Kinematic Geometry of Surface Machinin Episode Episode 7 pot

186 Kinematic Geometry of Surface Machining generates the plane P 21 1( ) and the circular cylinder P 21 2( ) . The plane portion T 22 of the generating surface generates the plane P 22 1( ) and the plane P 22 2( ) . Knowing the dimensions of the milling cutters and the parameters of their motion relative to the work, the above formulae can be used to derive equations of all of the machined part surfaces. Similarly, the above formulae work when necessary to determine the generating surface of a cutting tool for machining of a given part surface. 5.2.2 Kinematical Method for the Determining of Enveloping Surfaces For engineering applications, one more method for the determination of enveloping surfaces is helpful. This method is referred to as the kinematical method for the determination of enveloping surfaces. Initially, this method was proposed by Shishkov as early as in 1951 [31]. The kinematical method is based on the particular location of the vector V 1 2− of relative motion of the moving surface and of the enveloping surface. Vector V 1 2− is located within the common tangent plane to the surfaces. This condition immediately follows from the following consideration. Motions of only two kinds are feasible for the moving surface and the enveloping sur- face. The surfaces can roll over each other, and they can slide over each other. The component of the resultant relative motion V 1 2− in the direction perpen- dicular to the surfaces is always equal to zero (Figure 5.18). The cutting tool performs a certain motion relative to the work. The part surface P is generated as an enveloping surface to consecutive positions of the generating surface T of the cutting tool. Points of three different kinds can be distinguished on the moving surface T. Consider points of the rst kind, for example, the point A (Figure 5.18). The vector of resultant motion of the cutting tool with respect to the work at point A is designated as V Σ ( )A . Projection Pr n V Σ ( )A of the vector V Σ ( )A onto the unit C P T (C) V Σ B A (B) n T (C ) n T (B) Σ V (A) n T (A) V Σ (A) Pr n V Σ > 0 (B) Pr n V Σ = 0 (C) Pr n V Σ < 0 FIGURE 5.18 The concept of the kinematical method for determining the enveloping surface. © 2008 by Taylor & Francis Group, LLC Proling of the Form-Cutting Tools of Optimal Design 187 normal vector n T A( ) to the generating surface T is pointed to the interior of the work body ( Pr n V Σ ( )A > 0 ). Therefore, in the vicinity of point A, the cutting tool penetrates the work body. In this way, roughing portions of the tool-cut- ting edges cut out the stock. Further, consider points of the second kind, for example, the point B (Figure 5.18). The vector of resultant motion of the cutting tool with respect to the work at point B is designated as V Σ ( )B . Projection Pr n V Σ ( )B of the vector V Σ ( )B onto the unit normal vector n T B( ) to the generating surface T is perpendicular to this unit normal vector — that is, it is tangential to the part surface P ( ). ( ) Pr n V Σ B = 0 Therefore, in the vicinity of the point B, the cutting tool does not penetrate the part body. The tool-cutting edges do not cut out stock. The generating surface T of the cutting tool generates the part surface P in the vicinity of the point B. Ultimately, consider points of the third kind, for example, the point C (Fig- ure 5.18). The vector of resultant motion of the cutting tool with respect to the work at point C is designated V Σ ( )C . Projection Pr n V Σ ( )C of the vector V Σ ( )C onto the unit normal vector n T C( ) to the generating surface T is pointed outside the part body ( Pr n V Σ ( )C < 0 ). Therefore, in the vicinity of point C, the cutting tool departs from the machined part surface P. In the vicinity of points of the third kind, the tool-cutting edges do not cut out stock, and the generating surface T of the cutting tool does not generate the part surface P. The considered example unveils the nature of the kinematical method for the determination of the enveloping surface. Apparently, this method can be employed to solve problems of both kinds, to prole form-cutting tools for machining a given part surface, and to solve the inverse problem of the theory of surface generation. As an example, generation of the plane P with the cylindrical grinding wheel having the generating surface T is considered in Figure 5.19. When machining a plane P, the grinding wheel rotates about its axis of rotation with a certain angular velocity ω T . Simultaneously, the grinding wheel travels with a feed rate S T . At each of the points A, B, and C on the generating surface T of the grinding wheel, the speed of the resultant relative motion of the cutter with respect to the work is designated V Σ ( )A , V Σ ( )B , and V Σ ( )C , respectively. The speed of the resultant motion at each point A, B, and C is equal to the vector sum of the feed rate S T and the speed of cutting V cut A( ) , V cut B( ) , and V cut C( ) . The feed rate S T is the same value for all points A, B, and C. The velocities V cut A( ) , V cut B( ) , and V cut C( ) are equal to the linear speed of rotation necessary to perform cutting. It is easy to see that the vectors V cut A( ) , V cut B( ) , and V cut C( ) are of the same magni- tude ( | | | | | | ( ) ( ) ( ) V V V cut A cut B cut C = = ). They differ from each other only by directions. However, the vectors V Σ ( )A , V Σ ( )B , and V Σ ( )C are of different magnitude ( | | ( ) V Σ A ≠ | | ( ) V Σ B ≠ | | ( ) V Σ C ), and they have different directions V Σ ( )A ≠ V Σ ( )B ≠ V Σ ( )C . Projections of the vectors V Σ ( )A , V Σ ( )B , and V Σ ( )C onto the unit normal vectors n T A( ) , n T B( ) , and n T C( ) are as follows: Pr n V Σ ( )A > 0 , Pr n V Σ ( )B = 0 , and Pr n V Σ ( )C < 0 . Therefore, in the vicinity of point A, the grinding wheel cuts the stock off; in the vicinity of point B, the part surface P is generating; and in the vicinity of point C, the cutting tool departs from the machined plane P. Similarly, gen- eration of the plane surface P can be performed with the cylindrical milling © 2008 by Taylor & Francis Group, LLC Proling of the Form-Cutting Tools of Optimal Design 189 Ultimately, the equation of the characteristic curve E ∂ ∂ ∂ ∂ ⋅ ∂ ∂ − ∂ ∂ ⋅ ∂ ∂       − ∂ ∂ X U Y V Z Y Z V Y U 1 1 1 1 1 1 1 1 1 ω ω 11 1 1 1 1 1 1 1 1 1 ∂ ∂ ∂ ∂ − ∂ ∂ ∂ ∂       + ∂ ∂ ∂ ∂ X V Z X Z V Z U X V ω ω 11 1 1 1 1 0 ∂ ∂ − ∂ ∂ ∂ ∂       = Y X Y V ω ω (5.43) can be derived for the equation of contact N V r r V⋅ = ∂ ∂ × ∂ ∂ ⋅ = − −1 2 1 1 1 1 1 2 0 ( ) ( ) . ω ω U V (5.44) When using the kinematical method, the sufcient condition for the exis- tence of the enveloping surface can be obtained in the following way: Con- sider a smooth, regular surface r 1 that is given in a Cartesian coordinate system X Y Z 1 1 1 . The equation of the surface r 1 is represented in the form r r 1 1 1 1 2 = ∈( , )U V C . The family r 1 ω of these surfaces in a Cartesian coor- dinate system X Y Z 2 2 2 is given in the form r r 1 1 1 1 ω ω ω = ( , , )U V , where the inequality ω ω ω (min) (max) ≤ ≤ is observed. Then, if the conditions ∂ ∂ × ∂ ∂       ⋅ ∂ ∂ = r r r 1 1 1 1 1 ( ) ( ) ( ) ( ) ( ) [ ω ω ω ω ω ω U V f UU V f C 1 1 1 0( ), ( ), ] , ω ω ω = ∈ (5.45) or ∂ ∂ × ∂ ∂       ⋅ = − r r V 1 1 1 1 1 2 1 ( ) ( ) ( ) ( ) [ ( ω ω ω ω ω U V f U )), ( ), ] ,V f U f V 1 1 2 1 2 0 0 ω ω = ∂ ∂       + ∂ ∂       ≠ (5.46) g U V f U f V f U 1 1 1 1 1 1 1 [ ( ), ( ), ] ω ω ω ω = ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂      r  ∂ ∂       ⋅ ∂ ∂       ∂ ∂       ⋅ 2 1 1 1 1 1 1 1 r r r V U V U −− ∂ ∂       ⋅ ∂ ∂       ∂ ∂       ∂ 2 1 1 1 1 1 1 2 r r r r V U V 11 1 1 2 0 ∂       ⋅ ≠ − V V (5.47) © 2008 by Taylor & Francis Group, LLC 190 Kinematic Geometry of Surface Machining are satised at a certain point, then the enveloping surface exists and can be represented by equations in the forms r r 1 1 1 1 = ( , , )U V ω and ∂ ∂ = r 1 0 ω . Methods of determining enveloping surfaces based on the implementation of methods developed in differential geometry make it possible to determine points of local tangency of the moving surface with the enveloping surface under xed values of w. However, for a certain value of w = Const, global interference of the surfaces could occur. Differential methods for determining enveloping surfaces can be employed only when the equation of the moving surface is differentiable. Because sur- faces in engineering applications are not innite and could be represented by patches, and so forth, the part surface P can also be generated by special points on the surfaces. In the general theory of enveloping surfaces, the family of surfaces that changes their shapes is considered as well. Results of the research in this area can be used in the theory of surface generation, particularly for generation of surfaces with the cutting tools that have a changeable generating surface T [15,20,21,30]. Example 5.4 Consider a plane T that has a screw motion. The plane T makes a certain angle τ b with X 0 axis of the Cartesian coordinate