets stationary. However, the resultant motion of the cutting tool gets more complex. It is composed now of the initial motions of the cutting tool and the reversed motions of the work.

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Kinematic Geometry of Surface Machining

Implementation of the principle of inversion has proven convenient for the investigation of multiparametric surface generation. 2.1.1 Establishment of a Local Reference System A reference coordinate system is necessary for the development of an analytical description of the instant motion of the cutter relative to the work. A right-hand-oriented Cartesian coordinate system is used below as the local reference system. Several approaches can be employed to get the local coordinate system orthogonal. First, a convenient way to establish an appropriate local coordinate system is to use a coordinate system that is naturally embedded to the surface P. For this purpose, Darboux’s trihedron can be implemented. Darboux’s trihedron is a trihedron associated with a point on a surface P and defined by three vectors, given by the normal unit vector n P to the surface and two mutually orthogonal principal unit tangent vectors t1.P and t 2.P to the surface such that n P = t1.P × t 2.P. Properties of the surface can be described in terms of displacement of the Darboux’s trihedron when its base point moves over the surface [1]. The unit vectors t1.P, t 2.P, and n P (Figure 2.1) make up a right-hand-oriented coordinate system (otherwise the direction of one of the parametric curves must be reversed), which is naturally embedded to the surface P. The unit vectors t1.P, t 2.P, and n P yield the moving orthogonal Cartesian coordinate system xPyPzP with the origin at point K of contact of the surfaces P and T. Point K is referred to as the cutter-contact-point (CC-point). The axis of the coordinate system xPyPzP is directed along the corresponding zP ±ωz

T

P

±vy

±ωy

K

t2.P

t1.P

±vx

yP ±ωx ZP YP

rP XP

Figure 2.1 The principal instant kinematics of sculpture surface generation.

© 2008 by Taylor & Francis Group, LLC

xP

31

Kinematics of Surface Generation

unit vectors t1.P, t 2.P, and n P. Thus, the moving coordinate system xPyPzP is established at every point of the sculptured surface P. Most of the equations are significantly simplified when Darboux’s trihedron is used. However, computation of the unit tangent vectors of the principal directions is often a computation-consuming procedure. Second, in order to compose a right-hand Cartesian local coordinate system xPyPzP with the origin at point K of contact of the surfaces P and T, the unit tangent vectors uP, vP, and nP can be employed. Generally speaking, pairs of the vectors uP and nP, vP and nP are orthogonal to each other, while the unit tangent vectors uP and vP are not orthogonal to each other. Aiming the composing of the right-hand Cartesian coordinate system, the third unit vector v *P = u P × n P can be used. The three unit vectors uP, v *P , and nP make up a right-hand trihedron. The orthogonal trihedron u P, v *P , and n P can be constructed at any surface point. However, in order to construct the trihedron, computation of the vector v *P is necessary at every surface point. Third, the initial parameterization of the surface P can be changed when it becomes an orthogonal parameterization. Under such a scenario, the unit tangent vectors u P and vP are always orthogonal to each other. In order to change the initial surface P parameterization, it is necessary to replace the parameters UP and VP with new parameters U P* = U P* (U P , VP ) and VPP** = VP* (U P , VP ). First derivatives with respect to the new parameters U P* and VP* ∂ rP ∂ r ∂U ∂ r ∂V = P ⋅ P* + P ⋅ P* ∂U P* ∂U P ∂U P ∂VP ∂U P

∂ rP ∂ r ∂U ∂ r ∂V = P ⋅ P + P ⋅ P ∂VP* ∂U P ∂VP* ∂VP ∂VP*

(2.1) (2.2)

can be drawn from these equations so that A* = A ∙ J. Here it is designated

 ∂ XP  ∂U P A=  ∂ XP  ∂V  P

∂ YP ∂U P ∂ YP ∂VP

T

∂ ZP  ∂U P   =  ∂ rP ∂ ZP   ∂U P ∂VP 

∂ rP   ∂VP 

(2.3)

and

 ∂U P  ∂U * P J=  ∂VP  ∂U *  P

∂U P  ∂VP*   ∂VP  ∂VP* 

(2.4)

is called the Jacobian matrix of the transformation.

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Kinematic Geometry of Surface Machining

The new fundamental matrix is given by

EP* FP*

EP FP* = A * T ⋅ A * = JT ⋅ A T ⋅ A ⋅ J = JT ⋅ FP GP*

FP ⋅J GP

(2.5)

By the properties of determinants, it can be seen from Equation (2.5) that

EP* FP*

EP FP* =|J|2 ⋅ FP GP*

FP GP

(2.6)

It can be shown on the premises of Equation (2.1), Equation (2.2), and Equation (2.6) that the unit surface normal n P is invariant under the transformation, as it could be expected. The transformation of the second fundamental matrix can similarly be shown to be given by

L*P M P*

LP M P* =|J|2 ⋅ MP N P*

MP NP

(2.7)

by differentiating Equation (2.1) and Equation (2.2) and using the invariance of nP. It can be shown from Equation (2.6) and Equation (2.7) that the principal curvatures and the principal directions are invariant under the transformation. Equation (2.6) and Equation (2.7) yield the natural form of the surface P representation with the new U P* and VP* parameters. It can be concluded that the unit normal vector n P and the principal directions and curvatures are independent of the parameters used and are therefore geometric properties of the surface. They should be continuous if the surface is to be tangent and curvature continuous. At that point it is necessary to establish a set of constraints onto the relationships U P* = U P* (U P , VP ) and VP* = VP* (U P , VP ) under which an orthogonal parameterization of the sculptured surface P can be obtained. In order to obtain an orthogonal parameterization of the sculptured surface P, the relations U P* = U P* (U P , VP ) and VP* = VP* (U P , VP ) have to satisfy the following two conditions, FP ≡ 0 and MP ≡ 0, which are the necessary and sufficient conditions for the orthogonally parameterized sculptured surface P. Once reparameterized, the surface P yields easy computation of the orthogonal trihedron u P, vP, n P at every surface point. The presented consideration clearly illustrates the feasibility of composing the right-hand trihedron u P, vP, n P using one of the methods discussed above. The three unit vectors u P, vP, n P can be used as the direct vectors of the local Cartesian coordinate system xPyPzP with the origin at point K of contact of the sculptured surface P and the generating surface T of the cutting tool.

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Kinematics of Surface Generation 2.1.2 Elementary Relative Motions

Instant relative motion of the cutting tool can be interpreted as an instant screw motion. Depending on the actual configuration of the local reference system xPyPzP, the instant screw motion of the cutting tool can be decomposed on not more than six elementary motions — that is, on three translations along and onto three rotations about axes of the local coordinate system xPyPzP. Not all of the six elementary relative motions are feasible. The translational motion of the cutting tool along zP is not feasible. It is eliminated from the instant kinematics of sculptured surface generation because of two reasons. First, the elementary motion of the cutting tool in the +n P direction results in interruption of the surface-generating process, which is not allowed. Second, the motion of the cutting tool in the −n P direction results in unavoidable interference of the surfaces P and T. Any interference of the surfaces P and T is not allowed. Hence, the speed of the translational motion of the cutting tool along the common perpendicular must be equal to zero (Figure 2.1): ∂ zP ⋅ kP = 0 ∂t

Vz ≡ Vn =

(2.8)

Here time is designated as t. The speed of translational motion of the cutting tool along xP and yP axes is designated as Vx and Vy, respectively. Then x, y, and z designate rotations about axes of the local coordinate system xPyPzP. According to the principal instant kinematics of sculptured surface generation (Figure 2.1), the cutting tool instant screw motion relative to the surface P can be decomposed on not more than five elementary instant relative motions. This set of five feasible elementary relative motions includes two translations Vx =

∂ xP ∂2rP ⋅ uP = ; ∂t ∂U P ∂ t

Vy =

∂ yP ∂2rP ⋅ vP = ∂t ∂VP ∂ t

(2.9)

along axes of the local reference system xPyPzP, and three rotations

ωx =

∂ϕ x ⋅ uP ; ∂t

ωy =

∂ϕ y ⋅ vP ; ∂t

ωz =

∂ϕ z ⋅ nP ∂t

(2.10)

about axes of the local reference system xPyPzP. Here the angles of rotation of the cutting tool about axes of coordinate system xPyPzP are designated as f x, f y, and f z, respectively.

Further, attention will be focused on special cases of sculptured surface machining when a motion of the cutting tool along the zP axis is allowed.

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Kinematic Geometry of Surface Machining

2.2 Generating Motions of the Cutting Tool After being machined on a multi-axis NC machine, the sculptured surface is represented as a series of tool-paths. Generation of a sculptured surface by consequent tool-paths is a principal feature of sculptured surface machining on a multi-axis NC machine. The motion of the cutting tool along a tool-path can be considered as a permanent following motion, and a side-step motion of the cutting tool (in the direction that is orthogonal to the tool-path) can be considered as a discrete following motion. The surface P can be generated as an enveloping surface to consecutive positions of the moving surface T when the surfaces P and T make either linear or point contact. When linear contact of the surface is observed, then a onedegree-of-freedom relative motion of the cutting tool is sufficient for generating the surface P. Such relative motion is referred to as one-parametric motion of the cutting tool. When the surfaces make a point contact, then a two-degreesof-freedom relative motion of the cutting tool is required for generating the entire surface P. Such relative motion is referred to as two-parametric motion of the cutting tool. The number of available degrees of freedom can exceed 2 degrees. This results in multiparametric motion of the cutting tool. Known methods of developing machining operations do not provide a single solution to the problem of synthesis of the most efficient (i.e., optimal) machining operations. Known methods return a variety of solutions to the problem, of which the efficiency of each is not the highest possible. For the computing of parameters of relative motion, known methods are based mostly on the equation of contact: n P ∙ VΣ = 0. Here VΣ denotes the speed of the resultant motion of the cutting tool relative to the work. The equation of contact n P ∙ VΣ = 0 imposes restrictions onto only one component of the relative motion of the cutting tool and of the work. This means that projections of speed of the resultant motion onto the direction specified by the unit normal vector n P must be equal to zero PrnVΣ = 0 ≡ 0. No restrictions are imposed by the equation of contact onto other projections of the vector of resultant motion of the cutting tool and of the work. In compliance with the equation of contact, the magnitude and direction of VΣ within the common tangent plane can be arbitrary. Evidently, the infinite number of the vectors VΣ satisfies the equation of contact n P ∙ VΣ = 0. All are within the tangent plane to the sculptured surface P. This is the principal reason why the implementation of known methods returns an infinite number of solutions to the problem under consideration. No doubt, performance of a sculptured surface generation depends on direction of the vector VΣ. For a certain direction of VΣ, it is better; for another

To the best of the author’s knowledge, the condition of contact in the form that slightly differs from the equation of contact n P ∙ VΣ = 0 is known at least since publication of the monograph by Willis [39]. In the present form, n P ∙ VΣ = 0, the equation of contact is known from the late 1940s/early 1950s [38].

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Kinematics of Surface Generation

direction of VΣ, it is poor. This yields intermediate conclusions that the optimal direction of VΣ exists, that this direction satisfies the equation of contact n P ∙ VΣ = 0, and that the optimal parameters of the direction of VΣ can be computed. For the computation of the optimal parameters of the instant relative motion VΣ of the work and of the cutting tool, an appropriate criterion of optimization is necessary. The major purpose of the criterion of optimization is to select the optimal direction of the vector VΣ from the infinite number of feasible directions that satisfy the equation of contact n P ∙ VΣ = 0. In order to satisfy the equation of contact, vector VΣ of the resultant relative motion of the cutting tool must be within the tangent plane to the surfaces P and T at the CC-point K. This is the geometrical interpretation of the equation of contact. Consider a local reference system xPyPzP with the origin at CC-point K (Figure 2.2). In the coordinates system xPyPzP, vector VΣ can be described analytically by vector equation: VΣ =

∂ xP ∂ yP ∂z ⋅ uP + ⋅ v P + P ⋅ nP ∂t ∂t ∂t

(2.11)

In that same local coordinate system, the equality n P = k P is observed. Substituting Equation (2.11) and the relationship n P = k P along with Equation (2.8) into the equation of contact n P ∙ VΣ = 0, one can obtain n P ⋅ VΣ =

∂ zP =0 ∂t

(2.12)

P nP

uP

VΣ

vT

K

vP uT nT

ZP

rP XP YP

Figure 2.2 Feasible relative motions of the cutting tool.

© 2008 by Taylor & Francis Group, LLC

v

36

Kinematic Geometry of Surface Machining

In order to satisfy the equation of contact, projection of the vector VΣ of the resultant motion on the direction perpendicular to the surfaces P and T must equal zero. This is proof that the vector VΣ must be within the common tangent plane to the surfaces P and T. It is important to point out here that the condition prnVΣ < 0 can be considered as the condition of roughing. Portions of the surface T that perform such motion remove stock while machining the work. Condition prnVΣ = 0 that is equivalent to the condition of contact n P ∙ VΣ = 0 corresponds to generating the surface being machined. Finally, the condition prnVΣ > 0 relates to portions of the surface T that are departing from the machined surface P. These conditions are presented in more detail in Chapter 5. The equation of contact n P ∙ VΣ = 0 does not uniquely determine the instant kinematics of sculptured surface generation. In addition to the infinite number of feasible directions for the vector VΣ, one more reason can affect the indefiniteness of the equation of contact. Location and orientation of the common perpendicular n P are uniquely specified by the geometry of the surface P. Usually it cannot be changed. However, in special cases of machining, the orientation of the common perpendicular n P can be changed for manufacturing purposes. For example, when machining a thin-wall part (Figure 2.3), an elastic deformation can be applied to the work. Under the applied load, unit normal vectors n ma , nbm , and n cm to the part surface P become parallel to each other. In the deformed stage, the work is machining on a lathe with simple motion of the cutter relative to the work. The elastic deformation results in the plane machining instead of machining of the concave sculptured surface (for this purpose, the magnitude of the applied load may vary according to the cutter feed rate — a distributed load of variable magnitude can be applied as well). After being released, the work gets it original shape, and the machined plane

Load

a b c

nam

a

nbm

naP

b

ncm

nbP ncP

c Cutter

P

Feed

Figure 2.3 An example of the application of elastic deformation of the work for the purpose of machining the surface.

© 2008 by Taylor & Francis Group, LLC

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Kinematics of Surface Generation

transforms into a concave sculptured surface P. Unit normal vectors n aP , nbP , and n cP to the machined surface P are not parallel to each other. If elastic deformation is used for manufacturing purposes, then the equation of contact must be satisfied in the deformed stage of the surface being machined. Elastic deformation of a work for manufacturing purposes is observed when machining flex-spline that is an essential machine element of a harmonic drive, and in other applications. The capability to change the orientation of the unit normal vector to the surface is limited; however, such a capability exists, and it affects the generation of the surface P. The last is of principal importance. Capabilities of variation of orientation of the unit normal vector nT to the generating surface T of a cutting tool are significantly wider, especially when implementing cutting tools of special design with variable shape and parameters of the surface T for machining a given sculptured surface [28,29,33]. When machining a sculptured surface on a multi-axis NC machine, the cutting tool is performing a continuous follow motion along every tool-path [30,33]. Therefore, the generating motion can be considered as a continuous follow motion of the cutting tool relative to the work. This motion results in the CC-point traveling along the tool-path. After the machining of a certain tool-path is complete, then the cutting tool feeds across the tool-path in a new position. The machining of another toolpath begins from the new position of the cutting tool. Hence, the feed motion of the cutting tool can be represented as a discontinuous follow motion of the cutting tool relative to the work. This motion results in the CC-point traveling across the tool-path. The generating motion of the cutting tool can be described analytically. For this purpose, the elementary motions that make up the principal instant kinematics of surface generation are used (Figure 2.1). The elementary relative motions are properly timed (synchronized) with one another in order to produce the desired instant generation motion of the cutting tool (Figure 2.4). The following equations can easily be composed based on the premises of the analysis of the instant kinematics of sculptured surface generation:

|Vx|=|ω y|⋅ RP .x

and |Vy|=|ωx|⋅ RP . y

(2.13)

Here R P.x and R P.y designate the normal radii of curvature of the sculptured surface P. The radii of curvature R P.x and R P.y are measured in the plane sections through the unit tangent vectors u P and vP, respectively. Equation (2.13) yields a generalization of the following kind:

|VΣ|=|ω T - P|⋅ RP . Σ

(2.14)

where VΣ is the vector of the resultant motion of the CC-point along the toolpath; T−P is a vector of instant rotation of the surface T about an axis that is perpendicular to the normal plane through the vector VΣ; and R P.Σ is the radius of normal curvature of the surface P in the direction of VΣ.

© 2008 by Taylor & Francis Group, LLC

38

Kinematic Geometry of Surface Machining OT

nP

R

The Cutting Tool

VΣ

P

K

T

The Work ωT−P

RP

OP

Figure 2.4 Generating motions of the cutting tool.

After the relative motion of the cutting tool in the direction of the unit normal vector n P = k P is eliminated from further analysis, then the vector VΣ yields the following representation in projections on axes of the local coordinate system xPyPzP: VΣ =

∂ xP ∂ yP ⋅ uP + ⋅ vP ∂t ∂t

(2.15)

In order to satisfy the conditions (see Equation 2.13), both of the additives in Equation (2.15) which result in a continuous follow motion of the CC-point over the surface P are required. Otherwise, the relative motion of the work and of the cutting tool cannot be identified as the generating motion of the cutting tool. The condition (see Equation 2.13) is satisfied if

∂ xP ⋅ u P = ω y × ( RP . x ⋅ n P ) ∂t

and

∂ yP ⋅ v P = ω x × ( RP . y ⋅ n P ) ∂t

(2.16)

Under this result, Equation (2.15) casts into

VΣ = ω y × ( RP .x ⋅ n P ) + ωx × ( RP . y ⋅ n P )

(2.17)

The generating motion of the cutting tool satisfies both the equation of contact n P ∙ VΣ = 0 and Equation (2.17) at every instant of machining of the sculptured surface on a multi-axis NC machine.

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Kinematics of Surface Generation

39

The relative motion Vz of the cutting tool (Figure 2.1) is not completely eliminated from further analysis. Taking into consideration the tolerance on accuracy of machining of the surface P (Figure 1.5), the motion Vz along the unit normal vector n P is feasible if it is performing within the tolerance d = + d + d . Moreover, due to deviations of the desired cutting tool motion from the actual cutting tool motion, the motion Vz always observed is the actual machining operation. If necessary, the motion Vz can be incorporated into the principal instant kinematics of surface generation. This is one more example of the difference between the classical differential geometry of surfaces and the kinematic geometry of surface generation. The principal instant kinematics of surface generation includes five elementary relative motions. Thus, the surface P can be represented as an enveloping surface to consecutive positions of not more than five-parametric motion of the surface T of the cutting tool.

2.3 Motions of Orientation of the Cutting Tool As mentioned above, the machining of a sculptured surface on a multi-axis NC machine is the most general case of surface generation. This is because two surfaces, the sculptured surface P and the generating surface T of the cutting tool make point contact at every instant of machining. Among various kinds of relative motions of the cutting tool, one more motion can be distinguished. When performing relative motion of this kind, the CC-point does not change its position on the sculptured surface P being machined. This motion changes only the orientation of the cutting tool relative to the work. Motions of this kind are referred to as orientational motions of the cutting tool. When performing the orientational motion, the CC-point can remain in location on both, on the surface P as well as on the generating surface T of the cutting tool. Orientational motion of this kind is referred to as the orientational motion of the first kind. When machining a sculptured surface, the CC-point can remain in location on the surface P and change its location on the generating surface T of the cutting tool. Orientational motion of this kind is referred to as the orientational motion of the second kind. Speed of the orientational motions of the cutting tool is a function of variation of the principal curvatures of the surface P at the current CC-point, and of speed of the generating motion. Orientational motions of the cutting tool do not directly affect the stock removal capability of the cutting tool or the generation of the surface P. These motions change orientation of the cutting tool relative to the work as well as the relative direction of the generating motion of the cutting tool. In order to identify all feasible orientational motions of the cutting tool, it is helpful to consider all feasible groups of relative motions of the

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Kinematic Geometry of Surface Machining

cutting tool. All groups of the relative motions are represented with the singular relative motions and with the combined relative motions. The singular relative motions are composed of one elementary relative motion of the cutting tool. The combined relative motions are composed of two or more elementary relative motions of the cutting tool. There are only five groups of elementary relative motions of the cutting tool. The number of elementary relative motions at every group of relative motions is equal to the number of combinations of five elementary motions by i elementary motions. Here i = 1, 2, …, 5. The total number N of relative motions in the principal instant kinematics of surface generation can be computed from the following equation: 5

N=

∑ i =1

5

Ni =

∑C i =1

i 5

= 31

(2.18)

Analysis of all 31 relative motions reveals that only a few elementary relative motions and their combinations can be distinguished as the orientational motions of the cutting tool. They are as follows [30, 31, 33]: The first group of the motions: {n} The second group of the motions: {u, Vv}, {v, Vu} The third group of the motions: {u, n, Vv}, {v, n, Vu} The fourth group of the motions: {u, Vv, v, Vu} The fifth group of the motions: {u, Vv, n, v, Vu} Ultimately, one can come up with a set of the orientational motions of the cutting tool. One is the singular orientational motion, and six others are the combined orientational motions of the cutting tool. The orientational motions of the first kind are represented with the only singular orientational motion {n}. All other orientational motions are the orientational motions of the second kind. The orientational motion {u, Vv, n, v, Vu} is the most general. Other orientational motions can be considered as a particular motion of that one. Elementary motions that include a combined orientational motion of the cutting tool are timed (synchronized) with one another. The rotational elementary motion u about the xP axis is timed with the translational motion Vv along the yP axis. Similarly, the rotational elementary motion v about the yP axis is timed with the translational motion Vu along the xP axis. The timing of the elementary motions results in the surface T sliding over the sculptured surface P. The timing of that kind of elementary motions can be achieved when the following condition is satisfied. Orientational motion of the second kind can be considered as a superposition of the instant translational motion with a certain instant speed VT−P, and of the instant rotation  T−P of the cutting tool (Figure 2.5). In order to be an orientational

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Kinematics of Surface Generation

ωn OT RT

nP

The Cutting Tool ωT−P VT− P P

T

K TheWork RP

OP

Figure 2.5 Motions of orientation of a cutting tool.

motion of the cutting tool, the following equality must be satisfied: |VT - P|=|ω T - P|⋅ R T

(2.19)

where  T−P is the vector of the instant rotation of the cutting tool about the axis OT that crosses the vector VT−P at a right angle and RT is the radius of curvature of the surface T in the normal plane section through the vector VT−P. Vector VT−P can be represented in a form similar to that of Equation (2.15): VT - P =

∂ xP ∂ yP ⋅ uP + ⋅ vP ∂t ∂t

(2.20)

In order to satisfy the necessary condition (see Equation 2.19), both additives in Equation (2.15) must result in continuous following motion of the CC-point over the surface P. Otherwise, the relative motion cannot be distinguished as the orientational motion of the cutting tool. The condition (see Equation 2.19) is satisfied if

∂ xP ⋅ u P = ωv ⋅ RT .u ∂t

and

∂ yP ⋅ v P = ω u ⋅ RT .v ∂t

(2.21)

where RT.u and RT.v are normal radii of curvature of the generating surface T of the cutting tool in the plane sections through the unit vectors u P and vP, respectively; and u and v are rotations about the directions of u P and vP.

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Kinematic Geometry of Surface Machining

Equation (2.21) yields a representation of the combined orientational motion of the cutting tool Vorient in the form  ωorient = ω u + ω v

{ωu , Vv , ω v , Vu } ⇒ 

Vorient = ωu × ( R T .v ⋅ n P ) + ωv × ( R T .u ⋅ n P )

(2.22)

On the premises of the performed analysis, the following classification of the orientational motions of the cutting tool is developed (Figure 2.6). When representing a sculptured surface P as an enveloping surface to consecutive positions of the five-parametric motion of the generating surface T (see Section 2.2), the orientational motions of the cutting tool can be omitted from consideration. Orientational motions of the cutting tool make the machining of a sculptured surface more agile. The orientational motions of the cutting tool could increase the performance of the machining operation. The developed approach yields computation of the optimal parameters of all the motions of the cutting tool relative to the work. The solution to the problem of synthesis of the optimal kinematics of generation of a sculptured surface on a multi-axis NC machine can be drawn up from the analysis of kinematics of multiparametric motion of the cutting tool relative to the work.

2.4

Relative Motions Causing Sliding of a Surface over Itself

Surfaces that allow sliding over themselves are convenient for many applications in mechanical and manufacturing engineering. Surfaces of this kind can be generated by corresponding motion of a curve of an appropriate shape. The necessary motion can be easily performed on a machine tool. Relative motions causing sliding of a surface over itself are investigated in [30,32,35] and others. Reshetov and Portman [37] introduced the so-called hidden connections that must be performed on the machine tool of a motion that results in sliding of the surface over itself. For the analytical description of the hidden connections, it is necessary to compute the derivatives of rP with respect to the elementary motions Ωi (here i = 1, 2, …, n designates an integer number). The hidden connections exist if and only if pairs of colinear or triples of coplanar vectors are available among the derivatives ∂rP/∂Ωi of rP. Surfaces that allow sliding over themselves can be considered as the surfaces for which a resultant relative motion of a special kind is feasible. Relative motion of this kind results in the enveloping surface to consecutive positions of the moving surface P being congruent to the surface P. The same is true for the generating surface T of the cutting tool.

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Kinematics of Surface Generation

Motions of Orientation of a Cutting Tool

Of the Second Kind

Of the First Kind

Simple

{ωu, ωv, Vu, Vv}

{ωn}

{ωu, Vv}

{ωv, Vu}

{ωu, ωv, ωn, Vu, Vv}

{ωv, ωn, Vu}

{ωu, ωn, Vv}

Figure 2.6 A classification of the motions of orientation of a cutting tool.

Surfaces that allow sliding over themselves are of high importance for their application in mechanical and manufacturing engineering. It is of critical importance to discover all possible kinds of surfaces of that kind. It is proven [32] that screw surfaces of constant pitch represent the general kind of surfaces that allow sliding over themselves. It is convenient to represent a screw motion of constant pitch as a superposition of two motions: of a translation and of a rotation. Speeds of the motions are timed (synchronized) with one another. The last results in a screw motion being considered as one parametric motion — i.e., as a one-degree-of-freedom relative motion (Figure 2.7). Other feasible kinds of surfaces that allow sliding over themselves can be considered as a particular case of the corresponding screw surface of constant pitch px. When pitch of a screw surface is equal to zero (px = 0), then the screw surface reduces to a surface of revolution with a certain axial profile. Rotation of a surface of revolution about its axis of rotation causes sliding of the surface over itself. The rotation of a surface of revolution has to be considered as a one-parametric motion similar to that of a screw surface of constant pitch. When the pitch of a screw surface is equal to infinity (px = ∞), then the screw surface degenerates to a general cylinder. Translation of the cylinder along its straight generating line causes sliding of the cylinder over itself. The translation of a general cylinder has to be considered as a one-parametric motion similar to that of a screw surface of constant pitch. A translation and a rotation that are not timed (synchronized) with one another result in the generation of a cylinder of revolution. Both the translation as well as the rotation of the cylinder of revolution cause sliding of the circular cylinder over itself. The translation and the rotation of a cylinder

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Kinematic Geometry of Surface Machining

Two Synchronized Elementary Motions

One Elementary Motion

Two Nonsynchronized Elementary Motions

Three Nonsynchronized Elementary Motions

Cylinder of Revolution General Cylinder

YP

ZP

YP

YP

ZP

Surface of Revolution YP

ZP

ZP

XP Screw Surface of Constant Pitch

Plane

XP

XP

XP

Sphere Cylinder of Revolution

XP

XP

ZP YP

ZP

ZP

YP

XP

YP One Degree of Freedom

Two Degrees of Freedom

Three Degrees of Freedom

Figure 2.7 Part surfaces that are invariant with respect to a group of motions.

of revolution is considered as a two-parametric motion — that is, as a twodegrees-of-freedom relative motion. A smooth, regular surface P or T can perform three elementary relative motions that are not timed (synchronized) with one another. Only two threeparametric motions are feasible. The first three-parametric motion includes three independent rotations about axes of a certain Cartesian coordinate system. Three independent rotations cause the sphere to slide over itself. The second three-parametric motion includes two translations and one rotation. All of these elementary relative motions are independent of each other. Two translations and one rotation cause the plane to slide over itself. Three rotations in the first example as well as two translations and one rotation in the second example represent the three parametric motions — that is, the three-degrees-of-freedom relative motions. The performed analysis allows for another interpretation of surfaces that allow sliding over themselves. The surfaces of that kind can also be considered as the surfaces that are invariant with respect to a certain group of elementary motions.

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Kinematics of Surface Generation

45

According to the Bonnet theorem (see Section 1.1), the specification of the first and second fundamental forms determines a unique surface, and those two surfaces that have identical first and second fundamental forms must be congruent. Six fundamental magnitudes determine a surface uniquely, except as to position and orientation in space. This is the main theorem in surface theory. It is natural to assume that the property of surfaces that allows for sliding over can be interpreted in terms of six fundamental magnitudes of the first and the second fundamental forms if Gauss’ characteristic equation and the Codazzi-Mainardi’s relationships of compatibility are satisfied. The property of surfaces to allow sliding over themselves imposes an additional restriction onto relationships among the fundamental magnitudes. Thus, a representation of a surface in natural form will possess certain features. Comprehensive analysis and solution to the problem of analytical representation of surfaces that allow sliding over themselves can be found in [30–33,35]. For this purpose, analytical criteria expressed in terms of the fundamental magnitudes EP, FP, GP, and LP, MP, NP were composed for all feasible kinds of surfaces that allow for sliding over themselves. These criteria enable one to identify whether a certain surface allows for sliding over itself or not, and if it does, what type of surface it represents. The criteria reflect the following properties of a surface: For a screw surface: axial pitch is constant (px = Const) For a surface of revolution: axial pitch is equal to zero (px = 0) For a general cylinder: axial pitch is equal to infinity (px = ∞) For all the above-listed surfaces, the generating line is of nonchangeable shape, and its orientation with respect to the directing line of the surface remains the same. More particular cases of surfaces that allow sliding over themselves require additional restrictions to be imposed. The interested reader may wish to see References [32,35] for details on analytical descriptions of surfaces that allow sliding over themselves.

2.5 Feasible Kinematic Schemes of Surface Generation Kinematics of surface generation is a cornerstone of an infinite variety of various machining operations. Together with the shape and parameters of geometry of the surface P, the kinematic schemes of surface generation specify the generating surface of the cutting tool, as well as the principal part of the kinematic structure of a machine tool. Nonagile kinematics of surface generation is usually performed on conventional machine tools. Kinematics of this kind features relative motion of the cutting tool with constant parameters. It is used for the generation of surfaces P that allow for sliding over themselves. Various feasible versions of

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Kinematic Geometry of Surface Machining

nonagile kinematics of surface generation can be considered as a particular (degenerated) case of agile kinematics of surface generation, for example, as the kinematics of sculptured surface machining on a multi-axis NC machine (Figure 2.1). Only relative motions are investigated in kinematics of surface generation. A certain combination of elementary relative motions of the work and of the cutting tool makes up the corresponding nonagile kinematics of surface generation. Translational and rotational motions of the work and of the cutting tool along and about certain axes are implemented as the elementary relative motions. A combination of translations and of rotations of the cutting tool is referred to as the kinematic scheme of surface generation. Kinematics of surface generation that allows sliding of one or both surfaces P and T over themselves is usually referred to as the rigid kinematics of surface generation. Rigid kinematics of surface generation usually features constant relative translation motions Vi and rotations w i. No acceleration or deceleration usually occurs. Due to this, the kinematic schemes of surface generation can be composed. Similar to the multi-axis sculptured surface generation, the principle of inversion is used for the investigation of the nonagile kinematics of surface generation. In order to apply this fundamental principle of mechanics, both the work and the cutting tool are moved with the motions directed opposite to the motions that the work is performing in the machining operation. Ultimately, this results in the work becoming motionless, and the cutting tool performs all the required relative motions. Under such an assumption, a Cartesian coordinate system XPYPZP associated with the work is considered as the stationary coordinate system. Two principal problems are tightly connected with the kinematics of surface generation. One is referred to as the direct principal problem of surface generation, and the other is referred to as the inverse principal problem of surface generation. The direct principal problem of surface generation relates to the determination of the shape and parameters of the generating surface of the cutting tool for machining a given part surface. Therefore, the generation surface T of the cutting tool is the solution to the first principal problem of surface generation. The cutting tool surface T is expressed in terms of shape and parameters of the surface P and of parameters of the kinematic scheme of the machining operation. The inverse principal problem of surface generation relates to the determination of the shape and parameters of the actual machined surface P. The set of necessary and sufficient conditions of proper surface generation (further, conditions of proper PSG) are not always satisfied [29,31,33]. This causes unavoidable deviations of the actual machined surface Pact from the desired nominal surface Pdes. Therefore, the actual generated surface Pact is

The necessary and sufficient conditions of proper part surface generation are considered in Chapter 7.

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47

the solution to the second principal problem of surface generation. The surface Pact is expressed in terms of parameters of the generating surface T of the cutting tool and of parameters of the inverse kinematics scheme of the machining operation. On the premises of the performed analysis, it is convenient to distinguish between two different kinematic schemes. First, when a surface P is considered as the stationary surface, then the combination of elementary motions of the cutting tool relative to the work represents a true kinematic scheme of surface generation. Second, in another case, the cutting tool (or a coordinate system to which the cutting tool will be associated) can be considered as the stationary element. Then the combination of elementary motions of the work relative to the coordinate system of the cutting tool can be considered as the kinematic scheme of the cutting tool profiling. Kinematic schemes of cutting tool profiling are used for solving the direct principal problem of surface generation. The application of it yields the determination of the generating surface T of the cutting tool for the machining of a given surface P. In order to solve the inverse principal problem of surface generation, the true kinematic schemes of surface machining are used. This yields determination of the actual machined part surface Pact. Machining operations of actual surfaces often include motions that cause sliding of the surface P or the surface T over them. Motions of this kind simplify obtaining the required speed of cutting, make possible the generation of the entire surface P, and so forth. Relative motions that cause sliding of the surface P or the surface T over them are not a part of the kinematic schemes of surface generation. They are considered separate from the kinematic schemes of surface generation. Various kinematic schemes of surface generation can be composed. Aiming for the development of the easiest possible machining operations, it is recommended that a limited number of elementary motions of simple kinds (translations and rotations) be used. Usually, the total number of elementary relative motions in a kinematic scheme does not exceed five motions. Due to that, the kinematic schemes of surface generation composed of three or less translations and rotations are the most implemented in practice. Because of this, the total number of kinematic schemes of surface generation is not infinite and is relatively small. Kinematic schemes of surface generation having more than three elementary motions have limited implementation in industry. Therefore, the consideration below is limited to the analysis of kinematic schemes of surface generation having three or less elementary motions. Kinematic schemes with more complex structures are considered briefly. Instant motion of a rigid body in E3 space can be represented as a combination of an instant rotation and an instant translation along and about a particular line or axis. The combination of the translation and rotation of the rigid body in E3 space is referred to as an instant screw motion. Apparently, this is because the instant screw motion of the rigid body resembles in part the motion that is performed by a bolt or a screw. Consideration of instant

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Kinematic Geometry of Surface Machining

motion of a rigid body in E3 space is an appropriate starting point at which to begin the investigation of the kinematic schemes of surface generation. Relative motion of the cutting tool can be reduced to a corresponding kinematic scheme of surface generation only for nonagile kinematics of surface machining. The nonagile kinematics of surface machining features the elementary relative motions of the cutting tool with constant parameters of the motions. Under such a scenario, for the investigation of all feasible kinematic schemes of surface generation, axodes that are associated with the work and with the cutting tool can be employed. Implementation of the concept of axodes is useful for the analysis and visualization of mating surfaces in a certain surface machining operation. An axode associated with the work, and the axode associated with the cutting tool make line contact with one another and roll over each other without sliding. In many practical cases of surface generation (e.g., in gear-shaping operation), the axodes are congruent to the pitch surfaces. In the above example, the axodes are congruent to the pitch surfaces of the gear being machined and of the gear shaper. However, the congruence of the axodes and of the pitch surface is not the mandatory requirement for the surface generation. In a gear hobbing operation when axes of rotation of the gear and of the gear hob cross each other, the axodes do not coincide with the corresponding pitch surfaces. In the kinematic schemes of surface generation with more complex structure, axodes yield their sliding in their axial direction. However, there must exist at least one point within the line of the axodes’ contact at which there is no sliding in transverse direction. This point is usually referred to as the mean point of contact of axodes. It is reasonable to begin the analysis of kinematic schemes from the most general kinematic scheme of surface generation. The most general kinematic scheme of surface generation includes five elementary relative rotations. Kinematic schemes of surface generation with more general structure are theoretically feasible. However, they are not used in practice and therefore are not considered here. Various combinations of five or less elementary translations and rotations make up different kinematic schemes of surface generation. A detailed analysis of all feasible combinations of five elementary motions reveals that without loss of generality, it is sufficient to consider just one kinematic scheme of surface generation. This combination of five elementary motions is referred to as the kinematic scheme of the fifth 50 class. The kinematic scheme of the 50 class is made up of one translation and four rotations (Figure 2.8). For convenience, axodes of the different elementary motions in Figure 2.8 are shown separately. Axes of the two rotations are parallel to each other. The axis of the third rotation crosses the first two axes at a certain crossed-axis angle Σ1−2 (Figure 2.8). The axis of the fourth rotation aligns with the shortest distance of approach of the above axes of rotation. The translational motion is performed along the fourth axis of rotation.

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Kinematics of Surface Generation C1−3

ω3 2

O3

V1−2 1

C1−2 ω1

ω2

ω4

3

O2 Σ1−2 O1 Figure 2.8 The kinematic scheme of surface generation of the fifth class.

Axode 1 rotates about axis O1 with a certain angular velocity w 1. Axode 2 rotates about its axis O2 with an angular velocity w 2. Axes O1 and O2 are at a certain center distance C1−2. They intersect at a crossed-axis angle Σ1−2. (Due to lack of space, the crossed-axis angle Σ1−2 is not shown in Figure 2.8.) At that same time, axode 1 performs another rotation w 3 about axis O3. The axes O1 and O3 are parallel to each other and are at a center-distance C1−3. Axode 3 is an enveloping surface to consecutive positions of axode 2 in its rotation about the axis O3. Magnitudes and directions of the rotations w 1, w 2, w 3, and magnitudes and sign of the center-distances C1−2, C1−3 are timed (synchronized) with each other in a timely manner. The timing of the parameters enables only axial sliding of the axodes 1, 2, and 3 in the mean point. Under such a scenario, all three axodes are shaped in the form of corresponding one-sheet hyperboloids of rotation. Axis O4 of rotation w 4 aligns with the center-distance C1−2. This rotation changes the actual value of the crossed-axis angle Σ1−2. Translational motion V1−2 is performing in the direction that is parallel to the center-distance C1−2. The translation V1−2 causes change of magnitude and sign of the center-distances C1−2, C1−3. The kinematic scheme of the fifth class is generalized. It can be employed for the development of novel methods of surface machining. However, to the best of the author’s knowledge, not many methods of surface machining that are based on implementation of the fifth-class kinematic scheme (Figure 2.8) have been developed. This yields a conclusion that in the meantime, this kinematic scheme of surface generation is of importance mostly from a theoretical standpoint. Kinematic schemes of the fourth class can be obtained from the kinematic scheme of the fifth class (Figure 2.8). The interested reader may wish to go to

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Kinematic Geometry of Surface Machining

Reference [31] for details on the classification (Figure 2.9) of all feasible kinematic schemes of surface generation. The classification also incorporates the kinematic schemes having additional motion that causes sliding of the axode over itself. Motions of this kind can increase the capability of the principal kinematic schemes of surface generation.

Principal Kinematic Scheme of the 50 Class 41: ω1 = 0 312: ω2 = 0

313: ω3 = 0

315: V1–2 = 0

314: ω4 = 0

43: ω3 = 0 331=313 332 = 323 ω2

44: ω4 = 0

334: ω4 = 0

335: V1–2 = 0

ω2

V1–2 ω1

V1–2 ω1

−ω4

ω1

ω1

ω2

ω1

ω2 ω1

+ω 4

341= 314 342 = 324 343 = 334

−ω 4

+ω 4

ω3

ω1

ω2

ω2

ω1

ω2 ω1

ω1 ω2

345: V1–2 = 0

ω2 ω2

ω3 ω3

ω ω1

V

ω

V

ω1 = 0; ω2 = 0; ω3 = 0; ω4 = 0; V1–2 = 0

Figure 2.9 Kinematic schemes of surface generation.

© 2008 by Taylor & Francis Group, LLC

ω1 ω2

ω2

51

Kinematics of Surface Generation Principal Kinematic Scheme of the 50 Class 42: ω2 = 0 321= 312

324: ω4 = 0

323: ω3 = 0

325: V1–2 = 0

45: V1–2 = 0 341= 314 342 = 324 343 = 334 343 = 334 ω3 ω2

ω1

ω1

ω2

ω2

ω1

ω3 ω1

ω2

ω1

ω3 ω1

ω2

ω3

ω2

ω1

ω1

ω2

ω2

ω2

ω1

ω2

Figure 2.9 (Continued)

2.6

On the Possibility of Replacement of Axodes with Pitch Surfaces

Pitch surfaces in many cases of surface machining are identical to the axodes (i.e., the pitch surface is congruent to the corresponding axode). Such congruence is observed, for example, in a conventional gear-shaping operation. However, in many cases of surface machining, the pitch surface is not congruent to the corresponding axode (e.g., in all kinds of conventional gear shaving operations, in gear hobbing operations, and in others, the pitch surface is not congruent to the axode).

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Kinematic Geometry of Surface Machining

The analysis of kinematic schemes of surface generation (Figure 2.9) reveals that the axodes of the surface P and of the surface T can be of simple shape (plane, circular cylinder, circular cone), as well as of more complex shape (internal and external noncircular cylinders and cones, internal and external one-sheet hyperboloids of revolution, and even more complex surfaces). The implementation of axodes of complex shapes is usually inconvenient. It is preferred to replace axodes of complex shape with pitch surfaces having simple geometry. Many methods of surface machining feature pitch surface and axode that are not congruent to one another. Moreover, these two surfaces might have completely different geometries. The following example illustrates the difference between the axodes and the pitch surfaces. Consider the kinematic scheme of surface generation for the shaping of a face gear (Figure 2.10a). The pitch surface PP of the face gear is rotating about its axis with a certain angular velocity w P. The pitch surface PT of the gear shaper is rotating about its axis with the angular velocity w T. The rotations w P and w T are timed (synchronized) in a timely manner. Because of this, at some point within the line of contact of the pitch surfaces PP and PT, there is no relative sliding of the surfaces. However, sliding is observed at all other points of the line of contact. Knowing the locations of the axes of the pitch surfaces PP and PT, and knowing rotations w P and w T, it is possible to construct the corresponding axodes AP and AT, which roll over each other completely without sliding. For the operation of shaping the face gear, Figure 2.10a clearly illustrates the difference between the axodes AP and AT, and between the corresponding pitch surfaces PP and PT. Another example that also illustrates the difference between the axodes and between the pitch surfaces is shown in Figure 2.10b. The depicted P2

А2

The Axode АT The Pitch Surface PT The Axode АP The Pitch Surface PP

ωT

ω2

P1 А1

ωP

Е

The Characteristic Е

(a)

ω1

(b)

Figure 2.10 Examples of the axodes and pitch surfaces in kinematic schemes of surface generation.

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Kinematics of Surface Generation

53

principal kinematic scheme of surface generation relates to the operation of machining of bevel gears. Numerous impressive examples that clearly illustrate the difference between the axodes and the pitch surfaces are known from the monograph by Davidov [2]. Circular cylinders, circular cones, and planes are the most widely used kinds of pitch surfaces. The capabilities of the kinematic schemes of surface generation can be enhanced when used in addition to the axodes, and the possibility of implementing pitch surfaces PP and PT of the surfaces P and T is analyzed as well.

2.7

Examples of Implementation of the Kinematic Schemes of Surface Generation

This is a good point at which to provide examples of applications of the classification (see Figure 2.9) for the purpose of developing novel methods of surface machining, as well as for the purpose of designing cutting tools of novel design. Numerous impressive examples can be found in industry and are known by the proficient reader. Just a few examples, those not known by many, from the author’s application of the developed classification will be presented here. The kinematic schemes of zero class 012345 (see Figure 2.9) are made up of the only motion that results in the sliding of the surface P over itself. This kinematic scheme includes no surface-generating motions (w 1 = 0, w 2 = 0, w 3 = 0, w 4 = 0, V1−2 = 0). Machining operations with the kinematic schemes of zero class 012345 are used for the broaching of internal and external spur and helical gears, splines, the rotary broaching of bevel gears, and so forth. The kinematic schemes of the first class are made up of just one either translational or rotational motion. The kinematic schemes of those kinds are used for milling straight grooves, gears, and so forth. The kinematic schemes of the second class are made up of two motions. These kinematics schemes of surface generation are implemented, for example, for gear-shaping operations, both conventional and with modified orientation of the gear shaper axis of rotation, and so forth. The kinematic scheme of the conventional gear-shaping operation is made up of two rotations. One rotation is performed by the work, and another rotation is performed by the gear shaper. The rotations are timed (synchronized) with one another in a timely manner. The synchronization of rotations results in the axode of the gear rolling without sliding over the axode of the gear shaper. Reciprocation of the gear shaper does not affect the kinematic scheme, because this motion results in the surface P sliding over itself. A novel gear-shaping operation and a novel design of the gear shaper can be developed based on analysis of kinematic schemes of surface generation of

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54

Kinematic Geometry of Surface Machining αsh.c

Work

Osh.c

ωgear

Ogear

γsh.c

αsh.c

View A A

ωgear

Work

Osh.c

ωsh.c

Vsh.c

(a)

Ogear

ωsh.c

ξ sh.c = 1...3°

(b)

Figure 2.11 Machining an involute gear with the shaper cutter having a doubly inclined axis of rotation Osh.c . (The gear shaper is not shown in view A.)

the second class. For example, in the gear-shaping operation in Reference [24], the work and the gear shaper rotate about their axes Ogear and Osh.c (Figure 2.11) with certain angular velocities w gear and w sh.c. The rotations w gear and w sh.c are synchronized in a timely manner. The shaping cutter reciprocates with a velocity Vsh.c along the gear axis Ogear . The shaping cutter axis Osh.c is inclined and makes the clearance angle a sh.c with the axis Ogear of the gear. (Thus, the kinematic scheme of the method of gear shaping corresponds to the kinematic scheme of the class 2354. The last is shown in Figure 2.9.) The inclination of the shaping cutter axis makes a shaping cutter with zero clearance angle. The modified method of the gear-shaping operation features two important advantages. Implementation of the method yields an increase in accuracy of the machined gears and doubled tool-life of the gear shaper cutter. The additional inclination of the axis Osh.c on ξsh.c = 1…3° transforms the kinematic scheme of the method of gear shaping from the second class 2354 to the kinematic scheme 2453, also of the second class. Inclination of the axis Osh c on ξsh.c = 1…3° equalizes the clearance angle on the opposite flanks of the shaping cutter tooth. Such an inclination additionally increases the toollife of the shaping cutter. Many examples of novel methods of surface machining that have various kinematic schemes of the second class can be found in many sources [34,36, and others]. Kinematic schemes that include three or more elementary motions possess much wider kinematical capabilities. As an example, consider a method of gear finishing [13,26]. A pinion has longitudinally modified teeth (Figure 2.12). In Figure 2.12, d designates the actual value of the tooth flank modification. Pinion 1 with modified tooth surface (Figure 2.13a) is finishing with a gear-finishing tool 2 located inside bandage 3. Screw 4 that is coaxial with the gear-finishing tool drives all the pinions being finished. The crossed-axis angle Σ of axes of the

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Kinematics of Surface Generation

δ

Figure 2.12 A gear tooth with longitudinal modification.

gear-finishing tool OT and of the axes OP of the spur or helical pinions is equal to the right angle (see Figure 2.13a). In order to maintain the crossed-axis angle Σ equal to 90°, a corresponding synchronization of the pinion helix angle ψP and of the setting angles ζT and ζw (of the gear-finishing tool and of the driving worm, respectively) is required. The necessary synchronization of the angles ψP, ζT, and ζw can be derived from the following fundamental relationship [32]: db.T =

mZT cos α T 1 - cos 2 α T cos 2 ζ T

(2.23)

where db.T is the base diameter of the gear-finishing tool, m is the modulus, ZT is the number of starts of the gear-finishing tool, aT is a normal pressure angle, and zT is the setting angle of the gear-finishing tool. A similar relationship is valid for driving worm 4. A hand of threads of the driving worm is opposite to the hand of threads of the gear-finishing tool. Axis Ow of the driving worm aligns with axis OT of the gear-finishing tool (i.e., OT ≡ Ow, see Figure 2.13a). The pitch cone angle q T of the gear-finishing tool (Figure 2.13a) and a similar pitch cone angle q w of the driving worm are identical (q T ≡ q w). Shifting the conical driving worm up and down results in a change of width of the room between the gear-finishing tool and worm 4. In such a way, the pinions are feeding onto the gear-finishing tool. The supporting screens 5 subdivide the room between the gear-finishing tool and the driving worm into a number of chambers. While finishing, the pinion passes through the chamber. Due to Σ = 90°, the supporting screens have planar working surfaces. They are evenly distributed circumferentially inside the gear-finishing tool. Rotation is transmitting from the electric motor M to the gear-finishing tool and to the driving worm. The gear-finishing tool is rotating with a certain

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θT θw

R

1

5

θT

A

4

1

ωp

ωp

2

Vp

Op

Op

Vp

ωw

isynch

M

(a)

Figure 2.13 The concept of a method and apparatus for the finishing of a modified pinion with the tapered gear finishing tool. © 2008 by Taylor & Francis Group, LLC

Kinematic Geometry of Surface Machining

ωT 3

2

1

ωT

ωp

ωp

ωp

ωp

3

ωp

ωp

ωp

ωp

ωp

ωp

ωp

Kinematics of Surface Generation

View A

ωp

ωp

ωw

ωp

ωp

ωp ωp

ωp ωp

ωp ωp

ωp

ωp

ωp

8 4 9

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Figure 2.13 (Continued)

58

Kinematic Geometry of Surface Machining

angular velocity w T. The driving worm is rotating with the angular velocity w w. Rotation w w of the driving worm is synchronized with rotation w T of the gear-finishing tool in timely manner. The actual synchronization of the rotations w T and w w depends upon parameters of the generating surface T of the gear-finishing tool, upon the number of starts of the gear-finishing tool NT, and upon the number of starts of the driving worm Nw. While finishing, the pinions 1 are traveling through the chambers between the gear-finishing tool, the driving worm 4, and the supporting screens 9. The pinion is rotating about its axis OP with angular velocity w P, and is moving along the axes OT ≡ Ow with a certain speed VP. The pinion rotation w P is synchronized with the rotations w T and w w in the manner that allows at least one complete revolution of the pinion while traveling through the chamber. While finishing the pinion, the gear-finishing tool performs the required longitudinal modification of the pinion tooth surface. If necessary, profile modification of the pinion tooth surface can be performed following a conventional modification of the finishing tool profile. Similar to the finishing of pinions that have longitudinally modified teeth, the chamfering of the pinions can be performed as well [27]. An example of the chamfered pinion tooth is shown in Figure 2.14. In order to machine chamfers, face surfaces of the screens are offset from the axis of rotation OT of the tool (Figure 2.15). In order to increase the productivity of the gear finishing and chamfering operation, it is recommended that a multithread gear-machining tool and a multithread driving worm be applied. In this way, not one but several (dozens) of pinions could be machined in every chamber simultaneously, and an immense increase in productivity of the gear finishing operation could be observed. Numerous modifications of the disclosed method are known from References [5,10,11,15,16,18], and others.

Chamfers Figure 2.14 A chamfered gear tooth.

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Kinematics of Surface Generation View A 2

ωp

ωp

1

ωp

ωp

ωp

3

ωp

ωp

ωp ωp

H

ωp

ωp

ωw

ωp

ωp

ωp ωp ωp

ωp

8 4

ωp 9

Figure 2.15 The concept of a method and apparatus for the chamfering of pinion teeth.

The analysis of the kinematical capabilities of kinematic schemes (Figure 2.9) might be fruitful when solving complex problems concerned with surface machining. Interesting examples can be found from References [3,4,6–9,12,14,17, 19–23,25,26], as well as from many other sources. The interested reader may wish to apply the analysis of the discussed classification (Figure 2.9) for the purposes of the development of novel methods of surface machining, of novel design of cutting tools, of machine tools of novel design, and so on.

References [1] Darboux, G., Leçons la Théorie Générale des Surfaces et ses Applications Géométiques du Calcul Infinitésimal, Vol. 1, Gauthier-Villars, 1887. [2] Davidov, Ya.S., Noninvolute Gearing, Mashgiz, Moscow, 1950.

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[3] Pat. No. 831.546, USSR, A Method of Grinding of Relieved Surface of a Gear Hob./P.R. Rodin and S.P. Radzevich, Int. Cl. B24b 3/00, Filed 07.02.1979. [4] Pat. No. 880.589, USSR, A Gear Finishing Tool./S.P. Radzevich, Int. Cl. B21h 5/02, Filed 06.11.1979. [5] Pat. No. 921.727, USSR, A Gear Finishing Device./S.P. Radzevich, Int. Cl. B23f 19/00, Filed 28.09.1980. [6] Pat. No. 933.316, USSR, A Gear Cutting Tool./S.P. Radzevich, Int. Cl. B23f 21/00, Filed 18.10.1979. [7] Pat. No. 946.833, USSR, A Gear Hob./S.P. Radzevich, Int. Cl. B23f 21/16, Filed 13.06.1980. [8] Pat. No. 965.582, USSR, A Gear Finishing Tool./S.P. Radzevich, Int. Cl. B21h 5/00, Filed 02.10.1980. [9] Pat. No. 965.728, USSR, A Method of Grinding of Relieved Surface of a Tapered Gear Hob./S.P. Radzevich, Int. Cl. B24b 3/12, Filed 01.21.1980. [10] Pat. No. 969.395, USSR, A Device for Finishing Gears./S.P. Radzevich, Int. Cl. B21h 5/02, Filed 06.04.1981. [11] Pat. No. 984.744, USSR, A Device for Finishing Gears./S.P. Radzevich, Int. Cl. B23f 19/00, Filed 16.04.1981. [12] Pat. No. 996.016, USSR, A Device for Finishing Gears./S.P. Radzevich, Int. Cl. B21h 5/02, Filed 16.04.1981. [13] Pat. No. 1.000.186, USSR, A Device for Finishing Gears./S.P. Radzevich, Int. Cl. B23f 19/00, Filed 06.11.1981. [14] Pat. No. 1.004.029, USSR, A Gear Hob./S.P. Radzevich, Int. Cl. B23f 21/16, Filed 06.11.1979. [15] Pat. No. 1.028.450, USSR, A Device for Finishing Gears./S.P. Radzevich, Int. Cl. B23f 19/00, Filed 26.04.1982. [16] Pat. No. 1.055.578, USSR, A Device for Finishing Gears./S.P. Radzevich, Int. Cl. B21h 5/02, Filed 06.11.1981. [17] Pat. No. 1.087.309, USSR, A Method of Grinding of a Gear Hob./S.P. Radzevich, Int. Cl. B24b 3/12, Filed 17.10.1980. [18] Pat. No. 1.110.566, USSR, A Device for Finishing Gear./S.P. Radzevich, Int. Cl. B23f 19/00, Filed 07.06.1981. [19] Pat. No. 1.174.139, USSR, A Gear Finishing Tool./S.P. Radzevich, Int. Cl. B21h 5/00, Filed 10.10.1983. [20] Pat. No. 1.194.612, USSR, A Method of Grinding of Relieved Surface of a Gear Hob./S.P. Radzevich, Int. Cl. B24b 3/12, Filed 20.07.1984. [21] Pat. No. 1.196.232, USSR, A Method of Grinding of Relieved Surface of a Gear Hob./S.P. Radzevich, Int. Cl. B24b 3/00, Filed 03.08.1984. [22] Pat. No. 1.240.548, USSR, A Method of Grinding of a Gear Hob./S.P. Radzevich, Int. Cl. B24b 3/12, Filed 22.09.1984. [23] Pat. No. 1.365.550, USSR, A Method of Grinding of Relieved Surface of a Tapered Gear Hob./S.P. Radzevich, Int. Cl. B24b 3/12, Filed 04.29.1986. [24] Pat. No. 1.504.903, USSR, A Method of Machining of a Gear with Shaper Cutter./ S.P. Radzevich, Int. Cl. B23f 5/12, Filed 02.12.1987. [25] Pat. No. 1.743.810, USSR, A Method of Grinding of Relieved Surface of a Gear Hob./S.P. Radzevich, Int. Cl. B24b 3/12, Filed 13.09.1989. [26] Pat. No. 2.040.3761, Russia, A Gear Hob./S.P. Radzevich, Int. Cl. B23f 21/16, Filed 03.01.1992. [27] Radzevich, S.P., A Crowning Achievement for Automotive Applications, Gear Solutions, December 2004, 16–25.

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[28] Radzevich, S.P., Advanced Methods of Sculptured Surface Machining, VNIITEMR, Moscow, 1988. [29] Radzevich, S.P., Conditions of Proper Sculptured Surface Machining, ComputerAided Design, 34 (10), 727–740, 2002. [30] Radzevich, S.P., Differential-Geometric Method of Surface Generation, DrSci thesis, Tula Polytechnic Institute, Tula, Russia, 1991. [31] Radzevich, S.P., Fundamentals of Surface Generation, Kiev, Rastan, 2001. Copy of the monograph is available from the Library of Congress, call number: MLCM 2006/04297. [32] Radzevich, S.P., Part Surfaces Those Allowing Sliding over Them. In Research in the Field of Surface Generation, S.P. Radzevich (Ed.), UkrNIINTI, Kiev, No. 65– Uk89, 1989, pp. 29–53. [33] Radzevich, S.P., Sculptured Surface Machining on Multi-Axis NC Machine, Vishcha Schola, Kiev, 1991. Copy of the monograph is available from the Library of Congress, call number: TJ1189.R26 1991. [34] Radzevich, S.P., and Palaguta, V.A., Achievements in the Field of Finishing of Cylindrical Gears, VNIITEMR, Moscow, 1988. [35] Radzevich, S.P., and Petrenko, T.Yu., Part and Tool Surfaces That Allow Sliding over Themselves, Mechanika ta Mashinobuduvann’a, Kharkiv, Ukraine, 1, 231–240, 1999. [36] Radzevich, S.P., and Vinokurov, I.V., Advanced Methods of Dressing of Form Grinding Wheels, VNIITEMR, Moscow, 1991. [37] Reshetov, D.N., and Portman, V.T., Accuracy of Machine Tools, ASME Press, New York, 1988. [Russian edition, 1986]. [38] Shishkov, V.A., Generation of Surfaces Using Continuously Indexing Method, Mashgiz, Moscow, 1951. [39] Willis, R., Principles of Mechanism, Designed for the Use of Students in the Universities and for Engineering Students Generally, John W. Parker, London, J. & J. J. Deighton, West Stand, Cambridge, 1841.

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3 Applied Coordinate Systems and Linear Transformations It is a common practice to analytically describe a surface in a certain coordinate system associated with the surface. The majority of parts to be machined are bounded by two or more surfaces (see Chapter 1). And each of the surfaces is usually represented in reference systems associated with the corresponding surface. Ultimately, it is necessary to analytically describe the whole part in a coordinate system that is common to all the surfaces. In the theory of surface generation, preference is given to implementation of the left-hand-oriented orthogonal Cartesian coordinate system. Reference systems such as cylindrical coordinates, spherical coordinates, and others are occasionally used as well. Linear transformations are used mostly for transformation of an analytical description of a machining operation and its elements from one coordinate system to another coordinate system. Numerous examples of the coordinate system transformation can be found in the field of surface generation [11–13], machining of surfaces on a numerical control (NC) machine [1–3,6], robotics [9,10], computer-aided design (CAD) [4,5,7,8], and others.

3.1

Applied Coordinate Systems

The machining of surfaces on NC machines requires formal analytical representation of the machining operation of the surface being machined, of the applied cutting tool, and of their relative motion at every instant of the machining operation. For this purpose, a common coordinate system is required. A set of surfaces being machined is referred to as the prime element of the machining operation. Because of this, the consideration below begins from coordinate systems associated with a part being machined. 3.1.1 Coordinate Systems of a Part Being Machined A part to be machined can be represented as a number of surfaces — planes, cylinders, cones, sculptured surfaces, and so on — that are oriented relative to one another in a certain manner. In order to obtain the simplest analytical 63 © 2008 by Taylor & Francis Group, LLC

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Kinematic Geometry of Surface Machining

X1

ZP

b

Z1

Z1

ZP

Y1

Y1 X1

Y2 a YP

XP X2

Z2

YP

XP

Figure 3.1 An example of the coordinate systems associated with the part surfaces.

representation of a part surface, it is necessary to introduce the coordinate system that is properly associated to the surface. Otherwise, analytical representation of the surface would not be the simplest possible. Initially, numerous coordinate systems XiYiZi are associated with the corresponding part surfaces (Figure 3.1). In order to machine the part, it is necessary to machine a set of its surfaces. For this purpose, the initial analytical representation of all the part surfaces must be transformed to a common coordinates system XPYPZP. Finally, every part surface is represented not in a particular coordinate system XiYiZi, but in the common coordinate system XPYPZP. 3.1.2 Coordinate System of Multi-Axis Numerical Control (NC) Machine ISO-R 841 recommends implementation of the right-hand-oriented Cartesian coordinate system XNCYNCZNC for the machining of surfaces on a multi-axis NC machine. Setup parameters of a surface to be machined, of the cutting tool to be applied, as well as their relative motion must be represented in this coordinate system (Figure 3.2). Coordinate systems of at least three types — the coordinate system XPYPZP of the part, the coordinate system XTYTZT of the cutting tool, and the coordinate system XNCYNCZNC of the multi-axis NC machine are used for the analytical description of a procedure of generation of a part surface.

EIA Standard RS-267-B: Axis and Motion Nomenclature for Numerically Controlled Machines (ANSI/EIA RS-267-B-83), Electronic Institute Association, Washington, DC, June 1983. ISO Standard 841-1874: Axis and Motion Nomenclature for Numerically Controlled Machines, International Organization for Standardization, Switzerland, 1974.

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Applied Coordinate Systems and Linear Transformations +ZNC

+ZNC

−YNC

+С −В

+А −YNC

–XNC

−А −С

+XNC

+В

+YNC

–ZNC –XNC ZP XP

+XNC

YP

–ZNC +YNC Figure 3.2 Coordinate system of a multi-axis numerical control machine (ISO-R 841).

3.2

Coordinate System Transformation

Coordinate system transformation is a powerful tool for solving many geometrical and kinematical problems that relate to surface generation. Finite transformation is used to describe the motion of a point on a rigid body and the motion of the rigid body. Consequent coordinate systems transformations can be easily described analytically with implementation of matrices. The use of matrices for the coordinate system transformation can be traced back to the early 1950s [14]. Matrices have a natural application to many branches of engineering. The theory of surface generation can be formulated concisely by the use of matrices, and practical numerical results can be obtained by means of the theorems of matrix algebra. Coordinate system transformation is necessary because of two major reasons. First, as stated above, the implementation of coordinate system transformation

Matrices were introduced into mathematics by Cayley in 1857. They provided a compact and flexible notation particularly useful in dealing with linear transformations, and they presented an organized method for the solution of systems of linear differential equations.

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Kinematic Geometry of Surface Machining

is necessary for representation of all particular part surfaces in a common coordinate system. Coordinate system transformation yields a transition from the initial coordinate system XiYiZi that is associated with a particular part surface, to the coordinate system XPYPZP embedded to the part. Second, coordinate system transformation is necessary for representation of the cutting tool and its motion relative to the surface in that above common coordinate system. At every instant, the configuration (position and orientation) of the cutting tool relative to the work can be described analytically with the help of a homogeneous transformation matrix corresponding to the displacement of the cutting tool from its current location to its consecutive location. In this section, coordinate system transformation is briefly discussed from the standpoint of its implementation in the theory of surface generation. 3.2.1 Introduction Homogenous coordinates utilize a mathematical trick to embed three-dimensional coordinates and transformations into a four-dimensional matrix format. As a result, inversions or combinations of linear transformations are simplified to inversion or multiplication of the corresponding matrices. 3.2.1.1

Homogenous Coordinate Vectors

Instead of representing each point r(x, y, z) in three-dimensional space with a single three-dimensional vector, x   r = y   z 

(3.1)

homogenous coordinates allow each point r(x, y, z) to be represented by any of an infinite number of four-dimensional vectors: T ⋅ x  T ⋅ y   r= T ⋅ z     T 

(3.2)

The three-dimensional vector corresponding to any four-dimensional vector can be computed by dividing the first three elements by the fourth, and a four-dimensional vector corresponding to any three-dimensional vector can be created by simply adding a fourth element and setting it equal to one. 3.2.1.2

Homogenous Coordinate Transformation Matrices of the Dimension 4 × 4

Homogenous coordinate transformation matrices operate on four-dimensional homogenous vector representations of traditional three-dimensional

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Applied Coordinate Systems and Linear Transformations

67

coordinate locations. Any three-dimensional linear transformation (translation, rotation, etc.) can be represented by a 4 × 4 homogenous coordinate transformation matrix. In fact, because of the redundant representation of three-space in a homogenous coordinate system, an infinite number of different 4 × 4 homogenous coordinate transformation matrices are available to perform any given linear transformation. This redundancy can be eliminated to provide a unique representation by dividing all elements of a 4 × 4 homogenous transformation matrix by the last element (which will become equal to one). This means that a 4 × 4 homogenous transformation matrix can incorporate as many as 15 independent parameters. The generic format representation of a homogenous transformation equation for mapping the three-dimensional coordinate (x1, y1, z1) to the three-dimensional coordinate (x2, y2, z2) is

T ⋅ x2   T ⋅ a T ⋅ y   T ⋅ e 2  =  T ⋅ z2   T ⋅ i     T  T ⋅ n

T ⋅ b T ⋅ f T ⋅ j T ⋅ p

T ⋅ d  T ⋅ x2  T ⋅ h  T ⋅ y 2  ⋅ T ⋅ m   T ⋅ z2     T   T 

T ⋅ c T ⋅ g T ⋅ k T ⋅ q

(3.3)

If any two matrices or vectors of this equation are known, the third matrix (or vector) can be computed, and then the redundant T element in the solution can be eliminated by dividing all elements of the matrix by the last element. Various transformation models can be used to constrain the form of the matrix to transformations with fewer degrees of freedom. 3.2.2 Translations Translation of a coordinate system is one of the major linear transformations used in the theory of surface generation. Translations of the coordinate system X2Y2Z2 along axes of the coordinate system X1Y1Z1 are depicted in Figure 3.3. Translations can be analytically described by the homogenous transformation matrices of dimension 4 × 4. For an analytical description of translation along coordinate axes, the operators of translation Tr(ax, X), Tr(ay, Y), and Tr(az, Z) are used. The operators yield matrix representation in the form

1 0 Tr ( ax , X ) =  0  0

0 1 0 0

0 0 1 0

ax  0  , 0  1

1 0 Tr ( ay , Y ) =  0  0

0 1 0 0

0 0 1 0

0 ay  , 0  1

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Kinematic Geometry of Surface Machining 1 0 Tr ( az , Z) =  0  0

0 1 0 0

0 0 1 0

0 0  az   1

(3.4)

Here, ax, ay, az are signed values that denote the distance of translation along the corresponding axis. Consider two coordinate systems X1Y1Z1 and X2Y2Z2 shifted along X1 axis on ax (Figure 3.3a). A point M in the coordinate system X2Y2Z2 is given by the position vector r 2(M). In the coordinate system X1Y1Z1, that same point M can be specified by the position vector r 1(M). Then the position vector r1(M) can be expressed in terms of the position vector r 2(M) by the equation r1(M) = Tr(ax, X) ∙ r 2(M). Equations similar to that above are valid for other operators Tr(ay, Y) and Tr(az, Z) of the coordinate system transformation (Figure 3.3b and Figure 3.3c). Suppose that a point P on a rigid body goes through a translation describing a straight path from P1 to P2 with a change of coordinates of (ax, ay, az). This motion can be described with a resultant translation operator Tr(a, A): 1 0 Tr ( a, A) =  0  0

0 1 0 0

0 0 1 0

ax  ay  az   1

(3.5)

The operator Tr(a, A) can be interpreted as the operator of translation along an arbitrary axis A. An analytical description of translation of the coordinate system X1Y1Z1 along an arbitrary axis A to the position of X2Y2Z2 can be composed from Figure 3.4. Z2 Z1

Z2

Z1

Z1 az

Z2

ax

X2 a2 X1 Y1

Y2 X1

X2 Y1

Y2

X2

X1 Y1

Y2 (a)

(b)

(c)

Figure 3.3 Analytical description of the operators of translations Tr(ax, X), Tr(ay, Y), Tr(az, Z) along the coordinate axis.

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Applied Coordinate Systems and Linear Transformations Z2

Z1

Axis A

Tr(a, A)

Tr(az, Z ) Tr(ay, Y )

Y2

Y1

X2

X1

Tr(ax, X)

Figure 3.4 Analytical description of an operator Tr(a,A) of translation along an arbitrary axis.

The operator Tr(a, A) of translation of that kind can be expressed in terms of the operators Tr(ax, X), Tr(ay, Y), and Tr(az, Z) of elementary translations Tr ( a, A) = Tr ( az , Z) ⋅ Tr ( ay , Y ) ⋅ Tr ( ax , X ) (3.6) Evidently, the axis A is always the axis through the origin. Any coordinate system transformation that does not change the orientation of a geometrical object is an orientation-preserving transformation, or a direct transformation. Therefore, transformation of translation is an example of a direct transformation. 3.2.3 Rotation about a Coordinate Axis Rotation of a coordinate system about a coordinate axis is another major linear transformation used in the theory of surface generation. Rotation of the coordinate system X2Y2Z2 about the axis of the coordinate system X1Y1Z1 is illustrated in Figure 3.5. For analytical description of rotation about a coordinate axis, the operators of rotation Rt(j x, X), Rt(j y, Y), and Rt(j z, Z) are used. The operators yield representation in the form of homogenous matrices:

cos ϕ y  0 Rt (ϕ x , X ) =   sin ϕ y   0

1 0 Rt (ϕ y , Y ) =  0  0

 cos ϕ z − sin ϕ z Rt (ϕ z , Z) =   0   0

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0 1 0 0

0 cos ϕ x − sin ϕ x 0

− sin ϕ y 0 cos ϕ y 0

0 0  0  1

(3.7)

0 sin ϕ x cos ϕ x 0

0 0  0  1

(3.8)

0 0  0  1

(3.9)

sin ϕ z cos ϕ z 0 0

0 0 2 0

70

Kinematic Geometry of Surface Machining

Z2

Z1

Z1

X2

jy

X1

X1

Y2

Y1

Y2

Z2

X2

X2

jx Y1

Z1

Z2

jz Y1

(a)

X1

Y2 (c)

(b)

Figure 3.5 Analytical description of the operators of rotations Rt(j x, X), Rt(j y, Y), Rt(j z, Z) about a coordinate axis.

Here j x, j y, j z are signed values that denote angles of rotation about corresponding axis: j x is rotation around the X-axis (pitch); j y is rotation around the Y-axis (roll), and j z is rotation around the Z-axis (yaw). Consider two coordinate systems X1Y1Z1 and X2Y2Z2 turned about the X1 axis through the angle j x (Figure 3.5a). In the coordinate system X2Y2Z2, a point M is given by the position vector r 2(M). In the coordinate system X1Y1Z1, that same point M can be specified by the position vector r 1(M). Then the position vector r 1(M) can be expressed in terms of the position vector r 2(M) by the equation r 1(M) = Rt(j x, X) ∙ r 2(M). Equations similar to that above are valid for other operators Rt(j y, Y) and Rt(j z, Z) of the coordinate system transformation (Figure 3.5b and Figure 3.5c). 3.2.4 Rotation about an Arbitrary Axis through the Origin Analytical description of rotation of the coordinate system X1Y1Z1 about an arbitrary axis A0 through the origin to the position of X2Y2Z2 is illustrated in Figure 3.6. The operator Rt(jA, A0) of rotation of that kind can be expressed in terms of the operators Rt(j x, X), Rt(j y, Y), and Rt(j z, Z) of elementary rotations Rt(ϕ A , A0 ) = Rt(ϕ z, Z) ⋅ Rt(ϕ y , Y ) ⋅ Rt(ϕ x , X )

(3.10)

Evidently, the axis A0 is always the axis through the origin. Z1 jz

Axis A0

jA

Z1

Rt(jA, A0)

Rt(jz, Z)

X1 jy

Rt(jx, X )

jA

X2

jx

Rt(jy, Y ) Y1

Axis A0

Z2

Y1

X1 Y2

Figure 3.6 Analytical description of the operator Rt(jA,A 0) of rotation about an arbitrary axis through the origin.

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Applied Coordinate Systems and Linear Transformations

The operators of translation and of rotation also yield linear transformations of other kinds. 3.2.5 Eulerian Transformation Eulerian transformation is a well-known type of linear transformation used widely in mechanical engineering. The operator Eu(y,q,j) of Eulerian transformation can be expressed in terms of three rotations about three axes through the origin: Eu(ψ , θ , ϕ ) =  − sin ψ cos θ sin ϕ + cos ψ cos ϕ  − sin ψ cos θ cos ϕ − cos ψ sin ϕ   sin θ sin ϕ  0 

cos ψ cos θ sin ϕ + sin ψ cos ϕ cos ψ cos θ cos ϕ − sin ψ cos ϕ − cos ψ cos θ 0

sin θ sin ϕ sin θ cos ϕ cos θ 0

0 0  0  1 (3.11)

Here y, q, and j denote Eulerian angles, say the angle of nutation y, the angle of precession q, and the angle of pure rotation j. The operator Rt(yA, A) of rotation about an arbitrary axis through the origin can result in that same final orientation of the coordinate system X2Y2Z2 as implementation of the operator Eu(y, q, j) of Euler’s transformation. However, the operators Rt(jA, A) and Eu(y, q, j) are operators of completely different natures. They result in identical coordinate system transformation, but they are not equal to one another. 3.2.6 Rotation about an Arbitrary Axis Not through the Origin Rotation of the coordinate system X1Y1Z1 to the position of X2Y2Z2 can be performed about an arbitrary axis A not through the origin (Figure 3.7). The operator of transformation of this kind Rt(jA, A) can be expressed in terms of the operator Tr(a, A) of translation along and of the operator Rt(jA, A) of rotation about an arbitrary axis A through the origin:

Rt (ϕ A , A) = Tr (−b , B ) ⋅ Rt (ϕ A , A 0 ) ⋅ Tr (b , B)

(3.12)

where Tr(b, B) is the operator of translation along the shortest distance of approach of the axis of rotation and origin of the coordinate system and Tr(−b, B*) is the operator of translation in the direction opposite to Tr(b, B) after the rotation Rt(jA, A0) is completed. In order to determine the shortest distance B of approach of the axis of rotation B and origin of the coordinate system, consider the axis B through two given points r B.1 and r B.2.

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Kinematic Geometry of Surface Machining Z1

Z*

Z **

Z2

jA

Axis A Rt(jA, A)

Tr(b, B)

X1

Tr(–b, B *)

X2

X **

X* Axis B *

Y1 Y2

Y*

Y **

Axis B

Figure 3.7 Analytical description of the operator Rt(jA,A) of rotation about an arbitrary axis not through the origin.

The shortest distance between a certain point r 0 and the straight line through the points r B.1 and r B.2 can be computed from the following formula:

B=

|(r 2 − r1 ) × (r1 − r 0 )| |r 2 − r1|

(3.13)

For the origin of the coordinate system, the equality r 0 = 0 is observed. Then,

B =|r1|⋅ sin ∠ [r1 , (r 2 − r1 )]

(3.14)

Matrix representation of the operators of translation Tr(ax, X), Tr(ay, Y), Tr(az, Z) along the coordinate axis, together with the operators of rotation Rt(j x, X), Rt(j y, Y), Rt(j z, Z) about the coordinate axis is convenient for implementation in the theory of surface generation. Moreover, it is the simplest possible way to represent them. 3.2.7 Resultant Coordinate System Transformation The operators of translations Tr(ax, X), Tr(ay, Y), Tr(az, Z) together with the operators of rotation Rt(j x, X), Rt(j y, Y), Rt(j z, Z) are used for composing the operator Rs(1 → 2) of the resultant coordinate system transformation. The operator Rs(1 → 2) of the resultant coordinate system transformation analytically describes the transition from the initial coordinate system X1Y1Z1 to the coordinate system X2Y2Z2. For example, the expression Rs(1 → 5) = Tr(ax, X) ∙ Rt(j z, Z) ∙ Rt(j x, X) ∙ Tr(ay, Y) indicates that the transition from the coordinates system X1Y1Z1 to the coordinate system X5Y5Z5 (Figure 3.8) is performed in the following

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Applied Coordinate Systems and Linear Transformations Z1

ax1

Z2

Z3

X1, X2 Y1

Z2

Z3

Y2

Y3

jz2 Y2

Y2 Z5

Z4

Y5

Y4

Z1

Z5

Y4

ax1

r1(M) X5

X3

X4

r5(M)

M

X1

Y5

X4

jy3

Y3

X3

ay4

Z4

X5

Y1

Figure 3.8 An example of the resultant coordinate system transformation.

four steps: (a) translation Tr(ay,Y) followed by (b) rotation Rt(j x,X), followed by (c) second rotation Rt(j z,Z), and finally followed by (d) the translation Tr(ax,X). Ultimately, the equality r1(M) = Rs(5 → 1) ∙ r5(M) is observed. Each of the resultant coordinate system transformations can be analytically described by the generalized equation Operator of a Resultant Coordinate System ⇒ Rs (1 → t) = Trransformations

nt

∏ Cp [i → (i ± 1)] i

i =1

(3.15)

nt

=

aˆ , Xˆ ) ⋅ Rt (ϕˆ , Xˆ )] ∏ [1Tr4(44 4244443 k

i =1

xˆ

k

Cpi [ i →( i ±1)]

xˆ

In Equation (3.15), Cpi[i → (i ± 1)] denotes the local translation/rotation couple, k denotes the number of elementary coordinate system transformations in a given series of consequent transformations, and nt denotes the total number of consequent coordinate system transformations in that same series of elementary transformations. Formally, a couple Cpi[i → (i ± 1)] possesses the following property: One component of the couple Cpi[i → (i ± 1)] is always equal to the identity matrix I4×4, and another component is not equal to I4×4. So, if, for example, Tr(ax, X) ≠ I4×4, then the operator Rt(j x,X) = I4×4, and vice versa. The same is true with respect to all other operators of elementary coordinate system transformations. The couples Cpi[i → (i ± 1)] of translation/rotation proved to be useful in computeraided design (CAD) and computer-aided manufacturing (CAM) applications.

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When the operator Rs(1 → t) of the resultant coordinate system transformation is known, then the transition in the opposite direction can be performed by means of the operator Rs(t → 1) of the inverse coordinate system transformation. The following equality can easily be proven: Rs (t → 1) = Rs−1 (1 → t)

(3.16)

In the above example (Figure 3.8), the operator Rs(S5 → S1) of the inverse resultant coordinate system transformation can be expressed in terms of the operator Rs(S1 → S5) of the direct resultant coordinate system transformation. Following Equation (3. 16), one can come up with the equation Rs(S5 → S1) = Rs−1(S1 → S5). It is easy to show that the operator Rs(1 → t) of the resultant coordinate system transformation yields representation in the following form: Rs (1 → t) = Tr ( a, A) ⋅ Eu (δ , ζ , η)

(3.17)

The linear transformation Rs(1 → t) (see Equation 3.7) can also be expressed in terms of rotation about an axis Rt(jA, A), not through the origin (see Equation 3.5). 3.2.8 An Example of Nonorthogonal Linear Transformation Axes X1 and Y1 of a coordinate system X1Y1Z1 are at a certain angle w 1. Axis Z1 is orthogonal to the coordinate plane X1Y1. After being turned through a certain angle j about Z1, the coordinate system X1Y1Z1 occupies the location of the coordinate system X2Y2Z2. Linear transformation of this kind can be analytically described by the operator of the coordinate system rotation:

 sin(ω 1 + ϕ )  sin ω 1   sin ϕ Rtω (1 → 2) =  − sin ω 1   0   0

sin ϕ sin ω 1

0

sin(ω 1 − ϕ ) sin ω 1

0

0 0

1 0

 0   0  0  1 

(3.18)

In order to distinguish the operator of rotation in the orthogonal linear transformation Rt(1 → 2) from the similar operator of rotation in a nonorthogonal linear transformation Rtw (1 → 2), the superscript “w” is assigned to the last. When w 1 = 90°, then Equation (3.18) casts into Equation (3.9). 3.2.9 Conversion of the Coordinate System Orientation Application of the matrix method of coordinate system transformation presumes that both of the coordinate systems i and (i ± 1) are of the same hand.

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This means that from the very beginning it is assumed that both are either right-hand- or left-hand-oriented Cartesian coordinate systems. In the event the coordinate systems i and (i ± 1) are of opposite hand, say one is the righthand-oriented coordinate system and the other is a left-hand-oriented coordinate system, then one of the coordinate systems must be converted into the oppositely oriented Cartesian coordinate system. For conversion of a left-hand-oriented Cartesian coordinate system into a right-hand-oriented coordinate system or vice versa, the operators of reflection are used. In order to change the direction of the Xi axis of the initial coordinate system i to the opposite direction (in this case in the new coordinate system (i ± 1), the equalities Xi ± 1 = −Xi, Yi ± 1 ≡ Yi ≡ Zi ± 1 ≡ Zi are observed), the operator of reflection Rfx(YiZi) can be applied. The operator of reflection yields representation in matrix form: −1 0 Rfx (Yi Zi ) =  0  0

0 1 0 0

0 0 1 0

0 0  0  1

(3.19)

Similarly, implementation of the operators of reflections Rfy(XiZi) and Rfz(XiYi) change the directions of the Yi and Zi axes into opposite directions. The operators of reflections Rfy(XiZi) and Rfz(XiYi) can be expressed analytically in the following form: 1 0 Rfy (Xi Zi ) =  0  0

0 −1 0 0

0 0 1 0

0 1  0 0 and Rfz (Xi Yi ) =  0 0   1 0

0 1 0 0

0 0 −1 0

0 0  0  1

(3.20)

A linear transformation that reverses the direction of the coordinate axis is an opposite transformation. Transformation of reflection is an example of orientation-reversing transformation.

3.3

Useful Equations

The sequence of successive rotations can vary depending on the intention of the researcher. Several special types of successive rotations are known, including Eulerian, Cardanian, and two kinds of Euler-Krylov transformations. The sequence of successive rotations can be chosen from a total of 12 different combinations. Even though the Cardanian transformation is different from the Eulerian transformation in terms of the combination of rotations, they both use a similar approach to compute the orientation angles.

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3.3.1 RPY-Transformation A series of rotations can be performed in the order roll matrix by pitch matrix and finally by yaw matrix. The linear transformation of this kind is referred to as RPY-transformation. The resultant transformation of this kind can be represented by the homogenous coordinate transformation matrix: RPY(ϕ x , ϕ y , ϕ z ) =  cos ϕ y cos ϕ z + sin ϕ x sin ϕ y sin ϕ z  − cos ϕ x sin ϕ z   sin ϕ x cos ϕ y sin ϕ z − sin ϕ y cos ϕ z  0 

cos ϕ y sin ϕ z − sin ϕ x sin ϕ y cos ϕ z cos ϕ x cos ϕ z − sin ϕ x cos ϕ y cos ϕ z − cos ϕ y sin ϕ z 0

cos ϕ x sin ϕ y sin ϕ x cos ϕ x cos ϕ y 0

0 0  0  1 

(3.21)

3.3.2 Rotation Operator A spatial rotation operator for the rotational transformation of a point about a unit axis a0(cosa, cosb, cosg) passing through the origin of the coordinate system can be described as follows, with a0 = A0/|A0| designating the unit vector along the axis of rotation A0. Suppose the angle of rotation of the point about a0 is q, then the rotation operator is expressed by Rt (θ , a 0 ) = (1 − cos θ ) cos 2 α + cos θ (1 − cos θ ) cos α cos β − sin θ cos γ   (1 − cos θ ) cos 2 β + cos θ (1 − cos θ ) cos α cos β + sin θ cos γ (1 − cos θ ) cos α cos γ − sin θ cos β (1 − cos θ ) cos β cos γ + sin θ cos α  0 0 

(1 − cos θ ) cos α cos γ + sin θ cos β (1 − cos θ ) cos β cos γ − sin θ cos α (1 − cos θ ) cos 2 γ + cos θ 0

0  0 0  1 

(3.22)

3.3.3 A Combined Linear Transformation Suppose a point P on a rigid body rotates with an angular displacement q about a unit axis a0 passing through the origin of the coordinate system at first, and then followed by a translation at a distance B in the direction of a unit vector b. The linear transformation of this kind can be analytically described by the homogenous matrix: Rt (θa 0 , Bb ) = (1 − cos θ ) cos 2 α + cos θ (1 − cos θ ) cos α cos β − sin θ cos γ   (1 − cos θ ) cos 2 β + cos θ (1 − cos θ ) cos α cos β + sin θ cos γ (1 − cos θ ) cos α cos γ − sin θ cos β (1 − cos θ ) cos β cos γ + sin θ cos α  0 0 

(1 − cos θ ) cos α cos γ + sin θ cos β B cos α   (1 − cos θ ) cos β cos γ − sin θ cos α B cos β  B cos γ  (1 − cos θ ) cos 2 γ + cos θ  0 1 

(3.23)

More operators of particular linear transformations can be found in the literature.

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3.4

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Chains of Consequent Linear Transformations and a Closed Loop of Consequent Coordinate System Transformations

Consequent coordinate system transformations form chains of transformations. The elementary chain of coordinate system transformation is composed of two consequent transformations. Chains of linear transformations play an important role in the theory of surface generation. Two different kinds of chains of consequent coordinate system transformations are distinguished: Transition from the coordinate system XPYPZP associated with the surface P, to the local Cartesian coordinate system xPyPzP with the origin at a point K of contact of the surfaces P and T. This linear transformation is also made up of numerous intermediate coordinate systems XinYinZin and forms a chain of direct consequent coordinate system transformations (Figure 3.9a). The coordinate system xPyPzP is connected to the surface P. Operator Rs(P → K P) of the resultant coordinate system transformations for a direct chain of transformations can be computed from Equation (3.15). Second, transition from the coordinate system XPYPZP to the local Cartesian coordinate system xT yT zT with origin at a point K of contact of the surfaces P and T. The coordinate system xT yT zT is associated with the generating surface T of the cutting tool. This linear transformation is also made up of numerous intermediate coordinate systems XjYjZj (e.g., the coordinate system XNCYNCZNC of a multi-axis NC machine) and numerous intermediate coordinate systems XiYiZi. The linear transformation of this kind forms a chain of inverse consequent coordinate system transformations (Figure 3.9b). Operator Rs(P → KT) of the resultant coordinate system transformations for the inverse chain of transformations can be computed from Equation (3.15). Chains of the direct and of the reverse consequent coordinate system transformations together with the operator of transition from the local coordinate system xTyTzT to the local coordinate system xPyPzP form a closed loop of consequent coordinate system transformations (Figure 3.9c). When no deviation of configuration of the cutting tool relative to the work is considered, then the loop of consequent coordinate system transformation is closed. Otherwise, parameters of the actual relative configuration of the surfaces P and T must be incorporated in order to close the loop. For a complete closed loop of the consequent coordinate system transformations, Equation (3.15) returns a result that is identical to the input date. This means that analytical description of a machining process in the original coordinate system remains the same after implementation of the operator

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Rs(P → in) XinYinZin Rs(in → Kp) xP yP zP Rs(KT → Kp) xT yT zT (a)

Rs(P → j)

XP YP ZP

... Rs( j → NC ) xP yP zP

XNC YNC ZNC

Rs(KT → Kp)

Rs( N C → i) ... Rs( i → T)

xT yT zT XT YT ZT

Rs( T → KT )

(b)

Rs(P → N C)

XPYPZP

Rs(P → Kp) xP yP zP

XNC YNC ZNC

Rs(KT → Kp)

Rs( N C → T ) xT yT zT

XT YT ZT Rs(T → Kp) (c)

Figure 3.9 An example of direct chain (a), of reverse chain (b), and a closed loop (c) of consequent coordinate system transformations.

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of the resultant coordinate system transformations (see Equation 3.15). This condition is necessary and sufficient for a closed loop of consequent coordinate system transformations to exist. Implementation of the chains and the closed loops of consequent coordinate system transformation yield consideration of the surface P machining operation in each coordinate system that makes up the loop. Therefore, for consideration of a particular problem of surface generation, the most convenient coordinate system can be chosen. In order to complete the construction of a closed loop of a consequent coordinate system transformation, an operator of transformation from the local coordinate system xT yT zT to the local coordinate system xP yP zP must be composed. The coordinate systems xP yP zP and xT yT zT are usually semiorthogonal coordinate systems. This means that the axis zP is always orthogonal to the coordinate plane xP yP, and the axes xP and yP can be either orthogonal or not orthogonal to each other. The same is valid for the local coordinate system xT yT zT. Two possible ways to perform the required transformation of the local coordinate systems xPyPzP and xTyTzT are considered below. Following the first way, the operator Rtw (t → p) of the linear transformation of semiorthogonal coordinate systems (Figure 3.10) must be composed. The operator Rtw (t → p) can be represented in the form of the homogenous matrix:

 sin (ω T + α )  sin ω T   sin α Rtω (t → p) =  − sin ω T   0   0

−

sin (ω P − ω T − α ) sin ω T sin (ω P − α ) sin ω T

yP

ωP

ωT β K

xT

α

xP

Figure 3.10 Local coordinate systems with the origin at a cutter-contact-point K.

© 2008 by Taylor & Francis Group, LLC

0 −1 0

0 0

yT

0

 0   0   0 1 

(3.24)

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where w P is the angle that makes UP and VP coordinate lines on the surface P (see Equation 1.28), wT is the angle that makes UT and VT coordinate lines on the surface T (see Equation (1.28), and a is the angle that makes the axis xP and xT. The auxiliary angle b in Figure 3.10 is equal to b = w T + a. The inverse coordinate system transformation — that is, the transformation from the local coordinate system xPyPzP to the local coordinate system xTyTzT can be analytically described by the operator Rtw (p → t) of the inverse coordinate system transformation. The operator Rtw (p → t) can be represented in the form of the homogenous matrix:

 sin (ω P − α )   sin ω P  sin α Rtω ( p → t) =  sin ωP   0  0 

sin (ω P − ω T − α ) sin ω P sin (ω T + α ) sin ω P 0 0

0 0 −1 0

 0   0  0  1 

(3.25)

Following the second way of transformation of the local coordinate systems, the auxiliary orthogonal local coordinate system must be constructed. For the purposes of surface generation, it is proven to be convenient to represent an analytical description of a machining operation of the surface P in the moving local coordinate system xPyPzP. Implementation of the coordinate system xPyPzP having an orthogonal normalized basis is preferred. Use of the orthogonal local coordinate system xPyPzP results in significant simplification of the linear transformations, as well as makes it possible to obtain the final result of computation not hidden within bulky equations. Consider a way in which a closed loop of the consequent coordinate system transformations can be composed. In order to construct an orthogonal normalized basis of the coordinate system xPyPzP, an intermediate coordinate system x1y1z1 is used. Axis x1 of the coordinate system x1y1z1 is pointed out along the unit vector u P that is tangent to the UP coordinate curve (Figure 3.11). Axis y1 is directed along vector vP that is tangent to the VP coordinate line on P. Axis z1 aligns with unit normal vector n P and is pointed outward from the surface of the P body. For a surface P having orthogonal parameterization (for which FP = 0, and therefore w P = π/2), an analytical description of coordinate system transformations is significantly easier. Further simplification of the coordinate system transformation is possible when the coordinate UP and VP lines are congruent to the lines of curvature on P. With such a scenario, the local coordinate system is represented by Darboux’s trihedron. To construct Darboux’s trihedron, principal directions on the surface P must be computed. Determination of the unit tangent vectors t1.P and t1.P of the principal directions on the surface P is considered in Chapter 1.

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nP

P

uP

K

ZP

vP

y1

rP

P

XP YP Figure 3.11 Local coordinate system xP yPzP associated with the surface P.

In the common tangent plane, orientation of the unit vector t1.P of the first principal direction can be uniquely specified by the included angle x1.P that the vector t1.P makes with the UP coordinate curve. This angle depends on both the surface P geometry and the surface parameterization. Depicted in Figure 3.12 is the relationship between the tangent vectors UP and VP, and the included angle x1.P. From the law of sines,

GP EP FP = = sin ξ 1. P sin[π − ξ 1. P − (π − ω P )] sin(ω P − ξ 1. P )

GP =

(3.26)

VP

VP .VP

t1 .P ωP

ξ

EP =

UP  UP

.

UP

Figure 3.12 Differential relationships between the tangent vectors UP and VP, the fundamental magnitudes of the first order, the included angle x1.P, and the directions t 1.P.

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x3

x3

x4 = x1 × x2 x2

e1

x2

e4

x1

(a)

x4

e1

e4

x1

(b)

x5 = x4 × x1 e5

x2

e1

x1

(c)

x4 e4

e5

e1

x5

x1

(d)

Figure 3.13 An orthogonally normalized basis e1e4 e 5 that is constructed from an arbitrary basis x1x 2 x 3.

where

ωP =

FP . EP GP

Solving the expression above for the included angle x1.P results in

ξ 1. P = tan −1

EP GP − FP2 EP + FP

(3.27)

Another possible way of constructing an orthogonal local basis of the local Cartesian coordinate system xPyPzP is based on the following consideration. Consider an arbitrary nonorthogonal and not normalized basis x1x 2 x 3 (Figure 3.13a). Construct an orthogonal and normalized basis based on the initial basis x1x 2 x 3. The cross-product of any two of three vectors x1, x 2, x 3 (for example, the cross-product of the vectors x1 × x 2) determines a new vector x4 (Figure 3.13b). Evidently, the vector x4 is orthogonal to the coordinate plane x1x 2. Then, use the computed vector x4 and one of the two original vectors x1 or x 2, for instance, use the vector x 2. This yields computation of a new vector x 5 = x4 × x 2 (Figure 3.13c). The computed basis x1x4 x 5 is orthogonal. In order to convert it into a normalized basis, each of the vectors x1, x4, x 5 must be divided by its magnitude: e1 = x1/|x1|, e4 = x4/|x4|, e5 = x 5/|x 5|. The resultant basis e1e4e5 (see Figure 3.13d) is always orthogonal, as well as it is always normalized. In order to complete the analytical description of a closed loop of consequent coordinate system transformations, it is necessary to compose the operator Rs(KT → K P) of transformation from xTyTzT to xPyPzP (Figure 3.9c). In the case under consideration, the axes zP and zT align with the common unit normal vector n P. The axis zP is pointed out from the bodily side to the void side of the surface P. The axis zT is directed oppositely. Due to that, the following equality observes Rs(KT → K P) = Rt(j z, zT). The inverse coordinate system transformation can be analytically described by the operator Rs(K P → KT) = Rs−1(KT → K P) = Rt(−w z, zT).

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Implementation of the discussed results yields representation of the surfaces P and T, as well as their relative motion in a common coordinate system and consideration of the machining process of the part surface P in any desired coordinate systems that make up the chain or the closed loop of consequent coordinate system transformations. Transition from one coordinate system to another coordinate system can be performed in both of two feasible directions, say in direct as well as in inverse directions.

3.5

Impact of the Coordinate System Transformations on Fundamental Forms of the Surface

Every coordinate system transformation results in a corresponding change of equation of the surface P and the generating surface T of a cutting tool. Therefore, it is necessary to recalculate coefficients of the first f1.P and of the second f2.P fundamentals of the surface P as many times as the coordinate system transformation is performing. This routing and time-consuming operation can be eliminated if the operators of coordinate system transformations are used directly in the fundamental forms f1.P and f2.P. After being computed in an initial coordinate system, the fundamental magnitudes EP, FP, GP, LP, MP, and NP can be determined in any new coordinate system using for this purpose the operators of translation, of rotation, and of resultant coordinate system transformation. Transformation of such kind of the fundamental magnitudes f1.P and f2.P becomes possible due to implementation of a formula that is derived below. Consider a surface P given by the equation rP = rP (UP,VP), where ( U P ,V P ) ∈ G

For the analysis below, it is convenient to use the equation of the first fundamental form f1.P of the surface P represented in matrix form (see Equation 1.6):

[φ1.P ] = [dU P

dV P

E P F P 0 0] ⋅  0  0

FP GP 0 0

0 0 1 0

0  dU P  0   dV P  ⋅  0  0     1  0 

(3.28)

Similarly, the equation of the second fundamental form f 2.P of the surface P can be given by Equation (1.10):

[φ2.P ] = [dU P

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dV P

 LP M P 0 0] ⋅   0   0

MP NP 0 0

0 0 1 0

0  dU P  0   dV P  ⋅  0  0     1  0 

(3.29)

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Kinematic Geometry of Surface Machining

The coordinate system transformation with the operator of linear transformation Rs(1 → 2) transfers the equation rP = rP (UP,VP) of the surface P initially given in X1Y1Z1, to the equation r P = r P (U P , V P ) of that same surface P in a new coordinate system X2Y2Z2. It is clear that r P ≠ r P . In the new coordinate system, the surface P is analytically described by the following expression: r P (U P , V P ) = Rs (1 → 2) ⋅ r P (U P , V P )

(3.30)

The operator of resultant coordinate system transformation Rs(1 → 2) casts the column matrices of variables in Equation (3.28) and Equation (3.29) to the form [dU P

dV P

0 0] T = Rs (1 → 2) ⋅ [dU P

dV P

0 0] T

(3.31)

Substitution of Equation (3.31) into Equation (3.28) and Equation (3.29) yields expressions for φ1,P and φ2,P in the new coordinate system: [φ 1.P ] = [Rs (1 → 2) ⋅ [dU P dVP 0 0] T ] T ⋅ [φ1.P ] ⋅ Rs (1 → 2) ⋅ [dU P dVP 0 0] T

(3.32)

[φ 2.P ] = [Rs (1 → 2) ⋅ [dU P dVP 0 0] T ] T ⋅ [φ2.P ] ⋅ Rs (1 → 2) ⋅ [dU P dVP 0 0] T

(3.33) The following equation is valid for multiplication:

[Rs (1 → 2) ⋅ [dU P

dVP

0 0] T ] T = RsT (1 → 2) ⋅ [dU P

dVP

0 0]

(3.34)

dVP

0 0]

Therefore, [φ 1.P ] = [dU P

dVP

0 0] T ⋅ {RsT (1 → 2) ⋅ [φ1.P ] ⋅ Rs (1 → 2)} ⋅ [dU P

[φ 2.P ] = [dU P

(3.35)

dVP

0 0] T ⋅ {RsT (1 → 2) ⋅ [φ2.P ] ⋅ Rs (1 → 2)} ⋅ [dU P

dVP

0 0] (3.36)

It can be easily shown that the matrices [φ 1.P ] and [φ 2.P ] in Equation (3.35) and Equation (3.36) represent quadratic forms with respect to dUP and dVP. The operator of transformation Rs(1 → 2) of the surface P having the first f1.P and the second f 2.P fundamental forms from the initial coordinate system X1Y1Z1 to the new coordinate system X2Y2Z2 results in that in the new

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85

coordinate system, the corresponding fundamental forms are expressed in the following form:

[φ 1.P ] = RsT (1 → 2) ⋅ [φ1.P ] ⋅ Rs (1 → 2)

(3.37)

[φ 2.P ] = RsT (1 → 2) ⋅ [φ2.P ] ⋅ Rs (1 → 2)

(3.38)

Equation (3.37) and Equation (3.38) reveal that after the coordinate system transformation is completed, the first φ 1.P and the second φ 2.P fundamental forms of the surface P in the coordinate system X2Y2Z2 are expressed in terms of the first f1.P and the second f 2.P fundamental forms initially represented in the coordinate system X1Y1Z1. In order to do that, the corresponding fundamental form either f1.P or f 2.P must be premultiplied by Rs(1 → 2), and after that it must be postmultiplied by RsT(1 → 2). The implementation of Equation (3.37) and Equation (3.38) significantly simplifies formulae transformations. Equations similar to Equation (3.37) and Equation (3.38) are valid for the generating surface T of the cutting tool. If the third f 3.P and fourth f 4.P fundamental forms are used, their coefficients can be expressed in terms of the fundamental magnitudes of the first and of the second order.

References [1] Amirouche, F.M.L., Computer-Aided Design and Manufacturing, Englewood Cliffs, NJ, Prentice Hall, 1993. [2] Chang, Chao-Hwa, and Melkanoff, M.M., NC Machine Programming and Software Design, Englewood Cliffs, NJ, Prentice Hall, 1989. [3] Choi, B.K., and Jerard, R.B., Sculptured Surface Machining. Theory and Application, Kluwer Academic, Dordrecht/Boston/London, 1998. [4] Faux, L.D., and Pratt, M.J., Computational Geometry for Design and Manufacture, Ellis Horwood, Chichester, 1987. [5] Ferguson, R.S., Practical Algorithms for 3D Computer Graphics, A K Peters, Natick, MA, 2001. [6] Marciniak, K., Geometric Modeling for Numerically Controlled Machining, Oxford University Press, Oxford, 1991. [7] Mortenson, M.E., Geometric Modeling, John Wiley & Sons, New York, 1985. [8] Mortenson, M., Mathematics for Computer Graphics Application, 2nd ed., Industrial Press, New York, 1999. [9] Murray, R.M., Zexiang, L., and Sastry, S.S., A Mathematical Introduction to Robotic Manipulation, CRC Press, Boca Raton, FL, 1994. [10] Paul, P.R., Robot Manipulators: Mathematics, Programming, and Control. The Computer Control of Robot Manipulators, MIT Press, Cambridge, MA, 1981 (2nd printing, 1982).

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[11] Radzevich, S.P., Differential-Geometric Method of Surface Generation, DrSci thesis, Tula Polytechnic Institute, Tula, Russia, 1991. [12] Radzevich, S.P., Fundamentals of Surface Generation, Rastan, Kiev, 2001. Copy of the monograph is available from the Library of Congress, call number: MLCM 2006/04297. [13] Radzevich, S.P., Sculptured Surface Machining on Multi-Axis NC Machine, Vishcha Schola, Kiev, 1991. Copy of the monograph is available from the Library of Congress, call number: TJ1189.R26 1991. [14] Denavit, J., and Hartenberg, R.S., A Kinematics Notation for Lower-Pair Mechanisms Based on Matrices, ASME Journal of Applied Mechanics, 77, 215–221, 1955.

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Part II

Fundamentals

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4 The Geometry of Contact of Two Smooth, Regular Surfaces In the theory of surface generation, the part surface is considered as the prime element. Other important elements of the surface-generation process are kinematics of the surface generation, shape and geometry of the generating surface of a cutting tool, and so are considered the secondary elements of surface generation. This does not mean that the importance of the secondary elements is lower than the importance of the primary element — that is incorrect. This just means that the optimal parameters of the secondary elements can be expressed in terms of the geometrical elements of the prime element. Ultimately, the entire surface-generation process can be synthesized on the premises of just the prime element. In other words, having just the geometry of the part surface P to be machined, implementation of the Differential Geometry/ Kinematics (DG/K)-method of surface generation makes possible the solution to the problem of synthesis of the optimal machining operation. The surface P geometry is used for the purposes of synthesis of the optimal machining operation of a given part surface. The concept that establishes priority of the part surface to be machined over the rest of the elements of the machining process is the cornerstone concept of the DG/K-method of surface generation. In order to solve the problem of optimal surface generation, an appropriate analytical description of the geometry of contact of the surfaces P and T is necessary. The problem of analytical description of the geometry of contact of two smooth, regular surfaces in the first order of tangency is a sophisticated one. It can be solved on the premises of analysis of topology of the contacting surfaces in differential vicinity of the point of their contact. In sculptured surface machining on a multi-axis numerical control (NC) machine, the point of contact of the surfaces P and T is often referred to as the cutter-contact point (CC-point). Various methods for analytically describing the geometry of contact of smooth, regular surfaces have been developed. An overview of the methods can be found in the monograph by Radzevich [15]. The latest achievements in the field are discussed in the literature [12,15,16,18]. Detailed analysis of the known methods of analytical description of the geometry of contact of two smooth, regular surfaces uncovered poor capability of the methods for solving problems in the field of optimal surface generation. Therefore, it is necessary to develop an accurate analytical method for analytical description of the geometry of contact of two smooth, regular 89 © 2008 by Taylor & Francis Group, LLC

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surfaces in the first order of tangency that fits the needs of the theory of optimal surface generation. Such a method is worked out in this chapter. It is convenient to begin the analysis from the analytical description of local relative orientation of the surfaces P and T (in differential vicinity of the point of contact of the surfaces).

4.1 Local Relative Orientation of a Part Surface and of the Cutting Tool A surface P being machined and the generating surface T of the cutting tool are the conjugate surfaces in nature. This means that during the machining operation, the surfaces P and T are in permanent tangency to one another (Figure 4.1). The requirement to be permanently in tangency to each other imposes restrictions on the relative configuration (location and orientation) of the surfaces, and on their relative motion. In the theory of surface generation, a quantitative measure of the surfaces P and T relative orientation is established. Relative orientation of the surfaces P and T is specified by the angle μ of the surface’s local relative orientation. By definition, angle μ is equal to the angle that the unit tangent vector t1.P of the first principal direction of the surface P makes with the unit tangent vector t1.T of the first principal direction of the surface T. That same angle μ can also be determined as the angle that makes the unit tangent vectors t 2.P and t 2.T of the second principal directions of the surfaces P and T at a point K of their contact. This immediately zP

C1.T R2.T

Tangent Plane T

nP

t2.P

yP

R2.P

R1.T t2.T

t1.P R1.P

μ xP

xT

P

zT C2.P

yT t1.T

K

nT

C2.T

C1.P

Figure 4.1 The second-order analysis: tangent quadrics to the surfaces P and T. (From Radzevich, S.P., Mathematical and Computer Modeling, 39 (9–10), 1083–1112, 2004. With permission.)

The surface orientation is local in nature because it relates only to differential vicinity of the point K of contact of the surfaces P and T.

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The Geometry of Contact of Two Smooth, Regular Surfaces yields equations for the computation of the angle μ:

sin µ =|t 1. P × t 1.T |=|t 2. P × t 2.T |

(4.1)

cos µ = t 1. P ⋅ t 1.T = t 2. P ⋅ t 2.T

(4.2)

t 2. P × t 2.T

(4.3)

tan µ =

t 1. P × t 1.T t 1. P ⋅ t 1.T

≡

t 2. P ⋅ t 2.T

where t1.P , t 2 P are the unit vectors of the principal directions on the surface P; and t1.T, t 2.T are the unit vectors of the principal directions on the generating surface T of the cutting tool. In case of point contact of the surfaces P and T, the actual value of the angle μ is computed at the point K of contact of the surfaces. If the surfaces P and T are in line contact, then the actual value of the angle μ can be computed at every point of the line of contact. The line of contact of the surfaces P and T is referred to as the characteristic line E, or just as the characteristic E. Determination of the angle μ of the surfaces P and T local relative orientation is illustrated in Figure 4.2. In order to compute the actual value of the angle μ, unit vectors of the principal directions t1.P and t1.T are employed. С2. P

С1. P

С2. T

С1. T

t1. P t1. T

P nT

T rrPP

t2 . P t2 . T

μ

XP

YP Figure 4.2 Angle m of the surfaces P and T local relative orientation.

It is noteworthy that for a line contact, relative orientation of the surfaces P and T is predetermined in a global sense. However, the actual value of the angle m of the surface’s local relative orientation at different points of the characteristic E is different.

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Kinematic Geometry of Surface Machining C2.T

C2.P

ωT ωP

ε μ vT

t2.P t2.T

θ vP u T

C1.T

uP

μ

t1.T t1.P

K

C1.P

Figure 4.3 Local relative orientation of the surfaces P and T, represented in a common tangent plane.

Consider two surfaces P and T in point contact, which are represented in a common coordinate system. The surfaces make contact at a point K. For further analysis, an equation of the common tangent plane to the surfaces at K is necessary (Figure 4.1 and Figure 4.3): (r tp - r K ) ⋅ u P ⋅ v P = 0

(4.4)

where rtp is the position vector of a point of the common tangent plane, rk is the position vector of the contact point K, and u P and vP are unit vectors tangent to UP and VP coordinate lines on P at K. The actual value of the angle w P can be computed from one of the following equations (see also Equation 1.28): sin ω P =

EP GP - FP2 , EP GP

cos ω P =

FP , EP GP

EP GP - FP2

tan ω P =

FP

(4.5)

Equations similar to those in Equation (4.5) are also valid for the computation of angle w T of the generating surface T of the cutting tool. Tangent directions uP and vP to the UP and VP coordinate lines on the surface P, and tangent directions uT and vT to the UT and VT coordinate lines on the surface T are specified by the angles q and e. For computation of actual values of the angles q and e, the following equations can be used: cos q = uP ◊ vT and cos e = vP ◊ vT. Angle xP is the angle that the first principal direction t1.P on the surface P makes with the unit tangent vector uP (see Figure 4.3). The equation for the computation of the actual value of the angle xP was derived by Radzevich [12,13,17]: sin ξP =

ηP

2

ηP sin ω P - 2ηP cos ω P + 1

where hP designates the ratio hP = ∂UP/∂VP.

