P1: GDZ/SPH P2: GDZ CB672-21 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 2:1 21.9 Example of Design and Optimization of a Spiral Bevel Gear Drive 673 Figure 21.9.6: (a) and (b) bearing contact for design cases 1b and 1c, respectively, and (c) function of transmission errors for both cases of design. P1: GDZ/SPH P2: GDZ CB672-21 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 2:1 Figure 21.9.7: Contact and bending stresses for the pinion for design case 1c. Figure 21.9.8: Contact and bending stresses for the gear for design case 1c. 674 P1: GDZ/SPH P2: GDZ CB672-21 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 2:1 21.9 Example of Design and Optimization of a Spiral Bevel Gear Drive 675 Figure 21.9.9: Evolution of contact and bending stresses for the pinion for design cases 1a, 1b, and 1c. case 1c enables us to avoid the appearance of a small area of severe contact at the top edge of the pinion with a higher level of contact stresses as shown in Figure 21.9.9(a). On the contrary, bending stresses for design cases 1b and 1c are higher than they are for the existing case of design as shown in Fig. 21.9.9(b). A substantial reduction of contact stresses could be achieved as well for the gear for design cases 1b and 1c with respect to the existing design of the gear of the gear drive. The same results as previously discussed for the pinion have been obtained for the gear for design cases 1b and 1c as shown in Fig. 21.9.10. P1: GDZ/SPH P2: GDZ CB672-21 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 2:1 676 Spiral Bevel Gears Figure 21.9.10: Evolution of contact and bending stresses for the gear for design cases 1a, 1b, and 1c. 21.10 COMPENSATION OF THE SHIFT OF THE BEARING CONTACT Spiral bevel gear drives are very sensitive to the shortest distance between the axes of the pinion and the gear, E, when the pinion–gear axes are not intersected but crossed. However, the shift of the bearing contact due to error of alignment E can be compensated by the axial displacement A 1 of the pinion. Figure 21.10.1 shows an example of compensation of an error of alignment E = 0.02 mm for design case 1c. Figure 21.10.1(a) shows the path of contact for design case 1c when no errors of alignment occur. Figure 21.10.1(b) shows the path of contact for an error of alignment P1: GDZ/SPH P2: GDZ CB672-21 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 2:1 21.10 Compensation of the Shift of the Bearing Contact 677 Figure 21.10.1: Design case 1c: (a) path of contact when no errors of alignment occur, (b) path of contact for error of alignment E = 0.02 mm, (c) path of contact when error of alignment E = 0.02 mm is compensated by  A 1 =−0.05 mm, (d) function of transmission errors for conditions of item (c). P1: GDZ/SPH P2: GDZ CB672-21 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 2:1 678 Spiral Bevel Gears E = 0.02 mm. Figure 21.10.1(c) shows the path of contact for an error of alignment E = 0.02 mm and an axial displacement of the pinion A 1 =−0.05 mm. As shown in Fig. 21.10.1(c), an axial displacement of the pinion may compensate the shift of the path of contact caused by an error E of the shortest distance between axes. Figure 21.10.1(d) shows the function of transmission errors when an error of alignment E = 0.02 mm is compensated by an axial displacement of the pinion A 1 =−0.05 mm. The function of transmission errors is still of parabolic shape. P1: JDW CB672-22 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 2:3 22 Hypoid Gear Drives 22.1 INTRODUCTION Hypoid gear drives have found a broad application in the automotive industry for transformation of rotation between crossed axes. Enhanced design and generation of hypoid gear drives requires an approach based on the ideas discussed for spiral bevel gears (see Chapter 21). The contents of this chapter are limited to (i) design of pitch cones, (ii) pinion and gear machine-tool settings, and (iii) equations of pinion–gear tooth surfaces. Design of pitch cones for hypoid gears was the subject of research performed by Baxter [1961], Litvin et al. [1974, 1990] and Litvin [1994]. Details of determination of machine-tool settings for manufacture of hypoid gears are given in Litvin & Gutman [1981]. 