Gear Geometry and Applied Theory Episode 2 Part 10 doc

P1: JsY CB672-19 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 1:28 19.3 Design Parameters and Their Relations 553 worm pitch diameter may be chosen as d p = 2r p = q P ax . (19.3.1) The value of q depends on the number N 1 of threads of the worm and the number N 2 of gear teeth and may be picked up from a recommended set (7 ≤ q ≤ 25). Let us develop the pitch cylinder on a plane [Fig. 19.3.1(b)]. The helix for each worm thread is represented by a straight line. The distance p ax between the neighboring straight lines is p ax = H N 1 (19.3.2) where N 1 is the number of worm threads, and H is the lead. Considering as known r p and P ax , we can determine the lead angle on the pitch cylinder from the following equation [Fig. 19.3.1(b)]: tan λ (p) 1 = H πd p = p ax N 1 2πr p = N 1 2P ax r p . (19.3.3) Lead Angle on Worm Operating Pitch Cylinder The lead angles on the worm operating pitch cylinder and ordinary pitch cylinder are related as tan λ (o) 1 r o = tan λ (p) 1 r p = p (19.3.4) where p = H/(2π) is the screw parameter. Equations (19.3.3) and (19.3.4) yield tan λ (o) 1 = N 1 2P ax r o (19.3.5) where r o is the chosen radius of the operating pitch cylinder. The difference between r o and r p affects the shape of contact lines between the surfaces of the worm and the worm-gear. Relation Between Worm and Worm-Gear Pitches We emphasize that we now consider the worm and worm-gear pitches on the operating pitch cylinder (Fig. 19.3.2). The axial section of two neighboring teeth represents two parallel curves. Therefore, the worm axial pitch p ax is the same for the worm pitch cylinder and the operating pitch cylinder. The normal pitch p n is the same for the worm and the worm-gear and is represented by the equation p n = p ax cos λ (o) 1 . The worm-gear transverse pitch, p t , is represented by the equation (Fig. 19.3.2) p t = p n cos β (o) 2 = p ax cos λ (o) 1 cos  90 ◦ ±  λ (o) 1 − γ  =± p ax cos λ (o) 1 sin  γ − λ (o) 1   provided γ − λ (o) 1 = 0  . (19.3.6) P1: JsY CB672-19 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 1:28 554 Worm-Gear Drives with Cylindrical Worms Figure 19.3.2: Worm and worm-gear operating pitch cylinders. Here, β (o) 2 is the gear helix angle on the worm-gear operating pitch cylinder. The upper sign corresponds to the case where γ>λ (o) 1 , and the lower sign corresponds to γ<λ (o) 1 . Equation (19.3.6) provides the positive sign for p t . Similar derivations for the left-hand worm and worm-gear (Fig. 19.2.3) yield p t = p ax cos λ (o) 1 sin  γ + λ (o) 1  . (19.3.7) It is obvious that for the case of an orthogonal worm-gear drive (with γ = 90 ◦ )we obtain that p t = p ax . Radius of Worm-Gear Operating Pitch Cylinder We take into account that p t N 2 = 2π R o . (19.3.8) Equations (19.3.6), (19.3.7), and (19.3.8) yield the following: (i) R o is represented for the right-hand worm and worm-gear as R o =± p ax N 2 cos λ (o) 1 2π sin  γ − λ (1) o   provided γ − λ (o) 1 = 0  . (19.3.9) The upper sign corresponds to the case when γ>λ (o) 1 , and the lower sign corre- sponds to the case when γ<λ (o) 1 . (ii) For the left-hand worm and worm-gear, we have R o = p ax N 2 cos λ (o) 1 2π sin  γ + λ (o) 1  . (19.3.10) P1: JsY CB672-19 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 1:28 19.3 Design Parameters and Their Relations 555 Representation of m 21 in Terms of N 1 and N 2 The gear ratio m 21 was represented for the right-hand and left-hand worms and worm- gears by Eqs. (19.2.8) and (19.2.9), respectively. Equations (19.2.8), (19.2.9), (19.3.9), and (19.3.10) yield m 21 = N 1 N 2 . (19.3.11) Shortest Distance E The shortest distance E between the