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(4.6)

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The Geometry of Contact of Two Smooth, Regular Surfaces

In the event FP = 0, the equality tan x = hP is observed. Here the ratio hP is equal to the root of the quadratic equation

( FP LP - EP M P )ηP2 + (GP LP - EP N P )ηP + (GP M P - FP N P ) = 0

(4.7)

that immediately follows from Equation (1.13). The equation for the computation of the actual value of the angle xP yields another representation. Following the chain rule, drP can be represented in the form d r P = U P dU P + VP dVP

(4.8)

By definition, tan xP = sin xP/cos xP. The functions sin xP and cos xP yield representation in the form sin ξP =

|U P × d r P| U ⋅ dr and cos ξP = P P U P ⋅|d r P| |U P|⋅|drP|

(4.9)

The last expressions yield tan ξP =

|U P × d r P| sin ξP |U P × d r P| = = UP ⋅ d rP U P ⋅ (U P dU P + VP dVP ) cos ξP

|U P × d r P|⋅dVP = U P ⋅ U P dU P + U P ⋅ VP dVP

(4.10)

By definition (see Equation 1.7),

U P ⋅ U P = EP, U P ⋅ VP = FP, and

U P × VP = EP GP - FP2

(4.11)

Equation (4.8) through Equation (4.11) yield the following formula for the computation of the actual value of the angle xP :

tan ξP =

EP GP - FP2

ηP ⋅ EP + FP

(4.12)

Equations similar to Equation (4.6) and Equation (4.12) are also valid for the computation of the actual value of the angle xT that the first principal direction t1.T on the generating surface T makes with the unit tangent vector uT. The performed analysis yields the following equations for computation of the principal directions t1.P for the surface P t 2.P :

  π t 2. P = Rt  ξP +  , n P  ⋅ u P   2  

t 1. P = Rt (ξP , n P ) ⋅ u P,

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(4.13)

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And similar equations are yielded for computation of the principal directions t1.T , t 2.T for the generating surface T of the cutting tool:

  π t 2.T = Rt  ξT +  , n P  ⋅ uT   2  

t 1.T = Rt (ξT , n P ) ⋅ uT ,

(4.14)

Equation (3.10) for the operator Rt(jA, A0) is employed for the computation of the operators of rotation in Equation (4.13) and Equation (4.14).

4.2

The First-Order Analysis: Common Tangent Plane

Various methods can be implemented for the analytical description of the geometry of contact of two smooth, regular surfaces in the first order of tangency. First-order analysis is one of them. First-order analysis is the simplest method used for the analysis. Implementation of the first-order analysis returns limited information about the geometry of contact of the surfaces in differential vicinity of the point of contact. Accurate analytical description of the geometry of contact of two smooth, regular surfaces is possible only when the first-order analysis is used together with a kind of higher-order analysis. Under such a scenario, the first-order analysis is incorporated as the first step of the higher-order analysis. The common tangent plane can be defined as a plane through the contact point K, which is perpendicular to the common unit normal vector n P to the surfaces P and T at K. For analytical description, the common tangent plane vector equation (see Equation 1.3) can be employed:

(r tp - r K ) ⋅ u P ⋅ v P = 0

(4.15)

Any pair of vectors chosen from the vectors u P, vP, uT, vT, t1.P, t 2.P, t1.T, t 2.T can be used to replace the unit tangent vectors u P and vP in Equation (4.15). The first-order analysis provides very limited information about the geometry of contact of two smooth, regular surfaces in the first order of tangency. Because of this, the first-order analysis has limited application. However, the first-order analysis becomes valuable as it is incorporated into the second-order analysis and into the higher-order analysis.

4.3 The Second-Order Analysis For more accurate analytical description of the geometry of contact of the surfaces P and T, consideration of the second-order parameters is necessary. The second-order analysis incorporates elements of the first order as well as elements of the second order. For performing the second-order analysis,

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The Geometry of Contact of Two Smooth, Regular Surfaces W1

W2

W3 nP

Q δ

K

P

Dup(P)

Figure 4.4 Dupin’s indicatrix of the surface P.

familiarity with Dupin’s indicatrix is highly desired. Dupin’s indicatrix is a perfect starting point for consideration of the second-order analysis. 4.3.1

Preliminary Remarks: Dupin’s Indicatrix

Dupin’s indicatrices Dup(P) and Dup(T) of the surfaces P and T are of critical importance in the theory of surface generation. Generation of this planar characteristic curve is illustrated with a diagram as depicted in Figure 4.4. A plane W through the unit normal vector nP at the point K is rotating about nP. While rotating, the plane occupies consecutive positions W1, W2, W3, and others. Radii of normal curvature of the line of intersection of the surface P by normal planes W1, W2, W3, and so on, are equal to RP, 1, RP, 2, RP, 3, and so on. A plane Q intersects the surface P and is orthogonal to the unit normal vector nP. This plane is at a certain distance d from the point K. When δ → 0, and when the scale of the line of intersection of the surface P with the plane Q approaches infinity, then the line of intersection of P with Q approaches the planar characteristic curve Dup(P). In differential geometry of surfaces, Dupin’s indicatrices of the following five kinds are distinguished (Figure 4.5): elliptic (a), umbilic (b), parabolic (c), hyperbolic (d), and minimal (e). Dupin’s indicatrix for the plane local patch of the surface P does not exist. All its points are remote to infinity. Phantom branches of the characteristic curve Dup(P) in Figure 4.5d and Figure 4.5e are shown in dashed lines. An easy way to derive an equation of the characteristic curve Dup(P) is discussed below. Euler’s formula k1. P cos 2 ϕ + k 2. P sin 2 ϕ = k P yields representation in the form

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k1. P k cos 2 ϕ + 2. P sin 2 ϕ = 1 kP kP

(4.16)

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Kinematic Geometry of Surface Machining yP

R1.P

yP

Dup(P)

K

Dup(P)

K

xP

xP

RP

R2.P (a)

R1.P

(b)

yP

Dup(P)

K

yP

R2.P

xP

Dup (P)

xP

K

Dup(P)

R1.P (c)

(d)

R2.P

yP

Dup (P)

xP

K

R1.P (e) Figure 4.5 Dupin’s indicatrices Dup(P) of a smooth, regular surface.

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The Geometry of Contact of Two Smooth, Regular Surfaces

Transition from polar coordinates to Cartesian coordinates can be performed using well-known formulas xP = ρ cos ϕ and y P = ρ sin ϕ . These formulas yield

cos 2 ϕ =

xP2 ρ2

sin 2 ϕ =

y P2 ρ2

and

After substituting the last formulas into the above Equation (4.16), one can come up with the following: k1. P xP2 k 2. P y P2 ⋅ + ⋅ =1 kP ρ2 kP ρ2

(4.17)

It is convenient to designate ρ = k P-1 . Principal curvatures k1.P and k2.P are the roots of the following quadratic equation: LP - EP k P M P - FP k P

M P - FP k P =0 N P - GP k P

(4.18)

Substituting the computed values of k1.P and k2.P into Equation (4.17), and after performing the necessary formulae transformation equation for the Dupin’s indicatrix, Dup (P) can be represented as follows: k1. P xP2 + k 2. P y P2 = 1

(4.19)

The general form of the equation of Dupin’s indicatrix is often presented as Dup ( P) ⇒

2 MP LP 2 N xP + xP y P + P y P2 = 1 EP GP EP GP

(4.20)

Equation (4.19) describes a particular case of the Dupin’s indicatrix, which is represented in Darboux’s trihedron. 4.3.2 Surface of Normal Relative Curvature The surface of normal relative curvature is another good example of secondorder analysis. The idea of implementation of the surface of normal relative

The same equation of the Dupin’s indicatrix could be derived in another way. Coxeter [22] considers a pair of conics obtained by expanding an equation in Monge’s form z = z(x,y) in a Maclaurin series: z = z(0, 0) + z1 x + z2 y + 12 ( z11 x12 + 2 z12 xy + z22 y 2 ) + L = 12 (b11 x 2 + 2b12 xy + b22 y 2 ). This gives the equation (b11 x 2 + 2b12 xy + b22 y 2 ) = ±1 of the Dupin’s indicatrix.

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Kinematic Geometry of Surface Machining

curvature for the analytical description of the geometry of contact of two smooth, regular surfaces P and T can be traced back to publications by Hertz [5]. Consider a Taylor expansion of equations of the surfaces P and T at the point of their contact K. Implementation of the Taylor expansion is convenient for determining a surface of relative normal curvature. In the local coordinate system xPyPzP, the equation of the surface P in differential vicinity of the point K can be represented in the following form: P ⇒ zP =

y2 xP2 + P +K 2 R1. P 2 R2. P

(4.21)

where R1.P and R 2.P are the principal radii of curvature of the surface P at K. They are positive if the corresponding center of curvature is located within the body bounded by the surface P — that is, the center of curvature is located on the negative portion of the zP axis. Similarly, in the local coordinate system xT yT zT, the equation of the surface T in differential vicinity of the point K yields representation in the following form: T ⇒ zT =

y2 xT2 + T +K 2 R1.T 2 R2.T

(4.22)

The quantities R1.T and R 2.T have the same meaning for the generating surface T as the quantities R1.P and R 2.P for the part surface P. A surface for which the equality zR = zP - zT is observed is referred to as the surface of relative normal curvature. Further, the surface of relative normal curvature is designated as R . In order to employ the equation zR = zP - zT , it is necessary to represent both surfaces P and T in a common coordinate system with its origin at the point K and to align the positive direction of the zR axis with the unit normal vector n P to the surface P at K. The local coordinate system xPyPzP is suitable for the analytical description of the surface of normal relative curvature. In this reference system, the equation of the surface of relative curvature R yields representation in the form

R ⇒ zP =

xP2 y P2 + +K 2 R1.R 2 R2.R

(4.23)

The surface of relative normal curvature reflects the topology of the surfaces P and T and geometry of contact of this surface locally, just in differential vicinity of the point K. This yields only those terms in Equation (4.21) being taken into account. In Darboux’s trihedron, this result yields the simplified equation for the surface of relative normal curvature:

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R ⇒ 2 zP = k1.R xP2 + k 2.R y P2

(4.24)

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The Geometry of Contact of Two Smooth, Regular Surfaces

Relative normal curvature is the major analytical tool used in contemporary practice for analytical description of the geometry of contact in higher kinematic pairs. The relative normal curvature kR of two surfaces P and T at a given point K is defined as the summa of normal curvatures k P and kT of both surfaces P and T. It is taken in a common normal plane section of surfaces P and T and is equal to k R = k P + kT

(4.25)

Here kR designates normal curvature of the surface of relative curvature R, k P is normal curvature of the surface P, and kT is normal curvature of the generating surface T of the cutting tool. All three curvatures are computed in a common normal plane section of the surfaces P, T, and R through K. Consider a common normal plane section of the surfaces P and T through K. This plane section makes a certain angle j with the unit tangent vector t1.P. That same common normal plane section makes the angle (j + m) with the unit tangent vector t1.T. Recall that the angle m of the surfaces local relative orientation is the angle that makes the first t 1.P and t 1.T (or, the same, the second t 2.P and t 2.T ) principal directions of the surfaces P and T (Figure 4.2). Euler’s equation yields representation of the normal curvatures k P and kT in the form

k P = k 1. P cos ϕ + k 2. P sin ϕ

(4.26)

kT = k 1.T cos(ϕ + µ ) + k 2.T sin(ϕ + µ )

(4.27)

where k 1. P and k 2. P are the first and the second principal curvatures of the surface P, k 1.T and k 2.T are the first and the second principal curvatures of the surface T, j is the angular parameter, and m is the angle of the surfaces P and T local relative orientation. Note that the inequality k 1. P > k 2. P is always observed.** Thus, Equation (4.25) yields an equation for the normal curvature of the surface R :

k R = k1. P cos 2 ϕ + k 2. P sin 2 ϕ + k1.T cos 2 (ϕ + µ ) + k 2.T sin 2 (ϕ + µ )

(4.28)

This equation is expressed in terms of principal c curvatures k1. P (T ) , k 2. P (T ); angle m of the surfaces P and T local relative orientation; and the angular

Similarly, relative normal radius of curvature RR of two surfaces P and T at a given point K could be defined as the difference of the normal radii of curvature R P and RT. It is taken in a common normal plane section of surfaces P and T, and is equal to RR = R R - RT. ** This inequality is often represented in the form k 1.P(T) ≥ k2.P(T), which is incorrect. In case of equality — that is, if k1.P(T) = k2.P(T), all normal curvatures of the surface P and T at K are of the same value (and of the same sign) observed for umbilics, as well as for plane surface. For this reason, at an umbilic point, the principal directions are undefined. Therefore, principal curvatures are also undefined. This means that the inequality k1.P(T) > k2.P(T) properly reflects correspondence between the principal curvatures k1.P(T) and k2.P(T).

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Kinematic Geometry of Surface Machining

parameter j. Equation (4.28) can then be cast to

kR = a ⋅ cos 2 ϕ + b ⋅ sin(2ϕ ) + c ⋅ sin 2 ϕ

(4.29)

where the parameters a, b, and c can be computed from the following: a = k1. P + k1.T cos 2 µ + k 2.T sin 2 µ

b=

( k 2.T - k1.T ) ⋅ sin(2 µ ) 2

c = ( k 2. P + k1.T sin 2 µ + cos 2 µ )

(4.30) (4.31) (4.32)

Principal curvatures of the surface of relative normal curvature are the extreme values of function kR (ϕ ) (see Equation 4.29). The principal directions of the surface R are those equations for which the following equation is satisfied: ∂kR (ϕ ) =0 ∂ϕ

(4.33)

The last equation together with Equation (4.29) yield tan(2ϕ ) =

2b c-a

(4.34)

Equation (4.34) determines two solutions for the angle j: ϕ 1 and ϕ 2 = ϕ 1 + 90°. This means that there are two perpendicular directions for the principal directions of the surface of relative normal curvature. The principal curvatures of the surface of normal relative curvature can be computed from

k1.2 ,R =

( a + c ) ± ( a + c )2 + 4 b 2 2

(4.35)

The surface of relative normal curvature R is important in many engineering applications. Note again that all three normal curvatures k R , k P , and kT in Equation (4.25) are taken in a common normal plane section through the point of contact K of the surfaces P, T, and R . Based on the computed values of principal curvatures k 1.R and k 2.R , the implicit equation of the surface of relative curvature yields the following

In case of line contact of the surfaces P and T, point K is the point on the line of the surfaces contact at which the normal curvatures kR , kP, and kT are required to be computed.

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The Geometry of Contact of Two Smooth, Regular Surfaces representation: 2 +k 2 2Z R = k 1.R X R 2.R YR

(4.36)

A characteristic surface similar to the surface R of relative normal curvature can be defined as the surface for which the equality R R = RP - RT is observed. Evidently, this equality is similar in nature to Equation (4.25). 4.3.3 Dupin’s Indicatrix of Surface of Relative Curvature Consider an intersection of the surface of relative normal curvature with a plane, parallel to the tangent plane at the point K, but only at a small distance away from it. Then project the intersection on the tangent plane. In the coordinate plane xP y P , the principal part of the intersection will be given by the equation of Dupin’s indicatrix [2]:

Dup (R ) ≡Dup ( P / T ) ⇒

LR ER

xR2 +

2 MR ER GR

xR y R +

NR GR

yR2 = ±1

(4.37)

The Equation of Dupin’s indicatrix Dup (R ) [12,13,17] describes the distribution of normal relative curvature within differential vicinity of the point K. Here, ER , FR , and GR designate fundamental magnitudes of the first order; and LR , MR , and NR designate fundamental magnitudes of the second order of the surface R at the point K. If axes xR and yR of the local coordinate system xR yR align with the principal directions t1.R and t2.R of the surface of relative curvature R , then Equation (4.37) reduces to

Dup ( P / T ) ⇒ k1.R xR2 + k 2.R yR2 = ±1

(4.38)

An important intermediate conclusion immediately follows from Equation (4.37): The direction t1.R for the maximum k1.R and the direction t2.R for the minimum k1.R values of normal curvature of the surface of relative curvature are always orthogonal to one another; therefore, the condition t 1.R ⊥ t 2.R is always observed. The major axes of the Dupin’s indicatrix Dup (R ) make the angles ϕ min and ϕ max with the principal directions t 1.P and t 2.P .

To be more exact, Dupin’s indicatrix Dup (R ) ≡ Dup (P/ T) reflects distribution not of normal relative curvature RK , but distribution of normal relative radii of curvature RK . Thus, it could be designated as DupR (R ). However, equation of the indicatrix Dupk(R ) of a surface normal curvature could also be composed. Similar to that, corresponding equations for the normalized DupR (R ) indicatrix of relative normal radius of curvature and indicatrix of normal curvature Dupk (K ) could also be easily derived.

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Kinematic Geometry of Surface Machining

4.3.4 Matrix Representation of Equation of the Dupin’s Indicatrix of the Surface of Relative Normal Curvature Like any other quadratic form, Equation (4.37) of Dupin’s indicatrix of the surface of relative curvature R can be represented in matrix form:

Dup ( P / T ) ⇒ [ xP y P

 LR  E R   2M R 0 0] ⋅   ER G R  0   0 

2MR E R GR NR GR 0 0

 0 0   xP   y  P 0 0  ⋅   = ±1   0    m1 0   0  0 1 

(4.39)

In Darboux’s trihedron, this equation reduces to

Dup ( P / T ) ⇒ [ xP y P

 LR M R 0 0   xP  M N R 0 0   y P  R  ⋅ = ±1 0 0] ⋅  0 0 m1 0   0      0 0 1  0   0

(4.40)

It is convenient to implement the matrix representation of the equation of Dupin’s indicatrix (see above), for instance, when developing software for machining of a sculptured surface on a multi-axis NC machine when multiple coordinate system transformations are required. The equation of Dupin’s indicatrix can be represented in the form rDup (ϕ ) = |RP (ϕ )|⋅ sgn Φ2-.1P . The last equation reveals that the position vector of a point of the indicatrix Dup(P) in any direction is equal to the square root of the radius of curvature in that same direction. 4.3.5 Surface of Relative Normal Radii of Curvature The normal curvatures kR , k P, and k T can be represented in the form , k P = RP-1, and k T = RT-1 , respectively, where RR , RP , and R T are k R = R -1 R the corresponding radii of normal curvature of the surfaces P, T, and R . They are also taken in a common normal plane section through the point of contact K of the surfaces P, T, and R .

Similar to Dupin’s indicatrix Dup (P), a planar characteristic curve of another kind can be introduced. The equation of this characteristic curve can be postulated in the form rDup. k (ϕ ) = |k P (ϕ )|.sgn Φ -2.1P . Application of the curvature indicatrix in the form Dupk (j) yields the avoidance of uncertainty in cases of plane. For plane surface Dup (P), while rDup.k (j) exists, it shrinks to the point K.

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The Geometry of Contact of Two Smooth, Regular Surfaces

The radius of relative normal curvature is another widely known tool that is used in contemporary practice for analytical description of the geometry of contact when performing the second-order analysis. The radius RR of relative normal curvature can be defined by the following expression:

RR = R P - RT

(4.41)

In many applications, Equation (4.41) for the radius of relative normal curvature RR is equivalent to Equation (4.25) for the relative normal curvature kR . 4.3.6 Normalized Relative Normal Curvature Usually, it is preferred to operate with unitless values when performing the second-order analysis rather than with values that have units. In order to eliminate unit values, it is recommended that a normalized relative normal curvature k R of the surfaces P and T be used. Normalized relative normal curvature k R of the surfaces P and T is referred to as the value determined by

kR =

k P + kT |k1. P|

(4.42)

Similarly, the normalized radius of relative normal curvature R R of the surfaces P and T can be introduced here based on Equation (4.41). The normalized relative radius of normal curvature R R of the surfaces P and T is referred to as the value determined by RR =

R P - RT |R 1. P|

(4.43)

Implementation of the unitless parameters k R , R R , and others permits avoidance of operation with unit values. Equations made up of unitless parameters are often more convenient in application. Dupin’s indicatrix can be constructed for all of the above considered characteristic surfaces: (a) for a surface of normal relative radii of curvature DupR (R ), (b) for a normalized surface of normal relative curvature Dup(R ) , and (c) for a normalized surface of normal radii of relative curvature DupR (R ).

4.3.7 Curvature Indicatrix Five different types of the characteristic curve Dup ( P) are distinguished in differential geometry of surfaces (Figure 4.5): (a) elliptic (for which Gauss’ curvature is always positive [G P > 0] , (b) umbilic (GP > 0) , (c) parabolic (GP = 0) , (d) hyperbolic (GP < 0), and (e) minimal hyperbolic G P < 0 , |R 1. P|= R 2. P ). For a planar local patch of a surface P, the characteristic curve Dup(P) does not exist. All points of this characteristic curve for a planar local patch of a surface P are remote to infinity.

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As shown in Chapter 1, surfaces considered in engineering geometry differ from surfaces considered in differential geometry of a surface. In differential geometry of surfaces, Dupin’s indicatrix is implemented for the purpose of illustration of the distribution of the surface normal curvature. There are only five different kinds of Dupin’s indicatrices of a smooth, regular surface. All are depicted in Figure 4.5. When the DG/K-method of surface generation is used, the bodily and void sides of a surface P must be distinguished [11–13,17]. Unfortunately, the Dupin’s indicatrix for a convex surface P can be identical to the Dupin’s indicatrix for a concave surface P, if in both cases the bounding mathematical surface can be described with the same equation. Therefore, with the help of Dupin’s indicatrix, no difference can be found between the convex and concave surfaces. The above can be summarized by the statement: Dupin’s indicatrix Dup(P) of the surface P possesses no ability to distinguish whether the surface P is convex or the surface P is concave. For the purpose of distinguishing whether a surface P is convex or the surface P is concave, a characteristic image of novel kind can be used. This newly introduced characteristic image is referred to as the curvature indicatrix Crv ( P) of the surface P. The curvature indicatrix of the surface P can be described analytically by inequality Crv ( P) ⇒

LP 2 N 2 MP xP + xP y P + P y P2 ≥ 1 EP GP EP GP

(4.44)

% ≥ 0 , and by inequality when the surface mean curvature is nonnegative M P Crv ( P) ⇒

LP 2 N 2 MP xP + xP y P + P y P2 ≤ 1 EP GP EP GP

(4.45)

% ≤ 0 . The inequalities (4.44) when the surface mean curvature is nonpositive M P and (4.45) are composed on the premises of Dupin’s indicatrix Dup(P) of the surface P. The performed analysis shows that the total number of curvature indicatrices Crv(P) of the surface P is as many as ten different kinds. Equations similar to Equation (4.44) and Equation (4.45) are observed for the generating surface of the cutting tool T. Although the analytical description of the curvature indicatrix Crv(P) (see Equation 4.44 and Equation 4.45) resembles the analytical description of Dupin’s indicatrix Dup(P) (see Equation 4.20), these two characteristic images are different in nature. Dupin’s indicatrix is a planar curve of the

Dupin’s indicatrix Dup(P) is completely equivalent to the second fundamental form f 2.P of the surface P. The second fundamental form f 2.P is also known as an operator of the surface shape. Koenderink [6] recommends that the characteristic curve Dup(P) be considered as a rotation of operator of the surface shape f 2.P.

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second order. The curvature indicatrix is a portion of plane that is bounded by Dupin’s indicatrix and is located either inside the characteristic curve Dup(P) % % M (if M P (T ) ≤ 0 ). P (T ) ≥ 0), or outside the corresponding Dupin’s indicatrix (if When plotting the curvature indicatrix of a part surface P, the use of mean curvature of the surface along with its Gaussian curvature is helpful. Curvature indicatrices of a surface P of all possible kinds are depicted in Figure 4.6. For convenience, all possible kinds of curvature indicatrices

yP

R1 P

Dup(P)

xP

K

yP

K

xP

K

R1 P

(c)

yP

R1 P

Dup (P)

K

xP

xP

RP

(b) Dup (P)

Dup(P)

Crv(P)

R1 P

(a)

yP

Crv(P)

Dup(P)

K

Crv(P)

R2.P

yP

R2 P

yP

Dup (P)

K

xP

xP

RP Dup(P)

Crv(P) (d) R2 P

yP

Dup (P)

Crv(P)

(f)

(e) Dup (P)

K

R1 P

xP

Crv(P)

yP

Dup (P)

K

(g)

R2 P

R2 P

Crv(P)

(h)

Figure 4.6 Curvature indicatrices Crv(P) of a smooth, regular part surface P.

© 2008 by Taylor & Francis Group, LLC

yP

Dup (P)

K

xP

Crv(P) R1.P

Crv(P)

xP

R1 P

(i)

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Crv(P) of a smooth, regular surface P are listed along with the corresponding sign of the mean M P and of the Gaussian G P curvatures (Figure 4.6): • • • • • • • • •

Convex elliptic (M P > 0, G P > 0) in Figure 4.6a Concave elliptic (M P < 0, G P < 0) in Figure 4.6b Convex umbilic (M P > 0, G P > 0) in Figure 4.6c Concave umbilic (M P < 0, G P < 0) in Figure 4.6d Convex parabolic (M P > 0, G P = 0) in Figure 4.6e Concave parabolic (M P < 0, G P = 0) in Figure 4.6f Quasi-convex hyperbolic (M P > 0, G P < 0 ) in Figure 4.6g Quasi-concave hyperbolic (M P < 0, G P < 0) in Figure 4.6h Minimal hyperbolic (M P = 0, G P < 0 ) in Figure 4.6i

Phantom branches of the characteristic curve in Figure 4.6g through Figure 4.6i are indicated by dashed lines. For a plane local patch of a surface P, the curvature indicatrix does not exist. All points of this characteristic curve are remote to infinity. 4.3.8 Introduction of the Jr k(P/T ) Characteristic Curve For the purpose of analytical description of the distribution of normal curvature in differential vicinity of a point on a smooth, regular surface, Böhm recommends [1] that the following characteristic curve be employed. Setting h = dVP/ dUP at a given point of a sculptured surface P, one can rewrite the equation

kP =

Φ2. P LP dU P2 + 2 M P dU P dVP + N P dVP2 = Φ1. P EP dU P2 + 2 FP dU P dVP + GP dVP2

(4.46)

for normal curvature in the form of

kP =

LP + 2 M P η + N P η 2 EP + 2 FP η + GP η 2

(4.47)

In the particular case when LP : M P : N P = EP : FP : GP , the normal curvature k P is independent of h. Surface points with this property are known as umbilic points and flatten points. In general cases when k P changes as h changes, the function k P = k P (η) is a rational quadratic form, as illustrated in Figure 4.7. The extreme values k1. P and k 2. P of k P = k P (η) occur at the roots η 1 and η 2 of

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η2 EP LP

-η FP MP

1 GP = 0 NP

(4.48)

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The Geometry of Contact of Two Smooth, Regular Surfaces

kP(η)

0.50

k

0.35 kT (η) 0.20

k(P/T )

–10

−5

kR(η)

η

0.05 5

10

−0.10

Figure 4.7 An example of the characteristic curve Ir k ( P/T ).

It can be shown that η 1 and η 2 are always real. The quantities η 1 and η 2 define directions that align with the principal directions on the surface P. The characteristic curve k P = k P (η) specifies the distribution of normal curvature of the surface P (Figure 4.7) at the point K. Another characteristic curve kT = kT (η) specifies the distribution of normal curvature of the generating surface T that makes contact with P at K. For the surfaces P and T, the surface of relative curvature R can be constructed. The distribution of normal curvature of the surface R at K is described by the characteristic curve kR = kR (η) . Characteristic curve Ir k ( P/T ) of a novel kind is defined here as

Ir k ( P/T ) ⇒ kIr = k P (η) + kT (η + µ )

(4.49)

Similarly, a characteristic curve Ir R ( P/T ) of another sort is defined as

Ir R ( P/T ) ⇒ RIr = RP (η) - R T (η + µ )

(4.50)

The developed methods for analytical description of the geometry of contact of two smooth, regular surfaces in the first order of tangency are not limited to the methods disclosed above [14–16].

4.4 Rate of Conformity of Two Smooth, Regular Surfaces in the First Order of Tangency Accuracy of the discussed methods of the second-order analysis is not sufficient for the accurate analytical description of the geometry of contact of two smooth, regular surfaces in the first order of tangency. In order to increase the accuracy, higher-order analysis is necessary.

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The discussed methods of higher-order analysis target the development of an analytical description of the rate of conformity of the generating surface T of the cutting tool to the part surface P at the current point K of their contact. The higher the rate of conformity of the surfaces P and T, the closer are these surfaces to each other in differential vicinity of the point K. This qualitative (intuitive) definition of the rate of conformity of two smooth, regular surfaces needs a corresponding quantitative measure. 4.4.1

Preliminary Remarks

Consider two surfaces P and T in the first order of tangency that make contact at a point K. The rate of conformity of the surfaces P and T can be interpreted as a function of radii of normal curvature RP and R T of the surfaces. The radii of normal curvature RP and R T are taken in a common normal plane section through point K. For a given radius of normal curvature RP of the surface P, the rate of conformity of the surfaces depends on the corresponding value of radius of normal curvature R T of the generating surface T. In most cases of part surface generation, the rate of conformity of the surfaces P and T is not constant. It depends on orientation of the normal plane section through the point K and changes as the normal plane section is turning about the common perpendicular n P . This statement immediately follows from the above conclusion that the rate of conformity of the surfaces P and T yields interpretation in terms of radii of normal curvature RP and R T . Illustrated in Figure 4.8 is the change of the rate of conformity of the surfaces P and T due to the turning of the normal plane section about the common perpendicular n P . In Figure 4.8, only two-dimensional examples are shown, for which that same normal plane section of the surface P makes contact with different plane sections T (i ) of the generating surface T. (1) In the example shown in Figure 4.8a, the radius of normal curvature RT ( 1 ) of the convex plane section T (1) of the surface T is positive ( RT > 0). The convex normal plane section of the surface T makes contact with the convex normal plane section ( RP > 0 ) of the surface P. The rate of conformity of the generating surface T to the part surface P in Figure 4.8a is relatively low. Another example is shown in Figure 4.8b. The radius of normal curvature RT( 2) of the convex plane section T ( 2) of the surface T is also positive (RT( 2) > 0 ). However, its value exceeds the value RT(1) of radius of normal curvature in the ( 2) (1) first example ( RT > RT ). This results in the rate of conformity of the surface T to the surface P (Figure 4.8a) being higher compared to what is shown in Figure 4.8b. In the next example (Figure 4.8c), the normal plane section T ( 3) of the surface T is represented with a locally flattened section. The radius of normal curva( 3) ture RT of the flattened plane section T ( 3) approaches infinity ( RT( 3) → ∞). Thus, the inequality RT( 3) > RT( 2) > RT(1) is valid. Therefore, the rate of conformity of the surface T to the surface P in Figure 4.8c is also getting higher.

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(1)

RT P

nP

(2)

P

K

RT

nP K

T (1) RP

RP

(a)

(3)

RT P

T (2)

(b)

nP

nP

K

K

P

T (3) RP (c)

T (4)

(4)

RT

RP (d)

Figure 4.8 Plane sections of the surfaces P and T through the common perpendicular. (From Radzevich, S.P., Mathematical and Computer Modeling, 39 (9–10), 1083–1112, 2004. With permission.)

Finally, for concave normal plane section T ( 4 ) of the surface T (Figure 4.8d), ( 4) the radius of normal curvature RT is negative ( RT( 4 ) < 0 ). The rate of conformity of the surface T to the surface P is the highest of four examples considered in Figure 4.8. The examples shown in Figure 4.8 qualitatively illustrate what is known intuitively regarding the different rates of conformity of two smooth, regular surfaces in the first order of tangency. Intuitively, one can realize that in the examples shown in Figure 4.8a through Figure 4.8d, the rate of conformity of two surfaces P and T is increasing. A similar observation is made for a given pair of the surfaces P and T when different sections of the surfaces by a plane surface through the common perpendicular n P are considered (Figure 4.9a). When rotating the plane section about the common perpendicular , it can be observed that the rate of conformity of the surfaces P and T is different in different directions (Figure 4.9b). The above examples provide an intuitive understanding of what the rate of conformity of two smooth, regular surfaces P and T means. They cannot be employed directly for evaluation in quantities of the rate of conformity of two smooth, regular surfaces P and T. The next required step is to introduce an appropriate quantitative evaluation of the rate of conformity of two surfaces in the first order of tangency. In other words, how can a certain rate of conformity of two smooth, regular surfaces be described analytically?

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nP

RT

P

K

P

T

K

T

RP (a)

(b)

Figure 4.9 Analytical description of the geometry of contact of the surface P being machined and of the generating surface T of the cutting tool. (From Radzevich, S.P., Mathematical and Computer Modeling, 39 (9–10), 1083–1112, 2004. With permission.)

4.4.2 Indicatrix of Conformity of the Surfaces P and T Introduced in this section is a quantitative measure of the rate of conformity of two surfaces. The rate of conformity of two surfaces P and T indicates how the surface T is close to the surface P in differential vicinity of the point K of their contact, say how much the surface T is congruent to the surface P in differential vicinity of the point K. Quantitatively, the rate of conformity of a surface T to another surface P can be expressed in terms of the difference between the corresponding radii of normal curvature of the surfaces. In order to develop a quantitative measure of the rate of conformity of the surfaces P and T, it is convenient to implement Dupin’s indicatrices Dup(P) and Dup(T) of the surfaces P and T, respectively. It is natural to assume that the higher rate of conformity of the surfaces P and T is due to the smaller difference between the normal curvatures of the surfaces P and T in a common cross-section by a plane through the common normal vector n P . Dupin’s indicatrix Dup(P) indicates the distribution of radii of normal curvature of the surface P as it had been shown, for example, for a concave elliptic patch of the surface P (Figure 4.10). The equation of this characteristic curve for surface P (see Equation 4.37) in polar coordinates can be represented in the following form:

Dup ( P) ⇒ rP (ϕ P ) =

RP (ϕ P )

(4.51)

where rP is the position vector of a point of the Dupin’s indicatrix Dup(P) of the surface P, and ϕ P is the polar angle of the indicatrix Dup(P).

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The Geometry of Contact of Two Smooth, Regular Surfaces yP

yP t2.T

t2.P

Dup(T)

t1.T

μ

t1.P

xP

K

Dup(

Figure 4.10 Derivation of equation of the indicatrix of conformity Cnf R (P / T ) of two smooth, regular surfaces P and T in the first order of tangency.

The same is true with respect to the Dupin’s indicatrix Dup(T) of the surface T as it was shown, for instance, for a convex elliptical patch of the surface T (Figure 4.10). The equation of this characteristic curve in polar coordinates can be represented in the form Dup (T ) ⇒ rT (ϕT ) = |RT (ϕT )

(4.52)

where rT is the position vector of a point of the Dupin’s indicatrix Dup(T) of the surface T, and ϕT is the polar angle of the indicatrix Dup(T). In the coordinate plane xP y P of the local coordinate system xP y P zP, the equalities ϕ P = ϕ and ϕT = ϕ + µ are valid. Therefore, in the coordinate plane xP y P , Equation (4.51) and Equation (4.52) cast into

Dup ( P) ⇒ rP (ϕ ) =

Dup (T ) ⇒ rT (ϕ , µ ) =

(4.53)

RP (ϕ )

(4.54)

RT (ϕ , µ )

When the rate of conformity of the surface T to the surface P is higher, then the difference between the functions rP (ϕ ) and rT (ϕ , µ ) gets smaller. The last makes valid the following conclusion: Distance between the corresponding points of the Dupin’s indicatrices Dup(P) and Dup(T) of the surfaces P and T

Corresponding points of the Dupin’s indicatrices Dup(P) and Dup(T) share the same straight line through point K of the surfaces P and T contact and are located at the same side of point K.

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can be employed for indication of the rate of conformity of the surfaces P and T at point K. The equation of indicatrix of conformity Cnf R ( P/T ) of the surfaces P and T is postulated of the following structure: Cnf R ( P/T ) ⇒ rcnf (ϕ , µ ) =

RP (ϕ ) sgn RP (ϕ ) +

RT (ϕ , µ ) sgn RT (ϕ , µ )

= rP (ϕ ) sgn RP (ϕ ) + rT (ϕ , µ ) sgn RT (ϕ , µ )

(4.55)

where rP = |RP|is the position vector of a point of the Dupin’s indicatrix of the surface P and rT = |R T is a position vector of a corresponding point of the Dupin’s indicatrix of the surface T. Here, in Equation (4.55), the multipliers sgn RP (ϕ ) and sgn RT (ϕ , µ ) are assigned to each of the functions rP (ϕ ) = RP (ϕ ) and rT (ϕ , µ ) = RT (ϕ , µ ) just for the purpose of remaining the corresponding sign of the functions — that is, that same sign that the radii of normal curvature RP (ϕ ) and RT (ϕ , µ ) have. Because the position vector rP (ϕ ) defines location of a point aP of the Dupin’s indicatrix Dup(P), and the position vector rT (ϕ , µ ) defines location of a point aT of the Dupin’s indicatrix Dup(T), then the position vector rcnf (ϕ , µ ) defines location of a point aC (see Figure 4.10) of the indicatrix of conformity Cnf R ( P/T ) of the surfaces P and T. Therefore, the equality rcnf (ϕ , µ ) = KaC is observed, and the length of the straight-line segment KaC is equal to the distance aP aT . Ultimately one can conclude that position vector rcnf of a point of the indicatrix of conformity Cnf R ( P/T ) can be expressed in terms of position vectors rP and rT of the Dupin’s indicatrices Dup(P) and Dup(T). For the computation of current value of the radius of normal curvature RP (ϕ ), the equality RP(j) = f1.P/f 2.P can be used. Similarly, for the computation of current value of the radius of normal curvature RT (ϕ , µ ), the equality RT(j, m) = j1.T/f 2.T can be employed. Use of the angle m of the surfaces P and T local relative orientation indicates that the radii of normal curvature RP (ϕ ) and RT (ϕ , µ ) are taken in a common normal plane section through the point K. Further, it is well known that the inequalities φ1.P ≥ 0 and φ1.T ≥ 0 are always valid. Therefore, Equation (4.55) can be rewritten in the following form:

rcnf = rP (ϕ )sgn φ2-.P1 + rT (ϕ , µ )sgn φ2-.T1

(4.56)

For the derivation of equation of the indicatrix of conformity Cnf R ( P/T ), it is convenient to use Euler’s equation for RP (ϕ ) (see Equation 1.31): RP (ϕ ) =

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R 1. P ⋅ R 2. P R 1. P ⋅ sin 2 ϕ + R 2. P ⋅ cos 2 ϕ

(4.57)

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The Geometry of Contact of Two Smooth, Regular Surfaces

Here, the radii of principal curvature R 1. P and R 2. P are the roots of the quadratic equation: LP ⋅ RP - EP M P ⋅ RP - FP

M P ⋅ RP - FP =0 N P ⋅ R P - GP

(4.58)

Recall that the inequality R 1. P < R 2. P is always observed. Equation (4.57) and Equation (4.58) allow expression of the radius of normal curvature RP (ϕ ) of the surface P in terms of the fundamental magnitudes of the first order EP, FP , and GP , and of the fundamental magnitudes of the second order LP , M P , and N P. A similar consideration is applicable for the generating surface T of the cutting tool. Omitting routing analysis, one can conclude that the radius of normal curvature RT (ϕ , µ ) of the surface T can be expressed in terms of the fundamental magnitudes of the first order ET , FT , and GT , and of the fundamental magnitudes of the second order L T , MT , and N T . Finally, on the premises of the above-performed analysis, the following equation for the indicatrix of conformity Cnf R ( P/T ) of the surfaces P and T can be derived: rcnf (ϕ , µ ) =

+

EPGP sgn φ2-.P1 LPGP cos 2 ϕ - M P EPGP sin 2ϕ + N P EP sin 2 ϕ

LT GT cos (ϕ + µ ) - MT 2

ET GT sgn φ2-.T1 ET GT sin 2(ϕ + µ ) + N T ET sin 2 (ϕ + µ ) (4.59)

Equation (4.59) of the characteristic curve Cnf R ( P/T ) is published in [7] and (in a hidden form) in [8]. Analysis of Equation (4.59) reveals that the indicatrix of conformity Cnf R ( P/T ) of the surfaces P and T at the point K is represented with a planar centro-symmetrical curve of the fourth order. In particular cases, this characteristic curve also possesses a property of mirror symmetry. Mirror symmetry of the indicatrix of conformity observes, for example, when the angle m of the local relative orientation of surfaces P and T is equal m = ±p∙ n/2, where n designates an integer number. It is important to note that even for the most general case of surface generation, position vector rcnf (ϕ , µ ) of the indicatrix of conformity Cnf R ( P/T ) is not dependent on the fundamental magnitudes FP and FT . Independence of

An equation of this characteristic curve is also known from [7] and (in a hidden form) from [8].

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the characteristic curve Cnf R ( P/T ) of the fundamental magnitudes FP and FT is due to the following. The coordinate angle ω P can be calculated by the formula

ω P = arccos

FP EP GP

The position vector rcnf (ϕ , µ ) of a point of the indicatrix of conformity Cnf R ( P/T ) is not a function of the coordinate angle ω P. Although the position vector rcnf (ϕ , µ ) depends on the fundamental magnitudes EP , GP and ET , GT , the above analysis makes it clear why rcnf (ϕ , µ ) is not dependent on the fundamental magnitudes FP and FT. Two illustrative examples of the indicatrix of conformity Cnf R ( P/T ) are shown in Figure 4.11. The first example (Figure 4.11a) relates to the cases of contact of a saddle-like local patch of the part surface P and of a convex ellipticlike local patch of the generating surface T. The second one (Figure 4.11b) is for the case of contact of a convex parabolic-like local patch of the part surface P and of a convex, elliptic-like local patch of the generating surface T. For both cases (see Figure 4.11), the corresponding curvature indicatrices Crv(P) and Crv(T) of the surfaces P and T are depicted as well. The imaginary (phantom) branches of the Dupin’s indicatrix Dup(P) for the saddle-like local patch of the part surface P are represented by dashed lines (see Figure 4.11a). Surfaces P and T can make contact geometrically but physical conditions of their contact could be violated. Violation of the physical condition of contact results in the surfaces P and T interfering with one another. Implementation of the indicatrix of conformity Cnf R ( P/T ) immediately uncovers the interference of the surfaces, if there is any. Three illustrative examples of the violation of physical condition of contact are depicted in Figure 4.12. When the correspondence between radii of normal curvature is inappropriate, then the indicatrix of conformity Cnf R ( P/T ) either intersects itself (Figure 4.12a), or all of its diameters become negative (Figure 4.12b and Figure 4.12c). The value of the current diameter dcnf of the indicatrix of conformity Cnf R ( P/T ) indicates the rate of conformity of the surfaces P and T in the corresponding cross-section of the surfaces by normal plane through the common perpendicular. Orientation of the normal plane sections with respect to the surfaces P and T is defined by the corresponding central angle j. For the orthogonally parameterized surfaces P and T, the equation of Dupin’s indicatrices Dup(P) and Dup(T) simplifies to

LP xP2 + 2 M P xP y P + N P y P2 = ±1

(4.60)

LT xT2 + 2 MT xT yT + N T yT2 = ±1

(4.61)

The diameter of a centro-symmetrical curve can be defined as a distance between two points of the curve, measured along the corresponding straight line through the center of symmetry of the curve.

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The Geometry of Contact of Two Smooth, Regular Surfaces yP

CnfR(P/T )

CnfR(P/T )

nP

Crv(P)

dcnf

P

C1 P

Crv(P) nT

xP

K

C2.T

K

Crv(T )

T

C1.T

C2 P (a)

yP

CnfR(P/T )

nP Crv(P)

P

C2 P

K

K

C1.T dcnf

Crv(P) nT

xP

Crv(T ) μ

T

CnfR(P/T)

C1 P

C2.T

(b)

Figure 4.11 Examples of indicatrix of conformity Cnf R (P / T ) for two smooth, regular surfaces. (From Radzevich, S.P., Mathematical and Computer Modeling, 39 (9–10), 1083–1112, 2004. With permission.)

After being represented in a common coordinate system, Equation (4.60) and Equation (4.61) yield a simplified equation of the indicatrix of conformity Cnf R ( P / T ) of the surfaces P and T: rcnf (ϕ , µ ) = ( LP cos 2 ϕ - M P sin 2ϕ + N P sin 2 ϕ )

-

1 2

sgn φ2-1.P

1 + [ LT cos 2 (ϕ + µ ) - MT sin 2(ϕ + µ ) + N T sin 2 (ϕ + µ )] 2 sgn φ2-.T1

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(4.62)

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Kinematic Geometry of Surface Machining CnfR (P/T )

yP

CnfR (P/T ) C2.P

nP

C2.T

Crv(P)

T

C1.P

K

C1.T

Crv(P)

xP

K

Crv(T) P

nT (min)

dcnf

(a) yP C1.T

nP C1.P

K

T

nT

xP

C2.T K

Crv(T ) P

Crv(P ) (min)

dcnf yP

Crv(P)

nT

C2.T

K

<0

Crv(T )

C1.T C1.P

P

C2.P

T

Crv(T )

Crv(P) (b)

nP

CnfR (P/T )

C2.P

CnfR (P/T)

<0

CnfR (P/T)

xP

K (min)

dcnf

<0

Crv(P )

Crv(T )

(c) Figure 4.12 Examples of violation of condition of contact of two smooth, regular surfaces P and T. (From Radzevich, S.P., Mathematical and Computer Modeling, 39 (9–10), 1083–1112, 2004. With permission.)

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Equation (4.59) of the indicatrix of conformity Cnf R ( P/T ) yields an equation of one more characteristic curve. This characteristic curve is referred to as the curve of minimal values of the position vector rcnf , which is expressed in terms of j. In the general case, the equation of this characteristic curve can be represented (min) (min) in the form rcnf = rcnf ( µ ). For the derivation of the equation of the character(min) (min) istic curve rcnf = rcnf ( µ ) , the following method can be employed. A given relative orientation of the surfaces P and T is specified by the value of the angle m of the surfaces P and T local relative orientation. The minimal (min) value of rcnf is observed when the angular parameter j 2is equal to the root ∂ ϕ1 of equation ∂∂ϕ rcnf (ϕ , µ ) = 0 . The additional condition ∂ϕ 2 rcnf (ϕ , µ ) > 0 must be satisfied as well. In order to determine the necessary value of the angle ϕ1, the equation ∂∂ϕ rcnf (ϕ , µ ) = 0 must be solved with respect to m. After substituting the obtained solution µ (min) to Equation (4.48) of the indicatrix of (min) (min) conformity Cnf R ( P/T ), the equation rcnf = rcnf (ϕ ) of the curve of minimal diameters of the characteristic curve Cnf R ( P/T ) can be derived. In a similar way, one more characteristic curve, say the characteristic curve (max) (max) rcnf = rcnf (ϕ ) , can be derived. The last characteristic curve reflects the distribution of the maximal values of the position vector rcnf in terms of j. 4.4.3 Directions of the Extremum Rate of Conformity of the Surfaces P and T Directions, along which the rate of conformity of the surfaces P and T is extremum (that is, it reaches either its maximum or its minimum value), are of prime importance for many engineering applications. This issue is especially important when designing blend surfaces, for computation of parameters of optimal tool-paths for the machining of sculptured surfaces on a multi-axis NC machine, for improving the accuracy of the solution to the problem of two elastic bodies in contact, and for many other applications in applied science and in engineering. Directions of the extremal rate of conformity of the surfaces P and T (i.e., (min) (max) the directions pointed along the extremal diameters dcnf and dcnf ) can be determined from the equation of the indicatrix of conformity Cnf R ( P/T ). For convenience, Equation (4.48) of this characteristic curve is transformed and is represented in the form rcnf (ϕ , µ ) = |r1. P cos 2 ϕ + r2. P sin 2 ϕ |sgn φ2-.P1

+ |r1.T cos (ϕ + µ ) + r2.T sin (ϕ + µ )|sgn φ 2

2

(4.63) -1 2.T

Two angles ϕ min and ϕ max specify two directions within the common tangent plane, along which the rate of conformity of the surface T to the surface P reaches its extremal values. These angles are the roots of the following equation:

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∂ rcnf (ϕ , µ ) = 0. ∂ϕ

(4.64)

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It is easy to prove that in the general case of two sculptured surfaces in contact, the difference between the angles ϕ min and ϕ max is not equal to 0.5π . This means the equality ϕ min - ϕ max = ±0.5π n is not observed, and in most cases, the relationship ϕ min - ϕ max ≠ ±0.5π n is valid. (Here n is an integer number.) The condition ϕ min = ϕ max ± 0.5π n is satisfied only in cases when the angle μ of the surfaces P and T local relative orientation is equal to µ = ±0.5π n, and thus the principal directions t 1.P and t 2.P of the surface P, and the principal directions t 1.T and t 2.T of the surface T are either aligned or are directed oppositely. This enables one to make the following statement: In the general case of two sculptured surfaces in contact, directions along which the rate of conformity of two smooth, regular surfaces P and T is extremal are not orthogonal to each other. This conclusion is important for engineering applications. The solution to Equation (4.28) returns two extremal angles ϕ min and ϕ max = ϕ min + 90°. Equation (4.64) allows for two solutions ϕ min and ϕ max. Therefore, it is easy to compute the extremal difference ∆ϕ min = ϕ min - ϕ min, . as well as the extremal difference ∆ϕ max = ϕ max - ϕ max Generally speaking, neither the extremal difference ∆ϕ min nor the extremal difference ∆ϕ max is equal to zero. They are equal to zero only in particular cases, say when the angle μ of the surfaces P and T local relative orientation satisfies the relationship µ = ±0.5π n.

Example 1 As an illustrative example, let us describe analytically the geometry of contact of two convex parabolic patches of the surfaces P and T (Figure 4.13). In the example under consideration, the design parameters of the gear and of the shaving cutter together with the given gear and the cutter configuration yield the following numerical data for the computation. At the point K of the surfaces contact, principal curvatures of the surface P are equal: k1. P = 4 mm -1 and k 2. P = 0 . Principal curvatures of the surface T are equal: k1.T = 1mm -1 and k 2.T = 0 . The angle m of the surfaces P and T local relative orientation is equal to µ = 45°. Two approaches can be implemented for the analytical description of the geometry of contact of the surfaces P and T. The first one is based on implementation of Dupin’s indicatrix of the surface of relative curvature. Another is based on application of the indicatrix of conformity Cnf R ( P / T ) of the surfaces P and T at point K. The First Approach For the case under consideration, Equation (4.28) reduces to kR = k1. P cos 2 ϕ - k1.T cos 2 (ϕ + µ )

(4.65)

Therefore, the equality

∂ kR = -2k1. P sin ϕ cos ϕ + 2k1.T sin(ϕ + µ )cos(ϕ + µ ) = 0 ∂ϕ

is valid.

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(4.66)

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Crv (T ) Cnf

(im)

jˆ2

Crv (P

π 2

CnfR(P/T )

j2

j1 jˆ 1

K CnfR(P/T)

(min)

xP

(max) tcnf

dcnf

(im)

CnfR (P/T )

Figure 4.13 Example 1: Determination of the optimal instant kinematics for a gear shaving operation. (From Radzevich, S.P., Mathematical and Computer Modeling, 39 (9–10), 1083–1112, 2004. With permission.)

For the directions of the extremal rate of conformity of the surfaces P and T, Equation (4.66) yields computation of the extremal values ϕ min = 7° and ϕ max = ϕ min + 90° = 97° of the angles ϕ min and ϕ max . The direction specified by the angle ϕ min = 7° indicates the direction of the minimal diameter of Dupin’s indicatrix of the surface of relative curvature. That same direction corresponds to the maximal rate of conformity of the surfaces P and T. Another direction, which is specified by the angle ϕ max = 97° , indicates the direction of the minimum rate of conformity of the surfaces P and T. The Second Approach For the case under consideration, use of Equation (4.59) of the indicatrix of conformity Cnf R ( P/T ) of the surfaces P and T enables one computation of the extremal angles ϕ min = 19° and ϕ max = 118°. Imaginary branches of the indicatrix of conformity Cnf R ( P/T ) of the surfaces P and T in Figure 4.13 are depicted in dashed lines. Two issues are of importance here. First, the extremal angles ϕ min and ϕ max computed using the first approach are not equal to the corresponding extremal angles ϕ min and ϕ max that are computed using the second approach. The relationships ϕ min ≠ ϕ min and ϕ max ≠ ϕ max are generally observed. Second, the difference ∆ϕ of the extremal angles ϕ min and ϕ max is not equal to

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half of π. Therefore, the relationship ϕ max - ϕ min  90° between the extremal angles ϕ min and ϕ max is observed. In the general case, directions of the extremal rate of conformity of the surfaces P and T are not orthogonal to one another. The example reveals that in general cases of two smooth, regular sculptured surfaces in contact, the indicatrix of conformity Cnf R ( P/T ) can be implemented for the purpose of accurate analytical description of the geometry of contact of the surfaces. Dupin’s indicatrix of the surface of relative normal curvature can be implemented for this purpose only in particular cases of the surface’s configuration. Application of Dupin’s indicatrix of the surface of relative curvature enables only approximate analytical description of the geometry of contact of the surfaces. Dupin’s indicatrix of the surface of relative curvature could be equivalent to the indicatrix of conformity only in degenerated cases of contact of two surfaces. Advantages of the indicatrix of conformity over Dupin’s indicatrix of the surface of relative curvature are that this characteristic curve is a curve of the fourth order. 4.4.4 Asymptotes of the Indicatrix of Conformity CnfR (P/T) In the theory of surface generation, asymptotes of the indicatrix of conformity Cnf R ( P/T ) play an important role. The indicatrix of conformity could have asymptotes when a certain combination of parameters of shape of the surfaces P and T is observed. Straight lines that possess the property of becoming and staying infinitely close to the curve as the distance from the origin increases to infinity are referred to as the asymptotes. This definition of the asymptotes is helpful for derivation of the equation of asymptotes of the indicatrix of conformity of the surfaces P and T. In polar coordinates, the indicatrix of conformity Cnf R ( P/T ) is analytically described by Equation (4.59). For convenience, the equation of this characteristic curve is represented below in the form of rcnf = rcnf (ϕ , µ ). Derivation of the equation of the asymptotes of the characteristic curve rcnf = rcnf (ϕ , µ ) can be accomplished in just a few steps: For a given indicatrix of conformity rcnf = rcnf (ϕ , µ ), compose a function (ϕ , µ ) rcnf that is equal: (ϕ , µ ) = rcnf

1 rcnf (ϕ , µ )

(4.67)

(ϕ , µ ) = 0 with respect to j. The solution ϕ Solve the equation rcnf 0 to this equation specifies the direction of the asymptote. Calculate the value of the parameter m 0. The value of the parameter m 0 is ∂g (ϕ , µ ) equal m0 = ( ∂ϕ )-1 under the condition ϕ = ϕ 0.

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121

The asymptote is the line through point (m0 , ϕ 0 + 0.5π ), and with the direction ϕ 0. Its equation is r(ϕ ) =

m0 sin(ϕ - ϕ 0 )

(4.68)

In particular cases, asymptotes of the indicatrix of conformity Cnf R ( P/T ) can coincide either with the asymptotes of the Dupin’s indicatrix Dup(P) of the surface P, or of the Dupin’s indicatrix Dup(T) of the surface T, or finally with Dupin’s indicatrix Dup(P/T) of the surface of relative curvature R. 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 Both characteristic curves — that is, the indicatrix of conformity Cnf R ( P/T ) of the surfaces P and T, and Dupin’s indicatrix Dup(P/T) of the surface of relative curvature can be used with the same goal of analytical description of the geometry of contact of the surfaces P and T in the first order of tangency. Therefore, it is important to compare the capabilities of these characteristic curves. A detailed analysis of capabilities of the indicatrix of conformity Cnf R ( P/T ) of the surfaces P and T (see Equation 4.59) and of Dupin’s indicatrix of the surface of relative curvature Dup(P/T) (see Equation 4.37) is performed. This analysis allows the following conclusions to be made. From the viewpoint of completeness and effectiveness of analytical description of the geometry of contact of two surfaces in the first order of tangency, the indicatrix of conformity Cnf R ( P/T ) is more informative compared to Dupin’s indicatrix Dup(P/T) of the surface of relative curvature. It more accurately reflects important features of the geometry of contact in differential vicinity of the point K. Thus, implementation of the indicatrix of conformity Cnf R ( P/T ) for scientific and engineering purposes has advantages over Dupin’s indicatrix of the surface of relative curvature Dup(P/T). This conclusion is directly drawn from the following: Directions of the extremal rate of conformity of the surfaces P and T that are specified by Dupin’s indicatrix Dup(P/T) are always orthogonal to one another. Actually, in the general case of contact of two sculptured surfaces, these directions are not orthogonal to each other. They could be orthogonal only in particular cases of the surfaces’ contact. The indicatrix of conformity Cnf R ( P/T ) of the surfaces P and T properly specifies the actual directions of the extremal rate of conformity of the surfaces P and T. This is particularly (but not only) due to the fact that the characteristic curve Cnf R ( P/T ) is a curve of the fourth order, while the Dupin’s indicatrix Dup(P/T) of the surface of relative curvature is a curve of the second order.

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An accounting of the members of higher order in the equation of Dupin’s indicatrix Dup(P/T) of the surface of relative curvature does not enhance the capabilities of this characteristic curve and is useless. An accounting of the members of higher order in Taylor’s expansion of the equation of Dupin’s indicatrix gives nothing more for proper analytical description of the geometry of contact of two surfaces in the first order of tangency. Principal features of the equation of this characteristic curve cause a principal disadvantage of Dupin’s indicatrix Dup(P/T). The disadvantage above is inherent to Dupin’s indicatrix, and it cannot be eliminated. 4.4.6 Important Properties of the Indicatrix of Conformity CnfR (P/T) Analysis of Equation (4.59) of the indicatrix of conformity Cnf R ( P/T ) reveals that this characteristic curve possesses the following important properties: The indicatrix of conformity Cnf R ( P/T ) of the surfaces P and T is a planar characteristic curve of the fourth order. It possesses the property of central symmetry, and in particular cases it also possesses the property of mirror symmetry. The indicatrix of conformity Cnf R ( P/T ) is closely related to the surfaces’ P and T second fundamental forms φ 2.P and φ 2.T. This characteristic curve is invariant with respect to the kind of parameterization of the surfaces P and T, but its equation does. The last is similar in much to an indicatrix of conformity CnfR(P/T) is represented in different reference systems. A change in the surfaces’ P and T parameterization leads to changes in the equation of the indicatrix of conformity Cnf R ( P/T ), while the shape and parameters of this characteristic curve remain unchanged. The characteristic curve Cnf R ( P/T ) is independent of the actual value of the coordinate angle ω P that makes the coordinate lines U P and VP on the part surface P. It is also independent of the actual value of the coordinate angle ωT that makes the coordinate lines UT and VT on the generating surface T of the cutting tool. However, parameters of the indicatrix of conformity Cnf R ( P/T ) are dependent upon the angle m of the surfaces P and T local relative orientation. Therefore, for the given pair of surfaces P and T, the rate of conformity of the surface varies correspondingly to variation of the angle m, while the surface T is spinning around the unit vector of the common perpendicular. 4.4.7 The Converse Indicatrix of Conformity of the Surfaces P and T in the First Order of Tangency For Dupin’s indicatrix Dup(P/T) of the surface of relative curvature, a corresponding inverse Dupin’s indicatrix Dupk ( P/T ) exists. Similarly, for the indicatrix of conformity Cnf R ( P/T ) of the surfaces P and T, a corresponding converse indicatrix of conformity Cnf k ( P/T ) exists. This characteristic curve

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The Geometry of Contact of Two Smooth, Regular Surfaces

can be expressed directly in terms of the surfaces’ P and T normal curvatures k P and kT:

cnv Cnf k ( P / T ) ⇒ rcnf (ϕ , µ ) = |k P (ϕ )|⋅ sgn Φ -21. P - |kT (ϕ , µ )|⋅ sgn Φ -21.T

(4.69)

For derivation of the equation of the converse indicatrix of conformity Cnf k ( P/T ), the Euler’s formula for a surface normal curvature is used in the following representation:

k P (ϕ ) = k1. P cos 2 ϕ + k 2. P sin 2 ϕ kT (ϕ , µ ) = k1.T cos 2 (ϕ + µ ) + k 2.T sin 2 (ϕ + µ )

(4.70)

(4.71)

In Equation (4.70) and Equation (4.71), the principal curvatures of the part surface P are designated as k1. P and k 2. P , and k1.T and k 2.T designate the principal curvatures of the generating surface T of the cutting tool. After substitution of Equation (4.70) and Equation (4.71) into Equation (4.69), one can come up with the equation for the converse indicatrix of conformity Cnf k ( P/T ) of the surfaces P and T in the first order of tangency: cnv rcnf (ϕ , µ ) = |k1. P cos 2 ϕ + k 2. P sin 2 ϕ |sgn Φ2-.1P

- |k1.T cos (ϕ + µ ) + k 2.T sin (ϕ + µ )|sgn Φ 2

2

(4.72) -1 2.T

where principal curvatures k1. P , k 2. P and k1.T , k 2.T can be expressed in terms of the corresponding fundamental magnitudes EP , FP , GP of the first order and LP, M P, N P of the second order of the part surface P, and in terms of the corresponding fundamental magnitudes ET , FT , GT of the first order and LT , MT , N T of the second order of the generating surface T of the cutting tool. Following this, Equation (4.72) of the inverse indicatrix of conformity Cnf k ( P/T ) can be cast into the form similar to Equation (4.59) of the ordinary indicatrix of conformity Cnf R ( P/T ) of the surfaces P and T. Similar to the indicatrix of conformity Cnf R ( P/T ) , the characteristic curve Cnf k ( P/T ) also possesses the property of central symmetry. In particular cases of the surface contact, it also possesses the property of mirror symmetry. Directions of the extremal rate conformity of the surfaces P and T are orthogonal to one another only in degenerated cases of the surfaces contact. Equation (4.72) of the converse indicatrix of conformity Cnf k ( P/T ) is convenient for implementation when the surface P, or the surface T, or both have points or lines of inflection. In the points or lines of inflection, radii of normal curvature RP (T ) of the surface P(T) are equal to infinity. This causes indefiniteness when computing the position vector rcnf (ϕ , µ ) of the characteristic curve Cnf R ( P/T ). Equation (4.72) of the converse indicatrix of conformity Cnf k ( P/T ) is free of these disadvantages and is therefore recommended for practical applications.

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4.5

Kinematic Geometry of Surface Machining

Plücker’s Conoid: More Characteristic Curves

More characteristic curves for the analytical description of the geometry of contact of two smooth, regular surfaces in the first order of tangency can be derived on the premises of Plücker’s conoid [9]. 4.5.1

Plücker’s Conoid

Several definitions for Plücker’s conoid are known. First, Plücker’s conoid is a smooth, regular, ruled surface. A ruled surface is sometimes also called the cylindroid, which is the inversion of the cross-cap. Plücker’s conoid can also be considered as an example of a right conoid. A ruled surface is called a right conoid if it can be generated by moving a straight line intersecting a fixed straight line such that the lines are always perpendicular. As with the cathenoid, another ruled surface, Plücker’s conoid must be reparameterized to see the rulings. Illustrative examples of various Plücker’s conoids are considered in [10]. 4.5.1.1 Basics The ruled surface can be swept out by moving a line in space; therefore, it has a parameterization of the following form: x(u, v) = b(u) + v (u)

(4.73)

where b is the directrix (also referred to as the base curve) and v is the director curve. The straight lines are the rulings. The rulings of a ruled surface are asymptotic curves. Furthermore, the Gaussian curvature on a ruled, regular surface is nonpositive at all points. The surface is known for the presence of two or more folds formed by the application of a cylindrical equation to the line during this rotation. This equation defines the path of the line along the axis of rotation. 4.5.1.2 Analytical Representation For the Plücker’s conoid, von Seggern [20] gives the general functional form as a x2 + b y 2 - z x2 - z y 2 = 0

(4.74)

whereas Fischer [3] and Gray [4] give z=

2xy x2 + y 2

(4.75)

Plücker’s conoid is a ruled surface, bearing the name of famous German mathematician and physicist Julius Plücker (1802–1868), who is known for his research in the field of a new geometry of space [9].

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The Geometry of Contact of Two Smooth, Regular Surfaces

Another form of Cartesian equation for a two-folds Plücker’s conoid is known as well [21]: z=a

x2 - y 2 x2 + y 2

The equation above yields the following matrix representation of nonpolar parameterization of the Plücker’s conoid:

 rpc (u, v) = u 

2uv u2 + v 2

v

T

 0 

(4.76)

The Plücker’s conoid could be represented by the polar parameterization:

rpc (r , θ ) = [r cos θ r sin θ 2 cos θ sin θ 0] T

(4.77)

A more general form of the Plücker’s conoid is parameterized below, with “n” folds instead of just two. A generalization of Plücker’s conoid to n folds is given by the following [4]:

rpc (r , θ ) = [r cos θ r sin θ sin (nθ ) 0] T

(4.78)

The difference between these two forms is the function in the z axis. The polar form is a specialized function that outputs only one type of curvature with two undulations; the generalized form is more flexible with the number of undulations of the output curvature being determined by the value of n. Cartesian parameterization of the equation of the multifold Plücker’s conoid (see Equation 4.78) therefore gives [21] z

(

x2 + y 2

) = ∑ (-1) C n

k

0≤ k ≤

n 2

2k n

x n- 2 k y 2 k .

(4.79)

The surface appearance depends upon the actual number of folds [10]. In order to represent the Plücker’s conoid as a ruled surface, it is sufficient to represent Equation (4.78) in the form of Equation (4.79):

0  r cos θ   r cos θ    cos θ   r sin θ   r sin θ     sin θ  0        rpc (r , θ ) = = = +r sin (nθ ) 2 cos θ sin θ  2 cos θ sin θ   0          0 0  0       0 

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(4.80)

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Kinematic Geometry of Surface Machining

Taking the perpendicular plane as the xy plane and taking the line to be the x-axis gives the parametric equation [4]:  v ⋅ cos v(u)  v ⋅ sin v(u)   rpc =   h(u)    0  

(4.81)

The equation in cylindrical coordinates [21] is z = a cos (nθ ) , which simplifies to z = a cos 2θ if n = 2. 4.5.1.3 Local Properties Following Bonnet’s theorem (see Chapter 1), local properties of Plücker’s conoid could be analytically expressed in terms of the first and of the second fundamental forms of the surface. For practical application, some useful auxiliary formulas are also required. The first and the second fundamental forms [21] of Plücker’s conoid could be represented as

φ1 ⇒ d s2 = d ρ 2 + ( ρ 2 + n2 a2 sin 2 (n θ )) d θ 2

(4.82)

na [sin(n θ ) dρ - n ρ cos(n θ ) dθ ] dθ H

(4.83)

φ2 ⇒

Asymptotes are given by the equation ρ n = k an sin (n θ ) . They strictly correlate to Bernoulli’s lemniscates [21]. For the simplified case of Plücker’s conoid n = 2, the first and the second fundamental forms reduce to [21]

φ1 ⇒ d s2 = d ρ 2 + ( ρ 2 + 4 a2 cos 2 2θ d θ 2

E = 1, F = 0, G = ρ 2 + 4 a2 cos 2 2θ , H = G

φ2 ⇒ -

4a [sin 2θ dρ - n ρ cos 2θ dθ ] dθ H

L = 0, M = -

2 a cos 2θ 4 a ρ sin 2θ , N=H H

(4.84)

(4.85) (4.86) (4.87)

Because the discussion of auxiliary formulas that follows is limited to the case of n = 2, auxiliary formulas for further reference would be helpful.

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The Geometry of Contact of Two Smooth, Regular Surfaces 4.5.1.4 Auxiliary Formulas At u = u0 , v = v0 , the tangent to the surfaces is parameterized by

u + u0     v + v0   rpc (u, v) =  2 (-u u02 v0 + u v03 + u0 v02 (- v + v0 ) + u03 (v + v0 ))    u02 + v02     0

(4.88)

The surface normal is its double line [10,19]. The infinitesimal area of a patch on the surface is given by

φ1 ⇒ d s = 1 +

4 (u - v)2 (u + v 2 ) du dv (u2 + v 2 )3

(4.89)

Gaussian curvature of the Plücker’s conoid could be computed from

G (u, v) =

4 (u 4 - v 4 )2 (u + v ( 4 + v ) + u2v 2 (-8 + 3v 2 ) + u 4 ( 4 + 3v 2 ))2 6

4

2

(4.90)

The mean curvature of the Plücker’s conoid is equal to

M (u, v) =

4uv

(

(u2 + v 2 )2 1 +

)

3 4 ( u- v )2 ( u + v )2 2 ( u2 + v 2 )3

(4.91)

4.5.2 Analytical Description of Local Topology of the Smooth, Regular Surface P For this discussion, the following characteristics of a smooth, regular surface P are of prime importance: (a) tangent plane to the surface P, (b) unit normal n P , and (c) the surface P principal curvatures k1. P and k 2. P, as well as the surface P normal curvature k P at the prespecified direction. Plücker’s conoid could be used for visualization of the distribution of the surface P normal curvature at a given point. The corresponding Plücker’s conoid can be determined at every point of the smooth, regular surface P. The surface unit normal vector n P can be employed as the axis of the Plücker’s conoid. The rulings are the straight lines that intersect the z axis at a right angle. The generating straight-line segments of the Plücker’s conoid are always parallel to the tangent plane to the surface P at the point at which the Plücker’s conoid is erected. Other applications of the tangent plane to the

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surface P are also important. Consequently, the above yields the natural way to connect Plücker’s conoid to the surface P. 4.5.2.1 Preliminary Remarks An example of implementation of Plücker’s conoid is given by Struik [19]. He considers a cylindroid that is represented by the locus of the curvature vectors at a point P of a surface belonging to all curves passing through P: z( x 2 + y 2 ) = k1. P x 2 + k 2. P y 2

(4.92)

where k1. P and k 2. P designate the principal curvatures of the surface P (the inequality k1. P > k 2. P is always observed). The curvature vector is defined in the following way. According to [19], a proportionality factor k P can be introduced such that k P = d t P dS = k P n P

(4.93)

where the vector k P = d t P dS expresses the rate of change of the tangent when we proceed along the curve. It is called the curvature vector. The factor k P is called the curvature; |k P | is the length of the curvature vector. Although the sense of n P may be arbitrarily chosen, that of dt P dS is perfectly determined by the curve, independent of its orientation; when S changes sign, t P also changes sign. When n P is taken in the sense of S (as is often done), then κ P is always positive, but we will not adhere to this convention. 4.5.2.2 Plücker’s Conoid In order to develop an appropriate graphical interpretation of Plücker’s conoid Pl R ( P) of a surface P, consider a smooth regular surface P that is given by vector equation rP = rP (U P , VP ). From the prospective of naturally connecting Plücker’s conoid to the surface P, the axis of Plücker’s conoid Pl R ( P) aligns with the unit normal vector n P to the surface P at K. For further consideration, the normal radii of curvature RP = k P-1 of the surface P at point K must be computed. In order to simplify computations, the equation R P = f1.P/f 2.P can be reduced to the Euler’s formula for normal radii of curvature:

(

RP (ϕ ) = R -11. P cos 2 ϕ + R2-.1P sin 2 ϕ

)

-1

(4.94)

where R 1. P and R1. P are the principal radii of curvature of the surface P at point K, and j is the angle the normal plane section RP (ϕ ) makes with the first principal direction t 1.P.

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Point C1 coincides with the curvature center of the surface P in the first principal plane section of P at K. It is located on the axis of Plücker’s conoid Pl R ( P). The straight-line segment of length R 1. P extends from C1 in the direction of t 1.P . The unit tangent vector t 1.P indicates the first principal direction of the surface P at K. It makes a right angle with the axis of the surface Pl R ( P). The straight-line segment of the same length R 1. P extends from C1 in the opposite direction -t 1.P . Point C2 coincides with the curvature center of the surface P in the second principal plane section of P at K. It is remote from C1 at a distance ( R1. P - R2. P ). (Remember that the normal radius of curvature RP and the principal radii of curvature R 1. P and R 2. P are the algebraic values in nature.) The straight-line segment of length R 2. P extends from C2 in the direction of t 2.P. The unit tangent vector t 2.P indicates the second principal direction of the surface P at K. It also makes a right angle with the axis of Pl R ( P). The straight-line segment of the same length R 2. P extends from C2 in the direction -t 2.P . A certain point C coincides with the center of curvature of the surface P in the normal plane section of P at K in an arbitrary direction specified by the corresponding value of central angle j. The point C is located on the axis of the surface Pl R ( P). The normal radius of curvature R P (ϕ ) corresponds to the principal radii of curvature R 1. P and R2. P in the manner R 1. P < R P (ϕ ) < R 2. P. The straight-line segment of length R P = R P (ϕ ) rotates about and travels up and down the axis of Plücker’s conoid Pl R ( P). In such a way, Plücker’s conoid could be represented as a locus of consecutive positions of the straight-line segment R P = R P (ϕ ). Figure 4.14 reveals that Plücker’s conoid perfectly reflects the topology of the surface P in the differential vicinity of point K. Therefore, the surface Pl R ( P) could be implemented as a tool for visualization of the change of its local parameters. In order to plot Plücker’s conoid Rr = RP + RT together with the surface P (Figure 4.14), equations of both surfaces in a common coordinate system (for example, in the coordinate system XSYS ZS) must be represented. For this purpose, the operator of resultant coordinate system transformation Rs(S → P) must be composed (see Equation 3.15). After being constructed at a point of the smooth, regular surface P, the characteristic surface Pl R ( P) clearly indicates the actual values of principal radii of curvature R1. P and R2. P ; the location of the curvature centers O1. P and O2. P ; orientation of the principal plane sections C1. P and C2. P (that is, directions of the unit tangent vectors t 1.P and t 2.P of the principal directions); as well as the current value of normal radii of curvature R(ϕ ) , location of curvature center OP for any given section by normal plane CP through the given direction t P (ϕ ).

It is important to note that for convenience, Plücker’s conoid (Figure 4.14) is scaled along the axes of the local coordinate system (with the goal of better visualization of the local geometrical properties of surface P).

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−t2.P

vR2.P

t2.P

C2.P PIR(P)

t1.P

C2.P −t1.P

C1.P R2.P nP

M ZP

rP

XP

C1.P

R1.P

PIR(P)

P

t1.P t2.P

ZS XS

YP YS

Figure 4.14 Plücker’s conoid Pl R ( P) and the Plücker’s curvature indicatrix Pl R ( P), naturally connected to the concave patch of smooth, regular surface P. (From Radzevich, S.P., Mathematical and Computer Modeling, 42 (9–10), 999–1022, 2005. With permission.)

Therefore, Plücker’s conoid could be considered as an example of a characteristic surface that could potentially be used in the theory of surface generation. In addition to Plücker’s conoid Pl R ( P) described above (Figure 4.14), an inverse characteristic surface Pl k ( P) could be introduced. When constructing the Plücker’s conoid Pl k ( P) (Figure 4.14), the straight-line segment of the length Rp (ϕ ) does not have to be used, but the straight-line segment of length k p (ϕ ) can be used instead. (Here k p (ϕ ) = RP-1 (ϕ ) is the normal curvature of the surface P at point K in the given normal plane section through K.) This yields construction of the inverse characteristic surface Pl k ( P). The characteristic surfaces Pl R ( P) and Pl k ( P) resemble one another from many aspects. They also appear similarly, except when Rp (ϕ ) or k p (ϕ ) is equal either to zero (0) or to infinity (∞). The characteristic surface Pl R ( P) is referred to as the Plücker’s conoid of the first kind; the characteristic surface Pl k ( P) is referred to as the Plücker’s conoid of the second kind. The conoids Pl R ( P) and Pl k ( P) are inverse to each other (Pl R ( P) = Plinv k ( P ), and vice versa) . Plücker’s conoids — Pl R ( P) and Pl k ( P) — clearly indicate the change of parameters of local topology in the differential vicinity of a point of smooth, regular surface P.

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4.5.2.3 Plücker’s Curvature Indicatrix The boundary curve of Plücker’s conoid contains all the necessary information on the distribution of normal curvatures of the surface P in differential vicinity of the point K, and the surface Pl R ( P) represents additional information about the local topology of surface P. This additional information is beyond the scope of this book from the standpoint of its implementation in the theory of surface generation. We are again reminded of the principle of Occam’s razor. Thus, without loss of accuracy, Plücker’s conoid could be replaced with the boundary curve of the surface Pl R ( P). The boundary curve Pl R ( P) of the characteristic surface Pl R ( P) is referred to as Plücker’s curvature indicatrix of the first kind of the part surface P at point K. Plücker’s curvature indicatrix is represented therefore by the endpoints of the position vector of length of R P (ϕ ) that is rotating about and travels up and down the axis of the surface Pl R ( P). This immediately leads to the equation of this characteristic curve:

 RP (ϕ )cos ϕ   R (ϕ )sin ϕ  P  Pl R ( P) ⇒ rR (ϕ ) =   RP (ϕ )    0  

(4.95)

where R P (ϕ ) is given by the Euler’s formula RP (ϕ ) = ( R -11. P cos 2 ϕ + R2-.1P sin 2 ϕ )-1 . The performed analysis [10] reveals that for most kinds of smooth, regular surface P, Plücker’s curvature indicatrix Pl R ( P) of the first kind is a closed, regular three-dimensional curve. For the surface local patches of parabolic and of saddle-like type, the Plücker’s indicatrix Pl R ( P) separates onto two and four branches, correspondingly. In particular cases, it could be reduced to a planar curve — to a circle (for example, for umbilic local patches of the surface P). The following equation, similar to Equation (4.95), is valid for Plücker’s curvature indicatrix Pl k ( P) of the second kind:  k P (ϕ )cos ϕ   k (ϕ )sin ϕ  P  Pl k ( P) ⇒ rk (ϕ ) =   k P (ϕ )    0  

(4.96)

where k P (ϕ ) = k1. P cos 2 ϕ + k 2. P sin 2 ϕ .

William of Ockham, also spelled Occam (1285, Ockham, Surrey?, England 1347/49, Munich, Bavaria [now in Germany]), is remembered mostly because he developed the tools of logic. He insisted that we should always look for the simplest explanation that fits all the facts, instead of inventing complicated theories. The rule that said “plurality should not be assumed without necessity” is called Occam’s razor.

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Usually, the Plücker’s curvature indicatrix Pl k ( P) is represented with a closed curve. Further possible simplification of the analytical description of local topology of smooth, regular surface P could be based on the following consideration. 4.5.2.4 An R (P)-Indicatrix of the Surface P Aiming for further simplification of the analytical description of local topology of two smooth, regular surfaces in the first order of tangency, Plücker’s curvature indicatrix could be replaced with the planar characteristic curve of a novel kind. As follows from Equation (4.95), the first two elements RP (ϕ ) cos ϕ and RP (ϕ ) sin ϕ at the right-hand side portion of the above equation contain all the required information on the distribution of normal radii of curvature of P at K. Hence, instead of considering the Plücker’s curvature indicatrix Pl R ( P) (see Equation 4.95) for implementation for the purpose of analytical description of the geometry of contact of two smooth, regular surfaces, a planar characteristic curve An R ( P) of simpler structure could be used. An equation of this characteristic curve yields representation in the form

 RP (ϕ )cos ϕ   R (ϕ )sin ϕ  P  An R ( P) ⇒ r iR (ϕ ) =    0   0  

(4.97)

This planar characteristic curve is referred to as the An R ( P)-indicatrix of the first kind of the surface P at the point K. The distribution of normal curvature of the surface P at K could be given by another planar characteristic curve:

 k P (ϕ )cos ϕ   k (ϕ )cos ϕ  P  An k ( P) ⇒ r ik (ϕ ) =    0   0  

(4.98)

This planar characteristic curve (see Equation 4.98) is referred to as the An R ( P)-indicatrix of the second kind of the part surface P at K. An example of the An k ( P)-indicatrix is shown in Figure 4.15. The characteristic curve An R ( P) is computed for the surface P having principal radii of curvature equal to R1. P = 3 mm and R2. P = 15 mm. It is important to point out that the direction of minimal diameter dR aligns with the first principal direction t 1.P , and the direction of maximal diameter DR aligns with the second principal direction t 2.P on the surface P at K; therefore, the directions for dR and DR are always orthogonal to one another.

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100 120

80

15

60

12.86

3`R(P)

10.71

140

40

8.57

Dup (P)

6.43

160

20

4.29

DR

C1. P

2.14

180

0°

K

Ddup dR

200

220

t1.P

ddup 340

320

C2.P 300

240 260

280

Figure 4.15 The An R (P)-indicatrix of the part surface P at K ( R1. P = 3 mm, R2. P = 15 mm ) plotted together with the corresponding Dupin’s indicatrix Dup(P) that is magnified 10 times. (From Radzevich, S.P., Mathematical and Computer Modeling, 42 (9–10), 999–1022, 2005. With permission.

It is of interest to compare the An R ( P)-indicatrix with the corresponding Dupin’s indicatrix Dup(P). In order to make the comparison, the characteristic curve Dup(P) is computed for that same point K of the surface P (R1. P = 3 mm, R2. P = 15 mm). The characteristic curve Dup(P) is also plotted in Figure 4.15. For convenience, the characteristic curve Dup(P) is magnified ten times with respect to its original (computed) parameters. The direction of the minimal diameter dDup aligns with the first principal direction t 1.P , and the direction of maximal diameter D Dup aligns with the second principal direction t 2.P on the part surface P at K. Figure 4.15 makes it clear that both characteristic curves — that is, the An R ( P) and Dup(P) — indicate the same directions for the first R1. P as well as for the second R2. P radii of curvature of P at K. However, the difference in shape of the characteristic curves An R ( P) and Dup(P) is observed. Dupin’s indicatrix is a planar smooth, regular curve of the second order. In the case under consideration, it is always convex with a uniform change of curvature. The An R ( P)-indicatrix is also a planar smooth, regular curve. However,

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points of inflection are inherent to this curve in nature. This is because the An R ( P)- indicatrix is a curve of the fourth order. Higher order enhances the capabilities of the characteristic curve An R ( P) . Because of the higher order of the equation, the An R ( P)- indicatrix reflects the distribution of normal radii of curvature, and Dupin’s indicatrix Dup(P) reflects the distribution of the square root of normal radii of curvature of P at K. In order to make the difference clear, it is sufficient to represent the equation of Dupin’s indicatrix in the form that is similar to that for An R ( P)indicatrix (see Equation 4.97):

 |RP (ϕ )|cos ϕ sgn RP (ϕ )    |RP (ϕ )|sin ϕ sgn RP (ϕ )  Dup ( P) ⇒ rDup (ϕ ) =   0     0

(4.99)

It is evident from the above that Equation (4.97) and Equation (4.99) are similar. 4.5.3 Relative Characteristic Curves The considered properties of Plücker’s conoid can be employed for derivation of the equation of a planar characteristic curve for analytical description of the geometry of contact of two surfaces for the needs of the theory of surfaces generation. 4.5.3.1 On a Possibility of Implementation of Two of Plücker’s Conoids At a first glimpse, the implementation of two of Plücker’s conoids sounds promising for the purpose of solving the problem of analytical description of the geometry of contact of two smooth, regular surfaces. In order to develop an appropriate solution to the problem under consideration, the characteristic surface Pl R ( P/T ) that reflects summa of the corresponding normal radii of curvature of the surfaces P and T could be introduced. The following matrix representation of the equation of the surface Pl R ( P/T ) immediately follows from the above consideration:

( RP + RT )cos ϕ   ( R + R )sin ϕ  P T  Pl R ( P/T ) ⇒ R R* (ϕ ) =   2 sin ϕ cos ϕ    0  

(4.100)

Below, the characteristic surface Pl R ( P/T ) is referred to as the Plücker’s relative conoid. Because centers of principal curvature C1. P and C2. P of the surface P, as well as centers of principal curvatures C1.T and C2.T of the surface T in the

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general case do not coincide with one another, the actual reciprocation of the straight-line segment of the length ( RP - RT ) could be restricted by different pairs of the limiting points C1. P , C2. P , C1.T , C2.T. Various locations of the limiting points on the axis of rotation result in deformation of the surface Pl R ( P/T ) in its axial direction. Deformation of this kind does not affect the surface appearance in the direction of ( RP + RT ) that is of critical importance for the theory of surface generation. The characteristic surface Pl R ( P/T ) could be analytically described by Equation (4.100). This indicates that the Plücker’s relative conoid properly reflects the rate of conformity of the surfaces P and T at point K. However, the surface Pl R ( P/T ) is inconvenient for implementation in engineering geometry of surfaces. In order to fix this undesired particular problem, one may decide to follow the same method as disclosed above, and introduce the Plücker’s relative indicatrix Pl R ( P/T ) . The equation of this three-dimensional characteristic curve immediately follows from Equation (4.100):

( RP + RT )cos ϕ   ( R + R )sin ϕ  P T  Pl R ( P/T ) ⇒ R R (ϕ ) =   ( RP + RT )    0  

(4.101)

Further, the characteristic curve Pl R ( P/T ) could be reduced to a corresponding planar characteristic curve. In order to shorten the consideration below, the intermediate considerations are omitted, and one can wish to go directly to the An R ( P/T )-relative indicatrix of the surfaces P and T at K. 4.5.3.2 An R(P/T)- Relative Indicatrix of the Surfaces P and T With a goal of further simplifying the analytical description of the geometry of contact of the surfaces P and T, the Plücker’s relative indicatrix Pl R ( P/T ) could be replaced with the planar characteristic curve of a simpler structure. The equation of the two-dimensional An R ( P/T )- relative indicatrix of the surfaces P and T at K could be obtained on the premises of Equation (4.101):

( RP + RT )cos ϕ   ( R + R )sin ϕ  P T  An R ( P/T ) ⇒ R iR (ϕ ) =    0   0  

(4.102)

This planar characteristic curve is referred to as the An R ( P/T )-relative indicatrix of the first kind. The An R ( P/T )-relative indicatrix of the first kind analytically describes the distribution of summa of normal radii of curvature of the part surface P and of the generating surface T of the cutting tool at point K.

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t2.P 90

120

C2.P

30

ϑ

3`R(P/T)

C1.P

max dind

210 3`R(T)

t2.T

60

C1.T

150

180

μ

min

0

t1.P

dind < 0

C2.T

330

μ

3`R(P) 240

270

300

t1.T

Figure 4.16 An example of the An R ( P / T )-relative indicatrix of the surfaces P and T at K (R1. P = 2 mm , R2. P = 3 mm, R1.T = -2 mm, R2.T = -5 mm, m = 45˚) plotted together with the corresponding An R (P)-indicatrix and indicatrix An R (T ) . (From Radzevich, S.P., Mathematical and Computer Modeling, 42 (9–10), 999–1022, 2005. With permission.)

An example of the An R ( P/T )-relative indicatrix of the surfaces P and T is shown in Figure 4.16. The characteristic curve An R ( P/T ) is computed for the case of contact of the convex elliptic local patch of the surface P (R1. P = 3 mm and R2. P = 15 mm ) with the concave elliptic local patch of the surface T R1.T = -2 mm and R2.T = -5 mm). The surfaces P and T are turned through the angle μ = 45° relative to one another around the common perpendicular n P . The corresponding An R ( P/T )-indicatrix and the An R ( P / T )-indicatrix are also plotted in Figure 4.16. Note that the direction of the minimal diameter max min dind and the direction of the maximal diameter dind of the characteristic curve An R ( P/T ) do not align with the principal directions t 1.P and t 2.P on the part surface P or with the principal directions t 1.T and t 2.T on the generating surface T of the cutting tool. The extremal directions of the An R ( P/T )relative indicatrix are not orthogonal to each other. In the general case of the surfaces contact, they make a certain angle ϑ ≠ 90°. The following statement can be made at this point: In the general case of surface contact, directions of the extremal (i.e., of the maximal and of the minimal)

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rate of conformity of the surfaces P and T at the point K are not orthogonal to one another. The directions of the extremal rate of conformity of the surfaces P and T could be orthogonal to one another only in particular (degenerated) cases of the surfaces contact. Shape and parameters of the An R ( P/T )-relative indicatrix depend upon algebraic values of the principal radii of curvature k r = k P - kT , R2. P and R1.T , R2.T of the surfaces P and T, as well as on the actual value of the angle m of the surfaces P and T local relative orientation. Dupin’s indicatrix Dup(P/T) of the surface of relative curvature indicates orthogonality of the directions of the extremal rate of conformity of the surfaces P and T at K. The above consideration reveals that this is not correct and results in calculation errors. The characteristic curve An R ( P/T ) is of a simpler structure than that of Plücker’s relative indicatrix Pl R ( P/T ). The AnR ( P / T )-relative indicatrix is always a planar curve, and the Plücker’s relative indicatrix Pl R ( P/T ) is a three-dimensional curve. This makes the use of the characteristic curve An R ( P/T ) more preferred than the Plücker’s relative indicatrix Pl R ( P/T ). The distribution of differences between normal curvatures of the surfaces P and T at K can be analytically described by a planar characteristic curve of another kind:

( k P - kT )cos ϕ   ( k - k )sin ϕ  P T  An k ( P/T ) ⇒ R ik (ϕ ) =    0   0  

(4.103)

The characteristic curve (see Equation 4.103) is referred to as the An k ( P/T )relative indicatrix of the second kind. The difference between the An R ( P/T )-relative indicatrix and between Dupin’s indicatrix of the surface of relative curvature DupR ( P/T ) is clearly illustrated in Figure 4.17. The planar characteristic curves An R ( P) and An R ( P/T ), as well as the characteristic curves An k ( P) and An k ( P/T ), originate from the Plücker’s conoid. Equation (4.97), Equation (4.98), Equation (4.102), and Equation (4.103) of the corresponding indicatrices An R ( P), An R ( P/T ) and An k ( P), An k ( P/T ) are derived on the premises of Equation (4.80) of the surface of Plücker’s conoid (see [17], and section 4.9, pp. 257–260 in [13]). It is proven analytically that both planar characteristic curves, say the characteristic curve An R ( P/T ), as well as the indicatrix of conformity Cnf R ( P/T ) of two smooth, regular surfaces P and T specify that same direction t max at cnf which the rate of conformity of the surfaces P and T is the maximal. Both characteristic curves Cnf R ( P/T ) and An R ( P/T ) are powerful tools in the theory of surface generation. They are widely used for the analysis of the geometry of contact of two smooth surfaces P and T.

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90

120 150

180

t2.r

tmin

t1.r

60

240

30 d3`

D3` 0

K

Dup(P/T )

tmax

210

Dr

dr 3`k(P/T )

270

330

300

Figure 4.17 Comparison of the An R ( P / T )-relative indicatrix with the Dupin’s indicatrix Dup(P/T) of the surface of relative curvature of the surfaces P and T. (From Radzevich, S.P., Mathematical and Computer Modeling, 42 (9–10), 999–1022, 2005. With permission.)

4.6

Feasible Kinds of Contact of the Surfaces P and T

Analysis and classification of all feasible kinds of contact of a part surface P and of the generating surface T of the cutting tool are a critical issue in the theory of surface generation. Development of a scientific classification of kinds of contact of the surfaces P and T could be considered as the ultimate point in the analysis of the surfaces P and T contact. Prior to discussing this important issue, two more particular issues must be discussed. The first relates to a possibility of implementing the indicatrix of conformity for identification of the actual kind of contact of the surfaces P and T. The second issue relates to the impact of accuracy of the computations on the desired parameters of the indicatrix of conformity Cnf R ( P/T ) of the surfaces P and T. 4.6.1

On a Possibility of Implementation of the Indicatrix of Conformity for Identification of Kind of Contact of the Surfaces P and T

Two surfaces P and T can make contact either at a point or along a line (that is, along a characteristic curve E), or ultimately over a surface patch. An

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actual contact of the surfaces P and T results in certain features in the shape and in the parameters of the indicatrix of conformity Cnf R ( P/T ). From Equation (4.96), it is known that special features in the shape and in the parameters of the indicatrix of conformity are inherent to every kind of contact of the surfaces P and T. For example, when surfaces P and T make contact: (min) At a point K (Figure 4.18a), then the minimal diameter dcnf of the indicatrix of conformity Cnf R ( P/T ) (as well as all other diameters of this (min) characteristic curve) is always positive ( dcnf > 0 ). Along a characteristic curve E (Figure 4.18b), then the minimal diam(min) eter dcnf of the indicatrix of conformity Cnf R ( P/T ) is always iden(min) tical to zero ( dcnf ≡0 ), and all other diameters of this characteristic curve are of positive values ( dcnf > 0 ). Over a surface patch (Figure 4.18c), then the indicatrix of conformity of the surfaces P and T shrinks to a point that coincides with the point K of contact of the surfaces P and T.

The above examples are worked out for the surfaces P and T contact, when in local vicinity of the point K, both the surfaces P and T are smooth, regular surfaces of saddled type. The same is observed for all other types of local patches of the surfaces P and T. In cases when the surfaces P and T intersect — that is, they interfere with each other (Figure 4.12) — then the minimal diameter of the indicatrix of (min) conformity Cnf R ( P/T ) is always negative ( dcnf < 0 ). It is important to note the difference between partial and full interference of the surfaces P and T in differential vicinity of the point K. For instance, in differential vicinity of the point K, a convex elliptic local patch of the surface T can partially intersect a hyperbolic local patch of the surface P (min) (Figure 4.12a). In this case, the minimal diameter dcnf of the indicatrix of (min) conformity Cnf R ( P/T ) is negative ( dcnf < 0 ). For this reason, the indicatrix of conformity Cnf R ( P/T ) not only intersects itself, but intersection of each of its branches also occurs. Varying the angular parameter j within the interval 0 ≤ ϕ ≤ π , positive ( dcnf > 0 ), as well as negative ( dcnf < 0 ) values of the current diameter dcnf of the characteristic curve Cnf R ( P/T ) are observed. Another example to consider is in differential vicinity of the point K, the local patch of the surface T interferes with a local patch of the surface P, as shown for two hyperbolic local patches (Figure 4.12b) and for two parabolic (Figure 4.12c) local patches of the surfaces P and T. For the cases of total (min) local interference of the surfaces P and T, the minimal diameter dcnf of the (min) indicatrix of conformity Cnf R ( P/T ) is always negative ( dcnf < 0 ), and all other diameters dcnf are also of negative value regardless of the actual value of the angular parameter j. The examples above show that every type of contact of the surfaces P and T features important peculiarities of the shape and parameters of the indicatrix of conformity Cnf R ( P/T ). The shape and the parameters of the characteristic curve Cnf R ( P/T ) uniquely follow from the actual kind of

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Kinematic Geometry of Surface Machining yP C1.T

nP K

C2.P

Crv(P)

C2.T

nT

xP

K

Crv(P)

P

CnfR (P/T )

C1 P

CnfR (P/T )

Crv (T )

Crv (T )

(min)

dcnf

(a) yP

CnfR (P/T ) nP K

C1.T

E C2.P

Crv(P)

C2..T

nT

xP

K

Crv ( P )

P

CnfR (P/T )

C1 P

Crv(T)

(min)

dcnf

=0

Crv(T )

(b) yP C1 P

nP

C1.T

K

C2.P Crv (

P

nT

Crv(P)

CnfR (P/T )

K Crv(T )

T

Crv(T ) (min)

(c)

xP

C2.T

dcnf

dcnf

0

Figure 4.18 Correspondence between the shape of the indicatrix of conformity Cnf R (P/T ) and the kind of contact of the surfaces P and T: (a) point contact, (b) line contact, and (c) surface contact. (From Radzevich, S.P., Mathematical and Computer Modeling, 39 (9–10), 1083–1112, 2004. With permission.)

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contact of the surfaces P and T, and are completely determined by the actual kind of contact of the surfaces. In this concern, it is natural to assume that the inverse statement could also be true. The question can be formulated as follows: Are the features of shape and parameters of the indicatrix of conformity Cnf R ( P/T ) necessary and sufficient for making a conclusion regarding the kind of surfaces P and T contact: contact at a point, along a characteristic curve E, or over a surface patch? Or, in other words, could the value and sign of the minimal (min) diameter dcnf of the indicatrix of conformity and features of its shape be implemented as a criteria for determining uniquely the kind of contact of surfaces P and T? The detailed investigation of this particular problem enables one to make the following statement: The actual value and sign of the minimal diameter (min) of the indicatrix of conformity of surfaces P and T and features of its dcnf shape cannot be implemented as sufficient criteria for determining uniquely the actual kind of contact of surfaces P and T. (min) A positive value of the minimal diameter dcnf of the indicatrix of confor(min) mity Cnf R ( P/T ) (that is, dcnf > 0 ) is the sufficient but not necessary requirement for contact of the surfaces P and T at a point. The indicatrix of conformity Cnf R ( P/T ) shrunk to the point K is not sufficient to identify the kind of contact of the surfaces P and T over a surface patch. However, if the surfaces P and T are congruent within a certain surface patch, then the indicatrix of conformity shrinks to a point that coincides with the point K. The inverse statement is not correct. In the event the indicatrix of conformity Cnf R ( P/T ) shrinks to a point, then the surfaces P and T can be congruent to one another only locally. Thus, if the indicatrix of conformity Cnf R ( P/T ) shrinks to a point K, this indicates only necessary but not sufficient condition of the surfaces P and T contact over a certain surface patch. In the case under consideration, the surfaces P and T can make contact along a characteristic curve E and at a point K. (min) If the minimal diameter dcnf of the indicatrix of conformity Cnf R ( P/T ) is equal to zero, this does not necessarily mean that the surfaces P and T make contact along the characteristic curve E. This requirement is only necessary (min) but not sufficient for the line contact of surfaces. In the event dcnf = 0, the surfaces P and T can make contact at a point. As follows from the discussion above, if the surfaces P and T make contact along a characteristic curve E, then the direction along the minimal diameter (min) aligns with the tangent line to the characteristic curve E at point K. dcnf This issue is important for the theory of surface generation, because it follows directly from the statement made above, according to which in differential (min) vicinity of the point K, the direction along which the minimal diameter dcnf can be measured aligns with the direction along which the rate of conformity of the surfaces P and T reaches maximal value. Therefore, at point K, the (min) direction along the minimal diameter dcnf and the direction tangent to the characteristic curve E align. For this reason, point K is a point of tangency of the straight line that aligns with the direction in which the minimal diameter

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(min) is measuring and of characteristic curve E. These statements are also true dcnf for the inverse indicatrix of conformity Cnf k ( P/T ) of the surfaces P and T. It is critically important to stress that some conclusions could be made based on known approaches. This indirectly proves that the proposed novel approach of analytical description of the geometry of contact of two smooth, regular surfaces in the first order of tangency is in agreement with the earlier developed approaches in the field. If the contacting surfaces are of simple geometry, then implementation of the discussed approach as well as implementation of known approaches return identical results. If the contacting surfaces are of complex topology, then implementation of only the indicatrix of conformity (of both kinds) or the relative indicatrix (of both kinds), and not of any other characteristic curve, enables correct results of computation of the geometry of contact of two surfaces to be obtained.

4.6.2 Impact of Accuracy of the Computations on the Desired Parameters of the Indicatrices of Conformity CnfR(P/T) All computations performed by the NC system of a numerically controlled machine are performed with errors of computations. No computations by the NC system are performed with zero error. Errors of computations are unavoidable for many reasons. Accuracy of the computations affects the desired parameters of the indicatrix of conformity Cnf R ( P/T ) of the surfaces P and T. For a predetermined error of the computations, the optimal parameters of the characteristic curve Cnf R ( P/T ) can be computed. As known, the characteristic curve Cnf R ( P/T ) is a function of two surfaces P and T, and of their relative orientation. The part surface T to be machined is given, and it cannot be changed, except in the scenario illustrated in Figure 2.3. However, the parameters of the generating surface T and configuration of the surface T with respect to P are under control. It is possible to change both the geometry of the surface T as well as its orientation relative to the part surface T. All these alterations affect the parameters of the characteristic curve Cnf R ( P/T ). It is possible to compute such parameters of the indicatrix of conformity (say, it is possible to compute such design parameters of the generating surface T of the cutting tool and of its configuration) for which the indicatrix of conformity is less vulnerable to the errors of the computations. Figure 4.19 illustrates a portion of an indicatrix of conformity Cnf R ( P/T ) for a certain kind of contact of the surfaces P and T. The actual arc of the indicatrix of conformity within the direction through the points A and B along which the min(min) imal diameter dcnf of the indicatrix of conformity is measured is approximated with a circular arc. Radius of the circular arc ρr.cnf is equal to the radius of curvature of the Cnf R ( P/T ) at the point B. It can be computed from the equation

ρr.cnf (ϕ , µ ) =

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( ) + 2( ) -r  2 rcnf +

2 rcnf

∂ rcnf ∂ϕ

∂ rcnf ∂ϕ 2

2

3

2 

cnf

⋅

(4.104)

2

∂ r cnf ∂2ϕ

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The Geometry of Contact of Two Smooth, Regular Surfaces CnfR(P/T ) yK

(min)

Δd cnf

(min)

d cnf D A

K C

B

xK

ρcnf

CnfR(P/T )

Figure 4.19 A local vicinity of the indicatrix of conformity Cnf R (P / T ) of the surfaces P and T. (min) As follows from Figure 4.19, the error ∆d cnf of the computation of the (min) minimal diameter dcnf causes the deviation Δj of the direction along which (min) the minimal diameter dcnf of the indicatrix of conformity is measured. The deviation Δj can be computed from the equation

(min) (min) 2  0.25 ⋅ (d cnf + ∆d cnf ) + (0.5 ⋅ d c(min) + ρr.cnf )2  nf ∆ϕ = cos -1   (min) (min) (min) + ρr.cnf ) ⋅ (d cnf 2 ⋅ (0.5 ⋅ d cnf + ∆d cnf )  

(4.105)

An example of the function ∆ϕ = ∆ϕ ( ρr.cnf ) is plotted in Figure 4.20. It is important to note that, first, the optimal value of ρr.cnf is not equal to zero, and, second, the optimal value of ρr.cnf depends on the geometry of the generating surface T of the cutting tool, and on the tool configuration. This means that the proper design of the cutting tool and the proper configuration of the cutting tool could be helpful for minimization of the impact of the errors of the computations on the process of the optimal generation of the part surface P. Similar computations can be performed with respect to the computation of optimal parameters of the characteristic curves of other kinds. The consideration above can be employed for enhancement of the classification of the kinds of contact of the surfaces P and T. 4.6.3 Classification of Kinds of Contact of the Surfaces P and T Classification of possible kinds of contact of two smooth, regular surfaces is of importance for implementation of the methods developed in the theory of surface generation. The above discussed analysis yields the development of a scientific classification of kinds of contact of two smooth, regular surfaces P and T.

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0.15

2 0.1 Δj 2 15

3

0.05

1

ρr.cnf

1 0.5

–10

–5

0

0

0.05

1

5

2

0.1

3

0.15

0.2

ρr.cnf

10

–0.5

Figure 4.20 The impact of the errors of the computations on the direction of the maximal conformity of the generating surface T of the cutting tool to the part surface P.

The following statement: If a surface P to be machined and the machining surface of a cutting tool T make contact with one another, then there is at least one point of their contact.

This is implemented as a postulate: Surfaces P and T could make contact (a) at a point K (or at a certain number of points K i ), (b) along a characteristic E (or along several characteristics Ei ), or (c) within a certain surface patch. No other kinds of surface contact are feasible. This is mostly due to the fundamental properties of the three-dimensional space in which we live. The following three kinds of surface contact are evident. They are (a) point contact, (b) line contact, and (c) surface contact of two smooth, regular surfaces. These kinds of surface contact are not only evident, but they are trivial. We now focus on the following: 1. Consider the kinds of contact made on the surfaces P and T. When the surfaces P and T make contact at a point, three kinds of point contact can be recognized.

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1.1. There are no normal plane sections of the surfaces P and T at point K, at which normal curvatures k P and kT are of the same magnitude and of opposite sign. The equality k P = - kT is observed in no plane section of the surfaces P and T through the common unit normal vector n P . This kind of surface contact is referred to as the true-point contact of the surfaces. If two surfaces make true-point contact, then the expression k P (ϕ ) ≠ - kT (ϕ , µ ) is valid for any value of the angle j. 1.2. There is only one normal plane section of the surfaces P and T at point K, at which normal curvatures k P and kT are of the same magnitude and of opposite sign. Thus, the equality k P = - kT is observed in a single plane section of the surfaces P and T through the common unit normal vector n P . In this plane section, the surfaces P and T make contact along an infinitely short arc. Torsion of the surfaces P and T along the infinitely short arc of contact is identical — that us, geodesic (relative) torsion τ g . P ≡ τ g .T . This kind of surfaces contact is referred to as the local-line contact of the surfaces. When two surfaces are in local-line contact, then the expression k P (ϕ ) = - kT (ϕ , µ ) is valid for a certain value of j. As long as the second (and not higher) derivatives are considered, then the local-line contact of the surfaces is identical to the true-line contact of the surfaces. 1.3. Normal curvatures k P and kT are of the same magnitude and of opposite sign in all normal plane section of the surfaces P and T at point K. Thus, the identity k P ≡ - kT is observed in all plane sections of the surfaces P and T through the common unit normal vector n P . In the case under consideration, the surfaces P and T make contact within the infinitely small area. This kind of surfaces contact is referred to as the local-surface contact of the surfaces of the first kind. As long as the second (and not higher) derivatives are considered, then the local-surface contact of surfaces of the first kind is identical to the true-surface contact of the surfaces. 2. Consider line kind of contact of the surfaces P and T. When the surfaces P and T make contact along a characteristic, two kinds of line contact can be recognized. 2.1. There is the only normal plane section of the surfaces P and T at point K, at which normal curvatures k P and kT are of the same magnitude and of opposite sign. This normal plane section is congruent to the osculate plane to the characteristic E at K. Thus, the equality k P = - kT is observed in a single plane section of the surfaces P and T through the common unit normal vector n P. Torsion of the surfaces P and T along the arc of contact is identical — that is, geodesic (relative) torsion τ g . P ≡τ g .T . This kind

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Kinematic Geometry of Surface Machining of surfaces contact is referred to as the true-line contact of the surfaces. When two surfaces are in true-line contact, then the expression k P (ϕ ) = - kT (ϕ , µ ) is valid for a certain value of j. Geodesic (relative) torsion τ g . P of the characteristic curve E determines the rate of rotation of the tangent plane to the surface P about the E. It is assumed that the characteristic curve E and the surface P are regular, and the rate of rotation of the tangent plane is a function of length s of the characteristic curve E. Relative torsion can be defined by a point on the characteristic curve E and by a direction on the surface P. It is equal to the torsion of the geodesic curve in that same direction: d nP  dφ dr τ g.P =  E × n P ⋅ = τE + = ( k 1. P - k 2. P ) sin κ cos κ  ds ds ds  

(4.106)

where rE is the position vector of a point of the characteristic curve E, n P is the unit normal vector to the surface P, τ E is the regular torsion of the characteristic curve E, j is the angle that makes the osculating plane to E and the tangent plane to P, k 1. P and k 2. P are the principal curvatures of P at point K, and κ is the angle that makes the tangent to E at K and the first principal direction t 1.P. 2.2. Normal curvatures k P and kT are of the same magnitude and of opposite sign in all normal plane sections of the surfaces P and T at K. Thus, the identity k P ≡- kT is observed in all plane sections of the surfaces P and T through the common unit normal vector n P . In the case under consideration, the surfaces P and T make contact within an infinitely small area. This kind of surfaces contact is referred to as the local-surface contact of surfaces of the second kind. As long as the second (and not higher) derivatives are considered, then the local-surface contact of the surfaces of the second kind is identical to the true-surface contact of the surfaces. In differential vicinity of point K, the surfaces P and T are locally congruent to each other. 3. Consider the surface kind of contact of the surfaces P and T. When the surfaces P and T make contact within a surface patch, just one kind of contact can be recognized. 3.1. Normal curvatures k P and kT are of the same magnitude and of opposite sign in all normal plane sections of the surfaces P and T at K. Thus, the identity k P ≡- kT is observed in all plane sections of the surfaces P and T through the common unit normal vector n P . In the case under consideration, the surfaces P and T make contact within a surface patch. This kind of surfaces contact is referred to as the true-surface contact of surfaces.

