Gear Geometry and Applied Theory Episode 2 Part 8 pps

P1: JXR CB672-17 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:58 17.7 Generation of Double-Crowned Pinion by a Worm 493 Figure 17.7.3: For illustration of axodes of worm, pinion, and rack-cutter. the rack-cutter corresponds to rotation of the pinion with angular velocity ω (p) . The relation between v 1 and ω (p) is defined as v 1 = ω (p) r p (17.7.3) where r p is the radius of the pinion pitch cylinder. Step 3: An additional motion of surface  c with velocity v aux along direction t−t of skew rack-cutter teeth (Fig. 17.7.3) is performed and this motion does not affect the generation of surface  σ . Vector equation v 2 = v 1 + v aux allows us to obtain velocity v 2 of rack-cutter  c in a direction that is perpendicular to the axis of the worm. Then, we may represent the generation of worm surface  w by rack-cutter  c considering that the rack-cutter performs translational motion v 2 while the worm is rotated with angular velocity ω (w) . The relation between v 2 and ω (w) is defined as v 2 = ω (w) r w (17.7.4) where r w is the radius of the worm pitch cylinder. P1: JXR CB672-17 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:58 494 New Version of Novikov–Wildhaber Helical Gears Figure 17.7.4: Contact lines L cσ and L cw corresponding to meshing of rack-cutter  c with pinion and worm surfaces  σ and  w , respectively. Worm surface  w is generated as the envelope to the family of rack-cutter surfaces  c . Step 4: The discussion above enables us to verify simultaneous generation of profile- crowned pinion tooth surface  σ and worm thread surface  w by rack-cutter surface  c . Each of the two generated surfaces  σ and  w are in line contact with rack-cutter surface  c . However, the contact lines L cσ and L cw do not coincide but rather intersect each other as shown in Fig. 17.7.4. Here, L cσ and L cw represent the lines of contact between  c and  σ , and between  c and  w , respectively. Lines L cσ and L cw are obtained for a chosen value of related parameters of motion between  c ,  σ , and  w . Point N of intersection of lines L cw and L cσ (Fig. 17.7.4) is the common point of tangency of surfaces  c ,  σ , and  w . Profile Crowning of Pinion Profile-crowned pinion tooth surface  σ has been previously obtained by using rack- cutter surface  c . Direct derivation of generation of  σ by the worm  w may be ac- complished as follows: (a) Consider that worm surface  w and pinion tooth surface  σ perform rotation between their crossed axes with angular velocities ω (w) and ω (p) . It follows from previous discussions that  w and  σ are in point contact and N is one of the instantaneous points of contact of  w and  σ (Fig. 17.7.4). (b) The concept of direct derivation of  σ by  w is based on the two-parameter enveloping process (see Section 6.10). The process of such enveloping is based on application of two independent sets of parameters of motion [Litvin & Seol, 1996]: (i) One set of parameters relates the angles of rotation of the worm and the pinion as m wp = ω (w) ω (p) = N p (17.7.5) where the number N w of worm threads is considered as N w = 1, and N p is the number of teeth of the pinion. (ii) The second set of parameters of motion is provided as a combination of two components: (1) translational motion s w of the worm that is performed P1: JXR CB672-17 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:58 17.7 Generation of Double-Crowned Pinion by a Worm 495 Figure 17.7.5: Schematic of generation: (a) without worm plunging; (b) with worm plunging. collinear to the axis of the pinion [Fig. 17.7.5(a)], and (2) small rotational motion of the pinion about the pinion axis determined as ψ p = s w p (17.7.6) where p is the screw parameter of the pinion. Analytical determination of a surface generated as the envelope to a two-parameter enveloping process is presented in Section 6.10. The schematic generation of  σ by  w is shown in