Gear Geometry and Applied Theory Episode 2 Part 5 pptx

P1: GDZ/SPH P2: GDZ CB672-14 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:39 14.10 Nomenclature 403 parabolic function of transmission errors that is able to absorb the linear functions of transmission errors caused by misalignments. 14.10 NOMENCLATURE α n rack profile angle in normal section (Fig. 14.4.7) α t rack profile angle in transverse section (Fig. 14.4.7) β k (k = p,ρ) helix angle on pitch cylinder (k = p), on cylinder of radius ρ (k = ρ) (Figs. 14.2.1 and 14.4.7) λ i (i = p, b,ρ) lead angle on the pitch cylinder (i = p), on the base cylinder (i = b), and on the cylinder of radius ρ (Figs. 14.2.1, 14.4.5 and 14.4.7) µ 1 half of the angular width of the tooth space on the base circle of gear 1 (Fig. 14.3.2) θ, θ 1 , and θ 2 surface parameter of the screw involute surface (Figs. 14.3.2 and 14.3.3) φ, φ 1 , and φ 2 angle of gear rotation (Figs. 14.4.1 and 14.5.1) η 2 half of the angular tooth thickness on pitch circle of gear 2 E shortest axes distance (Fig. 14.5.1) F (12,n) normal component of contact force (Fig. 14.8.2) H lead (Fig. 14.2.1) l axial dimension of helical gear [Fig. 14.7.1(b)] m 12 gear ratio m c gear contact ratio N surface normal n surface unit normal p n circular pitch measured perpendicular to the direction of skew teeth of the rack [Fig. 14.4.7(c)] p t circular pitch in the cross section [Fig. 14.4.7(c)] P n and P t diametral pitches that correspond to p n and p t p = H /2π screw parameter q orientation angle of straight contact lines on rack tooth surface (Fig. 14.4.3) r b radius of base cylinder (Fig. 14.4.4) r o radius of operating pitch cylinder, axode r pi radius of pitch cylinder i (Figs. 14.3.2 and 14.3.3) s rack displacement (Fig. 14.4.1) s t tooth thickness on the pitch circle in the cross section u surface parameter of a screw involute surface w t space width measured on the pitch circle in cross section X (12) f , Y (12) f , Z (12) f components of contact force (Figs. 14.8.2 and 14.8.3) P1: GDZ/SPH P2: GDZ CB672-15 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:44 15 Modified Involute Gears 15.1 INTRODUCTION Involute gears, spur and helical ones, are widely used in reducers, planetary gear trains, transmissions, and many other industrial applications. The level of sophistication in the design and manufacture of such gears (by hobbing, shaping, and grinding) is impressive. The geometry, design, and manufacture of helical gears was the subject of research presented in the works of Litvin et al. [1995, 1999, 2001a, 2003], Stosic [1998], and Feng et al. [1999]. The advantage of involute gearing in comparison with cycloidal gearing is that the change of center distance does not cause transmission errors. However, the practice of design and the test of bearing contact and transmission errors show the need for modification of involute gearing, particularly of helical gears. Figure 15.1.1 shows a 3D model of a modified involute helical gear drive. The existing design and manufacture of involute helical gears provide instantaneous contact of tooth surfaces along a line. The instantaneous line of contact of conjugated tooth surfaces is a straight line L 0 that is the tangent to the helix on the base cylinder (Fig. 15.1.2). The normals to the tooth surface at any point of line L 0 are collinear and they intersect in the process of meshing with the instantaneous axis of relative motion that is the tangent to the pitch cylinders. The concept of pitch cylinders is discussed in Section 15.2. The involute gearing is sensitive to the following errors of assembly and manufacture: (i) the change γ of the shaft angle, and (ii) the variation of the screw parameter (of one of the mating gears). Angle γ is formed by the axes of the gears when they are crossed, but not parallel, due to misalignment (see Fig. 15.4.4). Such errors cause discontinuous linear functions of transmission errors which result in vibration and noise, and these errors may also cause edge contact wherein meshing of a curve and a surface occurs instead of surface-to-surface contact (see Section 15.9). In a misaligned gear drive, the transmission function varies in each cycle of meshing (a cycle for each pair of meshing teeth). Therefore the function of transmission errors is interrupted at the transfer of meshing between two pairs of teeth [see Fig. 15.4.6(a)]. This chapter covers (i) computerized design, (ii) methods for generation, (iii) simu- lation of meshing, and (iv) enhanced stress analysis of modified involute helical gears. 