P1: JXR CB672-18 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 1:5 18.7 Pointing of Face-Gear Teeth Generated by Involute Shaper 523 Figure 18.7.2: Cross section profiles of face-gear and shaper in plane  2 .  2 is designated by “A” (Fig. 18.7.2). Point A has to be located on the addendum line of the face-gear, and therefore its location with respect to axis y a is determined by r ps − 1/P d (Fig. 18.7.2). The goal is determination of magnitude L 2 defined by distance l between planes  1 and  2 (Fig. 18.6.2). Figures 18.6.2 and 18.7.2 illustrate the procedure of derivation of magnitude  l and L 2 . The computation of L 2 is based on the following procedure: Step 1: Determination of pressure angle α of pointed teeth (Fig. 18.7.2). We use vector equation (Fig. 18.7.2) O ∗ a N + NM + MA = O ∗ a A. (18.7.1) (See the location of point O ∗ a in Fig. 18.6.2.) Here, O ∗ a A = r ps − 1 P d = N s − 2 2P d (18.7.2) where P d is the diametral pitch; point M is the point of tangency of profiles of the shaper and the face-gear in plane  2 (Fig. 18.7.2); |MA|=λ s ; |NM|=r bs θ s . Vector equation (18.7.1) yields two scalar equations in two unknowns α and λ s : r bs (cos α + θ s sin α) − λ s cos α = N s − 2 2P d (18.7.3) r bs (sin α − θ s cos α) − λ s sin α = 0. (18.7.4) Here, r bs = (N s /(2P d )) cos α 0 ; θ s = α − θ 0s ; θ 0s = π/(2N s ) − inv α 0 . Eliminating λ s ,we obtain the following equation for determination of α: α − sinα N s − 2 N s cos α 0 = π 2N s − inv α 0 . (18.7.5) The sought-for angle α is obtained by solving the nonlinear equation (18.7.5). P1: JXR CB672-18 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 1:5 524 Face-Gear Drives Step 2: Determination of magnitude L 2 (Fig. 18.6.2). Figure 18.7.2 yields O ∗ a I = r bs cos α = N s cos α 0 2P d cos α . (18.7.6) Then, we obtain (Fig. 18.6.2) L 2 = O ∗ a I tan γ s = N s cos α 0 2P d cos α tanγ s . (18.7.7) Knowing the magnitudes of L 1 and L 2 (Fig. 18.6.2), it becomes possible to design a face-gear of the gear drive that is free of undercutting and pointing. 18.8 FILLET SURFACE Two types of fillet surfaces might be provided: (i) those generated by the generatrix G of the addendum cylinder [Fig. 18.4.3(a)], and (ii) those generated by the rounded top of the shaper (Fig. 18.8.1). Case 1: Generation of the fillet by edge G [Fig. 18.4.3(a)]. Using Fig. 18.5.1, we represent edge G [Fig. 18.4.3(a)] in coordinate system S s by vector function r s (u s ,θ ∗ s ) where θ ∗ s =  r 2 as −r 2 bs  0.5 r bs , r as = r ps + 1.25 P d = N s + 2.5 2P d . (18.8.1) The fillet surface is represented in S 2 by the equation r 2 (u s ,ψ s ) = M 2s (ψ s )r s (u s ,θ ∗ s ). (18.8.2) Case 2: Generation of the fillet by the rounded top of the shaper. The fillet is generated as the envelope to the family of circles of radius ρ (Fig. 18.8.1). The investigation of bending stresses shows that application of a shaper with a rounded top reduces bending stresses on approximately 10% with respect to those obtained by application of an edged top shaper. Figure 18.8.1: Rounded top of the shaper tooth. P1: JXR CB672-18 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 1:5 18.9 Geometry of Parabolic Rack-Cutters 525 Figure 18.9.1: Illustration of rack-cutter profiles; (b) and (c) parabolic profiles of the shaper and pinion rack-cutters, respectively. 