Kinematic Geometry of Surface Machinin Episode 3 pptx

Kinematics of Surface Generation 31 unit vectors t 1.P , t 2.P , and n P . Thus, the moving coordinate system x P y P z P is established at every point of the sculptured surface P. Most of the equations are signicantly simplied when Darboux’s trihedron is used. However, computation of the unit tangent vectors of the principal directions is often a computation-consuming procedure. Second, in order to compose a right-hand Cartesian local coordinate system x P y P z P with the origin at point K of contact of the surfaces P and T, the unit tangent vectors u P , v P , and n P can be employed. Generally speaking, pairs of the vectors u P and n P , v P and n P are orthogonal to each other, while the unit tangent vectors u P and v P are not orthogonal to each other. Aiming the composing of the right-hand Cartesian coordinate system, the third unit vector v u n P P P * = × can be used. The three unit vectors u P , v P * , and n P make up a right-hand trihedron. The orthogonal trihedron u P , v P * , and n P can be constructed at any surface point. However, in order to construct the trihedron, computation of the vector v P * is necessary at every surface point. Third, the initial parameterization of the surface P can be changed when it becomes an orthogonal parameterization. Under such a scenario, the unit tangent vectors u P and v P are always orthogonal to each other. In order to change the initial surface P parameterization, it is necessary to replace the parameters U P and V P with new parameters U U U V P P P P * * ( , )= and V P * = V V U V P P P P * * ( , )= . First derivatives with respect to the new parameters U P * and V P * ∂ ∂ = ∂ ∂ ⋅ ∂ ∂ + ∂ ∂ ⋅ ∂ ∂ r r r P P P P P P P P P P U U U U V V U * * * (2.1) ∂ ∂ = ∂ ∂ ⋅ ∂ ∂ + ∂ ∂ ⋅ ∂ ∂ r r r P P P P P P P P P P V U U V V V V * * * (2.2) can be drawn from these equations so that A * = A ∙ J. Here it is designated A = ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂      X U Y U Z U X V Y V Z V P P P P P P P P P P P P        = ∂ ∂ ∂ ∂       T P P P P U V r r (2.3) and J = ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂             U U U V V U V V P P P P P P P P * * * * (2.4) is called the Jacobian matrix of the transformation. © 2008 by Taylor & Francis Group, LLC 32 Kinematic Geometry of Surface Machining The new fundamental matrix is given by E F F G E F F G P P P P T T T T P P P P * * * * * * = ⋅ = ⋅ ⋅ ⋅ = ⋅ ⋅A A J A A J J J (2.5) By the properties of determinants, it can be seen from Equation (2.5) that E F F G E F F G P P P P P P P P * * * * | |= ⋅J 2 (2.6) It can be shown on the premises of Equation (2.1), Equation (2.2), and Equation (2.6) that the unit surface normal n P is invariant under the transformation, as it could be expected. The transformation of the second fundamental matrix can similarly be shown to be given by L M M N L M M N P P P P P P P P * * * * | |= ⋅J 2 (2.7) by differentiating Equation (2.1) and Equation (2.2) and using the invariance of n P . It can be shown from Equation (2.6) and Equation (2.7) that the principal curvatures and the principal directions are invariant under the transformation. Equation (2.6) and Equation (2.7) yield the natural form of the surface P representation with the new U P * and V P * parameters. It can be concluded that the unit normal vector n P and the principal directions and curvatures are independent of the parameters used and are therefore geometric properties of the surface. They should be continuous if the surface is to be tangent and curvature continuous. At that point it is necessary to establish a set of constraints onto the rela- tionships U U U V P P P P * * ( , )= and V V U V P P P P * * ( , )= under which an orthogonal parameterization of the sculptured surface P can be obtained. In order to obtain an orthogonal parameterization of the sculptured surface P, the rela- tions U U U V P P P P * * ( , )= and V V U V P P P P * * ( , )= have to satisfy the following two conditions, F P ≡ 0 and M P ≡ 0, which are the necessary and sufcient conditions for the orthogonally parameterized sculptured surface P. Once reparameterized, the surface P yields easy computation of the orthogonal trihedron u P , v P , n P at every surface point. The presented consideration clearly illustrates the feasibility of composing the right-hand