system X Y Z 0 0 0 . The reduced pitch p of the screw motion is given. Axis X 0 is the axis of the screw motion. The auxiliary coordinate system X Y 1 1 is rigidly connected to plane T (Figure 5.20). V Z 1 V 2 V T p·tanτ b −ω 2 ω 2 ω X 0 Z 0 Y 1 Y 0 X 1 E τ b FIGURE 5.20 Generation of a screw involute surface as an enveloping surface to consecutive positions of a plane that performs a screw motion. © 2008 by Taylor & Francis Group, LLC Proling of the Form-Cutting Tools of Optimal Design 191 The equation of plane T can be represented in the form Y X b1 1 = ⋅tan τ (5.48) The auxiliary coordinate system X Y Z 1 1 1 is performing the screw motion together with the plane T with respect to the motionless coordinate system X Y Z 0 0 0 . In the coordinate system X Y Z 1 1 1 , the unit normal vector n T to the plane T can be represented as n T b = −             1 0 1 tan τ (5.49) The position vector r T of an arbitrary point M of plane T is as follows: r T T T T X Y Z =             1 (5.50) The speed of the point M in its screw motion is v v R M = + ×[ ] ωω (5.51) where v is the speed of translation motion, and M is the speed of rotation motion. Determining the characteristic E direction of v M is important, but its magnitude is not of interest. Because of that, it can be assumed that | | ω = 1 . Therefore, ωω = = ⋅i v i, p (5.52) This yields v i i j k M p X Y Z = ⋅ + 1 0 0 1 1 1 (5.53) and v i j k M p Y Z= ⋅ − ⋅ + ⋅ 1 1 (5.54) © 2008 by Taylor & Francis Group, LLC 192 Kinematic Geometry of Surface Machining The dot product of the unit normal vector n T and of the speed v M equals n v T M b p Z⋅ = ⋅ − =tan τ 1 0 (5.55) Thus, the equation of contact can be represented in the form Z p b1 = ⋅tan τ (5.56) The above equation of contact together with the equation of plane T repre- sents the characteristic E. r E b b t y t p ( ) tan tan = ⋅ ⋅             τ τ 1 (5.57) where t designates the parameter of the characteristic E. The characteristic E is the straight line of intersection of these two planes. It is parallel to the coordinate plane X Z 1 1 and distanced at p b ⋅tan τ . For a given screw motion, the characteristic E remains at its location within the plane T in the initial coordinate system X Y Z 0 0 0 . The angle of rotation of the coordinate system X Y Z 1 1 1 about the X 0 axis is designated as e. The translation of the coordinate system X Y Z 1 1 1 with respect to X Y Z 0 0 0 that corresponds to the angle e is equal to p ∙ e. This yields composition of the equation of coordinate system transformation: Rs( ) cos sin sin cos 1 0 1 0 0 0 0 0 0 0 0 0 1 → = ⋅ −     p ε ε ε ε ε         (5.58) In order to represent analytically the enveloping surface P, it is necessary to consider the equation r E t( ) of the characteristic E together with the opera- tor Rs( )1 0→ of coordinate system transformation: r P b b X X p X p X ( , ) tan cos tan sin 1 1 1 ε ε τ ε τ ε = + ⋅ ⋅ ⋅ + ⋅ ⋅ − 11 1 ⋅ ⋅ + ⋅ ⋅             tan sin tan cos τ ε τ ε b b p (5.59) © 2008 by Taylor & Francis Group, LLC Proling of the Form-Cutting Tools of Optimal Design 193 Consider the intersection of the enveloping surface P by the plane X X p 0 1 0= + ⋅ = ε . The last equation yields X p 1 = − ⋅ ε . Therefore, r X b b p p p 0 0 ( ) tan (sin cos ) tan (cos ε τ ε ε ε τ = ⋅ ⋅ − ⋅ ⋅ ⋅ ⋅ εε ε ε + ⋅ ⋅             p sin ) 1 (5.60) Equation (5.60) represents the involute of a circle. The radius of the base circle of the involute curve is r p b b = ⋅tan τ (5.61) Therefore, the enveloping surface to consecutive positions of a plane T having a screw motion is a screw involute surface. The reduced pitch of the involute screw surface equals p, and the radius of the base cylinder equals r p b b = ⋅tan τ . The screw involute surface intersects the base cylinder. The line of intersection is a helix. The tangent to the helix makes the angle ω b with the axis of screw motion: tan ω b b r p = (5.62) From this, tan tan ω τ b b = , and ω τ b b = . The straight line characteristic E is tangent to the helix of intersection of the enveloping surface P with the base cylinder. This means that if a plane A is tangent to the base cylinder, a straight line E within a plane A makes the angle τ b with the axis of the screw motion, and the plane A is rolling over the base cylinder without sliding, then the enveloping surface P can be represented as a locus of consecutive positions of the straight line E that rolls without sliding over the base cylinder together with the plane A. The enveloping surface is a screw involute surface. The obtained screw involute surface (Figure 5.20) is as that shown in Figure 1.2 and as that analytically described by Equation (1.20). Another solution to the problem of determining the envelope of a plane having screw motion is given by Cormac [6]. 