18.9 GEOMETRY OF PARABOLIC RACK-CUTTERS Basic Concept The second version of the geometry of face-gear drives is based on the following ideas Litvin et al. [2002b]: (i) Two imaginary rigidly connected rack-cutters designated as A 1 and A s are applied for generation of the pinion and the shaper, respectively. Designation A 0 indicates a reference rack-cutter with straight-line profiles (Fig. 18.9.1). (ii) Rack-cutters A 1 and A s are provided with mismatched parabolic profiles that de- viate from the straight-line profiles of reference rack-cutter A 0 . Figure 18.9.1(a) shows schematically an exaggerated deviation of A 1 and A s from A 0 . The parabolic profiles of rack-cutters A 1 and A s for one tooth side are shown schematically in Figs. 18.9.1(b) and 18.9.1(c). (iii) The tooth surfaces  1 and  s of the pinion and the shaper are determined as envelopes to the tooth surfaces of rack-cutters A 1 and A s , respectively. (iv) The tooth surfaces of the face-gear  2 are generated by the shaper and are de- termined by a sequence of two enveloping processes wherein (a) the parabolic rack-cutter A s generates the shaper, and (b) the shaper generates the face-gear. The face-gear tooth surface  2 may also be ground (or cut) by a worm (hob) of a special shape (see Section 18.14). (v) The pinion and face-gear tooth surfaces are in point contact at every instant because: (i) rack-cutters A 1 and A s are mismatched [Fig. 18.9.1(a)] due to application of two P1: JXR CB672-18 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 1:5 526 Face-Gear Drives different parabola coefficients, and (ii) the pinion and the shaper are provided with a different number of teeth. Figures 18.9.1(b) and 18.9.1(c) show schematically the profiles of the rack-cutters of the pinion and the shaper, respectively. Application of both items, (i) and (ii), provides more freedom for observation of the desired dimensions of the instantaneous contact ellipse and for the predesign of a parabolic function of transmission errors. (vi) An alternative method of generation of face-gears is based on application of a worm of a special shape, which might be applied for grinding or cutting (Fig. 18.1.3). Grinding enables us to harden the tooth surfaces and to increase the permissible contact stresses. It is shown below that the derivation of the worm thread surface is based on simultaneous meshing of the shaper with the face-gear and the worm (see Section 18.14). Reference and Parabolic Rack-Cutters Reference rack-cutter A 0 has straight-line profiles [Fig. 18.9.1(a)]. Parabolic rack-cutters designated as A s and A 1 are in mesh with the shaper and the pinion. Parabolic profiles of A s and A 1 deviate from straight-line profiles of A 0 . Coordinate systems S q and S r are applied for derivation of equations of shaper rack- cutter A s . Parameters u r and parabola coefficient a r determine the parabolic profile of rack-cutter A s [Fig. 18.9.1(b)]. Respectively, coordinate systems S k and S e are applied for derivation of equations of rack-cutter A 1 . Parameters u e and parabola coefficient a e determine the parabolic profile of rack-cutter A 1 [Fig. 18.9.1(c)]. Origins O q and O k of coordinate systems S q and S k , respectively [Figs. 18.9.1(b) and 18.9.1(c)], coincide, and their location is determined by parameter f d . The profiles of the rack-cutter are considered for the side with profile angle α d [Fig. 18.9.1(a)]. The design parameters of reference rack-cutter A 0 [Fig. 18.9.1(a)] are w 0 , s 0 , and α d . Taking into account that w 0 + s 0 = p = π P (18.9.1) we obtain s 0 = p 1 +λ = π (1 +λ)P ; w 0 = λp 1 +λ = λπ (1 +λ)P . (18.9.2) Here, λ = w 0 /s 0 , and p and P are the circular and diametral pitches, respectively. The tooth surface of rack-cutter A s is