Kinematic Geometry of Surface Machinin Episode Episode 6 pptx

The Geometry of Contact of Two Smooth, Regular Surfaces 147 Without going into detail, mention will be made here that for the purposes of efficient surface generation in a machining operation, it is desired to maintain that kind of contact of the surfaces P and T which features the highest possible rate of conformity of the generating surface T of the cutting tool to the part surface P Actually, while machining a part surface, deviations in the cutting tool relative location and orientation with respect to the surface P are always observed The deviations of the cutting tool configuration are unavoidable Because of the deviations, the desired local-extremal kind of contact is replaced with another kind of contact of the surfaces P and T Such replacement can be achieved with the introduction of precalculated deviations of the cutting tool principal radii of curvature R 1.T and R 2.T If the precalculated deviations are small, then instead of the desired local-extremal kinds of contact of the surfaces, the “quasi-” kind of contact of the surfaces P and T may occur There are several kinds of quasi- contact, including quasi-line contact of the surfaces P and T, quasi-surface of the first kind, and quasisurface of the second kind contact of the surfaces P and T The required precomputed values of small deviations of the actual normal curvatures from their initially computed values can be determined on the premises of the following consideration When the maximal deviations in the actual cutting tool configuration (location and orientation of the cutting tool relative to the surface being machined) occur, the rate of conformity of the generating surface T with respect to the surface P must not exceed the rate of their conformity in one of the local-extremal kinds of surface contact When the actual deviations of the cutting tool configuration not exceed the corresponding tolerances, then one of the feasible kinds of quasi-contact of the surfaces P and T is observed, Evidently, bigger deviations in the cutting tool configuration result in bigger precomputed corrections in the normal curvature of the generating surface of the cutting tool, and vice versa In the ideal case, when there are no deviations in the cutting tool configuration, it is recommended to assign normal curvatures of the values that enable one of the local-extremal kinds of contact of the surfaces P and T Local-surface contact of the second kind is the preferred kind of contact of the surfaces P and T The local-surface contact of the second kind yields the minimal value (min) of the radius rcnf = of the indicatrix of conformity Cnf R ( P/T ) When machining an actual part surface, deviations in the cutting tool configuration are unavoidable The pure surface kind of contact of the surfaces when the (min) equality rcnf = is observed is not feasible Due to the deviations in the cutting tool configuration, maintenance of the pure surface contact of the surfaces P and T would unavoidably result in interference of the cutting tool beneath the part surface P Therefore, it is recommended that a pure surface contact not be maintained, but a kind of quasi-surface contact of the second kind be maintained instead A quasi-surface contact of the surfaces P and T yields avoidance of interference of the surface T within the interior of the surface P Moreover, the (min) minimal radius rcnf of the characteristic curve CnfR ( P/T ) could be as close to (min) (min) (min) zero as possible ( rcnf > 0, rcnf → , rcnf ≠ ) © 2008 by Taylor & Francis Group, LLC 148 Kinematic Geometry of Surface Machining Quasi-contact of the surfaces P and T is observed only when deviations of the cutting tool configuration are incorporated into consideration Definition 4.1: Quasi-line contact of the surfaces P and T is a kind of point contact of the surfaces under which actual tangency of the surfaces is within the true-point contact and the local-line contact, and it varies as a function of the deviations in the cutting tool configuration Definition 4.2: Quasi-surface (of the first kind) contact of the surfaces P and T is a kind of point contact of the surfaces under which actual tangency of the surfaces is within the true-point contact and the localsurface (of the first kind) contact, and it varies as a function of deviations in the cutting tool configuration Definition 4.3: Quasi-surface (of the second kind) contact of the surfaces P and T is a kind of point contact of the surfaces under which actual tangency of the surfaces is within the true-point contact to local-surface (of the second kind) contact, and it varies as a function of deviations in the cutting tool configuration The difference between various kinds of the quasi-contact of the surfaces P and T, as well as the difference between the corresponding kinds of localextremal contact of the surfaces can be recognized only under the limit values of the allowed deviations in the cutting tool configuration relative to the part surface P In the event the actual deviations are below the tolerances, then various possible kinds of quasi-contact of the surfaces cannot be distinguished from other nonquasi-kinds of their contact The only difference is in actual location of the point K of contact of the surfaces Due to the deviations, it shifts from the original position to a certain other location There are only nine principally different kinds of contact of the surfaces P and T In addition to the true-point, the true-line, and the true-surface contact, the following three local-extremal kinds of contact are possible: (a) the local-line, (b) the local-surface of the first kind, and (c) the local-surface of the second kind Three kinds of quasi-contact of the surfaces are also possible: the quasi-line, the quasi-surface of the first kind, and the quasi-surface of the second kind of the surfaces P and T Taking into consideration that there are only 10 different kinds of local patches of smooth, regular surfaces P and T (see Chapter 1, Figure 1.11), each of the kinds of surfaces contact can be represented in detail For this purpose, a square morphological matrix of size 10 × 10 = 100 is composed This matrix covers all possible combinations of the surfaces contact One axis of the morphological matrix is represented with 10 kinds of local patches of the part surface P, and the other axis is represented with 10 kinds of local patches of the generating surface T of the cutting tool The morphological 9! m matrix contains Σ =1C9 = m !( 9- m )! = 1002-10 + 10 = 55 different combinations of m the local patches of the surfaces P and T Only 55 of them are required to be investigated The analysis reveals that the following kinds of contact of the surfaces P and T are feasible: © 2008 by Taylor & Francis Group, LLC The Geometry of Contact of Two Smooth, Regular Surfaces 149 29 kinds of true-point contact 23 kinds of true-line contact kinds of true-surface contact 20 kinds of local-line contact kinds of local-surface (of the first kind) contact kinds of local-surface (of the second kind) contact 20 kinds of quasi-line contact kinds of quasi-surface (of the first kind) contact kinds of quasi-surface (of the second kind) contact This means that only 29 + 23 + + 20 + + + 20 + + = 128 kinds of contact of two smooth, regular surfaces P and T are possible in nature For some kinds of surfaces contact, no restrictions are imposed on the actual value of the angle m of the surfaces P and T local relative orientation For the rest of the types of surfaces contact, a corresponding interval of the allowed value of the angle m — [ µ ] ≤ µ ≤ [ µ max ] — can be determined For particular cases of the surfaces contact, the only feasible value µ = [ µ ] is allowed On the premises of the above