P1: JsY CB672-19 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 1:28 19.13 Prospects of New Developments 613 where E hg and E wg are the center distances between the hob and the worm-gear and between the worm and the worm-gear, respectively; r = r ph −r pw ; r ph and r pw are the radii of pitch cylinders of the hob and the worm, respectively; γ = λ w − λ h ; λ w and λ h are the lead angles of the worm and the hob, respectively. For instance, in the case of an involute worm-gear drive the hob and the worm are two involute helicoids. In the case of K worm-gear drives (see Section 19.7), the hob and the worm are generated by a cone with the same profile angle. Figure 19.13.2 shows the output of TCA for a K worm-gear drive wherein the worm- gear has been generated by an oversized hob [Seol & Litvin, 1996]. The path of contact is oriented across the worm-gear surface and is located around the center of the worm- gear surface [Fig. 19.13.2(a)]. The function of transmission errors is of a parabolic type [Fig. 19.13.2(b)]. For some cases of misalignment, an oversized hob that is too small fails to provide a continuous function of transmission errors. In the opinion of the authors of this book, localization of the bearing contact by double crowning of the worm is the approach with much greater potential. P1: GDZ/SPH P2: GDZ CB672-20 CB672/Litvin CB672/Litvin-v2.cls April 15, 2004 16:11 20 Double-Enveloping Worm-Gear Drives 20.1 INTRODUCTION The invention of the double-enveloping worm-gear drive is a breathtaking story with two dramatic characters, Friedrich Wilhelm Lorenz and Samuel I. Cone, each acting in distant parts of the world – one in Germany and the other in the United States [Litvin, 1998]. The double-enveloping worm-gear drive was invented by both Cone and Lorenz independently, and we have to credit them both for it [Litvin, 1998]. The invention of Samuel I. Cone in the United States has been applied by a company that bears the name of the inventor, known by the name Cone Drive. The invented gear drive is a significant achievement. The special shape of the worm increases the number of teeth that are simultaneously in mesh and improves the con- ditions of force transmission. The conditions of lubrication and the efficiency of the invented drive (in comparison with a worm-gear drive with a cylindrical worm) are substantially better due to the special shape of lines of contact between the worm and gear surfaces (see below). The theory of double-enveloping worm-gear drives has been the subject of intensive research by many scientists. This chapter is based on the work by Litvin [1994]. We consider in this chapter the Cone double-enveloping worm-gear drive. 20.2 GENERATION OF WORM AND WORM-GEAR SURFACES Worm Generation The worm surface is generated by a straight-lined blade (Fig. 20.2.1). The blade per- forms rotational motion about axis O b with the angular velocity Ω (b) = dΨ b /dt, while the worm rotates about its axis with the angular velocity Ω (1) = dΨ 1 /dt; ψ b and ψ 1 are the angles of rotation of the blade and the worm in the process for gener- ation (Fig. 20.2.2). The shortest distance between the axes of rotation of the blade and the worm is E c . The generating lines of the blade in the process of generation keep the direction of tangents to the circle of radius R o . The directions of rotation shown in Figs. 20.2.1 and 20.2.2 correspond to the case of generation of a right-hand worm. 614 P1: GDZ/SPH P2: GDZ CB672-20 CB672/Litvin CB672/Litvin-v2.cls April 15, 2004 16:11 20.2 Generation of Worm and Worm-Gear