472 Kinematic Geometry of Surface Machining of the cutting edge of the cutter at the current CC-point.* In this method of surface machining, the work having axial prole 1 of variable curvature is rotating about its axis of rotation O P with a certain rotation ω P (Figure 11.9). The cutting edge of the cutter, an arc segment of a curve 2 having perma- nently variable radius of curvature R T , is used for performing this machin- ing operation. The cutter is traveling along the axial prole of the part surface in the peripheral direction 3. On the lathe, this motion is obtained as the superposi- tion of the axial motion 4 of the cutter and its reciprocal motion toward the part axis of rotation 5 and in backward direction 6. In addition, the cutter is performing the orientational motion of the sec- ond kind (see Chapter 2). This motion of the cutter is performed either in the direction 7 or in the direction 8. The actual direction of the orientation motion of the cutter depends upon the actual geometry of the axial prole 1 of the part surface at two neighboring CC-points K i and K i+1 . Ultimately, due to the cutter motion either in the direction 7 or in the direction 8, the cutting edge is rolling with sliding over the axial prole 1 of the part surface being machined. Implementation of the orientational motion of the cutter allows for better t of the radius of curvature R T to the part surface radius of curvature R P at every CC-point K. In this way (see Equation 11.10), the surface generation output is increasing. It is important to note a possibility of machining form surfaces of revolu- tion in compliance with the method (Figure 11.9), when not the cutter, but a * SU Pat. No. 1171210, A Method of Turning of Form Surfaces of Revolution./S.P. Radzevich, Int. Cl. B23B 1/00, Filed November 3, 1984. 2 7 4 6 5 1 8 K i+1 K i 3 О P ω P FIGURE 11.9 Utilization of the orientational motion of the second kind of the cutter in the method of turning form surfaces of revolution (SU Pat. No. 1171210). © 2008 by Taylor & Francis Group, LLC Examples of Implementation of the DG/K-Based Method 473 milling cutter or grinding wheel having a corresponding axial prole of the generating surface of the tool is used instead. Similar to the utilization of the orientational motion of the second kind (Figure 11.9), the orientational motion of the rst kind of the cutter can be utilized in the turning of surfaces of revolution as well. A method of turning of form surfaces of revolution (Figure 11.10) is fea- turing in its kinematics the orientational motion of the rst kind [8]. This method of surface machining is similar to the earlier discussed method of surface machining shown in Figure 11.9. For convenience, designations of the major elements in Figure 11.10 are identical to the designations of the corresponding major elements in Figure 11.9. So, there is no reason to repeat all the details of the method under consideration. In the method of surface machining* (Figure 11.10), the orientational motion of the rst kind is utilized. The orientational motion of this kind allows for turning of the cutter about the axis O K i. along the unit normal vector n P either in the direction 9 or in the direction 10. The actual direc- tion of the orientation motion depends upon parameters of geometry of the surface P at the two neighboring CC-points K i and K i+1 . In the neighboring CC-point K i+1 , the orientation motion is designated as 11/12. * SU Pat. No. 1232375, A Method of Turning of Form Surfaces of Revolution./S.P. Radzevich, Int. Cl. B23B 1/00, Filed September 13, 1984. 8 10 9 3 О K.i+1 O P O K.i 2 11 1 6 5 4 12 P 7 ω P K i+1 K i FIGURE 11.10 Utilization of the orientational motion of the rst kind of the cutter in the method of turning of form surfaces of revolution (SU Pat. No. 1232375). © 2008 by Taylor & Francis Group, LLC 474 Kinematic Geometry of Surface Machining Evidently, the parameters of the orientational motion of the rst kind are strongly constrained by the limit values of the geometrical parameters of the cutting edge of the cutter to be used, rst of all by the clearance angle of the cutting edge. Implementation of the orientational motion of the cutter allows for better t of the radius of curvature R T to the part surface radius of curvature R P at every CC-point K. In this way (see Equation 11.10), the surface-generation output is increasing. It is the right point to stress that it may also be possible to machine form surfaces of revolution in compliance with the method (Figure 11.10) when not the cutter, but a milling cutter or grinding wheel having a corresponding axial prole of the generating surface of the tool is used instead. Under such a scenario, no constraints are imposed by the limit values of the geometrical parameters of the cutting edge of the cutting tool to be used. 11.2.2 Milling Operations The earlier discussed methods of