P1: GDZ/SPH P2: GDZ CB672-21 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 2:1 21.3 Derivation of Gear Tooth Surfaces 643 M 2a 2 =          sin γ m 2 0 −cos γ m 2 0 01 0 0 cos γ m 2 0 sinγ m 2 −X D 2 00 0 1          . Parameters V 2 , H 2 , X B 2 , X D 2 , and γ m 2 (Fig. 21.3.6) are the gear machine-tool settings. The upper and lower signs in front of V 2 correspond to right-hand and left- hand gears, respectively. The whole set of machine-tool settings for a formate-cut gear is presented in Table 21.3.2. Figure 21.3.6: Coordinate systems applied for cutting or grinding of a formate-cut gear: (a) for right- hand gear; (b) for left-hand gear. P1: GDZ/SPH P2: GDZ CB672-21 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 2:1 644 Spiral Bevel Gears Table 21.3.2: Machine-tool settings of a formate-cut gear Name Notation Reference Blade angle α g (Fig. 21.3.4) Blade parabolic coefficient a c [Eq. (21.3.8)] Parabola apex location s g o (Fig. 21.3.4) Cutter (grinding wheel) radius R u (Fig. 21.3.4) Point width P w 2 (Fig. 21.3.4) Cutter point radius R g = R u ± P w 2 /2 (Fig. 21.3.4) Horizontal setting H 2 (Fig. 21.3.6) Vertical setting V 2 (Fig. 21.3.6) Sliding base X B 2 (Fig. 21.3.6) Machine center to back X D 2 (Fig. 21.3.6) Machine root angle γ m2 (Fig. 21.3.6) Edge radius of head-cutter ρ w (Fig. 21.3.4) 21.4 DERIVATION OF PINION TOOTH SURFACE We limit the discussion to the generation of the pinion by straight-line blades of the head-cutter. However, application of blades of parabolic profile for pinion generation is beneficial in some cases, for instance for design of a gear ratio close to 1. Applied Coordinate Systems Coordinate systems applied for generation of the pinion are shown in Fig. 21.4.1. Co- ordinate systems S m1 , S a1 , S b1 are the fixed ones and they are rigidly connected to the cutting machine. The movable coordinate systems S 1 and S c1 are rigidly connected to the pinion and the cradle, respectively. They are rotated about the z b1 axis and the z m1 axis, respectively, and their rotations are related with a polynomial function ψ 1 ( ψ c1 ) wherein modified roll is applied (see below). The ratio of instantaneous angular ve- locities of the pinion and the cradle is defined as m 1c ( ψ 1 ( ψ c1 )) = ω (1) (ψ c1 )/ω (c) . The magnitude m 1c (ψ 1 )atψ c1 = 0 is called ratio of roll or velocity ratio. Parameters X D 1 , X B 1 , E m 1 , and γ m1 are the basic machine-tool settings for pinion generation. Coordinate system S p [Figs. 21.4.1(a) and 21.4.1(b)] is applied for illustration of installment of the head-cutter on the cradle and corresponds to generation of the right- hand and left-hand pinion, respectively. Head-Cutter Surfaces The pinion generating surfaces are formed by surface  (a) p and  (b) p generated by straight- line and circular arc parts of the blades. Surface  (a) p is represented as r (a) p (s p ,θ p ) =      (R p ∓ s p sin α p ) cos θ p (R p ∓ s p sin α p ) sin θ p −s p cos α p      (21.4.1) P1: GDZ/SPH P2: GDZ CB672-21 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 2:1 21.4 Derivation of Pinion Tooth Surface 645 Figure 21.4.1: Coordinate systems applied for pinion generation: (a) and (b) illustration of tool in- stallment for generation of right- and left-hand pinions; (c) illustration of installment of machine-tool settings. where s p and θ p are the surface coordinates, α p is the blade angle, and R p is the cutter point radius (Fig. 21.4.2). The upper and lower signs in Eq. (21.4.1) correspond