1、2000FTMS1 Effects of Helix Slope and Form Deviation on the Contact and FIllet Stresses of Helical Gears by: R. Guilbault, Laval University American Gear Manufacturers Association TECHNICAL PAPER Effects of Helix Slope and Form Deviation on the Contact and FIllet Stresses of Helical Gears Raynald Gui
2、lbault, Laval University The statements and opinions contained herein are those of the author and should not be construed as an official action or opinion of the American Gear Manufacturers Association. Abstract A cylindrical gear model, called the Displacement-Stress model, is developed to establis
3、h the load sharing between meshing gear teeth and along contact lines. The model uses of the Finite Strip Method combined to a model of the tooth base solved with the Finite Differences Method to produce tooth deflexion and fillet stresses. The accuracy of this procedure is established with 3D Finit
4、e Element models. The Displacement-Stress model is coupled to a contact cell discretization of the contact areas based on the Boussinesq and Cerruti solution for point normal traction acting on an elastic half-space. An investigation is conducted on the effects of helix slope and form deviations tol
5、erances specified for grades 5 and 7 of the ANWAGMA IS0 1328-1 for cylindrical gears. The results show an almost linear correspondence between deviation amplitude and tooth load and fillet stress increases: using grade 7 instead of grade 5 can double the tooth flank load increase and associated fill
6、et stress increase. Results also show that effects are even more significant on the maximum contact pressure. Copyright O 2000 American Gear Manufacturers Association 1500 King Street, Suite 201 Alexandria, Virginia, 22314 October, 2000 ISBN: 1-55589-774-6 1.0 Introduction The performance of a gear
7、pair depends largely on the manufacturing precision. The production cost consequently increases with accuracy. The gear designer is thus faced with the decision as to the precision level really needed for a given application. The ANWAGMA IS0 1328-1 for Cylindrical Gears i, which sets 13 gear accurac
8、y levels with allowable lead and form deviations for gearwheels, specifies mandatory inspection items and also contains criterions of quality given as useful information. This paper aims to study the effects of the tolerances on helix formfj and slopehp deviations which, while not individually subje
9、ct to mandatory tolerances, are recognised to have a significant influence on performance of a gear pair. The deviations are studied and compared for an helical gear of grades 5 and 7. The investigation is conducted from the perspective of the longitudinal load distribution in view of form and lead
10、deviations, and its influence on tooth fillet bending and surface contact stresses. 2.0 Notation 2 cc ce : angular approach of pinion and gear members : angular term related to contact deformation : angular term related to bending and shear deflexion of teeth and their flexible base : angular term r
11、elated to initial separation of the profiles zg : angular term related to surface deformation due to frictional traction p(Q : load distributions on a contact line D : number of tooth pairs in mesh n : unit normal to the contact plan in the pinion ceordinates fiame of reference II 7 II : radius modu
12、lus of the meshing position in the pinion ceordinates frame of reference. t %i - - : unit vector parallel to the tangential direction of the treated position in the pinion mordinates frame of reference. 1 3.0 Model and solution The longitudinal load distribution is governed by the stiffness of the t
13、ooth pair in contact. If we disregard other components such as shafts, bearings, housing and exclude the torsion and flexion of the gear blanks, the total stifless is made up of three components: the contact, bending and shear stiffness of the clampeckfree teeth and tooth tase. In this paper, the mo
14、delling is limited to relatively short facewidth gears. The determination of load distribution along meshing surfaces is obtained by the simultaneous solution of the well-known relations, written as: O Equation (1) expresses the contact condition for a point with ceordinates kc in the contact plane
15、of the meshing tooth flanks. Angular terms . and are combined pinion and gear factors, respectively initial separation of the profiles, contact deformation, surface deformation due to frictional traction, tooth bending and shear deflexion and their flexible base, and angular approach of the mating t
16、eeth. When equation (1) is an inequality, no load is supported by the surfaces at point kc; otherwise, point kc belongs to the loaded region of the meshing flanks. Equation (2) gives the static equilibrium condition between the applied torque (r) and the torque produced by the load distributions (o)
17、 along each contact line of the D tooth pairs in mesh. Terms n , 11 II and t are respectively the unit normal to the contact plane, the radius of the meshing position, and the unit vector parallel to the tangential direction of the treated position in the pinion reference frame. 2 The formulation pr
18、esented above leads to the knowledge of the load sharing between meshing gear teeth and along contact lines, when angle Ca is determined. Several analytical and numerical approaches have been proposed to solve the system 2- 151 from which the difficulty in obtaining a precise solution with low compu
19、tation is evidenced. The objective is now to represent helical gear pairs with as high a precision as can be obtained with the Finite Element Method (FEM), but without all the pre and post- processing work required. - O In 1996 and 1997 Gagnon and al. 16-17 extended the Finite Strip Method (FSM), in
20、troduced earlier for thick plates, to the analysis of spur and helical gear teeth. In the FSM, which can be considered as a particular case of the FEM, the neutral plane of the gear tooth is represented by twedimensional finite strip based on functions giving the exact tooth shape. The simple formul
21、ation involves less calculation without loss of precision. The bending displacement of the tooth is calculated to within 5 % of that with the FEM. In counterpart of its speed and 2D simplicity, the FSM lacks versatility in modelling the boundary conditions of the tooth base, and the displacement at
22、a given position is constant throughout the tooth thickness. Thus the FSM cannot account for the effects caused by local geometric variations such as the stress concentration in the fillet area. O In this paper, the FSM is combined to a model of the tooth base constructed by considering the part und
23、er the tooth as a series segments of equal width. In each segment, the equilibrium for the two- dimensional plane strain problem is written in Navier?s form (or in terms of displacements) and solved by the Finite Differences Method 181. The series of segments is a repetition of an original segment.