404 P1: GDZ/SPH P2: GDZ CB672-15 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:44 15.1 Introduction 405 Figure 15.1.1: Modified involute helical gear drive. The approaches proposed for modification of conventional involute helical gears are based on the following basic ideas: (i) Line contact of tooth surfaces is substituted by instantaneous point contact. (ii) The point contact of tooth surfaces is achieved by crowning of the pinion in the profile and longitudinal directions. The tooth surface of the gear is a conventional screw involute surface. Contact lines L 0 Base cylinder helix Figure 15.1.2: Contact lines on an involute helical tooth surface. P1: GDZ/SPH P2: GDZ CB672-15 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:44 406 Modified Involute Gears Screw Involute surface Profile-crowned pinion tooth surface Screw Involute surface Double-crowned pinion tooth surface (a) (b) Figure 15.1.3: Crowning of pinion tooth surface. (iii) Profile crowning provides localization of bearing contact, and the path of contact on the tooth surface of the pinion or the gear is oriented longitudinally (see Section 15.4). (iv) Longitudinal crowning enables us to provide a parabolic function of transmission errors of the gear drive. Such a function absorbs discontinuous linear functions of transmission errors caused by misalignment and therefore reduces noise and vibration (see Section 15.7). Figures 15.1.3(a) and 15.1.3(b) illustrate the profile- crowned and double-crowned pinion tooth surface. (v) Profile crowning of the pinion tooth surface is achieved by deviation of the gener- ating tool surface in the profile direction (see Section 15.2). Longitudinal crown- ing of the pinion tooth surface can be achieved by: (i) plunging of the tool, or (ii) application of modified roll (see Sections 15.5 and 15.6). (vi) The effectiveness of the procedure of stress analysis is enhanced by automatization of development of the contacting model of several pairs of teeth. The derivation of the model is based on application of the equations of the tooth surfaces; CAD codes for building the model are not required. Details of application of the proposed approaches are presented in Section 15.9. P1: GDZ/SPH P2: GDZ CB672-15 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:44 15.2 Axodes of Helical Gears and Rack-Cutters 407 - Figure 15.2.1: Axodes of pinion, gear, and rack-cutter: (a) axodes; (b) tooth surfaces of two skew rack-cutters. 15.2 AXODES OF HELIC AL GEARS AND RACK-CUTTERS The concept of generation of pinion and gear tooth surfaces is based on application of rack-cutters. The idea of the rack-cutters is the basis for design of such generating tools as disks and worms. The concept of axodes is applied when the meshing and generation of helical gears are considered. Figure 15.2.1(a) shows the case wherein gears 1 and 2 perform rotation about parallel axes with angular velocities ω (1) and ω (2) with the ratio ω (1) /ω (2) = m 12 where m 12 is the constant gear ratio. The axodes of the gears are two cylinders of radii r p1 and r p2 , and the line of tangency of the cylinders designated as P 1 –P 2 is the instantaneous axis of rotation (see Chapter 3). The axodes roll over each other without sliding. The rack-cutter and the gear being generated perform related motions: (i) translational motion with velocity v = ω (1) × O 1 P = ω (2) × O 2 P (15.2.1) where P belongs to P 1 –P 2 (ii) rotation with angular velocity ω (i ) (i = 1, 2) about the axis of the gear. P1: GDZ/SPH P2: GDZ CB672-15 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:44 408 Modified Involute Gears The axode of the rack-cutter that is meshing with gear i is plane  that is tangent to the gear axodes. In the existing design, one rack-cutter with a straight-line profile is applied for gen- eration of pinion and gear tooth surfaces. Then, the tooth surfaces contact each other along a line and edge contact in a misaligned gear drive is inevitable. Point contact in the proposed design (instead of line contact) is provided by applica- tion of two