1、05FTM05Computerized Design of Face Hobbed HypoidGears: Tooth Surfaces Generation, ContactAnalysis and Stress Calculationby: M. Vimercati, Politecnico di Milano and A. Piazza, CentroRicerche FIAT - ScpATECHNICAL PAPERAmerican Gear Manufacturers AssociationComputerized Design of Face Hobbed Hypoid Gea
2、rs:Tooth Surface Generation, Contact Analysis andStress CalculationMartino Vimercati, Politecnico di Milano and Andrea Piazza, Centro RicercheFIAT - ScpAThe statements and opinions contained herein are those of the author and should not be construed as anofficial action or opinion of the American Ge
3、ar Manufacturers Association.AbstractWhile face milled hypoid gears have been widely studied, about face hobbed ones only very few studies havebeen developed. Aim of this paper is just to propose an accurate tool for computerized design of face hobbedhypoid gears. Firstly, a mathematical model able
4、to compute detailed gear tooth surface representation will bederived; then, the obtained surfaces will be employed as input for an advanced contact solver that, using ahybrid method combining finite element technique with semianalytical solutions, is able to efficiently carry outcontact analysis und
5、er light and heavy loads and stress calculation of these gears.Copyright 2005American Gear Manufacturers Association500 Montgomery Street, Suite 350Alexandria, Virginia, 22314October, 2005ISBN: 1-55589-853-X1Computerized Design of Face Hobbed Hypoid Gears: Tooth Surfaces Generation, Contact Analysis
6、 and Stress Calculation M. Vimercati a, A. Piazza baPolitecnico di Milano Dipartimento di Meccanica, Italy, e-mail: martino.vimercatipolimi.it bCentro Ricerche FIAT ScpA, Italy, e-mail: andrea.piazzacrf.it Nomenclature NbNumber of blade groups RbRatio between Nband the number of being generated gear
7、 teeth rbBlade Radius bEccentric Angle bHook Angle hf Blade Height eRake Angle eBlade Angle reEdge Radius LTLength of Toprem Angle of TopremtBlade radius of curvature i Tilt Angle j Swivel Angle Sr Radial Setting q Cradle Angle Em Blank Offset m Machine Root Angle Xp Machine Center to Back Xb Slidin
8、g Base RaRatio of Roll 1. Introduction In the geared transmission design it is very useful to have a numerical tool able to simulate the real behavior of the gear drive, searching for proper contact pattern, low level of transmission error and acceptable fillet stress. In this paper a set of numeric
9、al procedure for performance analysis of face hobbed hypoid gear will be proposed. As well known, this kind of gear is largely applied when it is needed to transfer power and motion between intersecting and crossing axles 1. Hypoid pairs are mainly manufactured by face hobbing (FH) or face milling (
10、FM) cutting process 2; while this latter uses a single indexing cutting method, in FH process the being generated gear has continuous rotation and rotates in a timed relationship with the cutter. As the gear is being cut, successive cutter blades groups engage successive tooth slots guaranteeing a c
11、ontinuous indexing. This process is now spreading in automotive industry because of its fast manufacturing time. Since many decades, a lot of studies about tooth surface representation, contact and stress analysis of FM gears have been carried out 3-6. On the contrary, about FH process, which is con
12、siderably more complex, only a small number of papers have been published. Regarding tooth geometry, Litvin et al. derived a mathematical model able to describe tooth surfaces only of a non-generated Oerlikon gear member 7; Fong proposed a computerized universal generator able to simulate virtually
13、all primary spiral bevel and hypoid cutting methods without providing a detailed description of the FH case 8. Referring to performance analysis, the state of art is even worse: in the open literature any work has been found. Consequently, nowadays, it is possible to study this kind of gears only by
14、 using, basically as a “black box”, proprietary software which has been developed by manufacturing machine and tools suppliers. Goal of this paper is just to propose an integrated tool for computerized design of FH hypoid gears. The first step in order to build a reliable numerical model is to get a
15、 fine geometrical representation of gear tooth surfaces. With this aim, a series of algorithms able to compute tooth surfaces of FH gears starting from cutting process will be described 9. The geometry of real FH head cutter (Gleason Tri-Ac) will be firstly analyzed; many kinds of blade configuratio
16、n (straight and curve blades, with or without Toprem) will be considered. Then, according to the theory of gearing, FH cutting process (with and without generation motion) will be simulated and gear tooth surfaces equations will be computed. The proposed mathematical model is able to provide an accu
