Kinematic Geometry of Surface Machinin Episode Episode 12 pps

367 8 Accuracy of Surface Generation Accuracy of the machined part surfaces is a critical issue for many reasons. Deviations of the actual part surface from the desired part surface are investigated in this chapter from the prospective of capabilities of the theory of surface generation. Two major reasons often cause surface deviation: When machining a part surface, the entire generating surface of the cutting tool does not actually exist. In all cases of implementation of wedge cutting tools, the generating surface of the cutting tool is not represented entirely but by a limited number of cutting edges. In other words, the generating surface of the cutting tool is represented discretely. The discrete representation of the surface T of the cutting tool causes deviations of the actual machined part surface P ac from the desired (say, from the nominal) part surface P nom . Point contact of the part surface and of the generating surface of the cutting tool is usually observed when machining a sculptured surface on a multi-axis numerical control (NC) machine. When the surfaces make point contact, then articulation capabilities of the multi-axis NC machine can be utilized in full. From this prospective, point contact of the surfaces can be considered as the most general kind of surface contact. However, point contact of the surfaces P and T also causes deviations of the actual machined part surface P ac from the desired part surface P nom . Ultimately, when the generating surface T of a cutting tool is represented discretely, and the surfaces P and T make point contact, then the deviations of the actual machined part surface P ac from the desired part surface P nom are getting bigger. Sources for the deviations of the machined part surface from the desired part surface are limited to two major reasons only in a simplied case of surface machining. In the simplied cases of surface machining, no deviations in the surfaces P and T conguration are observed. Deviations in the conguration of surfaces P and T are unavoidable. Therefore, the impact of deviations of the conguration of surfaces P and T onto the resultant deviation of the surface P ac from the surface P nom must be investigated as well. © 2008 by Taylor & Francis Group, LLC 368 Kinematic Geometry of Surface Machining 8.1 Two Principal Kinds of Deviations of the Machined Surface from the Nominal Part Surface The discrete representation of the generating surface of the cutting tool as well as point contact of the surfaces P and T result in that during a certain limited period of time, it is impossible to generate the part surface precisely, without deviations of the actual machined part surface from the desired part surface. 8.1.1 Principal Deviations of the First Kind For proper generation of a part surface, the entire generating surface T must be represented by the cutting tool. Actually, the surface T of a cutting tool is represented as a certain number of cutting edges. The number of cutting edges of the cutting tools of conventional design is limited, and the total number could be easily counted. The generated surface T of a cutting tool of this type is discontinuous. The number of cutting edges of grinding wheels and of other abrasive tools is also limited. However, it is not that easy to count all the cutting edges of a grinding wheel as can be done with respect to wedge cutting tools. There- fore, in most cases of surface machining, the generating surface of abrasive cutting tools can be considered as a continuous surface T. When machining a part surface, for example, with a milling cutter (Figure 8.1), the cutting tool is rotating about its axis O T with a certain angular velocity ω T . In addition, the milling cutter is traveling across the surface P with a certain feed-rate F fr . The generating surface T of the milling cutter is contacting the nominal part surface P at a point K. The actual machined part surface P ac is formed as consecutive positions of trajectories of the cutting edge. Usu- ally, the trajectories can be represented by prolate cycloids. In particular cases, the trajectories are represented by pure cycloids and even by curtate cycloids. In any case, the actual surface P ac becomes wavy. The length of the waves is equal to the feed rate per tooth F fr of the milling cutter, and the wave height (cusp) is specied by h fr . The elementary surface deviation h fr (the surface waviness) is measured along the unit normal vector n P to the nominal part surface P nom and is equal to the distance between the surfaces P ac and P nom . If the part surface to be machined and the generating surface of the cutting tool are in line contact, then the cusp height h fr is the only source of the resultant deviation h Σ of the surface P ac from the surface P nom . Figure 8.1 reveals that the cusp height h fr strongly depends upon the feed rate per tooth ( F fr of the milling cutter. For milling cutters of most conventional designs, those that work under high rotation ω T , the cusp height h fr is negligibly small. However, this does not mean that the elementary deviation