axes of the worm and the worm-gear is E = r o + R o (19.3.12) where r o = N 1 p ax 2π tan λ (o) 1 , (19.3.13) and R o is represented by Eq. (19.3.9) or Eq. (19.3.10). For the case when γ = 90 ◦ and the operating pitch cylinders coincide with the ordinary pitch cylinders, we obtain E = p ax 2π  N 1 tan λ (o) 1 + N 2  . (19.3.14) Relations Between Profile Angles in Axial, Normal, and Transverse Sections Consider the transverse, normal, and axial sections of the worm surface. The transverse section is obtained by cutting of the surface by plane z = 0 [Fig. 19.3.3(a)]. The axial sec- tion is obtained by cutting of the surface by plane y = 0 [Fig. 19.3.3(d)]. Figure 19.3.3(b) shows the unit tangent a to the helix on the pitch cylinder at point P of the helix. The normal section [Fig. 19.3.3(c)] is obtained by cutting of the surface by plane  that passes through the x axis and is perpendicular to vector a [Fig. 19.3.3(b)]. The normal section is shown in Fig. 19.3.3(c), and the unit tangent to the profile at point P is b. The unit normal n to the worm surface at P is represented as n = a ×b (19.3.15) where a = [0 cos λ p sin λ p ] T b = [ cos α n sin α n sin λ p −sin α n cos λ p ] T , (19.3.16) and λ p is the lead angle of the helix at the pitch cylinder. Equations (19.3.15) and (19.3.16) yield n = [ −sin α n cos α n sin λ p −cos α n cos λ p ] T . (19.3.17) Projections of the unit normal are shown in Fig. 19.3.3. The orientations of the tangents to the profiles in the transverse, normal, and axial sections are represented by P1: JsY CB672-19 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 1:28 556 Worm-Gear Drives with Cylindrical Worms Figure 19.3.3: Sections of worm sur- face: (a) tooth cross section; (b) worm pitch cylinder in 3D-space; (c) section of pitch cylinder by plane ; (d) axial section of worm tooth. angles α t , α n , and α ax , respectively. It is evident from Fig. 19.3.3 that tan α t =− n x n y = tan α n sin λ p , tan α ax = n x n z = tan α n cos λ p . Thus, tan α n = tan α t sin λ p = tan α ax cos λ p . (19.3.18) Equation (19.3.18) relates the profile angles in normal, transverse, and axial sections. Let us now consider a particular case, an involute worm. We may express the radius r b of the base cylinder of an involute worm in terms of the screw parameter p, the lead angle on the pitch cylinder λ p , and the axial profile angle α ax . The derivations are based P1: JsY CB672-19 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 1:28 19.4 Generation and Geometry of ZA Worms 557 on the following considerations: cos α t = r b r p = tan λ p tan λ b . (19.3.19) Equation (19.3.18) yields tan α t = tan α ax tan λ p . (19.3.20) The radius of the base cylinder is represented as r b = p tan λ b = p tan λ p cos α t = p tan λ p (1 + tan 2 α t ) 1/2 . (19.3.21) Equations (19.3.20) and (19.3.21) yield the following final expression for r b : r b = p (tan 2 α ax + tan 2 λ p ) 1/2 . (19.3.22) 19.4 GENERATION AND GEOMETRY OF ZA WORMS The worm is generated by a straight-lined blade (Fig. 19.4.1). The cutting edges of the blade are installed in the axial section of the worm. Henceforth we consider two generating lines, I and II, that generate the surface sides I and II of the worm space, respectively (Fig. 19.4.2). The generating lines are represented in coordinate system S b that is rigidly connected to the blade. The respective surfaces of both sides of the worm thread are generated while coordinate system S b performs the screw motion about the worm axis (Fig. 19.4.3). The generated surface is represented in coordinate system S 1 by the matrix equation r 1 (u,θ) = M 1b (θ) r b (u). (19.4.1) Here, the coordinate system S 1 is rigidly