Surfaces 615 Figure 20.2.1: Worm generation. Worm-Gear Generation The generation of the worm-gear is based on simulation of meshing of the worm and the worm-gear in the process of worm-gear generation. A hob identical to the generated worm is in mesh with the worm-gear being generated on the cutting machine. The axes of rotation of the hob and the worm-gear are crossed; the shortest distance E between the axes is the same as in the designed worm-gear drive; the ratio m 21 between the angular velocities of the hob (worm) and the worm-gear is also the same. Here, m 21 = ω (2) ω (1) = N 1 N 2 (20.2.1) where N 1 and N 2 are the numbers of worm threads and gear teeth. Figure 20.2.2: Coordinate systems ap- plied for worm generation. P1: GDZ/SPH P2: GDZ CB672-20 CB672/Litvin CB672/Litvin-v2.cls April 15, 2004 16:11 616 Double-Enveloping Worm-Gear Drives Figure 20.2.3: Illustration of (a) applied coordinate systems S 1 , S 2 , and S f ; and (b) schematic of double-enveloping worm-gear drive. Applied Coordinate Systems We limit the discussion to the case of an orthogonal worm-gear drive, with a crossing angle of 90 ◦ . Moveable coordinate systems S 1 and S 2 are rigidly connected to the worm and the worm-gear, respectively (Fig. 20.2.3); S f is a fixed coordinate system that is rigidly connected to the housing of the worm-gear drive. In the process of meshing the worm rotates about the z 1 axis, while the gear rotates about the y 2 axis. Worm-Gear Surface The analytical determination of the worm-gear surface  2 is based on the following ideas: (i) Consider that the worm (hob) surface  1 is known. (ii) Using the method of coordinate transformation, we can derive a family of surfaces  1 that is represented in coordinate system S 2 . (iii) Surface  2 is the envelope to the family of surfaces  1 . Obviously,  1 and  2 are in line contact at every instant. P1: GDZ/SPH P2: GDZ CB672-20 CB672/Litvin CB672/Litvin-v2.cls April 15, 2004 16:11 20.2 Generation of Worm and Worm-Gear Surfaces 617 Figure 20.2.4: Schematic of (a) unmodified and (b) modified gear drives. Unmodified and Modified Gearing The conjugation of surfaces  1 and  2 requires that the hob surface be the same as the worm surface. The principle of conjugation will not be infringed if the same values of m b1 and E c are used for generation of the worm and the hob. Here, m b1 = dψ b dt ÷ dψ 1 dt (20.2.2) is the cutting ratio. However, m b1 and E c may differ from m 21 and E given for the designed worm-gear drive. Henceforth, we differentiate two types of gearing for double-enveloping worm-gear drives: (i) unmodified gearing when m b1 = m 21 , and E c = E; and (ii) modified gearing when E c = E (E c > E). The cutting ratio m b1 for the modified gearing may be chosen to be equal to m 21 or to differ from it. Surfaces  1 and  2 are conjugated in both cases, for unmodified and modified gearings, but there are some advantages when the modified gearing is used. Consider that E c = E is chosen. The decision regarding how to choose m b1 will affect the radius ρ of the throat of the worm (hob) and other worm dimensions. The following discussion provides an explanation of this statement. The unmodified and modified gearings are shown in Figs. 20.2.4(a) and 20.2.4(b), respectively. The gear ratio for an orthogonal drive satisfies the equation m 21 = ρ tan λ E − ρ = N 1 N 2 . (20.2.3) The cutting ratio m b1 may be determined considering an imaginary worm-gear drive that is represented in Fig. 20.2.4(b); the blade for worm cutting is considered as the worm-gear tooth. Then, we obtain m