Preface xix number of research and application papers and articles. Commonly, isolated theoretical and practical ndings 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 broad- based 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 prac- titioners 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 the- ory 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 synthe- sizing 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 oor environment. This makes this book not just another book on the subject. For the rst 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 ofsurfacegeneration,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 posi- tion 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 benet from this book: mechanical and manufacturing engineers involved in continuous process improvement, research workers whoareactiveorintendtobecomeactiveintheeld,andseniorundergraduate and graduate students of mechanical engineering and manufacturing. © 2008 by Taylor & Francis Group, LLC xx 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 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 sur- faces, as well as principal elements of the theory of multiparametric motion of a rigid body in E 3 space. The applied coordinate systems and linear transformations are briey considered. The selected mate- rial 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 classication of surfaces is discussed, and a scientic classica- tion of local patches of sculptured surfaces is proposed. Chapter 2. Kinematics of Surface Generation — The general- ized 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 rela- tive motions of the cutting tool are discovered, including gen- erating 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 kine- matics of surface generation are discussed as well. Chapter 3. Applied Coordinate Systems and Linear Transforma- tions — The denitions and determinations of major applied coordinate systems are introduced in this chapter. The matrix © 2008 by Taylor & Francis Group, LLC and practical implementation of the proposed theory (Part III) will be of Preface xxi approach for the coordinate system transformations is briey discussed. Here, useful notations and practical equations are provided. Two issues of critical importance are introduced here. The rst is chains of consequent linear transformations and a closed loop of consequent coordinate systems transformations. The impact of the coordinate systems transformations on funda- mental 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-sufcient unit suited for self-study. Part II: Fundamentals — Fundamentals of the theory of surface genera- tion 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 rst order of tangency; a novel kind of mapping of one surface onto another surface; a novel analyti- cal 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 rst part of the book. The following chapters are included in this section. Chapter 4. The Geometry of Contact of Two Smooth Regular Sur- faces — Local characteristics of contact of two smooth, regular surfaces that make tangency of the rst 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 sur- faces and the rst- and second-order analyses. The concept of conformity of two smooth, regular surfaces in the rst order of tangency is introduced and explained in this chapter. For the pur- poses of analyses, properties of Plücker’s conoid are implemented. Ultimately, all feasible kinds of contact of the part and of the tool surfaces ar e classied. Chapter 5. Proling of the Form-Cutting Tools of Optimal Design — A novel method of proling the form-cutting tools for sculp- tured 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 proling form-cutting tools for machining part surfaces on con- ventional machine tools are also considered. These methods are based on elements of the theory of enveloping surfaces. Numer- ous particular issues of proling form-cutting tools are discussed at the end of the chapter. © 2008 by Taylor & Francis Group, LLC xxii Preface Chapter 6. Geometry of Active Part of a Cutting Tool — The gen- erating body of the form-cutting tool is bounded by the generat- ing 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 computa- tions are also presented. Chapter 7. Conditions of Proper Part Surface Generation — The satisfactory conditions necessary and sufcient for proper part surface machining ar e 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 con- cludes with the global verication of satisfaction of the condi- tions of proper part surface generation. Chapter 8. Accuracy of Surface Generation — Accuracy is an impor- tant issue for the manufacturer of the machined part surfaces. Analytical methods for the analysis and computation of the devia- tionsofthemachinedpart surfacefromthedesiredpartsurfaceare discussed here. Two principal kinds of deviations of the machined surface from the nominal part surface ar e distinguished. Methods for the computation of the elementary surface deviations are pro- posed. 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 efcient methods of part surface generation. Numerous practical examples of implementation of the theory are disclosed in this part of the mono- graph. 