turning form surfaces of revolution (see Figure 11.9 and Figure 11.10) allow substitution of the cutter with a mill- ing cutter or with a grinding wheel having a corresponding prole of axial cross-section of the generating surface T. These methods of surface machin- ing indicate that efcient methods of milling of form surfaces of revolution can be developed. A method of milling of form surfaces of revolution on NC machine tools was developed by the author [26]. In compliance with the method (Figure 11.11), a form surface of revolution P having axial prole 1 is machined with the mill- ing cutter having a curved axial prole of the generating surface T. The work is rotating about its axis of rotation O P with a certain rotation ω P . The axis 6 4 2 1 View А (Turned) 7 8 K A 3 5 K O P O T R o.P R o.T d P d T d P d T ω P ω P ω T ω P FIGURE 11.11 A method of milling a form surfaces of revolution. (See also Radzevich, S.P., and Dmitrenko, G. V., Mashinostroitel’, No. 5, 17–19, 1987.) © 2008 by Taylor & Francis Group, LLC Examples of Implementation of the DG/K-Based Method 475 of rotation of the work O P and the axis of rotation of the milling cutter O T are crossing at a right angle. The milling cutter is traveling in the direction 3 along the axial prole 1 of the part with a certain peripheral feed rate. This motion is a superposition of the milling cutter motion in the axial direction 4 of the work, and of its motion 5 toward the work axis of rotation and in backward direction 6. In addition to the mentioned motions, the milling cutter is also performing the motion of orientation of the second kind. While traveling in the axial direc- tion 4 of the work, the milling cutter simultaneously performs linear motion along its axis of rotation O T . This motion is performing either in the direc- tion 7 or in the opposite direction 8 depending upon the geometry of the surfaces P and T at the current CC-point K. The orientational motion of the milling cutter provides a possibility for increasing the rate of conformity of the generating surface T of the milling cutter to the form surface of revolution P at every CC-point K. In this way, the surface generation output is increased. Grinding of form surfaces of revolution can be performed in the same way as shown in Figure 11.11. 11.2.3 Machining of Cylinder Surfaces Orientation motions of the cutting tool are also used for the improvement of machining of general cylinder surfaces. Such a possibility is illustrated below by the method of machining of a camshaft.* The method of machining of a camshaft [1] is targeting the maximal pos- sible material removal rate. In compliance with the method, a grinding wheel having conical generat- ing surface T is used for the machining of the surface P of a cam (Figure 11.12). The grinding wheel is rotating about its axis of rotation O T with a certain rotation ω T . The surface generation motions are performed by the work. The set of these motions includes the rotation ω P of the work about the axis of rotation O P and the reciprocal motion 1 in the direction of the common perpendicular to the axes O P and O T . The rotation ω P of the grinding wheel can be either uniform or nonuniform. The grinding wheel is performing an auxiliary straight motion 2. Direc- tion of the motion 2 is parallel to the axis of rotation of the work O P . The straight motion 2 is timed with work rotation ω P in the way under which the material removal rate is constant and equal to its greatest feasible value: Q L v b Const cr T max , [ ( )] ( )= ⋅ ⋅ ⋅ =0 5 2 ϕ ϕ (11.11) * SU Pat. No. 1703291, A Method of Machining of Form Surfaces./S.I. Chukhno and S.P. Radzevich, Int. Cl. B23C 3/16, Filed August 2, 1989. © 2008 by Taylor & Francis Group, LLC Examples of Implementation of the DG/K-Based Method 477 Reinforcement of form surfaces of revolution can be performed with the tool having a conical generating surface T [21,22]. In this method (Figure 11.13), the work is rotating about its axis O P with a certain rotation ω P . The axis O T of the conical indenter 1 (conical tool) is crossing the work axis of rotation O P at the right angle. The tool is moving along the axial prole of the part surface P with a certain peripheral feed rate. The indenter 1 is pressed into the part surface P by normal force P rnf . In the relative motion, the CC-point K traces the trajectory 2 on the machined part surface. Two congurations of the indenter 1 are possible. The rst conguration is shown in Figure 11.13. In such a tool conguration, its bigger diameter is below the smaller diameter. The inverse conguration of the tool, when the smaller diameter is below the bigger diameter, is feasible as well. When machining a form surface of revolution, the portion of the surface P having bigger diameter is machined with the portion of the tool having smaller diameter, and vice versa. In this way, it is possible