22.2 AXODES AND OPERATING PITCH CONES Spiral bevel gears perform rotation about intersected axes, and their axodes are two cones (Section 3.4). The line of tangency of these cones is the instantaneous axis of rotation in relative motion. In the case of standard spiral bevel gears, the gear axodes coincide with the pitch cones. Hypoid gears perform rotation about crossed axes, the relative motion is a screw motion, and instead of the instantaneous axis of rotation we have to consider the in- stantaneous screw axis s–s (Section 3.5). The gear axodes are two hyperboloids of revolution that are in tangency along the axis of screw motion s–s (Fig. 3.5.1). The hypoid pinion–gear axodes (the hyperboloids of revolution) perform in relative motion rotation about and translation along s–s. The concept of axodes of hypoid gears has found a limited application in design and is used merely for visualization of relative velocity. The main reason for this is that the location of axodes is out of the zone of meshing of hypoid gears. The design of blanks of hypoid gears is meant to determine operating pitch cones instead of hyperboloids of revolution, the hypoid gear axodes. The operating pitch cones (Fig. 22.2.1) must satisfy the following requirements: (a) The axes of the pitch cones form the prescribed crossing angle γ between the axes of rotation (usually, γ = 90 ◦ ). 679 P1: JDW CB672-22 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 2:3 680 Hypoid Gear Drives Figure 22.2.1: Operating pitch cones of hypoid gears. (b) The shortest distance E between the axes of the pitch cones is equal to the prescribed value of the hypoid gear drive. (c) The pitch cones are in tangency at the prescribed point P that is located in the zone of meshing of the pinion–gear tooth surfaces. (d) The relative (sliding) velocity at point P is directed along the common tangent to the “helices” of contacting pitch cones. The term “helix” is used to denote a curve obtained by intersection of the tooth surface by the pitch cone. 22.3 TANGENCY OF HYPOID PITCH CONES A cone is represented in coordinate system S i by the equations (Fig. 22.3.1) x i = u i sin γ i cos θ i y i = u i sin γ i sin θ i (i = 1, 2) z i = u i cos γ i (22.3.1) Figure 22.3.1: Operating pitch cone and its param- eters. P1: JDW CB672-22 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 2:3 22.3 Tangency of Hypoid Pitch Cones 681 where (u i , θ i ) are the surface coordinates (the Gaussian coordinates). The surface unit normal is represented by the equations n i = N i |N i | , N i = ∂r i ∂u i × ∂r i ∂θ i . (22.3.2) Equations (22.3.1) and (22.3.2) yield (providing u i sin γ i = 0) n i = [cos θ i cos γ i sin θ i cos γ i − sin γ i ] T . (22.3.3) To derive the equations of tangency of the pitch cones at the pitch point P , we represent the pitch cones in the fixed coordinate system S f . The location and orientation of coordinate systems S 1 and S 2 with respect to S f are shown in Fig. 22.4.1. Coordinate transformation from S 1 and S 2 to S f allows us to represent in coordinate system S f the pitch cones of the pinion and the gear and their unit normals by the following vector functions: r (1) f (u 1 ,θ 1 ) =   r 1 