220 Kinematic Geometry of Surface Machining which the surface C s passes through the computed cutting edge of the form- cutting tool. Third, the conguration of the dened clearance surface C s of the form-cutting tool must be of the sort for which the surface C s makes the optimal clearance angle a with the generating surface T of the form-cutting tool at a given point of the cutting edge. It is easy to see that clearance surfaces of the cutting tool cannot always be shaped in the form that is convenient for manufacturing the cutting tool. The cutting edge of a precision form-cutting tool can be considered as a line of intersection of the three surfaces, say of the generating surface T of the cutting tool, of the rake surface R s , and of the clearance surface C s . This requirement is compliant with three surfaces T, R s , and C s being the sur- faces through the common line, say through the cutting edge of the cutting tool, that could impose strong constraints on the actual shape of the clear- ance surface of the cutting tool. Under such restrictions, the clearance sur- face C s usually cannot allow sliding over itself. However, the desired surface C s can be approximated by a surface that allows sliding over itself; thus, the approximation could be more convenient for design and manufacture of the form-cutting tool. This means that in certain cases of implementation of the rst method, approximation of the desired clearance surface C s with a surface that features another geometry can be unavoidable. The approxi- mation of the desired surface C s results in the surface P being generated not with the precise surface T, but with an approximated surface T g of the cutting tool. The approximated surface T g deviates from the desired surface T. The deviation δ T is measuring along the unit normal vector n T to the surface T at a corresponding surface point. Application of the form-cutting tool having approximated the generated surface is allowed if and only if the resultant deviation δ T is within the corresponding tolerance [ ] δ T — that is, when the inequality δ δ T T ≤ [ ] is valid. Summarizing, one can come up with the following generalized procedure for designing the form-cutting tool in compliance with the rst method: 1. Determination of the generating surface T of the form-cutting tool (see Chapter 5). 2. Determination of the rake surface R s : The rake surface is selected within surfaces that are technologically convenient (a kind of rea- sonably practical surface). Conguration of the rake surface is spec- ied by the rake angle of the desired value. 3. Determination of the cutting edge: The cutting edge is represented with the line of intersection of the generating surface T of the form- cutting tool by the rake surface R s . 4. Construction of the clearance surface C s that passes through the cutting edge and makes the clearance angle of the desired value with the surface of the cut. The clearance surface is selected within surfaces that are technologically convenient (a kind of reasonably practical surface). © 2008 by Taylor & Francis Group, LLC The Geometry of the Active Part of a Cutting Tool 221 For the practicality, a normal cross-section of the clearance surface must be determined as well. Figure 6.1 illustrates an example of implementation of the rst method for the transformation of the generating body of the form-cutting tool into the workable-edge cutting tool. For illustrative purposes, the round form cutter for external turning of the part is chosen. In the case under consideration, the part surface P is represented by two separate portions P 1 and P 2 . An axial prole of the part is specied by the composite line through points a p , b p , and c p . For the particular case shown in Figure 6.1, the generating surface T of the form cutter is congruent with the part surface P being machined. This statement easily follows from the con- sideration that is based on the analysis of kinematics of the machining opera- tion (see Chapter 2). Thus, the identity T ≡ P is observed (to be more exact, two identities T 1 ≡ P 1 and T 2 ≡ P 2 are valid). Ultimately, the generating surface T of the form cutter is also represented with two portions T 1 and T 2 . C s.1 O c O p O p O c h c α c α c c γ b c b p c p C s.2 b c c c a c a p b p c p a c a p T 2 P 2 T 1 P 1 T 1 P 1 c c FIGURE 6.1 The concept of the rst method for the transformation of the generating body of the form- cutting tool into the workable edge cutting tool. © 2008 by Taylor & Francis Group, LLC 222 Kinematic Geometry of Surface Machining Due to the identity T ≡ P observed, the axial prole of the generating sur- face T of the form cutter is composed of two segments through the points a T , b T , and c T (not labeled in Figure 6.1); and the identities a T ≡ a p , b T ≡ b p , and c T ≡ c p are valid. Then, a plane is chosen as the rake surface R s of the round form