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Without going into detail, mention will be made here that for the purposes of efficient surface generation in a machining operation, it is desired to maintain that kind of contact of the surfaces P and T which features the highest possible rate of conformity of the generating surface T of the cutting tool to the part surface P. Actually, while machining a part surface, deviations in the cutting tool relative location and orientation with respect to the surface P are always observed. The deviations of the cutting tool configuration are unavoidable. Because of the deviations, the desired local-extremal kind of contact is replaced with another kind of contact of the surfaces P and T. Such replacement can be achieved with the introduction of precalculated deviations of the cutting tool principal radii of curvature R 1.T and R 2.T . If the precalculated deviations are small, then instead of the desired local-extremal kinds of contact of the surfaces, the “quasi-” kind of contact of the surfaces P and T may occur. There are several kinds of quasi- contact, including quasi-line contact of the surfaces P and T, quasi-surface of the first kind, and quasisurface of the second kind contact of the surfaces P and T. The required precomputed values of small deviations of the actual normal curvatures from their initially computed values can be determined on the premises of the following consideration. When the maximal deviations in the actual cutting tool configuration (location and orientation of the cutting tool relative to the surface being machined) occur, the rate of conformity of the generating surface T with respect to the surface P must not exceed the rate of their conformity in one of the local-extremal kinds of surface contact. When the actual deviations of the cutting tool configuration do not exceed the corresponding tolerances, then one of the feasible kinds of quasi-contact of the surfaces P and T is observed, Evidently, bigger deviations in the cutting tool configuration result in bigger precomputed corrections in the normal curvature of the generating surface of the cutting tool, and vice versa. In the ideal case, when there are no deviations in the cutting tool configuration, it is recommended to assign normal curvatures of the values that enable one of the local-extremal kinds of contact of the surfaces P and T. Local-surface contact of the second kind is the preferred kind of contact of the surfaces P and T. The local-surface contact of the second kind yields the minimal value (min) of the radius rcnf = 0 of the indicatrix of conformity Cnf R ( P/T ) . When machining an actual part surface, deviations in the cutting tool configuration are unavoidable. The pure surface kind of contact of the surfaces when the (min) equality rcnf = 0 is observed is not feasible. Due to the deviations in the cutting tool configuration, maintenance of the pure surface contact of the surfaces P and T would unavoidably result in interference of the cutting tool beneath the part surface P. Therefore, it is recommended that a pure surface contact not be maintained, but a kind of quasi-surface contact of the second kind be maintained instead. A quasi-surface contact of the surfaces P and T yields avoidance of interference of the surface T within the interior of the surface P. Moreover, the (min) minimal radius rcnf of the characteristic curve CnfR ( P/T ) could be as close to (min) (min) (min) zero as possible ( rcnf > 0, rcnf → 0 , rcnf ≠ 0 ).

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Quasi-contact of the surfaces P and T is observed only when deviations of the cutting tool configuration are incorporated into consideration. Definition 4.1: Quasi-line contact of the surfaces P and T is a kind of point contact of the surfaces under which actual tangency of the surfaces is within the true-point contact and the local-line contact, and it varies as a function of the deviations in the cutting tool configuration. Definition 4.2: Quasi-surface (of the first kind) contact of the surfaces P and T is a kind of point contact of the surfaces under which actual tangency of the surfaces is within the true-point contact and the localsurface (of the first kind) contact, and it varies as a function of deviations in the cutting tool configuration. Definition 4.3: Quasi-surface (of the second kind) contact of the surfaces P and T is a kind of point contact of the surfaces under which actual tangency of the surfaces is within the true-point contact to local-surface (of the second kind) contact, and it varies as a function of deviations in the cutting tool configuration.

The difference between various kinds of the quasi-contact of the surfaces P and T, as well as the difference between the corresponding kinds of localextremal contact of the surfaces can be recognized only under the limit values of the allowed deviations in the cutting tool configuration relative to the part surface P. In the event the actual deviations are below the tolerances, then various possible kinds of quasi-contact of the surfaces cannot be distinguished from other nonquasi-kinds of their contact. The only difference is in actual location of the point K of contact of the surfaces. Due to the deviations, it shifts from the original position to a certain other location. There are only nine principally different kinds of contact of the surfaces P and T. In addition to the true-point, the true-line, and the true-surface contact, the following three local-extremal kinds of contact are possible: (a) the local-line, (b) the local-surface of the first kind, and (c) the local-surface of the second kind. Three kinds of quasi-contact of the surfaces are also possible: the quasi-line, the quasi-surface of the first kind, and the quasi-surface of the second kind of the surfaces P and T. Taking into consideration that there are only 10 different kinds of local patches of smooth, regular surfaces P and T (see Chapter 1, Figure 1.11), each of the 9 kinds of surfaces contact can be represented in detail. For this purpose, a square morphological matrix of size 10 × 10 = 100 is composed. This matrix covers all possible combinations of the surfaces contact. One axis of the morphological matrix is represented with 10 kinds of local patches of the part surface P, and the other axis is represented with 10 kinds of local patches of the generating surface T of the cutting tool. The morphological matrix contains Σ 9m=1C9m = m !( 99-! m )! = 1002-10 + 10 = 55 different combinations of the local patches of the surfaces P and T. Only 55 of them are required to be investigated. The analysis reveals that the following kinds of contact of the surfaces P and T are feasible:

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29 kinds of true-point contact 23 kinds of true-line contact 6 kinds of true-surface contact 20 kinds of local-line contact 7 kinds of local-surface (of the first kind) contact 8 kinds of local-surface (of the second kind) contact 20 kinds of quasi-line contact 7 kinds of quasi-surface (of the first kind) contact 8 kinds of quasi-surface (of the second kind) contact This means that only 29 + 23 + 6 + 20 + 7 + 8 + 20 + 7 + 8 = 128 kinds of contact of two smooth, regular surfaces P and T are possible in nature. For some kinds of surfaces contact, no restrictions are imposed on the actual value of the angle m of the surfaces P and T local relative orientation. For the rest of the types of surfaces contact, a corresponding interval of the allowed value of the angle m — [ µ min ] ≤ µ ≤ [ µ max ] — can be determined. For particular cases of the surfaces contact, the only feasible value µ = [ µ ] is allowed. On the premises of the above analysis, a scientific classification of possible kinds of contact of the surfaces P and T is developed (Figure 4.21). The classification (Figure 4.21) is potentially complete. It can be further developed and enhanced. It can be used for the analysis and qualitative evaluation of the rate of effectiveness of a machining operation. The classification indicates perfect correlation with the earlier developed classification (see table 3.1 on pp. 230–243 in [13]). Replacement of the true-point contact of the surfaces P and T with their local-line contact, and further with the local-surface of the first and of the second kind, and finally with the true-surface contact results in significant alterations of the surface P generation. Only local-extremal kinds of contact of the surfaces P and T are considered here. If deviations in the cutting tool configuration are considered, then the above-mentioned local-extremal kinds of surfaces contact require being replaced with the corresponding quasi-kinds of contact of the surfaces P |/| T. In order to achieve the highest possible productivity of machining of the part surface P, it is recommended that the true-surface contact of the surfaces P and T be maintained. Under such a scenario, all portions of the surface P are machined in one instant. However, in those cases, large-scale surfaces P cannot be machined. The machining of the surface P while maintaining the true-point contact of the surfaces is least efficient. In this case, the generation of every strip on the surface P occurs in time. Depending on the kinds of contact of the surfaces P and T that are maintained when machining a part surface P, all possible kinds of contact of the surface can be ranked in the following order (from the least efficient to the most efficient): True-point contact Local-line or quasi-line contact

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Line Contact

Surface Contact

True-Point Contact

True-Line Contact

True-Surface Contact

Locally-Line Locally-Surface I

Locally-Surface II

Quasi-Line Quasi-Surface II

Quasi-Surface I 29

20

7

20

7

23

8

8

6

Figure 4.21 There are as many as 128 different kinds of contact of two smooth, regular surfaces P and T.

Local-surface of the first kind contact or quasi-surface contact of the first kind True-line contact Local-surface of the second kind contact or quasi-surface contact of the second kind True-surface contact of the surfaces P and T The productivity of machining of the surface P increases from (1) to (6). However, the productivity of surface machining is not the only criteria of effectiveness of a machining operation. The agility of the machining operation is another criterion of interest. The agility of the machining can be evaluated by a versatility of part surfaces P that can be machined with a given cutting tool. From this standpoint, the most effective is a machining operation under which the true-point contact of the surfaces P and T is continuously maintained. The least effective is a machining operation under which the true-surface contact of the surfaces P and T is maintained. Methods of surface machining under which the surfaces P and T contact other kinds, can be ranged in converse order — that is, from (6) to (1).

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Maintenance of the true-point contact of the surface P and T results in the highest possible agility of a machining operation. The true-point contact can also be regarded as the most general kind of surfaces contact. Under the true-point contact of the surfaces, the cutting tool can perform five-parametric motion with respect to the surface P. Under the true-surface contact, the cutting tool is capable of performing no motion relative to the work. A relative motion of the surfaces P and T is feasible only as an exclusion, say when the surfaces P and T yield for sliding over themselves. In those cases, an enveloping surface P to consecutive positions of the generating surface T of the cutting tool that is moving relative to the work, is congruent to P. Generally speaking, under such a scenario, the surfaces P and T are capable of performing a single-parametric motion, and not higher than a three-parametric motion (see Chapter 2, Section 2.4). The developed classification of all possible quasi-kinds of contact of the surfaces P and T can be extended and represented in more detail.

References [1] Boehm, W., Differential Geometry II. In Farin, G. Curves and Surfaces for Computer Aided Geometric Design. A Practical Guide, 2nd ed., Academic Press, Boston, 1990, pp. 367–383. [2] doCarmo, M.P., Differential Geometry of Curves and Surfaces, Prentice Hall, Englewood Cliffs, NJ, 1976. [3] Fisher, G. (Ed.), Mathematical Models, Friedrich Vieweg & Sohn, Braunachweig/ Wiesbaden, 1986. [4] Gray, A., Plücker’s Conoid. In Modern Differential Geometry of Curves and Surfaces with Mathematics, 2nd ed., CRC Press, Boca Raton, FL, 1997, pp. 435–437. [5] Hertz, H., Über die Berührung Fester Elastischer Körper (The Contact of Solid Elastic Bodies), Journal für die Reine und Angewandte Mathematik (Journal for Pure and Applied Mathematics), Berlin, 1981, pp. 156–171; Über die Berührung Fester Elastischer Körper und Über die Härte (The Contact of Solid Elastic Bodies and Their Harnesses), Berlin, 1882; Reprinted in: H. Hertz, Gesammelte Werke (Collected Works), Vol. 1, pp. 155–173 and pp. 174–196, Leipzig, 1895, or the English translation: Miscellaneous Papers, translated by D.E. Jones and G.A. Schott, pp. 146–162, 163–183, Macmillan, London, 1896. [6] Koenderink, J.J., Solid Shape, MIT Press, Cambridge, MA, 1990. [7] Pat. No. 1249787, A Method of Sculptured Surface Machining on Multi-Axis NC Machine./S.P. Radzevich, Int. Cl. B23c 3/16, Filed: December 27, 1984. [8] Pat. No. 1185749, A Method of Sculptured Surface Machining on Multi-Axis NC Machine./S.P. Radzevich, Int. Cl. B23c 3/16, Filed: October 24, 1983. [9] Plücker, J., On a New Geometry of Space, Phil. Trans. R. Soc. London, 155, 725– 791, 1865. [10] Radzevich, S.P., A Possibility of Application of Plücker’s Conoid for Mathematical Modeling of Contact of Two Smooth Regular Surfaces in the First Order of Tangency, Mathematical and Computer Modeling, 42, 999–1022, 2004.

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[11] Radzevich, S.P., Classification of Surfaces, Kiev, UkrNIINTI, No. 1440-Yk88, 1988. [12] Radzevich, S.P., Differential-Geometric Method of Surface Generation, DrSci thesis, Tula Polytechnic Institute, Tula, Russia, 1991. [13] Radzevich, S.P., Fundamentals of Surface Generation, Kiev, Rastan, 2001. Copy of the monograph is available from the Library of Congress. [14] Radzevich, S.P., Mathematical Modeling of Contact of Two Surfaces in the First Order of Tangency, Mathematical and Computer Modeling, 39, 1083–1112, 2004. [15] Radzevich, S.P., Methods for Investigation of the Conditions of Contact of Surfaces, Kiev, UkrNIINTI, No. 759–Uk88, 1987. [16] Radzevich, S.P., On Analytical Description of the Geometry of Contact of Surfaces in Highest Kinematic Pairs, Theory for Mechanisms and Machines (St. Petersburg Polytechnic Institute), 3 (5), 3–14, 2005. [17] Radzevich, S.P., Sculptured Surface Machining on Multi-Axis NC Machine, Vishcha Schola, Kiev, 1991. Copy of the monograph is available from the Library of Congress. [18] Shevel’ova, G.I., Theory of Surface Generation and of Contact of Moving Bodies, MosSTANKIN, Moscow, 1999. [19] Struik, D.J., Lectures on Classical Differential Geometry, 2nd ed., Addison-Wesley, Reading, MA, 1961. [20] von Seggern, D., CRC Standard Curves and Surfaces, CRC Press, Boca Raton, FL, 1993. [21] www.mathcurve.com/surfaces/plucker/plucker.shtml. [22] www.math.hmc.edu/faculty/gu/curves_and_surfaces/surfaces/plucker.html.

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5 Profiling of the Form-Cutting Tools of Optimal Design Generation of the part surface P practically is performed with the help of the cutting tools of appropriate design. The stock removal and generation of the surface P are the two major functions of the cutting. Profiling of the generating surface T is required for designing a high-performance cutting tool. As shown below, the shape and parameters of the generating surface T significantly affect the performance of the cutting. Cutting edges of a precision cutting tool are within the generating surface of the tool. This makes it clear that prior to developing a practical design of the cutting tool, profiling of the optimal generating surface is required. In this chapter, profiling of the form-cutting tools for all possible methods of part surface machining is considered. The consideration begins from the theory of profiling of the tools for machining sculptured surfaces on a multi-axis numerical control (NC) machine. This subject represents the most complex case in the theory of profiling of cutting tools.

5.1 Profiling of the Form-Cutting Tools for Sculptured Surface Machining The problem of profiling the form-cutting tool for machining of a sculptured surface on a multi-axis NC machine has not yet been investigated in detail. Not profiling of the form-cutting tool of optimal design but selecting a certain cutting tool among several available designs is often recommended instead. The selection of the cutting tool is usually based on minimizing machining time, reducing scallop height, and so forth. This yields a conclusion that a robust mathematical method for design of the optimal form-cutting tool for the maximally productive machining of a given sculptured surface on a multi-axis NC machine is needed. 5.1.1 Preliminary Remarks Many advanced sources are devoted to the investigation of sculptured surfaces generation on multi-axis NC machines. Without going into a detailed review of previous publications in the field, mention is made of a few monographs by Amirouch [1], Chang and Melkanoff [4], Choi and Jerard [5], 153 © 2008 by Taylor & Francis Group, LLC

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(a)

(b)

(c)

(d)

(e)

(f)

Figure 5.1 Examples of milling cutters of conventional design for the machining of sculptured surfaces on a multi-axis numerical control machine: cylindrical (a), conical (b), ball-end (c), filleted-end (d), and form-shaped (e), (f).

and Marciniak [11]. Unfortunately, the problem of profiling the form-cutting tools for sculptured surface machining has not yet been investigated. Most often, the generation of sculptured surfaces with the milling cutters of conventional designs (Figure 5.1) is considered. The following terms (some of which are not new) are introduced below to avoid ambiguities in later discussions: Definition 5.1: Sculptured surface P is a smooth, regular surface, the major parameters of local topology at a point of which are not identical to the corresponding parameters of local topology of any other infinitesimally close point of the surface.

It is instructive to point out here that sculptured surface P does not allow for sliding “over itself.” While machining a sculptured surface, the cutting tool rotates about its axis of rotation and moves relative to the sculptured surface P. When rotating or when performing relative motion of another kind, cutting edges of the cutting tool generate a certain surface. The surface represented by consecutive positions of cutting edges is referred to as the generating surface of the cutting tool [18, 19, 25]: Definition 5.2: The generating surface T of the cutting tool is a surface that is conjugate to the surface P being machined.

In fact, our terminology draws inspiration mostly from the Theory for Mechanisms and Machines, and from the Theory of Conjugate Surfaces.

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An infinite number of surfaces satisfy Definition 5.2. Use of all of the conjugate surfaces satisfies the equation of contact n P ⋅ VΣ = 0 (see Chapter 2). Although the unit normal vector n P is uniquely determined at a given surface point, the number of feasible vectors VΣ is equal to infinity: All vectors VΣ within the common tangent plane satisfy the equation of contact n P ⋅ VΣ = 0 . It is natural to assume that not all are equivalent to each other from the standpoint of efficiency of surface generation, and some could be preferred; moreover, an optimal direction of VΣ exists within the common tangent plane for which the efficiency of surface machining reaches it optimal rate. This makes the problem of profiling the form-cutting tool indefinite. However, this indefiniteness is successfully overcome below. The uniquely determined generating surface T is used in further steps of designing an optimal form-cutting tool for machining of a given part surface. In most cases of surface generation, the generating surface T of the cutting tool does not exist physically. Usually, it is represented as the set of consecutive positions of the cutting edges in their motion relative to the stationary coordinate system, embedded in the cutting tool. In most practical cases, the generating surface T allows for sliding over itself. The enveloping surface to consecutive positions of the surface T that performs such a motion is congruent to the surface T. When machining a surface P, the surface T is conjugate to the sculptured surface P. For simplification in programming machining operation, the APT cutting tool is proposed (Figure 5.2). Physically, the APT cutting tool does not exist. The generating surface T of the APT cutting tool is made up of a conical portion that has the cone angle a, a conical portion that has the cone angle b, and a portion of the surface of a torus. The last is specified by the radius r of the generating circle, and by the diameter d of the directing circle. The axial location of the torus surface with respect to the conical surfaces is specified by the parameter designated as f. For a certain combination of the parameters a, b, r, d, and f, the generating surface of the virtual APT cutting tool transforms to the generating surface T of the actual cutting tool. For example, assuming a = 0°, b = 0°, and r = 0, one can come up with the generating surface T of the cylindrical milling cutter (Figure 5.1a). If r = d2 and b = 0°, then the

The procedure of designing a form-cutting tool usually begins from determination of the generating surface of the cutting tool. This is a common practice. However, sometimes when the geometric structure of the surface to be machined is inconsistent, another procedure is used. Relieving hob clearance surfaces, cutting bevel gears with spiral teeth, machining noninvolute gears of the first and of the second kind are perfect examples of surface machining when the geometric structure of the surface to be machined is inconsistent. Under such circumstances, the generating surface of the cutting tool of an appropriate form is selected. Further, the actual shape and parameters of the machined part surface can be determined. Keep in mind that the part surface to be machined is the primary element of the machining operation, on the premises of which the determination of the optimal machining operation is possible. This includes profiling of both the optimal cutting tool and the optimal kinematics of the machining operation. Otherwise, only an approximate solution to the problem of optimal surface generation is possible.

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β d h

r f

α

Figure 5.2 Major design parameters of the APT cutting tool. (From Radzevich, S.P., Computer-Aided Design, 34 (10), 727–740, 2002. With permission.)

generating surface of the virtual APT cutting tool reduces to the generating surface T of the ball-end milling cutter (Figure 5.1c). This series of elementary transformations of the APT cutting tool to the generating surface T of the actual cutting tool can be extended. The major advantage of the APT cutting tool is that if a computer program for controlling a multi-axis NC machine is developed for implementation of the APT cutting tool, then this computer program can be easily adapted for implementation of a cutting tool of any other design. Actually, the major developments in the profiling of form-cutting tools for machining a given sculptured surface on a multi-axis NC machine are limited to selection of an appropriate cutting tool within available designs, and implementation of the APT cutting tool for the development of computer programs for controlling multi-axis NC machines. 5.1.2 On the Concept of Profiling the Optimal Form-Cutting Tool It is critical to clarify from the very beginning what the term optimal cutting tool means. In the consideration below, the term optimal cutting tool means that the design parameters of a certain cutting tool are those under which implementation of the form-cutting tool enables the required extremum (minimum/maximum) of the given criterion of optimization. Maximal productivity of the part surface machining, and minimal deviations of the actual part surface from the desired part surface are the perfect examples of the criteria of optimization. In the theory of surface generation, only those criteria of optimization that could be expressed in terms of geometry of the part surface P, geometry of

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I ωP

t

P I

[h]

T

S

t

ha ≤ [h]

a 1

a

ωP

P

b 1<

a 1

a 1 a

S

<

a

S

P

I

(b)

t

~ 2S =1,

hc < hb < ha

S

RT

S

T

(c)

d

P

ωP

I = 0˚

d

h =0

I d 1

t d

RT

b 1

S

T

I ωP

hb < ha

b b

S

(a)

t

[h]

= 0˚

T

= 0˚ S

[h]

S

d 1

(d)

Figure 5.3 Turning of an arbor on a lathe: the concept of profiling the optimal form-cutting tool. (From Radzevich, S.P., Mathematical and Computer Modeling, 36 (7–8), 921–938, 2002. With permission.)

the generating surface T of the cutting tool, and kinematics of the machining operation are applicable. The following example illustrates the actual meaning of the criterion of optimization in the sense of profiling of the form-cutting tool. Consider a trivial machining operation — a turning operation of an arbor on a lathe (Figure 5.3). When machining, the work rotates about its axis of rotation with an angular velocity ω P . The cutting tool travels along the work axis of rotation with a feed rate S. The feed rate S is of constant magnitude in the examples considered below. A stock t is removed in the turning operation. Use of the cutter with the tool cutting edge angle ϕ a , and the tool minor (end) cutting edge angle ϕ a1 causes cusps on the machined part surface P (Figure 5.3a). The actual height h a of the cusps must be less than the tolerance [h] on accuracy of the surface P. In order to satisfy the inequality h a ≤ [ h], a corresponding relationship between the parameters ϕ a, ϕ a1 , and S must be observed. Otherwise, the part cannot be machined in compliance with the part blueprint. That same arbor can be machined with the cutter having the tool cutting edge angle ϕ b < ϕ a , and the tool minor cutting edge angle ϕ b1 < ϕ a1 (Figure 5.3b). Cusps on the machined surface P are observed. Elementary computations of the actual cusp height h b in this case reveal that the inequality h b < h a is valid. Further, that same surface P can be machined with the cutter having the cutting edge that is shaped in the form of a circular arc of radius R (Figure 5.3c). Use of the cutter with the curvilinear cutting edge results in cusps on the machined surface P. However, if the radius R is chosen properly, then the

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actual cusp height h c can be smaller than h b . In other words, the inequality h c < h b could be observed. Ultimately, consider the turning operation of the surface P with the cutter that has an auxiliary cutting edge (Figure 5.3d). The auxiliary cutting edge is parallel to the direction of the feed rate S. The length of the auxiliary cutting edge exceeds the distance that the cutter travels per one revolution of the work. Geometrical parameters of the auxiliary cutting edge can be specified by the tool cutting edge angle ϕ d = 0° , and the tool minor (end) cutting edge angle ϕ d1 = 0°. Under such a scenario, no cusps are observed on the machined part surface P. The above consideration makes possible a conclusion: An appropriate alteration of shape of the cutting edge of the cutter can make possible a reduction of deviations of the machined part surface with respect to the desired part surface. This conclusion is critically important for further consideration. At this point, a more general example of surface generation that supports the above conclusion will be considered. Consider generation of a sculptured part surface P with the form-cutting tool having arbitrarily shaped the generation surface T. The intersection of the surfaces P and T by the plane through the unit normal vector n P is shown in Figure 5.4. This plane section is perpendicular to the tool-path on the surface

R aT > 0

RbT > RaT > 0

ST

a

K

Ta

ST

P hP

RP

Tb

(b) R cT

K Tc

∞ ST

P

(c)

P hdp < hcP

K

hcp < hbP RP

RP

hbp < haP

RP

(a)

ST

P

K

RdT < 0 (d)

Td

Figure 5.4 Examples of various rates of conformity of the generating surface T of the tool to the sculptured surface P in the plane section through the unit normal vector n P . (From Radzevich, S.P., Mathematical and Computer Modeling, 36 (7–8), 921–938, 2002. With permission.)

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P at point K. In Figure 5.4, ST designates the width of the tool-path. In all the examples considered, the width ST of the tool-path remains identical. The radius of normal curvature RP of the surface P at point K remains the same. The scallop’s height on the machined surface P is designated hP . The surface P can be generated with the generating surface T a of the cutting tool (Figure 5.4a). The radius of curvature of the generating surface T a is of a certain positive value R aT > 0 . Due to point contact of the surfaces P and T a , the desired surface P is not generated, but an approximation to it is generated instead. The generated surface has scallops. For the prespecified width ST of the tool-path, the scallop’s height is equal to a certain value hPa . The scallop’s height must be smaller than the tolerance [h] on the accuracy of the surface P. That same surface P can be generated with the generating surface T b of the cutting tool (Figure 5.4b). The radius of curvature of the generating surface T b in this case exceeds the value of the radius of curvature of the surface T a in the previous case ( RbT > R aT ). Because surfaces P and T b are in point contact, scallops on the generated surface are observed. When width ST of the tool-path is predetermined, then the scallop’s height is of a certain value hPb . Under such a scenario, a certain reduction of the scallop’s height occurs ( hPb < hPa ). The scallop height reduction occurs because in differential vicinity of point K, the surface T b is getting closer to surface P rather than to surface T a . Locally, surface T b is more congruent to surface P than to surface T a . The rate of conformity of surface T b to surface P is greater than the rate of conformity of the surface T a to that same surface P. Further, that same surface P can be generated with the generating surface T c of the cutting tool (Figure 5.4c). At point K, surface T c is flattened; therefore, the radius of curvature R cT is equal to infinity ( R cT → ∞). That value of the radius of curvature R cT exceeds the value of the radius of curvature RbT. Again, surfaces P and T c make contact at a point; therefore, scallops on the generated surface are observed. Because R cT > RbT , the scallop’s height hPc is smaller than the scallop’s height hPb . The scallop height reduction in this case is due to the increase of the rate of conformity of the generating surface T c of the tool to the part surface P compared to what is observed with respect to surfaces P and T b . Ultimately, consider generation of the surface P with the concave generating surface T d of the cutting tool (Figure 5.4d). The radius of curvature R dT of the cutting-tool surface T d is negative (R dT < 0). In this case, the rate of conformity of the generating surface T d of the cutting tool to the part surface P is the biggest of all considered cases (Figure 5.4). Thus, scallops of the smallest height hPd are observed on the generated surface P. Summarizing the analysis of Figure 5.4, the following conclusion can be formulated: An increase of the rate of conformity of the generating surface T of the cutting tool to the sculptured surface P causes a corresponding reduction of height of the residual scallop on the machined surface P. This conclusion is of critical importance for the development of methods of profiling of form-cutting tools, as well as for the theory of surface generation.

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The rate of conformity of the surface T to the surface P can be used as a mathematical criterion of efficiency of a machining operation. This issue is of prime importance in order to bypass all the major bottlenecks (imposed by the initial indefiniteness of the problem) in designing the optimal formcutting tool. 5.1.3 R-Mapping of the Part Surface P on the Generating Surface T of the Form-Cutting Tool The theory of surface generation offers a method for profiling form-cutting tools of optimal design for the machining of a given sculptured part surface on a multi-axis NC machine. The rate of conformity of the generating surface T of the cutting tool to the sculptured part surface P significantly affects the efficiency of the machining operation. A higher rate of conformity results in higher productivity of the machining operation, smaller residual scallops on the machined part surface, shorter machining time, and so forth [18,19,25,26]. In order to come up with the optimal design of a cutting tool, the generating surface of the tool must conform to the part surface to be machined as much as possible. For this purpose, the cutting tool surface T can be generated as a kind of mapping of the part surface P to be machined. The required kind of mapping of the surface P onto the generating surface T must ensure the required rate of conformity of the surfaces P and T at every point of their contact. This kind of mapping of the surface P onto the surface T was initially proposed by Radzevich [16,22,23,25]. It is referred to as the R-mapping of the sculptured part surface P onto the generating surface T of the cutting tool. Consider the generating surface T of a form-cutting tool that makes contact with a sculptured part surface P at point K. The unit normal vector to the surface P at K is designated n P . A pencil of planes can be constructed using vector n P as the directing vector of the axis of the pencil of planes. The maximal rate of conformity of the generating surface T of the tool to the sculptured surface P is observed when for every plane of the pencil of planes the equality R T = − RP

(5.1)

is valid (in some cases, not the equality R T = − RP , but the equivalent equality k T = − k P can be used instead). When the equality (see Equation 5.1) is satisfied, then the surfaces P and T make either a surface kind of contact or a kind of local-surface contact (either local-surface contact of the first kind, or local-surface contact of the second kind — see Chapter 4).

Pat. No. 4242296/08, USSR, A Method for Designing of the Optimal Form-Cutting-Tool for Machining of a Given Sculptured Surface on Multi-Axis NC Machine./S.P. Radzevich, Filed March 31, 1987.

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Actually, deviations in the configuration of the cutting tool with respect to the sculptured part surface P are unavoidable. Because of this, Equation (5.1) cannot be satisfied. This forces the replacement of Equation (5.1) with an equality of the sort

R T = R T ( RP )

(5.2)

The function R T ( RP ) can be expressed in terms of deviations or tolerances of the actual configuration of the cutting tool with respect to the work. It can be determined using either analytical or experimental methods. Ultimately, the problem of profiling a form-cutting tool of the optimal design for machining a given sculptured part surface on a multi-axis NC machine reduces to determination of the generating surface T that is maximally conformal to the given surface P and that satisfies Equation (5.2). Switching from the use of the function R T = − RP to the use of a function R T = R T ( RP ) results in the ideal local-extremal contact of the surfaces P and T being replaced with a kind of quasi-kind of contact (Chapter 4). Recall that quasi-kinds of contact of the surfaces P and T yield that same range of agility of the machining operation as that possessed by point contact of the surfaces. Moreover, the productivity of surface generation when maintaining the quasi-kind of contact of the surfaces P and T is practically identical to the productivity of surface generation when the surface-kind of contact of the surfaces is maintained. Further, consider a sculptured surface P that is analytically represented by an equation of the form of Equation (1.1). An equation of the sculptured surface yields computation of the fundamental magnitudes EP, FP, and GP of the first order (see Equation 1.7), and of the fundamental magnitudes LP , M P, and N P of the second order (see Equation 1.11) of the part surface P. Using R-mapping of the surfaces, it is convenient to derive an equation of the generating surface T of the form-cutting tool initially in its natural parameterization (see Equation 1.19). For this purpose, it is necessary to express the first φ1.T and the second φ2.T fundamental forms of the generating surface T in terms of the fundamental magnitudes EP , FP , GP and LP , M P, N P of the part surface P. The R-mapping of surfaces is capable of establishing the required correspondence between points of the surfaces P and T. This means that for every point on the surface P at which the precomputed principal radii of curvature are equal to R 1. P and R 2. P , a corresponding point on the generating surface T of the cutting tool, with the desired principal radii of curvature to R 1.T and R 2.T , can be computed (but is not mandatory vice versa). There could be one or more points on the surface P that correspond to that same point of the surface T. When two surfaces P and T are given, then one can easily compute the rate of conformity of the surfaces at the given cutter-contact-point (CC-point; see Chapter 4). In the case under consideration, a problem of another sort arises. This problem could be interpreted as an inverse problem to the problem of

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computation of the actual rate of conformity of surface T to surface P in the prescribed direction on P. The R-mapping establishes a functional relationship between principal radii of curvature of the surface P and of the generating surface T of the cutting tool in differential vicinity of the CC-point K. The R-mapping yields composition of two equations for the determining of six unknown fundamental magnitudes ET , FT , GT of the first φ1.T and L T, MT , N T of the second φ2.T fundamental forms of the surface T. The equation RT = RT ( RP ) can be split into a set of two equations: M T = M T (M P ; GP )  GT = GT(M P ; GP )

(5.3)

Here M P and M T designate the mean curvatures, and GP and GT are the Gaussian curvatures of the surfaces P and T at a CC-point K, respectively. Expression R T = R T ( RP ) gives insight into the significance of a correlation between the radii of normal curvature RP and RT . In order to make the function R T = R T ( RP ), the rate of conformity functions F 1, F 2 , and F 3 are implemented. The rate of conformity functions F 1, F 2, and F 3 are of principal importance for the determination of the function R T = R T ( RP ). The functions F 1, F 2, and F 3 specify the required rate of conformity of the generating surface T of the cutting tool to the sculptured surface P at every CCpoint. Because of that, they are referred to as the rate of conformity functions. In order to satisfy the set of two equation in Equation (5.3), it is necessary to satisfy the following equalities:

LT N T − MT2 = F 1 ( LP N P − M P2 )

ET N T − 2 FT MT + GT LT = F 2 (EP N P − 2 FP MP + GP LP )

ET GT − FT2 = F 3 ( EPGP − FP2 )

(5.4)

(5.5) (5.6)

In a particular case, Equation (5.4) through Equation (5.6) can be reduced to the following forms:

LT N T − MT2 = LP N P − M P2

(5.7)

ET N T − 2 FT MT + GT LT = EP N P − 2 FP M P + GP LP

(5.8)

ET GT − FT2 = EP GP − FP2

(5.9)

However, this does not require that the equality R T = − RP be valid.

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These expressions analytically describe the vital link between the optimal design parameters of the form-cutting tool and the actual process of sculptured surface machining. Equation (5.4) through Equation (5.6) allow incorporation into the design of the actual cutting tool all of the important features of the machining operation: cutting tool performance, tool wear, rigidity of a cutting tool, and so forth. The rate of conformity functions F 1, F 2, and F 3 are of prime importance for designing the form-cutting tool of optimal design for the machining of a given sculptured part surface. They can be determined, for instance, using the proposed [14] experimental method of simulating the machining of a sculptured surface. Other approaches for determining the rate of conformity functions F 1, F 2, and F 3 can be used as well. There is much room for research in this concern. Equation (5.4) through Equation (5.6) are necessary but not sufficient for the determination of six unknown fundamental magnitudes ET , FT , GT of the first φ1.T and L T , MT, N T of the second φ2.T fundamental forms of the generating surface T of the form-cutting tool. The equations of compatibility could be incorporated into the analyses in order to transform Equation (5.4) through Equation (5.6) to a set of six equations of six unknowns. Every smooth, regular generating surface T of the form-cutting tool mandatory satisfies Gauss’ equation of compatibility that follows from his famous theorema egregium [7,18,19,29,32]:  ∂2 FT 1  ∂2ET ∂2GT   GT ( ET GT − FT2 ) =  −  +  ⋅ ( ET GT − FT2 ) 2 ∂UT2    ∂UT ∂VT 2  ∂VT 0 +

∂FT 1 ∂GT − ∂VT 2 ∂UT

1 ∂GT 2 ∂VT

ET

FT

FT

GT

1 ∂ET 2 ∂UT ∂FT 1 ∂ET − ∂UT 2 ∂VT

0 −

1 ∂ET 2 ∂VT 1 ∂GT 2 ∂UT

1 ∂ET 2 ∂VT

1 ∂GT 2 ∂UT

ET

FT

FT

GT

(5.10)

Two other equations of compatibility were derived by Mainardi-Codazzi [7,18,19,29,32]:

∂LT ∂MT 2 2 − = LT Γ 112 + MT ⋅ ( Γ 12 − Γ 111 ) − N T Γ 11 ∂VT ∂UT

(5.11)

∂MT ∂N T 2 − = LT Γ 122 + MT ⋅ ( Γ 222 − Γ 112 ) − N T Γ 12 ∂VT ∂UT

(5.12)

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Kinematic Geometry of Surface Machining

In Equation (5.11) and Equation (5.12), Christoffel’s symbols of the second kind are used. Christoffel’s symbols can be computed from the following formulae [7,32]: ∂Eu ∂F ∂E − 2 Fu u + Fu u ∂Uu ∂Uu ∂Vu 2(EuGu − Fu2 )

G Γ 111 =

Γ 112 =

Γ 122 =

2 = Γ 11

2Eu

∂Fu ∂E ∂E − Eu u + Fu u ∂Uu ∂Vu ∂Uu 2(EuGu − Fu2 )

∂E ∂Gu ∂Eu ∂G Eu − Fu u − Fu u ∂Vu ∂U u ∂Vu ∂Uu 2 = = Γ 221 Γ 12 = Γ 121 2 2 E G F − 2 ( 2(EGu − Fu ) u u u)

Gu

2Gu

∂Fu ∂G ∂G − Gu u + Fu u ∂Vu ∂Uu ∂Vu 2(EuGu − Fu2 )

Γ 222 =

Eu

∂Gu ∂F ∂G − 2 Fu u + Fu u ∂Vu ∂Vu ∂Uu 2(EuGu − Fu2 )

(5.13)

(5.14)

(5.15)

Equations of compatibility are necessary in order to transform the set of three equations [Equation (5.4) through Equation (5.6)] to a set of six equations of six unknowns. The set of three equations (Equation 5.4 through Equation 5.6) together with three equations of compatibility (Equation 5.10 and Equation 5.11) completely describe the R-mapping of two smooth, regular surfaces (that is, they describe the R-mapping of the sculptured surface P onto the generating surface T of the cutting tool of optimal design). Thus, the fundamental magnitudes of the first f 1.T and of the second f 2.T fundamental forms of the generating surface T of the form-cutting tool can be determined using the R-mapping of the sculptured part surface P onto the generating surface T of the cutting tool. A routing procedure can be used to solve the set of six equations of six unknowns, say of Equation (5.4) through Equation (5.6) and Equation (5.10) and Equation (5.11) with six unknowns ET , FT , GT and L T , MT , N T . Ultimately, the determined generating surface T of the cutting tool is represented in natural parameterization similar to Equation (1.19). Below, the fundamental magnitudes ET , FT , GT and L T , MT , N T are considered as the known functions. 5.1.4 Reconstruction of the Generating Surface T of the Form-Cutting Tool from the Precomputed Natural Parameterization Analytical representation of the generating surface T in the form of Equation (1.19) is inconvenient for application in engineering practice when designing form-cutting tools. However, natural parameterization of the generating surface T can be converted to its parameterization in a convenient form, say to its representation in a Cartesian coordinate system.

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For the conversion of the natural parameterization of the surface T to its representation in a Cartesian coordinate, the set of two Gauss–Weingarten’s equations in tensor notation rij = Γ ijk rP + bij n P ⇐ kj of the Form Cutting Tool ni = −bik g rj

The Generating Surface T

(5.16)

must be solved. The solution to the set of equations in Equation (5.16) returns a matrix equation of the generating surface T of the form-cutting tool for the machining of a given sculptured part surface P on a multi-axis NC machine. The initial conditions of integration of Equation (5.16) must be selected properly. In Equation (5.16), ri = ∂r T/∂UT; rij = ∂2r T/∂UT∂VT; ni = ∂nT/∂UT; bij = rij ⋅ nT = − ni rj − n j ⋅ r;i gij is the metric tensor of the generating surface T of the form-cutting tool of optimal design; and g ij is a contra-variant tensor of the generating surface T of the cutting tool. To solve the set of Equation (5.16), known methods [9] are used. Initial conditions for integrating Equation (5.16) must be established. These conditions, for example, might include coordinates of two points on the surface T and direct cosines of the unit normal vector nT at one of these points. The set of two differential equations in tensor notation (see Equation 5.16) can be converted either to a set of five differential equations in vector notation, or to a set of fifteen differential equations in parametric notation. Conventional mathematical methods can be implemented to solve a set of five differential equations or a set of fifteen differential equations with a corresponding number of unknowns. This is a trivial mathematical problem that follows from the proposed theory of surface generation. 5.1.5 A Method for the Determination of the Rate of Conformity Functions F 1, F 2, and F 3 For determining the rate of conformity functions, various approaches can be employed. It is possible to implement a method of experimental simulation of a machining operation for determining the rate of conformity functions F 1, F 2, and F 3. The method of simulation is shown in Figure 5.5. It is proposed by Radzevich [14]. As an example of implementation of the method of simulation, consider machining a sculptured surface P on a multi-axis NC machine (Figure 5.5a). The part surface P is machined with a milling cutter that has the generating surface T. The method of simulation of machining sculptured surfaces is carried out with the equivalent models of the part surface P and of the generating surface T of the cutting tool (Figure 5.5b). Working surfaces of the

Pat. No. 1449246, USSR, A Method of Experimental Simulation of Machining of a Sculptured Surface on Multi-Axis NC Machine./S.P. Radzevich, Int. Cl. B 23 C, 3/16, Filed February 17, 1987.

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Kinematic Geometry of Surface Machining

I n(s) P

(s) С2.T

P

p (m)

OT

T

ωT

T (m) K

µ(s)

nP OT

(s) С1.T

(s) С1.P

(s) С2.P

n(s) T

K

nT

View t(s) 2P

I

t(s) 2.T

t(s) 1.T

t(s) 1P

K

Vf

µ ∆(s)

θT(s)

ω (s) Σ Vpr

V(s) Σ (a)

ω (s) Σ (b)

Figure 5.5 The experimental method for determining the desired rate of conformity functions F 1, F 2, and F 3. (See also Radzevich, S.P., USSR Patent 1449246, 1987.)

equivalent models that are used for the simulation are designated P ( s) and T ( s) , accordingly. The local topology of the surfaces P ( s) and T ( s) can be uniquely specified by two parameters — that is, by the mean curvature M P (T ) and the Gaussian curvature GP(T ) . Due to there being only two parameters of local topology, there are only ten kinds of surfaces P ( s) and T ( s) (Figure 1.11). The surface P ( s) , as well as the surface T ( s) , is a kind of quadric surface. Both surfaces P ( s) and T ( s) become tangent at point K. The local geometry of tangency of the surfaces P ( s) and T ( s) is identical to the local geometry of tangency of the sculptured surface P and the generating surface T of the cutting tool. Due to that, the unit tangent vectors t (1.sP) and t (2.s)P of the principal directions on the quadric surface P ( s) align with the corresponding unit tangent vectors t 1.P and t 2.P of the sculptured surface P. Moreover, the principal radii of curvature R1.(sP) and R2.(sP) of the quadric surface P ( s) at every CC-point K are equal to the corresponding principal radii of curvature R 1. P and R 2. P of the sculptured surface P — that is, the identities R(1s.)P ≡ R 1. P and R(2s.)P ≡ R 2. P are observed. Because of this, Euler’s formula yields the conclusion that in

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the differential vicinity of the CC-point K, the surfaces P ( s) and P are locally congruent to each other up to members of the second order. A similar conclusion is valid for the quadric surface T ( s) that is used for local simulation of the generating surface T of the form-cutting tool. At the CCpoint K, the unit tangent vectors t (1.sT) and t (2.sT) of the principal directions on the quadric surface T ( s) align with the corresponding unit vectors t 1.T and t 2.T of the generating surface T of the cutting tool. Principal radii of curvature R(1.s)T and R(2.s)T of the quadric surface T ( s) are also equal to the corresponding principal radii of curvature R 1.T and R 2.T of the surface T — that is, the identities R(1s.)T ≡ R 1.T and R(2s.)T ≡ R 2.T are observed. Therefore, in the differential vicinity of the CC-point K, the surfaces T ( s) and T are locally congruent to each other up to the members of the second order. Local orientation of the cutting edges on the quadric surface T ( s) remains the same with respect to the principal directions t (1.sT) and t (2.sT) as that for the actual form-cutting tool T [ t (1s.T) ≡ t 1.T and t (2s.T) ≡ t 2.T ]. For this purpose, it is required, at first, to determine the orientation of the principal plane sections C1.T and C2.T of the generating surface T of the cutting tool with respect to the coordinate UT and VT lines. Orientation of the plane section of a surface by a normal plane surface can be determined by the ratio ∂UT/∂VT (see Chapter 1). For the orthogonally (UT; VT)-parameterized generating surface T of the cutting tool, the ratio ∂UT/∂VT determines the value of tan ξT . Here, angle ξT designates the angle of inclination of the principal plane sections C1.T and C2.T relative to the coordinate UT and VT lines on the generating surface T. Usually, parameterization of the surface T is not orthogonal. In such a case, the angle ξT (not shown in Figure 5.5b) can be computed from the following formula [19]:

2  ∂V ∂V   ∂V  sin ξT = T   T  − 2 T cos ωT + 1 ∂UT ∂UT   ∂UT  

−

1 2

(5.17)

At the CC-point K, the cutting edge makes a certain angle ζ T with the UTcoordinate line. The angle ζ T can be computed from the equation [18,19]. The cutting edge makes a certain angle θT with the first principal plane section C1.T of the generating surface T of the cutting tool. The angle θT is equal to the algebraic sum of the angles ξT and ζ T — that is, θT = ξT + ς T . Therefore, orientation of the cutting edge (angle θT( m ) ≡ θT = ξT + ς T ) relative to the first principal plane section C1.(sT) of the quadric surface T ( s) makes local orientation of the cutting edge on the quadric surface T ( s) identical to the orientation of the corresponding cutting edge on the generating surface T of the cutting tool. The accuracy of the simulation is up to members of the second order or even exceeds that. The quadric surfaces P ( s) and T ( s) are turned about the unit normal vector ( s) n P relative to each other through an angle µ ( s) . The angle µ ( s) is the angle of the local relative orientation of surfaces P ( s) and T ( s) . The angle µ ( s) is

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identical to the angle μ of the local relative orientation of actual surfaces P and T [ µ ( s) ≡ µ ]. Angle μ makes the first t 1.P and t 1.T (or, the same, the second t 2.P and t 2.T ) principal directions of the surfaces at the CC-point [18,19]:

µ ( s) ≡ µ = tan −1

t 1. P × t 1.T t 1. P ⋅ t 1.T

≡tan −1

t 2. P × t 2.T t 2. P ⋅ t 2.T

(5.18)

The local relative orientation of the quadric surfaces P ( s) and T ( s) in differential vicinity of the CC-point K up to the members of the second order is identical to the local relative orientation of the actual sculptured surface P and the generating surface T of the cutting tool when machining the part surface P on a multi-axis NC machine. The trajectory of the cutting-edge point relative to the sculptured surface P can be represented as a vector sum of the motions that the surfaces P and T perform on a multi-axis NC machine. When simulating the machining operation of the surface P, the quadric surfaces P ( s) and T ( s) perform the relative motion with respect to one another. Resultant speed VΣ( s) of the relative motion can be represented as a vector sum of the speed of cutting Vc( s) and of all other partial motion Vi( s) , namely VΣ( s) = Vc( s) + Σ in=−11Vi( s). Here, n designates the total number of all of the partial motions Vi( s). The feed-rate motion V (f s) is an example of the partial motions Vi( s) . When machining a sculptured surface, instant relative motion of the surfaces P and T can be represented as an instant screw motion. Therefore, when simulating the machining operation, the quadric surfaces P ( s) and T ( s) perform rotation with the resultant angular velocity ω Σ( s) in addition to the resultant linear motion VΣ( s) . While simulating, the resultant relative motion VΣ( s) of the surfaces P ( s) and T ( s) is identical to the instant resultant relative screw motion VΣ of the actual sculptured surface P and the generating surface T of the cutting tool ( VΣ( s) ≡ VΣ ). For this purpose, the angle ∆ ( s) that the vector VΣ( m ) makes with ( s) the first principal plane section C1. P of the quadric surface P ( s) is identical to the similar angle Δ that the vector VΣ makes with the first principal plane section C1. P of the sculptured surface P (that is, ∆ ( s) ≡ ∆ ). Instant relative screw motion of the quadric surfaces P ( s) and T ( s) is identical to the instant relative screw motion of the sculptured surface P and of the generating surface T of the cutting tool. At every CC-point K, implementation of the method of experimental simulation of machining of a sculptured surface (Figure 5.5) provides the local identity of all geometrical and kinematical parameters of the machining operation, namely [14], • Quadric surface P ( s) and actual sculptured surface P • Quadric surface T ( s) and actual generating surface T of the cutting tool

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• Location of the cutting edge on the quadric surface T ( s) and cutting edge on the generating surface T of the actual cutting tool • Relative local orientation of the quadric surfaces P ( s) and T ( s) and the relative local orientation of the actual sculptured surface P and the generating surface T of the actual cutting tool • Instant relative motion while simulating and the instant relative motion while machining (that is, kinematics of machining remains identical) Ultimately, this results in high efficiency of the method of simulation, and in high accuracy of the determined rate of conformity functions F 1, F 2, and F 3. When simulating a machining operation, it is preferred to perform not the instant relative motions of the modeling quadrics P ( s) and T ( s) , but continuous relative motions instead. Implementation of the continuous motions leads to significant simplification of the procedure of modeling. In order to perform the desired continuous relative motions of the modeling quadrics P ( s) and T ( s) , use of the surfaces that allow for sliding over itself is helpful. A screw surface of constant pitch p = Const is the most general case of surfaces P ( s) and T ( s) that allow sliding over themselves (see Chapter 1). As a screw surface moves along and rotates about its axis with the same parameter of the screw motion as the instant screw parameter of the screw surface, the enveloping surface to consecutive positions of the screw surface is congruent to the screw surface. Particular cases of surfaces that allow for sliding over itself [surfaces of revolution (for which p = 0), general (not circular) cylinders (for which p = ∞)] are considered in Chapter 1. Cylinders of revolution, spherical surfaces, and the plane represent examples of the simplest and completely degenerated surfaces that allow for sliding over themselves. For simulation of the machining operation of the sculptured surface on a multi-axis NC machine, it is convenient to use a screw with external surface P ( s) , and with either a convex or a concave thread profile (Figure 5.6). Application of such a screw enables simulation of both convex and saddle-like local patches of a given sculptured surface P. For simulation of concave and saddle-like local patches of a given sculptured surface P, a screw with internal surface P ( s) and with either a convex or a concave thread profile can be used (Figure 5.7). In both cases (Figure 5.6, and Figure 5.7), the screw might be either single or multithreaded, as well as single or multistarted. In order to provide the required parameters of topology of the surface P ( s) (i.e., the parameters R(1s.)P ≡ R 1. P , R(2s.)P ≡ R 2. P ); the required radii of principal curvature of the surface T ( s) (i.e., the parameters R(1s.)T ≡ R 1.T , R(2s.)T ≡ R 2.T ); and their local relative orientation (i.e., the angle µ ( s) ≡ µ of the surfaces P ( s) and T ( s) local relative orientation), the parameters dP( s) and rP( s) of design of the screw have to be computed properly. For this purpose, Meusnier’s formula and Euler’s formula can be used. Mensnier’s formula establishes the correspondence between a radius of normal curvature RP of a surface P (or a surface T) through a certain direction t P

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Kinematic Geometry of Surface Machining rP(s)

P(s)

K

dP(s)

(a) P(s)

rP(s) K

dP(s)

(b) Figure 5.6 An external screw with (a) convex and (b) saddle-like local patches of the quadric surface P ( s) for the experimental simulation of machining of a sculptured surface on a multi-axis numerical control machine.

on the surface, and between the radius of curvature RP(ϑ ) of the surface P (or a surface T) through that same direction t P on the surface, which is inclined to the normal plane section at a known angle ϑ P . Usually, Meusnier’s formula is represented in the form

RP(ϑ ) = RP ⋅ cos ϑ P

(5.19)

Equation (1.31) represents a conventional form of Euler’s formula. For the purposes of simulation, it is much more convenient to model the surface T ( s) with external or internal surface of revolution that has either a convex or a concave axial profile (Figure 5.8). The same Meusnier’s and Euler’s formulae can be used for computing the parameters dT( s), and rT( s) of design of the cutting tool in order to provide the identities R(1s.)T ≡ R 1.T and R(2s.)T ≡ R 2.T .

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P(s)

rP(s)

rP(s) K

K

d(s) P

dP(m)

(a)

(b)

Figure 5.7 An internal screw with (a) concave and (b) saddle-like local patches of the quadric surface P(s) for the experimental simulation of machining a sculptured surface on a multi-axis numerical control machine. rT(s)

rT(s) K

rT(s)

K

(s)

K rT(s)

(s)

dT

dT

K (s)

dT

rT(s) T (s)

T (s)

T (s)

T (s)

(a)

(b)

(c)

(d)

Figure 5.8 External (a,b), and internal (c,d) surfaces of revolution with (a) convex, (b and c) saddle-like, and (d) concave local patches of the quadric surface T ( s) for the experimental simulation of machining a sculptured surface on a multi-axis numerical control machine.

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P(s)

(s)

OP

ωP(s)

(s)

VK K

θ (s)

(s)

FT

ω(s) T

(s)

OT

T (s) Figure 5.9 Simulation of machining a convex local patch of a sculptured part surface with the saddle-like local patch of the generating surface of the cutting tool.

Implementation of the screw surfaces P ( s) (Figure 5.6 and Figure 5.7) and the surfaces of revolution (Figure 5.8) allows one to reach the desired topology of the simulating surfaces P and T. Figure 5.9 illustrates an example of implementation of the disclosed method [14]. In the particular case (Figure 5.9) of a convex local patch of surface P with the saddle-like local patch of surface T, machining is simulated with the external worm having a convex profile of threads that is machining with the grinding wheel that has a concave axial profile. The design parameters of the worm and the design parameters of the grinding wheel are precomputed in tight correlation with the corresponding design parameters of the actual part surface P and the actual generating surface T of the tool. Rotation of the worm and of the grinding wheel are timed in order to get the resultant motion of the surfaces P ( s) and T ( s) identical to those of the relative motion of the surfaces in the machining operation to be simulated. In the case when one or both modeling quadric surfaces P ( s) and T ( s) allow for sliding over themselves, manufacturing of the specimens for simulation is simplified. At the same time, this results in two instant relative motions of the surfaces P ( s) and T ( s) being substituted with their continuous motion. The last is much more convenient for simulation and enables more precise and

Pat. No. 1449246, USSR, A Method of Experimental Simulation of Machining of Sculptured Surface on Multi-Axis NC Machine./S.P. Radzevich, Filed February 17, 1987.

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more reliable experimental results to be obtained. Ultimately, this allows for accurate determination of the rate of conformity functions F 1, F 2, and F 3. It is important to note that many experimental data necessary for determining the rate of conformity functions F 1, F 2, and F 3 can be collected from already published scientific papers in the field. For this purpose, it is necessary to analyze the published results of the research on efficiency of surface machining on a machine tool from the standpoint of the theory of surface generation. It is critical to mention here that the rate of conformity functions F 1, F 2, and F 3 also play another important role in the theory of surface generation. They serve as a bridge between the pure geometrical and kinematical theory, and between real machining processes including physical phenomena. 5.1.6 An Algorithm for the Computation of the Design Parameters of the Form-Cutting Tool Computation of the major design parameters of the generating surface T of the form-cutting tool for machining of a given sculptured surface on a multiaxis NC machine requires high-volume computations. Generally, computations of this kind can be performed with the help of computers. An algorithm for the computation of the design parameters of the generating surface T of the form-cutting tool is illustrated with the flow chart in Figure 5.10: 1. Compose an equation of the smooth, regular sculptured surface P. When the part surface P is made up of two or more portions, then a set of equations of all n surface patches Pi |in=1 must be composed. 2. Compute the first derivatives of equations of the sculptured part surface P. 3. Compute the fundamental magnitudes EP, FP, and GP of the first order of the surface P. 4. Compute the second derivatives of equations of the part surface P. 5. Compute the fundamental magnitudes Lp, M p, and N p of the second order of the surface P. Results (3) and (5) could be interpreted as the natural parameterization of the sculptured surface P. 6. Compose a set of three equations that describe the desired rate of conformity of the generating surface T of the form-cutting tool to the sculptured surface P. 7. Determine the rate of conformity functions F 1, F 2, and F 3. 8. Use the R-mapping of the sculptured surface P onto the generating surface T of the form-cutting tool to return a set of three equations (Equation 5.4, Equation 5.5, Equation 5.6) of six unknowns for the computation of the fundamental magnitudes of surface T.

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Kinematic Geometry of Surface Machining Start 1

2

3

4

5

E

6

7

8

10

9

11 13

I

II

12 End

Figure 5.10 Flow chart of the algorithm for the computation of the major design parameters of the formcutting tool of optimal design for machining a given sculptured surface P on a multi-axis numerical control machine.

9 . Incorporate Gauss’ equation of compatibility (Equation 5.10). 10. Incorporate two Mainardi-Codazzi’s equations of compatibility (Equation 5.11 and Equation 5.12) — ultimately, this yields a set of six equations (Equation 5.4 through Equation 5.12) of six unknowns ET , FT , GT and LT , MT , N T . 11. Compute the fundamental magnitudes ET , FT , and GT of the first, and LT , MT , and N T of the second order of the generating surface T of the form-cutting tool. The results (11) could be interpreted as the natural parameterization of the generating surface T of the form-cutting tool. 12. Solve the set of two Gauss–Weingarten’s equations in tensor notation. The output is the matrix equation of the generating surface T of the form-cutting tool in a Cartesian coordinate. 13. Incorporate initial conditions for integration of the set of two Gauss–Weingarten’s equations. In the general case, principal curvatures k 1. P = R −11. P and k 2. P = R −21. P of the sculptured surface P depend on surface definition (i.e., they depend on surface topology) and usually are not related by any function. Due to this fact, principal

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curvatures k 1.T and k 2.T for the designed generating surface T of the cutting tool will not be related by a function. In cases of milling cutter, grinding wheel, and so forth, the principal curvatures k 1.T and k 2.T are related by a function — in this case, the surface T is represented by a surface of revolution. For the machining of a sculptured surface P of any geometry, the above generalized solution (see Equation 5.16) yields an approximation of the surface T with optimal topology by a surface of revolution, and so producing a form-cutting tool having all necessary combinations of k 1.T and k 2.T . 5.1.7 Illustrative Examples of the Computation of the Design Parameters of the Form-Cutting Tool Two illustrative examples of the computation that do not require extensive computer application are presented in this section. Instead of a general demonstration of the developed approach, consider a case when the six fundamental magnitudes ET , FT , GT, LT , MT, N T of the form-cutting tool are already determined from Equation (5.4) through Equation (5.6) and from equations of compatibility Equation (5.10), Equation (5.11), and Equation (5.12). Example 5.1 Given two differential forms

φ1.T ⇒ dUT2 + cos 2 UT dVT2 and φ2.T ⇒ dUT2 + cos2 UT dVT2 , find the generating surface T of the form-cutting tool, for which φ 1.T and φ 2.T are the first and second fundamental forms. Because ET = 1 , FT = 0 , GT = cos 2 UT , and LT = 1 , MT = 0 , N T = cos 2 UT, then it could be determined that Christoffel’s symbols Γ 111 = Γ 222 = Γ 112 = Γ 222 = 0 , 2 Γ 12 = − tan UT , Γ 122 = sin UT cos UT satisfy the Gauss–Codazzi’s equations of compatibility, as the direct substitution shows. The set of Gauss–Weingarten’s equations (Equation 5.10 through Equation 5.12) returns the solution

r T = r 0T

cos VT cos UT   sin V cos U  T T  +   sin UT   1  

(5.20)

which is the equation of the sphere. The detailed derivation of Equation (5.20) is not covered here. However, it is covered in detail in the literature [17]). Here r0T designates the vector that specifies the location of the generating surface T of the cutting tool. By the choice of r0T, one can place the surface T in any position of space, selecting any orthogonal system of meridians and parallels for UT and VT curvilinear coordinates of an arbitrary point M on the tool surface T. The sphere can be used as the generating surface T of the cutting tool, for example, of the grinding wheel (Figure 5.11) for machining a sculptured surface P on a multi-axis NC machine.

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Kinematic Geometry of Surface Machining ωT

ZT

T

rT

UT YT Z

Y

rT

M

XT VT

OT

X

Figure 5.11 Example 5.1: The spherical generating surface T of a grinding wheel. (From Radzevich, S.P., Mathematical and Computer Modeling, 36 (7–8), 921–938, 2002. With permission.)

Example 5.2 In much the same way as done above, for the fundamental forms

φ1.T ⇒ (rT cos θT + RT )2 dUT2 + (− rT sin θT + RT )2 + rT2 coss θT2 dVT2

(5.21)

and the corresponding

φ2.T ⇒ LT dUT2 + 2 MT dUT dVT + N T dVT2 , one can derive an equation of the generating surface T of the form-cutting tool. Without going into detail here, on solving Equation (5.16), the following expression is derived:

(rT cos θT + RT ) cos ϕT   (r cos θ + R ) sin ϕ  T T T T  rT =    rT sin θT   1  

(5.22)

and the surface T is a torus surface (Figure 5.12). The presented solution to the problem under consideration is grounded on a novel kind of surfaces mapping — on that developed by the author,

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ZT

177

RT T

XT

rT YT

RT

M O

rT θT

Figure 5.12 Example 5.2: Toroidal portion of the generating surface T of the APT cutting tool.

R-mapping of a sculptured surface P onto the generating surface T of the cutting tool [18,19,25]. The disclosed method is tightly connected to the method of simulation of the interaction of the form-cutting tool and the work. The last method is vital for determining the rate of conformity functions of critical importance for the use of the R-mapping-based method of the formcutting tool design. The idea and the general concept of implementation of the R-mapping of surfaces in the field of designing an optimal cutting tool for machining a sculptured surface on a multi-axis NC machine has been proposed [16] by the author [23,24].

5.2

Generation of Enveloping Surfaces

When machining a part surface, the surfaces P and T are the conjugate surfaces. At every instant of the machining operation, surfaces P and T are tangent to each other. They make contact either at a point, or along a characteristic line. Tangency of the surfaces is a strong restriction on the parameters of their relative motion. No the surfaces P and T interference, no interruption of their contact is allowed. When machining a part surface, the interaction of surfaces P and T** is similar to the interaction of the working surfaces of a mechanism, for example, it is quite similar to the interaction of tooth flanks of a gear pair. Pat. No. 4242296/08, USSR, A Method for Designing of the Optimal Form-Cutting-Tool for Machining of a Given Sculptured Surface on Multi-Axis NC Machine./S.P. Radzevich, Filed March 31, 1987. ** If we are to consider parallels between the conjugate action of the surface in the theory of surface generation, and between the conjugate action of surfaces in a gear drive, then it is of critical importance to point out that in surface generation, the surface of action is always congruent to the part surface P to be machined.

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However, a principal difference between the conjugate action of the surfaces in the machining operation, and between the conjugate action of the surfaces in a corresponding mechanism certainly occur. The difference is due to the following reason: The input motion of the mechanism is predetermined. The interacting surfaces of a mechanism are determined by targeting the required parameters of the output motion. In the theory of surface generation, problems of two kinds are recognized. Problems of the first kind are usually referred to as the direct problems of surface generation. Problems of this kind consider that the surface P to be machined and the kinematics of the machining operation are known. The generating surface T of the cutting tool must be determined for machining a given part surface. Problems of the second kind are inverse to the direct problems of surface generation. Usually, problems of the second kind are referred to as the inverse problems of surface generation. When solving problems of this kind, the generating surface T of the cutting tool and kinematics of the machining operation are considered as known. The actual parameters of shape of the machined part surface P must be determined. The total number of problems to be solved in the theory of surface generation is not limited to the two mentioned above. Problems of another nature are considered as well. Mostly surfaces that allow for sliding over themselves can be machined on a conventional machine tool. Conjugate action of the part surface to be machined and of the generating surface of the cutting tool is insightful from the standpoint of implementation of the elements of theory of enveloping curves and enveloping surfaces for the purpose of profiling form-cutting tools. The use of elements of theory of envelopes could significantly simplify the solution to the problem of profiling form-cutting tools for machining surfaces that allow for sliding over themselves. 5.2.1 Elements of Theory of Envelopes The theory of enveloping curves and the theory of enveloping surfaces are widely used for profiling form-cutting tools. For convenience, brief presentations from differential geometry of curves and surfaces are made below. 5.2.1.1

Envelope to a Planar Curve

Consider a planar curve that moves within the plane of its location. If certain conditions are satisfied, then an enveloping curve to consecutive positions of the moving curve could exist [10]. As an example of a planar curve, a circle l of radius r is shown in Figure 5.13. All points of the circle l are within the coordinate plane XY. The circle l moves along the X axis. When moving, the circle l occupies consecutive positions.

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Profiling of the Form-Cutting Tools of Optimal Design Y

l

l1

X

r l2

Figure 5.13 Generation of the enveloping lines l1 and l2 to consecutive positions of a circle l of radius r that moves along the X axis.

In this way, a family of circles is generated. Two straight lines l1 and l2 are the enveloping lines to the family of circles of radius r. These straight lines are parallel to the X axis and are at a distance 2r. The enveloping curve to a family of lines is tangent at each of its points to one of the curves of the family of curves. Each circle shown in Figure 5.13 has a common point with the enveloping line l1 and with another enveloping line l2 . Consider a family r fm (u, ω ) of planar curves r(u) : X(u, ω )   r (u, ω ) =  Y(u, ω )   1  fm

(5.23)

where the parameter of the initially given curve r(u) is designated as u, and w designates the parameter of motion of the moving curve. In other words, w designates the parameter of the family of curves r fm (u, ω ). Parallelism of the tangent vectors ∂r/∂u and ∂rfm/∂w is the necessary condition for existence of the enveloping curve. This condition yields an analytical representation in the following form:

∂ r ∂ r fm × =0 ∂u ∂ω

(5.24)

∂ r ∂ r fm ∂ r ∂ r fm ⋅ − ⋅ ≠0 ∂u ∂ω ∂ω ∂u

(5.25)

The inequality

represents the sufficient condition of existence of the enveloping curve.

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Example 5.3 Consider a planar curve T that is given by the following equation: (2 ⋅ R + R ⋅ cos α )   rT (α , R) =  R ⋅ sin α    1

(5.26)

Partial derivatives of rT (α , R) with respect to a and R are equal: − R ⋅ sin α    =  R ⋅ cos α  ∂α   1

∂ rT

(5.27)

and  2 + cos α    = sin α  ∂R    1

∂ rT

(5.28)

Equation (5.27) and Equation (5.28) yield the equality

− R ⋅ sin α 2 + cos α

R ⋅ cos α =0 sin α

(5.29)

From the determinant Equation (5.29), it is easy to come up with the expression

− R ⋅ (1 + 2 ⋅ cos α ) = 0

(5.30)

Simple formulae transformations yield cos α = − 0.5 , and sin α = 23 . After substituting the last equalities into Equation (5.26) of a family of curves, and after excluding the parameter R, the equation of the enveloping curve can be represented in the form

Y(X ) =

1 ⋅X 3

(5.31)

Therefore, the enveloping curve is a straight line at the angle ± 30° to the X axis. The family of the curves was a family of circles with centers on the X axis (Figure 5.14). The family of curves can be generated by a circle having

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Profiling of the Form-Cutting Tools of Optimal Design P

Y

T

X

P Figure 5.14 Enveloping curve to a family of planar curves.

translation along the X axis, the radius of which increases accordingly to the distance from the origin of the coordinate system XY to the center of a movable circle. The considered example is of practical importance for machining a sheet metal workpiece with a milling cutter with a conical generating surface T (Figure 5.15). The milling cutter axis of rotation moves along the X axis with the feed rate V fr . Simultaneously, the milling cutter performs motion Vax in its axial direction along its axis of rotation Z axis (not shown in Figure 5.15). The actual timing of the motions V fr and Vax depends upon the shape of

Vfr Vax T

P Figure 5.15 Practical implementation of the solution to the problem of determining an enveloping to a family of planar curves.

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r

X Y Figure 5.16 Generation of the enveloping surface to consecutive positions of a sphere of radius r that moves along the X axis.

the part surface P. The functional relation between the motions V fr and Vax can be linear or nonlinear. 5.2.1.2

Envelope to a One-Parametric Family of Surfaces

Consider a one-parametric family of surfaces. The family of surfaces is dependent on a parameter of motion that is designated w. The enveloping surface becomes tangent with every surface of the family of surfaces [10]. For example, centers of all spheres of a family of spheres of radius r are located within the X axis of the Cartesian coordinate system XYZ (Figure 5.16). A circular cylinder of radius r with an X axis as the axis of its rotation represents the enveloping surface to the family of spheres of radius r. fm Consider a family r (U , V , ω ) of surfaces r(U , V ):

X(U , V , ω )  Y(U , V , ω )   r fm (U , V , ω ) =   Z(U , V , ω )    1  

(5.32)

for which the inequality ∂r/∂U × ∂r/∂V ≠ 0 is valid. The necessary condition of existence of an enveloping surface is as follows:

r fm = r fm (U , V , ω )

(5.33)

 ∂r ∂r ∂r   ∂U ∂V ∂ω  = 0

(5.34)

The line of tangency of a surface r(U, V) from a family of surfaces r fm (U , V , ω ) with the enveloping surface is referred to as the characteristic line E. The characteristic line E satisfies Equation (5.30) and Equation (5.31).

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The enveloping surface yields representation in the form of a family of the characteristic lines. When the enveloping surface is of the class of surfaces that allows for sliding over itself, then Equation (5.31) describes the profile of the enveloping surface. Satisfaction of the condition r ∈ω 2 of relationships (see Equation 5.33 and Equation 5.34) together with the conditions ∂ψ ∂U  ∂r    ∂U 

∂ψ ∂V 2

∂r ∂r ⋅ ∂U ∂V

∂ψ ∂ω

∂r ∂r ⋅ ∂U ∂V  ∂r    ∂V 

2

∂r ∂r ≠ 0, ⋅ ∂U ∂ω

∂ψ ∂ψ + ≠0 ∂U ∂V

∂r ∂r ⋅ ∂V ∂ω

(5.35)

is the sufficient condition for the existence of the profile of the enveloping surface. Violation of the first of conditions Equation (5.35) is usually due to the edge of inversion observed. Characteristic lines of the part surface P and of the generating surface T of the cutting tool satisfy the conditions

r1 = r1 (U1 , V1 , ω ),

f [U1 (ω ) , V1 (ω ) , ω ] = 0, ω = Const

(5.36)

r2 = r2 (U 2 , V2 , ω ),

f [U 2 (ω ), V2 (ω ), ω ] = 0, ω = Const

(5.37)

In a stationary coordinate system, for example in a coordinate system associated with the machine tool, the family of the characteristic lines can be represented by the following set of equations:

∂ r2 ∂ r2 = (U 2 , V2 , ω ), ∂f ∂f

f (U 2 , V2 , ω ) = 0

(5.38)

where the equality ∂∂rf2 (U 2 , V2 , ω ) = Rs (1 → 2) ⋅ r1 (U1 , V1 ) is observed. The operator Rs(1 → 2) of the resultant coordinate system transformation is a function

In a coordinate system associated with the cutting tool, the family of the characteristic lines E determines the generating surface T of the cutting tool. For this purpose, Equation (5.34) is necessary to consider together with the operator that describes the motion of the characteristic line E in the coordinate system associated with the cutting tool. In the event that the inverse problem, not the direct problem, of surface generation is considered, then the family of the characteristics E in a coordinate system associated with the work determines the actual machined part surface P.

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of the parameter of motion w. The theory of surface generation also deals with surfaces for which the enveloping surface is congruent to the moving surface. 5.2.1.3

Envelope to a Two-Parametric Family of Surfaces

The two-parametric enveloping surface can be expressed in terms of two parameters, say of the parameters ω 1 and ω 2 . At every point, the enveloping surface becomes tangent with one of the surfaces of the family of surfaces specified by the parameters ω 1 (U , V ) and ω 2 (U , V ) . The parameters ω1 and ω 2 have the same value at every point of every surface of the family of surfaces. However, they differ at different points of the enveloping surface. If the condition ∂r/∂U × ∂r/∂V ≠ 0 is satisfied, then the necessary condition for the existence of the enveloping surface to a family of surfaces r(U , V , ω 1 , ω 2 ) can be represented in the following form [10]:  ∂r ∂r ∂r   ∂r ∂r ∂r  ψ1 =  = 0, ψ 2 =  =0   ∂U ∂V ∂ω 2   ∂U ∂V ∂ω 1 

(5.39)

In order to obtain a sufficient set of conditions for the existence of the enveloping surface, the above conditions [see Equation (5.39)] must be considered together with the following conditions: ∂ψ 1 ∂u ∂ψ 2 ∂u  ∂r    ∂U 

2

∂r ∂r ⋅ ∂V ∂U

∂ψ 1 ∂v ∂ψ 2 ∂v

∂ψ 1 ∂A ∂ψ 2 ∂A

∂ψ 1 ∂B ∂ψ 2 ∂B

∂r ∂r ⋅ ∂U ∂V

∂r ∂r ⋅ ∂U ∂ω1

∂ r ∂r ⋅ ∂U ∂ω 2

2

∂r ∂r ⋅ ∂V ∂ω1

∂r ∂r ⋅ ∂V ∂ω 2

 ∂r    ∂V 

≠ 0,

D(ψ 1 , ψ 2 ) ≠0 D(ω1 , ω 2 )

(5.40)

If a surface r(U , V , ω 1 , ω 2 ) (a) is performing a two-parametric motion, (b) both the motions are independent from each other, and (c) the characteristics E1 and E2 occur for each of the motions, then the point of intersection of the characteristic lines E1 and E2 is a point of the enveloping surface. This point is referred to as the characteristic point. At the characteristic point, the (ω ) (ω ) conditions n ⋅ V1− 21 = 0 and n ⋅ V1− 22 = 0 are always satisfied. Here, n designates a unit normal vector to the enveloping surface. (ω ) (ω ) The conditions n ⋅ V1− 21 = 0 and n ⋅ V1− 22 = 0 are derived for the cases when the resultant relative motion of the surfaces VΣ is decomposed on the (ω ) (ω ) two components V1− 21 and V1− 22 , and both of the components are within the common tangent plane. Definitely, these conditions are sufficient, but they are not mandatory. It is feasible to decompose the resultant relative motion of

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the surfaces VΣ (which is within the common tangent plane) on two particu(ω ) (ω ) lar motions V1− 21 and V1− 22 that are not within the common tangent plane. However, the location of the motion VΣ within the common tangent plane is the essential factor. The discussed approach can be employed for determining the envelope (if any) of an arbitrary surface that has motion of any desired kind. The interested reader may wish to go to [6] for details on the solution to the problem of computation of an envelope of a sphere that has screw motion. Many practical examples of this approach are found in other sources. Elements of theory of enveloping surfaces can be employed for the purposes of profiling form-cutting tools (the direct problem of the theory of surface generation), as well as for the purposes of determining the actual machined part surface (the inverse problem of the theory of surface generation). The desired and the actual part surface may differ because of violation of the necessary condition of proper part surface generation (see Chapter 7). As an illustrative example of a problem that can be solved using the elements of the theory of enveloping surfaces, see Figure 5.17. The generating surface of the first milling cutter is composed of two portions, say of the cylindrical portion T11 and of the spherical portion T12. When performing circular motion, the cylindrical portion T11 of the gen(1) erating surface generates the part surface P11 . The spherical portion T12 of (1) the generating surface of the milling cutter generates the surface P12 , which is a torus. When the circular feed-rate motion is stopped, then the spherical ( 2) portion P12 is machined on the part. Similarly, the generating surface of the second milling cutter is composed of two portions, say of the cylindrical portion T21 and of the flat portion T22. The cylindrical portion T21 of the generating surface of the milling cutter

(2) P22 (1) P22

(2) P12

(1) P11

(2) P21

(1) P12

(1) P21

T21 T11 ωt1

ωt2

T22

T12

Figure 5.17 An example of problems that can be solved using elements of the theory of enveloping surfaces.

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(1) ( 2) generates the plane P21 and the circular cylinder P21 . The plane portion T22 (1) ( 2) of the generating surface generates the plane P22 and the plane P22 . Knowing the dimensions of the milling cutters and the parameters of their motion relative to the work, the above formulae can be used to derive equations of all of the machined part surfaces. Similarly, the above formulae work when necessary to determine the generating surface of a cutting tool for machining of a given part surface.

5.2.2 Kinematical Method for the Determining of Enveloping Surfaces For engineering applications, one more method for the determination of enveloping surfaces is helpful. This method is referred to as the kinematical method for the determination of enveloping surfaces. Initially, this method was proposed by Shishkov as early as in 1951 [31]. The kinematical method is based on the particular location of the vector V1− 2 of relative motion of the moving surface and of the enveloping surface. Vector V1− 2 is located within the common tangent plane to the surfaces. This condition immediately follows from the following consideration. Motions of only two kinds are feasible for the moving surface and the enveloping surface. The surfaces can roll over each other, and they can slide over each other. The component of the resultant relative motion V1− 2 in the direction perpendicular to the surfaces is always equal to zero (Figure 5.18). The cutting tool performs a certain motion relative to the work. The part surface P is generated as an enveloping surface to consecutive positions of the generating surface T of the cutting tool. Points of three different kinds can be distinguished on the moving surface T. Consider points of the first kind, for example, the point A (Figure 5.18). The vector of resultant motion of the cutting tool with respect to the work at point A is designated as V (ΣA) . Projection Prn V (ΣA) of the vector V (ΣA) onto the unit

(A) VΣ

A

(C)

VΣ

(C)

PrnVΣ < 0

(B)

VΣ

B

n(A) T (A)

(B)

PrnVΣ > 0

nT

C

(C )

nT

T P

(B)

PrnVΣ = 0

Figure 5.18 The concept of the kinematical method for determining the enveloping surface.

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Profiling of the Form-Cutting Tools of Optimal Design

187

( A) normal vector nT to the generating surface T is pointed to the interior of the work body ( Prn V (ΣA) > 0 ). Therefore, in the vicinity of point A, the cutting tool penetrates the work body. In this way, roughing portions of the tool-cutting edges cut out the stock. Further, consider points of the second kind, for example, the point B (Figure 5.18). The vector of resultant motion of the cutting tool with respect to the work at point B is designated as V (ΣB) . Projection Prn V (ΣB) of the vector V (ΣB) onto the unit normal vector n(TB) to the generating surface T is perpendicular to this unit normal vector — that is, it is tangential to the part surface P ( Prn V (ΣB) = 0). Therefore, in the vicinity of the point B, the cutting tool does not penetrate the part body. The tool-cutting edges do not cut out stock. The generating surface T of the cutting tool generates the part surface P in the vicinity of the point B. Ultimately, consider points of the third kind, for example, the point C (Figure 5.18). The vector of resultant motion of the cutting tool with respect to the work at point C is designated V (ΣC ) . Projection Prn V (ΣC ) of the vector V (ΣC ) onto the unit normal vector n(TC ) to the generating surface T is pointed outside the part body ( Prn V (ΣC ) < 0). Therefore, in the vicinity of point C, the cutting tool departs from the machined part surface P. In the vicinity of points of the third kind, the tool-cutting edges do not cut out stock, and the generating surface T of the cutting tool does not generate the part surface P. The considered example unveils the nature of the kinematical method for the determination of the enveloping surface. Apparently, this method can be employed to solve problems of both kinds, to profile form-cutting tools for machining a given part surface, and to solve the inverse problem of the theory of surface generation. As an example, generation of the plane P with the cylindrical grinding wheel having the generating surface T is considered in Figure 5.19. When machining a plane P, the grinding wheel rotates about its axis of rotation with a certain angular velocity ω T . Simultaneously, the grinding wheel travels with a feed rate ST . At each of the points A, B, and C on the generating surface T of the grinding wheel, the speed of the resultant relative motion of the cutter with respect to the work is designated VΣ( A) , VΣ( B) , and VΣ(C ), respectively. The speed of the resultant motion at each point A, B, and C ( A) is equal to the vector sum of the feed rate ST and the speed of cutting Vcut , ( B) (C ) Vcut , and Vcut . The feed rate ST is the same value for all points A, B, and C. ( A) ( B) (C ) The velocities Vcut , Vcut , and Vcut are equal to the linear speed of rotation necessary to perform cutting. ( A) ( B) (C ) It is easy to see that the vectors Vcut , Vcut , and Vcut are of the same magni( A) ( B) (C ) tude (|Vcut |=|Vcut |=|Vcut |). They differ from each other only by directions. However, the vectors VΣ( A), VΣ( B) , and VΣ(C ) are of different magnitude (|VΣ( A) |≠ |VΣ( B) |≠ |VΣ(C ) |), and they have different directions VΣ( A) ≠ VΣ( B) ≠ VΣ(C ). Projections of the vectors VΣ( A) , VΣ( B) , and VΣ(C ) onto the unit normal vectors (C ) ( A) nT , n(TB), and n(TC ) are as follows: Prn V (ΣA) > 0, Prn V (ΣB) = 0, and Prn V Σ < 0 . Therefore, in the vicinity of point A, the grinding wheel cuts the stock off; in the vicinity of point B, the part surface P is generating; and in the vicinity of point C, the cutting tool departs from the machined plane P. Similarly, generation of the plane surface P can be performed with the cylindrical milling

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Kinematic Geometry of Surface Machining ωT

T

(C)

PrnVΣ < 0

ST (C)

VΣ (A) PrnVΣ >

0

ST

ST

A

(C)

Vcut C

B (B)

(C)

VΣ

(A)

nT

(A)

VΣ

P

nT

(A)

Vcut (B)

nT

Figure 5.19 Analysis of grinding of a plane surface P from the standpoint of the kinematical method for determining the enveloping surface.

cutter. The analysis similar to this example of grinding of a plane P can be performed on any kind of surface machining on a machine tool. For a family of surfaces r = r(U,V,w), the equation of the characteristic curve E yields representation in the form r = r(U , V , ω ), n ⋅ V1− 2 = 0

(5.41)

A perpendicular N to the surface r1 can be expressed analytically as ∂r ∂r N= × . ∂U1 ∂V1 This means that dot product N ⋅ V1− 2 yields representation in the form of the triple product

∂ r1 ∂ r1 × ⋅ V1− 2 ∂U1 ∂V1

of three vectors. Here, the speed V1− 2 of the relative motion with a certain parameter w is equal:

V1(−ω2)

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 ∂ X1   ∂ω     ∂ Y1  ∂r = 1 =  ∂ω   ∂ω   ∂ Z1   ∂ω     1 

(5.42)

189

Profiling of the Form-Cutting Tools of Optimal Design Ultimately, the equation of the characteristic curve E ∂ X1 ∂ U1

 ∂ Y1 ∂ Z1 ∂ Y1 ∂ Z1  ∂ Y1  ∂ V ⋅ ∂ ω − ∂ ω ⋅ ∂ V  − ∂ U 1 1 1

 ∂ X1 ∂ Z1 ∂ X1 ∂ Z1   ∂ V ∂ ω − ∂ ω ∂ V  1 1

∂ Z1  ∂ X1 ∂ Y1 ∂ X1 ∂ Y1  + − =0 ∂ U1  ∂ V1 ∂ ω ∂ ω ∂ V1 

(5.43)

can be derived for the equation of contact N ⋅ V1(−ω2) =

∂ r1 ∂ r1 × ⋅ V1(−ω2) = 0. ∂U1 ∂V1

(5.44)

When using the kinematical method, the sufficient condition for the existence of the enveloping surface can be obtained in the following way: Consider a smooth, regular surface r1 that is given in a Cartesian coordinate system X1Y1Z1 . The equation of the surface r1 is represented in the form r1 = r1 (U1 , V1 ) ∈ C 2 . The family r1ω of these surfaces in a Cartesian coordinate system X 2 Y2 Z2 is given in the form r1ω = r1ω (U1 , V1 , ω ), where the inequality ω (min) ≤ ω ≤ ω (max) is observed. Then, if the conditions

 ∂ r1 (ω ) ∂ r1 (ω )  ∂ r1 (ω )  ∂U (ω ) × ∂V (ω )  ⋅ ∂ω = f [U1 (ω ) , V1 (ω ) , ω ] = 0, 1 1

f ∈ C1

(5.45)

or 2

2

 ∂ r1 (ω ) ∂ r1 (ω )  ∂f ∂f   ∂U (ω ) × ∂V (ω )  ⋅ V1− 2 = f [U1 (ω ) , V1 (ω ) , ω ] = 0,  ∂U  +  ∂V  ≠ 0 1 1 1 1

(5.46)

∂f ∂U1  ∂ r1   ∂U 

g1 [U1 (ω ) , V1 (ω ) , ω ] =

∂f ∂V1 2

 ∂ r1   ∂ r1   ∂U  ⋅  ∂V 

1

1

 ∂ r1   ∂ r1   ∂V  ⋅  ∂U  1

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∂f ∂ω

1

1

 ∂ r1   ∂V  1

2

 ∂ r1   ∂U  ⋅ V1− 2 ≠ 0 1

 ∂ r1   ∂V  ⋅ V1− 2 1

(5.47)

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Kinematic Geometry of Surface Machining

are satisfied at a certain point, then the enveloping surface exists and can be represented by equations in the forms r1 = r1 (U1 , V1 , ω ) and ∂ r1 = 0. ∂ω

Methods of determining enveloping surfaces based on the implementation of methods developed in differential geometry make it possible to determine points of local tangency of the moving surface with the enveloping surface under fixed values of w. However, for a certain value of w = Const, global interference of the surfaces could occur. Differential methods for determining enveloping surfaces can be employed only when the equation of the moving surface is differentiable. Because surfaces in engineering applications are not infinite and could be represented by patches, and so forth, the part surface P can also be generated by special points on the surfaces. In the general theory of enveloping surfaces, the family of surfaces that changes their shapes is considered as well. Results of the research in this area can be used in the theory of surface generation, particularly for generation of surfaces with the cutting tools that have a changeable generating surface T [15,20,21,30]. Example 5.4 Consider a plane T that has a screw motion. The plane T makes a certain angle τ b with X 0 axis of the Cartesian coordinate system X 0 Y0 Z 0. The reduced pitch p of the screw motion is given. Axis X 0 is the axis of the screw motion. The auxiliary coordinate system X1Y1 is rigidly connected to plane T (Figure 5.20). Z0

Z1 ω2 p·tanτb

−ω2

Y0

Y1

E

V2 ω

V T

X0

X1

V

τb

Figure 5.20 Generation of a screw involute surface as an enveloping surface to consecutive positions of a plane that performs a screw motion.

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191

Profiling of the Form-Cutting Tools of Optimal Design The equation of plane T can be represented in the form Y1 = X1 ⋅ tan τ b

(5.48)

The auxiliary coordinate system X1Y1Z 1 is performing the screw motion together with the plane T with respect to the motionless coordinate system X 0 Y0 Z 0 . In the coordinate system X1Y1Z 1, the unit normal vector n T to the plane T can be represented as  1  − tan τ  b nT =   0     1 

(5.49)

The position vector rT of an arbitrary point M of plane T is as follows:  XT  Y  T rT =   Z T     1 

(5.50)

The speed of the point M in its screw motion is v M = v + [ω × R]

(5.51)

where v is the speed of translation motion, and M is the speed of rotation motion. Determining the characteristic E direction of v M is important, but its magnitude is not of interest. Because of that, it can be assumed that |ω|= 1. Therefore,

ω = i, v = i ⋅ p

(5.52)

This yields

vM

i = i⋅p + 1 X1

j 0 Y1

k 0 Z1

(5.53)

and

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v M = i ⋅ p − j ⋅ Y1 + k ⋅ Z1

(5.54)

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Kinematic Geometry of Surface Machining

The dot product of the unit normal vector n T and of the speed v M equals n T ⋅ v M = p ⋅ tan τ b − Z 1 = 0

(5.55)

Thus, the equation of contact can be represented in the form Z 1 = p ⋅ tan τ b

(5.56)

The above equation of contact together with the equation of plane T represents the characteristic E. y    t ⋅ tan τ  b  r E (t) =   p ⋅ tan τ b    1  

(5.57)

where t designates the parameter of the characteristic E. The characteristic E is the straight line of intersection of these two planes. It is parallel to the coordinate plane X1Z 1 and distanced at p ⋅ tan τ b. For a given screw motion, the characteristic E remains at its location within the plane T in the initial coordinate system X 0 Y0 Z 0. The angle of rotation of the coordinate system X1Y1Z 1 about the X 0 axis is designated as e. The translation of the coordinate system X1Y1Z 1 with respect to X 0 Y0 Z 0 that corresponds to the angle e is equal to p ∙ e. This yields composition of the equation of coordinate system transformation:

1 0 Rs(1 → 0) =  0  0

0 cos ε − sin ε 0

0 sin ε cos ε 0

p⋅ε 0  0   1 

(5.58)

In order to represent analytically the enveloping surface P, it is necessary to consider the equation r E (t) of the characteristic E together with the operator Rs(1 → 0) of coordinate system transformation:

X1 + p ⋅ ε    X ⋅ tan τ ⋅ cos ε + p ⋅ tan τ ⋅ sin ε  1 b b  r P ( X1 , ε ) =  − X1 ⋅ tan τ b ⋅ sin ε + p ⋅ tan τ b ⋅ cos ε    1  

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(5.59)

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Profiling of the Form-Cutting Tools of Optimal Design

Consider the intersection of the enveloping surface P by the plane X 0 = X1 + p ⋅ ε = 0. The last equation yields X1 = − p ⋅ ε . Therefore,

0    p ⋅ tan τ ⋅ (sin ε − p ⋅ ε ⋅ cos ε ) b  r X0 (ε ) =   p ⋅ tan τ b ⋅ (cos ε + p ⋅ ε ⋅ sin ε )   1  

(5.60)

Equation (5.60) represents the involute of a circle. The radius of the base circle of the involute curve is

rb = p ⋅ tan τ b

(5.61)

Therefore, the enveloping surface to consecutive positions of a plane T having a screw motion is a screw involute surface. The reduced pitch of the involute screw surface equals p, and the radius of the base cylinder equals rb = p ⋅ tan τ b . The screw involute surface intersects the base cylinder. The line of intersection is a helix. The tangent to the helix makes the angle ω b with the axis of screw motion:

tan ω b =

rb p

(5.62)

From this, tan ω b = tan τ b , and ω b = τ b. The straight line characteristic E is tangent to the helix of intersection of the enveloping surface P with the base cylinder. This means that if a plane A is tangent to the base cylinder, a straight line E within a plane A makes the angle τ b with the axis of the screw motion, and the plane A is rolling over the base cylinder without sliding, then the enveloping surface P can be represented as a locus of consecutive positions of the straight line E that rolls without sliding over the base cylinder together with the plane A. The enveloping surface is a screw involute surface. The obtained screw involute surface (Figure 5.20) is as that shown in Figure 1.2 and as that analytically described by Equation (1.20). Another solution to the problem of determining the envelope of a plane having screw motion is given by Cormac [6].

5.3 Profiling of the Form-Cutting Tools for Machining Parts on Conventional Machine Tools Cutting tools for machining parts on conventional machine tools feature a property that allows the generating surface of the cutting tool to slide over itself. Those surfaces that allow for sliding over themselves are currently the

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most widely used surfaces in industry. This feature is of importance for the theory of surface generation. It can be used for simplification of the solution to the problem of profiling of the form-cutting tool. Certain simplification is feasible because in the case of surfaces that allow for sliding over themselves, it is not necessary to determine the entire generating surface of the cutting tool. It is sufficient to determine either the profile of the generating surface of the cutting tool or the characteristic line along which the generating surface of the cutting tool makes contact with the machined part surface. 5.3.1 Two Fundamental Principles by Theodore Olivier A solution to the problem of profiling the form-cutting tool for machining a part surface on a conventional machine tool can be derived much easier when using the fundamental principles of surface generation proposed by T. Olivier [12] as early as 1842. The R-mapping-based method for the profiling of form-cutting tools (see Section 5.1) is general. It is a powerful tool for solving the most general problems of cutting-tool profiling. However, in particular cases, simpler methods of profiling form-cutting tools are practical. It is common practice to design form-cutting tools on the premises of one of two Olivier’s principles: The First Olivier’s Principle: Both conjugate surfaces can be generated with an auxiliary generating surface. The generating surface in this case differs from both conjugate surfaces. The Second Olivier’s Principle: The auxiliary generating surface can be congruent to one of the conjugate surfaces.

Prior to solving a problem of profiling of a certain form-cutting tool, geometry of the part surface P (see Chapter 1) and the kinematics of the surface P generation (see Chapter 2) must be predefined. The operators of the coordinate system transformation (see Chapter 3) are used for the representation of all elements of the surface-generation process in a common coordinates system, use of which is preferred for a particular consideration. If we are not just to develop a workable cutting tool, but also to develop the design of the optimal cutting tool, then the methods of analytical description of the geometry of contact of the part surface P and of the generating surface T of the cutting tool (see Chapter 4) are also employed. Design of a form-cutting tool can be developed on the premises of its generating surface T. Derivation of the generating surface T is the starting point of designing the form-cutting tool. For generation of a given surface P, the cutting tool performs certain motions with respect to the work. The part surface P is given, and the generating surface T of the cutting tool is not yet known. Therefore, at the beginning, the actual relative motions of the surfaces P and T are not considered, but the corresponding motions of the part surface P and of a certain coordinate

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system XT YT ZT are analyzed. After being determined, the generating surface of the form-cutting tool would be described analytically in the coordinate system XT YT ZT . 5.3.2 Profiling of the Form-Cutting Tools for Single-Parametric Kinematic Schemes of Surface Generation Kinematic schemes of surface generation, those that feature just one relative motion of the part surface P and of the generating surface T of the cutting tool are referred to as the single-parametric kinematic schemes of surface generation. In the case under consideration, it is convenient to begin consideration of the procedure of profiling of a form-cutting tool for the generation of a surface P in the form of circular cylinder. When machining a circular cylinder of radius RP (Figure 5.21a) [28,29], the work rotates about its axis OP with a certain angular velocity ω P. A coordinate system XT YT ZT is rotating with a certain angular velocity ω T. Axis of this rotation OT crosses at a right angle the axis of rotation OP of surface P. Simultaneously, the coordinate system XT YT ZT travels along the axis OP with a feed rate ST . The generating surface T of the cutting tool in this case can be represented as an enveloping surface to consecutive positions of the surface P in the coordinate system XT YT ZT . Remember that the coordinate system XT YT ZT is the reference system at which the generating surface T could be determined. After being determined, the generating surface T of the cutting tool and the part surface P become tangent along the characteristic line E. In the case under consideration, the characteristic line E is represented with a circular arc ∪ABC of the radius RP.

ZT

RT

RP

OT

T OT H

ZT

H

XT ZP

ωT

E

B A

OP

YP

C

RP

P ωP (a)

ωT

K

T

| RT |> RP XT

YT ZP

RP H

XP

OP

P

ωP

YP (b)

(c)

Figure 5.21 Example of a single-parametric kinematic scheme for derivation of the generating surface of the cutting tool.

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Use of a single-parametric kinematic scheme of surface generation allows a simplification. The generating surface T of the cutting tool can be generated not only as an enveloping surface to consecutive positions of the part surface P in the coordinate system XT YT ZT , but also as a family of the characteristic lines E that rotate about the axis OT. In the example, the generating surface T of the cutting tool is shaped in the form of a torus surface. Radius R T of the generating circle of the torus surface T is equal to the radius RP of the surface P (R T = RP ). The radius of the directing circle of the torus surface T is equal to the closest distance of approach H of the axes OP and OT . The determined torus surface T (Figure 5.21a) can be used for the designing of various cutting tools for the machining of the surface P: milling cutters, grinding wheels, and so forth. The considered example of implementation of the single-parametric kinematic scheme of surface generation (see Figure 5.21a) returns a qualitative (not quantitative) solution to the problem of profiling a form-cutting tool. No optimal parameters of the kinematic scheme of surface generation are determined at this point. The closest distance of approach H of the axes OP and OT, and the optimal value of the cross-axis angle c are those parameters of interest (Figure 5.21b). Optimal values of the parameters H and c can be computed on the premises of analysis of the geometry of contact of the surfaces P and T. The optimal values of the parameters H and c can be drawn from the desired degree of conformity of the surface T to the surface P. Actually, in any machining operation, deviations of the actual cutting-tool configuration with respect to the desired configuration are unavoidable. Because of the deviations, it is practical to introduce appropriate alterations to the rate of conformity of surface T to surface P. Figure 5.21c illustrates an example when the rate of conformity of surface T to surface P is reduced. Reduction of the rate of conformity causes point contact of the surfaces P and T, instead of their line contact in the ideal case of surface generation (Figure 5.21a). The schematic (Figure 5.21c) allows for analysis of the impact of the rate of conformity of the surfaces T and P onto the accuracy and quality of surface generation. Varying values of the parameters H and c (including the case when c ≠ 90°), one can come up with the solution under which vulnerability of the machining process to the resultant deviations of configuration of the cutting tool with respect to surface P is the smallest possible. Use of single-parametric kinematic schemes of surface generation (Figure 5.21) is a perfect example of implementation of the second Olivier’s principle for the purposes of profiling form-cutting tools for the machining of a given part surface. 5.3.3 Profiling of the Form-Cutting Tools for Two-Parametric Kinematic Schemes of Surface Generation Kinematic schemes of surface generation, those that feature two relative motions of the part surface P and of the generating surface T of the cutting tool, are referred to as the two-parametric kinematic schemes of surface generation.

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The resultant motion of the cutting tool with respect to the part surface P can be of a complex nature. For simplification, it can be decomposed on two elementary motions. The elementary motions are usually represented with a rotational motion, and with a translational motion. No one of these motions caused sliding of the surface over itself. Under such a scenario, it is convenient to implement two-parametric schemes of surface generation for profiling the form-cutting tool. Similar to the analysis of implementation of single-parametric kinematic schemes (Figure 5.21), consider implementation of two-parametric kinematic schemes of surface generation for the case of machining a surface of a circular cylinder (Figure 5.22). The work is rotating about the axis OP (Figure 5.22a) with a certain angular velocity ω P. The cutting-tool coordinate system XT YT ZT is rotating with a certain angular velocity ω T about the axis OT. The axes of rotations OP and OT are at a right angle. The resultant generating motion VΣ is decomposed on two elementary motions V1 and V2. (Here the equality VΣ = V1 + V2 is observed.) At the beginning, an auxiliary generating surface R must be determined. The auxiliary surface R is an enveloping surface to consecutive positions of the part surface P in its motion with the velocity V1 of the first elementary motion. Further, the generating surface T of the cutting tool is represented in the coordinate system XT YT ZT with the enveloping surface to consecutive positions of the auxiliary generating surface R in its motion with the velocity V2 of the second elementary motion. Here, vector V2 designates linear velocity of rotational motion of the auxiliary surface R about the axis OP with the angular velocity ω T. The generating surface T of the cutting tool, which is determined following the two-parametric approach, usually makes point contact with the part surface P. This is observed because of the following reason: The line E1 is the characteristic line for the part surface P and the auxiliary generating surface R . The auxiliary generating surface R and the generating surface T of the cutting tool make line contact, and the line E2 is the characteristic line to the surfaces R and T. The characteristic lines E1 and E2 are within the auxiliary generating surface R . Generally speaking, the characteristic lines E1 and E2 intersect at a point that is the characteristic point. At least one common point of the characteristic lines E1 and E2 is necessary — otherwise surfaces P and T cannot make contact. Point K is the point of intersection of the characteristic lines E1 and E2 . In particular cases, the characteristic lines E1 and E2 can be congruent to each other. This results in the surfaces P and T making line contact instead of point contact. When the surfaces P and T move relative to each other with the velocity of V1 , then the characteristic line E1 occupies a stationary location on the part surface P, but it travels over the generating surface T of the cutting tool. When the plane R rotates about the axis OT, the characteristic curve E2 on the auxiliary surface R is stationary, but it is traveling over the generating surface T of the cutting tool. Therefore, in the general case of surface

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Kinematic Geometry of Surface Machining T

T

OT

OT

ωT

ωT

K E1

ωP

ωP

OP

E2 OP

Z

P

V1

K E1

E2

Z

Y

OT

OT

V1

X

P

X

Y (a)

(b)

V1

T1 T

T

OT

OT

K

OP

Z

E2

Z

Y

X (c)

OT E1

K

A

P Y

E2

E1

V1

θ

l

ωT

ωT

ωP

T2

C OP

ωP

P X (d)

Figure 5.22 Examples of two-parametric kinematic schemes for derivation of the generating surface of the cutting tool. (See also Rodin, P.R., Fundamentals of Theory of Cutting Tool Design, Mashgiz, Kiev, 1960; and Rodin, P.R., Questions of Theory of Cutting Tool Design, DrSci dissertation, Odessa Polytechnic Institute, Odessa, Ukraine, 1961.)

generation (Figure 5.22), point K of the intersection of the characteristic lines E1 and E2 is movable. The generating surface T of the cutting tool that is determined for the two-parametric kinematic scheme makes point contact with the part surface P. This means that no ideal generation of the surface P is feasible. Cusps on the machined part surface P are unavoidable under such circumstances.

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It is necessary that the cusp height h Σ be less than the tolerance [h] on the accuracy of generation of the surface P. The inequality h Σ ≤ [ h] must be satisfied. Location of the point K and the shape of the generating surface T of the cutting tool depend upon the direction of the motion V2. Depending on V2, the surface T can be shaped in the form of a cylinder (Figure 5.22a), a cone (Figure 5.22b), or a plane (Figure 5.22c). In particular cases, surfaces P and T make line contact. Line contact of the surfaces P and T is observed when the characteristic lines E1 and E2 are congruent. No cusps are observed on the machined part surface P when the surfaces P and T are in line contact. The auxiliary generating surface R is tangent to the part surface P. Therefore, the surface R can be employed not only as an auxiliary surface, but also as the generating surface T of the cutting tool. In the last case, it is necessary to consider the surface R as the generating surface of the cutting tool that is derived using the single-parametric scheme of surface generation. In order to generate the generating surface T of the cutting tool (Figure 5.22d), the generating surface T1 can be determined as the enveloping surface to consecutive positions of the part surface P in its motion relative to the coordinate system XT YT ZT . By doing this, the approach shown in Figure 5.22a can then be followed. Another motion V2 of the determined generating surface T1 is then considered [28,29]. Ultimately, the generating surface T2 of the cutting tool can be determined as the enveloping surface to consecutive positions of the surface T1 that is performing the motion V2. In the example shown in Figure 5.22d, the generating surface T2 of the cutting tool is shaped in the form of a noncircular cylinder. The generating surface T2 and the part surface P make point contact. Point of contact K of the surfaces P and T2 is traveling over both surfaces: over the part surface P and over the generating surface T2 . The closed three-dimensional curve l represents the tool-path on the part surface P. The solution to the problem of profiling the form-cutting tool using twoparametric kinematic schemes of surface generation yields qualitative (not quantitative) results. No optimal parameters of the kinematic scheme of surface generation or of the optimal cutting tool can be derived from the considered approach. An analytical solution to the problem of determining optimal parameters of the kinematic scheme of surface generation and of optimal parameters of the geometry of the generating surface of the form-cutting tool can be obtained on the premises of the comprehensive analysis of the geometry of contact of the surfaces P and T (see Chapter 4). The considered example (Figure 5.22) is insightful. It gives impetus to the investigation of impact of the rate of conformity of the surfaces P and T on the accuracy and quality of the machined part surface. It is also practical for the analysis of vulnerability of the surface-generation process to the deviations of configuration of the cutting tool with respect to the work. The use of two-parametric kinematic schemes of surface generation (Figure 5.22) is a perfect example of implementation of the first Olivier’s

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principle for the purposes of profiling form-cutting tools for machining a given part surface. 5.3.4 Profiling of the Form-Cutting Tools for Multiparametric Kinematic Schemes of Surface Generation Kinematic schemes of surface generation, those that feature more than two relative motions of part surface P and generating surface T of the cutting tool are referred to as the multiparametric kinematic schemes of surface generation. The resultant motion of the cutting tool with respect to the part surface P can be of a complex nature. It is not a common practice to implement kinematic schemes of surface generation that have more than two elementary motions. However, as an example, methods of hob relieving composed of up to six elementary motions are known. Speed VΣ of the resultant relative motion of the surfaces P and T can be decomposed on a finite number of elementary motions Vi (where i is an integer number, and i > 2 is considered): VΣ = Σ in=1Vi . The total number of elementary motions is designated as n. There are no principal restrictions on the number n of elementary relative motions Vi. In the event the kinematic scheme of surface generation is composed of n elementary relative motions, then it is possible to generate (n − 1) auxiliary generating surfaces R 1 , R 2, … , R n−1 . The last auxiliary generating surface R n is congruent to the generating surface T of the cutting tool ( R n ≡ T ). Extension of Olivier’s principles in this direction is supported by the proven feasibility of the generation of conjugate surfaces by means of two auxiliary generating surfaces [18,19]. This achievement can be recognized as the third principle of generation of conjugate surfaces. If the multiparametric kinematic scheme of surface generation is used, then the equation of the generating surface T of the form-cutting tool can be derived by solving the set of two equations:

rT = rT (U P , VP , ω 1 , ω 2 , K , ω i , K , ω n )  n P ⋅ VΣ = 0

(5.63)

In this equation, the resultant motion VΣ is represented in the summa VΣ = V1 + V2 + K + Vi + K + Vn. However, this does not mean that the particular equalities n P ⋅ V1 = 0, n P ⋅ V2 = 0, … , n P ⋅ Vi = 0 and others must be satisfied. Satisfaction of the particular equalities is the sufficient, but not necessary, condition. The elementary relative motions Vi are not mandatory within the common tangent plane. Location of the resultant motion VΣ within the common tangent plane is the only mandatory requirement in this concern.

Pat. No. 965.728, USSR, A Method of Hob Relieving./S.P. Radzevich, Filed January 21, 1980.

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Profiling of the Form-Cutting Tools of Optimal Design

However, an increase in the number of elementary motions Vi causes kinematic schemes of surface generation to become more complex. This is the major reason that the generating surface T of the form-cutting tool is generated in most cases by using either single-parametric or two-parametric kinematic schemes of surface generation. The problem of derivation of the equation of the generating surface T of the form-cutting tool when the implemented kinematic scheme of surface generation is multiparametric is not complex in principle. However, it often causes technical problems when performing routing computations.

5.4 Characteristic Line E of the Part Surface P and of the Generating Surface T of the Cutting Tool In this section, an advantage of implementing kinematic schemes of surface generation is provided. Because sliding of the generating surface of the cutting tool over itself is allowed, then it is necessary to derive not the equation of the entire surface T, but the equation of its profile instead. Such substitution results in significant simplification of the problem of profiling the formcutting tool. When the part surface P is given by equation r P = r P (U P , VP ), then the equation of a family of the surfaces P can be represented in the form r Pfm = r Pfm (U P , VP , ω 1 , ω 2 , K , ω i , K , ω n ). By definition, the unit normal vector n P to surface P is equal to n P = u P × v P . Then, the equation of the characteristic line E can be derived after solving the set of equations:

r Pfm = r Pfm (U P , VP , ω 1 , ω 2 , K , ω i , K , ω n )   ∂r P ∂r P ∂r P ∂r P = 0, K , n P × = 0, K , n P × = 0, n P × n P × ∂ω 2 ∂ω i ∂ω n ∂ω 1 

(5.64)

The set of Equation (5.64) is sufficient but not necessary (see above). Summa of all the equations at the bottom (see Equation 5.64) must be equal to zero. However, it is not mandatory that each expression equal zero. Another way of deriving the characteristic line is based on consideration of instant screw motion of the part surface P relative to the coordinate system XT YT ZT to which the generating surface T will be connected. The surface P performs the screw motion about the part axis of the instant screw motion (Figure 5.23). This axis of the screw motion yields analytical representation in the form

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P( p) = P0 + p ⋅ p

(5.65)

202

Kinematic Geometry of Surface Machining The Tool Axis

P0

c

p P0

ZP

YP

T0

t

The Part Axis

T0 XP

Figure 5.23 Interpretation of the instant screw motions of the part surface P and of a coordinate system XT YT ZT associated with the cutting tool.

where P0 designates the position vector of a point P0 on the part axis of the instant screw motion, unit vector p determines the direction of the part axis, and p designates the parameter of the part axis. Similarly, the coordinate system XT YT ZT performs the screw motion about the tool axis of the instant screw motion. This axis of the screw motion yields vector representation

T(t) = T0 + t ⋅ t

(5.66)

where T0 designates the position vector of a point T0 on the tool axis of the instant screw motion, unit vector t determines the direction of the tool axis, and t designates a parameter of the tool axis. One restriction for the candidate surface T is that the common perpendicular between the surfaces P and T in contact and the instant screw axis must satisfy Ball’s reciprocity relation [2,3] — namely,

a = h ⋅ tan ε

(5.67)

where a designates the shortest distance of approach between the axis of the instant screw motion and between the perpendicular to the surface P at a point of the characteristic line E, h designates the parameter (the reduced pitch) of the instant screw motion, and e is the angle between the instant screw axis and the line of action. If axes of the instant screw motions are not parallel to each other, then the line segment C joining the closest points of the part axis and of the tool axis

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203

is uniquely perpendicular to both axes at the same time. No other segment between the axes possesses this property — that is, the unit direction vector c is uniquely perpendicular to the line direction vectors p and t. This is equivalent to the vector c satisfying the two equations p ⋅ c = 0 and t ⋅ c = 0. In order to satisfy these equations, the unit vector c must be equal to c = p × t. Ultimately, for the computation of the closest distance of approach of the part axis and of the tool axis, Equation (5.67) can be employed. The direction of the line segment through the closest distance of approach is specified by the unit vector c. This yields computation of coordinates of points of the characteristic line E. The characteristic line E can be interpreted as the projection of the axis of the instant relative screw motion onto the part surface P. At every point of the characteristic line E, the unit normal vector n P to the part surface P makes a certain angle e with the axis of the screw motion (see Equation 5.67). Following the above consideration, the equation of the characteristic line E can be derived. Another approach to the determinination of the characteristic of a surface having a certain motion can be found in the monograph by Cormac [6]. Parameters of the kinematic scheme of surface generation can be time dependent. This means that the parameters of the characteristic line E could also be time dependent. Methods of surface machining with a cutting tool when the characteristic line E changes it shape in time are known [8].

