Accuracy of Surface Generation 403 At the point of intersection P of the straight line r h (see Equation 8.50) and to the torus surface Tr P (see Equation 8.51) the equality r r h Tr P Tr P Tr P K ( , ) [ ] . . . ( ) θ ϕ = is observed. In expended form, this equality casts into X t Y t Z Q h Tr P Tr P Q h Tr P Tr P − − cos cos cos sin . . . . θ ϕ θ ϕ QQ h Tr P t−             sin . 1 = ⋅ + ⋅ −( cos ) cos ( . . . . . . r R R r Tr P Tr P Tr P Tr P Tr P Tr θ ϕ PP Tr P Tr P Tr P Tr P R r⋅ + ⋅ ⋅ + ⋅cos ) sin cos sin . . . . θ ϕ µ θ TTr P Tr P Tr P Tr P Tr P r R . . . . . sin ( cos ) sin ⋅ − ⋅ + ⋅ ⋅ µ θ ϕ ssin sin cos . . µ θ µ + ⋅ ⋅             r Tr P Tr P 1 (8.52) which yields the computation of three parameters θ Tr P. , ϕ Tr P. , and t h that specify coordinates of the point P . Finally, an analytical expression for the vector r P can be derived. The computed vectors r Q and r P yield computation of the resultant sur- face deviation h Σ : h Q PΣ = −| |r r (8.53) With the above analysis, the nal 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 h ss is valid if and only if the inequality h h h Σ Σ Σ ∆− ≤ [ ] is observed. Here, [ ]∆ Σ h 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, Engle- wood 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, Computer- Aided 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 Ma chine. Part 1, Izvestiya VUZov. Mashinostroyeniye, 5, 138–142, 1985. © 2008 by Taylor & Francis Group, LLC 404 Kinematic Geometry of Surface Machining [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. © 2008 by Taylor & Francis Group, LLC Part III Application © 2008 by Taylor & Francis Group, LLC 407 9 Selection of the Criterion of Optimization For machining a part surface, a machine tool, a cutting tool, a xturing, and so forth are necessary. All these elements together are referred to as the tech- nological system. For lubrication and for cooling purposes, liquids and gas substances are often used. The coolants and the lubricants create the techno- logical 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 technologi- cal system and of the technological environment are used the most com- pletely, manufacturing processes of this kind are usually called the extremal manufacturing processes. Ultimately, use of the extremal manufacturing pro- cesses results in the most economical machining of part surfaces. The Differential Geometry/Kinematics (DG/K )-method of surface genera- tion (disclosed in previous chapters) is capable of synthesizing extremal meth- ods 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 efcient machining operation, appropriate input information is required. Capabili- ties of a theoretical approach can be estimated by the amount of input infor- mation the approach requires for its implementation, and by the amount of output information the method is capable of creating. A more powerful theo- retical 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 geo- metrical 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 oper- ations 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. © 2008 by Taylor & Francis Group, LLC 408 Kinematic Geometry of Surface Machining 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 pre- ferred 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 signicantly sim- pler structure can be proposed. The productivity of surface machining and productivity of surface genera- tion are the important criteria of optimization. Both are often used for creat- ing 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 implementa- tion 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 perspec- tive 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. Conse- quently, 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. © 2008 by Taylor & Francis Group, LLC Selection of the Criterion of Optimization 409 9.2 Productivity of Surface Machining Productivity of surface generation reects 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 sculp- tured 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 pro- ductivity of surface generation vary in time. This makes reasonable consid- eration 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 ( F fr and of the side-step ( F ss (here t desig- nates time). Usually, the vector F fr and the vector F ss are orthogonal to each other ( F F fr ss ⊥ ). In particular cases, the vectors F fr and F ss are at a certain angle θ to each other. Instantaneous productivity of surface generation can be computed by the following formula [7,8]: P ( ) | |t fr ss = ×F F (9.1) Equation (9.1) casts into [7,8] P ( ) sint F F fr ss = ⋅ ⋅ ( ( θ (9.2) Here, ( F fr is equal to | |F fr , and ( F ss is equal to | |F ss . Equation (9.1) and Equation (9.2) reveal that an increase of the feed-rate ( F fr , and an increase of the side-step ( F ss lead to an increase of the instanta- neous productivity of surface generation P ( )t . Deviation of the angle θ from θ = 90 results in a corresponding reduction of the instantaneous productiv- ity 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 ( F fr , ( F ss , 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: [ ] [ ]( , )h h U V P P = (9.3) © 2008 by Taylor & Francis Group, LLC 410 Kinematic Geometry of Surface Machining Within certain portions of a surface patch, the tolerance can be bigger, and within other portions, it can be smaller depending on the functional require- ments of the actual part surface. Because the resultant cusp height h Σ is made up of two components h fr and h ss , it is necessary to split the tolerance [ ]h on two corresponding por- tions: on the portion [ ]h fr for the elementary deviation h fr , and on the por- tion [ ]h ss for the elementary deviation h ss . The equality [ ] [ ] [ ]h h h fr ss = + (9.4) is always observed (see Equation 8.44). But, the equality h h h fr ssΣ ≅ + (9.5) is always approximate. For computations of the surface deviation h Σ , it is recommended that Equation (8.4) be used: h a h b h h fr h ssΣ ≅ ⋅ + ⋅ (9.6) Here, coefcients a h and b h are within the intervals 0 1≤ ≤a h and 0 1≤ ≤b h . The coefcients a h and b h 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 V P , the current values of the coefcients a h and b h also depend on coordinates of the point K on the surface P (that