438 Kinematic Geometry of Surface Machining the surface P. However, when the chip-removal output is included in the con- sideration, the maximal speed of the cutting tool travel is smaller when the concave portion of the surface P is machining, and it is bigger when machin- ing a convex portion of the surface P. For the orthogonally parameterized surface P, Equation (10.12) for [ ]r tp opt reduces to r tp P P P V V U V dV P P opt opt ⇒ = − ∫ cot[ ( )] . . ϕ 1 2 (10.13) In many particular cases of sculptured surface machining, both Equa- tion (10.12) and Equation (10.13) can be integrated analytically. In some particular cases of sculptured surface generation, the equation for the optimal tool-paths simplies to the differential equation r tp P P P P P P P P P P P E dU F dV F dU G dV L dU M dV opt ⇒ + + + PP P P P P M dU N dV+ = 0 (10.14) Equation (10.14) for the optimal tool-paths is applicable, for instance, when machining a sculptured surface P either with a ball-nose milling cutter, or with a at-end milling cutter, and so forth. Under such a scenario, the angle m of the local relative orientation of surfaces P and T vanishes. It is getting indenite: no principal directions can be identied on a sphere or on the plane surface. Therefore, the optimal tool-paths align with lines of curvature on the surface P (see Equation 10.14). When machining a part surface, the coordinate system X Y Z T T T associated with the cutting tool is rotating like a rigid body. This rotation is performing about a certain instant axis of rotation. The angular velocity of the rotation of the coordinate system X Y Z T T T is equal to | | .W = +k tp tp 2 2 τ The axis of instant rotation aligns with Darboux’s vector W = +k tp tp tp tp t b τ (here k tp and τ tp denote curvature and torsion of the trajectory of the CC- point, and t tp and b tp are the unit tangent vector and the binormal vector to the trajectory of the CC-point at a current point K). Darboux’s vector is located in the rectifying plane to the trajectory of the CC-point. It can be expressed in terms of the normal vector n tp and of the tangent vector t tp to the trajectory of the CC-point: W = + +k tp tp tp tp 2 2 τ θ θ ( cos sin )t n (10.15) where q is the angle that makes Darboux’s vector W and the tangent vector t tp to the trajectory at the CC-point. © 2008 by Taylor & Francis Group, LLC Synthesis of Optimal Surface Machining Operations 439 It is instructive to note that velocity |W| is a function of full curvature of the trajectory of the CC-point. 10.3 Synthesis of Optimal Surface Generation: The Global Analysis Synthesis of optimal global surface generation is the nal subproblem of the general problem of synthesis of optimal surface generation. The solution to the problem of optimal global surface generation is based much on the derived solutions to the problems of optimal local and of optimal regional surface generation. Minimal machining time is the major goal of the problem of synthesis of optimal global surface generation. In order to solve the problem under con- sideration, it is necessary to do the following: Minimize interference of the neighboring tool-paths of the cutting tool over the part surface being machined. Determine the optimal parameters of placing the cutting tool into contact with the part surface, and of its departing from the contact. This subproblem is referred to as the boundary problem of surface generation. Determine the location of the optimal starting point of the surface machining. 10.3.1 Minimization of Partial Interference of the Neighboring Tool-Paths The actual machined part surface is represented as a set of tool-paths that cover the nominal surface P. At a current surface point, the width of the tool-path is equal to the side-step ( F ss computed at that same CC-point (see Equation 9.46). The tool-path width varies along the trajectory of the CC-point over the sculptured surface, as well as across the trajectory. Because of this, neighboring tool-paths partially interfere with each other. Ultimately, some portions of the part surface P are double-covered by the tool-paths. Partial interference of the neighboring tool-paths causes reduction of the surface generation output. For the synthesis of optimal surface generation operation, the interference of the neighboring tool-paths must be minimized. The trajectory of the CC-point over the sculptured surface is a three-dimen- sional curve. For the analysis below, it is convenient to operate with the natu- ral parameterization of the trajectory: