Examples of Implementation of the DG/K-Based Method 483 for the coefcients L g , M g , and N g for the screw involute surface P g . Equation (1.4) returns n g b g g b g g b g V V = sin sin sin cos cos . . . ψ ψ ψ 1 (11.17) for the unit normal vector n g to the gear-tooth surface P g . Computations similar to those above must be performed for the generating surface T sh of the shaving cutter. Vector V Σ of the resultant relative motion of the surfaces P g and T sh passes through the point K, and it is located in a common tangent plane to the sur- faces P g and T sh . Consider a plane through the unit normal vector n g that is orthogonal to the direction of V Σ . Radii of curvature of the surfaces P g and T sh in this cross-section differ. The width of the tool-path over the lateral tooth surface P g depends upon the direction of the vector V Σ . By varying the direction of feed F diag , say timing in various manner angular velocities w g and w sh with feed F diag , tool-paths of various width ( F i could be obtained. The shortest shaving time, and the highest accuracy of the involute gear- tooth surface could be obtained if and only if the feed rate per tooth ( F i of the shaving cutter remains equal to its maximal value — that is, if ( ( F F i cnf = (max) . In order to make the equality ( ( F F i cnf = (max) valid, it is necessary to remain at the highest possible rate of conformity of the surface T sh to the surface P g . In general, the rate of conformity of an involute gear-tooth surface T sh to the involute tooth surface P g at the point K varies, as the normal plane sec- tion 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 P g and T sh is observed. The tangent plane to the gear-tooth surface P g 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 P g : ( ) . r r u v g K g gtang − × ⋅ = 0 , where r g.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 sin sin cos sin cos s µ φ φ ψ φ = ⋅ − ⋅ ( ) ⋅ − ⋅ n n g n Σ 1 1 2 2 2 iin 2 ψ sh ( ) (11.18) where φ n 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. © 2008 by Taylor & Francis Group, LLC 484 Kinematic Geometry of Surface Machining For the case under consideration, the equation of the indicatrix of confor- mity Cnf P T R g sh ( / ) can be derived from the general form of equation of this characteristic curve (see Equation 4.59). The rst f 1.g , and the second f 2.g , f 2.sh fundamental forms are initially com- puted in the coordinate systems X Y Z g g g and X Y Z sh sh sh , correspondingly (see Figure 11.18). It is necessary to convert these expressions to the common local coordinate system x y z g g g . Such a transformation can be performed by means of the formula of quadratic form transformation (see Equation 3.37 and Equation 3.38): [ ] ( ) [ ] ( ) ( ) ( ) φ φ 1 2 1 2 1 2 , .g sh k T , .g sh g sh = → ⋅ ⋅Rs Rs(( )1 2→ (11.19) C g C 2.sh C 2.g C sh z g Y sh O sh Z sh X sh r sh x g S diag z g Z g y g O g r g Y g X g C Σ T sh P g y g ω g ω sh t 2.sh t 2.g n g t t 2.g t 2.sh K K n g µ µ x g Figure 11.18 The major coordinate systems. (From Radzevich, S.P., International Journal of Advanced Manufac- turing Technology, 32 (11–12), 1170–1187, 2007. With permission.) © 2008 by Taylor & Francis Group, LLC Examples of Implementation of the DG/K-Based Method 485 where [ ] ( ) ( ) φ 1 2, .g sh g sh and [ ] ( ) φ 1 2, .g sh k are the fundamental forms of the sur- faces P T g sh ( ) , initially represented in the coordinate systems X Y Z g g g and X Y Z sh sh sh , and nally in the common coordinate system x y z g g g . In the local coordinate system x y z g g g , the equation for the Cnf P T R g sh ( / ) casts into Indicatrix