24、To avoid calculation redundancy, the solution to Navier?s equations is made once for unit loads, and is repeated for all segments in the series. There lies the main advantage of a 2D solution over a 3D in terms of computing time. A typical discretization of a helical gear tooth is shown in figure 1.
25、 Results obtained with the 2D model and FEM for the helical gear tooth are presented in figure 2. The 3D-FEM mesh uses parabolic Hex-20 elements. The geometric parameters of the gear are given in.table 1. Two different 1000 N loads are 3 applied normal to the surface at the tip edge of the pinion to
26、oth: in Load Case A, the load is uniformly distributed; in Load Case B, the load is concentrated at tooth center. Table 1 Helical gearset geometric parameters. Pressure Angle Helix Angle Helical Gear - 2w 15“ Module I 6,OO mm Tonth Number I 20 x 31 - - - . - -. . . - - - Addendum Factor ledendurn Fa
27、ctor 1 1.25 - -. - . . . - _. -. - X Factor I 0,163 -0,1631 Face Width 70.00 mm . . - - D;M E-V 15,OO mm 30,OO mm 200 GPa : 0.3 4 D Figure 1 : Typical discretization of a helical gear tooth. Case B Location for Results taken on the compressive side - 0.0045 E E a - g 0.0035 0.0025 0.0015 0.0005 -0.0
28、005 O 10 20 30 40 50 60 70 Posltion x3 (mm) (a) Load cases and results position. 4 FEM Case B -25 -*- Model Case A U0 O 10 20 30 40 50 60 70 Position x3 (mm) (c) Minimum principal (compressive side). (b) Displacement in the neutral plane. = 16 n 14 12 W e ti 10 6 6 4 2 n “O 10 20 30 40 50 60 70 Posl
29、tlon x3 (mm) (d) Maximum shear (compressive side). Figure 2 : Displacement and tooth fillet stresses for the pinion 5 O As is clearly shown, accurate results are obtained with the proposed Displacement-Stress Model. It is worth noting that the contact deformation is not included in the displacement,
30、 since results are taken on the compressive side. The final gear model is completed with the calculation of the contact deformation and pressure distribution done using a discretization of the tangent plane in cells. The method, developed by Hartnett 11 91 to solve three-dimensional non-Hertzian con
31、tact problems, is based on the Boussinesq and Cerruti solution for point normal traction acting on an elastic half-space. The initial Hartnett pressure cells, figure 3, are transformed for gear contact by accounting for tangential forces, and the elastic half- space is corrected to account for the f
32、ree faces of a real gear tooth by a relaxation procedure proposed by Vijayaksir 1141 and based on a method developed by de Mu1 20. ItC c Pi =F i =o Figure 3 : Contact Stress calculation zone 6 4.0 Helix slope and form deviations : effects of tolerance margins for AN WAGMA IS0 1328- 1 grades 5 and 7
33、on contact and fillet stresses Actual gear teeth must comply to standardized tolerance, or deviation, margins that specify the grade to which the gears belong. The deviations are normally measured in the direction of the transverse base tangent, and apply on an evaluation length that can be shorter
34、than the tooth facewidth. In this paper, the evaluation length is taken equal to the facewidth. A 200 N-m torque is applied to the helical pinion of table 1. The studied meshing position is shown schematically on the pinion member in figure 3, where two tooth pairs mesh simultaneously. Table 2 gives
35、 the radius and axial position of the endpoints of the contact lines. Figure 4 shows tooth meshing sequence: tooth pair O is the main meshing tooth pair whereas tooth pair -1 begins mesh. Table 2 Coordinates (R, Z3) of contact line endpoints (R, 16,800) (63,020 ; 70,000) Pair O I (63,297 ; 0,OOO) (6
36、8,899 ; 47,600) 2 Figure 4 : Contact lines of the studied meshing position The theoretical contact lines observed on the pinion teeth extend, for tooth pair -1, from the fillet to a radius juste above the pitch circle and, for tooth pair O, from a radius above the pitch circle to the tip circle (fig
37、ure 4). 7 4.1 Heb slope deviation (fh). The helix slope deviation measurement is realized between two helix traces that intersect the mean helix trace at the endpoints of the evaluation length (ANWAGMA IS0 1328-1 111). Table 3 lists the tolerance values for grades 5 and 7 pinion and gear members. Ta
