Gear Geometry and Applied Theory Episode 3 Part 6 pps

P1: JXT CB672-24 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 2:30 24.5 Generation by Disk-Shaped Tool: Workpiece Surface is Given 733 Solution (i) f (u p ,θ p ) = (u p − pθ p sin λ b ) sin(θ p + µ) −(E c + p cot γ c ) cos λ b + (E c cot γ c sin λ b +r b cos λ b ) cos(θ p + µ) = 0. (ii) x c = r b cos(θ p + µ) +u p cos λ b sin(θ p + µ) − E c y c = [r b sin(θ p + µ) −u p cos λ b cos(θ p + µ)] cos γ c + [−u p sin λ b + pθ p ] sin γ c z c = [−r b sin(θ p + µ) +u p cos λ b cos(θ p + µ)] sin γ c + [−u p sin λ b + pθ p ] cos γ c f (u p ,θ p ) = 0. Considering the previously represented system of equations and choosing θ p as the input parameter, we determine the coordinates of the tool axial profiles as x c (θ p ) =−(x 2 c + y 2 c ) 0.5 , z c (θ p ). P1: JXR CB672-25 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 2:32 25 Design of Flyblades 25.1 INTRODUCTION Flyblades are used for generation of worm-gears in small-scale production to avoid the manufacture of expensive hobs. However, the production of worm-gears by flyblades is less effective in comparison with production by hobs. The profiles of the flyblades are determined as profiles of the worm thread in the normal tooth section obtained by intersection of the thread by plane  (Fig. 25.1.1). The orientation of the plane  is determined with the lead angle λ p on the worm pitch cylinder. A symmetrical location of the profiles of the flyblade in coordinate system S  1 can be obtained if the x  1 axis is the axis of symmetry of the worm thread. In Chapter 19, we have derived equations of worm surfaces for the case when the x 1 axis of coordinate system S 1 is the axis of symmetry of the worm space. To obtain the desired location of axis x 1 (as the axis of tooth symmetry), it is necessary to displace the origin O 1 of coordinate system S 1 in the axial direction at the magnitude a o = p ax /2, where p ax is the axial distance between two neighboring threads of the worm. The process of generation of the worm-gear by a flyblade simulates the meshing of the worm with the worm-gear in such a specific case when the worm performs translational motion in an axial direction in addition to the worm rotational motion. The angle of rotation φ 2 of the worm-gear in the process for generation is a sum of the two following components: φ 2 = φ 1 N 1 N 2 + s tr p . (25.1.1) Here, φ 1 is the angle of rotation of the worm, N 1 is the number of worm threads, N 2 is the number of worm-gear teeth, p = r p tan λ p is the screw parameter, and s tr is the worm (flyblade) axial translation that is an imput parameter chosen from technological considerations. In the case of a drive with a multi-thread worm, the flyblade generates only those teeth of the worm-gear that are in mesh with the respective threads of the worm. Therefore, indexing of the worm-gear is required to generate the entire number of worm-gear teeth. Indexing can be avoided if N 1 and N 2 are prime numbers (they do not have a common multiplier), for instance, when we have N 1 = 3, N 2 = 32. In such a case the flyblade 734 P1: JXR CB672-25 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 2:32 25.2 Two-parameter Form Representation of Worm Surfaces 735 Figure 25.1.1: For determination of profiles of flyblade. after each revolution of the worm-gear will start to generate the worm-gear teeth that are in mesh with the next thread of the multi-thread worm. The following part of this chapter covers the determination of profiles of the flyblade for various types of worm geometry of worm-gear drives. The computational procedure represented below covers two cases of representation of worm geometry: (i) the two- parameter form, and (ii) the three-parameter form (but with related parameters) of worm surface representation. 