P1: GDZ/SPH P2: JXR CB672-26 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 2:33 26.4 Generation of a Surface with Optimal Approximation 763 Figure 26.4.6: Determination of maximal deviations along line L gk . MINIMIZATION OF DEVIATIONS δ i, j . Consider that deviations δ i, j (i = 1, ,n; j = 1, ,m)of g with respect to  p have been determined at the (n, m) grid points. The minimization of deviations can be obtained by corrections of previously obtained func- tion β (1) (θ p ). The correction of angle β is equivalent to the correction of the angle that is formed by the principal directions on surfaces  t and  g . The correction of angle β can be achieved by turning of the tool about the common normal to surfaces  t and  p at their instantaneous point of tangency M k . The minimization of deviations δ i, j is based on the following procedure: Step 1: Consider the characteristic L gk , the line of contact between surfaces  t and  g , that passes through current point M k of mean line L m on surface  p (Fig. 26.4.6). Determine the deviations δ k between  t and  p along line L gk and find out the maximal deviations designated as δ (1) kmax and δ (2) kmax . Points of L gk where the deviations are maximal are designated as N (1) k and N (2) k . These points are determined in regions I and II of surface  g with line L m as the border. The simultaneous consideration of maximal deviations in both regions enables us to minimize the deviations for the whole surface  g . Note. The deviations of  t from  p along L gk are simultaneously the deviations of  g from  p along L gk because L gk is the line of tangency of  t and  g . Step 2: The minimization of deviations is accomplished by correction of angle β k that is determined at point M k (Fig. 26.4.6). The minimization of deviations is performed lo- cally, for a piece k of surface  g with the characteristic L gk . The process of minimization is a computerized iterative process based on the following considerations: (i) The objective function is represented as F k = min  δ (1) k max + δ (2) k max  (26.4.46) with the constraint δ i, j ≥ 0. (ii) The variable of the objective function is β k . Then, considering the angle β (2) k = β (1) k + β k (26.4.47) and using the equation of meshing with β k , we can determine the new characteristic, the piece of envelope  (k) g , and the new deviations. The applied iterations provide P1: GDZ/SPH P2: JXR CB672-26 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 2:33 764 Generation of Surfaces by CNC Machines the sought-for objective function. The final correction of angle β k we designate as β (opt ) k . Note 1. The new contact line L (2) gk (determined with β (2) k ) slightly differs from the real contact line because the derivative dβ (1) k /ds but not dβ (2) k /ds is used for determination L (2) gk . However, L (2) gk is very close to the real contact line. Step 3: The discussed procedure must be performed for the set of pieces of surfaces  g with the characteristic L gk for each surface piece. Recall that the deviations for the whole surface must satisfy the inequality δ i, j ≥ 0. The procedure of optimization is illustrated with the flowchart in Fig. 26.4.7. Curvatures of Ground Surface Σ g The direct determination of curvatures of  g by using surface  g equations is a com- plicated problem. The solution to this problem can be substantially simplified using the following conditions proposed by the authors: (i) the normal curvatures and surface torsions (geodesic torsions) of surfaces  p and  g are equal along line L m , respectively; and (ii) the normal curvatures and surface torsions of surfaces  t and  g are equal along line L g . This enables us to derive four equations that represent the principal cur- vatures of surface  g in terms of normal curvatures and surface torsions of  p and  t . However, only three of these equations are independent (see below). Further derivations are based on the following equations: k n = k I cos 2 q + k II sin 2 q = 1 2 (k I + k II ) + 1 2 (k I − k II ) cos 2q (26.4.48) t = 0.5(k II − k I ) sin 2q. (26.4.49) Here, k I and k II are the surface principal curvatures and angle q is formed by unit vectors e I and e and is