represented in coordinate system S r [Fig. 18.9.1(a)] as r r (u r ,θ r ) =       (u r − f d ) sin α d −l d cos α d − a r u 2 r cos α d (u r − f d ) cos α d +l d sin α d + a r u 2 r sin α d θ r 1       . (18.9.3) P1: JXR CB672-18 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 1:5 18.10 Derivation of Tooth Surfaces of Shaper and Pinion 527 Parameter θ r is measured along the z r axis. Parameter l d is shown in Fig. 18.9.1(a). Normal N r to the shaper rack-cutter is represented as N r (u r ) =    cos α d + 2a r u r sin α d −sin α d + 2a r u r cos α d 0    . (18.9.4) Similarly, we may represent vector function r e (u e ,θ e ) of pinion rack-cutter A 1 and normal N e (u e ). 18.10 SECOND VERSION OF GEOMETRY: DERIVATION OF TOOTH SURFACES OF SHAPER AND PINION Shaper Tooth Surface We apply for derivation of shaper tooth surface  s : (i) movable coordinate sys- tems S r and S s that are rigidly connected to the shaper rack-cutter and the shaper, and (ii) fixed coordinate system S n [Fig. 18.10.1(a)]. Rack-cutter A s and the shaper perform related motions of translation and rotation determined by (r ps ψ r ) and ψ r [Fig. 18.10.1 (a)]. Figure 18.10.1: For generation of shaper of pinion by rack-cutters: (a) generation of the shaper, (b) installation of pinion rack-cutter, and (c) generation of the pinion. P1: JXR CB672-18 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 1:5 528 Face-Gear Drives The shaper tooth surface  s is determined as the envelope to the family of rack-cutter surfaces A s considering simultaneously the following equations: r s ( u r ,θ r ,ψ r ) = M sr (ψ r )r r (u r ,θ r ) (18.10.1) N r (u r ) ·v (sb) r = f sr ( u r ,ψ r ) = 0. (18.10.2) Here, vector function r s ( u r ,θ r ,ψ r ) represents in S s the family of rack-cutter A s tooth surfaces; matrix M sr (ψ r ) describes coordinate transformation from S r to S s ; vector function N r (u r ) represents the normal to the rack-cutter A s [see Eq. (18.9.4)]; v (sb) r is the relative (sliding) velocity. Equation (18.10.2) (the equation of meshing) yields f sr (u r ,ψ r ) = x r N yr − y r N xr r ps N yr − ψ r = 0. (18.10.3) Finally, we represent the surface of the shaper by vector function r s (u r (ψ r ),ψ r ,θ r ) = R s (ψ r ,θ r ). (18.10.4) The normal to the shaper is represented in coordinate system S s as N s = ∂R s ∂ψ r × ∂R s ∂θ r . (18.10.5) Pinion Tooth Surface Movable coordinate systems S e and S 1 are rigidly connected to the pinion rack-cutter and the pinion, respectively [Figs. 18.10.1(b) and 18.10.1(c)]; S ∗ n is the fixed coordinate system. The installation angle β [Fig. 18.10.1(b)] is provided for the improvement of the bearing contact between the pinion and the face-gear (see Section 18.13). Derivations of pinion tooth surfaces are similar to those applied for derivation of shaper tooth surfaces and are based on the following procedure: Step 1: We obtain the family of pinion rack-cutters represented in coordinate system S 1 as r 1 ( u e ,θ e ,ψ e ) = M 1e (ψ e )r e (u e ,θ e ) (18.10.6) where matrix M 1e describes coordinate transformation from S e via S ∗ n to S 1 [Figs. 18.10.1(b) and 18.10.1(c)]. Step 2: Using the equation of meshing between the rack-cutter and the shaper, we obtain u e (ψ e ) = x e N ye − y e N xe r p1 N ye − ψ e . (18.10.7) Step 3: We represent the pinion tooth surfaces by vector function r 1 (u e (ψ e ),ψ e ,θ e ) = R 1 (ψ e ,θ e ). (18.10.8) P1: JXR CB672-18 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 1:5 18.11 Derivation of Face-Gear Tooth Surface 529 18.11 SECOND VERSION OF