P1: GDZ/SPH P2: GDZ CB672-15 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:44 15.8 Undercutting and Pointing 433 The derivation of line L is based on the following considerations: (i) Equation (15.8.1) yields ∂r c ∂u c du c dt + ∂r c ∂θ c dθ c dt =−v (cσ) c . (15.8.3) Here, ∂r c /∂u c , ∂r c /∂θ c , and v (cσ) c are three-dimensional vectors represented in sys- tem S c of the pinion rack-cutter. (ii) Equation (15.8.2) yields ∂ f ∂u c du c dt + ∂ f ∂θ c dθ c dt =− ∂ f ∂ψ σ dψ σ dt . (15.8.4) (iii) Equations (15.8.3) and (15.8.4) represent a system of four linear equations in two unknowns: du c /dt and dθ c /dt. This system has a certain solution for the unknowns if matrix A =      ∂r c ∂u c ∂r c ∂θ c −v (cσ) c ∂ f ∂u c ∂ f ∂θ c − ∂ f ∂ψ σ dψ σ dt      (15.8.5) has the rank r = 2. This yields  1 =               ∂x c ∂u c ∂x c ∂θ c −v (cσ) xc ∂y c ∂u c ∂y c ∂θ c −v (cσ) yc ∂ f ∂u c ∂ f ∂θ c − ∂ f ∂ψ σ dψ σ dt               = 0 (15.8.6)  2 =               ∂x c ∂u c ∂x c ∂θ c −v (cσ) xc ∂z c ∂u c ∂z c ∂θ c −v (cσ) zc ∂ f ∂u c ∂ f ∂θ c − ∂ f ∂ψ σ dψ σ dt               = 0 (15.8.7)  3 =               ∂y c ∂u c ∂y c ∂θ c −v (cσ) yc ∂z c ∂u c ∂z c ∂θ c −v (cσ) zc ∂ f ∂u c ∂ f ∂θ c − ∂ f ∂ψ σ dψ σ dt               = 0 (15.8.8) P1: GDZ/SPH P2: GDZ CB672-15 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:44 434 Modified Involute Gears  4 =               ∂x c ∂u c ∂x c ∂θ c −v (cσ) xc ∂y c ∂u c ∂y c ∂θ c −v (cσ) yc ∂z c ∂u c ∂z c ∂θ c −v (cσ) zc               = 0. (15.8.9) Equation (15.8.9) yields the equation of meshing f (u c ,θ c ,ψ σ ) = 0 and is not ap- plied for investigation of singularities. The requirement that determinants  1 ,  2 , and  3 must be equal to zero simultaneously may be represented as  2 1 +  2 2 +  2 3 = 0. (15.8.10) Equation (15.8.10) enables us to obtain for determination of singularities the fol- lowing function: F (u c ,θ c ,ψ σ ) = 0 (15.8.11) NOTE. In most cases, it is sufficient for derivation of function F = 0 to use instead of (15.8.10) only one of the three following equations:  1 = 0, 2 = 0, 3 = 0. (15.8.12) An exceptional case, when application of (15.8.10) is required, is discussed in Section 6.3. Singularities of the pinion may be avoided by limitation by line L of the rack-cutter surface  c that generates the pinion. The determination of L [Fig. 15.8.1(a)] is based on the following procedure: (1) Using equation of meshing f (u c ,θ c ,ψ σ ) = 0, we may obtain in the plane of pa- rameters (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 is determined in the space of parameters (u c ,θ c )by simultaneous consideration of equations f = 0 and F = 0 [Fig. 15.8.1(a)]. Then, we can obtain the limiting line L on the surface of the rack-cutter [Fig. 15.8.1(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 enable 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 15.8.2(a) shows the cross sections of the pinion and the pinion rack-cutter. Point A c of the rack-cutter generates the limiting point A σ of the pinion when singularity of the pinion is still avoided. 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 P1: GDZ/SPH P2: GDZ CB672-15 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:44 15.9 Stress Analysis 435 (mm) (mm) Figure 15.8.1: Contact lines L cσ and limiting line L: (a) in plane (u c , θ c ); (b) on surface  c . location of limiting points A c and B c of the rack-cutter profiles. Figure 15.8.2(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 = 30 ◦ , parabola coefficient of pinion rack-cutter a c = 0.002 mm −1 , s a = 0.3 m, parameter s 12 = 1.0 [see Eq. (15.2.3)], and module m = 1 mm. 15.9 STRESS ANALYSIS This section covers stress analysis and investigation of formation of bearing contact of contacting surfaces. The performed stress analysis is based on the finite element method [Zienkiewicz & Taylor, 2000] and application of a general computer program [Hibbit, Karlsson & Sirensen, Inc., 1998]. An enhanced approach for application of finite element analysis is presented in Section 9.5. P1: GDZ/SPH P2: GDZ CB672-15 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:44 436 Modified Involute Gears - Figure 15.8.2: Permissible