h fr may always be eliminated from the analysis of the surface P accuracy. The elementary deviation h fr donates more or less to the resultant deviation h Σ of the actual part surface P ac from the nominal part surface P nom . © 2008 by Taylor & Francis Group, LLC 370 Kinematic Geometry of Surface Machining 8.1.3 The Resultant Deviation of the Machined Part Surface The resultant deviation h Σ of the actual part surface P ac from the nominal surface P nom is measured along the unit normal vector n P to the nominal part surface P nom and is equal to the distance between the surfaces P ac and P nom . The value of the resultant deviation h Σ depends upon the elementary deviations h fr and h ss . Consider a portion of the actual part surface P ac depicted in Figure 8.3. This portion of the surface is bounded by two neighboring arc segments m and ( )m + 1 , and by two arc segments n and ( )n + 1 . The distance between the arc segments m and ( )m + 1 is equal to the feed rate per tooth ( F fr of the cutting tool, while the distance between the arc segments n and ( )n + 1 is equal to the side- step ( F ss . The surface P portion that is bounded by the arc segments m, ( )m + 1 and n, ( )n + 1 is referred to as the elementary surface cell of the part surface P. The major parameters h fr , ( F fr , h ss , and ( F ss of the elementary surface cell are not constant within the part surface P. They vary in certain intervals within the sculptured surface. Current values of the major parameters of the elementary surface cell depend on (a) the principal radii or curvature P P1. , P P2. of the surface P; (b) the principal radii or curvature P T1. , P T2. of the surface T; (c) the angle µ of the local relative orientation of surfaces P and T; and (d) on F S S h SS K  ω T O T F SS FIGURE 8.2 Deviation of the machined part surface P ac from the desired part surface P nom that is caused by point kind of contact of the surfaces P and T. © 2008 by Taylor & Francis Group, LLC 372 Kinematic Geometry of Surface Machining The maximal resultant deviation h Σ max of the surface P ac from the surface P nom is often used for the quantitative evaluation of accuracy of the machined part surface. It is widely recognized that in sculptured surface machining on a multi- axis NC machine, the principle of superposition of the elementary deviations h fr and h ss is valid. Implementation of the principle of superposition to the elementary deviations h fr and h ss is questionable. This issue requires further investigation. If the principle of superposition of the elementary deviations h fr and h ss is assumed to be valid, then for the computation of the resultant deviation h Σ , the following equation can be used: h a h b h h fr h ssΣ = ⋅ + ⋅ (8.4) where a h and b h designate certain constants for a given point K. The constants a h and b h are within the intervals 0 1≤ ≤a h and 0 1≤ ≤b h . The resultant deviation of surface generation h Σ is getting its maximal value of h Σ max when the equality a b h h = = 1 is observed. In this particular case, the deviation h Σ max can be computed from h h h fr ssΣ max max max = + (8.5) Generally, the function h h h h fr ssΣ Σ = ( , ) is complex. In compliance with the sixth necessary condition of proper part surface generation (PSG) [3], the deviation h Σ must be within the tolerance of the surface accuracy of surface machining (see Section 7.2.6). The maximal value of the resultant deviation h Σ is limited by the tolerance [ ]h on accuracy of the surface machining. It is recommended that an operation of a sculptured surface machining be designed in a way so that the maximal deviation h Σ max of the surface P ac from the surface P nom is equal to the tolerance [ ]h . A signif- icant reduction in machining time can be achieved if the equality h h Σ max [ ]= is satised within the entire part surface P being machined. Both the elementary deviations h fr and h ss and the resultant deviation h Σ can be computed on the premises of methods developed in the theory of surface generation. 8.2 Local Approximation of the Contacting Surfaces P and T The major surface deviations h fr , h ss , and h Σ can be interpreted in terms of geometry of the nominal part surface P and of the surface P ac of the elementary surface cell. In order to solve the problem, an analytical local representation of the surfaces is helpful. © 2008 by Taylor & Francis Group, LLC Accuracy of Surface Generation 373 The nominal part surface is given. Locally, the surface P is specied by the principal radii of curvature R P1. and R P2. at point K, and by the surface torsion τ P . The actual part surface P ac within the elementary surface cell is congruent to the surface of the cut. When machining a part, the cutting edge of the cutting tool moves relative to the work. Consecutive positions of the moving cutting edge form the surface of cut S c . A portion of the surface of the cut that is located within the elementary surface cell is congruent to the actual part surface ( S P c ac ≡ ). The surface of cut S c can also be locally specied by the principal radii of curvature R c1. and R c2. at point K, and by the surface torsion τ c . For the computation of the parameters R c1. , R c2. , and τ c , the equation of the surface of cut S c is necessary. The equation can be derived on the premises of the geometry of the cutting edge of the cutting tool, the kinemat- ics of the relative motion of the cutting edge with respect to the work, and the operators of coordinate systems transformations (see Chapter 3 for details). Fortunately, for most kinds of surface machining, the surface of cut S c is very close to the generating surface T of the cutting tool as long as the elementary surface cell is considered. Therefore, beside the major parameters R c1. , R c2. , and τ c of the surface of cut S c cannot be computed, the similar major parameters R T1. , R T2. , and τ T of the generating surface T of the cutting tool can be computed instead. 8.2.1 Local Approximation of the Surfaces P and T by Portions of Torus Surfaces Actual surfaces P and T can be given in a complex analytical form that is not convenient for computations of the major parameters of the surfaces. Solu- tions to many geometrical problems can be more easily derived from local consideration of the surfaces rather than from consideration of the entire surfaces. For the local analysis, the surfaces are often represented by quadrics. As shown in our previous works [4,5,8], from the perspective of local approximation of surface patches, helical canal surfaces feature important advantages over other candidates. A helical canal surface (Figure 8.4) is a particular case of a swept surface. Monge was the rst to investigate the class of surfaces formed by sweeping a sphere, in 1850 [2]. He named them canal surfaces. In the particular case when the path on which the sphere is swept along is a helix, and the sphere has constant radius, the surface swept out is referred to as a helical canal surface. A surface of this kind is of particular interest for engineers. A canal surface is the envelope of a one-parametric family of spheres. The envelope is dened as the union of all circles of intersection of innitesimally neighboring pairs of spheres. These circles are referred to as the composing circles. Helical canal surfaces can t the principal curvatures and torsion of the local patch of sculptured surfaces, as well as of the generating surfaces of cutting tools. © 2008 by Taylor & Francis Group, LLC Accuracy of Surface Generation 375 A torus surface can be expressed in terms of radius r tr of its generating circle, and in terms of radius R tr of its directing circle. Depending on the actual ratio between the radii r tr and R tr , the torus radius r tr can be equal to the rst principal radius of curvature R P1. of the part surface ( r R tr P = 1. ), while the torus radius R tr in this case is equal to the difference R R R tr P P = − 2 1. . . For another ratio between the radii r tr and R tr , the equalities r R R tr P P = − 2 1. . and R R tr P = 1. are valid. At a current surface point, principal radii of curvature can be computed as discussed in Chapter 1. In the coordinate system X Y Z tr tr tr associated with the torus surface (Figure 8.6), the position vector r tr tr tr ( , ) θ ϕ of a point of the approximating torus surface can be represented in the following way: r R r tr tr tr tr tr tr ( , ) ( ) ( , ) θ ϕ θ θ ϕ = + . Here, r( , ) θ ϕ tr tr designates the position vector of a point on the generating circle of radius r tr in its current location (Figure 8.6), and R( ) θ tr designates the position vector of the center of the generating circle, which rotates about the Z tr axis. A routine transformation yields the following expression for r tr tr tr ( , ) θ ϕ : r tr tr tr P P tr P tr R R R ( , ) ( )cos cos c . . . θ ϕ θ ϕ = − − + 2 1 1 oos ( )sin cos sin . . . θ θ ϕ θ tr P P tr P tr tr R R R R − − + 2 1 1 1 sin P tr ϕ 1             (8.6) P P 1 Tr P Z tr X P Z P Y P * r TP1 r TP1 O t Y tr t 2 P t 1.P n P r P1 X tr FIGURE 8.5 Construction of the torus surface T P1 at the point P 1 of the part surface P. © 2008 by Taylor & Francis Group, LLC Accuracy of Surface Generation 377 Ultimately, the approximating torus Tr C through point C is specied by the radius r r tr C tr. ≡ of the generating circle of the torus, and by the radius [ ( cos ) ] . . R R r R tr C tr tr tr tr C = + ⋅ θ of the directing circle of the torus surface. (Here, the angle θ tr species the location of point C on the arc of the generating circle of radius r tr .) Note that all ten kinds of local patches of smooth, regular surfaces (see Chapter 1, Figure 1.11) can be found on the torus surface Tr . Figure 8.8 illustrates this important property of the torus surface. Consider points on the surface Tr that occupy various positions M 1 , M 2 , M 3 , M i , and so forth. The part body can be located either inside the torus surface Tr, or outside the surface Tr . Depending upon the chosen location of the point M i either within the convex surface Tr or within the concave surface Tr , all ten kinds of local patches of smooth, regular surface can be found on the torus surface Tr . The major advantage of implementation of the torus surface for local approximation of the sculptured surface is due to a patch of the torus