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/Kine- matics (DG/K)-based method of surface generation, an appropri- ate criterion of efciency 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 ef- ciency of machining operation are considered. Tight connection of the economical criteria of optimization with geometrical ana- logues (as established in Chapter 4) is illustrated. The part surface © 2008 by Taylor & Francis Group, LLC Preface xxiii generation output is expressed in terms of functions of confor- mity. The last signicantly simplies 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 sur- face 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 devel- oped on the premises of implementation of the proposed DG/K- based method surface generation. Addressed are novel methods of machining sculptured surfaces on a multi-axis NC machine, novel methods ofmachining surfaces ofrevolution, and a novel method of nishing involute gears. The proposed theory of surface generation is oriented on extensive appli- cation of a multi-axis NC machine of modern design. In particular cases, implementation of the theory can be useful for machining parts on conven- tional 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 exten- sive industrial experience in gear design and manufacture. He has devel- oped numerous software packages dealing with computer-aided design (CAD) and computer-aided manufacturing (CAM) of precise gear nishing for a variety of industrial sponsors. Dr. Radzevich’s main research inter- est 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 multiow gear trains, (d) design of special-purpose gear cutting and n- ishing tools, (e) design and machining (nishing) of precision gears for low- noise/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 scientic papers; and he holds more than 150 patents in the eld. 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 kind- ness and help must go unrecorded. © 2008 by Taylor & Francis Group, LLC Part I Basics © 2008 by Taylor & Francis Group, LLC 3 1 Part Surfaces: Geometry The generation of part surfaces is one of the major purposes of machin- ing operations. An enormous variety of parts are manufactured in various industries. Every part to be machined is bounded with two or more sur- faces.* 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 com- puter-aided design (CAD) and computer-aided manufacturing (CAM) appli- cations, 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, specically, (a) in the differential geometry of surfaces and (b) in the engineering geometry of surfaces. The second is based on the rst. For further consideration, it is convenient to briey 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 sec- tion 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 X P , Y P , and Z P , as functions of two Gaussian coor- dinates U P and V P in a certain closed interval: r r P P P P P P P P P P P P P U V X U V Y U V Z U V = =   ( , ) ( , ) ( , ) ( , ) 1           ≤ ≤ ≤ ≤; ( ; ) . . . . U U U V V P P P P P P1 2 1 2 V (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 Part Surfaces: Geometry 5 Signicance of the vectors u P and v P becomes evident from the following considerations. First, tangent vectors u P and v P yield an equation of the tan- gent plane to the surface P at M: Tangent plane t p P M P P ⇒ − ( )              r r u v . ( ) 1  = 0 (1.3) where r t.P is the position vector of a point of the tangent plane to the surface P at M, and r P M( ) is the position vector of the point M on the surface P. Second, tangent vectors yield an equation of the perpendicular N P , and of the unit normal vector n P to the surface P at M: N U V n N N U V U V P P P P P P P P P P = × = = × × and == ×u v P P (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 v P to a surface at a point are of critical impor- tance when solving practical problems in the eld 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 rst issue is the rst fundamental form of a surface P. The rst funda- mental form f 1.P of a smooth, regular surface describes the metric properties of the surface P. Usually, it is represented as the quadratic form: φ 1 2 2 2 2 .P P P P P P P P P ds E dU F dU dV G dV⇒ = + + (1.5) where s P is the linear element of the surface P (s P is equal to the length of a segment of a certain curve line on the surface P), and E P , F P , G P are funda- mental magnitudes of the rst order. Equation (1.5) is known from many advanced sources. In the theory of sur- face generation, another form of analytical representation of the rst funda- mental form f 1.P is proven to be useful: φ 1 2 0 0 0 0 0 0 0 0 1 0 0 0 0 1 . [ ] P P P P P P P P ds dU dV E F F G ⇒ = ⋅             ⋅             dU dV P P 0 0 (1.6) © 2008 by Taylor & Francis Group, LLC [...]... 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. .. 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 © 20 08 by Taylor & Francis Group, LLC 24 Kinematic Geometry of Surface. .. the most efficient machining operation of a given part surface, it is necessary to determine the optimal parameters of the relative 27 © 20 08 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... feasible kinds of local surface patches is limited Hence, local surface patches of every kind can be © 20 08 by Taylor & Francis Group, LLC 26 Kinematic Geometry of Surface Machining [13] Radzevich, S.P., Fundamentals of Surface Generation, Kiev, Rastan, 20 01 Copy of the monograph is available from the Library of Congress, call number: MLCM 20 06/0 429 7 [14] Radzevich, S.P., Sculptured Surface Machining on... LLC RP = 2 2 k 1 P + k 2 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... computation of the second fundamental form of the surface G can be obtained: 2 g ⇒ − dr g ⋅ d N g = −U g sin τ b g cos τ b g dVg2 © 20 08 by Taylor & Francis Group, LLC (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... equations: MP= k1 P + k 2 P EP N P − 2 FP M P + GP LP = 2 2 2 ⋅ ( EPGP − FP ) GP = k1 P ⋅ k 2 P = The formulae for M P = k1 P + k 2 P 2 2 LP N P − M P 2 EPGP − FP (1.16) and G P = k1 P ⋅ k 2 P yield a quadratic equation: 2 k P − 2 M 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... 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. .. 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. .. 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 . F = + = − + ⋅ − ( ) 1 2 2 2 2 2 . . (1.15) G P P P P P P P P P k k L N M E G F = ⋅ = − − 1 2 2 2 . . (1.16) The formulae for M P k k P P = + 1 2 2 . . and G P P P k k= ⋅ 1 2. . yield a quadratic. k = + 1 2 2 2 2 . . (1.33) © 20 08 by Taylor & Francis Group, LLC 14 Kinematic Geometry of Surface Machining The curvedness describes the scale of the surface P independent of its shape. These. 20 08 by Taylor & Francis Group, LLC 8 Kinematic Geometry of Surface Machining C 2. P is orthogonal to P at M and passes through the second principal direc- tion T 2. P . In the theory of surface