25.2 TWO-PARAMETER FORM REPRESENTATION OF WORM SURFACES Step 1: Consider that the worm thread surface (say, the surface side I ) is represented by the vector equation (see Chapter 19) r 1 (u,θ) = x 1 (u,θ) i 1 + y 1 (u,θ) j 1 + z 1 (u,θ) k 1 , (25.2.1) and the x 1 axis is the axis of symmetry of the worm space. To provide that the x 1 axis will be the axis of symmetry of the worm thread, it is necessary to displace the origin O 1 of coordinate system S 1 along the z 1 axis on a o = p ax /2. Thus, we obtain that z 1 = z 1 (u,θ) + a o . (25.2.2) Step 2: The profiles of the worm thread are considered in plane  of the normal section. Thus, we have y 1 + z 1 tan λ p = 0. (25.2.3) Equations (25.2.2) and (25.2.3) yield F (u,θ) = y 1 (u,θ) + tan λ p [z 1 (u,θ) + a o ] = 0. (25.2.4) Step 3: Consider that θ is the input parameter. Solving Eq. (25.2.4), we will obtain function u(θ) (provided ∂ F /∂θ = 0). The requirement ∂ F /∂θ = 0 follows from the theorem of implicit function system existence (see Korn & Korn [1968] and Litvin [1989]). P1: JXR CB672-25 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 2:32 736 Design of Flyblades Figure 25.2.1: (a) Illustration of flyblade profile; (b) for derivation of upper fillet. Step 4: We can determine now the profile of one side of the flying blade using the equations x 1 (u,θ) = x 1 (u(θ ),θ) = x 1 (θ), z  1 (θ) =− y 1 (u(θ ),θ) sin λ p . (25.2.5) Step 5: Functions x 1 (θ), z  1 (θ) determine the profile of the flyblade (Fig. 25.2.1). The range of θ is determined with the following conditions: (i) Point A of the normal section of the worm thread must belong to the worm cylinder of radius r a . Here,  [x 1 (θ)] 2 + [y 1 (θ)] 2  0.5 = r a (25.2.6) is the radius of the worm addendum cylinder. (ii) Point B of the normal section of the worm thread belongs to the worm cylinder of radius (r d + c) and is determined with the equation  [x 1 (θ)] 2 + [y 1 (θ)] 2  0.5 = r d + c. (25.2.7) Here, r d is the radius of the worm dedendum cylinder; c is the clearance between the worm and the worm-gear. P1: JXR CB672-25 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 2:32 25.3 Three-parameter Form Representation of Worm Surfaces 737 Step 6: The upper path of the profile of the flyblade must be complemented with the upper fillet [Fig. 25.2.1(a)]. To provide the tangency of the fillet with the segment AB,we will have to determine the normal a 1 to the planar curve AB at point A. Consider that normal N 1 to the worm thread surface is represented in coordinate system S 1 (x 1 , y 1 , z 1 ). Then, using the coordinate transformation from S 1 to S  1 (x 1 , y  1 , z  1 ), we will determine the normal N  1 and then obtain a 1 = N  x1 i  1 + N  z1 k  1 . (25.2.8) Step 7: Using Fig. 25.2.1(b), we can derive the following equation for the fillet ra- dius ρ: ρ = c 1 − sin δ (provided ρ cos δ ≤|z  1 (A)|) (25.2.9) where tan δ =     n  x1 n  z1      0 <δ< π 2  . (25.2.10) The upper fillet generates the bottom of the space of the worm-gear. The bottom fillet of the flyblade is to be obtained similarly. The described procedure can be applied for the ZA, ZN, and ZI worms (see Chapter 19). 