17、rate description of the whole tooth, including fillet region; it will also take into account undercutting occurrence, which is very common in FH gears due to uniform depth tooth 4. By means of this model, tooth surfaces of a real gear drive, which is mounted in a truck differential system, will be c
18、omputed and the results will be validated by comparing them with the ones calculated by a reference proprietary software and with the real surfaces. 2Then, the obtained tooth surfaces will be used as fundamental input for a powerful contact solver which is based on a unique semianalytical finite ele
19、ment formulation 10-11. Firstly, the gear drive it will study under light load by monitoring, for drive and coast side, the contact pattern and transmission error (Tooth Contact Analysis - TCA). After that, with the aim to find out gear drive performance in the real service conditions, a set of torq
20、ue values will be applied and the influence of the load on contact pattern, on transmission error and on load sharing will be accurately analyzed (Loaded Tooth Contact Analysis - LTCA). Contact pressure and stress distribution will be also evaluated. The obtained results will be compared with the on
21、es calculated by a reference software. 2. Theoretical Background of Face Hobbing Method As known 2, FH head cutter is provided with a proper number of blade groups Nb, each of them consists in an outer and an inner blade. As reported in Figure 1, in order to accomplish continuous indexing, the head
22、cutter and the being generated gear are rotating in opposite directions and the next group of blades will start to cut the next gear tooth after that the current group of blades has finished cutting the current tooth. Figure 1. Sketch of FH cutting process In this way, the angular velocity of the he
23、ad cutter bis related to the angular velocity of the work-piece waccording to the ratio between the number of blade groups Nband the number of being generated gear teeth Nw: bwwbbNNR = (1) It is evident that the edge of the blade, during cutting, tracks an epicycloid curve. In order to accommodate t
24、his path, unlike FM method, the effective cutting direction of the blade is not perpendicular to the cutter radius and the blade is moved in the head cutter tangentially to an offset position. Fig. is referred to the non-generated process (Formate); if a generated tooth is needed, the generation mot
25、ion, which relates cradle and work-piece rotation, has to be superimposed. Traditionally, FH gear drive has uniform depth tooth; it follows that FH tooth often shows undercut toe-section with sharp topland. This latter inconvenient can be eliminated by introducing a secondary face angle; undercuttin
26、g avoidance could be a difficult task and often FH gear drives work affected by it 2, 12. 3. Simulation of Face Hobbing Cutting Process: Tooth Surface Generation According to the theory of gearing 3-4, 13, in order to get the analytical representation of gear tooth surfaces, firstly cutting process
27、(i.e. head cutter, cutting blades and cutting machine) has to be described. It will be clear that 9, due to the complexity of FH cutting process, FH cutting blades require a more complicated representation than the ones usually illustrated in the literature (typically for FM method). In this paper a
28、 real FH process, Gleason Tri-Ac, will be studied. 3.1. Cutting Tools: Head Cutter and Cutting Blades As shown (Figure 1), FH head cutter carries a given number Nbof blade groups; each group contains an outer blade (OB) for cutting concave gear side and an inner one (IB) for convex side. Figure 2 re
29、ports, from two different viewing points, one blade group in a Gleason Tri-achead cutter. Referring, for example, to the outer blade, in order to correctly locate the blade in the head cutter, the pitch point P of cutting edge (see also Figure 3) has to be defined. The distance from this point to th
30、e head cutter center is the equal to rb; the angle bis introduced in order to take into account that FH process, unlike FM, shows blades that are not aligned to the cutter radius. It is also evident that the blade is not perpendicular to the head cutter plane, but it is mounted at an angle bwith res
31、pect to the head cutter rotation axis. The distance from the pitch point P to the tip of the blade is measured by hf. 3Figure 2. Sketch of a blade group mounted in the FH head cutter. Figure 3. Details of the FH blades. Figure 3 shows the details of outer and inner blades. It is evident that the cut
32、ting edge lies entirely on a plane, called Rake plane, which forms an angle ewith tool plane. It is also introduced the angle eas the angle between the vertical axis of the blade and the projection of the cutting edge on the tool plane. Once the blade geometry has been introduced, it is possible to
33、compute the analytical formulation of the cutting edge. Many blade profiles are available in commerce. In this paper, a complex blade shape - curved with Toprem- will be analyzed; simpler configurations as straight blade with and without Topremor curved blade without Topremcan be easily derived star