to the convex and concave sides of the pinion tooth, which are in mesh with the concave and convex sides of the gear, respectively. The unit normal to the pinion generating surface  (a) p is represented by the equations n (a) p (θ p ) = N p   N p   , N p = ∂r (a) p ∂s p × ∂r (a) p ∂θ p . (21.4.2) Equations (21.4.1) and (21.4.2) yield n (a) p (θ p ) =      cos α p cos θ p cos α p sin θ p ∓sin α p      . (21.4.3) P1: GDZ/SPH P2: GDZ CB672-21 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 2:1 646 Spiral Bevel Gears Figure 21.4.2: Blades and generating cones for pinion generating tool with straight blades: (a) con- vex side blade; (b) convex side generating cone; (c) concave side blade; (d) concave side generating cone. For surface  (b) p , we obtain r (b) p (λ f ,θ p ) =      (X f ∓ ρ f sin λ f ) cos θ p (X f ∓ ρ f sin λ f ) sin θ p −ρ f (1 − cos λ f )      , 0 ≤ λ f ≤ π 2 − α p (21.4.4) where X f = R p ± ρ f (1 − sin α p )/ cos α p and ρ f is the edge radius of the head-cutter for the pinion (Fig. 21.4.2). P1: GDZ/SPH P2: GDZ CB672-21 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 2:1 21.4 Derivation of Pinion Tooth Surface 647 The unit normal to the pinion generating surface of part (b) is represented by the equations n (b) p (θ p ) = N (b) p   N (b) p   , N p = ∂r (b) p ∂λ f × ∂r (b) p ∂θ p . (21.4.5) Equations (21.4.4) and (21.4.5) yield n (b) p (θ p ) =      sin λ f cos θ p sin λ f sin θ p ∓cos λ p      . (21.4.6) Families of Pinion Tooth Surfaces Such families are represented as r (a) 1 (s p ,θ p ,ψ c1 ) = M 1p (ψ c1 ) r (a) p (s p ,θ p ) (21.4.7) r (b) 1 (λ f ,θ p ,ψ c1 ) = M 1p (ψ c1 ) r (b) p (λ f ,θ p ) (21.4.8) where M 1p = M 1b 1 M b 1 a 1 M a 1 m 1 M m 1 c 1 M c 1 p M c 1 p =          100S r 1 cos q 1 010 S r 1 sin q 1 001 0 000 1          M m 1 c 1 =          cos ψ c 1 −sin ψ c 1 00 sin ψ c 1 cos ψ c 1 00 0010 0001          M a 1 m 1 =          100 0 010 E m 1 001−X B 1 000 1          P1: GDZ/SPH P2: GDZ CB672-21 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 2:1 648 Spiral Bevel Gears M b 1 a 1 =          sin γ m 1 0 −cos γ m 1 0 01 0 0 cos γ m 1 0 sinγ m 1 −X D 2 00 0 1          M 1b 1 =          cos ψ 1 sin ψ 1 00 −sin ψ 1 cos ψ 1 00 0010 0001          . When modified roll is applied in the process of generation, the rotation angles ψ 1 of the pinion and ψ c 1 of the cradle are related as ψ 1 = b 1 ψ c 1 − b 2 ψ 2 c 1 − b 3 ψ 3 c 1 (21.4.9) = b 1  ψ c 1 − b 2 b 1 ψ 2 c 1 − b 3 b 1 ψ 3 c 1  = m 1c  ψ c 1 − Cψ 2 c 1 − Dψ 3 c 1  where b 1 , b 2 , and b 3 are the modified roll parameters and C and D are the modified roll coefficients. The derivative of function ψ 1 ( ψ c1 ) taken at ψ c1 = 0 determines the so-called ratio of roll or velocity ratio, determined in Eq. (21.4.9) by b 1 or m 1c . Equation of Meshing The pinion tooth surface  1 is the envelope to the family of cutter surfaces. The modified roll is applied in the process of generation. The equation of meshing is represented as n (a) m 1 · v (p1) m 1 = f (a) 1p (s p ,θ p ,ψ c1 ) = 0 (21.4.10) where n (a) m 1 is the unit normal to the surface, and v (p1) m 1 is the velocity in relative motion. The vectors are represented in the fixed coordinate system S m 1 as follows: n (a) m 1 = L m 1 c 1 L c 1 p n (a) p (θ p ) (21.4.11) v (p1) m 1 =  ω (p) m 1 − ω (1) m 1  ×r m1  −  O m 1 O a 2 × ω (1) m 1  . (21.4.12) The 3 × 3 matrices L m 1 c 1 and L c 1 p in Eq. (21.4.11) and in similar derivations are the sub-matrices of the 4 × 4 matrices M m 1 c 1 and