P1: JTH CB672-16 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:51 16.5 Design of Crossed Helical Gears 463 The new shortest center distance is E o = r o1 +r o2 = 116.1537 mm. The new crossing angle is γ o = β o1 + β o2 = 91.0055 ◦ . The new radii of addendum and dedendum cylinders: r oa1 = r o1 + m on = 40.2473 mm r oa2 = r o2 + m on = 83.9760 mm r od1 = r o1 − 1.25m on = 31.1690 mm r od2 = r o2 − 1.25m on = 74.8977 mm. It is easy to verify that Eq. (16.B.10) is satisfied for the obtained parameters of non- standard crossed helical gears. Numerical Example 3: Approach 2 for Design of Nonstandard Crossed Helical Gears Numerical example 2 (Approach 1) of design of nonstandard gears has shown that the crossing angle of the drive is slightly changed in comparison with the crossing angle of a similar design of standard gears. The main goal of Approach 2 of design is to keep the same crossing angle that is applied in a similar design of standard gears. The approach is based on the following considerations: (i) The assigned crossing angle γ o = γ p and the gear ratio m 12 have to be observed. (ii) Module m pn and normal pressure angle α pn of the common rack-cutter are given. (iii) Settings of rack-cutter χ 1 and χ 2 for the pinion and the gear are applied respectively, and the tooth thicknesses of the pinion and gear must fit each other. The observation of the assigned crossing angle of the gear drive is satisfied by modifica- tion of the skew angles of the rack-cutters. The computational procedure is an iterative process accomplished as follows. Step 1: Determination of parameters on the pitch cylinders as a function of β p1 and β p2 : r pi = m pn N i 2 cos β pi (i = 1, 2) α pt i = arctan tan α pn cos β pi (i = 1, 2) s pt i = πm pn 2 cos β pi + 2χ i m pn tan α pt i (i = 1, 2). P1: JTH CB672-16 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:51 464 Involute Helical Gears with Crossed Axes Step 2: Determination of parameters on the base cylinders: r bi = r pi cos α pt i (i = 1, 2) λ bi = arctan 1 tan β pi cos α pt i (i = 1, 2) s bti = r bi  s pt i r pi + 2invα pt i  (i = 1, 2). Step 3: Determination of parameters on the operating pitch cylinders: cos α on = (cos 2 λ b1 ± 2 cos λ b1 cos λ b2 cos γ o + cos 2 λ b2 ) 0.5 sin γ o r oi = r bi sin λ bi  cos 2 α on − cos 2 λ bi (i = 1, 2) λ oi = arctan r bi tan λ bi r oi (i = 1, 2) α oti = arccos r bi r oi (i = 1, 2) s oti = r oi  s bi r bi − 2invα oti  (i = 1, 2) m on = 2r o1 sin λ o1 N 1 . Step 4: Determination of the following functions: f 1 = r b2 sin λ b2 r b1 sin λ b1 − m 12 f 2 = s ot1 sin λ o1 + s ot2 sin λ o2 − π m on . The iterative process for determination of β p1 and β p2 is applied as follows: (i) Initially, in the first iteration, the applied magnitudes β p1 and β p2 are the same as in standard design. Generally, the equations of Step 4 are not satisfied simultaneously. (ii) In the process of iterations, β p1 and β p2 are changed and steps 1, 2, and 3 are repeated until observation of Eqs. f 1 = 0 and f 2 = 0. The computations have been applied for the following example. The settings of the rack-cutters are χ 1 = 0.3m pn , χ 2 = 0.2m pn . The crossing angle γ o = γ p = 90 ◦ . The iterative process yields: β p1 = 46.9860 ◦ ,β p2 = 42.0010 ◦ . Using the equations from Step 1 to Step 3 all the parameters of the gear drive can be determined. The new center distance is E o = r o1 +r o2 = 115.1898 mm. The assigned crossing angle γ o = 90 ◦ is observed because γ o = 180 ◦ − λ o1 − λ o2 = 180 ◦ − 42.4631 ◦ − 47.5369 ◦ = 90.0000 ◦ . P1: JTH CB672-16 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:51 16.6 Stress Analysis 465 16.6 STRESS ANALYSIS The goal of stress analysis presented in this section is determination of contact and bending stresses and the investigation of formation of the bearing contact in a crossed helical gear drive formed by an involute helical worm