trihedron u P , v P , n P using one of the methods discussed above. The three unit vectors u P , v P , n P can be used as the direct vectors of the local Cartesian coordinate system x P y P z P with the origin at point K of contact of the sculptured surface P and the generating surface T of the cutting tool. © 2008 by Taylor & Francis Group, LLC Kinematics of Surface Generation 33 2.1.2 Elementary Relative Motions Instant relative motion of the cutting tool can be interpreted as an instant screw motion. Depending on the actual conguration of the local reference system x P y P z P , the instant screw motion of the cutting tool can be decomposed on not more than six elementary motions — that is, on three translations along and onto three rotations about axes of the local coordinate system x P y P z P . Not all of the six elementary relative motions are feasible. The translational motion of the cutting tool along z P is not feasible. It is eliminated from the instant kinematics of sculptured surface generation because of two reasons.* First, the elementary motion of the cutting tool in the +n P direction results in interruption of the surface-generating process, which is not allowed. Second, the motion of the cutting tool in the −n P direction results in unavoidable interference of the surfaces P and T. Any interference of the surfaces P and T is not allowed. Hence, the speed of the translational motion of the cutting tool along the common perpendicular must be equal to zero (Figure 2.1): V V k z n ≡ = ∂ ∂ ⋅ = z t P P 0 (2.8) Here time is designated as t. The speed of translational motion of the cutting tool along x P and y P axes is designated as V x and V y , respectively. Then  x ,  y , and  z designate rotations about axes of the local coordinate system x P y P z P . According to the principal instant kinematics of sculptured surface generation (Figure 2.1), the cutting tool instant screw motion relative to the surface P can be decomposed on not more than ve elementary instant relative motions. This set of ve feasible elementary relative motions includes two translations V u r x = ∂ ∂ ⋅ = ∂ ∂ ∂ x t U t P P P P 2 ; V v r y = ∂ ∂ ⋅ = ∂ ∂ ∂ y t V t P P P P 2 (2.9) along axes of the local reference system x P y P z P , and three rotations ω x x u= ∂ ∂ ⋅ ϕ t P ; ωω y y v= ∂ ∂ ⋅ ϕ t P ; ω z z n= ∂ ∂ ⋅ ϕ t P (2.10) about axes of the local reference system x P y P z P . Here the angles of rotation of the cutting tool about axes of coordinate system x P y P z P are designated as f x , f y , and f z , respectively. * Further, attention will be focused on special cases of sculptured surface machining when a motion of the cutting tool along the z P axis is allowed. © 2008 by Taylor & Francis Group, LLC 34 Kinematic Geometry of Surface Machining 2.2 Generating Motions of the Cutting Tool After being machined on a multi-axis NC machine, the sculptured surface is represented as a series of tool-paths. Generation of a sculptured surface by consequent tool-paths is a principal feature of sculptured surface machining on a multi-axis NC machine. The motion of the cutting tool along a tool-path can be considered as a permanent following motion, and a side-step motion of the cutting tool (in the direction that is orthogonal to the tool-path) can be considered as a discrete following motion. The surface P can be generated as an enveloping surface to consecutive positions of the moving surface T when the surfaces P and T make either linear or point contact. When linear contact of the surface is observed, then a one- degree-of-freedom relative motion of the cutting tool is sufcient for generating the surface P. Such relative motion is referred to as one-parametric motion of the cutting tool. When the surfaces make a point contact, then a two-degrees- of-freedom relative motion of the cutting tool is required for generating the entire surface P. Such relative motion is referred to as two-parametric motion of the cutting tool. The number of available degrees of freedom can exceed 2 degrees. This results in multiparametric motion of the cutting tool. Known methods of developing machining operations do not provide a single solution to the problem of synthesis of the most efcient (i.e., optimal) machining operations. Known methods return a variety of solutions to the problem, of which the efciency of each is not