analysis, a scientific classification of possible kinds of contact of the surfaces P and T is developed (Figure 4.21) The classification (Figure 4.21) is potentially complete It can be further developed and enhanced It can be used for the analysis and qualitative evaluation of the rate of effectiveness of a machining operation The classification indicates perfect correlation with the earlier developed classification (see table 3.1 on pp. 230–243 in [13]) Replacement of the true-point contact of the surfaces P and T with their local-line contact, and further with the local-surface of the first and of the second kind, and finally with the true-surface contact results in significant alterations of the surface P generation Only local-extremal kinds of contact of the surfaces P and T are considered here If deviations in the cutting tool configuration are considered, then the above-mentioned local-extremal kinds of surfaces contact require being replaced with the corresponding quasi-kinds / | of contact of the surfaces P | T In order to achieve the highest possible productivity of machining of the part surface P, it is recommended that the true-surface contact of the surfaces P and T be maintained Under such a scenario, all portions of the surface P are machined in one instant However, in those cases, large-scale surfaces P cannot be machined The machining of the surface P while maintaining the true-point contact of the surfaces is least efficient In this case, the generation of every strip on the surface P occurs in time Depending on the kinds of contact of the surfaces P and T that are maintained when machining a part surface P, all possible kinds of contact of the surface can be ranked in the following order (from the least efficient to the most efficient): True-point contact Local-line or quasi-line contact © 2008 by Taylor & Francis Group, LLC The Geometry of Contact of Two Smooth, Regular Surfaces 151 Maintenance of the true-point contact of the surface P and T results in the highest possible agility of a machining operation The true-point contact can also be regarded as the most general kind of surfaces contact Under the true-point contact of the surfaces, the cutting tool can perform five-parametric motion with respect to the surface P Under the true-surface contact, the cutting tool is capable of performing no motion relative to the work A relative motion of the surfaces P and T is feasible only as an exclusion, say when the surfaces P and T yield for sliding over themselves In those cases, an enveloping surface P to consecutive positions of the generating surface T of the cutting tool that is moving relative to the work, is congruent to P Generally speaking, under such a scenario, the surfaces P and T are capable of performing a single-parametric motion, and not higher than a three-parametric motion (see Chapter 2, Section 2.4) The developed classification of all possible quasi-kinds of contact of the surfaces P and T can be extended and represented in more detail References [1] Boehm, W., Differential Geometry II In Farin, G Curves and Surfaces for Computer Aided Geometric Design A Practical Guide, 2nd ed., Academic Press, Boston, 1990, pp 367–383 [2] doCarmo, M.P., Differential Geometry of Curves and Surfaces, Prentice Hall, Englewood Cliffs, NJ, 1976 [3] Fisher, G (Ed.), Mathematical Models, Friedrich Vieweg & Sohn, Braunachweig/ Wiesbaden, 1986 [4] Gray, A., Plücker’s Conoid In Modern Differential Geometry of Curves and Surfaces with Mathematics, 2nd ed., CRC Press, Boca Raton, FL, 1997, pp 435–437 [5] Hertz, H., Über die Berührung Fester Elastischer Körper (The Contact of Solid Elastic Bodies), Journal für die Reine und Angewandte Mathematik (Journal for Pure and Applied Mathematics), Berlin, 1981, pp 156–171; Über die Berührung Fester Elastischer Körper und Über die Härte (The Contact of Solid Elastic Bodies and Their Harnesses), Berlin, 1882; Reprinted in: H Hertz, Gesammelte Werke (Collected Works), Vol. 1, pp. 155–173 and pp. 174–196, Leipzig, 1895, or the English translation: Miscellaneous Papers, translated by D.E. Jones and G.A. Schott, pp 146–162, 163–183, Macmillan, London, 1896 [6] Koenderink, J.J., Solid Shape, MIT Press, Cambridge, MA, 1990 [7] Pat. No. 1249787, A Method of Sculptured Surface Machining on Multi-Axis NC Machine./S.P. Radzevich, Int. Cl B23c 3/16, Filed: December 27, 1984 [8] Pat. No. 1185749, A Method of Sculptured Surface Machining on Multi-Axis NC Machine./S.P. Radzevich, Int. Cl B23c 3/16, Filed: October 24, 1983 [9] Plücker, J., On a New Geometry of Space, Phil Trans R Soc London, 155, 725– 791, 1865 [10] Radzevich, S.P., A Possibility of Application of Plücker’s Conoid for Mathematical Modeling of Contact of Two Smooth Regular Surfaces in the First Order of Tangency, Mathematical and Computer Modeling, 42, 999–1022, 2004 © 2008 by Taylor & Francis Group, LLC Profiling of the Form-Cutting Tools of Optimal Design Generation of the part surface P practically is performed with the help of the cutting tools of appropriate design The stock removal and generation of the surface P are the two major functions of the cutting Profiling of the generating surface T is required for designing a high-performance cutting tool As shown below, the shape and parameters of the generating surface T significantly affect the performance of the cutting Cutting edges of a precision cutting tool are within the generating surface of the tool This makes it clear that prior to developing a practical design of the cutting tool, profiling of the optimal generating surface is required In this chapter, profiling of the form-cutting tools for all possible methods of part surface machining is considered The consideration begins from the theory of profiling of the tools for machining sculptured surfaces on a multi-axis numerical control (NC) machine This subject represents the most complex case in the theory of profiling of cutting tools 5.1 Profiling of the Form-Cutting Tools for Sculptured Surface Machining The problem of profiling the form-cutting tool for machining of a sculptured surface on a multi-axis NC machine has not yet been investigated in detail Not profiling of the form-cutting tool of optimal design but selecting a certain cutting tool among several available designs is often recommended instead The selection of the cutting tool is usually based on minimizing machining time, reducing scallop height, and so forth This yields a conclusion that a robust mathematical method for design of the optimal form-cutting tool for the maximally productive machining of a given sculptured surface on a multi-axis NC machine is needed 5.1.1 Preliminary Remarks Many advanced sources are devoted to the investigation of sculptured surfaces generation on multi-axis NC machines Without going into a detailed review of previous publications in the field, mention is made of a few monographs by Amirouch [1], Chang and Melkanoff [4], Choi and Jerard [5], 153 © 2008 by Taylor & Francis Group, LLC 154 Kinematic Geometry of Surface Machining (a) (b) (c) (d) (e) (f) Figure 5.1 Examples of milling cutters of conventional design for the machining of sculptured surfaces on a multi-axis numerical control machine: cylindrical (a), conical (b), ball-end (c), filleted-end (d), and form-shaped (e), (f) and Marciniak [11] Unfortunately, the problem of profiling the form-cutting tools for sculptured surface machining has not yet been investigated Most often, the generation of sculptured surfaces with the milling cutters of conventional designs (Figure 5.1) is considered The following terms (some of which are not new) are introduced below to avoid ambiguities in later discussions: Definition 5.1: Sculptured surface P is a smooth, regular surface, the major parameters of local topology at a point of which