M c 1 p , respectively. They are obtained by elimination of the last row and column of M m 1 c 1 and M c 1 p , respectively. Elements of matrices L m 1 c 1 and L c 1 p represent the direction cosines formed by the respective axes of coordinate systems S m 1 and S c 1 for L m 1 c 1 and coordinate systems S c 1 and S p for L c 1 p (see Chapter 1). P1: GDZ/SPH P2: GDZ CB672-21 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 2:1 21.5 Local Synthesis and Determination of Pinion Machine-Tool Settings 649 Position vector r m 1 in Eq. (21.4.12) is determined as r m1 = M m 1 c 1 M c 1 p r (a) p (s p ,θ p ) O m 1 O a 2 = [0 −E m 1 X B 1 ] T ω (1) m 1 = [cos γ m 1 0 sin γ m 1 ] T ω (p) m 1 = [0 0 m 1c (ψ c 1 )] T . The ratio m 1c (ψ c 1 ) is not constant because modified roll is applied and it can be repre- sented as m 1c (ψ c 1 ) = ω c 1 ω 1 = dψ c 1 /dt dψ 1 /dt = 1 dψ 1 /dψ c 1 = 1 m 1c  1 − 2Cψ c 1 − 3Dψ 2 c 1  = 1 m 1c − 2b 2 ψ c 1 − 3b 3 ψ 2 c 1 (21.4.13) where C and D are the modified roll coefficients. Finally, we obtain the equations for pinion tooth surface part (a)as r (a) 1 (s p ,θ p ,ψ c 1 ) = M 1p (ψ c 1 ) r (a) p (s p ,θ p ) (21.4.14) f 1p (s p ,θ p ,ψ c 1 ) = 0. (21.4.15) Using similar derivations, the fillet surface may be represented as r (b) 1 (λ f ,θ p ,ψ c 1 ) = M 1p (ψ c 1 ) r (b) p (λ f ,θ p ) (21.4.16) f 1p (λ f ,θ p ,ψ c 1 ) = 0. (21.4.17) 21.5 LOCAL SYNTHESIS AND DETERMINATION OF PINION MACHINE-TOOL SETTINGS The purpose of local synthesis is to obtain favorable conditions of meshing and contact at the chosen mean contact point M. Such conditions at M are defined by η 2 , a, and m  21 (Fig. 21.2.1). The gear machine-tool settings are considered as known and they may be adapted, for instance, from the manufacturing summary. The procedure of local synthesis is a part of the proposed integrated approach for the design of spiral bevel gears with localized bearing contact and reduced levels of vibration and noise based on application of a longitudinally directed path of contact and application of parabolic blades for generation of the gear to avoid hidden areas of severe contact. The procedure of local synthesis is represented as a sequence of three stages that provide: (i) the tangency at M of gear tooth surface  2 and gear head-cutter surface  g , (ii) the tangency at M of gear and pinion tooth surfaces  2 and  1 , and (iii) the tangency at M of pinion tooth surface  1 and pinion head-cutter surface  p . Finally, we obtain that all four surfaces  2 ,  g ,  1 , and  p are in tangency at M. At all stages, P1: GDZ/SPH P2: GDZ CB672-21 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 2:1 650 Spiral Bevel Gears the relationships between the principal curvatures and directions of meshing surfaces are applied (provided in Section 21.6). Then, it becomes possible to obtain the sought-for pinion machine-tool settings. The procedure of local synthesis is applied for both cases of design of spiral bevel gear drives: (i) face-milled generated gears, and (ii) formate-cut spiral bevel gears. The procedure for the case of face-milled generated spiral bevel gear drives is represented below. The procedure of local synthesis for formate-cut spiral bevel gear drives can be considered as a particular case of the one applied for face-milled generated spiral bevel gear drives and is discussed below. Local Synthesis of Face-Milled Generated Spiral Bevel Gear Drives The procedure of local synthesis of face-milled generated spiral bevel gear