cos θ 1 r 1 sin θ 1 r 1 cot γ 1 − d 1   (22.3.4) n (1) f (θ 1 ) =   cos γ 1 cos θ 1 cos γ 1 sin θ 1 −sinγ 1   (22.3.5) r (2) f (u 2 ,θ 2 ) =   r 2 cos θ 2 + E −r 2 cot γ 2 + d 2 r 2 sin θ 2   (22.3.6) n (2) f (θ 2 ) =   −cosγ 2 cos θ 2 −sinγ 2 −cosγ 2 sin θ 2   . (22.3.7) Here, d i (i = 1, 2) determines the location of the apex of the pitch cone. The pitch cones are in tangency at the pitch point P , and the equations of tangency are r (1) f (u 1 ,θ 1 ) = r (2) f (u 2 ,θ 2 ) = r (P) f (22.3.8) n (1) f (θ 1 ) = n (2) f (θ 2 ) = n (P) f (22.3.9) where r (P) f and n (P) f are the position vector and the common normal to the pitch cones at pitch point P. The mating pitch cones are located above and below the pitch plane. Therefore, their surface unit normals have opposite directions at P and the coincidence of the surface unit normals is provided with the negative sign in Eq. (22.3.9). Vector equations (22.3.8) and (22.3.9) provide the six scalar equations r 1 cos θ 1 = r 2 cos θ 2 + E = x (P) f (22.3.10) r 1 sin θ 1 =−r 2 cos θ 2 + d 2 = y (P) f (22.3.11) r 1 cot γ 1 − d 1 = r 2 sin θ 2 = z (P) f , (22.3.12) P1: JDW CB672-22 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 2:3 682 Hypoid Gear Drives where r i = u i sin γ i is the radius of the pitch cone at P, and cos γ i cos θ i =−cos γ 2 cos θ 2 = n (P) xf (22.3.13) cos γ i sin θ i =−sin γ 2 = n (P) yf (22.3.14) −sinγ 1 =−cos γ 2 sin θ 2 = n (P) zf . (22.3.15) Only two equations of the equation system (22.3.13) to (22.3.15) are independent because |n (1) f |=|n (2) f |=1. Eliminating cos θ i and sin θ i , we obtain after some transformation the following equa- tions: r 1 r 2 = (E/r 2 ) cosγ 1  cos 2 γ 1 − sin 2 γ 2 − cos γ 1 cos γ 2 (22.3.16) d 1 =− r 2 cos γ 2 sin γ 1 + E cos γ 1 cot γ 1  cos 2 γ 1 − sin 2 γ 2 (22.3.17) d 2 = r 2 cos γ 2 sin γ 2 − E sin γ 2  cos 2 γ 1 − sin 2 γ 2 (22.3.18) x (P) f = E − r 2  cos 2 γ 1 − sin 2 γ 2 cos γ 2 (22.3.19) y (P) f = r 2 tan γ 2 − E sin γ 2  cos 2 γ 1 − sin 2 γ 2 (22.3.20) z (P) f = r 2 sin γ 1 cos γ 2 (22.3.21) n (P) xf =  cos 2 γ 2 − sin 2 γ 1 (22.3.22) n (P) yf =−sin γ 2 (22.3.23) n (P) zf =−sin γ 1 . (22.3.24) The derived equations are the basis for the design of hypoid pitch cones (see Section 22.5). 22.4 AUXILIARY EQUATIONS The plane of tangency of the pitch cones is determined as the plane that passes through the cone apexes O 1 and O 2 and the pitch point P (Fig. 22.4.1). Unit vectors τ (1) and τ (2) represent the generatrices of the pitch cones that lie in the pitch plane and intersect each other at pitch point P . For further derivations we use the concept of the tooth longitudinal shape and the sliding velocity at the pitch point. [...]... (c) (c) (c) m41 = m 43 · m21 = (−1) N3 N4 · (−1) (N3 )(N 1 ) N1 = N2 (N 4 )(N 2 ) ( 23. 2.5) Equations ( 23. 2 .4) and ( 23. 2.5) yield 4 (c) = 1 − m41 ωc ( 23. 2.6) The ratio ( 4 /ωc ) represents the ratio between the angular velocities of link 4 (gear 4) and link c (carrier c) of the planetary gear train The negative or positive signs of the ratio ( 4 /ωc ) indicate that the rotation of gear 4 and the carrier... 23. 2 .3 The planetary mechanism shown in Fig 23. 2 .3 is designed with two pairs of gears, 1–2 and 3 4, being in internal meshing Gear 1 is fixed, planet gears 2 and 3 have a joint Figure 23. 2 .3: Planetary gear train with internal meshing of gears P1: JXR CB672- 23 CB672/Litvin CB672/Litvin-v2.cls February 27, 20 04 2:6 23. 2 Gear Ratio 701 shaft and are mounted on carrier c Due to internal meshing of gears... m41 is positive This statement is correct for both planetary gear mechanisms of Figs 23. 2.2 and 23. 