are not identical to the corresponding parameters of local topology of any other infinitesimally close point of the surface It is instructive to point out here that sculptured surface P does not allow for sliding “over itself.” While machining a sculptured surface, the cutting tool rotates about its axis of rotation and moves relative to the sculptured surface P When rotating or when performing relative motion of another kind, cutting edges of the cutting tool generate a certain surface The surface represented by consecutive positions of cutting edges is referred to as the generating surface of the cutting tool [18, 19, 25]: Definition 5.2: The generating surface T of the cutting tool is a surface that is conjugate to the surface P being machined  In fact, our terminology draws inspiration mostly from the Theory for Mechanisms and Machines, and from the Theory of Conjugate Surfaces © 2008 by Taylor & Francis Group, LLC Profiling of the Form-Cutting Tools of Optimal Design 155 An infinite number of surfaces satisfy Definition 5.2 Use of all of the conjugate surfaces satisfies the equation of contact n P ⋅ VΣ = (see Chapter 2) Although the unit normal vector n P is uniquely determined at a given surface point, the number of feasible vectors VΣ is equal to infinity: All vectors VΣ within the common tangent plane satisfy the equation of contact n P ⋅ VΣ = It is natural to assume that not all are equivalent to each other from the standpoint of efficiency of surface generation, and some could be preferred; moreover, an optimal direction of VΣ exists within the common tangent plane for which the efficiency of surface machining reaches it optimal rate This makes the problem of profiling the form-cutting tool indefinite However, this indefiniteness is successfully overcome below The uniquely determined generating surface T is used in further steps of designing an optimal form-cutting tool for machining of a given part surface. In most cases of surface generation, the generating surface T of the cutting tool does not exist physically Usually, it is represented as the set of consecutive positions of the cutting edges in their motion relative to the stationary coordinate system, embedded in the cutting tool In most practical cases, the generating surface T allows for sliding over itself The enveloping surface to consecutive positions of the surface T that performs such a motion is congruent to the surface T When machining a surface P, the surface T is conjugate to the sculptured surface P For simplification in programming machining operation, the APT cutting tool is proposed (Figure 5.2) Physically, the APT cutting tool does not exist The generating surface T of the APT cutting tool is made up of a conical portion that has the cone angle a, a conical portion that has the cone angle b, and a portion of the surface of a torus The last is specified by the radius r of the generating circle, and by the diameter d of the directing circle The axial location of the torus surface with respect to the conical surfaces is specified by the parameter designated as f For a certain combination of the parameters a, b, r, d, and f, the generating surface of the virtual APT cutting tool transforms to the generating surface T of the actual cutting tool For example, assuming a = 0°, b = 0°, and r = 0, one can come up with the generating surface T of the cylindrical milling cutter (Figure 5.1a) If r = d and b = 0°, then the  The procedure of designing a form-cutting tool usually begins from determination of the generating surface of the cutting tool This is a common practice However, sometimes when the geometric structure of the surface to be machined is inconsistent, another procedure is used Relieving hob clearance surfaces, cutting bevel gears with spiral teeth, machining noninvolute gears of the first and of the second kind are perfect examples of surface machining when the geometric structure of the surface to be machined is inconsistent Under such circumstances, the generating surface of the cutting tool of an appropriate form is selected Further, the actual shape and parameters of the machined part surface can be determined Keep in mind that the part surface to be machined is the primary element of the machining operation, on the premises of which the determination of the optimal machining operation is possible This includes profiling of both the optimal cutting tool and the optimal kinematics of the machining operation Otherwise, only an approximate solution to the problem of optimal surface generation is possible © 2008 by Taylor & Francis Group, LLC 157 Profiling of the Form-Cutting Tools of Optimal Design I I ωP t P I [h] T S t ≤ [h] a a ωP P b 1< a a a S < a S P I (b) t ~ =1, 2S hc < hb < S RT S T (c) d P ωP I = 0˚ d h =0 I d t d RT b S T I ωP hb < b b S (a) t [h] = 0˚ T = 0˚ S [h] S d (d) Figure 5.3 Turning of an arbor on a lathe: the concept of profiling the optimal form-cutting tool (From Radzevich, S.P., Mathematical and Computer Modeling, 36 (7–8), 921–938, 2002 With permission.) the generating surface T of the cutting tool, and kinematics of the machining operation are applicable The following example illustrates the actual meaning of the criterion of optimization in the sense of profiling of the form-cutting tool Consider a trivial machining operation — a turning operation of an arbor on a lathe (Figure 5.3) When machining, the work rotates about its axis of rotation with an angular velocity ω P The cutting tool travels along the work axis of rotation with a feed rate S The feed rate S is of constant magnitude in the examples considered below A stock t is removed in the turning operation Use of the cutter with the tool cutting edge angle ϕ a , and the tool minor (end) cutting edge angle ϕ a causes cusps on the machined part surface P (Figure 5.3a) The actual height h a of the cusps must be less than the tolerance [h] on accuracy of the surface P In order to satisfy the inequality h a ≤ [ h], a corresponding relationship between the parameters ϕ a, ϕ a , and S must be observed Otherwise, the part cannot be machined in compliance with the part blueprint That same arbor can be machined with the cutter having the tool cutting edge angle ϕ b < ϕ a , and the tool minor cutting edge angle ϕ b < ϕ a (Figure 5.3b) 1 Cusps on the machined surface P are observed Elementary computations of the actual cusp height h b in this case reveal that the inequality h b < h a is valid Further, that same surface P can be machined with the cutter having the cutting edge that is shaped in the form of a circular arc of radius R (Figure 5.3c) Use of the cutter with the curvilinear cutting edge results in cusps on the machined surface P However, if the radius R is chosen properly, then the © 2008 by Taylor & Francis Group, LLC 158 Kinematic Geometry of Surface Machining actual cusp height h c can be smaller than h b In other words, the inequality h c < h b could be observed Ultimately, consider the turning operation of the surface P with the cutter that has an auxiliary cutting edge (Figure 5.3d) The auxiliary cutting edge is parallel to the direction of the feed rate S The length of the auxiliary cutting edge exceeds the distance that the cutter travels per one revolution of the work Geometrical parameters of the auxiliary cutting edge can be specified by the tool cutting edge angle ϕ d = 0° , and the tool minor (end) cutting edge angle ϕ d = 0° Under such a scenario, no cusps are observed on the machined part surface P The above consideration makes possible a conclusion: An appropriate alteration of shape of the cutting edge of the cutter can make possible a reduction of deviations of the machined part surface with respect to the desired part surface This conclusion is critically important for further consideration At this