5.3 Profiling of the Form-Cutting Tools for Machining Parts on Conventional Machine Tools Cutting tools for machining parts on conventional machine tools feature a property that allows the generating surface of the cutting tool to slide over itself. Those surfaces that allow for sliding over themselves are currently the © 2008 by Taylor & Francis Group, LLC 194 Kinematic Geometry of Surface Machining most widely used surfaces in industry. This feature is of importance for the theory of surface generation. It can be used for simplication of the solution to the problem of proling of the form-cutting tool. Certain simplication is feasible because in the case of surfaces that allow for sliding over themselves, it is not necessary to determine the entire generating surface of the cutting tool. It is sufcient to determine either the prole of the generating surface of the cutting tool or the characteristic line along which the generating surface of the cutting tool makes contact with the machined part surface. 5.3.1 Two Fundamental Principles by Theodore Olivier A solution to the problem of proling the form-cutting tool for machining a part surface on a conventional machine tool can be derived much easier when using the fundamental principles of surface generation proposed by T. Olivier [12] as early as 1842. The R-mapping-based method for the proling of form-cutting tools (see Section 5.1) is general. It is a powerful tool for solving the most general prob- lems of cutting-tool proling. However, in particular cases, simpler methods of proling form-cutting tools are practical. It is common practice to design form-cutting tools on the premises of one of two Olivier’s principles: The First Olivier’s Principle: Both conjugate surfaces can be generated with an auxiliary generating surface. The generating surface in this case differs from both conjugate surfaces. Th e Second Olivier’s Principle: The auxiliary generating surface can be congruent to one of the conjugate surfaces. Prior to solving a problem of proling of a certain form-cutting tool, geom- etry of the part surface P (see Chapter 1) and the kinematics of the surface P generation (see Chapter 2) must be predened. The operators of the coordi- nate system transformation (see Chapter 3) are used for the representation of all elements of the surface-generation process in a common coordinates system, use of which is preferred for a particular consideration. If we are not just to develop a workable cutting tool, but also to develop the design of the optimal cutting tool, then the methods of analytical description of the geometry of contact of the part surface P and of the generating surface T of the cutting tool (see Chapter 4) are also employed. Design of a form-cutting tool can be developed on the premises of its gen- erating surface T. Derivation of the generating surface T is the starting point of designing the form-cutting tool. For generation of a given surface P, the cutting tool performs certain motions with respect to the work. The part surface P is given, and the generating surface T of the cutting tool is not yet known. Therefore, at the beginning, the actual relative motions of the surfaces P and T are not considered, but the corresponding motions of the part surface P and of a certain coordinate © 2008 by Taylor & Francis Group, LLC Proling of the Form-Cutting Tools of Optimal Design 195 system X Y Z T T T are analyzed. After being determined, the generating sur- face of the form-cutting tool would be described analytically in the coordi- nate system X Y Z T T T . 5.3.2 Profiling of the Form-Cutting Tools for Single-Parametric Kinematic Schemes of Surface Generation Kinematic schemes of surface generation, those that feature just one relative motion of the part surface P and of the generating surface T of the cutting tool are referred to as the single-parametric kinematic schemes of surface generation. In the case under consideration, it is convenient to begin consideration of the procedure of proling of a form-cutting tool for the generation of a sur- face P in the form of circular cylinder. When machining a circular cylinder of radius R P (Figure 5.21a) [28,29], the work rotates about its axis O P with a certain angular velocity ω P . A coordinate system X Y Z T T T is rotating with a certain