Fig. 17.7.5(a) wherein the shortest center distance is shown as an extended one for the purpose of better illustration. In the process of meshing of  w and  σ , the worm surface  w and the profile-crowned pinion surface perform rotation about crossed axes. The shortest distance is executed as E wp = r p +r w . (17.7.7) Surfaces  w and  σ are in point tangency. Feed motion of the worm is provided as a screw motion with the screw parameter of the pinion. Designations in Fig. 17.7.5(a) indicate (1) M 1 and M 2 are points on pitch cylinders (these points do not coincide with each other because the shortest distance is illustrated as extended); (2) ω (w) and ω (p) are the angular velocities of the worm and profile-crowned pinion in their rotation about P1: JXR CB672-17 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:58 496 New Version of Novikov–Wildhaber Helical Gears crossed axes; (3) s w and ψ p are the components of the screw motion of the feed motion; and (4) r w and r p are the radii of pitch cylinders. Double Crowning of Pinion We have presented above the generation by a worm of a profile-crowned surface  σ of the pinion. However, our final goal is the generation by a worm of a double-crowned surface  1 of the pinion. Two approaches are proposed for this purpose: WORM PLUNGING. Additional pinion crowning (longitudinal crowning) is provided by plunging of the worm with respect to the pinion which is shown schematically in Fig. 17.7.5(b). Plunging of the worm in the process of pinion generation is performed as variation of the shortest distance between the axes of the grinding worm and the pinion. The instantaneous shortest center distance E wp (s w ) between the grinding worm and the pinion is executed as [Fig. 17.7.5(b)]: E wp (s w ) = E (0) wp − a pl (s w ) 2 . (17.7.8) Here, s w is measured along the pinion axis from the middle of the pinion; a pl is the parabola coefficient of the function a pl (s w ) 2 ; and E (0) wp is the nominal value of the shortest distance defined by Eq. (17.7.7). Plunging of the worm with observation of Eq. (17.7.8) provides a parabolic function of transmission errors in the process of mesh- ing of the pinion and the gear of the proposed new version of the Novikov–Wildhaber helical gear drive. MODIFIED ROLL OF FEED MOTION. Conventionally, the feed motion of the worm is pro- vided by observation of relation (17.7.6) between components s w and ψ p . For the purpose of pinion longitudinal crowning, the following function ψ p (s w ) is observed: ψ p (s w ) = s w p + a mr (s w ) 2 (17.7.9) where a mr is the parabola coefficient of the parabolic function in Eq. (17.7.9). Modified roll motion is provided to the worm instead of worm plunging. Application of func- tion ψ p (s w ) enables us to modify the pinion tooth surface and provide a parabolic function of transmission errors of the proposed gear drive. The derivation of double-crowned surface  1 of the pinion by application of both previously mentioned approaches is based on determination of  1 as a two-parameter enveloping process: Step 1: We consider that surface  w is determined as the envelope to the rack-cutter surface  c . The determination of  w is a one-parameter enveloping process. Step 2: Double-crowned surface  1 of the pinion is determined as an envelope of a two-parameter enveloping process by application of the following equations: r 1 (u w ,θ w ,ψ w , s w ) = M 1w (ψ w , s w )r w (u w ,θ w ) (17.7.10) N w · v (w1,ψ w ) w = 0 (17.7.11) N w · v (w1,s w ) w = 0. (17.7.12) P1: JXR CB672-17 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:58 17.8 TCA of a Gear Drive with a Double-Crowned Pinion 497 Here, (u w ,θ w ) are the worm surface parameters, and (ψ w , s w ) are the generalized pa- rameters of motion of the two-parameter enveloping process. Vector equation (17.7.10) represents the family of surfaces  w of the worm in coordinate system S 1 of the pin- ion. Equations (17.7.11) and (17.7.12) represent