surface being capable of providing perfect approximation for bigger surface area compared to the approximation by quadrics, use of which is valid just within a differential vicinity of the surface point. C A C T B Z tr Y tr r tr R tr X tr O tr O T O T R 2.T (C) t T (B) t T (A) R 1.T (C) t T (C) θ tr (R tr + r tr .cos θ tr ) FIGURE 8.7 Analysis of the local geometry of the generating surface T of a lleted-end milling cutter. © 2008 by Taylor & Francis Group, LLC Accuracy of Surface Generation 379 The Darboux trihedron is implemented here for the purpose of construction of the local left-hand-oriented Cartesian coordinate system x y z P P P having origin at the point K. Conguration of the sculptured surface P as well as conguration of the generating surface T in the coordinate system X Y Z NC NC NC associated with the machine tool is known. Therefore, the corresponding operators of the coordinate systems transformation, the operator Rs( )NC Pa of the resultant transformation from the coordinate system X Y Z NC NC NC to the coordinate system X Y Z P P P and, further, the operator Rs( )P K P a of the resultant transformation from the coordinate system X Y Z P P P to the local coordinate system x y z P P P can be composed. Ultimately, the operator Rs( )NC K P a of the resultant coordinate systems transformation can be composed. The similar operators Rs( )NC Ta , Rs( )T K T a , and Rs( )NC K T a of the consequent coordinate systems transformations are composed for the generating surface T of the cutting tool. Ultimately, the operators of the direct Rs( )K K T P a and of the inverse Rs( )K K P T a coordinate systems transformations can be composed as well. The operators Rs( )K K T P a and Rs( )K K P T a complement the earlier composed operators of the coordinate Tr T Tr P Z TrT X Tr.T Y Tr.T n P u P v P x P (K) z P (K) y P (K) P K T X Tr.P Y Tr.P Z Tr.P FIGURE 8.9 Example of relative disposition of the approximating torus surfaces Tr P and Tr T . © 2008 by Taylor & Francis Group, LLC 380 Kinematic Geometry of Surface Machining systems transformation to a closed loop of the coordinate systems transformation (see Chapter 3). The derived operators of the coordinate systems transformations yield representation of the surfaces r P , r T , and of all major elements of their geometry in a common coordinate system. Implementation of the local coordinate system x y z P P P for this purpose is convenient. 8.3 Computation of the Elementary Surface Deviations The earlier performed analysis shows that the resultant deviation h Σ of the machined part surface P ac from its desired shape can be evaluated using the formula h a h b h h fr h ssΣ = ⋅ + ⋅ (see Equation 8.4). For the computation of the resultant deviation h Σ , actual values of the elementary deviations h fr and h ss are necessary. As will be shown, for the computation of both elementary deviations h fr and h ss , similar equations can be used. Therefore, it is not necessary to investigate both elementary deviations separately. It is sufcient to investigate just one of them, and afterwards to write similar equations for the computation of another. 8.3.1 Waviness of the Machined Part Surface Consider, for example, computation of the elementary deviation h fr . Fig- ure 8.10 illustrates a cross-section of a sculptured part surface P by a plane through the unit normal vector n P and through the feed-rate vector F fr . Depending on the chosen point of interest on the surface P, the cross-section of the surface P could have either straight prole KK 1 or convex prole K K 1 2 or concave prole K K 2 3 . It is convenient to mention here that the rate of conformity of the generating surface T of the cutting tool is the lowest at the convex point K 2 ; it is bigger at the point of inection K 1 (or at the similar point K); and it is highest at the concave point K 3 . This yields making a conclusion according to which when the higher rate of conformity of the surface T to the surface P observes, then the higher accuracy of the machined part surface and vice versa. For the computation of the elementary deviation h fr , the following equation is derived by Radzevich [6,7]: h R R R F R fr P fr P fr T fr fr P fr ≅ ⋅ + ⋅ − ⋅    . . . . ( ) cos1 2 (    − + ⋅ ⋅      R R R F R P fr P fr T fr fr P fr . . . . ( ) cos ( 2  (8.7) where radii of normal curvature of the surfaces P and T are designated as R P fr. and R T fr. , respectively, and the arc segment ( F fr designates the feed rate © 2008 by Taylor & Francis Group, LLC Accuracy of Surface Generation 381 per tooth of the cutting tool. The radii R P fr. and R T fr. are measured in the direction of the feed-rate vector F fr . For computation of the radius of normal curvature R P fr. , the following equation is derived in [6,7]: R E G G L M E G E N P fr P P P P P P P P P . sin sin cos = + + 2 2 2 ξ ξ ξ (8.8) where angle ξ species the direction of the feed-rate vector F fr relative to the principal directions t 1.P and t 2.P of the sculptured surface P. An equation similar to Equation (8.8) is derived in [6,7] for the computation of the radius of normal curvature R T fr. : R E G G L M E G E N T fr T T T T T T T T . sin ( ) sin ( ) ≅ + + + + 2 2 ξ µ ξ µ TT cos ( ) 2 ξ µ + (8.9) where m is the angle of the local relative orientation of surfaces P and T. It is assumed in Equation (8.9) that the radius of normal curvature of the surface O P (cv) h fr (cv) h fr P h fr (cx) O P (cx) F fr T F fr R T.fr R T.fr R T.fr R P.fr R P.fr F ˘ fr F ˘ fr O T.2 K 3 K 1 K 2 O T.3 O T.1 O T T 2 T 3 A E C K D B FIGURE 8.10 Computation of the elementary deviation h fr (the waviness) on the sculptured part surface P. © 2008 by Taylor & Francis Group, LLC [...]