25.3 THREE-PARAMETER FORM REPRESENTATION OF WORM SURFACES The worm thread surface is the envelope to the family of tool surfaces and is represented by the equations r 1 (u c ,θ c ,ψ) = x 1 (u c ,θ c ,ψ) i 1 + y 1 (u c ,θ c ,ψ) j 1 + z 1 (u c ,θ c ,ψ,a o ) k 1 (25.3.1) f (u c ,θ c ) = 0. (25.3.2) Here, u c ,θ c are the surface parameters of the tool; ψ is the parameter of motion in the process for generation of the worm by the tool surface; Eq. (25.3.2) is the equation of meshing. Equations (25.3.1) and (25.3.2) represent the thread surfaces of the ZK and ZF worms (see Chapter 19). The thread profile is located in plane  (Fig. 25.1.1), and y 1 (u c ,θ c ,ψ) + z 1 (u c ,θ c ,ψ) tan λ p = 0. (25.3.3) Equations (25.3.1) and (25.3.3) yield F (u c ,θ c ,ψ) = 0. (25.3.4) The system of Eqs. (25.3.2) and (25.3.4) represents two relations for the set of three parameters (u c ,θ c ,ψ). Choosing one of the three parameters as the input one, say θ c , we can determine the coordinates x 1 (θ c ), z  1 (θ c ) of the flyblade profile. The following is the application of the computational procedures described above for the determination of flyblade profiles. P1: JXR CB672-25 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 2:32 738 Design of Flyblades 25.4 WORKING EQUATIONS ZA (Archimedes) Worm The surface side I of the worm thread (dashed line in Fig. 25.1.1) for the right-hand worm is represented by the equations (see Section 19.4) x 1 = u cos α cos θ y 1 = u cos α sin θ z 1 =−u sin α +  r p tan α − s p 2  + pθ + p ax 2 (25.4.1) where the nominal value of s p = p ax /2. The normal to the worm thread is determined with the equations N x1 =−(p sin θ + u sin α cos θ) N y1 = ( p cos θ − u sin α sin θ) N z1 =−u cos α. (25.4.2) Following the previously described procedure of derivations, we obtain u(θ ) = −  r p tan α + s p 2 + pθ  (sin θ cot λ p − tan α) cos α (25.4.3) x 1 (θ) = u(θ ) cos α cos θ, z  1 =− u(θ ) cosα sin θ sin λ p . (25.4.4) Equations (25.4.3) and (25.4.4) enable us to determine the profile of the flyblade. The range of θ for computations is determined with Eqs. (25.2.6) and (25.2.7). The starting value of θ for computations by application of Eq. (25.4.3) is θ = 0. The upper fillet is determined as described in Section 25.2. ZN (Convolute) Worm The surface side I of the thread for the right-hand worms is represented as (see Section 19.5) x 1 = ρ sin(θ + µ) +u cos δ cos(θ + µ) y 1 =−ρ cos(θ + µ) + u cos δ sin(θ +µ) z 1 = ρ cos α cot λ p cos δ − u sin δ + pθ + p ax 2 (25.4.5) where the x 1 axis is the axis of symmetry of the thread. We recall that the worm surface is generated by a straight line that performs the screw motion about the worm axis. The straight line is tangent to the worm cylinder of radius ρ and the orientation of the P1: JXR CB672-25 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 2:32 25.4 Working Equations 739 generating straight line is determined with parameter α. Here, cos µ = cos α cos δ , cos δ = (cos 2 α + sin 2 α sin 2 λ p ) 1 2 ρ =  r p − s p 2 cot α  sin α sin λ p (cos 2 α + sin 2 α sin 2 λ p ) 1 2 (see the designations of s p and α in Fig. 19.5.3). The normal to the worm thread is represented by the equations N x1 =−[(p +ρ tan δ) sin(θ + µ) +u sin δ cos(θ + µ)] N y1 = ( p + ρ tan δ) cos(θ +µ) − u sin δ sin(θ + µ) N z1 =−u cos δ. (25.4.6) The profile of the flyblade is determined with the following systems of equations: u = ρ  cos(θ + µ) − cos α cos δ  cot λ p − pθ − p ax 2 [sin(θ + µ) cot λ p − tan δ] cos δ x 1 = ρ sin(θ + µ) +u cos δ cos(θ + µ) z  1 = ρ cos(θ + µ) − u cos δ sin(θ +µ) sin λ p . (25.4.7) The starting value of θ for computations is θ = 0. The range of θ is determined with Eqs. (25.2.6) and (25.2.7). The upper fillet is determined as described in