two equations of meshing. Vector N w is the normal to the worm tooth surface  w and is represented in system S w . Vector v (w1,ψ w ) w represents the relative velocity between the worm and pinion determined un- der the condition that parameter ψ w of motion is varied and the other parameter s w is held at rest. Vector v (w1,s w ) w is determined under the condition that parameter s w is varied and the other parameter of motion ψ w is held at rest. Both vectors of relative velocity are represented in coordinate system S w . Vector equations (17.7.10), (17.7.11), and (17.7.12) (considered simultaneously) determine a double-crowned pinion tooth surface obtained by a two-parameter enveloping process (see Section 6.10). 17.8 TC A OF A GEAR DRIVE WITH A DOUBLE-CROWNED PINION Simulation of meshing of a gear drive with a double-crowned pinion is investigated by application of the same algorithm discussed in Section 17.5 for a gear drive with a profile-crowned pinion and gear tooth surfaces. The TCA has been performed for the following cases: (1) The new version of the Novikov–Wildhaber helical gear drive. (2) The modified involute helical gear drive, whose design is based on the following ideas: (i) a pinion rack-cutter with a parabolic profile and a conventional gear rack- cutter with a straight profile are applied for the generation of the pinion and the gear, respectively; and (ii) the pinion of the gear drive is double-crowned. The applied design parameters are shown in Table 17.8.1 for both the new version of the Novikov–Wildhaber gear drive (case 1) and the modified involute helical gear Table 17.8.1: Design parameters Number of teeth of the pinion, N 1 17 Number of teeth of the gear, N 2 77 Module, m 5.08 mm Driving-side pressure angle, α d 25 ◦ Coast-side pressure angle, α c 25 ◦ Helix angle, β 20 ◦ Parameter of rack-cutter, b 0.7 Face width 90 mm Radius of the worm pitch cylinder, r w 98 mm Parabolic coefficient of pinion rack-cutter (a) , a c 0.016739 mm −1 Parabolic coefficient of gear rack-cutter (a) , a t 0.0155 mm −1 Parabolic coefficient of plunging (a) , a pl 0.00005 mm −1 Parabolic coefficient of pinion rack-cutter (b) , a c 0.016739 mm −1 Parabolic coefficient of gear rack-cutter (b) , a t 0.0mm −1 Parabolic coefficient of plunging (b) , a pl 0.0000315 mm −1 (a) Novikov–Wildhaber helical gear drive. (b) Modified involute helical gear drive. P1: JXR CB672-17 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:58 498 New Version of Novikov–Wildhaber Helical Gears (arc sec) (rad) φ φ Figure 17.8.1: Output of TCA for a gear drive wherein the pinion is generated by plunging of the grinding worm: (a) path of contact and (c) function of transmission errors for the new version of the Novikov–Wildhaber helical gear drive; (b) path of contact for the modified involute helical gear drive. drive (case 2). The same parabolic coefficient of profile crowning for the pinion rack- cutter a c has been used for both the new version of the Novikov–Wildhaber gear drive and the modified involute helical gear drive. The parabolic coefficient of longitudinal crowning a pl used in each case provides a limited error of 8 arcsec of the predesigned function of transmission errors for a gear drive without errors of alignment. Figures 17.8.1(a) and 17.8.1(b) show the path of contact for cases (1) and (2), respectively. Figure 17.8.1(c) shows the function of transmission errors for case (1). The function of transmission errors for case (2) is similar and also provides a maximum transmission error of 8 arcsec. The TCA output shows that a parabolic function of transmission errors is indeed obtained in the meshing of the pinion and the gear due to application of a double-crowned pinion. The approaches chosen for TCA cover application of (i) a disk-shaped tool (Sec- tion 17.6), (ii) a plunging worm (Section 17.7), and (iii) modified roll of feed motion P1: JXR CB672-17 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:58 