38、ble 3 : Helix slope deviation for grades 5 and 7 (ANWAGMA IS0 1328-1). Pinion Wheel -. grade 5 7,OO 7,50 grade7 I 14,OO 15,Oo For a given grade, the combined deviation of a tooth pair is a function of the pinion tooth deviation and the gear tooth deviation. In order to limit the combined pinion and
39、gear deviation range, individual deviations are considered to vary proportionally. The simulation domain defined for meshing tooth pairs can be represented by a two factor plane: combined deviation of pau -i and combined deviation,of pair O. Figure 5 draws the simulation plane. Combined deviation pa
40、ir O 1 2 3 positive deviation pinion and gear pair -1. negative deviation pinion and gear pair O. positive deviation pinion and gear pair -1. 4 negative deviation pinion and gear pair O. deviation Figure 5 : Simulation domain and studied points Using a two level experiment strategy 7, simulation of
41、the limit points I to IV will cover ail possible grade variations. However, points II and IV represent the most extreme operating conditions and are thus selected in order to limit the investigation. If we name points II and IV respectively case Aand case B, table 4 gives all conditions modelled wit
42、h helix slope deviation of teeth. In addition, to offer a better graphic comparison, the first simulation is done with the theoretical gear set. 8 Table 4 : Helix slope deviation, studied conditions Deviation (pm) Pair -i (pinion ; wheef) Pair O (pinion ; wheel) Case A grade 5 (-7,0 -7,5) (78 7,5) g
43、rade 7 (-14,O -15,O) (14,O ; 15,O) CaseB I made5 I (730 ; 775) (-7,0 -7,5) I grade7 I (14,O ; 15,O) (-14,O ; -15,O) The obtained results for the above operating conditions are shown in figures 6 to 16. In the following, directions 5 and q are respectively along and across the plane of contact. Helix
44、 slope deviation of a tooth, if not compensated by deviation of the other tooth in the meshing pair, results in a shift of the first contact point onto an endpoint of the theoretical line of contact. In Case A, the first contact point of tooth pair -1 shifts toward endpoint I, figure 4, and toward p
45、oint II for tooth pair O. For Case B, the first contact point moves toward point II for tooth pair -1 and toward point I for tooth pair O. Deviations, for Cases A and B, result in a loading reduction of the tooth pair for which the deviation is negative and an increase in load for the tooth pair wit
46、h a positive deviation. a The first series of results, figures 6 to 10 below, show the pressure distribution on the active profiles. For the theoretical gear pair, e.g. without deviations, pressure peaks related to tip contact can be observed, figure 6. For tooth pair -1 the pressure elevation is cl
47、early visible; for pair O the phenomenon is of lesser importance. The real contact pressure value at tooth edge cannot be predicted exactly since the solution gives average pressure values across each contact cell, and thus depends on cell size and its pressure center position relative to tooth edge
48、. Effects of helix slope deviation are observed in the pressure charts of figures 7 to 10. For the studied meshing position, Case A is worse since it increases edge contact pressure on tooth pair O. It is interesting to note that the increase appears to be proportionally dependent on the slope devia
49、tion: the maximum pressure increase is 1 162 MPa for grade 5 and 2017 MPa for grade 7 (figures 6 to 8) while grade 7 deviation is twice that of grade 5. 9 (A) Tooth pair - 1 (B) Tooth pair O Figure 6 : Pressure distribution in contact plane - theoretical teeth. (A) Tooth pair - 1 (B) Tooth pair O Figure 7 : Pressure distribution in contact plane- Case A grade 5. (A) Tooth pair - 1 (B) Tooth pair O Figure 8 : Pressure distribution in contact plane - Case A grade 7. 10 (A) Tooth pair -1 (B) Tooth pair O Figure 9: Pressure distribution in contact plane - Case B grade 5
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