1、92 FTM 7Difference in the Local Stressof the Gear Tooth Root Based OnHobbing Cutters and Pinion Cuttersby: H. Linke and J. B_SmerDresden University of Technology, GermanyAmerican Gear Manufacturers AssociationTECHNICAL PAPERDifference in the Local Stress of the Gear Tooth RootBased on Hobbing Cutter
2、s and Pinion CuttersH. Linke and J. BiSrnerDresden University of TechnologyThe statements and opinions contained herein are those of the author and should notbe construed as an official action oropinion of the American Gear Manufacturers Association.ABSTRACT:Differences of the tooth root geometry ar
3、e caused if a pinion shaped cutter is used in the gear production instead of ahob.These differences also lead to a different stress concentration in the tooth root. The tooth root stresses are exactlycalculated with the Singularity Method (variant of Boundary-Element-Method) for both production meth
4、ods and theirdifferences are discussed. The approximate calculation of stress concentration with the stress parameter 2pF n / s Fn(on 30 - tangent) is proved to be applicable.Copyright 1992American Gear Manufacturers Association1500 King Street, Suite 201Alexandria, Virginia, 22314October, 1992ISBN:
5、 1-55589-587-5Difference in the Local Stress of the Gear Tooth RootBased on Hobbing Cutters and Pinion CuttersH. Linke ; J. BSrnerDresden University of Technology, Germany_therefore a repeated investigation of these X = rwl. sin_- (_h/sin_+ _a0).cos8 2)problems appears to be worthwhile. The perpendi
6、cular distance Y from the rootcross-section, given by the tangent anglei. Fundamental Knowledge 8, to the gear centre, isI.I, .Tooth Root G_ometry Y = rwlCS_- (Ah/sin_+ _a0).sine (3)Tooth Root ThicknessFirst of all an easy method for the calcu- The equations (2 and (3) can be iterative-lation of the
7、 tooth root geometry by use of ly solved with the aid of the relationsany type of cutters is to be explained by given in table 1. It proved to be suitableway of introduction. With this method the to change the rolling angle in small stepsgeometry is computable for given and gene- A_ for a given tang
8、ent angle O_ startingrally variable tangent angles 8 to thetooth root fillet. The derivation for ex- from _min = n/Zl - z0/zl “ Aa (4)ternal gearings is carried out. Internalgearings are determined by negative numbers to _max = _/Zl + z0/zl “ (aM - ha (5)of teeth for both the manufactured gear and -
9、 arccos(rb0/rw0)the pinion cutter. For very large numbersof teeth of the pinion cutter (z0 -_) the until the pertinent tangent angle e(_)tooth root thickness as well as the radii agreed with the rated qUantity _* (withinof curvature of the root fillet curve permissible small deviations).asymptotical
10、ly pass into the pairing of ,Table 1 contains individual specificationgeared rack - cylindrical gear. quantities for eqUations (2) to (5). IndexThe tooth root thickness SFs is determi- 0 and index 1 designate the pinion cutterned according to fig. I. A dlfferent kind (tool) and the gear to be made (
11、gearof presentation was given by, e. g., blank), respectively. Figures 2 and 3 ex-Hirschmann /5/. For the first time the plain other quantities. The designationstooth root thickness for a tangent to the have been chosen in accordance with DINroot fillet was derived by Hofer /6/. 3960, DIN 3990 or IS
12、O 701.Figure I: Generation of the fillet curve bymeans of a pinion cutter - Deri-vation of the tooth root geometryThe shown geometry calculation at the basis Table i: Auxiliary quantities for the cal-of increasing _ is also suitable to create culation of root thickness SFn andequidistant points at t