b1 = ρ ∗ tan λ ∗ E c − ρ ∗ . (20.2.4) Here, λ and λ ∗ are the worm lead angles at M and M ∗ . P1: GDZ/SPH P2: GDZ CB672-20 CB672/Litvin CB672/Litvin-v2.cls April 15, 2004 16:11 618 Double-Enveloping Worm-Gear Drives According to the existing practice of design, the lead angle at M is chosen to be the same for both designs. We consider as given N 1 , N 2 , E, ρ, and E c . Our goal is to determine ρ ∗ and m b1 . Equations (20.2.3) and (20.2.4) with λ ∗ = λ yield m b1 (E c − ρ ∗ ) ρ ∗ = N 1 (E − ρ) ρ N 2 . (20.2.5) Equation (20.2.5) just relates parameters m b1 and ρ ∗ , and the solution for m b1 and ρ ∗ is not unique. We may consider the two following cases: (i) The cutting ratio m b1 is chosen to be equal to m 21 . Then, we obtain the following solution for ρ ∗ : ρ ∗ = E c E ρ. (20.2.6) This means that the worm of the modified worm-gear drive will have an increased throat radius ρ ∗ and other dimensions in comparison with the worm of the un- modified drive. The axial diametral pitch of the modified worm is P ∗ = ρ ρ ∗ P. (20.2.7) (ii) The radius of the throat is chosen to be the same for both designs. Thus, ρ ∗ = ρ and we obtain that m b1 = N 1 (E − ρ) N 2 (E c − ρ) (20.2.8) P ∗ = P . (20.2.9) The dimensions of the worm are the same for both designs, but m b1 = m 21 . There are other possible options for m b1 and ρ ∗ in addition to those discussed. 20.3 WORM SURFACE EQUATIONS We set up three coordinate systems for derivation of the worm surface (Fig. 20.2.2); S 1 and S b rigidly connected to the worm and the blade, respectively, and the fixed coordi- nate system S 0 rigidly connected to the machine for worm generation. The generating straight line AB is represented in S b by the equations (Fig. 20.3.1) x b = u cos δ + R o sin δ, y b = 0, z b = u sin δ − R o cos δ (20.3.1) where the variable parameter u determines the location of a current point on the blade, and δ = arcsin  R o R  − s p 2R . (20.3.2) Here, R is the radius of the reference circle where the thickness of the blade is given. P1: GDZ/SPH P2: GDZ CB672-20 CB672/Litvin CB672/Litvin-v2.cls April 15, 2004 16:11 20.3 Worm Surface Equations 619 Figure 20.3.1: Blade representation. The worm surface  1 is generated as the family of straight lines and is a ruled surface. We may derive the equations of the worm surface using Eqs. (20.3.1) and the coordinate transformation from S b to S 1 . Then we obtain x 1 = cos ψ 1 [u cos(δ + ψ b ) + R o sin(δ + ψ b ) − E c ] y 1 = sin ψ 1 [u cos(δ + ψ b ) + R o sin(δ + ψ b ) − E c ] z 1 = u sin(δ + ψ b ) − R o cos(δ + ψ b ) (20.3.3) where ψ b = ψ 1 m b1 . The generalized parameter ψ ≡ ψ 1 and parameter u represent the surface coordinates (Gaussian coordinates). Equations (20.3.3) with the fixed value of ψ represent on  1 the u-coordinate line, the generating straight line. Equations (20.3.3) with the fixed parameter u represent in  1 the ψ-coordinate line, that is, a spatial curve. This curve can be obtained by intersection of  1 by a torus. The axial section of the torus is the circle of radius (u 2 + R 2 o ) 1/2 . Equations (20.3.3) work for the modified and unmodified worms. For the case of the unmodified worm-gear drive, we have to take in these equations E c = E, and m b1 = m 21 . The surface normal is represented by vector equation N 1 = ∂r 1 /∂u × ∂r 1 /∂ψ, which yields N x1 = um b1 sin ψ 1 − sin(δ + ψ b ) cos ψ 1 [u cos(δ + ψ b ) + R o sin(δ + ψ b ) − E c ] = um b1 sin ψ 1 − x 1 sin(δ + ψ b ) N y1 =−um b1 cos ψ 1 − sin(δ + ψ b ) sin ψ 1 [u cos(δ + ψ b ) + R o sin(δ + ψ b ) − E c ] =−um b1 cos ψ 1 − y 1 sin(δ + ψ b ) N z1 = cos(δ + ψ b )[u cos(δ + ψ b ) + R o sin(δ + ψ b ) − E c ]. (20.3.4) P1: GDZ/SPH P2: GDZ CB672-20 CB672/Litvin CB672/Litvin-v2.cls April 15, 2004 16:11 620 Double-Enveloping Worm-Gear Drives Surface (20.3.3) is an undeveloped one, because the surface normals along the generating line are not collinear (the orientation of the surface normal depends on u). 