connected to the worm; θ is the angle of rotation in the screw motion; parameter u determines the location of a current point on the generating line and is measured from the point of intersection of the generating line with the z b axis. Thus u =|BB  | for the current point B  of the left generating line II. Similarly, u =| AA  | for the current point A  of the right generating line I. Figure 19.4.1: Installation of blade for generation of an Archimedes worm. P1: JsY CB672-19 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 1:28 558 Worm-Gear Drives with Cylindrical Worms Figure 19.4.2: Geometry of straight-lined blade. The unit surface normal is represented in coordinate system S 1 by the equations n 1 (u,θ) =±k N 1 =±k  ∂r 1 ∂u × ∂r 1 ∂θ  (19.4.2) where k = 1/|N 1 |. The upper or lower sign must be chosen with the condition that the surface unit normal will be directed toward the worm thread. Matrix M 1b is represented by the equation (Fig. 19.4.3) M 1b =        cos θ −sin θ 00 sin θ cos θ 00 001±pθ 0001        . (19.4.3) Figure 19.4.3: Coordinate transformation in the case of screw motion. P1: JsY CB672-19 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 1:28 19.4 Generation and Geometry of ZA Worms 559 Here, p is the screw parameter that is considered as an arithmetic value (p > 0). The upper and lower signs for pθ correspond to the cases when a right-hand worm and a left-hand worm are generated, respectively. Figure 19.4.3 shows the generation of a right-hand worm. The surface sides I and II for right-hand and left-hand worms are generated by generating line I and generating line II, respectively. Using Eqs. (19.4.1) and (19.4.2) we may represent the surface equations and the surface unit normals for both sides of the worm thread in S 1 as follows: (i) Surface side I, right-hand worm: x 1 = u cos α cos θ y 1 = u cos α sin θ z 1 =−u sin α +  r p tan α − s p 2  + pθ. (19.4.4) The surface unit normal is n 1 =−k[(p sin θ + u sin α cos θ) i 1 − ( p cos θ −u sin α sin θ) j 1 + u cos α k 1 ] (provided cos α = 0) (19.4.5) where k = 1/( p 2 + u 2 ) 0.5 . We recall that parameter u is measured along the gen- erating line I from point A of intersection of this line with axis z b (Fig. 19.4.2). Design parameter s p is equal to the axial width w ax of the worm space in the axial section. For standard worm gear drives we have w ax = π 2P ax (19.4.6) where P ax is the axial diametral pitch. (ii) Surface side II, right-hand worm: x 1 = u cos α cos θ y 1 = u cos α sin θ z 1 = u sin α −  r p tan α − s p 2  + pθ. (19.4.7) The surface unit normal is n 1 = k[(p sin θ −u sin α cos θ) i 1 − ( p cos θ +u sin α sin θ) j 1 + u cos α k 1 ] (provided cos α = 0) (19.4.8) where k = 1/(p 2 + u 2 ) 0.5 . (iii) Surface side I, left-hand worm: x 1 = u cos α cos θ y 1 = u cos α sin θ z 1 =−u sin α +  r p tan α − s p 2  − pθ. (19.4.9) The surface unit normal is n 1 =−k[(−p sin θ + u sin α cos θ) i 1 + ( p cos θ +u sin α sin θ) j 1 + u cos α k 1 ] (provided cos α = 0) (19.4.10) where k = 1/(p 2 + u 2 ) 0.5 . P1: JsY CB672-19 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 1:28 560 Worm-Gear Drives with Cylindrical Worms (iv) Surface side II, left-hand worm: x 1 = u cos α cos θ y 1 = u cos α sin θ z 1 = u sin α −  r p tan α − s p 2  − pθ. (19.4.11) The surface unit normal is n 1 = k[−(p sin θ +u sin α cos θ) i 1 + ( p cos θ −u sin α sin θ) j 1 + u cos α k 1 ] (provided cos α = 0) (19.4.12) where k = 1/(p 2 + u 2 ) 0.5 . Problem 19.4.1 The worm surface  1 is represented by Eqs. (19.4.7). Consider the axial section of  1 as the intersection of  1 by plane y 1 = 0. Equations (19.4.7) with y 1 = 0 provide two solutions: (i) Derive the equations of two axial