is, depend on U T and V T 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: a a U V U V h h P P T T = ( , , , , ) µ (9.7) b b U V U V h h P P T T = ( , , , , ) µ (9.8) Values of the feed-rate ( F fr per tooth of the cutting tool, and of the side-step [ ]h ss at a current point K depend on the partial tolerances [ ]h fr and [ ]h ss . One can immediately conclude from the above that both the feed-rate ( F fr and the side-step [ ]h ss 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 ( ( ( F F h F U V U V fr fr fr fr P P T T = =([ ]) ( , , , , , ) µ ϕ (9.9) ( ( ( F F h F U V U V ss ss ss ss P P T T = =([ ]) ( , , , , , ) µ ϕ (9.10) © 2008 by Taylor & Francis Group, LLC Selection of the Criterion of Optimization 411 reveal this relationship. Here the angle that species 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 coor- dinates 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 direc- tion of the relative motion of the surfaces P and T at point K: P P sg sg P P T T U V U V= ( , , , , , ) µ ϕ (9.11) Certainly, not just the tolerance [ ]h , but also the partial tolerances [ ]h fr and [ ]h ss can be constant within the part surface patch or can vary within the sculptured surface P. In the rst case, the actual values of the tolerances [ ]h , [ ]h fr , and [ ]h ss must be given. In the second case, the following func- tions must be known: [ ] [ ]( , , , )h h U V U V P P T T = (9.12) [ ] [ ]( , , , , )h h U V U V fr fr P P T T = µ (9.13) [ ] [ ]( , , , , )h h U V U V ss ss P P T T = µ (9.14) The principal radii of curvature R P1. and R P2. of the part surface are the functions of parameters U P and V P of the sculptured surface P, while the principal radii of curvature R T1. and R T2. of the generating surface T of the cutting tool are the functions of the parameters U T and V T . In special cases of sculptured surface machining, when, for example, elas- tic deformation is applied to the work for technological purposes as shown in Figure 2.3, or for a special-purpose cutting tool with changeable generat- ing surface that is used for the machining [3, 9, 13], then in addition to the parameters U P , V P , U T , V T , µ , ϕ some more parameters have to be incorpo- rated 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 W ps must be derived. Equation r wp of the surface W ps can be composed on © 2008 by Taylor & Francis Group, LLC 412 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 rst 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 V P P ( , ) . In the event the equation of the workpiece surface W ps 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 W ps yields a com- putation of the stock-distribution function b U V P P ( , ) : b U V P P wp P ( , ) | |= −r r (9.15) When the stock-distribution function b U V P P ( , ) is given, then the equa- tion of the workpiece surface W ps can be derived analytically. For this pur- pose, an equation of the nominal part surface r r P P P P U V= ( , ) is employed (Figure 9.1): r r n wp P P P P b U V= + ⋅ ( , ) (9.16) In Figure 9.1, point M wp on the surface of the workpiece W ps is shown at a distance b U V P P ( , ) from the point M on the nominal part surface P. Elements of local topology of the workpiece surface W ps (say, the rst Φ 1.ps and the second Φ 2.ps fundamental forms of the workpiece surface W ps ) P v P n P b(U P , V P ) Z P Y P X P r wp r P M wp W ps M u P FIGURE 9.1 On derivation of equation r wp of the workpiece surface W ps . © 2008 by Taylor & Francis Group, LLC Selection of the Criterion of Optimization 413 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 r P P P P U V= ( , ) of the part surface P and Equation (9.16) of the workpiece sur- face W ps must be represented in a common reference system. When neces- sary, an appropriate operator Rs( )W P ps a of the resultant coordinate system transformations can be composed for this purpose (see Chapter 3 for details). Modied Equation (9.16) can also be helpful for the computation of param- eters of uncut chip. Si milar to Equation (9.16), the equation of the surface of tolerance S h[ ] can be immediately written: r r n [ ] [ ]( , ) h P P P P h U V= + ⋅ (9.17) Equation (9.17) is used for the computation of parameters of the critical val- ues 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 W ps are necessary. Mean chip-removal output is used for the analysis of efciency of a global machining operation, say for the whole part surface P. The mean chip- removal output % P mr can be used as an index. By denition [5,6,10,12], % P mr mr V t = Σ (9.18) where V mr 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 efciency 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 denition [5,6,10,12], P mr mr t d v dt ( ) = (9.19) © 2008 by Taylor & Francis Group, LLC [...]... 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 © 2008 by Taylor & Francis Group, LLC 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... 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 © 2008 by Taylor & Francis Group, LLC 422 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... examples of the solutions to the problem of synthesis of local surface generation © 2008 by Taylor & Francis Group, LLC Kinematic Geometry of Surface Machining 432 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. .. 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 427 © 2008 by Taylor & Francis Group, LLC Kinematic Geometry of Surface Machining... 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. .. 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. .. 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... generating surface of the cutting tool   © 2008 by Taylor & Francis Group, LLC Synthesis of Optimal Surface Machining Operations 431 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... 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. .. 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  is at the optimal  principal plane-section C1.T of the cutting tool surface T   angle µopt of the local relative orientation of surfaces P and T  Vector Vopt  of the cutof the optimal direction of motion of 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. 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. 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