1、97FTM7 Bending Load on Internal Gears of Planetary Gear Sets by: Prof. Heinz Linke, and Dipt-lng. Claudia Jahn, Technische Universitat Dresden TECHNICAL PAPER Bending Load on Internal Gears of Planetary Gear Sets Prof. Heinz Linke and DipI.-Ing. Claudia Jahn, Technische UniversitiH Dresden The state
2、ments and opinions contajned herein arc those of the author and should not be construed as an official action or opinion of the American Gear Manufacturers Association. Abstract As compared to external toathings, the design fundamentals and standards for proving the load-carrying capacity of interna
3、l toothings have been little developed. Mostly only insufficiently considered is the influence of stress in the toothed gear rim, which is essential in almost all cases. Also for the theoretical borderline case of the infinitely thick toothed gear rim some design data are too inaccurate, since they
4、are not based on the actual tooth fillet and on precise notch-stress calculations. Even larger are the deviations in elastically designed toothed gear rims. This paper is to present results of invstigations for the more precise determination of stress in the tooth root or the toothed gear rim of int
5、ernal toothings specifically for their use in planetary gearings. Proposals for the practical calculation are included. The results of calculation are confirmed by tests. At first, the considerations concern borderline cases of load in an infinitely thick toothed gear rim, whose mounting in the casi
6、ng has no influence on the stresses examined here (very elastic connecting elements). On this basis, the case of the inner toothed gear rim with a cylindrical connection contour is treated (the internal too thing is positioned only on a part of the hollow cylinder). The influence of the connection c
7、ontours on the stress as compared with the version with a ground contour as successfully taken into account by approximation relations. The assumptions and results of calculation were well determined by measurements done on a planetary gearing. The stress calculations are carried out by a locus-depe
8、ndent superposition of the stress components resulting from the radial force and the tangential force. Proceeding from the moment curve and longitudinal-force curve determined in the tooth gear rim, the local stress in the tooth gullet is determined with the aid of the boundary element method. The p
9、ractical application of the findings is facilitated by generalized stress concentration factors, which are newly calculated for internal toothings. To sum up, it can be stated that the contribution gives an insight into general connections and proposes a refined method for practical usc. Copyright 1
10、997 American Gear Manufacturers Association 1500 King Street, Suite 201 Alexandria, Virginia, 22314 November, 1997 ISBN: 1-55589-701-0 BENDING LOAD ON INTERNAL GEARS OF PLANETARY GEAR SETS Prof. H. Linke; Dipl.-Ing. Cl. Jahn Dresden University of Technology, Germany INTRODUCTION I) In gear units wit
11、h internal gears the geometrical main dimensions can be calculated according to the same relations as for external gears. The number of teeth of the internal gears and the centrere distance have to be used with negative sign 1, 2. Compared with gear units with external gears, however, there are pe c
12、uliarities. These are first the kinds or possibilities of interferences. Some do not occur in gear units with external gears. Others are pos sible only in extreme cases. Instead of the tooth tip thickness, as in external gears, in internal gears the space width at the tooth root is one of the limit
13、criteria. In gear units with internal gears the flank contact pressure is lower than in gears with external gears of the same absolute length of diame ters and the same loading. This is due to the concavo-convex pairing of the tooth profiles as against the otherwise convexoconvex pairing. 11 often s
14、eems that the to“oth root load in internal gears is generally lower than in external gears. This impression is caused by the greater tooth root thickness on internal gears. It is overlooked here that at fatigue stressing (continuous loading) also the tooth root notches have an essential influence. T
15、he root fillets of the internal gears are usually smaller than the root fillets of the exter nal gears of the same module and with the same absolute number of teeth. Moreover, in the toothed rim of internal gears there is a con s!derable stress. Its decisive share is the bending stress. The stress i
16、n the toothed rim may lead to a considerable reduction of load capacity. This contribution deals with this problem of the tooth root load or too thed rim load. Continuous loading is assumed, i. e. a number of stress cycles N 106 First the method of the general proof is briefly pre sented. I) Symbols
17、 at end of paper This is followed by a concrete calculation for the possible limiting cases. They constitute the highest and the lowest load capacity. The maximum value of load capacity results if the stress in the toothed rim is neglected. This is tantamount to assuming a very great toothed rim thi
18、ckness. A minimum value of load capacity is obtained if it is as sumed that the defonnation of the toothed rim with internal gears to be calculated is not hindered by the mounting of the toothed rim in the casing or on other components. The toothed rim is thus assumed to be a free ring (figure I). T
19、he torque is applied or absorbed by an evenly distributed loading on the periphery. -Figure 1 Elastically designed toothed rims; negligible influence of the support The calculation of these limiting cases is followed by statements con cerning the approximated consideration of the influence of the mo