20.4 EQUATION OF MESHING We consider the meshing of surfaces  1 and  2 . Worm surface  1 may be generated as unmodified or modified. The worm and the gear perform rotational motions about crossed axes as shown in Fig. 20.2.3. Surface  2 is the envelope to the family of  1 that is represented in S 2 . The necessary condition of existence of an envelope (see Section 6.1) is represented by the equation of meshing, N 1 · v (12) 1 = f (u,ψ 1 ,φ) = 0. (20.4.1) The subscript “1” shows that vectors N 1 and v (12) 1 are represented in S 1 . Vector N 1 is the normal to  1 , and v (12) 1 is the sliding velocity that is determined in terms of con- stant parameters ω (1) , ω (2) , E, and m 21 , and varied parameter φ ≡ φ 1 , because v (12) 1 is represented in S 1 (see Section 2.1). Parameter φ is the generalized parameter of motion. We recall that angle φ 2 of rotation of worm-gear 2 is represented as φ 2 = m 21 φ 1 . (20.4.2) Vector N 1 is represented by Eqs. (20.3.4) in terms of varied surface parameters u and ψ 1 and constant parameters E c and m b1 . The designation f (u,ψ 1 ,φ) = 0 indicates the relation between the varied parameters. Using this relation, we are able to determine the lines of contact between  1 and  2 and represent the lines of contact in S 1 , S 2 , and S f . The equation of meshing is derived for two cases: unmodified and modified gearing. Unmodified Gearing We take in Eqs. (20.3.4) for the worm surface normal that m b1 = m 21 , and E c = E. Using Eq. (20.4.1), after transformations, we obtain u 2 [(1 −cos θ) cos(δ + ψ b ) +m 21 sin θ sin(δ + ψ b )] + u{R o [(1 −cos θ) sin(δ + ψ b ) −m 21 sin θ cos(δ + ψ b )] − E(1 − cos θ)[1 + cos 2 (δ + ψ b )]} + E cos(δ + ψ b )(1 −cos θ)[E − R o sin(δ + ψ b )] = 0 (20.4.3) where θ = ψ 1 − φ 1 . Equation (20.4.3) may be represented as 2 sin θ 2 (u 2 P + uQ+ M) = 0 (20.4.4) P1: GDZ/SPH P2: GDZ CB672-20 CB672/Litvin CB672/Litvin-v2.cls April 15, 2004 16:11 20.4 Equation of Meshing 621 where P = sin θ 2 cos(δ + ψ b ) +m 21 cos θ 2 sin(δ + ψ b ) (20.4.5) Q = R o  sin θ 2 sin(δ + ψ b ) −m 21 cos θ 2 cos(δ + ψ b )  − E sin θ 2  1 +cos 2 (δ + ψ b )  (20.4.6) M = E sin θ 2 cos(δ + ψ b )[E − R o sin(δ + ψ b )]. (20.4.7) Equation (20.4.4) is satisfied if at least one of the two following conditions is ob- served: (i) sin θ 2 = 0. (20.4.8) (ii) u 2 P + uQ+ M = 0. (20.4.9) This means that two types of contact lines may exist simultaneously on  1 : (i) a straight line (the generating line), and (ii) a spatial curve determined with Eq. (20.4.9). The existence on  1 of a contact line that coincides with the generat- ing line AB (Fig. 20.3.1) does not depend on the shape of the generating line. The contact line of type “i” will coincide with the generating line as well if the worm is generated by a curved blade. The existence of contact lines of type “ii” means that a part of surface  2 is generated as the envelope to the family of surfaces  1 . Modified Gearing The derivation of the equation of meshing in this case is also based on Eq. (20.4.1), but it is assumed that the worm surface is generated with E c = E. However, the cutting ratio m b1 may be equal to m 21 or may differ from it. The performed derivations yield the following equation