the highest possible. For the computing of parameters of relative motion, known methods are based mostly on the equation of contact*: n P ∙ V Σ = 0. Here V Σ denotes the speed of the resultant motion of the cutting tool relative to the work. The equation of contact n P ∙ V Σ = 0 imposes restrictions onto only one component of the relative motion of the cutting tool and of the work. This means that projections of speed of the resultant motion onto the direction specied by the unit normal vector n P must be equal to zero Pr n V Σ = 0 ≡ 0. No restrictions are imposed by the equation of contact onto other projections of the vector of resultant motion of the cutting tool and of the work. In compli- ance with the equation of contact, the magnitude and direction of V Σ within the common tangent plane can be arbitrary. Evidently, the innite number of the vectors V Σ satises the equation of contact n P ∙ V Σ = 0. All are within the tangent plane to the sculptured surface P. This is the principal reason why the implementation of known methods returns an innite number of solutions to the problem under consideration. No doubt, performance of a sculptured surface generation depends on direction of the vector V Σ . For a certain direction of V Σ , it is better; for another * To the best of the author’s knowledge, the condition of contact in the form that slightly differs from the equation of contact n P ∙ V Σ = 0 is known at least since publication of the monograph by Willis [39]. In the present form, n P ∙ V Σ = 0, the equation of contact is known from the late 1940s/early 1950s [38]. © 2008 by Taylor & Francis Group, LLC Kinematics of Surface Generation 35 direction of V Σ , it is poor. This yields intermediate conclusions that the optimal direction of V Σ exists, that this direction satises the equation of contact n P ∙ V Σ = 0, and that the optimal parameters of the direction of V Σ can be computed. For the computation of the optimal parameters of the instant relative motion V Σ of the work and of the cutting tool, an appropriate criterion of optimization is necessary. The major purpose of the criterion of optimization is to select the optimal direction of the vector V Σ from the innite number of feasible directions that satisfy the equation of contact n P ∙ V Σ = 0. In order to satisfy the equation of contact, vector V Σ of the resultant relative motion of the cutting tool must be within the tangent plane to the surfaces P and T at the CC-point K. This is the geometrical interpretation of the equation of contact. Consider a local reference system x P y P z P with the origin at CC-point K (Figure 2.2). In the coordinates system x P y P z P , vector V Σ can be described analytically by vector equation: V u v n Σ = ∂ ∂ ⋅ + ∂ ∂ ⋅ + ∂ ∂ ⋅ x t y t z t P P P P P P (2.11) In that same local coordinate system, the equality n P = k P is observed. Substituting Equation (2.11) and the relationship n P = k P along with Equa- tion (2.8) into the equation of contact n P ∙ V Σ = 0, one can obtain n V P P z t ⋅ = ∂ ∂ = Σ 0 (2.12) r P V Σ n P Z P Y P X P P v P K n T v T u T u P v FIGURE 2.2 Feasible relative motions of the cutting tool. © 2008 by Taylor & Francis Group, LLC 36 Kinematic Geometry of Surface Machining In order to satisfy the equation of contact, projection of the vector V Σ of the resultant motion on the direction perpendicular to the surfaces P and T must equal zero. This is proof that the vector V Σ must be within the common tangent plane to the surfaces P and T. It is important to point out here that the condition pr n V Σ < 0 can be considered as the condition of roughing. Portions of the surface T that perform such motion remove stock while machining the work. Condition pr n V Σ = 0 that is equivalent to the condition of contact n P ∙ V Σ = 0 corresponds to generating the surface being machined. Finally, the condition pr n V Σ > 0 relates to portions of the surface T that are departing from the machined surface P. These conditions are presented in more detail in Chapter 5. The equation of contact n P ∙ V Σ = 0 does not uniquely determine the instant kinematics of sculptured surface generation. In addition to the innite number of feasible directions for the vector V Σ , one more reason can affect the indeniteness of the equation of contact. Location and orientation of the common perpendicular n P