of conformity Cnf P T r R g sh cnf ( / ) (⇒ RR R R R g sh g sh 1 1 1 1 . . . . , , , ) sin sin( ) µ ϕ ϕ µ ϕ = + − (11.20) where r R R cnf g sh ( , , , ) . .1 1 µ ϕ is the position vector of a point of the characteristic curve Cnf P T R g sh ( / ) — for nishing of a given gear, the function r R R cnf g sh ( , , , ) . .1 1 µ ϕ reduces to r R cnf sh ( , , ) .1 µ ϕ ; and j is the polar angle (further the argument ϕ is employed for determining the optimal direction of resultant relative motion V Σ of the surfaces P g and T sh ). The characteristic curve Cnf P T R g sh ( / ) is depicted in Figure 11.19. The rate of conformity of the surfaces P g and T sh in the normal cross-section through the minimal diameter d cnf (min) (or, the same, through the direction t cnf (max) of the maximal rate of conformity of the surfaces P g and T sh ) is the highest pos- sible (Figure 11.20). This plane section of the surfaces P g and T sh is referred to as the optimal normal cross-section. V Σ y g y sh x g x sh R 2.sh R 2.g d cnf (im) (min) t cnf (max) –t cnf (min) d cnf (min) Cnf R (P g /T sh ) (im) Cnf R (P g /T sh ) Cnf R (P g /T sh ) (im) Cnf R (P g /T sh ) opt µ t 1.sh t 2.sh t 1.g t 2.g K 90° opt Figure 11.19 The indicatrix of conformity Cnf P T R g sh ( / ) of the tooth anks P g and T sh . (From Radzevich, S.P., International Journal of Advanced Manufacturing Technology, 32 (11–12), 1170–1187, 2007. With permission.) © 2008 by Taylor & Francis Group, LLC Examples of Implementation of the DG/K-Based Method 487 The same angle i opt makes vector V Σ with the perpendicular to the cutting edge (Figure 11.21). At every point of the tooth ank P g , the rst principal curvature k g1. is uniquely determined by the topology of the surface P g . The second princi- pal curvature k g2. of the screw involute surface P g is always equal to zero ( k g2 0 . ≡ ). Similarly, the second principal curvature k sh2. of the screw invo- lute surface T sh is also always equal to zero ( k sh2 0 . ≡ ). At this point, the rest of the parameters of the indicatrix of conformity Cnf P T R g sh ( / ) of the sur- faces P g and T sh (that is, the parameters R sh1. , j, and m) can be considered as the variable parameters. It is necessary to determine the optimal combina- tion of values of the parameters R R U V sh sh g g 1 1. . ( , ) opt opt = , ϕ ϕ opt opt = ( , )U V g g , and µ µ opt opt = ( , )U V g g . If the proper combination of the parameters R sh1. opt , ϕ opt , and µ opt 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 P T R g sh ( / ) reveals how close the tooth sur- face T sh of the shaving cutter is to the gear-tooth surface P g in every cross- section of the surfaces P g and T sh by normal plane through K. It enables specication of an orientation of the normal plane section, at which the sur- faces P g and T sh are extremely close to each other — that is, the normal plane section through the unit tangent vector t cnf (max) in the direction of the maximal rate of conformity of the surfaces P g and T sh . This normal plane section sat- ises the following conditions: ∂ ∂ = ∂ ∂ = ∂ ∂ = r R r r cnf sh cnf cnf 1 0 0 0 . , , . µ ϕ and Cnf R (P g /T sh ) Cnf R (P g /T sh ) z g x g n g i opt i opt y g d cnf (min) t cnf (max) opt opt K V Σ g sh 90° pt Figure 11.21 Elements of local topology of the tooth surfaces P g and T sh referred to the lateral plane of the auxiliary phantom rack R. (From Radzevich, S.P., International Journal of Advanced Manufactur- ing Technology, 32 (11–12), 1170–1187, 2007. With permission.) © 2008 by Taylor & Francis Group, LLC 488 Kinematic Geometry of Surface Machining Equation (11.20) of the indicatrix of conformity Cnf P T R g sh ( / ) yields the fol- lowing necessary conditions of the maximal rate of conformity of the shav- ing cutter tooth surface T sh to the involute gear-tooth surface P g : The Necessary Conditions for the Minimal Shavving Time and the Maximal Accuracy of the Sh aaved Involute Gear ⇒ ∂ ∂ = − r R R cnf sh sh 1 1 1 . . sin( µ ϕϕ µ µ ϕ µ ϕ ) cos( ) sin ( ) . = ∂ ∂ = − − − = ∂ 0 0 2 1 r R r cnf sh cnf ∂∂ = − − − − = ϕ ϕ ϕ µ ϕ µ ϕ cos sin cos( ) sin ( ) . . 