17.8 TCA of a Gear Drive with a Double-Crowned Pinion 499 Figure 17.8.2: Influence of errors of alignment in the shift of the path of contact for a Novikov– Wildhaber helical gear drive wherein the pinion is generated by plunging of the generating worm: (a) with error E [70 µm]; (b) with error γ [3 arcmin]; (c) with error λ [3 arcmin]; (d) with γ + λ 1 = 0 arcmin. (Section 17.7). These approaches yield almost the same output of TCA. The simulation of meshing is performed for the following errors of alignment: (i) change of center distance E = 70 µm, (ii) change of shaft angle γ = 3 arcmin, (iii) error λ = 3 arcmin, and (iv) combination of errors γ and λ as γ + λ = 0. The results of TCA accomplished for the design parameters represented in Table 17.8.1 are as follows: (1) Figures 17.8.1(a) and 17.8.1(b) show that the paths of contact of aligned gear drives are oriented longitudinally in both cases of design Novikov–Wildhaber gears and modified helical gears. Deviation from the longitudinal direction is less for modified involute helical gear drives in comparison with the new version of the Novikov–Wildhaber helical gear drive. However, the advantage of the new Novikov–Wildhaber gear drive is the reduction of stresses (see Section 17.10). (2) Figures 17.8.2(a), 17.8.2(b), and 17.8.2(c) show the shift of the paths of contact caused by errors of alignment E, γ , and λ, respectively. The shift of paths of contact caused by γ may be compensated by correction λ 1 of the pinion (or λ 2 of the gear). Figure 17.8.2(d) shows that the location of the path of contact can be restored by correction of λ 1 of the pinion by taking γ + λ 1 = 0. This means that correction of λ 1 can be used for the restoration of the location of the path of contact. Correction of λ 1 or λ 2 may be applied in the process of generation of the pinion or the gear, respectively. P1: JXR CB672-17 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:58 500 New Version of Novikov–Wildhaber Helical Gears It was previously mentioned (see Section 17.5) that double crowning of the pinion provides a predesigned parabolic function of transmission errors. Therefore, linear func- tions of transmission errors caused by γ , λ, and other errors are absorbed by the predesigned parabolic function of transmission errors φ 2 (φ 1 ). The final function of transmission errors φ 2 (φ 1 ) remains a parabolic one. However, increase of the magni- tude of errors γ and λ may result in the final function of transmission errors φ 2 (φ 1 ) becoming a discontinued one. In such a case, the predesigned parabolic function φ 2 (φ 1 ) has to be of larger magnitude or it becomes necessary to limit the range of γ , λ, and other errors. 17.9 UNDERCUTTING AND POINTING The pinion of the drive is more sensitive to undercutting than the gear because the pinion has a smaller number of teeth. Undercutting Avoidance of undercutting is applied to pinion tooth surface  σ and is based on the following ideas: (i) The appearance of singular points on generated surface  σ is the warning that the surface may be undercut in the process of generation [Litvin, 1989]. (ii) Singular points on surface  σ are generated by regular points on the generating surface  c when the velocity of a contact point in its motion over  σ becomes equal to zero [Litvin, 1989; Litvin, 1994]: v (σ ) r = v (c) r + v (cσ) = 0. (17.9.1) (iii) Equation (17.9.1) and differented equation of meshing d dt [ f (u c ,θ c ,ψ σ )] = 0 (17.9.2) allow us to determine a function F (u c ,θ c ,ψ σ ) = 0 (17.9.3) that relates parameters u c ,θ c , and ψ σ at a point of singularity of surface  σ . The limitation of generating surface  c for avoidance of singularities of generated surface  σ is based on the following procedure: (1) Using equation of meshing f σ c (u c ,θ c ,ψ σ ) = 0 between the rack-cutter and the pinion, we may obtain in plane of parameters (u c ,θ c ) the family of contact lines of the rack-cutter and the