are uniquely specied by the geometry of the surface P. Usually it cannot be changed. However, in special cases of machining, the orientation of the common perpendicular n P can be changed for manufacturing purposes. For example, when machining a thin-wall part (Figure 2.3), an elastic deformation can be applied to the work. Under the applied load, unit normal vectors n m a , n m b , and n m c to the part surface P become parallel to each other. In the deformed stage, the work is machining on a lathe with simple motion of the cutter relative to the work. The elastic deformation results in the plane machining instead of machining of the concave sculptured surface (for this purpose, the magnitude of the applied load may vary according to the cutter feed rate — a distributed load of variable magnitude can be applied as well). After being released, the work gets it original shape, and the machined plane a P n c P n b P n c m n b m n a m n Feed Load P Cutter a b c a b c FIGURE 2.3 An example of the application of elastic deformation of the work for the purpose of machining the surface. © 2008 by Taylor & Francis Group, LLC Kinematics of Surface Generation 37 transforms into a concave sculptured surface P. Unit normal vectors n P a , n P b , and n P c to the machined surface P are not parallel to each other. If elastic deformation is used for manufacturing purposes, then the equation of contact must be satised in the deformed stage of the surface being machined. Elastic deformation of a work for manufacturing purposes is observed when machining ex-spline that is an essential machine element of a harmonic drive, and in other applications. The capability to change the orientation of the unit normal vector to the surface is limited; however, such a capability exists, and it affects the generation of the surface P. The last is of principal importance. Capabilities of variation of orientation of the unit normal vector n T to the generating surface T of a cutting tool are signicantly wider, especially when implementing cutting tools of special design with variable shape and parameters of the surface T for machining a given sculptured surface [28,29,33]. When machining a sculptured surface on a multi-axis NC machine, the cutting tool is performing a continuous follow motion along every tool-path [30,33]. Therefore, the generating motion can be considered as a continuous follow motion of the cutting tool relative to the work. This motion results in the CC-point traveling along the tool-path. After the machining of a certain tool-path is complete, then the cutting tool feeds across the tool-path in a new position. The machining of another tool- path begins from the new position of the cutting tool. Hence, the feed motion of the cutting tool can be represented as a discontinuous follow motion of the cutting tool relative to the work. This motion results in the CC-point traveling across the tool-path. The generating motion of the cutting tool can be described analytically. For this purpose, the elementary motions that make up the principal instant kinematics of surface generation are used (Figure 2.1). The elementary relative motions are properly timed (synchronized) with one another in order to pro- duce the desired instant generation motion of the cutting tool (Figure 2.4). The following equations can easily be composed based on the premises of the analysis of the instant kinematics of sculptured surface generation: | | | | . V x y x = ⋅ ωω R P and | | | | . V y x y = ⋅ ω R P (2.13) Here R P.x and R P.y designate the normal radii of curvature of the sculptured surface P. The radii of curvature R P.x and R P.y are measured in the plane sec- tions through the unit tangent vectors u P and v P , respectively. Equation (2.13) yields a generalization of the following kind: | | | | . V Σ Σ = ⋅ - ω T P P R (2.14) where V Σ is the vector of the resultant motion of the CC-point along the tool- path;  T−P is a vector of instant rotation of the surface T about an axis that is perpendicular to the normal plane through the vector V Σ ; and R P.