that is in mesh with an involute helical gear. A similar approach may be applied for stress analysis in a gear drive formed by mating helical gears. The performed stress analysis is based on the finite element method [Zienkiewicz & Taylor, 2000] and application of a general purpose computer program [Hibbit, Karlsson & Sirensen, Inc., 1998]. The developed approach for the finite element models is described in Section 9.5. Numerical Example Finite element analysis has been performed for a gear drive formed by an involute worm and an involute helical gear. The applied design parameters are the same as those shown in Table 16.3.1, but an involute worm and not an Archimedes’ worm is considered in this case. Therefore, transmission errors do not occur. The output from TCA [see Figs. 16.6.1(a) and 16.6.1(b)] and the developed approach for the finite element models automatically builds one model for every point of contact. Figure 16.6.2 shows a three-tooth model of an involute worm. Figure 16.6.3 shows the finite element model of the whole worm gear drive. A three-tooth model has been applied for finite element analysis at each chosen point of the path of contact (Fig. 16.6.4). An angle of 60 ◦ has been applied to delimit the worm gear body. Elements C3D8I of first order (enhanced by incompatible modes to improve their bending behavior) [Hibbit, Karlsson & Sirensen, Inc., 1998] have been used to form the finite element mesh. The total number of elements is 59,866 with 74,561 nodes. The material is steel with the properties of Young’s Modulus E = 2.068 × 10 5 MPa and Poisson’s ratio of 0.29. A torque of 40 Nm has been applied to the worm. Figure 16.6.1: Paths of contact on (a) the worm and (b) the gear. P1: JTH CB672-16 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:51 1 2 3 Figure 16.6.2: Three-tooth model of an involute worm. 1 2 3 Figure 16.6.3: Whole worm gear drive finite element model. 466 P1: JTH CB672-16 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:51 16.6 Stress Analysis 467 1 2 3 Figure 16.6.4: Finite element model with three pairs of teeth. Figures 16.6.5 and 16.6.6 show the distribution of pressure on the worm and gear surfaces, respectively, in a chosen point of contact. The variation of contact and bending stresses along the path of contact has been also studied. Figures 16.6.7(a) and 16.6.7(b) illustrate the variation of contact stresses of the pinion and the gear, respectively, using the Von Mises criteria. Figures 16.6.8(a) and 16.6.8(b) show the evolution of bending stresses in the pinion and the gear, respectively. Areas of severe contact stresses are inevitable as a consequence of a crossed path of contact. The obtained results of stress analysis show that a gear drive formed by crossed helical gears should be applied as a light loaded gear drive only. APPENDIX 16.A: DERIVATION OF SHORTEST CENTER DISTANCE FOR C ANONIC AL DESIGN The goal is to derive the shortest center distance considering as given parameters r b1 , λ b1 , r b2 , λ b2 , and α on . Step 1: Derivation of the equation: cos λ oi = cos λ bi cos α on (i = 1, 2). (16.A.1) P1: JTH CB672-16 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:51 468 Involute Helical Gears with Crossed Axes (Ave. Crit.: 75%) S, Pressure -8.935e+01 +2.500e+01 +9.553e+01 +1.661e+02 +2.366e+02 +3.071e+02 +3.776e+02 +4.482e+02 +5.187e+02 +5.892e+02 +6.598e+02 +7.303e+02 +8.008e+02 +8.714e+02 1 2 3 (MPa) Figure 16.6.5: Distribution of pressure on the worm model. The derivation is based on two relations between the transverse profiles and normal profiles of a rack-cutter that yield (see Chapter 14) tan α oti = tan α on sin λ oi (i = 1, 2) (16.A.2) cos α oti = tan λ oi tan λ bi (i = 1, 2). (16.A.3) Equations (16.A.2) and (16.A.3) yield the following transformations: (a) 1 + tan 2 α on sin 2 λ oi = 1 +tan 2 α oti = 1 cos 2 α oti = tan 2 λ bi tan 2 λ oi . (16.A.4) Then