cutter. Denitely, the plane allows for sliding over itself. It is convenient to machine the plane in cutting tool production. The plane is parallel to the axis of rotation O T ≡ O p of the generating surface T of the form cutter. It makes the rake angle γ c perpendicular to the surface T at the base point a a T c ≡ . The piecewise line of intersection a b c c c c of the generating surface T of the round form cutter by the rake surface R s serves as the cutting edge of the form cutter. The clearance surface C s of the form cutter is shaped in the form of a surface of revolution. All surfaces of revolution allow for sliding over them- selves. The surface of revolution C s is represented with two separate portions C s.1 and C s.2 . The clearance surface of the round form cutter can be generated as a series of consecutive positions of the cutting edge a b c c c c when rotating the cutting edge a b c c c c about the surface C s axis of rotation O c . For practical needs, the axial prole of the clearance surface C s must be determined. The considered example (Figure 6.1) illustrates implementation of the rst method for the transformation of the generating body of the cutting tool into the workable edge cutting tool. This method has been known for many decades. The rst method for the transformation of the generating body of the cut- ting tool into the workable edge cutting tool is widely used in many indus- tries. Form edge cutting tools of most designs can be designed in compliance with the rst method. When the rst method is employed, this yields design- ing of the form-cutting tools that are convenient in manufacturing and in application. The major disadvantages of the rst method are twofold. First, in most cases of application of the rst method, no optimal values of the geo- metrical parameters of the form-cutting tool at every point of the cut- ting edge can be ensured. Optimization of the geometrical parameters at every point of the cutting edge is a challenging problem. The solution to the problem of optimization of the geometrical parameters of the cutting edge (if any) is often far from practical needs: It could be feasible, but it is often not practical. Second, unavoidable deviations of the actual approxi- mated generated surface of the cutting tool from its desired shape often cannot be eliminated when the rst method is employed to the design of the form-cutting tool. 6.1.2 The Second Method for the Transformation of the Generating Body of the Cutting Tool into the Workable Edge Cutting Tool Consider another scenario under which the generating surface of the cutting tool is also determined (see Chapter 5). The cutting tool clearance surface is chosen within the surfaces that are convenient for machining of the surface © 2008 by Taylor & Francis Group, LLC The Geometry of the Active Part of a Cutting Tool 223 when manufacturing the form-cutting tool, for inspection purposes, and so forth. In most cases, the clearance surface C s is within the surfaces that allow for sliding over themselves (see Section 2.4). The clearance surface C s is properly oriented with respect to the generat- ing surface T of the form-cutting tool. It makes an optimal clearance angle with the surface T at a given point. Following the second method for the transformation of the generating body of the cutting tool into the workable-edge cutting tool, cutting edges of the form-cutting tool are dened as the line of intersection of the generating surface T of the cutting tool by the clearance surface C s . Once the cutting edge is constructed, then the rake surface R s can be con- structed in compliance with the following routing. First, the rake surface is selected within the surfaces that allow for sliding over themselves (see Sec- tion 2.4). This requirement is highly desirable but not mandatory. Actually, any surface having reasonable geometry could serve as the rake surface of the form-cutting tool. Second, parameters of the chosen surface R s must be computed in compliance with the requirement under which the surface R s passes through the computed cutting edge of the form-cutting tool. Third, the conguration of the dened rake surface R s of the form-cutting tool must be of the sort for which the surface R s makes the optimal rake angle γ with respect to the perpendicular to the generating surface T of the form-cutting tool at a given point of the cutting edge. It is easy to understand that rake surfaces of a cutting tool cannot always be shaped in the form that is convenient for manufacturing the cutting tool. The cutting edge of a precision form-cutting tool can be considered as a line of intersection of three surfaces: of the generating surface T of the cutting tool, of the rake surface R s , and of the clearance surface C s . The requirement that three surfaces T, R s , and C s be the surfaces through the common line, say through the cutting edge