13、ooth root fillet tooth root fillet radius 8Fncurve for Singularity Method used in stresscalculations. A very efficient iterationwith Newtons Method is also possible for Designation Equationas was done in /7/ (_= _n in /7/).Pitch circles = a0 / (I + Zo/ZIradius rwo, rwl rwlrwo = rwl. zO/zIlentredista
14、nce a0 = Zl+ZO)-ml2.cosaOlcosawO I,fgenerating_airing a0 inVUw0 (Xl+X0).2.tans0= + inva0zl+z0_-_ Difference of 620= YM = rMcs_athe tip radiusXM YM rM = ra0 - _a0aa = Sb0/2-_a0+Spr /rb0-inv_MtossM = rb0/rM00 Auxiliary angle _ _= arctan -rw0-Sin designations accord, to DIN3960, DIN3990, ISOT01.SrRadii
15、 of Curvature of the Fillet CurveThe radii of curvature are to be calculatedon the basis of the Euler-Savary equation,which results according to figure 4 fromthe following deliberation:pitch circle 2 roll on pitch circle I.LetLet the straight line _ be closely con-nected with circle 2._ _. The radiu
16、s of curvature 8M of the rela-0_“ X tire path resulting during the rolling ofcircle 2 on circle 1 is to be calculated.From the ratio of the relative rotationFigure 3: Data on the pinion shaper with _ to the pole velocity uprotuberance - Tooth thickness atbase circle and pressure angle _/u = i/rwl +
17、I/rw2 (6)3with _M according to equ. (8). In table I,equations are given for Ah _ rwl and rw 0“Equations (8) and (9) apply also to inter-nal gears, if a negative number of teeth isused for them. For the manufacturing withhobs and rack cutters a very large number_, of teeth can be used for the tool, e
18、. g. zQ= 10,000, so that a sufficient degree oZapproximation will be obtained.1.2. Stress ConcentrationInvestigations of the stress concentrationeffect were carried out on a computationalbasis at the Dresden University of Techno-logy /3/ /4/. The Singularity Method chosenfor this purpose (BEM varian
19、t) has provedto be very good. Short computing times anda comparatively low expenditure for thepreparation of the numerical investigationsconstituted essential properties. Accuracywas tested with the solution by Neuber /2/for a tooth-like projection. Deviationslower than 2 % resulted.Fig. 5 shows str
20、ess curves, which werecalculated with the method mentioned. FromI these calculations and other investigations62 7 + follows that the stress maximum occurs inu Fwl tw2 the range of, 25_ 8 _ 50 in case of hob-bing with a0 a 0.25.m and by the use ofpinion shaped cutters with _a0 z 0.1-m0_ (inelasticgea
21、r bodies being assumed).The results of the calculations performedfor the numbers of teeth and addendum modi-fication factors occurring in the range ofFigure 4: Radius of curvature _M of the usual gearing geometries were applied inrelative path of the centre M of the former standard TGL 10545 (new dr
22、aft).the tool tip radius - Derivation In comparison with DIN 3990 and ISO DISof the Euler-Savary equation (M-P 6336/03, these values are mostly 10% to 15%rigidly connected with gear 2, lower; agreeing, however, with newer re-rolling on gear i) sults of other research institutions, too.In summary, th
23、e following could be stated:one obtain the Euler-Savary equation in a * The stress maximum mostly lies at tan-general form gent angles to the root fillet curvegreater than 30 (usually 8 = 30 .(i/i I 1/12).cos _ = i/rwl + 17rw2 (7) 50). The use of the nominal stress atthe 30-tangent to relate the max
24、imumthe minus sign being valid if the points M stress in the root fillet is notand M0 lie on the same side as seen from wrong, nevertheless. If YFS is repre-the pole P. With the quantities sented in dependence upon z, x andtool profile parameters, the resulting_M = 12 - ii ; i/rw = I/rwl + i/rw2 err
25、or is equal to zero. Since YFSis calculated with YFa(O=30o anaI/i I = 1/12 + i/(rwCOS_ ) YSa(2 _/s) (for instance with eq.(10)the resulting error is small in caseand of _0 _ 0.2.m, because the radius of12 = _h/sin_ curvature of the tooth fillet curvechanges only slightly.we get the radius of curvatu