of meshing when m b1 = m 21 : u 2 [(1 −cos θ) cos(δ + ψ b ) +m 21 sin θ sin(δ + ψ b )] + u{R o [(1 −cos θ) sin(δ + ψ b ) −m 21 sin θ cos(δ + ψ b )] − E c (1 −cos θ)[1 + cos 2 (δ + ψ b ) −(E − E c ) cos 2 (δ + ψ b )]} + cos(δ + ψ b )(E − E c cos θ)[E c − R o sin(δ + ψ b )] = 0 (20.4.10) where ψ b = m b1 ψ 1 . Taking in Eq. (20.4.10) E = E c , we obtain equation of meshing (20.4.3) for the unmodified gearing. P1: GDZ/SPH P2: GDZ CB672-20 CB672/Litvin CB672/Litvin-v2.cls April 15, 2004 16:11 622 Double-Enveloping Worm-Gear Drives 20.5 CONTACT LINES We consider the contact lines on worm surface  1 , on worm-gear surface  2 , and in the fixed coordinate system S f , respectively. Contact Lines on Σ 1 The contact lines on the worm surface  1 are represented by the equations r 1 = r 1 (u,ψ 1 ), f  u,ψ 1 ,φ (i )  = 0(i = 1, 2, ,n). (20.5.1) Equations (20.5.1) represent the worm surface and the equation of meshing, and these equations are considered simultaneously. The designation φ (i ) (i = 1, 2, ,n) indicates that the generalized parameter φ is fixed-in when the instantaneous contact line is con- sidered. Surface  1 is in tangency with  2 at every instant at two lines: one is the generating straight line, the other is the line of contact between surface  1 and those parts of surface  2 that are the envelope to the family of  1 . Contact Lines on Σ 2 The contact lines on  2 are represented by the equations r 2  u,ψ 1 ,φ (i )  = M 21 r 1  u,ψ 1  , f  u,ψ 1 ,φ (i )  = 0(i = 1, 2, ,n). (20.5.2) Matrix M 21 describes the coordinate transformation from S 1 to S 2 . Surface  2 is rep- resented by Eqs. (20.5.2) as the family of instantaneous contact lines. We may ex- pect that  2 is represented by two parts because two contact lines exist simultane- ously at every instant. In reality,  2 consists of three parts due to undercutting (see below). Contact Lines on the Surface of Action The totality of contact lines in coordinate system S f represents the surface of action, which we designate by  f . The surface of action is represented by the equations r f (u,ψ 1 ,φ) = M f 1 (φ)r 1 (u,ψ 1 ), f (u,ψ 1 ,φ) = 0. (20.5.3) Matrix M f 1 describes the coordinate transformation from S 1 to S f . 20.6 WORM-GEAR SURFACE EQUATIONS Using Eqs. (20.5.2), we may represent  2 in terms of three varied but related parameters (u,ψ 1 ,φ). We consider the cases of unmodified and modified gearings separately. [...]... equations (21 .3. 13) and (21 .3. 14) or (21 .3. 13) and (21 .3. 16) enables us to represent (a) 2 in two-parameter form as (a) (a) R2 (θg , 2 ) = r 2 (s g (θg , 2 ), θg , 2 ) (b) 2 Similar considerations for enable us to obtain (b) r 2 (λw , θg , 2 ) = M2g ( 2 ) r (b) (λw , θg ) g (b) (21 .3. 17) (b) ∂r ∂r 2 × 2 ∂λw ∂θg (21 .3. 18) (b) · ∂r 2 (b) = f 2g (λw , θg , 2 ) = 0 ∂ 2 (21 .3. 19) Equation of meshing (21 .3. 19)... (Fig 21 .3. 3) (Fig 21 .3. 3) (Fig 21 .3. 2) (Fig 21 .3. 2) (Fig 21 .3. 2) (Fig 21 .3. 2) (Fig 21 .3. 2) (Fig 21 .3. 2) [Eq (21 .3. 1)] (Fig 21 .3. 3) rotation of the head-cutter about the z g axis is necessary for the cutting or grinding process but does not affect the shape of gear tooth surfaces (a) (b) The surfaces 2 and 2 that represent the working part of the gear tooth surface and the fillet of a formate-cut gear. .. surface (a) 2 will be represented as follows: (a) r2 (s g , θg , 2 ) = M2g ( 2 )r (a) (s g , θg ) g (a) (a) (21 .3. 13) (a) ∂r ∂r 2 × 2 ∂s g ∂θg · ∂r 2 (a) = f 2g (s g , θg , 2 ) = 0 ∂ 2 (21 .3. 14) Here, 2 is the generalized parameter of motion; matrix M2g represents the coordinate transformation from S g to S 2 (Fig 21 .3. 