of the cutting tool, could impose strong con- straints on the actual shape of the clearance surface of the cutting tool. Under such restrictions, the rake surface R s usually cannot allow sliding over itself. However, the desired surface R s can be approximated by a surface that allows for sliding over itself; thus, the approximation could be more con- venient for the design and manufacture of the form-cutting tool. This means that in certain cases of implementation of the second method, approximation of the desired rake surface R s with a surface featuring another geometry can be unavoidable. The approximation of the desired surface R s results in that ultimately the surface P is generated not with the precise surface T, but with an approximated surface T g of the cutting tool. The approximated sur- face T g deviates from the desired surface T. The deviation δ T is measuring along the unit normal vector n T to the surface T at a corresponding surface point. Application of the form-cutting tool having approximated the gener- ated surface is allowed if and only if the resultant deviation δ T is within the corresponding tolerance [ ] δ T — that is, when the inequality δ δ T T ≤ [ ] is valid. © 2008 by Taylor & Francis Group, LLC 224 Kinematic Geometry of Surface Machining Summarizing, one can come up with the following generalized procedure for designing of the form cutting tool in compliance with the second method: 1. Determination of the generating surface T of the form-cutting tool (see Chapter 5). 2. Determination of the clearance surface C s : The clearance surface is selected within surfaces that are technologically convenient (a kind of reasonably practical surface). Conguration of the clearance sur- face is specied by the clearance angle of the desired value. 3. Determination of the cutting edge: The cutting edge is represented with the line of intersection of the generating surface T of the form- cutting tool by the clearance surface C s . 4. Construction of the rake surface R s that passes through the cutting edge and makes the rake angle of the desired value perpendicular to the surface of the cut. The rake surface is selected within surfaces that are technologically convenient (a kind of reasonably practical surface). For practicality, a typical cross-section of the clearance surface must be deter- mined as well. Figure 6.2 illustrates an example of implementation of the second method for the transformation of the generating body of the form-cutting tool into the workable edge cutting tool. For illustrative purposes, the form milling cutter for machining helical grooves is chosen. Consider that the generating surface T of the cutting tool is already deter- mined. Geometry of the chosen clearance surface C s is predetermined by O c O c C s e Cutting Edge e Cutting Edge T R s FIGURE 6.2 The concept of the second method for the transformation of the generating body of the form- cutting tool into the workable edge cutting tool. © 2008 by Taylor & Francis Group, LLC The Geometry of the Active Part of a Cutting Tool 225 kinematics of the operation of relieving the milling cutter teeth. The cutting edge is represented as the line of intersection of the generating surface T of the milling cutter by the clearance surface C s . Further, the constructed cutting edge is used for the generation of the rake face R s . For this purpose, either the cutting edge or its projection onto the transverse plane moves along the milling cutter axis of rotation O c . In the case under consideration, the face surface R s is represented as the locus of consecutive positions of the cutting edge in its motion along the axis O c . The rake surface R s is shaped in the form of a general cylinder. The considered example (Figure 6.2) illustrates implementation of the sec- ond method for the transformation of the generating body of the cutting tool into the workable edge cutting tool. This method is not as widely used in industry as is the rst method. The second method for the transformation of the generating body of the cutting tool into the workable edge cutting tool does not have wide imple- mentation in industry. Form edge cutting tools of most designs can be designed in compliance with the second method. When the second method is employed, this yields designing of the form-cutting tools that are conve- nient in manufacturing and in application. The major disadvantages of the second method are twofold: First, in most cases of implementation of the second method, no optimal values of the geo- metrical parameters of the form-cutting tool at every point of the cutting edge can be ensured. Optimization of the geometrical parameters at every point of the cutting edge is a challenging problem. The solution to the prob- lem of optimization of the geometrical parameters of the cutting edge (if any) is often far from practical: It could be feasible, but it