GEOMETRY: DERIVATION OF FACE-GEAR TOOTH SURFACE Preliminary Considerations The face-gear tooth surface is determined as the result of two enveloping processes wherein (i) a parabolic rack-cutter generates the shaper tooth surface (see Section 18.10), and (ii) the shaper generates the face-gear tooth surface. The second enveloping process is based on the algorithm presented in Section 18.5 wherein an involute shaper generates the face-gear tooth surface of the first version of geometry. Recall that the shaper tooth surface of the second version of geometry is represented in two-parameter form by vector function R s (ψ r ,θ r ) [see Eq. (18.10.4)]. The normal to the surface mentioned above is represented by vector function (18.10.5). Investigation of undercutting of surface  2 (of the second version of geometry) is based on the algorithm discussed in Section 18.6. Structure of Face-Gear Tooth Surface Σ 2 The type of a surface may be defined by the Gaussian curvature that represents the product of principal surface curvatures at the chosen surface point. Thus, the Gaussian curvature K at a surface point M is defined as K = K I K II (18.11.1) where K I and K II are the principal surface curvatures at M. The type of surface point (elliptical, parabolic, or hyperbolic) depends on the sign of Gaussian curvature K. Direct determination of Gaussian curvature for a surface represented by three, sometimes four, related parameters requires complex derivations and computations. The derivations and computations previously mentioned may be simplified using pro- posed relations between the curvatures of the generating and generated surfaces (see Chapter 8). Investigation shows that surface  2 has elliptical (K > 0) and hyperbolic (K < 0) points (Fig. 18.11.1). The common line of both sub-areas is the line of parabolic points. The dimensions of the area of surface elliptical points depend on the magnitude of the parabola coefficient a r of the shaper rack-cutter. Surface  2 of the first version of geometry contains only hyperbolic points. 18.12 DESIGN RECOMMENDATIONS The bending stresses in a face-gear drive depend on the unitless coefficient c = P d l = P d (L 2 − L 1 ). (18.12.1) (See the designations of L 2 and L 1 in Fig. 18.6.2.) Usually, the coefficient c is chosen as c = 10 for high-power transmissions. The coefficient c can be increased for face-gear drives by choosing a higher gear ratio and increasing the tooth number. This statement can be confirmed by the graphs shown in Fig. 18.12.1 for face-gear drives of the first type of geometry. P1: JXR CB672-18 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 1:5 530 Face-Gear Drives Figure 18.11.1: Areas of elliptical and hyperbolic points of face-gear tooth surface  2 for rack-cutter parabola coefficients (a) a r = 0.01 1/mm, (b) a r = 0.02 1/mm, and (c) a r = 0.03 1/mm. The investigation of the influence of coefficient c on the structure of the face-gear teeth is based on the following considerations: Assume that the outer radius L 2 is known (it has been determined from the conditions of avoidance of pointing). We are able to eliminate the portion of the tooth where the fillet exists (Figs. 18.6.2 and 18.11.1) by increasing the inner radius L 1 . This means that the coefficient c will be decreased [see Figure 18.12.1: Graphs of coefficient c for face-gears of the first type of geometry. P1: JXR CB672-18 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 1:5 18.13 Tooth Contact Analysis (TCA) 531 Figure 18.12.2: Illustration of influence of parabola