l l r tr tr tr tr = ( , ) τ . Here, length l tr of the arc of the trajectory is measured from a certain point within the trajectory. The length l tr is expressed in terms of the radius of curvature r tr at a current trajectory point, and of torsion τ tr of the trajectory at that same point. © 2008 by Taylor & Francis Group, LLC 440 Kinematic Geometry of Surface Machining At the current CC-point, the tool-path width can be expressed in terms of the length of the ith trajectory: ( ( F F l ss i ss i tr i ( ) ( ) ( ) = (10.16) During the innitesimal time dt, the cutting tool travels along the i-th tra- jectory at a distance dl th . A sculptured surface portion d F l dt tr i ss i tr i S ( ) ( ) ( ) = ⋅ ( (10.17) is generated in this motion of the cutting tool. The area of a single i-th tool-path is S tr i ss i tr i l F l dt tr i ( ) ( ) ( ) [ ] ( ) = ⋅ ∫ ( (10.18) The total area of all the tool-paths can be computed from S S tr tr i i n ss i tr i l F l dt tr i = = ⋅ = ∑ ( ) ( ) ( ) [ ( ) 1 ( ]] ∫ ∑ =i n 1 (10.19) where n denotes the total number of tool-paths necessary to cover the entire part surface. Due to partial interference of the neighboring tool-paths, the total area S tr (Equation 10.19) exceeds the area S sg of the actual generated part surface P (i.e., the inequality S S tr sg > is always observed). The rate of interference of the neighboring tool-paths is evaluated by the coefcient of interference K int : K F l dt tr sg sg ss i tr i l tr i int ( ) ( ) [ ( ) = − = ⋅ S S S ( ]] ∫ ∑ = − i n sg sg 1 S S (10.20) The coefcient of interference K int is a function of design parameters of the part surface being machined, of design parameters of the generating surface of the cutting tool, and of parameters of kinematics of the surface machin- ing operation. So, it can be minimized ( min) int K → using for this purpose conventional methods of minimization of analytical functions. 10.3.2 Solution to the Boundary Problem The generation of a part surface within the area next to the surface border differs from that when machining of a boundless surface is investigated. The shape and parameters of the surface contour affect the efciency of the sur- face generation process. Prior to searching for a solution to the boundary problem, it is necessary to determine the part surface region within which the boundary effect is signicant. © 2008 by Taylor & Francis Group, LLC Synthesis of Optimal Surface Machining Operations 441 Consider a sculpture surface P having tolerance [ ]h on the accuracy of the surface machining. The surface of tolerance S h[ ] is at the distance [ ]h from the surface P. The actual machined part surface is located within the interior between the surfaces P and S h[ ] . The generating surface T of the cutting tool is contacting the nominal part surface P at a certain point K 1 (Figure 10.4). The surface T intersects the sur- face of tolerance S h[ ] . The line 1 of intersection of the surfaces S h[ ] and T is a kind of closed ellipse-like curve. The curve has no common points with the part surface boundary. Therefore, no boundary effect is observed in this location of the cutting tool. At the point K 2 within the trajectory of the CC-point, the cutting tool sur- face T also intersects the surface of tolerance S h[ ] . The line 2 of intersection is also a kind of closed, ellipse-like curve. However, in this location of the cutting tool, the curve 2 makes tangency with the part surface boundary at the point A. This indicates that starting at the point K 2 , the boundary affects the efciency of surface generation. The impact of the boundary is getting stronger toward point D on the part surface boundary curve. At a certain point K 3 of the trajectory of CC-point, the line of intersection 3 of the surfaces S h[ ] and T is not a closed line. It intersects the part surface boundary at the points B and C. Departure of the cutting tool from the interaction with the surface P is over when the limit point L on the biggest diameter of the curve 3 reaches the part surface boundary curve at the point L * . The point K 2 is constructed for the point A of the part surface boundary curve. For every point A i of the part surface boundary curve, a point K i that is similar to the point K 2 can be constructed. All the points K i specify the limit contour. It is necessary to take into account the impact of the bound- ary effect for those arcs of CC-point trajectories, which are located between the part surface boundary curve and the limit contour. L * Part boundary c K 