KINEMATIC GEOMETRY OF SURFACE MACHINING 2008 by Taylor & Francis Group ppt

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 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 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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) 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 1.1  Elements of Differential Geometry of Surfaces 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 × 66 3.2.2 Translations 67 © 2008 by Taylor & Francis Group, LLC viii Contents 3.2.3 Rotation about a Coordinate Axis 69 3.2.4 Rotation about an Arbitrary Axis through the Origin 70 3.2.5 Eulerian Transformation 71 3.2.6 Rotation about an Arbitrary Axis Not through the Origin 71 3.2.7 Resultant Coordinate System Transformation 72 3.2.8 An Example of Nonorthogonal Linear Transformation 74 3.2.9 Conversion of the Coordinate System Orientation 74 3.3  Useful Equations 75 3.3.1 RPY-Transformation 76 3.3.2 Rotation Operator 76 3.3.3 A Combined Linear Transformation 76 3.4  Chains of Consequent Linear Transformations and a Closed Loop of Consequent Coordinate System Transformations 77 3.5  Impact of the Coordinate System Transformations on Fundamental Forms of the Surface 83 References 85 Part II  Fundamentals 4   The Geometry of Contact of Two Smooth, Regular Surfaces 89 4.1  Local Relative Orientation of a Part Surface and of the Cutting Tool 90 4.2  The First-Order Analysis: Common Tangent Plane 94 4.3  The Second-Order Analysis 94 4.3.1 Preliminary Remarks: Dupin’s Indicatrix 95 4.3.2 Surface of Normal Relative Curvature 97 4.3.3 Dupin’s Indicatrix of Surface of Relative Curvature 101 4.3.4 Matrix Representation of Equation of the Dupin’s Indicatrix of the Surface of Relative Normal Curvature 102 4.3.5 Surface of Relative Normal Radii of Curvature 102 4.3.6 Normalized Relative Normal Curvature 103 4.3.7 Curvature Indicatrix 103 4.3.8 Introduction of the Ir k(P/T) Characteristic Curve 106 4.4  Rate of Conformity of Two Smooth, Regular Surfaces in the First Order of Tangency 107 4.4.1 Preliminary Remarks 108 4.4.2 Indicatrix of Conformity of the Surfaces P and T 110 4.4.3 Directions of the Extremum Rate of Conformity of the Surfaces P and T 117 4.4.4 Asymptotes of the Indicatrix of Conformity Cnf R (P/T) 120 4.4.5 Comparison of Capabilities of the Indicatrix of 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 © 2008 by Taylor & Francis Group, LLC Contents ix 4.5  Plücker’s Conoid: More Characteristic Curves 124 4.5.1 Plücker’s Conoid 124 4.5.1.1 Basics 124 4.5.1.2 Analytical Representation 124 4.5.1.3 Local Properties 126 4.5.1.4 Auxiliary Formulas 127 4.5.2 Analytical Description of Local Topology of the Smooth, Regular Surface P 127 4.5.2.1 Preliminary Remarks 128 4.5.2.2 Plücker’s Conoid 128 4.5.2.3 Plücker’s Curvature Indicatrix 131 4.5.2.4 An R  (P)-Indicatrix of the Surface P 132 4.5.3 Relative Characteristic Curves 134 4.5.3.1 On a Possibility of Implementation of Two of Plücker’s Conoids 134 4.5.3.2 An R(P/T)-Relative Indicatrix of the Surfaces P and T 135 4.6  Feasible Kinds of Contact of the Surfaces P and T 138 4.6.1 On a Possibility of Implementation of the Indicatrix of Conformity for 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 165 5.1.6 An Algorithm for the Computation of the Design Parameters of the Form-Cutting Tool 173 5.1.7 Illustrative Examples of the Computation of the Design Parameters of the Form-Cutting Tool 175 5.2  Generation of Enveloping Surfaces 177 5.2.1 Elements of Theory of Envelopes 178 © 2008 by Taylor & Francis Group, LLC  Contents 5.2.1.1 Envelope to a Planar Curve 178 5.2.1.2 Envelope to a One-Parametric Family of Surfaces 182 5.2.1.3 Envelope to a Two-Parametric Family of Surfaces 184 5.2.2 Kinematical Method for the Determining of Enveloping Surfaces 186 5.3  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 © 2008 by Taylor & Francis Group, LLC Contents xi 6.2.3.5 Orthogonal Plane 242 6.2.3.6 Cutting Edge Normal Plane 242 6.2.4 Analytical Representation of the Geometric Parameters of the Cutting Edge of a Cutting Tool 243 6.2.5 Correspondence between Geometric Parameters of the Active Part of Cutting Tools That Are Measured in Different Reference Planes 244 6.2.6 Diagrams of Variation of the Geometry of the Active Part of a Cutting Tool 253 6.3  Geometry of the Active Part of Cutting Tools in the Tool-in-Use System 255 6.3.1 The Resultant Speed of Relative Motion in the Cutting of Materials 257 6.3.2 Tool-in-Use Reference System 258 6.3.3 Reference  Planes 261 6.3.3.1 The Plane of Cut Is Tangential to the Surface of Cut at the Point of Interest M 261 6.3.3.2 The Normal