Σ is the radius of normal curvature of the surface P in the direction of V Σ . © 2008 by Taylor & Francis Group, LLC Kinematics of Surface Generation 39 The relative motion V z of the cutting tool (Figure 2.1) is not completely eliminated from further analysis. Taking into consideration the tolerance on accuracy of machining of the surface P (Figure 1.5), the motion V z along the unit normal vector n P is feasible if it is performing within the tolerance d = d + + d - . Moreover, due to deviations of the desired cutting tool motion from the actual cutting tool motion, the motion V z always observed is the actual machining operation. If necessary, the motion V z can be incorporated into the principal instant kinematics of surface generation. This is one more example of the difference between the classical differen- tial geometry of surfaces and the kinematic geometry of surface generation. The principal instant kinematics of surface generation includes ve elementary relative motions. Thus, the surface P can be represented as an enveloping surface to consecutive positions of not more than ve-parametric motion of the surface T of the cutting tool. 2.3 Motions of Orientation of the Cutting Tool As mentioned above, the machining of a sculptured surface on a multi-axis NC machine is the most general case of surface generation. This is because two surfaces, the sculptured surface P and the generating surface T of the cutting tool make point contact at every instant of machining. Among various kinds of relative motions of the cutting tool, one more motion can be distinguished. When performing relative motion of this kind, the CC-point does not change its position on the sculptured surface P being machined. This motion changes only the orientation of the cutting tool relative to the work. Motions of this kind are referred to as orientational motions of the cutting tool. When performing the orientational motion, the CC-point can remain in location on both, on the surface P as well as on the generating surface T of the cutting tool. Orientational motion of this kind is referred to as the orientational motion of the rst kind. When machining a sculptured surface, the CC-point can remain in location on the surface P and change its location on the generating surface T of the cutting tool. Orientational motion of this kind is referred to as the orientational motion of the second kind. Speed of the orientational motions of the cutting tool is a function of variation of the principal curvatures of the surface P at the current CC-point, and of speed of the generating motion. Orientational motions of the cutting tool do not directly affect the stock removal capability of the cutting tool or the generation of the surface P. These motions change orientation of the cutting tool relative to the work as well as the relative direction of the generating motion of the cutting tool. In order to identify all feasible orientational motions of the cutting tool, it is helpful to consider all feasible groups of relative motions of the © 2008 by Taylor & Francis Group, LLC 40 Kinematic Geometry of Surface Machining cutting tool. All groups of the relative motions are represented with the singular relative motions and with the combined relative motions. The singular relative motions are composed of one elementary relative motion of the cutting tool. The combined relative motions are composed of two or more elementary relative motions of the cutting tool. There are only ve groups of elementary relative motions of the cutting tool. The number of elementary relative motions at every group of relative motions is equal to the number of combinations of ve elementary motions by i elementary motions. Here i = 1, 2, …, 5. The total number N of relative motions in the principal instant kinematics of surface generation can be computed from the following equation: N N C i i i i = = = = = ∑ ∑ 1 5 5 1 5 31 (2.18) Analysis of all 31 relative motions reveals that only a few elementary relative motions and their combinations can be distinguished as the orientational motions of the cutting tool. They are as follows [30, 31, 33]: The rst group of the motions: {  n } The second group of the motions: {  u , V v }, {  v , V u } The third group of the motions: {  u ,  n , V v }, {  v ,  n , V u } The fourth group of the motions: {  u , V v ,  v , V u } The fth group of the motions: {  u , V v ,  n ,  v , V u } Ultimately, one can come up with a set of the orientational motions of the cutting tool. One is the singular orientational motion, and six others are the combined orientational motions of the cutting tool. The