drives is illustrated by the following three stages: STAGE 1: TANGENCY OF SURFACES  2 AND  g AT CHOSEN POINT A . Point A on surface  2 is chosen as a candidate for the mean contact point M of pinion–gear tooth surfaces. Step 1: The meshing of surfaces  2 and  g is represented in coordinate system S m 2 (Fig. 21.3.2) by the following equations: r m 2 (s g ,θ g ,ψ 2 ) = M m 2 g (ψ 2 ) r g (s g ,θ g ) (21.5.1) f 2g (s g ,θ g ,ψ 2 ) = 0. (21.5.2) Equation (21.5.1) represents in S m 2 the family of surfaces  g , and Eq. (21.5.2) is the equation of meshing. The generated surface  2 is represented in S 2 by the matrix equa- tion r 2 (s g ,θ g ,ψ 2 ) = M 2g (ψ 2 ) r g (s g ,θ g ) (21.5.3) and the equation of meshing (21.5.2). Step 2: Mean point A on surface  2 is chosen by designation of parameters L A and R A (Fig. 21.5.1), where A is the candidate for the mean contact point M of surfaces  2 Figure 21.5.1: Representation of point A in coordinate system S 2 . P1: GDZ/SPH P2: GDZ CB672-21 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 2:1 21.5 Local Synthesis and Determination of Pinion Machine-Tool Settings 651 and  1 . Then we obtain the following system of two equations in three unknowns:    Z 2 (s ∗ g ,θ ∗ g ,ψ ∗ 2 ) = L A X 2 2 (s ∗ g ,θ ∗ g ,ψ ∗ 2 ) + Y 2 2 (s ∗ g ,θ ∗ g ,ψ ∗ 2 ) = R 2 A (21.5.4) where X 2 , Y 2 , Z 2 are the projections of position vector r 2 (s ∗ g ,θ ∗ g ,ψ ∗ 2 ) (see Eq. (21.5.3)). The third equation for determination of these three unknowns is the equation of meshing (21.5.2). Step 3: Equations (21.5.2), (21.5.3), and (21.5.4) considered simultaneously allow the determination of parameters (s ∗ g ,θ ∗ g ,ψ ∗ 2 ) for point A. Vector functions r g (s g ,θ g ) and n g (θ g ) determine the position vector and surface unit normal for a current point of surface  g . Taking in these vector functions s g = s ∗ g and θ g = θ ∗ g , we can determine the position vector r (A) g of point A and the surface unit normal at A. Step 4: Parameters s ∗ g and θ ∗ g and the unit vectors e g and e u of principal directions on surface  g are considered as known. For a head-cutter with blades of straight-line profiles: e g = ∂r (a) g ∂s g ÷      ∂r (a) g ∂s g      =      ±sin α g cos θ g ±sin α g sin θ g −cos α g      (21.5.5) e u = ∂r (a) g ∂θ g ÷      ∂r (a) g ∂θ g      =      −sin θ g cos θ g 0      . (21.5.6) In this case, the generating surface is a conical surface, and the principal curvatures k g and k u of  g can be determined by the following equations:      k g = 0 k u = cos α g R cg ± s g sin α g . (21.5.7) The upper and lower signs correspond to the concave and convex sides of the gear tooth, respectively. The approach discussed in Section 21.6 enables the determination at point A of (i) the principal curvatures k s and k q of  2 , and (ii) the unit vectors e s and e q of principal directions on surface  2 . The unit vectors e s and e q are represented in S m 2 . The general procedure presented in Section 8.4 can be applied for determination of principal curvatures k g and k u of the surfaces of the blades of parabolic profile. STAGE 2: TANGENCY OF SURFACES  2 ,  g ,AND 1 AT MEAN CONTACT POINT M Step 1: The derivations accomplished at Stage 1 enable the determination of the position vector r (A) 2 and the surface unit normal n (A) 2 of point A of tangency of surfaces  2 and  g . The goal now is to determine such a point M in the fixed coordinate system P1: GDZ/SPH P2: GDZ CB672-21 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 