measured counterclockwise from e I and e; e I is the principal direction with principal curvature k I ; e is the unit vector for the direction where the normal curvature is considered; t is the surface torsion for the direction represented by e. Equation (26.4.48) is known as the Euler equation. Equation (26.4.49) is known in differential geometry as the Bonnet–Germain equation (see Chapter 7). The determination of the principal curvatures and principal directions for  g is based on the following computational procedure (see Section 7.9): Step 1: Determination of k (1) n and t (1) for surface  g at the direction determined by the tangent to L m . The determination is based on Eqs. (26.4.48) and (26.4.49) applied to surface  p . Recall that  p and  g have the same values of k (1) n and t (1) along the previously mentioned direction. Step 2: Determination of k (2) n and t (2) . The designations k (2) n and t (2) indicate the normal curvatures of  g and the surface torsion along the tangent to L g . Recall that k (2) n and t (2) are the same for  t and  g along L g . We determine k (2) n and t (2) for surface  t using Eqs. (26.4.48) and (26.4.49), respectively. Step 3: We consider at this stage of computation that for surface  g the following are known: k (1) n and t (1) , and k (2) n and t (2) for two directions with tangents τ 1 and τ 2 that form the known angle µ (Fig. 26.4.8). Our goal is to determine angle q 1 (or q 2 ) P1: GDZ/SPH P2: JXR CB672-26 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 2:33 26.4 Generation of a Surface with Optimal Approximation 765 Figure 26.4.7: Flowchart for optimization. for the principal direction e (g) I and the principal curvatures k (g) I and k (g) II (Fig. 26.4.8). Using Eqs. (26.4.48) and (26.4.49), we can prove that k (i ) n and t (i ) (i = 1, 2) given for two directions represented by τ 1 and τ 2 are related with the following equation: t (1) + t (2) k (2) n − k (1) n = cot µ (26.4.50) P1: GDZ/SPH P2: JXR CB672-26 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 2:33 766 Generation of Surfaces by CNC Machines Figure 26.4.8: For determination of prin- cipal directions of generated surface  g . Step 4: Using Eqs. (26.4.48) and (26.4.49), we can derive the following three equa- tions for determination of q 1 , k (g) I , and k (g) II : tan 2q 1 = t (1) sin 2µ t (2) − t (1) cos 2µ (26.4.51) k (g) I = k (1) n − t (1) tan q 1 (26.4.52) k (g) II = k (1) n + t (1) cot q 1 . (26.4.53) Equation (26.4.51) provides two solutions for q 1 (q (2) 1 = q (1) 1 + 90 ◦ ) and both are cor- rect. We choose the solution with the smaller value of q 1 . Numerical Example: Grinding of an Archimedes Worm Surface The worm surface shown in Fig. 26.4.9 is a ruled undeveloped surface formed by the screw motion of straight line KN (|KN|=u p ). The screw motion is performed in coordinate system S p [Fig. 26.4.9(b)]. The to-be-ground surface  p is represented in S p as r p = u p cos α cos θ p i p + u p cos α sin θ p j p + ( pθ p − u p sin α) k p (26.4.54) where u p and θ p are the surface parameters. The surface unit normal is n p = N p |N p | , N p = ∂r p ∂u p × ∂r p ∂θ p . (26.4.55) Thus n p = 1  u 2 p + p 2  0.5      p sin θ p + u p sin α cos θ p −p cos θ p + u p sin α sin θ p u p cos α      (provided cos α = 0). (26.4.56) P1: GDZ/SPH P2: JXR CB672-26 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 2:33 26.4 Generation of a Surface with Optimal Approximation 767 Figure 26.4.9: Surface of an Archimedes worm. The design data are: number of threads N 1 = 2; axial diametral pitch P ax = 8/in; α = 20 ◦ ; the radius of the pitch cylinder is 1.25 in. The remaining design parameters are determined from the following equations: (i) The screw parameter is p = N 1 2P ax = 0.125 in. Longitudinal Number of Points Latitudinal Number of Points Normal Deviations (mm × 10 −3 ) 2 3 4 5 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 2 3 4 5 6 7 8 9 Figure 26.4.10: Deviations of the ground surface  g from ideal surface  p of an Archimedes worm. P1: GDZ/SPH P2: JXR CB672-26 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 2:33 768 Generation of Surfaces by CNC Machines (ii) The lead angle is tan λ p = p r p = 0.125 1.25 ,λ p = 5.7106 ◦ . The mean line is determined as r p (u m ,θ p ), u m =  r p + 1 P ax  +  r p − 1.25 P ax  2 cos α = r p − 0.125 P ax cos α = 1.3136 in. where 1/P ax and 1.25/P ax determine the addendum and dedendum of the worm. The worm is ground by a cone with the apex angle γ t = 30 ◦ , and outside diameter 8 in. The inside angle β (1) =−88.0121 ◦ provides the coincidence of both generatrices of the cone and the Archimedes worm. The maximal deviation of the ground surface  g from the ideal surface  p with the above value of β (1) is 3 µm. The optimal angle β (opt ) =−94.6788 ◦ has been determined by the developed opti- mization method. The deviations of the ground surface  g from  p with the optimal β (opt ) are positive and the maximal deviation has been reduced to 0.35 µm (Fig. 26.4.10). P1: GDZ/SPH P2: GDZ CB672-27 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 2:49 27 Overwire (Ball) Measurement 27.1 INTRODUCTION Indirect determination of gear tooth thickness by overwire (ball) measurement has found broad application. This topic has been the subject of research by many scientists. The earliest publications dealing with such measurement of worms and spatial gears are Litvin’s papers and books. Detailed references regarding the history of the performed research are given in Litvin [1968]. The application of computers and subroutines for the solution of systems of non- linear equations is a significant step forward in this area that was accomplished by Litvin et al. [1998b]. This chapter covers the following topics: (i) Algorithms for determination of location of a wire or a ball placed into the space of a workpiece with symmetric and nonsymmetric location of tooth surfaces (ii) Relation between the tooth thickness and overwire measurement – this relation enables us to use the developed algorithms for a workpiece with various tolerances. The developed theory was applied for measurement of tooth thickness of worms, screws, and involute helical gears. Computer programs for this purpose have been de- veloped. 27.2 PROBLEM DESCRIPTION Consider that a ball (a wire) is placed into the space of a workpiece (a worm, screw, or gear). The surfaces of the ball and the workpiece are in tangency at two points whose location depends on the geometry of the surfaces of the workpiece, the width of the space, and the diameter of the ball. The surface geometry of the workpiece is represented analytically and the width of the space and the ball (wire) diameter are given. Then, we can determine analytically (i) the distance of the center of the ball from the axis of the workpiece, or (ii) the shortest distance between the axes of the wire and the workpiece. The measurement over the ball (the wire) and the comparison of the obtained data with the analytically determined data enable us to find out if the space width satisfies the requirements. 769 P1: GDZ/SPH P2: GDZ CB672-27 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 2:49 770 Overwire (Ball) Measurement The following is a description of the analytical approach that has been developed for the determination of points of tangency of a ball (wire) with the surfaces of the space of a workpiece. In the most general case there is no symmetry in the location and orientation of the two surfaces that form the space of the workpiece. This is typical, for instance, wherein the workpiece is a spiral bevel pinion or a hypoid pinion. Thus, we have to consider in such a case the simultaneous tangency of the ball or the wire with both surfaces of the space. Consider that the two surfaces of the space and the unit normals to the surfaces are represented by vector equations, r (i ) = r (i )  u (i ) ,θ (i )  (i = 1, 2) (27.2.1) n (i ) = N (i ) |N (i ) | , N (i ) = r (i ) u × r (i ) θ (i = 1, 2) (27.2.2) where r (i ) u = ∂r (i ) /∂u (i ) , r (i ) θ = ∂r (i ) /∂θ (i ) , and (u (i ) ,θ (i ) ) are the Gaussian coordinates of the surface. The designation (i = 1, 2) indicates the surfaces for both sides of the space. It is assumed that the surface normal is directed outward from the space, toward the tooth, and such a direction can be provided by the proper order of cofactors in the cross- products for N (i ) . Consider