is often not practical. Sec- ond, unavoidable deviations of the actual approximated generated surface of the cutting tool from its desired shape often cannot be eliminated when the rst method is employed to design of the form-cutting tool. It is important to stress that both methods for the transformation of the generating body of the cutting tool into the workable edge cutting tool fea- ture a common disadvantage. This disadvantage results in the incapability of designing a form-cutting tool that has optimal value of the angle of incli- nation l. The actual value of the angle l at a current point within the cutting edge is a function of shape, of parameters, and of location of the rake R s or the clearance C s surfaces of the form-cutting tool with respect to the gener- ating surface T of the form-cutting tool. Due to this, the optimal values λ opt of angle of inclination of the cutting tool edge become impractical due to signicant difculties in manufacturing the form-cutting tool. 6.1.3 The Third Method for the Transformation of the Generating Body of the Cutting Tool into the Workable Edge Cutting Tool Ultimately, consider the third scenario under which the generating surface of the cutting tool is also determined (see Chapter 5). However, in this case, neither the rake surface nor the clearance surface of a desired geometry is © 2008 by Taylor & Francis Group, LLC 226 Kinematic Geometry of Surface Machining selected at the beginning, but the cutting edge is selected instead. Such an approach allows for optimization of the angle of inclination at every point of the cutting edge of the form-cutting tool. The method considered below is proposed by Radzevich [14,15,19]. For implementation of the third method, it is necessary to construct a spe- cial family of lines within the generating surface of the form-cutting tool. Lines of this family of lines represent the assumed trajectories of motion of the cutting edge points over the surface of the cut when the work is machin- ing (Figure 6.3). Below this family of lines within the generating surface T of the cutting tool is referred to as the primary family of lines. Analysis of a particular machining operation allows for analytical representation of the family of lines within the surface T. Further, after the primary family of lines is dened, it is necessary to construct a secondary family of lines. The secondary family of lines is also within the generating surface T, and it is isogonal to the primary family of lines. At every point of intersection of the lines of the primary and of the secondary families, the angle between the lines is equal to ( )90°- λ opt . Here, λ opt designates the optimal value of the angle of inclination of the cutting edge. Therefore, the angle of inclination is at its optimal value at every point of the cutting edge. This is due to the primary family of lines within the gen- erating surface T of the tool being isogonal to the secondary family of lines at every point of the cutting edge. Different segments of the cutting edge of a form-cutting tool are at different distances from the axis of the tool rotation. Because of this, they work with dif- ferent cutting speeds. This results in the optimal value of the angle of inclina- tion being different for different portions of the cutting tool edge. Under such e Trajector y e Cutting Edge M λ оpt V Σ c e FIGURE 6.3 The concept of the third method for the transformation of the generating body of the form- cutting tool into the workable edge cutting tool. © 2008 by Taylor & Francis Group, LLC The Geometry of the Active Part of a Cutting Tool 227 a scenario, the actual value of the angle of inclination could be either constant within the cutting edge (and thus equal to its average value), or the desired variation of the angle of inclination can be ensured. In the last case, the prob- lem of design of a form-cutting tool becomes more sophisticated. An appropriate number of lines from the second family of lines can be selected to serve as the cutting edges of the form-cutting tool to be designed. These lines are uniformly distributed and are at a certain distance t from one another. The distance t is equal to the tooth pitch of the form-cutting tool. Rake surface R s is a surface through the cutting edge of the form-cutting tool. The surface R s makes the rake angle g with the perpendicular n c to the surface of cut. Actually the perpendicular to the surface of cut deviates from the perpendicular n T to the generating surface T of the form cutting tool. Fortunately, this deviation is of negligibly small value. For practical needs of design of the form-cutting tool, the perpendicular to the surface of cut n c is not used, but the corresponding perpendicular n T to the generat- ing surface T is used instead. The clearance surface C s is also a surface through the cutting edge of the form-cutting tool. The surface