point, a more general example of surface generation that supports the above conclusion will be considered Consider generation of a sculptured part surface P with the form-cutting tool having arbitrarily shaped the generation surface T The intersection of the surfaces P and T by the plane through the unit normal vector n P is shown in Figure 5.4 This plane section is perpendicular to the tool-path on the surface Ra > T Rb > Ra > T T ST a K Ta ST P hP RP Tb (b) Rc T K Tc ∞ ST P (c) P hd < hc p P K hc < hb p P RP RP hb < p P RP (a) ST P K Rd < T (d) Td Figure 5.4 Examples of various rates of conformity of the generating surface T of the tool to the sculptured surface P in the plane section through the unit normal vector n P (From Radzevich, S.P., Mathematical and Computer Modeling, 36 (7–8), 921–938, 2002 With permission.) © 2008 by Taylor & Francis Group, LLC Profiling of the Form-Cutting Tools of Optimal Design 159 P at point K In Figure 5.4, ST designates the width of the tool-path In all the examples considered, the width ST of the tool-path remains identical The radius of normal curvature RP of the surface P at point K remains the same The scallop’s height on the machined surface P is designated hP The surface P can be generated with the generating surface T a of the cutting tool (Figure 5.4a) The radius of curvature of the generating surface T a is of a certain positive value R a > Due to point contact of the surfaces P T and T a , the desired surface P is not generated, but an approximation to it is generated instead The generated surface has scallops For the prespecified a width ST of the tool-path, the scallop’s height is equal to a certain value hP The scallop’s height must be smaller than the tolerance [h] on the accuracy of the surface P That same surface P can be generated with the generating surface T b of the cutting tool (Figure 5.4b) The radius of curvature of the generating surface T b in this case exceeds the value of the radius of curvature of the surface T a in the previous case ( Rb > R a ) Because surfaces P and T b are in point contact, scalT T lops on the generated surface are observed When width ST of the tool-path is b predetermined, then the scallop’s height is of a certain value hP Under such a b a scenario, a certain reduction of the scallop’s height occurs ( hP < hP ) The scallop height reduction occurs because in differential vicinity of point K, the surface T b is getting closer to surface P rather than to surface T a Locally, surface T b is more congruent to surface P than to surface T a The rate of conformity of surface T b to surface P is greater than the rate of conformity of the surface T a to that same surface P Further, that same surface P can be generated with the generating surface T c of the cutting tool (Figure 5.4c) At point K, surface T c is flattened; therefore, the radius of curvature R c is equal to infinity ( R c → ∞) That value T T of the radius of curvature R c exceeds the value of the radius of curvature Rb T T Again, surfaces P and T c make contact at a point; therefore, scallops on the c generated surface are observed Because R c > Rb , the scallop’s height hP is T T b smaller than the scallop’s height hP The scallop height reduction in this case is due to the increase of the rate of conformity of the generating surface T c of the tool to the part surface P compared to what is observed with respect to surfaces P and T b Ultimately, consider generation of the surface P with the concave generating surface T d of the cutting tool (Figure 5.4d) The radius of curvature R d T of the cutting-tool surface T d is negative (R d < 0) In this case, the rate of T conformity of the generating surface T d of the cutting tool to the part surface P is the biggest of all considered cases (Figure 5.4) Thus, scallops of the d smallest height hP are observed on the generated surface P Summarizing the analysis of Figure 5.4, the following conclusion can be formulated: An increase of the rate of conformity of the generating surface T of the cutting tool to the sculptured surface P causes a corresponding reduction of height of the residual scallop on the machined surface P This conclusion is of critical importance for the development of methods of profiling of form-cutting tools, as well as for the theory of surface generation   © 2008 by Taylor & Francis Group, LLC Profiling of the Form-Cutting Tools of Optimal Design 165 For the conversion of the natural parameterization of the surface T to its representation in a Cartesian coordinate, the set of two Gauss–Weingarten’s equations in tensor notation k rij = Γ ij rP + bij n P  ⇐ kj of the Form Cutting Tool  ni = −bik g rj The Generating Surface T (5.16) must be solved The solution to the set of equations in Equation (5.16) returns a matrix equation of the generating surface T of the form-cutting tool for the machining of a given sculptured part surface P on a multi-axis NC machine The initial conditions of integration of Equation (5.16) must be selected properly In Equation (5.16), ri = ∂r T/∂UT; rij = ∂2r T/∂UT∂VT; ni = ∂nT/∂UT; bij = rij ⋅ nT = − ni rj − n j ⋅ r; gij is the metric tensor of the generating surface T i of the form-cutting tool of optimal design; and g ij is a contra-variant tensor of the generating surface T of the cutting tool To solve the set of Equation (5.16), known methods [9] are used Initial conditions for integrating Equation (5.16) must be established These conditions, for example, might include coordinates of two points on the surface T and direct cosines of the unit normal vector nT at one of these points The set of two differential equations in tensor notation (see Equation 5.16) can be converted either to a set of five differential equations in vector notation, or to a set of fifteen differential equations in parametric notation Conventional mathematical methods can be implemented to solve a set of five differential equations or a set of fifteen differential equations with a corresponding number of unknowns This is a trivial mathematical problem that follows from the proposed theory of surface generation 5.1.5 A Method for the Determination of the Rate of Conformity Functions F 1, F 2, and F For determining the rate of conformity functions, various approaches can be employed It is possible to implement a method of experimental simulation of a machining operation for determining the rate of conformity functions F 1, F 2, and F The method of simulation is shown in Figure 5.5 It is proposed by Radzevich [14] As an example of implementation of the method of simulation, consider machining a sculptured surface P on a multi-axis NC machine (Figure 5.5a) The part surface P is machined with a milling cutter that has the generating surface T The method of simulation of machining sculptured surfaces is carried out with the equivalent models of the part surface P and of the generating surface T of the cutting tool (Figure 5.5b) Working surfaces of the  Pat. No 1449246, USSR, A Method of Experimental Simulation of Machining of a Sculptured Surface on Multi-Axis NC Machine./S.P. Radzevich, Int Cl B 23 C, 3/16, Filed February 17, 1987 © 2008 by Taylor & Francis Group, LLC 167 Profiling of the Form-Cutting Tools of Optimal Design the differential vicinity of the CC-point K, the surfaces P ( s) and P are locally congruent to each other up to members of the second order A similar conclusion is valid for the quadric surface T ( s) that is used for local simulation of the generating surface T of the form-cutting tool At the CC) s) point K, the unit tangent vectors t ( sT and t (2.T of the principal directions on the ( s ) align with the corresponding unit vectors t quadric surface T 1.T and t 2.T of s the generating surface T of the cutting tool Principal radii of curvature R(1.) T ( s) ( s ) are also equal to the corresponding prinand R 2.T of the quadric surface T cipal radii of curvature R 1.T and R 2.T of the surface T — that is, the identities R(1s.) ≡ R 1.T and R(2s.)T ≡ R 2.T are observed Therefore, in the differential vicinity T of the CC-point K, the surfaces T ( s) and T are locally congruent to each other up to the members of the second order Local orientation of the cutting edges on the quadric surface T ( s) remains the ) s) same with respect to the principal directions t ( sT and t (2.T as that for the actual ( s) ( s) form-cutting tool T [ t 1.T ≡ t 1.T and t 2.T ≡ t 2.T ] For this purpose, it is required, at first, to determine the orientation of the principal plane sections C1.T and C2.T of the generating surface T of the cutting tool with respect to the coordinate UT and VT lines Orientation of the plane section of a surface by a normal plane surface can be determined by the ratio ∂UT/∂VT (see Chapter 1) For the orthogonally (UT; VT)-parameterized generating surface T of the cutting tool, the ratio ∂UT/∂VT determines the value of tan ξT Here, angle ξT designates the angle of inclination of the principal plane sections C1.T and C2.T relative to the coordinate UT and VT lines on the generating surface T Usually, parameterization of the surface T is not orthogonal In such a case, the angle ξT (not shown in Figure 5.5b) can be computed from the following formula [19]:  ∂V ∂V   ∂V  sin ξT = T   T  − T cos ωT + 1 ∂UT ∂UT   ∂UT   − (5.17) At the CC-point K, the cutting edge makes a certain angle ζ T with the UTcoordinate line The angle ζ T can be computed from the equation [18,19] The cutting edge makes a certain angle θT with the first principal plane section C1.T of the generating surface T of the cutting tool The angle θT is equal to the algebraic sum of the angles ξT and ζ T — that is, θT = ξT + ς T ( Therefore, orientation of the cutting edge (angle θTm ) ≡ θT = ξT + ς T ) relative to ( s) the first principal plane section C1.T of the quadric surface T ( s) makes local orientation of the cutting edge on the quadric surface T ( s) identical to the orientation of the corresponding cutting edge on the generating surface T of the cutting tool The accuracy of the simulation is up to members of the second order or even exceeds that The quadric surfaces P ( s) and T ( s) are turned about the unit normal vector ( s) n P relative to each other through an angle µ ( s) The angle µ ( s) is the angle of the local relative orientation of surfaces P ( s) and T ( s) The angle µ ( s) is © 2008 by Taylor & Francis Group, LLC 168 Kinematic Geometry of Surface Machining identical to the angle μ of the local relative orientation of actual surfaces P and T [ µ ( s) ≡ µ ] Angle μ makes the first t 1.P and t 1.T (or, the same, the second t 2.P and t 2.T ) principal directions of the surfaces at the CC-point [18,19]: µ ( s) ≡ µ = tan −1 t P × t 1.T t P ⋅ t 1.T ≡tan −1 t P × t 2.T t P ⋅ t 2.T (5.18) The local relative orientation of the quadric surfaces P ( s) and T ( s) in differential vicinity of the CC-point K up to the members of the second order is identical to the local relative orientation of the actual sculptured surface P and the generating surface T of the cutting tool when machining the part surface P on a multi-axis NC machine The trajectory of the cutting-edge point relative to the sculptured surface P can be represented as a vector sum of the motions that the surfaces P and T perform on a multi-axis NC machine When simulating the machining operation of the surface P, the quadric surfaces P ( s) and T ( s) perform the relative motion with respect to one ( another Resultant speed VΣs) of the relative motion can be represented as a vector sum of the speed of cutting Vc( s) and of all other partial motion Vi( s) , ( namely VΣs) = Vc( s) + Σ in=−11Vi( s) Here, n designates the total number of all of the partial motions Vi( s) The feed-rate motion V (f s) is an example of the partial motions Vi( s) When machining a sculptured surface, instant relative motion of the surfaces P and T can be represented as an instant screw motion Therefore, when simulating the machining operation, the quadric surfaces P ( s) and T ( s) ( perform rotation with the resultant angular velocity ω Σs) in addition to the ( s) resultant linear motion VΣ ( While simulating, the resultant relative motion VΣs) of the surfaces P ( s) and T ( s) is identical to the instant resultant relative screw motion VΣ of the actual sculptured surface P and the generating surface T of the cutting tool ( ( ( VΣs) ≡ VΣ ) For this purpose, the angle ∆ ( s) that the vector VΣm ) makes with ( s) the first principal plane section C1 P of the quadric surface P ( s) is identical to the similar angle Δ that the vector VΣ makes with the first principal plane section C1 P of the sculptured surface P (that is, ∆ ( s) ≡ ∆ ) Instant relative screw motion of the quadric surfaces P ( s) and T ( s) is identical to the instant relative screw motion of the sculptured surface P and of the generating surface T of the cutting tool At every CC-point K, implementation of the method of experimental simulation of machining of a sculptured surface (Figure 5.5) provides the local identity of all geometrical and kinematical parameters of the machining operation, namely [14], • Quadric surface P ( s) and actual sculptured surface P • Quadric surface T ( s) and actual generating surface T of the cutting tool © 2008 by Taylor & Francis Group, LLC Profiling of the Form-Cutting Tools of Optimal Design 169 • Location of the cutting edge on the quadric surface T ( s) and cutting edge on the generating surface T of the actual cutting tool • Relative local orientation of the quadric surfaces P ( s) and T ( s) and the relative local orientation of the actual sculptured surface P and the generating surface T of the actual cutting tool • Instant relative motion while simulating and the instant relative motion while machining (that is, kinematics of machining remains identical) Ultimately, this results in high efficiency of the method of simulation, and in high accuracy of the determined rate of conformity functions F 1, F 2, and F When simulating a machining operation, it is preferred to perform not the instant relative motions of the modeling quadrics P ( s) and T ( s) , but continuous relative motions instead Implementation of the continuous motions leads to significant simplification of the procedure of modeling In order to perform the desired continuous relative motions of the modeling quadrics P ( s) and T ( s) , use of the surfaces that allow for sliding over itself is helpful A screw surface of constant pitch p = Const is the most general case of surfaces P ( s) and T ( s) that allow sliding over themselves (see Chapter 1) As a screw surface moves along and rotates about its axis with the same parameter of the screw motion as the instant screw parameter of the screw surface, the enveloping surface to consecutive positions of the screw surface is congruent to the screw surface Particular cases of surfaces that allow for sliding over itself [surfaces of revolution (for which p = 0), general (not circular) cylinders (for which p = ∞)] are considered in Chapter 1 Cylinders of revolution, spherical surfaces, and the plane represent examples of the simplest and completely degenerated surfaces that allow for sliding over themselves For simulation of the machining operation of the sculptured surface on a multi-axis NC machine, it is convenient to use a screw with external surface P ( s) , and with either a convex or a concave thread profile (Figure 5.6) Application of such a screw enables simulation of both convex and saddle-like local patches of a given sculptured surface P For simulation of concave and saddle-like local patches of a given sculptured surface