mismatched rack-cutters, as shown in Fig. 15.2.1(b), one of a straight-line profile for generation of the gear and the other of a parabolic profile for generation of the pinion. This method of generation provides a profile-crowned pinion. It is shown below (see Sections 15.5 and 15.6) that the pinion in the proposed new de- sign is double-crowned (longitudinal crowning is provided in addition to profile crown- ing). Double-crowning of the pinion (proposed in Litvin et al. [2001c]) allows edge contact to be avoided and provides a favorable function of transmission errors. Normal and Transverse Sections The normal section a−a of the rack-cutter is obtained by a plane that is perpendicular to plane  and whose orientation is determined by angle β [Fig. 15.2.1(b)]. The transverse section of the rack-cutter is determined as a section by a plane that has the orientation of b–b [Fig. 15.2.1(b)]. Mismatched Rack-Cutters Figure 15.2.2(a) shows the profiles of the normal sections of the mismatched rack- cutters. The profiles of the pinion and gear rack-cutters are shown in Figs. 15.2.2(b) and 15.2.2(c), respectively. Dimensions s 1 and s 2 are related by module m and parameter b as follows: s 1 + s 2 = π m (15.2.2) s 12 = s 1 s 2 . (15.2.3) Parameter s 12 , which might be chosen in the process of optimization, relates pinion and gear tooth thicknesses and it allows modification of the relative rigidity. In a conventional case of design, we choose s 12 = 1. The rack-cutter for gear generation is a conventional one and has a straight-line profile in the normal section. The rack-cutter for pinion generation is provided with a parabolic profile. The profiles of the rack-cutters are in tangency at points Q and Q ∗ [Fig. 15.2.2(a)] that belong to the normal profiles of the driving and coast sides of the teeth, respectively. The common normal to the profiles passes through point P that belongs to the instantaneous axis of rotation P 1 –P 2 [Fig. 15.2.1(a)]. Pinion Parabolic Rack-Cutter The parabolic profile of the pinion rack-cutter is represented in parametric form in an auxiliary coordinate system s a (x a , y a ) as (Fig. 15.2.3) x a = a c u 2 c , y a = u c (15.2.4) where a c is the parabola coefficient. The origin of s a coincides with Q. P1: GDZ/SPH P2: GDZ CB672-15 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:44 15.2 Axodes of Helical Gears and Rack-Cutters 409 Figure 15.2.2: Normal sections of pinion and gear rack-cutters: (a) mismatched profiles; (b) profiles of pinion rack-cutter in coordinate systems s a and S b ; (c) profiles of gear rack-cutter in coordinate systems S e and S k . The surface of the rack-cutter is denoted by  c and is derived as follows: (i) The mismatched profiles of pinion and gear rack-cutters are represented in Fig. 15.2.2(a). The pressure angles are α d for the driving profile and α c for the coast profile. The locations of points Q and Q ∗ are denoted by |QP|=l d and |Q ∗ P |=l c where l d and l c are defined as l d = πm 1 + s 12 · sin α d cos α d cos α c sin(α d + α c ) (15.2.5) l c = πm 1 + s 12 · sin α c cos α c cos α d sin(α d + α c ) . (15.2.6) (ii) Coordinate systems s a (x a , y a ) and S b (x b , y b ) are located in the plane of the normal section of the rack-cutter [Fig. 15.2.2(b)]. The normal profile is represented in S b by the matrix equation r b (u c ) = M ba r a (u c ) = M ba [a c u 2 c u c 01] T (15.2.7) P1: GDZ/SPH P2: GDZ CB672-15 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:44 410 Modified Involute Gears Figure 15.2.3: Parabolic profile of pinion rack-cutter in normal section. (iii) The rack-cutter surface  c is represented in coordinate system S c (Fig. 15.2.4) wherein the normal profile performs translational motion along c–c. Then we obtain that surface  c is determined by vector function r c (u c ,θ c ) = M cb (θ c )r b (u c ) = M cb (θ c )M ba r a (u c ). (15.2.8) Gear Rack-Cutter We apply coordinate systems S e and S k [Fig. 15.2.2(c)] and coordinate system S t [Fig. 15.3.1(b)]. The straight-line profile of the gear rack-cutter is represented in parametric Figure 15.2.4: For derivation of pinion rack-cutter. P1: GDZ/SPH P2: GDZ CB672-15 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:44 15.3 Profile-Crowned Pinion and Gear Tooth Surfaces 411 form in coordinate system S e (x e , y e ) as: x e = 0, y e = u t . (15.2.9) The coordinate transformation from S k to S t is similar to the transformation from S b to S c (Fig. 15.2.4), and the gear rack-cutter surface is represented by the following matrix equation: r t (u t ,θ t ) = M tk (θ t )M ke r e (u t ). (15.2.10) 15.3 PROFILE-CROWNED PINION AND GEAR TOOTH SURFACES Profile-crowned pinion and gear tooth surfaces are designated as  σ and  2 , respectively, wherein  1 indicates the pinion double-crowned surface. Generation of Σ σ Profile-crowned pinion tooth surface  σ is generated as the envelope to the pinion rack-cutter surface  c . The derivation of  σ is based on the following considera- tions: (i) Movable coordinate systems S c (x c , y c ) and S σ (x σ , y σ ) are rigidly connected to the pinion rack-cutter and the pinion, respectively (Fig. 15.3.1(a)). The fixed coordinate system S m is rigidly connected to the cutting machine. (ii) The rack-cutter and the pinion perform related motions, as shown in Fig. 15.3.1(a), Figure 15.3.1: Generation of profile-crowned tooth surfaces by application of rack-cutters: (a) for pinion generation by rack-cutter  c ; (b) for gear generation by rack-cutter  t . P1: GDZ/SPH P2: GDZ CB672-15 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:44 412 Modified Involute Gears where s c = r p1 ψ σ is the displacement of the rack-cutter in its translational motion, and ψ σ is the angle of rotation of the pinion. (iii) Using coordinate transformation from coordinate system S c to coordinate system S σ we obtain a family of generating surfaces  σ represented in S σ by the following matrix equation: r σ (u c ,θ c ,ψ σ ) = M σ c (ψ σ )r c (u c ,θ c ). (15.3.1) (iv) The pinion tooth surface  σ is determined as the envelope to the family of sur- faces r σ (u c ,θ c ,ψ σ ) and requires simultaneous application of vector function r σ (u c , θ c ,ψ σ ) and the equation of meshing represented as follows (see Zalgaller [1975], Litvin [1994], and Litvin et al. [1995]):  ∂r σ ∂u c × ∂r σ ∂θ c  · ∂r σ ∂ψ σ = f cσ (u c ,θ c ,ψ σ ) = 0. (15.3.2) Equation f cσ = 0 may be determined applying an alternative approach: N c · v (cσ ) c = 0. (15.3.3) Here, N c is the normal to  c represented in S c ; v (cσ ) c is the relative velocity repre- sented in S c . The coordinate transformation discussed above is based on application of homogeneous coordinates and 4x4 matrices (Chapter 1). Generation of Gear Tooth Surface Σ 2 The schematic of generation of  2 is shown in Fig. 15.3.1(b). Surface  2 is represented by the following two equations considered simultaneously: r 2 (u t ,θ t ,ψ 2 ) = M 2t (ψ 2 )r t (u t ,θ t ) (15.3.4) f t2 (u t ,θ t ,ψ 2 ) = 0. (15.3.5) Here, vector equation r t (u t ,θ t ) represents the gear rack-cutter surface  t ;(u t ,θ t ) are the surface parameters of  t ; matrix M 2t (ψ 2 ) represents the coordinate transformation from S t to S 2 ; ψ 2 is the generalized parameter of motion. It may be verified that the generated surface is a screw involute one. Equations (15.3.4) and (15.3.5) represent surface  2 by three related parameters. The gear tooth surface may be represented as well in two-parameter form describing it as a ruled surface generated by a tangent to the helix on the base cylinder. Necessary and Sufficient Conditions of Existence of an Envelope to a Parametric Family of Surfaces Such conditions in the case of profile-crowned pinion tooth surface  σ are formulated as follows (see Zalgaller [1975] and Litvin [1989, 1994]): (i) Vector function r σ (u c ,θ c ,ψ σ ) of class C 2 is considered. (ii) We designate by point M(u (0) c ,θ (0) c ,ψ (0) σ ) the set of parameters that satisfies the equation of meshing (15.3.2) at M and satisfies as well the following conditions [see items (iii)–(v)]. [...]... of vector equations: (σ ) (2) r f (u c , θc , ψσ , φσ ) − r f (u t , θt , 2 , 2 ) = 0 (σ ) (2) ( 15. 4 .2) N f (u c , ψσ , φσ ) − νN f (u t , 2 , 2 ) = 0 ( 15. 4.3) f cσ (u c , θc , ψσ ) = 0 ( 15. 4.4) f t2 (u t , θt , 2 ) = 0 ( 15. 4 .5) Here, f cσ = 0 and f t2 = 0 are the equations of meshing of the pinion and gear with the respective generating rack-cutters c and t ; φσ and 2 are the angles of rotation... Sections 15. 5, 15. 6, and 15. 7) Comparison of the output for both cases P1: GDZ/SPH CB6 72- 15 P2: GDZ CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 0:44 15. 