5.5

Selection of the Form-Cutting Tools of Rational Design

For machining sculptured surfaces, cutting tools of standard design are used. Form-cutting tools of the most widely used designs for sculptured surface machining are shown in Figure 5.1. When selecting a certain cutting tool for machining a given sculptured surface, the user usually follows conventional rules. This means that the user will want to select the cutting tool design that satisfies two requirements: First, any and all regions of the part surface must be reachable by the cutting tool without mutual interference of the part surface P and of the generating surface T of the cutting tool. In order to achieve the highest possible productivity of surface machining, in compliance with the second requirement, the diameter of the cutting tool must be the biggest feasible. Versatility of known designs of form-cutting tools for sculptured surface machining on a multi-axis NC machine is not limited to the designs schematically shown in Figure 5.1. Other designs of form-cutting tools can also be implemented. For machining sculptured surfaces, form-milling cutters are used. The machining surface T of the milling cutter is shaped in the form of a surface of revolution. For better performance of the milling cutter, it is highly desired to have the principal radii of curvature of the generating surface of

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the milling cutter equal to (or almost equal to) the extremal values of principal radii of curvature of the part surface taken with the opposite sign (see Chapter 4). For this purpose, the interval of variation of the radius of curvature of the generating curve of the milling cutter surface T and configuration of the generating curve with respect to the cutting tool axis of rotation must be determined in compliance with the interval of variation of principal radii of curvature of the part surface P. It is practical to assign a constant gradient of alteration of radius of curvature of the generating curve. For the constant gradient of the alteration, the linear relationship between the radius of curvature ρT of the generating curve and the length of its arc L T must be observed. Here, L T designates the length of the arc of the generating curve, which is measured from a certain point on the generating curve to its current point. Therefore, in the case under consideration, the following equality is valid:

ρT = c ⋅ L T

(5.68)

where the constant parameter c specifies the intensity of alteration of the radius of curvature ρT. Equation (5.68) is a perfect example of natural parameterization of a curve. In polar coordinates, the equation of the generating curve can be represented in the following way. Consider the equation of the generating curve in the form rT = rT (ψ ). Here, rT designates the position vector of a point of the generating curve, and ψ designates the parameter of the generating curve. The radius of curvature ρT at the current point of the planar curve rT = rT (ψ ) can be computed by 3

2   drT   2 rT2 +    dψ    ρT = 2  d 2 rT   drT  − ⋅ rT2 + 2  r T   dψ   dψ 2 

(5.69)

The length L T of the arc of the generating curve between two points that are specified by actual values ψ 1 and ψ 2 of polar angle ψ is equal to ψ2

LT =

∫

drT2 + rT2 dψ 2

ψ1

(5.70)

After substituting Equation (5.69) and Equation (5.70) into Equation (5.68), the following equation can be derived:

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R T = R T .0 exp (c ⋅ ψ )

(5.71)

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Profiling of the Form-Cutting Tools of Optimal Design ωT 120°

ωT

90°

θ

150° RT

180°

ОT

ψ

RT.0 330°

210° 240°

270°

300°

Figure 5.24 The generating surface T of a form-cutting tool for sculptured surface machining, which is shaped in the form of a surface of revolution of a logarithmic spiral curve. (From Radzevich, S.P., Mathematical and Computer Modeling, 36 (7–8), 921–938, 2002. With permission.)

where the position vector of a certain zero point (Figure 5.24) is designated R T.0 . The generating curve (Equation 5.71) is an isogonal curve to a bunch of straight lines through a point. This point is referred to as the pole of the curve. In order to verify the last statement, it is sufficient to substitute the equation ϕ ⋅ (Y − k ⋅ X ) = 0 of a bunch of straight lines to differential equation

∂ϕ ∂ϕ  ∂ϕ   ∂ϕ  cos θ − sin θ  dX +  sin θ + cos θ  dY = 0   ∂X   ∂X ∂Y ∂Y

(5.72)

for isogonal trajectories. Here, q designates the angle at which the curve (Equation 5.72) intersects all straight lines of the bunch of straight lines. It is important to mention here that the following equality θ = tan −1 (c) is observed. When rotating the generating curve (Equation 5.71) about a tool axis OT, the generating surface T of the cutting tool is generated (Figure 5.24). The surface T can be further used for design of a form-milling cutter, a formgrinding wheel, or a form-cutting tool of another design. The equation of the generating surface T is analytically described by

 (rt + R e c⋅ψ cos ψ ) ⋅ sin δ  (r + R e c⋅ψ cos ψ ) ⋅ cos δ  t  rT (ψ , δ ) =   rt tan ϕ + R e c ψ sin ψ    1  

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(5.73)

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Kinematic Geometry of Surface Machining

M

(a)

(b)

(c)

Figure 5.25 Examples of form-milling cutters (USSR Patent 1.271.680) for machining a sculptured part surface on a multi-axis numerical control machine.

The disclosed approach makes it possible to generate the generating surface T of the cutting tool that has either a convex (Figure 5.25a) or a concave generating curve (Figure 5.25b), as well as the generating curve with a point of inflection (Figure 5.25c), say the point M of tangency of two logarithmic spiral curves given by Equation (5.73). In addition, it makes it possible to create the generating surface of the cutting tool having internal tangency with the part surface to be machined. In the last case, the work is located inside the cutting tool [22]. The shape of the generating curve of the surface T enables us to fit to any desired value of radius of normal curvature of the part surface P. Any desired rate of conformity of the tool surface T to the part surface P can be reached with the cutting tool (Figure 5.25). Ultimately, use of the milling cutters (Figure 5.25) makes it possible to significantly reduce the height of the cusps, and in this way to improve the quality of the machined part surface. Fundamental magnitudes of the first order of the generating surface of the cutting tool (see Equation 5.73) are equal:

E T = R 20 ⋅ e 2⋅c⋅ψ ⋅ (1 + c 2 ),

FT = 0;

(

GT = rt + R 0 ⋅ e c⋅ψ ⋅ cos ψ

)

2

(5.74)

Fundamental magnitudes of the second order of the generating surface of the cutting tool (see Equation 5.73) are equal:

(

)

L T = − R 0 ⋅ e c⋅ψ ⋅ 1 + c 2 , MT = 0, N T = − rt + Ro ⋅ e c⋅ψ ⋅ cos ψ ⋅

c ⋅ sin ψ + cos ψ 1 + c2

(5.75)

Pat. No. 1.271.680, USSR, A Form Cutting Tool for Machining of Sculptured Surface on Multi-Axis NC Machine./S.P. Radzevich, Int. Cl. B 23 C 5/10, Filed August 9, 1984.

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DT

DT

dT

dT

2

1 min R1.T

min R2.T

yT

yT

Dup(Td*)

Md*

Dup(TD*) max R2.T

MD*

xT

Dup(Td**)

Dup(TD**)

min R2.T

xT

xT

min R1.T

max R1.T

lT

yT

yT

max R2.T

(a)

MD**

Md** lT

xT

max R1.T

(b)

Figure 5.26 Form-milling cutters (USSR Patent 1.355.378) for machining a sculptured part surface on a multi-axis numerical control machine.

The fundamental magnitudes of the first order (see Equation 5.74), and of the second order (see Equation 5.75) yield equation for the Dupin’s indicatrix of the generating surface T of the form cutting tool (see Equation 5.73) c ⋅ sin ψ + cos ψ 1 ⋅ y 2 = ±1 ⋅ x2 + R 0 e c⋅ψ ⋅ 1 + c 2 rt + R 0 ⋅ e c⋅ψ ⋅ 1 + c 2 ⋅ cos ψ

(5.76)

Equation (1.14) yields computation of the principal radii of curvature of the generating surface T of the form-cutting tool (see Equation 5.73). They can be computed from the following: R 1.T = −

(

2 ⋅ 1 + c 2 ⋅ rt + R 0 ⋅ e c⋅θ ⋅ cos θ c ⋅ sin θ + cos θ

),

R2.T = −2 R 0 ⋅ e c⋅θ ⋅ 1 + c 2

(5.77)

The computed values of principal radii of curvature of the generating surface T of the form-cutting tool (see Equation 5.77) were used for the development of novel designs of the form-milling cutters (Figure 5.26). The generating surface T of the form-milling cutter (Figure 5.26) is represented with a surface of revolution (see Equation 5.73). Principal radii of curvature R 1.T and R2.T of the generating curve of the surface T gradually change from one point

Pat. No. 1.355.378, USSR, A Form Cutting Tool for Machining of Sculptured Surface on Multi-Axis NC Machine./S.P. Radzevich, Int. Cl. B 23 C 5/10, Filed April 14, 1986.

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of the curve to another. At least two points are observed on a given generating surface T at which the first principal radius of curvature R 1.T reaches it maximal min R max 1.T and its minimal R 1.T values. There also exist at least two points on the surface T at which the second principal radius of curvature R 2.T reaches it maximal min R max 2.T and its minimal R 2.T values. In particular, points (say two pairs of points) at which the principal radii of curvature reach their extremal values could coincide. Two kinds of coincidence are distinguished: In the first case, the principal min radii of curvature reach their extremal values R min 1.T and R 2.T at one point of the max max surface T, while the extremal values R 1.T and R 2.T are observed in another surface point. In the second case, at a certain point of the surface T their values are max max min R min 1.T and R 2.T , while at another surface T point they equal R 1.T and R 2.T . Configuration of the generating line of the tool surface T depends upon the diameters dT and DT at the ends of the milling cutter, and upon the distance lT between the faces of the milling cutter. Two options are feasible for the correlation of the parameters dT , DT , and lT of the milling cutter, and the parameters of the generating curve of the surface T. Following the first option (Figure 5.26a), the configuration of the generating curve with respect to the milling cutter axis of rotation yields the principal min radii of curvature R min 1.T and R 2.T at the milling cutter face of smaller diameter max dT, and the principal radii of curvature R max 1.T and R 2.T were observed at the D milling cutter face of bigger diameter T . Dupin’s indicatrices Dup(Td ) and Dup(TD ) show the distribution of radii of normal curvature of the generating surface T of the milling cutter at the endpoints of the generating curve. In compliance with the second option (Figure 5.26b), the configuration of the generating curve with respect to the milling cutter axis of rotation yields the max principal radii of curvature R min 1.T and R 2.T at the milling cutter face of smaller min diameter dT, and the principal radii of curvature R max 1.T and R 2.T were observed D at the milling cutter face of bigger diameter T . Dupin’s indicatrices Dup(Td ) and Dup(TD ) show the distribution of radii of normal curvature of the generating surface T of the milling cutter at the endpoints of the generating curve. Other combinations of the configuration of the generating curve of the surface T relative to the milling cutter axis of rotation are evident. The smaller diameter dT of the milling cutter is equal to dT = 2 R min 2.T cos ϕ 2; the bigger diameter DT is equal to DT = 2 R max 2.T cos ϕ 1 . Here, ϕ 1 and ϕ 2 designate angles between the axis of rotation of the milling cutter and the tangents to the generating curve at its endpoints. It is convenient to show the angles ϕ 1 and ϕ 2 between the perpendiculars to the mentioned lines. Use of the milling cutter of the discussed design allows an increase of productivity of surface machining, as well as enhanced quality of the machined surface. A similar approach is applicable to the design of finishing tools for reinforcement of sculptured surface using the method of surface plastic deformation.

Pat. No. 1428563, USSR, A Tool for Finishing of Sculptured Surface on Multi-Axis NC Machine./ S.P. Radzevich, Int. Cl. B 24 B 39/00, Filed February 11, 1986.

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Pstr 2

5

Pstr

Pstr

9

8 O

RT.0

RT E

Δ

ωT 8

OT

Paxial

2

7

(a)

Paxial

9

Δ

4

3

Pstr

OT

Pbend

Pbend

M

6

1

3

7

4

(b)

Figure 5.27 A tool for finishing a sculptured surface on a multi-axis numerical control machine (USSR Patent 1.428.563).

The maximal radius of curvature R max of the generating curve of the T generating surface of the tool is observed at point 3. The minimal radius of curvature R min of the generating curve of the generating surface of T the tool is observed at point 4. Two options for the design of the finishing tool are considered: The first option relates to the design of the finishing tool that slides over the part surface (Figure 5.27a), and another option relates to the design of the finishing tool that rolls over the part surface (Figure 5.27b). When finishing a part surface, the tool (Figure 5.27) is performing a following motion. Due to the following motion, at every instant of finishing the desired rate of conformity of the machining surface T of the tool to the part surface P can be ensured. Use of the finishing tool (Figure 5.27) of the discussed design allows for an increase in productivity of surface machining, as well as enhanced quality of the machined surface. Similarly, the problem of profiling a form-cutting tool can be solved for a more general case, say when the correspondence between the parameters ρT and L T is not linear but is given by a certain equation in the form ρT = ρT (L T ).

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5.6

The Form-Cutting Tools Having a Continuously Changeable Generating Surface

The generating surface T of the most-known designs of the form-cutting tools is a rigid surface. No changes to the shape and parameters of the surface T are feasible. However, for the machining of sculptured surfaces on multi-axis NC machines, form-cutting tools of special designs are used as well. Cutting tools of the discussed design have continuously movable blades [15,20,21,30]. The continuous motion of the cutting tool blades is under numerical control. Wooden parts, as well as parts made of plastics, light alloys, and so forth, can be machined with form-cutting tools of the design. An equation of the generating surface T of the form-cutting tools of the design under consideration always contains at least one parameter that is under numerical control. A continuously changeable generating surface T of the form-cutting tool cannot be formed as an envelope to consecutive positions of the part surface P that is performing a motion relative to a coordinate system XT YT ZT of the form-cutting tool. Unknown kinematics of the surface generation is the major reason for this: That same sculptured surface can be machined with that same cutting tool under different kinematics of surface generation. For the development of design of the form-cutting tool having a continuously changeable generating surface T, as well as for determining the optimal kinematics of surface generation, implementation of the R-mapping of surfaces is vital.

5.7

Incorrect Problems in Profiling the Form-Cutting Tools

Use of the methods for derivation of the generating surface of the formcutting tool based on R-mapping of the surfaces and on elements of theory of enveloping surfaces returns an accurate solution to the problem of profiling the form-cutting tool. In the practice of surface machining, no surface is generated precisely. Deviations of the actual part surface from the desired part surface are unavoidable. Tolerances on accuracy of the machined part surface are helpful in resolving this issue. If deviations of the machined part surface are within the tolerance, then the surface P is generated properly. There are no principle restrictions on what portion of the resultant tolerance on accuracy should be allowed for deviations of the actual generating surface from the desired generating surface of the form-cutting tool. Moreover, in particular cases, the precise generating surface of the form-cutting tool cannot be generated, and only an approximation to the desired surface T exists. The approximation of the actual generating surface of the cutting tool to its desired generating surface can even be asymptotically accurate.

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The approximation to the desired generating surface T of the form-cutting tool is referred to as the approximate generating surface of the form-cutting tool, and it is designated Ta. Introduction of the approximate generating surface Ta of the form-cutting tool is necessary because of the following: • Use of approximate methods of profiling form-cutting tools for which deviations of the desired generating surface T from the actual generating surface are unavoidable. • Implementation of approximate working surfaces of the cutting tool by surfaces that are easy to generate. • Impossibility of generating geometrically precise sculptured working surfaces of the form-cutting tool. Deviation of profiling of the form-cutting tool is understood to be the distance between the generating surface T of the form-cutting tool and between the approximate generating surface Ta. This distance is measured along the unit normal vector nT to the surface T in the corresponding surface point. The following geometrical consideration is valid. Evidently, for a given part surface P, a particular surface can be found that can be used as an approximate generating surface Ta. The cutting tool designed on the premises of the surface Ta can be capable of generating part surface P with any given tolerance. Moreover, the tolerance can be asymptotically of zero value. Usually it is convenient to have a generating surface of the cutting tool that is easy to implement. For example, it is often desired that the generating surface possess the property of sliding over itself. Certainly, this is not mandatory. The approach under current consideration is not capable of generating precision generating surfaces T of the cutting tool. However, it is capable of generating the approximate generating surface that is asymptotically accurate. The following two illustrative examples are helpful for understanding the nature of the problem under consideration. In the design of the transmission for low noise/noiseless transmissions for cars and light trucks, helical gears with topologically modified tooth flanks are implemented. In high-volume production, for finishing the topologically modified gears, plunge shaving cutters of special design are used. The desired tooth flank of a topologically modified shaving cutter is a kind of sculptured surface (Figure 5.28a). As an example, consider the generation of the shaving cutter tooth flank in either a grinding or regrinding operation. In a grinding operation, the desired tooth flank of the shaving cutter is generated with a surface of revolution of the grinding wheel (Figure 5.28b). For precision generation of the tooth flank surface, multi-axis relative motion of the grinding wheel is necessary. However, because the actual number of the NC-axis is limited, the shaving cutter tooth flank cannot be generated

The desired clearance surfaces of hob teeth are a perfect example of surfaces of this kind.

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Ta (a) Grinding Wheel

ωgw 1

2

3

4

Shaving Cutter (b) Figure 5.28 Generation of the approximate generating surface T a of the shaving cutter for plunge shaving of topologically modified involute gears.

in compliance with the blueprint. Deviations of the surface generation in this case are unavoidable. Therefore, the generating surface T of the plunge shaving cutter cannot be generated in a shaving cutter grinding operation, but the approximate generating surface Ta is generated instead. Consider the generation of the surface of an internal circular cone, say generation of the rake surface of the broach for machining a hole (Figure 5.29). For this purpose, the approximate generation surface Ta of the grinding wheel can be used. Usually, grinding wheels having the generating surface T that is shaped in the form of an external circular cylinder are used. Design parameters of the grinding wheel, as well as its setup parameters are determined in accordance with accurate generation of the rake surface of the broach to be ground.

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T

ωgw

Ta

P

G Ogw

F D

D

E

C E2

B

A

E1

A

B

E1

OP Figure 5.29 A particular case of grinding of a broach with a form-grinding wheel (an example of surface generation when the surface characteristic splits on two parts).

The generating straight line AF of the grinding wheel aligns with the generating straight line AD of the rake face of the broach. Straight-line segment AD is the surface characteristic in this case. The necessity of satisfaction of the conditions of proper part surface generation (see Chapter 7 for details) imposes strict constraints on the maximal diameter of the grinding wheel. Broaches can be reground with the form-grinding wheels of an appropriate design. The form-grinding wheel rotates about its axis OT with a certain rotation ω T. The generating curve AC is used as the generating curve of the approximate generating surface Ta of the grinding wheel (Figure 5.29). The grinding wheel surface Ta and the rake surface of the broach become tangent along the characteristic curve BAC. The portion BA of this curve is the surface characteristic E1 , and the portion AC serves as the surface characteristic E2 . The approximate generating surface Ta of the form-grinding wheel is represented by a surface of revolution that is tangent to the rake surface of the broach along two symmetrically located surface characteristics E1 and E2. Parameters of the curvilinear generating curve AC satisfy the conditions under which deviations of the actually generated part surface P from the desired shape are within the tolerance on accuracy of the part surface P. Definitely, the number of examples can be extended. It is of interest to mention here about the problems of profiling gear-cutting tools having zero pressure angle, including rack cutters, hobs [27], and shaper cutters** [13]. In particular cases, the approximate generating surface of the cutting tool can be degenerated to a curve, say into a generating curve of the cutting tool. In cases like the last one, the surface generation is referred to as the edge-type surface generation. ** Pat. No. 1174187, USSR, A Gear Shaper Cutter./S.P. Radzevich, Int. Cl. B23f 21/10, Filed April 23, 1981.

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The profiling of skiving gear-cutting tools, of multistart disk-type hobs, and others are also strictly related to the incorrect problems of profiling form cutting tools. One more example of the problem under consideration is in the field of profiling form-cutting tools having partially completed motion relative to the part. In this case, two portions of the generating surface T are distinguished: The first portion is the ordinary enveloping surface (generating method), and the second portion is a “replica” of the part surface P — namely, TP. Two portions of the surface T are somehow conjugate to each other. The problem can be formulated as follows: how should the generating surface T that is “reduced” be compared to the conventional enveloping surface? The above-mentioned problems should be investigated in more detail.

5.8

Intermediate Conclusion

The R-mapping-based method of surface generation allows for the determination of the generating surface of the form-cutting tool in terms of the sculptured surface to be machined. Therefore, shape and parameters of the generating surface T can be expressed in terms of shape and parameters of the part surface P being machined. This statement can be expressed analytically in the form of function T = T(P). This means that the proposed R-mapping-based method of surface generation allows for the expression of the generating surface T of the form-cutting tool in terms of sculputered surface P to be machined. Designate the kinematics of relation motion of the surfaces P and T as K rm . Generally, the optimal kinematics of surface generation can be expressed as a function K rm = K rm [ P , T( P)] , and ultimately as a function K rm = K rm ( P) . After the generating surface T of the form-cutting tool is determined, and further, the kinematics of the surface generationg K rm is determined, problems of other kinds must be solved. In nature, problems of these kinds are inverse problems of surface generation. In order to briefly formulate problems of this sort, assume that the actual generated part surface is designated as Pa . Then the problems to be solved can be formulated as follows: ( a) To determine the function Pa = Pa (T , K rm ), taking into account deviations of the kinematics of surface generation, when the actual kine( a) matics of surface generation K rm differ from the desired kinematics ( a) of surface generation K rm , say when the inequality K rm ≠ K rm is observed. To determine the function Pa = Pa (Ta , K rm ), taking into account deviations of profiling the form-cutting tool, say when the inequality Ta ≠ T is observed.

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(

215

)

( a) To determine the function Pa = Pa Ta , K rm , taking into account deviations of the actual kinematics of surface generation, and deviations of ( a) profiling the form-cutting tool, say when the inequalities K rm ≠ K rm and Ta ≠ T are observed.

Another group of problems to be solved relates to derivation of the approximate generating surface Ta of the form-cutting tool. These problems can be formulated as follows: Parameters of the nominal part surface P are not given, but the parameters of the approximated part surface Pa are known instead, say when it is necessary to determine the function Ta = Ta ( Pa , K rm ). ( a) The actual kinematics of surface generation K rm differs from the nominal kinematics of the surface generation K rm , say when the function ( a) Ta = Ta ( P , K rm ) must be determined. The nominal part surface P and the nominal kinematics of surface generation K rm are not known, but the actual part surface Pa and the ( a) actual kinematics K rm are given. In this case, it is necessary to determine the function T = T P , K ( a) . a a a rm

(

)

The problems listed above can be associated with the inverse problem of surface generation. However, the nature of each of these problems is completely different [19].

References [1] Amirouch, F.M.L., Computer-Aided Design and Manufacturing, Englewood Cliffs, NJ, Prentice Hall, 1993. [2] Ball, R.S., A Treatise on the Theory of Screws, Cambridge University Press, Cambridge, 1990. [3] Ball, R.S., Treatise on the Theory of Screws: A Study in the Dynamics of Rigid Body, Hodges & Foster, Dublin, 1998. [4] Chang, C.H., and Melkanoff, M.A., NC Machining Programming and Software Design, Prentice Hall, Englewood Cliffs, NJ, 1989. [5] Choi, B.K., and Jerard, R.B., Sculptured Surface Machining. Theory and Application, Kluwer Academic, Dordrecht/Boston/London, 1998. [6] Cormac, P., A Treatise on Screws and Worm Gear, Their Mills and Hobs, Chapman & Hall, London, 1936. [7] doCarmo, M.P., Differential Geometry of Curves and Surfaces, Prentice Hall, Englewood Cliffs, NJ, 1976. [8] Improvement of Gear Cutting Tools, Collected papers, NIIMASh, Moscow, 1989. [9] Jeffreys, H., Cartesian Tensors, Cambridge University Press, Cambridge, 1961. [10] L’ukshin, V.S., Theory of Screw Surfaces in Cutting Tool Design, Mashinostroyeniye, Moscow, 1968. [11] Marciniak, K., Geometric Modeling for Numerically Controlled Machining, Oxford University Press, New York, 1991.

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[12] Olivier, T., Theorie Geometrique des Engrenages, Paris, 1842. [13] Pat. No. 1174187, USSR, A Gear Shaper Cutter./S.P. Radzevich, Int. Cl. B23f 21/10, Filed April 23, 1981. [14] Pat. No. 1449246, USSR, A Method of Experimental Simulation of Machining of a Sculptured Surface on Multi-Axis NC Machine./S.P. Radzevich, Int. Cl. B 23 C, 3/16, Filed February 17, 1987. [15] Radzevich, S.P., Advanced Technological Processes of Sculptured Surface Machining, VNIITEMR, Moscow, 1988. [16] Pat. No. 4242296/08, USSR, A Method for Designing of the Optimal FormCutting-Tool for Machining of a Given Sculptured Surface on Multi-Axis NC Machine./S.P. Radzevich, Filed March 31, 1987. [17] Radzevich, S.P., A Novel Method for Mathematical Modeling of a Form-CuttingTool of the Optimum Design, Applied Mathematical Modeling, 31 (12), 2639–2654, December 2007. [18] Radzevich, S.P., Differential-Geometric Method of Surface Generation, DrSci thesis, Tula Polytechnic Institute, Tula, Russia, 1991. [19] Radzevich, S.P., Fundamentals of Surface Generation, Rastan, Kiev, 2001. Copy of the monograph is available from the Library of Congress. [20] Radzevich, S.P., Methods of Milling of Sculptured Surfaces, VNIITEMR, Moscow, 1989. [21] Radzevich, S.P., New Achievements in the Field of Sculptured Surface Machining on Multi-Axis NC Machine, VNIITEMR, Moscow, 1987. [22] Radzevich, S.P., Profiling of the Form Cutting Tools for Machining of Sculptured Surface on Multi-Axis NC Machine, Stanki I Instrument, 7, 10–12, 1989. [23] Radzevich, S.P., Profiling of the Form Cutting Tools for Sculptured Surface Machining on Multi-Axis NC Machine. In Proceedings of the Conference: Advanced Designs of Cutting Tools for Agile Production and Robotic Complexes, MDNTP, Moscow, 1987, pp. 53–57. [24] Radzevich, S.P., R-Mapping Based Method for Designing of Form Cutting Tool for Sculptured Surface Machining, Mathematical and Computer Modeling, 36 (7– 8), 921–938, 2002. [25] Radzevich, S.P., Sculptured Surface Machining on Multi-Axis NC Machine, Vishcha Shkola, Kiev, 1991. Copy of the monograph is available from the Library of Congress. [26] Radzevich, S.P., and Goodman, E.D., Efficiency of Multi-Axis NC Machining of Sculpted Part Surfaces. In G.J. Olling, B.K. Choi, and R.B. Jerard (Eds.), SSM ’98: Proceedings of the IFIP International Conference on Sculptured Surface Machining, Kluwer, Deventer, The Netherlands, 1998, pp. 42–58. [27] Radzevich, S.P., and Palaguta, V.A., Advances in the Field of Finishing of Cylindrical Gears, VNIITEMR, Moscow, 1988. [28] Rodin, P.R., Fundamentals of Theory of Cutting Tool Design, Mashgiz, Kiev, 1960. [29] Rodin, P.R., Issues of Theory of Cutting Tool Design, PhD dissertation, Odessa Polytechnic Institute, Odessa, Ukraine, 1961. [30] Rodin, P.R., Linkin, G.A., and Tatarenko, G.A., Machining of Sculptured Surfaces on NC Machines, Technica, Kiev, 1976. [31] Shishkov, V.A., Generation of Surfaces Using Continuously Indexing Method, Mashgiz, Moscow, 1951. [32] Struik, D.J., Lectures on Classical Differential Geometry, 2nd ed., Addison-Wesley, Reading, MA, 1961.

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6 The Geometry of the Active Part of a Cutting Tool Generally speaking, the active part of a cutting tool appears as a wedge. In a machining operation, the cutting wedge is properly oriented with respect to the surface of the cut, as well as with respect to the direction of the speed of the resultant motion of the cutting wedge relative to the surface of the cut. The geometry of the cutting wedge strongly affects the performance of the cutting tool. It is critically important from the standpoint of achieving the required surface finish, the productivity of the machining operation, and so forth. Geometry of the active part of cutting tools has remained under careful investigation by researchers since 1870 [22] or even earlier. But many questions in this concern have no appropriate answers yet. For the specification of the geometry of the active part of cutting tools, the following geometric parameters are commonly used: (a) the rake angle g, (b) the clearance angle a, (c) the angle of inclination l, (d) the major cutting edge approach angle ϕ 1 , (e) the minor cutting tool approach angle ϕ 2 , (f) the cutting edge roundness r, and (g) the radius of curvature of a cutting edge rT. In particular cases, some other geometric parameters of the active part of cutting tools are used as well. Prior to designing a high-performance cutting tool, it is necessary to know the optimal (or the desired) values of the geometric parameters of the cutting edge. For the specification of the geometric parameters of the active part of cutting tools, corresponding reference systems are used. In the production of cutting tools, for inspection and other purposes, stationary reference systems are widely implemented. Reference systems of this sort are often referred to as the static reference systems. They are associated with the reference surface of the cutting tool. The tool-in-hand reference system is a good example of static reference systems. This reference system is recommended for application by the International Standard ISO 3002. The tool-in-machine system (or setting system) is another perfect example of a static reference system [4]. The last reference system incorporates the deviation of the actual configuration of the cutting tool in the machine tool with respect to its ideal configuration. Geometric parameters of the active part of cutting tools that are measured in a static reference system specify the static geometry of the active part of the cutting tool. For the purposes of analysis and optimization of the material removal process, moving reference systems are used. Reference systems of this sort are 217 © 2008 by Taylor & Francis Group, LLC

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often referred to as kinematic reference systems. They are associated with the surface of the cut and with the direction of the relative motion of the cutting wedge with respect to the surface of the cut. The tool-in-use reference system is a good example of a kinematic reference system. This reference system is also recommended for application by the International Standard ISO 3002. Geometric parameters of the active part of a cutting tool which are measured in a kinematic reference system specify the kinematic geometry of the active part of the cutting tool. Actual values of the corresponding geometric parameters of the active part of a cutting tool in static and in kinematic reference systems differ. However, once the geometry of the cutting edge is determined in a certain reference system, it can be converted to another reference system. Analytical methods for the conversion are considered below.

6.1 Transformation of the Body Bounded by the Generating Surface T into the Cutting Tool It was earlier determined that the generating surface T of the cutting tool (see Chapter 5) bounds the body made of tool material, say of high-speed-steel. This body is referred to as the generating body of the cutting tool. The cutting tool designer uses this body to design a workable cutting tool. In order to proceed with further analysis, it is necessary to distinguish two major functions of the cutting tool. The first major function of the cutting tool is to remove stock from the work. This function can be performed either by roughing cutting edges or by finishing cutting edges of the cutting tool. The major purpose of roughing cutting edges is to remove the stock. Roughing cutting edges of the cutting tool are located either within the generating body of the cutting tool or within the generating surface of the cutting tool. Roughing cutting edges do not generate the part surface P. Therefore, the roughing cutting edges can be constructed as a line of intersection of just two surfaces, say of the rake surface and of the clearance surface of the cutting tool. Both of these surfaces can be of a shape that is convenient for manufacturing the form-edge cutting tool. The second major function of the cutting tool is to generate the part surface. This function is performed only by finishing cutting edges of the cutting tool. Finishing cutting edges of the cutting tool are located within the generated surface T of the tool, and they cannot be located within the generating body of the cutting tool. Therefore, the finishing cutting edges can be constructed as a line of intersection of three surfaces, say of the generating surface T to the cutting tool, of the rake surface, and of the clearance surface of the cutting tool. The major purpose of finishing (clean-up) cutting edges is to generate the surface P. It is important to stress here that different cutting edges of the cutting tool (or their different segments) in different instants of time can perform

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different functions. An involute gear hob is a perfect example in this concern. Cutting teeth at the entering end of the hob just remove stock. They do not generate the teeth surface of the gear being machined. The hob teeth that are not far from the center distance “the hob—the work” remove the stock and generate the gear teeth surface. Ultimately, the hob teeth that are beyond the center distance remove almost no stock. These teeth mostly generate the gear teeth surface. More detailed analysis reveals that different segments of that same cutting edge can simultaneously serve as the roughing cutting edges, as well as the finishing cutting edges. The consideration below is limited to the analysis of only two major functions of the cutting tool. Other important functions of the cutting tool, say chip evacuation from the area of cutting, chip curling, chip braking and its transportation from the area of cutting, coolant supply, and others are not considered. In order to design a workable cutting tool on the premises of the generating body of the cutting tool, the last must be capable of removing the stock from the work. Three methods for transforming the generating body of the cutting tool into the workable edge cutting tool are considered below. All target making three surfaces: (a) the generating surface of the form-cutting tool, (b) the cutting tool rake surface, and (c) the cutting tool clearance surface passing through a common line. The line of intersections of three surfaces is referred to as the cutting edge of the form-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

Consider a scenario under which the generating surface of the cutting tool is already determined (see Chapter 5). The cutting tool rake surface is chosen within the surfaces that are convenient for machining the surface when manufacturing the form cutting tool, for inspection purposes, and so forth. In most cases, the rake surface Rs is within the surfaces that allow for sliding over themselves (see Section 2.4). The rake surface Rs is properly oriented with respect to the generating surface T of the form-cutting tool. It makes an optimal rake angle with respect to the perpendicular to the surface T at a given point. Following the first method for the transformation of the generating body of the cutting tool into the workable-edge cutting tool, cutting edges of the form-cutting tool are defined as the line of intersection of the generating surface T of the cutting tool by the rake surface R s . Once the cutting edge is constructed, then the clearance surface Cs can be constructed in compliance with the following routing. First, the clearance surface is selected within the surfaces that allow for sliding over themselves (see Section 2.4). This requirement is highly desirable, but it is not mandatory. Actually, any surface having reasonable geometry could serve as the clearance surface of the form-cutting tool. Second, parameters of the chosen surface Cs must be computed in compliance with the requirement under

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which the surface Cs passes through the computed cutting edge of the formcutting tool. Third, the configuration of the defined clearance surface Cs of the form-cutting tool must be of the sort for which the surface Cs makes the optimal clearance angle a with the generating surface T of the form-cutting tool at a given point of the cutting edge. It is easy to see that clearance surfaces of the cutting tool cannot always be shaped in the form that is convenient for manufacturing the cutting tool. The cutting edge of a precision form-cutting tool can be considered as a line of intersection of the three surfaces, say of the generating surface T of the cutting tool, of the rake surface R s , and of the clearance surface Cs . This requirement is compliant with three surfaces T, R s , and Cs being the surfaces through the common line, say through the cutting edge of the cutting tool, that could impose strong constraints on the actual shape of the clearance surface of the cutting tool. Under such restrictions, the clearance surface Cs usually cannot allow sliding over itself. However, the desired surface Cs can be approximated by a surface that allows sliding over itself; thus, the approximation could be more convenient for design and manufacture of the form-cutting tool. This means that in certain cases of implementation of the first method, approximation of the desired clearance surface Cs with a surface that features another geometry can be unavoidable. The approximation of the desired surface Cs results in the surface P being generated not with the precise surface T, but with an approximated surface T g of the cutting tool. The approximated surface T g deviates from the desired surface T. The deviation δ T is measuring along the unit normal vector n T to the surface T at a corresponding surface point. Application of the form-cutting tool having approximated the generated surface is allowed if and only if the resultant deviation δ T is within the corresponding tolerance [δ T ] — that is, when the inequality δ T ≤ [δ T ] is valid. Summarizing, one can come up with the following generalized procedure for designing the form-cutting tool in compliance with the first method: 1. Determination of the generating surface T of the form-cutting tool (see Chapter 5). 2. Determination of the rake surface R s: The rake surface is selected within surfaces that are technologically convenient (a kind of reasonably practical surface). Configuration of the rake surface is specified by the rake angle of the desired value. 3. Determination of the cutting edge: The cutting edge is represented with the line of intersection of the generating surface T of the formcutting tool by the rake surface R s . 4. Construction of the clearance surface Cs that passes through the cutting edge and makes the clearance angle of the desired value with the surface of the cut. The clearance surface is selected within surfaces that are technologically convenient (a kind of reasonably practical surface).

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P1

bp ac

αc

cp

Oc

ap

hc

Op bc T2

cc

αc

P2

Cs.2 cc

cp

bc

Cs.1

bp

ac T1

γc

P1

ap Oc

Op Figure 6.1 The concept of the first method for the transformation of the generating body of the formcutting tool into the workable edge cutting tool.

For the practicality, a normal cross-section of the clearance surface must be determined as well. Figure 6.1 illustrates an example of implementation of the first method for the transformation of the generating body of the form-cutting tool into the workable-edge cutting tool. For illustrative purposes, the round form cutter for external turning of the part is chosen. In the case under consideration, the part surface P is represented by two separate portions P1 and P2 . An axial profile of the part is specified by the composite line through points ap, bp , and c p . For the particular case shown in Figure 6.1, the generating surface T of the form cutter is congruent with the part surface P being machined. This statement easily follows from the consideration that is based on the analysis of kinematics of the machining operation (see Chapter 2). Thus, the identity T ≡ P is observed (to be more exact, two identities T1 ≡ P1 and T2 ≡ P2 are valid). Ultimately, the generating surface T of the form cutter is also represented with two portions T1 and T2.

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Due to the identity T ≡ P observed, the axial profile of the generating surface T of the form cutter is composed of two segments through the points aT , bT , and cT (not labeled in Figure 6.1); and the identities aT ≡ ap, bT ≡ bp, and cT ≡ cp are valid. Then, a plane is chosen as the rake surface R s of the round form cutter. Definitely, the plane allows for sliding over itself. It is convenient to machine the plane in cutting tool production. The plane is parallel to the axis of rotation OT ≡ Op of the generating surface T of the form cutter. It makes the rake angle γ c perpendicular to the surface T at the base point aT ≡ ac . The piecewise line of intersection ac bc cc of the generating surface T of the round form cutter by the rake surface R s serves as the cutting edge of the form cutter. The clearance surface Cs of the form cutter is shaped in the form of a surface of revolution. All surfaces of revolution allow for sliding over themselves. The surface of revolution Cs is represented with two separate portions Cs.1 and Cs.2 . The clearance surface of the round form cutter can be generated as a series of consecutive positions of the cutting edge ac bc cc when rotating the cutting edge ac bc cc about the surface Cs axis of rotation Oc . For practical needs, the axial profile of the clearance surface Cs must be determined. The considered example (Figure 6.1) illustrates implementation of the first method for the transformation of the generating body of the cutting tool into the workable edge cutting tool. This method has been known for many decades. The first method for the transformation of the generating body of the cutting tool into the workable edge cutting tool is widely used in many industries. Form edge cutting tools of most designs can be designed in compliance with the first method. When the first method is employed, this yields designing of the form-cutting tools that are convenient in manufacturing and in application. The major disadvantages of the first method are twofold. First, in most cases of application of the first method, no optimal values of the geometrical parameters of the form-cutting tool at every point of the cutting edge can be ensured. Optimization of the geometrical parameters at every point of the cutting edge is a challenging problem. The solution to the problem of optimization of the geometrical parameters of the cutting edge (if any) is often far from practical needs: It could be feasible, but it is often not practical. Second, unavoidable deviations of the actual approximated generated surface of the cutting tool from its desired shape often cannot be eliminated when the first method is employed to the design of the form-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 Consider another scenario under which the generating surface of the cutting tool is also determined (see Chapter 5). The cutting tool clearance surface is chosen within the surfaces that are convenient for machining of the surface

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when manufacturing the form-cutting tool, for inspection purposes, and so forth. In most cases, the clearance surface Cs is within the surfaces that allow for sliding over themselves (see Section 2.4). The clearance surface Cs is properly oriented with respect to the generating surface T of the form-cutting tool. It makes an optimal clearance angle with the surface T at a given point. Following the second method for the transformation of the generating body of the cutting tool into the workable-edge cutting tool, cutting edges of the form-cutting tool are defined as the line of intersection of the generating surface T of the cutting tool by the clearance surface Cs. Once the cutting edge is constructed, then the rake surface Rs can be constructed in compliance with the following routing. First, the rake surface is selected within the surfaces that allow for sliding over themselves (see Section 2.4). This requirement is highly desirable but not mandatory. Actually, any surface having reasonable geometry could serve as the rake surface of the form-cutting tool. Second, parameters of the chosen surface Rs must be computed in compliance with the requirement under which the surface Rs passes through the computed cutting edge of the form-cutting tool. Third, the configuration of the defined rake surface Rs of the form-cutting tool must be of the sort for which the surface Rs makes the optimal rake angle γ with respect to the perpendicular to the generating surface T of the form-cutting tool at a given point of the cutting edge. It is easy to understand that rake surfaces of a cutting tool cannot always be shaped in the form that is convenient for manufacturing the cutting tool. The cutting edge of a precision form-cutting tool can be considered as a line of intersection of three surfaces: of the generating surface T of the cutting tool, of the rake surface R s , and of the clearance surface Cs. The requirement that three surfaces T, R s , and Cs be the surfaces through the common line, say through the cutting edge of the cutting tool, could impose strong constraints on the actual shape of the clearance surface of the cutting tool. Under such restrictions, the rake surface R s usually cannot allow sliding over itself. However, the desired surface R s can be approximated by a surface that allows for sliding over itself; thus, the approximation could be more convenient for the design and manufacture of the form-cutting tool. This means that in certain cases of implementation of the second method, approximation of the desired rake surface R s with a surface featuring another geometry can be unavoidable. The approximation of the desired surface R s results in that ultimately the surface P is generated not with the precise surface T, but with an approximated surface T g of the cutting tool. The approximated surface T g deviates from the desired surface T. The deviation δ T is measuring along the unit normal vector n T to the surface T at a corresponding surface point. Application of the form-cutting tool having approximated the generated surface is allowed if and only if the resultant deviation δ T is within the corresponding tolerance [δ T ] — that is, when the inequality δ T ≤ [δ T ] is valid.

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Summarizing, one can come up with the following generalized procedure for designing of the form cutting tool in compliance with the second method: 1. Determination of the generating surface T of the form-cutting tool (see Chapter 5). 2. Determination of the clearance surface Cs: The clearance surface is selected within surfaces that are technologically convenient (a kind of reasonably practical surface). Configuration of the clearance surface is specified by the clearance angle of the desired value. 3. Determination of the cutting edge: The cutting edge is represented with the line of intersection of the generating surface T of the formcutting tool by the clearance surface Cs. 4. Construction of the rake surface R s that passes through the cutting edge and makes the rake angle of the desired value perpendicular to the surface of the cut. The rake surface is selected within surfaces that are technologically convenient (a kind of reasonably practical surface). For practicality, a typical cross-section of the clearance surface must be determined as well. Figure 6.2 illustrates an example of implementation of the second method for the transformation of the generating body of the form-cutting tool into the workable edge cutting tool. For illustrative purposes, the form milling cutter for machining helical grooves is chosen. Consider that the generating surface T of the cutting tool is already determined. Geometry of the chosen clearance surface Cs is predetermined by The Cutting Edge

The Cutting Edge Cs

T

Oc

Rs

Oc

Figure 6.2 The concept of the second method for the transformation of the generating body of the formcutting tool into the workable edge cutting tool.

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kinematics of the operation of relieving the milling cutter teeth. The cutting edge is represented as the line of intersection of the generating surface T of the milling cutter by the clearance surface Cs. Further, the constructed cutting edge is used for the generation of the rake face Rs. For this purpose, either the cutting edge or its projection onto the transverse plane moves along the milling cutter axis of rotation Oc . In the case under consideration, the face surface Rs is represented as the locus of consecutive positions of the cutting edge in its motion along the axis Oc . The rake surface Rs is shaped in the form of a general cylinder. The considered example (Figure 6.2) illustrates implementation of the second method for the transformation of the generating body of the cutting tool into the workable edge cutting tool. This method is not as widely used in industry as is the first method. The second method for the transformation of the generating body of the cutting tool into the workable edge cutting tool does not have wide implementation in industry. Form edge cutting tools of most designs can be designed in compliance with the second method. When the second method is employed, this yields designing of the form-cutting tools that are convenient in manufacturing and in application. The major disadvantages of the second method are twofold: First, in most cases of implementation of the second method, no optimal values of the geometrical parameters of the form-cutting tool at every point of the cutting edge can be ensured. Optimization of the geometrical parameters at every point of the cutting edge is a challenging problem. The solution to the problem of optimization of the geometrical parameters of the cutting edge (if any) is often far from practical: It could be feasible, but it is often not practical. Second, unavoidable deviations of the actual approximated generated surface of the cutting tool from its desired shape often cannot be eliminated when the first method is employed to design of the form-cutting tool. It is important to stress that both methods for the transformation of the generating body of the cutting tool into the workable edge cutting tool feature a common disadvantage. This disadvantage results in the incapability of designing a form-cutting tool that has optimal value of the angle of inclination l. The actual value of the angle l at a current point within the cutting edge is a function of shape, of parameters, and of location of the rake R s or the clearance Cs surfaces of the form-cutting tool with respect to the generating surface T of the form-cutting tool. Due to this, the optimal values λ opt of angle of inclination of the cutting tool edge become impractical due to significant difficulties in manufacturing the form-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 Ultimately, consider the third scenario under which the generating surface of the cutting tool is also determined (see Chapter 5). However, in this case, neither the rake surface nor the clearance surface of a desired geometry is

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The Cutting Edge

λоpt

VΣ

M ce The Trajectory

Figure 6.3 The concept of the third method for the transformation of the generating body of the formcutting tool into the workable edge cutting tool.

selected at the beginning, but the cutting edge is selected instead. Such an approach allows for optimization of the angle of inclination at every point of the cutting edge of the form-cutting tool. The method considered below is proposed by Radzevich [14,15,19]. For implementation of the third method, it is necessary to construct a special family of lines within the generating surface of the form-cutting tool. Lines of this family of lines represent the assumed trajectories of motion of the cutting edge points over the surface of the cut when the work is machining (Figure 6.3). Below this family of lines within the generating surface T of the cutting tool is referred to as the primary family of lines. Analysis of a particular machining operation allows for analytical representation of the family of lines within the surface T. Further, after the primary family of lines is defined, it is necessary to construct a secondary family of lines. The secondary family of lines is also within the generating surface T, and it is isogonal to the primary family of lines. At every point of intersection of the lines of the primary and of the secondary families, the angle between the lines is equal to (90° - λ opt ) . Here, λ opt designates the optimal value of the angle of inclination of the cutting edge. Therefore, the angle of inclination is at its optimal value at every point of the cutting edge. This is due to the primary family of lines within the generating surface T of the tool being isogonal to the secondary family of lines at every point of the cutting edge. Different segments of the cutting edge of a form-cutting tool are at different distances from the axis of the tool rotation. Because of this, they work with different cutting speeds. This results in the optimal value of the angle of inclination being different for different portions of the cutting tool edge. Under such

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a scenario, the actual value of the angle of inclination could be either constant within the cutting edge (and thus equal to its average value), or the desired variation of the angle of inclination can be ensured. In the last case, the problem of design of a form-cutting tool becomes more sophisticated. An appropriate number of lines from the second family of lines can be selected to serve as the cutting edges of the form-cutting tool to be designed. These lines are uniformly distributed and are at a certain distance t from one another. The distance t is equal to the tooth pitch of the form-cutting tool. Rake surface R s is a surface through the cutting edge of the form-cutting tool. The surface R s makes the rake angle g with the perpendicular n c to the surface of cut. Actually the perpendicular to the surface of cut deviates from the perpendicular nT to the generating surface T of the form cutting tool. Fortunately, this deviation is of negligibly small value. For practical needs of design of the form-cutting tool, the perpendicular to the surface of cut n c is not used, but the corresponding perpendicular nT to the generating surface T is used instead. The clearance surface Cs is also a surface through the cutting edge of the form-cutting tool. The surface Cs makes the clearance angle α with the generating surface T. It is possible to formulate the problem of design of the form-cutting tool in the way following which the rake angle g, as well as the clearance angle a, could be of optimal value at every point of the cutting edge of the formcutting tool. In order to satisfy this requirement, both the rake surface R s and the clearance surface Cs must be of special geometry. This problem could be solved analytically. When deriving equations of the surfaces R s and Cs , it is necessary to ensure optimal values γ opt , α opt , and λ opt for the parameters g, a, and l for the new form-cutting tool, as well as for the cutting tool after it is reground. The optimal values γ opt , α opt , and λ opt for the new form-cutting tool and for the reground cutting tool are not necessarily the same. Summarizing, one can come up with the following generalized procedure for designing the form-cutting tool in compliance with the third method: 1. Determination of the generating surface T of the form-cutting tool (See Chapter 5). 2. Determination of the cutting edge: The cutting edge is at the angle of inclination of an optimal value with respect to the direction of speed of the resultant motion of the cutting edge point relative to the surface of the cut. 3. Construction of the rake surface R s , and the clearance surface Cs simultaneously: The rake surface passes through the cutting edge and makes the rake angle of the desired value perpendicular to the surface of the cut. The clearance surface also passes through the cutting edge and makes the clearance angle of the desired value with the generating surface of the cutting tool. Both the

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Kinematic Geometry of Surface Machining rake surface and the clearance surface are selected based on their technological convenience and property of sliding over themselves (a kind of reasonably practical surface). See Section 2.4 for more detail.

The third method for the transformation of the generating body of the cutting tool into the workable edge cutting tool is a completely novel method [14,15,19]. It has not yet been comprehensively investigated. Therefore, more detailed explanation of the method is important. Consider the generating surface T that is shaped in the form of a surface of revolution. This assumption is practical, because, for example, milling cutters of all designs have the generating surface T in the form of a surface of revolution. Using the third method, it is easy to come up with an understanding that the cutting edge of milling cutters of all designs must be shaped in the form of loxodroma. By definition, loxodroma is a line that makes equal angles with a given family of lines on a surface. Actually, loxodroma can be easily defined with respect to coordinate lines on the surface [2]. In the case under consideration, loxodroma having special shape parameters is of particular interest. The loxodroma that makes the angle (90° - λ opt ) with the primary family of lines on the generating surface T can be employed as the cutting edge of the form-cutting tool. In a particular case, when parameterization of the generating surface T of the form-cutting tool yields the expression

φ 1.T ⇒ d ST2 = dUT2 + G T (U T ) dVT2

(6.1)

for the first fundamental form φ1.T , then the cutting edge having optimal value of the angle of inclination λ opt at every point can be described by the following equation: UT

VT cot λ opt = ±

∫

U 0. T

dU T G T (U T )

(6.2)

Equation (6.2) of the cutting edge is expressed in terms of UT and VT parameters of the generating surface T of the form-cutting tool. Using conventional mathematical methods, Equation (6.2) can be converted to a Cartesian coordinates. Example 6.1 Consider a ball-nose milling cutter of radius rT (Figure 6.4). The milling cutter is used for machining a sculptured surface on a multi-axis numerical control

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The Geometry of the Active Part of a Cutting Tool T

ZT (90° − λоpt) ce

M

rT θ XT

B M

YT

XT A

ζ YT

(a)

(b)

Figure 6.4 A ball-nose milling cutter having the optimal value of the angle of inclination λ opt at every point of the cutting edge.

(NC) machine. It is necessary to derive an equation of the cutting edge at every point of which the angle of inclination is equal to its optimal value λ opt . In the Cartesian coordinate system X T YT Z T associated with the milling cutter (Figure 6.4), the position vector of a point of the generating surface T of the milling cutter can be represented in matrix form:

rT sin ϕ cos θ   r sin ϕ sin θ  T  rT (ϕ , θ ) =   rT cos ϕ    1  

(6.3)

Under such a parameterization, the cutting edge can be interpreted as a line on the generating surface T of the milling cutter, which intersects the meridians at the angle (90° - λ opt ). A parametric equation of the form θ = θ (ϕ ) describes a line on the surface. When q = 0°, then the lines θ = θ (ϕ ) represent meridians on the surface T (see Equation 6.3). From another viewpoint, when q = -90°, then the lines θ = θ (ϕ ) represent parallels. The unit tangent vector c e to the curve θ = θ (ϕ ) on the surface T makes the angles α x, β y, and γ z with axes of the coordinate system X T YT Z T . The equation

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for the unit tangent vector c e (ϕ , θ ) can be derived from Equation (6.3):

c e (ϕ , θ ) =

d rT d ST

=

 cos ϕ cos θ dϕ - sin ϕ sin θ dθ  cos ϕ sin θ dϕ + sin ϕ cos θ dθ  1  ⋅  - sin ϕ dϕ dϕ 2 + sin 2 ϕ dθ 2    1  

(6.4)

where d S T denotes the differential of the arc segment of the cutting edge. Particularly, when θ = θ c = Const , then Equation (6.4) for the unit tangent vector c e reduces to cos ϕ cos θ c   cos ϕ sin θ  c c e (ϕ , θ c ) =   - sin ϕ    1  

(6.5)

Under the imposed constraint θ = θ c , the following equality is valid: dϕ dϕ + sin 2 ϕ dθ 2 2

(6.6)

= cos ℘

Equation (6.6) immediately yields dϕ = ± cot ℘ dθ sin ϕ

(6.7)

where ℘ designates a certain angle. After integration of Equation (6.7) is accomplished, one can come up with the solution tan

ϕ = e q(θ +C ) 2

(6.8)

where q = ± cot ℘ and C is an arbitrary constant value. Implementation of the trivial trigonometric formulae

ϕ ϕ 1 - tan 2 2 2 sin ϕ = , cos ϕ = ϕ ϕ 1 + tan 2 1 + tan 2 2 2 2 tan

(6.9)

yields an intermediate result sin ϕ =

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1 , cos ϕ = th q (θ + C) ch q (θ + C)

(6.10)

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Ultimately, under the assumption θ c = λ opt , the above analysis yields an equation for position vector r ce of a point of the cutting edge of the ball-nose milling cutter (Figure 6.4a):  rT cos λ opt   ch q (λ + C)  opt    rT sin λ opt   r ce =   ch q (λ opt + C)    rT th q (λ opt + C)   1

(6.11)

The angle of inclination for the milling cutter having a cutting edge that is shaped in compliance with Equation (6.11) is constant. At every point of the cutting edge, it is equal to its optimal value λ opt . The entire loxodroma (see Equation 6.11) is not used for design of the cutting edge of the ball-nose milling cutter. Only the arc segment AB is used for this purpose (Figure 6.4b). Other methods for derivation of Equation (6.11) can be implemented as well [14,15,19]. Consider another approach for derivation of an analytical description of the cutting edge of the ball-nose milling cutter. In this particular case, when parameterization of the equation of the generating surface T of the milling cutter yields the expression

φ1.T ⇒ dS2T = rT2 ( dUT2 + cos 2 UT dVT2 )

(6.12)

for the first fundamental form φ1.T , then the cutting edge having optimal value of the angle of inclination λ opt at every point can be described by the following equation:

π U  VT cot λ opt = R T ln tan  + T   4 2rT 

(6.13)

It is important to focus on the shape of the loxodroma. The loxodroma makes an infinite number of revolutions about its pole. It approaches the pole infinitely close. This curve approaches the pole similar to an asymptotic point. The last can cause some inconveniences while manufacturing cutting tools. However, several methods are developed for avoiding the inconveniences [15]. Feasibility of the optimization of the angle of inclination λ opt is not limited to ball-nose milling cutters. Form-cutting tools having the generating

The loxodroma’s pole is located at the point of intersection of the generating surface T of the milling cutter by the axis of rotation of the cutting tool.

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3

T2

C

T3

A

1 B

B T1 A

1 (a)

T2

3

T3

C

T1

2 (b)

Figure 6.5 The filleted-end milling cutter having an optimized value of the angle of inclination λ opt.

surface T of any feasible shape can be designed with the optimal value of the angle of inclination. The last statement encompasses composite generating surfaces T of the form-cutting tools as well. As an example, consider the optimization of the angle of inclination of a filleted-end milling cutter (Figure 6.5). The generating surface of the filletedend milling cutter is composed of three portions: the cylindrical portion T1 , the flat-end T2 , and the torus surface T3 . For the cylindrical portion T1 of the generating surface of the filleted-end milling cutter, the cutting edge AB having the optimal angle of inclination λ opt = Const reduces to a helix 1 of constant pitch. For the flat-end portion T2 of the generating surface, the cutting edge is represented in the form of a logarithmic spiral curve 2. Ultimately, the equation of the cutting edge segment BC within the portion T3 of the generating surface of the milling cutter can be derived on the premises of Equation (6.2). This segment of the cutting edge is represented by the arc segment 3 of the loxodroma. The loxodroma is within the torus surface T3. The generalized Equation (6.2) of the cutting edge having optimal value of the angle of inclination is valid for edge-cutting tools of any possible design. However, in particular cases of the filleted-end milling cutter, significant simplifications are possible. For example, for the flat-end portion T2 of the filleted-end milling cutter (Figure 6.5), an equation of the cutting edge can be derived following one of two possible ways. In compliance with the first of them, the flat-end surface is considered as a surface of revolution that is degenerated into the plane. Further, the equation of the cutting edge can be derived for the surface of revolution. Following the second possible way, it is preferred to employ the differential equation for isogonal trajectories. If a planar curve intersects all the curves of the initially given single-parametric family of planar curves

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ϕ ( x , y , θ ) = 0 at a given constant angle of intersection z, then the line satisfies the differential equation:

 ∂ϕ   ∂ϕ  ∂ϕ ∂ϕ  ∂x cos ς - ∂y sin ς  dx +  ∂x sin ς + ∂y cos ς  dy = 0

(6.14)

For the flat-end portion T2 of the generating surface of the filleted-end milling cutter, the initially given single-parametric family of planar curves ϕ ( x , y , θ ) = 0 is represented by the family of straight lines through the axis of rotation of the milling cutter. Here θ designates the angular parameter of the family of straight lines ϕ ( x , y , θ ) = 0. The equation of the family of straight lines can be represented in the form

ϕ ( x , y , θ ) = y - x tan θ = 0

(6.15)

Further, assume that ς = 90° - λ opt . After substituting Equation (6.15) into Equation (6.14), the Equation (6.14) casts into an equation of the cutting edge of the filleted-end milling cutter. This equation describes a logarithmic spiral curve. This means that in the particular case under consideration, the logarithmic spiral curve can be interpreted as the loxodroma for the family of straight lines within the plane. It is convenient to represent the equation of the cutting edge in polar coordinates:

ρ = ρ0 eϕ tan λ opt , ( ρ0 > 0,

- ∞< ϕ < +∞)

(6.16)

where r is the position vector of a point of the cutting edge; and r 0 is the position vector of a given point of the cutting edge, from which the angle j is measuring. The cutting edge (see Equation 6.16) intersects all straight lines through the point O at that same angle ς = 90° - λ opt . The pole of the logarithmic spiral curve coincides with the axis of rotation of the milling cutter. It represents an asymptotic point of this planar curve. Because of this, the cutting edge of the flat-end portion T2 of the generating surface cannot pass through the axis of the tool rotation. It is possible to design the cutting edge in the shape of the logarithmic spiral curve (see Equation 6.16) only within a certain portion, similar to the arc AB (Figure 6.4b). The cutting edge cannot be shaped in the form of the logarithmic spiral curve between a certain point A and the axis of rotation of the filleted-end milling cutter. In the case under consideration, representation of the equation of the cutting edge in the following form proved to be useful:

ρ = ρAeϕ tan λ opt ,

(ρA > 0, ϕ A = 0° < ϕ < ϕ B )

where rA is the position vector of the point A of the cutting edge AB.

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(6.17)

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A few other design parameters of the cutting edge of the form-cutting tool which can be drawn up from the geometrical analysis are as follows. Length SAB of the cutting edge AB can be computed from the equation SAB =

( ρB - ρA ) 1 + tan 2 λ opt tan λ opt

=

( ρB - ρ A ) sin λ opt

(6.18)

where ρB is the position vector of the point B of the cutting edge AB. The radius of curvature RT at a current point of the cutting edge AB can be computed from the equation RT ( ρ) = ρ 1 + tan 2 λ opt = ρA eϕ tan λ opt 1 + tan 2 λ opt =

ρA eϕ tan λ opt cos λ opt

(6.19)

Length SAB and the radius of curvature RT of the cutting edge are often required for optimization of performance of the filleted-end milling cutter. The third method for the transformation of the generating body of the cutting tool into the workable edge cutting tool can be implemented to design cutting tools for machining both sculptured surfaces on a multi-axis NC machine as well as for machining parts on conventional machine tools. In addition to loxodroma possessing useful properties for a tool designer, this curve can be evolved into two possible areas. First, a curve similar to loxodroma can be constructed on the generating surface T of a form-cutting tool that is shaped not only in the form of a surface of revolution, but also for the surface T of another topology, including surfaces T that allow for sliding over themselves. Second, the loxodroma can be evolved to a more general area, when the optimal value of the angle of inclination λ opt varies within the cutting edge. Under such a scenario, the desired current value of the angle λ opt can be expressed in terms of curvilinear coordinates UT and VT , say by equation λ opt = λ opt (UT , VT ). The desired function λ opt = λ opt (UT , VT ) of variation of the optimal value of angle of inclination λ opt can be determined experimentally. A form-cutting tool of any kind can be designed using any of three methods considered above. The tool design engineer makes his or her own decision of which method is preferred to use to design a particular form-cutting tool.

6.2 Geometry of the Active Part of Cutting Tools in the Tool-in-Hand System The active part of the cutting tool is composed of two surfaces intersecting each other to form the cutting edge. The surface over which the chip is flowing is known as the rake surface R s or more simply as the face. And that

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surface, which is faced to the machined surface, is known as the clearance surface Cs or the flank. In the simplest yet common case, both surfaces R s and Cs are planes. The cutting edge is represented as the line of intersection of the rake surface R s and of the clearance surface Cs. Cutting edges of two kinds can be distinguished: roughing cutting edges and finishing (clean-up) cutting edges. Roughing cutting edges do not generate the surface P being machined, but finishing cutting edges do. Finishing cutting edges are always within the generating surface T of the cutting tool. Roughing cutting edges are beneath the surface T and within the generating body of the cutting tool. The generating surface T of a cutting tool can make point contact with the part surface P. Under such a scenario, roughing portions of the cutting edges may be within the surface T as well. Major and minor cutting edges of the cutting tool are distinguished. A whole cutting edge or its portion that is faced toward the direction of the feed rate is referred to as the major cutting edge of the cutting tool. Another cutting edge or the rest of the whole cutting edge is referred to as the minor cutting edge of the cutting tool. The major cutting edge of a cutting tool contacts the chip being cut off. The minor cutting edge of a cutting tool contacts with the uncut portion of the stock. The regular (and not stochastic) residual roughness on the surface P is caused by both the major and the minor cutting edges. For a form-cutting tool having curved cutting edges (for example, for a milling cutter), an elementary cutting edge of infinitesimal length dl is considered below. Depending upon the actual problem under consideration, the infinitesimal cutting edge dl is considered either as a straightline segment or as a circular-arc segment of the corresponding radius of curvature. 6.2.1 Tool-in-Hand Reference System A references system associated with reference surfaces of the cutting tool is referred to as the tool-in-hand reference system. This reference system is often used when designing, manufacturing, regrinding, and inspecting the cutting tool. In order to accomplish design of a high-performance cutting tool, the geometry of the active part of the cutting tool in various cross-sections of the cutting wedge must be known. The tool-in-hand reference system is made up of planes that are tangent to the generating surface T of the cutting tool, to the rake surface R s , and to the clearance surface Cs. In particular cases, the surfaces T, R s , and Cs (all or some of them) degenerate to corresponding planes. The actual values of geometric parameters of the active part of a cutting tool are determined in a coordinate system associated with the cutting tool. This coordinate system is referred to as the static coordinate system. Various configurations of the static coordinate system with respect to the cutting tool are feasible.

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ZT 3

κr´

1

2 Vf

3´

1´

2´ Vf

YT

κr XT

Figure 6.6 Definition of the main reference plane Pr (ISO 3002).

The right-hand-oriented static coordinate system XTYTZT is recommended for application by the International Standard ISO 3002 (Figure 6.6). Axis ZT of the coordinate system XTYTZT aligns with the assumed direction of primary motion Vp . Axis X T aligns with the assumed direction of the cutting feed-rate motion V f . Axis YT complements the axes X T and Z T to a righthand-oriented Cartesian coordinate system XTYTZT. No principal constraints are imposed on the actual configuration of the static coordinate system. The convenience of performing computations is usually the only recommendation to follow when selecting the coordinate system X T YT Z T . By definition, the main reference plane Pr is perpendicular to the assumed direction of primary motion Vp in the tool-in-hand coordinate system X T YT Z T (Figure 6.6). In this figure, vector V f designates the assumed direction of the feed-rate motion. For computer-aided design and computer-aided manufacturing (CAD/ CAM) applications, analytical description of the reference planes is of critical importance. In order to describe analytically the reference planes, it is necessary to represent the surfaces T, R s , and Cs in the tool-in-hand coordinate system. In the coordinate systems associated with each of the surfaces T, R s , Cs , the surfaces yield analytical representation in the forms rT = rT (UT , VT ) , r rs = r rs (U rs , Vrs ), and r ce = r ce (U ce , Vce ). For the representation of these equations in the common coordinate system X T YT Z T , the

ISO 3002. Basic Quantities in Cutting and Grinding. Part 1: Geometry of the Active Part of Cutting Tools — General Terms, Reference Systems, Tool and Working Angles, Chip Breakers, 1982. ISO 3002-1/AMD1. Amendment 1 to ISO 3002-1, 1982. 1992. ISO 3002. Basic Quantities in Cutting and Grinding — Part 2: Geometry of the Active Part of Cutting Tools — General Conversion Formulae to Relate Tool and Working Angles, 1982.

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corresponding operators of resultant coordinate system transformations are used (see Chapter 3). The equations of the surfaces T, R s , Cs yield computation of unit tangent vectors u T and v T , u rs and v rs , and u cs and v cs . Ultimately, the following equations can be derived: For the tangent plane rTt to the generating surface T of the form-cutting tool: [rTt - r ( M ) ] × uT ⋅ vT = 0

(6.20)

For the tangent plane r rt to the rake surface Rs of the form-cutting tool: [r rt - r ( M ) ] × u rs ⋅ v rs = 0

(6.21)

For the tangent plane rct to the clearance surface Cs of the form-cutting tool: [r ct - r ( M ) ] × u cs ⋅ v cs = 0

(6.22)

where r ( M ) designates the position vector of a point M of interest within the cutting edge. The same unit tangent vectors yield computation of unit normal vectors n T , n rs, n cs to the surfaces T, R s, Cs at M. They are equal to n T = u T × v T , n rs = u rs × v rs , and n cs = u cs × v cs , correspondingly. All the unit normal vectors are pointed out from the bodily side to the void side of the cutting tool (see Chapter 1). Equations for the tangent planes rTt , r rt , r ct together with equations for the unit normal vectors n T , n rs, n cs yield derivation of equations of major reference planes at a current point M of the cutting edge. 6.2.2 Major Reference Planes: Geometry of the Active Part of a Cutting Tool Defined in a Series of Reference Planes Figure 6.6 shows a system of reference planes in the tool-in-hand system defined by the Standard ISO 3002. The system is made up of five basic planes, each of which is defined relative to the main reference plane Pr : • By definition, the main reference plane Pr is perpendicular to the assumed direction of primary motion Vp (Figure 6.7). In Figure 6.6, this reference plane is congruent to the XT YT coordinate plane. • Perpendicular to the main reference plane Pr and containing the assumed direction of feed-rate motion V f is the assumed working plane P f as shown in Figure 6.7.

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238

Kinematic Geometry of Surface Machining ZT Vf

Pfi Vp

Pr

M

Po Pf

pn Ps Pp

Figure 6.7 ISO 3002 system of reference planes in the tool-in-hand system for a major cutting edge.

• The tool cutting edge plane Ps is perpendicular to the main reference plane Pr and contains the side (main) cutting edge 1 – 2. For finishing cutting edges, the plane Ps is tangent to the generating surface T of a cutting tool. • The tool back plane Pp is perpendicular to the main reference Pr and the assumed working P f planes (Figure 6.7). • Perpendicular to the projection of the cutting edge onto the main reference plane Pr is the orthogonal plane Po . • The cutting edge normal plane Pn is perpendicular to the cutting edge. This reference plane can also be defined as the plane that is perpendicular to the rake R s and to the clearance Cs surfaces at M (Figure 6.8 and Figure 6.9). As an example, the ISO 3002 system of the reference planes for a lathe cutter is depicted in Figure 6.9. The orthogonal plane Pn passes through the unit normal vector n pc to the cutting edge located within the cutting edge plane Ps . Configuration of the reference plane Pn can be defined by a pair of unit vectors through M: either n rs and n cs or nT and n pc. Other combinations of the above unit vectors and a unit vector c e along the cutting edge (or tangent to the cutting edge) can be employed for definition of Pn as well. From another viewpoint, the unit tangent vector c e is orthogonal to the unit normal vectors n rs and n cs to the rake surface R s and to the clearance surface Cs of the cutting edge. This yields the equation for c e = n ce × n cs. The corresponding unit normal vectors n ce and n cs are equal to n ce = u ce × v ce and n cs = u cs × v cs.

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ZT

M

XT

Pfi

Po

Rs

ce

YT

Ps

Cs

Pn

Figure 6.8 Derivation of equation of the normal plane section Pn .

γf

Pr Cross-Section by the Plane Pf βf

Pp αf

Cross-Section by the Plane Pp Pf

S

αp

Pr

Ps αp

γp

γo −γ

βp

κr Pr

Cross-Section by the Plane Po

+γ βn

ψr

αn

Ps

View S

βp γn Pr

+λs λs

–λs

Figure 6.9 Example of the cutting tool geometry in the tool-in-hand system (ISO 3002).

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Cross-Section by the Plane Pn

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Kinematic Geometry of Surface Machining

In the tool-in-hand coordinate system XTYTZT, the equation of the orthogonal reference plane Pn through M yields representation in the following form: (r n - r ( M ) ) × c e = 0

(6.23)

where r n is the position vector of a point of the orthogonal reference plane Pn , and r ( M ) is the position vector of a point M. Similarly, a corresponding system of reference planes can be attributed to the minor cutting edge. The system of reference planes for the minor cutting edge is composed of five planes P ′f , Ps′ , Pp′ , Po′ , and Pn′ . These reference planes strictly correlate to the reference planes P f , Ps, Pp , Po , and Pn of the major cutting edge of a cutting tool. The ISO 3002 system of reference planes is not consistent for solving a variety of problems that relate to cutting-tool geometry. This is the reason why other reference planes are used in addition to the reference planes Pr , P f , Ps , Pp , Po , and Pn for computation of the geometry of the active part of cutting tools.

6.2.3 Major Geometric Parameters of the Cutting Edge of a Cutting Tool The geometry of the active part of a cutting tool is specified by angles and some other geometric parameters. The geometric parameters of the active part of a cutting tool are measured in corresponding reference planes. 6.2.3.1 Main Reference Plane In the main reference plane Pr, the following geometric parameters of the active part of a cutting tool are measured: The tool-cutting edge angle κ r (or j) is the acute angle that the reference plane Ps makes with the reference plane P f . In another words, the angle κ r is the acute angle between the projection of the major cutting edge onto the reference plane Pr and the P f reference plane (Figure 6.6). The tool-cutting edge angle κ r is measured in counterclockwise direction from the P f reference plane. The angle κ r is always positive. The tool minor (end) cutting edge angle κ r1 is the acute angle that the reference plane Ps′ makes with the reference plane P ′f . In other words, the angle κ r1 is the acute angle between the projection of the minor (end) cutting edge onto the reference plane Pr and the P ′f reference plane (Figure 6.6). The tool minor cutting edge angle κ r is measured in a clockwise direction from the P ′f reference plane. The angle κ r1 is always positive, including zero (Figure 6.6). The tool approach angle ψ r is the acute angle that the reference plane Ps makes with the reference plane Pp as shown in Figure 6.9. The angle ψ r can be computed from the expression ψ r = 90° - κ r .

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In addition to those specified by ISO 3002, angles that are listed above the following geometric parameters of the active part of the cutting tool are measured in the main reference plane Pr: The tool tip angle ε is the angle that makes the projections of the major and the minor cutting edges into the main reference plane. The tool tip angle ε can be computed from the equation e = 180° - (k r + k r1) = 90° + ψ r - κ r1 [or ε = 180° - (ϕ + ϕ 1 )]. The radius of normal curvature R T of the generating surface of a cutting tool T. The radius R T specifies the actual radius of curvature of the cutting edge R ce . 6.2.3.2 Assumed Reference Plane In the assumed reference plane P f , the following geometric parameters of the active part of a cutting tool are measured: The tool rake angle γ f is the angle between the reference plane Pr (the trace of which appears as the normal to the direction of primary motion) and the intersection line formed by the assumed working plane P f and the tool rake plane R s. The rake angle is defined as being acute and positive when looking across the rake face from the selected point and along the line of intersection of the face and the assumed working plane P f . The viewed line of intersection lies on the opposite side of the tool reference plane from the direction of primary motion. The sign of the rake angle is well defined (Figure 6.9). The tool clearance (flank) angle α f is defined in a way similar to the tool rake angle γ f, though here if the viewed line of intersection lies on the opposite side of the cutting edge plane Ps from the direction of feed-rate motion (assumed or actual as the case may be), then the clearance angle is positive. In other words, the clearance angle is the angle between the tool-cutting edge plane Ps and the tool flank plane Cs (Figure 6.9). The tool wedge angle β f is the angle between the two intersection lines formed as the reference plane P f intersects with the rake R s and flank Cs planes. The sum of algebraic values of the rake γ f , wedge β f , and clearance α f angles is equal to 90°:

γ f + β f + α f = 90°

(6.24)

Equations similar to Equation (6.24) are valid for other reference planes that cross the cutting edge. For the minor (side) cutting edge, geometric parameters are specified in a way similar to that considered above.

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6.2.3.3 Tool Cutting Edge Plane The orientation and inclination of the cutting edge are specified in the tool cutting edge plane Ps . In this reference plane, the cutting edge inclination angle λ s is measured. The angle of inclination is measured between the assumed direction of primary motion Vp and the unit normal vector n ce to the cutting edge. The angle λ s is defined as always being acute. It is positive if the cutting edge is turned in counterclockwise direction with respect to Vp . This angle can be defined at any point of the cutting edge. The sign of the inclination angle is well defined in Figure 6.9.

6.2.3.4 Tool Back Plane In the tool back plane Pp , the rake angle γ p, the clearance angle α p , and the tool wedge angle β p are measured. These angles are measured in a way similar to the way that the corresponding angles in the assumed working plane P f are measured (Figure 6.9). Like in the reference plane P f , the sum of algebraic values of the rake γ p , the wedge β p, and the clearance α p angles is equal to 90°:

γ p + β p + α p = 90°

(6.25)

6.2.3.5 Orthogonal Plane The rake angle γ o , the clearance angle α o , and the wedge angle βo are measured in the orthogonal plane Po . These angles are measured between lines of intersection of the rake surface R s, the clearance surface Cs , and the corresponding reference planes by the orthogonal plane Po . The equality for algebraic values of α o , βo , and γ o

γ o + βo + α o = 90°

(6.26)

is observed in the plane Po . 6.2.3.6 Cutting Edge Normal Plane In the cutting edge normal plane Pn , in addition to the rake angle γ n, the clearance angle α n and the wedge angle βn and the cutting wedge roundness ρn are measured. The sum of algebraic values of the angles γ n, α n , and βn is always equal to

γ n + βn + α n = 90°

(6.27)

The angles γ n, α n, and βn are measured between the lines of intersection of the rake surface R s, the clearance surface Cs , and the corresponding reference planes by the normal plane Pn .

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The Geometry of the Active Part of a Cutting Tool 6.2.4 Analytical Representation of the Geometric Parameters of the Cutting Edge of a Cutting Tool

It is the right point now to introduce equations for computation of actual values of the angles γ n, α n, and βn . Inclination of the rake surface R s with respect to the tool cutting edge plane Ps is specified by the normal rake angle γ n. The angle γ n is the angle that makes the unit normal vector n ps to the reference plane Ps , and the rake surface R s. Its value is accounted for from n ps to the line of intersection of the rake surface R s by the normal plane Pn. Angle γ n is positive if the unit normal vector n ps does not pass through the cutting wedge (Figure 6.9). Otherwise, the angle γ n is negative. The unit normal vector to the rake surface R s is designated as n rs. It is convenient to define the normal rake angle γ n as the angle that complements to 90° the angle between the unit normal vectors n ps and n rs (Figure 6.10). The above equations n T = u T × v T and n rs = u rs × v rs for n ps and n rs yield equations for the normal rake angle γ n :

γ n = 90° - arccos(n rs ⋅ n ps ) = arcsin(n rs ⋅ n ps )

n rs ⋅ n T γ n = 90° - ∠ (n rs , n ps ) = arctan |n rs × n T|

(6.28)

(6.29)

The cutting tool clearance surface Cs makes a certain normal clearance angle α n with the tool cutting edge plane Ps . The angle α n is the angle

Rs

γn βn nps

(nrs, ns)

Ps αn

nrs M Cs (ncs, nps)

(nrs, ncs) M

n cs

Figure 6.10 Cross-section of the cutting wedge by a normal plane section Pn .

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Kinematic Geometry of Surface Machining

between the unit normal vector n ps to the plane Ps , and the clearance surface Cs. Its value is accounted for from the reference plane Ps toward the clearance surface Cs. The normal clearance angle α n is always positive (α n > 0°). Only within a narrow chamfer on the clearance surface can the angle α n be equal to zero, or it can even be negative (up to the value of α n = -20° ÷ -25°). The unit normal vector to the clearance surface Cs is designated n cs . It is convenient to define the normal clearance angle α n as the angle that complements to 180° the angle between the unit normal vectors n ps and n cs (Figure 6.10). The above equations n T = u T × v T and n cs = u cs × v cs for n ps and n cs yield the following formulae:

α n = 180o - ∠ (n cs , n ps )

α n = - arctan

(6.30)

|n ps × n cs |

n ps ⋅ n cs

(6.31)

for the computation of the normal clearance angle α n. The wedge angle βn can be defined as the angle that complements to 180° the angle between the unit normal vectors n rs and n cs. This immediately leads to the equations

βn = 180o - ∠ (n rs , n cs )

(6.32)

|n rs × n cs | n rs ⋅ n cs

(6.33)

βn = - arctan

for the computation of βn . Remember that the equation β n = 90° - (α n + γ n ) is the simplest for the computation of βn. 6.2.5 Correspondence between Geometric Parameters of the Active Part of Cutting Tools That Are Measured in Different Reference Planes Certain geometric parameters of the active part of a cutting tool can be measured directly on the cutting tool of a given design. The following geometric parameters α n , β n, γ n, λ s, ρn, and ε ′ are among those that can be measured directly. Other geometric parameters of the active part of a cutting tool can be computed. For derivation of equations for the computation of geometry of the active part of cutting tools, implementation of elements of vector calculus is helpful. To the best of the author’s knowledge, Mozhayev [5] was the first

The angle e′ is measured within the rake surface R s. Projection of the angle e′ onto the main reference plane Pr is equal to the tool tip angle e.

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The Geometry of the Active Part of a Cutting Tool YT YT

γn

nce

A ZT

Ps

ZT λs

M

γc

A

Vp

A nce

M

1 prxy A

M

b

B

λs

XT

ce dl

Pn

A

XT ZT

ns

Rs

Pc

γn

YT

B

γc XT

ZT

b M

Figure 6.11 Geometry of the active part of a cutting wedge in cross-sections by various reference planes.

to use (1948) elements of vector calculus for solving problems that relate to the geometry of the active part of a cutting tool. For illustration of the capabilities of the method, consider an elementary cutting edge of the infinitesimally short length dl. The geometric parameters of the cutting edge in the normal reference plane Pn and the value of the cutting edge inclination angle λ s are given. It is necessary to determine the rake angle γ ce that is measured in the reference plane Pce. The reference plane Pce is the plane through the assumed direction of primary motion Vp perpendicular to the cutting edge plane Ps at M (Figure 6.11). The Cartesian coordinate system X T YT Z T is associated with the cutting wedge as shown in Figure 6.11. Construct a vector A (Figure 6.11). The vector A is tangent to the line of intersection of the rake surface R s by the normal reference plane Pn . The vector A is of the length for which its projection onto the X T YT coordinate plane is equal to Pr xy A = 1. Analytical expression for the vector A can be represented in matrix form:

 - sin λ s   cot γ  n  A= - cos λ s     q 

(6.34)

Construct a unit vector b. The unit vector b is tangent to the line of intersection of the rake surface R s by the reference plane Pce. In the coordinate

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Kinematic Geometry of Surface Machining

system XTYTZT, the unit vector b yields representation in the form 0   - cos γ  ce  b=  sin γ e    1  

(6.35)

Ultimately, construct a unit vector c e . The unit vector c e is directed along the cutting edge of the length dl. The vector c e is equal to  cos λ s  - sin λ  s ce =   0     q 

(6.36)

By construction, the vectors A, b, and c e are the coplanar vectors. All three are within the plane that is tangent to the cutting edge AB at point M. Due to this, the triple product [A, b, c e ] of the vectors A, b, and c e is identical to zero. Therefore, the following expression is valid:

- sin λ s A × b ⋅ ce = 0 cos λ s

cot γ n - cos γ ce 0

- cos λ s sin γ e ≡ 0 - sin λ s

(6.37)

After the necessary formulae transformations are performed, one can come up with the equation for γ ce:

γ

e

= cot -1 [cos λ s ⋅ cot γ n ]

(6.38)

Equation (6.38) is derived under the assumption that the cutting edge is infinitesimally short. This means that Equation (6.38) is applicable for the computation of the angle γ ce for the form-cutting tool of any design. Considerations similar to those above are valid for other geometrical parameters of a cutting edge of a cutting tool that are measured in other reference planes. The angle between the major and the minor cutting edges of a cutting tool is designated as ε ′ . This angle is measured in the rake surface R s. Angle ε ′ yields computation of the tool tip angle e. The tool tip angle e can be thought of as the projection of the angle ε ′ onto the main reference plane Pr . Conversely, angle ε ′ is the projection of the tool tip angle e onto the rake face.

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The Geometry of the Active Part of a Cutting Tool

For computation of the geometry of the active part of a cutting tool, the International Standard ISO 3002 recommends the following equations:

tan λ s = sin κ r tan γ p - cos κ r tan γ f

tan γ n = cos λ s tan γ o

tan γ o = cos κ r tan γ p + sin κ r tan γ f

cot α n = cos λ s cot α o

(6.39)

(6.40)

(6.41) (6.42)

Equation (6.39) through Equation (6.42) are derived under the assumptions that the tool side rake angle γ f , the tool back rake angle γ p , and the tool cutting edge angle κ r are the basic angles for the tool face, and the tool side clearance angle α f , the tool back clearance angle α p , and the tool cutting edge angle κ r, are the basic angles for the tool flank. Other equations used for the computation of the geometry of the active part of a cutting tool are as follows (ISO 3002):

cot α o = cos κ r cot α p + sin κ r cot α f

tan λ s = - sin κ r tan γ p - cos κ r tan γ f

tan γ n = cos λ s tan γ o

tan γ o = - cos κ r tan γ p + sin κ r tan γ f

cot α n = cos λ s cot α o

cot α o = - cos κ r cot α p + sin κ r cot α f

(6.43)

(6.44) (6.45)

(6.46) (6.47)

(6.48)

Implementation of the method of computation of the geometry of an active part of a cutting tool is illustrated with the following examples [16]. Example 6.2 Consider a twist drill. The core diameter d of the twist drill is 0.2 of the nominal drill diameter D — that is, d = 0.2D . The tool cutting edge angle is equal to 2 ⋅κ r = 120° . It is necessary to determine the cutting edge inclination angle λ (Figure 6.12). Based on definition, the cutting edge inclination angle λ s is equal to the angle that makes the unit vector of the assumed direction of primary motion v p and the unit vector n ce. Both vectors v p and n ce are within the tool cutting edge plane, and the unit vector n ce is perpendicular to the cutting edge.

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Kinematic Geometry of Surface Machining γn = +30 γn γn = 0

Vp

XT

γn = –30 dc

YT M

D

κ1

YT M

Vf

ZT

2·κ

~ 50 ∆λn =

λs < 0 Figure 6.12 Example of computation of the geometry of the cutting edge of a twist drill.

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The Geometry of the Active Part of a Cutting Tool

Therefore, the equality λ s = ∠ (n ce , v p ) = [90° - ∠ (c e , v p )] is valid. Here c e designates the unit vector along the cutting edge. The unit vectors c e and v p yield the following representation in the local coordinate system XT YT Z T :

 0   - sin κ  r ce =  - cos κ r     1 

cos µ   sin µ   vp =   0     1 

(6.49)

(6.50)

This immediately yields an equation for the computation of the inclination angle λ s:

 d λ s = - arcsin  sin κ r  d  i 

(6.51)

Equation (6.51) for the computation of angle λ s reveals that for a twist drill, the cutting edge inclination angle is always negative. The actual value of the angle λ s depends upon a distance at which the cutting edge current point is remote from the axis of rotation of the twist drill. It is easy to verify that the angle λ s is of the smallest magnitude at the outside diameter of the twist drill where λ s = -7.464° . The magnitude of the angle λ s increases toward the axis of rotation of the twist drill. The largest magnitude of the angle λ s occurs for the cutting edge point that is closest to the axis of rotation where λ s ≈ - 55°. The difference of the angle λ s for various cutting edge points is about D λ s ≈ 50°. Example 6.3 Consider the same twist drill as that considered in Example 6.2. The helix angle at the major diameter of the drill is ω = 30° . It is necessary to determine the tool normal rake angle γ n . By definition, for the normal rake angle γ n, the twist drill rake angle γ n can be specified in terms of two unit normal vectors n ps and n rs: tan γ n =

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n ps ⋅ n rs |n ps × n rs |

(6.52)

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Kinematic Geometry of Surface Machining

where n ps is the unit normal vector to the tool cutting edge reference plane Ps, and n rs is the unit normal vector to the twist drill rake face R s. The unit normal vector n ps can be computed as cross-product n ps = c e × v p:

 i  ns =  0  cos µ

j - sin κ r sin µ

k - cos κ r 0

   

(6.53)

This immediately yields the equation for the unit normal vector n ps :  cos κ r sin µ  - cos κ cos µ  r  n ps =   sin κ r cos µ    1  

(6.54)

The unit normal vector n rs can be computed as the cross-product n rs = c e × t rs. Here, t rs designates a unit vector that is tangent to the rake surface R s of the twist drill. It is convenient to employ the vector t rs that is tangent to a helix line of the rake surface. In this case, the following equation is valid:

- sin ω y sin µ   sin ω cos µ  y  t rs =   cos ω y    0  

(6.55)

where ω i designates the helix angle of the screw rake surface at a current point of the cutting edge of the twist drill. The above expressions for c e and t rs yield the equation for the unit normal vector n rs:

 i  n rs =  0 - sin ω y sin µ 

j sin κ r sin ω y cos µ

k   cos κ r  cos ω y 

(6.56)

This result immediately returns the equation for n rs:

sin κ r cos ω y - cos κ r cos µ sin ω y    - cos κ r sin µ sin ω y  n rs =    sin κ r sin µ sin ω y   1  

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(6.57)

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The Geometry of the Active Part of a Cutting Tool

Substituting the derived expressions for the unit vectors n ps and n rs into Equation (6.52) for tan γ n, one can come up with the equation for tan γ n:

tan γ n =

1 - sin 2 κ r sin 2 µ ⋅ tan ω y - cos κ r tan µ sin κ r cos µ

(6.58)

Example 6.4 Consider the same twist drill as that considered in Example 6.1 and Example 6.2. It is necessary to determine the tool normal clearance angle α n . By definition, for the normal clearance angle α n , the twist drill clearance angle α n can be specified in terms of two unit normal vectors n ps and n rs: tan α n =

n ps ⋅ n cs |n ps × n cs |

(6.59)

where n cs designates the normal unit vector to the clearance surface Cs . The unit normal vector n ps was defined in Equation (6.54). The unit normal vector n cs can be computed as the cross-product n cs = n ps × t cs. Here, t cs designates a vector that is tangent to the clearance surface Cs of the twist drill. In the coordinate system XT YT Z T , the unit tangent vector t cs yields representation in the following form:

t cs

- sin α i sin µ   sin α cos µ  i  =  cos α i    1  

(6.60)

In Equation (6.60), α i designates the clearance angle that is measured in a section of the twist drill by a cylinder coaxial with the twist drill. Following a method similar to that considered in Example 6.3, the equation for the computation of α n can be derived as well. The cutting edge radius of curvature is measured in the rake surface R s. In order to determine the curvature of the cutting edge, an additional reference plane is constructed. This reference plane Pce is perpendicular to the cutting edge plane Ps and is tangent to the cutting edge. In this reference plane, the radius of normal curvature of the generating surface T of a cutting tool corresponds to the cutting edge radius of curvature RT . Meusnier’s equation (see Equation 5.19) together with Euler’s equation (see Equation 1.31) yield the equation for R T in terms of RT : RT =

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RT cos γ n

=

R 1.T sin 2 λ s + R 2.T cos 2 λ s R 1.T R 2.T cos γ n

(6.61)

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Kinematic Geometry of Surface Machining

γn Rs

βn Ps

ρn

Cs

αn

Figure 6.13 Roundness of the cutting edge in the normal plane section Pn.

The cutting edge of a cutting tool is not absolutely sharp. Actually, there exists a transition surface that connects the rake surface R s and the flank Cs . This transition surface is supposed to have a circular profile of a certain radius that is considered as radius ρn of the cutting edge roundness (Figure 6.13). The roundness of the cutting wedge can be determined from experiments (for example, from etching tests). The roundness of the cutting wedge of a cutting tool made of HSS (high-speed steel) is usually in the range of ρn = 20 ÷ 50 µ m, while that of a cutting tool made of sintered carbide is as low as ρn = 10 ÷ 30 µ m. For diamond inserts, roundness drops down to ρn = 5 ÷ 8 µ m and can even be reduced to ρn ≅ 2 µ m . The cutting edge roundness ρn affects a material removal process in metal cutting. The effect of the cutting edge roundness is more substantial when a thin chip is removed, especially when stock to be removed is of the same range as the cutting edge roundness ρn . When the stock thickness is about 0.003 mm or less, then the rake angle γ n does not affect the material removal process; thus, it can be neglected. In the reference plane Po through Vp that is perpendicular to the cutting edge plane Ps , roundness r of the cutting edge can be expressed in terms of r n by Meusnier’s equation:

ρ = ρn cos λ s

(6.62)

Torsion of the cutting edge is one more geometric parameter of the active part of a cutting tool to be considered. The shape of the rake surface R s and of the clearance surface Cs affect the material removal process in metal cutting. This is because the geometry of the

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active part of a cutting tool in many cases is measured in reference planes configuration of which depends upon tangent planes to the surfaces R s and Cs . The effect of the cutting edge torsion onto the material removal process in metal cutting has not yet been profoundly investigated. A cutting edge can be considered as a line of intersection of three surfaces: the generating surface T of the cutting tool, the rake surface R s, and the clearance surface Cs. The equation of the cutting edge can be derived as a result of mutual consideration of the equation of one of three pairs of surfaces: (a) r rs (U rs , Vrs ) and r cs (U cs , Vcs ), (b) rT (U T , VT ) and r rs (U rs , Vrs ), or (c) rT (U T , VT ) and r cs (U cs , Vcs ) . The solution to any of three pairs of equations can be reduced to the equation of the cutting edge, which yields representation in matrix form:

X T (t ce )  Y (t )  T ce  r ce = r ce (t ce ) =   Z T (t ce )     1 

(6.63)

where t ce denotes the parameter of the cutting edge. In a particular case, length S ce of the cutting edge can be chosen as the parameter of the cutting edge (that is, t ce ≡ S ce ). Under such a scenario, torsion τ ce of the cutting edge can be computed from

τ ce

  d r ce d 2 r ce d 3 r ce   =  ρ 2ce  ⋅ ⋅    d S ce d S2ce d S3ce  

-1

(6.64)

where the sign of the torsion τ ce is not in compliance with the direction of the angle of inclination λ s. 6.2.6 Diagrams of Variation of the Geometry of the Active Part of a Cutting Tool Analytical methods for the computation of actual values of the geometry of the active part of a cutting tool are accurate. They are capable of computing the distribution of a geometrical parameter of a cutting tool both at a given point of the cutting edge in different reference cross-sections or within the active part of the cutting edge in similar cross-sections. Results of such computations are accurate and are of critical importance to a tool designer. For the preliminary analysis of the geometry of the active part of a cutting tool, the implementation of diagrams of variations of the geometrical parameters have proven useful. Distribution of the function tan γ of the rake angle in different reference planes through the point M within the cutting edge of a form-cutting tool is shown in Figure 6.14. Once the rake angle in two different reference planes is determined, then the distribution of the function tan γ follows the circle. The circle constructed on any two known vectors through the point M enables easy determination of the function tan γ i in any direction through

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Kinematic Geometry of Surface Machining

tan γf

M tan λs

1

tan γo

tan γp

O1

tan γmax

Figure 6.14 Example of the circular diagram of the distribution of the function tan g of a cutting tool rake angle g.

the point M. The magnitude of the vector is equal to the actual value of the function tan g in the corresponding direction. Similarly, the distribution of the function cot a of the clearance angle in different reference planes through the point M within the cutting edge of a formcutting tool can be constructed. The corresponding circular diagram of the distribution of the function cot a is depicted in Figure 6.15. Again, a circle constructed on two known values of the function cot a in two different directions reflects the distribution of the function cot a in all other reference planes.

M cot αp

cot αf 2 O2 cot αo

Figure 6.15 An example of the circular diagram of the distribution of the function cot a of a cutting tool clearance angle a.

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The Geometry of the Active Part of a Cutting Tool M2 γo

h=1

A2 cot γo h=1 M1

M

M3

cot γD

cot γo

D

C

A1 λs

cot γC

cot γE E

cot γF cot λs

A

F cot γB

cot λs

B B1

B3

Figure 6.16 Example of the diagram of distribution of the function cot g of a cutting tool rake angle g .

For analysis of the distribution of the function cot g, Figure 6.16 is helpful [4]. Two known values of the function cot g in known corresponding directions yield construction of the straight line AB. Ultimately, the actual value of the function cot g i in the reference plane through a current direction is specified by the corresponding point within the straight line AB. All the diagrams are in perfect correlation with the results of the analytical computations.

6.3 Geometry of the Active Part of Cutting Tools in the Tool-in-Use System When machining a part surface, the actual direction of the primary motion, as well as of the feed-rate motion, can differ from the assumed directions of these motions, say in the tool-in-hand system. Moreover, the actual kinematics of a machining operation can be made up not only of the primary and the feed-rate motions, but also of motions of another nature (for example, vibrations, orientation motions of the cutting tool [see Chapter 2], etc.). In order to

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precisely specify geometric parameters of the active part of a cutting tool, all the elementary motions that compose the resultant motion of the cutting tool relative to the work must be taken into consideration. There are two possible ways to represent the machined surface P. First, the machined surface P can be considered as an enveloping surface to consecutive positions of the generating surface T of the cutting tool when the cutting tool is moving relative to the blank. Second, the machined surface P can be considered as a set of discrete surfaces of cut Pse. At an instance of time when the surface P is generated, both the surface T and the surface Pse are tangent to P either at point K or along the characteristic curve. Because of this, the tool-in-hand reference system can be associated either with the assumed surface of the cut or with the cutting tool. The two options are identical in the sense of the tool-in-hand reference system. When machining a part surface, it is necessary to consider the kinematic geometric parameters of the active part of the cutting tool in a reference system associated with the surface of the cut. For this purpose, the tool-in-use reference system is used. Commonly, rake surface R s as well as clearance surface Cs of a cutting tool are shaped in the form of three-dimensional surfaces having complex geometry. Due to this, consideration of the surfaces R s and Cs at a distinct point of the cutting edge is required. Contact of the cutting wedge with the work is considered at a distinct point of the cutting edge. Because size of the area of contact of the cutting edge and the work is small, the rake surface as well as the clearance surface are locally approximated by corresponding planes, by the planes that are tangent to the surfaces R s and Cs at the point of interest of the cutting edge. Generally speaking, the geometry of the active part of a cutting tool must be determined for an elementary cutting edge of length dl (that is, in differential vicinity of the point M within the cutting edge). It is also necessary to consider the geometry of the active part at a given instant of time, say for the vector V Σ of known magnitude and direction. Such an approach would enable one to determine the distribution curves of geometric parameters within the cutting edge and the distribution curves of geometric parameters in time. In order to perform such an analysis, a generalized method of computation of geometry of the active part of a cutting tool is necessary. In particular cases, actual values of geometric parameters of the active part of a cutting tool can impose certain constraints onto parameters of kinematics of the machining operation. For example, variation in the actual value of geometric parameters either within the cutting edge or in time may impose restrictions on the parameters of feed-rate motion, of orientation motion of the cutting tool, and so forth. If parameters of kinematics of the machining operation exceed the limits, then the machining operation is not feasible. The capability to determine critical feasible values of parameters of geometry of the active part of a cutting tool is critically important for the tool designer.

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The Geometry of the Active Part of a Cutting Tool 6.3.1 The Resultant Speed of Relative Motion in the Cutting of Materials

As follows from the above analysis, the direction of the resultant speed V Σ of relative motion in the cutting of materials is a critical issue for establishing the tool-in-use reference system. Usually, relative motion of the cutting tool is of a complex nature. In the general case of surface machining, this motion is composed of the actual primary motion Vp , the surface generation motion Vgen, one or more feed-rate motions V f .i , the orientation motions of the first VorI and of the second VorII kinds, and of other motions. This yields the following equation for vector V Σ : n

VΣ = V + Vgen +

∑

m

V f .i + VorΙ + VorΙΙ + K =

i =1

∑V

(6.65)

j

j =1

where Vp is the vector of the primary motion, Vgen is the vector of the motion of surface generation, V f .i is the vector of the feed-rate motion, n is the total number of feed-rate motions, VorI is the vector of the orientation motion of the first kind of the cutting tool, VorII is the vector of the orientation motion of the second kind of cutting tool, Vj is the j elementary relative motion of the cutting tool, and m is the total number of elementary relative motions of the cutting tool. When determining the vector V Σ , vectors of all particular relative motions, those that significantly affect the V Σ must be taken into account. Relative motions, those that cause sliding of the surface P or the generating surface T of the cutting tool over itself must be incorporated as well. Motions VorI and VorII of orientation of the cutting tool, as well as the feedrate motions V f .i , are usually significantly smaller compared to the primary motion Vp . However, all must be incorporated for determination of the vector V Σ . In particular cases, some of these motions are comparable with the motion V Σ . Moreover, in special cases, they can even exceed the primary motion Vp . When cutting a material, vibration of the cutting tool is often observed. The vibration may result in positive and negative clearance angle (Figure 6.17a). For certain frequencies and magnitudes of the vibration, neglecting the vector of vibration Vvib is not allowed [1,3,13]. Due to vibrations, the rake and the clearance angles vary within a certain interval ± σ o . The current value of the angle

|V | σ o is σ o = arctan vib . |Vp|

When the vector Vvib is pointed toward the part surface P, then the corresponding rake angle γ o raises to the range of γ o′ = γ o + δ o . At this instant, the clearance angle α o reduces to α o′ = α o - δ o . If the vector Vvib is directed oppositely, then the corresponding rake angle γ o and the clearance angle α o can be computed from the equations γ o′′ = γ o - δ o and α o′′ = α o + δ o (see Figure 6.17b). When a partly worn cutting tool is used, then the clearance angle within a narrow land on the clearance surface next to the cutting edge reduces to 0°.

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258

Kinematic Geometry of Surface Machining γo

αo

γ"o γ'o

M δo α'o

Vp

M

VΣ

Vvibr

Vvibr VΣ

Pse M α"o

Vp

δo

(a) γ"o

γo γ'o αo

M δo α'o M

Vp VΣ

Vvibr

Pse δo

Vvibr VΣ M Vp α"o

(b) Figure 6.17 Effect of vibration onto the actual geometry of the active part of a cutting tool.

Cutting tool wear causes the effect of vibrations on cutting tool geometry to be more severe (see Figure 6.17b). The performed brief analysis of the impact of vibration onto the resultant instant motion of the cutting tool reveals the importance of careful analysis of all elementary motions, especially those the cutting tool is performing when machining a part surface. 6.3.2 Tool-in-Use Reference System The actual configuration of reference planes is a critical issue for the establishment of the tool-in-use reference system [17]. Because the efficiency of the chip removal process closely depends upon actual orientation of the cutting wedge with respect to the surface of the cut, then the surface of the cut as a base element for the determination of instant geometry of the active part of a cutting tool must be introduced.

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The Geometry of the Active Part of a Cutting Tool

259

The surface of cut Pse can be represented as a locus of consecutive positions of the cutting edge that travels with the resultant speed V Σ relative to the work. The plane of cut is tangent to the surface of cut Pse at the point of interest within the cutting edge. In a particular case, the surface of the cut and the plane of the cut are congruent. The last case is the degenerated one. The main reference plane Pre is perpendicular to the vector V Σ . The working plane Pfe is the plane through the directions of the primary motion, and of the feed-rate motion. Due to this, the working plane Pfe is perpendicular to the main reference plane Pre. The tool back plane Ppe is perpendicular to the reference planes Pre and Pfe. Other reference planes are of importance for the tool-in-use system. They are the plane of cut Pse , the rake surface plane R s , and the clearance plane Cs. For the purpose of determining the geometry of the active part of the cutting tool, it is convenient to employ three reference planes Pse , R s, and Cs in conjunction with the vector of the resultant cutting tool motion V Σ . The current orientation of the reference planes Pse, R s, and Cs is specified by unit normal vectors n rs, n cs, and c e. Prior to running the analysis, it is necessary to represent equation rP = r P = r P (U P of the part surface P as well as equation rT = rT (UT , VT ) of the generating surface T of the cutting tool in a common coordinate system XT YT Z T . For this purpose, implementation of the operator Rs(T → P) of the resultant coordinate system transformation is helpful. Equations of tangent planes r P .tp and rT .tp to the surfaces P and T at the point of interest M can be represented in vectorial form:

(r

P .tp

- r P( M ) × n P = 0

)

(6.66)

(r

T .tp

- rT( M ) × n T = 0

)

(6.67)

and

The kinematic method can be employed for the derivation of the equation of the surface of cut Pse . For this purpose, it is necessary to know the equation of the cutting edge and the parameters of the resultant relative motion of the cutting tool with respect to the work. The equation of the surface of cut Pse can be obtained in the following way. Consider a form-cutting tool. The cutting edge of the form-cutting tool is determined as the line of intersection of the face rake surface R s by clearance surface Cs . Therefore, in the coordinate system XT YT Z T , the cutting edge of the form-cutting tool can be described analytically by a set of two vectorial equations:

© 2008 by Taylor & Francis Group, LLC

r rs = r rs (U rs , Vrs )  r cs = r cs (U cs , Vcs )

(6.68)

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Kinematic Geometry of Surface Machining

An auxiliary Cartesian coordinate system XceYceZce is associated with the cutting edge. Initially, axes of the coordinate system XceYceZce align with corresponding axes of the coordinate system XTYTZT. Then consider the motion that the cutting edge together with the coordinate system XceYceZce is performing in the coordinate system XTYTZT. Parameters of this relative motion of the cutting edge are identical to the corresponding parameters of motion of the cutting tool relative to the work. The equation of the cutting edge in a current location of the coordinate system XceYceZce with respect to the coordinate system XTYTZT can be represented in the form r rs = r rs (U rs , Vrs , Ξ Σ )  r cs = r cs (U cs , Vcs , Ξ Σ )

(6.69)

where Ξ Σ designates the parameter of the resultant relative motion of the cutting tool. On the premises of Equation (6.69), one of the two curvilinear parameters, either the Urs or Vcs parameter can be expressed in terms of another parameter. For example, the Urs parameter is expressed in terms of the Vcs parameter. This relationship yields analytical representation in the form U rs = U rs (Vcs ). Ultimately, this results in the vectorial equation of the surface of cut Pse in the form

r se = r se [U cs (Vcs ), Vcs , Ξ Σ ] = r se [Vcs , Ξ Σ ]

(6.70)

Similarly, the equation of the surface of cut Pse can be expressed in terms of Vcs and ΞΣ parameters. For many purposes, the generating surface T of the formcutting tool can be considered as a good approximation to the surface of cut Pse. In order to compose the tool-in-use system for machining a surface on a conventional machine tool, two vectors are of principal importance: vector V Σ of resultant relative motion of the cutting tool with respect to the work and unit normal vector to the surface of cutting n se. The vector V Σ is computed from Equation (6.65). The unit normal n se can be computed as the cross-product n se = u se × v se . For the derivation of the unit tangent vectors u se and v se , Equation (6.70) of the surface of cut Pse can be used. That same unit normal vector n se can also be computed as the cross-product (Figure 6.18):

n se = v Σ × c e

(6.71)

where the unit vector v Σ is equal to vΣ = VΣ/|VΣ|. Equation (6.71) for the computation of the unit normal vector n se is convenient for performing computations. Other equations for the computation of the unit normal vector n se can be used as well:

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n se =

VΣ × [n rs × n cs ] |VΣ × [n rs × n cs ]|

(6.72)

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The Geometry of the Active Part of a Cutting Tool

nse

Rs Pse

dl

nrs VΣ

M ce ncs

Cs

Figure 6.18 Elementary cutting wedge of the infinitesimal length dl.

It is useful to keep in mind that the approximation n se ≅ nT is valid in most practical cases of the computations. Unit vectors v Σ (or V Σ ) and n se are helpful for the analytical representation of the tool-in-use system. 6.3.3 Reference Planes Investigation of the impact of kinematics of a machining operation on actual (kinematical) values of geometry of the active part of a cutting tool can be traced back to research done by Pankin [6] or even to earlier works. A proper tool-in-use system is necessary but not sufficient for determining geometric parameters of the active part of a cutting tool. The specification of the configuration of reference planes is also of critical importance. For free orthogonal cutting, the reference plane for the rake angle g, the clearance angle a, the cutting wedge angle b, and the angle of cutting d is the plane through the vector V Σ . This reference plane is orthogonal to the plane of cut Pse. For free oblique cutting, there are several reference planes for specification of the angles g, a, b, and d. The configuration of reference planes for nonfree cutting cannot be specified in general terms. The mechanics of non-free cutting has not yet been thoroughly investigated. 6.3.3.1 The Plane of Cut Is Tangential to the Surface of Cut at the Point of Interest M For specification of the configuration of the plane of cut r se.tp, the vector of the resultant motion V Σ of the cutting tool relative to the work, and the unit vector c e that is tangent to the cutting edge at M can be employed (Figure 6.19).

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Kinematic Geometry of Surface Machining

YT nse

Rs Pse

nrs

ZT

M VΣ

λs

ce

nce

ncs

XT

Cs

(a)

Pse –λs

+λs

VΣ

M

ZT λs nce XT

ce

(b) Figure 6.19 Definition of the angle of inclination l s of the cutting edge.

These two vectors immediately yield the vectorial equation for the tangent plane:

(r

se .tp

)

- r se( M ) × [c e × v Σ ] = 0

(6.73)

The angle of inclination of the cutting edge specifies the orientation of the cutting edge relative to the vector V Σ of the resultant motion of the

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The Geometry of the Active Part of a Cutting Tool

263

cutting edge. The angle of inclination λ se is measured within the plane of cut r se.tp. This is the angle between the vector V Σ and the unit normal vector n ce . The vector n ce is orthogonal to the cutting edge (Figure 6.19a), and is within the plane of cut r se.tp. If observing from the end of the unit normal vector n se to the surface of cut Pse, then the positive angle λ se is measured in a counterclockwise direction, and the negative angle λ se is measured in a clockwise direction (see Figure 6.19b). When the equality λ se = 0° is valid, then the cutting is the orthogonal cutting. Otherwise, when λ se ≠ 0° , then the more general case of cutting — the oblique cutting — is observed. Major frictions of the cutting tool (that is, chip deformation, direction of chip flow over the rake surface, etc.) depend upon the actual value of the angle of inclination λ se. The algebraic value of the angle of inclination λ se can be computed from the following equation (Figure 6.19b):

c ⋅v λ se = ∠ (c e , v Σ ) - 90° = - arctan e Σ |c e × v Σ |

(6.74)

For the cutting tools of various designs the optimal value of the angle of inclination λ se varies within the interval λ se = ± 80°. 6.3.3.2 The Normal Reference Plane Configuration of the normal reference plane Pne of a cutting tool in the toolin-use system is identical to its configuration in the tool-in-hand system. The normal plane is orthogonal simultaneously to the rake surface R s, to the clearance surface Cs of the cutting wedge, to the plane of cut Pse, and ultimately, to the cutting edge (Figure 6.20). The unit normal vector n ce to the cutting edge is within the normal reference plane Pne. Therefore, configuration of the normal reference plane Pne can be specified in terms of any two unit vectors n rs, n cs, n se, and n ce at the point M (Figure 6.20), or by the point M and the unit vector c e along the cutting edge. Evidently, there are many more options for the specification of configuration of the normal reference plane in the tool-in-use system rather than in the tool-in-hand system. 6.3.3.2.1 Normal Rake Angle Orientation of the rake surface of a cutting tool relative to the plane of cut depends upon the actual value of normal rake angle γ ne. The normal rake angle is measured in the normal reference plane. This is the angle that forms the unit normal vector n se to the plane of cut Pse and the rake surface R s . The value of the angle γ ne is measured from the vector n se toward the rake surface R s. The normal rake angle γ ne is positive when the unit normal vector n se does not pass through the cutting wedge of the tool, and it is negative when the vector n se is passing through the cutting wedge of the tool (Figure 6.20b). It is convenient to determine the normal rake angle γ ne as the angle that complements to 90° the angle between the unit normal vectors n se and n rs

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Kinematic Geometry of Surface Machining

YT ns

Rs nrs

Pse

A

ZT VΣ

ce nce

Pn

Cs XT

(a)

–γn

YT

+γn γn

Rs

βn

ns A

nrs

δn

Pse

+αn pl.XTZT

nce M ncs

Cs

(b) Figure 6.20 Geometry of the active part of a cutting tool in the normal reference plane Pn.