P, a screw with internal surface P ( s) and with either a convex or a concave thread profile can be used (Figure 5.7) In both cases (Figure 5.6, and Figure 5.7), the screw might be either single or multithreaded, as well as single or multistarted In order to provide the required parameters of topology of the surface P ( s) (i.e., the parameters R(1s.)P ≡ R P , R(2s.)P ≡ R P ); the required radii of principal curvature of the surface T ( s) (i.e., the parameters R(1s.) ≡ R 1.T , R(2s.)T ≡ R 2.T ); and T their local relative orientation (i.e., the angle µ ( s) ≡ µ of the surfaces P ( s) and ( ( T ( s) local relative orientation), the parameters dPs) and rPs) of design of the screw have to be computed properly For this purpose, Meusnier’s formula and Euler’s formula can be used Mensnier’s formula establishes the correspondence between a radius of normal curvature RP of a surface P (or a surface T) through a certain direction t P © 2008 by Taylor & Francis Group, LLC 172 Kinematic Geometry of Surface Machining Path of Contact P(s) (s) OP (s) ωP (s) VK K θ (s) (s) FT ω(s) T (s) OT T (s) Figure 5.9 Simulation of machining a convex local patch of a sculptured part surface with the saddle-like local patch of the generating surface of the cutting tool Implementation of the screw surfaces P ( s) (Figure 5.6 and Figure 5.7) and the surfaces of revolution (Figure 5.8) allows one to reach the desired topology of the simulating surfaces P and T Figure 5.9 illustrates an example of implementation of the disclosed method [14] In the particular case (Figure 5.9) of a convex local patch of surface P with the saddle-like local patch of surface T, machining is simulated with the external worm having a convex profile of threads that is machining with the grinding wheel that has a concave axial profile The design parameters of the worm and the design parameters of the grinding wheel are precomputed in tight correlation with the corresponding design parameters of the actual part surface P and the actual generating surface T of the tool Rotation of the worm and of the grinding wheel are timed in order to get the resultant motion of the surfaces P ( s) and T ( s) identical to those of the relative motion of the surfaces in the machining operation to be simulated In the case when one or both modeling quadric surfaces P ( s) and T ( s) allow for sliding over themselves, manufacturing of the specimens for simulation is simplified At the same time, this results in two instant relative motions of the surfaces P ( s) and T ( s) being substituted with their continuous motion The last is much more convenient for simulation and enables more precise and  Pat No 1449246, USSR, A Method of Experimental Simulation of Machining of Sculptured Surface on Multi-Axis NC Machine./S.P Radzevich, Filed February 17, 1987 © 2008 by Taylor & Francis Group, LLC Profiling of the Form-Cutting Tools of Optimal Design 173 more reliable experimental results to be obtained Ultimately, this allows for accurate determination of the rate of conformity functions F 1, F 2, and F It is important to note that many experimental data necessary for determining the rate of conformity functions F 1, F 2, and F can be collected from already published scientific papers in the field For this purpose, it is necessary to analyze the published results of the research on efficiency of surface machining on a machine tool from the standpoint of the theory of surface generation It is critical to mention here that the rate of conformity functions F 1, F 2, and F also play another important role in the theory of surface generation They serve as a bridge between the pure geometrical and kinematical theory, and between real machining processes including physical phenomena 5.1.6 An Algorithm for the Computation of the Design Parameters of the Form-Cutting Tool Computation of the major design parameters of the generating surface T of the form-cutting tool for machining of a given sculptured surface on a multiaxis NC machine requires high-volume computations Generally, computations of this kind can be performed with the help of computers An algorithm for the computation of the design parameters of the generating surface T of the form-cutting tool is illustrated with the flow chart in Figure 5.10: Compose an equation of the smooth, regular sculptured surface P When the part surface P is made up of two or more portions, then a n set of equations of all n surface patches Pi |=1 must be composed i Compute the first derivatives of equations of the sculptured part surface P Compute the fundamental magnitudes EP, FP, and GP of the first order of the surface P Compute the second derivatives of equations of the part surface P Compute the fundamental magnitudes Lp, M p, and N p of the second order of the surface P Results (3) and (5) could be interpreted as the natural parameterization of the sculptured surface P Compose a set of three equations that describe the desired rate of conformity of the generating surface T of the form-cutting tool to the sculptured surface P Determine the rate of conformity functions F 1, F 2, and F Use the R-mapping of the sculptured surface P onto the generating surface T of the form-cutting tool to return a set of three equations (Equation 5.4, Equation 5.5, Equation 5.6) of six unknowns for the computation of the fundamental magnitudes of surface T © 2008 by Taylor & Francis Group, LLC Profiling of the Form-Cutting Tools of Optimal Design 175 curvatures k 1.T and k 2.T for the designed generating surface T of the cutting tool will not be related by a function In cases of milling cutter, grinding wheel, and so forth, the principal curvatures k 1.T and k 2.T are related by a function — in this case, the surface T is represented by a surface of revolution For the machining of a sculptured surface P of any geometry, the above generalized solution (see Equation 5.16) yields an approximation of the surface T with optimal topology by a surface of revolution, and so producing a form-cutting tool having all necessary combinations of k 1.T and k 2.T 5.1.7 Illustrative Examples of the Computation of the Design Parameters of the Form-Cutting Tool Two illustrative examples of the computation that not require extensive computer application are presented in this section Instead of a general demonstration of the developed approach, consider a case when the six fundamental magnitudes ET , FT , GT, LT , MT, N T of the form-cutting tool are already determined from Equation (5.4) through Equation (5.6) and from equations of compatibility Equation (5.10), Equation (5.11), and Equation (5.12) Example 5.1 Given two differential forms 2 φ1.T ⇒ dUT + cos UT dVT2 and φ2.T ⇒ dUT + cos2 UT dVT2 , find the generating surface T of the form-cutting tool, for which φ 1.T and φ 2.T are the first and second fundamental forms Because ET = , FT = , GT = cos UT , and LT = , MT = , N T = cos UT, then it could be determined that Christoffel’s symbols Γ = Γ = Γ = Γ = , 11 22 12 22 Γ 12 = − tan UT , Γ = sin UT cos UT satisfy the Gauss–Codazzi’s equations of 22 compatibility, as the direct substitution shows The set of Gauss–Weingarten’s equations (Equation 5.10 through Equation 5.12) returns the solution   r T = r 0T cos VT cos UT   sin V cos U  T T  +   sin UT     (5.20) which is the equation of the sphere The detailed derivation of Equation (5.20) is not covered here However, it is covered in detail in the literature [17]) Here r0T  designates the vector that specifies the location of the generating surface T of the cutting tool By the choice of r0T, one can place the surface T in any position of space, selecting any orthogonal system of meridians and parallels for UT and VT curvilinear coordinates of an arbitrary point M on the tool surface T The sphere can be used as the