pinion. Each contact line is determined for a fixed parameter of motion ψ σ . (2) The sought-for limiting line L [Fig. 17.9.1(a)] that limits the rack-cutter surface is determined in the space of parameters (u c ,θ c ) by simultaneous consideration of equations f σ c = 0 and F = 0 [Fig. 17.9.1(a)]. Then we can obtain the limiting P1: JXR CB672-17 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:58 17.9 Undercutting and Pointing 501 (mm) θ (mm) Figure 17.9.1: Contact lines L σ c and limiting line L: (a) in plane (u c , θ c ), and (b) on surface  c . line L on the surface of the rack-cutter [Fig. 17.9.1(b)]. The limiting line L on the rack-cutter surface is formed by regular points of the rack-cutter, but these points will generate singular points on the pinion tooth surface. Limitations of the rack-cutter surface by L enables us to avoid singular points on the pinion tooth surface. Singular points on the pinion tooth surface can be obtained by coordinate transformation of line L on rack-cutter surface  c to surface  σ . Pointing Pointing of the pinion means that the width of the topland becomes equal to zero. Figure 17.9.2(a) shows cross sections of the pinion and the pinion rack-cutter. Point A c of the rack-cutter generates the point A σ that is the limiting point of the cross section of the pinion tooth surface which is still free of singularities. Point B c of the rack-cutter generates point B σ of the pinion profile. Parameter s a indicates the chosen width of the pinion topland. Parameter α t indicates the pressure angle at point Q. Parameters h 1 and h 2 indicate the limitation of location of limiting points A c and B c of the rack-cutter profiles. Figure 17.9.2(b) shows functions h 1 (N 1 ) and h 2 (N 1 )(N 1 is the pinion tooth number) obtained for the following data: α d = 25 ◦ , β = 20 ◦ , parabola P1: JXR CB672-17 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:58 502 New Version of Novikov–Wildhaber Helical Gears Figure 17.9.2: Permissible dimensions h 1 and h 2 of rack-cutter: (a) cross sec- tions of pinion and rack-cutter; (b) func- tions h 1 (N 1 ) and h 2 (N 1 ). coefficient of pinion rack-cutter a c = 0.016739 mm −1 , s a = 0.3m, parameter s 12 = 1.0 [see Eq. (17.3.2)], and module m = 1 mm. Functions h 1 (N 1 ) and h 2 (N 2 ) are obtained as discussed in Section 15.8. 17.10 STRESS ANALYSIS Stress analysis and investigation of formation of bearing contact have been performed: (i) for the proposed new version of Novikov–Wildhaber, and (ii) for a gear drive with modified involute helical gears. The second type of gearing has been proposed by patent [Litvin et al., 2001c] and is formed by a double-crowned helical pinion and a conven- tional involute helical gear. The second type of gear drive has been predesigned with a parabolic function of transmission errors, similar to the function of transmission errors of the proposed version of Novikov–Wildhaber gear drives (see Section 17.8). The difference between the two types of gear drives that have been investigated is that the Novikov–Wildhaber gear drives are generated by two parabolic rack-cutters that [...]... 1) and the face -gear (i = 2) The axodes are two cones of semiangles γ1 and 2 that are determined with the equations (see Section 3.4) cot γ1 = m 12 + cos γ , sin γ cot 2 = 1 + m 12 cos γ m21 + cos γ = sin γ m 12 sin γ ( 18 .2. 2) P1: JXR CB6 72- 18 CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 1:5 18 .2 Axodes, Pitch Surfaces, and Pitch Point 511 Figure 18 .2. 1: Axodes and pitch cones where m21 = 1 m 12. .. dimensions L1 and L2 of face -gear P1: JXR CB6 72- 18 CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 1:5 522 Face -Gear Drives equation 0.5 2 2 r as − r bs ( 18. 6.11) r bs where ras and r bs are the radii of the addendum circle and base circle of the shaper (b) Investigation shows that for determination of singularities of 2 it is sufficient to take 2 = 0 or 3 = 0 instead of Eq ( 18. 