(see Figure 6.20b):

n ⋅n γ ne = 90° - ∠ (n rs , n se ) = arctan rs se |n rs × n se |

(6.75)

For cutting tools of various designs, the optimal value of the normal rake angle γ ne is usually within the interval γ ne = -10° ÷ 30°.

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The Geometry of the Active Part of a Cutting Tool

6.3.3.2.2 Normal Clearance Angle Orientation of the clearance surface Cs with respect to the plane of cut Pse depends upon the normal clearance angle α ne . This angle is measured in the normal reference plane. The normal clearance angle α ne is the angle that the unit normal vector n se forms with the opposite direction of the unit normal vector — the clearance surface Cs. The value of the clearance angle α ne is measured from the plane of cut Pse toward the clearance surface Cs . The normal rake angle α ne is always positive (α ne > 0°) . Only within a narrow land along the cutting edge can the normal clearance angle α ne be equal to zero or even be negative (α ne ≤ 0°). It is convenient to determine the normal clearance angle α ne as the angle that complements to 180° the angle between the unit vectors n se and n cs (see Figure 6.20b):

|n × n se | α ne = 180° - ∠ (n cs , n se ) = - arctan cs n cs ⋅ n se

(6.76)

For cutting tools of various designs, the optimal value of the normal clearance angle α ne is usually within the interval α ne = 10° ÷ 30°. The uncut chip thickness a is the predominant factor that affects the optimal value of the clearance angle. On the premises of the analysis of impact of chip thickness a, Larin [23] proposed an empirical formulae

α ne = arcsin

0.13 a0.3

(6.77)

for the computation of reasonable value of the clearance angle. After a short period of cutting, a zero clearance angle α ne = 0° is observed within a narrow worn land along the cutting wedge. 6.3.3.2.3 The Mandatory Relationship For a workable cutting tool, satisfaction of the relationship N se ⋅ Nce < 0 (or the equivalent relationship n se ⋅ n ce = -1 ) is necessary (see Figure 6.20). Violation of the relationship is allowed only within a narrow land along the cutting wedge. The normal cutting wedge angle is measured in the normal reference plane. The normal cutting wedge angle is the angle that forms the rake plane R s and the clearance plane Cs . The value of the angle β ne can be computed from a simple equation (see Figure 6.20b):

β ne = 90° - (α ne + γ ne )

(6.78)

The normal cutting angle is measured in the normal reference plane. The normal cutting angle is the angle that forms the plane of cut Pse and the clearance plane Cs. The value of this angle δ ne is equal (see Figure 6.20b):

© 2008 by Taylor & Francis Group, LLC

δ ne = 90° - γ ne

(6.79)

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Kinematic Geometry of Surface Machining

Definitely, both the angles β ne and δ ne can be expressed in terms of unit normal vectors to the corresponding planes of the cutting wedge, and to the reference surfaces. 6.3.3.3 The Major Section Plane Configuration of the major section plane Pve is determined by two directions through the point M. One of the directions is specified by the unit normal vector n se to the plane of cut Pse, and another direction is specified by the vector of the resultant motion of the cutting tool V Σ with respect to the work (Figure 6.21a). The major section plane Pve is perpendicular to the plane of cut Pse. The equation of the major section plane Pve in terms of the vectors V Σ and n se yields representation in vectorial form:

(r

ve .tp

)

- r se( M ) × [n se × v Σ ] = 0

(6.80)

where r ve.tp designates the position vector of a point of the major section plane. The rake angle γ ve is measured in the major section plane Pve (Figure 6.21b). The rake angle γ ve is equal to the angle between the unit normal vector n se to the plane of cut, and the unit vector b is tangent to R s and is located within the reference plane Pve :

γ ve = ∠ (n se , b) = arctan

|n se × b| n se ⋅ b

(6.81)

The rake angle γ ve is positive when the vector n se does not penetrate the cutting wedge, and it is negative when it does (Figure 6.21b). The clearance angle α ve is the angle that the unit normal vector n se makes with the unit vector c. Here, the unit vector c is tangent to the line of intersection of the clearance surface Cs by the major section plane Pve (see Figure 6.21b):

α ve = ∠ (n se , c) = arctan

|n se × c| n se ⋅ c

(6.82)

The cutting wedge angle β ve is the angle between the unit vectors b and c (see Figure 6.21b):

β ve = ∠ ( b, c) = arctan

| b × c| b⋅c

(6.83)

In the major section plane Pve, the equality β ve = 90° - (α ve + γ ve ) is always observed. The angle of cutting δ ve is the angle that the unit vector b makes with the vector V Σ of the resultant motion of the cutting tool relative to the surface of

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The Geometry of the Active Part of a Cutting Tool

YT ns

Rs Pse

b

nrs

ZT VΣ

M ce

Pm ncs

XT

Cs

(a)

−γm

YT

+γm γm

βm

ns

Rs

b nrs ZT

δm Pse

c VΣ

+αm

M ncs

Cs (b)

Figure 6.21 Geometry of the active part of a cutting tool in the major plane Pm.

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Kinematic Geometry of Surface Machining

the cut (see Figure 6.21b):

|b × v Σ | δ ve = 180° - ∠ ( b, VΣ ) = - arctan b ⋅ vΣ

(6.84)

The equality δ ve = (90° - γ ve ) is always observed in the major section plane Pve.

6.3.3.4 Correspondence between the Geometric Parameters Measured in Different Reference Planes When geometric parameters of the active part of a cutting tool are known in the plane of cut Pse and in the normal reference plane Pne , then the corresponding geometric parameters can be computed in the major section plane Pve , and vice versa. Consider the computation of the rake angle γ ve as an example of the proposed approach. The origin of the Cartesian coordinate system XT YT Z T is at the point of interest M (see Figure 6.19) within the cutting edge. Construct a vector A that is tangent to the line of intersection of the rake surface R s by the normal reference plane Pne (see Figure 6.20). The projection length of vector A onto the coordinate plane XT Z T is equal to unity (Pr zx A = 1). This yields the following equation:

-i sin λ se   cot γ  ne  A=  - cos λ se    1  

(6.85)

The unit vector b is tangent to the line of intersection of the rake surface R s by the major section plane Pve (see Figure 6.21):

 0   cos γ  νe  b= - sin γ ν e     1 

(6.86)

The unit vector c e is tangent to the cutting edge at the point of interest M (see Figure 6.19). It is equal

© 2008 by Taylor & Francis Group, LLC

 0   cos λ  s  ce =  - sin λ se     1 

(6.87)

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The Geometry of the Active Part of a Cutting Tool

By construction all three vectors A, b, c e are within a certain plane that is tangent to the rake surface R s at the point of interest M. This means that the vectors A, b, c e represent a set of coplanar vectors. Therefore, the triple product of these vectors is identical to zero (A × b ⋅ c e ≡ 0) . Consequently, the following equality is valid: - sin λ se [ A , b, c e ] = 0 cos λ se

cot γ ne cos γ ve 0

- cos λ se - sin γ ve ≡ 0 - sin λ se

(6.88)

After the required formulae transformation are performed, one can come up with the equations for the computation of the rake angle γ ve: tan γ ve =

tan γ ne cos λ se

(6.89)

or in the form cot γ ve = cot γ ne cos λ se

(6.90)

Following the way similar to that disclosed above, the equation

tan α ve = tan α ne cos λ se

(6.91)

for the computation of the clearance angle α ve can be derived. Equation (6.89) through Equation (6.91) for the computation of the rake angle γ ve and the clearance angle α ve are known since the publication by Stabler [20]. The roundness r of the cutting edge in the major section plane Pve can be computed from the equation ρve = ρne ⋅ cos λ se. Here ρne denotes the roundness of the cutting edge in the normal reference plane Pne. The equation for ρve is another example of the correlation between the geometric parameters of the active part of a cutting tool measured in different reference planes. 6.3.3.5 The Main Reference Plane Pre is orthogonal to the vector V Σ of the resultant motion of the cutting tool with respect to the surface of the cut (Figure 6.22). This reference plane can also be determined as a plane through the unit normal vector n se to the surface of cutting Pse, and through the unit vector me that is orthogonal to the vector V Σ . The unit normal me belongs to the surface of cutting Pse (Figure 6.22). In the coordinate system XT YT Z T (see Figure 6.22), the unit normal vector me is identical to the unit vector i, i.e., of the XT-axis me = i .

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270

Kinematic Geometry of Surface Machining YT ns

Pre Ps

Rs

ZT

VΣ

M me

Cs XT

(a) ε YT

Pse nP

A C

nse P

e

a

t e

M

Vf

B

b

K

RP

XT RT

RTe

(b) Figure 6.22 Geometry of the active part of a cutting tool in the main reference plane Pr .

6.3.3.5.1 The Major Cutting Edge Approach Angle The vector V f of the feed-rate motion and projection of the major cutting edge of a cutting tool make the major cutting edge approach angle ϕ e (Figure 6.23). The angle ϕ e is an acute angle (0° < ϕ e ≤ 90°) . The major cutting edge approach angle ϕ e can be computed from the equation

ϕ e = arctan

© 2008 by Taylor & Francis Group, LLC

|V f(ϕ ) × C (ϕ e ) | V f(ϕ ) ⋅ C (ϕ e )

(6.92)

271

The Geometry of the Active Part of a Cutting Tool

RT C(κr)

The Cutting Edge Pnom

a

κr (κ ) Vf r

M(κr)

nP

Vf(κr1)

b M(κr1)

κr1

K

C(κr) Pact

RP Figure 6.23 Cutting edge angle k r and the minor (end) cutting edge angle k r 1 for a curved cutting edge in the tool-in-use reference system.

where C (ϕ e ) is a vector that aligns with the major cutting edge. In the case of a curved cutting edge, the vector C (ϕ ) is aligned with the tangent to the cutting edge at the point of interest. 6.3.3.5.2 The Minor Cutting Edge Approach Angle Similarly, the minor cutting edge approach angle ϕ 1e can be measured between the projection of the minor cutting edge and the vector V f of the feed-rate motion (Figure 6.23). The angle ϕ 1e is also an acute angle (0° ≤ ϕ1e ≤ 90°) . Moreover, usually the value of the angle ϕ 1e does not exceed the value of the corresponding angle ϕ e. For the computation of the minor cutting edge approach angle ϕ 1e, the following formula can be employed: (ϕ 1 e )

ϕ e1 = arctan

|V f

(ϕ 1 e )

Vf

× C (ϕ 1e ) | ⋅ C (ϕ 1e )

(6.93)

where C (ϕ1e ) denotes the vector that aligns with the minor cutting edge. In the case of a curved cutting edge, the vector C (ϕ1e ) is aligned with the tangent to the cutting edge at the point of interest. The angle ϕ 1e is computed for a portion of the cutting edge within the residual cusps. On the rest of the portion of the cutting edge, it does not affect the material removal process. In the event of small angles γ ne and λ se, the analysis of actual values of the angles ϕ e and ϕ 1e can be performed not in the main reference plane Pse but in the rake plane of the cutting tool. In this case, instead of actual values of the angles ϕ e and ϕ 1e, projections of these angles onto the rake plane can be used.

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Kinematic Geometry of Surface Machining

The tool tip (nose) angle ε e is determined for the tip of a cutting tool. The angle ε e can be computed from the equation

ε e = 180° - (ϕ e + ϕ 1e )

(6.94)

The tip of a form-cutting tool coincides with the point of contact K of the generating surface T of the cutting tool and the surface P being machined (see Figure 6.23). At the point K the tool tip angle ε e = 180°. The cutting edge approach angle ϕ e affects the parameters of the uncut chip, say of thickness a and width b of the uncut chip. If the depth of cut t, and the feed-rate V f (or S) are of constant value, then the following two equations a = S ⋅ sin ϕ e and b = t sin ϕ e are valid, and there will be a lower cutting edge approach angle ϕ e, a higher width b of the uncut chip, and a bigger tool-nose angle ε e. Both the angles ϕ e and ϕ 1e affect parameters of residual cusps on the machined part surface. Bigger ϕ e or ϕ 1e results in higher residual cusps on the machined part surface. 6.3.3.6 The Reference Plane of Chip Flow In the case of free orthogonal cutting (when the inclination angle λ se = 0° ), the vector of chip motion over the rake surface is orthogonal to the cutting edge. Kinematical geometric parameters of the cutting edge are specified in the plane that is orthogonal to the cutting edge. The correctness of that approach is comprehensively validated experimentally. Oblique cutting (when the angle of inclination λ se ≠ 0° ) is a much more complex phenomenon than orthogonal cutting. This is first because deformation of material does not occur in the major reference plane Pm , but within a certain volume, and thus deformation of material in a three-dimensional space occurs. Oblique cutting is much less understood than orthogonal cutting. However, approximate results of the investigation of orthogonal cutting can be adjusted for implementation for the analysis of oblique cutting as well. For oblique cutting, it is necessary to specify the rake angle taking into consideration the direction of chip flow over the rake face. Lots of research has been carried out to determine the actual direction of chip flow over the rake face. The research was summarized by Stabler [20]. Without going into detail, consider Stabler’s chip flow law. 6.3.3.6.1 Stabler’s Chip Flow Law It is convenient to specify the direction of chip flow over the rake surface in terms of the chip-flow angle h. The chip-flow angle η is measured within the rake plane. This is the angle that the vector Vcf of the chip flow makes with the perpendicular to the cutting edge within the rake plane (Stabler [20]).

© 2008 by Taylor & Francis Group, LLC

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The Geometry of the Active Part of a Cutting Tool

The chip-flow angle η can be expressed in terms of width of cut bcf, width b of the machined plane, and of inclination angle λ se

cos η =

bcf b

cos λ se

(6.95)

Equation (6.95) is derived under the assumption that there is no deformation within the chip width. It is proven that this assumption is valid for orthogonal cutting [21]. In compliance with the chip-flow law, the chip-flow angle η is approximately equal to the angle of inclination of the cutting edge λ se. This correlation can be analytically expressed by the following approximate equality η ≅ λ se . The equation is based on the assumption that bcf = b, and it is valid for all cutting tools having the inclination angle λ se < 45° . When the inclination angle λ se ≥ 45° , then the difference between the angles λ se and h remains within (λ se - η) ≤ 5 ÷ 6° . Stabler later modified the chip-flow law and represented it in the form η ≅ (1, 0 ÷ 0, 9) λ se . Along with the chip-flow angle h, the direction of chip flow over the rake face can be specified by the projection of this angle onto the main reference plane [20]. 6.3.3.6.2 The Chip-Flow Rake Angle In an attempt to specify (approximately) the poorly understood oblique cutting in terms of the comprehensively investigated orthogonal cutting, the term chip-flow reference plane was introduced. The chip-flow rake angle γ cf is measured in the chip-flow reference plane Pcf . The rake angle γ cf (as well as the depth of cut tcf ) in the chip-flow reference plane differs from the analogue parameters measured in other reference planes. The chip-flow reference plane Pcf is the plane through the vectors V Σ and Vcf . Here V Σ designates the vector of resultant motion of the cutting edge with respect to the surface of the cut, and Vcf designates the vector of chip flow over the rake surface. The vector Vcf is located within the rake plane. It forms the chip-flow angle h perpendicular to the cutting edge (Figure 6.24). The vector Vcf is orthogonal to the unit normal vector n rs to the rake surface R s. Therefore, the equality Vcf ⋅ n rs = 0 occurs. At the point of interest, the chip-flow reference plane Pcf is a plane through the vectors V Σ and Vcf at the point M. This yields

(r cf - r ( M ) ) ⋅ V Σ × V cf = 0

(6.96)

Here r cf designates the position vector of a point of the chip-flow reference plane Pcf , and r ( M ) designates the position vector of the point of interest M. The chip-flow rake angle γ cf is the angle that the vector Vcf of chip flow over the rake plane forms with the main reference plane Pre (Figure 6.20).

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274

Kinematic Geometry of Surface Machining

YT Pre Pse

ns

η γcf

Rs

Vcf

nrs ZT

VΣ Pcf

M

XT

Cs

Figure 6.24 Geometry of the active part of a cutting tool in the chip-flow reference plane Pcf .

Therefore, the rake angle γ cf can be computed from the equation |VΣ × Vcf | VΣ ⋅ Vcf - 90° = arctan γ cf = ∠(VΣ , Vcf ) - 90° = arctan VΣ ⋅ Vcf |VΣ × Vcf |

(6.97)

For the unit vector vΣ = VΣ/|VΣ|, one can derive that - sin λ se   cos λ  se  vΣ =   0     1 

(6.98)

The projection length of vector v cf onto the coordinate plane YZ is a unity vector. Therefore, the following equality is valid for the vector v cf :  tan η  - sin γ  ne  v cf =   cos γ ne     1 

(6.99)

Substituting Equation (6.98) and Equation (6.99) into Equation (6.97), one can come up with the equation for the computation of the chip-flow rake angle γ cf :

sin γ cf = sin η sin λ se + cos η cos λ se sin γ ne

© 2008 by Taylor & Francis Group, LLC

(6.100)

275

The Geometry of the Active Part of a Cutting Tool

Usually, the angle of inclination does not exceed λ se ≤ 45° . Under such a scenario, the following approximate equality η ≅ λ se is valid. This immediately leads to Stabler’s equation for the computation of the rake angle γ cf [20]: sin γ cf ≅ sin 2 λ se + cos 2 λ se sin γ ne ≅ 1 - cos 2 λ se (1 - sin γ ne )

(6.101)

Taking into account that tan γ ne = tan γ ve cos λ se

(6.102)

and sin γ n =

tan γ n = 1 + tan 2 γ n

tan γ ve cos λ se 1 + tan 2 γ ve cos 2 λ se

(6.103)

Equation (6.30) casts into sin γ cf = sin η sin λ se - cos η cos 2 λ se

tan γ ve cos λ se 1 + tan 2 γ ve cos 2 λ se

(6.104)

The similar equation

 tan γ ve cos λ se sin γ cf ≅ 1 - cos 2 λ se  1  1 + tan 2 γ ve cos 2 λ se

  

(6.105)

can be derived from Equation (6.101). The derived equations are valid for free oblique cutting. They enable computation of the chip-flow rake angle γ cf for any actual value of the inclination angle λ se. Computation of the chip-flow angle h is a challenging problem. A reliable value of the chip-flow angle h can be obtained experimentally. The disclosed vectorial method enables one derivation of all necessary equations for the computation of geometry of the active part of a cutting tool in any reference plane. The method works in the tool-in-hand as well as in the tool-in-use reference systems. The method can also be used for derivation of equations of some of the parameters that describe the material removal process (e.g., the angle of cutting, the shear angle j , the angle of

The effective angle of cutting d cf is the angle that can be measured between the opposite direction of the vector VΣ and between the vector Vcf of chip flow over the rake surface. For the case of free oblique cutting, the angle of cutting d cf depends upon the angle of inclination l se and upon the chip-flow angle h — that is, d cf = arccos[cos l se cos h cos d ne + sin l se sin h]. The last equation reveals that the angle of cutting d cf decreases as the inclination angle h se and the chip-flow angle h get smaller. This is in perfect correlation with the results of experimental investigation of the metal cutting process.

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Kinematic Geometry of Surface Machining

action y, the depth of cutting, etc.). All of the parameters can be determined in any reference plane of interest. 6.3.4 A Descriptive-Geometry-Based Method for the Determination of the Chip-Flow Rake Angle In addition to analytical methods, the descriptive-geometry-based method (DGB-method) can also be implemented for the determination of the chipflow rake angle. In order to come up with a DGB-solution to the problem of the determination of the chip-flow rake angle γ cf, the cutting wedge of a tool is depicted in the planes of projections HVF (Figure 6.25). Here and below, the subscript “1” is assigned to projections of all elements (points, lines, planes) onto the horizontal plane of projections H. The subscript “2” is assigned to projections of those same elements onto the vertical plane of projections V. Similarly, the subscript “3” is assigned to projections of those same elements onto the frontal plane of projections F. While cutting, cutting wedge 1 travels along the H/V axis. The speed V Σ of this motion is projected without any distortion onto both planes of projections, say on the planes of projections H and V. Due to this, no subscripts are assigned to the projections of the vector V Σ . Z Rs2

H Rs1

F

Rr3

3n

l3

s

γcf

VΣ V

V

l2

2n s

M2

a2

2V c

a3

b2

a1 1m c

VΣ

M1

λs 1nc Y

3V cf

M3

b3

l1 1V

a5

cf

b1

Rs5

M5 X

η5

b5

H W

M4

l5

5

4V cf 4

ns γn

Vcf U W

η 5

mc

П4

Figure 6.25 A descriptive-geometry-based method for the determination of the actual value of the rake angle g cf in the chip-flow reference plane Pcf.

© 2008 by Taylor & Francis Group, LLC

The Geometry of the Active Part of a Cutting Tool

277

The cutting edge ab inclination angle is designated conventionally as λ se. The reference plane Pse within which the inclination angle λ se can be measured is parallel to the horizontal plane of projections H. The inclination angle λ se is measured between the unit normal vector n c to the cutting edge and the vector V Σ of the cutting wedge motion. The unit normal vector n se to the surface of cut Pse is erected at the point of interest M. The two auxiliary planes of projections, say W and U are constructed for the determination of the vector Vcf of chip flow. Subscripts “4” and “5” are assigned to projections of all elements onto the planes of projections W and U, respectively. The axis of projections HW is at a right angle to the projection a1b1 of the cutting edge onto the horizontal plane of projections H. Onto the auxiliary plane of projections W, the cutting edge is projected into the point A4 ≡ B4 ≡ M 4 . The rake plane is projected onto the plane of projections W into the trace R s4 . Finally, the normal rake angle γ ne is projected onto the plane W without distortions. The projection of the rake angle γ ne onto the plane W is equal to the angle γ ne. The axis of projections W/U is parallel to the trace R s4 of the rake plane. Therefore, the rake plane R s is projected onto the auxiliary plane of projections U without distortions. Due to that, the projection of the chip-flow angle h onto the plane of projections U is equal to the angle h, and the vector Vcf of chip flow is at the angle h to the unit normal vector mc to the cutting wedge. Using conventional rules from descriptive geometry, one can construct projections of the vector Vcf of chip flow onto all the other planes of projections. Finally, the projection V cf1is turned about the axis that is parallel to the axis of projections V/H. In the orientation of the projection Vcf1 when it is parallel to the vertical plane of projections, the nondistorted value of the chipflow rake angle γ cf is obtained in the plane of projections V (Figure 6.25).

6.4 On Capabilities of the Analysis of Geometry of the Active Part of Cutting Tools Implementation of the vectorial method for the computation of geometry of the active part of a cutting tool is very helpful. In order to demonstrate the creative capabilities of the method, consider a few practical examples. 6.4.1 Elements of Geometry of Active Part of a Skiving Hob The geometry of active parts of skiving hobs for finishing or semifinishing hardened gears is significantly different compared to the geometry of hobs of conventional design. Only lateral cutting edges of the hob teeth remove the stock — the top edges do not cut the work material [9,10,12,13,18]. This is an important feature of skiving hobs. The negative rake angle γ t at the top

© 2008 by Taylor & Francis Group, LLC

278

Kinematic Geometry of Surface Machining γt

γt.a γt.b a

b

γt.c

a

γt

c

c

ωT dl.T

OT

(a)

γt

b

do.T

ωT dl.T

OT

do.T

(b)

Figure 6.26 Elements of geometry of the active part of skiving hob [US Patent 3.786.719 (Azumi)], having g t.a < g t.b < g t.c (a), and skiving hob [US Patent 1.114.543], having g c = Const (b).

edge of the hob varies within the interval γ t = (-30° ÷ - 60°) . The big negative rake angle γ t is another important feature of the design of skiving hobs. Due to the big negative rake angle γ t at the top edge of the skiving hob, the inclination angle l of the lateral cutting edges varies greatly. The estimation of the variation of the inclination angle l can be derived from Figure 6.26a that reveals that the inequality γ t. a < γ t.b < γ t.c is observed. The difference between actual values of the inclination angle l that are measured at the outside diameter of the hob do.T and at the limit diameter of the hob dl.T is in the range up to Dλ ≅ 30° . It is known that deviation of the inclination angle of a cutting tool just on Dλ ≅ 5° from its optimal value could cause a double reduction of the cutting tool life. This means that design of a skiving hob could be significantly improved if the inclination angle of the hob cutting edges would be of constant value within the cutting edge, and the angle of inclination would be of optimal value at every point of the cutting edge of the hob. For this purpose, it is recommended that the rake face be shaped with the help of a convex segment of logarithmic spiral curve having the pole at the hob axis of rotation OT as shown in Figure 6.26b. (For hobs having positive rake angle γ t , a concave segment of the logarithmic spiral curve is used.) The equation of the generating curve of the hob rake surface can be derived on the premises of a differential equation for isogonal trajectories

© 2008 by Taylor & Francis Group, LLC

279

The Geometry of the Active Part of a Cutting Tool (see Equation 6.14):

 ∂ϕ   ∂ϕ  ∂ϕ ∂ϕ  ∂x cos ς - ∂y sin ς  dx +  ∂x sin ς + ∂y cos ς  dy = 0

(6.106)

In the case under consideration, the solution to Equation (6.106) returns the equation of the logarithmic spiral curve (see Equation 6.9):

ρ = ρ0 eϕ tan λ opt

(6.107)

Actually, Equation (6.107) is a reasonably good approximation of the requirement λopt = Const. This is because the actual value of the inclination angle λ is measured not in the transverse cross-section of the hob cutting edge, but within the surface of the cut. However, it is proven that the derived solution (see Equation 6.107) is practical. 6.4.2 Elements of Geometry of the Active Part of a Cutting Tool for Machining Modified Gear Teeth Change in the shape of the rake surface of hob teeth could enhance capabilities of the cutting tool. It would be possible to use the hob having a modified rake surface of the teeth for machining of gears having a modified tooth profile. Modification of the gear tooth profile is recognized as a powerful tool for improving the performance of gear drives. For this purpose, the rake surface of the hob teeth is composed of two portions Rs1 and Rs2 [8,13]. These two portions of the rake surface (Figure 6.27) are at a certain angle j to each other. In order to ensure the required value of the gear tooth profile modification, it is necessary to accurately compute the actual value of the angle j. Rs1

Cs

Rs1

A2 αrs O2

Y2 ac

X2

c3

c2

B2

A3

αt

hm

O3

γt

αrm

B3 X3

π2

π3

VΣ

Rs2

γm Δγm

π1 Figure 6.27 Elements of geometry of active part of a gear-cutting tool having modified rake surface (USSR Patent 1.017.444).

© 2008 by Taylor & Francis Group, LLC

280

Kinematic Geometry of Surface Machining

Consider a Cartesian coordinate system XYZ associated with the hob tooth. Origin O of the coordinate system is within the line of intersection of the two portions of the hob rake surfaces. The axis Y aligns to the line of intersection of the two portions of the rake surface. The axis Z is parallel to the vector VΣ , and ultimately, the axis X complements the axes Y and Z to the right-hand-oriented Cartesian coordinate system XYZ. Consider a plane through the origin O. This plane is tangent to the hob teeth clearance surface Cs. Then, construct three vectors A, B, and c. These three vectors are constructed in such a way that all are within the common tangent plane to the clearance surface Cs of the hob teeth. Magnitudes of the vectors A and B are prespecified in order to get projections of these vectors onto the coordinate plane XZ equal to the unity. Vector c is a unit vector. In compliance with Figure 6.27, the following expressions for the vectors A, B, and c can be composed:

(6.108)

- cos γ t   tan α  rs  A=  sin γ t     1 

(6.109)

 cos γ m  - tan α  rm  B=   sin α t    1  

(6.110)

 sin α t   0   c= - cos α t     1 

where, in Equation (6.108) through Equation (6.110), γ t is the rake angle at the top of the hob tooth, γ m is the rake angle at the modified portion of the hob tooth, α t is the clearance angle of the hob tooth, α rs is the nominal pressure angle of the hob tooth, and α rm is the pressure angle of the modified portion of the hob tooth. By construction, three vectors A, B, and c make up a triple of coplanar vectors. Due to this, the triple product of the vectors is identical to zero: A × B ⋅ c ≡ 0. The last identity yields the equation

- cos γ t  [A, B, c] =  cos γ m  sin α t

© 2008 by Taylor & Francis Group, LLC

tan α rs - tan α rm 0

sin γ t   sin α t  ≡ 0 - cos α t 

(6.111)

281

The Geometry of the Active Part of a Cutting Tool Equation (6.111) casts into the formulae

 (1 - tan γ t tan α rm ) - tan α rm cos γ t  ϕ = γ t + γ m = γ t + tan -1  ⋅  tan α rs sin α t  

(6.112)

for the computation of the desired angle j between the portions Rs1 and Rs2 of the rake surface of the hob. In a particular case when the rake angle γ t is equal to zero (say when the equality γ t = 0o is observed), then Equation (6.112) reduces to tan ϕ =

tan α rs - tan α rm tan α rs sin α t

(6.113)

Even a brief look through the capabilities of analysis of the cutting tool geometry reveals how much room for improvement can be found there. 6.4.3 Elements of Geometry of the Active Part of a Precision Involute Hob Consider a precision involute hob having straight lateral cutting edges [7]. The concept of the design of the hob is based on the following considerations: Lateral cutting edges of one side of an involute hob tooth belong to the corresponding screw involute surface. Lateral cutting edges of the opposite side of the involute hob tooth belong to the opposite screw involute surface. The screw involute surfaces of the opposite sides of the involute hob tooth intersect, and the line of intersection is a helix. Two characteristics E l and E r are passing through every point of the helix. The two characteristics E l and E r through the common point of the helix intersect one another at that point and thus specify a plane. That plane is used as a rake face of the involute hob teeth. The steps listed above yield determination of orientation of the rake surface of the precision involute hob teeth. 6.4.3.1 An Auxiliary Parameter R Lateral tooth surfaces of the auxiliary rack R intersect each other along a straight line through the point A (Figure 6.28). This straight line is at a distance R from the hob axis. For the distance R, Figure 6.29 yields

© 2008 by Taylor & Francis Group, LLC

R = 0.5 ⋅ (dh + tc ⋅ cot φ n )

(6.114)

282

Kinematic Geometry of Surface Machining trf A

R

Yh

th

A

D

C B φr

El

f

The Rake Face

ζh Er

φn ξ

g

Zh D

E

db.h

Xh

Figure 6.28 Elements of geometry of the active part of a precision involute hob (USSR Patent 990.445): orientation of the rake surface.

6.4.3.2

The Angle f r between the Lateral Cutting Edges of the Hob Tooth

Prior to deriving the equation for the computation of the angle φ r that makes the lateral cutting edges of the gear hob tooth, it is convenient to derive an equation for the computation of projection n of the angle φ r onto the coordinate plane X h Yh. The projections of the lateral cutting edges of the involute tooth onto the coordinate plane X h Yh form an angle n. For the computation of the actual value of the angle n, the following expression can be used: tan ν =

db. h 4 ⋅ R 2 - db2. h

(6.115)

Then, consider three unity vectors A, B, and C. These vectors yield the following analytical representations:

© 2008 by Taylor & Francis Group, LLC

cos ζ h   0   A=  sin ζ h     1 

(6.116)

283

The Geometry of the Active Part of a Cutting Tool tr Yh

A A

Prxy C R

D

El

ν

f

El

Er

db.h

g

Oh

Er

Yh φr

0.5 . dh

Pr A

g

R

Xh Prxz E

Prxy C

f

db.h db.h . cotξ

Zh ξ

0.5 . dh φn

tc A

A

E Prxy C

B

A

0.5 . dh ζh Xh Er

φr

db.h

tr

A El

Figure 6.29 Elements of geometry of the active part of a precision involute hob (USSR Patent 990.445): the employed characteristic vectors.

(6.117)

 sin φn sin ζ h   - cos φ  n  B=  - sin φn cos ζ h    1  

(6.118)

- cos φ r tan ν   - cos φ  r  C=  - sin φ r cos ξ    1  

where ζ h designates the hob-setting angle of the involute hob [11,12].

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Kinematic Geometry of Surface Machining

The hob-setting angle ζ h required for further computations can be chosen by the designer of the gear hob. Usually, it is recommended that the actual value of the hob-setting angle ζ h equal to the pitch helix angle ψ h of the hob be assigned. As proven in our earlier work [12], in order to satisfy the equality ζ h = ψ h (this condition is the best possible), the actual value of the hob-setting angle must be computed from the equation tan ζ h =

m ⋅ Nh (do. h - 2 ⋅ 1.25 ⋅ m - D do. h )2 - m 2 N h2

(6.119)

where Dd0.h designates reduction of the hob outside diameter Dd0.h due to resharpening of the worn gear hob. Three vectors A, B, and C are within the common lateral surface of the auxiliary rack R ; therefore, the following identity A × B ⋅ C ≡0 is observed. The last expression yields a determinant:

cos ζ h sin φn sin ζ h - cos φ r tan ν

0 - cos φn - cos φ r

sin ζ h - sin φn cos ζ h = 0 - sin φ r cos ξ

(6.120)

After expending the determinant (see Equation 6.120), and after the necessary formulae transformations are performed, one can come up with the equation of two unknowns, namely — φ r and x. 6.4.3.3 The Angle x of Intersection of the Rake Surface and of the Hob Axis of Rotation The rake surface of the involute hob is inclined to the hob axis at a certain angle x. In order to determine the required value of the angle x, the following three unity vectors C, D, and E were used. For the vectors D and E, Figure 6.29 yields

D = [0

E = [sin ξ

-1 0

0

1] T

cos ξ

(6.121)

1] T

(6.122)

Three vectors C, D, and E are located within the rake surface of the hob tooth; therefore, the identity C × D ⋅ E ≡0 is observed. This yields a determinant:

- cos φ r tan ν 0 sin ξ

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- cos φ r -1 0

- sin φ r cos ξ 0 =0 cos ξ

(6.123)

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The Geometry of the Active Part of a Cutting Tool

After expending the determinant (see Equation 6.123) and after the necessary formulae transformations are performed, one can come up with one more equation of two unknowns — f r and x. Further, consider the set of two equations — Equation (6.120) and Equation (6.123) of the two unknowns f r and x . The solution to the set of the above equations can be represented in the form tan ξ =

cos ζ h ⋅ tan ν tan φ n + sin ζ h ⋅ tan ν tan φ r =

tan ν sin ξ

(6.124) (6.125)

The hob-setting angle zh specifies inclination of the gear hob axis of rotation Oh with respect to the auxiliary rack R . It is necessary to point out that the angle zh is a parameter of the gear hob design and is not a parameter of the gear hobbing operation. It could be either positive ( +ζ h > 0°) or negative (-ζ h < 0°) , as well as it could be of zero value (ζ h = 0°) . Under special conditions, the hob-setting angle could be equal to the gear hob pitch helix angle ψ R (i.e., the equality ζ h = ψ R could be observed). The above examples reveal capabilities of the vectorial method for the computation of the geometry of the active part of a cutting tool.

References [1] Ismail, F., Elbestawi, M.A., Du, R., and Urbasik, K., Generation of Milled Surfaces Including Tool Dynamics and Wear, ASME Journal of Engineering for Industry, 115, August, 245–252, 1993. [2] Koenderink, J.J., Solid Shape, MIT Press, Cambridge, MA, 1990. [3] Kovacic, I., The Chatter Vibrations in Metal Cutting — Theoretical Approach, University of Nis, The scientific journal Facta Universitatis, Series: Mechanical Engineering, 1 (5), 581–593, 1988. [4] Kunstetter, S., Narze˛ dzia skrawaja˛ ce do metali: Konstrukcja, 2nd ed., Wydawnictwa Naukowo-Techniczne, Warszawa, 1970 (1st ed., 1961). [5] Mozhayev, S.S., Analytical Theory of Twist Drills, Mashgiz, Moscow, 1948. [6] Pankin, A.V., Kinematic Acuting of the Cutting Wedge, Stanki i Instrumetnt, No.1, 1936. [7] Pat. No. 990.445, USSR, A Precision Involute Hob./S.P. Radzevich, Int. Cl. B23F 21/16, Filed October 8, 1981. [8] Pat. No. 1.017.444, USSR, An Involute Hob./S.P. Radzevich, Int. Cl. B 23f21/16, Filed April 26, 1982. [9] Pat. No. 1.114.543, USSR, A Gear Hob./S.P. Radzevich, Int. Cl. B 23f21/16, Filed September 7, 1982.

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[10] Pat. No. 3.786.719, US, A Gear Hob./AZUMI, January 22, 1974. [11] Radzevich, S.P., About Hob Idle Distance in Gear Hobbing Operation, Journal of Mechanical Design, 124 (4), 772–786, 2002. [12] Radzevich, S.P., Cutting Tools for Machining Hardened Gears, VNIITEMR, Moscow, 1992. [13] Radzevich, S.P., Design and Investigation of Skiving Hobs for Finishing Hardened Gears, PhD thesis, Kiev Polytechnic Institute, Kiev, 1982. [14] Radzevich, S.P., Differential-Geometric Method of Surface Generation, DrSci thesis, Tula Polytechnic Institute, Tula, Russia, 1991. [15] Radzevich, S.P., Fundamentals of Surface Generation, Rastan, Kiev, 2001. [16] Radzevich, S.P., Geometry of the Active Part of Cutting Tools (in the Tool-inHand System), SME Summit: Where Manufacturers, Technologies and Innovations Connect, August 3–4, 2005, Olympia Resort and Conference Center, Oconomowoc (Milwaukee, WI), SME Paper TP06PUB37, published May 15, 2006. [17] Radzevich, S.P., Geometry of the Active Part of Cutting Tools (in the Tool-inUse System), SME Midwest Summit: MIDWEST 2005 — Manufacturing Excellence Through Collaboration Conference, September 13–14, 2005, Rock Financial Showplace (Novi, MI), SME Paper TP06PUB36, published May 15, 2006. [18] Radzevich, S.P., Production of Hardened Gears, VNIITEMR, Moscow, 1985. [19] Radzevich, S.P., Sculptured Surface Machining on Multi-Axis NC Machine, Vishcha Shkola, Kiev, 1991. [20] Stabler, G.V., The Fundamental Geometry of Cutting Tools, Proceedings of Institution of Mechanical Engineers, 165, 14–21, 1951. [21] Shaw, M.C., Metal Cutting Principles, Clarendon Press, Oxford, 1984. [22] Time, I.A., Resistance of Metals and Wood to Cutting, Dermacow Press House, St. Petersburg, Russia, 1870. [23] Larin, M.N., Optimal Geometrical Parameters of the Metal Cutting Tools, Moscow, Oborongiz, 1953.

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7 Conditions of Proper Part Surface Generation When machining a part surface, the cutting tool is traveling with respect to the work. While traveling, the cutting tool removes the operating stock. The machined surface can be considered as an enveloping surface to consecutive positions of the cutting tool relative to the work. Due to peculiarities of shape of the part surface to be machined, of the generating surface of the cutting tool, and of the kinematics of the machining operation, the shape of the machined part surface could deviate from its desired shape. When a portion of stock on the part surface remains uncut, then an overcut is observed. When the cutting tool removes material beneath the part surface, then an undercut is observed. Both the undercut and the overcut are allowed if and only if the resultant deviations of the machined part surface from the desired part surface are within the tolerance on accuracy of the part surface. Practically, it is important to investigate the reasons why deviations of the actual machined part surface Pac from the desired nominal part surface Pnom are caused. This problem becomes more severe for computer-aided design and computer-aided manufacturing (CAD/CAM) applications when the entire machining process must be completely formalized. In order to ensure precise machining of the given part surface, it is necessary to properly orient the work on the worktable of a multi-axis numerical control (NC) machine and to satisfy a set of conditions of proper part surface generation. For the analysis and for analytical description of conditions of proper part surface generation, it is presumed that the part surface P is described analytically (see Chapter 1), kinematics of the machining operation is known (see Chapter 2), and the generating surface T of the cutting tool is determined (see Chapter 5).

7.1

Optimal Workpiece Orientation on the Worktable of a Multi-Axis Numerical Control (NC) Machine

There are many feasible configurations of a given sculptured part surface on the worktable of a multi-axis NC machine. It is natural to assume that not all of them are equivalent, and that a particular orientation of the sculptured surface is preferred for machining purposes — that is, this configuration is optimal in a certain sense. 287 © 2008 by Taylor & Francis Group, LLC

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Kinematic Geometry of Surface Machining

Consider the general case of machining a sculptured surface on a multiaxis NC machine. Optimal workpiece orientation is generally defined as orientation of the workpiece so as to minimize the number of setups in the multi-axis NC machining of a given sculptured surface P, or to allow the maximal number of surfaces to be machined in a single setup. Here, a method for computing such an optimal workpiece orientation is developed based on the geometry of the sculptured part surface P to be machined, on the geometry of the generating surface T of the tool, and on the articulation capabilities of the multi-axis NC machine. In addition, for the cases in which some freedom of orientation remains after conditions for machining in a single setup are satisfied, a second sort of optimality can also be considered: finding an orientation such that the cutting condition remains as close as possible to the conditions of machining at a surface point at which they are the most favorable. This second form of optimality is obtained by choosing an orientation (within the bounds of those allowing a single setup) in which the angle between the neutral axis of the milling tool and the area-weighted mean normal to the part surface is zero, or as small as possible. To find this solution, mapping of surfaces on a unit sphere sounds promising. Mapping of a surface on a unit sphere was initially proposed by Gauss [4]. He used this kind of surface mapping for the purpose of investigating the surface topology. Since Gauss’ publication [4], the mapping of a surface on a unit sphere is usually referred to as Gauss’ mapping of the surface. Later Gauss’ idea of surface mapping received wide implementation both in science [1] and engineering. As early as in 1987, Radzevich [9] applied Gauss’ idea to the sculptured part surface orientation problem. He developed the general approach to solve this important engineering problem [9,16,20,22]. The proposed method of finding the optimal workpiece orientation maintains minimal difference of the angle between the normal to sculptured part surface P at its central point and a milling tool axis at its optimal position, in order to minimize the number of setups in rough machining. By means of Gauss’ mapping of surfaces, the problem of optimal workpiece orientation can be formulated as a geometric problem on a unit sphere. The method incorporates the impact of the area-weighted mean normal to the surface P. This technique is applicable for cutting tools of any design. 7.1.1

Analysis of a Given Workpiece Orientation

Rational workpiece orientation is an important task in optimal sculptured surface machining on a multi-axis NC machine. When machining a sculptured surface, conditions of cutting strongly depend upon the location of the point on the surface P in the vicinity of *

Pat. No.1442371, USSR, A Method of Optimal Work-Piece Orientation on the Worktable of Multi-Axis NC Machine./S. P. Radzevich. Int. Cl. B23q15/007, Filed February 17, 1987.

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Conditions of Proper Part Surface Generation

nT

KJ

P

nP

T

nP

nP

T

P

KH

P

T

KS nT

nT P

ωT

ZP nT

nP

P

KB

T

R

S

J

nT A

T

nP

Ω

Axy YP

A*

P

αP

H KI

Q

γP

nP

T

P

M β P

B I * Bxy B

OT

XP C D

Figure 7.1 An arbitrary orientation of a sculptured surface P on the worktable of a multi-axis numerical control machine.

which machining is occurring and on the relative orientation of the surface P and the cutting tool while machining. If the actual workpiece orientation is far from being optimal, this leads to decreased tool performance or to a situation in which machining of the surface P is not feasible in one setup. As an example, consider configuration of a sculptured surface P on the worktable of a three-axis NC machine (Figure 7.1). The Cartesian coordinate system X PYP ZP is associated with the work. It is assumed that the ZP axis is parallel to the cutting tool axis of rotation OT . The shadowed planar region Axy Bxy CB A appears in Figure 7.1. Thus, for the given configuration, the portion ABCB A of the sculptured surface P cannot be machined in that setup. Consider an arbitrary curve B IBJHS within the sculptured surface P (Figure 7.1). Points I, B, J, H, and S belong to the curve B IBJHS. At every point of contact K, unit normal vector n P to the surface P and unit normal vector nT to the generating surface T of the tool are directed opposite to each other. When the generating surface T of the cutting tool makes contact with the sculptured surface P at the points B, J, H, and S, then the designation of these points changes to K B , K J , K H, and KS, respectively. Evidently, the angle that the tool axis of rotation OT makes with the unit normal vector n P through the points K B , K J, K H , and KS are of different values. Without going into details of the mechanics of metal cutting, one can conclude that conditions

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Kinematic Geometry of Surface Machining

of the material removal at the points K B, K J , K H , and KS are different. This is mostly due to two reasons: first, because the angles the tool axis of rotation OT makes with the unit normal vector n P at each of these points are of different values; and second, because the normal radii of curvature of the surfaces P and T, as well as their local relative orientation at the points K B, K J, K H , and KS are different. Moreover, in the vicinity of the points I and B , interference of the surfaces P and T occurs. Machining of the surface portion in the vicinity of the curve IBB is not feasible in such an orientation of the surface P. Sculptured surface P can be represented to be an infinite number of small (infinitesimal), flat patches. In this scenario, the criteria of optimization presented here are as follows: To maximize the number of flat patches that can be accessed in a single setup. To minimize the mean difference in the condition of machining over all of these small, flat patches. The last requirement yields interpretation in terms of the difference in angle between part surface unit normal n P and the cutting tool axis of rotation in its optimal configuration. The number of infinitesimal flat patches that approximate the surface P is infinite. It is not possible to simultaneously minimize the angles between the cutting tool axis of rotation and the unit normal vectors to each of the infinitesimal flat patches. To summarize the situation, the infinite number of the unit normal vectors to the surface P is not considered below, but the weighted normal to the sculptured part surface P is considered instead. For cutting condition optimization, and as a useful datum for the setup minimization evaluation, the orientation of the sculptured surface sought is that which minimizes not the maximal angle between the cutting tool axis of rotation and the normal to the surface P, but the angle between the surfacearea-weighted normal to the surface P and the cutting tool axis of rotation. Results of the computation following this approach are reasonably close to results of the computation performed in compliance with another procedure. In the last case, the angle between the surface-area-weighted normal and the average normal to the surface P is minimized. Use of both approaches reduces the mean difference in condition of machining of each small portion of the given sculptured surfaces P and thereby increases tool life. For evaluation of the degree of deviation of actual sculptured surface orientation from its desired orientation, a measure of the deviation is required. 7.1.2

Gaussian Maps of a Sculptured Surface P and of the Generating Surface T of the Cutting Tool

Gauss introduced the notion of mapping of surface normals onto the surface of a unit sphere by means of parallel normals, in which a point on a map is the result of the intersection of the surface normal vector, translated so as to emanate from the center of a unit sphere, with the surface of the unit sphere [4].

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291

Conditions of Proper Part Surface Generation ZP0

nP

P

MP0

M ZP YP

rP0

rP

XP0

XP

YP0

Figure 7.2 Spherical map of point M on the sculptured surface P.

The point on the surface of the unit sphere so produced has come to be known as the Gaussian map of the point on the given surface P. The map of all points on a given sculptured surface P is called its GMap ( P). The boundary of the GMap ( P), if it exists, of a given part region or surface P is referred to as its Gaussian spherical indicatrix GInd ( P). The measure proposed for the degree of deviation from optimal conditions of machining is the mean angle between the neutral axis of rotation of the cutting tool and the weighted normal to the sculptured surface P. For implementation of Gaussian mapping of a surface for the calculation of the parameters of optimal workpiece orientation, it is necessary to derive the corresponding equations of the Gaussian map GMap ( P) of the sculptured surface P, and of the Gaussian map GMap (T ) of the generating surface T of the cutting tool as well. Let position vector of a point on the GMap ( P) of the sculptured surface P be designated rP0 . A Cartesian coordinate system X P 0YP 0 Z P 0 is associated with a unit sphere (Figure 7.2). In the coordinate system X P 0YP 0 Z P 0 , the equality rP 0 = n P is valid (here |n P |= 1 ). Cosines of the angles α , β , γ that the unit normal vector n P makes with the axes of the coordinate system X PYP Z P are calculated by cos α = i ⋅ n P, cos β = j ⋅ n P , and cos γ = k ⋅ n P . The equations above for cos α , cos β , cos γ yield

rP0

 cos α    cos β  =  cos γ     1 

(7.1)

A similar equation holds for the generating surface T of the cutting tool. Exploration of the GMap (P ) of the sculptured surface P (see Equation 7.1), like study of an arbitrary surface, is facilitated by expressing it in canonical

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Kinematic Geometry of Surface Machining

form, say in terms of the first and the second fundamental forms of the GMap ( P) of the given surface P. The first fundamental form Φ1.P 0 of the GMap ( P) is given by expression [2]

Φ1.P 0

⇒ dsP2 0 = e P duP2 + 2 f P duP dvP + g P dvP2

(7.2)

where dsP0 is the differential of an arc of a curve on the unit sphere; eP, f P, gP, are the first-order Gaussian coefficients of the GMap(P); and uP, vP are the parametric coordinates of an arbitrary point of the GMap(P). Omitting bulky derivations, one can write the following equations for the first fundamental form Φ1.P 0 of the GMap ( P) of the part surface P:

Φ1.P 0

⇒ dsP2 0 = Φ2.PM P − Φ1.PGP

(7.3)

where M P designates mean curvature of the sculptured surface P (see Equation 1.15), and GP designates Gaussian curvature of that same surface P (see Equation 1.16). The second fundamental form Φ2.P 0 of the GMap ( P) of a given patch of the surface P is calculated as [2]

Φ2.P 0

⇒ − dr P 0 ⋅ dn P 0 = lP duP2 + 2 mP duP dvP + nP dvP2

(7.4)

and is derived in a similar manner. In Equation (7.4), the values lP , mP , nP are the second-order Gaussian coefficients of the GMap ( P) of the surface unit sphere. Skipping the proofs, some useful properties of the GMap ( P) and GMap (T ) can be noted: The GMap ( P) of an orthogonal net on a sculptured surface P for which mean curvature M P is not equal to zero ( M P ≠ 0 ) is also an orthogonal net if and only if the initial net is made up of lines of curvature. If the mean curvature M P of the surface P is equal to zero (M P = 0), then the net of coordinate lines on the GMap ( P) will be orthogonal as well. Points on the boundaries of the surface P and on its GMap ( P) are not necessarily in one-to-one correspondence. GMap ( P) is a many-to-one map: Each point on a smooth part surface P has a corresponding point on the GMap ( P), but each point on GMap ( P) may correspond to more than one point on the part surface P. This means that in particular cases, GMap ( P) can be interpreted as having more than one layer. GMap ( P) of this kind are often referred to as the multilayer GMap ( P). For example, GMap ( P) of a torus surface is of two layers.

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293

Conditions of Proper Part Surface Generation

Figure 7.3 Examples of the form-cutting tools of various designs.

The general solution of the problem of design of a form-cutting tool for machining sculptured surfaces on a multi-axis NC machine (see Chapter 5) reveals that the generating surface T of the cutting tool can be as complex as a sculptured surface P can be. The generating surface T of the cutting tool of any design has the corresponding GMap (T ). Figure 7.3 illustrates examples of GMap (T ) of cutting tools that are commonly used in industry. Computation of the parameters of the GMap (T ) of the generating surface T of the cutting tool is similar to the calculation of the parameters of the GMap ( P) of a sculptured surface P. 7.1.3 The Area-Weighted Mean Normal to a Sculptured Surface P The efficiency of machining a sculptured surface on a multi-axis NC machine can be extremely high when the workpiece orientation is optimal. It is convenient to calculate the parameters of the (two-criterion) optimal workpiece orientation taking into consideration the orientation of the area-weighted mean normal to the surface P. A point on the part surface P at which the surface normal is parallel to the area-weighted mean normal of the surface P is referred to as the central point of the surface P. To calculate the parameters of the area-weighted mean normal, the surface P can be subdivided into a large number of reasonably small patches SPi = ∆U Pi × ∆VPi . Here “i” indexes the small patches on the surface P. At the central point Mi inside of each small patch of the surface P, the parameters of the perpendicular N Pi to the surface P can be computed: N Pi =

∂ rP ∂r × P ∂ U P i ∂ VP

i

(7.5)

The perpendicular N Pi may be considered to be an area vector element with magnitude equal to the infinitesimal area of part surface P at a point i.

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Kinematic Geometry of Surface Machining

For the computation of parameters of orientation of the area-weighted % , the following formula is employed: mean normal vector N P n

% = N P

∑

n

∑N

N Pi ∆SPi

i=1

=

SP

Pi

∆U Pi ∆VPi

i=1

SP

(7.6)

where n designates the number of small patches on surface P, and SP designates the area of surface P to be machined. ( SP is equal to the sum of all the areas of the separate workpiece surfaces to be machined in one setup.) Allowing the number of small patches on the surface P to approach infinity yields

∫ N (U ; V ) dS ∫ N (U ; V ) dU P

% = N P

P

P

SP

SP

P

P

=

P

P

P

dVP

SP

SP

(7.7)

Differential of the surface P area is dSP = EPGP − FP2 dU P dVP . Accordingly, Equation (7.7) casts into

∫∫

% = N P

EPGP − FP2 N P (U P ; VP )dU P dVP

P

SP

(7.8)

In cases when several part surfaces Pi are to be machined on a multi-axis NC machine in one setup, Equation (7.8) yields the more general formula k

% = N P

∑ ∫∫ i=1

EPGP − FP2 N P (U P ; VP )dU P dVP

Pi

(7.9)

k

∑S

Pi

i=1

where k is the total number of the part surfaces Pi to be machined in one setup. In the latter case, the area-weighted mean normal to the part surface P is not considered, but the area-weighted mean normal to the several surfaces Pi is considered. The last is referred to as the area-weighted mean normal to all part surfaces Pi . In this case, instead of a central point of the surface, a central point of the entire part to be machined is considered. Definitely, this is a considerably more general approach.

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295

Conditions of Proper Part Surface Generation

ZP0

M*P0

γ

r*P0 rP0 α

β

YP0

MP0

XP0

Figure 7.4 Spherical map of a point of the surface P in the initial orientation of the work and after its optimal orientation.

The area-weighted mean normal to a flat portion of surface P is N Pi SPi . This result can be used in Equation (7.9). The parameters of the area-weighted mean normal to a sculptured surface P as calculated above allow alteration of the initial orientation of the sculptured surface P to the desired orientation in the coordinate system of the multi-axis NC machine. By rotations of the workpiece, for example, through the angle of nutation ψ , through the angle of precession θ , and through the angle of pure rotation ϕ , the workpiece can be reoriented to its optimal orientation. In its optimal orientation, the workpiece allows machining of all surfaces with a single setup. 7.1.4 Optimal Workpiece Orientation In the initial orientation of the workpiece, the angles that the area-weighted mean normal to the surface P makes with the coordinate axes of the NC machine are denoted α , β , γ (Figure 7.4). It is convenient to show these angles on the GMap ( P) of the part surface P (remembering that the areaweighted mean normal to the part surface P has the same direction as the position vector of the point on the GMap ( P) corresponding to the point on surface P at which the perpendicular is erected). In the case under consideration, the problem of optimal workpiece orientation reduces to a problem of coordinate system transformation. Consider two Cartesian coordinate systems X PYP ZP and X NCYNC ZNC . The first coordinate system is associated with the workpiece. Another is connected to the multi-axis NC machine. In the initial orientation of the workpiece, orientation of the coordinate system X PYP ZP relative to the coordinate system X NCYNC ZNC is defined

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Kinematic Geometry of Surface Machining

by the angles α , β , and γ . Actual values of these angles can be computed using one of the above-derived Equation (7.6) through Equation (7.9). The % immediately computed value of the area-weighted mean normal vector N P yields computation of the angles α , β , and γ . For this purpose, the follow% , cos β = j ⋅ N % , and cos γ = k ⋅ N % can be used. All ing formulae cos α = i ⋅ N P P P the computations must be performed in a common reference system. Use of the coordinate system X NCYNC ZNC is preferred. In the optimal workpiece orientation, corresponding axes of these coordinate systems are parallel to each other and are of the same direction. To put the workpiece into the optimal orientation means to make three successive rotations, for example, by the Euler angles — that is, to rotate the workpiece in the coordinate X NCYNC ZNC through the angle of nutation ψ , through the angle of precession θ , and finally, through the angle of pure rotation ϕ . The resultant coordinate system transformation using Euler’s angles can be analytically represented with the operator Eu (ψ , θ , ϕ ) of Eulerian transformation (see Equation 3.11). In the optimal workpiece orientation, it is possible to rotate the part sur% . Under such a rotation, face P about the area-weighted mean normal N P the optimal orientation of the workpiece is preserved, but the orientation of part surface P relative to the NC machine coordinate axes changes. This feasible rotation of the surface P can be used for satisfying additional requirements to the part surface orientation on the worktable of the multiaxis NC machine. For example, the workspace of the multi-axis NC machine is the bounded plane or volume within which the cutting tool and the workpiece can be positioned and through which controlled motion can be invoked. When NC instructions are generated by a part programmer, the geometry of the workpiece must be transformed into a coordinate system that is consistent with the workspace origin and coordinate reference frame. That is why after the workpiece is turned to a position at which its area-weighted mean normal has an optimal orientation, it is necessary to rotate it about the weighted normal to a position in which the projection of the part surface P (or of the part surfaces Pi ) to be machined is within the largest closed contour traced by the cutting tool on the plane of the NC machine worktable. In addition, the vertical position of the workpiece must conform to the capabilities of the NC machine to move in the vertical direction. Proper location of the workpiece on the worktable of a multi-axis NC machine can be specified in terms of (a) the joint space, which is defined by a vector whose components are the relative space displacements at every joint of a multi-axis NC machine; (b) the working envelope, which is understood as a surface or surfaces that bound the working space; (c) the working range, which means the range of any variable for normal operation of a multi-axis NC machine; and (d) the working space that includes totality of points that can be reached by the reference point of the cutting tool.

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Conditions of Proper Part Surface Generation 7.1.5

297

Expanded Gaussian Map of the Generating Surface of the Cutting Tool

An ordinary Gauss’ map can be constructed for any generating surface of a cutting tool. Examples of the GMap (T ) of the form-cutting tools of various designs are shown in Figure 7.3. However, when machining a sculptured surface, the cutting tool is traveling with respect to the surface P. Correspondingly, the GMap (T ) moves over the surface of the unit sphere covering in such a motion that the area exceeds the area of the original GMap (T ). Gauss’ map that is constructed for the moving generating surface T of the cutting tool in all its feasible positions is referred to as the expanded Gauss’ map GMape (T ) of the generating surface of the cutting tool. Actually, when machining a sculptured surface on a multi-axis NC machine, the workpiece and the cutting tool perform certain relative motions. For further analysis, it is convenient to implement the principle of inversion of relative motions. On the premises of the principle of inversion of relative motions, consider the resultant motion of the cutting tool relative to the stationary workpiece. At every point K of contact of the surfaces P and T, the unit normal vectors n P and nT to these surfaces are of opposite directions. (Remember that a normal to the part surface P is pointed outward from the part body, and a normal to the generating surface T of the cutting tool is pointed outward from the generating body of the cutting tool. Therefore, the equality n P = − nT must be satisfied.) Then, employ the concept of antipodal points [5]. Those points on the Gauss’ map are usually referred to as the antipodal points, which are the pairs of diametrically opposed points on the unit sphere. Implementation of the antipodal points yields introduction of the centro-symmetrical image of the GMap (T ) of cutting tool surface T. The last is referred to as the antipodal GMapa (T ) of the generating surface T of the cutting tool. Analysis of possible relative positions of the GMap ( P) of the sculptured surface P and of the antipodal GMapa (T ) of the generating surface T of the cutting tool yields the following intermediate conclusions: Conclusion 7.1: If GMap ( P) of the part surface P is entirely located within the antipodal GMapa (T ) of the generating surface T of the cutting tool (that is, the GMap ( P) contains no points outside GMapa (T ) ), then machining of the surface P is possible.

This is the necessary but not sufficient condition for the machinability of the part surface with the given cutting tool. Conclusion 7.2: If any portion of the GMap ( P) is located outside the antipodal GMapa (T ) , then machining of the surface P is impossible.

This is the sufficient condition that the part surface P cannot be machined with the given cutting tool.

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Kinematic Geometry of Surface Machining

ZP0 B2 A2 B1 A1

YP0

2

C2 D2 B

C1

D1

GMapae(T)

1

A

C

MT0 XP0 D

GMap a (T)

Figure 7.5 An example of the expanded antipodal of the generating surface T of the cutting tool.

When machining a sculptured surface on a multi-axis NC machine, the cutting tool is capable of moving along three axes of the coordinate system X NCYNC ZNC, and rotating about one or more of the axes. These additional degrees of freedom (rotations) allow the antipodal indicatrix GMapa (T ) of the generating surface T of the cutting tool to expand around the surface of the unit sphere, while the GMap ( P) remains fixed. Similar to the spherical indicatrix GInd (T ) of the surface T that serves as the boundary curve for the corresponding GMap (T ), the antipodal indicatrix GInda (T ) serves as the boundary curve to the antipodal GMapa (T ). For example, consider machining of a sculptured surface P on a three-axis NC machine. The antipodal GMapa (T ) of the generating surface T of the cutting tool occupies the fixed area ABCD (Figure 7.5). Then, assume that one more NC-axis is added somehow to the articulation capabilities of the NC machine. The additional fourth NC-axis (say, rotation of the cutting tool about an axis not coinciding with the axis of its cutter-speed rotation) causes the antipodal GMapa (T ) to extend in direction 1 from the initial location ABCD to encompass A1B1CD. If the fifth and the sixth NC-axes are added, then these additional NC-axes cause the antipodal GMapa (T ) to extend in direction 2 and to rotate about an axis through the center of the unit sphere and through a point within the antipodal GMapa (T ) of the generating surface T of the cutting tool. A surface patch on the unit sphere is covered by the antipodal GMapa (T ) such that its motion over the unit sphere is referred to as the expanded antipodal GMapae (T ) of generating surface T of the cutting tool.

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Conditions of Proper Part Surface Generation

299

The expanded antipodal indicatrix GMapae (T ) is a useful tool for the investigation of workpiece orientation on the worktable of a multi-axis NC machine. The boundary curve of the expanded antipodal GMapae (T ) of the generating surface T of the cutting tool serves as the expanded antipodal indicatrix GIndae (T ) of the tool surface T. Because the surface P is considered motionless, the expanded antipodal indicatrix GIndae (T ) of the generating surface T of the cutting tool as well as its expanded antipodal GMapae (T ) cannot rotate about an axis through the origin of the coordinate system X P 0YP 0 ZP 0. When the parameters of the relative motion of the given sculptured surface P and of the given generating surface T are known, then the parameters of the expanded initial GInde (T ) and of the expanded antipodal GIndae (T ) indicatrices of the tool surface T can be calculated using the developed methods of spherical trigonometry [5]. For machining a sculptured surface on a multi-axis (four or more axes) NC machine, the following two statements hold: Conclusion 7.3: If GMap ( P) of the part surface P is contained entirely inside the expanded antipodal GMapae (T ) of generating surface T of the cutting tool, then the surface P can be machined.

This is the necessary but not sufficient condition for the machinability of a sculptured surface on a give multi-axis NC machine with the given cutting tool. Conclusion 7.4: If GMap ( P) of the part surface P contains at least one point outside the expanded antipodal GMapae (T ) of generating surface T of the cutting tool, then machining of the surface P is not feasible.

This condition is sufficient for the sculptured surface that cannot be machined on a given multi-axis NC machine with the given cutting tool. 7.1.6 Important Peculiarities of Gaussian Maps of the Surfaces P and T In particular cases, a sculptured surface P, as well as the generating surface T of the form-cutting tool can have two or more points, at which unit normal vectors are parallel to each other and are pointed in that same direction. Points of this sort can be easily found out, for example, on the torus surface. When parallel and similarly directed unit normal vectors are observed, then the GMap ( P) of the sculptured surface P becomes “multilayered.” The number of layers of the the GMap ( P) is equal to the number of points with parallel and similarly oriented unit normal vectors. For example, parallel and similarly oriented unit normal vectors occur on the part surface P (Figure 7.6). The surface P is bounded by the bordering line ABCDEFG. Gauss’ map GMap ( P) for this portion of the surface P is represented by the portion A0 B0G0 D0 E0 F0 of the unit sphere. Figure 7.6 reveals that the area B0 C0D0 G 0 on the unit sphere corresponds to the GMap ( P) of the portion BCDG of the

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300

Kinematic Geometry of Surface Machining ZP

ZP0 C0

P

GMap (P)

nC C

A0 B0

E0

F0

GInd (P)

D nD

nA

nF

A

XP0

YP0

B

G

D0

G0

nB

nG

YP

GMapm (P)

nE

F

E

XP

Figure 7.6 An example of multilayer of the surface P.

surface P. This means that the portion A0B0G 0D 0 E 0 F0 of the GMap ( P) is covered twice. The number of layers of the GMap ( P) of a surface P can exceed two layers. Equation (7.4) through Equation (7.7) take into account that the Gauss’ maps of surfaces may have two and more layers. Multiple layers affect the weight of the multilayer portion of the GMap ( P). The weight of the multilayer portion of the GMap ( P) is getting larger. It can result in shadowed portions on part surface P if one strictly directs the rule of optimal orientation of the sculptured surface P in compliance with the direction of its area-weighted mean normal. (That is, it is not allowed to neglect considering the reachability constraint on the surface P orientation.) As an example, consider the machining of a sculptured surface on a threeaxis NC machine. Two cases are distinguished. In the first case, a sculptured surface P1 has a one-layer GMap ( P) shown in Figure 7.7. The GMap ( P) of GInd(P)

ZP0

H, J

MP0

GMap(P)

* M P0 E F

A D

G

C

J W

r*P0

B XP0

H

G YP0

ZP0

M*P0 r *P0 ξ

rP0

W

YP0 Figure 7.7 Gauss’ mapping of a sculptured surface P having multilayer (two-layer) when it is machining on a three-axis numerical control machine.

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Conditions of Proper Part Surface Generation

301

the surface P1 occupies a portion of the unit sphere within the closed contour ABDCFE. The area-weighted mean normal nP 0  r P 0 passes through the point MP0 . In such an orientation, the surface P1 can be machined on the threeaxis NC machine in one setup. In the second case, a sculptured surface P2 has a multilayer (two-layer) GMapm ( P). The subscript m here indicates that the Gauss’ map is multilayer. Likewise, in the first case, GMapm ( P) of the surface P 2 occupies the portion within the closed contour ABDCFE on the unit sphere. In addition to the portion ABDCFE , the GMapm ( P) is represented with the portion CDEF on the unit sphere (Figure 7.7). When the portion CDEF is added, then the Gauss’ map is a two-layer map, so it is twice as heavily weighted. Due to the increase in weight of the GMapm ( P) , the area-weighted mean normal n P 0  r P 0 of the surface P 2 turns about the center of the unit sphere through a certain angle ε . In this position, the unit normal vector passes through the point M P 0 . In such a position of the workpiece, it is infeasible to machine the sculptured surface P 2 on the three-axis NC machine in one setup. It is necessary to consider the trade-offs between the orientation of the part surface P 2 in compliance with the position of its area-weighted mean normal and the orientation of the surface P 2 to avoid shadowed areas. One could consider, for example, whether it is preferred to machine the sculptured part surface P 2 on a cheaper, three-axis NC machine with nonoptimal workpiece orientation vis-à-vis cutting conditions, or to machine the part surface P 2 in the optimal workpiece orientation but on a more costly four (or more) axis NC machine. Generally, machining of a part surface in a single setup with some loss of optimality of cutting condition is preferable to machining in two or more setups. Thus, the generally favored situation is to orient the workpiece such that the difference in angle between the areaweighted normal to the part surface to be machined and the tool axis of rotation changes as little as possible, without requiring more setups than necessary. After the analysis of Figure 7.7 is performed, it is important to focus again on the properties of Gauss’ mapping of the surfaces. Figure 7.6 provides a good example to illustrate the property (b) of the GMap ( P) (see Section 7.1.2). Gauss’ map of the bordering contour ABCDE of the surface P is represented by the circular arc A0B0C 0D 0 E 0 (Figure 7.6). In this case, all points of the bordering contour ABCDE of the surface P and all points of the boundary A0B0C 0D 0 E 0 of the Gauss’ map are in one-to-one correspondence. On the other hand, Gauss’ map of the bordering contour AFE of the surface P is represented by the circular arc A0 F0 E 0 . It is evident that the Gauss’ map A0 F0 E 0 of the bordering contour AFE is not a border for the GMap ( P) of the sculptured surface P. Moreover, boundary arc B0G 0D 0 of the GMap ( P) of the surface P is just an image of the curve BGD on the surface P. However, the BGD is not a boundary of the surface P. This example illustrates that a boundary of the GMap ( P) of a sculptured surface P may or may not be a boundary of its GMap ( P) , and vice versa.

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Kinematic Geometry of Surface Machining

7.1.7 Spherical Indicatrix of Machinability of a Sculptured Surface The above-considered Gauss maps of the sculptured part surface and of the generating surface of the cutting tool provide engineers with a powerful analytical tool. Among others, implementation of this tool is helpful for determining whether or not a given sculptured surface P can be machined with the given cutting tool on the NC machine with the given articulation. For this purpose, spherical indicatrices GInd ( P) and GInd (T ) can be used. It is inconvenient to treat simultaneously two separate characteristic curves GInd ( P) and GInd (T ) to determine whether the part surface P can or cannot be machined in one setup with the given cutting tool on the NC machine with the given articulation. For this purpose, a characteristic curve of another nature is proposed. This characteristic curve is referred to as the spherical indicatrix of machinability Mch ( P/ T ) of a given sculptured surface P with the given cutting tool T on the NC machine with the given articulation. For CAD/CAM application, it is necessary to represent this characteristic curve analytically. Without loss of generality, one can consider for simplicity the machining of a sculptured surface P with a ball-end milling cutter. For this case, the GMap (T ) of the generating surface of the cutting tool occupies a hemisphere of the unit sphere (Figure 7.8). GMap ( P) of the sculptured surface P is represented with a certain patch on the unit sphere. The great circle of the unit sphere serves as the GInd (T ) of the generating surface T of the cutting tool. Ultimately, GInd ( P) is represented by the boundary of the GMap ( P). An arbitrary point M P 0 is chosen within the GMap ( P) of the surface P. A cross-section of the unit sphere by the plane Σ i through the origin of the coordinate system X P 0YP 0 Z P 0 and the chosen point M P 0 is considered. The plane Σ i intersects the spherical indicatrices GInd ( P) and GInd (T ) at the points A0 i and B0 i , respectively. The angle between the position vector rAi of the point A0 i and the position vector rM of the chosen point M P 0 is designated as ς Ai . A similar angle between the position vector rBi of the point B0 i and the position vector rM is designated as ς Bi . The difference of the ς *Mi = ςMi

GMap a (T )

ZP0

ZP0 GMap(P) rMi rAi

MP0

rMi

Σi

B0i

XP0

Σi

* MP0

A0i

A0i

rBi YP0

rAi

ZP0

ςAi

MP0

rBi

ς Bi

ς Mi

rMi

B0i

(b)

rMi

XP0

YP0 Σ1

(a)

MP0

Σ2

Σ3

(c)

Figure 7.8 Derivation of equation of the spherical indicatrix of machinability of a surface P with the given cutting tool.

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303

Conditions of Proper Part Surface Generation

angles ς Ai and ς Bi is equal to (ς Ai − ς Bi ) = ς Mi . Angle ς Mi , which determines the location of the point M0 i on the indicatrix of machinability Mch ( P/ T ) =ς of the surface P using tool T, is equal to ς Mi Mi . This angle is measured within the plane Σ i through the vectors rM , rAi , rBi , either in the clockwise < 0ο direction if ς Ai > ς Bi (in this case, ς Mi ), or in the counterclockwise direc tion if ς Ai < ς Bi (in this case, ς Mi < 0ο). Rotating the plane Σ i about the position vector rM to positions Σ 1 , Σ 2 , Σ 3 , and so on, all points of the indicatrix of machinability Mch ( P/T ) can be obtained.

Conclusion 7.5: A sculptured surface P is machinable using the given generating surface T of the cutting tool if and only if the indicatrix of machinability Mch ( P/T ) has no negative diameters i.e., it is not a selfinteresting curve on the unit sphere.

The equation of the indicatrix of machinability Mch ( P/ T ) immediately follows from the analysis below. The position vector rM of the point M 0 P is  cos α M    cos β M  rM =   cos γ M     1 

(7.10)

The equation of the GMap (T ) of the generating surface T of the cutting tool yields two equations. The first equation is for the unit vector rAi :  cos α Ai    cos β Ai   rAi =  cos γ Ai     1 

(7.11)

and another equation is for the unit vector rBi :  cos α Bi    cos βBi   rBi =  cos γ Bi     1 

(7.12)

The vectors rM and rAi make an angle ς Ai . The actual value of the angle ς Ai can be computed from the formula

ς Ai = ∠(rM , rAi ) = arctan

© 2008 by Taylor & Francis Group, LLC

rM × rAi rM ⋅ rAi

(7.13)

304

Kinematic Geometry of Surface Machining

The vectors rM and rBi make an angle ς Bi . The actual value of the angle ς Bi is

ς Bi = ∠(rM , rBi ) = arctan

rM × rBi rM ⋅ rBi

(7.14)

With these results, the value of the angle ς Mi that is determining the location of an arbitrary point of the indicatrix of machinability Mch ( P/ T ) of part surface P using tool surface T is

ς Mi = ∠(rAi , rBi ) = ς Ai − ς Bi = arctan

rM × rAi rM ⋅ rBi

− arctan

rM × rBi rM ⋅ rBi

(7.15)

Equation (7.15) specifies the location of a current point of the spherical indicatrix of machinability Mch ( P/ T ). This characteristic curve is convenient for determining whether the sculptured part surface P can or cannot be machined using the given generating surface T of the cutting tool. As an illustration of implementation of the spherical indicatrix of machinability Mch ( P/ T ) , consider machining of various cylindrical local portions of a sculptured surface P on a three-axis NC machine (Figure 7.9). In the case under consideration, the antipodal indicatrix GInda (T ) of the generating surface T of the cutting tool is represented with a hemisphere. However, an arc of a great circle of the unit sphere serves as the spherical indicatrix GInd ( P) of the surface P. Analysis of the relative configuration of the spherical indicatrices GInd ( P) and GInd (T ) shows that the surface P can be machined in the first two cases, and it cannot be machined in the third case. One can come up with that same result via analysis of shape of the indicatrix of machinability Mch ( P/ T ) of the surface P using tool surface T. The analysis shows that in the first case, the surface P can be machined on the three-axis NC machine. Moreover, there remains some freedom in orienting the workpiece: the surface P can be turned about its axis in opposite directions by a certain angle of ξ > 0 . In the second case, the surface P can also be machined on the three-axis NC machine. However, in this case, no degree of freedom remains. The surface P cannot be turned about its axis, because the angle GMap ( P) is equal to zero (ξ = 0) . In the third case, the surface P cannot be machined on the three-axis NC machine. The arc of a great circle through the spherical map GMap ( P) includes points for which the angle ξ is negative ( ξ < 0 ). The last indicates that negative diameters of the indicatrix of machinability Mch ( P/ T ) are observed. For convenience in implementation, the indicatrix of machinability Mch ( P/ T ) can be depicted in Cartesian coordinates. No formulae transformations are required in this concern. The spherical parameters must be considered as the Cartesian coordinates.

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305

Conditions of Proper Part Surface Generation T

T

T P

P

P

P

P ξ > 0

ξ > 0

P

ξ = 0

ξ = 0 ξ > 0

ξ > 0

GMap(P)

GMap(P)

ZP0

GMapA(T )

GMap(P) ZP0

ZP0

GMapA(T)

GMapA(T)

XP0

XP0

XP0

YP0

YP0

YP0 ZP0

ZP0

MInd(P)

ξ > 0

ZP0

MInd (P)

ξ < 0

MInd (P) ξ = 0

ξ > 0

YP0

XP0

XP0 YP0

(a )

ξ > 0

XP0

YP0 (b)

(c )

Figure 7.9 Generation of various cylindrical local portions of a sculptured surface with a ball-nose milling cutter.

Actually, using the indicatrix of machinability Mch ( P/ T ) , we came up with the same result as that previously obtained, when two spherical indicatrices GInd ( P) and GInd (T ) were implemented. As follows from the above analysis, it is preferred to use one spherical indicatrix of machinability rather than two characteristic curves GInd ( P) and GInd (T ). Example 7.1 Consider machining of a portion P of a torus surface shown in Figure 7.10. The radius rP of the generating circle of the surface P is rP = 100 mm . The radius RP of the directing circle of the surface P is RP = 200 mm . The Gaussian (curvilinear) coordinates of a point on the surface P are designated as θ P and ϕ P , correspondingly. They vary in the intervals 0 ≤ θ P ≤ 90 and 0 ≤ ϕ P ≤ 90 . The goal is to define an optimal part orientation of the part surface P on the table of a three-axis NC milling machine. Position vector R of a point of the surface P can be expressed as the summa R = R + r of two position vectors. Here R is the position vector of

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306

Kinematic Geometry of Surface Machining ~ nP

ZP

P

~ M M rP P

XP

~

θP

P

YP

RP

Figure 7.10 An example of a part surface to be machined.

the center of the generating circle that rotates about the ZP axis, and r is the position vector of a point on the generating circle. Due to lack of space, the vectors R , R , and r are not depicted in Figure 7.10. Representation of the position vector R in the form R = R + r yields an expanded equation for the surface P:

 − RP cos θ P + rP cos ϕ P cos θ P    − R sin θ P + rP cos ϕ P sin θ P  R= P   rP sin ϕ P   1  

(7.16)

The perpendicular vector N P to the surface P at an arbitrary point M can be calculated as N P = U P × VP , where tangent vectors U P and VP are given by

(7.17)

 RP sin θ P − rP cos ϕ P sin θ P    ∂ R  − RP cos θ P + rP cos ϕ P cos θ P  = UP =  ∂θP  rP cos θ P   0  

(7.18)

 − rP sin ϕ P cos θ P    ∂ R  − rP sin ϕ P sin θ P  = VP =  ∂ϕP  0   0  

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307

Conditions of Proper Part Surface Generation Equation (7.17) and Equation (7.18) yield the formula i

j − RP cos θ P + rP cos ϕ P cos θ P − rP sin ϕ P sin θ P

N P = RP sin θ P − rP cos ϕ P sin θ P − rP sin ϕ P cos θ P

k rP cos θ P 0

(7.19)

for the computation of the normal vector N P to the surface P. The angles α P , βP , and γ P which the vector N P makes with the axes of the coordinate system X PYP ZP attached to the part to be machined ( cos α P = i ⋅ N P, cos β P = j ⋅ N P , and cos γ P = k ⋅ N P ) can be calculated as

α P = arccos

β P = arccos

γ P = arccos

rP sin θ P cos θ P rP2 cos 2 θ P + RP2 (1 − cos ϕ P )2

(7.20)

rP cos 2 θ P rP2 cos 2 θ P + RP2 (1 − cos ϕ P )2

(7.21)

RP (cos ϕ P − 1) r cos 2 θ P + RP2 (cos ϕ P − 1)2 2 P

(7.22)

For the computation, it is convenient to consider discrete values of the parameters θ P and ϕ P with certain increments δθ P and δϕ P . Under such a scenario, the surface P could be represented in δθ P = 1 and δϕ P = 1 increments by 90 ⋅ 90 = 8100 points, which provide sufficient accuracy for the computation (Figure 7.11).

P

(i + 1), j

i, j

1 2 3 4

Mi,j (i + 1), ( j + 1) i, ( j + 1) δ

P

= 1 δθP =1

1 2 3 4 5

i, j

88 90 δθP =1

(i + 1), j Mi,j

j 88 89 90

i

i, ( j + 1)

(i + 1), ( j + 1)

δ

Figure 7.11 A grid of nearly rectangular patches on the part surface P.

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P

= 1

308

Kinematic Geometry of Surface Machining

Thereby, the surface P is covered with nearly rectangular patches, vertices of which may be enumerated in the following manner: (i, j), (i + 1, j) , (i, j + 1) , and (i + 1, j + 1) . Each patch of the surface P can be considered as nearly a flat patch that can be inscribed in a circle. The optimal orientation of the part surface P is calculated in the following six steps: Step 1. Using the enumeration of the rectangular patches from above, Equation (7.6) yields n

% = N P

∑N

P.i , j

∆SP.i , j

i−1

SP

(7.23)

In Equation (7.23), the perpendicular N P.i , j is computed by i N P.i , j = RP sin(1 ⋅ j) − rP cos(1 ⋅ i)sin(1 ⋅ j) − rP sin(1 ⋅ i)cos(1 ⋅ j)

j − RP cos(1 ⋅ j) + rP cos(1 ⋅ i)cos(1 ⋅ j)

k rP cos(1 ⋅ j)

− rP sin(1 ⋅ i)sin(1 ⋅ j)

0

Equation (7.24) casts into the matrix form:

N P.i , j

(7.24)

 − rP2 sin(1 ⋅ i)sin(1 ⋅ j)cos(1 ⋅ j)    rP2 sin(1 ⋅ i)cos 2 (1 ⋅ j)  =  rP (− RP + rP cos(1 ⋅ i))sin(1 ⋅ i)    1  

(7.25)

Step 2. The length of each side of one of the nearly rectangular patches is

ai , j =|r( i+1), j − r i , j |

(7.26)

bi , j =|r( i+1),( j+1) − r( i+1), j |

(7.27)

ci , j =|r i ,( j+1) − r( i+1),( j+1) |

(7.28)

di , j =|r i , j − r i ,( j+1) |

(7.29)

Step 3. The semiperimeter of one of the nearly rectangular patches is

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p i, j =

ai , j + bi , j + ci , j + di , j 2

(7.30)

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Conditions of Proper Part Surface Generation

Step 4. Because each of the nearly rectangular patches is nearly flat and can be inscribed in a circle, area SP.i , j of the nearly rectangular patches can be calculated by SP.i , j = ( pi , j − ai , j ) ⋅ ( pi , j − bi , j ) ⋅ ( pi , j − ci , j ) ⋅ ( pi , j − di , j )

(7.31)

Step 5. The area of the part surface P is calculated by

SP =

1 ⋅ 4π 2 RP rP = 4.4674RP rP 16

Step 6. The angles α% P , β% P , and γ% P that the % makes with area-weighted mean normal N P the axes of the coordinate system X PYP ZP attached to the part surface P are calculated by the formulae cos α% = i ⋅ n% P , cos β% = j ⋅ n% P , and cos γ% = k ⋅ n% P . To orient the part surface P optimally for machining on a three-axis mill, it must be rotated about three axes as shown, such that the area% is in a vertical weighted mean normal vector N P orientation (Figure 7.12) (that is, aligned with the tool axis).

7.2

(7.32) P

~ nP

~ M

Figure 7.12 The optimal orientation of the part surface P on the worktable of a threeaxis numerical control milling machine.

Necessary and Sufficient Conditions of Proper Part Surface Generation

Once the optimal (or at least a feasible) workpiece orientation is defined, it is necessary to establish the rest of the necessary and sufficient conditions of proper part surface generation (conditions of proper PSG) [11,12,14–17,20]. 7.2.1 The First Condition of Proper Part Surface Generation When machining, the sculptured part surface is generated by the generating surface of the cutting tool. A cutting tool of a certain design is necessary for the machining of a given sculptured surface on a multi-axis NC machine. A cutting tool of any design can be designed on the premises of the generating surface T of the cutting tool. This means that existence of the generating surface T of the cutting tool is a prerequisite for the feasibility of machining a given sculptured part surface. The principal methods for generation of the generating surface T of the cutting tool are disclosed in Chapter 5.

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310

Kinematic Geometry of Surface Machining Z

Z

ωπ

π2

π4

π4

π5

π1

π3

π2

M Y

nπ

VM

X

VM

nπ

X

M

Y π1

( a)

π3

π5 ( b)

Z Z ωπ

π

Π

nπ

ωπ

nπ

M VM

Y

( c)

Π

X

π

M

Y

X VM ( d)

Figure 7.13 Examples of violation (a) and (b) and of satisfaction (c) and (d) of the first necessary condition of proper part surface generation. (From Radzevich, S.P., Computer-Aided Design, 34 (10), 727–740, 2002. With permission.)

Regardless of whether the generation of the surface T as an enveloping surface is convenient or not, the generating surface T of the cutting tool can be considered as the enveloping surface to consecutive positions of the P in their relative motion on a multi-axis NC machine. Due to this, the sculptured part surface P and the generating surface T of the form-cutting tool are conjugate surfaces. In particular cases, the enveloping surface does not exist. For instance, a plane surface π is passing through the axis Z of a stationary coordinate system XYZ. The plane π is rotating about this Z axis with a certain angular velocity ωπ (Figure 7.13a). The vector VM of linear velocity of an arbitrary point M of the surface π is aligned to the unit normal vector n π to π at the point M. Both vectors VM and n π are perpendicular to the plane surface π . Because of this, an enveloping surface to consecutive positions π 1 , π 2 , . . ., π i of the plane surface π does not exist.

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311

Conditions of Proper Part Surface Generation

A similar situation is observed when a plane π is performing straight motion VM in the direction of the perpendicular Nπ to the plane π (Figure 7.13b). In this case, the enveloping surface to the consecutive positions π 1, π 2 , π 3, π 4 , π 5 of the plane π does not exist. An enveloping surface Π exists if and only if the vector VM of linear velocity is perpendicular to the unit normal vector n π to the plane surface π . For example, a plane surface π is parallel to the axis Z (Figure 7.13c). The plane is rotating about the Z axis with a certain angular velocity. Thus, the vectors VM and n π are perpendicular to each other ( n π ⋅ VM = 0), and therefore, the enveloping surface Π exists — this is a cylindrical surface. The same condition ( n π ⋅ VM = 0 ) is satisfied when the plane surface π is perpendicular to the axis of rotation (Figure 7.13d). The enveloping surface is represented with a plane. The plane π can make a certain angle with the Z axis. In the last case, a circular cone surface would be the enveloping surface to consecutive positions of the plane π. It is convenient to illustrate the effectiveness of the first necessary condition of proper PSG with an example of machining of an involute working surface of a cam (Figure 7.14). Working surface P of the cam is shaped in the OT ωT

P

± VT

TG

P*

K

w* w w**

P P ** d ** w

± ωP

dw = db d *w

OP

Figure 7.14 Examples of satisfaction and of violation of the first necessary condition of proper part surface generation when machining the involute working surface P of a cam. (From Radzevich, S.P., Computer-Aided Design, 34 (10), 727–740, 2002. With permission.)

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Kinematic Geometry of Surface Machining

form of an involute surface that has the base circle of diameter db . When machining the surface P, the cam is swinging about the axis OP with an angular velocity ±ω P . The grinding wheel has a flat working surface. When machining the surface P, the grinding wheel is rotating about the axis OT with an angular velocity of ωT . In addition to the rotation ωT , the grinding wheel is reciprocating along the axis OT with a certain speed ±VT . The swinging motion ±ω P of the cam and the reciprocation ±VT of the grinding wheel are timed in a certain manner. Due to the timing, an imaginary (phantom) pitch line associated with the grinding wheel is rolling without sliding over an imaginary (phantom) pitch circle associated with the surface P of the cam. The location of the pitch line and the actual value of the pitch diameter depend upon the timing of the motions ±ω P and ±VT . This means that the pitch diameter can vary depending on the actual ratio of the motions ±ω P and ±VT when machining the surface P. If the motions ±ω P and ±VT are timed such that the pitch diameter is equal to ( dw = db ) or exceeds ( dw > db) the base diameter db of the involute surface P, then the first condition of proper part surface generation (PSG) is satisfied. In the first case under consideration, the pitch line w is tangent to the base cylinder db of the cam at the pitch point P . In the second case pitch, the line w makes contact with the pitch circle of diameter dw at the pitch point P . In both cases, the enveloping surface exists — this is the plane surface TG . Surfaces P and TG make contact at point K. If the motions ±ω P and ±VT are timed such that pitch diameter dw is less than the base diameter db (i.e., dw < db ), then the pitch line w makes contact with the pitch cylinder at the pitch point P . In this last case, the enveloping surface does not exist, and therefore, the machining of the involute surface P is impossible. The above analysis reveals that absence of the cutting tool leads to the machining operation of a sculptured surface P not being able to be performed. To design an appropriate form-cutting tool, the machining surface T of the cutting tool must exist. This allows formulation of the following: The First Condition of Proper PSG: Existence of the generating surface T of a form-cutting tool which is conjugate to a given part surface P to be machined (to be generated) is the first necessary condition of proper PSG.

In cases when the first condition of proper PSG is violated (that is, the generating surface T does not exist), a form-cutting tool cannot be designed; therefore, the sculptured surface P cannot be machined. When a form cutter has been selected or is given, then verification of the first condition of proper PSG might be omitted. Satisfaction of the first condition of proper PSG is necessary but not sufficient for correct handling of the machining operation. When the generating surface T of the cutting tool does not exist, then the first necessary condition of proper PSG is violated. In particular cases, not the

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Conditions of Proper Part Surface Generation

313

entire generating surface T, but just a certain portion does not exist. For example, partial violation of the first necessary condition of proper PSG is observed when hobbing an involute gear with a multistart hob of small diameter. If the base diameter of the hob exceeds the hob limit diameter ( db. h > dl. h ), then the portion of the generating surface T of the involute hob that should be located between the diameters db. h and dl. h does not exist. The screw involute surface cannot be extended inside the base cylinder db. h of the involute hob. This causes partial violation of the first necessary condition of proper PSG. In a similar way, the first necessary condition of proper PSG is violated when the involute hob is shifted too much in its axial direction, and thus, there are not enough hob threads within the line of action of the hob and of the gear being machined. The gear shaving operation is another good example of when the first necessary condition of proper PSG can be violated. For all methods of gear shaving, if face-width of the shaving cutter is narrow and, thus, insufficient for proper finishing of the gear, this causes violation of the first necessary condition of proper PSG. 7.2.2 The Second Condition of Proper Part Surface Generation When machining a part, the generating surface T of the form-cutting tool must be in contact with the part surface P to be machined. The surfaces P and T can be either in permanent contact with one another, or they could make contact just in a certain instant of time. In the first case, generation of the sculptured surface P is referred to as the continuous surface generation. In the second case, generation of the sculptured surface P is referred to as the instantaneous surface generation. In the instant of surface generation, the sculptured surface P and the generating surface T of the cutting tool must be in tangency to each other. For the analytical interpretation of this requirement, the equation of contact n P/T ⋅ VΣ = 0 must be satisfied. Here, in the equation of contact, n P/T designates a common unit normal vector. (The unit vector n P/T can be interpreted either as the n P/T  n P or as the n P/T  − nT .) And VΣ designates the vector of a resultant relative motion of the surfaces P and T. In order to satisfy the equation of contact, the unit normal vectors n P and nT to the surfaces P and T have to be aligned to each other and directed in opposite directions. Both of these two necessary requirements are satisfied in Figure 7.15a. Here, the equality n P + nT = 0 is observed. When the unit vectors n P and nT are aligned to each other but are of the same direction (Figure 7.15b), the surfaces P and T interfere with each other. Ultimately, when the unit normal vector n P is not aligned to the unit normal vector nT and they intersect each other at a certain angle ε ≠ 180 (Figure 7.15c), then the surfaces P and T cannot be in tangency, and interference of the surfaces is unavoidable. The above-mentioned equality n P + nT = 0 in common-sense engineering application is equivalent to the equality n P ⋅ nT = −1. Any of these two equations can be interpreted as the analytical representation of the second

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Kinematic Geometry of Surface Machining

nP

nT

T P

P

n

T

P

T

nP

K

K nT

nT (a )

P

K

(b)

ε

180

(c)

Figure 7.15 Examples of satisfaction (a), and of violation (b) and (c) of the second necessary condition of proper part surface generation. (From Radzevich, S.P., Computer-Aided Design, 34 (10), 727–740, 2002. With permission.)

necessary condition of proper PSG. Other forms of the equalities can be drawn up as well. The second necessary condition of proper PSG can be formulated in the following way: The Second Condition of Proper PSG: The unit normal vectors to the sculptured part surface P to be machined and to the generating surface T of a form-cutting tool at each point of the surface contact must be aligned to each other and be directed oppositely.

Usually, it is not difficult to satisfy the second condition of proper PSG when developing a software for machining a sculptured surface on a multiaxis NC machine. 7.2.3 The Third Condition of Proper Part Surface Generation To ensure proper contact of the surfaces P and T without penetration of one of the surfaces into another surface, it is also necessary to satisfy certain correspondence of their normal radii of curvature at every cross-section of the surfaces by a plane through the common perpendicular. Evidently, no problem arises when two convex surfaces P and T must be put in contact. It is also evident that it is not feasible to put in contact two concave surfaces P and T, or two surfaces, one of which is concave and another is saddle-like. These cases are evident, and thus they do not need careful analysis. The problem of critical importance for the machining of a sculptured surface on a multi-axis NC machine is to establish necessary and sufficient conditions of proper contact of the surfaces P and T in the vicinity of an arbitrary point K when one of the surfaces P and T is a convex surface and another is a concave surface, when one of the surfaces P and T is a convex surface and another is a saddle-like surface, and finally, when both of the surfaces P and T are saddle-like surfaces. Analytical interpretation of the condition of contact of the surfaces is the most critical for the machining of a sculptured surface on a multi-axis NC machine.

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Conditions of Proper Part Surface Generation

Table 7.1 A Classification of Kinds of Contact of Normal Cross-Sections of

the Surfaces P and T

Normal Plane Section of the Surface P (Local Representation) Normal plane section of the Convex Linear Concave generating surface T of a 1 R 2 3. P RP P nP cutting tool (Local RP nP P P nP representation) 1

11

12

RP

P

nP

RT

K 0 < RT < +∞

Rp > 0

Convex

RT

P

RT >0 nP

0 < RT < RP

RT

nP RT

P

T

nT

T

P

nP

RT

RP

K

K nT

T

RP > RT

nT RP

RP

K

nT T

RP = RT

nT

T

RT

nP

RP

P

K 2

RT

RT

RT

nP

P

nP

RT

Linear

nT RP 0 < RP < RT

nT

3.

RT

np

nT

RT RP

nP T

K

K T

P

RP < RT

RP

T 3.2

P

RP = RT

nT

T

T

K RP < 0 3.3

nT

K T RT > RP

Concave

−∞ < RT < 0

RT T

nT

Ru < 0

R

RP

nP

P

nP K

RT

nT

T

P RT = RP

nT

RT

nP

RT < RP

K T nT

RP < 0 RP

nP

RT

T

nT

P

K

K RP

RT < 0

RP

RT

nT T

P

RP

For the analysis of the actual correspondence between the radii of normal curvature of the surfaces, the normal cross-sections of all possible kinds of contact of the surfaces P and T have been analyzed (Table 7.1).

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Kinematic Geometry of Surface Machining

RP RT

T

RP

RT

T

RT

RP RT RP P

P

K

K

(a )

(b)

Figure 7.16 Examples of satisfaction (a) and of violation (b) of the third necessary condition of proper part surface generation. (From Radzevich, S.P., Computer-Aided Design, 34 (10), 727–740, 2002. With permission.)

When the proper correspondence is observed between the radii of normal curvature RP of the sculptured surface P and of the radii of normal curvature RT of the generating surface T of the form-cutting tool (i.e., when the inequality |RP |> RT is observed), then the sculptured surface P can be generated with the surface T in the vicinity of every point K of their contact (Figure 7.16a). Otherwise, when the inequality |RP |< RT is valid, an interference of the surfaces P and T occurs, and surface P cannot be generated in the differential vicinity of point K (Figure 7.16b). The inequality |RP |> RT can be used for the analytical description of satisfaction of the third necessary condition of proper PSG. To verify the satisfaction or violation of the third necessary condition of proper PSG, the indicatrix of conformity CnfR ( P/T ) of the surfaces P and T can be implemented. (See Chapter 4 for details on this characteristic curve.) In polar coordinates, the equation of the indicatrix of conformity CnfR ( P/T ) [7,8] of the surfaces P and T can be represented in the following form (see Equation 4.59):

Cnf R ( P / T ) ⇒ rcnf =

+

EPGP LPGP cos ϕ − MP EPGP sin 2ϕ + N P EP sin 2 ϕ 2

ET GT LT GT cos (ϕ + µ ) − MT ET GT sin 2 (ϕ + µ ) + NT ET sin 2 (ϕ + µ ) 2

(See Chapter 4 for more detail about Equation 7.33.)

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sgn Φ2 P (7.33)

sgn Φ2 T ≥ 0

Conditions of Proper Part Surface Generation

317

When the third necessary condition of proper PSG is satisfied, then all the diameters dcnf = 2 rcnf of the indicatrix of conformity CnfR ( P/T ) are nonnegative. Thus, within the common tangent plane in any direction through the point K, the inequality rcnf ≥ 0 is satisfied. The set of two equations sgn rcnf = 0 or sgn rcnf = +1 is equivalent to the inequality rcnf ≥ 0 . When the third necessary condition of proper PSG is satisfied, either the equation sgn rcnf = 0 or the equation sgn rcnf = +1 is valid. (min) When the minimal diameter dcnf of the indicatrix of conformity CnfR ( P/T ) (min) of the surfaces P and T is nonnegative, say when the inequality dcnf ≥ 0 is observed, all other diameters dcnf of this characteristic curve are nonnegative as well. Therefore, the third necessary condition of proper PSG is satisfied when (min) (min) the minimal diameter dcnf is equal to or exceeds zero [ dcnf ≥ 0 ]. The above analysis enables the following formulation: The Third Condition of Proper PSG: The condition of proper contact of a sculptured surface P to be machined and of the generating surface T of the form-cutting tool without their mutual penetration (that is, without their mutual interference in differential vicinity of the point of contact) is the third necessary condition of proper PSG.

As an illustration of the necessity of proper correspondence between the radii of normal curvature of the contacting surfaces, consider the contact of a concave portion of the sculptured surfaces P and of a convex portion of the generating surface T of a form-cutting tool (Figure 7.17a). In Figure 7.17, the principal plane sections C1. P and C1. P are passing through the principle directions t 1.P and t 2.P of the sculptured surfaces P. The principal plane sections C1.T and C2.T are passing through the principle directions t 1.T and t 2.P of the generating surface of the form-cutting tool T. The unit tangent vectors of the principal directions t 1.P , t 2.P and t 1.T , t 2.P are aligned with the axis of symmetry of Dupin’s indicatrix Ind ( P) and Ind (T ) of the surfaces P and T. Radii of normal curvature of the surfaces P and T at point K of their contact are measured in a plane section through the unit normal n P and through an arbitrary direction t within the common tangent plane. In every cross-section of the surfaces P and T by the plane through the unit normal vector n P within a certain central angle κ , interference of the surfaces P and T occurs (Figure 7.17a). The interference of the surfaces P and T is observed because the inequality RP < RT is true to these cross-sections (Figure 7.17b). The last is depicted in Figure 7.17c, where diameters of the indicatrix of conformity CnfR ( P/T ) within the central angle κ are negative (The corresponding segment of the indicatrix of conformity CnfR ( P/T ) is represented by a dashed line.) Many examples of satisfaction or violation of the third necessary condition of proper PSG can be found in practice. The regrinding of a broach is a good example in this concern.

A diameter of a centro-symmetrical curve can be defined as a distance between two points of the curve measured along a straight line through the center of symmetry of the curve.

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318

Kinematic Geometry of Surface Machining nP

P

κ

T

K (a) RT

nP

P

RP

K

T ( b)

κ

CnfR (P/T)

K C1 P

μ

C1.T Ind(T )

C2 P

μ

C2.T

Ind(P)

(c) Figure 7.17 Example of local penetration of the generating surface T of the form-cutting tool into a sculptured surface P in differential vicinity of the point of their contact.

In most practical cases, the rake surface of a broach is reground. The rake surface P of a broach is shaped in the form of an internal cone of revolution (Figure 7.18). The working surface T of the grinding wheel is represented by a surface of an external circular cone. The grinding wheel is rotating about

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319

ρT.c ρT.b

OP

γt

T

αt

P

ωP

β

ωT

RT.c

a

OT

b

c

ρT

ρP

a

rP.c

ρP.a

ρT.a

b

ρP.b

c

ρP.c

Conditions of Proper Part Surface Generation

Figure 7.18 Example of violation of the third necessary condition of proper part surface generation when regrinding a broach.

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Kinematic Geometry of Surface Machining

its axis OT with a certain angular velocity ω T . The rotation ω T provides the necessary speed of cut. The broach is rotating about its axis of rotation OP with an angular velocity ω P . The rotation ω P can be considered as the feed-rate motion. The surfaces P and T make contact along the straight line. Points a, b, and c are within the line of contact of the surfaces P and T. Radii of normal curvature ρP.i and ρT .i are not of the same value within the line of contact of the surfaces P and T. For the surface of internal cone P, all the radii of normal curvature are negative. For this surface, the inequality |ρP. a |>|ρP.b |>|ρP.c | is valid. For the generating surface T of the grinding wheel, all the radii of normal curvature are positive. For the surface T, the inequality ρT . a < ρT .b < ρT .c is observed. Without going into detail, it is evident that the point c is the first point at which the third necessary condition of proper PSG can be violated. If the third condition of proper PSG is satisfied at point c, then it is satisfied for all other points of the line of contact of the surfaces P and T. Radius of normal curvature ρP.c of the rake surface P can be computed using Meusnier’s theorem:

ρP. c =

rP.c sin γ t

(7.34)

where rP.c designates the distance of the point c from the axis of the broach OP , and γ t designates the rake angle of the broach. The radius of normal curvature ρT .c of the generating surface T of the grinding wheel can also be computed on the premises of Meusnier’s theorem:

ρT .c =

R T .c sin(β − γ t )

(7.35)

where R T .c designates the distance of the point c from the axis of the grinding wheel OT , and β designates the angle that forms the axes OP and OT . To satisfy the third necessary condition of proper PSG, satisfaction of the inequality ρT .c ≤|ρP.c | is necessary. Therefore, the inequality

R T .c rP.c ≥ sin γ t sin(β − γ t )

(7.36)

must be satisfied when regrinding the rake surface of the broach. The inequality (Equation 7.36) can be cast into the formula for computation of the maximal feasible outside radius R T .c (diameter) of the grinding wheel:

© 2008 by Taylor & Francis Group, LLC

R T .c ≤

rP.c sin( β −γ t ) sin γ t

(7.37)

Conditions of Proper Part Surface Generation

321

In a similar way, a formula for the computation of the maximal feasible outside radius R T .c (diameter) of the grinding wheel for grinding broaches having helical grooves can be derived. The indicatrix of conformity CnfR ( P/T ) of the surfaces P and T (see Equation 7.33) can be used for the purpose of verification of whether the third necessary condition of proper PSG is satisfied or violated. The equation of this characteristic curve can also be interpreted as an analytical representation of the third necessary condition of proper PSG. Implementation of this characteristic curve is of particular importance for the development of software for the machining of a sculptured part surface on a multi-axis NC machine. The scenario when the radii of normal curvature RP and RT of the surfaces P and T are of the same magnitude and of opposite sign (say, when the equality RP = − RT ) is of special interest for the theory of surface generation. Due to unavoidable deviations in relative configuration when machining a sculptured surface on a multi-axis NC machine, kinds of contact for which the condition RP = − RT is valid must be eliminated. However, the condition RP = − RT could be satisfied in one of quasi-kinds of contact of the surfaces P and T. For example, consider contact of a curve ABC having continuously reduced radii of curvature from the point A toward the point C. For this curve, the inequalities RA > RB > RC are satisfied. At point B, the curve ABC makes tangency with a circle of radius R. Evidently at the points a, b, and c of the circular arc abc, the equality Ra = Rb = Rc is valid. Figure 7.19a reveals that either the arcs AB and ab are in proper contact and the arcs BC and bc interfere with one another ( RA > Ra , RB = Rb and RC < Rc ), or vice versa, the arcs BC and bc are in proper contact, and the arcs AB and ab interfere with one another (RA < Ra , RB = Rb and RC > Rc ). The interference is observed regardless that at the point B the equality RB = Rb is valid. Because of this, when the equality RP = − RT is valid, then for the proper contact of the surfaces of P and T, it is necessary to properly orient them with respect to one another, and to ensure corresponding gradients of radii of normal curvature (Figure 7.19b). Otherwise, violation of the third necessary condition of proper PSG is unavoidable (Figure 7.19c). Violation of the third necessary condition of proper PSG under a scenario when the equality RP = − RT is valid, could be observed in the practice of sculptured surface machining on a multi-axis NC machine. Figure 7.20 illustrates just a few cases when the third condition is violated. The machining of a sculptured surface P in the vicinity of a point of inflection (Figure 7.20a) cannot be performed with a flat-end milling cutter (Figure 7.20a). The curvature k P of the surface P, and the curvature kT of the surface T at point K are equal to each other ( k P = kT = 0 ). However, because the equality RP = − RT is valid, satisfaction of the condition k P = kT = 0 is not sufficient for the satisfaction of the third necessary condition of proper PSG in this case. Similarly, when a gradient of increasing curvature k P of the surface P exceeds the limits, then the part surface P cannot be properly machined in

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322

Kinematic Geometry of Surface Machining Rb

RB

B

RA

b

RC

C

A

c

a

Rc

Ra

(a) RB

Rb Ra

a RA

P

Rc C

c

A B

b

T

RC

(b) RB C

RC

T

a

P

Rc

Rb Ra

B b

A

c RA

(c) Figure 7.19 Examples of satisfaction and of violation of the third necessary condition of proper part surface generation when the condition R P = –RT is observed.

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323

Conditions of Proper Part Surface Generation

T

T

RT = ∞

RT

P

I

P

K RP = ∞ (a)

RT = ∞

T I

K

RP

K

I P

K

RP = ∞ (b)

(c)

RT

P

T

K RP = ∞ (d)

Figure 7.20 Examples of particular cases of the violation of the third necessary condition of proper part surface generation.

the vicinity of the point of inflection (Figure 7.20b) or in the vicinity of the concave point (Figure 7.20c). Again, at the point K ratio between the curvatures k P and kT yields the assumption that the machining of the surface P is feasible. However, because of insufficient gradient of increasing curvature k P of the surface P, the third necessary condition of proper PSG in this case is not satisfied. A similar analysis can be performed for the case when at the point of contact K of the surfaces P and T, both of them are contacting with the point of inflection (Figure 7.20d). Evidently, in this case, the third necessary condition of proper PSG can be easily violated. 7.2.4 The Fourth Condition of Proper Part Surface Generation When no local interference of the surfaces P and T is observed, the surfaces can interfere with each other out of local vicinity of the point K of their contact (Figure 7.21). This kind of interference of the surfaces is referred to as the global interference of the surfaces P and T. T T

P P

Figure 7.21 Examples of violation of the fourth necessary condition of proper part surface generation. (From Radzevich, S.P., Computer-Aided Design, 34 (10), 727–740, 2002. With permission.)

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Kinematic Geometry of Surface Machining

In order to verify satisfaction or violation of the global interference of the sculptured surface P and of the generating surface T of the form-cutting tool, equation rP = rP (U P , VP ) of the part surface P and equation rT = rT (UT , VT ) of the generation surface T must be represented in a common reference system. For the satisfaction of the fourth necessary condition of proper PSG, no real solutions to the set of two equations rP = rP (U P , VP )   rT = rT (UT , VT )

(7.38)

must be observed out of point(s) at which the surfaces P and T make regular contact. The Fourth Condition of Proper PSG: The fourth necessary condition of proper PSG is satisfied if and only if no global interference of the sculptured part surface P to be machined and of the generating surface T of the form-cutting tool is observed.