Section 25.2. ZI (Involute) Worm The surface side I of the thread for the right-hand worm is represented as (see Section 19.6) x 1 = r b cos(θ + µ) +u cos λ b sin(θ + µ) y 1 = r b sin(θ + µ) −u cos λ b cos(θ + µ) z 1 =−u sin λ b + pθ + p ax 2 . (25.4.8) The surface normal is represented with equations, N x1 =−sin λ b sin(θ + µ) N y1 = sin λ b cos(θ + µ) N z1 =−cos λ b . (25.4.9) Here, axis x 1 is the axis of symmetry of thread in the y 1 = 0 plane, r b is the radius of the base cylinder, and λ b is the radius of the lead angle on the base cylinder. Angle µ is P1: JXR CB672-25 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 2:32 740 Design of Flyblades determined as µ = w t 2r p − inv α t where w t is the space width (see Fig. 19.6.3). The flyblade profile is determined with the following equations: u(θ ) = r b sin(θ + µ) cot λ p + pθ + p ax 2 [tan λ b + cos(θ + µ) cot λ p ] cos λ b x 1 = r b cos(θ + µ) +u cos λ b sin(θ + µ) z  1 =− r b sin(θ + µ) −u cos λ b cos(θ + µ) sin λ p . (25.4.10) The initial value θ for computations is θ = 0. The range of θ is determined with Eqs. (25.2.6) and (25.2.7). The upper fillet is determined as described in Section 25.2. ZK (Klingelnberg) Worm We remind the reader that the thread surface is generated by a cone surface (see Section 19.7). The surface side I of the thread of the right-hand worm is represented by the equations x 1 = u c (cos α c cos θ c cos ψ + cos α c cos γ c sin θ c sin ψ − sin α c sin γ c sin ψ) +a sin γ c sin ψ + E c cos ψ y 1 = u c (−cos α c cos θ c sin ψ + cos α c cos γ c sin θ c cos ψ − sin α c sin γ c cos ψ) +a sin γ c cos ψ − E c sin ψ z 1 = u c (sin α c cos γ c + cos α c sin γ c sin θ c ) − pψ − a cos γ c + p ax 2 . (25.4.11) Here, u c = a sin α c − (E c sin α c cot γ c + p sin α c ) tan θ c − (E c − p cot γ c ) cos α c cos θ c (25.4.12) where (25.4.12) is the equation of meshing of the tool and the thread surfaces. The normal to the thread surface is represented by the equations N x1 = cos ψ sin α c cos θ c + sin ψ(cos γ c sin α c sin θ c + sin γ c cos α c ) N y1 =−sin ψ sin α c cos θ c + cos ψ(cos γ c sin α c sin θ c + sin γ c cos α c ) N z1 = sin γ c sin α c sin θ c − cos γ c cos α c . (25.4.13) Usually, γ c = λ p . Equation y 1 + z 1 tan λ p = 0 P1: JXR CB672-25 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 2:32 25.4 Working Equations 741 yields F (ψ, u c ,θ c ) = cos ψ[u c (cos α c cos γ c sin θ c − sin α c sin γ c ) + a sin γ c ] − sin ψ(u c cos α c cos θ c + E c ) − p tan λ p ψ + tan λ p  u c (sin α c cos γ c + cos α c sin γ c sin θ c ) − a cos γ c + p ax 2  = 0. (25.4.14) The procedure of computations is as follows: Step 1: Consider simultaneously the system of nonlinear equations (25.4.12) and (25.4.14) in the unknowns (ψ, u c ,θ c ). Solve numerically the above system by func- tions u c (θ c ), ψ(θ c ), where θ c is the input variable. The first guess for the solution of Eqs. (25.4.12) and (25.4.14) is based on the following assumptions: θ c = π , sin ψ ≈ ψ, cos ψ ≈ 1. Then, using Eqs. (25.4.12) and (25.4.14), we obtain ψ = p ax 2 tan λ p E c + p tan λ p − u c cos α c (25.4.15) where u c = a sin α c + (E c − p cot γ c ) cos α c . (25.4.16) Step 2: Determine the coordinates x  1 ≡ x 1 and z  1 of the blade profile using Eqs. (25.4.17). Here, x  1 = x 1 (ψ, u c ,θ c ) is the first equation of equation system (25.4.11), and z  1 =−[u c (−cos α c cos θ c sin ψ + cos α c cos γ c sin θ c cos ψ − sin α c sin γ c cos ψ) +a sin γ c cos ψ − E c sin ψ]/ sin λ p . (25.4.17) The range of θ c for computations is determined with Eqs. (25.2.6) and (25.2.7). The