4 Tooth Contact Analysis (TCA) 417 Figure 15. 4.4: Illustration of installment of coordinate systems for simulation of misalignment (Sections 15. 4 and 15. 7) shows that double-crowning of the pinion reduces transmission errors and noise and vibration... ( 15. 6.1) where λ p and λw are the lead angles on the pitch cylinders of the pinion and the worm Figure 15. 6.1: Generation of pinion by grinding worm P1: GDZ/SPH CB6 72- 15 P2: GDZ CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 0:44 15. 6 Grinding of Double-Crowned Pinion by a Worm 4 25 Figure 15. 6 .2: Installment of grinding (cutting) worm Figure 15. 6 .2 shows that the pitch cylinders of the worm and. .. determined (Fig 15. 5.1) The axes of the disk and pinion tooth surface σ are crossed and the crossing angle γ Dp is equal to the lead angle on the pinion pitch cylinder [Fig 15. 5 .2( b)] The center distance E Dp [Fig 15. 5 .2( a)] is defined as E Dp = rd1 + ρ D ( 15. 5.1) where rd1 is the dedendum radius of the pinion and ρ D is the grinding disk radius P1: GDZ/SPH CB6 72- 15 P2: GDZ CB6 72/ Litvin CB6 72/ Litvin-v2.cls February... same magnitude: m 12 = ω(1) N2 = ω (2) N1 ( 15. 4.7) P1: GDZ/SPH CB6 72- 15 P2: GDZ CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 0:44 15. 5 Longitudinal Crowning of Pinion by a Plunging Disk 419 Figure 15. 4 .5: Shift of bearing contact caused by E for the following cases: (a) path of contact on pinion surface when no error of center distance is applied and (b) when an error E = 1 mm is applied; (c) path... connected to the generating disk [Fig 15. 5.3(c)] and is considered fixed P1: GDZ/SPH CB6 72- 15 P2: GDZ CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 0:44 15. 5 Longitudinal Crowning of Pinion by a Plunging Disk 423 Figure 15. 5.3: Generation of double-crowned pinion surface 1 by a plunging disk: (a) initial positions of pinion and disk; (b) schematic of generation; (c) applied coordinate systems (3) Coordinate... helicoids have to be related as ω (2) p1 = (1) p2 ω where ω(i ) (i = 1, 2) is the angular velocity of the helicoid ( 15. 4.1) P1: GDZ/SPH CB6 72- 15 P2: GDZ CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 0:44 15. 4 Tooth Contact Analysis (TCA) 4 15 Planar curves Figure 15. 4 .2: Illustration of formation of helicoid surface by screw motion of a cross-profile of the helicoid (5) The common normal to the cross-profiles... 1994] Figure 15. 5 .2( c) shows line Lσ D obtained on surface D Rotation of Lσ D about the axis of D enables representation of surface D as the family of lines Lσ D Step 2: It is obvious that screw motion of disk D about the axis of pinion tooth surface σ provides surface τ that coincides with σ [Fig 15. 5 .2( d)] P1: GDZ/SPH CB6 72- 15 P2: GDZ CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 422 0:44 Modified... arcmin and L = 0, and (iv) change of γ = 15 arcmin and L = 15 mm The results of computation are as follows: (1) Figure 15. 4 .5 illustrates the shift of bearing contact caused by error E (2) The path of contact is indeed oriented longitudinally (Figs 15. 4 .5, 15. 4.6(b), and 15. 4.6(c)) (3) Error E of shortest center distance does not cause transmission errors The gear ratio m 12 remains constant and of... pinion and gear tooth surfaces and the function of transmission errors We have applied for the simulation of meshing the following coordinate systems (Fig 15. 4.4): (i) Movable coordinate systems S σ and S 2 that are rigidly connected to the pinion and the gear, respectively [Figs 15. 4.4(a) and 15. 4.4(c)] (ii) The fixed coordinate system S f where the meshing of tooth surfaces σ and 2 of the pinion and gear . Section 15. 9. P1: GDZ/SPH P2: GDZ CB6 72- 15 CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 0:44 15 .2 Axodes of Helical Gears and Rack-Cutters 407 - Figure 15 .2. 1: Axodes of pinion, gear, and rack-cutter:. section X ( 12) f , Y ( 12) f , Z ( 12) f components of contact force (Figs. 14.8 .2 and 14.8.3) P1: GDZ/SPH P2: GDZ CB6 72- 15 CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 0:44 15 Modified Involute Gears 15. 1. magnitude: m 12 = ω (1) ω (2) = N 2 N 1 . ( 15. 4.7) P1: GDZ/SPH P2: GDZ CB6 72- 15 CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 0:44 15. 5 Longitudinal Crowning of Pinion by a Plunging Disk 419 Figure 15. 4 .5: Shift