26、re _M of thepath of the centre M of the tool tip radius * The Stress peak is the closer to theroot circle the shorter the bendingAh2 momentarm is.8M = sin_. (rw. Sin_ + Ah) (8) If the pressure angle a in the rangeFinally, the searched radius of curvature of 0 to 30 changes, the stress con-_Fn of the
27、 fillet curve is centration will change (at constant2 _Fn/SFm and hF/SFn) with respect to= O_Fn _M + 8a0 (9) a = _0 5y about IO %._ * For differentbendingmomentarms theIi Y_.3,24_“654 kl Ii _s462y_230 Jl following equation takes into accountI.-e561 I 8.47 the change of the stress concentration i=-20
28、e-0“ =.20-_-_e.0. (caused by the change of the shearV JZo-8O _-o I _ I,o.8O_,oi stress component)_ j ,. 50 _,.o., l _ Iz,-_o _,-o.,I_ _o-0,1.m 0.071_2+0.0657 I) hF/SFn 0.234I YS = + _ +0.973 (ii)_f_ 2 _Fn/SFn hF/SFn For estimations of Y-= Y- -Y-_ thefollowing equations are suztable whicha) b) -“ dep
29、end on the v_rtual number of teethZnx (Znx = z/conj,) and the addendummo-dific-ation factor x (profile accord.I _3.,3 _ D4.sB to DIN 867 or ISO 53):, i Y=-,.e7 I i Y=-1.56_ l e _s.: l _ I e334 - for tip radius of the hob a0: 0.2-mA j_.20“e.0“ j _ a-20 e.0“ = 4.08 + 0 iBx 2 + 7.63/Znx Iz0.80_.0 , iz0
30、.80%.0 YFS _A _z,-50x,-0,8 l _ jr,-20x,.0 - 15.94X/Znx (12)“ i_ _ /_ -fr tip radius f the hb 8a0=0“38“m YFS = 3.467 + 0.091x 2 + 13.17/Znx- - 27.91X/Znx (13)C) d) 2. Results of Numerical Investigations_ _4.22 ;_ Yv_.4.el 2-I- TOoth RoOt ThiCkness and stress COn-j Y_- %4e v.,.2.2_ centration Paramete
31、re.3o.3o 8.49.3 With regard to already existing results,I ,._. I _ _-2o e-o“ in the following comparisons of numericali _ Jz0-e0_-0 shown by way of intro- |Zo-SO _.0 investigations are/z,-2o x,-O . _ ,o. ,.-_%|z,-Soox,-o duction, which consider the manufacturing_ ,_ with hobbing and pinion shaped cu
32、tters. Thehob engages the rack cutter, too. This will not be pointed out repeatedly in the follo-w,oo- For the hob the tool tip radiusof _a0I = 0.25-m was assumed, and for the pinioncutter no tool tip radius was assumed, ase) |) _o_ usually done in many cases. The gearingscorrespondend to a standard
33、ized gearingaccording to DIN 867 or ISO 53. The clea-Figure 5: Stress curves calculated with the rance was c = 0.25.m or the tool addendumSingularity Method (BEM) for ma- haP0= 1.25-m (adequate to the dedendum ofnufacturing with a pinion cutter t_e gear blank). For the pinion cutter,with z0_=80 deno
34、ted by the index p (or 0p) in general,a)-c) _I =50 Xl=0“8 _a0 varied the zero addendum modification (x0, = 0) _au wasd!,e) zi=20 Xl=0 _ varied assumed. The volume of the calculation_J zi=500 Xl=0 , _a0:0.1“m results shown should be limited reasonablyby this.First, the tooth root thickness s_ , The i
35、nfluence of the neighbouring calculated at the 30-tangent (O = 30 _,tooth is lower than 10 %. was calculated for manufacturing with a hoband with a pinion shaped cutter. The tooth For tooth root radii _ e_ 0.2.m and root thickness is a little bit larger inload application to _ tooth tip, case of usi
36、ng a pinion cutter than in casewith tooth profiles according to DIN of using a hob. The difference between the867 or ISO 53 the stress concentration root thickness at the gear blank in case offactor (stress correction factor) YSa manufacturing with a hob and with a pinionis well approximated by the
37、relation cutter amounts to 3.5% at most in the real-, ly used range of addendum modification_. 1 (i0) factors up to 1.0. If the addendum modifi-YSa = 1.22 2 _Fn/SFn cation is extremly large with xl=l.5, thedifference amounts to 6% for a plnion cut-ter with Z0p=80. If one assumes that the5Iplnioncutt