2 1 2 1 0R R g sh (11.22) The sufcient conditions for the maximum of the function r R cnf sh ( , , ) .1 µ ϕ of three variables are also satised. The rst equality in Equation (11.22) consists in condensed form all the necessary information on the optimal design parameters of the shaving cut- ter. Analysis of this equality reveals that it could be satised when R 1.sh → ∞. Thus, for a conventional diagonal shaving operation when the gear and the shaving cutter are in external mesh, it is recommended to nish 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 P g and T sh . 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 R sh sh n sh1 1. . ( , )= φ ψ reveals that the rate of confor- mity of the surfaces P g and T sh increases when both normal pressure angle φ n and helix angle ψ sh are smaller — that is, φ n → 0 o and ψ sh → 0 o . 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 P g and T sh ( µ → 0 o , however, the inequality Σ ≠ 0 o is required) and on the optimal parameters of instant kinematics of diagonal shaving ( ϕ ϕ = opt ). The resultant relative motion V Σ of the surfaces P g and T sh is decom- posed on its projections onto directions of the motions to be performed on the gear-shaving machine. Vector V sl of the velocity of relative sliding of the surfaces P g and T sh is located in the common tangent plane. It is convenient to decompose the vector V sl at the point K onto two components V V V sl = + φ ψ . The rst component © 2008 by Taylor & Francis Group, LLC Examples of Implementation of the DG/K-Based Method 489 V φ represents sliding along the tooth prole, and the second component V ψ represents sliding in the longitudinal tooth direction. The feed F diag 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 resul- tant speed V Σ of cutting ( V V F Σ = + sl diag ). Varying parameters of the diago- nal shaving operation and of design parameters of the shaving cutter enable one to control the resultant speed V V F Σ = + sl diag 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 y z g g g (Figure 11.22), the vector V Σ of the resultant motion makes a certain angle ϕ Σ with the y g axis. Thus, V V V Σ Σ Σ Σ Σ = ⋅ ⋅ | | sin | | cos ϕ ϕ 0 1 (11.23) e Shaving Cutter Σ F diag Z k Z k Σ ω sh ω sh O sh O g e Work-Gear C V g = R w.g . w g V sh = R w sh . ω h * V yz = Pr(V Σ ) yz V sl = V g + V sh 0.5 Σ Figure 11.22 Timing of the feed F diag 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 V Σ in the global coordinate system X Y Z k k k (Figure 11.22), the operator Rs( )g k→ of the resultant coordinate system transformation is used: V Rs V V V V Σ Σ Σ Σ Σ * * * * ( ) | | | | | | = → ⋅ = g k 1 (11.24) Equation (11.24) yields the projection Pr ( ) * yz V Σ of the vector V Σ * onto the coordinate plane Y Z k k : Pr ( ) | | | | * * * yz V V V Σ Σ Σ = 0 1 (11.25) Relative sliding V sl of the tooth surfaces of the gear and the shaving cutter can be computed by V V V sl g sh w g g w sh sh R R= + = ⋅ + ⋅ . . ωω ω (11.26) where V g , V sh are the linear velocities of the rotations g and sh , respec- tively; and R w g. , R w sh. are radii of pitch cylinders of the gear and the shaving