sections as x 1 = x 1 (u), and z 1 = z 1 (u). (ii) Determine coordinates x 1 and z 1 for the point of intersection of the respective axial section with the pitch cylinder of radius r p . Solution (i) Solution 1 x 1 = u cos α, y 1 = 0, z 1 = u sin α −  r p tan α − s p 2  . Solution 2 x 1 =−u cos α, y 1 = 0, z 1 = u sin α −  r p tan α − s p 2  + pπ. (ii) Solution 1 θ = 0, x 1 = r p , z 1 = s p 2 . Solution 2 θ = π, x 1 =−r p , z 1 = s p 2 + pπ. Problem 19.4.2 The worm surface  1 is represented by Eqs. (19.4.7). Consider the cross section of  1 by plane z 1 = 0. Investigate the equation r 1 = r 1 (θ), where r 1 = (x 2 1 + y 2 1 ) 0.5 , and verify that it represents the Archimedes spiral. Solution (i) Equation z 1 = 0 yields u = r p tan α − s p 2 − pθ sin α = a − pθ sin α . P1: JsY CB672-19 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 1:28 19.5 Generation and Geometry of ZN Worms 561 Figure 19.4.4: Cross section of an Archimedes worm. (ii) The cross section is represented by equations x 1 = (a − pθ) cot α cos θ, y 1 = (a − pθ) cot α sin θ. (iii) Equation r 1 =  x 2 1 + y 2 1  0.5 yields r 1 = (a − pθ) cot α. The magnitude of the initial position vector for θ = 0isr 1 = a cot α. The increment and decrement of the magnitude of the position vector is proportional to θ, and this is the proof that the cross section is an Archimedes spiral. Figure 19.4.4 shows the cross section of the ZA worm with three threads. 19.5 GENERATION AND GEOMETRY OF ZN WORMS Generation ZA worms are used if the lead angle of the worm is small enough (λ p ≤ 10 ◦ ). In the case of generation of worms with large lead angles, the blade is installed as shown in Figs. 19.5.1(a) or (b) to provide better conditions of cutting. The first version of installation [Fig. 19.5.1(a)] provides straight-lined shapes in the normal section of the thread. Straight-lined shapes are provided in the normal section of the space with the second version of installation [Fig. 19.5.1(b)]. The surfaces of the worm will be generated by the blade performing a screw motion with respect to the worm. To describe the installation of the blade with respect to the worm, we use coordinate systems S a and S b that are rigidly connected to the blade and the worm. We start the discussion with the generation of the worm space (Fig. 19.5.2). Axis z b coincides with P1: JsY CB672-19 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 1:28 562 Worm-Gear Drives with Cylindrical Worms Figure 19.5.1: Blade installation for genera- tion of ZN worm: (a) for thread generation; (b) for space generation. Figure 19.5.2: Coordinate systems applied for blade in- stallation. [...]... −1 sin α sin λp (19.5 .10) (19.5.11) (19.5. 12) we obtain the following expressions for cos δ and sin δ: cos δ = (cos2 α + sin2 α sin2 λp )0.5 , Equations (19.5.7) are confirmed sin δ = (1 − cos2 δ)0.5 = sin α cos λp P1: JsY CB6 72- 19 CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 566 1 :28 Worm -Gear Drives with Cylindrical Worms Note 2: Derivation of Expressions for cos µ and sin µ Equations (19.5.9)... /2) cot α We consider Eqs (19.5.15) as a system of two linear equations in the unknowns u ∗ and ρ and represent them as a 11 ρ + a 12 u ∗ = d, a 21 ρ + a 22 u ∗ = 0 (19.5.16) The solution for the unknown ρ is 1 ρ= (19.5.17) where 1 = d a 12 0 a 22 = =∓ a 11 a 12 a 21 a 22 d cos δ sin µ sin λp =∓ cos δ sin λp (19.5.18) (19.5.19) Equations (19.5.16) to (19.5.19) yield ρ=d sin α sin λp (cos2 α + sin2... with axes 2 and 2 /sin λp The coordinate transformation from S a to S b is represented by the matrix Mba :  Mba 1 0   0 cos λp  =  0 ± sin λp  0 0 0 ∓ sin λp cos λp 0 0   0   0  (19.5.1) 1 The upper and lower signs correspond to