orientational motions of the rst kind are represented with the only singular orientational motion {  n }. All other orientational motions are the orientational motions of the second kind. The orientational motion {  u , V v ,  n ,  v , V u } is the most general. Other orientational motions can be considered as a particular motion of that one. Elementary motions that include a combined orientational motion of the cutting tool are timed (synchronized) with one another. The rotational elementary motion  u about the x P axis is timed with the translational motion V v along the y P axis. Similarly, the rotational elementary motion  v about the y P axis is timed with the translational motion V u along the x P axis. The timing of the elementary motions results in the surface T sliding over the sculptured surface P. The timing of that kind of elementary motions can be achieved when the following condition is satised. Orientational motion of the second kind can be considered as a superposition of the instant translational motion with a certain instant speed V T−P , and of the instant rotation  T−P of the cutting tool (Figure 2.5). In order to be an orientational © 2008 by Taylor & Francis Group, LLC 42 Kinematic Geometry of Surface Machining Equation (2.21) yields a representation of the combined orientational motion of the cutting tool V orient in the form ω ωω ω ωω ωω ω u v v u u v u V V V , , , ( . { } ⇒ = + = × orient orient T R vv v u n n⋅ + × ⋅      P T P R) ( ) . ω (2.22) On the premises of the performed analysis, the following classication of the orientational motions of the cutting tool is developed (Figure 2.6). When representing a sculptured surface P as an enveloping surface to consecutive positions of the ve-parametric motion of the generating surface T (see Section 2.2), the orientational motions of the cutting tool can be omitted from consideration. Orientational motions of the cutting tool make the machining of a sculptured surface more agile. The orientational motions of the cutting tool could increase the performance of the machining operation. The developed approach yields computation of the optimal parameters of all the motions of the cutting tool relative to the work. The solution to the problem of synthesis of the optimal kinematics of generation of a sculptured surface on a multi-axis NC machine can be drawn up from the analysis of kinematics of multiparametric motion of the cutting tool relative to the work. 2.4 Relative Motions Causing Sliding of a Surface over Itself Surfaces that allow sliding over themselves are convenient for many applications in mechanical and manufacturing engineering. Surfaces of this kind can be generated by corresponding motion of a curve of an appropriate shape. The necessary motion can be easily performed on a machine tool. Relative motions causing sliding of a surface over itself are investigated in [30,32,35] and others. Reshetov and Portman [37] introduced the so-called hidden connections that must be performed on the machine tool of a motion that results in sliding of the surface over itself. For the analytical descrip- tion of the hidden connections, it is necessary to compute the derivatives of r P with respect to the elementary motions Ω i (here i = 1, 2, …, n designates an integer number). The hidden connections exist if and only if pairs of colinear or triples of coplanar vectors are available among the derivatives ∂r P /∂Ω i of r P . Surfaces that allow sliding over themselves can be considered as the surfaces for which a resultant relative motion of a special kind is feasible. Rel- ative motion of this kind results in the enveloping surface to consecutive positions of the moving surface P being congruent to the surface P. The same is true for the generating surface T of the cutting tool. © 2008 by Taylor & Francis Group, LLC [...]