7.2.5 The Fifth Condition of Proper Part Surface Generation Mechanism and machine components are usually bounded not by the single surface P to be machined, but by several surfaces Pi, for example, having a sculptured surface P component (shown in the left portion of Figure 1.3) also bounded by walls next to the surface P. A sculptured surface Pi may have not only two but usually more neighboring surfaces Pi±1 . In order to machine the part, the form-cutting tool has to reproduce all the corresponding generating surfaces Ti . For the machining of the neighboring surface Pi±1 , the form-cutting tool has to be capable of generating corresponding surface Ti±1 . Various kinds of relative configurations of the neighboring generating surfaces Ti and Ti±1 are feasible. The generating surfaces Ti and Ti±1 of the form-cutting tool can be apart from each other (Figure 7.22a). In this case, boundaries ab and cd of the surfaces Ti and Ti±1 have no common points. The generating surfaces Ti and Ti±1 of the form-cutting tool may have a common boundary curve ab  cd (Figure 7.22b), or they can intersect each other (Figure 7.22c) at a segment ef of a curved line. Finally, the generating surface Ti of the form-cutting tool can be located within the interior of the cutting tool body, which is bounded by the generated surface Ti±1 (Figure 7.22d). Various feasible relative configurations of the generating surfaces Ti and Ti±1 of the cutting tool cause different conditions of generation of the sculptured surfaces Pi and Pi±1 . When the surfaces Ti and Ti±1 are apart from each other (Figure 7.22a), both the portions Ti and Ti±1 of the generating surface of the form-cutting tool can be reproduced by the cutting tool. Under such a scenario, the given

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325

Conditions of Proper Part Surface Generation Ti

Ti±1

Ti±1

Ti

Ti±1

b a

d c

d

b d a

(a)

c

c

(b)

Ti

e a

Ti±1

f b

d

c

Ti

b a

(c)

(d)

Figure 7.22 Feasible relative configurations of two neighboring portions of the generating surface of a formcutting tool. (From Radzevich, S.P., Computer-Aided Design, 34 (10), 727–740, 2002. With permission.)

sculptured surface can be machined without deviating from the desired geometry. The same occurs when the surfaces Ti and Ti±1 share a boundary curve ab  cd (Figure 7.22b). In this case, the given sculptured part surface can be machined without any deviations from its desired shape. In a particular case, the surfaces Ti and Ti±1 can intersect each other (Figure 7.22c). When the intersection is observed, then the entire generating surface of the cutting tool cannot be reproduced by the form-cutting tool. The portion abfe of the surface Ti and the portion cdef of the surface Ti±1 cannot be reproduced by the form-cutting tool edges. Therefore, a corresponding portion of the sculptured part surfaces Pi and Pi±1 , which have to be generated by the portions abfe and cdef of the surfaces Ti and Ti±1 , cannot be machined — this is caused by the form-cutting tool not being capable of reproducing the portions abfe and cdef of the surfaces Ti and Ti±1 . Portions of the surfaces Pi and Pi±1 , those that have to be generated by the portions abfe and cdef of the surfaces Ti and Ti±1 , are actually replaced with some kind of transition surface instead. As an example, consider two neighboring portions Pi and Pi±1 of a sculptured surface. The surfaces Pi and Pi±1 share the mutual boundary curve AB (Figure 7.23a). For proper generation of the surface Pi , the equation of contact n i ⋅ VΣ = 0 must be satisfied. Similarly, for proper machining of the surface Pi±1 , the equation of contact ni ± 1 ⋅ VΣ = 0 must be satisfied. For machining both surfaces Pi and Pi±1 simultaneously, both equations of contact n i ⋅ VΣ = 0 and n i ± 1 ⋅ VΣ = 0 must be satisfied simultaneously. When machining a convex portion of the sculptured surface (Figure 7.23b), the equations of contact n i ⋅ VΣ = 0 and n( i±1) ⋅ VΣ = 0 can easily be satisfied at every instant of machining. In this case, the surfaces Ti and Ti±1 can be located apart from each other (as shown in Figure 7.22a). The lines Ei and E i ±1 , and so forth, of contact of the surfaces Pi and Ti , Pi±1 and Ti±1 , and so forth, are known as characteristic lines. It is much more difficult to satisfy simultaneously the equations of contact n i ⋅ VΣ = 0 and n( i±1) ⋅ VΣ = 0 when machining concave portions of a sculptured surface (Figure 7.23c). The surfaces Ti and Ti±1 intersect each other along a certain curve AB as shown in Figure 7.23c). The entire surfaces of Pi

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326

Kinematic Geometry of Surface Machining

Pi B

ni±1

Ei

Pi±1

ni

M

B

Pi

Ti

Ti±1 Ei±1

A

A

Pi±1 (a) Ei±1

Ti

Ei

A

(b) Pi

Ti±1

B

Pi±1

Ei

Ti

Ti±1

Pi B

A

Pi±1

Ei±1

(c)

(d)

Figure 7.23 Various conditions of generating the two neighboring portions of the sculptured part surface. (From Radzevich, S.P., Computer-Aided Design, 34 (10), 727–740, 2002. With permission.)

and Pi±1 cannot be machined in this way. Some portions of the surfaces Pi and Pi±1 are replaced with a certain transition curved surface. For proper machining of the surface (Figure 7.23c), there is only one possible way to satisfy the equations of contact n i ⋅ VΣ = 0 and n( i±1) ⋅ VΣ = 0 simultaneously. For this purpose, it is necessary to eliminate crossing of the characteristic line Ei and Ei±1 as shown in Figure 7.23c, and to ensure the characteristic lines Ei and Ei±1 share common endpoints (Figure 7.23d). For the analytical description of the fifth necessary condition of proper PSG, the following set of two equations is considered:

rT( i ) = rT( i ) (UT( i ) ; VT( i ) )   rT( i±1) = rT( i±1) (UT( i±1) ; VT( i±1) )

(7.39)

( i ±1) where rT( i ) and rT designate position vectors of the surfaces Ti and Ti±1 , and UT( i ) , VT( i ) and UT( i±1) , VT( i±1) are the curvilinear (Gaussian) coordinates of a point on the surfaces Ti and Ti±1 , correspondingly.

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Conditions of Proper Part Surface Generation

The fifth necessary condition of proper PSG is satisfied if and only if the set of two Equations (7.39) has no real solution. A solution (if any) is allowed only along the common boundary curve (similar to the curve AB shown in Figure 7.23d). The Fifth Condition of Proper PSG: The fifth necessary condition of proper PSG could be satisfied if and only if the neighboring portions of the generating surface of the form-cutting tool do not intersect each other, and not one of them is located within the generating body of the cutting tool beneath the other neighboring surface portion.

In other words, in order to satisfy the fifth necessary condition of proper PSG, no transition surfaces are allowed to be observed on the machined part surface P. A particular case of surface generation occurs when the adjacent portions of the generating surface of the cutting tool overlap. Such a scenario is illustrated with an example. Consider machining a part having two adjacent surfaces P1 and P2 (Figure 7.24). The conical surface P1 and the cylindrical surface P2 of the part are machining in one setup with the cutting tool having conical the generating surface T. Both portions T1 and T2 of the generating surface of the cutting tool are congruent to each other (say, T1  T2 ). When machining the part, the axis of rotation of the work is parallel to the generating straight line of the cone surface T1  T2 . The work is rotating about the part axis of rotation, while the cutting tool is rotating about its axis of rotation. The cone angle of the surfaces T1  T2 is designated as θ , the rotation of the work is designated as ω P , and rotation of the cutting tool is designated as ω T . For the generation of the conical surface P1 having generated a straight line AD, the characteristic line E 1 is necessary. The cylindrical surface P2 generated by straight line BC. In order to be generated, the characteristic line T1 T2

Ξ = 2δ

D E1 B*

A

θ

ωT E2

F A*

O P1

B RP = RT

D*

ωP

C P2

Figure 7.24 Generation of cylindrical and of conical surfaces with a surface of internal cone.

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Kinematic Geometry of Surface Machining ωT

ωT

E2

T1

E1

E1

E2 P1

ωP

T2

ωP

P2

P2

T1

T2

P2

P1

The Transition Surface

E2

ωT

E1 e T1

ωP

P1

Figure 7.25 Machining of two surfaces P1 and P2 in one setup with the flat-end milling cutter.

E 2 is required. However, due to the symmetry, certain portions BD and B D of the characteristic lines E 1 and E 2 overlap. Because of this characteristics, E 1 and E 2 do not intersect each other, so no transition surface is generated. This is because the fifth necessary condition of proper PSG is not violated in this case. Violation of the fifth necessary condition of proper PSG is illustrated with an example. Consider milling a part having two adjacent surfaces P1 and P2 (Figure 7.25). The flat surface P 1 and the cylindrical surface P2 are machining in one setup with the flat-end milling cutter. When machining the part, the axis of rotation of the cutting tool is perpendicular to the axis of rotation of the part. The work is rotating about the part axis of rotation, while the cutting tool is rotating about its axis of rotation. The rotation of the work is designated ω P , and rotation of the cutting tool is designated ω T . For the generation of the flat portion P1 , the characteristic line E 1 is necessary. The cylindrical portion P2 is generated by the characteristic line E 2 . However, the characteristic lines E 1 and E 2 intersect each other. Because of this, neither the entire flat surface P1 , nor the entire cylindrical surface P2

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329

Conditions of Proper Part Surface Generation P(n)

P(t)

P(a)

K [h] r(t) P ZP

r(a) P

(n) nP hΣ

hΣ

r(n) P

YP

XP

Figure 7.26 Cusps on the machined sculptured part surface P. (From Radzevich, S.P., Computer-Aided Design, 34 (10), 727–740, 2002. With permission.)

can be generated. The form transition surface is machined instead. This is due to violation of the fifth necessary condition of proper PSG. 7.2.6 The Sixth Condition of Proper Part Surface Generation When machining a sculptured surface, point contact of the surfaces P and T is usually observed. Due to point contact of the surfaces, the discrete generation of the sculptured surface often occurs. Representation of the generating surface T by distinct cutting edges of the form-cutting tool is the other reason the discrete generation of the sculptured surface takes place. In an instant of time, it is physically impossible to generate the sculptured surface P by a single moving point. When the discrete surface generation occurs, the nominal smooth, regular sculptured surface P( n) and the actual machined surface P( a ) are not identical. The actual part surface P( a ) can be interpreted as the nominal sculptured surface P( n) that is covered by cusps (Figure 7.26) or may have other deviations from P( n) . The sixth necessary condition of proper PSG is formulated as follows: The Sixth Condition of Proper PSG: The actual part surface P with cusps, if any, must remain within the tolerance on surface accuracy.

Cusps on the machined sculptured surface P must be within the tolerance on the surface accuracy. Then maximal height hΣ of the cusps must not exceed the tolerance [ h] on the sculptured surface accuracy. Consider a Cartesian coordinate system X PYP ZP associated with the sculptured surface P. The sixth necessary condition of proper PSG is satisfied if and only if the following condition is satisfied at every point of the nominal sculptured surface P:

n(Pn) ⋅ hΣ = rP( a ) − rP( n) ≤ rP(t ) = n(Pn) ⋅ [ h]

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(7.40)

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Kinematic Geometry of Surface Machining

where the position vector of a point of a nominal sculptured surface P ( n ) is designated as rP( n) ; the position vector of the corresponding point of the actual part surface P (a ) is designated as rP( a ) , the position vector of a point of the surface of tolerance P(t ) is designated as rP(t ) , and the unit normal vector to surface P( n) is designated as n(Pn) . If the sixth necessary condition of proper PSG is satisfied, then the actual part surface P( a ) is entirely located within the nominal sculptured part surface P( n) and the surface of tolerance P(t ) . In the example (Figure 7.26), the surface of tolerance P(t ) is depicted over the surface P( n) at the distance n(Pn) ⋅ [ h] . Fulfillment of the set of six conditions of proper part surface generation is necessary and sufficient to insure machining of the part surface in compliance with the requirements indicated in the part blueprint.

7.3

Global Verification of Satisfaction of the Conditions of Proper Part Surface Generation

When machining a sculptured surface on a multi-axis NC machine, it is important to get to know whether the entire part surface can or cannot be machined on the given machine. It is also important to detect the sculptured surface regions, those that are not accessible by the cutting tool of a given design. In other words, it is necessary to detect regions on the sculptured surface P which the cutting tool cannot reach without being obstructed by another portion of the part. Certainly, such regions (if any) are due not just to the geometry of the sculptured surface P, but also to the geometry of the generating surface T of the cutting tool. The particular problem under consideration is now referred to as the cutting-tool-dependent partitioning of a sculptured surface onto the cutting-tool-accessible and onto the cutting-toolnot-accessible regions. 7.3.1 Implementation of the Focal Surfaces For solving the problem of cutting-tool-dependent partitioning (CT-dependent partitioning) of a sculptured surface, the third necessary condition of proper PSG is the most critical issue. The geometry of contact of the surfaces P and T in the infinitesimal vicinity of a cutter-contact-point (CC-point) K is a vital link for verification of whether the third necessary condition of proper PSG is globally satisfied or not. Within the cutting-tool-accessible portions of the sculptured surface, the proper correspondence is observed between the normal curvature k P of the

Two points on the surfaces rP( n) and rP( a ) are corresponding to each other if they share a common straight line, which aligned with the perpendicular n(Pn) to the surface rP( n).

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Conditions of Proper Part Surface Generation

331

surface P, and the normal curvature kT of the generating surface T of the cutting tool (Table 7.1). The normal curvatures k P and kT are measured in the same direction specified by the unit tangent vector t P . Otherwise, when the correspondence between the normal curvatures k P and kT is improper (Table 7.1), interference of the surfaces P and T occurs. Such regions of the surface P cannot be machined properly. Implementation of the indicatrix of conformity CnfR ( P/T ) (see Equation 4.59) enables detection of local, not global, interference of the surfaces P and T. If negative diameters dcnf of the indicatrix of conformity CnfR ( P/T ) are observed, this immediately indicates that a certain portion of the surface P is not machinable with the cutting tool of a given design. It is easy to conclude that within the bordering curve between the cutting-tool-accessible and the cutting-tool-not-accessible portions of the surface P, the identity dcnf  0 is observed. Ultimately, the problem of partitioning of a sculptured surface reduces to the problem of finding those lines on the part surface P within which the identity dcnf  0 is valid. For solving the problem, various approaches can be used. The implementation of focal surfaces is promising in this concern. 7.3.1.1

Focal Surfaces

The geometry of contact of the surfaces P and T in the infinitesimal vicinity of a CC-point K, turns our attention to the normal curvatures of the surfaces P and T, and to the location of centers of normal curvature of these surfaces. The direction of feasible tool approach to a surface point is defined as the direction along which a cutting tool can reach a part surface without being obstructed by another portion of the part. For a part design to be machinable, every feature of the part design should have at least one such feasible direction. For a sculptured surface, if a point on the surface does not have at least one such feasible direction, it is not machinable. Global analysis and detection of the surface P regions, those that are cuttingtool-accessible, as well as those that are cutting-tool-not-accessible, and a visual interpretation of the global accessibility of the surface can be performed by means of focal surfaces for the surfaces P and T. For generating the focal surfaces, it is necessary to recall that there are two principal plane sections C1. P and C2. P through a point M of smooth, regular sculptured surface P. Principle surfaces C1. P and C2. P are passing through the surface P unit normal vector n P , and through the directions specified by the principal unit tangent vectors t 1.P and t 2.P . Principal radii of curvature R1. P and R2. P of the surface P are measured in the principal plane sections C1. P and C2. P . Centers of curvature O1. P and O2. P of the sculptured surface at point M (Figure 7.27) are located within the straight line through the unit normal vector n P erected at the point M. Points of this kind are usually referred to as the focal points of a surface P at M.

The same is true with respect to the Ar R ( P / T )-indicatrix (see Chapter 4).

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Kinematic Geometry of Surface Machining nP

C1.P

C2.P

P M

rP

t1.P

f1.P

t2.P

R1.P R2.P

O1.P f2.P

ZP XP

O2.P

YP Figure 7.27 Determining the focal point at the current point of the sculptured surface P. (From Radzevich, S.P., Computer-Aided Design, 37 (7), 767–778, 2005. With permission.)

The locus of the focal points O1. P, which are determined for every point within the sculptured surface P, represents the first principal focal surface f1. P (U P , VP ) . Similarly, the locus of the focal points O2. P , which are also determined for every point within the sculptured surface P, represents the second principal focal surface f2. P (U P , VP ) . In particular cases, two focal surfaces f1. P (U P , VP ) and f2. P (U P , VP ) can be congruent to each other. Under such a scenario, both focal surfaces appear as the single surface. A focal surface can shrink to a curved line. The surface of a circular cylinder is a good example in this concern. The focal surface f1. P (U P , VP ) for the cylinder of revolution is degenerated to the straight line that aligns with the axis of rotation of the cylinder surface. Moreover, a focal surface can even shrink to a point. The last is observed for both the focal surfaces f1. P (U P , VP ) and f2. P (U P , VP ) of a spherical surface P. Under certain conditions, consecutive positions of a straight line through unit normal vector n P at points along lines of curvature of a surface P can form enveloping curves that are space curves (Figure 7.28). Focal surfaces can be considered as the family of consecutive positions of the corresponding space enveloping curves [17]. The considered geometrical property of focal surfaces is used below for derivation of the equation of the focal surfaces. The current point of the focal surface coincides with the centers of corresponding principal curvature of the surface P. Thus, position vector f1. P (U P , VP ) of a point of the first focal surface, as well as position vector and f2. P (U P , VP ) of a point of the second focal surface can be represented in terms of position vector rP (U P , VP ) of a point of the surface P; in

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333

Conditions of Proper Part Surface Generation P S M

nP

a

rP

M1

r1

S1 Figure 7.28 On representation of a focal surface as an enveloping surface to perpendiculars to the surface P. (From Radzevich, S.P., Computer-Aided Design, 37 (7), 767–778, 2005. With permission.)

terms of unit normal vector n P to the surface P; and in terms of corresponding radii of principal curvature, either R1. P or R2. P (Figure 7.27):

f1. P (U P , VP ) = r P (U P , VP ) − R 1. P ⋅ n P f2. P (U P , VP ) = r P (U P , VP ) − R 2. P ⋅ n P

(7.41) (7.42)

Elementary substitution R1. P = k1−.1P and R2. P = k2−.1P yields expression of the focal surfaces f1.P , f2.P (Equation 7.41 and Equation 7.42) in terms of principal curvatures:

f2. P (U P , VP ) = r P (U P , VP ) − k −21. P ⋅ n P f1. P (U P , VP ) = r P (U P , VP ) − k −11. P ⋅ n P

(7.43) (7.44)

Radii of principle curvatures R 1.P and R 2.P in Equation (7.41) and Equation (7.42) are computed using one of the equations represented in Chapter 1 (Equation 1.14 and Equation 1.19). Radii of principle curvatures R 1.P and R 2.P can be expressed in terms of the % and of the Gaussian curvature G% of the surface P: mean curvature M P P

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( % = (M

) − G% )

% + M % 2 − G% R 1. P = M P P P R 2. P

P

%2 − M P

P

−1

−1

(7.45) (7.46)

334

Kinematic Geometry of Surface Machining f2 P

С 1.T O2 P

С 1 P

T

М С2.P

nP

С 2.T

nT

t1.T

t2.T P

М

t2.P

t1.P

O1.T

f1.T

f2.T O2.T

f1 P ZP

YP

O1 P

ZP

YP

XP

XP ( b)

( a)

Figure 7.29 Examples of focal surfaces constructed for a local patch of the sculptured surface P (a), and for a local patch of the generating surface T of a cutting tool (b). (From Radzevich, S.P., ComputerAided Design, 37 (7), 767–778, 2005. With permission.)

Ultimately, equations for the focal surfaces f1. P (U P , VP ) and f2. P (U P , VP ) can be represented in the form f1. P (U P , VP ) = r P − f2. P (U P , VP ) = r P −

nP % + M % 2 − G% M P P P

(7.47)

nP % % 2 − G% MP − M P P

(7.48)

The focal surfaces f1.P and f2.P for a saddle-like patch of a sculptured surface P are plotted in Figure 7.29a. Such a patch of the surface P can be machined, for example, with the convex generating surface T of a cutting tool. Focal surfaces f1.T (UT , VT ) , and f2.T (UT , VT ) for this surface T are depicted in Figure 7.29b. ( ( In Figure 7.29, the respective lines of curvature are designated as C 1.P , C 2.P ( ( and C1.T, C2.T, correspondingly. Points O1.T and O2.T are the points of the corresponding focal surfaces f1.T and f2.T for the surface T at point M:

f1.T (UT , VT ) = rT (UT , VT ) − k −11.T ⋅ nT f2.T (UT , VT ) = rT (UT , VT ) − k2−.1T ⋅ nT

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(7.49) (7.50)

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Conditions of Proper Part Surface Generation

O2.T

f2.T

O2.P

f1.T

O1.T

f2.P

Č1.T

Č2.T

T

nP K

P Č1.P

nT

Č2.P

f1.P ZP

YP

O1.P

XP

Figure 7.30 Configuration of the focal surfaces f1.P, f 2.P for the sculptured surface P relative to the focal surfaces f1.T and f 2.T for the generating surface T of the cutting tool. (From Radzevich, S.P., Computer-Aided Design, 37 (7), 767–778, 2005. With permission.)

Focal surfaces f1.P and f2.P intersect the sculptured surface P along parabolic curved lines on it — that is, along lines at which Gaussian curvature G% P of the sculptured surface is equal to zero ( G% P  0 ). In order to use focal surfaces for the verification of whether or not the third necessary condition of proper PSG is satisfied globally, it is necessary to plot both of the focal surfaces f1.P and f2.P for the sculptured surface P and the similar focal surfaces f1.T and f2.T for the generating surface T of the cutting tool in a common coordinate system. An example of the relative configuration of the focal surfaces f1.P , f2.P and f1.T , f2.T at the point K of contact of the given surfaces P and T is illustrated in Figure 7.30. The saddle-like (G P < 0) local patch of a sculptured surface P is machined with a convex patch ( GT > 0, M T > 0 ) of the generating surface T of the cutting tool. In the case under consideration, angle µ of the local

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Kinematic Geometry of Surface Machining

relative orientation of surfaces P and T is equal to zero ( µ = 0 ). Inspecting Figure 7.30, it is easy to realize that the first principle planes C1. P  C1.T (the identity is ( due to µ ( = 0 ) intersect the surfaces P and T. The lines of the intersection C1. P and C1.T are convex lines ( k1. P > 0 ; k1.T > 0 ). Therefore, no problem is observed to satisfy the third necessary condition of proper PSG in this plane section. The second principle planes C2. P  C2.T (the identity is due to µ = 0 , these plane sections ( are also congruent to each other) intersect C the surfaces P and T. The line 2. P of the intersection is a concave curve. The ( line C2.T of the intersection is the convex curve. Because the distance KO2.T exceeds the distance KO2. P (i.e., KO2.T > KO2. P ), the principle curvatures k2. P and k2.T correspond to each other as |k2. P |> k2.T . Because the inequality |k2. P |> k2.T is valid, the third necessary condition of proper PSG in the second principal section of the surfaces P and T is not satisfied. Summarizing, one can conclude that the third necessary condition of proper PSG is not satisfied in the infinitesimal vicinity of the CC-point K (Figure 7.30). Analysis of Table 7.1 allows for analytical expression of the criterion for verification of whether the third necessary condition of proper PSG is satisfied or not:

0 sgn k P ⋅ sgn kT ⋅ sgn ( k P + kT ) =  −1

(7.51)

In order to globally satisfy the third necessary condition of proper PSG, it is necessary to ensure satisfaction of Equation (7.51) at every point K, and in every cross-section of the surfaces P and T by a plane through the unit normal vector n P . The third condition of proper PSG could be satisfied globally when each of the focal surfaces f1.T and f2.T is entirely located between the convex surface P and the corresponding focal surface f1.P or f2.P . Focal surfaces f1.T and f2.T can touch one or both focal surfaces f1.P or f2.P . In a similar way, location of the focal surfaces f1.T and f2.T , for concave and for saddle-like local patches of surface P can be specified. Focal surfaces f1.T and f2.T must not intersect the sculptured surface P and the corresponding focal surfaces f1.P and f2.P for the generating surface of the form-cutting tool. Otherwise, the third necessary condition of proper PSG would be violated. Focal surfaces f1.P and f2.P are the bounding surfaces of space, within which the centers of principal curvatures of the generating surface T of the cutting tool have been located. The portions of space bounded by the focal surfaces f1.P and f2.P are referred to as the cutting-tool-allowed (CT-allowed) zones. The rest of the space is referred to as the cutting-tool-prohibited (CT-prohibited) zones. 7.3.1.2

Cutting Tool (CT)-Dependent Characteristic Surfaces

When the third necessary condition of proper PSG is globally satisfied, then certain constraints are imposed on the actual configuration of the focal surfaces. For the purpose of verification of accessibility of the surface P by the

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337

Conditions of Proper Part Surface Generation

cutting tool, the CT-dependent characteristic surfaces can be used. It is convenient to illustrate the concept of the CT-dependent characteristic surfaces with an example of generation of a concave patch of the sculptured surface P. First, the current point of the first focal surface f1.T of the generating surface T of the cutting tool is located within the straight line along the unit normal vector n P that is erected at the corresponding point of the surface P. Second, within the straight line there exists a straight-line segment. Location of the current point of the focal surface f1.T is allowed within the straightline segment, as well as at its endpoints. Therefore, without loss of generality, instead of two) focal surfaces f1.P and f1.T , just one CT-dependent characteristic surface f1 (UT , VT ) can be employed. This surface features the summa (R 1. P + R 1.T ) of the first principal radii of curvature. The locus of points,)determined in the above way, forms the first CT-dependent characteristic surface f1 (UT , VT ) of the sculptured surface P and of the generating surface T of the cutting tool. The) position vector of a point of the first CT-dependent characteristic surface f1 can be expressed in terms of the parameters rP , n P , R1. P , and R1.T:

) f1 (UT , VT ) = r P (UT , VT ) − (R 1. P + R 1.T ) ⋅ n P

(7.52)

A similar analysis can be performed for the second focal surface f2.T of the generating surface T of the cutting tool. Ultimately, the position vector of a point of the second CT-dependent char) acteristic surfaces f2 can be expressed in terms of the parameters rP , n P , R2.P , and R2.T : ) f2 (UT , VT ) = r P (UT , VT ) − (R 2. P + R 2.T ) ⋅ n P (7.53) Summarizing, one can conclude that the CT-dependent characteristic surface is a surface, each point of which is remote from the sculptured surface P perpendicular to it at a distance that is equal to the algebraic sum of the corresponding radii of principal curvature of the surfaces ) )P and T. When the CT-dependent characteristic surfaces f1 and f2 do not intersect the sculptured surface P, then the third necessary condition of proper PSG is satisfied globally. Under such a scenario, the sculptured surfaces P can be machined properly in compliance with the ) surface ) blueprint. Otherwise, if the CT-dependent characteristic surfaces f1 and f2 intersect the surface P, or they are entirely located within the interior part of the body, the third necessary condition of proper PSG cannot be satisfied. In this case, the surface P cannot be machined properly. Application of the CT-dependent characteristic surfaces for the purposes of resolving the problem of partitioning the sculptured surface onto the cutting-tool-accessible and cutting-tool-not-accessible regions reduces the number of surfaces to be considered from four focal surfaces ) )( f1.P , f2.P and f1.T , f2.T ) to two CT-dependent characteristic surfaces ( f1 and f2 ).

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Kinematic Geometry of Surface Machining

The cutting-tool-accessible regions are separated from the cutting-toolnot-accessible regions of the sculptured surface P by a corresponding boundary curve. 7.3.1.3

Boundary Curves of the CT-Dependent Characteristic Surfaces

The boundary curve for cutting-tool-accessible region of the sculptured surface P is the line of intersection of the ) part)surface by the corresponding CT-dependent characteristic surfaces f1 and f2 . Therefore, every point of the boundary curve rbc satisfies the corresponding set of two equations:

)  f = fˆ (U , V ) = r − (R + R ) ⋅ n 1 P P P 1. P 1.T P  1  r P = r P (U P , VP ) )  f = fˆ (U , V ) = r − (R + R ) ⋅ n 2 P P P 2. P 2.T P  2  r P = r P (U P , VP ).

(7.54)

(7.55)

Equations for the two-surface intersection curve can be derived from the condition

r P − (R 1. P + R 1.T ) ⋅ n P = r P (U P , VP )

(7.56)

r P − (R 2. P + R 2.T ) ⋅ n P = r P (U P , VP )

(7.57)

The approach for determining the boundary curves which is based on the solutions to Equation (7.56) and Equation (7.57) can be significantly simplified taking into consideration Equation (7.51). After inserting the previously derived Equation (7.51) and rearranging Equation (7.56) and Equation (7.57) cast into

r P − sgn k 1. P ⋅ sgn ⋅ k 1.T ⋅ sgn ( k 1. P + k 1.T ) ⋅ n P = r P (U P , VP ) r P − sgn k 2. P ⋅ sgn ⋅ k 2.T ⋅ sgn ( k 2. P + k 2.T ) ⋅ n P = r P (U P , VP )

(7.58) (7.59)

Equation (7.58) and Equation (7.59) represent an analytical description of the boundary curves that separate the cutting-tool-accessible regions of the sculptured surface P from the cutting-tool-not-accessible regions on it. Derivation of the boundary curves of the CT-dependent characteristic surfaces is illustrated below with two examples. Consider generation of the torus surface P. A computer model of a torus surface is widely used as a convenient test case. It is proven [16,17,20] that the torus surface provides significantly higher accuracy of approximation and thus is preferred for local approximation of the surfaces P and T over quadrics. This is because the principal radii of curvature R1. P and R2. P of the

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339

Conditions of Proper Part Surface Generation The Boundary Curves T3

ωT

ωT

ωT

T2

T4

ZP

K2

K3

P

K4

K1 R ωT

YP

XP

r

T1 Figure 7.31 Partitioning of the torus surface P that is machining with a flat-end milling cutter.

surface P (and the similar principal radii of curvature R1.T and R2.T of the surface T) uniquely specify the torus surface. The first principal radius of curvature R1. P is equal to the radius of the generating circle of the torus surface, and the second principal radius of curvature R2. P is equal to the radius of the outside circle of the torus surface (and therefore, the radius R of the directing circle is equal to the difference R = R2. P − R1. P). A similar condition is valid with respect to the generating surface T of the cutting tool. For both examples below, Equation (7.16) of the torus surface P from Example 7.1 is implemented. Example 7.2 Consider machining of a torus surface P with the flat-end milling cutter (Figure 7.31). The radius r of the generating circle of the surface P is equal to r = 50 mm, and the radius R of the directing circle of the surface P is equal to R = 90 mm . Gaussian (curvilinear) coordinates θ P and ϕ P of a point on the surface P vary in the range of 0 ≤ θ P ≤ 180 and 0 ≤ ϕ P ≤ 360 . Using Equation (7.58) and Equation (7.59) in the commercial software MathCAD allows the equation

© 2008 by Taylor & Francis Group, LLC

 90 ⋅ sin θ P    90 ⋅ cos θ P  r bc (θ P ) =   ±50    1  

(7.60)

340

Kinematic Geometry of Surface Machining

for the position vector rbc (θ P ) of a point of the boundary curve for the CTdependent characteristic surface. When machining the torus surface P, the milling cutter rotates about its axis with a certain angular velocity ωT . The milling cutter is traveling with respect to the work occupying various positions T1, T2, T3, and T4 relative to the surface P. The milling cutter contacts the surface P at the corresponding CC-points K1 , K 2 , K 3 , and K 4 . The boundary curve rbc (θ P ) subdivides the surfaces P onto the cutting-tool-accessible and onto the cutting-tool-not-accessible ℜ (shadowed) regions. The boundary curve rbc (θ P ) (see Equation 7.60) indicates that the positions T1 and T2 of the milling cutter are feasible. The cutter position T3 is also allowed, and it is limited in its position. Due to the CC-point, K 4 is located within the cutting-tool-not-accessible region ℜ—the position T4 of the milling cutter is not feasible. In that position of the milling cutter, the third necessary condition of proper PSG is not satisfied; thus, the surface P cannot be machined properly. Example 7.3 Consider machining of a torus surface P with the cylindrical milling cutter (Figure 7.32). The same surface P as that in Example 7.2 could be machined with a cylindrical milling cutter of the radius 50 mm . When machining the torus surface P, the cutter rotates about its axis with a certain angular velocity ωT . The milling cutter is traveling with respect to the work. Using Equation (7.58) and Equation (7.59) in the commercial software MathCAD, allows the equation

ωT

T3

T2

ωT

ZP

P

K2

K1

R K3

R

XP

K4 YP

r The Boundary Curve

T1

ωT

ωT

T4

Figure 7.32 Partitioning of the torus surface P that is machining with a cylindrical milling cutter.

© 2008 by Taylor & Francis Group, LLC

341

Conditions of Proper Part Surface Generation  90 ⋅ sin θ P    90 ⋅ cos θ P  r bc (θ P ) =   ±21.793    1  

(7.61)

for the position vector rbc (θ P ) of a point of the boundary curve for the CTdependent characteristic surface. When machining the torus surface P, the milling cutter occupies various positions T1 , T2 , T3 , and T4 relative to the surface P. It is contacting the surface P at the CC-points K1 , K 2 , K 3 , and K 4 correspondingly. The boundary curve rbc (θ P ) subdivides the surfaces P onto the cutting-tool-accessible and onto the cutting-tool-not-accessible ℜ (shadowed) regions. The boundary curve rbc (θ P ) (see Equation 7.60) indicates that the positions T1 and T2 of the milling cutter are allowed. The position T3 of the cutting tool is also allowed, and it is the limited cutter location. Because the CC-point K 4 is located within the cutting-tool-not-accessible region ℜ, the position T4 of the milling cutter is not allowed. In that position of the milling cutter, the third necessary condition of proper PSG is not satisfied; thus, the surface P cannot be machined properly. 7.3.1.4

Cases of Local-Extremal Tangency of the Surfaces P and T

Possible kinds of contact of the surfaces P and T are investigated in Chapter 4. In the theory of surface generation, pure local-extremal tangency of the surfaces is out of practical interest. However, this kind of surface contact could be observed in the form of quasi-kinds of surface contact when relative displacements of the contacting surfaces are maximal. Local-extremal kinds of contact of the surfaces P and T are observed when the equality k P = − kT is valid. Under such a scenario, the focal surfaces f1.P ) ) , f2.P and f1.T , f2.T (or the two CT-dependent characteristic surfaces f1 and f2 ) are not helpful for solving the problem of verification of the global satisfaction of the third necessary condition of proper PSG. In the case under consideration, another tool must be implemented. On the premises of the above analysis, it is recommended to use derivatives of the corresponding functions. In this way, the derivative-focal-surfaces (DF-surfaces) are introduced [19]. The DF-surfaces are analytically described by the following equation: f1 ,2. P(T ) = r P(T ) −

∂n R 1,2. P(T ) ( ⋅ n P (T ) dC1n,2. P(T )

(7.62)

where n designates the smallest integer number under which any uncertainty in global satisfaction of the third necessary condition of proper PSG

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342

Kinematic Geometry of Surface Machining n

∂ R does not( arise, and ( n1,2. P(T ) designates the derivative of R 1,2. P(T ) in the direcdC1 , 2. P ( T ) tion of C1n,2. P(T ) . In the cases under consideration, it is necessary to determine the DF-surfaces for the sculptured surface P:

f1 . P = rP − f2 . P = rP −

∂n R 1. P ( ⋅ nP dC1n. P

∂n R 2. P ( ⋅ nP dC2n. P

(7.63)

(7.64)

It is also necessary to determine the similar DF-surfaces for the generating surface T of the cutting tool: f1 .T = rT − f2 .T = rT −

∂n R 1.T ( ⋅ nT dC1n.T ∂n R 2.T ( ⋅ nT dC2n.T

(7.65) (7.66)

In order to globally satisfy the third necessary condition of proper PSG, the shape, the parameters, and the relative disposition of the DF-surfaces f1. P , f2. P , and f1.T , f2.T must be correlated with the shape, the parameters, and the relative location of the surfaces P and T, say in the way similar to that considered above. Similarly, the derivative-cutting-tool-dependent (DCT-dependent) characteristic surfaces can be introduced: (7.67)

 ∂n R 1. P ∂n R 1.T  ) f1 = rP −  ( + (  ⋅ nP dC1n.T   dC1n. P

(7.68)

 ∂n R 2. P ∂n R 2.T  ) f2 = rP −  ( + (  ⋅ nP dC2n.T   dC2n. P

The surfaces above could be used in the way that the focal surfaces f1)P , f2.P , ) f1.T , and f2.T (and the CT-dependent characteristic surfaces f1 and f2 ) are used for the cases of regular tangency of the surfaces P and T. In cases of local-extremal tangency of the surfaces P and T, implementation of the DF-surfaces, and inplementation of the DCT-dependent characteristic surfaces is helpful for partitioning the sculptured surface P onto the cuttingtool-accessible and onto the cutting-tool-not-accessible regions.

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343

Conditions of Proper Part Surface Generation 7.3.2 Implementation of R-Surfaces

A proper correspondence between the normal curvatures k P of the part surface P and the corresponding normal curvatures kT of the generating surface T of the cutting tool is one of the major prerequisites for proper generation of the surface P in the differential vicinity of the CC-point. 7.3.2.1

Local Consideration

The geometry of contact of the surfaces P and T can be analytically described by the indicatrices of conformity CnfR ( P/T ) and Cnf k ( P/T ) [16,17,20] and by the An R ( P/T ) -indicatrix, or An k ( P/T ) -indicatrix [13] (see Chapter 4). For the purpose of verification of global satisfaction of the third necessary condition of proper PSG implementation, these characteristic curves have proven to be convenient in CAD/CAM applications. It is critically important to stress that all of the characteristic curves CnfR ( P/T ) , Cnf k ( P/T ) , An R ( P/T ) , and An k ( P/T ) specify the same directions of the extremal rate of conformity of the surfaces P and T at the current CC-point. This important property of the characteristic curves is illustrated by an example of machining of a bicubic Bezier surface P (Figure 7.33). The matrix equation for a bicubic Bezier patch P that is defined by a 4 × 4 array of points is as follows [6]:

rP (U P , VP ) = [(1 − U P )3

3U P (1 − U P )2

3U P2 (1 − U P )

 (1 − VP )3    3V (1 − VP )2  U P3 ] ⋅ PP ⋅  P2 (7.69)  3V (1 − V )   P 3 P  VP  

where [PP ] = [p i , j ]ij==11,, KK,, 44 , and position vectors of the control points are denoted as pi , j . In Equation (7.69), the bicubic patch is expressed in a form similar to the Hermite bicubic patch [6]. The matrix [P] contains the position vectors for points that define the characteristic polyhedron and, therefore, the Bezier surface patch. In the Bezier formulation, only four corner points p11 , p41 , p14 , and p44 actually lie on the surface patch. The points p21 , p31 , p12 , p13 , p42 , p43 , p24 , and p34 control the slope of the boundary curves. The remaining four points p22 , p32 , p23 , and p33 control the cross-slopes along the boundary curves in the same way as the twist vectors of the bicubic patch. The Bezier surface is completely defined by a net of design points describing two families of Bezier curves on the surface. min of the indicatrix In Figure 7.33, the direction of the minimal diameter dcnf max t of conformity CnfR ( P/T ) aligns with the direction cnf at which the rate of conformity of the surfaces P and T is maximal.

The equation of the characteristic curves An R ( P/T ), and An k ( P/T ) is derived by Radzevich [13] on the premises of the equation of the well-known surface—namely, of the surface of Plücker’s conoid (see Chapter 4).

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Kinematic Geometry of Surface Machining

6

T

4

OT

ωT

P

nP

2 01 1

K

2

3

90 * 0.2

150

180

210

t2.P K

t max cnf t1.P

90 02

120

60 150

30

01 t2.P

0

min d3`

180

d min cnf 330

CnfR(P/T ) 240

T

(a)

4

120

CnfR(P/T)

nT

4

3

2

P

K

*

60 30

max

t cnf

K

t1.P

210

0

330 3`R(P/T )

270 (b)

300

240

270

300

(c)

Figure 7.33 Characteristic curves Cnf R(P/T) and An R(P/T) identically determine the direction of the maximal rate of conformity of the surfaces P and T at the current cutter-contact-point K.

At a current CC-point, both planar characteristic curves CnfR ( P/T ) (Figure 7.33b) and An R ( P/T ) (Figure 7.33c) identify that same direction t max cnf at which the rate of conformity of the surfaces P and T is maximal. From this standpoint, the characteristic curves CnfR ( P/T ) and An R ( P/T ) are equivalent. This is because the actual value of the angle ϕ that is determined from CnfR ( P/T ) (Figure 7.33b) is identical to the actual value of the angle ϕ that is determined from An R ( P/T ) (Figure 7.33c). The consideration below is focused on implementation of the indicatrix of conformity CnfR ( P/T ) . However, that same result can be obtained using a An R ( P/T )- characteristic curve. It is well known that the larger the cutting tool, the smaller the resulting scallop height, or the excessive material not removed, might result for the same tool-path. The indicatrix of conformity CnfR ( P/T ) and the An R ( P/T )characteristic curve can be employed to directly bound the largest cutting tool radius that can be used for the machining of saddle-like and concave regions, and hence aid in tool selection.

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345

Conditions of Proper Part Surface Generation 90

120

* 02

150 nP

CnfR(P/T)

P

K

K

T CnfR(P/T)

90

90

120

60

t2.P

150

30

CnfR(P/T ) dmin cnf = 0 t1.P

0

330

210 CnfR(P/T ) 240

60

0.15

0.1

K

180

300

270 (b)

0.15 CnfR(P/T)

0

dmin cnf > 0 330

240

150

t1.P

210

(a )

120

30

t2.P

180

nT

60

270 (c )

300

0.1 t2.P K

180

210

30 min d cnf < 0

0

t1.P

330

CnfR(P/T ) 240

270

300

(d )

Figure 7.34 Examples of satisfaction and of violation of the third necessary condition of proper part surface generation. [The current cutter-contact-point K in (c) represents a point of the boundary curve that subdivides the surface P into the cutting-tool-accessible and the cutting-tool-notaccessible regions.]

In order to satisfy the third necessary condition of proper PSG, all diameters d cnf = 2 rcnf of this characteristic curve must be nonnegative — that is, in all directions through the point K, the relationship rcnf ≥ 0 must be satisfied. The indicatrix of conformity CnfR ( P/T ) (see Equation 4.59) yields a conclusion on the actual kind of contact of the surfaces P and T at a current CC-point (Figure 7.34a). When the surfaces makes a regular point contact, then the minimal diammin eter d min cnf of the indicatrix of conformity Cnf R ( P/T ) is positive ( d cnf > 0), as depicted in Figure 7.34b. A CC-point of that kind cannot be a point of the boundary curve r bc .

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346

Kinematic Geometry of Surface Machining

If the minimal diameter d min cnf of the indicatrix of conformity Cnf R ( P/T ) is max min equal to zero d min cnf = 0 (Figure 7.34c), then in the direction of t cnf along d cnf , radii of normal curvature of the surfaces P and T are of the same magnitude and of opposite sign ( R P = − R T ). A point at which the equality d min cnf = 0 is observed is a point of the boundary curve r bc . Finally, the minimal diameter d min cnf of the indicatrix of conformity can be of negative value ( d min cnf < 0 ) as shown in Figure 7.34d. Negative value of the diameter d min cnf indicates that at this CC-point, the surface T interferes with the surface P. This means that violation of the third necessary condition of proper PSG occurs at this CC-point. A CC-point at which the inequality d min cnf < 0 is observed is not within the boundary curve r bc . 7.3.2.2

Global Interpretation of the Results of the Local Analysis

For derivation of equations of the characteristic surfaces that enable one partitioning of a sculptured surface P onto the cutting-tool-accessible regions ℜ + and onto the cutting-tool-not-accessible regions ℜ − (if any), geometric properties of the characteristic curve CnfR ( P/T ) are employed below. 7.3.2.2.1 Characteristic Surface of the First Kind Consider unit normal vector n P erected at an arbitrary point of the sculpmin (U , V , U , V , µ ) tured surface P. A straight-line segment of the length rcnf P P T T aligns with the vector n P . min Further, consider a vector of the length rcnf , those that align with the vecmin tors n P . Positive vectors n P ⋅ rcnf > 0 are directed outward from the part body, min < 0 and negative vectors n P ⋅ rcnf are directed inward to the part body. min The loci of endpoints of the vectors n P ⋅ rcnf determine a characteristic R 1-surface of the first kind : ( 1) min The characteristic R 1- surface ⇒ rR (U P , VP , UT , VT , ϕ , µ ) = rP + n P ⋅ rcnf

(7.70)

Example 7.4 Consider a surface P (Figure 7.35) that is given by the equation

 3 ⋅ cos ϕ P cos θ P + 5 ⋅ cos θ P    3 ⋅ cos ϕ P sin θ P + 5 ⋅ sin θ P  π , rP (θ P , ϕ P ) =    0 ≤ θ ≤ , 0 ≤ ϕ ≤ π (7.71) 3 ⋅ sin ϕ P 2   1   This surface has convex and saddle-like regions.

Reminder: n P = n P(UP,VP), and rcnf = rcnf (UP,VP,U T,V T,m).

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347

Conditions of Proper Part Surface Generation

15

1−

Surface 0

10

– Boundary Curve +

–

5

– Portion P

− Portion

2

00 0 2

4 6

8

Figure 7.35 An example of the ℜ characteristic surface.

In the example under consideration, only the second principal radius of curvature R 2. P is of importance. The first principal radius of curvature R 1. P imposes no restrictions on the satisfaction or violation of the third necessary condition of proper PSG. An expanded equation for the unit normal vector n P can be easily derived from Equation (7.71) using the equality n P = u P × v P . For the machining of the surface P, the cylindrical milling cutter of diameter d T = 3 ′′ is used. The equation for the generating surface T of the cylindrical milling cutter is trivial. Equation (7.71) of the surface P along with the parameters of the second principal radius of curvature R 2. P , the unit normal vector n P , and the equation of the cylindrical surface T yields derivation of the equation of the R1 surface. A routing formulae transformation leads to the following expression for the R 1-surface:

  5 + 3 ⋅ cos ϕ P  ⋅ cos ϕ P cos θ P + 5 ⋅ cos θ P   3 +  |cos ϕ P |       + ⋅ ϕ 5 3 cos P  3+  n θ + ⋅ sin θ ϕ 5 ⋅ cos sin P P P  rR (θ P , ϕ P ) =   |cos ϕ P |     5 + 3 ⋅ cos ϕ P       3 + |cos ϕ |  ⋅ sin ϕ P   P     1

© 2008 by Taylor & Francis Group, LLC

(7.72)

348

Kinematic Geometry of Surface Machining

The surface rR (see Equation 7.72) is depicted in Figure 7.35. It is evident that the surfaces P and R 1 intersect one another. The line of intersection serves as the boundary curve ℜ 0 . The major feature of the boundary curve min ℜ 0 is that it is represented by a locus of points at which dcnf is identical to min zero ( dcnf  0 ). The boundary curve ℜ 0 subdivides the surface P onto two (or more) portions ℜ + and ℜ − . The region ℜ + of surfaces P is located over the boundary curve ℜ 0 , and thus it is the cutting-tool-accessible region. At all points within min the portion ℜ + , the minimal diameter dcnf of the indicatrix of conformity min + is positive ( dcnf > 0 ). The portion ℜ of the surface P can be machined on a multi-axis NC machine in full compliance with the blueprint. The portion ℜ − of the surfaces P is located below the boundary curve min ℜ 0 . At all points within the portion ℜ − , the minimal diameter dcnf of the min − indicatrix of conformity is negative ( dcnf < 0 ). The portion ℜ is the cuttingtool-not-accessible region of the surface P. Therefore, the portion ℜ − of the surface P cannot be machined on a multi-axis NC machine with the given cutting tool [10]. It is important to note that in the example under consideration, the R 1surface approaches infinity. This indicates that at some points of the surface P, the principal radii of curvature approach infinity ( R 2. P → ∞), and the corresponding points of the surface P are of parabolic kind. Figure 7.36 reveals that the parabolic line at ϕ = π2 is really observed on the surface P (Equation 7.71). At point(s) of parabolic kind on the surface P, the R 1-surface always approaches infinity. 150 R2.P 100

50

0

0

1

π 2

2

P

3

Figure 7.36 Parabolic lines of the surface P: the second principal radius of curvature versus curvilinear coordinate.

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349

Conditions of Proper Part Surface Generation

Use of the characteristic R 1-surface enables verification of whether the third necessary condition of proper PSG is globally satisfied or not. 7.3.2.2.2 Elements of Local Topology of the R 1-Surfaces For the implementation of the characteristic surfaces of the first kind, it is convenient to express all the elements of local topology of the R 1-surface in terms of the corresponding elements of local topology of the surface P and min of the minimal radius rcnf of the indicatrix of conformity CnfR ( P/T ) at the current CC-point. Consider a sculptured surface P that is given by vector equation rP = rP (U P , VP ). Components gR .ij of the metric tensor of the first order of the R 1-surface can be expressed in terms of the components of the fundamental tensors g ij of the first order and bij of the second order, and of the mean M P and full (Gaussian) G P curvatures of the surface P:

min )2  g − 2 r min  1 + M ⋅ r min  b gR .ij = 1 − G P ⋅ (rcnf cnf  P cnf  ij  ij

(7.73)

The mean M P and the Gaussian G P curvatures of a smooth, regular surface P can be computed from

MP =

E P N P − 2 FP M P + G P LP 2 (E P G P − FP2 )

(7.74)

LP N P − M P2 E P G P − FP2

(7.75)

GP =

The determinant of the metric tensor of the R 1-surface is equal to det( gR .ij ) = A2 ⋅ det( gij ) , where

min )2 + 2M ⋅ r min + 1 = (1 + k min min A = G P ⋅ (rcnf P cnf 1. P ⋅ rcnf ) ⋅ (1 + k 2. P ⋅ rcnf )

(7.76)

Singularities of the R 1-surface are observed at points that correspond to points of the surface P, at which one of the principal curvatures is equal min )−1 to −(rcnf . The unit normal vector n R to the R 1-surface can be analytically represented as nR =

A n | A| P

(7.77)

This derivation immediately follows from the definition of principal curvatures of a surface [17].

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350

Kinematic Geometry of Surface Machining

The second fundamental tensor of the R 1-surface can be expressed in terms of the components of the fundamental tensors g ij and bij, the mean M P and full (Gauss) G P curvature of the surface P: bR .ij =

A min ⋅ g + (1 + 2M ⋅ r min ) ⋅ b   G P ⋅ rcnf ij P cnf ij  |A|

(7.78)

As for the metric tensor, the determinant of the second fundamental tensor det(bR .ij ) = A ⋅ det(bij )

(7.79)

has singularities at points that correspond to points of the sculptured surface min )−1 P, at which one of the principal curvatures is equal to −(rcnf . Gaussian curvature GR and mean curvature M R of the R 1-surface can be computed from

GR =

MR =

GP

(7.80)

A

min M P + GP ⋅ rcnf

| A|

(7.81)

Principal curvatures k 1.R and k 2.R of the R 1-surface at a point, at which the principal curvatures k 1. P and k 2. P of the sculptured surface P are known, can be computed from k 1,2.R =

k 1,2. P A ⋅ min | | A| |1 + k 1,2. P ⋅ rcnf

(7.82)

The principal curvatures k 1. P and k 2. P can be expressed in terms of Gaussian G P and mean M P curvature of the surface P: k 1,2. P = M P ± M P2 − GP

(7.83)

or they can be computed as the roots of the quadratic equation: LP − EP ⋅ k P MP − FP ⋅ k P

MP − FP ⋅ k P =0 N P − GP ⋅ k P

Reminder: the following inequality k1.P > k2.P is always observed.

© 2008 by Taylor & Francis Group, LLC

(7.84)

351

Conditions of Proper Part Surface Generation

7.3.2.2.3 The Normalized R 1 -Surface For global satisfaction of the third necessary condition of proper PSG, the min actual magnitude of the minimal radius rcnf of the indicatrix of conformity min can be norCnfR ( P/T ) is not important. Therefore, the minimal radius rcnf malized. This leads to an equation of a characteristic surface of another kind: min rcnf min The normalized R 1 -surface ⇒ rR(1) (U P , VP ) = rP + n P ⋅ min ≡rP + n P ⋅ sgn rcnf |rcnf |

(7.85)

This characteristic surface is referred to as the normalized characteristic surface of the first kind, or simply as the R 1 -surface. Example 7.5 Consider that same surface P given by Equation (7.71). Equation (7.71) of the surface P along with the above-determined parameters R 2. P , n P , and the equation of the cylindrical surface T, yield the equation of the normalized R 1 -surface in matrix form:

  5 + 3 ⋅ cos ϕ  ⋅ cos ϕ P cos θ P + 5 ⋅ cos θ P   3 + sgn  |cos ϕ |       ϕ 5 + 3 ⋅ cos  3 + sgn  ⋅ ϕ s in θ + 5 ⋅ sin θ cos P P P rR (θ P , ϕ P ) =   |cos ϕ |      5 + 3 ⋅ cos ϕ    3 + sgn ⋅ ϕ sin P    |cos ϕ |    1  

(7.86)

The normalized R 1 -surface (see Equation 7.25) is depicted in Figure 7.37. The R 1 -surface is shown together with the surface P. Figure 7.37 reveals that the characteristic R 1 -surface intersects the surface P. The line of intersection is the boundary curve ℜ 0. The boundary curve ℜ 0 is identical to the boundary curve shown in Figure 7.35. This indicates that application of Equation (7.85) of the normalized R 1 -surface returns the results that are identical to the result obtained from Equation (7.70) of the ordinary R 1-surface. Use of the normalized characteristic R 1-surface enables verification of whether the third necessary condition of proper PSG is globally satisfied or not. 7.3.2.2.4 Equation of the Boundary Curve Boundary curve(s) ℜ 0 subdivides the surface P onto the cutting-tool-accessible regions ℜ + and onto the cutting-tool-not-accessible regions ℜ − . The equation of the boundary curve ℜ 0 is employed for marking these curves on the surface P to identify the surface regions that require additional care. Two approaches for the derivation of equation of the boundary ℜ 0-curve can be employed. First, the equation of the boundary curve ℜ 0 can be derived from consideration of the line of intersection of the characteristic R 1 -surface and of the surface P itself. Coordinates of points of the boundary

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Kinematic Geometry of Surface Machining

– − Surface 1 6 + – Portion

–

4

P

− Portion 2

0 0

2

4

6

0

2

4

6

8

0 – Boundary Curve

Figure 7.37 An example of the normalized characteristic ℜ surface of the first kind.

curve ℜ 0 satisfy equations of both the surfaces, say of the surface P and of the R 1 -surface simultaneously. Second, the equation of the boundary curve can be derived in quite a different way. At points of the boundary curve ℜ 0 , the minimal diameter of the min  0 ). The diameter d min is indicatrix of conformity is identical to zero ( dcnf cnf positive for small changes of ϕ (i.e., for value of the parameter ϕ changed min on ±d ϕ ). Then, at all points of the boundary curve, the direction of dcnf is 0 ℜ tangent to the -curve. Hence, the equation of the boundary curve ℜ 0 can be derived from the equation rcnf = rcnf (U P , VP , UT , VT , µ , ϕ ) of the indicatrix of conformity CnfR ( P/T ) . For this purpose, a set of four equations ∂ rcnf = 0 , ∂ UT

∂ rcnf = 0 , ∂ VT

∂ rcnf = 0 , and ∂µ

∂ rcnf =0 ∂ϕ

that represents the necessary conditions of the function extremum must be considered together with the well-known sufficient conditions for extremum of the function rcnf = rcnf (U P , VP , UT , VT , µ , ϕ ) of four variables (actually, for a given CC-point, the values of the parameters U P and VP are fixed). The premin requisite to the obtained solution is that it must satisfy the condition rcnf  0. Example 7.6 Consider the surface P that is given by that same Equation (7.71) as in previous examples. Equation (7.71) of the surface P along with the above-determined parameters R 2. P , n P , and the equation of the cylindrical surface T, yield

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353

Conditions of Proper Part Surface Generation 0 – Boundary Curve

+ – Portion

P

5

00

2

4

0

2

4

6

8

– – Portion

Figure 7.38 The boundary ℜ0-curve on the surface P constructed with implementation of the normalized characteristic ℜ– -surface of the first kind.

derivation of the computer code for the computation of parameters of the boundary curve ℜ 0 . Results of computer modeling are shown in Figure 7.38. The boundary curve ℜ 0 (Figure 7.38) is identical to the boundary curve shown in Figure 7.35 and to that shown in Figure 7.37. In particular cases, derivation of the equation of the ℜ 0 -boundary curve can be significantly simplified. The use of the boundary curve ℜ 0 enables verification of whether the third necessary condition of proper PSG is globally satisfied or not. 7.3.2.2.5 Algorithm for the Computation of Parameters of the Boundary Curve ℜ 0 Computation of parameters of the ℜ 0 boundary curve for machining of a given part surface on a multi-axis NC machine can be performed following the algorithm shown in Figure 7.39. The input information for the computations is available from the blueprint (1) of the sculptured part surface P. Equation rP = rP (U P , VP ) of the surface P is derived (2) based on the information available from the part blueprint (1). The derived equation of the surface P allows computation (3) of the first derivatives U P = ∂ rP ∂U P and VP = ∂ rP ∂VP of the surface P, as well as the computation (4) of the unit normal vector n P. Then, fundamental magnitudes EP = U P ⋅ U P , FP = U P ⋅ VP , and GP = VP ⋅ VP of the first order are computed (5), and the first fundamental form Φ1.P of the surface P is composed (6).

The equation for the boundary curve ℜ0 is not represented here due to space constraints.

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Kinematic Geometry of Surface Machining Start I

4

2

1

7

3

4

5

6

8

9

26

19

11

13

12

16

15

14

18

17

III

IV 20

13 2

21

22

23

24 2

II

10

25

End Figure 7.39 Flow chart of the algorithm for the computation of parameters of the boundary ℜ0-curve on the surface P.

Further, the computed first derivatives of the surface P (3) yield computation (7) of the second derivatives ∂2 rP ∂U P2 , ∂2 rP ∂U P ∂VP , and ∂2 rP ∂VP2 of the surface P. Fundamental magnitudes LP , MP, and N P of the second order are computed (8) based on the results obtained on the previous steps (3) and (7). The second fundamental form Φ2.P of the surface P is composed (9) using the derived equations for LP , MP , and N P . Steps (1) through (9) in Figure 7.39 form the sculptured surface P block I. Steps (10) through (18) (block II) of the generating surface T of the cutting tool are similar to the corresponding steps (1) through (9) of the sculptured surface P block I. The fundamental forms Φ1.P (6) and Φ2.P (9) represent the natural form of the sculptured surface P parameterization (block III). The similar block IV is represented by steps (15) and (18) on which the fundamental forms Φ1.T (15) and Φ2.T (18) are computed. The fundamental magnitudes of the first (5) and (14), and of the second (8) and (17) order of the surfaces P and T yield the composition (20) of the equation rcnf = rcnf (U P , VP , UT , VT , ϕ , µ ) of the indicatrix of conformity CnfR ( P/T ) min of the surfaces P and T. Further, the minimal value r cnf of the indicatrix of conformity CnfR ( P/T ) is computed at a current CC-point. The computed

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Conditions of Proper Part Surface Generation

min value of the unit normal vector n P (4) together with the minimal radius rcnf of the CnfR ( P/T ) yield the equation of the R 1-surface (21).

7.3.2.3 Characteristic Surfaces of the Second Kind It is important to note the perfect correlation between the characteristic R 1surface, the normalized R 1-surface, and the boundary ℜ0 -curve from one side, and between the spherical indicatrix of machinability Mch ( P/T ) of the surface P with the given cutting tool (see Section 7.1.7) [16,17,20]. The perfect correlation becomes more evident when in addition to the characteristic surfaces of the first kind, the characteristic surfaces of the second kind are considered. 7.3.2.3.1 Determination of the Characteristic Surface of the Second Kind In order to globally satisfy the third necessary condition of proper PSG, the characteristic R 1-surface of the first kind has to be located outside the part body, and it does not intersect the sculptured surface P. Only tangency of the surfaces P and T is allowed. Further, we go to a characteristic surface, which is determined by (2) ( 1) min The characteristic R 2- surface ⇒ rR (U P , VP , UT , VT , µ ) = rR − rP = n P ⋅ rcnf (7.87)

The characteristic surface given by Equation (7.87) is referred to as the characteristic surface of the second kind, or simply as the R 2-surface. Elementary analysis of Equation (7.87) reveals that location of the R 2surface yields an answer to the question of whether the third necessary condition of proper PSG is globally satisfied or not. Consideration of the R 2-surface alone is sufficient for making an appropriate conclusion. If the R 2-surface intersects the X PYP - coordinate plane, then one can make a conclusion that some portion of the surface P is not reachable for the cutting tool of the given design. The line of intersection of the R 2-surface with the X PYP is analogous to the boundary curve ℜ 0 . The following three scenarios could be observed depending on the sign of the ZP coordinate of the R 2-surface: (a) when ZP ≥ 0 , then the corresponding portion of the surface P represents the cutting-tool-accessible region ℜ + of the surface; (b) when ZP  0 , then the corresponding portion of the surface P represents the boundary ℜ0 -curve on the surface P; and finally, (c) when ZP < 0 , then the corresponding portion of the surface P represents the cutting-tool-not-accessible region ℜ − of the surface P. Example 7.7 Consider the surface P given by Equation (7.71) as in the previous examples. Substitute the required parameters from Equation (7.71), the above-determined parameters R 2. P , n P , and the equation of the cylindrical surface T into Equation (7.87). This yields an equation of the characteristic R 2-surface in

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Kinematic Geometry of Surface Machining 15

10

5 ZP 0 00 0

2−

1

2 2

Surface

3

3

Figure 7.40 An example of the characteristic ℜ2 -surface of the second kind.

matrix representation. A visualized image of the R 2-surface is depicted in Figure 7.40. The line of intersection of the R 2-surface exactly corresponds to the boundary ℜ 0 - curve on the surface P (Figure 7.38). So, application of only one R 2 -surface (and not two surfaces P and R 1 ) yields an answer to the question of whether the whole surface P can be machined with the cutting tool of a given design, or it cannot be machined under such a scenario. Use of the characteristic R 2-surface solely enables verification of whether the third necessary condition of proper PSG is globally satisfied or not. Elements of local topology of the characteristic R 2-surface can also be expressed in terms of the corresponding elements of local topology of the sculpmin (U , V , U , V , µ ) tured surface P and of the minimal radius rcnf of the indicaP P T T trix of conformity CnfR ( P/T ) at a current CC-point K as done with respect to the characteristic R 1-surfaces of the first kind (see Section 7.3.2.2.2). 7.3.2.3.2 Normalized Characteristic Surface of the Second Kind Following the method similar to that used to derive the equation of the normalized R1 -surface of the first kind, the equation of the normalized characteristic R 2-surface of the second kind can be derived. The equation of this characteristic surface is represented as follows: min rcnf ( 1) min (7.88) Normalized R 2-surface ⇒ rR( 2 ) (U P , VP ) = rR − rP = n P ⋅ min  n P ⋅ sgn rcnf |rcnf |

The equation of the characteristic R 2-surface is not represented here due to space constraints.

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The characteristic surface that is described by Equation (7.27) is referred to as the normalized characteristic surface of the second kind or simply as the R 2-surface. Analysis of Equation (7.88) reveals that the R 2-surface is equivalent to the spherical indicatrix of machinability Mch( P/T ) of the surface P with the given cutting tool proposed [16,17,20] by the author (see Section 7.1.7). The line of intersection of the R 2 -surface with X PYP -coordinate plane perfectly corresponds to the boundary curve of the indicatrix of machinability Mch ( P/T ). The characteristic surfaces of the first and of the second kinds can be employed for any feasible value of the angle µ of the local relative orientation of the surfaces P and T. Much room remains for further developments in the field of characteristic surfaces of the discovered kind. 7.3.3 Selection of the Form-Cutting Tool of Optimal Design For the verification of satisfaction of the third necessary condition of proper PSG, global use of K-mapping is useful [17,21]. Two kinds of K-mapping are of importance in this concern. First, the local KLR-mapping of the surfaces P and T that gives insight into the global KGR-mapping of the surfaces P and T. 7.3.3.1

Local KLR-Mapping of the Surfaces P and T

Analysis of the local interference of surfaces P and T can be performed with application of the local KLR-mapping of the surfaces [18]. Consider the coordinate plane k1. P k2. P (Figure 7.41). For a given smooth, regular sculptured surface P, the first R1. P and the second R2. P principal radii of curvature and the corresponding principal curvatures k1. P and k2. P can be determined from Equation (1.14). Principal curvatures are measured in the respective principal plane-sections C1. P and C2. P through the unit tangent vectors t 1.P and t 2.P of the principal directions on the surface P. Every point of the surface P has a corresponding point M ( k1. P, k2. P) in the coordinate plane k1. P k2. P (and not vice versa: each point of the coordinate plane k1. P k2. P may have one or more corresponding points on the sculptured surface P). A point in the coordinate plane k1. P k2. P represents the local KLRmap of the corresponding point of the sculptured surface P — that is, it represents the local KLR-map of the sculptured surface point. Location of the local KLR-map of the sculptured surface point within the coordinate plane k1. P k2. P depends on parameters of a surface P local topology in differential vicinity of the surface point M. All the local KLR-maps are located within the first, the third, and the fourth quadrants in the allowed region of the coordinate plane k1. P k2. P — that is, along the boundary straight line k2. P = k1. P and below this boundary line. Origin of the coordinate system k1. P k2. P represents the local KLR-map of the planar local region of a sculptured surface P. Within the planar local region of the surface, its normal curvature is equal to zero: k P  0. The mean

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Kinematic Geometry of Surface Machining k2.P

ZP

P

nP

P Convex Umbilic P

nP

XP YP Plane

Concave Elliptic

Concave Umbilic

nP

P

M Convex Parabolic

Convex Elliptic

P

M

α1 β1 α3

M

P M

nP

M

nP

P

nP

M

β2

nP α2

P

k1.P

M Quasi-Convex Hyperbolic

nP P

Concave Parabolic

M

nP

M

nP

P

P

M

Quasi-Concave Hyperbolic Hyperbolic Minimum

Figure 7.41 The local KLR-mapping of a sculptured surface P.

curvature M P and the full curvature (Gaussian curvature) GP are also equal to zero: M P = 0 , and GP = 0 . The local KLR-maps of umbilical local patches of a sculptured surface P are located within the boundary straight line k2. P = k1. P . The local KLR-maps of convex ( GP > 0 , M P > 0 ) patches are located in the first quadrant, and the similar local KLR-maps of concave ( G P > 0 , M P < 0 ) patches of the surface P are located in the fourth quadrant (see Figure 7.41). The allowed region is subdivided into three sectors α 1 , α 2 , and α 3 (see Figure 7.41). The local KLR-maps of convex elliptical local patches ( G P > 0 , M P > 0 ) of a surface P are located within the sector α 1 . Saddle-like hyperbolical local patches ( G P < 0 ) are locally KLR-mapped within the α 2 sector. Finally, the local KLR-maps of concave elliptical local patches ( G P > 0 , M P < 0 ) of a surface P are located within the α 3 sector. Within the boundary line k2. P = 0 that separates the sector α 1 from the sector α 2 — that is, within the axis of abscises — the local KLR-maps of convex parabolic local patches ( GP = 0 , M P > 0 ) of a surface P are located. The similar local KLR-maps of concave parabolic patches ( G P = 0 , M P < 0 ) are located

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within the boundary line k1. P = 0 that separates the sector P from the sector α 3 (that is, within the axis of ordinates). Mean curvature M P of a saddle-like local patch of a surface P can be positive or negative or it can be equal to zero. Due to this, the sector α 2 is subdivided into two symmetrical subsectors β1 and β2 . The local KLR-maps of quasi-convex hyperbolical local patches ( G P < 0 , M P > 0 ) of a surface P are located within the subsector β1 , and the local KLR-maps of quasi-concave hyperbolical local patches ( G P < 0 , M P < 0 ) of a surface P are located within the subsector β2 (see Figure 7.41). The boundary line k2. P = − k1. P that separates the subsectors β1 and β2 is the locus of the local KLR-maps of minimum hyperbolical local patches ( GP < 0 , M P = 0 ) of a surface P. Arrows that are coming out from each of the local KLR-maps (see Figure 7.41) indicate the directions in which parameters of a surface P local topology could be changed. While parameters of the surface topology are changing, the type of sculptured surface local patch remains the same. Evidently, the local KLR-mapping is also available for the generating surface T of the cutting tool. The difference in topology of local patches of a sculptured surface P within sectors α 1 , α 2 , α 3 , and within sectors β1 , β2 (Figure 7.41) can be demonstrated with implementation of the following analysis. 7.3.3.2

The Global KGR-Mapping of the Surfaces P and T

Application of the local KLR-mapping of the surfaces enables development of a global approach to verify whether or not the third necessary condition of proper PSG is satisfied within the entire sculptured surface P. The analysis below is based on application of the global KGR-mapping of surfaces. A global KGR-map of the entire sculptured surface can be obtained if every local region of the surface P is locally KLR-mapped onto the coordinate plane k1. P k2. P (Figure 7.42). It is evident that the following are true: The global KGR-map is located within the allowed region — that is, on the boundary straight line k2. P = k1. P , and below this straight line (see Figure 7.42). Similar to that above, the allowed region is subdivided into three sectors a1, a2, and a 3. Sector is subdivided into two subsectors b1, and b2 that are also similar to that above (see Figure 7.41). ~ ~ The global KGR-maps of concave patches ( G p > 0 , M p > 0 ) of a sculptured surface P are located within the sector a1; saddle-type local patches ~ ( G p < 0) of a sculptured surface P could be globally KGR-mapped ~within the sector a2; and finally, the global KGR-maps of concave patches ( G p > 0 , ~ M p < 0 ) of a sculptured surface P are located within the sector a3. Boundaries of the sectors a1, a 2, and a 3 represent the locus of global KGRmaps of local patches of sculptured surface P, Gaussian curvature of ~ which is equal to zero ( G = 0 ): global KGR-maps of convex patches p ~ ( M p > 0 ) are located ~along the axis of abscises, and global KGR-maps of concave patches ( M p < 0 ) are located along the axis of ordinates.

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Kinematic Geometry of Surface Machining Convex Spherical Surface k2.P

k max 2.P

P

P

Plane Surface

5

4

3

Concave Spherical Surface

Convex Cylindrical Surface P Convex Conical Surface

α1

P

β1

P

α2

α3

Concave Conical Surface

k1.P

Surface of Convex Torus

Minimal Surface

∞

P

∞

β2 P

6

Surface of Concave Torus

P

P

k min 2.P

P

2

The Global GR-map of a Sculptured Surface P Concave Cylindrical Surface 1 k min 1.P

7 k max 1.P

Figure 7.42 The global KGR-mapping of a sculptured surface P.

Equation (1.18) for the computation of the values of the principal curvatures k1.P and k2.P of a sculptured surface P can be generalized as

k1.P = k1.P(UP, VP)

(7.89)

k2.P = k2.P(UP, VP)

(7.90)

Application of a known method (see Chapter 1) allows computation of max extremal values of the curvatures — that is, maximum k1.max P , and k 2. P , min min and minimum k1. P , and k2. P , values of the principal curvatures k1.P and k2.P within intervals of variation of the parameters UP and VP (i.e., within the intervals U1.P < UP < U2.P and V1.P < VP < V2.P ). In the coordinate plane k1.P k2.P, two pairs of the straight lines k1.P = k1.max P , max , k min can be plotted (see Figure 7.42). k1.P = k1.min , and k = = k k 2.P 2.P P 2. P 2. P The first pair of straight lines is parallel to the axis of abscises, and the second pair of straight lines is parallel to the axis of ordinates of

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the coordinate system k1.P k2.P. These two pairs of straight lines form the allowed rectangle. In particular cases, the allowed rectangle can be opened from one, from two, from three, and (in a completely degenerated case) even from four of its sides. The global KGR-map of all sculptured surfaces P is located within the allowed rectangle. A boundary curve of the global KGR-map of a sculptured surface P shares at least one point with each side of the allowed rectangle above (or they are approaching infinity in the case of a corresponding side of the allowed rectangle being opened). Boundaries of the global KGR-map of a sculptured surface P can coincide in full or in part with the corresponding side of the allowed rectangle. Generally, there are just a few types of local patches of a sculptured surface P. They are convex, concave, and saddle-like local patches (see Chapter 1). For this reason, the global KGR-map of a sculptured surface P can be located within one or within two or even within three sectors a1, a 2, and a 3 simultaneously. A motion of a point over a sculptured surface can be represented with a respective motion within its global KGR-map. It is evident that transition from one of the sectors a1, a 2 and a 3 to another can be performed only with the crossing of a corresponding axis of the coordinate system k1.P k2.P, or with passing through the origin of this coordinate system. The global KGR-map of a sculptured surface can have multilayer portions (say, two or more layer portions) that are observed in case a sculptured surface P consists of two or more patches with the same values of principal curvatures. Points of the boundary curve of the global KGR-map of a sculptured surface can be or cannot be corresponding to points of boundary of the sculptured surface. This is because the extremal values of a sculptured surface P principal curvatures could not be observed on the boundary of a surface P. For this reason, there is no one-to-one correspondence between points of the global KGR-map of a sculptured surface and points of the surface P. In general, the global KGR-map of a sculptured surface is represented with a portion of the coordinate plane k1. P k2. P that is bounded by a closed or semiclosed line. An example of the global KGR-map of a sculptured surface P is shown in Figure 7.42. The global KGR-map depicted in Figure 7.42 satisfies all of the above-listed necessary requirements: (a) this global KGR-map is located in the allowed region; (b) it is located within the allowed rectangle; (c) it is located within the sectors a1, a 2, and a 3; (d) it is sharing common

Remember that points located on the axis of the coordinates of the coordinate system k1.P k2.P, represent the global KGR-maps of parabolic local patches of a sculptured surface P. The origin of the coordinate system k1.P k2.P reflects the global KGR-map of planar local patch of a sculptured surface to be machined. This statement is in agreement with the well-known statement proven in differential geometry of surfaces [2,3]: if a sculptured surface P consists of convex and concave local patches, there is a corresponding number of parabolic curves on it.

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Kinematic Geometry of Surface Machining

points with every side of the allowed rectangle (at points 1, 2, 3, and 4); and (e) it shares straight-line segments 4–5, 5–6, and 1–7 with the sides of the allowed rectangle. In particular cases, the global KGR-map of sculptured surface degenerates to a straight line or even to a point (see Figure 7.42). To derive equations of the boundary curves of the global KGR-map of a surface P, the following approach can be employed. Equations (7.89) and (7.90) can be rewritten in the form

% (U , V ) + M % 2 (U , V ) − G% (U , V ) k1. P (U P , VP ) = M P P P P P P P P P

% (U , V ) − M % 2 (U , V ) − G% (U , V ) k2. P (U P , VP ) = M P P P P P P P P P

(7.91) (7.92)

The first principal curvature k1.P is not constant within the surface P patch. As follows from Equation (7.91), it is a function of Gaussian coordinates — that is, k1.P = k1.P (UP,VP). The extremal values (the maximal and the minimal values) of the first principal curvature of the sculptured surface are equal to the roots of the set of two equations:

k1. P = k1. P (U P , VP )

∂ k (U , V ) = 0 ∂U P 1. P P P

(7.93)

(7.94)

Equation (7.94) can be solved with respect to the variable UP . The solution of the equation can be represented in the form UP = UP(k1.P,VP). Substitution of UP = UP(k1.P,VP) into Equation (7.93) after performing necessary formula transformation can result in the equation

k1min/max = k1min/max (VP ) .P .P

(7.95)

that is derived from Equation (7.93). Similar manipulations can be performed with Equation (7.92). Consequently, one can obtain

k2min/max = k2min/max (VP ) .P .P

(7.96)

Equation (7.93) can be solved with respect to the variable VP . The obtained solution has to be substituted into Equation (7.94). After performing necessary transformations, the following equations can be obtained:

k1min/max = k1min/max ( k2. P ) .P .P

(7.97)

k2min/max = k2min/max ( k1. P ) .P .P

(7.98)

or

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Conditions of Proper Part Surface Generation DT k2.T k2.T

2 DT

1 DT 2

k1.T

2 DT

1 r

k2.T

θ

2 DT

r 2 DT

DT

DT

DT k2.T

k1.T

2 DT

k1.T

dT 2 . cos θ dT

k1.T

2 . cos θ DT Figure 7.43 Examples of the global KGR-map of the generating surface T of the milling cutters.

Equation (7.97) and Equation (7.98) describe the boundary curves of the global KGR-map of the surface P. It is evident that the same result can be obtained in the case of differentiation of Equation (7.93) not with respect to UP, but at first, with respect to VP. Application of the same approach yields plotting the global KGR-map of the generating surface T of the cutting tool (Figure 7.43). As evident, the represented global KGR-maps of the surface T are in perfect correspondence to the global KGR-maps of similar surfaces (Figure 7.42). 7.3.3.3 Implementation of the Global KGR-Mapping KGR-mapping of surfaces is developed for the purpose of verifying whether the third necessary condition of proper PSG is globally satisfied or not (Figure 7.43). For this purpose, the allowed rectangle has to be plotted onto the coordinate plane k1.P k2.P. Size and location of the allowed rectangle are completely determax min min mined by the maximal k1.max P , k 2. P and by the minimal k1. P , k 2. P , values of the

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Kinematic Geometry of Surface Machining

principal curvatures of the surface P. The global KGR-map of the surface P is located inside the allowed rectangle and is sharing at least one point with each of its sides. The obtained KGR-map of the surface P helps to visualize major restrictions that are imposed on the parameters of the generating surface of the cutting tool by the surface P topology. The limited case for the satisfaction of the third necessary condition of proper PSG corresponds to the case of the maximal rate of conformity of the generating surface of the tool to the sculptured surface — when normal radii of curvature of these surfaces in a current section by normal plane are of the same value and of the opposed sign. 7.3.3.4

Selection of an Optimal Cutting Tool for Sculptured Surface Machining

Further, the maximal principal curvatures can be employed to directly bound the largest tool radius that can be used for the machining of concave and saddlelike regions, and hence to aid in the tool selection. Productivity of sculptured surface machining depends on the type of machine tool employed. To investigate the machine tool settings, the machine tool should first be studied. The larger the machining tool, the smaller the resulting scallop height, or the excessive material not removed, that might result for the same tool-path. Moreover, larger cutting tools reduce machining times. A smooth finish usually results by using flat-end tools. Machining using ball-end tools is slow because of a vanishing cutting speed at a tip of the tool — an impediment that shows up neither in the five-axis flat-end milling mode nor in five-axis side-milling approach proposed herein. However, a systematic approach for the selection of the cutting tool is yet to be attempted. Limited research has been conducted toward choosing an optimal cutting tool for four- and five-axes freeform surface machining. Application of the global KGR-mapping of a sculptured surface P to be machined and of the generating Ti surface of cutting tools to be applied (here i = 1 . . . N– is an integer number, and N– is a number of cutting tools available for machining a given sculptured surface P) allows one to select the proper cutting tool for machining a given sculptured surface (Figure 7.44). For this purpose, taken with opposite sign, the global KGR-map of all cutting tools available has to be plotted onto the same coordinate system on which the global KGR-map of the given sculptured surface P is already plotted. The global KGR-map of the best cutting tool (among available cutting tools) contains a point that is closest to the point with coordinates max ( k1.max P , k 2. P ) — that is, the one closest to the corner point of the allowed rectangle of the global KGR-map of the given sculptured surface P. A cutting tool of such design would have the highest rate of conformity to the sculptured surface P in every CC-point. Evidently, this would allow for the smallest cusp height and highest productivity of machining operation. Implementation of the obtained results enables one to increase efficiency of machining of a sculptured surface in die/mold production, in high-speed machining in conventional and rapid prototyping, and so forth.

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Conditions of Proper Part Surface Generation The Global GR-map of a Sculptured Surface P

ωT

max

max

k2.P = k2 P (k1 P)

k2 P 0.0189 0.0177

k 1min P = 0.025

max

k2 P

α1 β2 –0.0100 rT

α3 OT

min k*2.T >− k2.T

–0.0333

β1 α2

J

GR

J*

0.025

GR

max

k1P

0.050

Two-Layered Portion of the Global KGR-map of a Sculptured Surface P

k*2.T =−k2.T =–

k1.P k min 2P

DGR CGR

1 min min rT k 2.P = k 2 P (k1 P) min k1P max

The Inverse ( GR)–1–Map of the Tool Surface

= 0.050

k1P

QGR

Q*GR

BGR AGR

Allowed Rectangle

Figure 7.44 Example of implementation of the KGR-mapping of a sculptured surface P (Figure 7.43) for determining the allowed parameters of the generating surface T of a cutting tool.

It can be observed that application of the global KGR-mapping of a sculptured surface P and of the generating surface Ti of cutting tools allows one to select the design of the cutting tool most fitted for machining a given sculptured surface. The capabilities of the global KGR-mapping for selecting a proper cutting tool for machining a given sculptured surface were demonstrated.

References [1] B anchoff, T., Gaffney, T., and McCrory, C., Cusps of Gauss Mapping, Pitman Advanced Publishing Program, Boston, 1982. [2] doCarmo, M.P., Differential Geometry of Curves and Surfaces, Prentice Hall, Englewood Cliffs, NJ, 1976. [3] Favard, J., Course de Geometrie Differentialle Locale, Gauthier-Villars, Paris, 1957. [4] Gauss, K.-F., Disquisitions Generales Circa Superficies Curvas, Goettingen (1828). (English translation: General Investigation of Curved Surfaces, trans. by J.C. Moreheat and A.M. Hiltebeitel, Princeton, 1902, reprinted with introduction by Courant, Raven Press, Hewlett, New York, 1965.) [5] Kells, L.M., Kern, W.F., and Bland, J.R., Plane and Spherical Trigonometry, 3rd ed., McGraw-Hill, New York, 1951. [6] Mortenson, M.E., Geometric Modeling, John Wiley & Sons, New York, 1985.

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[7] Pat. No. 1185749, USSR, A Method of Sculptured Surface Machining on MultiAxis NC Machine/S.P. Radzevich, Int. Cl. B23C 3/16, Filed October 24, 1983. [8] Pat. No. 1249787, USSR, A Method of Sculptured Surface Machining on MultiAxis NC Machine./S.P. Radzevich, Int. Cl. B23C 3/16, Filed December 27, 1984. [9] Pat. No. 1442371, USSR, A Method of Optimal Work-Piece Orientation on the Worktable of Multi-Axis NC Machine/S.P. Radzevich, Int. Cl. B23q15/007, Filed February 17, 1987. [10] Radzevich, S.P., A Cutting-Tool-Dependent Approach for Partitioning of Sculptured Surface, Computer-Aided Design, 37 (7), 767–778, 2005. [11] Radzevich, S.P., A Generalized Analytical Form of the Conditions of Proper Part Surface Generation. Part 1. In Improvement of Efficiency of Metal Cutting, Volgograd, VolgPI, 1987, pp. 70–79. [12] Radzevich, S.P., A Generalized Analytical Form of the Conditions of Proper Part Surface Generation. Part 2, In Improvement of Efficiency of Metal Cutting, Volgograd, VolgPI, 1988, pp. 56–73. [13] Radzevich, S.P., A Possibility of Application of Plücker’s Conoid for Mathematical Modeling of Contact of Two Smooth Regular Surfaces in the First Order of Tangency, Mathematical and Computer Modeling, 42, 999–1022, 2005. [14] Radzevich, S.P., Basic Conditions of Proper Part Surface Generating While Machining on Conventional Machine Tool, Facta Universitatis, Series: Mechanical Engineering, 1 (6), 637–651, 1998. [15] Radzevich, S.P., Conditions of Proper Sculptured Surface Machining, ComputerAided Design, 34 (10), 727–740, 2002. [16] Radzevich, S.P., Differential-Geometric Method of Surface Generation, DrSci thesis, Tula polytechnic institute, 1991. [17] Radzevich, S.P., Fundamentals of Surface Generation, Rastan, Kiev, 2001. [18] Radzevich, S.P., K-mapping of Sculptured Part Surfaces and of the Machining Surface of a Cutting Tool, Proceedings of National Technical University of the Ukraine “Kiev Polytechnic Institute,” Series: Machine-Building, 33, 232–240, 1998. [19] Radzevich, S.P., On a Possibility of Application of the R-Surfaces for Partitioning of a Sculptured Surface, Mathematical and Computer Modeling, 46 (7–8), October 2007. [20] Radzevich, S.P., Sculptured Surface Machining on Multi-Axis NC Machine, Vishcha Shkola, Kiev, 1991. [21] Radzevich, S.P., Selection of the Cutting Tool of Optimum Design for Sculptured Surface Machining on Multi-Axis CNC Machine. In Automation & Assembly Summit 2005, April 18–20, 2005, St. Louis, MO, SME Technical Paper TP05PUB71. [22] Radzevich, S.P., and Goodman, E.D., Computation of Optimal Workpiece Orientation for Multi-Axis NC Machining of Sculptured Part Surfaces, Journal of Mechanical Design, 124 (2), 201–212, 2002.

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8 Accuracy of Surface Generation Accuracy of the machined part surfaces is a critical issue for many reasons. Deviations of the actual part surface from the desired part surface are investigated in this chapter from the prospective of capabilities of the theory of surface generation. Two major reasons often cause surface deviation: When machining a part surface, the entire generating surface of the cutting tool does not actually exist. In all cases of implementation of wedge cutting tools, the generating surface of the cutting tool is not represented entirely but by a limited number of cutting edges. In other words, the generating surface of the cutting tool is represented discretely. The discrete representation of the surface T of the cutting tool causes deviations of the actual machined part surface Pac from the desired (say, from the nominal) part surface Pnom . Point contact of the part surface and of the generating surface of the cutting tool is usually observed when machining a sculptured surface on a multi-axis numerical control (NC) machine. When the surfaces make point contact, then articulation capabilities of the multi-axis NC machine can be utilized in full. From this prospective, point contact of the surfaces can be considered as the most general kind of surface contact. However, point contact of the surfaces P and T also causes deviations of the actual machined part surface Pac from the desired part surface Pnom . Ultimately, when the generating surface T of a cutting tool is represented discretely, and the surfaces P and T make point contact, then the deviations of the actual machined part surface Pac from the desired part surface Pnom are getting bigger. Sources for the deviations of the machined part surface from the desired part surface are limited to two major reasons only in a simplified case of surface machining. In the simplified cases of surface machining, no deviations in the surfaces P and T configuration are observed. Deviations in the configuration of surfaces P and T are unavoidable. Therefore, the impact of deviations of the configuration of surfaces P and T onto the resultant deviation of the surface Pac from the surface Pnom must be investigated as well.

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8.1 Two Principal Kinds of Deviations of the Machined Surface from the Nominal Part Surface The discrete representation of the generating surface of the cutting tool as well as point contact of the surfaces P and T result in that during a certain limited period of time, it is impossible to generate the part surface precisely, without deviations of the actual machined part surface from the desired part surface. 8.1.1 Principal Deviations of the First Kind For proper generation of a part surface, the entire generating surface T must be represented by the cutting tool. Actually, the surface T of a cutting tool is represented as a certain number of cutting edges. The number of cutting edges of the cutting tools of conventional design is limited, and the total number could be easily counted. The generated surface T of a cutting tool of this type is discontinuous. The number of cutting edges of grinding wheels and of other abrasive tools is also limited. However, it is not that easy to count all the cutting edges of a grinding wheel as can be done with respect to wedge cutting tools. Therefore, in most cases of surface machining, the generating surface of abrasive cutting tools can be considered as a continuous surface T. When machining a part surface, for example, with a milling cutter (Figure 8.1), the cutting tool is rotating about its axis OT with a certain angular velocity ω T. In addition, the milling cutter is traveling across the surface P with a certain feed-rate F fr. The generating surface T of the milling cutter is contacting the nominal part surface P at a point K. The actual machined part surface Pac is formed as consecutive positions of trajectories of the cutting edge. Usually, the trajectories can be represented by prolate cycloids. In particular cases, the trajectories are represented by pure cycloids and even by curtate cycloids. In any case, the actual surface Pac becomes wavy. The length of the waves is equal to the feed rate per tooth Ffr of the milling cutter, and the wave height (cusp) is specified by h fr. The elementary surface deviation h fr (the surface waviness) is measured along the unit normal vector n P to the nominal part surface Pnom and is equal to the distance between the surfaces Pac and Pnom . If the part surface to be machined and the generating surface of the cutting tool are in line contact, then the cusp height h fr is the only source of the resultant deviation h Σ of the surface Pac from the surface Pnom . Figure 8.1 reveals ( that the cusp height h fr strongly depends upon the feed rate per tooth Ffr of the milling cutter. For milling cutters of most conventional designs, those that work under high rotation ω T , the cusp height h fr is negligibly small. However, this does not mean that the elementary deviation h fr may always be eliminated from the analysis of the surface P accuracy. The elementary deviation h fr donates more or less to the resultant deviation h Σ of the actual part surface Pac from the nominal part surface Pnom .

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ωT

OT Ffr

Pac hfr

K

Ffr Figure 8.1 Deviation of the machined part surface from the desired part surface that is caused by the limited number of cutting edges of the cutting tool.

8.1.2 Principal Deviations of the Second Kind Point contact of the desired part surface P and of the generating surface T of the cutting tool is observed when machining a sculptured surface on a multi-axis NC machine. Generation of the part surface is performed by a series of consequent tool-paths. After the generation of a certain tool-path is accomplished, then the cutting tool is moved in side-step in order to generate the next tool-path. Consider milling of a sculptured part surface on a multi-axis NC machine (Figure 8.2). The cutting tool is rotating about its axis OT with a certain angular velocity ω T . In addition, the milling cutter is shifting across the tool-path at a certain side-step Fss. The generating surface T of the milling cutter is contacting the nominal part surface P at a point K. The actual machined part surface Pac is formed as consecutive positions of axial profiles of the surface ( T of the milling cutter. The length of the facets is equal to the side-step Fss of the milling cutter, and the cusp height is specified by hss. The surface deviation hss is measured along the unit normal vector n P to the nominal part surface Pnom and is equal to the distance between the surfaces Pac and Pnom . The cusp height hss is another source of the resultant deviation h Σ of the machined surface Pac from the desired surface Pnom . Figure 8.2 reveals that ( the cusp height hss strongly depends upon the side-step Fss of the milling cutter. The elementary deviation hss donates to the resultant deviation h Σ of the actual part surface Pac from the nominal part surface Pnom .

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ωT FSS

OT

hSS

K

FSS

Figure 8.2 Deviation of the machined part surface Pac from the desired part surface Pnom that is caused by point kind of contact of the surfaces P and T.

8.1.3 The Resultant Deviation of the Machined Part Surface The resultant deviation h Σ of the actual part surface Pac from the nominal surface Pnom is measured along the unit normal vector n P to the nominal part surface Pnom and is equal to the distance between the surfaces Pac and Pnom . The value of the resultant deviation h Σ depends upon the elementary deviations h fr and hss . Consider a portion of the actual part surface Pac depicted in Figure 8.3. This portion of the surface is bounded by two neighboring arc segments m and (m + 1), and by two arc segments n and (n + 1). The distance between the arc ( segments m and (m + 1) is equal to the feed rate per tooth Ffr of the cutting tool, while(the distance between the arc segments n and (n + 1) is equal to the sidestep Fss. The surface P portion that is bounded by the arc segments m, (m + 1) and n, (n + 1) is referred to as (the elementary ( surface cell of the part surface P. The major parameters h fr, Ffr, hss, and Fss of the elementary surface cell are not constant within the part surface P. They vary in certain intervals within the sculptured surface. Current values of the major parameters of the elementary surface cell depend on (a) the principal radii or curvature P1. P, P2. P of the surface P; (b) the principal radii or curvature P1.T , P2.T of the surface T; (c) the angle µ of the local relative orientation of surfaces P and T; and (d) on

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(tr)

A local vicinity of the point of contact K of the approximating torus surfaces TrP and TrT

(tr) R2T

R1.T

R2T

TrT

R1.T TrP

nP

Tr[h] K

Rs (P

XT

T) ZT

YT Rs (T

ZTr.T n3 P

F˘fr

Pnom

rTr.T

n4 P rTr P

Rs (P

[h]

n2 P

hss TrP)

Pac

Fss

1P

4P h∑

XP Rs (T

R2 P

n1.P

K

m

R2(tr)P

R1 P

2P nP

3P

XTr.P

ZTr P

YTr.T

TrT)

n

(tr)

R1.P

XTr.T

hfr

(m +1) (n +1) Fˇss Ffr

P)

A local vicinity of the point of contact K of the sculptured surface P and of the generating surface T of the form cutting tool

YTr.P ZP

YP

Figure 8.3 Generation of an elementary surface cell on the machined part surface P.

instant parameters of kinematics of the surface machining. Therefore, the resultant deviation h Σ,i at a current i point of the surface P is equal:

h Σ ,i = h Σ ,i ( h fr ,i , hss ,i )

(8.1)

where the elementary deviations h fr and hss have to be considered as functions of coordinates of the point on the part surface P, of the corresponding point of the generating surface of the cutting tool, and of the angle µ of the local relative orientation of surfaces P and T. This relationship is expressed by two functions:

h fr = h fr (U P , VP , UT , VT , µ )

(8.2)

hss = hss (U P , VP , UT , VT , µ ) (8.3) ( ( The feed-rate Ffr and the side-step Fss are not incorporated into Equations (8.2) and (8.3).

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The maximal resultant deviation h max of the surface Pac from the surface Σ Pnom is often used for the quantitative evaluation of accuracy of the machined part surface. It is widely recognized that in sculptured surface machining on a multiaxis NC machine, the principle of superposition of the elementary deviations h fr and hss is valid. Implementation of the principle of superposition to the elementary deviations h fr and hss is questionable. This issue requires further investigation. If the principle of superposition of the elementary deviations h fr and hss is assumed to be valid, then for the computation of the resultant deviation h Σ , the following equation can be used: h Σ = ah ⋅ h fr + bh ⋅ hss

(8.4)

where ah and bh designate certain constants for a given point K. The constants ah and bh are within the intervals 0 ≤ ah ≤ 1 and 0 ≤ bh ≤ 1. The resultant deviation of surface generation h Σ is getting its maximal value of h max when the equality ah = bh = 1 is observed. In this particular Σ case, the deviation h max can be computed from Σ h max = h max + hssmax Σ fr

(8.5)

Generally, the function h Σ = h Σ ( h fr , hss ) is complex. In compliance with the sixth necessary condition of proper part surface generation (PSG) [3], the deviation h Σ must be within the tolerance of the surface accuracy of surface machining (see Section 7.2.6). The maximal value of the resultant deviation h Σ is limited by the tolerance [ h] on accuracy of the surface machining. It is recommended that an operation of a sculptured surface machining be designed in a way so that the maximal deviation h max Σ of the surface Pac from the surface Pnom is equal to the tolerance [ h]. A significant reduction in machining time can be achieved if the equality h max = [ h] Σ is satisfied within the entire part surface P being machined. Both the elementary deviations h fr and hss and the resultant deviation h Σ can be computed on the premises of methods developed in the theory of surface generation.

8.2

Local Approximation of the Contacting Surfaces P and T

The major surface deviations h fr , hss , and h Σ can be interpreted in terms of geometry of the nominal part surface P and of the surface Pac of the elementary surface cell. In order to solve the problem, an analytical local representation of the surfaces is helpful.

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373

The nominal part surface is given. Locally, the surface P is specified by the principal radii of curvature R 1. P and R 2. P at point K, and by the surface torsion τ P . The actual part surface Pac within the elementary surface cell is congruent to the surface of the cut. When machining a part, the cutting edge of the cutting tool moves relative to the work. Consecutive positions of the moving cutting edge form the surface of cut Sc . A portion of the surface of the cut that is located within the elementary surface cell is congruent to the actual part surface ( Sc ≡ Pac ). The surface of cut Sc can also be locally specified by the principal radii of curvature R 1.c and R 2.c at point K, and by the surface torsion τ c . For the computation of the parameters R 1.c , R 2.c , and τ c , the equation of the surface of cut Sc is necessary. The equation can be derived on the premises of the geometry of the cutting edge of the cutting tool, the kinematics of the relative motion of the cutting edge with respect to the work, and the operators of coordinate systems transformations (see Chapter 3 for details). Fortunately, for most kinds of surface machining, the surface of cut Sc is very close to the generating surface T of the cutting tool as long as the elementary surface cell is considered. Therefore, beside the major parameters R 1.c , R 2.c , and τ c of the surface of cut Sc cannot be computed, the similar major parameters R 1.T , R 2.T , and τ T of the generating surface T of the cutting tool can be computed instead. 8.2.1 Local Approximation of the Surfaces P and T by Portions of Torus Surfaces Actual surfaces P and T can be given in a complex analytical form that is not convenient for computations of the major parameters of the surfaces. Solutions to many geometrical problems can be more easily derived from local consideration of the surfaces rather than from consideration of the entire surfaces. For the local analysis, the surfaces are often represented by quadrics. As shown in our previous works [4,5,8], from the perspective of local approximation of surface patches, helical canal surfaces feature important advantages over other candidates. A helical canal surface (Figure 8.4) is a particular case of a swept surface. Monge was the first to investigate the class of surfaces formed by sweeping a sphere, in 1850 [2]. He named them canal surfaces. In the particular case when the path on which the sphere is swept along is a helix, and the sphere has constant radius, the surface swept out is referred to as a helical canal surface. A surface of this kind is of particular interest for engineers. A canal surface is the envelope of a one-parametric family of spheres. The envelope is defined as the union of all circles of intersection of infinitesimally neighboring pairs of spheres. These circles are referred to as the composing circles. Helical canal surfaces can fit the principal curvatures and torsion of the local patch of sculptured surfaces, as well as of the generating surfaces of cutting tools.

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Kinematic Geometry of Surface Machining

Phc

t1.hc Phc R2.hc nhc

t2.hc

R1.hc

Figure 8.4 Local portion of a helical canal surface.

Further simplification is due to the torsion of the surfaces P and T and is usually of negligibly small value. Therefore, for the purposes of local surface approximation, implementation of torus surfaces is of great practical importance. Torus surface can be considered as the above-discussed canal surface, the screw parameter of which is put equal to zero. Implementation of torus surfaces allows perfect approximation of a bigger surface area, not just in differential vicinity of a surface point like quadrics do. Local approximation of the part surface P by the torus surface TrP is illustrated in Figure 8.5. The surface P is given in a Cartesian coordinate system X P YP ZP . The position vector of an arbitrary point p 1 of the surface P is designated as rp1 . In Figure 8.5, the surfaces P and TrP share common unit tangent vectors t 1. hc ≡ t 1. P and t 2. hc ≡ t 2. P of the principal directions of the surfaces, as well as they share common unit normal vector n P. The vectors t 1.P, t 2.P, and n P together with the computed values of the principal radii of curvature R 1. P and R 2. P yield computation of the position vector rTP1. The last vector together with the position vector rp1 of the point p 1 allow for compu* tation of the position vector rTP1 in the coordinate system X P YP ZP associated with the surface P. Ultimately, a Cartesian coordinate system Xtr Ytr Ztr is associated with the torus surface TrP . For further consideration, it is important to stress that not any point of the approximating torus surface can be used for the local approximation of the surfaces P and T. For this purpose, only points that are within the circle either of the biggest meridian of the approximating torus or of the smallest meridian are used.

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Accuracy of Surface Generation TrP

Ztr Ot r*TP1

Xtr Ytr

rTP1 t2 P

ZP

XP

rP1 nP

YP Figure 8.5 Construction of the torus surface

P1

TP1

t1.P

P

at the point P1 of the part surface P.

A torus surface can be expressed in terms of radius rtr of its generating circle, and in terms of radius R tr of its directing circle. Depending on the actual ratio between the radii rtr and R tr, the torus radius rtr can be equal to the first principal radius of curvature R 1. P of the part surface ( rtr = R 1. P ), while the torus radius R tr in this case is equal to the difference R tr = R 2. P − R 1. P . For another ratio between the radii rtr and R tr, the equalities rtr = R 2. P − R 1. P and R tr = R 1. P are valid. At a current surface point, principal radii of curvature can be computed as discussed in Chapter 1. In the coordinate system Xtr Ytr Ztr associated with the torus surface (Figure 8.6), the position vector r tr (θ tr , ϕ tr ) of a point of the approximating torus surface can be represented in the following way: r tr (θ tr , ϕ tr ) = R(θ tr ) + r(θ tr , ϕ tr ). Here, r(θ tr , ϕ tr ) designates the position vector of a point on the generating circle of radius rtr in its current location (Figure 8.6), and R(θ tr ) designates the position vector of the center of the generating circle, which rotates about the Ztr axis. A routine transformation yields the following expression for r tr (θ tr , ϕ tr ):

 −( R 2. P − R 1. P ) cos θ tr + R 1. P cos ϕ tr cos θ tr   −( R − R ) sin θ + R cos ϕ sin θ  tr tr tr  2. P 1. P 1. P r tr (θ tr , ϕ tr ) =    R 1.. P sin ϕ tr   1  

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(8.6)

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Kinematic Geometry of Surface Machining Ztr

Rtr R(θtr,

Ytr

θtr

rtr tr

utr

tr)

rtr R(θtr) Vtr Rtr

Xtr

ntr O

M

tr

r(θtr,

tr)

Figure 8.6 Generation of a torus surface as a locus of consecutive position of the circle of radius rtr . (From Radzevich, S.P., Computer-Aided Design, 37 (7), 767–778, 2005. With permission.)

The unit normal vector n tr to the torus surface r tr can be calculated by the formula n tr = u tr × v tr, where u tr = U tr |U tr |, v tr = Vtr |Vtr |, and the tangent vectors U tr and Vtr are given by the equations U tr = ∂ r tr ∂U tr and Vtr = ∂ r tr ∂Vtr . Only points within the plane section of the torus surface by the coordinate plane Xtr Ytr are used for the local approximation of the surfaces P and T. To specify the configuration of the torus surface r tr , the unit tangent vectors t 1.tr and t 2.tr can be employed. At the point K, the unit tangent vectors t 1.tr and t 2.tr are identical to the unit tangent vectors t 1.P and t 2.P of the surface P. It is important to stress here that the patches of torus surfaces that locally approximate the surfaces P and T from one hand and the torus portion of the generating surface T of a cutting tool from another hand are completely different entities. The last is clearly illustrated in Figure 8.7 where a portion of a filleted-end milling cutter is shown. The torus portion of the generating surface of the cutting tool is specified by the radius rtr of the generating circle of the torus, and by the radius R tr of the directing circle of the torus surface. Three arbitrary points A, B, and C are chosen within the generating circle of the torus portion of the surface T. Approximating torus surfaces can be constructed at each of the points A, B, and C. The approximating torus surface TrA through the point A can be specified by the radius rtr. A → ∞ of the generating circle of the torus, and by the radius R tr. A → ∞ of the directing circle of the torus surface (the radius R tr. A → ∞ is not indicated in Figure 8.7). The approximating torus surface TrB through the point B is congruent to the toroidal portion of the generating surface of the milling cutter; thus, it can be specified by the radius rtr . B ≡ rtr of the generating circle of the torus, and by the radius R tr . B ≡ R tr of the directing circle of the torus surface.

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Accuracy of Surface Generation (C)

R2.T

Ztr (C)

R1.T

Otr

OT

C

T Rtr (B) tT

(C)

tT B rtr

OT Ytr (A)

tT

A

C

θtr

Xtr

(Rtr + rtr .cos θtr)

Figure 8.7 Analysis of the local geometry of the generating surface T of a filleted-end milling cutter.

Ultimately, the approximating torus TrC through point C is specified by the radius rtr . C ≡ rtr of the generating circle of the torus, and by the radius [Rtr .C = (R tr + rtr ⋅ cos θ tr )R tr . C ] of the directing circle of the torus surface. (Here, the angle θ tr specifies the location of point C on the arc of the generating circle of radius rtr .) Note that all ten kinds of local patches of smooth, regular surfaces (see Chapter 1, Figure 1.11) can be found on the torus surface Tr . Figure 8.8 illustrates this important property of the torus surface. Consider points on the surface Tr that occupy various positions M1, M2 , M3 , Mi , and so forth. The part body can be located either inside the torus surface Tr, or outside the surface Tr. Depending upon the chosen location of the point Mi either within the convex surface Tr or within the concave surface Tr, all ten kinds of local patches of smooth, regular surface can be found on the torus surface Tr. The major advantage of implementation of the torus surface for local approximation of the sculptured surface is due to a patch of the torus surface being capable of providing perfect approximation for bigger surface area compared to the approximation by quadrics, use of which is valid just within a differential vicinity of the surface point.

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Kinematic Geometry of Surface Machining ZTr Tr

dTr

M2

M4 M3 rTr

XTr

M1 Tr

RTr

YTr

θTr

DTr Tr M1

nTr

M2

Tr

nTr Tr

nTr

nTr

M3

Tr M3

Figure 8.8 Major elements of the local topology of a torus surface.

8.2.2 Local Configuration of the Approximating Torus Surfaces When a sculptured surface P and the generating surface T of a cutting tool contact each other at a certain point K, the approximating torus surfaces are also contacting each other at that same point K. Moreover, when parameterization of the approximating torus surfaces is chosen in the form of Equation (8.6), then the unit tangent vectors t 1.P and t 2.P of the surface P at K and the unit tangent vectors t 1.T and t 2.T of the surface T at K are identical to the corresponding unit tangent vectors of the approximating torus surfaces TrP and TrT . The last is convenient for the development of the analytical description of local configuration of the approximating torus surfaces. It is assumed that the sculptured surface P and the generating surface T of the cutting tool are in proper tangency at a certain point K (Figure 8.9). The unit tangent vectors u P and v P , as well as the unit normal vector n P for the surface P at point K are computed from the equation rP = rP (U P , VP ) of the part surface P (see Chapter 1.1). The equation of the surface P also yields computation of the unit tangent vectors t 1.P and t 2.P of the principal directions on P. The vectors t 1.P , t 2.P, and n P make up the Darboux trihedron.

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Accuracy of Surface Generation TrT

(K )

xP

ZTrT

YTr.T T

uP

nP K

P vP

XTr.T

(K )

yP

(K)

zP

XTr.P ZTr.P

YTr.P

TrP Figure 8.9 Example of relative disposition of the approximating torus surfaces Tr P and TrT.

The Darboux trihedron is implemented here for the purpose of construction of the local left-hand-oriented Cartesian coordinate system xP y P zP having origin at the point K. Configuration of the sculptured surface P as well as configuration of the generating surface T in the coordinate system X NC YNC ZNC associated with the machine tool is known. Therefore, the corresponding operators of the coordinate systems transformation, the operator Rs ( NC a P) of the resultant transformation from the coordinate system X NC YNC ZNC to the coordinate system X P YP ZP and, further, the operator Rs ( P a K P ) of the resultant transformation from the coordinate system X P YP ZP to the local coordinate system xP y P zP can be composed. Ultimately, the operator Rs ( NC a K P ) of the resultant coordinate systems transformation can be composed. The similar operators Rs ( NC a T ), Rs (T a KT ) , and Rs ( NC a KT ) of the consequent coordinate systems transformations are composed for the generating surface T of the cutting tool. Ultimately, the operators of the direct Rs ( KT a K P ) and of the inverse Rs ( K P a KT ) coordinate systems transformations can be composed as well. The operators Rs ( KT a K P ) and Rs ( K P a KT ) complement the earlier composed operators of the coordinate

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Kinematic Geometry of Surface Machining

systems transformation to a closed loop of the coordinate systems transformation (see Chapter 3). The derived operators of the coordinate systems transformations yield representation of the surfaces rP , rT, and of all major elements of their geometry in a common coordinate system. Implementation of the local coordinate system xP y P zP for this purpose is convenient.

8.3

Computation of the Elementary Surface Deviations

The earlier performed analysis shows that the resultant deviation h Σ of the machined part surface Pac from its desired shape can be evaluated using the formula h Σ = ah ⋅ h fr + bh ⋅ hss (see Equation 8.4). For the computation of the resultant deviation h Σ , actual values of the elementary deviations h fr and hss are necessary. As will be shown, for the computation of both elementary deviations h fr and hss , similar equations can be used. Therefore, it is not necessary to investigate both elementary deviations separately. It is sufficient to investigate just one of them, and afterwards to write similar equations for the computation of another. 8.3.1 Waviness of the Machined Part Surface Consider, for example, computation of the elementary deviation h fr. Figure 8.10 illustrates a cross-section of a sculptured part surface P by a plane through the unit normal vector n P and through the feed-rate vector F fr . Depending on the chosen point of interest on the surface P, the cross-section of the surface P could have either straight profile KK1 or convex profile K1K 2 or concave profile K 2 K 3 . It is convenient to mention here that the rate of conformity of the generating surface T of the cutting tool is the lowest at the convex point K 2 ; it is bigger at the point of inflection K1 (or at the similar point K); and it is highest at the concave point K 3 . This yields making a conclusion according to which when the higher rate of conformity of the surface T to the surface P observes, then the higher accuracy of the machined part surface and vice versa. For the computation of the elementary deviation h fr, the following equation is derived by Radzevich [6,7]: (  Ffr  RP. fr ⋅ (RP. fr + R T . fr ) ⋅  1 − cos  2 ⋅ RP. fr   (8.7) h fr ≅ (  Ffr  RP. fr − (RP. fr + R T . fr ) ⋅ cos    2 ⋅ RP. fr  where radii of normal curvature of the surfaces P( and T are designated as RP . fr and R T . fr, respectively, and the arc segment Ffr designates the feed rate

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Accuracy of Surface Generation (cv)

OP

Ffr

T3

Ffr

RT.fr

RP.fr

T OT

OT.3 D

(cv)

hfr

K3

A K

RT.fr

B

E

C

K1

RT.fr (cx) hfr

K2

F˘fr

P

F˘fr

hfr

T2

OT.2

OT.1

RP.fr

(cx)

OP Figure 8.10 Computation of the elementary deviation

h fr

(the waviness) on the sculptured part surface P.

per tooth of the cutting tool. The radii RP . fr and R T . fr are measured in the direction of the feed-rate vector F fr. For computation of the radius of normal curvature RP . fr, the following equation is derived in [6,7]: RP . fr =

EP GP GP LP sin 2 ξ + M P EP GP sin 2ξ + EP N P cos 2 ξ

(8.8)

where angle ξ specifies the direction of the feed-rate vector F fr relative to the principal directions t 1.P and t 2.P of the sculptured surface P. An equation similar to Equation (8.8) is derived in [6,7] for the computation of the radius of normal curvature R T . fr: R T . fr ≅

GT LT sin (ξ + µ ) + MT 2

ET GT (8.9) ET GT sin 2(ξ + µ ) + ET N T cos 2 (ξ + µ )

where m is the angle of the local relative orientation of surfaces P and T. It is assumed in Equation (8.9) that the radius of normal curvature of the surface

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of cut is approximately equal to the corresponding radius of normal curvature of the generating surface T of the cutting tool. In particular cases, Equation (8.7) can be significantly simplified. For example, when a flat portion of a part surface P is machined with the milling cutter of diameter dT, then the cusp height is equal to ( h fr = R T . fr − RT2 . f − 0, 25 FT2. f

(8.10)

Equation (8.10) is well known from practice.