6.4) Determinants 2 = 0 or... has a certain solution for the unknowns if the matrix   ∂rs ∂rs (2s ) −vs  ∂u s  ∂θs   ( 18. 6.4) A=   ∂ fs 2 ∂ fs 2 ∂ f s 2 dψs  − ∂u s ∂θs ∂ψs dt has the rank r = 2 Then, we obtain that 2 1 + 2 2 + 2 3 =0 ( 18. 6.5) P1: JXR CB6 72- 18 CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 1:5 520 Face -Gear Drives where i (i = 1, 2, 3) are three determinants obtained from matrix A The equality of... face -gear as shown in Fig 18. 5 .2, and  cos 2 − sin ψ 2  M2m =   0  sin 2 0 cos 2 0 0 1 0   0 0 0 1 0 0 ( 18. 5.9) The equation of meshing is determined as (see Section 6.1) ns · v(s 2) = f s 2 (u s , θs , ψs ) = 0 s ( 18. 5.10) (s 2) Computerized determination of relative velocity vs is based on the procedure presented in Section 2. 2 Designations in Figs 18. 5 .2( a) and 18. 5 .2( b) indicate the angle... f s 2 = 0 is satisfied at a point (u s , θs , ψs ), and at this point ∂ fs 2 = 0 ∂u s ( 18. 5.11) (ii) Then, equation f s 2 = 0 may be solved by function u s = u s (θs , ψs ) ∈ C 1 , ( 18. 5. 12) P1: JXR CB6 72- 18 CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 1:5 18. 6 Conditions of Nonundercutting of Face -Gear Tooth Surface and surface 2 519 may be represented as r2 (u s (θs , ψs ), θs , ψs ) = R2 (θs... face -gear teeth and the dimensions of the area of meshing (see Section 18. 3) The P1: JXR CB6 72- 18 CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 1:5 5 12 Face -Gear Drives Figure 18. 3.1: Face -gear generation relative motion at point P is pure rolling, and sliding and rolling at other points of the pitch line O M 18. 3 FACE -GEAR GENERATION The generation of a face -gear by a shaper is shown in Fig 18. 3.1... location and orientation of the instantaneous axes of rotation in the meshing of s , 2 , and 1 The instantaneous axes of rotation are designated as IAs 2 , IAs 1 , and IA 12 The subscripts “s 2, ” “s 1,” and “ 12 indicate that the respective meshings between “s ” and 2, ” “s ” and “1, ” and “1” and 2 are considered Angle γs that is formed between the shaper axis and IAs 2 is P1: JXR CB6 72- 18 CB6 72/ Litvin... The structure of a face -gear tooth is shown in Fig 18. 1.4(a) The surface of the tooth consists of two parts: (i) the working part formed by lines L2s of tangency of the shaper and the face -gear, and (ii) the fillet surface generated by the edge of the top of the 5 08 P1: JXR CB6 72- 18 CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 1:5 18. 1 Introduction 509 Figure 18. 1.1: Face -gear drive in 3D-space... Novikov–Wildhaber gear drive in comparison with the modified involute helical gear drive P1: JXR CB6 72- 18 CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 1:5 18 Face -Gear Drives 18. 1 INTRODUCTION A conventional face -gear drive is formed by an involute spur pinion and a conjugated face -gear (Fig 18. 1.1) Such a gear drive may be applied for transformation of rotation between intersected and crossed axes... shaper and the face -gear (Fig 18. 5 .2) , and (ii) fixed coordinate systems S a and S m Coordinate axis Oa xa passes through pitch point P (Fig 18. 5 .2) During the generation, the shaper and the face -gear perform rotations about axes z a and z m related as follows: ψs N2 = 2 Ns The family of shaper surfaces equation [Fig 18. 5 .2( b)] s ( 18. 5.5) is represented in coordinate system S 2 by the matrix r2 (u . γ 1 = m 12 + cos γ sin γ , cot γ 2 = m 21 + cos γ sin γ = 1 + m 12 cos γ m 12 sin γ ( 18 .2. 2) P1: JXR CB6 72- 18 CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 1:5 18 .2 Axodes, Pitch Surfaces, and. face -gear teeth and the dimensions of the area of meshing (see Section 18. 3). The P1: JXR CB6 72- 18 CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 1:5 5 12 Face -Gear Drives Figure 18. 3.1: Face -gear. of the shaper and the face -gear, and (ii) the fillet surface generated by the edge of the top of the 5 08 P1: JXR CB6 72- 18 CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 1:5 18. 1 Introduction