Reference Plane 263 6.3.3.3 The Major Section Plane 266 6.3.3.4 Correspondence between the Geometric Parameters Measured in Different Reference Planes 268 6.3.3.5 The Main Reference Plane 269 6.3.3.6 The Reference Plane of Chip Flow 272 6.3.4 A Descriptive-Geometry-Based Method for the Determination of the Chip-Flow Rake Angle 276 6.4  On Capabilities of the Analysis of Geometry of the Active Part of Cutting Tools 277 6.4.1 Elements of Geometry of Active Part of a Skiving Hob 277 6.4.2 Elements of Geometry of the Active Part of a Cutting Tool for Machining 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 © 2008 by Taylor & Francis Group, LLC xii Contents 7.1.3 The Area-Weighted Mean Normal to a Sculptured Surface P 293 7.1.4 Optimal Workpiece Orientation 295 7.1.5 Expanded Gaussian Map of the Generating Surface of the Cutting Tool 297 7.1.6 Important Peculiarities of Gaussian Maps of the Surfaces P and T 299 7.1.7 Spherical Indicatrix of Machinability of a Sculptured Surface 302 7.2  Necessary and 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 8.1  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 © 2008 by Taylor & Francis Group, LLC 484 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.) © 2008 by Taylor & Francis Group, LLC 485 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 g , R 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 g , R sh , µ , ϕ )   reduces to rcnf ( R 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.) © 2008 by Taylor & Francis Group, LLC 486 Kinematic Geometry of Surface Machining ng Rsh (max) VΣ = VΣ tcnf ng Shaving Cutter P[h] Pg (max) tcnf K Tsh ˘ (max) Fcnf [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 ) = sin ψ g sin φn d y g −d b g cos λb g sin ϕ + d y sh sin ψ sh sin φn − db sh cos λb sh sin ( 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   © 2008 by Taylor & Francis Group, LLC 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 g is uniquely determined by the topology of the surface Pg The second principal curvature k g of the screw involute surface Pg is always equal to zero ( k g ≡ ) Similarly, the second principal curvature k sh of the screw involute surface Tsh is also always equal to zero ( k sh ≡ ) 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 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 sh = R 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 , 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, and = 0, R sh â 2008 by Taylor & Francis Group, LLC 488 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 Shaving v Time and the Maximal a Accuracy of the Shaved Involute Gear  ∂ rcnf = =0  R sh sin( µ − ϕ )  ∂R sh  cos( µ − ϕ )  ∂ rcnf =− ⇒ R sh = ∂µ sin ( µ − ϕ )   ∂ rcnf cos ϕ cos( µ − ϕ ) =− R g − R sh =  ∂ϕ sin ϕ sin ( µ − ϕ )   (11.22) The sufficient conditions for the maximum of the function rcnf ( R 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 sh = R 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     © 2008 by Taylor & Francis Group, LLC 489 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 =        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.) © 2008 by Taylor & Francis Group, LLC 490 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 | |     (11.24)   Equation (11.24) yields the projection Pryz (VΣ ) of the vector VΣ onto the coordinate plane Yk Zk :    V  | | Σ  Pryz (VΣ ) =    |VΣ |     (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] © 2008 by Taylor & Francis Group, LLC 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 © 2008 by Taylor & Francis Group, LLC 492 Kinematic Geometry of Surface Machining [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, (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 © 2008 by Taylor & Francis Group, LLC 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 © 2008 by Taylor & Francis Group, LLC 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 © 2008 by Taylor & Francis Group, LLC 496 ( [ Ffr ] Ffr ( Fss ( [ Fss ] Fss F 1, F 2, F 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 © 2008 by Taylor & Francis Group, LLC 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 © 2008 by Taylor & Francis Group, LLC 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 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 11/8/07 1:54:37 PM ... 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. .. unrecorded © 2008 by Taylor & Francis Group, LLC Part I Basics © 2008 by Taylor & Francis Group, LLC Part Surfaces: Geometry The generation of part surfaces is one of the major purposes of machining. . .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,