8.3.2 Elementary Deviation hss of the Machined Surface For computation of the elementary surface deviation hss, a plane through the unit normal vector n P and through the vector Fss of the side-step of the cutting tool is employed. The plane through the vectors n P and Fss is orthogonal to the plane through the vectors n P and F fr [4–8]. Derivation of the equation for the computation of the elementary deviation hss is similar to the derivation of Equation (8.7). Therefore, without going into details of derivation, the final equation for the computation of the elementary deviation hss is represented below:

(  Fss  RP. ss ⋅ (RP. ss + R T . ss ) ⋅  1 − cos 2 ⋅ RP. ss   hss ≅ (  Fss  RP. ss − (RP. ss + R T . ss ) ⋅ cos    2 ⋅ RP. ss 

(8.11)

where RP . ss = R T . ss ≅

GP LP cos ξ + M P 2

EP GP EP GP sin 2ξ + EP N P sin 2 ξ

(8.12)

ET GT GT LT cos 2 (ξ + µ ) + MT ET GT sin 2(ξ + µ ) + ET N T sin 2 (ξ + µ )

(8.13)

In Equation (8.11), radii of normal curvature of the surfaces P and ( T are designated as RP . ss and R T . ss , respectively, and the arc segment Fss designates the side-step of the cutting tool. The radii RP . ss and R T . ss are measured in the direction of the side-step vector Fss .

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Accuracy of Surface Generation 8.3.3 An Alternative Approach for the Computation of the Elementary Surface Deviations

Reasonable assumptions yield simplification of equations for the computation of elementary surface deviations. As an example, an alternative approach for the computation of the elementary surface deviation h fr is illustrated in Figure 8.11. Elementary analysis of Figure 8.11 yields computation of coordinates of centers OT(1) and OT( 2) in two consecutive positions of the cutting tool relative to the work. Equations of the circular arcs of the radius R T . fr together with the equation of the circular arc of the radius R P . fr yield computation of cusp height h fr. For convenience, the equation for the computation of height h fr of the surface P waviness is expended in the Taylor’s series. Ultimately, this yields the approximate equation h fr ≅

ϕ2 RP . fr ( RP . fr − R T . fr ) 2 R T . fr

(8.14)

for the computation of the elementary surface deviation h fr. The interested reader may wish to go to [5] for details on the derivation of Equation (8.14). YP. fr OP XP. fr

(1)

(2)

OI

OT

RT. fr

RT. fr

RP. fr

P

K1

K2

T

hfr

Figure 8.11 Another approach for the computation of the surface waviness

© 2008 by Taylor & Francis Group, LLC

h fr.

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The considered approach can be enhanced to the situation when radii of normal curvature of the part surface P and of the generating surface T of the cutting tool are significantly different between two consequent tool-passes.

8.4 Total Displacement of the Cutting Tool with Respect to the Part Surface No absolute accuracy is observed in machining sculptured surfaces on a multi-axis NC machine. Both the NC machine and the cutting tool are the major sources of unavoidable deviations of the machined part surface from the desired sculptured surface. Actual relative motion of the cutting tool is performed with certain deviations of its parameters with respect to the desired relative motion of the cutting tool. The last is also a source of significant surface deviations. Displacements of the generating surface T of the cutting tool with respect to the desired part surface P are unavoidable. Problems of two kinds arise in this concern. First, it is important to compute how much the displacement of a cutting tool donates to the resultant deviation of the actual machined part surface from the desired part surface. Second, in order to avoid the cutter penetration into the part surface P, it is of critical importance to determine the maximal allowed dimensions of the cutting tool in order to avoid violation of the necessary conditions of proper surface generation (see Chapter 7). For solving problems of both kinds, computation of the closest distance of approach (CDA) of the surfaces P and T is necessary. The minimal separation between objects is a fundamental problem that has application in a variety of arenas. The problem of computation of the CDA of two surfaces is sophisticated. However, it can be solved using methods developed in the theory of surface generation. 8.4.1 Actual Configuration of the Cutting Tool with Respect to the Part Surface It is convenient to begin the analysis from the ideal case, when the surfaces P and T are in proper tangency at a certain point K (Figure 8.12). For the ideal case of surface generation, the closed loop of consequent coordinate systems transformation (Figure 3.9) can be constructed. The unit tangent vectors u P and v P , the unit normal vector n P , as well as the unit tangent vectors t 1.P and t 2.P of the principal directions can be computed for the sculptured surface P at point K. Equation r P = r P (U P , VP ) of the surface P is used for this purpose (see Chapter 1). The unit vectors t 1.P , t 2.P , and n P compose the Darboux trihedron that is used for the construction of the left-hand-oriented local Cartesian coordinate system xP y P zP having origin at point K (Figure 8.12).

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x*T y*T zP

K*T nT

TrT

∑

yT

nP

z*T

xT

TrP

K TrT

yP

nT

xP

zT Figure 8.12 Actual configuration of local patches of the torus surfaces Tr P and TrT .

The similar left-hand-oriented local Cartesian coordinate system xT yT zT is associated with the generating surface T of the cutting tool. The local coordinate system xT yT zT is turned about the zT axis with respect to the local coordinate system xP y P zP through the angle µ of the local relative orientation of surfaces P and T. The axes zP and zT are pointed oppositely to each other. A left-hand-oriented Cartesian coordinate system X NC YNC ZNC is associated with the multi-axis NC machine. Configuration of the sculptured surface P as well as configuration of the generating surface T of the cutting tool in the coordinate system X NC YNC ZNC is known. Therefore, corresponding operators of the coordinate systems transformations — the operator Rs ( NC a P) of the resultant transformation from the coordinate system X NC YNC ZNC to the coordinate system X P YP ZP and the operator Rs ( P a K P ) of the resultant transformation from the coordinate system X P YP ZP to the local coordinate system xP y P zP — can be composed. The operators Rs ( NC a P) and Rs ( P a K P ) of the resultant coordinate systems transformations are expressed in terms of the operators of elementary coordinate systems transformations (see Chapter 3). Ultimately, the operator Rs ( NC a K P ) of the resultant coordinate systems transformation can be composed as well. Similar operators Rs ( NC a T ) , Rs (T a KT ) , and Rs ( NC a KT ) of the consequent coordinate systems transformations are composed for the generating surface T of the cutting tool. Ultimately, the operators of the direct Rs ( KT a K P ) and of the inverse Rs ( K P a KT ) coordinate systems transformations can be composed as well. It is important to note that the

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equality Rs ( KT a K P ) = Rs −1 ( K P a KT ) is always observed. The operators Rs ( KT a K P ) and Rs ( K P a KT ) complement the earlier composed operators of the coordinate systems transformation to the closed loop of the coordinate systems transformations (see Figure 3.9). Equation rP = rP (U P , VP ) and equation rT = rT (UT , VT ) of the surfaces P and T together with the above-mentioned operators of the coordinate systems transformation yield representations of the surfaces rP and rT in a common coordinate system. Below, the local Cartesian coordinate system xP y P zP is used for this purpose. When the generating surface T of the cutting tool is in proper tangency with the sculptured surface P (Figure 8.12), then the origins of both local coordinate systems xP y P zP and xT yT zT coincide with the point of contact K of the surfaces P and T. In reality, the surfaces P and T do not make proper contact. Actually, the surfaces are either slightly apart, or the surface T penetrates into the surface P. This is due to the unavoidable deviations of configuration of the cutting tool with respect to the part surface P. The deviations cause a displacement of the local coordinate system xT yT zT from its desired position to the actual position xT* yT* zT* . Again, deviations of this kind are unavoidable. The resultant linear displacement δ Σ of the cutting tool with respect to the part surface P can be expressed in terms of the elementary linear displacements δ x , δ y , and δ z of the cutting tool along the axes xP , y P , zP :

δ x  δ  y δΣ =   δ z    1

(8.15)

In addition to the linear displacements δ x , δ y, and δ z , the elementary angular displacements θ x , θ y , and θ z of the local coordinate system xT yT zT with respect to the local coordinate system xP y P zP are observed. The resultant angular displacement θ Σ of the cutting tool with respect to the part surface P can be expressed in terms of the elementary angular displacements of the cutting tool through the angles θ x , θ y, and θ z about the axes xP , y P , zP :

θ x  θ  y θΣ =   θ z    1

( 8.16)

Ultimately, the local coordinate system xT yT zT associated with the cutting tool moves to a position xT* yT* zT* .

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Accuracy of Surface Generation

387

Because the displacements δ Σ and θ Σ are always observed, either a gap between local patches of the surfaces P and T occurs, or the surfaces cause P and T to interfere with each other. The resultant linear δ Σ and the resultant angular θ Σ displacements can be expressed in terms of the corresponding elementary displacements of all the local coordinate systems between the point K P ≡ K and the point KT . Here K P and KT designate origins of the local coordinate systems xP y P zP and xT* yT* zT* . No closed loop of the consequent coordinate systems transformations can be constructed at this point. The loop of the consequent coordinate systems transformations is not closed yet. In order to make the loop close, it is necessary to compose the operator Rs ( KT* a K P ) of the resultant coordinate systems transformation, and the operator Rs ( K P a KT* ) = Rs −1 ( KT* a K P ) of the inverse coordinate systems transformation. For the composing of the operators Rs ( KT* a K P ) and Rs ( K P a KT* ) , the earlier developed operators Rs ( P a K P ) and Rs ( P a KT ) are helpful:

Rs ( KT* a K P ) = Rs −1 ( P a KT ) ⋅ Rs ( P a K P )

(8.17)

In order to solve the problem, the CDA between the surfaces P and T must be computed. In the ideal case of surface generation when no displacement of the surface T with respect to the surface P occurs, the surfaces P and T make contact at a point K. Actually, it is allowed to interpret the ideal surfaces contact in the way that the point K P of the part surface P, and the point KT of the cutting tool surface T are snapped into a common point K. Therefore, the identity K P ≡ KT ≡ K is valid for the ideal case of surface generation. For convenience, the designation K P ≡ KT ≡ K is not used for the point of contact of the surfaces in the ideal case of surfaces generation, but the designation K is used instead. Because the identity K P ≡ KT ≡ K is valid, then the closest distance of approach between the surfaces P and T is identical to the closest distance of approach between the approximating torus surfaces TrP and TrT , and it is identical to zero. The closest distance of approach between the surfaces P and T can be interpreted as the distance between the points K P and KT. Therefore, for the ideal case of surfaces generation, the equality K P KT = 0 is valid. In reality, the generating surface T of the cutting tool is displaced with respect to the part surface P. The total linear displacement of the surface T with respect to the surface P is equal to the magnitude of the vector δ Σ (see Equation 8.15). The total angular displacement of the surface T with respect to the surface P is equal to the magnitude of the vector θ Σ (see Equation 8.16). The closest distance of approach of the surfaces P and T is not equal to zero. It can be positive or negative. In the first case, the cutting tool surface T is located apart from the part surface P. In the second case, the cutting tool surface T interferes with the part surface P.

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It is critical to realize that the following theorem is correct: Theorem: The closest distance of approach of two smooth, regular surfaces is perpendicular to both the surfaces simultaneously.

The theorem is proven analytically. We are not going into the details of the proof of the theorem here. The interested reader may wish to exercise this concern on his or her own. Due to the occurrence of displacements δ Σ and δ Σ , the closest distance of approach of the surfaces P and T is not equal to the distance between the points K *T and K (≡ K P ) . In-perpendicularity of the straight-line segment KK *T to the surfaces P and T is the major reason the equality is not observed. Once the inequality of the length of the straight-line segment KK *T to the closest distance of approach between two surfaces P and T is understood, then one can proceed with further analysis. The analysis below is based in much on the same presumption that the configuration of the local coordinate system xT* yT* zT* with respect to the local coordinate system xP y P zP is known. The configuration is specified by the operator Rs ( KT* a K P ) of the resultant coordinate systems transformation (see Equation 8.17). Use of the operator Rs ( KT* a K P ) together with the operator Rs ( K P a KT ) discussed earlier in this section yield introduction of the matrix Ds(T / P) of the displacement of the generating surface T of the cutting tool with respect to the part surface P. The displacement matrix Ds(T / P) specifies the actual configuration of the local coordinate system xT* yT* zT* associated with the cutting tool with respect to the local coordinate system xT yT zT associated with the cutting tool in its ideal configuration with respect to the part surface P. Actually, the matrix Ds(T / P) of the resultant displacements can be composed in the following way. Consider all n elements and joints between the elements, those that are involved in the closed loop of the consequent coordinate systems transformations. Elementary displacement of every element and at every joint donate to the resultant displacement of the cutting tool with respect to the part surface P. The elementary i-th displacement can be interpreted as the displacement of the actual elementary coordinate system X aci Yiac Z aci with respect to the nominal location of the corresponding elementary coordinate system X i Yi Z i . Implementation of the generalized formula for the resultant coordinate systems transformations (see Equation 3.15) yields derivation of the matrix ds i ( aci a nom i ) of a particular elementary displacement:

(i)  cos θ xx  (i)  cos θ xy ds i ( aci a nom i ) =  (i)  cos θ xz  0 

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(i) cos θ xy

(i) cos θ xz

cos(yyi)

(i) cos θ yz

(i) cos θ yz

cos(zzi)

0

0

δ x( i )   δ y( i )   δ z( i )  1 

(8.18)

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Accuracy of Surface Generation

In particular cases, the i-th displacement can be either just linear or just angular. Encompassing all the elementary displacements between the local coordinate systems xT* yT* zT* and xP y P zP , the following equation for the computation of the matrix Ds(T / P) of the resultant displacements can be obtained: n

Ds (T / P) =

∏ ds (ac i

i

nom i )

(8.19)

i =1

where aci and nom i designate the actual and the nominal location of the ith coordinate system. When the operators Rs ( KT* a K P ) and Rs ( K P a KT ) are known, then the displacement matrix Ds(T / P) can be expressed in terms of the operators Rs ( KT* a K P ) and Rs ( K P a KT ) of the resultant coordinate systems transformations:

Ds (T / P) = Rs ( K P a KT ) ⋅ Rs ( KT* a K P )

(8.20)

Ultimately, the displacement matrix Ds( P / T ) can be expressed in terms of the elementary linear and angular displacements of the local coordinate system xT* yT* zT* with respect to the desired location of the local coordinate system xT yT zT :

 cos θ xx  cos θ xy Ds( P / T ) =   cos θ xz   0

cos θ xy cos θ yy cos θ yz 0

cos θ xz cos θ yz cos θ zz 0

δx  δ y  δz   1

(8.21)

For the inverse transformation, the displacement matrix Ds( P / T ) can be used. The matrix Ds( P / T ) can be either composed similar to the way the displacement matrix Ds(T / P) is composed (see Equation 8.20), or it can be computed from the formulae Ds ( P / T ) = Ds−1 (T / P) . All the elements of the displacement matrix Ds(T / P) can be expressed in terms of the actual elementary displacements of all the elements that make up the closed loop of the consequent coordinate systems transformations. The elementary displacements include the linear and the angular displacements in all mechanical joints, all the deflections caused by elasticity of the material of the components involved in the closed loop of the consequent coordinate systems transformations, all the thermal extensions of the components involved in the closed loop of the consequent coordinate systems transformations, and so forth. The displacements of the surface T with respect to the surface P are not known. Theoretically, components of all of the above-listed matrices of the

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actual elementary displacements can be determined through the direct measurements of the system composed of Work/NC machine tool/Cutting tool. Actually, it is not practical to perform such complex measurements. Under such a scenario the displacement matrix Ds(T / P) cannot be employed for the computation of the actual configuration of the generating surface T of the cutting tool with respect to the part surface P, but the tolerances matrix Tl(T / P) can be used instead. The tolerances matrix Tl(T / P) is composed similarly to the displacements matrix Ds(T / P) . The only difference is that the elementary displacements ds i ( aci a nom i ) of the surface T with respect to the surface P are not employed for the computations, but the corresponding tolerances tli ( aci a nom i ) are used instead: n

Tl (T / P) =

∏ tl (ac a nom ) i

i

(8.22)

i

i=1

Based on the last statement, the following approximate equality occurs: Ds (T / P) ≅ Tl (T / P)

(8.23)

The required elementary tolerances for composing the tolerance matrix Tl(T / P) can be determined much easier. Therefore, if the displacements matrix Ds(T / P) is not known, the tolerance matrix Tl(T / P) can be used instead. The approximating torus surface TrT is associated with the local coordinate system xT* yT* zT* . Once the displacements matrix Ds(T / P) is composed, then the equation r tr.T (θ tr.T , ϕ tr.T ) (see Equation 8.6) of the approximating torus surface TrT can be represented in the local coordinate system xP y P zP :

r tr( P.T) (θ tr.T , ϕ tr.T ) = Ds (T / P) ⋅ r tr.T (θ tr.T , ϕ tr.T )

(8.24)

Equation r tr. P (θ tr. P , ϕ tr. P ) of the approximating torus surface TrP is initially determined in the local coordinate system xP y P zP . Equation (8.24) describes analytically the approximating torus surface TrT in that same local coordinate system xP y P zP . This yields making a conclusion that the actual configuration of the torus surfaces TrP and TrT is determined. 8.4.2 The Closest Distance of Approach between the Surfaces P and T Generally, the problem of the computation of the closest distance of approach between two smooth, regular surfaces is sophisticated and challenging. Per the author’s knowledge, no general solution to the problem of computation of the closest distance of approach between two smooth, regular surfaces has been published. For the purpose of computation of the deviation δ P of

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the actual part surface Pac with respect to the desired part surface Pnom , the problem under consideration can be reduced to the problem of computation of the CDA between two torus surfaces. Consider the surfaces P and T that initially are given in a common coordinate system X NC YNC ZNC (Figure 8.13) associated with the NC machine. The surfaces P and T are locally approximated by portions of torus surfaces TrP and TrT , respectively. Again, not all points of the torus surfaces TrP and TrT can be used for the local approximation of the surfaces P and T. Only points that are located either within the biggest meridian or within the smallest meridian of the torus surface are employed for this purpose. The points K P and K *T are chosen as the first guess points on the surfaces TrP and TrT . For the analysis below, it is convenient to relabel the points K P and K *T to pi and t i correspondingly.

TrT

ZNC (i)

θtr.T

YNC Otr.T

dT

ti

uT uT TrP

XNC

(i)

ti+1

vT

nP

T

pi+1 (i) dP

P vP

Pi

uP

Otr.P (i) θtr.P

Figure 8.13 Computation of the closest distance of approach of the surfaces P and T.

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For a given configuration of the torus surfaces TrP and TrT , the CDA between these surfaces can be used as a first approximation to the CDA between the surfaces P and T. The CDA between the torus surfaces TrP and TrT is measured along the common perpendicular to these surfaces. The following equations can be composed on the premises of this property of the CDA. Unit normal vector n Tr. P to the torus surface TrP is within a plane through the axis of rotation of the surface TrP . In the coordinate system Xtr. P Ytr. P Ztr. P that is associated with the surface TrP , the equation of a plane through the axis of rotation of the torus surface TrP can be expressed in the following form [1,4,5,8]: [rτ P − rtr( 0. P) ] × k tr. P × R tr. P = 0

(8.25)

where rτ P is the position vector of a point of the plane through the axis of rotation of the torus TrP , rtr( 0. P) is the position vector of a point within the plane rτ P (it is assumed below that this point coincides with the origin of the coordinate system Xtr. P Ytr. P Ztr. P ), and k tr. P is the unit vector of the Ztr.P axis. Equation (8.25) is expressed in terms of the radius R tr. P . This indicates that the set of all planes through the fixed Ztr.P axis forms a pencil of planes. The equation of the pencil of planes rτ P in the common coordinate system X NC YNC ZNC can be represented in the form

Vtr . P ⋅ cos θtr . P    V ⋅ sin θtr . P  rτ P (Ztr . P , Vτ r . P , θtr . P ) = Rs (TrP a NC) ⋅  tr . P   Ztr . P   1  

(8.26)

The unit normal vector n Tr.T to the torus surface TrP is within a plane through the axis of rotation of the surface TrP . In the coordinate system Xtr.T Ytr.T Ztr.T that is associated with the surface TrP , the equation of a plane through the axis of rotation of the torus surface TrP can be represented in the form

[rτ T − r tr( 0.)T ] × k tr.T × R tr.T = 0

(8.27)

where rτ T is the position vector of a point of the plane through the torus ( 0) TrP axis of rotation, rtr.T is the position vector of a point within the plane rτ T (it is assumed below that this point coincides with the origin of the coordinate system Xtr.T Ytr.T Ztr.T ), and k tr.T is the unit vector of the Ztr.P axis. Equation (8.27) is expressed in terms of the radius R tr.T . This indicates that the set of all planes through the fixed Ztr.P axis forms a pencil of planes.

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Accuracy of Surface Generation

The equation of this pencil of planes rτ T in the common coordinate system X NC YNC ZNC can be represented in the form

Vtr .T ⋅ cos θtr .T    V ⋅ sin θtr .T  rτ T (Ztr .T , Vtr .T , θtr .T ) = Rs (TrT a NC) ⋅  tr .T   Ztr .T   1  

(8.28)

A straight line through the points dP(i ) and dT(i ) along which the shortest (min) distance of approach dPT of the torus surfaces TrP and TrP is measured, (min) is the line of intersection of the planes rτ P and rτ T. Therefore, this line dPT must be aligned with both unit normal vectors n tr. P and n tr.T . In the coordinate system X NC YNC ZNC , the equation for the unit normal vector n tr. P to the surface TrP yields representation in matrix form:

n tr . P

(Ctr . P + cos ϕ tr . P ) ⋅ cos ϕ tr . P ⋅ cos θtr . P    (C + cos ϕ tr . P ) ⋅ cos ϕ tr . P ⋅ sin θtr . P  = Rs (TrP a NC) ⋅  tr . P   (Ctr . P + cos ϕ tr . P ) ⋅ sin ϕ tr . P   1  

where Ctr. P designates the parameter Ctr. P = 1 −

(8.29)

R 2. P

. R 1. P Similarly, in the coordinate system X NC YNC ZNC , the equation for the unit normal vector n tr.T to the surface TrT yields matrix representation in the form

n tr .T

(Ctr .T + cos ϕ tr .T ) ⋅ cos ϕ tr .T ⋅ cos θtr .T    (C + cos ϕ tr .T ) ⋅ cos ϕ tr .T ⋅ sin θtr .T  = Rs (TrT a NC) ⋅  tr .T   (Ctr .T + cos ϕ tr .T ) ⋅ sin ϕ tr .T   1  

(8.30)

where Ctr.T designates the parameter Ctr.T = 1 −

R 2.T . R 1.T

Evidently, the points Otr. P , Otr.T , dP(i ) , and dT(i ) (Figure 8.13) are within the straight line through the centers Otr. P and Otr.T . The position vector r cd of this straight line can be computed from the equation

(r cd − r cp ) × (r ct − r cp ) = 0

(8.31)

where r cp is the position vector of a point on the circle of radius R tr. P , and r ct is the position vector of a point on the circle of radius R tr.T .

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Kinematic Geometry of Surface Machining

It is necessary that the straight line r cd be along the unit normal vectors ntP and ntT to the torus surfaces TrP and TrT . Considered together, Equation (8.26), Equation (8.28), and Equation (8.31) make possible the computation of the CDA between the torus surfaces TrP (min) and TrT . Then, the line dPT intersects the surfaces P and T at the points pi+1 and t i+1, correspondingly. The points pi+1 and t i+1 serve as the second guess to the CDA between the surfaces P and T. The cycle of the recursive computations is repeated as many times as necessary for making the deviation of the computation of the CDA between the surfaces P and T smaller than the maximal allowed value. There is an alternative approach for the computation of the CDA between two torus surfaces. The direction of the unit normal vector to an offset surface to TrP is identical to the equation of the unit normal vector n tr. P to the torus surface TrP . This statement is also valid for the unit normal vector n tr.T to the torus surface TrT . This property of the unit normal vectors n tr. P and n tr.T can be used for the modification of the method of computation of the CDA between two torus surfaces. The equation of the circle of radius R tr. P yields matrix representation

 R tr . P ⋅ cos θtr . P     R tr . P ⋅ sin θtr . P  r cp (θtr . P ) = Rs ( P a NC) ⋅   0     1

(8.32)

The equation of the circle of radius R tr.T can be analytically described in a similar way:

 R tr .T ⋅ cos θtr .T     R tr .T ⋅ sin θtr .T  r ct (θtr .T ) = Rs (T a NC) ⋅   0     1

(8.33)

The distance dpt between two arbitrary points on the circles r cp (θ tr. P ) and r ct (θ tr.T ) is equal:

dpt (θ tr. P , θ tr.T ) =|r cp (θ tr. P ) − r ct (θ tr.T )|

(8.34)

The distance dpt is minimal for a specific (optimal) combination of the parameters θ tr. P and θ tr.T . The optimal values of the parameters θ tr. P and θ tr.T can be computed on solution of the set of two equations:

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Accuracy of Surface Generation

∂ r cp (θ tr. P ) = 0 ∂θ tr. P

(8.35)

∂ r ct (θ tr.T ) = 0 ∂θ tr.T

(8.36)

) On solution of Equation (8.35) and Equation (8.36), the optimal values θ tr( opt .P ( opt ) and θ tr.T can be computed. These angles specify the direction of the CDA of the torus surfaces TrP and TrT . Following this method, the three-dimensional problem of computation of the closest distance of approach of two torus surfaces is reduced to the problem of computation of the CDA between two circles. Under a certain scenario, the last approach could possess an advantage over the previous approach. Convergence of the disclosed algorithms for the computation of the CDA between two smooth, regular surfaces is illustrated in Figure 8.14. The computation procedure is convergent regardless of the actual location of the first guess points on the surfaces P and T. It is instructive to draw attention here to the similarities between the disclosed iterative method for the computation of the CDA between two smooth, regular surfaces, and between the Newton–Raphson’s method, the iterative method of chords, and so forth. Many similarities can be found in this comparison.

(i)

OT

(i+1)

OT

(i+1)

(i)

RT

RT

t1

p1

t2 p2

t3 p3

t0 p0

tT

T tP

P

(i+1)

(i)

RP

RP

(i)

OP

(i+1)

OP

Figure 8.14 Convergence of the methods of computation of the closest distance of approach of the surfaces P and T.

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8.5

Effective Reduction of the Elementary Surface Deviations

As follows from the above consideration, the resultant deviation of the machined part surface Pac from the desired surface Pnom depends mostly on two components: height h Σ of the residual cusps on the machined part surface is the first component and the displacement δ P of the surfaces is the second component. Therefore, the resultant deviation δ Σ of the machined part surface from the desired surface can be expressed by a simple formula:

δΣ = h Σ + δP

(8.37)

Recall, both the components hS and d P are signed values. Both the components h Σ and δ Σ must be reduced in order to increase the resultant accuracy of the machined part surface. However, only the component h Σ is under control using methods developed in the theory of surface generation. The component δ Σ is out of control when just the theory of surface generation is used. However, even under such restrictions, the methods developed in the theory of surface generation are helpful when aiming for reduction of the machining error. Consider just two opportunities in this concern. 8.5.1 Method of Gradient For the most intensive reduction of the resultant( surface deviation h Σ , gradient ( of the function h Σ = h Σ (RP. fr , RP. ss , R T . fr , RT . ss , Ffr , Fss , K) can be implemented. As an example, implementation of the gradient function grad ( h fr ) for a flattened portion of a sculptured part surface (Figure 8.10) is considered below. In the case under consideration, the height of the waviness h fr is ( h fr = R T . fr − R 2T . fr − 0, 25 ⋅ Ffr2

(8.38)

By definition, grad ( h fr ) =

∂h fr ∂R T . fr

∂h fr i+ ( j ∂Ffr

(8.39)

In the case under consideration, the solution to Equation (8.39) is obtained in parametric form:

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R −T3. fr = C ⋅ t 2 ⋅ (t 2 + 4)−3 ⋅ (t 2 + 12)2 ( (t 2 + 4) ⋅ Ffr = −8 ⋅ R T . fr ⋅ t

(8.40) (8.41)

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Accuracy of Surface Generation

10

RT. fr, mm

8 6 4

A

2

0

1

2

3

Ffr, mm Figure 8.15 Interpretation of the solution obtained by implementation of the gradient method.

Here, the constant parameter C specifies a curve of the family of curves (see Equation 9.30), and t designates the parameter. Figure 8.15 illustrates the obtained solution to the differential( Equation (8.39) where gradient curves for the function h fr = h fr (R T . fr , Ffr ) are depicted. The boundary curve A represents( the projection of the line h fr = 0.020 mm onto the coordinate plane R T . fr Ffr . For the most efficient reduction of the waviness height h fr, the parameters of the machining operation must be alternated in compliance with Equation (8.39) through Equation (8.41). If some parameter affects the cusp height the most, then it is recommended to alternate this parameter first. 8.5.2 Optimal Feed-Rate and Side-Step Ratio The resultant height of cusps h Σ is a function of two elementary surface deviations — it is a function of the height of waviness h fr and of height of the elementary surface deviation hss . Various ratios between the feed-rate |F fr | and the side-step |Fss | result in different total cusp height h Σ . It is natural to assume that the resultant cusp height h Σ reaches its minimal value under a certain ratio

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between the feed rate and the side step. This ratio is referred to as the optimal ratio between the feed rate and the side step. The maximal elementary surface deviation hss is smaller then the resultant tolerance [ h] on accuracy of the part surface. The deviation hss is equal to a portion of the tolerance [ h], say hss = c ⋅[ h]. Here, c designates the local parameter of distribution of the tolerance [ h]. The actual value of the parameter c is within the interval 0 ≤ c ≤ 1. At a current point of the surface P, there exists the optimal value of the parameter c. Therefore, the current value of the parameter c can be expressed in terms of Gaussian coordinates of the sculptured surface P, that is, c = c(U P , VP ). If it is assumed that h Σ = h fr + hss , then the equality h fr = (c − 1) ⋅ [ h] is valid for the elementary deviation h fr . Ultimately, the height of cusps h Σ can be expressed as a function of the parameter c, say h Σ = h Σ (c). When the parameter c is of optimal value, the following equality is observed: ∂ hΣ (c ) =0 ∂c

(8.42)

0,505

0,500

c

0,495

0,490

0,485

0,480 0,01

0,05

0,10

1,00

3,00

5,00

[h] Figure 8.16 ( Parameter c against the tolerance ( ( ( [ h] on accuracy of the surface P ( R P. fr = 100 mm , R P. ss = 20 mm , R T . fr = 50 mm and R P. ss = 50 mm ).

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This condition is necessary for the minimum of the function h Σ = h Σ (c). In addition, the inequality ∂ 2 h Σ (c )

∂ c2

(8.43)

>0

must be observed. For the computation, a computer code is developed [9]. An example of the results of the computations is depicted in Figure 8.16. The major conclusion to be made here is that for every point of the sculptured surface P there exists the optimal distribution of the tolerance [ h] between the elementary surface deviations h fr and hss. The optimal distribution of the tolerance [ h] is specified by the optimal value of the parameter c = c(U P , VP ). When the parameter c is equal to its optimal value, then the accuracy of the machined sculptured surface is within the prespecified tolerance [ h], and the machining time in this case is the shortest possible.

8.6 Principle of Superposition of Elementary Surface Deviations Resultant cusp height h Σ depends upon two components: waviness height h fr and the elementary surface deviation hss . When the waviness height is negligible, then the approximate equality h Σ ≈ hss is valid. Otherwise, for the computation of the resultant cusp height h Σ , a corresponding formula is necessary. For simplicity, the equation

h Σ = h fr + hss

(8.44)

is often used for this purpose. For more accurate computations, Equation (8.4) is used instead. In particular cases when the assumptions h fr = hmax fr , max hss = hmax are valid, then Equation (8.5) can be implemented ss , and h Σ = hΣ for the computation of the maximal value h max of the resultant cusp height. Σ All the approximate equations, Equation (8.44), Equation (8.4), and Equation (8.5) are derived on the premises of the principle of superposition of the elementary surface deviations. Actually, use of the principle of superposition is valid only for linear functions. Analysis of the earlier derived Equation (8.16) for the computation of the elementary deviation h fr and of Equation (8.11) for the computation of the ( elementary deviation hss reveals ( that the functions h fr = h fr (RP. fr , R T . fr , Ffr , K) and hss = hss (RP. ss , RT . ss , Fss , K) are the substantionally nonlinear functions. In this concern, the question “Under which conditions is implementation of the principle of superposition of the elementary surface deviations valid?” naturally arises. In order to answer this practical question, comparison of results of computations of the resultant cusp height, which are performed using Equation (8.44) (or using more general Equation 8.4), with the results of precise

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computations is vital. For this purpose, local approximation of the surfaces P and T by patches of torus surfaces is very helpful. At the point of contact K, the surfaces P and T are locally approximated by the patches of the torus surface TrP and of the torus surface TrT , correspondingly (Figure 8.3). Use of this scheme for the precise computation of the resultant cusp height h Σ does not require implementation of the principle of superposition of the elementary surface deviations h fr and hss . A torus surface Tr[ h ] is the offset surface to the torus surface TrP . The distance [ h] between the torus surfaces TrP and Tr[ h ] exceeds or is equal to the maximal allowed cusp height (Figure 8.3). The torus surface TrT is plunged into the space between the surfaces TrP and Tr[ h ]. When the torus surface TrT is contacting the torus surface TrP , then the elementary surface cell on the torus surface TrP can be constructed. Boundaries of the elementary surface cell on the surface TrP can be projected on the common tangent plane. The projection of the boundaries is located within the projection on the common tangent plane of the line of intersection of the torus surfaces Tr[ h ] and TrT . The maximal deviation h Σ is measured along the unit normal vector n P that passes through a vertex of the elementary surface cell on the machined part surface. For example, at a surface point Q (Figure 8.3), the maximal deviation h Σ is equal to h% Σ = P Q . The point P is within the torus surface TrT . Within the arc segments Ffr and Fss , radii of normal curvature R(1.trP) , R(2.trP) and R(1.trT) , R(2.trT) are of constant value. In Figure 8.3, points on the surface P that correspond to vertices of the elementary cell are designated as 1P , 2 P, 3 P , and 4 P . Unit normal vectors through the vertices 1P , 2 P , 3 P , and 4 P are designated as n 1.P , n 2.P , n 3.P , and n 4.P . The unit normal vectors n 1.P and n 2.P are the coplanar vectors. The same is true with respect to the pairs of the unit normal vectors n 1.P and n 3.P , n 1.P and n 4.P , n 2.P and n 3.P , n 2.P and n 4.P , n 3.P and n 4.P . When the parameters of machining of a sculptured surface are properly computed, then the resultant surface deviation h Σ is of the same value at all four( vertices of the elementary cell, and the point K is distance 0.5 Ffr and 0.5 Fss from the corresponding boundaries of the elementary cell. For the computation of the resultant surface deviation h Σ , it is necessary to compute coordinates of the point Q , to derive the equation of the unit normal vector to the torus surface TrP at the point Q , and to compute the coordinates of the point P of intersection of the torus surface TrT by the unit normal vector to the torus surface TrP at the point Q . The resultant surface deviation h Σ is equal to the distance between the points P and Q . For the analysis below, the local Cartesian coordinate system x(PK ) y (PK ) z(PK ) (see Figure 8.9) is used. In the local reference system x(PK ) y (PK ) z(PK ) , the unit tangent vectors t 1.P and t 2.P of the principal directions on the sculptured surface P align with the coordinate axes y (PK ) and z(PK ). The unit tangent vectors t 1.T and t 2.T of the principal directions on the generating surface T of the cutting tool are at the angle µ of local relative orientation of the surfaces P and T.

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The vector F fr of feed-rate motion of the cutting tool is at a certain angle ε with respect to the first principal direction t 1.P of the part surface. The vector Fss of the side-step motion is orthogonal to the vector F fr of the feedrate motion. The point Q * is within the common tangent plane through the point K. The actual location of the point Q * is specified by the position vector rQ * = 0.5 ⋅ (F fr + Fss ). In the local coordinate system x(PK ) y (PK ) z(PK ), the vectors F fr and Fss yield representation in matrix form: (  0.5 ⋅ Ffr ⋅ cos(ε + µ )    (  0.5 ⋅ Ffr ⋅ sin(ε + µ )  (8.45) F fr =   0     1   (  0.5 ⋅ Fss ⋅ sin(ε + µ )  (   −0.5 ⋅ Fss ⋅ cos(ε + µ )   (8.46) Fss =   0   1   Equation (8.45) and Equation (8.46) yield an expression for the position vector rQ * : ( (  0.5 ⋅ [ Ffr cos(ε + µ ) + Fss sin(ε + µ )]   ( (  0.5 ⋅ [ Ffr sin(ε + µ ) − Fss cos(ε + µ )] (8.47) rQ* =   0     1  

Further, consider a portion of the torus surface TrT within the elementary surface cell. Assume that “in small” the portion of the torus surface TrT is developable. Under such an assumption, location of the point Q on the torus surface TrT is predetermined by the location of the corresponding point Q * within the common tangent plane. Therefore, when the tangent plane is locally bending onto the torus surface TrT , the point Q * goes to the location of the point Q . Ultimately, in the local coordinate system x(PK ) y (PK ) z(PK ) , this yields computation of the parameters θQ and ϕQ of the point Q , which are equal to

θQ =

|rQ * sin σ | rTr. P

and

ϕQ =

|rQ * cos σ | . rTr. P

Here, the radius of the generating circle of the torus surface TrP is denoted as rTr. P , and the angle σ is computed from (  Ffr  σ = [90° − (ε + µ )] − arctan  (  .  Fss 

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The computed parameters θQ and ϕQ yield analytical expression of the position vector rQ of the point Q in the local coordinate system x(PK ) y (PK ) z(PK ). For computation of the unit normal vector nTr. P , the matrix equation of the approximating torus surface TrP can be used:

rTr. P

(rTr. P ⋅ cos θ Tr. P + R Tr. P ) ⋅ cos ϕ Tr. P   (r ⋅ cos θ  Tr . P Tr . P + R Tr . P ) ⋅ sin ϕ Tr . P  =   rTr. P ⋅ sin θ Tr. P   1  

(8.48)

This equation yields

nTr. P

 − cos θTr. P cos ϕ Tr. P   − cos θ  Tr . P sin ϕ Tr . P  =   − sin θTr. P   1  

(8.49)

Height h Σ of the resultant cusps is measured along the straight line rh . The straight line rh passes through the point rQ and is pointed along the unit normal vector nTr. P . Therefore, the equation of this straight line is

 XQ − th cos θTr. P cos ϕ Tr. P   Y − t cos θ  Q h Tr . P sin ϕ Tr . P  r h (θTr. P , ϕ Tr. P ) =    ZQ − th sinTr. P   1  

(8.50)

Equation (8.50) and Equation (8.48) of the torus surface TrP uniquely specify the coordinates of the point P (see Figure 8.3). For this purpose, the torus surface TrP must be represented in that same reference system x(PK ) y (PK ) z(PK ) as the straight line rh is represented. The required coordinate systems transformation can be performed by the operator Rs (TrP a K ) of the resultant coordinate systems transformation. Ultimately, the equation of the approximating torus surface TrP in the local reference system x(PK ) y (PK ) z(PK ) casts into matrix form: [rTr . P ]( K ) = Rs (TrP a K ) ⋅ rTr . P    (rTr . P ⋅ cos θ Tr . P + R Tr . P ) ⋅ cos ϕ Tr . P − R Tr . P    (rTr . P ⋅ cos θ Tr . P + R Tr . P ) ⋅ sin ϕ Tr . P ⋅ cos µ + rTr . P ⋅ sin θ Tr . P ⋅ sin µ  =   −(rTr . P ⋅ cos θ Tr . P + R Tr . P ) ⋅ sin ϕ Tr . P ⋅ sin µ + rTr . P ⋅ sin θ Tr . P ⋅ cos µ    1  

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(8.51)

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Accuracy of Surface Generation

At the point of intersection P of the straight line rh (see Equation 8.50) and to the torus surface TrP (see Equation 8.51) the equality r h (θTr. P , ϕ Tr. P ) = [rTr. P ]( K ) is observed. In expended form, this equality casts into XQ − th cos θTr. P cos ϕTr. P   Y − t cos θ sin ϕ  Tr . P Tr . P   Q h   ZQ − th sinTr. P   1   (rTr. P ⋅ cos θ Tr. P + R Tr. P ) ⋅ cos ϕ Tr. P − R Tr. P    (r  ⋅ cos θ + R ) ⋅ sin ϕ ⋅ cos µ + r ⋅ sin θ ⋅ sin µ Tr . P Tr . P Tr . P Tr . P Tr . P Tr . P  = −(rTr. P ⋅ cos θ Tr. P + R Tr. P ) ⋅ sin ϕ Tr. P ⋅ sin µ + rTr. P ⋅ sin θ Tr. P ⋅ cos µ    1  

(8.52)

which yields the computation of three parameters θTr. P , ϕ Tr. P , and th that specify coordinates of the point P . Finally, an analytical expression for the vector rP can be derived. The computed vectors rQ and rP yield computation of the resultant surface deviation h Σ :

h Σ =|rQ − r P |

(8.53)

With the above analysis, the final conclusion can be made with respect to the principle of superposition of the elementary surface deviations: The principle of superposition of the elementary surface deviations h fr and hss is valid if and only if the inequality h Σ − h Σ ≤ [ ∆h Σ ] is observed. Here, [ ∆h Σ ] designates the tolerance on accuracy of computation of the height of the resultant cusps.

References [1] doCarmo, M.P., Differential Geometry of Curves and Surfaces, Prentice Hall, Englewood Cliffs, NJ, 1967. [2] Monge, G., Application de l’analyse à la géométrie, Bachelier, 1850. [3] Radzevich, S.P., Conditions of Proper Sculptured Surface Machining, ComputerAided Design, 34 (10), 727–740, 2002. [4] Radzevich, S.P., Differential-Geometric Method of Surface Generation, DrSci thesis, Tula Polytechnic Institute, 1991. [5] Radzevich, S.P., Fundamentals of Surface Generation, Rastan, Kiev, 2001. [6] Radzevich, S.P., Generation of Actual Sculptured Part Surface on Multi-Axis NC Machine. Part 1, Izvestiya VUZov. Mashinostroyeniye, 5, 138–142, 1985.

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[7] Radzevich, S.P., Generation of Actual Sculptured Part Surface on Multi-Axis NC Machine. Part 2, Izvestiya VUZov. Mashinostroyeniye, 9, 141–146, 1985. [8] Radzevich, S.P., Sculptured Surface Machining on Multi-Axis NC Machine, Vishcha Shkola, Kiev, 1991. [9] Radzevich, S.P. et al., On the Optimization of Parameters of Sculptured Surface Machining on Multi-Axis NC Machine, In Investigation into the Surface Generation, UkrNIINTI, Kiev, No. 65-Uk89, pp. 57–72.

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Part III

Application

© 2008 by Taylor & Francis Group, LLC

9 Selection of the Criterion of Optimization For machining a part surface, a machine tool, a cutting tool, a fixturing, and so forth are necessary. All these elements together are referred to as the technological system. For lubrication and for cooling purposes, liquids and gas substances are often used. The coolants and the lubricants create the technological environment. When a surface machining process is designed properly, then capabilities of both the technological system and of the technological environment are used most completely. When capabilities of the technological system and of the technological environment are used the most completely, manufacturing processes of this kind are usually called the extremal manufacturing processes. Ultimately, use of the extremal manufacturing processes results in the most economical machining of part surfaces. The Differential Geometry/Kinematics (DG/K )-method of surface generation (disclosed in previous chapters) is capable of synthesizing extremal methods of machining of sculptured surfaces on a multi-axis numerical control (NC) machine, as well as synthesizing extremal methods of machining surfaces that have relatively simple geometry on conventional machine tools [5,6,10]. Machining the part surface in the most economical way is the main goal when designing a manufacturing process. For synthesizing the most efficient machining operation, appropriate input information is required. Capabilities of a theoretical approach can be estimated by the amount of input information the approach requires for its implementation, and by the amount of output information the method is capable of creating. A more powerful theoretical approach requires less input information to solve a problem, and use of it enables more output information in comparison with the less powerful theoretical approach. The DG/K-method requires a minimum of input information: just the geometrical information on the part surface to be machined. The geometrical information on the part surface to be machined is the smallest possible input information for solving a problem of synthesis of the optimal machining operation. Based only on the geometrical information on the part surface P, use of the DG/K-method yields computation of the optimal parameters of the machining process. No selection of parameters of the machining operation is required when the DG/K-method is used. This makes it possible to conclude that the DG/K-method of surface generation is the most powerful theoretical method capable of solving problems of synthesis of optimal machining operations on the premises of the smallest possible input information. No other theoretical method is capable of solving problems of this sort on only the premises of geometrical information on the part surface to be machined. 407 © 2008 by Taylor & Francis Group, LLC

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The selection of appropriate criterion of optimization is critical for the implementation of the DG/K-method of surface generation.

9.1

Criteria of the Efficiency of Part Surface Machining

The design of a sculptured surface machining process is an example of a problem having a multivariant solution. In order to solve a problem of this sort, a criterion of optimization is necessary. Various criteria of optimization are used in industry for the optimization of parameters of surface machining. The productivity of surface machining, tool life, accuracy, and quality of the machined surface are among them. Other criteria of optimization of parameters of machining operations are used as well. Economical criteria of optimization are the most general and the most preferred criteria of optimization of machining processes. However, analytical description of the economical criteria of optimization is complex and makes them very inconvenient for practical computations. For particular cases of surface machining, equivalent criteria of optimization of significantly simpler structure can be proposed. The productivity of surface machining and productivity of surface generation are the important criteria of optimization. Both are often used for creating more general criteria of optimization of surface machining. Therefore, it is reasonable to use the productivity of surface machining as the criterion of optimization for the purpose of demonstration of the potential capabilities of the DG/K-method of surface generation. Results of the synthesis of optimal surface machining operations can be generalized for the case of implementation of another criterion of optimization. There are many ways to increase the productivity of surface machining on machine tools. Here, mostly geometrical and kinematical aspects of the optimization of surface machining are considered. In the theory of surface generation, three aspects of the surface generation process are distinguished: the local surface generation, the regional surface generation, and the global surface generation [5,6,10]. The local analysis of the part surface generation encompasses generation of the surface P in differential vicinity of the point K of contact of the part surface P and of the generating surface T of the cutting tool. Generation of the part surface within a single tool-path is investigated from the perspective of the regional surface generation. Ultimately, partial interference of the neighboring tool-paths, coordinates of the starting point for the surface machining, and impact of shape of the contour of the surface P patch are investigated from the perspective of the global surface generation. Consequently, three kinds of productivity of surface machining are distinguished: local productivity of surface generation, regional productivity of surface generation, and global productivity of surface generation.

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9.2

Productivity of Surface Machining

Productivity of surface generation reflects the intensity of generation of the nominal part surface in time. It can be used for the purpose of synthesis of optimal surface machining operations (for example, of machining a sculptured part surface on a multi-axis NC machine). 9.2.1 Major Parameters of Surface Machining Operation It is natural to begin the investigation of major parameters of the surface machining operation from the local surface generation. When machining a sculptured surface on a multi-axis NC machine, all major parameters of the machining operation and the instantaneous productivity of surface generation vary in time. This makes reasonable consideration of instantaneous (current) values of the surface-generation process. Instantaneous productivity of surface generation P sg (t) (is determined by ( current values of the feed-rate Ffr and of the side-step Fss (here t designates time). Usually, the vector F fr and the vector Fss are orthogonal to each other ( F fr ⊥ Fss ). In particular cases, the vectors F fr and Fss are at a certain angle θ to each other. Instantaneous productivity of surface generation can be computed by the following formula [7,8]: (9.1)

P (t) =|F fr × Fss |

Equation (9.1) casts into [7,8]

( ( P (t) = Ffr ⋅ Fss ⋅ sin θ

(9.2)

( ( Here, Ffr is equal to |F fr |, and Fss is equal to |Fss |. that an increase of the feed-rate ( Equation (9.1) and Equation (9.2) reveal ( Ffr , and an increase of the side-step Fss lead to an increase of the instantaneous productivity of surface generation P (t) . Deviation of the angle θ from θ = 90 results in a corresponding reduction of the instantaneous productivity of surface generation P (t) . At a current point K of contact of the part surface P and of the generation ( ( surface T of the cutting tool, optimal values of the parameters Ffr , Fss, and θ depend upon local geometrical (differential) characteristics of the surfaces P and T, and upon the tolerance on accuracy [ h] of the machined part surface. The value of the tolerance on accuracy [ h] of surface machining is usually constant within the patch of the surface P. However, in a more general case of surface machining, the current value of the tolerance [ h] can vary within the surface patch:

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[ h] = [ h](U P , VP )

(9.3)

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Within certain portions of a surface patch, the tolerance can be bigger, and within other portions, it can be smaller depending on the functional requirements of the actual part surface. Because the resultant cusp height h Σ is made up of two components h fr and hss , it is necessary to split the tolerance [ h] on two corresponding portions: on the portion [ h fr ] for the elementary deviation h fr , and on the portion [ hss ] for the elementary deviation hss . The equality (9.4)

[ h] = [ h fr ] + [ hss ]

is always observed (see Equation 8.44). But, the equality (9.5)

h Σ ≅ h fr + hss

is always approximate. For computations of the surface deviation h Σ , it is recommended that Equation (8.4) be used: (9.6)

h Σ ≅ ah ⋅ h fr + bh ⋅ hss

Here, coefficients ah and bh are within the intervals 0 ≤ ah ≤ 1 and 0 ≤ bh ≤ 1. The coefficients ah and bh can be determined at a current point K of the sculptured surface P. At a current point K on the part surface P having coordinates U P and VP , the current values of the coefficients ah and bh also depend on coordinates of the point K on the surface P (that is, depend on UT and VT parameters) and on the angle µ of the local relative orientation of surfaces P and T at the point K. This relationship is expressed by two formulae:

ah = ah (U P , VP , UT , VT , µ )

bh = bh (U P , VP , UT , VT , µ )

(9.7) (9.8)

( Values of the feed-rate Ffr per tooth of the cutting tool, and of the side-step [ hss ] at a current point K depend on the partial tolerances [ h fr ] and( [ hss ]. One can immediately conclude from the above that both the feed-rate Ffr and the side-step [ hss ] are functions of coordinates of the point K on the surface P, of coordinates of the point K on the surface T, of the angle µ of the local relative orientation of surfaces P and T, and of the direction of motion of the surface T with respect to the surface P. The following expressions

( ( ( Ffr = Ffr ([ h fr ]) = Ffr (U P , VP , UT , VT , µ , ϕ ) ( ( ( Fss = Fss ([ hss ]) = Fss (U P , VP , UT , VT , µ , ϕ )

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(9.9) (9.10)

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reveal this relationship. Here the angle that specifies the direction of the feed-rate vector F fr is designated as ϕ . Substituting Equation (9.9) and Equation (9.10) in Equation (9.2), it is easy to conclude that productivity of surface generation P sg also depends on coordinates of the current point of contact K on both the surfaces P and T, on the angle µ of the local relative orientation of surfaces P and T, and on the direction of the relative motion of the surfaces P and T at point K:

P

sg

= P sg (U P , VP , UT , VT , µ , ϕ )

(9.11)

Certainly, not just the tolerance [ h] , but also the partial tolerances [ h fr ] and [ hss ] can be constant within the part surface patch or can vary within the sculptured surface P. In the first case, the actual values of the tolerances [ h] , [ h fr ] , and [ hss ] must be given. In the second case, the following functions must be known:

[ h] = [ h](U P , VP , UT , VT )

(9.12)

[ h fr ] = [ h fr ](U P , VP , UT , VT , µ )

(9.13)

[ hss ] = [ hss ](U P , VP , UT , VT , µ )

(9.14)

The principal radii of curvature R 1. P and R 2. P of the part surface are the functions of parameters U P and VP of the sculptured surface P, while the principal radii of curvature R 1.T and R 2.T of the generating surface T of the cutting tool are the functions of the parameters UT and VT . In special cases of sculptured surface machining, when, for example, elastic deformation is applied to the work for technological purposes as shown in Figure 2.3, or for a special-purpose cutting tool with changeable generating surface that is used for the machining [3, 9, 13], then in addition to the parameters U P , VP , UT , VT , µ , ϕ some more parameters have to be incorporated into Equation (9.11) for the computation of the productivity of surface generation P sg (see Chapter 8 in [6] for details). 9.2.2 Productivity of Material Removal When machining a part surface, the intensity of stock removal is evaluated by the productivity of material removal. The productivity of material removal is equal to the amount of stock removed from the work in a unit of time. 9.2.2.1 Equation of the Workpiece Surface For the analytical description of productivity of material removal in terms of parameters of the machining operation, an equation of the workpiece surface Wps must be derived. Equation r wp of the surface Wps can be composed on

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Kinematic Geometry of Surface Machining

the premises of numerical data obtained from measurements of the actual workpiece. The stock to be removed b can be of constant value, or its value can vary within the surface patch. In the first case, the thickness of the stock b must be known. In the second case, it is necessary to know the function of the stock distribution b(U P , VP ) . In the event the equation of the workpiece surface Wps is obtained on the basis of the surface measurements, then the equation r P of the part surface P together with the equation r wp of the workpiece surface Wps yields a computation of the stock-distribution function b(U P , VP ) : b(U P , VP ) =|r wp − r P |

(9.15)

When the stock-distribution function b(U P , VP ) is given, then the equation of the workpiece surface Wps can be derived analytically. For this purpose, an equation of the nominal part surface r P = r P (U P , VP ) is employed (Figure 9.1): (9.16)

r wp = r P + n P ⋅ b(U P , VP )

In Figure 9.1, point Mwp on the surface of the workpiece Wps is shown at a distance b(U P , VP ) from the point M on the nominal part surface P. Elements of local topology of the workpiece surface Wps (say, the first Φ 1.ps and the second Φ 2.ps fundamental forms of the workpiece surface Wps)

nP

P Wps

vP

Mwp

b(UP, VP) M

ZP

rwp

uP

rP XP YP

Figure 9.1 On derivation of equation r wp of the workpiece surface Wps.

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can be expressed in terms of the elements of local topology of the nominal part surface P. These derivations are similar to the derivations of major elements of local topology of the characteristic R 1 -surfaces (see Section 7.3.2.2.2 for details). For the computation of the productivity of material removal, equation r P = r P (U P , VP ) of the part surface P and Equation (9.16) of the workpiece surface Wps must be represented in a common reference system. When necessary, an appropriate operator Rs (Wps a P) of the resultant coordinate system transformations can be composed for this purpose (see Chapter 3 for details). Modified Equation (9.16) can also be helpful for the computation of parameters of uncut chip. Similar to Equation (9.16), the equation of the surface of tolerance S[ h] can be immediately written:

r[ h] = r P + n P ⋅ [ h](U P , VP )

(9.17)

Equation (9.17) is used for the computation of parameters of the critical values of the feed rate and of the side step. Elements of analysis of machine tool performance can be found in [12]. 9.2.2.2 Mean Chip-Removal Output For the computation of the chip-removal output, vectorial equations of the part surface to be machined P and of workpiece surface Wps are necessary. Mean chip-removal output is used for the analysis of efficiency of a global machining operation, say for the whole part surface P. The mean chipremoval output P %mr can be used as an index. By definition [5,6,10,12],

P %mr =

Vmr tΣ

(9.18)

where Vmr is the total volume of the stock to be removed, and t Σ is the total time required for the stock removal. 9.2.2.3 Instantaneous Chip-Removal Output For the local analysis of efficiency of a machining operation, instantaneous chip-removal output is used. The instantaneous chip-removal output P mr can also be used as an index. By definition [5,6,10,12],

© 2008 by Taylor & Francis Group, LLC

P mr (t) =

d vmr dt

(9.19)

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Kinematic Geometry of Surface Machining

The volume of chip d vmr to be removed in an instant of time is d vmr = b ⋅ d FP

(9.20)

where d FP is the vector area element, and b is the vector of the stock thickness (here, b = b(U P , VP ) ⋅ n P , see Equation (9.15) for details; |b|= MMwp in Figure 9.1). From another viewpoint, the following equation can be used for computation of the vector b: b = r wp − r P

(9.21)

The vector area element d FP is as follows [2,5,6,10,12]:  ∂ rP ∂ rP  d FP =  ×  dU P dVP  ∂U P ∂VP 

(9.22)

After integration with respect to the surface FP =|FP |, an expression for the volume [5,6,10,12] Vmr =



∂ rP

∫∫ b ⋅ d F = ∫∫  b ⋅ ∂U P

FP

FP

×

P

∂ rP   ⋅ dU P ⋅ dVP ∂VP 

(9.23)

can be derived. Generally, the curvilinear coordinates U P and VP depend upon time t according to the relations

U P = U P (w , t)

(9.24)

VP = VP (w , t)

(9.25)

where w is a new variable. Here, the Jacobian matrix of transformation J P for the implementation of Equation (9.24) and Equation (9.25) is as follows [1,5,6,10]:

∂U P ∂w JP = ∂VP ∂w

∂U P ∂t ∂VP ∂t

(9.26)

The region of the part surface within which the sign of the Jacobi transformation matrix J P is maintained constant is the one under consideration.

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Further, an expression for the computation of the instantaneous chipremoval output [5,6,10,12] w 2 (t )

P mr (t) =

∫

w1

 ∂ rP ∂ rP  × b⋅  ⋅ J ⋅ dU P ⋅ dVP ∂U P ∂VP  P  (t )

(9.27)

can be obtained. Here, w 1 (t) and w 2 (t) are the boundary values of the variable w on the coordinate curve t = Const that corresponds to the boundaries of the part surface P (Figure 9.2). An alternative approach for the computation of the instantaneous chipremoval output can be used. For this purpose, consider the surface of a cut. The surface of cut Sc is generated by the cutting edge of the cutting tool in its motion with respect to the work. The surface of cut Sc can be considered as a set of consecutive positions of the cutting edge of the cutting tool that is moving relative to the work. Such a consideration yields an equation for the position vector r sc of the surface of the cut. Implementation of a corresponding operator of the resultant coordinate system transformations could be helpful when performing derivations of this kind. Two different kinds of analytical representation of the instantaneous chipremoval output P mr (t) can be derived in this case. The first kind of analytical representation of the instantaneous chip-removal output P mr (t) relates to implementation of the cutting tools having the whole generating surface T. In other words, it relates to implementation of grinding wheels, shaving cutters, and so forth. In this case, the equation of the surface of cut Sc can be represented in the form r sc = r sc (UT , VT , ti )

dFP

P

dw

dUP dt Curves t = Const dVP ZP

YP

Curves w = Const

XP

Figure 9.2 On the definition of the chip-removal output P mr (t).

© 2008 by Taylor & Francis Group, LLC

(9.28)

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Kinematic Geometry of Surface Machining

Here, UT and VT denote curvilinear coordinates on the generating surface T of the cutting tool, and ti is a fixed moment of time. The second kind of analytical representation of the instantaneous chipremoval output P mr (t) relates to the implementation of the cutting tools having a discrete generating surface T (for example, it relates to the implementation of milling cutters, etc.). In this case, the equation of the surface of cut Sc can be represented in the form

r sc = r sc (U ce , t)

(9.29)

where U ce designates a coordinate along the cutting edge of the cutting tool. Vector area element d Fsc of the surface of cut Sc can be computed either from the formula

 ∂ r sc ∂ r sc  d Fsc =  ×  ⋅ dU P ⋅ dVP  ∂U P ∂VP 

(9.30)

or from the formula

 ∂ r sc ∂ r sc  d Fsc =  ×  ⋅ dU ce ⋅ d t ∂t   ∂U ce

(9.31)

Equation (9.30) is valid for the first kind, and Equation (9.31) is applicable for the second kind of analytical representation of the instantaneous chipremoval output. When machining a part surface, the vector area element d Fsc is traveling with a certain velocity w through the stock to be removed. In this way, a volume of the stock d vmr = ω ⋅ d Fsc is removed in a unit of time. Hence, for the computation of the instant chip-removal output, the following formula can be used:

P mr (t) =

∫∫

ω ⋅ d Fsc

(9.32)

( Sc )

When surface machining is performed with a cutting tool having multiple cutting edges, Equation (9.32) acquires the form [5,6,10,12] N

P mr (t) =

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∑ ∫∫ ω ⋅ d F i

i =1 ( Sc . i )

(9.33)

sc .i

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Selection of the Criterion of Optimization

where N is the total number of cutting edges of the cutting simultaneously taking part in the chip-removal process, and d Fsc.i is the vector area element of the surface of cut Sc.i that is created by the i-th cutting edge. 9.2.3 Surface Generation Output When machining a part surface, the rate of increase of the machined surface area reflects the surface generation output. The mean surface generation output P %sg can be analytically expressed by the following formula:

P %sg =

Ssg tΣ

(9.34)

where Ssg designates the machined part surface area. Instantaneous surface generation output P sg is another characteristic of surface machining performance. By definition, the instantaneous surface generation output P sg is

P

sg

=

d Ssg dt

(9.35)

It can be assumed that Ssg = c ⋅ t , where c is a certain constant value. This immediately results in

P

sg

= P %sg = c

(9.36)

This means that the instantaneous surface generation output P sg can be of constant value. In this case, it is equal to the mean surface generation output P %sg . Generally speaking, the instantaneous surface generation output is time dependent. An expression for the computation of P sg (t) in this case can be derived in the following way: For the computation of area Ssg of the surface P patch, a formula Ssg =

∫∫|d F |

( Ssg )

(9.37)

P

is used. The magnitude |d FP | of the vector d FP that determines the element of the surface area is |d FP |=

∂ rP ∂U P

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×

∂ rP ∂VP

⋅ dU P ⋅ dVP = EPGP − FP2 ⋅ dU P ⋅ dVP

(9.38)

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Kinematic Geometry of Surface Machining

This equation yields Ssg =

∫∫

( Ssg )

(9.39)

EPGP − FP2 ⋅ J P ⋅ dt ⋅ dw

Substituting Equation (9.39) into Equation (9.35), a formula [5,6,10,12] w 2 (t )

P sg (t) =

∫

w 1 (t )

(9.40)

EPGP − FP2 ⋅ J P ⋅ dt ⋅ dw

for the computation of the instantaneous surface generation output P %sg can be obtained (see Figure 9.2). 9.2.4 Limit Parameters of the Cutting Tool Motion When the part surface P and the generating surface T of the cutting tool are in point contact with each other, then the feed-rate motion and the side-step motion of the cutting tool must both be performed when machining a sculptured surface on a multi-axis NC machine. The maximal allowed displacements of the( cutting tool are constrained ( by the corresponding limit values [ Ffr ] and [ Fss ]. These limits specify many parameters of the elementary surface cell on the machined part surface. ( ( ( [ F ] [ F ] The limit values and of the cutting tool displacements Ffr and fr ss ( Fss can be computed. For this purpose, the tolerance [ h] on accuracy of surface machining has been taken into consideration. In compliance with [4], it is assumed below that the maximal resultant height of cusps h Σ is within the tolerance [ h] . This means that the inequality h Σ ≤ [ h] is valid, and, moreover, the elementary surface deviation δ P (see Equation 8.37) is not investigated here. 9.2.4.1 Computation of the Limit Feed-Rate Shift Milling cutters are widely used for machining sculptured part surfaces on multi-axis NC machines. The use of milling cutters causes waviness of the machined surface P. It is necessary to keep the waviness height h fr under the corresponding ( portion [ h fr ] of the total tolerance [ h]. The limit feed-rate displacement [ Ffr ] strongly depends on the allowed value of the partial tolerance [ h fr ] . In order to compute the limit feed-rate displacement [ h fr ] , it is necessary to investigate the topography of the machined part surface. In the direction of vector F fr of the feed-rate motion, the cusps profile is shaped in the form of prolate cycloids. The elementary machined surface

In special cases of surface machining, the profile of the machined surface in the direction of the feed-rate motion of the cutting tool can be shaped in the form of pure cycloid and even in the form of curtate cycloid.

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Selection of the Criterion of Optimization

cells on the part surface represent portions of the surface of cut Sc in its consecutive positions relative to the work. However, the linear speed of cutting edges of the rotating cutting tool is incomparably bigger than their linear speed in the feed-rate motion F fr. Therefore, in most practical cases of surface machining, not the surface of cut Sc , but its approximation by the generating surface T of the( cutting tool have to be considered when the limit feed-rate displacement [ Ffr ] is computed. In the vicinity of point K, the nominal sculptured surface P and the generating surface T of the cutting tool are locally approximated by torus surfaces. The approximation allows significant reduction of computations without considerable loss of accuracy of the computations. The approximation is based on the assumption, in compliance with which the following statement is made: The principal radii of curvature R 1. P and R 2. P of the part surface P, and the principal radii of curvature R 1.T and R 2.T of the generating surface T of the cutting tool do not change their values within the elementary surface cell on the machined part surface, and torsion of the surfaces P and T within the elementary surface cell is equal to zero, and thus it can be neglected. The above assumption is reasonable, because in the direction of the vector F fr (as well as in the direction of the vector Fss ), radii of normal curvature of the surfaces P and T are much bigger than the corresponding parameters of the( elementary surface cell on the machined part surface. The inequali( ties Ffr << RP. fr and Ffr << RT . fr are observed in the direction of the feed-rate ( ( motion F fr . (The similar inequalities Fss << RP. ss and Fss << RT . ss are valid for the direction of the side-step shift Fss .) Figure 9.3 reveals that for the computation of the instantaneous value of ( the feed-rate [ Ffr ] per tooth of the cutting tool, the approximate formulae [5,6,10,11] can be used:

RP2 . fr + RT . fr ⋅ (RP. fr + [ h fr ] ⋅ sgn RP. fr ) ( [ Ffr ] ≅ 2 RP. fr arccos (RP. fr + R T . fr ) ⋅ (RP. fr + [ h fr ] ⋅ sgn RP. fr )

(9.41)

( It is assumed in this equation that the inequality Ffr << RP. fr is valid; there( fore, Ffr ≅ AB = Ffr . It is assumed also that [ h fr ] 2 is a reasonably small value, and therefore it can be omitted from further analysis. In Equation (9.41), radii of normal curvature RP. fr and R T . fr of the surfaces P and T in the direction of the feed-rate motion F fr are equal

RP. fr = R T . fr =

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R 1. P R 2. P R 1. P sin ϕ + R 2. P cos 2 ϕ 2

(9.42)

R 1.T R 2.T R 1.T sin (ϕ + µ ) + R 2.T cos 2 (ϕ + µ ) 2

(9.43)

420

Kinematic Geometry of Surface Machining OT RT. fr S[h] P

K

A

[hfr]

T

B F˘fr

θ fr θ fr

RP. fr

OP Figure 9.3 ( Schematic for the computation of the limit feed-rate Ffr .

where, in Equation (9.42) and Equation (9.43), ϕ designates the angle that specifies the direction of the feed-rate vector F fr , and µ denotes the angle of the local relative orientation of surfaces P and T at point K. For the computation of radii of normal curvature RP. fr and R T . fr of the surfaces P and T in the direction of F fr , the following formulae can be helpful: RP. fr = R T . fr =

EPGP GP LP sin ξ + MP EPGP sin 2ξ + EP N P cos 2 ξ 2

(9.44)

ET GT GT LT sin (ξ + µ ) + MT ET GT sin[2 ⋅ (ξ + µ )] + ET NT cos 2 (ξ + µ ) 2

(9.45)

Here the angle ξ is equal to ξ = ϕ + 90. For further analysis, it is important to point out that Equation (9.44) and ( [ F ] Equation (9.45) yield representation of the relationships “ versus ϕ” fr ( ( ( and “ [ Ffr ] versus µ ” in the form [ F ] = [ F ]( ϕ , µ ) . The limit feed-rate dis( fr fr placement [ Ffr ] depends on the direction of the feed-rate motion F fr , as well as upon relative local orientation of the surfaces P and T at point K. 9.2.4.2 Computation of the Limit Side-Step Shift

( In the most practical cases, the feed-step displacement Fss affects the surface generation output P sg the most. Computation of the limit side-step displace( ( [ F ] ment ss is similar to the computation of the limit feed-rate displacement [ Ffr ].

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Selection of the Criterion of Optimization

Without going into the details of the derivations of equations, in this case, we will just rewrite the equation

RP2 . ss + R T . ss ⋅ (RP. ss + [ hss ] ⋅ sgn RP. ss ) ( [ Fss ] ≅ 2 RP. ss arccos (RP. ss + R T . ss ) ⋅ (RP. ss + [ hss ] ⋅ sgn RP. ss )

(9.46)

( for the computation of the limit side-step displacement [ Fss ] . This equation is derived in a way similar to the way Equation (9.41) was derived. This means that the approximate Equation (9.46) is derived under similar assumptions as those made ( for Equation (9.41). It is assumed in Equation (9.46) that the inequality Fss << RP. ss is valid. It is also assumed that [ hP. ss ] 2 is a reasonably small value, and it can be omitted from further analysis. In Equation (9.46), the radii of normal curvature RP. ss and R T . ss of the surfaces P and T in the direction of the side-step shift Fss are RP. ss = R T . ss =

R 1. P R 2. P R 1. P sin ϕ + R 2. P cos 2 ϕ 2

(9.47)

R 1.T R 2.T R 1.T sin 2 (ϕ + µ ) + R 2.T cos 2 (ϕ + µ )

(9.48)

For the computation of radii of normal curvature RP. ss and R T . ss of the surfaces P and T in the direction Fss , the following formulae can be helpful: RP. ss = R T . ss =

EPGP GP LP cos ξ + MP EPGP sin 2ξ + EP N P sin 2 ξ 2

(9.49)

ET GT GT LT cos (ξ + µ ) + MT ET GT sin[2 ⋅ (ξ + µ )] + ET NT sin 2 (ξ + µ ) 2

(9.50)

It is necessary to stress here that Equation (9.49) and Equation (9.50) yield ( ( [ F ] [ F ] representation of the relationships “ versus ϕ ” and “ versus µ ” in ss ss( ( ( the form [ Fss ] = [ Fss ](ϕ , µ ) . The limit side-step displacement [ Fss ] depends on the direction of the side-step shift Fss , as well as on the relative local orientation of the surfaces P and T at point K. 9.2.5 Maximal Instantaneous Productivity of Surface Generation The greatest part surface area coverage is an important output when machining a sculptured surface on a multi-axis NC machine. The optimal conditions of surface generation, those determined on the premises of implementation of

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Kinematic Geometry of Surface Machining

economical criteria of optimization, and those determined on the premises of implementation of the productivity of surface generation as the criterion, are close to each other. This is due to the impact of the high cost of a multi-axis NC machine prevailing over the impact of other factors onto the conditions of optimal sculptured surface machining. Determining the conditions under which the productivity of surface generation is maximal is a critical issue in sculptured surface machining. Use of the DG/K-method of surface generation enables an analytical solution to this challenging engineering problem to be found. The productivity of surface generation P sgmax reaches its maximal rate if and only if the instantaneous productivity of surface generation P sgmax (t) is maximal at every point of contact K of the surfaces P and T. For the computation of conditions of the maximal instantaneous productivity of surface generation, the modified Equation (8.1) can be used: ( ( P sgmax (t) = [ Ffr (t)] ⋅ [ Fss (t)] ⋅ sin θ

(9.51)

Evidently, for the maximal instantaneous productivity P sgmax (t), the vectors of the feed-rate motion F fr and of the side-step shift Fss must be orthogonal to each other; thus, the equality θ = 90 must be valid at every instant of the surface machining. Substitute Equation (9.41) and Equation (9.46) to Equation (9.51). Under the condition θ = 90, Equation (9.51) casts into

P

max sg

 RP2 . fr + R T . fr ⋅ (RP. fr + [ h fr ] ⋅ sgn RP. fr )  (t) ≅ 4 ⋅ RP. fr ⋅ RP. ss ⋅ arccos   (9.52)  (RP. fr + RP. fr ) ⋅ (R T . fr + [ h fr ] ⋅ sgn RP. fr )   RP2 . ss + R T . ss ⋅ (R P. ss + [ hss ] ⋅ sgn RP. ss )  ⋅ arccos    (RP. ss + R T . ss ) ⋅ (R P. ss + [ hss ] ⋅ sgn RP. ss ) 

Despite Equation (9.52) being bulky, the analytical representation of the instantaneous maximal productivity P sgmax (t) of surface generation is incomparably more compact than the analytical description of any economical criteria of optimization. Use of P sgmax (t) for the purpose of synthesis of optimal processes of sculptured surface machining on a multi-axis NC machine returns a result that is practically equivalent to the result when an economical criterion of optimization is used. Due to significantly simpler analytical representation, economical criteria of optimization do not have to be used for the purpose of synthesis of optimal machining operations, but the instantaneous productivity of surface generation P sgmax (t) has to be used instead.

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Selection of the Criterion of Optimization

9.3 Interpretation of the Surface Generation Output as a Function of Conformity Transition from economical criteria of optimization to the instantaneous maximal surface generation output is a good step toward simplification of analytical description of a surface machining process. More opportunities in this concern can be found when functions of conformity of two surfaces (see Chapter 4) are involved in the analysis. As follows from Equation (9.51), in order to increase the instantaneous sur( face generation output P( sgmax (t) , the feed-rate displacement Ffr as well as the side-step displacement Fss must be of maximal rate at every point K of contact of the surfaces P and T. The current configuration of the cutting tool with respect to the part surface being machined, and the instantaneous kinematics of the machining operation are the tools used to control current values of ( ( the parameters Ffr and Fss at every point of contact of the surfaces P and T. Consider a cross-section of a sculptured surface P and of the generating surface T of the cutting tool by a normal plane through point K. The crosssection is along the side-step vector Fss (Figure 9.4). The radius of normal curvature of the part surface P at the point K is equal to RP . Partial tolerance on the accuracy of the machined part surface P is denoted as [ hss ] . The part surface P can be machined with the cutting tools of different designs. Assume that at point K the surface P is machined with the cutting tool having radius of normal curvature of a(certain positive value R(T1.)s . The limit side-step shift in this case is equal to [ Fss(1) ] (Figure 9.4a). That same local portion of the sculptured surface P can be machined with the cutting tool having at point K the radius of normal curvature of a certain value R(T2.)s . The radius of normal curvature R(T2.)s is also positive, but its magnitude exceeds the magnitude of the radius of normal curvature R(T1.)s (2)

OT

(1)

OT

(1)

RT.ss

(2)

(1) F˘ss

T (1) hss

A(1) Rp.ss

K

P

RT.ss B(1)

(2) F˘ss

P

P

hss

OP (a)

A(2)

K Rp.ss

OP (b)

B(2) T (2)

hss

(3) F˘T.ss

A(3) Rp.ss

B(3)

K T (3)

OP

(3)

RT.ss (3)

OT (c)

Figure 9.4 Various rate of conformity of the generating surface T of a cutting tool to a sculptured surface P at a current cutter-contact-point K.

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Kinematic Geometry of Surface Machining

( (Figure 9.4b). The limit side-step shift in this case is equal [ Fss( 2 )(] . Because of the inequality R(T2.)s > R(T1.)s observed, the limit side-step shift [ Fss( 2 ) ] is bigger ( 1 ( ) F than the limit side-step shift ([ ss ] in ( the first case. In the second case (see Figure 9.4b), the inequality [ Fss( 2 ) ] > [ Fss(1) ] is observed. That same local portion of the sculptured surface P can be machined with the cutting tool having at point K the radius of normal curvature of a certain negative value R(T3.)s (Figure 9.4c). The limit side-step shift in this case is equal ( ( 3 ( ) F to [ ss (] . Evidently, the limit side-step shift [ Fss( 3) ] is bigger( than the ( limit( sidestep [ Fss( 2 ) ] in the second case. Therefore, the inequality [ Fss( 3) ] > [ Fss( 2 ) ] > [ Fss(1) ] is observed. The rate of conformity of the generating surface T of the cutting tool to the sculptured surface P at point K is the smallest in the first case (Figure 9.4a), is bigger in the second case (Figure 9.4b), and is the biggest in the third case (Figure 9.4c). This analysis reveals that the width of the limit side-step shift increases when the rate of conformity of the surface T to the surface P increases. A(similar conclusion can be derived with respect to the limit feed-rate shift [ Ffr ] . Because the instantaneous productivity of surface generation P sgmax (t (is a ( function of the limit feed-rate shift [ Ffr ] and of the limit side-step shift [ Fss ] , this statement immediately yields a conclusion: The bigger the rate of conformity of the generating surface T of the cutting tool to the sculptured surface P, the bigger is the instantaneous productivity of surface generation P sgmax (t) . It can be proven analytically that the function P sgmax (t) (see Equation 9.52) is a kind of function of conformity of two smooth, regular surfaces. All functions of conformity have extremes under the same values of the input arguments. Therefore, not just the function P sgmax (t) can be used for solving the problem of synthesis of optimal machining operations, but any corresponding function of conformity of the surfaces P and T can be used instead. For example, the function P sgmax (t) of the instantaneous productivity of surface generation can be substituted with the indicatrix of conformity CnfR ( P / T ) of the surfaces P and T at point K. Such a substitution is reasonable, because the analytical expression for the indicatrix of conformity CnfR ( P / T ) is simpler compared to the analytical expression for the function P sgmax (t) . The indicatrix of conformity CnfR ( P / T ) can be considered as a kind of geometrical analogue of the instantaneous productivity P sgmax (t) of surface generation. For the purposes of practical implementation of the discussed approach for the computation of sculptured surface machining output, a wide application of computers is necessary.

References [1] doCarmo, M.P., Differential Geometry of Curves and Surfaces, Prentice Hall, Englewood Cliffs, NJ, 1976. [2] Favard, J., Course de Geometrie Differentialle Locale, Gauthier-Villars, Paris, 1957.

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[3] Radzevich, S.P., Advanced Technological Processes of Sculptured Surface Machining, VNIITEMR, Moscow, 1988. [4] Radzevich, S.P., Conditions of Proper Sculptured Surface Machining, Computer-Aided Design, 34 (10), 727–740, 2002. [5] Radzevich, S.P., Differential-Geometric Method of Surface Generation, DrSci thesis, Tula, Polytechnic Institute, 1991. [6] Radzevich, S.P., Fundamentals of Surface Generation, Rastan, Kiev, 2001. [7] Radzevich, S.P., Generation of Actual Sculptured Part Surface on Multi-Axis NC Machine. Part 1, Izvestiya VUZov. Mashinostroyeniye, No. 5, pp. 138–142, 1985. [8] Radzevich, S.P., Generation of Actual Sculptured Part Surface on Multi-Axis NC Machine. Part 2, Izvestiya VUZov. Mashinostroyeniye, No. 9, pp. 141–146, 1985. [9] Radzevich, S.P., Methods of Milling of Sculptured Surfaces, VNIITEMR, Moscow, 1989. [10] Radzevich, S.P., Sculptured Surface Machining on Multi-Axis NC Machine, Vishcha Shkola, Kiev, 1991. [11] Radzevich, S.P., and Goodman, E.D., Efficiency of Multi-Axis NC Machining of Sculptured Part Surfaces. In Machining Impossible Shapes, IFIP TC5 International Conference on Sculptured Surface Machining (SSM’98), November 9–11, 1998, Chrysler Technology Center, MI, Kluwer Academic, Boston, MA, 1998, pp. 42–58. [12] Reshetov, D.N., and Portman, V.T., Accuracy of Machine Tools, ASME Press, New York, 1988. [13] Rodin, P.R., Linkin, G.A., and Tatarenko, G.A., Machining of Sculptured Surfaces on NC Machines, Technica, Kiev, 1976.

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10 Synthesis of Optimal Surface Machining Operations Synthesis of optimal surface machining here means a development of a procedure of computation of the optimal parameters of a surface machining operation. Minimum input information is required for solving the challenging problem of synthesis. The derived solution to the problem enables us to find the desired extremum of the criterion of optimization. The solution to the problem of synthesizing an optimal surface machining operation can be solved in three steps. The local surface generation is synthesized on the first step. Then, on the second step, the regional synthesis is performed on the premises of the results of the computations obtained on the first step. Ultimately, the solution to the problem of global synthesis of the optimal surface machining operation is derived on the final third step.

10.1

Synthesis of Optimal Surface Generation: The Local Analysis

Local surface generation is considered just within an elementary surface cell on the machined part surface. From the perspective of local consideration, the synthesis of optimal surface generation is targeted to determine the following: Coordinates of a point within the generation surface T of the cutting tool, local geometry of the surface T at which enables the desired geometry of contact of the surfaces P and T (see Chapter 4). For convenience, this point is designated below as KT . In the event the cutting tool is given, coordinates of the point on the surface T can be computed on the premises of proper correspondence between the local geometry of the surfaces P and T at the cutter-contact (CC)-point K. Otherwise, the computed desired parameters of local geometry of the surface T are used further to design the optimal cutting tool as the R-map of the part surface P (see Chapter 5). Local configuration of the cutting tool with respect to the part surface being machined.

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Optimal direction of the instant motion of the cutting tool relative to the work. Optimal parameters of the instant motions of orientation of the cutting tool (see Chapter 2). For illustrative purposes, the minimal diameter of the indicatrix of conformity of the part surface and of the generating surface of the cutting tool is used below as the criterion of optimization. Chip removal output, productivity of surface generation, or an economical criterion of optimization can also be used for this purpose. 10.1.1

Local Synthesis

For the analytical representation of the indicatrix of conformity Cnf R ( P/T ) of two smooth, regular surfaces P and T, the following expression is derived (see Equation 4.59): Cnf R ( P / T ) ⇒ rcnf =

+

LP GP cos ϕ − M P 2

LT GT cos (ϕ + µ ) − MT 2

EP GP sgn φ2−.P1 EP GP sin 2ϕ + N P EP sin 2 ϕ

ET GT sgn φ2−.T1 ET GT sin 2(ϕ + µ ) + N T ET sin 2 (ϕ + µ ) (10.1)

The current diameter dcnf of the characteristic curve Cnf R ( P/T ) is equal to dcnf = 2 ⋅ rcnf . Further, it is necessary to recall that (1) The fundamental magnitudes of the first and the second order of the part surface P can be expressed in terms of the curvilinear (Gaussian) coordinates U P and VP . This means that the analytical functions EP = EP (U P , VP ), FP = FP (U P , VP ), GP = GP (U P , VP ) and LP = LP (U P , VP ), M P = M P (U P , VP ), N P = N P (U P , VP ) are known (see Chapter 1). (2) The fundamental magnitudes of the first and the second order of the generating surface T of the cutting tool can be expressed in terms of the curvilinear (Gaussian) coordinates UT and VT. This means that analytical expressions for the functions ET = ET (UT , VT ), FT = FT (UT , VT ), GT = GT (UT , VT ) and LT = LT (UT , VT ), MT = MT (UT , VT ), NT = NT (UT , VT ) are also known (see Chapter 1).

Equation (10.1) together with items (1) and (2) yields a generalized form for the diameter dcnf of the indicatrix of conformity Cnf R ( P/T ):

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dcnf = dcnf (U P , VP , UT , VT , µ , ϕ )

(10.2)

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Once a point of interest K P on the sculptured surface P is chosen, then the Gaussian coordinates U P and VP of this point are known. For a point within the generating surface T of the cutting tool being capable of making optimal contact with the part surface P at the selected point of interest, the necessary conditions ∂ dcnf = 0 and ∂UT

∂ dcnf =0 ∂VT

(10.3)

have to be satisfied. In addition to the necessary conditions for the minimum of dcnf (see Equamin tion 10.3), the sufficient conditions for the dcnf ∂2 dcnf ∂UT2 ∂2 dcnf

∂UT ∂VT

∂2 dcnf ∂UT ∂VT > 0 and ∂2 dcnf ∂VT2

∂2 dcnf ∂UT2

>0

(10.4)

must be satisfied as well. The solution to the set of two equations in Equation (10.3) under those conditions (see Equation 10.4) returns Gaussian coordinates UTopt and VTopt of the optimal point KT within the generating surface T of the cutting tool. For the highest possible productivity of surface generation, it is necessary that at the point of interest K P , the generation surface T of the cutting tool contacts the part surfaces P with its optimal point KT . Once the points K P and KT are snapped together, their designation is further substituted with K (that is, K P ≡ KT ≡ K ). It is assumed here and below that all necessary and sufficient conditions of proper part surface generation [9,10,11,16] are satisfied (see Chapter 7). The computed optimal parameters UTopt and VTopt specify location of the point KT within the generation surface T of the cutting tool. The point contact of surfaces imposes strong restrictions on feasible motions of the surface T relative to the surface P. The only motion is allowed for the surfaces P and T while they are in contact at the fixed point K. Depending on the parameters of the actual surfaces P and T at point K, the cutting tool surface T is allowed either to rotate about the common unit normal vector n P, or just to turn through a certain angle about this normal unit vector. No other relative motions are feasible for the surfaces P and T in the case under consideration. Definitely, the local productivity of surface generation under various angular positions of the surface T with respect to the part surface P is different. Thus, optimal configuration of the generating surface T with respect to the part surface P exists, and moreover, it can be computed. For this purpose, the optimal local relative orientation of the surfaces P and T is necessary to be computed.

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Kinematic Geometry of Surface Machining

Current configuration of the surfaces P and T at the CC-point is specified by the angle m of their local relative orientation. In order to get the local relative orientation optimal, it is necessary to compute the optimal value µ opt of the angle of the local relative orientation of surfaces P and T. Equation (10.2) is helpful for solving this particular problem. The angle µ opt can be computed as a solution to the equation ∂ dcnf =0 ∂µ

(10.5)

The derived solution to Equation (10.5) must satisfy the sufficient condition ∂2 dcnf

∂µ 2

>0

(10.6)

min for the minimum of dcnf . The solution to Equation (10.5) under the condition (see Equation 10.6) specifies the optimal local configuration of the cutting tool with respect to the part surface being machined. In compliance with the derived solution, the cutting tool must be turned about the unit normal vector n P through a certain angle to its optimal configuration relative to the surface P which is specified by the computed angle µ opt . Once the optimal configuration of the generating surface of the cutting tool with respect to the sculptured part surface is determined, then optimal parameters of the instant kinematics of local surface generation can also be determined. The direction along which the current diameter dcnf is measured makes a certain angle j with the first principal direction t 1.P of the part surface P. For min the minimal diameter dcnf , the equality

∂ dcnf =0 ∂ϕ

(10.7)

is satisfied. Here, for Equation (10.7), the sufficient condition for the maximin mum of the diameter dcnf ∂2 dcnf

∂ϕ 2

>0

(10.8)

is also observed. The solution to Equation (10.7) under condition (see Equation 10.8) returns the optimal value ϕ opt of the angle j. Vector F fr of the cutting tool feed-rate motion is at the angle ξ opt = ϕ opt + 90° . Ultimately, the computed angle ξ opt specifies the optimal direction of the instant motion of the cutting tool relative to the work. The vector F fr also defines the direction at which the point K travels over the generating surface of the cutting tool.

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After the vector Ffr of the cutting tool feed-rate motion is determined, the problem of synthesis of local surface generation is over. As follows from the above consideration, when solving the problem of synthesis of local surface generation, computation of the first and of the second derivatives of the indicatrix of conformity Cnf R ( P/T ) (see Equation 10.2) is necessary. This trivial mathematical operation does not cause any inconveniences when spline functions are used for the local approximation of the sculptured surface P. Computation of the first and of the second derivatives of spline-functions can be performed easily. Two examples in Figure 10.1 illustrate the results of solutions to the problem of synthesis of local surface generation. When the saddle-like local patch of the sculptured surface P is machined with the convex portion of the generating surface T of the cutting tool (Figure 10.1a), a solution to the problem

Vopt

CnfR (P/T ) yP C2.P

nP

opt

K C1.T T

–wl

(b)

nP

P

Vopt

Dup (T ) K

T

+Swl –wl

nT (c)

Ind (P)

Ind (P)

(a)

Dup(P)

C2.T

Dcnf

Dup(P)

nT

C1.P xp

K

µ +Swl

min

dcnf

ξ

Ind (T )

P

Dup (T)

dcnf

yP

C2.P

opt

C1.P

ξ K

Dup(P) Ind (T )

xp

µ

CnfR (P/T ) C1.T

C2.T

Ind (P)

(d)

Figure 10.1 Two examples of the solutions to the problem of synthesis of local surface generation.

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of synthesis of local surface generation is as shown in Figure 10.1b. Here, the min minimal diameter dcnf of the indicatrix of conformity Cnf R ( P/T ) is at the opt optimal angle ϕ to the first principal cross-section C 1. P of the surface P at point K. The optimal direction of the vector V opt of the relative motion of surfaces P and T makes the angle ξ opt = ϕ opt + 90° with the first principal cross-section C 1. P . When the surface being machined is a sculptured surface, then the vector V opt is identical to the vector F opt of the optimal feed-rate fr motion of the cutting tool (V opt ≡ F opt . In other cases, the vector V opt yields fr ) different interpretations. Similarly, when the convex local patch of the sculptured surface P is machined with the convex portion of the generating surface T of the cutting tool (Figure 10.1c), the solution to the problem of synthesis of local surface min generation is as shown in Figure 10.1d. Again, the minimal diameter dcnf opt of the indicatrix of conformity Cnf R ( P/T ) is at the optimal angle ϕ to the first principal cross-section C 1. P of the surface P at point K. The optimal direction of the vector V opt of the relative motion of surfaces P and T makes the angle ξ opt = ϕ opt + 90° with the first principal cross-section C 1. P . When solving a problem of synthesis of local surface generation, some peculiarities could be observed. 10.1.2

Indefiniteness

Indefiniteness of the kind 00 , or of the kind ∞ ∞ , or finally, of the kind 0⋅∞ could occur when optimal values of the local surface generation are computing. To overcome this particular problem, substitution of the equation of the indicatrix of conformity Cnf R ( P/T ) with the equation of the corresponding indicatrix of conformity Cnf k ( P/T ) is usually helpful. 10.1.3

A Possibility of Alternative Optimal Configurations of the Cutting Tool

Particular cases of contact of the surfaces P and T could be observed when solving the problem of synthesis of local surface generation. One such problem is due to the possibility of two alternative optimal configurations of the cutting tool which actually are equivalent to each other. An example of two equivalent optimal configurations of the cutting tool is depicted in Figure 10.2. The example in Figure 10.2 relates to the generation of the saddle-like local portion of the sculptured surface P with the convex portion of the generating surface T of the cutting tool. The minimal radius of the indicatrix of conformity Cnf R ( P/T ) of the surface P and the cutting tool surface min in its first optimal configuration T is equal to zero (rcnf = 0). The first principal plane-section C1.T of the cutting tool surface T is at the optimal angle µopt of the local relative orientation of surfaces P and T . Vector Vopt of the optimal direction of motion of the generating surface T of the cutting tool relative to the part surface P is orthogonal to the direction along

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∆ξopt V** opt

V*opt π 2 yP

CnfR (P/T *)

π 2

Dup (T **)

Dup (P )

C1.P

t**max cnf

C *2.T CnfR (P/T * * )

µ*opt t2.P

max t*cnf

min r*cnf =0

Dup(P ) t1.P

C ** 2.T min r** cnf = 0

CnfR (P/T * )

K C*1.T

C*1*.T Dup (T* )

xp

C2.P

µ** opt CnfR (P/T * * )

Figure 10.2 An example of alternative configurations of the cutting tool.

min which the minimal radius rcnf of the indicatrix of conformity Cnf R ( P/T ) is measured. This direction is the direction of maximal rate of conformity of the generating surface of the cutting tool to the sculptured part surface. In Figure 10.2, it is labeled as t cnfmax. That same saddle-like local portion of the sculptured surface P can also be generated with that same convex portion of the generating surface T of the cutting tool under a different configuration of the cutting tool. The minimal radius of the indicatrix of conformity Cnf R ( P/T ) of the surface P and the cutting tool surface T in its second optimal configuration is also equal min to zero (rcnf = 0). The first principal plane-section C1.T of the cutting tool of the local relative orientation of sursurface T is at the optimal angle µopt faces P and T . Vector Vopt of the optimal direction of motion of the generating surface T of the cutting tool relative to the part surface P is orthogonal min to the direction along which the minimal radius rcnf of the indicatrix of conformity Cnf R ( P/T ) is measured. This direction is the direction of maximal rate of conformity of the generating surface of the cutting tool to the sculptured part surface. In Figure 10.2, it is labeled t max . cnf Ultimately, Figure 10.2 reveals that two optimal directions of the cutting tool relative motion are feasible. The first optimal direction Vopt is specified by the = ϕ ξ ° angle opt opt + 90 . The second optimal direction Vopt is specified by the

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= ϕ + 90o , ϕ angle ξopt . Due to lack of space, the angles ξopt opt opt and ξopt , ϕ opt are not depicted in Figure 10.2. The directions Vopt and Vopt are at a certain angle ∆ξopt to each other. However, no problem arises in this concern. One of two optimal directions is selected automatically depending upon the cutting tool conand Vopt Vopt figuration at the previous points of contact with the part surface P. Correlation between directions of the cutting tool motion along the tool-path is always observed. A situation that is shown in Figure 10.2 requires special attention when the is reasonably small angle ∆ξopt between the two directions Vopt and Vopt and its magnitude is comparable with the deviation δϕ opt of the computation of the angle ϕ opt . If the deviation δϕ opt exceeds the angle ∆ξopt , say when the inequality δϕ opt ≥ ∆ξopt is observed, then the numerical control (NC) system of the machine tool generates two contradictory commands to move the cutting tool in two different directions. Two ways to follow are possible for avoidance of such a scenario:

To replace the actual cutting tool with the cutting tool having appropriate parameters of the generating surface T. In this way, either the angle ∆ξopt can be increased to the value ∆ξopt > δϕ opt , or just one optimal direction Vopt of the cutting tool relative motion could actually exist. A multi-axis NC machine having higher resolution and capable of higher accuracy when performing the computations is recommended to be implemented. 10.1.4

Cases of Multiple Points of Contact of the Surfaces P and T

The scenario when the generating surface T of the cutting tool is contacting the part surface P at two or more points is actually not practical, but it could take place. Consider surfaces P and T that make contact at two distinct points K1 and K 2 . In order to make the machining operation feasible, the equation of contact n P ⋅ VΣ = 0 (see Equation 2.12) must be satisfied at both points K1 and K 2 . The equation of contact must be satisfied at both, at the point K1 (i.e., n P.1 ⋅ VΣ .1 = 0), as well as at the point K 2 , (i.e., n P.2 ⋅ VΣ .2 = 0 ). Here, n P.1 and n P.2 denote the unit normal vectors to the part surface P at the points K1 and K 2 , respectively (actually, the points K1 and K 2 could be within the different portions P1 and P2 of the composite part surface). This requirement imposes a strong restriction on the allowed direction of the relative motion VΣ( 2) of the cutting tool. When two points of contact K1 and K 2 of the surfaces P and T are observed, then

VΣ( 2) =|VΣ( 2) |⋅n P .1 × n P .2

is the only feasible direction for VΣ( 2).

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(10.9)

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If the number of points of contact exceeds two points of contact, then generation of the sculptured surface P under such a scenario is feasible if and only if the unit normal vector at the third and at all other i-th points of contact are coplanar to the unit normal vectors n P.1 and n P.2. Two important conclusions can be drawn from the solution to the problem of synthesis of the optimal local surface generation: For the computation of optimal directions of the tool-paths on the sculptured surface, the geometry of both the sculptured part surface and the generating surface of the cutting tool must be known. It is impossible to derive the optimal tool-paths only on the premises of the geometry of the part surface P. The last is possible only in particular cases of sculptured surface machining (e.g., when for machining, either the ball-nose or the flat-end milling cutters are used). In particular cases like that just mentioned, the optimal toolpaths align with the lines of curvature on the sculptured surface. Optimal tool-paths do not intersect each other: generally, they are not a kind of transversal curve on the sculptured part surface.

10.2 Synthesis of Optimal Surface Generation: The Regional Analysis The derived solution to the problem of synthesis of optimal local surface generation (see Section 10.1) returns the optimal parameters of the surface generation process that is valid just within a surface cell on the machined part surface. This solution is the key input for the synthesis of optimal regional surface generation. The synthesis of the optimal regional surface generation is a targeting solution to the problem of the most efficient generation along a toolpath on a sculptured surface. Once the problem of the synthesis of optimal local surface generation is solved, then the problem of synthesis of optimal regional surface generation could be under consideration when the local solution is obtained at every point of the tool-path. Consider generation of a sculptured surface P with the generating surface T of the form-cutting tool (Figure 10.3). At a given CC-point K within the part surfaces P, Gaussian coordinates UT and VT of the point of contact within the generating surface T of the cutting tool are of optimal values UTopt and VTopt . The optimal angle µ opt of the local relative orientation of the surfaces P and T is measured between the principal directions t 1.P and t 1.T . The min direction of minimal value of the diameter dcnf of the characteristic curve opt Cnf R ( P/T ) is at the angle ϕ with respect to the principal direction t 1.P. The optimal direction of traveling of the cutting tool over the sculptured surface P is specified by the angle ξ opt = ϕ opt + 90° .

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Kinematic Geometry of Surface Machining

436 ZP

XP

zp

YP Rs (P Rs (NC

P)

T

nP

P

C2.P

K

ZNC

ξ opt

t2.P

YNC

ZT

π 2

Q

t1.T t2.T

K

t2.P

Rs (T

yp

C1.T K)

µopt

C1.P

t1.P

XT

xp

π 2

t1.P

opt

T)

V opt

ξ opt

xp

nT XNC

Rs (NC

V opt

K)

opt

µopt

yp

YT Figure 10.3 Configuration and relative motion of the surfaces P and T in the Cartesian coordinate system X NCYNCZNC associated with the multi-axis numerical control machine. (From Radzevich, S.P., Mathematical and Computer Modeling, 43 (3–4), 222–243, 2006. With permission.)

The local Cartesian coordinate system xP y P zP is associated with part surface P. The origin of the local coordinate system is at the CC-point K. Darboux’s trihedron can be used for construction of the local coordinate system xP y P zP. In the local coordinate system xP y P zP , vector V opt of the optimal direction of the cutter travel relative to the surface P can be expressed in matrix form:

V opt

 V opt   V opt =   

sin ξ opt   cos ξ opt   0    1

(10.10)

A Cartesian coordinate system X NC YNC ZNC is associated with the multiaxis NC machine. The operator Rs(K a NC) of the resultant coordinate system transformation, say for the transition from the local coordinate system xP y P zP to the coordinate system X NC YNC ZNC , can be composed as it is disclosed in Chapter 3. Use of the operator Rs(K a NC) allows representation

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of the vector V opt in the coordinate system X NC YNC ZNC:

opt VNC = Rs (K a NC) ⋅ V opt

 ∂ x P  ∂ xP (t) sin ϕ opt (t)  (t) cos ϕ opt (t) −  ∂VP   ∂U P  ∂ y   ∂ yP P opt opt  ( ) cos ( ) − ( t ) sin ϕ ( t ) ϕ t t   =  ∂U P ∂VP    ∂ zP   ∂ zP opt opt   ∂U (t) cos ϕ (t) − ∂V (t) sin ϕ (t)  P P     1

(10.11)

Equation (10.11) can be interpreted as the differential form of the solution to the problem of synthesis of the optimal regional surface generation. A closed-form solution to the problem of the optimal tool-paths can be derived on the premises of Equation (10.11). A surface tool-path is a time-parameterized surface trajectory r(u(t), v(t)) = r(t), which corresponds to a certain curve u(t) , v(t) . Thus, the closed-form solution to the problem of the optimal tool-paths can be represented in the following form [4]:  t2  ∂ X   ∂ XP P opt opt   ⋅ ( )cos ϕ ( ) − ( ) s in ϕ ( t ) dt t t t  ∂VP   ∂U P   t1   t2    ∂ Y ∂ Y P P opt opt   ⋅ ( t )cos ϕ ( t ) − ( t )sin ϕ ( t ) dt   rtpopt  (t) =   ∂U P ∂ V P    t1   t2    ∂ ZP (t)cos ϕ opt (t) − ∂ ZP (t)sin ϕ opt (t) ⋅ dt     ∂U P  ∂VP  t1    1  

∫

∫

(10.12)

∫

which directly follows from Equation (10.11). Another form of representation of Equation (10.12) is known from the literature [10,11,16]. At a current CC-point, the maximal allowed speed of travel of the cutting tool along the optimal tool-path (see ( Equation 10.12) is restricted by the limit value of the feed rate per tooth [ Ffr ] of the cutting tool (see Equation 9.41). From the perspective of geometrical and kinematical consideration, the maximal speed of cutting tool travel is bigger when the concave portion of the surface P is machining, and it is smaller when machining a convex portion of

The author would like to mention here that many proficient researchers came up with the decision that no closed-form solution to the problem of optimal tool-paths is feasible.

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the surface P. However, when the chip-removal output is included in the consideration, the maximal speed of the cutting tool travel is smaller when the concave portion of the surface P is machining, and it is bigger when machining a convex portion of the surface P. For the orthogonally parameterized surface P, Equation (10.12) for [r tpopt ] reduces to

r tpopt  ⇒ U P = −  

∫

VP .2

VP .1

cot[ϕ opt (VP )] dVP

(10.13)

In many particular cases of sculptured surface machining, both Equation (10.12) and Equation (10.13) can be integrated analytically. In some particular cases of sculptured surface generation, the equation for the optimal tool-paths simplifies to the differential equation

E dU P + FP dVP rtpopt  ⇒ P   LP dU P + M P dVP

FP dU P + GP dVP =0 M P dU P + N P dVP

(10.14)

Equation (10.14) for the optimal tool-paths is applicable, for instance, when machining a sculptured surface P either with a ball-nose milling cutter, or with a flat-end milling cutter, and so forth. Under such a scenario, the angle m of the local relative orientation of surfaces P and T vanishes. It is getting indefinite: no principal directions can be identified on a sphere or on the plane surface. Therefore, the optimal tool-paths align with lines of curvature on the surface P (see Equation 10.14). When machining a part surface, the coordinate system XT YT ZT associated with the cutting tool is rotating like a rigid body. This rotation is performing about a certain instant axis of rotation. The angular velocity of the rotation of the coordinate system XT YT ZT is equal to |W|= ktp2 + τ tp2 .

The axis of instant rotation aligns with Darboux’s vector W = ktp t tp + τ tp btp (here ktp and τ tp denote curvature and torsion of the trajectory of the CCpoint, and t tp and btp are the unit tangent vector and the binormal vector to the trajectory of the CC-point at a current point K). Darboux’s vector is located in the rectifying plane to the trajectory of the CC-point. It can be expressed in terms of the normal vector ntp and of the tangent vector t tp to the trajectory of the CC-point:

W = ktp2 + τ tp2 (t tp cos θ + ntp sin θ )

(10.15)

where q is the angle that makes Darboux’s vector W and the tangent vector t tp to the trajectory at the CC-point.

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It is instructive to note that velocity |W| is a function of full curvature of the trajectory of the CC-point.

10.3 Synthesis of Optimal Surface Generation: The Global Analysis Synthesis of optimal global surface generation is the final subproblem of the general problem of synthesis of optimal surface generation. The solution to the problem of optimal global surface generation is based much on the derived solutions to the problems of optimal local and of optimal regional surface generation. Minimal machining time is the major goal of the problem of synthesis of optimal global surface generation. In order to solve the problem under consideration, it is necessary to do the following: Minimize interference of the neighboring tool-paths of the cutting tool over the part surface being machined. Determine the optimal parameters of placing the cutting tool into contact with the part surface, and of its departing from the contact. This subproblem is referred to as the boundary problem of surface generation. Determine the location of the optimal starting point of the surface machining. 10.3.1

Minimization of Partial Interference of the Neighboring Tool-Paths

The actual machined part surface is represented as a set of tool-paths that cover the nominal surface P. At a (current surface point, the width of the tool-path is equal to the side-step Fss computed at that same CC-point (see Equation 9.46). The tool-path width varies along the trajectory of the CC-point over the sculptured surface, as well as across the trajectory. Because of this, neighboring tool-paths partially interfere with each other. Ultimately, some portions of the part surface P are double-covered by the tool-paths. Partial interference of the neighboring tool-paths causes reduction of the surface generation output. For the synthesis of optimal surface generation operation, the interference of the neighboring tool-paths must be minimized. The trajectory of the CC-point over the sculptured surface is a three-dimensional curve. For the analysis below, it is convenient to operate with the natural parameterization of the trajectory: ltr = ltr (rtr , τ tr ). Here, length ltr of the arc of the trajectory is measured from a certain point within the trajectory. The length ltr is expressed in terms of the radius of curvature rtr at a current trajectory point, and of torsion τ tr of the trajectory at that same point.

© 2008 by Taylor & Francis Group, LLC

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At the current CC-point, the tool-path width can be expressed in terms of the length of the ith trajectory: ( ( Fss(i ) = Fss(i ) ltr(i )  (10.16) During the infinitesimal time dt, the cutting tool travels along the i-th trajectory at a distance dlth. A sculptured surface portion ( dS tr(i ) = Fss(i ) ltr(i )  ⋅ dt (10.17) is generated in this motion of the cutting tool. The area of a single i-th tool-path is (

S tr(i ) =

∫F

(i ) ss

ltr(i )  ⋅ dt

(10.18)

(i ) [ ltr ]

The total area of all the tool-paths can be computed from n

S tr =

∑ i =1

n

S tr(i ) =

∑∫F (

(i ) ss

i =1 [ l( i ) ] tr

ltr(i )  ⋅ dt

(10.19)

where n denotes the total number of tool-paths necessary to cover the entire part surface. Due to partial interference of the neighboring tool-paths, the total area S tr (Equation 10.19) exceeds the area S sg of the actual generated part surface P (i.e., the inequality S tr > S sg is always observed). The rate of interference of the neighboring tool-paths is evaluated by the coefficient of interference K int : n

K int =

S tr − S sg = S sg

∑∫F (

(i ) ss

i =1 [ l( i ) ] tr

ltr(i )  ⋅ dt − S sg

S sg

(10.20)

The coefficient of interference K int is a function of design parameters of the part surface being machined, of design parameters of the generating surface of the cutting tool, and of parameters of kinematics of the surface machining operation. So, it can be minimized (K int → min) using for this purpose conventional methods of minimization of analytical functions. 10.3.2 Solution to the Boundary Problem The generation of a part surface within the area next to the surface border differs from that when machining of a boundless surface is investigated. The shape and parameters of the surface contour affect the efficiency of the surface generation process. Prior to searching for a solution to the boundary problem, it is necessary to determine the part surface region within which the boundary effect is significant.

© 2008 by Taylor & Francis Group, LLC

Synthesis of Optimal Surface Machining Operations P

Trajectory of the CC-Point

Q Limiting contour

1

Ffr 2 K2

K1

b

K3

L*

C 3

441

D

B

A

a

c

Part boundary

Wi

Figure 10.4 Boundary effect when machining a sculptured surface P.

Consider a sculpture surface P having tolerance [ h] on the accuracy of the surface machining. The surface of tolerance S [ h ] is at the distance [ h] from the surface P. The actual machined part surface is located within the interior between the surfaces P and S [ h ]. The generating surface T of the cutting tool is contacting the nominal part surface P at a certain point K1 (Figure 10.4). The surface T intersects the surface of tolerance S [ h ]. The line 1 of intersection of the surfaces S [ h ] and T is a kind of closed ellipse-like curve. The curve has no common points with the part surface boundary. Therefore, no boundary effect is observed in this location of the cutting tool. At the point K 2 within the trajectory of the CC-point, the cutting tool surface T also intersects the surface of tolerance S [ h ]. The line 2 of intersection is also a kind of closed, ellipse-like curve. However, in this location of the cutting tool, the curve 2 makes tangency with the part surface boundary at the point A. This indicates that starting at the point K 2, the boundary affects the efficiency of surface generation. The impact of the boundary is getting stronger toward point D on the part surface boundary curve. At a certain point K 3 of the trajectory of CC-point, the line of intersection 3 of the surfaces S[ h ] and T is not a closed line. It intersects the part surface boundary at the points B and C. Departure of the cutting tool from the interaction with the surface P is over when the limit point L on the biggest diameter of the curve 3 reaches the part surface boundary curve at the point L . The point K 2 is constructed for the point A of the part surface boundary curve. For every point A i of the part surface boundary curve, a point K i that is similar to the point K 2 can be constructed. All the points K i specify the limit contour. It is necessary to take into account the impact of the boundary effect for those arcs of CC-point trajectories, which are located between the part surface boundary curve and the limit contour.