upper fillet is determined as described in Section 25.2. F-I (Flender Version I) Worm Recall that the worm thread surface is the envelope to the family of tool surfaces (see Section 19.8). The surface side I of the thread for right-hand worms is represented by the equations (see Section 19.8) x 1 = (ρ sin θ c + d)(−cos ν cos ψ + sin ν sin ψ cos γ c ) +(ρ cos θ c − b) sin ψ sin γ c + E c cos ψ y 1 = (ρ sin θ c + d)(cos ν sin ψ + sin ν cos ψ cos γ c ) +(ρ cos θ c − b) cos ψ sin γ c − E c sin ψ z 1 = (ρ sin θ c + d) sin ν sin γ c + (b − ρ cos θ c ) cos γ c − pψ +a o + p ax 2 (25.4.18) P1: JXR CB672-25 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 2:32 742 Design of Flyblades where tan θ c = E c − p cot γ c − d cos ν b cos ν − (E c cot γ c + p) sin ν . (25.4.19) Equation (25.4.19) is the equation of meshing of the worm thread surface with the tool surface. Parameter a o enables us to obtain that the axis of symmetry of the axial section of the worm space will coincide with the x 1 axis. Taking in Eqs. (25.4.18) y 1 = 0, x 1 = r p , and z 1 = p ax /4, we obtain a o =−(ρ sin θ c + d) sin ν sin γ c − (b − ρ cos θ c ) cos γ c + pψ − p ax 4 . (25.4.20) The normal to the worm thread surface is represented by the equations N x1 = sin θ c (−cos ν cos ψ + sin ν sin ψ cos γ c ) + cos θ c sin ψ sin γ c N y1 = sin θ c (cos ν sin ψ + sin ν cos ψ cos γ c ) + cos θ c cos ψ sin γ c N z1 = sin θ c sin ν sin γ c − cos θ c cos γ c . (25.4.21) Equation y 1 + z 1 tan λ p = 0 yields F (ψ, θ c ,ν) = cos ψ[(ρ sin θ c + d) sin ν cos γ c + (ρ cos θ c − b) sin γ c ] + sin ψ[(ρ sin θ c + d) cos ν − E c ] − p tan λ p ψ +  (ρ sin θ c + d) sin ν sin γ c + (b − ρ cos θ c ) cos γ c + a o + p ax 2  tan λ p = 0. (25.4.22) Nonlinear equations (25.4.19) and (25.4.22) relate three unknowns: ν, θ c , and ψ. The procedure for computations of the blade profile is as follows: Step 1: Using Eq. (25.4.19), we obtain numerically function θ c (ν). Equation (25.4.19) with the input value of ν provides two solutions for θ c , but only the solution 0 <θ c < 180 ◦ must be used (see Section 19.8). Step 2: Knowing the related parameters ν and θ c , we can solve Eq. (25.4.22) for ψ. The first guess for the solution is based on the following considerations: (i) Taking ν = 0, ρ = r p , and γ c = λ p , we obtain from Eq. (25.4.19) tan θ c = E c − p cot λ p − d b = tan α n . Thus, θ c = α n . [...]... flyblade for generation of the F-I worm -gear The design data of the worm are N1 = 3, r p = 46 mm, axial module max = 8 mm, Figure 25.4.1: Flyblade for F-I worm -gear P1: JXR CB672-25 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 2 :32 744 Design of Flyblades ρ = 46 mm, γc = λ p = 14◦ 37 15 , αn = 20◦ , a = r p + ρ sin αn = 61 . 733 mm, and b = ρ cos αn = 43. 2 26 mm F-II (Flender Version II) Worm Unlike... a gear Generation of conventional spiral bevel gears and hypoid gears by the CNC 7 46 P1: GDZ/SPH CB672- 26 P2: JXR CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 2 :33 26. 2 Execution of Motions of CNC Machines 747 Figure 26. 2.1: Schematic of “Phoenix” machine machine is the example of case 2 of generation Case 3 is the basic idea for a new method for surface generation discussed in Section 26. 4 26. 2... + ym sin ψ − z m cos φ cos ψ ( 26. 2 .6) 26 .3 GENERATION OF HYPOID PINION (G) Derivation of L pt and (Ot Op)(G) The generation of a hypoid pinion by a conventional generator was described in Chapter 22 The coordinate systems applied for the CNC machine are represented in Fig 26. 