angular velocity ω T . Axis of this rotation O T crosses at a right angle the axis of rotation O P of surface P. Simultaneously, the coordinate system X Y Z T T T travels along the axis O P with a feed rate S T . The generating surface T of the cutting tool in this case can be represented as an enveloping surface to consecutive positions of the surface P in the coordinate system X Y Z T T T . Remember that the coordinate system X Y Z T T T is the reference system at which the generating surface T could be determined. After being determined, the generating surface T of the cutting tool and the part surface P become tangent along the characteristic line E. In the case under consideration, the characteristic line E is represented with a circular arc ∪ABC of the radius R P . (b) H Y T X T Z T Y P X P Z P (a) Y P P R P E B C A Z P T H O T ω T ω P ω T ω P X T R T R P Z T O P (c) T P R P K H O T O P | R T |> R P FIGURE 5.21 Example of a single-parametric kinematic scheme for derivation of the generating surface of the cutting tool. © 2008 by Taylor & Francis Group, LLC 196 Kinematic Geometry of Surface Machining Use of a single-parametric kinematic scheme of surface generation allows a simplication. The generating surface T of the cutting tool can be generated not only as an enveloping surface to consecutive positions of the part surface P in the coordinate system X Y Z T T T , but also as a family of the characteristic lines E that rotate about the axis O T . In the example, the generating surface T of the cutting tool is shaped in the form of a torus surface. Radius R T of the generating circle of the torus surface T is equal to the radius R P of the surface P ( R R T P = ). The radius of the directing circle of the torus surface T is equal to the closest distance of approach H of the axes O P and O T . The determined torus surface T (Figure 5.21a) can be used for the designing of various cutting tools for the machining of the surface P: milling cutters, grinding wheels, and so forth. The considered example of implementation of the single-parametric kine- matic scheme of surface generation (see Figure 5.21a) returns a qualitative (not quantitative) solution to the problem of proling a form-cutting tool. No optimal parameters of the kinematic scheme of surface generation are deter- mined at this point. The closest distance of approach H of the axes O P and O T , and the optimal value of the cross-axis angle c are those parameters of interest (Figure 5.21b). Optimal values of the parameters H and c can be com- puted on the premises of analysis of the geometry of contact of the surfaces P and T. The optimal values of the parameters H and c can be drawn from the desired degree of conformity of the surface T to the surface P. Actually, in any machining operation, deviations of the actual cutting-tool conguration with respect to the desired conguration are unavoidable. Because of the deviations, it is practical to introduce appropriate alterations to the rate of conformity of surface T to surface P. Figure 5.21c illustrates an example when the rate of conformity of surface T to surface P is reduced. Reduction of the rate of conformity causes point contact of the surfaces P and T, instead of their line contact in the ideal case of surface generation (Figure 5.21a). The schematic (Figure 5.21c) allows for analysis of the impact of the rate of conformity of the surfaces T and P onto the accuracy and quality of surface generation. Varying values of the parameters H and c (includ- ing the case when c ≠ 90°), one can come up with the solution under which vul- nerability of the machining process to the resultant deviations of conguration of the cutting tool with respect to surface P is the smallest possible. Use of single-parametric kinematic schemes of surface generation (Figure 5.21) is a perfect example of implementation of the second Olivier’s principle for the purposes of proling form-cutting tools for the machining of a given part surface. 5.3.3 Profiling of the Form-Cutting Tools for Two-Parametric Kinematic Schemes of Surface Generation Kinematic schemes of surface generation, those that feature two relative motions of the part surface P and of the generating surface T of the cutting tool, are referred to as the two-parametric kinematic schemes of surface generation. © 2008 by Taylor & Francis Group, LLC [...]