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Kinematic Geometry of Surface Machining

Width bc of the part surface boundary affected region is not constant. Width Wi at a current point c is measured along the perpendicular to the part surface boundary curve. The point c is the endpoint ( of the arc ac of the trajectory of the CC-point. The feed rate per tooth Ffr of the cutting tool could be either constant within the arc ac of the trajectory of the CC-point, or it can vary in compliance with the current width of the tool-path. Particular features of impact of the boundary effect could be observed: When the stock thickness is bigger, this causes longer trajectories of the cutting tool to enter in contact with the part surface. A bigger tolerance [ h] on accuracy of the machined part surface results in longer trajectories of the cutting tool to exit from contact with the part surface. The smaller the area of the nominal part surface P, the more significant is the impact of the boundary effect on the efficiency of the machining of the whole part surface. The impact of the boundary effect could be more significant when machining long surfaces. 10.3.3 Optimal Location of the Starting Point The location of the point from which machining of the sculptured surface begins also affects the resultant surface generation output. One can conclude from this that the optimal location of the starting point exists, and it can be determined. Consider the machining of a sculptured part surface on a multi-axis NC machine (Figure 10.5). The boundary of the sculptured surface P is of arbitrary shape. The region of the boundary effect is shown as the shadowed strip along the boundary curve. The surface P can be covered by the infinite number of optimal trajectories of the CC-point. The equation of the optimal trajectories of the CC-point is the output of the subproblem of synthesis of optimal regional surface generation. Two of the infinite number of trajectories are tangent to the surface boundary curve of the sculptures at the points c1 and c2 (see Figure 10.5). Another two optimal trajectories of the CC-point are at the distance 0.5 Fss from the points c1 and c2 inward from the bounded portion of the sculptured surface P. These two last trajectories of the CC-point can be used as the trajectories for the actual tool-paths when machining the sculptured surface P. They intersect the sculptured surface boundary curve at the points a1 , a2 and a3 , a4, (respectively. The rest of the trajectories of CC-point are at the limit side-step [ Fss ] from each other (see Equation 9.46). It is important to point out here that length f ss of the arc through the points c1 and c2 usually is not ( divisible on the limit side-step [ Fss ] . However, no big problem arises in this concern, and it can be neglected at this point.

© 2008 by Taylor & Francis Group, LLC

Synthesis of Optimal Surface Machining Operations P

0, 5Fss A3

a3

b A2 2

P

A4 Fss

fss

A2 a2

c1

c1

a2

len Fss

(b)

a1

Wps

0, 5Fss RT

(a)

a1

lex

Fss A1

b1 A1

0, 5Fss

Fss

a4

Region of the boundary effect

443

[h]

t

S[h]

Pn

t

Wps

[h] S[h]

Pn

b1

RT

T A1

b2 A2

RP RP (c)

(d)

Figure 10.5 Location of the optimal starting point for sculptured surface machining.

Then, outside the bounded portion of the sculptured surface P, two points A1 and A2 are selected within the trajectory through the points a1 and a2 . The point A1 is at the distance len from the boundary curve of the surface P. The len distance is sufficient for entering the cutting tool in contact with the part surface. Another point A2 is at the distance lex from the boundary curve of the surface P. The lex distance is sufficient for exiting the cutting tool from contact with the part surface. Similarly, two more points A3 and A4 are selected within the trajectory through the points a3 and a4 . The machining of the surface P begins at the point A1. In most cases of sculptured surface machining, the inequality len > lex is observed. Therefore, if one wishes to begin the surface machining not from the point A1, but from the opposite end of the trajectory a1 a2, another four points A1, A2 , A3 , and A4 (the points A1, A2 , A3 , and A4 are not shown in Figure 10.5) can be constructed instead. The points A1, A2 , A3 , and A4 are constructed in the way the points A1, A2 , A3 , and A4 are constructed. The only difference here is that for all the points A1, A2 , A3 , and A4 , the arc segments of the length len are substituted with the arc segments of the length lex, and vice versa. When locating at the point A1 of the trajectory a1 a2, the generating surface T is contacting the workpiece surface Wps at the point b1. The workpiece surface Wps is an offset surface at the distance t to the part surface P. Here t designates the thickness of the stock to be removed. In the general case, a function t = t(U P , VP ) is observed (see Chapter 9).

© 2008 by Taylor & Francis Group, LLC

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Kinematic Geometry of Surface Machining

When locating at the point A2 of the trajectory a1 a2, the generating surface T is contacting the surface of tolerance S [ h ] at a point b2 . The surface of tolerance S [ h ] is an offset surface at the distance [ h] to the part surface P. Here [ h] denotes the tolerance on accuracy of the machined part surface P. In the general case, a function [ h] = [ h](U P , VP ) is observed (see Chapter 9). The distance len that is necessary for entering the cutting tool in contact with the sculptured part surface can be expressed in terms of thickness of the stock t, radius of normal curvature R T of the generating surface T of the cutting tool (here R T is measured in the direction tangent to the trajectory a1 a2 at the point A1 ), and radius of curvature Rtr of the trajectory at the point A1 through the points a1 and a2. The distance lex that is necessary for exiting the cutting tool from contact with the sculptured part surface can be expressed in terms of tolerance [ h] on accuracy of the part surface, radius of normal curvature R T of the generating surface T of the cutting tool (here R T is measured in the direction tangent to the trajectory a1 a2 at the point A2 ), and radius of curvature Rtr of the trajectory at the point A2 through the points a1 and a2 . Ultimately, either one of four points A1, A2 , A3 , A4 or one of four points A1, A2 , A3 , A4 is selected as the starting point of the sculptured surface machining. Practically, both sets of points are equivalent. Computation of coordinates of the chosen point is a trivial mathematical procedure. The interested reader may wish to exercise him- or herself in doing this. Prior to beginning the machining of the given part surface, a contact point within the generating surface T of the cutting tool is computed. This is the point local geometry of the surface T which corresponds to the local geometry of the surface P at the point a1. Then, the cutting tool contact point is snapped with the computed starting point, say with the point A1. Satisfaction of the conditions of proper part surface generation (see Chapter 7) is required. Much room for investigation is left in the synthesis of optimal global part surface generation.

10.4 Rational Reparameterization of the Part Surface The solution to the problem of optimal regional synthesis of part surface generation returns a set of optimal trajectories of the CC-point on the surface P. For the purposes of development of a computer program for sculptured surface machining on a multi-axis NC machine, it is convenient to use the computed optimal trajectories as a set of curvilinear coordinates on the sculptured surface P. For this purpose, it is necessary to change the initial parameterization of the surface P with a new parameterization — with the parameterization by means of the optimal trajectories of the CC-point on the surface P. For the reparameterization of the surface P, known methods [1,7,11,13] and others can be used.

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Synthesis of Optimal Surface Machining Operations 10.4.1

445

Transformation of Parameters

Consider a part surface P that is given by vector equation r P = r P (U P , VP ). It is assumed that the surface P is a smooth, regular surface. The required additional restrictions that must be imposed will be introduced later. The initial (UP,VP)–parameterization of the part surface can be transformed to another parameterization. The new parameterization of the surface P is denoted as (U P , VP )−parameterization. In the new parameters, the initial equation of the surface P is substituted with the equivalent equation r P = r P (U P , VP ). The new parameters U P and VP can be expressed in terms of original parameters U P and VP :

U P = U P (U P , VP ) VP = VP (U P , VP )

(10.21)

One of the curvilinear parameters in Equation (10.21) (for example, U P − coordinate curve) can be congruent to the optimal trajectories of the CCpoint (see Equation 10.12), while another curvilinear parameter VP can be directed orthogonally to the first one. Equations for the derivatives in the new parameters are as follows: ∂ rP ∂U P

∂ rP ∂V

P

=

∂ r P ∂U P ∂ r P ∂VP ⋅ + ⋅ ∂U P ∂U P ∂VP ∂U P

(10.22)

=

∂ r P ∂U P ∂ r P ∂VP ⋅ + ⋅ ∂U P ∂VP ∂VP ∂VP

(10.23)

Then, the cross-product of tangents is equal:

∂ rP ∂ rP U P × = ∂U P ∂VP U P

VP   ∂ rP ∂ rP  ⋅ × VP   ∂U P ∂VP 

(10.24)

In order to satisfy the restriction ∂ rP ∂ rP × ≠0 ∂U P ∂VP

(10.25)

for the part surface P expressed in the new parameters, the Jacobian matrix of transformation J must not be equal to zero:

U P J= U P

© 2008 by Taylor & Francis Group, LLC

∂U P VP  ∂U P =  VP  ∂VP ∂U P

∂U P ∂VP ≠0 ∂VP ∂VP

(10.26)

Kinematic Geometry of Surface Machining

446

The matrix [DP ] of the first derivatives of the surface P in its original parameterization is  ∂r ∂ rP  [ DP ] =  P ;  U VP  ∂ ∂  P

(10.27)

The similar matrix [DP ] can be composed for the new parameterization of the surface P:

 ∂r ∂ rP  [DP ] =  P ;  U VP  ∂ ∂  P

(10.28)

[DP ] = [DP ] ⋅ J

(10.29)

The following equality

is true. The matrices [DP ] and [DP ] enable computation of the first fundamental matrix [Φ 1 . P ] in the new parameters of the surface P:

[Φ 1 . P ] = [DP ]T ⋅ [DP ] = J T ⋅ [DP ]T ⋅ [DP ] ⋅ J = J T ⋅ [Φ 1. P ] ⋅ J

(10.30)

Similarly, the equation for the second fundamental matrix [Φ 2. P ] in the new parameters of the surface P: [Φ 2. P ] = J T ⋅ [Φ 2. P ] ⋅ J

(10.31)

can be derived. The discriminant of the first order H P is computed from H P = J ⋅ H P

(10.32)

The similar is true with respect to the discriminant of the second order TP :

TP = J ⋅ TP

The rest of the major parameters of geometry in the new parameterization of the part surface P can be computed on the premises of the above-discussed equations, particularly on the premises of Equation (10.30) and Equation (10.31). 10.4.2 Transformation of Parameters in Connection with the Surface Boundary Contour Boundary contour C of the sculptured surface P is made up of four smooth arcs C11 , C12 , C21, and C22 as an example (Figure 10.6). The plane P0 serves as

© 2008 by Taylor & Francis Group, LLC

Synthesis of Optimal Surface Machining Operations P

m

β1 Ω

P0

M

r02

α2 α2 = const

P0

Ωf α1 = const

0

M0

δ2

r01

0

C 12

0

e1f

Mf rf

e2f

rf

r1 C 22

n0

δ1

C22

C12 H(ai )

0 C 11

r

r2 β2

P

C21 Ω0

447

C 21

α1 = α1k α1

α2 = α2k

Figure 10.6 Transformation of parameters in connection with the boundary contour of the sculptured surface.

a coordinate plane. Consider a certain region W within the part surface P. The region W0 is the projection of W onto the coordinate plane P0 . The distance H of P to P0 is measured in the direction of the unit normal vector n 0 to P0 . In a certain reference system, the region W0 within the plane P0 is represented by r f = r f (α i )

(10.33)

The projection W0 is not a canonical region: the contour lines Ci0j of the W0 do not align with the coordinate lines α i = Const. The problem of the reparameterization of the surface P can be solved in two steps. Following this method, the solution to the problem would be represented as the superposition of the two consequent mappings. On the first step, a canonical region W f is constructed within the reference plane P0 . The region is bounded by the coordinate curves α i = α ki (Figure 10.6). For this purpose, the equation [2]

r(α i ) = r f (α i ) + Fk (α i ) r fk = r f (α i ) + F k (α i ) rkf

(10.34)

is used for mapping the region onto the region W0. Here, r kf and r fk are the covariant and contravariant basis vectors on the surface P, and Fk and F k

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Kinematic Geometry of Surface Machining

448

are the covariant and contravariant components of the vector of the fictive displacements. The components Fk and F k in Equation (10.34) must be constructed based on the requirements of one-to-one correspondence between the contours. After the necessary formulae transformations are accomplished, then the equation

r 0 (α 1 , α 2 ) = r f (α 1 , α 2 ) + Fi (α 1 , α 2 ) r if

(10.35)

can be obtained. Here, r f (α i ) is the position vector of a point M f that is mapped into the point M0 having position vector r 0; r if are the reciprocal basis vectors at the point M f; and F1 and F2 are the components of the vector of the fictive displacements M f , those that can be constructed depend upon the shape of the region W f. At every point of the region W0, the constructed functions Fi together with Equation (10.33) yield computation of the following: The position vector r i :

(

)

(

)

)

r i = δ ik + eifk rkf = aik + eikf r fk

(10.36)

0 The major basis vectors r i at the point M0:

(

)

(

r i0 = δ ik + eifk rkf = aikf + eikf r fk

(10.37)

These vectors are tangent to the coordinate curves specified by the mapping (see Equation 10.35). In Figure 10.6 they are designated by lines δ i = const . The covariant components of the first metric tensor:

Φ 1.P ⇒ aik0 = aikf + 2ε ikf

(10.38)

Christoffel’s symbols of the second kind

Γ ij0 k = Γ ijfk + Aijfk

(10.39)

at the point M0 . In Equation (10.37), the parameters eifk and eikf can be computed by formulae eifk = ∇if F k and eiff = ∇if Fk . For the computation of the parameter 2ε ikf , the formula 2ε ikf = ri ⋅ rk − ri f ⋅ rkf = eikf + e kif + a jsf eijf e ksf is used. Ultimately, the parameter Aik0 j is computed from

© 2008 by Taylor & Francis Group, LLC

Aik0 j = a0jn Pnf,ik

(10.40)

Synthesis of Optimal Surface Machining Operations

449

Here, for the case under consideration, the equalities a011 = a0 0 a 0 − ( a 0 )2 , and a012 = − a120 , a0 = a11 22 12 Pj ,fik = ∇if ε jkf + ∇ kf ε ijf − ∇ jf ε ikf

0 a22 a0

, a022 =

0 a11 a0

,

(10.41)

are valid. On the second step, the region Ω0 (see Equation 10.35) is mapped onto the sculptured part surface P. For the mapping, the vectorial equality r(α i ) = r 0 (α i ) + H(α i ) n 0

(10.42)

is used. In this case, with the help of Equation (10.35), the above Equation (10.42) casts into

r (α i ) = r (α i ) + F (α i ) r + H(α i ) n 0

(10.43)

The point M0 is the projection of the point M onto the reference plane P0 . Therefore, Gaussian coordinates of a certain point M f serve as the Gaussian coordinates of the point M (see Equation 10.35). In the region M0, for the basis vectors given by Equation (10.37), as well as for r 0i = a0ik r k0 , the following two equations rik0 = Γ ik0 j rj0

and r k0i = − Γ 0kji r0j

(10.44)

0j

are observed. Christoffel’s symbols Γ ik in Equation (10.44) were determined in Equation (10.39). If the parameters α 1 and α 2 in Equation (10.33) determine two families of orthogonal coordinate curves within the plane P0 , then not Equation (10.35), but the equality

r 0 (α i ) = r f (V ) + F1 (α i ) e1f + F2 (α i ) e2f

(10.45)

is used instead. Here, r1f and r 2f are the unit vectors of the coordinate curves α 1 = Const and α 2 = Const at the point M f. Under such a scenario, not Equation (10.37) through Equation (10.41) are used, but the relationships

(

)

r i0 = A if δ is + eisf e sf

ai0 = A if Akf δ ki + 2ε kif

© 2008 by Taylor & Francis Group, LLC

(

(10.46)

)

(10.47)

Kinematic Geometry of Surface Machining

450

2ε ikf = eikf + e kif + eisf e ksf

(

)(

)

(10.48)

(

)

(

)

 1 + 2ε 22f 1 + e11f − 2ε12f e21f  e1f +  1 + 2ε1f2 e12f − 2ε12f 1 + e22f  e2f    (10.49) r 10 =  2 f f  f  f A1 1 + e11 − e12 e21  

(

)

(

)

are used instead. The relationships

r i0 = A if δ is + eisf e sf

(10.50)

aik0 = A if Akf δ ki + 2ε kif

)

(10.51)

2ε ikf = eikf + e kif + eisf e ksf

(10.52)

(

(

)(

)

(

)

(

)

 1 + 2ε 22f 1 + e11f − 2ε12f e21f  e1f +  1 + 2ε1f2 e12f − 2ε12f 1 + e22f  e2f    r = (10.53) 2 f f  f  f A1 1 + e11 − e12 e21   1 0

(

)

used in Equation (10.37) through Equation (10.41) are still valid. Again, in these equations, the following equalities f e 11 =

f e 12 =

F1, 1 A 1f F2 , 1 A

f 1

+

−

F2 A 1f , 2 A 1f A 2f

(10.54)

uuur 1, 2

F1 A 1f , 2 A 1f A2f

(

)

2

(

a0 =  1 + e11f − e12f e21f  A1f A2f  

(10.55)

) 2

(10.56)

remain valid. In Equation (10.44), Christoffel’s symbols are computed using expended formulae:

aΓ 111 = aΓ 112 =

a22 a11, 1 2

a11, 2   − a12  a12 , 1 − 2  

a22 a11, 2 − a12 a22 , 1 2

(10.58)

a22 , 1  a12 a22 , 2  aΓ 122 = a22  a12 , 2 − − 2  2 

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(10.57)

uuuur 1, 2

(10.59)

Synthesis of Optimal Surface Machining Operations

451

For the computations, the expressions (see Equation 10.45) are substituted into the last equations.

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 The machining of a sculptured surface on a multi-axis NC machine is a challenging engineering problem. Implementation of the DG/K-method of surface generation makes it possible to develop a CAD/CAM system for optimal sculptured surface machining on a multi-axis NC machine. 10.5.1 Major Blocks of the DG/K-Based CAD/CAM System The proposed concept of the DG/K-based CAD/CAM system for optimal sculptured surface machining on a multi-axis NC machine is composed of seven major parts. In Figure 10.7, the major parts are depicted as blocks of the DG/K-based CAD/CAM system. Only data on part surface geometry are used as the input information for the functioning of the DG/K-based CAD/CAM system. The synthesis of the optimal machining operation begins from an analytical description of the sculptured part surface being machined (I).

Start

II

I

VI

III

VII

IV

V

End Figure 10.7 Principal blocks of the DG/K-based CAD/CAM system for optimal sculptured surface machining on a multi-axis numerical control machine.

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Kinematic Geometry of Surface Machining

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The initially given representation of the sculptured surface P is converting to the natural parameterization of the surface P, when the surface P is expressed in terms of the first Φ1.P and of the second Φ2.P fundamental forms (see Chapter 1). The converted analytical representation of the sculptured surface is used for solution (II) to the problem of optimal orientation of the surface P on the worktable of the NC machine (see Chapter 7). It also yields a solution to the problem of designing the optimal cutting tool (see Chapter 5) for machining the sculptured surface P (III). For solving the above problems, the analytical description of the geometry of contact between the sculptured surface P and between the generating surface T of a cutting tool is employed (see Chapter 4). Further, implementation of the analytical description of the geometry of contact of the surfaces P and T enables computation of optimal parameters of kinematics of sculptured surface machining. Ultimately, this yields a closedform solution (IV) to the problem of optimal tool-path generation, computation of coordinates of the optimal starting point for surface machining, and verification of satisfaction or violation of the necessary conditions of proper part surface generation (V). The cutting tool for machining the sculptured surface can be either designed or it can be chosen (VI) within the available cutting tools. In particular cases of sculptured surface machining, it is allowable (VII) to maintain the desired kind of contact of the surfaces P and T, and not make mandatory their optimal contact. Finally, when the optimization is accomplished, the shortest possible machining time of sculptured surface machining, as well as the lowest possible cost of the machining operation can be achieved. More detail about the major blocks of the DG/K-based CAD/CAM system (see Figure 10.7) are disclosed in the following sections. 10.5.2 Representation of the Input Data A sculptured surface P to be machined is initially represented either analytically or discretely. For application of the DG/K-method of surface generation, the initial representation of the surface P is converting to the natural representation in terms of the fundamental magnitudes EP, FP , GP of the first order Φ 1.P, and of the fundamental magnitudes LP , M P , N P of the second order Φ 2.P . Conversion of the initial surface P representation to its representation in the natural form is performed by the first block (I) of the CAD/CAM system (Figure 10.7). For analytically represented surface P, computation of the fundamental magnitudes of the first Φ 1.P and of the second Φ 2.P order turns to a routing mathematical procedure (see Chapter 1). The sequence of the required steps of computation is as follows (Figure 10.8):

(1) → (2) → (3) → ( 4) → (5) → (6) → L

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(10.60)

Synthesis of Optimal Surface Machining Operations

453

Start I

1

Yes

2

No

7

Yes Yes 3

8 9

No No

10

14 11 12 33

5

28

34

6

29

35

13 4

II

VII

VI 27

No

15

17 Yes

25 Yes

16

18

21

26

24

19

22

30

No

III

31 23

32

20

IV 36

42

46

51

37

43

47

52

56

58

53

57

59

49

54

62

60

50

55

48 Yes

38 40 45

No No

39 Yes

44

41 Yes

No

V

61 End

Figure 10.8 The generalized flow chart of the DG/K-based CAD/CAM system for optimal sculptured surface machining on a multi-axis numerical control machine.

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Kinematic Geometry of Surface Machining

454

In case of discrete representation (7), the surface P is approximated either by a common analytical function

K → (7 ) → (8) → (10) → (3) → ( 4) → (5) → (6) → K

(10.61)

or piecewise approximation K → (7 ) → (9) → (10) → (11) → (12) → (13) → (3) → ( 4) → (5) → (6) → K

(10.62)

is implemented. Fundamental magnitudes of the first and of the second order of the surface P can be determined directly from the specification of the surface P in discrete form (8). Following this method, the determined partial derivatives (14) are used as input to

K → (8) → (14) → (5) → (6) → K

(10.63)

(The interested reader may wish to go to [11] for more information in this concern.) The derived natural representation of the surface P is the output of the first block (1) of the DG/K-based CAD/CAM system (Figure 10.7). It is used below as the input to the cutting tool block (III). 10.5.3 Optimal Workpiece Configuration Computed (6) values of the fundamental magnitudes of the first Φ 1.P and of the second Φ 2.P order are used (II) for the computation (15) of parameters of the optimal orientation of the surface P on the worktable of a multi-axis NC machine. For the optimization, the approach earlier developed by the author [3,5,11,16,17] is used (see Chapter 7). The computed parameters of optimal orientation of the workpiece on the worktable of the multi-axis NC machine are the output (16) of this subsystem of the CAD/CAM system (Figure 10.8). 10.5.4 Optimal Design of the Form-Cutting Tool Design parameters of the form-cutting tool for the optimal machining of the surface P on a multi-axis NC machine are computed in the block (III). Again, the criterion of the optimization is the lowest possible cost of the machining operation. The key problem at that point is to determine the geometry of the generating surface T of the cutting tool [6,8,14,15]. The equation of the surface T is derived in natural parameterization — that is, in terms of the fundamental magnitudes ET , FT , GT of the first order Φ 1.T , and the fundamental magnitudes L T , MT, N T of the second order Φ 2.T of the surface T. A method for the computation of geometry of the cutting tool surface T is disclosed in Chapter 5. Shown in Figure 10.7, the cutting tool

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455

block (III) utilizes the technique briefly disclosed there. The computation of parameters of design of the optimal form-cutting tool encounters principal steps (17) through (24). The cutting tool block (III) yields selection of a design of a form-cutting tool that best fits the requirements of machining of the given surface P (the principal steps (27) through (29) in Figure 10.8). Ultimately, machining of a given surface P can be performed with a given form-cutting tool T:

K → (25) → (26) → (30) → (31) → (20) → K

(10.64)

Under such a scenario, the parameters of design of the form-cutting tool that are required for further computations could be computed following the routing

K → (33) → (34) → (35) → (32) → (20) → K

(10.65)

The computed parameters of design of the form-cutting tool for machining a given sculptured surface P are the output (20) of the cutting tool block (III). 10.5.5 Optimal Tool-Paths for Sculptured Surface Machining The major purpose of the block (IV) is to compute the parameters of the optimal tool-paths for the shortest machining time of a sculptured surface on a multi-axis NC machine. For this purpose, unit normal vectors n P and n T to the surfaces P and T are computed in (36). Then, contact of the surfaces P and T is numerically simulated in (37). At the CC-point K, unit normal vectors n P and n T are directed oppositely to each other. Further, verification of satisfaction or violation of the necessary conditions of proper part surface generation [9, 11] is performed (38). If the necessary conditions of proper part surface generation (PSG) are violated (39), then an appropriate correction of the cutting tool configuration relative to the work is performed in (40). After verification of satisfaction of the necessary conditions of proper PSG (39), machining of the sculptured surface is performed (41) along the tool-paths of the widest possible width. This is because the parameters of the optimal tool-paths are computed (42) as the curves on the surface P that are equidistant (parallel) to the longest geodesic curve on P. Aiming derivation of the best results of the computations, boundary conditions could be incorporated (43) into the procedure of computation. Ultimately, the optimal parameters of tool-paths transfer to (44), which is the output of the block (IV). Usually, the sculptured surface P and the generating surface T of the cutting tool make point contact (41). This kind of sculptured surface machining is the most commonly used in various industries. Under such a scenario, for the computation of the optimal tool-paths, the indicatrix of conformity Cnf R ( P/T ) [11,12,16] of the surfaces P and T at the CC-point is implemented.

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After the parameters of the indicatrix of conformity Cnf R ( P/T ) are computed (45), then the cutting tool configuration could be optimized. For this purpose, orientation motions of the first and of the second kind are performed by the cutting tool. The orientation motion of the second kind of the cutting tool results in that the point at which local topology of the surface T is optimal is put into contact (46) with the current CC-point of the surface P. The orientation motion of the first kind of the cutting tool yields optimization (47) of the local orientation of the surfaces P and T. For grinding sculptured surfaces when the cutting tool represents (48) the entire surface T, the characteristic curve Cnf R ( P/T ) yields computation (49) of the optimal direction of tool-paths at every point of the surface P. Further, this yields computation (50) of the optimal tool-paths. A closed-form solution to the problem of optimal tool-path generation for sculptured surface machining on a multi-axis NC machine was developed by Radzevich [4]. Following the derived solution to the problem, the optimal tool-paths are directed orthogonally of the minimal diameter of the characteristic curve Cnf R ( P/T ) at every CC-point. So, the indicatrix of conformity yields directions tangent to the optimal tool-paths. Further, elementary integration returns the equation of the optimal tool-paths for sculptured surface machining on a multi-axis NC machine. Without going into detail of the derived solution to the problem, the final equation of the optimal tool-paths for sculptured surface machining is represented in matrix form (see Equation 10.12). Targeting computation of the most accurate solution to the problem of synthesis of the optimal machining operation, the following factors can be incorporated into the computations: the boundary effect (51), partial interference of the neighboring tool-paths (52), and so forth. Finally, the computed parameters of the optimal tool-paths represent (44) the output of the block (IV). When the sculptured surface P is machined with an edge-cutting tool (for example, with a milling cutter), then (48) the generating surface T is represented discretely by a certain number of distinct cutting edges. In this case, the surface T cannot be represented as a continuous surface. The DG/Kbased CAD/CAM system is capable of treating (53) all the restrictions that are imposed by the discretely represented surface T. The encountered restrictions include (and are not limited to) the optimal distribution (54) of the resultant tolerance on accuracy of the surface P onto two portions, the effect of the critical values of the feed-rate F fr, and of the side-step Fss onto optimization (55) of the kinematics of the machining operation, and so forth. Further computations in the CAD/CAM system are performed following the above-considered route:

L → (55) → ( 49) → (50) → (51) → (52) → ( 44) → L

(10.66)

The optimal tool-paths of sculptured surface machining on a multi-axis NC machine are the output of the block (IV) (Figure 10.8).

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457

10.5.6 Optimal Location of the Starting Point The coordinates of the optimal starting point for machining of the sculptured surface are computed in the block (V). Prior to starting the machining operation, the cutter-location (CL)-point must be coincident with the starting point. For the computation of coordinates of the optimal starting point, a family of nonintersecting curves on the surface P is determined (56). The family of infinite number of curves on P is the same as that within which the optimal tool-paths have to be selected. Two curves from the family are selected (57). Depending on the surface P geometry and on the shape of its boundary curve, these two curves are tangent to the contour of the portion of the surface P to be machined, or they align with the surface P border, or they share with the surface P border at least one common point each. Then, the selected two curves are shifted (58) inside the surface P area. The distance at which the curves are shifted is half the side-step |Fss|. One of four optimal starting points is located on one of two curves defined just above. Further, four distances that are necessary for the cutting tool entering into the machining operation and exiting from the machining operation are computed (59). At this point, one can come up with coordinates of four points that are remote from the boundary curve of the surface P at the computed distances. Selection (60) of one of them returns the optimal starting point for machining the sculptured surface. The computed coordinates of the optimal starting point transfer (61) to output of the block (V). The procedure of computing the optimal starting point is capable of incorporating (62) actual features of the cutting tool entering into and exiting from machining, and so forth. The DG/K-based CAD/CAM system (Figure 10.8) can not only be used for sculptured part surfaces, but can also be adjusted for the machining of surfaces having simpler geometry.

References [1] Faux, L.D., and Pratt, M.J., Computational Geometry for Design and Manufacture, Ellis Horwood, New York, 1987. [2] Galimov, K.Z., and Paimushin, V.N., Theory of Shells of Complex Geometry, Kazan’ State University Press, Kazan’, 1985. [3] Pat. No. 1442371, USSR, A Method of Optimal Orientation of a Sculptured Surface on the Worktable of Multi-Axis NC Machine./S.P. Radzevich, Int. Cl4 B23Q 15/007, Filed February 17, 1987. [4] Radzevich, S.P., A Closed-Form Solution to the Problem of Optimal Tool-Path Generation for Sculptured Surface Machining on Multi-Axis NC Machine, Mathematical and Computer Modeling, 43 (3–4), 222–243, 2006.

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Kinematic Geometry of Surface Machining

[5] Radzevich, S.P., A Method of Optimal Orientation of a Sculptured Surface on the Worktable of Multi-Axis NC Machine, Izvestiya VUZov. Mashinostroyeniye, No. 2, 140–145, 1990. [6] Pat. No. 4242296/08, USSR, A Method of Profiling of a Form Cutting Tool./S.P. Radzevich, Filed March 31, 1987. [7] Radzevich, S.P., Advanced Technological Processes of Sculptured Surface Machining, VNIITEMR, Moscow, 1988. [8] Radzevich, S.P., A Novel Method for Mathematical Modeling of a FormCutting-Tool of the Optimum Design, Applied Mathematical Modeling 46 (7–8), October 2007. [9] Radzevich, S.P., Conditions of Proper Sculptured Surface Machining, ComputerAided Design, 34 (10), 727–740, 2002. [10] Radzevich, S.P., Differential-Geometric Method of Surface Generation, DrSci thesis, Tula, Polytechnic Institute, 1991. [11] Radzevich, S.P., Fundamentals of Surface Generation, Rastan, Kiev, 2001. [12] Radzevich, S.P., Mathematical Modeling of Contact of Two Surfaces in the First Order of Tangency, Mathematical and Computer Modeling, 39 (9–10), 1083–1112, 2004. [13] Radzevich, S.P., Methods of Milling of Sculptured Surfaces, VNIITEMR, Moscow, 1989. [14] Radzevich, S.P., Profiling of a Form Cutting Tool for Sculptured Surface Machining on Multi-Axis NC Machine, Stanki i Instrument, No. 7, 10–12, 1989. [15] Radzevich, S.P., R-Mapping Based Method for Designing of Form Cutting Tool for Sculptured Surface Machining, Mathematical and Computer Modeling, 36 (7–8), 921–938, 2002. [16] Radzevich, S.P., Sculptured Surface Machining on Multi-Axis NC Machine, Vishcha Shkola, Kiev, 1991. [17] Radzevich, S.P., and Goodman, E.D., Computation of Optimal Workpiece Orientation for Multi-Axis NC Machining of Sculptured Part Surfaces, Journal of Mechanical Design, 124 (2), 201–202, 2002.

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11 Examples of Implementation of the Differential Geometry/Kinematics (DG/K)-Based Method of Surface Generation The DG/K-based approach for surface generation is discussed in the previous chapters. This approach can be used for the development of novel advanced methods of surface machining. Below, numerous examples that illustrate the capabilities of the DG/K-based approach are briefly considered. In the examples below, it is assumed that all necessary conditions of proper part surface generation [19,21,22,25] are satisfied (see Chapter 7).

11.1

Machining of Sculptured Surfaces on a Multi-Axis Numerical Control (NC) Machine

Numerous methods of sculptured surface machining on a multi-axis NC machine are developed by now. A review of known methods of sculptured surface machining is available from the literature [18,23]. Below, a method of sculptured surface machining that is developed on the premises of the DG/K-based approach of surface generation is considered. Consider machining a sculptured part surface on a multi-axis NC machine (Figure 11.1). Prior to machining the part surface, the workpiece has to be properly oriented on the worktable of the NC machine [12]. The workpiece is oriented in compliance with the method of optimal workpiece orientation on the worktable of the multi-axis NC machine (SU Pat. No. 1442371). Optimal part surface orientation is discussed in detail in Chapter 7. For machining the sculptured surface P, a form milling cutter is used. Parameters of geometry of the generating surface T of the cutting tool are computed on the basis of the method of design of a form-cutting tool for sculptured surface machining on a multi-axis NC machine (SU Pat. No. 4242296/08). This method of form-cutting tool design [5] widely employs the R-mapping of the sculptured surface P onto the generating surface T of the form-cutting tool [24]. When machining the sculptured surface P (Figure 11.1), the cutting tool is rotating about its axis OT with a certain angular velocity ωT. The cutting tool is traveling with the optimal feed-rate F fr along the optimal tool-paths. The optimal tool-paths are given by Equation (10.12). After the machining of a tool-path is accomplished, then the cutting tool moves across the trajectory 459 © 2008 by Taylor & Francis Group, LLC

460

Kinematic Geometry of Surface Machining P

OP

А

Ffr

OT

T

View А

ωP

F˘ss

nP K

OT ±Swl ±Swn

Ffr

Wps Generating Curve of the Surface T OT

±Swl

OT

K

Swivel Motion of the Cutting Tool

ωT

Swinging Motion of the Cutting Tool

P

OP (a)

(b)

Figure 11.1 Optimization of machining of a sculptured part surface on a multi-axis numerical control machine (SU Pat. No. 1185749; SU Pat. No. 1249787).

of the cutter-contact-point (CC-point) in the direction of side-step Fss at a ( distance Fss . In the new position of the cutting tool, machining of the next tool-path begins. For the most efficient surface machining, the width of the tool-path must be the maximal possible at every instance of the surface machining. In order to maintain the maximal width of the tool-path, two more motions are performed by the cutting tool. These two motions are the motions of orientation of the cutting tool. One of the orientation motions is swiveling of the cutting tool [7]. Another orientation motion** is swinging ±Swn of the cutting tool [9]. It is convenient to consider the swinging motion of the cutting tool before considering its swivel motion. The swinging motion ±Swn is the orientation motion of the second kind of the cutting tool (see Chapter 2). Consider the cross-section of the part surfaces and of the generating surface of the cutting tool with a plane surface through the common perpendicular n P ≡ − nT at a CC-point K (Figure 11.2). The CC-point K that belongs to the surface P is designated there as K P . Similarly, the CC-point K that belongs to the surface T, is designated as KT. When machining the sculptured surface, various points KT(1), KT( 2), KT( 3), and so forth, of the generating surface T of the cutting tool can make contact with the surface P at the point K P . Machining of the sculptured surface P is performed with cutting tool of different radii of normal curvature RT(1) , RT( 2),

SU Pat. No.1185749, A Method of Sculptured Surface Machining on Multi-Axis NC Machine./ S.P. Radzevich, Int. Cl. B23C 3/16, Filed October 24, 1983. ** SU Pat. No.1249787, A Method of Sculptured Surface Machining on Multi-Axis NC Machine./ S.P. Radzevich, Int. Cl. B23C 3/16, Filed December 27, 1984.

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461

Examples of Implementation of the DG/K-Based Method RP

(1)

n P RT

(2)

KT

P + Swn KP

nT

(3)

KT

T

(1)

KT

(a)

RP +Swn P (1) KT

(2)

nP RT –Swn

(3)

KT KP

nT (b)

(2)

KT

RP T

T

(1)

KT (2) KT

(3)

nP RT –Swn

KP

nT

P (3)

KT

(c)

Figure 11.2 Swinging motion ±Swn of the cutting tool: the orientation motion of the second kind of the cutting tool (SU Pat. No. 1249787).

RT( 3) , and so forth, of its surface T. When the radii of normal curvature are different, say the inequality RT(1) < RT( 2) < RT( 3) is observed, this gives a possibility for controlling the rate of conformity of the surface T to the sculptured surface P at a current CC-point. In order to maintain the maximal possible rate of conformity of the surface T to the surface P at every CC-point K, it is necessary to perform a swinging motion of the cutting tool either in the direction +Swn or in the opposite direction −Swn. The actual direction of the swinging motion ±Swn depends upon the current configuration of the cutting tool relative to the part surface, as well as it depends upon parameters of geometry of the surfaces P and T at the current CC-point K. The swinging motion ±Swn of the cutting tool can be decomposed onto the rotation w c about a certain axis, and onto translation Vc in the direction orthogonal to the axis of rotation w c . Direction of the translation Vc is orthogonal to the unit normal vector n P to the surface P. The axis of the rotation w c is orthogonal to the direction of the translational motion. Moreover, magnitudes of the rotation w c and the translation Vc are timed with each other in compliance to |VP ||ω P |= RP( c ) , where RP( c ) is the radius of curvature of the surface P in the normal plane surface through the Vc. Because the cutting tool is performing the swinging motion, the location of the CC-point on the sculptured surface P remains the same. At that same time, location of the CC-point on the generating surface T of the cutting tool changes. The orientation motion of the second kind enables the controlling of the rate of conformity of the surface T to the surface P at a current CC-point, and, in such a way reduces the machining time. For precise positioning of the cutting tool, the swinging motion is performed at a certain angle to the direction specified by t max cnf (SU Pat. No. 1336366). The oblique trajectory of the swinging motion reduces the impact of the deviations inherent to the multi-axis NC machine of this particular design [10].

SU Pat. No.1336366, A Method of Sculptured Surface Machining on Multi-Axis NC Machine./ S.P. Radzevich, Int. Cl. B23C 3/16, Filed October 21, 1985.

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Kinematic Geometry of Surface Machining

The swivel motion ±Swl is the orientational motion of the first kind of the cutting tool (see Chapter 2). This orientational motion results in rotation w n (or turning through a certain angle) of the cutting tool about a common perpendicular n P ≡ − nT. Evidently, the speed of the orientational motion ±Swl is equal to ω n = ∂ µ ∂t. The swivel motion is performed by the cutting tool simultaneously with its swinging motion. The radii of principal curvature R1.T and R2.T of the generating surface T of the cutting tool are not equal to each other, and the inequality R1.T < R2.T is always observed. This provides an additional opportunity for controlling the rate of conformity of the surface T to the sculptured surface P at every CC-point. In order to reach the maximal possible rate of conformity of the surfaces P and T, it is necessary to perform a swivel motion ±Swl of the cutting tool on the multi-axis NC machine. Figure 11.3 reveals that the swivel motion ±Swl of the cutting tool is the orientation motion of the first kind. It also makes clear that the swivel motion ±Swl of the cutting tool provides an additional opportunity to control the rate of conformity of the surface T to the surface P at every CC-point and, in such a way, to reach the highest rate of conformity of the surfaces. Direction and speed of both orientational motions ±Swl and ±Swn are timed with each other, and are synchronized with other geometrical and kinematical parameters of the machining operation. This yields permanent mainte(min) nance of the minimal possible value of dcnf ⇒ min , as well as the minimal (max) value of dcnf ⇒ min of the indicatrix of conformity Cnf R ( P / T ) of the surfaces P and T at the current CC-point (both of these diameters are functions (min) (min) of the angle m of the local orientation of surfaces P and T; dcnf = dcnf (µ) (max) (max) and dcnf = dcnf ( µ ). The orientational motion ±Swl (Figure 11.3a) reduces the minimal diameter of the characteristic curve Cnf R ( P/T ) and changes (min) (min) its direction from dcnf to dcnf (Figure 11.3b). The orientational motion ±Swn allows additional reduction of the minimal diameter of the indicatrix of conformity Cnf R ( P/T ) of the surfaces P and T at the current CC-point and makes additional changes in its direction (Figure 11.3c). Both orientation motions ±Swl and ±Swn allow for the optimal cutting tool posture as well as the optimal direction of the resultant motion of the cutting tool relative to the sculptured surface being machined. The orientational motions of the cutting tool enable changing directions of the relative motion along the tool-path from V to V ( opt ). At every CC-point K, the optimal toolpaths are directed orthogonally to the direction of normal plane surface through K, at which the rate of conformity of the surfaces P and T is the high(min) est possible (Figure 11.3) — that is, it is directed orthogonally to dcnf . The resultant relative motion of the part and of the cutting tool could be decomposed onto several rotations and translations. The rotations and translations (they are not labeled in Figure 11.1) are performed by corresponding servo-drives of a multi-axis NC machine. The direction along which the rate of conformity of the generating surface of the cutting tool to the part surface is maximal aligns with the direction along which the diameter of the indicatrix of conformity Cnf R ( P/T ) is

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463

Examples of Implementation of the DG/K-Based Method nP

P K

T

Swivel Motion of the Cutter

nT

Swinging Motion of the Cutter

±Swl ±Swn (a) V*

±Swl CnfR (P/T)

Vopt (min) dcnf =

0

Ind * (T )

yp

C1.T

Dup(T )

CnfR (P/T )

opt

Dup (P)

Ind(P)

µ

Cnf *R (P/T )

Cnf *R (P/T ) C*1.T

K

C2 P

C2.T

µ*

Ind(P) (min) d*cnf

Ind(T ) Dup(P)

Dup*(T )

C1.P

C*2.T

xp

(b) V* V(opt)

Ind**(T )

(opt) Ind (T ) µ

**

(opt)

(opt)

**

Ind (P) K Cnf **R (P/T )

Dup(T ) Dup**(T )

µ**

CnfR (P/T )

(c)

Figure 11.3 Swivel motion ±Swl of the cutting tool: the orientational motion of the first kind of the cutting tool (SU Pat. No. 1185749).

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464

Kinematic Geometry of Surface Machining

minimal. The direction of the maximal rate of conformity of the surface T to the surface P is specified by the unit tangent vector t (max) . cnf Similarly, the direction of the minimal rate of conformity of the surface T to the surface P can be specified by the unit tangent vector t (min) . cnf In the general case of sculptured surface machining, the directions t (max) cnf and t (min) are not orthogonal to each other. Usually, the feed-rate motion F fr cnf is orthogonal to t (max) . Thus, this motion F fr does not align with the direccnf tion of t (min) . Accordingly, if the direction F fr aligns with the unit tangent cnf vector t (min) , it does not align orthogonally to the direction specified by the cnf unit tangent vector t (max) . This is because the unit tangent vectors t (max) and cnf cnf (min) t cnf are not orthogonal to each other. Therefore, in the general case of sculptured surface machining, it is impossible to generate the sculptured part surface under the widest tool-path and with the highest possible feed rate per tooth of the cutting tool. This issue becomes critical when the magnitude of the limit feed-rate |[F fr ]| and magnitude of the limit side-step |[Fss ]| are of comparable value. Under such a scenario, the surface generation output P sg must be used for determining the optimal parameters of sculptured surface machining. The method of sculptured surface machining on a multi-axis NC machine [11] is illustrated in Figure 11.4. In the common tangent plane through the CC-point K, two principal crosssections C1. P and C2. P of the sculptured surface P are along the unit tangent vectors t 1.P and t 2.P . The unit tangent vector t (max) is at the optimal angle cnf

Ffr

opt

Ffr

∆ C2.P

ξopt

t2.P (min)

tcnf K

t1.P

opt

(max)

tcnf

C1.P Figure 11.4 A method of sculptured surface machining (SU Pat. No. 1367300).

SU Pat. No. 1367300, A Method of Sculptured Surface Machining on Multi-Axis NC Machine./ S.P. Radzevich, Int. Cl. B23C 3/16, Filed January 30, 1986.

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Examples of Implementation of the DG/K-Based Method

(max) ϕ opt with respect to the unit tangent vector t 1.P . The unit tangent vector t cnf specifies the direction of the feed-rate motion F fr . The last is orthogonal to the unit tangent vector t (max) . cnf The direction of the minimal rate of conformity of the surfaces P and T at the point K is specified by the unit tangent vector t (min) . In the case under cnf (min) consideration, the unit tangent vectors t (max) and are not orthogonal to t cnf cnf each other. Instant part surface generation output P sg (see Equation 9.1) is as follows:

P sg =|F fr × Fss|

(11.1)

Without going into details of analysis here, a generalized expression (Equation 9.11)

P

sg

= P sg (U P , VP , UT , VT , µ , ϕ )

(11.2)

for the instant surface generation output P sg can be used in the analysis below. When the feed-rate motion F fr is directed orthogonally to the minimal (min) diameter dcnf of the indicatrix of conformity Cnf R ( P/T ), then the width of the tool-path is maximal; however, the limit value of the feed rate per tooth ( [ Ffr ] of the cutting ( tool is getting smaller. A smaller limit value of the feed rate per tooth [ Ffr ] of the cutting tool causes smaller instant surface gen( eration output P sg. Put another way, an increase of the limit feed-rate [ Ffr ] ( requires a corresponding decrease of the limit side-step [ Fss ]. The above consideration reveals that there must be ( an optimal correspon( dence between the feed-rate Ffr and the side-step Fss. For this purpose, the direction of the feed-rate motion F fr must be properly determined. For the maximal surface generation output, optimal direction of the feedrate motion F opt satisfies the condition fr ∂P

sg

∂ξ

=0

(11.3)

where the auxiliary angle x is linearly dependent on the angle j. When the solution to Equation (11.3) returns multiple solutions, only those that satisfy the sufficient condition for the maximum of P sg is acceptable: ∂2P

∂ξ 2

sg

<0

(11.4)

The computed optimal value ξ opt of the auxiliary angle x specifies the optimal direction of the feed-rate motion F opt . This direction is at a certain fr angle Dx with respect to the direction t (max) at which the minimal diameter cnf (min) of the indicatrix of conformity Cnf R ( P/T ) is measured. dcnf

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466

Kinematic Geometry of Surface Machining hΣ

P

opt hss

opt

hfr

P

opt

hΣ

opt

F˘ *fr

Fss

(a)

K F˘ ss*

Ffr* (b)

h** ss

h** fr

K opt F˘ ss

Ffr

h*ss

h*fr

K opt F˘ fr

P

hΣ

Fss*

F˘ fr**

˘ ** F ss F** ss

Ffr** (c)

Figure 11.5 Cusps on the machined sculptured surface.

The efficiency of sculptured surface machining can be increased in that way when the feed rate and the side step at the current CC-point K are properly timed with each other. Use of the method of sculptured surface machining [17] makes it possible to achieve this goal. Figure 11.5 illustrates residual cusps on the machined part surface. When parameters(of the sculptured surface ( machining are optimal, then cusps of the width Fssopt and of the length Ffropt appear on the machined part surface (Figure 11.5a). ( ( If the value of the feed rate is increased from Ffropt to Ffr , then it is neces( opt ( sary to decrease the side step from Fss to Fss (Figure 11.5b). Ultimately, the part surface generation output in this case is smaller compared to that in the first case (Figure 11.5a). ( ( If the value of the feed rate is decreased from Ffropt to Ffr , then it is neces( opt ( sary to increase the side step from Fss to Fss (Figure 11.5b). Ultimately, the surface generation output in this case is also smaller compared to that in the first case (Figure 11.5a). As previously discussed (see Section 8.5.2), the limit value [ hss ] of the deviation hss is equal to a portion of the tolerance [ h] , say [ hss ] = c ⋅ [ h] . Here, c designates the local parameter of distribution of the tolerance [ h] . Due to the equality [ h] = [ h fr ] + [ hss ], the following expression [ h fr ] = (c − 1) ⋅ [ h] is valid for the limit value [ h fr ] of the waviness h fr . Substituting the derived formula for [ h fr ] to Equation (9.41), the following ( equation for the computation of the limit value [ Ffr ] of the feed rate

( RP2 . fr + R T . fr ⋅ ( RP . fr + (c − 1) ⋅ [ h] ⋅ sgn RP . fr ) [ Ffr ](c) ≅ 2 RP . fr arccos (11.5) ( RP . fr + R T . fr ) ⋅ ( RP . fr + (c − 1) ⋅ [ h] ⋅ sgn RP . fr )

can be obtained.

RU Pat. No. 2050228, A Method of Sculptured Surface Machining on Multi-Axis NC Machine./S.P. Radzevich, Int. Cl. B23C 3/16, Filed December 25, 1990.

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Similarly, substituting the derived formula for ([ hss ] to Equation (9.46), the equation for the computation of the limit value [ Fss ] of the side step

( RP2 . ss + R T . ss ⋅ ( RP . ss + c ⋅ [ h] ⋅ sgn RP . ss ) [ Fss ](c) ≅ 2 RP . ss arccos ( RP . ss + R T . ss ) ⋅ ( RP . ss + c ⋅ [ h] ⋅ sgn RP . ss )

(11.6)

can be obtained. For the maximal( possible ( surface-generating output [P sg ] , Equation (11.1) yields [P sg ](c) = [ Ffr ](c) ⋅ [ Fss ](c) . When the parameter c is of optimal value, the following equality

∂ [P sg ](c) = 0 ∂c

(11.7)

is observed. This condition (see Equation 11.7) is necessary for the minimum of the function h Σ = h Σ (c) . In addition, the inequality

∂2 [P sg ](c) > 0 ∂ c2

(11.8)

must be observed. Details on the solution to Equation (11.7) are available from the paper [31] by the author. The(timing of the feed rate per tooth Ffr of the cutting tool and the sidestep Fss in compliance with Equation (11.7) ensures an increase of the part surface generation output P sg . It is possible to achieve further improvement of the surface-generation output P sg by means of optimization of configuration of the neighboring elementary surface cells on the machined sculptured surface. Usually, the elementary surface cells are shaped in a form that is close to that of a rectangle. Under proper synchronization of the feed rate per tooth Ffr and the side-step ( Fss the rectangular elementary surface cells (Figure 11.5) could be substituted with the approximately hexagonal elementary surface cells (Figure 11.6). When other conditions of surface generation remain the same, the area of hexagonal elementary surface cells is bigger in comparison with the area of rectangular elementary surface cells. Therefore, the surface-generation output when machining a sculptured surface with the hexagonal elementary surface cells could be bigger. Proper configuration of the neighboring elementary surface cells is an efficient way to increase the surface-generation output P sg. An appropriate method of sculptured surface machining on a multi-axis NC machine was proposed by the author as early as in 1991. When the feed-rate motion F fr and the side-step motion Fss are properly timed with each other, then the cusp height at all vertices 1, 2, …, 6 of the elementary surface cell are equal to each other, and they equal the resultant

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Kinematic Geometry of Surface Machining

hss

Pact

5

Pnom

4 3

6

hfr

F˘fr

K

K 1

2 Ffr

Fss

Pact Fss

Figure 11.6 Hexagonal elementary surface cells on the machined part surface.

cusp height h Σ (i.e., the equality h 1 = h 2 = h 3 = h 4 = h 5 = h 6 = h ). All elementary surface cells are shaped in the form of nonregular hexagons. For an arbitrary correspondence between the motions F fr and Fss at a ( ( current CC-point (e.g., when Ffr = f ⋅ Fss ), the actual cusp heights h i at all six vertices of the elementary surface cell are different. Evidently, the cusp height h i can be expressed in terms of the parameter f, say as h i = h i ( f ). Further, the differences Dh i ( f ) = h Σ − h i ( f ) are computed. The desired value of the parameter f is that under which all the differences Dh i ( f ) are of zero value. The solution to the problem of the computation of the optimal values of the parameter f is not provided here due to space constraints. For details on the solution to this problem, the interested reader may wish to go to [3] where an example of the solution is presented. Conditions of the material removal process in conventional and in climb milling are different. When the difference is significant, then it is easier to remove the stock, for instance, by climb milling and not by conventional milling. Under such a scenario, the cutting tool is traveling not along each consequent tool-path, but it travels over one path. For example (Figure 11.7), machining of the sculptured surface starts at point 1. After machining of the first tool-path is over, then the tool-path through the 2 is machined. The second tool-path is not the neighboring one to the first tool-path. Such a scheme of tool-paths allows for a decrease of the load on the cutting tool when conditions of machining are inconvenient (conventional milling), and an increase of the load on the cutting tool when conditions of machining are convenient (climb milling). Such a method of sculptured surface machining has proven to be practical. The considered examples of methods of sculptured ( surface machining clearly show that the increase of the feed rate per tooth Ffr of the cutting tool is the reliable way to increase the efficiency of sculptured surface machining on a multi-axis NC machine. For advanced methods of machining, the feed rate per tooth of the cutting tool and the side step are values of the same range. It is necessary to mention here that the DG/K-based method of surface generation is fruitful for solving problems of synthesis of optimal sculptured

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Examples of Implementation of the DG/K-Based Method Ffr P

5

6

3

2 4 1

Fss 2F˘ ss

Uncut stock

Stock, to be removed on even tool-path P

i+2

i+1

i

Figure 11.7 An over-path method of sculptured surface machining.

surface machining on a multi-axis NC machine not only by using cutting tools or a grinding wheel, but by using technology of surfaces reinforcement by plastic subsurface deformation of the work. A method of sculptured surface reinforcement is a good example in this regard [14].

11.2 Machining of Surfaces of Revolution Use of the DG/K-based method of surface generation is fruitful for the development of novel advanced methods of machining, not only of sculptured part surfaces shown above, but also of novel advanced methods of machining part surfaces of simpler geometry. Examples can be found in the field of machining of surfaces of revolution, cylindrical surfaces, gears, and in many other fields. 11.2.1

Turning Operations

In compliance with the conventional method of turning of a form surface of revolution, the work is rotating about its axis of rotation. The cutter travels with a certain feed rate along the axis of rotation of the work. In addition to

SU Pat. No. 1533174, A Method of Reinforcement of Sculptured Surface on Multi-Axis NC Machine./S.P. Radzevich, Int. Cl. B24B 39/00, Filed December 2, 1987.

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Kinematic Geometry of Surface Machining

this motion, the cutter is performing a motion toward the work axis of rotation or in backward direction depending upon the actual shape of the axial contour of the surface being machined. This is an example of a trivial turning operation that is often used for machining form surfaces of revolution. Targeting an increase of the part surface generation output, a method of turning form surfaces of revolution is developed [2]. In this method, the work is rotating about its axis of rotation. The cutter travels along the axial profile of the form part surface with a constant peripheral feed rate. The use of this method of surface machining ensures perfect results when the radius of curvature of the axial profile of the form part surface is of constant value or when variation of the radius of curvature is reasonably small and, thus, could be neglected. For machining of form surfaces of revolution having significant variation of curvature of the axial profile, a method of turning of surfaces of revolution is proposed [16]. In this method of surface machining,** the current value of the peripheral feed rate is synchronized with the radius of curvature of the axial profile of the part surface being machined. When machining the surface P, the work is rotating about its axis of rotation OP with a certain rotation ω P (Figure 11.8). The radii of curvature RP P

OP ωP 1

K3

RP K1

K2

RT

2 RT Fprl

Figure 11.8 A method of turning of form surfaces of revolution with cusps of smallest constant height (SU Pat. No. 1708522). US Pat. No. 4.415.977, Method of Constant Peripheral Speed Control./F. Hiroomi and I. Shinichi, Int. Cl. B23 15/10, 05B 19/18, National Cl. 364/474, Filed June 30, 1980, No.243928; Priority June 30, 1979, No.54-82779, Japan. ** SU Pat. No. 1708522, A Method of Turning of Form Surfaces of Revolution./S.P. Radzevich and L.V. Bondarenko, Int. Cl. B23B 1/00, Filed December 13, 1988.

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471

of the axial profile 1 of the part surface P vary from one point of the profile to another point. The cutting edge of the cutter is represented as the arc segment of a curve 2. Radii of curvature R T of the cutting edge can be either of the same value at all points of the cutting edge or can vary from one point to another point of the cutting edge. The cutter is traveling along the axial profile 1 with the peripheral feedrate Fprl . In different instants of the part surface machining, the value of the feed-rate Fprl is different. This means that the current value of the feed-rate Fprl at the points K1 , K 2 , and K 3 of the axial profile 1 of the surface P is not of the same value. When the cutter having constant radius R T = Const is used for the machining of the part surface, then the value of the peripheral feed-rate Fprl at the current point of contact of the axial profile 1 and of the cutting edge can be computed from the formula

 ( RP + R T )2 + ( RP + [ h])2 − R 2T  Fprl = 2 ⋅ RP ⋅ cos −1   2( RP + R T ) ( RP + [ h])  

(11.9)

In a more general case of machining of form surfaces of revolution — say, when the cutter having variable radius R T = Var is used for the machining of the part surface — then the value of the peripheral feed-rate Fprl at the current point of contact of the axial profile 1 and of the cutting edge can be computed from the formula  ( RP + R T .i )2 + ( RP + [ h])2 − R 2T .i  Fprl = RP cos −1   2 ⋅ ( RP + R T .i ) ⋅ ( RP + [ h])  

(11.10)

 [ RP + R T .(i +1)) ]2 + { RP + [ h]}2 − R 2T .(i +1)  + cos −1   2 ⋅ [ RP + R T .(i +1) ] ⋅ { RP + [ h]}  

where R T .i and R T .(i +1) designate radii of curvature of the cutting edge of the cutter at two neighboring contact points K i and K i+1. Due to the optimization of the current value of the peripheral feed rate of the cutter at every CC-point K, the surface generation output in this method is bigger. In the method of surface machining (Figure 11.8), the radius of curvature of the cutting edge R T of the cutter is out of the control of the user. Analysis of Equation (11.10) reveals that the impact of variation of radius of curvature R T onto the surface-generation output can be significant. Therefore, the efficiency of turning of form surfaces of revolution can be increased if the proper control of the radius R T is introduced. Kinematics of surface machining in the method of turning of form surfaces of revolution [6] is capable of properly controlling the radius of curvature R T

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Kinematic Geometry of Surface Machining

1 ωP ОP Ki+1 6

Ki

3

5 4

2

7 8

Figure 11.9 Utilization of the orientational motion of the second kind of the cutter in the method of turning form surfaces of revolution (SU Pat. No. 1171210).

of the cutting edge of the cutter at the current CC-point. In this method of surface machining, the work having axial profile 1 of variable curvature is rotating about its axis of rotation OP with a certain rotation ω P (Figure 11.9). The cutting edge of the cutter, an arc segment of a curve 2 having permanently variable radius of curvature R T, is used for performing this machining operation. The cutter is traveling along the axial profile of the part surface in the peripheral direction 3. On the lathe, this motion is obtained as the superposition of the axial motion 4 of the cutter and its reciprocal motion toward the part axis of rotation 5 and in backward direction 6. In addition, the cutter is performing the orientational motion of the second kind (see Chapter 2). This motion of the cutter is performed either in the direction 7 or in the direction 8. The actual direction of the orientation motion of the cutter depends upon the actual geometry of the axial profile 1 of the part surface at two neighboring CC-points K i and K i+1. Ultimately, due to the cutter motion either in the direction 7 or in the direction 8, the cutting edge is rolling with sliding over the axial profile 1 of the part surface being machined. Implementation of the orientational motion of the cutter allows for better fit of the radius of curvature R T to the part surface radius of curvature RP at every CC-point K. In this way (see Equation 11.10), the surface generation output is increasing. It is important to note a possibility of machining form surfaces of revolution in compliance with the method (Figure 11.9), when not the cutter, but a

SU Pat. No. 1171210, A Method of Turning of Form Surfaces of Revolution./S.P. Radzevich, Int. Cl. B23B 1/00, Filed November 3, 1984.

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Examples of Implementation of the DG/K-Based Method

9 11

OK.i

P

10

12 1

OP ωP

Ki+1

3

Ki

5 6

2

ОK.i+1 4

8 7

Figure 11.10 Utilization of the orientational motion of the first kind of the cutter in the method of turning of form surfaces of revolution (SU Pat. No. 1232375).

milling cutter or grinding wheel having a corresponding axial profile of the generating surface of the tool is used instead. Similar to the utilization of the orientational motion of the second kind (Figure 11.9), the orientational motion of the first kind of the cutter can be utilized in the turning of surfaces of revolution as well. A method of turning of form surfaces of revolution (Figure 11.10) is featuring in its kinematics the orientational motion of the first kind [8]. This method of surface machining is similar to the earlier discussed method of surface machining shown in Figure 11.9. For convenience, designations of the major elements in Figure 11.10 are identical to the designations of the corresponding major elements in Figure 11.9. So, there is no reason to repeat all the details of the method under consideration. In the method of surface machining (Figure 11.10), the orientational motion of the first kind is utilized. The orientational motion of this kind allows for turning of the cutter about the axis OK .i along the unit normal vector n P either in the direction 9 or in the direction 10. The actual direction of the orientation motion depends upon parameters of geometry of the surface P at the two neighboring CC-points K i and K i+1. In the neighboring CC-point K i+1, the orientation motion is designated as 11/12.

SU Pat. No. 1232375, A Method of Turning of Form Surfaces of Revolution./S.P. Radzevich, Int. Cl. B23B 1/00, Filed September 13, 1984.

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Kinematic Geometry of Surface Machining

Evidently, the parameters of the orientational motion of the first kind are strongly constrained by the limit values of the geometrical parameters of the cutting edge of the cutter to be used, first of all by the clearance angle of the cutting edge. Implementation of the orientational motion of the cutter allows for better fit of the radius of curvature R T to the part surface radius of curvature RP at every CC-point K. In this way (see Equation 11.10), the surface-generation output is increasing. It is the right point to stress that it may also be possible to machine form surfaces of revolution in compliance with the method (Figure 11.10) when not the cutter, but a milling cutter or grinding wheel having a corresponding axial profile of the generating surface of the tool is used instead. Under such a scenario, no constraints are imposed by the limit values of the geometrical parameters of the cutting edge of the cutting tool to be used. 11.2.2 Milling Operations The earlier discussed methods of turning form surfaces of revolution (see Figure 11.9 and Figure 11.10) allow substitution of the cutter with a milling cutter or with a grinding wheel having a corresponding profile of axial cross-section of the generating surface T. These methods of surface machining indicate that efficient methods of milling of form surfaces of revolution can be developed. A method of milling of form surfaces of revolution on NC machine tools was developed by the author [26]. In compliance with the method (Figure 11.11), a form surface of revolution P having axial profile 1 is machined with the milling cutter having a curved axial profile of the generating surface T. The work is rotating about its axis of rotation OP with a certain rotation ω P. The axis

1

Ro.P dP

4

ωT

dT

3

OT

2

dP K

Ro.T

K

5 6

ωP

ωP

ωP

OP A

View А (Turned)

7

8

dT

Figure 11.11 A method of milling a form surfaces of revolution. (See also Radzevich, S.P., and Dmitrenko, G. V., Mashinostroitel’, No. 5, 17–19, 1987.)

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Examples of Implementation of the DG/K-Based Method

of rotation of the work OP and the axis of rotation of the milling cutter OT are crossing at a right angle. The milling cutter is traveling in the direction 3 along the axial profile 1 of the part with a certain peripheral feed rate. This motion is a superposition of the milling cutter motion in the axial direction 4 of the work, and of its motion 5 toward the work axis of rotation and in backward direction 6. In addition to the mentioned motions, the milling cutter is also performing the motion of orientation of the second kind. While traveling in the axial direction 4 of the work, the milling cutter simultaneously performs linear motion along its axis of rotation OT . This motion is performing either in the direction 7 or in the opposite direction 8 depending upon the geometry of the surfaces P and T at the current CC-point K. The orientational motion of the milling cutter provides a possibility for increasing the rate of conformity of the generating surface T of the milling cutter to the form surface of revolution P at every CC-point K. In this way, the surface generation output is increased. Grinding of form surfaces of revolution can be performed in the same way as shown in Figure 11.11. 11.2.3 Machining of Cylinder Surfaces Orientation motions of the cutting tool are also used for the improvement of machining of general cylinder surfaces. Such a possibility is illustrated below by the method of machining of a camshaft. The method of machining of a camshaft [1] is targeting the maximal possible material removal rate. In compliance with the method, a grinding wheel having conical generating surface T is used for the machining of the surface P of a cam (Figure 11.12). The grinding wheel is rotating about its axis of rotation OT with a certain rotation ω T. The surface generation motions are performed by the work. The set of these motions includes the rotation ω P of the work about the axis of rotation OP and the reciprocal motion 1 in the direction of the common perpendicular to the axes OP and OT. The rotation ω P of the grinding wheel can be either uniform or nonuniform. The grinding wheel is performing an auxiliary straight motion 2. Direction of the motion 2 is parallel to the axis of rotation of the work OP . The straight motion 2 is timed with work rotation ω P in the way under which the material removal rate is constant and equal to its greatest feasible value:

Qcr

max

= 0, 5 ⋅ [L(ϕ )]2 ⋅ vT (ϕ ) ⋅ b = Const

(11.11)

SU Pat. No. 1703291, A Method of Machining of Form Surfaces./S.I. Chukhno and S.P. Radzevich, Int. Cl. B23C 3/16, Filed August 2, 1989.

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Kinematic Geometry of Surface Machining D2

D1

View A

T ωT

OT A

T

2 1

1

ωP

OP

OP

ωP

P (a)

P

(b)

Figure 11.12 A method of grinding of a cam (SU Pat. No. 1703291).

where L(ϕ ) is the length of the line of contact of the grinding wheel with the stock to be removed, vT (ϕ ) is the rate of change of the leading coordinate, b is the width of the part surface being machined, and j is the angular coordinate of the currently machining portion of the part surface P. In order to stabilize the maximal feasible material removal output, the corresponding change of the length L(ϕ ) is provided by the auxiliary straight motion 2. In addition, the grinding wheel rotation ω T is timed with the auxiliary straight motion 2. Timing of the motions is performed in inverse order to the change of working diameter of the grinding wheel. When working diameter of the grinding wheel is D1, then the corresponding rotation is ω T.1. When working diameter of the grinding wheel is D2, then the corresponding rotation is ω T.2 . The following equality ω T .1 ⋅ D1 = ω T .2 ⋅ D2 is observed. The method of machining of a camshaft (Figure 11.12) allows for the highest possible material removal output and high quality of the machined parts. 11.2.4 Reinforcement of Surfaces of Revolution The reinforcement of part surfaces by plastic deformation is often used for the finishing of parts. Optimal parameters of such a machining operation as well as optimal design parameters of the tool that is used for these purposes can be determined on the premises of the DG/K-based method of surface generation. Examples below illustrate the capabilities of the DG/K-based method of surface generation for the improvement of the finishing of surface of revolution.

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Examples of Implementation of the DG/K-Based Method 1 OT 3

K

4

2

ωP

OP

P

Figure 11.13 A method of finishing of a form surface of revolution with a conical indenter.

Reinforcement of form surfaces of revolution can be performed with the tool having a conical generating surface T [21,22]. In this method (Figure 11.13), the work is rotating about its axis OP with a certain rotation ω P . The axis OT of the conical indenter 1 (conical tool) is crossing the work axis of rotation OP at the right angle. The tool is moving along the axial profile of the part surface P with a certain peripheral feed rate. The indenter 1 is pressed into the part surface P by normal force Prnf . In the relative motion, the CC-point K traces the trajectory 2 on the machined part surface. Two configurations of the indenter 1 are possible. The first configuration is shown in Figure 11.13. In such a tool configuration, its bigger diameter is below the smaller diameter. The inverse configuration of the tool, when the smaller diameter is below the bigger diameter, is feasible as well. When machining a form surface of revolution, the portion of the surface P having bigger diameter is machined with the portion of the tool having smaller diameter, and vice versa. In this way, it is possible to maintain that same pressure when machining portions of the part with different geometry of the surface P. For this purpose, the indenter is performing an auxiliary straight motion either downward 3 or upward 4, depending on the geometry of the surface P being machined. The auxiliary motion requires in corresponding compensation of center distance between the axes OP and OT . A component of the auxiliary straight motion creates the orientational motion of the second kind of the tool. Reinforcement of the part surfaces under the optimal pressure that is of the same value at every CC-point K enables an increase of the quality of the surface finish. For reinforcement of form surfaces of revolution, not only a conical tool but a cylindrical tool can be used as well. In the method of reinforcement of a

SU Pat. No. 1463454, A Method of Reinforcement of Surfaces./S.P. Radzevich, Int. Cl. B24B 39/00, 39/04, Filed May 5, 1987.

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Kinematic Geometry of Surface Machining 1

(i)

dP

ωP

O** i

α*i

O*i

ωn

α** i

K

OP

2

P

Figure 11.14 A method of reinforcement of a form surface of revolution with a cylindrical tool (SU Pat. No. 1463454).

form surface of revolution [13], finishing of the part surface is performed with the cylindrical indenter. When machining the surface P, the work is rotating about its axis OP with a certain rotation ω P (Figure 11.14). The cylindrical indenter 1 is pressed into the part surface P by normal force Prnf . The tool 1 is traveling along the axial profile of the part surface P with a certain peripheral feed rate. Simultaneously, the tool 1 is performing the orientational motion of the first kind w n about unit normal vector n P to the part surface P. The ori(i ) entational motion of the tool is timed with part diameter dP at the current CC-point K. Due to the orientational motion of the tool, the angle that the axis OP of the part makes with the axis Oi at the current ith point is under the control of the user. At every CC-point K, the angle of crossing α i is of its optimal value. When the diameter dP(i ) is bigger, then the cross-axis angle α i is also bigger, and vice versa. In this way, the optimal pressure that is of the same value at every CC-point K is maintained. In particular cases, two paths of the indenter 1 are required to be performed. On the second tool-path, the angle that the axis OP of the part makes with the axis Oi at the current ith point is reduced to a value α i. On the second tool-path, angle α i at the current CC-point K is always smaller than that angle α i on the first tool-path (α i < α i). Reinforcement of the part surfaces under the optimal pressure that is of the same value at every CC-point K enables for an increase in the quality of the surface finish. Similarly, reinforcement of part surfaces of revolution can be performed with a form roller. For example, a method of reinforcement of a surface of revolution is featuring the implementation of a form tool [15]. The method of reinforcement of form surfaces of revolution is illustrated with an example of finishing of a cylindrical part surface P (Figure 11.15). However, the method of surface finishing can be implemented for the reinforcement form surfaces of revolution as well.

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Examples of Implementation of the DG/K-Based Method ωP

K1 *

R(min)

T

(max) ** RT

T

ωT

K2

T

Prnf

ωT

K1 θ

RT(min) Prnf

Figure 11.15 A method of reinforcement of a surface of revolution with a form tool (SU Pat. No. 1636196).

In compliance with the method of surface machining, the work is rotating about its axis of rotation with a certain rotation ω P (Figure 11.15). For the machining, a roller having a form axial profile is used. The radius of curvature R T of the axial profile of the roller varies (R T = Var ). The roller is pressed into the part surface P by normal force Prnf . When machining the part surface, the roller is traveling along the axial profile of the part surface P. In this motion, the roller rotates about its axis of rotation with a certain rotation ωT. The roller is driven due to friction between the part surface P and the working surface of the roller. On a rough tool-path, the roller is contacting the part surface with the point K1 of the generating profile of its working surface. The radius of curvature of the axial profile of the tool at the point K1 is minimal R(min) . On the finT ishing tool-path, the roller is contacting the part surface with the point K 2 of the generating profile of its working surface. The radius of curvature of the axial profile of the tool at the point K 2 is maximal R(max) . On the rough T tool-path, the tool approach angle ϕ T is bigger than that ϕ T on the finishing tool-path. The turn of the roller through a certain angle q can be interpreted as the degenerated kind of orientational motion of the second kind of the tool. Utilization of the degenerated orientational motion of the second kind in the method of surface finishing (Figure 11.15) makes it possible to maintain optimal conditions of surface reinforcement at every CC-point K on both the rough tool-path of the roller, as well as on its finishing tool-path. Ultimately, this improves the quality of the finished part surface.

SU Pat. No. 1636196, A Method of Reinforcement of Surfaces./S.P. Radzevich and V.V. Novodon, Int. Cl. B24B 39/00, Filed January 30, 1991.

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Kinematic Geometry of Surface Machining

11.3 Finishing of Involute Gears Various methods of shaving are widely used for finishing spur and helical involute gears [28]. Most gear shaving operations are not optimized. Computation of the optimal parameters of a diagonal shaving operation provides a perfect example of implementation of the DG/K-based method of surface generation. In compliance with the method, it is possible to compute the desired design parameters of the shaving cutter best suited for finishing the given involute gear. It is also possible to compute the optimal parameters of the relative motions of the shaving cutter with respect to the gear to be finished. For this purpose, the indicatrix of conformity Cnf R ( Pg /Tsh ) of the generating surface Tsh of the shaving cutter to the screw involute tooth surface Pg of the gear is commonly employed. In diagonal shaving (Figure 11.16), the work-gear rotates about its axis O g with a certain angular velocity ω g . The shaving cutter rotates about its axis Osh with an angular velocity ω sh that is timed with the ω g — that is, ω sh = u ⋅ ω g , where u is the tooth ratio ( u = N g N sh ; here N g is the number of the gear teeth, and N sh is the number of the shaving cutter teeth). Axes of rotation O g of the gear and Osh of the shaving cutter are at a centerdistance C, and cross each other at an angle Σ . The angle Σ is as follows: Σ = ψ g + ψ sh. Here ψ p is the gear helix angle. It is positive (+) to the righthand gear and negative (−) to the left-hand gear to be machined. The same is observed with respect to the shaving cutter helix angle ψ sh. In addition, the

Shaving Cutter

Shaving Cutter Fdiag

ωsh Osh

L

Fdiag C

ωg

θ

Work Gear

Osh

C

Σ

ωg

K1

Work Gear

θ

K2

Og

Og

ωsh

Bg

Figure 11.16 Schematic of a diagonal shaving method. (From Radzevich, S.P., International Journal of Advanced Manufacturing Technology, 32 (11–12), 1170–1187, 2007. With permission.)

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shaving machine table reciprocates relative to the shaving cutter with feed Fdiag. The axis of rotation O g of the gear and direction of the feed Fdiag make a certain angle q. The traverse path of the feed Fdiag is at a certain angle q to the gear axis of rotation O g (Figure 11.16). The relationship between the face width of the gear Bg and the shaving cutter Bsh is an important consideration. It defines the value of the diagonal traverse angle. The surface of tolerance P[ h ] is at a distance of the tolerance [ h] to the geartooth surface Pg. After tooth surface Pg of a gear and tooth surface Tsh of a shaving cutter are put into contact at point K, then the surface Tsh intersects the surface P[ h ]. The line of intersection is a certain closed three-dimensional curve C pt shown in Figure 11.17. It bounds the spot of contact of the gear and the shaving cutter tooth. It is recommended that the area of the spot of contact C pt be kept as small as possible (Figure 11.17). Due to the tooth surfaces Pg and Tsh making contact at a distinct point K, only discrete generation of the gear flank is feasible. In order to increase productivity of the gear finishing operation, it is required to maintain the tool-paths on the gear-tooth flank Pg as wide as possible. For this purpose, the major axis of the spot of contact C pt has to be as long as possible, and relative motion VΣ of the surfaces Pg and Tsh has to be directed orthogonally to the major axis of the spot of contact C pt. Tooth of the Shaving Cutter sh

K g

Tooth of the Work-Gear VΣ g

χ ≠ 90

g

K pt

Figure 11.17 The problem at hand. (From Radzevich, S.P., International Journal of Advanced Manufacturing Technology, 32 (11–12), 1170–1187, 2007. With permission.)

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Kinematic Geometry of Surface Machining

Fortunately, it is possible to control the shape, size, and orientation of the spot of contact C pt. For this purpose, an optimal combination of the design parameters of the shaving cutter, of direction and speed of the feed Fdiag , of rotation of the gear ω g, and of rotation of the shaving cutter ω sh must be computed. This also makes possible the control of the direction of relative motion of the surfaces Pg and Tsh , and in such a way as to increase the gear accuracy and to cut the shaving time. For the analysis below, equations of the tooth flank surfaces Pg and Tsh are necessary. The equation of the gear-tooth surface Pg can be represented in the form of the column matrix (see Equation 1.20): rb. g cos Vg + U g cos ψ b. g sin Vg   r sin V − U sin ψ sin V  b. g g g b. g g  rg =   rb. g tan ψ b. g − U p sin ψ b. g    1  

(11.12)

where the gear base cylinder diameter db. g = 2 rb. g can be computed from db. g =

m ⋅ N g ⋅ cos φn 1 − cos 2 φn sin 2 λ b. g

=

25.4 ⋅ N g ⋅ cos φn Pg ⋅ 1 − cos 2 φn sin 2 λ b. g

(11.13)

where m is the gear modulus, N g is the number of gear teeth, φn is the normal pressure angle, λ b. g is the gear base lead angle ( λ b. g = 90o − ψ b. g ), ψ b. g is the gear base lead angle, and Pg is the diametral pitch. The Ug parameter in Equation (11.12) can be expressed in terms of parameters of the gear design [22,27]: Ug =

dy2. g − db2. g 2 sin ψ b. g

=

dy2. g − db2. g 2 sin ψ g sin φn

(11.14)

where the diameter of a cylinder that is coaxial to the gear is designated as dy . g , and ψ g is the gear pitch helix angle. Equation (1.7) yields computation of the fundamental magnitudes of the first order Eg = 1, Fg = −

rb. g cos ψ b. g

, Gg =

U g2 cos 4 ψ b. g + rb2. g cos 2 ψ b. g

(11.15)

for the screw involute surface Pg . For the fundamental magnitudes of the second order, use of Equation (1.11) returns expressions

Lg = 0, M g = 0, N g = −U g ⋅ sin τ b. g ⋅ cos τ b. g

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(11.16)

483

Examples of Implementation of the DG/K-Based Method for the coefficients Lg, M g, and N g for the screw involute surface Pg. Equation (1.4) returns  sin ψ b. g sin Vg  sin ψ cos V  b. g g ng =    cos ψ b. g   1  

(11.17)

for the unit normal vector n g to the gear-tooth surface Pg . Computations similar to those above must be performed for the generating surface Tsh of the shaving cutter. Vector VΣ of the resultant relative motion of the surfaces Pg and Tsh passes through the point K, and it is located in a common tangent plane to the surfaces Pg and Tsh . Consider a plane through the unit normal vector n g that is orthogonal to the direction of VΣ . Radii of curvature of the surfaces Pg and Tsh in this cross-section differ. The width of the tool-path over the lateral tooth surface Pg depends upon the direction of the vector VΣ . By varying the direction of feed Fdiag , say timing in various manner ( angular velocities w g and w sh with feed Fdiag , tool-paths of various width Fi could be obtained. The shortest shaving time, and the highest accuracy of the involute ( gearF tooth surface could be obtained if and only if the feed rate per tooth of the ( (i(max) shaving cutter remains equal( to its maximal value — that is, if . In F F = i cnf ( (max) order to make the equality Fi = Fcnf valid, it is necessary to remain at the highest possible rate of conformity of the surface Tsh to the surface Pg. In general, the rate of conformity of an involute gear-tooth surface Tsh to the involute tooth surface Pg at the point K varies, as the normal plane section rotates around the common unit vector n g. The direction of the major axis of the spot of contact aligns with the direction at which the highest rate of conformity of the involute tooth surfaces Pg and Tsh is observed. The tangent plane to the gear-tooth surface Pg at the point K is the plane through two unit tangent vectors u g and v g . These yield the equation for the tangent plane through point K (i.e., through the point r K) on the gear-tooth surface Pg : (rg .tang − rK ) × u g ⋅ v g = 0,

where rg.tang is the position vector of a point of the tangent plane. The angle of the gear and of the local relative orientation of the shaving cutter tooth surfaces (see Equation 4.1 through Equation 4.3) is equal: sin µ =

(1 − cos

sin φn ⋅ sin Σ

2

)

φn ⋅ sin 2 ψ g ⋅ (1 − cos 2 φn ⋅ sin 2 ψ sh )

(11.18)

where φn is the normal pressure angle, ψ g is the gear helix angle, ψ sh is the shaving cutter helix angle, and Σ is the gear and shaving cutter crossed axes angle.

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Kinematic Geometry of Surface Machining

For the case under consideration, the equation of the indicatrix of conformity Cnf R ( Pg /Tsh ) can be derived from the general form of equation of this characteristic curve (see Equation 4.59). The first f1.g, and the second f 2.g, f 2.sh fundamental forms are initially computed in the coordinate systems X g Yg Z g and X sh Ysh Zsh, correspondingly (see Figure 11.18). It is necessary to convert these expressions to the common local coordinate system x g y g z g. Such a transformation can be performed by means of the formula of quadratic form transformation (see Equation 3.37 and Equation 3.38): [φ1,2.g ( sh ) ]k = Rs T (1 → 2) ⋅ [φ1,2.g ( sh ) ]g ( sh ) ⋅ Rs((1 → 2)

Σ

Ysh

(11.19)

Osh

Zsh

ωsh

Xsh

rsh

Tsh

yg zg C

xg

t2.g

ng

Sdiag

K

µ

t t2.sh

Pg

rg Xg Zg

zg Yg

ωg

Og

ng

C2.g µ

K

xg t2.sh t2.g

Cg

C2.sh

Csh yg

Figure 11.18 The major coordinate systems. (From Radzevich, S.P., International Journal of Advanced Manufacturing Technology, 32 (11–12), 1170–1187, 2007. With permission.)

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Examples of Implementation of the DG/K-Based Method

where [φ1,2.g ( sh ) ]g ( sh ) and [φ1,2.g ( sh ) ]k are the fundamental forms of the surfaces Pg (Tsh ), initially represented in the coordinate systems X g Yg Z g and X sh Ysh Zsh , and finally in the common coordinate system x g y g z g. In the local coordinate system x g y g z g, the equation for the Cnf R ( Pg /Tsh ) casts into

Indicatrix of conformity Cnf R ( Pg / Tsh ) ⇒ rcnf ( R1. g , R1. sh , µ , ϕ ) (11.20)

R1. g

R1. sh = + sin ϕ sin( µ − ϕ )

where rcnf ( R 1. g , R 1. sh , µ , ϕ ) is the position vector of a point of the characteristic curve Cnf R ( Pg /Tsh ) — for finishing of a given gear, the function rcnf ( R 1. g , R 1. sh , µ , ϕ ) reduces to rcnf ( R 1. sh , µ , ϕ ) ; and j is the polar angle (further the argument ϕ is employed for determining the optimal direction of resultant relative motion VΣ of the surfaces Pg and Tsh ). The characteristic curve Cnf R ( Pg /Tsh ) is depicted in Figure 11.19. The rate of conformity of the surfaces Pg and Tsh in the normal cross-section through (min) the minimal diameter dcnf (or, the same, through the direction t (max) of the cnf maximal rate of conformity of the surfaces Pg and Tsh ) is the highest possible (Figure 11.20). This plane section of the surfaces Pg and Tsh is referred to as the optimal normal cross-section.

ysh

VΣ

R2.g

t1.sh

(min)

µ

CnfR (Pg /Tsh)

(Pg /Tsh)

opt

90°

dcnf

(im)

CnfR

yg

(max) tcnf

xsh

CnfR (Pg /Tsh)

t1.g t2.sh K

t2.g

xg opt

(min) –tcnf (min)

dcnf (im)

(im)

CnfR

(Pg /Tsh)

R2.sh

Figure 11.19 The indicatrix of conformity Cnf R ( Pg /T sh ) of the tooth flanks Pg and Tsh. (From Radzevich, S.P., International Journal of Advanced Manufacturing Technology, 32 (11–12), 1170–1187, 2007. With permission.)

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486

Kinematic Geometry of Surface Machining ng

Rsh

(max)

VΣ = VΣ . tcnf

ng

Shaving Cutter

P[h] Pg

(max)

tcnf

K

Tsh

(max) F˘cnf

[h]

Rg

Involute Pinion

Figure 11.20 The cross-section of the tooth surfaces Pg and Tsh by optimally oriented normal plane. (From Radzevich, S.P., International Journal of Advanced Manufacturing Technology, 32 (11–12), 1170–1187, 2007. With permission.)

The equation of the indicatrix of conformity Cnf R ( Pg /Tsh ) can be expressed in terms of design parameters of the involute gear and of the shaving cutter: rcnf (ϕ , µ ) =

2 sin ψ g sin φn d

2 y. g

−d

2 b. g

cos λb. g sin ϕ 2

+

d

2 y . sh

2 sin ψ sh sin φn − db2. sh cos λb. sh sin 2 ( µ − ϕ ) (11.21)

which contains in condensed form all information necessary for the computation of optimal design parameters of the shaving cutter, and of optimal parameters of the diagonal shaving operation. Elements of local topology of the surfaces Pg and Tsh relate to the lateral surface of the auxiliary phantom rack R of the shaving cutter. Location and relative orientation of the characteristic curve Cnf R ( Pg /Tsh ) are illustrated in Figure 11.21. The major axis of the spot of contact of the involute surfaces Pg and (min) Tsh aligns with the minimal diameter dcnf of the characteristic curve Cnf R ( Pg /Tsh ). This axis is within the angle that makes the characteristic E g of the surface Pg and the characteristic Esh of the surface Tsh . The characteristics Eg and Esh (Figure 11.17) are the straight lines along which the involute surfaces Pg and Tsh make contact with the corresponding lateral plane surface of the auxiliary phantom rack R . It is important to stress here that the major axis and the minor axis of the spot of contact are not orthogonal to each other. Generally, they are at an angle χ ≠ 90o. The involute gear and the shaving cutter relative motion VΣ at the point K is directed orthogo(min) nally to the diameter dcnf (to the unit tangent vector t (max) ). The cutting cnf edge of the shaving cutter makes an angle of inclination i with the direction of t (max) . It is important to maintain this angle equal to its optimal value iopt . cnf

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487

Examples of Implementation of the DG/K-Based Method yg CnfR (Pg /Tsh)

VΣ opt

sh

iopt

iopt

(min) dcnf

90° ng

pt

K

xg

(max) tcnf

g

opt

zg CnfR (Pg /Tsh) Figure 11.21 Elements of local topology of the tooth surfaces Pg and Tsh referred to the lateral plane of the auxiliary phantom rack R. (From Radzevich, S.P., International Journal of Advanced Manufacturing Technology, 32 (11–12), 1170–1187, 2007. With permission.)

The same angle iopt makes vector VΣ with the perpendicular to the cutting edge (Figure 11.21). At every point of the tooth flank Pg, the first principal curvature k 1. g is uniquely determined by the topology of the surface Pg . The second principal curvature k 2. g of the screw involute surface Pg is always equal to zero ( k 2. g ≡ 0 ). Similarly, the second principal curvature k 2. sh of the screw involute surface Tsh is also always equal to zero ( k 2. sh ≡ 0 ). At this point, the rest of the parameters of the indicatrix of conformity Cnf R ( Pg /Tsh ) of the surfaces Pg and Tsh (that is, the parameters R 1. sh, j, and m) can be considered as the variable parameters. It is necessary to determine the optimal combinaopt opt opt opt tion of values of the parameters R 1. sh = R 1. sh (U g , Vg ), ϕ = ϕ (U g , Vg ) , and opt opt µ = µ (U g , Vg ). If the proper combination of the parameters Ropt ϕ opt , 1. sh, opt and µ is determined, then computation of the optimal design parameters of the shaving cutter and of the optimal parameters of kinematics of the diagonal shaving operation turns to the routing engineering calculations. The indicatrix of conformity Cnf R ( Pg /Tsh ) reveals how close the tooth surface Tsh of the shaving cutter is to the gear-tooth surface Pg in every crosssection of the surfaces Pg and Tsh by normal plane through K. It enables specification of an orientation of the normal plane section, at which the surfaces Pg and Tsh are extremely close to each other — that is, the normal plane section through the unit tangent vector t (max) in the direction of the maximal cnf rate of conformity of the surfaces Pg and Tsh . This normal plane section satisfies the following conditions:

∂ rcnf ∂ rcnf ∂ rcnf = 0. = 0, and = 0, ∂ϕ ∂µ ∂R 1. sh

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Kinematic Geometry of Surface Machining

Equation (11.20) of the indicatrix of conformity Cnf R ( Pg /Tsh ) yields the following necessary conditions of the maximal rate of conformity of the shaving cutter tooth surface Tsh to the involute gear-tooth surface Pg:

The Necessary Conditions for the Minimal Shav ving Time and the Maximal Accuracy of the Shaaved Involute Gear

 ∂ rcnf 1 = =0  R 1. sh sin( µ − ϕ )  ∂R 1. sh  cos( µ − ϕ )  ∂ rcnf =− ⇒ R 1. sh = 0 2 (µ − ϕ ) ∂ µ sin   ∂ rcnf cos ϕ cos( µ − ϕ ) =− R 1. g − R 1. sh = 0  2ϕ 2 (µ − ϕ ) ∂ ϕ sin sin   (11.22)

The sufficient conditions for the maximum of the function rcnf ( R 1. sh , µ , ϕ ) of three variables are also satisfied. The first equality in Equation (11.22) consists in condensed form all the necessary information on the optimal design parameters of the shaving cutter. Analysis of this equality reveals that it could be satisfied when R1.sh → ∞. Thus, for a conventional diagonal shaving operation when the gear and the shaving cutter are in external mesh, it is recommended to finish the gear with the shaving cutter of the maximal possible pitch diameter. In the ideal case, the gear can be shaved with a rack-type shaving cutter. Application of the shaving cutter of larger pitch diameter increases the difference between pitch diameters of the gear and of the shaving cutter. This yields a larger rate of conformity of the surfaces Pg and Tsh. Actually, the pitch diameter of the shaving cutter to be applied for a rotary shaving operation is restricted by the design of a shaving machine. Analysis of the function R 1. sh = R 1. sh (φ n , ψ sh ) reveals that the rate of conformity of the surfaces Pg and Tsh increases when both normal pressure angle φn and helix angle ψ sh are smaller — that is, φn → 0o and ψ sh → 0o . The interested reader may wish to refer to [20] for details of the analysis. The second and the third equalities in Equation (11.22) together enable one to give an answer to the question on the optimal relative orientation of the surfaces Pg and Tsh ( µ → 0o , however, the inequality Σ ≠ 0o is required) and on the optimal parameters of instant kinematics of diagonal shaving ( ϕ = ϕ opt ). The resultant relative motion VΣ of the surfaces Pg and Tsh is decomposed on its projections onto directions of the motions to be performed on the gear-shaving machine. Vector Vsl of the velocity of relative sliding of the surfaces Pg and Tsh is located in the common tangent plane. It is convenient to decompose the vector Vsl at the point K onto two components Vsl = Vφ + Vψ . The first component

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Examples of Implementation of the DG/K-Based Method

Vφ represents sliding along the tooth profile, and the second component Vψ represents sliding in the longitudinal tooth direction. The feed Fdiag is directed parallel to the plane surface that is tangent to pitch cylinders of the gear and of the shaving cutter. It also affects the resultant speed VΣ of cutting ( VΣ = Vsl + Fdiag). Varying parameters of the diagonal shaving operation and of design parameters of the shaving cutter enable one to control the resultant speed VΣ = Vsl + Fdiag of cutting. For this purpose, the speed and direction of the shaving machine reciprocation and shaving cutter rotation have to be timed with each other. In the local coordinate system x g y g z g (Figure 11.22), the vector VΣ of the resultant motion makes a certain angle ϕ Σ with the y g axis. Thus, |VΣ |⋅ sin ϕ Σ  |V |⋅ cos ϕ  Σ Σ VΣ =    0   1  

The Shaving Cutter

(11.23)

Zk

ωsh

Fdiag

Osh Og

Zk

C

Σ Vg = Rw.g . wg

Vsh = Rw sh . ωh

ωsh

Vyz = Pr(VΣ* )yz

The Work-Gear Σ

0.5 Σ

Vsl = Vg + Vsh Figure 11.22 Timing of the feed Fdiag with rotations of the involute gear and of the shaving cutter. (From Radzevich, S.P., International Journal of Advanced Manufacturing Technology, 32 (11–12), 1170–1187, 2007. With permission.)

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Kinematic Geometry of Surface Machining

To represent the vector VΣ in the global coordinate system X k Yk Zk (Figure 11.22), the operator Rs( g → k ) of the resultant coordinate system transformation is used: |VΣ | |V | Σ VΣ = Rs ( g → k ) ⋅ VΣ =   |VΣ |    1 

(11.24)

Equation (11.24) yields the projection Pryz (VΣ ) of the vector VΣ onto the coordinate plane Yk Zk :  0  |V | Σ Pryz (VΣ ) =   |VΣ |    1 

(11.25)

Relative sliding Vsl of the tooth surfaces of the gear and the shaving cutter can be computed by

Vsl = Vg + Vsh = R w. g ⋅ ω g + R w. sh ⋅ ω sh

(11.26)

where Vg , Vsh are the linear velocities of the rotations g and sh, respectively; and R w. g, R w. sh are radii of pitch cylinders of the gear and the shaving cutter. And,

|Vsl|= 2⋅|ω g|⋅ R w. g ⋅ cos( 0.5 ⋅ Σ) = 2⋅|ω sh|⋅ R w. sh ⋅ cos(0.5 ⋅ Σ)

(11.27)

In the coordinate plane Yk Zk, the resultant motion Vyz of the gear and the shaving cutter can be represented as follows:

Vyz = Vsl + Fdiag

(11.28)

Fdiag = Vsl − Vyz

(11.29)

Thus, reciprocation is equal to

This is the way the values of the shaving cutter rotation and its reciprocation are timed with each other. The synthesized method of diagonal shaving of involute gears is disclosed in detail in [4,20,29,30].

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Examples of Implementation of the DG/K-Based Method

491

References [1] Pat. No. 1703291, USSR, A Method of Machining of Form Surfaces./S.I. Chukhno and S.P. Radzevich, Int. Cl. B23C 3/16, Filed August 2, 1989. [2] Pat. No. 4.415.977, US, Method of Constant Peripheral Speed Control./F. Hiroomi and I. Shinichi, Int. Cl. B23 15/10, 05B 19/18, National Cl. 700/188, 318/571, Filed March 2, 1981; No. 243928, Priority June 30, 1979; No.54-82779, Japan. [3] Ligun, A.A., Shumeiko, A.A., Radzevich, S.P., and Goodman, E.D., Asymptotically Optimal Disposition of Tangent Points for Approximation of Smooth Convex Surfaces by Polygonal Functions, Computer Aided Geometric Design, 14, 533–546, 1997. [4] Palaguta, V.A., The Development and Investigation of Methods for Increasing Productivity of Shaving of Cylindrical Gears, PhD thesis, Kiev Polytechnic Institute, Kiev, 1995. [5] Pat. No. 4242296/08, USSR, A Method of Design of a Form Cutting Tool for Sculptured Surface Machining on Multi-Axis NC Machine./S.P. Radzevich, Filed 31.03.1987. [6] Pat. No. 1171210, USSR, A Method of Turning of Form Surfaces of Revolution./ S.P. Radzevich, B23B 1/00, Filed November 24, 1984. [7] Pat. No. 1185749, USSR. A Method of Sculptured Surface Machining on MultiAxis NC Machine./S.P. Radzevich, B23C 3/16, Filed October 24, 1983. [8] Pat. No. 1232375, USSR, A Method of Turning of Form Surfaces of Revolution./ S.P. Radzevich, B23B 1/00, Filed September 13, 1984. [9] Pat. No. 1249787, USSR, A Method of Sculptured Surface Machining on MultiAxis NC Machine./S.P. Radzevich, B23C 3/16, Filed December 27, 1984. [10] Pat. No. 1336366, USSR, A Method of Sculptured Surface Machining on MultiAxis NC Machine./S.P. Radzevich, B23C 3/16, Filed October 21, 1985. [11] Pat. No. 1367300, USSR, A Method of Sculptured Surface Machining on MultiAxis NC Machine./S.P. Radzevich, B23C 3/16, Filed January 30, 1986. [12] Pat. No. 1442371, USSR, A Method of Optimal Workpiece Orientation on the Worktable of Multi-Axis NC Machine./S.P. Radzevich, Filed February 17, 1987. [13] Pat. No. 1463454, USSR, A Method of Reinforcement of Surfaces./S.P. Radzevich, Int. Cl. B24B 39/00, 39/04, Filed May 5, 1987. [14] Pat. No. 1533174, USSR, A Method of Reinforcement of Sculptured Surface on MultiAxis NC Machine./S.P. Radzevich, Int. Cl. B24B 39/00, Filed December 2, 1987. [15] Pat. No. 1636196, USSR, A Method of Reinforcement of Surfaces./S.P. Radzevich and V.V. Novodon, Int. Cl. B24B 39/00, Filed January 30, 1991. [16] Pat. No. 1708522, USSR, A Method of Turning of Form Surfaces of Revolution./ S.P. Radzevich, B23B 1/00, Filed December 13, 1988. [17] Pat. No. 2050228, Russia, A Method of Sculptured Surface Machining on MultiAxis NC Machine./S.P. Radzevich, B23C 3/16, Filed December 25, 1990. [18] Radzevich, S.P., Advanced Technological Processes of Sculptured Surface Machining, VNIITEMR, Moscow, 1988. [19] Radzevich, S.P., Conditions of Proper Sculptured Surface Machining, ComputerAided Design, 34 (10), 727–740, 2002. [20] Radzevich, S.P., Diagonal Shaving of an Involute Pinion: Optimization of the Geometric and Kinematic Parameters for the Pinion Finishing Operation, International Journal of Advanced Manufacturing Technology, 46 (7–8), October 2007.

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[21] Radzevich, S.P., Differential-Geometric Method of Surface Generation, DrSci thesis, Tula, Polytechnic Institute, 1991. [22] Radzevich, S.P., Fundamentals of Surface Generation, Rastan, Kiev, 2001. [23] Radzevich, S.P., Methods of Milling of Sculptured Surfaces, VNIITEMR, Moscow, 1989. [24] Radzevich, S.P., R-Mapping Based Method for Designing of Form Cutting Tool for Sculptured Surface Machining, Mathematical and Computer Modeling, 36 (7–8), 921–938, 2002. [25] Radzevich, S.P., Sculptured Surface Machining on Multi-Axis NC Machine, Vishcha Shkola, Kiev, 1991. [26] Radzevich, S.P., and Dmitrenko, G.V., Machining of Form Surfaces of Revolution on NC Machine Tool, Mashinostroitel’, No. 5, 17–19, 1987. [27] Radzevich, S.P., Goodman, E.D., and Palaguta, V.A., Tooth Surface Fundamental Forms in Gear Technology, University of Niš, the Scientific Journal Facta Universitatis, Series: Mechanical Engineering, 1 (5), 515–525, 1998. [28] Radzevich, S.P., and Palaguta, V.A., Advanced Methods in Gear Finishing, VNIITEMR, Moscow, 1988. [29] Radzevich, S.P., and Palaguta, V.A., CAD/CAM System for Finishing of Cylindrical Gears, Mekhanizaciya i Avtomatizaciya Proizvodstva, No. 10, 13–15, 1988. [30] Radzevich, S.P., and Palaguta, V.A., Synthesis of Optimal Gear Shaving Operations, Vestink Mashinostroyeniya, No. 8, 36–41, 1997. [31] Radzevich, S.P. et al., On the Optimization of Parameters of Sculptured Surface Machining on Multi-Axis NC Machine, In Investigation into the Surface Generation, UkrNIINTI, Kiev, No. 65-Uk89, pp. 57–72, 1988.

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Conclusion A novel method of surface generation for the purposes of surface machining on a multi-axis numerical control machine, as well as on a machine tool of conventional design is disclosed in this monograph. The method is developed on the premises of wide use of Differential Geometry of surfaces, and of elements of Kinematics of multiparametric motion of rigid body in Euclidian space. Due to this, the proposed method is referred to as the DG/K-based method of surface generation. The DG/K-based method is targeting synthesizing of optimal methods of part surface machining, and of optimal form-cutting tools for machining of surfaces. A minimal amount of input information is required for the implementation of the method. Potentially, the method is capable of synthesizing optimal surface machining processes on the premises of just the geometry of the part surface to be machined. However, any additional information on the surface machining process, if any, can be incorporated as well. Ultimately, the use of the DG/K-based method of surface generation enables one to get a maximal amount of output information on the surface machining process while using for this purpose a minimal amount of input information. The last illustrates the significant capacity of the disclosed method of surface generation. The developed DG/K-based method of surface generation is a cornerstone of the subject theoretical machining/production technology to study by university students.

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Notation Ank (P) Ank (P/ T) Anl k (T) Anl R (P) An R (P/ T) Anl R (T) CC–point Cnf k(P/ T) Cnf R(P/ T) Cpi[i a (i ± 1)] Crv(P) Crv(T) C1.P , C2.P C1.T, C2.T Ds(P/ T) Dup(P) Dup(P/ T) Dup(R ) Dup(P) E EP, FP , GP ET, FT, GT Eu(y, q, j) ( Ffr

Andrew’s indicatrix of normal curvature of the surface P Andrew’s indicatrix of normal curvature of the surfaces P and T Andrew’s indicatrix of normal curvature of the generating surface T of the cutting tool Andrew’s indicatrix of radii of normal curvature of the surface P Andrew’s indicatrix of normal radii of curvature of the surfaces P and T Andrew’s indicatrix of radii of normal curvature of the generating surface T of the cutting tool Cutter contact point Indicatrix of conformity of the part surface P and of the generating surface T of the cutting tool at the current contact point K (normal curvatures) Indicatrix of conformity of the part surface P and of the generating surface T of the cutting tool at the current contact point K (radii of normal curvatures) Couple of elementary coordinate system transformation Curvature indicatrix of the surface P Curvature indicatrix of the generating surface T of the cutting tool The first and the second principal plane sections of the part surface P The first and the second principal plane sections of the generating surface T of the cutting tool Matrix of the resultant displacement of the cutting tool with respect to the part surface P Dupin’s indicatrix of the surface P Dupin’s indicatrix of the surface of relative curvature R Dupin’s indicatrix of the surface of relative curvature R Dupin’s indicatrix of the generating surface T of the cutting tool A characteristic line Fundamental magnitudes of the first order of the surface P Fundamental magnitudes of the first order of the generating surface T of the cutting tool Operator of the Eulerian transformation Feed rate per tooth of the cutting tool 495

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496 ( [ Ffr ] F(fr Fss( [ Fss ] Fss F 1, F 2, F 3 Glnd(P) Glnd(T) GMap(P) GMap(T) G P, G T HP HT Jr k(P/ T) Jr R(P/ T) K LP, MP, NP LT, MT, NT Mch(P/ T) M P, M T NP NT P P mr P sg P l k(P) P l k(T) P l R(P) P l R(T) R Rfx(YiZi)

Kinematic Geometry of Surface Machining Limit feed rate per tooth of the cutting tool Vector of the feed-rate motion of the cutting tool Magnitude of the side step of the cutting tool Limit magnitude of the side step of the cutting tool Vector of the side-step motion of the cutting tool The rate of degree of conformity functions Gauss’ indicatrix of the surface P Gauss’ indicatrix of the generating surface T of the cutting tool Gauss’ map of the surface P Gauss’ map of the generating surface T of the cutting tool Full (Gaussian) curvature of the surface P, and of the generating surface T of the cutting tool Discriminant of the first fundamental form of the surface P Discriminant of the first fundamental form of the generating surface T of the cutting tool A planar curvature characteristic curve A planar radii of curvature characteristic curve Point of contact of the surfaces P and T (or a point within the line of contact of the surfaces P and T) Fundamental magnitudes of the second order of the surface P Fundamental magnitudes of the second order of the generating surface T of the cutting tool Indicatrix of machinability of the surface P with the cutting tool T Mean curvature of the surface P, and of the generating surface T of the cutting tool Perpendicular to the surface P Perpendicular to the generating surface T of the cutting tool Part surface to be machined Chip (material) removal output Part surface generation output Plücker’s indicatrix of normal curvature of the surface P Plücker’s indicatrix of normal curvature of the generating surface T of the cutting tool Plücker’s indicatrix of normal radii of curvature of the surface P Plücker’s indicatrix of radii of normal radii of curvature of the generating surface T of the cutting tool Surface of relative curvature Operator of reflection with respect to YiZi– coordinate plane

© 2008 by Taylor & Francis Group, LLC

Notation

497

Rfy(ZiXi)

Operator of reflection with respect to YiZi– coordinate plane Operator of reflection with respect to XiYi– coordinate plane Operator of “Roll/Pitch/Yaw” transformation Operator of the resultant coordinate system transformation, say from the coordinate system A to the coordinate system B Operator of rotation through an angle j x about the X axis Operator of rotation through an angle j y about the Y axis Operator of rotation through an angle j z about the Z axis Operator of rotation through an angle jA about the A axis not through the origin of the coordinate system Operator of rotation through an angle jA about the A0axis through the origin of the coordinate system Operator of nonorthogonal coordinate system trans- formation The first and the second principal radii of curvature of the surface P The first and the second principal radii of curvature of the generating surface T of the cutting tool The generating surface of the cutting tool Discriminant of the second fundamental form of the surface P Discriminant of the second fundamental form of the generating surface T of the cutting tool Resultant matrix of tolerances on relative configuration of the cutting tool with respect to the part surface P Operator of translation at a distance ax along the X axis Operator of translation at a distance ay along the Y axis Operator of translation at a distance ax along the Z axis Tangent vectors of the principal directions on the surface P Tangent vectors of the principal directions on the generating surface T of the cutting tool Curvilinear (Gaussian) coordinates of a point of the surface P Curvilinear (Gaussian) coordinates of a point of the generating surface T of the cutting tool Tangent vectors to the curvilinear coordinate lines on the surface P Tangent vectors to the curvilinear coordinate lines on the generating surface T of the cutting tool Vector of the resultant motion of the generating surface T of the cutting tool with respect to the part surface P

Rfz(XiYi) RPY(j x, j y, j z) Rs(A a B) Rt(j x, X) Rt(j y, Y) Rt(j z, Z) Rt(jA, A) Rt(jA, A0) Rtw (AaB) R1.P, R 2.P R1.T , R 2.T T TP TT TI(P/ T) Tr(ax, X) Tr(ay, Y) Tr(az, Z) T1.P, T2.P T1.T, T2.T

UP, VP UT, VT UP, VP UT, VT VΣ

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498 XNC,YNC,ZNC XP,YP,ZP XT,YT,ZT dsi[i a (i ± 1)] [h] hfr hss h Σ k1.P, k2.P k1.T, k2.T n P nT rcnf rP rT tli[i a (i ± 1)] t1.P, t 2.P t1.T, t 2.T u P, vP uT, vT xPyPzP

Kinematic Geometry of Surface Machining Cartesian coordinates of a point in the coordinate system associated with the multi-axis numerical control machine Cartesian coordinates of a point of the surface P Cartesian coordinates of a point of the generating surface T of the cutting tool Matrix of an elementary i-th displacement of the cutting tool with respect to the part surface P Tolerance on accuracy of the machined part surface P Height of the surface waviness Height of the surface cusps in the direction of vector Fss of the side-step motion Resultant deviation of the machined surface from the desired part surface The first and second principal curvatures of the surface P The first and second principal curvatures of the generating surface T of the cutting tool Unit normal vector to the surface P Unit normal vector to the generating surface T of the cutting tool Position vector of a point of the indicatrix of conformity Cnf R(P/ T) Position vector of a point of the surface P Position vector of a point of the generating surface T of the cutting tool Matrix of the i-th element of the resultant tolerance on configuration of the cutting tool with respect to the part surface P Unit tangent vectors of the principal directions on the surface P Unit tangent vectors of the principal directions on the generating surface T of the cutting tool Unit tangent vectors to the curvilinear coordinate lines on the surface P Unit tangent vectors to the curvilinear coordinate lines on the generating surface T of the cutting tool Local Cartesian coordinate system with the origin at the point of contact of the surfaces P and T

Greek Symbols f1.P, f 2.P The first and the second fundamental forms of the surface P f1.T, f2.T The first and the second fundamental forms of the generating surface T of the cutting tool

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499

Notation

Σ a b g g cf d e j e j e1 f n l m t P t T zT w P wT

Crossed-axis angle The clearance (flank) angle of the cutting tool The tool wedge angle The rake angle of the cutting tool The rake angle of the cutting tool in the chip-flow direction The cutting angle The tool-tip (nose) angle The major cutting edge approach angle The minor cutting edge approach angle Normal pressure angle of a gear-cutting tool The angle of inclination of the cutting edge Angle of the local relative orientation of surfaces P and T Torsion of the surface P Torsion of the generating surface T of the cutting tool Setting angle of the gear finishing tool Coordinate angle on the part surface P Coordinate angle on the part-generating surface T of the cutting tool

Subscripts R cnf max min opt P T

Surface of relative curvature Conformity Maximal Minimal Optimal Part surface being machined Generating surface of the form-cutting tool

© 2008 by Taylor & Francis Group, LLC 63405_C013.indd 499

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