dimensions h 1 and h 2 of rack-cutter: (a) cross sections of pinion and rack- cutter; (b) functions h 1 (N 1 ) and h 2 (N 1 ). Using the developed approach for stress analysis, the following advantages can be obtained: • Finite element models of the gear drive can be automatically obtained for any position of pinion and gear obtained from TCA. Stress convergence is assured because there is at least one point of contact between the contacting surfaces. • Assumption of load distribution in the contact area is not required because the contact algorithm of the general computer program [Hibbit, Karlsson & Sirensen, Inc., 1998] is used to get the contact area and stresses by application of torque to the pinion while the gear is considered at rest. P1: GDZ/SPH P2: GDZ CB672-15 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:44 1 2 3 Figure 15.9.1: Whole gear drive finite element model. 1 2 3 Figure 15.9.2: Contacting model of five pairs of teeth derived for stress analysis. 437 P1: GDZ/SPH P2: GDZ CB672-15 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:44 (Ave. Crit.: 75%) S, Mises +2.724e-03 +7.653e+01 +1.531e+02 +2.296e+02 +3.061e+02 +3.827e+02 +4.592e+02 +5.357e+02 +6.122e+02 +6.888e+02 +7.653e+02 +8.418e+02 +9.184e+02 1 2 3 Bending Stress: 136.8 MPa (MPa) Figure 15.9.3: Contact and bending stresses in the middle point of the path of contact on the pinion tooth surface for a modified involute helical gear drive wherein the generation is performed by plunging of the grinding worm. Contact Stresses (MPa) (rad) Bending Stresses (MPa) (rad) φ φ Figure 15.9.4: Contact and bending stresses during the cycle of meshing of the pinion. 438 P1: GDZ/SPH P2: GDZ CB672-15 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:44 15.9 Stress Analysis 439 (Ave. Crit.: 75%) S, Mises +1.569e+03 +4.956e-04 +7.500e+01 +1.500e+02 +2.250e+02 +3.000e+02 +3.750e+02 +4.500e+02 +5.250e+02 +6.000e+02 +6.750e+02 +7.500e+02 +8.250e+02 +9.000e+02 1 2 3 Bending Stress: 76.9 MPa (MPa) Figure 15.9.5: Contact and bending stresses in the middle point of the path of contact on a conventional involute helical pinion with error γ = 3 arcmin: edge contact with high stresses occurs. • Finite element models of any number of teeth can be obtained. As an example, Fig. 15.9.1 shows a whole gear drive finite element model. However, such a model is not recommended if an exact definition of the contact ellipse is required. Three- or five-tooth models are more adequate in such a case. Figure 15.9.2 shows a contacting model of five pairs of teeth derived for stress analysis. The use of several teeth in the models has the following advantages: (i) Boundary conditions are far enough from the loaded areas of the teeth. (ii) Simultaneous meshing of two pairs of teeth can occur due to the elasticity of sur- faces. Therefore, the load transition at the beginning and at the end of the path of contact can be studied. Numerical Example Stress analysis has been performed for the gear drive with the design parameters shown in Table 15.7.1. A finite element model of three pairs of contacting teeth has been applied for each chosen point of the path of contact. Elements C3D8I [Hibbit, Karlsson & Sirensen, Inc., 1998] of first order (enhanced by incompatible modes to improve their P1: GDZ/SPH P2: GDZ CB672-15 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:44 440 Modified Involute Gears (Ave. Crit.: 75%) S, Mises +1.598e-03 +9.215e+01 +1.843e+02 +2.764e+02 +3.686e+02 +4.607e+02 +5.529e+02 +6.450e+02 +7.372e+02 +8.293e+02 +9.215e+02 +1.014e+03 +1.106e+03 1 2 3 Bending Stress: 135.6 MPa (MPa) Figure 15.9.6: Contact and bending stresses in the middle point of the path of contact on the pinion tooth surface for a modified involute helical gear drive wherein an error γ = 3 arcmin is considered: edge contact is avoided. bending behavior) have been used to form the finite element mesh. The total number of elements is 45,600 with 55,818 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 500 Nm has been applied to the pinion. Figure 15.9.3 shows the contact and bending stresses obtained at the mean contact point for the pinion. The variation of contact