coefficient a r and gear ratio m 2s on coefficient c. Eq. (18.12.1)]. However, observing a sufficient value of c enables us to obtain a more uniform structure, eliminating the weaker part of the face-gear tooth. Figure 18.12.2 shows the influence of the parabola coefficient a r of the parabolic profile of the rack-cutter and the gear ratio on the possible tooth length of the face-gear of the second type of geometry. Results of the investigation of undercutting and pointing are shown in Fig. 18.12.2, which represents the influence of gear ratio m 2s and parabola coefficient a r on the coefficient c represented in Eq. (18.12.1). 18.13 TOOTH CONTACT ANALYSIS (TCA) Tooth contact analysis is directed at simulation of meshing and contact of surfaces  1 and  2 and enables investigation of the influence of errors of alignment on transmission errors and the shift of bearing contact. The algorithm for simulation of meshing is based on equations that describe the continuous tangency of surfaces  1 and  2 and is presented in Section 9.4. Applied Coordinate Systems The following coordinate systems are applied for TCA: (a) coordinate system S f , rigidly connected to the frame of the face-gear drive [Fig. 18.13.1(a)]; (b) coordinate sys- tems S 1 [Fig. 18.13.1(a)] and S 2 [Fig. 18.13.2(b)], rigidly connected to the pinion and the face-gear respectively; and (c) auxiliary coordinate systems S d , S e , and S q , ap- plied for simulation of errors of alignment of the face-gear drive [Figs. 18.13.2(a) and 18.13.2(b)]. All misalignments are referred to the gear. Parameters E, B, and B cot γ determine the location of origin O q with respect to O f [Fig. 18.13.1(b)]. Here, E is the shortest distance between the pinion and the face-gear axes when the axes are crossed but not intersected. The location and orientation of coordinate systems S d and S e with respect to S q are shown in Fig. 18.13.2(a). The misaligned face-gear performs rotation about the z e axis [Fig. 18.13.2(b)]. P1: JXR CB672-18 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 1:5 532 Face-Gear Drives Figure 18.13.1: Coordinate systems applied for simu- lation of meshing, I. Figure 18.13.2: Coordinate systems ap- plied for simulation of meshing, II. [...]... CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 19 .2 Pitch Surfaces and Gear Ratio Figure 19 .2. 2: Velocity polygon for righthand worm -gear drive Figure 19 .2. 3: Operating pitch cylinders for left-hand worm -gear drive 1 :28 551 P1: JsY CB6 72- 19 CB6 72/ Litvin 5 52 CB6 72/ Litvin-v2.cls February 27 , 20 04 1 :28 Worm -Gear Drives with Cylindrical Worms Figure 19 .2. 4: Velocity polygon for lefthand worm -gear drive It is... tangent is τ f ; and λ1 P1: JsY CB6 72- 19 CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 1 :28 19 .2 Pitch Surfaces and Gear Ratio 5 49 Figure 19 .2. 1: Illustration of (a) operating pitch cylinders of the worm and the gear, and (b) the worm helix is the lead angle on the worm operating pitch cylinder Figure 19 .2. 1(a) corresponds to the case when the worm and the worm -gear are right-handed The direction... (1) (2) vf − vf ( 12) · mf = vf · mf = 0 ( 19 .2. 6) because m f is perpendicular to τ f For further derivations we use Eqs ( 19 .2. 1), ( 19 .2. 4), and ( 19 .2. 5), which yield (o) (o) ω(1)r o sin λ1 = ω (2) R o sin γ − λ1 ( 19 .2. 7) (o) For the case where γ > λ1 , ω (2) is positive and ω (2) has the same direction as k2 (o) [Fig 19 .2. 1(b)] The negative sign for ω (2) when γ < λ1 indicates that in this case ω (2) ... 