generating surface T of the cutting tool, for example, of the grinding wheel (Figure 5.11) for machining a sculptured surface P on a multi-axis NC machine © 2008 by Taylor & Francis Group, LLC Profiling of the Form-Cutting Tools of Optimal Design ZT 177 RT T XT rT YT RT M O rT θT Figure 5.12 Example 5.2: Toroidal portion of the generating surface T of the APT cutting tool R-mapping of a sculptured surface P onto the generating surface T of the cutting tool [18,19,25] The disclosed method is tightly connected to the method of simulation of the interaction of the form-cutting tool and the work The last method is vital for determining the rate of conformity functions of critical importance for the use of the R-mapping-based method of the formcutting tool design The idea and the general concept of implementation of the R-mapping of surfaces in the field of designing an optimal cutting tool for machining a sculptured surface on a multi-axis NC machine has been proposed [16] by the author [23,24] 5.2 Generation of Enveloping Surfaces When machining a part surface, the surfaces P and T are the conjugate surfaces At every instant of the machining operation, surfaces P and T are tangent to each other They make contact either at a point, or along a characteristic line Tangency of the surfaces is a strong restriction on the parameters of their relative motion No the surfaces P and T interference, no interruption of their contact is allowed When machining a part surface, the interaction of surfaces P and T** is similar to the interaction of the working surfaces of a mechanism, for example, it is quite similar to the interaction of tooth flanks of a gear pair Pat No 4242296/08, USSR, A Method for Designing of the Optimal Form-Cutting-Tool for Machining of a Given Sculptured Surface on Multi-Axis NC Machine./S.P Radzevich, Filed March 31, 1987 ** If we are to consider parallels between the conjugate action of the surface in the theory of surface generation, and between the conjugate action of surfaces in a gear drive, then it is of critical importance to point out that in surface generation, the surface of action is always congruent to the part surface P to be machined  © 2008 by Taylor & Francis Group, LLC 178 Kinematic Geometry of Surface Machining However, a principal difference between the conjugate action of the surfaces in the machining operation, and between the conjugate action of the surfaces in a corresponding mechanism certainly occur The difference is due to the following reason: The input motion of the mechanism is predetermined The interacting surfaces of a mechanism are determined by targeting the required parameters of the output motion In the theory of surface generation, problems of two kinds are recognized Problems of the first kind are usually referred to as the direct problems of surface generation Problems of this kind consider that the surface P to be machined and the kinematics of the machining operation are known The generating surface T of the cutting tool must be determined for machining a given part surface Problems of the second kind are inverse to the direct problems of surface generation Usually, problems of the second kind are referred to as the inverse problems of surface generation When solving problems of this kind, the generating surface T of the cutting tool and kinematics of the machining operation are considered as known The actual parameters of shape of the machined part surface P must be determined The total number of problems to be solved in the theory of surface generation is not limited to the two mentioned above Problems of another nature are considered as well Mostly surfaces that allow for sliding over themselves can be machined on a conventional machine tool Conjugate action of the part surface to be machined and of the generating surface of the cutting tool is insightful from the standpoint of implementation of the elements of theory of enveloping curves and enveloping surfaces for the purpose of profiling form-cutting tools The use of elements of theory of envelopes could significantly simplify the solution to the problem of profiling form-cutting tools for machining surfaces that allow for sliding over themselves 5.2.1 Elements of Theory of Envelopes The theory of enveloping curves and the theory of enveloping surfaces are widely used for profiling form-cutting tools For convenience, brief presentations from differential geometry of curves and surfaces are made below 5.2.1.1 Envelope to a Planar Curve Consider a planar curve that moves within the plane of its location If certain conditions are satisfied, then an enveloping curve to consecutive positions of the moving curve could exist [10] As an example of a planar curve, a circle l of radius r is shown in Figure 5.13 All points of the circle l are within the coordinate plane XY The circle l moves along the X axis When moving, the circle l occupies consecutive positions © 2008 by Taylor & Francis Group, LLC 180 Kinematic Geometry of Surface Machining Example 5.3 Consider a planar curve T that is given by the following equation: (2 ⋅ R + R ⋅ cos α )   rT (α , R) =  R ⋅ sin α      (5.26) Partial derivatives of rT (α , R) with respect to a and R are equal: − R ⋅ sin α    =  R ⋅ cos α  ∂α     ∂ rT (5.27) and  + cos α    = sin α  ∂R      ∂ rT (5.28) Equation (5.27) and Equation (5.28) yield the equality − R ⋅ sin α + cos α R ⋅ cos α =0 sin α (5.29) From the determinant Equation (5.29), it is easy to come up with the expression − R ⋅ (1 + ⋅ cos α ) = (5.30) Simple formulae transformations yield cos α = − 0.5 , and sin α = 23 After substituting the last equalities into Equation (5.26) of a family of curves, and after excluding the parameter R, the equation of the enveloping curve can be represented in the form Y(X ) = ⋅X (5.31) Therefore, the enveloping curve is a straight line at the angle ± 30° to the X axis The family of the curves was a family of circles with centers on the X axis (Figure 5.14) The family of curves can be generated by a circle having © 2008 by Taylor & Francis Group, LLC 181 Profiling of the Form-Cutting Tools of Optimal Design P Y T X P Figure 5.14 Enveloping curve to a family of planar curves translation along the X axis, the radius of which increases accordingly to the distance from the origin of the coordinate system XY to the center of a movable circle The considered example is of practical importance for machining a sheet metal workpiece with a milling cutter with a conical generating surface T (Figure 5.15) The milling cutter axis of rotation moves along the X axis with the feed rate V fr Simultaneously, the milling cutter performs motion Vax in its axial direction along its axis of rotation Z axis (not shown in Figure 5.15) The actual timing of the motions V fr and Vax depends upon the shape of Vfr Vax T P Figure 5.15 Practical implementation of the solution to the problem of determining an enveloping to a family of planar curves © 2008 by Taylor & Francis Group, LLC 182 Kinematic Geometry of Surface Machining Z r X Y Figure 5.16 Generation of the enveloping surface to consecutive positions of a sphere of radius r that moves along the X axis the part surface P The functional relation between the motions V fr and Vax can be linear or nonlinear 5.2.1.2 Envelope to a One-Parametric Family of Surfaces Consider a one-parametric family of surfaces The family of surfaces is dependent on a parameter of motion that is designated w The enveloping surface becomes tangent with every surface of the family of surfaces [10] For example, centers of all spheres of a family of spheres of radius r are located within the X axis of the Cartesian coordinate system XYZ (Figure 5.16) A circular cylinder of radius r with an X axis as the axis of its rotation represents the enveloping surface to the