2 .3 (c) The gear ratio m 14 is negative for a planetary gear mechanism with mixed-type meshing of gears 1–2 and 3 4, as the combination of internal and external meshing of contacting gears In this case, a large reduction of angular velocity 4 cannot be obtained Planetary Gear Train of Fig 23. 2 .4 The gear. .. driving links is determined from the equation 4 − ωc (c) = m41 −ωc ( 23. 2.7) (c) The gear ratio m41 of the inverted mechanism is determined as follows: (c) (c) (c) m41 = m 43 · m21 = (+1) N3 N4 · (+1) N1 (N3 )(N 1 ) = N2 (N 4 )(N 2 ) ( 23. 2.8) Then we obtain (N3 )(N 1 ) 4 =1− ωc (N 4 )(N 2 ) ( 23. 2.9) Large reduction of angular velocity of driven gear 4 with respect to angular velocity (c) of carrier... The lower and upper signs in Eqs ( 23. 2.2) and ( 23. 2 .3) correspond to planetary gear trains shown in Figs 23. 2.1(a) and (b), respectively Planetary Mechanism of Fig 23. 2.2 The planetary mechanism shown in Fig 23. 2.2 is formed by two pairs of gears that are in external meshing Gear 1 is the fixed one The relation between the angular velocities of gears 4 and 1 is ω 4 − ωc (c) = m41 −ωc ( 23. 2 .4) where (c)... mechanism (c) (c) (c) m 13 = m12 · m 23 = (−1) N2 N1 · (+1) N3 N2 = (−1) N3 N1 ( 23. 2.11) Equations ( 23. 2.10) and ( 23. 2.11) yield N1 ωc = ω1 N1 + N3 ( 23. 2.12) The reduction of angular velocity ωc is obtained wherein gear 1 and carrier c are the driving and driven links, respectively Bevel Gear Differential of Fig 23. 2.5 The bevel gear differential shown in Fig 23. 2.5 is applied for addition or subtraction... train is applied in helicopter transmissions and other cases of design Gear 3 (called the ring gear) is fixed and the carrier c carries n planet gears (n = 5 is shown in Fig 23. 2 .4) The relation between angular velocities ω1 of the sun gear and the carrier c is based on the following equation: ω1 − ωc (c) = m 13 − ωc ( 23. 2.10) (c) Here, m 13 is the gear ratio of the inverted mechanism (c) (c) (c) m 13 = m12... 23. 2 GEAR RATIO A planetary gear mechanism has at least one gear whose axis is movable in the process of meshing Planetary Mechanisms of Figs 23. 2.1 (a) and (b) Figures 23. 2.1(a) and (b) represent two simple planetary gear mechanisms formed by two gears 1 and 2 that are in external or internal meshing, respectively, and a carrier c on which the gear with the movable axis is mounted Gear 1 is fixed and. .. planet gear is not necessary in a kinematic sense The discussed mechanism is a coaxial differential – the axes of rotation of gears 1, 3, and the carrier coincide, but they may rotate with different angular velocities ω1 , 3 , and ωc It is obvious that gears 1 and 3 have equal numbers of teeth N 1 and N 3 Angular velocities of the links of the differential are related as (c) m31 = 3 − ωc ω1 − ωc ( 23. 2. 13) ... four movable P1: JXR CB672- 23 CB672/Litvin CB672/Litvin-v2.cls February 27, 20 04 2:6 702 Planetary Gear Trains Figure 23. 2 .4: Schematic representation of planetary gear train applied in helicopter transmissions links: (a) the carrier c, (b) two sun gears 1 and 3, and (c) the planet gear 2 which is mounted on carrier c Usually, the discussed differential contains two planet gears mounted on the carrier, . γ 1 ] T (22 .4. 2) τ (2) = O 2 P |O 2 P | = ∂r (2) f ∂u 2      ∂r (2) f ∂u 2      = [sin γ 2 cos θ 2 − cos γ 2 sin γ 2 sin θ 2 ] T . (22 .4 .3) Using Eqs. (22 .4. 1), (22 .4. 2), and (22 .4 .3) , we. February 27, 20 04 2:1 21.9 Example of Design and Optimization of a Spiral Bevel Gear Drive 6 73 Figure 21.9.6: (a) and (b) bearing contact for design cases 1b and 1c, respectively, and (c) function. 20 04 2 :3 22.5 Design of Hypoid Pitch Cones 689 Here, N c = N 2 sin γ 2 (22.5.17) where N c and N 2 , are the numbers of teeth of the crown gear and the hypoid gear, respectively; γ 2 is the gear