34、ting from the following discussion. The particular disposition of the cutting edge on the Rake plane requires to set up the reference frame St(Figure 2 and 3) and to search firstly for the analytical description of projection on XtZtplane of the cutting edge. Figure 4 shows the projection on this pl
35、ane of a curved blade with Toprem; the following sections, which are function of the curvilinear coordinate s, have been defined: I) Bottom: straight horizontal segment; II) Fillet: circular arc of radius reand center at point R (xR, zR); III) Toprem: inclined straight segment characterized by the l
36、ength LTand the angle ; IV) Curved blade: circular arc with radius of curvature tand center at point O (xO, zO). At pitch point P the segment tangent to the blade is inclined by an angle equal to t. 4Figure 4. Projection on XtZtplane of curved blades with Toprem. The analytical representation of the
37、 blade profile has been computed as follows: for s 0 +=0)()(Rttxsszsx(2.a) for 0 s L1=)/cos(1()/sin()()(eeeeRttrsrrsrxszsx(2.b) for L1 s L1+ L2=)cos()()cos(1()sin()()sin()()(11LsrLsrxszsxeeRtt(2.c) for L1+ L2 s +=)/)(sin()/)(cos()()(2121ttOttOttLLszLLsxszsx (2.d) The value of the parameters re, LT,
38、and tdepends upon the chosen cutter blades. In order to calculate (xR, zR), (xO, zO), , L1and L2it is necessary to develop some simply geometrical considerations based on Figure 4; the value of the angle tis derived from e. Blade profile ytcomponent is computed by imposing that an arbitrary point of
39、 the blade )(),(),()( szsysxsttt=tr and the blade tip Ot0,0,0 lie on the Rake plane. Starting from vector )(str , by means of some coordinates transformations, it is useful to measure the blade profile also in the head cutter reference frame Sh(Fig. 2), obtaining vector )(shr . This latter represent
40、ation will be the starting point for computation of tooth surfaces equations. 3.2. Cutting Process and Tooth Surface Generation In order to numerically compute gear tooth surfaces, the classic theory of gearing requires to compute a proper set of coordinate transformations able to simulate cutting p
41、rocess and to represent the cutting edge in reference frame of being generated gear 13. Consider Fig. 5, where a FH cutting machine set up to cut a generated pinion is shown. In order to describe the machine settings, a set of reference frames have to be introduced. Firstly, the system Sm, which is
42、fixed and rigidly connected to the cutting machine, is defined; it has the origin Omin the center of cradle and Zmaxis coinciding with the cradle rotation axis. Then, reference frame Sh, which has been introduced in the previous section for blade profile computation, is considered; it allows to meas
43、ures the head cutter rotation and to take into account the tilt and swivel angles. The origin Ohis located by means of the distance Srand the angle q. Finally, system Sw, which is rigidly connected to the work-piece, is introduced; its origin Owis placed using the following blank settings: Em, m, Xp
44、and Xb. Zwaxis is coinciding with the gear rotation axis. In order to accomplish the peculiarity of FH process (i.e. continuous indexing), the system Swhas to rotate about Zwaxis by an angle equal to Rb (see Eq. (1). If, as in this case, a generated tooth is needed, it is necessary to add to the wor
45、k-piece rotation a term equal to Ra where Rais the ratio of roll and is the cradle rotation angle (i.e. of the system Sc). 5Figure 5. Sketch of a cutting machine set up for manufacturing a generated FH gear drive. By properly computing the matrices for coordinates transformations (see Appendix A), t
46、he cutting edge representation in the system Swis derived: )(),( ssh1h2132c3mc4m54w5wrMMMMMMMMr = (3)As well known, Equation (3) represents a family of surfaces; with the aim to compute tooth surfaces, one has to search for the envelope of this family by solving equation of meshing: 0),(),(),( =ssss
47、wwwrrr (4)By using values of , and s that satisfy Eq. (3), Eq. (4) and the tooth geometric boundaries, a surface for each section of the blade is generated*. For example, considering previously *When representation of a non-generated gear is needed, cradle rotation is null and tooth surfaces are dis
48、cussed blade, four surfaces, corresponding to the four blade parts (bottom, fillet, Topremand curved blade), are obtained. In order to compute final tooth surfaces, it is necessary to handle properly these four surfaces. Due to the severe analytical complexity of these equations, a convenient way to
49、 accomplish this step is to slice the tooth by means of several cross sections and to compute numerically the tooth profiles belonging to these cross sections; in other words, by solving a non-linear problem, four profiles cut by the four parts of the blades and belonging to the selected cross section are firstly calculated. Then, these four profiles have to be correctly merg
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