2) and is given by M2g ( 2 ) = M2b2 Mb2 a 2 Ma 2 m2 Mm2 c 2 Mc 2 g where  Mc 2 g 1... 0 0  Sr 2 cos q2   Sr 2 sin q2     0   1  Mm2 c 2   cos ψc 2   sin ψc 2  =  0   0 − sin ψc 2 cos ψc 2 0 0 1 0 0  0 1 0  = 0 0 1   0 0 0 0 0  0 0   1 0   0 1   Ma 2 m2 (21 .3. 15) 0   E m2    − XB2    1 P1: GDZ/SPH CB6 72- 21 P2: GDZ CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 2: 1 21 .3 Derivation of Gear Tooth Surfaces 641   Mb2 a 2  sin γm2   0 ... parameters Sr 2 and q2 , which are called the radial distance and the basic cradle angle The installments of the head-cutter on the cradle for generation of right-hand and left-hand gears are shown in Figs 21 .3. 2( a) and 21 .3. 2( b), respectively Parameters XB2 , E m2 , XD2 , and γm2 represent the settings of a generated spiral bevel gear [Fig 21 .3. 2( c)] P1: GDZ/SPH CB6 72- 21 P2: GDZ CB6 72/ Litvin 636 CB6 72/ Litvin-v2.cls... coordinate system S c 2 The cradle and the gear perform related rotations about the z m2 axis and the z b2 axis, respectively, for the case of a generated spiral bevel gear Angles ψc 2 and 2 are related P1: GDZ/SPH CB6 72- 21 P2: GDZ CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 2: 1 21 .3 Derivation of Gear Tooth Surfaces 635 Figure 21 .3. 2: Coordinate systems applied for gear generation: (a) and (b) illustration... zero The solution of Eq (21 .2. 1) for 2 = 0 is obtained by application of the secant method [Press et al., 19 92] that is illustrated by Fig 21 .2 .3 Designations (i ) 2 (i = 1, 2, 3, ) (Fig 21 .2 .3) indicate the magnitude of 2 obtained in the process of P1: GDZ/SPH CB6 72- 21 P2: GDZ CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 2: 1 630 Spiral Bevel Gears Figure 21 .2. 2: (a) Representation of various... 6 32 Spiral Bevel Gears (1) Transformation of function 2 (φ1 ) into ficients b2 and b3 of function (21 .2 .3) Here, 2 2 (φ1 ) = −a 2 φ1 , | 2 (φ1 )|max = a 2 2 (φ1 ) is obtained by variation of coef− π π ≤ φ1 ≤ N1 N1 π N1 (21 .2. 4) 2 = (21 .2. 5) The variation of b2 and b3 is performed independently and is illustrated by Fig 21 .2. 4 Figure 21 .2. 4(a) illustrates variation of coefficient b3 of modified roll,... obtain Figure 21 .2. 4: Schematic representation of computational procedure for determination of coefficients b2 and b3 of modified roll P1: GDZ/SPH CB6 72- 21 P2: GDZ CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 2: 1 21 .3 Derivation of Gear Tooth Surfaces 633 coefficient a 3 = 0 in function (21 .2. 2) Function a 3 (b3 ) is determined from the output of TCA by variation of modified roll Figure 21 .2. 4(b) illustrates... formate-cut gear are represented as follows, respectively, (a) r2 (s g , θg ) = M2g r (a) (s g , θg ) g (21 .3. 22 ) (b) (21 .3. 23 ) r2 (λw , θg ) = M2g r(b) (λw , θg ) g Here, M2g = M2a 2 Ma 2 m2 Mm2 g (21 .3. 24 ) where  Mm2 g 1 0 0  0 1 0  = 0 0 1   0 0 0  H2   ±V2    0    1   Ma 2 m2 1 0 0  0 1 0  = 0 0 1   0 0 0 0    0   − XB2    1 . Figs. 20 .2. 1 and 20 .2. 2 correspond to the case of generation of a right-hand worm. 614 P1: GDZ/SPH P2: GDZ CB6 72- 20 CB6 72/ Litvin CB6 72/ Litvin-v2.cls April 15, 20 04 16:11 20 .2 Generation of Worm and. unmodified and modified gearings separately. P1: GDZ/SPH P2: GDZ CB6 72- 20 CB6 72/ Litvin CB6 72/ Litvin-v2.cls April 15, 20 04 16:11 20 .6 Worm -Gear Surface Equations 6 23 Figure 20 .6.1: Three parts of worm -gear. parameters η 2 and a applied for local synthesis. P1: GDZ/SPH P2: GDZ CB6 72- 21 CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 2: 1 21 .2 Basic Ideas of the Developed Approach 629 of 6–8 arcsec