the generation of a right-hand worm and left-hand worm, respectively P1: JsY CB6 72- 19 CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 564 1 :28 Worm -Gear Drives... sin2 α sin2 λp )0.5 where d = rp − sp cot α 2 (19.5 .20 ) P1: JsY CB6 72- 19 CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 1 :28 19.5 Generation and Geometry of ZN Worms 567 Figure 19.5.5: Worm thread generation [Fig 19.5.1(a)]: representation of generating lines in Sa For the case where the blades are installed as shown in Fig 19.5.1(a), we obtain that (Fig 19.5.5) d = rp + wp cot α 2 (19.5 .21 ) Here,... not by a blade P1: JsY CB6 72- 19 CB6 72/ Litvin 5 82 CB6 72/ Litvin-v2.cls February 27 , 20 04 1 :28 Worm -Gear Drives with Cylindrical Worms Figure 19.7.1: Cutter for milling of K worms: (a) illustration of the cutter; (b) illustration of parameters a, s c /2, and r c of the cutter Applied Coordinate Systems We use coordinate systems S c and S 1 rigidly connected to the cutter (tool) and the worm S o is a fixed... system S b that P1: JsY CB6 72- 19 CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 580 1 :28 Worm -Gear Drives with Cylindrical Worms Figure 19.6.6: Installation of grinding wheel and applied coordinate systems: initial installation of grinding wheel with (a) position l 1 of vector c, (b) position l 2 of c, and (c) position l 3 of c is rigidly connected to the grinding head, and (c) fixed coordinated system... chosen Step 2: Determination of q The unique solution for q is determined with the following equations: sin q = = cos q = = sin λb sin αt cos γ cos τ − sin γ cos λb 1 − cos2 γ sin2 τ sin αn cos γ cos τ − sin γ cos λb (19.6 .22 ) 1 − sin2 λp cos2 αn cos γ cos τ cos λb + sin γ sin λb sin αt 1 − cos2 γ sin2 τ cos γ cos τ cos λb + sin γ sin αn 1 − sin2 λp cos2 αn (19.6 .23 ) The profile angles αt and αn, in... CB6 72- 19 CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 1 :28 19.5 Generation and Geometry of ZN Worms 565 follow this section) ρ = cos δ, |Ta | sin µ(I ) = cos µ(II ) = − cos α , cos δ cos µ(I ) = cos α cos δ sin α sin λp = tan δ tan λp cos δ sin µ(II ) = (19.5.6) sin α sin λp = tan δ tan λp cos δ Here, cos δ = (cos2 α + sin2 α sin2 λp )0.5 , sin δ = sin α cos λp (19.5.7) The designations “I” and. .. line have the same orientation, and the worm surface is a ruled developed one (We recall that the surfaces of ZA worms and ZN worms are ruled but undeveloped surfaces.) P1: JsY CB6 72- 19 CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 576 1 :28 Worm -Gear Drives with Cylindrical Worms Figure 19.6 .2: Derivation of screw involute surface for the surface side II of a right-hand worm It is easy to verify... Surfaces and surface unit normals of ZN worms are represented as follows: (i) Surface side I, right-hand worm: x1 = ρ sin(θ + µ) + u cos δ cos(θ + µ) y1 = −ρ cos(θ + µ) + u cos δ sin(θ + µ) cos α cot λp − u sin δ + pθ z1 = ρ cos δ (19.5 .24 ) P1: JsY CB6 72- 19 CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 568 1 :28 Worm -Gear Drives with Cylindrical Worms Here, cos µ = cos α , cos δ sin µ = cos δ = (cos2 . m 21 in Terms of N 1 and N 2 The gear ratio m 21 was represented for the right-hand and left-hand worms and worm- gears by Eqs. (19 .2. 8) and (19 .2. 9), respectively. Equations (19 .2. 8), (19 .2. 9),. left-hand worm and worm -gear, we have R o = p ax N 2 cos λ (o) 1 2 sin  γ + λ (o) 1  . (19.3 .10) P1: JsY CB6 72- 19 CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 1 :28 19.3 Design Parameters and. α sin λ p (cos 2 α + sin 2 α sin 2 λ p ) 0.5 (19.5 .20 ) where d = r p − s p 2 cot α. P1: JsY CB6 72- 19 CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 1 :28 19.5 Generation and Geometry of ZN