cutter. And, | | | | cos( . ) | | . . V sl g w g sh w sh R R= ⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅2 0 5 2 ω ωω Σ ccos( . )0 5⋅Σ (11.27) In the coordinate plane Y Z k k , the resultant motion V yz of the gear and the shaving cutter can be represented as follows: V V F yz sl diag = + (11.28) Thus, reciprocation is equal to F V V diag sl yz = − (11.29) 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 492 Kinematic Geometry of Surface Machining [21] Radzevich, S.P., Differential-Geometric Method of Surface Generation, DrSci thesis, Tula, Polytechnic Institute, 1991. [2 2] Radzevich, S.P., Fundamentals of Surface Generation, Rastan, Kiev, 2001. [2 3] Radzevich, S.P., Methods of Milling of Sculptured Surfaces, VNIITEMR, Moscow, 1989. [2 4] 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. [ 2 5] Radzevich, S.P., Sculptured Surface Machining on Multi-Axis NC Machine, Vishcha Shkola, Kiev, 1991. [2 6] Radzevich, S.P., and Dmitrenko, G.V., Machining of Form Surfaces of Revolu- tion on NC Machine Tool, Mashinostroitel’, No . 5, 17–19, 1987. [27] Radzevich, S.P., Goodman, E.D., and Palaguta, V.A., Tooth Surface Fundamen- tal Forms in Gear Technology, University of Niš, the Scientic Journal Facta Universitatis, Series: Mechanical Engineering, 1 (5), 515–525, 1998. [ 2 8] Radzevich, S.P., and Palaguta, V.A., Advanced Methods in Gear Finishing, VNI- ITEMR, Moscow, 1988. [2 9] Radzevich, S.P., and Palaguta, V.A., CAD/CAM System for Finishing of Cylin- drical Gears, Mekhanizaciya i Avtomatizaciya Proizvodstva, No. 10, 13–15, 1988. [3 0] Radzevich, S.P., and Palaguta, V.A., Synthesis of Optimal Gear Shaving Opera- tions, Vestink Mashinostroyeniya, No. 8, 36–41, 1997. [3 1] Radzevich, S.P. et al., On the Optimization of Parameters of Sculptured Surface Machining on Multi-Axis NC Machine, In Investigation into the Surface Genera- tion, 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 devel- oped on the premises of wide use of Differential Geometry of surfaces, and of elements of Kinematics of multiparametric motion of rigid body in Euclid- ian 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 implementa- tion 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 signicant 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 uni- versity students. © 2008 by Taylor & Francis Group, LLC 495 Notation An k (P) Andrew’s indicatrix of normal curvature of the surface P An k (P/ T) Andrew’s indicatrix of normal curvature of the surfaces P and T Anl k (T) Andrew’s indicatrix of normal curvature of the generat- ing surface T of the cutting tool Anl R (P) Andrew’s indicatrix of radii of normal curvature of the surface P An R (P/ T) Andrew’s indicatrix of normal radii of curvature of the surfaces P and T Anl R (T) Andrew’s indicatrix of radii of normal curvature of the generating surface T of the cutting tool CC–point Cutter contact point Cnf k (P/ T) 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) Cnf R (P/ T) 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) Cp i [i a(i ±1)] Couple of elementary coordinate system transformation Crv(P) Curvature indicatrix of the surface P Crv(T) Curvature indicatrix of the generating surface T of the cutting tool C 1.P , C 2.P The rst and the second principal plane sections of the part surface P C 1.T , C 2.T The rst and the second principal plane sections of the generating surface T of the cutting tool Ds(P/ T) Matrix of the resultant displacement of the cutting tool with respect to the part surface P Dup(P) Dupin’s indicatrix of the surface P Dup(P/ T) Dupin’s indicatrix of the surface of relative curvature R Dup(R) Dupin’s indicatrix of the surface of relative curvature R Dup(P) Dupin’s indicatrix of the generating surface T of the cutting tool E A characteristic line E P , F P , G P Fundamental magnitudes of the rst order of the sur- face P E T , F T , G T Fundamental magnitudes of the rst order of the gener- ating surface T of the cutting tool Eu(y,q,j) Operator of the Eulerian transformation ( F fr Feed rate per tooth of the cutting tool © 2008 by Taylor & Francis Group, LLC [...]