and bending stresses along the path of contact has been also studied. Figure 15.9.4 illustrates the variation of contact and bending stresses for the pinion. Stress analysis has also been performed for a conventional helical involute drive with an error of the shaft angle of γ = 3 arcmin (Fig. 15.9.5). Recall that the tooth surfaces of an aligned conventional helical gear drive are in line contact, but they are in point contact with error γ . The results of computation show that error γ causes an edge contact and an area of severe contact stresses. Figure 15.9.6 shows the results of finite element analysis for the pinion of a modi- fied involute helical gear drive wherein an error γ = 3 arcmin occurs. As shown in Fig. 15.9.6, a helical gear drive with modified geometry is indeed free of edge contact and areas of severe contact stresses. P1: JTH CB672-16 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:51 16 Involute Helical Gears with Crossed Axes 16.1 INTRODUCTION Involute helical gears are widely applied in the industry for transformation of rotation between parallel and crossed axes. Figure 16.1.1 shows an involute helical gear drive with crossed axes in 3D-space. A gear drive formed by a helical gear and a worm gear is a particular case of a gear drive with crossed axes (Figure 16.1.2). Gear tooth surfaces are in line contact for involute helical gear drives with parallel axes and in point contact for involute helical gear drives with crossed axes. The theory of involute gears and research in this area have been presented by Litvin [1968], Colbourne [1987], Townsend [1991], and Litvin et al. [1999, 2001a, 2001c, 2001d] and the theory of shaving and honing technological processes are discussed in the works of Townsend [1991] and Litvin et al. [2001a]. Despite the broad in- vestigation that has been accomplished in this area, the quality of misaligned invo- lute helical gear drives is still a concern of manufacturers and designers. The main defects of such misaligned gear drives are (i) appearance of edge contact, (ii) high levels of vibration, and (iii) the shift of the bearing contact far from the central location. To overcome the defects mentioned above, some corrections of gear geometry have been applied in the past: (i) correction of the lead angle of the pinion (requires regrind- ing), and (ii) crowning in the areas of the tip of the profile and the edge of the teeth (based on the experience of manufacturers). A more general approach for localization of bearing contact has been proposed in Litvin et al. [2001c]. The conditions of meshing of crossed involute gears are represented in this chapter as follows: (1) It is shown that a special design (called the canonical one) provides a central location of bearing contact. (2) Modification of the representation of lines of action (as the sets of contact points) allows the following: (i) representation of an edge contact as the result of the shift of lines of action in a misaligned gear drive (ii) relation of the sensitivity to an edge contact with the nominal value of the crossing angle. 441 P1: JTH CB672-16 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:51 442 Involute Helical Gears with Crossed Axes Figure 16.1.1: Involute helical gears with crossed axes in 3D-space. Worm Helical gear Figure 16.1.2: Gear drive formed by a worm and a helical gear. [...]... Appendices 16. A and 16. B are observed Numerical Example 1: Design of Standard Gears The input parameters are: N1 = 12; N 2 = 29 ; β p1 = 47.5◦ ; β p2 = 42. 5◦ ; γ p = 90◦ ; m pn = 4.0 mm; and α pn = 25 ◦ Transverse pressure angles: α pt1 = arctan tan α pn cos β p1 = 34 .61 43◦ α pt2 = arctan tan α pn cos β p2 = 32. 3 122 ◦ P1: JTH CB6 72- 16 CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 0:51 16. 5 Design... vector of point M2 of the line of action (located in plane 2 ) as ∗ ∗ r f (h 2 , r b2 , λb2 , αot2 , γo , E o ) (2) ( 16 .2. 9) P1: JTH CB6 72- 16 CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 0:51 16 .2 Analysis and Simulation of Meshing of Helical Gears 451 (iii) Taking into account that points M1 and M2 belong to the same line of action, (1) (2) vector equation r f (M1 ) = r f (M2 ) yields a system... Gears Figure 16 .2. 