2 K 3 K 1 W i C Q B A P 3 2 1 Limiting contour Trajectory of the CC-Point F fr D b a FIGURE 10.4 Boundary effect when machining a sculptured surface P. © 2008 by Taylor & Francis Group, LLC 442 Kinematic Geometry of Surface Machining Width bc of the part surface boundary affected region is not constant. Width W i at a current point c is measured along the perpendicular to the part surface boundary curve. The point c is the endpoint of the arc ac of the trajectory of the CC-point. The feed rate per tooth ( F fr of the cutting tool could be either constant within the arc ac of the trajectory of the CC-point, or it can vary in compliance with the current width of the tool-path. Particular features of impact of the boundary effect could be observed: When the stock thickness is bigger, this causes longer trajectories of the cutting tool to enter in contact with the part surface. A bigger tolerance [ ]h on accuracy of the machined part surface results in longer trajectories of the cutting tool to exit from contact with the part surface. The smaller the area of the nominal part surface P, the more signi- cant is the impact of the boundary effect on the efciency of the machining of the whole part surface. The impact of the boundary effect could be more signicant when machining long surfaces. 10.3.3 Optimal Location of the Starting Point The location of the point from which machining of the sculptured surface begins also affects the resultant surface generation output. One can conclude from this that the optimal location of the starting point exists, and it can be determined. Consider the machining of a sculptured part surface on a multi-axis NC machine (Figure 10.5). The boundary of the sculptured surface P is of arbi- trary shape. The region of the boundary effect is shown as the shadowed strip along the boundary curve. The surface P can be covered by the innite number of optimal trajectories of the CC-point. The equation of the optimal trajectories of the CC-point is the output of the subproblem of synthesis of optimal regional surface generation. Two of the innite number of trajectories are tangent to the surface boundary curve of the sculptures at the points c 1 and c 2 (see Figure 10.5). Another two optimal trajectories of the CC-point are at the distance 0 5. F ss from the points c 1 and c 2 inward from the bounded portion of the sculp- tured surface P. These two last trajectories of the CC-point can be used as the trajectories for the actual tool-paths when machining the sculptured surface P. They intersect the sculptured surface boundary curve at the points a 1 , a 2 and a 3 , a 4 , respectively. The rest of the trajectories of CC-point are at the limit side-step [ ] ( F ss from each other (see Equation 9.46). It is important to point out here that length f ss of the arc through the points c 1 and c 2 usually is not divisible on the limit side-step [ ] ( F ss . However, no big problem arises in this concern, and it can be neglected at this point. © 2008 by Taylor & Francis Group, LLC Synthesis of Optimal Surface Machining Operations 443 Then, outside the bounded portion of the sculptured surface P, two points A 1 and A 2 are selected within the trajectory through the points a 1 and a 2 . The point A 1 is at the distance l en from the boundary curve of the surface P. The l en distance is sufcient for entering the cutting tool in contact with the part surface. Another point A 2 is at the distance l ex from the boundary curve of the surface P. The l ex distance is sufcient for exiting the cutting tool from contact with the part surface. Similarly, two more points A 3 and A 4 are selected within the trajectory through the points a 3 and a 4 . The machining of the surface P begins at the point A 1 . In most cases of sculptured surface machining, the inequality l l en ex > is observed. Therefore, if one wishes to begin the surface machining not from the point A 1 , but from the opposite end of the trajectory a a 1 2 , another four points A 1 * , A 2 * , A 3 * , and A 4 * (the points A 1 * , A 2 * , A 3 * , and A 4 * are not shown in Figure 10.5) can be constructed instead. The points A 1 * , A 2 * , A 3 * , and A 4 * are constructed in the way the points A 1 , A 2 , A 3 , and A 4 are constructed. The only difference here is that for all the points A 1 * , A 2 * , A 3 * , and A 4 * , the arc seg- ments of the length l en are substituted with the arc segments of the length l ex , and vice versa. When locating at the point A 1 of the trajectory a a 1 2 , the generating