that U = [ XY Z ] T (27.2.3) is the position vector of center C of the ball, or the point of intersection of both normals with the wire axis. It is evident from the conditions of force transmissions by the mea- surement, that both normals to the wire intersect the wire axis at the same point. We consider that z is the axis of the workpiece, and Z is chosen to determine the location of point C in a plane that is perpendicular to the axis of the workpiece. The tangency of the wire (ball) with the surfaces of the space is represented by the equation (Fig. 27.2.1) U + ρn (i ) = r (i )  u (i ) ,θ (i )  (27.2.4) where U = [XY Z] T . This equation yields X − x (i )  u (i ) ,θ (i )  n (i ) x  u (i ) ,θ (i )  = Y − y (i )  u (i ) ,θ (i )  n (i ) y  u (i ) ,θ (i )  = Z − z (i )  u (i ) ,θ (i )  n (i ) z  u (i ) ,θ (i )  =−ρ (i = 1, 2) (27.2.5) where ρ is the radius of the wire (ball). P1: GDZ/SPH P2: GDZ CB672-27 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 2:49 27.2 Problem Description 771 Figure 27.2.1: Measurement by a single ball. Our goal is to determine the distance R = (X 2 + Y 2 ) 1/2 (27.2.6) of point C from the axis of the workpiece. Then, knowing R, we may determine the overwire (ball) measurement M. Three wires are used for the measurement of worms and screws, and in this case M = 2(R + ρ). (27.2.7) Equation (27.2.7) works for the measurement of a gear with an even number of teeth. The equation for M, when a gear with an odd number of teeth is measured by two balls, is represented as (Fig. 27.2.2) M = 2R  cos 90 ◦ N  + 2ρ (27.2.8) where N is the number of gear teeth. There is a special approach for determination of M when a gear with an odd number of teeth is measured by two wires (see Section 27.3). Procedure of Computation The procedure of computation for overwire (ball) measurement may be represented as follows: Step 1: Equations (27.2.5) represent a system of six equations in six unknowns: X, Y, u (i ) , θ (i ) (i = 1, 2). We may represent this system of equations by a subsystem of four non-linear equations, and a system of two linear equations. The subsystem of four P1: GDZ/SPH P2: GDZ CB672-27 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 2:49 772 Overwire (Ball) Measurement Figure 27.2.2: Measurement by two balls. nonlinear equations is Z − z (1)  u (1) ,θ (1)  n (1) z  u (1) ,θ (1)  =−ρ (27.2.9) Z − z (2)  u (2) ,θ (2)  n (2) z  u (2) ,θ (2)  =−ρ (27.2.10) x (1)  u (1) ,θ (1)  − ρn (1) x  u (1) ,θ (1)  = x (2)  u (2) ,θ (2)  − ρn (2) x  u (2) ,θ (2)  (27.2.11) y (1)  u (1) ,θ (1)  − ρn (1) y  u (1) ,θ (1)  = y (2)  u (2) ,θ (2)  − ρn (2) y  u (2) ,θ (2)  . (27.2.12) Equations (27.2.9) to (27.2.12) considered simultaneously provide the solution for four unknowns: u (1) , θ (1) , u (2) , and θ (2) . A subroutine for the solution of the above system of four nonlinear equations is required [for instance, we can use the one contained in the IMSL library [Visual Numerics, Inc., 1998]]. The unknowns X and Y may be determined from the following two linear equations: X = x (i )  u (i ) ,θ (i )  − ρn (i ) x  u (i ) ,θ (i )  (i = 1 or 2) (27.2.13) Y = y (i )  u (i ) ,θ (i )  − ρn (i ) y  u (i ) ,θ (i )  (i = 1or2). (27.2.14) Step 2: We have considered the radius ρ of the ball (the wire) as known. In reality, we have to determine a value of ρ that satisfies the equation (R + ρ) −r a = δ. (27.2.15) Here, r a is the radius of the addendum circle in the plane Z = d, and δ is the desired difference between (R + ρ) and r a . The proper value of ρ can be determined by variation [...]... ( 27 .3. 14) φ = tan2 β R sin(γ − φ) ( 27 .3. 15) Equation ( 27 .3. 14) yields Equations ( 27 .3. 13) and ( 27 .3. 15) yield that the extreme value of C is γ −φ 2 C = 2R sin 1 + tan2 β R cos2 γ −φ 2 1/2 ( 27 .3. 16) and M = C + 2ρ ( 27 .3. 17) The derivation of equation (1) (2) r 2 − r2 · c2 = C ( 27 .3. 18) is based on the following considerations: (1) (2) (i) Unit vectors a2 and a2 of the axes of two wires are crossed and. .. Industry, 96, 33 0 33 4 Litvin, F L., Krylov, N N., & Erikhov, M L 1 975 Generation of Tooth Surfaces by TwoParameter Enveloping Mechanism and Machine Theory, 10(5), 36 5 37 3 Litvin, F L., & Gutman, Y 1981 Methods of Synthesis and Analysis for Hypoid Gear Drives of Formate and Helixform Parts 1, 2, and 3 ASME Journal of Mechanical Design, 1 03( 1), 83 1 13 Litvin, F L., & Tsay, C.