6: Lines of action in gear drive with error of shaft angle Figure 16 .2. 7: Illustration of (a) plane 1 and contact lines on 1 , and (b) line of action A1 and parameters h 1 and m1 γ 447 P1: JTH CB6 72- 16 CB6 72/ Litvin 448 CB6 72/ Litvin-v2.cls February 27 , 20 04 0:51 Involute Helical Gears with Crossed Axes Figure 16 .2. 8: Illustration of (a) plane 2 and contact lines on 2 , and (b) line... crossing angle, and the error γ of the crossing angle P1: JTH CB6 72- 16 CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 0:51 16 .2 Analysis and Simulation of Meshing of Helical Gears 449 ∗ Step 1: The pressure angle αon in the normal section of rack-cutters (see Appendix 16. B) is determined as ∗ cos2 αon = ∗ cos2 λb1 ± 2 cos λb1 cos λb2 cos γo + cos2 λb2 ∗ sin2 γo ( 16 .2. 1) Here and below, the superscript... Gears 461 Transverse modules: m pn = 5. 920 7 mm cos β p1 m pn = = 5. 425 4 mm cos β p2 m pt1 = m pt2 Radii of standard pitch cylinders: m pt1 N 1 = 35. 524 5 mm 2 m pt2 N 2 = = 78 .66 78 mm 2 r p1 = r p2 Radii of base cylinders: r b1 = r p1 cos α pt1 = 29 .23 65 mm r b2 = r p2 cos α pt2 = 66 .4859 mm Lead angles of base cylinders: λb1 = arctan 1 cos α pt1 tan β p1 = 48.0717◦ λb2 = arctan 1 cos α pt2 tan β p2... tan β p2 = 52. 2445◦ Shortest center distance: E p = r p1 + r p2 = 114.1 923 mm Tooth thicknesses on pitch cylinders: π = 9.3003 mm N1 π = r p2 = 8. 522 1 mm N2 s pt1 = r p1 s pt2 Radii of addendum and dedendum cylinders: r pa1 = r p1 + m pn = 39. 524 5 mm r pa2 = r p2 + m pn = 82. 66 78 mm r pd1 = r p1 − 1 .25 m pn = 30. 524 5 mm r pd2 = r p2 − 1 .25 m pn = 73 .66 78 mm It is easy to verify that Eq ( 16. B.10) (see... intersected and crossed lines of action of crossed helical gears in Figs 16 .2. 5 and 16 .2. 6, respectively In addition to this presentation, it is important to represent the lines of action in the plane that is tangent to the base cylinder of one of the crossed helical gears, say the P1: JTH CB6 72- 16 CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 0:51 16 .2 Analysis and Simulation of Meshing of Helical Gears... equations for determination of h 1 and h 2 and the auxiliary parameter m1 Parameters h 1 and h 2 determine the shifts of the line of action in planes 1 and 2 by Eqs ( 16 .2. 4) and ( 16 .2. 5), respectively The axial displacements of the line of action with respect to the pinion and the gear are determined as follows: h1 ( 16 .2. 10) Z1 = sin λb1 Z2 = h2 sin λb2 ( 16 .2. 11) Case 2: Error ∆E of the Center Distance... below) ∗ cos αot2 ∗ − E o cos αot2 ∗ cos αot1 sin λb1 ∗ ∗ sin αot2 sin γo r b2 + r b1 h1 = h2 = A B ( 16 .2. 4) ( 16 .2. 5) where A = r b1 + r b2 ∗ cos αot1 ∗ − E o cos αot1 ∗ cos αot2 ( 16 .2. 6) ∗ ∗ ∗ ∗ ∗ ∗ B = sin λb2 sin αot1 sin γo + cos λb2 (cos αot1 sin αot2 − cos αot2 sin αot1 cos γo ) ( 16 .2. 7) The derivation of equations above is based on the following considerations: (1) Figure 16 .2. 7 shows coordinate... Appendix 16. B) that relates the crossing angle and the normal pressure angle for canonical design is satisfied for the standard gear drive P1: JTH CB6 72- 16 CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 4 62 0:51 Involute Helical Gears with Crossed Axes Numerical Example 2: Approach 1 for Design of Nonstandard Crossed Helical Gears The discussed approach to design of nonstandard crossed helical gears . February 27 , 20 04 0:44 (Ave. Crit.: 75%) S, Mises +2. 724 e-03 +7 .65 3e+01 +1.531e+ 02 +2. 296e+ 02 +3. 061 e+ 02 +3. 827 e+ 02 +4.592e+ 02 +5.357e+ 02 +6. 122 e+ 02 +6. 888e+ 02 +7 .65 3e+ 02 +8.418e+ 02 +9.184e+ 02 1 2 3 Bending.  2 )as r (2) f (h 2 , r b2 ,λ b2 ,α ∗ ot2 ,γ ∗ o , E o ). ( 16 .2. 9) P1: JTH CB6 72- 16 CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 0:51 16 .2 Analysis and Simulation of Meshing of Helical Gears. P2: GDZ CB6 72- 15 CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 0:44 440 Modified Involute Gears (Ave. Crit.: 75%) S, Mises +1.598e-03 +9 .21 5e+01 +1.843e+ 02 +2. 764 e+ 02 +3 .68 6e+ 02 +4 .60 7e+ 02 +5. 529 e+ 02 +6. 450e+ 02 +7.372e+ 02 +8 .29 3e+ 02 +9 .21 5e+ 02 +1.014e+03 +1.106e+03 1 2 3 Bending