surface T is contacting the workpiece surface W ps at the point b 1 . The workpiece surface W ps is an offset surface at the distance t to the part surface P. Here t designates the thickness of the stock to be removed. In the general case, a function t t U V P P = ( , ) is observed (see Chapter 9). a 4 c 1 b 2 b 1 A 4 A 2 S [h] S [h] W ps W ps P n P n R P R T R P R T A 3 0, 5F ss t t P Region of the boundary effect a 3 a 1 a 2 F ss F ss f ss F ss A 1 A 2 A 1 [h] [h] b 2 P A 2 A 1 l ex l en 0, 5F ss b 1 a 1 F ss c 1 a 2 (b) (c) (a) (d) 0, 5F ss T FIGURE 10.5 Location of the optimal starting point for sculptured surface machining. © 2008 by Taylor & Francis Group, LLC 444 Kinematic Geometry of Surface Machining When locating at the point A 2 of the trajectory a a 1 2 , the generating surface T is contacting the surface of tolerance S h[ ] at a point b 2 . The surface of toler- ance S h[ ] is an offset surface at the distance [ ]h to the part surface P. Here [ ]h denotes the tolerance on accuracy of the machined part surface P. In the general case, a function [ ] [ ]( , )h h U V P P = is observed (see Chapter 9). The distance l en that is necessary for entering the cutting tool in contact with the sculptured part surface can be expressed in terms of thickness of the stock t, radius of normal curvature R T of the generating surface T of the cutting tool (here R T is measured in the direction tangent to the trajectory a a 1 2 at the point A 1 ), and radius of curvature R tr of the trajectory at the point A 1 through the points a 1 and a 2 . The distance l ex that is necessary for exiting the cutting tool from contact with the sculptured part surface can be expressed in terms of tolerance [ ]h on accuracy of the part surface, radius of normal curvature R T of the gen- erating surface T of the cutting tool (here R T is measured in the direction tangent to the trajectory a a 1 2 at the point A 2 ), and radius of curvature R tr of the trajectory at the point A 2 through the points a 1 and a 2 . Ultimately, either one of four points A 1 , A 2 , A 3 , A 4 or one of four points A 1 * , A 2 * , A 3 * , A 4 * is selected as the starting point of the sculptured surface machining. Practically, both sets of points are equivalent. Computation of coordinates of the chosen point is a trivial mathematical procedure. The interested reader may wish to exercise him- or herself in doing this. Prior to beginning the machining of the given part surface, a contact point within the generating surface T of the cutting tool is computed. This is the point local geometry of the surface T which corresponds to the local geometry of the surface P at the point a 1 . Then, the cutting tool contact point is snapped with the computed starting point, say with the point A 1 . Satisfaction of the conditions of proper part surface generation (see Chapter 7) is required. Much room for investigation is left in the synthesis of optimal global part surface generation. 10.4 Rational Reparameterization of the Part Surface The solution to the problem of optimal regional synthesis of part surface generation returns a set of optimal trajectories of the CC-point on the sur- face P. For the purposes of development of a computer program for sculp- tured surface machining on a multi-axis NC machine, it is convenient to use the computed optimal trajectories as a set of curvilinear coordinates on the sculptured surface P. For this purpose, it is necessary to change the initial parameterization of the surface P with a new parameterization — with the parameterization by means of the optimal trajectories of the CC-point on the surface P. For the reparameterization of the surface P, known methods [1,7,11,13] and others can be used. © 2008 by Taylor & Francis Group, LLC Synthesis of Optimal Surface Machining Operations 445 10.4.1 Transformation of Parameters Consider a part surface P that is given by vector equation r r P P P P U V= ( , ) . It is assumed that the surface P is a smooth, regular surface. The required addi- tional restrictions that must be imposed will be introduced later. The initial (U P ,V P )–parameterization of the part surface can be trans- formed to another parameterization. The new parameterization of the sur- face P is denoted as ( , ) * * U V P P − parameterization. In the new parameters, the initial equation of the surface P is substituted with the equivalent equation r r P P P P U V= ( , ) * * . The new parameters U P * and V P * can be expressed in terms of original parameters U P and V P : U U U