to maintain that same pressure when machining portions of the part with different geome- try of the surface P. For this purpose, the indenter is performing an auxiliary straight motion either downward 3 or upward 4, depending on the geometry of the surface P being machined. The auxiliary motion requires in corre- sponding compensation of center distance between the axes O P and O T . A component of the auxiliary straight motion creates the orientational motion of the second kind of the tool. Reinforcement of the part surfaces under the optimal pressure that is of the same value at every CC-point K enables an increase of the quality of the surface nish. For reinforcement of form surfaces of revolution, not only a conical tool but a cylindrical tool* can be used as well. In the method of reinforcement of a * SU Pat. No. 1463454, A Method of Reinforcement of Surfaces./S.P. Radzevich, Int. Cl. B24B 39/00, 39/04, Filed May 5, 1987. O P 2 1 P 4 3 K O T ω P FIGURE 11.13 A method of nishing of a form surface of revolution with a conical indenter. © 2008 by Taylor & Francis Group, LLC 478 Kinematic Geometry of Surface Machining form surface of revolution [13], nishing of the part surface is performed with the cylindrical indenter. When machining the surface P, the work is rotating about its axis O P with a certain rotation ω P (Figure 11.14). The cylindrical indenter 1 is pressed into the part surface P by normal force P rnf . The tool 1 is traveling along the axial prole of the part surface P with a certain peripheral feed rate. Simultaneously, the tool 1 is performing the orientational motion of the rst kind w n about unit normal vector n P to the part surface P. The ori- entational motion of the tool is timed with part diameter d P i( ) at the current CC-point K. Due to the orientational motion of the tool, the angle that the axis O P of the part makes with the axis O i * at the current ith point is under the control of the user. At every CC-point K, the angle of crossing α i * is of its optimal value. When the diameter d P i( ) is bigger, then the cross-axis angle α i * is also bigger, and vice versa. In this way, the optimal pressure that is of the same value at every CC-point K is maintained. In particular cases, two paths of the indenter 1 are required to be per- formed. On the second tool-path, the angle that the axis O P of the part makes with the axis O i ** at the current ith point is reduced to a value α i ** . On the second tool-path, angle α i ** at the current CC-point K is always smaller than that angle α i * on the rst tool-path ( α α i i ** * < ). Reinforcement of the part surfaces under the optimal pressure that is of the same value at every CC-point K enables for an increase in the quality of the surface nish. Similarly, reinforcement of part surfaces of revolution can be performed with a form roller. For example, a method of reinforcement of a surface of revolution is featuring the implementation of a form tool [15]. The method of reinforcement of form surfaces of revolution is illustrated with an example of nishing of a cylindrical part surface P (Figure 11.15). However, the method of surface nishing can be implemented for the rein- forcement form surfaces of revolution as well. P 2 α i ** α i * O i * O i O P ** d P (i) 1 ω P ω n K FIGURE 11.14 A method of reinforcement of a form surface of revolution with a cylindrical tool (SU Pat. No. 1463454). © 2008 by Taylor & Francis Group, LLC 480 Kinematic Geometry of Surface Machining 11.3 Finishing of Involute Gears Various methods of shaving are widely used for nishing spur and helical involute gears [28]. Most gear shaving operations are not optimized. Com- putation of the optimal parameters of a diagonal shaving operation pro- vides a perfect example of implementation of the DG/K-based method of surface generation. In compliance with the method, it is possible to compute the desired design parameters of the shaving cutter best suited for nishing the given involute gear. It is also possible to compute the optimal parameters of the relative motions of the shaving cutter with respect to the gear to be nished. For this purpose, the indicatrix of conformity Cnf P T R g sh ( / ) of the generating surface T sh of the shaving cutter to the screw involute tooth sur- face P g of the gear is commonly employed. In diagonal shaving (Figure 11.16), the work-gear rotates about its axis O g with a certain angular velocity ω g . The shaving cutter rotates about its axis O sh with an angular velocity ω sh that is timed with the ω g — that is, ω ω sh g u= ⋅ , where u is the tooth ratio ( u N N g sh = ; here N g is the number of the gear teeth, and N sh is the number of the shaving cutter teeth). Axes of rotation O g of the gear and O sh of the shaving cutter are at a center- distance C, and cross each other at an angle Σ . The angle Σ is as follows: Σ = + ψ ψ g sh . Here ψ p is