... approximation of the surfaces P and T by patches of torus surfaces is very helpful At the point of contact K, the surfaces P and T are locally approximated by the patches of the torus surface TrP and of the torus surface TrT , correspondingly (Figure 8.3) Use of this scheme for the precise computation of the resultant cusp height h Σ does not require implementation of the principle of superposition of the... (i) θtr.P Figure 8.13 Computation of the closest distance of approach of the surfaces P and T © 2008 by Taylor & Francis Group, LLC 392 Kinematic Geometry of Surface Machining For a given configuration of the torus surfaces TrP and TrT , the CDA between these surfaces can be used as a first approximation to the CDA between the surfaces P and T The CDA between the torus surfaces TrP and TrT is measured... derivation of Equation (8.14) YP fr OP XP fr (1) (2) OI OT RT fr RT fr RP fr P K1 K2 T hfr Figure 8.11 Another approach for the computation of the surface waviness © 2008 by Taylor & Francis Group, LLC h fr 384 Kinematic Geometry of Surface Machining The considered approach can be enhanced to the situation when radii of normal curvature of the part surface P and of the generating surface T of the cutting...382 Kinematic Geometry of Surface Machining of cut is approximately equal to the corresponding radius of normal curvature of the generating surface T of the cutting tool In particular cases, Equation (8.7) can be significantly simplified For example, when a flat portion of a part surface P is machined with the milling cutter of diameter dT, then the cusp height is... method of chords, and so forth Many similarities can be found in this comparison (i) OT (i+1) OT (i+1) (i) RT RT t1 p1 t2 p2 t3 p3 t0 p0 tT T tP P (i+1) (i) RP RP (i) OP (i+1) OP Figure 8.14 Convergence of the methods of computation of the closest distance of approach of the surfaces P and T © 2008 by Taylor & Francis Group, LLC 396 Kinematic Geometry of Surface Machining 8.5 Effective Reduction of the... implementation of the principle of superposition of the elementary surface deviations valid?” naturally arises In order to answer this practical question, comparison of results of computations of the resultant cusp height, which are performed using Equation (8.44) (or using more general Equation 8.4), with the results of precise © 2008 by Taylor & Francis Group, LLC 400 Kinematic Geometry of Surface Machining... magnitude of the vector θ Σ (see Equation 8.16) The closest distance of approach of the surfaces P and T is not equal to zero It can be positive or negative In the first case, the cutting tool surface T is located apart from the part surface P In the second case, the cutting tool surface T interferes with the part surface P © 2008 by Taylor & Francis Group, LLC 388 Kinematic Geometry of Surface Machining... perpendicular to these surfaces The following equations can be composed on the premises of this property of the CDA Unit normal vector n Tr P to the torus surface TrP is within a plane through the axis of rotation of the surface TrP In the coordinate system Xtr P Ytr P Ztr P that is associated with the surface TrP , the equation of a plane through the axis of rotation of the torus surface TrP can be expressed... the ideal case of surfaces generation, the equality K P KT = 0 is valid In reality, the generating surface T of the cutting tool is displaced with respect to the part surface P The total linear displacement of the surface T with respect to the surface P is equal to the magnitude of the vector δ Σ (see Equation 8.15) The total angular displacement of the surface T with respect to the surface P is equal... the desired part surface Second, in order to avoid the cutter penetration into the part surface P, it is of critical importance to determine the maximal allowed dimensions of the cutting tool in order to avoid violation of the necessary conditions of proper surface generation (see Chapter 7) For solving problems of both kinds, computation of the closest distance of approach (CDA) of the surfaces P and . 8.13 Computation of the closest distance of approach of the surfaces P and T. © 2008 by Taylor & Francis Group, LLC 392 Kinematic Geometry of Surface Machining For a given conguration of the torus surfaces. LLC 370 Kinematic Geometry of Surface Machining 8.1.3 The Resultant Deviation of the Machined Part Surface The resultant deviation h Σ of the actual part surface P ac from the nominal surface. terms of geometry of the nominal part surface P and of the surface P ac of the elementary surface cell. In order to solve the problem, an analytical local representation of the surfaces