2:1 652 Spiral Bevel Gears Figure 21.5.2: Coordinate systems S 2 , S  , and S 1 applied for local synthesis. S  (Fig. 21.5.2) where three surfaces,  2 ,  g , and  1 , will be in tangency with each other. It can be imagined that surface  g is rigidly attached to  2 at point A and that both surfaces,  g and  2 , perform motion as a rigid body turning about the gear axis on a certain angle φ (0) 2 . Using the coordinate transformation from S 2 to S  (Fig. 21.5.2), we may obtain r (A)  and n (A)  . The new position of point A in S  will be the point of tangency of  2 and  1 (designated as M), if the following equation of meshing between  2 and  1 is observed: n (A)   φ (0) 2  · v (21,A)   φ (0) 2  = 0 (21.5.8) Here, n (A)  ≡ n (M)  and v (21,A)  ≡ v (21,M)  ; v (21,A)  is the relative velocity at point A deter- mined with the ideal gear ratio m (0) 21 = ω (2) ω (1) . (21.5.9) The solution of Eq. (21.5.8) for φ (0) 2 provides the value of the turning angle φ (0) 2 .Itis evident that three surfaces,  2 ,  g , and  1 , are now in tangency with each other at point M. We emphasize that the procedure in Step 1 enables us to avoid the tilt of the head-cutter for generation of the pinion. Step 2: We consider as known at point M the principal curvatures k s and k q of surface  2 , and the unit vectors e s and e q of principal directions on  2 . The unit vectors e s and e q are represented in S  . The goal is to determine at point M the principal curvatures k f and k h of surface  1 , and the unit vectors e f and e h of principal directions on  1 . This goal can be achieved by application of the procedure described in Section 21.6. It is shown in Section 21.6 that the determination of k f , k h , e f , and e h becomes possible if parameters m  21 , η 2 (or η 1 ), and a/δ are assumed to be known or are used as input data. STAGE 3: TANGENCY OF SURFACES  2 ,  g ,  1 ,AND p AT MEAN CONTACT POINT M. We consider in this stage two sub-stages: (a) derivation of basic equations of surface tan- gency, and (b) determination of pinion machine-tool settings that satisfy the equations [...]... 0.0000 3. 667770 −0.001180 0.006460 Concave 136 .1114 22.0000 0. 635 0 5.4860 −2.1264 −10.9672 79.4459 13. 8 833 55.5190 0.0000 0.0000 4.5 431 30 0.00704 −0.14949 Convex 114 .34 83 22.0000 0. 635 0 −1.97 93 −0 .33 52 −2.9984 62.2550 13. 8 833 59 .37 51 0.00000 0.00000 3. 667770 −0.001180 0.006460 Concave Case 1c 138 .9061 22.0000 0. 635 0 6 .30 96 −2 .32 41 −9.4557 79 .34 30 13. 8 833 55 .30 37 0.0000 0.0000 4.546945 0.006 53 −0.15951... = −a 13 v q a 33 + (12) tan η2 + a 33 + a 13 v s (12) a 23 v q − (12) vs tan η2 (21.6.14) Choosing η2 at point M, we can determine η1 Procedure of Determination of k f , k h , and σ (12) Step 1: Determine η1 choosing η2 Step 2: a 33 (1) vs = a 13 + a 23 tan η1 (21.6.15) a 33 tan η1 a 13 + a 23 tan η1 (1) vq = (21.6.16) Step 3: A= δ a2 (21.6.17) Step 4: 2 2 a 13 + a 23 (1) 2 + vs (1) a 13 v s +... CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 2:1 658 Spiral Bevel Gears represented as follows: tan 2σ (g2) = 2 c 23 −2c 13 c 23 − − (kg − ku )c 33 2 c 13 −2c 13 c 23 c 33 sin 2σ (g2) kq − ks = kq + ks = kg + ku + (21.6.7) 2 2 c 13 + c 23 c 33 Here, (g2) + n × ω (g2) · e g (g2) + n × ω (g2) · eu c 13 = −kg v g c 23 = −ku v u (g2) 2 c 33 = −kg v g (g2) 2 − ku v u + n × ω (g2) · v(g2) (21.6.8) (g) (2)... Simulation of Meshing and Contact 661 following equations are obtained: tan 2σ (1 p) = kt − k p = 2 d 23 −2d 13 d 23 − − (k f − kh )d 33 2 d 13 −2d 13 d 23 d 33 sin 2σ (1 p) kt + k p = k f + kh + (21.6.27) 2 2 d 13 + d 23 d 33 Here, (1 p) + n × ω (1 p) · e f (1 p) + n × ω(1 p) · eh d 13 = −k f v f d 23 = −kh v h (1 p) 2 d 33 = −k f v f −n· (1 p) 2 − kh v h ( p) ω (1) × vtr + n × ω (1 p) · v(1 p) (21.6.28) (1) − ω ( p)... (deg.) Mean spiral angle (deg.) Hand of spiral Face width (mm) Mean cone distance (mm) Whole depth (mm) Pitch angles (deg.) Root angles (deg.) Face angles (deg.) Clearance (mm) Addendum (mm) Dedendum (mm) 9 4. 