... Principal Kinematic Scheme of the 50 Class 41: ω1 = 0 31 2: ω2 = 0 31 3: 3 = 0 31 5: V1–2 = 0 31 4: ω4 = 0 43: 3 = 0 33 1 =31 3 33 2 = 32 3 ω2 44: ω4 = 0 33 4: ω4 = 0 33 5: V1–2 = 0 ω2 V1–2 ω1 V1–2 −ω4 ω1 ω2 ω1 ω1 +ω 4 ω1 −ω 4 +ω 4 3 ω1 ω1 ω2 ω1 ω1 ω2 34 5: V1–2 = 0 ω2 ω2 ω1 ω2 34 1= 31 4 34 2 = 32 4 34 3 = 33 4 ω2 ω2 3 3 ω ω1 V ω V ω1 = 0; ω2 = 0; 3 = 0; ω4 = 0; V1–2 = 0 Figure 2.9 Kinematic schemes of surface. .. ω2 ω2 51 Kinematics of Surface Generation Principal Kinematic Scheme of the 50 Class 42: ω2 = 0 32 1= 31 2 32 4: ω4 = 0 32 3: 3 = 0 32 5: V1–2 = 0 45: V1–2 = 0 34 1= 31 4 34 2 = 32 4 34 3 = 33 4 34 3 = 33 4 3 ω2 ω1 ω1 ω2 ω2 ω1 3 ω1 ω2 ω1 3 ω1 ω2 3 ω2 ω1 ω1 ω2 ω2 ω2 ω1 ω2 Figure 2.9 (Continued) 2.6 On the Possibility of Replacement of Axodes with Pitch Surfaces Pitch surfaces in many cases of surface machining... the kinematic scheme of the cutting tool profiling Kinematic schemes of cutting tool profiling are used for solving the direct principal problem of surface generation The application of it yields the determination of the generating surface T of the cutting tool for the machining of a given surface P In order to solve the inverse principal problem of surface generation, the true kinematic schemes of surface. .. infinite variety of various machining operations Together with the shape and parameters of geometry of the surface P, the kinematic schemes of surface generation specify the generating surface of the cutting tool, as well as the principal part of the kinematic structure of a machine tool Nonagile kinematics of surface generation is usually performed on conventional machine tools Kinematics of this kind... principal problem of surface generation The cutting tool surface T is expressed in terms of shape and parameters of the surface P and of parameters of the kinematic scheme of the machining operation The inverse principal problem of surface generation relates to the determination of the shape and parameters of the actual machined surface P The set of necessary and sufficient conditions of proper surface generation... cutting tool can be reduced to a corresponding kinematic scheme of surface generation only for nonagile kinematics of surface machining The nonagile kinematics of surface machining features the elementary relative motions of the cutting tool with constant parameters of the motions Under such a scenario, for the investigation of all feasible kinematic schemes of surface generation, axodes that are associated... Group, LLC 50 Kinematic Geometry of Surface Machining Reference [31 ] for details on the classification (Figure 2.9) of all feasible kinematic schemes of surface generation The classification also incorporates the kinematic schemes having additional motion that causes sliding of the axode over itself Motions of this kind can increase the capability of the principal kinematic schemes of surface generation... motion of the cutting tool with constant parameters It is used for the generation of surfaces P that allow for sliding over themselves Various feasible versions of © 2008 by Taylor & Francis Group, LLC 46 Kinematic Geometry of Surface Machining nonagile kinematics of surface generation can be considered as a particular (degenerated) case of agile kinematics of surface generation, for example, as the kinematics... as the elementary relative motions A combination of translations and of rotations of the cutting tool is referred to as the kinematic scheme of surface generation Kinematics of surface generation that allows sliding of one or both surfaces P and T over themselves is usually referred to as the rigid kinematics of surface generation Rigid kinematics of surface generation usually features constant relative... Hob./S.P. Radzevich, Int. Cl. B24b 3/ 12, Filed 20.07.1984 [21] Pat. No. 1.196. 232 , USSR, A Method of Grinding of Relieved Surface of a Gear Hob./S.P. Radzevich, Int. Cl. B24b 3/ 00, Filed 03. 08.1984 [22] Pat. No. 1.240.548, USSR, A Method of Grinding of a Gear Hob./S.P. Radzevich, Int. Cl. B24b 3/ 12, Filed 22.09.1984 [ 23] Pat. No. 1 .36 5.550, USSR, A Method of Grinding of Relieved Surface of a Tapered Gear Hob./S.P. Radzevich, . 0 V ω 1 ω 2 ω 1 ω 2 ω 2 ω 1 ω 4 −ω 4 +ω 4 +ω ω 1 2 ω 4 −ω 1 ω 2 ω 1 ω 2 ω 2 ω 1 ω 1 ω 2 ω 3 ω 2 ω 1 ω 3 ω 2 ω 1 ω 2 ω 1 ω 2 ω 1 ω 3 ω ω V 3 31 =3 13 3 32 =3 23 3 41 =3 14 3 42 =3 24 3 43 =3 34 3 34 : ω 4 = 0 3 35 : V 1–2 = 0 V 1–2 V 1–2 3 45 : V 1–2 = 0 ω 1 = 0; ω 2 = 0; ω 3 = 0; ω 4 = 0; V 1–2 = 0 FiguRE 2.9 Kinematic. 0 3 21 =3 12 3 41 =3 14 3 42 =3 24 3 43 =3 34 3 43 =3 34 3 23 : ω 3 = 0 3 24 : ω 4 = 0 3 25 : V 1–2 = 0 4 5 : V 1–2 = 0 FiguRE 2.9 (Continued) © 2008 by Taylor & Francis Group, LLC 52 Kinematic. LLC 52 Kinematic Geometry of Surface Machining The analysis of kinematic schemes of surface generation (Figure 2.9) reveals that the axodes of the surface P and of the surface T can be of simple shape