... 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. .. 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... Identification of Kind of Contact of the Surfaces P and T 13 8 4.6.2 Impact of Accuracy of the Computations on the Desired Parameters of the Indicatrices of Conformity Cnf R(P/T) 14 2 4.6.3 Classification of Kinds of Contact of the Surfaces P and T 14 3 References 15 1 5 Profiling of the Form-Cutting Tools of the Optimal Design 15 3 5 .1 Profiling of the Form-Cutting Tools for Sculptured Surface. .. 10 3 4.3.8 Introduction of the Ir k(P/T) Characteristic Curve 10 6 4.4 Rate of Conformity of Two Smooth, Regular Surfaces in the First Order of Tangency 10 7 4.4 .1 Preliminary Remarks 10 8 4.4.2 Indicatrix of Conformity of the Surfaces P and T 11 0 4.4.3 Directions of the Extremum Rate of Conformity of the Surfaces P and T 11 7 4.4.4 Asymptotes of the Indicatrix of Conformity... Sculptured Surface Machining 15 3 5 .1. 1 Preliminary Remarks 15 3 5 .1. 2 On the Concept of Profiling the Optimal Form-Cutting Tool 15 6 5 .1. 3 R-Mapping of the Part Surface P on the Generating Surface T of the Form-Cutting Tool 16 0 5 .1. 4 Reconstruction of the Generating Surface T of the Form-Cutting Tool from the Precomputed Natural Parameterization 16 4 5 .1. 5 A Method... Olivier 19 4 5.3.2 Profiling of the Form-Cutting Tools for Single-Parametric Kinematic Schemes of Surface Generation 19 5 5.3.3 Profiling of the Form-Cutting Tools for Two-Parametric Kinematic Schemes of Surface Generation 19 6 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... 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... Indicatrix 13 1 4.5.2.4 An R (P)-Indicatrix of the Surface P 13 2 4.5.3 Relative Characteristic Curves 13 4 4.5.3 .1 On a Possibility of Implementation of Two of Plücker’s Conoids 13 4 4.5.3.2 An R(P/T)-Relative Indicatrix of the Surfaces P and T 13 5 4.6 Feasible Kinds of Contact of the Surfaces P and T 13 8 4.6 .1 On a Possibility of Implementation of the Indicatrix of Conformity... for the Determination of the Rate of Conformity Functions F 1, F 2, and F 3 16 5 5 .1. 6 An Algorithm for the Computation of the Design Parameters of the Form-Cutting Tool 17 3 5 .1. 7 Illustrative Examples of the Computation of the Design Parameters of the Form-Cutting Tool 17 5 5.2 Generation of Enveloping Surfaces 17 7 5.2 .1 Elements of Theory of Envelopes 17 8 © 2008 by Taylor... Group, LLC Contents 5.2 .1. 1 Envelope to a Planar Curve 17 8 5.2 .1. 2 Envelope to a One-Parametric Family of Surfaces 18 2 5.2 .1. 3 Envelope to a Two-Parametric Family of Surfaces 18 4 5.2.2 Kinematical Method for the Determining of Enveloping Surfaces 18 6 5.3 Profiling of the Form-Cutting Tools for Machining Parts on Conventional Machine Tools 19 3 5.3 .1 Two Fundamental Principles . Technology, 32 (11 12 ), 11 70 11 87, 2007. With permission.) © 2008 by Taylor & Francis Group, LLC 488 Kinematic Geometry of Surface Machining Equation (11 .20) of the indicatrix of conformity. Classication 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 Classication of Sculptured Surfaces 24 References 25 2 Kinematics. 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