V V V U V P P P P P P P P * * * * ( , ) ( , )= = (10.21) One of the curvilinear parameters in Equation (10.21) (for example, U P * − coordinate curve) can be congruent to the optimal trajectories of the CC- point (see Equation 10.12), while another curvilinear parameter V P * can be directed orthogonally to the rst one. Equations for the derivatives in the new parameters are as follows: ∂ ∂ = ∂ ∂ ⋅ ∂ ∂ + ∂ ∂ ⋅ ∂ ∂ r r r P P P P P P P P P P U U U U V V U * * * (10.22) ∂ ∂ = ∂ ∂ ⋅ ∂ ∂ + ∂ ∂ ⋅ ∂ ∂ r r r P P P P P P P P P P V U U V V V V * * * (10.23) Then, the cross-product of tangents is equal: ∂ ∂ × ∂ ∂ = ⋅ ∂ ∂ × ∂r r r r P P P P P P P P P P P U V U V U V U * * * * ∂∂ V P (10.24) In order to satisfy the restriction ∂ ∂ × ∂ ∂ ≠ r r P P P P U V * * 0 (10.25) for the part surface P expressed in the new parameters, the Jacobian matrix of transformation J must not be equal to zero: J U V U V U U U V V U V P P P P P P P P P P P = = ∂ ∂ ∂ ∂ ∂ ∂ ∂ * * * * * ∂∂ ≠ V P * 0 (10.26) © 2008 by Taylor & Francis Group, LLC 446 Kinematic Geometry of Surface Machining The matrix [ ]D P of the rst derivatives of the surface P in its original parameterization is [ ] ;D U V P P P P P = ∂ ∂ ∂ ∂ r r (10.27) The similar matrix [ ] * D P can be composed for the new parameterization of the surface P: [ ] ; * * * D U V P P P P P = ∂ ∂ ∂ ∂ r r (10.28) The following equality [ ] [ ] * D D J P P = ⋅ (10.29) is true. The matrices [ ]D P and [ ] * D P enable computation of the rst fundamental matrix [ ] . * Φ 1 P in the new parameters of the surface P: [ ] [ ] [ ] [ ] [ ] [ . * * * Φ Φ 1 1P P T P T P T P T D D J D D J J= ⋅ = ⋅ ⋅ ⋅ = ⋅ ] P J⋅ (10.30) Similarly, the equation for the second fundamental matrix [ ] . * Φ 2 P in the new parameters of the surface P: [ ] [ ] . * . Φ Φ 2 2P T P J J= ⋅ ⋅ (10.31) can be derived. The discriminant of the rst order H P * is computed from H J H P P * = ⋅ (10.32) The similar is true with respect to the discriminant of the second order T P * : T J T P P * = ⋅ The rest of the major parameters of geometry in the new parameterization of the part surface P can be computed on the premises of the above-discussed equa- tions, particularly on the premises of Equation (10.30) and Equation (10.31). 10.4.2 Transformation of Parameters in Connection with the Surface Boundary Contour Boundary contour C of the sculptured surface P is made up of four smooth arcs C 11 , C 12 , C 21 , and C 22 as an example (Figure 10.6). The plane P 0 serves as © 2008 by Taylor & Francis Group, LLC 448 Kinematic Geometry of Surface Machining are the covariant and contravariant components of the vector of the ctive displacements. The components F k and F k in Equation (10.34) must be con- structed based on the requirements of one-to-one correspondence between the contours. After the necessary formulae transformations are accomplished, then the equation r r r 0 1 2 1 2 1 2 ( , ) ( , ) ( , ) α α α α α α = + f i f i F (10.35) can be obtained. Here, r f i ( ) α is the position vector of a point M f that is mapped into the point M 0 having position vector r 0 ; r f i are the reciprocal basis vectors at the point M f ; and F 1 and F 2 are the components of the vec- tor of the ctive displacements M f , those that can be constructed depend upon the shape of the region W f . At every point of the region W 0 , the constructed functions F i together with Equation (10.33) yield computation of the following: The position vector r i : r r r i i k i fk k f ik ik f f k e a e= + ( ) = + ( ) δ (10.36) The major basis vectors r i 0 at the point M 0 : r r r i i k i fk k f ik f ik f f k e a e 0 = + ( ) = + ( ) δ (10.37) These vectors are tangent to the coordinate curves specied by the mapping (see Equation 10.35). In Figure 10.6 they are designated by lines δ i const= . The covariant components of the rst metric tensor: Φ 1 0 2 .P ik ik f ik f a a⇒ = + ε (10.38) Christoffel’s symbols of the second kind Γ Γ ij k ij fk ij fk A 0 = + (10.39) at the point M 0 . In Equation (10.37), the parameters e i fk and e ik f can be computed by formu- lae e F i fk i f k = ∇ and e F if f i f k = ∇ . For the computation of the parameter 2 ε ik f , the formula 2 ε ik f i k i f k f ik f ki f f js ij f ks f e e a e e= ⋅ − ⋅ = + +r r r r is used. Ultimately, the param- eter A ik j0 is computed from A a P ik j jn n ik f0 0 = , (10.40) © 2008 by Taylor & Francis Group, LLC [...]