the gear helix angle. It is positive (+) to the right- hand gear and negative (−) to the left-hand gear to be machined. The same is observed with respect to the shaving cutter helix angle ψ sh . In addition, the Work Gear Work Gear F diag F diag C L Σ Shaving Cutter K 2 K 1 C Shaving Cutter ω sh ω sh O sh O sh O g O g B g ω g ω g θ θ FIGURE 11.16 Schematic of a diagonal shaving method. (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 481 shaving machine table reciprocates relative to the shaving cutter with feed F diag . The axis of rotation O g of the gear and direction of the feed F diag make a certain angle q. The traverse path of the feed F diag is at a certain angle q to the gear axis of rotation O g (Figure 11.16). The relationship between the face width of the gear B g and the shaving cutter B sh is an important consideration. It denes the value of the diagonal traverse angle. The surface of tolerance P h[ ] is at a distance of the tolerance [ ]h to the gear- tooth surface P g . After tooth surface P g of a gear and tooth surface T sh of a shaving cutter are put into contact at point K, then the surface T sh intersects the surface P h[ ] . The line of intersection is a certain closed three-dimensional curve C pt shown in Figure 11.17. It bounds the spot of contact of the gear and the shaving cutter tooth. It is recommended that the area of the spot of con- tact C pt be kept as small as possible (Figure 11.17). Due to the tooth surfaces P g and T sh making contact at a distinct point K, only discrete generation of the gear ank is feasible. In order to increase productivity of the gear nishing operation, it is required to maintain the tool-paths on the gear-tooth ank P g as wide as possible. For this purpose, the major axis of the spot of contact C pt has to be as long as possible, and relative motion V Σ of the surfaces P g and T sh has to be directed orthogo- nally to the major axis of the spot of contact C pt . Tooth of the Shaving Cutter Tooth of the Work-Gear K K V Σ sh g g g pt χ ≠ 90 FIGURE 11.17 The problem at hand. (From Radzevich, S.P., International Journal of Advanced Manufacturing Technology, 32 (11–12), 1170–1187, 2007. With permission.) © 2008 by Taylor & Francis Group, LLC 482 Kinematic Geometry of Surface Machining Fortunately, it is possible to control the shape, size, and orientation of the spot of contact C pt . For this purpose, an optimal combination of the design parameters of the shaving cutter, of direction and speed of the feed F diag , of rotation of the gear ω g , and of rotation of the shaving cutter ω sh must be computed. This also makes possible the control of the direction of relative motion of the surfaces P g and T sh , and in such a way as to increase the gear accuracy and to cut the shaving time. For the analysis below, equations of the tooth ank surfaces P g and T sh are necessary. The equation of the gear-tooth surface P g can be represented in the form of the column matrix (see Equation 1.20): r g b g g g b g g b g g g r V U V r V U = + − . . . cos cos sin sin sin ψ ψ bb g g b g b g p b g V r U . . . . sin tan sin ψ ψ −             1 (11.12) where the gear base cylinder diameter d r b g b g. . = 2 can be computed from d m N N b g g n n b g g . . cos cos sin . cos = ⋅ ⋅ − = ⋅ ⋅ φ φ λ 1 25 4 2 2 φφ φ λ n g n b g P ⋅ −1 2 2 cos sin . (11.13) where m is the gear modulus, N g is the number of gear teeth, φ n is the nor- mal pressure angle, λ b g. is the gear base lead angle ( λ ψ b g b g. . = −90 o ), ψ b g. is the gear base lead angle, and P g is the diametral pitch. The U g parameter in Equation (11.12) can be expressed in terms of param- eters of the gear design [22,27]: U d d d d g y g b g b g y g b g g = − = − . . . . . sin sin sin 2 2 2 2 2 2 ψ ψ φφ n (11.14) where the diameter of a cylinder that is coaxial to the gear is designated as d y g. , and ψ g is the gear pitch helix angle. Equation (1.7) yields computation of the fundamental magnitudes of the rst order E F r G U r g g b g b g g g b g b g = = − = + 1 2 4 2 , cos , cos co . . . . ψ ψ ss . 2 ψ b g (11.15) for the screw involute surface P g . For the fundamental magnitudes of the second order, use of Equation (1.11) returns expressions L M N U g g g g b g b g = = = − ⋅ ⋅0 0, , sin cos . . τ τ (11.16) © 2008 by Taylor & Francis Group, LLC . 472 Kinematic Geometry of Surface Machining of the cutting edge of the cutter at the current CC-point.* In this method of surface machining, the work having axial prole 1 of variable. cross-section of the generating surface T. These methods of surface machin- ing indicate that efcient methods of milling of form surfaces of revolution can be developed. A method of milling of form surfaces. Group, LLC 478 Kinematic Geometry of Surface Machining form surface of revolution [13], nishing of the part surface is performed with the cylindrical indenter. When machining the surface P, the