family of spheres of radius r fm Consider a family r (U , V , ω ) of surfaces r(U , V ): X(U , V , ω )  Y(U , V , ω )   r fm (U , V , ω ) =   Z(U , V , ω )      (5.32) for which the inequality ∂r/∂U × ∂r/∂V ≠ is valid The necessary condition of existence of an enveloping surface is as follows: r fm = r fm (U , V , ω ) (5.33)  ∂r ∂r ∂r   ∂U ∂V ∂ω  =   (5.34) The line of tangency of a surface r(U, V) from a family of surfaces r fm (U , V , ω ) with the enveloping surface is referred to as the characteristic line E The characteristic line E satisfies Equation (5.30) and Equation (5.31) © 2008 by Taylor & Francis Group, LLC 183 Profiling of the Form-Cutting Tools of Optimal Design The enveloping surface yields representation in the form of a family of the characteristic lines. When the enveloping surface is of the class of surfaces that allows for sliding over itself, then Equation (5.31) describes the profile of the enveloping surface Satisfaction of the condition r ∈ω of relationships (see Equation 5.33 and Equation 5.34) together with the conditions ∂ψ ∂U  ∂r    ∂U   ∂ψ ∂V ∂r ∂r ⋅ ∂U ∂V ∂ψ ∂ω ∂r ∂r ⋅ ∂U ∂V  ∂r    ∂V   ∂r ∂r ≠ 0, ⋅ ∂U ∂ω ∂ψ ∂ψ + ≠0 ∂U ∂V ∂r ∂r ⋅ ∂V ∂ω (5.35) is the sufficient condition for the existence of the profile of the enveloping surface Violation of the first of conditions Equation (5.35) is usually due to the edge of inversion observed Characteristic lines of the part surface P and of the generating surface T of the cutting tool satisfy the conditions r1 = r1 (U1 , V1 , ω ), f [U1 (ω ) , V1 (ω ) , ω ] = 0, ω = Const (5.36) r2 = r2 (U , V2 , ω ), f [U (ω ), V2 (ω ), ω ] = 0, ω = Const (5.37) In a stationary coordinate system, for example in a coordinate system associated with the machine tool, the family of the characteristic lines can be represented by the following set of equations: ∂ r2 ∂ r2 = (U , V2 , ω ), ∂f ∂f f (U , V2 , ω ) = (5.38) where the equality ∂ rf2 (U , V2 , ω ) = Rs (1 → 2) ⋅ r1 (U1 , V1 ) is observed The opera∂ tor Rs(1 → 2) of the resultant coordinate system transformation is a function  In a coordinate system associated with the cutting tool, the family of the characteristic lines E determines the generating surface T of the cutting tool For this purpose, Equation (5.34) is necessary to consider together with the operator that describes the motion of the characteristic line E in the coordinate system associated with the cutting tool In the event that the inverse problem, not the direct problem, of surface generation is considered, then the family of the characteristics E in a coordinate system associated with the work determines the actual machined part surface P © 2008 by Taylor & Francis Group, LLC 184 Kinematic Geometry of Surface Machining of the parameter of motion w The theory of surface generation also deals with surfaces for which the enveloping surface is congruent to the moving surface 5.2.1.3 Envelope to a Two-Parametric Family of Surfaces The two-parametric enveloping surface can be expressed in terms of two parameters, say of the parameters ω and ω At every point, the enveloping surface becomes tangent with one of the surfaces of the family of surfaces specified by the parameters ω (U , V ) and ω (U , V ) The parameters ω1 and ω have the same value at every point of every surface of the family of surfaces However, they differ at different points of the enveloping surface If the condition ∂r/∂U × ∂r/∂V ≠ is satisfied, then the necessary condition for the existence of the enveloping surface to a family of surfaces r(U , V , ω , ω ) can be represented in the following form [10]:  ∂r ∂r ∂r   ∂r ∂r ∂r  ψ1 =   = 0, ψ =  ∂U ∂V ∂ω  =   ∂U ∂V ∂ω  2 (5.39) In order to obtain a sufficient set of conditions for the existence of the enveloping surface, the above conditions [see Equation (5.39)] must be considered together with the following conditions: ∂ψ ∂u ∂ψ ∂u  ∂r    ∂U   ∂ψ ∂v ∂ψ ∂v ∂r ∂r ⋅ ∂V ∂U ∂ψ ∂A ∂ψ ∂A ∂ψ ∂B ∂ψ ∂B ∂r ∂r ⋅ ∂U ∂V ∂r ∂r ⋅ ∂U ∂ω1 ∂ r ∂r ⋅ ∂U ∂ω 2 ∂r ∂r ⋅ ∂V ∂ω1 ∂r ∂r ⋅ ∂V ∂ω  ∂r    ∂V   ≠ 0, D(ψ , ψ ) ≠0 D(ω1 , ω ) (5.40) If a surface r(U , V , ω , ω ) (a) is performing a two-parametric motion, (b) both the motions are independent from each other, and (c) the characteristics E1 and E2 occur for each of the motions, then the point of intersection of the characteristic lines E1 and E2 is a point of the enveloping surface This point is referred to as the characteristic point At the characteristic point, the (ω ) (ω ) conditions n ⋅ V1− 21 = and n ⋅ V1− 22 = are always satisfied Here, n designates a unit normal vector to the enveloping surface (ω ) (ω ) The conditions n ⋅ V1− 21 = and n ⋅ V1− 22 = are derived for the cases when the resultant relative motion of the surfaces VΣ is decomposed on the (ω ) (ω ) two components V1− 21 and V1− 22 , and both of the components are within the common tangent plane Definitely, these conditions are sufficient, but they are not mandatory It is feasible to decompose the resultant relative motion of © 2008 by Taylor & Francis Group, LLC 185 Profiling of the Form-Cutting Tools of Optimal Design the surfaces VΣ (which is within the common tangent plane) on two particu(ω ) (ω ) lar motions V1− 21 and V1− 22 that are not within the common tangent plane However, the location of the motion VΣ within the common tangent plane is the essential factor The discussed approach can be employed for determining the envelope (if any) of an arbitrary surface that has motion of any desired kind The interested reader may wish to go to [6] for details on the solution to the problem of computation of an envelope of a sphere that has screw motion Many practical examples of this approach are found in other sources Elements of theory of enveloping surfaces can be employed for the purposes of profiling form-cutting tools (the direct problem of the theory of surface generation), as well as for the purposes of determining the actual machined part surface (the inverse problem of the theory of surface generation) The desired and the actual part surface may differ because of violation of the necessary condition of proper part surface generation (see Chapter 7) As an illustrative example of a problem that can be solved using the elements of the theory of enveloping surfaces, see Figure 5.17 The generating surface of the first milling cutter is composed of two portions, say of the cylindrical portion T11 and of the spherical portion T12 When performing circular motion, the cylindrical portion T11 of the gen(1) erating surface generates the part surface P11 The spherical portion T12 of (1 the generating surface of the milling cutter generates the surface P12) , which is a torus When the circular feed-rate motion is stopped, then the spherical (2 portion P12) is machined on the part Similarly, the generating surface of the second milling cutter is composed of two portions, say of the cylindrical portion T21 and of the flat portion T22 The cylindrical portion T21 of the generating surface of the milling cutter   (2) P22 (1) P22 (2) P12 (1) P11 (2) P21 (1) P12 (1) P21 T21 T11 ωt1 ωt2 T22 T12 Figure 5.17 An example of problems that can be solved using elements of the theory of enveloping surfaces © 2008 by Taylor & Francis Group, LLC ... well as for the theory of surface generation   © 2008 by Taylor & Francis Group, LLC 160 Kinematic Geometry of Surface Machining The rate of conformity of the surface T to the surface P can be used... 184 Kinematic Geometry of Surface Machining of the parameter of motion w The theory of surface generation also deals with surfaces for which the enveloping surface is congruent to the moving surface. .. 182 Kinematic Geometry of Surface Machining Z r X Y Figure 5. 16 Generation of the enveloping surface to consecutive positions of a sphere of radius r that moves along the X axis the part surface