we obtain 1 + tan 2 α on sin 2 λ oi = tan 2 λ bi tan 2 λ oi . (16.A.5) P1: JTH CB672-16 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:51 16.6 Stress Analysis 469 (Ave. Crit.: 75%) S, Pressure -8.180e+01 +2.500e+01 +9.731e+01 +1.696e+02 +2.419e+02 +3.142e+02 +3.865e+02 +4.588e+02 +5.311e+02 +6.035e+02 +6.758e+02 +7.481e+02 +8.204e+02 +8.927e+02 1 2 3 (MPa) Figure 16.6.6: Distribution of pressure on the gear model. (b) Equation (16.A.5) yields the following transformations: sin 2 λ oi + tan 2 α on = cos 2 λ oi tan 2 λ bi (16.A.6) 1 − cos 2 λ oi + tan 2 α on = cos 2 λ oi tan 2 λ bi (16.A.7) 1 + tan 2 α on = cos 2 λ oi (1 + tan 2 λ bi ) (16.A.8) 1 cos 2 α on = cos 2 λ oi cos 2 λ bi . (16.A.9) Finally, we obtain relation (16.A.1). Step 2: Consider as given Eq. (16.A.1) and derive the relation between r oi and r bi taking into account that r oi tan λ oi = r bi tan λ bi = p i (16.A.10) where p i is the screw parameter. P1: JTH CB672-16 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:51 470 Involute Helical Gears with Crossed Axes Contact Stresses (MPa) Contact Stresses (MPa) Figure 16.6.7: Variation of contact stresses on (a) the worm surface and (b) the gear surface. P1: JTH CB672-16 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:51 16.6 Stress Analysis 471 Bending Stresses (MPa) Bending Stresses (MPa) Figure 16.6.8: Variation of bending stresses on (a) the worm surface and (b) the gear surface. P1: JTH CB672-16 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:51 472 Involute Helical Gears with Crossed Axes Then we obtain r oi = r bi tan λ bi tan λ oi (16.A.11) and E o = r o1 +r o2 = r b1 tan λ b1 tan λ o1 + r b2 tan λ b2 tan λ o2 . (16.A.12) Taking into account Eq. (16.A.1), we obtain the following final equation for E o : E o = r b1 sin λ b1 (cos 2 α on − cos 2 λ b1 ) 0.5 + r b2 sin λ b2 (cos 2 α on − cos 2 λ b2 ) 0.5 . (16.A.13) APPENDIX 16.B: DERIVATION OF EQUATION OF C ANONIC AL DESIGN f (γ o ,α on ,λ b1 ,λ b2 ) = 0 Consider as given the equation cos α on = cos λ bi cos λ oi (i = 1, 2) [see Eq. (16.A.1)], (16.B.1) which yields cos λ o1 cos λ o2 = cos λ b1 cos λ b2 (16.B.2) and the equation γ o =|β o1 ± β o2 |. (16.B.3) Step 1: Equations of cos γ o and sin γ o using Eq. (16.B.3) are represented as cos γ o = cos β o1 cos β o2 ∓ sin β o1 sin β o2 (16.B.4) sin γ o = sin β o1 cos β o2 ± cos β o1 sin β o2 . (16.B.5) Step 2: The transformation of Eqs. (16.B.4) and (16.B.5) taking into account that β oi = 90 ◦ − λ oi yields cos γ o = sin λ o1 sin λ o2 ∓ cos λ o1 cos λ o2 (16.B.6) sin γ o = cos λ o1 sin λ o2 ± sin λ o1 cos λ o2 . (16.B.7) Step 3: The further transformation is based on Eq. (16.B.6) which yields sin λ o1 sin λ o2 = cos γ o ± cos λ o1 cos λ o2 (16.B.8) sin 2 γ o cos 2 α on = cos 2 λ b1 sin 2 λ o2 ± 2 cos λ b1 cos λ b2 sin λ o1 sin λ o2 + cos 2 λ b2 sin 2 λ o1 = cos 2 λ b1 ± 2 cos λ b1 cos λ b2 sin λ o1 sin λ o2 + cos 2 λ b2 − 2 cos 2 λ b1 cos 2 λ b2 cos 2 α on . (16.B.9) [...]... pt2 + 2( invα pt2 − invαot2 ) r o2 r p2 (16.D.3) where Ni moti (i = 1, 2) 2 mon = (i = 1, 2) sin λoi r oi = moti Equations (16.D.1) to (16.D.5) yield Eq (16.5.5) (16.D.4) (16.D.5) P1: JTH CB6 72 - 16 CB6 72 / Litvin CB6 72 / Litvin-v2.cls February 27 , 20 04 474 0:51 Involute Helical Gears with Crossed Axes APPENDIX 16.E: DERIVATION OF ADDITIONAL RELATIONS BETWEEN αot1 AND αot2 The goal is to prove Eq (16.5.6) and. .. are related by module m and parameter s 12 as follows: s1 + s2 = π m s1 s 12 = s2 ( 17. 3.1) ( 17. 3 .2) Here, s 12 is chosen in the process of optimization, relates pinion and gear tooth thicknesses, and can be varied in the design to modify the relative rigidity In a conventional case of design, we have s 12 = 1 P1: JXR CB6 72 - 17 CB6 72 / Litvin 480 CB6 72 / Litvin-v2.cls February 27 , 20 04 0:58 New Version of... noise 17 .2 AXODES OF HELICAL GEARS AND RACK-CUTTER The concept of axodes is applied when meshing and generation of helical gears are considered Figure 17 .2. 