-B 1985 Helical Gears with Circular... The relation between dM and dR may be obtained by differentiation of Eqs ( 27 .3. 16) and ( 27 .3. 17) that yields dM = 2R C − tan2 β R sin2 (γ − φ) dR R C ( 27 .3. 22) where dR is represented by ( 27 .3. 3) The derivation of ( 27 .3. 22) is based on the following auxiliary relations: dφ = cos2 2 tan β R sin(γ − φ) dβ R β R [1 + tan2 β R cos(γ − φ)] dβ R = sin β R cos β R dR R ( 27 .3. 23) ( 27 .3. 24) Numerical Example:...P1: GDZ/SPH CB 672 - 27 P2: GDZ CB 672 /Litvin CB 672 /Litvin-v2.cls February 27, 2004 2:49 27 .3 Measurement of Involute Worms, Involute Helical Gears, and Spur Gears 7 73 of ρ in Eqs ( 27. 2.9) to ( 27. 2.15) until the desired value of δ is obtained Then, an even value of ρ may finally be chosen, and we can start the computations using Eqs ( 27. 2.9) to ( 27. 2.14) Step 3: The width wt of the space on...  2   γ −φ  − tan β R cos 2 =0 ( 27 .3. 12) The shortest distance C between the wires is represented as 2 sin (1) (2) C(φ) = r2 − r2 · c2 = 1+ γ −φ 2 tan2 βR + φ cos cos2 γ −φ 2 γ −φ 2 1/2 R ( 27 .3. 13) P1: GDZ/SPH CB 672 - 27 P2: GDZ CB 672 /Litvin CB 672 /Litvin-v2.cls February 27, 2004 2:49 27 .3 Measurement of Involute Worms, Involute Helical Gears, and Spur Gears 77 7 (See derivations below.) The overwire... Worms, Screws and Gears Mechanism and Machine Theory, 33 (6), 851– 877 Litvin, F L., Lu, J., Townsend, D P., & Howkins, M 1999 Computerized Simulation of Meshing of Conventional Helical Involute Gears and Modification of Geometry Mechanism and Machine Theory, 34 (1), 1 23 1 47 Litvin, F L., Chen, Y.-J., Heath, G F., Sheth, V J., & Chen, N 2000a Apparatus and Method for Precision Grinding Face Gears USA Patent... a2 − λ1 a2 ( 27 .3. 20) We multiply both sides of Eq ( 27 .3. 20) by the unit vector (1) c2 = (2) (1) (2) a2 × a2 a2 × a2 and take into account that C · c2 = C, (1) (1) (2) a2 · a2 × a2 Then, we obtain Eq ( 27 .3. 18) (2) (1) (2) = a2 · a2 × a2 = 0 ( 27 .3. 21) P1: GDZ/SPH CB 672 - 27 P2: GDZ CB 672 /Litvin CB 672 /Litvin-v2.cls February 27, 2004 2:49 77 8 Overwire (Ball) Measurement Relation Between dM and dR The relation... pφ in the screw motion Figure 27 .3. 2: For derivation of location and orientation of wire 2 in S 2 P1: GDZ/SPH CB 672 - 27 P2: GDZ CB 672 /Litvin CB 672 /Litvin-v2.cls February 27, 2004 2:49 77 6 Overwire (Ball) Measurement Figure 27 .3. 3: For derivation of shortest distance between two wires Equations ( 27 .3. 9) yield (2) r2 = R[cos(γ − φ) i2 − sin(γ − φ) j2 + φ cot β R k2 ] ( 27 .3. 10) a2 = sin β R sin(γ − φ)... wire and the worm Relation betweend R and dsp dp Pa x α1 α2 N1 = = = = = 1.125 in 10.0 (1/in.) 7 45◦ 8 λp sp p H = = = = 35 .4 170 55◦ 0.1 570 80 in 0.400000 in 2.5 132 7 in ρ = 0. 071 000 in R = 0.598 838 in dR = −0.829 939 dsp P1: JXT CB 672 -28 CB 672 /Litvin CB 672 /Litvin-v2.cls February 27, 2004 2:51 28 Minimization of Deviations of Gear Real Tooth Surfaces 28.1 INTRODUCTION Coordinate measurements of gear real... Eqs ( 27. 2.16) and take Z = 0 27 .3 MEASUREMENT OF INVOLUTE WORMS, INVOLUTE HELICAL GEARS, AND SPUR GEARS Basic Equations Equations of involute worms are represented in Section 19.6 Equation ( 27. 2.16) with Z = 0 yields the following computational procedure: ρ wt − ( 27 .3. 1) inv (θ + µ) = inv αt + r b sin λb 2r p rb ( 27 .3. 2) R= cos(θ + µ) dwt cos αt dR = − ( 27 .3. 3) 2 sin(θ + µ) P1: GDZ/SPH CB 672 - 27 P2: . λ b − w t 2r p ( 27 .3. 1) R = r b cos(θ + µ) ( 27 .3. 2) dR =− dw t cos α t 2 sin(θ + µ) . ( 27 .3. 3) P1: GDZ/SPH P2: GDZ CB 672 - 27 CB 672 /Litvin CB 672 /Litvin-v2.cls February 27, 2004 2:49 77 4 Overwire (Ball). φ 2  1/2 R. ( 27 .3. 13) P1: GDZ/SPH P2: GDZ CB 672 - 27 CB 672 /Litvin CB 672 /Litvin-v2.cls February 27, 2004 2:49 27 .3 Measurement of Involute Worms, Involute Helical Gears, and Spur Gears 77 7 (See derivations. C(φ) reaches its extreme value, that is, dC(φ) dφ = 0. ( 27 .3. 14) Equation ( 27 .3. 14) yields φ = tan 2 β R sin(γ − φ). ( 27 .3. 15) Equations ( 27 .3. 13) and ( 27 .3. 15) yield that the extreme value of C is C =