C s makes the clearance angle α with the generating surface T. It is possible to formulate the problem of design of the form-cutting tool in the way following which the rake angle g, as well as the clearance angle a, could be of optimal value at every point of the cutting edge of the form- cutting tool. In order to satisfy this requirement, both the rake surface R s and the clearance surface C s must be of special geometry. This problem could be solved analytically. When deriving equations of the surfaces R s and C s , it is necessary to ensure optimal values γ opt , α opt , and λ opt for the parameters g, a, and l for the new form-cutting tool, as well as for the cutting tool after it is reground. The optimal values γ opt , α opt , and λ opt for the new form-cutting tool and for the reground cutting tool are not necessarily the same. Summarizing, one can come up with the following generalized procedure for designing the form-cutting tool in compliance with the third method: 1. Determination of the generating surface T of the form-cutting tool (See Chapter 5). 2. Determination of the cutting edge: The cutting edge is at the angle of inclination of an optimal value with respect to the direction of speed of the resultant motion of the cutting edge point relative to the surface of the cut. 3. Construction of the rake surface R s , and the clearance surface C s simultaneously: The rake surface passes through the cutting edge and makes the rake angle of the desired value perpendicular to the surface of the cut. The clearance surface also passes through the cutting edge and makes the clearance angle of the desired value with the generating surface of the cutting tool. Both the © 2008 by Taylor & Francis Group, LLC 228 Kinematic Geometry of Surface Machining rake surface and the clearance surface are selected based on their technological convenience and property of sliding over them- selves (a kind of reasonably practical surface). See Section 2.4 for more detail. The third method for the transformation of the generating body of the cut- ting tool into the workable edge cutting tool is a completely novel method [14,15,19]. It has not yet been comprehensively investigated. Therefore, more detailed explanation of the method is important. Consider the generating surface T that is shaped in the form of a sur- face of revolution. This assumption is practical, because, for example, milling cutters of all designs have the generating surface T in the form of a surface of revolution. Using the third method, it is easy to come up with an understanding that the cutting edge of milling cutters of all designs must be shaped in the form of loxodroma. By denition, loxodroma is a line that makes equal angles with a given family of lines on a surface. Actu- ally, loxodroma can be easily dened with respect to coordinate lines on the surface [2]. In the case under consideration, loxodroma having special shape param- eters is of particular interest. The loxodroma that makes the angle ( )90°- λ opt with the primary family of lines on the generating surface T can be employed as the cutting edge of the form-cutting tool. In a particular case, when parameterization of the generating surface T of the form-cutting tool yields the expression φ 1 2 2 2 .T T T T T T dS dU G U dV⇒ = + ( ) (6.1) for the rst fundamental form φ 1.T , then the cutting edge having optimal value of the angle of inclination λ opt at every point can be described by the following equation: V dU G U T opt T T T U U T T cot ( ) . λ = ± ∫ 0 (6.2) Equation (6.2) of the cutting edge is expressed in terms of U T and V T param- eters of the generating surface T of the form-cutting tool. Using conven- tional mathematical methods, Equation (6.2) can be converted to a Cartesian coordinates. Example 6.1 Consider a ball-nose milling cutter of radius r T (Figure 6.4). The milling cutter is used for machining a sculptured surface on a multi-axis numerical control © 2008 by Taylor & Francis Group, LLC 230 Kinematic Geometry of Surface Machining for the unit tangent vector c e ( , ) ϕ θ can be derived from Equation (6.3): c r e T T d dS d d d ( , ) sin cos cos sin ϕ θ ϕ ϕ θ ϕ θ ϕ = = + ⋅ - 1 2 2 2 ϕϕ θ θ ϕ θ ϕ ϕ θ θ ϕ ϕ sin cos sin sin cos sin d d d d + - 1 (6.4) where dS T denotes the differential of the arc segment of the cutting edge. Particularly, when θ θ = = c Const , then Equation (6.4) for the unit tangent vector c e reduces to c e c c c ( , ) cos cos cos sin sin ϕ θ ϕ θ ϕ θ ϕ = - 1 (6.5) Under the imposed constraint θ θ = c , the following equality is valid: d d d ϕ ϕ ϕ θ 2 2 2 + = sin cos℘ (6.6) Equation (6.6) immediately yields d d ϕ ϕ θ sin cot= ± ℘ (6.7) where ℘ designates a certain angle. After integration of Equation (6.7) is accomplished, one can come up with the solution tan ( ) ϕ θ 2 = + e q C (6.8) where q = ±cot℘ and C is an arbitrary constant value. Implementation of the trivial trigonometric formulae sin tan tan , cos tan tan ϕ ϕ ϕ ϕ ϕ ϕ = + = - + 2 2 1 2 1 2 1 2 2 2 2 (6.9) yields an intermediate result sin ( ) , cos ( ) ϕ θ ϕ θ = + = + 1 ch th q C q C (6.10) © 2008 by Taylor & Francis Group, LLC [...]