1, 2) (i = 1, 2) (i ) n f (u i , θi , φi ) (2) Continuous tangency of 1 and 2 (18.13.1) (18.13 .2) is represented by vector equations (1) (2) r f (u 1 , θ1 , φ1 ) − r f (u 2 , 2 , 2 ) = 0 (18.13.3) (1) (18.13.4) (2) n f (u 1 , θ1 , φ1 ) − n f (u 2 , 2 , 2 ) = 0 Here, (u i , θi ) (i = 1, 2) are the surface parameters of 1 and 2 , φ1 and 2 are the angles of rotation of the pinion and the face -gear. .. ( 19 .2. 1) where r f = Of P and E = Of O2 Velocities v(1) and v (2) lie in plane that is perpendicular to the x f axis; this plane is tangent to the operating pitch cylinders at point P Thus, v(1) · i f = v (2) · i f = 0 where i f is the unit vector of axis xf ( 19 .2. 2) P1: JsY CB6 72- 19 CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 550 1 :28 Worm -Gear Drives with Cylindrical Worms To determine the gear. .. Modulus E = 2. 068 × 108 mN/mm2 and Poisson’s Pinion Face -Gear 1 3 2 Figure 18.15.1: Three-pairs-of-teeth face -gear drive finite element model P1: JXR CB6 72- 18 CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 1:5 18.15 Stress Analysis S, Mises (Ave Crit.: 75%) +1.453e+06 +5.000e+05 +4.545e+05 +4. 091 e+05 +3.636e+05 +3.182e+05 +2. 727 e+05 +2. 273e+05 +1.818e+05 +1.364e+05 +9. 092 e+04 +4.546e+04 +8.056e+00... following gear ratio: (o) m21 = (o) r o sin λ1 (o) R o sin γ + λ1 ( 19 .2. 9) The magnitude of λ1 is considered as a positive value The derivations yield that for the chosen direction of ω (1) (Fig 19 .2. 3), vector ω (2) is opposite to k2 The velocity polygon is shown in Fig 19 .2. 4 In the most common case, the crossing angle γ is 90 ◦ and ro (o) m21 = tan λ1 ( 19 .2. 10) Ro P1: JsY CB6 72- 19 CB6 72/ Litvin CB6 72/ Litvin-v2.cls... face -gear Outer radius of the face -gear N1 = 25 Ns = 28 N2 = 160 m = 6.35 mm αd = 25 .0o αc = 25 .0o γm = 90 .0o 471.0 mm 5 59. 0 mm P1: JXR CB6 72- 18 CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 1:5 5 42 Face -Gear Drives Table 18.15 .2: Design parameters of face -gear of second type of geometry Number of teeth of the pinion Number of teeth of the shaper Number of teeth of the face -gear Module Driving-side pressure... , u 2 , 2 , 2 (18.13.8) that satisfies system of equations (18.13.3) and (18.13.4) (4) The solution by functions (18.13.7) enables us to obtain: (a) transmission function 2 (φ1 ) and function of transmission errors 2 (φ1 ) = 2 (φ1 ) − N1 φ1 ; N2 (18.13 .9) P1: JXR CB6 72- 18 CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 1:5 534 Face -Gear Drives (b) the paths of contact on surfaces as 1 and 2 that... stresses, and σ Pma x is the magnitude of maximal stress P1: JsY CB6 72- 19 CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 1 :28 19 Worm -Gear Drives with Cylindrical Worms 19. 1 INTRODUCTION There are two types of worm -gear drives: (i) those with cylindrical worms (Fig 19. 1.1) (single-enveloping worm -gear drives), and (ii) those with hourglass worms (see Chapter 20 ) (double-enveloping worm -gear drives) . 90 .0 o Inner radius of the face -gear 471.0 mm Outer radius of the face -gear 5 59. 0 mm P1: JXR CB6 72- 18 CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 1:5 5 42 Face -Gear Drives Table 18.15 .2: . (18.7.5). P1: JXR CB6 72- 18 CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 1:5 524 Face -Gear Drives Step 2: Determination of magnitude L 2 (Fig. 18.6 .2) . Figure 18.7 .2 yields O ∗ a I = r bs cos. tooth. P1: JXR CB6 72- 18 CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 1:5 18 .9 Geometry of Parabolic Rack-Cutters 525 Figure 18 .9. 1: Illustration of rack-cutter profiles; (b) and (c) parabolic