20、un ting of the toothed rim with adjacent cylindrical shells (figure 2). Finally, experimental investigations are referred to. , r Figure 2 Elastic toothed rim with cylindrical adjacent contour; sup porting effect The objective of this paper is to point out influences and dependencies and to present
21、an efficient approximation method for the ca1culationu sing pes. 2 GENERAL METHOD FOR PROVING THE LOAD CAPA CITY Due to the greater elasticity of internal gears, in general the stress distribution is considerably different from that for external gears (figu re 3). As it is shown in principle in figu
22、re 3, there is an essential stress not only during the tooth contact. On internal gears the whole gear periphery or the whole cycle (I Y. It bY 1,/ r; plimit (e, ,O) -t - I -t - -1 -2 nurrber of teeth zn tip factor for internal gears cao.Jale:l t.n tha 00 4 pressure angle on = 20Q -: /, I i I V , ,
23、i! , , , , I I ! ! ! I I I 10 -io intemal gear: root fillet radius Pf I mn = 0,25 tooth dedendum hr I mn = 1,25 Figure 8b The concentration factor Y FS for internal gears; root fillet radius Qlmn = 0,25 cos (* -II) 2 SID (*) sin(* - !II) 2 SID (*) _ p (IR,I - hK ) (!. 2n p - ili)sgnl ifW;?: 0, then
24、sgnftl “ +1; if W .!_ other quantities !S ISO 6336 8 -6 Mb(F“,) YSMb (tab. 2) od iMb(FB)!) “ 2 fSMb b SR , _ 6 Mb(F,B,FIB) Krn KFll = K A KlFClnKFn 0n(,Mb(F,B,FIB)I) - 2 YSMb b SR Orn(IMb(Fb)I) “ 6 Mb(Fb) KFUl 2 YSMb b SR Note The stress distribution factors of the loaded tooth (KFal, KFI) differ fr
25、om the factos of the neighboring teeth (KFctll KF!3l1) and from the factors outside the zone of contact (KaJl! K(3111) For the present, KFal “ KFuIl KF!31 “ KF!3Il and KFcdll = KFf31l! “ 1 may be applied as a rough approx1matlon Tab!e 2 Stress concentration factors for the stress shares at the 60 ta
26、ngent basic equation: stress concentration factor Y SFt _ .!E!L _ O,5 with: a 0,290 b-2,727 c 1,092 d - 0,4188 3.3 Tooth Root Stress with the Rim Influence in the Case of an Adiacent Contour which cannot be neglected The investigations start from a model with cylindrical shells which are symmetrical
27、ly adjacent to the toothed rim (figure 2). As against the freely deformable ring treated in section 3.2, lower stresses result due to the support. With the aid of calculations according to the FEM supporting factors f. could be determined on an empirical basis. They approximately represent the reduc
28、tion of stresses compared with the freely deforma ble ring. 7 stress concentration factor Y 5Mb with: a - 0,147 b=-1,094 c 1,220 d - 0,3908 The equations (table 1) for detennining the local stress proportions according to eqs. (11) to (14) have to be modified in order to take the supporting effect i
29、nto account (Table 3). The supporting factors influence only the share of the local stresses which is caused by the toothed rim. At stress due to the tangential force component (Ft) this is included in the second addend of the equation O“I(Ft in table 3. The supporting factors are given in the equat
30、ions (15) to (J 9). T able quatiOns or e ennmmg D d I 3 E e oca s resses, s ar h es Ih I I t stress 01 (F,) FtKA YFaYSaYtYp + = -bm“ FtKFl YF.(YS“ - Ys.) /r(FI) Y,Y, f.-.-bm“ 6 M,(F,.) fl(F Ib“ 250 NIrrm e.: ho.:; bsee fiQ.re2 4t:;:;:=e:=j:l:;:;:;: Zo - fl.ITi:lao“ct teath ct tro prien o.ttff W 40 S
31、J 60 70 60 ro 00 70 6:) SJ 40 30 k + ai:Ion:im IlC1Jdicrlfacto“ tangent angle in degrees dtlrsyrrtrds see tS06336 61 Figure 13 Comparison of experimental results with calculated values according to the set-up stated References I 2J 3 4 5 6 7 8J DIN 3960: Begriffe und BestimmungsgroBen fUr Stirnrader
32、 (Zylinderr1ider) und Stirnradpaare (Zylinderpaare) mit Evolven tenverzahnung. 1987 Linke, H: Stirnradverzahnung. Munchen, Wien: Hanser 1996 Jahn, Cl.: Untersuchungen zur ZahnfuJ3tragfahigkeitsberech nung bei lnnenverzahnungen. Als Dissertation an der FakulUit Maschinenwesen der TU Dresden eingereic
33、hte Arbeit, 1997 Derzhavetz, 1: Getriebe def Energiemaschinen. Verlag Mas chinostrojenige, Leningrad 1985 A GMA 908-B89: Geometry factors for determining the pitting resistance and bending strength of spur, helical and her ringbone gear teeth. 1989 Linke, H und Borner, J: Diffence in the Local Stres
34、s of the Gear Toot root based on Hobbing Cutters and Pinion Cutters. AGMA FALL Techn. Meeting. 1992 Linke, H und Borner, J: Prazisierte Ergebnisse zur Span nungskonzentration am ZahnfuB. Dresden: International Con ference on Gears, April 1996; VDIBerichte 1230 ISO 6336: Calculation of load capacity
35、of spur and helical gears, 1996 SYMBOLS b face width da tip diameter e fn tooth gap Fb tooth force Fr radial component of the tooth force Ft tangential component of the tooth force f. supporting factors hCS thickness of the adjacent contour hK distance of the applied load to the neutral rim section
36、hf tooth dedendum K A KFr, KFa, KFp = Kv stress distribution factors (as ISO 6336 8) k addendum reduction factor L, Q, M reactions in the rim lcs damping length Mb bending moment of the rim modul number of planet gears radius of the external reaction load radius of the neutral rim section safety fac
37、tor tooth root thickness (calculated on the 60-tangent) rim thickness applied torque resistance moment addendum modification factor 10 an p “F “F(-!) apE O“PEm “Fm Qf eFn i Index I II III B c t tooth style factor tip factor stress concentration factors with rim influence stress concentrations factor
38、 without rim influence over lap factor number of teeth pressure angle helix angle peak to peak amplitude (tooth root stress) fatigue strength pulsating fatigue strength fatigue strength in dependent on the mean stress mean stress root fillet radius Icalculated on the 600tangent) angle coordinate loaded tooth neighboring tooth place with the maximum tensile stress occurring offset by a larger angle highest point of single-tooth contact compressive side tensile side other symbols see ISO 6336 8J