833 8 90.0000 32 .0000 RH 27.5000 68.9200 9. 430 0 15.2551 13. 8 833 20.4167 1. 030 0 6.6400 2.7900 33 4. 833 8 90.0000 32 .0000 LH 27.5000 68.9200 9. 430 0 74.7449 69.5 833 76.1167 1. 030 0 1.7600 7.6700 • Case 1a:... 656 Spiral Bevel Gears In addition, the three following curvature equations are applied: tan 2σ (1 p) = kt − k p = 2 d 23 −2d 13 d 23 − − (k f − kh )d 33 2 d 13 −2d 13 d 23 d 33 sin 2σ (1 p) kt + k p = k f + kh + (21.5.28) 2 2 d 13 + d 23 d 33 Equation (21.5.24) is equivalent to two independent scalar equations; Eq (21.5.25) is equivalent to three scalar equations; and Eqs (21.5.26), (21.5.27), and (21.5.28) represent... +3. 653e+06 +1.400e+06 +1.283e+06 +1.167e+06 +1.050e+06 +9 .33 3e+05 +8.167e+05 +7.000e+05 +5. 833 e+05 +4.667e+05 +3. 500e+05 +2 .33 3e+05 +1.167e+05 +1.746e+01 Bending stress: 4 53 MPa Area of severe contact Figure 21.9 .3: Contact and bending stresses for the gear for design case 1a 671 P1: GDZ/SPH CB672-21 P2: GDZ CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 2:1 Figure 21.9.4: Contact and bending stresses... 55 .30 37 0.0000 0.0000 4.546945 0.006 53 −0.15951 Convex CB672/Litvin-v2.cls 138 .6840 22.0000 0. 635 0 1.2 739 −6.2067 1.6201 66 .32 76 13. 8 833 59.4500 28. 630 0 4.1200 3. 779475 0.0000 0.0000 Convex Case 1b CB672/Litvin Case 1a CB672-21 Table 21.9 .3: Parameters and installment settings of the pinion head-cutter for design cases 1a, 1b, and 1c P1: GDZ/SPH P2: GDZ February 27, 2004 2:1 669 P1: GDZ/SPH CB672-21... generate the gear and the pinion The author’s approach is based on application of parabolic blades for generation of the gear whereas the pinion is still generated by blades of straight-line profile However, if more mismatch is necessary, parabolic blades can also be applied for pinion generation Concave 116.8400 22.0000 0. 635 0 −0.7749 −6.0857 −0.6288 62.74 93 13. 8 833 53. 0400 −20.8800 −5.4400 3. 837 460 0.00000... 0.00000 Applied settings Cutter point diameter (mm) Pressure angle (deg.) Root fillet radius (mm) Machine center to back (mm) Sliding base (mm) Blank offset (mm) Radial distance (mm) Machine root angle (deg.) Cradle angle (deg.) Swivel angle (deg.) Tilt Angle (deg.) Velocity ratio Modified Roll Coefficient C Modified Roll Coefficient D 114 .34 83 22.0000 0. 635 0 −1.97 93 −0 .33 52 −2.9984 62.2550 13. 8 833 59 .37 51 . Spiral Bevel Gears represented as follows: tan 2σ (g2) = −2c 13 c 23 c 2 23 − c 2 13 − (k g − k u )c 33 k q − k s = −2c 13 c 23 c 33 sin 2σ (g2) k q + k s = k g + k u + c 2 13 + c 2 23 c 33 . (21.6.7) Here, c 13 =−k g v (g2) g +  n. and Contact 661 following equations are obtained: tan 2σ (1p) = −2d 13 d 23 d 2 23 − d 2 13 − (k f − k h )d 33 k t − k p = −2d 13 d 23 d 33 sin 2σ (1p) k t + k p = k f + k h + d 2 13 + d 2 23 d 33 . (21.6.27) Here, d 13 =−k f v (1p) f +  n. 2:1 656 Spiral Bevel Gears In addition, the three following curvature equations are applied: tan 2σ (1p) = −2d 13 d 23 d 2 23 − d 2 13 − (k f − k h )d 33 k t − k p = −2d 13 d 23 d 33 sin 2σ (1p) k t +