... is located at the point of intersection of the generating surface T of the milling cutter by the axis of rotation of the cutting tool © 20 08 by Taylor & Francis Group, LLC 232 Kinematic Geometry of Surface Machining 3 T2 C T3 A 1 B B T1 A 1 (a) T2 3 T3 C T1 2 (b) Figure 6.5 The filleted-end milling cutter having an optimized value of the angle of inclination λ opt surface T of any feasible shape can... Taylor & Francis Group, LLC The Geometry of the Active Part of a Cutting Tool 235 surface, which is faced to the machined surface, is known as the clearance surface Cs or the flank In the simplest yet common case, both surfaces R s and Cs are planes The cutting edge is represented as the line of intersection of the rake surface R s and of the clearance surface Cs Cutting edges of two kinds can be distinguished:... and the axis of rotation of the filleted-end milling cutter In the case under consideration, representation of the equation of the cutting edge in the following form proved to be useful: ρ = ρAeϕ tan λ opt , (ρA > 0, ϕ A = 0° < ϕ < ϕ B ) where rA is the position vector of the point A of the cutting edge AB © 20 08 by Taylor & Francis Group, LLC (6.17) 234 Kinematic Geometry of Surface Machining A few... the lines of intersection of the rake surface R s, the clearance surface Cs , and the corresponding reference planes by the normal plane Pn © 20 08 by Taylor & Francis Group, LLC 243 The Geometry of the Active Part of a Cutting Tool 6.2.4 Analytical Representation of the Geometric Parameters of the Cutting Edge of a Cutting Tool It is the right point now to introduce equations for computation of actual... ′ yields computation of the tool tip angle e The tool tip angle e can be thought of as the projection of the angle ε ′ onto the main reference plane Pr Conversely, angle ε ′ is the projection of the tool tip angle e onto the rake face © 20 08 by Taylor & Francis Group, LLC 247 The Geometry of the Active Part of a Cutting Tool For computation of the geometry of the active part of a cutting tool, the... , roundness r of the cutting edge can be expressed in terms of r n by Meusnier’s equation: ρ = ρn cos λ s (6.62) Torsion of the cutting edge is one more geometric parameter of the active part of a cutting tool to be considered The shape of the rake surface R s and of the clearance surface Cs affect the material removal process in metal cutting This is because the geometry of the © 20 08 by Taylor &... value of the angle of inclination The last statement encompasses composite generating surfaces T of the form-cutting tools as well As an example, consider the optimization of the angle of inclination of a filleted-end milling cutter (Figure 6.5) The generating surface of the filletedend milling cutter is composed of three portions: the cylindrical portion T1 , the flat-end T2 , and the torus surface. .. cutting tool of a given design The following geometric parameters α n , β n, γ n, λ s, ρn, and ε ′ are among those that can be measured directly. Other geometric parameters of the active part of a cutting tool can be computed For derivation of equations for the computation of geometry of the active part of cutting tools, implementation of elements of vector calculus is helpful To the best of the author’s... generating surface of the milling cutter can be derived on the premises of Equation (6.2) This segment of the cutting edge is represented by the arc segment 3 of the loxodroma The loxodroma is within the torus surface T3 The generalized Equation (6.2) of the cutting edge having optimal value of the angle of inclination is valid for edge-cutting tools of any possible design However, in particular cases of the... Geometry of Surface Machining γn Rs βn Ps ρn Cs αn Figure 6.13 Roundness of the cutting edge in the normal plane section Pn The cutting edge of a cutting tool is not absolutely sharp Actually, there exists a transition surface that connects the rake surface R s and the flank Cs This transition surface is supposed to have a circular profile of a certain radius that is considered as radius ρn of the cutting . of intersection of the generating surface T of the milling cutter by the axis of rotation of the cutting tool. © 20 08 by Taylor & Francis Group, LLC 232 Kinematic Geometry of Surface Machining surface. clearance angle of the desired value with the generating surface of the cutting tool. Both the © 20 08 by Taylor & Francis Group, LLC 2 28 Kinematic Geometry of Surface Machining rake surface and. this case, neither the rake surface nor the clearance surface of a desired geometry is © 20 08 by Taylor & Francis Group, LLC 226 Kinematic Geometry of Surface Machining selected at the beginning,