... technology of surfaces reinforcement by plastic subsurface deformation of the work A method of sculptured surface reinforcement is a good example in this regard [14] 11.2 Machining of Surfaces of Revolution Use of the DG/K-based method of surface generation is fruitful for the development of novel advanced methods of machining, not only of sculptured part surfaces shown above, but also of novel advanced... part surface with a constant peripheral feed rate The use of this method of surface machining ensures perfect results when the radius of curvature of the axial profile of the form part surface is of constant value or when variation of the radius of curvature is reasonably small and, thus, could be neglected For machining of form surfaces of revolution having significant variation of curvature of the... user Analysis of Equation (11.10) reveals that the impact of variation of radius of curvature R T onto the surface- generation output can be significant Therefore, the efficiency of turning of form surfaces of revolution can be increased if the proper control of the radius R T is introduced Kinematics of surface machining in the method of turning of form surfaces of revolution [6] is capable of properly... axial profile, a method of turning of surfaces of revolution is proposed [16] In this method of surface machining,** the current value of the peripheral feed rate is synchronized with the radius of curvature of the axial profile of the part surface being machined When machining the surface P, the work is rotating about its axis of rotation OP with a certain rotation ω P (Figure 11.8) The radii of curvature... methods of machining part surfaces of simpler geometry Examples can be found in the field of machining of surfaces of revolution, cylindrical surfaces, gears, and in many other fields 11.2.1 Turning Operations In compliance with the conventional method of turning of a form surface of revolution, the work is rotating about its axis of rotation The cutter travels with a certain feed rate along the axis of. .. ±Swl of the cutting tool: the orientational motion of the first kind of the cutting tool (SU Pat. No 1185749) © 2008 by Taylor & Francis Group, LLC 464 Kinematic Geometry of Surface Machining minimal The direction of the maximal rate of conformity of the surface T to the surface P is specified by the unit tangent vector t (max) cnf Similarly, the direction of the minimal rate of conformity of the surface. .. Chapter 7) 11.1 Machining of Sculptured Surfaces on a Multi-Axis Numerical Control (NC) Machine Numerous methods of sculptured surface machining on a multi-axis NC machine are developed by now A review of known methods of sculptured surface machining is available from the literature [18,23] Below, a method of sculptured surface machining that is developed on the premises of the DG/K-based approach of surface. .. computation of optimal parameters of kinematics of sculptured surface machining Ultimately, this yields a closedform solution (IV) to the problem of optimal tool-path generation, computation of coordinates of the optimal starting point for surface machining, and verification of satisfaction or violation of the necessary conditions of proper part surface generation (V) The cutting tool for machining the... shape of the axial contour of the surface being machined This is an example of a trivial turning operation that is often used for machining form surfaces of revolution Targeting an increase of the part surface generation output, a method of turning form surfaces of revolution is developed [2] In this method, the work is rotating about its axis of rotation The cutter travels along the axial profile of. .. Chapter 7 For machining the sculptured surface P, a form milling cutter is used Parameters of geometry of the generating surface T of the cutting tool are computed on the basis of the method of design of a form-cutting tool for sculptured surface machining on a multi-axis NC machine (SU Pat. No 4242296/08) This method of form-cutting tool design [5] widely employs the R-mapping of the sculptured surface . coefcient of interference K int is a function of design parameters of the part surface being machined, of design parameters of the generating surface of the cutting tool, and of parameters of kinematics. LLC 452 Kinematic Geometry of Surface Machining The initially given representation of the sculptured surface P is convert- ing to the natural parameterization of the surface P, when the surface. used. Param- eters of geometry of the generating surface T of the cutting tool are computed on the basis of the method of design of a form-cutting tool for sculptured surface machining on a multi-axis