1(a) shows that 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 gear ratio The axodes of the gears are two cylinders of radii r p1 and r p2 [Fig 17 .2. 1(a)]... relative motion between the disk and the pinion P1: JXR CB6 72 - 17 CB6 72 / Litvin CB6 72 / Litvin-v2.cls February 27 , 20 04 0:58 17. 7 Generation of Double-Crowned Pinion by a Worm 491 Figure 17. 7.1: Generation of pinion by worm 17. 7 GENERATION OF DOUBLE-CROWNED PINION BY A WORM Figure 17. 7.1 shows a single-thread generating worm and the pinion in 3D-space The worm surface shown in Fig 17. 7.1 belongs to the same thread...P1: JTH CB6 72 - 16 CB6 72 / Litvin CB6 72 / Litvin-v2.cls February 27 , 20 04 0:51 16.6 Stress Analysis 473 Step 4: Equations (16.B.8) and (16.B.9) yield the following final expression: cos2 αon sin2 γo = cos2 λb1 ± 2 cos λb1 cos λb2 cos γo + cos2 λb2 (16.B.10) The advantage of Eq (16.B.10) is that it allows us to obtain the relation between αon and γo considering as input parameters λb1 and λb2 One parameter... Equations (16.E.4) and (16.A.1) yield the relation sin αot1 sin λb2 = sin αot2 sin λb1 (16.E.5) sin α pt1 sin λb2 = sin α pt2 sin λb1 (16.E.6) (v) A similar approach yields (vi) Equation (16.E.1) follows from Eqs (16.E.5) and (16.E.6) P1: JXR CB6 72 - 17 CB6 72 / Litvin CB6 72 / Litvin-v2.cls February 27 , 20 04 0:58 17 New Version of Novikov–Wildhaber Helical Gears 17. 1 INTRODUCTION Wildhaber [1 926 ] and Novikov... the surface parameters of t ; matrix M2t ( 2 ) represents the coordinate transformation P1: JXR CB6 72 - 17 CB6 72 / Litvin CB6 72 / Litvin-v2.cls February 27 , 20 04 484 0:58 New Version of Novikov–Wildhaber Helical Gears from S t to S 2 ; and 2 is the generalized parameter of motion Equations ( 17. 4.4) and ( 17. 4.5) represent surface 2 by three related parameters Necessary and Sufficient Conditions of Existence... 0:58 New Version of Novikov–Wildhaber Helical Gears Figure 17. 1.1: Previous design of Novikov gears with one zone of meshing Figure 17. 1 .2: Profiles of rack-cutter for Novikov gears with two zones of meshing P1: JXR CB6 72 - 17 CB6 72 / Litvin CB6 72 / Litvin-v2.cls February 27 , 20 04 17. 1 Introduction 0:58 477 Figure 17. 1.3: 3D model of new version of Novikov–Wildhaber gears are now based on application of a double-crowned... The P1: JXR CB6 72 - 17 CB6 72 / Litvin 4 92 CB6 72 / Litvin-v2.cls February 27 , 20 04 0:58 New Version of Novikov–Wildhaber Helical Gears Figure 17. 7 .2: Installment of generating worm velocity polygon at M satisfies the relation v(w) − v( p) = µit ( 17. 7 .2) Here, v(w) and v( p) are the velocities of the worm and the pinion at M; it is the unit vector directed along the common tangent to the helices; and µ is the... been applied Novikov–Wildhaber gears have been the subject of intensive research [Wildhaber, 1 926 ; Novikov, 1956; Niemann, 1961; Winter & Jooman, 1961; Litvin, 19 62; Wells & Shotter, 19 62; Davidov, 1963; Chironis, 19 67; Litvin & Tsay, 1985; Litvin, 1989; Litvin & Lu, 1995; Litvin et al., 20 00c] New designs of helical gear drives 475 P1: JXR CB6 72 - 17 CB6 72 / Litvin 476 CB6 72 / Litvin-v2.cls February 27 , 20 04 . 469 (Ave. Crit.: 75 %) S, Pressure -8.180e+01 +2. 500e+01 +9 .73 1e+01 +1.696e+ 02 +2. 419e+ 02 +3.142e+ 02 +3.865e+ 02 +4.588e+ 02 +5.311e+ 02 +6.035e+ 02 +6 .75 8e+ 02 +7. 481e+ 02 +8 .20 4e+ 02 +8. 9 27 e+ 02 1 2 3 (MPa) Figure. Helical Gears with Crossed Axes (Ave. Crit.: 75 %) S, Pressure -8.935e+01 +2. 500e+01 +9.553e+01 +1.661e+ 02 +2. 366e+ 02 +3. 071 e+ 02 +3 .77 6e+ 02 +4.482e+ 02 +5.187e+ 02 +5.892e+ 02 +6.598e+ 02 +7. 303e+ 02 +8.008e+ 02 +8 .71 4e+ 02 1 2 3 (MPa) Figure. λ o2 (16.B.8) sin 2 γ o cos 2 α on = cos 2 λ b1 sin 2 λ o2 ± 2 cos λ b1 cos λ b2 sin λ o1 sin λ o2 + cos 2 λ b2 sin 2 λ o1 = cos 2 λ b1 ± 2 cos λ b1 cos λ b2 sin λ o1 sin λ o2 + cos 2 λ b2 − 2