38、er : 7o20 xo- 0 e -“ Olmenufectureda) , _. - - t -e-r -_0.8.E hbfing cutter: _“ 025“m1 t _ 0.26 -_hobblng cutter: _ I _/_ XL_.o. 0.26.m , _)0 “looo- _ ?,I4-.3“-k-d.:G“-4-“Y_.D-“4.-I-=ooo., I I -=Z“.mlil il_ii _, -_-_J_l-_i_ 600-0.5 -OolB 0 0:25 0=_6 0,_75 1 1.25 1.5 -1.o“-0.5-028 0 0.26 0.5 0.76 1 1
39、.25 1.5R) addendummodlfloetlonfactor xI a) addendummodlfkietlonfactor xImanufactured pinioncutter : 4“ 00 _- 0 Q_ 0 manufacturedQ 0.8- Ilnlon cutter : zo- 20 %- 0 _,o“ 0 -gear zi _ 1 _bbl_ out_= _o“ 0.26om _ar z10.76 /-80600 = i-12o_ -200 _ 2_/ -200“ _,o= _“_“_ ._, _-aoo-12020.4_ -,._,._“ :/o _ -60
40、_0.25 _ _-6000.2 I # =_“_-0-0.6-0.26 0 0.25 0.6 0.75 1 1._ 1.6 -_6-0.26 0 0.26 _6 0.76 1 1.26 1.6b) _de_um modlfic_lon factor x_ b) addendummodlf_n _ctmFigure 6: Root fillet radius at the 30- Figure 7: Referred deviations of the stresstangent (hap0 =1.25.m) concentration parameter 2 _n/sF_al product
41、ion with a hob for prodhction with a ho_ an_b) production with a pinion pinion cutters (haP0 = 1.25.m)cutter (Z0p = 20) a) pinion cutter wih z0_ = 20b) pinion cutter with Z0p = 80root thickness influences the local root cations investigated here the centre Mstress about quadratically, for really use
42、d always has a finite distance to the pitchaddendum modifications there are possible circle. Only if the addendum modificationmaximum root stress deviations of 7%. By it of the pinion cutter amounts to xO = -x 1the nominal stress of gears manufactured =(hap 0 - _a0)/m the centre M would lie onwith p
43、inion shaped cutters is lower than the pitch _ircle. But this modification isthat of hobbed gears by the amount mentio- without practical use.ned. The radii of curvature of the filletThe curve of the tooth fillet radii in curve become the shorter the larger thecase of production with a hob is repres
44、en- number of teeth of the pinion cutter is andted in fig. 6a, and that for production the closer the pitch circle of the gearwith a pinion cutter (z_ = 20) is plotted blank approaches its root circle. This isin figure 6b. Courses wHlch are different, the case, starting from the equal addendumin pri
45、nciple, are found. Whereas at an gearing with increasing addendum modifica-addendum modification factor x = I in the tion factor of the gear blank xl and de-of production with a ho_ a minimum creasing addendum modification factor _caseoccurs, in the case of production with the of the pinion cutter.
46、This decreasing _pinion cutter the curvature radius decrea- dendum modification factor x0_ occurs dueses steadily. This is caused by the to regrinding of the plane s_face of thedistance of the pitch circle from the root pinion cutter. Therefore, smaller (morecircle. When a hob is used and x$ =(hap 0
47、 unfavourable) radii of curvature of the- 8a0)/m (=I for the profile investigated), root curve at the gear to be produced re-the center of the tool tip radius lies on sult in the course of the using of a pinionthe pitch circle, the tip radius being shaped cutter.reflected directly as the tooth fille
48、t. In the following, the stress concentra-With pinion cutters, however, in the whole tion parameter 2 _ F_/SFn is to be exa-range of tooth numbers and addendum modifi- mined. It can be seen _rom fig. 7a) and b)0that at great addendum modifications thereis a significant change of 2_Fn/SFn. Thereferre
49、d stress concentration parameter 5.c0.6- manufacturedgear zI-100difference ; 0.46 _,_ t hbblngcutter: Q,o“0-26“m i(2_Fn/SFn)h- (2_Fn/SFn)D (14)A2_Fn/SF n = _ 0.4(2_Fn/SFn) h _ _ _is represented. The greatest influence is ,_given at the large number of teeth of the o 0.8-ooo_ pinion cutter z0 = 80 which is used for “_ _smaller modules. _he great influence of thenumber of teeth of the pinion cutter on the