2.2 P1: GDZ/SPH CB672- 26 P2: JXR CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 2 :33 26 .3 Generation of Hypoid Pinion 751... be chosen P1: GDZ/SPH CB672- 26 P2: JXR CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 752 2 :33 Generation of Surfaces by CNC Machines Step 2: Determination of ψ cos φ sin ψ = a 23 (q), cos φ cos ψ = a 33 (q) ( 26 .3. 5) These equations provide a unique solution for ψ, considering φ as given Step 3: Determination of µ cos µ cos φ = a 11 (q), − sin µ cos φ = a 12 (q) ( 26 .3. 6) These equations provide... normal of t at Ob ; db = nb × tb ; vectors tb and db form the tangent plane to t at Ob Trihedron S f moves along the mean line Lm (Fig 26. 4 .3) ; t f is the tangent Figure 26. 4 .3: Orientation of trihedron S b with respect to S f P1: GDZ/SPH CB672- 26 P2: JXR CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 7 56 2 :33 Generation of Surfaces by CNC Machines Figure 26. 4.4: Surface of grinding tool cone to the... =  t py  t pz d px npx    npy   npz d py d pz ( 26. 4 . 36 ) where ∂ (M) r ∂θ p p tp = ∂ (M) r ∂θ p p ( 26. 4 .37 ) is the unit tangent to the mean line Lm , ∂r p ∂r p × ∂up ∂θ p np = ± ∂r p ∂rp × ∂up ∂θ p dp = np × tp ( 26. 4 .38 ) ( 26. 4 .39 ) The sign chosen in Eq ( 26. 4 .38 ) must provide the direction of n p toward the surface “body.” Equations ( 26. 4 .35 ) represent in Sp the generated surface g in three-parameter... (Figs 26. 4.1 and 26. 4 .3) ; (iii) determination of deviations of g from p with the initial function β (1) (θ p ); and (iv) optimal minimization of deviations GRID ON SURFACE Σ p Figure 26. 4.5(a) shows the grid on surface (n, m) points, where the deviations of g from p p , the net of are considered The position vector P1: GDZ/SPH CB672- 26 P2: JXR CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 2 :33 26. 4... c, where [ c1 c2 T T c 3 ] = A [ b1 b2 T b3 ] ( 26. 4.28) Using the above considerations and eliminating ds /dt, the final expression of the equation of meshing can be represented as (t) (tg) nf · vf = f (u t , θt , θ p ) = u t nT A(s ) et − l t nT B(s ) ib + nT LT t t f = 0 t b t f ( 26. 4.29) P1: GDZ/SPH CB672- 26 P2: JXR CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 2 :33 26. 4 Generation of a Surface... a3  −a 2 ds (s ) = LT b Ω f L f b f dt  −a 3 a 2   0 −a 1   a1 0 B(s ) ( 26. 4 .30 ) ( 26. 4 .31 )   dβ t cos β sin γt − kn sin β sin γt + kg + cos γt  a1  ds          a 2  = −  ( 26. 4 .32 ) t sin β + kn cos β       dβ a3 t cos β cos γt − kn sin β cos γt − kg + sin γt ds       t cos β − kn sin β b1 0 −b3 b2            −t sin β − kn cos β    (s ) B =  b3  ( 26. 4 .33 )... of g to p is obtained by variation of angle β (Fig 26. 4.1) P1: GDZ/SPH CB672- 26 P2: JXR CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 2 :33 26. 4 Generation of a Surface with Optimal Approximation 7 53 Figure 26. 4.1: Installment and orientation of tool surface t with respect to ideal surface p (5) The continuous tangency of tool surface t with p and properly varied orientation of t can be obtained . forging of a gear. Generation of conventional spiral bevel gears and hypoid gears by the CNC 7 46 P1: GDZ/SPH P2: JXR CB672- 26 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 2 :33 26. 2 Execution. (Fig. 25.1.1), and y 1 (u c ,θ c ,ψ) + z 1 (u c ,θ c ,ψ) tan λ p = 0. (25 .3. 3) Equations (25 .3. 1) and (25 .3. 3) yield F (u c ,θ c ,ψ) = 0. (25 .3. 4) The system of Eqs. (25 .3. 2) and (25 .3. 4) represents. S e and S d (Fig. 26. 2.2). P1: GDZ/SPH P2: JXR CB672- 26 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 2 :33 748 Generation of Surfaces by CNC Machines Figure 26. 2.2: Coordinate systems applied