... Group, LLC 200 Kinematic Geometry of Surface Machining principle for the purposes of profiling form-cutting tools for machining a given part surface 5.3.4 Profiling of the Form-Cutting Tools for Multiparametric Kinematic Schemes of Surface Generation Kinematic schemes of surface generation, those that feature more than two relative motions of part surface P and generating surface T of the cutting tool... determining the optimal kinematics of surface generation, implementation of the R-mapping of surfaces is vital 5 .7 Incorrect Problems in Profiling the Form-Cutting Tools Use of the methods for derivation of the generating surface of the formcutting tool based on R-mapping of the surfaces and on elements of theory of enveloping surfaces returns an accurate solution to the problem of profiling the form-cutting... problem of determining optimal parameters of the kinematic scheme of surface generation and of optimal parameters of the geometry of the generating surface of the form-cutting tool can be obtained on the premises of the comprehensive analysis of the geometry of contact of the surfaces P and T (see Chapter 4) The considered example (Figure 5.22) is insightful It gives impetus to the investigation of impact... surface, the tool (Figure 5. 27) is performing a following motion Due to the following motion, at every instant of finishing the desired rate of conformity of the machining surface T of the tool to the part surface P can be ensured Use of the finishing tool (Figure 5. 27) of the discussed design allows for an increase in productivity of surface machining, as well as enhanced quality of the machined surface. .. impact of the rate of conformity of the surfaces P and T on the accuracy and quality of the machined part surface It is also practical for the analysis of vulnerability of the surface- generation process to the deviations of configuration of the cutting tool with respect to the work The use of two-parametric kinematic schemes of surface generation (Figure 5.22) is a perfect example of implementation of. .. designs of form-cutting tools can also be implemented For machining sculptured surfaces, form-milling cutters are used The machining surface T of the milling cutter is shaped in the form of a surface of revolution For better performance of the milling cutter, it is highly desired to have the principal radii of curvature of the generating surface of © 2008 by Taylor & Francis Group, LLC 204 Kinematic Geometry. .. Equation 5 .77 ) were used for the development of novel designs of the form-milling cutters (Figure 5.26) The generating surface T of the form-milling cutter (Figure 5.26) is represented with a surface of revolution (see Equation 5 .73 ) Principal radii of curvature R 1.T and R2.T of the generating curve of the surface T gradually change from one point  Pat No. 1.355. 378 , USSR, A Form Cutting Tool for Machining... mentioned lines Use of the milling cutter of the discussed design allows an increase of productivity of surface machining, as well as enhanced quality of the machined surface A similar approach is applicable to the design of finishing tools for reinforcement of sculptured surface using the method of surface plastic deformation    Pat No 1428563, USSR, A Tool for Finishing of Sculptured Surface on Multi-Axis... Geometry of Surface Machining the milling cutter equal to (or almost equal to) the extremal values of principal radii of curvature of the part surface taken with the opposite sign (see Chapter 4) For this purpose, the interval of variation of the radius of curvature of the generating curve of the milling cutter surface T and configuration of the generating curve with respect to the cutting tool axis of rotation... interval of variation of principal radii of curvature of the part surface P It is practical to assign a constant gradient of alteration of radius of curvature of the generating curve For the constant gradient of the alteration, the linear relationship between the radius of curvature ρT of the generating curve and the length of its arc L T must be observed Here, L T designates the length of the arc of the . Two-Parametric Kinematic Schemes of Surface Generation Kinematic schemes of surface generation, those that feature two relative motions of the part surface P and of the generating surface T of the cutting. solution to the problem of determining optimal parameters of the kinematic scheme of surface generation and of optimal parameters of the geometry of the generating surface of the form-cutting tool. two-parametric kinematic schemes of surface generation. The problem of derivation of the equation of the generating surface T of the form-cutting tool when the implemented kinematic scheme of surface